Prediction of Wind Farm Power Ramp Rates: A Data ... - CiteSeerX

Haiyang Zheng Andrew Kusiak e-mail: [email protected] Department of Mechanical and Industrial Engineering, 3131 Seamans Center, University of Iowa, Iowa City, IA 52242-1527

Prediction of Wind Farm Power Ramp Rates: A Data-Mining Approach In this paper, multivariate time series models were built to predict the power ramp rates of a wind farm. The power changes were predicted at 10 min intervals. Multivariate time series models were built with data-mining algorithms. Five different data-mining algorithms were tested using data collected at a wind farm. The support vector machine regression algorithm performed best out of the five algorithms studied in this research. It provided predictions of the power ramp rate for a time horizon of 10–60 min. The boosting tree algorithm selects parameters for enhancement of the prediction accuracy of the power ramp rate. The data used in this research originated at a wind farm of 100 turbines. The test results of multivariate time series models were presented in this paper. Suggestions for future research were provided. 关DOI: 10.1115/1.3142727兴 Keywords: power ramp rate prediction, wind farm, data-mining algorithms, multivariate time series model, parameter selection

1

Introduction

Wind power generation is rapidly expanding and is becoming a noticeable contributor to the electric grid. The fact that most largescale wind farms were developed in recent years has made studies of their performance overdue. Given the changing nature of the wind regime, wind farm power varies across all time scales. The fluctuating power of wind farms is usually balanced by the power produced by the traditional power plants to meet the grid requirements. The change of power output in time is referred to as ramping and it is measured with the power ramp rate 共PRR兲. The prediction of PRR at 10 min intervals is of interest to the wind industry due to the tightening electric grid requirements 关1兴. Though the power prediction research has a long tradition in the wind industry, the interest in prediction of power ramps is emerging. There is no industry standard for PRR prediction. Power ramp rate on 10 min intervals is to benefit the gird management and power scheduling in the wind industry. The literature related to power ramps is discussed next. Svoboda et al. 关2兴 proposed a Lagrangian relaxation method to solve hydrothermal generation scheduling problems. Three PRR constraints were considered and illustrated with a numerical example. Ummels et al. 关3兴 presented a simulation method to evaluate the integration of large-scale wind farm power with the conventional power generation sources from a cost, reliability, and environmental perspective. Based on the PRR constraints for the reserve activation and generation schedule, the capability of a thermal generation system for balancing a wind power was investigated. Potter and Negnevitsky 关4兴 applied an adaptive-neuron-fuzzy inference approach to forecast short-term wind speed and direction. Torres et al. 关5兴 used transformed data to build the autoregressive moving average 共ARMA兲 time series model for prediction of mean hourly wind speed of up to 10 h into the future. Sfetsos 关6兴 presented a novel method for forecasting mean hourly wind speed based on the time series analysis data and showed that the developed model outperformed the conventional forecasting models. Lange and Focken 关7兴 presented various models for short-term wind power prediction, including physics-based, fuzzy, and neu-

rofuzzy models. Using meteorological data, Barbounis et al. 关8兴 constructed a local recurrent neural network model for long-term wind speed and power forecasting. Hourly wind farm forecasts of up to 72 h were produced. Developing power and PRR prediction models for wind farms is challenging, as power output is known to undergo rapid variations due to changes in the wind speed, e.g., due to gusts. The power output strongly depends on the wind conditions and the changing environment of the wind farm. The stochastic nature of a wind farm environment calls for new modeling approaches to accurately predict the power ramp rate. Data mining is a promising approach for modeling wind farm performance. Numerous applications of data mining in manufacturing, marketing, medical informatics, and energy industry proved successful 关9–14兴. In this paper, a data-mining approach was applied to build a multivariate time series model to predict power ramp rates of a wind farm over 10 min intervals. Five different data-mining algorithms for the PRR prediction were employed. The boosting tree algorithm was used to reduce the dimensionality of the input and to enhance prediction accuracy. The models were built using historical data collected by the supervisory control and data acquisition 共SCADA兲 system installed at a wind farm.

2

Basic Methodologies for PRR Prediction

2.1 Time Series Prediction Modeling. Time series prediction 关15兴 focuses on determining future events based on known observations, measured typically at successive time intervals 共often uniform兲. Time series models are generally applicable to monitoring industrial processes and tracking time-based business metrics. There are two types of time series models: univariate and multivariate models. The univariate time series model consists of observations of a single parameter recorded sequentially over equal time increments. In the multivariate time series model, observations are fixed-dimension vectors of different parameter values. The univariate time series prediction model 关15,16兴 is expressed as follows: yˆ 共t + wT兲 = f共y共t兲,y共t − T兲, . . . ,y共t − mT兲兲

Contributed by the Solar Energy Engineering Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received August 10, 2008; final manuscript received March 6, 2009; published online July 9, 2009. Review conducted by Spyros Voutsinas.

Journal of Solar Energy Engineering

共1兲

where T is the sampling time 共interval兲, wT is the prediction horizon 共for example, for w = 2 and T = 10 min, the prediction horizon is 20 min兲, yˆ 共t + wT兲 is the predicted parameter, y共t兲 , y共t

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− T兲 , . . . , y共t − mT兲 are the current and past observed parameters, and m + 1 is the number of inputs 共predictors兲 of the model. The multivariate time series model 关15兴 is formulated as follows: yˆ 共t + wT兲 = f共y共t兲,y共t − T兲, . . . ,y共t − mT兲;x1共t兲,x1共t − T兲, . . . , x1共t − mT兲;x2共t兲,x2共t − T兲, . . . ,x2共t − mT兲; . . . ; xn共t兲,xn共t − T兲, . . . ,xn共t − mT兲兲

共2兲

where T is the sampling time 共interval兲, wT is the prediction horizon, x1 . . . , xn , y and n + 1 are the observations of the time series forming the n + 1 dimensional vector, yˆ 共t + wT兲 is the predicted parameter, y共t兲 , y共t − T兲 , . . . , y共t − mT兲 are the current and past observed values of y, x1共t兲 , x1共t − T兲 , . . . , x1共t − mT兲 are the current and past observed values of parameters x1 , . . . , xn, and 共m + 1兲 ⫻ 共n + 1兲 is the number of inputs 共predictors兲 of the model. To obtain an accurate prediction model with the data-mining approach, appropriate parameters 共predictors兲 need to be selected. Data mining offers different algorithms to perform this task. For example, the boosting tree algorithm 关17,18兴 and the wrapper approach 关19,20兴, utilizing the genetic or the first best search algorithm 关13,21兴 select the important predictors. The total number of all possible predictors 共m + 1兲 ⫻ 共n + 1兲 forms a high-dimensional input to the time series model, and therefore the performance of the resultant model is likely to be inferior. To maximize performance of the prediction model, a boosting tree algorithm is employed to select a set of the most important predictors among the 共m + 1兲 ⫻ 共n + 1兲 ones in Eq. 共2兲: 兵y共t兲,y共t − T兲, . . . ,y共t − mT兲;x1共t兲,x1共t − T兲, . . . , x1共t − mT兲; . . . ;xn共t兲,xn共t − T兲, . . . ,xn共t − mT兲其 2.2 Prediction Accuracy Metrics. Two main metrics, the mean absolute error 共MAE兲 and the standard deviation 共Std兲 of the absolute error 共AE兲, were used to measure prediction accuracy of different data-mining algorithms. The small value of MAE and Std imply the superior prediction performance of the models extracted by data-mining algorithms. In fact, MAE and Std based on absolute error are widely used in the wind industry. Their definitions are expressed as AE = yˆ 共t + wT兲 − y共t + wT兲

共3兲 Fig. 1 Typical power, power ramp rate, and wind speed plots: „a… wind farm power, „b… power ramp rate, and „c… wind speed

N

兺 AE共i兲 MAE =

Std =

冑

i=1

N

共4兲

N

兺共AE共i兲 − MAE兲 i=1

N−1

共5兲

where yˆ 共t + wT兲 is the predicted PRR, y共t + wT兲 is the observed 共measured兲 PRR, and N is the number of test data points for the prediction model. The data set used by the PRR prediction models is divided into training and test data sets. 2.3 Data Description. The data used in this research were generated at a wind farm with 100 turbines. Though the data were sampled at high frequency, e.g., 2 s, it was averaged and stored at 10 min intervals 共referred to as the 10 min average data兲. The data used in this research were collected over a period of 1 month for all turbines of the wind farm. Some data contained many missing values or abnormal values outside of the normal physical range, and thus 89 turbines were selected for the study. For example, the SCADA recorded wind speed should be in the range 0–20 m/s, and the power should be in the range 0–1600 kW. As the rated power 031011-2 / Vol. 131, AUGUST 2009

of each turbine is 1.5 MW, the capacity of the wind farm is 133.5 MW. The power ramp rate used in this paper is defined as the rate of change of wind farm power during a 10 min interval 共the standard time interval in wind energy industry兲 and is expressed in kW/ min: PRR =

兩P共t + 10兲 − P共t兲兩 10

共6兲

where P共t + 10兲 is the wind farm power at time t + 10 共time t plus 10 min兲 and P共t兲 is the wind farm power at time t. The power ramp rate expresses the rate of change of the wind farm power due to the stochastic nature of the wind. Figure 1共a兲 illustrates the power produced by a wind farm over 10 min intervals. Figure 1共b兲 shows the power ramp rate corresponding to the power presented in Fig. 1共a兲. Figure 1共c兲 shows the wind speed for the time period considered in Figs. 1共b兲 and 1共c兲. Ignoring the power consumed by the wind farm, the power produced is always positive 共Fig. 1共a兲兲; however, the PRR can be positive or negative. The positive PRR indicates increasing power over time, while the negative PRR value means that the wind farm power is decreasTransactions of the ASME


Table 1 List of parameters Parameter Mean Std Max Min Power PRR

Table 3 The importance index of predictors generated by the boosting tree algorithm for t + 10 model

Description

Unit

Mean wind speed of a turbine Standard deviation of the wind speed of a turbine Maximum wind speed of a turbine Minimum wind speed of a turbine Wind farm power Power ramp rate of the wind farm

m/s m/s m/s m/s kW kW/min

Table 2 The data set description Data set 1 2 3

Start time stamp

End time stamp

Description

Total data set; 4455 observations Training data set; 3568 1/1/07 1:40 a.m. 1/25/07 8:00 p.m. observations Test data set; 887 1/25/07 8:10 p.m. 1/31/07 11:50 p.m. observations 1/1/07 1:40 a.m.

1/31/07 11:50 p.m.

ing. The larger the absolute value of PRR, the faster the power surge 共or drop兲. The wind speeds of 89 turbines, the wind speed statistics, and the power collected by the SCADA system were used in data mining. In this paper, six different parameters were used to build the multivariate time series model. The mean, Std, max, min, and power are the first five parameters x1 , . . . , x5 and the PRR is the sixth parameter y of model 共2兲. Table 1 lists all the parameters used in this paper. The number of parameters is limited by the data available in this research. The model accuracy could be enhanced if more data were available. The six parameters recorded at 10 min intervals resulted in 4455 instances 共data set 1 in Table 2兲, beginning from “1/1/07 at 1:40 a.m.” and continuing to “1/31/07 at 11:50 p.m.” During this time period, the overall wind farm performance was considered to be normal. Data set 1 was divided into two subsets: data set 2 and data set 3. Data set 2 contains 3568 data points and were used to develop a prediction model with data-mining algorithms. Data set

Predictor

Variable rank

Importance

PRR-1 PRR-2 PRR-3 PRR-4 PRR-5 Mean-1 Mean-2 Mean-3 Mean-4 Mean-5 Min-1 Min-2 Min-3 Min-4 Min-5 Max-1 Max-2 Max-3 Max-4 Max-5 Std-1 Std-2 Std-3 Std-4 Std-5 Power-1 Power-2 Power-3 Power-4 Power-5

100 100 66 53 71 44 49 38 41 37 67 52 49 44 42 45 48 37 42 40 43 51 45 43 36 40 54 48 41 39

1.00 1.00 0.66 0.53 0.71 0.44 0.49 0.38 0.41 0.37 0.67 0.52 0.49 0.44 0.42 0.45 0.48 0.37 0.42 0.40 0.43 0.51 0.45 0.43 0.36 0.40 0.54 0.48 0.41 0.39

3 includes 887 data points and were used to test the prediction performance of the model extracted from data set 2. For the test data set, the MAE 共Eq. 共4兲兲 and Std 共Eq. 共5兲兲 were the metrics used to evaluate the data-mining algorithms applied to learn multivariate time series model of Sec. 2.1. 2.4 Parameter Selection. Due to the high-dimensionality of the input vector of predictors of the multivariate time series model, the number of inputs was reduced. The quality of the mod-

Fig. 2 The importance of predictors generated by the boosting tree algorithm for the t + 10 model


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Fig. 3 Illustration of the multiperiod multivariate time series prediction model: „a… the t + 10 min PRR prediction and „b… the t + 20 min PRR prediction

els learned from high- and reduced-dimensionality data were compared in Secs. 3.1 and 3.2. The most significant predictors were determined by the boosting tree algorithm 关17,18兴. The same approach was shown to be successful in a previous research 关14兴. The basic idea of the boosting tree algorithm is to build a number of trees 共e.g., binary trees兲 splitting the data set and to approximate the underlying function. The importance of each predictor is measured by its contribution to the prediction accuracy of the training data set. To build a multivariate t + 10 time series model 共for 10 min ahead predictions兲, the value of m = 5 used in the multivariate model is selected, which means that four values observed in the past and one current value of each parameter are considered. In total, six different parameters of the multivariate model were considered and thus it contains 共5 ⫻ 6兲 = 30 predictors. The 30dimensional input is reduced by the boosting tree algorithm. Table 3 shows the importance index of 30 predictors computed by the boosting tree algorithm based on data set 2 of Table 2. The index “-1” in Table 3 indicates the observation sampled 10 min earlier, “-2” indicates the observation sampled 20 min earlier, and “-3, -4, and -5” indicate the observations sampled 30 min, 40 min, and 50 min earlier, respectively. Note that all the parameter values used in this paper were all average values over the 10 min interval. Figure 2 shows the importance of all 30 predictors for the t + 10 min models ranked from the largest to the smallest one. To maximize prediction accuracy it is important to select important predictors among the ones on the list

Fig. 4 Prediction results produced by the t + 10 model without parameter selection: „a… prediction performance of the five different algorithms for the test data set of Table 2 and „b… the observed and predicted PPRs by the SVM algorithm

predictors that could degrade performance of the models due to the “curse of dimensionality” principle 关19,22兴, which means that high-dimension input could negatively impact performance of the model built by the data-mining algorithm.

A threshold value of 0.50 was established heuristically to select the predictors for the time series models. The predictors selected by the boosting tree algorithm for the t + 10 min PRR are PPR-1, PPR-2, PPR-5, Min-1, PPR-3, Power-2, PRR-4, Min-2, and Std-2. The number of predictors was reduced from 30 to 9. The threshold value of 0.50 used in the computation produced good quality results. A lower threshold value would lead to more

2.5 Multiperiod Predictions With a Multivariate Time Series Model. The t + 10 min prediction model is not sufficient for integration of the wind farm with the power grid. Six different multivariate time series models are needed to predict the PRR at t + 10– t + 60 min intervals. For t + 10 interval prediction, data set 2 in Table 2 is used for parameter selection and building time series models with data-mining algorithms, and the test data 共data set 3 in Table 2兲 were used to validate performance of the models. For t + 20– t + 60 predictions, the training data set remains the same; however, the test data set containing 887 points is reduced by one for each of the next 10 min period predictions. Figure 3 illustrates the concept of a multiperiod prediction for PRR over 10 min intervals. In this model, the sampling time period T is 10 min. Using the 10 min average measured values 共including mean, Std, max, min, power, and PRR in Table 1兲 at the intervals 关t = −50, t = −40兲 , . . . , 关t = −10, t = 0−兲, the average PRR value at the subsequent interval t + 10 is predicted 共Fig. 3共a兲兲. In

Table 4 Prediction error of the t + 10 models without parameter selection generated by the five different algorithms

Table 5 Prediction error of the t + 10 model with selected parameters generated by five different algorithms

Absolute error 共kW/min兲

Absolute error 共kW/min兲

兵y共t兲,y共t − T兲, . . . ,y共t − mT兲;x1共t兲,x1共t − T兲, . . . , x1共t − mT兲; . . . ;xn共t兲,xn共t − T兲, . . . ,xn共t − mT兲其

MLP SVM

Random forest tree Pace regression

C&R

MAE

Std

Maximum

Minimum

340.66 298.94 360.19 396.62 312.44

448.19 323.32 407.56 396.62 342.33

5119.73 2512.34 2657.89 4236.02 3516.80

0.03 0.15 0.15 0.38 0.03

031011-4 / Vol. 131, AUGUST 2009

MLP SVM


C&R

MAE

Std

Maximum

Minimum

280.13 243.14 307.97 356.79 290.57

309.38 276.39 335.56 323.92 318.37

3248.12 2817.77 3860.94 3516.65 3270.62

0.16 0.03 0.61 0.15 0.03

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Table 6 The importance index of predictors generated by the boosting tree algorithm for t + 20 model

Fig. 5 The prediction results of the t + 10 model with parameter selection: „a… prediction performance of the five algorithms for the test data set of Table 2 and „b… observed and predicted PRRs by the SVM algorithm

Fig. 3共b兲, based on the measured values 共including mean, Std, max, min, power, and PRR in Table 1兲 at the intervals 关t = −50, t = −40兲 , . . . , 关t = −10, t = 0兴, the average PRR value at the subsequent interval t + 20 is predicted. Similarly, with the same input and different models, the 10 min average PRR values at intervals t + 30, t + 40, and t + 50 are predicted.

3

Industrial Case Study

3.1 The t + 10 min PRR Prediction Without Parameter Selection. To compare the accuracy of models built before and after parameters selection, the original 30 predictors were used as inputs to construct a multivariate time series model. Five different data-mining algorithms were applied to build PRR prediction

Predictor

Variable rank

Importance

Mean-1 Mean-2 Mean-3 Mean-4 Mean-5 Std-1 Std-2 Std-3 Std-4 Std-5 Max-1 Max-2 Max-3 Max-4 Max-5 Min-1 Min-2 Min-3 Min-4 Min-5 PRR-1 PRR-2 PRR-3 PRR-4 PRR-5 Power-1 Power-2 Power-3 Power-4 Power-5

54 50 41 39 31 40 46 48 46 32 68 61 42 47 36 33 46 31 32 28 100 72 26 49 38 68 57 46 47 40

0.54 0.50 0.41 0.39 0.31 0.40 0.46 0.48 0.46 0.32 0.68 0.61 0.42 0.47 0.36 0.33 0.38 0.31 0.32 0.28 1.00 0.72 0.26 0.52 0.38 0.68 0.57 0.50 0.51 0.40

models for a wind farm based on data set 2 of Table 2. These algorithms include the multilayer perceptron algorithm 共MLP兲关23,24兴, the support vector machine 共SVM兲 regression 关25,26兴, the random forest 关27,28兴, the classification and regression 共C&R兲 tree 关13,29兴, and the pace regression algorithm 关13,30兴. The five algorithms used in this research are representative of different classes of data-mining algorithms. The MLP algorithm is usually used in nonlinear regression and classification modeling. The SVM is a supervised learning algorithm used in classification and regression. It constructs a linear discriminant function that separates instances as widely as possible. The C&R tree builds a decision tree to predict either classes 共classification兲 or Gaussians 共regression兲. The random forest algorithm grows many classification trees to classify a new object from an input vector. Each tree

Fig. 6 The importance of predictors computed by the boosting tree algorithm


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Table 7 Prediction error for the t + 20 models generated by the five different algorithms Absolute error 共kW/min兲 MLP SVM


C&R

MAE

Std

Maximum

Minimum

362.52 301.31 364.28 336.25 336.79

360.21 319.48 366.12 340.41 347.08

3960.36 3635.03 4067.49 4473.17 4023.24

1.27 0.10 0.88 1.34 0.65

votes for every class, and finally the forest chooses the classification having the most votes over all the trees in the forest. The pace regression algorithm consists of a group of estimators that are either optimal overall or optimal under certain conditions. It is a new approach to fitting linear models in high-dimensional spaces. To test the accuracy of these algorithms, models trained from data set 2 of Table 2 were tested on data set 3 from Table 2. Table 4 shows the prediction accuracy of the models generated by the five algorithms. Figure 4共a兲 illustrates the absolute error of different algorithms. The first 100 observed PPRs and those predicted by the SVM algorithm for data set 3 were shown in Fig. 4共b兲. It can be seen from Table 4 and Fig. 4 that the SVM algorithm outperforms the other four algorithms. The C&R tree algorithm produces the worst predictions, and the pace regression algorithm performs quite well. The model can be updated to reflect the process change over time. The update frequency could be, e.g., 3 weeks. Alternatively, a separate routine could monitor the model performance and refresh the model once its performance would degrade. 3.2 The t + 10 min Prediction With Parameter Selection. In this section, the predictors as input for the multivariate time series model are selected by the boosting tree algorithm. As described in Sec. 2.3, 9 out of 30 predictors were selected to build the time series model. The nine selected predictors are PPR-1, PPR-2, PPR-5, Min-1, PPR-3, Power-2, PRR-4, Min-2, and Std-2. To test the difference between t + 10 min prediction models built with and without parameter selection, the five data-mining algorithms in Sec. 3.1 were used. Multivariate models were retrained from data set 2 of Table 2 and were tested on data set 3 from Table 2. Table 5 shows the prediction accuracy of the models generated by the five algorithms. Figure 5共a兲 illustrates the absolute error of the five algorithms, while Fig. 5共b兲 shows the first 100 observed PPRs and those predicted by the SVM algorithm for data set 3. The results in Tables 4 and 5, and Figs. 4 and 5 demonstrate that the prediction accuracy of all five algorithms was improved after parameter selection by the boosting tree algorithm. The SVM algorithm outperformed the other four algorithms in both scenarios, i.e., with and without parameter selection. 3.3 The t + 20 min Prediction With Parameter Selection. To build a multivariate time series model for t + 20 min PRR prediction, parameter selection is performed by the boosting tree algorithm. Table 6 shows the importance of 30 predictors computed by the boosting tree algorithm based on data set 2 in Table 2 and t + 20 prediction horizons. In Table 6, -1 denotes the observation sampled 10 min earlier, ⫺2 denotes the observation sampled 20 min earlier, and -3, -4, and -5 denote the observations sampled 30 min, 40 min, and 50 min in the past, respectively. Figure 6 shows the importance index of the 30 predictors for t + 20 PRR predictions ranked from the largest to the smallest one. When comparing the results in Figs. 6 and 2, and Tables 6 and 3, the importance of predictors varies for the t + 10 and t + 20 models. Similar to Sec. 2.4, 0.5 was established as a threshold to select significant predictors for t + 20 model. The boosting tree algorithm selected seven predictors and provided the following ranking: PPR-1, PPR-2, Max-1, Power-1, Max-2, Power-2, and Mean-1. 031011-6 / Vol. 131, AUGUST 2009

Fig. 7 Observed and predicted PRRs from the t + 20 models with selected parameters: „a… MLP algorithm, „b… SVM algorithm, „c… random forest algorithm, „d… C&R tree algorithm, and „e… pace regression algorithm



Table 8 Absolute error statistics for multiperiod models Absolute error 共kW/min兲 t + 30 t + 40 t + 50 t + 60

min min min min

prediction prediction prediction prediction

MAE

Std

Maximum

Minimum

329.83 347.92 387.45 458.70

347.03 418.41 404.92 469.24

4109.27 4600.32 4566.47 4972.20

0.59 1.94 0.02 0.62

Table 7 shows the prediction error of the models generated by the five algorithms 共the same as in Sec. 3.2兲. Figure 7 shows the first 100 observed and predicted PRR values for data set 3 in Table 2. The SVM algorithm outperformed the other four; however, the accuracy decreased compared with the t + 10 results reported in Sec. 3.2. 3.4 Multiperiod Prediction With Parameter Selection. As the SVM algorithm performed better for both t + 10 and t + 20 predictions. Therefore, it was selected to build multivariate time series PRR models for t + 30– t + 60 min intervals. After parameter selection with the same parameter importance threshold of 0.5, the 30 predictors were reduced to a seven-dimensional input with the boosting tree algorithm. For the t + 30 min model, the seven predictors were ranked as follows: Min-3, Min-1, Min-2, PRR-2, PRR-3, Max-3, and PRR-1. For the t + 40 min model, the ranking is PRR-2, PRR-4, PRR-1, Max-1, Power-1, PRR-3, and Mean-1. For the t + 50 min model, the ranking is PRR-1, Max-1, Mean-1, PRR-3, Std-1, PRR-4, and Power-5. And for the t + 60 min model, the ranking is Std-2, PRR-2, Mean-2, Max-2, Power-4, Power-5, and Max-3. The boosting tree algorithm selects different parameters over different periods of the PRR prediction, i.e., the results depend on the data set properties. Using the selected parameters, multiperiod prediction models were built by the SVM algorithm. The test data set used for the t + 10 min model of Sec. 3.2 containing 887 points was reduced by 1 for each of the next 10 min period predictions. Table 8 shows the absolute error statistics for the multivariate time series prediction over four different 10 min intervals. Figures 8共a兲–8共d兲 show the first 100 observed and predicted PRRs over t + 30 min, t + 40 min, t + 50 min, and t + 60 min intervals, respectively. The mean, the standard deviation, and the maximum error all increase as the prediction horizon lengthens. However, the minimum error remains relatively stable. The multivariate model provides accurate PRR prediction at the t + 10 to t + 40 intervals; however, the accuracy at the t + 50 and t + 60 intervals deteriorates. It appears that for longer horizon predictions, weather forecasting data may be useful.

4

Conclusion

In this paper, multivariate time series models for power ramp rate prediction at different time horizons, from 10 min to 60 min, were constructed. Five different data-mining algorithms were used to build the PRR prediction models. The boosting tree algorithm selected important predictors. After parameter selection, the original 30-dimensional input was significantly reduced, and thus the accuracy of the multivariate time series model was improved. The SVM algorithm outperformed the other four algorithms studied in this paper. The multivariate time series model for PRR prediction built by the SVM algorithm turned out to be accurate and robust. The models constructed in the paper predicted the power ramp at t + 10– t + 60 min intervals. A comprehensive comparative analysis of the multivariate models built with different data-mining algorithms was reported in this paper. The time series models accurately predicted the power ramp rate of the wind farm at t + 10– t + 40 horizons; however, the accuracy at t + 50 min and t + 60 min horizons degrades. The extracted Journal of Solar Energy Engineering

Fig. 8 Observed and predicted PRRs for different periods for the first 100 test data points: „a… the t + 30 min PRR model, „b… the t + 40 min PRR model, „c… the t + 50 min PRR model, and „d… the t + 60 min PRR model

models are essential in power grid integration and management. The multivariate time series prediction model may become a basis for predictive control aimed at optimizing the power ramp rate. The current wind farm power prediction models usually estimate the power at 1 h or 3 h intervals based on weather forecastAUGUST 2009, Vol. 131 / 031011-7


ing data. These predictions reveal power ramps over long time horizons. Prediction of power ramp rates at shorter intervals, e.g., 10 min, is of importance to the electric grid. The model built in this research does not use weather forecasting data, and it provides valuable ramp rate prediction on 10 min intervals. One avenue to be pursued in future research is the transformation of the time series data, e.g., using wavelets or Kalman filters. One disadvantage of the proposed approach is that the multivariate time series model used different parameters, and therefore updating the model with most current data is important. As the number of prediction steps increases, the error increases. The models investigated in this research were intended for predicting the power ramp rate at relatively short horizons. One possible mitigation strategy is to incorporate weather forecasting and additional off-site observation data, all at additional computational cost. Other research questions, including the seasonal performance of the proposed approach, could be addressed, provided that the appropriate data would be available.

Acknowledgment The research reported in the paper has been partially supported by funding from the Iowa Energy Center Grant No. 07-01.

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