A MIP framework for non-convex uniform price day-ahead electricity auctions

Abstract

It is well known that a market equilibrium with uniform prices often does not exist in non-convex day-ahead electricity auctions. We consider the case of the non-convex, uniform-price Pan-European day-ahead electricity market “PCR” (Price Coupling of Regions), with non-convexities arising from so-called complex and block orders. Extending previous results, we propose a new primal-dual framework for these auctions, which has applications in both economic analysis and algorithm design. The contribution here is threefold. First, from the algorithmic point of view, we give a non-trivial exact (i.e., not approximate) linearization of a non-convex ‘minimum income condition’ that must hold for complex orders arising from the Spanish market, avoiding the introduction of any auxiliary variables, and allowing us to solve market clearing instances involving most of the bidding products proposed in PCR using off-the-shelf MIP solvers. Second, from the economic analysis point of view, we give the first MILP formulations of optimization problems such as the maximization of the traded volume, or the minimization of opportunity costs of paradoxically rejected block bids. We first show on a toy example that these two objectives are distinct from maximizing welfare. Third, we provide numerical experiments on realistic large-scale instances. They illustrate the efficiency of the approach, as well as the economics trade-offs that may occur in practice.

Appendix: Omitted proofs in main text

Let us first consider the equality of primal and dual objective functions of Sect. 2.2:

Observation 1

By strong duality for linear programs, for a pair of primal and dual feasible points corresponding to a block bid selection and a MIC bid selection, i.e., satisfying, respectively, (2)–(13) and (15)–(22), the complementarity constraints (23)–(37) hold if and only if we have the equality (1) $=$ (14).

1.1 Proof of Theorem 1

Proof

We emphasize again, and use below, the fact that according to Lemma 2, we can assume without loss of generality $du^a_{c_a} = 0, \forall c_a \in C_a$ in (14)–(22).

(I) Let $MCS=(x,y,n,u,\pi ,v, s,d^a,d^r, du^a, du^r)$ be a feasible point of UMFS and let us define $J_r := \{j \in J | y_j = 0\}, J_a := \{j \in J | y_j = 1\}$ and likewise for $C_r, C_a$ with respect to the values of the variables $u_c$. Consider the projection ${\tilde{MCS}}=(x,y,n,u, \pi ,v, s,d^a_{j_a \in J_a},d^r_{j_r \in J_r}, du^a_{c_a \in C_a}, du^r_{c_r \in C_r})$. Constraints (50)–(55) ensure that ${\tilde{MCS}}$ satisfies constraints (17)–(20): constraints (52)–(55) are ‘dispatching’ constraints (50)–(51) to constraints (17)–(20). Therefore ${\tilde{MCS}}$ satisfies primal conditions (2)–(13) and dual conditions (15)–(22). Condition (54) ensures that $d^a_j=0$ for $j \in J_r$, and with (55), it shows that condition (39) implies the equality (1) $=$ (14). By Observation 1, we can then replace this equality by the needed complementarity conditions (23)–(37).

(II) Conversely let ${\tilde{MCS}}=(x,y,n,u, \pi ,v, s,d^a_{j_a \in J_a},d^r_{j_r \in J_r}, du^a_{c_a \in C_a}, du^r_{c_r \in C_r})$ be a point satisfying primal conditions (2)–(13), dual conditions (15)–(22), and complementarity conditions (23)–(37), associated to a given block and MIC bid selection $J=J_a \cup J_r, C = C_a \cup C_r$. Observation 1 ensures that this point also satisfies the equality (1) $=$ (14). Let us set additional values $d^r_j=0$, for $j \in J_a$, also $d^a_j=0$ for $j \in J_r$, and similarly $du^a_c = 0$ for $c \in C_r$, $du^r_c = 0$ for $c \in C_a$, giving a point $MCS=(x,y,n, u, \pi ,v, s,d^a,d^r, du^a, du^r)$. The new point satisfies condition (39), since only terms $d^a_j=0, j \in J_r, du^a_c=0$ are added to the equality (1) $=$ (14). It remains to verify that all the remaining constraints defining UMFS are satisfied as well. All these additional values trivially satisfy constraints (52)–(55). Therefore, it is needed to show that conditions (50)–(55) are also satisfied for all $j \in J, c \in C$. Due to (24), in condition (17), $s_{j_r}=0$ and we can set $d^r_{j_r }:=P_{j_r} \lambda ^{j_r}-P_{j_r} \pi$ without altering the satisfaction of any condition. Due to the price range condition and the choice of the parameters $M_j$, these $d^r_j,j \in J_r$ satisfy conditions (52) which therefore hold for all $j \in J$. In condition (18), $s_{j_a},d^a_{j_a}$ can be redefined without modifying the values of $(s_{j_a}-d^a_{j_a})$ and hence without altering satisfaction of any other constraint. Due to the large values of the parameters $M_j$, this again can be done so as to satisfy conditions (54) for $j \in J_a$, hence for all $j \in J$. Then, (17, 18), and the ‘dispatcher conditions’ (52)–(54) imply (50). Finally, concerning the analogue constraints related to the MIC bids, and first using Lemma 1 to set $du^a_{c_a} = 0$ for all $c_a \in Ca$, it is straightforward to show in a similar way that (19, 20) together with the $M_c$ and the additional null values for part of the $du^r$ (resp. $du^a$) given above allow satisfying (51, 53). $\square$