Investing and Pricing with Supply Uncertainty in ... - IEEE Xplore

SMART GRID COMMUNICATIONS

Investing and Pricing with Supply Uncertainty in Electricity Market: A General View Combining Wholesale and Retail Market LI Xiaobo1, GAO Li2, WANG Gongpu3, GAO Feifei4, WU Qingwei1 Beijing University of Posts and Telecommunications, Beijing 100876, China School of Digital Media and Design Arts, Beijing University of Posts and Telecommunications, China 3 School of Computer and Information Technology, Beijing Jiaotong University, China 4 Department of Automation,Tsinghua University, State Key Lab of Intelligent Technologies and Systems, Tsinghua National Laboratory for Information Science and Technology (TNList) Beijing, China 1 2

Abstract: Renewable energy, such as wind and solar energy, may vary significantly over time and locations depending on the weather and the climate conditions. This leads to the supply uncertainty in the electricity (power) market with renewable energy integrated to power grid. In this paper, electricity in the market is classified into two types: stablesupply electricity (SSE) and unstablesupply electricity (USE). We investigate the investment and pricing strategies under the electricity supply uncertainty in wholesale and retail electricity market. In particular, our model combines the wholesale and retail market and capture the dominant players, i.e., consumers, power plant (power operator), and electricity supplier. To derive the market behaviors of these players, we formulate the market decision problems as a multistage Stackelberg game. By solving the game model, we obtain the optimal, with closedform, wholesale investment and retail pricing strategy for the operator. We also obtain the energy supplier's best price mechanism numerically under certain assumption. We find the price of SSE being about 1.4 times higher than that of USE will benefit energy supplier China Communications • March 2015

optimally, under which power plant's optimal strategy of investing is to purchase USE about 4.5 times much more than SSE. Keywords: electricity supply with uncertainty; electricity investment; electricity pricing; wholesale market; retail market; Stackelberg game

I. INTRODUCTION 1.1 Background The main characteristic of electricity is its limited storability. Though power can be stored, such as potential energy can be stored in the form of water behind dams, the capacity to quickly convert potential energy to electricity remains limited. As a consequence, the electricity prices are volatile because inventories cannot be used to smooth supply or demand shocks. In some extreme cases, even negative prices occur [1]. Another characteristic of electricity is that different energy sources have different cost structures [2]. Nuclear power and hydro energy have high fix costs, but the cost to generate electricity is low. On the other hand, coal and gas power plants have relative-

20

I n t h i s p a p e r, t h e electricity market with supply uncertainty is studied theoretically in a general view.

21

ly low fixed costs but high variable costs for burning fossil fuel. Besides, CO2 allowances is also a component of electricity prices [3]. Consequently, nuclear power and hydro energy are usually used to cover the base demand, while thermo energy is used to cover peak demand. The operation of electric power systems involves a complicated process of forecasting the demand for electricity, and scheduling and operating a large number of power plants to meet that varying demand. In the last decade, it has become apparent that the wholesale electricity prices fluctuate hour by hour, but the retail prices have been almost fixed and adjusted only very a few times per year. It is supposed to be economically inefficient when retail prices do not reflect wholesale acquisition costs. This gives impetus to dynamic retail pricing method considering demand-side participation to reflect the marginal cost of electricity, such as time-of-use (TOU) pricing, real-time pricing (RTP). In TOU pricing, both prices and time periods are known ex ante and are fixed for some duration. In RTP, prices change on an hour basis and are fixed and known only on a day-ahead or hour-ahead basis. [4] presents an overview and analysis of possible approaches to dynamic pricing. [5] shows enhancing the ability of the demand for electricity to respond to price signals could benefit not only the consumers, but also help market operate more efficiently and satisfactorily. [6] gives a computable equilibrium model to estimate ex ante TOU prices for a retail electricity market. In [7], it demonstrates that in the long-run the magnitude of efficiency gains from RTP is significant and much higher than TOU pricing. Since the liberalization of electricity market, the power industry worldwide has experienced big changes. On the one hand, it has been widely recognized that the creation of mechanisms for wholesale and retail electricity market to supply consumer energy need is at the core of the changes. For example, since 1990, a large number of electricity exchanges have opened in Europe, where suppliers and

consumers can freely negotiate the purchase and sale of electricity and the electricity prices are purely determined by supply and demand [8]. On the other hand, renewable energy resources (RER), such as wind and solar, are very promising to release the dependence on fossil fuels and reduce greenhouse gas emissions. For more comprehensive analysis of the changes, we recommend [9] and [10].

1.2 Motivation and related works It is well known that electricity generation by RER can be intermittent and unpredictable. As a consequence, the integration of RER to power grid will introduce uncertainty into the market. On the side of industry, to enhance efficiency and reliability of electrical power grid, the new concept of smart grid has emerged, where modern energy management techniques on demand side is a great challenge and attracts much attentions. To this end, many works have been done on RTP in smart grid, which can be classified into two types: those without regard to the supply uncertainty of renewable electricity, such as [11], [12]; those consider this uncertainty, such as [13], [14], [15]. [13] considers the best output strategy for a wind power producer and formulates it into a linear programming problem of moderate size. [14] considers the optimal RTP problem. The model in [15] focus the dayahead procurement and demand response on the users' side. Supply uncertainty of renewable electricity has a vital influence on the electricity market. A few recent works have been done on the framework of electricity market to study the behaviors of market players. [16] proposes a two-stage two-level model faced by power plant in the retail market where the power price in the wholesale market is not distinguished and is determined from the market clearing process. [17] proposes a similar framework as ours but doesn't go deeper theoretically as we do. [18] develops a very detailed framework of two-layer agents considering diverse players and market interactions as well as many kinds of uncertainty in Smart China Communications • March 2015

Grid, which is too complicated to make any analysis and only simulations are provided. However, as far as the authors' knowledge, there is no existing work trying an particularly clear modeling and analysis of market dynamics under this uncertainty, meanwhile capturing all the behaviors of the dominant market players, i.e., power supplier, power plant and the vast consumers. To this end, we take a supply-demand view and propose a game-theoretical model combining the wholesale market and retail market, and investigate the pricing and investment behaviors in the wholesale market as well as the pricing behavior in the retail market.

1.3 Contributions Our contributions are of three-fold: ● We propose a game-theoretical model capturing the dominant market players, i.e. generators company (supplier), consumers company (operator) and the vast consumers, to analysis the market dynamics under supply uncertainty. ● We derive the close-form of optimal investment for the operator, which is the unique equilibrium of our proposed model. ● We propose an assumption to guarantee the uniqueness of equilibrium for the supplier-operator wholesale pricing game, and investigate the pricing relationship for wholesale and retail market, and finally give a numerical solution. The rest of this paper is organized as follows: Section II introduces the market game model and the method to derive equilibrium solutions. Section III solves the decision problems on the operator's side. Section IV solves the decision problems on the energy supplier's side. Section V discusses the solutions and analyzes them with simulation. Section VI concludes this paper.

II. SYSTEM MODEL 2.1 Market game model The system model is given in Fig. 1. We asChina Communications • March 2015

sume there are three rational entities: one supplier, one operator (power plant), and a set of consumers. The supplier has two kinds of electricity to provide for the power plant: stably-supplied electricity (SSE) and unstably-supplied electricity (USE). We assume that these two kinds of electricity are priced at Cc and Cu respectively. Power operator purchases electricity from the supplier and then sell them to the consumers, by which the wholesale market and retail market are generated. The quantity of the prospective investments on SSE and USE are denoted by Ec and Eu, respectively. The retail price operator charges for the consumers on per unit power is denoted by p. Furthermore, the power operator has to take into consideration the uncertainty of USE, which means it may get a lower quantity than its desired E u. We denote the actual obtained quantity of USE by αEu, where α follows some probabilistic distribution in [0,1]. Without loss of generality, in this paper, we assume α follows an uniform distribution in [0,1]. To make the notations Cu and αEu easy to understand, we give an example here. Suppose a supplier own many distributed wind farms, and a power plant decides to purchase the electricity generated by two of these farms. In our model, we assume the supplier will charge the power plant by the generating capacity rather than the installed capacity, and the unit price charged is denoted by Cu. The prospective investment quantity is indeed the installed capacity of the two wind farms, denoted by Eu. The obtained quantity is the real generating capacity of the two wind farms which is no greater than their installed capacity and varies greatly due to the weather. To depict the unstability of real generating wind electricity, we introduce α and endow it with an assumption that α follows some probabilistic distribution in [0,1]. Therefore, The actual obtained quantity of USE during a period of time is denoted by αEu. For an arbitrary consumer i, we assume that its demand of power is di. The demand di will

22

Stable electricity Supply

Unstable electricity Supply

(Ec , Cc )

(E Eu , Cu ) αEu

P ower Operator

p

Consumer 1

d1 p

d2 p

Consumer 2

di p

dN

··· ··· ···

Consumer N

Fig.1 Market model combining wholesale and retail market

be influenced by price p, as a demand response to the retail price. We assume the game model is based on RTP where prices changes on an hour basis, and the wholesale and retail prices as well as the investment are average. Both the prices and investment are decided hour-ahead and fixed during the basis period and the delivery occurs at the end of basis. The decision problems are summarized as follows: the operator will find the optimal investment quantity of Ec and Eu as well as the optimal retail price p; The energy supplier will design the best wholesale price mechanism (Cc,Cu) for the operator.

2.2 Methods to solve the model We denote the utility functions of consumer i, power operator and energy supplier by Uc,i, Up, UE, respectively. The formulated multi-level game can be summarized as follows: consumer : max Uc,i di

power plant : max U p (p,Ec ,Eu )

energy supplier : max U E

retail market, whereas Ec, Eu, Cc, Cu are for the wholesale market. We find the players in the two markets are of sequential moves in nature: power plant's investments Ec, Eu follow the power prices Cc, Cu set by the supplier; user's demand di follows the market retail price p set by the power plant. Thus, the leader-follower relationships make it quite suitable to model the behaviors of the users, power plant and supplier in a framework of multistage Stackelberg game. Therefore, we combine the wholesale market with the retail market, and formulate these decision problems as a Stackelberg game model of three levels. What's more, we assume the game model is of complete and perfect information. Complete information means each player's utility function is common knowledge among all the players. Perfect information means that at each move, the player with the move knows the full history of the play of the game thus far [19]. To get the solution of this stackelberg game, we use the standard method: backward induction. The backward induction of supplier-plant Stackelberg game can be described as follows: move 1 supplier : announce Cc and Cu to plant move 2 plant : (p∗ , Ec∗ , Eu∗ ) = arg max U p (p,Ec ,Eu )

move 3 supplier : (C , C ) = arg max U E ∗ c

∗ u

(Cc ,Cu )

where move 2 includes another simple backward induction of plant-consumer Stackelberg game, which is move 1 plant : anounance p to consumer move 2 consumer i : di∗ = arg max Uc,i di

move 3

plant : p = arg max U p ∗

p

We find the problem for the power plant is complicated, where it involves optimizing three parameters as well as the situation of uncertainty. Therefore, to solve the decision problem for power plant, we put it into a fourstage decision structure [20], as shown in Fig. 2, with which this complicated decision problem can be solved by backward induction. The complete formulation of backward induction will be given in Section III.

(Cc ,Cu )

It is clear that di and p are considered in the

23

China Communications • March 2015

2.3 Utility Functions 2.3.1 Consumer's utility function The definition of consumer's utility function is rather flexible. Generally, we consider the law of diminishing marginal utility works on the consumptions of electricity by consumers. We assume the marginal utility function of power demands, denoted by mi, is [21]  ω     ω − βdi , 0 ≤ di ≤ β mi =  (1)   , di > ω  0 β where ω is the maximal marginal utility of electricity to consumer and β can be viewed the rate of diminishing marginal utility, as shown in Fig. 3. Therefore, for consumer i, its utility of using the amount of electricity up to di is the integral of mi from 0 to di, denoted by gainc,i, which is  β 2  ω    ωdi − 2 di , 0 ≤ di ≤ β gainc,i =  (2)   1 ω2  , di > ω β 2 β We assume the consistent of ω and β for all consumers and regard them as constants. As illustrated in Fig. 3, gainc,i is a proper concave and increasing function of electricity demand di. Apparently, the cost of the demand di for consumer i is dip. Therefore, the utility function of consumer i, denoted by Uc,i(di,p), is defined as:  β 2  ω    ωdi − 2 di − di p , 0 ≤ di ≤ β Uc,i (di , p) =    1 ω2 − di p  , di > ω β 2 β (3) 2.3.2 Power plant's utility function The income and expenditure of power plant are defined respectively by (4) and (5) : gain p = p min( di , Ec + αEu ) (4) i∈I

where I represents the set of consumers. The min function means power plant's sales of electricity are either i∈I di if in an oversupply market or Ec+αEu if in a tight market. cost p = (EcCc + αEuCu ) (5) where αEu means power plant will consider China Communications • March 2015

Fig.2 Four-stage decision structure for power plant

gainc,i ω

mi

(0, 0)

di = ω β

di

Fig.3 Marginal utility and utility of consume against its electricity demand

the uncertainty supply and pay for the amount it actually obtains from the scheduled investment Eu. Therefore, from (4) and (5), the utility function of power plant is defined by : U p ( di , p, Ec , Eu , α) i∈I

= p min( di , Ec + αEu ) − (EcCc + αEuCu ) i∈I

(6) In addition, it is worth noting that, in [17], the cost of power plant is defined by EcCc + EuCu .

24

Table I Notations of Symbols Symbol

Meaning

Ec

Amount invested on stable supplying power

Cc

Price of stable supplying power

Eu

Amount invested on unstable supplying power

Cu

Price of instable supplying power

α∈[0,1]

Realization factor of unstable investment

β

Demands constraint factor

p

Price set for the power consuming market

ω

Value of power to consumers

di

Demand of power from individual consumer

I={1,2,...,N}

Set of power consumers

f (di )

III. FORMULATION OF BACKWARD INDUCTION

pdi pdi p=ω

pdi

pp>Cc>Cu>0 holds. If ω≤p, there will be no demands on the consumer's side. If p≤Cc, the power plant will get no profit and bankrupt. We shall have CuC c>C u, we learn Δ √ N(ω − Cc ) N(ω − Cc ) ω − Cc Thus ψ = 0 is excluded, and ψ = 4π 3 is the right one.

References [1] F. Genoese, M. Genoese, and M. Wietschel, “Occurrence of negative prices on the german spot market for electricity and their influence on balancing power markets,” in Energy Market (EEM), 2010 7th International Conference on the European, pp. 1–6, IEEE, 2010. [2] H. Weigt and C. v. Hirschhausen, “Price formation and market power in the german wholesale electricity market in 2006,” Energy policy, vol. 36, no. 11, pp. 4227–4234, 2008. [3] J. Reinaud, “Co2 allowance & electricity price interaction,” Impact on industrys electricity purchasing strategies in Europe. IEA information paper, 2007. [4] S. Borenstein, M. Jaske, and A. Rosenfeld, “Dynamic pricing, advanced metering, and demand response in electricity markets,” Center for the Study of Energy Markets, 2002. [5] D. S. Kirschen, “Demand-side view of electricity markets,” Power Systems, IEEE Transactions on, vol. 18, no. 2, pp. 520–527, 2003. [6] E. Celebi and J. D. Fuller, “A model for efficient consumer pricing schemes in electricity markets,” Power Systems, IEEE Transactions on, vol. 22, no. 1, pp. 60–67, 2007. [7] S. Borenstein, “The long-run efficiency of real-time electricity pricing,” The Energy Journal, pp. 93–116, 2005. [8] C. Kl¨uppelberg, T. Meyer-Brandis, and A. Schmidt, “Electricity spot price modelling with a view towards extreme spike risk,” Quantitative Finance, vol. 10, no. 9, pp. 963–974, 2010. [9] T. Jamasb and M. Pollitt, “Electricity market reform in the european union: review of progress toward liberalization & integration,” The Energy

33

Journal, pp. 11–41, 2005. [10] P. Joskow, “Lessons learned from electricity market liberalization,” The Energy Journal, vol. 29, no. 2, pp. 9–42, 2008. [11] P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V.W.Wong, and J. Jatskevich, “Optimal real-time pricing algorithm based on utility maximization for smart grid,” in Smart Grid Communications (SmartGridComm), 2010 First IEEE International Conference on, pp. 415–420, IEEE, 2010. [12] L. P. Qian, Y. J. A. Zhang, J. Huang, and Y. Wu, “Demand response management via real-time electricity price control in smart grids,” Selected Areas in Communications, IEEE Journal on, vol. 31, no. 7, pp. 1268–1280, 2013. [13] J. M. Morales, A. J. Conejo, and J. Perez-Ruiz, “Short-term trading for a wind power producer,” Power Systems, IEEE Transactions on, vol. 25, no. 1, pp. 554–564, 2010. [14] P. Tarasak, “Optimal real-time pricing under load uncertainty based on utility maximization for smart grid,” in Smart Grid Communications (SmartGridComm), 2011 IEEE International Conference on, pp. 321–326, IEEE, 2011. [15] L. Jiang and S. Low, “Multi-period optimal energy procurement and demand response in smart grid with uncertain supply,” in Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pp. 4348–4353, IEEE, 2011. [16] W. Wei, L. Feng, and M. Shengwei, “Energy prcing and dispatch for smart grid retailers under demand response and market price uncertainty,” Smart Grid, IEEE Transactions on, vol. accepted for inclusion in a future issue, 2014. [17] S. Bu and F. Yu, “A game-theoretical scheme in the smart grid with demand-side management: Towards a smart cyber-physical power infrastructure,” 2013. [18] S.-k. Miadreza and P. C. Joao, “A stochastic multi-layer agent-based model to study electricity market participants behavior,” Power Systems, IEEE Transactions on, vol. 30, no. 2, pp. 867–881, 2015. [19] R. Gibbons, Game theory for applied economists. Princeton University Press, 1992. [20] L. Duan, J. Huang, and B. Shou, “Investment and pricing with spectrum uncertainty: a cognitive operator’s perspective,” Mobile Computing, IEEE Transactions on, vol. 10, no. 11, pp. 1590–1604, 2011. [21] M. Fahrioglu and F. L. Alvarado, “Designing cost effective demand management contracts using game theory,” in Power Engineering Society 1999 Winter Meeting, IEEE, vol. 1, pp. 427–432, IEEE, 1998. [22] S. Borenstein, “The trouble with electricity markets: Understandering california’s restructuring disaster,” Journal of Economic Perspectives, vol. 16, no. 1, pp. 191–211, 2002.


[23] S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004. [24] I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. M¨uhlig, Handbook of mathematics. Springer, 2007.

Biographies LI Xiaobo, received the B.Eng. degree from Beijing University of Posts and Telecommunications, Beijing, China in 2011. He is now pursuing his M.Sc. degree from Beijing University of Posts and Telecommunications, Beijing, China. His research interests include wireless communication theory, game theory, cognitive radio networks, Smart Grid. GAO Li, received the B.Eng. degree in 1995 and the Ph.D. degree in 2014 from Beijing University of Posts and Telecommunications (BUPT), Beijing, China. She is now an Associate Professor in School of Digital Media and Design Arts, BUPT. Her current research interests include signal processing, Smart Grid, wireless communication theory, coherent optical communications. WANG Gongpu, received the B.Eng. degree in communication engineering from Anhui University, Hefei, Anhui, China, in 2001, the M.Sc. degree from Beijing University of Posts and Telecommunications, Beijing, China, in 2004. From 2004 to 2007, he was a teacher in Beijing University of Posts and Telecommunications. He received Ph.D. degree from University of Alberta, Edmonton, Canada, in 2011, and then joined


the School of Computer and Information Technology, Beijing Jiaotong University, China. His research interests include wireless communication theory and signal processing algorithms. GAO Feifei, received the B.Eng. degree from Xian Jiaotong University, Xi’an, China in 2002, the M.Sc. degree from McMaster University, Hamilton, ON, Canada in 2004, and the Ph.D. degree from National University of Singapore, Singapore in 2007. He was a Research Fellow with the Institute for Infocomm Research (I2R), A*STAR, Singapore in 2008 and an Assistant Professor with the School of Engineering and Science, Jacobs University, Bremen, Germany from 2009 to 2010. In 2011, he joined the Department of Automation, Tsinghua University, Beijing, China, where he is currently an Associate Professor. Prof. Gao’s research areas include communication theory, signal processing for communications, array signal processing, and convex optimizations, with particular interests in MIMO techniques, multi-carrier communications, cooperative communication, and cognitive radio networks. He has authored/coauthored more than 60 refereed IEEE journal papers and more than 80 IEEE conference proceeding papers, which have been cited more than 2500 times from Google Scholar. WU Qingwei, is an undergraduate student of Beijing University of Posts and Telecommunications, Beijing, China. His research interests include game theory, Smart Grid.

34