Sensitivity analysis for the outages of nuclear power plants

Nuclear power plants must be regularly shut down in order to perform refueling and maintenance operations. The scheduling of the outages is the first problem to be solved in electricity production management. It is a hard combinatorial problem for which an exact solving is impossible. Our approach consists in modelling the problem by a two-level problem. First, we fix a feasible schedule of the dates of the outages. Then, we solve a low-level problem of optimization of elecricity production, by respecting the initial planning. In our model, the low-level problem is a deterministic convex optimal control problem. Given the set of solutions and Lagrange multipliers of the low-level problem, we can perform a sensitivity analysis with respect to dates of the outages. The approximation of the value function which is obtained could be used for the optimization of the schedule with a local search algorithm. Key-words: Sensitivity analysis, nuclear power plants, optimal control, Pontryagin’s principle. ∗ EDF R&D, 92141 Clamart, France ([email protected]) † INRIA-Saclay and CMAP, École Polytechnique, 91128 Palaiseau, France ([email protected]) ‡ INRIA-Saclay and CMAP, École Polytechnique, 91128 Palaiseau, France ([email protected]) ha l-0 06 71 18 6, v er si on 1 16 F eb 2 01 2 Analyse de sensibilité pour les arrêts des centrales nucléaires Résumé : Les centrales nucléaires doivent être régulièrement arrêtées afin de réaliser des opérations de maintenance et de rechargement en combustible nucléaire. La planification de ces arrêts constitue le premier problème à résoudre en gestion de la production d’électricité. C’est un problème combinatoire difficile qui ne peut être résolu exactement. Notre approche consiste à modéliser ce problème par un problème à deux niveaux. Tout d’abord, nous fixons un calendrier admissible des dates des arrêts des centrales. Puis, nous résolvons un sous-problème de production d’électricité, en respectant le calendrier initial. Dans notre modèle, ce sous-problème est un problème de contrôle optimal déterministe et convexe. Etant donnés les solutions et multiplicateurs de Lagrange du sous-problème, nous pouvons réaliser une analyse de sensibilité par rapport aux dates des arrêts. L’approximation de la fonction valeur que nous obtenons devrait permettre de mettre en place un algorithme de recherche locale pour l’optimisation de ces dates d’arrêts. Mots-clés : Analyse de sensibilité, centrales nucléaires, contrôle optimal, principe de Pontryagine. ha l-0 06 71 18 6, v er si on 1 16 F eb 2 01 2 Sensitivity analysis for the outages of nuclear power plants 3 Introduction Energy generation in France is a competitive market, whereas transportation and distribution are monopolies. Electric utilities generate electricity from hydro reservoirs, fossil energy (coal, gas), atom (nuclear fission process) and to a small extent from wind farms, solar energy or run of river plant without pondage. This energy mix provides enough power and flexibility to match energy demand in any circumstances. Hydro power stations are managed in order to remove peaks on the load curve during peak-hours, whereas thermal power stations supply base load energy. Due to their capacity generation and their production cost as well, the base load part is mainly supported by nuclear power stations. Nuclear facilities are subject to various constraints, which induces a variation of the availability of nuclear energy. Some events may occur randomly during the operating period and cause forced outages. This is why outages must be planned by the producer in order to perform maintenance and refuelling operations of the fleet of nuclear power stations and in order to avoid a dramatical decrease of the nuclear availability. Thermal power stations, using expensive resources such as coal or gas, enable to compensate a lack of nuclear energy. These supplementary costs, due to the nuclear unavailability must be minimized when a schedule of the outages is planned. Each power station has its scheduling variables, which are submitted to local and coupling constraints as well. There are different constraints in the scheduling of outages of power plants: on the minimum spacing, on the maximum overlapping between outages, and on the number of outages in parallel. For operating purposes, the decision to stop a power station for maintenance has to be forecast far ahead. Hence scheduling decisions are modelled as “open-loop” decisions, which means that they do not depend on the consumption scenario. Given the planning of outages, the low-level problem of electricity production can be described by a discrete time dynamic and stochastic optimization problem. The overall optimization problem is a large scale, mixed integer stochastic problem. We refer to [2, 5, 6, 7] for precise descriptions of this problem. At Electricité de France, the numerical resolution of this problem uses local search algorithms in order to improve the current planned program. Numerous slight modifications are performed around the current program and the most profitable determines the next program. The computational burden to solve this problem is heavy, reducing it is a challenging task. In this paper, we perform a sensitivity analysis of the electricity production problem when the integer parameters defining the scheduling of the outages are set. We provide a first-order expansion of the value of this low-problem, with respect to the dates of the outages. For the sake of simplicity, the low-level problem is a convex deterministic optimal control problem with continuous time. We do not consider the combinatorial side of the problem. In the first section, we discuss the structure of solutions to the low-level problem, which are not unique in general. In the second section, we realize the sensitivity analysis by using a well suited time reparameterization. We obtain a formula for the directional derivatives of the value function using the opposite of the jumps of the true Hamiltonian at the times of beginning or end of the outages. It is based on the set of Lagrange multipliers, which we describe precisely. The result is an application of a theorem of [1]. The technical aspects related to the theorem such as the proof of qualification or the proof of convergence of the solutions to the perturbed problems are postponed in the third section. RR n° 7884 ha l-0 06 71 18 6, v er si on 1 16 F eb 2 01 2 4 Barty, Bonnans & Pfeiffer 1 Study of the reference problem In this first part, we study the low-level problem of production management and therefore consider that the dates of the outages are fixed. In our model, we only consider one outage for each plant. Applying Pontryagin’s principle, we study the particular structure of the optimal controls, which are not unique in general. 1.1 Notations, model and mathematical hypotheses The main notations for the problem are the following: [0, T ] the time period c(x) the cost of production of an amount x with thermal power stations d(t) the demand of electricity at time t S the set of nuclear power plants n the number of nuclear power plants s(t) the amount of available fuel of plant i at time t s0 the initial level of plant i u(t) the rate of production of plant i at time t u the maximum rate of production of plant i at time t U the bounds on controls, equal to ∏ i∈S[0, u ] τ i b the date of the beginning of the outage of plant i τ i e the date of the end of the outage of plant i W (t) the set of working plants at time t, defined by W (t) = {i ∈ S, t / ∈ [τ i b , τ i e]} a(t) the rate of refuelling of plant i at time t (for all t, a(t) ≥ 0) φ(s(T )) a decreasing convex function of the final state V (τb, τe) the value of the optimal control problem in function of τb and τe. The optimal control problem (P(τb, τe)) is V (τb, τe) = min u,s ∫ T