Optimal Siting and Sizing of Battery Energy Storage System in Distribution System in View of Resource Uncertainty

Abstract

The integration of intermittent Distributed Generations (DGs) like solar photovoltaics into Radial Distribution Systems (RDSs) reduces system losses but causes voltage and power instability issues. It has also been observed that seasonal variations affect the performance of such DGs. These issues can be resolved by placing optimum-sized Battery Energy Storage (BES) Systems into RDSs. This work proposes a new approach to the placement of optimally sized BESSs considering multiple objectives, Active Power Losses, the Power Stability Index, and the Voltage Stability Index, which are prioritized using the Weighted Sum Method. The proposed multi-objectives are investigated using the probabilistic and Polynomial Multiple Regression (PMR) approaches to account for the randomness in solar irradiance and its effect on BESS sizing and placements. To analyze system behavior, simultaneous and sequential strategies considering aggregated and distributed BESS placement are executed on IEEE 33-bus and 94-bus Portuguese RDSs by applying the Improved Grey Wolf Optimization and TOPSIS techniques. Significant loss reduction is observed in distributed BESS placement compared to aggregated BESSs. Also, the sequentially distributed BESS stabilized the RDS to a greater extent than the simultaneously distributed BESS. In view of the uncertainty, the probabilistic and PMR approaches require a larger optimal BESS size than the deterministic approach, representing practical systems. Additionally, the results are validated using Improved Particle Swarm Optimization–TOPSIS techniques.

1. Introduction

The addition of Renewable Energy Resources (RERs), like wind generators and solar photovoltaics (PV), into Radial Distribution Systems (RDSs) minimizes system losses and reduces the dependence on fossil fuels. The increased integration of irregular and uncontrollable RERs into RDSs leads to voltage and power instability. This variability and instability can be reduced by properly placing optimally sized Battery Energy Storage (BES) Systems into RDSs for smooth power generation from RERs. So, as the level of PV penetration increases, BES becomes more important [1]. Numerous researchers have addressed challenges associated with PV-integrated RDSs and proposed BES as a potential solution for various objective functions with several IEEE standard RDSs [1,2,3,4] and real-time case studies [1,2,3,4,5].

The BES System (BESS) is an essential asset that contributes significantly to various operational domains such as stationary and mobile applications [6]. In the electrical power system, it supports distribution services such as upgrade deferral and voltage stability. In addition, the BESS plays a vital role in ancillary services, encompassing congestion management, frequency regulation, and the provision of both spinning and non-spinning reserves. The BESS also enhances energy services through its supply and arbitrage and assists with the limitation of RE outputs. Moreover, the BESS is effective in energy management services by enhancing power quality and ensuring reliability [7,8]. The most common and beneficial utilization of the BESS can be improving system voltage [1,9], reducing system losses [1,10], congestion management [3,11], load leveling, and peak shaving using different optimization techniques [1,5,9,10,11,12]. To provide all the above-mentioned services satisfactorily, the OPSBESS is crucial. Installing the BESS in suboptimum places may lead to increased costs as well as system losses, thereby reducing system reliability.

While most of the research has centered on the sizing of the BESS, there is relatively less research on siting. The research on siting has focused on a single-objective function [13,14]. Few researchers have used multiple objectives for the OPSBESS, such as voltage deviations and power losses [5,15,16], voltage improvement, and congestion management [11]. Zhengmao et al. proposed a methodology for voltage restoration and load current sharing in multi-bus microgrids over directed networks [17]. It is reported in the literature that losses, power stability, and voltage stability are affected by the integration of PV and BESSs into RDSs [15,18,19]. Therefore, in this work, a new approach based on the Power Stability Index (PSI), along with objectives indicating Active Power Losses (APLs) and the Voltage Stability Index (VSI), is considered and optimized simultaneously. Hence, this work focuses on the multiple objectives of siting and sizing the BESS while minimizing the APL and PSI and maximizing the VSI for a grid-connected PV-BES System. These objectives are assigned with suitable weights using the Weighted Sum Method (WSM) and are then optimized simultaneously using the Improved Grey Wolf Optimization (IGWO) technique and ‘The Technique for Order of Preference by Similarity to the Ideal Solution’ (TOPSIS).

Earlier studies showed that the OPSBESS problem for RDSs has been predominantly addressed using a deterministic approach without considering variability in the resource. However, considering uncertainties in RESs is crucial for analyzing system behavior under real-world conditions. In the literature, it is found that the uncertainties associated with PV are addressed using the Beta Probability Density Function (β-PDF) [20,21]. In the proposed work, the uncertainties in PV irradiance for ten years are addressed with the β-PDF and Polynomial Multiple Regression (PMR) techniques [22]. The results of deterministic along with β-PDF and PMR approaches are compared to predict solar PV irradiance to determine the OPSBESS for a modified IEEE 33-bus RDS (MBRDS) and a modified real 94-bus Portuguese RDS (MRBPRDS)[23].

In addition to this, the problem of optimal-sized BESS placement can be analyzed using the simultaneous and sequential placement of BESSs in RDSs. It is observed that the comparison of simultaneously and sequentially distributed BESS placement is less addressed in the literature. Hence, considering lithium-ion battery chemistry, the logic of the simultaneous and sequential placement of optimally sized distributed BESSs is implemented and compared using two RDSs.

The significant contributions of this paper are as follows:

• Proposal of a multiple-objective function (MOF) based on minimizing the APL and PSI and maximizing the VSI for the OPSBESS.
• Incorporation of uncertainties in the resource for the OPSBESS using probabilistic and PMR approaches for MBRDSs.
• Analysis of simultaneously and sequentially distributed BESS placement to understand the system behavior using IGWO and TOPSIS (IGT). The results are validated using Improved Particle Swarm Optimization–TOPSIS (IPT) techniques.

This paper is structured as follows: Section 2 describes the grid-connected PV-BES System with its mathematical modeling. Section 3 formulates a multi-objective problem along with the constraints. Section 4 details IGWO, the optimization technique. Section 5 presents a detailed analysis of the results derived from the proposed work, whereas Section 6 offers the concluding remarks of this paper.

2. Grid-Connected PV-BES System

The problem of the OPSBESS is solved for a grid-connected PV-BES System for the MOF by minimizing the APL and PSI and maximizing the VSI simultaneously using the WSM-IGWO-TOPSIS methods. The MBRDSs test the logic of the simultaneous and sequential placements of the distributed BESS. The schematic of the system under study consists of a solar PV system, utility grid, BESS, inverter, and load (Figure 1).

A detailed description of the system components with mathematical modelling is provided below.

2.1. Solar PV System

This study evaluated the strategic placement of three PV systems at designated locations, focusing on the criteria of loss minimization and voltage enhancement for two RDSs. In the case of the standard IEEE 33-bus RDS, these PV systems are placed at bus numbers 18, 25, and 30, with the size of each PV system set as 600 kW. Therefore, the PV system has a total power rating of 1.8 MW.

In the case of the real 94-bus Portuguese RDS, these PV systems are placed at bus numbers 19, 58, and 84, with the size of each PV system set as 800 kW. Therefore, the PV system has a total power rating of 2.4 MW. The system load is supplied by PV, utility grid, and BESS power. Three approaches, deterministic, probabilistic, and PMR, are considered to determine the profiles of solar irradiance and calculate the output of the PV systems for 24 h. The output power of each PV panel and the whole PV system can be found by taking into account the solar irradiance received by the PV panels at that site. The solar irradiance for ten years is considered for a specific location (Portugal—latitude 39.99° N and longitude 8.22° W) [22]. The detailed mathematical modeling for these approaches is as follows.

2.1.1. Deterministic Approach

This approach focuses on matching the estimated PV generation with the expected demand under the assumption that all input variables are known with certainty. It does not need past power measurements and detailed information regarding the PV panel set-up, such as geographic position, electrical configuration, etc. In the deterministic approach, the individual PV panel’s output power is ascertained from Equation (1), while the aggregate power generated by the entire PV system can be determined using Equation (2).

$${\mathrm{p}}_{\mathrm{P}\mathrm{V}-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}=\left\{\begin{array}{c}{\mathrm{P}}_{\mathrm{R}\mathrm{S}}\left({\displaystyle \frac{{\mathrm{s}}^2}{{\mathrm{R}}_{\mathrm{S}\mathrm{R}\mathrm{S}}{\mathrm{R}}_{\mathrm{C}\mathrm{R}}}}\right) \mathrm{if} 0 \le s\le {\mathrm{R}}_{\mathrm{C}\mathrm{R}} \\ {\mathrm{P}}_{\mathrm{R}\mathrm{S}}\left({\displaystyle \frac{\mathrm{s}}{{\mathrm{R}}_{\mathrm{S}\mathrm{R}\mathrm{S}}}}\right) \mathrm{i}\mathrm{f} {\mathrm{R}}_{\mathrm{C}\mathrm{R}} \le s\le {\mathrm{R}}_{\mathrm{S}\mathrm{R}\mathrm{S}} \\ {\mathrm{P}}_{\mathrm{R}\mathrm{S}} \mathrm{i}\mathrm{f} {\mathrm{R}}_{\mathrm{S}\mathrm{R}\mathrm{S}}\le s \end{array}\right.$$

(1)

$${\mathrm{P}}_{\mathrm{P}\mathrm{V}}={\mathrm{p}}_{\mathrm{P}\mathrm{V}-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}\times {\mathrm{N}}_{\mathrm{P}\mathrm{V}}$$

(2)

where the following variables are used:

P_RS: rated PV power (W);

s: solar irradiance (W/m²);

R_SRS: solar irradiance under the standard environment set as 1000 W/m²;

R_CR: certain solar irradiance point (W/m²);

p__(PV-single): power rating of a PV panel (W);

Ppv: total power output of all PV panels (W);

N_PV: number of PV panels.

2.1.2. Probabilistic Approach

Probabilistic models offer valuable insights into the influence of various factors on outcomes, enabling the identification of data patterns and relationships. This approach is applied to account for uncertainties in solar PV irradiance to study the system dynamics for the OPSBESS. Detailed information about the PV system, such as the geographic position, voltage and current temperature coefficients, and electrical configuration (voltage and current under maximum power point conditions, short circuit, open circuit, etc.), is needed for a probabilistic approach. After ten years of solar PV irradiance data collection in Portugal [22], statistical analysis was conducted utilizing EasyFit 5.5 (EasyFit 5.5, Informer Technologies, Inc., Los Angeles, CA, USA) to ascertain the most suitable PDF for the observed data pattern. The analysis revealed that the Beta PDF (β-PDF) is the most appropriate for the dataset in question. Furthermore, the uncertainties associated with PV irradiance were modeled using the β-PDF, as outlined in Equations (3)–(14), ultimately providing the output for a single PV panel [5,8,20,21]. The overall output of the PV system is assessed based on the number of PV panels installed in the system.

The β-PDF for solar irradiance (s), established within the limits of 0 ≤ s ≤ 1, is characterized by the shape parameters α and β, both of which are greater than zero. This function is represented as beta (s) and defined by Equation (3). Equations (3)–(14) are derived from [8,21]. These equations incorporate the effects of seasonal changes as well as daily variations in temperature and solar irradiance. The various parameters of solar irradiance (α, β: shape, mean (μ), and standard deviation (σ)) are computed for each hour of each day over a span of ten years.

$$\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}\left(\mathrm{s}\right)=\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{\Gamma }\left(\mathrm{\alpha }+\mathrm{\beta }\right)}{\mathrm{\Gamma }\left(\mathrm{\alpha }\right)\mathrm{\Gamma }\left(\mathrm{\beta }\right)}}{.s}^{\mathrm{\alpha }-1}.{\left(1-s\right)}^{\mathrm{\beta }-1} 0 \le s \le 1, \alpha , \beta ⩾ 0\\ 0 \text{otherwise}\end{array}\right.$$

(3)

The mean (µ) and standard deviation (σ) are used to derive the shape parameters (α and β) of solar irradiance over a period of ten years, as below.

$$\mathrm{\beta }=(1-\mathrm{\mu })[{\displaystyle \frac{\mathrm{\mu }\left(1+\mathrm{\mu }\right)}{\left({\mathrm{\sigma }}^2\right)}}-1]$$

(4)

$$\mathrm{\alpha }=[{\displaystyle \frac{(\mathrm{\mu }\times \mathrm{\beta })}{(1-\mathrm{\mu })}} ]$$

(5)

For any specific hour, the probability of solar irradiance is calculated as

$$\mathrm{p}(\mathrm{s})={\int }_{s1}^{s2}[(\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}(\mathrm{s}).\mathrm{d}\mathrm{s})]$$

(6)

where the two solar irradiance limits are s1 and s2.

Equations (7)–(14) are used to calculate the PV panel’s output for a given time (h).

$${\int }_0^1[\mathrm{s}(\mathrm{p}(\mathrm{s}).\mathrm{d}\mathrm{s})]={\int }_0^1[[\mathrm{s}{\displaystyle \frac{\mathrm{\Gamma }(\mathrm{\alpha }+\mathrm{\beta })}{\mathrm{\Gamma }(\mathrm{\alpha })\mathrm{\Gamma }(\mathrm{\beta })}}{.\mathrm{s}}^{\mathrm{\alpha }-1}.{(1-\mathrm{s})}^{\mathrm{\beta }-1}]\mathrm{d}\mathrm{s}]$$

(7)

The probabilistic output of a single PV panel for an hour (h) is found by

$${\mathrm{P}\mathrm{V}}_{\mathrm{O}\mathrm{U}\mathrm{T}}\left(\mathrm{h}\right)={\int }_0^1\left[\left({\mathrm{P}\mathrm{V}}_{\mathrm{N}\mathrm{E}\mathrm{T}}\right)\times \left(\mathrm{p}\left(\mathrm{s}\right).\mathrm{d}\mathrm{s}\right)\right] \mathrm{w}\mathrm{a}\mathrm{t}\mathrm{t}$$

(8)

The net output of a single PV panel (PV_NET) can be found by multiplying the Fill Factor (FF), net voltage (V_NET), and net current (I_NET), as given by Equation (9).

PV_NET = [FF × V_NET × I_NET] watt

(9)

The FF serves as an indicator of the efficiency of a PV module, as defined by Equation (10).

FF = {[V_MPP × I_MPP]/[V_OC x I_SC]}

(10)

V_NET = [V_OC + K_v (Tc)] volts

(11)

I_NET = s [I_SC + K_i (Tc − 25)] amperes

(12)

$${\int }_0^1[\mathrm{s}(\mathrm{p}(\mathrm{s}).\mathrm{d}\mathrm{s})]={\int }_0^1\mathrm{s}[(\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}(\mathrm{s}).\mathrm{d}\mathrm{s})]$$

(13)

T_C = T_A + s [(T_NOCT − 20)/0.8] °C

(14)

where the following variables are used:

s1 and s2: solar irradiance limits (W/m²);

α and β: shape parameters of solar irradiance (W/m²);

µ and σ: mean and standard deviation of solar irradiance (W/m²);

p(s): probability of solar irradiance (W/m²);

s: solar irradiance (W/m²);

Γ: gamma function;

T_C: cell temperature (°C);

T_A: ambient temperature (°C);

T_NOCT: temperature at Nominal Operating Cell Temperature (NOCT) (°C).

The following specifications of the solar PV panel [24] are used to calculate the probabilistic output of a single PV panel for an hour as ‘PV_OUT (h)’ given by Equation (8).

The total probabilistic output power of the whole PV system is found by multiplying Equation (8) with the daily hours of available solar irradiance and the number of PV panels installed in the system, as given by Equation (2) [25]. The solar irradiance for one day is calculated from the total probabilistic PV power output given by Equation (2) and by knowing the area of the PV panel and its efficiency (

Table 1. Specifications of 550Wp solar PV panel [24].

Sr. No.	Parameter (Symbol)	Value	Unit
1	Rated power of solar PV panel (PRS)	550	Wp
2	Ambient temperature (TA)	19.19	°C
3	Cell temperature (TC)	25	°C
4	Normal Operating Cell Temperature (TNOCT)	45	°C
5	Short-Circuit Current (ISC)	11.29	A
6	Current at Maximum Power Point (IMPP)	10.51	A
7	Current temperature coefficient (Ki)	0.05	A/°C
8	Open-Circuit Voltage (Voc)	46.82	V
9	Voltage at Maximum Power Point (VMPP)	39.14	V
10	Voltage temperature coefficient (Kv)	−0.265	V/°C
11	Efficiency of PV panel	21.3	%
12	Length of PV panel	2.278	M
13	Width of PV panel	1.134	M

).

The solar irradiance for a specific time (t) is expressed in Equation (15).

s(t) = Ppv(t)/(η_PV × A)

(15)

where the following variables are used:

• s(t): solar irradiance (W/m²) at ‘t’;
• Ppv(t): total power output of PV panels at ‘t’(W);
• η_PV: efficiency of PV panel (%);
• A: area of PV panel (m²).

2.1.3. Polynomial Multiple Regression Approach (PMRA)

The PMRA is one of the proven ways to account for the randomness in system behavior. It is a regression analysis that finds one dependent variable against multiple independent variables which share non-linearity among them. In this case, solar irradiance is the dependent variable and is estimated against three independent variables: the wind velocity, cell temperature, and ambient temperature over 10 years. To establish the PMR model among these variables, the solar irradiance of Portugal is considered for ten years [22]. From the given solar irradiance data, the mean square error (mse) and root mean square error (rmse) are found using the sklearn.metrics library of Python programs (Polynomial Regression In Python, n.d.).

The best-fit curve of solar irradiance with the PMRA is obtained for Portugal for 24 h. Utilizing this best-fitted curve of solar irradiance, the output of three solar PV systems is determined with an accuracy level of 0.99 and compared with deterministic and probabilistic approaches for finding the OPSBESS.

The solar irradiance obtained using deterministic, probabilistic, and PMR approaches is depicted in Figure 2. The total output of three PV systems for three sets of solar irradiances is calculated to determine the OPSBESS.

2.2. Load

The load demand data for Portugal are derived by calculating the average annual load demand values, which are subsequently averaged daily, as depicted in Figure 3. The load demand curves considered for the IEEE 33-bus and 94-bus Portuguese RDSs are shown in Figure 3a,b, respectively [23].

2.3. BESS

Depending on the State of Charge (SOC), the BESS stores excess power if the generated power surpasses the load demand (charge) and supplies the load when there is a shortage of electricity (discharge). In this work, lithium-ion battery chemistry is considered with an SOC range of 20% to 100%. The charging mode energy and discharging mode energy of the BESS at a specific time ‘t’ are determined by Equations (16) and (17) [25].

Charging mode:

$${\mathrm{E}}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}\left(\mathrm{t}-1\right)\times \left(1-\mathrm{\rho }\right)+\left[{\mathrm{E}}_{\mathrm{p}\mathrm{v}}\left(\mathrm{t}\right)-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)}{{\mathrm{ɳ}}_{\mathrm{i}\mathrm{n}\mathrm{v}}}}\right]\times {\mathrm{ɳ}}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$$

(16)

Discharging mode:

$${\mathrm{E}}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}\left(\mathrm{t}\right)={\mathrm{E}}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}\left(\mathrm{t}-1\right)\times \left(1-\mathrm{\rho }\right)-\left[{\displaystyle \frac{{\mathrm{E}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)}{{\mathrm{ɳ}}_{\mathrm{i}\mathrm{n}\mathrm{v}}}}-{\mathrm{E}}_{\mathrm{p}\mathrm{v}}\left(\mathrm{t}\right)\right]\times {\mathrm{ɳ}}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$$

(17)

where the following variables are used:

• E_PV(t): energy generated by the solar PV at instant t (kWh);
• E_load(t): energy required by the load at instant t (kWh);
• EBESS(t): energy stored by the BESS at instant t (kWh);
• ρ: self-discharge rate of the BESS;
• η_BESS: charging and discharging efficiencies of the BESS;
• η_inv: efficiency of the inverter.

3. Multi-Objective Problem Formulation and Constraints

Integrating BESSs, along with their optimal placement and sizing within the RDS, significantly impacts the system’s power flow, losses, voltage stability, and overall power stability. Therefore, this study aims to simultaneously optimize three key objectives: the minimization of the APL and PSI and the maximization of the VSI. Simultaneous and sequential strategies for BESS placement with their optimal sizes are executed on two MBRDSs considering lithium-ion battery chemistry.

3.1. Minimization of APL (F1)

In conventional power systems, significant power losses are primarily observed in the delivery phase, thereby adversely affecting the annual revenue generated by utilities within Distribution Systems (DSs). Therefore, the primary objective is to deliver power at minimum losses, which is expressed by Equation (18) and referred from [12].

$$\mathrm{F}1=\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\sum }_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{\alpha }}_{\mathrm{i}\mathrm{j}}\left({\mathrm{P}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{j}}+{\mathrm{Q}}_{\mathrm{i}}{\mathrm{Q}}_{\mathrm{j}}\right)+{\mathrm{\beta }}_{\mathrm{i}\mathrm{j}}({\mathrm{Q}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{j}}-{\mathrm{P}}_{\mathrm{i}}{\mathrm{Q}}_{\mathrm{j}})$$

(18)

where the following variables are used:

−α_ij = [R_ij cos(δ_i − δ_j)/V_iV_j] and β_ij = [R_ij sin(δ_i − δ_j)/V_iV_j]

(19)

• N: number of buses in the system;
• R_ij: resistance of branch between bus i and bus j (Ω);
• P_i: injected active power at the ith bus (watt);
• P_j: injected active power at the jth bus (watt);
• Q_i: injected reactive power at the ith bus (VAR);
• Q_j: injected reactive power at the jth bus (VAR);
• V_i: voltage magnitude at sending end voltage at the ith bus (volts);
• V_j: voltage magnitude at receiving end voltage at the jth bus (volts);
• δ_i: phase angle at the ith bus (radians);
• δ_j: phase angle at the jth bus (radians).

3.2. Minimization of PSI (F2)

The PSI serves as a measure of line voltage stability within the system. It assists in identifying the most vulnerable bus, which may be susceptible to line voltage instability when an increase in load demand occurs [26]. Better line voltage stability of the system is achieved if the PSI value is closer to zero (PSI ≤ 0). When a PV system is added to the RDS, it affects the line voltage stability. Therefore, to maintain line voltage stability, the PSI is incorporated in this study for BESS placement. It is also equally important as the APL and VSI. Initially, the PSI values for all buses are calculated without the integration of the BESS and are subsequently arranged in descending order. The optimization process is designed to minimize this index. Therefore, the BESS is installed on the bus, which demonstrates the highest PSI value.

The PSI framework is developed based on a two-bus system that operates with a stability margin of less than one to ensure proper voltage stability, as outlined in Equation (20) [26].

$$\mathrm{P}\mathrm{S}\mathrm{I}={\displaystyle \frac{[{4\mathrm{R}}_{\mathrm{i}\mathrm{j}}({\mathrm{P}}_{\mathrm{L}}-{\mathrm{P}}_{\mathrm{G}})]}{ {[|{\mathrm{V}}_{\mathrm{i}}|\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\theta }-\mathrm{\delta })]}^2}}\le 1$$

(20)

$$\mathrm{F}2=\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathrm{P}\mathrm{S}\mathrm{I}=\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} {\displaystyle \frac{[{4\mathrm{R}}_{\mathrm{i}\mathrm{j}}({\mathrm{P}}_{\mathrm{L}}-{\mathrm{P}}_{\mathrm{G}})]}{ {[|{\mathrm{V}}_{\mathrm{i}}|\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\theta }-\mathrm{\delta })]}^2}}\le 1$$

(21)

where the following variables are used:

• R_ij: resistance of branch between bus i and bus j (Ω);
• P_L: active power at the load bus (watt);
• P_G: injected active power into the system (watt);
• V_i: sending end voltage (volts);
• θ: angle of line impedance (radians);
• δ: phase angle (radians).

3.3. Maximization of VSI (F3)

Maximizing the VSI of a system branch with the minimum VSI value improves the VSM of the complete system. The system is stable when the VSI is less than one and unstable when it is more than one. A VSI closer to one indicates higher voltage stability. The proposed objective function for the maximization of the VSI and, hence, the maximization of the VSM can be expressed by Equation (22). It is developed according to a quadratic equation for the bus ‘j’ in a power system, which is referred from [12,27] as

VSI_j = [|V_i|⁴ − 4(P_jR_ij + Q_jX_ij)|V_i|² − 4(P_jX_ij − Q_jR_ij)²]

(22)

where the following variables are used:

• VSI_i: Voltage Stability Index of bus ‘j’;
• P_j: (sum of the active power loads of all the buses beyond bus j) + (active power load of bus j itself) + (sum of the Active Power Losses of all the branches beyond bus j) (watt);
• Q_j: (sum of the reactive power loads of all the buses beyond bus j) + (reactive power load of bus j itself) + (sum of the reactive power losses of all the branches beyond bus j) (VAR);
• R_ij: resistance of the branch connecting bus i and bus j (Ω);
• X_ij: reactance of the branch connecting bus i and bus j (Ω);
• V_i: sending end voltage (volts).

If the calculated VSI for all buses (except the slack bus) to the last bus is greater than zero, the studied DS has stable performance conditions. The bus with the lowest VSI is more sensitive to voltage collapse and is the most suitable bus for installing the BESS to achieve better stability conditions for the system. The bus with the lowest VSI has the worst stability; hence, this index is proposed inversely, as given in Equation (23). Therefore, the objective function becomes VSIj.

F3 = Maximization of VSI = Maximization of VSIj

(23)

Therefore, by combining Equations (18)–(23), the minimum of the MOF can be formulated. The OPSBESS is calculated after satisfying all the constraints mentioned below.

3.4. Multi-Objective Problem Formulation

The problem of the OPSBESS in grid-connected or islanded energy systems is important. A minor adjustment in the number and placement of BESSs can influence the overall sizing and efficiency of the system. Such changes have a significant impact on performance, investment, and operating costs.

In this work, two Multiple-Criteria Decision-Making (MCDM) techniques are implemented: the WSM and TOPSIS [28]. The WSM prioritizes the objectives, while the combination of the TOPSIS and IGWO techniques yields better results.

The MOF is formulated as illustrated in Equations (24)–(27). The weights can be determined as per the priority of the objective function. After systematically adjusting the weights W₁, W₂, and W₃ from 0 to 1 in steps of 0.1, it was determined that variations in the APL exhibit greater sensitivity to minor changes in the size of the BESS compared to the PSI and VSI. As a result, the APL is assigned the highest priority in this analysis, leading to the final weights of W₁ = 0.4 and W₂ = W₃ = 0.3 for the optimal sizing of the BESS. The MOF is utilized to ascertain the most advantageous location for the BESS, as mathematically represented in Equation (27). Accordingly, the bus associated with the lowest value of the MOF is identified as the most appropriate site for the installation of the BESS.

$${\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{W}}_{\mathrm{i}}\right)={\sum }_{\mathrm{i}=1}^3\left({\mathrm{W}}_{\mathrm{i}}\right)=1$$

(24)

where W₁, W₂, and W₃ represent the weights associated with the objective functions.

(W₁ + W₂ + W₃) = (0.4 + 0.3 + 0.3) = 1

(25)

Equation (25) is used to find the optimum solution.

Minimization of MOF = Min [(W₁ × F1) + (W₂ × F2) + (W₃/F3)]

(26)

where F1 and F2 represent the minimization of the APL and PSI, respectively, and F3 signifies the maximization of the VSI.

Therefore, from Equation (25), Equation (26) becomes

Minimization of MOF = Min [(0.4 × APL) + (0.3 × PSI) + (0.3/VSI)]

(27)

Constraints: The following constraints are applied to Equation (27):

Equations (28) and (29) represent the active and reactive power balance at the ith bus.

$$\mathrm{P}\mathrm{i}=\mathrm{V}\mathrm{i}{\sum }_{j=1}^N{\mathrm{V}}_{\mathrm{j}}{\mathrm{Y}}_{\mathrm{i}\mathrm{j}}\mathrm{c}\mathrm{o}\mathrm{s}({\mathrm{\theta }}_{\mathrm{i}\mathrm{j}}+{\mathrm{\delta }}_{\mathrm{j}}-{\mathrm{\delta }}_{\mathrm{i}})\forall \mathrm{i}$$

(28)

$$\mathrm{Q}\mathrm{i}=-\mathrm{V}\mathrm{i}{\sum }_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{V}}_{\mathrm{j}}{\mathrm{Y}}_{\mathrm{i}\mathrm{j}}\mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{\theta }}_{\mathrm{i}\mathrm{j}}+{\mathrm{\delta }}_{\mathrm{j}}-{\mathrm{\delta }}_{\mathrm{i}})\forall \mathrm{i}$$

(29)

The bounds on the maximum installation size of an individual PV panel at the ith bus are

0 ≤ N_PV ≤ N_PVmax ∀ i

(30)

The constraints pertaining to the maximum allowable installation size of an individual BESS at the ith bus have been defined.

S_BESSi ≤ S_{BESS max} ∀ i

(31)

The integration of the solar PV and BESS into the RDS causes an inrush current and reverse power flow into the system, leading to a rise in voltage at the buses. Therefore, the minimum and maximum voltage at each bus for all cases should satisfy the following limits:

V_i(min) ≤ V_i ≤ V_i(max) ∀ i

(32)

where V_i represents the voltage magnitude at the ith bus. V_i(min) and V_i(max) are assumed to be 0.95 pu and 1.05 pu, respectively, considering the allowable variation as ±5% of the rated voltage.

Consequently, after fulfilling all stipulated constraints, Equation (27) is solved by applying the IGWO algorithm to determine the optimum size of the BESS to be implemented at the maximum MOF value. The optimization methodology is elaborated upon in the subsequent section.

4. IGWO for OPSBESS

The multiple objectives of minimizing the APL and PSI and maximizing the VSI, each with distinct weights, are systematically analyzed and implemented in the context of the MBRDS to identify the OPSBESS. To solve this multi-objective problem, it is necessary to run the load flow many times with all objective functions and their constraints. Therefore, the use of the optimization algorithm is necessary. The IGWO method is selected for the said problem because it is suitable for solving various problems of engineering design [29] as well as electrical power systems. IGWO is implemented for the optimal allocation of Distributed Generation [30], economic dispatch [31], intelligent energy management and battery sizing [32], the OPSBESS [33], etc. In addition to this, IGWO has the distinct advantages of simple principles, fewer parameters, easy implementation, convergence speed, convergence precision, and robustness compared with other algorithms and the ability to resolve the problem of instability compared with other algorithms [34,35]. To decide the preferences of objective functions, the WSM assigns different weights to them. To analyze system behavior, simultaneous and sequential strategies of BESS locations are executed on the MBRDS by combining IGWO with the TOPSIS for better-optimized results. Hwang and Yoon invented the TOPSIS in 1981 [36]. The technique identifies the most favorable alternatives based on their proximity to the positive ideal solution while maintaining the greatest distance from the negative ideal solution. It is combined with different optimization techniques for solving various engineering problems [12,37] for better system performance. This technique emphasizes the significance of proximity to the positive ideal solution, and as such, it has been incorporated into this work.

The proposed IGWO algorithm aims to achieve the optimized solution in the shortest time possible with the most accuracy based on local and global searches. The flow chart for determining the OPSBESS in a grid-connected PV-BES System using IGWO is described in Figure 4.

This study applies IGWO-TOPSIS to solve the OPSBESS by satisfying the MOF along with the constraints. This study is implemented in the MBRDS and MRBPRDS, considering deterministic, probabilistic, and PMR approaches. In this algorithm, each wolf (α, β, δ, ω) is the feasible solution for the optimal BESS size and its placement. After obtaining optimal values for the size and placement of the BESS, its simultaneous and sequential placements are carried out using IGWO-TOPSIS techniques. The system behaviour is analyzed for different cases, which are explained in Section 5.

5. Results and Discussion

The randomness in the solar PV irradiance is considered for ten years, and its effect on the OPSBESS is analyzed for the probabilistic and PMR approaches. The results of the OPSBESS of these approaches are compared with the deterministic approach. The size of the BESS and the performance of the system will vary according to the approach used. The OPSBESS has been optimized for the MOF and executed on two RDSs: standard IEEE 33-bus and real 94-bus Portuguese, considering all relevant constraints through the application of the IGWO-TOPSIS techniques.

5.1. IEEE 33-Bus RDS

The proposed work of the OPSBESS is analyzed for various cases with placements of the PV and BESS (Case I to Case V), as explained below.

Case I: standard IEEE 33-bus RDS.

The problem of the OPSBESS is framed and solved using the standard IEEE 33-bus RDS. It is a 12.66 kV, 100 MVA system with active and reactive power demands of 3.715 MW and 2.3 MVAR, respectively [23].

Case II: standard IEEE 33-bus RDS with placements of PV systems (modified IEEE 33-bus RDS—MBRDS without BESS).

With voltage improvement and loss reduction as prime objectives, three PV systems are placed on specific buses (18, 25, and 30) of the IEEE 33-bus RDS. Thus, the system formed is termed a modified IEEE 33-bus system (MBRDS). Each PV system has a power rating of 600 kW. Consequently, the aggregate capacity of the three PV systems is 1.8 MW. The addition of three PV systems to the standard IEEE 33-bus RDS changed the load flow, reduced the system losses, and improved the voltage profiles of the buses.

Case III: MBRDS with aggregated BESS.

In this case, the load flow is run to satisfy the MOF and the constraints to find the optimal BESS size and candidate bus for the BESS location after placing three PV systems. Following the initial load flow analysis, Bus 26 was identified as the priority site for the deployment of the aggregated BESS. This strategic placement effectively minimized both the APL and PSI while concurrently maximizing the VSI at the optimal capacity. Bus 26 was identified as the candidate bus for placing an aggregated BESS with a size of 2.3 MW for the deterministic approach, 2.42 MW for the B-PDF approach, and 2.4 MW for the PMR approach. Case III is depicted in Figure 5.

5.2. Analysis of MBRDS Using Simultaneous and Sequential BESS Placements

The optimally sized BESS is placed in the MBRDS simultaneously and sequentially. For aggregated BESS placement (Case III), as only one unit of the BESS needs to be placed after the first load flow run in the system after PV placement, there is no difference between the logic for the simultaneous and sequential placements of the BESS. The logic of the OPSBESS changes for Case IV and Case V. In these cases, the optimal size of the aggregated BESS unit is distributed optimally into multiple smaller units based on the system’s requirements. The BESS placement process varies between the simultaneous and sequential strategies.

5.2.1. Simultaneous BESS Placements

In the simultaneous placement strategy, all BESS units are placed at once, with load flow calculations run only once, irrespective of the number of BESS units required within the system. This analysis is essential for determining key parameters, including the APL, PSI, and VSI. In a specific case involving the placement of two BESS units following an initial load flow analysis with three PV systems, the first BESS unit was strategically located at the weakest bus (Bus 26), and at the same time, the second BESS unit, optimally sized, was subsequently positioned at the next weakest bus. This deliberate placement is intended to significantly enhance the overall performance and reliability of the system.

5.2.2. Sequential BESS Placements

In the sequential placement strategy, the BESS units are placed one by one, with the load flow recalculated after each placement to determine the next weakest bus for BESS allocation. For example, when placing two BESS units—one at a specific bus (Bus 26) and the other BESS unit at the optimal capacity—the process begins with a load flow analysis that accounts for the integration of three PV systems. The first BESS unit is strategically placed at the weakest bus, identified as Bus 26. Following this placement, the load flow analysis is repeated to identify the next weakest bus, which is Bus 18, for the installation of the second BESS unit. This sequential approach is repeated until all required BESS units are effectively integrated into the system.

For illustration purposes, Case IV and Case V are explained for the sequential OPSBESS.

Case IV: MBRDS with two BESS units.

To improve the system performance, the aggregated BESS (optimal size) is distributed optimally into two suitable sizes of BESSs. The major portion of the optimal BESS size is strategically allocated to the first location, with the residual capacity assigned to the second location. The first location of the BESS is the weakest bus in the system (Bus 26), and the second location of the BESS is the second weakest bus in the system after running the load flow by placing the first BESS at Bus 26.

Case V: MBRDS with three BESS units.

For further improvement in the system performance, the aggregated BESS (optimal size) is distributed into three suitable sizes of BESSs. The optimal BESS size is placed at the first location (weakest bus in the system—Bus 26). Then, the load flow is run, and the second optimal size of the BESS is placed at the second weakest bus (Bus 18). Again, the load flow is run with these two BESSs to obtain the third weakest bus. The remaining optimal size of the BESS is placed at this third weakest bus of the system. Thus, this process is repeated as per the BESS units required to be installed in the system.

Similarly, the BESS can be distributed into four (Case VI—four BESS units of optimal size), five (Case VII—five BESS units of optimal size), or six (Case VIII—six BESS units as of optimal size) units or as per the requirement for BESS units in the system. For Case IV to Case VIII, the OPSBESS logic is applied sequentially, optimizing the placement of two to six BESS units across different buses. Each BESS unit is placed at the weakest bus in the system based on load flow analysis. The system’s performance improves as more BESS units are optimally placed in sequence, following the priorities of the APL, PSI, and VSI.

In the same way, the results for these five cases (Case I to Case V) are obtained for probabilistic and PMR approaches with simultaneous and sequential BESS placements. The outcomes covering all three approaches are summarized in

Table 2. Simultaneous and sequential OPSBESS for MBRDS.

Case Number	Deterministic Approach				Probabilistic Beta PDF Approach				Polynomial Multiple Regression Approach
	Simultaneous OPSBESS		Sequential OPSBESS		Simultaneous OPSBESS		Sequential OPSBESS		Simultaneous OPSBESS		Sequential OPSBESS
	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)
Case I	No PV and No BESS (W1 = 1, W2 = W3 = 0)
IGT	-	2.649	-	2.649	-	2.649	-	2.649	-	2.649	-	2.649
IPT	-	2.649	-	2.649	-	2.649	-	2.649	-	2.649	-	2.649
Case II	With 3 PV Systems and No BESS (W1 = 1, W2 = W3 = 0)
IGT	No BESS	2.170	No BESS	2.170	No BESS	2.064	No BESS	2.064	No BESS	2.162	No BESS	2.162
IPT	No BESS	2.170	No BESS	2.170	No BESS	2.064	No BESS	2.064	No BESS	2.162	No BESS	2.162
Case III A	With 3 PV Systems and 1 BESS Unit (Aggregated BESS) (W1 = 1, W2 = W3 = 0)
IGT	30(2.326)	2.212	30(2.326)	2.212	29(2.414)	2.017	29(2.414)	2.017	30(2.324)	2.186	30 (2.324)	2.186
IPT	30(2.270)	2.168	30(2.270)	2.168	29(2.356)	1.980	29(2.356)	1.980	30(2.266)	2.140	30 (2.266)	2.140
Case III B	With 3 PV Systems and 1 BESS Unit (Aggregated BESS) (W1 = 0.4, W2 = W3 = 0.3)
IGT	26 (2.30)	1.890	26 (2.30)	1.890	26 (2.42)	1.785	26 (2.42)	1.785	26 (2.40)	1.871	26 (2.40)	1.871
IPT	26 (2.26)	1.899	26 (2.26)	1.899	26 (2.36)	1.867	26 (2.36)	1.867	26 (2.27)	1.882	26 (2.27)	1.882
Case IV	With 3 PV Systems and 2 BESS Units (W1 = 0.4, W2 = W3 = 0.3)
IGT	14 (1.60)	1.832	26 (1.89)	1.810	7 (0.65)	1.769	26 (2.09)	1.735	7 (0.98)	1.835	26 (1.99)	1.812
	30 (0.70)		18 (0.43)		26 (1.77)		18 (0.33)		25 (1.51)		18 (0.45)
IPT	14 (0.55)	1.867	26 (1.98)	1.860	15 (0.55)	1.850	26 (1.80)	1.845	14 (0.49)	1.860	26 (1.98)	1.855
	27 (1.71)		18 (0.28)		27 (1.81)		30 (0.56)		19 (1.77)		18 (0.29)
Case V	With 3 PV Systems and 3 BESS Units (W1 = 0.4, W2 = W3 = 0.3)
IGT	18 (0.58)	1.737	26 (1.67)	1.719	12 (0.64)	1.724	26 (1.51)	1.631	7 (0.67)	1.741	26 (1.60)
	26 (1.16)		18 (0.44)		26 (1.36)		18 (0.63)		26 (1.24)		18 (0.50)	1.702
	30 (0.57)		25 (0.21)		32 (0.41)		30 (0.26)		30 (0.51)		32 (0.21)
IPT	15 (0.47)	1.755	26 (1.54)	1.741	15 (0.39)	1.735	26 (1.75)	1.685	25 (0.43)	1.750	26 (1.66)	1.739
	26 (0.94)		18 (0.49)		19 (0.83)		18 (0.33)		26 (1.09)		18 (0.39)
	30 (0.85)		25 (0.23)		26 (1.13)		30 (0.28)		30 (0.74)		30 (0.22)

IGT: Improved Grey Wolf Optimization—Technique for Order of Preference by Similarity to the Ideal Solution, IPT: Improved Particle Swarm Optimization—Technique for Order of Preference by Similarity to the Ideal Solution. Deterministic Aggregated Optimal BESS Size = 2.3 MW (IGT), 2.26 MW (IPT), Probabilistic Beta PDF Aggregated Optimal BESS Size = 2.42 MW (IGT), 2.36 MW (IPT), Polynomial Multiple Regression Aggregated Optimal BESS Size = 2.4 MW (IGT), 2.27 MW (IPT).

for comprehensive comparison and the performance assessment of the modified IEEE 33-bus RDS.

In these five cases, Table 2 shows a significant loss reduction in sequentially distributed BESS placement compared to simultaneously distributed BESS placement for all three approaches, which is graphically presented in Figure 6.

The loss reduction is obtained for the five cases (Case I to Case V) using deterministic, probabilistic, and PMR approaches with simultaneous and sequential BESS placements. It is presented in Table 2 and graphically shown in Figure 6. The comprehensive loss comparison for the modified IEEE 33-bus RDS is given in Figure 6.

It is noted from Figure 6a–c that the loss reduction in the case of sequential BESS placements for all approaches is more significant compared to the simultaneous BESS placements. It is observed from Figure 6d that the loss reduction in the case of sequential BESS placements for the probabilistic approach is greater compared to the other two approaches due to the larger BESS size. The observed reduction in losses may seem marginal; however, it represents a daily decrease in megawatt-hours (MWh) with an aggregated BESS capacity of 2.3 MW for the deterministic approach, 2.42 MW for the probabilistic (β-PDF) approach, and 2.4 MW for the PMR approach. Over the course of a year, this loss reduction will accumulate significantly, particularly with larger BESS capacities implemented in more extensive system configurations.

5.3. Analysis of System Parameters for Probabilistic Approach (Beta PDF) for MBRDS

This work analyzes the optimization of multiple objectives, minimizing the APL and PSI and maximizing the VSI for the OPSBESS for the MBRDS. Probabilistic (Beta PDF) and PMR approaches account for uncertainties in solar irradiance, and then they are evaluated against a deterministic approach. The analysis encompasses a variety of system parameters, including the APL, PSI, VSI, voltage profiles, overall system power profiles, and convergence curve of IGWO, for all three approaches. However, for the purpose of understanding, the results are shown for the aggregated OPSBESS (Case III) for the probabilistic approach in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. These figures illustrate three scenarios: the base case (standard IEEE 33-bus RDS—Case I), the PV case (modified IEEE 33-bus RDS with three PV placements—Case II), and the PV + BESS case (MBRDS with aggregated BESS integration—Case III).

Figure 7 shows that the system losses are highest in the base case. In the PV case, the system losses decrease with the availability of PV power and reach their lowest when both the PV and battery are utilized. Placing the PV systems near the maximum load or at the farthest end reduced the system losses. Adding three PV systems into the standard IEEE 33-bus RDS (MBRDS) changed the load flow. The load flow is run considering the APL minimization, PSI minimization, and VSI maximization criteria to identify the weakest bus for aggregated BESS placement. This addition of an aggregated BESS into the MBRDS reduced system losses, irrespective of its charging–discharging cycles.

Figure 7 shows a sudden rise in the system losses when the BESS acts as a load and is charged by the PV or grid power in the morning. The system losses decrease when the BESS is discharged in the evening. The net result of the charging and discharging of the BESS throughout the day, along with the PV, reduces the system losses to a greater extent.

A PSI value approaching zero (PSI ≤ 0) signifies enhanced line voltage stability within the system. The bus with the highest PSI value is the most suitable for BESS placement. The IGWO-TOPSIS techniques minimized this index after placing an optimum-sized BESS. As depicted in Figure 8, the buses with the PV + BESS result in the lowest PSI values, while the base case (Case I) and the case with PV placement (Case II) exhibit changes in PSI values at a few buses.

Figure 8 indicates that the combination of the PV and BESS provides greater system stability compared to the base case or the PV case alone, as the maximum PSI value obtained is 0.01 pu.

Figure 9 indicates that the PV + BESS has higher voltage stability than the base case (Case I) and the system with the PV case (Case II). The system is stable when the VSI is less than one and unstable when it is more than one. A VSI closer to one indicates higher voltage stability. After PV placement, the VSI values for all buses (except the slack bus) are computed without the integration of the BESS and arranged in ascending order. The bus exhibiting the lowest VSI demonstrates an increased sensitivity to voltage collapse and is the most appropriate candidate bus for the installation of the BESS. This installation is expected to enhance the stability conditions of the overall system effectively. The IGWO-TOPSIS process maximized this index after placing an optimum-sized BESS.

Figure 10 shows the system’s voltage profiles for three scenarios: the base case (Case I), the case involving PV placements (Case II), and the case incorporating both PV and BESS placements (Case III). In the base case, which utilizes the standard IEEE 33-bus RDS, it has been observed that the system voltage is at its lowest across all buses. After adding three PV systems to the standard IEEE 33-bus RDS, the system voltage at all buses is improved. The system voltage is further enhanced by adding an aggregated BES System. A notable enhancement in the system voltage profile has been observed when utilizing the combined PV and BESS, in comparison to the other two scenarios.

Figure 11 presents a comprehensive overview of the power flow within the system, demonstrating how the PV power, grid power, and BESS power collectively fulfill the system load profile. During the first hour to the sixth hour (initial six hours), when PV power is unavailable, the BESS is charged utilizing the power from the grid. In the period from the 6th hour to the 7th hour, both the grid and BESS contribute to managing the system load in the absence of PV power. From the 7th hour to the 17th hour, the combination of solar PV power and grid power sufficiently meets the load demands while simultaneously charging the BESS. Between the 17th hour and the 19th hour, the system load is adequately supported by PV power, grid power, and BESS power. Notably, the majority of the load from the 20th hour to the 24th hour is handled by the BESS in conjunction with grid power. After the 24th hour, from the 1st hour to the 6th hour (initial 6 hours), the load fulfillment sequence is repeated for the complete day. This indicates a reduced reliance on grid power when PV power and BESS power are available.

Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 suggest that although the introduction of PV reduces power losses and enhances the voltage profile, the integration of both PV and BESS leads to even more significant improvements in system performance.

The IGWO technique used to determine the OPSBESS has considered 50 populations and 100 iterations to reach the optimal value of the MOF. Figure 12a,b display the convergence curves of the IGWO and IPSO techniques for Case III for the MBRDS.

The above five cases (Case I to V) are presented for the modified IEEE 33-bus RDS. The results of the real 94-bus Portuguese RDS are presented in the next section.

5.4. Real 94-Bus Portuguese RDS

The proposed work of the OPSBESS is implemented in the real 94-bus Portuguese RDS and analyzed for various cases with placements of the PV and BESS (Case I to Case V), as explained below.

Case I: real 94-bus Portuguese RDS.

A single-line diagram of a 15 kV, 100 MVA, 94-bus real RDS located in Portugal is shown in Figure 13 [23,38]. It has a base active and reactive load of 4.797 MW and 2.323 MVAR, respectively [39].

Case II: real 94-bus Portuguese RDS with placements of PV systems (modified real 94-bus Portuguese RDS—MRBPRDS without BESS).

With voltage improvement and loss reduction as prime objectives, three PV systems are placed at specific buses (19, 58, 84) of the real 94-bus Portuguese RDS described in Case I. Thus, the system formed is termed as a modified real 94-bus Portuguese RDS (MRBPRDS). Each PV system has a power rating of 800 kW. Consequently, the aggregate capacity of the three PV systems is 2.4 MW. The addition of three PV systems to the real 94-bus Portuguese RDS changed the load flow, reduced the system losses, and improved the voltage profile of each bus.

Case III: MRBPRDS with aggregated BESS.

In this case, load flow is run to satisfy the MOF and the constraints to find the optimal BESS size and candidate bus for BESS placement after placing three PV systems. Upon conducting the initial load flow analysis, it was determined that Bus 20 emerged as the highest priority location for the installation of the aggregated BESS. This strategic placement of the optimally sized BESS effectively minimized both the APL and PSI while simultaneously maximizing the VSI. Bus 20 was identified as the candidate bus for placing an aggregated BESS with a size of 1.8 MW for the deterministic approach, 1.96 MW for the B-PDF approach, and 1.84 MW for the PMR approach. Figure 14 shows the MRBPRDS with the placement of an aggregated BESS—Case III.

5.4.1. Analysis of MRBPRDS Using Simultaneous and Sequential BESS Placements

Two strategies, simultaneous and sequential BESS placements, are applied to the MRBPRDS, considering their optimal size. For aggregated BESS placement (Case III), as only one unit of the BESS needs to be placed after the first load flow run in the system, there is no difference between the logic for simultaneous and sequential placements of the BESS. This logic of BESS placement is explained in Section 5.2.1 and Section 5.2.2 for the modified IEEE 33-bus RDS. Similarly, it is applied to the MRBPRDS. In these cases, the optimal size of the aggregated BESS unit is distributed optimally into multiple smaller units based on the system’s requirements. The logic of the OPSBESS changes for Case IV and Case V. This logic is implemented in the MRBPRDS for deterministic, probabilistic (Beta PDF), and PMR approaches. The outcomes are presented in

Table 3. Simultaneous and sequential OPSBESS for MRBPRDS.

Case Number	Deterministic Approach				Probabilistic Beta PDF Approach				Polynomial Multiple Regression Approach
	Simultaneous OPSBESS		Sequential OPSBESS		Simultaneous OPSBESS		Sequential OPSBESS		Simultaneous OPSBESS		Sequential OPSBESS
	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)	BESS Location (Size MW)	System Losses (MWh)	BESS Location (Size, MW)	System Losses (MWh)
Case I	No PV and No BESS (W1 = 1, W2 = W3 = 0)
IGT	-	6.772	-	6.772	-	6.772	-	6.772	-	6.772	-	6.772
IPT	-	6.772	-	6.772	-	6.772	-	6.772	-	6.772	-	6.772
Case II	With 3 PV Systems and No BESS (W1 = 1, W2 = W3 = 0)
IGT	No BESS	5.382	No BESS	5.382	No BESS	5.108	No BESS	5.108	No BESS	5.378	No BESS	5.378
IPT	No BESS	5.382	No BESS	5.382	No BESS	5.108	No BESS	5.108	No BESS	5.378	No BESS	5.378
Case III A	With 3 PV Systems and 1 BESS Unit (Aggregated BESS) (W1 = 1, W2 = W3 = 0)
IGT	19 (1.8)	4.837	19 (1.8)	4.838	19 (1.96)	4.552	19 (1.96)	4.552	19 (1.84)	4.846	19 (1.84)	4.846
IPT	19 (1.8)	4.838	19 (1.8)	4.838	19 (1.96)	4.553	19 (1.96)	4.553	19 (1.84)	4.847	19 (1.84)	4.847
Case III B	With 3 PV Systems and 1 BESS Unit (Aggregated BESS) (W1 = 0.4, W2 = W3 = 0.3)
IGT	20 (1.8)	4.902	20 (1.8)	4.902	20 (1.96)	4.580	20 (1.96)	4.617	20 (1.84)	4.869	20 (1.84)	4.869
IPT	20 (1.8)	4.902	20 (1.8)	4.902	20 (1.96)	4.580	20 (1.96)	4.553	20 (1.84)	4.870	20 (1.84)	4.870
Case IV	With 3 PV Systems and 2 BESS Units (W1 = 0.4, W2 = W3 = 0.3)
IGT	77 (1.09)	4.639	20 (1.00)	4.620	77 (1.21)	4.356	20 (1.37)	4.325	77 (1.38)	4.681	20 (1.67)	4.621
	61 (0.75)		58 (0.8)		88 (0.74)		58 (0.58)		61 (0.45)		58 (0.17)
IPT	58 (1.44)	4.640	20 (1.09)	4.639	58 (1.17)	4.356	20 (1.25)	4.339	77 (1.38)	4.703	20 (1.37)	4.693
	77 (0.40)		58 (0.75)		77 (0.79)		58 (0.71)		61 (0.47)		58 (0.48)
Case V	With 3 PV Systems and 3 BESS Units (W1 = 0.4, W2 = W3 = 0.3)
IGT	20 (1.02)	4.613	20 (0.87)	4.616	58 (0.68)	4.391	20 (1.26)	4.335	20 (0.92)	4.632	20 (1.56)	4.612
	58 (0.1)		58 (0.72)		77 (0.75)		58 (0.54)		58 (0.52)		58 (0.20)
	77 (0.72)		84 (0.23)		84 (0.52)		84 (0.15)		77 (0.40)		84 (0.07)
IPT	20 (0.45)	4.631	20 (1.058)	4.615	58 (1.22)	4.365	20 (1.35)	4.352	20 (0.87)	4.704	20 (1.35)	4.703
	27 (0.78)		50 (0.04)		77 (0.72)		58 (0.53)		59 (0.58)		58 (0.42)
	59 (0.61)		58 (0.75)		84 (0.02)		92 (0.08)		84 (0.40)		92 (0.08)

IGT: Improved Grey Wolf Optimization—Technique for Order of Preference by Similarity to the Ideal Solution, IPT: Improved Particle Swarm Optimization—Technique for Order of Preference by Similarity to the Ideal Solution. Deterministic Aggregated Optimal BESS Size = 1.8 MW (IGT, IPT), Probabilistic Beta PDF Aggregated Optimal BESS Size = 1.96 MW (IGT, IPT), Polynomial Multiple Regression Aggregated Optimal BESS Size = 1.84 MW (IGT, IPT).

.

The results of the simultaneous and sequential OPSBESS for the MRBPRDS under resource uncertainty are shown in Table 3. The base case load flow is run using the forward–backward sweep method to obtain the system losses. IGT and IPT are applied to multi-objective functions. It is observed that IGT Case III B converges earlier than IPT. It shows a reduction in losses compared to IPT, whereas in the single-objective function (Case III A), IPT gives better results than IGT. This shows that IGT handles the MOF better than IPT, and the same can be visualized from the convergence curves shown in Figure 12a,b.

The loss reduction is obtained for five cases (Case I to Case V) using deterministic, probabilistic, and PMR approaches with simultaneous and sequential BESS placements. It is presented in Table 3 and graphically shown in Figure 15. The comprehensive loss comparison for the modified IEEE 33-bus RDS is given in Figure 15.

It is noted from Figure 15a–c that the loss reduction in the case of sequential BESS placements for all approaches is more significant compared to the simultaneous BESS placements. It is observed from Figure 15d that the loss reduction in the case of sequential BESS placements for the probabilistic approach is greater compared to the other two approaches due to the larger BESS size. The observed reduction in losses may seem marginal; however, it represents a daily decrease in megawatt-hours (MWh) with an aggregated BESS capacity of 1.8 MW for the deterministic approach, 1.96 MW for the probabilistic (β-PDF) approach, and 1.84 MW for the PMR approach. Over the course of a year, this loss reduction will accumulate significantly, particularly with larger BESS capacities implemented in more extensive system configurations. It is observed from Figure 15d that the loss reduction in the case of sequential BESS placements for the probabilistic approach is greater compared to the other two approaches due to the larger BESS size.

5.4.2. Analysis of System Parameters for Probabilistic (Beta PDF) Approach for Real 94-Bus Portuguese RDS

This work analyzes the optimization of multiple objectives, minimizing the APL and PSI and maximizing the VSI for the OPSBESS for the MRBPRDS. The probabilistic approach, specifically the Beta PDF and PMR approach, effectively accounts for uncertainties in solar irradiance and is subsequently compared with a deterministic method. The results include a comprehensive analysis of various system parameters, such as the APL, PSI, VSI, voltage profile, and overall system profile for all three approaches, along with the convergence curve of IGWO. However, for the purpose of understanding, the results are shown for the aggregated OPSBESS (Case III) for the probabilistic approach in Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20.

These figures illustrate three scenarios: the base case (real 94-bus Portuguese RDS—Case I), the PV case (real 94-bus Portuguese RDS with three PV placements—Case II), and the PV + BESS case (MRBPRDS with aggregated BESS integration—Case III).

Figure 16 shows that the system losses are highest in the base case. In the PV case, the system losses decrease with the availability of PV power and reach their lowest when both the PV and BESS are utilized. Placing the PV systems near the maximum load or the farthest end reduced the system losses. Adding three PV systems into the standard real 94-bus Portuguese RDS (MRBPRDS) changed the load flow. The load flow is run considering the APL minimization, PSI minimization, and VSI maximization criteria to identify the weakest bus for aggregated BESS placement. This addition of an aggregated BESS into the MBRDS reduced system losses, irrespective of its charging–discharging cycles. Figure 16 shows a sudden rise in the system losses when the BESS acts as a load and is charged by the PV or grid power in the morning. The system losses decrease when the BESS is discharged in the evening. The net result of the charging and discharging of the BESS throughout the day, along with the PV, reduces the system losses to a greater extent.

The above five cases (Case I to V) for the RDS are analyzed for solar PV irradiance in Portugal. However, depending upon seasonal variations in the solar PV irradiance of a particular location, there may be significant differences in the profiles of solar irradiance. This results in a change in the OPSBESS, which considerably changes the performance of the PV-BES System.

A PSI value nearer to zero (PSI ≤ 0) reflects the improved stability of the line voltage within the system. The bus with the highest PSI value is the most suitable for BESS placement. The IGWO-TOPSIS techniques tried to reduce the PSI values to near to zero along with reducing the APL and maximizing the VSI values. The nature of the PSI changes as per the system characteristics. In the case of the IEEE 33-bus system (Figure 8), the PSI values decreased with PV placement (Case II) and PV + BESS placements (Case III). The PSI values are quite small (PSI ≤ 0) for the base case (Case I); however, these values are further reduced with PV placement and PV + BESS placements.

From Figure 17, it is seen that in the case of the real 94-bus Portuguese RDS, the PSI values are quite small (PSI ≤ 0), and at a few buses, the PSI values merge with the values of the PV and PV + BESS cases. But the PSI values increase at a few buses where the PV and BESS are installed. The combination of the PV and BESS provides greater system stability, with a maximum PSI value of 0.01 p.u.

Figure 18 indicates that the PV + BESS has higher voltage stability than the base case (Case I) and the system with the PV case (Case II). The system is stable when the VSI is less than one and unstable when it is more than one. A VSI closer to one indicates higher voltage stability. After PV placement, the VSI values for all buses (except the slack bus) are computed without the integration of the BESS and arranged in ascending order. The bus that demonstrates the lowest VSI is particularly sensitive to voltage collapse. Consequently, it is identified as the most suitable candidate for the installation of the BESS, which can facilitate improved stability conditions within the system. The IGWO-TOPSIS process maximized this index after placing an optimum-sized BESS.

Figure 19 outlines the system’s voltage profiles for three scenarios: the base case (Case I), the case involving PV placements (Case II), and the case incorporating both PV and BESS placements (Case III). For the base case (94-bus Portuguese RDS), the system voltage is found to be the lowest for all buses. After adding three PV systems to the 94-bus Portuguese RDS, the system voltage at all buses is improved. The system voltage is further enhanced by adding an aggregated BES System. A notable enhancement in the system voltage profile has been observed when utilizing the combined PV and BESS, in comparison to the other two scenarios.

Figure 20 gives the complete power flow of the system and illustrates how solar PV power, grid power, and BESS power collectively meet the system load profile. The load is supplied by the grid and BESS in the first hour and the second hour (initial two hours).

During the third hour to the fifth hour (initial three hours), when PV power is unavailable, the grid supplies power to the load and simultaneously charges the BESS. In the period from the 5th hour to the 8th hour, both the grid and BESS contribute to managing the system load in the absence of PV power. From the 8th hour to the 17th hour, the combination of solar PV power and grid power sufficiently meets the load demands while simultaneously charging the BESS. Between the 17th hour and the 19th hour, the system load is adequately supported by PV power, grid power, and BESS power. Notably, the majority of the load during the 20th hour to the 24th hour and the initial 2 hours is handled by the BESS in conjunction with grid power.

After the third hour to the fifth hour (initial three hours), the load fulfillment sequence is repeated for the complete day. This indicates a reduced reliance on grid power when PV and BESS power are available.

Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 suggest that although the introduction of the PV reduces power losses and enhances the voltage profile, the integration of both the PV and BESS leads to even more significant improvements in system performance.

The IGWO technique considered 50 populations and 100 iterations to reach the optimal value of the MOF. Figure 21a,b display the convergence curves of the IGWO and IPSO techniques for Case III for the 94-bus Portuguese RDS.

The above five cases (Case I to Case V) implemented in the MBRDS and MRBPRDS indicate that the system losses are minimal for the probabilistic approach (β-PDF). This study shows that the integration of both PV and BES Systems results in more pronounced enhancements in overall system performance.

Resource uncertainty in relation to the optimal planning of Distributed Generations (DGs) or BESSs aimed at minimizing system losses remains relatively underexplored by researchers. This study presents a comparative analysis of existing research outcomes with the proposed work, which specifically focuses on the IEEE 33-bus and 94-bus Portuguese systems, as detailed in

Table 4. Table of comparison for resource uncertainty.

Sr. No.	Paper	Resource Uncertainty	MOF	Optimization Techniques	APL (kW)	PSI (pu)	VSI (pu)	DG/BESS Size (MW) and Location
IEEE 33-bus RDS
1	Arya Abdolahi [5]	PV by Beta PDF; wind by Weibull PDF	Congestion alleviation and procurement cost minimization	MOPSO	Not mentioned	-	-	2 PVs @ Bus 11, 22; 2 WTs @ Bus 6, 32; 2 CHP @ Bus 16, 26; 6 BESSs @ Bus 5, 10, 14, 20, 24,31. Total DG Size = not mentioned.
2	Mohd Nor [7]	PV by Beta PDF	ELI and VD	GA	14,773.75	-	-	PV of 3.3 @ Bus 6; BESS of 1.84 @ Bus 6. Total DG Size = 3.3 MW.
3	Elkadeem [36]	PV by Beta PDF	Loss, VSI, VD, and AEL	HHO	83.0964	-	(28.828) or 0.9 *	3 PVs (0.761 @ Bus 14; 1.0947 @ Bus 24; 1.0684 @ Bus 30); 3 WTs (0.8204 @ Bus 14; 1.173 @ Bus 24; 1.481 @ Bus 30). Total DG Size = 6.3985 MW.
4	Proposed Work	PV by Beta PDF and PMR	APL, PSI, and VSI (W1 = 0.4, W2 = W3 = 0.3)	IGWO	74.375	0.01	0.8	PV of 0.6 MW @ Bus 18, 25, 30; BESS of 2.42 @ Bus 26. Total DG Size = 1.8 MW.
Real 94-bus Portuguese RDS
5	Elkadeem [36]	PV by Beta PDF	Loss, VSI, VD, and AEL	HHO	213.661	-	(76.384) or 0.82 *	1 PV (2.636 @ Bus 19); 1 WT (2.968 @ Bus 19). Total DG Size = 5.604 MW.
6	Proposed Work	PV by Beta PDF and PMR	APL, PSI, and VSI (W1 = 0.4, W2 = W3 = 0.3)	IGWO	192.375	0.01	0.62	PV of 0.8 MW @ Bus 19, 58, 84; BESS of 1.96 @ Bus 20. Total DG Size = 2.4 MW.

APL: Active Power Loss; PSI: Power Stability Index; VSI: Voltage Stability Index; DG/BESS: Distributed Generation/Battery Energy Storage System; MOPSO: Multi-Objective Particle Swarm Optimization; WT: wind turbine; CHP: Combined Heat Power; ELI: Energy Loss Index; VD: voltage deviation; GA: Genetic Algorithm; AEL: Annual Economic Loss; HHO: Harris Hawk Optimizer; PMR: Polynomial Multiple Regression; IGWO: Improved Grey Wolf Optimization. * Overall VSI (OVSI) = (${\sum }_{i=2}^{NB}\left|OVSI\right|$) at all buses except the slack bus.

. The proposed work is examined in comparison to prior studies in several key areas, including test systems, the modeling of resource uncertainty, objective functions, optimization techniques, system losses (APLs), the VSI, and the sizing and locations of DGs or BESSs. Furthermore, while the PSI has received limited attention in the literature, this work advocates for its inclusion due to its innovative potential.

These VSI values are notably high for both the IEEE 33-bus and the actual 94-bus Portuguese systems, which can be attributed to the larger PV installations, measuring 6.3985 MW and 5.604 MW, respectively. The dimensions of Distributed Generations (DGs) or BESSs and the corresponding system parameters depicted in this table are associated with the differing objective functions and the varying percentage levels of PV penetration. It is observed that the variations in PV sizes have a significant impact on the sizes of the BESS and influence system parameters such as the APL and VSI.

In this paper, the results derived from the probabilistic approach utilizing the Beta PDF for the MBRDS and the MRBPRDS, which take into account resource uncertainty, are comprehensively discussed. However, the findings associated with the deterministic and PMR approaches are presented in the Appendix A for a better understanding of the importance of the probabilistic approach.

6. Conclusions

The impact of the randomness in solar irradiance and its effect on BESS sizing and placements is analyzed with the help of two RDSs. This work proposes a new approach to the placement of optimally sized BESSs with the prioritized multiple-objective functions of the integrated APL, PSI, and VSI. The function is prioritized using the WSM and simultaneously optimized using the IGT technique. Additionally, the results are validated using the IPT technique.

This study presents five cases (Case I to Case V) for the modified IEEE 33-bus and modified real 94-bus Portuguese RDSs. The presented RDS results are analyzed for solar PV irradiance in Portugal using three approaches. The system losses are minimal for the probabilistic approach compared to the other two approaches due to the largest BESS size. This study suggests that although the introduction of the PV reduces power losses and enhances the voltage profile, the integration of both the PV and BESS leads to even more significant improvements in system performance. In addition to this, the effectiveness and robustness of the sequential and simultaneous BESS placements have been comparatively tested and validated on both RDSs. The sequentially distributed BESS stabilizes the system more than the simultaneously distributed BESS. The distributed BESS placement reduces the system loss significantly compared to the aggregated BESS.

It is observed that system losses are minimal for the probabilistic approach compared to the other two approaches due to the largest BESS size. However, the deterministic approach reflects the results without accounting for variability in the resource. The results indicate that the probabilistic and PMR approaches need larger BESS sizes but give practical solutions to PV-BES Systems, considering the uncertainty in solar irradiance. Therefore, this study shows the importance of using probabilistic and PMR approaches over a deterministic approach for the OPSBESS. The results emphasize that, in practice, for a large BESS capacity, significant loss reduction can be observed along with improved system performance. This study can be extended to various energy resources, such as PV–wind turbine–BESS or PV–wind turbine–fuel cell, for different RDSs.