AbstractâThis paper presents a methodology based on game theory for power system static reserve planning with large-scale integration of wind power to ...
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Robust Optimization of Static Reserve Planning With Large-Scale Integration of Wind Power: A Game Theoretic Approach Shengwei Mei, Senior Member, IEEE, De Zhang, Yingying Wang, Feng Liu, and Wei Wei
Abstract—This paper presents a methodology based on game theory for power system static reserve planning with large-scale integration of wind power to ensure the generating capacity adequacy. First, a min–max game model is proposed for decisionmaking problems with uncertainties. Then, it is applied to the static reserve capacity planning problem. In the proposed model, the system planner (as one player) aims to find the minimum static reserve capacity to meet the total load demand while keeping the system reliability index within a desired value. Nature (as the other player), which determines the wind power output, is modeled as an attacker who wants to worsen the system reliability level due to its uncertainty and inaccurate prediction. Then, a two-stage relaxation algorithm is introduced to solve the min–max game. Finally, the proposed model for static reserve planning is applied to the IEEE Reliability Test System (RTS), and its robustness and high efficiency are demonstrated by comparing it to the traditional expectation method and Monte Carlo method. Index Terms—Game theory, generating capacity adequacy, static reserve, system reliability index, wind power.
I. INTRODUCTION HE development of wind power has drawn more and more attention all over the world in the past decade [1]. The introduction of variable renewable energy generation into a power system will have impacts on system operation in multiple time scales [2]. The first can be labeled as the system reliability impact, which relates to the requirement that there should be enough generation capacity to meet the peak load demand in the long term (e.g., several years). The second is referred to as the balancing impact that relates to the management of load demand fluctuations in a time scale from seconds to hours. In this paper, the first aspect-related static reserve planning to meet the peak load demand or to ensure generating capacity adequacy in the long term will be studied.
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Manuscript received May 05, 2013; revised July 13, 2013, September 6, 2013, and November 27, 2013; accepted January 09, 2014. This work was supported in part by the National Natural Science Foundation of China (51007041), in part by the Special Fund of the National Basic Research Program of China (2012CB215103), and in part by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51321005). (Corresponding author: W. Wei.) S. Mei, D. Zhang, F. Liu, and W. Wei are with the Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail: wei-wei04@ mails.tsinghua.edu.cn). Y. Wang is with Electric Power Planning and Engineering Institute (EPPEI), Beijing 100120, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSTE.2014.2299827
The research on power system reserve requirements with only traditional generators is quite mature [3] and recently reserve requirement for power systems incorporating wind power has been a hot topic because the penetration level of variable wind power generation is increasing. However, most literature is devoted to estimating reserve requirements in a relatively short time period; thus, both load demand and wind power forecast are necessary. Reference [4] presents an analysis of the interaction between the variability characteristics of the utility load, wind power generation, solar power generation, and ocean wave power generation. Their impacts on the reserve requirements are analyzed, where renewable resource forecasting is implemented by the 1-h persistence method. In [5], a probabilistic reserve setting tool is presented to support the transmission system operator (TSO) in defining the operating reserve requirements for the coming day or hours, using as input the probabilistic wind power predictions. Reference [6] presents methods to determine the required operating reserve in recent wind integration research and the impact of wind power forecast horizon time scale and forecast errors is discussed. In [7], a methodology for calculating the dynamic operating balancing reserve in a 1- to 48-h-ahead time horizon to ensure a high-level reliability and security is presented, addressing the effects of wind variability and forecast errors. In [8]–[11], the adaptive robust optimization is applied to the problem of unit commitment as well as look-ahead energy and reserve dispatch, which is helpful for real-time operation. As it is well known, generating capacity adequacy is an important aspect of system reliability evaluation. Reference [12] presents a Monte Carlo simulation approach for generating capacity adequacy evaluation of power systems incorporating wind power. Furthermore, a multistate wind power generation model is proposed for system generating adequacy assessment or reliability evaluation in [13] and [14]. Based on these works, this paper aims to quantify the optimal desired static reserve capacity to ensure system generating adequacy from a long-term viewpoint, where the time-scale considered is from 1 year to several years. In this situation, the two-parameter Weibull distribution model is widely used to represent the variations of wind speed [15] and is also adopted in this paper. The main topic is to develop a new effective method to deal with wind power uncertainty and find the optimal static reserve capacity to keep the system reliable with respect to generating capacity adequacy. To this end, a robust optimization model based on game theory is proposed in this paper. Game theory is essentially an advanced type of optimization technique [16]. Therefore, many power system problems can be
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modeled in the framework of game. In fact, game theory has been applied to solve power system decision-making problems in various research fields [17]–[19], mainly including power system planning and power markets. The authors have also used game theory for economic dispatch [20] and hybrid power system planning [21] with renewable energy generation. In this paper, based on game theory, the robust optimization model of the static reserve planning considering the uncertainty of load demand and wind power is considered as a min–max game between the system planner and nature. The model can produce an optimal static reserve capacity, which is able to deal with the worst wind speed distribution scenario. The rest of the paper is organized as follows. The robust optimization model based on game theory for static reserve planning is built in Section II. Then, a two-stage relaxation algorithm that can solve the min–max game is introduced in Section III. Case study on IEEE Reliability Test System (RTS) is carried out to illustrate the application of the proposed min–max game model in Section IV, and its comparison with both the traditional expectation method and Monte Carlo method is also given. Finally, conclusions and future study are given in Section V. II. PROBLEM FORMULATION In this section, first, a general robust optimization model based on game theory to deal with decision-making problems with uncertainties is presented. Then, in the framework of the proposed model, by taking the system planner and nature as two players, the min–max game model for static reserve planning with wind power integration is formulated. A. Min–Max Game Model for Robust Decision-Making Under Uncertainties Different from some popular models in stochastic optimization [22], such as the expectation model and chance-constraints model, here the robust optimization model for decision-making problems with uncertainties will be built based on game theory. By taking the decision-maker and nature (which represents the environmental uncertainty, such as wind power output) as the two players, the min–max game model for robust decisionmaking under uncertainties can be constructed as follows:
where is the state variable, is the decisions controlled by the decision-maker (or the system planner), is the decisions controlled by nature, is the payoff, represents constraints, and and are the strategy sets of the decision-maker and nature, respectively. Taking the problem of power system static reserve planning incorporating wind power as an example, the system planner’s decision is the amount of static reserve, the decision of nature is the set of variables used to represent the variations in wind speed, and the state is the set of variables including conventional generation capacity, load demand, wind power capacity, etc.
Taking nature as a “virtual” player stems from the idea of “games against nature” [23]. Specifically, in (1), the decisionmaker aims to minimize through its decision variable and oppositely nature tries to maximize through its decision variable . Thus, from the viewpoint of game theory, and can be seen as the strategies of the two players and the decision-making problem can be described by a two-person zero-sum game. It should be noted that not every two-person zero-sum game has a pure strategy Nash equilibrium. Therefore, a min–max game model derived from (1) is widely used in practical engineering as follows:
It is proved that there always exists a solution for (2). The model in (2) is a min–max problem, which is known as a kind of two-person zero-sum game by Von Neumann and Morgenstern [24]. The engineering meanings of model (2) are stated as follows. 1) The Best Strategy of the Decision-Maker in Engineering is the One Capable of Dealing With the Worst-Case Scenario of Nature: From (2), it can be seen that the strategy of the decisionmaker (denoted by ) is designed according to the worst-case scenario of nature (denoted by ). Thus, the strategy is also capable of dealing with all the possible strategies of nature. Actually, the idea of strategy choice to defeat the worst-case scenario is widely accepted in power systems, such as robust unit commitment incorporating wind power [8]–[10]. 2) The Decision-Making Order of the Two Players is Nature First With the Decision-Maker Following: In engineering, it is assumed that nature first makes its strategy to maximize the cost function and then the decision-maker tries to find the best strategy to minimize the cost function. Due to this kind of decision-making order, the decision-making problem is formulated as a “min–max” game model in (2). This decisionmaking order is unfair to the decision-maker. However, since the strategy of nature is not clear or the information of nature is not sufficient, the best strategy is to find a way to defeat the most threatening strategy of nature, although it is a relatively conservative one. 3) Rationality Analysis of Taking Nature as a Player: Rationality means that all players intend to maximum their own payoff in the game. Thus, the game equilibrium among the players will be achieved when not a single player can improve his or her own profit unilaterally. In min–max game (2), the decision-maker is clearly rational and always tries to minimize his cost. Nature, taken as a “virtual” player, can also be considered “rational” in the sense that considering the most threatening strategy of nature, its uncertainty can be modeled as a maximizer of the cost. It should be noted that in reality, an actual cost increase might depend on the correlation type of the wind and load. However, since what will happen in the future is not known in the planning stage, it is reasonable to consider the worst distribution of wind, and the cost in this circumstance (worst-case cost) is usually increased. This idea is conservative
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. MEI et al.: ROBUST OPTIMIZATION OF STATIC RESERVE PLANNING WITH LARGE-SCALE INTEGRATION OF WIND POWER
to some extent but preferred when the decision maker is more risk-aversive. 4) The Relationship Between the Strategies of Nature and Its Prediction: The strategy set of nature (denoted by ) is actually the prediction of possible behavior of nature. The more accurate the prediction is, the smaller the strategy set becomes. So the conservativeness of model (2) decreases when the prediction of nature becomes more accurate. B. Min–Max Game Model for Static Reserve Planning Incorporating Wind Power For the problem of static reserve planning incorporating wind power, the system planner aims to find the minimum static reserve capacity to meet the total load demand while keeping the system reliability index within a desired value. Nature, which determines the wind power output, can be seen as an attacker who wants to worsen the system reliability level (REL) due to its uncertainty and inaccurate prediction. Thus, for the two players, i.e., the system planner and nature, one’s payoff is the other’s loss. In this sense, the static reserve planning problem can be modeled as a two-person zero-sum game. Specifically, the proposed min–max game model for static reserve planning contains three essential elements: player set, strategy set, and payoff function [25], which are described as follows. 1) Player Set: This problem involves two players, i.e., the system planner and nature, respectively. The system planner, which is a real decision-maker in practice, aims to find the minimum reserve capacity to satisfy the generation capacity adequacy with respect to all nature conditions. Nature is considered to have the opposite objective due to wind speed uncertainty. 2) Strategy Set: The strategy of the system planner is the reserve capacity, denoted by . Due to limited available resources, an upper limit of the reserve capacity is always predefined, denoted by . So the strategy set of the system planner will be
Nature will determine the characteristics of wind speed, which depend on many factors such as climate, geography, etc., and can be estimated by the historical wind speed data in the targeted region. In this paper, the two-parameter Weibull distribution is adopted to describe the wind speed random characteristics as follows:
