Optimal Planning of Renewable Microgrids for Loss-Aware Integration of Distributed Energy Resources Using the Geese V-Formation Algorithm

Abstract

This research introduces a unique optimization framework centered on the Geese V-Formation Algorithm to enhance the technical planning of distributed energy resources in renewable microgrid-oriented radial distribution systems. The proposed methodology addresses the optimal placement and sizing of photovoltaic panels, wind turbines, battery energy storage systems, and capacitor banks to provide comprehensive voltage support, minimize active power losses, and refine overall grid functionality. Drawing inspiration from the aerodynamic efficiency of migratory geese, the Geese V-Formation Algorithm integrates dynamic leader-follower coordination, adaptive role rotation, and cooperative information exchange mechanisms. These features allow the algorithm to effectively balance global exploration and local exploitation, making it uniquely suited to address the complex, nonlinear, and multi-objective nature of modern microgrid design. The effectiveness of this approach was evaluated through rigorous simulations on the IEEE-33 and IEEE-69 bus distribution systems utilizing the Python programming language. The empirical results indicate that the Geese V-Formation Algorithm achieves substantial power loss reductions, reaching 91.62% and 92.45%, respectively, when integrating solar and wind resources with energy storage and reactive power compensation. Furthermore, the optimized configurations significantly improved bus voltage profiles and enhanced substation power factors, confirming the technical effectiveness of the framework under the considered benchmark constraints. By providing a technical decision-support approach for engineers and utility planners, this framework facilitates the deployment of reliable, decentralized renewable energy systems that align with global energy transition objectives and promote sustainable infrastructure development.

1. Introduction

Modern distribution systems are undergoing significant transformation due to the increasing integration of renewable energy sources and distributed energy resources. Photovoltaic systems, wind turbines, battery energy storage systems, and reactive power compensation devices such as capacitor banks are increasingly deployed in distribution feeders to improve local energy utilization, reduce technical losses, and support voltage regulation. However, the benefits of these resources strongly depend on their locations and rated capacities. Improper placement or oversizing of distributed resources may lead to voltage violations, inefficient power-flow patterns, increased losses, or poor utilization of network assets.

The renewable microgrid optimization problem considered in this study is a planning-oriented DER allocation problem applied to radial distribution systems. The main objective is to determine the optimal placement and sizing of multiple DER technologies, including PV, WT, BESS, and capacitor banks, to reduce active power losses and improve the voltage profile. The IEEE-33 and IEEE-69 bus systems [1] are used as benchmark radial distribution feeders because they are widely adopted for evaluating DER planning methods and voltage improvement strategies.

In this work, the load demand is treated as a static operating condition. Therefore, the optimization problem focuses on identifying the best DER configuration for a representative load-flow condition rather than performing time-series dispatch or dynamic energy management. Under this static-load formulation, each candidate solution defines the installation buses and capacities of the DER units. The performance of each solution is then evaluated through distribution load-flow analysis by calculating total active power loss and bus voltage magnitudes.

The problem considered is constrained by several technical requirements. Bus voltages must remain within acceptable operating limits, DER sizes must remain within predefined capacity bounds, and the network must satisfy active and reactive power-balance conditions. In addition, the selected DER locations must correspond to feasible buses in the radial feeder. Capacitor banks are included to provide reactive power support, while PV, WT, and BESS units are considered as active power support resources for reducing feeder loading and improving voltage magnitudes.

This type of DER allocation problem is nonlinear, constrained, and combinatorial because it involves both discrete decision variables, such as installation buses, and continuous decision variables, such as DER capacities. As the number of candidate buses and DER technologies increases, the search space becomes large and difficult to solve using conventional deterministic methods. Therefore, metaheuristic optimization algorithms are often used to search for high-quality solutions within reasonable computational effort.

To address this problem, this study proposes the Geese V-Formation Algorithm for optimal placement and sizing of PV, WT, BESS, and capacitor banks in IEEE-33 and IEEE-69 radial distribution systems. The GVFA is inspired by the cooperative V-formation flight behavior of migrating geese, where leader–follower coordination, information exchange, and adaptive movement are used to balance exploration and exploitation during the search process. In the proposed framework, GVFA generates candidate DER configurations, while the load-flow model evaluates each configuration according to power loss reduction, voltage-profile improvement, and constraint satisfaction.

The main contribution of this study is the application of GVFA to a coordinated multi-DER planning problem under static-load conditions. Unlike approaches that optimize a single DER type or focus only on one performance index, the proposed framework simultaneously considers PV, WT, BESS, and capacitor banks for improving radial distribution network performance. The effectiveness of the method is validated on both IEEE-33 and IEEE-69 systems, and the results obtained are compared with related studies and representative bio-inspired algorithms to demonstrate the enhancement achieved by the proposed GVFA approach.

This paper proceeds as follows: Section 2 provides a focused literature review on optimal placement and sizing of PV, WT, BESS, and reactive-power support in radial distribution networks and microgrids, highlighting the limitations of existing metaheuristic and planning frameworks and motivating the proposed GVFA-based approach; Section 3 introduces the mathematical formulation of the optimization problem, its objective function, decision variables, and operational constraints; and elaborates on the mechanism of the Geese V-Formation Algorithm (GVFA) and its update mechanism and explains its constraint handling; Section 4 presents the simulation results for different DER scenarios from both the IEEE-33 and IEEE-69 distribution networks, including a thorough investigation into the performance of power losses, voltage profiles, power factor behavior, and a power sharing analysis; and finally, Section 5 concludes the paper with key findings and future research direction.

2. Literature Review

Literature on optimal planning and management of renewable energy microgrids has evolved along two complementary directions. The first direction is planning-centric and focuses on network-constrained sitting and sizing DERs in radial distribution systems to reduce losses and improve voltage profiles. The second direction is operation-centric and emphasizes microgrid energy management (EMS), where intermittent renewables and storage are coordinated—often under uncertainty—to achieve economical and reliable operation. To clarify how prior studies cover (or omit) the key elements of source–load–network–storage coordination, we organize the state of the art into the main methodology families summarized in

Table 1. Taxonomy of coordinated source–load–network–storage planning and management methodologies, with representative studies and key strengths/limitations.

Methodology Family	Typical Scope	Rep. Refs.	Strengths	Limitations/Gaps
Deterministic feeder-constrained siting/sizing (snapshot)	PV/DG allocation for loss reduction and voltage improvement under limited operating points	[2,3,4,5]	Captures feeder physics; clear siting guidance	Often single-period; limited uncertainty; weak linkage to SOC-feasible multi-period operation and resilience
Metaheuristic feeder planning (nonconvex/mixed-integer) and economic analysis	DER siting/sizing using PSO/GA/ALO variants; planning-centric objectives	[1,6,7,8,9,10,11]	Handles nonlinearity/ discrete decisions; flexible multi-objective design	Sensitive to tuning/constraint handling; frequently planning-metric benchmarking; limited EMS/SOC realism
Uncertainty-aware planning (probabilistic/scenario/stochastic)	Planning under uncertain PV/load/EV; probabilistic multi-objective siting/sizing	[12,13,14]	Improves robustness; more realistic inputs	Often partial uncertainty coverage; scenario-limited; may not enforce SOC-feasible multi-period dispatch and resilience recovery
Long-horizon planning with load growth/expansion	Investment planning with demand growth/seasonality and long-term relevance	[15,16,17]	Strategic planning relevance; investment timing insight	Operational feeder detail and short-timescale EMS/SOC coupling often simplified; resilience indices limited
Storage-inclusive planning with dispatch feasibility (PV–BESS dominant)	BESS siting/sizing + daily charge/discharge feasibility under feeder constraints	[18,19,20,21]	Coupled placement with dispatch feasibility and SOC constraints	Often PV–BESS focused; limited multi-DER portfolios and interconnected microgrid coordination/resilience
Operational EMS/ DSM-centric optimization	Multi-period dispatch, DSM scheduling, uncertainty-aware EMS; typically fixed siting	[22,23,24,25]	Operational realism; scheduling under uncertainty	Usually does not optimize feeder-constrained siting/sizing; network/resilience modeling can be simplified
Integrated planning–management co-optimization	Joint sizing + EMS (cost/CO2), sometimes DR/reconfiguration and reliability tradeoffs	[26,27]	Unified investment-operation view; sustainability alignment	MILP/linearization may reduce AC fidelity and discrete siting realism; resilience/SOC detail may be simplified
Sustainability/hybrid supply design (LCA, rural electrification, production coupling)	Hybrid supply sizing, life-cycle sustainability, coupling with production planning	[28,29,30,31]	Broad sustainability perspective beyond losses	Often outside AC feeder-constrained planning; interconnected microgrid resilience and operational constraints not central
Multi-energy/multi-carrier microgrids (energy hubs)	Electric/thermal/multi-carrier coordination; networked microgrids	[32,33]	High modeling richness; sector coupling	Feeder voltage/loss constraints and discrete siting in radial networks often simplified
Decentralized/agent-based planning	Decentralized coordination and interaction modeling	[34]	Captures decentralized behavior and coordination	Global optimality under MINLP is challenging; feeder-accurate AC feasibility and SOC-coupled scheduling often require additional layers
Geese-inspired optimizers/ domain transfer	Wild-geese-inspired optimizers; cross-domain demonstrations	[35,36]	Motivates leader–follower and rotation dynamics	Needs power-system-specific constraint handling (AC feasibility, discrete siting, SOC coupling) and rigorous benchmarking
Applied distribution-sector studies (reconfiguration/DG placement)	Practical feeder reconfiguration and DG planning case studies	[37]	Real-world context and constraints	Not necessarily integrated microgrid planning + uncertainty-aware EMS with explicit resilience metrics

, highlighting for each family the typical modeling scope, strengths, and limitations (notably: snapshot vs. multi-period realism, SOC-feasible storage scheduling, uncertainty coverage, and the inclusion of reliability/resilience metrics).

Despite substantial progress in each stream, the review reveals a persistent integration gap: many feeder-constrained planning studies remain largely snapshot-driven, while EMS/DSM-oriented works typically assume fixed siting/sizing and simplify feeder power-flow constraints. Even when uncertainty is incorporated, it is frequently scoped to a limited set of sources or scenarios, and resilience-oriented performance (e.g., disturbance recovery and networked microgrid support) is not consistently co-optimized with siting/sizing and SOC-aware dispatch feasibility. Motivated by these gaps—and by the need for scalable solvers capable of handling mixed-integer nonconvexity with domain-specific feasibility requirements—this work proposes a GVFA-based coordinated framework that combines feeder-constrained DER placement and sizing with post-optimization time-series validation of SOC-aware storage under representative profiles. The proposed approach focuses on technical loss minimization and voltage-profile improvement, while multi-period uncertainty-aware and resilience-oriented optimization are left as future extensions.

2.1. Advanced Techniques for Microgrid Planning and Management

Microgrid planning and management have been widely addressed using deterministic, stochastic, robust, and metaheuristic optimization methods. Recent uncertainty-aware studies have emphasized the need to represent renewable-generation and load-demand variability. The authors of [38] proposed a robust opportunistic energy-management framework for mixed microgrids under asymmetric uncertainties, combining machine-learning prediction with multiband robust optimization to capture favorable and unfavorable deviations in PV generation and demand. The authors in [39] developed a two-stage robust optimization model for renewable distributed generation and load uncertainty, highlighting the role of recourse-based adjustment in balancing economy and robustness. Ref. [40] reviewed uncertainty-modeling techniques for renewable generation and microgrid loads, showing that stochastic optimization, scenario generation, and probabilistic modeling improve realism but may increase computational burden. Robust optimization offers an alternative by avoiding full probability distributions; for example, the budgeted robust framework of [41] controls conservatism through an uncertainty budget. Robust energy-management studies for renewable microgrids [42,43] and multi-microgrid energy transactions [44] further demonstrate the relevance of uncertainty-aware scheduling for reliable operation under variable renewable and load conditions. In parallel, metaheuristic and hybrid metaheuristic algorithms remain widely used in power-system optimization because DER planning problems are nonlinear, constrained, and mixed discrete–continuous. Ref. [45] showed that hybrid moth–flame optimization with particle swarm optimization can improve solution quality and convergence speed in transmission and distribution applications. Within this context, the present study develops a deterministic GVFA-based planning framework for optimal PV, WT, BESS, and capacitor-bank placement and sizing in IEEE-33 and IEEE-69 radial distribution systems. The formulation focuses on active power-loss minimization and voltage-profile improvement using Backward/Forward Sweep load-flow evaluation, while stochastic or robust GVFA-based DER planning is identified as an important extension beyond the current static-load optimization scope.

2.2. Thematic Synthesis

The literature clearly confirms that DER siting and sizing can reduce losses and enhance voltage profiles in distribution networks and microgrids, and that metaheuristics remain a practical tool for the resulting mixed-integer, nonconvex optimization landscape. However, many contributions remain planning-dominant and evaluated under snapshot conditions or limited scenarios, which can understate the operational implications of hourly variability and storage SOC coupling. Energy storage studies emphasize the value of combining placement decisions with dispatch feasibility, yet resilience is often treated indirectly. Uncertainty-aware EMS and uncertainty-aware planning have both matured, but they are frequently pursued as separate problems rather than as a unified planning–operation framework. Sustainability extensions (CO₂ minimization, LCA-aware sizing, multi-energy integration) improve environmental relevance but may require simplifying assumptions that weaken feeder constraint fidelity and resilience modeling. Finally, although reliability and reconfiguration studies highlight the importance of topology and cost–reliability trade-offs, an explicit resilience-oriented perspective—capturing disturbance response and recovery—remains insufficiently integrated with DER siting/sizing and multi-period operational management.

2.3. Research Gap

Synthesizing the existing literature indicates several unresolved issues that continue to limit coordinated source–load–network–storage planning in radial distribution feeders and microgrids. First, many studies treat DER components in a partially coupled manner—optimizing PV or WT placement alone or adding storage and reactive support as secondary stages—so the resulting solutions may not reflect the true interaction among PV/WT generation variability, SOC-coupled storage operation, and feeder voltage–loss behavior under operational limits. Second, multi-objective formulations are often incompletely integrated: loss reduction and voltage-profile improvement are commonly addressed, but substation power-factor improvement and reactive-power planning (e.g., capacitors) are frequently simplified, fixed, or handled outside the main optimization loop, which can obscure trade-offs among technical performance criteria. Third, operational feasibility is not always ensured under time-varying intermittency and SOC-aware constraints; several planning models rely on steady-state snapshots or limited periods, leaving uncertainty in whether the selected capacities and locations remain feasible across representative operating conditions. Finally, while metaheuristic approaches are widely used for mixed-integer, nonlinear DER planning, many lack transparent coordination structures that systematically manage exploration–exploitation trade-offs and robustness, which remains a practical limitation when the decision space expands to simultaneously include siting, sizing, storage scheduling constraints, and feeder operating limits. These unresolved issues motivate further research toward planning methodologies that are simultaneously feeder-constrained, multi-objective, and operationally credible under renewable variability and storage dynamics [1,6,10].

2.4. Positioning of the Present Study and Its Contributions

This study addresses the above gap by proposing a Geese V-Formation Algorithm (GVFA)-based framework for technical DER placement and sizing in renewable microgrid-oriented radial distribution systems. Inspired by the efficiency of V-shaped flight formation, GVFA employs dynamic leader–follower coordination, adaptive leader rotation, and cooperative information sharing to improve search efficiency and maintain a balanced exploration–exploitation trade-off in nonlinear, multi-constraint nonlinear design spaces. Unlike many prior approaches that emphasize either siting/sizing or EMS in isolation, the proposed framework jointly optimizes the placement of PV, WT, BESS, and shunt capacitors to deliver coordinated improvements in voltage support, loss reduction, and power-factor enhancement. The framework is implemented in Python 3.12 within the PyCharm Community 2026 integrated development environment (IDE) and validated on IEEE-33 and IEEE-69 test systems.

3. Mathematical Methodology and Objective Function

This section presents the mathematical formulation of the proposed distributed energy resource (DER) planning problem for radial distribution networks. The objective is to determine the optimal placement and sizing of photovoltaic (PV) units, wind turbines (WTs), battery energy storage systems (BESSs), and capacitor banks in the IEEE-33 and IEEE-69 bus systems using the Geese V-Formation Algorithm (GVFA). The optimization is formulated for a representative steady-state operating snapshot of the distribution feeder. Therefore, the main optimization stage is based on static load-flow evaluation, while the time-dependent behavior of the BESS and renewable generation is assessed separately through post-optimization validation.

As illustrated in Figure 1a, the proposed methodology follows an iterative planning process that begins with load and feeder-data analysis, followed by renewable-resource and DER-parameter definition, GVFA-based placement and sizing, load-flow evaluation, constraint checking, and final performance assessment. Figure 1b shows the four main DER components considered in the optimization framework, namely PV systems, wind turbines, BESS units, and capacitor banks, and highlights their technical interaction with the distribution feeder. PV and WT units provide active power support, BESS units provide active power injection or absorption within their rated limits, and capacitor banks provide reactive power compensation for voltage-profile improvement.

3.1. Problem Description and System Components

The distribution network considered is represented as a radial graph:

$$G=\left(\mathcal{N},\mathcal{L}\right)$$

(1)

where $\mathcal{N}$ is the set of buses and $\mathcal{L}$ is the set of distribution lines. Each bus $j\in \mathcal{N}$ is characterized by static active and reactive load demands, $P_{L,j}$ and $Q_{L,j}$, respectively. Each distribution line $i\in \mathcal{L}$ is characterized by resistance $R_i$, reactance $X_i$, current magnitude $I_i$, and active/reactive power flow. The slack bus is used as the reference bus, while the remaining buses are considered possible locations for DER installation.

The feasible candidate buses for DER placement are defined as

$${\mathcal{N}}_{\rfloor }\subseteq \mathcal{N}$$

(2)

where ${\mathcal{N}}_{\rfloor }$ excludes the slack bus and includes only buses that can technically host DER components.

The set of DER technologies considered in this work is defined as

$$\mathcal{D}=\left\{\mathrm{PV},\mathrm{WT},\mathrm{BESS},\mathrm{CAP}\right\}$$

(3)

where PV and WT represent renewable active-power generation units, BESS represents storage-based active-power support, and CAP represents capacitor-bank reactive-power compensation.

The technical role of each component is summarized as follows. PV and WT units reduce the active power drawn from the substation by injecting local renewable generation. BESS units support the feeder by injecting or absorbing active power according to their rated power and energy capacity. Capacitor banks inject reactive power to improve voltage magnitudes and reduce reactive power flow through the feeder. Therefore, the optimization problem involves both active-power planning and reactive-power compensation.

3.2. Optimization Problem Statement

Given a radial distribution feeder $G=\left(\mathcal{N},\mathcal{L}\right)$ with known branch parameters $R_i,X_i$, static load demands $P_{L,j},Q_{L,j}$, voltage limits $V_j^{min},V_j^{max}$, candidate installation buses ${\mathcal{N}}_{\rfloor }$, and capacity bounds for PV, WT, BESS, and capacitor banks, the proposed optimal DER planning problem consists of determining the installation buses and rated capacities of the DER components.

The objective is to minimize the total active power losses of the feeder while satisfying active and reactive power-flow equations, voltage operating limits, DER capacity limits, BESS planning limits, capacitor-bank limits, and feeder hosting-capacity restrictions. The problem solved by GVFA can therefore be expressed compactly as

$$\underset{X}{min}F\left(X\right)$$

(4)

subject to

$$g\left(X\right)=0$$

(5)

$$h\left(X\right)\le 0$$

(6)

where X is the decision vector representing DER placement and sizing, $F\left(X\right)$ is the objective function, $g\left(X\right)$ represents equality constraints associated with active and reactive power balance, and $h\left(X\right)$ represents inequality constraints associated with voltage limits, DER capacity limits, BESS limits, capacitor constraints, and feeder hosting-capacity restrictions.

Thus, the considered optimization problem can be stated as follows: determine the optimal buses and capacities of PV, WT, BESS, and capacitor banks in IEEE-33 and IEEE-69 radial distribution systems so that total active power losses are minimized and the feeder operates within acceptable technical limits.

3.3. Decision Variables and Encoding

The GVFA search process uses a hybrid decision vector that includes both discrete placement variables and continuous or discretized sizing variables. The placement variables indicate whether a DER component is installed at a candidate bus, while the sizing variables determine the corresponding rated capacities.

For each candidate bus $j\in {\mathcal{N}}_{\rfloor }$, the placement variables are defined as:

$$x_{\mathrm{PV},j}\in \{0,1\}$$

(7)

$$x_{\mathrm{WT},j}\in \{0,1\}$$

(8)

$$x_{\mathrm{BESS},j}\in \{0,1\}$$

(9)

$$x_{\mathrm{CAP},j}\in \{0,1\}$$

(10)

where $x_{\mathrm{PV},j}=1$, $x_{\mathrm{WT},j}=1$, $x_{\mathrm{BESS},j}=1$, and $x_{\mathrm{CAP},j}=1$ indicate the installation of PV, WT, BESS, and capacitor-bank units at bus j, respectively. A value of zero indicates that the corresponding component is not installed on that bus.

The sizing variables are defined as

$$P_{\mathrm{PV},j},P_{\mathrm{WT},j},P_{\mathrm{BESS},j},E_{\mathrm{BESS},j},Q_{\mathrm{CAP},j}$$

(11)

where $P_{\mathrm{PV},j}$ and $P_{\mathrm{WT},j}$ are the rated active-power capacities of PV and WT units, $P_{\mathrm{BESS},j}$ is the rated active-power capacity of the BESS, $E_{\mathrm{BESS},j}$ is the BESS energy capacity, and $Q_{\mathrm{CAP},j}$ is the reactive-power capacity of the capacitor bank.

The complete decision vector is expressed as

$$\begin{array}{cc}\hfill X=[& x_{\mathrm{PV},j},x_{\mathrm{WT},j},x_{\mathrm{BESS},j},x_{\mathrm{CAP},j},\hfill \\ & P_{\mathrm{PV},j},P_{\mathrm{WT},j},P_{\mathrm{BESS},j},E_{\mathrm{BESS},j},Q_{\mathrm{CAP},j}{]}_{j\in {\mathcal{N}}_{\rfloor }}\hfill \end{array}$$

(12)

This vector allows GVFA to optimize both the topological allocation and the physical sizing of the DER components.

Since GVFA operates in a continuous search space, the binary placement variables are obtained using a rounding-based repair mechanism:

$$x_{\mathrm{PV},j}=\mathrm{round}\left(x_{\mathrm{PV},j}^{\left(s\right)}\right)$$

(13)

$$x_{\mathrm{WT},j}=\mathrm{round}\left(x_{\mathrm{WT},j}^{\left(s\right)}\right)$$

(14)

$$x_{\mathrm{BESS},j}=\mathrm{round}\left(x_{\mathrm{BESS},j}^{\left(s\right)}\right)$$

(15)

$$x_{\mathrm{CAP},j}=\mathrm{round}\left(x_{\mathrm{CAP},j}^{\left(s\right)}\right)$$

(16)

where the superscript s denotes the corresponding search-space variable generated by GVFA.

To maintain feasible DER penetration, a hosting-capacity repair mechanism is applied. If the total active DER capacity exceeds the maximum allowable feeder hosting capacity, the active DER capacities are uniformly scaled as follows:

$$\sum _{j\in {\mathcal{N}}_{\rfloor }}\left(P_{\mathrm{PV},j}+P_{\mathrm{WT},j}+P_{\mathrm{BESS},j}^{+}\right)>P_{\mathrm{host}}^{max}$$

(17)

$$P_{\mathrm{DER},j}^{\mathrm{new}}=\alpha P_{\mathrm{DER},j}^{\mathrm{old}}$$

(18)

$$\alpha ={\displaystyle \frac{P_{\mathrm{host}}^{max}}{{\sum }_{j\in {\mathcal{N}}_{\rfloor }}{P}_{\mathrm{DER},j}^{\mathrm{old}}}}$$

(19)

where $P_{\mathrm{host}}^{max}$ is the maximum allowable DER hosting capacity of the feeder, $P_{\mathrm{BESS},j}^{+}$ denotes the discharging contribution of the BESS, and $\alpha$ is the penetration scaling factor. This repair mechanism ensures that all candidate solutions satisfy the feeder hosting-capacity limit before load-flow evaluation.

3.4. Objective Function

The proposed GVFA-based optimization is formulated as a planning-oriented DER placement and sizing problem under a representative steady-state operating snapshot. Accordingly, the main optimization stage minimizes active power losses and enforces voltage-profile constraints through static Backward/Forward Sweep load-flow evaluation. The time-dependent behavior of renewable generation and BESS operation is not embedded directly as a multi-period constraint within the primary GVFA search. Instead, the optimized BESS configuration is subsequently assessed through a post-optimization validation stage to verify the feasibility of charging, discharging, and state-of-charge limits over representative operating profiles. This separation reduces computational complexity and preserves the focus of the present work on optimal sitting and sizing. However, it also represents a limitation of the current formulation, since direct incorporation of time-series load demand, renewable intermittency, and multi-period BESS scheduling would provide a more detailed operational representation.

The primary objective of the optimization problem is to minimize total active power losses in the distribution network. The total active power loss is calculated as

$$P_{\mathrm{loss}}\left(X\right)={\displaystyle \sum _{i=1}^{N_l}}{I}_i^2{R}_i$$

(20)

where $N_l$ is the number of distribution lines, $I_i$ is the current magnitude through line i, and $R_i$ is the resistance of line i. The current $I_i$ depends on the feeder operating condition and the net power injections from PV, WT, BESS, and capacitor banks. These quantities are calculated using the Backward/Forward Sweep load-flow method.

The fitness function minimized by GVFA is defined as

$$F\left(X\right)=P_{\mathrm{loss}}\left(X\right)+\mathsf{\Omega }\left(X\right)$$

(21)

where $\mathsf{\Omega }\left(X\right)$ is a penalty function used to handle infeasible solutions. The penalty term is formulated as

$$\begin{array}{cc}\hfill \mathsf{\Omega }\left(X\right)=& {\lambda }_V\sum _{j\in \mathcal{N}}\left[max{\left(0,V_j^{min}-V_j\right)}^2+max{\left(0,V_j-V_j^{max}\right)}^2\right]\hfill \\ \hfill & +{\lambda }_{H}max{\left(0,P_{\mathrm{DER}}^{\mathrm{tot}}-P_{\mathrm{host}}^{max}\right)}^2\hfill \end{array}$$

(22)

where $V_j$ is the voltage magnitude at bus j, $V_j^{min}$ and $V_j^{max}$ are the minimum and maximum voltage limits, $P_{\mathrm{DER}}^{\mathrm{tot}}$ is the total active DER capacity, $P_{\mathrm{host}}^{max}$ is the feeder hosting limit, and ${\lambda }_V$ and ${\lambda }_H$ are penalty coefficients.

Although the primary objective is active power-loss minimization, voltage-profile improvement is enforced through the voltage-constraint penalty in Equation (22). In addition, the voltage deviation index is used as a performance indicator to quantify the improvement in voltage profile:

$$VDI=\sum _{j\in \mathcal{N}}\left|V_j-V_{\mathrm{ref}}\right|$$

(23)

where $V_{\mathrm{ref}}$ is the nominal reference voltage, usually taken as 1.0 p.u.

Accordingly, the present model does not explicitly include CAPEX/OPEX terms in the objective function, nor does it formulate a fully stochastic optimization under renewable and load uncertainty. These aspects are recognized as important practical extensions and are discussed as future work. To partially bridge the gap between static planning and real operation, the selected solutions are further assessed through time-series validation under representative daily, weekly, and annual operating profiles.

3.5. Feasibility Constraints

The optimization problem is subject to equality and inequality constraints to ensure technically feasible operation of the distribution feeder.

3.5.1. Active and Reactive Power-Flow Constraints

For each bus $j\in \mathcal{N}$, the active power balance is expressed as

$$\begin{array}{c}\hfill P_{\mathrm{in},j}-P_{L,j}+P_{\mathrm{PV},j}+P_{\mathrm{WT},j}+P_{\mathrm{BESS},j}^{\mathrm{dis}}-P_{\mathrm{BESS},j}^{\mathrm{ch}}=\sum _{k\in \mathcal{N}\left(j\right)}{P}_{jk}\end{array}$$

(24)

where $P_{\mathrm{in},j}$ is the active power entering bus j, $P_{L,j}$ is the active load demand, $P_{\mathrm{PV},j}$ and $P_{\mathrm{WT},j}$ are the active power contributions from PV and WT units, $P_{\mathrm{BESS},j}$ is the net active-power contribution of the BESS, $P_{jk}$ is the active power flow from bus j to bus k, and $\mathcal{N}\left(j\right)$ is the set of buses directly connected to bus j.

In the static-load optimization stage, $P_{\mathrm{BESS},j}$ is modeled as a net active-power injection or absorption term. A positive value represents discharging, while a negative value represents charging.

The reactive power balance is expressed as

$$\begin{array}{c}\hfill Q_{\mathrm{in},j}-Q_{L,j}+Q_{\mathrm{PV},j}+Q_{\mathrm{WT},j}+Q_{\mathrm{BESS},j}+Q_{\mathrm{CAP},j}=\sum _{k\in \mathcal{N}\left(j\right)}{Q}_{jk}\end{array}$$

(25)

where $Q_{\mathrm{in},j}$ is the reactive power entering bus j, $Q_{L,j}$ is the reactive load demand, $Q_{\mathrm{PV},j}$, $Q_{\mathrm{WT},j}$, and $Q_{\mathrm{BESS},j}$ represent reactive-power support if inverter-based reactive power control is considered, and $Q_{\mathrm{CAP},j}$ is the reactive power injected by the capacitor bank.

The capacitor-bank contribution is defined as

$$Q_{\mathrm{CAP},j}=x_{\mathrm{CAP},j}{C}_j$$

(26)

where $C_j$ is the installed capacitor size at bus j.

3.5.2. Voltage Constraints

The voltage magnitude at each bus must remain within the allowable operating range:

$$V_j^{min}\le V_j\le V_j^{max},\forall j\in \mathcal{N}$$

(27)

where $V_j^{min}$ and $V_j^{max}$ are the lower and upper voltage limits. These constraints ensure that the optimized DER configuration improves the voltage profile without violating standard operating limits.

3.5.3. DER Placement Constraints

DER components can only be installed at feasible candidate buses:

$$x_{\mathrm{PV},j},x_{\mathrm{WT},j},x_{\mathrm{BESS},j},x_{\mathrm{CAP},j}=0,\forall j\notin {\mathcal{N}}_{\rfloor }$$

(28)

The maximum number of installed units may be limited according to the planning scenario:

$$\sum _{j\in {\mathcal{N}}_{\rfloor }}{x}_{\mathrm{PV},j}\le N_{\mathrm{PV}}^{max}$$

(29)

$$\sum _{j\in {\mathcal{N}}_{\rfloor }}{x}_{\mathrm{WT},j}\le N_{\mathrm{WT}}^{max}$$

(30)

$$\sum _{j\in {\mathcal{N}}_{\rfloor }}{x}_{\mathrm{BESS},j}\le N_{\mathrm{BESS}}^{max}$$

(31)

$$\sum _{j\in {\mathcal{N}}_{\rfloor }}{x}_{\mathrm{CAP},j}\le N_{\mathrm{CAP}}^{max}$$

(32)

where $N_{\mathrm{PV}}^{max}$, $N_{\mathrm{WT}}^{max}$, $N_{\mathrm{BESS}}^{max}$, and $N_{\mathrm{CAP}}^{max}$ denote the maximum allowable numbers of PV units, WT units, BESS units, and capacitor banks, respectively.

3.5.4. DER Capacity Constraints

The rated capacity of each PV unit is bounded by

$$0\le P_{\mathrm{PV},j}\le x_{\mathrm{PV},j}{P}_{\mathrm{PV},j}^{max},\forall j\in {\mathcal{N}}_{\rfloor }$$

(33)

The rated capacity of each WT unit is bound by

$$0\le P_{\mathrm{WT},j}\le x_{\mathrm{WT},j}{P}_{\mathrm{WT},j}^{max},\forall j\in {\mathcal{N}}_{\rfloor }$$

(34)

The BESS active-power rating is bound by

$$-x_{\mathrm{BESS},j}{P}_{\mathrm{BESS},j}^{max}\le P_{\mathrm{BESS},j}\le x_{\mathrm{BESS},j}{P}_{\mathrm{BESS},j}^{max},\forall j\in {\mathcal{N}}_{\rfloor }$$

(35)

The BESS energy capacity is bound by

$$x_{\mathrm{BESS},j}{E}_{\mathrm{BESS},j}^{min}\le E_{\mathrm{BESS},j}\le x_{\mathrm{BESS},j}{E}_{\mathrm{BESS},j}^{max},\forall j\in {\mathcal{N}}_{\rfloor }$$

(36)

The capacitor-bank capacity is bound by

$$0\le Q_{\mathrm{CAP},j}\le x_{\mathrm{CAP},j}{Q}_{\mathrm{CAP},j}^{max},\forall j\in {\mathcal{N}}_{\rfloor }$$

(37)

These constraints ensure that no capacity is assigned to a DER component unless the corresponding placement variable is activated.

3.5.5. Feeder Hosting-Capacity Constraint

To prevent excessive DER penetration, the total installed active DER capacity is constrained by the feeder hosting limit:

$$\sum _{j\in {\mathcal{N}}_{\rfloor }}\left(P_{\mathrm{PV},j}+P_{\mathrm{WT},j}+P_{\mathrm{BESS},j}^{+}\right)\le P_{\mathrm{host}}^{max}$$

(38)

where $P_{\mathrm{BESS},j}^{+}$ represents the positive discharging contribution of BESS and $P_{\mathrm{host}}^{max}$ is the maximum allowable active DER hosting capacity.

3.6. BESS Planning and Post-Optimization Validation

In the main GVFA optimization stage, the BESS is modeled under a representative steady-state operating snapshot. Therefore, it is included in the active-power balance as a net injection or absorption term, while its power and energy ratings are constrained by Equations (35) and (36). This formulation maintains computational efficiency and is consistent with the static-load planning objective.

However, because BESS operation is inherently time-dependent, the selected BESS configuration is further evaluated after optimization through a post-optimization validation stage. This validation is not part of the primary GVFA search process. Its purpose is to verify that the optimized BESS size can operate over a representative time horizon without violating charging, discharging, or state-of-charge limits.

For the post-optimization validation, the time horizon is defined as

$$\mathcal{T}=\{1,2,\dots ,T\}$$

(39)

where $T=24$ h for daily validation.

To prevent simultaneous charging and discharging, a binary operating-state variable $u_j\left(t\right)$ is introduced:

$$P_{\mathrm{BESS},j}^{\mathrm{ch}}\left(t\right)\le u_j\left(t\right)P_{\mathrm{BESS},j}^{\mathrm{ch},\max }$$

(40)

$$P_{\mathrm{BESS},j}^{\mathrm{dis}}\left(t\right)\le \left[1-u_j\left(t\right)\right]P_{\mathrm{BESS},j}^{\mathrm{dis},\max }$$

(41)

$$u_j\left(t\right)\in \{0,1\}$$

(42)

where $P_{\mathrm{BESS},j}^{\mathrm{ch}}\left(t\right)$ and $P_{\mathrm{BESS},j}^{\mathrm{dis}}\left(t\right)$ are the charging and discharging powers of the BESS at time t.

The stored energy is updated as

$$E_j\left(t+1\right)=E_j\left(t\right)+\left[{\eta }_{\mathrm{ch}}{P}_{\mathrm{BESS},j}^{\mathrm{ch}}\left(t\right)-{\displaystyle \frac{P_{\mathrm{BESS},j}^{\mathrm{dis}}\left(t\right)}{{\eta }_{\mathrm{dis}}}}\right]\Delta t$$

(43)

where $E_j\left(t\right)$ is the stored energy at time t, ${\eta }_{\mathrm{ch}}$ and ${\eta }_{\mathrm{dis}}$ are the charging and discharging efficiencies, and $\Delta t$ is the simulation time step.

The state of charge is calculated as

$$SO{C}_j\left(t\right)={\displaystyle \frac{E_j\left(t\right)}{E_{\mathrm{BESS},j}}}$$

(44)

The BESS operation is considered feasible only if

$$SO{C}_j^{min}\le SO{C}_j\left(t\right)\le SO{C}_j^{max},\forall t\in \mathcal{T}$$

(45)

This post-optimization validation ensures that the BESS capacity selected during the static GVFA planning stage remains physically meaningful when evaluated under representative operational profiles.

3.7. Geese V-Formation Algorithm (GVFA) Mechanism

GVFA is a bio-inspired optimization method that simulates the cooperative flight pattern of migratory geese [35,36]. As illustrated in Figure 2, GVFA is a bio-inspired optimization method. The steps of the method are shown in Figure 3. The GVFA mechanism initiates by establishing a formation with a designated leader and followers, while computing optimal spacing (analogous to wingtip spacing in natural flocks). Within the main control loop, shown in Figure 4, each agent calculates its positional error relative to its desired offset, adjusts its velocity and orientation using a tailored control law, and updates its state accordingly. A critical decision point evaluates the leader’s energy level using an if-condition; if the leader’s energy falls below a preset threshold, a rotation is triggered by selecting a follower with higher remaining energy to assume the leader role. This leader–follower switching mechanism ensures that the energy cost is distributed evenly, maintaining the integrity of the V formation. To clarify the distinction between GVFA and conventional swarm-based optimizers, the proposed GVFA is formulated as a formation-guided cooperative search rather than a generic leader–follower heuristic. While PSO relies on velocity-memory terms and GWO uses a hierarchical encircling mechanism, GVFA introduces a formation-guided structure in which the leader tracks the best feasible DER configuration, the co-leader provides a secondary search direction, and followers update through cooperative formation-based information exchange. This is particularly useful for the present mixed discrete–continuous DER planning problem because bus-location decisions and capacity-sizing variables must be adjusted jointly after each feasibility repair. In contrast to PSO, GVFA does not depend only on individual/global memory attraction, and in contrast to GWO, it avoids relying solely on a fixed hierarchy of dominant agents; instead, the leader–co-leader–follower structure helps preserve diversity while refining feasible DER placement and sizing combinations.

In this planning framework, each agent encodes a mixed-integer vector representing DER locations and capacities. The mathematical value of this architecture lies in its structured separation of exploitation and exploration: the leader drives the search toward minimal active power losses, while co-leader and follower perturbations prevent premature convergence. Following each update, deterministic feasibility repair, elite exploitation, and diversity maintenance mechanisms enforce admissible bus indices, capacity bounds, voltage limits, and hosting capacity constraints before executing the Backward/Forward Sweep load flow. The GVFA parameter settings adopted in this work are summarized in

Table 2. Main GVFA parameters used in the scripts.

Parameter	IEEE-33	IEEE-69
Population size	200	50
Max iterations	200	500
Re-seeding/restart	Not used	Deterministic partial re-seeding triggered after 50 non-improving iterations
Penalty coefficient	$1\times {10}^4$	$1\times {10}^4$
Stall threshold	150	50

. For the IEEE-33 feeder, the algorithm was executed with a population size of 200 and a maximum of 200 iterations, using a penalty coefficient of 1 × 10⁴ and a stall threshold of 150 iterations. For the IEEE-69 feeder, the corresponding settings were a population size of 50, a maximum of 500 iterations, and the same penalty coefficient of 1 × 10⁴. Unlike the IEEE-69 implementation, which employs a deterministic partial re-seeding mechanism activated after 50 consecutive non-improving iterations, the IEEE-33 code does not use probability-based restart or re-seeding. Instead, the IEEE-33 implementation relies on deterministic scenario-level seeding for reproducibility and uses an improvement-based early-stopping criterion: the GVFA search continues while meaningful objective-function improvement is observed and terminates once the improvement falls below the predefined stall limit of 150 consecutive iterations. This distinction ensures that the IEEE-33 results remain reproducible while avoiding an additional restart mechanism that is not implemented in the script.

Stability, Anti-Stagnation, and Exploitation Control

To balance exploration and exploitation while reducing the risk of premature convergence, GVFA employs a leader–follower coordination mechanism with stagnation-gated re-initialization. During the exploitation phase, three stabilizing mechanisms are applied to promote reliable convergence:

1.

Adaptive Gain Annealing: The search step is progressively annealed using a decreasing gain $\alpha \left(k\right)$ to suppress oscillations near the optimum. The update magnitude $\Delta x\left(k\right)$ shrinks with the iteration index k:

$$\Delta x\left(k\right)=\alpha \left(k\right)\Delta x_0,\alpha \left(k\right)\to 0.$$

(46)

2.

Stagnation-Gated Trigger: The algorithm monitors the best-so-far fitness for improvements. A stagnation counter is activated if the fitness does not improve beyond a tolerance of ${10}^{-6}$. To ensure sufficient time for local search, stagnation is monitored through a non-improvement counter. In the IEEE-69 implementation, deterministic partial re-seeding is triggered after 50 consecutive non-improving iterations. In the IEEE-33 implementation, no re-seeding is applied; instead, the search terminates when the stall threshold of 150 consecutive non-improving iterations is reached.

3.

Re-Seeding (Diversity Preservation): In the IEEE-69 case, when the stagnation gate is triggered, a fraction $F{r}_{\mathrm{res}}=0.5$ of the lower-performing candidates is re-initialized. Elite individuals are preserved to maintain monotonic improvement, while new candidates explore under-sampled regions of the distribution network.

3.8. Stepwise Repair and Penalty Handling

To maintain the physical feasibility of the distribution system throughout the optimization process, a boundary-repair and penalty-evaluation mechanism is applied at each iteration:

• Boundary Clamping: Any candidate variable $x_i$ representing DER sizing that exceeds its admissible penetration bounds is clamped to the search-space limits:

$$x_i=max\left(min\left(x_i,x_{max}\right),x_{min}\right).$$

(47)

3.8.1. Convergence and Robust Considerations

The proposed GVFA is a stochastic population-based metaheuristic applied to a mixed-integer, highly nonconvex DER planning problem; therefore, deterministic global convergence cannot be guaranteed in the general case. Instead, GVFA is designed to enhance global search capability and reduce premature convergence through two main mechanisms: (i) leader–follower coordination, where the leader guides the population toward promising regions while follower candidates explore and refine alternative solutions, and (ii) adaptive leader rotation and diversity maintenance, which help refresh the search process when stagnation is detected.

In response to the complexity of the constrained search space, the present implementation adopts elitism, where the best feasible solution is preserved at each iteration, together with feasibility repair for DER bus indices, penetration limits, voltage constraints, and device-size bounds. Rather than claiming formal global convergence in probability, which would require a dedicated ergodic or stochastic-process proof, the robustness of GVFA is evaluated empirically. Specifically, repeated independent trials are used to assess the consistency of the final objective values, including the best, mean, and standard deviation of active power losses, together with the convergence trends shown in Figure 5 and Figure 6. These empirical robustness indicators provide a more appropriate and defensible basis for evaluating GVFA performance in the present mixed-integer, constrained, and multimodal DER placement and sizing problem.

3.8.2. Computational Complexity and Runtime Benchmarking

The computational cost of the proposed Geese V-Formation Algorithm (GVFA) is dominated by the repeated Backward/Forward Sweep (BFS) power-flow evaluations embedded in the fitness function. For a radial feeder with $N_b$ buses, the computational complexity of a single BFS iteration scales linearly with respect to N, expressed as $O\left(N_b\right)$. This linear scaling is a result of the recursive nature of the BFS method, which avoids the $O\left(N^3\right)$ or $O\left(N^2\right)$ overhead associated with Jacobian-based matrix inversions.

If $n_{sw}$ (denoted as G) represents the average number of sweeps required for convergence in a single load-flow solution, the computational cost of one fitness evaluation is $O(n_{sw}·N_b)$. Accordingly, for a population size $N_{pop}$ (denoted as P) and a maximum iteration count $T_{max}$ (denoted as i), the overall runtime complexity of the GVFA is dominated by

$$C_{GVFA}=O(N_b·i·P·G)$$

(48)

The computational burden of the GVFA position-update rules, the sequential repair mechanism (clamping → rounding → evaluation), and the penalty-based constraint handling remains secondary relative to the BFS evaluations.

In the present study, these complexity considerations are directly linked to the proposed GVFA formulation, including its solution encoding and parameter settings. The optimization is performed at a nominal loading level of 1.0 p.u., while time-varying validation utilizes load multipliers in the range of 0.50–1.10. Network feasibility is strictly enforced through a penalty-based formulation subject to bus-voltage limits of $0.94\le V\le 1.06$ p.u., with the BFS solver permitted a maximum of 100 sweeps per evaluation. These settings are summarized in

Table 3. Simulation settings and constraints used in the proposed GVFA framework.

Parameter	Value
Load level	Main optimization at $1.0$ p.u.; hourly assessment with 24-h multipliers in the range $0.50$–$1.10$
DG penetration limit	PV/WT/BESS: $\le 100\times$ local active load; CAP: $\le 100\times$ local reactive load
Voltage limits	$0.94$–$1.06$ p.u.
Power-flow solver	Backward/Forward Sweep (BFS)
BFS maximum sweeps	100
Penalty coefficient	$1\times {10}^4$

.

• Workflow of the Proposed Algorithm for IEEE-33 and IEEE-69 DER Placement and Sizing

• Step 1: Feeder and System Initialization
  Initialize the IEEE-33 or IEEE-69 radial distribution system using the network representation, candidate-bus set, and DER set defined in Equations (1)–(3).
• Step 2: Optimization Problem Definition
  Formulate the DER planning problem according to the compact optimization model in Equations (4)–(6).
• Step 3: Candidate Solution Encoding
  Generate the initial population of candidate solutions. Each goose represents a mixed discrete–continuous DER solution using the placement variables in Equations (7)–(10), the sizing variables in Equation (11), and the complete decision vector in Equation (12).
• Step 4: Feasibility Repair
  Apply the rounding-based repair mechanism and hosting-capacity repair using Equations (13)–(19). The candidate solution is also checked against the DER placement and capacity limits in Equations (28)–(38).
• Step 5: Candidate Evaluation
  For each candidate solution, apply PV, WT, and BESS as active-power support and capacitor banks as reactive-power compensation. Then solve the load flow using the active and reactive power-balance equations in Equations (24)–(26).
• Step 6: Fitness Calculation
  Calculate the active power loss using Equation (20) and evaluate the penalized fitness function using Equations (21) and (22). The voltage profile is checked using Equation (27), while the voltage deviation index is computed using Equation (23).
• Step 7: Leader Selection
  Rank all candidate solutions according to the penalized fitness value in Equation (21). The candidate with the lowest fitness value is selected as the leader, and the second-best candidate is selected as the co-leader.
• Step 8: Formation Update
  Update follower geese using the customized GVFA movement according to the equation below:

  $$X_i^{\mathrm{new}}=X_i+\alpha \left(X_{\mathrm{leader}}-X_i\right)+\beta {\Delta }_i.$$

  (49)
• Step 9: Local Exploitation and Diversity Maintenance
  Apply local search to elite candidate solutions. After each local modification, repeat the feasibility repair using Equations (13)–(19) and re-evaluate the candidate using Equations (20)–(27).
• Step 10: Anti-Stagnation Control
  Monitor the improvement in the best fitness value. For IEEE-69, activate deterministic partial re-seeding after 50 consecutive non-improving iterations. For IEEE-33, terminate the search when the stall threshold of 150 iterations is reached.
• Step 11: Termination and Final Validation
  Terminate the GVFA search when the maximum iteration number, target loss-reduction level, or stall threshold is reached. The final BESS feasibility is checked using the post-optimization validation Equations (39)–(45). Figure 7 shows a conceptualization of the energy optimization framework for the integration of DER using GVFA. Finally, the detailed flowchart is shown in Figure 8.

4. Results and Discussion

This section presents the outcomes generated by the Geese V-Formation Algorithm (GVFA) applied to the IEEE 33-bus and IEEE-69 distribution networks using Python. The primary objective is to minimize real power losses and enhance the voltage profile by judiciously locating and sizing distributed energy resources (DERs)—namely, photovoltaic (PV) systems, wind turbines (WTs), and battery energy storage systems (BESSs)—in conjunction with capacitors (CAPs) as shown in configuration of Figure 9a for IEEE-33 and Figure 9b for IEEE-69. Five scenarios are investigated: (1) the base case with no DERs or capacitors, (2) PV with capacitors, (3) WT with capacitors, (4) BESS with capacitors, and (5) the combined use of PV, WT, and BESS plus capacitors. The following subsections detail the overall results, DER and capacitor placements, as well as the resulting improvements in both system losses and voltage profiles.

4.1. Planning and Management of IEEE-33

GVFA delivers substantial efficiency and voltage improvements on the IEEE-33 feeder. Relative to the base case loss of 202.68 kW, all optimized scenarios in

Table 4. Planning and management result of using Geese V-Formation Algorithm (GVFA) of IEEE-33.

Scenario	Base Loss (kW)	Scenario Loss (kW)	Red. (%)	Min ${\mathit{V}}_{\mathbf{before}}$	Min ${\mathit{V}}_{\mathbf{after}}$	PV (Bus → kW)	WT (Bus → kW)	BESS (Bus → kW)	CAP (Bus → kVAR)
PV + CAP	202.68	20.14	90.06	0.9166	0.9830	14 → 550.15; 27 → 1620.14; 25 → 758.55	–	–	16 → 196.60; 6 → 815.82; 30 → 701.46
WT + CAP	202.68	19.86	90.20	0.9166	0.9936	–	30 → 1005.06; 23 → 942.42; 12 → 865.47	–	17 → 292.56; 30 → 1053.98; 10 → 140.53
BESS + CAP	202.68	15.77	92.22	0.9166	0.9917	–	–	25 → 932.34; 13 → 847.68; 32 → 862.60	30 → 1036.60; 12 → 419.19; 3 → 386.09
PV + WT + BESS + CAP	202.68	17.41	91.41	0.9166	0.9893	25 → 704.34; 17 → 335.69; 10 → 0.00	27 → 1088.52; 30 → 562.04; 27 → 0.00	4 → 545.68; 6 → 0.00; 26 → 0.00	10 → 201.25; 31 → 878.79; 13 → 284.84

cut losses by $\approx 90–92\%$—BESS + CAP achieves the best value ($15.77\mathrm{kW};-92.22\%$), followed by PV–WT–BESS + CAP ($17.41\mathrm{kW};-91.41\%$), WT + CAP ($19.86\mathrm{kW};-90.20\%$), and PV + CAP ($20.14\mathrm{kW};-90.06\%$). This ranking is visualized in Figure 10. Voltage recovery is equally decisive: the base minimum of 0.9166 p.u. is lifted to 0.9830–0.9936 p.u. across scenarios, with WT + CAP yielding the highest nadir (0.9936 p.u.). The spatial flattening of the peak-hour profile is clear in Figure 11, where traces for all optimized designs cluster near unity along the radial path. The siting pattern in Table 1—DERs and capacitors pushed toward distal, weak buses (e.g., PV at 14/25/27; WT at 30/23/12; BESS at 25/13/32; CAPs at 30/16/6/12/3/17/10/31/13)—explains both phenomena: downstream injections shrink upstream currents and local VARS raise voltages where (R/X) is high.

4.1.1. Temporal Adequacy and Power-Factor Behavior for IEEE-33

Hourly envelopes confirm these gains hold over the day. Figure 12a–d show near-unity voltages with the tightest band for BESS + CAP, reflecting dispatchable support that tracks evening peaks; PV + CAP exhibits mild midday uplift and wider off-sun spread; WT + CAP approaches the integrated case around buses 20–26 during windy windows; and PV–WT–BESS + CAP narrows PV’s spread but does not surpass BESS-only on loss.

The hourly power-factor plots (Figure 13a–d) reveal how active/reactive coordination underpins compliance: BESS + CAP approaches unity when discharging; PV/WT + CAP reach high PF during resource availability yet can show leading PF at low-load hours when capacitors dominate Q; the integrated case exhibits the most variability as mixes shift. These signatures indicate standard operational controls—staged capacitor switching and inverter Volt-VAR/Volt-Watt droops, complemented by BESS P–Q dispatch—are sufficient to keep PF within regulatory bounds while preserving the voltage and loss benefits visible in Figure 10, Figure 11 and Figure 12.

4.1.2. Power Sharing, Control Implications, and Planning Guidance

The power-sharing plots (Figure 14a–d) juxtapose total feeder load against aggregate DER output for each scenario. BESS + CAP maintains a nearly flat DER contribution aligned with high-load hours, making it the most effective for loss minimization (compare Figure 14a with Figure 10). PV + CAP and WT + CAP exhibit resource-driven timing; when their injections coincide with peaks (e.g., midday PV or windy periods), the peak-hour minimum voltage improves markedly (see Figure 11), but off-peak control depends more on capacitors. In multi-DER portfolios, GVFA prunes redundant assets (zero-sized entries in Table 4) to avoid Q oversupply and counter-flows, producing compact, non-redundant designs. Practically, the results recommend (i) placing controllable active power and local VARs at downstream buses, (ii) enforcing modest PF floors with simple droops and switched banks, and (iii) coordinating storage with stochastic DERs to target high-loss windows. This combination reliably yields $\approx 90–92\%$ loss reductions with compliant voltages across the day on radial feeders akin to IEEE-33.

4.2. Planning and Management of IEEE-69

The Geese V-Formation Algorithm (GVFA) yields substantial efficiency and voltage improvements on the IEEE-69 feeder. From

Table 5. Planning and management result of using Geese V-Formation Algorithm (GVFA) of IEEE-69.

Scenario	Base Loss (kW)	Optimized Loss (kW)	Reduction (%)	Min V (Before, p.u.)	Min V (After, p.u.)	PV Placements (Bus: kW)	WT Placements (Bus: kW)	BESS Placements (Bus: kW)	CAP Placements (Bus: kVAR)
PV + CAP	224.99	10.64	95.27%	0.9092 (bus 65)	0.9941 (min)	18: 0.00; 62: 1768.17; 21: 507.27	–	–	62: 1320.06; 26: 294.49; 19: 0.00
WT + CAP	224.99	15.91	92.93%	0.9092 (bus 65)	0.9832 (bus 65, min)	–	14: 568.93; 62: 1313.06; 30: 0.00	–	26: 264.78; 62: 1214.10; 25: 10.00
BESS + CAP	224.99	18.70	91.69%	0.9092 (bus 65)	0.9829 (buses 26–27, min)	–	–	62: 1883.51; 37: 604.05; 67: 801.45	46: 0.00; 4: 0.00; 62: 1373.48
PV + WT + BESS + CAP	224.99	17.32	92.30%	0.9092 (bus 65)	0.9943 (min)	53: 430.00; 62: 1798.77; 24: 466.45	59: 0.00; 10: 209.20; 41: 120.00	28: 2600.00; 36: 1809.26; 56: 10.00	59: 1014.24; 9: 568.07; 13: 550.00

, feeder losses drop from 224.99 kW (base) to 10.64 kW ($\mathrm{PV}+\mathrm{CAP};-95.27\%$), 15.91 kW ($\mathrm{WT}+\mathrm{CAP};-92.93\%$), 18.70 kW $(\mathrm{BESS}+\mathrm{CAP};-91.69\%)$, and 17.32 kW ($\mathrm{PV}–\mathrm{WT}–\mathrm{BESS}+\mathrm{CAP};-92.30\%$), mirroring the ranking in Figure 15. The minimum peak-hour voltage rises from 0.9092 p.u. at bus 65 to 0.9941 p.u. with PV + CAP and 0.9943 p.u. with the integrated portfolio; WT + CAP and BESS + CAP reach $\approx 0.983\mathrm{p}.\mathrm{u}.$ The spatial traces in Figure 16 reveal how all optimized cases flatten the deep sag along the 50–66 bus corridor—most prominently for PV + CAP and the integrated case. This is consistent with GVFA’s siting pattern in Table 5, which concentrates high-capacity DERs and capacitors at or near bus 62, a high-leverage downstream pivot: e.g., PV 62: 1768 kW/CAP 62: 1320 KVAR (PV + CAP), WT 62: 1313 kW/CAP 62: 1214 KVAR (WT + CAP), and BESS 62: 1884 kW/CAP 62: 1373 KVAR (BESS + CAP).

4.2.1. Temporal Adequacy and Power-Factor Behavior for IEEE-69

The 24-h envelopes (Figure 17a–d show that voltages remain close to unity through the day: PV + CAP and WT + CAP elevate the distal section ($\approx 60–66$) whenever resources are available, BESS + CAP tightens the envelope by time-shifting energy into evening peaks, and the integrated portfolio combines corridor relief from PV, opportunistic WT, and BESS dispatch to maintain a consistently elevated profile, explaining its highest peak-hour nadir in Figure 17d.

Hourly power-factor plots (Figure 18a–d clarify P–Q coordination: $\mathrm{BESS}+\mathrm{CAP}$ approaches unity when discharging and can swing leading when charging at light load; $\mathrm{PV}+\mathrm{CAP}$ achieves high PF across most daylight hours with brief leading periods when capacitors dominate early-morning Q; $\mathrm{WT}+\mathrm{CAP}$ sustains $\ge 0.85–0.95$ PF for most of the day, with transient dips when wind and capacitor steps misalign; and the integrated case shows the largest PF variability (signed), reflecting shifting shares among PV/WT/BESS—yet voltage compliance is preserved in Figure 18d.

4.2.2. Power Sharing and Planning Guidance

Power-sharing curves (Figure 19a–d) juxtapose total feeder load with aggregated DER output and explain the objective trade-offs. $\mathrm{BESS}+\mathrm{CAP}$ supplies a nearly flat contribution aligned with high-loss windows, making it the most controllable option even if PV wins on pure loss reduction here. $\mathrm{PV}+\mathrm{CAP}$ and $\mathrm{WT}+\mathrm{CAP}$ deliver strong relief when injections coincide with demand peaks (improving the weakest-bus voltage in Figure 16), while off-peak control relies more on capacitors. GVFA also prunes redundant assets (0-sized entries in Table 5) to avoid VAR oversupply and back-flows, yielding compact, non-redundant portfolios. For radial feeders with a dominant weak corridor like IEEE-69, the results recommend: (i) placing large PV/WT and capacitors at the downstream pivot $(\approx \mathrm{bus}62)$, (ii) using BESS to shape peaks and cover non-sunny/windy hours without over-dispersing units, and (iii) enforcing simple Volt-VAR droops, switched capacitor steps, and BESS P–Q dispatch to maintain PF targets—consistently achieving $\sim 93–95\%$ loss cuts with near-unity voltages across the day.

4.3. Long-Horizon and Time-Series Validation of GVFA-Based PV–WT–BESS–CAP Planning

Figure 20 demonstrates that, over a one-year planning horizon, the base-case monthly energy losses for both IEEE-33 and IEEE-69 exhibit pronounced seasonality with summer peaks (around July–August) and consistently higher magnitudes for IEEE-69 due to its larger network, whereas the GVFA-optimized PV + WT + BESS + CAP configuration compresses these trajectories to a low, nearly flat band (10–13 MWh/month), evidencing robust loss reduction under seasonal variability. To reconcile static planning with time-varying operation, we further validated the fixed GVFA-derived locations and rated sizes using a 168 h (one-week) load trajectory, evaluating out-of-sample performance via standard distribution sensitivities (loss scaling approximately with and voltage-drop roughly linear with anchored to BFS reference points; the resulting weekly profiles (Figure 21) confirm sustained low hourly losses and worst-bus maintained well above 0.95 p.u. throughout the week. Consistently,

Table 6. Weekly aggregate gains summary.

Test System	Case	Total Energy Losses (kWh) over 1 Week	Max Loss (kW)	Mean Loss (kW)	Min Voltage (Worst-Hour, p.u.)	Mean Min Voltage (p.u.)	Voltage-Violation Hours (<0.95 pu) (%)
IEEE-33	Base case	33,429.9	312.892	198.987	0.896	0.918	100
IEEE-33	With PV + WT + BESS + CAP	2871.594	26.877	17.092	0.986	0.989	0
IEEE-69	Base case	37,109.698	347.333	220.891	0.887	0.911	100
IEEE-69	With PV + WT + BESS + CAP	2856.749	26.738	17.004	0.992	0.994	0

quantifies the week-aggregated gains: total energy losses drop from 33.43 → 2.87 MWh (IEEE-33) and 37.11 → 2.86 MWh (IEEE-69), peak loss reduces from 312.89 → 26.88 kW (IEEE-33) and 347.33 → 26.74 kW (IEEE-69), and the minimum worst-hour voltage improves from 0.896 → 0.987 p.u. (IEEE-33) and 0.887 → 0.993 p.u. (IEEE-69), while voltage-violation hours (<0.95 p.u.) are eliminated (100% → 0%) in both feeders.

4.4. Sensitivity Analysis

Table 7. Detailed numerical sensitivity analysis of the GVFA algorithm for IEEE-33 and IEEE-69.

System	Scenario	Base Loss (kW)	Optimized Loss (kW)	Reduction (%)	Min V Before (p.u.)	Min V After (p.u.)
IEEE-33	PV + CAP	202.68	20.14	90.06	0.9166	0.9830
IEEE-33	WT + CAP	202.68	19.86	90.20	0.9166	0.9936
IEEE-33	BESS + CAP	202.68	15.77	92.22	0.9166	0.9917
IEEE-33	PV + WT + BESS + CAP	202.68	17.41	91.41	0.9166	0.9893
IEEE-69	PV + CAP	224.99	10.64	95.27	0.9092	0.9941
IEEE-69	WT + CAP	224.99	15.91	92.93	0.9092	0.9832
IEEE-69	BESS + CAP	224.99	18.70	91.69	0.9092	0.9829
IEEE-69	PV + WT + BESS + CAP	224.99	17.32	92.30	0.9092	0.9943

and Figure 22 show that the GVFA is sensitive to the adopted DER/CAP configuration, and that this sensitivity is feeder dependent. For the IEEE-33 system, the BESS + CAP case provides the best loss minimization, reducing the active-power loss from 202.68 kW to 15.77 kW (92.22%), while WT + CAP yields the best post-optimization voltage support with a minimum voltage of 0.9936 p.u. For the IEEE-69 system, the strongest loss-reduction performance is achieved by PV + CAP, where the loss decreases from 224.99 kW to 10.64 kW (95.27%), whereas the combined PV + WT + BESS + CAP configuration gives the best minimum-voltage recovery of 0.9943 p.u. These results indicate that increasing the number of integrated resources does not always guarantee the lowest loss, because the optimal GVFA response depends on feeder topology, load distribution, and the spatial interaction between active- and reactive-power compensation. Overall, all tested scenarios satisfy the voltage-security requirement after optimization and confirm the robustness of the GVFA for both medium-scale and larger radial distribution systems.

4.5. Comparative Analysis

As shown in Figure 5 and Figure 6, the proposed GVFA demonstrates competitive convergence behavior compared with representative bio-inspired optimization algorithms, including GWO [46], HGWO [47], and ABC [48]. The comparison indicates that GVFA reaches a low final objective value while maintaining a stable convergence trajectory, reflecting its effective balance between exploration and exploitation. This behavior is mainly attributed to the leader–follower coordination mechanism, V-formation-based information exchange, local exploitation step, and diversity-maintenance strategy. Therefore, the proposed GVFA provides a clear performance enhancement for the optimal placement and sizing of PV, WT, BESS, and capacitor banks in the IEEE-33 and IEEE-69 distribution systems. Unlike methods that may converge rapidly but stagnate at higher objective values, GVFA maintains a smoother improvement trend and achieves a lower final objective value, indicating better search stability and enhanced exploitation of promising DER placement regions.

To address the need for algorithmic benchmarking, the convergence performance of GVFA was compared with GWO, HGWO, and ABC under the same DER planning scenario, objective-function formulation, voltage-constraint handling, and Backward/Forward Sweep load-flow evaluation.

The comparison

Table 8. Comparison of GVFA results with related studies on IEEE-33 and IEEE-69 feeders.

Test System	Study	Assets Optimized	Base Loss	Best/Reported Loss	Reduction (%)	Min V (Before → After) (p.u.)	Optimal Placements (Bus → Size)
IEEE-33	This work (GVFA)—PV + CAP	PV + CAP	202.68 kW	20.14 kW	90.06	0.9166 → 0.9830	PV: 14 → 550.15; 27 → 1620.14; 25 → 758.55 |CAP: 16 → 196.60; 6 → 815.82; 30 → 701.46
IEEE-33	This work (GVFA)—WT + CAP	WT + CAP	202.68 kW	19.86 kW	90.20	0.9166 → 0.9936	WT: 30 → 1005.06; 23 → 942.42; 12 → 865.47 |CAP: 17 → 292.56; 30 → 1053.98; 10 → 140.53
IEEE-33	This work (GVFA)—BESS + CAP	BESS + CAP	202.68 kW	15.77 kW	92.22	0.9166 → 0.9917	BESS: 25 → 932.34; 13 → 847.68; 32 → 862.60 |CAP: 30 → 1036.60; 12 → 419.19; 3 → 386.09
IEEE-33	This work (GVFA)—PV + WT + BESS + CAP	PV + WT + BESS + CAP	202.68 kW	17.41 kW	91.41	0.9166 → 0.9893	PV: 25 → 704.34; 17 → 335.69 |WT: 27 → 1088.52; 30 → 562.04 |BESS: 4 → 545.68 |CAP: 10 → 201.25; 31 → 878.79; 13 → 284.84
IEEE-33	Chakraborty et al. (HHO) [49]	PV-DG (unity pf)	202.67 kW	72.10 kW	64.42	NR	PV-DG: 14 → 0.813; 30 → 1.092; 24 → 1.098 (MW)
IEEE-33	Kwangkaew et al. (SSA) [51]	RDG (+reactive comp.)	202.677 kW	72.078 kW	64.43	NR	RDG: 13 → 0.9774; 24 → 1.0879; 30 → 0.9772 (MW)
IEEE-69	This work (GVFA)—PV + CAP	PV + CAP	224.99 kW	10.64 kW	95.27	0.9092 → 0.9941	PV: 62 → 1768.17; 21 → 507.27 |CAP: 62 → 1320.06; 26 → 294.49
IEEE-69	This work (GVFA)—WT + CAP	WT + CAP	224.99 kW	15.91 kW	92.93	0.9092 → 0.9832	WT: 14 → 568.93; 62 → 1313.06 |CAP: 26 → 264.78; 62 → 1214.10; 25 → 10.00
IEEE-69	This work (GVFA)—BESS + CAP	BESS + CAP	224.99 kW	18.70 kW	91.69	0.9092 → 0.9829	BESS: 62 → 1883.51; 37 → 604.05; 67 → 801.45 |CAP: 62 → 1373.48
IEEE-69	This work (GVFA)—PV + WT + BESS + CAP	PV + WT + BESS + CAP	224.99 kW	17.32 kW	92.30	0.9092 → 0.9943	PV: 53 → 430.00; 62 → 1798.77; 24 → 466.45 |WT: 10 → 209.20; 41 → 120.00 |BESS: 28 → 2600.00; 36 → 1809.26; 56 → 10.00 |CAP: 59 → 1014.24; 9 → 568.07; 13 → 550.00
IEEE-69	Chakraborty et al. (HHO)  [49]	PV-DG (unity pf)	224.9 kW	71.8 kW	68.074	NR	PV-DG: 61 → 1.872; 17 → 0.380; 11 → 0.526 (MW)
IEEE-69	Abdul Kadir et al. (IGSA)  [50]	Renewable DG	0.2298 MW	0.0199 MW	91.34	NR	DG: 11 → 1.0531; 61 → 0.8638; 64 → 0.6085 (MW)
IEEE-69	Radosavljević et al. (PPSO–GSA hybrid)  [10]	DG (unity pf; benchmark)	224.946 kW	69.397 kW (3-DG best)	69.15	NR	Best 3-DG loss reported; bus allocations are tabulated in the paper’s IEEE-69 benchmark table.

Note: NR stands for Not Reported.

benchmarks the proposed GVFA-based co-optimization against representative DG-allocation studies on the IEEE-33 and IEEE-69 radial systems, highlighting both loss reduction and voltage-support impacts. For IEEE-33, GVFA achieves deep loss mitigation across all scenarios—reducing losses from 202.68 kW to 20.14 kW (PV + CAP, 90.06%), 19.86 kW (WT + CAP, 90.20%), and 15.77 kW (BESS + CAP, 92.22%)—while simultaneously lifting the minimum bus voltage from 0.9166 p.u. to 0.9830–0.9936 p.u.; the combined PV+WT+BESS + CAP case sustains a 91.41% reduction (17.41 kW) with min-voltage improved to 0.9893 p.u. In contrast, placement-only PV/RDG studies such as Chakraborty et al. and Kwangkaew et al. report ∼64% loss reduction on IEEE-33 ($\approx 72.1$ kW residual loss) and do not tabulate minimum-voltage values, underscoring the advantage of jointly sizing reactive support and multiple DER types in a unified search. A similar pattern is observed for IEEE-69: GVFA reduces losses from 224.99 kW to 10.64 kW (PV + CAP, 95.27%), 15.91 kW (WT + CAP, 92.93%), 18.70 kW (BESS + CAP, 91.69%), and 17.32 kW (PV+WT+BESS + CAP, 92.30%), while improving the weakest-voltage level from 0.9092 p.u. to 0.9829–0.9943 p.u. Relative to PV-DG allocation with unity power factor ($\approx 68\%$ reduction) [49] and hybrid DG benchmarks ($\approx 69\%$ reduction) [10], the proposed approach attains markedly lower residual losses and explicitly quantifies voltage recovery, and it is competitive with advanced renewable-DG sizing results on IEEE-69 ($\approx 91\%$ reduction) [50], with the added benefit of integrated capacitor support and multi-asset coordination that strengthens voltage regulation alongside loss minimization.

It should be noted that the literature-based comparison presented in Table 8 should be interpreted as a contextual benchmark, rather than as a universal proof of superiority over all previously published DER allocation methods. This is because the reported studies differ in several modeling aspects, including DER technology types, unity or non-unity power-factor assumptions, objective-function definitions, voltage-constraint handling, load-flow solvers, and stopping criteria. Nevertheless, Table 8 provides useful insight into the relative performance of the proposed GVFA compared with representative IEEE-33 and IEEE-69 DER allocation studies reported in the literature. To provide a fairer algorithmic assessment, the controlled convergence comparison shown in Figure 5 and Figure 6 is emphasized as the primary benchmark, since GVFA, GWO, HGWO, and ABC were evaluated under identical DER allocation assumptions, the same objective-function formulation, the same voltage operating limits, the same constraint-handling strategy, and the same Backward/Forward Sweep load-flow evaluation. Therefore, Figure 5 and Figure 6 provide the main evidence for assessing the convergence behavior and solution quality of the proposed GVFA, while Table 8 serves as a supporting literature-based comparison.

5. Conclusions and Future Work

This study demonstrates that the application of the Geese V-Formation Algorithm (GVFA) offers a highly effective approach for optimizing the placement and sizing of distributed energy resources (DERs) and capacitors in both IEEE-33 and IEEE-69 bus systems. By drawing inspiration from the natural formation patterns of geese, GVFA mimics their collective behavior to dynamically cluster and coordinate resources. The results show that whether PV, wind, BESS, or a hybrid of these resources is deployed, the algorithm consistently achieves dramatic reductions in power losses—exceeding $90\%$—and markedly improves voltage profiles across the network. In addition, the integration of capacitors with the DERs, as optimized by GVFA, enhances reactive power management and helps maintain power factor stability, ultimately contributing to a more efficient distribution system.

Overall, these figures demonstrate that deploying various DERs—alone or in combination—with capacitive support can substantially reduce the system’s net load on the main grid. The GVFA optimizes where and how these resources are placed, ensuring each scenario yields improved performance metrics. For distribution network planners and operators, these results emphasize the value of multi-technology integration, guided by an intelligent optimization algorithm, to meet load demands effectively and reliably.

Although the proposed GVFA-based framework demonstrates strong performance in reducing active power losses and improving voltage profiles in the IEEE-33 and IEEE-69 benchmark systems, several limitations should be acknowledged from a real-world implementation perspective. The present study is based on standard radial test feeders and a static-load planning formulation; therefore, its direct deployment in practical distribution networks requires further validation using actual utility feeder data and detailed operational constraints. In real systems, additional factors such as time-varying demand, renewable-generation variability, protection coordination, inverter operating constraints, equipment availability, site-specific installation restrictions, and techno-economic feasibility may affect the implementation of the optimized DER and capacitor placements. These limitations define the practical scope of the current study and should be considered when transferring the proposed GVFA-based planning framework from benchmark systems to field applications.

Future Work

Future research will extend the present technically oriented framework toward techno-economic multi-objective planning, including CAPEX/OPEX terms, degradation-aware BESS costs, and stochastic or scenario-based treatment of renewable and load uncertainty.

• Dynamic and Real-Time Optimization: Future research should include variations in time-varying loads and incorporate real-time generated electricity from renewable resources to further improve the GVFA approach. The development of adaptive scheduling approaches should give the processes the ability to deal with rapidly changing conditions in this manner, improving the operational flexibility of the distribution network.
• Economic and Environmental Impact Analysis: Utilizing an integrated cost-benefit analysis, as well as an environmental impact assessment, could provide a more holistic assessment of GVFA performance than what was possible in WP2. Future work could quantify the trade-offs between economic performance and sustainability for various DERS, and would assist in future investment decisions.
• Stochastic Modeling and Uncertainty Management: Considering that both renewables and load demand have uncertainty, using stochastic models within the GVFA framework would allow for more effective planning. Future work may implement probabilistic methods or machine learning, as a means of forecasting the uncertainty of risk with taking into account reliability considerations.
• Scalability to Larger Networks:
  Although this research has primarily examined two system examples, IEEE-33 and IEEE-69, it is completely reasonable to utilize the GVFA in even larger and more complex networks. It should be considered important to evaluate the GVFA’s scalability and computational efficiency in large grid contexts. For the GVFA to be applied practically in contemporary smart grids, it will be important to understand and evaluate the algorithm scalability and computational efficiency.