In this paper we consider a supply function model of an electricity market where strategic firms have capacity constraints. We show that if firms have heterogeneous cost functions and capacity constraints then the differential equation approach to finding the equilibrium supply function may not be effective by itself because it produces supply functions that fail to be non-decreasing. Even when the differential equation approach yields solutions that satisfy the non-decreasing constraints, many of the equilibria are unstable, restricting the range of the equilibria that are likely to be observed in practice. We analyze the non-decreasing constraints and characterize piece-wise continuously differentiable equilibria. To find stable equilibria, we numerically solve for the equilibrium by iterating in the function space of allowable supply functions. Using a numerical example based on supply in the England and Wales market in 1999, we investigate the potential for multiple equilibria and the interaction of capacity constraints, price caps, and the length of the time horizon over which bids must remain unchanged.

. We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, first-level constraints are linear, and second-level (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, ill-conditioning, convexity of the objective, monotonicity and symmetry of the second-level problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.maths.mu.OZ.AU/~danny/qpecgendoc.h...

Abstract — In this paper, I use an example from the literature to compare Cournot and supply function equilibrium models of bid-based electricity markets both with and without transmission constraints. I will demonstrate that the parametrization of the supply function model has a significant effect on the calculated results. In particular, several results reported in the literature are artifacts of assumptions in the parametrization of the model.

Abstract. This paper studies algorithms for equilibrium problems with equilibrium constraints (EPECs). We present a generalization of Scholtes’s regularization scheme for MPECs and extend his convergence results to this new relaxation method. We propose a sequential nonlinear complementarity (SNCP) algorithm to solve EPECs and establish the convergence of this algorithm. We present numerical results comparing the SNCP algorithm and diagonalization (nonlinear Gauss-Seidel and nonlinear Jacobi) methods on randomly generated EPEC test problems. The computational experience to date shows that both the SNCP algorithm and the nonlinear Gauss-Seidel method outperform the nonlinear Jacobi method. 1

In this paper we consider the formulation and uses of electricity market equilibrium models.

In this report, two different approaches to analyzing firm-based market power consideringtransmissionconstraintsareproposed. Oneisanapplicationofthetransmissionconstrained residual demand Jacobian, while the other is a generalization of the “residualsupplyindex”tothecaseoftransmissionconstraints. Thesetwoapproachesprovide complementary evaluations of market power. Medium- and large-scale system examples are provided to demonstrate computational efficiency, and both approaches could be applied to real-world electricity markets.