Local Search for Mixed-Integer Nonlinear Optimization: A Methodology and an Application

A methodology is presented for tackling mixed-integer nonlinear optimization problems by local search, in particular large-scale real-life problems. This methodology is illustrated through the localsearch heuristic implemented for solving an energy management problem posed by the EDF company in the context of the ROADEF/EURO Challenge 2010, an international competition of applied optimization. Our local-search approach is pure and direct: the problem is tackled frontally, without decomposition nor hybridization. In this way, both combinatorial and continuous decisions can be modified by a move during the search. Then, our work focuses on the diversification by the moves and on the performance of the incremental evaluation machinery. Exploring millions of feasible solutions within one hour of running time, the resulting local search allows us to obtain among the best results of the competition, in which 44 teams from 25 countries were engaged. 1 Presentation of the Problem Électricité de France (EDF) is the historical French energy producer and supplier, whose operations and participations span worldwide today. The EDF power generation facilities in France stand for a total installed capacity of nearly 100 GW. Most of the French electricity is produced by thermal power plants: 90% in 2009 among which 82% by nuclear power plants. The subject of the ROADEF/EURO Challenge 2010, an international competition organized by the French Operational Research and Decision Support Society (ROADEF) and the Association of European Operational Research Societies (EURO), was focused on the medium-term (5 years) management of the EDF French thermal power park, and especially of nuclear plants which have to be repeatedly shut down for refueling and maintenance. Before describing our contributions (methodology and application), the main characteristics of this problem (decision variables, objectives, constraints) are outlined. For the sake of concision and readability, the presentation remains voluntarily informal. The interested reader is referred to the detailed technical specification provided by EDF in the context of the 2 F. Gardi, K. Nouioua ROADEF/EURO Challenge 2010 [9], which can be downloaded on the ROADEF website. The thermal power park is composed of two kinds of plants. Type-1 (shortly T1) plants can be supplied in fuel continuously. They correspond to coal, fuel oil, or gas power facilities, or even virtual power stations allowing to import energy. On the other hand, Type-2 (shortly T2) plants have to be shut down for refueling and maintenance regularly. They correspond to nuclear power plants. Indeed, the fuel stock of these facilities is consumed as power is produced. Whenever a T2 plant is supplied with new fuel, it has to be offline and cannot produce during the length of this outage period, generally several weeks. Thus, the operation of a T2 plant is organized in a succession of cycles, namely an offline period (outage) followed by an online period (production campaign). These production assets are used to satisfy a customer demand over a specific time horizon. This horizon is discretized with a homogeneous time step. Customer load is uncertain and known only through an available set of uncertainty scenarios. These scenarios are assumed to be the realization of some stochastic processes (in particular weather conditions). Production at T1 plants incurs a cost proportional to the power output and also depends on the load scenario and the time step. For each T2 facility, the initial fuel stock at the beginning of the time horizon is known. Then, refueling of T2 plants leads to costs proportional to the amount of loaded fuel, also depending on the time step. Because refueling and maintenance are heavy operations immobilizing many resources, the order of cycles is fixed for each T2 power station. The earliest outages over the horizon cannot be canceled, and must be planned in given time intervals. For the latest outages, whose dates are not forced and which can be postponed, the following rule applies: if an outage is canceled, all following outages must be canceled too. Ultimately, the objective is to minimize the expected cost of production over the given horizon. More precisely, the decision variables of the problem are: the starting dates of outages for all T2 plants, the refuel quantities for all outages of all T2 plants, the production levels for all T1 and T2 plants, all time steps, and all scenarios. Note that the first kind of decision variables are discrete, whereas the second and third ones are continuous. The objective function to minimize is composed of two terms: the expected production costs for all T1 plants (that is, the production costs over all scenarios divided by the number of scenarios), and the refuel costs for all T2 plants minus the expected stock values at the end of the period (to avoid end-of-side effects). The constraints can be classified into three categories; all these constraints are listed and numbered from CT1 to CT21 in the EDF specification. The first category (CT1) corresponds to constraints for coupling the production of the plants: for each scenario and each time step, the sum of productions for all T1 and T2 plants must be equal to the demand. The second category (CT2-CT12) concerns how plants can produce. The power of T1 plants must remain between minimum and maximum values depending on time steps and scenarios. When a T2 plant is offline, its power is equal to zero. When a T2 plant is online, its power must be non negative, lower than a maximum 3 http://challenge.roadef.org/2010/index.en.htm Local Search for Mixed-Integer Nonlinear Optimization 3 value depending on time steps. During the production campaign, the fuel level dynamics couples fuel quantity and production output over the time horizon: the fuel stock at time step t+ 1 is equal to the fuel stock at t minus the energy produced at t, namely the product of the power delivered at t by the duration of a time step. However, the T2 power must respect an imposed profile when the fuel stock becomes lower than a given limit; this profile follows a decreasing piecewise affine function (with only a few pieces). If there is no longer enough fuel stock to produce, the power production is equal to zero. A nuclear plant produces generally at maximal power; otherwise, the T2 plant is said to be in modulation. Because the difference between the maximum power of the plant and the actual production leads to an undesirable wear of the equipment involved, the quantity of energy modulated (that is, not produced at maximal power) between two outages cannot be lower than a given value. For T2 plants, additional constraints are set on refueling operations. The refuel quantity must be in a given interval, the stocks before and after refueling must be lower than given limits. Note that at each refueling operation, a given percentage of stock is lost. Finally, the third category (CT13-CT21) corresponds to constraints on outage scheduling of T2 plants: earliest and latest starting dates of outages, minimum spacing or maximum overlapping of two outages, minimum spacing between the starting or ending dates of two outages, resource constraints (the number of maintenance teams able to perform refueling operations is limited), maximum number of outages containing a given week, maximum cumulated offline power during a given period. This optimization problem can be classified as mixed-integer nonlinear. One can observe that it includes two dependent subproblems, nested according to a master/slave scheme. The master subproblem consists in determining a schedule of outages of T2 plants, satisfying constraints induced by limitations on resources which are necessary to perform refueling and maintenance operations (CT13CT21). In summary, this subproblem involves combinatorial decision variables, subject to constraints related to intervals on the integer line. Having scheduling outages, the slave subproblem consists in planning production to satisfy the demand at the minimum cost, that is determining the stock refuels for each T2 plant and each outage, and the quantity of energy to produce by each plant (T1 and T2) at each time step for each scenario. This subproblem involves continuous decision variables subject to classical flow conservation and capacity constraints, but also to nonlinear constraints activated under certain logical conditions (CT1-CT12). The master subproblem of outage scheduling is theoretically NP-complete, because it corresponds to scheduling tasks with date and resource constraints [5]. On the other hand, the slave subproblem of production planning seems to be NP-hard too because of the nonlinear constraints (CT6, also called imposition constraints). Moreover, the instances to tackle may be very large: 8 outages to schedule over 300 weeks for 70 T2 plants, the production levels to determine for 170 plants (T1 and T2) over 10000 time steps for 500 scenarios. The execution time of the algorithm is limited to 1 hour on a standard computer. 4 F. Gardi, K. Nouioua We insist on the fact that the economic issues subjacent to this problem are considerable. The economic function to minimize (obfuscated in EDF data) contains in effect nine digits, representing nearly one billion euros per year of operating costs for EDF [6]. Therefore, a gap of only 0.1% between two solutions represents savings of the order of one million euros a year. 2 Methodology and Outcomes To the best of our knowledge, no work has been published yet in the literature which addresses the energy management model exposed above. Nevertheless, several softwares have been implemented by EDF researchers these last ten years for solving this problem. These solutions consists essentially in decomposing and reducing the problem so as to approach it with mixed-integer programming solvers [7, pp. 116–118]. The software currently exploited at EDF is based on a decomposition of the problem site by site, each subproblem being heuristically solved by integer linear programming according to the master/slave decomposition. Unfortunately, this solution does not guaranty the satisfiability of all the constraints of the problem. A large neighborhood search approach based on constraint programming (for scheduling outages) and integer linear programming (for planning production) was recently proposed by [8] (see also [7]), but the proposed solution does not take in input a set of scenarios but only one. The solution approach that we have implemented in the context of the ROADEF/EURO Challenge 2010 is a randomized local search, technique which is rarely used in mixed-integer nonlinear optimization. The design and engineering of our local-search algorithm follow a precise methodology, inspired by the previous works of the authors [1–4]. This methodology is outlined below. The first particularity of our local search is to be pure and direct. Indeed, no decomposition is done; the problem is tackled frontally. The search space explored by our algorithm is close to the original solution space. In particular, the combinatorial and continuous parts of the problem are treated together: combinatorial and continuous decisions can be simultaneously modified by a move during the search. By avoiding decompositions or reductions, no solution is lost and the probability to find good-quality ones is increased. Then, no hybridization is done: no particular metaheuristic is used, no tree-search technique is used. The diversification of the search is obtained by exploring a large variety of randomized neighborhoods. The second specificity of our local search is to be very aggressive: millions of feasible solutions are visited within the time limit. Indeed, randomized local search is a non deterministic, incomplete exploration of the search space. Therefore, exploring a huge number of (feasible) solutions during the allocated time augments the probability to find good-quality solutions. Then, our local-search heuristic is composed of three layers: general heuristic, moves, evaluation machinery. The evaluation machinery forms the engine of the local search; it computes the impacts of moves on constraints and objectives during the search. The time spent to engineer each layer during the project follows the following distribution: 10% on general heuristic, 20% on moves, 70% on Local Search for Mixed-Integer Nonlinear Optimization 5 evaluation machinery. In summary, our work was focused on: designing randomized moves allowing a diversified exploration of the search space despite strong constraints, and speeding up the evaluation of these moves notably by implementing an incremental randomized combinatorial algorithm for solving approximately but very efficiently the continuous subproblem. Numerical experiments show that this combinatorial algorithm is 10 000 times faster than state-of-theart linear programming solvers (without imposition constraints), while providing near-optimal production plans. Note that the same approach was recently applied by one of the authors for solving a real-life inventory routing problem [1], which can be viewed as a mixed-integer linear optimization problem. Benchmarks are divided into three categories A, B, X containing each one 5 instances. The instances A were communicated at the beginning of the qualification phase of the ROADEF/EURO Challenge 2010. Teams were selected for the final stage based on the results obtained on these instances. Then, instances B, much larger and very realistic, were given as test bed for the final stage. Ultimately, finalists were ranked according to their results on instances B (known from competitors) and instances X (unknown from competitors, communicated after the announcements of the final ranking). The final results were announced during EURO 2010, the 24th European Conference on Operational Research. Our algorithm was ranked 1st on instances A and B (among 44 teams engaged, 16 finalists), before falling to the 8th place due to a late-working-hours bug appearing on some instances X (note that only 4 teams among the 16 finalists have been able to provide all the solutions to instances X). Once corrected, our algorithm provides state-of-the-art results on instances X in conditions similar to ones of the Challenge. The results on instances B as computed by the Challenge’s organizers show an average gap greater than 1% (resp. 10%) between our solutions and the ones of the team ranked 3rd (resp. 6th). As evoked above, such gaps are important because corresponding from dozens to hundreds million euros of savings. One can observe that the majority of approaches proposed by the other competitors corresponds to MIP/CP-based decomposition heuristics. The presentation of the local-search algorithm is done through three sections, each one corresponding to one layer of the local search. The last section is devoted to numerical experiments.