Multiobjective Particle Swarm Optimization for Microgrids Pareto Optimization Dispatch

Department of Electrical Engineering and Automation, Anhui University, Hefei 230039, China Engineering Research Centre of Power Quality, Ministry of Education, Anhui University, Hefei 230039, China Anhui Electric Power Science Research Institute, Hefei 230601, China Electrical Engineering in the Faculty of Science, Engineering, and Technology, Swinburne University of Technology, Melbourne 3059, Australia State Grid Anhui Electric Power Co. Ltd., Hefei 230061, China


Introduction
e increasing concerns over environment have led to a growing demand for renewable energy (RE). More and more presence of RE generations achieves the benefits of greenhouse gas emission reduction and diversity of supply. Due to the intermittent nature of RE, the high penetration of RE generation into power grids has brought challenges to the management of power networks [1][2][3]. e implementation of multiobjective optimization (MOO) dispatch for microgrids (MGs), which consist of distributed generations (DGs), load systems, storages, and communication systems, can help maximize the benefits while mitigating the negative impacts of DGs on power grids. e MG can operate in both islanded [4][5][6] and gridconnected modes [7][8][9][10][11]. It is able to improve the stability and reliability of power supply in factory [8], residential districts [9][10][11], and military zone [12]. e management goal of the MG is not only to satisfy the basic demand of power supply but also to utilize DGs efficiently to improve economic efficiency and environmental protection. ere are many approaches to optimize the MG operation. ey are generally divided into single objective optimization (SOO) approaches [11,[13][14][15][16] and MOO approaches [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. One certain object is concerned though the SOO method, such as energy cost [13,14]and quality of service [16]. In [11], a hybrid price-based demand response (HPDR) is proposed, in which the MG profit is maximized as a single-objective function. However, the SOO approaches ignore certain benefits, which may be acceptable in some situations. e better solution can be achieved by the MOO approaches for more comprehensive consideration. e MOO provides many alternative solutions with conflicting nature of different objectives for a decision maker. Weighted sum mode [18] is commonly used to deal with the relationship of different objectives. In [32], the cost of power system operation is formulated as the sum of three parts. It fails to provide an alternative solution of two conflicting objectives. In practice, there are many naturally conflicting goals in the application of MG: minimize the operation cost, maximize the user's comfort, minimize the greenhouse gas emission, minimize the risk, maximize the electricity generation capacity, and minimize the power system losses. A trade-off is unavoidable between these naturally conflicting objectives. Many other methods are introduced to handle differences between the optimal values of various objectives. Fuzzy logic algorithms are proposed to perform social criteria technique in [5,19,20]. A modified multiobjective hybrid big Bang-Big Crunch algorithm is employed to cope with the objective functions of voltage overshoot, undershoot, rise time, settling time, and integral time absolute error in [21]. A mixed integer linear programming (MILP) is applied in [22].
Besides these aforementioned methods, the Pareto solution is widely considered as an efficient way to trade-off between the naturally different objectives.
e weighted factor as a tuning operator to illustrate the preferences has been modified systematically to obtain the Pareto frontiers. In [26], a technoeconomic approach based on nondominated sorting genetic algorithm-II (NSGA-II) for the MOO is proposed to the design of the photovoltaicwind hybrid system. NSGA-II is also utilized to the calculation of Pareto optimal fronts in [27]. e traditional PSO shows promising performance on single-objective, nonlinear optimization and gains popularity in the optimal operation of MG. However, it is hard to deal with the multiobjects. e chaos optimization algorithm is introduced to the particle initialization in [20,23].
is amelioration is useful to discrete space optimization. In [24], a fuzzy self-adaptive particle swarm optimization algorithm is proposed and implemented to dispatch the generations in a typical microgrid considering economy and emission as competitive objectives. Furthermore, an efficient decision-making approach, which adopts the fuzzy adaptive particle swarm optimization algorithm, is suggested to find the best compromise strategy of MG energy management in [25]. ree objective functions, namely, annualized cost of the system, loss of load expected, and loss of energy expected, are considered in [29]. e archiving mechanism is introduced to deal with the multiobjects obtaining a kind of MOPSO. In [30], this algorithm is applied to the optimization of a hybrid wind-PV-battery system for reducing cost and increasing reliability. But it fails to provide an efficient way to the selection and maintenance of the nondominate solutions. In our research, we proposed the external archiving mechanism, in which the maintenance procedure and global best particle selection are improved.
is paper contents are as follows. In Section 2, we build the components and economic models for the illustrated MG system on the basis of optimization dispatch. In Section 3, the MOO problem is proposed with three objects and several constraints for the MG optimization dispatch. In Section 4, an advanced MOPSO algorithm is constructed for the Pareto frontier calculation of the obtained constrained MOO problem after the benchmark function test. In Section 5, the Pareto frontiers and final trade-off solutions are obtained. As conclusion in Section 6, the simulation results demonstrate the proposed modelling method is able to describe the dispatch problem of MG. Besides, the advanced MOPSO has better searching ability than the traditional NSGA-II in Pareto frontier distribution and running time for the MG system under investigation in this study.

Modelling of PV-Battery-Diesel MG System
e MG in this study is located in Anhui University, Hefei, Anhui, China, as shown in Figure 1. It is the MG with the PV-diesel-battery system, which includes 400 V/100 kW MG and 400 V/20 kW MG. e MG mainly consists of the PV power system, the diesel generating unit, and the lithium ion (Li ion) battery pack as energy storage. e 400 V/ 100 kW MG is selected in our simulation and analysis. On the basis of generation dispatch, we will build the characteristic and economic models of the components in the selected MG. en, the MOO including two objectives and three objectives are carried out under different scenarios.

Model of PV Array.
In the simplified PV steady-state power output model [31], the PV output is only related to solar radiation and ambient temperature: e value of k pv is − 0.45% (°C), and the value of T STC is 25°C.

Model of Diesel
Generator. e fuel consumption cost of the diesel generator is mainly related to its output power [33]. It can be expressed by a quadratic polynomial as follows: where a, b, and c are related to the specific type of diesel generator.

Model of Li Ion Battery
Pack. e state of charge (SOC), which is an important parameter for the battery management, describes the remaining capacity of the Li ion battery pack [13]. It is defined as

Model of Load considering Plug-In Electric
Vehicles. e selected MG mainly supplies a laboratory building, 15 kW power electronic load, and ten electric vehicle (EV) charging piles shown in Figure 1. e daily EV traveling distance approximately satisfies with lognormal distribution, namely, S ∼ log N(μ t, σ 2 t ). Its probability density function f s (x) is expressed by us, the SOC of the EV can be obtained from the daily traveling distance S, which lead to the charging duration as e daily load of a certain number of the EV charging is obtained by the Monte Carlo method:

e First Objective Function: Operation
Cost. e optimization objective F 1 contains energy consumption cost C dg , the operation management cost C om , the depreciation cost of battery C bd , and the energy interaction cost between the grid and MG C grid . MG absorbed power from the power grid with the purchase price and output power to the power grid with the electricity price: where N DG is a total number of DGs, P i (t) is the energy output of the i th DG, and P grid (t) is the electricity exchange between the MG and power grid. C fi , K i , and C bw are the ECC, OMC, and BDC coefficients of the i th DG. c p (t) is the electricity purchasing price when the MG absorbs energy  Mathematical Problems in Engineering from the power grid, which varies with the peak and valley periods. c s (t) is the electricity selling price when the MG delivers power to the power grid, which is higher than the electricity purchasing price because of the subsidization from the government. ese coefficients of the studied MG are summarized in Tables 1 and 2.

e Second Objective Function: Environmental
Cost. e environmental cost is as important as the economic one. It can be divided into two parts [34]. e first component is the environmental value of power generation emissions, which represents the environmental influence in the operation of MG. e other part is the pollution fine for the emissions, which determined by the jurisdiction. erefore, the environmental protective cost is considered in the dispatch model, which is defined as follows and in Table 3:

e ird Objective Function: Loss of Load Probability.
Reliability is one of the important factors in the MG planning and operation. In [34], the adequacy of power to meet load demand is expressed by loss of load probability (LOLP). In this study, the LOLP is only considered in the islanded operation mode. For the convenience of calculation, an auxiliary quantity-Boolean quantity, is defined, that is, where H LOLP(t) is a Boolean and Δ t is the quantity of power imbalance at time t. e LOLP can be expressed as e third optimization objective F 3 of the MG is defined as

Constraints of MG.
e natural power generation constraints impact the flexibility of MG system and participate in generation units on reserve and energy markets [35]. e constraints in this study are listed as follows: (1) Power balance constraints: (2) DG capacity constraints: (3) Transmission capacity constraints: (4) Capacity constraints of Li ion battery: (5) Capacity constraint of service transformer:

Constrained Multiobjective Optimization
Problem. e constrained MOO problem deals with the dispatch of the DGs to satisfy the multiple objectives of the MG, limited by the constraints and operating limits. It can be formulated as (17)-(21).

Pareto Particle Swarm Optimization Algorithm
In recent years, Pareto optimality is applied in engineering and social sciences [36]. It is defined as allocating goods among individuals where no individual can improve its situation without worsening another's. When two or more conflicting objectives are sought to be minimized, there will not be a single solution that is optimal for every objective function; instead, the optimal solution set comprises multiple solutions with a trade-off between the objectives, namely, the Pareto-optimal front.

Pareto Domination.
Let F: R n ⟶ R m be the function that assigns a criteria space point F(X) to each design space point X ∈ Θ ⊆ R n , Θ is the search space, where the vector X is the set of all feasible solutions in R n , and the objective vector Y � F(X) represents their measured values in R m . We consider n real parameters and m objectives in a constrained MOO problem to define a Pareto optimal front. Definition 1. Let a solution vector be X 0 , X 0 is said to dominate a particle X 1 , if and only if J(X 0 ) ≤ J(X 1 ). X 0 is said to be Pareto optimal if ∃X 1 dominates X 0 , X 1 ≻ X 0 .
Definition 2. Pareto optimal set P s of all Pareto optimal decision vectors is defined as P s � X 0 ∈ Θ | ∃X 1 ∈ Θ, X 1 ≻ X 0 }. All Pareto optimal values corresponding to the decision vectors in P s constitute the Pareto optimal front P F . e standard PSO is a kind of the heuristic optimization algorithm, which attracts significant attention since its introduction in 1995.
In the standard PSO algorithm, the i th particle X i � x i1 , . . . , x in represented by its position in the n-dimensional space is a potential solution to the problem. Within the search space, the i th particle flies under a velocity V i � v i1 , . . . , v in . In each flying step, say the t th step, the best position of the i th particle Pbest i (t) is Pbest i1 (t), . . . , Pbest in (t) , and the best position of the entire swarm Gbest(t) is Gbest 1 (t), . . . , Gbest n (t) . At the (t + 1) th step, the new position for the i th particle can be determined by where rand is a uniform random value in the range of [0, 1].

Advanced MOPSO.
To achieve the Pareto front in this constrained MOO problem, there are two critical steps: archive maintenance and Pareto selection. e external archive is used to store some of the nondominated solutions produced in the searching process of the MOO algorithm. e selection of Gbest is crucial in the PSO algorithm. In the single-objective PSO, all the particles have the sameGbest. us, we can choose the best solution of fitness function as Gbest in SOO progress. However, in the MOO problem, several nondominate solutions will be obtained from the whole population in each evolution iteration. e archive maintenance aims to get diversified Gbest, while the best choice of Gbest for each particle helps to improve or maintain the diversity of external archives.
In this paper, the advanced MOPSO with a mixed process of performing external archive maintenance and Gbest selection is proposed. e new nondominated solution becomes a member of the external archives, if it satisfies one of the following three conditions: (i) the new solution dominates some members of the external archive; (ii) the external archive size is less than its maximum size; (iii) the new solution has better density when the external archive already has the maximum size. e Manhattan distance D ij is introduced to describe the distance between two solutions X i and X j as e combination procedure of archive maintenance and Gbest selection is analysed in three different conditions. For a new nondominate solution X i , the following conditions apply: (i) e first possibility is X i dominates some solutions in external archive N p , then X i takes the place of dominated solutions and will be chosen as Gbest.
where N a is the size of current external archive while N is its maximum size, remove a best member X l with minimum crowding measure under (24) and (25). If X l ≠ X k , substitute X l for the global best position of all particles in N a . (iii) If N a < N, crowding measure fromX i and replace Gbest as e cardinality of the set F is denoted by |F|. e flowchart of the advanced MOPSO is demonstrated in Figure 2. e terminating condition in the program is (i) if the maximum number of iterations is meet or (ii) the desired change of fitness function is meet.

Benchmark Function Test.
ree representative benchmark functions associated with their dimensions and   Table 4 are used to test the effectiveness of the advanced MOPSO. e testing results are compared against those obtained by the traditional NSGA-II, which is well known and widely used in the optimization dispatch and energy management of MGs. e spread of the solutions in the nondenominated fronts is measured by the variance of the nearest distance between neighbor solutions [36]. e spacing matric S p is introduced as a quantitative measure to evaluate the spread of the solutions to the multiobjective algorithm. S p is defined below.
where nis the number of solutions in Pareto optimization set, D ij is the Manhattan Distance defined in (27) and (28), and d represents the average value of all d i . e values of S p is lower when the solutions in the Pareto frontier set spread more even.
Based on the prior research [37], the advanced MOPSO is run multiple times for the selection of parameters. Finally, in our test, the parameters in advanced MOPSO are set in Table 5. e parameters of NSGA-II are set following Ref. [38] as follows: population size is 100, maximum iteration is 500, crossover probability is 0.5, and mutation probability is 0.3. e advanced MOPSO and NSGA-II are run independently 200 times under the same conditions.
To display the difference of optimal Pareto frontier between the given method and the real one clearly, the best optimal Pareto frontiers of MOPSO and NSGA-II are shown in Figures 3-5.
Figures 3-5 display that the optimal Pareto frontiers of MOPSO for ZDT1, ZDT2, and ZDT3 are almost same as the real Pareto frontiers. Compared with NSGA-II, MOPSO obtains better solution set in both convergence and diversity. e optimal Pareto frontiers obtained by MOPSO are almost the same with the real Pareto frontiers, which are significantly superior to the traditional NSGA-II in both convergence and diversity.
is owes to the combination of the archive maintenance and Pareto selection. e external archive helps to store enough nondominated solutions, and the replacement strategy is able to seek the coincident and continuous dominate solutions. While in NSGA-II and the other optimization algorithms, the possible solutions are less likely to survive when they are crowding in multiobjective dimensions.
For the purpose of further comparison, the worst, best, mean, and standard deviations (Std.) of S p with MOPSO and NSGA-II for benchmark tests are shown in Table 6. Meantime, the average running times are also listed in Table 6.
e results show that MOPSO has better solutions of S p and less operating time than the traditional NSGA-II. is

Initialization of particle swarm
Calculate the target vector of each particle in the particle swarm Add the noninferior solution to the external archive Determine the initial optimal position (pbest) of each particle Select the global optimal position (gbest) of each particle Update the velocity and position of the particle Calculate the target vector of each particle and update the pbest of the particle Update and maintain the external archive and select their gbest for each particle Whether to meet the terminating conditions End The first particle is directly stored in the external archive The particle i does not join the external archive Marking and deleting all the particles dominated by i Deleted the worst individual of the density

Experiments and Analysis
e benchmark function tests demonstrate the effectiveness and advantages of the proposed MOPSO algorithm.
en, we apply this method to the constrained MOO problem in the optimization dispatch for the MG with the PV-battery-diesel system. e MG in Figure 1 can operate in both grid-connected and islanded modes (latitude: 31°51′S and longitude: 117°16′W). According to different operation modes and solar radiation intensity, four scenarios are investigated, as shown in Table 7. When the MG works in the grid-connected mode, the      When the MG works in the islanded mode, three objectives are considered in the Scenarios 3 and 4, as shown in Table 7.
e flexibility of load in the laboratory building varies over one day or one season and has a different value for each time period. In this paper, the flexibility of buildings for various time periods has been extracted as the typical days shown in Figure 6. In winter, a day with weak solar radiation (January 5, 2018) is chosen as Scenarios 2 and 4. When PV power is low, it is important to dispatch the energy from the main grid or other distributed generation. In summer, a day with strong solar radiation (July 28, 2018) is chosen as Scenarios 1 and 3. Under this situation, the PV power and load demand are both high, and the battery management is crucial for the dispatch optimization.

Pareto Frontiers.
Finally, in the experiment, parameters in advanced MOPSO are set as below: c 1 � c 2 � 2, linear decreasing inertial weight with w max � 0.9 and w min � 0.4, the population size of swarm N � 200, and the maximum number of iteration T � 600. With the lowest economic and environmental costs as the objective function when the MG operates in the grid-connected mode under Scenarios 1 and 2, the Pareto frontiers with the two objectives are calculated by using the MOPSO and NSGA-II, and the results are shown in Figures 7 and 8     When the MG operates in the islanded mode under Scenarios 3 and 4, there are three objective functions in optimization dispatch. e three objective functions are lowest economic cost, environmental cost, and LOLP. In the islanded mode, the diesel generator starts to work together with the PV power, supplying the load demand and recharging the Li ion battery pack. e Pareto frontiers with three objectives are calculated by using the MOPSO and NSGA-II, and the results are shown in Figures 9 and 10, respectively. ere are no comparison results offered by the traditional NSGA-II because it fails to get the Pareto frontiers of the three-object problems. Meantime, the MOPSO provides the Pareto frontiers as shown before. e average running times in the four scenarios are listed below. e superiority of the proposed method is obvious for comparison.
It is demonstrated that the proposed MOPSO algorithm has better performance in variable domains of economic and environmental cost.

Final Trade-Off Solution.
After the obtaining of the Pareto optimal solutions using the proposed methods, it comes to the stage of choosing one best compromise solution for real applications. When the MG works in the gridconnected mode, the fuzzy set [39] is introduced to handle the challenges of inaccurate nature for the MG system. e membership function η is defined below. e solution with the largest membership function is chosen as the final tradeoff solution: Each nondominated solution is a possible scheduling scheme of the MG. e final trade-off solution is selected according to the membership function formula (26). e solution with the largest membership can be considered as the final trade-off solution. e optimal dispatch plans of distributed generations and power grid corresponding to this optimal solution are shown in Figures 11  and 12 under Scenarios 1 and 2, respectively. Under Scenario 1, PV output focuses on the daytime period, especially from 9 to 15. Besides satisfying the system load, the excess power from the PV generation is not only charged to the lithium ion battery but also to the large power grid for the economic benefit.
Under the weak solar radiation in winter, the output of PV power generation is low. e load of the system is mainly provided by the output of the large power grid. In the evening, the load is relatively small, and the lithium ion battery is in the discharge state. e battery subsystem works together with the large power grid to meet the load demand. Under Scenario 2, PV power generation is insufficient to meet the load demand of the system, so the power grid is in the state of power generation. Diesel generators have high economic and environmental costs; thus, it does not work either in Scenario 1 or Scenario 2. e final solution of the proposed Pareto MOO problem using the MOPSO algorithm is reasonable. e economic and environmental costs are listed in Table 8. For comparison, the solutions of NSGA-II under these two scenarios are also listed in the same table.
e economic and environmental costs are (58.17$, 42.16$) and (176.64$, 84.14$) in Scenario 1 or Scenario 2 using the MOPSO algorithm, with the comparison of (58.21$, 43.38$) and (177.29$, 84.37$). It is shown that the proposed method is able to obtain better solution in the economic and environmental aspects than the traditional NSGA-II method.
When the MG is operated in an isolated mode, it is critical to consider the probability of load loss of the system. In these situations, the trade-off solution is chosen as the one of the minimum load loss probabilities. Although the corresponding MG system runs the maximum economic and environmental costs under the minimum load loss probability, the primary goal of the operation of the MG system should be to meet as many load requirements as possible, so the trade-off solution of the minimum load loss probability is still selected as the three target optimization operation of the MG system.
Under the Scenario 3, the optimal dispatch plans of distribution generations are shown in Figure 13. e diesel generator is used to meet the load demands in the islanded mode of MG. In summer under strong solar radiation, PV can provide the load in day time. Meanwhile, the lithium ion battery package charges power in this period. During 17 : 00-19 : 00, the diesel plays as a supplementary role in energy supply, and lithium ion battery package works in discharge state as well. In the evening, the output power of lithium ion battery package reaches its maximum. e optimized economic, environmental costs, and LOLP is (105.61$, 71.05$, 0.02).
Under the Scenario 4, the solar radiation is weak in the winter typical daily. e diesel is the main energy supply in the islanded MG system in day time, while lithium ion battery package discharges during 19 : 00-22 : 00. e optimal dispatch plan of the MG is illustrated in Figure 14, and the optimized economic, environmental costs, and LOLP is (188.34$, 142.95$, 0.01). Because of the long-term running of diesel, the environmental cost is highest among the four scenarios. On the contrary, with lower dependence on PV, the LOLP is down from 0.02 to 0.01.
ere is no comparison of three objectives optimization between NSGA-II, as it fails to find the Pareto solution. e economic and environmental costs during one day corresponding to the extreme solution and the trade-off solution under different optimization scenarios are shown in Table 8.
e average running times are listed as well.
From the simulation results presented in Table 8, the numerical results are better than the comparison algorithm but not by much. is is mainly because the simulation results are obtained for a small-scale MG system, and the cost savings are only for 1 day. Compared with NSGA-II, the economic and environmental costs can be further reduced. In the meantime, the running time of the proposed method is reduced to nearly 30% of the NSGA-II. It is beyond the request of day-ahead Pareto optimization dispatch. us, it is demonstrated that the proposed optimal method has superiority in practice.

Conclusion
In this paper, the modelling of a PV-diesel-battery MG system is accomplished with the precise economic and environmental parameters. e multiobjective Pareto optimization dispatch is constructed as a MOO problem. An advanced MPSO is developed on the basis of archive maintenance and Pareto selection. In this algorithm, the external archive helps to maintain the diversity of Pareto solutions. Four scenarios with various operation modes and illumination conditions are considered in the simulation experiment. e Pareto frontiers with two objects and three objects for the dispatch MOO problem are obtained by the proposed algorithm. e proposed algorithm shows the advantages on the nondominate Pareto frontiers and running time than the NSGA-II algorithm in benchmark tests and simulation experiment. e final trade-off solutions consisting of economic cost, environmental cost, and LOLP are obtained. It is proved that the model and algorithm researched in this paper are effective for the MOO dispatch of MG, which contain photovoltaic, diesel generator, Li ion battery, and electric vehicle charging points.
It is worthy pointing out that the stochastic process and power flow are not involved in this paper. In our future work, we will take the uncertainties of microgrids into account and introduce the stochastic process. e scenariobased stochastic optimal operation with load flow will be studied in the advance research. Actual sunlight intensity G STC : Sunlight intensity under the STC k pv : Power temperature coefficient Surface temperature of the PV array at time t T air (t): Ambient temperature T STC : Surface temperature of the PV array under STC T max : Maximum temperature of the day T min : Minimum temperature of the day t p : Moment of average temperature V w : Current wind speed C DE : Fuel cost of diesel generator

P DE (t):
Output power of diesel generator a, b, c: Coefficients of fuel cost function S load , S N : Capacity of load and service transformer SOC(t): Battery SOC at the time t I e (t): Charging/discharging currents of battery at the time t P bat (t): Output power of battery at the time t η ch : Battery charging efficiency η dis : Battery discharging efficiency C N : Nominal capacity of battery E 0 : Battery internal EMF K e , A, B: Fitting parameters of battery C max : Battery maximum capacity Q e : Battery discharged capacity f s (x): Probability density function of daily EV traveling distance μ s : Mean of f s (x) σ 2 s : Variance of f s (x) S: Daily traveling distance of EV W 100 : Power consumption of an EV traveling 100 kilometres P c : Charging power of single EV η C_EV : EV charging efficiency P EVi : Charging power of an EV P EVload (t): Total charge power of electric vehicles, t � 1, 2, 3, . . ., 24 N EV : Total number of adjustable electric vehicles F k (X i ): Value of fitness function for the k th subobjective with the particle X i α i (t): Standard coefficients of the environmental value for the i th -type pollutant β i (t): Standard coefficients of the pollution penalty for the i th -type pollutant Q i (t): Emission of the i th type of pollutant N P : Number of considered types of pollutant P tot (t): Total power supply by the MG at time t P load (t): Total load in the MG at time t P PV (t): Output power of the PV at time t P gen (t): Output power of the diesel generator at time t P bat (t): Output power of the Li ion battery pack at time t P i (t): Output power of the scheduled DG unit P i,min : e minimum output power of the scheduled DG unit P i,max : e maximum output power of the scheduled DG unit P grid,min : e minimum transmission power between MG and power grid P grid,max : e maximum transmission power between MG and power grid SOC min : e minimum allowable values of the SOC SOC max : e maximum allowable values of the SOC β: Load rate X 0 : Solution vector P s : Pareto optimal set P F : Pareto optimal front X i : e i th particle in MOPSO V i : Velocity of the i th particle in MOPSO Pbest i (t): e best position of the i th particle Gbest i (t): e best position of entire swarm c 1 , c 2 : e acceleration constants w: Inertia weight in MOPSO w max : e maximum value of w w min : e minimum value of w. D ij : Manhattan distance M: Number of the objectives in AMOPSO u k : Constant and chosen to make all u k F k close to each other N p : External archive N a : Size of current external archive hp(X i ): Number of particles whose global best position is X i |F|: Cardinality of the set F S p : Spacing matric f i : Membership function value of the i th Pareto solution f imin : e minimum value of the i th objective function f imax : e maximum value of the i th objective function.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.