Optimal generation maintenance scheduling considering ﬁnancial return and unexpected failure of distributed generation

Generation maintenance scheduling (GMS) is an important factor that can improve the reliability of power systems and decrease revenue achieved by generation companies (GenCos). Several GMS models, therefore, are based on these two crucial values. Nonethe-less, there is no GMS problem that simultaneously considers unexpected failure of distributed generator (DG), ﬁnancial return of GenCo, and reserve of system. This paper proposes the GMS model based on a global criterion approach to compromise functions that maximise the GenCo’s annual return and probability that no DG fails unexpectedly. The system reserve (SR) is considered as a reliability constraint, while surplus reserve is exchanged with the main grid. To support alternative energy sources that have uncertain outputs and ensure continuous operation of DG, short-term GMS model including power from wind farm, photovoltaic system, energy system storage, and demand response (DR) is also run by adding the SR and inconstant cost of DR. Effectiveness of the proposed model is examined using the IEEE 6 and IEEE 18-bus test systems. Results show that not only the proposed model provides a better GMS solution for the GenCo, resulting in appropriate values of the two objectives, but also alternative energy sources are useful for the short-term GMS.


Motivation and literature review
Generation maintenance scheduling (GMS), in restructured power system, is one of the most important factors used to improve system reliability. It can increase performance and prevent unexpected failure of distributed generator (DG). However, if the GMS model is solved improperly, the financial return of a generation company (GenCo) can be decreased as a result of costs required to support DG maintenance [1,2], an outage of DG might occur unexpectedly [3], and the system reserve (SR) determined by the independent system operator (ISO) may not be satisfied [4]. Therefore, to ensure the stability of these three crucial values, a multi-objective GMS model should be developed and proposed.
description, to maximise the profit of GenCo, an ageing effect of components depending on seasonal conditions is added as a constraint, and the GMS problem is solved by using an equivalent operating hour approach [5]. In [6], GMS solutions of several GenCos are coordinated by adapting a multiple-leader-common-follower game. In [7], the GMS model is proposed by considering a failure rate (FR) of DG in views of renewal cost. The FR is calculated by utilising a superposed power low-process and bathtub-shaped failure behaviour of DG, whereas the problem is solved with genetic algorithm and linear programming method. In [8], the GMS is modelled to maximise the profit of several GenCos in the competitive market by using a nash equilibriums and energy efficiency plants. Also, the constraints about the SR and emission monitored by the ISO are considered. In [9], a smart GMS model with a cost of energy not supplied is proposed. The cost is run by using a sequential monte carlo simulation, while the problem is solved by applying the accelerated quantum particle swarm optimization technique. In [10], the GMS plan for GenCo in the oligopolistic market is run by considering the GenCo's profit and the SR. The short-term GMS is solved after the ISO denounces the long-term plan, which is received by using game theory. The risk analysis is adapted to find the results under the uncertainties of market price and GenCo's behaviours. The FR of DG is used to impose the state of the DG, and calculate the SR. To minimise the sum of squared residuals, optimisation models considering crew constraints are proposed. The problems are solved using simulated annealing [11], and ant colony optimisation [12]. To minimise cost of GenCo, a cost reduction index is proposed to minimise expenditures of operation and maintenance [13]. The authors of [14] present a short-term GMS by connecting energy system storage (ESS), hydropower, biomass plant, and wind power plant (WPP) to the smart power system. The FR of DG is also added as one of the reliability constraints. In [15], the FR based on the ageing momentum of thermal unit is considered to minimise costs of DG outage, labour, materials and security. In [16], operating hours of units are calculated in a GMS model. The problem is solved by using simulated annealing and ant colony optimisation. A co-generation power plant considering boilers and turbines to maximise the objective is proposed by [17]. Hybrid particle swarm optimisation, genetic algorithm and shuffled frog leaping method are developed by [18], while mathematical approaches of assisted differential evolution algorithm and clonal selection algorithm are presented in [19] and [20], respectively. In [21], pump storage units are connected to the system. Lost profit and penalty factors are described in the objective function, while benders decomposition is adapted to determine the results. In [22], the GMS model is run by considering power exchanged with the main grid, while a short-term GMS (ST-GMS) is proposed by including alternative energy sources, namely WPP, ESS, photovoltaic generator (PV), and demand response (DR). A risk analysis is adapted to give several GMS solutions to the GenCo. In [23], the maintenance scheduling for the transformer is proposed to minimize risk costs, which are calculated from collective maintenance and energy not supplied. The maintenance plan can be rescheduled by adapting the time-varying maintenance threshold solution. The model is formulated as non-linear and solved by using the genetic algorithm. In [24], by considering the cost from energy not supplied, the GMS plan is based on the reliability-constrained adaptiverobust multi-resolution. The energy not supplied is calculated by adapting the uncertainty of load, wind power generation, and equipment unavailability, whereas the bender decomposition is used to solve the problem. In [25], a multiscale multiresolution GMS model is based on the uncertainties of weekly and hourly loads. The weekly and hourly loads are forecasted by using the probability distribution function and polyhedral uncertainty, respectively. The problem is run by stochastic affinely adjustable robust technique. In [26], costs of transmission and generation maintenance, and risk are considered in the GMS model. The risk cost is based on the punishment coefficient for the overflow of the line. The problem is solved by using lagrangian relaxation and bender decomposition. To minimise the loss of load probability, a hybrid algorithm combining particle swarm optimisation with simulated annealing is presented in [27], while hybrid improved binary particle swarm optimisation is adapted in [28]. To minimise the unexpected failure of DG, the GMS model is considered in the perspective of maximising the probability that no DG fails during the period between when the DG was placed back into normal mode after its previous maintenance and when its next maintenance is run (PnF), which is solved by using the maintenance interval, FR, and weighted capacity of each DG [3]. To minimize the energy not supplied, in [29], the GMS model is based on the isolated power system connected to a wind turbine. The problem is formulated as a stochastic model and optimised by using the particle swarm technique compared with the genetic algorithm. For multi-objective GMS problems, the aims of minimising the sum of squared residuals and cost of GMS are determined by considering crew availability as a constraint and using particle swarm optimisation as a mathematical solution [30]. In [31], the benefit of GenCo and system reliability are maximised considering unserved demand based on the satisfaction of customers. In [32], a hydro-generator is included in the power system to minimise cost of GMS and maximise future water value. A relaxation induced method is adapted to maximise profit of GenCo and system reliability index [33]. Transmission outage scheduling is coordinated with the GMS problem for minimising transmission and generation maintenance scheduling costs [34]. A coordination mechanism considering incentives and disincentives among GenCos is proposed to maximise system reliability and GenCo's profits [35]. In [36], a lexicographic technique is modified to minimise system emission, minimise GMS cost and maximise average net reserve value by including DR in the objective functions. In [37], a dominance-based multi-objective SA approach is proposed to demonstrate the trade-off between minimising cost of GenCo and maximising system reliability. A risk-based analysis is used with the aim of maximising the GenCo's profit and minimising financial risk. Penalty cost caused by the FR of DG is proposed in the GMS model [38]. The global criterion is adapted to maximise the return of a GenCo and the SR, simultaneously [4]. In [8], the GMS model is based on energy efficiency to satisfy the profit of several GenCos and system reliability. The model is solved by using the Nash equilibrium and non-corporative game theory.
To the best of our knowledge, there is no GMS model based on maximising the GenCo's yearly financial return (annual return) and the PnF, simultaneously, while also maintaining the SR constraints determined by the ISO. Moreover, A ST-GMS, that considers the alternative energy resources to support a trend of using clean energy to generate the power, has been also formulated to maintain DGs in a day without consideration of the SR and inconstant cost of DR.

Main contributions of this paper
From previous studies, if the GMS model is formulated to maximise the SR and financial return [4], there is a chance that the reserve is higher than the value required by the ISO. It can result in a redundant capacity, which is not beneficial to the GenCo.
At the same time, DG outage may occur unexpectedly resulting in unsatisfied profit and SR for the GenCo and ISO, respectively. On the other hand, if the GMS problem is only based on the PnF maximisation [3], the annual revenue achieved by the GenCo can be greatly decreased.
To decrease unexpected outage of DG, satisfy the financial management of Genco, and maintain SR of system, this paper proposes a multi-objective GMS model based on the global criterion approach to compromise functions that maximise the GenCo's annual return and PnF. This is expressed as the formulation of mixed-integer non-linear programming by considering the proposed SR, which is solved after the GenCo sells the electricity to the main grid, as one of the operational and security constraints. Moreover, to decrease the depreciation of DG [22] and support a trend of using alternative energy sources, the ST-GMS model in [22] is adapted to maximise the annual return by addressing the inconstant cost of DR and SR as expenditure factor and reliability constraint, respectively. The alternative energy sources [39][40][41][42] are included in the ST-GMS because they use clean energy to produce the electricity resulting in decreasing emission for the environment and saving fuel cost for the GenCo, whereas their intermittent power outputs cannot necessarily affect the GenCo's profit and SR. Therefore, the main contributions of this paper are as follows: 1. A GMS model is constructed based on maximising the annual return of GenCo and PnF and solved by adding the proposed SR as the ISO's constraint.

A GMS solution based on the global criterion approach is
proposed to compromise the perspectives of maximising the GenCo's annual return and PnF. Results of single and multiobjective problems are compared to demonstrate the effectiveness of the proposed model. The organisation of this paper is as follows: In Section 2, the problem formulations and constraints are formulated. Section 3 describes the GMS model based on the global criterion, while numerical studies are demonstrated in Section 4. Finally, in Section 5, the conclusion is summarised.

PROBLEM FORMULATIONS
In this section, objective functions and constraints related to the GenCo's annual return and PnF are described.

The GenCo return and PnF maximisation
The definition of annual return for GenCo expressed in Equation (1) can be represented by the difference between revenue and cost.
In Equation (2), the revenue is composed of the energy sold to the main grid and the GenCo's consumers.
(2) In Equation (3), the cost [22] is denoted by expenditure factors as follows: The PnF, as expressed in Equation (4), is based on the FR, duration between previous and present maintenance intervals, and capacity of the DG [3]. It can be equivalent to minimise terms demonstrated in Equations (5) and (6). Therefore, when using the property of natural logarithm to linearize the model, the objective can be expressed as Equations (7) and (8).

Annual return maximisation for the ST-GMS
In the ST-GMS, the annual return and revenue of GenCo are the same equations as described in Section 2.1, whereas the costs (Cost ST ,t ) depend on energies purchased from the main grid, WPP, PV, and ESS [22]. The inconstant cost of DR is additionally considered as determined by Equation (9).

Constraints
Two categories of technical constraint are considered as follows:

System reserve constraints
Power generation, power loss, load, and power exchanged with the main grid are considered for both normal and ST-GMS, while power from alternative energy sources is included only for the ST-GMS. These relationships are demonstrated by Equations (10)- (12).
To preserve the minimum system reserve (MSR) determined by the ISO, terms in Equations (13) and (14) must be satisfied. Furthermore, to calculate the reserve before selling the electricity to the main grid, the normal system reserve (NSR) for normal and ST-GMS are also expressed as Equations (15) and (16), respectively.

Operational constraints
Several constraints related to DG operation, system security and alternative sources [22,35] are described below. DG maintenance duration is calculated by Equation (17), while outage continuity is expressed as Equation (18). The startup, shutdown and maintenance states are determined by Equations (19), (20), and (21) respectively.
The output of DG and the power flow on the system are limited as Equations (22) and (23), respectively.
The outputs of WPP and PV are expressed as Equations (24) and (25), respectively.
The remaining power of ESS is demonstrated by Equation (26). Limitations of charging/discharging are determined by Equations (27) and (28). In Equation (29), two states of ESS cannot be simultaneously operated.  The maximum value of DR is imposed by Equation (30)

THE MULTI-OBJECTIVE GMS MODEL
In multi-objective optimisation problem, there is a chance that only one goal can be reached while others are ignored. Therefore, if the GenCo needs to compromise on several objectives, mathematical methods must be adapted.
In this paper, the global criterion approach is used to solve the GMS problem with the aims of maximising the GenCo's annual return and PnF. The main function of this technique can be written as Equations (31) and (32).
In Equations (31) and (32), the problem is considered as a scalar optimisation. The objectives are compromised among each other resulting in the appropriate values, while the Z is between 0 and 1. The n can be assumed as 1. This method has advantages since information concerning weight and priority ranking of the GenCo is not required [4,43].
The global criterion method is adapted to the proposed GMS model with the solution depicted in Figure 1, while the mathematical equation is demonstrated as Equation (33).

NUMERICAL EXAMPLES
In this section, to demonstrate the effectiveness of the proposed GMS models, the IEEE 6-bus and IEEE 18-bus test systems are examined. The problem is carried out by using the MAT-LAB program.

Case I: The IEEE 6-bus test system
The modified IEEE 6-bus test system [22] is shown in Figure 2, with characteristics of DGs and transmission lines given in Tables 1 and 2, respectively. The F LMP is assumed to be 1.

The normal GMS solution
To avoid the overlap between time intervals for maintaining DGs in normal and ST-GMS, the normal GMS must be solved in this section before running the ST-GMS plan. For the normal GMS plan, the total number of the time intervals in the maintenance scheduling window is 52 weeks, whereas the required maintenance duration is 1 week for all DGs. Then, the h is 24 × 7. The FR i shown in Table 1 are extrapolated from [3] using the linear regression method [44]. The m ′ i of all DGs are based on the assumption that their previous maintenance periods are run during low market price intervals.
The weekly load [45,46] and the market price estimated from [22] are illustrated in Figures 3 and 4 respectively. The f DR l is 20% [22], whereas the MSR is 15% [4].

GenCo's return maximisation
In this case, the GMS model is only based on the perspective of maximising GenCo's annual return. Results of SR and NSR are illustrated in Figure 5 and indicate that if the GMS model does not consider the power exchanged with the main grid, the difference between the SR and NSR, which is high value, cannot

FIGURE 4
Case I: Market price benefit the GenCo. In Table 3, the GMS solution shows that DG 1-5 are maintained in weeks 38,11,36,13, and 15, respectively. Maintenance periods with low market price are chosen, consistent with the previous study [4]. The GenCo's annual return is $5,681,621.75, while the objective of PnF is 75.42.

FIGURE 3 Case I: Weekly load
The PnF maximisation In this case, only the perspective of maximising PnF is considered. Figure 5 shows SR and NSR. From Table 3, DG 1-5 are maintained in weeks 4, 5, 2, 1, and 3, respectively. The annual return of GenCo is $5,581,218.84, whereas the objective of PnF is 64.11. The GMS solution demonstrates that DG with maximum capacity and FR is maintained firstly to decrease its unexpected outage.

The multi-objective GMS model
The perspectives of maximising both annual return and PnF are compromised by using the solution proposed in Section 3. The SR and NSR are also shown in Figure 5. In Table 3, DG 1-5 are maintained in weeks 11, 12, 13, 9, and 10, respectively. The GenCo's annual return is $5,670,901.88, whereas the objective  For return maximisation, the annual return of GenCo is return * , while the objective of PnF is as 17.6467%, which is calculated by ((75.42 − 64.11)/ 64.11)×100%, higher than the PnF * . For the PnF maximisation, the annual return of GenCo is 1.767152415% lower than the return * , while the objective of PnF is the PnF * value. In multi-objective GMS model, the annual return of GenCo is less than the return * by 0.188676359%, while the difference between the objective of PnF and the PnF * is 6.87%. These results reveal that the annual return from the multi-objective model is very close to the maximum value, while the objective of PnF is lower than the largest value by 10%, which is consistent to the benefit demonstrated in [3]. An effective GMS solution achieved can satisfy both financial and reliability goals of the GenCo. Also, the SR complements the security of the power system, whereas the difference between the SR and NSR is of benefit to the GenCo.

The ST-GMS solution
In this section, the total number of time intervals for the maintenance schedule is 24 h. The required maintenance durations of DG 1-5 are 5, 4, 6, 5, and 8 h, respectively. Hourly demand is shown in Figure 6 [46]. The market and inconstant DR prices [22,47] are depicted in Figure 7. The capacities of WPP and PV are given in Table 4. The P C s,t and P D s,t are 2 and 1.5, respectively [22]. The WPP, PV and ESS costs are 10.18, 20.83, and 4.2 $/MWh, respectively [48]. The C s and D s are assumed to be one. The MSR is 15%. In Figure 8, power outputs of WPP, PV, ESS and DR are illustrated. These show that  inconstant DR is of greater benefit than the main grid in just 3 hours, and consistent with prices shown in Figure 7. To receive maximum return and satisfy the minimum reserve constraint, one day in week 43, which is not planned for normal GMS, is chosen to be optimised in the ST-GMS model. In Table 5, the maximum return is $3467.245. From these results, they can be

Case II: The IEEE 18-bus test system
The IEEE 18-bus test system [47] adapted in this section is shown in Figure 9. It is based on 33 kV section of the IEEE 30-bus system. The characteristics DG Units [22] are demonstrated in Table 6. The FLMP of all main grids are assumed to be one.

The normal GMS solution
Total time intervals in the maintenance scheduling window and the required maintenance durations for all DGs are the same as

The GenCo's annual return maximisation
In Figure 10, the SR and NSR are illustrated. The GMS solution presented in Table 7 shows that DG 1-7 are maintained in weeks 40,36,31,15,38,13, and 11, respectively. The annual return and the objective of PnF are $32,680,612.56, and 96.65, respectively.

The PnF maximisation
The SR and NSR values are shown in Figure 10. In Table 7, DG 1-7 are maintained in weeks 7, 2, 6, 3, 5, 1, and 4, respectively. The GenCo's annual return is $32,329,117.65, while the objective of PnF is 76.26. Figure 10 shows the SR and NSR. In Table 7, DG 1-7 are maintained in weeks 14,15,9,10,13,11, and 12, respectively. The annual return of GenCo and the objective of PnF are $32,645,665.44 and 83.74, respectively. For the annual return maximisation, Table 7 shows that the objective of PnF is as 26.74% higher than the PnF * . For PnF maximisation, the gap between the annual return and the return * is 1.075545688%. In multi-objective model, the GenCo's annual return is less than the return * at 0.106935343%, while the difference between the objective PnF and the PnF * is 9.88%.

The multi-objective GMS model
Results show compromised values for the GenCo, while differences among the three GMS solutions are consistent with those of case I. It is worth noting that our proposed model is effective with the general power system. In this section, the total load is shown in Figure 11. The market and inconstant DR prices [22,47] are shown in Figure 12, the values of WPP, PV, and ESS are the same as descripted in section 4.1.2. The d i of DG 1-7 are 4, 6, 6, 6, 6, 6, and 6 h, respectively. In Figure 13, WPP, PV, ESS and DR outputs are demonstrated.  One day in week 42 is chosen to be run in the ST-GMS model. In Table 8, the maximum return is $12,583.48. These results concur with those of case I.

CONCLUSION
In this paper, a multi-objective GMS model is proposed to determine compromised results among perspectives of maximising the GenCo's annual return and PnF.