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

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. We also recover directly a previously noted property of an alternative market model. 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

Revisiting minimum profit conditions in uniform price day-ahead electricity auctions

We examine the problem of clearing day-ahead electricity market auctions where each bidder, whether a producer or consumer, can specify a minimum profit or maximum payment condition constraining the acceptance of a set of bid curves spanning multiple time periods in locations connected through a transmission network with linear constraints. Such types of conditions are for example considered in the Spanish and Portuguese day-ahead markets. This helps describing the recovery of start-up costs of a power plant, or analogously for a large consumer, utility reduced by a constant term. A new market model is proposed with a corresponding MILP formulation for uniform locational price day-ahead auctions, handling bids with a minimum profit or maximum payment condition in a uniform and computationally-efficient way. An exact decomposition procedure with sparse strengthened Benders cuts derived from the MILP formulation is also proposed. The MILP formulation and the decomposition procedure are similar to computationally-efficient approaches previously proposed to handle so-called block bids according to European market rules, though the clearing conditions could appear different at first sight. Both solving approaches are also valid to deal with both kinds of bids simultaneously, as block bids with a minimum acceptance ratio, generalizing fully indivisible block bids, are but a special case of the MP bids introduced here. We argue in favour of the MP bids by comparing them to previous models for minimum profit conditions proposed in the academic literature, and to the model for minimum income conditions used by the Spanish power exchange OMIE

A new formulation of the European day-ahead electricity market problem and its algorithmic consequences

A new formulation of the optimization problem implementing European market rules for non- convex day-ahead electricity markets is presented, that avoids the use of complementarity constraints to express market equilibrium conditions, and also avoids the introduction of auxiliary binary variables to linearise these constraints. Instead, we rely on strong duality theory for linear or convex quadratic optimization problems to recover equilibrium constraints imposed by most of European power exchanges facing indivisible orders. When only so-called stepwise preference curves are considered to describe continuous bids, the new formulation allows to take full advantage of state-of-the-art solvers, and in most cases, an optimal solution together with market clearing prices can be computed for large-scale instances without any further algorithmic work. The new formulation also suggests a very competitive Benders-like decomposition procedure, which helps to handle the case of interpolated preference curves that yield quadratic primal and dual objective functions, and consequently a dense quadratic constraint. This procedure essentially consists in strengthening classical Benders cuts locally. Computational experiments on real data kindly provided by main European power exchanges (Apx-Endex, Belpex and Epex spot) show that in the linear case, both approaches are very efficient, while for quadratic instances, only the decomposition procedure is tractable and shows very good results. Finally, when most orders are block orders, and instances are combinatorially very hard, the new MILP approach is substantially more efficient

Linear prices for non-convex electricity markets: models and algorithms

Strict Linear Pricing in non-convex markets is a mathematical impossibility. In the context of electricity markets, two different classes of solutions have been proposed to this conundrum on both sides of the Atlantic. We formally describe these two approaches in a common framework, review and analyze their main properties, and discuss their shortcomings. In US, some orders are not settled at the market price, but at their bidding price, deviating from uniform pricing (all orders are financially settled at the same prices). This creates a disincentive to bid one’s own true cost, and creates a missing money problem for the clearing house of the market. In Europe, all accepted orders are in-the-money are settled at the uniform market price. This implies that the welfare-maximizing solution is considered infeasible and also that the optimization problem is much less convex and more difficult to solve. This also creates fairness issues for orders of small volume, and the solution obtained does not implement a Walrasian equilibrium. Based on this analysis we propose a new model that draws on both approaches and retains their best theoretical properties. We also show how the different approaches compare on classical toy problem

Algorithms for single item constant capacity lotsizing problems

The main result of this paper is to provide an O(n3) algorithm for the single item constant capacity lotsizing problem with backlogging and a general number of installable batches, i.e in each time period t we may install up to mt multiples of the batch capacity, where the mt are given and are time-dependent. This generalizes earlier results [12, 16] as we consider backlogging and a general number of installable batches. We also give faster algorithms for three special cases of this general problem. When backlogging is not allowed and the costs satisfy the Wagner-Whitin property the problem is solvable in O(n2logn) time. In the discrete case it is possible to solve the problem with and without backlogging in O(n2) and O(n2logn) time respectively

Efficient approximation algorithms for the economic lot-sizing in continuous time

We consider a continuous-time variant of the classical Economic Lot-Sizing (ELS) problem. In this model, the setup cost is a continuous function with lower bound $K_min > 0$, the demand and holding costs are integrable functions of time and the replenishment decisions are not restricted to be multiples of a base period. Starting from the assumption that certain operations involving the setup and holding cost functions can be carried out efficiently, we argue that this variant admits a simple approximation scheme based on dynamic programming: if the optimal cost of an instance is OPT, we can find a solution with cost at most $(1 + ε)OPT$ using no more than $O ({1\over ε^2} {OPT \over K_min} log {OPT \over K_min}$ of these operations. We argue, however, that this algorithm could be improved on instances where the setup costs are “generally” very large compared with $K_min$. This leads us to introduce a notion of input-size σ that is significantly smaller than $OPT/K_min$ on instances of this type, and then to define an approximation scheme that executes $O({1\over ε^2}σ^2 ({OPT \over K_min}))$ operations. Besides dynamic programming, this second approximation scheme builds on a novel algorithmic approach for Economic Lot Sizing problems

A solution approach of production planning problems based on compact formulations for single-item lot-sizing models

We survey the main results presented in the author’s PhD Thesis presented in June 2003 at the Université catholique de Louvain and supervised by Y. Pochet and L.A. Wolsey. The dissertation is written in English and is available from the author. In the first part of the thesis, we investigate the complexity and the polyhedral structure of various extensions of the uncapacitated single-item lot-sizing problem (Barany et al. 1984). In particular, we study models involving fixed charges on stocks, constant capacity and backlogging, and lower bounds on production. We describe algorithms, extended formulations, (facet-defining) valid inequalities and separation algorithms. Emphasis is placed on compact (i.e. of polynomial size) exact extended formulations. In a second part, we show how such extended reformulations for single-item problems can help to improve the solution of much more general production planning problems

Algorithms for single-item lot-sizing problems with constant batch size

The main result of this paper is an O(n(3)) algorithm for the single-item lot-sizing problem with constant batch size and backlogging. We consider a general number of installable batches, i.e., in each time period t we may produce up to m, batches, where the m, are given and time-dependent. This generalizes earlier results as we consider backlogging and a general number of maximum batches. We also give faster algorithms for three special cases of this general problem. When backlogging is not allowed and the costs satisfy the Wagner-Whitin property, the problem is solvable in O(n(2)log n) time. When the production in each period is required to be either zero or equal to the installed capacity, it is possible to solve the problem with and without backlogging in O(n(2)) and O(n log n) time, respectively