A novel hybrid multi operator evolutionary algorithm for dynamic distributed generation optimization and optimal feeder reconfiguration

Abstract

This study addresses the integration of distributed generations (DG) and network reconfiguration in distribution networks, that has not been thoroughly investigated in prior research. The importance of technical objectives, such as power loss, voltage deviation, and voltage stability index, is emphasized in improving distribution network planning and operation. The study investigates the impact of changing sun irradiation and load demand on the IEEE 33 and 69-bus test systems. The issue at hand pertains to a mixed integer non-linear configuration, and four distinct research cases have been constructed in order to address and resolve it. Traditional evolutionary algorithms (EAs) are effective for such problems, but the study notes that using a single operator can limit performance. Hence, an innovative approach combines genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO) to tackle multiperiod large-scale DG and network reconfiguration issues. Dealing with infeasible solutions during optimization poses a challenge, so penalty functions are often used in the literature. The penalty function can be limited by the selection of the penalty parameter, however; a large value of this parameter slows down the process, but a smaller value is stuck in infeasible space. Therefore, in the proposed hybrid method representative constraint handling techniques are incorporated to make a trade-off between exploration and exploitation. The simulation results illustrate the capability of the suggested strategy to converge towards the global optimal solution. Furthermore, taking into account the voltage stability index greatly improves the loading capacity as compared to the base situation. The hybrid multi-operator EA suggested in this study demonstrates a nearly global optimal solution for large-scale mixed integer non-linear problems, as evidenced by the comparison of simulation results with existing EAs. Moreover, the results demonstrate a substantial decrease in power loss by over 86%, a significant improvement in voltage deviation by more than 90%, and an increase in load capacity by over 700% through the effective integration of DGs with the voltage stability index as the objective function.

Introduction

The world has vowed to shift reliance from the conventional energy fuels to the renewable energy sources to with the aim to reduce expenses and carbon emissions. Several researchers have proposed several optimization methodologies to address the formulated problem, encompassing conventional, artificial intelligence, and hybrid intelligent system techniques for optimizing the given difficulties¹. In the available literature, standard mathematical expressions, such as the analytical approach are shown in references^2–6. This allocation aims to improve the voltage profile or voltage stability index. The aforementioned objectives have been incorporated with the aim of attaining the desired results. Furthermore, a multitude of research investigations have utilized heuristic, metaheuristic, and hybrid optimization methodologies to ascertain the optimal locations and dimensions for individual Distributed Generations (DGs) in distribution networks. Previous research primarily focused on utilizing single-objective heuristic, metaheuristic, or hybrid evolutionary algorithms to address DG allocation problems with a single objective. This list encompasses the following: the EGA (enhanced genetic algorithm)⁷, the GWO (grey wolf optimization)⁸, the EAEO (improved artificial eco-system-based optimization)⁹, the GSA (gravitational search algorithm)¹⁰, the IBPSO (improved binary PSO)¹¹, the SHO (spotted hyena optimizer)¹², and the BOMINLP (basic open source MINLP)¹³. When trying to optimize for multiple goals at once, current methods run into problems. Using the weighted sum method slows down the optimization procedure and reduces the likelihood of the de-sired results being attained.

As an alternative, the application of Pareto optimality proves effective in efficiently addressing optimization challenges involving various objective functions. To do this, a weighted sum was used in order to determine the best DG and SC site and size, and multi-objective techniques that integrated technical, economic, and environ-mental aspects. Multiple optimization algorithms were employed¹⁴, CFPSO¹⁵, GWO¹⁶, DE¹⁷, EGWO-PSO¹⁸, battery energy storage system (BESS), and reconfiguration along with DG and SC allocation using PSO in¹⁹, MRFO were implemented in²⁰, and WOA²¹. However, the complexity of the weighted sum approach stems from the difficulty in choosing the right weights for various objective functions. Additionally, this approach provides a single non-dominated solution for each weight like JSA¹⁴ and WOA²¹. On the other hand, Pareto Front (PF) based approach generates numerous non-dominated solutions in a single simulation, exemplified by CSSA²², PSO in²³, I-DBEA²⁴, MOEA/D²⁵, MOMVO²⁶, IGWO-PSO²⁷, BiCo²⁸, MALO²⁹, MSLF³⁰, hybrid GA-DE and SPEA called GA-DE-SPEA2³¹, NSGAII³², INSGAII³³, and IMOHS³⁴, have recently been employed in solving DG allocation problems. The examined papers explore diverse conflicting objective functions in addressing DG allocation problems, yet none of them incorporate multiperiod DG allocation alongside optimal network reconfiguration. The current body of research predominantly concentrates on technical goals, specifically power loss, voltage deviation, and voltage stability index, in the context of PF-based multi-objective optimization. By employing uniform optimization of multiple objectives, the Pareto optimality technique generates Pareto sets of superior solutions for optimal outcomes.

Integrating DGs into electricity networks has positive effects. Furthermore, network reconfiguration can be used as an alternate method to reduce losses in the distribution network. Managing the open/close statuses of sectionalizing and tie-switches is necessary to establish an optimal network architecture that meets operational restrictions and optimizes numerous metrics. To enhance reliability, load distribution, and voltage control, distribution systems incorporate switches within each branch, which can be closed or open. Strategically modifying the positions of these switches to alter the distribution network’s topology constitutes optimal feeder reconfiguration (OFR). The central aim of OFR is to curtail active power losses by redistributing load from heavily loaded nodes to lightly loaded ones, promoting load balance, and enhancing the overall distribution system efficiency³⁵. OFR presents a complex challenge encompassing multiple objectives, constraints, and a blend of continuous and discrete variables^36,37. The optimal network configuration (ONR) should meet operational requirements and can be achieved by strategically controlling the open/close status of sectionalizing and tie-switches during the process of optimal network reconfiguration. This allows for the optimization of multiple variables simultaneously. Authors conducted the first network reconfiguration to mitigate active power losses. Various research literature employed in the past years to address the network reconfiguration problem by integrating extra parameters for optimization, which comprise voltage profile, system reliability indices, etc. Numerous optimization techniques, including genetic algorithms (GAs)^38–41 are employed to perform distribution network reconfiguration to solve the multi-objective problems. Nevertheless, combining these two sub-problems will be a better choice for the whole power system. There is not enough literature that addresses network reconfiguration along with optimal DG allocation and sizing. The majority of the authors regard a single objective function to minimize power loss in the distributed networks. Researchers in^42–47 point out that comparatively less power was lost when the DG reconfiguration and integration were combined and optimized to find the optimal solution. Only a few researchers have attempted to optimize this complex problem with extra objectives. To address the issue of active power loss, augment the balance of feeder loading, and optimize the voltage profile of the system, the authors⁴⁸ employed a fuzzy-ACO (ant colony optimization) approach that is based on Pareto optimality. The authors⁴⁹ used the multi-objective bang-big crunch technique to determine the ideal size of DGs, without taking into account their appropriate placement. However, some EAs are used to find the solution of more than three objective functions discussed in^50–52. A study in⁵³ solved the integration of ideally distributed generators (DGs) and the resolution of coupled reconfiguration multi-objective problems. The study employed the Improved Particle Swarm Optimization (IPSO) algorithm. Pareto optimality did not achieve equitable optimization of the objectives as these methods evaluated the weighted sum of the objective functions. Prior research focused on seeing DGs as deterministic models, despite the fact that in practice they exhibit intermittent behavior, particularly in the case of renewable energy-based DGs. EAs have proven to be highly effective in solving several practical problems involving mixed-integer non-linear programming (MINLP) shown in Table 1.

Table 1.

Literature review in terms of objective function (OF), DV and Constraints.

Ref/Year/Method	Optimization			OF	Decision Vector					DG			Time Frame
	S	WS	M		N	Site	Size	VG	ONR	WT	PV	NG	t1	tn
2020/EAEO/9	☑	☒	☑	PL, VD, VSI	☒	☑	☑	☒	☒	☑	☒	☒	☑	☑
2020/EGA/7	☑	☒	☒	PL, QL, VD	☒	☑	☑	☒	☒	☑	☒	☒	☑	☒
2022/JSA/14	☑	☒	☒	C, PL, QL, VD, SI	☒	☑	☑	☒	☒	☑	☑	☒	☑	☒
2022/GWO/16	☑	☒	☒	PL, QL, VD, SI	☒	☑	☑	☒	☒	☑	☒	☒	☑	☒
2018/ALO/54	☑	☑	☒	C, PL, VD	☒	☑	☑	☒	☒	☑	☒	☒	☑	☒
2020/PSO/19	☑	☑	☒	PL, VD	☒	☑	☑	☑	☑	☒	☒	☒	☑	☑
2018/WCA/55	☑	☒	☒	C, PL, VD, SI	☒	☑	☑	☒	☒	☑	☑	☒	☑	☒
2020/MRFO/20	☑	☒	☒	PL, VD, SI	☒	☒	☒	☒	☒	☒	☒	☒	☒	☒
2021/WOA/21	☑	☒	☒	PL, VD	☒	☑	☑	☒	☒	☑	☒	☒	☑	☑
2022/MSLF/30	☑	☒	☑	C, CEL, PL	☒	☑	☑	☒	☒	☑	☒	☒	☒	☑
2022/MALO/29	☑	☒	☑	PL, QL, VD, SI	☒	☑	☑	☒	☒	☑	☑	☒	☒	☑
2022/IGWOPSO/27	☑	☒	☑	PL, VD, SI	☒	☑	☑	☒	☒	☑	☑	☒	☑	☒
2021/MOMVO/26	☒	☒	☑	C, VD	☒	☑	☑	☒	☒	☑	☑	☒	☒	☑
2021/I-DBEA/24	☒	☒	☑	PL, SI, VD	☒	☑	☑	☒	☒	☑	☑	☒	☑	☒
2020/CSSA/22	☒	☒	☑	PL, SI, VD	☒	☑	☑	☒	☒	☑	☑	☒	☒	☑
2023/AHA/56	☑	☒	☒	E, VD, VSI	☒	☑	☑	☒	☒	☑	☑	☒	☒	☑
2022/CPLAX/57	☑	☒	☒	C	☒	☑	☑	☒	☑	☑	☑	☒	☒	☑
Proposed	☑	☑	☑	PL, VD, VSI	☑	☑	☑	☑	☑	☑	☑

Opt: Optimization; N-Disp: Non-Dispatchable; P, Q, PQ: Shows integration of active only, reactive only or both; Disp: Dispatchable; S: Single objective; WS: Weighted sum; M: Multi-objective; DR: Distributed Resources; ONR: Optimal Network Reconfiguration; : Single Period; : Multi-period; D: Deterministic; P: Probabilistic; N: Number; : generator voltage set point; Is landed main grid or sub-station; Di: Diesel; NG: Natural Gas type DG; O: other.

Table1 addresses several critical research gaps identified in recent literature on DG integration and ONR optimization. Unlike existing studies that predominantly focus on limited decision variables such as site and size, this work introduces a comprehensive decision-making framework incorporating number of |DGs, site, size and ONR positions of normally open and closed switches. Also, the coordinated scheduling of both dispatchable and non-dispatchable DGs are still not properly searched. Thess combination of decision variables is rarely explored in existing research, yet it provides significant operational flexibility and system reliability benefits. Furthermore, the proposed model adopts a true multi-objective optimization approach that simultaneously minimizes total power loss, voltage deviation, and voltage st6ability index—a combination not collectively addressed in prior studies. The optimization is performed over an extended time horizon (from ), capturing the temporal dynamics of load demand and renewable generation, which are often neglected in short-term or static models. By integrating a broader set of control variables and objective functions, this paper offers a more realistic and effective solution framework for modern power distribution systems with high DER penetration. The proposed method is validated through comparative analysis against established metaheuristic techniques, demonstrating superior performance in operational efficiency and power quality enhancement. Furthermore, the selection of EAs is a crucial factor when solving presented optimization problems. GA, DE, and PSO were chosen to meet the given problems because of their complementing attributes and demonstrated efficacy in resolving a diverse array of optimization problems. These algorithms have been highly researched in academic literature and have steadily shown their resilience and capacity to adjust to different situations. Within the current literature, a common method used to solve highly intricate MINLP issues is the use of a single operator. Using many operators in conjunction can improve efficiency and robustness when dealing with complex situations, as different operators have different search patterns. This paper solves the challenge of multi-period optimal DG allocation and reconfiguration using a novel hybrid multi-operator evolutionary technique. The method being suggested utilizes a feasibility criterion to remove options that are not possible, in order to improve the process of exploring and exploiting alternatives. Main contribution of this work is summarized as:

• Simultaneous Optimization DG integration and network reconfiguration in distribution networks, considering most predominant technical objective functions like power loss, voltage deviation, and voltage stability index.

• Incorporating both dispatchable and non-dispatchable DGs in dynamic time frame with varying load demand.

• A novel hybrid multi-operator evolutionary algorithm combining genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO) to tackle large-scale time varying DG allocation and network reconfiguration problem, improving performance over traditional single-operator evolutionary algorithms.

• Introduces a feasibility criterion to eliminate infeasible solutions and improve the exploration and exploitation processes, ensuring convergence towards a global optimal solution.

The rest of the article is as follows: Section II presents the reconfiguration, DG sizing, and allocation mathematical formulation. Section III describes the proposed multi-operator EA. Section IV covers time-varying device modeling and simulation results. Section V concludes the article.

Problem formualiton

Optimization problem

OFR and DG allocation problems are characteristically complex optimization problems that encompass various objectives, non-linear relationships, and a combination of discrete and continuous decision factors. Multi-objective constrained mixed integer non-linear programming (MINLP) issues can be defined as follows, without any assumptions limiting their generality.

1

where, are the proposed m real-valued conflicting objective functions, and are and non-linear equality and inequality constraints for the time slot and is the n-dimensional mixed integer decision vector that is comprised of continuous and integer decision vector. In the proposed constrained multi-objective optimization problem (CMOP) the ith degree of constraint violation at a given decision vector can be computed as;

2

whereas is the tolerance value used to relax the equality constraints. Usually, in most of MOEAs the degree of overall constraint violation (CV) for all the constraints is computed as.

3

Decision vector is a feasible search space if is zero, else it is an infeasible solution. In recent decades, evolutionary algorithms (EAs) have attracted great attention for solving practical large-scale multi-objective mixed integer non-linear programming (MINLP) problems. This article presents three objective functions to improve distribution system performance. The optimal integration of DGs with OFR helps to minimize energy loss, voltage deviation (VD), and maximization of Voltage Stability Index (VSI) by determining the optimal distribution network and DG locations and sizes. Generally, in a power system load varies w.r.t time, therefore, it is desirable to estimate the site of DG under varying load is not appropriate. In this article, two types of DGs are allocated in the proposed distribution network, which are solar PV type DGs (non-dispatchable because they do not regulate the connected bus voltage) and dispatchable because they can control the bus voltage where they are connected. The best DG allocation should be achieved without violating any constraints and these constraints are to be checked through load flow analysis at each iteration. Fundamental objective functions like line losses, voltage profile enhancement, and voltage stability index are examined. In subsequent sub-sections, the power flow technique, objective functions, constraints, and decision variables are described.

Power flow problem

Distribution networks are often structured in a radial small ratio of reactance to resistance. This makes traditional load flow methods like Gauss–Seidel and Newton Raphson⁵⁸ less suitable. Therefore, in this article, a direct approach-based forward–backward seep load flow technique is applied.

The suggested methodology involves the creation of three matrices, namely Bus Injection to Branch Current (BIBC), Branch Current to Bus Voltage (BCBV), and Distribution Load Flow (DLF). These matrices are utilized to determine the magnitude of the voltage and phase angle at load and PV buses. Figure 1 depicts a typical single-line diagram of a six-bus radial distribution network⁵⁹. As seen in Fig. 1, a backward sweep is applied to compute branch flow matrix B. Where B is the branch flow current, and it is calculated by using KCL at each bus as

4

Fig. 1.

Single line diagram of distribution network.

However, the voltage drop at each bus (forward sweep) with respect to reference bus is calculated by using Kirchhoff’s voltage law (KVL) as:

5

Substitute the value of matrix B from Eq. (4) in Eq. (5) we get:

6

where, DLF is the distribution load flow matrix, which is used to determine the voltage drop at each bus relative to the reference bus. The two simple steps shown below can be used to calculate the DLF matrix:

Step 1: The branch impedance vector can be derived from the provided branch data.

Step 2: The branch impedance vector can be transformed into a diagonal matrix, denoted as , where all the members above and below the main diagonal are set to zero. This diagonal matrix can then be multiplied with the matrix I to obtain the product ∆V.

Objective functions

In this paper three objective functions are formulated to find the decision variables of OFR and DG allocation considering multiperiod time slots. The selection of objective functions is based on economical, technical, and reliability points of view. Usually, maximum power loss in a power system appears across the distribution network which affects the annual revenue. Therefore, the reduction of real line losses across a distribution side is the primary objective considered in this study. After the load flow analysis power loss is calculated by using bus current injection⁶⁰.

7

where Bi and Ri are the branch current and resistance respectively, [R] is the branch resistance matrix containing the resistance of all the branches. The first objective function is based on total power loss, and it aims to minimize power loss over the entire time range using Eq. (8).

8

Nodal voltage magnitude is an important indicator to evaluate system security and power quality (PQ). The minimization of voltage deviation can help to guarantee better voltage levels in the power distribution systems. The second objective function is related to the voltage quality of the distribution network, which is achieved by the minimization of voltage deviation (VD) between bus voltage and reference bus voltage. It can be calculated as;

9

where, is load bus voltage and are the total number of load buses in the network. is the voltage at slack bus and it is set to 1 p.u in this paper. is the voltage at a specific bus. The security level of the distribution system is not sufficient only by considering the voltage deviation alone. Therefore, in order to achieve an improved voltage profile, it is recommended to include VSI as one of the objective functions in this study⁶¹. The maximum VSI of a distribution system refers to the bus’s capacity to maintain its voltage profile within an acceptable range under varying load situations. VSI of connecting buses and expressed as

10

where and are the active and reactive power injection at bus j, line resistance, and inductive reactance respectively linking between bus and . During the operation, it is necessary to boost VSI of a certain bus, which has the lowest value among all other buses, in order to improve the voltage level of the entire network. Hence, in order to optimize the VSI, the objective function VSI is defined as stated in reference⁶².

11

Constraints

The equality constraints refer to the power balancing equations, which ensure that the total active and reactive power generated within the network is equal to the combined load demand and losses in the network.

12

13

where, and are the active and reactive power supplied by the sub-station, and are the active and reactive demand, and is the voltage angle of the branch connected between bus and . are the transfer conductance and susceptance of the branch between bus and .

The inequality constraints are the operating limits of the equipment, components in the power system, and security constraints on the line and load buses.

14

15

16

17

18

19

20

21

22

Equation (14)-(15) shows the active and reactive limit of the main substation which is the reference bus in this case, Eq. (16) represents the active and reactive power constraints of dispatchable and non-dispatchable distributed generators, Eq. (17)-(18) are the cumulative power constraints of DGs which ensure that overall active and reactive power generated by DG must be less than or equal to total demand, Eq. (18)-(19) is for the voltage constraints of PV and PQ buses respectively, Eq. (20) is for the MVA branch flow limit. Equation (21)-(22) ensures the radiality constraints of the distribution system. Whereas, matrix A is the connection matrix also called the branch and bus incident matrix, in which shows that branch is linked with from bus j however shows that the ith branch is linked with to bus and ensures that the ith branch is not connected to the jth bus. All the equality constraints are satisfied during the forward–backward sweep load flow calculation. However, inequality constraints are handled using representative constrained techniques discussed in section III of this paper.

Decision vector

In this paper, the Decision vector is composed of continuous and integer variables. Mathematically, the vector is given as

whereas is the grid station’s active electricity generated continuously over the whole time range, the continuous active power generated by distributed generation (DG) throughout the designated time period is being considered in , represents the integer locations of all the DGs between 2 to buses in the entire time period, and continuous voltage levels of the substation and dispatchable DGs in the entire time period, shows the integer tie switches in loop L which also ensures the radiality of the distribution network. There are three solar PV-type DGs installed to inject variable active power at the 10, 21, and 28 buses. The maximum active power produced by each solar PV type DG is 0.8 MW and cumulative 2.4 MW. In this work, decision variable is composed of mixed integers in which , , and are the continuous variables whereas, and are the discrete variables. The most widely used distribution network, the IEEE 33-bus network, is selected to conduct this study. This network has one substation and five Loops () and three dispatchable DGs are integrated to inject active power, therefore, in the entire time range, the total length of the decision vector is 388. The topic at hand can be classified as a large-scale optimization problem. However, the technique proposed in this study effectively addresses this problem and successfully identifies a significant number of non-dominated solutions that are uniformly distributed.

Implementation of proposed hybrid algorithm

Evolutionary algorithms (EAs) are critical in dealing with complex optimization problems known as COPs. The success of the search algorithm depends on effectively managing variety, convergence, constraints, and objective functions during the evolutionary process. Evolutionary algorithms, including GA⁶³, Particle Swarm Optimization (PSO)⁶⁴, and Differential Evolution (DE)⁶⁵, have been used expansively to tackle the issues associated with Distributed Generation (DG) and Smart Grid (SC) allocation concerns. GA demonstrates proficiency in identifying optimal solutions, whilst PSO showcases faster convergence. Additionally, IMODE⁶⁶ achieves a harmonious equilibrium between exploration and variety. In order to tackle these issues, a novel approach is proposed in the form of a hybrid-constrained evolutionary algorithm. This approach integrates diverse constraint management techniques and includes operators from Genetic Algorithms (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO) to create vector strategies with distinctive and advantageous attributes. Figure 2 displays the flow diagram of the proposed algorithm.

Fig. 2.

Applied algorithm.

In this research, a parallel optimization strategy was implemented in which the three solvers, namely GA, DE, and PSO, worked concurrently on equally partitioned candidate solution populations. This approach allowed each solver to operate independently and simultaneously on its designated subset of solutions. Furthermore, as depicted in Fig. 2, the offspring generated by each solver were interconnected through the use of two representative constraint handling techniques: the Feasibility Rule (FR) and the Epsilon Constrain Method (ECM). These constraint-handling techniques were integrated into the proposed hybrid search method to effectively manage infeasible solutions.

Furthermore, while the solvers work independently, a global coordination mechanism was implemented to share information about the best solutions found by each solver. This coordination helps ensure that promising solutions identified by one solver can be explored by the others, facilitating global convergence. Figure 2 illustrates the parallel.

operation of the three solvers, showing how they evolve solutions in their respective portions of the solution space. The graphical representation was included for clarity and visualization of the parallelization process. The GA operator employs binary tournament selection as a means of determining the individuals that will be included in the mating pool. Following the genetic algorithm (GA) operations, a distinct subset encompassing one-third of the population, mentioned to as"particles,"is selected at random to undertake the particle swarm optimization (PSO) operator⁶⁶. The PSO algorithm is a stochastic method for global optimization that is based on the coordinated movement of bird flocks. It functions by qualifying the exchange of information among individuals, referred to as particles, within the population. In PSO, every particle undergoes stochastic adjustments to its.

trajectory, aiming to improve its performance based on two factors: its own previous best performance (pbest) and the best performance achieved by either its neighboring particles or the entire swarm (gbest). The equations governing the update of velocity and location at the (t + 1) iteration are formulated as follows.

23

24

The remaining one-third of the population members reproduce by utilizing tactics that are similar to those described in reference⁶⁶. The following three DE operators are employed to generate offspring, these are:

25

26

27

where and are randomly selected distinct target vectors, is the best target vector, and rand is a number [0, 1]. In the proposed formulation of DG and ONR, large number of constraints are handled during optimization as discussed in Eq. (12)-(13). Therefore, two representative constraint handling technique such as Feasibility rule (FR) and epsilon constrain method (ECM) are incorporated in the proposed hybrid search method in order to handle infeasible solutions. FR is a procedure that involves the randomly selection of two solution vectors say and , and assume that survive for the next generation if and only if the following two conditions are satisfied.

28

whereas CV is defined in Eq. (3). Only FR integration has the drawback that during selection process FR prominence to feasible solutions only, whereas high quality infeasible solutions are discarded. Therefore, in order to make trade0ff. between exploration and exploitation to emphasis high quality infeasible solutions at selection stage, another constraint handling technique such as epsilon constraint method (ECM) is employed. In ECM randomly select two solutions and and u survive for the next generation if:

29

30

31

whereas parameter , t is the current iteration and T is the maximum number of iterations. shows the initial threshold and is equal to the maximum CV. Exponent cp is computed as;

32

The parameter λ is assigned a value of 6, whereas the parameter p governs the degree of exploitation in the goal function. Furthermore, the use of a restarting technique is proposed as a means to aid the population in overcoming local optima that are located within the infeasible domain. This particular situation is frequently encountered in complex distribution grid (DG) allocation problems. The operator selection technique of this proposed algorithm is influenced by the IMODE⁶⁶ approach. In the following section, we apply the proposed algorithm to solve various mathematical benchmark functions and the modified IEEE 33 and 69-bus test networks. Additionally, the parameters for each algorithm have been selected based on their specifications in the original papers.

Simulation results and comparisons

Before applying the proposed algorithm to the modified IEEE 33-bus test network to solve the intended problem, it is desirable to validate the method by solving state-of-the-art mathematical constrained benchmark functions. In this paper, we use 12 benchmark functions from CEC-2017⁶⁷, as listed in Table 2 to optimize using the proposed methodology. The simulation results are then compared with those of state-of-the-art evolutionary algorithms to demonstrate the superiority and performance of the proposed hybrid algorithm. In Table 2, represents the main decision vector, while and are the shifting and rotating variables, respectively. Detailed information on these variables is provided in⁶⁷. For each benchmark function, each method is run independently 25 times. The best, worst, mean, standard deviation from these runs are selected for comparison among various evolutionary algorithms (EAs). Table 3 presents the comparison of these statistical parameters for the state-of-the-art EAs and the proposed method. Table 3 demonstrates that the proposed algorithm and IMODE outperform other EAs in minimizing and with 100% feasibility rates (FR). While PSO yields better results for minimizing , the overall performance of the proposed algorithm is superior compared to other EAs. None of the methods achieve the optimal solution for the CEC benchmark function. However, GA optimizes better in terms of worst-case, mean, and standard deviation values, while the proposed algorithm obtains the minimum value of the objective function. The best run is identified from the statistical results, and the comparison of the convergence curves of the best runs of the EAs is shown in Fig. 3. This figure illustrates the convergence curve comparisons for all twelve benchmark functions. The convergence curves clearly indicate that the proposed algorithm outperforms in most functions, though it fails to find a better optimal solution for , proposed algorithm fails to find the better optimal solution. For and the proposed algorithm achieves near-global optimal solutions with a 100% FR within 10,000 function evaluations. The simulation results in Table 3 clearly validate that the proposed algorithm can find near-global optimal solutions. Therefore, in the next subsection, the proposed algorithm is applied to solve large-scale (more than 200 decision variables) complex mixed-integer nonlinear DG allocation along with the optimal feeder reconfiguration problem.

Table 2.

Details of proposed mathematical benchmark functions.

Objective functions and Equality and Inequality Constraints	Type	D
;	Non-separable	10
;	Non-Separable, and Rotated	10
; ;	Non-Separable	10
;	Separable	25
; ;	Non-Separable	25
; ; ; ; ; ;	Separable	25
; ;	Separable	50
; ;	Separable	50
; where	Separable	50
; ;	Separable	100
; ;	Separable	100
; ;	Separable	100

Table 3.

Statistical results of state-of-the-art EAs through 25 independent runs of all the proposed benchmark functions.

function	Alg	GA63	PSO64	DE65	IMODE66	Proposed
	Best	0.12859	2.19E-25	2.22E-10	0	0
	Worst	0.675818	5.03E-12	6.98E-10	0	0
	Mean	0.402508	5.98E-13	3.61E-10	0	0
	STD	0.210803	1.57E-12	1.45E-10	0	0
	%FR	100	100	100	100	100
	Best	0.076443	2.44E-25	4.60E-10	0	0
	Worst	0.815984	9.73E-12	2.68E-09	0	0
	Mean	0.339467	1.39E-12	1.42E-09	0	0
	STD	0.265489	3.19E-12	8.67E-10	0	0
	%FR	100	100	99.9	100	100
	Best	11,106.7	380.66	5164.492	6660.103	7975.244
	Worst	411,576.8	7589.741	19,138.61	778,201.4	436,330.9
	Mean	94,866.48	2194.273	14,299.9	237,601.5	140,756.6
	STD	123,718.6	2184.794	3897.538	220,456.3	132,817.2
	%FR	90	11	0	10	90
	Best	13.57462	50.74261	248.7607	24.69309	13.57279
	Worst	13.68051	83.57606	293.648	47.40895	13.57279
	Mean	13.62264	63.27902	270.5668	33.09219	13.57279
	STD	0.034431	10.70776	14.92495	6.839402	2.37E-15
	%FR	100	100	100	100	100
	Best	0.034706	3.389605	5.40E-10	0	0
	Worst	6.739193	4.085362	6.83E-06	0	0
	Mean	5.291205	3.87859	7.87E-07	0	0
	STD	2.158817	0.226618	2.14E-06	0	0
	%FR	100	100	99.8	100	100
	Best	2484.236	240.7464	1634.275	1523.479	1613.051
	Worst	6031.597	367.9567	2927.834	1954.533	3896.891
	Mean	3658.203	301.8707	2286.525	1733.994	2628.884
	STD	1252.75	44.17834	497.8476	141.7224	724.352
	%FR	0	87.8	0	0	100
	Best	−147.237	−257.794	−426.547	−442.659	−263.186
	Worst	296.2267	287.0413	−320.027	154.561	73.09104
	Mean	5.656956	−15.6784	−372.397	−195.229	−70.9883
	STD	147.6926	171.1773	37.55813	179.381	123.1889
	%FR	80	1.3	0	27	90
	Best	4.489423	0.709155	0.709443	−0.00013	0.000117
	Worst	11.03104	2.939181	1.54796	−0.00013	0.000571
	Mean	8.392441	1.472646	1.17483	−0.00013	0.000311
	STD	1.723969	0.782013	0.258249	2.00E-07	0.000123
	%FR	0	0	0	100	100
	Best	0.014635	2.802473	6.61641	−0.00204	−0.00204
	Worst	18.22211	10.74783	9.231994	−0.00204	−0.00204
	Mean	4.682085	4.854125	8.171059	−0.00204	−0.00204
	STD	6.885091	2.628286	0.751585	2.81E-16	3.31E-08
	%FR	70	97.6	15	100	100
	Best	3.554166	0.054003	2.57434	8.21E-05	0.00075
	Worst	7.730253	0.429661	5.813388	0.000191	0.005072
	Mean	5.339893	0.167299	3.756313	0.000147	0.001976
	STD	1.436366	0.11649	0.922698	3.40E-05	0.001847
	%FR	0	0	0	70	100
	Best	−1106.06	−244.392	−2133.59	−190.611	−14.3836
	Worst	1294.241	389.9865	−321.534	172.1193	54.39232
	Mean	355.8247	−3.6592	−1014.87	−15.3742	32.36237
	STD	797.8293	198.5486	533.4628	112.4385	17.89627
	%FR	0	0	0	0	0
	Best	6.558192	9.997097	26.27628	9.995661	5.034115
	Worst	15.88731	31.58281	44.93823	31.57779	31.66285
	Mean	9.495143	26.87858	34.02718	15.69906	18.87137
	STD	3.900335	7.946334	8.15891	7.107889	9.902189
	%FR	100	81.7	93.5	100	100

Fig. 3.

Convergence curve of best run of all the algorithms to solve 12 mathematical benchmark function.

Whereas, , is the random numbers between usually shifting decision vector, is the rotation matrix⁶⁷.

Modelling of time-varying devices and study cases

In this article, multiperiod OFR along with the optimal DG allocation problem is optimized using a hybrid Evolutionary.

algorithm. Both dispatchable and non-dispatchable DGs are considered. Non-dispatchable solar PV type DGs are implemented to integrate active power only (operated at unity pf). However, dispatchable DGs locally inject reactive power.

that highly supports the voltage profile of the distribution system and reduces active power loss to some extent. The simulations are performed on the modified IEEE 33-bus system, as depicted in Fig. 4.

Fig. 4.

Base configuration proposed network.

The IEEE 33 bus refers to a radial distribution network operating at a voltage level of 12.66 kV. This network is characterized by the presence of five open switches, specifically identified by branch numbers 33, 34, 35, 36, and 37. The maximum and minimum active and reactive power generation associated with the installed dispatchable DGs are [0.19, 2] MW and [−1.9, 1. 9] MVAr respectively, taken from ⁶⁸.

Three Solar PVs with a total capacity of 2.4 MW are installed at buses 10, 21, and 28²⁸. The solar PV power generation is a function of random solar irradiances⁶⁸. In this research paper, a simultaneous solution is provided to the optimization problem of network reconfiguration and optimal allocation of both dispatchable and non-dispatchable DGs. The time-varying nature of the load is taken into consideration to recognize that the optimization problem in existing literature typically assumes a constant power generation model for renewable DGs. However, non-dispatchable DGs, such as solar PV systems, are subject to intermittency in real-world scenarios, influenced by environmental and geographical factors. The aim of this study is to improve the precision and authenticity of network reconfiguration by taking into account the variations in both load and distributed generation (DG) output over time. This will enable us to determine the ideal network design and appropriate sizes for DG systems. The analysis focuses on the illustrative scenario of changes in hourly load variance throughout a typical summer day. The actual active and reactive demand and hourly power generation of all the solar PV units are illustrated in Fig. 5. The load demand’s variation is depicted in Fig. 5 with a fixed load for each time period set equal to the maximum load during that period. For each period’s outset, the optimal configuration for the subsequent period is determined. The scheduled DGs’ cost exceeds that of power supplied by the grid station. This study assumes small-scale solar PV DGs, or non-dispatchable DGs, ranging from 0 to 1MW each, connected to the system at medium voltage. DG profiles, mirroring load profiles, span 24 h. Solar PV DG power.

Fig. 5.

Time-Varying active and reactive load demand, MW Output power of cumulative solar PV.

production hinges on meteorological factors like solar irradiance and temperature forecasts. These meteorological predictions enable the network manager to gauge the renewable DGs’ power output fluctuations within the designated study period, aiding the pursuit of an optimal solution. Fluctuations in solar PV DG power output, coupled with load variations, directly impact the power system’s load flow.

This dynamic load flow nature significantly influences the search process to pinpoint network configuration, DG placements, and sizes that optimize energy loss reduction,

Voltage Deviation c(VD) minimization, and VSI enhancement. In this study, the power production of renewable DGs, specifically the solar generator that relies on photo-voltaic modules, is determined by the utilization of a specialized power generating function, which is specified as follows:

33

where, is the solar irradiance in a standard environment set as 800 is a certain irradiance point set as 120 is the rated output power of the solar unit. Given that distribution network buses are situated within a limited geographic area, it can be inferred that the solar photovoltaic (PV) systems deployed at various buses in the investigation are subject to identical solar irradiances. The increasing incorporation of renewable-based DGs in recent.

years has resulted in compromised voltage stability within the network. Voltage stability refers to the stability of the distribution system to maintain an acceptable voltage level at each bus of the system under various operating conditions. The voltage stability of the distribution network is highly deteriorated by a high level of Solar PV integration. During periods of high solar irradiance and low load demand, solar PV systems can generate more power than the local demand. This excess generation can lead to overvoltage conditions,

potentially causing voltage instability and damaging customer equipment. The intermittent nature of solar PV generation due to cloud cover or time-of-day variations can lead to voltage fluctuations in the distribution system. Rapid changes in voltage can impact the stability of the system and affect sensitive equipment. Solar PV systems can inject power into the distribution system, causing reverse power flow in feeders. This can result in voltage regulation challenges and require the adjustment of control strategies to maintain stable operation. Solar PV systems do not contribute to system inertia in the same way that conventional rotating generators do. This reduction in system inertia can lead to faster voltage instability responses during disturbances. Implementing microgrids with solar PV and local energy resources can enhance control over voltage stability within specific areas of the distribution system. Therefore, in this work, dispatchable conventional thermal-generator-based DGs are designed to be integrated into the proposed study network; these DGs have the capability to support the voltage profile by injecting reactive power into the system at the time of heavy loading periods. The proposed formulated problem is solved using the hybrid multi-operator evolutionary algorithm and considered to minimize three conflicting objective functions as given in Eq. (8), (9), and (11) which are total energy loss, voltage deviation, and voltage stability index in the entire time range. Four cases of single and multi-objective are considered to test the proposed method for finding the realistic multiperiod optimal network reconfiguration and DG allocation problem.

Case 1: minimization of active power loss

Case 2: minimization of VD

Case 3: minimization of VSI

Case 4: minimization of power loss, VD and VSI

During the optimization period, the position of dispatchable DGs can computed, however, it is not practical to change the location of DGs under different periods. Therefore, the proposed problem formulation and all the cases are run by considering the fixed site of all three dispatchable DGs at 13, 14, and 28 buses. From a computational perspective, the simulations were executed on a PC with Intel Core i7 CPU @3.20 GHz and 8 GB RAM. The proposed model was solved using hybrid EA in the MATLAB platform.

Comparison of state of art ea with the proposed algorithm of 33-bus network

Over the past two decades, EAs have been increasingly important in addressing the challenges associated with determining optimal sites and sizes for Distributed Generation (DG) allocation, which are subject to constraints. In the proposed problem formulation, each time period’s DG, and OFR problems are combined to create a unified master problem, thereby increasing the dimensionality of decision variables. Within a single time period, this encompasses three dimensions for DG sizing, four dimensions for voltage levels at each generator’s bus, and variables pertaining to the OFR. As such, a single time period comprises twelve decision variables, while the entire time range entails a total of 288 decision variables. The performance of DG allocation and optimal network reconfiguration within a multiperiod problem are deeply intertwined. Relying solely on individual optimal decision variables, considering power loss, VD, and VSI as objective functions, fails to yield satisfactory results. Consequently, a holistic approach involving all time slots becomes pivotal, yielding superior outcomes. However, due to the substantial number of decision variables, the performance of optimization algorithms significantly deteriorates.

To address this challenge, a multi-operator EA is employed in this study to effectively solve the extensive DG and OFR problem on a large scale. The convergence curve for all cases, as depicted in Fig. 6, illustrates the comparison between the proposed approach and state-of-the-art EAs. Nevertheless, these algorithms tend to become trapped in locally optimal solutions. The rate of convergence exhibited by IMODE is very sluggish, although it effectively examines the entirety of the solution space. However, in all of the specified study instances, the hybrid algorithm and genetic algorithm demonstrate superior performance. The Genetic Algorithm (GA) demonstrates superior convergence in comparison to other advanced EAs when used to the optimization of the optimal site and size of Distributed Generation (DG) allocation and OFR problem. In scenarios 1 and 4, the suggested algorithms demonstrate a marginal improvement in the objective function values when compared to the Ga and IMODE algorithms. Whereas, in cases 2 and 3 the proposed algorithm finds the best solution and hence objective function values compared to GA⁶³ and IMODE⁶⁶. To achieve this objective, this study presents a novel approach involving a hybrid constrained evolutionary algorithm that incorporates key constraint management techniques, effectively addressing the DG allocation and OFR challenges. A combination of operators GA⁶³, PSO⁶⁴, and DE⁶⁵ is harnessed to generate off-spring with enhanced exploration and exploitation abilities, catering to the solution requirements of the intricate and large-scale constraint optimization problem (COP) proposed in this research. In recent years, numerous strategies combining multiple methods and operators have been proposed to address optimization issues. Generally, their performance surpasses that of single operator or single-algorithm-based approaches.

Fig. 6.

Comparison of convergence curve of all the cases of the entire time range.

Nevertheless, their effectiveness can vary across different problems examined in the existing literature⁶⁶. Hence, the suggested approach employs a composite of three operators, namely GA, DE, and PSO. The aforementioned operators are utilized to produce three distinct vector techniques, each holding unique advantages. The utilization of crossover operators in GAs is employed to enhance the convergence rate in a more elaborate manner. In contrast, DE utilizes two distinct techniques for generating trail vectors in order to strike a balanced equilibrium between variation and convergence. In addition, the velocity, local best, and glob-al best components of PSO operators are impacted by the inputs derived from GA and DE operators with the aim of improving the rate at which convergence occurs. During the course of evolution, it has been observed that these three operators generate three offspring for each input vector. Afterward, the feasibility rule is employed to choose Np offspring from a pool including 2Np individuals, with a focus on convergence. The selected NP offspring are further subjected to comparison with the parents using the Evolutionary Computation Method (ECM) in order to achieve a harmonious equilibrium between convergence and exploration. This technique ultimately identifies the ultimate members of the NP population for subsequent iterations. The Fitness Ranking (FR) is employed in this particular situation to selectively identify the top Np offspring generated by different variation operators. Furthermore, the Elitist Crowding Mechanism (ECM) is utilized to choose the most optimal solution by a comparison between the preselected solutions and the parent solutions.

Table 4 presents the decision variables derived from the conclusive results of the OFR for each study scenario across all time periods. The OFR decision variable shown in Table 4 reveals the switches most frequently activated under different scenarios. In Case 1, which aims to minimize active power loss, switches 6, 8, 9, 13, and 31 are most frequently operated. In Case 2, where the focus is on minimizing overall voltage deviation, switches 5, 9, 10, 13, and 32 emerge as the most frequently utilized. For Case 3, spanning all time periods, switches 7, 13, 17, 24, and 31 are most frequently operated. In Case 4, where the objective functions encompass power loss, Voltage Deviation (VD), and VSI optimizations, the switches most commonly operated are 6, 8, 13, 24, and 31. These findings illustrate the specific switches that play a crucial role under various optimization conditions Table 5 displays the voltage set levels of all DGs and the substation throughout the optimization process for all cases.

Table 4.

Simulation results of OFR decision variables of all the study cases.

Time	OFR Case 1	OFR Case 2	OFR Case 3	OFR Case 4
1	6, 9, 13, 27, 31	4, 9, 13, 27, 31	4, 9, 13, 28, 31	6, 9, 13, 27, 32
2	7, 9, 13, 27, 31	9, 13, 17, 18, 28	5, 11, 12, 23, 31	7, 9, 13, 27, 31
3	6, 9, 13, 24, 31	5, 9, 13, 24, 32	14, 15, 20, 24, 33	7, 9, 13, 25, 31
4	6, 13, 21, 26, 31	7, 13, 26, 32, 33	7, 9, 13, 28, 31	13, 20, 25, 31, 33
5	6, 9, 13, 27, 32	5, 9, 13, 24, 32	2, 8, 26, 29, 34	5, 9, 13, 24, 31
6	7, 9, 13, 17, 23	9, 13, 20, 23, 30	6, 13, 17, 28, 35	6, 9, 13, 17, 24
7	5, 10, 13, 24, 36	5, 11, 13, 23, 31	4, 10, 24, 34, 36	5, 10, 13, 24, 31
8	8, 13, 19, 25, 32	6, 10, 13, 26, 32	9, 13, 20, 24, 32	8, 13, 19, 28, 32
9	7, 8, 12, 24, 31	5, 10, 12, 24, 30	7, 13, 21, 28, 31	7, 8, 12, 24, 31
10	6, 8, 25, 30, 34	7, 8, 27, 32, 34	7, 10, 12, 23, 32	7, 8, 25, 30, 34
11	6, 10, 13, 17, 26	3, 10, 13, 17, 25	5, 10, 13, 27, 31	6, 10, 13, 17, 26
12	8, 13, 19, 23, 31	10, 13, 18, 25, 31	7, 14, 17, 27, 33	8, 13, 19, 23, 31
13	5, 10, 13, 23, 32	3, 10, 13, 25, 31	5, 8, 25, 30, 34	6, 10, 13, 24, 32
14	6, 13, 21, 25, 30	4, 10, 13, 27, 30	5, 11, 13, 17, 27	5, 10, 13, 25, 30
15	8, 20, 26, 32, 34	8, 14, 18, 27, 32	11, 13, 17, 19, 24	8, 19, 25, 32, 34
16	6, 13, 21, 25, 30	6, 11, 13, 27, 31	6, 13, 17, 21, 24	6, 13, 21, 26, 30
17	6, 11, 12, 25, 30	5, 11, 12, 27, 30	7, 10, 12, 24, 32	6, 11, 12, 25, 30
18	6, 21, 26, 30, 34	4, 10, 28, 32, 34	6, 8, 13, 24, 31	6, 21, 26, 30, 34
19	6, 10, 13, 17, 28	11, 13, 17, 18, 28	5, 8, 13, 27, 30	13, 17, 19, 28, 33
20	2, 13, 27, 32, 35	2, 13, 28, 32, 35	10, 13, 20, 28, 30	2, 13, 28, 32, 35
21	5, 8, 17, 26, 34	5, 8, 17, 26, 34	6, 13, 17, 21, 24	5, 8, 17, 26, 34
22	7, 8, 26, 31, 34	7, 8, 13, 28, 31	20, 28, 31, 34, 35	7, 8, 13, 24, 31
23	9, 13, 20, 24, 31	11, 13, 20, 24, 30	7, 8, 13, 24, 31	9, 13, 20, 24, 31
24	8, 20, 25, 32, 34	9, 14, 18, 25, 36	20, 23, 31, 33, 34	8, 20, 24, 32, 34

Table 5.

Voltage set point of all the dispatchable DGs and substation of all the cases where dispatchable DGs are integration at buses 13, 14, and 25.

Time	case 1		case 2		case 3		case 4
1	1.046	1.050, 1.049, 1.044	1.010	1.011, 1.021, 1.005	1.098	1.099, 1.099, 1.083	1.054	1.056, 1.051, 1.053
2	1.044	1.046, 1.040, 1.040	1.025	1.007, 0.999, 1.010	1.100	0.986, 0.994, 1.042	1.045	1.048, 1.041, 1.044
3	1.091	1.089, 1.092, 1.096	1.004	1.005, 1.007, 1.010	1.094	1.088, 1.082, 1.100	1.092	1.091, 1.097, 1.091
4	1.016	1.012, 1.014, 1.019	1.004	1.003, 1.005, 1.007	1.099	1.100, 1.100, 1.098	1.010	1.002, 1.006, 1.013
5	1.059	1.062, 1.063, 1.058	1.003	1.005, 1.006, 1.008	1.055	1.009, 1.007, 1.044	1.039	1.040, 1.042, 1.046
6	0.997	1.003, 1.004, 0.993	1.001	1.006, 1.016, 1.004	1.095	1.100, 1.093, 1.100	1.009	1.014, 1.015, 1.004
7	1.042	1.043, 1.038, 1.043	1.003	1.006, 1.011, 1.008	1.054	0.973, 0.971, 0.981	1.037	1.045, 1.033, 1.039
8	1.022	1.040, 1.038, 1.021	1.004	1.004, 1.008, 1.006	1.100	1.100, 1.100, 1.100	1.006	1.031, 1.027, 1.002
9	1.004	0.978, 0.977, 1.014	1.010	0.990, 0.988, 1.020	1.100	1.100, 1.100, 1.090	1.011	0.988, 0.988, 1.014
10	1.044	1.013, 1.009, 1.046	1.013	0.998, 1.002, 1.009	1.100	1.040, 1.047, 1.059	1.044	1.004, 0.999, 1.043
11	1.079	1.084, 1.081, 1.069	1.034	0.999, 1.004, 1.012	1.020	1.050, 1.046, 1.023	1.075	1.078, 1.076, 1.063
12	1.079	1.084, 1.075, 1.076	1.003	1.008, 1.018, 1.005	1.079	1.078, 1.071, 1.051	1.071	1.077, 1.068, 1.067
13	1.009	1.010, 1.004, 0.987	1.006	1.007, 1.021, 0.996	1.092	1.097, 1.100, 1.073	1.011	1.015, 1.011, 1.014
14	1.028	1.044, 1.040, 1.025	1.001	1.012, 1.029, 1.000	1.027	1.044, 1.074, 1.020	1.034	1.041, 1.041, 1.029
15	0.958	1.029, 1.028, 0.951	1.009	1.041, 1.052, 1.005	1.100	1.095, 1.099, 1.090	0.961	1.037, 1.037, 0.950
16	1.016	1.017, 1.013, 1.010	1.008	1.000, 1.020, 1.005	1.100	1.074, 1.070, 1.086	1.010	1.010, 1.003, 1.006
17	0.999	1.017, 1.014, 0.988	1.012	1.042, 1.034, 0.997	1.100	1.014, 1.020, 1.046	0.998	1.023, 1.019, 0.985
18	1.068	1.033, 1.027, 1.053	1.056	1.004, 1.004, 1.002	1.100	1.095, 1.093, 1.071	1.082	1.040, 1.035, 1.066
19	1.024	1.024, 1.029, 0.990	1.029	1.020, 1.006, 0.991	1.031	1.080, 1.075, 1.001	1.030	1.045, 1.056, 0.997
20	1.082	1.058, 1.065, 0.954	1.074	1.050, 1.071, 0.954	1.050	1.099, 1.064, 1.027	1.077	1.050, 1.065, 0.952
21	1.043	0.996, 0.995, 1.007	1.044	1.007, 1.003, 1.007	1.100	1.054, 1.042, 1.042	1.043	1.004, 1.001, 1.006
22	0.998	0.990, 0.988, 0.982	1.023	0.997, 1.022, 0.995	1.056	1.018, 1.023, 1.010	1.000	1.011, 0.999, 0.998
23	0.992	0.997, 0.998, 0.989	1.010	1.008, 1.029, 1.008	1.100	1.094, 1.100, 1.080	0.998	1.001, 1.006, 0.994
24	1.045	1.015, 1.013, 1.031	1.052	1.001, 0.992, 1.011	1.100	1.078, 1.081, 1.068	1.042	1.019, 1.018, 1.037

Table 5 provides a clear depiction of voltage set points over the entire time span. Among the cases, Case 2 stands out for showcasing the most optimal voltage set points, particularly when Voltage Deviation (VD) serves as the objective function. However, in Case 3, with the VSI as the objective function, the voltage set point is positioned close to its upper limit, regardless of load demand. In a comparative analysis of Case 2 and Case 3, Case 2 boasts significantly enhanced voltage quality. Conversely, an increase in load demand in Case 3 leads to voltage collapse at critical buses. In this context, Case 3, emphasizing VSI, raises voltage levels at the substation and generator buses, bolstering the voltage collapse threshold at buses with minimal voltage levels. This proactive approach enhances overall power system stability. Meanwhile, Case 1 features voltage set points for DGs, and substations situated between those of Case 2 and Case 3. To visually represent these dynamics, Fig. 7 presents both the substation voltage set point and DG voltage levels across all cases. Given the complexity of reaching conclusions solely from voltage set level simulations, Fig. 7 employs a line graph to depict substation voltage set points, along with a statistical box plot that offers a comprehensive overview of DG voltage set points for all cases. This presentation approach ensures clarity and facilitates the analysis of these intricate relationships.

Fig. 7.

Voltage set point of (a) Sub -station and (b) all the DGs in the entire time period.

Figure 7 (a) and (b) provide clear insights into the impact of different objective functions on the optimization process. When power loss is the objective function (Case 1), the optimization focuses on reducing losses, which intuitively enhances the voltage level at the substation. On the other hand, when voltage deviation is prioritized (Case 2), the optimization aims to bring the voltage set points closer to unity, resulting in improved voltage quality across the network. In contrast, when the VSI is set as the objective function (Case 3), it adjusts the voltage levels of all generators and the substation to a higher value, such as near 1.1 per unit (p.u.) in this instance. However, a critical observation emerges when considering VSI as an individual objective function (Case 3). This approach inadvertently leads to elevated voltage levels at the generator buses, consequently causing an increase in voltage at the load buses. This phenomenon significantly impacts voltage-sensitive loads. This underscores that individual technical objective functions might have unintended consequences due to their focus on particular solutions that may not align with other technical aspects. For instance, emphasizing VSI can lead to higher voltages in cases involving voltage-sensitive load models. To mitigate these challenges and arrive at more balanced solutions, Case 4 adopts a comprehensive approach by simultaneously optimizing all three objective functions. Figure 7 (a) and (b) in the graph make it evident that Case 4 yields high-quality solutions that equally emphasize all technical objectives. Furthermore, the active power generation capacity of DGs across all cases is presented in Table 6.

Table 6.

Active power integration of all the DGs in all cases at buses 13, 14, and 25.

Time	Case 1	Case 2	Case 3	Case 3
1	0.522 + 1.236 + 1.666 = 3.424	1.816 + 0.755 + 1.159 = 3.730	1.386 + 1.451 + 0.969 = 3.806	0.540 + 0.881 + 1.970 = 3.391
2	0.404 + 0.746 + 1.208 = 2.357	0.802 + 0.198 + 0.500 = 1.500	0.240 + 0.479 + 1.187 = 1.906	0.437 + 0.742 + 1.315 = 2.494
3	0.270, 0.846, 1.091 = 2.207	1.094 + 0.930 + 0.910 = 2.934	0.297 + 0.263 + 1.313 = 1.873	0.253 + 0.795 + 1.229 = 2.276
4	0.296 + 0.740 + 1.302 = 2.338	0.263 + 0.846 + 1.302 = 2.410	0.576 + 0.824 + 0.750 = 2.150	0.233 + 0.750 + 1.420 = 2.403
5	0.358 + 0.607 + 1.203 = 2.168	0.671 + 0.612 + 1.316 = 2.599	1.698 + 0.253 + 0.908 = 2.859	0.268 + 0.697 + 1.087 = 2.052
6	0.285 + 0.492 + 1.449 = 2.226	1.137 + 0.736 + 1.097 = 2.971	0.224 + 0.327 + 1.311 = 1.862	0.332 + 0.584 + 1.072 = 1.987
7	0.318 + 0.419 + 1.380 = 2.116	1.083 + 0.196 + 0.979 = 2.257	0.632 + 0.193 + 1.355 = 2.180	0.403 + 0.650 + 1.144 = 2.197
8	0.520 + 0.437 + 1.541 = 2.497	0.743 + 0.194 + 1.452 = 2.389	0.714 + 0.640 + 1.417 = 2.771	0.524 + 0.455 + 1.568 = 2.546
9	0.263 + 0.419 + 1.490 = 2.172	0.304 + 0.269 + 1.342 = 1.915	0.254 + 1.161 + 1.700 = 3.115	0.294 + 0.435 + 1.406 = 2.135
10	0.384 + 0.543 + 1.554 = 2.481	0.746 + 0.190 + 1.752 = 2.688	0.374 + 0.191 + 1.683 = 2.248	0.430 + 0.304 + 1.473 = 2.207
11	0.337 + 0.347 + 1.633 = 2.316	1.624 + 0.190 + 0.970 = 2.785	1.349 + 0.483 + 1.454 = 3.285	0.296 + 0.399 + 1.546 = 2.241
12	0.241 + 0.661 + 1.727 = 2.629	1.487 + 0.287 + 1.672 = 3.445	1.950 + 0.305 + 0.611 = 2.865	0.374 + 0.526 + 1.690 = 2.591
13	0.387 + 0.633 + 1.823 = 2.843	1.787 + 0.664 + 0.987 = 3.438	0.372 + 1.816 + 0.605 = 2.793	0.280 + 0.533 + 1.586 = 2.399
14	0.440 + 1.367 + 1.296 = 3.102	1.293 + 1.696 + 1.069 = 4.058	0.768 + 0.891 + 1.920 = 3.580	0.404 + 1.353 + 1.187 = 2.944
15	1.016 + 0.534 + 1.845 = 3.394	1.175 + 1.590 + 2.000 = 4.765	0.934 + 1.133 + 1.809 = 3.876	1.153 + 0.505 + 1.785 = 3.443
16	0.505 + 1.345 + 1.468 = 3.318	0.392 + 0.696 + 1.811 = 2.899	0.502 + 0.483 + 1.869 = 2.854	0.573 + 1.267 + 1.489 = 3.329
17	1.106 + 1.080 + 1.656 = 3.842	0.934 + 0.716 + 1.021 = 2.671	0.540 + 0.346 + 0.601 = 1.487	1.071 + 1.103 + 1.544 = 3.717
18	1.590 + 0.451 + 1.589 = 3.630	1.770 + 0.219 + 0.522 = 2.511	1.587 + 0.782 + 0.965 = 3.334	1.393 + 0.615 + 1.595 = 3.603
19	0.566 + 1.205 + 2.000 = 3.771	1.328 + 1.266 + 1.888 = 4.482	1.481 + 1.385 + 0.717 = 3.582	0.959 + 1.572 + 1.993 = 4.524
20	1.324 + 2.000 + 2.000 = 5.323	0.190 + 1.821 + 1.986 = 3.997	1.995 + 1.901 + 1.114 = 5.009	1.210 + 2.000 + 2.000 = 5.210
21	0.478 + 0.491 + 1.992 = 2.961	0.579 + 0.302 + 1.969 = 2.851	1.739 + 1.462 + 1.897 = 5.097	0.587 + 0.470 + 1.996 = 3.052
22	1.307 + 0.552 + 1.992 = 3.851	0.296 + 1.135 + 1.073 = 2.504	1.261 + 1.286 + 0.266 = 2.814	1.043 + 1.074 + 1.998 = 4.115
23	0.729 + 1.549 + 1.702 = 3.981	1.293 + 1.640 + 1.134 = 4.067	1.374 + 1.379 + 1.205 = 3.958	0.698 + 1.661 + 1.719 = 4.078
24	0.839 + 0.327 + 1.999 = 3.165	0.659 + 0.194 + 0.680 = 1.533	0.550 + 0.653 + 1.377 = 2.580	0.917 + 0.337 + 2.000 = 3.255
Total (MW)	72.115	71.399	71.886	72.189

This Table 6 provides insights into the optimal generated power output from each DG concerning the load demand. However, when considering the entire time range, the cumulative power supply from all DGs remains consistent across all cases, totaling 72.115 MW, 71.399 MW, 71.886 MW, and 72.189 MW respectively. Conversely, notable disparities emerge in cumulative reactive power generation: 58.437 MVAr in Case 1, 55.23 MVAr in Case 2, 37.42 MVAr in Case 3, and 58.22 MVAr in Case 4 across the entire period. These outcomes lead to the inference that, under constant PQ load demand (as assumed in this study), higher voltage set points, such as in Case 3, yield lower reactive power generation. The minimum voltage set level is observed in Case 2, correlating with the least relative power production. Moreover, elevating the voltage set level of the substation or DGs subsequently reduces the reactive power reserve capacity. Notably, lower reactive power generation corresponds to the appearance of a voltage collapse point at higher load levels. In Fig. 8, x-axis comprises a total of 792 buses (33 * 24) load buses. The visualization clearly demonstrates that Case 2 exhibits superior voltage quality, while Case 3 significantly increases the VSI and thus reactive power reserve capacity. Moreover, this figure shows that load bus voltage in the entire time horizon with in the desirable limit. Cases 1 and 4 exhibit moderate voltage levels compared to Cases 2 and 3. Figure 8 also underscores that the proposed algorithm, combined with the integration of the Feasibility Rule (FR) and ε Constrained Method (ECM), produces feasible solutions, maintaining voltage levels within desirable limits. Furthermore, Table 7 shows the comparison of all the objective functions in all the cases in the entire time range.

Fig. 8.

Voltage profile of all the buses of the entire time range.

Table 7.

Simulation results of case 4, OFR, and dispatchable DGs are integrated at buses 13, 14, and 25.

Time	Case 1			Case 2			Case 3			Case 4
	(kW)	VD	min (VSI) (bus)	(kW)	VD	min (VSI) (bus)	(kW)	VD	(kW)	Loss (kW)	VD	min (VSI) (bus)
1	33.1	1.102	1.115 (33)	187.3	0.022	0.914 (13)	92.6	5.199	1.333 (32)	30.6	1.534	1.161 (33)
2	20	0.955	1.100 (33)	77.5	0.045	0.972 (13)	210.6	2.129	0.939 (33)	19.5	1.04	1.107 (33)
3	16.6	5.359	1.348 (33)	76.8	0.003	0.945 (13)	36.7	5.378	1.364 (22)	13.4	5.592	1.376 (33)
4	14.6	0.093	1.000 (33)	26.1	0.003	0.965 (15)	25.9	6.317	1.396 (33)	16.1	0.029	0.968 (33)
5	11.9	2.26	1.213 (33)	33.4	0.002	0.973 (13)	192	0.719	0.874 (31)	15.1	0.935	1.119 (33)
6	16.3	0.071	0.921 (33)	84.8	0.006	0.945 (13)	24	6.109	1.399 (33)	19	0.036	0.965 (33)
7	19.9	0.905	1.124 (33)	142.3	0.005	0.948 (13)	260.1	0.63	0.869 (33)	21.3	0.692	1.071 (33)
8	25.4	0.404	1.030 (33)	54.4	0.005	0.966 (13)	68.3	6.47	1.400 (33)	28.7	0.137	0.953 (33)
9	47.5	0.256	0.835 (33)	67.1	0.217	0.843 (32)	42.9	6.109	1.358 (33)	44.2	0.13	0.872 (33)
10	63.7	0.694	0.902 (32)	65.3	0.02	0.965 (33)	141.3	3.55	1.172 (33)	71	0.71	0.864 (32)
11	39.8	3.619	1.202 (33)	286.5	0.115	0.883 (13)	125.8	0.472	1.054 (32)	41.7	3.133	1.174 (33)
12	36.7	3.657	1.231 (33)	195.5	0.013	0.928 (13)	310.9	2.647	1.119 (33)	39.8	2.928	1.195 (33)
13	47.1	0.107	0.887 (33)	220.9	0.036	0.905 (13)	106.9	4.73	1.293 (32)	46.6	0.033	0.985 (33)
14	54.5	0.475	1.015 (32)	133.4	0.034	0.930 (20)	90.7	0.962	0.984 (33)	54.1	0.554	1.018 (32)
15	104.4	1.241	0.749 (33)	351.9	0.068	0.940 (8)	170.1	5.643	1.295 (33)	113.7	1.275	0.746 (33)
16	63.4	0.106	0.897 (32)	114.6	0.018	0.957 (33)	83.5	4.44	1.272 (33)	65.2	0.141	0.860 (32)
17	90.4	0.18	0.879 (32)	180.7	0.048	0.951 (32)	268.3	2.503	1.048 (33)	93.9	0.174	0.895 (32)
18	109.8	1.274	0.930 (32)	325.3	0.32	0.915 (33)	152.7	4.513	1.260 (32)	112.5	1.986	0.962 (32)
19	128.1	0.319	0.838 (33)	216.4	0.221	0.844 (33)	210.7	0.904	0.969 (31)	140.7	0.578	0.866 (33)
20	319.9	1.588	0.743 (33)	538.9	1.415	0.745 (33)	180.8	1.417	1.075 (31)	317.7	1.417	0.740 (33)
21	190.6	0.273	0.900 (33)	186.7	0.26	0.899 (33)	520.3	2.394	1.069 (8)	182.8	0.259	0.896 (33)
22	82.6	0.398	0.844 (33)	153.5	0.055	0.929 (32)	259.1	0.378	0.978 (33)	85.2	0.151	0.887 (33)
23	68.1	0.284	0.882 (33)	150.6	0.028	0.944 (13)	135.5	5.152	1.310 (32)	68.3	0.136	0.912 (33)
24	63.5	0.381	1.011 (33)	198	0.191	0.950 (33)	117	3.976	1.257 (33)	61.8	0.4	1.030 (33)
Total	1667.9	26		4067.9	3.15		3826.7	82.74		1702.9	24

Table 7 illustrates that the lowest power loss is observed in Case 1, amounting to 1667.99 kW, which signifies an approximate 87% reduction compared to the base case condition. In the base case, the total power loss throughout the entire time period is 12,788.9 kW. Across all cases, power loss experiences reductions of around 68% and 70% in Case 2 and Case 3 respectively. Notably, the lowest VSI values are consistently present at buses 31, 32, and 33 in most cases. However, in Case 3, bus 8 reaches a minimum voltage level with an increase in network load demand, potentially leading to voltage collapse. For improved clarity, comparisons of power loss, Voltage Deviation (VD), and VSI in each case relative to the base case are depicted in Fig. 9 (a), (b), (c) across the entire time range.

Fig. 9.

Comparison of (a) Power loss, (b) VD, and (c) VSI of the entire time range of all the study cases.

In Fig. 9 (a), the comparison between the power loss bar in Case 1 and the base case clearly illustrates a substantial reduction. Similarly, in Cases 2 and 3, their respective objective functions, namely Voltage Deviation (VD) and VSI, exhibit significant improvement. However, it’s important to note that in Case 1, during periods 5 and 6, power loss is slightly higher than the base load condition. This anomaly is attributed to the minimum load during these periods, where the VSI objective function emphasizes raising the voltage set point of the substation and PV buses, consequently leading to an increased power loss. Figure 9 (b) presents the distribution of VD across all cases throughout the entire time period. Notably, during periods of low load, Cases 1, 3, and 4 result in elevated VD compared to the base case. This phenomenon can be attributed to the fact that during low load conditions, the power minimization and VSI maximization objectives result in solutions that prioritize increasing voltage levels beyond unity. Figure 9 (c) highlights the distribution of the VSI index, with Case 3 exhibiting the most favorable performance compared to all other cases.

The higher voltage set level in Case 3 ultimately leads to an increase in the voltage collapse threshold. These figures are derived by considering the minimum voltage and bus location from the Table 7, gradually increasing the total load from 0 to 700%. Figure 9 shows the voltage collapse level of all the study cases compared to the base case condition in the entire time period. Figure 9 is designed to show how various study cases affect the voltage collapse level in a power distribution network. The comparison is made against a ‘base case,’ which likely represents the network’s standard or initial operating condition without any modifications or interventions. In this figure x-axis represents the total MVA (Mega Volt Ampere) load on the distribution network, this is a measure of the power demand on the network. Whereas y-axis shows the voltage at the bus (a node in the power distribution network) that has the minimum Voltage Stability Index (VSI). The VSI is a metric used to assess the voltage stability at a bus; a lower VSI indicates a higher risk of voltage instability or collapse. In Case 3, the primary objective is to maximize the VSI, which translates to improving voltage stability. The paragraph highlights that in this case, the point of voltage collapse—which is the point where the network can no longer maintain a stable voltage—is significantly higher compared to other cases. Specifically, it can sustain up to 700% of the network’s load demand, which is a substantial increase. This significant improvement in voltage stability in Case 3 is achieved by incrementally increasing the load demand at all buses by 1%. This methodical increase helps in assessing the robustness of the network under rising load conditions. The paragraph contrasts Case 3 with other study cases, where the maximal load demand increases before reaching a voltage collapse does not exceed 600%. This comparison is used to underscore the effectiveness of the strategies employed in Case 3. Therefore, Case 3 emphasizes a critical observation from the figure – as the network’s loading capacity is pushed higher, the bus with the lowest voltage (and hence the lowest VSI) experiences a significant improvement in its ability to withstand higher loads without collapsing. This is an indicator of enhanced network robustness and stability, especially in Case 3. Figure 10 clearly illustrate the impact of various strategies (especially in Case 3) on improving the voltage stability of a power distribution network, highlighting the network’s enhanced ability to handle increased load demands without succumbing to voltage collapse.

Fig. 10.

Comparison of Voltage collapse point (a) base case vs case 1, (b) base case Vs case 2 (c) base Case vs Case 3 (d) base Case vs Case 4 of the entire time range.

Comparison of state of art ea with the proposed algorithm of 69-bus network

The convergence curve for all cases, as depicted in Fig. 11. Nevertheless, proposed algorithms finds the faster convergence in most of Cases.

Fig. 11.

Comparison of convergence curve of all the cases of the entire time range.

In this figure left subplot represents Case 1 and Case 2, while the right subplot illustrates Case 3 and Case 4. In the left subplot, the parameters shown are power losses in kilowatts and voltage deviation (VD) in per unit. The convergence curve for starts at a high value of approximately 60,000 kW and decreases steadily as the iterations progress and finally reaches to some fractions of kW. It stabilizes after around 200 iterations at a value close to 1,000 kW in the entire time horizon, indicating the optimization process achieves a significant reduction in power losses. Similarly, the VD curve begins at around 75 (p.u.) and rapidly decreases within the first 100 iterations, stabilizing at approximately 35 (p.u.), reflecting an optimal VD. The right subplot focuses on Case 3 and Case 4, showcasing the voltage stability index (VSI) in per unit and a weighted parameter. The VSI curve starts near −22.35 (p.u.) and quickly converges to −22.3 (p.u.) within the first 100 iterations, demonstrating an improvement in voltage stability. The weighted parameter begins at a high value of approximately 35,000 and decreases rapidly, stabilizing at around 5000 after the first 50 iterations. In both subplots, all the curves exhibit rapid convergence within the initial iterations, highlighting the efficiency of the optimization algorithm. The stabilization of all parameters after convergence signifies that the system reaches an optimized and steady state. Moreover, Table 8 presents the OFR decision variables derived from the conclusive results of the OFR for each study scenario across all time periods. In OFR Case 1, the most frequently activated switch is Switch 60, appearing 10 times throughout the day. This is followed closely by Switch 44, which appears 9 times, suggesting it plays a major role in network reconfiguration. Switches 10 and 52 each occur 8 times, reflecting their high importance, while Switch 70 appears 7 times. These switches likely contribute significantly to loss minimization and voltage stability enhancement in this configuration.

Table 8.

Simulation results of OFR decision variables of all the study cases.

Time	OFR Case 1	OFR Case 2	OFR Case 3	OFR Case 4
1	8, 13, 20, 48, 60	12, 15, 38, 47, 61	20, 25, 35, 45, 57	13, 20, 41, 58, 62
2	10, 44, 53, 60, 70	10, 13, 17, 48, 59	9, 11, 19, 62, 72	9, 19, 45, 53, 63
3	10, 12, 19, 48, 60	7, 19, 57, 61, 71	8, 14, 16, 24, 53	5, 20, 43, 60, 72
4	10, 19, 44, 56, 60	12, 20, 39, 58, 59	10, 12, 15, 49, 63	7, 16, 57, 60, 71
5	10, 24, 43, 58, 70	20, 37, 45, 54, 60	6, 19, 22, 71, 72	18, 35, 43, 53, 59
6	7, 20, 26, 43, 55	5, 15, 47, 59, 71	16, 22, 38, 43, 58	18, 40, 44, 46, 60
7	8, 18, 24, 44, 48	16, 42, 43, 58, 60	14, 16, 38, 55, 60	14, 17, 22, 35, 57
8	18, 44, 52, 60, 69	8, 13, 19, 47, 59	5, 18, 23, 44, 55	20, 26, 35, 43, 49
9	8, 44, 53, 59, 70	10, 18, 45, 53, 60	10, 19, 44, 47, 62	3, 14, 18, 21, 48
10	7, 11, 26, 53, 70	7, 14, 46, 62, 70	26, 40, 44, 49, 70	11, 16, 40, 54, 59
11	14, 20, 40, 46, 59	12, 15, 21, 39, 72	13, 16, 39, 49, 60	16, 39, 54, 59, 71
12	8, 17, 44, 54, 60	13, 16, 40, 58, 60	3, 12, 17, 55, 62	6, 15, 21, 45, 57
13	18, 44, 56, 62, 69	13, 18, 40, 49, 59	17, 37, 43, 46, 59	18, 38, 45, 49, 61
14	8, 12, 18, 48, 60	9, 17, 25, 45, 72	4, 11, 15, 57, 64	4, 17, 44, 54, 60
15	18, 42, 43, 54, 60	3, 12, 19, 24, 57	12, 23, 42, 46, 70	19, 26, 40, 49, 71
16	9, 18, 43, 52, 59	13, 18, 41, 58, 63	10, 12, 17, 25, 49	16, 36, 44, 54, 59
17	9, 43, 48, 60, 70	7, 17, 26, 43, 55	8, 13, 17, 25, 52	15, 39, 54, 59, 71
18	13, 15, 58, 62, 69	9, 12, 18, 25, 55	10, 17, 23, 44, 47	3, 16, 26, 43, 54
19	10, 18, 43, 52, 73	4, 17, 25, 45, 58	17, 37, 43, 56, 61	19, 58, 61, 69, 71
20	10, 15, 52, 71, 73	16, 44, 55, 59, 69	9, 43, 57, 59, 70	10, 11, 26, 58, 70
21	10, 26, 44, 52, 70	6, 12, 15, 60, 72	8, 12, 17, 55, 60	4, 19, 45, 59, 72
22	10, 44, 52, 60, 70	11, 21, 52, 69, 70	9, 12, 17, 53, 60	10, 16, 43, 57, 59
23	11, 15, 52, 63, 69	12, 18, 37, 48, 60	9, 12, 16, 24, 53	10, 12, 55, 60, 70
24	13, 18, 42, 52, 59	6, 16, 44, 55, 64	20, 35, 57, 59, 71	16, 42, 44, 60, 72

On the other hand, in Case 2, Switch 60 once again emerges as the most active, being used 9 times, demonstrating its consistent importance across different scenarios. Switch 12 follows with 7 activations, while Switch 59 is used 6 times. Switches 13 and 57 appear 5 times each, indicating their moderate but stable role in the optimal network configuration for this case. In Case 3, the switching pattern shifts slightly, with Switches 12, 17, and 59 each being utilized 6 times, marking them as equally dominant in this scenario. Switch 57 appears 5 times, while Switch 44 also shows significant involvement, also with 5 occurrences. The appearance of Switch 44 in multiple cases underlines its versatility and operational importance in maintaining an efficient feeder topology. Lastly, in Case 4, Switch 59 is the most recurrent, appearing 7 times, while Switches 60, 16, 18, and 44 each occur 6 times. This balanced distribution indicates a broader spread of switching actions in this case, suggesting a more dynamic or complex reconfiguration strategy. The recurrence of Switches 59 and 44 across all cases further reinforces their strategic relevance in ensuring optimal power distribution and reliability. Table 8 clearly shows that each case adapts the switching schedule based on its unique optimization criteria, certain switches—most notably 60, 44, and 59—appear consistently among the top positions, signifying their critical function in feeder reconfiguration decisions.

Table 9 presents the voltage set points and active power integration of distributed generators at buses 16, 59, and 61 across four different study cases, with each case evaluated over a 24-h horizon. Each study case includes two columns—one for voltage and one for power—indicating that both are continuous decision variables optimized during the simulation process. This comparison finds how each case manages distributed generation (DG) in terms of voltage stability and power injection levels.

Table 9.

Voltage set point and active power integration of DGs in of all cases at buses 16, 59, and 61.

Time	case 1		case 2		case 3		case 4
1	1.009, 1.010, 1.006	0.445, 1.096, 1.627, = 3.168	1.017, 1.074, 1.081	0.190, 1.472, 1.721, = 3.384	0.995, 1.007, 1.005	0.192, 1.196, 0.789, = 2.177	1.023, 1.061, 1.057	0.922, 0.699, 0.896, = 2.517
2	1.044, 1.041, 1.050	0.344, 0.190, 1.338, = 1.873	1.051, 1.055, 0.994	0.767, 0.364, 1.922, = 3.052	1.041, 1.025, 1.013	0.553, 1.533, 0.345, = 2.432	1.008, 1.044, 1.050	0.463, 1.510, 1.201, = 3.174
3	0.997, 1.004, 0.999	0.279, 0.770, 1.125, = 2.175	1.049, 0.972, 0.969	0.893, 0.746, 1.092, = 2.731	1.058, 1.001, 0.996	1.342, 0.570, 1.525, = 3.437	0.954, 0.978, 1.004	0.194, 1.607, 1.466, = 3.267
4	1.017, 1.021, 1.016	0.249, 0.193, 1.011, = 1.453	1.055, 1.077, 1.002	0.745, 1.141, 0.497, = 2.384	1.050, 0.983, 0.979	0.827, 1.282, 1.035, = 3.145	1.061, 1.059, 1.002	1.171, 0.769, 0.793, = 2.733
5	1.037, 1.043, 1.042	0.273, 0.383, 0.883, = 1.539	1.004, 1.037, 1.060	0.267, 0.899, 1.462, = 2.629	1.036, 1.009, 1.011	0.826, 0.547, 1.865, = 3.237	1.039, 1.021, 1.039	0.396, 0.656, 1.026, = 2.078
6	1.025, 1.029, 1.028	0.884, 1.562, 0.243, = 2.689	1.065, 1.060, 1.012	1.479, 0.861, 0.441, = 2.781	1.061, 1.062, 1.060	0.335, 1.120, 0.273, = 1.728	1.058, 1.044, 1.032	1.514, 1.395, 0.190, = 3.100
7	0.979, 0.971, 0.969	0.359, 0.749, 0.910, = 2.018	1.062, 1.040, 1.059	0.974, 1.129, 0.936, = 3.039	1.004, 1.010, 1.060	1.237, 1.370, 1.289, = 3.896	1.029, 1.001, 0.993	1.118, 0.713, 0.515, = 2.346
8	1.015, 1.016, 1.011	0.270, 0.308, 0.930, = 1.508	0.994, 0.979, 0.955	0.945, 0.333, 1.211, = 2.489	1.046, 1.037, 1.039	0.281, 1.477, 0.928, = 2.686	1.057, 1.023, 1.021	1.647, 0.193, 1.291, = 3.132
9	0.989, 0.989, 0.982	0.308, 0.314, 0.979, = 1.602	1.013, 1.002, 0.963	1.412, 0.800, 0.843, = 3.055	0.967, 1.031, 1.038	0.579, 1.411, 1.477, = 3.466	1.039, 1.005, 1.002	1.181, 0.996, 0.397, = 2.574
10	1.021, 1.020, 1.021	0.190, 0.419, 1.339, = 1.948	0.989, 1.053, 1.057	0.434, 1.402, 1.319, = 3.154	1.050, 1.074, 1.078	1.448, 0.515, 1.225, = 3.187	1.036, 0.990, 1.016	1.054, 0.213, 1.476, = 2.743
11	1.051, 1.034, 1.039	0.714, 0.808, 1.186, = 2.708	1.026, 0.969, 0.963	0.894, 1.579, 1.854, = 4.327	1.016, 1.045, 1.020	1.108, 1.295, 1.490, = 3.893	1.059, 1.057, 1.046	0.426, 1.030, 0.711, = 2.167
12	1.025, 1.023, 1.021	0.190, 0.331, 0.901, = 1.422	1.039, 1.020, 0.999	0.210, 0.597, 1.297, = 2.105	1.008, 1.028, 1.025	1.489, 1.038, 1.088, = 3.615	1.062, 1.011, 1.006	1.127, 1.157, 0.323, = 2.608
13	0.978, 0.966, 0.964	0.382, 0.362, 0.628, = 1.373	1.055, 1.007, 0.980	1.433, 0.824, 0.542, = 2.800	1.000, 1.039, 0.959	0.288, 1.769, 1.357, = 3.414	0.990, 1.016, 1.024	0.993, 0.432, 1.426, = 2.852
14	1.041, 1.050, 1.056	0.773, 1.473, 0.892, = 3.138	0.999, 1.049, 1.045	0.411, 1.100, 1.630, = 3.142	1.051, 0.988, 0.985	1.318, 0.737, 1.923, = 3.978	1.054, 1.059, 1.055	0.292, 1.434, 1.131, = 2.857
15	1.049, 1.040, 1.040	0.457, 0.338, 1.106, = 1.901	1.038, 0.955, 0.955	0.926, 0.823, 0.921, = 2.670	1.051, 1.008, 1.009	1.482, 0.249, 0.984, = 2.716	1.060, 1.047, 1.049	1.088, 1.382, 1.338, = 3.807
16	1.004, 1.006, 1.008	0.191, 0.438, 1.501, = 2.130	1.028, 1.028, 1.028	1.820, 0.309, 1.137, = 3.265	1.028, 0.974, 0.960	1.036, 1.050, 0.262, = 2.348	1.016, 1.025, 0.998	1.800, 0.486, 1.707, = 3.993
17	1.035, 1.047, 1.029	0.386, 1.515, 1.615, = 3.516	1.038, 1.047, 1.047	0.281, 0.787, 1.275, = 2.343	1.055, 1.032, 1.025	1.360, 0.375, 0.607, = 2.342	1.007, 1.055, 0.953	1.188, 0.626, 0.432, = 2.246
18	1.061, 1.059, 1.057	0.807, 0.827, 1.308, = 2.942	1.023, 1.060, 1.060	0.940, 1.990, 1.828, = 4.758	0.979, 0.974, 0.966	1.105, 1.176, 0.397, = 2.677	1.041, 1.045, 1.046	0.980, 0.444, 1.875, = 3.299
19	0.989, 0.986, 0.983	0.673, 0.876, 1.776, = 3.326	1.023, 1.063, 1.055	0.360, 1.741, 1.228, = 3.329	0.989, 1.071, 1.073	0.922, 1.323, 0.873, = 3.119	0.993, 1.083, 1.087	1.270, 1.141, 1.836, = 4.247
20	1.033, 1.010, 0.986	1.255, 1.827, 0.920, = 4.003	1.020, 1.014, 0.978	1.783, 1.429, 1.274, = 4.485	1.033, 1.010, 0.986	1.255, 1.827, 0.920, = 4.003	1.024, 1.044, 1.039	1.708, 1.354, 1.106, = 4.169
21	0.960, 1.061, 1.042	1.238, 1.563, 1.429, = 4.230	1.043, 1.012, 1.011	1.611, 0.745, 1.964, = 4.320	0.960, 1.061, 1.042	1.238, 1.563, 1.429, = 4.230	1.058, 0.997, 1.028	1.839, 0.558, 1.977, = 4.374
22	0.992, 1.017, 1.066	0.604, 1.934, 1.151, = 3.689	1.023, 1.021, 1.017	0.569, 1.928, 1.805, = 4.302	0.992, 1.017, 1.066	0.604, 1.934, 1.151, = 3.689	1.047, 1.028, 1.025	0.314, 1.154, 1.420, = 2.887
23	0.998, 0.975, 0.965	0.872, 0.988, 1.011, = 2.871	1.060, 1.053, 1.046	1.963, 0.926, 1.106, = 3.995	0.998, 0.975, 0.965	0.872, 0.988, 1.011, = 2.871	1.039, 0.992, 1.045	1.466, 1.715, 1.775, = 4.955
24	1.035, 1.080, 0.986	0.302, 0.577, 0.226, = 1.105	0.952, 0.975, 0.965	0.884, 0.378, 0.772, = 2.034	1.035, 1.080, 0.986	0.302, 0.577, 0.226, = 1.105	1.054, 1.056, 1.042	1.474, 0.324, 1.949, = 3.747

In Case 1, voltage set points mostly fluctuate within the narrow range of approximately 0.98 to 1.05 p.u., maintaining voltage regulation close to nominal values. The cumulative active power integration of all DGs varies notably, reaching values as high as 2.688 MW in first hour, showing that this case allows significant penetration of DGs when beneficial. Case 2 also maintains voltage values in a relatively controlled band, between 0.94 and 1.06 p.u., showcasing stable voltage support similar to Case 1. However, its DG power integration is more moderate, with several hours showing values around 1.5 to 3.0 MW, and fewer extreme values compared to Case 1. The integration curve appears smoother, suggesting a more balanced distribution of DG operation throughout the day, possibly prioritizing grid stability or emission constraints over maximizing power injection. In contrast, Case 3 shows more aggressive voltage regulation strategies, with some voltage values dropping below 0.95 p.u. (e.g., hour 9), suggesting flexibility in voltage control under varying load/generation conditions. The power integration reaches relatively high values, frequently exceeding 3 MW, and even hitting peaks above 4 MW (e.g., hour 10 with 4.327 MW). This indicates a strong emphasis on DG utilization, likely aiming to reduce grid dependency or losses. However, this aggressive strategy may challenge voltage profiles in weak parts of the system. Finally, Case 4 generally maintains voltage levels within 0.92 to 1.06 p.u., showing controlled but slightly wider voltage variability compared to Case 2. What distinguishes Case 4 is its high cumulative DG integration, frequently exceeding 3 MW, and peaking as high as 4.769 MW (hour 24), the highest among all cases. This suggests Case 4 is optimized for maximum utilization of DG resources, possibly under an economic or emission reduction objective. The trade-off appears in more frequent excursions near voltage limits, requiring careful control mechanisms.

Table 10 presents a comparative analysis of four study cases based on key objective functions such as power loss (), voltage deviation (VD), and voltage stability index (VSI), with DGs integrated at buses 16, 59, and 61. Among all cases, Case 4 consistently offers the most balanced performance. It achieves smallest total power loss (8787.1 kW)—much lower than Cases 2 (9798.1 kW) and Case 3 (12,465.5 kW)—while maintaining a low voltage deviation of 34.229, close to Case 1 (31.726), which has the least deviation but compromises on voltage stability.

Table 10.

Simulation results of case 4, OFR, and dispatchable DGs are integrated at buses 16, 59, and 61.

Time	Case 1			Case 2			Case 3			Case 4
	(kW)	VD	min (VSI) (bus)	(kW)	VD	min (VSI) (bus)	(kW)	VD	min (VSI) (bus)	(kW)	VD	min (VSI) (bus)
1	17.6	0.081	0.997 (49)	589.4	1.004	1.003 (62)	32.8	0.047	0.992 (24)	166.3	2.201	1.011 (63)
2	3.9	2.99	1.040 (50)	494.1	2.878	0.994 (60)	580.3	3.431	1.013 (62)	359.9	0.788	0.977 (52)
3	8.3	0.015	0.994 (69)	478	1.877	0.969 (61)	438.8	3.368	0.993 (25)	822.8	1.365	0.954 (20)
4	3.4	0.581	1.014 (65)	117.9	3.942	1.002 (60)	471.4	1.019	0.974 (49)	315.4	1.101	1.002 (61)
5	2.8	2.46	1.036 (69)	264.5	0.992	1.004 (20)	642.3	2.236	1.008 (23)	273.2	0.793	1.006 (35)
6	271	0.541	0.978 (35)	444.3	1.824	0.989 (35)	12.3	5.451	1.056 (39)	778	1.141	0.990 (35)
7	10.8	1.091	0.960 (49)	654.3	1.391	0.999 (35)	388.4	0.441	0.994 (49)	43.5	0.323	0.989 (23)
8	4.5	0.358	1.010 (64)	312.4	1.986	0.954 (52)	100.1	3.405	1.035 (24)	673.4	3.775	1.017 (50)
9	6.6	0.245	0.981 (64)	528	1.163	0.963 (61)	548.3	1.323	0.965 (63)	81	1.576	0.990 (49)
10	8.9	0.605	1.014 (69)	420.3	0.588	0.976 (63)	793.4	5.09	1.048 (13)	149.2	0.416	0.989 (55)
11	202.1	1.428	1.014 (35)	1834.3	0.738	0.957 (22)	1023.1	3.014	1.016 (16)	52.2	3.492	1.041 (35)
12	9.9	0.95	1.019 (64)	191.4	0.387	0.999 (61)	307.2	0.819	0.995 (63)	253.7	0.755	0.984 (35)
13	19.4	1.326	0.963 (63)	356.1	1.32	0.980 (60)	769.4	0.977	0.959 (60)	518.1	0.186	0.981 (62)
14	478.7	2.181	0.965 (52)	630.7	2.526	0.999 (17)	708.8	2.631	0.982 (64)	280.2	2.313	1.010 (35)
15	11.4	3.046	1.038 (50)	232.2	1.408	0.950 (25)	301.8	3.355	0.988 (47)	761.4	1.406	0.979 (49)
16	9.8	0.066	1.003 (50)	152	0.201	0.993 (64)	373.5	1.66	0.955 (26)	705.2	0.241	0.997 (64)
17	29.8	2.381	1.027 (64)	355.3	1.075	1.007 (35)	161.6	1.72	1.020 (26)	197.7	0.908	0.953 (60)
18	23.6	5.055	1.041 (63)	73.3	1.769	1.016 (69)	674.4	1.446	0.949 (48)	372.8	1.031	1.009 (35)
19	15.5	0.339	0.978 (65)	63	2.664	1.022 (18)	552.3	0.835	0.962 (62)	617.8	0.774	0.971 (62)
20	18	0.508	0.975 (65)	397.1	1.21	0.978 (60)	579.5	0.932	0.986 (60)	266	0.427	1.000 (55)
21	13.2	4.702	1.048 (69)	99.8	0.308	0.997 (50)	1222.7	1.312	0.958 (69)	355.9	0.728	0.997 (59)
22	13.1	0.242	1.000 (64)	331.7	2.8	1.010 (22)	977.5	1.828	0.992 (17)	236.5	3.642	1.025 (64)
23	21	0.407	0.971 (64)	276.7	3.346	1.036 (49)	82.7	0.951	0.960 (25)	231	0.847	0.980 (50)
24	11.6	0.128	0.986 (19)	501.3	1.888	0.951 (65)	722.9	2.244	0.986 (60)	275.9	4	1.040 (64)
Total	1214.9	31.726		9798.1	39.285		12,465.5	49.535		8787.1	34.229

In terms of VSI, Case 4 stands out with consistently high minimum VSI values, indicating strong voltage stability throughout the day, unlike Case 3 which drops to as low as 0.835. While Case 1 excels in loss and VD, its VSI is weaker compared to Case 4. Case 2 offers moderate values across all metrics but lacks the consistency and stability of Case 4. Overall, Case 4 provides the smoothest and most reliable operation, striking an effective trade-off between minimizing losses, controlling voltage deviation, and ensuring strong voltage stability—making it the most efficient and technically robust among all cases. Furthermore, Fig. 12 shows the comparison of VSI of bus where voltage is minimum.

Fig. 12.

Comparison of Voltage collapse point (a) base case vs case 1, (b) base case Vs case 2 (c) base Case vs Case 3 (d) base Case vs Case 4 of the entire time range.

Figure 12 clearly illustrate the impact of various strategies (especially in Case 3) on improving the voltage stability of a power distribution network, highlighting the network’s enhanced ability to handle increased load demands without succumbing to voltage collapse.

Conclusion

This study introduces an innovative methodology for allocating and sizing DGs and OFR, considering multiperiod technical objective functions. The distribution system encompasses both dispatchable and uncertain solar photovoltaic (PV) non-dispatchable DGs. The main goal is to improve the distribution network’s VSI while also lowering Voltage Deviation (VD) and power loss. The augmentation of the voltage profile is achieved by minimizing the squared differences in voltage magnitudes between adjacent buses. The proposed methodology is applied to radial distribution networks with 33 and 69 buses, utilizing real-time series data pertaining to loads and DGs across a 24-h timeframe. The study’s findings demonstrate that individual technical objective functions yield improved solutions, characterized by minimal power loss, enhanced voltage quality, and heightened voltage stability index. However, it is also observed that within multiperiod analysis, certain objective functions deteriorate in select time periods due to excessive emphasis on specific solutions. Thus, a judicious trade-off among diverse objective functions becomes crucial to maximize the benefits derived from DG allocation and OFR. This encompasses aspects such as loss reduction, voltage quality enhancement, and the optimization of voltage stability collapse points.

Notably, the study reveals that maximizing VSI as an objective function optimally distributes the reactive power generation of dispatchable DGs. Conversely, minimizing power loss and voltage deviation inadvertently leads to increased reactive power integration during lower loading conditions. Moreover, the simulation results are benchmarked against the base case values to gauge the magnitude of benefits achieved through the proposed algorithm. The results indicate that the formulation effectively improves the voltage profile of the system. This is supported by a substantial decrease in power loss of over 86% and a large enhancement in voltage deviation of more than 90%. The load capacity experiences a significant increase of over 700% through the effective integration of DGs while considering the VSI as the goal function. The proposed system demonstrates an appropriate adjustment of the power factor in response to reactive load needs, which is a notable characteristic.

Future work and limitation of proposed algorithm

Future work will involve applying the proposed hybrid algorithm to larger distribution networks, such as the 118-bus systems, to validate its scalability and effectiveness. Subsequent research could integrate economic considerations, like system costs and investment analysis, into the optimization framework for a more holistic approach. Additionally, optimizing other renewable energy sources, such as wind and battery storage systems, alongside DGs could enhance network stability and efficiency.

Hybrid GAs, DE, and PSO approaches can offer significant benefits in solving large-scale distributed generation and optimal network reconfiguration problems. However, like any optimization method, Hybrid EAs have the following limitations. Typically, it involves a range of parameters that need to be carefully tuned to achieve good performance; finding the right combination of parameters can be a time-consuming and challenging task, and the performance of the algorithms may be sensitive to parameter settings. Hybrid EAs have larger computational complexity compared to individual ones, so may get stuck in local optimal solutions, and converge more slowly than single algorithms.

Abbreviations

BIBC

Bus Injection to Branch Current

BCBV

Branch Current to Bus Voltage

CV

Constraint Violation

COP

Constrained Optimization Problem

CB

Circuit Breaker

DE

Differential Evolution

DLF

Distribution Load Flow

DG

Distributed Generation

EAs

Evolutionary Algorithms

ECM

Epsilon Constraint method

FR

Feasibility Ratio

GA

Genetic Algorithm

MINLP

Mixed Integer Nonlinear Problem

OFR

Optimal Feeder Reconfiguration

PV

Photovoltaic

PSO

Particle Swarm Optimization

RES

Renewable Energy resource

VD

Voltage Deviation

VSI

Voltage Stability Index

Indices/Variables/Parameters

Span length of time

Current time index

Total number of DGs

Total number of online feeders

Active and reactive power injection at bus j,

Resistance and inductive reactance of branch connected between bus and

Location of DG

Normally open CB/tie line breaker

Active and reactive power demand

Active and reactive power injection of DG

MVA branch flow

Bus incident matrix

Total number of buses

Branch current matrix

Velocity vector in PSO and Mutant vector in DE

Decision or target vector

Objective functions, power loss, VD and VSI respectively

Number of equality and inequality constraints

Reference bus voltage set point

Load bus voltage

Number of buses

∆V

Change in bus voltage from reference bus

VSI of j^th (to) bus

Active and reactive power supplied by substation

Objective functions

Equality and inequality constraint functions

Uncertain Solar PV active power injection

Local and global best vector of PSO method

Author contributions

Aamir Ali: Conceptualization; validation; writing—original draft; writing—review and editing. A. S. Saand: Conceptualization; formal analysis; resources; software; writing— original draft; writing—review and editing. Shoaib Ali: Conceptualization; funding acquisition; methodology; project administration. Rizwan A. Siddiqui: Investigation; software; validation; visualization; writing—original draft; writing—review and editing. Mohsin Kondhar: Investigation; software; validation; visualization; writing—original draft; writing—review and editing. Lutfi Albasha: Investigation; methodology; resources; validation; visualization. Rizwan A. Siddiqui: Conceptualization; writing—original draft. Faisal Alsaif: Validation; visualization.

Funding

This research has required no external funding.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.