Linear Supply Function Equilibrium Research Articles

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.

Existence and uniqueness of conjectured supply function equilibria

Supply Function Equilibrium (SFE) and Conjectured Supply Function Equilibrium (CSFE) are some of the approaches most used to model electricity markets in the medium and long term.SFE represents the generators’ strategies with functions that link prices and quantities, but leads to systems of differential equations hard to solve, unless linearity is assumed (Linear Supply Function Equilibrium, LSFE). CSFE also assumes linearity of the supply functions but only around the equilibrium point, also avoiding the system of differential equations.This paper analyzes the existence and uniqueness of G-CSFE (a CSFE previously proposed by the Authors) for both elastic and inelastic demands. In addition, it also proves that the iterative algorithm proposed to compute G-CSFE has a fixed point structure and is convergent, and that LSFE is a particular case of G-CSFE when demand and marginal costs are linear. Selected examples show the performance of G-CSFE and how it can be applied to market power analysis with meaningful results.

A new enhanced bat-inspired algorithm for finding linear supply function equilibrium of GENCOs in the competitive electricity market

This paper proposes a new enhanced bat-inspired algorithm to find out linear supply function equilibrium of Generating Companies (GENCOs) in a network-constrained electricity market where they have incomplete information about other rivals. The model enables a GENCO to link its bidding price with the bidding quantity of its product. In this regard, the social welfare maximization is applied to clearing the market and nodal pricing mechanism is utilized to calculate the GENCO’s profit. It is formulated as a bi level optimization problem, where the higher level problem maximizes GENCO’s payoff and the lower level problem solves the independent system operator’s market clearing problem based on the maximization of social welfare. Due to non-convexity nature of the proposed bi level optimization problem, the mathematical-based optimization approach is incapable to solve the problem and obtain the nearly global optima. In order to overcome the obstacle of the conventional approaches, this study suggests a new meta-heuristic Bat-inspired Algorithm (BA) to achieve the nearly global solution of the bi level optimization problem. In addition a novel self-adaptive learning mechanism is utilized on the original BA to improve the population diversity and global searching capability. Numerical examples are applied to three test systems in order to evaluate the performances of the presented framework.

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.

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.

Electricity market modelling with photovoltaic active and reactive power generation

Renewable energy systems are increasingly employed all over the world in order to provide alternative sources of electricity production and reduce CO 2 emissions from conventional power plants. However, as the density of renewable systems increases, certain issues arise that need to be addressed. In particular, for photovoltaic (PV) systems connected to the grid, one parameter of importance is the variation of the PV power output with respect to the intensity of solar irradiance. This paper investigates the impact of the active and reactive power injected to the grid by PV systems, on the electricity market equilibrium, using the linear Supply Function Equilibrium (SFE) model. Numerical results for a 5-bus and the IEEE 57-bus systems, which are based on real PV output performance data collected from an experimental PV park, show the effects on the electricity market equilibrium outcome.

Theory and Application of Linear Supply Function Equilibrium in Electricity Markets

We consider a supply function equilibrium (SFE) model of interaction in an electricity market. We assume a linear demand function and consider a competitive fringe and several strategic players having capacity limits and affine marginal costs. The choice of SFE over Cournot equilibrium and other models and the choice of affine marginal costs is reviewed in the context of the existing literature. We assume that bid rules allow affine or piecewise affine non-decreasing supply functions by firms and extend results of Green and Rudkevitch concerning the linear SFE solution. An incentive compatibility result is proved. We also find that a piecewise affine SFE can be found easily for the case where there are non-negativity limits on generation. Upper capacity limits, however, pose problems and we propose an ad hoc approach. We apply the analysis to the England and Wales electricity market, considering the 1996 and 1999 divestitures. The piecewise affine SFE solutions generally provide better matches to the empirical data than previous analysis.

Choosing duopoly solutions by consistent conjectures and by uncertainty

Abstract Breshnahan's consistent conjectures criterion and Robson's linear supply function equilibrium under uncertainty are equivalent in the sense that they single out the same Bowley conjectural variations as the unique conjectural variations satisfying the given consistency requirements.