A Modified Frilled Lizard Optimizer for simultaneous allocation of DG, capacitor, and reconfiguration in the presence of EV stations in distribution systems

Introduction A crucial strategy for maximizing the performance of large-scale, intricate Distribution Systems (DSs) is the merging of Distributed Generation (DG), Capacitor Banks (CBs), and Distribution System Reconfiguration (DSR). When combined with DSR, DG and CBs improve network resilience, reduce active and reactive power losses, and improve voltage fluctuations. The goal of DSR is to determine the optimal topology of DSR based on the target functions that has been considered. Since there are multiple switching configurations from which the ideal network topology must be determined, optimization issues related to the DSR have been described as a non-differentiable, combinatorial, and restricted dilemma. The two primary groups of methods for tackling optimal DSR challenges are classical and metaheuristic algorithms. DSR with classical methods include mixed integer programming 1 , modified simplex method 2 , and mixed integer convex programming 3 . Recently developed algorithms have been developed for solving DSR problem. In 4 , the ant colony optimization has been applied for DSR; however, the method offered strong global search capability but required high computational effort and longer convergence time. Moreover, the DS with varying DSR has been considered using a Genetic Algorithm (GA) with adaptive population size as illustrated in 5 . The GA has improved diversity and avoids premature convergence, yet it has remained computationally intensive. In order to achieve lower line losses and better bus voltage profiles, a teaching–learning harmony search optimization algorithm has been demonstrated in 6 for optimum DG locations with DSR. The coyote algorithm has been employed in 7 , taking into account two scenarios: DSR alone and simultaneous DSR and DG deployment, and applied on IEEE 69- and 119-bus DSs in order to reduce the active power line losses. In 8 , the ant colony optimization algorithm (ACOA) has been developed for the best locations and sizes of DGs with RN and studied on IEEE 33- and IEEE 69-bus systems, where the ACOA dramatically lowers power loss, improves system reliability and increases voltage stability. In 9 , a Harmony Search Algorithm (HSA) has been used to minimize power loss, where it has focused on the sensitivity of the network to switch changes and DG placement of IEEE 33-bus and 69-bus DS systems. However, HSA often requires many iterations and can get trapped in local optima for larger, more complex systems. a Discrete Teaching–Learning-Based Optimization (DTLBO) algorithm has been introduced in 10 to improve the network topology of IEEE 33-bus, 69-bus, and 119-bus systems. However, the "teaching" and "learning" phases require careful balancing to prevent the algorithm from becoming computationally expensive. In 11 , an Adaptive Cuckoo Search (ACS) has been adopted to find the best DG locations and switch states of IEEE 33-bus and 69-bus systems. Nevertheless, ACS struggles to find the best solution because of the high dependency on the initial random population. In 12 Improved Binary Particle Swarm Optimizer (IBPSO) has been proposed to handle the "on/off" nature of switches and capacitor steps in the IEEE 33-bus and 69-bus systems for power loss reduction. A Harmony Search Algorithm (HSA) has been presented in 13 . It treats the search for the best grid configuration of switch positions and capacitor sizes in the DS 33-bus and 12.66 kV. An improved DSR method has been introduced in 14 for 33-bus, 69-bus, and a large 136-bus system. This made the simultaneous capacitor placement much faster by ignoring irrelevant parts of the grid. However, the pre-processing step to identify loops becomes extremely difficult and mathematically messy in highly interconnected urban grids. The influence of simultaneous optimum DG and CB sizes in DS with DSR has been explored by several researchers, who observed that the combination is more advantageous than the previously mentioned techniques. In 15 , The multi-criteria decision-making has been emerged with the non-dominated sorting genetic algorithm II (MCDM-NSGAII) in order to obtain the best location for DGs and CBs with DSR for 33- and 69-bus DSs. In 16 , the allocation of DGs and CBs is optimized in conjunction with DS with DSR using a modified artificial ecosystem optimizer (AEO) technique and applied on the Cairo DS. A comparative study using the grey wolf optimizer (GWO) approach for DG and CB implantation with DSR has been carried out in 17 to lower power loss and improve voltage profiles in a DS. In 18 , the DSR and optimal sizing of DGs and CBs in the DS have been demonstrated for 33-bus and 69-bus systems using a combination of bacterial foraging optimization and a fuzzy multi-objective approach to enhance voltage profiles and reduce power losses. Nevertheless, these problems are inherently high-dimensional and sensitive to load uncertainty, system constraints, and network topology, which complicates the optimization process. In 19 , a multi-objective optimization for the concurrent allocation of DGs, CBs, and (EV) charging stations within reconfigured DSs was presented. It employed a fuzzified objective function aimed at reducing line losses, improving power factor, minimizing voltage deviations, and maintaining DG penetration limits. The solution utilized an enhanced artificial hummingbird algorithm with centroid-based oppositional learning (COLAHA), tested on IEEE 69-bus and 118-bus systems. Results indicated that fuzzy-COLAHA outperforms traditional algorithms like the basic AHA arithmetic optimization algorithm (AOA), genetic algorithm (GA), and whale optimization algorithm (WOA) in speed and overall network performance, especially under complex conditions. However, the computational-budget fairness of the applied COLAHA comparison was not adequately established. The COLAHA generated additional opposition solutions and then evaluated them together with the original population. Therefore, equal population size and equal iteration number did not necessarily imply equal computational effort. In 20 , simultaneous optimal allocation of DGs and distribution static compensators (DSTATCOMs) while integrating DSR and EV charging stations was addressed using a hybrid African vulture optimizer (AVO) with GA operators (HAVOGO). It combined the exploration strength of the AVO with genetic operators to balance technical metrics, involving power loss reduction, voltage deviation minimization, voltage stability enhancement, and investment costs minimization. It was applied to large- scale 118-bus and 415-bus networks to mitigate the impacts of EV charging stations. However, several practical constraints were missing, such as branch thermal limits and explicit line-current or branch-apparent-power constraints, which are crucial for effective EV charging station planning due to significant localized loading effects. Complementing device placement studies, a systematic review analyzed smart inverters' role in coordinating DGs for optimal integration with PV-BESS and voltage stability 21 . It highlighted advancements in smart inverter functionalities such as Volt/VAr control, including smart inverters' effectiveness in reducing voltage violations and enhancing PV hosting capacity, with critical enablers being optimal inverter settings and advanced control algorithms. Also, unfair active power curtailment in high-PV-penetration networks was treated in 22 by focusing on operational control. In this work, a multi-objective optimization was handled that aimed at refining smart inverter Volt-Watt control parameters to mitigate the side effects of the conventional Volt-Watt control. Water Cycle Algorithm (WCA) was employed and compared to GA and PSO. However, the comparisons conducted were limited to the small IEEE 33-bus system. Furthermore, the established fairness may contradict the purpose of Volt-Watt control, which is location-dependent due to varying voltage sensitivity along a radial feeder. A PV unit situated at a weak, distant bus may require greater curtailment than a PV unit near the substation to address a specific voltage issue, indicating that uniform PV outputs could lead to physical inefficiencies. Moreover, a modified frilled lizard optimizer (MFLO) was designed in 23 for the efficient allocation of DGs, CBs, and DSR in radial DSs. In this study, six scenarios were tested on the IEEE 69-bus and a 141-bus system, showing superior convergence speed and solution quality, achieving substantial reduction in power losses on the larger system while significantly enhancing voltage stability and overall efficiency. Although this recent work has demonstrated the effectiveness of the MFLO, that framework was developed primarily for conventional radial DS operating conditions and did not explicitly formulate the additional demand and uncertainty associated with EV charging. Based on the literature review, significant research efforts have explored various combinations of DG and CB placement together with DSR to enhance the performance of DS. However, comprehensive investigations that simultaneously optimize DG and CB sizing and placement along with DSR, particularly under the uncertainty introduced by stochastic EV charging, remain limited. The stochastic nature of EV charging behaviour, characterized by random arrival times, charging durations, and state-of-charge levels, further increases the complexity of the optimization problem 24 . Consequently, employing a robust metaheuristic algorithm is essential to obtain accurate, stable, and computationally efficient solutions. The Frilled Lizard Optimizer (FLO), a bio-inspired metaheuristic introduced in 25 , is developed based on the adaptive survival and locomotion characteristics of frilled lizards. In this study, an enhanced version, termed Modified Frilled Lizard Optimizer (MFLO), is proposed to optimally determine the sizes and locations of DGs and CBs, in conjunction with DSR, while considering stochastic EV charging demand. Unlike the conventional FLO, which primarily relies on hunting and tree-climbing strategies, the MFLO integrates additional biologically inspired mechanisms that improve population diversity, convergence stability, and exploitation precision. Specifically, it incorporates a Defensive Strategy Phase, which emulates survival responses under uncertain operating conditions such as EV load fluctuations, and an Adaptive Local Search Phase, which models agile movement and refinement of candidate solutions. The present study advances optimization frameworks for EV-integrated DSs by integrating EV charging demands into network models. It examines how EV charging stations affect feeder loading, potentially worsening voltage deviations and altering the optimal placements for DGs and CBs, as well as network topology. Consequently, the contribution of the present study is not simply the application of MFLO to another test system; rather, it lies in formulating and solving a more comprehensive coordinated optimization problem in which DG deployment, reactive-power compensation, network topology, and EV charging demand are considered within a unified framework. The IEEE 69-bus and large-scale IEEE 141-bus studies are subsequently used to quantify the effect of EV charging on the optimal network configuration and to evaluate the capability of MFLO to maintain acceptable distribution-system performance under the resulting operating conditions. Thus, the main contributions of this study can be summarized as follows: An EV-aware problem formulation is presented with incorporation of EV charging demand into the radial DS optimization problem. A unified coordinated optimization of simultaneous optimization of DG placement/sizing, CB placement/sizing, and DSR are suggested while explicitly accounting for EV charging demand. Uncertainty/operating-condition is represented with consideration of different EV charging conditions rather than evaluating DG/CB/DSR only under conventional feeder loading. The proposed MFLO-based coordinated DG–CB–DSR framework is evaluated under EV penetration levels from 0 to 100% on the IEEE 69-bus and IEEE 141-bus systems. The analysis demonstrates its ability to maintain satisfactory voltage profiles under substantially different EV loading conditions and highlights the system-dependent relationship between EV penetration and network losses. The proposed framework is investigated under load-growth levels of 2.5–15% on the IEEE 141-bus system. MFLO adaptively determines the DG/CB allocations and feeder configuration for each loading condition, achieving substantial loss reduction while maintaining acceptable voltage magnitudes. This demonstrates the capability of the proposed framework to accommodate progressive demand growth in large distribution networks. Performance robustness is implemented through repeated stochastic optimization runs and statistical comparison of MFLO against competing algorithms under the EV-integrated cases. Extensive and statistically rigorous benchmarking: MFLO is compared with ten established and recent optimization algorithms, including Horned Lizard Optimization Algorithm (HLOA) 2024 26 , Particle Swarm Optimization (PSO) 27 , Pied Kingfisher Optimizer (PKO) 2024 28 , Harris Hawks Optimization (HHO) 29 , Equilibrium Optimizer (EO) 30 , Parrot optimizer 2024 31 , Birds of prey-based optimization (BPBO) 2025 32 , Crocodile Ambush Optimization Algorithm (CAOA) 2025 33 , and Baboon Algorithm 2026 34 , using identical population size, iteration number, and independent-run settings. In addition to conventional performance indicators, Levene, Kruskal–Wallis, Friedman, Wilcoxon signed-rank, Mann–Whitney U, Holm-adjusted significance, and effect-size analyses are employed. The results demonstrate that MFLO achieves the best numerical performance and high stochastic consistency, while statistically outperforming seven of the ten competing algorithms. The remainder of this paper is organized as follows. Section " DGs and capacitors with DSR integration in DS " presents the detailed problem formulation, including system objectives, constraints, and stochastic EV load modelling. Section " Proposed MFLO for DGs and CBs allocation and size with DSR integration " provides a comprehensive description of the conventional FLO and the proposed MFLO algorithms and their application to the optimal allocation problem. Section " Simulation results " discusses the simulation results and comparative performance analysis. Finally, Sect. 5 summarizes the main findings and conclusions of the study. DGs and capacitors with DSR integration in DS The primary objective of the optimization problem for the simultaneous allocation of DG and CB units with DSR integration in DS is to minimize total active power losses across the network. This objective is achieved by optimally determining the locations and ratings of DG and CB units and DSR while satisfying operational and technical constraints. The objective function can be expressed as: $$OBF = \sum\limits_{m = 1}^{{N_{Ln} }} {Losses_{m} } = \sum\limits_{\begin{subarray}{l} m = 1 \\ m = uv \end{subarray} }^{{N_{Ln} }} {G_{uv} \left( {V_{u}^{2} + V_{v}^{2} - V_{u} V_{v} \cos \theta_{uv} } \right)}$$ (1) where G uv provides the mutual conductance among buses ( u) and ( v) ; N Ln characterizes the cumulative distribution lines; θ uv illustrates the voltage angle difference between corresponding buses; V u and V v symbolize the voltage amplitudes at buses ( u) and ( v) . Furthermore, Losses m clarifies the active power losses associated with distribution line ( m ). Inequality constraints The optimization problem involves three categories of control variables: CB, DG units, and network DSR switches. CBs constraints The installation locations and sizes of CB units are subject to practical limitations. The candidate buses for capacitor installation are represented as integer decision variables, whereas capacitor ratings are represented by discrete variables according to commercially available capacitor sizes. These constraints can be formulated as: $$N_{Nodes} \ge CB_{Bus,v} \ge 1, v = 1,2,....N_{CB}$$ (2) $$CB_{\max } \ge CB_{S,v} \ge 0, v = 1,2,...N_{CB}$$ (3) where ( N CB ) represents the number of installed CBs, N Nodes denotes the total number of buses in DS, and ( CB Bus ) identifies the candidate buses for capacitor placement. Moreover, ( CB S ) and ( CB max ) denote the capacitor size and the maximum allowable capacitor rating, respectively. DGs constraints The placement and sizing of DG units are also constrained. The candidate buses for DG installation are represented by integer variables, while the DG power ratings are modeled as continuous decision variables. These constraints are expressed as: $$N_{DG} \ge DG_{Bus,u} \ge 1, u = 1,2,....N_{DG}$$ (4) $$DG_{\max } \ge DG_{S,u} \ge 0, u = 1,2,...N_{DG}$$ (5) where N DG signifies the number of DG units, ( DG Bus ) denotes the candidate buses for DG installation, and ( DGS ) and ( DG ma x ) represent the generated power and maximum allowable capacity ( u ) of each DG unit, respectively. DSR constraints The status of distribution switches is represented by integer variables. To maintain the radial structure of the DSs, a predetermined number of branches must remain open ( Open Ln ). This constraint is expressed as 35 : $$1 \le Open_{Ln,m} \le N_{Ln} , m = 1,2,...N_{Open}$$ (6) where N Open represents the number of branches that must remain open to preserve radiality, while N Ln denotes the total number of distribution lines in the system. Equality and inequality constraints To ensure secure and reliable operation of the DSs, several operational constraints must be satisfied, including bus-voltage limits, branch-current limits, and DG penetration restrictions 36 , 37 . These constraints are formulated as follows: $${\text{V}}_{m}^{{{\text{min}}}} \le {\text{V}}_{m} \le {\text{V}}_{m}^{{{\text{max}}}} {\text{ m}} = {1 , 2, }...{\text{ N}}_{{{\text{Nodes}}}}$$ (7) $$\left| {{\text{I}}_{m} } \right| \le {\text{I}}_{m}^{{{\text{max}}}} {\text{ m}} = {1 , 2, }...{\text{ N}}_{{{\text{Ln}}}}$$ (8) $$\sum\limits_{{u = {1}}}^{{{\text{N}}_{{{\text{DG}}}} }} {{\text{DG}}_{u} } \le {\text{KP}}\sum\limits_{{v = {1}}}^{{{\text{N}}_{{{\text{Nodes}}}} }} {{\text{(PD}}_{v} } {)}$$ (9) where V m represents the voltage magnitude at bus ( m ), while I and I max denote the branch current and its maximum thermal limit, respectively. The PD v stands for the active power demand at bus ( v ), and KP denotes the maximum allowable DG penetration level, which is typically limited to 60% of the total system demand 38 . In addition to that, active and reactive power balance constraints must be satisfied throughout the network. These equality constraints are expressed as: $${\text{P}}_{Sub} + \sum\limits_{{m = {1}}}^{{{\text{N}}_{{{\text{DG}}}} }} {{\text{DG}}_{m} } = \sum\limits_{{v = {1}}}^{{{\text{N}}_{{{\text{Nodes}}}} }} {{\text{PD}}_{v} } + \sum\limits_{{i = {1}}}^{{{\text{N}}_{{{\text{EVCS}}}} }} {{\text{P}}_{{avg,cs_{i} }} }$$ (10) $${\text{Q}}_{Sub} + \sum\limits_{{m = {1}}}^{{{\text{N}}_{CB} }} {{\text{CB}}_{m} } = \sum\limits_{{v = {1}}}^{{{\text{N}}_{{{\text{Nodes}}}} }} {{\text{QD}}_{v} }$$ (11) where QD v stands for the reactive power demand at bus (v), while P Sub and Q Sub denote the active and reactive power supplied by the substation, respectively; \({P}_{avg,c{s}_{i}}\) indicates the average hourly charging demand for station (i) while N EVCS refers to the number of EV stations. Equation ( 7 ) ensures that the voltage magnitude at each bus remains within the permissible operating limits, whereas Eq. ( 8 ) guarantees that the current flowing through each distribution branch does not exceed its thermal capacity. Equation ( 9 ) restricts the total DG penetration to the predefined allowable limit, thereby maintaining system security and operational stability 9 . Furthermore, Eqs. ( 10 ) and ( 11 ) enforce active and reactive power balance by ensuring that the combined power supplied by the substation, DG units, and capacitor banks adequately satisfies the total system demand. For the reason that no switching actions are considered under load variants, Eq. ( 12 ) indicates that the DSR problem is treated as a static one. In other words, based on this constraint, the adopted method does not perform any switching operations during an entire day. However, this limitation can also be interpreted as an additional objective aimed at reducing dynamic DS operations. Furthermore, maintaining the radial topology of the network is essential for operational reliability. Therefore, Eq. ( 12 ) 39 describes the formation of the branch–bus incidence matrix. $${\text{A}}_{{{\text{uv}}}} = \left\{ {\begin{array}{*{20}c} {\,\,\,\,\,\,\,{0, }\,\,\,\,\,\,\,\,\,\,L{\text{n u not connected with bus v }}} \\ { - {1, }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ Ln u enter to bus v }}} \\ {{1, }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L{\text{n u exits from bus v }}} \\ \end{array} } \right.$$ (12) EV charging demand and power requirements of charging stations (CSs) To incorporate EV-CSs into the DSs analysis, a set of CSs with predefined capacities is connected to selected buses. Since EV chargers are assumed to operate at unity power factor, only active power demand is considered. The charging demand and associated power requirements are estimated through three sequential steps. Step 1: estimation of charging demand The charging demand is first determined based on the total EV population and the charging participation ratio. Let ( \(totn\) ) denote the total number of EVs in the city and ( \(evR\) ) represent the fraction of EVs requiring charging under a given operating scenario. The daily number of EVs requiring charging is calculated as 40 : $$\begin{array}{cccc}& n{P}_{day}=totn\times evR& & \end{array}$$ (13) where different values of ( \(evR\) ) are used to represent various charging conditions and user behaviors. The EV charging period is assumed to extend from 12:00 to 18:00, following the charging-demand modeling approach adopted from 40 . A uniform distribution of charging activities over this six-hour period is used to provide a consistent representation of the aggregate charging demand, while a fast-charging duration of 20 min is considered for individual charging events. This assumption represents a simplified aggregate charging profile for distribution-system planning rather than an exact representation of every individual EV user's behavior. Actual charging patterns may vary with user behavior and operating conditions; therefore, the proposed framework is additionally evaluated under different EV penetration levels, with the planning solution considering the worst-case EV loading condition to provide adequate network support under high charging demand. Accordingly, the hourly charging demand is given by $$\begin{array}{cccc}& n{P}_{h}=\frac{n{P}_{day}}{{D}_{CH}}& & \end{array}$$ (14) where \({D}_{CH}\) =6 h. Considering fast chargers with a charging duration of ( \({t}_{CH}\) =20 min) ((1/3) h), the number of simultaneously charging EVs is estimated as 40 : $$\begin{array}{cccc}& nSim=n{P}_{h}\times {t}_{CH}& & \end{array}$$ (15) This quantity represents the simultaneous charging demand that must be supplied by the charging infrastructure. Step 2: allocation of charging demand among CSs Because CSs have different capacities, an equal allocation of EVs would lead to unequal utilization levels. Therefore, the charging demand is distributed proportionally according to station capacities. The loading ratio ( \({\alpha}_{i}\) ) of CS ( i ) is defined as 40 : $$\begin{array}{*{20}c} {} & {\alpha_{i} = \frac{{Cap_{i} }}{{\mathop \sum \nolimits_{j = 1}^{{N_{CS} }} Cap_{j} }}} & {} & {} \\ \end{array}$$ (16) where \(Ca{p}_{i}\) is the capacity of CS \(i\) and \({N}_{CS}\) is the total number of CSs. The number ( \({{N}_{CS}}_{i}\) ) of simultaneously charging EVs assigned to CS ( i ) is then obtained as $$\begin{array}{cccc}& {{N}_{CS}}_{i}=round\left({\alpha}_{i}\times nSim\right)& & \end{array}$$ (17) where nSim represents the entire number of charged EVs derived. The associated station utilisation is determined by $$\begin{array}{cccc}& {U}_{i}=\frac{{N}_{i}}{Ca{p}_{i}}& & \end{array}$$ (18) This capacity-proportional allocation ensures balanced loading among CS, preventing overloading of smaller stations while avoiding underutilization of larger ones. Step 3: CS-power demand After determining the number of charging EVs assigned to each station, the charging power demand can be evaluated. Each EV is assumed to receive a recharge equivalent to the usable battery capacity. Considering the battery capacity, the usable energy fraction, and the charging efficiency, the energy required from the grid ( \({E}_{EV}\) ) per charging event as 40 : $$\begin{array}{cccc}& {E}_{EV}=\frac{{E}_{bat}\times {UF}_{bat}}{{\eta}_{CH}}& & \end{array}$$ (19) where, \({E}_{bat}\) is the battery capacity considering 40 kWh, while \({UF}_{bat}\) is the usable battery fraction regarding the depth of discharge, considering 80% 41 . The average charging demand of station (i) is assessed hourly; therefore, as it occupies only one-third of the hourly interval, leading to the average hourly charging demand for station (i) as manifested in 40 , 41 : $$\begin{array}{cccc}& {{P}_{avg,CS}}_{i}={{N}_{CS}}_{i}\times {E}_{EV}\times {t}_{CH}& & \end{array}$$ (20) while the total charging demand of the charging network is obtained as 40 , 41 : $$\begin{array}{cccc}& {P}_{avg,tot}=\sum_{i=1}^{5}{P}_{avg,C{S}_{i}}& & \end{array}$$ (21) Since EV chargers operate at unity power factor, the reactive power demand is neglected as 40 , 41 : $$\begin{array}{cccc}& {Q}_{EV}=0& & \end{array}$$ (22) In the present formulation, EV uncertainty is represented at the aggregate charging-demand level. The charging participation ratio is varied to represent different EV charging conditions and user behaviors, and the resulting number of simultaneously charging EVs is used to determine the charging-station demand. The model assumes a 40-kWh battery capacity, 80% usable battery fraction, and 20-min fast-charging duration. Individual EV arrival time, departure time, and state-of-charge (SOC) are not modeled as separate stochastic variables; instead, their effects are aggregated into the charging-demand representation. The Hong point estimate method (PEM) 40 , 41 is used to characterize the uncertainty of the resulting EV charging demand, and the coordinated planning is performed considering the worst-case EV loading condition. Explicit joint modeling of arrival time, departure time, SOC, and charging demand will be considered in future work to provide a more detailed behavioral representation of EV charging uncertainty. Proposed MFLO for DGs and CBs allocation and size with DSR integration Mathematical representation and operational procedure of the FLO The algorithm abstracts the behavioral intelligence of these reptiles into a mathematical optimization framework capable of balancing global exploration and local exploitation throughout the search process. In FLO, each frilled lizard corresponds to a candidate solution within the multidimensional search domain, while the collective movement of the population represents the evolutionary search for the optimal solution. The optimization mechanism of FLO is primarily governed by two fundamental behavioral modes observed in frilled lizards: (i) Hunting behavior, and (ii) Tree-climbing behavior. Through the interaction of these two complementary mechanisms, the algorithm maintains an effective balance between diversification and intensification during the optimization process. Population initialization At the beginning of the optimization process ( Iter = 0), FLO randomly generates a population consisting of N f frilled lizards that act as search agents distributed throughout the search space. Each lizard is mathematically represented by a multidimensional position vector: $$F_{i} = \, \left[ {\begin{array}{*{20}c} {{\text{f}}_{i,1} ,} & {{\text{f}}_{i,2} ,} & {..........,} & {{\text{f}}_{i,Dim} } \\ \end{array} } \right]$$ (23) where, \({F}_{i}\) denotes the position vector of the \({i}^{th}\) frilled lizard; \(Dim\) represents the dimensionality of the optimization problem; \({f}_{i,j}\) corresponds to the value of the \({j}^{th}\) decision variable associated with the \({i}^{th}\) lizard; and \({f}_{i,j}\in \left[L{o}_{j},U{p}_{j}\right]\) indicates that each variable is constrained between predefined lower and upper search limits. $$F_{i} \left( {Iter = 0} \right){\text{ = Lo}} + \mathop {RD_{1} }\limits^{ \to } \times \left( {Up \, - \, Lo} \right); \, i = 1,2,....,N_{f}$$ (24) where, \(\overrightarrow{R{D}_{1}}\) represents a randomized vector with dimension \(Dim\) having random numbers between 0 and 1. After initialization, the quality of each lizard’s position is evaluated using an objective function \(f({F}_{i})\) that measures habitat suitability based on food resources, safety, and strategic advantage. FLO identifies the lizard with the best fitness value, known as the alpha lizard, which leads the group. Other lizards adjust their movements based on this dominant individual, facilitating collective learning and guiding the search towards the global optimum. Hunting behavior In the hunting stage, frilled lizards explore their environment for prey, adjusting their position toward the best discovered solution or a neighboring lizard with better fitness performance. The hunting movement is mathematically formulated as: $$F_{i}^{A} = F_{i} + z_{a} \times \left( {F_{\text{P}} - \left( {I \times F_{i} } \right)} \right)$$ (25) where, \({{F}_{i}}^{A}\) denotes the updated position generated through the hunting mechanism; F i represents the current position of the \({i}^{th}\) lizard; F P refers to the prey position selected from fitter candidate solutions; \({z}_{a}\in \left[\text{0,1}\right]\) is a stochastic coefficient controlling the movement strength and hunting responsiveness; \(I=round\left(1+rand\right)\in \left\{\text{1,2}\right\}\) is a behavioral control parameter that determines the pursuit style adopted by the lizard. A stochastic component \({z}_{a}\) adds variability to the movement, reflecting uncertain terrains, prey motion, and the lizard's visual perception and alertness. Following a position update, decision variables are checked against allowable boundaries. Variables exceeding limits are corrected to the nearest feasible bound. The fitness value associated with the new position, \(f({F}_{i}^{A})\) , is evaluated, and if it improves upon the current solution, the new position is accepted. $$F_{i} \left( {Iter + 1} \right) = \left\{ {\begin{array}{*{20}c} {F_{i}^{A} } \\ {F_{i} } \\ \end{array} } \right. \, \begin{array}{*{20}c} {if{\text{ f}}\left( {F_{i}^{A} } \right) \le f\left( {F_{i} } \right)} \\ {Else} \\ \end{array}$$ (26) This selection mechanism ensures that only beneficial hunting movements contribute to the evolutionary progress of the population. Tree-climbing behavior When a frilled lizard is unable to successfully capture prey or when it seeks protection from external threats, it transitions into a tree-climbing mode. In the optimization context, this behavior represents a localized exploitation strategy designed to refine previously discovered promising regions within the search space. The mathematical formulation of the tree-climbing phase is expressed as: $$F_{i}^{B} = F_{i} + \left( {1 - 2z_{b} } \right) \times \left( {\left( {Up - Lo} \right)/Iter} \right)$$ (27) During the climbing phase, \(F_{i}^{B}\) represents the lizard's new position, with \(z_{b}\) controlling climbing orientation and displacement. The term \(\left( {12z_{b} } \right)\) enables movement in both directions, simulating tree climbing behavior. The adaptive step size \(\left( {\left( {Up - Lo} \right)/Iter} \right)\) decreases over iterations, allowing for finer adjustments near promising solutions. Initially, the lizard makes larger exploratory movements, but as the search progresses, it becomes more controlled for accurate local exploitation, akin to gathering visibility and improving hunting or evading predators. After the climbing movement, the new position's fitness value \(f\left( {F_{i}^{B} } \right)\) is calculated and compared to the current solution. If it is superior, the new position replaces the previous one. $$F_{i} \left( {Iter + 1} \right) = \left\{ {\begin{array}{*{20}c} {F_{i}^{B} } \\ {F_{i} } \\ \end{array} } \right. \, \begin{array}{*{20}c} {if{\text{ f}}\left( {F_{i}^{B} } \right) \le f\left( {F_{i} } \right)} \\ {Else} \\ \end{array}$$ (28) The hunting and tree-climbing phases are repeatedly executed for all lizards until the stopping condition is satisfied, typically when the maximum allowable number of iterations \({Iter}_{m}\) is reached. Therefore, FLO balances diversification and intensification in optimization through iterative interactions between hunting-based exploration and tree-climbing-based exploitation. Proposed MFLO Within each iteration, frilled lizards probabilistically choose a behavioral mode based on a stochastic variable ( Rf ∈ [0,1]). The value of this variable determines whether the lizard performs hunting ( \({R}_{f}\le 0.25\) ), climbing ( \(0.25<{R}_{f}\le 0.5\) ), defensive action ( \(0.5<{R}_{f}\le 0.75\) ), or adaptive refinement ( \({R}_{f}>0.75\) ) during the current iteration. This mechanism allows lizards to adapt their behavior dynamically in response to environmental factors, making the MFLO search process behaviorally flexible and promoting varied movement patterns among search agents during optimization. Proposed defensive strategy The suggested MFLO introduces a defense phase inspired by frilled lizards' anti-predator behaviors, which utilize aggressive and evasive tactics such as frill expansion and camouflage. This phase adapts these natural behaviors into mechanisms that avoid poor search areas and escape suboptimal solutions by identifying the worst-performing search agent within the population. $$F_{enemy} \, = \, \arg \max_{i} f\left( {{\text{F}}_{i} } \right)$$ (29) where, \({F}_{enemy}\) denotes the position of the worst solution in the population, and \(f\left({F}_{i}\right)\) represents the fitness value of the \({i}^{th}\) lizard. This worst solution is interpreted biologically as a predator zone, hazardous area, or failed hunting region that should be avoided by the remaining lizards. To estimate the level of danger faced by each lizard, the Euclidean distance between the current lizard and the enemy position is calculated as: $$Dist_{i,enemy} \, = \, \left\| {{\text{F}}_{i} - {\text{F}}_{enemy} } \right\| \,$$ (30) where, \({Dist}_{{\varvec{i}},{\varvec{e}}{\varvec{n}}{\varvec{e}}{\varvec{m}}{\varvec{y}}}\) measures the threat proximity, while smaller values indicate higher danger levels and larger values indicate safer regions. Depending on this threat distance, the lizard activates one of three biologically inspired defensive responses. Frill display mechanism When the lizard detects severe danger and is extremely close to the enemy region, it performs a sudden frill expansion to intimidate predators and rapidly reposition itself. This behavior is mathematically modeled as a large random exploratory burst: $$F_{i}^{C} = F_{i} + \frac{1}{4} \times \left( {\mathop {RD_{2} }\limits^{ \to } - \frac{1}{2}} \right)\left( {Up - Lo} \right){\text{ if }}Dist_{i,enemy} < 0.25 \times \frac{{\left\| {Up - Lo} \right\|}}{3} \,$$ (31) where, \(\overrightarrow{R{D}_{2}}\) represents a randomized vector with dimension \(Dim\) having random numbers between 0 and 1. The sudden displacement reflects the defensive visual display and aggressive expansion behavior exhibited by frilled lizards under direct threat. This mechanism introduces strong diversification into the population and enables the algorithm to abruptly escape poor local regions. Erratic escape mechanism For moderate threat levels, the lizard executes rapid and unpredictable escape movements to avoid predators. This behavior is formulated as: $$F_{i}^{C} = F_{i} + \left( {\frac{Up - Lo}{{10}}} \right) \times \left( {1 + 0.2z_{c} } \right)\left( {\frac{{F_{i} - F_{enemy} }}{{\left\| {F_{i} - F_{enemy} } \right\| + \varepsilon }}} \right){\text{ if }}Dist_{i,enemy} < 0.5 \times \frac{{\left\| {Up - Lo} \right\|}}{3}$$ (32) The repulsive direction away from the enemy simulates instinctive predator avoidance behavior, while the stochastic perturbation mimics chaotic sprinting and unpredictable directional changes observed in real frilled lizards. This mechanism helps MFLO avoid premature convergence by actively steering agents away from unfavorable search regions. Camouflage mechanism When the perceived threat level is relatively low, the lizard adopts camouflage behavior by minimizing movement and performing only slight positional adjustments. This is represented mathematically by: $$F_{i}^{C} = F_{i} + 0.01 \times \left( {Up - Lo} \right) \times \mathop {RD_{3} }\limits^{ \to } {\text{ f }}Dist_{i,enemy}> 0.5 \times \frac{{\left\| {Up - Lo} \right\|}}{3}$$ (33) where, \(\overrightarrow{R{D}_{3}}\) represents a randomized vector with dimension \(Dim\) having random numbers between 0 and 1. Camouflage corresponds to hiding behavior and body stillness used by frilled lizards to reduce visibility and avoid unnecessary exposure to predators. This mechanism performs delicate local refinements while preserving stability near promising regions. Proposed adaptive local search The proposed MFLO utilizes an adaptive local search mechanism inspired by frilled lizards, improving agility in both hunting and defensive phases. It adapts its methods according to iteration progress, population diversity, search stability, and convergence state, transitioning from broad exploration in early iterations to focused exploitation in later stages. The iteration progress ratio ( p ) as: \(p=Iter/{Iter}_{m}\) , regulates the balance between exploration and exploitation during optimization. For average-performing lizards, the refinement strategy involves leveraging the globally best solution, nearby teammates, and directional movement adjustments. $$F_{i}^{D} = \left\{ {\begin{array}{*{20}c} {F_{i} + \gamma \times \left( {\frac{1}{2}\left( {F_{P} + F_{r} } \right) - F_{i} } \right) + \eta \times \left( {\mathop {RD}\limits^{ \to }_{4} - 0.5} \right) \times \left( {Up - Lo} \right)} \\ {F_{P} + \lambda \times \left( {F_{r} - F_{i} } \right)} \\ {F_{P} + \rho \times \left( {F_{r} - F_{i} } \right) - \eta_{tail} \times \left( {F_{enemy} - F_{i} } \right)} \\ {F_{P} + z_{e} \times \left( {F_{r} - F_{i} } \right) + \beta \times \Delta F_{p} } \\ \end{array} } \right. \, \begin{array}{*{20}c} {p \le 0.3{\text{ \& z}}_{{\text{d}}} \le {0}{\text{.5}}} \\ {p \le 0.3{\text{ \& z}}_{{\text{d}}} {> 0}{\text{.5}}} \\ {0.3 < R_{f} \le 0.7} \\ {Else} \\ \end{array}$$ (34) where, \({F}_{i}\) , \({F}_{P}\) , \({F}_{r}\) and \({F}_{enemy}\) are the current, best, random, and worst solution lizard positions, respectively; where, \(\overrightarrow{R{D}_{4}}\) represents a randomized vector with dimension \(Dim\) having random numbers between 0 and 1; γ and η control agility and burst magnitude; \(\lambda\) is a random pursuit vector to display stochastic variations; \(\rho\) denotes the attraction gain representing the forward acceleration factor; \({\eta}_{tail}\) is the tail correction factor which tracks consecutive rejections of unsuccessful moves; z e symbolizes a random pursuit rate representing the stochastic \(\text{v}\text{a}\text{r}\text{i}\text{a}\text{t}\text{i}\text{o}\text{n} \text{i}\text{n} \text{s}\text{p}\text{r}\text{i}\text{n}\text{t} \text{s}\text{p}\text{e}\text{e}\text{d}\) ; and \(\beta\) is the tail inertia coefficient, maintaining stability while accelerating toward the prey. This formulation of Eq. ( 34 ) allows the lizard to adjust its trajectory through cooperative information and directional diversity. It encompasses rapid trajectory adjustments (First condition in Eq. ( 34 )), body stabilization via tail dynamics (Second condition in Eq. ( 34 )), agile pursuit maneuvers (Third condition in Eq. ( 34 )), and adaptive locomotion (Last condition in Eq. ( 34 )). The algorithm switches among refinement variants based on the progress ratio p . Early in the search, agility-driven movements enhance exploration, while later stages focus on tail-assisted stabilization and momentum control for precise local exploitation. This reflects the behavioral evolution of frilled lizards, transitioning from energetic motion in uncertainty to controlled pursuit near a target. In the designed MFLO, the proposed strategies include four key mechanisms. The first condition strategy emulates the frilled lizard's swift interception of prey through acceleration bursts and body redirection. It integrates the best prey position, teammate information, and random acceleration, guiding the lizard toward the optimal prey location while allowing for agile deviations. This simulates the lizard's explosive sprinting behavior in uncertain situations. In contrast, the second condition strategy models the lizard's zigzag movement during high-speed chases, incorporating sinusoidal oscillations for directional changes and momentum terms for tail-assisted balance, promoting effective exploration and smooth convergence. The third mechanism helps lizards avoid unfavorable areas by using a tail-mediated directional correction based on past movements. It combines attraction toward beneficial regions with repulsion from poor ones, mimicking the lizard's instinct to adjust direction after failed attempts. This increases population diversity and prevents stagnation in optimization. The fourth strategy, inspired by frilled lizards, enhances movement continuity and stabilizes transitions during prey capture. It combines momentum and attraction to neighboring lizards to improve efficiency near optimal areas. Biologically, it reflects how frilled lizards stabilize during pursuit, while in optimization, it boosts convergence precision. Boundary maintenance In the proposed MFLO, the position of each lizard is dynamically updated using one of the four behavioral mechanisms introduced previously. Depending on the selected behavioral mode, the \({i}^{th}\) lizard may generate a new candidate position through the hunting strategy, the tree-climbing strategy, the defensive mechanism, or the adaptive local search phase. Accordingly, the newly generated position of the lizard can generally be represented as: $$F_{i}^{New} \in \left\{ {F_{i}^{A} ,F_{i}^{B} ,F_{i}^{C} ,F_{i}^{D} } \right\}$$ (35) where, \({F}_{i}^{A}\) denotes the position generated using Eq. ( 25 ), \({F}_{i}^{B}\) represents the obtained position from Eq. ( 27 ), \({F}_{i}^{C}\) corresponds to the defensive movement generated through Eqs. ( 31 )–( 33 ), \({F}_{i}^{D}\) refers to the adaptive local search position generated using Eq. ( 24 ). The selection is implemented via F i A . $$F_{i} ^{D} = \left\{ {\begin{array}{*{20}c} {F_{i} + \gamma \times \left( {\frac{1}{2}\left( {F_{P} + F_{r} } \right) - F_{i} } \right) + \eta \times \left( {\mathop {RD}\limits^{ \to } _{4} - 0.5} \right) \times \left( {Up - Lo} \right)} \\ {F_{P} + \lambda \times \left( {F_{r} - F_{i} } \right)} \\ {F_{P} + \rho \times \left( {F_{r} - F_{i} } \right) - \eta _{{tail}} \times \left( {F_{{enemy}} - F_{i} } \right)} \\ {F_{P} + z_{e} \times \left( {F_{r} - F_{i} } \right) + \beta \times \Delta F_{p} } \\ \end{array} } \right.{\text{ }}\begin{array}{*{20}c} {p \le 0.3{\text{ }}\& {\text{ }}z_{d} \le 0.5} \\ {p \le 0.3{\text{ }}\& {\text{ }}z_{d}> 0.5} \\ {0.3 < R_{f} \le 0.7} \\ {Else} \\ \end{array}$$ (36) This adaptive switching mechanism allows the lizard population to continuously alternate between exploration-oriented and exploitation-oriented behaviors according to the optimization state and environmental conditions. Since the generated positions may occasionally exceed the feasible search limits, a boundary maintenance procedure is employed to preserve the validity of all candidate solutions. After each movement update, every decision variable is examined independently, and any violated component is corrected by projecting it back into the allowable search interval. The boundary correction mechanism is mathematically expressed as: $$F_{i,d}^{new} = \left\{ {\begin{array}{*{20}c} {{\text{Lo}}_{d} } \\ {{\text{UB}}_{d} } \\ {F_{i,d}^{new} } \\ \end{array} } \right. \, \begin{array}{*{20}c} {if \, F_{i}^{new} \le {\text{Lo}}_{d} } \\ {if \, F_{i}^{new} \ge {\text{Up}}_{d} } \\ {Else} \\ \end{array}$$ (37) After generating and correcting the new candidate position, the fitness value associated with the updated lizard location is evaluated using the objective function as \(f\left({F}_{i}^{new}\right)\) . The greedy selection mechanism is defined as: $$F_{i} \left( {Iter + 1} \right) = \left\{ {\begin{array}{*{20}c} {F_{i}^{new} } \\ {F_{i} } \\ \end{array} } \right. \, \begin{array}{*{20}c} {if{\text{ f}}\left( {F_{i}^{new} } \right) \le f\left( {F_{i} } \right)} \\ {Else} \\ \end{array}$$ (38) This selection mechanism ensures that only beneficial movements contribute to the evolutionary progress of the population while preserving high-quality solutions discovered during previous iterations. The complete operational flowchart of the proposed MFLO algorithm is illustrated in Fig. 1 . Fig. 1 Proposed MFLO steps. Theoretical distinction and contributions of the proposed MFLO The original FLO establishes its search process through two principal behavioral mechanisms: hunting and tree-climbing. Hunting primarily provides global exploration by moving candidate solutions toward promising prey locations while retaining stochastic perturbations, whereas tree-climbing provides local exploitation through progressively smaller movements around promising regions. Consequently, the search dynamics of conventional FLO are mainly governed by the interaction between a global prey-oriented movement and a gradually intensified local refinement mechanism. The tree-climbing step uses a decreasing movement amplitude to transition from relatively broad movements during the early iterations to finer adjustments during the later iterations. In the proposed MFLO, two modifications are involved, which address different failure modes of population-based optimization. The defensive strategy phase primarily addresses loss of diversity and entrapment in unfavorable regions, whereas the adaptive local search phase primarily addresses insufficient exploitation and inaccurate refinement around promising regions. Consequently, MFLO does not simply replace FLO's original mechanisms. Instead, it augments them with two complementary search dimensions. The defensive strategy phase aims to prevent stagnation and promote diversification in population dynamics by addressing the risks of local attraction in conventional FLO. It identifies the worst-performing member of the population and uses its neighborhood as a negative search indicator. This method distinguishes between safe and unfavorable regions, mitigating premature contraction in multimodal search spaces by redistributing agents concentrated in undesirable areas. The defensive phase contains three different responses according to the estimated threat level. For severe threat, the Frill Display mechanism generates a large stochastic displacement. Its theoretical purpose is macro-diversification, i.e., increasing the probability of crossing the boundary of the current attraction basin. For moderate threat, the Erratic Escape mechanism combines repulsion from the unfavorable region with stochastic perturbation, providing an intermediate diversification level. For relatively low threat, the Camouflage mechanism produces only a small positional adjustment and therefore behaves as a conservative local perturbation. Thus, the three defensive responses indicate a hierarchy of perturbation amplitudes, serving as an adaptive diversification mechanism instead of relying on fixed random mutations. Large perturbations facilitate significant displacement, while small perturbations maintain promising areas when only minor adjustments are needed. The second major modification is the adaptive local search phase, which is designed to strengthen exploitation without eliminating directional diversity. In contrast with the original FLO tree-climbing mechanism, which progressively reduces the movement amplitude as the optimization proceeds, the proposed adaptive local search phase combines several sources of information and uses different refinement mechanisms according to the optimization state. The proposed local-search mechanism identifies several adaptive factors, including agility, burst magnitude, pursuit behavior, attraction gain, tail correction, pursuit rate, and tail inertia. From an optimization perspective, these components provide four complementary functions. First, best-solution attraction introduces intensification, which increases the probability of exploiting the best region identified so far. Second, neighborhood information prevents every individual from following exactly the same trajectory. The inclusion of randomly selected teammates produces directional diversity and reduces excessive dependence on a single global-best solution. Third, stochastic directional perturbation allows the local search to investigate neighboring regions rather than performing deterministic gradient-like movement toward the current best solution. Fourth, tail correction and inertia introduce a form of short-term search memory. The correction mechanism responds to unsuccessful movements, whereas the inertia component preserves directional continuity. The manuscript specifically describes the tail correction as tracking consecutive rejected movements and the tail inertia as maintaining stability during acceleration toward promising regions. This is different from merely reducing the step size of the original tree-climbing operator. The original FLO mainly obtains exploitation through progressively smaller movements, whereas the adaptive local search phase introduces directionally informed and state-dependent local refinement. The four refinement conditions described in the manuscript provide different search behaviors. The first combines the global best position, teammate information, and stochastic acceleration; the second introduces oscillatory directional changes and momentum; the third uses attraction toward beneficial regions together with correction away from unfavorable regions; and the fourth combines momentum and neighboring-agent attraction to improve stability near promising solutions. The main difference between MFLO and recent metaheuristic algorithms lies in its information architecture rather than its biological metaphor. While global-best attraction algorithms enhance search intensity but risk population collapse, and random mutation algorithms maintain diversity yet may lack mechanisms for avoiding poor regions, MFLO integrates these attributes. It utilizes positive information from optimal and neighboring solutions alongside negative information from the worst solution, adapting based on threat proximity and optimization progress. This approach combines population-state and search-history information, focusing on a multifaceted search direction rather than a singular attractor or fixed perturbation rule. Simulation results Two DSs, namely the IEEE 69-bus system 42 and a practical large-scale 141-bus system located in Caracas 43 , Venezuela, are utilized to evaluate the performance of the proposed MFLO approach. The IEEE 69-bus system serves as a benchmark for medium-scale optimization problems, whereas the practical 141-bus system is employed to examine scalability and effectiveness under higher system complexity. To evaluate the impact of EV charging stations on DS performance, different EV penetration levels and charging capacities are considered for both the IEEE 69-bus and the practical 141-bus systems 44 . In this analysis, Table 1 shows the investigated parameters, where four CS with capacities of 15, 20, 25, and 30 electric vehicles (EVs) are analyzed. The evaluation focuses on a worst-case scenario where the charging participation ratio is maximized ( \(evR\) =1), indicating that all EVs require charging throughout the day. The findings illustrate the utilization and charging power demand under this scenario, highlighting the criticality of assessing the adequacy of CS capacities and their effects on the DSs. Table 1 Parameters of the CSs used in the analysis. Full size table Table 2 presents the utilization and power demand of the CSs under the worst-case operating scenario. As shown, the total CS capacity is 90 EVs, with a simultaneous charging demand of 84 EVs. A capacity-proportional allocation strategy distributes this demand across four stations, allowing for 14, 19, 23, and 28 EVs at CS1, CS2, CS3, and CS4, respectively. Utilization levels are 93.33%, 95.00%, 92.00%, and 93.33%, with an average factor of 93.42%, indicating effective use without overload. Table 2 Utilization and power demand of the CSs for the Worst-Case Scenario. Full size table Results indicate that the largest CS (CS4) handles about double the demand of the smallest station (CS1), illustrating the proportional relationship between capacity and demand. This confirms the effectiveness of the allocation strategy in promoting fair utilization, preventing smaller stations from becoming overloaded while larger ones are underused, thus enhancing infrastructure efficiency and service quality. Also, Table 2 outlines the charging power demand in a worst-case scenario, where demand rises with the number of EVs charging at each station. Specific contributions are 497.78 kW for CS1, 675.56 kW for CS2, 817.78 kW for CS3, and 995.56 kW for CS4, with the largest station accounting for about 33.3% of total demand. Overall, the maximum total charging demand reaches 2.987 MW, crucial for DS planning and reinforcement. These test scenarios are designed to examine the effectiveness, scalability, and robustness of the proposed MFLO algorithm in minimizing power losses and improving system performance in the presence of EV charging demand. In this study, three test cases are implemented for both the IEEE 69-bus DS and the 141-bus DS in Caracas as follows: Case 1: Simultaneous DSR with DG allocations Case 2: Simultaneous DSR with CB allocations Case 3: Simultaneous DSR with DG and CB allocations Case 4: Analysis of EV Penetration Levels For all simulation cases, a population of 50 search agents and a maximum of 200 iterations were adopted for each optimization run. To ensure statistically reliable performance assessment, each algorithm was independently executed 25 times using different random seeds, and the best, mean, worst, and standard deviation of the objective-function values were subsequently evaluated. The planning problem considered the simultaneous installation of up to five DG units and five CBs. The maximum permissible capacity of each DG unit was set to 5000 kW, whereas the maximum capacity of each CB was limited to 3600 kVAr. The DG and capacitor locations were restricted to the permissible buses of the corresponding radial DS, while the resulting solutions were required to satisfy the prescribed bus-voltage operating limits of [0.95]– [1.05] p.u. throughout the DS. These settings were kept identical for all compared optimization algorithms to ensure a fair and consistent performance comparison. IEEE 69-bus system The IEEE 69-bus DS consists of 69 buses interconnected through 68 distribution branches as shown in Fig. 2 , supplying a total active load of approximately 3.80 MW and a reactive load of about 2.69 MVAr. Owing to its long feeder structure and relatively high load concentration, the network experiences considerable voltage drops and elevated power losses, making voltage regulation and loss minimization challenging tasks. Furthermore, the radial topology and non-uniform load distribution intensify operational difficulties compared with smaller benchmark systems. Consequently, the IEEE 69-bus system is widely employed as a standard test network for evaluating DG allocation, CB placement, DSR, and other loss reduction techniques. Its size and operational complexity provide an effective platform for assessing the performance, robustness, and effectiveness of advanced optimization algorithms designed to enhance DS efficiency and reliability. Fig. 2 IEEE 69-bus DS. Simultaneous DSR with DG allocations for IEEE 69-bus DS (Case 1) In this case, the results of DSR combined with DG placement and sizing using the standard FLO and proposed MFLO algorithms in the presence of EV stations are manifested in Table 3 . As shown, the proposed MFLO approach significantly reduces the losses to 46.93009 kW, where the initial system losses are 489.0540 kW. The proposed MFLO determines the optimal DSR switches at branches 10, 13, 71, 55, and 73, while the DG units are optimally allocated at buses 17, 12, 22, 60, and 61 with ratings of 447, 411, 454, 692, and 2022 kW, respectively. On the other side, the standard FLO improves the performance as well, with losses of 58.61127 kW. Based on that outcome, the proposed MFLO shows better performance against the FLO by 24.89%. Table 3 DSR with DGs allocations of FLO and MFLO for IEEE 69-bus DS (Case 1). Full size table Figure 3 illustrates the convergence characteristics of the proposed MFLO algorithm compared with the conventional FLO. The results clearly demonstrate that MFLO achieves better convergence behaviour. At the initial iterations, both algorithms start with high loss values close to the base-case loss of approximately 225 MW. However, the proposed MFLO rapidly decreases the losses and continues improving steadily throughout the optimization process until reaching the final minimum loss value of about 46.93 kW. This indicates that MFLO possesses strong exploration and exploitation capabilities, enabling it to avoid local optima and achieve better-quality solutions. In contrast, the standard FLO shows slower convergence speed without achieving progress for a long time and stabilizing near 58 kW. The convergence curves confirm that the proposed MFLO algorithm provides superior convergence speed, improved stability, and better optimization accuracy compared to FLO, making it a more effective approach for DSR and DG allocation problems in DSs. Fig. 3 Convergence rates of MFLO against FLO for IEEE 69-bus DS (Case 1). In order to assess the effectiveness of the designed MFLO, comparisons are implemented against several well-known and recent algorithms, including HLOA 2024 26 , PSO 27 , PKO 2024 28 , HHO, EO, Parrot optimizer 2024 31 , BPBO 2025 32 , CAOA 2025 33 , and Baboon Algorithm 2026 34 . All algorithms are executed using 50 search agents and 200 iterations under identical conditions. Based on the statistical results obtained from 25 independent runs for each algorithm, a clear comparison of performance and robustness can be established. Figure 4 displays the box and whiskers plot of the MFLO against several well-known and recent algorithms for IEEE 69-bus DS (Case 1). Table 4 provides a comprehensive statistical comparison between the proposed MFLO algorithm and the other optimization approaches. Fig. 4 Box and Whiskers plot of the MFLO against several well-known and recent algorithms for IEEE 69-bus DS (Case 1). Table 4 Statistical assessment obtained of the MFLO against several well-known and recent algorithms for IEEE 69-bus DS (Case 1). Full size table As shown, the designed MFLO achieves the lowest average losses, 56.785, followed very closely by PKO at 57.143. The difference is 0.358 or approximately 0.63%. Against the other algorithms, however, the mean-loss differences become much more substantial, at approximately 11.9%, 17.2%, 19.6%, 28.9%, 47.6%, 50.2%, 60.3%, 66.9%, and 71.8% lower than agents PSO, EO, Parrot, BPBO, HHO, FLO, CAOA, HLOA, and Baboon, respectively. These numerical differences are consistent with the statistical results. The proposed MFLO also demonstrates excellent robustness. The standard deviation is particularly important for stochastic optimization. MFLO has an STD of 8.486, which is slightly better than that of PKO of 8.598. Also, MFLO has the lowest coefficient of variation among all algorithms at 14.943%. PKO has 15.047%, while PSO has 18.859%. Several other algorithms have substantially larger variation, particularly FLO at 39.271% and EO at 33.796%. Therefore, MFLO does not merely obtain a low average loss; it also provides highly consistent performance across independent runs. Additionally, the Levene test is performed as shown in Table 5 , which produces p < 0.001, indicating a statistically significant difference among the variances of the algorithms. This finding is consistent with the substantial differences in STD and CV observed in Table 4 . For example, MFLO has an STD of only 8.486, whereas FLO and HLOA exhibit STD values of 44.780 and 53.188, respectively. Table 5 Levene's test for equality of variances among the investigated algorithms. Full size table Both non-parametric global tests are performed as shown in Table 6 , which reject the null hypothesis of equivalent algorithmic performance. The Kruskal–Wallis test produces p < 0.001, confirming statistically significant differences among the performance distributions. Similarly, the Friedman test also yields p < 0.001, demonstrating significant differences when the results are compared across runs using their within-run ranks. Table 6 Global non-parametric statistical tests for comparing the optimization algorithms. Full size table Moreover, Table 7 displays the pairwise Wilcoxon signed-rank analysis that provides the most direct statistical assessment of MFLO relative to each competing algorithm. After controlling for multiple comparisons using the Holm procedure, MFLO significantly outperforms Parrot, BPBO, HHO, FLO, CAOA, Baboon, and HLOA, with all adjusted p-values below 0.05. In contrast, no statistically significant difference is found among MFLO, PKO, PSO, and EO after Holm correction. The comparison between MFLO and PKO shows similar mean losses, with pHolm = 0.717220, indicating no significant superiority. While MFLO's mean loss is about 11.9% lower than PSO, the adjusted p-value is 0.272990, and it also has a lower mean than EO with p = 0.395670. This highlights the importance of considering both statistical significance and numerical improvement together. Table 7 Pairwise Wilcoxon signed-rank test comparing MFLO with the competing algorithms. Full size table Moreover, the Mann–Whitney U test was applied to MFLO and each competing algorithm as shown in Table 8 . The results exhibit a pattern consistent with the Wilcoxon analysis. MFLO is statistically superior to Parrot, BPBO, HHO, FLO, CAOA, Baboon, and HLOA after Holm correction. The comparison with PSO is particularly close to the significance threshold, with pHolm​ = 0.054048, but it remains slightly above the adopted 0.05 significance level. Table 8 Mann–Whitney U test comparing MFLO with the competing algorithms. Full size table Furthermore, the effect-size analysis in Table 9 provides an important complement to the hypothesis tests. The comparison with PKO produces a rank-biserial effect size of only 0.095, confirming that the practical distinction between these two algorithms is very small. The effect sizes against PSO and EO are moderate, whereas the comparisons against Parrot and BPBO produce very large effects of 0.811 and 0.821, respectively. The largest effects occur for HHO, FLO, CAOA, Baboon, and HLOA, with values ranging from 0.989 to 1.000. Table 9 Rank-biserial effect sizes for the pairwise comparison of MFLO with the competing algorithms. Full size table Finally, the overall ranking in Table 10 confirms that MFLO obtains the lowest average power loss, followed by PKO and PSO. The ranking also reveals a pronounced deterioration in performance after the first few algorithms, particularly for HHO, FLO, CAOA, HLOA, and Baboon. Table 10 Overall ranking of the optimization algorithms according to their mean power loss. Full size table Based on the above analysis, the MFLO outperforms most investigated algorithms in terms of numerical performance and stochastic robustness, achieving the lowest mean loss and low variability, with significant statistical differences confirmed by Kruskal–Wallis and Friedman tests (p < 0.001). Post-hoc analyses show MFLO significantly surpasses seven out of ten competitors, while differences with PKO, PSO, and EO are not significant. Overall, MFLO is identified as the best algorithm but not universally superior to all others. Simultaneous DSR with CB allocations for IEEE 69-bus DS (Case 2) In this case, the integration of DSR with optimal CB allocation and sizing using various optimization algorithms is examined. Post-optimization, significant reductions in losses were noted across all methods. The proposed MFLO algorithm yielded the best results as shown in Table 11 , achieving the lowest losses of 161.7528 kW. In comparison, the HLOA reduced losses to 195.5067 kW, while PSO and FLO reached values of 167.4797 kW and 196.49 kW, respectively. The PKO demonstrated a competitive performance with losses reduced to 162.0981 kW, yet MFLO was the most effective. The MFLO efficiently determined the optimal switching configuration for branches 69, 20, 13, 57, and 61 and identified ideal CB placements at buses 61, 24, and 50 with ratings of 1200, 600, and 900 kVAr, respectively. Table 11 DSR with CBs allocations for IEEE 69-bus DS (Case 2). Full size table The convergence behaviour of the considered algorithms for this case, in terms of power losses (kW) versus iterations, is illustrated in Fig. 5 . At the initial stage, all algorithms start with high loss values ranging approximately from 200 to 240 kW. PSO shows a rapid initial reduction, reaching around 60 kW within the first 10 iterations, indicating strong early exploration capability. PKO demonstrates a similarly fast convergence, achieving slightly lower loss values than PSO and stabilizing around 50 kW. FLO follows a smoother convergence pattern but stabilizes at a relatively higher level (around 58 kW), reflecting moderate performance. In contrast, HLOA exhibits slower and unstable convergence, remaining at higher loss levels (above 140 kW) for a significant number of iterations before dropping sharply and eventually stagnating around 55–60 kW, which indicates premature convergence and limited optimization ability. The proposed MFLO algorithm shows a different convergence pattern: although it starts with relatively high losses and decreases gradually in the early stages, it experiences a significant improvement after approximately 40–50 iterations, followed by a steady and consistent reduction. MFLO ultimately converges to the lowest loss value (around 45 kW), outperforming all other algorithms. This figure clearly demonstrates that MFLO achieves the best balance between convergence speed and solution quality, providing the lowest final power losses. PKO ranks second in performance, followed by PSO and FLO, while HLOA shows the weakest convergence behavior. Fig. 5 Convergence rates of MFLO against FLO and other approaches for IEEE 69-bus DS (Case 2). Figure 6 illustrates a radar-chart comparison of optimization techniques after 25 independent runs, highlighting the MFLO approach as the most effective in optimization performance, stability, and robustness. The MFLO algorithm occupies the smallest radar area, signifying lower loss values and greater reliability during runs. In contrast, HLOA displays the largest area, indicating poorer optimization capabilities. Table 12 presents the statistical evaluation of HLOA, PSO, PKO, FLO, and the proposed MFLO algorithm based on 25 independent runs for each optimization technique. In terms of average losses, MFLO records an average value of 186.8327 kW, outperforming HLOA, PKO, and FLO, which achieve 229.2933 kW, 194.0029 kW, and 206.4593 kW, respectively. PSO shows a relatively close performance with an average value of 191.9464 kW; however, MFLO still provides the best overall average performance, indicating higher consistency across repeated runs. Regarding the maximum obtained losses, MFLO again demonstrates improved robustness with a maximum value of 202.9881 kW, which is considerably lower than HLOA, PKO, and FLO. This reflects the ability of MFLO to avoid poor-quality solutions and maintain reliable convergence behaviour. Furthermore, the statistical stability of the investigated methods can be evaluated using the standard deviation values. FLO achieves the lowest STD value of 5.111949, indicating highly stable convergence behaviour. Nevertheless, MFLO also shows strong robustness with an STD of 14.19392 while still maintaining the best optimization performance and lowest losses among all methods. In contrast, HLOA exhibits the highest STD value of 30.89368, confirming its unstable performance and weak convergence characteristics. Fig. 6 Radar chart for MFLO, FLO, PKO, HLOA, and PSO for IEEE 69-bus DS (Case 2). Table 12 Statistical assessment obtained from MFLO, FLO, PKO, PSO, and HLOA for IEEE 69-bus DS (Case 2). Full size table Simultaneous DSR with DG and CB allocations for IEEE 69-bus DS (Case 3) This case investigates the simultaneous optimal allocation and sizing of DG units, CBs, and DSR in the DS, with the corresponding results summarized in Table 13 . Regarding network reconfiguration, FLO selects branches 41, 19, 44, 57, and 63 as the open switches, whereas MFLO selects 40, 18, 71, 57, and 63. Thus, both algorithms retain two common open switches (57 and 63) but identify different configurations for the remaining three switches. For DG allocation, FLO places the DGs at buses 65, 62, 67, and 63, with ratings of 914, 721, 787, and 491 kW, respectively. MFLO, in contrast, selects buses 61, 7, 22, and 63, with ratings of 860, 2307, 252, and 50 kW, respectively. For CB allocation, FLO installs CB units at buses 65, 62, 67, and 63, with capacities of 300, 300, 600, and 600 kVAr, respectively, whereas MFLO installs DG units at buses 61 and 7, with capacities of 1200 and 600 kVAr, respectively. The results demonstrate a substantial improvement when MFLO is employed, whereas FLO reduces the loss to 27.29 kW, corresponding to a reduction of approximately 94.42%. MFLO further decreases the loss to only 13.82 kW, corresponding to a 97.17% reduction relative to the initial case and an additional 49.37% reduction compared with FLO. This clearly indicates that the proposed MFLO is able to identify a substantially more effective coordinated solution. Table 13 DSR with CBs and DGs allocations for IEEE 69-bus DS (Case 3). Full size table Furthermore, Fig. 7 presents the convergence behaviour of the proposed MFLO algorithm in comparison with FLO. As shown, the proposed MFLO significantly reduces loss, achieving a final minimum of approximately 13.82 kW, demonstrating effective exploration and exploitation capabilities. In contrast, the standard FLO shows slower convergence, stabilizing near 27.9 kW without significant improvement. The convergence curves indicate that MFLO offers superior speed, stability, and optimization accuracy, making it a more effective solution for DSR and DG allocation in DSs. Fig. 7 Convergence rates of MFLO against FLO for IEEE 69-bus DS (Case 3). Figure 8 and Table 14 present the statistical comparison among HLOA, PSO, PKO, FLO, and the proposed MFLO algorithm based on 25 independent runs. The obtained results clearly demonstrate the superiority of the proposed MFLO algorithm over the other investigated approaches. In terms of the minimum loss value, MFLO achieves the best result with losses of 13.81892 kW, which is significantly lower than those achieved by HLOA, PSO, PKO, and FLO, corresponding to 61.00344 kW, 27.38769 kW, 15.02303 kW, and 27.2929 kW, respectively. Although PKO provides competitive performance, MFLO still attains the lowest power losses, highlighting its enhanced optimization capability. Regarding the average loss values, MFLO again provides the best performance with an average loss of 33.89825 kW. In comparison, HLOA records a very high average value of 188.6535 kW, indicating weak optimization performance and instability. PSO and PKO achieve average losses of 50.67228 kW and 50.25378 kW, respectively, whereas FLO records 68.45874 kW. These results confirm the consistent superiority of MFLO over the competing techniques. Furthermore, MFLO demonstrates improved robustness by maintaining a relatively low maximum loss value of 74.59722 kW compared with HLOA, PKO, and FLO, which produce maximum losses of 320.5565 kW, 142.7524 kW, and 154.5777 kW, respectively. Although PSO achieves a comparable maximum value of 81.50855 kW, MFLO still provides better overall performance. The standard deviation results further verify the stability of the proposed algorithm. MFLO achieves an STD value of 16.91429, indicating stable and reliable convergence behaviour across repeated runs. In contrast, HLOA exhibits the highest STD value of 74.8215, reflecting significant fluctuations and poor robustness. Similarly, FLO and PKO show higher deviations of 41.13204 and 33.8972, respectively, while PSO achieves a slightly lower STD value of 15.5946. Overall, the statistical assessment demonstrates that the proposed MFLO algorithm offers superior optimization accuracy, faster convergence, stronger robustness, and more stable performance compared with HLOA, PSO, PKO, and FLO. Fig. 8 Radar chart for MFLO, FLO, PKO, HLOA, and PSO for IEEE 69-bus DS (Case 3). Table 14 Statistical assessment obtained from MFLO, FLO, PKO, PSO, and HLOA for IEEE 69-bus DS (Case 3). Full size table The plotted voltage profile clearly compares the impact of different optimization cases on a 69-bus system as demonstrated in Fig. 9 . In the initial case, the voltage drops significantly, especially in the mid- and far-end buses, reaching values as low as around 0.87 p.u., which indicates poor voltage regulation and potential stability issues. After applying the MFLO-based optimization, all cases show noticeable improvement. MFLO-Case 1 reduces the voltage drop moderately, maintaining levels mostly above 0.98 p.u., while MFLO-Case 2 further enhances performance with a flatter and more stable profile close to 0.99 p.u. across most buses. The best performance is observed in MFLO-Case 3, where the voltage remains almost ideal (very close to 1.0 p.u.) throughout the entire system, with only minimal deviations. Overall, the results demonstrate that the MFLO approach significantly improves voltage stability and minimizes drops, with Case 3 providing the most effective solution. Fig. 9 Voltage profiles obtained by MFLO for Cases 1–3 versus the initial scenario for IEEE 69-bus DS. Analysis of EV penetration levels for IEEE 69-bus DS (Case 4) To further investigate the capability of the proposed MFLO-based framework to accommodate the different penetration of EVs, a sensitivity analysis was conducted on the IEEE 69-bus DS by considering six EV penetration levels ranging from 0 to 100%. The corresponding active power losses, minimum bus voltage, and maximum bus voltage are presented in Figs. 10 and 11 , respectively. As shown, the results demonstrate that the proposed coordinated optimization of DG allocation, CBs, and DSR is capable of maintaining acceptable voltage conditions over the entire investigated EV penetration range. Fig. 10 Active power losses obtained by MFLO for different EV penetration levels for IEEE 69-bus DS. Fig. 11 Minimum and maximum voltages obtained by MFLO for different EV penetration for IEEE 69-bus DS. At 0% EV penetration, the active power loss in the system is 29.417 kW, with bus voltages at 0.9985 p.u. (minimum) and 1.0197 p.u. (maximum). As EV penetration increases to 20%, the power loss decreases to 22.1974 kW, a reduction of approximately 24.54%. The minimum voltage drops marginally to 0.9980 p.u., while the maximum voltage decreases slightly to 1.0161 p.u., indicating that increased EV demand does not significantly deteriorate voltage levels. At 40% penetration, active power loss further declines to 16.9898 kW (a 42.24% reduction from 0%), with the minimum voltage maintaining close to unity at 0.9975 p.u. and a maximum of 1.0125 p.u. At 60% penetration, power loss is recorded at 13.8319 kW, signifying a 52.98% reduction from the 0% case. The minimum voltage is slightly lower at 0.9969 p.u. and the maximum at 1.0089 p.u., with a narrowing voltage range indicating effective load compensation without severe voltage degradation. The minimum loss performance occurs at 80% EV penetration, with power loss minimized to 12.7617 kW—56.61% lower than the initial state. Both minimum and maximum voltages are sustained around 0.9962 p.u. and 1.0052 p.u., respectively, underscoring the system's stability and effective load distribution. At 100% EV penetration, active power loss increases to 13.8189 kW, yet this reflects a 53.03% decrease from 0%. Minimum voltage drops to 0.9950 p.u., the lowest observed, while the maximum at 1.0015 p.u. remains close to nominal levels. As shown, the maximum voltage remains extremely close to the nominal value, and the minimum voltage remains within the prescribed operating range. Also, the overall trend is non-monotonic for active power losses because the optimization is designed for the worst-case EV loading condition. In particular, the DG capacities and locations, CB ratings and locations, and feeder configuration are optimized considering the maximum EV penetration/loading condition. Therefore, when the actual EV penetration is lower than the worst-case level, the same optimized configuration provides more active-power generation from DG units and more reactive-power support from CBs than is required by the prevailing load. This creates a surplus active-power injection and reactive-power compensation relative to the lower EV-demand condition. Consequently, the optimized solution for the worst-case condition is not necessarily the loss-minimizing solution for lower EV penetration levels, which explains why the losses can be higher at 0–60% EV penetration despite the lower load for this system. As the EV penetration increases, the additional charging demand progressively absorbs the surplus DG generation and reactive-power support, resulting in improved power-flow conditions and a continuous reduction in losses up to 80% penetration. At 100% penetration, the EV demand becomes sufficiently high that the additional feeder loading begins to offset these benefits, producing the observed slight increase in losses from 12.7617 kW at 80% to 13.8189 kW at 100%. Large-scale radial 141-bus DS The large-scale 141-bus radial DS corresponds to the 12.47-kV AES-Venezuela located in the metropolitan area of Caracas. This benchmark system consists of 141 buses and serves a total peak load demand of approximately 12.19 MW and 6.2894 MVAr, as reported in the literature 45 , 46 . Owing to its extensive network structure, high loading conditions, and large number of decision variables, the 141-bus system represents a challenging test case for evaluating the effectiveness of optimization algorithms. The single-line diagram of the system is illustrated in Fig. 12 . Fig. 12 A large-scale radial DS with 141 buses. Simultaneous DSR with DG allocations for 141-bus DS in Caracas (Case 1) In this case, Table 15 presents the results of DSR combined with optimal DG placement and sizing of the 141-bus DS in the presence of EV charging station loads. The initial system experiences active power losses of 869.8742 kW. After applying the investigated optimization techniques, the proposed MFLO method provides the best performance by reducing the losses to 104.9009 kW. In contrast, the conventional FLO records significantly higher losses of 312.3088 kW, indicating relatively poor optimization capability for this large-scale system. Regarding DG allocation, MFLO determines the optimal installation buses at 46, 16, 57, 95, and 141 with corresponding DG ratings of 3038, 592, 451, 2791, and 2232 kW, respectively. In addition, the optimal DSR scheme obtained by MFLO includes the switches 53, 87, 36, 12, 85, 139, 44, 67, 128, 150, 107, 103, 117, 22, and 16. Table 15 DSR with DGs allocations for 141-bus DS in Caracas (Case 1). Full size table The convergence behaviour of the proposed MFLO algorithm is manifested in Fig. 13 and compared with the standard FLO. The results clearly indicate achieves lower objective values with faster and more stable convergence characteristics throughout the optimization process. It shows great ability in addressing the stagnation problem of the conventional FLO, which doesn’t provide progress for more than half of the iterations. Fig. 13 Convergence rates of MFLO against FLO and other approaches for 141-bus DS in Caracas (Case 1). To further validate the performance of the proposed MFLO, a comprehensive statistical analysis was conducted for the second test system using the results obtained from several independent runs of each optimization algorithm. The algorithms. The analysis included descriptive statistical measures, Levene's test for variance homogeneity, Kruskal–Wallis and Friedman non-parametric tests, pairwise Wilcoxon signed-rank tests with Holm correction, Mann–Whitney U tests, and rank-biserial effect-size analysis. These tests provide complementary evidence regarding the statistical significance, robustness, and practical magnitude of the differences among the competing algorithms. Figure 14 displays the box and Whiskers plot of the MFLO against several well-known and recent algorithms for 141-bus DS (Case 1). Table 16 provides a comprehensive statistical comparison between the proposed MFLO algorithm and the other optimization approaches. Fig. 14 Box and Whiskers plot of the MFLO against several well-known and recent algorithms for 141-bus DS (Case 1). Table 16 Statistical assessment obtained of the MFLO against several well-known and recent algorithms for 141-bus DS (Case 1). Full size table As shown, MFLO achieves the lowest mean power loss of 121.57, thereby obtaining the first rank among all investigated algorithms. The next-best mean values are obtained by PSO and Parrot, both with 124.62, followed by PKO with 125.55. Hence, the difference between MFLO and the closest competitors is relatively small compared with the considerably larger differences observed for EO, BPBO, HHO, HLOA, Baboon, and particularly FLO. The mean loss obtained by MFLO is approximately 2.44% lower than PSO and Parrot, and approximately 3.17% lower than PKO. The improvement becomes substantially larger for the remaining algorithms, reaching approximately 16.55% over EO, 19.49% over BPBO, 40.95% over HHO, 58.61% over HLOA, 67.49% over Baboon, and 73.91% over FLO. MFLO also exhibits good run-to-run consistency, with an STD of 9.769 and a CV of only 8.036%. Although PSO and Parrot have slightly lower variability, with STD = 7.8 and CV = 6.25%, MFLO provides a better mean loss. PKO has a larger STD of 15.650 and CV of 12.465%. The substantially larger CV values of HHO (46.727%), Baboon (38.319%), and HLOA (22.420%) indicate considerably weaker stochastic stability. Therefore, MFLO achieves a favorable compromise between low objective value and high robustness across independent runs. Additionally, the Levene test is performed as shown in Table 17 , which produces p < 0.001. These findings highlight significant differences in STD among the algorithms: MFLO has an STD of 9.769, while HHO and Baboon show STDs of 96.195 and 143.280, respectively. Although MFLO meets the normality assumption, the equal-variance assumption for conventional ANOVA is not satisfied across all algorithms. Both non-parametric global tests in Table 18 indicate significant differences in algorithmic performance, with the Kruskal–Wallis test resulting in p < 0.001 and the Friedman test also yielding p < 0.001, confirming differences across performance distributions and comparisons of within-run ranks. Table 17 Levene's test for equality of variances among the investigated algorithms for 141-bus DS (Case 1). Full size table Table 18 Global non-parametric statistical tests for comparing the optimization algorithms for 141-bus DS (Case 1). Full size table Moreover, Table 19 displays the pairwise Wilcoxon signed-rank analysis. The Wilcoxon signed-rank analysis provides particularly strong evidence concerning the superiority of MFLO. After Holm correction for multiple comparisons, MFLO is statistically superior to EO, BPBO, HHO, FLO, CAOA, Baboon, and HLOA, with all adjusted p-values below 0.05. Table 19 Pairwise Wilcoxon signed-rank test comparing MFLO with the competing algorithms for 141-bus DS (Case 1). Full size table Moreover, the Mann–Whitney U test was applied to MFLO and each competing algorithm as shown in Table 20 . It leads to a highly consistent conclusion. MFLO is statistically different from BPBO, HHO, FLO, CAOA, Baboon, and HLOA and, given its lower loss, superior to EO, BPBO, HHO, FLO, CAOA, Baboon, and HLOA, with all Holm-adjusted p-values well below 0.05. Table 20 Mann–Whitney U test comparing MFLO with the competing algorithms for 141-bus DS (Case 1). Full size table Furthermore, the effect-size analysis in Table 21 provides an important complement to the hypothesis tests. The comparisons of MFLO with PKO, PSO, and Parrot produce relatively small rank-biserial effects of 0.181, 0.276, and 0.276, respectively, which is consistent with the absence of statistically significant differences in the Wilcoxon and Mann–Whitney analyses. In contrast, the comparisons with EO and BPBO produce effect sizes of 0.943, while HHO produces an effect size of 0.952. The comparisons with FLO, Baboon, and HLOA reach an effect size of 1.000, indicating an extremely strong separation between MFLO and these algorithms. CAOA also exhibits a very large effect size of 0.800. Table 21 Rank-biserial effect sizes for the pairwise comparison of MFLO with the competing algorithms for 141-bus DS (Case 1). Full size table Finally, the overall ranking in Table 22 places MFLO in the first position with the lowest mean loss of 121.57. PSO and Parrot jointly occupy the second position with an identical mean loss of 124.62, followed by PKO at 125.55. The ranking then exhibits a pronounced deterioration in performance for CAOA, EO, BPBO, HHO, HLOA, Baboon, and FLO. Table 22 Overall ranking of the algorithms according to their mean power loss for 141-bus DS (Case 1). Full size table Simultaneous DSR with CB allocations for 141-bus DS in Caracas (Case 2) In this case, the simultaneous DSR and optimal CB allocation in the presence of EV charging station loads are considered in the 141-bus DS. The initial system experiences active power losses of 869.8742 kW due to the increased loading conditions introduced by EV charging demand. After applying the investigated optimization methods as shown in Table 23 , significant reductions in power losses are achieved. Among all compared techniques, the proposed MFLO algorithm achieves the best performance with minimum power losses of 277.9189 kW. In comparison, HLOA reduces the losses to 401.7765 kW, while PSO and PKO achieve better results with losses of 286.2345 kW and 287.6173 kW, respectively. The conventional FLO exhibits the weakest performance, resulting in losses of 422.8365 kW. These results clearly demonstrate the superiority of MFLO in handling the additional operational challenges caused by EV charging station loads. Regarding capacitor allocation, MFLO identifies the optimal CB installation buses at 69, 46, 141, 118, and 130 with corresponding ratings of 1800, 900, 600, 900, and 1500 kVAr, respectively. Furthermore, the optimal DSR strategy determined by MFLO includes switches 54, 88, 5, 73, 85, 139, 58, 48, 133, 104, 107, 100, 123, 18, and 24. These optimal settings effectively enhance reactive power compensation, improve voltage support, and reduce active power losses under EV charging conditions. Table 23 DSR with CBs allocations for 141-bus DS in Caracas (Case 2). Full size table Figure 15 illustrates the convergence behaviour of the proposed MFLO algorithm compared with the standard FLO, PKO, PSO, and HLOA. The results clearly indicate that MFLO surpasses all competing algorithms, including the conventional FLO, in terms of convergence performance, solution quality, and power loss reduction capability. The proposed MFLO achieves lower objective values with faster and more stable convergence characteristics throughout the optimization process. These findings demonstrate that the implemented enhancements significantly improve the optimizer’s search capability, convergence accuracy, and robustness. Fig. 15 Convergence rates of MFLO against FLO and other approaches for 141-bus DS in Caracas (Case 2). Table 24 presents the statistical performance assessment of HLOA, PSO, PKO, FLO, and the proposed MFLO algorithm in the presence of EV charging station loads, based on 25 independent runs for each optimization technique. The evaluation includes the minimum, average, maximum, and standard deviation values of the obtained active power losses. Regarding the average loss values, MFLO again provides the best overall performance with an average loss of 299.0625 kW. In comparison, HLOA records the highest average losses of 642.2627 kW, indicating weak optimization capability and unstable convergence behaviour. PSO and PKO achieve average losses of 304.4773 kW and 325.6955 kW, respectively, while FLO produces 521.0177 kW. These findings confirm the effectiveness of MFLO in consistently generating high-quality solutions across repeated runs. Furthermore, MFLO demonstrates excellent robustness and stability with the lowest standard deviation value of 5.299528, indicating highly reliable convergence characteristics and reduced fluctuations between independent runs. Although PSO also exhibits good stability with an STD value of 7.442245, its overall optimization performance remains inferior to MFLO. In contrast, HLOA and FLO show significantly larger STD values of 173.785 and 130.0332, respectively, reflecting poor stability and inconsistent optimization behaviour. The maximum loss values further highlight the robustness of MFLO, which records a maximum loss of 300.8507 kW, substantially lower than those obtained using HLOA, PKO, and FLO. Table 24 Statistical assessment of MFLO, FLO, PKO, PSO, and HLOA for 141-bus DS in Caracas (Case 2). Full size table Simultaneous DSR with DG and CB allocations for 141-bus DS in Caracas (Case 3) Table 25 presents the results for this case, which considers simultaneous DSR along with optimal allocation of CBs and DGs in the presence of EV charging station loads. The performance of the proposed MFLO algorithm is compared with the standard FLO technique and the initial case for the 141-bus DS. The obtained results show that MFLO achieves a superior allocation strategy for both CBs and DGs, resulting in a significant reduction in active power losses. In the case of CB placement, the MFLO method installs capacitor banks at buses 15, 54, 37, and 28 with ratings of 1200, 2400, 2400, and 600 kVAr, respectively, whereas the FLO approach allocates five CBs at buses 81, 126, 34, 81, and 82. Similarly, for DG allocation, MFLO identifies buses 46, 138, 49, and 119 with ratings of 4779, 1113, 23, and 3187 kW, respectively, while FLO allocates DGs at buses 25, 26, 43, 27, and 117. Furthermore, the optimal tie-switch configuration determined by MFLO differs from that obtained using FLO, reflecting better DSR performance. As a result, the proposed MFLO reduces the total active power losses from 603.821 kW in the initial case to 61.9585 kW, outperforming the FLO algorithm, which achieves losses of 146.4118 kW. These results confirm the effectiveness and robustness of the proposed MFLO method in improving DS performance. Table 25 DSR with CBs and DGs allocations for 141-bus DS in Caracas (Case 3). Full size table Figure 16 illustrates the convergence behaviour of the proposed MFLO algorithm compared with the standard FLO, PKO, PSO, and HLOA. The results clearly indicate that MFLO surpasses all competing algorithms, including the conventional FLO, in terms of convergence performance, solution quality, and power loss reduction capability. The proposed MFLO achieves lower objective values with faster and more stable convergence characteristics throughout the optimization process. These findings demonstrate that the implemented enhancements significantly improve the optimizer’s search capability, convergence accuracy, and robustness. In particular, the proposed MFLO shows a superior ability to handle highly nonlinear, complex, and large-scale DS optimization problems, especially in the presence of EV charging demand. Fig. 16 Convergence rates of MFLO against FLO and other approaches for 141-bus DS in Caracas (Case 3). Table 26 presents the statistical evaluation of the proposed MFLO algorithm compared with FLO, PKO, PSO, and HLOA. The results demonstrate that the proposed MFLO algorithm provides the best overall performance among all compared techniques. Specifically, MFLO achieves the lowest minimum power loss of 61.9585 kW, outperforming FLO, PKO, PSO, and HLOA, which record minimum losses of 146.4118, 68.19266, 74.47828, and 99.4422 kW, respectively. In addition, MFLO attains the lowest average loss value of 77.19895 kW, indicating superior convergence characteristics and solution quality. Moreover, the maximum loss obtained by MFLO is only 91.65945 kW, which is significantly lower than the corresponding values achieved by the other optimization methods. Although PKO and PSO show relatively competitive standard deviation values, the MFLO algorithm still maintains a low standard deviation of 10.37631, confirming its robustness and stability in reaching near-optimal solutions consistently. On the other hand, FLO and HLOA exhibit larger variations and poorer convergence behaviour. Figure 17 illustrates the radar chart comparison among MFLO, FLO, PKO, PSO, and HLOA under Scenario No. 6. The radar chart clearly highlights the superiority of the proposed MFLO approach in all statistical performance indices, demonstrating its effectiveness in reducing power losses and enhancing optimization performance for the studied DS. Table 26 Statistical assessment of MFLO, FLO, PKO, PSO, and HLOA for 141-bus DS in Caracas (Case 3). Full size table Fig. 17 Radar chart for MFLO, FLO, PKO, HLOA, and PSO for 141-bus DS in Caracas (Case 3). The voltage profile results for 141 buses in the initial and optimized cases using MFLO under the three cases studied are depicted in Fig. 18 , which clearly demonstrate the effectiveness of the proposed approach. In the initial scenario, significant voltage drops are observed across several buses, with values falling to around 0.92 p.u., indicating poor voltage stability and deviation from the nominal level. In contrast, all MFLO-based cases show a substantial improvement in voltage profiles, maintaining values close to 1.0 p.u. across nearly all buses. Among these, Case 3 provides the best performance, with voltages consistently staying within approximately 0.99–1.0 p.u., indicating excellent regulation and minimal deviation. Case 2 also shows strong performance, slightly below Case 3 but still maintaining a stable and acceptable voltage range. Case 2, while slightly lower than the other MFLO cases, still significantly outperforms the initial condition. The results indicate that the MFLO effectively enhances voltage stability and reduces voltage deviations across different operational cases, with Case 3 achieving the best performance among the tested cases. Fig. 18 Voltage profiles obtained by MFLO for Cases 1–3 versus the initial scenario for 141-bus DS in Caracas. Analysis of EV Penetration Levels for IEEE 141-bus DS (Case 4) The impact of EV penetration on the IEEE 141-bus distribution system is presented in Figs. 19 and 20 . Unlike the IEEE 69-bus system, the 141-bus system exhibits a monotonic increase in active power losses with increasing EV penetration. The loss increases from 44.6841 kW at 0% EV penetration to 61.9585 kW at 100%, corresponding to an increase of approximately 38.65%. Specifically, the losses increase gradually from 44.8775 kW at 20% to 46.687 kW at 40%, 50.1271 kW at 60%, 55.2126 kW at 80%, and finally 61.9585 kW at 100%. This behavior indicates that, for the larger and more complex 141-bus feeder, the additional EV charging demand progressively increases feeder currents and associated I 2 R losses. Fig. 19 Active power losses obtained by MFLO for different EV penetration levels for 141-bus DS. Fig. 20 Minimum and maximum voltages obtained by MFLO for different EV penetration for 141-bus DS. The voltage profile remains generally well controlled throughout the investigated penetration levels. The minimum voltage decreases from 0.9937 p.u. at 0% to 0.9894 p.u. at 100% penetration, while the maximum voltage decreases from 1.0034 p.u. to approximately 1.000 p.u. from 40% onward. Thus, despite the increasing EV demand and corresponding increase in losses, the proposed coordinated optimization maintains the bus voltages close to the nominal value. In contrast to the IEEE 69-bus system, the IEEE 141-bus feeder exhibits a continuous increase in losses as EV penetration increases. This behavior can mainly be attributed to the substantially larger size and loading level of the 141-bus system, which contains approximately twice the number of buses and a considerably higher overall demand than the 69-bus feeder. As EV penetration increases, the limited capacity of the five DG units and five CBs leads to significant additional charging demand. At lower EV penetration levels, the available DG and CB capacities are more adequate relative to the total system demand, leading to lower losses. Nevertheless, the voltage profile remains very well controlled throughout all penetration levels, with the minimum voltage remaining above 0.989 p.u. and the maximum voltage not exceeding 1.0034 p.u. This demonstrates that, despite the increased losses associated with the larger system and higher EV demand, the proposed coordinated DG–CB–reconfiguration framework successfully maintains acceptable voltage conditions. Therefore, the different loss trends observed for the IEEE 69-bus and IEEE 141-bus systems primarily reflect their different network sizes, total demand levels, and the relative capacity of the limited number of DG and CB devices, rather than a deterioration in the effectiveness of the proposed MFLO. Effects of load growth on the coordinated DG–CB–DSR framework for IEEE 141-bus DS To investigate the effect of increasing load demand on the coordinated DG–CB–DSR framework, six load-growth levels of 2.5%, 5%, 7.5%, 10%, 12.5%, and 15% were considered for the IEEE 141-bus distribution system. At each load-growth level, the proposed MFLO was independently applied to determine the optimal locations and ratings of DGs and CBs, together with the optimal DSR. The resulting configurations and operating performance are summarized in Table 26 , while the convergence characteristics and corresponding voltage profiles are presented in Figs. 21 and 22 , respectively. Fig. 21 Convergence rates of MFLO under different load growth for 141-bus DS in Caracas. Fig. 22 Voltage profiles obtained by MFLO under different load growth for 141-bus DS in Caracas. The results in Table 26 demonstrate that the initial active power losses increase progressively with load growth, from 910.3326 kW at 2.5% load growth to 1120.778 kW at 15%. This behavior is expected because increasing the demand increases the power transferred through the feeder branches and consequently increases the associated I 2 R losses. However, after the coordinated optimization, the losses are substantially reduced to 78.17667, 79.52999, 84.77274, 100.7881, 105.7348, and 115.3135 kW for 2.5%, 5%, 7.5%, 10%, 12.5%, and 15% load growth, respectively. Thus, MFLO achieves substantial loss reduction under all investigated loading conditions. Although the optimized losses increase with load growth, this increase is considerably smaller than the corresponding increase in the initial losses, demonstrating the effectiveness of coordinated DG placement, CB allocation, and feeder reconfiguration in mitigating the adverse effects of increased demand. Figure 21 illustrates the convergence characteristics of MFLO for the six load-growth conditions. In all cases, the objective function sharply decreases, indicating MFLO's effective exploration of the search space. As iterations progress, the reduction slows, highlighting a shift from global exploration to local exploitation, with the curves stabilizing in later iterations. Despite higher-load cases starting from larger loss values requiring broader searches, MFLO successfully converges across all loading levels within 200 iterations. The voltage profiles shown in Fig. 22 demonstrate that the coordinated optimization maintains acceptable voltage conditions despite the increasing load. The voltage magnitudes remain within a relatively narrow range, approximately between 0.980 and 1.000 p.u., throughout the 141-bus feeder. The minimum voltage is 0.982834 p.u. at 2.5% load growth, increases to 0.985657 p.u. and 0.987063 p.u. at 5% and 7.5%, respectively, and then remains close to 0.98 p.u. at the higher loading levels. The minimum values are 0.980377 p.u., 0.980217 p.u., and 0.983390 p.u. for 10%, 12.5%, and 15% growth, respectively. The maximum voltage is maintained at 1.0 p.u. for all cases reported in Table 26 . These results indicate that the increased loading does not lead to unacceptable voltage deviations because the optimization simultaneously adjusts DG injection, CB reactive-power support, and feeder topology (Table 27 ). Table 27 DSR with CBs and DGs allocations based on MFLO for several load growth for IEEE 141-bus DS. Full size table Practical deployment considerations and robustness under changing operating conditions Although the proposed MFLO-based coordinated DG–CB–DSR framework is evaluated through numerical simulations, its practical implementation can be integrated with the existing monitoring and distribution-management infrastructure of modern active distribution networks. The optimization framework can operate at the planning or supervisory level, receiving network measurements and operating information from existing SCADA, AMI, and other monitoring devices. The resulting optimal DG dispatch/allocation, CB switching, and DSR decisions can subsequently be transmitted to controllable field devices through the Distribution Management System (DMS). From a computational perspective, the proposed MFLO is a population-based optimization algorithm and therefore requires greater computational effort than deterministic power-flow calculations; however, the optimization can be executed on a conventional engineering workstation or DMS server, with the computational burden depending primarily on the population size, number of iterations, number of decision variables, and computational cost of the embedded distribution load-flow calculation. Significantly, MFLO does not replace the existing protection system. Any feeder reconfiguration or DG integration obtained from the optimization must be checked against protection coordination, fault-current levels, relay settings, voltage constraints, and equipment ratings before implementation. Communication latency and measurement availability should also be considered if the framework is extended from planning to supervisory operation. Furthermore, practical distribution networks are subject to uncertainties associated with load demand, EV charging behavior, renewable generation, and equipment availability. The EV-penetration and load-growth analyses presented in this study provide an initial assessment of the robustness of the proposed framework under changing operating conditions. A practical deployment would further benefit from probabilistic or scenario-based uncertainty modeling and real-time state estimation. Therefore, the proposed framework should be viewed as a decision-support and optimization layer that can complement, rather than replace, existing DMS, SCADA, protection, and field-control systems. Beyond the conventional benchmark optimization cases, the proposed framework was evaluated under substantial variations in EV penetration and system loading. EV penetration was varied from 0 to 100% for the IEEE 69-bus and IEEE 141-bus systems, while load growth from 2.5% to 15% was investigated for the IEEE 141-bus system. These analyses demonstrate that the proposed coordinated DG–CB–DSR framework is not restricted to a single nominal operating condition. Instead, MFLO adapts the locations and capacities of DGs and CBs and the feeder configuration to changing network conditions. The results also demonstrate an important practical characteristic: the relationship between EV penetration and losses is network-dependent, particularly because the number and capacity of installed DG and CB units are limited. Nevertheless, the optimized configurations maintain acceptable voltage profiles over the investigated operating range. These sensitivity studies therefore provide an intermediate validation step between a single deterministic benchmark and future validation using chronological real-network data or real-time experimental platforms. Conclusion This study proposed an MFLO for the coordinated optimization of DG allocation, CB placement, and DSR in modern distribution networks with increasing EV penetration. The proposed MFLO extends the original FLO through the incorporation of a defensive strategy phase and an adaptive local search phase, which improve population diversity, exploration of promising regions, and exploitation around high-quality solutions. The resulting framework provides a coordinated optimization mechanism in which DG active-power support, CB reactive-power compensation, and feeder topology are optimized simultaneously rather than independently. EV charging uncertainty is modeled to capture the stochastic characteristics of EV demand, while the coordinated DG–CB–DSR planning is designed considering the worst-case EV loading condition to ensure adequate network support under high-demand scenarios. The simulation results on the IEEE 69-bus and IEEE 141-bus distribution systems demonstrate the effectiveness of the proposed approach. In the IEEE 69-bus system, MFLO reduced the active power loss from 489.0540 kW in the initial configuration to 13.81892 kW in Case 3, corresponding to a reduction of approximately 97.17%. The optimized solution simultaneously determined the feeder switching configuration, DG locations and capacities, and CB locations and ratings, demonstrating the importance of their coordinated optimization. For the EV-penetration analysis, the IEEE 69-bus system achieved its lowest loss of 12.7617 kW at 80% EV penetration, while the loss at 100% penetration remained only 13.8189 kW, approximately 53% lower than the corresponding no-EV case. For the IEEE 141-bus system, losses increased from 44.6841 kW to 61.9585 kW as EV penetration increased from 0 to 100%, reflecting the considerably larger network demand and the finite capacities of the five DG and five CB units. Nevertheless, the minimum voltage remained above 0.989 p.u., confirming effective voltage regulation. The load-growth analysis further demonstrated the adaptability of MFLO under progressively increasing demand. For the IEEE 141-bus system, load growth from 2.5% to 15% increased the initial losses from 910.3326 kW to 1120.778 kW, whereas the optimized losses were limited to 78.17667–115.3135 kW. At the same time, the minimum voltage remained approximately 0.98 p.u. or higher, while the maximum voltage was maintained at 1.0 p.u. These results demonstrate that MFLO can adapt the DG/CB allocation and feeder topology to changing loading conditions while maintaining satisfactory voltage performance. The superiority and robustness of MFLO were further examined against ten established and recent optimization algorithms, namely FLO, PSO, HHO, EO, PKO, Parrot Optimizer, BPBO, CAOA, HLOA, and Baboon Algorithm. Under identical conditions, MFLO achieved the lowest mean loss of 56.78524 and a low coefficient of variation of 14.943%. Statistical analysis using Levene, Kruskal–Wallis, Friedman, Wilcoxon signed-rank, and Mann–Whitney U tests confirmed significant performance differences among the investigated methods. In particular, MFLO statistically outperformed Parrot, BPBO, HHO, FLO, CAOA, Baboon, and HLOA, while its differences from PKO, PSO, and EO were not statistically significant after multiple-comparison correction. Thus, the results support MFLO as a highly competitive and statistically robust optimizer rather than suggesting universal superiority over every competing method. Future investigations can extend the proposed framework from technical loss and voltage optimization to a comprehensive multi-objective planning problem incorporating investment and operating costs, carbon emissions, reliability indices, and the hosting capacity of renewable generation and EV charging infrastructure. Also, future work can extend the proposed MFLO framework from benchmark-based steady-state simulations to real distribution-network data and time-series operating conditions, including chronological load, EV charging, and renewable-generation profiles. Further validation can be conducted using hardware-in-the-loop (HIL) and real-time digital simulation (RTDS) platforms to evaluate computational performance, communication requirements, switching dynamics, protection coordination, and interaction with existing DMS infrastructure.