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Stochastic scheduling of energy storage systems in harmonic polluted active distribution networks

Stochastic scheduling of energy storage systems in harmonic polluted active distribution networks INTRODUCTIONMotivationThere are many non‐linear loads such as power electronic devices in modern active distribution networks (ADN), which propagate current and voltage harmonics in the network. Eliminating voltage and current harmonics is essential, because, these harmonics have considerable influence on the distribution system devices and their functions. The harmonic distortions can increase losses, distort capacitors, cause electrical and electronic components failure, and interfere with communication, control, and protection devices [1].ADNs have a significant portion of renewable energy sources (RESs) due to their undeniable technical and environmental advantages. These sources encounter ADNs with more and more uncertainties due to their unpredictable nature. The other uncertainties such as load variation besides these RES make the state of the ADN completely intermittent. Consequently, operational decisions making for these ADNs are more difficult because their conditions are not deterministic [2].Energy storage systems (ESS) with their programmable power dispatch characteristics can efficiently overcome the intermittent difference between uncertain generation and uncertain load consumption, to a great extent. They are connected to the power system via power electronic converters. The main function of these converters is exchanging energy between the network and the ESS. However, there is a possibility of voltage and/or current harmonic compensation by these converters besides their main function [3, 4].In addition to employing ESSs for efficient operating of ADNs, some other conventional routines such as structure reconfiguration and reactive power scheduling can be used without applying any more costs to the system operator [5, 6].On the other hand, appropriate techniques should be used for handling uncertainties that arise from RESs and loads.The Monte Carlo simulation (MCS) method is the most accurate and simple technique for handling uncertainties but it requires a very large calculation burden. There is a serious restriction in applying this method to online requirements. Instead, data clustering techniques such as the K‐means technique present acceptable results with reasonable speed compared to the MCS method. Also, there are some efficient data sampling techniques such as the Latin hypercube sampling (LHS) technique, which make a reasonable trade‐off between accuracy and speed [7, 8].Optimal reconfiguration, reactive power, and ESS scheduling during a 24‐hour interval considering the uncertainties of the RESs and load demands to reduce total harmonic distortion (THD) and operate costs is a non‐linear optimization problem that can be solved efficiently by multi‐objective evolutionary algorithms. Multi‐objective particle swarm optimization (MOPSO) has a superior ability to solve these optimization problems [9].Literature reviewESS has a crucial role in ANDs' optimal operating scheduling where it can be useful from the economic and technical point of view, and they have been taken into consideration in many studies, up to now.The optimal location and size of ESS in a distribution network were obtained considering the system's reliability in [10]. Ref. [11] considered ESSs and a demand response program to reduce the total operation cost of the distribution network. Ref. [12] proposed a control strategy for ESS connection to the distribution networks by minimizing the real‐time voltage‐tracking mismatch as the objective function.Despite the efficient allocation and scheduling of the ESS in distribution systems, these efforts did not consider the RESs and their uncertainty in power generation, which is one of the essential elements of modern distribution networks.Ref. [13] improved the voltage profile and reduced the active power losses by the ESS active and reactive power injection. The authors in [14] applied a strategy for optimal dispatching of ESS based on peak load shifting to enhance the voltage profile in the distribution network.Similar to [10, 11, 12], these studies did not consider the uncertainty of RESs.Ref. [15] implements ESSs and RESs to minimize the energy not supplied index as a reliability index in a distribution network. Also, an optimal 24‐h operating program by obtaining the active and reactive powers of the ESS has been presented in ref. [16]. This reference focused on minimizing the electricity cost and power losses as well as the safe and reliable operation of the network. This study has also considered the photovoltaics (PVs) in the network. The authors in [17] presented an optimized model for the ESS in peak shaving along with active power losses and voltage fluctuation reduction in distribution networks. This reference considered the stochastic behaviour of the photovoltaic generation output. These studies have not considered non‐linear loads and their harmonic pollution effects. However, non‐linear loads are one of the crucial issues in the present networks.Although, there are some efficient schemes for ESSs in which they can compensate harmonic components besides active and reactive power exchange with the network [18, 19] the scheduling of the ESS has not been considered in harmonic polluted networks, significantly.In ref. [20], the ESS due to the flexible adjusting ability can reduce the harmonic distortion, power losses, and voltage unbalance of the distribution network. Ref. [21] applies the ESS, RES, and electric vehicles parking lot to the distribution network and formulates the optimal harmonic power flow to minimize the total harmonic distortion, voltage deviation, and energy cost.Although refs. [20, 21] paid attention to the harmonic reducing capability of ESSs besides their main function, they did not use reconfiguration and reactive power scheduling besides ESS scheduling.From the optimization algorithm point of view, the evolutionary‐based optimization techniques can efficiently be applied to this problem because of the non‐analytic, non‐linear and non‐convex nature of the problem. There have been various types of evolutionary‐based optimization algorithms. Among them, multi‐objective particle swarm optimization has been attractively taken into consideration, up to now [22, 23, 24]. Multi‐objective techniques present a set of solutions for the problem instead of one final solution. These solutions can be ranked based on the decision maker's evaluation. The technique for order of preference by similarity to ideal solution (TOPSIS) approach, which is one of the multi‐criteria ranking techniques has been used to rank solutions by the preference of decision‐makers, widely [25, 26].From the probabilistic assessment of ADNs in the presence of various uncertainties, the MCS technique is the most accurate and simple method for uncertainty evaluation [9]. However, there is a serious restriction in using the MCS method inside the evolutionary‐based optimization algorithms [27]. Instead, some approximate methods such as the K‐means data clustering technique or some efficient sampling techniques such as the LHS method have been efficiently used in combination with the evolutionary‐based algorithms [28, 29].As said, there is no comprehensive study for ESSs scheduling considering the non‐linear and uncertain nature of the ADN, simultaneously. This paper presents ESSs operating scheduling considering the uncertainty and non‐linearity of the AND. The THD and operating costs are two objectives in a multi‐objective optimization framework solved by MOPSO. Also, the TOPSIS is used for selecting the final solution.Paper contributionThis study presents an optimal scheduling program for ESSs, reactive power compensators and reconfiguration for reducing the THD and operating costs as objectives. Uncertainty of RESs and load demands are considered using two K‐means and LHS techniques. This optimization problem is solved in a multi‐objective optimization problem framework and finally solved by the MOPSO algorithm.The main contribution of this study is as follows;Considering the non‐linear loads and uncertain variables simultaneously in operating scheduling of ADNs.Using ESSs to eliminate the voltage/current harmonic components besides their power exchange with the network.Obtaining the trade‐off between the THD and the operating costs.Proposing the best operating scheduling using the TOPSIS approach.Also, other key characteristics of this study can be summarized as follows.Reactive power scheduling and reconfiguration are considered besides ESSs scheduling during a 24‐h interval.Uncertainty of RESs and load demands are considered to make results more realistic.Uncertainties of stochastic variables are handled by two K‐means and LHS methods with a comparison between them.The MOPSO is used as a powerful multi‐objective evolutionary‐based optimization algorithm.This method presents a more efficient operating scheduling for ESSs. The conventional operating scheduling only takes the active power exchange between ESSs and the network; however, this scheme reduces the voltage harmonic components besides their primary function, that is, active power exchange. This makes it possible to simultaneously reduce the operating costs and the THD. However, these two objectives may have a conflicting relationship, and it would be necessary to take an appropriate trade‐off between operating costs and the THD. The MOPSO algorithm minimizes the objectives and finally keeps a trade‐off region between these two objectives. A distribution system operator can have its own choice between operating costs and the THD based on its preferences.Paper structureThis paper is organized as follows. In Sections 2 and 3, the network and uncertainty modelling are described. The probabilistic analysis consisting of LHS and K‐means methods is presented in Section 4. In Section 5, the solution method is discussed in detail. The objective functions, control variables, and related constraints are presented in Section 6. Finally, the simulation results and conclusion are presented in Sections 7 and 8.NETWORK MODELLINGFigure 1 illustrates the radial diagram of the distribution system. A non‐linear load, an ESS, and a capacitor have been connected to ith, jth, and kth bus, respectively. However, there can be one or more of these components in one specific bus, but in this figure, we connected each component to separate buses to easily explain the modeling details.1FIGUREThe radial distribution network diagramFigure 2 illustrates the equivalent scheme of the distribution system in hth$ht{h}$ harmonic frequency.2FIGUREThe equivalent diagram of the distribution system in hth harmonic frequencyNon‐linear loadsLoads are assumed to have two linear and non‐linear parts. The ratio of linear loads in the ith bus is (1−wi)$( {1 - {w}_i} )$. Besides, the values of the fundamental and hth harmonic currents are determined using the Equations (1) and (2).1Iloadi1=Pi−jQiVi1∗\begin{equation} Iload_i^1 = \frac{{{P}_i - j{Q}_i}}{{V_i^{1*}}} \end{equation}2Iloadih=wiIloadi1hh=1,2,…,H\begin{eqnarray} Iload_i^h &=& {w}_i \frac{{Iload_i^1}}{h}\nonumber\\ h &=& 1,2, \ldots , H \end{eqnarray}where Iloadi1$Iload_i^1$ and Iloadih$Iload_i^h$ are the current value of the ith bus at the fundamental frequency and hth harmonic frequency.Pi${P}_i$ and Qi${Q}_i$ are the active and reactive load demands in the ith bus.Vi1$V_i^1$ is the voltage of the ith bus at the fundamental frequency.wi${w}_i$ is the non‐linear loads proportion in the ith bus.h, and H are harmonic index, and the highest harmonic frequency.Admittances of the loads at hth harmonic frequency is expressed as (3) [30].3Yloadih=(1−wi)Pi−jQi/h|Vi1|2\begin{equation} Yload_i^h = ( 1 - {w}_i)\frac{{P}_i - j{Q}_i/ h}{{| {V_i^1} |}^2} \end{equation}where Yloadih$Yload_i^h$ is the admittance of the ith load at hth harmonic frequency.In addition, the feeder impedance in hth harmonic is obtained by Equation (4).4Yseijh=1/Rseij+j·h·Xseij\begin{equation} Yse_{ij}^h = 1/\left( {Rs{e}_{ij} + j \cdot h \cdot Xs{e}_{ij}} \right) \end{equation}where Yseijh$Yse_{ij}^h$ is the admittance of the line between ith bus and jth bus at hth harmonic frequency.Rseij$Rs{e}_{ij}$ and Xseij$Xs{e}_{ij}$ are the resistance and reactance of the line between ith bus and jth bus.ESSHere, a voltage source converter (VSC)‐based model for ESS [31] is used, in which VSC can control both active power and harmonic currents.The main function of these converters is exchanging active power between the network and the ESS. This variable is shown by PjESS$P_j^{ESS}$ in Figure 2. However, it has the possibility of current harmonic compensation which is shown by Iessjh$Iess_j^h$ in Figure 2. According to Equation (5), this study assumes that the ESS can completely absorb the current harmonic components in the local bus to which is connected [19].5Iessih=Iloadih\begin{equation} Iess_i^h = Iload_i^h \end{equation}where Iessih$Iess_i^h$ is the current value of the ESS at hth harmonic frequency.Reactive power compensationAn admittance model was used for the capacitor in each harmonic frequency. The admittance at the fundamental and the hth harmonic frequency is obtained by Equations (6) and (7).6Yck1=jQkVk12\begin{equation} Yc_k^1 = \frac{{j{Q}_k}}{{{{\left| {V_k^1} \right|}}^2}} \end{equation}7Yckh=h·Yck1\begin{equation} Yc_k^h = h \cdot Yc_k^1 \end{equation}where Yck1$Yc_k^1$ and Yckh$Yc_k^h$ are the admittances of the capacitors in base frequency and hth harmonic.Qk${Q}_k$ is the reactive power injection of the kth capacitor.Harmonic load flowEquation (8) represents the voltage hth harmonic component. The total root‐mean‐square (RMS) voltage considering H harmonics can be calculated by equation (9).8V1hV2h⋮Vih=YBush−1I1hI2h⋮Iih\begin{equation} \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {V_1^h}\\[4pt] {V_2^h} \end{array} }\\[4pt] \vdots \\[4pt] {V_i^h} \end{array} } \right] = {\left[ {Y_{Bus}^h} \right]}^{ - 1} \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {I_1^h}\\[4pt] {I_2^h} \end{array} }\\[4pt] \vdots \\[4pt] {I_i^h} \end{array} } \right] \end{equation}9Vi=∑h=1HVih2\begin{equation} \left| {{V}_i} \right| = \sqrt {\mathop \sum \limits_{h = 1}^H {{\left| {V_i^h} \right|}}^2} \end{equation}where Vih$V_i^h$ is the hth harmonic voltage at ith bus.YBush$Y_{Bus}^h$ is network admittance bus at hth harmonic frequency.Vi${V}_i$ is the RMS value of the voltage in ith bus.Iih$I_i^h$ is the nodal hth harmonic current at ith bus [32].UNCERTAINTY MODELLINGLoad demands uncertainty modellingThe load demand is explained by the normal distribution. The probability density function (PDF) of the load demand following the normal distribution is introduced as (10) and (11) [33].10fPi=1σPi2πe−Pi−EPi22σPi2\begin{equation} f \left( {{P}_i} \right) = \frac{1}{{\left( {\sigma \left[ {{P}_i} \right]} \right)\sqrt {2\pi } }}\ {e}^{ - \frac{{{{\left( {{P}_i - E\left[ {{P}_i} \right]} \right)}}^2}}{{2\sigma {{\left[ {{P}_i} \right]}}^2}}} \end{equation}11fQi=1σQi2πe−Qi−EQi22σQi2\begin{equation} f \left( {{Q}_i} \right) = \frac{1}{{\left( {\sigma \left[ {{Q}_i} \right]} \right)\sqrt {2\pi } }}\ {e}^{ - \frac{{{{\left( {{Q}_i - E\left[ {{Q}_i} \right]} \right)}}^2}}{{2\sigma {{\left[ {{Q}_i} \right]}}^2}}} \end{equation}where f(·)$f( \cdot )$ is the PDF of the load demand.E[] is the expected value.σ[] is the standard deviation.Equation (10) indicates the active power demand in ith bus and Equation (11) denotes the reactive power demand in ith bus.WTs power generation uncertainty modellingUsually, the Weibull distribution is used to model the wind speed to obtain probabilistic values of the WT generation output. As exact WT capacity is added at one bus, a Weibull distribution can obtain WT generation output without considering geographical factors. In consequence, the declared parameters are required for Weibull distribution. The Weibull distribution is expressed as (12) [34].12fv=ABvBA−1e−vBAv≥00v<0\begin{equation} f \left( v \right) = \left\{ \def\eqcellsep{&}\begin{array}{ll} \displaystyle\frac{A}{B}{{\left( {\displaystyle\frac{v}{B}} \right)}}^{A - 1} \ {e}^{ - {{\left( {\frac{v}{B}} \right)}}^A} & v \ge 0\\[12pt] 0 & v &lt; 0 \end{array} \right. \end{equation}where A and B are the shape and scale components.v is the wind speed.The generated power of the WT can be computed as (13).13Pwv=0v≤vin∥v≥voutv−vinvrated−vinPrwvin≤v≤vratedPrwvrated≤v≤vout\begin{equation} {P}^w \left( v \right) = \left\{ \def\eqcellsep{&}\begin{array}{ll} 0&{v \le {v}_{in}\parallel v \ge {v}_{out}}\\[9pt] {\displaystyle\frac{{v - {v}_{in}}}{{{v}_{rated} - {v}_{in}}}P_r^w}&{{v}_{in} \le v \le {v}_{rated}}\\[12pt] {P_r^w}&{{v}_{rated} \le v \le {v}_{out}} \end{array} \right. \end{equation}where vin${v}_{in}$ and vout${v}_{out}$ are cut in and cut out speeds.vrated${v}_{rated}$ and Prw$P_r^w$ are rated speed and power of the WT.PROBABILISTIC ANALYSISIt is essential to use the probabilistic load flow (PLF) instead of deterministic load flow (DLF) due to the ever‐increasing uncertainties of modern electrical networks. The network state can be broadly studied considering all probable range of uncertain variables [35, 36]. These uncertain variables can be defined by the PDF) [37].Although, the MCS technique is commonly and simply used to take samples of uncertain variables it suffers from a large computational burden. Efficient methods to take samples of uncertain variables are attractive from the computational burden point of view. These sampling techniques use less number of samples for each uncertain variable, but they still keep the appropriate accuracy. The LHS technique and the K‐means data clustering technique can efficiently reduce the number of samples without losing the accuracy very much. These techniques are discussed in Sections 4.1 and 4.2.Latin hypercube sampling techniqueMckay et al. introduced the LHS as a sampling method to calculate quasi‐random samples of multi‐dimensional probability distribution functions [38]. This method partitions the parameter spaces into regions of equal probability and picks a sample from each region to attain a more even distribution of sample points.The LHS can be explained as follows:X=[X1,X2,…,XN]$X = [ {{X}_1,{X}_2, \ldots ,{X}_N} ]$ is supposed to be an N dimensional input random variable. The cumulative distribution function (CDF) of Xn${X}_n$ is according to (14).14CDFXn=CDFXnXn,n=1,2,…,N\begin{equation} CD\ {F}_{{X}_n} = CD{F}_{{X}_n} \left( {{X}_n} \right),\quad n = \left[ {1, 2, \ldots , N} \right] \end{equation}where Xn${X}_n$ denotes for the nth input random variable.CDFXn$CD{F}_{{X}_n}$ and CDFXn(Xn)$CD{F}_{{X}_n}( {{X}_n} )$ are the CDF of the Xn${X}_n$.N is the number of random input variables.The range of CDFXn$CD{F}_{{X}_n}$ is from 0 to 1. CDFXn$CD{F}_{{X}_n}$ range is divided into N equal and non‐overlapping intervals, where 1/K$1/K$ is the length of each interval for a sample size of K. One value is chosen from every interval in a pre‐determined way (for example the midpoint of each interval) or randomly. Then the inverse function of the (14) is adopted to calculate the corresponding samples of Xn${X}_n$. The kth sample of Xn${X}_n$ is determined by Equation (15).15Xkn=FXn−1k−0.5K,k=1,2,…,K\begin{equation} {X}_{kn} = {F}_{{X}_n}^{ - 1} \left( {\frac{{k - 0.5}}{K}} \right),\quad k = \left[ {1, 2, \ldots , K} \right] \end{equation}where Xkn${X}_{kn}$ is the kth sample of the Xn${X}_n$.K is the number of samples.Repeating Equation (15) for k=[1,2,…,K]$k = [ {1, 2, \ldots , K} ]$ and n=[1,2,…,N]$n = [ {1, 2, \ldots , N} ]$ makes the sampling matrix. The nth column of this matrix, [X1n,X2n,…,XKn]T${[ {{X}_{1n},{X}_{2n}, \ldots ,{X}_{Kn}} ]}^T$ represents the samples of Xn${X}_n$ and the kth row of this matrix Ak=[Xk1,Xk2,…,XkN]${A}_k = [ {{X}_{k1},{X}_{k2}, \ldots ,{X}_{kN}} ]$ represents the kth scenario or observation.In the end of this step, a K×N$K \times N$ sampling matrix with K samples and N input random variable is available [39].Each column is independently permutated to make a completely random combination of variables. The size of the sampling matrix is still a K×N$K \times N$ matrix.Now, the kth row of this matrix Ak=[Xk1,Xk2,…,XkN]${A}_k = [ {{X}_{k1},{X}_{k2}, \ldots ,{X}_{kN}} ]$ that represents kth the scenario or observation is evaluated according to (16).16Fk=FAk\begin{equation} {F}_k = F\left( {{A}_k} \right) \end{equation}where Fk${F}_k$ is the kth output for scenario or observation.The ith statistical moment of F is computed as (17).17EFi=1K∑k=1KFAki\begin{equation} {\rm{E }}\left[ {{F}^i} \right] = \frac{1}{K} \mathop \sum \limits_{k = 1}^K F{\left( {{A}_k} \right)}^i \end{equation}where E[Fi]${\rm{E}}[ {{F}^i} ]$ is ith statistical moment of F.K‐means data clustering algorithmThe concept of clustering was introduced in 1935. The large multi‐dimensional objects or data are separated into subsets called clusters in the clustering process. In a cluster, objects or data are more similar than those in other clusters in particular criteria. So, various criteria such as distance can be employed to determine the similarity of the data.By using data clustering, only limited datasets are examined instead of analysing a large amount of information. Different techniques for data clustering have been presented until now.Mac‐Queen introduced the K‐means algorithm in 1967 [40]; this algorithm is one of the simplest and most popular unmanaged learning algorithms for data clustering. This method can categorize a large number of data and minimizes the total distance between all data from the nearest centre of the cluster [41].The following steps are the procedure for running the K‐means algorithm [42]:The number of clusters (K) is specified.Initial agents Ak${A}_k$ for each cluster are determined randomly with N dimension according to (18).18Ak=Xk1,Xk2,…,XkN,k=1,2,…,K\begin{equation} {A}_k = \left[ {{X}_{k1},{X}_{k2}, \ldots ,{X}_{kN}} \right] , k = \left[ {1, 2, \ldots , K} \right] \end{equation}whereN is the number of random input variables.3.Observations are assigned to clusters by calculating the minimum distance to each agent as:if ds−Ak<|ds−Al|⇒dsεGk${d}_s - {A}_k &lt; | {{d}_s - {A}_l} | \Rightarrow {d}_s\epsilon {G}_k$ (19)wheres=1,2,…,S$s = 1, 2, \ldots , S$ and k,l=1,2,…,K(l≠k)$k,l = 1, 2, \ldots , K ( {l \ne k} )$.S is the total number of data in each Xi${X}_i$.ds${d}_s$ is the sth observation or scenario.Ak${A}_k$ and Al${A}_l$ are the kth and lth agents.Gk${G}_k$ is the kth cluster.Various algorithms such as Euclidean, City block, and Mikowski can be used for calculating the distance of the data.4.Centre point of each cluster is calculated and selected as the new agent:20Ak=∑s∈GkdsNGkk,l=1,2,…,K\begin{equation} {A}_k = \frac{{\mathop \sum \nolimits_{s \in {G}_k} {d}_s}}{{{N}_{{G}_k}}} k,\quad l = 1, 2, \ldots , K \end{equation}NGk${N}_{{G}_k}$ is the number of data in the Gkth${G}_k{\rm{th}}$ cluster.5.If no data point is specified to a new cluster, the proceeding is ended; the 3–5 steps are repeated to discover new data points.6.The probability of each sample is equal to Equation (21):21PAk=NGkS\begin{equation} P \left( {{A}_k} \right) = \frac{{{N}_{{G}_k}}}{S} \end{equation}Also, the ith statistical moment of F is computed as (22).22EFi=∑k=1KPAk·FAki\begin{equation} {\rm{E }}\left[ {{F}^i} \right] = \mathop \sum \limits_{k = 1}^K P\left( {{A}_k} \right) \cdot F{\left( {{A}_k} \right)}^i \end{equation}whereF(Ak)$F( {{A}_k} )$ is obtained from (16).SOLUTION METHODMulti‐objective optimization methodIn multi‐objective optimization methods, the problems are defined as multi objectives and corresponding related constraints. The constraints have to be met in all possible solutions. Multi‐objective optimization is formulated as:23fnx,n=1,2,…,N;\begin{equation} {f}_n\left( x \right), n = 1,2, \ldots ,N; \end{equation}24subjecttogjx≤0,j=1,2,…,J;\begin{equation} {\rm{subject\ to}}\ {g}_j\left( x \right) \le 0,j = 1,2, \ldots ,J; \end{equation}25hkx=0,k=1,2,…,K;\begin{equation} {h}_k \left( x \right) = 0, k = 1,2, \ldots ,K; \end{equation}26xiL≤xi≤xiU,i=1,2,…,M\begin{equation} x_i^L \le {x}_i \le x_i^U, i = 1,2, \ldots ,M \end{equation}Equation (23) defines the objective functions; equality constraints and some inequality constraints are represented in Equations (24, 25). Where x is an indicator of m control variables. Inequality Equation (26) is variable limits to restrict every control variable to take a value within a lower xiL$x_i^L$ and an upper xiU$x_i^U$ limit.In multi‐objective problems, the objectives may be defined in conflict aspects. In these problems set of compromised solutions known as Pareto solutions are obtained, which means there is no single optimal solution.Mathematical and evolutionary approaches are two multi‐objective optimization methods. Mathematical methods have limitations in solving some non‐convex problems, besides the final solution may depend on the initial solution and get trapped in a sub‐optimal solution.The evolutionary algorithms are presented to solve mentioned drawbacks. These algorithms can escape from local optimum solutions due to considering the population of feasible solutions, and they don't need the derivatives of the objective functions, for this reason, they can solve any type of problem.Despite these advantages, some issues related to these algorithms must be mentioned, such as, they reach different solutions in every execution, the obtained solutions cannot guarantee to be global optimum, and they need more computation burden. It must be mentioned these solutions are still beneficial, applicable, and efficient [27].The MOPSO method [43] is used to designate the weight of each factor or criterion properly. The MOPSO algorithm has been used in solving various problems, among which are the utilization related to handling the distribution networks [44, 45]. The collection of possible results is obtained in multi‐objective optimization algorithms. Then, the TOPSIS technique is employed to rank the results based on their suitability. This approach relies on the desired and undesired points' intervals.The MOPSO algorithmIn the particle swarm optimization (PSO) algorithm, a search area‐id is considered with n particles and d dimensions, where Xi(xi1,xi2,…,xid)${X}_i( {{x}_{i1},{x}_{i2}, \ldots ,{x}_{id}} )$, and Vi(vi1,vi2,…,vid)${V}_i( {{v}_{i1},{v}_{i2}, \ldots ,{v}_{id}} )$ are position and velocity of the ith moving particle. Furthermore, Pi(pi1,pi2,…,pid)${P}_i( {{p}_{i1},{p}_{i2}, \ldots ,{p}_{id}} )$, and Gi(gi1,gi2,…,gid)${G}_i( {{g}_{i1},{g}_{i2}, \ldots ,{g}_{id}} )$ are particle best and global best while their own best position specifies them, and general best performance of the particles. Each particle modifies its location by current velocities and positions, the interval between the pbest$pbest$ and current position, and the gbest$gbest$ and current position.27Vit+1=wVit+c1r1xpbest−Xit+wVit+c2r2xgbest−Xit\begin{eqnarray} V_i^{t + 1} &=& wV_i^t + {c}_1{r}_1\left( {{x}_{pbest} - X_i^t} \right) + wV_i^t\nonumber\\ && +\ {c}_2{r}_2\left( {{x}_{gbest} - X_i^t} \right) \end{eqnarray}28Xit+1=Xit+Vit+1\begin{equation} X_i^{t + 1} = X_i^t + V_i^{t + 1} \end{equation}wherew is the inertia weight.c1, and c2 are cognitive acceleration and social acceleration coefficients.xpbest${x}_{pbest}$ and xgbest${x}_{gbest}$ are the individual and global best of the particle.r1 and r2 are the random variables.Vit$V_i^t$, and Xit$X_i^t$ are the current velocity and position in iteration t of ith particle.The particle's velocity and position are renewed similarly at the PSO and MOPSO algorithms. The PSO and MOPSO algorithms are different in some aspects: Global and individual leaders' selection and update methods are different. Besides, the Pareto optimal solution set is employed to obtain the solution in the MOPSO algorithm. Beyond, the MOPSO algorithm needs a repository for saving the non‐dominated solutions obtained by the input variables to be established. Several researchers have studied the selection and updating of global and individual leaders, the parameter settings, and the establishment of a repository and have achieved some results [46–48]. According to these results, new strategies can be integrated with the MOPSO algorithm and improve the efficiency of this algorithm in solving complex problems and reducing execution time. In addition, this algorithm is still encountered challenges such as the ineffectiveness of exploring the gbest$gbest$ solution and prone to premature convergence. The diagram of the MOPSO algorithm according to dominance criteria is illustrated in Figure 3.3FIGUREMOPSO algorithm flowchartThe TOPSIS methodThe number of results has to be compared and deliberated by some criteria in multi‐objective optimization problems. Plus, the TOPSIS method aims to assist the decision‐maker in the trade‐off between solutions. So, possible results are usually specified by different criteria, and maybe none of the results satisfies all the criteria. Hence, the result is a relative result based on the decision maker's preference. TOPSIS method is proposed by Chen (1992) [49]. The basic concept is that the determined result must have the maximum interval from the negative‐best result and the minimum interval from the best result. TOPSIS has some advantages, including:TOPSIS is faster and simpler than fuzzy Delphi analytic, hierarchy process, analytic hierarchy process, and simple additive weighting, several criteria are permissible in the decision process, ease of decision‐making involving positive as well as negative criteria [50], and TOPSIS is easy to understand, simple in the calculation, and flexible to utilize, as is to be used in many technological and social areas [51].The following steps are the process of the TOPSIS method:Compute the rij${r}_{ij}$ as a normalized decision matrix by Equation (29).29rij=fij/∑j=1Jfij2j=1,…,J;i=1,…,n\begin{equation} {r}_{ij} = {f}_{ij}/\sqrt {\mathop \sum \limits_{j = 1}^J f_{ij}^2} j = 1, \ldots ,J;i = 1, \ldots ,n \end{equation}Compute the vij$v{ }_{ij}$ as weighted normalized decision matrix by Equation (30).30vij=wirijj=1,…,J;i=1,…,n\begin{equation} v{ }_{ij} = {w}_i\ {r}_{ij}\quad j = 1, \ldots ,J; i = 1, \ldots ,n \end{equation}Specify the negative‐best and best results.31A−=v1−,…,vn−=minvij|i∈I′,maxvij|i∈I′′\begin{eqnarray} {A}^ - = \left\{ {v_1^ - , \ldots ,v_n^ - } \right\} = \left\{ {\left( {\min {v}_{ij}|i \in I^{\prime}} \right),\left( {\max \ {v}_{ij}|i \in I^{\prime\prime}} \right)} \right\} \nonumber\\ \end{eqnarray}32A∗=v1∗,…,vn∗=maxvij|i∈I′,minvij|i∈I′′\begin{eqnarray} {A}^* = \left\{ {v_1^*, \ldots ,v_n^*} \right\} = \left\{ {\left( {\max {v}_{ij}|i \in I^{\prime}} \right),\left( {\min \ {v}_{ij}|i \in I^{\prime\prime}} \right)} \right\} \nonumber\\ \end{eqnarray}where I′$I^{\prime}$ and I′′$I^{\prime\prime}$ are associated with benefit criteria, and cost criteria, respectively.4.Compute the distance measures by the n‐dimensional Euclidean distance. The distance of every result from the best result is calculated as:33Dj∗=∑i=1nvij−vi∗2j=1,…,J.\begin{equation} D_j^* = \sqrt {\mathop \sum \limits_{i = 1}^n {{\left( {{v}_{ij} - v_i^*} \right)}}^2} \quad j = 1, \ldots ,J. \end{equation}Uniformly, the distance from the negative‐best result is calculated as:34Dj−=∑i=1nvij−vi−2j=1,…,J.\begin{equation} D_j^ - = \sqrt {\mathop \sum \limits_{i = 1}^n {{\left( {{v}_{ij} - v_i^ - } \right)}}^2} \quad j = 1, \ldots ,J. \end{equation}5.The relative closeness of the result aj${a}_j$ is determined by:35Cj∗=Dj−/Dj∗+Dj−j=1,…,J.\begin{equation} C_j^* = D_j^ - /\left( {D_j^* + D_j^ - } \right)\quad j = 1, \ldots ,J. \end{equation}6.Rate the preference result.PROBLEM FORMULATIONObjective functionThe main objectives of this study are reducing the total operating costs and the voltage THD of the ADN, which are formulated in detail in the following subsections.The first objective function is the operational costs. The operational costs which are denoted by Cope${C}^{ope}$ consists of: (a) The operational cost of distributed generations (DGs), (b) power losses cost, (c) the maintenance cost of ESS, and (d) the cost of purchasing electric power from the upper network, according to (36).36Cope=∑tPtCtp+PtlossCtp+∑dCdESS+CgDG∑t∑gPg,tDG/24\begin{eqnarray} {C}^{ope} &=& \mathop \sum \limits_t \left( {{P}_tC_t^p + P_t^{loss}C_t^p} \right) + \mathop \sum \limits_d C_d^{ESS}\nonumber\\ && +\ C_g^{DG}\left( {\left( {\mathop \sum \limits_t \mathop \sum \limits_g P_{g,t}^{DG}} \right)/24} \right) \end{eqnarray}where Pt${P}_t$ is the purchased electric power from the upper network at time t.Ptloss$P_t^{loss}$ denotes the network power loss at time t.Pg,tDG$P_{g,t}^{DG}$ is the active power generation of the gth DG at time t.Ctp$C_t^p$ is the purchased electric power cost at time t.CdESS$C_d^{ESS}$ and CgDG$C_g^{DG}$ are the daily maintenance cost of the dth ESS and gth DG.The annual maintenance cost (1, 2, or 3 times per year) can be divided into 365 (number of days in one year) as daily cost and this cost can be considered in day‐ahead studies besides the daily operating costs [52–54]The active power losses of the network at time t are obtained by Equation (37).37Ptloss∑brbI1.b.t2+I2.b.t2+⋯IH.b.t2\begin{equation} P_t^{loss}\mathop \sum \limits_b {r}_b\left( {I_{1.b.t}^2 + I_{2.b.t}^2 + \cdots I_{H.b.t}^2} \right) \end{equation}whereb is branch index.rb${r}_b$ is the resistance of the bth branch.I1.b.t${I}_{1.b.t}$, I2.b.t${I}_{2.b.t}$, and IH.b.t${I}_{H.b.t}$ are fundamental, second, and the Hth harmonic frequency currents of the bth branch at time t.The voltage THD reduction is the second objective function that enhances the power quality of the ADN. The voltage THD is formulated as the RMS value of the voltage harmonic components over the RMS value of the fundamental component of the voltage in IEEE standards. So, the THDv$TH{D}_v$ is represented as a percentage of the fundamental voltage.38THDv=1T×N∑t=1T∑i=1NV2.i.t2+V3.i.t2+⋯VH.i.t2V1.i.t×100\begin{eqnarray} TH{D}_v &=& \frac{1}{{T \times N}} \mathop \sum \limits_{t = 1}^T \left( {\mathop \sum \limits_{i = 1}^N \left( {\frac{{\sqrt {V_{2.i.t}^2 + V_{3.i.t}^2 + \cdots V_{H.i.t}^2} }}{{{V}_{1.i.t}}}} \right)} \right)\nonumber\\ && \times\ 100 \end{eqnarray}whereV1.i.t${V}_{1.i.t}$ denotes the fundamental voltage of ith bus at time t.V2.i.t${V}_{2.i.t}$, V3.i.t${V}_{3.i.t}$, and VH.i.t${V}_{H.i.t}$ are second, third and the Hth harmonic frequency voltages of the ith bus at time t.H is the highest harmofnic frequency.Although adding a constant value in the objective function related to ESS daily cost does not affect the optimization results, considering this constant cost in the objective function might lead to more actual values.Control variablesThe control variables consist of:Reactive power of capacitor banks at time t,Charging/discharging power of the ESSs at time t,Corresponding sectionalizing switches state.ConstraintsBuses voltage constraint:The buses voltages should have remained in the allowed range according to (39)39Vimin≤V1.i.t2+V2.i.t2+⋯VH.i.t2≤Vimax\begin{equation} V_i^{min} \le \sqrt {V_{1.i.t}^2 + V_{2.i.t}^2 + \cdots V_{H.i.t}^2} \le V_i^{max} \end{equation}Vimin$V_i^{min}$ and Vimax$V_i^{max}$ are the minimum and maximum voltage limit of the ith bus.Branches capacity constraint:Feeder currents should be remained in their allowable ratings to preserve the feeders and cables against excessive currents; in this regard, the following constraint should be satisfied:Ibmax$I_b^{max}$ is the maximum current limit of the bth branch.40I1.b.t2+I2.b.t2+⋯IH.b.t2≤Ibmax\begin{equation} \sqrt {I_{1.b.t}^2 + I_{2.b.t}^2 + \cdots I_{H.b.t}^2} \le I_b^{max} \end{equation}DGs and capacitor banks constraints:The DG units and reactive power compensators (capacitors) limits must be satisfied every hour in all scenarios.41Pg.minDG≤Pg,tDG≤Pg.maxDG\begin{equation} P_{g.min}^{DG} \le P_{g,t}^{DG} \le P_{g.max}^{DG} \end{equation}42Qm.minC≤Qm,tC≤Qm.maxC\begin{equation} Q_{m.min}^C \le Q_{m,t}^C \le Q_{m.max}^C \end{equation}Pg.minDG$P_{g.min}^{DG}$ and Pg.maxDG$P_{g.max}^{DG}$ are minimum and maximum active power generation of the gth DG.Qm.minC$Q_{m.min}^C$ and Qm.maxC$Q_{m.max}^C$ are minimum and maximum reactive power of the mth capacitor.Equation (41) is the DG dispatch constraints, and Equation (42) is the reactive power compensator adjusting constraint.Network reconfiguration constraints:The primary constraint in network reconfiguration is its radiality. Usually, the network structure is designed as a poor ring system, so network operations are radial to preserve integrity in distribution networks' protection, stability, and other aspects. Hence, the radiality circumstance must be preserved in the reconfiguration network structure. The tie switches number is determined using the following equation [55]:43NTie−switches=NBranches−NBuses+1\begin{equation} {N}_{Tie - switches} = {N}_{Branches} - {N}_{Buses} + 1 \end{equation}whereNTie−switches${N}_{Tie - switches}$ is the number of the tie switches.NBranches${N}_{Branches}$ is the number of branches.NBuses${N}_{Buses}$ is the number of the buses.The radiality of the distribution networks is preserved by using the graph rules [56]. A tree is a subgraph with (N−1)$( {N - 1} )$ lines. According to these rules, the structure of a network is maintained radial if the following equation is satisfied.44NBrachesRe=Nbuses−1\begin{equation} N_{Braches}^{Re} = {N}_{buses} - 1 \end{equation}NBrachesRe$N_{Braches}^{Re}$ is the number of branches after reconfiguration of the network structure.Eventually, all the new loads must avoid the isolated load chain or isolated load to satisfy the network structure limits (Figure 4).ESS operation constraints:4FIGURETopology structure constraint: The circles are the loads; the solid lines are the present branches; dashed lines are alternative branches. (a) Initial network. (b) The topology that is leading to isolated load and loop. (c) The topology that is leading to isolated load chainThe energy and active power of an ESS are constrained according to (45–48)45Ed.minESS≤Ed.tESS≤Ed.maxESS\begin{equation} E_{d.min}^{ESS} \le E_{d.t}^{ESS} \le E_{d.max}^{ESS} \end{equation}46Ed.tESS=Ed.t−1ESS+Δt×Pd.tESS,Ch×η×ψCh−Δt×Pd.tESS,Dischη×ψDisch\begin{eqnarray} E_{d.t}^{ESS} &=& E_{d.t - 1}^{ESS} + \left( {\Delta t \times P_{d.t}^{ESS,Ch} \times \eta } \right) \times {\psi }^{Ch}\nonumber\\ && -\, \left( {\Delta t \times \frac{{P_{d.t}^{ESS,Disch}}}{\eta }} \right) \times {\psi }^{Disch} \end{eqnarray}47Ed.0ESS=Ed.iniESS\begin{equation} E_{d.0}^{ESS} = E_{d.ini}^{ESS} \end{equation}48−RDdESS≤Pd.tESS−Pd.t−1ESS≤RUdESS\begin{equation} - RD_d^{ESS} \le P_{d.t}^{ESS} - P_{d.t - 1}^{ESS} \le RU_d^{ESS} \end{equation}where Ed.tESS$E_{d.t}^{ESS}$ is the energy of the dth ESS at time t.Pd.tESS,Ch$P_{d.t}^{ESS,Ch}$ and Pd.tESS,Disch$P_{d.t}^{ESS,Disch}$ are the charge and discharge active power of the dth ESS at time t.Pd.tESS$P_{d.t}^{ESS}$ in (48) denotes both Pd.tESS,Ch$P_{d.t}^{ESS,Ch}$ and Pd.tESS,Disch$P_{d.t}^{ESS,Disch}$.Ed.minESS$E_{d.min}^{ESS}$ and Ed.maxESS$E_{d.max}^{ESS}$ are the minimum and maximum energy of dth ESS.Δt$\Delta t$ is the time interval.η is charging and discharging efficiency.ψCh${\psi }^{Ch}$ and ψDisch${\psi }^{Disch}$ are binary factors that represent charging and discharging modes, respectively.Ed.iniESS$E_{d.ini}^{ESS}$ is the initial energy of the dth ESS.RDdESS$RD_d^{ESS}$ and RUdESS$RU_d^{ESS}$ are maximum ramp‐down and ramp‐down rate of the dth ESS.Constraint (45) is the maximum and minimum energy of ESS. Equations (46) and (47) are energy storage constraints between two adjacent hours. Also, (48) is the ramp‐up and ramp‐down limits of the ESS power.THD constraint:The following constraint is defied to remain THDv$TH{D}_v$ in the allowed range that the system operator determines.49THDv≤THDvmax\begin{equation} TH{D}_v \le THD_v^{max} \end{equation}THDvmax$THD_v^{max}$ is the maximum amount of THDv$TH{D}_v$.Proposed methodFigure 5 represents a flowchart of the proposed approach. The flowchart shows that control variables values are generated, and the optimization process begins. Stochastic data is generated, and the desired data are determined using the K‐means or LHS methods. Control variables are applied to the distribution network, and the harmonic load‐flow process determines the wanted components. The values of objective functions are calculated based on each cluster (by applying the K‐means method) or each sample (by applying the LHS method), and the end requirements of the algorithm process are investigated.5FIGURESteps of the proposed methodThe estimated values of the defined objective functions are computed according to the obtained values of the objective functions and the probable coefficients of clusters or samples.At the end of the process and obtain different results for the objective functions, the ranking method based on the weight coefficient of each objective function will be performed (using the TOPSIS method).SIMULATION RESULTSIn order to test the proposed methods, the modified IEEE 33‐bus and 118‐bus distribution networks have been used. The obtained results are represented in Sections 7.2 and 7.3, respectively.AssumptionsThe load curve depicts the changes of load power demand throughout a particular period. In this paper, the residential daily load curve is utilized as in [57]. The hourly price of purchased energy is obtained from the Nord Pool grid [58]. Moreover, DGs operational cost data is obtained from [59], and the cost of the DG unit using the discount rate is converted to current year cost values.The employed DG generators are WT generators with stochastic nature in producing electric power. This means the WTs power generation are related to wind speed, while Weibull distribution can be utilized to model the wind speed behaviour. So, the Weibull distribution parameters, k, and λ are equal to 3 and 8. The rest of the data relevant to WTs is presented in Table 1.1TABLEThe related parameters of the WTsParameterValueCut‐in speedvinc(m/s)$v_{in}^c ( {{\rm{m}}/{\rm{s}}} )$3Cut‐out speedvoutc(m/s)$v_{out}^c ( {{\rm{m}}/{\rm{s}}} )$25Rated speedvrated(m/s)${v}_{rated} ( {{\rm{m}}/{\rm{s}}} )$13The power demand loads are considered uncertain loads, and active and reactive power loads models are based on the normal distribution. Their expected and standard deviation values are considered to be equal to the nominal active power value of buses at the base state and 10% of their expected value [60].The WT units, capacitor banks, and ESSs discharge process are simulated as negative loads. Besides, the ESS charge process is assumed as a positive load. According to mentioned assumptions, there is no control on voltage.The number of generated data in the MCS method is 10,000 samples for each uncertain input variable, which are reduced to 10 and 50 datasets by applying the K‐means and LHS methods, respectively.In this paper, the MOPSO algorithm calculates the minimum values for total cost and THD index to determine a non‐inferior solution. Then, non‐dominated solutions are ranked using the proper approach to help the operator specify the best solution. Decision‐making between the results obtained in the Pareto‐front is challenging for the operator. In this regard, the priority and viewpoint of the active distribution operator will ascertain the final proper solution. Therefore, the TOPSIS approach can help the operator rank and distinguish the ideal solutions. For that reason, the network operator determines the proper compromise solution using weight coefficients. In this paper, both weight coefficient values for estimated THD and estimated total cost are considered equal to 0.5.The proposed method is conducted on a PC with an Intel(R) Core (TM) i7–8500 K CPU and 16.00 GB of memory. The simulation time by implementing the K‐means and LHS methods with an iteration number of 100 and population size of 100 in both 33‐bus (with 77 control variables) and 118‐bus (with 159 control variables) distribution networks are given in Table 2.2TABLEThe simulation time by implementing the K‐means and LHS methods in the 33‐bus and 118‐bus distribution networksDistribution network33‐bus118‐busK‐meansClusters number1010Simulation time (s)620792LHSSamples number5050Simulation time (s)28743094The 33‐bus distribution networkIn order to test the proposed methods, the modified IEEE 33‐bus distribution network has been used. This network's total active and reactive loads are 3.715 MW and 2.3 MVAr, respectively; the voltage rating is 12.66 kV. The test system comprises 32 sectionalizing switches and five tie switches presented in Figure 6 [61].6FIGUREThe modified IEEE 33‐bus distribution networkFour WT units are installed on buses number 10, 16, 17, and 30 with active power rating equal to 500 kW, 200 kW, 150 kW, and 200 kW according to [62].Figure 7 illustrates the scatter plot with two marginal histograms to represent the samples of WT generations at bus 17 and active power demands at bus 10. This scatter plot is drawn based on corresponding data at 9 PM.7FIGUREScatter plot of the active power demand samples in bus 10 and the WT generation samples at bus 17Also, two capacitor banks are installed at bases number 3 and 28 with the reactive power capacity of 1050 kVAr (7 capacitors with 150 kVAr capacity), according to [63].One ESS unit is already located in bus 22 [64] with an energy capacity of 1000 kWh. The considered values for ESS parameters are represented in Table 3. The daily maintenance cost for the ESS is approximately according to [52] updated considering the interest rate of 7%.3TABLEAssumed data for operation specifications of the ESS with 1000 kWh energy ratingParametersValueESS daily maintenance cost23.4 €/dayESS ramping up/down rate60 kW/hESS charging/discharging efficiency95 %The harmonic orders are assumed to be 5, 7, 11, 13, 17, 19, 23, and 25 in both scenarios similarly (0.4 coefficient in buses 5–8, 0.3 coefficient in buses 11–15, 0.5 coefficient in buses 25–31, and 0.5 coefficient in bus number 20 are considered for modelling the non‐linear loads). Two scenarios are defined to test the proposed strategy.Scenario 1: In this scenario, the ESS unit and capacitor banks are not managed, and the network structure is not modified. So, considering the uncertainties of the WT units and demand loads, the objective functions are calculated. It should be noted that scenario 1 is the base scenario and is considered to compare with the proposed strategy. The obtained objective functions with consideration of the K‐means and LHS samples are presented in Table 4.4TABLEValues of objective functions in scenario 1, obtained by executing the K‐means and LHS methods in the modified IEEE 33‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS5.105.132526.162534.17Scenario 2: In this scenario, charging/discharging power of the ESS unit, and reactive power of the capacitor banks are obtained by the MOPSO algorithm, besides the reconfiguration of the network structure is applied. A set of the optimal objective functions values is executed using the MOPSO algorithm.Figure 8 indicates the estimated Pareto optimal solution points consisting of the THD and the total cost in the objective area. The presented result points in Figure 8 could assist the system operator in making decisions from economic and performance points of view.8FIGUREComparison of K‐means and LHS method based on obtained non‐dominated Pareto front solutions in the 33‐bus distribution networkThe best solutions are selected by the TOPSIS technique and are presented in Table 5. The presented optimal values are selected using an equal 0.5 coefficient value.5TABLEValues of objective functions in scenario 2, obtained by executing the K‐means and LHS methods in the modified IEEE 33‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS4.124.082444.152454.20According to Tables 4 and 5, the estimated THDs are reduced by 19.21% and 20.46%, and the estimated total costs are decreased by about 3.24% and 3.15% using the K‐means and the LHS methods, respectively. These results verify the effectiveness of the proposed strategy using both probabilistic methods, that each method can be used based on operator needs, that is, speed or reliability.The data of capacitor banks that are defined as control variables are presented in Table 6.6TABLECalculated values for control variables in the modified IEEE 33‐bus distribution networkK‐meansLHStC1 (kVAr)C2 (kVAr)C1 (kVAr)C2 (kVAr)175030030045023003004503003150150450150445045045030053006003004506300600600600745045045045083006007504509600450450450109003007506001145045045045012450750750600134506004506001460060060060015600600450600166007507506001775060030060018450600600750196007504509002060030090060021450450750450224506006006002375075045045024450450450150The network reconfigurations are shown in Figure 9. It can be observed that, in the K‐means and LHS methods, the network structure reconfiguration is implemented, and the radiality of the ADN is preserved. Also, Figure 10 shows the charging /discharging powers of the ESS unit.9FIGUREThe modified IEEE 33‐bus distribution network structure after reconfiguration using the (a) K‐means method (b) LHS method10FIGUREThe charging/discharging powers of the ESS unit during 24 h in the 33‐bus distribution network (Positive values represent the charging mode, and negative values represent the discharging mode of the ESS). (a) K‐means method. (b) LHS methodThe ESS unit is charged and discharged at different hours, considering the electricity price to make a profit for the AND in a financial aspect. Also, the power converter associated with the ESS unit injects the harmonic currents into the network to reduce the THD. In Figure 11, corresponding THDs of some buses are indicated. According to this figure, the ESS reduces the THD and can influence the total THD of the network. It can be noticed in Figure 11 that the capacitor bank located in bus 28 helps reduce the THD even more.11FIGUREThe THD of some buses of the IEEE 33‐bus distribution network using (a) the K‐means method and (b) the LHS method. (a) K‐means method. (b) LHS methodThe 118‐bus distribution networkIn order to test the proposed methods in a large‐scale distribution network, the modified IEEE 118‐bus distribution network has been used. The test system comprises 107 sectionalizing switches and 15 tie switches presented in Figure 12 [65].12FIGUREThe modified IEEE 118‐bus distribution networkFour WT units are already located at 22, 51, 81, and 115 buses with active power rating of 2052 kW, 4487 kW, 4641 kW, and 3282 kW, respectively [66]. Also, three capacitor banks with 15 capacitors (each capacitor capacity is 150 kVAr) are installed at buses 70, 78, and 108 [63]. Three ESS units with 1000 kWh energy capacities are located in buses 26, 58, and 100, as assumed by the authors. The ESSs data are given in Table 3.A scatter plot with two marginal histograms illustrates the MCS samples in Figure 13. This figure indicates the active power demand in bus 109 and the active power generation of the WT unit located in bus 81.13FIGUREScatter plot of the WT generation samples at bus 81 and the active power demand samples in bus 109The harmonic orders of 5, 7, 11, 13, 17, 19, 23, and 25 are modelled in this distribution network, 0.35 coefficient in buses 2–18, 0.4 coefficient in buses 24–80, 0.5 coefficient in buses 85–100, and 0.35 coefficient in bus number 105 are considered for modelling the non‐linear loads.The performance of both K‐means and LHS methods were verified in the 33‐bus distribution network; however, both methods are applied to the 118‐bus system.Scenario 1: As defined scenario1 in Section 7.2, obtained values of the objective functions based on the K‐means and LHS methods are presented in Table 7.7TABLEValues of objective functions in scenario 1, obtained by executing the K‐means and LHS methods in the modified IEEE 118‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS5.505.7025,275.5725,341.22Scenario 2: In this scenario, three ESS units and three capacitor banks of the network are scheduled, and the network structure is reconfigured. In this scenario, optimal control variables are executed by both defined probabilistic methods. The estimated values of the objective functions are represented in Table 8.8TABLEValues of objective functions in scenario 2, obtained by executing the K‐means and LHS methods in the modified IEEE 118‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS4.985.0024,746.4424,907.69According to Tables 7 and 8, the estimated THDs are reduced by 9.45% and 12.28%, and the estimated total costs are decreased by about 2.09% and 1.71% using the K‐means and the LHS methods, respectively. These results prove that the presented approach is reliable and applicable even in large‐scale distribution networks. The optimal values of capacitor banks are given in Table 9.9TABLEDetermined values of the control variables by executing the K‐means and LHS methods in the modified IEEE 118‐bus distribution networkK‐meansLHStC1 (kVAr)C2 (kVAr)C3 (kVAr)C1 (kVAr)C2 (kVAr)C3 (kVAr)116509001050150090075021800105090010501650105039001650105016501350900415001200180013501350105051200165030090010501350612006001500105018001200775090018001350135010508900165012001350900900912001350105010501350105010105012001800120016501500111350165012007501350165012300105075015001050135013105010501050135013501350146001650165010501200180015750120015001050120012001613501200180012009001200171050900750450120013501813501050105013501350900191350900600135010501200201050150010501200120090021120012004501500120010502290075010501200120010502312009009001200750150024165018006009007501200Furthermore, Figures 14 and 15 illustrate the power profiles of the ESS units in the IEEE 118‐bus distribution network that the K‐means and LHS methods optimally obtain.14FIGUREThe charging/discharging powers of the ESS units during 24 h in the 118‐bus distribution network using the K‐means method (Positive values represent the charging mode and negative values represent the discharging mode of the ESS). (a) The ESS unit in bus 26 (ESS1). (b) The ESS unit in bus 58 (ESS2). (c) The ESS unit in bus 100 (ESS3)15FIGUREThe charging/discharging powers of the ESS units during 24 h in the 118‐bus distribution network using the LHS method (Positive values represent the charging mode and negative values represent the discharging mode of the ESS). (a) The ESS unit in bus 26 (ESS1). (b) The ESS unit in bus 58 (ESS2). (c) The ESS unit in bus 100 (ESS3)The reconfigurations of the 118‐bus distribution network are drawn in Figure 16 (K‐means method) and Figure 17 (LHS method). This figure indicates that the applied approach preserved the radiality structure of the network even in large‐scale networks, and no isolated bus or buses exist.16FIGUREThe modified IEEE 118‐bus distribution network structure after reconfiguration using the K‐means method17FIGUREThe modified IEEE 118‐bus distribution network structure after reconfiguration using the LHS methodCONCLUSIONThis paper has presented an optimal operating scheduling (energy storage system(s) charge and discharge scheduling, reactive power scheduling of capacitor banks as well as structure reconfiguration of the network) for active distribution networks, considering the uncertain and harmonically polluted nature of these networks. The uncertain variables include load demands and renewable energy generations. The energy storage system units have been employed to provide the electric power during peak times and reduce harmonic distortion in the charging/discharging process by using the capabilities of the power‐electronic converter employed to connect the ESS to the grid.Also, the objective function has composed of two independent components including the operating costs and the total harmonic distortion. The uncertainty has been handled by two K‐means and Latin hypercube sampling techniques and the optimization problem has been solved using the multi‐objective particle swarm optimization algorithm. Two IEEE 33‐node and IEEE 118‐node test systems have been studied to show the performance of the proposed method and results validation.According to Figure 8, the operating costs and the total harmonic distortion have a conflicting relationship. Also, the K‐means presents some superior characteristics rather than the LHS technique. This figure shows that it is possible to decrease the THD, but it will lead to more operating costs. A solution has been selected using the TOPSIS technique among these non‐dominated solutions. The results for evaluating this specific solution are as follow.For the IEEE 33‐node test system, according to Tables 4 and 5, the estimated THDs are reduced by 19.21% and 20.46%, and the estimated total costs are decreased by about 3.24% and 3.15% using the K‐means and the LHS methods, respectively. Also, the estimated THD is decreased by 9.45% and 12.28%, and the estimated total cost is reduced by about 2.09% and 1.71% using the K‐means and LHS methods, for the IEEE 118‐node test system. However, it is possible to decrease the THD more than these amounts, but it will lead to an increase in the operating cost considerably.NOMENCLATUREAShape componentsBScale componentsNThe number of random input variablesKThe number of samplesNTie−switches${N}_{Tie - switches}$The number of the tie switchesNBranches${N}_{Branches}$The number of the branchesNBuses${N}_{Buses}$The number of the busesNBrachesRe$N_{Braches}^{Re}$The number of branches after reconfiguration of the network structureVimin${\rm{V}}_{\rm{i}}^{{\rm{min}}}$The minimum voltage limit of the ith busVimax${\rm{V}}_{\rm{i}}^{{\rm{max}}}$The maximum voltage limit of the nth busIbmax$I_b^{max}$The maximum current limit of the bth branchA∗${A}^*$The best resultA−${A}^ - $The negative‐best resultDj∗$D_j^*$The distance of every result from the best resultDj−$D_j^ - $The distance from the negative‐best resultCj∗$C_j^*$The relative closeness of the result aj${a}_j$I′$I^{\prime}$Benefit criteriaI′′$I^{\prime\prime}$Cost criteriarij${r}_{ij}$Normalized decision matrixvij$v_{ij}$Weighted normalized decision matrixwThe inertia weightc1Cognitive acceleration coefficientc2social acceleration coefficientr1, r2random variablesxpbest${x}_{pbest}$The individual best of the particlexgbest${x}_{gbest}$The global best of the particleVit$V_i^t$The current velocity in iteration t of ith particleXit$X_i^t$The current position in iteration t of ith particleGk${G}_k$The kth clusterNGk${N}_{{G}_k}$The number of data in the Gkth${G}_k^{th}$ clusterSThe total number of data in each Xi${X}_i$ds${d}_s$The sth observation or scenarioAk${A}_k$The kth agentAl${A}_l$The lth agentE[Fi]${\rm{E}}[ {{F}^i} ]$The ith statistical moment of FXn${X}_n$The nth input random variableXkn${X}_{kn}$The kth sample of the Xn${X}_n$CDFXn$CD{F}_{{X}_n}$The CDF of the Xn${X}_n$CDFXn(Xn)$CD{F}_{{X}_n}( {{X}_n} )$The CDF of the Xn${X}_n$Fk${F}_k$The kth output for scenario or observationf(·)$f( \cdot )$PDF of the load demandHThe highest harmonic frequencywi${w}_i$The equivalent non‐linear loads proportion in the ith busIloadi1$Iload_i^1$The current value of the ith bus at the fundamental frequencyIloadih$Iload_i^h$The current value of the ith bus at the hth harmonic frequencyIessih$Iess_i^h$The current value of the ESS at hth harmonic frequencyIih$I_i^h$The nodal hth harmonic current at ith busVi${V}_i$The RMS value of the voltage in ith busVi1$V_i^1$The voltage of the ith bus at the fundamental frequencyVih$V_i^h$The hth harmonic voltage at ith busYBush$Y_{Bus}^h$network admittance bus at hth harmonic frequencyCtp$C_t^p$The electric power cost at time tPiThe active load demands in the ith busQiThe reactive load demands in the ith busYloadih$Yload_i^h$The admittance of the ith load at hth harmonic frequencyRseij$Rs{e}_{ij}$The resistance of the line between ith bus and jth busXseij$Xs{e}_{ij}$The reactance of the line between ith bus and jth busYseijh$Yse_{ij}^h$The admittance of the line between ith bus and jth bus at hth harmonic frequencyYck1$Yc_k^1$The admittance of the capacitors in base frequencyYckh$Yc_k^h$The admittance of the capacitors at hth harmonic frequencyQk${Q}_k$The reactive power injection of the kth capacitorE[]Expected valueσ[]Standard deviationvWind speedvin${v}_{in}$Cut in speed of the WTvout${v}_{out}$Cut out speed of the WTvrated${v}_{rated}$Rated speed of the WTPrw$P_r^w$Rated power value of the WTCdESS$C_d^{ESS}$The maintenance cost of dth ESSrb${r}_b$The resistance of the bth branchCgDG$C_g^{DG}$The daily maintenance cost of the gth DGPg.minDG$P_{g.min}^{DG}$The minimum active power of the gth DGQm.minC$Q_{m.min}^C$The minimum reactive power of the mth capacitorPg.maxDG$P_{g.max}^{DG}$The maximum active power of the gth DGQm.maxC$Q_{m.max}^C$The maximum reactive power of the mth capacitorEd.minESS$E_{d.min}^{ESS}$The minimum energy of dth ESSRDdESS$RD_d^{ESS}$The maximum ramp‐down rate of the dth ESSRUdESS$RU_d^{ESS}$The maximum ramp‐up rate of the dth ESSEd.maxESS$E_{d.max}^{ESS}$The maximum energy of dth ESSEd.iniESS$E_{d.ini}^{ESS}$Energy storage in the dth ESS on the initial hourηCharging and discharging efficiencyψCh${\psi }^{Ch}$A binary factor that represents charging modesψDisch${\psi }^{Disch}$A binary factor that represents discharging modesTHDvmax$THD_v^{max}$The maximum amount of THD of the voltageΔt$\Delta t$the time intervalEd.tESS$E_{d.t}^{ESS}$Energy value of the dth ESS at time tPd.tESS,Ch$P_{d.t}^{ESS,Ch}$Charge power of the ESS at time tPd.tESS,Disch$P_{d.t}^{ESS,Disch}$Discharge power of the ESS at time tPg,tDG$P_{g,t}^{DG}$The active power generation of the gth DG at time tPt${P}_t$The purchased electric power from the upper network at time tPtloss$P_t^{loss}$The network power loss at time tQm,tC$Q_{m,t}^C$Reactive power of the mth capacitor at time tPg,tDG$P_{g,t}^{DG}$Active power of the gth DG at time tI1.b.t${I}_{1.b.t}$The fundamental current of the bth branch at time tI2.b.t${I}_{2.b.t}$The second harmonic frequency current of the bth branch at time tIH.b.t${I}_{H.b.t}$The Hth harmonic frequency current of the bth branch at time tV1.i.t${V}_{1.i.t}$The fundamental voltage of ith bus at time tV2.i.t${V}_{2.i.t}$The second harmonic frequency voltage of the ith bus at time tV3.i.t${V}_{3.i.t}$The third harmonic frequency voltage of the ith bus at time tVH.i.t${V}_{H.i.t}$The Hth harmonic frequency voltage of the ith bus at time tCope${C}^{ope}$The operational costTHDv$TH{D}_v$The THD of the voltagefn(·)${f}_n( \cdot )$The nth objective functionhk(·)${h}_k( \cdot )$The kth equality equationxi${x}_i$The ith inequality equationxiL$x_i^L$Lower limit of the ith inequality equationxiU$x_i^U$Upper limit of the ith inequality equationFUNDING INFORMATIONThe authors received no specific funding for this work.CONFLICT OF INTERESTThe authors declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported.AUTHOR CONTRIBUTIONSH.E.: Conceptualization; Formal analysis; Software; Writing – original draft. 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Stochastic scheduling of energy storage systems in harmonic polluted active distribution networks

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Abstract

INTRODUCTIONMotivationThere are many non‐linear loads such as power electronic devices in modern active distribution networks (ADN), which propagate current and voltage harmonics in the network. Eliminating voltage and current harmonics is essential, because, these harmonics have considerable influence on the distribution system devices and their functions. The harmonic distortions can increase losses, distort capacitors, cause electrical and electronic components failure, and interfere with communication, control, and protection devices [1].ADNs have a significant portion of renewable energy sources (RESs) due to their undeniable technical and environmental advantages. These sources encounter ADNs with more and more uncertainties due to their unpredictable nature. The other uncertainties such as load variation besides these RES make the state of the ADN completely intermittent. Consequently, operational decisions making for these ADNs are more difficult because their conditions are not deterministic [2].Energy storage systems (ESS) with their programmable power dispatch characteristics can efficiently overcome the intermittent difference between uncertain generation and uncertain load consumption, to a great extent. They are connected to the power system via power electronic converters. The main function of these converters is exchanging energy between the network and the ESS. However, there is a possibility of voltage and/or current harmonic compensation by these converters besides their main function [3, 4].In addition to employing ESSs for efficient operating of ADNs, some other conventional routines such as structure reconfiguration and reactive power scheduling can be used without applying any more costs to the system operator [5, 6].On the other hand, appropriate techniques should be used for handling uncertainties that arise from RESs and loads.The Monte Carlo simulation (MCS) method is the most accurate and simple technique for handling uncertainties but it requires a very large calculation burden. There is a serious restriction in applying this method to online requirements. Instead, data clustering techniques such as the K‐means technique present acceptable results with reasonable speed compared to the MCS method. Also, there are some efficient data sampling techniques such as the Latin hypercube sampling (LHS) technique, which make a reasonable trade‐off between accuracy and speed [7, 8].Optimal reconfiguration, reactive power, and ESS scheduling during a 24‐hour interval considering the uncertainties of the RESs and load demands to reduce total harmonic distortion (THD) and operate costs is a non‐linear optimization problem that can be solved efficiently by multi‐objective evolutionary algorithms. Multi‐objective particle swarm optimization (MOPSO) has a superior ability to solve these optimization problems [9].Literature reviewESS has a crucial role in ANDs' optimal operating scheduling where it can be useful from the economic and technical point of view, and they have been taken into consideration in many studies, up to now.The optimal location and size of ESS in a distribution network were obtained considering the system's reliability in [10]. Ref. [11] considered ESSs and a demand response program to reduce the total operation cost of the distribution network. Ref. [12] proposed a control strategy for ESS connection to the distribution networks by minimizing the real‐time voltage‐tracking mismatch as the objective function.Despite the efficient allocation and scheduling of the ESS in distribution systems, these efforts did not consider the RESs and their uncertainty in power generation, which is one of the essential elements of modern distribution networks.Ref. [13] improved the voltage profile and reduced the active power losses by the ESS active and reactive power injection. The authors in [14] applied a strategy for optimal dispatching of ESS based on peak load shifting to enhance the voltage profile in the distribution network.Similar to [10, 11, 12], these studies did not consider the uncertainty of RESs.Ref. [15] implements ESSs and RESs to minimize the energy not supplied index as a reliability index in a distribution network. Also, an optimal 24‐h operating program by obtaining the active and reactive powers of the ESS has been presented in ref. [16]. This reference focused on minimizing the electricity cost and power losses as well as the safe and reliable operation of the network. This study has also considered the photovoltaics (PVs) in the network. The authors in [17] presented an optimized model for the ESS in peak shaving along with active power losses and voltage fluctuation reduction in distribution networks. This reference considered the stochastic behaviour of the photovoltaic generation output. These studies have not considered non‐linear loads and their harmonic pollution effects. However, non‐linear loads are one of the crucial issues in the present networks.Although, there are some efficient schemes for ESSs in which they can compensate harmonic components besides active and reactive power exchange with the network [18, 19] the scheduling of the ESS has not been considered in harmonic polluted networks, significantly.In ref. [20], the ESS due to the flexible adjusting ability can reduce the harmonic distortion, power losses, and voltage unbalance of the distribution network. Ref. [21] applies the ESS, RES, and electric vehicles parking lot to the distribution network and formulates the optimal harmonic power flow to minimize the total harmonic distortion, voltage deviation, and energy cost.Although refs. [20, 21] paid attention to the harmonic reducing capability of ESSs besides their main function, they did not use reconfiguration and reactive power scheduling besides ESS scheduling.From the optimization algorithm point of view, the evolutionary‐based optimization techniques can efficiently be applied to this problem because of the non‐analytic, non‐linear and non‐convex nature of the problem. There have been various types of evolutionary‐based optimization algorithms. Among them, multi‐objective particle swarm optimization has been attractively taken into consideration, up to now [22, 23, 24]. Multi‐objective techniques present a set of solutions for the problem instead of one final solution. These solutions can be ranked based on the decision maker's evaluation. The technique for order of preference by similarity to ideal solution (TOPSIS) approach, which is one of the multi‐criteria ranking techniques has been used to rank solutions by the preference of decision‐makers, widely [25, 26].From the probabilistic assessment of ADNs in the presence of various uncertainties, the MCS technique is the most accurate and simple method for uncertainty evaluation [9]. However, there is a serious restriction in using the MCS method inside the evolutionary‐based optimization algorithms [27]. Instead, some approximate methods such as the K‐means data clustering technique or some efficient sampling techniques such as the LHS method have been efficiently used in combination with the evolutionary‐based algorithms [28, 29].As said, there is no comprehensive study for ESSs scheduling considering the non‐linear and uncertain nature of the ADN, simultaneously. This paper presents ESSs operating scheduling considering the uncertainty and non‐linearity of the AND. The THD and operating costs are two objectives in a multi‐objective optimization framework solved by MOPSO. Also, the TOPSIS is used for selecting the final solution.Paper contributionThis study presents an optimal scheduling program for ESSs, reactive power compensators and reconfiguration for reducing the THD and operating costs as objectives. Uncertainty of RESs and load demands are considered using two K‐means and LHS techniques. This optimization problem is solved in a multi‐objective optimization problem framework and finally solved by the MOPSO algorithm.The main contribution of this study is as follows;Considering the non‐linear loads and uncertain variables simultaneously in operating scheduling of ADNs.Using ESSs to eliminate the voltage/current harmonic components besides their power exchange with the network.Obtaining the trade‐off between the THD and the operating costs.Proposing the best operating scheduling using the TOPSIS approach.Also, other key characteristics of this study can be summarized as follows.Reactive power scheduling and reconfiguration are considered besides ESSs scheduling during a 24‐h interval.Uncertainty of RESs and load demands are considered to make results more realistic.Uncertainties of stochastic variables are handled by two K‐means and LHS methods with a comparison between them.The MOPSO is used as a powerful multi‐objective evolutionary‐based optimization algorithm.This method presents a more efficient operating scheduling for ESSs. The conventional operating scheduling only takes the active power exchange between ESSs and the network; however, this scheme reduces the voltage harmonic components besides their primary function, that is, active power exchange. This makes it possible to simultaneously reduce the operating costs and the THD. However, these two objectives may have a conflicting relationship, and it would be necessary to take an appropriate trade‐off between operating costs and the THD. The MOPSO algorithm minimizes the objectives and finally keeps a trade‐off region between these two objectives. A distribution system operator can have its own choice between operating costs and the THD based on its preferences.Paper structureThis paper is organized as follows. In Sections 2 and 3, the network and uncertainty modelling are described. The probabilistic analysis consisting of LHS and K‐means methods is presented in Section 4. In Section 5, the solution method is discussed in detail. The objective functions, control variables, and related constraints are presented in Section 6. Finally, the simulation results and conclusion are presented in Sections 7 and 8.NETWORK MODELLINGFigure 1 illustrates the radial diagram of the distribution system. A non‐linear load, an ESS, and a capacitor have been connected to ith, jth, and kth bus, respectively. However, there can be one or more of these components in one specific bus, but in this figure, we connected each component to separate buses to easily explain the modeling details.1FIGUREThe radial distribution network diagramFigure 2 illustrates the equivalent scheme of the distribution system in hth$ht{h}$ harmonic frequency.2FIGUREThe equivalent diagram of the distribution system in hth harmonic frequencyNon‐linear loadsLoads are assumed to have two linear and non‐linear parts. The ratio of linear loads in the ith bus is (1−wi)$( {1 - {w}_i} )$. Besides, the values of the fundamental and hth harmonic currents are determined using the Equations (1) and (2).1Iloadi1=Pi−jQiVi1∗\begin{equation} Iload_i^1 = \frac{{{P}_i - j{Q}_i}}{{V_i^{1*}}} \end{equation}2Iloadih=wiIloadi1hh=1,2,…,H\begin{eqnarray} Iload_i^h &=& {w}_i \frac{{Iload_i^1}}{h}\nonumber\\ h &=& 1,2, \ldots , H \end{eqnarray}where Iloadi1$Iload_i^1$ and Iloadih$Iload_i^h$ are the current value of the ith bus at the fundamental frequency and hth harmonic frequency.Pi${P}_i$ and Qi${Q}_i$ are the active and reactive load demands in the ith bus.Vi1$V_i^1$ is the voltage of the ith bus at the fundamental frequency.wi${w}_i$ is the non‐linear loads proportion in the ith bus.h, and H are harmonic index, and the highest harmonic frequency.Admittances of the loads at hth harmonic frequency is expressed as (3) [30].3Yloadih=(1−wi)Pi−jQi/h|Vi1|2\begin{equation} Yload_i^h = ( 1 - {w}_i)\frac{{P}_i - j{Q}_i/ h}{{| {V_i^1} |}^2} \end{equation}where Yloadih$Yload_i^h$ is the admittance of the ith load at hth harmonic frequency.In addition, the feeder impedance in hth harmonic is obtained by Equation (4).4Yseijh=1/Rseij+j·h·Xseij\begin{equation} Yse_{ij}^h = 1/\left( {Rs{e}_{ij} + j \cdot h \cdot Xs{e}_{ij}} \right) \end{equation}where Yseijh$Yse_{ij}^h$ is the admittance of the line between ith bus and jth bus at hth harmonic frequency.Rseij$Rs{e}_{ij}$ and Xseij$Xs{e}_{ij}$ are the resistance and reactance of the line between ith bus and jth bus.ESSHere, a voltage source converter (VSC)‐based model for ESS [31] is used, in which VSC can control both active power and harmonic currents.The main function of these converters is exchanging active power between the network and the ESS. This variable is shown by PjESS$P_j^{ESS}$ in Figure 2. However, it has the possibility of current harmonic compensation which is shown by Iessjh$Iess_j^h$ in Figure 2. According to Equation (5), this study assumes that the ESS can completely absorb the current harmonic components in the local bus to which is connected [19].5Iessih=Iloadih\begin{equation} Iess_i^h = Iload_i^h \end{equation}where Iessih$Iess_i^h$ is the current value of the ESS at hth harmonic frequency.Reactive power compensationAn admittance model was used for the capacitor in each harmonic frequency. The admittance at the fundamental and the hth harmonic frequency is obtained by Equations (6) and (7).6Yck1=jQkVk12\begin{equation} Yc_k^1 = \frac{{j{Q}_k}}{{{{\left| {V_k^1} \right|}}^2}} \end{equation}7Yckh=h·Yck1\begin{equation} Yc_k^h = h \cdot Yc_k^1 \end{equation}where Yck1$Yc_k^1$ and Yckh$Yc_k^h$ are the admittances of the capacitors in base frequency and hth harmonic.Qk${Q}_k$ is the reactive power injection of the kth capacitor.Harmonic load flowEquation (8) represents the voltage hth harmonic component. The total root‐mean‐square (RMS) voltage considering H harmonics can be calculated by equation (9).8V1hV2h⋮Vih=YBush−1I1hI2h⋮Iih\begin{equation} \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {V_1^h}\\[4pt] {V_2^h} \end{array} }\\[4pt] \vdots \\[4pt] {V_i^h} \end{array} } \right] = {\left[ {Y_{Bus}^h} \right]}^{ - 1} \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {I_1^h}\\[4pt] {I_2^h} \end{array} }\\[4pt] \vdots \\[4pt] {I_i^h} \end{array} } \right] \end{equation}9Vi=∑h=1HVih2\begin{equation} \left| {{V}_i} \right| = \sqrt {\mathop \sum \limits_{h = 1}^H {{\left| {V_i^h} \right|}}^2} \end{equation}where Vih$V_i^h$ is the hth harmonic voltage at ith bus.YBush$Y_{Bus}^h$ is network admittance bus at hth harmonic frequency.Vi${V}_i$ is the RMS value of the voltage in ith bus.Iih$I_i^h$ is the nodal hth harmonic current at ith bus [32].UNCERTAINTY MODELLINGLoad demands uncertainty modellingThe load demand is explained by the normal distribution. The probability density function (PDF) of the load demand following the normal distribution is introduced as (10) and (11) [33].10fPi=1σPi2πe−Pi−EPi22σPi2\begin{equation} f \left( {{P}_i} \right) = \frac{1}{{\left( {\sigma \left[ {{P}_i} \right]} \right)\sqrt {2\pi } }}\ {e}^{ - \frac{{{{\left( {{P}_i - E\left[ {{P}_i} \right]} \right)}}^2}}{{2\sigma {{\left[ {{P}_i} \right]}}^2}}} \end{equation}11fQi=1σQi2πe−Qi−EQi22σQi2\begin{equation} f \left( {{Q}_i} \right) = \frac{1}{{\left( {\sigma \left[ {{Q}_i} \right]} \right)\sqrt {2\pi } }}\ {e}^{ - \frac{{{{\left( {{Q}_i - E\left[ {{Q}_i} \right]} \right)}}^2}}{{2\sigma {{\left[ {{Q}_i} \right]}}^2}}} \end{equation}where f(·)$f( \cdot )$ is the PDF of the load demand.E[] is the expected value.σ[] is the standard deviation.Equation (10) indicates the active power demand in ith bus and Equation (11) denotes the reactive power demand in ith bus.WTs power generation uncertainty modellingUsually, the Weibull distribution is used to model the wind speed to obtain probabilistic values of the WT generation output. As exact WT capacity is added at one bus, a Weibull distribution can obtain WT generation output without considering geographical factors. In consequence, the declared parameters are required for Weibull distribution. The Weibull distribution is expressed as (12) [34].12fv=ABvBA−1e−vBAv≥00v<0\begin{equation} f \left( v \right) = \left\{ \def\eqcellsep{&}\begin{array}{ll} \displaystyle\frac{A}{B}{{\left( {\displaystyle\frac{v}{B}} \right)}}^{A - 1} \ {e}^{ - {{\left( {\frac{v}{B}} \right)}}^A} & v \ge 0\\[12pt] 0 & v &lt; 0 \end{array} \right. \end{equation}where A and B are the shape and scale components.v is the wind speed.The generated power of the WT can be computed as (13).13Pwv=0v≤vin∥v≥voutv−vinvrated−vinPrwvin≤v≤vratedPrwvrated≤v≤vout\begin{equation} {P}^w \left( v \right) = \left\{ \def\eqcellsep{&}\begin{array}{ll} 0&{v \le {v}_{in}\parallel v \ge {v}_{out}}\\[9pt] {\displaystyle\frac{{v - {v}_{in}}}{{{v}_{rated} - {v}_{in}}}P_r^w}&{{v}_{in} \le v \le {v}_{rated}}\\[12pt] {P_r^w}&{{v}_{rated} \le v \le {v}_{out}} \end{array} \right. \end{equation}where vin${v}_{in}$ and vout${v}_{out}$ are cut in and cut out speeds.vrated${v}_{rated}$ and Prw$P_r^w$ are rated speed and power of the WT.PROBABILISTIC ANALYSISIt is essential to use the probabilistic load flow (PLF) instead of deterministic load flow (DLF) due to the ever‐increasing uncertainties of modern electrical networks. The network state can be broadly studied considering all probable range of uncertain variables [35, 36]. These uncertain variables can be defined by the PDF) [37].Although, the MCS technique is commonly and simply used to take samples of uncertain variables it suffers from a large computational burden. Efficient methods to take samples of uncertain variables are attractive from the computational burden point of view. These sampling techniques use less number of samples for each uncertain variable, but they still keep the appropriate accuracy. The LHS technique and the K‐means data clustering technique can efficiently reduce the number of samples without losing the accuracy very much. These techniques are discussed in Sections 4.1 and 4.2.Latin hypercube sampling techniqueMckay et al. introduced the LHS as a sampling method to calculate quasi‐random samples of multi‐dimensional probability distribution functions [38]. This method partitions the parameter spaces into regions of equal probability and picks a sample from each region to attain a more even distribution of sample points.The LHS can be explained as follows:X=[X1,X2,…,XN]$X = [ {{X}_1,{X}_2, \ldots ,{X}_N} ]$ is supposed to be an N dimensional input random variable. The cumulative distribution function (CDF) of Xn${X}_n$ is according to (14).14CDFXn=CDFXnXn,n=1,2,…,N\begin{equation} CD\ {F}_{{X}_n} = CD{F}_{{X}_n} \left( {{X}_n} \right),\quad n = \left[ {1, 2, \ldots , N} \right] \end{equation}where Xn${X}_n$ denotes for the nth input random variable.CDFXn$CD{F}_{{X}_n}$ and CDFXn(Xn)$CD{F}_{{X}_n}( {{X}_n} )$ are the CDF of the Xn${X}_n$.N is the number of random input variables.The range of CDFXn$CD{F}_{{X}_n}$ is from 0 to 1. CDFXn$CD{F}_{{X}_n}$ range is divided into N equal and non‐overlapping intervals, where 1/K$1/K$ is the length of each interval for a sample size of K. One value is chosen from every interval in a pre‐determined way (for example the midpoint of each interval) or randomly. Then the inverse function of the (14) is adopted to calculate the corresponding samples of Xn${X}_n$. The kth sample of Xn${X}_n$ is determined by Equation (15).15Xkn=FXn−1k−0.5K,k=1,2,…,K\begin{equation} {X}_{kn} = {F}_{{X}_n}^{ - 1} \left( {\frac{{k - 0.5}}{K}} \right),\quad k = \left[ {1, 2, \ldots , K} \right] \end{equation}where Xkn${X}_{kn}$ is the kth sample of the Xn${X}_n$.K is the number of samples.Repeating Equation (15) for k=[1,2,…,K]$k = [ {1, 2, \ldots , K} ]$ and n=[1,2,…,N]$n = [ {1, 2, \ldots , N} ]$ makes the sampling matrix. The nth column of this matrix, [X1n,X2n,…,XKn]T${[ {{X}_{1n},{X}_{2n}, \ldots ,{X}_{Kn}} ]}^T$ represents the samples of Xn${X}_n$ and the kth row of this matrix Ak=[Xk1,Xk2,…,XkN]${A}_k = [ {{X}_{k1},{X}_{k2}, \ldots ,{X}_{kN}} ]$ represents the kth scenario or observation.In the end of this step, a K×N$K \times N$ sampling matrix with K samples and N input random variable is available [39].Each column is independently permutated to make a completely random combination of variables. The size of the sampling matrix is still a K×N$K \times N$ matrix.Now, the kth row of this matrix Ak=[Xk1,Xk2,…,XkN]${A}_k = [ {{X}_{k1},{X}_{k2}, \ldots ,{X}_{kN}} ]$ that represents kth the scenario or observation is evaluated according to (16).16Fk=FAk\begin{equation} {F}_k = F\left( {{A}_k} \right) \end{equation}where Fk${F}_k$ is the kth output for scenario or observation.The ith statistical moment of F is computed as (17).17EFi=1K∑k=1KFAki\begin{equation} {\rm{E }}\left[ {{F}^i} \right] = \frac{1}{K} \mathop \sum \limits_{k = 1}^K F{\left( {{A}_k} \right)}^i \end{equation}where E[Fi]${\rm{E}}[ {{F}^i} ]$ is ith statistical moment of F.K‐means data clustering algorithmThe concept of clustering was introduced in 1935. The large multi‐dimensional objects or data are separated into subsets called clusters in the clustering process. In a cluster, objects or data are more similar than those in other clusters in particular criteria. So, various criteria such as distance can be employed to determine the similarity of the data.By using data clustering, only limited datasets are examined instead of analysing a large amount of information. Different techniques for data clustering have been presented until now.Mac‐Queen introduced the K‐means algorithm in 1967 [40]; this algorithm is one of the simplest and most popular unmanaged learning algorithms for data clustering. This method can categorize a large number of data and minimizes the total distance between all data from the nearest centre of the cluster [41].The following steps are the procedure for running the K‐means algorithm [42]:The number of clusters (K) is specified.Initial agents Ak${A}_k$ for each cluster are determined randomly with N dimension according to (18).18Ak=Xk1,Xk2,…,XkN,k=1,2,…,K\begin{equation} {A}_k = \left[ {{X}_{k1},{X}_{k2}, \ldots ,{X}_{kN}} \right] , k = \left[ {1, 2, \ldots , K} \right] \end{equation}whereN is the number of random input variables.3.Observations are assigned to clusters by calculating the minimum distance to each agent as:if ds−Ak<|ds−Al|⇒dsεGk${d}_s - {A}_k &lt; | {{d}_s - {A}_l} | \Rightarrow {d}_s\epsilon {G}_k$ (19)wheres=1,2,…,S$s = 1, 2, \ldots , S$ and k,l=1,2,…,K(l≠k)$k,l = 1, 2, \ldots , K ( {l \ne k} )$.S is the total number of data in each Xi${X}_i$.ds${d}_s$ is the sth observation or scenario.Ak${A}_k$ and Al${A}_l$ are the kth and lth agents.Gk${G}_k$ is the kth cluster.Various algorithms such as Euclidean, City block, and Mikowski can be used for calculating the distance of the data.4.Centre point of each cluster is calculated and selected as the new agent:20Ak=∑s∈GkdsNGkk,l=1,2,…,K\begin{equation} {A}_k = \frac{{\mathop \sum \nolimits_{s \in {G}_k} {d}_s}}{{{N}_{{G}_k}}} k,\quad l = 1, 2, \ldots , K \end{equation}NGk${N}_{{G}_k}$ is the number of data in the Gkth${G}_k{\rm{th}}$ cluster.5.If no data point is specified to a new cluster, the proceeding is ended; the 3–5 steps are repeated to discover new data points.6.The probability of each sample is equal to Equation (21):21PAk=NGkS\begin{equation} P \left( {{A}_k} \right) = \frac{{{N}_{{G}_k}}}{S} \end{equation}Also, the ith statistical moment of F is computed as (22).22EFi=∑k=1KPAk·FAki\begin{equation} {\rm{E }}\left[ {{F}^i} \right] = \mathop \sum \limits_{k = 1}^K P\left( {{A}_k} \right) \cdot F{\left( {{A}_k} \right)}^i \end{equation}whereF(Ak)$F( {{A}_k} )$ is obtained from (16).SOLUTION METHODMulti‐objective optimization methodIn multi‐objective optimization methods, the problems are defined as multi objectives and corresponding related constraints. The constraints have to be met in all possible solutions. Multi‐objective optimization is formulated as:23fnx,n=1,2,…,N;\begin{equation} {f}_n\left( x \right), n = 1,2, \ldots ,N; \end{equation}24subjecttogjx≤0,j=1,2,…,J;\begin{equation} {\rm{subject\ to}}\ {g}_j\left( x \right) \le 0,j = 1,2, \ldots ,J; \end{equation}25hkx=0,k=1,2,…,K;\begin{equation} {h}_k \left( x \right) = 0, k = 1,2, \ldots ,K; \end{equation}26xiL≤xi≤xiU,i=1,2,…,M\begin{equation} x_i^L \le {x}_i \le x_i^U, i = 1,2, \ldots ,M \end{equation}Equation (23) defines the objective functions; equality constraints and some inequality constraints are represented in Equations (24, 25). Where x is an indicator of m control variables. Inequality Equation (26) is variable limits to restrict every control variable to take a value within a lower xiL$x_i^L$ and an upper xiU$x_i^U$ limit.In multi‐objective problems, the objectives may be defined in conflict aspects. In these problems set of compromised solutions known as Pareto solutions are obtained, which means there is no single optimal solution.Mathematical and evolutionary approaches are two multi‐objective optimization methods. Mathematical methods have limitations in solving some non‐convex problems, besides the final solution may depend on the initial solution and get trapped in a sub‐optimal solution.The evolutionary algorithms are presented to solve mentioned drawbacks. These algorithms can escape from local optimum solutions due to considering the population of feasible solutions, and they don't need the derivatives of the objective functions, for this reason, they can solve any type of problem.Despite these advantages, some issues related to these algorithms must be mentioned, such as, they reach different solutions in every execution, the obtained solutions cannot guarantee to be global optimum, and they need more computation burden. It must be mentioned these solutions are still beneficial, applicable, and efficient [27].The MOPSO method [43] is used to designate the weight of each factor or criterion properly. The MOPSO algorithm has been used in solving various problems, among which are the utilization related to handling the distribution networks [44, 45]. The collection of possible results is obtained in multi‐objective optimization algorithms. Then, the TOPSIS technique is employed to rank the results based on their suitability. This approach relies on the desired and undesired points' intervals.The MOPSO algorithmIn the particle swarm optimization (PSO) algorithm, a search area‐id is considered with n particles and d dimensions, where Xi(xi1,xi2,…,xid)${X}_i( {{x}_{i1},{x}_{i2}, \ldots ,{x}_{id}} )$, and Vi(vi1,vi2,…,vid)${V}_i( {{v}_{i1},{v}_{i2}, \ldots ,{v}_{id}} )$ are position and velocity of the ith moving particle. Furthermore, Pi(pi1,pi2,…,pid)${P}_i( {{p}_{i1},{p}_{i2}, \ldots ,{p}_{id}} )$, and Gi(gi1,gi2,…,gid)${G}_i( {{g}_{i1},{g}_{i2}, \ldots ,{g}_{id}} )$ are particle best and global best while their own best position specifies them, and general best performance of the particles. Each particle modifies its location by current velocities and positions, the interval between the pbest$pbest$ and current position, and the gbest$gbest$ and current position.27Vit+1=wVit+c1r1xpbest−Xit+wVit+c2r2xgbest−Xit\begin{eqnarray} V_i^{t + 1} &=& wV_i^t + {c}_1{r}_1\left( {{x}_{pbest} - X_i^t} \right) + wV_i^t\nonumber\\ && +\ {c}_2{r}_2\left( {{x}_{gbest} - X_i^t} \right) \end{eqnarray}28Xit+1=Xit+Vit+1\begin{equation} X_i^{t + 1} = X_i^t + V_i^{t + 1} \end{equation}wherew is the inertia weight.c1, and c2 are cognitive acceleration and social acceleration coefficients.xpbest${x}_{pbest}$ and xgbest${x}_{gbest}$ are the individual and global best of the particle.r1 and r2 are the random variables.Vit$V_i^t$, and Xit$X_i^t$ are the current velocity and position in iteration t of ith particle.The particle's velocity and position are renewed similarly at the PSO and MOPSO algorithms. The PSO and MOPSO algorithms are different in some aspects: Global and individual leaders' selection and update methods are different. Besides, the Pareto optimal solution set is employed to obtain the solution in the MOPSO algorithm. Beyond, the MOPSO algorithm needs a repository for saving the non‐dominated solutions obtained by the input variables to be established. Several researchers have studied the selection and updating of global and individual leaders, the parameter settings, and the establishment of a repository and have achieved some results [46–48]. According to these results, new strategies can be integrated with the MOPSO algorithm and improve the efficiency of this algorithm in solving complex problems and reducing execution time. In addition, this algorithm is still encountered challenges such as the ineffectiveness of exploring the gbest$gbest$ solution and prone to premature convergence. The diagram of the MOPSO algorithm according to dominance criteria is illustrated in Figure 3.3FIGUREMOPSO algorithm flowchartThe TOPSIS methodThe number of results has to be compared and deliberated by some criteria in multi‐objective optimization problems. Plus, the TOPSIS method aims to assist the decision‐maker in the trade‐off between solutions. So, possible results are usually specified by different criteria, and maybe none of the results satisfies all the criteria. Hence, the result is a relative result based on the decision maker's preference. TOPSIS method is proposed by Chen (1992) [49]. The basic concept is that the determined result must have the maximum interval from the negative‐best result and the minimum interval from the best result. TOPSIS has some advantages, including:TOPSIS is faster and simpler than fuzzy Delphi analytic, hierarchy process, analytic hierarchy process, and simple additive weighting, several criteria are permissible in the decision process, ease of decision‐making involving positive as well as negative criteria [50], and TOPSIS is easy to understand, simple in the calculation, and flexible to utilize, as is to be used in many technological and social areas [51].The following steps are the process of the TOPSIS method:Compute the rij${r}_{ij}$ as a normalized decision matrix by Equation (29).29rij=fij/∑j=1Jfij2j=1,…,J;i=1,…,n\begin{equation} {r}_{ij} = {f}_{ij}/\sqrt {\mathop \sum \limits_{j = 1}^J f_{ij}^2} j = 1, \ldots ,J;i = 1, \ldots ,n \end{equation}Compute the vij$v{ }_{ij}$ as weighted normalized decision matrix by Equation (30).30vij=wirijj=1,…,J;i=1,…,n\begin{equation} v{ }_{ij} = {w}_i\ {r}_{ij}\quad j = 1, \ldots ,J; i = 1, \ldots ,n \end{equation}Specify the negative‐best and best results.31A−=v1−,…,vn−=minvij|i∈I′,maxvij|i∈I′′\begin{eqnarray} {A}^ - = \left\{ {v_1^ - , \ldots ,v_n^ - } \right\} = \left\{ {\left( {\min {v}_{ij}|i \in I^{\prime}} \right),\left( {\max \ {v}_{ij}|i \in I^{\prime\prime}} \right)} \right\} \nonumber\\ \end{eqnarray}32A∗=v1∗,…,vn∗=maxvij|i∈I′,minvij|i∈I′′\begin{eqnarray} {A}^* = \left\{ {v_1^*, \ldots ,v_n^*} \right\} = \left\{ {\left( {\max {v}_{ij}|i \in I^{\prime}} \right),\left( {\min \ {v}_{ij}|i \in I^{\prime\prime}} \right)} \right\} \nonumber\\ \end{eqnarray}where I′$I^{\prime}$ and I′′$I^{\prime\prime}$ are associated with benefit criteria, and cost criteria, respectively.4.Compute the distance measures by the n‐dimensional Euclidean distance. The distance of every result from the best result is calculated as:33Dj∗=∑i=1nvij−vi∗2j=1,…,J.\begin{equation} D_j^* = \sqrt {\mathop \sum \limits_{i = 1}^n {{\left( {{v}_{ij} - v_i^*} \right)}}^2} \quad j = 1, \ldots ,J. \end{equation}Uniformly, the distance from the negative‐best result is calculated as:34Dj−=∑i=1nvij−vi−2j=1,…,J.\begin{equation} D_j^ - = \sqrt {\mathop \sum \limits_{i = 1}^n {{\left( {{v}_{ij} - v_i^ - } \right)}}^2} \quad j = 1, \ldots ,J. \end{equation}5.The relative closeness of the result aj${a}_j$ is determined by:35Cj∗=Dj−/Dj∗+Dj−j=1,…,J.\begin{equation} C_j^* = D_j^ - /\left( {D_j^* + D_j^ - } \right)\quad j = 1, \ldots ,J. \end{equation}6.Rate the preference result.PROBLEM FORMULATIONObjective functionThe main objectives of this study are reducing the total operating costs and the voltage THD of the ADN, which are formulated in detail in the following subsections.The first objective function is the operational costs. The operational costs which are denoted by Cope${C}^{ope}$ consists of: (a) The operational cost of distributed generations (DGs), (b) power losses cost, (c) the maintenance cost of ESS, and (d) the cost of purchasing electric power from the upper network, according to (36).36Cope=∑tPtCtp+PtlossCtp+∑dCdESS+CgDG∑t∑gPg,tDG/24\begin{eqnarray} {C}^{ope} &=& \mathop \sum \limits_t \left( {{P}_tC_t^p + P_t^{loss}C_t^p} \right) + \mathop \sum \limits_d C_d^{ESS}\nonumber\\ && +\ C_g^{DG}\left( {\left( {\mathop \sum \limits_t \mathop \sum \limits_g P_{g,t}^{DG}} \right)/24} \right) \end{eqnarray}where Pt${P}_t$ is the purchased electric power from the upper network at time t.Ptloss$P_t^{loss}$ denotes the network power loss at time t.Pg,tDG$P_{g,t}^{DG}$ is the active power generation of the gth DG at time t.Ctp$C_t^p$ is the purchased electric power cost at time t.CdESS$C_d^{ESS}$ and CgDG$C_g^{DG}$ are the daily maintenance cost of the dth ESS and gth DG.The annual maintenance cost (1, 2, or 3 times per year) can be divided into 365 (number of days in one year) as daily cost and this cost can be considered in day‐ahead studies besides the daily operating costs [52–54]The active power losses of the network at time t are obtained by Equation (37).37Ptloss∑brbI1.b.t2+I2.b.t2+⋯IH.b.t2\begin{equation} P_t^{loss}\mathop \sum \limits_b {r}_b\left( {I_{1.b.t}^2 + I_{2.b.t}^2 + \cdots I_{H.b.t}^2} \right) \end{equation}whereb is branch index.rb${r}_b$ is the resistance of the bth branch.I1.b.t${I}_{1.b.t}$, I2.b.t${I}_{2.b.t}$, and IH.b.t${I}_{H.b.t}$ are fundamental, second, and the Hth harmonic frequency currents of the bth branch at time t.The voltage THD reduction is the second objective function that enhances the power quality of the ADN. The voltage THD is formulated as the RMS value of the voltage harmonic components over the RMS value of the fundamental component of the voltage in IEEE standards. So, the THDv$TH{D}_v$ is represented as a percentage of the fundamental voltage.38THDv=1T×N∑t=1T∑i=1NV2.i.t2+V3.i.t2+⋯VH.i.t2V1.i.t×100\begin{eqnarray} TH{D}_v &=& \frac{1}{{T \times N}} \mathop \sum \limits_{t = 1}^T \left( {\mathop \sum \limits_{i = 1}^N \left( {\frac{{\sqrt {V_{2.i.t}^2 + V_{3.i.t}^2 + \cdots V_{H.i.t}^2} }}{{{V}_{1.i.t}}}} \right)} \right)\nonumber\\ && \times\ 100 \end{eqnarray}whereV1.i.t${V}_{1.i.t}$ denotes the fundamental voltage of ith bus at time t.V2.i.t${V}_{2.i.t}$, V3.i.t${V}_{3.i.t}$, and VH.i.t${V}_{H.i.t}$ are second, third and the Hth harmonic frequency voltages of the ith bus at time t.H is the highest harmofnic frequency.Although adding a constant value in the objective function related to ESS daily cost does not affect the optimization results, considering this constant cost in the objective function might lead to more actual values.Control variablesThe control variables consist of:Reactive power of capacitor banks at time t,Charging/discharging power of the ESSs at time t,Corresponding sectionalizing switches state.ConstraintsBuses voltage constraint:The buses voltages should have remained in the allowed range according to (39)39Vimin≤V1.i.t2+V2.i.t2+⋯VH.i.t2≤Vimax\begin{equation} V_i^{min} \le \sqrt {V_{1.i.t}^2 + V_{2.i.t}^2 + \cdots V_{H.i.t}^2} \le V_i^{max} \end{equation}Vimin$V_i^{min}$ and Vimax$V_i^{max}$ are the minimum and maximum voltage limit of the ith bus.Branches capacity constraint:Feeder currents should be remained in their allowable ratings to preserve the feeders and cables against excessive currents; in this regard, the following constraint should be satisfied:Ibmax$I_b^{max}$ is the maximum current limit of the bth branch.40I1.b.t2+I2.b.t2+⋯IH.b.t2≤Ibmax\begin{equation} \sqrt {I_{1.b.t}^2 + I_{2.b.t}^2 + \cdots I_{H.b.t}^2} \le I_b^{max} \end{equation}DGs and capacitor banks constraints:The DG units and reactive power compensators (capacitors) limits must be satisfied every hour in all scenarios.41Pg.minDG≤Pg,tDG≤Pg.maxDG\begin{equation} P_{g.min}^{DG} \le P_{g,t}^{DG} \le P_{g.max}^{DG} \end{equation}42Qm.minC≤Qm,tC≤Qm.maxC\begin{equation} Q_{m.min}^C \le Q_{m,t}^C \le Q_{m.max}^C \end{equation}Pg.minDG$P_{g.min}^{DG}$ and Pg.maxDG$P_{g.max}^{DG}$ are minimum and maximum active power generation of the gth DG.Qm.minC$Q_{m.min}^C$ and Qm.maxC$Q_{m.max}^C$ are minimum and maximum reactive power of the mth capacitor.Equation (41) is the DG dispatch constraints, and Equation (42) is the reactive power compensator adjusting constraint.Network reconfiguration constraints:The primary constraint in network reconfiguration is its radiality. Usually, the network structure is designed as a poor ring system, so network operations are radial to preserve integrity in distribution networks' protection, stability, and other aspects. Hence, the radiality circumstance must be preserved in the reconfiguration network structure. The tie switches number is determined using the following equation [55]:43NTie−switches=NBranches−NBuses+1\begin{equation} {N}_{Tie - switches} = {N}_{Branches} - {N}_{Buses} + 1 \end{equation}whereNTie−switches${N}_{Tie - switches}$ is the number of the tie switches.NBranches${N}_{Branches}$ is the number of branches.NBuses${N}_{Buses}$ is the number of the buses.The radiality of the distribution networks is preserved by using the graph rules [56]. A tree is a subgraph with (N−1)$( {N - 1} )$ lines. According to these rules, the structure of a network is maintained radial if the following equation is satisfied.44NBrachesRe=Nbuses−1\begin{equation} N_{Braches}^{Re} = {N}_{buses} - 1 \end{equation}NBrachesRe$N_{Braches}^{Re}$ is the number of branches after reconfiguration of the network structure.Eventually, all the new loads must avoid the isolated load chain or isolated load to satisfy the network structure limits (Figure 4).ESS operation constraints:4FIGURETopology structure constraint: The circles are the loads; the solid lines are the present branches; dashed lines are alternative branches. (a) Initial network. (b) The topology that is leading to isolated load and loop. (c) The topology that is leading to isolated load chainThe energy and active power of an ESS are constrained according to (45–48)45Ed.minESS≤Ed.tESS≤Ed.maxESS\begin{equation} E_{d.min}^{ESS} \le E_{d.t}^{ESS} \le E_{d.max}^{ESS} \end{equation}46Ed.tESS=Ed.t−1ESS+Δt×Pd.tESS,Ch×η×ψCh−Δt×Pd.tESS,Dischη×ψDisch\begin{eqnarray} E_{d.t}^{ESS} &=& E_{d.t - 1}^{ESS} + \left( {\Delta t \times P_{d.t}^{ESS,Ch} \times \eta } \right) \times {\psi }^{Ch}\nonumber\\ && -\, \left( {\Delta t \times \frac{{P_{d.t}^{ESS,Disch}}}{\eta }} \right) \times {\psi }^{Disch} \end{eqnarray}47Ed.0ESS=Ed.iniESS\begin{equation} E_{d.0}^{ESS} = E_{d.ini}^{ESS} \end{equation}48−RDdESS≤Pd.tESS−Pd.t−1ESS≤RUdESS\begin{equation} - RD_d^{ESS} \le P_{d.t}^{ESS} - P_{d.t - 1}^{ESS} \le RU_d^{ESS} \end{equation}where Ed.tESS$E_{d.t}^{ESS}$ is the energy of the dth ESS at time t.Pd.tESS,Ch$P_{d.t}^{ESS,Ch}$ and Pd.tESS,Disch$P_{d.t}^{ESS,Disch}$ are the charge and discharge active power of the dth ESS at time t.Pd.tESS$P_{d.t}^{ESS}$ in (48) denotes both Pd.tESS,Ch$P_{d.t}^{ESS,Ch}$ and Pd.tESS,Disch$P_{d.t}^{ESS,Disch}$.Ed.minESS$E_{d.min}^{ESS}$ and Ed.maxESS$E_{d.max}^{ESS}$ are the minimum and maximum energy of dth ESS.Δt$\Delta t$ is the time interval.η is charging and discharging efficiency.ψCh${\psi }^{Ch}$ and ψDisch${\psi }^{Disch}$ are binary factors that represent charging and discharging modes, respectively.Ed.iniESS$E_{d.ini}^{ESS}$ is the initial energy of the dth ESS.RDdESS$RD_d^{ESS}$ and RUdESS$RU_d^{ESS}$ are maximum ramp‐down and ramp‐down rate of the dth ESS.Constraint (45) is the maximum and minimum energy of ESS. Equations (46) and (47) are energy storage constraints between two adjacent hours. Also, (48) is the ramp‐up and ramp‐down limits of the ESS power.THD constraint:The following constraint is defied to remain THDv$TH{D}_v$ in the allowed range that the system operator determines.49THDv≤THDvmax\begin{equation} TH{D}_v \le THD_v^{max} \end{equation}THDvmax$THD_v^{max}$ is the maximum amount of THDv$TH{D}_v$.Proposed methodFigure 5 represents a flowchart of the proposed approach. The flowchart shows that control variables values are generated, and the optimization process begins. Stochastic data is generated, and the desired data are determined using the K‐means or LHS methods. Control variables are applied to the distribution network, and the harmonic load‐flow process determines the wanted components. The values of objective functions are calculated based on each cluster (by applying the K‐means method) or each sample (by applying the LHS method), and the end requirements of the algorithm process are investigated.5FIGURESteps of the proposed methodThe estimated values of the defined objective functions are computed according to the obtained values of the objective functions and the probable coefficients of clusters or samples.At the end of the process and obtain different results for the objective functions, the ranking method based on the weight coefficient of each objective function will be performed (using the TOPSIS method).SIMULATION RESULTSIn order to test the proposed methods, the modified IEEE 33‐bus and 118‐bus distribution networks have been used. The obtained results are represented in Sections 7.2 and 7.3, respectively.AssumptionsThe load curve depicts the changes of load power demand throughout a particular period. In this paper, the residential daily load curve is utilized as in [57]. The hourly price of purchased energy is obtained from the Nord Pool grid [58]. Moreover, DGs operational cost data is obtained from [59], and the cost of the DG unit using the discount rate is converted to current year cost values.The employed DG generators are WT generators with stochastic nature in producing electric power. This means the WTs power generation are related to wind speed, while Weibull distribution can be utilized to model the wind speed behaviour. So, the Weibull distribution parameters, k, and λ are equal to 3 and 8. The rest of the data relevant to WTs is presented in Table 1.1TABLEThe related parameters of the WTsParameterValueCut‐in speedvinc(m/s)$v_{in}^c ( {{\rm{m}}/{\rm{s}}} )$3Cut‐out speedvoutc(m/s)$v_{out}^c ( {{\rm{m}}/{\rm{s}}} )$25Rated speedvrated(m/s)${v}_{rated} ( {{\rm{m}}/{\rm{s}}} )$13The power demand loads are considered uncertain loads, and active and reactive power loads models are based on the normal distribution. Their expected and standard deviation values are considered to be equal to the nominal active power value of buses at the base state and 10% of their expected value [60].The WT units, capacitor banks, and ESSs discharge process are simulated as negative loads. Besides, the ESS charge process is assumed as a positive load. According to mentioned assumptions, there is no control on voltage.The number of generated data in the MCS method is 10,000 samples for each uncertain input variable, which are reduced to 10 and 50 datasets by applying the K‐means and LHS methods, respectively.In this paper, the MOPSO algorithm calculates the minimum values for total cost and THD index to determine a non‐inferior solution. Then, non‐dominated solutions are ranked using the proper approach to help the operator specify the best solution. Decision‐making between the results obtained in the Pareto‐front is challenging for the operator. In this regard, the priority and viewpoint of the active distribution operator will ascertain the final proper solution. Therefore, the TOPSIS approach can help the operator rank and distinguish the ideal solutions. For that reason, the network operator determines the proper compromise solution using weight coefficients. In this paper, both weight coefficient values for estimated THD and estimated total cost are considered equal to 0.5.The proposed method is conducted on a PC with an Intel(R) Core (TM) i7–8500 K CPU and 16.00 GB of memory. The simulation time by implementing the K‐means and LHS methods with an iteration number of 100 and population size of 100 in both 33‐bus (with 77 control variables) and 118‐bus (with 159 control variables) distribution networks are given in Table 2.2TABLEThe simulation time by implementing the K‐means and LHS methods in the 33‐bus and 118‐bus distribution networksDistribution network33‐bus118‐busK‐meansClusters number1010Simulation time (s)620792LHSSamples number5050Simulation time (s)28743094The 33‐bus distribution networkIn order to test the proposed methods, the modified IEEE 33‐bus distribution network has been used. This network's total active and reactive loads are 3.715 MW and 2.3 MVAr, respectively; the voltage rating is 12.66 kV. The test system comprises 32 sectionalizing switches and five tie switches presented in Figure 6 [61].6FIGUREThe modified IEEE 33‐bus distribution networkFour WT units are installed on buses number 10, 16, 17, and 30 with active power rating equal to 500 kW, 200 kW, 150 kW, and 200 kW according to [62].Figure 7 illustrates the scatter plot with two marginal histograms to represent the samples of WT generations at bus 17 and active power demands at bus 10. This scatter plot is drawn based on corresponding data at 9 PM.7FIGUREScatter plot of the active power demand samples in bus 10 and the WT generation samples at bus 17Also, two capacitor banks are installed at bases number 3 and 28 with the reactive power capacity of 1050 kVAr (7 capacitors with 150 kVAr capacity), according to [63].One ESS unit is already located in bus 22 [64] with an energy capacity of 1000 kWh. The considered values for ESS parameters are represented in Table 3. The daily maintenance cost for the ESS is approximately according to [52] updated considering the interest rate of 7%.3TABLEAssumed data for operation specifications of the ESS with 1000 kWh energy ratingParametersValueESS daily maintenance cost23.4 €/dayESS ramping up/down rate60 kW/hESS charging/discharging efficiency95 %The harmonic orders are assumed to be 5, 7, 11, 13, 17, 19, 23, and 25 in both scenarios similarly (0.4 coefficient in buses 5–8, 0.3 coefficient in buses 11–15, 0.5 coefficient in buses 25–31, and 0.5 coefficient in bus number 20 are considered for modelling the non‐linear loads). Two scenarios are defined to test the proposed strategy.Scenario 1: In this scenario, the ESS unit and capacitor banks are not managed, and the network structure is not modified. So, considering the uncertainties of the WT units and demand loads, the objective functions are calculated. It should be noted that scenario 1 is the base scenario and is considered to compare with the proposed strategy. The obtained objective functions with consideration of the K‐means and LHS samples are presented in Table 4.4TABLEValues of objective functions in scenario 1, obtained by executing the K‐means and LHS methods in the modified IEEE 33‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS5.105.132526.162534.17Scenario 2: In this scenario, charging/discharging power of the ESS unit, and reactive power of the capacitor banks are obtained by the MOPSO algorithm, besides the reconfiguration of the network structure is applied. A set of the optimal objective functions values is executed using the MOPSO algorithm.Figure 8 indicates the estimated Pareto optimal solution points consisting of the THD and the total cost in the objective area. The presented result points in Figure 8 could assist the system operator in making decisions from economic and performance points of view.8FIGUREComparison of K‐means and LHS method based on obtained non‐dominated Pareto front solutions in the 33‐bus distribution networkThe best solutions are selected by the TOPSIS technique and are presented in Table 5. The presented optimal values are selected using an equal 0.5 coefficient value.5TABLEValues of objective functions in scenario 2, obtained by executing the K‐means and LHS methods in the modified IEEE 33‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS4.124.082444.152454.20According to Tables 4 and 5, the estimated THDs are reduced by 19.21% and 20.46%, and the estimated total costs are decreased by about 3.24% and 3.15% using the K‐means and the LHS methods, respectively. These results verify the effectiveness of the proposed strategy using both probabilistic methods, that each method can be used based on operator needs, that is, speed or reliability.The data of capacitor banks that are defined as control variables are presented in Table 6.6TABLECalculated values for control variables in the modified IEEE 33‐bus distribution networkK‐meansLHStC1 (kVAr)C2 (kVAr)C1 (kVAr)C2 (kVAr)175030030045023003004503003150150450150445045045030053006003004506300600600600745045045045083006007504509600450450450109003007506001145045045045012450750750600134506004506001460060060060015600600450600166007507506001775060030060018450600600750196007504509002060030090060021450450750450224506006006002375075045045024450450450150The network reconfigurations are shown in Figure 9. It can be observed that, in the K‐means and LHS methods, the network structure reconfiguration is implemented, and the radiality of the ADN is preserved. Also, Figure 10 shows the charging /discharging powers of the ESS unit.9FIGUREThe modified IEEE 33‐bus distribution network structure after reconfiguration using the (a) K‐means method (b) LHS method10FIGUREThe charging/discharging powers of the ESS unit during 24 h in the 33‐bus distribution network (Positive values represent the charging mode, and negative values represent the discharging mode of the ESS). (a) K‐means method. (b) LHS methodThe ESS unit is charged and discharged at different hours, considering the electricity price to make a profit for the AND in a financial aspect. Also, the power converter associated with the ESS unit injects the harmonic currents into the network to reduce the THD. In Figure 11, corresponding THDs of some buses are indicated. According to this figure, the ESS reduces the THD and can influence the total THD of the network. It can be noticed in Figure 11 that the capacitor bank located in bus 28 helps reduce the THD even more.11FIGUREThe THD of some buses of the IEEE 33‐bus distribution network using (a) the K‐means method and (b) the LHS method. (a) K‐means method. (b) LHS methodThe 118‐bus distribution networkIn order to test the proposed methods in a large‐scale distribution network, the modified IEEE 118‐bus distribution network has been used. The test system comprises 107 sectionalizing switches and 15 tie switches presented in Figure 12 [65].12FIGUREThe modified IEEE 118‐bus distribution networkFour WT units are already located at 22, 51, 81, and 115 buses with active power rating of 2052 kW, 4487 kW, 4641 kW, and 3282 kW, respectively [66]. Also, three capacitor banks with 15 capacitors (each capacitor capacity is 150 kVAr) are installed at buses 70, 78, and 108 [63]. Three ESS units with 1000 kWh energy capacities are located in buses 26, 58, and 100, as assumed by the authors. The ESSs data are given in Table 3.A scatter plot with two marginal histograms illustrates the MCS samples in Figure 13. This figure indicates the active power demand in bus 109 and the active power generation of the WT unit located in bus 81.13FIGUREScatter plot of the WT generation samples at bus 81 and the active power demand samples in bus 109The harmonic orders of 5, 7, 11, 13, 17, 19, 23, and 25 are modelled in this distribution network, 0.35 coefficient in buses 2–18, 0.4 coefficient in buses 24–80, 0.5 coefficient in buses 85–100, and 0.35 coefficient in bus number 105 are considered for modelling the non‐linear loads.The performance of both K‐means and LHS methods were verified in the 33‐bus distribution network; however, both methods are applied to the 118‐bus system.Scenario 1: As defined scenario1 in Section 7.2, obtained values of the objective functions based on the K‐means and LHS methods are presented in Table 7.7TABLEValues of objective functions in scenario 1, obtained by executing the K‐means and LHS methods in the modified IEEE 118‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS5.505.7025,275.5725,341.22Scenario 2: In this scenario, three ESS units and three capacitor banks of the network are scheduled, and the network structure is reconfigured. In this scenario, optimal control variables are executed by both defined probabilistic methods. The estimated values of the objective functions are represented in Table 8.8TABLEValues of objective functions in scenario 2, obtained by executing the K‐means and LHS methods in the modified IEEE 118‐bus distribution networkEstimated THD%Estimated total costEuroK‐meansLHSK‐meansLHS4.985.0024,746.4424,907.69According to Tables 7 and 8, the estimated THDs are reduced by 9.45% and 12.28%, and the estimated total costs are decreased by about 2.09% and 1.71% using the K‐means and the LHS methods, respectively. These results prove that the presented approach is reliable and applicable even in large‐scale distribution networks. The optimal values of capacitor banks are given in Table 9.9TABLEDetermined values of the control variables by executing the K‐means and LHS methods in the modified IEEE 118‐bus distribution networkK‐meansLHStC1 (kVAr)C2 (kVAr)C3 (kVAr)C1 (kVAr)C2 (kVAr)C3 (kVAr)116509001050150090075021800105090010501650105039001650105016501350900415001200180013501350105051200165030090010501350612006001500105018001200775090018001350135010508900165012001350900900912001350105010501350105010105012001800120016501500111350165012007501350165012300105075015001050135013105010501050135013501350146001650165010501200180015750120015001050120012001613501200180012009001200171050900750450120013501813501050105013501350900191350900600135010501200201050150010501200120090021120012004501500120010502290075010501200120010502312009009001200750150024165018006009007501200Furthermore, Figures 14 and 15 illustrate the power profiles of the ESS units in the IEEE 118‐bus distribution network that the K‐means and LHS methods optimally obtain.14FIGUREThe charging/discharging powers of the ESS units during 24 h in the 118‐bus distribution network using the K‐means method (Positive values represent the charging mode and negative values represent the discharging mode of the ESS). (a) The ESS unit in bus 26 (ESS1). (b) The ESS unit in bus 58 (ESS2). (c) The ESS unit in bus 100 (ESS3)15FIGUREThe charging/discharging powers of the ESS units during 24 h in the 118‐bus distribution network using the LHS method (Positive values represent the charging mode and negative values represent the discharging mode of the ESS). (a) The ESS unit in bus 26 (ESS1). (b) The ESS unit in bus 58 (ESS2). (c) The ESS unit in bus 100 (ESS3)The reconfigurations of the 118‐bus distribution network are drawn in Figure 16 (K‐means method) and Figure 17 (LHS method). This figure indicates that the applied approach preserved the radiality structure of the network even in large‐scale networks, and no isolated bus or buses exist.16FIGUREThe modified IEEE 118‐bus distribution network structure after reconfiguration using the K‐means method17FIGUREThe modified IEEE 118‐bus distribution network structure after reconfiguration using the LHS methodCONCLUSIONThis paper has presented an optimal operating scheduling (energy storage system(s) charge and discharge scheduling, reactive power scheduling of capacitor banks as well as structure reconfiguration of the network) for active distribution networks, considering the uncertain and harmonically polluted nature of these networks. The uncertain variables include load demands and renewable energy generations. The energy storage system units have been employed to provide the electric power during peak times and reduce harmonic distortion in the charging/discharging process by using the capabilities of the power‐electronic converter employed to connect the ESS to the grid.Also, the objective function has composed of two independent components including the operating costs and the total harmonic distortion. The uncertainty has been handled by two K‐means and Latin hypercube sampling techniques and the optimization problem has been solved using the multi‐objective particle swarm optimization algorithm. Two IEEE 33‐node and IEEE 118‐node test systems have been studied to show the performance of the proposed method and results validation.According to Figure 8, the operating costs and the total harmonic distortion have a conflicting relationship. Also, the K‐means presents some superior characteristics rather than the LHS technique. This figure shows that it is possible to decrease the THD, but it will lead to more operating costs. A solution has been selected using the TOPSIS technique among these non‐dominated solutions. The results for evaluating this specific solution are as follow.For the IEEE 33‐node test system, according to Tables 4 and 5, the estimated THDs are reduced by 19.21% and 20.46%, and the estimated total costs are decreased by about 3.24% and 3.15% using the K‐means and the LHS methods, respectively. Also, the estimated THD is decreased by 9.45% and 12.28%, and the estimated total cost is reduced by about 2.09% and 1.71% using the K‐means and LHS methods, for the IEEE 118‐node test system. However, it is possible to decrease the THD more than these amounts, but it will lead to an increase in the operating cost considerably.NOMENCLATUREAShape componentsBScale componentsNThe number of random input variablesKThe number of samplesNTie−switches${N}_{Tie - switches}$The number of the tie switchesNBranches${N}_{Branches}$The number of the branchesNBuses${N}_{Buses}$The number of the busesNBrachesRe$N_{Braches}^{Re}$The number of branches after reconfiguration of the network structureVimin${\rm{V}}_{\rm{i}}^{{\rm{min}}}$The minimum voltage limit of the ith busVimax${\rm{V}}_{\rm{i}}^{{\rm{max}}}$The maximum voltage limit of the nth busIbmax$I_b^{max}$The maximum current limit of the bth branchA∗${A}^*$The best resultA−${A}^ - $The negative‐best resultDj∗$D_j^*$The distance of every result from the best resultDj−$D_j^ - $The distance from the negative‐best resultCj∗$C_j^*$The relative closeness of the result aj${a}_j$I′$I^{\prime}$Benefit criteriaI′′$I^{\prime\prime}$Cost criteriarij${r}_{ij}$Normalized decision matrixvij$v_{ij}$Weighted normalized decision matrixwThe inertia weightc1Cognitive acceleration coefficientc2social acceleration coefficientr1, r2random variablesxpbest${x}_{pbest}$The individual best of the particlexgbest${x}_{gbest}$The global best of the particleVit$V_i^t$The current velocity in iteration t of ith particleXit$X_i^t$The current position in iteration t of ith particleGk${G}_k$The kth clusterNGk${N}_{{G}_k}$The number of data in the Gkth${G}_k^{th}$ clusterSThe total number of data in each Xi${X}_i$ds${d}_s$The sth observation or scenarioAk${A}_k$The kth agentAl${A}_l$The lth agentE[Fi]${\rm{E}}[ {{F}^i} ]$The ith statistical moment of FXn${X}_n$The nth input random variableXkn${X}_{kn}$The kth sample of the Xn${X}_n$CDFXn$CD{F}_{{X}_n}$The CDF of the Xn${X}_n$CDFXn(Xn)$CD{F}_{{X}_n}( {{X}_n} )$The CDF of the Xn${X}_n$Fk${F}_k$The kth output for scenario or observationf(·)$f( \cdot )$PDF of the load demandHThe highest harmonic frequencywi${w}_i$The equivalent non‐linear loads proportion in the ith busIloadi1$Iload_i^1$The current value of the ith bus at the fundamental frequencyIloadih$Iload_i^h$The current value of the ith bus at the hth harmonic frequencyIessih$Iess_i^h$The current value of the ESS at hth harmonic frequencyIih$I_i^h$The nodal hth harmonic current at ith busVi${V}_i$The RMS value of the voltage in ith busVi1$V_i^1$The voltage of the ith bus at the fundamental frequencyVih$V_i^h$The hth harmonic voltage at ith busYBush$Y_{Bus}^h$network admittance bus at hth harmonic frequencyCtp$C_t^p$The electric power cost at time tPiThe active load demands in the ith busQiThe reactive load demands in the ith busYloadih$Yload_i^h$The admittance of the ith load at hth harmonic frequencyRseij$Rs{e}_{ij}$The resistance of the line between ith bus and jth busXseij$Xs{e}_{ij}$The reactance of the line between ith bus and jth busYseijh$Yse_{ij}^h$The admittance of the line between ith bus and jth bus at hth harmonic frequencyYck1$Yc_k^1$The admittance of the capacitors in base frequencyYckh$Yc_k^h$The admittance of the capacitors at hth harmonic frequencyQk${Q}_k$The reactive power injection of the kth capacitorE[]Expected valueσ[]Standard deviationvWind speedvin${v}_{in}$Cut in speed of the WTvout${v}_{out}$Cut out speed of the WTvrated${v}_{rated}$Rated speed of the WTPrw$P_r^w$Rated power value of the WTCdESS$C_d^{ESS}$The maintenance cost of dth ESSrb${r}_b$The resistance of the bth branchCgDG$C_g^{DG}$The daily maintenance cost of the gth DGPg.minDG$P_{g.min}^{DG}$The minimum active power of the gth DGQm.minC$Q_{m.min}^C$The minimum reactive power of the mth capacitorPg.maxDG$P_{g.max}^{DG}$The maximum active power of the gth DGQm.maxC$Q_{m.max}^C$The maximum reactive power of the mth capacitorEd.minESS$E_{d.min}^{ESS}$The minimum energy of dth ESSRDdESS$RD_d^{ESS}$The maximum ramp‐down rate of the dth ESSRUdESS$RU_d^{ESS}$The maximum ramp‐up rate of the dth ESSEd.maxESS$E_{d.max}^{ESS}$The maximum energy of dth ESSEd.iniESS$E_{d.ini}^{ESS}$Energy storage in the dth ESS on the initial hourηCharging and discharging efficiencyψCh${\psi }^{Ch}$A binary factor that represents charging modesψDisch${\psi }^{Disch}$A binary factor that represents discharging modesTHDvmax$THD_v^{max}$The maximum amount of THD of the voltageΔt$\Delta t$the time intervalEd.tESS$E_{d.t}^{ESS}$Energy value of the dth ESS at time tPd.tESS,Ch$P_{d.t}^{ESS,Ch}$Charge power of the ESS at time tPd.tESS,Disch$P_{d.t}^{ESS,Disch}$Discharge power of the ESS at time tPg,tDG$P_{g,t}^{DG}$The active power generation of the gth DG at time tPt${P}_t$The purchased electric power from the upper network at time tPtloss$P_t^{loss}$The network power loss at time tQm,tC$Q_{m,t}^C$Reactive power of the mth capacitor at time tPg,tDG$P_{g,t}^{DG}$Active power of the gth DG at time tI1.b.t${I}_{1.b.t}$The fundamental current of the bth branch at time tI2.b.t${I}_{2.b.t}$The second harmonic frequency current of the bth branch at time tIH.b.t${I}_{H.b.t}$The Hth harmonic frequency current of the bth branch at time tV1.i.t${V}_{1.i.t}$The fundamental voltage of ith bus at time tV2.i.t${V}_{2.i.t}$The second harmonic frequency voltage of the ith bus at time tV3.i.t${V}_{3.i.t}$The third harmonic frequency voltage of the ith bus at time tVH.i.t${V}_{H.i.t}$The Hth harmonic frequency voltage of the ith bus at time tCope${C}^{ope}$The operational costTHDv$TH{D}_v$The THD of the voltagefn(·)${f}_n( \cdot )$The nth objective functionhk(·)${h}_k( \cdot )$The kth equality equationxi${x}_i$The ith inequality equationxiL$x_i^L$Lower limit of the ith inequality equationxiU$x_i^U$Upper limit of the ith inequality equationFUNDING INFORMATIONThe authors received no specific funding for this work.CONFLICT OF INTERESTThe authors declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported.AUTHOR CONTRIBUTIONSH.E.: Conceptualization; Formal analysis; Software; Writing – original draft. 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