This is driving the widespread integration of renewable energy sources and places variable wind-based DG at the heart of contemporary distribution networks. Its inherent variability, however, is the source of significant uncertainty that complicates network planning and operation [1,2]. Independently, load demand varies temporally; together, these factors further increase operational difficulty [3]. Classic deterministic approaches to planning are poorly suited to this indeterminate environment, since it risks either inadequate asset utilization or actual constraint violation due to realistic variability [4].

Similarly, optimal wind DG allocation should reconcile competing objectives, such as minimizing energy losses to enhance efficiency and improving reliability, often measured via the Expected Energy Not Supplied (EENS), to limit outage impacts. These objectives often conflict, hence requiring multi-objective optimization to find viable trade-offs. Moreover, the potential of DG is magnified when coordinated with network reconfiguration, a cost-effective strategy to adapt network topology to changing conditions. Despite their synergy, most studies optimize these elements in isolation, overlooking the compounded benefits of integrated planning. A comprehensive synthesis of these multi-objective planning challenges is provided by Al-Shamma’a et al. [5], who present a systematic review of optimization techniques, global insights, and smart grid implications for DG sizing and allocation. Their analysis of algorithmic trends and modeling innovations reinforces the importance of developing advanced metaheuristic approaches to address the increasing complexity of modern distribution systems.

Thus, the core challenge is to develop a stochastic multi-objective framework that simultaneously determines the optimal allocation of wind DG and dynamic network reconfiguration across multiple loading levels while accounting for uncertainties in both wind generation and load demand. This requires searching for a high-dimensional, constrained search space of discrete and continuous variables, which remains computationally demanding for conventional optimization methods [6,7].

In the context of distribution system performance, the driving force for DG integration is a major impetus. Optimizing key technical objectives, such as minimizing active power losses [3,8–11], has been one of the early and continuous areas of research. Improving voltage profiles, stability, and voltage deviation has also been widely investigated using various optimization frameworks for DG placement and sizing in distribution networks [8,9,12–15]. Several recent studies further explore advanced metaheuristic and evolutionary algorithms to enhance voltage stability and operational performance in both balanced and unbalanced systems [16–18].

In recent times, the scope has expanded toward economic and environmental considerations, thereby including other important objectives of multi-objective optimal planning, such as minimizing investment and operational costs [3,4,8,12]. These approaches increasingly incorporate techno-economic evaluations together with environmental and operational constraints to provide more practical DG planning strategies [15,19]. Additionally, reducing pollutant emissions and improving sustainability have been emphasized through advanced optimization techniques and renewable-based DG integration [20,21]. The review presented in [6] synthesizes these multi-objective planning challenges and underlines the inherent trade-offs that require more advanced optimization frameworks.

Another critical development is the transition from deterministic models to stochastic ones that consider genuine uncertainties. The intermittency of renewable DG, especially wind and solar, has become a key driver. Probabilistic models such as the Weibull distribution of wind speed and scenario-based approaches like the point estimation method to discretize this uncertainty for optimization have been undertaken [1,2]. Moreover, temporal variations in load are captured by time-varying loads [2] and multiple loading levels (e.g., high, medium, low) [3,22]. The incorporation of nonlinear, voltage dependent load models has also shown to have significant consequences on optimal DG sitting and sizing decisions [4]. This integration of generation and load uncertainty creates a more robust planning structure and is echoed in studies that put together these elements for a holistic uncertainty space [1,23–25].

The high-dimensional, non-linear, non-convex, and often multi-objective optimization problems emerging have, in turn, driven the development and application of lots of meta-heuristic algorithms. Conventional algorithms like GA [7,16] and PSO [8,9,11,15] are used broadly. This quest for better performance has welcomed and enhanced newer algorithms such as AOA [18], GWO along with its hybrid variants [8], SSA [2,24], and Jaya Algorithm [21]. In turn, a common thread is the modification of base algorithms to transcend these simple variants’ shortcomings, like premature convergence and/or poor population diversity. Enhancements include quasi-oppositional learning [20], hybrid strategies that combine different algorithms [8], and adaptive coefficient mechanisms as the most common proposals to improve exploration and exploitation capabilities and solution quality [17,20,26].

Given the synergies between different grid management tools, recent workplaces emphasis on integrated planning. Of note is that which considers the optimal allocation of DG and simultaneous network reconfiguration-the optimization of tie-switch status to minimize losses and improve voltage profiles for a range of different conditions [27] to exploit the flexibility of active distribution networks to maximize DG benefits. Further integration extends to coordinating DG with other devices, such as CBs [3,4,17], DSTATCOMs [14], SOPs [10], BESS [25,28] and AVRs [12]. Various studies, such as those by Elseify et al. [17] and the review by Kerur et al. [7], have underlined that coordinated planning gives better technical and economic results than optimizing each component separately.

The modern context of DG integration is represented by the paradigm of active distribution networks. Modern research incorporates active management schemes, such as coordinated voltage control and demand response (DR), to handle uncertainties and improve hosting capacity [25,28]. In addition to loss minimization, reliability assessment, quantified by metrics such as EENS, has become an important objective [1,13]. It has been demonstrated that strategic placement of DG, especially island-capable DG, significantly increases system reliability, since they provide local supply in case of upstream outages, a consideration of key importance in modern planning frameworks.

Recent studies by Cikan [29] focus on multi-objective solutions for optimizing unbalanced power distribution networks with reconfiguration. The results on the 37-bus system indicate that network reconfiguration alone can lead to substantial improvements in multiple technical and reliability criteria, thereby justifying the inclusion of reconfiguration as a decision variable in distribution system optimization problems. Alongside these planning-oriented solutions, operational tools such as Dynamic Thermal Line Rating (DTLR) and electric vehicle scheduling with Vehicle-to-Grid (V2G) capacity have been identified to make substantial contributions to the improvement of the reliability of wind-based networks by offering real-time flexibility to compensate for wind power variability in real-time [30]. While DTLR offers dynamic expansion of line capacities according to real-time weather conditions to mitigate congestion, and EVs offer distributed storage capacity to compensate for wind power variability, our framework proposes solutions to the underlying planning-oriented decisions of wind DG allocation and network reconfiguration, which form the infrastructure upon which such operational solutions can be effectively leveraged.

Despite the vast amount of literature on DG allocation and network reconfiguration, some critical gaps still exist that this work aims to fill:

Gap 1: Although some works have considered either the wind uncertainty or the load variability, few have built a comprehensive stochastic model that systematically combines probabilistic wind scenarios with different deterministic load levels via the Cartesian product. This is particularly important since the optimal network topology for high wind scenarios can be vastly different depending on whether it is aligned with peak or off-peak loading.

This paper fills these gaps by making the following contributions:

Contribution 1 (Addressing Gaps 1 & 2): Formulation of a dual-objective stochastic model that optimizes wind DG allocation and load-level-dependent dynamic network reconfiguration under combined wind-load uncertainty, formulated as the Cartesian product of wind scenarios generated by PEM and different levels of loading.

To further clarify the novelty of this work relative to existing literature, Table 1 provides a systematic comparison of key features across representative studies. As shown in Table 1, the combination of features in this research study, namely: (i) joint optimization of wind DG and network reconfiguration based on load conditions, (ii) analytical calculation of EENS with islanding function, (iii) overall uncertainty modeling by Cartesian product of PEM scenarios and different load levels, and (iv) customized improved metaheuristic algorithm for this MIP stochastic problem, is a new contribution to the existing literature.

The rest of the paper is organized as: Section 2 presents the uncertainty modeling and problem formulation, Section 3 explains the mathematical formulation, Section 4 describes the reliability evaluation methodology, Section 5 proposes the I-MGWO algorithm, Section 6 presents the simulation results, and finally Section 7 concludes the paper.

Wind speed is inherently stochastic and is commonly modeled using the Weibull distribution, which has been widely adopted in wind energy studies due to its flexibility in representing various wind regimes. The probability density function (PDF) of the Weibull distribution is given by [31,32]:(1)f(v∣λ,k)=(kλ)(vλ)k−1e−(vλ)kwhere v is the wind speed (m/s), k is the shape parameter (dimensionless), λ is the scale parameter (m/s).

The cumulative distribution function (CDF) is expressed as:(2)F(v)=1−e−(vλ)k

Although the Weibull distribution serves as a parametric basis for wind speed modeling, the recent development of deep learning-based forecasting methods provides additional data-driven solutions that can be further utilized to improve the accuracy of wind speed forecasting. For example, the hybrid approach of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), Sample Entropy (SE), Variational Mode Decomposition (VMD), and Bidirectional Temporal Convolutional Network-Transformer (BiTCN-TR) has been shown to achieve better results in modeling the multi-scale temporal and non-stationary patterns of wind power time series [33]. The combination of data-driven wind speed forecasting models and the PEM-based scenario generation method may provide more accurate input data for stochastic optimization, which can alleviate the conservatism of planning under uncertainty and potentially allow for higher wind penetration while meeting reliability criteria.

The electrical power output of a wind turbine is a function of wind speed and can be approximated using a piecewise linear model [22,31,32]:(3)Pw(v)={0if v≤vci or v>vcoPr⋅v−vcivr−vciif vci<v≤vrPrif vr<v≤vcowhere Pr is the rated power of the wind turbine, vci is the cut-in wind speed, vr is the rated wind speed, and vco is the cut-out wind speed.

To incorporate wind uncertainty into the optimization model, a scenario-based approach is employed using the Point Estimation Method (PEM). This method discretizes the continuous wind power distribution into a finite set of scenarios, each represented by a specific power output level and associated probability [31,32].

The steps for generating wind power scenarios are as follows:

Determine boundary probabilities for zero and rated power outputs:

Define the conditional PDF for power output in the linear region (vci≤v≤vr): (6)f~P(p)=fv(v(p))⋅dvdp1−P0−Prwhere v(p) is the inverse of the linear power-wind speed relationship.

The resulting set of power outputs {Pi} with probabilities {πi}, combined with the boundary points P0 and Pr, form the discrete wind power scenario set used in the stochastic optimization model.

In addition to wind power uncertainty, load variability is modeled using multiple loading levels (e.g., high, medium, low) based on typical daily or seasonal profiles. Each loading level is assigned to a duration factor, and the combined uncertainty space is constructed as the Cartesian product of wind scenarios and loading levels [22]. The coupling of stochastic wind modeling with network reconfiguration can be viewed as a topology-conscious approach to planning. In contrast to traditional planning approaches that assume the network topology to be fixed, the developed approach understands that optimal network performance under uncertainty can be achieved only through adaptive network topologies that are capable of adjusting to changing wind generation scenarios. This is in line with the latest flexibility-focused planning paradigms, where the network’s capability to reconfigure its topology offers an additional degree of freedom to adapt to wind variability [22]. The Cartesian product formulation of the uncertainty space, which integrates wind scenarios with multiple levels of loading, allows the optimization problem to determine topology settings that are robust over the entire range of operating conditions, thus incorporating flexibility into the planning solution.

The proposed method of constructing scenarios based on the Cartesian product of discretized wind power scenarios and loading levels provides a computationally efficient solution compared to more complex stochastic programming models. Recent advances in coordinated renewable and flexibility resource management have considered different techniques for modeling uncertainties, such as: (i) robust optimization models that ensure feasibility for the worst-case scenario realization [1,23–25], (ii) chance-constrained models that enforce reliability constraints in a probabilistic manner [1,23–25], and (iii) distributionally robust models that account for ambiguity in the probability distributions [1,23–25]. In contrast to these models, our PEM-based scenario generation method strikes a balance between accuracy and computational efficiency, producing a discrete set of representative scenarios that retain the statistical moments of the original continuous distributions. This is particularly advantageous for the integration of metaheuristic optimization algorithms that require multiple objective function evaluations.

This paper proposes a dual-objective stochastic optimization approach to achieve the optimal allocation of wind-based DG in active distribution networks, considering network dynamic reconfiguration under multiple loading conditions. It seeks to find a set of Pareto-optimal solutions that will provide the best trade-off between system energy losses reduction and system reliability enhancement. Although reductions in EENS and network energy losses may be positively correlated under certain operating conditions, they are not equivalent objectives. Loss minimization typically favors DG placement near load centers and low-impedance paths, whereas reliability enhancement (EENS reduction) promotes more spatially distributed DG placement to improve supply continuity during contingencies. Therefore, a Pareto-based multi-objective formulation is required to properly capture the trade-off between these objectives, rather than collapsing them into a single objective function. Unlike the commonly used single-objective methods, the proposed multi-objective formulation acknowledges the intrinsic conflict between technical performance and reliability. Thus, it allows decision makers to choose among the obtained solution according to the operational priorities. The overall structure of the proposed stochastic multi-objective framework is shown in Fig. 1.

The multi-objective formulation is valid due to the conflict between the two objectives. Minimizing energy losses and maximizing reliability enhancement could be in conflict because the first objective might suggest DG placement near the load centers with low-impedance paths, while the second objective could require DG placement in a distributed manner to serve islanding during outages. The weighted sum method is not suitable for this trade-off because it requires assigning weights to the objectives, and it might not be able to identify the non-convex part of the Pareto front. The proposed multi-objective method is able to identify the entire Pareto front.

The system load is represented by three different loading levels: High-H, Medium-M, and Low-L, which correspond to certain percentages of the peak load and for which specific annual duration is provided. That would ensure that the real consumption pattern in residential, commercial, and industrial categories is represented accordingly in daily and seasonal variations [22]. Furthermore, the probabilities of each loading level are determined based on its annual duration about the total hours in a year.

Let LL be loading level index, LL∈{H,M,L}. Each loading level is featured by a load scaling factor λLL, which states the percentage of peak load, and by an annual duration TLL in hours. In this way, the probability of each loading level may be calculated as:(13)ρLL=TLL8760

The active and reactive load at bus i for the loading level LL is calculated as:(14)Piload,LL=λLL⋅Pipeak (15)Qiload,LL=λLL⋅Qipeak

It allows the optimization model to consider real load profiles for different operating conditions and, by doing so, enhances the practical relevance of the obtained solutions.

The distribution network operation should be satisfied with the fundamental laws of power flow for all considered scenarios and loading levels. Physical feasibility and operational stability are ensured with the following constraints.

The active power balance of each bus is maintained through Eq. (19), which considers power inflows, outflows, and losses:(19)Ps,LL,inet=∑j∈δ(i)Ps,LL,ijline−∑k∈π(i)(Ps,LL,kiline−rki⋅Is,LL,ki2)+gi⋅Vs,LL,i2

Analogously, the balance of reactive power is imposed to guarantee voltage stability and adequate management of reactive power:(20)Qs,LL,inet=∑j∈δ(i)Qs,LL,ijline−∑k∈π(i)(Qs,LL,kiline−xki⋅Is,LL,ki2)+bi⋅Vs,LL,i2

The voltage drop equation, which includes both resistive and inductive effects, describes the relationship between voltage magnitudes at connected buses:(21)Vs,LL,j2=Vs,LL,i2−2(rij⋅Ps,LL,ijline+xij⋅Qs,LL,ijline)+(rij2+xij2)⋅Is,LL,ij2

Finally, apparent power flow constraint links current and voltage as well as flows in order to ensure coherence in the computation of power flows is expressed as follows:(22)Is,LL,ij2⋅Vs,LL,i2=(Ps,LL,ijline)2+(Qs,LL,ijline)2

In this respect, several technical limits must be obeyed by the distribution network under all operating conditions to ensure safe and reliable operation.

Voltage magnitude at each bus should be within permissible limits to avoid damaging the equipment and to maintain power quality:(23)Vmin≤Vs,LL,i2≤Vmax∀i∈Ωbus,∀s,∀LL

Similarly, line currents cannot be in excess of their thermal ratings to prevent overheating and possible failure:(24)Is,LL,ij2≤Imax,ij2∀(i,j)∈Ωline,∀s,∀LL

The net power injection at each bus is defined as the summation of wind DG output, existing DG contribution, and load demand:(25)Ps,LL,inet=Ps,LL,iWDG+Ps,LL,iDG−Ps,LL,iload (26)Qs,LL,inet=Qs,LL,iWDG+Qs,LL,iDG−Qs,LL,iload

Wind DG integration at the distribution level must face several physical and operational constraints reflecting capabilities and limitations of the wind generation technology itself.

The active power output of a wind DG unit is bound by its installed capacity and the available wind resource, which varies by scenario and loading level:(27)0≤Ps,LL,iWDG≤Prated,iWDG⋅γs,LLwind⋅ziWDG

Practical considerations impose limits on the number and size of the wind DG that can be installed in the network. These are captured by the following constraints:(28)∑i∈ΩbusziWDG≤NWDGmax (29)PWDGmin⋅ziWDG≤Prated,iWDG≤PWDGmax⋅ziWDG

Network reconfiguration enables the dynamic modification of the network topology through opening and closing switches, thus providing a better load balancing and reduction of active power losses [27]. Therefore, this reconfiguration process has been modeled independently for each loading level to preserve the adaptative nature of modern distribution systems. The status of each line is modelled using a binary variable, where a value of one represents a line that is closed and conducting power:(30)αijLL∈{0,1}∀(i,j)∈Ωline,∀LL

To maintain the radial structure of the distribution network, which is a must for protection coordination and fault management, the number of closed lines must be one less than the number of buses:(31)∑(i,j)∈ΩlineαijLL=Nbus−1∀LL

Connectivity of the network is guaranteed by flow direction variables, which avoid the formation of isolated buses and ensure all the loads are supplied:(32)βijLL+βjiLL=αijLL∀(i,j)∈Ωline,∀LL (33)∑j∈δ(i)βijLL=1∀i∈Ωbus\{substation},∀LL

Moreover, voltage feasibility needs to be guaranteed in any switching state to make sure that voltage limits are not violated no matter what the network topology is:(34)Vs,LL,i2≤Vmax⋅αijLL+M⋅(1−αijLL) (35)Vs,LL,i2≥Vmin⋅αijLL−M⋅(1−αijLL)

In this section, a systematic methodology to assess the reliability of the distribution systems in terms of a particular calculation of EENS is introduced. As explained before, EENS represents the second objective function in our methodology. The strategy presented in this section allows an analytical calculation of reliability without simulative methods, which will make it efficient in terms of CPU time, taking into consideration that this calculation will need to be performed a large number of times during our search in optimized algorithms, such as those presented in future chapters.

The approach thematically integrates the uncertainty modeling of wind power generation presented in Section 2 and takes into consideration the reliability improvements achievable by wind DG under fault conditions because of localized generation as investigated in [34]. The approach considers realistic modeling of the support capability of wind resources for islanded mode operations, thus assessing in a realistic way the mitigation capacity achievable by strategic planning of wind DG in case of outages.

The EENS calculates repercussions of interruptions in terms of total energy not supplied to consumers based on all possible failure situations and their probabilistic chances in a way that restoration strategies incorporated in EENS are considered. The EENS model puts into focus both a probabilistic model of wind generation output in different scenarios, along with a multi-level load model.

The reliability objective function is formulated as:(36)f2=EENSTotal=∑s=1Nsρs⋅[∑LL=1NLLρLL⋅(∑i∈ΩbusPs,LL,iload⋅Ds,LL,ieff)]where Ns is the total number of wind generation scenarios, ρs is the probability of wind scenario s (from Section 2), NLL is the number of loading levels (High, Medium, Low), ρLL is the probability (duration fraction) of loading level LL, Ωbus is the set of all buses in the network, Ps,LL,iload is the load demand at bus i under scenario s and loading level LL, Ds,LL,ieff is the effective annual interruption duration at bus i.

The effective interruption duration accounts for the reduction in outage time enabled by wind DG-supported island operation and is defined as follows [34]:(37)Ds,LL,ieff=DSs,LL,i+(1−ζs,LL,i)⋅(DRs,LL,i−DSs,LL,i)where DSs,LL,i is the annual switching-only interruption duration (temporary outage during fault isolation), DRs,LL,i is the annual repair-and-switching interruption duration (outage requiring repair), and ζs,LL,i is the wind DG islanding capability indicator.

The core of reliability calculation relies on the radially connected network topology formed by the binary connectivity variables βijLL presented in Section 3.7. The main idea of these variables is to represent the orientation of power flow and parent-child relationships among the buses, which make up an important part of calculating fault-propagation paths to the end consumers.

For each branch (i,j)∈Ωline, a binary variable βijLL takes a value of 1 if the load bus j gets power from bus i for loading level LL, otherwise 0. The topology constraint for a radial system enforces each load bus, except for the “substation”, to have exactly one “parent”:(38)∑j∈δ(i)βijLL=1∀i∈Ωbus\{substation},∀LLwith this topological characteristic, a reliable analysis can be performed because it determines paths for each load bus. The failure of a component affects all buses below it, leaving all buses above unaffected. Such analysis makes it possible to recursively compute downtime in a systematic manner along each feeder.

The DG islanding capability index ζs,LL,i will indicate if a bus can be self-powered by local winds in case of an outage upstream. The presence of this capability can greatly lower the actual outage duration if adequate wind power is available in the local area to satisfy local load.

The indicator is defined as:(42)ζs,LL,i={1if Ps,LL,iWDG≥Ps,LL,iload⋅ξ and ziWDG=10otherwisewhere Ps,LL,iWDG is the available wind DG generation at bus i under scenario s and loading level LL, Ps,LL,iload is the load demand at bus i, ξ is a security factor (typically 0.9–1.0) accounting for operational margins and reserve requirements, ziWDG is the binary decision variable indicating wind DG installation at bus i.

The available wind generation is determined using the scenario-based uncertainty model from Section 2:(43)Ps,LL,iWDG=Prated,iWDG⋅γs,LLwind⋅ziWDGwhere γs,LLwind is the scenario- and load-level-dependent wind power output factor.

The full procedure for EENS calculation for each candidate DG allocation and network configuration considered during the optimization consists of the following steps:

The GWO algorithm is a swarm intelligence technique based on social hierarchy and hunting behavior of grey wolves [35]. The search space is organized in a social hierarchy with an alpha (α) individual symbolizing the global optimum solution. The beta (β) and delta (δ) denote the second and third best solutions, respectively, and all other solutions are omega (ω) candidates. The search strategy in this optimization algorithm is driven by α, β, and δ wolves in a three-step manner [35]:

Social Hierarchy and Encircling Prey: The wolves adjust their positions to surround their prey (the best-known solution). This strategy can be described using these equations:(45)D→=∣C→⋅X→p(t)−X→(t)∣ (46)X→(t+1)=X→p(t)−A→⋅D→here, t represents the current iteration, X→ symbolizes the position vector of a wolf, and X→p represents the position vector of a prey. The coefficient vectors A→ and C→ are calculated using the following equations:(47)A→=2a→⋅r1→−a→,C→=2⋅r2→where r1→,r2→ are random vectors in [0, 1]. The components of vector a→ linearly decrease from 2 to 0 in order to regulate exploration and exploitation trade-off.

Hunting: The hunting procedure is controlled by the α, β, and the ones who are assumed to have knowledge of where the prey can be found. The ω wolves will update their positions based on:(48)D→α= ∣C1→⋅X→α−X→∣,D→β= ∣C2→⋅X→β−X→∣,D→δ= ∣C3→⋅X→δ−X→∣X1→=X→α−A1→⋅D→α,X2→=X→β−A2→⋅D→β,X3→=X→δ−A3→⋅D→δ X→(t+1)=X1→+X2→+X3→3

The conventional GWO algorithm is intended for single objective optimization. To address this problem which contains bi-objectives—minimizing energy loss function (f1), and improving reliability function (f2)—the I-MGWO algorithm incorporates an external archive and a Pareto selection approach [36]. With this approach, it is possible for the algorithm to search and store different non-dominated solutions simultaneously to construct a Pareto optimal front.

The step-by-step process of I-MGWO for solving the problem of optimized allocation of wind DG is explained below and shown in Fig. 2.

The choice of the metaheuristic strategy for solving this problem is based on the nature of the wind DG allocation and network reconfiguration optimization problem:

Problem complexity: The problem has mixed-integer decision variables (continuous DG capacity, integer geographical location, and binary switch status at different load levels), non-linear power flow equations, and non-convex objective functions. Gradient-based optimization algorithms are not applicable because of the presence of discrete decision variables and non-convexities, and branch-and-bound algorithms are computationally infeasible for large problem sizes.

In terms of convergence, although metaheuristics cannot ensure deterministic convergence to the global optimum, they can fulfill probabilistic convergence criteria under certain conditions. The original GWO algorithm has been proved to possess properties of Markov chains, and its convergence characteristics can be studied using the framework of stochastic processes [36]. The proposed modifications to I-MGWO maintain these theoretical underpinnings while offering improved performance:

Quasi-oppositional seeding helps to enhance the initial distribution of the population within the search space, thus lowering the chances of premature convergence to local optima [20].

It is worth pointing out that, in the context of NP-hard problems like the current ones, the aim is not to achieve global optimality (which is not computationally feasible for larger problem sizes) but to achieve high-quality approximate solutions within a reasonable computational time. The statistical results shown in Section 6.3.3 clearly show that I-MGWO performs significantly better in finding high-quality solutions in independent runs with very low standard deviations. The Friedman rank tests validate that the results are statistically significant.

Four different operational cases are analyzed with the objective of understanding separate and integrated impacts of network reconfiguration and wind DG allocation. Each case identifies a different planning or operational strategy that might be followed by distribution system operators:

Case 1 (Base Case): Original network configuration without wind DG or reconfiguration.

In this work, each case is analyzed for single-objective optimization-that is, minimizing energy losses or EENS separately-and multi-objective optimization, which minimizes both objectives simultaneously. Such a comprehensive analysis can allow deep comparison of different operational strategies and optimization approaches.

The optimization algorithms were run under identical computational conditions to ensure statistical robustness and a non-biased comparison. Each algorithm was run independently 25 times with a population size of 50 individuals and a maximum number of iterations of 100. Specific control parameters of the implemented algorithms in this study, namely the proposed Improved Grey Wolf Optimizer (IGWO), standard Grey Wolf Optimizer (GWO) [35], Particle Swarm Optimization (PSO) [39], and Genetic Algorithm (GA) [40] were set to their standard values as recommended in their original references, respectively, summarized in Table 5. This experimental setting provides a fair basis for testing the performance and reliability of algorithmic solutions across the four defined operational cases.

The next subsections describe the specific results obtained for the minimization of total system energy losses and for the minimization of the EENS, analyzed individually. Values reported correspond to the best solution found in all 25 runs of each case.

The multi-objective optimization results manifest a complete view of trade-offs and synergistic effects between energy loss minimization and reliability enhancement (EENS reduction). All algorithms, such as MOIGWO and standard MOGWO [36], MOPSO [41], and NSGA-II [42], were executed with a population size of 50 and a maximum limit of iterations as 100 for fair comparison. The control parameters of competitive algorithms (MOGWO, MOPSO, NSGA-II) were set as per the standard values recommended in their respective foundational references to ensure a balanced and equitable performance assessment of all the algorithms.

Fig. 7 reports the Pareto fronts obtained by each compared metaheuristic. A comparison of the results clearly points out the different algorithmic behaviors. MOIGWO was able to produce the most spread out and widespread Pareto front in all the comparisons, with the highest Maximum Spread values in Cases 2 and 4, respectively, equal to 1.3245 and 1.2960. This means that it can explore the most extreme portions of the objective space. In addition, MOIGWO reached the best Hypervolume values in Cases 2 and 4, respectively, equal to 0.8916 and 0.8252, confirming that its solution set is not only diverse but also closer to the true Pareto-optimal front. NSGA-II showed an excellent spacing between solutions, especially in Cases 2 and 3 with SP values equal to 0.0378 and 0.0839. Nevertheless, considering the overall convergence, measured through HV, it generally resulted in being worse than that ensured by MOIGWO. MOGWO showed a good diversity in the population, but it got the poorest convergence with the worst HV values for each case. MOPSO has an incoherent trend, too often obtaining the smallest spread and the worst spacing, with the lowest ranking index. This explains why the Pareto fronts in Fig. 7 do not collapse to a single minimum point, even though the objectives may exhibit partial correlation.

The best compromise solutions, chosen from the Pareto front using a fuzzy decision-making method, are presented in Tables 13 and 14. These solutions provide the most balanced trade-off between the two conflicting objectives to achieve the maximum overall improvement with respect to the base case. MOIGWO found the most balanced compromise solutions in the integrated Case 4: this case yields a total improvement of 38.00% (31.35% loss reduction and 44.66% EENS reduction). MOIGWO was also able to provide the best overall solution for Case 2, yielding an overall gain of 38.50%. Additionally, for Case 3-Wind DG Only-a total gain of 21.66% was achieved by the best compromise solution offered by NSGA-II. This again points towards the very important implication of having only wind DG-based allocations, which decreases the energy losses significantly, but does not improve the reliability as much as it would with the adaptability provided by the network reconfiguration. Indeed, the compromises obtained for Case 4 outperform the results by Case 2 or Case 3 alone in terms of better/balanced improvements in both objectives, confirming that the co-optimization between generation placement and network topology benefits from synergies.

While the results reveal very promising potential for the proposed integrated framework, some practical limitations and implementation considerations are in order. The computational burden of the stochastic approach presents scalability challenges when applying the framework to larger distribution systems, as each additional bus increases the dimensionality of the search space and the complexity of power flow calculations, potentially leading to quadratic increases in computation time. In particular, the optimization model relies on perfect forecasting of wind power scenarios and deterministic load levels, while actual distribution system operations must deal with forecast errors and continuous rather than discretized load variations. In particular, the optimization model relies on perfect forecasting of wind power scenarios and deterministic load levels, while actual distribution system operations must deal with forecast errors and continuous rather than discretized load variations. This points to the need for robust or adaptive real-time strategies to complement the offline planning solution. Furthermore, this work focuses on steady-state analysis and does not explicitly model transient stability and protection coordination challenges that may arise in dynamic network reconfiguration. The latter aspects are quite critical for practical implementation since switching operations need to be performed in a way that system stability is guaranteed and coordination among protective devices is maintained. Although the economic dimension has been recognized, it has not yet been completely included in the present technical optimization. The planning decision would necessitate a rigorous cost-benefit study that involves investment cost of wind DGs, switching operation and maintenance cost, and economic benefit due to improvements in reliability. The proposed model applied three fixed loading levels, which, though representative, simplifies real load profile continuous temporal variability. Higher time resolution such as hourly or daily segments may result in higher precision and more relevance to operation for the model but at greater computational complexity. The final assumption in the optimization is that all candidate buses and switches are indeed available for intervention, which might not be held true in practice because of physical constraints, land availability, or regulatory restrictions. These spatial and regulatory limitations should be part of future work with the aim of improving practical applicability based on the proposed methodology. In any case, the framework developed here provides a sound basis from a methodological point of view for strategic planning, where substantial technical benefits can be attained via coordinated wind DG allocation and network reconfiguration.