Bidding in sequential electricity markets: The Nordic case

For electricity market participants trading in sequential markets with differences in price levels and risk exposure, it is relevant to analyze the potential of coordinated bidding. We consider a Nordic power producer who engages in the day-ahead spot market and the hour-ahead balancing market. In both markets, clearing prices and dispatched volumes are unknown at the time of bidding. However, in the balancing market, the market participant faces an additional risk of not being dispatched. Taking into account the sequential clearing of these markets and the gradual realization of market prices, we formulate the bidding problem as a multi-stage stochastic program. We investigate whether higher risk exposure may cause hesitation to bid into the balancing market. Furthermore, we quantify the gain from coordinated bidding, and by deriving bounds on this gain, assess the performance of alternative bidding strategies used in practice.     

Electricity market clearing with improved scheduling of stochastic production

In this paper, we consider an electricity market that consists of a day-ahead and a balancing settlement, and includes a number of stochastic producers. We first introduce two reference procedures for scheduling and pricing energy in the day-ahead market: on the one hand, a conventional network-constrained auction purely based on the least-cost merit order, where stochastic generation enters with its expected production and a low marginal cost; on the other, a counterfactual auction that also accounts for the projected balancing costs using stochastic programming. Although the stochastic clearing procedure attains higher market efficiency in expectation than the conventional day-ahead auction, it suffers from fundamental drawbacks with a view to its practical implementation. In particular, it requires flexible producers (those that make up for the lack or surplus of stochastic generation) to accept losses in some scenarios. Using a bilevel programming framework, we then show that the conventional auction, if combined with a suitable day-ahead dispatch of stochastic producers (generally different from their expected production), can substantially increase market efficiency and emulate the advantageous features of the stochastic optimization ideal, while avoiding its major pitfalls.     

Optimal bidding of a virtual power plant on the Spanish day-ahead and intraday market for electricity

We develop a multi-stage stochastic programming approach to optimize the bidding strategy of a virtual power plant (VPP) operating on the Spanish spot market for electricity. The VPP markets electricity produced in the wind parks it manages on the day-ahead market and on six staggered auction-based intraday markets. Uncertainty enters the problem via stochastic electricity prices as well as uncertain wind energy production. We set up the problem of bidding for one day of operation as a Markov decision process (MDP) that is solved using a variant of the stochastic dual dynamic programming algorithm. We conduct an extensive out-of-sample comparison demonstrating that the optimal policy obtained by the stochastic program clearly outperforms deterministic planning, a pure day-ahead strategy, a benchmark that only uses the day-ahead market and the first intraday market, as well as a proprietary stochastic programming approach developed in the industry. Furthermore, we study the effect of risk aversion as modeled by the nested Conditional Value-at-Risk as well as the impact of changes in various problem parameters.     

Minimizing grid capacity in preemptive electric vehicle charging orchestration: Complexity,exact and heuristic approaches

Unlike refueling an internal combustion engine vehicle, charging electric vehicles is time-consuming and results in higher energy consumption. Hence, charging stations will face several challenges in providing high-quality charging services when the adoption of electric vehicles increases. These charging infrastructures must satisfy charging demands without overloading the power grid. In this work, we investigate the problem of scheduling the charging of electric vehicles to reduce the maximum peak power while satisfying all charging demands. We consider a charging station where the installed chargers deliver a preemptive constant charging power. These chargers can either be identical or non-identical. For both cases, we address two optimization problems. First, we study the problem of finding the minimum number of chargers needed to plug a set of electric vehicles giving different arrival and departure times and required energies. We prove that this problem belongs to the complexity class P, and we provide polynomial-time algorithms. Then, we study the problem of minimizing the power grid capacity. For identical chargers, we prove that the problem is polynomial, whereas it is NP-hard in the case of non-identical chargers. We formulate these problems as a mixed-integer linear programming model for both cases. To obtain near-optimal solutions for the NP-hard problem, we propose a heuristic and an iterated local search metaheuristic. Through computational results, we demonstrate the effectiveness of the proposed approaches in terms of reducing the grid capacity.     

The computational complexity of the pooling problem

The pooling problem is an extension of the minimum cost flow problem defined on a directed graph with three layers of nodes, where quality constraints are introduced at each terminal node. Flow entering the network at the source nodes has a given quality, at the internal nodes (pools) the entering flow is blended, and then sent to the terminal nodes where all entering flow streams are blended again. The resulting flow quality at the terminals has to satisfy given bounds. The objective is to find a cost-minimizing flow assignment that satisfies network capacities and the terminals’ quality specifications. Recently, it was proved that the pooling problem is NP-hard, and that the hardness persists when the network has a unique pool. In contrast, instances with only one source or only one terminal can be formulated as compact linear programs, and thereby solved in polynomial time. In this work, it is proved that the pooling problem remains NP-hard even if there is only one quality constraint at each terminal. Further, it is proved that the NP-hardness also persists if the number of sources and the number of terminals are no more than two, and it is proved that the problem remains hard if all in-degrees or all out-degrees are at most two. Examples of special cases in which the problem is solvable by linear programming are also given. Finally, some open problems, which need to be addressed in order to identify more closely the borderlines between polynomially solvable and NP-hard variants of the pooling problem, are pointed out.     

Strong NP-hardness of AC power flows feasibility

We prove that testing feasibility for an AC power flow system is a strongly NP-hard problem.     

A robust and informative method for solving large-scale power flow problems

Solving power flow problems is essential for the reliable and efficient operation of an electric power network. However, current software for solving these problems have questionable robustness due to the limitations of the solution methods used. These methods are typically based on the Newton–Raphson method combined with switching heuristics for handling generator reactive power limits and voltage regulation. Among the limitations are the requirement of a good initial solution estimate, the inability to handle near rank-deficient Jacobian matrices, and the convergence issues that may arise due to conflicts between the switching heuristics and the Newton–Raphson process. These limitations are addressed by reformulating the power flow problem and using robust optimization techniques. In particular, the problem is formulated as a constrained optimization problem in which the objective function incorporates prior knowledge about power flow solutions, and solved using an augmented Lagrangian algorithm. The prior information included in the objective adds convexity to the problem, guiding iterates towards physically meaningful solutions, and helps the algorithm handle near rank-deficient Jacobian matrices as well as poor initial solution estimates. To eliminate the negative effects of using switching heuristics, generator reactive power limits and voltage regulation are modeled with complementarity constraints, and these are handled using smooth approximations of the Fischer–Burmeister function. Furthermore, when no solution exists, the new method is able to provide sensitivity information that aids an operator on how best to alter the system. The proposed method has been extensively tested on real power flow networks of up to 58k buses.     

Day-ahead market bidding for a Nordic hydropower producer: taking the Elbas market into account

In many power markets around the world the energy generation decisions result from two-sided auctions in which producing and consuming agents submit their price-quantity bids. The determination of optimal bids in power markets is a complicated task that has to be undertaken every day. In the present work, we propose an optimization model for a price-taker hydropower producer in Nord Pool that takes into account the uncertainty in market prices and both production and physical trading aspects. The day-ahead bidding takes place a day before the actual operation and energy delivery. After this round of bidding, but before actual operation, some adjustments in the dispatched power (accepted bids) have to be done, due to uncertainty in prices, inflow and load. Such adjustments can be done in the Elbas market, which allows for trading physical electricity up to one hour before the operation hour. This paper uses stochastic programming to determine the optimal bidding strategy and the impact of the possibility to participate in the Elbas. ARMAX and GARCH techniques are used to generate realistic market price scenarios taking into account both day-ahead price and Elbas price uncertainty. The results show that considering Elbas when bidding in the day-ahead market does not significantly impact neither the profit nor the recommended bids of a typical hydro producer.     

Bilevel optimization applied to strategic pricing in competitive electricity markets

In this paper, we present a bilevel programming formulation for the problem of strategic bidding under uncertainty in a wholesale energy market (WEM), where the economic remuneration of each generator depends on the ability of its own management to submit price and quantity bids. The leader of the bilevel problem consists of one among a group of competing generators and the follower is the electric system operator. The capability of the agent represented by the leader to affect the market price is considered by the model. We propose two solution approaches for this non-convex problem. The first one is a heuristic procedure whose efficiency is confirmed through comparisons with the optimal solutions for some instances of the problem. These optimal solutions are obtained by the second approach proposed, which consists of a mixed integer reformulation of the bilevel model. The heuristic proposed is also compared to standard solvers for nonlinearly constrained optimization problems. The application of the procedures is illustrated in case studies with configurations derived from the Brazilian power system.     

An optimal capacity assignment for the robust design problem in capacitated flow networks

This paper proposes an exact algorithm to solve the robust design problem in a capacitated flow network in which each edge has several possible capacities. A capacitated flow network is popular in our daily life. For example, the computer network, the power transmission network, or even the supply chain network are capacitated flow networks. In practice, such network may suffer failure, partial failure or maintenance. Therefore, each edge in the network should be assigned sufficient capacity to keep the network functioning normally. The robust design problem (RDP) in a capacitated flow network is to search for the minimum capacity assignment of each edge such that the network still survived even under the edge’s failure. However, how to optimally assign the capacity to each edge is not an easy task. Although this kind of problem was known of NP-hard, this paper proposes an efficient exact algorithm to search for the optimal solutions for such a network and illustrates the efficiency of the proposed algorithm by numerical examples.     

Some no-wait shops scheduling problems: Complexity aspect

We investigate the computational complexity of no-wait shops scheduling problems. The problem of finding optimal finish time schedules is shown to be NP-hard for flowshops with two machine centres where each machine centre has one or more parallel machines for processing tasks. The complexity results are also reported for no-wait shops scheduling when all nonzero tasks have unit or identical processing time requirement. A polynomial time algorithm for 3-machine flowshops is proposed for optimal finish time schedules. Finding optimal finish time schedules in 2-machine jobshops in NP-hard. Also we establish NP-hard results for 3-machine jobshops for both optimal finish time and mean flow time schedules. Some results are deduced with the present work and with those reported earlier.     

Some inequalities related to the analysis of electricity auctions

Most balancing markets of electric power are organized as uniform-price auctions. In 2001, the balancing market of England and Wales switched to a pay-as-bid auction with the intention of reducing wholesale electricity prices. Numerical simulations of an electricity auction model have indicated that this should lead to decreased average prices. In this work we prove two inequalities which give an analytic proof of this claim in the same model.     

A note on the complexity of flow shop scheduling with transportation constraints

This note investigates two-machine flow shop scheduling with transportation constraints to minimize makespan. Recently, Soukhal et al. [A. Soukhal, A. Oulamara, P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research 161 (2005) 32–41] proved that this problem is strongly NP-hard when the capacity of the truck is limited to two or three parts. The considered problem with blocking constraints is also proved to be strongly NP-hard by Soukhal et al. Unfortunately, their proofs contain mistakes. We point out their proofs’ invalidity and then show that, when the capacity of the truck is limited to two parts, the problem is binary NP-hard, and when the capacity of the truck is limited to three parts the problem is strongly NP-hard even if the jobs have a common processing time on machine one and all jobs have the same transportation time. We show also that the last result can be generalized to any fixed c (c ? 3) parts.     

Complexity results and approximation algorithms for the two machine no-wait flow-shop with limited machine availability

The scheduling problems studied in this paper concern a two-machine no-wait flow shop problem with limited machine availability. In this model, we assume that machines may not always be available, for example because of preventive maintenance. We only consider the deterministic case where the unavailable periods are known in advance. The objective function considered is the maximum completion time (C_max). We prove that the problem is NP-hard even if only one non-availability period occurs on one of machines, and NP-hard in the strong sense for arbitrary numbers of non-availability periods. We also provide heuristic algorithms with error bounding analysis.     

Complexity of the job insertion problem in multi-stage scheduling

The job insertion problem in multi-stage scheduling is: given a schedule for n jobs and an additional job, find a feasible insertion of the additional job into the schedule that minimizes the resulting makespan. We prove that finding the optimal job insertion is NP-hard for flow shops and open shops.     

Three is easy, two is hard: open shop sum-batch scheduling problem refined

For the two-machine open shop sum-batch problem to minimize the makespan an optimal schedule is known to contain one, two or three batches on each machine, and finding a two-batch optimal schedule is NP-hard. We adapt the open shop algorithm by de Werra for finding a three-batch optimal schedule in linear time.     

Parallel machine scheduling problems with a single server

In this paper, we consider the problem of scheduling jobs on parallel machines with setup times. The setup has to be performed by a single server. The objective is to minimize the schedule length (makespan), as well as the forced idle time. The makespan problem is known to be NP-hard even for the case of two identical parallel machines. This paper presents a pseudopolynomial algorithm for the case of two machines when all setup times are equal to one. We also show that the more general problem with an arbitrary number of machines is unary NP-hard and analyze some list scheduling heuristics for this problem. The problem of minimizing the forced idle time is known to be unary NP-hard for the case of two machines and arbitrary setup and processing times. We prove unary NP-hardness of this problem even for the case of constant setup times. Moreover, some polynomially solvable cases are given.     

Balancing U-lines in a multiple U-line facility

U-shaped production lines and facilities consisting of many such lines are important parts of modem manufacturing systems. The problem of balancing and rebalancing U-line facilities is studied in this paper. Like the traditional line balancing problem this problem is NP-hard. The objective is to assign tasks to a minimum number of regular, crossover, and multiline stations while satisfying cycle time, precedence, location, and station-type constraints. A secondary objective is to concentrate the idle time in one station so that improvement efforts can be focused there in accordance with modern just-in-time principles. A reaching dynamic programming algorithm is presented for determining optimal balances. It is effective for balancing and rebalancing facilities with any number of U-lines, provided that individual U-lines do not have more than 22 tasks and do not have wide, sparse precedence graphs.     

Two-Machine Flow Shop Scheduling with Nonregular Criteria

We consider a two-machine flow shop problem with a common due date where the objective is to minimize the sum of functions which penalize early as well as tardy completion of jobs. Since the problem is NP-hard in the strong sense, we investigate some general properties of optimal schedules for the problem, we develop lower and upper bounds, derive dominance criteria, and propose an enumerative algorithm for finding an optimal schedule. The performance of the proposed algorithm together with the influence of the individual components is thoroughly discussed.     

A Decomposition Methodology Applied to the Multi-Area Optimal Power Flow Problem

This paper describes a decomposition methodology applied to the multi-area optimal power flow problem in the context of an electric energy system. The proposed procedure is simple and efficient, and presents some advantages with respect to other common decomposition techniques such as Lagrangian relaxation and augmented Lagrangian decomposition. The application to the multi-area optimal power flow problem allows the computation of an optimal coordinated but decentralized solution. The proposed method is appropriate for an Independent System Operator in charge of the electric energy system technical operation. Convergence properties of the proposed decomposition algorithm are described and related to the physical coupling between the areas. Theoretical and numerical results show that the proposed decentralized methodology has a lower computational cost than other decomposition techniques, and in large large-scale cases even lower than a centralized approach.