# Performance of Various Voltage Stability Indices in a Stochastic Multiobjective Optimal Power Flow Using Mayfly Algorithm

Authors: Rebeccah Kyomugisha; Christopher Maina Muriithi; George Nyauma Nyakoe
Journal: Journal of Electrical and Computer Engineering
 (2022)

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## Abstract

The performance of voltage stability indices in the multiobjective optimal power flow of modern power systems is presented in this work. Six indices: the Voltage Collapse Proximity Index (VCPI), Line Voltage Stability Index (LVSI), Line Stability Index (Lmn), Fast Voltage Stability Index (FVSI), Line Stability Factor (LQP), and Novel Line Stability Index (NLSI) were considered as case studies on a modified IEEE 30-bus consisting of thermal, wind, solar and hybrid wind-hydro generators. A multiobjective evaluation using the multiobjective mayfly algorithm (MOMA) was performed in two operational scenarios: normal and contingency conditions, using the MATLAB–MATPOWER toolbox. Fuzzy Decision-Making technique was used to determine the best compromise solutions for each Pareto front. To evaluate the computational efficiency of the case studies, a preference selection index was used. The results indicate that VCPI and NLSI yielded the best-optimized system performance in minimizing generation costs, transmission loss reduction, and simulation time for normal and contingency conditions. The best-case studies also promoted the most scheduled reactive power generation from renewable energy sources (RES). On average, the VCPI index contributed the highest penetration level from RES (13.40%), while the Lmn index had the lowest. Overall, VCPI and Lmn index provided the best and worst average performance in both operating scenarios, respectively. Also, the MOMA algorithm demonstrated superior performance against the multiobjective harris hawks algorithm (MHHO), multiobjective Jaya algorithm (MOJAYA), multiobjective particle swarm algorithm (MOPSO), and nondominated sorting genetic algorithm III (NSGA-III) algorithms. In all, the proposed approach yields the lowest system cost and loss compared to other methods.


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## Body

## 1. Introduction

Over the recent past, increasing the load demand has brought about different power systems problems in terms of power transmission congestion and constraints. The challenges are mainly attributed to maintaining the stability of the power system at its permissible level [1, 2]. One of the inexpensive ways to ensure system stability and security is to incorporate voltage stability indices in the conventional optimal power flow (OPF) problem [3]. The traditional OPF goal includes coal-fired power plants that use fossil fuels. With the current drive to combat global warming and climate change, there is increasing global use of renewable energy sources (RES) in modern power systems [4]. Therefore, an OPF analysis that considers these nonconventional energy units is necessary [5]. The addition of voltage stability indices (VSI) in optimal flow problems has already proven to be a significant voltage stability enhancement for fossil-based power systems, especially during normal and contingency operating conditions [3, 6–8]. However, optimization and stability remain challenging with the high penetration levels of variable renewable energy power [9, 10]. Therefore, to ensure system stability in modern power systems, the voltage stability constrained optimal power flow (VSC-OPF) must be carried out in a system integrated with renewable energy sources (RES) such as wind, solar, and hybrid RES.Due to the fact that the OPF problem is multimodal, nonlinear, and nonconvex, classical approaches are not always appropriate and cannot guarantee a global solution. Researchers in [11] provide a comprehensive review of these approaches. The use of sensitivity analysis and gradient-based search [12] reveals that several of these techniques in [11], such as the Newton approach [13], linear programming [14], quadratic programming [15], sequential programming [16], among others, exhibit strong convergence characteristics. However, they struggle with integer and discrete variables. Furthermore, it is challenging to develop these algorithms while considering multiobjective OPF problems [17].To address the drawbacks of the classical techniques, numerous heuristic solution methods for tackling the OPF problem have been devised. In [18], the Heap Optimization Algorithm (HOA) was used to solve the OPF of wind-solar hybrid IEEE bus systems. Firstly, the authors used HOA to optimally place RESs on buses where the lowest cost is observed for 24 hours. The authors then evaluated the OPF considering RES penetration and varying loading conditions. A MOPSO/Fuzzy Membership Function (FMF) method was used in [19] to solve the multiobjective stochastic optimal power flow of a modified IEEE 30-bus system. Three objectives: fuel cost (with and without valve-effect), power loss, and carbon emissions were considered. To deal with the intermittent nature of RESs, PDF and stochastic models are used to calculate the available power. The method yielded the least cost and emissions compared to multiobjective mayfly (MOFA), Nondominated Sorting Genetic Algorithm II (NSGA-II), and multiobjective evolutionary algorithm based on decomposition (MOEA/D), among others. However, it had the highest losses of 13.77%. Syed et al., 2021, [20] used machine learning algorithms to firstly predict wind and solar penetration levels on the Indian 124-bus system. The OPF was then solved using the Improved Wind-Driven Algorithm (IWDA) to provide the hourly and long-term power output, cost, and loss using 12 cases. The technique minimizes the complexities involved in RES integration in grids. A Modified Rao-2 Algorithm was used in [4] to optimize the control variables of the OPF problem without RES, with RES, and with RES under a contingency state. Four objectives were assessed: fuel cost, transmission loss, emission, and voltage profile improvement. The method was tested on modified IEEE 30- and IEEE 118-bus systems. The RES considered were biomass, wind, solar and hydrogenerators. The MRao-2 algorithm demonstrated dominant performance over other algorithms such as ASO, TFWO, and MPSO. In [21], a novel hybrid Mayfly algorithm-Aquila Optimizer was used to solve the OPF of a modified IEEE 30-bus system consisting of wind, solar, hydro, and thermal generators. The objectives minimized included fuel cost, power loss, VSI, and emissions. Similar studies have employed metaheuristic techniques such as evolutionary multiobjective algorithm [22], hybrid differential evolution–symbiotic organisms search (HSOS) [23], Jellyfish Search Optimization [24], multiobjective adaptive guided differential evolution (MOAGDE) algorithm [25], surrogate assisted multiobjective differential evolution method [26], Levy Interior Search Algorithm [27] to solve stochastic OPF problems. Other techniques such as the two-stage approach utilizing MOPSO algorithm, fuzzy c-means (FCM) clustering and Grey relation projection [28], and the bi-criterion evolution indicator-based evolutionary algorithm (BCE-IBEA) have also been employed to solve multiobjective problems [29]. Very few of these studies consider OPF of renewable energy integrated grid considering the system stability. Yet, the system’s stability becomes increasingly challenging with the intermittency of RES.Static voltage stability analysis remains the mainstream method for assessing proximity to collapse of power systems. This is because this type of analysis method can calculate the distance between each node’s voltage and the voltage collapse point relatively accurately. Thus, it can be used to improve and optimize the power system’s stability [30].Recent works reveal three main techniques for evaluating the effectiveness of voltage stability indices in optimal power flow problems of grids integrated with RES. The most commonly used method in recent literature evaluates VSIs as independent objective functions. A case study involving the minimization of the L-index in [31] yielded the greatest voltage stability improvement on a modified IEEE 30-bus incorporating RES. The improvement was 1.231% higher than other case studies that did not include VSIs. Additionally, the objective function of the VSI study also yielded the least emissions, up to 36.780% less than other case studies. However, these advantages came at the highest generation cost. Similar studies in [32], focusing on minimizing L-index as the objective function, were tested on the RE-integrated IEEE 30-bus. It was observed that the voltage stability improved by 92.93% compared to the conventional cost-function minimization case. Also, in [33], similar improvements were observed by using the voltage stability improvement as the objective function (using L-index). The VSI case study still provided the greatest stability improvement of 2.066% compared to the base case (cost minimization). From these studies using VSI as the objective function, it is observed that the main advantages of this technique are its simplicity and simulation speed, as the focus is only on one objective function.The second technique, commonly referred to as the weighted-sum approach, employs a multiobjective approach. In this method, several objective functions are treated as one by applying weights to each function and computing the summation of the functions. Authors in [34] employed this technique to optimally place thyristor-controlled series compensators (TCSC) on IEEE 30- and IEEE 57-buses with RES by considering the minimization of two indices: a rapid voltage stability index (RVSI) and a novel line stability index (NLSI) in the OPF problem. In [35], a case study involving the minimization of the L-index, considering a combination of traditional generators and controllable PV energy sources, was also evaluated. The analysis indicated that the addition of the L-index provided the most significant voltage stability improvement of 1.127%, higher than cases without VSI, on the IEEE 30-bus system. It is important to note that the VSC-OPF case study also provided the greatest loss and emissions reduction of 47.491%, 31.644%, compared to the case study incorporating the emissions minimization function. Thus, VSC-OPF has significant advantages of reducing dependency on fossil-based generators by lowering the emissions on thermal generators. Although this technique provides greater improvement than the first approach, it remains a challenge to assign weights to each objective function [36].The majority of the literature discussed involves incorporating the L-index in the VSC-OPF. Voltage stability indices have a significant impact on the generation cost and general system performance of the power system. For example, comparative studies considering various voltage stability indices in [7] revealed that the Voltage Collapse Proximity Index (VCPI) and Line voltage stability index (LVSI) provided the lowest transmission loss and emissions, respectively, across all tests systems, compared to other indices. Additionally, the L-index provided the greatest maximum loadability improvement. With these varying advantages of the different indices, it is not straightforward to select the best-performing index for the ideal operation of the power system. More studies carried out in [8] compare the effectiveness of three VSIs: LVSI index, Lmn, and the Fast Voltage Stability Index (FVSI) on IEEE 30-, IEEE 57-, and IEEE 118-bus test systems. Still, the different VSIs had varying benefits in terms of power generation cost reduction, loss minimization, and voltage stability improvement. The studies showed that the LVSI index provided better system performance for IEEE 30- and IEEE 57-buses. However, for the larger system (IEEE 118-bus), the FVSI provided the most remarkable results. Such comparative studies must continue to be assessed, especially for modern power systems, including the use of clean energy technologies. This ensures the continued secure, reliable, and cost-effective operation of power systems. However, the studies carried out employed methods that do not consider the simultaneous objectives of the power systems. That is, power system operation involves multiple parameters such as stability, cost, and loss minimization that must be considered simultaneously. Thus, a multiobjective optimization involving the use of the Pareto-optimality principle is paramount. Additionally, the challenges of the power system face a set of decision problems affiliated to different parts (e.g., scheduling, investment, and operation) where decision-makers must distinguish all alternatives from cost, revenue, or risk point of view [37]. The lack of informed decision-making can compromise the stability of systems, leading to collapse in some instances. Therefore, a selection index is required to advise on the best choice of selection of voltage stability indices in modern power systems.From the literature assessment carried out and to the best of the authors’ knowledge, no studies have compared the performance of various voltage stability indices in the multiobjective optimal power of power systems considering the stochastic nature of renewable energy technologies integration. Yet, comprehensive studies have proven that multiobjective VSC-OPF provides more significant voltage stability improvement than the two approaches used in literature in normal (28.13%) and contingency conditions (by 13.60%) [6]. More so, OPF studies considering a recently developed Mayfly algorithm are scarce; yet this algorithm has proven a dominant performance due to its global search capabilities [38].Therefore, this paper focuses on the evaluation of six voltage stability indices: VCPI, LVSI, Lmn, FVSI, line stability factor (LQP), and the novel line stability index (NLSI) in the multiobjective OPF of a system incorporating intermittent renewable energy sources. A modified IEEE 30-bus consisting of wind, solar photovoltaic (PV), and hybrid wind- small hydro generators taken from [39] is utilized to assess the performance of the VSIs based on voltage stability improvement, generation cost minimization, transmission loss reduction, and simulation speed for each VSI. A multiobjective Mayfly algorithm (MOMA) is used to obtain the Pareto optimal solutions from all case studies. Results obtained using MOMA are validated against previous literature and four other algorithms: multiobjective harris hawks algorithm (MHHO), multiobjective Jaya algorithm (MOJAYA), multiobjective particle swarm algorithm (MOPSO), and nondominated sorting genetic algorithm III (NSGAIII).The key contributions of this paper include the following:(i)
Evaluation of the impact of voltage stability indices in the optimal power flow of a renewable energy integrated system in normal and contingency conditions(ii)
A ranking of the performance of six voltage stability indices in a multiobjective optimal power flow, using the preference selection index (PSI).(iii)
Development and assessment of the multiobjective mayfly algorithm in the optimal power flow considering renewable energy sources(iv)
Comparison of the performance multiobjective mayfly algorithm against other algorithms in a MO-OPF in terms of generation cost and loss reduction, voltage stability improvement, and simulation speed.(v)
Impact of voltage stability constrained optimal power flow on the penetration level of renewable energy sourcesThe rest of the article is organized as follows: Section2 describes the objective functions to be considered for optimization. The details of the formulations also cost functions for the RES and the general constraints for the study. Section 3 provides an overview of the stochastic modeling of the RES, while Section 4 presents the computational procedure of the MOMA algorithm. Sections 5 and 6 detail the discussion of the results obtained in the study and their validation, respectively. Lastly, Section 7 highlights general deductions from all the studies performed in this paper.


## 2. Problem Formulation

The multiobjective optimization problem considers three objective functions: generation cost, transmission power loss, and voltage stability index.


### 2.1. Generation Cost

This objective consists of minimizing the total cost of thermal generators and that of RESs, including the penalty and reserve costs.


#### 2.1.1. Thermal Power Generator Cost

The objective focuses on minimizing the total fuel generation cost. The cost function in terms of the valve-point effect is expressed through the following equation:(1)fTh_costx=∑i=1ngai+biPgi+ciPgi2+di×sinei×Pgimin−Pgi$h,where fTh_costx and ng are the total fuel cost and the number of thermal generators, respectively. ai, bi and ci are the i th generator cost coefficients, Pgi is the i-th generator active power injection, di and ei are the fuel cost coefficients modeling the valve-point effect, and Pgimin is the minimum active power generated by the i th generating unit. The cost function’s extra sinusoidal term reflects valve-point effects. Valve-point effects make the economic dispatch problem nonconvex and nondifferentiable.


#### 2.1.2. The Cost Model of Renewable Energy Sources

RES have no need for conventional fuel to generate power. However, only operation and maintenance costs are incurred if an independent system operator owns the RES. Therefore, the ISO must pay according to the contractually agreed scheduled power [39]. Hence, there is a direct cost associated with each renewable generator, expressed in equations (2)–(4) for wind, solar PV, and hybrid wind-small hydro.(2)CWPW=gwPW,where Pw represents the scheduled power and gw, the wind power direct cost coefficient.Equation (3) is the solar PV direct cost with scheduled power PPV and cost coefficient hpv:(3)CPVPPV=hpvPPV.Equation (4) presents the direct cost for scheduled power for the hybrid wind-small hydro generation plants. The maximum output of the small hydro plant is 5 MW which is dependent on the river’s ﬂow rate.(4)CWHPWH=gwhPWH=gwhPWH,w+gwhPWH,h,where PWH is the scheduled active power output from the hybrid plant, PWH,w is the output power from the wind unit and PWH,h from the hydro plant. gw and gh are the wind and hydro-unit direct coefficients, respectively.Due to the variable nature of wind energy, the actual power produced may be less or more than the scheduled power. As such, the ISO must have reserve generating capacity to meet the demand in cases of overestimation. The Reserve cost for the wind unit is computed as in the following equation [40]:(5)CRW,iPWsh,i−PWac,i=krw,iPWsh,i−PWac,i=krw,i∫0PWsh,iPWsh,i−pW,ifwpw,idpw,i.Suppose the power output is underestimated (there is excess power from the RES). The ISO instead pays a penalty cost by decreasing the thermal generator power output. The wind generator penalty cost is calculated as in the following equation:(6)CPW,iPWac,i−PWsh,i=kpw,iPWac,i−PWsh,i=kpw,i∫PWsh,iPWr,ipW,i−PWsh,ifwpw,idpw,i,where PWsh,i, PWac,i and PWr,i denote the scheduled, available, and rated powers from wind plants, respectively. fwpw,i represents the probability density function (PDF) of wind. In the same manner, the reserve cost, penalty of solar and hybrid wind hydro generators can be estimated. Table 1 indicates the coefficients for direct, reserve, and penalty costs for the different RESs considered.Table 1
Cost coefficients for RES.
RESDirect cost coefficientReserve cost coefficientPenalty cost coefficientWindgw=1.6krw=3kpw=1.5Solargs=1.6krs=3kps=1.5Small hydrogh=1.5krh=3kph=1.5Therefore, the cumulative generation cost function of the system can be computed as follows:(7)f1x=fTh_costx+fRES_directx+fRES_penaltyx+fRES_reservex$h.


### 2.2. Transmission Power Loss

The second objective minimizes the transmission power loss in MW. It is given by the following equation:(8)f2x=∑k=1NlineGkVi2+Vj2−2ViVjcosθi−θj,where Gk is the kth line conductance. Vi and Vj are the voltage magnitudes for sending and receiving bus respectively, along line k. θi and θj are the sending and receiving bus voltage.


### 2.3. Voltage Stability Improvement

The third objective function focuses on enhancing the system voltage stability by minimizing the voltage stability index (VSI) value, as shown in equation (9). The maximum value of the VSI is considered since it represents the weakest line in the system.(9)f3x=maxVSIi.The VSIs considered in this study are the VCPI, LVSI, Lmn, FVSI, LQP, and NLSI, whose formulations are described in Table2. VSI increases with power flow across a transmission line.Table 2
Details of the VSIs in the study.
VSIRef.VSI computationUnstable conditionStable conditionVoltage Collapse Proximity Index (VCPI)[41]VCPIpower=Pr/PrmaxVCPI>1VCPI≤1WherePrmax=Vs2/Zcos∅/4cos2θ−∅/2Line Voltage Stability Index (LVSI)[42]LVSI=4RP/Vscosθ−δ2LVSI>1LVSI≤1Line Stability Index (Lmn)[43]Lmn=4XQr/Vssinθ−δ2Lmn>1Lmn≤1Fast Voltage Stability Index (FVSI)[44]FVSI=4Z2Qr/Vs2XFVSI>1FVSI≤1Line Stability Factor (LQP)[45]LQP=4X/Vi2X/Vi2Pi2+QjLQP>1LQP≤1Novel Line Stability Index (NLSI)[46]NLSI=PjR+QjX/0.25V2NLSI>1NLSI≤1


### 2.4. System Constraints

There are two types of constraints to be handled: equality and inequality constraints. Equations (10)–(16) describe the details of these constraints.(i)
Equality ConstraintsThese consist of load flow equations that provide the system’s active and reactive power balance. They are formulated as in the following equations:(10)Pgi−Pdi=Vi∑j=1NVjGijcosθij+Bijsinθij,i=1,…,N,(11)Qgi−Qdi=Vi∑j=1NVjGijsinθij+Bijcosθij,i=1,…,N,wherePgi is the real power injection at the i th generator, Qgi is the reactive power outputs of the i th generator. Pdi is the load bus active power and Qdi the load reactive power at bus i. Vi and Vj are the i th and j th bus voltage magnitude. Gij denotes the conductance and Bij the susceptance between buses i and j. θij represents the buses i and j phase angle diﬀerence. N is the total number of system buses.(ii)
Inequality ConstraintsThe inequality constraints represent the control variable limits described in equations (12)–(17):Generator limits:(12)Pgimin≤Pgi≤Pgimax,i=1,…,Ng,(13)Qgimin≤Qgi≤Qgimax,i=1,…,Ng,(14)Vgimin≤Vgi≤Vgimax,i=1,…,Ng.Transmission line limits:(15)SLi≤SLimax.Load bus voltage magnitude limits:(16)Vdimin≤Vdi≤Vdimax,i=1,…,Nd,wherePgimin and Pgimax are the bus i minimum and maximum active powers (thermal/RES). Qgimin and Qgimax are the minimum and maximum reactive power (thermal/RES) generated at bus i. Vgimin and Vgimax are the minimum and maximum generator voltages at bus i. SLi and SLimax are the apparent power ﬂow and its maximum at branch i. Vdimin and Vdimax are the minimum and maximum load voltages at bus i. Timin and Timax are the minimum and maximum transformer tap ratios at bus i.


## 2.1. Generation Cost

This objective consists of minimizing the total cost of thermal generators and that of RESs, including the penalty and reserve costs.


### 2.1.1. Thermal Power Generator Cost

The objective focuses on minimizing the total fuel generation cost. The cost function in terms of the valve-point effect is expressed through the following equation:(1)fTh_costx=∑i=1ngai+biPgi+ciPgi2+di×sinei×Pgimin−Pgi$h,where fTh_costx and ng are the total fuel cost and the number of thermal generators, respectively. ai, bi and ci are the i th generator cost coefficients, Pgi is the i-th generator active power injection, di and ei are the fuel cost coefficients modeling the valve-point effect, and Pgimin is the minimum active power generated by the i th generating unit. The cost function’s extra sinusoidal term reflects valve-point effects. Valve-point effects make the economic dispatch problem nonconvex and nondifferentiable.


### 2.1.2. The Cost Model of Renewable Energy Sources

RES have no need for conventional fuel to generate power. However, only operation and maintenance costs are incurred if an independent system operator owns the RES. Therefore, the ISO must pay according to the contractually agreed scheduled power [39]. Hence, there is a direct cost associated with each renewable generator, expressed in equations (2)–(4) for wind, solar PV, and hybrid wind-small hydro.(2)CWPW=gwPW,where Pw represents the scheduled power and gw, the wind power direct cost coefficient.Equation (3) is the solar PV direct cost with scheduled power PPV and cost coefficient hpv:(3)CPVPPV=hpvPPV.Equation (4) presents the direct cost for scheduled power for the hybrid wind-small hydro generation plants. The maximum output of the small hydro plant is 5 MW which is dependent on the river’s ﬂow rate.(4)CWHPWH=gwhPWH=gwhPWH,w+gwhPWH,h,where PWH is the scheduled active power output from the hybrid plant, PWH,w is the output power from the wind unit and PWH,h from the hydro plant. gw and gh are the wind and hydro-unit direct coefficients, respectively.Due to the variable nature of wind energy, the actual power produced may be less or more than the scheduled power. As such, the ISO must have reserve generating capacity to meet the demand in cases of overestimation. The Reserve cost for the wind unit is computed as in the following equation [40]:(5)CRW,iPWsh,i−PWac,i=krw,iPWsh,i−PWac,i=krw,i∫0PWsh,iPWsh,i−pW,ifwpw,idpw,i.Suppose the power output is underestimated (there is excess power from the RES). The ISO instead pays a penalty cost by decreasing the thermal generator power output. The wind generator penalty cost is calculated as in the following equation:(6)CPW,iPWac,i−PWsh,i=kpw,iPWac,i−PWsh,i=kpw,i∫PWsh,iPWr,ipW,i−PWsh,ifwpw,idpw,i,where PWsh,i, PWac,i and PWr,i denote the scheduled, available, and rated powers from wind plants, respectively. fwpw,i represents the probability density function (PDF) of wind. In the same manner, the reserve cost, penalty of solar and hybrid wind hydro generators can be estimated. Table 1 indicates the coefficients for direct, reserve, and penalty costs for the different RESs considered.Table 1
Cost coefficients for RES.
RESDirect cost coefficientReserve cost coefficientPenalty cost coefficientWindgw=1.6krw=3kpw=1.5Solargs=1.6krs=3kps=1.5Small hydrogh=1.5krh=3kph=1.5Therefore, the cumulative generation cost function of the system can be computed as follows:(7)f1x=fTh_costx+fRES_directx+fRES_penaltyx+fRES_reservex$h.


## 2.1.1. Thermal Power Generator Cost

The objective focuses on minimizing the total fuel generation cost. The cost function in terms of the valve-point effect is expressed through the following equation:(1)fTh_costx=∑i=1ngai+biPgi+ciPgi2+di×sinei×Pgimin−Pgi$h,where fTh_costx and ng are the total fuel cost and the number of thermal generators, respectively. ai, bi and ci are the i th generator cost coefficients, Pgi is the i-th generator active power injection, di and ei are the fuel cost coefficients modeling the valve-point effect, and Pgimin is the minimum active power generated by the i th generating unit. The cost function’s extra sinusoidal term reflects valve-point effects. Valve-point effects make the economic dispatch problem nonconvex and nondifferentiable.


## 2.1.2. The Cost Model of Renewable Energy Sources

RES have no need for conventional fuel to generate power. However, only operation and maintenance costs are incurred if an independent system operator owns the RES. Therefore, the ISO must pay according to the contractually agreed scheduled power [39]. Hence, there is a direct cost associated with each renewable generator, expressed in equations (2)–(4) for wind, solar PV, and hybrid wind-small hydro.(2)CWPW=gwPW,where Pw represents the scheduled power and gw, the wind power direct cost coefficient.Equation (3) is the solar PV direct cost with scheduled power PPV and cost coefficient hpv:(3)CPVPPV=hpvPPV.Equation (4) presents the direct cost for scheduled power for the hybrid wind-small hydro generation plants. The maximum output of the small hydro plant is 5 MW which is dependent on the river’s ﬂow rate.(4)CWHPWH=gwhPWH=gwhPWH,w+gwhPWH,h,where PWH is the scheduled active power output from the hybrid plant, PWH,w is the output power from the wind unit and PWH,h from the hydro plant. gw and gh are the wind and hydro-unit direct coefficients, respectively.Due to the variable nature of wind energy, the actual power produced may be less or more than the scheduled power. As such, the ISO must have reserve generating capacity to meet the demand in cases of overestimation. The Reserve cost for the wind unit is computed as in the following equation [40]:(5)CRW,iPWsh,i−PWac,i=krw,iPWsh,i−PWac,i=krw,i∫0PWsh,iPWsh,i−pW,ifwpw,idpw,i.Suppose the power output is underestimated (there is excess power from the RES). The ISO instead pays a penalty cost by decreasing the thermal generator power output. The wind generator penalty cost is calculated as in the following equation:(6)CPW,iPWac,i−PWsh,i=kpw,iPWac,i−PWsh,i=kpw,i∫PWsh,iPWr,ipW,i−PWsh,ifwpw,idpw,i,where PWsh,i, PWac,i and PWr,i denote the scheduled, available, and rated powers from wind plants, respectively. fwpw,i represents the probability density function (PDF) of wind. In the same manner, the reserve cost, penalty of solar and hybrid wind hydro generators can be estimated. Table 1 indicates the coefficients for direct, reserve, and penalty costs for the different RESs considered.Table 1
Cost coefficients for RES.
RESDirect cost coefficientReserve cost coefficientPenalty cost coefficientWindgw=1.6krw=3kpw=1.5Solargs=1.6krs=3kps=1.5Small hydrogh=1.5krh=3kph=1.5Therefore, the cumulative generation cost function of the system can be computed as follows:(7)f1x=fTh_costx+fRES_directx+fRES_penaltyx+fRES_reservex$h.


## 2.2. Transmission Power Loss

The second objective minimizes the transmission power loss in MW. It is given by the following equation:(8)f2x=∑k=1NlineGkVi2+Vj2−2ViVjcosθi−θj,where Gk is the kth line conductance. Vi and Vj are the voltage magnitudes for sending and receiving bus respectively, along line k. θi and θj are the sending and receiving bus voltage.


## 2.3. Voltage Stability Improvement

The third objective function focuses on enhancing the system voltage stability by minimizing the voltage stability index (VSI) value, as shown in equation (9). The maximum value of the VSI is considered since it represents the weakest line in the system.(9)f3x=maxVSIi.The VSIs considered in this study are the VCPI, LVSI, Lmn, FVSI, LQP, and NLSI, whose formulations are described in Table2. VSI increases with power flow across a transmission line.Table 2
Details of the VSIs in the study.
VSIRef.VSI computationUnstable conditionStable conditionVoltage Collapse Proximity Index (VCPI)[41]VCPIpower=Pr/PrmaxVCPI>1VCPI≤1WherePrmax=Vs2/Zcos∅/4cos2θ−∅/2Line Voltage Stability Index (LVSI)[42]LVSI=4RP/Vscosθ−δ2LVSI>1LVSI≤1Line Stability Index (Lmn)[43]Lmn=4XQr/Vssinθ−δ2Lmn>1Lmn≤1Fast Voltage Stability Index (FVSI)[44]FVSI=4Z2Qr/Vs2XFVSI>1FVSI≤1Line Stability Factor (LQP)[45]LQP=4X/Vi2X/Vi2Pi2+QjLQP>1LQP≤1Novel Line Stability Index (NLSI)[46]NLSI=PjR+QjX/0.25V2NLSI>1NLSI≤1


## 2.4. System Constraints

There are two types of constraints to be handled: equality and inequality constraints. Equations (10)–(16) describe the details of these constraints.(i)
Equality ConstraintsThese consist of load flow equations that provide the system’s active and reactive power balance. They are formulated as in the following equations:(10)Pgi−Pdi=Vi∑j=1NVjGijcosθij+Bijsinθij,i=1,…,N,(11)Qgi−Qdi=Vi∑j=1NVjGijsinθij+Bijcosθij,i=1,…,N,wherePgi is the real power injection at the i th generator, Qgi is the reactive power outputs of the i th generator. Pdi is the load bus active power and Qdi the load reactive power at bus i. Vi and Vj are the i th and j th bus voltage magnitude. Gij denotes the conductance and Bij the susceptance between buses i and j. θij represents the buses i and j phase angle diﬀerence. N is the total number of system buses.(ii)
Inequality ConstraintsThe inequality constraints represent the control variable limits described in equations (12)–(17):Generator limits:(12)Pgimin≤Pgi≤Pgimax,i=1,…,Ng,(13)Qgimin≤Qgi≤Qgimax,i=1,…,Ng,(14)Vgimin≤Vgi≤Vgimax,i=1,…,Ng.Transmission line limits:(15)SLi≤SLimax.Load bus voltage magnitude limits:(16)Vdimin≤Vdi≤Vdimax,i=1,…,Nd,wherePgimin and Pgimax are the bus i minimum and maximum active powers (thermal/RES). Qgimin and Qgimax are the minimum and maximum reactive power (thermal/RES) generated at bus i. Vgimin and Vgimax are the minimum and maximum generator voltages at bus i. SLi and SLimax are the apparent power ﬂow and its maximum at branch i. Vdimin and Vdimax are the minimum and maximum load voltages at bus i. Timin and Timax are the minimum and maximum transformer tap ratios at bus i.


## 3. Modeling the Uncertainty of RES

This paper utilizes the uncertainty modeling of the stochastic nature of the RESs carried out in various recent literature [19, 21, 24, 47].Wind energy production is proportionate to wind speed, and wind speed probability is best represented by the Weibull PDF [48]. Solar PV power is determined by solar irradiance, and the probability distribution of solar irradiance is represented by lognormal PDF [49]. The flow rate and effective pressure head determine the power production of a small hydro plant. From previous works, it was established that Gumbel distribution is effective in the estimation of the river ﬂow rate [48, 50]. A summary of the wind, solar and hydro PDF parameters used in this study are presented in Table 3 [19].Table 3
Probability Density Function parameters for the RES stochastic models.
Wind at bus 5Solar at bus 11Wind + Small hydro at bus 13Number of turbines25Rated power (MW)50Number of turbines15Total rated power (MW)75Lognormal PDF parametersμ=6Rated wind power45Weibull PDF parametersα=9σ=0.6Weibull PDF parametersα=10β=2β=2Small hydro rated power (MW)5Gumbel PDF parametersλ=15γ=1.2Figure1 presents the probability density functions of the RES for the parameters stated in Table 3.Figure 1
PDFs at (a) Bus 5 Wind, (b) Bus 13 Wind, (c) Bus 11 Solar, and (d) Bus 13 Hydro.
(a)(b)(c)(d)


## 4. Methodology

Figure2 shows the methodological flow process adopted in this work for each of the two operating scenarios evaluated, normal and contingency conditions. After the RES uncertainty modeling, the MOMA algorithm parameters are set to ensure convergence to local optima. Six tests were performed to achieve the best performing control parameters for the algorithm. Then, 20 independent trials were evaluated for each case study performed. Fuzzy decision-making (FDM) tool was then employed to obtain the best performing trial run in each case. Details of MOMA and the FDM are described in this section. To obtain the overall best performing VSI, a computationally efficient index, the PSI is utilized, which aggregates the best alternative from the six case studies without deciding any relative importance between attributes.Figure 2
Methodology flow process chart.


### 4.1. Multiobjective Mayfly Algorithm (MOMA)

Zervoudakis and Tsafarakis developed Mayfly Algorithm in 2020 by mimicking the group behavior of mayflies, particularly their mating behavior [38]. Later in 2021, a multiobjective version was developed by Liu et al. in [51]. Unlike males, female mayflies do not gather in swarms; they instead fly toward males to breed. At the start of the algorithm, the mayflies are randomly divided into two separate groups: male and female populations. The mayflies are randomly scattered in a d-dimensional space, considered candidate solutions.Similar to the individuals in swarms of the PSO algorithm, the mayflies update the positions according to their current positionxt and velocity vt+1, as in the following equation:(17)xijt+1=xit+vijt+1.Since there is no single best solution in multiobjective problems, the selection ofgbest performed by picking a random solution from the repository of nondominated solutions.If the male mayfly is dominated by thegbest, (18) updates its velocity. Otherwise, the mayflies update from the current pbest with a random dance coefficient d.(18)vijt+1=vijt+1=g∗vijt+a1e−βrp2pbestij−xijt+a2e−βrg2gbestj−xijt,ifgbest>pbestg∗vijt+d∗r1,otherwise.Similarly, the movement of the female mayflies is also updated using Equation19.(19)vijt+1=g∗vijt+a2e−βrmf2xijt−yijt,if male dominates female,g∗vijt+d∗r2,otherwise,where pbest and gbest are the local optimal and global optimal, respectively; g is the gravity coefficient; a1, a2, are constants to balance the values; β is the fixed visibility coefficient; rp and rg are Cartesian distances from the i−th mayfly to the optimal local solution and the optimal global solution, respectively; r1 and r2 is the random number in uniform distribution and selected from the domain [−1, 1].Figure3 summarizes the flow chart of the multiobjective MayFly algorithm. The rest of the steps of the MOMA algorithm are detailed in [38, 51]Figure 3
Flow chart of multiobjective mayfly algorithm (MOMA).


### 4.2. Fuzzy Decision-Making Technique

To find out the best compromise solution among the nondominated solutions from the Pareto front is vital in the decision-making process. Fuzzy set theory has been widely utilized to effectively select a candidate Pareto-optimal solution from among the numerous alternative solutions on the Pareto front [6]. Due to the inherent irrationality nature of the decision-makers, the i−th objective function of a solution in the Pareto-optimal set, Fi, is represented by a membership function μi defined as follows [52]:(20)μi=1,Fi≤Fimin,Fimax−FiFimax−Fimin,Fimin≤Fi≤Fimax,0,Fi≥Fimax,where Fimax and Fimin are maximum and minimum values of the i−th objective function, respectively. The membership function values designate the achievement level of the objective functions of a problem, and these values are between 0 and 1. For each nondominated solution k, the normalized membership function μk is calculated as follows:(21)μk=∑i=1Nobjμik∑j=1M∑i=1Nobjμij.The number of nondominated solutions isM. The best compromise solution is the one having the highest value of μk.


## 4.1. Multiobjective Mayfly Algorithm (MOMA)

Zervoudakis and Tsafarakis developed Mayfly Algorithm in 2020 by mimicking the group behavior of mayflies, particularly their mating behavior [38]. Later in 2021, a multiobjective version was developed by Liu et al. in [51]. Unlike males, female mayflies do not gather in swarms; they instead fly toward males to breed. At the start of the algorithm, the mayflies are randomly divided into two separate groups: male and female populations. The mayflies are randomly scattered in a d-dimensional space, considered candidate solutions.Similar to the individuals in swarms of the PSO algorithm, the mayflies update the positions according to their current positionxt and velocity vt+1, as in the following equation:(17)xijt+1=xit+vijt+1.Since there is no single best solution in multiobjective problems, the selection ofgbest performed by picking a random solution from the repository of nondominated solutions.If the male mayfly is dominated by thegbest, (18) updates its velocity. Otherwise, the mayflies update from the current pbest with a random dance coefficient d.(18)vijt+1=vijt+1=g∗vijt+a1e−βrp2pbestij−xijt+a2e−βrg2gbestj−xijt,ifgbest>pbestg∗vijt+d∗r1,otherwise.Similarly, the movement of the female mayflies is also updated using Equation19.(19)vijt+1=g∗vijt+a2e−βrmf2xijt−yijt,if male dominates female,g∗vijt+d∗r2,otherwise,where pbest and gbest are the local optimal and global optimal, respectively; g is the gravity coefficient; a1, a2, are constants to balance the values; β is the fixed visibility coefficient; rp and rg are Cartesian distances from the i−th mayfly to the optimal local solution and the optimal global solution, respectively; r1 and r2 is the random number in uniform distribution and selected from the domain [−1, 1].Figure3 summarizes the flow chart of the multiobjective MayFly algorithm. The rest of the steps of the MOMA algorithm are detailed in [38, 51]Figure 3
Flow chart of multiobjective mayfly algorithm (MOMA).


## 4.2. Fuzzy Decision-Making Technique

To find out the best compromise solution among the nondominated solutions from the Pareto front is vital in the decision-making process. Fuzzy set theory has been widely utilized to effectively select a candidate Pareto-optimal solution from among the numerous alternative solutions on the Pareto front [6]. Due to the inherent irrationality nature of the decision-makers, the i−th objective function of a solution in the Pareto-optimal set, Fi, is represented by a membership function μi defined as follows [52]:(20)μi=1,Fi≤Fimin,Fimax−FiFimax−Fimin,Fimin≤Fi≤Fimax,0,Fi≥Fimax,where Fimax and Fimin are maximum and minimum values of the i−th objective function, respectively. The membership function values designate the achievement level of the objective functions of a problem, and these values are between 0 and 1. For each nondominated solution k, the normalized membership function μk is calculated as follows:(21)μk=∑i=1Nobjμik∑j=1M∑i=1Nobjμij.The number of nondominated solutions isM. The best compromise solution is the one having the highest value of μk.


## 5. Results and Discussion

To assess the performance of the voltage stability indices in a multiobjective OPF problem, a MOMA algorithm was executed on a modified IEEE 30-bus consisting of wind, solar PV, and small hydro sources. It was already established in [6] that three a three-objective VSC-OPF provided better system performance than the two ones. Therefore, each VSI is formulated as part of the triobjective function alongside generation cost and loss minimization objectives in this work. The tests were considered in two operating conditions, normal and contingency. The Best Compromise Solution was obtained using the Fuzzy decision-making theory [53], whereas to obtain a ranking of the case studies, a Preference Selection Index (PSI) was employed [6, 54]. The overall computational effectiveness of the VSIs is tested in terms of cost and transmission loss reduction, voltage stability improvement, simulation time, and support for RES penetration level.


### 5.1. Modified IEEE 30-Bus

The modified IEEE 30-bus presented in Figure4 and Table 4 was obtained from [39, 55]. The system comprises thirty buses, six generators, forty-one branches, and four transformers. Buses 1, 2, 5, 8, 11, and 13, indicate the locations of the generators. The active and reactive power loads are 283.4 MW and 126.2 MVAR, respectively.Figure 4
Modified IEEE 30-bus consisting of wind, solar, hydro, and thermal generators.Table 4
Details of the modified IEEE 30-bus.
BusTypePminPmaxGenerator cost coefficientsabcde1Thermal501400.0037520180.0372Thermal20800.017501.750160.0385Wind generator075—————8Thermal10350.008343.250120.04511Solar PV060—————13Wind generator + small hydro unit050—————


### 5.2. Case Studies

For all analyses, the MATLAB MATPOWER toolbox was employed on an Intel (R) Core (TM) i7-2640M CPU @2.8 GHz and (RAM) 8 GB computer. Six case studies were assessed to investigate the network’s performance under different conditions, as shown in Table5. The goal is to assess the network’s performance when different voltage stability indices are incorporated into the multiobjective function. The best performing VSI in each operating scenario is selected based on the Preference Selection index (PSI), whose formulations of PSI are detailed in [54]. The Best Compromise Solutions (BCS) generated from multiobjective studies using the Fuzzy Decision-Making technique are utilized to further evaluate the system performance.Table 5
Case studies and scenarios considered for the study.
Scenarios (SC)DetailsObjective FunctionsCase 1Case 2Case 3Case 4Case 5Case 6VCPILVSILmnFVSILQPNLSINormal operating conditions (SC-1)Baseload conditionsGeneration Cost ($/h)☑☑☑☑☑☑Power Loss (MW)VSILine outage contingency conditions (SC-2)Line 1-2 disconnected (it has the highest VSI, therefore weakest line)Generation Cost ($/h)☑☑☑☑☑☑Power Loss (MW)VSI


### 5.3. Multiobjective Optimization

This section presents the results of the triobjective optimization using the multiobjective Mayfly algorithm in two operating conditions.


#### 5.3.1. MOMA Convergence Characteristics

Heuristic techniques are sensitive to the proper selection of the control parameters. These parameters are selected through experiment. With the chosen parameters, an investigation was done to optimize the population sizes for the system. A total of 20 independent trials were carried out for each of the test runs. The population size in the present work is fixed at 20 particles in order to keep the computational requirements low. From Table6, it is clear that a population of 20 particles in 50 iterations provides satisfactory convergence characteristics. The selection of best parameters was established using fuzzy decision making, considering the contradicting objectives of cost, loss, and voltage stability. Table 6 shows that test run four (in bold) control parameters achieve the best compromise cost, loss, and voltage stability index for Case 1.Table 6
Parameter setting for MOMA.
TestMaxItPop No.Pop No. (males & females)Repository SizeCost ($/h)Loss (MW)VSI120202050858.97443.19750.3406220202080857.74733.52380.3336320202020850.83053.77960.3958450202020862.47962.96610.32855100202020856.5113.34310.3517620202020852.59793.36680.3635To verify the convergence characteristics of the MOMA with the selected parameters, OPF simulation was carried out with the optimum number of particles for each system for 50 iterations; the repository population variation vs. iterations is shown in Figure5. It can be seen that MOMA converges well in less than 50 iterations.Figure 5
Convergence characteristics at different iterations.To ensure consistency, the MOMA algorithm is run in 20 independent trials for each of the case studies of the multiobjective OPF. The best compromise solutions obtained in each test run trial undergo another best-case selection using the fuzzy decision technique. Figure6 shows the selection of the best compromise solution for case 1 in normal operating conditions. Details of the selected solution are provided in Table 7.Figure 6
20 independent trials run for case 1 SC-1.Table 7
Best compromise solutions from the different case studies in SC-1.
BusControl variablesBaseCase 1Case 2Case 3Case 4Case 5Case 6MinMaxVCPILVSILmnFVSILQPNLSI1PTH (MW)5014090.3991.29100.5264.93114.4491.732PTH (MW)208051.0457.2450.5638.1656.3058.365PW (MW)07570.3262.2847.5758.4858.9568.578PTH (MW)103522.3115.1018.7725.8622.9318.8711PPV (MW)06042.5044.8041.5054.8954.2640.6813PW+H (MW)05042.7338.9134.3335.9936.4340.23Pgen (MW)80440319.29309.62293.24278.31343.30318.441QTH (MVAR)−2015021.33116.2259.37117.82−0.3821.342QTH (MVAR)−2060−12.4523.5822.6141.7041.5527.375QW (MVAR)−30356.46−12.00−5.9720.60−8.06−7.298QTH (MVAR)−154020.810.7514.7830.2112.32−1.6811QPV (MVAR)−2025−1.290.14−1.5110.1918.548.5613QW+H (MVAR)−2530−2.881.37−3.70−9.44−9.008.19Qgen (MVAR)−13034031.97130.0585.58211.0854.9656.491VTH (p.u)0.951.11.11.08550.96721.0581.10.952VW (p.u)0.951.10.951.11.08041.07481.11.15VTH (p.u)0.951.11.11.09211.05661.07551.11.18VPV (p.u)0.951.11.11.11.05781.07551.11.111VW+H (p.u)0.951.10.950.98731.00120.99030.950.9513VTH (p.u)0.951.11.11.0861.09711.09881.11.1Cost ($/h)——852.46839.66809.68838.27846.88838.72Loss (MW)——2.553.034.262.862.612.93VSI——0.29430.08430.11980.13590.30690.2023Simulation Time (s)——45.70448.72547.23743.13244.95147.899PSI——0.95620.84700.79030.83160.89520.8547PSI ranking——146523


#### 5.3.2. Normal Operating Conditions (SC-1)

With the system under normal operating conditions, the results of the multiobjective optimization and their corresponding BCS (marked in red star) are presented in Figure7.Figure 7
SC-1 Pareto-optimal solutions for the six case studies.Results in Table7 indicate that the lowest costs were obtained in Case 3 by the Lmn index, followed by the NLSI index in Case 6. However, the case studies with the lowest costs also encounter the highest loss, as evidenced by the highest loss of 4.26 MW with Lmn index.The lowest costs (809.68 $/h), losses (2.55 MW), and simulation time (43.132 s) are realized in Case 3 (Lmn), Case 1 (VCPI), and Case 4 (FVSI), respectively. However, it is noteworthy that the Lmn index with the lowest costs also encounters the highest losses of 4.26 MW, 66.85% more than that of Case 1. Conversely, the VCPI index with the lowest losses also produces the highest costs. This is due to the fact that for losses to reduce, the system must generate more reactive or have the least consumption. Table3 shows that the VCPI index (Case1) has the lowest net consumption of reactive power (Qgen), hence the lowest losses (see Figure 8).Figure 8
SC-1 Reactive power generation from the different generators.The overall PSI ranking shows that VCPI is the most computationally efficient index, ranking highest with a PSI of 0.9562 in normal operating conditions for optimized system performance in terms of minimizing generation cost, loss, simulation time, and voltage stability improvement.


#### 5.3.3. Contingency Conditions (SC-2)

This study assesses voltage stability indices’ impact when line 1-2 is out of service. Contingency conditions are known to change the power flows in the system, thus resulting in increased loading of lines in some instances, leading to increased losses and costs. Figure9 shows the Pareto optimal solutions obtained from the different voltage stability indices.Figure 9
SC-2 Pareto-optimal solutions for the six case studies.From the graph in Figure9, it can be observed that VCPI and LQP indices have the highest generation costs. However, the Pgen, Qgen, and lowest losses are realized with VCPI, NLSI, and LVSI, respectively, as seen in Table 8. The FVSI index is still the index with the shortest simulation time, whereas the greatest voltage stability improvement of 89.9% is seen with the LQP index (compared to the base SC-2 LQP index of 3.7567).Table 8
Best compromise solutions from the different case studies in SC-2.
BusControl variablesBaseCase 1Case 2Case 3Case 4Case 5Case 6MinMaxVCPILVSILmnFVSILQPNLSI1PTH (MW)5014061.08113.14119.1084.4988.54100.362PTH (MW)208060.2366.3355.5851.8765.5144.925PW (MW)07556.4463.6242.6656.2162.4068.438PTH (MW)103522.5927.3517.1727.9018.4822.0211PPV (MW)06051.8843.7747.5450.0141.9349.7913PW+H (MW)05041.7439.8829.4340.1338.4436.96Pgen (MW)80440293.94354.08311.48310.60315.30322.481QTH (MVAR)−2015062.3474.8673.9242.8259.9427.182QTH (MVAR)−206045.1522.164.8425.6936.8733.665QW (MVAR)−30358.39−9.799.8714.75−18.11−10.188QTH (MVAR)−15404.2310.293.6515.543.296.8611QPV (MVAR)−2025−5.450.35−3.8612.6410.63−3.8613QW+H (MVAR)−25309.167.84−6.24−6.971.13−18.54Qgen (MVAR)−130340123.82105.7182.19104.4793.7635.121VTH (p.u)0.951.11.11.11.09971.09971.09711.12VW (p.u)0.951.11.11.11.07341.06251.09841.08655VTH (p.u)0.951.11.11.11.05921.06591.07561.06978VPV (p.u)0.951.11.11.11.04521.0621.11.064311VW+H (p.u)0.951.10.950.951.03890.9550.96281.092313VTH (p.u)0.951.11.11.11.08271.09751.09691.0979Cost ($/h)——851.13867.80832.23848.86852.22850.05Loss (MW)——3.542.927.683.933.893.92VSI——0.36410.08320.13550.13000.37670.1528Simulation Time (s)——43.60743.4141.73940.86745.26442.958PSI——0.86000.86330.81850.86530.85520.9290PSI ranking——436251Since different indices have varying advantages in reducing generation cost, loss and simulation time, and voltage stability improvement, the PSI index helps aggregate and rank the efficiency and performance of all case studies. Therefore, in contingency conditions, the index with the best-optimized system performance is the NSLI index with a PSI value of 0.9290. Figure10 affirms that the best performing index, NLSI, generates the most reactive power from renewable energy sources.Figure 10
SC-2 Reactive power generation from the different generators.


### 5.4. Impact on Voltage Stability

#### 5.4.1. Bus Voltage Magnitudes

Figures11 and 12 indicate the bus voltage magnitude performance resulting from the different case studies. FVSI and LQP indices achieve the highest average voltage profile improvement of 0.36% compared to the base case in normal operating conditions. On the other hand, the system voltage profile improves the most in the VCPI case study (by 1.42%) during line outage conditions.Figure 11
SC-1 bus voltage magnitudes for the different case studies.Figure 12
SC-2 bus voltage magnitudes for the different case studies.


#### 5.4.2. PV Curves

The best compromise solution obtained from the Pareto curves was used further to assess the maximum loadability of the different case studies.In normal operating conditions, it is observed from Figure13 that Case 6 (NLSI) gives the most significant load margin improvement of 10.17%. The rest of the indices improve the maximum loadability by 9.96%, 9.58%, 9.76%, 9.02%, 8.91% in Case 1, Case 2, Case 3, Case 4, and Case 5, respectively. The lowest loadability margin in SC-1 is realized with the FVSI index.Figure 13
Maximum loadability in SC-1 by PV curves at Bus 30.In contingency conditions, the maximum loadability of the system is improved by over 700% in all cases, as presented in Figure14. The greatest improvement was achieved with the LVSI index and the lowest with the Lmn index.Figure 14
Maximum loadability in SC-1 by PV curves at Bus 30.


### 5.5. Contribution of Renewable Energy Sources

Optimization of generation scheduling alters the system control variables, leading to varying generation power contributions from the renewables. High penetration of renewables reduces negative environmental impact compared to conventional fossil fuel-based energy generation.Figure15 presents the average contribution of power from the different RES for all cases studies in both operating conditions.Figure 15
Contribution of power from the RES.The detailed results in Table9 indicate that the NLSI, FVSI, and VCPI index led to the largest scheduled power of wind, solar PV, and hybrid wind/small hydro of 91.33%, 87.4%, and 84.5%, respectively. However, on a system average, the highest RES power-contributing VSI was the VCPI index, with the highest RES penetration level contribution of 16.61%.Table 9
Scheduled Power level of RES.
VSIWindPVWind and HydroAverage OverallContribution (MW)Scheduled Power (%)Penetration (%)Contribution (MW)Scheduled Power (%)Penetration (%)Contribution (MW)Scheduled Power (%)Penetration (%)Contribution (MW)Penetration (%)VCPI63.3884.5120.6747.1978.715.3942.23584.513.7782.5416.61LVSI62.9583.9318.9744.28573.813.3439.39578.811.8778.8414.73Lmn45.11560.1514.9244.5274.214.7231.8863.810.5466.0413.40FVSI57.34576.4619.4752.4587.417.8138.0676.112.9380.0016.74LQP60.67580.9018.4348.09580.214.6137.43574.911.3778.6414.80NLSI68.591.3321.3845.23575.414.1238.59577.212.0481.3115.84


## 5.1. Modified IEEE 30-Bus

The modified IEEE 30-bus presented in Figure4 and Table 4 was obtained from [39, 55]. The system comprises thirty buses, six generators, forty-one branches, and four transformers. Buses 1, 2, 5, 8, 11, and 13, indicate the locations of the generators. The active and reactive power loads are 283.4 MW and 126.2 MVAR, respectively.Figure 4
Modified IEEE 30-bus consisting of wind, solar, hydro, and thermal generators.Table 4
Details of the modified IEEE 30-bus.
BusTypePminPmaxGenerator cost coefficientsabcde1Thermal501400.0037520180.0372Thermal20800.017501.750160.0385Wind generator075—————8Thermal10350.008343.250120.04511Solar PV060—————13Wind generator + small hydro unit050—————


## 5.2. Case Studies

For all analyses, the MATLAB MATPOWER toolbox was employed on an Intel (R) Core (TM) i7-2640M CPU @2.8 GHz and (RAM) 8 GB computer. Six case studies were assessed to investigate the network’s performance under different conditions, as shown in Table5. The goal is to assess the network’s performance when different voltage stability indices are incorporated into the multiobjective function. The best performing VSI in each operating scenario is selected based on the Preference Selection index (PSI), whose formulations of PSI are detailed in [54]. The Best Compromise Solutions (BCS) generated from multiobjective studies using the Fuzzy Decision-Making technique are utilized to further evaluate the system performance.Table 5
Case studies and scenarios considered for the study.
Scenarios (SC)DetailsObjective FunctionsCase 1Case 2Case 3Case 4Case 5Case 6VCPILVSILmnFVSILQPNLSINormal operating conditions (SC-1)Baseload conditionsGeneration Cost ($/h)☑☑☑☑☑☑Power Loss (MW)VSILine outage contingency conditions (SC-2)Line 1-2 disconnected (it has the highest VSI, therefore weakest line)Generation Cost ($/h)☑☑☑☑☑☑Power Loss (MW)VSI


## 5.3. Multiobjective Optimization

This section presents the results of the triobjective optimization using the multiobjective Mayfly algorithm in two operating conditions.


### 5.3.1. MOMA Convergence Characteristics

Heuristic techniques are sensitive to the proper selection of the control parameters. These parameters are selected through experiment. With the chosen parameters, an investigation was done to optimize the population sizes for the system. A total of 20 independent trials were carried out for each of the test runs. The population size in the present work is fixed at 20 particles in order to keep the computational requirements low. From Table6, it is clear that a population of 20 particles in 50 iterations provides satisfactory convergence characteristics. The selection of best parameters was established using fuzzy decision making, considering the contradicting objectives of cost, loss, and voltage stability. Table 6 shows that test run four (in bold) control parameters achieve the best compromise cost, loss, and voltage stability index for Case 1.Table 6
Parameter setting for MOMA.
TestMaxItPop No.Pop No. (males & females)Repository SizeCost ($/h)Loss (MW)VSI120202050858.97443.19750.3406220202080857.74733.52380.3336320202020850.83053.77960.3958450202020862.47962.96610.32855100202020856.5113.34310.3517620202020852.59793.36680.3635To verify the convergence characteristics of the MOMA with the selected parameters, OPF simulation was carried out with the optimum number of particles for each system for 50 iterations; the repository population variation vs. iterations is shown in Figure5. It can be seen that MOMA converges well in less than 50 iterations.Figure 5
Convergence characteristics at different iterations.To ensure consistency, the MOMA algorithm is run in 20 independent trials for each of the case studies of the multiobjective OPF. The best compromise solutions obtained in each test run trial undergo another best-case selection using the fuzzy decision technique. Figure6 shows the selection of the best compromise solution for case 1 in normal operating conditions. Details of the selected solution are provided in Table 7.Figure 6
20 independent trials run for case 1 SC-1.Table 7
Best compromise solutions from the different case studies in SC-1.
BusControl variablesBaseCase 1Case 2Case 3Case 4Case 5Case 6MinMaxVCPILVSILmnFVSILQPNLSI1PTH (MW)5014090.3991.29100.5264.93114.4491.732PTH (MW)208051.0457.2450.5638.1656.3058.365PW (MW)07570.3262.2847.5758.4858.9568.578PTH (MW)103522.3115.1018.7725.8622.9318.8711PPV (MW)06042.5044.8041.5054.8954.2640.6813PW+H (MW)05042.7338.9134.3335.9936.4340.23Pgen (MW)80440319.29309.62293.24278.31343.30318.441QTH (MVAR)−2015021.33116.2259.37117.82−0.3821.342QTH (MVAR)−2060−12.4523.5822.6141.7041.5527.375QW (MVAR)−30356.46−12.00−5.9720.60−8.06−7.298QTH (MVAR)−154020.810.7514.7830.2112.32−1.6811QPV (MVAR)−2025−1.290.14−1.5110.1918.548.5613QW+H (MVAR)−2530−2.881.37−3.70−9.44−9.008.19Qgen (MVAR)−13034031.97130.0585.58211.0854.9656.491VTH (p.u)0.951.11.11.08550.96721.0581.10.952VW (p.u)0.951.10.951.11.08041.07481.11.15VTH (p.u)0.951.11.11.09211.05661.07551.11.18VPV (p.u)0.951.11.11.11.05781.07551.11.111VW+H (p.u)0.951.10.950.98731.00120.99030.950.9513VTH (p.u)0.951.11.11.0861.09711.09881.11.1Cost ($/h)——852.46839.66809.68838.27846.88838.72Loss (MW)——2.553.034.262.862.612.93VSI——0.29430.08430.11980.13590.30690.2023Simulation Time (s)——45.70448.72547.23743.13244.95147.899PSI——0.95620.84700.79030.83160.89520.8547PSI ranking——146523


### 5.3.2. Normal Operating Conditions (SC-1)

With the system under normal operating conditions, the results of the multiobjective optimization and their corresponding BCS (marked in red star) are presented in Figure7.Figure 7
SC-1 Pareto-optimal solutions for the six case studies.Results in Table7 indicate that the lowest costs were obtained in Case 3 by the Lmn index, followed by the NLSI index in Case 6. However, the case studies with the lowest costs also encounter the highest loss, as evidenced by the highest loss of 4.26 MW with Lmn index.The lowest costs (809.68 $/h), losses (2.55 MW), and simulation time (43.132 s) are realized in Case 3 (Lmn), Case 1 (VCPI), and Case 4 (FVSI), respectively. However, it is noteworthy that the Lmn index with the lowest costs also encounters the highest losses of 4.26 MW, 66.85% more than that of Case 1. Conversely, the VCPI index with the lowest losses also produces the highest costs. This is due to the fact that for losses to reduce, the system must generate more reactive or have the least consumption. Table3 shows that the VCPI index (Case1) has the lowest net consumption of reactive power (Qgen), hence the lowest losses (see Figure 8).Figure 8
SC-1 Reactive power generation from the different generators.The overall PSI ranking shows that VCPI is the most computationally efficient index, ranking highest with a PSI of 0.9562 in normal operating conditions for optimized system performance in terms of minimizing generation cost, loss, simulation time, and voltage stability improvement.


### 5.3.3. Contingency Conditions (SC-2)

This study assesses voltage stability indices’ impact when line 1-2 is out of service. Contingency conditions are known to change the power flows in the system, thus resulting in increased loading of lines in some instances, leading to increased losses and costs. Figure9 shows the Pareto optimal solutions obtained from the different voltage stability indices.Figure 9
SC-2 Pareto-optimal solutions for the six case studies.From the graph in Figure9, it can be observed that VCPI and LQP indices have the highest generation costs. However, the Pgen, Qgen, and lowest losses are realized with VCPI, NLSI, and LVSI, respectively, as seen in Table 8. The FVSI index is still the index with the shortest simulation time, whereas the greatest voltage stability improvement of 89.9% is seen with the LQP index (compared to the base SC-2 LQP index of 3.7567).Table 8
Best compromise solutions from the different case studies in SC-2.
BusControl variablesBaseCase 1Case 2Case 3Case 4Case 5Case 6MinMaxVCPILVSILmnFVSILQPNLSI1PTH (MW)5014061.08113.14119.1084.4988.54100.362PTH (MW)208060.2366.3355.5851.8765.5144.925PW (MW)07556.4463.6242.6656.2162.4068.438PTH (MW)103522.5927.3517.1727.9018.4822.0211PPV (MW)06051.8843.7747.5450.0141.9349.7913PW+H (MW)05041.7439.8829.4340.1338.4436.96Pgen (MW)80440293.94354.08311.48310.60315.30322.481QTH (MVAR)−2015062.3474.8673.9242.8259.9427.182QTH (MVAR)−206045.1522.164.8425.6936.8733.665QW (MVAR)−30358.39−9.799.8714.75−18.11−10.188QTH (MVAR)−15404.2310.293.6515.543.296.8611QPV (MVAR)−2025−5.450.35−3.8612.6410.63−3.8613QW+H (MVAR)−25309.167.84−6.24−6.971.13−18.54Qgen (MVAR)−130340123.82105.7182.19104.4793.7635.121VTH (p.u)0.951.11.11.11.09971.09971.09711.12VW (p.u)0.951.11.11.11.07341.06251.09841.08655VTH (p.u)0.951.11.11.11.05921.06591.07561.06978VPV (p.u)0.951.11.11.11.04521.0621.11.064311VW+H (p.u)0.951.10.950.951.03890.9550.96281.092313VTH (p.u)0.951.11.11.11.08271.09751.09691.0979Cost ($/h)——851.13867.80832.23848.86852.22850.05Loss (MW)——3.542.927.683.933.893.92VSI——0.36410.08320.13550.13000.37670.1528Simulation Time (s)——43.60743.4141.73940.86745.26442.958PSI——0.86000.86330.81850.86530.85520.9290PSI ranking——436251Since different indices have varying advantages in reducing generation cost, loss and simulation time, and voltage stability improvement, the PSI index helps aggregate and rank the efficiency and performance of all case studies. Therefore, in contingency conditions, the index with the best-optimized system performance is the NSLI index with a PSI value of 0.9290. Figure10 affirms that the best performing index, NLSI, generates the most reactive power from renewable energy sources.Figure 10
SC-2 Reactive power generation from the different generators.


## 5.3.1. MOMA Convergence Characteristics

Heuristic techniques are sensitive to the proper selection of the control parameters. These parameters are selected through experiment. With the chosen parameters, an investigation was done to optimize the population sizes for the system. A total of 20 independent trials were carried out for each of the test runs. The population size in the present work is fixed at 20 particles in order to keep the computational requirements low. From Table6, it is clear that a population of 20 particles in 50 iterations provides satisfactory convergence characteristics. The selection of best parameters was established using fuzzy decision making, considering the contradicting objectives of cost, loss, and voltage stability. Table 6 shows that test run four (in bold) control parameters achieve the best compromise cost, loss, and voltage stability index for Case 1.Table 6
Parameter setting for MOMA.
TestMaxItPop No.Pop No. (males & females)Repository SizeCost ($/h)Loss (MW)VSI120202050858.97443.19750.3406220202080857.74733.52380.3336320202020850.83053.77960.3958450202020862.47962.96610.32855100202020856.5113.34310.3517620202020852.59793.36680.3635To verify the convergence characteristics of the MOMA with the selected parameters, OPF simulation was carried out with the optimum number of particles for each system for 50 iterations; the repository population variation vs. iterations is shown in Figure5. It can be seen that MOMA converges well in less than 50 iterations.Figure 5
Convergence characteristics at different iterations.To ensure consistency, the MOMA algorithm is run in 20 independent trials for each of the case studies of the multiobjective OPF. The best compromise solutions obtained in each test run trial undergo another best-case selection using the fuzzy decision technique. Figure6 shows the selection of the best compromise solution for case 1 in normal operating conditions. Details of the selected solution are provided in Table 7.Figure 6
20 independent trials run for case 1 SC-1.Table 7
Best compromise solutions from the different case studies in SC-1.
BusControl variablesBaseCase 1Case 2Case 3Case 4Case 5Case 6MinMaxVCPILVSILmnFVSILQPNLSI1PTH (MW)5014090.3991.29100.5264.93114.4491.732PTH (MW)208051.0457.2450.5638.1656.3058.365PW (MW)07570.3262.2847.5758.4858.9568.578PTH (MW)103522.3115.1018.7725.8622.9318.8711PPV (MW)06042.5044.8041.5054.8954.2640.6813PW+H (MW)05042.7338.9134.3335.9936.4340.23Pgen (MW)80440319.29309.62293.24278.31343.30318.441QTH (MVAR)−2015021.33116.2259.37117.82−0.3821.342QTH (MVAR)−2060−12.4523.5822.6141.7041.5527.375QW (MVAR)−30356.46−12.00−5.9720.60−8.06−7.298QTH (MVAR)−154020.810.7514.7830.2112.32−1.6811QPV (MVAR)−2025−1.290.14−1.5110.1918.548.5613QW+H (MVAR)−2530−2.881.37−3.70−9.44−9.008.19Qgen (MVAR)−13034031.97130.0585.58211.0854.9656.491VTH (p.u)0.951.11.11.08550.96721.0581.10.952VW (p.u)0.951.10.951.11.08041.07481.11.15VTH (p.u)0.951.11.11.09211.05661.07551.11.18VPV (p.u)0.951.11.11.11.05781.07551.11.111VW+H (p.u)0.951.10.950.98731.00120.99030.950.9513VTH (p.u)0.951.11.11.0861.09711.09881.11.1Cost ($/h)——852.46839.66809.68838.27846.88838.72Loss (MW)——2.553.034.262.862.612.93VSI——0.29430.08430.11980.13590.30690.2023Simulation Time (s)——45.70448.72547.23743.13244.95147.899PSI——0.95620.84700.79030.83160.89520.8547PSI ranking——146523


## 5.3.2. Normal Operating Conditions (SC-1)

With the system under normal operating conditions, the results of the multiobjective optimization and their corresponding BCS (marked in red star) are presented in Figure7.Figure 7
SC-1 Pareto-optimal solutions for the six case studies.Results in Table7 indicate that the lowest costs were obtained in Case 3 by the Lmn index, followed by the NLSI index in Case 6. However, the case studies with the lowest costs also encounter the highest loss, as evidenced by the highest loss of 4.26 MW with Lmn index.The lowest costs (809.68 $/h), losses (2.55 MW), and simulation time (43.132 s) are realized in Case 3 (Lmn), Case 1 (VCPI), and Case 4 (FVSI), respectively. However, it is noteworthy that the Lmn index with the lowest costs also encounters the highest losses of 4.26 MW, 66.85% more than that of Case 1. Conversely, the VCPI index with the lowest losses also produces the highest costs. This is due to the fact that for losses to reduce, the system must generate more reactive or have the least consumption. Table3 shows that the VCPI index (Case1) has the lowest net consumption of reactive power (Qgen), hence the lowest losses (see Figure 8).Figure 8
SC-1 Reactive power generation from the different generators.The overall PSI ranking shows that VCPI is the most computationally efficient index, ranking highest with a PSI of 0.9562 in normal operating conditions for optimized system performance in terms of minimizing generation cost, loss, simulation time, and voltage stability improvement.


## 5.3.3. Contingency Conditions (SC-2)

This study assesses voltage stability indices’ impact when line 1-2 is out of service. Contingency conditions are known to change the power flows in the system, thus resulting in increased loading of lines in some instances, leading to increased losses and costs. Figure9 shows the Pareto optimal solutions obtained from the different voltage stability indices.Figure 9
SC-2 Pareto-optimal solutions for the six case studies.From the graph in Figure9, it can be observed that VCPI and LQP indices have the highest generation costs. However, the Pgen, Qgen, and lowest losses are realized with VCPI, NLSI, and LVSI, respectively, as seen in Table 8. The FVSI index is still the index with the shortest simulation time, whereas the greatest voltage stability improvement of 89.9% is seen with the LQP index (compared to the base SC-2 LQP index of 3.7567).Table 8
Best compromise solutions from the different case studies in SC-2.
BusControl variablesBaseCase 1Case 2Case 3Case 4Case 5Case 6MinMaxVCPILVSILmnFVSILQPNLSI1PTH (MW)5014061.08113.14119.1084.4988.54100.362PTH (MW)208060.2366.3355.5851.8765.5144.925PW (MW)07556.4463.6242.6656.2162.4068.438PTH (MW)103522.5927.3517.1727.9018.4822.0211PPV (MW)06051.8843.7747.5450.0141.9349.7913PW+H (MW)05041.7439.8829.4340.1338.4436.96Pgen (MW)80440293.94354.08311.48310.60315.30322.481QTH (MVAR)−2015062.3474.8673.9242.8259.9427.182QTH (MVAR)−206045.1522.164.8425.6936.8733.665QW (MVAR)−30358.39−9.799.8714.75−18.11−10.188QTH (MVAR)−15404.2310.293.6515.543.296.8611QPV (MVAR)−2025−5.450.35−3.8612.6410.63−3.8613QW+H (MVAR)−25309.167.84−6.24−6.971.13−18.54Qgen (MVAR)−130340123.82105.7182.19104.4793.7635.121VTH (p.u)0.951.11.11.11.09971.09971.09711.12VW (p.u)0.951.11.11.11.07341.06251.09841.08655VTH (p.u)0.951.11.11.11.05921.06591.07561.06978VPV (p.u)0.951.11.11.11.04521.0621.11.064311VW+H (p.u)0.951.10.950.951.03890.9550.96281.092313VTH (p.u)0.951.11.11.11.08271.09751.09691.0979Cost ($/h)——851.13867.80832.23848.86852.22850.05Loss (MW)——3.542.927.683.933.893.92VSI——0.36410.08320.13550.13000.37670.1528Simulation Time (s)——43.60743.4141.73940.86745.26442.958PSI——0.86000.86330.81850.86530.85520.9290PSI ranking——436251Since different indices have varying advantages in reducing generation cost, loss and simulation time, and voltage stability improvement, the PSI index helps aggregate and rank the efficiency and performance of all case studies. Therefore, in contingency conditions, the index with the best-optimized system performance is the NSLI index with a PSI value of 0.9290. Figure10 affirms that the best performing index, NLSI, generates the most reactive power from renewable energy sources.Figure 10
SC-2 Reactive power generation from the different generators.


## 5.4. Impact on Voltage Stability

### 5.4.1. Bus Voltage Magnitudes

Figures11 and 12 indicate the bus voltage magnitude performance resulting from the different case studies. FVSI and LQP indices achieve the highest average voltage profile improvement of 0.36% compared to the base case in normal operating conditions. On the other hand, the system voltage profile improves the most in the VCPI case study (by 1.42%) during line outage conditions.Figure 11
SC-1 bus voltage magnitudes for the different case studies.Figure 12
SC-2 bus voltage magnitudes for the different case studies.


### 5.4.2. PV Curves

The best compromise solution obtained from the Pareto curves was used further to assess the maximum loadability of the different case studies.In normal operating conditions, it is observed from Figure13 that Case 6 (NLSI) gives the most significant load margin improvement of 10.17%. The rest of the indices improve the maximum loadability by 9.96%, 9.58%, 9.76%, 9.02%, 8.91% in Case 1, Case 2, Case 3, Case 4, and Case 5, respectively. The lowest loadability margin in SC-1 is realized with the FVSI index.Figure 13
Maximum loadability in SC-1 by PV curves at Bus 30.In contingency conditions, the maximum loadability of the system is improved by over 700% in all cases, as presented in Figure14. The greatest improvement was achieved with the LVSI index and the lowest with the Lmn index.Figure 14
Maximum loadability in SC-1 by PV curves at Bus 30.


## 5.4.1. Bus Voltage Magnitudes

Figures11 and 12 indicate the bus voltage magnitude performance resulting from the different case studies. FVSI and LQP indices achieve the highest average voltage profile improvement of 0.36% compared to the base case in normal operating conditions. On the other hand, the system voltage profile improves the most in the VCPI case study (by 1.42%) during line outage conditions.Figure 11
SC-1 bus voltage magnitudes for the different case studies.Figure 12
SC-2 bus voltage magnitudes for the different case studies.


## 5.4.2. PV Curves

The best compromise solution obtained from the Pareto curves was used further to assess the maximum loadability of the different case studies.In normal operating conditions, it is observed from Figure13 that Case 6 (NLSI) gives the most significant load margin improvement of 10.17%. The rest of the indices improve the maximum loadability by 9.96%, 9.58%, 9.76%, 9.02%, 8.91% in Case 1, Case 2, Case 3, Case 4, and Case 5, respectively. The lowest loadability margin in SC-1 is realized with the FVSI index.Figure 13
Maximum loadability in SC-1 by PV curves at Bus 30.In contingency conditions, the maximum loadability of the system is improved by over 700% in all cases, as presented in Figure14. The greatest improvement was achieved with the LVSI index and the lowest with the Lmn index.Figure 14
Maximum loadability in SC-1 by PV curves at Bus 30.


## 5.5. Contribution of Renewable Energy Sources

Optimization of generation scheduling alters the system control variables, leading to varying generation power contributions from the renewables. High penetration of renewables reduces negative environmental impact compared to conventional fossil fuel-based energy generation.Figure15 presents the average contribution of power from the different RES for all cases studies in both operating conditions.Figure 15
Contribution of power from the RES.The detailed results in Table9 indicate that the NLSI, FVSI, and VCPI index led to the largest scheduled power of wind, solar PV, and hybrid wind/small hydro of 91.33%, 87.4%, and 84.5%, respectively. However, on a system average, the highest RES power-contributing VSI was the VCPI index, with the highest RES penetration level contribution of 16.61%.Table 9
Scheduled Power level of RES.
VSIWindPVWind and HydroAverage OverallContribution (MW)Scheduled Power (%)Penetration (%)Contribution (MW)Scheduled Power (%)Penetration (%)Contribution (MW)Scheduled Power (%)Penetration (%)Contribution (MW)Penetration (%)VCPI63.3884.5120.6747.1978.715.3942.23584.513.7782.5416.61LVSI62.9583.9318.9744.28573.813.3439.39578.811.8778.8414.73Lmn45.11560.1514.9244.5274.214.7231.8863.810.5466.0413.40FVSI57.34576.4619.4752.4587.417.8138.0676.112.9380.0016.74LQP60.67580.9018.4348.09580.214.6137.43574.911.3778.6414.80NLSI68.591.3321.3845.23575.414.1238.59577.212.0481.3115.84


## 6. Validation of Results

### 6.1. Validation of Algorithm

The results from the PSI ranking indicate the best voltage stability index to use in normal, and contingency conditions are VCPI and NLSI, respectively. Using these two indices, the results in normal conditions and contingency conditions were then validated against four other algorithms: multiobjective harris hawks algorithm (MHHO), multiobjective Jaya algorithm (MOJAYA), multiobjective particle swarm algorithm (MOPSO), and nondominated sorting genetic algorithm III (NSGA-III).Results from the two operating scenarios (See Table10 for SC-1 and Table 11 for SC-2) indicate that the multiobjective mayfly algorithm is the fastest, optimizing the system variables at an average speed of 106.72% faster than all other four algorithms. Additionally, the MOMA algorithm provided the greatest voltage stability improvement of 9.53%, 6.98%, 47.99%, 11.37% better than the MHHO, MOJAYA, MOPSO, and NSGAIII algorithms, respectively, in both normal and contingency operating conditions. On average, MOMA showed a dominant system performance during the two conditions (see Figure 16). This is attributed to the algorithm’s nuptial dance and random flight processes that enhance the balance between its exploration and exploitation properties and assist its escape from local optima [38].Table 10
Validation of SC-1 Case 1 MOMA results against other algorithms.
AlgorithmAlgorithm ParametersBest compromise SolutionsSimulation Time (s)PopulationIterationsPareto Solutions at Max. IterationCost ($/h)Loss (MW)VSIMOMA205020852.4642.5540.29445.704MHHO505050849.9072.3500.327122.035MOJAYA505025844.8262.7190.323567.395MOPSO505050863.5542.1890.311103.688NSGAIII505050842.7413.3700.35768.073MOPSO_FMF [19]50500—710.313.77——MOMA [21]———848.64865.1847——GROM [56]50100—851.237.017540.13789—TBLO [56]50100—854.04337.113660.13820—MOMVO [39]50500—797.20854.5205——Table 11
Validation of SC-2 Case 6 MOMA results against other algorithms.
AlgorithmAlgorithm ParametersBest compromise SolutionsSimulation Time (s)PopulationIterationsPareto Solutions at Max. IterationCost ($/h)Loss (MW)NLSIMOMA205020850.0533.9250.15342.958MHHO505050909.3281.9300.165124.217MOJAYA505027859.6363.3570.15979.673MOPSO505050859.6952.3360.29196.097NSGAIII505050922.6621.8300.15570.947Figure 16
Dominance of MOMA algorithm in average system performance during SC-1 and SC-2.


### 6.2. Validation of Proposed Methodology

The proposed multiobjective approach provides the lowest system losses and generation costs by comparing the system’s performance with previous research, as shown in Table12 and Figure 17. Therefore, it demonstrates better system performance.Table 12
Validation of the proposed approach.
ApproachDescriptionObjective FunctionAlgorithmVSI comparisonBest VSICost ($/h)Loss (MW)Maximum VSIRES?Proposed methodMulti-objective based on Pareto optimalityCost, Loss, VSIMOMAVCPI, LVSI, Lmn, FVSI, LQP, NLSIVCPI851.133.540.3641YesApproach 1 [7]Single objectiveCostImproved Particle SwarmL-index, FVSI, Lmn, LVSI, VCPI, FVSIFVSI845.1896.12—NoApproach 1 [7]Single objectiveLossImproved Particle SwarmL-index, FVSI, Lmn, LVSI, VCPI, FVSIVCPI978.003.269—NoApproach 1 [8]Single objectiveLVSIDA-PSOFVSI, Lmn, LVSILVSI971.554.720.8072NoApproach 1 [57]Single objectiveL-indexSymbiotic organisms search algorithm—L-index926.9234.55690.13671YesApproach 2 [27]Multi-objective based on weighted sum approach—Modified Interior Search——914.045.543—YesFigure 17
Comparison of the proposed approach against other methods.


## 6.1. Validation of Algorithm

The results from the PSI ranking indicate the best voltage stability index to use in normal, and contingency conditions are VCPI and NLSI, respectively. Using these two indices, the results in normal conditions and contingency conditions were then validated against four other algorithms: multiobjective harris hawks algorithm (MHHO), multiobjective Jaya algorithm (MOJAYA), multiobjective particle swarm algorithm (MOPSO), and nondominated sorting genetic algorithm III (NSGA-III).Results from the two operating scenarios (See Table10 for SC-1 and Table 11 for SC-2) indicate that the multiobjective mayfly algorithm is the fastest, optimizing the system variables at an average speed of 106.72% faster than all other four algorithms. Additionally, the MOMA algorithm provided the greatest voltage stability improvement of 9.53%, 6.98%, 47.99%, 11.37% better than the MHHO, MOJAYA, MOPSO, and NSGAIII algorithms, respectively, in both normal and contingency operating conditions. On average, MOMA showed a dominant system performance during the two conditions (see Figure 16). This is attributed to the algorithm’s nuptial dance and random flight processes that enhance the balance between its exploration and exploitation properties and assist its escape from local optima [38].Table 10
Validation of SC-1 Case 1 MOMA results against other algorithms.
AlgorithmAlgorithm ParametersBest compromise SolutionsSimulation Time (s)PopulationIterationsPareto Solutions at Max. IterationCost ($/h)Loss (MW)VSIMOMA205020852.4642.5540.29445.704MHHO505050849.9072.3500.327122.035MOJAYA505025844.8262.7190.323567.395MOPSO505050863.5542.1890.311103.688NSGAIII505050842.7413.3700.35768.073MOPSO_FMF [19]50500—710.313.77——MOMA [21]———848.64865.1847——GROM [56]50100—851.237.017540.13789—TBLO [56]50100—854.04337.113660.13820—MOMVO [39]50500—797.20854.5205——Table 11
Validation of SC-2 Case 6 MOMA results against other algorithms.
AlgorithmAlgorithm ParametersBest compromise SolutionsSimulation Time (s)PopulationIterationsPareto Solutions at Max. IterationCost ($/h)Loss (MW)NLSIMOMA205020850.0533.9250.15342.958MHHO505050909.3281.9300.165124.217MOJAYA505027859.6363.3570.15979.673MOPSO505050859.6952.3360.29196.097NSGAIII505050922.6621.8300.15570.947Figure 16
Dominance of MOMA algorithm in average system performance during SC-1 and SC-2.


## 6.2. Validation of Proposed Methodology

The proposed multiobjective approach provides the lowest system losses and generation costs by comparing the system’s performance with previous research, as shown in Table12 and Figure 17. Therefore, it demonstrates better system performance.Table 12
Validation of the proposed approach.
ApproachDescriptionObjective FunctionAlgorithmVSI comparisonBest VSICost ($/h)Loss (MW)Maximum VSIRES?Proposed methodMulti-objective based on Pareto optimalityCost, Loss, VSIMOMAVCPI, LVSI, Lmn, FVSI, LQP, NLSIVCPI851.133.540.3641YesApproach 1 [7]Single objectiveCostImproved Particle SwarmL-index, FVSI, Lmn, LVSI, VCPI, FVSIFVSI845.1896.12—NoApproach 1 [7]Single objectiveLossImproved Particle SwarmL-index, FVSI, Lmn, LVSI, VCPI, FVSIVCPI978.003.269—NoApproach 1 [8]Single objectiveLVSIDA-PSOFVSI, Lmn, LVSILVSI971.554.720.8072NoApproach 1 [57]Single objectiveL-indexSymbiotic organisms search algorithm—L-index926.9234.55690.13671YesApproach 2 [27]Multi-objective based on weighted sum approach—Modified Interior Search——914.045.543—YesFigure 17
Comparison of the proposed approach against other methods.


## 7. Conclusion

A comparison of the performance of six voltage stability indices in a multiobjective optimal power flow of a renewable energy integrated grid has been presented in this paper. The multiobjective mayfly algorithm minimized three objective functions: generation cost, active power loss, and voltage stability index. From the studies performed, the following deductions were made:(i)
The best VSI for improving system performance in normal operating conditions was the VCPI index, whereas it was the NLSI index in contingency conditions. However, an aggregated system performance in the two conditions indicated the VCPI is a superior index with an average PSI index of 0.9081, followed by the NSLI index of 0.8919.(ii)
The PSI best-performing indices are those with the least reactive power consumption in each operating scenario(iii)
The Lmn index ranked last in both operating conditions, resulting in the worst system performance in the highest generation cost, loss, and longest simulation time.(iv)
The VCPI index provided the largest average RES scheduled power of 82.54% to foster a high penetration of renewable energy sources on the system of 16.6%.(v)
There is a positive correlation between the increase in RES Scheduled Power and the average voltage stability enhancement of the system, as seen in the superior performance of the VCPI (highest average PSI rank) vs. its RES penetration levels.(vi)
The MOMA algorithm is superior to MOPSO, MOJAYA, NSGAIII, and MHHO algorithms in the studies carried out.(vii)
The proposed approach yields the lowest system costs and losses.Incorporating voltage stability in the optimal power flow problem enhances the voltage stability of power systems with the integration of renewable energy sources. Specifically, the VCPI and NLSI indices show dominant performance in normal and contingency conditions.The findings presented in this paper have substantial applicability in the planning of voltage stability enhancement for grids that incorporate renewable energy sources. Specifically, given the numerous voltage stability index computations, the approach in this work can serve as a practical guide for decision-makers when selecting indices that promote RES expansion and system stability improvement in modern energy management systems. Moreover, where cost and loss minimization are critical objectives, the voltage stability-constrained multiobjective optimal power flow assures that the aggregated system performance is improved across conflicting objectives.With the increasing smart technologies, the proposed approach in this work is computationally efficient, achieving fast enhanced system performance in offline applications. However, techniques that incorporate stability prediction and training for real-time or online security monitoring would be significant. For future studies, a comparison of stability indices using such machine learning techniques for the real-time operation of power systems can be studied. Also, considering the numerous distributed energy resources in practical systems, it is unknown how the various voltage stability indices would perform in networks with increasing plug-in electric vehicles or battery energy storage systems. Therefore such studies can also be performed.



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*Source: 7456333-2022-04-29.xml*