Supply Function Equilibrium Model Research Articles

An environmental/techno-economic approach for bidding strategy in security-constrained electricity markets by a bi-level harmony search algorithm

In this paper, a new approach is presented for developing optimal double-sided bidding strategy in security-constrained electricity markets by considering emission of pollutants, as further objectives. In the proposed methodology, both Generation Companies (GenCos) and Distribution Companies (DisCos) try to maximize their profit by implementation of optimal strategies, whiles they have incomplete information about their rivals and market mechanism of payment is locational marginal pricing. In addition, each participant provides its strategic bids based on supply function equilibrium model and it modifies its bidding strategies until Nash equilibrium points are computed. The proposed approach is modeled as a bi-level optimization problem with the upper sub problem addressing individual GenCos and DisCos and the lower sub problem addressing the Independent System Operator (ISO). The upper level maximizes the individual market participant's profit and the lower one solves the ISO's market clearing problem for maximizing Community Welfare Function (CWF). A Self-adaptive Global-based Harmony Search Algorithm (SGHSA) is used to obtain optimal bidding strategies. The proposed methodology has been implemented on the 6-machine 8-bus system test system to demonstrate the feasibility and effectiveness of the proposed approach. Simulation results illustrate the profitableness of the newly developed approach in the obtaining optimal bidding strategies.

Cooperative and non-cooperative Nash solution for linear supply function equilibrium game

In this paper, the competition among suppliers in an oligopolistic market is studied using supply function equilibrium model. Since the decision of each supplier affects other suppliers’ profit, we have a game problem rather than optimization. In the proposed supply function equilibrium game, both of the cooperative and non-cooperative behaviors of the players are studied. In non-cooperative case, the Nash equilibrium point of the game is obtained for n-player in constant and price sensitive demand cases. In cooperative game, the concept of Nash bargaining solution is utilized to study the collusive behavior of the suppliers. The convexity and closeness of the players’ payoff set in supply function equilibrium model is proved as the necessary and sufficient condition for existence and uniqueness of the solution and then, the mathematical formulation of Nash bargaining solution is derived. In addition, the cooperative and non-cooperative behavior of players is compared over a wide range of parameters in a case study.

New indices of capacity withholding in power markets

Summary Market power evaluation and mitigation influence the efficiency of the electricity markets. The traditional indices cannot analyze market power, which caused by capacity withholding of generation companies. This paper describes a new approach to assess capacity withholding. The method is an improvement of a strategy previously proposed by the authors. The contributions of the new approach can be summarized as the following. First, the supply function equilibrium model and Cournot model are used to develop the concept of capacity withholding in electricity markets. Then, distortion–withheld index is calculated according to capacity constraints, reliability, and demand elasticity. Based on distortion–withheld index, the value of capacity withholding index, which shows the cost that independent system operator could spend for market power mitigation, can be obtained. It has been proved that the new approach is able to assess capacity withholding exactly and identify the proper market power mitigation program. The finding in this paper is helpful for market designers and regulators. Copyright © 2013 John Wiley & Sons, Ltd.

Pivotal Suppliers and Market Power in Experimental Supply Function Competition

We use experiments to study market power with supply function competition, akin to the competition in electricity markets. Our treatments vary the distribution of demand levels as well as the amount and symmetry of the allocation of production capacity between different suppliers. We relate our results to a descriptive power index and to the predictions of two models: a supply function equilibrium (SFE) model and a multi-unit auction (MUA) model. Observed behaviour is consistent with the equilibria of the unrestricted SFE model and inconsistent with the unique equilibria of two refinements of the SFE model and of the MUA model.

Analysis of the Restructured Energy Market in the Framework of Game Theory: Iran's Power Industry

Restructuring of electricity market in Iran has been accepted and operationalized following the power market in industrial countries. Increasing of efficiency through insertion of economic considerations and transition from monopoly to more competitive conditions in order to become closer to the optimal status are the basis to accept restructuring. Given to the market structure, imperfect competition is ultimately fulfilled in this market that games theory solutions should be used to analyze it. On the other side, given to bidding and pay-as-bid (PAB) in Iran's electricity market, supply function equilibrium (SFE) models is suitable for analysis. Therefore, having proposed function and the suitable theoretical model and estimated marginal cost function and uncertainty demand parameters, in the framework of SFE approach, the optimal supply function equilibrium of one of the firms of the electricity market in Isfahan was extracted by means of MATLAB software. Then it was compared with the proposed supply function. The obtained results revealed that the current proposed supply function does not conform to the theoretical optimal supply function in Nash equilibrium.

Parameterization of Linear Supply Functions in Nonlinear AC Electricity Market Equilibrium Models—Part I: Literature Review and Equilibrium Algorithm

The linear supply function equilibrium (SFE) model can be used to investigate bid-based electricity pool markets. Several studies assume that the market players construct optimal strategic bids by parameterizing the linear marginal cost function of their generating units. Four different types of parameterization are possible, for which the slope and/or the intercept of the marginal cost functions are varied. This paper reviews the existing literature and proposes a primal-dual nonlinear interior point algorithm to solve a bi-level market problem with AC network representation, using any of the parameterization methods. Part II of this paper uses the proposed algorithm to examine the effects of the different parameterizations on the resulting SFE solutions, the role of network complexity and the nature of the equilibrium points.

Supply function equilibria: Step functions and continuous representations

Abstract In most electricity markets generators must submit step-function offers to a uniform price auction. These markets are often modelled as simpler pure-strategy Supply Function Equilibria (SFE) with continuous supply functions. Critics argue that the discreteness and discontinuity of the required steps drastically change Nash equilibria, invalidating predictions of the SFE model. We prove that there are sufficient conditions, offered quantities can be continuously varied, offered prices are selected from a finite set, and the density of the additive demand shock is not too steep, where the resulting stepped SFE converges to the continuous SFE as the number of steps increases, reconciling the apparently very disparate approaches to modelling electricity markets.

Impacts of Large-Scale Integration of Intermittent Resources on Electricity Markets: A Supply Function Equilibrium Approach

In this paper, the impact of large-scale integration of intermittent generation resources on electricity markets is studied through a supply function equilibrium model. A new formulation for determining supply function equilibria in uniform electricity markets is presented, where uncertainty of intermittent generation resources is taken into account. In the case of an unconstrained market model where generation limits are ignored, a closed form solution for market equilibrium is derived and its uniqueness is proved. In the case of a generation-constrained market model, an algorithm is proposed for determining supply function equilibria and the existence and uniqueness of the equilibrium is discussed. Case studies are presented for demonstrating the proposed approaches.

Market power assessment in electricity markets: supply function equilibrium-based model

SUMMARY Because of the oligopoly structure of the electricity markets, the Generation Companies (GenCos) may exert market power by using bidding strategies such as capacity withholding. This brings the market clearing price higher and the quantity generated of the market lower. In this article, capacity withholding in an oligopolistic electricity market in that all GenCos bid in the supply function equilibrium model is analyzed and the capacity withheld index, the capacity distortion index and the price distortion index are formulated. Demand elasticity and forward contracts can significantly affect the GenCos' capacity withholding in the market. The potential ability of market for capacity withholding is measured by the distortion–withheld index proposed in this article. The finding in this article is helpful for market power monitoring and mitigating. A two-settlement market including a forward market and a spot market is used to describe GenCos' strategic forward contracting and spot market competition. Copyright © 2012 John Wiley & Sons, Ltd.

Computing All Nash Equilibria of Multiplayer Games in Electricity Markets by Solving Polynomial Equations

The analysis of the Nash equilibrium (NE) in electricity markets with imperfect competition is subject to two major challenges: the treatment of multiplayer games and the determination of the existence of multiple market equilibria. To resolve these obstacles, a solution method, based on the payoff matrix approach and polynomial equations, is proposed in this paper to calculate all the Nash equilibria of multiplayer games in the electricity markets. First, the proposed method decomposes the game by means of a set of all pure strategies assigned with positive probabilities (support). For each possible support, the NE condition can be characterized by a set of polynomial equations with inequality constraints. Next, the homotopy continuation algorithm is employed to obtain all solutions of the polynomial system. Finally, all Nash equilibria can be found by verifying each solution of all polynomial systems. Two example systems, one involving the Cournot model and the other based on the supply function equilibrium (SFE) model, are applied to test the proposed method, respectively. The results show that the proposed method has the ability to identify all NE under certain conditions, indicating that the proposed algorithm is useful for providing insights into the theoretical analysis of electricity markets.

Bidding strategy of generation companies using PSO combined with SA method in the pay as bid markets

This paper proposes a new method that uses the combination of particle swarm optimization (PSO) and simulated annealing (SA) to predict the bidding strategy of Generating Companies (Gencos) in an electricity market where they have incomplete information about their opponents and market mechanism of payment is pay as bid. In the proposed methodology, Gencos prepare their strategic bids according to Supply Function Equilibrium (SFE) model and they change their bidding strategies until Nash equilibrium points are obtained. Nash equilibrium points constitute a central solution concept in game theory and they are computed with solving a global optimization problem. In this paper a new computational intelligence technique is introduced that can be used to solve the Nash optimization problem. This new procedure, is based on the PSO algorithm, which uses SA method to avoid becoming trapped in local minima or maxima and improve the velocity’s function of particles. The performance of this procedure is compared with results of other computational intelligence techniques such as PSO, Genetic Algorithm (GA), and a mathematical method (GAMS/DICOPT). The IEEE 39-bus test system is employed to illustrate and verify the results of the proposed method.

Pivotal Suppliers and Market Power in Experimental Supply Function Competition

In the process of regulatory reform in the electric power industry, the mitigation of market power is one of the basic problems regulators have to deal with. We use experimental data to study the sources of market power with supply function competition, akin to the competition in wholesale electricity markets. An acute form of market power may arise if a supplier is pivotal; that is, if the supplier's capacity is required in order to meet demand. To be able to isolate the impact of demand and capacity conditions on market power, our treatments vary the distribution of demand levels as well as the amount and symmetry of the allocation of production capacity between different suppliers. We relate our results to a descriptive power index and to the predictions of two alternative models: a supply function equilibrium (SFE) model and a multi-unit auction (MUA) model. We find that pivotal suppliers do indeed exercise their market power in the experiments. We also find that observed behavior is consistent with the range of equilibria of the unrestricted SFE model and inconsistent with the unique equilibria of two refinements of the SFE model and of the MUA model.

Supply Function Equilibria of Pay-as-Bid Auctions

This paper characterizes the Nash equilibrium in a pay-as-bid (discriminatory), divisible-good, procurement auction. Demand by the auctioneer is uncertain as in the supply function equilibrium model. A closed form expression is derived. Existence of an equilibrium is ensured if the hazard rate of the perfectly inelastic demand is monotonically decreasing and sellers have non-decreasing marginal costs. Multiple equilibria can be ruled out for markets, for which the auctioneer’s demand exceeds suppliers’ capacity with a positive probability. The derived equilibrium can be used to model strategic bidding behaviour in pay-as-bid electricity auctions, such as the balancing mechanism of United Kingdom. Offer curves and mark-ups of the derived equilibrium are compared to results for the SFE of a uniform-price auction.

Supply curve bidding of electricity in constrained power networks☆

This paper presents a Supply Curve Bidding (SCB) approach that complies with the notion of the Standard Market Design (SMD) in electricity markets. The approach considers the demand-side option and Locational Marginal Pricing (LMP) clearing. It iteratively alters Supply Function Equilibria (SFE) model solutions, then choosing the best bid based on market-clearing LMP and network conditions. It has been argued that SCB better benefits suppliers compared to fixed quantity-price bids. It provides more flexibility and better opportunity to achieving profitable outcomes over a range of demands. In addition, SCB fits two important criteria: simplifies evaluating electricity derivatives and captures smooth marginal cost characteristics that reflect actual production costs. The simultaneous inclusion of physical unit constraints and transmission security constraints will assure a feasible solution. An IEEE 24-bus system is used to illustrate perturbations of SCB in constrained power networks within the framework of SDM. By searching in the neighborhood of SFE model solutions, suppliers can obtain their best bid offers based on market-clearing LMP and network conditions. In this case, electricity producers can derive their best offering strategy both in the power exchange and the long-term contractual markets within a profitable, yet secure electricity market.

Normal boundary intersection method for suppliers’ strategic bidding in electricity markets: An environmental/economic approach

In this paper the problem of developing optimal bidding strategies for the participants of oligopolistic energy markets is studied. Special attention is given to the impacts of suppliers’ emission of pollutants on their bidding strategies. The proposed methodology employs supply function equilibrium (SFE) model to represent the strategic behavior of each supplier and locational marginal pricing mechanism for the market clearing. The optimal bidding strategies are developed mathematically using a bilevel optimization problem where the upper-level subproblem maximizes individual supplier payoff and the lower-level subproblem solves the Independent System Operator’s market clearing problem. In order to solve market clearing mechanism the multiobjective optimal power flow is used with supplier emission of pollutants, as an extra objective, subject to the supplier physical constraints. This paper uses normal boundary intersection (NBI) approach for generating Pareto optimal set and then fuzzy decision making to select the best compromise solution. The developed algorithm is applied to an IEEE 30-bus test system. Numerical results demonstrate the potential and effectiveness of the proposed multiobjective approach to develop successful bidding strategies in those energy markets that minimize generation cost and emission of pollutants simultaneously.

An optimization-based conjectured supply function equilibrium model for network constrained electricity markets

This paper proposes a model to compute nodal prices in oligopolistic markets. The model generalizes a previous model aimed at solving the single-bus problem by applying an optimization procedure. Both models can be classified as conjectured supply function models. The conjectured supply functions are assumed to be linear with constant slopes. The conjectured price responses (price sensitivity as seen for each generating unit), however, are assumed to be dependent on the system line's status (congested or not congested). The consideration of such a dependence is one of the main contributions of this paper. Market equilibrium is defined in this framework. A procedure based on solving an optimization problem is proposed. It only requires convexity of cost functions. Existence of equilibrium, however, is not guaranteed in this multi-nodal situation and an iterative search is required to find it if it exists. A two-area multi-period case study is analysed. The model reaches equilibrium for some cases, mainly depending on the number of periods considered and on the value of conjectured supply function slopes. Some oscillation patterns are observed that can be interpreted as quasi-equilibria. This methodology can be applied to the study of the future Iberian electricity market.

Electricity market equilibrium model with resource constraint and transmission congestion

Electricity market equilibrium model not only helps Independent System Operator/Regulator analyze market performance and market power, but also provides Market Participants the ability to build optimal bidding strategies based on Microeconomics analysis. Supply Function Equilibrium (SFE) is attractive compared to traditional models and many efforts have been made on it before. However, most past research focused on a single-period, single-market model and did not address the fact that GENCOs hold a portfolio of assets in both electricity and fuel markets. This paper first identifies a proper SFE model, which can be applied to a multiple-period situation. Then the paper develops the equilibrium condition using discrete time optimal control considering fuel resource constraints. Finally, the paper discusses the issues of multiple equilibria caused by transmission network and shows that a transmission constrained equilibrium may exist, however the shadow price may not be zero. Additionally, an advantage from the proposed model for merchant transmission planning is discussed.

A Supply Function Equilibrium Model with Forward Contracts - An Application to Wholesale Electricity Markets

Research into modeling electricity markets is continuing and the subject of many debates. All types of competition (Cournot, Bertrand, supply function) are utilized and have their advantages and disadvantages for electricity markets. It is well-recognized that models cannot address all questions of interest; however they appear as an interesting tool for gleaning insights into the complexity of electricity markets and whether electricity markets may deliver the expected benefits of liberalization. In particular, proving the existence of market power is a very complex task. Market simulation models should not be seen as the ultimate solution but as one powerful tool. While it is extremely difficult to prove if any market participants were manipulating markets, simulation models can show (under certain assumptions) if it would have been profitable to do so. Such models can be used in addition to traditional competition analysis. For instance, a model can estimate different benchmarks (competitive, supply function) against which actual market prices may be compared. This paper provides a practical application of the SFE concept and how such theoretical approach can be used in practice. The model combines the supply function equilibrium approach in an expanded version of the Baldick et al. model (2000) with forward contracting based on the model of Newbery (1998). We also discuss the different options for market modeling with respect to strategic variables and forward contracting. Finally, we present an application of this model to the Dutch electricity market.

Analyzing Two-Settlement Electricity Market Equilibrium by Coevolutionary Computation Approach

Forward contracts play an important role for market power mitigation and risk hedging in electricity markets. In this paper, a two-settlement electricity market including a forward market and a spot market is formulated as a two-stage game. The linear supply function equilibrium (LSFE) model and Cournot model are used to model strategic bidding for the spot market, while the forward market is modeled by Cournot model. A coevolutionary genetic algorithm (CGA) is employed to determine the market equilibrium. The paper then examines the question whether generation companies (GenCos) would voluntarily enter forward markets due to economic inspiration and studies theoretically and numerically the factors which can affect the bidding behaviors of GenCos. The effectiveness of CGA in determining the market equilibrium is investigated and demonstrated based on two test examples. The studies show that GenCos' decisions on the participation in forward markets depend significantly on the type of competition that they have in the spot market. Moreover, GenCos' bidding behaviors in the forward market can be affected by their cost parameters and the slope of system demand function significantly.

Supply function equilibria of pay-as-bid auctions

This paper characterizes the Nash equilibrium in a pay-as-bid (discriminatory), divisible-good, procurement auction. Demand by the auctioneer is uncertain as in the supply function equilibrium model. A closed form expression is derived for a one shot game. Existence of an equilibrium is ensured if the hazard rate of the demand distribution is monotonically decreasing with respect to the shock outcome and sellers have non-decreasing marginal costs. Multiple equilibria can be ruled out for markets, for which the auctioneer’s demand exceeds suppliers’ capacity with a positive probability. The derived equilibrium can be used to model strategic bidding behavior in pay-as-bid electricity auctions, such as the balancing mechanism of United Kingdom. Offer curves and mark-ups of the derived equilibrium are compared to results for the SFE of a uniform-price auction.