Generation Maintenance Scheduling for Generation Expansion Planning Using a Multi-Objective Binary Gravitational Search Algorithm

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Figure 2. Method of reserve capacity leveling using in Eq. (3)

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2.2. Levelized Reserve Capacity

For convenience in scheduling, the whole maintenance period is divided into N_st stages that each stage is either one week, ten days or one month. The levelized reserve method is a straight and instinctive method that draws up a maintenance schedule by leveling the system's net reserve capacity throughout the time intervals. Actually, sum of the system load and capacity of generating unit's maintenance outages throughout the time subintervals are made uniform in this method. Not that, the time subintervals are specified during the time intervals, considering the starting day and the length of generating units maintenance outages. This is a method conventionally used in GMS problem and it has found wide application in electric utilities owing to its easy and understandable concept. As aforementioned, reserve capacity leveling is one of the objective function of the proposed method. In doing so, an index is defined to evaluate this objective function. Figure 2 and Eq. (3) are used to calculate this index, taking into account the generating unit's maintenance outages.

(3)

As can be seen, is equal to 0, as long as the maximum capacity is the same throughout the time intervals. N_int is total number of time subintervals, that is dependent on GMS problem solution.

2.3. O&M Cost

O&M cost is one of the objectives used in proposed fuzzy multi-objective approach. O&M costs in each year are composed of fuel costs and non-fuel O&M costs of generation, which can be evaluated as:

(4)

,and ,respectively, denote the energy generated in base and peak capacity, and total energy generated by the type-k i^th plant unit located at bus b, in period f of year t of the study period. In this study, the energy generated by each unit in the system is calculated by probabilistic production simulation. In this approach the forced outages of thermal units are convolved with the inverted equivalent load duration curve (ELDC) and, consequently, the effect of unexpected outages of thermal units upon other units is accounted in a probabilistic way. The net effect is that the expected generating costs of the system is increased by an increase of peaking units generation in order to make up the reduction of base units generation due to scheduled outages for maintenance and unit failures ^[24]. Indices of , and are calculated using probabilistic production simulation considering an ELDC. Eq. (5) shows the equation of the ELDC for fixed and candidate units:

(5)

Where P_i is the operation rate of generating unit i and q_i=1-p_i is equal to the unit’s Forced Outage Rate (FOR);is generating capacity for unit i in p.u; energy generated by each unit is determined using Eq. (6)

(6)

Where x_i is equal to .

2.4. Expected Energy not Supplied

As already mentioned, EENS is another objective function of the proposed method. A generation unit maybe tripped at a rate given by its FOR, which represents the percentage of time, when unit maybe unavailable due to unexpected outages. Some portion of the energy demand may not be served owing to generating unit's FORs and maintenance outages. Probabilistic production simulation involving ELDC is also implemented to compute EENS as in Eq. (7).

(7)

whereis the total capacity of all of the generating units during the time intervals.

2.5. Constraints of the Proposed Method

In the proposed GMS problem, two types of constraints are imposed: technical constraints comprising of GEP constraints and basic constraints consisting of fuel constraints, maintenance constraints, maintenance crew constraints and transmission state constraints. The transmission state constraints are some special limits imposed on the generating unit maintenance plan in a certain area by the transmission network operation state. Some of these constraints are described at the following:

2.5.1. Technical Constraints

Generation capacity should be sufficient in meeting the load requirements, though there are uncertainties that may cause generation units to trip unexpectedly at any time as in Eq. (8). Eq. (8) simply states that the installed capacity in the critical period must lie between the given maximum and minimum reserve margins—a_t and b_t, respectively—above the peak demand (D_t,c) during the critical period of the year.

(8)

2.5.2. Basic Constraints

a) The continuity of maintenance activity: Maintenance must be completed in a continuous time interval once started as in Eq. (9)

(9)

b) The minimum time interval between two maintenances to each generating unit: Suppose that the time interval is considered as N stages; then this constraint for unit k is presented as follows:

(10)

c) Maintenance crew constraint: The maintenance crew constraint means that there are limitations due to the maintenance manpower availability; Normally, two generating units cannot be scheduled for maintenance together in the same power plant and at the same time, i.e. V_r,st=1. Only a few power plants with considerable maintenance resources can allow V_r,st>1. However, the following equation must still be met ^[24]:

(11)

Figure 3. The flowchart of BGSA

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3. Solution Methodology

As aforementioned, the purpose of this problem is to find the optimum maintenance schedule of generating units. The objectives are described in Section 2. In this section, firstly, BGSA is described, and then it is employed to the proposed method.

3.1. Binary Gravitational Search Algorithm (BGSA)

In the BGSA ^{[17, 18]}, agents are considered as objects and their performance is measured by their masses. All these objects attract each other by the gravity force, and a global movement of all objects towards the objects with heavier masses, is generated by this force. As a result, masses cooperate using a direct form of communication, through gravitational force. The heavy masses– which correspond to good solutions – guarantees the exploitation step of the algorithm, owing to their more slowly movement than lighter ones. The BGSA could be considered as a small artificial world of masses obeying the Newtonian laws of gravitation and motion system of masses. The flow diagram of BGSA is depicted in Figure 3. worst(t) and best(t) in this figure are defined as follows (for a minimization problem):

(12)

(13)

Where, fit_i(t) represent the fitness value of the agent i at time t. In BGSA a system with N agents (masses) is considered. The position of the i^thagent is defined by:

(14)

The step of Fitness evaluation is performed using following equations:

(15)

(16)

To update agents’ position Eqs. (17-24) are used:

(17)

(18)

The total force that acts on agent i in a dimension d () is supposed to be a randomly weighted sum of d^th components of the forces exerted from other agents, giving a stochastic characteristic to the algorithm:

(19)

Where r and j is a random number in the interval [0,1]^{[, 1]}.

(20)

(21)

is limited to an appropriate interval for a better convergence as Eq. (22).

(22)

In BGSA a probability function is defined for each agent as:

(23)

Position change of each agent in a dimension of binary space will cause its value change from 0 to 1 or vice versa. Following the above calculation of probability function each agent moves in each dimension as Eq. (24).

(24)

Figure 4. The data structure of each agent position of proposed BGSA

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Table 1. Integer structure and corresponding description for the position of each agent in each dimension

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3.2. Applying BGSA to the Proposed Method

To solve GMS problem using BGSA, problem variables are combined and represented as a mixed integer coding in each agent position. The data structure of each agent position is depicted in Figure 4. As can be seen from this figure, the binary to decimal conversion of each five bits of position of each agent in each dimension refers to the starting day of the time stage st when generating unit u is allowed to be scheduled for maintenance in year t of the study period, as shown in Table 1. In the proposed BGSA, candidate solutions of initial population are randomly selected between all solutions which satisfy the problem constraints and also new solutions obtained through the BGSA’s procedure are checked to sustain feasibility.

Flowchart of the main GEP problem along with the proposed method to solve GMS problem is presented in Figure 5. As can be seen from this figure GMS problem is solved as a sub-problem of the main GEP problem. Generation investment cost and transmission enhancement cost presented in this figure are other objective functions of the main GEP problem. To handle the problem complexity, it is assumed that, the transmission enhancement requirements are approximately proportional to the length-based overloads calculating by running an AC power flow during critical operating period. It is worthwhile to note that, as GEP problem is not the main subject of this paper and GMS problem is solved for a typical solution of the main GEP problem, so objectives of generation and transmission enhancement costs are not considered in section of simulation results of this paper.

Figure 5. Flowchart of the main GEP problem along with the proposed method to solve GMS problem

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4. Simulation

In order to demonstrate the effectiveness of the proposed approach, it is applied to the 4-bus test system from Grainger & Stevenson for a planning horizon of one year and an IEEE-RTS 24-bus test system for a planning horizon of two years with growing complexity. The best solution of the main GEP problem applying to case studies without considering maintenance scheduling of generating units, which GMS problem is solved for, is presented in Table 2. In this study, for convenience in scheduling, the whole maintenance year is divided into 12 stages (N_st=12); i.e. every stage is one month. Candidate plants for GEP problem are selected from four different types of natural gas units, coal units, oil units and nuclear units.

Table 2. The best solution of the main GEP problem applying to both case studies

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Two case studies and their results are described at the following. In the first case, the problem is applied to a 4-bus, 1-generator case from Grainger & Stevenson ^[25]. The figure of this system is presented in Figure 6. As can be seen, there is only one fixed nuclear unit of 318 MW nominal capacity, and 8% FOR; total network load is 500 MW. For this case study, GMS problem is solved for a planning horizon of one year. An IEEE-RTS 24-bus test system is selected as the second case study to which the GMS problem is applied for a planning horizon of two years with growing complexity ^[26]. There are 32 fixed units of 3,405 MW total nominal generating capacity; Total network load is 2,850 MW. The proposed BGSA that is validated for case study 1 is also employed to solve the GMS problem for this case study. The maximum load during the stages for both Case studies are presented in Table 3.

Figure 6. Case study 1: 4 bus test system from Grainger & Stevenson

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Table 3. Maximum load (MW) during the stages for both case studies (Y (Year), M (Month))

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First, BGSA is used as the optimization tool, then BGA ^[22] is used to solve the problem for feasible solutions, satisfying the problem constraints. As the performance of heuristic search methods relies on their parameters, so the algorithm parameters are tuned for both BGSA and BGA to obtain a better solution. In both cases, population size is set to 100 (N= 100) and maximum iteration is 60. In this study, equation () is used to set parameter G of BGSA, where G₀ is set to 100 and α is set to 12, and T is the total number of iterations. Total objective function is evaluated for this case study as described in Section 2. A probabilistic production simulation is also used to calculate the energy generated by each unit and also the expected energy not served in each period for these case studies, taking into account the generating units maintenance schedule.

4.1. Simulation Results

Figure 7 shows the ELDC of the probabilistic production simulation for the proposed maintenance plan by the BGSA for case study 1. As can be seen, the equivalent load duration curve is computed for six fixed and candidate units with respect to their FOR. Table 4 shows proposed maintenance plan by BGSA for both case studies. The maintenance schedule for each generating unit is shown in this table by denoting month and day of the maintenance starting during in each year of the scheduling horizon. Two scheduling turns are considered for units which are fixed or be added at the first year of planning horizon. It is noteworthy that the proposed maintenance plan by the BGSA and BGA applying to case study 1 are the same. Total generated energy and O&M costs of generation of each unit during the first stage for maintenance plan of case study 1 are presented in Table 5. The value of three objective functions, their membership function values and total objective function for two case studies are presented in Table 6. As can be seen from this table, discounted values of total operating costs are respectively equal to 57380.291 K$ and 982775.935 K$ for case studies 1 and 2. In calculating the total objective function indexes k₁, k₂ and k₃ are assumed to be equal (i.e. k₁= k₂= k₃=1/3). Member ship function values are computed from the value of objective functions using Eq. (2) and Figure 1. Eq. (1) is used for calculating the total objective function. Using reserve capacity levelizing method shown in Figure 2 and Eq. (3), index of I_resv is specified by consisting the capacity figure. Doing so, sum of areas under maximum capacity line of time intervals and also the average capacity and the area under average capacity line is calculated as shown in Table 6.

Table 4. Proposed maintenance plan by BGSA for case studies (Y (Year), M (Month), D (Day), FT (First Turn), ST (Second Turn))

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Table 5. Generated Energy and O&M costs of generation of each unit during first stage for proposed maintenance plan by BGSA for Case study 1

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Table 6. Simulation results of proposed BGSA-based method for both case studies

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Figure 7. ELDC for case study 1

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4.2. BGSA Versus BGA

In order to evaluate the performance of proposed BGSA-based method, a stochastic heuristic search, BGA, is also implemented to solve GMS problem ^[20]. The convergence rate of search algorithm and the final results of both method are compared in this subsection. In BGA, the tuned values of C_r and C_m are 0.92 and 0.07. The results show that, proposed maintenance schedules by BGSA and BGA are the same for case study 1, but have some differences for case study 2. As can be seen from Figure 8, BGSA has provided a better solution in comparison with BGA for case study 2. Table 7 shows the average of best fitness, final best fitness and number of iterations to convergence for BGSA and BGA for both case studies. As this table illustrates BGSA provides better results and has a higher convergence rate than BGA. In addition, the good convergence rate of BGSA for both case studies could be concluded from Figure 8. According to this figure, BGSA tends to find the global optimum faster than BGA.

Figure 8. BGSA and BGA Convergence manner for case studies

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Table 7. Comparison of BGSA and BGA performance for both case studies

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