Universidad de Zaragoza

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maintaining eup = 2%, a reduction of pESS to about 27–33% can be achieved for a 1-h forecast and 64–70% for a 24-h forecast. In Table 3.1 results for eup ...
Universidad de Zaragoza Departamento de Ingenier´ıa El´ectrica

Tesis doctoral Reduction of the uncertainty of wind power predictions using energy storage Reducci´on de la incertidumbre de la predicci´on de potencia e´olica mediante almacenamiento de energ´ıa

Hans Bludszuweit

Director: Dr. Jos´e Antonio Dom´ınguez

June 2009

Tribunal de la defensa de esta tesis doctoral: Presidente Secretario Vocal 1 Vocal 2 Vocal 3

Inmaculada Zamora Belver Jos´e Mar´ıa Yusta Loyo Hussein Mustapha Khodr Cl´audio Domingos Martins Monteiro Javier Contreras Sanz

Suplente 1 Suplente 2

´ Angel Javier Maz´on Sainz-Maza Jos´e Luis Bernal Agust´ın

Expertos europeos que han valorado positivamente esta tesis: Jo˜ao Abel Pe¸cas Lopes (INESC Porto) Gianfranco Chicco (Politecnico di Torino) Pierre Pinson (Technical University of Denmark)

Esta tesis doctoral fue apoyada por la fundaci´on CIRCE.

“Und wer nun noch immer meint, zuh¨ oren sei nichts Besonderes, der mag nur einmal versuchen, ob er es auch so gut kann.” 1 Momo Michael Ende (1929–1995) German writer

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Translation: “Those who still think listening isn’t an art should see if they can do it half as well.”

Agradecimientos En primer lugar quiero agradecer a mis padres que me han permitido emprender el camino en el cual me encuentro. Sin su apoyo y la preparaci´on que me han dado, no estar´ıa donde estoy. Estando lejos de ellos, Alicia y su familia han sido la mejor garant´ıa de que todo me haya salido bien. En segundo lugar siento un profundo agradecimiento por la c´alida acogida en el seno del CIRCE. Desde el primer momento me sent´ı querido e integrado, lo cual para m´ı era lo m´as importante, aparte de la calidad investigadora que atesora. He llegado a tener dos directores de tesis. Primero Andr´es Llombart me acogi´o muy bien. A ´el le debo en gran parte la buena impresi´on que tuve desde el principio. Luego Jos´e Antonio Dom´ınguez tom´o el relevo y me gui´o con muy buen criterio y mucha dedicaci´on, estando siempre disponible para cualquier duda que ten´ıa. Tampoco puedo olvidarme de los diversos impulsos muy estimulantes que me ha dado el incansable Mariano Sanz. Tambi´en yo me considero Marianista. No podr´e nombrar aqu´ı todos que me han acompa˜ nado en este camino, pero quiero mencionar a Jes´ us Sall´an por su ejemplo de humildad y saber hacer. Aprend´ı mucho de ´el, aunque seguramente no lo sabe. Otro personaje que espero que me haya influido es Jos´e Mar´ıa Yusta. Es de estas personas que est´a en todo pero siempre saca tiempo para algo m´as. Agradezco sus comentarios cr´ıticos y las conversaciones que he podido tener con ´el. En la u ´ltima fase del doctorado he tenido la suerte de conocer el INESC Porto. Agradezco la acogida por Jo˜ao Pe¸cas Lopes. Valoro mucho el tiempo que me dedic´o personalmente en estos 3 meses y los datos de predicci´on e´olica que me di´o. Pero lo m´as valioso que me he llevado de Oporto es el haber conocido a Ricardo Bessa. Tuvimos largas conversaciones – primero en ingl´es y luego cada vez m´as en portugu´es, que me ayudaron mucho. Tambi´en fue una gran ayuda en la redacci´on de la tesis en temas de predicci´on e´olica. Durante todo este tiempo he estado entre dos grupos bien diferentes. Por un lado agradezco a los e´olicos del proyecto AIRE su colaboraci´on con los datos y las diversas jamonadas y huevadas que he podido compartir con ellos. Por otro lado estaban los telecos del GTC que me permitieron desconectar a la hora de comer. Pero sobre todo agradezco a M´onica y Antonio su ayuda con el LATEX. Al final tampoco puedo olvidarme del ambiente internacional que tuve en mi laboratorio. La alegr´ıa mexicana mezclada con la brasile˜ na, condimentada con sabores colombianos, cubanos, paname˜ nos y et´ıopes fue una buena combinaci´on para superar las inclemencias de calor en verano y fr´ıo en invierno. Me alegro mucho de haber conocido a Luz, Carlitos, Durval y todos los dem´as que han pasado por este laboratorio.

Abstract The central subject of this thesis is to find out, what is the required size of an energy storage system (ESS) which is able to compensate completely or partially wind power forecast errors and to estimate its cost. Three probabilistic instruments are developed to enable the proposed ESS sizing method. The first one is a persistence model which permits the generation of large time series of forecast data from measured series of wind power. This model is verified with real world wind power forecast data. Secondly, probability density functions (pdf) of forecast errors are estimated with an algorithm based on the Beta distribution. In this context, an existing model is refined, thanks to the large quantity of persistence forecast data available. As a third tool, an online bias correction method based on moving averages is developed, which guarantees that forecast errors have zero mean. Unserved energy is chosen as sizing objective parameter. It is defined as the cumulative energy of uncompensated forecast errors. This parameter is an indicator of the grade of partial compensation if ESS size is reduced. The reference ESS size is derived from the case where unserved energy becomes zero. Starting from this reference, it is possible to obtain a relationship between the grade of reduction and unserved energy. It is shown that forecast error pdf can be used directly to determine ESS power rating and associated unserved energy. The process for determining ESS energy capacity is more complex. Saturation times are estimated as a function of ESS capacity reduction and unserved energy is calculated using the energy throughput ratio. A case study with real world forecast data shows the importance of bias correction for ESS sizing. A life-cycle cost model is developed which allows the comparison of different ESS technologies. Regions of ESS sizes for minimum costs are identified for five different technologies. Lowest costs are obtained with pumped hydro and highest with lead-acid and flow batteries. Further, large cost reduction potentials are detected for flow batteries and hydrogen storage. Despite its high power-related costs and low efficiency, hydrogen storage appears to be an economically interesting option for the case studied here. Finally, ESS costs are put into a market context. It is assumed that wind energy is sold on a liberalised market and deviations are compensated on the regulation market. In particular, the Spanish electricity market is analysed. As a result of the analysis, daily profiles of regulation costs are identified. Deviation costs are obtained, introducing these profiles in a probabilistic bidding strategy. It is shown the importance of having a good regulation price forecast method with this type of strategies. The results presented in this work lead to the conclusion that the installation of large storage capacities is needed to compensate forecast errors, which cannot be justified by costs derived from forecast deviations. Nevertheless, expected ESS cost reductions combined with added value of storage capacity for the power system may lead to profitable solutions in the near future.

Resumen El objetivo central de esta tesis es abordar la problem´atica de dimensionamiento de un sistema de almacenamiento de energ´ıa (ESS) capaz de compensar total o parcialmente los errores de la predicci´on e´olica y estimar su coste. Se desarrollan tres instrumentos probabil´ısticos para el dimensionamiento de ESS. El primero consiste en la generaci´on de largas series temporales de datos de predicci´on mediante un m´etodo basado en la persistencia. Este modelo se verifica con datos reales de predicci´on. En segundo lugar, se contempla la aproximaci´on de las funciones de densidad de probabilidad (pdf) del error de predicci´on con un algoritmo basado en la distribuci´on Beta. En ese contexto, se mejora un modelo existente gracias a la gran cantidad de datos disponibles. Como tercera herramienta, se desarrolla un m´etodo de correcci´on del sesgo o “bias” de la predicci´on con medias m´oviles que garantiza un promedio de error nulo. Se ha elegido la energ´ıa no servida como par´ametro objetivo del dimensionamiento. ´ Esta se define como la energ´ıa acumulada de errores no compensados cuando se reduce el tama˜ no del EES, indicando el grado de compensaci´on parcial. El tama˜ no de referencia se deriva del caso para el que la energ´ıa no servida se anula. Partiendo de esta referencia es posible obtener una relaci´on entre el nivel de reducci´on del tama˜ no del ESS y la energ´ıa no servida. Se demuestra que la pdf del error de predicci´on sirve para determinar la potencia nominal y la energ´ıa no servida asociada. La determinaci´on de la capacidad de energ´ıa del ESS es un proceso m´as complejo que consiste en la estimaci´on del tiempo de saturaci´on en funci´on de la reducci´on del ESS y el posterior c´alculo mediante la tasa de ciclaje de energ´ıa. La metodolog´ıa propuesta es aplicada a un caso de estudio que revela la importancia de la correcci´on del bias (o sesgo) en el proceso de dimensionamiento. Se desarrolla un modelo de coste de ciclo de vida que permite comparar diferentes tecnolog´ıas de ESS. Se identifican regiones de coste m´ınimo para cinco tecnolog´ıas de almacenamiento. Los costes m´as bajos se obtienen con el bombeo de agua, mientras que los m´as altos corresponden a las bater´ıas de plomo-´acido y de flujo. Por otra parte, los sistemas basados en bater´ıas de flujo y almacenamiento de hidr´ogeno presentan grandes potenciales de reducci´on de coste. A pesar de su alto coste de potencia y baja eficiencia, el hidr´ogeno aparece como opci´on econ´omicamente interesante para el caso contemplado en este trabajo. Por u ´ltimo, los costes del ESS son ubicados en un contexto de mercado. Se asume que la energ´ıa e´olica se vende en un mercado liberalizado y las desviaciones se compensan en el mercado de regulaci´on. En concreto, se analiza el caso de Espa˜ na. Como resultado del an´alisis se identifican perfiles diarios de precios de regulaci´on. Los costes de desviaci´on son obtenidos introduciendo dichos perfiles en una estrategia probabil´ıstica de puja. Se demuestra la importancia de disponer de un buen m´etodo de predicci´on de precios de desv´ıos para este tipo de estrategias. Los resultados presentados en este trabajo permiten concluir que se necesitan grandes sistemas de almacenamiento de energ´ıa para compensar el error de predicci´on e´olica, cuya instalaci´on no se puede justificar con los costes de desviaci´on. Sin embargo, las futuras reducciones de coste de los ESS, junto con el valor a˜ nadido que suponen para el sistema el´ectrico podr´ıan conducir a soluciones rentables a medio plazo.

Contents 1 Introduction 1.1 Context and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Analysis of the uncertainty of wind power forecasts 2.1 State of the art of uncertainty analysis of wind power forecasting . 2.1.1 Parametric probability density functions . . . . . . . . . . 2.1.2 Non-parametric probability density functions . . . . . . . . 2.1.3 Why using conditional distributions? . . . . . . . . . . . . 2.1.4 Forecast Quality: Risk Index and Skill scores . . . . . . . . 2.2 Modelling forecast scenarios using persistence . . . . . . . . . . . 2.3 Data basis – Datasets A, B and C . . . . . . . . . . . . . . . . . . 2.4 Application of the Beta distribution – Betafit . . . . . . . . . . . 2.4.1 Approximation of a Beta pdf for each power forecast bin . 2.4.2 The relationship of mean measured power and forecast . . 2.4.3 Obtaining forecast error pdf from Beta distributions . . . . 2.5 Bias correction of real world forecasts . . . . . . . . . . . . . . . . 2.5.1 Bias correction “a posteriori” . . . . . . . . . . . . . . . . . 2.5.2 Online bias correction with moving averages . . . . . . . . 2.6 Verification of persistence forecast simulation . . . . . . . . . . . . 2.6.1 Goodness of Betafit approximation . . . . . . . . . . . . . 2.6.2 Forecast model comparison with pdf and cdf . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 ESS sizing based on forecast uncertainty 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 State of the art of probabilistic ESS sizing . . . . . . . . . . . . . 3.1.2 ESS sizing approach proposed in this thesis . . . . . . . . . . . . 3.2 Sizing rated ESS Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Estimation of unserved energy eup for reduced ESS power . . . . . 3.3 Sizing ESS energy capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Energy Throughput Ratio – ETR . . . . . . . . . . . . . . . . . . 3.3.2 Statistics of SOC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Saturation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Estimation of unserved energy eue for reduced ESS energy capacity

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CONTENTS 3.3.5 Results from time step simulation: tsat and eue 3.3.6 Postprocessing of the results: deviation factors 3.3.7 Energy balance in case of ESS saturation . . . 3.3.8 Summary of ESS energy sizing . . . . . . . . . 3.4 Case study with real world forecast data . . . . . . . 3.4.1 Sizing of ESS power . . . . . . . . . . . . . . 3.4.2 Sizing of ESS energy capacity . . . . . . . . . 3.4.3 Conclusions from case study . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . .

4 Cost analysis of energy storage systems 4.1 A review on the benefits of grid-connected ESS 4.2 State of the art of energy storage technologies . 4.2.1 Pumped hydro storage – PHS . . . . . . 4.2.2 Compressed air energy storage – CAES . 4.2.3 Battery energy storage systems – BESS . 4.2.4 Vehicle to Grid – V2G . . . . . . . . . . 4.2.5 Redox Flow Battery – VRB and others . 4.2.6 Hydrogen storage and fuel cell – HSFC . 4.2.7 Thermal Energy Storage – TES . . . . . 4.3 Cost structure of ESS technologies . . . . . . . 4.3.1 ESS initial cost (Capital cost) . . . . . . 4.3.2 Cost related to ESS efficiency . . . . . . 4.3.3 Variable costs – O&M and replacement . 4.3.4 Cost of stored energy – COE . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . .

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5 Uncertainty of wind power forecast and electricity markets 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Wind power in European balancing markets – State of the art 5.2.1 Imbalance prices on the regulating market . . . . . . . 5.2.2 Forecasting of spot and imbalance prices . . . . . . . . 5.2.3 Energy storage and optimised bidding strategies . . . . 5.3 Statistical analysis of market prices in Spain . . . . . . . . . . 5.3.1 General description . . . . . . . . . . . . . . . . . . . . 5.3.2 Daily and weekly price profiles . . . . . . . . . . . . . . 5.3.3 The impact of wind power on market prices in Spain . 5.4 Optimised bidding strategy for wind energy . . . . . . . . . . 5.4.1 Formulation of the problem . . . . . . . . . . . . . . . 5.4.2 The reference case . . . . . . . . . . . . . . . . . . . . 5.4.3 Bidding with annual and monthly means . . . . . . . . 5.4.4 Bidding with daily profiles . . . . . . . . . . . . . . . . 5.4.5 Total vs. penalised deviations . . . . . . . . . . . . . . 5.4.6 Cost of wind forecast uncertainty . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

6 Conclusions and Future work 161 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 A Statistical evaluation of the forecast error A.1 Bias and Median . . . . . . . . . . . . . . A.2 MAE and RMSE . . . . . . . . . . . . . . A.3 Skewness and Kurtosis . . . . . . . . . . . A.4 The Beta distribution function . . . . . . . A.5 Goodness of fit tests . . . . . . . . . . . . A.5.1 Pearson’s Chi-Square χ2 test . . . . A.5.2 Kolmogorov-Smirnov K-S test . . . A.5.3 Anderson-Darling A2 test . . . . .

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B Results of ESS sizing for datasets A04, A13, A33 and B1 B.1 Cumulative histograms (cdf) of SOC . . . . . . . . . . . . . B.2 Proof of constant mean value of SOC . . . . . . . . . . . . . B.3 ESS sizes from histograms and predefined saturation times . B.4 Saturation times tsat . . . . . . . . . . . . . . . . . . . . . . B.5 Estimation of unserved energy eue . . . . . . . . . . . . . . . B.6 Deviation factors fsat . . . . . . . . . . . . . . . . . . . . . . B.7 Deviation factors fue . . . . . . . . . . . . . . . . . . . . . . B.8 Deviation factors fup . . . . . . . . . . . . . . . . . . . . . . B.9 Energy balance in case of ESS saturation . . . . . . . . . . .

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C ESS cost parameters and model results 189 C.1 Cost parameters from SNL . . . . . . . . . . . . . . . . . . . . . . . . . . 190 C.2 Cost structure of NaS battery . . . . . . . . . . . . . . . . . . . . . . . . 191 C.3 ESS cost for worst and best case scenarios . . . . . . . . . . . . . . . . . 193 D Bidding results for 2008 D.1 Bidding with annual and monthly means D.2 Bidding with daily profiles . . . . . . . . D.3 Total vs. penalised deviations . . . . . . D.4 Cost of forecast uncertainty . . . . . . .

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E List of acronyms

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F List of symbols

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Bibliography

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Curriculum Vitæ

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List of Figures 2.1 2.2 2.3

2.4 2.5 2.6 2.7

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2.12 2.13 2.14 2.15 2.16

Schematic principle of modern wind power forecast systems. . . . . . . . Kurtosis κ as a function of forecast interval T of persistence forecasts for datasets A, B and C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of a histogram of 24-h forecast error data (kurtosis 4.8) with Gaussian and Laplace pdf having the same standard deviation as the forecast error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantile forecast parameters. . . . . . . . . . . . . . . . . . . . . . . . . Wind power time series (thin line) with persistence forecast (bold line), forecast delay k and prediction time interval T (T = k = 24 h). . . . . . . Approximated Beta pdf of measured wind power for 9 different forecast power classes; dataset A, T×0 forecast 1 h. . . . . . . . . . . . . . . . . . Comparison of the histogram of measured wind power (dots) with Beta pdf (solid) and normal pdf (dashed); forecast power bin: 5 of 50 (forecast: 0.1 p.u.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two examples for the approximation of parameter pairs (µ, σ) from persistence forecast together with σ(µ) curves from Bofinger and Fabbri; dataset A, T×0 forecast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation parameters c1 and c2 of σ(µ) as a function of T for forecast scenarios T×0, T×1, T×2 and datasets A, B and C. . . . . . . . . . . . . Linear approximation of the mean forecast and measured mean within a forecast bin, examples T×2 forecast, dataset A; left: T = 1 h and 336 h (14 days), right: approximations from 1 h up to 30 d. . . . . . . . . . . . Comparison of the autocorrelation coefficient ak and slope u of the linear fit as a function of prediction interval T ; forecast scenario T×1, datasets A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superposed forecast errors for 9 different forecast power bins for a 1-h T×2 forecast using dataset B. . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of wind power histograms (bin width 0.01) of 10-min means (thin line) and 24-h means (bold line). . . . . . . . . . . . . . . . . . . . Comparison of observed error distribution and its Betafit approximation (bold line); left: 1-h right: 24-h T×2 forecast, dataset B. . . . . . . . . . Fitted wind power distribution in bin 0.8 p.u., 24-h T×2 forecast, dataset B. Normalised state of charge for time series from bias corrected MSE and MCC forecast compared with T×1 scenario. . . . . . . . . . . . . . . . .

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LIST OF FIGURES

2.17 Standard deviation of SOC, total bias, MAE and RMSE for different moving averages as a function of window width τ for correction of MSE and MCC data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Time series of SOC from bias corrected MSE and MCC data for three moving averages (simple – ma0, linear – ma1 and exponential – ema) with window width τ = 7 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Time series of SOC from bias corrected MSE and MCC data for simple moving average ma0 and three values of window width τ . . . . . . . . . . 2.20 Fitting the forecast error distribution of a 24-h forecast with persistence approach (above) and real forecast data (below). . . . . . . . . . . . . . . 2.21 Linear approximation of mean forecast and mean generated wind power per forecast bin for 24-h forecasts with persistence model (above) and advanced forecast models (below). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22 Approximation of parameter pairs (µj , σj ) for 24-h forecasts with persistence model (above) and advanced models (below). . . . . . . . . . . . . 2.23 Comparison of pdf (upper row) and cdf (lower row) of MSE (left) and MCC (right) forecast with simulated persistence scenarios. . . . . . . . . 3.1

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3.7 3.8 3.9

3.10

Minimum ESS power as a function of unserved energy for three forecast intervals, calculated with observed histograms (“hist”) and Betafit approximation (“betafit”) for datasets A, B, C and forecast scenarios T×0 and T×1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal energy throughput ratio ETR0 as a function of the forecast interval T for the three forecast scenarios T×0, T×1 and T×2 and datasets A, B and C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quotient between normalised mean forecast error and ETR for different datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative histograms (cdf) of SOC for different forecast intervals T (1– 720 h) and three scenarios (T×0, T×1 and T×2) from datasets A (above), B (center) and C (below). . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum, minimum and mean SOC from time step simulation for datasets A, B, C and 3 individual turbines, forecast scenarios T×0, T×1 and T×2. ESS sizes for 100% error compensation from time step simulation for datasets A, B, C and 3 individual turbines, forecast scenarios T×0, T×1 and T×2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ETR (or MAE normalised by long term mean wind generation) for datasets A, B, C and 3 individual turbines, forecast scenarios T×0, T×1 and T×2. Cumulative distribution (cdf) of SOC (T×1, T = 12 h) with representation of saturation time tsat for a reduced ESS size. . . . . . . . . . . . . . . . Advanced method for obtaining minimised ESS size from cdf of SOC (T×1, T = 14 days). Left: splitting of saturation time. Right: ex as a function of splitting coefficient cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . ESS sizes from cumulative histogram of SOC for different predefined saturation times between 0.1 and 50%, datasets A, B, C and forecast scenarios T×0, T×1 and T×2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

33 33 35

37 37 39

47

50 51

53 55

55 55 56

57

58

LIST OF FIGURES 3.11 Saturation times from 2–50% for datasets A, B and C estimated from cdf of SOC (empty symbols) and obtained from time step simulation (filled symbols). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Unserved energy as a function of forecast interval T for different saturation times; comparison of estimation (empty symbols) and time step simulation (filled symbols) for datasets A, B and C. . . . . . . . . . . . . . . . . . . 3.13 ESS energy capacity as a function of unserved energy for three forecast intervals, estimated from cdf of SOC (“estim.”) and time step simulation (“step”) for datasets A, B, C and forecast scenarios T×0 and T×1. . . . . 3.14 ESS energy capacities with improved estimation of unserved energy (“estim.”) and time step simulation (“step”) for datasets A, B, C and forecast scenarios T×0 and T×1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Deviation factor between estimated and simulated saturation time. . . . 3.16 Deviation factor between estimated and simulated mean unserved power. 3.17 Deviation factor between estimated and simulated mean unserved energy. 3.18 Energy balance expressed by percentage of unserved energy during discharge (saturation case “ESS empty”); comparison of aggregated datasets A (32 year data) and B (three wind farms) and dataset C (single wind farm, one year data). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Schematic of energy capacity sizing of an ESS and definition of unserved energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Minimum ESS power as a function of unserved energy from dataset C for real world 24-h forecasts MSE and MCC compared with persistence scenario T×1, calculated with observed histograms (“hist”) and Betafit approximation (“betafit”). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Cumulative histograms (cdf) of SOC for different 24-h forecast scenarios from datasets C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22 ESS energy capacity obtained from cdf of SOC for predefined saturation times between 0.1 and 50%. . . . . . . . . . . . . . . . . . . . . . . . . . 3.23 Correlation plot for comparison of saturation time and unserved energy obtained from cdf and simulation. . . . . . . . . . . . . . . . . . . . . . . 3.24 Minimum ESS energy capacity as a function of unserved energy from dataset C for real world 24-h forecasts MSE and MCC compared with persistence scenario T×1, estimated from cdf of SOC (“estim.”) and time step simulation (“step”). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.25 Deviation factors of saturation time, unserved power and unserved energy as a function of predefined saturation time. . . . . . . . . . . . . . . . . . ESS operating ranges of installed systems (November 2008). Source: Electricity Storage Association. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Compressed air energy storage system – CAES. . . . . . . . . . . . . . . 4.3 Battery energy storage system – BESS. . . . . . . . . . . . . . . . . . . . 4.4 Ragone chart for traction batteries, Source: Bossche et al. (2006). . . . . 4.5 Roadmap of German passenger car market until 2035, EV: electric vehicles, CV: conventional vehicles, PHEV30/90: 30/90 km plug-ins, Source: Engel (2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

61

62

64

65 67 69 70

71 72

73 76 77 78

78 80

4.1

90 94 95 96

100

xviii 4.6

4.7 4.8 4.9 4.10 4.11 4.12

4.13 4.14 4.15 4.16 5.1 5.2 5.3 5.4 5.5

5.6 5.7 5.8

5.9 5.10 5.11 5.12

LIST OF FIGURES Impact of availability of V2G technologies on global car technology market share, climate policy scenario; ICEV – internal combustion engine vehicle, HEV – hybrid-electric vehicle, FCV – fuel cell vehicle. (Source: Turton and Moura 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Redox flow battery system – RFB. . . . . . . . . . . . . . . . . . . . . . Hydrogen storage system with electrolyzer and fuel cell – HSFC. . . . . . Initial cost for the reference system of 1 MW and four ESS technologies. ESS energy balance without losses (above) and with losses (below). . . . Energy loss as a function of ESS efficiency; comparison of the two cases of maintaining input or output energy equal to δ. . . . . . . . . . . . . . . . 2D-interpolation of the energy throughput and corresponding annual full cycles as a function of ESS size, reference case of 1 MW installed power and 24-h forecast horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . Energy loss cost for the reference system of 1 MW and four technologies, (a) absolute cost, (b) cost share in % of the total system cost. . . . . . . O&M plus replacement costs for the reference system of 1 MW/24 MWh and four technologies, (a) annual cost, (b) percentage of total cost. . . . . Base cost scenario for reference system of 1 MW/24 MWh and four technologies, (a) annualised total system cost, (b) revenue requirement (COE). ESS revenue requirement (COE) for the reference system of 1 MW/24 MWh and four technologies, (a) worst case, (b) best case scenario. . . . . . . .

101 102 104 110 112 113

114 116 118 121 122

Timing of bidding on the NordPool spot market. . . . . . . . . . . . . . . 129 Marginal (PM) and regulation prices in the Spanish electricity market; hourly prices within one week in 2009 (left) and weekly means (right). . . 137 Hourly, weekly, monthly and quarterly mean values of regulation costs in the Spanish electricity market between 1-1-2007 and 31-3-2009. . . . . . . 138 Monthly means of the asymmetry factor between up and down regulation costs in the Spanish electricity market between 1-1-2007 and 31-3-2009. . 139 Probability density (left) and cumulative distribution (right) of marginal prices (PM) and deviation cost in the Spanish electricity market between 1-1-2007 and 31-3-2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Daily price profiles on the Spanish electricity market for the year 2007; annual mean week (left) and mean day (right). . . . . . . . . . . . . . . . 141 Daily regulation cost profile on the Spanish electricity market for the years 2007 and 2008; annual mean week (left) and mean day (right). . . . . . . 141 Annual mean and seasonal variability of hourly prices of a mean day for marginal spot market price (above) and regulation cost (below) for 2007 (left) and 2008 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Daily wind generation profiles in Spain for the years 2007 (left) and 2008 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Wind generation (left) and market prices (right) on 1st May 2007. . . . . 146 Loss function for the Spanish electricity market, based on mean values of regulation unit costs for the years 2007, 2008 and first quarter of 2009. . 149 Performance ratio PR as a function of MAPE (above) and penalised regulation energy (below), data from 2007. . . . . . . . . . . . . . . . . . . . 156

LIST OF FIGURES 5.13 Cost of (wind power) forecast uncertainty πd (above) and cost of penalised deviations πpen (below) as a function of MAPE, data from 2007. . . . . .

xix

158

A.1 Bias in % of Pinst as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC. . . . . . . . . . . . . . 172 A.2 Median in % of Pinst as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC. . . . . . . . . . . . . . 172 A.3 Mean absolute percentage error MAP E as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC. . 173 A.4 NRMSE in % of Pinst as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC. . . . . . . . . . . . . . 173 A.5 Skewness coefficient y as a function of T for datasets A, B, C, MSE and MCC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.6 Kurtosis κ as a function T for datasets A, B, C, MSE and MCC. . . . . . 174 A.7 Beta pdf as a function of its parameters a and b. . . . . . . . . . . . . . . 175 B.1 Cumulative distribution function cdf of SOC, datasets A and B, forecast scenarios T×0, T×1, and T×2. . . . . . . . . . . . . . . . . . . . . . . . . 180 B.2 ESS sizes from cdf of SOC for predefined saturation times between 0.1 and 50%, datasets A and B, forecast scenarios T×0, T×1, and T×2. . . . . . . 182 B.3 Saturation times from 2–50% estimated from cdf of SOC (empty symbols) and obtained from time step simulation (filled symbols). . . . . . . . . . 183 B.4 Unserved energy as a function of forecast interval T and ESS reduction level, comparison of time step simulation and estimation based on ETR0 . 184 B.5 Deviation factor of saturation time as a function of forecast interval T and ESS reduction level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 B.6 Deviation factor of unserved energy as a function of forecast interval T and ESS reduction level, estimating p¯up with ETR0 (NMAE). . . . . . . . . . 186 B.7 Deviation factor of mean unserved power as a function of forecast interval T and ESS reduction level, estimating p¯up with ETR0 (NMAE). . . . . . 187 B.8 Energy balance – percentage of energy loss during ESS discharge (saturation case “ESS empty”). . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 C.1 Base scenario with intermediate prices for the 1 MW/24 MWh reference system of NaS technology. . . . . . . . . . . . . . . . . . . . . . . . . . . 192 C.2 Comparison of worst and best case scenario with NaS technology, worst case (a), best case scenario (b). . . . . . . . . . . . . . . . . . . . . . . . 193 C.3 Share of cost due to efficiency, worst case (a), best case scenario (b). . . . 194 C.4 Share of O&M and replacement cost, (a) worst case, (b) best case scenario. 195 D.1 Performance ratio PR as a function of MAPE (above) and penalised regulation energy (below), data from 2008. . . . . . . . . . . . . . . . . . . . D.2 Cost of (wind power) forecast uncertainty πd (above) and cost of penalised deviations πpen (below) as a function of MAPE, data from 2008. . . . . .

201 202

List of Tables 1.1

Typical run times and applications of energy storage. Source: McDowall (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 2.2

Basic statistical parameters of datasets A, B and C. . . . . . . . . . . . . Comparison of forecast error parameters of different scenarios, all based on 24-h forecast of dataset C. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Anderson-Darling (A2 ), Kolmogorov-Smirnov (K-S) and Chi-square (χ2 ) test results for goodness of Betafit approximation of error pdf. . . . . . . 2.4 Two-sample Anderson-Darling test for all combinations of persistence and advanced forecast error cdf. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4

3.5

3.6 3.7 3.8 3.9

3.10 3.11

ESS nominal power [p.u.] for eup = 2% from empirical pdf, Betafit pdf and the deviation in [%]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviation of ESS power estimation from Betafit for several levels of unserved energy eup for 24-h persistence forecast and scenario T×1. . . . . . Definition of the zero error reference e00 for ESS energy capacity (size necessary for compensation of all forecast errors). . . . . . . . . . . . . . ESS energy capacity [p.u.] for eue = 2% from time step simulations and estimations from cdf of SOC; Case I: simple estimation, Case II: improved estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviation of ESS energy capacity estimation from time step simulation for several levels of unserved energy eue for 24-h persistence forecast and scenario T×1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviation of ESS nominal power from Betafit for several levels of unserved energy eue for 24-h persistence forecast and scenario T×1. . . . . . . . . . Parameters obtained from SOC time series and ETR for several forecast scenarios from dataset C. . . . . . . . . . . . . . . . . . . . . . . . . . . . ESS energy capacity obtained from cdf of SOC for predefined saturation times between 0.1 and 50%. . . . . . . . . . . . . . . . . . . . . . . . . . Deviation of ESS energy capacity estimation from time step simulation for several levels of unserved energy eue for real world forecasts and 24-h forecast scenarios T×1 and T×2. . . . . . . . . . . . . . . . . . . . . . . . Deviation factors of saturation time, unserved power and unserved energy for ESS sizes corresponding to tsat = 10% (e10 ). . . . . . . . . . . . . . . Deviation factors of saturation time, unserved power and unserved energy for ESS sizes corresponding to eue = 5%. . . . . . . . . . . . . . . . . . .

3 19 34 36 38 48 48 53

66

66 74 76 77

79 81 81

xxii 4.1 4.2 4.3 4.4

5.1 5.2 5.3 5.4

5.5 5.6

5.7 5.8 5.9

LIST OF TABLES Storage technologies and applications. Source: Barton 2004. . . . . . . . ESS capital cost coefficients related to energy and power assumed in this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions for the market environment. . . . . . . . . . . . . . . . . . . Selected parameters of the cost model; adopted value and the range found in the literature (Kaldellis 2007, Schoenung 2003/2008, Chen 2009) are shown for each parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . Annual means of marginal electricity price and regulation costs for the Spanish market from 1-1-2007 until 31-3-2009. . . . . . . . . . . . . . . . The first four moments of marginal prices and regulation costs for the Spanish market from 1-1-2007 until 31-3-2009. . . . . . . . . . . . . . . . Correlation coefficients for different combinations of wind generation and demand with market prices in Denmark and Spain. . . . . . . . . . . . . Bidding with point forecast; simulation results for 2007 Spanish energy market prices. Wind power data from 3 different sites (A, B and C) and 3 different forecast methods (persistence Tx1 and two real world forecasts). Bidding reference case for 2008 Spanish energy market. . . . . . . . . . . Bidding with point forecast and uncertainty, using mean penalty of one year (1a) and one month (30d); simulation results for 2007 Spanish energy market. Wind power data from 3 different sites (A, B and C) and 3 forecast methods (Tx1, MSE, MCC). . . . . . . . . . . . . . . . . . . . . . . . . . Advanced bidding, using daily profiles from one year (1a dp) and seasonal variation (3m dp); simulation results for 2007 Spanish market. . . . . . . Total and penalised up and down regulation as percent of total generation; simulation results for 2007 Spanish energy market (datasets A and B). . Total and penalised up and down regulation as percent of total generation; simulation results for 2007 Spanish energy market (dataset C). . . . . . .

A.1 Kolmogorov-Smirnov (K-S) test results for goodness of Betafit approximation of error pdf with 5% confidence level; dataset C, 24-h forecast. . . .

91 108 123

123 139 140 143

151 151

152 153 154 155 177

C.1 ESS capital cost coefficients related to energy, power and balance of plant (BoP), as published by Sandia National Labs in 2008. . . . . . . . . . . . 190 C.2 System specifications and assumptions for NGK NAS r . Source: Norris 2007. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 D.1 Bidding with point forecast and uncertainty, using mean penalty of one year (1a) and one month (30d); simulation results for 2008 Spanish energy market. Wind power data from 3 different sites (A, B and C) and 3 forecast methods (Tx1, MSE, MCC). . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Advanced bidding, using daily profiles from one year (1a dp) and seasonal variation (3m dp); simulation results for 2008 Spanish energy market. . . D.3 Total and penalised up and down regulation energy as percent of total generation; simulation results for 2008 Spanish energy market. . . . . . .

198 199 200

Chapter 1 Introduction “Begin at the beginning”, the King said, very gravely, “and go on till you come to the end: then stop.” Alice’s Adventures in Wonderland Lewis Carroll (1832–1898) English writer and mathematician

In this first chapter a short introduction is given for the present thesis. The first part explains the context and motivation of this work. The second part describes the objectives of the thesis. In the last part an overview of the contents of the work is given.

2

Chapter 1. Introduction

1.1

Context and motivation

The increasing penetration of wind power in electricity systems leads to the need for a better understanding of the wind power forecast error and, if possible, its reduction. It is assumed that many problems of integration of wind energy in the electrical power system (EPS) are due to its uncertainty. If this is removed, or at least reduced below an acceptable level, wind power could reach higher penetration levels. This thesis aims at giving answers about how much storage would be needed to achieve this reduction and what would this cost. Although it does not form part of the study presented in this thesis, the physical impact of wind power in the EPS will be exposed in the next paragraph, because it represents the main motivation of this work, which is to reduce those impacts. After that, a short introduction is given on the classification and benefits of grid-connected energy storage. The physical impact of wind power in the EPS has been widely researched an many studies have been published on this subject. In this introduction only a small selection is considered, within which one interesting work [1] presents a comparative analysis of the impact of several wind generator technologies and electrical grid configurations. The advantage of variable speed turbines is demonstrated, as they permit voltage control. In [2] the positive effect of a wind farm on the voltage level of a weak grid is shown. It is highlighted the importance of the capability of reactive power generation, which enables the wind farm helping to maintain the voltage level. On the other side, in [3] negative effects of wind power are analysed, such as: • Steady state voltage level • Voltage fluctuations • Wave form quality To compensate these impacts, the following solutions are discussed: • Installation of new power lines • Reactive power control • Load management • Energy dissipation (e.g. water heating) • Energy storage (water pumping, batteries, fly wheels) While the installation of new power lines is too costly, the other solutions are considered viable, but always depending on the local conditions. It is widely accepted that high penetration levels (far beyond 10% of the short circuit power at the point of common coupling) only can be achieved with additional technical measures [1]. These measures can be situated in the wind turbine itself or in the surrounding power system. Here it is commonly stated that the EPS must have an elevated flexibility. This means more regulation reserve which can be achieved by relatively small gas fired power stations and

3

1.1 Context and motivation

new energy storage systems such as advanced batteries, hydrogen (storage and fuel cell), double layer capacitors (“Ultra-Caps”) and others. In [4] a method is presented to quantify the need of reserve in case of an elevated wind power penetration. Anyway, if the EPS cannot absorb the generated energy, large amounts of wind energy may be lost, which is especially common in island grids, for example in Greece [5]. Renewable electricity generation demands additional flexibility of the EPS, which creates a great challenge for the established centralised system control. Here certainly distributed generation and control can be a good solution. In [6, 7] the application of wide area measurement systems (WAMS) to distributed generation control is shown. The basic idea of this approach is the creation of a control hierarchy having several layers. Autonomous agents take local decisions and only limited information is exchanged which is relevant for higher control layers. As a result, the complexity in each layer is limited and a stable and very flexible system control is achieved. Synchronised measurements are very important for this method, which are available from so called phasor measurement units (PMU). This thesis will concentrate on the sizing of energy storage to deal with the forecast error (see chapter 3). The added value of energy storage for increased system reserve (or spinning reserve) is not subject of this work, but should be mentioned here to complete the scenario. Placing an ESS near the wind generator has a number of benign effects on the energy system. In [8] a good introduction is given about possible applications of energy storage in relation to high wind penetration in weak grids. As shown in Table 1.1, ESS can be divided in three main categories, according to their typical discharge or run times [8] which correspond to specific applications. The typical run time is obtained dividing energy capacity by rated power. It is assumed here, that the power level in utility applications is usually in the megawatt range. A more detailed classification is given in chapter 4. Run time

Energy capacity

Seconds

kWh

Voltage stabilization Power quality

Minutes

several MWh

Output smoothing Stabilizing weak grids

Hours

> 10 MWh

Application

Renewable energy time shifting (peak shaving) Ancillary services (spinning reserve)

Table 1.1. Typical run times and applications of energy storage. Source: McDowall (2006)

Apart from physical impacts, the uncertainty of wind power forecasts cause deviations which are penalised when wind energy is traded in short term electricity markets. It is widely agreed that these penalization may ascend to approximately 10% of the total revenue from generated wind energy. This figure is confirmed in this thesis for the Spanish electricity market, although the strong dependence on market conditions is shown as well (see chapter 5).

4

Chapter 1. Introduction

1.2

Objectives

The central question of this thesis is how much energy storage capacity is needed to reduce wind power forecast uncertainty and how much does it cost. The fundamental instrument to find answers to this question is a probabilistic description of the forecast error, namely its probability distribution. It will be shown that this is a powerful tool for ESS sizing, cost estimations and short term trading in energy markets. It is the aim of this thesis to provide a complete framework which starts with the probabilistic description of the input variable, energy storage sizing and cost modelling. In addition the market environment is investigated to put results from the cost model in relation to market prices. As a consequence, four main objectives can be formulated, which are described below. 1. Development of statistical instruments as a basis for all further studies – In order to develop the sizing method, error distributions are obtained from several measured wind power time series. For the case that no forecast data is available, a method is proposed to generate synthetic forecasts in order to study the impact of variations in the forecast horizon and forecast quality. 2. Development of a methodology for ESS sizing – Two separate pathways will be needed to obtain required power and energy ratings. At the end, an answer should be given to the question, how much energy capacity is needed to compensate forecast errors and at which rated power. In addition it is desired to quantify, what happens if ESS capacity is reduced below these values. It was decided that the cumulative energy of uncompensated forecast errors is a useful measure to quantify the achievable grade of compensation. This parameter is denoted as unserved energy. Thus, the purpose of the sizing model is Establishing a reference ESS size which is able to compensate 100% of the forecast error and then obtaining a relationship between ESS size reduction and unserved energy. 3. Elaboration of an ESS cost model – The optimal energy capacity cannot be defined without introducing the cost of such a system. Therefore a model is developed to calculate the cost of the energy, provided by the ESS. The cost model should provide technology-specific ESS costs depending on energy throughput, provided by the sizing model. 4. Quantification of the cost of forecast errors under market conditions – At last, costs have to be compared with market prices of electrical energy and alternative reserve options such as gas turbines. The cost of these alternatives is indirectly given by regulation prices. On the example of the Spanish electricity market, regulation prices are analysed.

1.3 Outline of this thesis

1.3

5

Outline of this thesis

In this thesis a probabilistic method is proposed for energy storage system (ESS) sizing. The sizing method is applied to a case where the ESS is only used to compensate wind power forecast errors. By solving this task, the developed methodology can be applied to any other ESS sizing problems with stochastic power input. The outline of the present thesis is as follows: • Chapter 1: Introduction • Chapter 2: Probabilistic description of the wind forecast uncertainty • Chapter 3: Sizing of an ESS based only on forecast uncertainty • Chapter 4: Development of a cost model for ESS • Chapter 5: Statistical analysis of Spanish electricity market prices • Chapter 6: Conclusions and future work Chapter 2 In chapter 2 the statistical analysis of wind power forecast is treated. The aim is to obtain detailed information about the uncertainty of a given forecast. As a result, probability density functions (pdf) are generated for a number of forecast scenarios. Three stochastic instruments are developed in chapter 2: • A simple algorithm to generate time series of wind power forecast errors • Correction and refinement of a parametric method for error pdf estimation • Online bias correction of real world forecast data Chapter 3 In chapter 3 a method is proposed for ESS sizing in order to reduce forecast errors. Nominal power and energy capacity are determined separately, as their statistical characteristics are fundamentally different. The required nominal ESS power and unserved energy in case of its reduction, can be estimated directly from the forecast error pdf. For a determination of the ESS energy capacity, first the statistical behaviour of the state of charge (SOC) of the ESS is analysed. The impact of the forecast time horizon in combination with the forecast method is investigated. From empirical cumulative distribution functions (cdf) of SOC, a first estimate can be obtained for the saturation time of a given energy capacity of the ESS. To obtain the unserved energy from saturation times, the energy throughput ratio ETR is introduced. At the end of this chapter, curves are available which describe the relationships of unserved energy related to reduced power rating on the one hand, and related to reduced energy capacity on the other. A detailed discussion of estimation errors for ESS energy capacity is included in order to prepare further improvements of the model.

6

Chapter 1. Introduction

Chapter 4 In chapter 4 the cost of energy storage is examined. First the usefulness of ESS in power systems is highlighted and then current and emerging storage technologies are reviewed. In the second part this chapter, a cost model is developed in order to learn more about the cost structure of some selected ESS technologies such as pumped hydro storage (PHS), hydrogen storage, and battery storage, namely lead-acid, sodium-sulfur (NaS) and flow batteries. The cost model is based on an existing model, but important modifications are introduced. Fixed costs are modelled as a sum of energy and power related initial costs. The cost coefficients are assumed to be independent on ESS size, which is an important simplification, as unity costs for larger installations tend to be smaller, due to economies of scale. As variable costs, three elements are considered: losses due to ESS efficiency, operation and maintenance (O&M) and finally replacement of major system components. Costs of energy losses are related to ETR and are assumed to be charged at mean marginal market price, which was 50 e/MWh in Spain for 2007 and 2008 together. O&M costs are assumed to be proportional to initial costs (percentages are obtained from the literature). Replacement costs are also assumed to be a percentage of initial costs and possible cost reductions or increments in time are included in the model. Results are presented in form of cost surfaces as a function of reduction grade of power and energy of a reference ESS. Thus, regions of minimum costs can be identified. The most important model output is the cost of energy (COE) or revenue requirement, expressed in e/MWh. This parameter represents the price of energy served by the ESS which covers all costs for operating and owing the installation. Chapter 5 To compare the obtained COE of storage with current electricity prices, in chapter 5 the Spanish electricity market is studied. Special emphasis is given to the regulation market. Hourly spot and regulation prices are compared and analysed. The impact of wind power on market prices (spot and regulation prices) is investigated. Based on the results, a probabilistic bidding strategy is applied, assuming that wind energy is sold on the liberalised market and deviations are compensated on the regulation market. The cost of forecast errors is expressed in a performance ratio. If this ratio reaches 90%, the cost for deviations is 10% of total revenues. The performance ratio is calculated for several bidding scenarios, assuming more or less detailed knowledge of regulation prices. Conclusions of economical viability of ESS dedicated to forecast error reduction are drawn from the results of this final study. The impact of the applied bidding strategy on imbalance induced by wind energy is analysed. Penalised and non-penalised deviations are separated in order to distinguish positive and negative effects of advanced bidding strategies.

Chapter 2 Analysis of the uncertainty of wind power forecasts “We demand rigidly defined areas of doubt and uncertainty!” The Hitchhiker’s Guide to the Galaxy Douglas Adams (1952–2001) English writer

The aim of this chapter is to provide the statistical tools for probabilistic energy storage sizing. First a simple but novel algorithm is proposed to simulate forecast data. Further a probabilistic method is developed to describe wind forecast errors. The contribution of this thesis consists in the correction and refinement of an existing method. The third tool consists in a novel online bias correction method for real world forecast data. This method is of great importance if real world forecast data is available. This chapter is organised as follows. In section 2.1 the state of the art of wind forecasting methods is reviewed. Special emphasis is given on how the uncertainty of a forecast is quantified (probabilistic forecasting). Probabilistic forecasts provide probability density functions describing its uncertainty. The difference between parametric and non-parametric approaches is explained. In section 2.2 a method is suggested which permits the generation of synthetical forecast data, when no forecast data is available. In section 2.3 real measurements of wind power time series and forecasts, which are the basis of this work, are briefly introduced. In section 2.4 a parametric method based on the Beta distribution is developed. Advantages and limitation of this approach are shown and possible applications are discussed. With real world forecasts available, in section 2.5 a method for online bias correction is proposed. Finally in section 2.6 statistics of synthetical persistence forecasts are compared with data from real world forecasts. The usefulness of the proposed persistence approach and the impact of online bias correction are demonstrated.

8

Chapter 2. Analysis of the uncertainty of wind power forecasts

2.1

State of the art of uncertainty analysis of wind power forecasting

In [9,10] a good introduction to the state of the art of short-term wind power forecasting is given. In order to compare the performance of different predictions a reference is needed. The most extended reference is the persistence forecast. The persistence approach assumes that in the future exactly the same as in the past will happen, i.e. no change is expected. So, the value measured at time t serves as prediction for a future time t + k, being k the forecast horizon. For very short horizons up to 3–6 h, the persistence approach is very difficult to beat with more sophisticated methods. But when the horizon is beyond this, other methods become far better. In [11] a new reference is proposed, which adds to the persistence the long term mean, weighted by the autocorrelation coefficient of the wind power at the specific site. This modification is motivated by the observation that at forecast horizons beyond 12–15 h, the long term mean becomes a better forecast than the persistence. It exists a broad variety of prediction methods in the field of wind generation. They can be classified basically as physical models, which are based on solving basic equations of fluid dynamics and statistical models, based on available observation data. In [12] a historical review of the development of wind power short-term prediction is given. Nowadays common commercial forecast models take data from numerical weather prediction (NWP) from the national weather service [13, 14]. An excellent description of the recently developed portuguese model EPREV is given in [15]. In [16] another portuguese forecasting system (FORECAS) is described, where NWP data is combined with SCADA1 measurements. The forecast output is a combination of an auto-regressive (AR) model applied to measured wind power data and a power curve model (PCM) applied to the input from NWP. The PCM is represented by a neural network (NN). FORECAS applies classical mean squared error (MSE) criterion to train the NN. Bessa et al. [17] propose NN-training with minimum entropy error (MEE) or maximum correntropy criterion (MCC) instead. Thus, all moments of the pdf are taken into account, which is of special interest in wind power forecasting. Data from NWP has in the best case a resolution of 5×5 km and a spatial refinement is necessary to obtain a forecast of a specific site. The wind speed at hub height is calculated modelling the boundary layer with regard to roughness, orography and wake effects of the local site. After that, a statistical correction of systematic errors is done – the so called model output statistics (MOS). Once the wind speed at hub height is calculated, the aggregated power curve of the whole wind farm is used to predict the actual power production. In Fig. 2.1 the schematic principle of modern wind power forecasting is depicted. Until the year 2004, very few works can be found in the literature dealing with wind power forecast error distributions. Due to the lack of information, most authors assumed standard distributions [4, 18, 19]. In these cases, the error is described only with the RMSE (Root Mean Squared Error) which is nothing else than the standard deviation σ [11]. This indicator is very useful to compare different forecast methods. An early work on forecast error distributions, applied to wind energy was published 1

Supervisory Control And Data Acquisition

2.1 State of the art of uncertainty analysis of wind power forecasting

9

NWP Model

wind speed and direction forecast, ensemble forecasts

Local refinement roughness, stability, orography

SCADA

time series of wind power

Power curve model (PCM)

Autoregressive model (AR)

Aggregation model of PCM and AR forecasts Probabilistic model (skill forecast) Wind power forecast

Figure 2.1. Schematic principle of modern wind power forecast systems.

by Watson et al. in the year 1994 [20]. The method counts with NWP input and MOS to obtain the point forecast. But in addition, a Gaussian error for the wind speed is assumed, which in turn is “transformed” into power error using the power curve of the wind farm. Unfortunately no details are given about the transformation method. Already in this early work the value of an estimate of the forecast uncertainty is highlighted. The effect of uncertainty estimation is illustrated on the example of fossil fuel savings which higher wind power penetrations would be able to achieve. The best result is obtained with uncertainty estimation because less reserve power is needed to back up wind generation. It has to be mentioned that this work was not based on real measurements of wind generation but on the assumption of the installation of a large number of wind turbines spread throughout the UK. The first well documented work in this field, based on real measured data was published in 2000 by Focken et al. [21,22]. The authors describe the reduction of wind power prediction errors by spatial smoothing effects. This fundamental work is based on real forecast and generation data from all over Germany from 1996 until 1999. It was carried out by the research groups headed by Hans-Peter Waldl from the Carl von Ossietzky University of Oldenburg and Hans-Georg Beyer from the University of Applied Science (FH) Magdeburg. Two years later both groups were implicated in [13] where – still on the same data – results were refined. But the only parameter for quantifying the forecast quality is the standard deviation σ. Nothing is said about the pdf of the error. From the same circle of researchers, in the year 2001 at EWEC’01 conference, results

10

Chapter 2. Analysis of the uncertainty of wind power forecasts

from the forecast model Previento are presented [23]. and two different methods for a probabilistic analysis of the forecast error are proposed [24, 25] based on this data. More recently, a growing number of publications can be found about the description of wind power forecast uncertainty. Especially the working groups of the European project ANEMOS and its follow-up ANEMOS.plus are giving more detailed information about the uncertainty of the available forecast models. An excellent summary of the ANEMOS project is given in [14]. Modern probabilistic approaches take advantage of the so called Meteorological Ensemble Forecast, available from numerical weather forecast. Ensemble forecasts are giving information about the uncertainty of the forecast by supplying several possible scenarios. In [26] the use of such scenarios for probabilistic wind power forecasting is presented and in [27] different probabilistic forecast models are compared and classified, giving a good overview about forecast models available today. In the following paragraphs, the most important publications about forecast error pdf are presented. The works can be divided into two groups: parametric and non-parametric approaches. In the parametric framework, a distribution family is chosen (e.g. Gaussian, Weibull, Beta etc.) and its parameters are estimated from the available data. The main advantage of this approach is that it requires smaller data sets than the non-parametric one to obtain equivalent estimations. In the non-parametric framework the pdf is directly estimated from the data. In this case, weaker or no hypotheses are necessary about the underlying distribution which reduces possible errors due to incorrect hypotheses.

2.1.1

Parametric probability density functions

General Considerations In this section, possible parametric distributions for the estimation of the forecast error pdf are discussed. The parametric approach assumes that the distribution of the investigated variable has a well defined shape which can be described with an analytical function. This function is adjusted to the distribution by a set of parameters. The most common is the well known Gaussian “bell shape curve”. The advantage of such a function is that the distribution only depends on few parameters, in the case of the Gaussian function there are only two: standard deviation σ and mean µ. In this thesis it will be shown, that the tail of the error pdf is of special interest for the ESS sizing. Therefore, the normalised fourth moment, the kurtosis has been chosen as statistical parameter to evaluate the tail of the studied pdf. By calculating the kurtosis of the forecast error data, it can be shown that the normal pdf is not appropriate. The kurtosis κ of a distribution of random variable ε, is given by E(ε − ε¯)4 (2.1) σ4 where E denotes the expectation operator, ε the random variable (here the forecast error), ε¯ the mean error (or bias) and σ the standard deviation. κ=

If κ is larger than 3 (the value associated to normal distribution), the distribution is called leptokurtic or fat-tailed. For more details see appendix A.3.

2.1 State of the art of uncertainty analysis of wind power forecasting

11

In Fig. 2.2 kurtosis is depicted as a function of forecast interval length T . Results are presented for persistence forecasts based on three datasets (see more details on persistence forecast data in section 2.2). A broad range of kurtosis is observed from over 10 for very short forecasts (T < 2 h) down to around 3 for forecasts above 48 h. Interestingly, the overall tendency does not show significant differences between the datasets, except for very short forecast intervals. 20 A B C

Kurtosis

15 10 5 0

1

3

6

12 24 48

168

720

Forecast interval T [h] Figure 2.2. Kurtosis κ as a function of forecast interval T of persistence forecasts for datasets A, B and C. 2

Frequency [%]

10

Forecast error Gaussian pdf Laplace pdf

0

10

-2

10

-4

10

-1

-0.5

0

0.5

1

Forecast error ε [p.u.] Figure 2.3. Comparison of a histogram of 24-h forecast error data (kurtosis 4.8) with Gaussian and Laplace pdf having the same standard deviation as the forecast error.

In Fig. 2.3 the histogram of an example of 24 h forecast data with a kurtosis of 4.8 is shown. In the same plot the Gaussian and Laplace pdf are depicted, which have the same standard deviation as the sample data. The Laplace pdf (with κ = 6) was chosen as an example of a fat-tailed distribution. As expected, the tail of the forecast error pdf is situated between the Gaussian and the Laplace pdf, being 3 < κ < 6. Gaussian and Laplace pdf have constant κ for all parameters, while wind power forecast errors exhibit a wide range of κ. Hence, it can be stated that the forecast error pdf is fat-tailed with variable kurtosis, so that Gaussian or Laplace distributions are not suitable.

12

Chapter 2. Analysis of the uncertainty of wind power forecasts

Beta distribution as power forecast error pdf (Luig et al.) There are well known distributions with variable kurtosis, but for wind power only the Beta distribution has been proposed in the literature. Luig et al. [25] (University of Magdeburg) propose the application of the Beta distribution. The parameters are derived from expected power and standard deviations of the forecast error. This approach is refined one year later (2002) by the same authors in [28]. In 2005 Fabbri et al. [29] apply the approach from [28] to calculate the cost associated to forecast errors. Unfortunately, in [28] a mistake was committed. The standard deviation σ was modelled as a linear function of the mean forecast. As shown in this thesis (see section 2.4) and other publications [24, 30, 31], this is not true. The linear approach from [28] was adopted in [29] and the version of [29] in turn was used by Bouffard and Galiana [32] to obtain values of σ for the wind power forecast error. It may be mentioned that in [28,29] σ and mean values are calculated as parameters for a Beta distribution, while in [32] a Gaussian distribution of the forecast error is assumed. It is outside of the scope of this thesis to determine the consequences of this mistake on the results obtained in [29] and [32] but it would be an interesting field of further investigation. Fundamental ideas from Bofinger et al. [28] are adopted and further developed (and corrected!) in [31] and the present PhD thesis. It will be shown that the error does not follow strictly a Beta distribution. Nevertheless the Beta distribution is chosen because it adjusts quite well to basic characteristics of the wind power forecast error. According to [33], the kurtosis of Beta pdf is given by κbeta = 6 ·

a(a + 1)(a + 2b) + b(b + 1)(b + 2a) +3 ab(a + b + 2)(a + b + 3)

(2.2)

where a and b are the parameters of Beta pdf. Besides its variable kurtosis, the advantage of Beta pdf consists in its simplicity (it is defined by only two parameters) and that its values are limited to the interval [0, 1], as well as the normalised wind power. Transformation of wind into power forecast error pdf (Lange) Another approach for parametric pdf assumes a Gaussian distribution of wind speed forecast error. To obtain the wind power, the power curve of the wind farm is introduced. Lange and Waldl [24] (University of Oldenburg), quantify the uncertainty of wind power forecast using the well known distribution of wind speed forecasts in conjunction with the power curve of the wind energy conversion system (WECS). Similar to [20], the method starts with the assumption that wind speed forecast errors in a wide range can be approximated quite well with a normal distribution. In [24], contrary to [20], the transformation from wind to power forecast error pdf is shown in detail. In addition a very good explanation is given why wind power forecast errors cannot be Gaussian. It is highlighted that the power curve of the wind turbine amplifies or attenuates the error depending on its slope at the wind speed in question. This approach is refined by Lange in his PhD thesis in 2003 [34] and another publication in 2005 [35].

2.1 State of the art of uncertainty analysis of wind power forecasting

2.1.2

13

Non-parametric probability density functions

General considerations The advantage of non-parametric distributions is that no assumption about the shape is needed. This is an important advantage as wind generation represents a non-linear and bounded process and its forecast error distributions may be skewed and fat-tailed [35–37]. In [37] it is stated that: “They may even be multi-modal, owing to the cut-off discontinuity in the power curve, or simply because distributions [...] may themselves be multi-modal.” Another typical property of these complex distributions is that the mean and median may differ significantly. In the case of non-parametric methods, the distribution is described by a set of quantiles, instead of using a continuous pdf. In this case, the distribution is given in its cumulative form (cdf). As proposed in [37], a non-parametric forecast error cdf F can be defined by gathering a set of m given quantiles such that F = {qj , j = 1, ..., m|0 ≤ α1 < α2 < ... < αm ≤ 1}

(2.3)

where qj are the quantiles and αj associated nominal proportions (or probabilities).

Frequency [p.u.]

The definition in (2.3) is referring to the unit interval for both, proportions and quantiles. In case of forecast error cdf, qj represents the forecast error εj and its range is [–1,1]. Proportions (or probabilities) αj can be expressed in percent. In this case, quantiles are usually termed as percentiles. In Fig. 2.4 the most important parameters of a quantile forecast as described in [37] are shown. An interval forecast I is given by lower and upper quantile forecasts ql and qu . The nominal coverage rate (1 − β) of the forecast is defined as the difference between the lower and upper proportions αl and αu . In the illustrated case the prediction interval I is centered on the median of its pdf. In the example of Fig. 2.4 a typical wind power cdf is depicted, where the mean predicted power pˆmean is not equal to the median pˆmedian .

αu pˆmean

(1 − β)

pˆmedian

0.5

αl (1 − β) : Nominal coverage rate I = [ql , qu ] : Interval forecast

I

ql

qu Forecast [p.u.]

Figure 2.4. Quantile forecast parameters.

14

Chapter 2. Analysis of the uncertainty of wind power forecasts

Note that in this thesis the term “forecast interval” refers to the time interval T over which the forecast is issued. It might be more correct to say “forecast time interval”, but it was decided to use the abbreviated version due to the frequent use of this term in this manuscript. It is considered that confusions with the above mentioned interval forecast I are not likely as the concept of quantile forecasts is not further treated in this work. Recently, several probabilistic, non-parametric approaches for wind power forecasting have evolved due to their interest in relation with large-scale wind power integration. Next, the different approaches are presented. Markov-Chains (Bathurst) A technique to describe the variability of wind power in relationship with the electric energy market is proposed in [38]. In this work Markov-Chains are employed. This implies the creation of probability tables for different forecast power classes (similar to [24, 28]) and forecast horizons. Kernel density estimation KDE (Juban et al.) The approach described in [39] provides complete predictive wind power distributions for every forecast time t + k. A mutual information criterion from information theory is applied to select input variables and model order. The mutual information is a measure of information contained in one random variable about another, where the information is defined by the entropy of the variables. Wind speed and, in much less extent, wind direction are identified as most important meteorological input variables. Thus, the forecast pdf is mostly influenced by the wind speed forecast. Using KDE the actual shape of the pdf is obtained with a discrete-continuous mixed model. Spot forecasts, quantile forecasts and interval forecasts can be derived from the pdf. Adapted resampling (Pinson et al.) In 2004 first articles appeared from the european project ANEMOS, which highlighted the importance of detailed assessment of the uncertainty of wind power forecasts [30, 40– 42]. Pinson and Kariniotakis [40, 42] proposed for the first time the adapted resampling approach as non-parametric method to compute confidence intervals of the forecast. In this method, errors are classified using fuzzy sets derived from the power curve (three power levels and cut-off risk are included). Confidence intervals are fine-tuned (reduction of interval width) in case of stable weather conditions. For this purpose the “Prediction Risk Index” (PRI) was introduced, which describes the spread of multi-scenario NWP (based on the “poor man’s ensemble approach”). This method was introduced in detail in the PhD Thesis of Pinson [36]. The concept of the skill score has been further refined in [37,43,44]. In [43] measures of forecast quality were defined and its operational value in the electricity market explored. In [37] results were presented comparing two methods: adapted resampling and adaptive quantile regression (see below for more details). The advantages of the availability of a pdf for the forecast error for trading wind power in short-term markets was already emphasised in [30] and later refined in [45].

2.1 State of the art of uncertainty analysis of wind power forecasting

15

The same method described in [45] for data from the Dutch Electricity Pool is applied to the Spanish market in chapter 5 of this thesis. Quantile regression Local quantile regression – LQR (Bremnes) Bremnes introduced Local quantile regression (LQR) in [46] for the estimation of a finite number of quantiles of the conditional wind power distribution. One drawback of this method is that only one quantile can be estimated at a time and, hence, must be repeated if several quantiles are considered. In quantile regression, the dependence of quantiles on the random variable x are specified as in linear regression. For wind power, quantile regression can be applied only to data points close to x. This is achieved by weighting the data points such that data points close to x have more impact than those further away. This method is denominated as local quantile regression. In [47] LQR was compared with two other quantile estimators (Local Gaussian and Nadaraya-Watson). It was concluded that these three methods perform very similarly. The local Gaussian estimator only produces mono-modal distributions, which may be a disadvantage in some cases. The Nadaraya-Watson method is recommended if simplicity is required, as it is the case in real-time applications as online monitoring for example. Time-adaptive quantile regression (Nielsen and Møller et al.) Nielsen et al. [48] proposed quantile regression as a technique to extend an existing wind power forecasting system with probabilistic forecasts. The model estimated the 25% and 75% quantiles from data provided by the Danish wind power forecasting system (Zephyr/WPPT2 ). Results can be updated each time a new forecast is available. This way the uncertainty estimation is situation-specific. In [49] Møller et al. proposed time-adaptive quantile regression to account for variations in time of the distribution of data. The so-called simplex approach is applied to get the solution at time t+1, given the solution at time t, which leads to important savings in computational time. Møller described this technique in detail in his Master’s thesis [50].

2

WPPT: Wind Power Prediction Tool

16

2.1.3

Chapter 2. Analysis of the uncertainty of wind power forecasts

Why using conditional distributions?

In all methods described above conditional distributions are considered. This means that the pdf will depend on input variables of the model, such as wind speed, wind direction or directly on the result of a point forecast of wind power. In some cases forecast classes or bins are created and the distribution in every bin is analysed. In the adapted resampling approach the classification is obtained by a set of fuzzy rules. In the case of quantile regression, conditional distributions are obtained without the explicit definition of forecast bins. Kernel density estimation techniques are creating “smooth bins” extracting samples from the data with the help of kernel functions which have its maximum at the desired value of the random variable. The idea of conditional distributions makes sense because it can be observed that the distribution of the error changes dramatically depending on the value of the forecast. If for example a wind power of 0.9 p.u. (90% of the nominal installed power) is predicted, the probability for the actual power to be 0.7 p.u. (error: +0.2 p.u.) would be quite high, while an error of −0.2 p.u. would not even be possible, because the maximum output is limited to 1 p.u. (100%). On the other side, if the forecast is 0.5 p.u., errors of ±0.2 p.u. would have almost the same probability.

2.1.4

Forecast Quality: Risk Index and Skill scores

Already in 2004 a meteorological risk index “meteo-risk-index” MRI obtained from NWP ensemble forecasts was proposed by Pinson and Kariniotakis in [40]. MRI was aimed to enhance traditional point prediction by some information about its uncertainty. This concept was further developed by Pinson [36] to a prediction risk index PRI and so called “skill forecasts” are obtained, i.e. forecasts of the distributions of expected prediction errors. This approach was further refined in [44], where a normalised PRI (NPRI) is introduced. It is shown for a Danish offshore wind farm how NPRI may be related to potential energy imbalances. Energy imbalance is defined as mean absolute error (MAE) multiplied by the forecast interval. In [43] the quality of probabilistic forecasts was evaluated with an overall skill score, including sharpness and resolution. The operational value of probabilistic forecasts was obtained from simulation of a participation of a wind power producer in the Dutch electricity market. The overall skill score introduced in [43] was discussed in [37] more in detail. In this work, complete framework for the evaluation of the quality of probabilistic forecasts was given, based on the concepts of reliability, sharpness and resolution. This framework is applied for evaluating the quality of adapted re-sampling and adaptive quantile regression method. Both methods produce full predictive distributions from point predictions of wind power. The skill score could be a good method to quantify the quality of modelled persistence forecast scenarios, proposed in the next section.

17

2.2 Modelling forecast scenarios using persistence

2.2

Modelling forecast scenarios using persistence

As seen in the preceding section, there are many different wind power forecast methods in use. But every method has to be compared always with the simplest one: the persistence forecast. Therefore this simple approach was chosen to simulate three different scenarios of forecast quality. This allows us to investigate the changes in the error distribution with changing forecast scenario. For better understanding of the persistence method used here, in Fig. 2.5 a measured wind power time series (thin line) and the persistence forecast (step function, bold line) are depicted for a dataset of 1 second mean wind power. The instant t shown in the figure represents the time when the forecast is done, while T accounts for the forecast time interval length, during which the mean wind power is predicted. This time interval has to start later than t, so that a forecast delay k has to be defined. This delay describes the time gap between the instant, when the forecast is done and the beginning of T . In short term energy markets k is termed market closure delay [38].

Wind power [p.u.]

1 0.8

instant of forecast t

0.6 0.4 k

T

0.2 0 0

48

96

144

192

Time [h] Figure 2.5. Wind power time series (thin line) with persistence forecast (bold line), forecast delay k and prediction time interval T (T = k = 24 h).

In Fig. 2.5, the mean wind power, obtained from time interval h0, 24i is the forecast for the interval h48, 72i. Therefore, it can be stated that the persistence forecast in this case is 2T time-shifted relative to the interval, where the mean value was calculated. Using the the nomenclature proposed in [51, 52], persistence forecasts can be written as n−1 1X ˆ P (t − j ∆t) P (t + k|t) = T j=0

(2.4)

where Pˆ (t + k|t) is the wind power forecast for time t + k made at time origin t, k the forecast delay, T the prediction interval length (here T = k), P (t − j ∆t) the measured wind power for time t and the previous j time steps within T , n the number of time steps within T and ∆t the time step length of the measured time series (T = n ∆t). In the following, three forecast scenarios are defined. This way, different levels of forecast accuracy are simulated without using a specific forecast model. If a forecast model is determined, it can be classified according to the scenarios. The forecast is only updated once for each prediction interval, so the forecast value is constant during the

18

Chapter 2. Analysis of the uncertainty of wind power forecasts

whole prediction time interval T (see Fig. 2.5). In this thesis, time intervals T of up to 720 h are considered. This is a difference to forecasts used in practice, where values in 15-min up to hourly steps are provided for time intervals of up to 72 h. The worst case scenario, denoted as T×2, is based on a persistence forecast with k = T . This is the worst case because no forecast model should perform worse than this. The T×1 scenario is simulating an intermediate case. The calculated mean value of each interval is shifted in time by the width of the prediction interval T . This is equal to a persistence forecast with k = 0, which means increased accuracy in comparison with scenario T×2. The best-case scenario, termed T×0 is obtained by assigning the measured mean value in the interval T as forecast for same interval (k = −T ). As a result, a perfect forecast of the mean value for each interval is simulated. The error only consists in the power fluctuations within the interval T . Note that forecasts are constant values over the entire forecast interval T . Very recent publications [53, 54] suggest the application of exponential moving averages (EMA) for forecast simulations (for more details on EMA see section 2.5). Hence, forecasts would no longer be constant within T and errors would decrease. The suggestions in [53, 54] would result in a scenario situated between T×0 and T×1, thus owing to simplicity it has been discarded the incorporation of additional scenarios based on moving averages. The normalised prediction error ε is calculated as the difference between two time series of the same length. Following [51], it can be written as ε(t + k|t) =

1 

Pinst

P (t + k) − Pˆ (t + k|t)



(2.5)

where Pinst is the installed wind power, k is the forecast delay, Pˆ (t + k|t) is the forecast for time t + k for the prediction made at the origin t and P is the measured wind power. Results from forecast quality simulation are verified with data from real world forecast, described in [16,17] (see section 2.6). It is shown that the real forecast error distribution is situated between the T×0 and T×1 scenario. The T×2 scenario appears far too pessimistic. It is concluded that T×1 may be considered as worst case scenario, taking into account advanced forecast methods which combine a NWP mesoscale forecast model and a statistical power curve model (e.g. neural network).

2.3

Data basis – Datasets A, B and C

One-year time series of generated wind power from three sites form the data basis of the investigation presented in this thesis. The three datasets will be referred as datasets A, B and C. Basic statistical parameters of the datasets are shown in Table 2.1. Dataset A contains 10 min means from 32 constant speed 900 kW induction machines from the same wind farm, measured in 2004. The dataset is treated as if it were only one turbine but 32 years of data. The result is a database of around 1 million data values. The great advantage of this dataset lies in its large number of data points which results in very smooth probability distributions. In dataset B, 15 min means were available for three spatially very close wind farms with variable speed turbines. Analogue to dataset A, with the aggregation of data simu-

19

2.4 Application of the Beta distribution – Betafit

Dataset

Mean [p.u.]

NMAE [p.u.]

Std. dev. σ [p.u.]

A B C

0.278 0.302 0.261

0.262 0.295 0.245

0.313 0.340 0.299

Table 2.1. Basic statistical parameters of datasets A, B and C.

lating three consecutive years from one wind farm, around 100 thousand data values are available for statistical analysis. Dataset C contains 30-min means from one wind farm during one year (363 days) which results in 17 424 values. The wind farm comprises 12 equal variable speed wind turbines of 1.8 MW (total Pinst = 21.6 MW). It is situated in northern Portugal, in a complex mountainous region. For this third dataset results from two forecast models were available. According to [17], adaptive forecast systems are trained using the classical Mean Squared Error (MSE) criterion. A new approach, proposed in [17] and further explained in [16] applies a Maximum Correntropy Criterion (MCC) as cost function to train a neural network. Data from these two advanced forecast methods is available in dataset C and will be termed in this thesis as forecast scenarios MSE and MCC. These two real world scenarios are compared with results from the persistence model, presented in section 2.2.

2.4

Application of the Beta distribution – Betafit

In this section, a method based on a parametric probability density function (pdf) is developed. The principal idea was adopted from Bofinger et al. (2002) [28] but essential changes in the method are introduced, so that the result can be considered as a new contribution of this thesis. In this work, a relatively high number of up to 50 bins is used. This leads to more accurate results than in [28], where only 4 bins were considered. Next, the steps of the proposed method are explained and results for the forecast error pdf are presented. This method has been published by the author of this thesis in [31]. The development and evaluation of the proposed pdf approximation is tested with the three datasets A, B and C introduced in section 2.3. For datasets A and B no forecast data was available. Therefore, only forecast errors from the persistence approach (see section 2.2) were used. The performance of the parametric method to model the forecast error pdf is analysed. The application of the Beta pdf is developed in section 2.4.1, where a new, nonlinear function is proposed for the approximation of the parameters standard deviation σ and mean value µ. In section 2.4.2 the relationship between the persistence forecast and the mean measured power related to this forecast is discussed and the findings in [11] could be verified. In section 2.4.3 the calculation of the aggregate forecast error distribution is explained. The obtained forecast error pdf is applied in chapter 3 for ESS sizing and in chapter 5 in a probabilistic bidding strategy for participation in short-term markets.

20

2.4.1

Chapter 2. Analysis of the uncertainty of wind power forecasts

Approximation of a Beta pdf for each power forecast bin

The indirect approach, adopted to obtain the error pdf is based on [28]. First, forecast results are sorted into power classes or bins and the distribution of the measured power within each forecast bin is assumed to follow a Beta pdf. Knowing the pdf associated to each forecast bin, the error pdf of this bin can be calculated. Adding up the error pdfs of all bins, the overall forecast error pdf can be obtained. Consider a time series with data pairs of wind power p and associated forecast pˆ. Data pairs (ˆ p, p) are sorted by forecast values pˆ and assigned to forecast power bins. The bin width must be chosen depending on the number of data available. In this case 50 bins with 0.02 p.u. bin width seem to be a good choice, but if the database is smaller, the number of bins can be reduced down to 10. Even with 5 bins acceptable results are obtained, but only if knowledge from larger, similar time series is included. It is assumed, that all measured values of one forecast bin have the same forecast pˆj , which is the mean value of all the calculated forecasts corresponding to bin j. The distribution within bin j is called fj . The Beta pdf is given by pa−1 · (1 − p)b−1 B (a, b) Z1 B (a, b) = pa−1 · (1 − p)b−1 dp fj (p) =

(2.6) (2.7)

0

where p is the normalised measured wind power in p.u., a and b are the distribution parameters and B(a, b) is the Beta function. As can be seen in (2.7), Beta function B(a, b) is simply the integral of the term in the numerator of (2.6), which normalizes the integral of fj (p) in the interval [0, 1]. Therefore in [28], the following form was chosen to represent Beta pdf: fj (p) = pa−1 · (1 − p)b−1 · n

(2.8)

where n is the normalization factor 1/B(a, b).

In Fig. 2.6 an example is given for a possible result of approximated wind power pdf fj (p) for 9 different predicted power classes j with mean forecast pˆj . As mentioned in [29], the parameters a and b are related to the parameters σ 2 (variance) and µ (mean). Equations (2.9) and (2.10) show these relationships in both directions. σ2 =

a=

a·b (a + b + 1) · (a + b)2

µ=

a a+b

(1 − µ) · µ2 −µ σ2

b=

1−µ ·a µ

(2.9)

(2.10)

Next we show the advantages of using the Beta pdf instead of the Gaussian pdf for approximating the forecast error distribution. As an example, in Fig. 2.7 the histogram

21

2.4 Application of the Beta distribution – Betafit

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Frequency [%]

12 10 8 6 4 2 0 0

0.2

0.4 0.6 0.8 Measured power p [p.u.]

1

Figure 2.6. Approximated Beta pdf of measured wind power for 9 different forecast power classes; dataset A, T×0 forecast 1 h. Dataset A - Tx1, 60min, Bin Nº 5 (0.1)

2

Frequency [%]

10

Bin(0.1) Betafit Gaussian

0

10

-2

10

-4

10

0

0.2

0.4 0.6 0.8 Measured power [p.u.]

1

Figure 2.7. Comparison of the histogram of measured wind power (dots) with Beta pdf (solid) and normal pdf (dashed); forecast power bin: 5 of 50 (forecast: 0.1 p.u.).

of forecast power bin number 5 (ˆ p5 = 0.1 p.u.) is compared to the fitted Beta pdf and the corresponding Gaussian. The distribution of data in this example bin is very fat-tailed with a kurtosis of around 17. The fitted Beta reaches around 5, which is significantly better than the normal pdf (κ = 3). In Fig. 2.7 this can be appreciated for wind powers above 0.5 p.u., where the normal pdf tends much faster to zero than the Beta pdf. However, both tend faster to zero than the actual measured data does. Note that the Gaussian and Beta distributions in this example have the same values of µ and σ as the measured data. The previous example showed that the tail of the wind power distribution cannot always be perfectly modelled with the Beta pdf. As a consequence, the purpose of this chapter is to evaluate its fitting performance, as no better parametric distribution has been proposed yet in literature. After the approximation process, parameter pairs (aj , bj ) and (µj , σj ) are obtained for every forecast bin j. As in [28], the (µ, σ) pair is chosen for the representation of the

22

Chapter 2. Analysis of the uncertainty of wind power forecasts

results. In Fig. 2.8 examples of fitting curves σ(µ) for 1-h and 48-h persistence forecasts are shown in comparison with those published in [29] and [28]. A non linear behavior can be observed in all the investigated forecast scenarios and prediction horizons, while in [29] and [28] a linear relationship between µ and σ was assumed. This nonlinearity can be explained with the power curve of WECS. As shown in [35], the first order derivative of the power curve is strongly related to the uncertainty of the power forecast. At zero and nominal power (0 and 1 p.u.) the derivative is zero, while at around 0.5 p.u. it passes a maximum. The same can be observed for the standard deviation in Fig. 2.8. Dataset A - Tx0

Std. dev. σ [p.u.]

0.4

Persist. 48h Betafit 48h Persist. 1h Betafit 1h Bofinger 48h Fabbri 48h Fabbri 1h

0.3 0.2 0.1 0 0

0.2 0.4 0.6 0.8 Measured power [p.u.]

1

Figure 2.8. Two examples for the approximation of parameter pairs (µ, σ) from persistence forecast together with σ(µ) curves from Bofinger and Fabbri; dataset A, T×0 forecast.

The difference in results may be caused by the different procedures used to obtain the data. In [28], the forecast delay k is 48 h with T = 24 h. Forecasts of one day mean values lead to statistical database of only 365 values. The bias correction, performed on this data reduces the data further to 100 values. As this is a very poor amount of data, the linear approximation was based on only 4, non equally distributed forecast bins. In [29] no information is given about how the linear approximation was obtained from the measured data. In the present work 50 bins with equal bin widths were used. So possibly, the non linear behavior was not revealed in [28] for the lack of power bin resolution. The higher standard deviation of the presented examples is due to the power fluctuations within the forecast interval which were not taken into account in [28]. Considering, for example, a 1 h forecast, in the present work the six corresponding 10-min measured values of this interval were used to obtain the standard deviation, while in [28], every forecast interval only has one forecast associated to the measured mean value. Equation (2.11) is proposed as a new approximation function. σ=

p

µ (1 − µ) · (−c2 µ2 + c1 µ)

(2.11)

where coefficients c1 and c2 are the approximation parameters. The curves in Fig. 2.8 named “Betafit” show two examples of this approximation. The special form of the function is due to the fact that the parameter a cannot be negative.

23

2.4 Application of the Beta distribution – Betafit

From (2.10) follows (2.12) which shows clearly that σ 2 must be zero at the boundaries of the interval σ = [0, 1]. (1 − µ) µ2 − µ ≥ 0 ⇒ (1 − µ) µ ≥ σ 2 (2.12) σ2 Therefore it is practical to use the term (1 − µ) µ as a factor in the polynomial approximation of σ 2 . The analysis of different prediction horizons up to 30 days (T = 720 h), have shown that good fitting results can be obtained with only two polynomial coefficients. In equation (2.11) the coefficient c2 is given a negative sign in order to obtain positive values for both, c1 and c2 . In Fig. 2.9 coefficient values are shown as a function of T for the three forecast scenarios. It can be observed, that the three datasets show quite different behavior, although the global tendency is similar. This can be explained by the power curves as described in [35]. Dataset A was taken from constant speed wind turbines and datasets B and C from variable speed machines. a=

Dataset A

Dataset B

Dataset C

3

3

3

2

2

2

1

1

1

0 0.5 2

6

24 72

T [h]

720

0 0.5 2 6

24 72

720

T [h]

0 0.5 2

c2 Tx2 c1 Tx2 c2 Tx1 c1 Tx1 c2 Tx0 c1 Tx0

6

24 72

720

T [h]

Figure 2.9. Approximation parameters c1 and c2 of σ(µ) as a function of T for forecast scenarios T×0, T×1, T×2 and datasets A, B and C.

Very interesting is that the coefficients in dataset B are almost identical. Equal values of c1 and c2 mean that the curve σ(µ) is symmetric. If the symmetric case is assumed, the approximation function can be simplified as expressed in (2.13). The only remaining parameter c can be calculated as the square root of the former parameters c1 = c2 . σ = c · µ (1 − µ)

(2.13)

The observed symmetric σ(µ) from dataset B does not seem to be the normal case, as the other two datasets do not show this characteristic. The analysis of more datasets is needed to determine if the case of dataset B is an exception.

24

Chapter 2. Analysis of the uncertainty of wind power forecasts

2.4.2

The relationship of mean measured power and forecast

In [28] and [29] σj within a forecast bin j is represented as a function of the mean predicted power pˆj of this bin. It must be noted that these values can only be interpreted directly as parameters of the Beta pdf if pˆj is equal to the mean measured power µj in this bin. Indeed in [28] and [29] µj = pˆj is assumed, but without emphasizing that the parameter µj of the Beta pdf must be derived from the measured data, and not from the forecast data. In [28] a simple neural network is applied to correct the initial forecast with the aim to approach µj = pˆj , while in [29] it is only stated that the forecast will be “centered in the mean value”. However, it will be shown now that this assumption is not valid for the persistence forecast. If a linear relationship between pˆ and µ is considered and it is further assumed that all curves cross the point of the long term mean, the fitting function µ(ˆ p) can be written as µ = u · pˆ + (1 − u) · p¯

(2.14)

where pˆ is the mean forecast within a forecast bin, u is the slope of the linear fit, and p¯ is the normalised long term mean. In Fig. 2.10 linear approximations following (2.14) for several prediction intervals T are shown for the example of T×2 scenario of dataset A. As shown in the left plot, for short forecast delays up to 1 h, pˆ = µ is almost true, but this is not the case when the forecast interval T becomes larger. Dataset A - Tx2

Dataset A - Tx2

Mean measured µ [p.u.]

1 0.8

1

1h data 1h fit 14d data 14d fit

0.8

0.6

0.6

0.4

0.4

0.2 0 0

0.2

Longterm mean 0.2

0.4

1h 6h 24h 3d 30d

0.6

Mean forecast [p.u.]

0.8

1

0 0

0.2

0.4

0.6

0.8

1

Mean forecast [p.u.]

Figure 2.10. Linear approximation of the mean forecast and measured mean within a forecast bin, examples T×2 forecast, dataset A; left: T = 1 h and 336 h (14 days), right: approximations from 1 h up to 30 d.

It can be seen that the slope u of the linear fit depends strongly on the forecast interval length T . While the linearity is maintained, the slope of the curve tends to zero for very large values of T . In fact, for the 14-day forecast, the mean value of measured power is almost constant for all forecasts and equal to the long term mean (in this case the total mean of the entire dataset). In other words, long term mean p¯ value becomes the best forecast for large forecast intervals. Therefore in [11] a modified persistence model has been proposed as a new reference (cited also in [10] and [51]), which takes into account this effect. This new reference can be written as

25

2.4 Application of the Beta distribution – Betafit

PˆNR (t + k |t ) = ak · Pˆ + (1 − ak ) · P

(2.15)

where PˆNR is the new reference forecast, ak is the correlation coefficient between P (t) and P (t + k), Pˆ is the persistence forecast and P is the long term mean. Equations (2.14) and (2.15) have the same structure of a linear function, with the only difference that in (2.14) the slope is u and in (2.15) it is the autocorrelation coefficient ak . Using (2.14), slopes u are calculated for all considered prediction intervals T . In Fig. 2.11 u(T ) and ak (T ) for the datasets A and B forecast scenario T×1 are shown. Results for T×2 are similar and u for T×0 must obviously be unity. Forecast scenario Tx1 1 0.8 0.6 0.4 0.2 0

u (dataset A) ak (dataset A) u (dataset B) ak (dataset B) 0.5 1

3

6 12 24 48

168

720

Prediction interval T [h] Figure 2.11. Comparison of the autocorrelation coefficient ak and slope u of the linear fit as a function of prediction interval T ; forecast scenario T×1, datasets A and B.

From Fig. 2.11 the similarity between u and ak is confirmed, but also some differences can be found. While for small values of T the slope u is almost equal to the autocorrelation coefficient ak , it is greater than ak for higher values of T . This can be explained by the persistence forecast approach chosen in this work. For large values of T , the forecast is the average value over a large time interval. The autocorrelation does not take into account this averaging while u includes it. In dataset A, u is almost equal to ak up to around T = 10 h, but in dataset B the similarity only holds up to T = 2 h. This can be due to the smaller database size of dataset B (10 times smaller) which makes the calculation of u less reliable. Interestingly, the autocorrelation of dataset A and B is very similar, which shows that it is less sensitive to the specific properties of the data available.

2.4.3

Obtaining forecast error pdf from Beta distributions

Knowing the pdf of the measured wind power within every forecast bin, the error distribution of each bin can be obtained easily by subtracting the mean predicted power of the bin from the amplitude axis of the pdf. The result for 9 of the 50 bins chosen in this case can be seen in Fig. 2.12. Once the pdf of the forecast error fj (ε) is known for all bins, the total forecast error pdf f (ε) can be calculated by adding up all fj (ε). The pdf for each bin is normalised to 100%, which requires a weighted sum. The weight wj of each bin is equal to the

26

Chapter 2. Analysis of the uncertainty of wind power forecasts

T=1h, Tx2 - Dataset B

Frequency [%]

8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

6 4 2 0 -1

-0.5

0

0.5

1

Forecast error ε [p.u.] Figure 2.12. Superposed forecast errors for 9 different forecast power bins for a 1-h T×2 forecast using dataset B.

probability that a forecast will belong to it. A weighted sum is necessary because the probability that a forecast belongs to the bin with pˆ = 0.1 p.u. may be 10 times higher than it is for the bin with pˆ = 0.9. The weighted sum can be written as f (ε) =

n X j=1

wj · fj (ε)

(2.16)

where f (ε) is the total forecast error pdf, j the bin number, n the total number of bins, wj is the weight and fj (ε) is the forecast error pdf in bin j. Values of wj can be obtained from a histogram of long term forecast data with n bins. This histogram can be understood as a discrete weight function w(j). The shape of w(j) depends strongly on the forecast interval length T as shown in Fig. 2.13, where the histogram of long term forecast data is represented for 10-min and 24-h intervals. The frequency values shown for the bins from 0.8 to 1 p.u exhibit differences of one order of magnitude (factor 10) or more. 100

Frequency [%]

10min 24h 10 1 0.1 0.01 0

0.2

0.4

0.6

0.8

1

Mean forecast [p.u.] Figure 2.13. Comparison of wind power histograms (bin width 0.01) of 10-min means (thin line) and 24-h means (bold line).

27

2.4 Application of the Beta distribution – Betafit

The forecast error pdf by itself is an important result of this study. In the following some examples are shown and its accuracy is discussed. As mentioned before, the tail of the error pdf is of special relevance to this study. Therefore all results are shown in semi logarithmic plots for a better visibility of the tail region. In Fig. 2.14 observed forecast error pdf (thin line) and its Betafit approximation (bold line) are shown for T×2 forecasts. For the 1-h forecast (left side), the Betafit method is not able to model the positive tail of the forecast error. For the 24-h forecast (right side) the algorithm performs very well because it is able to approximate unexpected distributions like the one shown in Fig. 2.15. T = 60min, Tx2 (79914 values)

T = 24h, Tx2 (82856 values)

Frequency [%]

100

100 observed Betafit

10

10

1

1

0.1

0.1

0.01 -1

-0.5

0

0.5

observed Betafit

1

0.01 -1

Forecast error ε [p.u.]

-0.5

0

0.5

1

Forecast error ε [p.u.]

Figure 2.14. Comparison of observed error distribution and its Betafit approximation (bold line); left: 1-h right: 24-h T×2 forecast, dataset B.

Dataset B - Tx2, 24h, Bin Nº 40 (0.8) 100

Frequency [%]

Bin(0.8) Betafit 10 1 0.1 0.01 0

0.2

0.4

0.6

0.8

1

Measured wind power [p.u.] Figure 2.15. Fitted wind power distribution in bin 0.8 p.u., 24-h T×2 forecast, dataset B.

28

2.5

Chapter 2. Analysis of the uncertainty of wind power forecasts

Bias correction of real world forecasts

In general, wind power forecasts should be bias-free (zero mean error). Indeed most state-of-the-art forecasting tools include dynamic correction, correcting seasonal trends. But to the author, no data from such forecasts was available and dataset C shows that in real world applications is not guaranteed that, that forecasts are properly corrected. The two forecasts available from dataset C are a good example of the spread of forecast error distributions, even from the same wind farm. It will be shown in this thesis that bias corrected data is necessary for for ESS sizing. Forecast error compensation with energy storage (ESS) demands zero mean error in order to establish a stationary regime of the state of charge (SOC). Otherwise infinite capacity would be needed. In addition it will be shown in section 2.6 that bias-corrected data is needed for a proper approximation of error distributions. This will be achieved because non-linear relationships between forecast power class and mean wind power per bin are corrected and approximation quality is improved significantly.

2.5.1

Bias correction “a posteriori”

For ESS sizing a posteriori bias correction may be possible. It is assumed that forecast errors are available for a long period of time (not less than one year). Thus, a first trivial approach for bias correction would be calculating the mean error ε¯ over the whole observation period and subtract it from every forecast value, as in (2.17). pˆc (t) = pˆ(t) − ε¯

(2.17)

Note that the corrected forecast time series pˆc (t) may contain impossible forecasts with negative values or values beyond one. In our case negative forecasts are set to zero and forecasts beyond one are set to one, which leads to a residual bias. In general this is a minor problem, but an iterative process might be applied to obtain zero mean. I will be shown, that thanks to the proposed bias correction forecast error distributions are easier to approximate with Betafit and shapes are approaching those observed from persistence forecast scenario T×1. For ESS sizing another condition is important which is not that trivial as the total bias. The forecast error should have zero-mean also in sufficiently small time-steps, such as one day or at least one week. While persistence models provide this condition naturally (see appendix B.2), observed real world forecasts are sensitive to meteorological situations. In case of dataset C in summer, with prevailing low wind speeds, an over-prediction is observed, while in spring and autumn high wind speeds lead to a tendency of underprediction. The trivial approach in (2.17) is not able to correct seasonal bias or error trends. Error trends become visible if error time series ε(t) are integrated over time. This new time series represents the state of charge of an ESS system with 100% storage efficiency (see section 3.3.2). In Fig. 2.16 time series of normalised SOC are depicted for bias corrected MSE and MCC forecast from dataset C, using (2.17). For comparison, the curve obtained from scenario T×1 is included in the same figure. The seasonal trend of forecast error causes large deviations of SOC, while with the persistence scenario T×1, only values of SOC between zero and 1 p.u. are observed.

29

2.5 Bias correction of real world forecasts

SOC [p.u.]

2

Tx1 MCC MSE

0 -2 -4 -6 -8

0

60

120

180

240

300

360

Days of the year Figure 2.16. Normalised state of charge for time series from bias corrected MSE and MCC forecast compared with T×1 scenario.

2.5.2

Online bias correction with moving averages

A correction of seasonal trends a posteriori is difficult to define. Forecast errors could be reduced by any degree, only choosing sufficiently small windows for trend correction. The main questions that arise are: • What is the adequate window width for trend correction? • How this correction can be implemented in an operating scheme of an ESS, attached to a wind farm? The answer to these questions leads us to online bias correction which takes into account real world operating conditions. For online bias correction, moving averages (MA) of past forecast errors can be used. In addition a time shift, analogue to the T×2 persistence scenario (see section 2.2) is applied. This method can be implemented easily in the operation of the ESS and can even be included in the bidding process in a market situation (see chapter 5). The result is a persistence forecast of the expected bias. Hence the mathematical formulation of this approach shown in (2.18) is analogue to the persistence forecast, only that in this case bias of the forecast error is predicted. Of course more sophisticated statistical forecast methods such as auto-regression or neural networks can be considered, but this is outside the scope of this thesis. n−1

1X εˆ(t + k|t) = ε(t − j ∆t) τ j=0

(2.18)

where εˆ(t + k|t) is the bias forecast for time t + k made at time origin t, k the forecast delay, τ the window width of averaging, ε(t − j ∆t) the observed forecast error for time t and the previous j time steps within τ , n the number of time steps within τ and ∆t the time step length of the measured time series (τ = n ∆t). As a consequence, the online-corrected forecast time series pˆoc (t) is obtained from the difference between wind power forecast and expected bias at any time step t. pˆoc (t) = pˆ(t) − εˆ(t)

(2.19)

30

Chapter 2. Analysis of the uncertainty of wind power forecasts

Weighted moving averages (WMA) Weighted moving averages (WMA) are calculated including multiplying factors to give different weights to different data points. If weights are considered, from (2.18) we obtain (2.20). n−1

εˆ(t + k|t) =

1X w(j) ε(t − j · ∆t) τ j=0

(2.20)

where w(j) is the weight function for every data point within the averaging window. When w(j) is considered as a function of j, any functional relationship can be assumed. But most commonly the following formulation is adopted: w(j) =

(n − j)λ n−1 X (j + 1)λ

j = 0, 1, ..., n − 2, n − 1

(2.21)

j=0

where w(j) is the weight function for every data point within the averaging window and λ is the weighting exponent. Following the definition in (2.21), for positive values of λ WMA gives more weight to recent values. It can be seen that the simple moving average with equal weight of every data point is a special case, when λ = 0. With λ = 1 the linear moving average is obtained. Other special cases are the square root WMA (λ = 0.5) and the square WMA (λ = 2). Both are not relevant in our case. Another variation is the exponential moving average (EMA). It is calculated in a slightly different way. Here the window width (τ or number of data points n) is used to calculate the weighting exponent λ. The main difference is that values outside the defined window are included in the calculation. Matlab r uses the following definition to calculate the EMA of time series of x. 2 (2.22) n+1 ˆ1 = x1 has been adopted. Note that the first value x ˆ1 has to be defined. In this case x The corresponding equation for the calculation of εˆ is given in (2.23). xˆt = λ xt + (1 − λ) xˆt−1

with

λ=

εˆ(t + k|t) = λ ε(t) + (1 − λ) · εˆ(t + k|t − ∆t)

(2.23)

Evaluation of averaging window width τ The critical parameter is averaging window width τ . Due to the time shift k, small values of τ will affect negatively the forecast performance. Overall forecast errors increase and the frequency of “impossible forecasts” will be high. Larger averaging windows result in smaller values of εˆ. This reduces the impact on overall performance of the forecast but bias correction is less effective. The impact of τ on standard deviation of SOC, total bias, MAE and RMSE is depicted in Fig. 2.17 for τ between 1–14 days. In the legend “T×1” refers to results from persistence scenario T×1, “ma0” refers to simple MA (λ = 0), “ma1” is linear MA (λ = 1) and “ema” is EMA bias correction.

31

2.5 Bias correction of real world forecasts

MCC

std(SOC) [p.u.]

MSE 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

Total Bias [%]

0

MAE [%]

4

7

10

14

0

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

1 2

4

7

10

14

-0.4

20

20

19

19

18

18

17

17

16

RMSE [%]

1 2

1 2

4

7

10

14

16

28

28

27

27

26

26

25

25

24

24

23

1 2

4

7

10

Window width τ [days]

14

23

Tx1 ma0 ma1 ema

Tx1 ma0 ma1 ema

1 2

4

7

10

14

Tx1 ma0 ma1 ema

Tx1 ma0 ma1 ema

1 2

4

7

10

14

Tx1 ma0 ma1 ema orig bias

Tx1 ma0 ma1 ema orig bias

1 2

4

7

10

14

Tx1 ma0 ma1 ema orig

Tx1 ma0 ma1 ema orig

1 2

4

7

10

14

Window width τ [days]

Figure 2.17. Standard deviation of SOC, total bias, MAE and RMSE for different moving averages as a function of window width τ for correction of MSE and MCC data.

32

Chapter 2. Analysis of the uncertainty of wind power forecasts

Original (uncorrected) values “orig” are given for MAE and RMSE. Finally MAE is also given for initial bias correction a posteriori, denoted as “bias” in the legend. Standard deviation of SOC was chosen to visualize the capacity of bias correction to minimize the range of variability of SOC. All moving average methods show a minimum of ≈ 0.45 between 3–7 days. The value of 0.24 from T×1 scenario is never reached. If window width is smaller than 3 days, values of εˆ are not representing well enough the error trend due to the time shift of online-correction (48 h here). Windows larger than 7 days leave uncorrected trends with smaller time constants and deviations of SOC will grow. From this parameter, an optimum might be at τ = 4 days. The second parameter is the total bias, which represents the effectiveness of bias correction itself. No iterative method is applied and as a consequence total bias is not eliminated completely due to the correction of “impossible forecasts”. Interestingly, results from MSE and MCC data differ considerably. With MSE data the bias is in almost all studied cases negative. MCC data shows similar trends but values are shifted by 0.2 percentage points, so that for τ > 4 days bias is positive. For the selection of appropriate values of τ a residual bias of ±0.2% is considered acceptable, but other limits may be adopted. Following this criterion, values of τ of 3–7 days obtained from the standard deviation of SOC are acceptable. The third and fourth parameters are MAE and RMSE of the resulting forecast after bias correction. Here it can be seen clearly that with growing τ , errors are reduced. Taking into account acceptable ranges from the first two parameters, an averaging window width of τ = 7 days seems reasonable. Larger values would attain better forecasts but deviations of SOC would become too large. Note that for τ = 7 days, moving averages obtain values of MAE even below those from bias correction a posteriori. For the case of MSE data, this means that moving average correction achieves even better results of MAE than original forecast. Comparing the three moving averages, the exponential moving average EMA seems to achieve best results, especially for window widths below 7 days. But for 7-day windows simple MA (with constant weights) performs equally. Higher order weighted averages such as the linear MA do not improve the performance over simple MA. In Fig. 2.18 time series of SOC are shown for the three different moving averages with window width τ = 7 days. Simple and linear averages ma0 and ma1 perform very similarly, although ma1 produces slightly lower maximum deviations during the peaks around days 290 and 340. Exponential MA ema in general produces slightly lower values of SOC. For 7-day averaging window, linear moving average ma1 performs best in terms of deviations of SOC. In Fig. 2.19 time series of SOC are shown for simple MA ma0 for different values of τ . For 1-day averaging window the negative tendency shows that SOC is not stable. Total bias is not eliminated correctly. For larger windows of 7 and 14 days, bias is eliminated well. It is also seen clearly the larger deviations of SOC produced with 14-day window. Again peaks of days 290 and 340 are good indicators. As a conclusion, simple moving average with averaging window τ = 7 days is considered as the best choice for bias correction.

33

2.5 Bias correction of real world forecasts

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34

Chapter 2. Analysis of the uncertainty of wind power forecasts

2.6

Verification of persistence forecast simulation

Dataset C is used for comparison of results from persistence forecast and real world forecasts MSE and MCC (see section 2.3 for more details on dataset C). Forecast interval T is fixed at 24 h and five different forecast scenarios are evaluated: T×0, T×1, T×2, MSE and MCC. It may be mentioned that MCC forecast data shows a remarkable bias (mean forecast error ε¯) of 7.4% of nominal power. Therefore bias-corrected versions of MSE and MCC, denoted as MSEc and MCCc are included in the analysis. The bias correction algorithm with simple moving average and 7-day averaging window has been applied (see section 2.5 for more details). After bias correction, MSEc and MCCc forecasts are very similar. For MSE overall forecast in terms of MAPE (mean absolute percentage error) is improved from 18.1% to 17.6% of Pinst . The MCC forecast becomes slightly worse with an increase of MAPE from 16.6% to 17.5%. In section 2.5 it was already shown that bias correction with moving averages reduces seasonal bias trends. In this section it will be seen that other statistical parameters also improve significantly. Only to give an example, skewness is reduced from approximately 0.9 to 0.7 for both forecasts MSE and MCC. A summary of basic forecast error parameters is given in Table 2.2 for the seven scenarios considered here.

values in % of Pinst Forecast method

MAPE RMSE Bias Median Kurtosis Skewness

Persistence T×0 Persistence T×1 Persistence T×2

11.6 19.7 23.4

16.9 27.5 32.1

–0.00 0.17 0.34

–1.00 –1.12 –1.16

5.1 3.9 3.5

0.45 0.27 0.26

MSE MSEc MCC MCCc

18.1 17.6 16.6 17.5

23.4 23.7 23.4 23.8

–0.72 –0.03 7.37 0.20

–6.76 –3.28 –0.07 –3.71

4.0 4.1 4.4 4.3

0.86 0.68 0.91 0.70

Table 2.2. Comparison of forecast error parameters of different scenarios, all based on 24-h forecast of dataset C.

From Table 2.2 it may already be concluded that real forecast distributions can be situated between scenarios T×0 and T×1. Values of MAPE, RMSE and kurtosis lie between these two scenarios. Nevertheless, important differences can be observed for the parameters bias, median and skewness. While persistence forecasts are almost unbiased, real forecasts can be biased. The median of persistence forecasts is also near zero. The real forecasts available here are all significantly more asymmetric than persistence forecasts. This asymmetry can be measured with the skewness coefficient (see appendix A.3 for more details).

35

2.6 Verification of persistence forecast simulation

2.6.1

Goodness of Betafit approximation

Distributions for six forecast scenarios of dataset C are depicted in Fig. 2.20. A semilogarithmic plot has been chosen to show the accuracy of the approximation with the Beta distribution fitting (named “Betafit”), especially in the tail region. It can be observed that the simulated persistence forecasts in the upper row are approximated very well, while real world forecast data shows some problems in the approximation. The Betafit model performs rather well with traditional MSE and bias corrected MCCc forecast data, while the approximation of MCC forecast error pdf is not very satisfying. The pdf of MSEc scenario is not included, because it is almost identical to MCCc.

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Figure 2.20. Fitting the forecast error distribution of a 24-h forecast with persistence approach (above) and real forecast data (below).

In order to evaluate the goodness of Betafit approximation for the different forecast scenarios, the Anderson-Darling (A-D) test [55–57] is chosen. It is a modification of the Kolmogorov-Smirnov (K-S) test, giving more weight to the tails than does the K-S test. Both tests are distribution-free which means that no distribution parameters have to be estimated. The K-S test has the advantage that its critical values are known for any distribution. The A-D test in this case is not able to give an answer to the null hypothesis, but relative comparisons between the forecast scenarios are possible with more precision in the tail region of the distributions. The well known Pearson’s chi-square goodness of fit test (χ2 ) is not suitable for our case study, because for large sample sizes it is almost certain that the null hypothesis is rejected [33]. Reducing the sample size would mean reducing the bin number which makes it impossible evaluating the tails of the distribution. More details on the calculation of the goodness-of-fit tests is given in Appendix A.5. In Table 2.3 results of the A-D, K-S and the χ2 test are shown. The critical value

36

Chapter 2. Analysis of the uncertainty of wind power forecasts

for the K-S test is ≈ 0.095, thus, the K-S test rejects the null hypothesis for the Betafit approximation only for the MCC scenario. Hence, it can be stated that reasonably good approximations are obtained for all other scenarios. See appendix A.5.2 for more details on K-S test results. The A-D test shows that best approximations are obtained for the T×0 scenario with test score of 25. MSEc, MSE, T×2 and MCCc have intermediate scores between 122−181. For T×1 Betafit performs slightly worse with 241, while for MCC clearly the worst result is obtained with 1635. Here the sensitivity in the pdf-tails of the A-D test statistic compared to K-S and χ2 tests becomes visible. It may be noted that all approximations are rejected by the χ2 test. It produces large numbers due to the large sample sizes and tough is inadequate here. As a conclusion, the A-D test is far more appropriate to evaluate the goodness of fit in our case. Forecast scenario Persistence T×0 Persistence T×1 Persistence T×2 MSE MSEc MCC MCCc

A2

K-S

χ2

25 241 170

0.070 0.067 0.045

3400 8919 5425

148 122 1635 181

0.083 0.067 0.122 0.065

7522 5900 46072 6957

Table 2.3. Anderson-Darling (A2 ), Kolmogorov-Smirnov (K-S) and Chi-square (χ2 ) test results for goodness of Betafit approximation of error pdf.

The poor performance of the Betafit model in case of MCC forecast can be attributed widely to the assumption that a linear relationship should exist between mean measured power µj and mean forecast pˆj per bin (see section 2.4.2 for more details). It can be observed in Fig. 2.21 that this assumption holds well for persistence forecasts (upper row). In the lower row of the same figure, real forecast data reveal the general tendency of a slight curvature (with negative first derivative). For high values of wind power, both real world forecast models are unable to predict more than 80% of the nominal power (0.8 p.u.). Even though, approximations for MSE and MCCc data are quite good while the MCC forecast data shows a remarkable deviation from the linear assumption. The curve not only is non-linear, in addition the bias can be seen clearly (almost all measured points lie on the left side of the approximation line). It is interesting to note that in this representation MCC is the only case where total mean p¯ is not predicted well, being at the same time the only non-zero mean predictor. Another critical issue is the approximation of the relationship between mean measured power per bin µj and standard deviation σj (see section 2.4.1 for more details). In Fig. 2.22, results for the same six forecast scenarios are represented. Again, the persistence model data achieves best approximation results. It can be seen also, that good results can be achieved even with only 10 forecast bins. As can be seen with the scenarios T×1 and T×2 the approximation method is very robust against

37

2.6 Verification of persistence forecast simulation

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Figure 2.21. Linear approximation of mean forecast and mean generated wind power per forecast bin for 24-h forecasts with persistence model (above) and advanced forecast models (below).

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Figure 2.22. Approximation of parameter pairs (µj , σj ) for 24-h forecasts with persistence model (above) and advanced models (below).

38

Chapter 2. Analysis of the uncertainty of wind power forecasts

outliers. This way, even with small statistical database (not more than one year), stable results can be obtained. Looking at the advanced models in the lower row of Fig. 2.22, contrary to the observation in Fig. 2.21, the accuracy of the approximation is very similar for the three cases considered. In all three cases the standard deviation σ is underestimated for low and high wind power values and overestimated for intermediate values. The same effect has been described in section 2.4.1 (Fig. 2.8, p. 22). This is a systematical shortcoming of the method. Results can be improved applying a higher order polynomial as approximation function, which may be able to adapt to this typical characteristic of σj (µj ). Finally it may be noted that MCCc data does not show the anomalies in the highest wind power bins. This is due to the elimination of the saturation effect as already seen in Fig. 2.21. As a conclusion for practical applications, biased (non-zero mean) distributions can be approximated well, correcting the bias before the approximation procedure.

2.6.2

Forecast model comparison with pdf and cdf

In this paragraph probability distributions of simulated and real forecasts are compared by its probability density (pdf) or cumulative distribution functions (cdf). The pdf representation permits an examination of the tail region of the distributions, while the cdf permits an evaluation of its quantiles or percentiles respectively. The aim of this comparison is to analyse the capability of the persistence forecast simulation, to generate useful statistical information about possible forecast errors. As explained in detail in section 2.2, the three simulated forecast scenarios are T×0, T×1 and T×2, where T×0 is the best case and T×2 the worst case, being T×1 an intermediate case. A good measure for the comparison of two empirical distributions is the two-sample Anderson-Darling test statistic A2mn (see Appendix A.5.3 for more details). As can be seen in Table 2.4, best approximations are obtained with the combinations MSEc-T×1 and MCCc-T×1. This means that the T×1 scenario is the best of the persistence models to simulate distributions of bias corrected time series MSEc and MCCc. It can also be seen clearly, that the distribution of the MCC forecast cannot be modelled properly with any of the persistence scenarios. Forecast method

T×0

T×1

T×2

MSE MSEc MCC MCCc

808 684 1191 721

580 323 1123 325

1281 980 1887 988

Table 2.4. Two-sample Anderson-Darling test for all combinations of persistence and advanced forecast error cdf.

The simulated scenarios can be considered as useful if real forecast data can be described as bounded by two of the established scenarios. To clarify this concept, in Fig. 2.23

39

2.6 Verification of persistence forecast simulation

Frequency [%]

pdf and cdf of MSE (left) and MCC forecast (right) are depicted together with the persistence model. The pdf of MSE forecast can be considered as bounded by the T×1 scenario, although the negative errors (overprediction) are only slightly higher than for T×0. The MCC forecast pdf is very asymmetric. While the negative error is even lower than for T×0, the positive error (underprediction) even exceeds in some cases the T×2 scenario. As a conclusion, especially the tail region can be classified very well with distributions obtained from the persistence model. Roughly, tails exceeding ε = ±0.25 p.u. can be considered as bounded by the scenarios T×1 and T×0. In Fig. 2.23 it can be seen very well how the cdf of MSE and MCCc forecasts are situated between T×0 and T×1 in a wide range of ε. In addition a great similarity of MSE and MCCc cdf is observed. The MCC forecast in turn shows an even better performance than T×0 for negative errors, but is hardly better than T×2 for positive errors. Bounds from the persistence model are not useful here. Given the similarity between bias corrected MCCc and MSE data, it is reasonable to think that adding the bias to the T×1 scenario, as good results as those achieved for MCCc can be obtained for MCC. Results for MSEc are not shown because they are almost identical to MCCc. As a consequence it can be concluded that in case of strongly biased forecast models, the bias must be known for the successful application of the persistence model.

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40

Chapter 2. Analysis of the uncertainty of wind power forecasts

2.7

Summary

In this chapter possibilities of the analysis of forecast uncertainty have been discussed. In the literature, parametric and non-parametric distributions have been proposed to describe the uncertainty of wind power forecasts. For this thesis a parametric approach is adopted. The higher moments skewness and kurtosis of the distributions of wind power predictions have been studied. These shape factors indicate clearly that the normal distribution, as proposed in a number of publications, is not valid. Recent works especially around the ANEMOS project are confirming this observation. It is even stated that parametric distributions are unlikely to be able to describe properly wind power distributions. Three important novel developments have been presented in this chapter: • A simple algorithm to generate time series of wind power forecast errors • Correction and refinement of a parametric method for error pdf estimation • Online bias correction of real world forecast data As usually no long-term data of forecasts is available, a method has been developed to simulate several forecast qualities with the persistence approach. Three scenarios are considered: T×0 as best case, T×1 as intermediate and T×2 as worst case. Based on the simulated forecast data, the novel Betafit algorithm has been created to obtain a parametric probability density function (pdf) of wind power forecast error. The principle of approximating a Beta distribution, conditioned to the predicted power level is not new. Nevertheless, in this thesis fundamental conceptual corrections and refinements have been introduced. The Beta pdf was chosen, because it models fairly well the conditional distributions of wind power. One important advantage of the Beta pdf is that it has variable shape factors skewness and kurtosis, which makes it suitable to model the fat-tailed and skewed forecast error distributions. But even though, it should be mentioned that for forecasts below 24 h, Betafit is not always able to model sufficiently well observed distributions. In these cases tails of the resulting pdf are too short which leads to an underestimation of the frequency of the largest errors. Important discrepancies with former publications were found by analysing the relationship of mean and standard deviation of the measured wind power. No linearity could be found and a polynomial approximation is proposed. This approximation may be further improved with a polynomial of higher order or piecewise approximation. As a result, substantial improvements have been achieved for an existing approach which already has been used in other publications. Possible consequences on the results of these publications have not been studied, but are an interesting field of further investigation. Finally, results are verified with real world 24-h forecast data. Real world forecasts may be biased, which is not desirable for many reasons, as will be seen later in this thesis. Therefore, a simple bias correction algorithm, based on moving averages has been proposed. Two different forecast models were compared with the persistence

2.7 Summary

41

approach. The method based on traditional MSE criterion showed very good agreement with the T×1 scenario. If the strong bias of the novel MCC forecast (based on maximum correntropy criterion) is removed, results are very similar to MSE. If forecast data is biased (such as MCC), Betafit is unable to estimate properly error distributions. Further, these distributions cannot be simulated with the persistence approach. Statistical goodness of fit tests show that the proposed online bias correction is very effective in order to make sure that the Betafit method approximates real 24-h forecast errors with similar success as for the simulated T×1 forecast data. As a conclusion, the proposed persistence approach and Betafit method are only valid if real forecast data is unbiased. This can be guaranteed with the suggested online bias correction. Regarding ESS sizing, it is concluded that taking into account bias-free advanced forecast methods, time series obtained with the proposed persistence method are valid. In the next chapter it will be shown how the forecast error pdf obtained from Betafit for example, can be used for ESS sizing. Further the importance of bias correction for ESS sizing will be demonstrated. The method for online bias correction proposed in this chapter is very simple. Further improvements with auto-regressive methods or neural networks may obtain even better results. In some sense this procedure may me termed “bias-forecasting”, because all statistical forecast methods can be applied to the bias reduction. Bias-forecasting is certainly another promising field of further investigation, although outside of the scope of this thesis.

Chapter 3 ESS sizing based on forecast uncertainty “The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.” Albert Einstein (1879-1955) German scientist

In this chapter a novel probabilistic energy storage (ESS) sizing methodology is developed. The aim of this method is to determine the ESS size needed for reducing wind forecast errors. Thus, deviations from wind power forecasts constitute the input of the ESS under study. First, energy and power requirements are calculated for complete compensation of forecast errors. It is shown that this leads to very large ESS, therefore sizing is focussed on finding reduced values of ESS energy and power, obtaining at the same time the associated quantities of unserved energy as an indicator for the residual forecast error. In section 3.1 a short overview is given of the state of the art of probabilistic ESS sizing and the method proposed in this thesis is outlined. In section 3.2 unserved energy due to reduced ESS power is obtained from the forecast error pdf. The Betafit algorithm from chapter 2 is applied for pdf estimation and its accuracy is verified with simulated and real world forecast data. Unserved energy due to reduced ESS energy capacity cannot be obtained straightforward. In section 3.3, saturation times are defined first and then, corresponding ESS sizes (capacities) are obtained from cumulative distribution functions (cdf) of the state of charge (SOC). With the help of time step simulations, estimates based on saturation times are adjusted. Further investigation is necessary to refine the proposed estimation of unserved energy. In section 3.4 a case study is carried out to verify the developed methodology with real world forecast data. Results show that persistence forecast models are suitable for ESS sizing, but only with additional knowledge of its weak points. Bias correction of real world forecast data is identified as crucial for successful ESS sizing. Bias correction improves sizing accuracy and reduces energy and power requirements of the ESS.

44

3.1 3.1.1

Chapter 3. ESS sizing based on forecast uncertainty

Introduction State of the art of probabilistic ESS sizing

The relationship of the statistical behavior of the forecast error and ESS sizing was mentioned in [4, 18, 19], but little emphasis was given to the actual wind power forecast error pdf. In all cases a standard distribution was assumed as the main statistical property. In [18] the schedule and operation of ESS with wind power in electricity markets was simulated. System reserve was estimated in [4] depending on wind power penetration and in [19] the combination of water pumping and wind generation was presented, showing that it is possible to reduce the uncertainty of wind generation using appropriate operation schedules. In [58,59] a probabilistic approach for ESS sizing for wind and solar output smoothing was proposed, but without considering any forecasting. Energy storage for reduction of wind power fluctuations was proposed in [54]. In this work, no forecasting was assumed, but wind power output was smoothed with a first order low pass filter. It is shown that the discrete version of this filter is the well known exponential moving average (EMA, see also section 2.5). The ESS size was obtained for different time constants. Data with sampling intervals of ∆t = 1 s and 1 h was used. Datasets with 1 s resolution only covered 3–7 h, while hourly data covered one year. All turbines were of constant speed type, although different technologies were present and ESS capacity was obtained from time step simulations. It is concluded that smoothing out 10% of the fluctuations on a yearly time scale would need 2–3 MWh/MW of storage capacity. The suggested smoothing approach is similar to the persistence scenarios proposed in this thesis, with EMA being situated between T×0 and T×1 (see section 2.2). A similar approach was proposed by Koeppel and Korp˚ as in [53], where exponential moving averages were used to generate forecast simulations. In this work, the forecast error increase within the forecast interval (12–36 h ahead), was modelled based on empirical observations, with a linear increase from 20–40%. A pre-simulation was done to adjust the weighting exponent λ (see definition in 2.5) to obtain a predefined value of RMSE. Then, simulations were run with RMSE value as input and energy storage capacity as output. ESS efficiency was included by a usage factor which was updated for each planning period. Two case studies were carried out: one for wind power time series and another for solar photovoltaic (PV) power. This very recent work resembles in many parts with the method proposed in this thesis. The main difference in the forecast simulation resides in the averaging approach (T×0, T×1, T×2 vs. RMSE-adjusted EMA). ESS sizing in this thesis is focussed on unserved energy as a function of ESS sizing parameters such as nominal power and energy capacity, similar to the “energy index of unreliability” defined in [53]. and functional relationships between statistical parameters and ESS size are identified, while in [53] results are not further analysed. In another recent work [60], simulated wind power data was used to evaluate the possibility of output smoothing in 30-min intervals with individual batteries in 2-MW DFIG wind turbine towers. A 600-kW power rating (equal to grid-side converter of DFIG) and 600-kWh energy capacity was proposed for the battery pack. No real generation time series were used and simulated time series of output power show far to high frequencies of power variation. This leads to unrealistic estimations of needed battery capacity and possible reductions in nominal output power.

45

3.2 Sizing rated ESS Power

3.1.2

ESS sizing approach proposed in this thesis

The application of non-gaussian distributions to describe wind forecast uncertainty is not new [29, 30, 38]. But these studies are oriented to risk estimation in short-term energy markets and not to ESS sizing. The contribution of the present work is to employ a non-gaussian approach for ESS sizing. The ESS sizing method proposed in this thesis comprises basically two steps. First the rated ESS power is obtained from the forecast error pdf, based on the Beta distribution as described in chapter 2. The second step consists in defining the energy capacity of the ESS. Here the error pdf is not helpful and a hybrid method of time step simulation and statistical analysis is developed. No references have been found in the literature about the statistical behaviour of the state of charge (SOC) of an ESS which is compensating wind forecast errors. As sizing criterion the unserved energy Eu is adopted. The unserved energy is defined as the energy that cannot be absorbed or supplied by the ESS. In the following, it is expressed as a percentage of total generated wind energy Etotal in the observed time interval (e.g. one year), therefore the lower case variable eu is used to emphasize the normalised character of the value. Thus, eu may be defined by eu = 100% ·

Eu [kWh] Etotal [kWh]

(3.1)

where Eu is the unserved energy by the ESS and Etotal the total generated wind energy. In the next sections eu is analysed for two separated cases: first the nominal power of the ESS and then the energy capacity are reduced. Therefore two different results for eu are obtained. The unserved energy due to reduced ESS power will be denoted as eup and the unserved energy due to reduced ESS energy capacity as eue . It may be mentioned that it is not an easy task to fix the appropriate value for eu . It may be found from the trade-off between the investment in the ESS and the forecast error cost. In chapters 4 and 5 some aspects are investigated but further work has to be done on this issue, as additional services such as power quality (PQ) and peak shaving (PS) represent high value for grid integration, which are not considered in this thesis.

3.2

Sizing rated ESS Power

Within the ESS sizing process, the first step is defining the nominal power of ESS termed PESS . The rated power of an ESS defines its capacity to react quickly on events with very short time constants and is often decisive for the determination of the most appropriate ESS technology. The first guess for PESS is the installed wind power Pinst . In the following the normalised ESS power pESS = PESS /Pinst will be used. Thus, pESS = 1 can serve as a reference because it guarantees any power demand to be served. But as already demonstrated, large power errors are rare and if a certain amount of unserved energy is allowed, nominal ESS power can be reduced. If a functional relationship between unserved energy eup and pESS is known, pESS can be obtained easily for any value of eup .

46

3.2.1

Chapter 3. ESS sizing based on forecast uncertainty

Estimation of unserved energy eup for reduced ESS power

Once the forecast error pdf is known, the unserved energy of an ESS designed to reduce the forecast error can be calculated. If f (ε) is the probability density function of the forecast error ε and pESS the normalised power of the ESS, eup can be calculated as eup

1 = p¯

Z1

pESS

f (ε) · (ε − pESS ) dε

(3.2)

When f (ε) is given in [%] and ε in [p.u.], the integral in (3.2) represents the unserved energy in percent of Pinst multiplied by 8760 h (if one year is assumed as observation interval). In order to get the energy loss as percent of the total generated energy, the integral has to be divided by the normalised long term mean p¯ = P¯ /Pinst . From (3.2) it can be seen that eup is proportional to the integral of the tail of the pdf, starting from pESS up to 1. Therefore, to obtain satisfying results, the pdf must be especially precise in its tail region. To illustrate the usefulness and the limitations of the pdf obtained from Betafit approximation (see section 2.4), pESS has been calculated for datasets A, B and C. Results for scenarios T×0 and T×1 are shown in Fig. 3.1 and Table 3.1. As it could be expected, the best case scenario T×0 permits lowest values of pESS . Considering a 1-h forecast and unserved energy eup = 2%, pESS would be around 12% of the installed wind power Pinst for dataset A and below 10% for datasets B and C. ESS power increases with forecast horizon and reaches about 40–42% of Pinst for 24-h forecasts. Required ESS power is also higher if the forecast quality is worse. With scenario T×1, maintaining eup = 2%, a reduction of pESS to about 27–33% can be achieved for a 1-h forecast and 64–70% for a 24-h forecast. In Table 3.1 results for eup = 2% are shown for scenarios T×0 and T×1 and forecast intervals of 1, 6 and 24 h. The upper part contains ESS nominal power obtained from histograms (empirical pdf) of the forecast error. These figures are considered as the best approximation possible. In the middle part corresponding results from Betafit algorithm are presented. The lower part contains deviations between results from empirical pdf and Betafit. The deviation ∆pESS is calculated as expressed in (3.3). Deviations for a 24-h forecast are presented in Table 3.2 for several values of eup up to 30%. ∆pESS = 100%

pESS,beta − pESS,hist pESS,hist

(3.3)

where pESS,beta [p.u.] is ESS nominal power obtained from Betafit approximation and pESS,hist [p.u.] obtained from histograms of forecast error (empirical pdf). Results in Table 3.1 show that Betafit underestimates pESS in general (negative deviations). Only for the 1-h forecast of scenario T×0 a slight overestimation can be observed. A good performance of Betafit can be seen for the T×0 scenario. Deviations compared to empirical pdf are below 10%. Deviations for T×1 are over 20%. This is caused by too small kurtosis of the Beta pdf and thus underestimated tail. In Table 3.2 it can be observed how the influence of poorly approximated pdf tails diminish for larger values of unserved energy. While for datasets A and C the deviation decays rapidly down to 4–5%, for dataset B results only improve slightly. In general the

47

3.2 Sizing rated ESS Power

Dataset A - Tx0

Dataset A - Tx1

ESS power pESS [p.u.]

1

1 hist 24h hist 6h hist 1h betafit 24h betafit 6h betafit 1h

0.8 0.6

0.8 0.6

0.4

0.4

0.2

0.2

0

0

2

4

6

8

10

0

0

2

ESS power pESS [p.u.]

Dataset B - Tx0 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

2

4

6

8

10

0

0

2

ESS power pESS [p.u.]

Dataset C - Tx0 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

2

4

6

8

10

4

6

8

10

8

10

Dataset C - Tx1

1

0

6

Dataset B - Tx1

1

0

4

8

Unserved energy eup [%]

10

0

0

2

4

6

Unserved energy eup [%]

Figure 3.1. Minimum ESS power as a function of unserved energy for three forecast intervals, calculated with observed histograms (“hist”) and Betafit approximation (“betafit”) for datasets A, B, C and forecast scenarios T×0 and T×1.

Betafit algorithm performs well for 24-h forecasts in the T×1 scenario, with deviations of not more than 15%, compared to the reference results from empirical pdf. The forecast error pdf has been used to define only the ESS power, but without considering the energy capacity, the ESS sizing is not complete. Therefore, in the next section a sizing strategy is developed for the energy capacity EESS .

48

Chapter 3. ESS sizing based on forecast uncertainty

Dataset

1h

Scenario T×0 6h 24 h

1h

Scenario T×1 6h 24 h

Empirical pdf A B C

0.119 0.076 0.094

0.279 0.243 0.245

0.399 0.401 0.419

0.324 0.268 0.304

0.522 0.538 0.574

0.638 0.697 0.664

0.264 0.208 0.266

0.444 0.432 0.470

0.578 0.613 0.586

–14.9 –19.6 –18.1

–9.5 –12.1 –11.8

Betafit pdf A B C

0.119 0.080 0.095

0.256 0.225 0.236

0.379 0.375 0.404

Deviation ∆pESS [%] A B C

–0.4 5.5 1.3

–8.4 –7.4 –3.8

–5.1 –6.6 –3.6

–18.5 –22.4 –12.7

Table 3.1. ESS nominal power [p.u.] for eup = 2% from empirical pdf, Betafit pdf and the deviation in [%].

Dataset A B C

1%

2%

5%

10%

20%

30%

−10.2 −11.9 −12.3

−9.5 −12.1 −11.8

−8.2 −12.1 −10.0

−6.7 −11.6 −8.1

−4.8 −10.2 −5.7

−3.7 −9.8 −4.3

Table 3.2. Deviation of ESS power estimation from Betafit for several levels of unserved energy eup for 24-h persistence forecast and scenario T×1.

3.3

Sizing ESS energy capacity

Once the nominal ESS power is known, the energy capacity has to be calculated. This is more difficult, as the state of charge (SOC) of the ESS depends always on preceding events. On top of that, SOC will depend on the efficiency and the charge control algorithm. The energy balance always has to be maintained, which means that the mean of SOC has to be constant. The first simplification in order to reduce the complexity of the problem is to consider a 100% efficiency of the ESS. This approach has the advantage that statistical results will be independent of the specific ESS technology which basically will be characterised by its round-trip efficiency. How to include these aspects a posteriori will be explained in the next chapter (see section 4.3.2). Here, it will be assumed that the energy loss due to ESS efficiency is compensated by the charge controller. In [53] an interesting approach is suggested about how ESS losses can be treated in a real world operational situation.

49

3.3 Sizing ESS energy capacity

When the ESS is used to compensate forecast errors, the energy balance is only given if the mean forecast error is zero (zero mean assumption). Although some real forecast models may have a bias (non-zero long term mean), this bias – as it represents a systematic error – can be easily corrected. This is one of the tasks of the so called model output statistics (MOS) [13]. So zero mean error is assumed for the input data of the ESS model. If later the ESS efficiency is taken into account, a controlled bias can be introduced, in order to assure that always more energy is charged to the system than it is discharged. Here it becomes very clear that when the efficiency is introduced, things become more complex. As 100% efficiency is assumed for the moment, the values of SOC can be calculated very easily integrating the forecast error values over time. As measurements are discrete values, this is done by a cumulative sum. In this section a detailed statistical analysis of the state of charge over time is presented, always assuming that only the forecast error has to be compensated and storage capacity is unlimited. After that, with the knowledge of SOC statistics, it will be analysed what would be the effect if energy capacity is reduced, similar to the approach with the nominal power. The unserved energy eue – energy which cannot be compensated due to reduced capacity – is analysed. The described process is based on the assumption that the ESS has no losses.

3.3.1

Energy Throughput Ratio – ETR

It seems convenient to introduce at first a parameter which is very useful for ESS sizing, as will be seen later. The Energy Throughput Ratio ETR is defined in (3.4) as the energy throughput Etp of the ESS relative to the total generated energy Etotal . A similar definition is used in [61] as described in section 4.3.2 (p. 111), but here the following formulation is preferred: ETR [p.u.] =

Etp [kWh] Etotal [kWh]

(3.4)

where Etp is the annual ESS energy throughput and Etotal the total generated wind energy. Note that ETR is proportional to the number of equivalent full cycles (see equation (4.9) on page 111), but only if ESS size is constant. Therefore, it describes a property of the energy input variable (in this case forecast errors) and is independent of the ESS size or technology. The value of Etp is the integral of absolute energy input and output of the ESS, which means that charged and discharged energy is summed. For an idealised ESS the input and output energy is equal, as the energy balance has to be maintained always. Because this idealised case is of special interest here, the idealised ETR will be denoted as ETR0 . In this section, only the idealised ETR0 is considered. In this case Etotal is the wind energy generation but it could represent any reference such as total consumption in case of demand forecast, for example. For some more insight into the meaning and importance of ETR0 the definition in (3.4) will be analysed more in detail now. Energy throughput Etp can be written as the integral of the absolute forecast error and the total wind generation Etotal as the integral of wind power generation. As both integrals are calculated over the same time span (e.g.

50

Chapter 3. ESS sizing based on forecast uncertainty

one year), the integrals can be substituted by the mean absolute forecast error |ε| and normalised long term mean wind generation p¯. R Pinst |ε (t)| dt |ε| MAE R = ETR0 = (3.5) = ¯ p¯ Pinst p(t) dt P where Pinst is the installed wind power, ε the normalised forecast error, p the normalised measured wind power, MAE the mean absolute error [kW] and P¯ the annual mean wind power [kW]. MAE and P¯ are parameters whose values are easy to obtain and it does not matter if they are normalised or not. If normalised MAE (NMAE) is given, it should be verified if the reference for normalization is Pinst or mean wind power P¯ . In the second case, NMAE gives directly the value of ETR0 . Therefore equation (3.5) is very important for ETR0 estimation, for example for energy loss calculations (see section 4.3.2). It may be mentioned that this equation is valid for any forecast method, as it was obtained from a general formulation of the problem. In Fig. 3.2 ETR0 is depicted for datasets A, B and C and the three forecast scenarios. ETR0 could theoretically reach a value of 2, in the case that all generated energy is cycled through the ESS. This is because energy is counted twice – first as it is charged into the ESS and secondly when it is discharged. But in reality, as it can be observed in Fig. 3.2, ETR0 saturates near 1. This means that in the case of forecast error compensation it can be expected that a maximum of 50% of all generated energy would be cycled through the ESS. Another interesting interpretation of this observation is that the mean absolute error MAE is tending to the mean wind power. This is a logical consequence of the persistence forecast as it is performed in this work (see section 2.2). Scenario Tx0 1.2

Scenario Tx2 1.2

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

A B C

1

ETR0 [p.u.]

Scenario Tx1 1.2

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

720

T [h]

Figure 3.2. Ideal energy throughput ratio ETR0 as a function of the forecast interval T for the three forecast scenarios T×0, T×1 and T×2 and datasets A, B and C.

In order to validate the theoretical values of ETR0 time step simulations have been performed. Then the simulated ETR0 has been divided by the theoretical value obtained from (3.4). It can be seen in Fig. 3.3 that the largest deviations between the simulated and theoretical ETR0 occur when one year time series of individual turbines (A04, A18, A33) are used. Aggregated time series of sites A and B show deviations below 1%. One year time series from wind farm C also shows deviations below 1% up to forecast intervals of 7 days (168 h). Note that results are identical for all three scenarios.

51

3.3 Sizing ESS energy capacity

1.06 1.04

A B C A04 A18 A33

1.02 1 0.98 1

3

6

12

24

48

168

720

T [h] Figure 3.3. Quotient between normalised mean forecast error and ETR for different datasets.

The reason for the deviations lie in the fact that total used data points are reduced in order to include only entire forecast intervals. For the calculation of the theoretical ETR0 mean wind generation P¯ of the whole dataset was applied, while the observed throughput ratio is based on the reduced dataset. For large forecast intervals T the reduction of data points becomes significant, if only one year data is available (individual turbines from dataset A). As the aggregation of available data was performed as if several years of the same dataset would have been available, aggregated time series A and B show much lower deviations. The effect of short time series is identical for all scenarios, as only the mean wind generation is affected. Therefore deviations do not depend on the forecast scenario. As a conclusion it can be stated that ETR0 is sensitive to estimation of mean wind generation P¯ . Therefore it is advisable to use the 10 year mean value of expected wind generation for sizing the ESS.

3.3.2

Statistics of SOC

For the assessment of SOC statistics, one-year power output measurements from three different wind farms are used to generate forecast data and three forecast qualities (scenarios T×0, T×1 and T×2) based on the persistence approach are simulated. These persistence scenarios are a valuable reference frame for the evaluation of real world forecast methods. An example is given in a case study in section 3.4. Based on the forecast data, time series of the forecast error ε(t) are generated. Considering ε(t) as power input to the ESS, time series of the state of charge (SOC) are obtained, integrating ε over time. Thus, SOC can be defined mathematically as in (3.6). It may be mentioned that SOC is assumed to be zero at the beginning of the observation. Z SOC(t) = ε(t) dt (3.6) With this, cumulative histograms (empirical cumulative density function cdf) of SOC are generated and the influence of the forecast quality degree and interval length on the distribution of SOC can be observed. To understand the following figures, the meaning of normalised SOC, expressed in [p.u.] should be explained. For convenience, in this work SOC is normalised by the

52

Chapter 3. ESS sizing based on forecast uncertainty

nominal power of the wind farm Pinst and the forecast interval T as written in (3.7). SOC [p.u.] =

SOCkW h [kWh] Pinst [kW] · T [h]

(3.7)

As a consequence of this definition, SOC = 1 p.u. represents the energy amount charged at nominal power during one forecast interval T and does not represent the event “storage full”. As a consequence, values below zero and beyond one are possible. In forecast scenario T×2, SOC can reach 2, because the forecast delay k (in this case: k = T ) has to be added to the forecast interval T . This approach has been chosen because ESS storage capacity is unknown and the aim is to calculate it. Maximum values of SOC this way give directly information about the ESS capacity needed. In Fig. 3.4, cumulative histograms of normalised SOC are shown for datasets A, B and C under the three forecast quality scenarios. Results for individual turbines A04, A13, A33 and single wind farm B1 are shown in appendix B.1. The best case scenario T×0 shows a symmetrical distribution around zero and SOC stays below ±0.2. Looking at the scenarios T×1 and T×2, it is notable that there are no negative values and the maximum values of 1 and 2 can be seen clearly. The absence of negative values can be explained directly with the persistence model. Starting at zero, the power forecast will be zero until the first interval has concluded. In the worst case, the error was +Pinst (ε = 1) during the entire interval and SOC will reach 1. Then, in the next interval, the ESS cannot be discharged more than this, because now the worst case would be −Pinst (ε = −1) during the entire interval and SOC would return to zero. It should be remembered, that all the considerations assume an ESS efficiency of 100%. A real ESS will have losses and the balance of charge and discharge would not be given as in the idealised case. But it is assumed, that an appropriate charge control algorithm can establish this balance artificially so that the idealised case would approach reality to a reasonable extent. From simple observation of Fig. 3.4, a zero error reference e00 can be defined for each forecast scenario as shown in Table 3.3. This first reference represents an ESS with enough energy capacity to compensate any deviation from the forecast. This reference is only a first orientation but for preliminary sizing it may give valuable information about the expected ESS size. The most important drawback of e00 is that it does not depend on the forecast interval T . A closer look at Fig. 3.4 reveals that the range of SOC not only changes with the forecast scenario, but also with T . Therefore, in order to have some deeper insight into the behavior of SOC, minimum, maximum and mean values of SOC have been obtained from the time series and a new zero error value e0 is defined as e0 = emax − emin

(3.8)

where emax and emin are the maximum and minimum values of SOC observed. In figures 3.5 and 3.6 results are shown for the datasets from sites A, B and C and from three selected individual turbines from site A. In Fig. 3.5 emin , emax and SOC (mean value of SOC) are depicted and in Fig. 3.6 the corresponding zero error size e0 can be seen. For the persistence forecast, it can be shown that normalised mean state of charge SOC can be calculated with (3.9).

53

3.3 Sizing ESS energy capacity

Frequency [%]

Scenario Tx0 - A

Scenario Tx1 - A 100

100

80

80

80

1 24 72 168 720

60 40 20 0

-0.2

-0.1

0

0.1

0.2

1 24 72 168 720

60 40 20 0

0

Frequency [%]

Scenario Tx0 - B

0.2

0.4

0.6

0.8

1

40 20 0

80

80

80

1 24 72 168 720

20 -0.2

-0.1

0

0.1

0.2

1 24 72 168 720

60 40 20 0

0

Scenario Tx0 - C

0.2

0.4

0.6

0.8

1

20 0

80

80

80

20 0

-0.2

-0.1

0

0.1

0

0.5

0.2

State of Charge [p.u.]

1 24 72 168 720

40 20 0

0

0.2

0.4

0.6

1

1.5

2

Scenario Tx2 - C 100

40

2

1 24 72 168 720

Scenario Tx1 - C

60

1.5

40

100

1 24 72 168 720

1

60

100

60

0.5

Scenario Tx2 - B 100

40

0

Scenario Tx1 - B 100

60

1 24 72 168 720

60

100

0

Frequency [%]

Scenario Tx2 - A

100

0.8

1

State of Charge [p.u.]

1 24 72 168 720

60 40 20 0

0

0.5

1

1.5

2

State of Charge [p.u.]

Figure 3.4. Cumulative histograms (cdf) of SOC for different forecast intervals T (1–720 h) and three scenarios (T×0, T×1 and T×2) from datasets A (above), B (center) and C (below).

Observed SOC Scenario range [p.u.] T×0 T×1 T×2

–0.2 ... 0.2 0 ... 1 0 ... 2

Zero error reference e00 [p.u.] 0.4 1 2

Table 3.3. Definition of the zero error reference e00 for ESS energy capacity (size necessary for compensation of all forecast errors).

54

Chapter 3. ESS sizing based on forecast uncertainty

P¯ Pinst

(3.9a)

SOCkW h = (k + 1) · P¯ · T

(3.9b)

SOC = (k + 1) ·

where k is the dimensionless forecast delay, P¯ [kW] the long term mean wind power, Pinst [kW] the nominal power of the wind farm (or turbine) and T [h] the forecast interval. The expression k + 1 can be interpreted as the time shift factor of the persistence forecast model, which is 0, 1 and 2 for the scenarios T×0, T×1 and T×2. Interestingly, SOC does not depend on T , but as SOC is normalised by T (see eq. (3.7)), SOCkW h is directly proportional to T . The proof of (3.9) is provided in appendix B.2. Comparing e00 and e0 , the first observation is that e0 surpasses e00 on site A (aggregated and individual turbines). This is due to the installation of fixed speed wind turbines with a maximum power output beyond the nominal value Pinst . On sites B and C with variable speed turbines installed, Pinst is never surpassed. Here e00 limits are maintained very well. The second observation is that e00 limits are only valid for forecasts up to approximately 24 h. This means that linear proportionality between ESS size and forecast interval T is maintained up to 24 h. For larger forecast intervals, e0 starts to diminish drastically. Since forecast error ε represents the first derivative of SOC (see eq. (3.6)), the tendency of extreme values of SOC (emin and emax ) can be explained with the behaviour of ε. As a representation of ε, here the energy throughput ratio ETR is is chosen. ETR can be used as representation for the forecast error ε because it is equal to the mean absolute error MAE normalised by long term mean wind generation P¯ , as described in (3.5). It can be seen in Fig. 3.7 that ETR grows far less than T given the logarithmic scaling of T . Furthermore, a saturation can be observed in the scenarios T×1 and T×2 approximately at the same forecast intervals where e0 starts to diminish. Scenario T×0 does not reach the saturation and as a consequence, no sudden reduction of e0 can be observed. The third important observation is that emin and emax are almost symmetrical with SOC = 0 for T×0, but the case of T×1 and T×2 is different. Here for short forecasts emin is zero, but for forecasts beyond 24 h, emin starts to increase, tending also to SOC. It was shown above that the mean state of charge is directly proportional to T , while the deviations (represented by ETR) saturate. As a result, with growing T the mean state of charge will dominate more and more the value of SOC, as deviations become less important. For T = 24 h the initially charged energy begins to be higher than the maximum ever discharged and emin is greater than zero. In the extreme case of 30-day forecast, the oscillations, due to forecast error compensation are approximately ±50% of SOC (compare Fig. 3.5). As a conclusion, using a constant value as zero error reference for ESS size as in Table 3.3 may not be the best choice. It was shown that the assumption of a linear proportionality between ESS size and forecast interval T is only a good estimation for forecast intervals below 24 h. A better reference may be the saturation time of the ESS, as will be described in the next section.

55

SOC (min,mean,max) [p.u.]

3.3 Sizing ESS energy capacity

Scenario Tx0

Scenario Tx1

0.3

Scenario Tx2 2.5

1.2

0.2

1

0.1

0.8

0

0.6

2 A B C A04 A18 A33

0.4

-0.1

1 0.5

0.2

-0.2

0 -0.3

1

3 6 12 24 48 168

720

A B C A04 A18 A33

1.5

0 1

T [h]

3 6 12 24 48 168

720

1

T [h]

3 6 12 24 48 168

720

T [h]

Figure 3.5. Maximum, minimum and mean SOC from time step simulation for datasets A, B, C and 3 individual turbines, forecast scenarios T×0, T×1 and T×2.

Scenario Tx0

ESS size e0 [p.u.]

1.2

Scenario Tx1 A B C A04 A18 A33

1 0.8 0.6

Scenario Tx2 2.5

1.2 1

2

0.8

1.5

0.6 1

0.4

0.4

0.2

0.2

0

1

3 6 12 24 48 168

720

0

0.5

1

T [h]

3 6 12 24 48 168

720

0

1

T [h]

3 6 12 24 48 168

720

T [h]

Figure 3.6. ESS sizes for 100% error compensation from time step simulation for datasets A, B, C and 3 individual turbines, forecast scenarios T×0, T×1 and T×2.

Scenario Tx0 1.2

Scenario Tx2 1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

A B C A04 A18 A33

1

ET R [p.u.]

Scenario Tx1 1.2

0.8 0.6

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

720

T [h]

Figure 3.7. ETR (or MAE normalised by long term mean wind generation) for datasets A, B, C and 3 individual turbines, forecast scenarios T×0, T×1 and T×2.

56

3.3.3

Chapter 3. ESS sizing based on forecast uncertainty

Saturation time

If the size of the ESS is reduced below e00 , it may not be possible to compensate the entire forecast error (i.e. to serve the demand) at any time because ESS will be empty or full. The time span of empty or full ESS is denoted as saturation time tsat . It will be expressed as a percentage of the total observation time (e.g. one year). The saturation time tsat can be estimated from the cumulative histogram of SOC, because the cumulative frequency Fsoc of any value of SOC represents the percentage of time, the state of charge will be below this value. If ESS size is reduced to a value ex , 100 − Fsoc (ex ) is the time, SOC will be greater than ex which is nothing else than tsat . As a first approximation, equation (3.10) can be formulated to estimate tsat . tsat = 100 − Fsoc (ex )

(3.10)

where Fsoc is the cumulative distribution (or histogram) of SOC and ex the reduced ESS size. An example how to obtain tsat is shown in Fig. 3.8 for the case of a T×1 forecast of 12 h. When the ESS size is reduced to 54% (ex = 0.54), a frequency value of 80% is obtained. This means that during 20% of all the operating time, SOC is above ex and the ESS will probably not serve the demand. −1 Fsoc (100 − tsat )

100

tsat

Frequency [%]

80 60 40

ex = 0.54

20 0 0

0.2

0.4

0.6

0.8

1

SOC [p.u.] Figure 3.8. Cumulative distribution (cdf) of SOC (T×1, T = 12 h) with representation of saturation time tsat for a reduced ESS size.

As will be shown later in section 3.3.5, unserved energy due to saturation can be assumed equal for charge and discharge demand. But this balance does not exist for saturation times. Therefore, from the histogram it cannot be determined how much time the storage will be empty and how much time it will be full. As already seen in the last section, the distribution of SOC varies heavily with the forecast interval and scenario. As a consequence, the simple approach in (3.10) for obtaining saturation times does not lead to satisfactory results. Firstly, the same reduction of ESS size can have very different consequences for forecasts of 6 h or 72 h. Therefore, instead of fixing the ESS size ex , it is more appropriate to

57

3.3 Sizing ESS energy capacity

invert the procedure: fixing arbitrary values of tsat and obtaining ex . Another advantage of this approach is that no reference such as e00 needs to be known in advance. Doing this, another issue arises. Again due to the great variety of SOC distribution shapes, the simple approach shown in Fig. 3.8 does not always work properly. Therefore in this thesis a more sophisticated method is proposed which can be formulated mathematically as shown in (3.11). h  i −1 −1 100 − cs tsat − Fsoc (1 − cs ) tsat , eESS = min Fsoc cs

cs ∈ [0, 1]

(3.11)

−1 where eESS is the minimised ESS size for a given saturation time tsat , Fsoc is the inverted cumulative distribution (cdf) of SOC and cs is the saturation time splitting coefficient.

The idea of this method consists in finding the minimum ESS size eESS . Minimization of ex is needed to reproduce the behaviour of charge and discharge processes in case of saturation. To do so, a splitting coefficient cs is introduced to divide tsat into two parts. With cs = 0, the entire saturation time is applied to the left side of the cdf, while for cs = 1 it is applied to the right side. The case cs = 1 represents the initial approach proposed in (3.10). In Fig. 3.9 the method is illustrated for an example with tsat = 25% where the minimum of ex is obtained with cs = 0.78. With no minimization (cs = 1), an ESS size of eESS = 0.35 would be obtained, while the minimum is 0.22. −1 Fsoc (100 − tsat )

100

1

0.78 tsat

tsat

60

0.8

ex

0.6

ex

Frequency [%]

80

40

0.4

20 0 0

0.2

0.22 tsat 0.2

0.4

0.6

SOC [p.u.]

0.8

min(ex )

1

0 0

0.2

0.4

0.6

0.8

1

cs

Figure 3.9. Advanced method for obtaining minimised ESS size from cdf of SOC (T×1, T = 14 days). Left: splitting of saturation time. Right: ex as a function of splitting coefficient cs .

The proposed method is valid for any cdf shape. If the distribution is symmetric (T×0 scenario), cs will be 0.5. In case of strongly asymmetric distributions (short forecast intervals of T×1 and T×2), cs can become one. An example of cs = 1 is given in Fig. 3.8. In the following, the method of minimised ex is applied to obtain values of eESS for tsat = 0.1, 2, 10, 25 and 50%. The ESS sizes eESS are denoted as e0.1 , e2 , e10 and so on. In Fig. 3.10 ESS sizes are shown for the three datasets and forecast scenarios for forecast time intervals T up to 30 days (720 h).

58

Chapter 3. ESS sizing based on forecast uncertainty

In order to make visible the importance of very rare events, in Fig. 3.10 curves for tsat = 0% and 0.1% are included. For scenarios T×1 and T×2 e0.1 is very similar to e0 , while for T×0 larger differences can be observed. For T×0, e0 is rather constant near 0.4−0.5 for T < 12 h. Contrarily, e0.1 starts at about 0.2 and grows with T until reaching almost 0.4 at T = 12 h. This means that for T < 12 h, values of SOC > 0.2 are very seldom, as a saturation time of 0.1% only represents 9 h per year. In addition it may be noted that the tendency observed for e0.1 is present for all saturation times. This shows that the curves of e0 in fact mask the normal behavior of the system. ESS sizes for individual turbines A04, A13, A33 and single wind farm B1 are given in appendix B.3.

ESS size eESS [p.u.]

Scenario Tx0 - A

Scenario Tx1 - A e0 e0.1 e2 e10 e25 e50

1 0.8 0.6

1

1

0.2

0.2 3 6 12 24 48 168

720

1.5

0.6 0.4

1

2

0.8

0.4

0

0

0.5 1

ESS size eESS [p.u.]

Scenario Tx0 - B

0.8 0.6

720

ESS size eESS [p.u.]

0.8 0.6

0

1.5

0.5 1

1

720

3 6 12 24 48 168

720

2 1.5

0.6 1 0.2

T [h]

0

Scenario Tx2 - C

0.8

0.2 3 6 12 24 48 168

720

1

0.4

1

3 6 12 24 48 168 Scenario Tx1 - C

0.4

0

720

2

Scenario Tx0 - C e0 e0.1 e2 e10 e25 e50

3 6 12 24 48 168

1 0.2

1

1

0.6

0.2 3 6 12 24 48 168

0

Scenario Tx2 - B

0.8

0.4

1

720

1

0.4

0

3 6 12 24 48 168 Scenario Tx1 - B

e0 e0.1 e2 e10 e25 e50

1

Scenario Tx2 - A

0

0.5 1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

720

T [h]

Figure 3.10. ESS sizes from cumulative histogram of SOC for different predefined saturation times between 0.1 and 50%, datasets A, B, C and forecast scenarios T×0, T×1 and T×2.

59

3.3 Sizing ESS energy capacity

3.3.4

Estimation of unserved energy eue for reduced ESS energy capacity

Unserved energy eue is defined according to (3.1) as the percentage of energy demand which cannot be served by the ESS, due to a reduced energy capacity. In the following paragraph, eue will be analysed for different saturation times tsat . Furthermore it is assumed that the ESS can serve any power demand, so that eue is exclusively caused by limitations of energy capacity, in contrast to eup which is the unserved energy due to power limitations, as discussed in subsection 3.2.1. Analogue to subsection 3.2.1, where eup (pESS ) was calculated, at the end of this paragraph, the aim is to establish the relationship of unserved energy as a function of ESS energy capacity eue (eESS ). In paragraph 3.3.3 curves of eESS (tsat ) have been obtained from cdf of SOC. Though if another estimation is available for eue (tsat ), the combination of these two results leads us to the final result of eue (eESS ). The missing link between tsat and eue is the unserved mean power demand P¯ue during saturation. Thus, derived from (3.1), eue can be defined as P¯ue eue = tsat ¯ (3.12) P where tsat is the saturation time, P¯ue the unserved mean power demand due to limited ESS energy capacity and P¯ the long term mean wind power. The total generated wind energy Etotal in (3.1) is equal to the mean wind power P¯ multiplied by the total observation time and Eue is tsat,h [h]·P¯ue . By definition, tsat is already given as percentage of total observation time, this way eue is obtained in percent. The expression P¯ue /P¯ can be seen as a normalised unserved power which will be termed p¯ue . Note that here the normalization reference is the mean wind generation P¯ and not the nominal installed power Pinst . Note that P¯ue is the mean power without consideration of the flow direction. Charged and discharged power is added up analogue to the definition of unserved energy. So as a first rough approximation it may be assumed that P¯ue is equal to the mean absolute forecast error MAE. This means that the normalised value p¯ue would be equal to ETR0 (see subsection 3.3.1). MAE (3.13) = ETR0 P¯ where MAE is the mean absolute forecast error, P¯ the long term mean wind power and ETR0 the energy throughput ratio. p¯ue ≈

Taking into account the approximation in (3.13), unserved energy eue may be estimated in a first approach by eue ≈ tsat · ETR0

(3.14)

where tsat [%] is the saturation time and ETR0 [p.u.] the energy throughput ratio.

60

Chapter 3. ESS sizing based on forecast uncertainty

Example of improved estimation of eue A simple method to improve the approximation of p¯ue is introducing an empirical factor ν. This factor might be different for each scenario T×0, T×1 and T×2. The resulting equation is presented below. eue ≈ ν · tsat · ETR0

(3.15)

Other modifications of the estimation might include the RMSE instead of MAE for example. In addition, optimization methods may be applied to find the best values of ν. This optimization is outside of the scope of this thesis but represent a promising field of further investigation. In order to quantify the accuracy of the estimations of tsat and eue a time step simulation was performed. Results are presented in the next subsection.

3.3.5

Results from time step simulation: tsat and eue

In the stepwise simulation of the state of charge, time series of forecast error ε(t) are directly used as input to the ESS with limited size eESS . ESS sizes eESS (tsat ) from Fig. 3.10 are used for this purpose. Every value of eESS corresponds to a predefined saturation time. For the simulation, saturation times of 0.1, 2, 10, 25 and 50% are assumed. When SOC becomes larger than eESS or smaller than zero, saturation occurs and tsat and eue are registered. This method simulates real charge and discharge processes, but still assuming an ESS efficiency of 100%. In the following paragraphs results for saturation time and unserved energy are presented separately. Saturation time tsat The first result of time step simulation is the saturation time. This way, predefined and simulated saturation times can be compared. In Fig. 3.11 results for the aggregated datasets A and B and single wind farm C are depicted. For T×0 the predefined values match quite well with the simulation at least for tsat < 20% and forecast intervals between T = 3 − 24 h. The other scenarios exhibit larger deviations. These effects are discussed more in detail in section 3.3.6. Results for individual turbines A04, A13, A33 and single wind farm B1 are shown in appendix B.4. Unserved energy eue As the saturation time is widely overestimated, the same should be expected for the unserved energy. But another uncertainty is introduced here with the approximation of p¯ue by ETR0 . In Fig. 3.12 the unserved energy, obtained from time step simulation is compared to the result of estimation from equation (3.14). Results for individual turbines A04, A13, A33 and single wind farm B1 are shown in appendix B.5. Results in Fig. 3.12 show that no general trend can be defined for the deviations between time step simulation and estimation. For the T×0 scenario, contrary to what would be expected, eue is underestimated, except for quite long forecast intervals of

61

3.3 Sizing ESS energy capacity

Scenario Tx0 - A

Scenario Tx1 - A

tsat [%]

80

80

60

60

60

40

40

40

20

20

20

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - B

tsat [%]

3 6 12 24 48 168

720

0

60

40

40

40

20

20

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3 6 12 24 48 168

720

0

1

Scenario Tx0 - C

3 6 12 24 48 168

720

0

60

40

40

40

20

20

20

3 6 12 24 48 168

T [h]

720

0

3 6 12 24 48 168

720

80

60

1

1

Scenario Tx2 - C

80

60

0

720

e2 e10 e25 e50

Scenario Tx1 - C

80

3 6 12 24 48 168

80

60

1

1

Scenario Tx2 - B

80

60

0

e2 e10 e25 e50

Scenario Tx1 - B

80

tsat [%]

Scenario Tx2 - A

80

1

3 6 12 24 48 168

T [h]

720

0

e2 e10 e25 e50

1

3 6 12 24 48 168

720

T [h]

Figure 3.11. Saturation times from 2–50% for datasets A, B and C estimated from cdf of SOC (empty symbols) and obtained from time step simulation (filled symbols).

62

Chapter 3. ESS sizing based on forecast uncertainty

Scenario Tx0 - A e2 e10 e25 e50 estim.

50

eue [%]

Scenario Tx1 - A 50

50

40

40

30

30

30

20

20

20

10

10

10

40

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3 6 12 24 48 168

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eue [%]

3 6 12 24 48 168

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30

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20

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10

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Scenario Tx0 - C

720

0

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20

10

10

10

0

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3 6 12 24 48 168

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720

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T [h]

720

3 6 12 24 48 168

720

Scenario Tx2 - C

50

40

1

Scenario Tx1 - C

e2 e10 e25 e50 estim.

50

3 6 12 24 48 168

3 6 12 24 48 168 Scenario Tx2 - B

50

40

1

Scenario Tx1 - B

e2 e10 e25 e50 estim.

50

eue [%]

Scenario Tx2 - A

720

0

1

3 6 12 24 48 168

720

T [h]

Figure 3.12. Unserved energy as a function of forecast interval T for different saturation times; comparison of estimation (empty symbols) and time step simulation (filled symbols) for datasets A, B and C.

63

3.3 Sizing ESS energy capacity

T > 72 h. Results for T×1 and T×2 are similar, but quite different from T×0. First, an overestimation can be observed, except for shorter forecast intervals in case of tsat = 50% Secondly, estimations match simulation results better for larger saturation times, but only for forecasts up to 48 − 72 h. After that limit, the simulation shows a pronounced reduction of eue , while estimations fall only slightly. ESS size eESS (eue ) For ESS sizing purposes, it is convenient to represent ESS energy capacity eESS as a function of unserved energy eue , analogue to Fig. 3.1 from ESS power sizing. In Fig. 3.13 this representation was chosen. Improved estimations with empirical scaling factors of ν = 1.35 for T×0 and ν = 0.92 for T×1 are shown in Fig. 3.14. Comparing figures 3.13 and 3.14, the generalised underestimation of energy capacity for T×0 can be seen very well. Improvements with scaling factor 1.35 are evident. Especially the estimate for 24-h forecasts becomes very good. For scenario T×1 overestimation of energy capacity is evident. The best estimation results are obtained with dataset C and only slightly worse with dataset A. The similarity between these two cases is interesting because they represent different turbine technologies and averaging levels. Dataset A contains time series of 32 individual constant speed induction machines. Individual time series are aggregated as if they would come from a single wind turbine, but measured over 32 years. Dataset C represents a time series of only one year, but from a wind farm of 12 variable speed turbines. The averaging effect of the wind farm smoothes the output while in dataset A a higher variability is given. For a better appreciation of the improvements obtained with scaling factor ν, results are given in tables 3.4 and 3.5 on p. 66. In the first table estimations and results from time step simulation are shown for eue = 2%. Below, in the second table, deviations between estimations and simulation are exposed for a 24-h forecast in T×1 scenario. Results of simple estimation of mean unserved power with MAE is marked as “Case I”. Improved results obtained with scaling factor ν are marked as “Case II”. Deviations of energy capacity ∆eESS in the lower part of the table are calculated analogue to ∆pESS as shown in (3.16). ∆eESS = 100%

eESS,estim − eESS,step eESS,step

(3.16)

where eESS,estim [%] is the estimated and eESS,step [%] the simulated ESS energy capacity. Negative deviations show that for scenario T×0 ESS energy capacity is underestimated. Comparing case I and II, especially for 24-h forecast dramatic improvements are achieved for all datasets. Positive deviations for scenario T×1 indicate a generalised overestimation of ESS energy requirement. Here improvements from scaling factor ν are very limited. The effect of the scaling factor in scenario T×1 is more evident in Table 3.5. Here the two cases are compared for several levels of unserved energy. In the extreme case of 30% unserved energy the effect is so strong that even negative deviations appear – here the ESS size is slightly underestimated. But deviations below 5% must be considered as very good results.

64

Chapter 3. ESS sizing based on forecast uncertainty

Dataset A - Tx0

Dataset A - Tx1

ESS size eESS [p.u.]

1

1 step 24h step 6h step 1h estim. 24h estim. 6h estim. 1h

0.8 0.6

0.8 0.6

0.4

0.4

0.2

0.2

0

0

2

4

6

8

10

0

0

2

ESS size eESS [p.u.]

Dataset B - Tx0 1

0.8

0.8

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0.2

0.2

0

2

4

6

8

10

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ESS size eESS [p.u.]

Dataset C - Tx0 1

0.8

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0

2

4

6

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10

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Dataset C - Tx1

1

0

6

Dataset B - Tx1

1

0

4

8

Unserved energy eue [%]

10

0

0

2

4

6

Unserved energy eue [%]

Figure 3.13. ESS energy capacity as a function of unserved energy for three forecast intervals, estimated from cdf of SOC (“estim.”) and time step simulation (“step”) for datasets A, B, C and forecast scenarios T×0 and T×1.

65

3.3 Sizing ESS energy capacity

Dataset A - Tx0

Dataset A - Tx1

ESS size eESS [p.u.]

1

1 step 24h step 6h step 1h estim. 24h estim. 6h estim. 1h

0.8 0.6

0.8 0.6

0.4

0.4

0.2

0.2

0

0

2

4

6

8

10

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0

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ESS size eESS [p.u.]

Dataset B - Tx0 1

0.8

0.8

0.6

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0.2

0.2

0

2

4

6

8

10

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ESS size eESS [p.u.]

Dataset C - Tx0 1

0.8

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0.2

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0

2

4

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Dataset C - Tx1

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0

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Dataset B - Tx1

1

0

4

8

Unserved energy eue2 [%]

10

0

0

2

4

6

Unserved energy eue2 [%]

Figure 3.14. ESS energy capacities with improved estimation of unserved energy (“estim.”) and time step simulation (“step”) for datasets A, B, C and forecast scenarios T×0 and T×1.

66

Chapter 3. ESS sizing based on forecast uncertainty

Dataset

Scenario T×0 1h 6h 24 h

Scenario T×1 1h 6h 24 h

Time step simulation A B C

0.083 0.051 0.030

0.161 0.152 0.143

0.207 0.219 0.208

0.684 0.630 0.652

0.801 0.802 0.807

0.807 0.826 0.798

Estimation Case I

A B C

0.043 0.020 0.008

0.124 0.102 0.098

0.192 0.202 0.194

0.871 0.894 0.804

0.903 0.924 0.889

0.853 0.901 0.828

Case II

A B C

0.049 0.028 0.013

0.139 0.120 0.115

0.202 0.213 0.204

0.857 0.885 0.784

0.894 0.919 0.880

0.847 0.897 0.822

Deviation ∆eESS [%] Case I

A B C

−47.6 −60.8 −73.7

−22.8 −33.0 −31.4

−7.0 −7.7 −6.8

27.4 41.8 23.3

12.8 15.3 10.2

5.7 9.1 3.7

Case II

A B C

−41.2 −46.0 −56.8

−13.8 −20.8 −19.3

−2.3 −2.7 −1.8

25.3 40.4 20.4

11.7 14.7 9.1

4.9 8.6 3.0

Table 3.4. ESS energy capacity [p.u.] for eue = 2% from time step simulations and estimations from cdf of SOC; Case I: simple estimation, Case II: improved estimation.

Dataset

1%

2%

5%

10%

20%

30%

Case I

A B C

7.48 6.72 4.49

5.66 9.11 3.71

12.66 15.06 9.77

9.53 18.88 8.72

8.45 11.60 8.85

9.48 11.09 8.12

Case II

A B C

6.52 6.22 3.73

4.91 8.61 2.97

10.40 13.65 7.54

6.41 14.76 5.71

2.92 5.14 3.14

−2.37 −3.36 −4.05

Table 3.5. Deviation of ESS energy capacity estimation from time step simulation for several levels of unserved energy eue for 24-h persistence forecast and scenario T×1.

67

3.3 Sizing ESS energy capacity

3.3.6

Postprocessing of the results: deviation factors

Deviation factor fsat In order to provide a better comparison, a deviation factor fsat is defined in (3.17) to describe the discrepancy between estimated and simulated values of tsat . fsat =

tsat,sim tsat,cdf

(3.17)

where tsat,sim is the saturation time obtained from time step simulation and tsat,cdf the saturation time obtained from the cumulative distribution of SOC. Results for fsat are shown in Fig. 3.15. Values of fsat below unity mean that tsat has been overestimated. With the exception of tsat = 2% saturation times are overestimated. The overestimation is inherent in the approach based on cdf of SOC. Results for individual turbines A04, A13, A33 and single wind farm B1 are shown in appendix B.6. Scenario Tx0 - A

Scenario Tx1 - A

2

2

1.5

1.5

1

1

1

0.5

0.5

0.5

e2 e10 e25 e50

fsat

1.5

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3 6 12 24 48 168

720

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Scenario Tx0 - B

720

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e2 e10 e25 e50

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e2 e10 e25 e50

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720

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T [h]

3 6 12 24 48 168

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3 6 12 24 48 168

720

Scenario Tx2 - C

2

0

1

Scenario Tx1 - C

2 1.5

1

Scenario Tx2 - B

2

1.5

fsat

3 6 12 24 48 168 Scenario Tx1 - B

2

fsat

Scenario Tx2 - A

2

720

0

1

3 6 12 24 48 168

T [h]

Figure 3.15. Deviation factor between estimated and simulated saturation time.

720

68

Chapter 3. ESS sizing based on forecast uncertainty

An example may be the best way to explain the effect: Let a reference system with no saturation have the energy capacity 1 p.u. and the examined ESS have a reduced size of 0.8 p.u. From the cdf it is known how much time SOC of the reference system is above 0.8. The simplification consists in the assumption that the ESS with reduced energy capacity will be saturated the same time as the SOC of the reference system is above 0.8. It is true that the reduced ESS cannot be charged if its SOC has reached 0.8 (i.e. it is saturated), but it can be discharged and thus be available for subsequent charge demand. As long as the discharge is not saturating the ESS in the opposite direction (SOC = 0), all discharge and some of the charge demand is served, although the reference system would have been all the time above 0.8. Hence, fluctuations with relatively high frequency and low amplitude are reducing the real saturation time. In addition, for forecasts beyond 24 h the diminishing variability of SOC (see Fig. 3.5) amplifies this tendency even more. But why saturation times are underestimated for small values of T and tsat ? In case of T×0 and very short forecast intervals (0.5 – 1 h), one possible reason may be that for saturation times of 25 or 50% the ESS size becomes very small (for example: e50 (T = 0.5) = 0.003 and e50 (T = 1) = 0.007). In these cases, the ESS size is at the limit of calculus accuracy. The underestimation with T×1 and T×2 forecasts and low saturation times such as 2% cannot be justified with the same argument. Deviation factors fup and fue From Fig. 3.12 it is difficult to see any tendency of the estimation deviations. As already mentioned, mean unserved power p¯ue is the link between tsat and eue . As p¯ue has been approximated with ETR0 , a deviation factor fup can be defined as fup =

p¯ue P¯ue = ETR0 MAE

(3.18)

where p¯ue and P¯ue are the normalised and not normalised mean unserved power, MAE is the mean absolute forecast error and ETR0 the energy throughput ratio. Depending on the data available, it may be more comfortable to calculate fup by one or the other way. Results for fup are shown in Fig. 3.16. Results for individual turbines A04, A13, A33 and single wind farm B1 are shown in appendix B.8. Analogue to fsat , values of fup below unity represent an overestimation. While the behaviour of fsat was at least similar for all forecast scenarios, very complex patterns of fup can be observed in Fig. 3.16. While for T×0 the mean unserved power p¯ue is always underestimated, for T×1 and T×2 a wide spread of fup as a function of tsat is observed. The only common tendency is that for large forecast intervals fup is tending to unity. Further investigation is needed to identify the reasons for these discrepancies.

69

3.3 Sizing ESS energy capacity

Scenario Tx0 - A

Scenario Tx1 - A

4

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

fup

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720

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3 6 12 24 48 168

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Scenario Tx2 - C

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0

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4 3

1

Scenario Tx2 - B

4

3

fup

3 6 12 24 48 168 Scenario Tx1 - B

4

fup

Scenario Tx2 - A

4

720

T [h]

0

1

3 6 12 24 48 168

720

T [h]

Figure 3.16. Deviation factor between estimated and simulated mean unserved power.

At this point, the estimation of eue in (3.14) can be completed with the empirical deviation factors fsat and fup to formulate equation (3.19). eue = tsat · ETR0 · fsat · fup

(3.19)

where tsat [%] is the saturation time, ETR0 [p.u.] the energy throughput ratio, fsat the deviation factor of tsat and fup the deviation factor of p¯ue . Multiplying the deviation factors fup and fsat , the total deviation factor fue for the unserved energy eue is obtained. Results for datasets A, B and C are shown in Fig. 3.17. The complex structure of fue leads to the conclusion that it is not trivial to develop an analytical expression for the estimation of unserved energy eue .

70

Chapter 3. ESS sizing based on forecast uncertainty

Scenario Tx0 - A

Scenario Tx1 - A

4

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

fue

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720

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720

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3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

3 6 12 24 48 168

720

3 6 12 24 48 168

720

Scenario Tx2 - C

4

0

1

Scenario Tx1 - C

4 3

1

Scenario Tx2 - B

4

3

fue

3 6 12 24 48 168 Scenario Tx1 - B

4

fue

Scenario Tx2 - A

4

720

0

1

T [h]

3 6 12 24 48 168

720

T [h]

Figure 3.17. Deviation factor between estimated and simulated mean unserved energy.

A first approximation of unserved energy is formulated using saturation time from cdf curves and the energy throughput ratio. Based on this, two fundamentally different sources of deviation have been identified. On one side deviations in saturation time and on the other side, deviations in mean unserved power. With this study, the basis is provided for further refinement of the model, which is not pursued here.

3.3.7

Energy balance in case of ESS saturation

As a final comment concerning the accuracy of the statistical data, the energy balance of the ESS is analysed. The energy balance of the ESS has to be maintained, which means that energy inflow must be equal to outflow. As a consequence, if at some moment the ESS saturates because it is fully charged, the energy “lost” to the system will cause later another saturation, when the ESS is empty. In the idealised case of 100% efficiency, the

71

3.3 Sizing ESS energy capacity

values of not served energy demand due to the events “ESS full” and “ESS empty” must be equal. In appendix B.9 (Fig. B.8) it is shown that this assumption is only true if large time series are used (minimum 3 years, as in aggregated dataset B). Especially in the case of forecast scenarios T×1 and T×2, when large forecast horizons are assumed, the initially charged energy becomes so large that at the end of the observation time the energy balance is not given as assumed. In this case more energy will not be served due to “ESS full” than due to “ESS empty”. As a consequence, only the aggregated time series of dataset A and B are useful, where an equalised energy balance can be assumed up to 168 h of forecast horizon. In Fig. 3.18 the percentage of energy not served due to the saturation case “ESS empty” is depicted. The 50%-line represents equalised energy balance, which starts to be lost with forecast horizons of the above mentioned 168 h. It may be stressed here that in case of one-year time series the assumption of equalised energy balance only holds for forecasts up to 24 h (see also appendix B.9, Fig. B.8). eue,d (% of total) eue,d (% of total)

60

eue,d (% of total)

Scenario Tx0 - A 60

60

Scenario Tx1 - A

40 e2 e10 e25 e50

20 0

1

3 6 12 24 48 168 Scenario Tx0 - B

720

40 e2 e10 e25 e50

20 0

1

3 6 12 24 48 168 Scenario Tx0 - C

720

40 e2 e10 e25 e50

20 0

1

3 6 12 24 48 168

720

Scenario Tx2 - A

60

60

40

40

20

20

0

1

3 6 12 24 48 168 Scenario Tx1 - B

720

0

60

60

40

40

20

20

0

1

3 6 12 24 48 168 Scenario Tx1 - C

720

0

60

60

40

40

20

20

0

1

3 6 12 24 48 168

720

0

1

3 6 12 24 48 168 Scenario Tx2 - B

720

1

3 6 12 24 48 168 Scenario Tx2 - C

720

1

3 6 12 24 48 168

720

T [h] T [h] T [h] Figure 3.18. Energy balance expressed by percentage of unserved energy during discharge (saturation case “ESS empty”); comparison of aggregated datasets A (32 year data) and B (three wind farms) and dataset C (single wind farm, one year data).

72

3.3.8

Chapter 3. ESS sizing based on forecast uncertainty

Summary of ESS energy sizing

As the second step of ESS sizing, in this section a strategy has been developed for estimating the required energy capacity of an ESS under the condition of the reduction of wind power forecast errors. The method is based on the estimation of saturation times from cdf of SOC of the ESS. Results will be validated in the next section in a case study with real world forecast data. In Fig. 3.19 a scheme is depicted which summarizes the steps proposed in this section, to estimate the ESS energy capacity eESS and unserved energy eue based on predefined saturation times tsat . In a post-processing step, two empirical correction factors fsat and fup are extracted from a time step simulation. Thus, estimation errors are divided in errors from saturation time estimation and mean unserved power estimation. As shown in the next section, especially in combination with results from real world forecasts, deviation factors can give valuable information for improving the sizing algorithm. A further refinement of estimation methods is outside of the scope of this thesis, but represents an interesting issue for future work.

Forecast error: ε(t)

ETR

State of Charge: SOC(t)

SOC

cdfSOC

Saturation times: tsat

ESS sizes: eESS (tsat )

Time step simulation with ε(t) and eESS (tsat )

Saturation times: tsat,sim (eESS )

Unserved Energy: eue (eESS )

Post processing of the results

Pue =

eue tsat,sim

fup =

Pue ETR

fsat =

tsat,sim tsat

Figure 3.19. Schematic of energy capacity sizing of an ESS and definition of unserved energy.

73

3.4 Case study with real world forecast data

3.4

Case study with real world forecast data

In this section, the ESS sizing methods proposed in this chapter are verified with real world forecast data. For this purpose, a case study is carried out based on dataset C (see section 2.3 for more details on dataset C). Results are presented for up to six scenarios. Two forecast scenarios (T×1, T×2) are obtained from the persistence model. The T×0 scenario is not considered as it is far too optimistic in any case. Two scenarios are original forecasts MSE and MCC and finally two additional scenarios are obtained from online bias correction of original forecasts, denominated MSEc and MCCc (see section 2.5 for more details on bias correction). All results are based on a 24-h forecast. Following the methodology described in the previous sections of this chapter, first the ESS power is calculated. As concluded in section 2.6, persistence scenario T×1 can be considered as worst case scenario for ESS power sizing. Hence, only T×1 is chosen as reference scenario in this first sizing step. Therefore, for ESS power sizing only five scenarios are evaluated: T×1, MSE, MSEc, MCC and MCCc. Secondly, ESS energy capacity is calculated. Here, no assumption can be made a priori about the best reference model. Observations from bias correction in section 2.5 are indicating that possibly T×1 is too optimistic. Hence persistence scenarios T×1 and T×2 are included as reference in the second sizing step. As a consequence, six scenarios are evaluated for ESS energy capacity sizing: T×1, T×2, MSE, MSEc, MCC and MCCc.

3.4.1

Sizing of ESS power

The first step of ESS sizing is the calculation of its nominal power. Unserved energy eup which results from reduced nominal ESS power pESS can be obtained from the pdf of the forecast error ε, as described in section 3.2. If forecast data is available for at least one year, histograms of ε (empirical pdf) can be used. Otherwise an estimation of the pdf with Betafit for example is needed. In Fig. 3.20 results from T×1 scenario are compared with original and bias-corrected time series from MSE and MCC forecast. MSE - 24h

MCC - 24h

ESS power pESS [p.u.]

1

1 hist Tx1 hist MSE hist MSEc betafit Tx1 betafit MSE betafit MSEc

0.8

0.6

0.8

0.6

0.4

0.4

0.2

0.2

0 0

2

4

6

8

Unserved energy eup [%]

10

hist Tx1 hist MCC hist MCCc betafit Tx1 betafit MCC betafit MCCc

0 0

2

4

6

8

10

Unserved energy eup [%]

Figure 3.20. Minimum ESS power as a function of unserved energy from dataset C for real world 24-h forecasts MSE and MCC compared with persistence scenario T×1, calculated with observed histograms (“hist”) and Betafit approximation (“betafit”).

74

Chapter 3. ESS sizing based on forecast uncertainty

At a glance it can be seen that the T×1 scenario produces very similar results compared to real forecasts. If ESS power is calculated with Betafit estimation from T×1, very good results are obtained for both, MSE and MCC forecasts. Best results are obtained with bias corrected data MSEc and MCCc. In case of MCC, bias correction reduces the estimated ESS power by approximately 10%. This means in fact that the proposed online bias correction reduces the power requirements of the ESS by 10% in case of MCC. In Table 3.6 a summary of obtained values of ESS nominal power pESS is given for levels of unserved energy in a range of eup = 1 ... 30%. Results using empirical pdf, Betafit approximation and deviation ∆pESS are shown. Deviation ∆pESS is defined in (3.3) (see section 3.2, p. 46). Dataset

1%

2%

5%

10%

20%

30%

0.273 0.202 0.201 0.193 0.223

0.193 0.138 0.137 0.134 0.140

0.247 0.192 0.188 0.204 0.125

0.177 0.130 0.126 0.144 0.068

−9.5 −5.1 −6.7 5.7 −43.8

−8.2 −6.3 −7.4 7.5 −51.7

Empirical pdf T×1 MSEc MCCc MSE MCC

0.673 0.592 0.615 0.586 0.668

0.604 0.522 0.541 0.510 0.592

0.495 0.409 0.424 0.395 0.470

0.392 0.310 0.317 0.297 0.354

Betafit pdf T×1 MSEc MCCc MSE MCC

0.609 0.517 0.524 0.523 0.469

0.545 0.461 0.466 0.469 0.407

0.444 0.372 0.372 0.382 0.310

0.354 0.289 0.286 0.299 0.223

Deviation ∆pESS [%] T×1 MSEc MCCc MSE MCC

−9.5 −12.8 −14.8 −10.8 −29.8

−9.8 −11.8 −13.9 −8.0 −31.3

−10.1 −9.1 −12.2 −3.4 −34.0

−9.8 −7.0 −9.9 0.7 −37.1

Table 3.6. Deviation of ESS nominal power from Betafit for several levels of unserved energy eue for 24-h persistence forecast and scenario T×1.

Deviation ∆pESS reveals that Betafit performs fairly well for T×1 and bias corrected scenarios MSEc and MCCc, with deviations ranging from −15 ... − 5%. Uncorrected forecasts show significant differences. In case of MCC forecast, the strong bias leads to poor approximations with large deviations up to −52%. An interesting observation is that Betafit approximation of MSE forecast does not seem to improve with bias correction as expected. In section

3.4 Case study with real world forecast data

75

2.6.1 it was shown that the Anderson-Darling (A-D) test statistic could be reduced with bias correction from 148 for MSE to 122 for MSEc, indicating a better approximation performance of the Betafit method. Also the K-S test statistic lowered from 0.083 for MSE to 0.067 for MSEc. The reduction of A-D test statistic is –17.6% while the reduction of K-S test is slightly more pronounced with –19.3%. Taking into account that the A-D test gives more importance to the tail region of the pdf, it can be concluded that the improvement of Betafit is mainly obtained in the central part of the pdf and not in the tail. In fact, the approximation of the last 50% (absolute forecast error |ε| > 0.5 p.u.) is slightly worse for MSEc. The improvement of Betafit approximation due to bias correction becomes visible only for larger values of unserved energy. For eup > 10% the deviation becomes positive (ESS power is overestimated) and absolute deviation for 20 and 30% is already larger for MSE than for MSEc. Finally it can be concluded that bias correction should be preferred, as the observed impact for small values of unserved energy is very limited and for large values even a slight improvement is observed. As final conclusion of ESS power sizing it can be stated that the T×1 scenario is a very good tool for sizing if no real forecasts are available. Even taking into account the underestimation of about 10%, Betafit can be applied if data basis is poor (one year provides already a critically small amount of data). If real forecasts are available, bias correction is strongly recommended to improve approximation on one side and reduce ESS power requirements on the other.

3.4.2

Sizing of ESS energy capacity

The second step of ESS sizing is calculating the energy capacity depending on the permitted unserved energy eue . As already seen in section 3.3, this task is more difficult and additional parameters such as saturation times have to be introduced. The T×2 scenario is added to the evaluation, so that six scenarios are simulated. In addition, for the estimation of eue two versions are considered. As shown in section 3.3.4, eue may be estimated as a product of energy throughput ratio ETR0 and saturation time tsat . This will be the reference in this case study. The impact of a correction factor of ν = 0.92 as proposed in (3.15) (p. 60) will be examined as well. Cumulative distribution (cdf ) of SOC As a first step of ESS energy capacity sizing, the cumulative distributions (cdf) of SOC are obtained. In Fig. 3.21 cdf of SOC are represented separately for scenarios T×1, T×2 and bias corrected MSEc and MCCc (left) and uncorrected forecasts MSE and MCC (right), due to the large difference in scale. The plots of cdf already show that real world forecasts, even with bias correction demand significantly more ESS capacity for error compensation than scenarios T×1 or T×2. Parameters obtained from this first step are presented in Table 3.7. Looking at “zero error” reference e0 for complete error compensation, with persistence scenarios T×1 and T×2 values of 0.96 and 1.78 p.u. are obtained. Bias corrected real world scenarios MSEc and MCCc demand still 2.96 and 3.40 p.u. respectively. In case of MCCc this means almost duplicating the ESS capacity compared to the worst case scenario T×2. If MSE and MCC forecasts are not bias corrected, extremely large values of

76

Chapter 3. ESS sizing based on forecast uncertainty

Frequency [%]

bias corrected

original data

100

100

80

80

60

60

40

40

20 0

-1

0

1

MSEc MCCc Tx2 Tx1

20

2

0 -10

MSE MCC

State of Charge [p.u.]

0

10

20

State of Charge [p.u.]

Figure 3.21. Cumulative histograms (cdf) of SOC for different 24-h forecast scenarios from datasets C.

Scenario min(SOC) T×1 T×2 MSEc MCCc MSE MCC

max(SOC)

e0

ETR0

0.00 0.00

0.96 1.78

0.96 1.78

0.754 0.898

–1.43 –0.95 –9.71 0.05

1.53 2.45 1.36 26.76

2.96 3.40 11.07 26.71

0.675 0.672 0.693 0.637

Table 3.7. Parameters obtained from SOC time series and ETR for several forecast scenarios from dataset C.

ESS capacity are obtained, which demonstrates the mandatory need for bias correction. For comparison, energy throughput ratio ETR0 is included in Table 3.7. Here it can be seen clearly that advanced forecast models produce much lower energy throughput in the ESS, in contrast to the total ESS capacity needed to compensate the forecast error completely. Note that MCC presents the lowest value of ETR0 with 63.7% of total generation but ESS energy capacity is with 26.7 p.u. 15 times larger than for the worst case persistence forecast scenario T×2. Estimation of ESS capacity eESS from cdf of SOC The second step of ESS energy sizing consists in deriving values of ESS capacity eESS from curves of cdf of SOC for predefined saturation times tsat as described in section 3.3.3. Saturation times of 0.1, 2, 10, 25 and 50% have been considered and the corresponding values of eESS , denoted as e0.1 , e2 ,... etc. have been calculated. Results are presented in Fig. 3.22 and Table 3.8. Reduced ESS capacities follow the same tendencies as observed for zero error reference e0 . Curves for original forecast data MSE and MCC are represented in a separate plot, due to its large difference in scale.

77

3.4 Case study with real world forecast data

ESS capacity eESS [p.u.]

bias corrected

original data MSEc MCCc Tx2 Tx1

3

2

30

MSE MCC

25 20 15 10

1

5 0

2

10

25

50

0

2

10

25

tsat [%]

50

tsat [%]

Figure 3.22. ESS energy capacity obtained from cdf of SOC for predefined saturation times between 0.1 and 50%.

Scenario T×1 T×2 MSEc MCCc MSE MCC

e0

e0.1

e2

e10

e25

e50

0.96 1.78

0.95 1.75

0.84 1.61

0.64 1.18

0.43 0.19 0.82 0.40

2.96 2.91 2.22 1.53 1.08 0.65 3.40 3.35 2.46 1.50 1.00 0.62 11.07 11.03 10.49 9.77 8.21 4.10 26.71 26.68 25.08 20.08 12.26 5.55

Table 3.8. ESS energy capacity obtained from cdf of SOC for predefined saturation times between 0.1 and 50%.

Estimation of unserved energy eue for reduced ESS energy capacity As third step, unserved energy eue due to reduced ESS energy capacity is estimated. In this case study two versions will be adopted. The first estimation applies the simple assumption that mean excess power will be equal to mean absolute error MAE. Hence, as described on page 59 in (3.14), eue is estimated simply multiplying ETR0 with tsat . The second estimate, termed eue2 , includes an adjustment factor ν as in (3.15). From persistence forecasts of datasets A, B and C an empirical value of ν = 0.92 has been found to be a good choice. As already mentioned in section 3.3.4, an optimization of ν is outside of the scope of this thesis. Results of the two estimations will be presented together with results from time step simulation. Results from time step simulation: tsat and eue In order to verify the estimations done in the preceding steps, the fourth step is a time step simulation of the state of charge of the ESS. As a result, simulated values of tsat and eue are obtained. In Fig. 3.23 simulated and estimated values of tsat and eue are represented in correlation plots. Points on the diagonal line represent perfect matching of estimate and simulation.

78

Chapter 3. ESS sizing based on forecast uncertainty

The left plot shows saturation times. The generalised overestimation of tsat from cdf of SOC observed with persistence scenarios T×1 and T×2 is even more pronounced for real world forecasts. This may be due to the larger ESS sizes combined with lower throughput rates. Bias correction improves slightly the estimation. The most interesting effect is that results of tsat from bias corrected time series are very similar, which makes is easier to find adjustment factors, valid for more than one particular forecast scenario. Results for estimated and simulated unserved energy are shown in the right plot of Fig. 3.23. Estimated values of unserved energy are represented for the improved case of eue2 . A good performance of scenarios T×1 and T×2 can be seen. With original MCC data up to approximately eue = 16%, estimates are also very close to simulation results. For bias corrected data estimations are good only for small values of eue up to 6%. With MSE forecast unserved energy is overestimated widely in the whole range. As final result of ESS sizing, the relationship eESS (eue ) is represented in Fig. 3.24. The same results are also shown in Table 3.9. 50

50

25

Tx1 Tx2 MSEc MCCc MSE MCC

10 2 0

10

20

30

40

eue2,cdf [%]

tsat,cdf [%]

40 30 Tx1 Tx2 MSEc MCCc MSE MCC

20 10 0 0

50

10

20

30

40

50

eue,sim [%]

tsat,sim [%]

Figure 3.23. Correlation plot for comparison of saturation time and unserved energy obtained from cdf and simulation. original data

ESS size eESS [p.u.]

bias corrected step MSEc step MCCc step Tx2 step Tx1 estim. MSEc estim. MCCc estim. Tx2 estim. Tx1

3

2

step MSE step MCC estim. MSE estim. MCC

30 25 20 15 10

1

5 0 0

5

10

15

20

25

Unserved energy eue [%]

30

0 0

5

10

15

20

25

30

Unserved energy eue [%]

Figure 3.24. Minimum ESS energy capacity as a function of unserved energy from dataset C for real world 24-h forecasts MSE and MCC compared with persistence scenario T×1, estimated from cdf of SOC (“estim.”) and time step simulation (“step”).

79

3.4 Case study with real world forecast data

The similarity between Fig. 3.24 and Fig. 3.22 is due to the relationship between tsat and eue . Indeed for the estimation of eue a linear dependence is assumed. Hence estimated curves in Fig. 3.24 correspond directly with curves of eESS shown in Fig. 3.22. The left plot shows bias corrected scenarios together with T×1 and T×2. It is seen that T×2 scenario can be used for ESS energy sizing only for values of unserved energy eue > 10%. If lower values are desired, real world forecasts demand significantly larger storage capacities. Scenario T×1 is inadequate in this case. Excellent estimations are obtained for both persistence scenarios with deviations well below 10%. Estimations for MSEc and MCCc are very good for eue ≤ 5%, but for larger eue over-estimations of up to 50% are observed. Results for original forecast data are shown in the right plot of Fig. 3.24. Estimations for MCC are very good up to 15% unserved energy, while for uncorrected MSE data estimations are surprisingly very poor. Dataset

1%

2%

5%

10%

20%

30%

Time step simulation T×1 T×2 MSEc MCCc MSE MCC

0.84 1.60

0.80 1.53

0.66 1.30

0.55 1.08

0.38 0.80

0.27 0.61

2.37 2.61 10.51 25.81

2.11 2.29 9.68 24.87

1.63 1.63 7.40 22.03

1.10 1.09 4.70 17.29

0.64 0.63 3.47 7.83

0.50 0.49 2.77 4.53

Estimation T×1 T×2 MSEc MCCc MSE MCC

0.88 1.67

0.82 1.58

0.71 1.39

0.58 1.13

0.39 0.84

0.26 0.63

2.37 2.64 10.61 25.33

2.12 2.32 10.38 24.20

1.70 1.73 9.96 21.00

1.35 1.29 9.18 16.40

0.96 0.89 7.16 9.81

0.68 0.64 4.58 5.44

Deviation ∆eESS [%] T×1 T×2

3.73 3.99

2.97 3.65

7.54 7.03

5.71 5.03

3.14 6.07

–4.05 3.16

MSEc MCCc MSE MCC

−0.2 1.2 1.0 −1.9

0.6 1.3 7.3 −2.7

4.4 6.4 34.7 −4.7

22.5 19.1 95.1 −5.2

50.1 40.7 106.3 25.3

35.5 30.8 65.5 20.1

Table 3.9. Deviation of ESS energy capacity estimation from time step simulation for several levels of unserved energy eue for real world forecasts and 24-h forecast scenarios T×1 and T×2.

80

Chapter 3. ESS sizing based on forecast uncertainty

Postprocessing of the results: deviation factors Some more insight into the deviations of estimated parameters is obtained with deviation factors, as they are defined in section 3.3.6, which is the fifth and last step of ESS energy sizing. Deviation factors are defined here as the factor, the estimate should be multiplied by, to obtain the simulated result. Deviation factors are represented in Fig. 3.25 and tables 3.10 and 3.11. Note that deviation factors for eue exhibit larger deviations than those shown for ESS energy capacity eESS in Table 3.9. Hence, deviations of estimates of unserved energy translate to smaller deviations in energy capacity. Thus, the sizing method is relatively robust against estimation errors. This effect can be explained with the characteristic curves of eESS (tsat ), as shown in Fig. 3.22. Due to the small slope of the curves, large deviations in tsat cause small deviations in eESS . For example, in scenario MSEc, for eue = 10% a deviation factor of fue = 0.907 is obtained (compare Table 3.11), which means a deviation of approx. +10.3%. This translates to a deviation in eESS of only +4.4% (compare Table 3.9). 1.5

fsat

1

Tx1 Tx2 MSEc MCCc MSE MCC

4 3

fup2

Tx1 Tx2 MSEc MCCc MSE MCC

2

0.5 1

0

2

10

25

0

50

2

10

25

tsat,cdf [%]

tsat,cdf [%] Tx1 Tx2 MSEc MCCc MSE MCC

2 1.5

fue2

50

1 0.5 0

2

10

25

50

tsat,cdf [%] Figure 3.25. Deviation factors of saturation time, unserved power and unserved energy as a function of predefined saturation time.

As described in section 3.3.6, saturation times are strongly overestimated, while mean unserved power is underestimated. Hence, even with large deviations in estimated saturation time and unserved power, estimates for unserved energy are quite good.

81

3.4 Case study with real world forecast data

An exception is uncorrected forecast scenario MSE. In this case estimated saturation time is especially far overestimated, yielding a deviation factor of fsat = 0.311 for estimated saturation time 10%. This factor means that the estimate is more than three times higher than the obtained value from time step simulation. On the other hand, the estimate of mean unserved power is almost perfect for the whole range of saturation times. Hence, no compensation takes place and unserved energy is overestimated by almost the same factor as saturation time. Interestingly this behaviour disappears completely with bias correction. This special case may be a valuable example for further improvements of estimation of saturation times and unserved energy. Scenario

fsat

fup

fup2

fue

fue2

T×1 T×2

0.763 0.671

0.952 1.085

1.035 1.179

0.726 0.729

0.789 0.792

MSEc MCCc MSE MCC

0.593 0.559 0.311 0.548

1.402 1.482 0.887 2.019

1.523 1.611 0.965 2.195

0.831 0.829 0.276 1.107

0.903 0.901 0.300 1.204

Table 3.10. Deviation factors of saturation time, unserved power and unserved energy for ESS sizes corresponding to tsat = 10% (e10 ).

Scenario

fsat

fup

fup2

fue

fue2

T×1 T×2

0.766 0.684

0.946 1.045

1.028 1.136

0.725 0.715

0.788 0.777

MSEc MCCc MSE MCC

0.600 0.561 0.270 0.599

1.390 1.480 0.894 1.901

1.511 1.608 0.971 2.066

0.834 0.831 0.241 1.139

0.907 0.903 0.262 1.238

Table 3.11. Deviation factors of saturation time, unserved power and unserved energy for ESS sizes corresponding to eue = 5%.

Deviation factors can be expressed as a function of saturation time, as in Fig. 3.25 and Table 3.10 or as a function of unserved energy as done in Table 3.11. The deviation factors as a function of saturation time are more convenient for graphical representation, as saturation times are well defined and identical for all scenarios. Each value of tsat is associated to an ESS energy capacity, obtained from cdf of SOC. Thus, estimated and simulated values are available straightforward. If a representation as a function of eue is desired, results have to be interpolated. First the corresponding eESS for each value of eue has to be calculated. Results from time step simulation are adopted as a reference. Then estimated and simulated values of saturation

82

Chapter 3. ESS sizing based on forecast uncertainty

time and unserved energy can be obtained with interpolation for example. This second representation is convenient for comparison of deviation factors and results of Table 3.9. As final conclusion of the examination of deviation factors, the following can be stated. The overall tendency of the deviations of the proposed estimation method are the same for persistence scenarios T×1 and T×2 and real world forecasts. Overestimation of saturation times is more pronounced in real world scenarios. Underestimation of mean unserved power is also more pronounced in real world scenarios, with exception of uncorrected MSE forecast. Finally, similar deviation factors of unserved energy are obtained for persistence and real world scenarios. It should be highlighted the positive effect of bias correction on the accuracy of estimations. While the uncorrected scenarios MSE and MCC show large differences in their behaviour, the bias corrected versions MSEc and MCCc are very similar and closer to the persistence scenarios.

3.4.3

Conclusions from case study

In this case study, a complete ESS sizing routine has been elaborated comparing results from two persistence and four real world forecast scenarios. First, ESS nominal power has been calculated as a function of permitted unserved energy eup caused by limited nominal power. In the second part ESS energy capacity was calculated as a function of unserved energy eue , caused by limited energy capacity. Unserved energy is understood as the accumulated uncompensated forecast error, in percent of total generated wind power. As central conclusion from ESS power sizing it can be stated that the T×1 scenario is a very good tool for sizing if no real forecasts are available. Even taking into account the underestimation of about 10%, Betafit can be applied if data basis is poor (one year provides already a critically small amount of data). If real forecasts are available, the proposed bias correction improves approximation accuracy and reduces ESS power requirements. The most important conclusion from sizing of ESS energy capacity is that persistence scenario T×2 is a good approximation for values of unserved energy eue > 10%. If lower values are required, real world forecasts demand significantly larger storage capacities, up to 60% more for eue = 1%. The proposed estimation method tends to overestimate the required ESS energy capacity. An improvement of the critical issue of saturation time estimations would enhance significantly the overall performance of the method. Also for energy capacity sizing, bias correction revealed very positive effects on the accuracy of estimations. While the uncorrected scenarios MSE and MCC show large differences in their behaviour, the bias corrected versions MSEc and MCCc are very similar and closer to the persistence scenarios. More sophisticated bias correction methods might even amplify this tendency. Hence, ESS sizing based on the persistence model would become more accurate. The final conclusion of this case study is that persistence scenarios T×1 and T×2 are suitable for ESS sizing with some restrictions. But knowing the shortcomings, adjustment factors may be included to improve the performance. If real world forecast data is available, it is necessary to make sure that the forecast is bias free. A simple bias correction method, based on moving averages can improve estimations and reduce significantly requirements of ESS power and energy capacity.

3.5 Summary

3.5

83

Summary

In the literature, several probabilistic sizing methods have been proposed. The novelty of the approach suggested in this thesis consists in a detailed statistical study of forecast errors and state of charge of the ESS. Most existing studies have assumed gaussian distributions to describe the uncertainty of wind power, only few of the most recent works consider more realistic data. None of them proposes sizing based on non-gaussian probability density functions (pdf). The advantage of using pdf is that time step simulations can be avoided, once the statistical parameters are well understood. The parametric study presented here, covers forecast intervals from 30 min up to 30 days. None of the existing methods covers such a large range of forecast intervals. Existing methods rely strongly on time step simulations and no relationships are established between statistical parameters and ESS sizing parameters. Finally, none of the methods proposed in the literature allows probabilistic sizing of both, power and energy requirements of the ESS. The problem treated in this chapter can be formulated in three steps: • What energy and power ratings would have an ESS that compensates all forecast errors? • If rated power is reduced, what is the amount of residual errors? • If nominal energy capacity is reduced, what is the amount of residual errors? Before solving the questions above, the concept of “residual error” needs to be defined. Therefore, the term of unserved energy is introduced. It is defined as the cumulated energy of deviations from forecast which cannot be compensated by the ESS, and thus represents the residual error. The unserved energy eup is obtained for reduced rated power pESS , under the assumption that there is no energy capacity limit. In a next step unserved energy eue is estimated for reduction of eESS , assuming that no power limit is present. Main achievements of this chapter are: • Functional relationship between ESS power and unserved energy based on forecast error pdf • Functional relationship between ESS energy capacity and saturation times based on cdf of SOC • Estimation of unserved energy based on saturation times • Characterisation of estimation errors for future model improvement • Case study with real world forecast data, demonstrating the importance of bias free forecasts

84

Chapter 3. ESS sizing based on forecast uncertainty

ESS sizing It is concluded that the sizing of nominal power and energy cannot be done at once. Therefore, two independent probabilistic methods have been developed to estimate values of unserved energy eup and eue as a function of reduced nominal power pESS and normalised energy capacity eESS , respectively. A simple relationship between unserved energy eup and rated power can be formulated if the pdf of the forecast error is known. The only problem consists in a precise estimation of the pdf if data quality is low. Here the Betafit method proposed in chapter 2 can be helpful. Sizing of energy capacity eESS is far more complex. Due to the integrating effect of the state of charge of the ESS (SOC), the forecast error pdf cannot be used. Therefore, a more empirical strategy was adopted comprising two steps. First saturation times are estimated as a function of reduced ESS size. This is done with cumulative distributions of SOC, obtained from forecast error data. In this thesis data from three wind sites of different characteristics were analysed. In a second step unserved energy eue is estimated from saturation times taking into account the variability of the forecast error. The estimation of unserved energy eue is not an easy task, because the relationship between ESS energy capacity and unserved energy is strongly nonlinear. Here a first approach has been proposed and it is shown that empirical scaling factors lead to better results. Further investigation is needed to improve the model of energy capacity sizing. Therefore a detailed study on estimation errors has been performed in a postprocessing of the results. A major contribution of this study is the identification of the twofold character of the estimation error for eue . First the saturation time is estimated from cumulative distribution functions (cdf) of SOC. From estimated saturation times tsat , ESS sizes eESS are defined. Running a time step simulation with these reduced ESS sizes, saturation times and unserved energy are obtained. Correction factor fsat is obtained from estimated and simulated saturation times. The second source of estimation error has been identified as the mean unserved power p¯ue , which represents the link between eue and tsat . Hence, the second correction factor fup can be defined. It is assumed, that the estimation error of tsat is independent from the mean power. This way, improvements of estimation can be done separately for tsat and p¯ue . With this study, future work is prepared to identify statistical parameters which explain the structure of these two factors and thus may lead to a better analytical estimation of eue . The identification of these parameters was outside the scope of this thesis.

3.5 Summary

85

Case study Another contribution of this chapter is obtained from a case study with real world forecast data. Three main conclusion are derived from this study. • Persistence scenarios T×1 (for power) and T×2 (for energy) are valid for ESS sizing as they represent to a large extent statistics of real world forecasts • Scenario T×0 is far too optimistic and thus not useful for ESS sizing • The proposed sizing method performs well for 24-h forecasts with both, persistence and real world data • Bias correction improves model precision and reduces ESS capacity requirements It could be demonstrated that forecast errors from the persistence scenarios can be used for ESS sizing. The sizing method is verified with real world forecast data. Especially for 24-h forecasts, the method presents relatively low errors which is interesting for dayahead forecasts in electricity markets. Finally, online bias correction revealed a great value for ESS sizing. Estimations are more precise and statistic parameters resemble more those from persistence scenarios.

Chapter 4 Cost analysis of energy storage systems “Man muss das Unm¨ ogliche versuchen, um das M¨ ogliche zu erreichen.” 1 Hermann Hesse (1877–1962) German-Swiss writer

In this chapter the cost structure of a selection of energy storage systems is analysed. The objective of this chapter is twofold. First a cost model for ESS is developed which accounts for specific properties of different ESS technologies. Secondly, the model is applied in a case study to evaluate results from the ESS sizing method proposed in chapter 3. The life-cycle cost analysis includes initial cost (divided into power and energy related costs), costs of energy losses due to system efficiency and cost for replacement of major components. The model represents a synthesis of several existing models, eliminating their shortcomings. The main contribution of this chapter is a probabilistic formulation of the cost model for energy losses due to system efficiency. The proposed model gives an orientation about revenue requirements for several storage technologies. The final conclusion is that energy storage technologies imply important additional costs for the power system. Emerging technologies have promising potentials for cost reductions, but realistic estimations are difficult. In section 4.1 a short overview is given on grid-connected energy storage systems and in 4.2 a selection of ESS is presented more in detail. In section 4.3 a life-cycle cost model is developed and applied in a case study assuming a 24-h forecast of 1 MW wind power.

1

Translation: “To achieve the possible, you have to try the impossible.”

88

Chapter 4. Cost analysis of energy storage systems

4.1

A review on the benefits of grid-connected ESS

A fundamental work on energy storage in electricity systems was published in 1994 by A.G. Ter-Garzarian [62]. In this book, an exhaustive analysis is given on the subject of energy storage in electricity systems. The analysis ranges from the demand side characteristic to the description of the ESS technologies. In [8] a good introduction can be found about the possible application of ESS in combination with wind generation, especially in weak grid environments. For the case of Denmark with its high wind penetration, in [63] the introduction of battery energy storage systems (BESS) was discussed. An overview was given about the present status of BESS technology and methods of assessing their economic viability and impact on power system operation. In particular, the possible role of electric vehicles (EV) in this context was emphasised. See more details on integration of EVs into the grid in section 4.2.4. In [64] a summary about recent advancements and trends in storage technologies was given. In this work it was stated that energy storage has become an enabling technology for renewable energy applications, enhancing power quality in the power systems. The main difference between ESS and regulating power stations is the capability of absorbing energy which at a later time instant can be released. The main fields of investigation are concentrated in the following areas: • Backup systems for critical loads (Uninterruptible Power Supply – UPS) • Stand-alone systems (with renewable energies) • Improvement of power quality issues near industrial facilities • Transmission line reinforcement • Improvement of penetration limit of renewable energies Each application requires a different sizing strategy. There are many publications for UPS and stand-alone systems which are not of particular interest here. The improvement of power quality in the proximity of industrial facilities, as for example electrical furnaces is a very interesting application of storage systems, but also does not enter in the focus of this thesis. In some cases, an ESS has been installed to reenforce a distribution line [65]. In this case, the ESS cuts the daily power peaks and postpones the construction of additional line capacity. This can be an important benefit in case of the integration of renewable energies, but it is not considered here. In this thesis the sizing of the ESS is conditioned by the forecast error, in this case of wind generation. In other words, penetration limits for wind power might be higher if the uncertainty of the generation is reduced. The economic value of the reduction of uncertainty will be studied in chapter 5 of this work. Many studies about possible economical benefits of grid-connected utility scale energy storage are available. ESS as spinning reserve was analysed by the Electric Power Research Institute (EPRI) already in 1987 [66] and it was found to be very beneficial – technically and economically. Later, several studies have been carried out by Sandia National Laboratories (SNL) where benefits were studied separately for each application [67–69].

4.1 A review on the benefits of grid-connected ESS

89

During the last decade, especially SNL was very active developing a benefit/cost framework for the evaluation of ESS technologies [70–73]. In 2006 EPRI published a feasibility study for ESS in the state of New York [74], later refined in [75], which concludes that flywheels and NaS batteries are best suited for the New York electricity market. The most important benefits of energy storage are: • Spinning reserve • Peak power supply (water pumping) • Transmission line reinforcement (peak shaving) • Load side management and power quality In [76] a modular system is proposed, based on power electronic converters which permits the integration of any type of generation and storage units. Such type of control units would be a key element in the distributed generation scenario with high renewable energy penetration. As mentioned before, storage of electrical energy can be used as spinning reserve. In general spinning reserve is assigned to an existing generator in the system. In the case of a newly installed ESS it would be an additional reserve, which means that system flexibility is improved. The additional benefit for the power system would be the same as expected from traditional spinning reserve. There are several publications about the increasing need of reserve in the power system as the penetration of wind power grows [4, 32, 77]. Additional reserve demand was calculated considering forecast errors of the demand forecast and wind power forecast in [4] for the Irish case and in [32] for a hypothetical scenario. In [77] the effectiveness of storage and relocation options to increase system flexibility was assessed, in order to mitigate the impact of increased wind power penetration in Denmark. As the best solution, in this work heat and cold storage with heat pumps was proposed. In [58] the following conclusions were obtained for the capability of ESS to improve the integration of wind power in weak grids: 1. Energy storage over 10 min (e.g. flywheels) allows 10% more wind energy to be absorbed without grid reinforcement and appears to be economically beneficial. 2. Energy storage over 24 h (e.g. redox flow cells) allows up to 25% more wind energy and 30% more revenue to be earned but does not appear to be economically justified. 3. Even with 24-h energy storage, wind energy curtailment may still be necessary in a weak grid connection. 4. Energy management incorporating energy storage over 24 h and energy curtailment can allow up to three times the amount of wind energy to be absorbed by a weak grid compared to conventional grid connection of wind farms.

90

Chapter 4. Cost analysis of energy storage systems

4.2

State of the art of energy storage technologies

The main part of balancing is done with spinning reserve and interconnections with other systems. But still, energy storage is crucial for balancing any power system. The recent development of intermittent renewable energies (foremost wind power) is demanding increased flexibility of the power system, which can be obtained with more storage capacity. At utility scale almost exclusively pumped hydro storage (PHS) is in use, but a number of alternatives have been developed successfully. Some singular installations of battery energy storage (BESS) are running already since many years. But the majority of proposed solutions such as compressed air energy storage (CAES), Flow Batteries (“Regenesysr ”) or hydrogen storage are still in the phase of prototypes. Interestingly, the oil price peak in 2008 has revived the development of electric vehicles (EV). Switching from fossil fuels to electricity is a major challenge of the 21st Century. The concept of integrating electric vehicles into the power system has been named“Vehicle to Grid” or just “V2G” and is described in section 4.2.4. The U.S. based Electricity Storage Association (ESA2 ) [78] suggests the application ranges of ESS technologies as shown in Fig. 4.1. In Fig. 4.1 only existing installations for large scale permanent energy storage applications are represented. Three main operational categories are indicated: • Power Quality (PQ): only used for a few seconds or less • Bridging (Buffer): ensures service continuity when switching from one source of electricity to another • Energy (or Network) Management: decouples power generation and consumption (load levelling).

3 weeks

Energy Management PHS

CAES

NaS

2h

Ion

Ni

20 min

RFB

M

Ni

H

2 min

A L/

Discharge Time

8h Li-

CAES EC FW L/A Li-Ion NaS NiCd NiMH PHS RFB

E > 8 MWh

3 days Bridging Power

Cd

FW

8M

W

20 s

1 kW

EC

2s

h

NaS

Power Quality & UPS

20

kW

h

E < 20 kWh 50 kW

Compressed Air Electrochemical Cap. Flywheel Lead-Acid Lithium-Ion Sodium-Sulfur Nickel-Cadmium Nickel-Metal-Hydride Pumped Hydro Redox Flow Battery

1 MW

50 MW

1 GW

Rated Power Figure 4.1. ESS operating ranges of installed systems (November 2008). Source: Electricity Storage Association. 2

Not to be confused with the European Space Agency

91

4.2 State of the art of energy storage technologies

Smoothing weather effects loads, PV, wind and small hydro

• •



3 days

Weekly smoothing of loads and most weather variations

• •











8 hours

Daily load cycle PV, wind, Transmission line repair

• •













2 hours

Peak load lopping, standing • • reserve, wind power smoothing, Minimisation of trading penalties













20 min

Spinning reserve, wind power smoothing, clouds on PV

















3 min

Spinning reserve, wind power smoothing of gusts











20 s

Line or local faults, Voltage and Frequency control, Governor controlled generation









Applications

Conv. Cap/Ind

3 weeks

Supercapacitor (EC)

Redox Flow Battery



SMES

Pumped Hydro

• •

BESS Flywheel

Hot/Cold Storage

Annual smoothing of loads, PV, wind and small hydro

Biomass HSFC Large Hydro

4 months

Discharge time

CAES

Note that every storage system has its specific range of application. For technical and economical reasons, not all technologies are scalable to any arbitrary size. Barton and Infield (2004) [58] give an excellent overview of ESS types and its specific applications. In [58] applications of ESS were classified by its energy capacity, normalised with nominal power. The unit of the result is a measure of time, therefore it is denoted as full power duration or typical discharge time. Table 4.1 is taken from [58] and represented here, because it is a very concentrated representation of the diversity of ESS applications.







HSFC: Hydrogen-FuelCell, CAES: Compressed Air Energy Storage, BESS: Battery Energy Storage Systems SMES: Superconducting Magnetic Energy Storage, EC: Electrochemical Capacitor, Conv. Cap/Ind: conventional capacitors/inductors

Table 4.1. Storage technologies and applications. Source: Barton 2004.

As can be seen in Table 4.1 flywheels are limited to 20 min storage and technologies such as SMES (Superconducting Magnetic Energy Storage) and Supercapacitors (or double layer capacitors) only can cover very short time periods (up to 20 seconds). These

92

Chapter 4. Cost analysis of energy storage systems

technologies are not further considered in this thesis, but it is acknowledged that they may be very useful when operated in parallel with larger ESS. These high power/high efficiency units can reduce the impact of events with very short time constants, such as wind gusts and any type of faults in the power system. This way, high power peaks are reduced and system lifetime is increased. A number of reviews on ESS technologies have been published recently [79–83]. A very fundamental analysis of electrical energy storage is done in [83], where it is concluded that all known technologies store energy in one of these seven forms: 1. Gravitational potential energy (pumped hydro) 2. Compressed gas (CAES) 3. Electrochemical energy (BESS) 4. Chemical energy in the feedstock of a fuel cell 5. Kinetic energy (flywheel) 6. Magnetic field energy (inductor) 7. Electric field energy (capacitor) It is further stated, that selecting a particular system for bulk storage of grid energy, leads at least to a nine-dimensional objective function for each system considered. The nine objectives identified in [83] are: 1. Robustness 2. Longevity 3. “Greenness” 4. Simplicity 5. Cycle efficiency 6. Energy density 7. Minimum discharge time (h) 8. Response time (s) 9. Acceptance (proven, mature, etc.) It is suggested that a decision on a suitable ESS technology for a particular application can be made by setting up a rough 7x9 utility matrix. In the following sections, some ESS technologies are presented more in detail.

4.2 State of the art of energy storage technologies

4.2.1

93

Pumped hydro storage – PHS

Pumped hydro storage (PHS) is already widely used in the electricity systems worldwide and it represents a mature and profitable technology for peak shaving. But PHS cannot act in the millisecond range and water turbines are not designed for repeated rapid power fluctuations. Castronuovo presented in [19, 84] the benefits of combined wind-hydro power plants. In this work, an optimization for a combined wind-hydro daily operation strategy was presented. Given the stochastic behaviour of wind generation, the PHS can reduce the uncertainty of power generation in addition to the traditional function which is to store energy during the demand valley and deliver energy during peak hours. This approach was further developed in the following years by Angarita and Usaola in [85,86], where an optimised bidding strategy was proposed (see also section 5.2.3). Another possible application of PHS was discussed in [87] for the example of Alberta (Canada). Here the impact of installing up to 5.6 GW of wind power was evaluated, concluding that a new gas-fired peak power plant of up to 157 MW would be needed to integrate the wind generation. Increasing the existing hydro power capacity to 1.5 GW (+50%) and adding hydro pumping would reduce the required capacity of an additional peak load plant by almost 60%. Interestingly, if two uncorrelated wind profiles are assumed, the additional peak load plant could be eliminated completely. In [88] the implantation of PHS on the Canary Islands was proposed. A case study for Gran Canaria Island is presented and an optimised system with energy output price of 84 e/MWh was calculated, slightly below mean conventional peak hour generation costs. For the Greek island of Crete, in [89] simulations were carried out for a size optimization of PHS conceived to recover rejected energy from wind farms.

4.2.2

Compressed air energy storage – CAES

The CAES cycle is very similar to a standard gas turbine generation cycle [81, 90, 91]. As shown in Fig. 4.2, the CAES is charged with a compressor using off-peak or excess power. Air is pressurised into an appropriate underground cavern (rock caverns, depleted gas fields, saline aquifers or salt caverns). To discharge, air is released and expanded in a gas turbine which in turn drives the electric generator. Before being expanded, air is preheated in a recuperator and heated in the combustion chamber (not shown in the figure). This final step implies that this storage system is consuming natural gas or any other fuel. Indeed, in [90] it was mentioned that the fuel consumption is 30–40% of that for a standard gas turbine. CAES, like PHS, demands favorable sites and geological formations suitable for underground storage. In addition, the fuel cost could be in near future a limiting factor for this technology, as oil prices are likely to rise. As a result of this, as mentioned in [80] some improved CAES systems are proposed or under investigation, including the Small-Scale CAES with air storage in vessels, Advanced Adiabatic CAES (AACAES) with Thermal Energy Storage (TES) and Compressed Air Storage with Humidification (CASH). Greenblatt (2007) investigated in [92] the possibilities of CAES combined with wind power. Three base cases of electricity generation were compared: only gas, wind/gas and wind/CAES. It was shown that today the only-gas scenario is the cheapest (ca. 45 $/MWh) and wind/CAES the most expensive (60 $/MWh). But it was highlighted,

94

Chapter 4. Cost analysis of energy storage systems

that the wind/CAES system is the most robust scenario against price increase of natural gas. An increase of the gas price by 20% increases COE (cost of energy) in the all-gas scenario by 15%, while in case of wind/CAES the increase would be only 3%. This is due to the drastically reduced cost share of gas fuel. It is found that 80% or more of gas consumption can be replaced by wind power using CAES.

∼ Air inlet compressed air

M

G



Motor/Compressor



Turbine/Generator

Air Storage Fuel

Figure 4.2. Compressed air energy storage system – CAES.

As described by Cavallo [93], “CAES is the least cost utility scale bulk storage system available”. Cavallo estimated an installed capital cost of about 890 $/kW (50-h storage in a solution mined salt cavern). In this work it is argued that large energy storage capacity is needed to eliminate the intermittency of wind power and that only the most inexpensive storage media, like air or water, could be used in such an application. Salgi and Lund (2008) examined in [94] the energy-balance effects of adding CAES to the Western Danish energy-system, in order to cope with growing wind power penetration. It was found that even with very large CAES plant capacity it would be impossible to eliminate all excess wind energy. The potential of power regulating was not included in the analysis. This was done in a follow-up work [95]. Here the authors performed an hourly time step simulation to assess the benefits of CAES, including spot markets and regulating power markets. Seven years of historical data and future price estimations were considered. Again it was concluded that CAES is not economically feasible in Denmark. Savings are much higher for other options like heat pumps or electric boilers, as shown by Blarke and Lund (2008) [77] (see section 4.2.7 about thermal storage). In Greece, large amounts of wind energy are rejected by the grid. Zarfirakis and Kaldellis (2009) investigated the option of CAES to recover rejected wind energy [81,96]. Electricity prices are much higher than in Denmark due to the typical island configuration. In the case of Crete, electricity costs of 120 e/MWh were reported, while in Denmark even for 2030 only 54 e/MWh are expected [95]. In [81] it was concluded from a technoeconomic analysis that pumped hydro and CAES are most suitable for Greek islands. In [96] electricity generation costs for CAES of approximately 200 e/MWh were reported. This is still above maximum production costs of gas turbines on the island, but it was concluded that CAES may be an interesting option in the near future.

95

4.2 State of the art of energy storage technologies

4.2.3

Battery energy storage systems – BESS

A good overview of installations of large battery energy systems (BESS) and its applications was given by Parker in [97]. The installed power ranged from 200 kW up to 40 MW and the energy capacities are situated between 170 kWh and 40 MWh. As shown in Fig. 4.3, the components of BESS are basically the power conditioning system (PCS), including the control system, the DC-bus and the battery pack itself. BESS is coupled to the AC system by a bidirectional AC/DC converter. Especially in combination with wind and solar PV it might be interesting to set up a DC grid where all generating and storage units are connected. In this case the BESS would only need a DC/DC converter as charging unit. Batteries are the most versatile storage system. Applications range from small isolated systems up to utility scale installations with typical discharge times of several minutes. The most promising technologies are the“mature”lead-acid, the advanced sodium-sulphur (NaS) and lithium-ion (Li-Ion) batteries. Recently, NiCd batteries were still installed [8], but emerging technologies such as Li-Ion are likely to substitute them. Li-Ion systems are only considered as a viable solution for very small islands with maximum power rate of less than 500 kW [81]. Most important operational parameters of BESS are the depth of discharge (DOD), the temperature of operation, charge control and maintenance. BESS represents an optimal spinning reserve with the advantage of the absence of kinetic parts, reducing operation and maintenance (O&M) cost. Even though, BESS still represents one of the most expensive storage alternatives. The strong dynamics on the private car market pushing electric and hybrid models will have a huge impact on battery prices and the “Vehicle to Grid” concept (see section 4.2.4) may deeply change the idea of energy storage. AC grid



PCS

= ∼

DC bus

Battery Pack

Figure 4.3. Battery energy storage system – BESS.

For comparison of different battery types in [98] the so-called Ragone chart Fig. 4.4 is presented for the five technologies under investigation. With the Ragone chart one can compare easily the different batteries suitable for use in either battery-electric vehicles (which need high energy) and hybrid vehicles (which need high power). According to results from the SUBAT-project [98] the environmental impact of the five eminent battery chemistries for electric vehicles in descending order is: lead-acid, NiCd, NiMH, Li-Ion and NaNiCl (ZEBRA). Similar impact scores are obtained for leadacid, NiCd, and NiMH. Li-Ion and ZEBRA cells have lowest impacts (half of the other three chemistries).

96

Specific power [W/kg]

Chapter 4. Cost analysis of energy storage systems

Specific energy [Wh/kg] Figure 4.4. Ragone chart for traction batteries, Source: Bossche et al. (2006).

Lead-Acid battery Lead-acid batteries can be considered as a mature technology with known performance characteristics. Some examples of utility-scale installations between 1980 and 2000 are exposed in [97]. One interesting installation from 1996 was documented in [99, 100]. Here a 5 MVA/3.5 MWh valve-regulated lead-acid battery system (VRLA) was installed near to a lead recycling facility. In this case the ESS provided multiple services such as peak shaving, power quality and uninterrupted power supply for critical loads up to 1 h. Monthly savings of US$4000 in electricity were reported. In [101] it was emphasised that there have been considerable improvements in product and manufacturing technology. Batteries have become cheaper, more reliable and maintenance needs have been reduced to a minimum. In the EU project INVESTIRE [102] it was estimated that costs (investment and O&M) may still be reduced by 10–30%. Important improvements in efficiency were not expected. A more recent study from the same project [103] compared several battery systems and concluded that owing to its low cost, lead-acid is expected to have a large share of the future deployment of large grid connected storage systems. This conclusion could not be confirmed in this thesis (see results of COE in section 4.3.4). The low self-discharge and the low maintenance requirements are some of the main advantages, while some of the main drawbacks are the limited service period (5–10 years), the environmentally unfriendly content and the recommended low depth of discharge.

4.2 State of the art of energy storage technologies

97

Sodium nickel chloride battery – NaNiCl (ZEBRA) The sodium nickel chloride battery, better known as the ZEBRA battery, is a high temperature system (300◦ C). It was invented in 1985 by the Zeolite Battery Research Africa Project (ZEBRA) in South Africa. It uses nickel chloride as its positive electrode and operates across a broad ambient temperature range (−40 ... + 70◦ C) without cooling. As described in [80], this battery type, is only produced by one company (Beta R&D). Different versions for hybrid electric vehicles, storing renewable energy and a load-levelling battery for industrial applications are being developed at present. The disadvantage of this battery chemistry is its low energy density (120 Wh/kg) and power density (150 W/kg). Although these figures represent a considerable improvement over the lead acid battery, other technologies such as NiMH, NaS or Li-Ion are preferred in transport applications, due to their higher performance. Sodium-Sulfur battery – NaS The sodium-sulfur (NaS) battery has liquid electrodes (positive: sulfur, negative: sodium) and a solid electrolyte (beta alumina of sodium ion conductive ceramic which prevents any self-discharge). The battery is hermetically sealed and its working temperature is at approximately 300◦ C. Very recently, several NaS batteries have been installed in the USA, with modules from the japanese manufacturer NGK Insulators Ltd. [104]. One module has 50 kW nominal power with an energy capacity of 360 kWh. The module occupies 3 cubic metres of space and weighs approximately 3.5 metric tons. This results in a volumetric energy density of 120 kWh/m3 and a gravimetric density of 100 kWh/t. While in [81] efficiencies of up to 85% were reported, NGK gives in its specifications a system efficiency of 75% [104]. An EPRI report from 2006 [74] concluded that NaS batteries have the best economics among the battery technologies for MW size utility applications. In [105] results from a 1.2 MW system were presented while in [106] a 100 kW test system was described. The reported total system cost is consistent with estimations in [105]. For more details of technical specifications of the system described in [106] see appendix C.2. Together with the relatively long calendar life of 15 years (2500 cycles) this system represents a very interesting option. In publications [105, 106] economical viability for peak shaving (PS) and power quality (PQ) services was demonstrated. Lithium-Ion battery – Li-Ion In [8] it was stressed the huge impact of the development of electric vehicles (EV) on stationary battery systems. In fact, the duty cycle of an EV is very similar to the highpower cycling in a wind energy system. The latest announcements of car manufacturers are showing that the Li-Ion battery is already the best choice for electric and hybrid vehicles (see section 4.2.4 for more details). The main advantages of Li-Ion technology are the high efficiency (≈ 95%), the ability for the provision of very high currents and deep discharges [78] and almost zero maintenance requirements. The limitations are the high cost, lifetime (no long-term studies exist) and the required protection circuits to maintain voltage and current within safety limits (temperature issues at the membrane).

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Chapter 4. Cost analysis of energy storage systems

Li-Ion polymer batteries, which are already commercialised in mobile phones and laptops, promise cost reductions, but no major breakthrough has been achieved with this system [107]. The polymer battery is a battery with a polymer electrode and/or electrolyte. The main advantage of this battery type is the elimination of liquid electrolytes which increases the safety and reduces cell thickness and weight [108]. New materials (e.g. polymer or ceramic) for solid electrolyte systems may eliminate the safety problems, mentioned above [109]. World leader in high energy and high power Li-Ion batteries is the french manufacturer SAFT S.A. [110]. Metal-Air battery The metal-Air battery can be regarded as a special type of fuel cell using metal as fuel and air as oxidant. They are the most compact and, potentially, the least expensive batteries. Theoretical energy densities exceed those of Li-Ion batteries by a factor of 5– 10 [80]. The main disadvantage is that electrical recharging is very difficult and inefficient. Rechargeable metal-air batteries that are under development have a life of only a few hundred cycles and the efficiency is below 50%.

4.2.4

Vehicle to Grid – V2G

The idea of V2G is not new, but recently growing attention has been paid to it. The concept of connecting the transport sector with the electrical power system was first proposed 1997 by Kempton and Lentendre [111]. The name “vehicle-to-grid” or V2G was created in the research group headed by Willet Kempton at the University of Delaware (USA) and for the first time published in 2002 [112]. He has published since late nineties a number of papers and may be by the moment the most active author in this field. Already in the year 2000 a study for Japan was published [113], where several electric drive vehicles (EDV or EV) were analysed. In [114, 115] V2G is defined and explained in detail. Example calculations with the “Th!nk City vehicle” and the Toyota RAV4 were presented in [116]. On one side capacity calculation and net revenue was presented and on the other side possible services to the grid were evaluated. It was shown that V2G could provide important spinning reserve due to the very quick response of battery systems. In fact it was concluded that V2G might accelerate the implementation of cleaner vehicles and at the same time provide inexpensive backup power for renewable energies. In [114] for example it was shown that the mechanical power of the U.S. light vehicle fleet (passenger cars, vans, and light trucks) exceeds the electric power generation of the country by a factor of 24. On average, 96% of this fleet is parked and so, potentially available for V2G. The positive interaction of EV and renewable energy generation is summarised in three main items: • Electricity will displace gradually liquid fuel as an energy carrier • The vehicle fleet will provide storage and spinning reserve to the electric grid • Automated controls will optimize power transfers between these two systems The development of plug-in hybrid and electric vehicles (PHEV/EV) is opening new options for distributed energy storage systems (DESS). A large number of EVs, connected

4.2 State of the art of energy storage technologies

99

to the power system would have the same potential benefits as stationary DESS, providing power quality (PQ) and peak shaving (PS) services. The benefits of the DESS concept was analysed by Ali Nourai in [105]. It was concluded that most likely the future of battery storage is distributed, rather than centralised. Fischer et al. (2009) [117] proposed an interesting measure of effective energy density to compare batteries with liquid fuels for passenger cars. Instead of energy content, effective mechanical work was considered. In addition to the mass of fuel or battery, the entire drive train was included. As a result, gravimetric energy densities for EVs are higher for short distances, because of the lower mass of electric motors and drive trains. A cross-over range was calculated, which determines the range where battery electric vehicles (BEV) have a higher energy density than those with internal combustion engines (ICEV). Cross-over ranges from 115 − −190 km were reported, which would satisfy more than 80% of U.S. american automobile travel needs. The same authors showed how batteries can be more economic than gasoline [118]. It was concluded that EVs with up to 150-km range are already cost competitive. Andersen et al. (2009) presented in [119] a strategy for the creation of intelligent EV charging networks. It was suggested to create the business model of a so called Electric Recharge Grid Operator (ERGO), which is meant to accelerate the introduction of EVs in private transport. The model brings together consumption with renewable energy generation. Israel, Denmark, Australia and some regions of the USA have signed on to this model and are working to introduce it in 2009–2011. Germany: One million EV/PHEV by 2020 and 40 million by 2035 In November 2008 the German government announced a strategic plan to achieve one million electric vehicles circulating in Germany by 2020. This goal is based on a study of the German “Bundesverband Solare Mobilit¨at” BSM and the “Deutsche Gesellschaft f¨ ur Sonnenenergie” DGS [120] which showed the viability of this scenario. It is estimated that 1 million vehicles would represent only 0.3% of the total electricity consumption by 2020. A summary of this study is available online in german language [121]. It is highlighted that 40 million electric and hybrid electric vehicles would increase the german electricity demand only by 10%, which means 60 TWh. At the same time the oil consumption in the transport sector would be reduced by 50%. It is estimated that a complete switch to EV/PHEV would take 25 years. In Fig. 4.5 a possible roadmap for Germany is shown, according to [121]. Note that a 13% share of luxury vehicles is supposed not to be replaced by electric technology. This is a simplification in order to give a conservative estimate. Another simplification is that only small cars are replaced by all-electric vehicles (EV), as they are supposed to be used only for short distances within cities. Market shares of small, medium and luxury cars is assumed to be unchanged during the next 25 years. Each vehicle carries a considerable storage capacity of approximately 20–50 kWh with a nominal power between 50–100 kW. For 1 million vehicles, these figures represent GWh/GW. For Germany (assuming 82 million inhabitants and a motorization rate of 55%), one million EVs means that little less than 2.5% of all automobiles would be electric. Nevertheless, if 50 kW is assumed for one EV, the installed power of one million cars would be 50 GW, which is equivalent to 40% of the installed power in Germany, which was in 2006 125 GW (according to Eurostat [122]). In other words, with 2.5% of

100

Chapter 4. Cost analysis of energy storage systems

Market share [%]

100 80

EV-small CV-small PHEV90-medium PHEV30-medium CV-medium CV-luxury

60 40 20 0 2010

2015

2020

2025

2030

2035

Time Figure 4.5. Roadmap of German passenger car market until 2035, EV: electric vehicles, CV: conventional vehicles, PHEV30/90: 30/90 km plug-ins, Source: Engel (2007).

the vehicle fleet being electric, a regulating power of 20% of total installed power would be available, if 50% of these vehicles were offering balancing services. It becomes evident, that a large fleet of EVs would be a serious option of distributed energy storage, but also a challenge for its network integration. If battery charging is coordinated with the power system, grid-connected EVs can play an important role in balancing and power quality. Simulation of energy systems including V2G In [123] two national energy systems were modelled (including the V2G concept), one for Denmark, with combined heat and power (CHP) and the other, similarly sized without CHP (the latter being more typical for other industrialised countries). In the case of Denmark, even a combination of heat/cold storage and V2G was considered. Results showed that excess energy from wind generation can be reduced significantly with V2G. Another work considering V2G for the Danish case was published in [63]. Here the advantages of battery storage are highlighted and V2G was given as an example. In [124] a combination of stationary hydrogen generation and fuel cell vehicles is proposed, being batteries and fuel cells complementary elements rather than competitors. A global analysis of the potential of V2G over the long-term (until 2100) was described in [125], using the energy-system model ERIS. As described in [126], ERIS is conceived to simulate the global energy system and identify the effects of different climate change mitigation policies. As shown in Fig. 4.6, within a climate policy scenario it was estimated that 80% of the fleet of internal combustion engine vehicles (ICEV) will be replaced by 2050, being considered only hybrid (HEV) and fuel cell vehicles (FCV). (Note the difference with the German roadmap!) Battery electric vehicles (BEV) are judged to be too costly, based on results presented in [127]. A very important conclusion in [125] is that V2G favours more efficient technologies, i.e. FCV in this case. In Fig. 4.6 two climate policy scenarios are shown: ‘ClimateN’ stands for a policy without considering V2G and ‘ClimateV’ includes V2G. It can be seen that V2G accelerates the implementation of fuel

4.2 State of the art of energy storage technologies

101

cell vehicles, being the most efficient technology considered in this model. But why BEV cannot compete even in a V2G scenario? This is because energy losses of hydrogen production were neglected, fuel cell efficiency was very optimistic (> 70%) and fossil fuel prices were assumed to be extremely low (< 0.50 $/L). It has not been taken into account that mineral oil extraction possibly already reached its maximum in 2008 [128]. Therefore, despite the high value of this study, it is unlikely that the transition will last until the year 2100 and will take place without BEV.

Figure 4.6. Impact of availability of V2G technologies on global car technology market share, climate policy scenario; ICEV – internal combustion engine vehicle, HEV – hybrid-electric vehicle, FCV – fuel cell vehicle. (Source: Turton and Moura 2008).

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Chapter 4. Cost analysis of energy storage systems

4.2.5

Redox Flow Battery – VRB and others

In flow batteries – also known as redox flow cells – the energy is stored and released by means of a reversible electro-chemical reaction, similar to any rechargeable battery. But contrary to solid state batteries, the energy and power ratings are independent from one another. The storage capacity exclusively depends on the quantity of the electrolyte stored, while the power rating is determined by the active area of the cell stack. The need of circulating the electrolytes with pumps bring some parasitic losses (2–3% [129]) but at the same time it represents an excellent cooling mechanism for the system. A principle scheme of a flow battery system is depicted in Fig. 4.7. The charge and discharge takes place at an ion-exchange membrane, similar to a fuel cell 3 . The two electrolytes are pumped from the tanks through the cell stack. As in a fuel cell, at the membrane one electrolyte is oxidised and the other is reduced. Most common redox couples which can be found in literature are: • Only Vanadium (VRB) [129] • Zinc/Bromine (ZBB) [130, 131]

• Na Polysulphide/Na Bromide (Regenesysr ) The Vanadium Redox Battery (VRB) exploits the ability of vanadium to exist in solution in four different oxidation states. As only one electroactive element is used, the problem of cross-contamination is eliminated and the solutions have an indefinite life [129]. The technical feasibility of very large electrolytic tanks brings this technology in a scale where only PHS and CAES can compete. Special attention has to be paid to the containment of aggressive chemical solutions4 in the storage tanks. Electrolyte

AC grid



PCS

= ∼

DC bus

Pump

Cell Stack

Membrane

Pump Electrolyte

Figure 4.7. Redox flow battery system – RFB.

3 4

Sometimes flow batteries are termed as reversible or regenerative fuel cells. The vanadium redox system contains sulfuric acid at ca. 20 wt%.

Positive Electrolyte

Storage Tanks Negative Electrolyte

4.2 State of the art of energy storage technologies

103

A short history of the Vanadium Redox Battery As the VRB technology is the most extended and approved flow battery technology, here a short history is given. Based on the work of the group headed by Maria Skyllas-Kazacos [129, 132], in 1989 the University of New South Wales (Australia) was able to give commercial licenses on the vanadium redox concept to several companies. In 1998 the Australian Pinnacle VRB bought the basic patents and licensed them to Sumitomo Electric Industries Inc. (Japan) which until 2001 built several pilot plants in Japan [133]. In 2001 Pinnacle VRB became a subsidiary of the canadian VRB Power Systems Inc. and pilot plants were installed on King Island (Australia) and in Castle Valley (Utah, USA) [65, 134]. Independently, in 2000 a small company was founded in Austria. First under the name “Energy on Demand Production and Sales GmbH”, now it can be found under “cellstrom GmbH”. This company offers small scale stand-alone systems based on VRB technology [135]. In 2008 it started commercial series production of a 15 kW/100 kWh module. Today it is the only company with independent patents on VRB technology besides JD Holdings Inc. In the year 2000 there was a great optimism about the application of redox flow batteries. As a consequence Regenesysr Technologies Ltd. was founded (with the german RWE Group as parent company represented by its subsidiary Innogy). It had very ambitious plans of building large scale plants in England and later in the USA [136]. The first plant was designed with a storage capacity of 120 MWh and 15 MW (to be installed in Little Barford, UK). But at the end of 2003 the RWE group decided to stop funding this project. Apparently it was not ready for such large scale applications. Although Regenesysr is based on the Na polysulphide/Na bromide chemistry, it is mentioned here because it was the largest project with flow batteries and is still cited in many publications. In addition, VRB Power Systems Inc. purchased in late 2004 an exclusive global license for the Regenesysr electricity storage technology and controlled until late 2008 over 20 patents. In late November 2008 VRB Power Systems Inc. declared insolvency and was sold in January 2009 to JD Holdings Inc., the parent company of Prudent Energy, based in Beijing, China. In 2005 Sumitomo Electric Industries completed on Hokkaido Island (Japan) the installation of a 4 MW/6 MWh demonstration plant with VRB technology coupled to a wind farm. After some years of relative silence – from 2002 until 2007 only three VRB ESS plants were installed – Sustainable Energy of Ireland (SEI5 ) published in 2007 a feasibility study [137] proposing a 2 MW/6 h (12 MWh) VRB ESS in the proximity of 32 MW Sorne Hill wind farm (Donegal, Ireland). According to publications in the daily press, the contract had a volume of 9.4 million US$. It may be mentioned that this system permits a pulse power of 3 MW for 10-min periods every hour which is very useful in order to deal with short term volatility in wind generation. The study in [137] calculates internal rates of return for this project of up to 17.5%. This shows the great potential value of energy storage for wind generation, although it is unclear if the Flow Battery will now be installed, because VRB Power Systems Inc. had been contracted to install the system. 5

Not to be confused with Sumitomo Electric Industries.

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Chapter 4. Cost analysis of energy storage systems

4.2.6

Hydrogen storage and fuel cell – HSFC

Hydrogen storage, similar to CAES counts with three separate units: the electrolyzer for charging, the storage tank and the fuel cell for discharging (see Fig. 4.8). As a consequence, the storage capacity only depends on the volume of the hydrogen tank, while the maximum power is defined by the electrode surface. Here the proximity with the flow battery becomes clear. So it can be stated that hydrogen storage shares the advantage of separated sizing of power and energy ratings with PHS, CAES and flow batteries. When the storage is charged, the electrolyzer generates hydrogen. Due to the low density of hydrogen it is usually compressed and stored in pressurised storage tanks. For large-scale stationary storage systems, this would be the best option [138]. When high pressure electrolyzers are employed, compression losses can be reduced. The two main alternatives, cryogenic storage (liquid hydrogen at very low temperatures) and the absorbtion in solids (metal hydrides, carbon materials) are possible but until today they are too costly and not suitable for large-scale applications. PCS

= AC grid



∼ = ∼

DC bus Electrolyzer

FuelCell

Hydrogen Compressor

Hydrogen

Hydrogen Tank

Figure 4.8. Hydrogen storage system with electrolyzer and fuel cell – HSFC.

Finally, the fuel cell (FC) permits electrochemical combustion of hydrogen at relatively low temperatures, depending on the FC type. In principle, a fuel cell operates like a battery. However, electro-active materials (i.e. hydrogen and oxygen) are continuously supplied, so that a fuel cell never discharges. There are several types of fuel cells commonly in use. The selection of the type depends on the application [107]. For large stationary applications where high working temperatures are not a problem, the most suitable types are the Molten Carbonate (MCFC) and the Solid Oxide fuel cell (SOFC), having the highest efficiencies. On the other hand, for mobile applications and small isolated systems with renewable energy generation, the Proton Exchange Membrane FC (PEMFC) is the best choice, due to its compact design and low working temperatures. An additional advantage of hydrogen storage is the possibility of transporting the hydrogen gas in pipelines or tanks to the centres of consumption, where hydrogen not necessarily has to be converted into electricity. Indeed, today hydrogen is almost exclusively employed in refineries and the chemical and food industry.

4.2 State of the art of energy storage technologies

105

As stated in [81], the main disadvantage of this technology is the overall efficiency, when it is used for electrical storage, where hydrogen is converted completely into electricity. Round trip efficiencies of around 30–40% are not likely to be improved notably in the near future. Losses occur during electrolysis, storage (compressor, diffusion) and finally in the fuel cell. As will be shown in section 4.3, the relatively low energyrelated storage costs make the hydrogen option competitive for very large ESS applications where no CAES or PHS is viable.

4.2.7

Thermal Energy Storage – TES

A good overview of thermal storage systems is given in [80]. Systems are divided basically in low and high temperature TES. Low temperature TES is cold storage. Two types are mentioned: storage in ice and cryogenic energy storage (CES). CES is a new approach which is based on liquefying air. Electricity is recovered with a cryogenic heat engine, but round-trip efficiencies are still low (40–50%). High temperature storage is more usual. Especially in concentrating solar power stations molten salt storage is used. A recent very promising development is concrete storage. First commercial devices are estimated to be available before 2015. The last TES mentioned in [80] are Phase Change Materials (PCM). Latent heat of phase changing permits storage of heat without temperature increase. This way, energy can be stored long time without self discharge. Stadler (2008) [139] examined the potential of thermal storage to help increasing renewable energy penetration levels beyond 25%. In this work, the case of Germany was analysed modelling thermal storage devices and with laboratory tests. In fact, heat storage is done in Germany since the 1960s, when utilities began to distribute night storage room heating facilities. Another potential thermal storage is represented by ventilation systems. A positive control power (discharge) is obtained by reducing ventilation power and negative control power (charge) is available when ventilation power is increased. Several GW of control power are thus potentially available in Germany. A third storage system was identified in refrigeration. Again a potential several GW of control power is obtained. Another suggestion was the automatization of electric household devices which may be remotely controlled depending on system needs. It was concluded that control power between 10–25 GW can be obtained in Germany by load side management on the one hand and combination of CHP systems with thermal energy storage on the other. It was shown that the majority of balancing can be obtained with load side management, but additional capacity of heat and cold storage devices is required. Heat and Cold store – Heat pump Lund (2005) described in [140] and later in [77], coupling combined heat and power (CHP) stations with electric boilers and heat pumps as an interesting solution to provide thermal storage. The heat pump in addition opens the possibility of cold storage. The paper discussed the principle of “storage and relocation”, where relocation refers to the transformation of electrical energy into heat (boiler, heat pump) or cooling (heat pump). Thus, CHP becomes more flexible and is able to deliver valuable balancing services to the market. All this is possible maintaining and even increasing efficiency. In [77] an improvement of fuel efficiency from 92% to 97% was reported.

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Chapter 4. Cost analysis of energy storage systems

4.3

Cost structure of ESS technologies

The central objective of this section is to study the effect of reduced ESS energy and power ratings on the cost structure. To achieve this, a novel method is applied to estimate the number of equivalent full charge-discharge cycles. In the following paragraphs, first a short introduction is given to the state of the art of ESS cost calculation and the basic cost structure is defined. Then the cost components are explained in detail and tables of the assumed parameters are provided. Results are illustrated with a reference system of 1 MW nominal power and 24 MWh effective energy capacity. Fundamental work has been done by Ter-Gazarian (1994) [62] on storage in power systems and related cost calculations. In the last decade especially the Sandia National Laboratories (SNL) [141] published extensive work on ESS cost calculation [61,70–73,106] although maintaining the basic principles described in [62]. Schoenung and Hassenzahl recommended in [71] a life-cycle cost analysis including efficiency, replacement frequency, and operational factors. In [61] Kaldellis and Zafirakis proposed a detailed ESS cost model which has been repeatedly used for studies on the Greek power system [81, 142, 143]. Another detailed work on ESS costs has been performed in the framework of the EU project INVESTIRE (“Investigation on Storage Technologies for Intermittent Renewable Energies”) [102]. The INVESTIRE project detected a critical lack of reliable technical and economical data for emerging storage technologies. An economic performance index was developed for evaluating the adequacy of each technology for different types of applications. Valuable data was given about future expectations in terms of cost and efficiency for several storage technologies. The drawback of this approach is that ESS initial costs are not divided into power and energy related costs. Balance of plant (BoP) and power conditioning system (PCS) costs were not mentioned explicitly. In this thesis, the basic approach of cost calculation is adopted from [62]. The aspects of electricity price escalation and replacement of major components of the storage system have been added as proposed in [61]. As a difference to [61], in the present study no public subsidies are assumed and the energy input cost to the ESS is set to zero. It may be mentioned here, that cost models in [61,71] do not represent energy loss costs properly, as they are considered in [62]. Thus, as a synthesis of the existing approaches, the annual cost structure of a generic ESS is described in (4.1) as a sum of three basic elements: initial cost IC (capital cost), cost of energy losses Closs and other variable costs V C.

where IC Closs VC CRF i y

CESS = (IC + Closs + V C) CRF

(4.1)

i (1 + i)y CRF = (1 + i)y − 1

(4.2)

Initial or capital cost [e] Energy loss cost related to the ESS efficiency [e] Other variable costs [e] Capital recovery factor Annual interest rate of the local market [p.u.] Years of operation for the proposed configuration

4.3 Cost structure of ESS technologies

107

As proposed in [62], initial cost IC is obtained from nominal energy and power ratings of the ESS and associated cost coefficients (see section 4.3.1). Variable costs are expressed in two separate components as energy loss cost Closs and other variable costs V C. This separation is convenient due to the special character of Closs . As energy loss costs are related to the efficiency of the ESS, they depend on the operation of the ESS. Therefore, a quantification is only possible if the number of full charge-discharge cycles nc is known. The contribution of this thesis is a probabilistic approach for calculating nc as a basis to obtain Closs (see section 4.3.2). The introduction of the capital recovery factor CRF for annualised costs is inspired by cost calculations in [144] and in NREL’s simulation program HOMER [145, 146]. In order to understand the concept of CRF consider the following example: For i = 5% and y = 20 years, the capital recovery factor is equal to 0.08. A loan of 1000 e at 5% interest could therefore be paid back with 20 annual payments of 80 e. The present value of the 20 annual payments of 80 e is 1000 e. In the following paragraphs, each element of the cost structure is explained. The cost calculations will be illustrated with results from a reference case. Wind generation data from dataset A was used to obtain the typical curves of unserved energy (see chapter 3 for details). As forecast, the persistence scenario T×1 was chosen (see chapter 2 for more details). The nominal power of the reference wind energy generation system is set to 1 MW. In chapter 3 it is shown that an ESS which is able to compensate 100% of all forecast errors of the T×1 scenario would have a nominal power of PESS = 1 MW and an energy capacity of EESS = 24 MWh (24 h rated discharge time). Surface plots showing all possible combinations of ESS configurations with reduced power and energy ratings will be presented next. This way, the effect of reduced ESS ratings on the cost structure can be observed.

4.3.1

ESS initial cost (Capital cost)

In [71] it is stated that in most ESS the initial cost (or capital cost) is the sum of the cost of a power conversion system (PCS), the actual storage and the balance of plant (BoP). Balance of plant includes the auxiliary components outside of the storage subsystem. It is concluded that for many (but not all) systems the cost of the PCS is proportional to the rated power of the ESS, while costs of storage and BoP are proportional to the energy capacity. Only in some cases BoP are fixed costs or proportional to power rating. A selection of values published in the latest report (2008) from Sandia National Laboratories (SNL) [73] are shown in appendix C.1. For simplification, in this thesis BoP costs are not considered separately. They are assumed to be included in the energy-related cost. With this assumption, the initial cost of an ESS can be modelled as the sum of only two components, as described in [62]: one related to energy capacity EESS and the other to nominal power PESS . Hence, the initial cost IC can be calculated as follows. IC = ce EESS + cp PESS

(4.3)

where ce [e/kWh] and cp [e/kW] are the cost coefficients related to nominal energy capacity EESS [kWh] and the rated power PESS [kW].

108

Chapter 4. Cost analysis of energy storage systems

Battery energy storage systems (BESS) such as lead-acid or NaS are typically high power applications (discharge times of minutes). Indeed, existing installations rarely exceed d0 = 8 h (see Fig. 4.1, [78]). In high energy applications, large discharge times (d0 > 2 h) are achieved by limiting the ESS power with the power conditioning system (PCS). As a consequence, BESS are operated far below its technical power limit. The advantage is that lifetime is increased but costs are high due to the actual oversizing in terms of battery power. This way, for high energy applications, a virtual separation of energy and power rating is given. The energy is defined by the battery capacity and power by the PCS. Thus related costs are separated as well: the energy cost coefficient represents the battery itself and the power cost coefficient belongs to the PCS. If these coefficients are well defined, the “unused” power of a battery system with large discharge time (e.g. 24 h) will be reflected in elevated system costs. Discharge times below 1 h are not considered here, so that limitations in C-ratings are not relevant. Definition of energy and power coefficients Coefficients considered in this thesis are shown in Table 4.2. Three scenarios are defined: ‘base’, ‘best’ and ‘worst’. Values for the base scenario represent intermediate prices found in the literature. Lowest reported prices are used in the best case and highest in the worst case scenario. Note the relatively small range of values for lead-acid. The great consensus indicates the maturity of this technology and the relative independence on the site of installation. Although being a mature technology, prices of pumped hydro show a large range. Especially energy related costs (the water reservoir) depend strongly on the system size and specific conditions of the installation site. Emerging technologies such as NaS, redox flow batteries (RFB) and hydrogen storage show large ranges as only a small number of demonstration plants exist. Highest prices are related to realistic costs of singular existing projects. Most optimistic prices (especially for hydrogen storage) are often future projections rather than current market prices. ce [e/kWh]

Lead-Acid NaS Redox flow (RFB) Pumped hydro Hydrogen storage

cp [e/kW]

base

best

worst

base

best

worst

240 300 600 40 20

210 200 100 5 2

270 500 1000 80 30

170 200 1000 1000 4000

140 125 600 500 1000

200 300 1500 2000 10 000

Table 4.2. ESS capital cost coefficients related to energy and power assumed in this thesis.

Note the extremely low energy cost coefficient for hydrogen storage. Storage costs for hydrogen are often given per kg of hydrogen, rather than per kWh. A realistic value for

109

4.3 Cost structure of ESS technologies

compressed hydrogen is 400 e/kg (only for the storage tank) [147]. Assuming a specific energy of 40 kWh/kg [148], and a discharge efficiency of 50%, a storage cost of 20 e/kWh is obtained. The relatively low cost of hydrogen storage compared to PHS is due to the higher energy density. Compressed hydrogen may contain 160 kWh/m 3 (pressure: 5 MPa, temperature: 25◦ C, density: 4 kg/m3 [148]). For comparison, assuming a pumping height of 360 m, the potential energy content of water is only 1 kWh/m 3 . The cost advantage is even larger if a hydrogen tank is compared to BESS. While in case of BESS cost relevant materials are needed in the storage volume, in case of vessels, material is only needed at the surface. Thus, the high power-related cost is compensated if the required energy capacity is sufficiently large. As a consequence, hydrogen storage is especially indicated for high energy applications. Definition of nominal ESS power and energy Nominal power and energy values PESS and EESS can be defined as in (4.4 – 4.6). The effective energy capacity Eeff and power ratio ζ are obtained from the statistical models presented in chapter 3. The typical discharge time d0 is calculated dividing Eeff by PESS . PESS = ζPinst

(4.4)

Eeff DOD

(4.5)

EESS =

Eeff = d0 PESS

(4.6)

where Pinst [kW] is the nominal installed power of the wind farm, ζ [p.u.] is the power ratio covered by the ESS, Eeff [kWh] the gross energy capacity of the ESS, DOD [p.u.] the maximum depth of discharge (depending on technology) and d0 [h] the typical discharge time (or energy autonomy) of the ESS. It is assumed in this thesis, that energy related cost coefficient ce reflects the cost of nominal dischargeable energy, because this value should be always given as nominal ESS energy capacity. Not included in this cost is the operational limit for depth of discharge, so that ESS size is increased according to the maximum permitted depth of discharge. In [80] for example it was suggested to divide energy-specific prices by the ESS efficiency and in [61, 70] it was proposed to divide ESS capacity by the efficiency. In this thesis, none of these approaches are regarded as very fortunate. In the first case it is assumed that cost coefficients in general do not account for ESS efficiency which is not likely. In the second case, the physical size of the ESS is increased based on the roundtrip efficiency. Here, even assuming that energy capacity is given without taking into account the efficiency, only the discharge-related efficiency would apply. Consider the example of hydrogen storage, where storage costs are typically given per kg of hydrogen, often without including discharge losses. Given a desired dischargeable energy Eeff , storage costs should be divided by the efficiency of the fuel-cell for example, i.e. multiplied approximately by a factor two. This problem is intentionally left outside of the model, in order to avoid confusions. It is considered that this aspect is not essential for the model and should be treated carefully in every specific application, for example to calculate the volume of a hydrogen tank.

110

Chapter 4. Cost analysis of energy storage systems

Results for initial costs of the reference system are presented in Fig. 4.9. Note the high cost of flow batteries. Results shown here are based on a cost of 600 e/kWh, while most optimistic estimations are assuming a potential reduction to only 100 e/kWh. Lowest initial costs are observed for hydrogen and pumped hydro storage. Initial costs of NaS batteries (not shown in the figure) are only slightly higher than lead-acid (see appendix C.2). A - Tx1 (24h forecast) - Lead-Acid (base)

A - Tx1 (24h forecast) - FlowBattery (base) 30

30 30

20

20 10

10

0 1 12 0 0

20

20 10

10

0 1

24

0.5

PESS [MW]

IC [Me]

IC [Me]

30

24

0.5

0

12

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (base)

0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (base) 30

30 6

20

4 2

10

0 1

24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

0

IC [Me]

IC [Me]

6

20

4 2

10

0 1

24

0.5

PESS [MW]

12 0 0

0

Eeff [MWh]

Figure 4.9. Initial cost for the reference system of 1 MW and four ESS technologies.

111

4.3 Cost structure of ESS technologies

4.3.2

Cost related to ESS efficiency

According to Ter-Gazarian [62], the cost related to ESS efficiency can be expressed as a function of the number of equivalent full cycles nc . The novelty in this thesis is the approach to obtain nc . Contrary to the approach in [62], in [61, 70] it was proposed to include ESS efficiency in the capital cost, increasing ESS energy capacity, but this approach does not represent properly the nature of this cost component. It is related to the volume of energy which is cycled through the ESS and not to its nominal energy capacity. Therefore, here the approach from [62] is preferred. It will be assumed that the cost related to the efficiency is proportional to the mean market price π and the energy loss Eloss , so that energy loss costs Closs are given by Closs = Eloss · π

j y  X 1 + gπ j=1

1+i

(4.7)

where Eloss [kWh] is the energy loss due to ESS efficiency, π [e/kWh] the mean market price of electricity, gπ the electricity price escalation rate, y years of system life and i the capital cost of the local market. As a consequence, Closs is a cost which is variable and depending on the market price. It is very important to keep in mind this fact, especially when future projections of the feasibility of ESS implementation are made. While the mean market price can be estimated straightforward, the estimation of the expected energy loss Eloss is more difficult. To quantify Eloss , the energy cycled through the ESS is needed. For our case (forecast error reduction), the energy throughput ratio ETR as defined in section 3.3.1 is very useful. The energy throughput of an ESS depends on its operational schedule. Given the same nominal power and energy values, peak shaving will result in different throughput values than power quality operation. In this thesis the operational schedule is defined by the compensation of wind power forecast errors of a given wind farm, as described in chapter 3. Note that the proposed procedure can be applied to any other schedule. In order to maintain the analogy with the parameters used in [61], in (4.8) the energy ratio δ covered directly by the ESS is introduced. δ =1−

Edir ETR0 = Etotal 2

(4.8)

where Edir [kWh] is the energy directly injected into the grid, Etotal [kWh] the total energy production of the wind farm and ETR0 [p.u.] the idealised Energy Throughput Ratio (100% ESS efficiency). With the definitions of ETR0 and δ, the number of equivalent full cycles is given by nc =

Etotal ETR0 Etotal · δ = 2 Eeff Eeff

(4.9)

112

Chapter 4. Cost analysis of energy storage systems

In this expression the relationship between δ, ETR0 and nc can be observed very well. According to [62], Eloss can be calculated with   1−η Eloss = Etotal · δ (4.10a) η   1−η Eloss = Etotal · ETR0 (4.10b) 2η where Etotal [kWh] is the total generated wind energy, ETR0 is the ideal energy throughput ratio, η [p.u.] the ESS round-trip efficiency and δ [p.u.] is the ESS energy ratio. Because of the great importance of this equation, its derivation will be presented in the next pages with the energy throughput ratio as starting point. In the statistical model presented in chapter 3, ideal throughput ratios ETR0 are obtained assuming 100% ESS efficiency. The advantage of this approach is that the statistical model is easier to develop and results are independent of the ESS technology.

ESS ein = δ

Round-trip efficiency: η = 1

eout = δ

ETR0 = |ein | + |eout | = 2δ

ESS ein =

δ η

Round-trip efficiency: η < 1   δ 1 ETR = + δ = δ +1 η η

eout = δ

Figure 4.10. ESS energy balance without losses (above) and with losses (below).

The introduction of ESS efficiency below 100% is illustrated in Fig. 4.10. The values of input and output energy ein and eout are expressed in [p.u.], normalised by total generated energy Etotal . In the ideal case (upper part), ein and eout are equal and the throughput ratio is 2δ. In a real world situation (lower part, η < 1), in order to be able to supply the same quantity of energy, 1/η more energy has to be charged to the system. Here it can be seen clearly that it is not the ESS capacity which has to be increased by the factor 1/η, but the energy input. As illustrated in Fig. 4.10, real world ETR is given by η+1 η+1 = ETR0 (4.11) η 2η The same result is obtained if the normalised energy loss eloss is defined first. It can be easily expressed as the difference between input and output energy.   δ 1 eloss = ein − eout = − δ ⇒ eloss = δ −1 (4.12) η η ETR = δ

113

4.3 Cost structure of ESS technologies

As can be seen, (4.12) is equivalent to (4.10a), and thus, the origin of (4.10a) has been demonstrated. It may be mentioned, that here a slightly different approach was proposed. In [62], round-trip efficiency η is divided in three components: ηc for charge, ηs (t) for storage (self discharge) and ηd for discharge. As reference storage capacity ein /ηc is considered, instead of eout . Here, it is assumed in a first approach that only η is known. If more information is available, the model can easily be refined as proposed in [62]. Now ETR can be described as ETR0 plus the additional energy input which is needed to compensate the energy loss eloss and the same result as in (4.11) is obtained.   1 η+1 −1 =δ (4.13) ETR = ETR0 + eloss = 2δ + δ η η It has to be highlighted that the energy loss does not grow linearly with decreasing efficiency. This is due to the definition of maintaining the output energy of the ESS as if 100% efficiency would be present. If the input energy was maintained, the relationship would be linear, but with the inconvenience of lower served energy by the ESS. The two situations are depicted in Fig. 4.11.

Energy loss [p.u.]

1

eout = δ ein = δ

0.8 0.6 0.4 0.2 0 0.5

0.6

0.7

0.8

0.9

1

Efficiency η [p.u.] Figure 4.11. Energy loss as a function of ESS efficiency; comparison of the two cases of maintaining input or output energy equal to δ.

At this point, we only need to define the ideal energy throughput ratio ETR0 , to obtain finally Eloss . It is important to keep in mind, that in the next paragraph always the idealised case of 100% ESS efficiency is assumed. If ESS size is reduced, the throughput is reduced exactly by the amount of unserved energy eu , so that a reduced throughput ratio ETR00 can be defined, in case that not all storage demand is served, i.e. in our case, not the entire forecast error is compensated. ETR00 = ETR0 − eu

(4.14)

where ETR0 [p.u.] is the ideal throughput ratio if all errors are compensated and eu [p.u.] is the unserved energy. In chapter 3 curves of ETR0 were presented for different wind farm sites and forecast scenarios. Further it was demonstrated how to calculate the unserved energy eu (noncompensated forecast errors), if the ESS rated power PESS or the energy capacity Eeff are reduced. These curves were obtained maintaining the rated power at 1 p.u. (ζ = 1) while Eeff was reduced or Eeff was maintained at the maximum value, while PESS was reduced (ζ < 1).

114

Chapter 4. Cost analysis of energy storage systems

In order to estimate the effect of a simultaneous reduction of rated power and energy values, a discrete two-dimensional interpolation algorithm will be applied here. For unserved energy in chapter 3 the notation eup for power and eue for energy reduction was introduced to distinguish these two cases. In (4.15) the 2D-interpolation algorithm is formulated for the case that relationships of eup (PESS ) and eue (Eeff ) are given in discrete form.    eup,j eue,k 0 ETR0,jk = ETR0 1 − 1− (4.15) ETR0 ETR0 where j is the index of the unserved energy vector eup (PESS ) and k is the index of vector eue (Eeff ). Note that if one of the two factors is zero, the whole expression becomes zero. This occurs when nominal power or energy capacity are zero (eup,j = ETR0 or eue,k = ETR0 , respectively). In these cases no energy can be absorbed by the ESS. On the other hand, the factors become unity if all energy is compensated. In those cases the equation returns the original values obtained from (4.14). The result of the interpolation described in (4.15) is illustrated in Fig. 4.12. The reference size in this example is PESS = 1 MW and effective energy capacity Eeff = 24 MWh (discharge time d0 = 24 h). This size would guarantee a complete compensation of the forecast error, smaller values produce unserved energy as described above. The surface on the left represents energy throughput for any combination of reduced nominal power and energy capacity. On the right, corresponding equivalent full cycles per year are depicted. It can be observed that highest numbers of full cycles are obtained with systems between 0.5–1 MW (ζ > 0.5) and Eeff < 5 MWh. But even highest cycling rates are relatively small, with 100–110 cycles per year. A - Tx1 (24h forecast)

A - Tx1 (24h forecast)

ETR [p.u.]

1

120

0.8 0.6

0.5

0.4 0 1

24

0.5

PESS [MW]

12 0 0

Full cycles nc

1

100 100

80 60

50

40

0 1

0.2

24

0.5

0

12

PESS [MW]

Eeff [MWh]

0 0

20 0

Eeff [MWh]

Figure 4.12. 2D-interpolation of the energy throughput and corresponding annual full cycles as a function of ESS size, reference case of 1 MW installed power and 24-h forecast horizon.

Once the energy throughput has been obtained, the energy loss cost Closs can be calculated straightforward. Note that in (4.16) total losses of y years of operation are represented. Closs =

Etotal ETR00



1−η 2η



·π

j y  X 1 + gπ j=1

1+i

(4.16)

115

4.3 Cost structure of ESS technologies

where Etotal [kWh] is the total of annual generated wind energy, ETR00 the ideal throughput ratio considering unserved energy, η [p.u.] the ESS round-trip efficiency, π [e/kWh] the mean market price of electricity, gπ the electricity price escalation rate and i the capital cost of the local market. In order to close the circle from the beginning of this section, energy loss cost Closs may be expressed as a function of the number of equivalent full cycles nc . Introducing the relationship given by (4.9) into (4.16) we obtain Closs = nc · Eeff



1−η η



·π

j y  X 1 + gπ j=1

1+i

(4.17)

where nc is the number of equivalent full cycles and Eeff is the effective ESS energy capacity. For the reference system of 1 MW/24 MWh and mean market price of 50 e/MWh, results for lead-acid battery, redox flow battery, hydrogen and pumped hydro storage are depicted in Fig. 4.13 (see appendix C.2 for NaS battery). In the left column (a), Closs is expressed in absolute annual values. Lead-acid, NaS, PHS, flow battery show similar losses as their efficiencies are similar (70–75%). Hydrogen storage has three times larger losses. Round-trip efficiency of 40% is assumed here. In the right column (b) Closs is expressed as percentage of total annualised system cost CESS . Here lead-acid, NaS (not shown), and flow batteries (upper rows) exhibit the lowest values, with Closs below 10% of the total annualised cost. The maximum is reached if energy capacity is reduced to 5 MWh and nominal power is above 0.5 MW. Note the great similarity of the surface shape with full cycles in Fig. 4.12. This can be explained by the fact that losses are proportional to throughput and system cost is dominated by energy-related costs, i.e. total system costs are almost proportional to Eeff . Hydrogen fuel cell systems (HSFC) reach cost shares of energy losses over 50%, if the system power is reduced to below 0.4 MW, maintaining high energy capacities above 12 MWh. Note also the drastic change in shape of the surface. This is due to the predominance of power-related costs, contrary to BESS or flow batteries. With pumped hydro storage the maximum share of costs due to energy losses of up to 30% is reached in an area of intermediate values of energy and power reduction. In this case the surface is more equilibrated. Thus, costs due to efficiency are located between 20–30% of total costs for a wide range of system configurations. Note that cost surfaces expressed in absolute costs in column (a) are representing directly the amount of lost energy. Shapes in column (b) include the influence of system cost. It will be seen later that maximum shares of Closs are related to minimum system costs. A a consequence, larger systems (with larger absolute losses) do not necessarily have a higher share of Closs in total system costs.

116

Chapter 4. Cost analysis of energy storage systems

(a)

(b)

A - Tx1 (24h forecast) - Lead-Acid (base)

A - Tx1 (24h forecast) - Lead-Acid (base) 200 150

40 100

20 0 1

10

Closs [%]

Closs [ke]

60

100

12

PESS [MW]

0 0

60

5

40 0 1

50 24

0.5

80

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (base)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (base)

Closs [ke]

150

40 100

20 0 1

100

Closs [%]

200 60

50 12

PESS [MW]

0 0

30

80

20

60

10

40

0 1

24

0.5

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (base)

12 0 0

0

A - Tx1 (24h forecast) - HSFC (base)

100

0 1

100 100

Closs [%]

Closs [ke]

150

100

50 12

PESS [MW]

0 0

80 60

50

40 0 1

24

0.5

24

0.5

0

12

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (base)

0 0

0

A - Tx1 (24h forecast) - PHS (base)

40 100

0 1

50 24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

0

100 100

Closs [%]

Closs [ke]

150

20

20

Eeff [MWh]

200 60

20

Eeff [MWh]

200 200

20

80 60

50

40 0 1

24

0.5

PESS [MW]

12 0 0

20 0

Eeff [MWh]

Figure 4.13. Energy loss cost for the reference system of 1 MW and four technologies, (a) absolute cost, (b) cost share in % of the total system cost.

4.3 Cost structure of ESS technologies

4.3.3

117

Variable costs – O&M and replacement

For the calculation of variable costs, the approach in [61] was adopted without changes. Variable costs are divided into two parts: Operation and maintenance (O&M) and replacement costs. O&M costs are assumed to be proportional to the initial cost, and are given by O&M cost coefficient cm . The second part considers the replacement of major elements within projected system lifetime (not included in maintenance). The frequency of replacements depends on the different service life spans of ESS elements, such as battery stacks, pumps, membranes or power electronics. The cost of every replaced element is assumed to be proportional to the initial cost and expressed with the replacement coefficient ck , where k stands for the k th element to be replaced. Cost escalation rates for maintenance and replaced elements are included by the parameters gm and gk respectively. If no specific information is available, these two parameters can be assumed to be equal to the expected inflation rate. Technological improvements of replaced elements may introduce cost reductions, which are represented by the technological improvement rate %k . Thus, the sum of O&M and replacement costs V C can be formulated as " l  ( j X l·yk #) y  k0 k X X 1 + gm (1 + gk )(1 − %k ) + ck (4.18) V C = IC cm 1 + i 1 + i j=1 k=1 l=0 

y−1 lk = yk where IC cm y i gm , gk k0 ck lk %k yk



(4.19)

Initial cost of the ESS [e] O&M cost coefficient Years of operation for the proposed configuration Capital cost of the local market Mean annual inflation rate for O&M and replaced major components Major components to be replaced during the system’s service period Replacement cost coefficient for k th component to be replaced Number of replacements of k th component (integer number) Mean annual technological cost reduction of k th component Lifetime in years of the k th component

While O&M cost estimations are widely documented, it is much more difficult to find parameters for the replacement of major elements. In this thesis the replacement cost was only used to reflect the fact that some ESS have service lives yss shorter than the required 20–25 years, so that k0 = 1 and yk = yss are assumed. In these cases it is taken into account that not the whole installation of the ESS has to be replaced and that some technology improvement can be expected. Results for the reference case are shown in Fig. 4.14. Lead-acid batteries, due to their short cycle life show highest values of V C (almost 50%), and PHS the lowest (no replacements). It can be also observed that the V C-percentage of total costs is almost independent on system configuration.

118

Chapter 4. Cost analysis of energy storage systems

(a)

(b)

A - Tx1 (24h forecast) - Lead-Acid (base)

A - Tx1 (24h forecast) - Lead-Acid (base)

600

80 80

400

400 200

V C [%]

V C [ke]

600

200

0 1 12

PESS [MW]

0 0

40

40

20 0 1

24

0.5

60

60

20 24

0.5

0

12

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (base)

0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (base)

600

80 40

400 100

V C [%]

V C [ke]

200

200

0 1 12

PESS [MW]

0 0

20

40

0 1

24

0.5

60

20 24

0.5

0

12

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (base)

0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (base)

600

80 40

400

40 20

V C [%]

V C [ke]

60

200

0 1

PESS [MW]

12 0 0

20

40

0 1

24

0.5

60

20 24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (base)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (base)

600

80 40

400 5 200

0 1

24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

0

V C [%]

V C [ke]

10

60

20

40

0 1

20 24

0.5

PESS [MW]

12 0 0

0

Eeff [MWh]

Figure 4.14. O&M plus replacement costs for the reference system of 1 MW/24 MWh and four technologies, (a) annual cost, (b) percentage of total cost.

119

4.3 Cost structure of ESS technologies

4.3.4

Cost of stored energy – COE

In [71] the cost of energy (COE) of an ESS is described as a revenue requirement, which is defined as the price “that an energy provider [...] would need to charge for each kWh of energy delivered, to cover all costs for operating and owning the system.” In [61] the same concept is termed energy production cost, but here the denomination cost of energy COE is preferred, as for example in [144, 146]. When total cost CESS and the energy ratio δ covered by the ESS are known, COE can be calculated. Schoenung and Hassenzahl (2003) [71] proposed the formulation given in (4.20). Kaldellis and Zafirakis (2007) [61] suggested formulation (4.21) which takes into account the electricity price escalation ratio gπ . Schoenung and Hassenzahl [71] :

COE =

Kaldellis and Zafirakis [61] :

COE =

where CESS δ Etotal gπ i y CRF

CESS δEtotal

(4.20)

CESS  j 1 + gπ δEtotal CRF 1+i j=1 y P

(4.21)

Total levelized annual cost of the ESS [e] Energy ratio covered directly by the ESS [p.u.] Total annual wind energy [MWh] Electricity price escalation rate Capital cost of the local market Years of operation for the proposed configuration Capital recovery factor

In this thesis, formulation (4.21) is adopted to include energy price escalation. Note that in [61] equation (4.21) is written with the factor (1 + i)y instead of CRF . In order to obtain annualised values, CRF should be used in the same way as in equation (4.1) on page 106. The result is the same since this factor is canceled in this fraction, so that the revenue requirement COE can be written as COE =

IC + Closs + V C  j y P 1 + gπ δEtotal 1+i j=1

(4.22)

If gπ > i (electricity prices increase at a higher rate than the capital cost), the net income per kWh will rise during the lifetime of the ESS. In this case, taking into account the escalation of electricity prices, the initial revenue requirement becomes lower. In (4.20) a constant revenue is assumed which results in a higher initial price although after 20 years the same income is achieved. Especially for the rapid implementation of ESS, it might be interesting to reduce the revenue requirement as in (4.22) because it permits the ESS technologies to become competitive earlier. Results for the reference case are shown in Fig. 4.15. In column (a) the annualised total system cost CESS is represented and in column (b) the revenue requirement or cost

120

Chapter 4. Cost analysis of energy storage systems

of energy (COE). COE is expressed in e/MWh which permits a direct comparison with electricity market prices. It can be seen that lead-acid and flow batteries represent the most costly options. Lead acid although with lower initial price is costly in operation due to the short service life of only 7 years. Both types show a minimum for storage capacities of 5 MWh and nominal powers between 0.5–1 MW. As seen before, this minimum is caused by the elevated number of full cycles, as total costs are mainly determined by the energy capacity. High energy-related costs (200–600 e/kWh) of these technologies make them very expensive in high energy applications as it is the case here. Similar results are obtained for NaS batteries (not shown), although improved cycle life permits a cost reduction by approximately 30%. Minimum values of COE are near 150 e/MWh for lead-acid, 100 e/MWh for NaS (see appendix C.2) and 170 e/MWh flow batteries. As a third option, hydrogen storage with fuel cell (HSFC) is represented. It shows a relatively wide range of system configurations with COE between 90–110 e/MWh. Lowest costs are observed for low power and high energy capacities. The largest system with nominal power of 1 MW has a COE of 140 e/MWh. The unexpected low cost is mainly due to the very small cost coefficient of energy capacity cp (20 e/MWh) which is even lower than that for pumped hydro storage (PHS). But PHS remains with approximately 30–50 e/MWh the least costly option, mainly for its long service life and the sensibly higher system efficiency compared to HSFC. Comparison of worst and best case assumptions As mentioned before, values of cost parameters reported in literature [61, 71–73, 78, 80] show a wide spread, especially for the cost coefficients for energy capacity ce and installed power cp . Most optimistic figures may be interpreted as projections in the future, while most pessimistic values represent costs of technologies which exist only in demonstration plants. In order to account for the wide range of values, worst and best case scenarios have been created. The worst case scenario assumes that all parameters are such that final COE will be highest. In the best case scenario, just the opposite is done. In Fig. 4.16 results for COE are shown for four of the five ESS technologies considered. More results for NaS and parameters closs and V C can be found in appendix C.3. For lead-acid batteries, the most critical parameter is cycle life. Most pessimistic values lead to very high costs with a minimum above 200 e/MWh. The best case scenario yields minimum costs below 100 e/MWh. Similar results are obtained for NaS batteries (see appendix C.3.) The most dramatic difference is observed for flow batteries. This is a result of the contrast between very optimistic perspectives of this technology and the fact that until today no large scale series production exist. Lowest worst case prices are around 350 e/MWh, while best case prices are between 50–75 e/MWh for a wide range of configurations. A similar case is HSFC, but with the difference that even assuming worst case parameters, minimum values of COE between 150–200 e/MWh are obtained for configurations with low power (< 0.5 MW) and high energy (> 12 MWh). Best case prices are between 50–60 e/MWh for a wide range of configurations. Assuming worst case conditions, pumped hydro storage can be quite expensive. For most configurations, worst case COE lies between 70–100 e/MWh. Under optimal conditions, PHS might have a minimum COE below 15 e/MWh.

121

4.3 Cost structure of ESS technologies

(a)

(b)

A - Tx1 (24h forecast) - Lead-Acid (base)

A - Tx1 (24h forecast) - Lead-Acid (base)

CESS [Me]

2

1.5

1

1

0 1

0.5

400 400 300 200 200 0 1

24

0.5

PESS [MW]

500

COE [e/MWh]

2

12 0 0

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (base)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (base)

1.5

1

1

0 1

0.5 24

0.5

PESS [MW]

500

COE [e/MWh]

CESS [Me]

2 2

12 0 0

400 400 300 200 200 0 1

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (base)

12 0 0

0

A - Tx1 (24h forecast) - HSFC (base)

1

0 1

0.5 24

0.5

PESS [MW]

500

COE [e/MWh]

CESS [Me]

1.5

0.5

12 0 0

400 400 300 200 200 0 1

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (base)

12 0 0

0

A - Tx1 (24h forecast) - PHS (base)

1

0 1

0.5 24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

0

500

COE [e/MWh]

CESS [Me]

1.5

0.5

100

Eeff [MWh]

2 1

100

Eeff [MWh]

2 1

100

200

400 300

100

200 0 1

24

0.5

PESS [MW]

12 0 0

100 0

Eeff [MWh]

Figure 4.15. Base cost scenario for reference system of 1 MW/24 MWh and four technologies, (a) annualised total system cost, (b) revenue requirement (COE).

122

Chapter 4. Cost analysis of energy storage systems

(a)

(b)

A - Tx1 (24h forecast) - Lead-Acid (worst)

A - Tx1 (24h forecast) - Lead-Acid (best)

400 400 300 200 200 0 1

24

0.5

PESS [MW]

12 0 0

100

500

COE [e/MWh]

COE [e/MWh]

500

400 400 300 200 200 0 1

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (worst)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (best)

400 400 300 200 200 24

0.5

PESS [MW]

12 0 0

100

500

COE [e/MWh]

COE [e/MWh]

500

0 1

400 400 300 200 200 0 1

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (worst)

12 0 0

0

A - Tx1 (24h forecast) - HSFC (best)

300 200 200

0.5

PESS [MW]

12 0 0

100

500

COE [e/MWh]

COE [e/MWh]

400 400

24

400 400 300 200 200 0 1

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (worst)

12 0 0

0

A - Tx1 (24h forecast) - PHS (best)

300 200

0 1

24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

100 0

500

COE [e/MWh]

COE [e/MWh]

400

100

100

Eeff [MWh]

500 200

100

Eeff [MWh]

500

0 1

100

200

400 300

100

200 0 1

24

0.5

PESS [MW]

12 0 0

100 0

Eeff [MWh]

Figure 4.16. ESS revenue requirement (COE) for the reference system of 1 MW/24 MWh and four technologies, (a) worst case, (b) best case scenario.

123

4.3 Cost structure of ESS technologies

In Table 4.3 assumed market parameters are shown. The mean electricity marginal price represents the mean value on the Spanish market in 2007 and 2008. The annual inflation rates gm (O&M) and gk (replacement of elements) are considered to be equal. In Table 4.4 ESS-specific parameters of the cost model are presented. Here the variety of values reported in the literature is reflected by best and worst case values. As base case, intermediate values have been assumed. Replacement cost coefficients ck and price reduction due to technology improvement %k have been estimated by the author. Here further investigation is needed to obtain more realistic values. π gπ i gm , gk

50 e/MWh 10% 5% 3%

Mean annual electricity price (marginal price) Electricity escalation rate Capital cost of the local market Annual inflation rate (for O&M and replacement) Table 4.3. Assumptions for the market environment.

Parameter

Scenario

Lead-acid

NaS

RFB

HSFC

PHS

ce [e/kWh]

base best–worst

240 210–400

300 200–500

600 100–1000

20 2–30

40 5–100

cp [e/kW]

170 140–600

200 125–300

1000 250–1500

4000 1000–10,000

1000 500–2000

DOD [%]

65 70–60

70 80–60

100

90

95

η [%]

75 80–75

80 85–75

65 70–60

40 45–35

75 80–60

cm [%]

0.75 0.5–1.0

0.75 0.5–1.0

1.0 0.7–1.3

0.75 0.5–1.0

0.4 0.25–0.5

7 10–5

13 15–10

13 15–10

15 20–10

40 50–30

0.8 0.7– 0.9

0.4 0.5–0.6

0.4 0.5–0.6

0.4 0.5–0.6



3 5–1

10 20–5

10 20–5

10 20–5



Replacement parameters yk = yss [years] rk %k [%]

ce , cp : Specific energy and power cost, DOD: max. depth of discharge, η: Round-trip efficiency, cm : O&M coefficient, yss : System service life, rk : Replacement cost coefficient (only one element), %k : Technological improvement

Table 4.4. Selected parameters of the cost model; adopted value and the range found in the literature (Kaldellis 2007, Schoenung 2003/2008, Chen 2009) are shown for each parameter.

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4.4

Chapter 4. Cost analysis of energy storage systems

Summary

In this chapter, first a short overview on grid-connected energy storage systems was given. The main conclusions of the review may be summarised as follows: • ESS can be of high value for electrical power systems • Pumped hydro is the most extended and mature storage technology • Batteries and flywheels are operating in UPS and power quality applications • Electric vehicles have important potential for distributed ESS → V2G • Emerging technologies for utility scale ESS have still many cost uncertainties • Cost estimations depend strongly on specific conditions of application This chapter aimed at two main objectives. First, the development of a cost model for ESS which accounts for specific properties of different ESS technologies and second, the application of this model to results of the ESS sizing method proposed in chapter 3. Reducing the ESS size, depending on the technology, regions of minimum cost of energy (COE) could be identified. This way, important economical information is provided for the selection of adequate ESS technologies and size. A reference case is considered for error compensation of a 24-h forecast of a 1-MW wind farm. Applying the T×1 persistence scenario, an ESS with nominal power of 1-MW and energy capacity of 24 MWh (24-h discharge time) would guarantee 100% error compensation. Results from chapter 3 are used to determine the energy served by the ESS. The proposed cost model represents a life-cycle cost analysis which provides important information which is not available from the initial cost (capital cost) alone. The model has been developed as a synthesis of methods proposed mainly in [62] and [61]. The main differences between the method suggested in [61] and the method proposed here are: • Energy loss costs are added as variable cost according to [62] • EESS , PESS and nc are determined with probabilistic methods • Annualised costs are obtained with capital recovery factor CRF • Energy input cost to the ESS is zero (ESS is considered as an integral part of the wind farm) • No fuel cost is supposed for the ESS (CAES is not considered) • No public subsidy is considered on the initial cost The contribution of this thesis to the cost model is first that several models have been brought together, eliminating shortcomings. While in [62] no replacement costs and no price escalation are considered, in [61] the energy loss cost was not properly modelled. The second contribution is a probabilistic formulation for the definition of the number of equivalent full cycles, as a classical parameter for calculating energy losses.

4.4 Summary

125

The main conclusion of the cost analysis is that energy storage is almost always considerably more expensive than the mean electricity price of 50 e/MWh. In fact, today only pumped hydro storage provides energy at a cost below 100 e/MWh. Future price projections, as simulated in a best case scenario show that emerging technologies such as flow batteries, NaS or hydrogen storage may reduce costs to 50–100 e/MWh. This price region can be considered as very interesting for commercial applications. According to available cost data, lead-acid batteries are not likely to be able to reduce costs below 100 e/MWh, mainly due to its reduced cycle life. For the case study of reducing forecast errors of a 1-MW wind turbine, regions of least costs have been detected for the five ESS technologies considered here. Battery systems (including flow battery) have minimum costs of energy (COE) for systems with nominal power > 0.5 MW and energy capacity < 5 MW. This translates to reduction factors of < 0.5 for power and > 0.2 for energy capacity. This reflects the nature of BESS as typically best suited for high power applications (discharge times below 1 h). If flow batteries succeed in reducing energy related costs, the optimal region extends also to high energy (up to the maximum of 24 h). Hydrogen storage with fuel cell is best for low power and high energy applications. In spite of the low efficiency and high power related costs, very low energy related costs make this technology interesting. Hence, lowest costs are detected for maximum energy capacities (Eeff > 12 MWh) and low power (PESS < 0.5 MW). This exactly inverse characteristic compared to BESS may lead to hybrid solutions of combined battery-hydrogen systems. Similar to BESS, if power related costs are reduced as most optimistic data suggests, the optimal region extends to high power applications. Pumped hydro storage is in all cases best suited for high power/high energy applications. Nevertheless, in this case study, lowest costs are achieved for intermediate power and energy ratings (reduction factors of 0.5). This is widely caused by the number of equivalent full cycles, which are lower for the largest system configuration. Results can be summarised as follows: • BESS: highest costs, optimum at 100–250 e/MWh best suited for high power/low energy • Flow battery: high costs with large reduction potential, optimum at 50–350 e/MWh best suited for high power and potentially also high energy • Hydrogen-Fuel cell: Still high power related costs, optimum at 50–180 e/MWh best suited for high energy and potentially also high power • Pumped hydro: lowest costs, optimum at 15–100 e/MWh best suited for high power/high energy The final conclusion is that energy storage technologies imply important additional costs for the power system. Emerging technologies have promising potentials for cost reductions, but realistic estimations are difficult. The proposed model gives an orientation about revenue requirements for several storage technologies. Many sources of revenues from ESS are possible. An exhaustive study of these revenues is outside the scope of this thesis. In the next chapter only one possible application is

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Chapter 4. Cost analysis of energy storage systems

considered: mitigation of forecast deviation costs on regulation markets. It will be shown that costs of forecast deviations are below 10% of total revenues, which would not justify any energy storage.

Chapter 5 Uncertainty of wind power forecast and electricity markets “A business that makes nothing but money, is a poor business.” Henry Ford (1863–1947) US-american inventor and businessman

The main purpose of this chapter is the quantification of the cost of deviations caused by wind power forecast errors. The study is based on data from the Spanish electricity market. The analysis comprises three parts: (1) analysis of market prices and possible impact of wind energy, (2) identification of possible scenarios of price forecasting and (3) application of a probabilistic bidding strategy for the quantification of deviation costs. Section 5.2 contains a survey of recent literature which covers modelling of liberalised electricity markets, price forecasting and the role of energy storage in advanced bidding strategies. Section 5.3 is dedicated to a detailed statistical analysis of prices on the Spanish electricity market, with special emphasis on regulation costs. The impact of wind power is evaluated with correlation analysis. Comparing wind power with market prices, a negative correlation is observed. It is shown that daily patterns of wind power generation might contribute to this negative correlation. Several scenarios for price forecasting are derived from the analysis of market prices. Deviation prices are far less influenced by external factors, such as oil prices, than spot prices. Therefore, annual mean regulation prices might be a valid forecast. As contribution of this thesis, annual and seasonal mean daily profiles are identified. In section 5.4, forecast error pdf obtained in this thesis and price forecast scenarios are applied for optimised bidding strategies. Here the main contribution of the thesis is the detailed evaluation of different estimation approaches for regulation costs. Revenues can be improved with probabilistic bidding strategies by 1–3%, which is very encouraging. The adopted strategy increases overall imbalance, which reveals a conflict of interests.

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5.1

Chapter 5. Uncertainty of wind power forecast and electricity markets

Introduction

After the analysis of energy storage size and revenue requirements in chapters 3 and 4, the objective of this chapter is to quantify the cost of wind forecast errors. Comparing deviation costs with ESS costs from chapter 4, the economical viability of energy storage for forecast error compensation can be assessed. Deviation costs can be minimised with advanced bidding strategies, even without any energy storage if market prices are estimated in advance. Three aspects have major impact on deviation costs: a) Stochastic behavior of imbalance prices on the regulating market b) Stochastic behavior of energy imbalance, induced by forecast uncertainty c) Bidding rules on the market (intra-day market, closure delay) It is assumed that wind power participates in a liberalised electricity market environment, so that forecast error costs are generated by deviations of the contracted energy one day ahead. Note that intra-day market options are not taken into account, but may reduce considerably the forecast error and thus the deviation cost. A recent study [149] shows that wind generation costs have increased by more than 20% over the last 3 years, mainly due to a rise of the price of raw materials. With this background it becomes even more important for wind power producers to optimize revenues on the market. Imbalance prices are of stochastic nature due to uncertainties in generation (mainly wind power) and demand forecast. Thus, the actual cost of imbalances induced by wind power can be analysed best with statistical methods. In the presented study, two main assumptions are done: • Wind generators are price takers: do not affect the spot market or imbalance price • Imbalance price e/MWh is independent of the volume of imbalance Even if these two assumptions may not always hold, they still represent a quite good approximation and simplify the model considerably. Especially the widely considered condition of the wind generators as price takers is already changing in electricity markets with high wind penetration such as Denmark [77,150]. Today this influence on the energy prices is still small in most markets outside Denmark but in the future, especially with the large offshore wind projects in place, models should account for this effect. A general conclusion from the study of the literature is that the imbalance cost is a stochastic variable very difficult to predict. In this thesis, imbalance prices are not predicted. In [45] it is shown that even yearly means of regulation prices can be valuable information. Therefore, historical data is used to identify daily and seasonal patterns and it is assumed that any forecasting method may achieve similar accuracy. Energy storage can mitigate deviations and provide ancillary services but its viability depends on its cost compared to prices to be paid on the regulating market. It is not within the scope of this thesis to develop an optimised scheduling plan for wind power coordination with energy storage. Thus, the market is analysed without energy storage.

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5.2 Wind power in European balancing markets – State of the art

5.2

Wind power in European balancing markets – State of the art

To get a rapid overview of the situation of wind power balancing in Europe, the “European Balancing Act” published in 2007 by Thomas Ackermann et al. [151] can be recommended. In this work, issues like increased reserve requirements, balancing cost and fault-ride-through (FRT) capability were treated. The cases of Spain, Ireland and Germany were discussed in more detail. In addition, results of studies from Finland, Sweden and the UK were presented. The authors concluded that wind power integration has an economical and a technical dimension. The economical dimension is mainly related to the allocation of additional reserves which may increase costs for power system operation. To reduce these costs, three aspects are identified: (a) aggregating wind generation over large geographical areas, (b) larger balancing areas and (c) more flexible power system operation, with a reduced closure delay. Referring the technical dimension, FRT and frequency control are mentioned as essential for wind penetration levels beyond 15% (of total energy consumption). First results from Horns Rev offshore wind farm were also presented in [151], highlighting the integration of the main controller of the wind farm into the central control of the Danish power system as a model for the future. One of the first places worldwide, where wind power reached very high penetration is Denmark. As described in [152], already in 2003, wind power covered occasionally almost 100% of total power consumption in Western Denmark. Nowadays it is common, that wind power exceeds sometimes the regional consumption. As stated in [153] in 2001 installed wind power exceeded off peak load level and wind power production accounted for 16% of the total demand. In 2007 this figure grew to almost 20% and with the planned offshore wind farms it is estimated to reach 50% by the year 2025 [151]. For this reason, many publications are based on data from Denmark. This country belongs to the NordPool power exchange which is geographically bound to Norway, Sweden, Finland and Denmark. A good introduction to the NordPool market is given in [152] and more in detail it is described in [150]. Two energy markets are of importance for wind power: the spot market and the regulating (or balancing) market. The spot market is the central energy market, and the regulating market only comes into force if the bids on the spot market are not fulfilled. The spot price or marginal price is determined at the daily spot market by supply and demand. On the NordPool market for example, producers and customers give their bids to the market 12–36 h in advance, as illustrated in Fig. 5.1, providing price and quantity of electricity to buy or sell. At the power exchange (e.g. Elspot (NordPool) in Scandinavia, MIBEL in Spain and Portugal, EEX in Germany) marginal prices (i.e. prices that clears the market) are fixed for each hour. Current day t 12:00

Following day (Forecasting period) t + 12 t + 36 00:00

12:00

00:00

Figure 5.1. Timing of bidding on the NordPool spot market.

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Chapter 5. Uncertainty of wind power forecast and electricity markets

Only a minor portion of total electricity production is traded at the spot market. The majority is sold on long-term contracts, but the determined spot prices have a considerable impact on prices agreed in these contracts. In general only big customers and producers act on the market. Smaller entities as wind farms may form trading cooperatives. If contracts from the spot market are not fulfilled, the regulating market comes into play. This is especially important for wind power. In the NordPool, bids on the regulating market are placed 1–2 hours before the actual production hour. Availability of production is confirmed 15 min before the delivery (notice period). As a consequence, regulating power stations must respond fast.

5.2.1

Imbalance prices on the regulating market

In general, imbalances are defined as the deviation between generation and demand. The origin of imbalances lies in generation and demand forecast errors. From the point of view of wind power producers, imbalances are deviations between contracted and delivered energy. If bidding with point forecasts, these deviations are nothing more than wind power forecast errors, which have been examined in the preceding chapters of this thesis. If production is higher than the bid, down-regulation is required, i.e. surplus energy is sold on the regulating market. In the opposite case up-regulation is needed, i.e. lacking energy is bought. Prices for buying and selling energy on the regulating market are termed as imbalance prices. Generally, selling prices are below and buying prices are above the contracted price. As a result a penalty for deviations can be defined. This concept will be treated more in detail in section 5.4.1. Note that commonly imbalance prices are equal to contracted prices if the imbalance is contrary to the system imbalance. Mothorst (2003) analysed in [152] the impact of wind generation on market prices. Spot prices are slightly influenced in a manner that more wind generation tends to reduce spot prices. A stronger relationship is found at the regulating market. High wind power produces excess and low wind power lack of energy. As mentioned before, bids on the spot market in the NordPool are placed 12–36 h ahead. Especially for wind power this is a problem because forecasts up to two days ahead means having to correct 30–40% of wind production. Holttinen (2005/2006) examined in [153, 154] the reduction of deviation costs if the market was more flexible and bidding 6–12 h ahead was allowed. This would improve substantially forecast precision and reduce deviation costs by 8%. A more extensive work on the impact of large scale wind power on the Nordic electricity system, including technical and market issues has been published in the PhD thesis of Hannele Holttinen [155]. Holttinen mentioned in [153] also the scenario of an after-sales market where deviating production is traded 2 h before delivery (intra-day market). Such a market (Elbas) is already operating in Finland and Sweden. As a result, the income of wind energy producers would increase almost 7%. The simulation is carried out with prices from West Denmark and Elbas market (2001), real forecast (WPPT) and generation data. It is concluded that there are no technical barriers to make the electricity market more flexible. Furthermore a well working after sales market could help reducing cost and amount of deviations from wind power generation. Bathurst (2002) [38] described the NETA (New Electricity Trading Arrangements) trading system of the UK. Note that since April 2005 the follow up arrangement BETTA

5.2 Wind power in European balancing markets – State of the art

131

(British Electricity Trading and Transmission Arrangements) is implemented. However, in [38] conditions of NETA were considered. Pricing is done in half-hour intervals and wind power producers can make their bids only 4 h before delivery (4-h market closure time). In this system, it is on behalf of the producer, how many trades he makes per day. So that he decides the forecast window length. A fee per trade is fixed to minimize the number of trades per day. Four trades per day would lead to a required forecast length of 4–10 h. Compared to the NordPool these conditions are very favourable for wind power. Fabbri (2005) used in [29] estimates of forecast error pdf for calculating costs incurred due to deviations. The method for pdf estimation was adopted from [28]. It was based on the same fundamental idea as exposed in this thesis (see chapter 2). Historical prices from the Spanish tertiary reserve energy market were used to calculate costs. It was found that imbalance costs can reach up to 10% of total revenue of a wind farm and that aggregating several wind farms reduces forecast errors and thus imbalance costs. Another interesting study on the influence of wind power on market-clearing prices is presented by El-Fouly et al. (2008) in [156] using a single auction market model. In general a decrease of overall electricity prices was detected with increasing wind power penetration. In addition, control strategies of wind generators such as voltage control can reduce market prices, especially in high demand situations. One year later the same authors published further studies based on the same configuration [157]. Here especially the analysis of the impact of forecast accuracy on the market prices was of interest. Singh et al. (2008) proposed in [158] pricing mechanisms for wind energy as a guideline for policy makers. Clearing prices with and without wind energy were obtained for block and linear bidding markets. It was argued that with adequate pricing no imbalance penalties are needed for wind power.

5.2.2

Forecasting of spot and imbalance prices

In the last decade, increasing activity in the development of forecast models for regulation market prices can be detected. In the following only some models, mostly related to the impact of wind power are presented. Saint-Drenan analysed in [159] prices of the Dutch power system operator TenneT for the years 2001 and 2002. The TenneT imbalance pricing system is very complex and will not be considered here. Prices depend on the volume of deviations and incentives are applied with the goal to minimize the volume of imbalance energy. As a result negative imbalance prices can occur. This means that producers are actually paying for injecting energy to the system. With harmonic analysis daily and monthly patterns have been identified in [159]. Different daily patterns for the weekend and the rest of the week were observed. On the basis of these findings, a price prediction model was developed. The model output comprised expected mean, minimum and maximum values of up and down regulation prices for every hour in a given day of the year. It was claimed that this model predicts the range of imbalance prices with a confidence level of 85%. An hourly spot price forecasting model was proposed by Crespo et al. (2004) in [160]. The performance was evaluated based on data from the German Leipzig Power Exchange (LPX) which merged in 2002 with European Energy Exchange (EEX). The dataset comprised spot market prices from June 2000 until October 2001. This work is especially interesting given the importance of this power exchange market, being one of

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Chapter 5. Uncertainty of wind power forecast and electricity markets

the largest European markets. An auto-regressive model was applied in the time series model. Prices were estimated for each hour of the day including knowledge of typical daily profiles. That means 24 separated models, one for each hour of the day. Also on the EEX spot market, Diongue et al. [161] proposed a probabilistic forecast technique, using a GIGARCH process. GIGARCH stands for “Gegenbauer Integrated Generalised Autoregressive Conditional Heteroscedasticity” and is a creation of the cited authors. The model provides conditional mean and variance forecasts. Conejo et al. compared in [162] different forecasting techniques to predict the 24 hourly market-clearing prices of a pool-based day-ahead market. Time series analysis, neural networks and wavelets were considered. The analysis was conducted using data from the PJM Interconnection (Regional Transmission Organization of 13 states in the Eastern USA) of the year 2002. Daily and weekly seasonalities were observed. It was concluded that time series techniques perform better than wavelet-transform or neural networks. Among time series techniques, dynamic regression and transfer function algorithms were found to be more effective than autoregressive integrated moving average (ARIMA) models. It was highlighted that the na¨ıve technique achieves reasonable results in some cases and that a combination of wavelet transform and time series algorithms may improve results. Mateo Gonz´alez et al. proposed in [163] the application of hidden Markov models for electricity spot price forecasting. The proposed model can be interpreted as a piecewise dynamic regression model where the most suitable model is activated by an underlying Markov chain. A conditional probability transition matrix was established describing the probability of changing to another state. The model has been applied to clearing prices in the Spanish spot market from January to September 2001. The training set lasts from January to June and the test set from July to September. As model input (physical explanatory variables) hourly spot price, demand and production are considered. Production is classified into hydro, nuclear and thermal generation. No wind power is considered. The model produces a mean absolute percentage error (MAPE) of 15.8% for the testing set. Note that the output of the model is probabilistic, i.e. uncertainty of the point forecast is provided. This price forecast model is used in [164] for optimization of a combined wind-pumped hydro facility, applying Monte Carlo simulation. J´onsson proposed in his Master’s thesis [150] a probabilistic approach for price forecasting. Historical market data was used from NordPool, obtained mainly from the website of the Danish system operator (Energinet.dk). Wind power production forecasts were made by the Wind Power Prediction Tool (WPPT). The dataset of hourly values wind power, energy consumption and market prices covered almost two years from January 2006 until October 2007. It was emphasised that the distributions of energy prices are skewed and fat tailed, thus no normal distribution can be assumed. Adaptive probabilistic models were developed separately for spot prices and regulation prices. Analogue to probabilistic wind forecasting, the uncertainty is given together with point forecasts of the price. A severe impact of the predicted level of wind penetration on spot prices was detected. With increased wind power penetration, spot prices decrease and intra-day price variations diminish slightly. Spot price forecasts were judged to be satisfactory, with values of RMSE between 8–9 e/MWh for look ahead times up to 44 h, which means 20–22% assuming a mean price of 40 e/MWh. The model for deviation prices on the regulating market was characterised as “preliminary”, although results are promising.

5.2 Wind power in European balancing markets – State of the art

5.2.3

133

Energy storage and optimised bidding strategies

Energy storage can be used to reduce the forecast error and thus deviation penalties on short-term markets. Therefore, bidding strategies have to be developed which take into account market prices, wind power uncertainty and availability of energy storage. In the following paragraph a short survey is given of studies on the improvement of income from intermittent energy sources (mainly wind). The survey is limited to the last decade. Optimised bidding based on Markov Tables – Case: UK/Ireland Bathurst (2002) proposed in [38] a probabilistic bidding strategy for trading wind energy in the restructured electricity market of the UK. The model was applied to the NETA trading system which had been in force until 2005. Forecasts were obtained from persistence and Markov tables were employed to describe the wind forecast uncertainty. Imbalance prices were calculated with persistence as well. A minimum expected imbalance cost policy was suggested to maximize overall revenues. It was found that depending on up and down regulation prices the best point forecast not always corresponds to maximal income. One year later the same authors included the aspect of energy storage in the bidding strategy [165]. In this work, it was assumed that the ESS can take advantage of low or high imbalance prices to charge or discharge. The round-trip efficiency of the ESS was set to 75%. The added value of ESS was calculated subtracting the independent annual revenues of the wind farm and ESS arbitrage from the combined dispatch revenue. Different spreads of deviation prices are assumed: ±50%, ±100% and ±150%. Income from wind energy is lower if the spread of deviation prices is larger, while revenue from ESS arbitrage shows an opposite tendency. ESS size has been varied from 2–10 MW and 2–10 h of capacity. An optimum for added value was detected for an ESS with 6 MW and 36 MWh (6 hours of storage) for a 10-MW wind farm. Finally, grouping two small wind farms (9 and 10 MW) with one ESS has been found to increase total revenue considerably as the optimal ESS size is approximately the same as for a single wind farm. Results were used in a feasibility study for the implementation of a VRB-ESS at Sorne Hill Wind Farm (Ireland) [137]. ESS with local loads and congestion – Case: NordPool Korp˚ as et al. (2003) [18] proposed an algorithm to determine the optimal energy exchange with the market. Energy storage with 75% efficiency was used to smooth variations in wind power production in order to follow the scheduling plan (24 h). A daily profile of a local load, close to the wind farm was included in order to simulate possible congestions in a transmission line. Only the resulting balance of energy consumption and production was traded on the market. Wind velocity forecasts were simulated assuming normal distribution of forecast errors. Wind power was obtained straightforward from a simple power curve of a given wind generator. The case study was applied to the NordPool market, assuming a mean spot price of 30 $/MWh, following a daily profile. Profiles were equal for all days of the year. Up regulation prices were set 25% above and down regulation prices 10% below the spot price. Results with available price estimates suggest that energy storage devices such as reversible fuel cells are likely to be a more expensive

134

Chapter 5. Uncertainty of wind power forecast and electricity markets

alternative than grid expansions. Simulations were repeated for nine combinations of power and energy capacity of the ESS. As the cost of the ESS was not included in the calculation of annual revenue, highest incomes were obtained with the largest ESS. ´ Wind-Hydro coordination with real forecast – Case: SIPREOLICO, Spain Angarita et al. (2007/2009) proposed in [85, 86] coordinated bidding strategies for independent wind and hydro companies, without water pumping. Strategies were designed for a pool market, where bids must be made once a day and corrected on intraday markets. Two auction scenarios were considered: One single and six daily auctions. This results in forecasts of 14–38 h ahead in the first and 4–8 h ahead in the second case. The case study was applied to aggregated wind generation data from 13 Spanish wind farms (together almost 800 MW). Forecast scenarios considered minimum, maximum and ´ mean accuracy from SIPREOLICO model and persistence for comparison. Conditional probability distributions (pdf) of wind generation were obtained from historical data of 8 months depending on observed power at the moment of the forecast. This is very similar to the persistence approach adopted in this thesis, but only empirical pdfs were used. In [85] perfect price forecasting was assumed and imbalance penalties were considered to be proportional to spot prices. Independent and combined wind-hydro scheduling methods were compared. The advantage of combined scheduling was higher if forecast errors or imbalance penalties were greater. In [86] the procedure was further refined and the case study was applied to a 250-MW wind farm with considerably larger hydro power (beyond 3000 MW) in all cases. Results showed that between 15–45% of the penalties can be saved, considering a MAPE around 25–30% for a single wind farm and using penalties around 27% of the market price. Wind-Hydro coordination and congestion with Monte Carlo – Case: Ume river Sweden Matevosyan and S¨oder (2006,2009) proposed in several works algorithms for minimization of imbalance costs [166] and optimised combined wind-hydro power planning in areas with congestion problems [167, 168]. In [166] probabilistic methods were employed for obtaining optimal bids for wind power on the market. One year of forecast data was used to calibrate the probabilistic model for wind speed forecast errors. Then a MonteCarlo approach was used to represent possible outcomes of forecast errors. Wind speed was converted to wind power with a power curve model (including wake effects) and the forecast error model was validated with data from West Denmark. Imbalance prices were estimated with simple persistence. In [167] the methodology was applied to coordinate wind generation and hydro power in order to prevent congestions in a common power line. The scheduling plan was based only on spot market prices and the wind speed forecast error was adopted from [166]. It was shown that wind energy curtailment can be reduced by approx. 50% by preventing congestion of the transmission line. At the end of the year, both, hydro power and wind power producers improved their income and transmission capacity is increased. This study was refined in [168], including trading on the regulating market into the optimization procedure. Uncertainty of spot prices was simulated with a set of price scenarios in a scenario tree (50 price scenarios). This tree is split into more scenarios each hour as forecast horizon grows. Regulation prices were simulated with

5.2 Wind power in European balancing markets – State of the art

135

combined ARIMA and Markov processes as described in [169, 170]. Results showed that wind energy curtailment was reduced by approx. 75% in the refined model. Wind-Pumped Hydro coordination and congestion with Monte Carlo – Case: Portugal Another ESS operating approach was suggested by Castronuovo et al. (2004) in [19, 84] for a pumped hydro station (75% efficiency) in combination with a wind farm. Here the storage was used to maintain the output power of the combined wind-hydro (W-H) power plant within a given range. Two possible operational scenarios were investigated: (A) power limits in the entire scheduling period and (B) restricted output during peak hours (8–22 h). A 48 h planning period was chosen in order to optimize the ESS state of charge after 24 h. The optimum daily operational strategy was obtained from an hourly discretised optimization algorithm. Wind power forecast errors were supposed to be normally distributed and all W-H production was remunerated as wind energy following a fixed tariff scheme. Instead of market participation, an hourly profile with two feedin tariffs was considered. The stochastic characteristics of the wind generation were simulated with the Monte-Carlo approach and the algorithm tested for a hypothetical 12-MW wind farm and a 3-MW/24-MWh storage. Assuming Portuguese feed-in tariffs of 54 e/MWh (off-peak) and 103.84 e/MWh (peak), yearly average gains of 425–717 ke were obtained. Wind-Pumped Hydro coordination with probabilistic forecasts Case 1: probabilistic ESS sizing Pinson et al. (2007) suggested in [45] a method of defining advanced strategies for market participation based on probabilistic wind power forecasts. This method has been adopted in this thesis for analysing the Spanish regulating market (see section 5.4). Probabilistic forecasts provided predictive distributions in addition to the point forecast. In [45] imbalance prices were analysed and a loss function was formulated to describe asymmetries in up and down regulation penalties. Bids were calculated considering forecast error pdf and loss function. The regulating market was modelled with historical quarterly and annual mean values and revenues obtained from bids based on point forecasts were compared with the proposed advanced strategy. The case study was carried out for a multi-MW wind farm in the Dutch electricity pool. For 2002 mean penalty was 12 e/MWh for down regulation and 7 e/MWh for up regulation. It was shown that due to this difference it may be economically beneficial having more up regulation demand, even if the total imbalance is increased. Best results were obtained with quarterly mean prices, which shows that the proposed technique performs better if price estimates are better. Energy storage was added to this bidding strategy in [171]. Several scenarios of possible outcomes of wind generation were simulated, following a method proposed by the same group of authors [172]. The problem of the integrating behaviour of storage is tackled by analysing the interdependence structure of prediction errors. Generated outcomes were shown to respect the interdependence of forecast errors from different forecast steps ahead. Required storage capacity was calculated for each outcome and its statistical distribution was obtained. The quantiles of this distribution provide an information about the risk that in the next delivery period deviations cannot be compensated completely.

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Chapter 5. Uncertainty of wind power forecast and electricity markets

Note the similarity of this approach with the interpretation of cdf of SOC suggested in section 3.3, page 53 of this thesis. In the case study, one year of combined wind-hydro operation was simulated for a multi-MW wind farm in Denmark. Round-trip efficiency of the ESS (pumped hydro station) was assumed to be 75%. Nominal ESS power was set to 25% of nominal wind farm power. A cost of 1.5 e/MWh was assumed for charging the ESS. As price forecast, yearly means of a daily profile were assumed. Regulation prices were assumed to be proportional to spot prices. For each day ESS capacity was defined based on different scenarios, ranging from no storage to infinite storage, with different risk parameters in between (representing more or less storage requirement). It was concluded that energy storage should be rented in order to have variable storage capacity for every delivery period. In this case, storage may improve total revenue by 17% compared to the base scenario without storage. Maximum required ESS capacity (100% compensation of deviations) was found to be relatively low with 12% of total generation. Special issues of storage charge and discharge were treated in the PhD thesis of Kl¨ockl (2007) [173], one of the authors of the studies mentioned above. Wind-Pumped Hydro coordination with probabilistic forecasts Case 2: deterministic ESS sizing with risk index A similar approach to [45] was proposed by Bourry et al. [174]. The basis for decision making was formulated as in [45] by probabilistic wind forecasts and loss functions for imbalance penalties. The main difference consists in the measure of risk. It was considered that different attitudes toward risk may lead to different bid decisions, even with identical input data. The Conditional Value at Risk (CVaR) known from financial analysis and electricity markets was adopted and was obtained from the loss distribution (pdf of expected imbalance penalties). The optimal bid finally is such that a linear combination of mean loss and CVaR is minimised. For the case study, a simulation of the market was performed for one year (2002) for a 21-MW wind farm in Denmark. The forecast model was trained with data from the two preceding years. Results showed that in the reference case (without any bidding strategy), 86% of maximum revenue (zero imbalance penalty) can be achieved. When the mean imbalance prices from the year before were taken as forecast, an optimum was found for the risk attitude of the decision-maker. In this case regulation costs were reduced by 1.5%, corresponding to an improvement of total revenue of 0.24%. Interestingly, if the decision-maker is extremely risk-averse, imbalance costs may become worse than for the reference case (i.e. risk indifference). It was concluded that price forecasting is very important for obtaining benefits from the proposed method. The same group of authors included in [175] energy storage (pumped hydro) in the decision making process. In [174] decisions were independent in time, but if storage comes into play, decisions are coupled in time, also known as sequential decision problem. This means that possible decisions depend on the previous ones and condition later ones. This problem was solved by a dynamic programming approach in which the SOC of the ESS describes the state of the wind-hydro power plant at a given time step. The scheduling period was 24 h with the target SOC = 50% at the end of each day (starting at 50% the first day). Wind power was predicted with a probabilistic forecast model and spot prices with persistence. Results showed that revenues were almost unchanged by different scenarios and that ESS reduced deviations but did not increase revenues in this case.

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5.3 Statistical analysis of market prices in Spain

5.3

Statistical analysis of market prices in Spain

The first step towards probabilistic bidding strategies to increase revenues, is a statistical analysis of market prices. Statistical descriptions have been published for the Dutch [159], the Danish [45,77,153] and the German market [160,161]. For the Spanish market no such analysis has been done. Therefore, to complete the image, here a detailed analysis of prices on the Spanish electricity market MIBEL is presented. Marginal Prices and deviation prices are available since first of January 2007 from the web-based information system esios of the Spanish system operator Red El´ectrica de Espa˜ na (REE) [176]. Hourly values have been used for this study for the time span from 1-1-2007 until 31-3-2009, covering more than two years.

5.3.1

General description

For a global overview, in Fig. 5.2 marginal prices (PM) and regulation prices are represented in different time scales. On the left, one week starting from monday, 2 of February 2009 is represented. In this representation the hourly variability of market prices can be seen very well. Regulation prices are equal to marginal prices (PM) depending on systemwide deviations. Therefore, one of the two prices (up/down) always is equal to PM. No clear tendency for differences between weekend and working days can be observed in the representation. On the right side, weekly mean values of the entire dataset are depicted (from January 2007 till March 2009) and the long term tendency can be appreciated very well. Weekly mean prices

Prices [e/MWh]

Hourly mean prices 100

100

80

80

60

60

40

40

20

20

0

F02

F03

F04

F05

F06

F07

F08

0

PM up down

2007

2008

Figure 5.2. Marginal (PM) and regulation prices in the Spanish electricity market; hourly prices within one week in 2009 (left) and weekly means (right).

A remarkable increase of electricity prices can be seen for 2008. The mean marginal price stepped up from approximately 40 e/MWh to 60 e/MWh. In the first three months of 2009 prices are tending again to similar values observed in 2007. These price trends are similar to mineral oil prices in the same period. The annual mean spot market price per barrel of Brent Oil in 2007 was approximately 70$, rising in 2008 to almost 100$ and falling in the first three months of 2009 to less than 50$. This is an impressive demonstration of the dependence of the Spanish electricity market on energy imports. According to data from Eurostat [122], in 2006 approximately 54% of the electricity in Spain was generated from Natural Gas (30%) and Mineral Oil (24%). This share might

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Chapter 5. Uncertainty of wind power forecast and electricity markets

be similar for 2007 and 2008, so that it is not surprising to see such an impressive impact. Of course, there are many more factors influencing the market. This example has been given only for the demonstration of the importance of external factors. Therefore, market price estimations have to be done carefully. Note that on the Spanish market there are no negative regulation prices. Negative prices mean that the energy producer is actually paying for selling electricity on the regulating market or getting money for buying. Examples of markets where negative prices are possible are NordPool [153] in Scandinavia (although rare), the TenneT system of the Netherlands [159] or the balancing market operated by ELEXON in the UK [38]. For the evaluation of deviation costs, absolute regulation prices are of secondary interest. Therefore, the regulation cost πreg is introduced here. The regulation cost is obtained from the difference of the contracted spot market price πc (or marginal price) and the regulation price πud . Note that “cost” is used to express the price penalty, represented by the calculated difference. Prices are absolute values, while costs are price differences. πreg = πc − πud

(5.1)

where πc is the marginal price (spot price or market clearing price) and πud is the regulation price (up or down). To demonstrate the evolution of market prices over the last two years, in Fig. 5.3 hourly, weekly, monthly and quarterly mean values of πreg are represented. Regulation cost [e/MWh]

Hourly mean prices 250 200

Weekly mean prices 15

up down

10

150

5

100

0

50

-5

0

-10

-50

-15

-100

2007

2008

-20

up down

2007

Regulation cost [e/MWh]

Monthly mean prices 15 10

Quarterly mean prices 15

up down

10

5

5

0

0

-5

-5

-10

-10

-15

-15

-20

2008

2007

2008

-20

up down

2007

2008

Figure 5.3. Hourly, weekly, monthly and quarterly mean values of regulation costs in the Spanish electricity market between 1-1-2007 and 31-3-2009.

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5.3 Statistical analysis of market prices in Spain

It can be observed, that hourly variations of up regulation costs are larger than down regulation costs, although variations of weekly means are already lower. In Table 5.1 annual means of 2007, 2008 and the first quarter of 2009 are shown. Prices in e/MWh

2007

2008

2009∗

Marginal (spot) price Up regulation cost Down regulation cost

39.34 2.97 –7.99

64.47 4.04 –8.02

43.47 3.20 –8.35



only first quarter.

Table 5.1. Annual means of marginal electricity price and regulation costs for the Spanish market from 1-1-2007 until 31-3-2009.

From the figures presented before, a strong asymmetry is seen between up and down regulation costs in Spain. This asymmetry is important for probabilistic bidding strategies, as will be presented in section 5.4. To give an impression of this asymmetry, the asymmetry ratio ξ is introduced here. ξ=

−πdn πup − πdn

(5.2)

where πdn is the mean down regulation cost and πup is the mean up regulation cost. In Fig. 5.4 ξ of monthly mean regulation costs is depicted. Maximum values are obtained in January and February 2007, due to very low up regulation costs combined with high costs for down regulation. Almost equal regulation prices are observed in 2008 for the months May and December. Comparing months from one year to another, no general tendencies are observed. Monthly mean prices

Asymmetry ratio ξ

1 0.8 0.6 0.4 0.2 0

2007

2008

Figure 5.4. Monthly means of the asymmetry factor between up and down regulation costs in the Spanish electricity market between 1-1-2007 and 31-3-2009.

In order to get a better picture of the statistical behaviour of market prices, in Fig. 5.5 probability density (pdf) and cumulative distribution functions (cdf) are represented. The

Frequency [%]

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Chapter 5. Uncertainty of wind power forecast and electricity markets 5

100

4

80

3

60

2

40

1

20

0 0

20

40

60

80

100

0 0

up down PM 20

Prices [e/MWh]

40

60

80

100

Prices [e/MWh]

Figure 5.5. Probability density (left) and cumulative distribution (right) of marginal prices (PM) and deviation cost in the Spanish electricity market between 1-1-2007 and 31-3-2009.

pdf (left plot) is clearly not Gaussian. Distributions of regulation costs are skewed and bounded to the left (no negative prices) with a high spike at zero. In the cdf plot (on the right), probabilities of zero regulation cost can be seen very well. Up regulation cost is zero during 59% percent of the time, while down regulation cost is zero 48% of the time. This means that there is a considerable time of 7% where both are zero, assuming that always one of either values is zero. From this first observation it can be concluded that down regulation situations occur with a probability of 52% and thus are more frequent than up regulation situations which have a probability of only 41%. To provide some more parameters of the distributions, the first four moments (mean, standard deviation, skewness and kurtosis) of marginal prices and regulation costs are shown in Table 5.2. Note that standard deviations are normalised by the mean values.

Marginal price Up regulation cost Down regulation cost

Mean

Std. Dev. [p.u.]

Skewness

Kurtosis

50.93 3.48 –8.04

0.35 2.06 1.30

0.30 5.83 1.35

2.35 86.07 4.51

Table 5.2. The first four moments of marginal prices and regulation costs for the Spanish market from 1-1-2007 until 31-3-2009.

Total mean values show that up regulation cost is less than half of down regulation cost. This shows the tendency of the market to penalise more those deviations which occur more often. It can also be seen that mean values of regulation costs do not exceed 15% of the mean marginal price. On the other side, standard deviations beyond unity and in case of up regulation even beyond 2, show the great variability of regulation costs. Marginal prices in comparison are much less variable. The higher moments Skewness and Kurtosis show in addition the asymmetry and peakedness of regulation costs in contrast to marginal prices.

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5.3 Statistical analysis of market prices in Spain

5.3.2

Daily and weekly price profiles

The importance of daily profiles is emphasised in the literature on price forecasting [160]. It is widely known that electricity consumption follows daily profiles. In addition significant differences can be detected between weekend and working days. As spot market prices are correlated with consumption (see section 5.3.3), similar profiles may be expected for prices. Therefore in this section a short analysis of this subject is presented for the Spanish electricity market. Results are valuable information for optimised bidding strategies as will be shown in section 5.4. In figures 5.6 and 5.7 daily profiles of market prices are shown. In Fig. 5.6 annual mean hourly values are calculated for one week (left) and one day (right), representing typical weekly and daily profiles. In Fig. 5.7 it can be seen that regulation costs do not show significant differences between weekend and working days. Therefore, it seems justified to calculate only one daily profile for 24 h, as shown in Fig. 5.7 (right). Comparing years 2007 and 2008, first the great similarity can be highlighted. Annual means in Table 5.1 already indicated, that the sharp increase in marginal prices were not Mean week - 2007

Mean day - 2007 PM up down

πc [e/MWh]

80 60

60

40

40

20

20

0 0

1

2

3

4

5

6

PM up down

80

7

0 0

6

12

18

24

Hours of day

Days of week

Figure 5.6. Daily price profiles on the Spanish electricity market for the year 2007; annual mean week (left) and mean day (right).

πreg [e/MWh]

Mean week

Mean day

20

20

10

10

0

0

-10

-10

-20

0

1

2

3

4

Days of week

5

6

7

-20

0

up2007 down2007 up2008 down2008

6

12

18

24

Hours of day

Figure 5.7. Daily regulation cost profile on the Spanish electricity market for the years 2007 and 2008; annual mean week (left) and mean day (right).

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Chapter 5. Uncertainty of wind power forecast and electricity markets

πc [e/MWh]

Mean day - 2007

Mean day - 2008

80

PM 80 mar-may

60

jun-aug 60 sep-nov dec-feb

40

40

20

20

0 0

6

12

18

24

πreg [e/MWh]

10

0 0

-10

10 up 5down mar-may 0jun-aug -5sep-nov dec-feb -10

-15

-15

-20

-20

5 0 -5

0

6

12

Hours of day

18

24

0

PM mar-may jun-aug sep-nov dec-feb

6

12

18

24 up down mar-may jun-aug sep-nov dec-feb

6

12

18

24

Hours of day

Figure 5.8. Annual mean and seasonal variability of hourly prices of a mean day for marginal spot market price (above) and regulation cost (below) for 2007 (left) and 2008 (right).

reflected by the same extent in regulation costs. The daily profile changed only slightly. In 2008, the up regulation cost profile is more equilibrated: 3–6 e/MWh compared to 1–5 e/MWh in 2007. Especially the very low costs during off peak hours disappeared, which leads to a significant overall increase of mean costs from 3 to 4 e/MWh. The profile of down regulation costs remained almost the same. Only the daily minimum increased slightly from approximately 3.50 to 5 e/MWh. In general, annual profiles of regulation costs were more equilibrated in 2008. In Fig. 5.8 the seasonal variability of daily profiles is illustrated for the years 2007 (left column) and 2008 (right column). Means of three months were calculated to obtain an image of every season of the year. Large differences in marginal prices can be observed especially for peak demand hours (see upper plots). In spring (March to May) lowest peak hour prices are detected (around 40 e/MWh in 2007 and 70 e/MWh in 2008). In winter (December to February) the mean peak price reaches around 65 e/MWh in 2007 and 85 e/MWh in 2008. As explained before, marginal prices are strongly influenced by external factors like oil prices, which may have influences on seasonal behaviour. In the lower part of Fig. 5.8, regulation costs are represented. First it is interesting to note that the up regulation cost exhibits almost no seasonality. This observation can be made in 2007 and 2008 equally. On the other side, down regulation costs are strongly influenced, although the effect is less pronounced in 2008. In spring and summer, down regulation costs tend to be lower and less variable with values between −4 e/MWh at off peak and −11 e/MWh at peak hours. In winter the highest three-month mean can be found between 19–20 h with −22 e/MWh, while values at off peak hours do not differ significantly from other seasons. Therefore, it can be concluded that bidding strategies should include the seasonal variation of regulation costs.

143

5.3 Statistical analysis of market prices in Spain

5.3.3

The impact of wind power on market prices in Spain

When wind power participation in liberalised markets is analysed, in general it is assumed that it has no market power to influence prices. In the literature, bidding models assume that bidding of wind power producers has no significant effect on market prices. So when the participation of wind power in the liberalised market is discussed, it is treated as a “price taker”. Nevertheless, very recently it has been shown that in the case of Denmark, where wind penetration is very high, this assumption no longer holds [77]. For this reason, data available from the web based information service e-sios [176] of the Spanish electricity system operator REE was analysed. Data of wind power forecast and demand forecast for a time period from January 2007 until March 2009 were used to calculate correlation coefficients, similar to the procedure presented in [77]. Results from [77] (“Denmark 2006”) and results from the present study are shown in Table 5.3. In the first column, correlation coefficients are shown between the system-wide demand Pd and market prices. The second column contains correlation coefficients between wind power generation Pw and market prices. The last column shows price correlations with the difference of demand and wind power Pd − Pw , treating wind generation as negative demand. Demand Pd

Wind power Pw

Pd − Pw

Denmark 2006

0.55

–0.30

0.68

Spain Spain Spain Spain

0.72 0.64 0.61 0.50

–0.20 –0.24 –0.19 –0.08

0.76 0.71 0.68 0.52

0.64 0.59 0.55 0.47

–0.25 –0.27 –0.17 –0.11

0.70 0.67 0.61 0.50

0.52 0.45 0.46 0.37

–0.40 –0.36 –0.29 –0.19

0.62 0.57 0.57 0.43

Data Spot price

2007 2008 2009* 2007–09

Up regulation price Spain Spain Spain Spain

2007 2008 2009* 2007–09

Down regulation price Spain Spain Spain Spain ∗

2007 2008 2009* 2007–09

only first quarter.

Table 5.3. Correlation coefficients for different combinations of wind generation and demand with market prices in Denmark and Spain.

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Chapter 5. Uncertainty of wind power forecast and electricity markets

Correlation of spot market prices versus Pd , Pw and Pd − Pw First, the correlation coefficient between marginal spot market prices and electricity demand was calculated. In 2007 with 0.72 a high correlation can be found: during high demand periods, prices tend to be high. In the following years, this value decreases down to 0.61 for the first three months of 2009. If the whole dataset from 2007 until March 2009 is used, even lower correlation as reported in [77] (Denmark 2006) is detected. Every year separately exhibits higher correlation factors than the entire time series. This is a result of long term variations induced by external factors such as oil prices, as described in section 5.3.1. Secondly, the correlation between market prices and wind power production was calculated. Here negative and rather low values are obtained which indicates a tendency for spot market prices going down when wind generation is going up. No trend can be seen comparing individual years. In general slightly lower values are observed compared to Denmark 2006. But taking the entire dataset, correlation disappears completely. Daily patterns of wind speed may create coincidental correlations. If for example higher wind speeds are usual during off-peak hours, a negative correlation coefficient could be observed without any causality between them. In Fig. 5.9 daily profiles of wind power are shown, based on hourly wind power forecasts published by REE in [176]. Yearly means and seasonal profiles are depicted. First, an overall increase of ca. 0.5 GW can be seen, most probably due to the growing installed power in Spain. In winter no daily profile is observed but the rest of the year there is a clear minimum between 10–12 h and a maximum between 17–21 h. Especially in summer there is a strong pattern, possibly caused by thunderstorms which develop typically in the evening. The characteristic mean day between June and August had a minimum wind power of around 2 GW between 10–12 h and a maximum of more than 3 GW between 20–21 h. Comparing the wind profile with market prices, there is indeed some tendency for higher wind power during off-peak hours. The minimum of the wind power profile is near to the first price peak, which occurs between 12–13 h. Finally the maximum of the wind profile is very close to the evening peak of market prices at 21–22 h. Thus, it is not easy to determine how much of the observed correlation is due to coincidence and how much is caused by the impact of wind power on market prices.

Wind power [GW]

Mean day - 2007

Mean day - 2008

5

5

4

4

3

3

2 1 0 0

2

Mean mar-may jun-aug sep-nov dec-feb 6

12

Hours of day

18

Mean mar-may jun-aug sep-nov dec-feb

1

24

0 0

6

12

18

24

Hours of day

Figure 5.9. Daily wind generation profiles in Spain for the years 2007 (left) and 2008 (right).

5.3 Statistical analysis of market prices in Spain

145

Finally, wind power is treated as negative demand and the difference between wind power and demand was compared with market prices. In all cases higher correlations are observed than for the case of demand without wind power. The trend from the first column is maintained. Again, for 2007 with 0.76 the highest correlation coefficient is observed and for the entire dataset the lowest one. As a conclusion, the impact of wind power on spot market prices in Spain is still limited and further investigation is needed to separate coincident correlations from causalities. In case of Denmark, the increase of the correlation coefficient is larger. This can be interpreted as a sign that wind power has more impact on spot prices, due to its higher penetration in the danish power system. Correlation of regulation market prices versus Pd , Pw and Pd − Pw In [77] only spot market prices were analysed. Here in addition regulation prices were correlated applying the same procedure. Results are shown in the lower part of Table 5.3. A general observation is that the correlation between regulation prices and electricity demand is lower than that observed with marginal prices (compare middle and lower part of the first column with the upper part). Up regulation prices are slightly stronger correlated with demand than down regulation prices. Comparing values in the second column reveals that wind power shows increased correlations for both, up and down regulation. Especially for down regulation prices (energy excess in the system), there can be observed a significant increase of negative correlation compared with values from spot market prices. For up regulation prices no significant influence is observed and for the first 3 months of 2009, even a slight decrease is observed. Looking at the values in the last column (wind power as negative demand) the tendency is again analogue to spot prices. In all cases higher correlations are obtained. The increase is slightly higher for down regulation prices. In 2008 the increase from 0.45 to 0.57 is the highest observed difference between correlation coefficients with and without wind power. Zero down regulation price – Case study: 1st May 2007 As shown in Fig. 5.8, imbalance prices in the Spanish electricity market exhibit strong seasonal variations. While the up regulation cost is almost independent from the season of the year, down regulation costs show significant differences between summer and winter. One possible explanation may be the elevated wind energy available in winter and spring. As shown in Fig. 5.9 highest wind generation can be expected in the months between March and May. In this season it is more frequent to observe situations, when the down regulation price (not the cost!) becomes zero. This may occur when high wind generation comes together with low demand. Such a situation occurred on 1st of May 2007. An additional circumstance was that low system demand (it is a public holiday in Spain) came together with a rapid increase of wind generation in the evening. In Fig. 5.10 time series of hourly wind power generation (left plot) and market prices (right plot) are depicted for this day. In only 7 hours (7–14 h), wind generation (bold black line) doubled from 2500 MW to 5000 MW, reaching at around 16 h a national wide penetration of 25% of the total electricity demand. System deviation

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Chapter 5. Uncertainty of wind power forecast and electricity markets

(bold grey), wind forecast (thin black) and forecast error (thin grey) are included in the same figure. It can be observed that during the wind generation peak, an underestimation of wind power took place. It is likely that at 25% penetration this slight underprediction contributed to down regulation price zero from 11–17 h, as a large unexpected amount of surplus energy was available at this moment. Hourly means of wind forecast error switched from positive to negative at 11 h, exactly when down regulation price dropped to zero. At 17 h, when the forecast error switched back to positive values, down regulation prices started to approach again the spot price. 10

100

Power [GW]

8 6

Price [e/MWh]

Pred. Gen. Error Syst.

4 2 0 -2

0

3

6

9

12

15

Hours of day

18

21

24

PM Up Down

80 60 40 20 0

0

3

6

9

12

15

18

21

24

Hours of day

Figure 5.10. Wind generation (left) and market prices (right) on 1st May 2007.

A strong influence of wind power deviations (forecast errors) on system deviation can be seen in general this day. Without wind energy, system deviations would have been negative at a rather constant value oscillating around –700 MW. The mean contribution of the deviation during the six-hour time interval 11–17 h is with 36% significantly higher than the total share of wind generation of 23% at the same time. On the other hand, that same day, positive wind forecast errors (overestimation) are contributing to reduce system deviations during 9 hours. As a result, for the Spanish market it can be concluded, that wind generation still has a reduced influence on the marginal (spot market) prices but it has already an impact on regulation prices. Price forecast models should account for this impact in order to obtain a better bidding strategy for wind energy. In this thesis, no dependence of market prices and wind generation is considered, as no price forecasting is included.

5.4 Optimised bidding strategy for wind energy

5.4

147

Optimised bidding strategy for wind energy

The purpose of this section is the application of a probabilistic bidding strategy, in order to quantify the cost of forecast uncertainty. In liberalised electricity markets, (wind) energy producers can sell their energy directly by proposing a bid in an auction of an energy exchange. The aim of the bidding strategy is to maximize revenues taking into account market fluctuations and forecast uncertainty of wind generation. Estimated pdf of wind forecast errors from chapter 2 are used together with different market price forecast scenarios, derived from section 5.3. The main contribution of the thesis in this chapter consists in the detailed analysis of the impact of different price forecast scenarios on revenue improvement and imbalances for market prices in Spain. From the study of literature it is concluded that the revenue from wind power production can be improved by applying an adequate bidding strategy. Best results are obtained if probabilistic forecasts for wind power and imbalance prices are available. This means that the uncertainty of the forecasts is known. While for wind forecasting in recent years probabilistic forecasts have been evolved and are state of the art, price forecasting for regulating markets is still a field of investigation. In this study no price forecasting is done, but several scenarios are developed which simulate different types of possible forecast results. For wind power, probabilistic forecasts are considered, applying the Betafit pdf estimation developed in chapter 2. The approach in [45] for bidding with probabilistic forecasts appears to be the most suitable and is adopted here. With estimated mean values of imbalance costs, loss functions are created. Then, with the pdf of wind power forecast errors, the optimal bid is obtained.

5.4.1

Formulation of the problem

It is assumed that the market participant places a bid for every hour of the day in one single auction (no intra-day market is simulated), which requires a 24-h forecast. From the persistence model a single mean wind power is obtained per day. Assuming a market closure delay of 12 h, real world forecasts have to be available 12 h before delivery, thus the forecast horizon is 12–36 h. For dataset C forecasts are available at 7 AM, so that the 24-h forecast window is in fact a 17–41 h ahead forecast. In order to simplify the formulation of the problem, market closure delay k is not explicitly included. It is assumed that for every hour t, a forecast pˆt and its pdf fp,t are available. Further, normalised wind power pt , spot price πc,t and regulation cost πreg,t are known. For more simplicity, in the following, variables will be written without the index t. The bidding time step ∆t (also denoted as program time unit PTU) is set to 1 h. Thus, for one year a time series of 8760 steps (365 days) is obtained. The deviation ED from contracted energy can be defined as in (5.3). It is convenient to work with normalised variables. Thus normalised energy deviation eD is introduced in (5.4). ED = (p − pˆ) · ∆t · Pinst = Eg − Ec

(5.3)

eD =

(5.4)

eD =ε ∆t Pinst

where ED is the deviation from contracted energy [MWh], p and pˆ are mean power and forecast in [p.u.], ∆t bidding time step [h], Pinst the installed wind power in [MW], Eg

148

Chapter 5. Uncertainty of wind power forecast and electricity markets

and Ec generated and contracted energy [MWh] and ε is the forecast error [p.u.]. Normalization is done with ∆t Pinst . As a result, deviation eD adopts the same value as the forecast error ε (see definition (2.5) on p. 18), but note that eD is an energy quantity and ε is a power. Revenue R, for every time step can be written as R = Eg πc − |ED πreg |

(5.5)

where Eg is the actually generated energy [MWh], πc the spot price, ED the deviation from contracted energy [MWh] and πreg is the regulation cost. Further, as suggested in [45] a performance ratio PR is defined in (5.6), which makes it easier to compare different bidding strategies. It is the ratio between the revenue R obtained with a given strategy and the revenue obtained with perfect forecast (no deviations are paid). This ratio may be expressed in p.u. or %. P P Rt |εt πreg,t | t t PR = P =1− P (5.6) Eg,t πc,t pt πc,t t

t

where for every time step t, Rt [e] is the revenue, Eg,t [MWh] is the generated energy, πc,t the spot price, εt the forecast error [p.u.], πreg,t the regulation cost and pt [p.u.] the actually generated power. The loss function

Pinson (2007) [45] proposes the application of a loss function based on the regulation unit cost. Note the difference between regulation cost πreg and regulation unit costs π + and π − (see definitions below). If eD is the deviation from the contracted energy at any time instant and πreg is the regulation cost as defined in (5.1), then the loss function can be formulated as follows. g(eD ) = |eD πreg |

(5.7)

In order to account for a possible asymmetry between mean up and down regulation costs, it is convenient to write the loss function as proposed in [45]: ( |eD | · π + , eD > 0 (down regulation) (5.8) g(eD ) = |eD | · π − , eD < 0 (up regulation) ( with π + = −πreg , eD > 0 (down regulation) (5.9) π − = +πreg , eD < 0 (up regulation) where eD [p.u] is the deviation from contracted energy, π + the down regulation unit cost and π − the up regulation unit cost, both in e/MWh. Note that π + and π − are both positive, in contrast to πreg . Symbols π + and π − for regulation unit costs are adopted to maintain the analogy to the method suggested in [45], where the index ‘+’ refers to positive and ‘−’ to negative deviations. This might be somewhat misleading about the actual sign of the underlying regulation cost πreg . In

149

5.4 Optimised bidding strategy for wind energy

fact, πreg is negative for π + and positive for π − . In Fig. 5.11 loss functions are shown for the Spanish electricity market, based on yearly mean values of the regulation unit costs from 2007–2009. 10

g [e/MWh]

8 6

up2007 up2008 up2009 down2007 down2008 down2009

4 2 0 -1

-0.5

0

0.5

1

Deviation eD [p.u.] Figure 5.11. Loss function for the Spanish electricity market, based on mean values of regulation unit costs for the years 2007, 2008 and first quarter of 2009.

The curve for 2009 only represents mean values of the first three months, but already shows very similar behaviour compared with preceding years. While up regulation unit costs π − (continuous lines) show variations between 3 and 4 e/MWh, down regulation unit costs π + (dashed lines) have been very stable at 8 e/MWh during the observation period of over two years. Note the pronounced asymmetry: regulation unit costs (or penalties) for down regulation are approximately two times higher than for up regulation. Remember that on the Spanish market, the regulation cost is equal to the spot price if the deviation is contrary to the system imbalance. Or in other words, if the imbalance helps to balance the system, the imbalance penalty is zero. These zero-penalties contribute to to the mean values of π + and π − . Following the method proposed by [45], with loss functions and forecast power cdf Fp optimal bids Ec,opt can be obtained with (5.10) for every hour of the year. −1 Ec,opt (t) = Fp,t (αopt,t )

with

αopt,t =

πt+ πt+ + πt−

(5.10)

−1 (·) is the inverse cumulative where Ec,opt is the optimal bid [MWh] for hour t and Fp,t distribution function (cdf) of the power forecast pˆt and αopt,t is the optimal quantity (probability) depending on regulation unit costs πt+ and πt− .

For a proof of the equation for αopt see Bremnes (2004) [46]. If the loss function is symmetric, αopt is 0.5, i.e. the 50% percentile is the best bid. If positive deviations are more penalised than negative ones (π + > π − ), the best bid will be at higher percentiles. Bidding higher production values makes overprediction more probable, which is penalised less in this case. This is a straightforward calculation which can be done for each time step, if the uncertainty of the forecast is given by predictive pdf. Here the value of probabilistic forecasts is evident. If a price forecast model is used, αopt is also a function of time, which is symbolised with the index t in (5.10). But as will be shown, even annual mean values of regulation unit costs (constant αopt over the entire simulation period) yields considerable improvements in terms of performance ratio PR.

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Chapter 5. Uncertainty of wind power forecast and electricity markets

The reference case

To create a reference, in a first step imbalance costs are calculated assuming bidding based on point forecasts. This would be the strategy if no information is available about the loss function or the pdf of the forecast error. For this study, the following forecast scenarios are considered: 1) Persistence (T×1) – worst case simulation for datasets A, B and C 2) Real world forecast MSEc (bias corrected) 3) Real world forecast MCCc (bias corrected) 4) Real world forecast MSE (original data) 5) Real world forecast MCC (original data) The simulation time interval is one year. For all datasets per unit power values are used and a normalised installed power of 1 MW is considered to calculate total generated and contracted energy quantities Etotal and Ec , given in GWh/a. Note that thereby values 1 in GWh/a are representing 1000 of equivalent hours of generation. Real world forecasts are available only for dataset C. From dataset A, the time series of an individual turbine is used. In this series ca. 13% of data points are missing, but time stamps permitted the identification of the exact moment of generation. Deviations and costs are calculated for all existing points and it is assumed the result is representative for the whole year. For a description of the datasets see section 2.3. Parametric forecast error distributions (pdf) are estimated with the Betafit algorithm as presented in section 2.6 of this thesis. Results for the reference case are presented in Table 5.4 for 2007 and in Table 5.5 for 2008. In the first column, the performance ratio PR of the bidding strategy is given. It is obtained from the quotient between reference and observed revenue per MWh, given in the second and third column. The reference price πref is obtained dividing the revenue if energy is sold for the spot price (without penalization) by total generated energy Etotal . This is actually a weighted mean, with energy as weighting factor. Therefore πref depends on the wind power time series. The mean spot price in 2007 for example was π ¯ c = 39.34 e/MWh and for 2008 is was 64.47 e/MWh. For 2007 πref results slightly above and for 2008 slightly below π ¯c . The observed price πobs is the weighted mean revenue obtained including regulation costs. Columns 4 to 6 show the quantity of regulation energy needed (up, down and total), as a percentage of total generated energy Etotal , given in column 7. Note that total Ereg (column 6) is equivalent to the mean absolute percentage error (MAPE). The last column contains the sum of contracted energy Ec . The difference between the last two columns represents the mean forecast error. For bias free predictions Etotal and Ec are equal. Performance ratios PR are similar for all scenarios. Note that the case with lowest MAPE (MCC: 63.7%) generates one of the lowest performance ratios with only 88.2%. This is a result of the strong bias towards underprediction, producing a large demand for down regulation energy. Due to the higher down regulation cost, the very low need for up regulation cannot compensate the high costs for down regulation.

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5.4 Optimised bidding strategy for wind energy

Note that the performance ratio PR for bias corrected scenarios MSEc and MCCc are slightly improved, although in case of MCC total Ereg (MAPE) is lower for original forecast data. Here the more equilibrated error distribution has economical benefits. Results in Table 5.4 show that persistence forecasts and bias corrected MSEc and MCCc are indeed bias free, thus total produced energy Etotal and contracted energy Ec are almost equal. The MSE forecast shows a slight overweight in overprediction (leading to more up than down-regulation) while the MCC forecast shows a strong underprediction. For comparison, in Table 5.5 the reference case is shown for market data from 2008. The different price configuration causes some differences in the results. Marginal electricity prices in 2008 were 50% higher than 2007. At the same time, regulation costs did not increase at the same rate. Therefore performance ratios are approximately 4 percentage points higher, although deviations did not change. Price [e/MWh]

Ereg [%]

[GWh/a]

Data

PR [%]

πref

πobs

Up

Down

total

Etotal

Ec

A T×1 B T×1 C T×1

89.0 89.4 88.8

40.50 39.74 39.56

36.05 35.54 35.12

37.2 35.8 37.4

38.6 36.7 38.0

75.8 72.5 75.4

2.20 2.68 2.28

2.23 2.65 2.27

C C C C

90.0 89.8 89.6 88.2

39.56 39.56 39.56 39.56

35.60 35.54 35.47 34.89

33.8 33.2 36.0 17.7

33.7 34.0 33.3 46.0

67.5 67.2 69.3 63.7

2.28 2.28 2.28 2.28

2.29 2.27 2.35 1.64

MSEc MCCc MSE MCC

Table 5.4. Bidding with point forecast; simulation results for 2007 Spanish energy market prices. Wind power data from 3 different sites (A, B and C) and 3 different forecast methods (persistence Tx1 and two real world forecasts).

Price [e/MWh]

Ereg [%]

[GWh/a]

Data

PR [%]

πref

πobs

Up

Down

total

Etotal

Ec

A T×1 B T×1 C T×1

93.0 92.9 92.2

63.17 63.99 63.52

58.72 59.47 58.58

37.2 35.8 37.4

38.6 36.7 38.0

75.8 72.5 75.4

2.20 2.68 2.28

2.23 2.65 2.27

C C C C

93.8 93.6 93.7 93.2

63.52 63.52 63.52 63.52

59.57 59.48 59.51 59.17

33.8 33.2 36.0 17.7

33.7 34.0 33.3 46.0

67.5 67.2 69.3 63.7

2.28 2.28 2.28 2.28

2.29 2.27 2.35 1.64

MSEc MCCc MSE MCC

Table 5.5. Bidding reference case for 2008 Spanish energy market.

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5.4.3

Chapter 5. Uncertainty of wind power forecast and electricity markets

Bidding with annual and monthly means

In 2007 the mean values of π + and π − in Spain were 3 and 8 e/MWh, resulting in a mean αopt = 0.723 or 72.3%. In section 5.3 it was shown that deviation costs have a great seasonal variability. Here, in order to evaluate the impact of a bidding strategy with price forecast, perfect forecasts of the annual and monthly mean prices are assumed. Results are shown in Table 5.6. Improvements of PR for all scenarios are ca. 1 percentage point (p.p.) for the annual mean and 2 p.p. for the monthly means. As perfect forecasts are unlikely, achievable improvements will be below these values. Note the increased amount of regulation energy Ereg , when these strategies are applied. Results for 2008 are shown in appendix D.1. Price [e/MWh] Data

Ereg [%]

[GWh/a]

PR [%]

πref

πobs

Up

Down

total

Etotal

Ec

A T×1 A T×1 1a A T×1 30d

89.0 89.8 90.8

40.50 40.50 40.50

36.05 36.35 36.76

37.2 65.4 67.4

38.6 24.6 24.6

75.8 90.0 91.9

2.20 2.20 2.20

2.23 3.18 3.23

B T×1 B T×1 1a B T×1 30d

89.4 90.4 91.0

39.74 39.74 39.74

35.54 35.94 36.14

35.8 64.4 64.9

36.7 22.3 23.6

72.5 86.7 88.5

2.68 2.68 2.68

2.65 3.81 3.78

C T×1 C T×1 1a C T×1 30d

88.8 89.5 90.3

39.56 39.56 39.56

35.12 35.39 35.73

37.4 69.6 70.3

38.0 23.2 24.4

75.4 92.8 94.7

2.28 2.28 2.28

2.27 3.34 3.33

C MSEc C MSEc 1a C MSEc 30d

90.0 90.7 91.4

39.56 39.56 39.56

35.60 35.89 36.14

33.8 60.8 61.7

33.7 19.8 20.6

67.5 80.5 82.3

2.28 2.28 2.28

2.29 3.22 3.22

C MCCc C MCCc 1a C MCCc 30d

89.8 90.6 91.2

39.56 39.56 39.56

35.54 35.83 36.10

33.2 59.7 60.7

34.0 20.4 21.1

67.2 80.1 81.8

2.28 2.28 2.28

2.27 3.18 3.19

C MSE C MSE 1a C MSE 30d

89.6 90.5 91.2

39.56 39.56 39.56

35.47 35.80 36.08

36.0 64.5 64.8

33.3 18.5 19.4

69.3 83.1 84.3

2.28 2.28 2.28

2.35 3.34 3.32

C MCC C MCC 1a C MCC 30d

88.2 90.0 90.6

39.56 39.56 39.56

34.89 35.60 35.83

17.7 33.1 33.8

46.0 31.5 32.1

63.7 64.6 65.9

2.28 2.28 2.28

1.64 2.32 2.32

Table 5.6. Bidding with point forecast and uncertainty, using mean penalty of one year (1a) and one month (30d); simulation results for 2007 Spanish energy market. Wind power data from 3 different sites (A, B and C) and 3 forecast methods (Tx1, MSE, MCC).

153

5.4 Optimised bidding strategy for wind energy

5.4.4

Bidding with daily profiles

To improve the bidding strategy, daily profiles of deviation costs as presented in section 5.3.2 are considered. Perfect forecast of annual (‘1a dp’) and seasonal daily profiles (‘3m dp’) have been assumed in this simulation run. Results in Table 5.7 show that annual daily profiles almost show the same performance as monthly means. This is very interesting, as annual means are more reliable, since they change only slowly from one year to another. As shown in section 5.3, monthly means are varying considerably from one year to another and thus, are more difficult to predict. Results from 2008 are shown in appendix D.2.

Price [e/MWh] Data

Ereg [%]

[GWh/a]

PR [%]

πref

πobs

Up

Down

total

Etotal

Ec

A T×1 A T×1 1a dp A T×1 3m dp

89.0 90.3 91.1

40.50 40.50 40.50

36.05 36.57 36.91

37.2 67.4 64.2

38.6 25.0 26.5

75.8 92.4 90.6

2.20 2.20 2.20

2.23 3.22 3.11

B T×1 B T×1 1a dp B T×1 3m dp

89.4 91.0 91.4

39.74 39.74 39.74

35.54 36.16 36.32

35.8 65.4 64.0

36.7 22.3 24.3

72.5 87.7 88.4

2.68 2.68 2.68

2.65 3.83 3.74

C T×1 C T×1 1a dp C T×1 3m dp

88.8 90.0 91.0

39.56 39.56 39.56

35.12 35.60 36.00

37.4 71.2 69.1

38.0 24.4 24.8

75.4 95.6 93.9

2.28 2.28 2.28

2.27 3.35 3.30

C MSEc C MSEc 1a dp C MSEc 3m dp

90.0 91.1 91.6

39.56 39.56 39.56

35.60 36.03 36.26

33.8 62.3 61.9

33.7 20.7 21.5

67.5 83.0 83.3

2.28 2.28 2.28

2.29 3.23 3.21

C MCCc C MCCc 1a dp C MCCc 3m dp

89.8 91.0 91.6

39.56 39.56 39.56

35.54 36.01 36.24

33.2 61.0 60.7

34.0 21.3 22.0

67.2 82.3 82.7

2.28 2.28 2.28

2.27 3.19 3.17

C MSE C MSE 1a dp C MSE 3m dp

89.6 90.8 91.7

39.56 39.56 39.56

35.47 35.94 36.28

36.0 66.4 64.2

33.3 19.8 20.0

69.3 86.2 84.2

2.28 2.28 2.28

2.35 3.35 3.30

C MCC C MCC 1a dp C MCC 3m dp

88.2 90.4 91.1

39.56 39.56 39.56

34.89 35.76 36.04

17.7 34.4 33.7

46.0 32.0 32.2

63.7 66.4 65.8

2.28 2.28 2.28

1.64 2.34 2.32

Table 5.7. Advanced bidding, using daily profiles from one year (1a dp) and seasonal variation (3m dp); simulation results for 2007 Spanish market.

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5.4.5

Chapter 5. Uncertainty of wind power forecast and electricity markets

Total vs. penalised deviations

As seen in the preceding sections, advanced bidding strategies are increasing required regulation energy, so that a conflict between economical and technical interests arises here. To examine this issue more in detail, a new parameter is introduced: the penalised deviation Epen , which represents the penalised regulation energy in % of Etotal . This indicator informs about imbalances which are increasing the system imbalance. Hence it can be used as an additional criterion to find the optimal bidding strategy, as only the penalised imbalance represents a negative impact on the system. A good strategy should do both: increase the revenue and minimise penalised deviations. In tables 5.8 and 5.9 PR and Ereg are represented together with the new parameter Epen . Compared to the reference case (first line of each scenario), Epen is slightly higher when annual mean prices (‘1a’) are used. Introducing the annual daily profile (‘1a dp’) reduces Epen by approximately 1 p.p. but still lies above the reference case. Improving the bidding strategies, Epen is further reduced, but only if daily profiles with seasonal variation (‘3m dp’) are applied Epen becomes less than in the reference case. A general exception is forecast scenario MCC (Table 5.9, last scenario). Here with all advanced bidding strategies lower quantities of Epen are achieved. This effect is due to the strong bias of the MCC forecast, as discussed before. Note that bias correction (see MCCc) reduces Epen in the reference case, but the lowest value (28.6%) is obtained with original MCC data and seasonal daily profiles. For MCC the proposed bidding strategy is equilibrating up and down regulation energies. In all other cases, the same strategy leads to an overweight of up regulation. The reason why this does not happen for MCC forecasts may be the limited approximation accuracy of the pdf with Betafit (see section 2.6). A better approximation of the pdf would most likely lead to higher performance ratios for MCC. Ereg [%] Data A A A A A

T×1 T×1 T×1 T×1 T×1

B B B B B

T×1 T×1 T×1 T×1 T×1

Epen [%]

PR [%]

Up

Down

total

Up

Down

total

1a 1a dp 30d 3m dp

89.0 89.8 90.3 90.8 91.1

37.2 65.4 67.4 67.4 64.2

38.6 24.6 25.0 24.6 26.5

75.8 90.0 92.4 91.9 90.6

14.2 24.6 23.7 22.7 20.6

20.6 12.9 12.4 11.8 12.3

34.8 37.5 36.1 34.5 32.9

1a 1a dp 30d 3m dp

89.4 90.4 91.0 91.0 91.4

35.8 64.4 65.4 64.9 64.0

36.7 22.3 22.3 23.6 24.3

72.5 86.7 87.7 88.5 88.4

12.8 23.0 21.9 21.6 20.1

20.2 12.2 11.5 12.1 11.8

33.0 35.2 33.5 33.7 31.9

Table 5.8. Total and penalised up and down regulation as percent of total generation; simulation results for 2007 Spanish energy market (datasets A and B).

155

5.4 Optimised bidding strategy for wind energy

The penalised deviation has been represented divided in up and down regulation. For the reference cases it can be observed that even with unbiased forecasts, down regulation energy has a higher volume than up regulation. This reaffirms the observation from section 5.3 that in the Spanish electricity market down regulation is more frequent and more penalised. Hence, it is logical that the advanced strategy is inverting the situation towards more up regulation energy. Results from 2008 are shown in appendix D.3.

Ereg [%] Data

Epen [%]

PR [%]

Up

Down

total

Up

Down

total

88.8 89.5 90.0 90.3 91.0

37.4 69.6 71.2 70.3 69.1

38.0 23.2 24.4 24.4 24.8

75.4 92.8 95.6 94.7 93.9

13.1 24.9 23.7 22.6 21.1

21.6 13.5 13.1 13.4 12.6

34.7 38.3 36.8 36.0 33.7

1a 1a dp 30d 3m dp

90.0 90.7 91.1 91.4 91.6

33.8 60.8 62.3 61.7 61.9

33.7 19.8 20.7 20.6 21.5

67.5 80.5 83.0 82.3 83.3

12.8 22.6 22.0 20.7 20.1

19.7 11.7 11.3 11.8 11.4

32.6 34.3 33.3 32.5 31.4

1a 1a dp 30d 3m dp

89.8 90.6 91.0 91.2 91.6

33.2 59.7 61.0 60.7 60.7

34.0 20.4 21.3 21.1 22.0

67.2 80.1 82.3 81.8 82.7

12.5 22.1 21.3 20.2 19.5

20.0 12.2 11.7 12.1 11.7

32.4 34.3 33.1 32.4 31.2

1a 1a dp 30d 3m dp

89.6 90.5 90.8 91.2 91.7

36.0 64.5 66.4 64.8 64.2

33.3 18.5 19.8 19.4 20.0

69.3 83.1 86.2 84.3 84.2

13.6 24.1 23.3 21.8 20.8

19.8 11.3 11.1 11.5 10.7

33.4 35.4 34.4 33.3 31.5

1a 1a dp 30d 3m dp

88.2 90.0 90.4 90.6 91.1

17.7 33.1 34.4 33.8 33.7

46.0 31.5 32.0 32.1 32.2

63.7 64.6 66.4 65.9 65.8

6.7 12.4 12.1 11.3 10.7

27.2 18.9 18.3 18.8 17.9

33.9 31.3 30.3 30.0 28.6

C C C C C

T×1 T×1 T×1 T×1 T×1

C C C C C

MSEc MSEc MSEc MSEc MSEc

C C C C C

MCCc MCCc MCCc MCCc MCCc

C C C C C

MSE MSE MSE MSE MSE

C C C C C

MCC MCC MCC MCC MCC

1a 1a dp 30d 3m dp

Table 5.9. Total and penalised up and down regulation as percent of total generation; simulation results for 2007 Spanish energy market (dataset C).

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Chapter 5. Uncertainty of wind power forecast and electricity markets

5.4.6

Cost of wind forecast uncertainty

In the preceding sections, performance ratio PR was very useful for evaluating relative performances of several bidding scenarios because it indicates directly, how much of the maximum revenue can be obtained. In Fig. 5.12 results from 2007 are summarised. Persistence scenarios T×0 and T×2 were included to illustrate better the impact of forecast uncertainty. All results presented here for 2007 are also shown for the year 2008 in appendix D.4. 96 Ref. 1a 1a dp 1m 3m dp

94 Tx0

PR [%]

92 MSEc 90

MSE MCCc Tx1

88

MCC Tx2

86 40

50

60

70

80

90

100

110

MAPE [%] 96 Ref. 1a 1a dp 1m 3m dp

94 Tx0 MSEc MCCc

PR [%]

92

90 MSE Tx1 MCC

88

Tx2 86 15

20

25

30

35

40

45

50

Epen [%] Figure 5.12. Performance ratio PR as a function of MAPE (above) and penalised regulation energy (below), data from 2007.

5.4 Optimised bidding strategy for wind energy

157

As already seen in the tables before, higher values of MAPE (i.e. more uncertainty) lead to lower performance ratios. In this representation it can be seen very well the difference of forecast data MCC in comparison with all other datasets. Especially the impact of different price forecast scenarios can be seen. The greatest changes occur between the reference case (‘Ref.’) and annual mean prices (‘1a’). Here especially the forecast error (MAPE) is increased dramatically, but also PR is improved. The impact of more sophisticated price forecasting scenarios, such as annual (‘1a dp’) and seasonal (‘3m dp’) daily profiles, can be seen better in the lower representation of Fig. 5.12, where PR is shown as a function of penalised deviations Epen . But using PR also has a drawback. Being a relative measure, no direct comparisons with ESS costs are possible. Therefore, it is convenient to define a deviation unit cost πd , which describes the cost of wind power forecast uncertainty in e/MWh. A first definition can be formulated as πd =

πref − πobs πref − πobs = ereg NMAE

(5.11)

where πref and πobs are the reference and observed selling price and ereg is the relative regulation energy [p.u.] normalised by total generated energy which is equal to NMAE of the wind power forecast. The value of πd describes the cost of the forecast error, including the fact that one portion of the deviations are not penalised. This is a useful measure to quantify the cost of forecast errors in general. But if it is assumed that energy storage would reduce deviations, one could argue, that only penalised regulation energy would be compensated by an ESS. Thus, the cost of penalised deviations πpen may be defined by πpen =

πref − πobs epen

(5.12)

where epen is the relative penalised regulation energy [p.u.] normalised by Etotal . As penalised regulation energy is only 50% (or even less) of overall deviations, unit costs πpen are higher than πd by a factor between 2–3. Results from different forecast and bidding scenarios are represented in Fig. 5.13. Forecast deviation costs πd and πpen are represented as a function of MAPE. Remember that in our case MAPE is equal to regulation energy Ereg . From Fig. 5.13 it can be seen that these costs are almost independent of forecast uncertainty. There can even be observed that unitary costs are slightly lower for larger values of MAPE. Additionally, results show that the most important effect on costs is achieved with bidding based on annual means of deviation penalties. The most important conclusion which can be drawn from costs between 4–14 e/MWh is, that ESS will always be significantly more costly and its installation cannot be justified only with deviation costs.

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Chapter 5. Uncertainty of wind power forecast and electricity markets

10 Ref. 1a 1a dp 1m 3m dp

9

πd [e/MWh]

8 MCC MCCc

7 6

MSE

Tx0

Tx1

Tx2

5 MSEc 4 40

50

60

70

80

90

100

Ref. 1a 1a dp 1m 3m dp

16

πpen [e/MWh]

15 MCC MCCc

14 13

110

Tx1

Tx0 MSE

Tx2

12 11

MSEc

10 40

50

60

70

80

90

100

110

MAPE [%] Figure 5.13. Cost of (wind power) forecast uncertainty πd (above) and cost of penalised deviations πpen (below) as a function of MAPE, data from 2007.

5.5 Summary

5.5

159

Summary

In this chapter, the cost of deviations caused by forecast errors has been analysed for the example of wind energy in the Spanish electricity market. This issue is of great interest these days, as wind power is reaching significant penetration levels and is starting to have an impact on system balancing. The following conclusions can be drawn from a review of recent literature on modelling of liberalised electricity markets, price forecasting and the role of energy storage for advanced bidding strategies: • Electricity markets in Europe have similar architectures → market models can be easily adapted for any country • Flexible markets which allow intra-day trading (more than one auction per day) which benefits participation of wind energy by improving forecasts • Price forecasting is already yielding good results for spot prices with similar techniques as in probabilistic wind power forecasting • Regulation prices are very difficult to predict and are still object of investigation • Several studies have suggested a combined operation of hydro-electric stations with wind farms considering – Load shifting/balancing – Congestion of transmission lines – Optimised bidding Coordinating energy storage with wind power in electricity markets have not been touched in this thesis, but it represents a promising future field of investigation. Main contributions of this chapter are: • Statistical evaluation of prices on the Spanish electricity market • Identification of daily price profiles including seasonal variations • Study of the impact of wind power on regulation prices • Application of daily price profiles and forecast error pdf in an advanced bidding strategy • Identification of a conflict of interests between revenue and global system imbalances → minimization of penalised deviations Time series of marginal prices (or spot prices) and regulation costs have been analysed. Long term variations, induced by external factors such as oil and gas prices have been detected. Daily profiles are identified averaging hourly prices over one year or three months. Especially for down regulation costs, a strong seasonal variability has been observed. Largest seasonal variations are observed for the evening peak. As a consequence, it is concluded that seasonal variations in daily profiles should be included in

160

Chapter 5. Uncertainty of wind power forecast and electricity markets

price forecasts. Only data from 2 years has been analysed. Including more years would permit more precise conclusions on seasonal effects. Next, the impact of wind power on market prices is studied. A negative correlation between regulation prices and wind power is observed. If wind power is subtracted from demand, highest correlations with market prices are observed. Therefore, it can be concluded that wind power acts upon market prices as negative load. The causality of the observed correlations could not be determined clearly, as daily patterns of wind power are contributing to this correlation. Here further investigation is needed to separate accidental correlations from real impacts on the market. The last part of this chapter is dedicated to the application of parametric probability distributions with optimised bidding strategies, in order to quantify the cost of wind forecast errors. Here the main contribution of the thesis is the detailed evaluation of different estimation approaches for regulation costs (daily profiles) and forecast scenarios. In concordance with results reported in the literature, when bidding with point forecasts (i.e. the reference case), a reduction of revenues due to regulation costs by approximately 10% is observed. Or in other words, the cost of forecast errors amounts to 10% of total income from the wind farm. In 2008 this reduction was only 6% due to a large increase of market prices with only a moderate increase in regulation costs, which means that advanced strategies only can reach 6–10 percentage points (p.p.) of improvement. With this premise, obtained revenue improvements of up to 3 p.p. are very encouraging. In 2008 only up to 1 p.p. is reached. A central observation of the simulations is that probabilistic bidding strategies increase regulation energy. But only deviations that have the same direction as system-wide imbalance are penalised, as opposite deviations are helping to balance the system. As a consequence, only the penalised part of regulation energy should be analysed. It is observed that only price estimations which take into account daily profiles and seasonal variations are able to reduce penalised imbalances. In other words, while already quite simple price forecasting scenarios (e.g. annual mean prices) yield revenue improvements, only more sophisticated models are able to prevent an increase of undesirable imbalances in the system. It is concluded that only those strategies should be adopted which increase the revenue and minimise deviations.

Chapter 6 Conclusions and Future work “Science is always wrong, it never solves a problem without creating ten more.” George Bernard Shaw (1856–1950) Irish writer

As its title indicates, in this chapter the main conclusions of this thesis are summarised. Results are presented in chronological order as they appear in the thesis. Therefore, four groups of conclusions are formed: probabilistic instruments, ESS sizing methodology, life-cycle cost analysis of energy storage and analysis of the market environment. In the second part of the chapter, possible subjects of future investigation are outlined.

162

6.1

Chapter 6. Conclusions and Future work

Conclusions

With the present thesis, a probabilistic methodology for electrical energy storage sizing has been proposed. The main objective was to quantify the requirements of storage power and energy capacity for the mitigation of wind power forecast errors. In order to complete the scenario, costs of such storage systems have been evaluated and set into the context of a market environment. At the beginning of the research in this field, probabilistic wind power forecasting was still not state of the art. An additional problem was that wind power forecasting is a competitive business and obtaining real world data is not easy. During the development of the thesis a growing number of publications on probabilistic forecasting showed that the approach was worth to be further developed. Finally, the dramatic rise of oil prices in 2008 motivated many countries to set up plans for the accelerated implantation of electric vehicles. This was an additional motivation to pursue this work, although it turned out that centralised energy storage solutions at utility scale may remain very expensive for the next decade. The massive surge of electrical vehicles could provide at very low cost the energy storage which will be needed to integrate intermittent renewable energies such as wind or solar energy in the power system. Central contribution of this thesis is the development of a framework for ESS sizing which includes: • Realistic simulation of forecast errors based on wind power measurements • Estimation of forecast error pdf with the parametric Betafit method • Separated sizing of ESS power and energy requirements • Detailed insight in statistics of state of charge • Quantification of costs due to ESS efficiency within a global ESS cost model Not part of the sizing methodology, but important for the evaluation of the economic viability of ESS was the quantification of the cost of forecast errors. The novelty of the presented study consists in a detailed analysis of daily profiles of market prices and the impact of different forecast scenarios on imbalances and wind farm revenues. The most important findings will be exposed in the following paragraphs sorted by chapters. In chapter 2 statistical tools are developed which are used in the other chapters. In chapter 3 a novel energy storage sizing method is proposed, based on forecast uncertainty and statistics of the state of charge of the storage. Chapter 4 is dedicated to the quantification of the cost of storage systems. Most promising technologies are included. Compressed air storage (CAES) is not considered due to its dependence on natural gas. Storage in electrical vehicles (V2G) is also not considered, as it presents a totally different cost structure and could therefore not be included in the model. Finally, in chapter 5 the market environment is analysed. Mean market prices of electricity are shown for the Spanish case in the years 2007 and 2008. A detailed study of prices shows that energy storage is in general considerably more expensive than electricity generation. But market risks such as the dependence on limited reserves of fossil fuels have been detected which makes energy storage more interesting every year.

6.1 Conclusions

163

Three novel developments presented in Chapter 2: • A simple algorithm to generate time series of wind power forecast errors • Correction and refinement of a parametric method for error pdf estimation • Online bias correction of real world forecast data Three basic instruments are developed in chapter 2, which are the basis of this thesis. First time series of forecast errors are generated from wind power measurements. A simple persistence approach is chosen, to simulate three different forecast qualities, denoted as T×0, T×1 and T×2. This way, three forecast scenarios are created – best, intermediate and worst – corresponding to the time shift applied to the persistence forecast. Time shift zero (T×0) means that mean wind power is predicted perfectly. The forecast error only consists in the deviations around the mean. T×1 and T×2 represent time shifts of once and twice the forecast interval length. It could be shown that resulting probability distributions (pdf) are very similar to those from real world forecasts. As forecast data is very difficult to obtain, this simple technique is very useful for the creation of large time series of forecast errors, which are essential for any statistical analysis. The second contribution is the refinement of a parametric approach to describe forecast uncertainty with Beta distributions. The original method was insufficient due to a too small amount of data points. Improvements could be introduced due to the large number of data points (up to one million). The methodology was developed based on the synthetical forecast data, generated with the persistence approach described above and verified with real world forecast data from one year. The approximation method, denominated Betafit was tested with a Kolmogorov-Smirnov (K-S) goodness of fit test. The K-S test rejects the null hypothesis for the Betafit approximation only for one scenario, which showed a strong bias. Hence it can be stated, that reasonably good approximations are obtained for all other scenarios. It could be shown that Betafit approximation performs well if forecast data is unbiased. This leads us to the third contribution of this chapter – an online bias correction method, based on moving averages. The idea could be described as a persistence forecast of the bias. Empirical tests showed that an averaging window τ = 7 days gives best results. Exponential moving averages may improve results, but for simplicity a simple moving average was applied. The proposed method can be applied as post-processing algorithm to any forecasting system and provides important improvements of statistical properties of the forecast error. It could be demonstrated that statistical parameters such as skewness and kurtosis of bias-corrected data approximated better those from the persistence model. This gives additional value to the proposed bias correction, as the persistence approach gains validity. The great value of bias-correction has been shown in chapters 3 and 5. It can be concluded that forecasts ideally should be bias free and that this can be guaranteed with the proposed algorithm.

164

Chapter 6. Conclusions and Future work

Five main achievements obtained in Chapter 3: • Functional relationship between ESS power and unserved energy based on forecast error pdf • Functional relationship between ESS energy capacity and saturation times based on cdf of SOC • Estimation of unserved energy based on saturation times and energy throughput ratio ETR • Characterisation of estimation errors for future model improvement • Case study with real world forecast data, demonstrating the importance of bias-free forecasts for ESS sizing Objective of this chapter was to quantify the ESS size which would be needed to compensate forecast errors. Further, the cumulated energy of uncompensated errors in case of reduced ESS size is estimated. This energy is denoted as unserved energy and is obtained separately for power and energy reduction. Wind forecast data was used, but note that the same methodology can be applied with any forecast data, such as solar power or demand forecasts. The sizing method concentrates on effective power and energy requirements. This means that technology-specific parameters such as depth of discharge or efficiency are not included. These parameters can be added later, as shown in chapter 4. The methodology is divided in two independent sizing methods. The first determines ESS power requirements and the second ESS energy capacity. The most important contribution of this chapter is the quantification of unserved energy as a function of ESS size. Functional relationships are established to estimate the energy which cannot be compensated by the ESS if its size is reduced. Sizing of ESS rated power is rather simple. Unserved energy is obtained from pdf of forecast errors. The Betafit approximation method proved to be a valuable instrument to obtain smooth distributions from forecast data of only one year. It is shown that unserved energy is proportional to the integral of the tail of the pdf, starting from pESS up to 1. Therefore, to obtain satisfying results, the pdf must be especially precise in its tail region. The case study, performed with real world forecast data, confirmed that the T×1 scenario is a very good tool for sizing, if no real forecasts are available. If real forecasts are available, bias correction is strongly recommended to improve approximation on the one hand and reduce ESS power requirements on the other. To give an example, the proposed online bias correction may reduce the power requirements of the ESS by 10% in case of strongly biased forecasts. Sizing of energy storage capacity must be separated from power sizing because the integrating effect of storage requires different means of estimation. The most important parameter for ESS energy capacity sizing is the saturation time, which represents the accumulated time when the ESS is either full or empty. The relationship between saturation time and ESS size has been found to be strongly nonlinear and no statistical

6.1 Conclusions

165

parameter could be identified to describe it. For this reason, an empirical approach is suggested, which consists in the creation of SOC time series using forecast errors as input. Note that 100% ESS efficiency is assumed in this process. An algorithm is developed to obtain an estimation of saturation times from cumulative distributions of SOC. As second very important parameter energy throughput ratio ETR has been identified. In case of forecast error compensation, it represents the mean absolute error MAEnormalised by long term mean wind generation P¯ . A multiplication of saturation time with ETR gives already a fairly good estimate of unserved energy. An empirical factor ν is introduced to improve results. From the case study it is concluded that T×2 scenario can be used for ESS energy sizing only for values of unserved energy eue > 10%. If lower values are desired, real world forecasts demand significantly larger storage capacities. It should be highlighted the positive effect of bias correction on the accuracy of estimations. While uncorrected scenarios show large differences in their behaviour, bias corrected versions are very similar and closer to the persistence scenarios. Therefore, bias correction is again strongly recommended to obtain a good performance of the proposed sizing method. Results of the case study in Chapter 3: • Persistence scenarios T×1 (for power) and T×2 (for energy) are valid for ESS sizing as they represent to a large extent statistics of real world forecasts • Scenario T×0 is far too optimistic and thus not useful for ESS sizing • The proposed sizing method performs well for 24-h forecasts with both, persistence and real world data • Bias correction improves model precision and reduces ESS capacity requirements Postprocessing of sizing results is performed to evaluate estimation errors. For this purpose, obtained ESS sizes are introduced in a time step simulation and real saturation times and unserved energies are calculated. The aim of this postprocessing is the identification of the nature of estimation errors. Errors for estimation of saturation times and mean unserved power have been studied. Results may be very valuable for further refinements of the model which could not be included in this thesis.

166

Chapter 6. Conclusions and Future work

Three contributions of Chapter 4: • Combination of several cost models, eliminating shortcomings • Integration of probabilistic ESS sizing in a life-cycle cost model • Identification of technology-specific optimal cost regions with cost surfaces The objective of chapter 4 was to integrate the sizing model into a life-cycle cost analysis, in order to obtain information of ESS costs. At this moment, different storage technologies have to be considered. A large variety of electrical storage systems are being used for different applications, depending on their cost structures and technological limitations. Here most promising technologies for utility-scale storage have been analysed. Compressed air storage (CAES), as one of the most promising emerging technologies is not included due to its dependence on natural gas, but it may easily be included. Flywheels have been excluded due to its high self discharge and low energy capacity, which makes them unsuitable for high energy applications as are considered here. One contribution of this thesis is the combination of several existing life-cycle cost models, eliminating some shortcomings. As a result, energy loss costs, replacement costs and electricity price escalation are considered in one model. The probabilistic approach permits the calculation of the number of equivalent full cycles, which in general is not an easy task. In the literature, energy efficiency is introduced, increasing ESS energy capacity or related costs. In this thesis it is considered that the efficiency does not affect ESS size, but the volume of throughput. It is assumed that name-plate energy capacity should give the dischargeable energy, as common practice in batteries. Of course, manufacturers may not always fulfill this assumption, but it is assumed that in any case this parameter should be available. As most representative result of the cost model, cost of energy COE may be mentioned. Possibly a more adequate denomination is revenue requirement, as it represents the price per kWh (or MWh) the owner of the ESS must sell the stored energy to recover the investment plus variable costs. Results of COE can be summarised as follows: • BESS: highest costs, optimum at 100–250 e/MWh best suited for high power/low energy • Flow battery: high costs with large reduction potential, optimum at 50–350 e/MWh best suited for high power and potentially also high energy • Hydrogen-Fuel cell: Still high power related costs, optimum at 50–180 e/MWh best suited for high energy and potentially also high power • Pumped hydro: lowest costs, optimum at 15–100 e/MWh best suited for high power/high energy Pumped hydro storage (PHS) is the least expensive option nowadays, but does not provide rapid response times as the others. Largest potentials for cost reduction can be expected for flow batteries and hydrogen storage. Battery storage (BESS) is limited to

6.1 Conclusions

167

low energy applications and cost reductions seem to be rather low. Possible technological breakthroughs are not considered here. Although hydrogen storage still has important uncertainties about its power related costs, there is a clear consensus in the literature about its low energy storage related cost. As a consequence, hydrogen storage is especially indicated for high energy applications. Even with round-trip efficiencies as low as 40%, hydrogen storage yields second lowest values of COE after PHS. In addition it may be kept in mind that cycle life is less critical for stationary fuel cells, because in this case typically solid oxide (SOFC) or molten carbonate fuel cells (MCFC) may be installed. Four main contributions in Chapter 5: • Statistical evaluation of prices on the Spanish electricity market • Identification of daily price profiles including seasonal variations • Study of the impact of wind power on regulation prices • Application of daily price profiles and forecast error pdf in an advanced bidding strategy → quantification cost of forecast errors Studies have been published for important wind producing countries such as Denmark, Germany and Netherlands but no detailed description of the statistics of the Spanish regulation market has been available yet. Therefore the study itself is the first contribution of this chapter. As most important results of the study, two aspects are worth to be highlighted. Seasonal variations of daily profiles have been described for regulation costs. These profiles represent valuable information for the advanced bidding strategy, applied in this chapter. The second aspect is the analysis of correlations between wind power and market prices. The method is not new, but has not been applied yet to the Spanish market. In addition regulation prices have been included, which was not the case in other studies. A weak negative correlation between market prices and wind power has been found. Existing strong correlations between market prices and demand are increased, if wind power is subtracted from demand. Thus, wind power can be regarded as negative load. It is likely that wind power in Spain has an impact on market prices, especially on down regulation prices, as wind power typically causes excess energy in the system. But at the same time, wind power exhibits a daily pattern which may cause part of the observed correlations, without any influence on market prices. Therefore, it is not easy to determine how much of the observed correlation is due to coincidence and how much is actually due to interaction of wind power with market prices. For the Spanish market it can be concluded, that wind generation still has a reduced influence on the marginal (spot market) prices, but it has already an impact on regulation prices. Price forecast models should account for this impact in order to obtain a better bidding strategy for wind energy. In this thesis, no dependence of market prices and wind generation is considered, as no price forecasting is included. Finally, a probabilistic bidding strategy is applied for the Spanish market, in order to quantify the cost of forecast errors. If wind power is sold without any strategy, for 2007 and 2008, costs due to forecast errors of 6–10% of total revenue are obtained. In 2008 market prices were 50% higher than in 2007 but regulation costs did not rise at the same

168

Chapter 6. Conclusions and Future work

rate. As a consequence, deviation costs, relative to total revenues were 40% lower than those observed in 2007. The applied bidding strategy improves revenues in 2007 by up to 3 p.p. (percentage points), while in 2008 only 1 p.p. could be obtained. A central observation of the simulations is that probabilistic bidding strategies increase regulation energy. Given that all tested scenarios obtained increased revenues, a clear conflict of interests arises here. The wind farm owner wants to maximize his income, while the system operator wants to minimize deviations. To be correct, only deviations that have the same direction as system-wide imbalance are penalised, as opposite deviations are helping to balance the system. As a consequence, only the penalised part of regulation energy should be analysed, in order to decide, if a conflict of interest exists. It is observed that only price estimations which take into account daily profiles and seasonal variations are able to reduce penalised imbalances. In other words, while already quite simple price forecasting scenarios yield revenue improvements, only more sophisticated models are able to prevent an increase of undesirable imbalances in the system. It is concluded that only those strategies should be adopted which increase the revenue and minimize penalised deviations at the same time. Additionally, it may be mentioned that also imbalance costs are slightly reduced if the online bias correction is applied to the real world forecast data which was available for this study. The improvement was higher, if bias was larger, so that the revenue improvement can be more than 1 p.p., depending on forecast quality and market prices. The final conclusion of this thesis is that large storage capacities are needed to compensate forecast errors. The elevated cost of electrical storage systems cannot be payed off by saved costs from forecast deviations. Nevertheless, expected ESS cost reductions combined with added value of such large storage systems for the power system may lead to profitable solutions. The framework provided in this thesis can be used in further investigations to identify cost targets for ESS to become profitable.

6.2

Future work

Several fields for further investigation have been identified during this thesis. Ideas for future work are listed below in the same order as subjects appear in the chapters. The persistence approach for the simulation of forecast data may be compared with moving average methods. Most interesting would be the evaluation of the exponential moving average. The advantage of applying moving averages is that more realistic forecast time series are obtained. Forecasts which are used in practice for decision-making usually have time steps of 15 min up to 1 h, with forecast horizons ranging from zero to 48–72 h ahead. Possible improvements for the simulation of real world forecasts may be studied, applying statistical instruments as proposed in this thesis. The Betafit approximation method may be improved. The approximation of the relationship between mean and standard deviation of wind forecasts within one forecast bin σ(µ) has been identified as a weak point. Improvements can be obtained by using polynomials of higher order or piecewise approximation methods. These refinements should be done including wind power data from more sites. Of special value would be an introduction of power curves into the construction of the characteristic relationship. Modifications of the beta distribution itself might improve the behaviour in the tail regions.

6.2 Future work

169

Here mixtures of distribution functions may lead to better results. The difficulty of this approach is to maintain simplicity. Another option would be to consider non-parametric methods. As mentioned in chapter 2, the Betafit method in its original, erroneous version has been used in at least one study on the cost of forecast errors. Calculations may be repeated applying the improved method proposed in this thesis and this way, the impact of improvements may be evaluated. Further improvements of the sizing methodology for ESS energy capacity are prepared by the post-processing presented in this thesis. Mechanisms of saturation times may be investigated with frequency analysis for example. In addition, empirical adjustment factors may be refined with optimization methods. Additional time step simulations can be performed to investigate the effect of simultaneous reduction of energy and power ratings of the ESS. In chapter 4 a two-dimensional interpolation method was developed to estimate this effect. Simulations may evaluate the correctness of this approach. The life-cycle cost model may be exploited for further investigations. Cost surfaces may be developed assuming more scenarios and sensitivity of model parameters may be assessed more in detail. The uncertainty of energy and power related costs could be studied, in order to make more precise estimations on future price scenarios. Learning curves might be introduced to establish a time scale for cost projections of emerging technologies. Further investigation is needed to include additional revenues for energy storage into the cost model. It was demonstrated in this thesis that the cost of deviations on the regulation market cannot justify the installation of energy storage, because the penalisation is relatively low and ESS costs too high. Studies in the literature have shown that even relatively high costs of energy storage may be justified if revenues from ancillary services are included, such as power quality, ride-through capacity, frequency and voltage control. The introduction of these services obviously would have an important impact on the energy throughput of the ESS. The probabilistic approach pursued in this thesis is likely to be useful to describe also these effects. Energy storage may be included into bidding strategies in liberalised markets. Storage reserves can be used to provide regulation energy. A sizing method which takes into account this option would have certainly a probabilistic character and findings from this thesis would be useful. Finally, price forecasting is a very interesting field of investigation. In a first stage, more work is needed to assess the impact of wind power on market prices. Historical time series may be analysed in order to separate the impact of wind generation from other factors such as fossil fuel prices or climate impacts as hot summers and very cold winters. Results may be used to include the effect of wind power on market prices in advanced bidding strategies. This is clearly a very complex task, but represents a promising field of investigation.

Appendix A Statistical evaluation of the forecast error

In this appendix statistical parameters and evaluation methods which are applied to the forecast error are treated more in detail. The values of bias, median, MAE, RMSE, skewness and kurtosis are represented for the three datasets A, B, and C. In addition the available real forecasts MSE and MCC are included. A short introduction of the properties of the Beta distribution is given and several goodness of fit tests are shortly described.

172

Appendix A. Statistical evaluation of the forecast error

A.1

Bias and Median

In Fig. A.1 and Fig. A.2 bias (mean forecast error ε¯) and median are depicted for the three datasets and forecast scenarios. In addition the available real forecasts MSE and MCC are included. Scenario Tx0

Scenario Tx1

Bias [%]

15 A B C C MSE C MCC

10 5 0 -5

1

3 6 12 24 48 168

720

Scenario Tx2

15

15

10

10

5

5

0

0

-5

Forecast interval T [h]

1

3 6 12 24 48 168

720

-5

Forecast interval T [h]

1

3 6 12 24 48 168

720

Forecast interval T [h]

Figure A.1. Bias in % of Pinst as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC.

Median [%]

Scenario Tx0

Scenario Tx1

Scenario Tx2

5

5

5

0

0

0

-5

-5

-10

-10

A B C C MSE C MCC

-5 -10 -15

1

3 6 12 24 48 168

720

Forecast interval T [h]

-15

1

3 6 12 24 48 168

720

Forecast interval T [h]

-15

1

3 6 12 24 48 168

720

Forecast interval T [h]

Figure A.2. Median in % of Pinst as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC.

173

A.2 MAE and RMSE

A.2

MAE and RMSE

In (A.1) and (A.2) mean absolute forecast error MAE and root mean squared error RMSE are defined in their normalised forms. NMAE = |ε|

(A.1)

p NRMSE = σ = E(ε − ε¯)2

(A.2)

where E denotes the expectation operator, ε the random variable (here the forecast error), ε¯ the mean error (or bias) and σ the standard deviation (or NRMSE). NMAE and NRMSE may be represented in percentage. Because of its practical relevance, NMAE given in percent is often denoted as MAPE (mean absolute percentage error). In Fig. A.3 and Fig. A.4 MAPE and NRMSE (standard deviation σ) are depicted for the three datasets and forecast scenarios. In addition the available real forecasts MSE (triangle) and MCC (circle) are included. Scenario Tx0

Scenario Tx1

MAPE [%]

50

Scenario Tx2

50

50

40

40

30

30

20

20

20

10

10

10

A B C C MSE C MCC

40 30

0

1

3 6 12 24 48 168

720

0

Forecast interval T [h]

1

3 6 12 24 48 168

720

0

Forecast interval T [h]

1

3 6 12 24 48 168

720

Forecast interval T [h]

Figure A.3. Mean absolute percentage error MAP E as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC.

Scenario Tx0

Scenario Tx1

NRMSE [%]

50

Scenario Tx2

50

50

40

40

30

30

20

20

20

10

10

10

A B C C MSE C MCC

40 30

0

1

3 6 12 24 48 168

720

Forecast interval T [h]

0

1

3 6 12 24 48 168

720

Forecast interval T [h]

0

1

3 6 12 24 48 168

720

Forecast interval T [h]

Figure A.4. NRMSE in % of Pinst as a function of T of persistence forecasts for datasets A, B, C and real world forecasts MSE and MCC.

174

Appendix A. Statistical evaluation of the forecast error

A.3

Skewness and Kurtosis

As noted in [33], the normalised third and fourth moment of a random variable are also known as the first and second shape factors of a probability distribution. In Fig. A.5 and Fig. A.6 skewness and kurtosis are depicted for persistence and real world forecasts MSE and MCC. General definitions of central third and fourth moment can be written as γ=

E(ε − ε¯)3 σ3

(Skewness)

(A.3)

E(ε − ε¯)4 (Kurtosis) (A.4) κ= σ4 where E denotes the expectation operator, ε the random variable (here the forecast error), ε¯ the mean error (or bias) and σ the standard deviation (or RMSE). Skewness describes the symmetry of the distribution. A perfectly symmetric pdf, such as the Gaussian has skewness zero. Positive values, such as observed for wind power forecast errors, indicate that the left “negative” tail is shorter than the right “positive” tail. The Weibull pdf, typically used to describe wind speeds is positively skewed. Kurtosis is a measure of “peakedness” of the distribution. The Gaussian pdf has a value of 3, therefore it is used as a reference. Distributions with κ > 3 (typical for wind forecast errors) are relatively high-peaked or fat-tailed and are called leptokurtic. If κ < 3, the pdf is flat-topped and is called platykurtic and if κ = 3 it is called mesokurtic. Scenario Tx0 A B C C MSE C MCC

1.5

Skewness

Scenario Tx1

Scenario Tx2

1.5

1.5

1

1

0.5

0.5

0.5

0

0

0

1

1

3 6 12 24 48 168

720

1

Forecast interval T [h]

3 6 12 24 48 168

720

1

Forecast interval T [h]

3 6 12 24 48 168

720

Forecast interval T [h]

Figure A.5. Skewness coefficient y as a function of T for datasets A, B, C, MSE and MCC. Scenario Tx0

Scenario Tx1

Kurtosis

20 A B C C MSE C MCC

15 10 5 0

1

3 6 12 24 48 168

720

Forecast interval T [h]

Scenario Tx2

20

20

15

15

10

10

5

5

0

1

3 6 12 24 48 168

720

Forecast interval T [h]

0

1

3 6 12 24 48 168

720

Forecast interval T [h]

Figure A.6. Kurtosis κ as a function T for datasets A, B, C, MSE and MCC.

175

A.4 The Beta distribution function

A.4

The Beta distribution function

The Beta probability density function f for a given value x and given pair of parameters a and b is f (x) =

1 · xa−1 · (1 − x)b−1 · I(0,1) (x) B(a, b)

B(a, b) =

Z1 0

(A.5)

xa−1 · (1 − x)b−1 dx

(A.6)

where B(a, b) is the Beta function and I(x) is the Indicator function. The Indicator function ensures that only values of x in the interval [0, 1] have nonzero probability. As can be seen in (A.6), Beta function B(a, b) is simply the integral of the term in the numerator of (A.5), which normalizes the integral of f (x) in the interval [0, 1]. Parameters a and b of the Beta pdf can be calculated directly from the mean µ and standard deviation σ 2 and vice versa. A summary of the corresponding equations from section 2.4.1 is given below. σ2 =

a=

a·b (a + b + 1) · (a + b)2

µ=

a a+b

(A.7)

(1 − µ) · µ2 −µ σ2

b=

1−µ ·a µ

(A.8)

Frequency

To have an impression of the variability of shapes that can be adopted by the Beta pdf, in Fig. A.7 several parameter combinations are represented. If a = b symmetrical distributions are obtained, with the special case of the standard uniform distribution for a = b = 1. With a < b the pdf is positively skewed and if a > b it is negatively skewed. 5

5

4

4 a=b=8

3 a = b = 0.5

a=9 b=2 a=3 b=6

3

2

a=6 b=3

2 a=b=1

1 0 0

a=2 b=9

0.2

0.4

0.6

x [p.u.]

1 0.8

1

0 0

0.2

0.4

0.6

x [p.u.]

Figure A.7. Beta pdf as a function of its parameters a and b.

0.8

1

176

A.5

Appendix A. Statistical evaluation of the forecast error

Goodness of fit tests

The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Here three measures are used to test the statistical hypothesis that wind power forecast errors can be approximated with conditional Beta distributions. The selected test statistics are Kolmogorov-Smirnov and Anderson-Darling. Pearson’s chi-square test is included only for comparison purposes. The basis of statistical hypothesis testing is to decide, weather or not the null hypothesis H0 : F = F0 can be rejected or not. F is the observed, empirical distribution and F0 is the theoretical, expected distribution.

A.5.1

Pearson’s Chi-Square χ2 test

Pearson’s χ2 goodness of fit test is chosen for comparison purposes. Contrary to the statistics described below, it is based on the probability density function (pdf) and not on the cumulative distribution function (cdf). The χ2 statistic is defined in (A.9). The null hypothesis is rejected if p ≤ α. 2 m X f (x ) − f (x ) m i o i (A.9) χ2 = fo (xi ) i=1

where fm (x) is the empirical pdf with m bins and f0 (x) the Betafit approximation.

This test statistic depends strongly on the sample size. In addition the number of observations per bin have to be controlled in order to obtain reasonable results. In [33] it is recommended a minimum number of observations per bin of five. This criterion is implemented in the χ2 goodness of fit test in the statistics toolbox of Matlabr (chi2gof). If less than 5 observations are detected in the tail of the expected pdf f0 , bins are pooled. By default, the parameter that controls the pooling function of chi2gof is set to 5 (minimum observations per bin). In our case, only normalised pdfs were available from the Betafit algorithm and it was desired to represent the pdf properly (without pooling) down to frequencies of 0.01%. This was achieved by multiplication of the normalised pdf with a factor 50 000. As a result, the tails are pooled for frequencies below 0.01%, applying the 5-observations criterion. The same pooling method is applied for the Anderson-Darling test (see below).

A.5.2

Kolmogorov-Smirnov K-S test

This well known test statistic can be found in standard statistical literature [33] and it can be formulated as in (A.10). Dm = max |Fm (x) − F0 (x)| x

(A.10)

where Fm (x) is the empirical cdf with m bins and F0 (x) the Betafit approximation. Although this test do not give special importance to the tail of the distributions, it has the advantage to be distribution-free, with critical values available for any distribution from very small up to very large samples. Using the K-S test function kstest in

177

A.5 Goodness of fit tests

Matlabr test statistic Dm , p-value (power value), hypothesis and critical values can be obtained. Results are shown in Table A.1. With 5% confidence level (α = 0.05), only the approximation for the MCC scenario is rejected (test result ‘1’). P-values for all persistence scenarios are high, especially for T×2. Bias corrected MSEc and MCCc scenarios yield similar Dm and p-values compared with T×1. Forecast scenario

Dm

p-value

test result

Persistence T×0 Persistence T×1 Persistence T×2

0.070 0.067 0.045

0.265 0.322 0.798

0 0 0

MSE MSEc MCC MCCc

0.083 0.067 0.122 0.065

0.117 0.316 0.005 0.349

0 0 1 0

Table A.1. Kolmogorov-Smirnov (K-S) test results for goodness of Betafit approximation of error pdf with 5% confidence level; dataset C, 24-h forecast.

It is recommended rejecting the null hypothesis if p < α, even if the critical value of the test statistic is not reached. In this case study with m = 200 and α = 0.05 the critical value is ≈ 0.095. Note that the kstest function does not admit directly the cdf of observed data as input. Observed data input is required as a vector of observed quantities and kstest computes the cdf internally. As in our case study observed cdf were available, kstest was slightly modified in order to skip the internal cdf generation and accept the observed cdf as input.

178

A.5.3

Appendix A. Statistical evaluation of the forecast error

Anderson-Darling A2 test

The Anderson-Darling (A-D) test is a modification of the Kolmogorov-Smirnov (K-S) test, described above. It gives more weight to the tails than does the K-S test, which is of special interest here. Scholtz and Stephens proposed in [55] the k-sample Anderson-Darling test. It permits the evaluation of any number of samples if they share the same distribution, where this distribution is unspecified. Although the k-sample test in this context is not needed, [55] gives a good summary of the goodness of fit statistic, introduced by Anderson and Darling [56] and its two-sample version, proposed by Darling [57]. Note that due to its structure, pooling of the extreme tails may be necessary in order to avoid division by zero. For convenience the same approach as for the χ2 test is adopted. Extreme values below 0.01% of frequency are pooled together. Therefore the cdf is converted in pdf for pooling and re-converted in cdf for the A-D test. Anderson-Darling goodness of fit test Anderson and Darling [56] introduced the “goodness of fit” statistic A2m to test the hypothesis that a random sample X1 , ..., Xm , with empirical distribution Fm (x), comes from a continuous population with distribution function F0 (x). Here Fm (x) is defined as the proportion of the sample X1 , ..., Xm which is not greater than x. A2m

=m

Z∞

−∞

2 Fm (x) − F0 (x)  dF0 (x) F0 (x) 1 − F0 (x)

(A.11)

where Fm (x) is the empirical cdf and F0 (x) the Betafit approximation. Two-sample Anderson-Darling test The two-sample A-D test is used to test the hypothesis that F = G without specifying the common continuous cumulative distribution function. In this thesis it is used to compare the cdfs of errors of the persistence approach with those from advanced forecast models MSE and MCC. The two-sample version of Anderson-Darling test was proposed by Darling [57] and cited by Scholtz and Stephens [55] as shown in (A.12). A2mn

mn = N

Z∞

−∞

2 Fm (x) − Gn (x)  dHN (x) HN (x) 1 − HN (x)

(A.12)

mFm (x) + nGn (x) N where Fm (x) and Gn (x) are the empirical cdf of two independent samples with sizes m and n, HN (x) with N = m + n, is the empirical distribution function of the joint sample. HN (x) =

Appendix B Results of ESS sizing for datasets A04, A13, A33 and B1

This appendix is related to chapter 3. Results are represented for time series of datasets A and B. First three rows show single wind turbines A04, A13 and A33. The last row shows time series B1 from a single wind farm. In section B.1 cumulative distributions of SOC are represented. In section B.2 a proof is given of the fact that mean SOC obtained from persistence forecast is constant for any forecast interval length T . In sections B.3–B.8 simulation results are represented and in section B.9, the balance between charged and discharged energy is shown.

180

Appendix B. Results of ESS sizing for datasets A04, A13, A33 and B1

B.1

Cumulative histograms (cdf ) of SOC

In Fig. B.1 the cumulative histogram or cumulative density function cdf of SOC is shown for different datasets and forecast scenarios.

Scenario Tx0 - A04

Scenario Tx1 - A04

Frequency [%]

100 80

100

80 1 24 72 168 720

60 40 20 0

-0.2

-0.1

0

0.1

0.2

80 1 24 72 168 720

60 40 20 0

0

Scenario Tx0 - A13

Frequency [%] Frequency [%]

0.4

0.6

0.8

1

80

40 20 -0.2

-0.1

0

0.1

20 0

0.2

60 40 20 0

0

0.2

0.4

0.6

0.8

1

0

-0.1

0

0.1

0.2

1 24 72 168 720

60 40 20 0

0

Scenario Tx0 - B1

0.2

0.4

0.6

0.8

1

20 0

80

80

80

20 0

-0.2

-0.1

0

0.1

0

0.2

State of Charge [p.u.]

1 24 72 168 720

40 20 0

0

0.2

0.4

0.6

0.8

0.5

1

1.5

2

Scenario Tx2 - B1 100

40

2

1 24 72 168 720

Scenario Tx1 - B1

60

1.5

40

100

1 24 72 168 720

1

60

100

60

0.5

Scenario Tx2 - A33 80

-0.2

0

Scenario Tx1 - A33 80

0

2

1 24 72 168 720

20

80

20

1.5

40

100

40

1

60

100

1 24 72 168 720

0.5

80 1 24 72 168 720

100

60

0

100

80 1 24 72 168 720

60

40

Scenario Tx2 - A13

100

Scenario Tx0 - A33

Frequency [%]

0.2

1 24 72 168 720

60

Scenario Tx1 - A13

100

0

Scenario Tx2 - A04

100

1

State of Charge [p.u.]

1 24 72 168 720

60 40 20 0

0

0.5

1

1.5

2

State of Charge [p.u.]

Figure B.1. Cumulative distribution function cdf of SOC, datasets A and B, forecast scenarios T×0, T×1, and T×2.

181

B.2 Proof of constant mean value of SOC

B.2

Proof of constant mean value of SOC

The persistence forecast has always a phase error with a time shift fixed at ∆t = (k +1)T . Without this phase error (as in scenario T×0) SOC would be zero, because the forecast is assumed to be bias free (zero mean error). This is necessary because otherwise, it would be impossible to compensate the forecast error, as an infinite ESS size would be needed. As SOC is represented in p.u., a normalised time shift ∆t0 = k + 1 is defined. Let at the beginning of the simulation be SOC0 = 0 and the forecast for the first interval be zero too (ˆ p1 = 0). Then after that interval the state of charge SOC1 will be the mean wind power p¯1 during this time multiplied by ∆t0 . In the next time interval, the first forecast is available which is equal to the mean power from the first interval (ˆ p2 = p¯1 ). As the forecast error is charged to the ESS, the new state of charge SOC2 will be equal to the mean wind power p¯2 during this time interval multiplied by ∆t0 . Hence, the initial situation is repeated with every new forecast. As a consequence, the mean value of SOC is nothing else than the mean value p¯ of all p¯i multiplied by ∆t0 . The value of p¯ represents the normalised long term mean wind power P¯ /Pn . ∆t0 =

(k + 1)T =k+1 T

SOC0 = 0

(normalised time interval)

pˆ1 = 0

(initial conditions)

p1 − pˆ1 ) = ∆t0 p¯1 (initial charge) SOC1 = SOC0 + ∆t0 (¯ 0 SOC2 = SOC1 + ∆t (¯ p2 − pˆ2 ) 0 0 p2 − p¯1 ) = ∆t0 p¯2 SOC2 = ∆t p¯1 + ∆t (¯ ... SOCn = ∆t0 (¯ pn−1 ) + ∆t0 (¯ pn − p¯n−1 ) = ∆t0 p¯n n

SOC =

(B.1)

n

1X 1X SOCj = (k + 1) p¯j n j=1 n j=1

SOC = (k + 1) p¯

(n → ∞)

In the more general case of any forecast method, it is not guaranteed that the forecast pˆn is equal to p¯n−1 . In this case a residual ∆p is generated in every forecast interval. As a result, SOCn at forecast step n can be written as ! n X 0 ∆pj (B.2) SOCn = ∆t p¯n + j=1

It can be shown that ∆p = 0 if the mean forecast error is zero. As this is our initial condition, equation (B.2) collapses to the corresponding expression in (B.1). If the residual ∆p has a mean different from zero, SOCn would tend to ±∞ for n → ∞. Here the importance of the condition of bias free forecast error becomes evident.

182

Appendix B. Results of ESS sizing for datasets A04, A13, A33 and B1

B.3

ESS sizes from histograms and predefined saturation times

In Fig. B.2 ESS sizes (expressed in SOC [p.u.]) are shown for several saturation times.

ESS size eESS [p.u.]

Scenario Tx0 - A04

Scenario Tx1 - A04 e0 e0.1 e2 e10 e25 e50

1 0.8 0.6

1

1

0.2

0.2 3 6 12 24 48 168

720

1.5

0.6 0.4

1

2

0.8

0.4

0

0

0.5 1

ESS size eESS [p.u.]

Scenario Tx0 - A13

0.8 0.6

3 6 12 24 48 168

720

ESS size eESS [p.u.]

0.6

0

1.5

0.5 1

3 6 12 24 48 168

720

ESS size eESS [p.u.]

0.8 0.6

1

0

0.5 1

3 6 12 24 48 168

720

0

1

720

2

0.8

1.5

0.6 1 0.4

0.2

0.2 720

3 6 12 24 48 168

Scenario Tx2 - B1

1

0.4

3 6 12 24 48 168

720

1.5

Scenario Tx1 - B1 e0 e0.1 e2 e10 e25 e50

1

3 6 12 24 48 168

2

Scenario Tx0 - B1

T [h]

1

0.6

0.2

1

0

Scenario Tx2 - A33

0.8

0.2

0

720

1

0.4

1

3 6 12 24 48 168 Scenario Tx1 - A33

0.4

0

720

2

Scenario Tx0 - A33 e0 e0.1 e2 e10 e25 e50

3 6 12 24 48 168

1 0.2

0.8

1

0.6

0.2

1

0

Scenario Tx2 - A13

0.8

0.4

1

720

1

0.4

0

3 6 12 24 48 168 Scenario Tx1 - A13

e0 e0.1 e2 e10 e25 e50

1

Scenario Tx2 - A04

0

0.5 1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

720

T [h]

Figure B.2. ESS sizes from cdf of SOC for predefined saturation times between 0.1 and 50%, datasets A and B, forecast scenarios T×0, T×1, and T×2.

183

B.4 Saturation times tsat

B.4

Saturation times tsat

Scenario Tx0 - A04

Scenario Tx1 - A04

tsat [%]

80

80

60

60

60

40

40

40

20

20

20

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A13

tsat [%]

720

0

60

40

40

40

20

20

20

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A33

3 6 12 24 48 168

720

0

60

40

40

40

20

20

20

3 6 12 24 48 168

720

0

1

Scenario Tx0 - B1

3 6 12 24 48 168

720

0

60

40

40

40

20

20

20

3 6 12 24 48 168

T [h]

720

0

3 6 12 24 48 168

720

80

60

1

1

Scenario Tx2 - B1

80

60

0

720

e2 e10 e25 e50

Scenario Tx1 - B1

80

3 6 12 24 48 168

80

60

1

1

Scenario Tx2 - A33

80

60

0

720

e2 e10 e25 e50

Scenario Tx1 - A33

80

3 6 12 24 48 168

80

60

1

1

Scenario Tx2 - A13

80

60

0

tsat [%]

3 6 12 24 48 168

e2 e10 e25 e50

Scenario Tx1 - A13

80

tsat [%]

Scenario Tx2 - A04

80

1

3 6 12 24 48 168

T [h]

720

0

e2 e10 e25 e50

1

3 6 12 24 48 168

720

T [h]

Figure B.3. Saturation times from 2–50% estimated from cdf of SOC (empty symbols) and obtained from time step simulation (filled symbols).

184

Appendix B. Results of ESS sizing for datasets A04, A13, A33 and B1

B.5

Estimation of unserved energy eue

Estimation of eue with eue ≈ tsat · ETR0 , where tsat [%] is the saturation time and ETR0 [p.u.] the energy throughput ratio. Scenario Tx0 - A04 e2 e10 e25 e50 estim.

50

eue [%]

Scenario Tx1 - A04 50

50

40

40

30

30

20

20

20

10

10

10

40 30

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A13

eue [%]

720

0

50

40

40

30

30

20

20

20

10

10

10

40 30

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A33

3 6 12 24 48 168

720

0

50

40

40

30

30

20

20

20

10

10

10

30

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - B1

720

0

50

40

40

30

30

20

20

20

10

10

10

30

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

T [h]

720

3 6 12 24 48 168

720

3 6 12 24 48 168

720

Scenario Tx2 - B1

50

40

1

Scenario Tx1 - B1

e2 e10 e25 e50 estim.

50

3 6 12 24 48 168

3 6 12 24 48 168

Scenario Tx2 - A33

50

40

1

Scenario Tx1 - A33

e2 e10 e25 e50 estim.

50

1

Scenario Tx2 - A13

50

0

eue [%]

3 6 12 24 48 168 Scenario Tx1 - A13

e2 e10 e25 e50 estim.

50

eue [%]

Scenario Tx2 - A04

720

0

1

3 6 12 24 48 168

720

T [h]

Figure B.4. Unserved energy as a function of forecast interval T and ESS reduction level, comparison of time step simulation and estimation based on ETR0 .

185

B.6 Deviation factors fsat

B.6

Deviation factors fsat

Scenario Tx0 - A04

Scenario Tx1 - A04

2

2

1.5

1.5

1

1

1

0.5

0.5

0.5

e2 e10 e25 e50

fsat

1.5

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A13

fsat

720

0

2

1.5

1.5

1

1

1

0.5

0.5

0.5

0

e2 e10 e25 e50

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A33

3 6 12 24 48 168

720

0

2

1.5

1.5

1

1

1

0.5

0.5

0.5

e2 e10 e25 e50

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - B1

3 6 12 24 48 168

720

0

2

1.5

1.5

1

1

1

0.5

0.5

0.5

e2 e10 e25 e50

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

T [h]

720

3 6 12 24 48 168

720

3 6 12 24 48 168

720

Scenario Tx2 - B1

2

1.5

1

Scenario Tx1 - B1

2

3 6 12 24 48 168

Scenario Tx2 - A33

2

0

1

Scenario Tx1 - A33

2 1.5

1

Scenario Tx2 - A13

2

1.5

fsat

3 6 12 24 48 168 Scenario Tx1 - A13

2

fsat

Scenario Tx2 - A04

2

720

0

1

3 6 12 24 48 168

720

T [h]

Figure B.5. Deviation factor of saturation time as a function of forecast interval T and ESS reduction level.

186

Appendix B. Results of ESS sizing for datasets A04, A13, A33 and B1

B.7

Deviation factors fue

Scenario Tx0 - A04

Scenario Tx1 - A04

4

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

fue [%]

3

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A13

fue [%]

720

0

4

3

3

2

2

2

1

1

1

0

e2 e10 e25 e50

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A33

3 6 12 24 48 168

720

0

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - B1

3 6 12 24 48 168

720

0

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

T [h]

720

3 6 12 24 48 168

720

3 6 12 24 48 168

720

Scenario Tx2 - B1

4

3

1

Scenario Tx1 - B1

4

3 6 12 24 48 168

Scenario Tx2 - A33

4

0

1

Scenario Tx1 - A33

4 3

1

Scenario Tx2 - A13

4

3

fue [%]

3 6 12 24 48 168 Scenario Tx1 - A13

4

fue [%]

Scenario Tx2 - A04

4

720

0

1

3 6 12 24 48 168

720

T [h]

Figure B.6. Deviation factor of unserved energy as a function of forecast interval T and ESS reduction level, estimating p¯up with ETR0 (NMAE).

187

B.8 Deviation factors fup

B.8

Deviation factors fup

Scenario Tx0 - A04

Scenario Tx1 - A04

4

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

fup

3

0

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A13

fup

720

0

4

3

3

2

2

2

1

1

1

0

e2 e10 e25 e50

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A33

3 6 12 24 48 168

720

0

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - B1

3 6 12 24 48 168

720

0

4

3

3

2

2

2

1

1

1

e2 e10 e25 e50

0

1

3 6 12 24 48 168

T [h]

720

0

1

3 6 12 24 48 168

T [h]

720

3 6 12 24 48 168

720

3 6 12 24 48 168

720

Scenario Tx2 - B1

4

3

1

Scenario Tx1 - B1

4

3 6 12 24 48 168

Scenario Tx2 - A33

4

0

1

Scenario Tx1 - A33

4 3

1

Scenario Tx2 - A13

4

3

fup

3 6 12 24 48 168 Scenario Tx1 - A13

4

fup

Scenario Tx2 - A04

4

720

0

1

3 6 12 24 48 168

720

T [h]

Figure B.7. Deviation factor of mean unserved power as a function of forecast interval T and ESS reduction level, estimating p¯up with ETR0 (NMAE).

188

Appendix B. Results of ESS sizing for datasets A04, A13, A33 and B1

B.9

Energy balance in case of ESS saturation

In Fig. B.8 the energy balance of a saturated ESS is shown for different time series.

eue (% discharge)

Scenario Tx0 - A04

60

40

40

40

20

20

e2 e10 e25 e50

20

1

3 6 12 24 48 168 Scenario Tx0 - A13

720

0

1

720

0

60

60

40

40

40

20

20

e2 e10 e25 e50

20

1

3 6 12 24 48 168

720

0

1

Scenario Tx0 - A33

eue (% discharge)

3 6 12 24 48 168 Scenario Tx1 - A13

60

0

3 6 12 24 48 168

720

0 60

40

40

40

20

20

e2 e10 e25 e50 1

3 6 12 24 48 168 Scenario Tx0 - B1

720

0

1

3 6 12 24 48 168 Scenario Tx1 - B1

720

0

60

60

60

40

40

40

20

20

e2 e10 e25 e50

20

0

1

3 6 12 24 48 168

T [h]

1

720

0

1

3 6 12 24 48 168

T [h]

3 6 12 24 48 168 Scenario Tx2 - A13

720

3 6 12 24 48 168

720

Scenario Tx2 - A33

60

20

1

Scenario Tx1 - A33

60

0

eue (% discharge)

Scenario Tx2 - A04

60

0

eue (% discharge)

Scenario Tx1 - A04

60

720

0

1

3 6 12 24 48 168 Scenario Tx2 - B1

720

1

3 6 12 24 48 168

720

T [h]

Figure B.8. Energy balance – percentage of energy loss during ESS discharge (saturation case “ESS empty”).

Appendix C ESS cost parameters and model results

This appendix is related to chapter 4. Section C.1 contains model parameters published recently by Sandia National Laboratories (SNL). Note that the parameters used in this thesis are not always the same. More detailed model results are given for the five technologies considered in this thesis: Lead-acid, NaS, Redox flow battery (RFB), hydrogen (HSFC) and pumped hydro (PHS) storage. In section C.2 model results from base scenario are presented for Nas batteries. In section C.3 results from worst and best case scenarios are represented which are not shown in chapter 4.

190

Appendix C. ESS cost parameters and model results

C.1

Cost parameters from SNL

In Table C.1 initial cost parameters are shown as published by Sandia National Laboratories (SNL) in 2008 [73]. Prices for hydrogen storage from SNL seem very optimistic, although in other studies such as Aguado et al. (2009) [147] even lower prices are considered. In contrast, in [80] power related costs of over 10,000 $/kW are reported. Note that in [147] instead of a fuel cell an ICE (internal combustion engine) is suggested for generating electricity from hydrogen. ESS initial cost coefficients Energy-related [$/kWh]

Power-related [$/kW]

BoP [$/kWh]

Lead-Acid (VRLA) NiCd ZnBr NaS Li-Ion

200 600 400 250 500

175 175 175 150 175

50 50 0 0 0

VRB 1 Regenesysr CAES (surface) 1 CAES (aquifer)

350 100 120 3

175 275 550 425

30 50 50 50

1

Pumped hydro Pb/C asym. caps

10 500

1000 350

4 50

High-speed flywheel Low-speed flywheel

1000 380

300 280

0 0

Hydrogen storage

15 (9)2

1800 (1420)2

0

Storage tank Electrolyzer Fuel cell ICE

15 (9)2 0 0 0

0 300 (1100)2 1500 (1320)2

0 0 0 0

1 2

SNL data from 2003 Prices [e] assumed by Aguado 2009

Table C.1. ESS capital cost coefficients related to energy, power and balance of plant (BoP), as published by Sandia National Labs in 2008.

191

C.2 Cost structure of NaS battery

C.2

Cost structure of NaS battery

The cost structure of NaS battery systems is similar to lead-acid and flow batteries. Compared to lead-acid, slightly higher unit costs per energy and per installed power are overcompensated by the higher efficiency and longer cycle life. In Table C.2 technical specifications and price estimations for a projected 250 kW system (5 modules) are shown [106]. The specific energy related cost (modules and BoP) sum to 302 $/kWh and the power related cost (PCS) is given with 202 $/kW.

Energy capacity (kWh) Power rating (kW) 30 s power rating (kW) Price ($)

Module

System (5 modules)

360 50 150 75 000

1800 250 750 605 000*

Module BoP (Balance of Plant) PCS (Power Cond. Syst.) O&M (per year)

208 $/kWh 100 $/kWh 202 $/kW 26 $/kW

AC/AC round-trip efficiency Calendar life Cycle life

77% 15 years 2500 cycles

Discharge duration Recharge duration

7.2 h 8.6 h

* Installed system including PCS and BoP (2006 pricing)

Table C.2. System specifications and assumptions for NGK NASr . Source: Norris 2007.

192

Appendix C. ESS cost parameters and model results

In Fig. C.1 results for the base scenario are shown: initial (capital) cost IC, revenue requirement or cost of energy (COE), absolute and relative cost of energy losses Closs and O&M plus replacement V C. Note that assumed prices for base case cost calculations are similar but not exactly as reported in Table C.2. Specific costs for energy and power are set to 300 e/kWh and 200 e/kW. (a)

(b) A - Tx1 (24h forecast) - NaS (base)

A - Tx1 (24h forecast) - NaS (base)

500

IC [Me]

30 20

20 10

10

0 1

400 400 300 200

24

0.5

PESS [MW]

COE [e/MWh]

30

12 0 0

200 0 1

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (base)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (base) 200 150

40 100

20

100 10

closs [%]

Closs [ke]

60

0 1

50

PESS [MW]

12 0 0

80 60

5

40 0 1

24

0.5

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (base)

12 0 0

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (base) 80 40

400 100 200 24

PESS [MW]

12 0 0

Eeff [MWh]

0

V C [%]

V C [ke]

200

0.5

20 0

600

0 1

100

60

20

40

0 1

20 24

0.5

PESS [MW]

12 0 0

0

Eeff [MWh]

Figure C.1. Base scenario with intermediate prices for the 1 MW/24 MWh reference system of NaS technology.

193

C.3 ESS cost for worst and best case scenarios

C.3

ESS cost for worst and best case scenarios

In this appendix, results from worst and best case scenario are compared, all based on the 1 MW/24 MWh reference system. In Fig. C.2 for NaS batteries the following parameters are represented (from top to bottom): revenue requirement or cost of energy (COE), relative cost of energy losses (closs ) and O&M plus replacement cost (V C). In figures C.3 and C.4 closs and V C are shown for lead-acid, flow battery, hydrogen/fuel-cell (HSFC) and pumped hydro (PHS) storage systems. (a)

(b)

A - Tx1 (24h forecast) - NaS (worst)

A - Tx1 (24h forecast) - NaS (best)

400 400 300 200 200 0 1

24

0.5

PESS [MW]

12 0 0

100

500

COE [e/MWh]

COE [e/MWh]

500

400 400 300 200 200 0 1 0.5

0

12

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (worst)

0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (best) 100

100 10

80 60

5

40 24

0.5

PESS [MW]

12 0 0

closs [%]

closs [%]

10

0 1

80 60

5

40 0 1

20

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - NaS (worst)

12 0 0

0

A - Tx1 (24h forecast) - NaS (best) 80 40

60

20

40 20 24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

0

V C [%]

V C [%]

40

20

Eeff [MWh]

80

0 1

100

24

60

20

40

0 1

20 24

0.5

PESS [MW]

12 0 0

0

Eeff [MWh]

Figure C.2. Comparison of worst and best case scenario with NaS technology, worst case (a), best case scenario (b).

194

Appendix C. ESS cost parameters and model results

(a)

(b)

A - Tx1 (24h forecast) - Lead-Acid (worst)

A - Tx1 (24h forecast) - Lead-Acid (best) 100 10

80 60

5

40 0 1

24

0.5

PESS [MW]

12 0 0

closs [%]

closs [%]

10

100 80 60

5

40 0 1

20

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (worst)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (best)

30

80

20

60 40

0 1

24

0.5

PESS [MW]

12 0 0

100

closs [%]

closs [%]

100

10

20

30

80

20

60

10

40

0 1

20

24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (worst)

12 0 0

20 0

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (best) 100 100

80 60

50

40 0 1

24

0.5

PESS [MW]

12 0 0

closs [%]

closs [%]

100

100 80 60

50

40 0 1

20 0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (worst)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (best) 100

100 100

80 60

50

40 0 1

24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

20 0

closs [%]

closs [%]

100

20

24

0.5

80 60

50

40 0 1

24

0.5

PESS [MW]

12 0 0

20 0

Eeff [MWh]

Figure C.3. Share of cost due to efficiency, worst case (a), best case scenario (b).

195

C.3 ESS cost for worst and best case scenarios

(a)

(b)

A - Tx1 (24h forecast) - Lead-Acid (worst)

A - Tx1 (24h forecast) - Lead-Acid (best) 80 80

60

60 40

40

20 0 1

V C [%]

V C [%]

80

80

24

PESS [MW]

12 0 0

40

40

20 0 1

20 0.5

60

60

20 24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (worst)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - FlowBattery (best) 80 40

60

20

40

0 1

V C [%]

V C [%]

40

80

24

PESS [MW]

12 0 0

20

40

0 1

20 0.5

60

20 24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (worst)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - HSFC (best) 80 40

60

20

40

0 1

V C [%]

V C [%]

40

80

24

PESS [MW]

12 0 0

20

40

0 1

20 0.5

60

20 24

0.5

0

PESS [MW]

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (worst)

12 0 0

0

Eeff [MWh]

A - Tx1 (24h forecast) - PHS (best) 80 40

60

20

40

0 1

20 24

0.5

PESS [MW]

12 0 0

Eeff [MWh]

0

V C [%]

V C [%]

40

80 60

20

40

0 1

20 24

0.5

PESS [MW]

12 0 0

0

Eeff [MWh]

Figure C.4. Share of O&M and replacement cost, (a) worst case, (b) best case scenario.

Appendix D Bidding results for 2008

This appendix contains results from bidding strategies presented in chapter 5 (section 5.4), applied to market data from 2008. The main difference is that marginal electricity prices in 2008 were 50% higher than 2007. At the same time, regulation costs did not increase equally. Therefore performance ratios are higher in all considered scenarios. In addition, improvements due to advanced bidding strategies are smaller than those for market prices of 2007.

198

D.1

Appendix D. Bidding results for 2008

Bidding with annual and monthly means

Price [e/MWh] Data

Ereg [%]

[GWh/a]

PR [%]

πref

πobs

Up

Down

total

Etotal

Ec

A T×1 A T×1 1a A T×1 30d

93.0 93.2 93.4

63.17 63.17 63.17

58.72 58.87 59.03

37.2 52.5 53.9

38.6 30.1 31.2

75.8 82.6 85.1

2.20 2.20 2.20

2.23 2.77 2.77

B T×1 B T×1 1a B T×1 30d

92.9 93.1 93.3

63.99 63.99 63.99

59.47 59.56 59.72

35.8 52.4 53.3

36.7 27.6 29.2

72.5 80.0 82.5

2.68 2.68 2.68

2.65 3.34 3.32

C T×1 C T×1 1a C T×1 30d

92.2 92.4 92.5

63.52 63.52 63.52

58.58 58.70 58.78

37.4 53.9 54.8

38.0 29.9 31.6

75.4 83.7 86.4

2.28 2.28 2.28

2.27 2.83 2.81

C MSEc C MSEc 1a C MSEc 30d

93.8 93.9 94.1

63.52 63.52 63.52

59.57 59.65 59.74

33.8 48.6 48.6

33.7 24.7 26.1

67.5 73.3 74.6

2.28 2.28 2.28

2.29 2.83 2.80

C MCCc C MCCc 1a C MCCc 30d

93.6 93.7 93.9

63.52 63.52 63.52

59.48 59.53 59.63

33.2 47.6 47.8

34.0 25.4 26.8

67.2 73.0 74.6

2.28 2.28 2.28

2.27 2.79 2.76

C MSE C MSE 1a C MSE 30d

93.7 93.9 94.0

63.52 63.52 63.52

59.51 59.64 59.72

36.0 50.9 51.4

33.3 23.8 25.3

69.3 74.8 76.6

2.28 2.28 2.28

2.35 2.90 2.88

C MCC C MCC 1a C MCC 30d

93.2 93.7 93.8

63.52 63.52 63.52

59.17 59.52 59.57

17.7 25.6 26.0

46.0 37.3 38.5

63.7 62.9 64.5

2.28 2.28 2.28

1.64 2.02 2.00

Table D.1. Bidding with point forecast and uncertainty, using mean penalty of one year (1a) and one month (30d); simulation results for 2008 Spanish energy market. Wind power data from 3 different sites (A, B and C) and 3 forecast methods (Tx1, MSE, MCC).

199

D.2 Bidding with daily profiles

D.2

Bidding with daily profiles

Price [e/MWh] Data

Ereg [%]

[GWh/a]

PR [%]

πref

πobs

Up

Down

total

Etotal

Ec

A T×1 A T×1 1a dp A T×1 3m dp

93.0 93.6 93.7

63.17 63.17 63.17

58.72 59.10 59.19

37.2 52.4 53.2

38.6 31.8 32.0

75.8 84.1 85.3

2.20 2.20 2.20

2.23 2.72 2.74

B T×1 B T×1 1a dp B T×1 3m dp

92.9 93.6 93.7

63.99 63.99 63.99

59.47 59.88 59.96

35.8 51.2 52.5

36.7 28.5 28.9

72.5 79.7 81.4

2.68 2.68 2.68

2.65 3.29 3.31

C T×1 C T×1 1a dp C T×1 3m dp

92.2 92.7 92.8

63.52 63.52 63.52

58.58 58.85 58.95

37.4 54.6 55.8

38.0 32.7 33.2

75.4 87.3 89.0

2.28 2.28 2.28

2.27 2.78 2.80

C MSEc C MSEc 1a dp C MSEc 3m dp

93.8 94.0 94.2

63.52 63.52 63.52

59.57 59.73 59.84

33.8 49.9 50.2

33.7 27.1 27.4

67.5 77.1 77.6

2.28 2.28 2.28

2.29 2.81 2.81

C MCCc C MCCc 1a dp C MCCc 3m dp

93.6 94.0 94.1

63.52 63.52 63.52

59.48 59.68 59.80

33.2 48.0 48.5

34.0 27.6 27.8

67.2 75.6 76.3

2.28 2.28 2.28

2.27 2.75 2.76

C MSE C MSE 1a dp C MSE 3m dp

93.7 94.0 94.2

63.52 63.52 63.52

59.51 59.72 59.82

36.0 52.4 53.2

33.3 26.6 26.8

69.3 79.0 80.0

2.28 2.28 2.28

2.35 2.88 2.89

C MCC C MCC 1a dp C MCC 3m dp

93.2 94.0 94.1

63.52 63.52 63.52

59.17 59.68 59.78

17.7 26.0 26.4

46.0 39.1 39.2

63.7 65.1 65.5

2.28 2.28 2.28

1.64 1.99 1.99

Table D.2. Advanced bidding, using daily profiles from one year (1a dp) and seasonal variation (3m dp); simulation results for 2008 Spanish energy market.

200

Appendix D. Bidding results for 2008

D.3

Total vs. penalised deviations

Ereg [%] Data

PR [%]

Up

92.2 92.4 92.7 92.5 92.8

37.4 53.9 54.6 54.8 55.8

38.0 29.9 32.7 31.6 33.2

1a 1a dp 30d 3m dp

93.8 93.9 94.0 94.1 94.2

33.8 48.6 49.9 48.6 50.2

1a 1a dp 30d 3m dp

93.6 93.7 94.0 93.9 94.1

Epen [%] Up

Down

total

75.4 83.7 87.3 86.4 89.0

17.8 25.3 24.3 23.8 24.1

18.4 14.3 14.8 14.3 14.8

36.2 39.7 39.1 38.1 38.9

33.7 24.7 27.1 26.1 27.4

67.5 73.3 77.1 74.6 77.6

15.6 22.5 22.0 21.2 21.4

14.7 10.7 11.1 10.6 11.1

30.3 33.2 33.1 31.8 32.5

33.2 47.6 48.0 47.8 48.5

34.0 25.4 27.6 26.8 27.8

67.2 73.0 75.6 74.6 76.3

15.6 22.4 21.5 21.1 21.0

15.0 11.2 11.5 11.2 11.4

30.6 33.6 33.0 32.3 32.5

1a 1a dp 30d 3m dp

93.7 93.9 94.0 94.0 94.2

36.0 50.9 52.4 51.4 53.2

33.3 23.8 26.6 25.3 26.8

69.3 74.8 79.0 76.6 80.0

16.6 23.6 23.2 22.3 22.8

14.2 9.8 10.4 9.8 10.3

30.8 33.4 33.5 32.0 33.1

1a 1a dp 30d 3m dp

93.2 93.7 94.0 93.8 94.1

17.7 25.6 26.0 26.0 26.4

46.0 37.3 39.1 38.5 39.2

63.7 62.9 65.1 64.5 65.5

8.4 12.2 11.8 11.7 11.6

20.0 15.9 16.0 15.7 15.8

28.3 28.1 27.8 27.4 27.4

C C C C C

T×1 T×1 T×1 T×1 T×1

C C C C C

MSEc MSEc MSEc MSEc MSEc

C C C C C

MCCc MCCc MCCc MCCc MCCc

C C C C C

MSE MSE MSE MSE MSE

C C C C C

MCC MCC MCC MCC MCC

1a 1a dp 30d 3m dp

Down total

Table D.3. Total and penalised up and down regulation energy as percent of total generation; simulation results for 2008 Spanish energy market.

201

D.4 Cost of forecast uncertainty

D.4

Cost of forecast uncertainty

Below, results from 2008 for performance ratio PR, deviation cost πd and penalised deviation cost πpen are shown. Performance ratio PR is higher as in 2007 (compare Fig. 5.12 on page 156), the impact of forecast uncertainty and bidding strategy is lower. Deviation unit costs πd and πpen are higher in 2008 than in 2007, and also a reduced sensitivity to bidding strategies is observed, while the impact of forecast errors is unchanged. 97 Ref. 1a 1a dp 1m 3m dp

Tx0 95

PR [%]

MSEc

MSE 93

MCC

MCCc Tx1

91 40

Tx2 50

60

70

80

90

100

110

MAPE [%] 97 Ref. 1a 1a dp 1m 3m dp

Tx0 95

PR [%]

MCCc MSEc

MSE 93

MCC Tx1

91 15

Tx2 20

25

30

35

40

45

50

Epen [%] Figure D.1. Performance ratio PR as a function of MAPE (above) and penalised regulation energy (below), data from 2008.

202

Appendix D. Bidding results for 2008

10 Ref. 1a 1a dp 1m 3m dp

9

πd [e/MWh]

8 MCC MCCc

7

Tx1

Tx0

Tx2 MSE

6 5

MSEc 4 40

50

60

70

80

90

100

Ref. 1a 1a dp 1m 3m dp

16 MCC 15

πpen [e/MWh]

MCCc 14

110

Tx1 Tx0

Tx2

MSE

13 12

MSEc

11 10 40

50

60

70

80

90

100

110

MAPE [%] Figure D.2. Cost of (wind power) forecast uncertainty πd (above) and cost of penalised deviations πpen (below) as a function of MAPE, data from 2008.

Appendix E List of acronyms AACAES A-D ANEMOS

AR ARIMA BESS BETTA BEV BoP CAES CASH cdf CES CHP COE CRF CVaR DESS DFIG DOD EC EEX EMA EPRI EPS

Advanced Adiabatic CAES Anderson-Darling Development of A NExt Generation Wind Resource Forecasting SysteM for the Large-Scale Integration of Onshore and OffShore Wind Farms Autoregressive Autoregressive integrated moving average Battery Energy Storage System British Electricity Trading and Transmission Arrangements Battery Electric Vehicle Balance of Plant Compressed Air Energy Storage Compressed Air Storage with Humidification Cumulative Distribution Function Cryogenic Energy Storage Combined Heat and Power Cost of Energy Capital Recovery Factor Conditional Value at Risk Distributed Energy Storage System Doubly Fed Induction Generator Depth of Discharge Electrochemical Capacitor (Supercapacitor) European Energy Exchange Exponential Moving Average Electric Power Research Institute Electrical Power System

204 ERGO ESA ESS ETR EV / EDV FC FCV FRT FW GIGARCH HEV HSFC ICEV KDE K-S L/A Li-Ion LPX LQR MA MAE MAPE MCC MCCc MCFC MEE MIBEL MOS MRI MSE MSEc NaNiCl NaS NETA NiCd NiMH NMAE NN

Appendix E. List of acronyms Electric Recharge Grid Operator Electricity Storage Association Energy Storage System Energy Throughput Ratio Electric (Drive) Vehicle Fuel Cell Fuel Cell Vehicle Fault-Ride-Through Flywheel Gegenbauer Integrated Generalized Autoregressive Conditional Heteroscedasticity Hybrid EV Hydrogen Storage with Fuel Cell Internal Combustion Engine Vehicle Kernel Density Estimation Kolmogorov-Smirnov Lead-Acid Lithium-Ion Leipzig Power Exchange Local Quantile Regression Moving Average Mean Absolute Error Mean Absolute Percentage Error Maximum Correntropy Criterion bias-corrected MCC Molten Carbonate FC Minimum Entropy Error Mercado Ib´erico de Energ´ıa (Spanish electricity spot market) Model Output Statistics Meteo Risk Index Mean Squared Error bias-corrected MSE Sodium Nickel Chloride (see ZEBRA) Sodium-Sulfur New Electricity Trading Arrangements Nickel-Cadmium Nickel-Metal-Hydride Normalized MAE Neural Network

205 NPRI NRMSE NWP O&M PCM PCS pdf PEMFC PHEV PHS PJM PM PMU PQ PR PRI PS PTU PV REE RFB RMSE SCADA SMES SNL SOC SOFC TES UPS V2G VRB VRLA WAMS WECS WMA WPPT ZBB ZEBRA

Normalized PRI Normalized RMSE Numerical Weather Prediction Operation and Maintenance Power Curve Model Power Conditioning System Probability Density Function Proton Exchange Membrane FC Plug-in Hybrid EV Pumped Hydro Storage Pennsylvania New Jersey Maryland Interconnection LLC (Regional Transmission Organization of 13 states in the Eastern USA) Precio Marginal (marginal price) Phasor Measurement Units Power Quality Performance Ratio Prediction Risk Index Peak Shaving Program Time Unit Photovoltaics Red El´ectrica de Espa˜ na (Spanish transmission system operator) Redox Flow Battery Root Mean Squared Error Supervisory Control And Data Acquisition Superconducting Magnetic Energy Storage Sandia National Laboratories State Of Charge Solid Oxide FC Thermal Energy Storage Uninterruptible Power Supply Vehicle to Grid Vanadium Redox Battery Valve Regulated Lead-Acid (Battery) Wide Area Measurement System Wind Energy Conversion System Weighted Moving Averages Wind Power Prediction Tool Zinc-Bromine (Flow) Battery Zeolite Battery Research Africa Project (see NaNiCl)

Appendix F List of symbols Symbol A2 a, b ak B(·) CESS Closs closs CRF c, c1 , c2 ce ck cm cp cs COE d0 Dm DOD e00 e0 Ec Ec,opt ED eD Edir Eeff

Unit

[e] [e] [%]

[e/kWh]

[e/kWh] [e/MWh] [h] [p.u.],[%] [p.u.] [p.u.] [MWh] [MWh] [MWh] [p.u.] [MWh] [MWh]

Description Anderson-Darling (A-D) test statistic Parameters of the Beta pdf Correlation coefficient Beta function Annualized ESS cost Cost related to the efficiency of the storage system Normalized Closs Capital Recovery Factor Approximation parameters Energy related ESS cost coefficient Replacement cost coefficient O&M cost coefficient Power related ESS cost coefficient Saturation time splitting coefficient Annualized cost of energy or revenue requirement Typical discharge time (or energy autonomy) of the ESS Kolmogorov-Smirnov (K-S) test statistic Depth of discharge Zero error reference of ESS energy capacity Zero error ESS energy capacity Contracted energy Optimized contracted energy (optimal bid) Deviation from contracted energy Normalized ED Energy directly injected into the grid Effective ESS energy capacity

208

Appendix F. List of symbols

Symbol

Unit

EESS eESS ∆eESS Eg Eg,t ein , eout Eloss eloss emax emin Epen epen Ereg Etotal Etp Eu eu eue eup ex

[p.u.] [p.u.] [p.u.] [MWh] [MWh] [p.u.] [MWh] [p.u.] [p.u.] [p.u.] [%] [p.u.] [%] [MWh] [MWh] [MWh] [p.u.] [p.u.] [p.u.] [p.u.]

E ETR ETR0 ETR00 f (·) f0 (·) fm (·) F (·) F0 (·) Fm (·) Fsoc (·) −1 Fsoc (·) −1 Fp,t (·) fsat fup fue

[p.u.] [p.u.] [p.u.]

Description Nominal (rated) ESS energy capacity Normalized EESS Deviation of eESS estimation Generated energy Generated energy at time step t Normalized input and output energy of the ESS Lost energy due to ESS efficiency Normalized Eloss Maximum observed SOC Minimum observed SOC Penalised regulation energy Penalised regulation energy Regulation energy Total generated (wind) energy Energy throughput of an ESS Unserved energy Normalized unserved energy eu due to reduced ESS energy capacity eu due to reduced ESS rated power Reduced ESS energy capacity corresponding to x% saturation time Expectation operator Energy throughput ratio Idealized ETR (100% ESS efficiency) Reduced throughput ratio Probability density function (pdf) Continuous pdf Empirical pdf with m bins Cumulative probability distribution function (cdf) Continuous cdf Empirical cdf with m bins cdf of SOC Inverse cdf of SOC Inverse cdf of wind power forecast pˆ at time t Deviation factor of saturation time estimation Deviation factor of unserved power estimation Deviation factor of unserved energy estimation

209

Symbol g(·) ge gk gm IC i I j k k0 lk MAE m nc P p pt P¯ p¯ Pˆ pˆ pˆc pˆoc PESS pESS ∆pESS PˆNR Pinst P¯ue p¯ue PR q ql qu R

Unit [p.u.],[%] [p.u.],[%] [p.u.],[%] [e] [p.u.],[%] [p.u.] [h]

[kW],[MW]

[kW],[MW] [p.u.] [p.u.] [kW],[MW] [p.u.] [kW],[MW] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [kW],[MW] [kW],[MW] [kW],[MW] [p.u.] [%] [p.u.],[%] [p.u.],[%] [p.u.],[%] [e]

Description Loss function Electricity price escalation rate Mean annual inflation rate for replaced major components Mean annual inflation rate for O&M Initial cost (capital cost) annual interest rate Interval of interval forecast Counting index Forecast delay (horizon) or market closure delay Major components to be replaced during the system’s service period Number of replacements of k th ESS component Mean absolute error Number of bins of an empirical pdf Number of equivalent full charge-discharge cycles Measured (wind) power Normalized P Normalized P at time step t (Long-term) mean (wind) power Normalized (long-term) mean (wind) power (Wind) power forecast Normalized (wind) power forecast Corrected normalized (wind) power forecast Online corrected normalized (wind) power forecast Nominal (rated) ESS power Normalized PESS Deviation of pESS estimation New reference (wind) forecast Installed (wind) power Unserved mean power demand due to limited ESS energy capacity Normalized P¯ue Performance ratio Quantile / Percentile Lower quantile / percentile (interval forecast) Upper quantile / percentile (interval forecast) Revenue

210

Appendix F. List of symbols

Symbol

Unit

Rt SOC SOCkW h t tsat ∆t T u VC w y yk yss

[e] [p.u.] [kWh] [s],[min],[h] [%] [s],[min],[h] [h] [e] [a] [a] [a]

Description Revenue for time step t Normalized state of charge Absolute state of charge Time Saturation time Time step length of the measured time series Forecast time interval Slope of linear fit Variable cost Weighting coefficient Years of operation of an ESS configuration Lifetime of the k th ESS component ESS lifetime depending on technology Greek Letters

α αl αu αopt,t β γ δ ε ε¯ εˆ ζ η κ κbeta λ µ ν ξ π

[p.u.],[%] [p.u.],[%] [p.u.],[%] [p.u.],[%] [p.u.],[%]

[e/MWh]

πc πc,t πd

[e/MWh] [e/MWh] [e/MWh]

[p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.],[%]

[p.u.]

Quantities or probabilities of a cdf Lower quantities (interval forecast) Upper quantities (interval forecast) Optimal quantity or probability at time step t Coverage rate (interval forecast) Skewness Energy ratio covered directly by the ESS Normalized forecast error Mean forecast error (bias) Forecast error forecast Power ratio covered by the ESS ESS round-trip or cycle efficiency Kurtosis Kurtosis of Beta pdf Weighting exponent of weighted moving average (WMA) Mean Empirical scaling factor for estimation of eue Asymmetry Ratio Mean market price (marginal price or spot price) of electricity Contracted electricity price (spot price) πc at time step t Cost of (wind power) forecast uncertainty

211

Symbol

Unit

πobs πpen πreg πreg,t πref πud πup , πdn π+ πt+ π− πt− %k

[e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [p.u.],[%]

σ τ χ2

[p.u.],[%] [h],[d]

Description Observed price Cost of penalised deviations Regulation cost πreg at time step t Reference price Regulation price (up or down) Up and down regulation price Down regulation unit cost (positive deviations) π + at time step t Up regulation unit cost (negative deviations) π − at time step t Mean annual technological cost reduction of k th ESS component Standard deviation (NRMSE) Window width of moving average Pearson’s chi square test statistic

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Curriculum Vitæ

Hans Bludszuweit was born in Jena, Germany, in 1974. He received the Dipl.Ing. degree in Electrical Engineering from the Technical University of Ilmenau (Germany) in 2001 and carried out Ph.D. research at the University of Zaragoza (Spain) from 2003 until 2009. He spent a year working as a field test engineer for the german photovoltaic manufacturer ANTEC Solar GmbH, Arnstadt and the german subsidy of US-american First Solar Inc. He was for one year with the hydrogen research group at the Instituto de Investicaciones El´ectricas (IIE) in Cuernavaca, Mexico, before joining in 2003 the CIRCE foundation in Zaragoza, Spain, as a Ph.D. student. In 2008 he carried out Ph.D. research for three months with INESC Porto (Portugal). His interests include renewable energy and energy storage in distributed generation systems.