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Solution of constrained mixed‐integer multi‐objective optimal power flow problem considering the hybrid multi‐objective evolutionary algorithm

Solution of constrained mixed‐integer multi‐objective optimal power flow problem considering the... INTRODUCTIONPreliminary knowledge of OPF and literature reviewThe goal of the Optimal Power Flow (OPF) is to find the setting of control variables for the safe and economical operation of the power systems, which optimize a certain objective function while satisfying a set of operational constraints. Control variables consist of the active power of generators except for the generator at a slack bus, a voltage level of all the generators, transformer tap ratio, and shunt VAR injections. These variables are continuous (generator's power and their voltages) and discrete (tap ratio of transformer and shunt Var injection). In earlier days, the implementation of OPF included a single objective function, primarily reducing fuel cost (FC). In fact, along with the FC, there are some other objectives such as reduction of active power loss (PLoss${P}_{Loss}$), load bus voltage deviation (VD), emission (E) and enhancement of Lindex have simultaneously been used in the recent literature for finding the optimal values of control variable of OPF problem [1].In search of global optimal values of objective functions, the OPF program must satisfy the system constraints. In the literature, various equality and inequality constraints have been utilized to solve single and multi‐objective OPF problems. These are active and reactive power balanced equality constraints (this type of constraint must be satisfied during the convergence of load flow) and the most commonly used inequality constraint are MVA branch capacity, bus angles, and voltage magnitudes, generator active and reactive power injections, MVA flow of transmission line, voltages level at each bus.OPF is a constrained type, mixed‐integer, non‐linear, and multi‐objective optimization problem. Conventional methods such as linear programming (LP), quadratic programming (QP) and non‐linear programming (NLP) have been implemented to address the solution of OPF problems subject to minimizing the cost of generation only. During the optimization process, these techniques are considered some theoretical assumptions such as the problem must be convex, continuous, and differential. However, these assumptions may not be satisfied in the multi‐objective OPF problem. Besides, the convergence to the global or local optimal solution of these techniques is highly dependent on the selected initial guess [2]. Continuous LP, QP, and NLP formulations cannot accurately model discrete control variables, such as transformer tap ratios or switched capacitor banks. Mixed‐integer linear programming (MILP) [3] techniques were introduced to solve this problem. However, the non‐linearity of the power system cannot be fully represented by MILP formulations, and therefore cause inherent inaccuracy [1].Having overcome these drawbacks, Evolutionary algorithms (EA) such as genetic algorithm (GA) [4], evolutionary programming (EP) [5], developed Gray wolf optimization (DGWO) [6]; tabu search (TS) [7], improved krill herd algorithm (IKHA) [8]; particle swarm optimization (PSO) [9], jaya algorithm [10], differential evolution (DE) [11, 12], and modified coyote optimization algorithm (MCOA) [13] have been employed for the solution to the single objective OPF problem. The results reported in the previous literature were promising and encouraged further research in this area. Practically, OPF is a multi‐objective optimization problem. Multi‐objective OPF (MOPF) problem is solved either by considering the weighted sum approach or by a Pareto‐based non‐dominated one. In the weighted sum approach, multi‐objective functions are transformed into a single objective function by assigning a predetermined weight factor to each objective function. Whereas, in the Pareto‐based method all the objective functions are optimized simultaneously and the primary reason is to emphasize this technique because it can find multiple Pareto optimal solutions in a single simulation run. However, in weighted sum method leads to only one Pareto optimal solution with a set of specific weight factors for each objective function. A set of non‐dominated solutions (PF) in a single simulation run is preferred as compared to a single solution. Finding the Pareto front (PF) of the MOPF problem while complying with the system constraints is a difficult task for power system engineers. Solution of MOPF problem based on weighted sum approach includes differential search algorithm (DSA) [14], DE [11], moth swarm algorithm (MSA) [15], improved artificial bee colony (IABC) algorithm [16], improved colliding bodies optimization (ICBO) [17] and backtracking search algorithm (BSA) [18] all of these give a single final non‐dominated solution with the specific weight factor. As compared to a single non‐dominated solution, the decision‐maker accentuates the set of non‐dominated (Pareto frontier) solutions in a single simulation run.A limited number of papers in the literature are found on solving constrained multi‐objective evolutionary algorithms (CMOEAs). These were multi‐objective evolutionary algorithm‐based decomposition (MOEA/D) [9],multi‐objective firefly algorithm with the constraint‐prior Pareto dominance approach (MOFA‐CPD) [19], hybrid dragonfly algorithm (DA) and PSO called DA‐PSO [20], and enhanced self‐adaptive DE with mixed cross over (ESDE‐MC) [21] technique were considered for the better exploration and exploitation, modified shuffled frog leaping algorithm (MSFLA) [22] proposed the mutation operator to enhance quality and speed of original SFLA, modified teaching‐learning based optimization (MTLBO) [23] applied to solve MOPF problem in which wavelet technique was applied to form mutation pool and fuzzy clustering technique has been applied to select the population for the next generation. Moreover, modified gaussian bare‐bones multi‐objective imperialist competitive algorithm (MGBICA) [24], improved strength Pareto evolutionary algorithm II (I‐SPEA 2) [25] considers Euclidian distance of kth nearest neighbour for the selection of the next population and external archive are employed to solve MOPF problem. The original gravitational search algorithm (GSA) along with non‐dominated sorting and opposition‐based learning called (NSMOOGSA) [26], wherein, gravitational speed and better distribution of PF, non‐dominated sorting with crowding distance technique and opposition‐based learning concept have been employed. Multi‐objective DE (MDE) [27] modifies the DE variant (DE/ best/1) in which the best individual was selected by considering fuzzy‐based beast compromise solution (BCS) in each generation for mutation operator. Enhanced GA (EGA) [4], for finding the non‐dominated solution to the MOPF problem, combines GA operator for offspring generation and domination principle from SPEA for final PF. Hybrid quasi‐oppositional‐based learning (QOBL) and original teaching learning‐based optimization (TLBO) called (QOTLBO) [28] to improve the convergence speed and distribution of final PF. Imperialist competitive algorithm (ICA) often traps into local optima, therefore combined modified ICA (CMICA) [29] and multi‐objective modified ICA (MOMICA) [30] were used to solve constraints MOPF problem. Moreover, for better solution quality interaction effects of colonies on each other and policy rule of ICA method is modified. Pareto‐based domination principle along with fuzzy grouping has been adapted to enhance the exploration and exploitation of CMICA. Bio inspired modified multi‐objective flower pollination algorithm (B‐MMOFPA) [31] in which normal boundary intersection (NBI) with original FPA has been considered to find the nondominated solutions. Further three statistical techniques such as fuzzy decision making, median and centroid methods were used to select the best compromise solution. Single objective optimization Jaya algorithm combined with the Quasi‐oppositional technique called Quasi‐oppositional modified Jaya (QOMJaya) [32] to handle the MOPF problems. In this algorithm convergence and quality of solutions were increased through an intelligence strategy called quasi‐oppositional based learning. Whereas, non‐dominated sorting principle along with crowding distance selection scheme was used for finding the well‐distributed PF. An interior search algorithm (ISA) [33] together with non‐dominated sorting was applied to find the PF. A fuzzy membership function is considered to find the best compromise solution (BCS).Multi‐objective dimension‐based firefly algorithm (MODFA) and multi‐objective PSO (MOPSO) [34] are compared with the superior feasibility (SF) constraint technique to find the solution MOPF problem. Semi‐definite programming (SDP) [35] and improved normalized norm‐constrained (INNC) [36] methods incorporate the epsilon constraint method (ECM) for MOPF problem. Original NNC cannot find the entire PF, therefore improved NNC (INNC) has been applied to fulfil the shortcoming of NNC by converting the utopia hyperplane into a utopia line and converting the multi‐objective into three single and three bi‐objective optimization problems. Modified pigeon‐inspired optimization algorithm (MPIO) [37] integrated with constraint‐objective sorting rule (COSR) called (MPIO‐COSR) was considered for finding the PF of MOPF problem.Due to a large number of function evaluations to sort the constraint violation of COSR technique, it is computationally expensive. Chen et al. proposed novel hybrid bat algorithm along (NHBA) with constrain‐prior Pareto‐dominance method (CPM) called (NHBA‐CPM) [38] and hybrid firefly‐bat algorithm with constraints‐prior Pareto‐dominant rule (CPR) combined HFBA‐COFS [39] to solve MOPF problem considering three objective functions. Furthermore, Table 1 summarizes the discussed papers because of selection of objective functions, CHT, weighted sum, or Pareto‐based approach and test systems.1TABLEMulti‐objective optimization studies for the solution of MOPFMethod RefObjective functionsCHTApproachTest systemDSA [14]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionWeighted sum9, 30, 57IABC [16]FC, E and PLoss${P}_{Loss}$Penalty functionWeighted sum30, 57 and 300DE [11]FC, E, VD, PLoss${P}_{Loss}$and VSISF, SP, ECHTWeighted sum30, 57, 118MSA [15]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionWeighted sum30, 57, 118ICBO [17]FC, VD and VSIPenalty functionWeighted sum30, 57, 118BSA [18]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionWeighted sum30, 57, 118MOEA/D [9]FC, E, VD and PLoss${P}_{Loss}$Penalty functionPareto based30ESDE‐MC [21]FC, E, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30, 57, 59SFLA [22]FC, E,Penalty functionPareto based30MTLBO [23]FC, E,Penalty functionPareto based30, 57,MGBICA [24]FC, EPenalty functionPareto based30, 57,I‐SPEA2[25]FC, EPenalty functionPareto based30, 57,NSMOOGSA [26]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30,MDE [27]FC, VD, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based57, 118EGA [4]FC, PLoss${P}_{Loss}$and VSIPenalty functionPareto based30,QOTLBO [28]FC, E, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30, 118CMICA [29]FC, VD, PLoss${P}_{Loss}$Penalty functionPareto based57MOMICA [30]FC, E, VD, PLoss${P}_{Loss}$Penalty functionPareto based30, 57B‐MMOFPA [31]FC, VD, PLoss${P}_{Loss}$Penalty functionPareto based30QOMJaya [32]FC, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30ISA [33]FC, E, VD, PLoss${P}_{Loss}$and VSIPenalty functionPareto based30 and 57MODFA [34]FC, E, PLoss${P}_{Loss}$SFPareto based30, 57, 118DA‐PSO [20]FC, E, PLoss${P}_{Loss}$Penalty functionPareto based30, 57SDP [35]FC, E,ECMPareto based30, 57, 118INNC [36]FC, PLoss${P}_{Loss}$, VDECMPareto based30, 118NHBA‐CPFD [38]FC, E, PLoss${P}_{Loss}$CPDPareto based30, 57, 118MOFA‐CPD [19]FC, E, PLoss${P}_{Loss}$CPDPareto based30, 57HFBA‐COFS [39]FC, E, PLoss${P}_{Loss}$CPRPareto based30, 57, 118MPIO [37]FC, E and PLoss${P}_{Loss}$COSRPareto based30, 57, 118CHT = constraint handling technique, FC = Fuel cost, E = emission, VD = voltage deviation, PLoss${P}_{Loss}\ $= active power loss, Lindex = voltage stability indicator, SF = superiority of feasibility, ECM = ε‐constraint method, Constraint‐objective sorting rule (COSR).World global environment regulations have constituted a greater challenge to the thermal power generation industry because it emits harmful gases into the environment. European Union (EU) and G8 aim to reduce greenhouse gas (GHG) emissions by at least 80% below 1990 levels by 2050 [40].Therefore, RES has drawn momentous attention from both industry and academia because of its low emissions. In recent years, integration of sustainable energy generation has risen significantly due to being environmentally friendly compared to thermal resources and is forecasted to rapidly expand in the future. These sources generate variable power based on the availability of wind speed and solar irradiance. The uncertain nature of these sources represents the challenging constraints to the multi‐objective OPF problems in terms of supply‐demand mismatch. In such power systems, a deterministic system study such as optimal power flow (OPF) evaluation cannot reveal the state of system accurately; therefore, probabilistic evaluation is of significant interest. Recently, most of the authors quantify the solution of single objective OPF problem with the integration of probabilistic wind and solar generation. These include Success history‐based adaptive differential evolution (SHADE) [41], symbiotic organisms search algorithm [42], Lévy coyote optimization algorithm (LCOA) [43] and Fuzzification method to reduce the operating cost and increase the minimum voltage magnitude [44]. Increasing the integration of uncertain RES power, voltage stability of power system may go to violate (voltage stability limit is based on a theory of maximum power transfer between two buses [44]). Therefore, the consideration of RES should be adequately analyzed the voltage stability detection [45]. In most of the papers, security constraints such as voltage deviation and voltage stability index are not studied. To the knowledge of author, these constraints are not studied in probabilistic multiobjective OPF problems considering uncertain wind and solar generation.The computation of probabilistic multiobjective OPF (PMOPF) is one of the major requirements in power system planning and operation. Performing PMOPF study helps system planning engineers in making judgments concerning investments. But, the total number of probable combinations of uncertain variables such as the loads and generation units are high. As a result, performing PMOPF is a computational burden issue; therefore, a probabilistic model of wind, solar and load demand with acceptable accuracy is needed for the power system studies.ContributionMost of the MOEAs discussed in the literature review incorporate one popular constraint handling technique (CHT) such as the penalty function approach to managing the constraint violation (CV). The performance of the penalty function approach is widely dependent on the selection of constant penalty parameters. With a small value of penalty coefficient, the algorithm may trap in the infeasible space whereas a large value of this parameter may over‐explore the feasible space and may get stuck in local optima. Inappropriate penalty parameter selection may also give an infeasible solution in the final non‐dominated PF. Therefore, in a realistic MOPF problem, appropriate CHT must be selected to guide the MOEAs, search the entire feasible space and extricate out of the infeasible region, and find the near global and widely distributed PF. Moreover, in the given tabulated literature, all the authors have been selecting predefined parameters for their MOEAs and claiming that their MOEAs have been robust and converged which gives better PF than that of other MOEAs. Comparisons between various MOEAs, however, are challenging tasks, as the collection and pre‐defined parameters for each MOEAs dominate the result. Furthermore, according to no‐free‐lunch (NFL) theorem [46], a single MOEA is not found that solves all the problems (study cases) that are simply superior to priors.Inspired by aforesaid points, here, various new recent constrained MOEAs are considered to find the solution to MOPF problem that has not been tested yet for the solution of MOPF problem, these include ToP [47], CCMO [48], the decomposition‐based multi‐objective evolutionary algorithm with detect‐and‐escape constraint technique (MOEA/D‐DAE) [49] and tri‐goal evolution (TiGE_2) [50]. The very first time these algorithms are applied to find the nondominated solutions considering simultaneous optimization of 2, 3, 4, and 5 objective functions of a comprehensive set of ten study cases. Out of which 1–6 and 8 have already existed in the literature whereas 7, 9, and 10 are unprecedented ones. Moreover, IEEE standard 30‐bus (small network) and 300‐bus (vary large) test systems are considered to analyze the simulation results of proposed CMOEAs. Simulation results show that the ToP and CCMO can find the global optimal solution for proposed study cases compared to other MOEAs in the literature. ToP has better PF (near global and evenly distributed) compared to other algorithms in cases 1–6 and case 10, whereas, CCMO gives better results in cases 7–9. The computational complexity of ToP is less compared to other algorithms. In the first phase of the ToP, weighted sum approach is applied to find high‐quality feasible solutions) after that in phase II, NSGAII [51] has been applied to find the final evenly distributed and near‐global PF.Moreover, probabilistic multi‐objective OPF (PMOPF) is formulated with the integration of uncertain wind and solar PV. Various combinations of two and three objective functions are considered for the analysis and comparison of probabilistic MOPF problems. Furthermore, several supplementary constraints along with two unprecedented constraints are incorporated to find secure non‐dominated solutions to MOPF problem. In probabilistic MOPF, SVC and transformer tap settings are discrete. In most papers in the literature, round off operator has been applied to find the discrete decision variables for mixed integer problems. Due to the rounding operator, algorithm is stuck in the search area of local optima. However, in the proposed algorithm integer constraints are handled by measuring functions same as given in [52], to avoid the trap in the local optima.The main contributions of this work are as follows:The comprehensive ten study cases comprised of 2, 3, 4, and 5 objective functions are formulated on IEEE 30‐bus and 300‐bus test systems to solve the deterministic MOPF problem.Two unprecedented security constraints such as voltage deviation and local stability indicator Lj are considered to narrow down the feasible search space to show the effectiveness of the proposed algorithm.Various recent constrained MOEAs, yet not employed to solve MOPF problem, are implemented.Probabilistic wind and solar power generators are integrated considering appropriate distribution functions.A hybrid two‐phase (ToP) algorithm (combination of single and multi‐objective evolutionary algorithms) integrated with the constraint domination principal technique is implemented to address the solution of deterministic and probabilistic OPF problem.Simulation results of the proposed algorithms are analyzed and compared with the other recent available methods in the literature.In the rest of the paper, we first formulate the deterministic and probabilistic MOPF problem in Section 2 comprised of various objective functions and operational constraints. Then in Section 3, we elaborate on the CHT and framework of proposed the algorithm. Afterward, in Section 4detailed analysis and comparison of simulation results are presented. Finally, the conclusion and future work is outlined in Section 5.PROBLEM FORMULATIONConventional OPF problemMathematically multi‐objective problems having the number of objective functions (M≥2$M \ge 2$) are expressed as:1minFx⃗=f1x⃗,f2x⃗,…,fMx⃗s.t:ϕ(x⃗)=0ψx⃗≤0$$\begin{eqnarray} min\ F\left( {\vec{x}} \right) &=& \left[ {{f}_1\left( {\vec{x}} \right),{f}_2\left( {\vec{x}} \right), \ldots ,{f}_M\left( {\vec{x}} \right)} \right]\nonumber\\ s.t: \phi ( {\vec{x}} ) &=& 0\nonumber\\ && \psi \left( {\vec{x}} \right) \le 0 \end{eqnarray}$$where x⃗$\vec{x}$ is the decision vector,F(x⃗)$\ F( {\vec{x}} )$is the objective function, ϕ(x⃗)$\phi ( {\vec{x}} )$ are the equality constraints and ψ(x⃗)$\psi ( {\vec{x}} )$ is the inequality constraint for the OPF problem. The decision vector x⃗${{\vec{\rm x}}}$ for the OPF problem is given as:x⃗=PG2…PGN,VG1..VGN,τ1T…τTN,QC1…QCN$$\begin{equation*}\ {\bm{\vec{x}}} = \ \left[ {{P}_{G2} \ldots {P}_{GN},\ {V}_{G1}..{V}_{GN},\ {\tau }_{1T} \ldots {\tau }_{TN},\ {Q}_{C1} \ldots {Q}_{CN}} \right]\end{equation*}$$where P and V are the output power of generators and voltage at the generator bus,τ is transformer tap ratio andQ MVAr injections of shunt VAR compensator (SVC); GN, TN, and CN are the number of generators, transformer tap settings, and SVC. In the proposed study cases, two or more objective functions F(x⃗)${\rm{F}}( {{{\vec{\rm x}}}} )$ from (2) is considered to solve simultaneously.2Fx⃗=CPg⃗,EPg⃗,VDVL⃗,LindexVL⃗,APLV⃗$${\fontsize{9}{11}{\selectfont{ \begin{eqnarray} F\ \left( {\vec{x}} \right) = \left[ {C\left( {\overrightarrow {{P}_g} } \right),E\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right),\ VD\left( {\overrightarrow {{V}_L} } \right),\ {L}_{index}\left( {\overrightarrow {{V}_L} } \right),APL\left( {\vec{V}} \right)} \right]\nonumber\\ \end{eqnarray}}}}$$where C(Pg⃗)$C( {\overrightarrow {{P}_g} } )$ and E(Pg⃗)$E( {\overrightarrow {{P}_g} } )$are the quadratic fuel cost and emission of the thermal generator [15], mathematically proposed objective function described as:3CPg⃗=∑i=1NGai+biPGi+ciPGi2$$\begin{equation}C\ \left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right) = \sum _{i = 1}^{NG} \left( {{a}_i + {b}_i{P}_{Gi} + {c}_iP_{Gi}^2} \right)\ \end{equation}$$4EPg⃗=∑i−1NG[αi+βiPGi+γiPGi2+ωieμiPG]$$\begin{equation}E\ \left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right) = \sum _{i - 1}^{NG} [\ \left( {{\alpha }_i + {\beta }_i{P}_{Gi} + {\gamma }_iP_{Gi}^2} \right) + {\omega }_i{e}^{\left( {{\mu }_i{P}_G} \right)}]\end{equation}$$5VDVL⃗,Ybus=∑L=1LNVL−1$$\begin{equation}VD\ \left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} ,{Y}_{bus}} \right) = \left( {\sum _{L = 1}^{LN} \left| {{V}_L - 1} \right|} \right)\ \end{equation}$$6LindexVL⃗=maxLjwherej=1,2,…,NL$$\begin{equation}{L}_{index}\ \left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} } \right) = \ max\ \left( {{L}_j} \right)\hbox{where}\ j\ = \ 1,2,\ \ldots ,\ NL\end{equation}$$7PLossV⃗=∑i=1Nb∑j=i+1NbGijVi2+Vj2−2ViVjcosδij$$\begin{equation}{P}_{Loss}\left( {{\bm{\vec{V}}}} \right)\ = \sum _{i = 1}^{Nb} \ \sum _{j\ = \ i + 1}^{Nb} {G}_{ij}\left[ {V_i^2 + V_j^2 - 2{V}_i{V}_j\ cos\ \left( {{\delta }_{ij}} \right)} \right]\end{equation}$$where ai,bi,ci,αi,βi,γi,ωi${a}_i,\ {b}_i,\ {c}_i,\ {\alpha }_i,\ \ {\beta }_i,\ \ {\gamma }_i,\ {\omega }_i$andμia${{{\mu}}}_{\rm{i}}{\rm{\ a}}$re the constants for the cost and emission of the thermal generators taken from [15]. Where δij=δi−δj${\delta }_{ij} = {\delta }_i\ - {\delta }_j$, V⃗$\vec{V}$ is the complex voltage of all the buses and VL shows the voltage at load (PQ) buses. Voltage magnitudes of load buses are somewhat less than the slack and PV buses; the deviation of load bus voltage depends upon the reactive power demand (QL). Gij${G}_{ij}$ is the conductance of π equivalent model of transmission line between bus i and j. Estimate of voltage stability is an issue that is receiving growing attention from power system researchers due to system voltage collapses in the past because of voltage instability. Voltage stability index (Lindex${L}_{index}$) indicator has developed in [53] which can be defined based on Lj local indicator. Let NG and NL be the number of generator and load buses respectively then local indicator Lj can be calculated as;8Lj=1−∑i=1NLFjiViVj,wherej=1,2,…,NLandFji=−[YLL]−1YLG$$\begin{eqnarray} {L}_j &=& \left| {1 - \sum _{i = 1}^{NL} {F}_{ji}\frac{{{V}_i}}{{{V}_j}}} \right|\ ,\ \hbox{where}\ j\ = \ 1,\ 2, \ldots ,NL\quad \hbox{ and }\nonumber\\ {F}_{ji} &=& \ - {[{Y}_{LL}]}^{ - 1}\left[ {{Y}_{LG}} \right] \end{eqnarray}$$Where sub‐matrices YLL${Y}_{LL}$ and YLG${Y}_{LG}$ are calculated from Ybus${Y}_{bus}$matrix after separating PV and PQ buses. The equality constraint ϕ(x⃗)$\phi ( {{{\vec{\rm x}}}} )$ is the balanced power flow constraints, given as:9SiV+SLi−Sgi=0$$\begin{equation}{S}_i\left( {\bm{V}} \right) + {S}_{Li} - \ {S}_{gi} = \ 0\end{equation}$$Let V and I be the vectors of complex bus voltage and bus current injection, then mathematically I injection for each bus can be calculated by using I = YV, where Y is the bus admittance matrix than complex MVA bus injection is Si=VI∗⇒V(YV)∗${S}_i = \ V{I}^* \Rightarrow V{( {YV} )}^*$ and SLi=PLi+jQLi${S}_{Li} = {P}_{Li}\ + j{Q}_{Li}$ and Sgi=Pgi+jQgi${S}_{gi} = {P}_{gi}\ + j{Q}_{gi}$. On the other hand, inequality constraint ψ(x⃗)${{\psi}}( {{{\vec{\rm x}}}} )$ for the OPF problem consists of operational limits of all the generators and security constraints on the buses and lines.10Pgimin<Pgi<Pgimax∀i∈GN$$\begin{equation}P_{gi}^{min} &lt; {P}_{gi} &lt; P_{gi}^{max}\ \forall i \in GN\end{equation}$$11Vgimin≤Vgi≤Vgimax∀i∈GN$$\begin{equation}V_{gi}^{\ min\ } \le {V}_{gi} \le V_{gi}^{\ max\ }\forall i \in GN\end{equation}$$12VLmin≤VL≤VLmax∀L∈LN$$\begin{equation}V_L^{min\ } \le {V}_L \le V_L^{\ max\ }\forall L \in LN\end{equation}$$13Sl≤Slmax∀l∈nl$$\begin{equation}{S}_l \le S_l^{\ max\ }\forall l \in nl\end{equation}$$14τTmin≤τT≤τTmin∀T∈TN$$\begin{equation}\tau _T^{min\ } \le {\tau }_T \le \tau _T^{min\ }\forall T \in TN\end{equation}$$15QCmin≤QC≤QCmax∀T∈TN$$\begin{equation}Q_C^{min\ } \le {Q}_C \le Q_C^{max}\ \forall T \in TN\end{equation}$$16Lindex<Γ$$\begin{equation}{L}_{index} &lt; \ \Gamma \end{equation}$$17VD<ε$$\begin{equation}\left| {VD} \right| &lt; \epsilon \end{equation}$$where Pgi${P}_{gi}$ and Qgi${Q}_{gi}$ are the active and reactive output of generator at bus i, Vgi${V}_{gi}$ and VL${V}_L$ are the generator and load bus voltages, Sl${S}_l$ is the MVA branch flow, τT${\tau }_T$ is transformer tap ratio, QC${Q}_C$ is shunt Var compensator, Lindex${L}_{index}$ is the maximum value of Lj${L}_j$ indicator for all the load buses. Lindex is calculated by using Equation (6), whereas the constant Γ is local indicator Lj which is set by the decision‐maker. For the smaller value of Γ, the distance to voltage instability is larger and hence the system has more stability margin. In this paper, the constant Γ is less than 0.2 (minimum attainable value of Lindex), which is obtained by considering the single objective optimization of Lindex. Whereas, the constant ε is set to 1.2 for the IEEE 30 bus system and can be computed by considering either upper bound or lower bound of load bus voltage. In this paper, load bus voltage is set between 0.95 and 1.05 p.u. The acceptable variation of load bus voltage is ±0.05 p.u, cumulatively 0.05×24 is 1.2 therefore, and the overall load bus voltage violation must be less than 1.2 p.u.Probabilistic OPF problemSolution of probabilistic multi‐objective OPF (PMOPF) is a challenging task to satisfy technical, economic and environmental issues with the high penetration of uncertain wind and solar PV. Therefore, various combinations of two and three objective functions are considered for the analysis and comparison of probabilistic MOPF problems. Furthermore, several supplementary constraints along with two unprecedented constraints are incorporated to find secure non‐dominated solutions to MOPF problem. In the subsequent subsection, we will formulate the mathematical model of uncertain renewable generation and demand and objective functions.Modelling of uncertain renewable generation and demandIn the field of an electric power system, load demand is always uncertain. Further complexity of power system planning is increased with the integration of uncertain wind, solar PV and small hydro power. Therefore, here, appropriate PDFs are used to model the demand and output power of renewable generation. In the literature, Weibull, lognormal, and gamble probability distribution functions (PDFs) have been used for the modelling of wind, solar and small hydro power generation, whereas, normal PDF are used for load modelling [54–56]. The aforementioned PDFs are mathematically defined as:Weibull PDF for the wind velocity (v) estimating:18Δνv=bavab−1×e−vab$$\begin{equation}{\Delta }_\nu \ \left( v \right) = \left( {\frac{b}{a}} \right)\ {\left( {\frac{v}{a}} \right)}^{\left( {b - 1} \right)} \times {e}^{\left[ { - {{\left( {\frac{v}{a}} \right)}}^b} \right]}\end{equation}$$Lognormal PDF for the solar irradiance (G) predicting:19ΔGG=1G×d2πe−(lnG−c)22d2$$\begin{equation}{\Delta }_G\ \left( G \right) = \frac{1}{{G \times d\sqrt {2\pi } }}\ {e}^{\left[ {\frac{{ - {{(lnG - c)}}^2}}{{2{d}^2}}} \right]}\end{equation}$$where a, b are the shape and scale parameters Weibull PDF; c and d are the mean and standard deviation of lognormal PDF.Probabilistic available wind power pgW${p}_{gW}$ is the function of wind velocity (v), both are highly nonlinear with each other, mathematically given as [57, 58]:20pgWv=0,forνvinandvνouTpwrv−νinνr−vinforvin≤v≤vrpwrforvr<ν≤vouf$$\begin{equation} {p}_{gW}\left( v \right)\ = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{l}@{}} 0, for\ \nu \left\langle {{v}_{in}\ and\ v} \right\rangle {\nu }_{ouT}\\[15pt] {p}_{wr}\left( {\dfrac{{v - {\nu }_{in}}}{{{\nu }_r - {v}_{in}}}} \right)\ for\ {v}_{in} \le v \le {v}_r\\[15pt] {{p}_{wr}\ for\ {v}_r &lt; \nu \le {v}_{ouf}} \end{array} \right. \end{equation}$$In our study 3 MW, Enercon E82‐E4 wind turbine is adopted [59]. The probabilistic solar PV power pgS${p}_{gS}$ is the function of available solar irradiance (G) [60]21pgSGS=PsrGs2GsTdRcfor0<Gs<RcPsrGsGsTdforGs≥Rc$$\begin{equation} {p}_{gS}\left( {{G}_S} \right)\ = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{l}@{}} {{P}_{sr}\left( {\dfrac{{G_s^2}}{{{G}_{sTd}{R}_c}}} \right)\ for\ 0 &lt; {G}_s &lt; {R}_c}\\[15pt] {{P}_{sr}\left( {\dfrac{{{G}_s}}{{{G}_{sTd}}}} \right)\ for\ {G}_s \ge {R}_c} \end{array} \right. \end{equation}$$Normal PDF for estimating the percentage of active and reactive load demand (l):22ΔDl=1σ2Π×e−(l−μ)22σ2$$\begin{equation}{\Delta }_D\left( l \right)\ = \frac{1}{{\sigma \sqrt {2\Pi } }}\ \times {\rm{\ e}}^{ - \left( {\frac{{{{(l - \mu )}}^2}}{{2{\sigma }^2}}} \right)}\end{equation}$$Table 2 shows the parameters of PDFs, that are realistically selected and based upon installed capacity.2TABLEParameters of weibull, lognormal and vormal PDFUncertaintyPDFParametersPower and load demandWind speed vWeibullShape (a = 9); Scale (b = 2)25 turbines of each 3 MW (rated 75 MW)Solar irradiance GLognormalMean (c = 5.5); SD (d = 0.5)Rated 50 MWLoad lNormalMean (μL = 97); SD (σL = 5)Percentage of loadFigure 1 shows the Weibull, lognormal and normal PDF fittings and their associated uncertain power generation and demand. Moreover, for realistic OPF problems, 1‐h load demand uncertainty is considered to analyse the integration of uncertain generation.1FIGUREPDF of wind‐solar units and load demandIntegration of probabilistic wind and solar power affects the cost computation only. Therefore, cost function with the integration of RES is given in the next section.Combined operating costThe intermittent and unpredictable nature of RES makes grid integration challenging. Typically, private operators—An organisation that enters into purchase agreements with independent system operators (ISO)—are the owners of wind, and solar PV generation. The ISO is responsible for mitigating the deficit amount by retaining spinning reserves if demand increases and the wind and solar generation are unable to provide the scheduled electricity owing to non‐availability or insufficiency of renewable sources. Such events are termed as overestimation of renewable power and for which maintaining spinning reserve adds to the power generation cost. On the other hand, a situation can occur where more RES power is generated than is planned. If the excess power is not used in such a case of underestimate, it is wasted, and ISO is responsible for the associated costs.Therefore, with the integration of RES, objective function vector has been modified and given as:23minx⃗Fx⃗=CTotalPg⃗,EPg⃗,VDVL⃗,τTN,LindexVL⃗,PLossV⃗$$\begin{eqnarray} && \mathop {\min }\limits_{{\bm{\vec{x}}}} {\bm{F}}\left( {{\bm{\vec{x}}}} \right) = \left[ {C}_{Total}\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right), E\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right),\right.\nonumber\\ &&\left.\quad VD\left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} ,{{\bm{\tau }}}_{{\bm{TN}}}} \right),{L}_{index}\left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} } \right),\ {P}_{Loss}\left( {{\bm{\vec{V}}}} \right) \right]\ \end{eqnarray}$$24CTotalPg⃗=∑tNint∑i,j,k,mNTG,NWG,NSG,NDCiPgT+CjPgW+CkPgS+CmPD$$\begin{eqnarray} && {C}_{Total}\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right)\ = \sum _t^{{N}_{int}} \sum _{i,j,k,m}^{{N}_{TG},\ {N}_{WG},\ {N}_{SG},\ {N}_D}\nonumber\\ && \left[ {{C}_i\left( {{P}_{gT}} \right) + {C}_j\left( {{P}_{gW}} \right) + {C}_k\left( {{P}_{gS}} \right) + {C}_m\left( {{P}_D} \right)} \right]\ \end{eqnarray}$$Whereas, Ci(PgT)${C}_i( {{P}_{gT}} )$ is given in Equation (3), cost of wind generation Cj(PgW)${C}_j( {{P}_{gW}} )$ computed as:25CjPgW=gw,jPSchW,j︸Directcost+KRW,jPschW−pgW︸ReservecostPgW<Psched+KPW,jpgW−PschW︸Penalitycost(PgW>Psched)$$\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{C}_j\ \left( {{P}_{gW}} \right) = \underbrace {{g}_{w,j}{P}_{SchW,j}}_{Direct\ cost}\ + \ \underbrace {{K}_{RW,j}\left( {{P}_{schW} - {p}_{gW}} \right)}_{Reserve\ cost\ \left( {{P}_{gW} &lt; \ {P}_{sched}} \right)}}\\[35pt] { + \underbrace {{K}_{PW,j}\left( {{p}_{gW} - {P}_{schW}} \right)}_{Penality\ cost\ ({P}_{gW} &gt; \ {P}_{sched})}} \end{array} \end{equation}$$Mathematically reserve and penalty costs are computed as:CRPschW−pgW=KRW∫0PschW(PschW−pgW)×$$\begin{equation*}{C}_R\ \left( {{P}_{schW} - {p}_{gW}} \right) = {K}_{RW}\ \int _0^{{P}_{schW}} ({P}_{schW} - {p}_{gW}) \times \end{equation*}$$26πwpgWdpgW$$\begin{equation} {\pi }_w\left( {{p}_{gW}} \right)d{p}_{gW}\end{equation}$$27CPPgW−PschW=KPw∫PschWPgWrpgWr−PschWπWpgWdpgW$${\fontsize{9.7}{11.7}{\selectfont{ \begin{eqnarray} {C}_P\ \left( {{P}_{gW} - {P}_{schW}} \right) = {K}_{Pw}\ \int _{{P}_{schW}}^{{P}_{gWr}} \left( {{p}_{gWr} - {P}_{schW}} \right){\pi }_W\left( {{p}_{gW}} \right)d{p}_{gW}\nonumber\\ \end{eqnarray}}}}$$Similarly, operating cost of kth solar PV,Ck(PgS)${C}_k( {{P}_{gS}} )$, defined as [54]:28CkPgS=gS,kPgS,k︸Directcost+KR,kPschS−pgS︸ReservecostPgS<Psched+KP,kpgS−PschS︸Penalitycost(PgW>Psched)$$\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{C}_k\ \left( {{P}_{gS}} \right) = \underbrace {{g}_{S,k}{P}_{gS,k}}_{Direct\ cost} + \ \underbrace {{K}_{R,k}\left( {{P}_{schS} - {p}_{gS}} \right)}_{Reserve\ cost\ \left( {{P}_{gS} &lt; \ {P}_{sched}} \right)} }\\[35pt] {+\,\underbrace {{K}_{P,k}\left( {{p}_{gS} - {P}_{schS}} \right)}_{Penality\ cost\ ({P}_{gW} &gt; \ {P}_{sched})}} \end{array} \end{equation}$$Reserve cost and penalty cost of solar PV$PV$ is computed as [60]:29CRsPschS−pgS=KRs∫minpgSPschW(PschS−pgS)×πSpgSdpgS$$\begin{eqnarray}{C}_{Rs}\left( {{P}_{schS} - {p}_{gS}} \right)\ = {K}_{Rs}\ \int _{{\rm{min}}\left( {{p}_{gS}} \right)}^{{P}_{schW}} ({P}_{schS} - {p}_{gS}) \times {\pi }_S\left( {{p}_{gS}} \right)d{p}_{gS}\nonumber\\ \end{eqnarray}$$30CPspgS−PschS=KPs∫PschWmaxpgS(PschS−pgS)×πSpgSdpgS$${\fontsize{9.5}{11.5}{\selectfont{ \begin{eqnarray}{C}_{Ps}\left( {{p}_{gS} - {P}_{schS}} \right)\ = {K}_{Ps}\ \int _{{P}_{schW}}^{{\rm{max}}\left( {{p}_{gS}} \right)} ({P}_{schS} - {p}_{gS}) \times {\pi }_S\left( {{p}_{gS}} \right)d{p}_{gS}\nonumber\\ \end{eqnarray}}}}$$Last term in the combined operating cost function is the cost of load demand variability, it is comprised of reserve and penalty cost only and can be given as:31CmPD=CR,mPdsch−pD︸Reservecost(PD<Psched)+CP,mpD−Pdsch︸Penalitycost(PgW>Psched)$$\begin{equation}{C}_m\ \left( {{P}_D} \right) = \underbrace {{C}_{R,m}\left( {{P}_{dsch} - {p}_D} \right)}_{Reserve\ cost\ ({P}_D &lt; \ {P}_{sched})}\ + \underbrace {{C}_{P,m}\left( {{p}_D - {P}_{dsch}} \right)}_{Penality\ cost\ ({P}_{gW} &gt; \ {P}_{sched})}\end{equation}$$Reserve and penalty cost of load demand is computed as:CRDPdsch−pD=KRd∫PDminPdsch(Pdsch−pD)×$$\begin{equation*}{C}_{RD}\ \left( {{P}_{dsch} - {p}_D} \right) = {K}_{Rd}\ \int _{{P}_{Dmin}}^{{P}_{dsch}} ({P}_{dsch} - {p}_D) \times \end{equation*}$$32fd(pD<Pdsch)dpD$$\begin{equation}{f}_d({p}_D &lt; {P}_{dsch})d{p}_D\end{equation}$$CPDpD−Pdsch=KPd∫PdschPDmax(Pdsch−pD)×$$\begin{equation*}{C}_{PD}\ \left( {{p}_D - {P}_{dsch}} \right) = {K}_{Pd}\ \int _{{P}_{dsch}}^{{P}_{Dmax}} ({P}_{dsch} - {p}_D) \times \end{equation*}$$33fd(pD>Pdsch)dpD$$\begin{equation}{f}_d({p}_D &gt; {P}_{dsch})d{p}_D\end{equation}$$where fd(pD<Pdsch)${f}_d( {{p}_D &lt; {P}_{dsch}} )$andfd(pD<Pdsch)${f}_d({p}_D &lt; {P}_{dsch})$are the probability of over and under estimation respectively of load demand,PDmin${P}_{Dmin}$ and PDmax${P}_{Dmax}$ are the minimum and maximum demand that appeared during normal distribution of load uncertainty. Numerical values of direct, penalty, and reserve cost coefficients for stochastic wind‐solar and load demand are provided in Table 3.3TABLECost parameter for the uncertainty in wind–solar and loadCost co‐efficientWindSolarLoadBus511All PQDirect1.71.6–Reserve333Penalty1.41.41.4The direct cost coefficients are set up so that solar and wind power have the highest costs, respectively [56, 60]. The respective direct cost coefficient is lower than the reserve cost coefficient for sustaining spinning reserves. The direct cost is higher than the penalty for not using the available electricity, though.CONSTRAINT HANDLING TECHNIQUES AND MOEASMOPF problem is a practical constrained multi‐objective optimization problem (CMOP). To solve CMOPs, state‐of‐the‐art MOEAs possess some limitations [1, 47], that is, incapable of balancing constraints and objective functions with small feasible search space or combined discrete and real feasible search space that leads to the local optimal solution or bad convergence and exploration of final PF. Unfortunately, in MOPF problem, the control variable, that is, voltage of generators, transformer tap ratios, and values of shunt VAr compensators (SVC) are within small feasible regions and possess stiff challenges to existing MOEAs. Additionally, constraints in the objective space such as VD and Lindex further narrow the objective function space.Therefore, this paper addresses the recent most effective MOEAs such as Two‐phase (ToP) [47] and coevolutionary constrained multi‐objective (CCMO) [48] for finding the global PF and evenly distributed non‐dominated solutions to MOPF problem. In both of the algorithms, NSGAII is used to find the best PF and constraint domination principle (CDP) is used for handling the constraints of an optimization problem, but different strategies are used to find the global optimal PF. ToP and CCMO choosing the NSGAII algorithm for the exploration and exploitation of the final non‐dominated solution, because of its strong performance in the solution of MOPF problem. The findings of the analysis should also be checked. In the later simulation result section, it is proved that ToP and CCMO can find the global PF of constrained practical MOPF problems. According to the simulation results, the proposed MOEA outperforms as compared to several state‐of‐the‐art MOEAs to solve various cases of MOPF problems. In the subsequent subsections, the ToP and CCMO algorithms with their associated CHTs are explained.Constraint handling technique (CHT)Before discussing the CHT, first discuss the basic definitions of constrained CMOP.Pareto dominanceConsider the two decision vectors x⃗u${\vec{x}}_u$ and x⃗v${\vec{x}}_v$, if for all objective functions f(x⃗u)≤f(x⃗v)$f( {{{\vec{x}}}_u} ) \le f( {{{\vec{x}}}_v} )$ and there exists at least one objective function, say j, fj(x⃗u)<fj(x⃗v)${f}_j( {{{\vec{x}}}_u} ) &lt; {f}_j( {{{\vec{x}}}_v} )$ then x⃗u≺x⃗v${\vec{x}}_u \prec {\vec{x}}_v$ and can be read as, x⃗u${\vec{x}}_u$ is non‐dominated or x⃗u${\vec{x}}_u$ dominates x⃗v${\vec{x}}_v$ or x⃗v${\vec{x}}_v$ is dominated by x⃗u${\vec{x}}_u$.Pareto set and Pareto frontA region is said to be feasible if the overall degree of constraint violation (CV) is equal to zero for all the set of decision vector x⃗${{\vec{\rm x}}}$. All the solutions that belong to feasible regions are called Pareto optimal solutions and the image of Pareto optimal set‐in objective space is called Pareto front (PF).Constraint domination principleSimple and efficient CHT to handle constraints proposed in [51] is considered here, which compares pair‐wise individuals based on the following rules:If both solutions x⃗u${\vec{x}}_u$ and x⃗v${\vec{x}}_v$ are infeasible, select x⃗u${\vec{x}}_u$ if CV(x⃗u)<CV(x⃗v)$CV( {{{\vec{x}}}_u} ) &lt; CV( {{{\vec{x}}}_v} )$.x⃗u${\vec{x}}_u$ is feasible and x⃗v${{{\vec{\rm x}}}}_{\rm{v}}$ is infeasible, select the feasible one, that is,x⃗u${\vec{x}}_u$.If both x⃗u${\vec{x}}_u$ and x⃗v${\vec{x}}_v$ are feasible, then select x⃗u${\vec{x}}_u$ if for all the objective functions fi(x⃗u)≤fi(x⃗v)${f}_i( {{{\vec{x}}}_u} ) \le {f}_i( {{{\vec{x}}}_v} )$.Two phase MOEAHere, the cost of active power generation, emission, voltage deviation (VD), active power loss (PLoss${P}_{Loss}$), and maximum value of Lindex considered the objective functions. Moreover, the complexity of a problem is increased with the addition of objective constraints such as max of Lindex${L}_{index}$, whereas, minimum attainable value of Lindex${L}_{index}$ is obtained by the individual solution of OPF and VD. That is the reason why it is hard for an MOEA to find the promising feasible area of MOPF problem. Moreover, CMOEA needs to balance all the objective functions infeasible region, the convergence speed of the population is inevitably slow. In the proposed algorithm, first, find high‐quality feasible solutions, and then the Pareto optimal solutions (exploitation and exploration). A detailed flow chart of the proposed algorithm is shown in Figure 2.2FIGUREFlow diagram of ToP‐NSGAII‐CDP algorithmIn the first phase, transform the multi‐objective functions into weighted sum constrained single objective function as:34minFx=1m∑i=1mfix$$\begin{equation}min\ F\left( x \right)\ = \frac{1}{m}\ \sum _{i\ = \ 1}^m {f}_i\left( x \right)\end{equation}$$The goal of the first phase is to provide high‐quality feasible solutions for applicants for the next phase. Furthermore, in search engine for the generation of offspring, two popular trail vector strategies of differential evolution (DE) are considered, these are:DE/current‐to‐rand/l:35u⃗i=x⃗i+F∗x⃗r1−x⃗i+F∗x⃗r2−x⃗r3$$\begin{equation}{\vec{u}}_i = {{\bm{\vec{x}}}}_i\ + F*\left( {{{{\bm{\vec{x}}}}}_{r1} - {{{\bm{\vec{x}}}}}_i} \right) + F*\left( {{{{\bm{\vec{x}}}}}_{r2} - {{{\bm{\vec{x}}}}}_{r3}} \right)\end{equation}$$DE/rand‐to‐best/l/bin:36v⃗i=x⃗r1+F∗x⃗best−x⃗r1+F∗x⃗r2−x⃗r3$$\begin{equation}{{\bm{\vec{v}}}}_i = {{\bm{\vec{x}}}}_{r1}\ + F*\left( {{{{\bm{\vec{x}}}}}_{best} - {{{\bm{\vec{x}}}}}_{r1}} \right) + F*\left( {{{{\bm{\vec{x}}}}}_{r2} - {{{\bm{\vec{x}}}}}_{r3}} \right)\end{equation}$$37u⃗i,j=v⃗i,jifrandj<CRorj=jrand1,…,D.x⃗i,jotherwise$$\begin{equation} {{\bm{\vec{u}}}}_{i,j} = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{l}@{}} {{{{\bm{\vec{v}}}}}_{i,j}\ if\ ran{d}_j &lt; CR\ or\ j = {j}_{rand}\ 1,\ \ldots ,\ D.}\\[9pt] {{{{\bm{\vec{x}}}}}_{i,j}\ otherwise} \end{array} \right. \end{equation}$$where subscript i∈[1,Np]$i \in [ {1,Np} ]$ and j∈[1,D]$j \in [ {1,\ D} ]$; D is the decision vector; vi=(vi,1,vx,2,…,vi,D)T${v}_i = \ {({v}_{i,1},\ {v}_{x,2},\ \ldots ,\ {v}_{i,D})}^T$ is the ith mutant vector; ui=(ui,1,ui,2,…,ui,D)T${u}_i = \ {({u}_{i,1},\ {u}_{i,2},\ \ldots ,\ {u}_{i,D})}^T$ is the ith trial vector; r1,r2${r}_1,{r}_2$ and r3 are random integers between [1, Np],xbest$Np],{x}_{best}$ is the best individual in the current population, randj arbitrary number [0, 1], jrand is a random number [1,D],F$[ {1,\ D} ],F$ and CR$CR$ are the scaling and crossover control parameter and these are randomly selected from the Fpool = [0.6; 0.8; 1.0] and CRpool = [0.1; 0.2; 1.0].Then feasibility rule CHT is implemented to choose the best solution between the x⃗i${\vec{x}}_i$ and u⃗i${\vec{u}}_i$ for the next generation. It should be necessary to terminate the first phase when the high‐quality solutions are obtained before the entire population converges to a single point. We design the following two conditions to achieve such high‐quality solutions:1ConditionThe feasibility proportion, that is, Pf>1/3$Pf\ &gt; 1/3$, promises that the feasible region has been got.2ConditionCompute the normalized weighted sum single objective functionf∼(x)$\widetilde {\ f}( x )$ and then add them as:38f∼ix=fix−fminxfmaxx−fminxm$$\begin{equation}{\tilde{f}}_i\ \left( x \right) = \frac{{{f}_i\left( x \right) - {f}_{min}\left( x \right)}}{{{f}_{max}\left( x \right) - {f}_{min}\left( x \right)}}\ m\end{equation}$$39f¯x=∑i=1mf∼ix$$\begin{equation}{\bar{f}} \left( x \right) = \sum _{i = 1}^m {\tilde{f}}_i\left( x \right)\ \end{equation}$$When some high‐quality feasible solutions have appeared in the first phase, it is desirable to design some terminating criteria to jump into the second phase for finding the widely distributed non‐dominated solutions. In the proposed algorithm, for finding the terminating criteria, sort out all the solutions that appeared in the first phase f¯(x)$\bar{f}( x )$ and compute the biggest difference (δ) between the 33% of feasible solutions. If δ is less than 0.2, the second condition is to be satisfied. The second condition declares that some high‐quality solutions along with converging to the small area have been obtained. The ultimate goal of the first phase is to provide high‐quality solutions for the next phase. High‐quality candidate solutions obtained in the first phase are neither well distributed nor well converged efficiently. Therefore, in phase II a well‐known MOEA, that is, non‐dominated sorting genetic Algorithm (NSGAII) has been applied to find the well‐distributed and near‐global PF. The steps of phase II are given in the flow chart as shown in Figure 2.Constrained coevolutionary MOEACCMO starts with the random generation of two populations say Pop1 and Pop2 each of size Np as shown in Figure 3. After that fitness of each population say fitness1 and fitness2 has been computed by using the domination rule of strength Pareto evolutionary algorithm (SPEA2) [61]. In this domination rule, each population assigns a fitness based on the number of solutions they dominated. If a large number of solutions are dominated by a particular solution, less the fitness of that solution and vice versa. This assignment of fitness makes sure that the search is directed toward non‐dominated solutions.3FIGUREFlow diagram of CCMO‐CDP algorithmFurthermore, these fitness values are considered to select the mating pools (parents) of each population by using a tournament selection operator. Then, in each generation offspring populations (i.e. offspring 1 and offspring 2) are generated from the mating pools of associated parents of each population.After that updated fitness values (i.e. fitness1 and fitness2) of the combined population and offspring1 and offspring2 are computed. The size of each updated fitness is reduced to Np by considering the environmental selection. The above process is repeated until the termination criteria are met.SIMULATION RESULTS AND DISCUSSIONDeterministic MOPFHere, the IEEE standard 30‐bus test is considered to evaluate the effectiveness of proposed algorithms. Detailed data of the test is considered from [62]. Various ten study cases as shown in Table 4 of simultaneous optimization of 2, 3, 4, and 5 objective functions are considered to solve IEEE 30‐bus standard test system. The IEEE 30 bus system has six thermal generators, 24 load buses with a cumulative load of 283.4+j126.2 MVA, nine shunt VAR compensators, and four transformers connected. User‐defined parameters of NSGAII [51], MOEADDAE [49], and TiGE‐2 [50], which demonstrate high performance, are adopted from their original papers.4TABLEVarious multi‐objective study casesCasesCost ($/h)Emission (t/h)VDLindexPLoss1✓✓2✓✓3✓✓4✓✓5✓✓✓6✓✓✓7✓✓✓8✓✓✓✓9✓✓✓✓10✓✓✓✓✓However, the parameters and offspring generation strategies of ToP and CCMO algorithms are given in Sections 3.2 and 3.3. Population size (Np) for the two, three, four, and five objective functions are 100, 200, 300, and 400 respectively; and gen_max for all the cases is 1000.Selection of best PF and the best compromise solutionConvergence and diversity preservation (combined evenness and spread) of the final solution play a vital role during the design of MOEAs. Here, state‐of‐the‐art MOEAs are applied to find the best solutions to the MOPF problem. Various study cases are designed to check the superiority and performance of various constrained‐type MOEAs. Each case is independently run twenty times, and twenty PFs are attained. For the selection of the best PF in terms of quality assessments such as diversity preservation and convergence, a well‐organized hyper volume performance indicator (HVI) [63] technique is applied. In all the cases, the values of objective functions are different; therefore in making a fair comparison, all the objective functions are normalized first to get uniform rage between 0 and 1. Afterward, a reference point (1, 1, …1)M is considered for the computation of HVI. When comparing the PFs of different runs of each algorithm, the PF with the maximum value of HVI attained by the algorithm is considered the best one. Table 5 summarizes the statistical results of HVI (max, min, mean, and SD) of all the MOEAs for each study case over the twenty independent runs, and the best result is highlighted.5TABLEStatistical data based on HVI of all the study casesCase #AlgorithmMaxMinMeanSD1CCMO0.8280.8240.8270.001MOEADDAE0.7940.5760.6820.086NSGAII0.8280.8250.8280.001TiGE_20.7870.7590.7740.009ToP0.8320.8160.8250.0072CCMO0.8920.8710.8800.007MOEADDAE0.8650.8350.8520.009NSGAII0.7760.2710.5930.197TiGE_20.7950.7270.7720.025ToP0.9050.8890.8980.0063CCMO0.5530.0000.1100.193MOEADDAE0.6120.0000.1030.196NSGAII0.7910.2140.5670.195TiGE_20.8320.7580.8440.052ToP0.8560.0000.2350.3514CCMO0.77460.7710.7730.001MOEADDAE0.7710.6060.7490.051NSGAII0.7720.7690.7710.001TiGE_20.7730.7590.7660.005ToP0.77520.7330.7660.0145CCMO0.8130.7920.8010.007MOEADDAE0.7830.7150.7530.023NSGAII0.8120.7740.7980.013TiGE_20.8300.8110.8220.005ToP0.8460.7790.8130.0216CCMO0.8320.7730.8030.020MOEADDAE0.7870.7140.7590.026NSGAII0.8210.7910.8060.010TiGE_20.7870.6620.7360.040ToP0.8360.7210.7860.0457CCMO0.6930.6870.6910.002MOEADDAE0.6760.6640.6700.004NSGAII0.6920.6860.6900.002TiGE_20.6670.6480.6610.005ToP0.6900.5990.6390.0368CCMO0.6890.6780.6840.003MOEADDAE0.6530.6040.6380.015NSGAII0.6530.6350.6460.006TiGE_20.6640.6540.6590.003ToP0.6740.5060.5830.0519CCMO0.6720.6350.6510.012MOEADDAE0.6580.5990.6460.017NSGAII0.6660.6350.6540.011TiGE_20.6660.4610.5710.081ToP0.6690.3170.5740.11010CCMO0.6000.0000.5300.186MOEADDAE0.5480.0000.4810.169NSGAII0.5890.0000.5190.182TiGE_20.5540.0000.4650.166ToP0.6310.0000.4990.177Table 5 clearly shows that ToP outperforms as compared to all other algorithms for cases 1–6 and 10 to HVI, whereas CCMO performs better in cases 7–9. Furthermore, as per the NFL theorem [46], there could not exist any MOEA that solves all the problems that are simply superior to priors. Moreover, the comparative performance of all the runs of different MOEAs based on HVI is as shown in Figure 4. In each box plot, twenty values of each run are computed by HVI, and a red central line in each box indicates the median, bottom and top edges of the box indicate the 25th and 75th percentile, and symbol ‘+’ are the outliers. The box plot summarizes that the single MOEA is unable to find the global PF with better convergence and diversity for all the MOPF cases.4FIGUREBox plot based on HVI of all the independent runsAlso, in most of the cases, NSGAII, MOEADDAE, and TiGE_2 are trapped in local optima, whereas, ToP and CCMO escape from local optima that are unsafe for solving MOPF problems. Both statistical values and box plots show that in most cases ToP gives better results where the VD and VSI are being taken as the objective functions. Because these functions are also influencing the constraint region.In cases 1–6 and 10, ToP outperforms, whereas, in cases 7–9, CCMO outperforms. Further, the average performance of the NSGAII algorithm in terms of SD is good compared to all the other algorithms but it does not find the global PF in any case. In the literature, all the discussed MOEAs, however, tend to be computationally expensive and required more function evaluation to find global PF, to solve MOPF problems which will make them less robust. To overcome this, ToP and CCMO algorithms proposed a strategy to avoid local optima.The ToP algorithm is comprised of two phases; the first phase is implemented to find the promising feasible area by transforming a MOPF problem into a weighted sum constrained single‐objective OPF problem. In the first phase, the weighted sum approach with different weight vectors can jump from the local optimal solutions. Most of the MOEA will face this problem of trap into local optima. In the second phase, an NSGAII is implemented to obtain the final PF. Therefore, the computational time can be significantly reduced to decrease the maximum function evaluation. However, in CCMO algorithm starts with the random initialization of two populations with size N. Moreover, a comparison between the BCS of various MOEAs based upon convergence and diversity preservation to find these solutions will be discussed. The BCS is derived from the best Pareto front using a fuzzy decision approach [35]. In this approach, membership function (μmk)$( {\mu _m^k} )$ of objective is computed first as:40μmk=1forfmk≤fmminfmmax−fmkfmmax−fmminforfmmin<fmk<fmmax0forfmk≥fmmax$$\begin{equation}\mu _m^k = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{2}{l}@{}} 1&\quad {for\ f_m^k \le f_m^{\ min\ }}\\[15pt] {\dfrac{{f_m^{\ max\ } - f_m^k}}{{f_m^{\ max\ } - f_m^{\ min\ }}}} &\quad {for\ f_m^{\ min\ } &lt; f_m^k &lt; f_m^{\ max\ }}\\[25pt] 0&\quad {for\ f_m^k \ge f_m^{\ max\ }} \end{array} \right. \end{equation}$$where m and k are the numbers of objective functions and final non‐dominated solutions; fmk$f_m^k$ is the fitness value. After that, μmk$\mu _m^k$ is normalized to find normalized membership function μk${\mu }^k$.41f¯x=∑i=1mf∼ix$$\begin{equation}\bar{f}\ \left( x \right) = \sum _{i = 1}^m {\tilde{f}}_i\left( x \right)\ \end{equation}$$where Nd${N}_d$ is the number of solutions in the final PF. The BCS is the index of the highest μk${\mu }^k$ value. Table 6 shows the results of BCS of two objective functions of all the algorithms considering fuzzy decision‐making rules.6TABLEBCS of various MOEAsCase #AlgorithmBCSμkMin (f1)Min (f2)1CCMO831.38810.2482510.011357800.77630.20483FC ($/h) vs E (t/h)MOEADDAE820.53310.2670090.01124800.58350.21601NSGAII834.84240.2446270.011303800.66610.20485TiGEA‐2829.28360.2505540.011398804.52680.21053ToP832.2610.2473440.011499800.64530.204892CCMO800.9060.2809120.011023800.55370.09671FC ($/h) vs VD (p.u)MOEADDAE802.19350.1595420.010909800.78430.11892NSGAII800.84870.2913920.011586800.81080.09899TiGEA‐22802.18830.1569470.011689800.87670.13666ToP802.39780.1373240.011849800.61620.096163CCMO800.68580.1373730.011534800.66010.13704FC ($/h) vs VSIMOEADDAE800.51240.137980.080723800.51180.13763NSGAII800.60630.1374310.010972800.57860.13741TiGEA‐2800.81740.137150.011382800.80980.13712ToP801.06440.1369360.012889800.54560.136774CCMO842.95515.028480.011171800.75213.12850FC ($/h) vs PLoss${P}_{Loss}$ (MW)MOEADDAE840.04845.1209630.011081800.60793.14033NSGAII842.85755.0415640.011165800.60133.12805TiGEA‐2832.16295.4317450.011024804.29373.71035ToP841.02495.101370.011172800.80263.12451The fourth column of Table 6 shows the values of normalized fuzzy membership function, from these values it is clearly shown that in all the two objective cases ToP algorithm outperforms. A minimum value of each objective function individually is shown in the last two columns of Table 6. From these values’ diversity of PF of each algorithm can be computed. In cases 1–4, ToP attains a maximum value of μk${\mu }^k$ as compared to all the other algorithms. In case 1, the BCS of all the algorithms is non‐dominated by each other. Whereas in case two individual objective function values of ToP dominate the MOEA/D‐DAE [49], NSGAII [51], and TiGE‐2 [50]. In case 3, an individual minimum value of ToP dominates the CCMO, NSGAII [51] and TiGE‐2 [50] whereas non‐dominated by MOEA/D‐DAE. In case 4, ToP dominates only TiGE‐2 [50] whereas it gives minimum value of power loss of 3.1245 MW compared to CCMO, MOEA/D‐DAE [49], and NSGAII [51]. The minimum value of individual objective functions shows that the ToP finds better diversity and convergence compared to other algorithms.Detailed results analysis of BCS of ToP and CCMOAccording to HVI, ToP competes for all the algorithms in cases 1–6 and case 10, whereas, CCMO gives better solutions to cases 7–9. Therefore, only the results of the best algorithm such as ToP and CCMO of all the study cases are shown in Table 7.7TABLESimulation results of BCS of all the study cases based on HVIParameterMinMaxCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10PG2$P{G}_2$ (MW)208060.05548.63448.07353.18862.13861.36859.85665.33958.02965.500PG5$P{G}_5$ (MW)155026.67721.30221.51233.56726.48228.17139.99433.53836.66433.178PG8$P{G}_8$(MW)103535.00021.32621.36235.00034.15134.71435.00034.88134.99326.144PG11$P{G}_{11}$ (MW)103026.47312.00012.13829.98122.93929.22429.99829.85029.97120.009PG13$P{G}_{13}$ (MW)124024.46812.0011222.82921.22626.82034.68834.05127.31526.562VG1$V{G}_1$ (p.u)0.951.11.0721.0851.0761.0701.0481.0691.0621.0261.0611.059VG2$V{G}_2$ (p.u)0.951.11.0621.0651.0591.0581.0301.0521.0541.0171.0541.044VG5$V{G}_5$ (p.u)0.951.11.0301.0301.0311.0361.0161.0201.0360.9861.0371.015VG8$V{G}_8$ (p.u)0.951.11.0431.0361.0401.0441.0011.0301.0420.9971.0461.015VG11$V{G}_{11}$ (p.u)0.951.11.0891.0501.1001.0991.0291.0391.0691.0631.0791.055VG13$V{G}_{13}$ (p.u)0.951.11.0301.0171.0811.0441.0241.0821.0611.0161.0621.038QC10$Q{C}_{10}$ (MVAr)0.0052.8081.7782.0624.4801.21003.0913.1371.4903.809QC12$Q{C}_{12}$ (MVAr)050.2000.10404.3651.3192.6303.9010.7811.8253.387QC15$Q{C}_{15}$ (MVAr)051.0334.5200.1851.3263.34754.0974.1273.4522.868QC17$Q{C}_{17}$ (MVAr)054.7864.4400.8334.6914.1670.5414.7943.1423.5593.797QC20$Q{C}_{20}$ (MVAr)050.6334.3634.1292.4185.0004.8424.0894.0814.7033.221QC21$Q{C}_{21}$ (MVAr)054.2804.9990.9124.7680.0691.0044.9714.7324.8881.770QC23$Q{C}_{23}$ (MVAr)053.7212.8191.0754.9645.00003.6554.6623.7053.029QC24$Q{C}_{24}$ (MVAr)054.7474.9931.0214.9342.8551.6314.9964.9994.8734.090QC29$Q{C}_{29}$ (MVAr)051.3322.4540.0323.0735.0000.0002.5992.1512.0960.510T11 (p.u)0.901.11.0811.1001.0331.0941.0381.0031.0551.0731.0371.067T12 (p.u)0.901.10.9090.9420.9010.9140.9030.9080.9170.9090.9160.927T15 (p.u)0.901.10.9831.0091.0150.9780.9761.0251.0090.9711.0151.004T36 (p.u)0.901.10.9901.0020.9540.9790.9850.9410.9780.9580.9720.929PG1$P{G}_1$ (MW)50.00200116.402177.269177.495113.936123.045108.55787.95190.595101.058118.088QG1$Q{G}_1$ (MVAr)‐20150‐2.0088.520‐3.716‐1.28114.67512.142‐4.505‐0.735‐10.7867.958QG2$Q{G}_2$ (MVAr)‐206028.66526.92922.22414.8979.94814.0189.57623.50912.10418.936QG5$Q{G}_5$ (MVAr)‐1562.523.09725.67229.08426.49345.42924.27228.76021.97729.99026.707QG8$Q{G}_8$ (MVAr)‐1548.734.24930.46540.25327.71022.16337.11430.18826.46237.64723.647QG11$Q{G}_{11}$(MVAr)‐104031.79724.50126.69134.59714.3264.13818.05832.43918.45523.704QG13$Q{G}_{13}$(MVAr)‐1544.7‐1.8052.89924.359‐2.7115.03726.9749.2431.70811.35610.052C(Pg)$C( {{P}_g} )$ ($/h)832.261800.849801.064841.025827.123842.712882.926871.333858.478838.173E (t/h)0.2470.3670.3670.2420.2580.2380.2160.2210.2280.251Lindex${L}_{index}$(p.u)0.1430.1480.1370.1380.1500.1380.1390.1490.1370.140VD (p.u)0.5810.2910.8700.9030.1570.6940.8890.1350.9520.361PLoss${P}_{Loss}$(MW)5.6759.1329.1795.1016.5825.4544.0884.8534.6306.079All the values of the decision vector in Table 7 are within upper and lower limits in all the cases. Besides, constraints in the objective space are also satisfied, the maximum value of Lindex is 0.2 (must be between 0 and 0.2) and VD must not be greater than 1.2 (0.05×24 = 1.2) in all the load buses. In all the cases, minimization of fuel cost of real power is mandatory, also, in the literature, solution of MOPF problem cost function is considered as common. In comparison to all the cases, the least value of cost function 800.84$/h has appeared in case 2, where the cost and VD are minimized simultaneously. The minimum value of a second objective function (Emission) is appeared in case 7, in this case, cumulative active and reactive power generation is minimum whereas a value of SVC is maximum. Also, in case 7, the minimum value of active power loss has appeared, and the maximum value of cost function is obtained in this case. The value of Lindex is better in Case 7 as compared to cases 1, 2, 5, 8, and 10. The minimum value of Lindex is appeared in cases 3 and 9, whereas the least value of VD appears in case 8. The values of Cost, emission, and power losses are opposite in case 3 and case 7, that is, Loss is maximum in case 3 whereas it is minimum in case 7, Cost is near the minimum in case 3 and it is maximum in case 7.Figure 5 shows the optimal cumulative values of active and reactive power injection of generators and reactive power injection of shunt VAr compensators (SVC) in all the study cases. In Figure 5, it can be noticed that maximum values of PG and QG appeared in case 3 and minimum values of these decision variables appeared in case 7. However, the cumulative value of SVC is minimum in case3 and maximum in case 7. From this, it can be concluded that the more the SVC injection less the active power losses will be and vice versa. Case 10, where all the five objective functions are minimized simultaneously, is the special case in which average values of all the functions of case 1–9 give the approximate values of case 10. It shows that when all the five objective functions are considered simultaneously the average values of case 1–9 objective functions. Moreover, voltage profiles of all ten study cases are shown in Figure 6.5FIGUREOptimal cumulative values of PG, QG and QC6FIGURELoad bus voltage of all the casesThe voltage curve for all the cases shows that the voltage at each load bus has the desirable limit, whereas, a waveform of cases 2, 5, 8, and 10 are near the unity compared to other cases. However, it is located far away in cases 3, 4, 7, and 9. Therefore, cumulative VD objective constraint is considered in this paper to narrow the feasible search space to find the load bus voltage near the unity (ideal). The minimum value of cumulative VD ensures that the load bus voltage is near unity. In cases 2, 5, 8 and 10 cumulative VD is less than 0.3 p.u and hence betters the load bus voltage profile. On the other hand, in cases 3, 4, 7, and 9 cumulative VD is near the worst value (1.2 p.u) and hence poor the voltage waveform in such cases. Table 8 presents a comparison of BCS of ToP and CCMO in their associated best cases with the other recent MOEAs.8TABLEComparison of BCS of cases 1, 2, 3 and 4BCS of case1BCS of case 2BCS of case 3BCS of case 4AlgorithmC(PgT)($/h)E (t/h)C(PgT)($/h)VD (p.u)C(PgT)($/h)LindexC(PgT)($/h)PLoss(MW)CCMO831.3880.248800.9060.281800.6860.13737842.9555.028MOEADDAE820.5330.267802.1940.160800.5120.13798840.0485.121NSGAII834.8420.245800.8490.291800.6060.13743842.8575.042TiGEA_2829.2840.251802.1880.157800.8170.13715832.1635.432ToP832.2610.247802.3980.136801.0640.13694841.0255.101MOEA/D‐SF [64]829.5150.2501802.4060.136MOEA/D [9]833.720.2438799.99a0.354MOMICA [30]865.0660.2221800.03a0.4422ESDE [21]833.4740.254804.960.0952MSFLA [22]823.2780.29078ashow infeasible solution and Bold values show best values.In case 1, the cost of active power generation and emission is considered to minimize, in this case, BCS is 832.26098 ($/h) and 0.24734 (t/h) attained by ToP. Whereas in the literature minimum value of a cost function is achieved by MSFLA [22], whose both the objective functions are dominated by MOEADDAE, as compared to ToP and CCMO, PF of MOEADDAE algorithm stuck in the local optima as can be seen in Figure 7 of PF of case 1. Furthermore, the author in ref [21] deliberates the load bus voltage limit between 0.95 and 1.1 p.u, which is much larger than the actual load bus voltage limit between 0.95 and 1.05 p.u considered in the literature as well as here. The choice of BCS is highly dependent upon the distribution of non‐dominated solutions of final PF. Moreover, the overall comparison between the PF of various algorithms, and extreme values of objective functions show the convergence and spread of the final PF. Table 9 shows comparison of the extreme value of objective functions for case 1.7FIGUREBest PF of all the algorithms of case 1 to case 4 with the extreme values of ToP9TABLEComparison of extreme values of case 1ParameterMTLBO[23]MGBICA[24]MOICA[65]MOEA/D [9]MOEA/D‐SF [64]ToPC(Pg)$C( {{P}_g} )$($/h)801.89945.19801.14942.8801.14943.74799.29a944.3800.6932.96800.6944.31E (t/h)0.36650.20490.32960.2040.32960.20480.35930.2040.3670.20560.3650.2048In the literature, MOPF simultaneous optimization of more than two objective functions is relatively uncommon. While in the field of MOEAs the attention has been drawn to many objective optimizations. Table 10 shows the simulation results of various three objective functions of cases 5, 6, and 7. In the literature, only the BCS of case 1 is given, therefore, in Table 9 extreme values of PF of case, 1 are only compared, whereas, for other study cases extreme values of ToP algorithm are pointed out in Figure 7. Simulation results in Table 10 clearly show that the ToP achieves minimum cost and emission values and hence better converged the size of PF as compared to other algorithms.10TABLEComparison of BCS of cases 5, 6 and 7BCS of case 5BCS of case 6BCS of case 7AlgorithmC(PgT)$C( {{P}_{gT}} )$($/h)E (t/h)VD (p.u)C(PgT)$C( {{P}_{gT}} )$$/h)E (t/h)VSIC(PgT)$C( {{P}_{gT}} )$ ($/h)E (t/h)PLoss${P}_{Loss}$(MW)CCMO842.66320.23920.1210839.91560.23940.1374882.92620.21624.0880MOEADDAE836.47630.24860.1287834.73330.25180.1377868.63980.22234.5618NSGAII830.32640.25960.1115843.54890.23820.1374881.47470.21744.1105TiGEA_2836.55610.24560.0947829.77750.25250.1245859.15090.22624.5841ToP827.12310.25770.1568842.71240.23770.1383883.45040.21574.2078MOEA/D‐SF [64]842.4460.24060.1092881.01200.21644.1441MOEA/D [9]850.280.23320.1155902.540.21073.4594MOPSO [9]846.930.23860.2188891.480.21443.9557Figure 7 reveals that the PF of ToP algorithm provides better concentration for both of the objective functions. Consequently, the BCS is favourable to both of the objective functions. Simultaneous optimization of two objective functions in case 1–4, ToP leads to evenly distributed and good diversity of PF. Moreover, compared to other algorithms, ToP produces high‐quality solutions and evenly distributed a wide range of PF in all four cases.Besides, it can be clearly shown in Table 10 that a single solution cannot dominate all the objective functions of proposed algorithm. Among all the best compromise solutions of case 5, the value of cost objective is minimum in ToP, while minimum emission is achieved in case 6. In case 5 load bus voltage profile is near the unity, whereas in cases 6 and 7, touches the maximum value of load bus voltage, because the smallest value of reactive power is injected into the system in cases 6 and 7, also, the generated real power injection in case 7 is smallest compared to all other cases that are because of smallest active power loss appeared in case 7. The regular PF, according to HVI, the best index of ToP and CCMO out of twenty independent runs of cases 5–7 is as shown in Figure 8.8FIGUREPF of cases 5, 6, 7 of ToP and CCMOIn cases 5 and 6, the final non‐dominated objective function values (i.e. PF) of ToP algorithm are uniformly distributed in the entire feasible region and dominates most of the solutions of CCMO, whereas in case 7, CCMO finds the global PF and dominates large no of solutions of final PF of ToP algorithm. Moreover, simulation results of cases 8–10, where optimizing four and five objective functions simultaneously are shown in Table 11.11TABLEComparison of BCS of cases 8, 9 and 10BCS of Case 8BCS of Case 9BCS of case 10AlgorithmC(Pg) ($/h)E (t/h)VD (p.u)PLOSS (MW)FC ($/h)E (t/h)LindexPL (MW)FC ($/h)E (t/h)VSIVD (p.u)PL (MW)CCMO871.3330.22050.13484.853858.4780.22800.13754.6297868.5910.22700.14070.23165.6240MOEADDAE895.5040.21810.15004.379920.6860.21060.13753.5582900.3800.22070.14130.23514.3825NSGAII865.0260.22550.15415.466862.5210.22680.13814.5122864.4440.22510.14570.17325.3894TiGEA_2863.9290.22470.11414.906860.9750.22620.11434.6863860.0160.22710.13980.21425.1321ToP879.4960.21790.19314.663874.3640.21850.13784.4847838.1730.25100.14010.36146.0794MOMICA[30]830.1880.25230.29785.585MOEA/D‐SF [64]883.3220.21870.13224.4527In case 8, BCS of ToP has three objectives dominating the comparable algorithm MOMICA [30] and two objectives of MOEA/D‐SF [64]. The cost objective is much better in MOMICA [30]. As compared to MOMICA [30] and MOEA/D‐SF [64], ToP achieves the minimum value of emission which is 0.2179 (t/h). Moreover, ToP and CCMO obtain well‐distributed and diverse PF of cases 8, 9, and 10. In the literature, high‐dimensional data, that is, PF of more than three objective functions are visualized by using parallel coordinates plots in which the x‐axis point out the name of objective function and y‐axis is the values of the associated distribution of the objective function. Figure 9 shows the parallel plot coordinates of final non‐dominated solutions of cases 8 and 9 of CCMO algorithm, and case 10 of ToP algorithm. In Figure 9, all the objective functions are normalized and the distribution of y‐axis shows that the ToP and CCMO can find a good distribution of PF of MOPF problem considering more than three objective functions.9FIGUREParallel plot of normalized objective functions of final PFVery large scale 300‐bus test systemTo quantify the superiority and performance of proposed algorithm, a very large‐scale IEEE 300‐bus network is adopted to solve MOPF problem. This network consists of 69 generators, 117 tap changing transformers, 411 lines and 14 SVCs. Active and reactive power demand is 23527+j7728 MVA. In base case, 257th bus is the reference bus, system active power loss is 408.3 MW. Network data is taken from [66]. 258 total number of decision variables. A total of 300,000 function evaluations, 50 population sizes (with execution time of about 70 min) are performed.Two study cases are considered.Case 1: Minimization of cost and PLoss${P}_{Loss}$Case 2: minimization of cost and VD.Statistical values and objective functions of twenty independent runs of both cases are as shown in Table 12.12TABLEStatistical value of HVI and objective functions of cases 1 and 2QuantityCase 1Case 2max0.70690.7224min0.61710.6452mean0.644680.6972Standard Deviation0.3417380.373293C(Pg)$C( {{P}_g} )$ ($/h)720846.6720548.5PLoss${P}_{Loss}$(MW)322.04333.8285VD5.39612.4770It is observed from the results in Table 12 that the minimum, mean, maximum, and standard deviation values of each optimization objective are better in case 2. Minimum cost obtained in case 2 (720548.5$/h) compared to case 1. Power loss is minimum in case 1, however voltage deviation is much decreased in case 2. Final convergence and distribution of non‐dominated solution of both the cases are shown Figure 10. Where, green diamond colour shows the best compromise solution.10FIGUREFinal non‐dominated of 300‐bus for cases 1 and 2It can observe that non‐dominated solutions in the figure have better exploration and exploitation. The decision maker can select a single solution of interest from a large number of non‐dominated solution. The load bus voltage profile of both the study cases is shown in Figure 11.11FIGUREVoltage profile of cases 1 and 2 for 300 bus systemThe load bus voltage profile of both study cases is within upper and lower bound. As compared to case 1, voltage profile of case 2 is better (near unity), because of VD minimization.Probabilistic MOPFFrom detailed analysis of deterministic MOPF in most of the study cases, proposed algorithm outperforms the other most recent algorithms. Therefore, for the solution of probabilistic MOPF problem, only ToP algorithm is implemented. As mentioned before, most of the literature did not consider network security constraints and voltage stability constraints in MOPF study.Lindex${L}_{index}$ is calculated by using Equation (8), whereas the constant Γin Equation (16) is the maximum value of local indicator of Lj which is set by decision maker. For the smaller value of Γ, the distance to voltage instability is larger and hence the system has more stability margin. Therefore, for secure integration of RES, constant Γ is set to 0.2 (minimum attainable value of Lindex obtained in deterministic OPF). Whereas, the constant εin Equation (17) is set to 1.2 and can be computed by considering either upper bound or lower bound of load bus voltage. Here, load bus voltage is set between 0.95 and 1.05 p.u. The acceptable variation of load bus voltage is ±0.05 p.u, cumulatively 0.05×24 is 1.2. therefore, overall load bus voltage violation must be less than 1.2 p.u.Moreover, three objective functions, that is, cost, emission, and PLoss${P}_{Loss}$ on modified IEEE 30‐bus test system to examine the effectiveness of PMOPF problem. Moreover, real‐time dispatch is considered after every 10 min over the 1hr period with an increase of 2% of load in each interval of 10 min. In this 1hr scheduling period dynamic approach to load demand and renewable generation is considered the solution PMOPF problem. Furthermore, transformer tap ratio and the rating of SVC are treated as discrete variables that ensure more realistic OPF problem. The transformer tap setting is adjusted at the steps of 0.02 p.u, whereas, SVCs are switched at the discrete steps of 200 kVAr.For the superiority and performance of proposed algorithm, three different study cases of conflicting objective functions, are considered.Case1: Minimization of cost and emission, relaxing the Lindex${L}_{index}$ and VD constraintCase 2: Minimization of cost and PLoss${P}_{Loss}$, relaxing the Lindex and VD constraintCase 3: Minimization of cost, emission and power loss considering Lindex${L}_{index}$ and VD constraintSimulation results of probabilistic MOPFFor the simulation of Probabilistic MOPF, IEEE 30‐bus network is modified. It has 3 thermal generators (at buses 1, 2 and 8). Output of a wind farm is connected to bus 5 while solar PV unit is supplying power to buses 11 and 13. As a noticeable fact, output power from wind and solar are all variables and any deficit in total output from these units must be mitigated by spinning reserve. Table 13 depicts the simulation results of PMOPF problem at maximum load demand.13TABLESimulation results of PMOPFQuantityMinMaxCase 1Case 2Case 3PG2$P{G}_2$208046.02823.18746.291PG5$P{G}_5$ (wind)07549.36456.66168.306PG8$P{G}_8$103513.1712.4130.817PG11$P{G}_{11}$ (Solar)103049.98449.99749.954PG13$P{G}_{13}$ (Solar)124048.47148.4648.115VG1$V{G}_1$0.951.11.0661.0711.055VG2$V{G}_2$0.951.11.041.0561.042VG5$V{G}_5$0.951.11.0251.0311.031VG8$V{G}_8$0.951.11.0261.0371.038VG11$V{G}_{11}$0.951.11.0921.0851.091VG13$V{G}_{13}$0.951.11.0541.051.044QC10$Q{C}_{10}$050.830.6QC12$Q{C}_{12}$0514.62.4QC15$Q{C}_{15}$052.642.6QC17$Q{C}_{17}$052.83.22.6QC20$Q{C}_{20}$051.82.85QC21$Q{C}_{21}$0544.60.8QC23$Q{C}_{23}$05144.2QC24$Q{C}_{24}$051.44.23.4QC29$Q{C}_{29}$054.22.64.6T110.901.10.981.081.08T120.901.11.060.90.9T150.901.11.060.981T360.901.10.980.960.98PG1$P{G}_1$50.00200110.26126.3471.615QG1$Q{G}_1$‐2015025.7273.7569.618QG2$Q{G}_2$‐2060‐18.05420.998‐5.024QG5$Q{G}_5$‐1562.533.0124.26929.248QG8$Q{G}_8$‐1548.731.99236.26840.956QG11$Q{G}_{11}$‐104030.37331.44637.329QG13$Q{G}_{13}$‐1544.730.5072.039.092C(PgT)($/h)457.3413425.1C(PgW)($/h)$C( {{P}_{gW}} )( {{\rm{\$ }}/{{\bf h}}} )$157.4186.5236.4C(PgS)($/h)$C( {{P}_{gS}} )( {{\rm{\$ }}/{{\bf h}}} )$ (bus 11)208208207.8C(PgS)($/h)$C( {{P}_{gS}} )( {{\rm{\$ }}/{{\bf h}}} )$ (bus13)194.1194.1192.5C(PD)($/h)$C( {{P}_D} )( {{\rm{\$ }}/{{\bf h}}} )$3.73.73.7CTotal($/h)${C}_{Total}( {{\rm{\$ }}/{{\bf h}}} )$1020.41005.31065.5E (t/h)0.4341.0520.119PLoss${P}_{Loss}$ (MW)5.545.323.358VD (p.u)0.3470.78660.52454Lindex${L}_{index}$ (p.u)0.16260.157730.15343In the table, bold letters are the objective functions. Simulation results are self‐explanatory. In cases 1 and 2, VD and Lindex constraints are relaxed. All decision variables are within the range in all the intervals. With the increase in load demand, cost of thermal, wind and solar generation is increased. Transformer tap ratios and SVC are in discrete form. Maximum active power of 110.26 MW from the slack bus is taken in case 2, where cost and power loss are the objective functions. The minimum cost of thermal generation is 413 ($/h) in case 2. Out of cases 1–3, VD and Lindex${L}_{index}$of case 3 are better compared to cases 1 and 2. That is because of minimum value of Lindex${L}_{index}$ and VD constraints. In case 3, all the scheduled power of generators are uniformly selected because considering the security constraints. On the other hand, in cases 1 and 2, final solution emphases taking more power from slack bus. The load bus voltage profile of all the study cases is as shown in Figure 12. Load bus voltage level for all the intervals of all cases is within upper and lower bound. Relaxing the VD and Lindex${L}_{index}$ constraint gives poor voltage profile in cases 1 and 2. However, voltage profile of case 3 is near unity. This is due to the consideration of VD and Lindex${L}_{index}$ local stability index constraints. Figure 13 shows the PF (Final non‐dominated solutions) of all the study cases of all time intervals.12FIGUREVoltage profile of all the intervals of cases 1, 2 and 313FIGUREPF of PMOPF of all the casesPF shows that the proposed algorithm can find a better trade‐off between conflicting objective functions and gives better exploration and exploitation to solve complex PMOPF problems. PF of PMOPF problem consists of feasible final non‐dominated solutions. The probabilistic MOPF problem has non‐linear constraints and, for the MOEAs, it is very difficult to handle these constraints. Most of the authors in the literature did not consider security constraints such as voltage stability constraintsLindex$\ {L}_{index}$, and VD. With the consideration of these constraints all stability of power system is increased. In Figure 13 green diamond shows the best compromise solution obtained by using fuzzy weight function. In all the solutions overall constraint violation is zero.CONCLUSIONSMOEAs are computationally expensive to solve nonlinear, mixed‐integer, and highly constrained type MOPF problems. From the tabulated literature review it was found that most of the authors either used a weighted sum approach or Pareto‐based optimization algorithms along with the penalty function approach. The limitation of the former method is that it gives a single solution. Whereas, Pareto‐based method gives a large number of solutions for the decision‐maker with higher computational complexity. Therefore, the present study proposed a Two‐Phase (ToP) algorithm and new efficient constrained type MOEAs yet not been applied for the solution of MOPF problem. The proposed algorithm implements a hybrid flavour of weighted sum (less computational expensive in phase I) and Pareto based (gives non‐dominated solutions in phase II) approaches. To solve MOPF problem, various study cases of 2, 3, and 4 conflicting objectives functions are formulated on IEEE 30 and 300‐bus, to show the effectiveness and performance of proposed algorithm. Each case is independently run twenty times, for the selection of the best PF. HVI technique has been applied to find the best PF. In most of the cases, proposed algorithm outperforms. Figures of PF clearly show that the proposed algorithm can find well‐distributed and near‐global PF. Moreover, from the best PF, a single best non‐dominated solution is found by using a fuzzy membership function. Simulation results clearly show that the decision vector and operational constraints of MOPF problem are within the limits. Overall constraint violation in all the cases is zero.Furthermore, IEEE 30‐ bus test system is modified to inject uncertain wind and solar generation along with variable load demand. The probabilistic nature of wind, solar and load are modelled using appropriate PDFs. Increasing the integration of uncertain RES power, voltage stability of power system is violated. Therefore, two security constraints, Lindex${L}_{index}$ and VD must be within desirable limits and be formulated. Three study cases of 2 and 3 objective functions are implemented on modified IEEE‐30 bus system. Simulation results show that the proposed method efficiently solves the PMOPF problem.NOMENCLATUREAbbreviationsCHTConstraint handling techniqueCVConstraint VolitionEEmission in (t/h)FCFuel costHVIHyper volume indicatorMINLPMixed integer nonlinear programmingMOEAMultiobjective evolutionary algorithmMOPFMultiobjective optimal power flowNLPNonlinear ProgrammingOPFOptimal Power FlowPDFProbability Distribution FunctionPFPareto FrontPMOPFProbabilistic multiobjective OPFPVPhotovoltaicRESRenewable energy sourcesSVCShunt VAR compensatorToPTwo PhaseVDVoltage deviation of systemVSIVoltage stability indexWFWind farmSymbolLjlocal indicatorNint${N}_{int}$Number of time intervalsF(x⃗)$F( {\vec{x}} )$Vector of objective functionx⃗$\vec{x}$Decision vectorϕ,ψ$\phi ,\ \psi $Equality and inequality constraintsC, E, VD$VD$Cost, Emission, voltage deviationPLoss${P}_{Loss}$Power loss in MWLindex${L}_{index}$voltage stability indexi,j$i,\ j$From and to busNG, NLNumber of generator and load busesτT${\tau }_T$Transformer tapingsQC${Q}_C$SVC injection in MVArε, ΓAcceptable VD and VSI parametervin,vr${v}_{in},{v}_r$ and νout${\nu }_{out}$Cut‐in, rated and cut‐out wind speedsGstd, and RcSolar irradiance and certain irradiancegw,j${g}_{w,j}$ , pgW${p}_{gW}$Direct cost of wind and solar PV.PSchW,j${P}_{SchW,j}$, PschS${P}_{schS}$Scheduled power of wind and solar farmKRW,j${K}_{RW,j}$, KPW,j${K}_{PW,j}$, and KR,k${K}_{R,k}$, KPW,j${K}_{PW,j}$Reserve and penalty cost parameters of wind and solar PV generationΔν${\Delta }_\nu $, ΔG${\Delta }_G$, ΔD${\Delta }_D$Probability of wind velocity, solar irradiance and load demandpgW${p}_{gW}$, pgS${p}_{gS}$, pD${p}_D$Available uncertain wind‐solar generation and load demandC(Pg),$C( {{P}_g} ),$Cost of thermal generationC(PgW),C(PgS)$C( {{P}_{gW}} ),\ C( {{P}_{gS}} )$ and C(PD)$C( {{P}_D} )$variable cost of wind, solar and load demandAUTHOR CONTRIBUTIONSA.A., G.A.: Conceptualization; Formal analysis; Investigation; Methodology; Software; Validation; Visualization; S.M., M.U.K.: Writing—review and editing. 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Solution of constrained mixed‐integer multi‐objective optimal power flow problem considering the hybrid multi‐objective evolutionary algorithm

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Abstract

INTRODUCTIONPreliminary knowledge of OPF and literature reviewThe goal of the Optimal Power Flow (OPF) is to find the setting of control variables for the safe and economical operation of the power systems, which optimize a certain objective function while satisfying a set of operational constraints. Control variables consist of the active power of generators except for the generator at a slack bus, a voltage level of all the generators, transformer tap ratio, and shunt VAR injections. These variables are continuous (generator's power and their voltages) and discrete (tap ratio of transformer and shunt Var injection). In earlier days, the implementation of OPF included a single objective function, primarily reducing fuel cost (FC). In fact, along with the FC, there are some other objectives such as reduction of active power loss (PLoss${P}_{Loss}$), load bus voltage deviation (VD), emission (E) and enhancement of Lindex have simultaneously been used in the recent literature for finding the optimal values of control variable of OPF problem [1].In search of global optimal values of objective functions, the OPF program must satisfy the system constraints. In the literature, various equality and inequality constraints have been utilized to solve single and multi‐objective OPF problems. These are active and reactive power balanced equality constraints (this type of constraint must be satisfied during the convergence of load flow) and the most commonly used inequality constraint are MVA branch capacity, bus angles, and voltage magnitudes, generator active and reactive power injections, MVA flow of transmission line, voltages level at each bus.OPF is a constrained type, mixed‐integer, non‐linear, and multi‐objective optimization problem. Conventional methods such as linear programming (LP), quadratic programming (QP) and non‐linear programming (NLP) have been implemented to address the solution of OPF problems subject to minimizing the cost of generation only. During the optimization process, these techniques are considered some theoretical assumptions such as the problem must be convex, continuous, and differential. However, these assumptions may not be satisfied in the multi‐objective OPF problem. Besides, the convergence to the global or local optimal solution of these techniques is highly dependent on the selected initial guess [2]. Continuous LP, QP, and NLP formulations cannot accurately model discrete control variables, such as transformer tap ratios or switched capacitor banks. Mixed‐integer linear programming (MILP) [3] techniques were introduced to solve this problem. However, the non‐linearity of the power system cannot be fully represented by MILP formulations, and therefore cause inherent inaccuracy [1].Having overcome these drawbacks, Evolutionary algorithms (EA) such as genetic algorithm (GA) [4], evolutionary programming (EP) [5], developed Gray wolf optimization (DGWO) [6]; tabu search (TS) [7], improved krill herd algorithm (IKHA) [8]; particle swarm optimization (PSO) [9], jaya algorithm [10], differential evolution (DE) [11, 12], and modified coyote optimization algorithm (MCOA) [13] have been employed for the solution to the single objective OPF problem. The results reported in the previous literature were promising and encouraged further research in this area. Practically, OPF is a multi‐objective optimization problem. Multi‐objective OPF (MOPF) problem is solved either by considering the weighted sum approach or by a Pareto‐based non‐dominated one. In the weighted sum approach, multi‐objective functions are transformed into a single objective function by assigning a predetermined weight factor to each objective function. Whereas, in the Pareto‐based method all the objective functions are optimized simultaneously and the primary reason is to emphasize this technique because it can find multiple Pareto optimal solutions in a single simulation run. However, in weighted sum method leads to only one Pareto optimal solution with a set of specific weight factors for each objective function. A set of non‐dominated solutions (PF) in a single simulation run is preferred as compared to a single solution. Finding the Pareto front (PF) of the MOPF problem while complying with the system constraints is a difficult task for power system engineers. Solution of MOPF problem based on weighted sum approach includes differential search algorithm (DSA) [14], DE [11], moth swarm algorithm (MSA) [15], improved artificial bee colony (IABC) algorithm [16], improved colliding bodies optimization (ICBO) [17] and backtracking search algorithm (BSA) [18] all of these give a single final non‐dominated solution with the specific weight factor. As compared to a single non‐dominated solution, the decision‐maker accentuates the set of non‐dominated (Pareto frontier) solutions in a single simulation run.A limited number of papers in the literature are found on solving constrained multi‐objective evolutionary algorithms (CMOEAs). These were multi‐objective evolutionary algorithm‐based decomposition (MOEA/D) [9],multi‐objective firefly algorithm with the constraint‐prior Pareto dominance approach (MOFA‐CPD) [19], hybrid dragonfly algorithm (DA) and PSO called DA‐PSO [20], and enhanced self‐adaptive DE with mixed cross over (ESDE‐MC) [21] technique were considered for the better exploration and exploitation, modified shuffled frog leaping algorithm (MSFLA) [22] proposed the mutation operator to enhance quality and speed of original SFLA, modified teaching‐learning based optimization (MTLBO) [23] applied to solve MOPF problem in which wavelet technique was applied to form mutation pool and fuzzy clustering technique has been applied to select the population for the next generation. Moreover, modified gaussian bare‐bones multi‐objective imperialist competitive algorithm (MGBICA) [24], improved strength Pareto evolutionary algorithm II (I‐SPEA 2) [25] considers Euclidian distance of kth nearest neighbour for the selection of the next population and external archive are employed to solve MOPF problem. The original gravitational search algorithm (GSA) along with non‐dominated sorting and opposition‐based learning called (NSMOOGSA) [26], wherein, gravitational speed and better distribution of PF, non‐dominated sorting with crowding distance technique and opposition‐based learning concept have been employed. Multi‐objective DE (MDE) [27] modifies the DE variant (DE/ best/1) in which the best individual was selected by considering fuzzy‐based beast compromise solution (BCS) in each generation for mutation operator. Enhanced GA (EGA) [4], for finding the non‐dominated solution to the MOPF problem, combines GA operator for offspring generation and domination principle from SPEA for final PF. Hybrid quasi‐oppositional‐based learning (QOBL) and original teaching learning‐based optimization (TLBO) called (QOTLBO) [28] to improve the convergence speed and distribution of final PF. Imperialist competitive algorithm (ICA) often traps into local optima, therefore combined modified ICA (CMICA) [29] and multi‐objective modified ICA (MOMICA) [30] were used to solve constraints MOPF problem. Moreover, for better solution quality interaction effects of colonies on each other and policy rule of ICA method is modified. Pareto‐based domination principle along with fuzzy grouping has been adapted to enhance the exploration and exploitation of CMICA. Bio inspired modified multi‐objective flower pollination algorithm (B‐MMOFPA) [31] in which normal boundary intersection (NBI) with original FPA has been considered to find the nondominated solutions. Further three statistical techniques such as fuzzy decision making, median and centroid methods were used to select the best compromise solution. Single objective optimization Jaya algorithm combined with the Quasi‐oppositional technique called Quasi‐oppositional modified Jaya (QOMJaya) [32] to handle the MOPF problems. In this algorithm convergence and quality of solutions were increased through an intelligence strategy called quasi‐oppositional based learning. Whereas, non‐dominated sorting principle along with crowding distance selection scheme was used for finding the well‐distributed PF. An interior search algorithm (ISA) [33] together with non‐dominated sorting was applied to find the PF. A fuzzy membership function is considered to find the best compromise solution (BCS).Multi‐objective dimension‐based firefly algorithm (MODFA) and multi‐objective PSO (MOPSO) [34] are compared with the superior feasibility (SF) constraint technique to find the solution MOPF problem. Semi‐definite programming (SDP) [35] and improved normalized norm‐constrained (INNC) [36] methods incorporate the epsilon constraint method (ECM) for MOPF problem. Original NNC cannot find the entire PF, therefore improved NNC (INNC) has been applied to fulfil the shortcoming of NNC by converting the utopia hyperplane into a utopia line and converting the multi‐objective into three single and three bi‐objective optimization problems. Modified pigeon‐inspired optimization algorithm (MPIO) [37] integrated with constraint‐objective sorting rule (COSR) called (MPIO‐COSR) was considered for finding the PF of MOPF problem.Due to a large number of function evaluations to sort the constraint violation of COSR technique, it is computationally expensive. Chen et al. proposed novel hybrid bat algorithm along (NHBA) with constrain‐prior Pareto‐dominance method (CPM) called (NHBA‐CPM) [38] and hybrid firefly‐bat algorithm with constraints‐prior Pareto‐dominant rule (CPR) combined HFBA‐COFS [39] to solve MOPF problem considering three objective functions. Furthermore, Table 1 summarizes the discussed papers because of selection of objective functions, CHT, weighted sum, or Pareto‐based approach and test systems.1TABLEMulti‐objective optimization studies for the solution of MOPFMethod RefObjective functionsCHTApproachTest systemDSA [14]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionWeighted sum9, 30, 57IABC [16]FC, E and PLoss${P}_{Loss}$Penalty functionWeighted sum30, 57 and 300DE [11]FC, E, VD, PLoss${P}_{Loss}$and VSISF, SP, ECHTWeighted sum30, 57, 118MSA [15]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionWeighted sum30, 57, 118ICBO [17]FC, VD and VSIPenalty functionWeighted sum30, 57, 118BSA [18]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionWeighted sum30, 57, 118MOEA/D [9]FC, E, VD and PLoss${P}_{Loss}$Penalty functionPareto based30ESDE‐MC [21]FC, E, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30, 57, 59SFLA [22]FC, E,Penalty functionPareto based30MTLBO [23]FC, E,Penalty functionPareto based30, 57,MGBICA [24]FC, EPenalty functionPareto based30, 57,I‐SPEA2[25]FC, EPenalty functionPareto based30, 57,NSMOOGSA [26]FC, E, VD, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30,MDE [27]FC, VD, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based57, 118EGA [4]FC, PLoss${P}_{Loss}$and VSIPenalty functionPareto based30,QOTLBO [28]FC, E, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30, 118CMICA [29]FC, VD, PLoss${P}_{Loss}$Penalty functionPareto based57MOMICA [30]FC, E, VD, PLoss${P}_{Loss}$Penalty functionPareto based30, 57B‐MMOFPA [31]FC, VD, PLoss${P}_{Loss}$Penalty functionPareto based30QOMJaya [32]FC, PLoss${P}_{Loss}$ and VSIPenalty functionPareto based30ISA [33]FC, E, VD, PLoss${P}_{Loss}$and VSIPenalty functionPareto based30 and 57MODFA [34]FC, E, PLoss${P}_{Loss}$SFPareto based30, 57, 118DA‐PSO [20]FC, E, PLoss${P}_{Loss}$Penalty functionPareto based30, 57SDP [35]FC, E,ECMPareto based30, 57, 118INNC [36]FC, PLoss${P}_{Loss}$, VDECMPareto based30, 118NHBA‐CPFD [38]FC, E, PLoss${P}_{Loss}$CPDPareto based30, 57, 118MOFA‐CPD [19]FC, E, PLoss${P}_{Loss}$CPDPareto based30, 57HFBA‐COFS [39]FC, E, PLoss${P}_{Loss}$CPRPareto based30, 57, 118MPIO [37]FC, E and PLoss${P}_{Loss}$COSRPareto based30, 57, 118CHT = constraint handling technique, FC = Fuel cost, E = emission, VD = voltage deviation, PLoss${P}_{Loss}\ $= active power loss, Lindex = voltage stability indicator, SF = superiority of feasibility, ECM = ε‐constraint method, Constraint‐objective sorting rule (COSR).World global environment regulations have constituted a greater challenge to the thermal power generation industry because it emits harmful gases into the environment. European Union (EU) and G8 aim to reduce greenhouse gas (GHG) emissions by at least 80% below 1990 levels by 2050 [40].Therefore, RES has drawn momentous attention from both industry and academia because of its low emissions. In recent years, integration of sustainable energy generation has risen significantly due to being environmentally friendly compared to thermal resources and is forecasted to rapidly expand in the future. These sources generate variable power based on the availability of wind speed and solar irradiance. The uncertain nature of these sources represents the challenging constraints to the multi‐objective OPF problems in terms of supply‐demand mismatch. In such power systems, a deterministic system study such as optimal power flow (OPF) evaluation cannot reveal the state of system accurately; therefore, probabilistic evaluation is of significant interest. Recently, most of the authors quantify the solution of single objective OPF problem with the integration of probabilistic wind and solar generation. These include Success history‐based adaptive differential evolution (SHADE) [41], symbiotic organisms search algorithm [42], Lévy coyote optimization algorithm (LCOA) [43] and Fuzzification method to reduce the operating cost and increase the minimum voltage magnitude [44]. Increasing the integration of uncertain RES power, voltage stability of power system may go to violate (voltage stability limit is based on a theory of maximum power transfer between two buses [44]). Therefore, the consideration of RES should be adequately analyzed the voltage stability detection [45]. In most of the papers, security constraints such as voltage deviation and voltage stability index are not studied. To the knowledge of author, these constraints are not studied in probabilistic multiobjective OPF problems considering uncertain wind and solar generation.The computation of probabilistic multiobjective OPF (PMOPF) is one of the major requirements in power system planning and operation. Performing PMOPF study helps system planning engineers in making judgments concerning investments. But, the total number of probable combinations of uncertain variables such as the loads and generation units are high. As a result, performing PMOPF is a computational burden issue; therefore, a probabilistic model of wind, solar and load demand with acceptable accuracy is needed for the power system studies.ContributionMost of the MOEAs discussed in the literature review incorporate one popular constraint handling technique (CHT) such as the penalty function approach to managing the constraint violation (CV). The performance of the penalty function approach is widely dependent on the selection of constant penalty parameters. With a small value of penalty coefficient, the algorithm may trap in the infeasible space whereas a large value of this parameter may over‐explore the feasible space and may get stuck in local optima. Inappropriate penalty parameter selection may also give an infeasible solution in the final non‐dominated PF. Therefore, in a realistic MOPF problem, appropriate CHT must be selected to guide the MOEAs, search the entire feasible space and extricate out of the infeasible region, and find the near global and widely distributed PF. Moreover, in the given tabulated literature, all the authors have been selecting predefined parameters for their MOEAs and claiming that their MOEAs have been robust and converged which gives better PF than that of other MOEAs. Comparisons between various MOEAs, however, are challenging tasks, as the collection and pre‐defined parameters for each MOEAs dominate the result. Furthermore, according to no‐free‐lunch (NFL) theorem [46], a single MOEA is not found that solves all the problems (study cases) that are simply superior to priors.Inspired by aforesaid points, here, various new recent constrained MOEAs are considered to find the solution to MOPF problem that has not been tested yet for the solution of MOPF problem, these include ToP [47], CCMO [48], the decomposition‐based multi‐objective evolutionary algorithm with detect‐and‐escape constraint technique (MOEA/D‐DAE) [49] and tri‐goal evolution (TiGE_2) [50]. The very first time these algorithms are applied to find the nondominated solutions considering simultaneous optimization of 2, 3, 4, and 5 objective functions of a comprehensive set of ten study cases. Out of which 1–6 and 8 have already existed in the literature whereas 7, 9, and 10 are unprecedented ones. Moreover, IEEE standard 30‐bus (small network) and 300‐bus (vary large) test systems are considered to analyze the simulation results of proposed CMOEAs. Simulation results show that the ToP and CCMO can find the global optimal solution for proposed study cases compared to other MOEAs in the literature. ToP has better PF (near global and evenly distributed) compared to other algorithms in cases 1–6 and case 10, whereas, CCMO gives better results in cases 7–9. The computational complexity of ToP is less compared to other algorithms. In the first phase of the ToP, weighted sum approach is applied to find high‐quality feasible solutions) after that in phase II, NSGAII [51] has been applied to find the final evenly distributed and near‐global PF.Moreover, probabilistic multi‐objective OPF (PMOPF) is formulated with the integration of uncertain wind and solar PV. Various combinations of two and three objective functions are considered for the analysis and comparison of probabilistic MOPF problems. Furthermore, several supplementary constraints along with two unprecedented constraints are incorporated to find secure non‐dominated solutions to MOPF problem. In probabilistic MOPF, SVC and transformer tap settings are discrete. In most papers in the literature, round off operator has been applied to find the discrete decision variables for mixed integer problems. Due to the rounding operator, algorithm is stuck in the search area of local optima. However, in the proposed algorithm integer constraints are handled by measuring functions same as given in [52], to avoid the trap in the local optima.The main contributions of this work are as follows:The comprehensive ten study cases comprised of 2, 3, 4, and 5 objective functions are formulated on IEEE 30‐bus and 300‐bus test systems to solve the deterministic MOPF problem.Two unprecedented security constraints such as voltage deviation and local stability indicator Lj are considered to narrow down the feasible search space to show the effectiveness of the proposed algorithm.Various recent constrained MOEAs, yet not employed to solve MOPF problem, are implemented.Probabilistic wind and solar power generators are integrated considering appropriate distribution functions.A hybrid two‐phase (ToP) algorithm (combination of single and multi‐objective evolutionary algorithms) integrated with the constraint domination principal technique is implemented to address the solution of deterministic and probabilistic OPF problem.Simulation results of the proposed algorithms are analyzed and compared with the other recent available methods in the literature.In the rest of the paper, we first formulate the deterministic and probabilistic MOPF problem in Section 2 comprised of various objective functions and operational constraints. Then in Section 3, we elaborate on the CHT and framework of proposed the algorithm. Afterward, in Section 4detailed analysis and comparison of simulation results are presented. Finally, the conclusion and future work is outlined in Section 5.PROBLEM FORMULATIONConventional OPF problemMathematically multi‐objective problems having the number of objective functions (M≥2$M \ge 2$) are expressed as:1minFx⃗=f1x⃗,f2x⃗,…,fMx⃗s.t:ϕ(x⃗)=0ψx⃗≤0$$\begin{eqnarray} min\ F\left( {\vec{x}} \right) &=& \left[ {{f}_1\left( {\vec{x}} \right),{f}_2\left( {\vec{x}} \right), \ldots ,{f}_M\left( {\vec{x}} \right)} \right]\nonumber\\ s.t: \phi ( {\vec{x}} ) &=& 0\nonumber\\ && \psi \left( {\vec{x}} \right) \le 0 \end{eqnarray}$$where x⃗$\vec{x}$ is the decision vector,F(x⃗)$\ F( {\vec{x}} )$is the objective function, ϕ(x⃗)$\phi ( {\vec{x}} )$ are the equality constraints and ψ(x⃗)$\psi ( {\vec{x}} )$ is the inequality constraint for the OPF problem. The decision vector x⃗${{\vec{\rm x}}}$ for the OPF problem is given as:x⃗=PG2…PGN,VG1..VGN,τ1T…τTN,QC1…QCN$$\begin{equation*}\ {\bm{\vec{x}}} = \ \left[ {{P}_{G2} \ldots {P}_{GN},\ {V}_{G1}..{V}_{GN},\ {\tau }_{1T} \ldots {\tau }_{TN},\ {Q}_{C1} \ldots {Q}_{CN}} \right]\end{equation*}$$where P and V are the output power of generators and voltage at the generator bus,τ is transformer tap ratio andQ MVAr injections of shunt VAR compensator (SVC); GN, TN, and CN are the number of generators, transformer tap settings, and SVC. In the proposed study cases, two or more objective functions F(x⃗)${\rm{F}}( {{{\vec{\rm x}}}} )$ from (2) is considered to solve simultaneously.2Fx⃗=CPg⃗,EPg⃗,VDVL⃗,LindexVL⃗,APLV⃗$${\fontsize{9}{11}{\selectfont{ \begin{eqnarray} F\ \left( {\vec{x}} \right) = \left[ {C\left( {\overrightarrow {{P}_g} } \right),E\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right),\ VD\left( {\overrightarrow {{V}_L} } \right),\ {L}_{index}\left( {\overrightarrow {{V}_L} } \right),APL\left( {\vec{V}} \right)} \right]\nonumber\\ \end{eqnarray}}}}$$where C(Pg⃗)$C( {\overrightarrow {{P}_g} } )$ and E(Pg⃗)$E( {\overrightarrow {{P}_g} } )$are the quadratic fuel cost and emission of the thermal generator [15], mathematically proposed objective function described as:3CPg⃗=∑i=1NGai+biPGi+ciPGi2$$\begin{equation}C\ \left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right) = \sum _{i = 1}^{NG} \left( {{a}_i + {b}_i{P}_{Gi} + {c}_iP_{Gi}^2} \right)\ \end{equation}$$4EPg⃗=∑i−1NG[αi+βiPGi+γiPGi2+ωieμiPG]$$\begin{equation}E\ \left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right) = \sum _{i - 1}^{NG} [\ \left( {{\alpha }_i + {\beta }_i{P}_{Gi} + {\gamma }_iP_{Gi}^2} \right) + {\omega }_i{e}^{\left( {{\mu }_i{P}_G} \right)}]\end{equation}$$5VDVL⃗,Ybus=∑L=1LNVL−1$$\begin{equation}VD\ \left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} ,{Y}_{bus}} \right) = \left( {\sum _{L = 1}^{LN} \left| {{V}_L - 1} \right|} \right)\ \end{equation}$$6LindexVL⃗=maxLjwherej=1,2,…,NL$$\begin{equation}{L}_{index}\ \left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} } \right) = \ max\ \left( {{L}_j} \right)\hbox{where}\ j\ = \ 1,2,\ \ldots ,\ NL\end{equation}$$7PLossV⃗=∑i=1Nb∑j=i+1NbGijVi2+Vj2−2ViVjcosδij$$\begin{equation}{P}_{Loss}\left( {{\bm{\vec{V}}}} \right)\ = \sum _{i = 1}^{Nb} \ \sum _{j\ = \ i + 1}^{Nb} {G}_{ij}\left[ {V_i^2 + V_j^2 - 2{V}_i{V}_j\ cos\ \left( {{\delta }_{ij}} \right)} \right]\end{equation}$$where ai,bi,ci,αi,βi,γi,ωi${a}_i,\ {b}_i,\ {c}_i,\ {\alpha }_i,\ \ {\beta }_i,\ \ {\gamma }_i,\ {\omega }_i$andμia${{{\mu}}}_{\rm{i}}{\rm{\ a}}$re the constants for the cost and emission of the thermal generators taken from [15]. Where δij=δi−δj${\delta }_{ij} = {\delta }_i\ - {\delta }_j$, V⃗$\vec{V}$ is the complex voltage of all the buses and VL shows the voltage at load (PQ) buses. Voltage magnitudes of load buses are somewhat less than the slack and PV buses; the deviation of load bus voltage depends upon the reactive power demand (QL). Gij${G}_{ij}$ is the conductance of π equivalent model of transmission line between bus i and j. Estimate of voltage stability is an issue that is receiving growing attention from power system researchers due to system voltage collapses in the past because of voltage instability. Voltage stability index (Lindex${L}_{index}$) indicator has developed in [53] which can be defined based on Lj local indicator. Let NG and NL be the number of generator and load buses respectively then local indicator Lj can be calculated as;8Lj=1−∑i=1NLFjiViVj,wherej=1,2,…,NLandFji=−[YLL]−1YLG$$\begin{eqnarray} {L}_j &=& \left| {1 - \sum _{i = 1}^{NL} {F}_{ji}\frac{{{V}_i}}{{{V}_j}}} \right|\ ,\ \hbox{where}\ j\ = \ 1,\ 2, \ldots ,NL\quad \hbox{ and }\nonumber\\ {F}_{ji} &=& \ - {[{Y}_{LL}]}^{ - 1}\left[ {{Y}_{LG}} \right] \end{eqnarray}$$Where sub‐matrices YLL${Y}_{LL}$ and YLG${Y}_{LG}$ are calculated from Ybus${Y}_{bus}$matrix after separating PV and PQ buses. The equality constraint ϕ(x⃗)$\phi ( {{{\vec{\rm x}}}} )$ is the balanced power flow constraints, given as:9SiV+SLi−Sgi=0$$\begin{equation}{S}_i\left( {\bm{V}} \right) + {S}_{Li} - \ {S}_{gi} = \ 0\end{equation}$$Let V and I be the vectors of complex bus voltage and bus current injection, then mathematically I injection for each bus can be calculated by using I = YV, where Y is the bus admittance matrix than complex MVA bus injection is Si=VI∗⇒V(YV)∗${S}_i = \ V{I}^* \Rightarrow V{( {YV} )}^*$ and SLi=PLi+jQLi${S}_{Li} = {P}_{Li}\ + j{Q}_{Li}$ and Sgi=Pgi+jQgi${S}_{gi} = {P}_{gi}\ + j{Q}_{gi}$. On the other hand, inequality constraint ψ(x⃗)${{\psi}}( {{{\vec{\rm x}}}} )$ for the OPF problem consists of operational limits of all the generators and security constraints on the buses and lines.10Pgimin<Pgi<Pgimax∀i∈GN$$\begin{equation}P_{gi}^{min} &lt; {P}_{gi} &lt; P_{gi}^{max}\ \forall i \in GN\end{equation}$$11Vgimin≤Vgi≤Vgimax∀i∈GN$$\begin{equation}V_{gi}^{\ min\ } \le {V}_{gi} \le V_{gi}^{\ max\ }\forall i \in GN\end{equation}$$12VLmin≤VL≤VLmax∀L∈LN$$\begin{equation}V_L^{min\ } \le {V}_L \le V_L^{\ max\ }\forall L \in LN\end{equation}$$13Sl≤Slmax∀l∈nl$$\begin{equation}{S}_l \le S_l^{\ max\ }\forall l \in nl\end{equation}$$14τTmin≤τT≤τTmin∀T∈TN$$\begin{equation}\tau _T^{min\ } \le {\tau }_T \le \tau _T^{min\ }\forall T \in TN\end{equation}$$15QCmin≤QC≤QCmax∀T∈TN$$\begin{equation}Q_C^{min\ } \le {Q}_C \le Q_C^{max}\ \forall T \in TN\end{equation}$$16Lindex<Γ$$\begin{equation}{L}_{index} &lt; \ \Gamma \end{equation}$$17VD<ε$$\begin{equation}\left| {VD} \right| &lt; \epsilon \end{equation}$$where Pgi${P}_{gi}$ and Qgi${Q}_{gi}$ are the active and reactive output of generator at bus i, Vgi${V}_{gi}$ and VL${V}_L$ are the generator and load bus voltages, Sl${S}_l$ is the MVA branch flow, τT${\tau }_T$ is transformer tap ratio, QC${Q}_C$ is shunt Var compensator, Lindex${L}_{index}$ is the maximum value of Lj${L}_j$ indicator for all the load buses. Lindex is calculated by using Equation (6), whereas the constant Γ is local indicator Lj which is set by the decision‐maker. For the smaller value of Γ, the distance to voltage instability is larger and hence the system has more stability margin. In this paper, the constant Γ is less than 0.2 (minimum attainable value of Lindex), which is obtained by considering the single objective optimization of Lindex. Whereas, the constant ε is set to 1.2 for the IEEE 30 bus system and can be computed by considering either upper bound or lower bound of load bus voltage. In this paper, load bus voltage is set between 0.95 and 1.05 p.u. The acceptable variation of load bus voltage is ±0.05 p.u, cumulatively 0.05×24 is 1.2 therefore, and the overall load bus voltage violation must be less than 1.2 p.u.Probabilistic OPF problemSolution of probabilistic multi‐objective OPF (PMOPF) is a challenging task to satisfy technical, economic and environmental issues with the high penetration of uncertain wind and solar PV. Therefore, various combinations of two and three objective functions are considered for the analysis and comparison of probabilistic MOPF problems. Furthermore, several supplementary constraints along with two unprecedented constraints are incorporated to find secure non‐dominated solutions to MOPF problem. In the subsequent subsection, we will formulate the mathematical model of uncertain renewable generation and demand and objective functions.Modelling of uncertain renewable generation and demandIn the field of an electric power system, load demand is always uncertain. Further complexity of power system planning is increased with the integration of uncertain wind, solar PV and small hydro power. Therefore, here, appropriate PDFs are used to model the demand and output power of renewable generation. In the literature, Weibull, lognormal, and gamble probability distribution functions (PDFs) have been used for the modelling of wind, solar and small hydro power generation, whereas, normal PDF are used for load modelling [54–56]. The aforementioned PDFs are mathematically defined as:Weibull PDF for the wind velocity (v) estimating:18Δνv=bavab−1×e−vab$$\begin{equation}{\Delta }_\nu \ \left( v \right) = \left( {\frac{b}{a}} \right)\ {\left( {\frac{v}{a}} \right)}^{\left( {b - 1} \right)} \times {e}^{\left[ { - {{\left( {\frac{v}{a}} \right)}}^b} \right]}\end{equation}$$Lognormal PDF for the solar irradiance (G) predicting:19ΔGG=1G×d2πe−(lnG−c)22d2$$\begin{equation}{\Delta }_G\ \left( G \right) = \frac{1}{{G \times d\sqrt {2\pi } }}\ {e}^{\left[ {\frac{{ - {{(lnG - c)}}^2}}{{2{d}^2}}} \right]}\end{equation}$$where a, b are the shape and scale parameters Weibull PDF; c and d are the mean and standard deviation of lognormal PDF.Probabilistic available wind power pgW${p}_{gW}$ is the function of wind velocity (v), both are highly nonlinear with each other, mathematically given as [57, 58]:20pgWv=0,forνvinandvνouTpwrv−νinνr−vinforvin≤v≤vrpwrforvr<ν≤vouf$$\begin{equation} {p}_{gW}\left( v \right)\ = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{l}@{}} 0, for\ \nu \left\langle {{v}_{in}\ and\ v} \right\rangle {\nu }_{ouT}\\[15pt] {p}_{wr}\left( {\dfrac{{v - {\nu }_{in}}}{{{\nu }_r - {v}_{in}}}} \right)\ for\ {v}_{in} \le v \le {v}_r\\[15pt] {{p}_{wr}\ for\ {v}_r &lt; \nu \le {v}_{ouf}} \end{array} \right. \end{equation}$$In our study 3 MW, Enercon E82‐E4 wind turbine is adopted [59]. The probabilistic solar PV power pgS${p}_{gS}$ is the function of available solar irradiance (G) [60]21pgSGS=PsrGs2GsTdRcfor0<Gs<RcPsrGsGsTdforGs≥Rc$$\begin{equation} {p}_{gS}\left( {{G}_S} \right)\ = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{l}@{}} {{P}_{sr}\left( {\dfrac{{G_s^2}}{{{G}_{sTd}{R}_c}}} \right)\ for\ 0 &lt; {G}_s &lt; {R}_c}\\[15pt] {{P}_{sr}\left( {\dfrac{{{G}_s}}{{{G}_{sTd}}}} \right)\ for\ {G}_s \ge {R}_c} \end{array} \right. \end{equation}$$Normal PDF for estimating the percentage of active and reactive load demand (l):22ΔDl=1σ2Π×e−(l−μ)22σ2$$\begin{equation}{\Delta }_D\left( l \right)\ = \frac{1}{{\sigma \sqrt {2\Pi } }}\ \times {\rm{\ e}}^{ - \left( {\frac{{{{(l - \mu )}}^2}}{{2{\sigma }^2}}} \right)}\end{equation}$$Table 2 shows the parameters of PDFs, that are realistically selected and based upon installed capacity.2TABLEParameters of weibull, lognormal and vormal PDFUncertaintyPDFParametersPower and load demandWind speed vWeibullShape (a = 9); Scale (b = 2)25 turbines of each 3 MW (rated 75 MW)Solar irradiance GLognormalMean (c = 5.5); SD (d = 0.5)Rated 50 MWLoad lNormalMean (μL = 97); SD (σL = 5)Percentage of loadFigure 1 shows the Weibull, lognormal and normal PDF fittings and their associated uncertain power generation and demand. Moreover, for realistic OPF problems, 1‐h load demand uncertainty is considered to analyse the integration of uncertain generation.1FIGUREPDF of wind‐solar units and load demandIntegration of probabilistic wind and solar power affects the cost computation only. Therefore, cost function with the integration of RES is given in the next section.Combined operating costThe intermittent and unpredictable nature of RES makes grid integration challenging. Typically, private operators—An organisation that enters into purchase agreements with independent system operators (ISO)—are the owners of wind, and solar PV generation. The ISO is responsible for mitigating the deficit amount by retaining spinning reserves if demand increases and the wind and solar generation are unable to provide the scheduled electricity owing to non‐availability or insufficiency of renewable sources. Such events are termed as overestimation of renewable power and for which maintaining spinning reserve adds to the power generation cost. On the other hand, a situation can occur where more RES power is generated than is planned. If the excess power is not used in such a case of underestimate, it is wasted, and ISO is responsible for the associated costs.Therefore, with the integration of RES, objective function vector has been modified and given as:23minx⃗Fx⃗=CTotalPg⃗,EPg⃗,VDVL⃗,τTN,LindexVL⃗,PLossV⃗$$\begin{eqnarray} && \mathop {\min }\limits_{{\bm{\vec{x}}}} {\bm{F}}\left( {{\bm{\vec{x}}}} \right) = \left[ {C}_{Total}\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right), E\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right),\right.\nonumber\\ &&\left.\quad VD\left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} ,{{\bm{\tau }}}_{{\bm{TN}}}} \right),{L}_{index}\left( {\overrightarrow {{{\bm{V}}}_{\bm{L}}} } \right),\ {P}_{Loss}\left( {{\bm{\vec{V}}}} \right) \right]\ \end{eqnarray}$$24CTotalPg⃗=∑tNint∑i,j,k,mNTG,NWG,NSG,NDCiPgT+CjPgW+CkPgS+CmPD$$\begin{eqnarray} && {C}_{Total}\left( {\overrightarrow {{{\bm{P}}}_{\bm{g}}} } \right)\ = \sum _t^{{N}_{int}} \sum _{i,j,k,m}^{{N}_{TG},\ {N}_{WG},\ {N}_{SG},\ {N}_D}\nonumber\\ && \left[ {{C}_i\left( {{P}_{gT}} \right) + {C}_j\left( {{P}_{gW}} \right) + {C}_k\left( {{P}_{gS}} \right) + {C}_m\left( {{P}_D} \right)} \right]\ \end{eqnarray}$$Whereas, Ci(PgT)${C}_i( {{P}_{gT}} )$ is given in Equation (3), cost of wind generation Cj(PgW)${C}_j( {{P}_{gW}} )$ computed as:25CjPgW=gw,jPSchW,j︸Directcost+KRW,jPschW−pgW︸ReservecostPgW<Psched+KPW,jpgW−PschW︸Penalitycost(PgW>Psched)$$\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{C}_j\ \left( {{P}_{gW}} \right) = \underbrace {{g}_{w,j}{P}_{SchW,j}}_{Direct\ cost}\ + \ \underbrace {{K}_{RW,j}\left( {{P}_{schW} - {p}_{gW}} \right)}_{Reserve\ cost\ \left( {{P}_{gW} &lt; \ {P}_{sched}} \right)}}\\[35pt] { + \underbrace {{K}_{PW,j}\left( {{p}_{gW} - {P}_{schW}} \right)}_{Penality\ cost\ ({P}_{gW} &gt; \ {P}_{sched})}} \end{array} \end{equation}$$Mathematically reserve and penalty costs are computed as:CRPschW−pgW=KRW∫0PschW(PschW−pgW)×$$\begin{equation*}{C}_R\ \left( {{P}_{schW} - {p}_{gW}} \right) = {K}_{RW}\ \int _0^{{P}_{schW}} ({P}_{schW} - {p}_{gW}) \times \end{equation*}$$26πwpgWdpgW$$\begin{equation} {\pi }_w\left( {{p}_{gW}} \right)d{p}_{gW}\end{equation}$$27CPPgW−PschW=KPw∫PschWPgWrpgWr−PschWπWpgWdpgW$${\fontsize{9.7}{11.7}{\selectfont{ \begin{eqnarray} {C}_P\ \left( {{P}_{gW} - {P}_{schW}} \right) = {K}_{Pw}\ \int _{{P}_{schW}}^{{P}_{gWr}} \left( {{p}_{gWr} - {P}_{schW}} \right){\pi }_W\left( {{p}_{gW}} \right)d{p}_{gW}\nonumber\\ \end{eqnarray}}}}$$Similarly, operating cost of kth solar PV,Ck(PgS)${C}_k( {{P}_{gS}} )$, defined as [54]:28CkPgS=gS,kPgS,k︸Directcost+KR,kPschS−pgS︸ReservecostPgS<Psched+KP,kpgS−PschS︸Penalitycost(PgW>Psched)$$\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{C}_k\ \left( {{P}_{gS}} \right) = \underbrace {{g}_{S,k}{P}_{gS,k}}_{Direct\ cost} + \ \underbrace {{K}_{R,k}\left( {{P}_{schS} - {p}_{gS}} \right)}_{Reserve\ cost\ \left( {{P}_{gS} &lt; \ {P}_{sched}} \right)} }\\[35pt] {+\,\underbrace {{K}_{P,k}\left( {{p}_{gS} - {P}_{schS}} \right)}_{Penality\ cost\ ({P}_{gW} &gt; \ {P}_{sched})}} \end{array} \end{equation}$$Reserve cost and penalty cost of solar PV$PV$ is computed as [60]:29CRsPschS−pgS=KRs∫minpgSPschW(PschS−pgS)×πSpgSdpgS$$\begin{eqnarray}{C}_{Rs}\left( {{P}_{schS} - {p}_{gS}} \right)\ = {K}_{Rs}\ \int _{{\rm{min}}\left( {{p}_{gS}} \right)}^{{P}_{schW}} ({P}_{schS} - {p}_{gS}) \times {\pi }_S\left( {{p}_{gS}} \right)d{p}_{gS}\nonumber\\ \end{eqnarray}$$30CPspgS−PschS=KPs∫PschWmaxpgS(PschS−pgS)×πSpgSdpgS$${\fontsize{9.5}{11.5}{\selectfont{ \begin{eqnarray}{C}_{Ps}\left( {{p}_{gS} - {P}_{schS}} \right)\ = {K}_{Ps}\ \int _{{P}_{schW}}^{{\rm{max}}\left( {{p}_{gS}} \right)} ({P}_{schS} - {p}_{gS}) \times {\pi }_S\left( {{p}_{gS}} \right)d{p}_{gS}\nonumber\\ \end{eqnarray}}}}$$Last term in the combined operating cost function is the cost of load demand variability, it is comprised of reserve and penalty cost only and can be given as:31CmPD=CR,mPdsch−pD︸Reservecost(PD<Psched)+CP,mpD−Pdsch︸Penalitycost(PgW>Psched)$$\begin{equation}{C}_m\ \left( {{P}_D} \right) = \underbrace {{C}_{R,m}\left( {{P}_{dsch} - {p}_D} \right)}_{Reserve\ cost\ ({P}_D &lt; \ {P}_{sched})}\ + \underbrace {{C}_{P,m}\left( {{p}_D - {P}_{dsch}} \right)}_{Penality\ cost\ ({P}_{gW} &gt; \ {P}_{sched})}\end{equation}$$Reserve and penalty cost of load demand is computed as:CRDPdsch−pD=KRd∫PDminPdsch(Pdsch−pD)×$$\begin{equation*}{C}_{RD}\ \left( {{P}_{dsch} - {p}_D} \right) = {K}_{Rd}\ \int _{{P}_{Dmin}}^{{P}_{dsch}} ({P}_{dsch} - {p}_D) \times \end{equation*}$$32fd(pD<Pdsch)dpD$$\begin{equation}{f}_d({p}_D &lt; {P}_{dsch})d{p}_D\end{equation}$$CPDpD−Pdsch=KPd∫PdschPDmax(Pdsch−pD)×$$\begin{equation*}{C}_{PD}\ \left( {{p}_D - {P}_{dsch}} \right) = {K}_{Pd}\ \int _{{P}_{dsch}}^{{P}_{Dmax}} ({P}_{dsch} - {p}_D) \times \end{equation*}$$33fd(pD>Pdsch)dpD$$\begin{equation}{f}_d({p}_D &gt; {P}_{dsch})d{p}_D\end{equation}$$where fd(pD<Pdsch)${f}_d( {{p}_D &lt; {P}_{dsch}} )$andfd(pD<Pdsch)${f}_d({p}_D &lt; {P}_{dsch})$are the probability of over and under estimation respectively of load demand,PDmin${P}_{Dmin}$ and PDmax${P}_{Dmax}$ are the minimum and maximum demand that appeared during normal distribution of load uncertainty. Numerical values of direct, penalty, and reserve cost coefficients for stochastic wind‐solar and load demand are provided in Table 3.3TABLECost parameter for the uncertainty in wind–solar and loadCost co‐efficientWindSolarLoadBus511All PQDirect1.71.6–Reserve333Penalty1.41.41.4The direct cost coefficients are set up so that solar and wind power have the highest costs, respectively [56, 60]. The respective direct cost coefficient is lower than the reserve cost coefficient for sustaining spinning reserves. The direct cost is higher than the penalty for not using the available electricity, though.CONSTRAINT HANDLING TECHNIQUES AND MOEASMOPF problem is a practical constrained multi‐objective optimization problem (CMOP). To solve CMOPs, state‐of‐the‐art MOEAs possess some limitations [1, 47], that is, incapable of balancing constraints and objective functions with small feasible search space or combined discrete and real feasible search space that leads to the local optimal solution or bad convergence and exploration of final PF. Unfortunately, in MOPF problem, the control variable, that is, voltage of generators, transformer tap ratios, and values of shunt VAr compensators (SVC) are within small feasible regions and possess stiff challenges to existing MOEAs. Additionally, constraints in the objective space such as VD and Lindex further narrow the objective function space.Therefore, this paper addresses the recent most effective MOEAs such as Two‐phase (ToP) [47] and coevolutionary constrained multi‐objective (CCMO) [48] for finding the global PF and evenly distributed non‐dominated solutions to MOPF problem. In both of the algorithms, NSGAII is used to find the best PF and constraint domination principle (CDP) is used for handling the constraints of an optimization problem, but different strategies are used to find the global optimal PF. ToP and CCMO choosing the NSGAII algorithm for the exploration and exploitation of the final non‐dominated solution, because of its strong performance in the solution of MOPF problem. The findings of the analysis should also be checked. In the later simulation result section, it is proved that ToP and CCMO can find the global PF of constrained practical MOPF problems. According to the simulation results, the proposed MOEA outperforms as compared to several state‐of‐the‐art MOEAs to solve various cases of MOPF problems. In the subsequent subsections, the ToP and CCMO algorithms with their associated CHTs are explained.Constraint handling technique (CHT)Before discussing the CHT, first discuss the basic definitions of constrained CMOP.Pareto dominanceConsider the two decision vectors x⃗u${\vec{x}}_u$ and x⃗v${\vec{x}}_v$, if for all objective functions f(x⃗u)≤f(x⃗v)$f( {{{\vec{x}}}_u} ) \le f( {{{\vec{x}}}_v} )$ and there exists at least one objective function, say j, fj(x⃗u)<fj(x⃗v)${f}_j( {{{\vec{x}}}_u} ) &lt; {f}_j( {{{\vec{x}}}_v} )$ then x⃗u≺x⃗v${\vec{x}}_u \prec {\vec{x}}_v$ and can be read as, x⃗u${\vec{x}}_u$ is non‐dominated or x⃗u${\vec{x}}_u$ dominates x⃗v${\vec{x}}_v$ or x⃗v${\vec{x}}_v$ is dominated by x⃗u${\vec{x}}_u$.Pareto set and Pareto frontA region is said to be feasible if the overall degree of constraint violation (CV) is equal to zero for all the set of decision vector x⃗${{\vec{\rm x}}}$. All the solutions that belong to feasible regions are called Pareto optimal solutions and the image of Pareto optimal set‐in objective space is called Pareto front (PF).Constraint domination principleSimple and efficient CHT to handle constraints proposed in [51] is considered here, which compares pair‐wise individuals based on the following rules:If both solutions x⃗u${\vec{x}}_u$ and x⃗v${\vec{x}}_v$ are infeasible, select x⃗u${\vec{x}}_u$ if CV(x⃗u)<CV(x⃗v)$CV( {{{\vec{x}}}_u} ) &lt; CV( {{{\vec{x}}}_v} )$.x⃗u${\vec{x}}_u$ is feasible and x⃗v${{{\vec{\rm x}}}}_{\rm{v}}$ is infeasible, select the feasible one, that is,x⃗u${\vec{x}}_u$.If both x⃗u${\vec{x}}_u$ and x⃗v${\vec{x}}_v$ are feasible, then select x⃗u${\vec{x}}_u$ if for all the objective functions fi(x⃗u)≤fi(x⃗v)${f}_i( {{{\vec{x}}}_u} ) \le {f}_i( {{{\vec{x}}}_v} )$.Two phase MOEAHere, the cost of active power generation, emission, voltage deviation (VD), active power loss (PLoss${P}_{Loss}$), and maximum value of Lindex considered the objective functions. Moreover, the complexity of a problem is increased with the addition of objective constraints such as max of Lindex${L}_{index}$, whereas, minimum attainable value of Lindex${L}_{index}$ is obtained by the individual solution of OPF and VD. That is the reason why it is hard for an MOEA to find the promising feasible area of MOPF problem. Moreover, CMOEA needs to balance all the objective functions infeasible region, the convergence speed of the population is inevitably slow. In the proposed algorithm, first, find high‐quality feasible solutions, and then the Pareto optimal solutions (exploitation and exploration). A detailed flow chart of the proposed algorithm is shown in Figure 2.2FIGUREFlow diagram of ToP‐NSGAII‐CDP algorithmIn the first phase, transform the multi‐objective functions into weighted sum constrained single objective function as:34minFx=1m∑i=1mfix$$\begin{equation}min\ F\left( x \right)\ = \frac{1}{m}\ \sum _{i\ = \ 1}^m {f}_i\left( x \right)\end{equation}$$The goal of the first phase is to provide high‐quality feasible solutions for applicants for the next phase. Furthermore, in search engine for the generation of offspring, two popular trail vector strategies of differential evolution (DE) are considered, these are:DE/current‐to‐rand/l:35u⃗i=x⃗i+F∗x⃗r1−x⃗i+F∗x⃗r2−x⃗r3$$\begin{equation}{\vec{u}}_i = {{\bm{\vec{x}}}}_i\ + F*\left( {{{{\bm{\vec{x}}}}}_{r1} - {{{\bm{\vec{x}}}}}_i} \right) + F*\left( {{{{\bm{\vec{x}}}}}_{r2} - {{{\bm{\vec{x}}}}}_{r3}} \right)\end{equation}$$DE/rand‐to‐best/l/bin:36v⃗i=x⃗r1+F∗x⃗best−x⃗r1+F∗x⃗r2−x⃗r3$$\begin{equation}{{\bm{\vec{v}}}}_i = {{\bm{\vec{x}}}}_{r1}\ + F*\left( {{{{\bm{\vec{x}}}}}_{best} - {{{\bm{\vec{x}}}}}_{r1}} \right) + F*\left( {{{{\bm{\vec{x}}}}}_{r2} - {{{\bm{\vec{x}}}}}_{r3}} \right)\end{equation}$$37u⃗i,j=v⃗i,jifrandj<CRorj=jrand1,…,D.x⃗i,jotherwise$$\begin{equation} {{\bm{\vec{u}}}}_{i,j} = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{l}@{}} {{{{\bm{\vec{v}}}}}_{i,j}\ if\ ran{d}_j &lt; CR\ or\ j = {j}_{rand}\ 1,\ \ldots ,\ D.}\\[9pt] {{{{\bm{\vec{x}}}}}_{i,j}\ otherwise} \end{array} \right. \end{equation}$$where subscript i∈[1,Np]$i \in [ {1,Np} ]$ and j∈[1,D]$j \in [ {1,\ D} ]$; D is the decision vector; vi=(vi,1,vx,2,…,vi,D)T${v}_i = \ {({v}_{i,1},\ {v}_{x,2},\ \ldots ,\ {v}_{i,D})}^T$ is the ith mutant vector; ui=(ui,1,ui,2,…,ui,D)T${u}_i = \ {({u}_{i,1},\ {u}_{i,2},\ \ldots ,\ {u}_{i,D})}^T$ is the ith trial vector; r1,r2${r}_1,{r}_2$ and r3 are random integers between [1, Np],xbest$Np],{x}_{best}$ is the best individual in the current population, randj arbitrary number [0, 1], jrand is a random number [1,D],F$[ {1,\ D} ],F$ and CR$CR$ are the scaling and crossover control parameter and these are randomly selected from the Fpool = [0.6; 0.8; 1.0] and CRpool = [0.1; 0.2; 1.0].Then feasibility rule CHT is implemented to choose the best solution between the x⃗i${\vec{x}}_i$ and u⃗i${\vec{u}}_i$ for the next generation. It should be necessary to terminate the first phase when the high‐quality solutions are obtained before the entire population converges to a single point. We design the following two conditions to achieve such high‐quality solutions:1ConditionThe feasibility proportion, that is, Pf>1/3$Pf\ &gt; 1/3$, promises that the feasible region has been got.2ConditionCompute the normalized weighted sum single objective functionf∼(x)$\widetilde {\ f}( x )$ and then add them as:38f∼ix=fix−fminxfmaxx−fminxm$$\begin{equation}{\tilde{f}}_i\ \left( x \right) = \frac{{{f}_i\left( x \right) - {f}_{min}\left( x \right)}}{{{f}_{max}\left( x \right) - {f}_{min}\left( x \right)}}\ m\end{equation}$$39f¯x=∑i=1mf∼ix$$\begin{equation}{\bar{f}} \left( x \right) = \sum _{i = 1}^m {\tilde{f}}_i\left( x \right)\ \end{equation}$$When some high‐quality feasible solutions have appeared in the first phase, it is desirable to design some terminating criteria to jump into the second phase for finding the widely distributed non‐dominated solutions. In the proposed algorithm, for finding the terminating criteria, sort out all the solutions that appeared in the first phase f¯(x)$\bar{f}( x )$ and compute the biggest difference (δ) between the 33% of feasible solutions. If δ is less than 0.2, the second condition is to be satisfied. The second condition declares that some high‐quality solutions along with converging to the small area have been obtained. The ultimate goal of the first phase is to provide high‐quality solutions for the next phase. High‐quality candidate solutions obtained in the first phase are neither well distributed nor well converged efficiently. Therefore, in phase II a well‐known MOEA, that is, non‐dominated sorting genetic Algorithm (NSGAII) has been applied to find the well‐distributed and near‐global PF. The steps of phase II are given in the flow chart as shown in Figure 2.Constrained coevolutionary MOEACCMO starts with the random generation of two populations say Pop1 and Pop2 each of size Np as shown in Figure 3. After that fitness of each population say fitness1 and fitness2 has been computed by using the domination rule of strength Pareto evolutionary algorithm (SPEA2) [61]. In this domination rule, each population assigns a fitness based on the number of solutions they dominated. If a large number of solutions are dominated by a particular solution, less the fitness of that solution and vice versa. This assignment of fitness makes sure that the search is directed toward non‐dominated solutions.3FIGUREFlow diagram of CCMO‐CDP algorithmFurthermore, these fitness values are considered to select the mating pools (parents) of each population by using a tournament selection operator. Then, in each generation offspring populations (i.e. offspring 1 and offspring 2) are generated from the mating pools of associated parents of each population.After that updated fitness values (i.e. fitness1 and fitness2) of the combined population and offspring1 and offspring2 are computed. The size of each updated fitness is reduced to Np by considering the environmental selection. The above process is repeated until the termination criteria are met.SIMULATION RESULTS AND DISCUSSIONDeterministic MOPFHere, the IEEE standard 30‐bus test is considered to evaluate the effectiveness of proposed algorithms. Detailed data of the test is considered from [62]. Various ten study cases as shown in Table 4 of simultaneous optimization of 2, 3, 4, and 5 objective functions are considered to solve IEEE 30‐bus standard test system. The IEEE 30 bus system has six thermal generators, 24 load buses with a cumulative load of 283.4+j126.2 MVA, nine shunt VAR compensators, and four transformers connected. User‐defined parameters of NSGAII [51], MOEADDAE [49], and TiGE‐2 [50], which demonstrate high performance, are adopted from their original papers.4TABLEVarious multi‐objective study casesCasesCost ($/h)Emission (t/h)VDLindexPLoss1✓✓2✓✓3✓✓4✓✓5✓✓✓6✓✓✓7✓✓✓8✓✓✓✓9✓✓✓✓10✓✓✓✓✓However, the parameters and offspring generation strategies of ToP and CCMO algorithms are given in Sections 3.2 and 3.3. Population size (Np) for the two, three, four, and five objective functions are 100, 200, 300, and 400 respectively; and gen_max for all the cases is 1000.Selection of best PF and the best compromise solutionConvergence and diversity preservation (combined evenness and spread) of the final solution play a vital role during the design of MOEAs. Here, state‐of‐the‐art MOEAs are applied to find the best solutions to the MOPF problem. Various study cases are designed to check the superiority and performance of various constrained‐type MOEAs. Each case is independently run twenty times, and twenty PFs are attained. For the selection of the best PF in terms of quality assessments such as diversity preservation and convergence, a well‐organized hyper volume performance indicator (HVI) [63] technique is applied. In all the cases, the values of objective functions are different; therefore in making a fair comparison, all the objective functions are normalized first to get uniform rage between 0 and 1. Afterward, a reference point (1, 1, …1)M is considered for the computation of HVI. When comparing the PFs of different runs of each algorithm, the PF with the maximum value of HVI attained by the algorithm is considered the best one. Table 5 summarizes the statistical results of HVI (max, min, mean, and SD) of all the MOEAs for each study case over the twenty independent runs, and the best result is highlighted.5TABLEStatistical data based on HVI of all the study casesCase #AlgorithmMaxMinMeanSD1CCMO0.8280.8240.8270.001MOEADDAE0.7940.5760.6820.086NSGAII0.8280.8250.8280.001TiGE_20.7870.7590.7740.009ToP0.8320.8160.8250.0072CCMO0.8920.8710.8800.007MOEADDAE0.8650.8350.8520.009NSGAII0.7760.2710.5930.197TiGE_20.7950.7270.7720.025ToP0.9050.8890.8980.0063CCMO0.5530.0000.1100.193MOEADDAE0.6120.0000.1030.196NSGAII0.7910.2140.5670.195TiGE_20.8320.7580.8440.052ToP0.8560.0000.2350.3514CCMO0.77460.7710.7730.001MOEADDAE0.7710.6060.7490.051NSGAII0.7720.7690.7710.001TiGE_20.7730.7590.7660.005ToP0.77520.7330.7660.0145CCMO0.8130.7920.8010.007MOEADDAE0.7830.7150.7530.023NSGAII0.8120.7740.7980.013TiGE_20.8300.8110.8220.005ToP0.8460.7790.8130.0216CCMO0.8320.7730.8030.020MOEADDAE0.7870.7140.7590.026NSGAII0.8210.7910.8060.010TiGE_20.7870.6620.7360.040ToP0.8360.7210.7860.0457CCMO0.6930.6870.6910.002MOEADDAE0.6760.6640.6700.004NSGAII0.6920.6860.6900.002TiGE_20.6670.6480.6610.005ToP0.6900.5990.6390.0368CCMO0.6890.6780.6840.003MOEADDAE0.6530.6040.6380.015NSGAII0.6530.6350.6460.006TiGE_20.6640.6540.6590.003ToP0.6740.5060.5830.0519CCMO0.6720.6350.6510.012MOEADDAE0.6580.5990.6460.017NSGAII0.6660.6350.6540.011TiGE_20.6660.4610.5710.081ToP0.6690.3170.5740.11010CCMO0.6000.0000.5300.186MOEADDAE0.5480.0000.4810.169NSGAII0.5890.0000.5190.182TiGE_20.5540.0000.4650.166ToP0.6310.0000.4990.177Table 5 clearly shows that ToP outperforms as compared to all other algorithms for cases 1–6 and 10 to HVI, whereas CCMO performs better in cases 7–9. Furthermore, as per the NFL theorem [46], there could not exist any MOEA that solves all the problems that are simply superior to priors. Moreover, the comparative performance of all the runs of different MOEAs based on HVI is as shown in Figure 4. In each box plot, twenty values of each run are computed by HVI, and a red central line in each box indicates the median, bottom and top edges of the box indicate the 25th and 75th percentile, and symbol ‘+’ are the outliers. The box plot summarizes that the single MOEA is unable to find the global PF with better convergence and diversity for all the MOPF cases.4FIGUREBox plot based on HVI of all the independent runsAlso, in most of the cases, NSGAII, MOEADDAE, and TiGE_2 are trapped in local optima, whereas, ToP and CCMO escape from local optima that are unsafe for solving MOPF problems. Both statistical values and box plots show that in most cases ToP gives better results where the VD and VSI are being taken as the objective functions. Because these functions are also influencing the constraint region.In cases 1–6 and 10, ToP outperforms, whereas, in cases 7–9, CCMO outperforms. Further, the average performance of the NSGAII algorithm in terms of SD is good compared to all the other algorithms but it does not find the global PF in any case. In the literature, all the discussed MOEAs, however, tend to be computationally expensive and required more function evaluation to find global PF, to solve MOPF problems which will make them less robust. To overcome this, ToP and CCMO algorithms proposed a strategy to avoid local optima.The ToP algorithm is comprised of two phases; the first phase is implemented to find the promising feasible area by transforming a MOPF problem into a weighted sum constrained single‐objective OPF problem. In the first phase, the weighted sum approach with different weight vectors can jump from the local optimal solutions. Most of the MOEA will face this problem of trap into local optima. In the second phase, an NSGAII is implemented to obtain the final PF. Therefore, the computational time can be significantly reduced to decrease the maximum function evaluation. However, in CCMO algorithm starts with the random initialization of two populations with size N. Moreover, a comparison between the BCS of various MOEAs based upon convergence and diversity preservation to find these solutions will be discussed. The BCS is derived from the best Pareto front using a fuzzy decision approach [35]. In this approach, membership function (μmk)$( {\mu _m^k} )$ of objective is computed first as:40μmk=1forfmk≤fmminfmmax−fmkfmmax−fmminforfmmin<fmk<fmmax0forfmk≥fmmax$$\begin{equation}\mu _m^k = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{2}{l}@{}} 1&\quad {for\ f_m^k \le f_m^{\ min\ }}\\[15pt] {\dfrac{{f_m^{\ max\ } - f_m^k}}{{f_m^{\ max\ } - f_m^{\ min\ }}}} &\quad {for\ f_m^{\ min\ } &lt; f_m^k &lt; f_m^{\ max\ }}\\[25pt] 0&\quad {for\ f_m^k \ge f_m^{\ max\ }} \end{array} \right. \end{equation}$$where m and k are the numbers of objective functions and final non‐dominated solutions; fmk$f_m^k$ is the fitness value. After that, μmk$\mu _m^k$ is normalized to find normalized membership function μk${\mu }^k$.41f¯x=∑i=1mf∼ix$$\begin{equation}\bar{f}\ \left( x \right) = \sum _{i = 1}^m {\tilde{f}}_i\left( x \right)\ \end{equation}$$where Nd${N}_d$ is the number of solutions in the final PF. The BCS is the index of the highest μk${\mu }^k$ value. Table 6 shows the results of BCS of two objective functions of all the algorithms considering fuzzy decision‐making rules.6TABLEBCS of various MOEAsCase #AlgorithmBCSμkMin (f1)Min (f2)1CCMO831.38810.2482510.011357800.77630.20483FC ($/h) vs E (t/h)MOEADDAE820.53310.2670090.01124800.58350.21601NSGAII834.84240.2446270.011303800.66610.20485TiGEA‐2829.28360.2505540.011398804.52680.21053ToP832.2610.2473440.011499800.64530.204892CCMO800.9060.2809120.011023800.55370.09671FC ($/h) vs VD (p.u)MOEADDAE802.19350.1595420.010909800.78430.11892NSGAII800.84870.2913920.011586800.81080.09899TiGEA‐22802.18830.1569470.011689800.87670.13666ToP802.39780.1373240.011849800.61620.096163CCMO800.68580.1373730.011534800.66010.13704FC ($/h) vs VSIMOEADDAE800.51240.137980.080723800.51180.13763NSGAII800.60630.1374310.010972800.57860.13741TiGEA‐2800.81740.137150.011382800.80980.13712ToP801.06440.1369360.012889800.54560.136774CCMO842.95515.028480.011171800.75213.12850FC ($/h) vs PLoss${P}_{Loss}$ (MW)MOEADDAE840.04845.1209630.011081800.60793.14033NSGAII842.85755.0415640.011165800.60133.12805TiGEA‐2832.16295.4317450.011024804.29373.71035ToP841.02495.101370.011172800.80263.12451The fourth column of Table 6 shows the values of normalized fuzzy membership function, from these values it is clearly shown that in all the two objective cases ToP algorithm outperforms. A minimum value of each objective function individually is shown in the last two columns of Table 6. From these values’ diversity of PF of each algorithm can be computed. In cases 1–4, ToP attains a maximum value of μk${\mu }^k$ as compared to all the other algorithms. In case 1, the BCS of all the algorithms is non‐dominated by each other. Whereas in case two individual objective function values of ToP dominate the MOEA/D‐DAE [49], NSGAII [51], and TiGE‐2 [50]. In case 3, an individual minimum value of ToP dominates the CCMO, NSGAII [51] and TiGE‐2 [50] whereas non‐dominated by MOEA/D‐DAE. In case 4, ToP dominates only TiGE‐2 [50] whereas it gives minimum value of power loss of 3.1245 MW compared to CCMO, MOEA/D‐DAE [49], and NSGAII [51]. The minimum value of individual objective functions shows that the ToP finds better diversity and convergence compared to other algorithms.Detailed results analysis of BCS of ToP and CCMOAccording to HVI, ToP competes for all the algorithms in cases 1–6 and case 10, whereas, CCMO gives better solutions to cases 7–9. Therefore, only the results of the best algorithm such as ToP and CCMO of all the study cases are shown in Table 7.7TABLESimulation results of BCS of all the study cases based on HVIParameterMinMaxCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10PG2$P{G}_2$ (MW)208060.05548.63448.07353.18862.13861.36859.85665.33958.02965.500PG5$P{G}_5$ (MW)155026.67721.30221.51233.56726.48228.17139.99433.53836.66433.178PG8$P{G}_8$(MW)103535.00021.32621.36235.00034.15134.71435.00034.88134.99326.144PG11$P{G}_{11}$ (MW)103026.47312.00012.13829.98122.93929.22429.99829.85029.97120.009PG13$P{G}_{13}$ (MW)124024.46812.0011222.82921.22626.82034.68834.05127.31526.562VG1$V{G}_1$ (p.u)0.951.11.0721.0851.0761.0701.0481.0691.0621.0261.0611.059VG2$V{G}_2$ (p.u)0.951.11.0621.0651.0591.0581.0301.0521.0541.0171.0541.044VG5$V{G}_5$ (p.u)0.951.11.0301.0301.0311.0361.0161.0201.0360.9861.0371.015VG8$V{G}_8$ (p.u)0.951.11.0431.0361.0401.0441.0011.0301.0420.9971.0461.015VG11$V{G}_{11}$ (p.u)0.951.11.0891.0501.1001.0991.0291.0391.0691.0631.0791.055VG13$V{G}_{13}$ (p.u)0.951.11.0301.0171.0811.0441.0241.0821.0611.0161.0621.038QC10$Q{C}_{10}$ (MVAr)0.0052.8081.7782.0624.4801.21003.0913.1371.4903.809QC12$Q{C}_{12}$ (MVAr)050.2000.10404.3651.3192.6303.9010.7811.8253.387QC15$Q{C}_{15}$ (MVAr)051.0334.5200.1851.3263.34754.0974.1273.4522.868QC17$Q{C}_{17}$ (MVAr)054.7864.4400.8334.6914.1670.5414.7943.1423.5593.797QC20$Q{C}_{20}$ (MVAr)050.6334.3634.1292.4185.0004.8424.0894.0814.7033.221QC21$Q{C}_{21}$ (MVAr)054.2804.9990.9124.7680.0691.0044.9714.7324.8881.770QC23$Q{C}_{23}$ (MVAr)053.7212.8191.0754.9645.00003.6554.6623.7053.029QC24$Q{C}_{24}$ (MVAr)054.7474.9931.0214.9342.8551.6314.9964.9994.8734.090QC29$Q{C}_{29}$ (MVAr)051.3322.4540.0323.0735.0000.0002.5992.1512.0960.510T11 (p.u)0.901.11.0811.1001.0331.0941.0381.0031.0551.0731.0371.067T12 (p.u)0.901.10.9090.9420.9010.9140.9030.9080.9170.9090.9160.927T15 (p.u)0.901.10.9831.0091.0150.9780.9761.0251.0090.9711.0151.004T36 (p.u)0.901.10.9901.0020.9540.9790.9850.9410.9780.9580.9720.929PG1$P{G}_1$ (MW)50.00200116.402177.269177.495113.936123.045108.55787.95190.595101.058118.088QG1$Q{G}_1$ (MVAr)‐20150‐2.0088.520‐3.716‐1.28114.67512.142‐4.505‐0.735‐10.7867.958QG2$Q{G}_2$ (MVAr)‐206028.66526.92922.22414.8979.94814.0189.57623.50912.10418.936QG5$Q{G}_5$ (MVAr)‐1562.523.09725.67229.08426.49345.42924.27228.76021.97729.99026.707QG8$Q{G}_8$ (MVAr)‐1548.734.24930.46540.25327.71022.16337.11430.18826.46237.64723.647QG11$Q{G}_{11}$(MVAr)‐104031.79724.50126.69134.59714.3264.13818.05832.43918.45523.704QG13$Q{G}_{13}$(MVAr)‐1544.7‐1.8052.89924.359‐2.7115.03726.9749.2431.70811.35610.052C(Pg)$C( {{P}_g} )$ ($/h)832.261800.849801.064841.025827.123842.712882.926871.333858.478838.173E (t/h)0.2470.3670.3670.2420.2580.2380.2160.2210.2280.251Lindex${L}_{index}$(p.u)0.1430.1480.1370.1380.1500.1380.1390.1490.1370.140VD (p.u)0.5810.2910.8700.9030.1570.6940.8890.1350.9520.361PLoss${P}_{Loss}$(MW)5.6759.1329.1795.1016.5825.4544.0884.8534.6306.079All the values of the decision vector in Table 7 are within upper and lower limits in all the cases. Besides, constraints in the objective space are also satisfied, the maximum value of Lindex is 0.2 (must be between 0 and 0.2) and VD must not be greater than 1.2 (0.05×24 = 1.2) in all the load buses. In all the cases, minimization of fuel cost of real power is mandatory, also, in the literature, solution of MOPF problem cost function is considered as common. In comparison to all the cases, the least value of cost function 800.84$/h has appeared in case 2, where the cost and VD are minimized simultaneously. The minimum value of a second objective function (Emission) is appeared in case 7, in this case, cumulative active and reactive power generation is minimum whereas a value of SVC is maximum. Also, in case 7, the minimum value of active power loss has appeared, and the maximum value of cost function is obtained in this case. The value of Lindex is better in Case 7 as compared to cases 1, 2, 5, 8, and 10. The minimum value of Lindex is appeared in cases 3 and 9, whereas the least value of VD appears in case 8. The values of Cost, emission, and power losses are opposite in case 3 and case 7, that is, Loss is maximum in case 3 whereas it is minimum in case 7, Cost is near the minimum in case 3 and it is maximum in case 7.Figure 5 shows the optimal cumulative values of active and reactive power injection of generators and reactive power injection of shunt VAr compensators (SVC) in all the study cases. In Figure 5, it can be noticed that maximum values of PG and QG appeared in case 3 and minimum values of these decision variables appeared in case 7. However, the cumulative value of SVC is minimum in case3 and maximum in case 7. From this, it can be concluded that the more the SVC injection less the active power losses will be and vice versa. Case 10, where all the five objective functions are minimized simultaneously, is the special case in which average values of all the functions of case 1–9 give the approximate values of case 10. It shows that when all the five objective functions are considered simultaneously the average values of case 1–9 objective functions. Moreover, voltage profiles of all ten study cases are shown in Figure 6.5FIGUREOptimal cumulative values of PG, QG and QC6FIGURELoad bus voltage of all the casesThe voltage curve for all the cases shows that the voltage at each load bus has the desirable limit, whereas, a waveform of cases 2, 5, 8, and 10 are near the unity compared to other cases. However, it is located far away in cases 3, 4, 7, and 9. Therefore, cumulative VD objective constraint is considered in this paper to narrow the feasible search space to find the load bus voltage near the unity (ideal). The minimum value of cumulative VD ensures that the load bus voltage is near unity. In cases 2, 5, 8 and 10 cumulative VD is less than 0.3 p.u and hence betters the load bus voltage profile. On the other hand, in cases 3, 4, 7, and 9 cumulative VD is near the worst value (1.2 p.u) and hence poor the voltage waveform in such cases. Table 8 presents a comparison of BCS of ToP and CCMO in their associated best cases with the other recent MOEAs.8TABLEComparison of BCS of cases 1, 2, 3 and 4BCS of case1BCS of case 2BCS of case 3BCS of case 4AlgorithmC(PgT)($/h)E (t/h)C(PgT)($/h)VD (p.u)C(PgT)($/h)LindexC(PgT)($/h)PLoss(MW)CCMO831.3880.248800.9060.281800.6860.13737842.9555.028MOEADDAE820.5330.267802.1940.160800.5120.13798840.0485.121NSGAII834.8420.245800.8490.291800.6060.13743842.8575.042TiGEA_2829.2840.251802.1880.157800.8170.13715832.1635.432ToP832.2610.247802.3980.136801.0640.13694841.0255.101MOEA/D‐SF [64]829.5150.2501802.4060.136MOEA/D [9]833.720.2438799.99a0.354MOMICA [30]865.0660.2221800.03a0.4422ESDE [21]833.4740.254804.960.0952MSFLA [22]823.2780.29078ashow infeasible solution and Bold values show best values.In case 1, the cost of active power generation and emission is considered to minimize, in this case, BCS is 832.26098 ($/h) and 0.24734 (t/h) attained by ToP. Whereas in the literature minimum value of a cost function is achieved by MSFLA [22], whose both the objective functions are dominated by MOEADDAE, as compared to ToP and CCMO, PF of MOEADDAE algorithm stuck in the local optima as can be seen in Figure 7 of PF of case 1. Furthermore, the author in ref [21] deliberates the load bus voltage limit between 0.95 and 1.1 p.u, which is much larger than the actual load bus voltage limit between 0.95 and 1.05 p.u considered in the literature as well as here. The choice of BCS is highly dependent upon the distribution of non‐dominated solutions of final PF. Moreover, the overall comparison between the PF of various algorithms, and extreme values of objective functions show the convergence and spread of the final PF. Table 9 shows comparison of the extreme value of objective functions for case 1.7FIGUREBest PF of all the algorithms of case 1 to case 4 with the extreme values of ToP9TABLEComparison of extreme values of case 1ParameterMTLBO[23]MGBICA[24]MOICA[65]MOEA/D [9]MOEA/D‐SF [64]ToPC(Pg)$C( {{P}_g} )$($/h)801.89945.19801.14942.8801.14943.74799.29a944.3800.6932.96800.6944.31E (t/h)0.36650.20490.32960.2040.32960.20480.35930.2040.3670.20560.3650.2048In the literature, MOPF simultaneous optimization of more than two objective functions is relatively uncommon. While in the field of MOEAs the attention has been drawn to many objective optimizations. Table 10 shows the simulation results of various three objective functions of cases 5, 6, and 7. In the literature, only the BCS of case 1 is given, therefore, in Table 9 extreme values of PF of case, 1 are only compared, whereas, for other study cases extreme values of ToP algorithm are pointed out in Figure 7. Simulation results in Table 10 clearly show that the ToP achieves minimum cost and emission values and hence better converged the size of PF as compared to other algorithms.10TABLEComparison of BCS of cases 5, 6 and 7BCS of case 5BCS of case 6BCS of case 7AlgorithmC(PgT)$C( {{P}_{gT}} )$($/h)E (t/h)VD (p.u)C(PgT)$C( {{P}_{gT}} )$$/h)E (t/h)VSIC(PgT)$C( {{P}_{gT}} )$ ($/h)E (t/h)PLoss${P}_{Loss}$(MW)CCMO842.66320.23920.1210839.91560.23940.1374882.92620.21624.0880MOEADDAE836.47630.24860.1287834.73330.25180.1377868.63980.22234.5618NSGAII830.32640.25960.1115843.54890.23820.1374881.47470.21744.1105TiGEA_2836.55610.24560.0947829.77750.25250.1245859.15090.22624.5841ToP827.12310.25770.1568842.71240.23770.1383883.45040.21574.2078MOEA/D‐SF [64]842.4460.24060.1092881.01200.21644.1441MOEA/D [9]850.280.23320.1155902.540.21073.4594MOPSO [9]846.930.23860.2188891.480.21443.9557Figure 7 reveals that the PF of ToP algorithm provides better concentration for both of the objective functions. Consequently, the BCS is favourable to both of the objective functions. Simultaneous optimization of two objective functions in case 1–4, ToP leads to evenly distributed and good diversity of PF. Moreover, compared to other algorithms, ToP produces high‐quality solutions and evenly distributed a wide range of PF in all four cases.Besides, it can be clearly shown in Table 10 that a single solution cannot dominate all the objective functions of proposed algorithm. Among all the best compromise solutions of case 5, the value of cost objective is minimum in ToP, while minimum emission is achieved in case 6. In case 5 load bus voltage profile is near the unity, whereas in cases 6 and 7, touches the maximum value of load bus voltage, because the smallest value of reactive power is injected into the system in cases 6 and 7, also, the generated real power injection in case 7 is smallest compared to all other cases that are because of smallest active power loss appeared in case 7. The regular PF, according to HVI, the best index of ToP and CCMO out of twenty independent runs of cases 5–7 is as shown in Figure 8.8FIGUREPF of cases 5, 6, 7 of ToP and CCMOIn cases 5 and 6, the final non‐dominated objective function values (i.e. PF) of ToP algorithm are uniformly distributed in the entire feasible region and dominates most of the solutions of CCMO, whereas in case 7, CCMO finds the global PF and dominates large no of solutions of final PF of ToP algorithm. Moreover, simulation results of cases 8–10, where optimizing four and five objective functions simultaneously are shown in Table 11.11TABLEComparison of BCS of cases 8, 9 and 10BCS of Case 8BCS of Case 9BCS of case 10AlgorithmC(Pg) ($/h)E (t/h)VD (p.u)PLOSS (MW)FC ($/h)E (t/h)LindexPL (MW)FC ($/h)E (t/h)VSIVD (p.u)PL (MW)CCMO871.3330.22050.13484.853858.4780.22800.13754.6297868.5910.22700.14070.23165.6240MOEADDAE895.5040.21810.15004.379920.6860.21060.13753.5582900.3800.22070.14130.23514.3825NSGAII865.0260.22550.15415.466862.5210.22680.13814.5122864.4440.22510.14570.17325.3894TiGEA_2863.9290.22470.11414.906860.9750.22620.11434.6863860.0160.22710.13980.21425.1321ToP879.4960.21790.19314.663874.3640.21850.13784.4847838.1730.25100.14010.36146.0794MOMICA[30]830.1880.25230.29785.585MOEA/D‐SF [64]883.3220.21870.13224.4527In case 8, BCS of ToP has three objectives dominating the comparable algorithm MOMICA [30] and two objectives of MOEA/D‐SF [64]. The cost objective is much better in MOMICA [30]. As compared to MOMICA [30] and MOEA/D‐SF [64], ToP achieves the minimum value of emission which is 0.2179 (t/h). Moreover, ToP and CCMO obtain well‐distributed and diverse PF of cases 8, 9, and 10. In the literature, high‐dimensional data, that is, PF of more than three objective functions are visualized by using parallel coordinates plots in which the x‐axis point out the name of objective function and y‐axis is the values of the associated distribution of the objective function. Figure 9 shows the parallel plot coordinates of final non‐dominated solutions of cases 8 and 9 of CCMO algorithm, and case 10 of ToP algorithm. In Figure 9, all the objective functions are normalized and the distribution of y‐axis shows that the ToP and CCMO can find a good distribution of PF of MOPF problem considering more than three objective functions.9FIGUREParallel plot of normalized objective functions of final PFVery large scale 300‐bus test systemTo quantify the superiority and performance of proposed algorithm, a very large‐scale IEEE 300‐bus network is adopted to solve MOPF problem. This network consists of 69 generators, 117 tap changing transformers, 411 lines and 14 SVCs. Active and reactive power demand is 23527+j7728 MVA. In base case, 257th bus is the reference bus, system active power loss is 408.3 MW. Network data is taken from [66]. 258 total number of decision variables. A total of 300,000 function evaluations, 50 population sizes (with execution time of about 70 min) are performed.Two study cases are considered.Case 1: Minimization of cost and PLoss${P}_{Loss}$Case 2: minimization of cost and VD.Statistical values and objective functions of twenty independent runs of both cases are as shown in Table 12.12TABLEStatistical value of HVI and objective functions of cases 1 and 2QuantityCase 1Case 2max0.70690.7224min0.61710.6452mean0.644680.6972Standard Deviation0.3417380.373293C(Pg)$C( {{P}_g} )$ ($/h)720846.6720548.5PLoss${P}_{Loss}$(MW)322.04333.8285VD5.39612.4770It is observed from the results in Table 12 that the minimum, mean, maximum, and standard deviation values of each optimization objective are better in case 2. Minimum cost obtained in case 2 (720548.5$/h) compared to case 1. Power loss is minimum in case 1, however voltage deviation is much decreased in case 2. Final convergence and distribution of non‐dominated solution of both the cases are shown Figure 10. Where, green diamond colour shows the best compromise solution.10FIGUREFinal non‐dominated of 300‐bus for cases 1 and 2It can observe that non‐dominated solutions in the figure have better exploration and exploitation. The decision maker can select a single solution of interest from a large number of non‐dominated solution. The load bus voltage profile of both the study cases is shown in Figure 11.11FIGUREVoltage profile of cases 1 and 2 for 300 bus systemThe load bus voltage profile of both study cases is within upper and lower bound. As compared to case 1, voltage profile of case 2 is better (near unity), because of VD minimization.Probabilistic MOPFFrom detailed analysis of deterministic MOPF in most of the study cases, proposed algorithm outperforms the other most recent algorithms. Therefore, for the solution of probabilistic MOPF problem, only ToP algorithm is implemented. As mentioned before, most of the literature did not consider network security constraints and voltage stability constraints in MOPF study.Lindex${L}_{index}$ is calculated by using Equation (8), whereas the constant Γin Equation (16) is the maximum value of local indicator of Lj which is set by decision maker. For the smaller value of Γ, the distance to voltage instability is larger and hence the system has more stability margin. Therefore, for secure integration of RES, constant Γ is set to 0.2 (minimum attainable value of Lindex obtained in deterministic OPF). Whereas, the constant εin Equation (17) is set to 1.2 and can be computed by considering either upper bound or lower bound of load bus voltage. Here, load bus voltage is set between 0.95 and 1.05 p.u. The acceptable variation of load bus voltage is ±0.05 p.u, cumulatively 0.05×24 is 1.2. therefore, overall load bus voltage violation must be less than 1.2 p.u.Moreover, three objective functions, that is, cost, emission, and PLoss${P}_{Loss}$ on modified IEEE 30‐bus test system to examine the effectiveness of PMOPF problem. Moreover, real‐time dispatch is considered after every 10 min over the 1hr period with an increase of 2% of load in each interval of 10 min. In this 1hr scheduling period dynamic approach to load demand and renewable generation is considered the solution PMOPF problem. Furthermore, transformer tap ratio and the rating of SVC are treated as discrete variables that ensure more realistic OPF problem. The transformer tap setting is adjusted at the steps of 0.02 p.u, whereas, SVCs are switched at the discrete steps of 200 kVAr.For the superiority and performance of proposed algorithm, three different study cases of conflicting objective functions, are considered.Case1: Minimization of cost and emission, relaxing the Lindex${L}_{index}$ and VD constraintCase 2: Minimization of cost and PLoss${P}_{Loss}$, relaxing the Lindex and VD constraintCase 3: Minimization of cost, emission and power loss considering Lindex${L}_{index}$ and VD constraintSimulation results of probabilistic MOPFFor the simulation of Probabilistic MOPF, IEEE 30‐bus network is modified. It has 3 thermal generators (at buses 1, 2 and 8). Output of a wind farm is connected to bus 5 while solar PV unit is supplying power to buses 11 and 13. As a noticeable fact, output power from wind and solar are all variables and any deficit in total output from these units must be mitigated by spinning reserve. Table 13 depicts the simulation results of PMOPF problem at maximum load demand.13TABLESimulation results of PMOPFQuantityMinMaxCase 1Case 2Case 3PG2$P{G}_2$208046.02823.18746.291PG5$P{G}_5$ (wind)07549.36456.66168.306PG8$P{G}_8$103513.1712.4130.817PG11$P{G}_{11}$ (Solar)103049.98449.99749.954PG13$P{G}_{13}$ (Solar)124048.47148.4648.115VG1$V{G}_1$0.951.11.0661.0711.055VG2$V{G}_2$0.951.11.041.0561.042VG5$V{G}_5$0.951.11.0251.0311.031VG8$V{G}_8$0.951.11.0261.0371.038VG11$V{G}_{11}$0.951.11.0921.0851.091VG13$V{G}_{13}$0.951.11.0541.051.044QC10$Q{C}_{10}$050.830.6QC12$Q{C}_{12}$0514.62.4QC15$Q{C}_{15}$052.642.6QC17$Q{C}_{17}$052.83.22.6QC20$Q{C}_{20}$051.82.85QC21$Q{C}_{21}$0544.60.8QC23$Q{C}_{23}$05144.2QC24$Q{C}_{24}$051.44.23.4QC29$Q{C}_{29}$054.22.64.6T110.901.10.981.081.08T120.901.11.060.90.9T150.901.11.060.981T360.901.10.980.960.98PG1$P{G}_1$50.00200110.26126.3471.615QG1$Q{G}_1$‐2015025.7273.7569.618QG2$Q{G}_2$‐2060‐18.05420.998‐5.024QG5$Q{G}_5$‐1562.533.0124.26929.248QG8$Q{G}_8$‐1548.731.99236.26840.956QG11$Q{G}_{11}$‐104030.37331.44637.329QG13$Q{G}_{13}$‐1544.730.5072.039.092C(PgT)($/h)457.3413425.1C(PgW)($/h)$C( {{P}_{gW}} )( {{\rm{\$ }}/{{\bf h}}} )$157.4186.5236.4C(PgS)($/h)$C( {{P}_{gS}} )( {{\rm{\$ }}/{{\bf h}}} )$ (bus 11)208208207.8C(PgS)($/h)$C( {{P}_{gS}} )( {{\rm{\$ }}/{{\bf h}}} )$ (bus13)194.1194.1192.5C(PD)($/h)$C( {{P}_D} )( {{\rm{\$ }}/{{\bf h}}} )$3.73.73.7CTotal($/h)${C}_{Total}( {{\rm{\$ }}/{{\bf h}}} )$1020.41005.31065.5E (t/h)0.4341.0520.119PLoss${P}_{Loss}$ (MW)5.545.323.358VD (p.u)0.3470.78660.52454Lindex${L}_{index}$ (p.u)0.16260.157730.15343In the table, bold letters are the objective functions. Simulation results are self‐explanatory. In cases 1 and 2, VD and Lindex constraints are relaxed. All decision variables are within the range in all the intervals. With the increase in load demand, cost of thermal, wind and solar generation is increased. Transformer tap ratios and SVC are in discrete form. Maximum active power of 110.26 MW from the slack bus is taken in case 2, where cost and power loss are the objective functions. The minimum cost of thermal generation is 413 ($/h) in case 2. Out of cases 1–3, VD and Lindex${L}_{index}$of case 3 are better compared to cases 1 and 2. That is because of minimum value of Lindex${L}_{index}$ and VD constraints. In case 3, all the scheduled power of generators are uniformly selected because considering the security constraints. On the other hand, in cases 1 and 2, final solution emphases taking more power from slack bus. The load bus voltage profile of all the study cases is as shown in Figure 12. Load bus voltage level for all the intervals of all cases is within upper and lower bound. Relaxing the VD and Lindex${L}_{index}$ constraint gives poor voltage profile in cases 1 and 2. However, voltage profile of case 3 is near unity. This is due to the consideration of VD and Lindex${L}_{index}$ local stability index constraints. Figure 13 shows the PF (Final non‐dominated solutions) of all the study cases of all time intervals.12FIGUREVoltage profile of all the intervals of cases 1, 2 and 313FIGUREPF of PMOPF of all the casesPF shows that the proposed algorithm can find a better trade‐off between conflicting objective functions and gives better exploration and exploitation to solve complex PMOPF problems. PF of PMOPF problem consists of feasible final non‐dominated solutions. The probabilistic MOPF problem has non‐linear constraints and, for the MOEAs, it is very difficult to handle these constraints. Most of the authors in the literature did not consider security constraints such as voltage stability constraintsLindex$\ {L}_{index}$, and VD. With the consideration of these constraints all stability of power system is increased. In Figure 13 green diamond shows the best compromise solution obtained by using fuzzy weight function. In all the solutions overall constraint violation is zero.CONCLUSIONSMOEAs are computationally expensive to solve nonlinear, mixed‐integer, and highly constrained type MOPF problems. From the tabulated literature review it was found that most of the authors either used a weighted sum approach or Pareto‐based optimization algorithms along with the penalty function approach. The limitation of the former method is that it gives a single solution. Whereas, Pareto‐based method gives a large number of solutions for the decision‐maker with higher computational complexity. Therefore, the present study proposed a Two‐Phase (ToP) algorithm and new efficient constrained type MOEAs yet not been applied for the solution of MOPF problem. The proposed algorithm implements a hybrid flavour of weighted sum (less computational expensive in phase I) and Pareto based (gives non‐dominated solutions in phase II) approaches. To solve MOPF problem, various study cases of 2, 3, and 4 conflicting objectives functions are formulated on IEEE 30 and 300‐bus, to show the effectiveness and performance of proposed algorithm. Each case is independently run twenty times, for the selection of the best PF. HVI technique has been applied to find the best PF. In most of the cases, proposed algorithm outperforms. Figures of PF clearly show that the proposed algorithm can find well‐distributed and near‐global PF. Moreover, from the best PF, a single best non‐dominated solution is found by using a fuzzy membership function. Simulation results clearly show that the decision vector and operational constraints of MOPF problem are within the limits. Overall constraint violation in all the cases is zero.Furthermore, IEEE 30‐ bus test system is modified to inject uncertain wind and solar generation along with variable load demand. The probabilistic nature of wind, solar and load are modelled using appropriate PDFs. Increasing the integration of uncertain RES power, voltage stability of power system is violated. Therefore, two security constraints, Lindex${L}_{index}$ and VD must be within desirable limits and be formulated. Three study cases of 2 and 3 objective functions are implemented on modified IEEE‐30 bus system. Simulation results show that the proposed method efficiently solves the PMOPF problem.NOMENCLATUREAbbreviationsCHTConstraint handling techniqueCVConstraint VolitionEEmission in (t/h)FCFuel costHVIHyper volume indicatorMINLPMixed integer nonlinear programmingMOEAMultiobjective evolutionary algorithmMOPFMultiobjective optimal power flowNLPNonlinear ProgrammingOPFOptimal Power FlowPDFProbability Distribution FunctionPFPareto FrontPMOPFProbabilistic multiobjective OPFPVPhotovoltaicRESRenewable energy sourcesSVCShunt VAR compensatorToPTwo PhaseVDVoltage deviation of systemVSIVoltage stability indexWFWind farmSymbolLjlocal indicatorNint${N}_{int}$Number of time intervalsF(x⃗)$F( {\vec{x}} )$Vector of objective functionx⃗$\vec{x}$Decision vectorϕ,ψ$\phi ,\ \psi $Equality and inequality constraintsC, E, VD$VD$Cost, Emission, voltage deviationPLoss${P}_{Loss}$Power loss in MWLindex${L}_{index}$voltage stability indexi,j$i,\ j$From and to busNG, NLNumber of generator and load busesτT${\tau }_T$Transformer tapingsQC${Q}_C$SVC injection in MVArε, ΓAcceptable VD and VSI parametervin,vr${v}_{in},{v}_r$ and νout${\nu }_{out}$Cut‐in, rated and cut‐out wind speedsGstd, and RcSolar irradiance and certain irradiancegw,j${g}_{w,j}$ , pgW${p}_{gW}$Direct cost of wind and solar PV.PSchW,j${P}_{SchW,j}$, PschS${P}_{schS}$Scheduled power of wind and solar farmKRW,j${K}_{RW,j}$, KPW,j${K}_{PW,j}$, and KR,k${K}_{R,k}$, KPW,j${K}_{PW,j}$Reserve and penalty cost parameters of wind and solar PV generationΔν${\Delta }_\nu $, ΔG${\Delta }_G$, ΔD${\Delta }_D$Probability of wind velocity, solar irradiance and load demandpgW${p}_{gW}$, pgS${p}_{gS}$, pD${p}_D$Available uncertain wind‐solar generation and load demandC(Pg),$C( {{P}_g} ),$Cost of thermal generationC(PgW),C(PgS)$C( {{P}_{gW}} ),\ C( {{P}_{gS}} )$ and C(PD)$C( {{P}_D} )$variable cost of wind, solar and load demandAUTHOR CONTRIBUTIONSA.A., G.A.: Conceptualization; Formal analysis; Investigation; Methodology; Software; Validation; Visualization; S.M., M.U.K.: Writing—review and editing. 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