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 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

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

Feasibility of Xpert Ebola Assay in Médecins Sans Frontières Ebola Program, Guinea

Rapid diagnostic methods are essential in control of Ebola outbreaks and lead to timely isolation of cases and improved epidemiologic surveillance. Diagnosis during Ebola outbreaks in West Africa has relied on PCR performed in laboratories outside this region. Because time between sampling and PCR results can be considerable, we assessed the feasibility and added value of using the Xpert Ebola Assay in an Ebola control program in Guinea. A total of 218 samples were collected during diagnosis, treatment, and convalescence of patients. Median time for obtaining results was reduced from 334 min to 165 min. Twenty-six samples were positive for Ebola virus. Xpert cycle thresholds were consistently lower, and 8 (31%) samples were negative by routine PCR. Several logistic and safety issues were identified. We suggest that implementation of the Xpert Ebola Assay under programmatic conditions is feasible and represents a major advance in diagnosis of Ebola virus disease without apparent loss of assay sensitivity

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

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

Identifying when thresholds from the Paris Agreement are breached : the minmax average, a novel smoothing approach

Identifying when a given threshold has been breached in the global temperature record has become of crucial importance since the Paris Agreement. However there is no formally agreed methodology for this. In this work we show why local smoothing methodologies like the moving average and other climate modeling based approaches are fundamentally ill-suited for this purpose, and propose a better one, that we call the minmax average. It has strong links with the isotonic regression, is conceptually simple and is arguably closer to the intuitive meaning of ”breaching the threshold” in the climate discourse, all favorable features for acceptability. When applied to the global mean surface temperature anomaly (GMSTA) record from Berkeley Earth, we obtain the following conclusions. First, the rate of increase has been ~+0.25◦C per decade since 1995. Second, based on this new estimate alone, we should plausibly expect the GMSTA to reach 1.49◦C in 2023 and not go below that on average in the medium-term future. When taking into account the record temperatures of the second half of 2023, not having breached the 1.5◦C threshold already in July 2023 is only possible with record long and/or deep La Nina in the following years