Optimal planning of inverterbased renewable energy sources towards autonomous microgrids accommodating electric vehicle charging stations
Abstract
Renewable energy sources have recently been integrated into microgrids that are in turn connected to electric vehicle (EV) charging stations. In this regard, the optimal planning of microgrids is challenging with such uncertain generation and stochastic charging/discharging EV models. To achieve such ambitious goals, the best sites and sizes of photovoltaic and wind energy units in microgrids with EV are accurately determined in this work using an optimization technique. This proposed technique considers 1) generation profile uncertainty in photovoltaic and wind energy units as well as the total load demand, 2) photovoltaic and wind generation units' DSTATCOM operation capability, and 3) various branch and node constraints in the microgrid. Most importantly, the possible EV requirements are also taken into account, including initial and predetermined state of charge (SOC) arrangements, arrival and departure hours, and diverse regulated and unregulated charging strategies. A bilevel metaheuristicbased solution is established to address this complex planning model. The outer level and innerlevel functions optimize renewable energy sources and EV decision variables. Subobjectives to be optimized voltage deviations as well as grid power. The results demonstrate the effectiveness of the introduced method for planning renewable energy sources and managing EV to effectively achieve autonomous microgrids.
1 INTRODUCTION
Renewable energy sources (RESs) are being more widely used around the world. Global policies to minimize greenhouse gas emissions drive this trend, but innovations in future electrical power generation technologies are anticipated [13]. Due to their elasticity and costeffectiveness, photovoltaic (PV) and wind turbine (WT) are the most prominent RES variants. Remarkably, RESs have the potential to have a positive impact on distribution system efficiency. Such PV and WT systems, in particular, could improve supply efficiency, solve voltage problems, reduce power losses, improve power quality, and reduce the loading on traditional controlling units, thanks to their smart functions [4, 5]. However, the PV and WT units' extremely intermittent generation could cause a slew of functional and operational issues, limiting their authorized accommodating capacities in the grid [68]. In line with the rising popularity of PV and WT, interest in electric vehicles (EVs) has been growing rapidly around the world [9, 10].
There is a growing interest in the recent literature for determining the sites and sizes of various RESs in distribution systems to enhance various grid performance indices. The authors of [11] have proposed a twostage datadriven robust optimization model (for placement RESs, considering both load and generation uncertainties to minimize the total installation and operational costs. In [12], a probabilistic model has been introduced to allocate different RESs in distribution systems to maximize energy loss mitigation while still meeting system limits. A novel fast yet accurate approach for optimally sizing PV in distribution systems based on machine learning has been proposed in [13] to minimize energy losses. Different analytically based approaches are proposed in [14, 15] that can deliver fast results in terms of the optimal sizes and sites of multiple PV and WT systems by considering expected load/generation conditions. In [16], a multiobjective method for efficiently planning PV and WT systems in distribution networks has been proposed, taking into account their probabilistic frameworks while minimizing emissions and total costs. Wide varieties have been used for RES planning in distribution systems, particularly for multiobjective structures, resulting from recent revolutions in implementing and sustaining effective metaheuristic optimization problem solvers. Crow search algorithm autodrive particle swarm optimization method [17], gravitational search algorithm [18, 19], tabu search optimization solver [20], geneticbased optimization method [21], artificial ecosystembased optimization method [22], equilibrium optimizer [23], simulated annealing optimization method [24], and ant colony optimization method [25] are some examples of optimization solvers. The authors of [26] have proposed an adaptive robust cooptimization method for capacity allocation and bidding approach of a prosumer interconnected with PV, WT, and a battery energy storage system. The importance of energy storage systems and EVs in increasing RES hosting flexibility has been investigated in [27]. In [28], a new planning model for windbased DGs and fastcharging stations has been proposed considering residential loads and renewable power generation.
In [29], the longterm capacity expansion planning model is investigated in microgrids interconnected to EVs and various RES types. The authors of [30] have examined the generation planning problem in island microgrids with RES to reduce environmental impact. In [31], a technique for optimum design of a DC microgrid for electric vehicle supply infrastructure has been proposed with considering various converter types and topologies. In [32], a twolevel problem is presented for isolated microgrids with EVs. The authors of [33] have proposed a method for planning microgrids with EV charging demand to determine the most economical configuration for maximizing RES utilization. In [34], a comprehensive framework for optimally planning and operating EV batteries has been proposed for microgrids. In [35], a study for the impacts of EV integration approaches has been performed on utility grids' operation. Accordingly, most of the suggested work is based on assumptions to simplify the RES with the EVs planning model. Some of these approaches assume a single RES allocation or use deterministic RES and load models, but they do not take EVs into account. Even techniques that take into account the intermittent and unpredictable nature of RES and loads ignore the presence of EVs. The present techniques in the RES allocation problem do not take into account the various control systems of EVs units, their stochastic character, or their detailed model. As a result, further research and development are still needed to solve this RES allocation model in microgrids.
As demonstrated above, with the uncertain generation of photovoltaic and wind energy systems and stochastic charging/discharging EV models, microgrids' economic and optimal operation is problematic. In this work, the optimal locations and sizes of photovoltaic and wind energy units in microgrids with EV charging stations are precisely calculated using an optimization technique to achieve autonomous microgrids. The novelty of the suggested approach includes taking into account RES and load uncertainty, as well as the stochastic character of EVs. Besides, the optimization model considers EV operation restrictions such as various charging control schemes of EVs, including controlled and uncontrolled charging strategies. This technique takes into account generation profile intermittency in photovoltaic and wind energy systems, as well as overall load demand, DSTATCOM operating ability of inverterbased photovoltaic and wind generation stations, and various branch and node constraints in the microgrid. EV conditions, such as initial and predetermined state of charge (SOC) arrangements, arrival and departure hours, and various controlled and uncontrolled charging strategies, are also considered. To solve this multipart planning model accurately, we established a bilevel metaheuristicbased optimization approach. Renewable energy sources and EV decision variables are optimized using the outer level (main problem) and inner level (subproblem) functions, respectively. Total voltage deviations, as well as grid energy, are treated as subobjectives to be optimized. A series of tests and case studies are used to assess the viability of the proposed solution. The conclusions show that the proposed approach for planning renewable energy sources and controlling EVs effectively achieves autonomous microgrids with photovoltaic, wind, and EV charging stations.
2 PROPOSED RES PLANNING MODEL
The planning model for planning PV and WT units in microgrids integrated with EV charging stations is described in this section. Owing to the incorporation of PV and WT in the microgrid, the tieline control between the utility network and the microgrid may have high volatility, which may trigger various technical issues such as voltage fluctuations and severe voltage deviations. Besides, reducing tieline control reduces the microgrid's autonomy. As a result, the tieline power and energy deficits are used here as subobjectives in the planning problem to be reduced. Below, we describe in detail the panning model of PV and WT units considering the various constraints.
2.1 Objective function
2.1.1 Voltage deviations
2.1.2 Tieline power
in which ${P}_{TL,s}^{t}$ represents the amount of tieline power fed by the main grid to the microgrid.
2.1.3 Constraints
3 MULTIOBJECTIVE GREY WOLF OPTIMIZER (MOGWO)
3.1 Optimization algorithm
The grey wolf optimization (GWO) algorithm is a new metaheuristic algorithm and it was developed in [39]. The main inspiration of this algorithm is the social leadership and hunting technique of grey wolves. The GWO is firstly developed to solve a singleobjective optimization problem. The main operators needed for the reproduction process include social hierarchy, encircling prey, hunting and attacking prey, and searching for prey.
The GWO algorithm's optimization starts by creating a set of random solutions as the first population. The three best solutions which are obtained during the optimization should be saved as α, β, and δ solutions. The other search agents (ω wolves) are updating their positions according to Equations (28)–(34). In the meantime, a and A parameters are decreased in a linear manner over the iteration. Hence, the search agents diverge from the victim when $\stackrel{\u20d7}{A}$ is greater than one while they tend to converge towards the victim $\stackrel{\u20d7}{A}$ is less than one. In the end, the score and position of α solution are returned as the best solutions obtained throughout optimization when an end condition is satisfied.
To perform multiobjective optimization based on GWO, the MultiObjective Grey Wolf Optimizer (MOGWO) algorithm was proposed and utilized in [40]. The process is like its single objective, as mentioned above. Nevertheless, to employ this algorithm for solving a multiobjective optimization problem, the concept of dominance is utilized. A Pareto archive is responsible for storing and updating a set of nondominated Pareto optimal solutions obtained so far. A leader selection strategy is used as the second component to assist in choosing α, β, and δ solutions as the archive's hunting process leaders. The archive is normally set to have a limited size due to computer memory usage. In MOGWO, the adaptive grid (or hypercube segments) algorithm is used to deal with this task. This algorithm should work in which the generated adaptive grids can cover all the current nondominated solutions. In the most crowded hypercube, the members would be selected randomly and removed from the archive until the number of residual members is equal to the maximum archive size. For choosing α, β, and δ, a roulette wheel is applied in MOGWO to select a segment in which its probability is inversely proportional to its number of members. Three members in the selected segment will be assigned as the leaders. Suppose the segment contains members less than the three leaders. In that case, the roulette wheel selection will be executed for selecting another segment, and the process is repeated until having the three leaders. The MOGWO starting with the initial population and Pareto archive. Then, the roulette wheel and adaptive grid operators are used to select the three leaders. After that, all the search area agents are updated based on Equations (28)–(34). The nondominated solutions sorted from the combination of members in the previous archive and the new agents’ positions are used to update the Pareto archive. The three leaders α, β, and δ are then updated, and the next generation is generated. This process is repeated until a termination criterion is met. It is important to mention that this work's focus is to build a planning model for inverterbased RES not to improve the MOGWO. Consequently, any other optimization technique can be used for solving this RES planning model. The MOGWO has been employed due to its superior performance in wide applications to get global solutions [40].
3.2 NonDominated sort technique
3.3 Best compromise solution
4 SOLUTION PROCESS
The proposed methodology for optimal planning of inverterbased RES in the microgrids accommodating charging stations of EVs is described in Figure 3. This figure shows that the process has three main phases. In the first phase, reading and preparing the input data, the microgrid data, and the historical dataset of the solar irradiance, wind speed, and load demand have been read. These data are employed to estimate the probabilistic models of the RES and load demand. Furthermore, this phase defines the zones in the microgrid, constraints of the microgrid (e.g. RES constraints, EVs constraints, diesel constraints, and constraints of the inverters), and the EV charging stations' location and charging schemes of the EVs.
In the second phase, developed bilevel optimization, an optimization model is established to solve the planning problem. This model consists of two levels called outer level (main problem) and inner level (subproblem). MOGWO is employed in the two levels to solve the main and subproblems of the model. The outer level suggests optimal sizes and locations of the RES and passes them to the inner level. Based on these locations and sizes of the RES, the inner level analysis of the microgrid for 24h using a power flow solver along with the optimizer to determine the optimal charging/discharging power of the EVs, optimal power of the diesel, and optimal reactive power of the interfacing inverters of the RES. Therefore, the inner level calculates the objective functions (tieline power and VD) for each state according to their occurrence probability. The total sum of each objective function is utilized as an objective function for the outer level. This indicates that the outer level includes the inner level. Hence, for each iteration of the outer level, the inner level should be completely executed until its convergence.
In the end, the optimal sizes and locations of the RES, optimal charging/discharging power of the EVs, optimal reactive power of the RES inverters, and optimal power of the diesel generator are shown in the third phase. It is important to note that the capital costs and maintenance expenses of RES are not considered in the planning model. Differently, we show the benefits of EVs charging and reactive power capabilities of the interfacing RES inverters.
5 RESULTS AND DISCUSSIONS
5.1 Microgrid and dataset
The IEEE 69bus distribution system is chosen as a microgrid to examine the proposed methodology. The complete line and bus data of the distribution system are obtained from [44, 45]. However, the proposed model in Section 2 is generally formulated to be applied for any other distribution test systems. In this work, this microgrid is split into six different zones (Zone 1, Zone 2,…, Zone 6) as shown in Figure 4. The PV units can be allocated in Zone 1, Zone 2, and Zone 3 while the WT can be allocated in Zone 4, Zone 5, and Zone 6. The microgrid accommodates four charging stations connected to buses 33, 36, 52, and 65 as depicted in Figure 4. The maximum capacity of each charging station is sixty EVs. The arriving times of these EVs during a day follow Figure 5, while the initial SOC for each EV at that time is depicted in Figure 6.
Tesla Model S batteries are used for EVs. The capacity of the utilized battery is 85 kWh [46]. According to the daily nature of the EV’ owners, the EVs normally leave the charging station in the morning and back home after working hours. Therefore, it is expected that each EV can connect to the microgrid for 12 h (i.e. 7:00 AM06:00 PM). Vehicletogrid (V2G) technology can be employed during the connecting of the EVs to the microgrid with guaranteeing enough SOC for daily trips at the departure time of each EV. The maximum charging and discharging rate of each EV is 20% of the battery capacity ($0.2\phantom{\rule{0.28em}{0ex}}\times \phantom{\rule{0.28em}{0ex}}85$) [47]. Two different types of RES are optimally planned in the microgrid for minimizing the total voltage deviation and the tieline power between the microgrid and the utility.
Threeyears historical datasets of solar irradiance, wind speed, and load demand are used for calculating their probability distribution functions. Here, a day within the three years is used to represent by these years in which it is divided into the 24h time period. Considering that each year has 365 days, hence, each time period has 1095 data points for solar irradiance, wind speed, and load demand (3 years × 365 days per year). By utilizing these data, at each time period, the mean and standard deviation can be determined, and so the Beta, Weibull, and Normal probability density functions can be generated for each time period. The historical data of solar irradiance, wind speed, and load demand are taken from [48, 49], and [50], respectively. Two different situations are followed for the planning of the microgrid. In the first situation, the RES (PV and WT) are optimally planned the microgrids with applicability for transferring an amount of the power between the microgrid and the utility. In contrast, the second situation the planning of the RES is done considering autonomous microgrid in which the tieline power between the microgrid and the utility is zero. The proposed methodology is performed in the cases of disabling and enabling the reactive power capability of the interfacing inverters of the RES. We have written the code of the optimization problem (MOGWO optimization algorithm and the planning model of the inverterbased RES) in MATLAB 2017b, and this program has been carried out on a Core I5 PC with 8GB RAM. The maximum iterations and populations of the optimizer for both levels are 100 and 50, respectively.
5.2 Case studies
 Case 1: In this case, the RES are optimally planned in the microgrid without considering neither reactive power support from the RES inverters nor EVs.
 Case 2: This case considers only the EV charging/discharging.
 Case 3: In this case, the EV charging/discharging, and the reactive power support of the RES inverters are considered.
 Case 4: This case is like the previous case (Case 3) while a diesel generator at bus 6 is considered here.
In this work, we have carried out ten runs for each simulated case study while the best solution among them is chosen as the final solution. Further, it is assumed that all EVs are equipped with Tesla Model S, while the proposed model can adopt any other battery models.
5.3 Analysis of the cases
In Cases 1 and 2, the inverters of the RES are working at unity power factor while in Cases 3 and 4, the RES inverters have the DSTATCOM functionality in which they can inject/absorb the reactive power to/from the microgrid based on the amount of generated active power by the RES. Furthermore, the effect of EVs charging/discharging on the planning problem is considered in Cases 2, 3, and 4 while the EVs are considered to charge with a fixed rate of 20% of the total capacity regardless of the system state is considered in Case 1.
The set of dominant points (f_{1} and f_{2}) for the four different cases are depicted in Figures 710. The optimal sites and sizes of the PV units and WT units which are computed by employing the different cases along with the values of the corresponding average voltage deviation and average tieline power are given in Table 1. Figures 710 and Table 1 show that Case 4 gives better results compared to the other cases in terms of voltage magnitude deviation and tieline power in which they are significantly decreased. For instance, the voltage deviation reductions in Case 2, Case 3, and Case 4 compared to Case 1 are 1.89%, 40%, and 57%, respectively. On the other hand, the values of the tieline power are decreased by 5%, 73%, and 81% in the case of Case 2, Case 3, and Case 4, respectively, compared to the Case1. It can be noted that the optimal locations and sizes of the RES differ according to the applied case as illustrated in Table 1. Figure 11 shows the Paretooptimal solutions obtained in Case 4 including all solutions (dominated and nondominated solutions). Multiobjective ant lion optimizer (MOALO) is employed to solve the proposed approach and compared with MOGWO in Figure 12. This figure shows the superiority of MOGWO over MOALO to get best optimal solutions.
Item  Case 1  Case 2  Case 3  Case 4  

PV locations  47  61  66  47  63  66  50  62  66  48  61  67 
WT locations  14  32  40  12  29  40  20  34  45  23  29  42 
PV sizes (MW)  0.67  2.61  0.10  0.78  1.98  0.76  0.10  2.21  0.40  0.20  2.01  0.80 
WT sizes (MW)  1.97  1.54  0.58  2.44  1.42  0.75  0.81  0.92  2.24  0.82  1.00  0.27 
Average voltage deviation (pu)  0.53  0.52  0.32  0.23  
Voltage deviation reduction (%)  —  1.89  40  57  
Average tieline power (MVA)  2.61  2.48  0.71  0.50  
Reduction of tieline power (%)  —  5  73  81 
The hourly active tieline power between the utility and the microgrid and the voltage magnitude deviation are displayed in Figures 13 and 14, respectively. These figures show that enabling the reactive power capability of the RES inverters can significantly reduce the tieline power and the voltage magnitude deviation as in Case 3 and Case 4. However, the hourly tieline power and the voltage magnitude deviation in Case 4 are lower than those in the other cases. This is thanks to simultaneous optimization of reactive power of the interfacing inverters, EV charging/ discharging, and the active power of the diesel generator. The optimal reactive powers of the RES inverters injected/absorbed to/from the microgrid for Case 3 and Case 4 are given in Figures 15 and 16, respectively. It is worth mentioning that the positive values mean that the RES inverters inject the reactive powers to the microgrid, while the negative values indicate that the inverters absorb the reactive powers from the microgrid. On the other hand, the optimal hourly active power of the diesel generator which is employed in Case 4 is depicted in Figure 17. The optimized reactive powers of RES inverters and active power of the diesel generator along with the charging/discharging power of the EVs can greatly contribute for minimizing the objective functions.
Regarding the charging/discharging power of the EVs, Figures 18 and 19 show the charging power and SOC of the EVs in Case 1. In this case, EVs are considered to charge once they arrived at the charging station with a fixed rate of 20% of the total capacity regardless of the system. To avoid the repetition of the result, only the charging/discharging powers and SOC of the EVs connected to the charging station at bus 36 are depicted here. It is important to mention that, in this work, the EVs that arrive at the end of the previous day have been considered, in which they have to continue charging at the beginning of the next day. Figure 19 shows that the SOC of all EVs at departure time is 100%. The charging and discharging powers for Cases 2, 3, and 4, in which the V2G technology is employed, are shown in Figure 20. The positive values indicate that the EVs are charging from the microgrid while the negative powers indicate that they are discharging to the microgrid. Figure 21 illustrates the SOC of the EVs at the charging station connected to bus 36 for Case 4. The SOC in Cases 2 and 3 follow the same trend. Therefore, they have not been shown here. It is clear that the EVs can charge and discharge to the microgrid (i.e. the SOC are increasing and decreasing) while the SOC of all EVs at parting time is high enough for daily trips.
5.4 Application to autonomous microgrid
In this subsection, the proposed methodology is applied for the autonomous operation of the microgrid. For this purpose, Case 4 is applied here with giving the full priority to the objective function of the tieline power between the microgrid and the utility. The optimal locations, optimal sizes, average voltage magnitude deviation, and average tieline power by employing this case are given in Table 2. This table shows that the average tieline power is zero. This indicates that the microgrid is autonomous in which the load demand in the microgrid is fed only by the RES and diesel generator. It is important to note that the optimal locations and sizes of PV units and WT units are changed compared to Case 4 in the case of the gridconnected microgrid. For instance, the optimal locations of the PV units are buses 50, 54, and 66, while the optimal locations of the WT units are buses 26, 28, and 44. On the other hand, the optimal sizes of the PV units are 0.36, 0.54, and 0.16 MW, respectively. The optimal sizes of the WT units are 0.85, 0.11, and 3.55 MW, respectively. The hourly voltage magnitude deviation and tieline power are shown in Figure 22, while the active power of the diesel generator is depicted in Figure 23. Moreover, the optimal reactive powers of the interfacing inverters are displayed in Figure 24.
Item  Case 5  

PV locations  50  54  66 
WT locations  26  28  44 
PV sizes (MW)  0.36  0.54  0.16 
WT sizes (MW)  0.85  0.11  3.55 
Average voltage deviation (pu)  1.26  
Average tieline power (MVA)  0.0 
6 CONCLUSIONS
The optimal planning of microgrids is challenging with the uncertain generation of PV and WT systems and stochastic EV models. To address these issues, the best locations and sizes of photovoltaic and wind energy units in microgrids with EV charging stations are accurately calculated in this study using an optimization methodology to achieve autonomous microgrids. In particular, this introduced method considers generation profile intermittency in PV and wind energy systems, total load requirement, DSTATCOM operational capacity of inverterbased photovoltaic and wind generation stations, and microgrid branch and node restrictions. EV requirements are also taken into account, including initial and predetermined SOC schedules, arrival and departure hours, and various regulated and unregulated charging techniques. Accordingly, we have developed a bilevel metaheuristicbased optimization approach to efficiently deliver accurate solutions for this multipart planning model. The outer level (main problem) and inner level (subproblem) functions are used to maximize renewable energy sources and EV decision variables, respectively. Total voltage deviations and grid energy are considered subobjectives that must be optimized. The feasibility of the potential solution is determined by a sequence of experiments and case studies. The voltage deviation and the tieline power reductions by the proposed method are 57% and 81%, respectively, which is higher than those of the existing approaches. The findings indicate that the suggested method for designing renewable energy sources and monitoring EVs is successful in achieving autonomous microgrids with EV charging stations interconnected to photovoltaic and wind generation units. In this work, we have not considered the microgrid islanding mode, and will be considered in a future study.
Nomenclature

 ${\stackrel{\u20d7}{X}}_{p}$

 position vector of the victim

 $\stackrel{\u20d7}{{r}_{1}}$, $\stackrel{\u20d7}{{r}_{2}}$

 random vectors in [0,1]

 $\stackrel{\u20d7}{A},\phantom{\rule{0.28em}{0ex}}\stackrel{\u20d7}{C}$

 coefficient vectors

 ${C}_{PV,i}$

 capacity of PV unit

 ${C}_{PV,i}^{max}$

 highest capacity of PV unit at bus i

 ${C}_{PV,i}^{min}$

 lowest capacity of PV unit at bus i

 ${C}_{WT,i}$

 capacity of WT unit

 ${C}_{WT,i}^{max}$

 highest capacity of WT unit at bus i

 ${C}_{WT,i}^{min}$

 lowest capacity of WT unit at bus i

 ${I}_{ij,s}^{t}$

 the current flows in line ij

 ${I}_{ij}^{max}$

 maximum allowed current rate

 ${N}_{WTi}^{max}$

 maximum allowed number of WT units at bus i

 ${N}_{WTi}^{min}$

 minimum allowed number of WT units at bus i

 ${P}_{C,j,s}^{t}$

 charging station power at bus j for state s

 ${P}_{D,i,s}^{t}$

 load demand at bus i for state s during segment t

 ${P}_{Diesel,i,s}^{t}$

 active power output of the diesel generator

 $P{M}^{t}({\lambda}_{s})$

 amalgamated probability model of the solar irradiance, wind speed, and demand load

 ${P}_{PV,i,s}^{t}$

 power output of PV unit

 ${P}_{TL,s}^{t}$

 the amount of tieline power fed by the main grid

 ${P}_{WT,i,s}^{t}$

 power output of WT unit

 ${P}_{c,i,s}^{max,t}$

 maximum charging station power

 ${P}_{c,i,s}^{min,t}$

 minimum charging station power

 ${Q}_{D,i,s}^{t}$

 reactive power demand at bus i for state s

 ${Q}_{Diesel,i,s}^{max,t}$

 highest reactive power of diesel generator at bus i

 ${Q}_{Diesel,i,s}^{min,t}$

 lowest reactive power of diesel generator at bus i

 ${Q}_{I,PV,i,s}^{max}$

 highest reactive power of PV at bus i

 ${Q}_{I,PV,i,s}^{min}$

 lowest reactive power of PV at bus i

 ${Q}_{I,PV,i,s}^{t}$

 reactive power of the PV interfacing inverter

 ${Q}_{I,WT,i,s}^{max}$

 highest reactive power of WT at bus i

 ${Q}_{I,WT,i,s}^{min}$

 lowest reactive power of WT at bus i

 ${Q}_{I,WT,i,s}^{t}$

 reactive power of the WT interfacing inverter

 ${R}_{PV}^{max}$

 maximum total PV power

 ${R}_{PV}^{max}$

 maximum total WT power

 $V{D}_{s}^{t}$

 total voltage deviations of distribution system for the state s during time segment t

 $\stackrel{\u20d7}{X}$

 position vector of a wolf

 $\stackrel{\u20d7}{a}$

 an element that linearly decreased from 2 to 0 over the whole iterations

 ${f}_{i}^{max},{f}_{i}^{min}$

 maximum and minimum values of i^{th} objective function

 $pro{b}_{R}^{t}({G}_{x})$

 solar irradiance probability of state x

 $pro{b}_{l}^{t}({G}_{z})$

 load demand probability of state z

 $pro{b}_{w}^{t}({G}_{y})$

 wind speed probability of state y

 ${\delta}_{ij,s}^{t}$

 voltage angles variance at i^{th} and j^{th} nodes

 ${\mu}_{i}$

 membership function

 ${\varphi}_{b}$

 set of the system buses

 B_{ij}

 susceptance of line ij

 G_{ij}

 conductance of line ij

 N_{B}

 number of nodes

 N_{nd}

 number of nondominated solutions

 N_{obj}

 number of objective functions

 n_{s}

 number of the states

 n_{t}

 number of time segments

 N_{WTi}

 number of WT units at bus i

 S_{I,PV, i}

 inverter rating of PV

 S_{I,WT, i}

 inverter rating of WT

 SOC_{n,d,s}

 SOC of n^{th} battery at departure time

 SOC_{n,min}

 minimum SOC set by the vehicle's owner

 V_{i}

 voltage magnitude at bus i

 V^{max}

 highest voltage limit

 V^{min}

 lowest voltage limit

 V_{n}

 normal voltage magnitude

 x_{1}

 nondominated solution

 x_{2}

 dominated solution

 α

 the best solution

 β

 second best solution

 δ

 third best solution

 λ

 a matrix with two columns which comprises all possible amalgamations of PV power, WT power and the load demand states

 $\psi $

 complete probability model