Author's personal copy Direct approach in computing robust Nash strategies for generating companies in electricity markets Damoun Langary a , Nasser Sadati a,b,⇑ , Ali Mohammad Ranjbar b a Intelligent Systems Laboratory, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran b Center of Excellence in Power System Management and Control (CEPSMC), Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran article info Article history: Received 7 February 2013 Received in revised form 24 July 2013 Accepted 25 July 2013 Keywords: Game theory Supply function equilibrium (SFE) Strategic bidding Nash equilibrium (NE) Electricity market abstract Supply function equilibrium (SFE) is often used to describe the behavior of generating companies in electricity markets. However, comprehensive analytical description of supply function models is rarely available in the literature. In this paper, using some analytical calculations, a novel direct approach is pro- posed to compute the Nash equilibrium (NE) of the supply function model under uniform marginal pric- ing mechanism. An explicit mathematical proof for its existence and uniqueness is also presented. The proposed methodology is then generalized to accommodate practical market constraints. In addition, a new concept of robust NE is introduced and calculated based on this approach. Finally, numerical simu- lations demonstrate the applicability and effectiveness of the proposed solution scheme. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, the trend of electricity industry in many countries has been towards a less regulated and more competitive energy market. Several methods and theories have been intro- duced and utilized to model different aspects of deregulated pool-based electricity markets. Agent-based modeling [1–3], statistical methods [4,5], artificial intelligence based approaches [6–8], optimization theory [9], and of course game-theoretic approaches [10–14] have been all frequently used in the literature. Among these approaches, game theoretic methods not only have better ways of realistically simulating the oligopolistic competition in electric power markets, but also are perfectly applicable to a wide range of market models such as Bertrand, Cournot, Stackel- berg and supply function models [15–17]. Among all aforementioned market models, the Cournot and SFE models seem to fit better and hence are more frequently exploited in the analysis of electricity markets. The Cournot models are sim- pler in essence and provide more intuition into the actual behavior of the market through analytical results. On the other hand, supply function models grant much more consistency with what really goes on in most electricity markets, and as a result, many of recent publications in the literature have implemented SFE in their simulations. In spite of certain well-known advantages of the sup- ply function models [18], there are some known limitations in properly applying SFE to analyze strategies of generating compa- nies in power markets [19]. The first and probably most important drawback of supply function models is the uniqueness issue; it is straightforward to show that an infinite number of NEs could exist [20,21]. To tackle this issue, Baldick [21] and others have examined some arbitrary parameterization methods to restrict the results to a unique NE point, which is frequently used in the literature. This simplification process of course leads to some other questions on how to set those parameters. Another downside to supply function models is that the NE is usually calculated through some iterative algo- rithms that not only do not have any guarantees of convergence, but also provide little insight into the characteristics of the calcu- lated NE. In this paper, we present a new approach to directly calculate the NE of electricity markets using the supply function model. The solution is then generalized to support all aforementioned parameterization methods [21]. This direct computation approach not only guaranties the existence and uniqueness of the NE, but also opens way to analyze the effect of parameterization tech- niques in the resulted equilibrium point. Given this new perspec- tive into the NE calculation problem, we then proceed to some more advanced ideas to obtain a more eligible assessment of inde- pendent power producers’ behavior according to the characteris- tics of the actual market. As it will be shown, the proposed method is also applicable to computing NE of Cournot models. 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.07.031 ⇑ Corresponding author at: Center of Excellence in Power System Management and Control (CEPSMC), Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran. Tel.: +98 (21)6616 4365; fax: +98 (21)6616 5939. E-mail addresses: sadati@sharif.ir, sadati@ece.ubc.ca (N. Sadati). Electrical Power and Energy Systems 54 (2014) 442–453 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes