Aurélien Froger

Michel Gendreau

Jorge E Mendoza

Eric Pinson

Louis-Martin Rousseau

Maintenance scheduling in the electricity industry: a literature review

come

Introduction

The production of movement, heat, or light needs a common input: energy. Energy can be produced from fuel (e.g., oil, gasoline, uranium, gas, coal, wood) or natural forces (e.g., wind, water). The consumption of energy is growing with the development of countries and the increasing world population, and the production must meet this demand. Therefore, the reliability of power plants, and wind and solar farms is extremely important. In this context, equipment maintenance management is a major economic issue. Just to cite few examples, equipment maintenance management in electric power systems is concerned with decisions such as: when to stop a generation unit for maintenance, when to re-start it again, and how much resources (e.g., technicians) are to be assigned to the maintenance of a given unit during a given period. These decisions are taken under complex environments and constraints such as resource availability, demand satisfaction, and reliability thresholds.

One of the most successful contributions of operations research to improve decision making in equipment maintenance management is the application of optimization techniques to solve maintenance planning and scheduling problems. In the particular case of electric power systems, these problems range from simple technician-equipment assignments to complex problems considering interactions between different stakeholders and uncertainty in the problem parameters. In this paper, we build on the work of [START_REF] Yamayee | Maintenance scheduling: Description, literature survey, and interface with overall operations scheduling[END_REF]; [START_REF] Kralj | Optimal preventive maintenance scheduling of thermal generating units in power systems -A survey of problem formulations and solution methods[END_REF]; [START_REF] Dahal | A review of maintenance scheduling approaches in deregulated power systems[END_REF]; [START_REF] Khalid | A survey of generator maintenance scheduling techniques[END_REF] to update the state-of-the-art and provide a global overview of the current stream of research in the field. To make the document easier to read, the various acronyms used in this paper are summarized in Table 1. The paper is organized as follows. Section 1 presents a brief description of the energy industry, Section 2 reviews maintenance scheduling problems rising in regulated and deregulated environments, Section 3 discusses existing solution methods for these problems, and Section 4 concludes the paper and outlines research perspectives.

The energy industry

The energy industry carries out three activities: production, transmission, and distribution. Traditionally the industry is organized in a centralized, vertically integrated way (see Figure 1): a single company has a monopoly of the entire system in its area of operation. However, the government regulates the situation directly or indirectly: the entity must not take advantage of the end consumer. Therefore, the term regulated monopoly utilities is also used. With the deregulation of the electricity industry from the end of the 1990s, competition has been replacing monopolies in most places.

Deregulation of the power industry

The deregulation (or liberalization) of the power industry has opened up the electricity market to competition.

Several companies can now produce or distribute energy; it is, however, more difficult to introduce competition for the transmission management. Energy prices are no longer regulated by the government (hence the terms deregulation and liberalization) but are subject to market interactions. Regulations remain (sometimes the term restricted power system is used) but monopolies are no longer acceptable. Given the success of this system in the aeronautics, gas, and telephone industries, this reform is promoted as a benefit for the sector.

It is intended to favor innovation, to lower prices, and to lead to better service. This new system introduces challenges such as the organization of the electricity market, the price-setting mechanism, and the coordination of the various actors.

Indeed, the introduction of market players leads to the emergence of new actors or redefines the role or activities of existing actors. An independent system operator (ISO) is responsible for the reliability and security of the system. It dispatches all or part of the energy transactions and can decrease loads on the network to avoid congestion. The ISO is the leading entity in a power market, and it must be fair. It manages the interactions between three key entities: the generating companies (GENCOs), transmission companies (TRANSCOs), and distribution companies (DISCOs). When a single TRANSCO owns the entire transmission network, the ISO operates the transmission lines. The TRANSCO is then paid for the use of its lines and the maintenance of its network [START_REF] Shahidehpour | Market Operations in Electric Power Systems[END_REF]. Retail energy service companies (RETAILCOs) act as intermediaries between GENCOs and consumers by buying energy from the former to sell to the latter. Other actors exist but their roles are relatively minor.

Energy transactions of different natures can take place in this new market structure. In a power exchange model, GENCOs and RETAILCOs negotiate bilateral contracts defining prices and quantities independently of the ISO. However, the availability of the transmission lines must be checked with the system operator to maintain security. This decentralized approach is opposed to the centralized approach (pool-based model) where market participants share extensive information (e.g., energy offer, start-up costs, generation costs, ramp-rate for each generator) with the ISO, which is responsible for ensuring the social and economic welfare of the market while keeping the system safe. Two kinds of bids are submitted to the ISO: producers' bids consist of energy blocks and their selling prices, and buyers' bids consist of energy blocks and their buying prices.

The power price is determined by the balance between supply and demand using a market clearing process.

Several markets such as day-ahead, intra-day, real-time or a combination can be encountered. Although they are different, both pool-based and power exchange models can coexist. Moreover, a transmission market deals with the purchase and sale of transmission rights. For a more detailed explanation of all these specificities, the reader is referred to [START_REF] Shahidehpour | Market Operations in Electric Power Systems[END_REF]. Figure 2 summarizes the various interactions between the actors. It is however difficult to define a typical organization because several structures are possible.

Liberalization modifies and sometimes complicates power industry issues. GENCOs, TRANSCOs, and DISCOS mainly serve their own interests, which may call into question the stability of energy production and/or energy distribution. Regulations are therefore required.

After this brief presentation of the electricity industry, we discuss, in the next section, optimization problems in maintenance scheduling of generation units and transmission lines rising in both regulated and deregulated power systems. We focus on network constraints, on data uncertainty, and on fuel consumption and supply management. To provide a global overview, Table .2 classifies the references according to the problem they solve, the power system they target and the features they deal with. Maintenance represents the actions required to ensure that a product provides reliable service. Maintenance can be split into two categories: corrective and preventive. Corrective maintenance is performed after a breakdown. Preventive maintenance is performed at predetermined intervals or according to prescribed criteria and intended to reduce the probability of failure. Maintenance in the electricity industry concerns generation units and transmission lines; the horizon can be long-term or short-term. In this paper, we do not discuss failure prediction or maintenance policies that will manage the risks of equipment failure in the most effective way. We refer the reader to [START_REF] Doyle | On the application of stochastic models in nuclear power plant maintenance[END_REF]; [START_REF] Yssaad | Reliability centered maintenance optimization for power distribution systems[END_REF] for literature dealing with those problems. We consider defining time intervals for preventive maintenance for the equipment, given financial and reliability considerations.

Maintenance scheduling of generating units

The maintenance scheduling of generation units has been widely studied. On its basic version, the maintenance scheduling problem consists in defining when to stop the generating units for preventive maintenance in order to maintain the system reliability and to reduce the general operational costs. We refer to it as the generator maintenance scheduling (GMS) problem. Additional constraints include, but are not limited to:

• maintenance tasks: maintenance window (possible time for maintenance), sequence, incompatibility, spacing, and overlapping of tasks.

• generating units: highest/lowest production levels, ramp-rate 1 .

• manpower: availability for each period, requirements by maintenance tasks.

• resources: availability for each period, requirements, consumption by maintenance tasks.

• network: transmission-line capacity (see Section 2.2), voltage.

1 Output gap limitation between two successive periods for a generating unit

• demand: fully satisfied or not, meeting of demand, energy-not-served (ENS) threshold.

• reliability: minimum reserve required by period, risk levels, ENS.

This optimization problem is generally NP-hard and may be nonlinear and nonconvex. Moreover, the power production is strongly impacted by maintenance decisions. To include load constraints, especially demand satisfaction, in the GMS problem, it can be necessary to simultaneously decide the production levels of the generating units and the maintenance scheduling. The solutions obtained can then be used as guidelines for unit commitment (UC) with a short time horizon. UC aims to schedule generating units level to meet forecasted load and reserve requirements.

In the next two sections we discuss the GMS problem in regulated and deregulated power systems.

Regulated power systems

Monopolies still operate in some regions. In a vertically integrated utility, the maintenance is scheduled in a centralized way, and all the information is available (costs, network, etc.). The various studies can be classified according to the nature of the considered objective function:

• Reliability-based

References : [START_REF] Baskar | Genetic algorithms solution to generator maintenance scheduling with modified genetic operators[END_REF][START_REF] Canto | A model for the preventive maintenance scheduling of power plants including wind farms[END_REF][START_REF] Chen | Optimal generating unit maintenance scheduling for multi-area system with network constraints[END_REF][START_REF] Dahal | Generator maintenance scheduling of electric power systems using genetic algorithms with integer representations[END_REF][START_REF] Dahal | Generator maintenance scheduling using a genetic algorithm with a fuzzy evaluation function[END_REF][START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF][START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF][START_REF] El-Amin | A tabu search algorithm for maintenance scheduling of generating units[END_REF][START_REF] Fetanat | Generation maintenance scheduling in power systems using ant colony optimization for continuous domains based 01 integer programming[END_REF][START_REF] Foong | Ant colony optimization for power plant maintenance scheduling optimizationa five-station hydropower system[END_REF][START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF][START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF][START_REF] Schlünz | An investigation into the effectiveness of simulated annealing as a solution approach for the generator maintenance scheduling problem[END_REF][START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF][START_REF] Volkanovski | Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system[END_REF][START_REF] Wang | A new genetic algorithm for preventive unit maintenance scheduling of power systems[END_REF][START_REF] Yare | Optimal maintenance scheduling of generators using multiple swarms-MDPSO framework[END_REF].

For reliability, the main optimization criterion is the leveling of the net reserves along the planning horizon. For a given period t, the net reserves correspond to the maximal power that can be produced by the available generating units not in maintenance at t minus the estimated demand during t. The most common approach is to minimize the sum of squares of the net reserves by period [START_REF] Dahal | Generator maintenance scheduling of electric power systems using genetic algorithms with integer representations[END_REF][START_REF] Dahal | Generator maintenance scheduling using a genetic algorithm with a fuzzy evaluation function[END_REF][START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF][START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF][START_REF] Foong | Ant colony optimization for power plant maintenance scheduling optimizationa five-station hydropower system[END_REF][START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF][START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF][START_REF] Schlünz | An investigation into the effectiveness of simulated annealing as a solution approach for the generator maintenance scheduling problem[END_REF][START_REF] Yare | Optimal maintenance scheduling of generators using multiple swarms-MDPSO framework[END_REF].

In contrast, [START_REF] Chen | Optimal generating unit maintenance scheduling for multi-area system with network constraints[END_REF] maximized the reserve margin when isolated power systems are tackled, whereas they leveled the reserve margin for each area when a multi-area system is considered. [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF] and [START_REF] Wang | A new genetic algorithm for preventive unit maintenance scheduling of power systems[END_REF] leveled the reserve margin by minimizing an objective function based on the deviation between a reserve rate and its average (i.e. the average reserve rate over the planning horizon). [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF] defined the reserve rate as the ratio of the net reserve to the sum of the generation capacity plus the predicted maximum load while [START_REF] Wang | A new genetic algorithm for preventive unit maintenance scheduling of power systems[END_REF] defined the reserve rate as as the ratio of the difference between the sum of the generation capacity minus the maximum predicted load to the maximum predicted load. El-Amin et al.

( 2000) considered the deviation between the reserve by period and the average reserve along the horizon. [START_REF] Baskar | Genetic algorithms solution to generator maintenance scheduling with modified genetic operators[END_REF] considered the square of this deviation and study the impact of a crew constraint (manpower availability at each period) on the results. Canto and Rubio-Romero (2013) maximized the sum by period of the ratio of the net power reserves to the gross power reserves. They introduced geographical, seasonal, and coordination constraints for a problem with wind farm turbines and thermal and hydroelectric power plants. [START_REF] Volkanovski | Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system[END_REF] minimized the annual value of the loss of load expectation (LOLE)2 taking into account the forecasted outage rate of the generating units.

Finally, [START_REF] Fetanat | Generation maintenance scheduling in power systems using ant colony optimization for continuous domains based 01 integer programming[END_REF] defined an objective function such as the generating units have to be maintained as promptly as possible to reduce the expenses related to damaged machines. [START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF] and [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF] gave outlines to schedule the power production. [START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF] also considered ramp-rate constraints for the generating units. Other studies ensure for each period that the generating capacities that are not in maintenance are sufficient to cover the demand plus sometimes a reserve constraint.

• Cost-based

References : [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF][START_REF] Al-Khamis | Unit maintenance scheduling with fuel constraints[END_REF][START_REF] Anghinolfi | A matheuristic algorithm for a large-scale energy management problem[END_REF][START_REF] Baskar | Genetic algorithms solution to generator maintenance scheduling with modified genetic operators[END_REF][START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF][START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF][START_REF] Burke | Hybrid evolutionary techniques for the maintenance scheduling problem[END_REF][START_REF] Canto | Application of Benders' decomposition to power plant preventive maintenance scheduling[END_REF][START_REF] Charest | Preventive maintenance scheduling of power generating units[END_REF][START_REF] Chattopadhyay | A practical maintenance scheduling program mathematical model and case study[END_REF][START_REF] Digalakis | A multipopulation cultural algorithm for the electrical generator scheduling problem[END_REF][START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF][START_REF] El-Amin | A tabu search algorithm for maintenance scheduling of generating units[END_REF][START_REF] El-Sharkh | A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling[END_REF][START_REF] El-Sharkh | Clonal selection algorithm for power generators maintenance scheduling[END_REF][START_REF] Fattahi | A new approach for maintenance scheduling of generating units in electrical power systems based on their operational hours[END_REF][START_REF] Fourcade | Optimizing nuclear power plant refueling with mixed-integer programming[END_REF][START_REF] Frost | Optimizing with constraints: A case study in scheduling maintenance of electric power units[END_REF][START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF][START_REF] Gardi | Local search for mixed-integer nonlinear optimization: A methodology and an application[END_REF][START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF][START_REF] Gorge | Stochastic nuclear outages semidefinite relaxations[END_REF][START_REF] Jost | A 01 integer linear programming approach to schedule outages of nuclear power plants[END_REF][START_REF] Khemmoudj | When constraint programming and local search solve the scheduling problem of Electricit de France nuclear power plant outages[END_REF][START_REF] Kovács | A system for the detailed scheduling of wind farm maintenance[END_REF][START_REF] Kralj | A multiobjective optimization approach to thermal generating units maintenance scheduling[END_REF][START_REF] Leou | A new method for unit maintenance scheduling considering reliability and operation expense[END_REF][START_REF] Lusby | A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system[END_REF][START_REF] Marwali | A deterministic approach to generation and transmission maintenance scheduling with network constraints[END_REF], 1999a, 2000a;[START_REF] Mollahassani-Pour | Application of a novel cost reduction index to preventive maintenance scheduling[END_REF][START_REF] Mytakidis | Maintenance scheduling by using the bi-criterion algorithm of preferential antipheromones[END_REF][START_REF] Rozenknop | Solving the electricity production planning problem by a column generation based heuristic[END_REF][START_REF] Saraiva | A simulated annealing based approach to solve the generator maintenance scheduling problem[END_REF][START_REF] Satoh | Maintenance scheduling by using simulated annealing method[END_REF][START_REF] Silva | Transmission constrained maintenance scheduling of generating units: A stochastic programming approach[END_REF][START_REF] Silva | Generation maintenance scheduling considering transmission constraints[END_REF][START_REF] Yellen | A decomposition approach to unit maintenance scheduling[END_REF].

The other common objective is to minimize the general operational costs. These are production costs (e.g., fuel consumption), maintenance costs (e.g., loss of profit), and sometimes unit start-up costs [START_REF] Canto | Application of Benders' decomposition to power plant preventive maintenance scheduling[END_REF]. The production costs depend on the generators' power output, so it is necessary to schedule their production level. An economic dispatch problem is usually solved with an objective of satisfying the demand at a minimum cost. The units with the lowest marginal costs are used to meet the system requirements; the other units produce only during the peak periods. Some studies address UC jointly to the GMS problem or/and take into account the transmission network (see Section 2.2). Other authors deal with a more specific problem including an accurate fuel management [START_REF] Anghinolfi | A matheuristic algorithm for a large-scale energy management problem[END_REF][START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF][START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF][START_REF] Fourcade | Optimizing nuclear power plant refueling with mixed-integer programming[END_REF][START_REF] Gardi | Local search for mixed-integer nonlinear optimization: A methodology and an application[END_REF][START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF][START_REF] Gorge | Stochastic nuclear outages semidefinite relaxations[END_REF][START_REF] Khemmoudj | When constraint programming and local search solve the scheduling problem of Electricit de France nuclear power plant outages[END_REF][START_REF] Jost | A 01 integer linear programming approach to schedule outages of nuclear power plants[END_REF][START_REF] Lusby | A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system[END_REF][START_REF] Rozenknop | Solving the electricity production planning problem by a column generation based heuristic[END_REF]. We discuss this issue in more detail in Section 2.4.

• Reliability-and-cost-based

References : [START_REF] Huang | Generator maintenance scheduling: A fuzzy system approach with genetic enhancement[END_REF][START_REF] Kralj | A multiobjective optimization approach to thermal generating units maintenance scheduling[END_REF][START_REF] Muñoz-Moro | Goal programming approach to maintenance scheduling of generating units in large scale power systems[END_REF].

The literature reports on some studies dealing with objective functions that are based on both reliability and cost. For instance, [START_REF] Huang | Generator maintenance scheduling: A fuzzy system approach with genetic enhancement[END_REF]; [START_REF] Kralj | A multiobjective optimization approach to thermal generating units maintenance scheduling[END_REF]; Muñoz-Moro and Ramos (1999) proposed multiobjective approaches to try to find the best trade-off between these two conflicting objectives.

Among these examples, Muñoz-Moro and Ramos (1999) proposed a two-stage approach that first considers the operational costs and then the reserve margins.

Deregulated power systems

Deregulation changes the maintenance scheduling problem. The GENCOs and TRANSCOs are now usually responsible for maintaining their equipment. The ISO ensures the smooth running of the system in terms of reliability and security. Risk is managed by guaranteeing sufficient reserves of energy for each period to meet uncertainties in, for example, the demand or the generator deterioration. The different actors may have conflicting interests: GENCOs and TRANSCOs want to maximize their profits, whereas the ISO is concerned with demand satisfaction and congestion avoidance. For example, GENCOs tend to perform maintenance when the energy price is at its lowest, which may make it difficult to meet the demand. Thus, in an iterative way (see Figure . 3), the GENCOs and TRANSCOs submit their preferred maintenance schedules to the ISO, which verifies the acceptable behavior of the system on the basis of all the market-player information. If the ISO is not satisfied, it will request modifications (e.g., the rescheduling of one or more maintenance tasks). The coordination procedure may vary from one system to another.

ISO GENCOs TRANSCOs

Required modifications Since the opening up to competition, the GMS problem in deregulated power systems has been widely studied [START_REF] Badri | Preventive generation maintenance scheduling considering system reliability and energy purchase in restructured power systems[END_REF][START_REF] Barot | Security coordinated maintenance scheduling in deregulation based on genco contribution to unserved energy[END_REF][START_REF] Billinton | Short-term generating unit maintenance scheduling in a deregulated power system using a probabilistic approach[END_REF][START_REF] Bisanovic | A profit-based maintenance scheduling of thermal power units in electricity market[END_REF]Chattopadhyay, 2004b,a;[START_REF] Conejo | Generation maintenance scheduling in restructured power systems[END_REF][START_REF] Elyas | A novel method for maintenance scheduling of generating units considering the demand side[END_REF][START_REF] Eshraghnia | A new approach for maintenance scheduling of generating units in power market[END_REF][START_REF] Feng | Optimal maintenance scheduling of power producers considering unexpected unit failure[END_REF][START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF][START_REF] Han | ISO coordination of generator maintenance scheduling in competitive electricity markets using simulated annealing[END_REF][START_REF] Kim | Generating unit maintenance scheduling under competitive market environments[END_REF]Marwali andShahidehpour, 1999a,b, 2000b,a;[START_REF] Min | Game-theory-based generation maintenance scheduling in electricity markets[END_REF][START_REF] Wu | GENCO's risk-based maintenance outage scheduling[END_REF][START_REF] Zhan | Generation Maintenance Scheduling Based on Multiple Objectives and Their Relationship Analysis[END_REF]. The problem has globally the same constraints as for the vertically integrated case. [START_REF] Bisanovic | A profit-based maintenance scheduling of thermal power units in electricity market[END_REF] introduced bilateral contracts (defined through prices and power quantities to supply) in their model. [START_REF] Chattopadhyay | Life-cycle maintenance management of generating units in a competitive environment[END_REF]; Marwali andShahidehpour (1999b, 2000a) considered fuel based constraints. Indeed, restrictions on the maximum fuel supplied each week, month, and year may arise due to contractual agreements with fuel suppliers. As noted for regulated power system, the output levels of the generating units may be computed for each period. It allows for the more accurate estimation of the revenues and operating costs and can be required, among others, by the introduction of network constraints. In some studies [START_REF] Badri | Preventive generation maintenance scheduling considering system reliability and energy purchase in restructured power systems[END_REF][START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF]Marwali andShahidehpour, 1999a,b, 2000b,a), cost minimization is still the objective. These studies generally deal with a security-constrained GMS problem operating under a pool-based model or a regulated monopoly. However, profit maximization is a much more common objective [START_REF] Barot | Security coordinated maintenance scheduling in deregulation based on genco contribution to unserved energy[END_REF][START_REF] Bisanovic | A profit-based maintenance scheduling of thermal power units in electricity market[END_REF]Chattopadhyay, 2004b,a;[START_REF] Conejo | Generation maintenance scheduling in restructured power systems[END_REF][START_REF] Elyas | A novel method for maintenance scheduling of generating units considering the demand side[END_REF][START_REF] Eshraghnia | A new approach for maintenance scheduling of generating units in power market[END_REF][START_REF] Feng | Optimal maintenance scheduling of power producers considering unexpected unit failure[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF][START_REF] Kim | Generating unit maintenance scheduling under competitive market environments[END_REF][START_REF] Min | Game-theory-based generation maintenance scheduling in electricity markets[END_REF][START_REF] Wu | GENCO's risk-based maintenance outage scheduling[END_REF][START_REF] Zhan | Generation Maintenance Scheduling Based on Multiple Objectives and Their Relationship Analysis[END_REF].

Reliability and security checks

Maintenance planning submission

Required modifications

Note that these two objectives are different because profits depend on both costs and revenues. A very recent study by [START_REF] Dahal | Modelling generator maintenance scheduling costs in deregulated power markets[END_REF] explains how maintenance costs can be precisely modeled in deregulated markets considering, among others: failures, contractual compensations, rescheduling, and market opportunities. The impermissible maximum power for the maintenance of the generating units in each period. In the same way, [START_REF] Barot | Security coordinated maintenance scheduling in deregulation based on genco contribution to unserved energy[END_REF] coordinated the decisions through corrective signals sent by the ISO to the GENCOs, indicating the maximal capacities that can be in maintenance during critical periods. These signals are calculated according to the responsibility of each GENCO for not supplying the load. These capacity-based signals can be replaced by penalties and/or incentive signals. [START_REF] Conejo | Generation maintenance scheduling in restructured power systems[END_REF]; Geetha and Shanti Swarup (2009); [START_REF] Min | Game-theory-based generation maintenance scheduling in electricity markets[END_REF] penalized periods of maintenance during peak periods or when the reliability of the system is uncertain. The objective function associated with the GENCO problem is modified at each iteration to represent the ISO's recommendations. In [START_REF] Conejo | Generation maintenance scheduling in restructured power systems[END_REF], GENCOs that adjust their maintenance plans are paid to offset their losses compared to their initial plans; the cost is paid by the customers. [START_REF] Han | ISO coordination of generator maintenance scheduling in competitive electricity markets using simulated annealing[END_REF] proposed an ISO coordination procedure to adjust the individual generator-maintenance schedules according to the preferences of each GENCO, while guaranteeing system reliability. [START_REF] Elyas | A novel method for maintenance scheduling of generating units considering the demand side[END_REF] gave the ISO the responsibility for maintenance scheduling. They took into account consumer satisfaction by maximizing the annual social welfare and also consider the profit-seeking GENCOs. They suggested a maintenance bidding approach to model the coordination mechanism. Finally, [START_REF] Zhan | Generation Maintenance Scheduling Based on Multiple Objectives and Their Relationship Analysis[END_REF] analyzed the relationship between the ISO and the GENCOs in GMS using a multiobjective approach.

Transmission maintenance scheduling and network considerations

Along with generator maintenance, transmission-line maintenance must be scheduled. This problem, usually called the transmission maintenance scheduling (TMS) problem, has received less attention than the GMS problem [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF][START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF][START_REF] Langdon | Scheduling maintenance of electrical power transmission networks using genetic programming[END_REF][START_REF] Lv | Short-term transmission maintenance scheduling based on the Benders decomposition[END_REF][START_REF] Marwali | A deterministic approach to generation and transmission maintenance scheduling with network constraints[END_REF][START_REF] Marwali | Long-term transmission and generation maintenance scheduling with network, fuel and emission constraints[END_REF][START_REF] Marwali | Short-term transmission line maintenance scheduling in a deregulated system[END_REF]. It is necessary to ensure that taking a line out for maintenance does not impact the network reliability and security. The TMS constraints are globally the same as those for GMS (e.g., time windows for maintenance tasks, resource requirements, demand satisfaction).

These are also constraints on the line capacity and may include voltage considerations. The network can be modeled as either a transportation model [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF]Marwali andShahidehpour, 1998, 1999b) or a more complex but more realistic DC power flow model 3 [START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF][START_REF] Langdon | Scheduling maintenance of electrical power transmission networks using genetic programming[END_REF][START_REF] Lv | Short-term transmission maintenance scheduling based on the Benders decomposition[END_REF][START_REF] Marwali | Short-term transmission line maintenance scheduling in a deregulated system[END_REF].

The TMS problem can be addressed independently from the GMS problem [START_REF] Langdon | Scheduling maintenance of electrical power transmission networks using genetic programming[END_REF][START_REF] Lv | Short-term transmission maintenance scheduling based on the Benders decomposition[END_REF][START_REF] Marwali | Short-term transmission line maintenance scheduling in a deregulated system[END_REF]. In this approach, the state of the network appears as a constraint during the resolution of the GMS problem. Although this approach is especially valid for regulated systems it may also apply to deregulated power systems. For instance, [START_REF] Marwali | Short-term transmission line maintenance scheduling in a deregulated system[END_REF] looked for a trade-off between maintenance costs and loss of revenue over a short-term horizon.

When the TMS problem is tackled jointly with the GMS problem [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF][START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF]Marwali andShahidehpour, 1998, 1999b), it becomes more complex. The maintenance must take into account economic considerations while minimizing the unsatisfied demand. The GMS and TMS problems have been solved generally on a monthly or weekly basis. [START_REF] Marwali | A deterministic approach to generation and transmission maintenance scheduling with network constraints[END_REF] coordinated maintenance decisions over a long-term horizon. [START_REF] Marwali | Long-term transmission and generation maintenance scheduling with network, fuel and emission constraints[END_REF] included fuel and emission constraints in the problem, considering local transmission lines within a GENCO. [START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF] proposed an optimal coordination approach between generation and transmission outages, midterm maintenance outage and hourly security constrained generation scheduling. The technique can be used by a company in a monopoly position or by an ISO. [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF] solved the integrated maintenance scheduling problem on hourly basis and consider partial maintenance for transmission lines. When deregulated power systems are based on the power exchange model, the GENCOs and TRANSCOs are profit-oriented and do not have global information about the state of the system. As explained earlier, the ISO has to coordinate 3 Linearization of an AC power flow model the submitted schedules; the cheapest transmission lines and generators might be overloaded. To our knowledge, only one study applies to this case ; [START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF] solved the problem for every actor (ISO, GENCOs, and TRANSCOs) and coordinate the decisions through penalties.

Network considerations and especially coordination between GMS and TMS problems are important to maintain the security as well as the efficiency of the global system. If the TMS problem is not solved jointly with the GMS problem, network constraints can be introduced when the latter problem is solved [START_REF] Badri | Preventive generation maintenance scheduling considering system reliability and energy purchase in restructured power systems[END_REF][START_REF] Barot | Security coordinated maintenance scheduling in deregulation based on genco contribution to unserved energy[END_REF][START_REF] Chattopadhyay | A practical maintenance scheduling program mathematical model and case study[END_REF][START_REF] Chen | Optimal generating unit maintenance scheduling for multi-area system with network constraints[END_REF][START_REF] El-Sharkh | A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling[END_REF][START_REF] Leou | A new method for unit maintenance scheduling considering reliability and operation expense[END_REF]Marwali andShahidehpour, 2000a, 1999a;[START_REF] Silva | Transmission constrained maintenance scheduling of generating units: A stochastic programming approach[END_REF][START_REF] Silva | Generation maintenance scheduling considering transmission constraints[END_REF][START_REF] Wu | GENCO's risk-based maintenance outage scheduling[END_REF]. These constraints can implicitly include the maintenance tasks planned for the network. Maintenance and unit commitment decisions must never exceed the line capacities. To our knowledge, [START_REF] Chen | Optimal generating unit maintenance scheduling for multi-area system with network constraints[END_REF] were the first to consider these constraints in a multi-area problem, but they did not handle unexpected breakdowns. The transportation model is widely used, except by [START_REF] Silva | Generation maintenance scheduling considering transmission constraints[END_REF] who modeled a DC power flow.

Management of uncertainty

Uncertainty in GMS and TMS can be significant and it requires specific management. Indeed, the load curve and the fuel and energy prices may be difficult to estimate precisely. Furthermore, corrective maintenance, characterized by unexpected breakdowns, has a real impact on the production. Uncertainty has therefore to be handled with care.

Reserve constraints can help to deal with these risks. Reliability objectives, as discussed earlier, can also be used. However, using only deterministic strategies may be inappropriate in the case of large disturbances.

To explicitly consider unexpected breakdowns, researchers associate a forced outage rate (FOR) with generating units or transmission lines. The FOR represents the probability that equipment will not be available for service when required. It impacts the quantity of energy that can be supplied. Thus, it prevents unsuitable maintenance schedules when load constraints are considered. One can also (artificially) reduce the capacity of the generators and solve a deterministic problem, but the not-supplied energy can be overestimated. However, the use of stochastic reliability indices such as the expected energy not served or EENS [START_REF] Baskar | Genetic algorithms solution to generator maintenance scheduling with modified genetic operators[END_REF][START_REF] Chattopadhyay | Life-cycle maintenance management of generating units in a competitive environment[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF][START_REF] Lv | Short-term transmission maintenance scheduling based on the Benders decomposition[END_REF]Marwali and Shahidehpour, 1999a,b;[START_REF] Silva | Transmission constrained maintenance scheduling of generating units: A stochastic programming approach[END_REF][START_REF] Yellen | A decomposition approach to unit maintenance scheduling[END_REF] and the loss of load probability (LOLP) [START_REF] Billinton | Short-term generating unit maintenance scheduling in a deregulated power system using a probabilistic approach[END_REF][START_REF] Han | ISO coordination of generator maintenance scheduling in competitive electricity markets using simulated annealing[END_REF][START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF][START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF][START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF][START_REF] Volkanovski | Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system[END_REF] is better suited to tackle problems in this scenario. The EENS is minimized or a threshold for acceptability is defined. Satisfying EENS within a specific threshold results in an acceptable LOLP. Authors such as [START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF]; [START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF]; [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF] considered the LOLP reliability index in a stochastic levelized risk method. The first step of this technique is to build an outage capacity probability table (COPT) by associating a probability to every generation capacity level taking into account the forced outage rate of each generating unit. The table can be computed with a convolution algorithm. The second step consists in finding the system's risk characteristic coefficient, defined as the change of the generating units outage capacity in MW when the systems risk changes by a certain factor.

Each unit capacity is then replaced by an effective load carrying capacity based on the system's risk characteristic coefficient and the FOR. It represents the actual capacity of the units which is used for meeting the load demand. Similarly, the load of each interval is replaced by a value, called the equivalent load, which takes into account the peak variation during this interval and the system's risk characteristic coefficient. Leveling the risk may be finally realized by minimizing the sum of the squares of the reserves in the planning period.

The reserve in each interval is obtained by subtracting the effective load carrying capacity of the available generating units and the equivalent load. In [START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF], the risk associated with the resulting plan is evaluated by giving a confidence interval for the LOLP. Suresh and Kumarappan (2013) presented a coordination scheme to minimize the LOLP and the deviations of annual supply reserve ratio by considering the system's risk characteristic coefficient as a control parameter of their approach. We refer the reader to the previous given references for more details. As a fairly closed approach, [START_REF] Volkanovski | Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system[END_REF] chose to minimize directly the annual value of the LOLE. [START_REF] Billinton | Short-term generating unit maintenance scheduling in a deregulated power system using a probabilistic approach[END_REF] discussed a health levelization technique over a short-term horizon. Incorporated in a probabilistic framework, the objective is to maximize the health/security of the system, defined as the probability that the available reserves are greater than the required reserves. [START_REF] Chattopadhyay | Life-cycle maintenance management of generating units in a competitive environment[END_REF] simulated random outages using the Monte Carlo technique and proposes a stochastic optimization framework based on a game theory model to tackle the GMS problem.

In [START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF]; [START_REF] Lv | Short-term transmission maintenance scheduling based on the Benders decomposition[END_REF]; Marwali andShahidehpour (1999a,b, 2000a); [START_REF] Silva | Transmission constrained maintenance scheduling of generating units: A stochastic programming approach[END_REF], a probabilistic approach takes FOR into account in a problem with network constraints.

Last but not least, [START_REF] Feng | Optimal maintenance scheduling of power producers considering unexpected unit failure[END_REF] analyzed the impact of unexpected unit failures on the GMS solution and especially on: maintenance periods, producer benefits, maintenance costs and the costs of repairing or replacing some generating units. A modified superposed power law process models the unit failure rate. Its parameters are determined via the Gauss-Newton algorithm.

Fuzzy logic theory is also used to handle some data uncertainties in [START_REF] Huang | Generator maintenance scheduling: A fuzzy system approach with genetic enhancement[END_REF][START_REF] Dahal | Generator maintenance scheduling using a genetic algorithm with a fuzzy evaluation function[END_REF][START_REF] El-Sharkh | A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling[END_REF][START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF]. It allows the representation and the use of linguistic knowledge.

This concept is mainly used through fuzzy fitness function in a genetic algorithm (GA). [START_REF] Huang | Generator maintenance scheduling: A fuzzy system approach with genetic enhancement[END_REF] used triangular and trapezoidal membership functions for the multiple objectives (reserve margin, production cost) and for the soft constraints (manpower, time windows, geographical constraint). They used a GA to tune the membership functions. [START_REF] Dahal | Generator maintenance scheduling using a genetic algorithm with a fuzzy evaluation function[END_REF] introduced a fuzzy evaluation function combining the reliability objective function, the manpower constraint and a penalty factor associated with the inflexible demand satisfaction constraint. Based on their experience, they used triangular and trapezoidal membership functions to define the fuzzy sets. [START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF] incorporated uncertainty associated to the data and the forced outage rate by evaluating the reliability of power plants using fuzzy theory. They used a fuzzy LOLP to assess the quality of the method. The drawback of the fuzzy logic approach is the intensive need of expert knowledge and substantial amounts of data.

The power demand may also be uncertain. Its stochastic nature can be explicitly considered. A set of scenarios that model alternative demands is used in [START_REF] Anghinolfi | A matheuristic algorithm for a large-scale energy management problem[END_REF][START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF][START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF][START_REF] Canto | Application of Benders' decomposition to power plant preventive maintenance scheduling[END_REF][START_REF] Canto | A model for the preventive maintenance scheduling of power plants including wind farms[END_REF][START_REF] Gardi | Local search for mixed-integer nonlinear optimization: A methodology and an application[END_REF][START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF][START_REF] Gorge | Stochastic nuclear outages semidefinite relaxations[END_REF][START_REF] Jost | A 01 integer linear programming approach to schedule outages of nuclear power plants[END_REF][START_REF] Lusby | A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system[END_REF][START_REF] Rozenknop | Solving the electricity production planning problem by a column generation based heuristic[END_REF]. The maintenance decisions ensure that the demand is met in all the scenarios. [START_REF] El-Sharkh | A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling[END_REF] simulated the demand and the cost uncertainties using triangular membership functions defining fuzzy sets. [START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF] presented an effective method, called model predictive control, that detects demand disturbances and makes appropriate corrections.

When the objective is profit-based, it may be necessary to take into account the volatility of market prices. [START_REF] Wu | GENCO's risk-based maintenance outage scheduling[END_REF] used a stochastic model based on an hourly price-based unit commitment. The hourly electricity and fuel prices are modeled as a set of scenarios determined via a Monte-Carlo method.

Fuel management and maintenance scheduling

Thermal production represents around 80% of the total global electricity production4 . Fuel is fundamental for the effective functioning of these plants, therefore it can be subject to a specific management. In [START_REF] Badri | Preventive generation maintenance scheduling considering system reliability and energy purchase in restructured power systems[END_REF], the fuel consumption is limited by period for every generating unit. If fuel shortages occur, energy can be purchased externally. In [START_REF] Wu | GENCO's risk-based maintenance outage scheduling[END_REF], the fuel allocations, which depend on a predetermined contract with a supplier, are limited by group of units.

A bigger challenge comes with refueling considerations. Few studies handle this problem. In some cases, refueling can be done continuously without significantly affecting production [START_REF] Al-Khamis | Unit maintenance scheduling with fuel constraints[END_REF][START_REF] Chattopadhyay | A practical maintenance scheduling program mathematical model and case study[END_REF][START_REF] Chattopadhyay | Life-cycle maintenance management of generating units in a competitive environment[END_REF]Marwali andShahidehpour, 1999b, 2000a;[START_REF] Muñoz-Moro | Goal programming approach to maintenance scheduling of generating units in large scale power systems[END_REF], but sometimes (e.g., for nuclear reactors) it can occur only when the generators are offline [START_REF] Anghinolfi | A matheuristic algorithm for a large-scale energy management problem[END_REF][START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF][START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF][START_REF] Fourcade | Optimizing nuclear power plant refueling with mixed-integer programming[END_REF][START_REF] Gardi | Local search for mixed-integer nonlinear optimization: A methodology and an application[END_REF][START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF][START_REF] Gorge | Stochastic nuclear outages semidefinite relaxations[END_REF][START_REF] Khemmoudj | When constraint programming and local search solve the scheduling problem of Electricit de France nuclear power plant outages[END_REF][START_REF] Jost | A 01 integer linear programming approach to schedule outages of nuclear power plants[END_REF][START_REF] Lusby | A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system[END_REF][START_REF] Rozenknop | Solving the electricity production planning problem by a column generation based heuristic[END_REF]. The introduction of fuel management into the GMS problem increases its complexity, but also makes it more realistic. [START_REF] Al-Khamis | Unit maintenance scheduling with fuel constraints[END_REF] focused on the case of fuel constraints for each generating unit. Muñoz-Moro and Ramos (1999) are concerned with the maximum fuel storage capacities of thermal plants. Marwali andShahidehpour (1999b, 2000a) solved a fuel dispatch problem with multiple suppliers. The fuel consumption is limited by week, month, and year and is linked with the output level of the generators.

The coordination of long-term and short-term decisions is discussed in (Marwali and Shahidehpour, 2000a). [START_REF] Chattopadhyay | A practical maintenance scheduling program mathematical model and case study[END_REF] discussed coal supply management with different transport modes from the mine to the power stations. In [START_REF] Chattopadhyay | Life-cycle maintenance management of generating units in a competitive environment[END_REF], the author introduced fuel contracts with suppliers (with fixed fuel prices and volumes) in a GMS model and describes a successive linearization scheme to approximate the fuel consumption as a linear function. Two other studies [START_REF] Fourcade | Optimizing nuclear power plant refueling with mixed-integer programming[END_REF][START_REF] Khemmoudj | When constraint programming and local search solve the scheduling problem of Electricit de France nuclear power plant outages[END_REF] are concerned with planning shutdowns in production to carry out refueling and maintenance operations; the fuel quantity to supply is known in advance.

A challenge submitted jointly by EURO 5 and ROADEF 6 in collaboration with EDF 7 has renewed interest in this latter problem. It presents a large-scale energy management problem with many constraints [START_REF] Porcheron | Challenge ROADEF/EURO 2010: A large-scale energy management problem with varied constraints[END_REF]. The time horizon is long and fine-grained (up to 277 weeks with 7 or 21 timesteps per week). Two types of production units are considered. Non-nuclear plants can refuel continuously whereas nuclear plants must be shut down when fuel is supplied. In contrast to previous studies [START_REF] Fourcade | Optimizing nuclear power plant refueling with mixed-integer programming[END_REF][START_REF] Khemmoudj | When constraint programming and local search solve the scheduling problem of Electricit de France nuclear power plant outages[END_REF], the amount of fuel that is supplied for every nuclear plant is left as a decision variable. Furthermore, the production levels for the plants have to be planned under demand uncertainty modeled by a set of scenarios (up to 500). The objective is to plan the production and refueling while minimizing the production costs of non-nuclear plants and the refueling costs of nuclear plants. The problem has been proved to be NP-hard [START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF]. Almost twenty teams participated in this challenge, and some of them [START_REF] Anghinolfi | A matheuristic algorithm for a large-scale energy management problem[END_REF][START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF][START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF][START_REF] Gardi | Local search for mixed-integer nonlinear optimization: A methodology and an application[END_REF][START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF][START_REF] Jost | A 01 integer linear programming approach to schedule outages of nuclear power plants[END_REF][START_REF] Lusby | A solution approach based on Benders decomposition for the preventive maintenance scheduling problem of a stochastic large-scale energy system[END_REF][START_REF] Rozenknop | Solving the electricity production planning problem by a column generation based heuristic[END_REF] published their results. This problem has also been considered by [START_REF] Gorge | Stochastic nuclear outages semidefinite relaxations[END_REF] with some simplifications. The problem is often decomposed into several components: planning of the refueling, computation of the refuel amounts, and planning of the production.

Benchmarks

Publicly accessible data to test optimization algorithms for maintenance scheduling in electricity systems are rather scarce. Probably the most classical instance is the IEEE 8 Reliability Test System (IEEE-RTS) published in 1979 (Reliability Test System Task Force of the Application of Probability Methods Subcommittee, 1979) and released in 1996 (Reliability Test System Task Force of the Application of Probability Methods Subcommittee, 1999). The IEEE-RTS includes data on the network, the generating units, the demand, and the costs. This benchmark has been used in several articles [START_REF] Badri | Preventive generation maintenance scheduling considering system reliability and energy purchase in restructured power systems[END_REF][START_REF] Billinton | Short-term generating unit maintenance scheduling in a deregulated power system using a probabilistic approach[END_REF][START_REF] Elyas | A novel method for maintenance scheduling of generating units considering the demand side[END_REF][START_REF] Eshraghnia | A new approach for maintenance scheduling of generating units in power market[END_REF][START_REF] Fattahi | A new approach for maintenance scheduling of generating units in electrical power systems based on their operational hours[END_REF][START_REF] Feng | Optimal maintenance scheduling of power producers considering unexpected unit failure[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF][START_REF] Marwali | A deterministic approach to generation and transmission maintenance scheduling with network constraints[END_REF], 1999a,b, 2000a;[START_REF] Mollahassani-Pour | Application of a novel cost reduction index to preventive maintenance scheduling[END_REF][START_REF] Schlünz | An investigation into the effectiveness of simulated annealing as a solution approach for the generator maintenance scheduling problem[END_REF] [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF][START_REF] Zhan | Generation Maintenance Scheduling Based on Multiple Objectives and Their Relationship Analysis[END_REF]. Another commonly used set of instances contributed by the IEEE9 represent portions of the North American electricity system; this set served as a benchmark in [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF][START_REF] El-Sharkh | A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling[END_REF]; El-Sharkh (2014); [START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF]; [START_REF] Marwali | Short-term transmission line maintenance scheduling in a deregulated system[END_REF]. Additionally, an instance with 21 generation units described by [START_REF] Yamayee | A computationally efficient optimal maintenance scheduling method[END_REF] regularly serves as a test case [START_REF] Baskar | Genetic algorithms solution to generator maintenance scheduling with modified genetic operators[END_REF][START_REF] Dahal | Generator maintenance scheduling using a genetic algorithm with a fuzzy evaluation function[END_REF][START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF][START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF][START_REF] Schlünz | An investigation into the effectiveness of simulated annealing as a solution approach for the generator maintenance scheduling problem[END_REF][START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF][START_REF] Yare | Optimal maintenance scheduling of generators using multiple swarms-MDPSO framework[END_REF]. Data associated with real cases are often used to validate proposed techniques. However, to our knowledge, the only publicly available data is that published of the EURO-ROADEF-EDF challenge [START_REF] Porcheron | Challenge ROADEF/EURO 2010: A large-scale energy management problem with varied constraints[END_REF].

Solution methods

Various heuristic and exact approaches have been proposed for the GMS and/or TMS problems. The solution techniques mainly focus on metaheuristics and mathematical programming. This section provides details about all these techniques and discusses their applicability to the problems defined in the previous sections. To provide a global overview, Table .2 classifies the references according to the solution method they apply.

Mathematical programming approaches

Mathematical programming methods are essentially based on dynamic programming, pure mixed-integer programming, branch-and-bound, Lagrangian relaxation, and Benders decomposition.

Until the 90s, dynamic programming has been often used for solving the GMS problem because of its sequential decision process. For instance, [START_REF] Huang | Generator maintenance scheduling: A fuzzy system approach with genetic enhancement[END_REF] combined dynamic programming with fuzzy logic in a multiobjective problem. However, the "curse of dimensionality" limits the application of this method [START_REF] Yamayee | Maintenance scheduling: Description, literature survey, and interface with overall operations scheduling[END_REF].

Alternatively, many mixed integer programming models are proposed for the GMS and TMS problems.

Objectives and constraints widely vary from one study to another; a clear indicator of the difficulty to point out a general model. Only linear models are likely to be handled by commercial solvers and only small or medium-size instances can be efficiently solved. [START_REF] Badri | Preventive generation maintenance scheduling considering system reliability and energy purchase in restructured power systems[END_REF]; [START_REF] Barot | Security coordinated maintenance scheduling in deregulation based on genco contribution to unserved energy[END_REF]; [START_REF] Bisanovic | A profit-based maintenance scheduling of thermal power units in electricity market[END_REF][START_REF] Canto | A model for the preventive maintenance scheduling of power plants including wind farms[END_REF]; [START_REF] Chen | Optimal generating unit maintenance scheduling for multi-area system with network constraints[END_REF]; [START_REF] Conejo | Generation maintenance scheduling in restructured power systems[END_REF]; [START_REF] Fourcade | Optimizing nuclear power plant refueling with mixed-integer programming[END_REF]; [START_REF] Kovács | A system for the detailed scheduling of wind farm maintenance[END_REF][START_REF] Mollahassani-Pour | Application of a novel cost reduction index to preventive maintenance scheduling[END_REF]; Muñoz-Moro and Ramos (1999); [START_REF] Wu | GENCO's risk-based maintenance outage scheduling[END_REF] formulated mixed integer linear programming (MILP) models and solved them combining branch-and-bound with the simplex method or interior-point methods. Among studies based on multiobjective optimization, Kralj andPetrovic (1995) designed a custom branch-and-bound, and[START_REF] Muñoz-Moro | Goal programming approach to maintenance scheduling of generating units in large scale power systems[END_REF][START_REF] Muñoz-Moro | Goal programming approach to maintenance scheduling of generating units in large scale power systems[END_REF] proposed a two-stage goal programming approach solved by branch-and-bound.

The direct use of mixed integer programming is sometimes unsuitable as the computational time grows prohibitively with problem size. To overcome this drawback, decomposition techniques can be applied. The most studied technique is the Benders decomposition which allow to decouple a large-scale problem into a master problem and one or several independent small-scale subproblems which are easier to solve. The master problem works only with a subset of the variables and the constraints. Infeasibility cuts are generated in the master problem to ensure that the subproblem constraints are satisfied. Optimality cuts are also introduced if some parts of the objective are related to the subproblems. The resulting algorithm iteratively solves the master problem and the subproblems until it converges or concludes that there is no solution. For more details, the reader is invited to consult the original paper [START_REF] Benders | Partitioning procedures for solving mixed-variables programming problems[END_REF]. This decomposition technique applies particularly well to the GMS problem because of the problem's intrinsic two-stage structure. The master problem is concerned only with the constraints regarding the scheduling of the maintenance tasks as well as the resources requirement if needed. Load and network constraints, as well as fuel management, are moved to the subproblems. Benders decomposition is applied for both regulated [START_REF] Al-Khamis | Unit maintenance scheduling with fuel constraints[END_REF][START_REF] Canto | Application of Benders' decomposition to power plant preventive maintenance scheduling[END_REF][START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF]Marwali andShahidehpour, 1998, 1999a;[START_REF] Silva | Transmission constrained maintenance scheduling of generating units: A stochastic programming approach[END_REF][START_REF] Silva | Generation maintenance scheduling considering transmission constraints[END_REF][START_REF] Yellen | A decomposition approach to unit maintenance scheduling[END_REF] and deregulated [START_REF] Fu | Security-constrained optimal coordination of generation and transmission maintenance outage scheduling[END_REF][START_REF] Geetha | Coordinated preventive maintenance scheduling of GENCO and TRANSCO in restructured power systems[END_REF]Marwali andShahidehpour, 1999b, 2000a) power systems. [START_REF] Lv | Short-term transmission maintenance scheduling based on the Benders decomposition[END_REF] introduced modified Benders feasibility cuts. They also define an index of critical lines-related to the system reliability-to reduce the computational complexity and the solution time. Marwali and Shahidehpour (2000a) coordinated long-term and short-term generation decisions with a dynamic scheduling algorithm. They used a Benders decomposition to define the maintenance decisions and an augmented Lagrangian relaxation to solve the underlying unit commitment. The same authors proposed also a deterministic [START_REF] Marwali | A deterministic approach to generation and transmission maintenance scheduling with network constraints[END_REF] and a probabilistic (Marwali and Shahidehpour, 1999a) Benders based approach to jointly schedule the generation and transmission maintenance.

Mathematical programming may be coupled with heuristic approaches. For instance, linear programming has been used in combination with local search [START_REF] Gardi | Local search for mixed-integer nonlinear optimization: A methodology and an application[END_REF], genetic algorithms [START_REF] Feng | Optimal maintenance scheduling of power producers considering unexpected unit failure[END_REF], and custom heuristics [START_REF] Jost | A 01 integer linear programming approach to schedule outages of nuclear power plants[END_REF]. Examples of the latter include the work of [START_REF] Rozenknop | Solving the electricity production planning problem by a column generation based heuristic[END_REF] who combined column generation with customized heuristics, and that of [START_REF] Gorge | Stochastic nuclear outages semidefinite relaxations[END_REF] who applied a technique based on semidefinite programming, followed by a randomized rounding procedure.

It is worth noting that mathematical programming does not seem to be better adapted to one or other of problems discussed in the previous sections. However, all the methods presented in this part are mainly suitable when only linear objectives and constraints are considered in the GMS and/or TMS problems. Therefore, they cannot be used with the main reliability criterion (i.e. the minimization of the square of the reserves) or with quadratic cost functions.

Heuristics and metaheuristics

Due to the NP-hardness of the maintenance problems faced in power systems and the size of real-word instances, heuristic techniques have been largely developed. These methods also allow for more flexibility to deal with non linear or very complex constraints and/or objectives.

Genetic algorithms

GAs are widely used for solving the GMS problem due, mainly, to the structure of the decisions that are made. These algorithms are inspired by natural evolution. A population of abstract representations of solutions, called individuals, evolves through an iterative process toward better solutions. Solutions are usually encoded to facilitate the application of the several techniques used in GAs such as selection, mutation and crossover.

In the particular case of GSM, different coding methods have been considered. In the binary representation, an individual has G x T genes (where G is the number of generating units and T is the number of periods) and the value of gene (g,t) is set to 1 if unit g starts maintenance at time t, and 0 otherwise. In the integer representation, a individual has G genes and the value of gene g corresponds to the maintenance starting period of generating unit g. Binary or gray encoding of the latter values may also be used. [START_REF] Dahal | Generator maintenance scheduling of electric power systems using genetic algorithms with integer representations[END_REF] showed that integer-coding is the most efficient encoding strategy since it generates the smallest search space. Moreover, the integer coding reduces the probability of infeasibility during the process, and it avoids the overhead necessary to code and decode a solution. [START_REF] Baskar | Genetic algorithms solution to generator maintenance scheduling with modified genetic operators[END_REF] revisited the binary for integer representation, the real encoding (using real values instead of integer values), and the classic integer encoding.

For their experiments, integer coding proves to be the best independently of the problem size. [START_REF] Wang | A new genetic algorithm for preventive unit maintenance scheduling of power systems[END_REF] and [START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF] reached the same conclusions about the efficiency of integercoding. Despite these findings, binary coding is still used in some approaches [START_REF] Eshraghnia | A new approach for maintenance scheduling of generating units in power market[END_REF][START_REF] Leou | A new method for unit maintenance scheduling considering reliability and operation expense[END_REF][START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF][START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF]. Finally, [START_REF] Volkanovski | Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system[END_REF] introduced a completely different encoding strategy that uses real values between 0 and 1 obtained by dividing the maintenance starting time value by the number of periods of the horizon.

Needless to say, contrary to solution encoding, the fitness function associated to every individual is problem dependent. Usually, it includes penalties associated with the violations of some constraints. Fuzzy models may be developed to handle data uncertainties (see Section 2.3) through the use of fuzzy evaluation functions.

Being population-based metaheuristics, GAs require initial solutions to built the initial population. The literature reports on randomly generated solutions [START_REF] Dahal | Generator maintenance scheduling using a genetic algorithm with a fuzzy evaluation function[END_REF][START_REF] Huang | Generator maintenance scheduling: A fuzzy system approach with genetic enhancement[END_REF][START_REF] Leou | A new method for unit maintenance scheduling considering reliability and operation expense[END_REF][START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF][START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF] and heuristically generated solutions [START_REF] Burke | Hybrid evolutionary techniques for the maintenance scheduling problem[END_REF][START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF][START_REF] Volkanovski | Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system[END_REF]. The latter are often built using constructive methods that are mainly based on the following process: ranking generating units in order of decreasing capacity, and iteratively schedule the maintenance of the units when the demand is at its lowest level, while satisfying the constraints of the model. GAs try to improve solutions by applying the classic operators such as tournament or roulette wheel selection, one or two-point crossovers, random mutation and a replacement policy based usually on elitism. To reduce the probability of trapping in local optima, [START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF]; [START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF] accepted non-improving solutions at each iteration according to the probabilistic acceptance criterion of simulated annealing. [START_REF] Langdon | Scheduling maintenance of electrical power transmission networks using genetic programming[END_REF] designed a GA to schedule the maintenance of the transmission lines combined with some greedy heuristics. Some GAs include a local search algorithm for improving the quality of the solutions. Neighborhoods are usually defined by changing the maintenance starting time of a randomly selected generator. [START_REF] Burke | Hybrid evolutionary techniques for the maintenance scheduling problem[END_REF] tested the combination of a genetic algorithm with several local search methods : a basic hill climbing technique, a simulated annealing, and a tabu search.

Hybridization with tabu search proved to be the most efficient approach in their experiments. [START_REF] El-Sharkh | A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling[END_REF] maintained feasibility using a hill climbing technique during the solution process. [START_REF] Leou | A new method for unit maintenance scheduling considering reliability and operation expense[END_REF] executed simulated annealing for each individual solution of the population. [START_REF] Reihani | Reliability based generator maintenance scheduling using hybrid evolutionary approach[END_REF] designed an hybrid algorithm based on extremal optimization and a GA.

GAs are mostly used in regulated power systems; they allow dealing with the non linearity of the main reliability-based objectives.

Particle swarm optimisation

Recent studies [START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF][START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF][START_REF] Yare | Optimal maintenance scheduling of generators using multiple swarms-MDPSO framework[END_REF] applied the particle swarm optimisation (PSO) to the GMS problem. PSO is another population-based metaheuristic which bares many similarities with GAs. It simulates the social behaviour of birds within a flock, or even fishes within a school evolving by information exchange. The population is composed of particles moving in the search space of the optimization problem. The position of a particle represents a candidate solution.

Each particle is guided according to the best solution (fitness) it has achieved so far, and according to the current best particle, or particles if multiple swarms are considered. On its original version, PSO handles only continuous variables; however, [START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF] introduced a penalty function to deal with the discrete nature of the variables involved in the GMS problem. [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF] chose to test a particular binary version of PSO. They defined the particle according to the capacity outage probability table. To improve the effectiveness of the algorithm, they applied crossover and mutation operators to some particles during the iterative process. [START_REF] Yare | Optimal maintenance scheduling of generators using multiple swarms-MDPSO framework[END_REF] considered only the latter operator in a multiple swarms-modified discrete PSO where information sharing is enhanced by using multiple populations.

Non linear reliability-based objectives [START_REF] Suresh | Hybrid improved binary particle swarm optimization approach for generation maintenance scheduling problem[END_REF][START_REF] Yare | Optimal maintenance scheduling of generators using multiple swarms-MDPSO framework[END_REF] and cost-based objectives [START_REF] Ekpenyong | An improved robust model for generator maintenance scheduling[END_REF] of regulated power systems have been tackled using PSO.

Other populations-based methods

Apart from GAs and PSO, other population-based techniques have been explored in the literature. The first three studies deal with regulated power systems and are concerned with the maintenance and generation costs. [START_REF] Digalakis | A multipopulation cultural algorithm for the electrical generator scheduling problem[END_REF] applied a parallel co-operating cultural algorithm to solve the GMS problem. Contrary to GAs, this method considers multiple populations, but it uses similar operators (selection, crossover, mutation). The exchange of individuals between the populations allows to guide the algorithm towards the promising areas of the search space. The authors used local search to improve the quality of the solutions after the initialization and the application of the genetic operators.

El-Sharkh (2014) tested a clonal selection algorithm (CSA) to schedule the maintenance in a regulated power system. CSA imitates the mechanisms of the adaptive immune system. From a population of individuals randomly generated, all infeasible individuals are repaired using a hill climbing algorithm. The best individuals are then cloned and a mutation operator is applied to these copies. The hill climbing technique is used again to repair them. According to the outages planning, an economic load dispatch problem is solved within the process. This algorithm is also used for deregulated power systems in [START_REF] Elyas | A novel method for maintenance scheduling of generating units considering the demand side[END_REF].

Recently, [START_REF] Abirami | Source and transmission line maintenance outage scheduling in a power system using teaching learning based optimization algorithm[END_REF] presented a teaching-learning based optimization algorithm (TLBO) for solving the GMS problem. This technique is inspired by the transfer of knowledge between teacher and students in the classroom. TLBO is a population-based iterative learning algorithm. However, it does not use genetic operations like selection, crossover, and mutation but tries to improve individuals based on their interaction with the teacher and the communication with the other individuals. The population is initialized by randomly setting the starting period for the maintenance of the generators and the transmission lines. The best individual is deemed as the teacher. Other individuals are modified to move towards the teacher.

Finally, [START_REF] Zhan | Generation Maintenance Scheduling Based on Multiple Objectives and Their Relationship Analysis[END_REF] proposed an approach based on a novel evolutionary algorithm called group search optimizer inspired by animal searching behaviour. They computed a set of Pareto-optimal solutions for a multiobjective problem based on reliabiliy, costs, and profits.

Simulated annealing

Simulated annealing (SA) is a stochastic metaheuristic inspired by the annealing process in metallurgy. At each iteration, SA randomly generates a neighbor of the current solution. The algorithm accepts non-improving solutions according to some probability, allowing the search to escape local optima. These probability decrease as the number of iterations increases. SA has been used as the main approach to solve the GMS problem in [START_REF] Burke | Hybrid evolutionary techniques for the maintenance scheduling problem[END_REF][START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF][START_REF] Fattahi | A new approach for maintenance scheduling of generating units in electrical power systems based on their operational hours[END_REF][START_REF] Han | ISO coordination of generator maintenance scheduling in competitive electricity markets using simulated annealing[END_REF][START_REF] Saraiva | A simulated annealing based approach to solve the generator maintenance scheduling problem[END_REF]. [START_REF] Fattahi | A new approach for maintenance scheduling of generating units in electrical power systems based on their operational hours[END_REF] used in addition custom heuristics to solve the underlying UC. [START_REF] Saraiva | A simulated annealing based approach to solve the generator maintenance scheduling problem[END_REF] built new solutions by randomly selecting a generating unit and setting a new random maintenance starting time for it. At each iteration, [START_REF] Han | ISO coordination of generator maintenance scheduling in competitive electricity markets using simulated annealing[END_REF] selected first the unit with a weighted roulette wheel where weights are fixed according to the LOLP associated with each generator, and then change randomly its maintenance starting time. SA has been also combined with other techniques. For instance, [START_REF] Anghinolfi | A matheuristic algorithm for a large-scale energy management problem[END_REF] hybridized SA and linear programming; [START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF] coupled a SA and constraint programming; and [START_REF] Burke | Hybrid evolutionary techniques for the maintenance scheduling problem[END_REF]; [START_REF] Dahal | Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches[END_REF]; [START_REF] Mohanta | Fuzzy reliability evaluation of captive power plant maintenance scheduling incorporating uncertain forced outage rate and load representation[END_REF][START_REF] Mohanta | Deterministic and stochastic approach for safety and reliability optimization of captive power plant maintenance scheduling using GA/SA-based hybrid techniques: A comparison of results[END_REF]; [START_REF] Leou | A new method for unit maintenance scheduling considering reliability and operation expense[END_REF] embedded a SA into a GA. SA has been mainly used to solve problems arising in the regulated power systems.

Tabu search

Tabu search (TS) is a local search-based metaheuristic that avoids revisiting solutions by recording the recent history of the search in a short-time memory called tabu list. To escape local optimum, non improving solutions are accepted during the algorithm. To our knowledge, TS has only been applied to solve the GMS problem in regulated power systems. [START_REF] Burke | Hybrid evolutionary techniques for the maintenance scheduling problem[END_REF] found better computational results when TS is used as a local search embedded in a GA. The neighborhoods are those described in the Section 3.2.1. [START_REF] El-Amin | A tabu search algorithm for maintenance scheduling of generating units[END_REF] applied TS by randomly selecting a generator and modifying its maintenance starting time.

Ant colony optimization

Ant colony optimization (ACO), a constructive metaheuristic inspired by the behavior of ant colonies, has been also applied to solve the GMS problem in regulated power systems with a cost-based objective [START_REF] Foong | Ant colony optimization for power plant maintenance scheduling optimizationa five-station hydropower system[END_REF][START_REF] Fattahi | A new approach for maintenance scheduling of generating units in electrical power systems based on their operational hours[END_REF][START_REF] Mytakidis | Maintenance scheduling by using the bi-criterion algorithm of preferential antipheromones[END_REF]. The algorithm builds solutions incrementally by selecting a maintenance starting period for each generator. The selection process is based on the combination of pheromone level, related to the number of times the component has been selected, and some greedy heuristics. [START_REF] Fattahi | A new approach for maintenance scheduling of generating units in electrical power systems based on their operational hours[END_REF] combined ACO with some sub-algorithms to deal with a complex problem based on operational hours. According to the outages fixed in the previous phase, UC is solved by a custom heuristic which proves to be better in their experiments than a GA.

Constraint programming

Constraint programming (CP) is particularly useful for highly constrained problems. It is a powerful and flexible tool which makes express complex constraints relatively easy. It applies particularly well to the ROADEF challenge where maintenance tasks faced a variety of constraints such as overlapping, spacing, and incompatibility [START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF][START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF][START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF]. [START_REF] Brandt | A constraint programming-based approach to a large-scale energy management problem with varied constraints[END_REF] coupled CP to a greedy heuristic to solve the unit commitment and the refueling problems. [START_REF] Buljubasic | Orchestrating constrained programming and local search to solve a large scale energy management problem[END_REF] presented a heuristic approach combining a constraint satisfaction problem to both a local search based on the marginal cost and a constructive optimization algorithm. [START_REF] Godskesen | Solving a real-life, large-scale energy management problem[END_REF] used it with a local search and a greedy heuristic in a 3-phase algorithm. For a quite similar problem, [START_REF] Khemmoudj | When constraint programming and local search solve the scheduling problem of Electricit de France nuclear power plant outages[END_REF] proposed an approach combining constraint programming and local search.

CP is less suitable when the main objective of the problem is to find near-optimal solutions. Therefore, it has been scarcely used for solving the GMS problem. [START_REF] Frost | Optimizing with constraints: A case study in scheduling maintenance of electric power units[END_REF] iteratively applied constraint programming to solve a cost bound problem; learning constraints are added to improve the efficiency of the algorithm.

Game theory

Game theory-based approaches have been explored on the problem arising in deregulated power systems (Chattopadhyay, 2004a;[START_REF] Kim | Generating unit maintenance scheduling under competitive market environments[END_REF][START_REF] Min | Game-theory-based generation maintenance scheduling in electricity markets[END_REF]. These approaches are especially suitable since every GENCO tries to predict its competitors actions so as to stay one step ahead. The strategies adopted by the GENCOs are defined by a Nash equilibrium of the game.

Conclusion and perspectives

The GMS and TMS problems are the two main maintenance scheduling problems in the electricity industry.

The constraints concern the maintenance tasks (time windows, incompatibility, sequence), the resource requirements, the reliability, and the demand satisfaction. Sometimes, e.g., for nuclear power plants, fuel consumption management is required. The GMS and TMS problems can be solved jointly or network constraints can be introduced into the former. Production planning is often incorporated into GMS, especially over a short-term horizon. This results in a complex problem that is generally NP-hard.

Maintenance scheduling is a major challenge in the electricity industry, especially since the liberalization of the electricity market. The objectives of regulated power systems are based mainly on the reliability (leveling, maximization of net reserves) and the costs (minimization of the operational costs). These objectives are not necessarily suitable for deregulated systems. It may be more appropriate to maximize the profits of the GENCOs and to coordinate the decisions of the various actors. The objectives of regulated systems remain relevant to the ISO-the actor that must ensure system reliability and security-but may conflict with the goals of the other actors (GENCOs, TRANSCOs, DISCOs). A Multiobjective optimization is thus a future solution framework.

Many solution methods have been proposed for the GMS and/or TMS problems. They include heuristics, metaheuristics (genetic algorithms, particle swarm optimization, simulated annealing, tabu search,ant colony optimization), hybrid approaches, mathematical programming (dynamic programming, MILP, branch-andbound, Benders decomposition), constraint programming, game theory. As the problem complexity increases making frontal resolution impracticable, the use of decomposition techniques become more and more relevant.

To the best of our knowledge, some problems have not yet been investigated. These include load uncertainty and price volatility when the TMS problem is solved jointly with the GMS problem or where coordination is needed between the GENCOs and the ISO. Apart from that, the growing renewable energy industries and its stochastic nature have an impact on the planning in power systems. Maintenance decisions in wind industry are very complex since the weather has a huge impact on the possible concrete realization of the scheduled tasks.

Moreover, wind predictions can only be established in a very short term horizon. To explicitly handle these uncertainties via stochastic programming approaches may lead to substantial energy and cost savings. Taking into account numerous unexpected breakdowns, short-term rescheduling could be investigated. In conclusion, future research will have to improve the handling of uncertainty, the coordination of decisions as regards the different actors of the power systems as well as to answer the new challenge power systems will face especially with renewable energies.

A Classification of the bibliographical references • the problem it solves : GMS, TMS (a checkmark in both GMS and TMS columns means that the two problems are addressed in the paper), GMS with network constraints (GMS+N)

• the power system it targets : regulated (Rg) or/and deregulated (Dg)

• the objective function it handles : reliability-based (R), cost-based (C), profits-based (P).

• some features it deals with : forced outage rate (FOR), stochastic demand (D), refueling management (F).

• the solution method it proposes.

It is not however exhaustive to cover the wide range of features that can occur for the maintenance scheduling in electricity industry. Some classifications may also be debated since some studies do not fit well into the boxes, but we aim at being the most consistent. goal programming based on MILP (multiobjective) [START_REF] Mytakidis | Maintenance scheduling by using the bi-criterion algorithm of preferential antipheromones[END_REF]

Figure 2 :

 2 Figure 1: Interactions in a vertically regulated utility

Figure 3 :

 3 Figure 3: Coordination procedure in a deregulated power system

Table 1 :

 1 List of key terms and their acronyms

	Acronym	Term
	GENCO	generation company
	TRANSCO transportation company
	DISCO	distribution company
	RETAILCO retail company
	ISO	independent system operator
	GMS	generator maintenance scheduling
	TMS	transportation maintenance scheduling

  van Vuuren,

5 Association of European Operational Research Societies 6 Société française de recherche opérationnelle et d'aide à la décision 7 Electricité de France 8 Institute of Electrical and Electronics Engineers 2013;

  Table.2 classifies each reference according to :

expected time during which the demand is greater than the available capacity of the power system

The shift project data portal -Browse energy and climate data, http://www.tsp-data-portal.org/Breakdown-of-Electricity-Generation-by-Energy-Source, last accessed : 2015-05-07

University of Washington Electrical Engineering, Power systems case archive, http://www.ee.washington.edu/research/pstca/, last accessed : 2015-04-08

authors demonstrated the importance and the impact of market related costs in maintenance schedules.

Under deregulation, the GENCOs have limited information about the system. The coordination of the decisions and information exchange between the GENCOs and the ISO are important. The interactions between these two actors have received intensive interest (Barot and Bhattacharya, 2008;Conejo et al., 2005;Elyas et al., 2013;Eshraghnia et al., 2006;Geetha and Shanti Swarup, 2009;Han et al., 2011;Min et al., 2013;Zhan et al., 2014). Reliability-based objectives are considered when the problem is solved from the point of view of the ISO: maximizing the reserve throughout the horizon (Eshraghnia et al., 2006), maximizing the sum of the ratio of the net reserves to the gross reserves by period (Conejo et al., 2005), minimizing the standard deviation of this last ratio (Zhan et al., 2014), or minimizing a risk penalty factor related to an adequate level of reliability (Geetha and Shanti Swarup, 2009). Cost minimization is considered in (Barot and Bhattacharya, 2008). Barot and Bhattacharya (2008); Conejo et al. (2005); Eshraghnia et al. (2006); Geetha and Shanti Swarup (2009); Min et al. (2013) presented iterative coordination methods based on rescheduling signals. Eshraghnia et al. (2006) suggested a coordination procedure where at every iteration the ISO indicates permissible and ACO Reihani et al. (2012) GA combined with LS (extremal optimization) Rozenknop et al. (2013) column generation combined with heuristics Saraiva et al. (2011) SA Satoh and Nara (1991) SA Schlünz and van Vuuren (2013) SA and local search (hybridisation) Silva (2000) Benders decomposition Silva and Morozowski (1995) Benders decomposition Suresh and Kumarappan (2013) hybrid improved binary PSO / GA Volkanovski and Mavko (2008) GA Wang and Handschin (2000) GA Wu et al. (2008) Lagrangian relaxation combined with Dantzig-Wolf technique Yare and Venayagamoorthy (2010) multiple swarms-modified PSO Yellen and Al-Khamis (1992) Benders decomposition Zhan et al. (2014) GSO (multiobjective)