Research Article  Open Access
Multiobjective Particle Swarm Optimization for Microgrids Pareto Optimization Dispatch
Abstract
Multiobjective optimization (MOO) dispatch for microgrids (MGs) can achieve many benefits, such as minimized operation cost, greenhouse gas emission reduction, and enhanced reliability of service. In this paper, a MG with the PVbatterydiesel system is introduced to establish its characteristic and economic models. Based on the models and three objectives, the constrained MOO problem is formulated. Then, an advanced multiobjective particle swarm optimization (MOPSO) algorithm is proposed to obtain Pareto optimization dispatch for MGs. The combination of archive maintenance and Pareto selection enables the MOPSO algorithm to maintain enough nondominated solutions and seek Pareto frontiers. The final tradeoff solutions are decided based on the fuzzy set. The benchmark function tests and simulation results demonstrate that the proposed MOPSO algorithm has better searching ability than nondominated sorting genetic algorithmII (NSGAII), which is widely used in generation dispatch for MGs. The proposed method can efficiently offer more Pareto solutions and find a tradeoff one to simultaneously achieve three benefits: minimized operation cost, reduced environmental cost, and maximized reliability of service.
1. Introduction
The 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 [1–3].
The 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. The MG can operate in both islanded [4–6] and gridconnected modes [7–11]. It is able to improve the stability and reliability of power supply in factory [8], residential districts [9–11], and military zone [12].
The 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. There are many approaches to optimize the MG operation. They are generally divided into single objective optimization (SOO) approaches [11, 13–16] and MOO approaches [17–31]. One certain object is concerned though the SOO method, such as energy cost [13, 14]and quality of service [16]. In [11], a hybrid pricebased demand response (HPDR) is proposed, in which the MG profit is maximized as a singleobjective function. However, the SOO approaches ignore certain benefits, which may be acceptable in some situations. The better solution can be achieved by the MOO approaches for more comprehensive consideration.
The 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 tradeoff 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 BangBig 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 tradeoff between the naturally different objectives. The Pareto frontier of the MOO dispatch for the MG system is calculated using various metaheuristic algorithms [24–31]. The 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 algorithmII (NSGAII) for the MOO is proposed to the design of the photovoltaicwind hybrid system. NSGAII is also utilized to the calculation of Pareto optimal fronts in [27].
The traditional PSO shows promising performance on singleobjective, nonlinear optimization and gains popularity in the optimal operation of MG. However, it is hard to deal with the multiobjects. The chaos optimization algorithm is introduced to the particle initialization in [20, 23]. This amelioration is useful to discrete space optimization. In [24], a fuzzy selfadaptive 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 decisionmaking approach, which adopts the fuzzy adaptive particle swarm optimization algorithm, is suggested to find the best compromise strategy of MG energy management in [25]. Three objective functions, namely, annualized cost of the system, loss of load expected, and loss of energy expected, are considered in [29]. The 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 windPVbattery 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.
This 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 tradeoff 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 NSGAII in Pareto frontier distribution and running time for the MG system under investigation in this study.
2. Modelling of PVBatteryDiesel MG System
The MG in this study is located in Anhui University, Hefei, Anhui, China, as shown in Figure 1. It is the MG with the PVdieselbattery system, which includes 400 V/100 kW MG and 400 V/20 kW MG. The MG mainly consists of the PV power system, the diesel generating unit, and the lithium ion (Li ion) battery pack as energy storage. The 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. Then, the MOO including two objectives and three objectives are carried out under different scenarios.
2.1. Model of PV Array
In the simplified PV steadystate power output model [31], the PV output is only related to solar radiation and ambient temperature:The value of is (°C), and the value of is 25°C.
2.2. Model of Diesel Generator
The 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.
2.3. Model of Li Ion Battery Pack
The 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
2.4. Model of Load considering PlugIn Electric Vehicles
The selected MG mainly supplies a laboratory building, 15 kW power electronic load, and ten electric vehicle (EV) charging piles shown in Figure 1. The daily EV traveling distance approximately satisfies with lognormal distribution, namely, . Its probability density function is expressed by
Thus, the SOC of the EV can be obtained from the daily traveling distance S, which lead to the charging duration as
The daily load of a certain number of the EV charging is obtained by the Monte Carlo method:
3. Problem Formulation
3.1. The First Objective Function: Operation Cost
The optimization objective contains energy consumption cost , the operation management cost , the depreciation cost of battery , and the energy interaction cost between the grid and MG . MG absorbed power from the power grid with the purchase price and output power to the power grid with the electricity price:where is a total number of DGs, is the energy output of the DG, and is the electricity exchange between the MG and power grid. , , and are the ECC, OMC, and BDC coefficients of the DG. is the electricity purchasing price when the MG absorbs energy from the power grid, which varies with the peak and valley periods. 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. These coefficients of the studied MG are summarized in Tables 1 and 2.


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

3.3. The Third 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 is a Boolean and is the quantity of power imbalance at time . The LOLP can be expressed as
The third optimization objective of the MG is defined as
3.4. Constraints of MG
The natural power generation constraints impact the flexibility of MG system and participate in generation units on reserve and energy markets [35]. The 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:
3.5. Constrained Multiobjective Optimization Problem
The 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
The objects and constraints are defined as (8)–(22).
4. 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 tradeoff between the objectives, namely, the Paretooptimal front.
4.1. Pareto Domination
Let be the function that assigns a criteria space point to each design space point , is the search space, where the vector is the set of all feasible solutions in , and the objective vector represents their measured values in . We consider real parameters and objectives in a constrained MOO problem to define a Pareto optimal front.
Definition 1. Let a solution vector be , is said to dominate a particle , if and only if . is said to be Pareto optimal if dominates , .
Definition 2. Pareto optimal set of all Pareto optimal decision vectors is defined as . All Pareto optimal values corresponding to the decision vectors in constitute the Pareto optimal front .
4.2. The Standard PSO
The 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 particle represented by its position in the ndimensional space is a potential solution to the problem. Within the search space, the particle flies under a velocity . In each flying step, say the step, the best position of the particle is , and the best position of the entire swarm is . At the step, the new position for the particle can be determined bywhere rand is a uniform random value in the range of [0, 1].
4.3. Advanced MOPSO
To achieve the Pareto front in this constrained MOO problem, there are two critical steps: archive maintenance and Pareto selection. The external archive is used to store some of the nondominated solutions produced in the searching process of the MOO algorithm.
The selection of is crucial in the PSO algorithm. In the singleobjective PSO, all the particles have the same. Thus, we can choose the best solution of fitness function as in SOO progress. However, in the MOO problem, several nondominate solutions will be obtained from the whole population in each evolution iteration. The archive maintenance aims to get diversified , while the best choice of 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 selection is proposed. The 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. The Manhattan distance is introduced to describe the distance between two solutions and as
The combination procedure of archive maintenance and selection is analysed in three different conditions. For a new nondominate solution , the following conditions apply:(i)The first possibility is dominates some solutions in external archive , then takes the place of dominated solutions and will be chosen as .(ii)Otherwise, if , where is the size of current external archive while is its maximum size, remove a best member with minimum crowding measure under (24) and (25). If , substitute for the global best position of all particles in .(iii)If ,(a)For all the , set .(b), , is a constant, and . .(c)Select a solution with minimum crowding measure from and replace as . , , and .(d)If and , turn to (c). If and , turn to (b). If , insert into external archive. End if.
The cardinality of the set is denoted by . The flowchart of the advanced MOPSO is demonstrated in Figure 2. The 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.
4.4. Benchmark Function Test
Three representative benchmark functions associated with their dimensions and bounds of the variables shown in Table 4 are used to test the effectiveness of the advanced MOPSO. The testing results are compared against those obtained by the traditional NSGAII, which is well known and widely used in the optimization dispatch and energy management of MGs. The spread of the solutions in the nondenominated fronts is measured by the variance of the nearest distance between neighbor solutions [36]. The spacing matric is introduced as a quantitative measure to evaluate the spread of the solutions to the multiobjective algorithm. is defined below.where is the number of solutions in Pareto optimization set, is the Manhattan Distance defined in (27) and (28), and represents the average value of all . The values of 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. The parameters of NSGAII 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. The advanced MOPSO and NSGAII 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 NSGAII 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 NSGAII, MOPSO obtains better solution set in both convergence and diversity. The optimal Pareto frontiers obtained by MOPSO are almost the same with the real Pareto frontiers, which are significantly superior to the traditional NSGAII in both convergence and diversity. This owes to the combination of the archive maintenance and Pareto selection. The external archive helps to store enough nondominated solutions, and the replacement strategy is able to seek the coincident and continuous dominate solutions. While in NSGAII 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 with MOPSO and NSGAII for benchmark tests are shown in Table 6. Meantime, the average running times are also listed in Table 6.

The results show that MOPSO has better solutions of and less operating time than the traditional NSGAII. This indicates better performance of the proposed algorithm in the aspect of searching of nondominated front.
5. Experiments and Analysis
The benchmark function tests demonstrate the effectiveness and advantages of the proposed MOPSO algorithm. Then, we apply this method to the constrained MOO problem in the optimization dispatch for the MG with the PVbatterydiesel system. The MG in Figure 1 can operate in both gridconnected 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 gridconnected mode, the third objective of the LOLP is neglected, and only the first two objectives are considered in the Scenarios 1 and 2. When the MG works in the islanded mode, three objectives are considered in the Scenarios 3 and 4, as shown in Table 7.

The 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.
5.1. Pareto Frontiers
Finally, in the experiment, parameters in advanced MOPSO are set as below: , linear decreasing inertial weight with and , the population size of swarm , and the maximum number of iteration . With the lowest economic and environmental costs as the objective function when the MG operates in the gridconnected mode under Scenarios 1 and 2, the Pareto frontiers with the two objectives are calculated by using the MOPSO and NSGAII, and the results are shown in Figures 7 and 8, respectively.
For the constrained MOO problem in the optimization dispatch for the MG, NSGAII algorithm obtains 14 Pareto solutions with the following spread: (55.3$, 48.5$) and (71.8$, 26.9$). However, the advanced MOPSO algorithm obtains 67 solutions spreading in the range of (54.22$, 48.68$) and (72.29$, 26.03$), as shown in Figure 7. In Figure 8, it is shown that the traditional NSGAII method only finds 19 solutions in Scenario 2, while the MOPSO method gets 118 possible solutions in Pareto frontier sets. The advanced MOPSO algorithm gives a value as low as 174.3$ in economic cost and 73.9$ in environmental cost. For comparison, the traditional NSGAII method gives a value as low as 175.3$ in economic cost and 75.3$ in environmental cost.
When the MG operates in the islanded mode under Scenarios 3 and 4, there are three objective functions in optimization dispatch. The 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. The Pareto frontiers with three objectives are calculated by using the MOPSO and NSGAII, and the results are shown in Figures 9 and 10, respectively. There are no comparison results offered by the traditional NSGAII because it fails to get the Pareto frontiers of the threeobject problems. Meantime, the MOPSO provides the Pareto frontiers as shown before. The average running times in the four scenarios are listed below. The 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.
5.2. Final TradeOff 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. The membership function is defined below. The solution with the largest membership function is chosen as the final tradeoff solution:
Each nondominated solution is a possible scheduling scheme of the MG. The final tradeoff solution is selected according to the membership function formula (26).
The solution with the largest membership can be considered as the final tradeoff solution. The 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. The 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. The 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. The final solution of the proposed Pareto MOO problem using the MOPSO algorithm is reasonable. The economic and environmental costs are listed in Table 8. For comparison, the solutions of NSGAII under these two scenarios are also listed in the same table. The 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 NSGAII 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 tradeoff 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 tradeoff 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. The 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. The 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. The 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. The 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 longterm 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.
There is no comparison of three objectives optimization between NSGAII, as it fails to find the Pareto solution. The economic and environmental costs during one day corresponding to the extreme solution and the tradeoff solution under different optimization scenarios are shown in Table 8. The 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. This is mainly because the simulation results are obtained for a smallscale MG system, and the cost savings are only for 1 day. Compared with NSGAII, 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 NSGAII. It is beyond the request of dayahead Pareto optimization dispatch. Thus, it is demonstrated that the proposed optimal method has superiority in practice.
6. Conclusion
In this paper, the modelling of a PVdieselbattery MG system is accomplished with the precise economic and environmental parameters. The 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. The Pareto frontiers with two objects and three objects for the dispatch MOO problem are obtained by the proposed algorithm. The proposed algorithm shows the advantages on the nondominate Pareto frontiers and running time than the NSGAII algorithm in benchmark tests and simulation experiment. The final tradeoff 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. The scenariobased stochastic optimal operation with load flow will be studied in the advance research.
Nomenclature
:  PV array derating factor 
:  Maximum output power under standard test conditions (STCs) 
:  Actual sunlight intensity 
:  Sunlight intensity under the STC 
:  Power temperature coefficient 
:  Surface temperature of the PV array at time 
:  Ambient temperature 
:  Surface temperature of the PV array under STC 
:  Maximum temperature of the day 
:  Minimum temperature of the day 
:  Moment of average temperature 
:  Current wind speed 
:  Fuel cost of diesel generator 
:  Output power of diesel generator 
a, b, c:  Coefficients of fuel cost function 
:  Capacity of load and service transformer 
:  Battery SOC at the time t 
:  Charging/discharging currents of battery at the time t 
:  Output power of battery at the time t 
:  Battery charging efficiency 
:  Battery discharging efficiency 
:  Nominal capacity of battery 
:  Battery internal EMF 
, , :  Fitting parameters of battery 
:  Battery maximum capacity 
:  Battery discharged capacity 
:  Probability density function of daily EV traveling distance 
:  Mean of 
:  Variance of 
S:  Daily traveling distance of EV 
:  Power consumption of an EV traveling 100 kilometres 
:  Charging power of single EV 
:  EV charging efficiency 
:  Charging power of an EV 
:  Total charge power of electric vehicles, t = 1, 2, 3, …, 24 
:  Total number of adjustable electric vehicles 
:  Value of fitness function for the subobjective with the particle 
:  Standard coefficients of the environmental value for the type pollutant 
:  Standard coefficients of the pollution penalty for the type pollutant 
:  Emission of the type of pollutant 
:  Number of considered types of pollutant 
:  Total power supply by the MG at time 
:  Total load in the MG at time 
:  Output power of the PV at time 
:  Output power of the diesel generator at time 
:  Output power of the Li ion battery pack at time 
:  Output power of the scheduled DG unit 
:  The minimum output power of the scheduled DG unit 
:  The maximum output power of the scheduled DG unit 
:  The minimum transmission power between MG and power grid 
:  The maximum transmission power between MG and power grid 
:  The minimum allowable values of the SOC 
:  The maximum allowable values of the SOC 
:  Load rate 
:  Solution vector 
:  Pareto optimal set 
:  Pareto optimal front 
:  The particle in MOPSO 
:  Velocity of the particle in MOPSO 
:  The best position of the particle 
:  The best position of entire swarm 
, :  The acceleration constants 
:  Inertia weight in MOPSO 
:  The maximum value of 
:  The minimum value of . 
:  Manhattan distance 
:  Number of the objectives in AMOPSO 
:  Constant and chosen to make all close to each other 
:  External archive 
:  Size of current external archive 
:  Number of particles whose global best position is 
:  Cardinality of the set 
:  Spacing matric 
:  Membership function value of the Pareto solution 
:  The minimum value of the objective function 
:  The maximum value of the objective function. 
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Key R&D Program of China under 2016YFB0900400, National Natural Science Foundations (NNSF) of China under Grants 51507001, 51637001, and 61673001, and Doctoral research foundation of Anhui University under Grant J01001929.
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