Pareto Front-Based Multiobjective Optimization of Distributed Generation Considering the Effect of Voltage-Dependent Nonlinear Load Models

Single objective constant PQ load models were extensively considered for site and size of distributed generation (DG) and shunt capacitor (SC) allocation. Which may lead to single non-dominated solution of unpredictable and misleading results about their site and size, loss reduction and payback period. Therefore, primary objective of this study is to investigate the effects of seven nonlinear voltage-dependent load models for the siting and sizing of DG and SC considering various conflicting Multiobjective functions. These objective functions are minimization of active power loss, voltage deviation, cost of energy loss per year, the total cost of installed DG and SC, and emission. Three study cases of simultaneous optimization of two and three objective functions are intended to find optimal integration of DG and SC in the standard 33 and 118-bus radial distribution network considering seven nonlinear voltage-dependent load models. A new Multiobjective evolutionary algorithm (MOEA) called Bidirectional Coevolutionary (BiCo) is applied to the proposed study cases to demonstrate the impact of load model on DG and SC allocation. Further, to show the superiority and performance of proposed algorithm, six state-of-the-art MOEAs are implemented and statistically compared with proposed algorithm using a representative hypervolume indicator (HVI). The maximum savings in annual cost of annual energy loss reaches 58.99% in case1 of PQ load model, 59.4% in case 2 and 64.96% in case 3 of industrial load model considering only DG allocation, whereas, 93.673% in constant PQ load model of case1, 78.908% in constant current load model of case2 and 93.403% of case3 of PQ load model considering simultaneous DG and SC. Simulation results show that the proposed algorithm is adept and suitable to find a better trade-off between various conflicting objective functions compared to other recently designed MOEAs.


I. INTRODUCTION A. LITERATURE REVIEW
The electricity grid's biggest issues right now are the scheduling of available power supply, growing load demand, The associate editor coordinating the review of this manuscript and approving it for publication was Qiang Li . and the expanding number of consumers. In developing countries, where the losses may approach 15-20% of the total power generated, distribution networks experience even more power losses, frequently in the range of 10-13% of the energy produced and the cost of these losses is millions of dollars per year [1], [2]. Furthermore, radial nature of distribution system results in inefficiency and poor grid voltage regulation that VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ results in bus voltage may deviate from the prespecified limit and power losses being more significant, which can harm the effectiveness of the supply. These impacts of bus voltage drop and increased losses are ultimately due to shortage and unbalancing of active and reactive power. Transmission and distribution networks are very costly to build and upgrade. Moreover, current changes in the electric utility structure with the integration of distributed generation (DG) have created opportunities for many technological innovations to achieve a variety of technical, environmental and economic benefits. A maximum of benefits in the distribution network are obtained by integrating DG of appropriate site and size that add local active and reactive power directly near distribution system. DG resources are categorized into four categories [3] these are: Category-I (DG injects active power only i.e., Solar PV), Category-II (DG injective reactive power only i.e., Synchronous condenser/SC), Category-III (DG injections active power but consumes reactive power i.e., Doubly fed induction generator) and Category-IV (Injects both active and reactive power i.e., Voltage source invertor-based DG, Synchronous generator) For better efficiency, security, reliability, and performance of distribution systems, active and reactive demand and supply must be balanced. Load across the distribution system is highly variable and accurately balancing active and reactive power demand is extremely complex from transmission system. Active power balanced locally can be accomplished by injecting DG with a unity power factor, that ultimately enhance the voltage level and reduces active power losses. By adding reactive power to the system, SC banks help with reactive power correction. The ideal placement of SC banks lowers the need for reactive power, optimizes net savings, and improves voltage levels within acceptable bounds. DGs and SC banks, when placed in appropriate sites with the best sizes, may significantly improve quality of power, efficiency, and stability in addition to economic factors [4]. Laterally, SC at optimal site and size can also reduce maximum MVA capacity from the main grid [5]. Added to that, the voltage fluctuations caused by some types of DGs can be reduced by using (fixed-switched) capacitor banks [6]. Thus, hybrid DG and SC integration can ultimately reduce the distribution system losses, improve the voltage profile, and can effectively enhance the performance of distribution system. The location and size of various types of DG and SC banks have been integrated into the distribution networks using a variety of techniques during the past few decades. These strategies can be broadly categorized as analytical, evolutionary-based, and hybrid methods.
In the literature, many analytical expressions are formulated, called analytical metho (AM), to find the optimal size of DG or DG along with SC, which includes [7], [8], [9], [10], [11], [12] with different objectives, test systems, and analyses as shown in Table 1. The penalty function approach was applied in most AM techniques for the selection of candidate buses. In AM, many research attempts were made to perform optimum DG allocation with single objective minimization of active power loss (P L ) as the objective functions for the site and size of DG, however, most of them were not considered the Multiobjective optimization except in [8] selects the site and size of DG seeing weighted sum Multiobjective technical objective functions that include minimization of active power loss and voltage deviation (VD) and maximization of voltage stability index (VSI). Yet, the formulation of analytical expressions including the impacts of nonlinear load models along with Multiobjective DG and SC allocation were not considered. Interested readers can read the reference [12] for a detailed discussion of AM.
Numerous literary works that used heuristic or metaheuristic and hybrid optimization techniques were implemented to find the optimal site and size of individual DG, SC, or a combination of both DG-SC in the distribution network. Most research work based on a single objective heuristic, metaheuristic, or hybrid evolutionary algorithm is available in the literature to solve single objective DG and DG-SC allocation problems. That includes, particle swarm optimization (PSO) [3], intersect mutation differential evolution (IMDE) [4], grey wolf optimization (GWO) in [13], an improved artificial eco-system based optimization (EAEO) [14], genetic algorithm (GA) in [15], enhanced GA (EGA) [16], gravitational search algorithm (GSA) in [17], Mixed integer nonlinear programming (MINLP) in the platform of general algebraic modeling system (GAMS) [18], improved binary PSO (IBPSO) [19], a spotted hyena optimizer (SHO) [20], and basic open source MINLP (BOMINLP) [21].
In most of the research work, only constant PQ load models were considered to find DG site and size which may lead to unpredictable and misleading results about site and size DG and SC [22]. Load modeling has received more attention in recent years because of renewable integration, demand-side management, and smart metering devices [23]. Further, single objective EAs produce single solution, and the site and size of DG highly depend on the selected objective functions. By changing the objective functions DG site and hence size are changed, therefore, weighted sum Multiobjective approaches were implemented considering various simultaneous technical, economic and environmental objective functions to find the optimal site and size of DG and SC. That includes: ant lion optimizer (ALO) [24], Jellyfish Search Algorithm [1], Constriction Factor PSO (CFPSO) [2], Grey Wolf Optimization (GWO) [25], Differential Evolutions (DE) [26], hybrid enhanced GWO and PSO (EGWO-PSO) [27], battery energy storage system (BESS) and reconfiguration along with DG and SC allocation using PSO in [28], water cycle algorithm (WCA) [29], Manta Ray foraging optimization (MRFO) were implemented in [30], whale optimization algorithms (WOA) [31], hybrid ant colony optimization (ACO) and artificial bee colony (ACO-ABC) [32]. In weighted sum approach, selection of weights for the particular objective functions is highly complex and difficult. Moreover, weighted sum approach produces a single non-dominated solution for a particular weight. Most of the work using weighted sum approach considers constant PQ load models except JSA [1], and WOA [31] consider the various load models, but they do not consider the Pareto Front (PF) based Multiobjective optimization. The main advantage of PF based Multiobjective approach is that it produces large number of non-dominated solutions in a single simulation run, where the decision maker can select the best compromise solution of their interest.
Recently, PF-based Multiobjective approaches were accomplished to solve DG and SC allocation problems, as shown in Table 1, for instance, chaotic salp swarm algorithm (CSSA) [33], adaptive and exponential PSO in [34], improved decomposition-based EA (I-DBEA) in [35], multi-objective evolutionary algorithm based on decomposition (MOEA/D) [36], non-dominated sorting GA (NSGA) in [37], multi-objective multiverse optimization method (MOMVO) [38], hybrid improved GWO and PSO (IGWO-PSO) [39], Multi-objective Chaotic Mutation Immune Evolutionary Programming (MOCMIEP) [40], Multiobjective ALO (MALO) [41], modified shuffled frog leaping (MSLF) [42], hybrid GA-DE and strength Pareto evolutionary algorithm (SPEA) (GA-DE-SPEA2) [43] and hybrid MINLP and multi-criteria evaluation method [44]. In these papers, various conflicting objective functions are considered to find the DG or SC allocation problems. But none of them considers the impacts of realistic load models for DG or SC allocation. Most of the research conducted based on PF-based Multiobjective optimization considers technical objective functions (power loss, voltage deviation and voltage stability index). Few of the studies were investigated simultaneous minimization of cost and emission. Cost of energy loss (C EL ) is investigated in [42] only, but nonlinear load models were not take into account. The Tabulated data shown in Table 1 explicitly shows the crux of the literature review.

B. MOTIVATIONS
Based on the literature review, shown in Table 1, single objective optimization given constant non-realistic PQ type VOLUME 11, 2023 load models is widely studied for the optimal site and size of DG and SC allocation problems. The wide-ranging review of the load model has been presented in [22] and [23] it is shown that the load models can significantly affect the optimal location and sizing of DG and SC resources in distribution systems. An exhaustive review of load model to be used for power flow and dynamic studies has been presented in [23] and [45]. Moreover, optimizing one objective in the allocation problem could negatively impact the other objective functions. Therefore, DG and SC allocation problems are multiobjective to assess the trade-off between various conflicting objective functions to get maximum benefits of renewable DG integration to the distribution system. From the Table, it is obvious that sufficient work has been done concerning the allocation of DG and SC units in the distribution system considering various composite load models; unfortunately, little work was addressed on the PF based Multiobjective optimal site and size of DG and SC considering PQ load models. Furthermore, it is observed from the table that load models are not included in most of the studies except in [1] and [31]. To the best of the authors' knowledge, no work is implemented the PF-based Multiobjective considering trade-off between technical, economic, and environmental objective functions for the optimal allocation of DG and SC because of various nonlinear load models. From the critical literature review, it is demonstrated that DG and SC allocation is Multiobjective constrained optimization problem. Different objective functions are emphasized the optimal site and size of DG and SC with respect to their function values. For example, power loss minimization selects those candidate buses in which DG integration results of reduction in line current on the other hand VD reduction emphasize the selection of those buses which are the good candidate through which DG injection voltage reaches to unity and so on. Multiobjective optimization algorithms generate number of non-dominated solutions in a single simulation run and the decision maker can select the appropriate solution of its interest. Recently, new MOEAs, i.e., CAEAD [46], CCMO [47], ToP [48], CMOEA_MS [49], cDPEA [50] were designed to efficiently solve the mathematical constrained Multiobjective optimization problems (CMOPs) and yet those are not employed to test for DG and SC allocation problems. According to the no-free lunch theorem [51] ''any algorithm, any elevated performance over one class of problems is offset by performance over another class''. Therefore, there is always a gap to design and implement new MOEAs for specific types of problems.

C. CONTRIBUTIONS
In this work, seven types of nonlinear load models, including constant active and reactive power (PQ), constantcurrent (CI), constant-impedance (CZ), industrial (IND), commercial (COM), residential (RES) and realistic MIX loads are employed to find optimal site and size of DG and SC devices. These load models depend on a nonlinear relationship between the demand and the bus voltage. Recently state-of-the-art Multiobjective evolutionary algorithms (MOEAs) were employed efficiently to solve complex mathematical constrained type problems. Due to the presence of constraints, Pareto-optimal solutions of CMOPs' are very likely lying on constraint boundaries. Most of the MOEAs, optimize the problem to emphasize feasible solutions, these methods may bring about stuck at some locally feasible regions or locally optimal feasible regions and the driving force may be limited since the population only evolves from the feasible side of the search space. Therefore, bidirectional coevolution (BiCo) strategy-based MOEA is employed for better diversity and convergence of final on-dominated solutions. The proposed algorithm searches the space according to coevolving feasible and infeasible population. It has also a better capability to find the convergence and diversity in DG and SC allocation problems. To show the superiority and performance of proposed method small to large-scale distribution systems (33, and 118 bus test systems) are considered to solve multiple DG and SC allocation problems. Further, technical, economic, and environmental objective functions are considered to find the optimal site and size of DG and SC allocation. From the perspective of objective functions, this study includes analysis and comparison of trade-offs among active power loss, voltage deviation, cost of DG and SC, cost of energy loss, and emission. Finally, measuring the convergence and diversity of proposed algorithm with the recent state-of-theart MOEAs is done through representative hyper volume indicator (HVI) metric. The following is a summary of this paper's main contribution.
1. An efficient radial distribution network-based modified bus and branch order FBS load flow algorithm is applied. 2. The effects of different nonlinear composite load models are intended to investigate the optimal site and size of DG and SC. 3. The study includes optimizing and calculating the trade-off between conflicting objective functions such as yearly cost of energy loss, cost of installed DG units and SC banks, and the technical objective functions power loss and voltage deviation. 4. A new bidirectional coevolutionary algorithm is employed to solve the proposed formulated problem given two and three conflicting technical, economic, and environmental objective functions. 5. Representative HVI metric is employed to compare the diversity and convergence of proposed algorithm with the recently designed MOEAs.
In the remaining sections, insection II problem is formulated where proposed algorithm is defined in section III.
In Section IV analysis and comparison of Proposed algorithm to solve DG and SC allocation problems is accomplished. The main conclusion points are drawn in section V.

II. PROBLEM FORMULATION A. LOAD FLOW METHOD AND VOLTAGE-DEPENDENT COMPOSITE LOAD MODEL
Due to high resistance to reactance ratio and radial nature of distribution system, frequently used load flow methods, i.e., newton Raphson and fast decoupled, are poorly converged to obtain the solution of a radial distribution system. Therefore, many power flow solution methods have been developed to account for the specific nature of distribution systems, and the most widely used are the backward/forward sweep (FBS) methods. For the fast FBS load flow method, it is desirable to modify branch and bus order i.e., branch numbers (k) equal to the receiving bus numbers (k) and sending bus numbers (i) should be smaller than the receiving bus (k) number as shown in Figure 1.
For the illustration of FBS method a branch k connecting bus i and k of modified radial distribution network, as shown in Figure 1, is considered. In this work DG and SC are modeled as negative PQ load, supposed to be integrated into bus k of active power of −P DG k and reactive power of −Q SC k , as shown in Figure 1. There are five steps in the FBS method. Initially in step 1, set all bus voltages to 1p.u and iteration counter t = 1. In step 2, receiving end branch flow (S t k ) is given as: whereas; S t k is the power drawn by receiving end bus k, S Lk is the load on bus k, P DGk and jQ SCk are the DG and SC injections at bus k, (Yk) * × v 2 k is the power injected into the shunt branch component. Backward sweep is the third step, in which first to compute power at far end of the bus S f k [52] as; whereas, k = n, n − 1, n − 2, . . . ., 2. After that, perform power summation, starting from the branch with the biggest index and heading towards the branch whose index is equal to 1. The sending power of branch k is added to the receiving power of the branch whose index is equal to i = fk as: In the fourth step, forward sweep, sending end bus voltage V i is computed as; In the fifth step, increase the iteration counter and check the termination condition as If Eq. (5) is true, terminate the condition, else go to step 2 and continue. Parameters of voltage-dependent load models, i.e., constant active and reactive power (PQ), constant current (CI), and constant impedance (CZ), industrial (IND), commercial (COM), residential (RES) taken form [23], [45] have been adopted for investigations. To quantify the effect of various load models for the DG and SC planning, in each iteration t of step 2 of FBS load bus injection S Li = P Li +jQ Li is modified according to magnitude of bus voltage V i as whereas, P 0i and Q 0i real and reactive load demand at bus i at 1p.u voltage, parameters (c 1 , , and (c 6 , d 6 ) are the compositions of PQ, CI, CZ, IND, COM and RES non-linear voltage dependent load models respectively. While in all composite load models, 6 i=1 c i and 6 i=1 d i must be equal to 1. During investigations, the comparison of constant power (PQ) load model assumption with the other practical load models are emphasized. While investigating the effects of constant current (CI), system is assumed to be supplying constant current consumers only (all loads are constant current type) and for constant impedance (CZ) load, it is assumed that all the loads are constant impedance type and so on. In a practical situation MIX composite load class model may be seen; therefore, in this paper, a composition of voltage-dependent load models is considered as; 25% of constant power (c 1 whereas, share of all the other loads is 15% i.e., (c 1

B. OBJECTIVE FUNCTIONS
The optimal site and size of DG and SC allocation are constrained Multiobjective problems (CMOPs). Without loss of generality, a minimization CMOP can be formulated as [53] min whereas, x is the decision variable such as site and size of DG and SC, L and U are the lower and upper bounds, f (x) is the objective function that includes power loss, voltage deviation, cost of energy loss, cost of DG and SC and emission of thermal generation, g j (⃗ x) and h j (⃗ x) are the l and m inequality and equality constraints whose numbers are q and p − q, respectively. The constraint violation (CV j ) of x against the j th constraint is usually calculated as And the overall constraint violation CV (x) is computed as; If a given decision vector, say x, is feasible iff CV is zero else the solution is infeasible. For two feasible solutions, for ∀h ∈ {1, . . . , m} and f g (x 1 ) < f g (x 2 ) for ∃g ∈ {1, . . . , m}. A solution is called Pareto-optimal when no other feasible solution dominates it. The image of all Paretooptimal solutions in the objective space is the Pareto front (PF). Proper placement and sizing of DG and SC allocation in the distribution network following objective functions are optimized, including: The active power loss P loss of branch between sending bus i and receiving bus k is given as; where Z ik denotes branch impedance; P k and Q k are the active and reactive injection into the bus k, |V k | denotes absolute value of voltage at k th . The total power loss (P L ) of the system is the summation of all branch losses [7] as: The objective function for power loss is:

2) MINIMIZATION OF TOTAL VOLTAGE DEVIATION (VD)
The voltage deviation indicates the difference of any bus voltage magnitude from the unity, and hence the total voltage deviation (TVD) is calculated as: The objective function for the TVD is: 3

) COST OF ENERGY LOSS C EL
The price of energy losses per year (C EL ) [1] is given as: where P L represents the total power loss of the system; k c p , k c e , and k f L represent the cost of P L in $/kW/year, energy cost in $/kWh/year, and the loss factor, respectively.

4) TOTAL ANNUAL COST OF DG C DG AND SC C SC
The price of the DG output power is calculated from the second-order quadratic expression as: where P DG is the output power of the DG unit; k 1 , k 2 and k 3 are constants taken from [1]. The cost of the shunt capacitor banks SC is calculated [1] as; where, k f , and k c are constants; N sc is the number of capacitors, and Q SC is the reactive power of capacitor in kVAr. The total operating cost (C ToT ) [29] is estimated as: where k 4 and k 5 are constants; N DG is the number of DG units.

5) MINIMIZATION OF EMISSION (E)
Minimization of Generation Units emissions' such as CO 2 , SO 2 , and NO x are considered as most effective pollutants in power generation sources. The mathematical formulation of this objective function can be described as follows [9]: whereas, (21) whereas, P gGrid is the active power supplied by grid station. In our case, only non-dispatchable types of DG are integrated, therefore, E DGi = 0.

C. CONSTRAINT FUNCTIONS 1) EQUALITY CONSTRAINTS
Power balance constraints are satisfied during the load flow [52] and is described as: P G and Q G are active and reactive power generation, and P D and Q D are the active and reactive demands.

2) INEQUALITY CONSTRAINTS
The bus voltage, branch flow, and active and reactive power generated from the DGs and installed SC limits must satisfy the following logical conditions: where, V min i and V max i are the minimum and maximum allowable voltage limits for any bus i, S ij is the thermal capacity of the feeder. P max DG and Q max C are the maximum selected ratings of DG and SCB respectively, N DG and N c are the number of DGs and SCs.

III. MULTIOBJECTIVE BIDIRECTIONAL COEVOLUTIONARY ALGORITHM
The optimal site and size of DG and SC is a constrained type Multiobjective optimization problem (CMOP) that involves both conflicting objective functions and various constraints. Due to the presence of constraints, CMOPs' Pareto-optimal solutions are very likely lying on constraint boundaries. The ultimate goal of MOEAs is to obtain well-converged and evenly distributed PF. However, it is not an easy task to achieve that goal, especially because of the existence of constraints. Most of the MOEAs, optimize the problem to emphasize feasible solutions, these methods may bring about the following two issues.
1) It can lead to the population being stuck at some locally feasible regions or local optimal feasible regions.
2) In addition, the driving force may be limited since the population only evolves from the feasible side of the search space. Thus, the method derived from feasible sides is unable to push the population toward the PF promptly. To alleviate the above issues, the proposed algorithm searches the space according to coevolving feasible (main) and infeasible (archive) population and hence the name bidirectional coevolutionary (BiCo) [53] algorithm. BiCo can conveniently and effectively drive the solutions toward the PF from both the feasible main population (P t ) and infeasible archive population (A t ) sides of the search space, which is of essential importance in CMOP. Moreover, A novel angle-based density (AD) selection scheme is designed to update the archive population. This scheme can not only maintain the diversity of the search, facilitating the discovering of more feasible regions; but can also retain the infeasible solutions close to the PF, speeding up the search for the Pareto-optimal solutions. To coordinate the interactions between the main and archive populations and make use of their complementary information of them, a brand-new restricted mating selection mechanism is developed in the proposed algorithm.
The proposed algorithm is comprised of four steps. In the first steprandomly generate initial population after that evaluate the population to compute objective functions and overall constraint violation (CV) of each population. In the second step, generation of offspring population (Q t ). For better convergence and diversity of the Pareto Front, it is desirable to make interaction and collaboration between main (driving from feasible search space) and archive (driving from promising infeasible solutions) population to generate highquality offspring. Selection of parents for the mating pool is nominated by employing Binary tournament selection. If the length of archive population ∥A t ∥ is less than population size N then parents for mating pool are selected from combined population of main P t and archive A t . If not, mating pool is filled by alternative selections of parents by comparing main and archive populations based on constraint violation given in Eq. (9) and angle-based-density (AD) respectively. For the selection of parent p 1 , first randomly selects two solutions x 1 and a 1 from P t and A t , respectively; afterward, their CV values are compared and the one with the smaller CV is selected. To select p 2 , it also first randomly chooses x 2 and a 2 from P t and A t , respectively; thereafter, their AD values are compared and the one with the larger AD is chosen. In the proposed algorithm AD is computed as: First, normalize the objective function space, say j th solutions of objective functions . . , f ′ m v j ) using ideal Z i min and nadir Z i max points in the combined population U t according to; After that vector angle between F ′ x j and F ′ (x k ) solutions selected from U t is computed as Then rank each solution according to an angle between them, larger the angle higher the rank of solution and it is the promising candidate for the mating selection. By mating one solution with a good CV value and other with good AD value, it is expected that the generated offspring can not only be close to the PF but also with good diversity. After the selection of the mating parents, the popular simulated binary crossover (SBX) and polynomial mutation is applied to generate the offspring Q t . In the SBX operator, two parents are randomly selected from the mating pool, such as x j and x k and generate two offspring v j and v k as where β is called the spread factor and is computed as; Parameter η c is the distribution index of SBX, u i ∈ [0, 1] is the uniformly distributed random number. The polynomial mutation operator is applied to randomly selected offspring x j as where, x L and x U are the lower and upper bounds, and the parameter δ i can be computed as; else (35) where, η m is the distribution index of polynomial mutation and rand is a uniformly random number between [0,1]. In the third step, main population is updated P t+1 . The main population is the main driving force toward the PF from the feasible side of the search space. In this step, combine main and offspring populations and divide the feasible S 1 and infeasible S 1 solutions. If the number of feasible solutions is smaller than population size N , sort the infeasible solutions S 2 and from that select the first (N − S 1 ) solutions. On the other hand, if the number of feasible solutions S 1 are larger than population size N , apply non-dominated sorting [54] on S 1 to compute PF of different rank, say F 1 , . . . , F k (F 1 is the highest rank, F 2 is next highest rank and so on) and assign the highest rank PF to P t+1 than 2 nd highest and process is continuous until size of P t+1 is equal to N or greater than N . If the size of P t+1 is greater than N , than some of the solutions in the last front are eliminated using crowding distance (CD) operator [54].
Finally, in the fourth step, archive population for the next iteration A t+1 is updated. In the proposed algorithm archive population is responsible to generate non-dominated infeasible solutions which are promising for the better diversity of PF [53]. Herein constraint violation is considered as the M + 1 th objective function and transform the original constrained problem shown in Eq. (8) into unconstrained Multiobjective problem as, Afterward, an ND sort [54] is employed to rank the solutions for the PF and choose the capable infeasible solutions with the help of CV (m + 1 objective function) and AD given in Eq. (30). Here step copy all infeasible non-dominated solutions in archive A t+1 , if it exactly fit or less than size of archive goes into the next step, else truncation operator is applied to delete extra infeasible solutions according to AD and CV. In truncation, select two solutions according to smallest angle among them, and delete the one with higher constraint violation. The flow diagram of proposed algorithm is shown in Fig 2.

IV. SIMULATION RESULTS AND COMPARISON
In this paper, an experimental study for the DG and SC allocation is conducted on 33, and 118 bus radial distribution systems. Total complex power load demand (P 0 + jQ 0 ) for the 33 and 118-bus system are 3.715 + j2.300 MVA, and 22.7097+j17.04107 MVA respectively [52]. Herein all seven load models are simulated and resolved through the FBS method without DG and DG-SC. After the implementation of FBS method bus injectionP i + jQ i , minimum bus voltage V min , active power losses P L and cost of energy loss C EL without DG and DG-SC for the 33 and 118-bus distribution systems is computed. Table 2 shows the FBS load floe simulation results of proposed test system without DG or DG-SC at 1MVA base value. Careful examination of the Table 2 gives that the extreme value of MVA sys and active power losses are given in PQ load model. Minimum voltage V min has appeared in PQ load models and maximum voltage is given in COM load model. The lowest active power losses are specified in CZ load model. These variations of the system parameters ensure the critical consideration of load types for the site and size of DG allocation, where the demand depends significantly on the voltage values of the system.
In the following subsections, first, the convergence and diversity of proposed algorithm are analyzed and compared with the most recent MOEAs using representative statistical hyper volume performance indicator (HVI). After that, proposed algorithm is applied to IEEE 33-bus, and 118-bus systems to show the impacts of practical non-linear load models on the various parameters of distribution systems with the optimal integration of DG, and simultaneous DG-SC.

A. STATISTICAL COMPARISON OF PROPOSED ALGORITHM WITH THE STATE-OF-ART MOEAs
The ultimate goal of Multiobjective evolutionary algorithms (MOEAs) is to find high-quality non-dominated solutions in a single simulation run. Quality measurement of nondominated (ND) solutions consists of three objectives:   VOLUME 11, 2023 fair comparison between the MOEAs, various performance metrics have been proposed in the past. In this work, convergence and diversity measure of different MOEAs has been done by using a well-known hypervolume indicator (HVI) performance metric.
HVI metric requires a reference point (preferably a point close to the nadir point) which is obtained from the true PF. In a real-world DG and SC allocation problem, no true Pareto optimal solution is known. Without knowing the true Pareto front, the nadir point information is not available. In this paper, for the computation of ideal and nadir points, each algorithm run 20 times independently then all final obtained ND solutions are merged and filter-dominated solutions from this by applying the ND sort technique [54]. We refer to this merged filtered set of nondominated solutions as a true Pareto optimal set and it is used to compute the ideal and nadir points for the comparison of MOEAs. On a given problem, when comparing different PFs, the PF with the largest HVI is considered the best. To access the performance of proposed algorithm BiCo with the recent implemented MOEAs, four study cases of DG and DG-SC of small to large-scale radial distribution systems at constant PQ load model are considered. The parameters of all the compared algorithms were kept identical to their original papers. Whereas, population size N in all the cases is 50, and the number of objective functions (M) and maximum number of iterations (G) are shown in Table 3.
All experiments in this article were conducted on the platform developed by Tian et al. (PlatEMO) [55]. All the codes are run on corei7 PC MATLAB 2021b version 9.10. Statistical performance based on HVI of proposed algorithm compared with the other recently designed state-of-the-art MOEAs (i.e., NSGAII [54], CAEAD [46], CCMO [47], ToP [48], CMOEA_MS [49], cDPEA [50]) of all the cases and the simulation results are summarized in Table 3. It is worthwhile to mention that no constraint violation is observed in any trial run of all case studies. Simulation results shown in above Table 3 demonstrated that in most of the cases proposed algorithm outperforms or is marginally equivalent to existing algorithms based on best and worst HVI values. In Table 3, dark and highlighted results are the best and half-dark is the second-best value.
At first glance, BiCo can achieve the best performance on most test instances concerning HVI. As for the other algorithms, the proposed algorithm can obtain overall better performance than the other recently designed algorithms. In general, proposed algorithm obtained the best HVI results on all test instances except for case 3 of 33-bus DG only. However, for CAEAD, CMOEA_MS, and cDPEA, as observed in Table 3, none of them could obtain any best results on all the cases in both of the test systems. Whereas, cDPEA obtain second best value in DG-SC allocation problem of 33-bus. CCMO outperforms only in case 3 of 33bus DG only, whereas proposed algorithm beats CCMO in most of the other cases. Table 3 clearly shows that overall performance of proposed algorithm is best compared to other state-of-the-art MOEAs. Moreover, ToP consistently fails to global PF. It is shown from the simulation results that the proposed algorithm finds better or similar results compared with other EAs. On the other hand, in a large-scale 118-bus distribution system, proposed algorithm outperforms compared to rest of the EAs. Moreover, to visualize the simulation results of all the study cases, the final non-dominated populations of best PF based on HVI of each algorithm are shown in Fig 3. Pareto Fronts as shown in Fig 3, clearly show that in most of the cases nondominated solutions of proposed algorithm are converged and evenly distributed.
Whereas, in case 1 of DG-SC allocation problem, PF of ToP is trap into local optima, and compared to other algorithms PF of proposed algorithm is well distributed. Moreover, Fig 4, shows the PF of all the cases of 118-bus test system of DG and DG-SC allocation problem. Figure 5 shows that in the large-scale 118-bus system in all the cases of DG and DG-SC allocation problems, proposed algorithms obtain the widely distributed, evenly spaced, and well-converged PF compared to most of the algorithms. ToP and CAEAD in most cases stuck in locally optimal solutions. From the visualization of figures, it is proved that proposed algorithm can find widely distributed and evenly spaced non-dominated solutions. For most of the cases of optimal site and size of DG and DG-SC allocation problem proposed algorithm outperforms. Therefore, in the next section, the impact of realistic load models on the optimal site and size of DG and DG-SC allocation problems are considered to show the superiority and performance of proposed algorithm.

B. EFFECT OF LOAD MODELS CONSIDERING DG, AND CUMULATIVE DG-SCC
The following test cases are developed in which DG, and DG-SC optimal locations and sizes are computed considering various combinations of objective functions given in Eq. (9)- (11) implemented on small to large-scale radial distribution systems. The size of DG and SC is decided by 75% of total demand. It is considered that DG generates active power only (unity pf). Three study cases of 2 and 3 conflicting objective functions are formulated to find the optimal site and size of DG and SC allocation in small 33-bus to a large 118-bus test network. Study cases are: Case 1: P L vs VD Case 2: C ToT vs C EL Case 3: C ToT vs C EL vs E In the first case technical objective functions of power loss P L and VD is considered to find the optimal site and size of DG. The second case shows the trade-off between the cost of DG and SC (C ToT ) with the cost of energy loss C EL . In the third study case, Pareto Front (PF) among the economical total cost of DG and SC (C ToT ), energy loss C EL and environmental emission (E) is considered. Furthermore, in all the cases of seven non-linear load models, DG and

1) OPTIMAL ALLOCATION OF DG, AND DG-SC EFFECT OF LOAD MODELS ON 33-BUS
In this section, impact of seven nonlinear realistic load models on the optimal site and size of DG and DG-SC allocation is analyzed and compared. Figure 5 shows the PF of the final non-dominated solutions of DG and DG-SC allocation of seven non-linear load models of all three cases. An orange square shows the best compromise (BCS) solution. The BCS is derived from the final non-dominated solutions using fuzzy weight functions [56]. In this technique, first normalize the objective functions using ideal Z min i and nadir Z max i points for the computation of membership function µ k i , according to whereas, i = 1, 2, . . . , M is the number of objective functions, k = 1, 2, . . . , N solution number, and f k i is the i th objective function of k th non-dominated solution. The normalized membership function µ k for each non-dominated solution is computed as; (38) The solution with the highest µ k value is the BCS. Fig 5  clearly shows that objective functions in final non-dominated solutions are conflicting. PF of only DG integration shows that objective functions in ll cases of various load models are widely distant, however, DG integration along with SC reduces the gap between the objective functions of various load models. It is predicted that more gap in PF space also increases the gap in decision variable search space. Load models are highly impacted on only DG integration compared to DG-SC allocation. A decision maker can select a promise solution from the widely spread and well-converged nondominated solutions. Table 4, shows the results of BCS of DG allocation using the fuzzy weight function.
The Table 4 clearly shows that the impact of the nonlinear load model has highly affect the optimal site and size of DG and SC allocation. In the table, bold numbers are the objective functions, and highlighted gray cell shows the best values of parameters. In case 1, objective functions are P L and VD, the best sites for the DG's are located at [14], [25], and [28] buses of different ratings for the PQ, CI, COM and MIX load models. Minimum power loss of 69.6 kW was obtained in   2038t/h has seemed in CI load model. In case 2, the cost of energy loss and Total operating cost is the objective functions. Minimum values of C ToT and C EL are shown in IND load model are 32929.3$ and 5283$. Bus numbers [13], [25], [28] are the strong candidate for the DG siting in all the load models. The worst voltage appeared at bus 33 that 0.961p.u in most of the load models. Minimum emission E is 3958.1(t/h) has appeared in MIX load model. In case 3, where cost of DG, cost of energy loss, and emission are simultaneously emphasized for finding the optimal siting and sizing of DG. In this case, bus numbers 28 and 10 are the strongest candidate for the selection of DG in PQ, CI and VOLUME 11, 2023 CZ and MIX load models. However, IND, COM and RES load models differ in the selection of DG siting. Better V min is found in CZ load models, and minimum power loss is computed in IND load model. The best values of objective functions are obtained in RES, IND, and MIX load models these are 48390.8$, 4561.1$ and 2007.1(t/h) respectively. Simulation results of practical MIX load models in all the cases are between the worst and best values of different load models. Simulation results of all the cases proved that the load models highly impact the site and size of DG. Table 5 gives the simulation results of all the cases of DG-SC integration in 33-bus tests system.
The table clearly shows that the impact of the nonlinear load model has highly affected the optimal site and size of DG-SC allocation. In the table highlighted bold numbers show the objective functions and gray cell shows the best values of parameters. In case 1, objective functions are P L and VD, in most of the load models siting at 23 and 13 bus of different ratings are the candidate for the DG and SC allocation. Minimum power loss of 12.3 kW was obtained in COM load model and minimum VD 0.0067 is computed in CZ load model. Minimum voltage without DG-SC appeared at bus 18 th which is 0.91p.u which is enhanced to 0.99p.u in all the load models. Minimum C ToT is appeared in IND load model, this result shows that minimum DG power is injected in this load model. The cost of energy loss is highly reduced compared to only DG allocation that is 988.6$ in COM load model, minimum emission of 1885.9 t/h appears in CZ load model. In case 2, cost of energy loss and Total operating cost is the objective functions. Minimum values of C ToT and C EL are shown in CI load model are 27235.1$ and 2998.7$. Compared to case 1, values of PL and VD are shown higher in case1, that is because of objective functions. In case2, a tradeoff between the objective functions such as C ToT and C EL emphasize not only PL or VD reduction but this resultant PF gives solutions at minimum cost and energy loss along with an optimal reduction in loss and VD. In case 2, in each load model site and size of DG and SC is different and no single set of buses are strong candidate for the DG siting in all the load models. The worst voltage appears at bus 33 that 0.97p.u in most of the load models which is highly improved compared to base case and, in the case, where only DG integrates power. Minimum emission E is 5124.3(t/h) as seemed in CI load model. In case3, where total cost of DG-SC, cost of energy loss and emission are simultaneously minimized for finding the optimal siting and sizing of DG. In this case, selection of candidate bus for the DG and SC is highly dependent on load models and also different compared to cases 1 and 2. There is no single set of strong candidate buses for the selection of DG for all the load models. Better V min is 0.992p.u at bus 2 found in PQ load models, and minimum power loss of 13.2 kW is computed in PQ and IND load models. The best values of objective functions are obtained in CZ, PQ and RES load models these are 49080.9$, 1075.1$ and 1917.5(t/h) respectively. Simulation results of practical MIX load models in all the cases are between the worst and best values of different load models. Simulation results of all the cases show that the optimal site and size are highly dependent on the load models. Form the simulation results it can also be concluded that performance of distribution is highly increased with the integration of DG and SC simultaneously, also with the consideration of Multiobjective functions elasticities a trade-off between the various conflicting objective functions. However, single objective optimization or weighted sum approach produces a single solution for the decision maker. Furthermore, a comparison between all the cases of DG and DG-SC according to percentage loss reduction, overall MVA injection (MVS_SYS), cumulative DG and SC integration, active and reactive of grid injections and MVS sys increase or decrease with and without DG and DG-SC is as shown in Fig 6. In this figure first 21 stacked bar shows case 1-3 of only DG allocation and 22-42 bar shows the performance parameters of DG-SC. Blue color stacked bar shows the MVS sys and brown color with a circled marker focuses the % variation in blue color bar and that is approximately (3-5) % of MVA sys is reduced in the constant PQ load models in all the cases. That results in a decrease in MVA sys (positive values of MVA ± sys ) with placement of DGs and SCs compared to base case (without DG and SC). Furthermore, MVA of the DISCO also gets reduced in PQ load models. Hence, when a constant load model is assumed, reduction in overall system MVA gives an erroneous indication of higher benefits of DG placement. However, negative values of MVA ± sys concerning base case condition is as shown Figure 9 indicates that an extra percentage of MVA sys is required to meet nonlinear load models. In case of DG and DG-SC allocation maximum value of MVA sys is more than 5% computed in CZ and COM load model. It means that 5% of extra power is injected into the CZ and COM load models to meet the load demand. At the time of optimal allocation of DG and SC care must be taken into account for obtaining maximum benefits of DG and SC allocation. Else inappropriate allocation of DG and SC, where the impact of real-time voltage-dependent load model is not taken into account, causes an increase in power losses and violates system constraints. Compared to case1 and 2 of DG allocation, cumulative DG injection is increased in case 3. However, optimization of C ToT and C EL emphasize the high injection from DISCOs (P intake and Q intake ) compared to case 1 which increases the maximize Q intake only form DISCOs. In DG allocation 50 to 65% power loss is reduced in only DG allocation, but cumulative DG-SC integration reduces the power loss by more than 90% in case 1 and 75 to 90 % in case 2 and 3. Figure 9, shows that maximum technical benefits can be obtained in case 1 of DG and SC allocation. Figure 7 shows the voltage profile of the 33-bus test system after allocating optimal DG, and DG-SC.  Figure 7 clearly shows that all the voltage profiles are within upper and lower limits in all the cases. In case 1, P L and VD is the objective function, the voltage profile is better such as near unity compared to case 2 and 3. Figure 10(b), shows that DG along with SC increases the performance of distribution system and the voltage profile is improved compared to only DG allocation. However, over all MVA, MW and MVAr capacities are nearly equal in DG and DG-SC allocation. The main difference is that in DG-SC problem both MW and MVAr are locally injected into the system and performance of system is highly enhanced. Simulation results clearly show that the proposed algorithm can solve complex constrained DG and SC allocation problems.

2) EFFECT OF LOAD MODELS ON THE OPTIMAL ALLOCATION OF DG AND DG-SC ON 118-BUS
In this section, the impact of seven nonlinear realistic load models on 118-bus system given optimal site and size of DG and DG-SC allocation is analyzed and compared. Figure 8 shows the PF of final non-dominated solutions of DG and DG-SC allocation considering all the realistic non-linear load models of all three cases. The orange square shows the best compromise (BCS) solution. The BCS is derived from the final non-dominated solutions using fuzzy weight functions Eq. (38). Fig 8 shows that the PF of only DG integration is widely separated, however, DG integration along with SC reduces the gap between the objective functions of various load models. More the gap in PF of different load models results in a higher gap in decision space (such as site and size of DG and SC). Load models are highly impacted on site and size of only DG integration compared to DG-SC. A decision maker can select a promise solution from the widely spread and well-converged non-dominated solutions. Table 6, shows the results of BCS of integration of DG only.
The table clearly shows that the impact of nonlinear load model has highly affected the optimal site and size of DG and SC allocation. In case 1, the minimum voltage has appeared at 46 th bus which is 0.9583 p.u. Minimum P L is 452.7 computed in IND load model, whereas, the smallest VD      a tradeoff between the objective functions such as C ToT and C EL emphasize not only PL or VD reduction but this resultant PF gives solutions at minimum cost and energy loss along with an optimal reduction in loss and VD. Further, in each load model site and size of DG and SC is different and no single set of buses are strong candidate for the DG and SC siting in all the load models. The worst voltage appears at bus 54 which is 0.9542p.u in most of the load models which is highly improved compared to base case and, in the case, where only DG integrates power. Minimum emission E is 38560.4.3(t/h) seems in CZ load model. In case 3, where total cost of DG-SC, cost of energy loss, and emission are simultaneously minimized for finding the optimal siting and sizing of DG. In this case, selection of candidate buses for the DG and SC are highly dependent on load models and also different compared to case 1 and 2. There is no single set of strong candidate buses for the selection of DG for all the load models. Smallest V min is 0.9638p.u at bus 46 found in COM load model, and minimum powerloss157.4 kW is computed in PQ load model. The best values of objective functions are obtained in CZ, PQ and RES load models these are 296564.1$, 12670$ and 11747.9(t/h) respectively. Simulation results of practical MIX load models in all the cases are between the worst and best values of different load models. Simulation results of all the cases show that the optimal site and size are highly dependent on the load models. From the simulation results, it can also be concluded that the performance of distribution system is highly increased with the integration of DG and SC simultaneously, also with the consideration of Multiobjective functions gives a trade-off between the various conflicting objective functions. However, single objective optimization or weighted sum approach produces a single solution for the decision maker. Furthermore, a comparison between all the cases of DG and DG-SC according to percentage loss reduction, overall MVA injection (MVS SYS ), cumulative DG and SC integration, active and reactive of grid injections and MVS sys increase or decrease is as shown in Fig 9. In this figure first 21 stacked bar shows case 1-3 of only DG allocation and 22-42 bar shows the performance parameters of DG-SC. Blue color stacked bar shows the MVS sys and brown color with circled marker focuses the % variation and that is approximately (3-5) % of MVA sys is reduced in the constant PQ load models in all the cases. That results in a decrease in MVA sys (positive values of MVA ± sys ) with placement of DGs and SCs compared to base case (without DG and SC). Furthermore, MVA sys of the DISCO also gets reduced in PQ load models. Hence, when a constant PQ load model is assumed, reduction in overall system MVA gives an erroneous indication of higher benefits of DG placement. However, negative values of MVA ± sys concerning base, case condition is as shown Figure 9 indicates that an extra percentage of MVA sys is required to meet nonlinear load models. In case of DG and DG-SC allocation maximum value of MVA sys is more than 5% is computed in CZ and COM load model. It means that 5% of extra power is injected into the CZ and COM load models to meet the load demand. At the time of optimal allocation of DG and SC care must be taken into account for obtaining maximum benefits of DG and SC allocation. Else inappropriate allocation of DG and SC, where the impact of real-time voltage-dependent load model is not taken into account, causes an increase in power losses and violates system constraints. Compared to cases 1 and 2 of DG allocation, cumulative DG injection is increased in case 3. However, the optimization of C ToT and C EL emphasize the high injection from DISCOs (P intake and Q intake ) compared to case 1 which increases the maximize Q intake only form DISCOs. In DG allocation 50 to 65% power loss is reduced in only DG allocation, but cumulative DG-SC integration reduces the power loss by more than 90% in case 1 and 75 to 90 % in cases 2 and 3. Figure 9, shows that maximum technical benefits can be obtained in case 1 of DG and SC allocation. Figure 10 shows the voltage profile of the 118-bus test system after allocating optimal DG, and DG-SC. Figure 10 clearly shows that the voltage profiles of each case are within upper and lower limits. In case 1, P L and VD is the objective function, voltage profile is better such as near unity compared to cases 2 and 3. Figure 10 (b), shows that DG along with SC increases the performance of distribution system and the voltage profile is improved compared to only DG allocation. However, overall MVA, MW and MVAr capacity is nearly equal in DG and DG-SC allocation. The main difference is that in DG-SC problem both MW and MVAr are locally injected into the system and performance of system is highly enhanced. Simulation results clearly show that the proposed algorithm can solve efficiently complex constrained DG and SC allocation problems.

V. CONCLUSION
Pareto Front-based Multiobjective approach produces a large number of non-dominated solutions in a single simulation run, from which decision-makers can select the best compromise solution of their interest. Therefore, in this paper, various conflicting Multiobjective functions are formulated for the DG and SC allocation problem to show the impacts of seven nonlinear voltage-dependent load models on 33 and 118-bus distribution networks. The seven load models include the constant-power, constant current, constant impedance, industrial, commercial, residential, and mix. Various conflicting objective functions of three study cases of simultaneous optimization of two and three objective functions are intended to find optimal integration of DG and SC. The Multiobjective bidirectional coevolution algorithm (BiCo) is adopted to optimally allocate the compensating devices DG and SC. To show the superiority and performance of proposed algorithm, six state-of-the-art MOEAs are employed and statistically compared with the proposed algorithm by using a representative hypervolume indicator (HVI). Simulation results show that the proposed algorithm is adept and suitable to find a better trade-off between various conflicting objective functions compared to other recently designed MOEAs. The maximum savings in annual cost of energy loss reaches 93.673% in constant PQ load model of case 1, 78.908% in constant current load model of case 2, and 93.403% in case3 of PQ load model considering simultaneous DG and SC. The obtained results clarify that the proposed methodology achieves significant reductions in active power losses, cost of energy loss, total operating cost and emissions. In the future, proposed methodology can be applied to consider probabilistic wind and solar-type DG along with the energy storage systems given technical, economical, and environmental objective functions.