A barrel theory-based optimization of stochastic PV-DG integration in radial distribution networks under load and solar uncertainties

Abstract

The increasing penetration of photovoltaic distributed generation (PV-DG) in Radial Distribution Systems (RDSs) plays a vital role in achieving sustainable energy transition objectives; however, the inherent uncertainty associated with solar irradiance and load demand poses significant challenges to optimal planning and operation. This paper presents a stochastic optimization framework for PV-DG allocation in RDSs using the Barrel Theory-Based Optimizer (BTO). Uncertainties in solar irradiance and load demand are explicitly modeled using appropriate probability density functions and efficiently represented through a higher-order Point Estimate Method (PEM), which captures the essential statistical characteristics with a limited number of representative scenarios. The proposed framework simultaneously optimizes the location and capacity of PV-DG units to minimize real power losses and enhance voltage profile performance while ensuring system operational constraints are satisfied. The effectiveness of the proposed approach is validated on the 85-bus and the IEEE 118-bus RDSs, where the BTO exhibits superior convergence characteristics and enhanced solution robustness when compared with several benchmark optimization techniques, including the well-established Differential Evolution Algorithm (DEA), the recent Crocodile Ambush Optimization (CAO, 2025), and the Schrödinger Optimizer Algorithm (SOA, 2025). For the 85-bus RDS, the impact of integrating different numbers of PV units is systematically investigated. Simulation results confirm that the proposed BTO-based stochastic planning strategy significantly improves energy efficiency, voltage regulation, and loss reduction, thereby enhancing the overall sustainability of the RDS. For the 85-node RDS, the BTO achieves a noticeable reduction in average real power losses, outperforming DEA, CAO, and SOA by 2.55%, 4.10%, and 6.74%, respectively, when three PV units are installed. Additionally, for the case of four PV units, the proposed BTO yields even greater improvements, with loss reductions of 5.12%, 7.50%, and 14.12%, respectively, compared with the same benchmark algorithms. Furthermore, for five PV units, the BTO achieves much greater reduction, outperforming DEA, CAO, and SOA by 13.05%, 6.45%, and 32.31%, respectively, when three PV units are installed.

Introduction

Traditional power system planning and control, which traditionally relied on a unidirectional flow of energy from centralized plants to consumers, have changed due to the emergence of Distributed Generation (DG) in electrical distribution networks¹. Different DG resources, such as solar systems, wind turbines, combined heat and power units, microturbines, diesel generators, and battery storage, are now being directly integrated into distribution networks².

Importance of DG integration

DG integration is driven by the growing demand for dependable and sustainable energy, as well as advancements in technology and supportive regulatory frameworks. Understanding the impact, benefits, and constraints of DG in distribution systems is critical for utilities, researchers, and politicians. This study aims to thoroughly investigate these aspects, providing analytical insights and practical recommendations to aid in the proper integration of DG into modern power distribution networks.

DG enhances modern power systems by reducing transmission and distribution losses, mitigating feeder loading, and postponing network improvements. It improves power quality, voltage profiles, and provides ancillary services such as peak load shaving and frequency support. Additionally, DG bolsters system resilience by ensuring supply continuity during emergencies and enabling the formation of microgrids that support essential loads during outages.

By lowering reliance on centralized power plants and promoting regional energy markets, DG improves both economic efficiency and environmental sustainability. It supports decarbonization objectives by helping in the integration of renewable resources. However, DG presents operational difficulties such voltage rise, bidirectional power flow, and harmonic distortions, requiring sophisticated grid management and intelligent technology for secure operation.

Literature review

The positioning and sizing of DG units in distribution networks is critical due to its significant effects on system efficiency, reliability, and power quality³. The traditional design of distribution systems for unidirectional power flow requires careful planning to maximize benefits like loss reduction and reliability, while minimizing challenges such as voltage rise and congestion. Consequently, extensive research is focused on naming optimal locations and capacities for DG units within these systems.

The best locations and sizes for DG have been found using a range of methods. Mathematical indices such as voltage stability indices^4,5, loss sensitivity factors^2,3,4,6,7, and dynamic fault tree using dynamic loss evaluation indicator⁸ were first used in analytical and deterministic approaches⁶. Classical optimization techniques were adopted to formally model DG allocation as system complexity increased. However, because of the nonlinear nature of power systems and competing goals, metaheuristic and evolutionary algorithms became popular due to their adaptability and ability to traverse challenging search spaces⁹.

The No Free Lunch (NFL) Theorem asserts that no machine learning or optimization algorithm is superior in all scenarios¹⁰. Each method relies on specific assumptions about data structures, such as linearity or smoothness, and performs optimally when these match the dataset’s actual characteristics. Conversely, an algorithm’s strengths in some problem areas may be offset by weaknesses in others. Therefore, success in model performance is influenced by understanding the problem, the quality and amount of available data, and domain ability. Due to the latter mentioned, the room for improvement is still open based on this NFL theorem. To find the best location and size for DG units in distribution networks, metaheuristic optimizers have been used extensively^3,11. By improving system performance while meeting operational and technological restrictions, they are utilized to solve complicated, nonlinear, and multi-objective optimization issues¹². Minimizing power losses^3,13, improving voltage profiles^2,3, boosting reliability^13,14, expanding renewable energy hosting capacity¹⁵, lowering operating costs^16,17, decreasing emissions, and preserving system stability under changing load¹⁸ and generating circumstances are examples of typical uses.

Research has increasingly focused on multi-objective^{13,16,19,20,21} and probabilistic optimization frameworks to tackle realistic conditions such as load uncertainty^22,23 and renewable DG fluctuations²⁴. These frameworks produce Pareto-optimal solutions that balance goals like loss and cost minimization, voltage enhancement, emission reduction, and improved reliability^16,17. Stochastic methods, including probabilistic power flow²⁵ and Monte Carlo simulation (MCS)^25,26, facilitate precise evaluations of DG performance in uncertain conditions^22,27,28. Moreover, advancements in computational intelligence and data availability have promoted the use of AI and machine learning techniques for adaptive DG allocation strategies^{5,29,30,31,32,33}.

Additional comprehensive reviews on DG placement, associated technologies, planning strategies, and solution methodologies are available in referenced recent literature^1,4,34,35,36. This brief literature demonstrates a transition from simple deterministic techniques to sophisticated stochastic, multi-objective, and intelligent optimization frameworks.

Research gap

Current research on the optimal allocation of PV-DG units in RDSs reveals several critical gaps. First, a large portion of prior studies adopts deterministic optimization frameworks, where load demand and solar irradiance are assumed to be fixed or represented by limited scenarios. Such assumptions oversimplify real operating conditions and may lead to suboptimal or unreliable planning decisions when subjected to inherent uncertainties in renewable generation and load variability.

Second, although stochastic methods exist, they often involve complex techniques like MCS, which increase computational demands and hinder practical applicability. Additionally, some probabilistic models compromise between accuracy and efficiency, either oversimplifying uncertainties or requiring excessive simulations.

Third, commonly used metaheuristic algorithms like Particle Swarm Optimization (PSO), Genetic Algorithms (GA), and Grey Wolf Optimizer (GWO) face challenges such as premature convergence and inadequate exploration-exploitation balance when tackling high-dimensional, nonlinear, and constrained problems in DG allocations. Although some recent algorithms have sought to resolve these issues, their reliability in stochastic environments is not always confirmed. Moreover, the majority of reported works are validated on small or medium-scale benchmark systems, with limited attention given to scalability and performance on larger distribution networks. This raises concerns about the generalizability and applicability of these methods to large-scale systems with increased complexity and higher penetration of renewable energy sources.

Finally, there is a noticeable lack of integrated frameworks that simultaneously merge efficient stochastic modeling techniques, advanced optimization algorithms, and thorough performance evaluations, which encompass convergence behavior and statistical robustness. In light of these limitations, there is a clear need for a computationally efficient, robust, and scalable stochastic optimization framework capable of accurately capturing uncertainties while ensuring high-quality, stable, and reliable solutions for PV-DG allocation in modern distribution systems.

Study overview, paper’s contributions and outline

This study develops a stochastic optimization framework based on the Barrel Theory-Based Optimizer (BTO)³⁷ for the optimal sizing and siting of photovoltaic distributed generation (PV-DG) units in radial distribution systems. Inspired by the principle that the overall strength of a barrel is limited by its weakest plank, BTO models decision variables as planks and candidate solutions as barrels. The algorithm enhances search efficiency by guiding exploration toward promising regions using elite solutions, while simultaneously improving weaker solutions through a dedicated barrel adjustment mechanism. This balance between exploration and exploitation enables BTO to effectively solve complex, nonlinear, and high-dimensional engineering optimization problems³⁷. It views variables as planks and solutions as barrels, leveraging elite solutions to direct exploration and exploitation while trying to enhance low-fitness solutions through a “Barrel Adjustment” stage. According to recent research, BTO may provide engineering design optimization more quickly and accurately than techniques like PSO and GWO.

The primary objective of the proposed framework is to minimize real power losses while improving voltage profile performance, subject to operational constraints including voltage limits, power balance, and DG capacity bounds. To realistically capture system uncertainties, the framework explicitly incorporates stochastic variations in solar irradiance and load demand through a higher-order Point Estimate Method (PEM), which provides an efficient and accurate probabilistic representation without excessive computational burden.

The effectiveness and scalability of the proposed approach are confirmed through comprehensive simulations on both the IEEE 85-bus and the large-scale IEEE 118-bus radial distribution systems, under multiple PV penetration scenarios. In addition to achieving superior loss minimization and voltage profile enhancement, the results on the 118-bus system further confirm the robustness, consistency, and reliability of BTO, as evidenced by its fast convergence behavior and significantly lower statistical dispersion compared to benchmark algorithms.

To the best of the authors’ knowledge, this study represents the first attempt to employ and use this BTO algorithm in the context of power system applications, particularly for stochastic PV-DG allocation in radial distribution systems. The main contributions of this work can be summarized as follows:

• A stochastic optimization framework is developed for optimal PV-DG allocation in RDSs, explicitly accounting for uncertainties in solar irradiance and load demand,

• A novel integration of the BTO with a higher-order PEM is proposed, enabling efficient and accurate uncertainty modeling,

• A simultaneous optimization formulation is established for deciding the optimal locations, sizes, and operating conditions of PV-DG units while satisfying system operational constraints,

• Extensive simulations on the IEEE 85-bus system demonstrate the superior convergence characteristics, robustness, and effectiveness of the proposed BTO compared to several classical and recent metaheuristic algorithms, and.

• The applicability and scalability of the proposed approach are further validated on the large IEEE 118-bus system, where BTO achieves lower power losses, improved voltage profiles, and significantly enhanced robustness, as confirmed by statistical and boxplot analyses.

Apart from the introduction, the remainder of this paper is organized as follows: Sect.  2 discusses PV-DG integration in radial distribution networks, with particular focus on the uncertainties associated with PV generation, load demand, and their probabilistic modeling. Section  3 describes the operational procedure of the BTO algorithm. Section  4 presents the numerical results, analyzed scenarios, and comparative performance evaluations. Finally, Sect.  5 provides the conclusions and key findings of the study.

Probabilistic formulation of PV-DG integration in RDSs

Probabilistic modeling of PV generation and electrical load

The study focuses on uncertainties in solar PV generation and electrical load demand, which complicates RDS planning²⁵. It utilizes the PEM to efficiently represent these uncertainties, employing a lognormal distribution for solar irradiance. The study illustrates the probability density from MCS and the analytical PDF, with PEM concentration points and mean value marked for clarity. Therefore, the solar irradiance PDF can be modelled as follows:

$$\:{f}_{Ir}\left(Ir\right)=\frac{1}{Ir\times\:{\sigma\:}_{Ir}\times\:\sqrt{2\pi\:}}{e}^{\left(-\frac{{\left({ln}\left(Ir\right)-{\mu\:}_{Ir}\right)}^{2}}{2{\left({\sigma\:}_{Ir}\right)}^{2}}\right)}$$

(1)

As a result, two concentrated locations and a mean point as standardized sites accurately depict the stochastic pattern of solar energy output as follows³⁸:

$$\:{\xi\:}_{Ir,k}=\frac{{S}_{Ir}}{2}+{\left(-1\right)}^{3-k}\sqrt{\left(\frac{{K}_{Ir}-3{{S}_{Ir}}^{2}}{4}\right)}\:,\:\forall \text{k = 1,2}$$

(2)

Thus, the locations of the solar irradiance can be specified as:

$${G}_{k}={\mu\:}_{Ir}+\left({\xi\:}_{Ir,k}\times\:{\sigma\:}_{Ir}\right),\:\:\forall \text{k}=\text{1,2}$$

(3)

An irradiance-to-power conversion model is used to compute the probabilistic PV output while accounting for solar production variation. This framework transforms sporadic solar radiation into the PV plant’s electrical output³⁹ as follows:

$$\:{P}_{PV,s}\left(Ir\right)=\left\{\begin{array}{c}{P}_{PV\_Rated}\times\:I{r}^{2}/\left(I{r}_{c}\times\:I{r}_{STD}\right),\\\:{P}_{PV\_Rated}\times\:\left(Ir/I{r}_{STD}\right),\end{array}\right.\:\begin{array}{c}\forall\:\:0<Ir<I{r}_{c}\\\:\forall\:\:Ir\ge\:I{r}_{c}\end{array}\text{}$$

(4)

The amount of electricity generated is determined via piecewise function model³⁹, that distinguishes between nonlinear and linear operating areas. In addition, load ratio uncertainty is expressed using a normal PDF, as seen in Fig. 1. The following is the definition of the load ratio PDF³⁹ as follows:

$$\:{f}_{LD}\left(LD\right)=\frac{1}{\sqrt{2\pi\:{\left({\sigma\:}_{LD}\right)}^{2}}}{e}^{\left(-\frac{{\left(LD-{\mu\:}_{LD}\right)}^{2}}{2{\left({\sigma\:}_{LD}\right)}^{2}}\right)}$$

(5)

Fig. 1

Normal PDF modelling of loading demand.

Consequently, the following formula is used to determine the load concentration points in prescribed positions:

$${\xi\:}_{LD,k}=\frac{{S}_{LD}}{2}+{\left(-1\right)}^{3-k}\sqrt{\left(\frac{{K}_{LD}-3{{S}_{LD}}^{2}}{4}\right)},\forall \text{k}=\text{1,2}$$

(6)

After that, they are converted to actual loading levels as

$${L}_{k}={\mu\:}_{LD}+\left({\xi\:}_{LD,k}\times\:{\sigma\:}_{LD}\right),\forall\text{k}=\text{1,2}$$

(7)

In order to meet moment-matching requirements with no sacrificing overall probability mass, their weights have ultimately been determined. Different scenarios with various levels of uncertainty are used to generate a combined point estimate matrix that represents load and PV output uncertainties.

Probabilistic losses minimization objective

The aim function in this work is to minimize the anticipated power losses across all identified PEM situations. As a result, it is represented as the following weighted aggregate of the estimated losses in every particular PEM scenario:

$$\:f={\sum\:}_{s=1}^{2m+1}\left({\omega\:}_{s}\times\:P{L}_{s}\right),\:\:\:\:\text{m}=2\text{variables}$$

(8)

Probabilistic constraints

The following equality constraints include the reactive and active power balance formulae for every PEM scenario:

$$\:{\left({\sum\:}_{k=1}^{{N}_{PV}}{P}_{PV,k}+{P}_{Supply}\right)}^{\left(s\right)}={\left(PL+{\sum\:}_{j=1}^{{N}_{bus}}P{D}_{j}\right)}^{\left(s\right)}$$

(9)

$$\:{\left({\sum\:}_{k=1}^{{N}_{PV}}{Q}_{PV,k}+{Q}_{Supply}\right)}^{\left(s\right)}={\left(QL+{\sum\:}_{j=1}^{{N}_{bus}}Q{D}_{j}\right)}^{\left(s\right)}$$

(10)

The reactive power supply for every PV-DG is regulated by the prescribed power factor outlined below:

$$\:{Q}_{PV,k}={P}_{PV,k}\times\:{tan}\left({{cos}}^{-1}\left(P{F}_{k}\right)\right),\text{k}=1:{\text{N}}_{PV}$$

(11)

Each PV-DG generated actual power has to match the subsequent specifications:

$$\:0\le\:{\left({P}_{PV,k}\right)}^{\left(s\right)}\le\:{P}_{PV\_size},\:\forall \text{k}\in{\text{N}}_{PV}$$

(12)

Furthermore, the items below restrict the device’s operational power factor:

$$\text{0.85}\le\:P{F}_{k}\le\:1,\forall \text{k}\in{\text{N}}_{PV}$$

(13)

The voltage at every bus should always be inside the permitted operating limits⁴⁰:

$$\:{V}_{min}\le\:{\left({V}_{k}\right)}^{\left(s\right)}\le\:{V}_{max}\text{,}\text{}\forall\text{}\text{k}\text{}\in{\text{N}}_{bus}$$

(14)

The current flowing through every single distribution path can’t exceed the thermal capacity limitation⁴¹:

$$\:{\left({I}_{br}\right)}^{\left(s\right)}\le\:{I,\text{}\forall\text{}\text{br}\text{}\in{\text{N}}_{Line}}_{br,max}$$

(15)

To avoid excessive reverse power flow, the total PV-DG being penetrated for every scenario is restricted as⁴²:

$$\:{\left({\sum\:}_{k=1}^{{N}_{PV}}{P}_{PV,k}\right)}^{\left(s\right)}={\phi\:}_{P}\times\:\left({\sum\:}_{j=1}^{{N}_{bus}}P{D}_{j}\right)$$

(16)

where: $\:{\phi\:}_{P}$ is fixed at 60%⁴³.

A barrel theory-based optimization

The BTO is inspired by Barrel Theory, which models system performance as constrained by the weakest element. Barrel theory (also known as Liebig’s Law of the Minimum) states that “The capacity of a system is limited by its weakest component”³⁷. In the BTO, a barrel is composed of vertical wooden planks³⁷. The barrel can only hold water up to the height of its shortest plank. Improving taller planks does not increase capacity unless the shortest plank is strengthened. This metaphor is translated into optimization by identifying weak solution dimensions or individuals, strengthening them through targeted improvement, and allowing stronger components to guide the overall population. Barrel is mapped as population while plank represents solution variable (dimension)³⁷. The shortest plank corresponds to the weakest variable/low-fitness solution while the water level indicates the overall solution quality. Instead of only refining the best solutions, BTO actively strengthens weak solutions and weak dimensions, preventing stagnation and improving population-wide performance.

Population initialization

Each individual Y_i is initialized uniformly within bounds:

$$\:{Y}_{i,j}\left(t=0\right)=l{b}_{j}+{Z}_{1}\times\:\left(u{b}_{j}-l{b}_{j}\right);\text{}\forall\text{}i\in\:{N}_{S},\:j\in\:D$$

(17)

The fitness of each individual is evaluated as f(Y_i) and the global best solution ($\:{Y}_{GBest}$) is extracted which achieves the minimum fitness score.

Adaptive elite selection, probability based on weakness and harmonic disturbance

First, the elite size is determined where the number of elite solutions (K) increases adaptively:

$$\:K={max}\left(1,\beta\:\times\:\left(1-CE\right)\right)$$

(18)

where: β is the elite ratio parameter (β = 10)⁴⁴, and CE is a time-decreasing convergence factor that controls the search behavior:

$$\:CE={\left(\sqrt{\frac{t}{T}}\right)}^{\nu\:}$$

(19)

This model ensures that large values of CE activate exploration in the early stage, while small values of CE activate exploitation in the later stage.

The elite set ($\:\mathfrak{I}$) consists of the best K individuals which are sorted by ascending fitness:

$$\:\mathfrak{I}=\left\{\begin{array}{cccc}{Y}_{\left(1\right)},&\:{Y}_{\left(2\right)},&\ldots &{Y}_{\left(K\right)}\end{array}\right\}$$

(20)

Based on that model, early iterations rely on fewer elites (diversity), while later iterations exploit more high-quality solutions.

Also, each individual receives an update probability ($\:{\rho\:}_{i}$):

$$\:{\rho\:}_{i}={\lambda\:\left({\lambda\:min}_{max}\times\:{{f}_{i}}^{norm}\right)}_{min}$$

(21)

$\:{\lambda\:}_{\text{m}\text{i}\text{n}}\:$ and $\:{\lambda\:}_{\text{m}\text{a}\text{x}}\:$ control how frequently the decision variables of a solution are adjusted during the optimization process. This adaptive range allows the algorithm to vary the intensity of updates according to the relative quality of each individual.

The term $\:{f}_{i}^{\text{norm}}$ represents the normalized fitness of the $\:i$-th individual, reflecting its performance relative to the current population on a scale of 0 to 1 Through this normalization, individuals with poorer fitness receive higher update probabilities, while better-performing solutions are modified less aggressively. The normalized fitness is computed as follows:

$$\:{{f}_{i}}^{norm}=\frac{{f}_{i}-{f}_{B}}{{f}_{W}-{f}_{B}+\varepsilon\:}$$

(22)

$\varepsilon$ avoids division by zero. This model demonstrates that poor solutions will have higher update probability, while strong solutions will be protected from excessive modification, which directly reflects barrel theory.

Moreover, a disturbance vector introduces controlled randomness:

$$\:F=2\times\:\left({r}_{2}-\frac{1}{2}\right)\times\:sin\left(\frac{20\pi\:{r}_{2}t}{T}\right)\times\:\left(\frac{T-t}{T}\right)$$

(23)

This form encourages exploration early and gradually vanishes to allow exploitation.

Plank update strategies

In the BTO algorithm, two plank update strategies are adopted which are the explorative adjustment via elite-based learning strategy or the exploitative refinement via directional adjustment strategy. Each variable is updated conditionally. The first one can be activated if a random number (rand) is less than the convergence factor (CE). Else, the second strategy is implemented.

Explorative adjustment via elite-based learning strategy

In this strategy, the updating approach focusses on acquiring knowledge from elite performers in order to better guide future upgrades. For every dimension j, the new component ($\:{{Y}_{i,j}}^{new}$ ) gets modified as follows:

$$\:{{Y}_{i,j}}^{new}={Y}_{R,j}+{F}_{j}\times\:\left(\left({Y}_{E,j}-{Y}_{R,j}\right)+{\eta\:}_{A}\cdot\:{Z}_{2}\cdot\:\left(1-CE\right)\cdot\:\left(u{b}_{j}-l{b}_{j}\right)\right)$$

(24)

η_A regulates the widest perturbation amplitude (η_A = 0.25).

Exploitative refinement via directional adjustment strategy

This method aims to improve exploration, particularly at the start of the optimization process. Each unique individual and dimension is updated depending on direction and step size as follows:

$$\:{{Y}_{i,j}}^{new}={Y}_{i,j}+{\delta\:}_{i,j}\times\:\left(\left(\left(1-\left|{F}_{j}\right|\right)\cdot\:CE\cdot\:{Z}_{2}\cdot\:{Y}_{i,j}\right)+{\eta\:}_{B}\cdot\:{F}_{j}\cdot\:\left(u{b}_{j}-l{b}_{j}\right)+\varepsilon\:\right)$$

(25)

η_B regulates the fundamental step size (η_B = 0.3). δ_{i, j} indicates the direction term that determines if the quantity across dimension j grows or declines as follows:

$$\:{\delta\:}_{i,j}=-\left({Y}_{E,j}-{Y}_{i,j}+\gamma\:\right)$$

(26)

where γ refers to the correction value which is specified as:

$$\:\gamma\:=\left(2\times\:\varUpsilon\:\left({Z}_{4}<CE\right)-1\right)\times\:\varUpsilon\:\left({Y}_{E,j}={Y}_{i,j}\right)$$

(27)

$\:{\Upsilon\:}\left(.\right)$ is the indicator function which is defined as:

$$\:\varUpsilon\:\left(A\right)=\left\{\begin{array}{cc}1&\:If\text{ A is true}\\\:0&\:Otherwise\end{array}\right.$$

(28)

This mechanism ensures that, even when the elite and current solution components coincide, a positive or negative movement direction is randomly assigned. As a result, stagnation is avoided and exploratory behavior is preserved throughout the optimization process. The update rule integrates deterministic and stochastic elements: the deterministic part provides a baseline step linked to the search range, while the stochastic component adaptively modulates the update magnitude based on the current solution state and the convergence progress.

Boundary control

After the update, some dimensions of the new individual may violate the predefined lower and upper bounds. To handle this, a boundary control scheme is applied to each dimension by smoothly pulling out-of-range values back into the feasible region through averaging with the corresponding bound as follows:

$$\:{{Y}_{i,j}}^{new}=\left\{\begin{array}{c}0.5\times\:\left({{Y}_{i,j}}^{new}+u{b}_{j}\right)\\\:0.5\times\:\left({{Y}_{i,j}}^{new}+l{b}_{j}\right)\\\:{{Y}_{i,j}}^{new}\end{array}\right.\:\begin{array}{c}If\:{{Y}_{i,j}}^{new}>u{b}_{j}\\\:If\:{{Y}_{i,j}}^{new}<l{b}_{j}\\\:Otherwise\end{array}$$

(29)

This strategy preserves solution continuity and avoids abrupt clipping, leading to more stable convergence.

Selection and best update

The new generated solution is evaluated against the current one, and it replaces the current solution only if it yields an equal or better objective value.

$$\:{{Y}_{i}}^{\left(t+1\right)}=\left\{\begin{array}{cc}{{Y}_{i}}^{new}&\:If \text{f}\left({{Y}_{i}}^{new}\right)\le\:f\left({{Y}_{i}}^{\left(t\right)}\right)\\\:{{Y}_{i}}^{\left(t\right)}&\:Otherwise\end{array}\right.$$

(30)

Similarly, the best global solution is updated whenever an improved solution is found.

$$\:{Y}_{GBest}=\left\{\begin{array}{cc}{{Y}_{i}}^{new}&\:If \text{f}\left({{Y}_{i}}^{new}\right)\le\:f\left({Y}_{GBest}\right)\\\:{Y}_{GBest}&\:Otherwise\end{array}\right.$$

(31)

This greedy selection mechanism ensures monotonic improvement while preserving the best solution throughout the search. The steps of the BTO can be summarized in the flowchart in Fig. 2.

Fig. 2

Key steps of the BTO.

Simulation results and discussions

The study examines DG units powered by renewable PV energy, the irradiance PDF is defined by µ_Ir of 6 and a σ_Ir of 0.6^38,45 as illustrated in Fig. 3. Load demand uncertainty follows a normal PDF with a mean load ratio of 70% and a standard deviation of 10%, as illustrated in Fig. 1. Utilizing Hong’s probabilistic PEM method, the study condenses the stochastic behavior of both variables into five representative scenarios, significantly reducing computational complexity. The performance of the proposed BTO is rigorously evaluated through comparative analysis against a set of representative benchmark metaheuristic algorithms. These include the widely adopted Differential Evolution Algorithm (DEA)⁴⁶, as well as two recently developed optimization techniques, namely the Crocodile Ambush Optimization (CAO, 2025)⁴⁷ and the Schrödinger Optimization Algorithm (SOA, 2025)⁴⁸. This selection ensures a balanced comparison with both classical and state-of-the-art approaches in the field of evolutionary optimization.

Fig. 3

Solar radiation using Lognormal PDF model.

To ensure fairness and consistency in the evaluation process, all algorithms are implemented using identical computational settings. Specifically, the population size is fixed at 30 individuals, and the maximum number of iterations is set to 200 for each algorithm. These unified parameters eliminate potential bias arising from unequal computational effort and enable a direct and objective assessment of convergence behavior, solution quality, and robustness. The effectiveness of the proposed approach is validated on the 85-bus and the IEEE 118-bus RDSs.

Applications on the 85-bus RDS

In this study, the 85-bus RDS is addressed. Figure 4 displays its topology⁴⁶ which has a total load of 2.661 MW and 2.3788 MVAr at a standard voltage of 12.66 kV. The proposed framework is examined under three distinct planning scenarios, differentiated by the number of PV-DG units integrated into the RDS. In Case 1, three PV-DG units are optimally allocated; Case 2 extends the analysis to four PV-DG units; and Case 3 considers the installation of five PV-DG units. This progressive increase in the number of PV-DG installations allows for a comprehensive evaluation of the scalability and adaptability of the proposed BTO under varying levels of renewable penetration.

Fig. 4

85-bus topology.

Load flow analysis without PV insertion at loading scenarios

Using the PEM, three loading scenarios of 60.56, 70 and 85.45% loading are addressed indicating low, mean and high loading levels while three irradiance scenarios of 252.38, 482.99 and 1194.36 W/m² are addressed indicating low, mean and high PV irradiance. To assess these loading scenarios, the load flow analysis is simulated without PV insertion. Table 1; Fig. 5 illustrate the impact of load uncertainty on the 85-bus RDS in the absence of PV-DG integration. As the loading level increases from low (60.6%) to mean (70%) and high (85.5%), the real power losses rise significantly from 104.32 kW to 142.43 kW and 220.28 kW, corresponding to increases of approximately 36.5% (mean vs. low) and 54.6% (high vs. mean). Compared to the low loading condition, the losses under high loading increased by nearly 111.2%, highlighting the severe degradation in network efficiency under stressed operating conditions.

Fig. 5

Voltage Profile of 85-Bus RDS without PV insertion under low, mean and high loadings.

Table 1 Losses and minimum voltages of 85-bus RDS without PV insertion under low, mean and high loadings.

Similarly, the minimum bus voltage drops from 0.9263 pu at low loading to 0.9138 pu at mean loading and further to 0.8927 pu at high loading, reflecting voltage reductions of about 1.35% and 2.31%, respectively. The cumulative voltage decline of approximately 3.63% from low to high loading results in voltage violations below the acceptable limit of 0.9 pu at several buses, particularly in heavily loaded sections of the network. These findings clearly demonstrate the vulnerability of the system to load growth and justify the necessity of PV-DG integration to mitigate losses and improve voltage profiles under uncertain operating conditions.

Case 1: PV insertion with three units

In this case study, three PV-based DG units are optimally allocated within the 85-bus RDS in order to minimize the expected real power losses under stochastic load and solar irradiance conditions. Table 2 presents a comprehensive comparison of the optimal installation buses, rated capacities, and operating power factors obtained using the proposed BTO and CAO, DEA, and SOA. Also, the average convergence behavior of the compared algorithms is illustrated in Fig. 6. As indicated, the proposed BTO identifies a coordinated allocation of PV-DG units at buses 40, 48, and 68, with appropriately distributed capacities and controlled power factors around 0.85.

Fig. 6

Average convergences of BTO, CAO, DEA and SOA for 85-bus RDS (Case 1).

From a performance perspective, BTO achieves the minimum expected probabilistic power loss of 82.0647 kW, outperforming CAO, DEA, and SOA by approximately 0.40%, 2.05%, and 1.17%, respectively. These results clearly demonstrate the superior search capability of BTO in effectively exploring the solution space and identifying well-coordinated PV-DG placements under uncertainty.

Table 2 Optimal allocations of three PV-DG units using BTO, CAO, DEA and SOA of 85-bus RDS.

To address the concern regarding the convergence behavior and the selection of the maximum number of iterations, additional simulations were conducted by extending the maximum iteration limit of the BTO algorithm to 500 and 1000 iterations. The corresponding average convergence characteristics are presented in Fig. 7 for the 85-bus RDS (Case 1). As observed, the BTO exhibits a rapid reduction in probabilistic losses during the early stages of the search process, with the solution approaching near-optimal values within the first 150–200 iterations. Beyond this point, the convergence curves for 200, 500, and 1000 iterations show only marginal improvements, where the objective function value stabilizes around approximately 83.33 kW, 82.47 kW, and 82.08 kW, respectively. This indicates that extending the number of iterations yields only negligible gains (less than 1.5%), despite a substantial increase in computational effort. Therefore, although BTO continues fine-tuning the solution beyond 200 iterations, the improvements are not significant enough to justify the additional computational cost. This behavior confirms that the selection of 200 iterations provides an effective trade-off between solution quality and computational efficiency, while also demonstrating the strong convergence capability and fast search dynamics of the proposed algorithm.

Fig. 7

Average convergences of BTO under 200, 500 and 1000 iterations for 85-bus RDS (Case 1).

To further assess solution reliability, a robust analysis is conducted based on multiple independent runs, and the results are summarized in Table 3 and illustrated using boxplots in Fig. 8. The BTO demonstrates the lowest standard deviation of probabilistic losses (0.617 kW), which corresponds to robustness improvements of approximately 80.1%, 31.7%, and 91.6% compared with CAO, DEA, and SOA, respectively. Additionally, BTO maintains a narrow range between its minimum and maximum loss values, confirming its strong consistency and repeatability under stochastic operating conditions. In terms of the average real power losses, the BTO achieves a noticeable reduction, outperforming DEA, CAO, and SOA by 2.55%, 4.10%, and 6.74%, respectively.

Fig. 8

Boxplot of BTO, CAO, DEA and SOA for 85-bus RDS (Case 1).

Therefore, the findings of Case 1 clearly verify that the proposed BTO-based planning strategy not only achieves lower expected power losses but also delivers significantly enhanced robustness compared with both classical and recent metaheuristic algorithms.

Table 3 Robustness analysis of BTO, CAO, DEA and SOA for 85-bus RDS (Case 1).

Case 2: PV insertion with four units

In this case, the integration of four PV-DG units is investigated where the optimal allocation results obtained by the competing algorithms are summarized in Table 4. As shown, the proposed BTO achieves an expected probabilistic real power loss of 81.95 kW, which is lower than the values obtained using CAO, DEA and SOA by approximately 1.08%, 2.23% and 3.35%, respectively. These results indicate that BTO is more effective in coordinating both the locations and capacities of multiple PV units when the dimensionality of the optimization problem increases.

The average convergence characteristics illustrated in Fig. 9 further reinforces these findings. The BTO convergence curve shows a rapid initial descent, indicating efficient exploration of the search space, followed by a smooth and stable convergence pattern in later iterations. Unlike the benchmark algorithms, which exhibit slower convergence, BTO steadily approaches its final solution without premature stagnation.

Fig. 9

Average convergences of BTO, CAO, DEA and SOA for 85-bus RDS (Case 2).

This is further evidenced by the robustness analysis in Table 5; Fig. 10, where BTO yields the lowest standard deviation of 0.48 kW, corresponding to robustness improvements of about 89.5%, 59.4%, and 97.3% over CAO, DEA, and SOA, respectively. In terms of the average real power losses, the proposed BTO yields even greater improvements, with loss reductions of 5.12%, 7.50%, and 14.12%, respectively, compared with the same benchmark algorithms.

Fig. 10

Boxplot of BTO, CAO, DEA and SOA for 85-bus RDS (Case 2).

Table 4 Optimal allocations of four PV-DG units using BTO, CAO, DEA and SOA of 85-bus RDS.

Table 5 Robustness analysis of BTO, CAO, DEA and SOA for 85-bus RDS (Case 2).

Case 3: PV insertion with five units

In this case, the integration of five PV-DG units is investigated. As reported in Table 6, the proposed BTO achieves the lowest expected probabilistic power losses of 81.20 kW when five PV-DG units are optimally integrated into the 85-bus RDS. Compared with CAO, DEA, and SOA, the BTO reduces losses by approximately 3.77%, 4.38%, and 2.35%, respectively, confirming its superior optimization capability at higher PV penetration levels.

Table 6 Optimal allocations of five PV-DG units using BTO, CAO, DEA and SOA of 85-bus RDS.

The convergence curves in Fig. 11 show that BTO maintains a faster and more stable convergence behavior than benchmark algorithms, avoiding premature stagnation. Furthermore, the cropped results in Table 7; Fig. 12 indicate that BTO attains the lowest standard deviation (0.73 kW), representing robustness improvements of about 87.2%, 49.4%, and 98.1% compared with CAO, DEA, and SOA, respectively. These results demonstrate that BTO not only enhances loss minimization but also provides highly consistent and reliable solutions under stochastic operating conditions, particularly as the number of integrated PV units increases. In terms of the average real power losses, the BTO achieves a noticeable reduction, outperforming DEA, CAO, and SOA by 13.05%, 6.45%, and 32.31%, respectively, when three PV units are installed.

Fig. 11

Average convergences of BTO, CAO, DEA and SOA for 85-bus RDS (Case 3).

Fig. 12

Boxplot of BTO, CAO, DEA and SOA for 85-bus RDS (Case 3).

Table 7 Robustness analysis of BTO, CAO, DEA and SOA for 85-bus RDS (Case 3).

Load flow analysis with PV insertion (Case 3) under PEM scenarios

To further evaluate the effectiveness of the proposed BTO-based planning strategy under uncertainty, a detailed load flow analysis is conducted for Case 3, where five PV-DG units are integrated into the 85-bus RDS. The analysis considers the five PEM scenarios, which jointly capture the stochastic variations of load demand (60.56%, 70%, and 85.45%) and solar irradiance levels (252.38, 482.99, and 1194.36 W/m²).

Figure 13 illustrates the voltage profiles of the 85-bus system under all PEM scenarios with PV insertion. As shown, the voltage magnitudes at all buses are markedly improved, with the minimum voltage recorded at bus 54 being 0.9461 pu. Compared with the minimum voltage of 0.8927 pu in the no-PV case, this represents a voltage enhancement of approximately 6.0%, while completely eliminating voltage violations below the permissible limit of 0.9 pu. This improvement confirms the effectiveness of PV-DG units in providing local voltage support, particularly in heavily loaded sections of the network.

Fig. 13

Voltage profile of 85-bus RDS with PV insertion (Case 3) under PEM scenarios.

$$\:V{D}^{\left(s\right)}={\sum\:}_{k=1}^{{N}_{bus}}\left|1-{{V}_{k}}^{\left(s\right)}\right|\:$$

(32)

Also, a comparative assessment with other applied algorithms confirms the superiority of BTO under stochastic conditions is reported in Table 8. As shown, BTO consistently achieves lower real power losses across most PEM scenarios. On average, BTO reduces losses by approximately 4.1%, 4.6%, and 3.3% when compared with CAO, DEA, and SOA, respectively. These improvements reflect the ability of BTO to effectively coordinate PV locations and capacities while accounting for probabilistic variations in load and generation. When compared with the base case without PV integration at high loading, where the real power losses reach 220.28 kW, the maximum recorded loss with PV insertion is reduced to 71.77 kW. This corresponds to a significant loss reduction of approximately 67.4%, highlighting the strong capability of optimally placed PV-DG units to relieve feeder loading and reduce current flow throughout the network.

Table 8 Losses of BTO, CAO, DEA and SOA for 85-bus RDS with PV (Case 3) under PEM scenarios.

The calculated values, presented in Table 9, show that BTO consistently yields the lowest voltage deviation indices among the compared algorithms. The proposed BTO reduces the voltage deviation index by approximately 13.6%, 14.2%, and 6.8% compared with CAO, DEA, and SOA, respectively. The reduced voltage deviation indicates a more uniform voltage profile and improved overall power quality under uncertain operating conditions.

Table 9 Voltage deviations of BTO, CAO, DEA and SOA for 85-bus with PV (Case 3) under PEM scenarios.

Applications on the large IEEE 118-bus RDS

To further validate the effectiveness of the proposed BTO-based stochastic planning framework, an extended analysis is conducted on a larger-scale distribution system under the five PEM scenarios that jointly capture the uncertainty in load demand and solar irradiance. In this section, the IEEE 118-bus RDS is addressed. Figure 14 displays its topology which has a total load of 22.7097 MW and 17.0411 MVAr⁴⁹ at a standard voltage of 11 kV. The detailed data is taken from⁵⁰, and the bus and branches data are indicated in the Appendix.

Fig. 14

IEEE 118-bus topology.

The load flow analysis is first simulated under the high, medium, and mild loading situations without PV-DG integration. The effect of load uncertainty on the IEEE 118-bus RDS without PV-DG integration is shown in Table 10; Fig. 15. The true power losses increase dramatically with loading level, from 433.55 kW to 602.19 kW and 922.23 kW, or around 28% (mean vs. low) and 34.7% (high vs. mean). The losses under high loading rose by around 52.98% in comparison to the low loading condition, demonstrating a significant decline in network efficiency under stressful operating circumstances.

Fig. 15

Voltage profile of IEEE 118-bus RDS without PV insertion under low, mean and high loadings.

Table 10 Losses and minimum voltages of IEEE 118-bus RDS without PV insertion under low, mean and high loadings.

In the same direction, the minimum bus voltage at bus 76 decreases by around 1.38% and 2.34%, respectively, from 0.9246 pu at low loading to 0.9118 pu at middle loading and then to 0.8901 pu at heavy loading. Voltage breaches below the permissible limit of 0.9 pu occur at multiple buses due to the cumulative voltage reduction of around 3.73% from low to high loading, especially in severely loaded network segments.

In this case study, five PV-based DG units are optimally allocated within the IEEE 118-bus RDS in order to minimize the expected real power losses under stochastic load and solar irradiance conditions. Table 11 presents a comprehensive comparison of the optimal installation buses, rated capacities, and operating power factors obtained using the proposed BTO and CAO, DEA, and SOA. Also, the average convergence behavior of the compared algorithms is illustrated in Fig. 16.

Fig. 16

Average convergences of BTO, CAO, DEA and SOA for IEEE 118-bus RDS with PV units.

Table 11 Optimal allocations of five PV-DG units using BTO, CAO, DEA and SOA of IEEE 118-bus RDS.

As shown in Table 11, the proposed BTO selects installation buses that are well distributed across the network where the placement at buses such as 41, 74, 96, 107, and 118 reflects a strategic balance between upstream and downstream locations. With respect to the sizing of PV-DG units, BTO consistently assigns capacities that are relatively high and closely spaced. On the other hand, CAO, DEA, and SOA demonstrate more variability in sizing decisions, which may reflect less stable search behavior and weaker coordination among decision variables. From a performance perspective, BTO demonstrates superior performance in minimizing expected real power loss, achieving 430.30 kW, which is significantly lower than CAO, DEA, and SOA. This indicates BTO’s enhanced ability to navigate complex, high-dimensional search spaces. Its convergence behavior illustrated in Fig. 16, marked by a quick initial descent and stable transition to optimal solutions, contrasts with the slower convergence rates of the other algorithms, highlighting BTO’s robustness and efficiency in overcoming local optima.

To further evaluate the reliability and consistency of the BTO, a robustness analysis is conducted over multiple independent runs. The statistical distribution of the obtained solutions is illustrated in Fig. 17 using boxplot representations, while the corresponding numerical indicators are summarized in Table 12.

Fig. 17

Boxplot of BTO, CAO, DEA and SOA for IEEE 118-bus RDS with PV units.

The boxplot in Fig. 17 demonstrated that the BTO shows a compact distribution with a narrow interquartile range and minimal spread, indicating high stability. In contrast, CAO and SOA have wider boxplots with pronounced whiskers, signifying greater variability and less reliable convergence. DEA has moderate dispersion but is still more variable than BTO. These observations are quantitatively confirmed in Table 12, which reports several statistical performance metrics. The proposed BTO shows a minimum probabilistic loss of 430.2964 kW, and its mean loss of 433.1243 kW is very close to this minimum, indicating that the algorithm consistently converges near the global optimum. In terms of worst-case performance, BTO records the lowest maximum loss of 439.6264 kW among all compared algorithms, reflecting its ability to maintain stable performance and avoid poor-quality solutions. In contrast, SOA demonstrates a significantly larger maximum loss of 740.5957 kW, suggesting a risk of convergence to suboptimal solutions, while CAO exhibits a similar spread in performance. Further evidence of robustness is found in the standard deviation values, where BTO achieves a low standard deviation of 2.7999 kW, much smaller than CAO’s 24.88 kW, DEA’s 7.09 kW, and SOA’s 67.49 kW. This confirms that BTO provides highly consistent results with minimal sensitivity to stochastic variations in the optimization process.

Table 12 Robustness analysis of BTO, CAO, DEA and SOA for IEEE 118-bus RDS with PV units.

Additionally, Table 13 presents the real power losses obtained using BTO, CAO, DEA, and SOA for the IEEE 118-bus RDS with PV-DG integration under the five PEM scenarios. The proposed BTO algorithm consistently achieves competitive and often superior performance in minimizing real power losses, with significant outperformance in higher loading and lower irradiance conditions (Scenarios 2 and 3), reducing losses to 272.09 kW and 336.41 kW, respectively. In these scenarios, BTO outperforms CAO by approximately 15.4% and 21.6%. In moderate conditions (Scenarios 4 and 5), BTO also yields the lowest loss values among all algorithms, demonstrating robust and reliable performance across scenarios, despite CAO performing slightly better in Scenario 1.

Table 13 Losses of BTO, CAO, DEA and SOA for IEEE 118-bus RDS with PV units under PEM scenarios.

In addition to loss minimization, voltage quality is quantitatively assessed using the system voltage deviation index $\:V{D}^{\left(s\right)}$, defined in Eq. (32). Table 14 summarizes the voltage deviation values obtained by the compared algorithms across all PEM scenarios. The results indicate that the proposed BTO achieves the lowest voltage deviation in most scenarios, leading to a more uniform voltage profile, with values of 2.3566 and 2.7047 in Scenario 2 and Scenario 3, respectively. While DEA shows a lower deviation in Scenario 1, BTO consistently performs better across all scenarios, reducing voltage deviation by about 10–15% compared to CAO. This confirms that BTO minimizes real power losses and enhances voltage regulation performance, thereby improving power quality in the distribution network.

Table 14 Voltage deviations of BTO, CAO, DEA and SOA for IEEE 118-bus with PV units under PEM scenarios.

Moreover, Fig. 18 illustrates the voltage profiles of the IEEE 118-bus RDS after the optimal integration of PV-DG units under the considered PEM scenarios. It clearly demonstrates that the integration of optimally allocated PV units leads to a substantial enhancement in voltage magnitudes across all buses. Under low loading conditions, the voltage profile remains near nominal levels, while higher loading leads to some voltage decline along the feeder, although still within acceptable limits. Importantly, even under high loading conditions, PV-DG units reduce voltage degradation, resulting in significantly better minimum voltage values compared to scenarios without PV integration, with no violations below permissible limits.

Fig. 18

Voltage profile of 85-bus RDS with PV insertion with PV units under PEM scenarios.

Computational efficiency and memory utilization comparisons

The summarized results in Table 15 provide a clear comparison of the computational efficiency and memory utilization of the investigated algorithms across both test systems. It is evident that the proposed BTO demonstrates the best overall computational performance, achieving the lowest execution times in both the 85-bus and 118-bus systems, with values of 152.73 s and 161.83 s, respectively. In contrast, CAO exhibits the highest computational burden, requiring nearly double the execution time of BTO, which reflects its relatively slower convergence characteristics.

Table 15 Computational efficiency and memory utilization of BTO, CAO, DEA and SOA.

In terms of scalability, all algorithms show a moderate increase in execution time when transitioning from the 85-bus to the 118-bus system; however, BTO maintains its computational advantage, indicating strong scalability and efficiency for larger networks. DEA and SOA exhibit intermediate performance, with SOA being closer to BTO in terms of execution time but still consistently slower.

Regarding memory utilization, the differences among the algorithms are relatively small; however, BTO consistently requires slightly lower memory resources compared to CAO and SOA, while remaining comparable to DEA. This indicates that the improved performance of BTO is not achieved at the expense of increased memory demand, but rather through more efficient search dynamics.

Conclusions

This paper presented an optimal stochastic integration of PV-DG units in RDSs under load and solar irradiance uncertainties. In this work, a higher-order PEM was employed, enabling accurate probabilistic assessment with significantly reduced computational burden. The PV-DG integration problem, formulated to minimize expected real power losses while satisfying operational constraints, was effectively solved using the recently developed BTO. Extensive simulations were conducted on the IEEE 85-bus RDS considering different PV-DG numbers of units. The obtained results consistently demonstrated the superior performance of BTO in identifying well-coordinated PV locations, capacities, and operating power factors. Compared with established and state-of-the-art metaheuristic algorithms, including DEA, CAO, and SOA, the proposed BTO achieved lower expected power losses, faster and smoother convergence behavior, and significantly enhanced robustness under stochastic operating conditions. Furthermore, statistical robustness analyses confirmed the reliability and stability of BTO, as reflected by its low dispersion in solution quality across multiple runs.

Main findings

To further emphasize the effectiveness and scientific rigor of the proposed framework, the key quantitative findings obtained from the investigated cases can be summarized as follows:

• (i) Loss reductions exceeding 67% were achieved relative to the no-PV case at high loading for both investigated systems, while voltage profiles were substantially improved, with all bus voltages maintained well above the permissible limits.

• (ii) The proposed BTO consistently achieved the lowest probabilistic power losses, reaching 430.30 kW in the large-scale 118-bus system, outperforming the compared methods across all scenarios;

• (iii) The framework demonstrated high solution stability, with a very low standard deviation of 2.80 kW, compared to significantly higher variations observed in CAO (24.88 kW) and SOA (67.49 kW);

• (iv) The mean loss value remained very close to the best-obtained solution (433.12 kW), confirming reliable convergence toward near-global optima;

• (v) The maximum deviation from the best solution was limited to less than 10 kW, indicating strong robustness and avoidance of poor-quality solutions; and.

• (vi) Consistent improvements were also observed in voltage deviation indices and voltage profile enhancement across all PEM scenarios, ensuring compliance with operational limits even under high loading conditions.

These quantitative indicators collectively validate the superior performance, robustness, and practical suitability of the proposed BTO-based stochastic optimization framework.

Limitations and future work

Despite achieving promising results, the study’s framework is tested mainly on RDSs, necessitating further investigation into its applicability to more complex topologies, unbalanced loading conditions, and dynamic operational constraints. The analysis emphasizes technical performance metrics like power loss reduction and voltage profile enhancement, but does not address economic factors, protection coordination, or inverter control dynamics. The uncertainty modeling is based on the PEM, which, although computationally efficient, may not fully capture extreme or highly non-linear probabilistic behaviors. Future research directions may include extending the framework to multi-objective optimization considering economic and environmental aspects, integrating advanced uncertainty modeling techniques such as scenario-based or machine learning-driven approaches, and validating the methodology on real distribution networks with high-resolution temporal data. Moreover, the incorporation of hybrid energy resources, such as energy storage systems and hydrogen-based technologies, adaptive control strategies, and real-time implementation aspects would further enhance the practical applicability and robustness of the proposed approach.