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

Abstract

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.

Appendices

Appendix A

This appendix is devoted to prove some numerical results used in Section 2, which can be easily generalized to obtain the results for several areas of Section 3.

Lemma

• Let C(P) be a convex function and θ, α⩾0. So, the following expressions where ΔC _P (ΔP) denotes C(P+ΔP)−C(P), are equivalent:

  • a) πΔP⩽θΔP+ΔC _P (ΔP), ∀ΔP

  • b) πΔP⩽θΔP+ΔC _P (ΔP)+α(ΔP)², ∀ΔP

Proof

• (a) ↠ (b) is obvious, so we only need to prove (b) ↠ (a), that is, πΔP⩽θΔP+ΔC _P (ΔP)+α(ΔP)², ∀ΔP implies πΔP⩽θΔP+ΔC _P (ΔP), ∀ΔP

  First, because C(P) is a convex function of P, ΔC _P (ΔP) is a convex function of ΔP. So, ΔC _P (ΔP)+θΔP+α(ΔP)² is also a convex function. Expression (b) implies that π is a subgradient of this function. Therefore, π is the addition of subgradients of every added term, evaluated at ΔP=0, that is

  

  This means that π is a subgradient of convex function θΔP+ΔC _P (ΔP), which implies (a), and the proof is finished. □

Proposition

• Let C(P) be a convex function and θ⩾0. So, the following three expressions where ΔC _P (ΔP) denotes C(P+ΔP)−C(P), are equivalent:

  • a) πΔP⩽θΔP+ΔC _P (ΔP), ∀ΔP

  • b) πΔP⩽θΔP+ΔC _P (ΔP)+θ(ΔP)², ∀ΔP

  • c) πΔP⩽ΔC _P (ΔP)+Δ(½θP ²) = Δ(C(P)+½θP ²), ∀ΔP

Proof

• Applying the previous lemma is immediate because both expressions differ from the first one in terms ½θ(ΔP)² and θ(ΔP)². □

Appendix B

This appendix contains a result needed in Section 3 that proves the equivalence between the equilibrium constraints and the optimization problem and which is not included in the main text to make the reading easier.

Theorem

• P(S ^*) and V(S ^*) are equivalent.

Proof

• To prove that every solution of V(S ^*) is a minimum of problem P(S ^*), let us add the first set of inequalities in V(S ^*), that is

  

  On the other hand, from the demand constraints

  

  So

  

  from the absence of transport arbitrage condition, and therefore

  

  which proves that the solution of V(S ^*) is actually an optimal solution of P(S ^*).

  To prove that every solution of P(S ^*) fulfils V(S ^*) let us consider the Lagrangian:

  

  As P(S ^*) is a convex problem, in the optimum L has a saddle point, which is a minimum with respect to each generated power, that is

  

  By considering situations where a generating company only changes its output, the first set of inequalities in V(S ^*) is obtained. In contrast, it must be as well to be a minimum with respect to any feasible increment, that is

  

  which is nothing else but a non-arbitrage condition. □