Metaheuristic-Based PV–BESS Planning for Radial Distribution Networks with Inverter-Based Voltage Support

Abstract

High photovoltaic (PV) penetration can reduce grid-energy imports but may worsen voltage performance in radial distribution feeders. This paper develops a unified mixed-discrete PV–BESS planning framework that jointly optimizes siting, sizing, hourly storage operation, and a common inverter voltage-support gain subject to inverter capability limits. Five metaheuristic algorithms are compared over 30 independent runs on the 33-bus system using an explicitly converged under-relaxed fixed-point iteration. Adaptive Differential Evolution achieves the lowest best and median fitness values. Its best solution reduces active-loss energy by 38.0%, grid-energy import by 14.9%, peak grid power by 12.5%, voltage deviation by 53.7%, voltage-violation duration by 100%, and annual cost by 8.4% relative to the no-PV/no-BESS case. A separately optimized comparison shows substantial additional voltage and loss improvements from inverter reactive-power support. Objective-weight, voltage-penalty, BESS energy-CAPEX, and representative-day analyses clarify key planning trade-offs. On the 69-bus feeder, the 30-run ADE validation reduces active-loss energy by 51.2%, grid-energy import by 19.5%, peak grid power by 10.5%, voltage deviation by 39.5%, voltage-violation duration by 98.8%, and annual cost by 12.2%. A supplementary analysis with a higher PV upper bound further clarifies the influence of the adopted planning limit.

1. Introduction

Distribution networks are increasingly moving beyond the traditional fit-and-forget approach. As the penetration of distributed energy resources grows, active distribution networks (ADNs) must accommodate bidirectional power flows, photovoltaic (PV) generation, flexible loads, and battery energy storage systems (BESSs). Although PV integration can reduce grid-energy consumption and support local decarbonization, high PV penetration may also lead to voltage rise, reverse power flow, increased voltage fluctuations, and uneven loss patterns along radial feeders [1,2,3]. These challenges are particularly important in weak downstream feeder sections, where line impedances are higher and local voltage support is more limited [4,5].

BESSs are valuable planning resources because they can shift energy over time, reduce peak grid power, mitigate voltage deviations, and improve local PV utilization [6,7]. In addition, modern PV and BESS units are connected through power electronic inverters that can provide reactive-power support within their apparent-power limits [8,9,10]. Therefore, PV–BESS planning should not be restricted to active-power sizing alone. A more realistic framework should jointly consider siting, sizing, storage operation, and inverter-enabled voltage support.

From a broader perspective of an ADN, PV inverters and BESS units can also be viewed as spatially and temporally controllable flexibility resources. Recent studies have examined robust flexible-interconnection planning under source–load uncertainty and joint energy–flexibility market clearing in distribution systems [11,12]. The present work is complementary to that literature: it focuses on long-term PV–BESS asset allocation and feeder-level techno-economic performance rather than inter-area interconnection design or market clearing while explicitly representing the voltage-support flexibility available from inverter-connected resources.

The resulting planning problem involves both discrete and continuous decision variables and is therefore mixed-discrete, non-linear, and non-convex. The PV and BESS bus locations are discrete variables, whereas PV capacity, BESS power rating, BESS energy rating, and a common inverter voltage-support gain are continuous variables. The hourly BESS charge and discharge decisions are also coupled through the stored-energy dynamics and normalized state-of-charge (SOC) limits, while the available inverter reactive-power capability changes with active-power dispatch. These characteristics make the problem difficult to solve using conventional gradient-based methods and motivate the use of population-based metaheuristic algorithms. However, many existing studies report only the best result obtained by a single algorithm or provide limited statistical evidence when comparing different methods. A more reliable assessment of stochastic optimization algorithms requires multiple independent runs, convergence analysis, summary statistics, run-to-run variability, and non-parametric significance tests [13,14].

This study proposes a coordinated PV–BESS planning model for ADNs that incorporates inverter-based reactive-power support. The model includes both planning and operational elements. These are mixed-discrete PV–BESS allocation, time-coupled storage scheduling, and optimization of a common inverter voltage-support gain, with reactive-power commands constrained by the available inverter capability, replacement-aware economic modeling, and a statistically supported comparison of representative stochastic optimizers evaluated under the same computational budget. The main analysis is carried out on the 33-bus radial distribution system over 24 h. The effect of inverter-based reactive-power support is assessed by separately optimizing one case with the common voltage-support gain fixed at $K_q=0$ and another in which $K_q$ is optimized while keeping the ADE computational budget unchanged. The solution’s sensitivity according to the objective weights is investigated across cost-oriented, balanced, and network-oriented settings. The influence of BESS energy-CAPEX assumptions is assessed by cross-evaluating the plans obtained under the different CAPEX scenarios. Robustness across representative operating conditions is investigated using four equally weighted synthetic daily profiles, fixed asset decisions, day-specific BESS scheduling, and cross-evaluation of the resulting schedules across all profiles. The applicability of the formulation to a larger system is further examined on the 69-bus radial distribution network using ADE. In addition, a supplementary case with a 2500 kW PV upper bound is included to assess the sensitivity of the results to the principal 2000 kW planning limit.

Contributions

The main contributions of this study can be summarized as follows:

1.

A unified mixed-discrete PV–BESS planning formulation is developed for radial distribution networks. The model jointly considers PV siting and sizing, BESS siting and sizing, hourly charge/discharge scheduling, and optimization of a common inverter voltage-support gain, with reactive-power commands constrained by the available inverter capability.

2.

A normalized techno-economic objective is formulated by combining annualized investment cost, discounted BESS replacement cost, grid-energy cost, active-loss energy, voltage deviation, peak grid power, and voltage-violation penalties.

3.

Five representative evolutionary and swarm-based algorithms are implemented under the same population size, iteration limit, and number of independent runs. The comparison includes convergence behavior, run-to-run statistics, a Kruskal–Wallis test, and Holm-adjusted pairwise Wilcoxon rank-sum tests.

4.

An inverter-support contribution analysis separately optimizes the planning problem with the common inverter voltage-support gain fixed at $K_q=0$ and with $K_q$ treated as a decision variable. Comparing these two optimized cases isolates the additional technical benefits provided by inverter reactive-power support from those achieved through PV–BESS allocation and storage scheduling.

5.

Objective-weight sensitivity, voltage-penalty sensitivity, BESS energy-CAPEX sensitivity, and representative-day robustness analyses are used to distinguish planning-preference effects from operating-profile effects.

6.

Each algorithm is evaluated over 30 independent runs on the 33-bus radial distribution system. The best-performing algorithm is then applied to the larger 69-bus radial distribution system to evaluate the formulation on a feeder of greater scale.

2. Related Work

Optimal allocation and sizing of distributed energy resources have been widely studied in radial distribution systems. Early distribution-planning formulations mainly focused on loss minimization, voltage improvement, and feeder reconfiguration [15]. As renewable generation became more widespread, PV-allocation studies increasingly considered voltage constraints, hosting capacity, reverse power flow, and uncertainty [1,2,3]. More recent studies have incorporated BESSs to improve local PV utilization, reduce grid-energy imports, and provide flexibility under time-varying load and generation conditions [6,7,16]. Related allocation and operating strategies have also been investigated [17,18,19]. Recent contributions have further explored coordinated PV–BESS planning under different technical and economic assumptions [20,21,22,23]. Flexibility-oriented planning and operation studies additionally consider robust interconnection investment and joint energy–flexibility market mechanisms [11,12]; these works broaden the context in which inverter-connected PV and BESS assets can be viewed as controllable flexibility resources.

Distribution-level voltage-control studies have also shown that tap-changing transformers, capacitors, and PV inverters can be coordinated to improve feeder voltage profiles. In addition, nodal-sensitivity-based smart-inverter control can provide localized voltage regulation in PV-rich distribution networks [24,25]. These studies demonstrate the technical value of inverter-based voltage support, but this capability is not always incorporated directly into long-term PV–BESS planning models.

Metaheuristic algorithms are well suited to PV and BESS planning problems because they can handle discrete siting decisions together with continuous sizing and control variables, which are challenging for conventional gradient-based optimization methods [3,17,18,19]. Differential Evolution (DE), Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), and Moth-Flame Optimization (MFO) are widely used in engineering optimization [26,27,28,29]. Adaptive variants of DE are particularly useful because they reduce the dependence on fixed, manually selected mutation and crossover parameters [30,31,32].

Table 1. Positioning of the present study relative to representative PV–BESS planning themes.

Study Group	Joint PV–BESS Planning	Time-Coupled BESS Operation	Inverter Reactive Support	Replacement-Aware Economics	Multi-Run Statistical Comparison
PV/BESS allocation and sizing studies [17,18,19]	Yes	Varies	Varies	Varies	Varies
Recent coordinated planning studies [20,21,22,23]	Yes	Varies	Varies	Varies	Varies
Smart-inverter voltage-control studies [8,9,24,25]	Not the main focus	Not the main focus	Yes	No	Varies
This study	Yes	Yes	Yes	Yes	Yes

summarizes the main features addressed in the reviewed literature and highlights how the present study brings together joint PV–BESS planning, time-coupled storage operation, inverter reactive-power support, replacement-aware economics, and multi-run statistical comparison within a unified framework.

Despite this progress, three research gaps remain. First, PV and BESS planning studies do not always represent inverter-based reactive-power support explicitly. Prior studies and the relevant interconnection standard nevertheless support smart-inverter voltage regulation in PV-rich feeders [5,8,9,10], while nodal-sensitivity-based coordination studies demonstrate localized voltage-support capability [24,25]. Second, BESS replacement costs are not always represented explicitly in multi-year planning models, although storage-cost assumptions can materially affect the preferred capacity and the resulting annualized cost [33]. Third, comparisons of stochastic optimization algorithms are often based on the best result obtained from a limited number of runs, which may provide an incomplete picture of robustness. To address these gaps, this study combines inverter-based voltage support, replacement-aware BESS economics, and a 30-run statistical comparison framework supported by non-parametric significance tests [13,14].

3. System Modeling

3.1. Radial Distribution Network Model

The distribution feeder is modeled as a balanced radial network with $N_b$ buses and $N_l$ branches. The slack-bus voltage is fixed at $1\angle 0^{\circ }$ p.u. At each hour t, the load flow is solved using the backward/forward sweep method, which is well suited to radial distribution feeders [34]. Throughout this section, $t=1,\dots ,T$ denotes the time-step index, where T is the number of time steps in the daily simulation. The indices i and j refer to buses, while $\ell =1,\dots ,N_l$ denotes a branch. Unless otherwise stated, voltages, currents, and branch impedances are expressed in per unit. The net complex power demand at bus i and hour t is defined as

$$S_{i,t}^{\mathrm{net}}=\left(P_{i,t}^{\mathrm{L}}-P_{i,t}^{\mathrm{PV}}-P_{i,t}^{\mathrm{BESS}}\right)+j\left(Q_{i,t}^{\mathrm{L}}-Q_{i,t}^{\mathrm{PV}}-Q_{i,t}^{\mathrm{BESS}}\right).$$

(1)

Here, $S_{i,t}^{\mathrm{net}}$ is the net complex power demand at bus i and hour t, $P_{i,t}^{\mathrm{L}}$ and $Q_{i,t}^{\mathrm{L}}$ are the active and reactive load demands, $P_{i,t}^{\mathrm{PV}}$ and $Q_{i,t}^{\mathrm{PV}}$ are the active and reactive power contributions of the PV unit, and $P_{i,t}^{\mathrm{BESS}}$ and $Q_{i,t}^{\mathrm{BESS}}$ are the active and reactive power contributions of the BESS. The symbol j denotes the imaginary unit. With this sign convention, load demand is positive, while PV and BESS injections reduce the net demand seen by the feeder. For compactness, the device-level variables are mapped to the corresponding buses as follows. The PV contributions are equal to $P_t^{\mathrm{PV}}$ and $Q_t^{\mathrm{PV}}$ at the selected PV bus $b_{\mathrm{PV}}$ and zero at all other buses. Similarly, the BESS contributions are equal to $P_t^{\mathrm{BESS}}$ and $Q_t^{\mathrm{BESS}}$ at the selected BESS bus $b_{\mathrm{BESS}}$ and zero elsewhere. Positive reactive power denotes injection into the feeder, while negative reactive power denotes absorption.

Using the bus voltage from the current iteration, the corresponding net current demand at bus i and hour t is calculated as

$$I_{i,t}={\left({\displaystyle \frac{S_{i,t}^{\mathrm{net}}}{V_{i,t}}}\right)}^{*}.$$

(2)

Here, $I_{i,t}$ is the complex current associated with the net bus demand, $V_{i,t}$ is the complex bus voltage, and ${(·)}^{*}$ denotes the complex conjugate.

In the backward sweep, branch currents are accumulated from the terminal buses toward the slack bus. In the forward sweep, bus voltages are updated from the slack bus toward the downstream buses according to

$$V_{j,t}=V_{i,t}-Z_{ij}{I}_{ij,t},$$

(3)

where $V_{j,t}$ and $V_{i,t}$ are the complex voltages at the receiving and sending buses of branch $(i,j)$, respectively, $I_{ij,t}$ is the branch current flowing from bus i to bus j, and $Z_{ij}=R_{ij}+j{X}_{ij}$ is the branch impedance. Here, $R_{ij}$ and $X_{ij}$ denote the branch resistance and reactance.

The daily active-loss energy is then calculated as

$$E_{\mathrm{loss}}={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{\ell =1}^{N_l}}{R}_{\ell }{\left|I_{\ell ,t}\right|}^2{S}_{\mathrm{base}}\Delta t.$$

(4)

In this expression, $I_{\ell ,t}$ is the per-unit current magnitude of branch ℓ at hour t, $R_{\ell }$ is the per-unit branch resistance, $S_{\mathrm{base}}$ is the three-phase system apparent-power base expressed in kVA, and $\Delta t$ is the simulation time step in hours. Therefore, $E_{\mathrm{loss}}$ is obtained in kWh. The summation over all branches and time steps gives the total daily active-loss energy.

3.2. PV Model

The PV unit is connected to the selected bus $b_{\mathrm{PV}}$. Its active-power output at hour t is modeled as

$$P_t^{\mathrm{PV}}=P_{\mathrm{rated}}^{\mathrm{PV}}{\varphi }_t^{\mathrm{PV}},$$

(5)

where $P_{\mathrm{rated}}^{\mathrm{PV}}$ is the optimized PV capacity and ${\varphi }_t^{\mathrm{PV}}\in [0,1]$ is the normalized PV generation profile. The generated active power is injected at the selected PV bus, while it is zero at all other buses.

The reactive-power capability of the PV inverter is limited by its apparent-power rating:

$$|Q_t^{\mathrm{PV}}|\le \sqrt{{\left(S_{\mathrm{inv}}^{\mathrm{PV}}\right)}^2-{\left(P_t^{\mathrm{PV}}\right)}^2},$$

(6)

where $S_{\mathrm{inv}}^{\mathrm{PV}}={\gamma }_{\mathrm{PV}}{P}_{\mathrm{rated}}^{\mathrm{PV}}$ and ${\gamma }_{\mathrm{PV}}\ge 1$ is the PV inverter oversizing factor. This constraint ensures that the active and reactive power outputs remain within the inverter apparent-power capability, which is consistent with inverter-based DER interconnection and voltage-support practices [10].

3.3. BESS Model

The BESS is connected to the selected bus $b_{\mathrm{BESS}}$ and is described by its power rating $P_{\mathrm{rated}}^{\mathrm{BESS}}$, energy rating $E_{\mathrm{rated}}^{\mathrm{BESS}}$, and hourly active-power command $P_t^{\mathrm{BESS}}$. A positive value of $P_t^{\mathrm{BESS}}$ represents discharging, while a negative value represents charging. The BESS power command is constrained as

$$-P_{\mathrm{rated}}^{\mathrm{BESS}}\le P_t^{\mathrm{BESS}}\le P_{\mathrm{rated}}^{\mathrm{BESS}}.$$

(7)

During simulation, the requested hourly BESS power is clipped to the feasible range determined by the stored-energy limits. A penalty is added when the requested charge or discharge command cannot be fully implemented because of these limits.

The stored-energy dynamics of the BESS are modeled as

$$E_{t+1}^{\mathrm{BESS}}=\left\{\begin{array}{cc}{E}_t^{\mathrm{BESS}}-{\displaystyle \frac{P_t^{\mathrm{BESS}}\Delta t}{{\eta }_{\mathrm{dis}}}},\hfill & P_t^{\mathrm{BESS}}\ge 0,\hfill \\ E_t^{\mathrm{BESS}}-{\eta }_{\mathrm{ch}}{P}_t^{\mathrm{BESS}}\Delta t,\hfill & P_t^{\mathrm{BESS}}<0,\hfill \end{array}\right.$$

(8)

where $E_t^{\mathrm{BESS}}$ is the stored energy in kWh, ${\eta }_{\mathrm{ch}}$ and ${\eta }_{\mathrm{dis}}$ are the charging and discharging efficiencies, and $\Delta t$ is the simulation time step in hours. Since $P_t^{\mathrm{BESS}}<0$ during charging, the second line increases the stored energy according to the charging efficiency. The normalized state of charge is defined separately as

$$SO{C}_t={\displaystyle \frac{E_t^{\mathrm{BESS}}}{E_{\mathrm{rated}}^{\mathrm{BESS}}}}.$$

(9)

The stored-energy limits are

$$E^{min}\le E_t^{\mathrm{BESS}}\le E^{max},$$

(10)

where $E^{min}=SO{C}^{min}{E}_{\mathrm{rated}}^{\mathrm{BESS}}$ and $E^{max}=SO{C}^{max}{E}_{\mathrm{rated}}^{\mathrm{BESS}}$.

The stored-energy feasibility penalty accumulates the normalized mismatch between the requested and implemented BESS power commands together with any residual stored-energy-limit violation:

$$\begin{array}{cc}\hfill J_E={\displaystyle \sum _{t=1}^T}[& {\left({\displaystyle \frac{P_{t,\mathrm{req}}^{\mathrm{BESS}}-P_{t,\mathrm{impl}}^{\mathrm{BESS}}}{P_{\mathrm{rated}}^{\mathrm{BESS}}}}\right)}^2\hfill \\ \hfill & +{\left({\displaystyle \frac{max\left(0,E^{min}-E_{t+1}^{\mathrm{BESS}}\right)}{E_{\mathrm{rated}}^{\mathrm{BESS}}}}\right)}^2+{\left({\displaystyle \frac{max\left(0,E_{t+1}^{\mathrm{BESS}}-E^{max}\right)}{E_{\mathrm{rated}}^{\mathrm{BESS}}}}\right)}^2],\hfill \end{array}$$

(11)

where $P_{t,\mathrm{req}}^{\mathrm{BESS}}$ is the power command requested by the optimization schedule and $P_{t,\mathrm{impl}}^{\mathrm{BESS}}$ is the feasible power implemented after enforcing the stored-energy limits. For zero- or near-zero BESS ratings, the implemented power is set to zero and non-zero requested operation is penalized through a numerical safeguard to avoid division by zero.

A terminal-energy penalty is included to avoid unrealistic one-day depletion or excessive charging of the BESS:

$$J_{\mathrm{E},\mathrm{end}}=\left|{\displaystyle \frac{E_{T+1}^{\mathrm{BESS}}-E_1^{\mathrm{BESS}}}{E_{\mathrm{rated}}^{\mathrm{BESS}}}}\right|.$$

(12)

For zero-capacity BESS candidates, the terminal-energy penalty is set to zero to avoid division by zero. This term encourages a daily operation that does not artificially use the BESS as a one-time energy source or sink.

The reactive-power capability of the BESS inverter is similarly limited by its apparent-power rating:

$$|Q_t^{\mathrm{BESS}}|\le \sqrt{{\left(S_{\mathrm{inv}}^{\mathrm{BESS}}\right)}^2-{\left(P_t^{\mathrm{BESS}}\right)}^2}.$$

(13)

Here, $S_{\mathrm{inv}}^{\mathrm{BESS}}$ is the BESS inverter apparent-power rating. In this study, it is set equal to the BESS power rating, as described in the planning-parameter settings. For numerical robustness, the square-root argument in the inverter reactive-power capability calculation is lower-bounded by zero in the implementation.

3.4. Autonomous Inverter Voltage Support

A bounded droop-like voltage-support rule is used for the PV and BESS inverters, following the common use of inverter reactive-power capability for voltage regulation in PV-rich distribution networks [5,8,9,10]. For an inverter-connected device located at bus k, the reactive-power command at hour t is defined as

$$Q_{k,t}^{\mathrm{cmd}}=K_q{\displaystyle \frac{1-|V_{k,t}|}{0.05}}{Q}_{k,t}^{max},$$

(14)

where $K_q\in [0,1]$ denotes the common inverter voltage-support gain optimized by the planning model, $|V_{k,t}|$ is the local voltage magnitude, and $Q_{k,t}^{max}$ is the available reactive-power capability of the corresponding inverter at that hour. The denominator 0.05 normalizes the response with respect to a 5% voltage deviation from the nominal value. Accordingly, $K_q=1$ requests the full available reactive capability at a 5% voltage deviation, subject to the inverter capability limit. The command is saturated within $[-Q_{k,t}^{max},Q_{k,t}^{max}]$.

Equation (14) defines the steady-state Volt/Var relationship used in the hourly load-flow simulations. The interval $K_q\in [0,1]$ is adopted as the reference range in the main benchmark. Here, $K_q=0$ disables inverter voltage support, whereas $K_q=1$ requests the full available reactive-power capability at a 5% voltage deviation, subject to the inverter capability limit. A supplementary re-optimization extends the admissible range to $K_q\in [0,1.5]$ to assess the sensitivity of the planning results to the selected upper bound. Values above unity do not increase the available inverter reactive-power capability; instead, they cause the capability limit to be reached at voltage deviations smaller than 5%.

A common value of $K_q$ is applied to the PV and BESS inverters to maintain a coordinated voltage-support setting while limiting the number of continuous planning variables. Using separate gains would provide additional control flexibility but would add another decision variable to the planning problem.

For each hourly operating point, the backward/forward sweep load flow is coupled with the inverter reactive-power calculation through an under-relaxed fixed-point iteration. The PV and BESS reactive-power commands are initialized at zero. After each load-flow solution, the target command vector is calculated from Equation (14) and updated as

$${\mathbf{Q}}^{(m+1)}={\mathbf{Q}}^{\left(m\right)}+\omega \left({\mathbf{Q}}_{\mathrm{target}}^{\left(m\right)}-{\mathbf{Q}}^{\left(m\right)}\right).$$

(15)

Here, m denotes the outer-iteration index, $\mathbf{Q}$ contains the reactive-power commands of the PV and BESS inverters, and $\omega$ is the under-relaxation factor. A fixed value of $\omega =0.70$ is used in all simulations to damp successive command updates and promote stable convergence. Within each outer iteration, the backward/forward sweep load flow is solved using a voltage tolerance of ${10}^{-8}$ p.u. and a maximum of 100 iterations. Convergence of the outer fixed-point iteration is checked from the second update onward. The iteration terminates when the normalized reactive-power residual is no greater than ${10}^{-4}$ and the maximum voltage-magnitude change is no greater than ${10}^{-6}$ p.u. A maximum of 50 outer iterations is allowed.

4. Problem Formulation

4.1. Decision Vector

The planning problem includes siting and sizing decisions, hourly BESS operation, and inverter voltage-support control. The decision vector is defined as

$$\mathbf{x}=\left[b_{\mathrm{PV}},b_{\mathrm{BESS}},P_{\mathrm{PV}},P_{\mathrm{BESS}},E_{\mathrm{BESS}},K_q,u_1,\dots ,u_T\right],$$

(16)

where $b_{\mathrm{PV}}$ and $b_{\mathrm{BESS}}$ are the selected PV and BESS bus locations, $P_{\mathrm{PV}}$ is the PV capacity, $P_{\mathrm{BESS}}$ and $E_{\mathrm{BESS}}$ are the BESS power and energy ratings, $K_q$ is the common inverter voltage-support gain, and $u_t\in [-1,1]$ is the normalized hourly BESS command. During fitness evaluation, the bus-location variables are rounded to the nearest feasible non-slack bus index. The normalized BESS command is converted into active power as

$$P_t^{\mathrm{BESS}}=u_t{P}_{\mathrm{BESS}},$$

(17)

subject to the stored-energy and SOC feasibility checks described in the BESS model.

4.2. Economic Model

The economic assessment follows a discounted life-cycle-cost formulation in which capital expenditures and future replacement costs are converted into equivalent annual values [17,18,19]. The capital recovery factor is given by

$$CRF={\displaystyle \frac{r{(1+r)}^n}{{(1+r)}^n-1}},$$

(18)

where r is the discount rate and n is the project lifetime.

The annualized investment and operation-and-maintenance cost is modeled as

$$C_{\mathrm{inv}}^{\mathrm{ann}}=CRF\left(C_{\mathrm{PV}}{P}_{\mathrm{PV}}+C_{\mathrm{BESS}}^P{P}_{\mathrm{BESS}}+C_{\mathrm{BESS}}^E{E}_{\mathrm{BESS}}\right)+C_{\mathrm{OM}}^{\mathrm{PV}}+C_{\mathrm{OM}}^{\mathrm{BESS}}.$$

(19)

Here, $C_{\mathrm{PV}}$, $C_{\mathrm{BESS}}^P$, and $C_{\mathrm{BESS}}^E$ denote the PV power, BESS power, and BESS energy investment costs, respectively. The terms $C_{\mathrm{OM}}^{\mathrm{PV}}$ and $C_{\mathrm{OM}}^{\mathrm{BESS}}$ represent the corresponding annual O&M costs. In the numerical implementation, these O&M costs are calculated as fixed annual fractions of the corresponding PV and BESS investment costs.

To account for battery degradation and replacement over the 15-year planning horizon, the present value of the future BESS energy-module replacement cost is modeled as

$$C_{\mathrm{rep}}^0={\displaystyle \frac{{\rho }_{\mathrm{rep}}{C}_{\mathrm{BESS}}^E{E}_{\mathrm{BESS}}}{{(1+r)}^{y_{\mathrm{rep}}}}},$$

(20)

where $C_{\mathrm{rep}}^0$ denotes the present value of the future replacement cost, ${\rho }_{\mathrm{rep}}$ is the replacement-cost fraction, and $y_{\mathrm{rep}}$ is the replacement year. The annualized replacement cost is then calculated as

$$C_{\mathrm{rep}}^{\mathrm{ann}}=CRF{C}_{\mathrm{rep}}^0.$$

(21)

The annual grid-energy cost is calculated by scaling the representative daily grid-energy import over one year:

$$C_{\mathrm{grid}}^{\mathrm{ann}}=365c_e{\displaystyle \sum _{t=1}^T}{P}_{\mathrm{grid},t}^{+}\Delta t,$$

(22)

where $c_e$ is the energy tariff and $P_{\mathrm{grid},t}^{+}=max(0,P_{\mathrm{grid},t})$ is the positive grid power imported from the upstream grid. The use of the factor 365 reflects the representative-day benchmark assumption.

The annual total cost is

$$C_{\mathrm{tot}}^{\mathrm{ann}}=C_{\mathrm{inv}}^{\mathrm{ann}}+C_{\mathrm{rep}}^{\mathrm{ann}}+C_{\mathrm{grid}}^{\mathrm{ann}}.$$

(23)

4.3. Voltage, Grid Power, and Feasibility Metrics

The voltage-deviation index is defined as

$$VD={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{N_b}}\left||V_{i,t}|-1\right|\Delta t.$$

(24)

The voltage-violation severity is calculated using squared deviations outside the allowable voltage band:

$$J_V={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{N_b}}\left[max(0,|V_{i,t}|-V^{max}{)}^2+max(0,V^{min}-|V_{i,t}{\left|\right)}^2\right]\Delta t.$$

(25)

This metric penalizes the depth of voltage-limit violations rather than directly counting the number of violated bus-hour intervals.

The daily grid-energy import and peak grid power are defined as

$$E_{\mathrm{grid}}={\displaystyle \sum _{t=1}^T}{P}_{\mathrm{grid},t}^{+}\Delta t,$$

(26)

and

$$P_{\mathrm{grid}}^{max}=\underset{t}{max}{P}_{\mathrm{grid},t}^{+}.$$

(27)

The stored-energy feasibility penalty $J_E$ and the terminal-energy penalty $J_{\mathrm{E},\mathrm{end}}$ are defined in Equations (11) and (12), respectively. A separate BESS-duration penalty is used to discourage unrealistic combinations of BESS power and energy ratings. For candidates with positive BESS power and energy ratings, the nominal duration is

$$D_{\mathrm{BESS}}={\displaystyle \frac{E_{\mathrm{BESS}}}{P_{\mathrm{BESS}}}},$$

(28)

and the duration penalty is defined as

$$J_D=\left\{\begin{array}{cc}{\left[max\left(0,1-D_{\mathrm{BESS}}\right)\right]}^2+0.1{\left[max\left(0,D_{\mathrm{BESS}}-6\right)\right]}^2,\hfill & P_{\mathrm{BESS}}>ϵ,E_{\mathrm{BESS}}>ϵ,\hfill \\ 10,\hfill & P_{\mathrm{BESS}}>ϵ,E_{\mathrm{BESS}}\le ϵ,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(29)

where $ϵ={10}^{-6}$ is the numerical threshold used to identify negligible ratings. Thus, durations between 1 and 6 h receive no duration penalty, while shorter and longer designs are penalized according to Equation (29).

4.4. Objective Function

The optimization minimizes a normalized weighted objective that combines economic and technical performance terms:

$$\begin{array}{cc}\hfill minF\left(\mathbf{x}\right)=& w_C{\displaystyle \frac{C_{\mathrm{tot}}^{\mathrm{ann}}}{C_{\mathrm{tot},\mathrm{base}}^{\mathrm{ann}}}}+w_L{\displaystyle \frac{E_{\mathrm{loss}}}{E_{\mathrm{loss},\mathrm{base}}}}+w_{VD}{\displaystyle \frac{VD}{V{D}_{\mathrm{base}}}}\hfill \\ \hfill & +w_P{\displaystyle \frac{P_{\mathrm{grid}}^{\max }}{P_{\mathrm{grid},\mathrm{base}}^{\max }}}+w_E{\displaystyle \frac{E_{\mathrm{grid}}}{E_{\mathrm{grid},\mathrm{base}}}}+{\lambda }_V{J}_V+{\lambda }_E{J}_E+{\lambda }_{end}{J}_{\mathrm{E},\mathrm{end}}+{\lambda }_D{J}_D.\hfill \end{array}$$

(30)

The weights used in the base simulation are $w_C=0.40$, $w_L=0.20$, $w_{VD}=0.20$, $w_P=0.10$, and $w_E=0.10$. They are planning-preference parameters rather than universal coefficients. Because all five principal terms are normalized by their base-case values, these weights express the relative emphasis placed on annual cost, active-loss energy, voltage deviation, peak grid power, and imported energy. The base setting gives annualized cost the largest individual share while reserving 60% of the aggregate weight for network-performance terms. Section 8.1 compares cost-oriented, balanced, and network-oriented settings; therefore, the base solution represents one balanced decision preference rather than a universally optimal design.

5. Metaheuristic Algorithms

We compared five metaheuristic algorithms. DE is used as a classical evolutionary baseline [26]. ADE is implemented as a JADE-inspired adaptive DE variant with current-to-pbest mutation and adaptive mutation and crossover parameters [30,31,32]. PSO is used as a swarm intelligence baseline [27]. GWO is implemented with explicit historical alpha, beta, and delta wolves, preventing elite solutions from being overwritten by subsequent population movements [28]. MFO is implemented with flame memory by merging previous flames and current moths, retaining the best candidates as the new flame set [29].

6. Simulation Setup

6.1. Test Systems and Operating Profiles

The proposed framework is evaluated on the 33-bus radial distribution system, which consists of 33 buses and 32 radial branches. The branch and load data are taken from the benchmark system reported by Baran and Wu [15]. The base experiment uses a 24 h normalized load profile and a normalized PV-generation profile with a simulation step of $\Delta t=1$ h. The profiles are synthetic benchmark profiles constructed for controlled algorithm comparison. The load profile is specified directly by 24 normalized hourly values, while the PV profile is generated as a smooth normalized sinusoidal curve between hours 7 and 18 and is zero at night. Their exact numerical values are provided in Supplementary Data S1 load_pv_profiles.csv. These synthetic profiles are used as benchmark inputs for the controlled comparisons. The normalized profiles are shown in Figure 1.

To assess robustness beyond the base representative day, four synthetic operating profiles are considered in Section 8.4: winter/high-load, spring/moderate, summer/high-PV, and intermittent/cloudy. The best ADE asset variables—PV and BESS locations and ratings together with $K_q$—are fixed across all profiles, while a separate 24 h BESS schedule is optimized for each profile. The schedules obtained for the four profiles are combined into a common candidate set and cross-evaluated under each operating profile. For each profile, the schedule yielding the best performance among the available candidates is selected for reporting. All four profiles are assigned equal weights of 0.25 in the aggregate robustness assessment. The exact hourly multipliers are provided as Supplementary Data S2 in representative_day_profiles.csv. The intermittent profile represents an hourly planning stress case with variable PV output rather than sub-hourly cloud dynamics.

The proposed framework is also tested on the 69-bus radial distribution system using branch and load data from the benchmark system reported by Baran and Wu [35]. This feeder is used to examine the applicability of the model to a larger feeder rather than to repeat the complete five-algorithm comparison. Unless otherwise stated, the same normalized base profiles are used to maintain consistent operating assumptions between the two feeder tests.

All simulations were implemented in MATLAB R2024b Update 6 and executed on a computer with an Intel Core i5-1135G7 processor and 16 GB of RAM. Reported runtimes were measured under the same computational environment.

6.2. Planning Parameters

For the 33-bus benchmark, the PV, BESS power, and BESS energy bounds are set to 0–1500 kW, 0–1000 kW, and 0–3000 kWh, respectively. Relative to the no-PV/no-BESS peak grid demand of 4346.181 kW, the PV and BESS power upper bounds correspond to approximately 34.5% and 23.0%, while the energy-to-power bounds allow a maximum nominal duration of 3 h. For the larger 69-bus validation feeder, the principal bounds are 0–2000 kW, 0–1500 kW, and 0–4500 kWh. Relative to its base-case peak grid demand of 4459.481 kW, these correspond to approximately 44.8%, 33.6%, and 3 h, respectively. Because the 2000 kW solution approaches the PV limit, a supplementary sensitivity increases only the PV upper bound to 2500 kW, corresponding to approximately 56.1% of base-case peak demand; the BESS and voltage-support gain bounds remain unchanged.

The initial stored energy is set to 50% of rated energy, and the allowable SOC range is set to 15–95%. Charging and discharging efficiencies are both set to 95%. The PV inverter oversizing factor is 1.10, while the BESS inverter rating is equal to the BESS power rating.

The inverter apparent-power ratings are not treated as additional decision variables in this study. Instead, the PV inverter oversizing ratio and the BESS inverter rating are fixed according to common planning assumptions and commercially practical sizing conventions. This choice keeps the focus on PV–BESS siting, sizing, storage scheduling, and common inverter voltage-support gain optimization while avoiding an additional inverter-sizing layer that would require a separate techno-economic component model.

The economic assumptions include a 15-year project horizon, an 8% discount rate, PV CAPEX of 700 USD/kW, BESS power CAPEX of 160 USD/kW, BESS energy CAPEX of 250 USD/kWh in the base case, and an electricity tariff of 0.15 USD/kWh. A BESS replacement cost is included in year 8 and set to 60% of the initial BESS energy-module cost. The adopted storage-cost assumptions are consistent with a recent utility-scale battery cost projection [33].

6.3. Algorithm Parameters

The base experiment uses 30 independent runs, a population size of 40, and 300 iterations. The DE mutation factor and crossover rate are set to 0.60 and 0.90, respectively. PSO uses a linearly decreasing inertia weight from 0.90 to 0.40 and acceleration coefficients $c_1=c_2=1.80$.

Table 2. Main algorithm parameters used in the comparative study.

Algorithm	Parameter	Value
All algorithms	Population size	40
All algorithms	Maximum iterations	300
All algorithms	Independent runs	30
DE	Mutation factor F	0.60
DE	Crossover rate $CR$	0.90
PSO	Inertia weight	Linearly decreased from 0.90 to 0.40
PSO	Acceleration coefficients	$c_1=c_2=1.80$
ADE	Initial adaptive parameters	${\mu }_F=0.50$, ${\mu }_{CR}=0.80$
ADE	Adaptation rate	$c=0.10$
ADE	Top-solution fraction	$p=0.20$
GWO	Control parameter a	Linearly decreased from 2 to 0
MFO	Spiral constant b	1

summarizes the main algorithm parameters used in the comparative study. The same population size, number of iterations, and number of independent runs are used for all algorithms to ensure a fair comparison.

7. Results

7.1. Base Case

Table 3. Base-case results without PV and BESS.

Metric	Value
Daily active-loss energy	3706.318 kWh
Daily grid-energy import	78,675.018 kWh
Peak grid power	4346.181 kW
Voltage-deviation index	36.0642
Voltage violation duration	364 bus-h
Annual total cost	4,307,457.23 USD/year

presents the no-PV/no-BESS base-case results. The system exhibits significant voltage-limit duration, with 364 bus-hours of voltage violations over the daily horizon.

7.2. Algorithm Statistics

Table 4. Algorithm statistics over 30 independent runs using the inverter fixed-point implementation.

Algorithm	Best	Mean	Worst	Std.	Median	Mean Runtime (s)
DE	0.8595	1.3910	1.8960	0.2481	1.3869	427.85
ADE	0.7557	0.9564	3.5983	0.6966	0.7673	507.42
PSO	0.7787	2.6960	6.1657	2.0794	1.4136	399.24
GWO	0.7626	0.9448	3.5087	0.5033	0.8278	485.16
MFO	0.8035	1.5483	5.5458	1.0192	1.2079	640.88

summarizes the final-fitness statistics over 30 independent runs. ADE obtains the lowest best and median fitness values, whereas GWO achieves the lowest mean fitness. DE has the smallest standard deviation, indicating the lowest run-to-run variability, although its best, mean, and median fitness values are higher than those of ADE and GWO. PSO exhibits the largest variability, while MFO has the longest mean runtime. Overall, ADE provides the best individual and median outcomes, GWO achieves the strongest mean performance, and DE produces the most consistent results across independent runs.

The fixed-point iteration converged for all 120 hourly operating points associated with the five best algorithm-specific solutions. The maximum number of outer iterations was 23, and all final residuals satisfied the prescribed convergence tolerances.

Figure 2 and Figure 3 further illustrate the convergence behavior and final-solution distribution of the tested algorithms.

7.3. Best Planning Solutions

Table 5. Best PV–BESS planning solution obtained by each algorithm using the inverter fixed-point implementation.

Algorithm	PV Bus	PV (kW)	BESS Bus	BESS P (kW)	BESS E (kWh)	${\mathit{K}}_{\mathit{q}}$	Fitness	Annual Cost (USD/year)
DE	13	1320.96	33	642.27	2511.56	0.901	0.8595	3,974,484.17
ADE	13	1498.30	31	888.43	2784.62	1.000	0.7557	3,946,672.12
PSO	14	1167.46	32	836.01	1579.11	0.924	0.7787	3,983,698.92
GWO	13	1470.83	31	971.65	2474.96	0.991	0.7626	3,940,490.83
MFO	13	1488.70	33	652.37	2103.29	0.995	0.8035	3,915,794.05

reports the best planning solution obtained by each algorithm. The selected PV locations are concentrated at buses 13 and 14, with capacities ranging from 1167.46 to 1498.30 kW, whereas the BESS is consistently placed at downstream buses 31–33, with power and energy ratings ranging from 642.27 to 971.65 kW and from 1579.11 to 2784.62 kWh, respectively. The optimized common voltage-support gain remains close to its upper bound in all solutions, with $K_q$ values between 0.901 and 1.000. ADE achieves the lowest objective value of 0.7557, followed closely by GWO at 0.7626, whereas MFO produces the lowest annual cost of 3,915,794.05 USD/year. These results show that the lowest weighted objective and the lowest annual cost are obtained by different plans, reflecting the trade-off between economic and network-performance objectives.

7.4. Statistical Significance Analysis

To assess whether the performance differences are statistically meaningful, non-parametric tests were applied to the 30 independent final best-fitness values for each algorithm. Because the algorithm-specific runs use independent random seeds, the resulting samples are treated as independent. Therefore, a Kruskal–Wallis test followed by pairwise Wilcoxon rank-sum (Mann–Whitney U) tests is used instead of paired non-parametric tests [13,14]. The Kruskal–Wallis test gives $H=78.961$ with 4 degrees of freedom and $p=2.89\times {10}^{-16}$, indicating an overall significant difference. Holm–Bonferroni adjustment is applied to the ten pairwise comparisons. The adjusted pairwise results are reported in

Table 6. Pairwise Wilcoxon rank-sum test results with Holm–Bonferroni correction for the final best-fitness values.

Algorithm Pair	Raw p	Holm-Adjusted p	Significant at 5%?
DE–ADE	$8.48\times {10}^{-9}$	$6.79\times {10}^{-8}$	Yes
DE–PSO	0.9470	0.9470	No
DE–GWO	$7.12\times {10}^{-9}$	$6.41\times {10}^{-8}$	Yes
DE–MFO	0.2062	0.6186	No
ADE–PSO	$2.67\times {10}^{-9}$	$2.67\times {10}^{-8}$	Yes
ADE–GWO	$1.09\times {10}^{-5}$	$4.36\times {10}^{-5}$	Yes
ADE–MFO	$1.20\times {10}^{-8}$	$8.42\times {10}^{-8}$	Yes
PSO–GWO	$1.49\times {10}^{-6}$	$7.46\times {10}^{-6}$	Yes
PSO–MFO	0.3255	0.6511	No
GWO–MFO	$5.60\times {10}^{-7}$	$3.36\times {10}^{-6}$	Yes

. ADE differs significantly from DE, PSO, GWO, and MFO after Holm–Bonferroni correction. These results indicate that the final-fitness distribution obtained by ADE differs significantly from those of the other four algorithms. The descriptive statistics additionally show that ADE achieves the lowest best and median values, GWO achieves the lowest mean value, and DE has the smallest standard deviation.

7.5. Technical Performance of the Best Solutions

Table 7. Technical performance of the best planning solutions.

Algorithm	Loss (kWh)	Grid Energy (kWh)	Peak Grid Power (kW)	Voltage Deviation	Voltage Violation Duration (bus h)
Base	3706.318	78675.018	4346.181	36.0642	364
DE	2249.955	68106.503	4077.752	18.8408	21
ADE	2296.986	66983.346	3803.905	16.6862	0
PSO	2220.910	69199.891	3976.939	18.6928	6
GWO	2274.028	67130.502	4183.781	16.6012	1
MFO	2347.842	67073.459	4012.931	17.5743	14

compares the technical performance of the best planning solution obtained by each algorithm with that of the base case. All five optimized solutions improve the reported technical metrics relative to the base case. PSO achieves the lowest active-loss energy, whereas ADE produces the lowest grid-energy import and peak grid power and eliminates all voltage-limit violations. GWO achieves the lowest voltage-deviation index and retains only 1 bus h of voltage violations. These results demonstrate a clear trade-off among the technical performance metrics, as no single planning solution achieves the best value for every measure.

Figure 4 shows the daily voltage profile obtained with the best ADE solution. Under the fixed-point implementation, the ADE solution eliminates all voltage-limit bus-hour violations over the representative day.

Figure 5 and Figure 6 show the storage operation and grid-power behavior associated with the best ADE solution.

For the best ADE solution, the rated BESS energy is 2784.62 kWh and the initial stored energy is 1392.3114 kWh. The final, minimum, and maximum stored-energy values are 1392.3100, 418.7451, and 2330.4417 kWh, respectively. The terminal mismatch is 0.00144 kWh, or $5.16\times {10}^{-5}$% of rated energy. The minimum and maximum normalized SOC values are 15.0378% and 83.6897%, respectively, confirming compliance with the 15–95% limits.

Compared with the base case, the best ADE solution reduces daily active-loss energy from 3706.318 kWh to 2296.986 kWh (38.0%), grid-energy import from 78,675.018 kWh to 66,983.346 kWh (14.9%), peak grid power from 4346.181 kW to 3803.905 kW (12.5%), voltage deviation from 36.0642 to 16.6862 (53.7%), and voltage-violation duration from 364 bus h to 0 bus h (100%). Annual total cost decreases from 4,307,457.23 to 3,946,672.12 USD/year (8.4%).

Table 8. Percentage improvement of the best ADE solution relative to the base case.

Metric	Base Case	ADE Best	Improvement
Daily active-loss energy	3706.318 kWh	2296.986 kWh	38.0%
Daily grid-energy import	78,675.018 kWh	66,983.346 kWh	14.9%
Peak grid power	4346.181 kW	3803.905 kW	12.5%
Voltage-deviation index	36.0642	16.6862	53.7%
Voltage violation duration	364 bus-h	0 bus-h	100%
Annual total cost	4,307,457.23	3,946,672.12	8.4%

summarizes these improvements.

7.6. Contribution of Inverter-Based Voltage Support

To quantify the specific contribution of inverter-based reactive-power support, the planning problem is solved separately under two settings: (i) the common voltage-support gain is fixed at $K_q=0$, for which PV and BESS inverters operate without autonomous reactive-power support, and (ii) $K_q$ is optimized as a planning variable. All PV and BESS siting, sizing, and hourly scheduling variables are re-optimized in each setting using ADE under the same computational budget. Consequently, the comparison does not merely disable reactive power in an already optimized plan; it compares the best-found plan available to each control setting. The resulting plans and performance metrics are reported in

Table 9. Comparative assessment of PV–BESS planning with and without inverter-based reactive-power support.

Case	PV Bus	PV (kW)	BESS Bus	BESS P (kW)	BESS E (kWh)	${\mathit{K}}_{\mathit{q}}$	Annual Cost (USD/year)	Loss (kWh)	Voltage Deviation	Violation Duration (bus h)
No PV/no BESS	–	0	–	0	0	–	4,307,457.23	3706.318	36.0642	364
PV–BESS, no inverter support	17	1499.98	18	511.53	2999.80	0	3,980,696.19	2955.621	28.0976	235
PV–BESS, optimized inverter support	13	1498.30	31	888.43	2784.62	1.000	3,946,672.12	2296.986	16.6862	0

.

Relative to the separately optimized $K_q=0$ case, the separately optimized case with inverter support reduces daily active-loss energy from 2955.621 to 2296.986 kWh (22.3%), grid-energy import from 67,581.025 to 66,983.346 kWh (0.9%), the voltage-deviation index from 28.0976 to 16.6862 (40.6%), voltage-violation duration from 235 to 0 bus h (100%), and annual total cost from 3,980,696.19 to 3,946,672.12 USD/year (0.9%). Peak grid power increases slightly from 3789.747 to 3803.905 kW (0.4%). Thus, allowing inverter support within the planning optimization provides substantial improvements in active-loss energy and voltage performance, although it does not improve every individual metric. The best-found asset allocation also changes: the $K_q=0$ solution places the PV and BESS at buses 17 and 18, respectively, whereas the optimized-support solution places them at buses 13 and 31. These results show that including inverter-based voltage support affects both the resulting system performance and the preferred PV–BESS planning configuration. The corresponding minimum-voltage envelopes are shown in Figure 7.

Because the best solutions frequently approach $K_q=1$, an additional ADE experiment re-optimizes all siting, sizing, voltage-support gain, and hourly BESS-scheduling variables after extending the feasible range to $K_q\in [0,1.5]$. The same 30 random seeds, population size, and iteration limit used for the ADE benchmark are retained.

Table 10. Sensitivity to the feasible range of the inverter voltage-support gain.

${\mathit{K}}_{\mathit{q}}$ Range	Best ${\mathit{K}}_{\mathit{q}}$	Fitness	Voltage Deviation	Violation Severity	Violation Duration (bus h)
$[0,1]$	0.9997	0.7557	16.6862	0	0
$[0,1.5]$	1.4747	0.7459	15.8716	0	0

summarizes the best solutions obtained under the two gain ranges.

The extended range produces a best value of $K_q=1.4747$, indicating a preference for a higher voltage-support gain when the allowable range is extended beyond the normalized base interval. Relative to the best $K_q\le 1$ plan, the best extended-range plan improves fitness by 1.29%, annual total cost by 1.57%, active-loss energy by 1.96%, grid-energy import by 0.12%, and voltage deviation by 4.88%. Both plans eliminate voltage violations, while peak grid power increases by 3.50% in the extended-range case. The matched-seed median fitness decreases from 0.7673 to 0.7599, but the paired Wilcoxon signed-rank test gives $p=0.4165$; moreover, the extended search has a higher mean fitness and greater run-to-run variability. Therefore, the wider range provides a modest best-case improvement but does not establish a statistically consistent advantage across the 30 matched runs. The main benchmark retains $K_q\in [0,1]$ as a conservative normalized design range, and the $[0,1.5]$ result is reported as an explicit sensitivity check rather than replacing the five-algorithm benchmark. Since $K_q>1$ causes the reactive-power command to reach saturation at a voltage deviation smaller than 5%, the check also clarifies the physical meaning of the upper gain bound. Figure 8 compares the resulting minimum-voltage envelopes.

7.7. Discussion of Algorithm Behavior

The results do not indicate that a single algorithm is superior across all statistical and performance measures. ADE obtains the best individual solution and the lowest median fitness, GWO achieves the lowest mean fitness, and DE has the smallest standard deviation. The Holm-adjusted rank-sum tests show significant differences between ADE and each of the other algorithms, including GWO. Among the best planning solutions, MFO produces the lowest annual cost but retains 14 bus h of voltage violations, whereas ADE eliminates all violations and achieves the lowest peak grid power. These findings highlight the distinction between performance under the weighted planning objective and performance according to individual economic or technical metrics.

The optimized common inverter voltage-support gain is close to unity in almost all best solutions. The contribution analysis confirms the value of inverter-based voltage support. The $K_q\in [0,1.5]$ re-optimization gives a modestly better best solution with $K_q=1.4747$ but no significant matched-run advantage and greater variability. This supports retaining the normalized $[0,1]$ range for the principal benchmark while reporting the wider range transparently as a sensitivity analysis of the selected upper bound on $K_q$. The BESS is repeatedly located near downstream buses, especially buses 31–33, confirming that storage placement near voltage-sensitive feeder sections has strong planning value.

The five selected optimizers provide representative evolutionary and swarm-based baselines evaluated under a common computational budget. Within this benchmark, ADE achieves the lowest best and median fitness values, GWO achieves the lowest mean fitness, and DE exhibits the smallest standard deviation. Strong differential-evolution benchmarks such as L-SHADE and jSO may be included in future full-budget comparative studies [32,36].

7.8. Validation on the 69-Bus System

The 69-bus study applies the same explicitly converged under-relaxed inverter fixed-point model, objective formulation, and base weight setting used for the 33-bus benchmark. ADE is evaluated over 30 independent runs, with a population size of 40 and 300 iterations per run. The study assesses the applicability of the planning formulation to a larger radial feeder using the algorithm selected from the 33-bus benchmark; accordingly, it constitutes an ADE-based larger-feeder validation rather than a second five-algorithm comparison. The selected plan is reported in

Table 11. ADE-based PV–BESS planning solution for the 69-bus validation case under the adopted 2000 kW PV upper bound.

System	PV Bus	PV (kW)	BESS Bus	BESS P (kW)	BESS E (kWh)	${\mathit{K}}_{\mathit{q}}$
69-bus	62	1993.90	64	674.35	2841.34	0.7720

, and its performance relative to the no-PV/no-BESS base case is summarized in

Table 12. 69-bus validation results obtained using ADE under the adopted 2000 kW PV upper bound.

Metric	Base Case	ADE PV–BESS Planning	Improvement
Daily active-loss energy	3946.305 kWh	1924.693 kWh	51.23%
Daily grid-energy import	80,668.445 kWh	64,896.875 kWh	19.55%
Peak grid power	4459.481 kW	3991.087 kW	10.50%
Voltage-deviation index	36.7271	22.2172	39.51%
Voltage violation duration	166 bus-h	2 bus-h	98.80%
Annual total cost	4,416,597.37 USD/year	3,875,962.53 USD/year	12.24%

.

Across the 30 ADE runs, the best, median, and mean final objective values are 0.739807, 0.757021, and 0.825653, respectively, with a standard deviation of 0.198340. The selected plan retains only two isolated bus–hour voltage violations, with an aggregate squared violation severity of $3.85\times {10}^{-7}\mathrm{p}.\mathrm{u}{.}^2·\mathrm{bus}-\mathrm{h}$. All 24 hourly fixed-point calculations converge. The maximum number of outer iterations is 8, while the maximum final normalized reactive-power residual and voltage change are $8.36\times {10}^{-5}$ and $8.09\times {10}^{-7}$ p.u., respectively, both below the adopted convergence tolerances. The BESS SOC changes from 50% at the beginning of the daily horizon to 50.0002% at the end, corresponding to a terminal stored-energy mismatch of 0.0059 kWh. The corresponding daily voltage envelope is shown in Figure 9.

The 69-bus results show that the mixed-discrete planning formulation and fixed-point simulation framework can be successfully applied to a larger radial feeder, yielding substantial improvements over the base case across all reported metrics. Because the selected PV capacity of 1993.90 kW approaches the adopted 2000 kW upper bound, Section 8.5 presents an additional upper-bound sensitivity analysis. The 69-bus study therefore supports the applicability of the formulation and simulation workflow to a larger feeder, while the comparative ranking of the five optimizers remains specific to the 33-bus benchmark.

8. Sensitivity Analysis

The objective-weight sensitivity is performed using ADE and the same explicitly converged under-relaxed fixed-point implementation as in the main benchmark. The cost-oriented and network-oriented settings are evaluated using 10 matched seeds, a population of 40, and 300 iterations, while the balanced setting uses the corresponding ADE runs from the main benchmark. For each non-balanced setting, the best available feasible ADE solution evaluated under the corresponding weight vector is included in the initial population, together with seed-specific randomly generated candidates. This initialization ensures that each run begins with at least one verified feasible solution under its own objective setting.

The voltage-penalty sensitivity is evaluated using the same fixed-point implementation. For the BESS energy-CAPEX sensitivity, the candidates obtained under all CAPEX settings are combined into a common candidate set and cross-evaluated under each setting, thereby avoiding dependence on scenario-processing order. The representative-day robustness analysis also uses the same fixed-point implementation. The 69-bus validation includes an additional sensitivity analysis using a 2500 kW PV upper bound.

8.1. Objective-Weight Sensitivity

The weighted-sum formulation reflects planning preferences. To examine how those preferences affect the selected design, three normalized weight settings are evaluated: cost-oriented, balanced, and network-oriented. In every case, the five principal weights sum to one; feasibility-penalty coefficients are held fixed. The weight vectors are listed in

Table 13. Objective-weight settings used in the planning-preference analysis.

Scenario	${\mathit{w}}_{\mathit{C}}$	${\mathit{w}}_{\mathit{L}}$	${\mathit{w}}_{\mathit{V}\mathit{D}}$	${\mathit{w}}_{\mathit{P}}$	${\mathit{w}}_{\mathit{E}}$
Cost-oriented	0.60	0.15	0.10	0.075	0.075
Balanced	0.40	0.20	0.20	0.10	0.10
Network-oriented	0.20	0.25	0.35	0.10	0.10

, and the resulting best plans are reported in

Table 14. Effect of objective-weight preferences on the best ADE-based planning solutions.

Scenario	PV Bus	PV (kW)	BESS Bus	BESS P (kW)	BESS E (kWh)	${\mathit{K}}_{\mathit{q}}$	Annual Cost (USD/year)	Loss (kWh)	Voltage Deviation	Peak Power (kW)	Violation Duration (bus h)
Cost-oriented	13	1490.79	31	702.56	2213.54	0.9266	3,916,168.46	2262.080	17.7084	3791.634	0
Balanced	13	1498.30	31	888.43	2784.62	0.9997	3,946,672.12	2296.986	16.6862	3803.905	0
Network-oriented	14	1466.04	31	888.90	2774.84	0.9881	3,951,068.53	2256.255	16.6939	3774.340	0

.

Figure 10 compares the corresponding technical and economic metrics after normalization by the balanced solution.

Relative to the balanced plan, the cost-oriented plan reduces annual cost by 0.77%, active-loss energy by 1.52%, and peak grid power by 0.32% while increasing the voltage-deviation index by 6.13%. It also selects a smaller BESS, with the power and energy ratings decreasing from 888.43 kW and 2784.62 kWh to 702.56 kW and 2213.54 kWh, respectively. The network-oriented plan reduces active-loss energy by 1.77% and peak grid power by 0.78%, with a 0.11% increase in annual cost. All three plans yield zero voltage-violation duration. When evaluated using the balanced weight vector, the network-oriented candidate attains an objective value of 0.7536, compared with 0.7557 for the best ADE solution obtained in the equal-budget main benchmark. Because the former candidate was identified during the additional objective-weight sensitivity runs, it is reported within the sensitivity analysis and does not alter the five-algorithm comparison conducted under the common computational budget. Overall, the results show that the selected weight vector affects both the asset-sizing decisions and the resulting economic and technical performance, confirming the preference-dependent nature of the planning solution.

8.2. Voltage-Penalty Sensitivity

The squared voltage-violation penalty was examined from zero to the adopted base value, ${\lambda }_V\in \{0,50,500\}$, using 10 matched ADE runs per setting, a population of 40, and 300 iterations. All three settings used the same inverter fixed-point implementation and verified feasible-initialization procedure. The remaining objective weights and feasibility-penalty coefficients were held fixed.

Table 15. Verified low-to-base voltage-penalty sensitivity using ADE.

${\mathit{\lambda }}_{\mathit{V}}$	Fitness	Annual Cost (USD/Year)	Loss (kWh)	Peak Grid Power (kW)	Voltage Deviation	Violation Severity ($\mathbf{p}.\mathbf{u}{.}^2$ bus h)	Violation Duration (bus h)
0	0.754804	3,946,138.87	2247.092	3808.236	17.0078	$1.982\times {10}^{-5}$	7
50	0.754225	3,935,811.40	2087.256	3854.504	17.9296	$5.694\times {10}^{-5}$	18
500	0.751672	3,949,058.11	2267.681	3772.925	16.3512	0	0

reports the best plan obtained directly under each setting, while Figure 11 compares the corresponding normalized economic and network metrics.

With ${\lambda }_V=0$, the best directly obtained plan retains 7 bus h of voltage violations. The best plan obtained with ${\lambda }_V=50$ yields lower annual cost and active-loss energy but retains 18 bus h of small-magnitude voltage-limit violations, reflecting a different techno-economic trade-off. In contrast, the best plan obtained with the adopted ${\lambda }_V=500$ setting has zero squared violation severity and zero violation duration and also achieves the lowest peak grid power and voltage-deviation index among the three reported plans. The non-monotonic change between ${\lambda }_V=0$ and 50 reflects the stochastic-search variability of the independently optimized multi-term problems rather than a direct deterioration caused by the penalty coefficient. Cross-evaluation supports this interpretation: when both candidates are evaluated using ${\lambda }_V=50$, the zero-violation candidate obtained with ${\lambda }_V=500$ achieves a lower objective value than the candidate obtained directly with ${\lambda }_V=50$. Accordingly, ${\lambda }_V=500$ is retained as a conservative feasibility-oriented setting that provides sufficient selection pressure to obtain a violation-free plan in the studied benchmark, rather than as a universal threshold.

8.3. BESS Energy-CAPEX Sensitivity

The BESS energy-module CAPEX is varied from 200 to 300 USD/kWh around the base value of 250 USD/kWh, while all other economic, technical, and penalty settings are held fixed. Ten matched ADE runs are conducted for each CAPEX level using the same fixed-point implementation. To avoid dependence on the order in which the CAPEX scenarios are processed, all plans obtained from the main ADE benchmark and the CAPEX-specific runs are combined into a common candidate set. After duplicate plans are removed, 56 unique candidates are re-evaluated under each of the three CAPEX values, and the candidate with the lowest objective value is selected for each setting. The resulting cross-evaluated plans therefore represent the best solutions identified within the common candidate set. The results are reported in

Table 16. Cross-evaluated BESS energy-CAPEX sensitivity results based on the common ADE candidate set.

BESS Energy CAPEX	PV (kW)	BESS P (kW)	BESS E (kWh)	Fitness	Annual Cost (USD/year)	Loss (kWh)	Voltage Violation Duration (bus h)
200 USD/kWh	1497.744	999.793	2755.248	0.746891	3,917,924.36	2210.645	2
250 USD/kWh	1497.744	999.793	2755.248	0.749126	3,941,991.60	2210.645	2
300 USD/kWh	1497.744	999.793	2755.248	0.751361	3,966,058.85	2210.645	2

.

Figure 12 presents the corresponding economic and technical metrics normalized by the base CAPEX case.

The same plan is selected from the common candidate set at all three CAPEX levels: a 1497.74 kW PV system at bus 13, a 999.79 kW/2755.25 kWh BESS at bus 30, and $K_q=0.9980$. Consequently, active-loss energy, grid-energy import, peak grid power, voltage deviation, and violation duration remain unchanged across the three CAPEX settings. Relative to the 250 USD/kWh case, annual cost decreases by 0.61% at 200 USD/kWh and increases by 0.61% at 300 USD/kWh. The corresponding objective value decreases and increases by approximately 0.30%, respectively. The selected plan retains 2 bus h of voltage violations, with a very small aggregate squared violation severity of $6.445\times {10}^{-8}\mathrm{p}.\mathrm{u}{.}^2·\mathrm{bus}-\mathrm{h}$. Thus, within the tested $\pm 20\%$ CAPEX range, the change in the BESS energy cost does not alter the ranking of the plans in the common candidate set or the selected asset configuration. Wider storage-cost ranges or different tariff and degradation assumptions may lead to different planning decisions.

8.4. Robustness Across Representative Days

The best ADE asset configuration is retained across four synthetic representative days: winter/high-load, spring/moderate, summer/high-PV, and intermittent/cloudy. Accordingly, the PV location and capacity, BESS location and power and energy ratings, and common inverter voltage-support gain are fixed at bus 13 and 1498.30 kW, bus 31 and 888.43 kW/2784.62 kWh, and $K_q=0.9997$, respectively. Only the 24 h BESS schedule is re-optimized for each operating profile. Ten matched ADE runs are conducted for each profile, after which all generated schedules are combined into a common candidate set and cross-evaluated under all four profiles. For each profile, the schedule yielding the lowest objective value within this common set is selected. This procedure avoids dependence on the order in which the profile-specific searches are performed and allows a schedule generated under one profile to be selected under another when it provides better performance. The exact hourly profile multipliers are provided in Supplementary Data S2, and the resulting profile-specific performance is reported in

Table 17. Performance of the fixed ADE asset configuration using cross-evaluated, profile-specific BESS schedules across four representative operating days.

Representative Day	Weight	Annual Cost (USD/Year)	Grid Energy (kWh)	Loss (kWh)	Peak Power (kW)	Voltage Deviation	Voltage Violation Duration (bus h)
Winter/high-load	0.25	4,869,098.65	83,831.332	3170.388	4400.717	21.0939	38
Spring/moderate	0.25	3,595,410.02	60,567.597	1815.592	3253.012	15.1217	0
Summer/high-PV	0.25	3,999,140.58	67,941.675	2469.055	4412.543	17.0209	19
Intermittent/cloudy	0.25	4,164,982.17	70,970.748	2337.516	3574.040	17.8490	1
Equal-weight aggregate	1.00	4,157,157.86	70,827.838	2448.138	3910.078	17.7714	14.5

.

Figure 13 compares the normalized performance across the four representative operating days.

Relative to the corresponding no-PV/no-BESS cases, the equal-weight aggregate annualized cost decreases by 5.34%, active-loss energy by 37.84%, grid-energy import by 11.70%, peak grid power by 15.37%, and voltage deviation by 51.71%. The equal-weight average violation duration decreases from 359 to 14.5 bus h, corresponding to a reduction of approximately 96.0%. The spring/moderate profile is voltage-feasible, with zero bus-hour violations. The intermittent/cloudy profile retains only 1 bus h of violations, with a very small aggregate squared severity of $6.38\times {10}^{-8}\mathrm{p}.\mathrm{u}{.}^2·\mathrm{bus}-\mathrm{h}$. The more demanding winter/high-load and summer/high-PV profiles retain 38 and 19 bus h of violations, respectively, with minimum bus-voltage magnitudes of approximately 0.94795 and 0.94682 p.u. over the 24 h horizon. Thus, retaining the base-day asset configuration while adapting only the daily BESS schedule provides substantial equal-weight aggregate benefits across the four profiles, although it does not eliminate every voltage excursion under the most demanding synthetic conditions. All hourly fixed-point calculations converge, the maximum final voltage change remains below ${10}^{-6}$ p.u., and the end-of-day SOC returns to approximately its initial 50% value for every profile.

The four profiles are assigned equal weights solely for the aggregate comparison and do not constitute a chronological annual representation. The analysis therefore supports the robustness of the fixed asset configuration across the tested operating profiles, but it does not replace an 8760 h simulation, probabilistic profile modeling, sub-hourly cloud-transient analysis, or cycle-dependent battery-aging assessment.

8.5. Selected Upper-Bound Sensitivity Checks

Because the optimized 33-bus voltage-support gain and the selected 69-bus PV capacity lie close to their adopted upper bounds, supplementary upper-bound sensitivity checks are conducted.

Table 18. Effects of selected upper-bound extensions on the best-found planning solutions.

Case	Original Bound	Extended Bound	Best Decision	Main Performance Change
33-bus voltage-support gain	$K_q\le 1.0$	$K_q\le 1.5$	$K_q=1.4747$	Best fitness $-1.29\%$; voltage deviation $-4.88\%$; peak $+3.50\%$
69-bus PV capacity	$P_{\mathrm{PV}}\le 2000$ kW	$P_{\mathrm{PV}}\le 2500$ kW	$P_{\mathrm{PV}}=2490.21$ kW	Best fitness $-3.80\%$; annual cost $-3.51\%$; grid import $-5.34\%$; peak $-8.05\%$

summarizes the original and extended bounds, the resulting best-found decisions, and the principal changes in performance.

For the 33-bus voltage-support gain check, extending the range from $K_q\le 1$ to $K_q\le 1.5$ reduces the best objective value by 1.29%, while both best-found plans retain zero voltage violations. The extended-range solution selects $K_q=1.4747$, which lies close to the new upper bound. The original $K_q\le 1$ setting is therefore best interpreted as a conservative limit on the allowable voltage-support-gain range rather than as a constraint on inverter reactive-power capability. For the 69-bus PV upper-bound check, increasing the limit from 2000 to 2500 kW yields a best-found plan comprising a 2490.21 kW PV system, a 1008.75-kW/2856.67 kWh BESS, and $K_q=0.5661$. Relative to the principal 2000 kW result, the extended-bound plan reduces the best objective value by 3.80%, annual total cost by 3.51%, grid-energy import by 5.34%, peak grid power by 8.05%, and voltage deviation by 4.08%. Active-loss energy increases slightly by 0.43%, while the voltage-violation duration remains at 2 bus h. Because the selected PV capacity reaches 99.61% of the extended upper bound, this check does not identify an unconstrained PV optimum and shows that the 69-bus economic and technical results are sensitive to the adopted PV planning limit. The 2000 kW case is retained as the principal 30-run larger-feeder validation, whereas the 2500 kW case is reported as a 10-run upper-bound sensitivity analysis.

9. Discussion

The results demonstrate that PV–BESS co-planning can substantially improve radial-feeder performance. The base case has 364 bus h of voltage violations, whereas the best ADE solution eliminates them over the representative day. The best solutions obtained by the other four algorithms retain between 1 and 21 bus h of violations, illustrating optimizer-specific trade-offs under the same weighted objective. The verified penalty analysis further distinguishes continuous squared violation severity from count-based bus-hour violation duration. Low or zero penalty settings may yield best-found plans that retain small-magnitude voltage-limit violations in exchange for improvements in other objective terms, whereas ${\lambda }_V=500$ provides sufficient selection pressure to obtain a zero-severity, zero-duration solution in the studied benchmark. Cross-evaluation shows that the non-monotonic lower-penalty results are partly attributable to stochastic-search variability. The analysis therefore supports retaining ${\lambda }_V=500$ as a conservative feasibility-oriented setting for this benchmark, rather than identifying it as a universal threshold.

The separately optimized $K_q=0$ comparison shows that PV–BESS planning without reactive-power support still improves the base feeder but retains 235 bus h of voltage violations. Enabling optimized inverter support eliminates these violations, reduces voltage deviation by 40.6% and active-loss energy by 22.3%, and lowers annual cost by 0.9% relative to the no-support plan. Peak grid power increases slightly by 0.4%, showing that the benefit represents a multi-metric trade-off rather than a uniform improvement in every metric. Extending the feasible range to $K_q\le 1.5$ reduces the best objective value by 1.29% and voltage deviation by 4.88% while increasing peak grid power by 3.50%. Because the matched-run difference is not statistically significant ($p=0.4165$) and the extended-range runs exhibit greater variability, the result is interpreted as sensitivity to the allowable voltage-support-gain range rather than as a basis for replacing the original five-algorithm benchmark.

The economic results require a conditional interpretation. MFO obtains the lowest annual cost among the best solutions, whereas ADE obtains the lowest weighted objective value, reflecting a stronger balance between the economic and voltage-performance terms included in the adopted objective. The objective-weight sensitivity supports this interpretation. Increasing the cost emphasis reduces the selected BESS rating and annual cost but increases the voltage-deviation index by 6.13%. Increasing the network emphasis reduces active-loss energy and peak grid power while producing small increases in annual cost and grid-energy import. Thus, the preferred design depends on the decision maker’s priorities, even though all three best-found plans satisfy the adopted voltage limits over the representative day.

The BESS energy-CAPEX analysis indicates local design invariance over the tested range of 200–300 USD/kWh. The same plan achieves the lowest objective value within the common set of 56 candidates at all three CAPEX levels. Consequently, the CAPEX variation changes the annual cost and objective value without changing the selected asset configuration or its technical performance. This result demonstrates robustness within the tested CAPEX range and candidate set but does not imply that BESS sizing is generally insensitive to storage cost. Wider cost ranges, different electricity tariffs, or cycle-dependent degradation assumptions may produce different planning decisions.

The representative-day analysis shows that the fixed ADE asset configuration remains beneficial when only the daily BESS schedule is adapted to four distinct hourly operating profiles. The equal-weight aggregate reductions remain substantial, including 37.84% lower active-loss energy, 51.71% lower voltage deviation, and an approximately 96.0% reduction in violation duration. However, the residual 38 and 19 bus h of violations under the winter/high-load and summer/high-PV profiles, respectively, show that an asset configuration optimized for the base day is not guaranteed to eliminate every voltage excursion under more demanding operating conditions. This result supports the inclusion of multiple operating profiles in planning studies and motivates future chronological and uncertainty-aware formulations.

The repeated selection of downstream BESS locations indicates that storage siting affects more than energy-arbitrage performance. In radial feeders, the BESS location also influences voltage profiles and network losses [6,7,17,18]. The 69-bus ADE validation supports the applicability of the formulation to a larger radial feeder, with substantial reductions in all six reported base-case metrics. The extended-bound result further shows that the economic and grid-energy-import outcomes are sensitive to the allowable PV planning range. Accordingly, the reported 69-bus designs should be interpreted as conditional scenario solutions and do not establish either an unconstrained PV-capacity optimum or a system-size-invariant optimizer ranking.

10. Limitations and Future Work

This study uses balanced single-phase-equivalent representations of the 33-bus and 69-bus test systems. The base economic analysis annualizes the grid-energy import obtained from a representative operating day, while the robustness analysis considers a limited set of synthetic representative profiles. These choices are appropriate for controlled methodological and algorithmic comparisons, but the resulting costs should not be interpreted as a site-specific chronological annual forecast. A site-specific application would require measured feeder data, chronological multi-season operation, uncertainty modeling, the applicable tariff structure, and cycle-dependent battery degradation.

The inverter voltage-support rule is represented as a bounded steady-state relation within a quasi-static hourly load-flow framework. The model does not resolve the dynamic inner control loops of grid-following or grid-forming inverters. Small-signal stability, fast controller interactions, control oscillations, and sub-hourly PV fluctuations are therefore outside the scope of the present analysis. Examining these effects would require dynamic electromagnetic-transient or phasor-domain models together with a substantially finer time resolution.

The 69-bus ADE experiment shows that the formulation can be applied to a larger radial feeder, but it does not establish that the relative performance of the five algorithms remains unchanged with system size. In addition, the 2500 kW sensitivity selects 2490.21 kW of PV, corresponding to 99.61% of the extended upper bound. This result suggests that the adopted weighted objective favors additional PV capacity within the tested planning range. The reported PV upper bounds should therefore be interpreted as planning assumptions rather than estimates of an unconstrained technical optimum. Future studies could introduce site-specific hosting-capacity, export, land-use, inverter-rating, protection, and curtailment constraints and repeat the full multi-algorithm comparison on larger, unbalanced three-phase feeders. Established differential-evolution variants such as L-SHADE and jSO may be included in future full-budget comparative studies.

The present formulation uses a weighted-sum objective. The completed weight-sensitivity analysis demonstrates that the selected planning solution depends on the adopted preferences, while Pareto-based multi-objective methods could provide a more complete representation of the trade-offs among cost, voltage quality, active losses, and imported energy. Other useful extensions include separate voltage-support gains for the PV and BESS inverters, explicit inverter-rating decisions, electric-vehicle charging, demand response, network reconfiguration, and machine-learning surrogate models to reduce the computational cost of repeated load-flow evaluations.

11. Conclusions

This paper presented an integrated PV–BESS planning and benchmarking framework for radial distribution networks. The model jointly optimizes PV location and size, BESS location and size, hourly storage operation, and a common inverter voltage-support gain, with reactive-power commands constrained by the available inverter capability while accounting for annualized investment, replacement cost, grid-energy cost, active losses, voltage deviation, peak grid power, and voltage violations.

For the 33-bus benchmark, ADE achieved the lowest best and median objective values over 30 independent runs, while GWO achieved the lowest mean objective value and DE exhibited the smallest standard deviation. The best ADE solution reduced daily active-loss energy by 38.0%, grid-energy import by 14.9%, peak grid power by 12.5%, voltage deviation by 53.7%, voltage-violation duration by 100%, and annual total cost by 8.4% relative to the no-PV/no-BESS base case. Among the best algorithm-specific solutions, MFO produced the lowest annual cost, whereas ADE achieved the lowest weighted objective value. This distinction shows that the preferred solution depends on whether the decision criterion emphasizes minimum cost alone or the balanced techno-economic performance represented by the adopted objective.

The separately optimized comparison with and without inverter reactive-power support provides a controlled assessment of the inverter contribution. Relative to the optimized $K_q=0$ case, enabling inverter support reduces active-loss energy by 22.3%, voltage deviation by 40.6%, voltage-violation duration by 100%, and annual cost by 0.9% while increasing peak grid power slightly by 0.4%. Extending the feasible range to $K_q\le 1.5$ yields a best-found value of $K_q=1.4747$ and a best objective value 1.29% lower than that obtained under the original $K_q\le 1$ bound. Both best-found solutions retain zero voltage violations, and the matched-run difference is not statistically significant. Thus, the extended range indicates a modest best-case benefit from allowing a higher voltage-support gain without changing the principal planning conclusion.

The objective-weight analysis further shows that, relative to the balanced plan, the cost-oriented plan reduces annual cost by 0.77% but increases the voltage-deviation index by 6.13%. The network-oriented plan reduces active-loss energy by 1.77% and peak grid power by 0.78% while increasing annual cost by 0.11%. All three best-found plans retain zero voltage-violation duration.

The low-to-base voltage-penalty analysis shows that the best-found plans obtained directly with ${\lambda }_V=0$ and 50 retain 7 and 18 bus h of small-magnitude voltage-limit violations, respectively, whereas ${\lambda }_V=500$ yields zero violation severity and zero violation duration. Cross-evaluation indicates that the apparent non-monotonicity between the lower-penalty results is partly attributable to stochastic-search variability. Accordingly, ${\lambda }_V=500$ is retained as a conservative feasibility-oriented coefficient for the studied benchmark rather than identified as a universal threshold.

In the BESS energy-CAPEX analysis, the same asset configuration—a 1497.74 kW PV system and a 999.79 kW/2755.25 kWh BESS—is selected from the common candidate set at 200, 250, and 300 USD/kWh. Relative to the 250 USD/kWh case, annual cost decreases by 0.61% at 200 USD/kWh and increases by 0.61% at 300 USD/kWh, while the technical performance remains unchanged. The selected configuration therefore shows local robustness to the tested $\pm 20\%$ variation in BESS energy CAPEX within the generated candidate set.

With the asset configuration fixed and only the daily BESS schedule re-optimized, the four-profile analysis yields equal-weight aggregate reductions of 5.34% in annualized cost, 37.84% in active-loss energy, 11.70% in grid-energy import, 15.37% in peak grid power, 51.71% in voltage deviation, and approximately 96.0% in voltage-violation duration relative to the corresponding no-PV/no-BESS cases. The spring/moderate profile has zero voltage-violation duration, whereas the winter/high-load and summer/high-PV profiles retain 38 and 19 bus h of violations, respectively. These results indicate substantial cross-profile benefits while also showing that the fixed asset configuration does not eliminate every voltage excursion under the most demanding operating profiles.

On the 69-bus feeder, the 30-run ADE validation under the principal 2000 kW PV upper bound reduces active-loss energy by 51.23%, grid-energy import by 19.55%, peak grid power by 10.50%, voltage deviation by 39.51%, voltage-violation duration by 98.80%, and annual total cost by 12.24%. Extending the PV upper bound to 2500 kW reduces the best objective value by a further 3.80% and annual cost by 3.51%. However, the selected PV capacity increases to 2490.21 kW, corresponding to 99.61% of the extended upper bound. The larger-feeder results therefore support the applicability of the formulation, while the upper-bound sensitivity shows that the 69-bus economic outcome remains conditional on the adopted PV planning limit. Taken together, the results support the joint consideration of PV–BESS allocation, time-coupled storage operation, and an explicitly converged fixed-point representation of inverter reactive-power support without implying an unconstrained PV-capacity optimum.