BESS Sizing Optimization Combined with Optimal Scheduling Method Considering the Battery Degradation Using PSO

Alioune Diouf ^1,*, Yasuhiro Noro ², Fujita Goro ³

1. Department of Electrical Engineering, Shibaura Institute of Technology, Tokyo, Japan

2. Department of Electrical and Electronics Engineering, Kogakuin University, Tokyo, Japan

3. Electrical Engineering and Robotics Course, Electrical and Electronics Engineering, College of Engineering, Shibaura institute of Technology, Tokyo, Japan

* Correspondence: Alioune Diouf

Academic Editor: Tek Tjing Lie

Collection: Optimal Energy Management and Control of Renewable Energy Systems

Received: June 17, 2025 | Accepted: October 14, 2025 | Published: October 24, 2025

Journal of Energy and Power Technology 2025, Volume 7, Issue 4, doi:10.21926/jept.2504015

Recommended citation: Diouf A, Noro Y, Goro F. BESS Sizing Optimization Combined with Optimal Scheduling Method Considering the Battery Degradation Using PSO. Journal of Energy and Power Technology 2025; 7(4): 015; doi:10.21926/jept.2504015.

© 2025 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

Renewable energy source (RES) projects are increasing worldwide [1,2] because of global warming and carbon neutrality challenges. For example, to achieve a 46% reduction in CO₂ emissions by 2030, Japan has significantly increased its PV generation over the last decade [3]. In the USA, solar power is expected to grow to 250 GW by 2029 [4]. In 2021, the first Eco-Park SolarMicrogrid (EPSMG) in Sub-Saharan Africa, in the international industrial park of Diamniadio (P2ID), was installed in Senegal. It is an innovative PV solar plant coupled with a battery energy storage system (BESS) along with an electric vehicle charge station [5].

Although RES have become attractive, their variability and unpredictability are the main challenges for better integration with existing conventional generators [6]. Similar to traditional generator operators, RES operators must identify strategies to store their surplus production during better weather conditions or release their supply under specific conditions for ancillary services. This strategy allows the RES power plant to participate: i.) technically, in power balance support, frequency and voltage regulation support, power generation planning, etc., and ii.) economically, for energy arbitrage, participation in electricity markets, etc. As RESs fluctuate over time and are difficult to predict, BESSs have garnered significant attention for coping with the challenges above. Therefore, several studies have been conducted to integrate BESS into RES plants. In [7,8,9,10], BESS was employed for frequency regulation services; Haessig et al. [11] studied forecast error reduction to enhance RES planning; Zhang et al. [12] addressed electricity market participation based on BESS; and Diouf and Noro [13] proposed electricity cost reduction for time-of-use (TOU) customers employing BESS.

When a technical issue prompts RES plants to employ BESS, the size of the BESS might not receive considerable attention, as long as this technical challenge is satisfied. In addition, from an economic viewpoint, an optimization approach for BESS sizing is crucial to achieve cost-effectiveness owing to its considerable investment cost. Moreover, when a BESS undergoes frequent operation cycles, such as charging and discharging, its aging can be accelerated, resulting in high operation costs due to frequent replacements [14]. Therefore, when a RES-based power plant installs a BESS to reduce electricity charges, the optimal BESS size choice is essential and can be achieved by maximising revenue with minimum investment cost.

There are several methods to achieve this goal. Naces et al. [15] minimised the overall cost through the BESS lifetime. However, the disadvantage of this approach is that the period of return on investment is not considered, which might be long even if the overall electricity cost is minimised. Shin and Hur [16] maximised the net present cash flow to determine the optimal BESS size. An analytical algorithm was proposed to solve the BESS operation during the BESS sizing process, and the degradation effect was addressed with a BESS augmentation scheme. However, BESS operation algorithms are not optimised, which can result in an inappropriate BESS size. Another method is to maximise the rate of return (RoR) of the investment. The total operation cost for the BESS-based customer depends on the battery size; specifically, its SoC and output power are limited by the battery's rated energy and power capacity, which in turn determines the BESS investment cost. Therefore, the total operating and investment costs should be considered simultaneously in the BESS sizing optimisation process. Additionally, the total energy saved (ES) represents the difference between the two operational costs corresponding to the cases when the customer operates without and with BESS. The ratio of the ES to the investment cost determines the investment RoR. Specifically, maximising the RoR allows the customer to minimise the operational cost as well as shorten the payback period. Hu et al. [17] demonstrated that the maximum revenue does not necessarily imply a short payback period. However, regardless of the approach considered, a comprehensive sizing method is required to achieve cost-effective BESS integrations.

The comprehensive BESS sizing method must consider at least one decision vector of four parameters: charging and discharging scheduling, state of charge (SoC), BESS rate energy and power capacity, and degradation effect in the State of Health (SoH). The problem can be divided into two levels: BESS size optimisation and operation schedule optimisation. Furthermore, the SoC and SoH can be considered as constraints for the second level. These levels interact during the optimisation process, and the SoH variation is nonlinear. Most BESS size studies either omit one of these decision variables, exclude the interaction of the two optimization levels, and neglect the degradation effect, or consider the second level but solve it via a non-optimized approach. Yoo et al. [18] optimised the BESS size and its operation simultaneously, but did not focus on the effect of degradation. Saez-de-Ibarra et al. [6] addressed BESS sizing while considering health effects. However, the applied linear programming algorithm could not adequately address the nonlinearity of the problem.

Since the aforementioned decision variable cannot be solved simultaneously using a linear programming approach due to its nonlinear behavior, particularly in SoH modeling, this study proposes using the particle swarm optimization (PSO) algorithm to address these issues. The PSO algorithm is widely used in power engineering, known for its simplicity and ability to reach near-global optimal solutions more efficiently than other methods, with lower calculation time [13]. The second level of this optimisation problem (i.e. operation scheduling) has been covered by several studies using diverse algorithms. For instance, dynamic programming (DP) [17,19,20] and genetic algorithms [21,22] address optimal operations for BESS. However, one of the authors’ recent studies conducted in [13] showed some limitations of these optimal operation approaches while addressing robust BESS optimization scheduling for cost reduction based on PSO.

Therefore, based on its results, the same PSO algorithm is used in the current study to manage the operation schedule simultaneously during the BESS sizing process, which was also solved with PSO. In addition, an RoR-based approach was used.

The contribution of this study is summarised as follows:

• Determine the appropriate BESS size that maximizes the RoR during the first year of operation for different BESS Technologies using PSO.
• During the BESS sizing optimisation process, the BESS Operation was simultaneously optimised using PSO.
• An algorithm was proposed to maintain the SoH above a minimum level by adjusting the operating schedule selected using the PSO algorithm.
• Annual energy savings for the selected BESS from the optimization process for different BESS technologies are achieved over the life cycle to determine the shortest payback period.
• Finally, the BESS with the shortest payback is the best choice for this study.

The remainder of this paper is organised as follows: Section II outlines the motivation for this study. Section III discusses the problem formulation and the BESS optimal sizing procedure. The results and discussions are presented in Section IV. The conclusions are given in Section V.

2. Case Study and Motivation

In August 2020, the first eco-park solar microgrid (EPSMG) in sub-Saharan Africa was built in the international industrial park of Diamniadio (P2ID) [5]. It began functioning in late 2021. The idea was to enable APROSI, a government agency that manages P2ID, to reduce the energy cost of manufacturers while improving environmental impact.

2.1 EPSMG Framework

The system framework is illustrated in Figure 1. It consists of a PV plant of 150 kW connected to a 97 kWh/50 kW storage system based on Li-ion and an electric vehicle charge station. The EPSMG is connected downstream to the primary grid (SENELEC) and coexists with the medium-voltage (MV) 30 kV AC of the P2ID, along with 4 MV/low-voltage substations. Each substation contained a high-voltage switchgear (HVSG). A step-up auto-transform of 150 kVA-0.4 kV/30 kV was connected to both the PV plant and the storage of the 0.4 kV AC buses. Smart meters (SM) measure the energy flowing in AC and DC buses.

Figure 1 P2ID Microgrid system with EPSMG in 2021.

An associated power management system (PMS) and energy management system (ESM) ensure that the EPSMG meets the microgrid energy production and storage requirements via four functions: 1-Injection mode with the possibility of sending energy back to the primary grid (SENELEC), 2-Self consumption with storage, in which the PV production is only consumed by the P2ID and the surplus is stored in the BESS, 3-Solar self-consumption mode, with only local consumption without storage and main grid injection, and 4-Solar self-consumption with injection mode, which includes sending energy back to the grid without storage.

2.2 Pricing System

The public energy tariff, controlled solely by SENELEC and maintaining a sales monopoly at that time, is relatively high for industrial customers, such as P2ID. The electricity tariff for medium-voltage (MV) customers based on the TOU pricing system is illustrated in Table 1 [23].

Table 1 Senegal electricity tariff list in 2021 for MV customer.

In addition, customers are subject to penalties when the contracted power demand is exceeded. These penalties are expressed as follows:

\[ \begin{aligned}Penalty&=1.5\times\frac{PF}{30}\times\, \textit{billing period (day)}\\&\times(Max\textit{ power consumed}-\textit{contracted demand})\end{aligned} \tag{1} \]

2.3 Study Motivation and Assumptions

This study aims to develop a powerful approach to managing investment cost issues by accurately choosing storage capacity and replacement cost, considering the degradation effect in operation scheduling. Therefore, owing to some limitations regarding DATA access in the P2ID and case study pricing systems, the following assumptions are considered:

1. PV and load data were collected from the Tokyo Electric Power Company (TEPCO) for 2023 [24] and reduced to the scale of 1/10000 to fit the MG.
2. Based on the load DATA, only 10% is used to have a situation surplus of PV production [13] to meet the P2ID configuration in 2021, where the EPSMGPV plant produced more than the demand.
3. The P2ID is assumed to be sold back to SENELEC.
4. The selling price was set at 50% of the off-peak tariff, based on TOU pricing.
5. Forecast DATA were evaluated using historical TEPCO data by applying a forecast error.

3. Project Approach

3.1 Problem Formulation

The PSO solving process aims to maximize the ROR as follows:

\[ \mathrm{Max}\,ROR=\frac{ECS_y}{CAP_y} \tag{2} \]

The revenue or energy cost saving $ECS_y$ during a year is the difference between the total annual operation cost when the customer does not operate the BESS ($TOC_y$) and the total annual operation cost when operating the BESS ($TOCB_y$) [19].

Hence:

\[ ECS_y=TOC_y-TOCB_y \tag{3} \]

where:

\[ TOC_y=\sum_{i=1}^{365}\sum_{j=1}^{24}(P_{grid\,i,j}\times Tar_j)+Pen_i \tag{4} \]

\[ TOCB_y=\sum_{i=1}^{365}\sum_{j=1}^{24}(P_{grid\,i,j}^{BESS}\times Tar_j)+Pen_i+MC \tag{5} \]

\[ P_{grid\,i,j}=P_{Load\,i,j}^{for}-P_{PV\,i,j}^{for} \tag{6} \]

\[ P_{grid\,i,j}^{BESS}=P_{Load\,i,j}^{for}-P_{PV\,i,j}^{for}-P_{BESS\,i,j}^{PSO} \tag{7} \]

The BESS maintenance cost MC is defined as 3% of IC:

\[ MC=0.03\times IC \tag{8} \]

According to [25]:

\[ CAP_y=IC\times CRF \tag{9} \]

where:

\[ IC=E_{rate}\times cost_{kWh}+P_{rate}\times cost_{kW} \tag{10} \]

\[ CRF=\frac{d(1+d)^{Pl}}{(1+d)^{Pl}-1} \tag{11} \]

In Equation (7), the upper subscripts (i.e. PSO) are employed to emphasize the second-level suboptimization process using PSO to determine the optimal BESS operation daily. The approach for generating this optimal day-ahead scheduling was based on that used in [13]. Unlike for a time interval of 5 min, as in [13], this study considers a 1-hour time interval, which entails 24 intervals in total. Additionally, the applicable tariffs ($Tar_j$) and the penalties ($Pen_i$) are defined as follows:

\[ Tar_j=\begin{cases}TOU,&\quad P_{grid\,i,j}\geq0\\T_{sold},&\quad P_{grid\,i,j}<0&\end{cases} \tag{12} \]

\[ Pen=\begin{cases}0,&\quad\Delta P\leq0\\1.5\times\frac{PF}{30}\times\Delta P,&\quad\Delta P>0&\end{cases} \tag{13} \]

where:

\[ \Delta P=\max_{t\in \mathit{ \mathcal{T} }}P_{Grid}(t)-P_{cd} \tag{14} \]

In (12), the sold tariff ($T_{sold}$) is applied when an industrial customer sells energy back to the grid ($P_{grid\,i,j}<0$). In this study, it was assumed to be 50% of the TOU during off-peak times. Moreover, the TOU is based on Table 1, and the category concerns general-use-type customers.

The penalty tariff represents 1.5 times the contracted demand price observed in Table 1 (which is divided into 30 days). In addition, unlike (1), the penalties in (13) are evaluated daily, not monthly. This is for simplification purposes, as the second-level operation scheduling proceeds with a day-ahead forecasting approach [13].

Furthermore, the objective function defined in (1) is optimized subject to the following constraints.

3.2 Component Constraint

P2ID can determine the limit of the BESS with respect to its budget capacity, such that the BESS investment cost remains below the maximum investment cost ($IC_{max}$).

Hence:

\[ IC(E_{rate},P_{rate})\leq IC_{max}(E_{rate}^{Max},P_{rate}^{Max}) \tag{15} \]

where:

\[ 0\leq E_{rate}\leq E_{rate}^{Max} \tag{16} \]

\[ 0\leq P_{rate}\leq P_{rate}^{Max} \tag{17} \]

3.3 State of Charge SOC Constraint

\[ SOC_{i,j}=\begin{cases}(1-\mu)SOC_{i,j-1}-\frac{P_{BESS\,i,j}^{PSO}\times\mathrm{\eta}}{E_{rate}};&P_{BESS\,i,j}\leq0\\\\(1-\mu)SOC_{i,j-1}-\frac{P_{BESS\,i,j}^{PSO}}{E_{rate}\times\mathrm{\eta}};&P_{BESS\,i,j}>0&\end{cases} \tag{18} \]

Subjected to this limitation:

\[ SOC_{min}\leq SOC_{i,j}\leq SOC_{max} \tag{19} \]

Relation (18) includes the self-discharging ($\mu$). This was considered to be 0.6% per hour in this study.

3.4 State of Health (SoH) Constraint

SoH is defined as the ratio of the current storage capacity to the energy rate capacity [21]. For every cycle, i.e., charging and discharging mode, the total BESS remaining life cycle decreases. This results in degradation of the BESS storage capacity. Consequently, according to [14], the current BESS storage capacity $C_{i,j}$ is as follows:

\[ C_{i,j}=E_{rate}\left(1-C_d\sum_{j=1}^{24\times(i-1)+j}\frac{0.5}{N_{cycl\,i,j}^{life}}\right) \tag{20} \]

where:

\[ N_{cycl\,i\,j}^{life}=\left|\frac{SOC_{i,j-1}-SOC_{i,j}}{\Delta j}\right|^{-1.453}\times e^{-3.6\left(\frac{SOC_{i,j-1}-SOC_{i,j}}{2}\right)}\times14003 \tag{21} \]

where $\Delta j$ represents the 1 hour time length. Moreover, according to [21]:

\[ SoH_{i,j}=\frac{C_{i,j}}{E_{rate}} \tag{22} \]

Subjected to this limitation:

\[ SoH_{i,j}\geq SOH_{\min i,j} \tag{23} \]

where:

\[ SoH_{\min i,j}=1-\frac{C_d}{365\times24\times Pl}[24\times(i-1)+j] \tag{24} \]

In this study, we propose a SoH lower limit, Equation (24), which must be under the real-time SoH during the optimized process. When the BESS reaches project life $Pl$, its remaining capacity becomes at least $1-C_d$. The proposed SoH algorithm is presented in the following section.

3.5 Charging/Discharging Constraint

The charging and discharging powers are limited by the BESS power rate as follows:

\[ -P_{BESS}^{Max}\leq P_{BESS\,i,j}^{PSO}\leq P_{BESS}^{Max} \tag{25} \]

3.6 Optimization Process

The optimization process is depicted in Figure 2.

Figure 2 Proposed BESS sizing algorithm using PSO.

The proposed algorithm is subjected to the following steps:

3.6.1 Step1: Gathering the Annual Forecast Data and Set the PSO Parameters

The annual historical data were used to estimate the forecast data, as follows:

\[ P_{Load,year}^{for}=P_{Load,year}^{his}(1-e_{Load}) \tag{26} \]

\[ P_{PV,year}^{for}=P_{PV,year}^{his}\left(1-e_{PV}\right) \tag{27} \]

where $e_{Load}$ and $e_{PV}$ represent the PV and Load forecast errors, respectively.

3.6.2 Step 2: Create Random Initial Population

The initial population of BESS sizes was randomly selected. Given k^th particle as a vector with two elements denoted by the rated energy $E_{rate,k}$ and rated power capacity $P_{rate,k}$, we can model the particle population as follows:

where K stands for the search space or total number of particles. Hence, the initial population, $[E_{rate,k}^{Init}\,P_{rate,k}^{Init}]_{1\leq k\leq K}$, was randomly generated as follows:

\[ E_{rate,k}^{Init}=\mathrm{rand}(E_{rate}^{max}) \tag{29} \]

\[ P_{rate,k}^{Init}=\mathrm{rand}(P_{rate}^{max}) \tag{30} \]

This assumption is based on the maximum allowable budget for the BESS, as defined in Equations (15), (16), and (17).

3.6.3 Step3: Determining the Daily BESS Scheduling over a Year for the Initial Particle Population Generated from Step 2, Based on [13]

Each k^th particle, which represents a specific BESS size, should be optimally dispatched daily for one year as follows:

\[ \begin{bmatrix}P_{BESS,i,j,k}^{PSO,init}\end{bmatrix}_{1\leq k\leq K}=\begin{pmatrix}P_{BESS,i,1,k}^{PSO,init},\,P_{BESS,i,2,k}^{PSO,init}\,...P_{BESS,i,24,k}^{PSO,init}\end{pmatrix} \tag{31} \]

3.6.4 Step4: Evaluate the Initial Population $[E_{rate,k}^{Init}{P}_{rate,k}^{Init}{]}_{1\le <!--\; \le \; -->k\le <!--\; \le \; -->K}$, Based on [13]

The initial particle population is evaluated to define $p_{gbest}^{init}$ and $p_{best\,k}^{init}$ using the fitness function $\mathrm{F}_k$ defined as follows:

\[ \mathrm{F}_k\left([E_{rate,k}^{Init}\,P_{rate,k}^{Init}]\right)=\frac{TOC_y-TOCB_{y,k}\left([P_{BESS,i,j,k}^{PSO,Init}]\right)}{\left(E_{rate,k}^{Init}\times cost_{kWh}+P_{rate,k}^{Init}\times cost_{kW}\right)\times\mathrm{CRF}} \tag{32} \]

Hence, from (32), $p_{gbest}^{init}$ and $p_{best\,k}^{init}$ can be expressed as follows:

\[ \mathrm{Max}\left\{\mathrm{F}_k\left(\left[E_{rate,k}^{Init}\,P_{rate,k}^{Init}\right]_{1\leq k\leq K}\right)\right\}=[ind,\,maxval] \tag{33} \]

\[ p_{gbest,t}^{init}=[E_{rate,ind}^{Init}\,P_{rate,ind}^{Init}] \tag{34} \]

\[ p_{best\,i,t}^{init}=[E_{rate,k}^{Init}\,P_{rate,k}^{Init}]_{1\leq k\leq K} \tag{35} \]

3.6.5 Step5: Update the Velocity and Position and Apply Their Limits

This process is explained in detail in [13]. However, the position limits are based on the component limits defined by the customer budget, as expressed in (16) and (17).

3.6.6 Step 6: Determining the Daily BESS Schedule over a Year for the New Population Based on [13]

Similar from Step 2, the daily schedule , considering the iteration (iter), is a vector defined as follows:

\[ \begin{bmatrix}P_{BESS,i,j,k}^{PSO,iter}\end{bmatrix}_{1\leq k\leq K}=\begin{pmatrix}P_{BESS,i,1,k}^{PSO,iter},\,P_{BESS,i,2,k}^{PSO,iter}\,...P_{BESS,i,24,k}^{PSO,iter}\end{pmatrix} \tag{36} \]

After finding the optimal schedule on day i, the SoH constraint is applied based on section 2, using equations (23) and (24), to relieve premature BESS degradation. The proposed SoH algorithm runs through eight stages. An adjustment coefficient (adj) is defined to redistribute the optimally dispatched BESS output. Further, any coefficient is appropriate as long as the SoH limitation requirement is satisfied, such that the SoH during the optimization process remains above the minimum as set in (24). The SOH algorithm is given as follows:

for $\boldsymbol{p}=\boldsymbol{1}$: Number of particles

for $\boldsymbol{j}=\boldsymbol{1}:\boldsymbol{24}$

if $\boldsymbol{P_{BESS\,i,j,p}^{PSO}}>\boldsymbol{0}$

$\boldsymbol{SOC_{i,j}}=\boldsymbol{SOC_{i,j-1}*(1-\mu)-P_{BESS\,i,j,p}^{PSO}/(E_{rate}*\eta)}$;

else

$\boldsymbol{SOC_{i,j}}=\boldsymbol{SOC_{i,j-1}*(1-\mu)-P_{BESS\,i,j,p}^{PSO}/(E_{rate}*\eta)}$;

end

$\boldsymbol{ N_{cycl\,i,j,p}^{life} } =\boldsymbol{ abs((SOC_{i,j,p}-SOC_{i,j-1,p}))\wedge(-1.453)*\exp(-3.6*}$

$\boldsymbol{(SOC_{i,j,p}+ SOC_{i,j-1,p})*0.5)*14003}$;

$\boldsymbol{Deg_{i,j,p}^{BESS}}=\boldsymbol{Deg_{i,j-1,p}^{BESS}+0.5/N_{cycl\,i,j,p}^{life}}$;

$\boldsymbol{SOH_{i,j}}=\boldsymbol{SOH_{i-1}^{end}-C_d*Deg_{i,j,p}^{BESS}}$;

$\boldsymbol{SOH_{\min\,i,j}}=\boldsymbol{1-C_d/(15*365*24)*(24*(\mathrm{i-1})+\mathrm{j})}$;

if $\boldsymbol{SOH_{i,j}}<\boldsymbol{SOH_{min}}$

$\boldsymbol{P_{BESS\,i,j,p}^{PSO}}=\boldsymbol{\mathbf{adj}*P_{BESS\,i,j,p}^{PSO}}$;

end

end

end

The SOH algorithm stages is structed as follows:

• 1: Intialize the SOC, SOH and BESS degradation: $SOC:10\%$, $SOH:100\%$, $Deg_{i,j,p}^{BESS}:0\%$
• 2: Evaluation of the SOC
• 3: Evaluation of remaining cycle of life
• 4: Evaluation of the total BESS degradation
• 5: Evaluation of the State of Health
• 6: Evaluation of the minimum State of Health
• 7: Checking of SOH and readjusting the BESS output
• 8: Repeat 2, 3, 4 and 5 For determining the new SOH until $SOH_{i,j}\geq SOH_{min}$

3.6.7 Step 7: Evaluation of the New Population

$[P_{BESS,i,j,k}^{PSO,iter}]_{1\leq k\leq K}$, becomes the input for the fitness function. Hense $p_{best\,k}^{iter}$ is given as follows:

for $\boldsymbol{k}=\boldsymbol{1:K}$

if $\boldsymbol{\mathbf{F}_k([E_{rate,k}^{iter}P_{rate,k}^{iter}])}>\boldsymbol{\mathbf{F}_k([E_{rate,k}^{iter-1}P_{rate,k}^{iter-1}])}$

then $\boldsymbol{p_{best\,k}^{iter}}=\boldsymbol{[E_{rate,k}^{iter}P_{rate,k}^{iter}]}$

else $\boldsymbol{p_{best\,k}^{iter}}=\boldsymbol{[E_{rate,k}^{iter-1}P_{rate,k}^{iter-1}]}$

end

end

Moreover, $p_{gbest}^{iter}$ is defined with particle k which maximizes the function.

3.6.8 Step 8: Generate the Optimal BESS Size

In the last iteration ($iter\_max$), the optimal BESS size corresponds to:

4. Results and Discussion

In this section, an appropriate BESS size is simulated for two different BESS technologies: vanadium redox (VRB) and Lithium-ion (Li-ion). The simulations were conducted using MATLAB 2024a to compile the code. Because the P2ID customer fixes itself with the maximum BESS size based on its budget, the energy rate and power-rated capacity of its technologies can be chosen accordingly with respect to the allowable budget. A budget of 50000000 JPY is considered in this study. Based on [26], two different BESS technologies can be identified considering their price and efficiency. Table 2 lists the BESS parameters.

Table 2 BESS specification for VRB and Li-ion.

4.1 Assumption

• To maintain safe BESS operation, an initial SOC of 10% was considered every day during the simulation process, and its maximum was limited to 90%.
• The RoR was maximised only during the first year. Moreover, the BESS degradation is considered through SoH estimation and is maintained as greater than the minimum SoH defined in Equations (23) and (24). Considering a life-span of $Pl=15$ years (which corresponds to the project duration) and a degradation ratio $C_d$ of 60% (indicating that at least 40% of SoH should be achieved after 15^th year at the end of the life), we can determine the minimum SoH at the end of the first year ($SOH_{\min\_end}$) using equation (24) as:

• This implies that the optimal BESS size must exhibit an SoH of at least 96% at the end of the first year.
• The self-discharging rate is assumed to be equal to 0.6% per hour.
• 50 particles of the BESS initial population are considered, and 100 iterations as well, to find the optimal solution using the PSO as explained previously.
• The second-level subproblem regarding the BESS schedule was conducted with 50 iterations and 50 particles.
• The constant coefficient of adjustment (adj) of the BESS output during the scheduling process using PSO was assumed to be 85%.
• The forecast errors in (26) and (27) are assumed to be 2%.
• The contracted demand is assumed to be 240 kW over the timespan.
• The BESSS cost and electricity tariffs are assumed to be fixed over the timespan.

4.2 Validation Approach

The process for validating one of the two BESS types is performed as follows:

• Determining independently the optimal BESS sizes for both VRB and Li-ion, utilizing the proposed optimal sizing method shown in Figure 2.
• Considering the optimal size found and evaluating the annual revenue over 15 years for both BESS utilizing the PSO-based optimal scheduling addressed in [13].
• Finally, the validation is based on the BESS technology with the shortest payback period.

After determining the optimal size for each BESS technology, the energy saved is accumulated over the lifespan to achieve the payback period. The shortest value was the final candidate for the optimal size. Thus, Cases 1 and 2 are shown below.

4.3 Case1: Optimal Size Generation

In this case, the purpose was to determine the optimal BESS rate energy and power capacity for each type of technologylisted in Table 2. In addition, the maximum RoR, annual energy saved, and SoH with respect to the minimum are shown during the iteration process. The resutls are tabulated in Table 3.

Table 3 Summary of VRB and Li-ion optimal sizing.

Figure 3 and Figure 4 outlet the BESS sizing optimisation results for both the Li-ion and VRB types. After the 100^th iteration, the RoR for both battery technologies stabilised near the maximum value. As illustrated in Figure 3b, Figure 3c, Figure 4b, and Figure 4c, we observe that 55 kWh/20 kW and 75 kWh/20 kW are the optimal size for Li-ion and VRB, respectively. Table 3 summarises the related total capital costs (calculated using (9)), annual electricity costs saved in the first year, and final SOH. The SOH is subject to its constraint formulated in (23), which relies on the minimum SoH fixed at 96% at the end of the first year and the SoH algorithm defined previously. As shown in Figure 3e and Figure 4e, the SoH for both the Li-ion and VRB is above the lower limit of 96%, which confirms the performance of the SoH algorithm in minimising the degradation effect. Owing to its considerable efficiency, Li-ion furnishes more annual cost savings (1422000 JPY) than VRB, which has 771735 JPY in yearly savings. About the total capital cost, the Li-ion shows a greater amount than the VBR, i.e., 11001000 JPY and 7594300 JPY, respectively.

Figure 3 Optimal BESS Sizing for VRB.

Figure 4 Optimal BESS Sizing for Li-ion.

PSO can generate an optimal size using the previously described process. However, even if the customer limit budged of 50000000 JPY is greater than the investment cost for both battery types, paying attention to the payback periods is crucial. The shortest payback period was selected as the best option, as shown in Case 2.

4.4 Case 2: BESS Selection Using the Payback Period

After selecting the optimal BESS rate, energy, and capacity for each technology, as summarised in Table 3, this case aims to determine which technology meets the shortest payback period requirement during a project lifetime of 15 years. The total energy savings per year were calculated using Equation (3). Its total is compared to the total investment cost, considering the bank interest rate, which is 4% in this study, and using relations (9), (10), and (11). Although the PSO sizing process is deactivated daily, the operation schedule must be considered based on [13]. It was assumed that the final SoH would not be below 40% at the end of the project life. The strategy in this study is to maintain the minimum SoH defined in (24) every time constraint (23) is applied.

Figure 5 and Figure 6 outline the results for this case.

Figure 5 Cost saved and payback period for VRB.

Figure 6 Cost saved and payback period for Li-ion.

For both the Li-ion and VRB, the annual energy cost saved and its cumulative and end-day SoH over 15 years are shown in Figure 5 and Figure 6. Because the BESS total capacity degrades over time, and assuming that the customer operates in the same manner as its load demand, one can observe a decreasing annual saving, as shown in Figure 5a and Figure 6a. However, it has been proven from Case 1 that the SoH limit for the first years was respected, that is, 96%, owing to the SoH constraints conceived during the optimization process. If this limit is maintained, the SoH must be above 40% of the proposed SoH algorithm by the end of year 15. As shown in Figure 5c and Figure 6c, this limit is respected for both Li-ion- and VRB-based BESS technologies with 40% and 50% of the final SOH, respectively. This demonstrates that the degradation can be maintained within an acceptable range, saving sufficient cost, as shown in Figure 5b and Figure 6b.

To determine the shortest payback period for each BESS type, the cumulative energy cost saved per year is depicted in Figure 5b and Figure 6b. It can be observed that the Li-ion type reaches a payback period of 13 years because its total capital cost is exceeded in that period. Similarly, for the VRB, which shows a payback period of more than 15 years, the total capital cost does not exceed the expected value.

Accordingly, the best BESS for the customer is the Li-ion technology, and its optimal capacity is as found from previous simulation as 75 kWh/20 kW in this study.

5. Conclusion

Two optimisation problems were combined using the particle swarm optimisation (PSO) algorithm. The first-stage optimization problem managed the optimal size, energy rate, and power capacity of the BESS. The second stage managed the operation schedule, taking into account the health effects of BESS degradation. The scope of the sizing-solving process was detailed in this study, which used seven steps to research an acceptable solution by maximising the rate of return on investment. The second stage was conceived based on a recent survey of robust PSO-based scheduling and its assessment. Because the BESS gradually loses its energy-storage capacity over time, it has been proven that the SoH limit requirement is satisfactory during the first year of operation. Therefore, for both the Li-ion and VRB technologies, the SoHs were above the fixed limit in this study, that is, 96% at the end of the first year. Additionally, by maintaining the same approach throughout the 15-year project lifespan, we found that the proposed algorithm performed well. The SoHs for both BESS types remained above the final limit of 40% which aligns with the degradation behavior

up to 60% of degradation rate. However, short-term payback criteria are essential and require considerable attention for the proper choice of BESS. Therefore, cumulative energy cost savings were performed in this study for both optimal BESS sizes determined by PSO. It has been proven that the Li-ion suit is superior in this study, with an optimal size of 75 kWh/20 kW, and a payback period of 13 years.

As a future perspective:

• Although the study of a suitable forecast algorithm was beyond the scope of this investigation, it might be interesting to combine the proposed algorithm with a prediction method to determine the forecast data instead of using a fixed forecast error.
• The sensitivity of the algorithm to the market variables, such as pricing and battery costs, has not been covered in this study. This can be a relevant issue to explore in the future.

Nomenclature

Acknowledgments

The main author is grateful for his respective supervisors who have been supporting this research. Special recognition for Dr. Yasuhiro Noro, his former Professor and master research supervisor at Kogakuin University who had a decisive role by supervising and validating a part of the work. As well as Dr. Goro Fujita, Professor at Shibaura Institute of Technology who is currently supervising his PhD research. Finally, the author is grateful to Shibaura Institute of Technology for providing funds to support his research.

Author Contributions

Alioune Diouf was responsible to the conceptualization, methodology and original draft; Dr. Yasuhiro Noro performed the first supervision and validation and Dr. Goro Fujita completed the article review.

Competing Interests

The authors have declared that no competing interests exist.