Optimal Capacity and Cost Analysis of Battery Energy Storage System in Standalone Microgrid Considering Battery Lifetime

Abstract

In standalone microgrids, the Battery Energy Storage System (BESS) is a popular energy storage technology. Because of renewable energy generation sources such as PV and Wind Turbine (WT), the output power of a microgrid varies greatly, which can reduce the BESS lifetime. Because the BESS has a limited lifespan and is the most expensive component in a microgrid, frequent replacement significantly increases a project’s operating costs. This paper proposes a capacity optimization method as well as a cost analysis that takes the BESS lifetime into account. The weighted Wh throughput method is used in this paper to estimate the BESS lifetime. Furthermore, the well-known Particle Swarm Optimization (PSO) algorithm is employed to maximize battery capacity while minimizing the total net present value. According to simulation results, the optimal adjusting factor of 1.761 yields the lowest total net present value of US$200,653. The optimal capacity of the BESS can significantly reduce the net present value of total operation costs throughout the project by extending its lifetime. When applied to larger power systems, the proposed strategy can further reduce total costs.

1. Introduction

Recently, Renewable Energy Resources (RESs) have become attractive sources in the electrical power system due to various advantages, such as being sustainable sources and being environmentally friendly [1,2]. The primary RESs are solar and wind energy generation sources. However, power generation from the RESs is intermittent because of its behavior. Therefore, renewable energy generation sources cannot be directly connected to the existing system or used without the appropriate management system [3,4]. A microgrid is a small power system constructed to manage Distributed Generators (DGs) from renewable energy and load clusters. The microgrid that connects to the bulk power system is called to be in “on-grid mode”, and when it disconnects from the bulk power system in an emergency, it is called to be in “islanded mode”. The microgrid that operates in the islanded mode can be called a standalone microgrid [5,6]. The standalone microgrid is common in remote or rural areas. Thus, the standalone microgrid’s stability is low due to the lack of assistance from the bulk power system. The primary goal in the microgrid is to balance the power load and power generation without interrupting the load. Hence, it is necessary to have manageable power generation sources, such as diesel engine generators or the BESS, to control the microgrid’s power [7,8]. A standalone microgrid consisting of renewable energy generation sources, load clusters, and BESS is widely used [9] and avoids using diesel generators for environmental reasons. One of the significant problems for BESS applications is finding optimal capacity that considers the lifetime of BESS. Because of the high cost of the BESS, BESSs with a short life have been widely used in a microgrid. The performance assessment of the grid is primarily based on cost and reliability related to the system’s lower greenhouse gas emissions [10]. The optimal size of PV and BESS in a microgrid for maintaining the longest BESS lifetime using the Loss of Power Supply Probability (LPSP) and only the initial cost as the indicators was presented in [11]. The cost reduction analysis was estimated for the battery State of Health (SOH) and artificial intelligence was applied to help with the battery management system. A proper battery health assessment can extend the battery’s lifetime and reduce the cost of the battery [12]. An accurate SOH is essential to optimizing battery lifespan. Battery deterioration tracking and SOH estimates are based on electrochemical impedance spectroscopy and the distribution of relaxation time measurements. They were used to look for indicators linked to deterioration [13]. The BESS lifetime estimation in a PV system using a practical model was presented by [14]. The authors also considered the impact of BESS sizes and types on its lifetime. The BESS lifetime increases with increased BESS size, and upfront costs also increase. The authors also introduced a strategy to optimise the total cost, including upfront costs and the replacement cost of BESS. The lifetime prediction method and sizing of lead-acid BESS in microgrids were applied by varying the BESS’s size and the weighted Wh throughput method to estimate the BESS lifetime. However, the authors did not consider the cost analysis to obtain the optimal size [15]. This paper proposes an optimization of BESS on a microgrid with PV and WT as the power generation source. The objective function of this study is to reduce total costs. According to the literature review, the articles in the references [11,14] used a microgrid system with only PV and BESS. References [12,13] considered the SOH estimation method to reduce costs, but this paper uses the optimization method. The work in [14] only studied the BESS lifetime estimation in a PV system. Moreover, the authors in [15] did not consider the cost analysis to obtain the optimal size. Therefore, in this paper, both PV and WT generation systems are analyzed. The sizing and optimal NPV of the BESS were analyzed with the optimization method. Implementing a microgrid system requires forecasting investment costs and profits, as well as maintenance over its lifecycle because the batteries are expensive components of the microgrid system. If the battery is replaced prematurely, the cost of the system will increase. Forecasting and estimation methods are generally used for the life cycle and the replacement of the battery. However, this paper proposes optimization to get the best results and reduce the total cost of the BESS system. The weighted Wh method and the PSO algorithm are applied for optimizing the cost of BESS. In a standalone microgrid system, prolonging the life of the equipment is necessary to reduce the cost of its replacement. However, the size and installation costs of the storage systems must be appropriate. Therefore, this paper provides an appropriate weighting to minimize the cost of the microgrid system. The PSO algorithm is applied to determine the optimum weighting parameter, which results in faster and more accurate analysis results than statistical analysis.

The major contributions of the paper are summarized as follows: The capacity optimization method and cost analysis of the BESS by taking into account the BESS lifetime are proposed in this paper. The BESS lifetime is estimated using the weighted Wh throughput method, which is fully described in [16]. The total cost of the microgrid and the notion of Net Present Value ($NPV$) constructs serve as the objective function. The objective function is solved using the PSO algorithm to determine the BESS’s ideal capacity. Due to its dependability and affordability, the lead-acid battery is used in this study. According to an economic analysis, it is a better choice than a modern battery like a lithium-ion battery [17,18].

The rest of this paper is structured as follows: Section 2 explains the mathematical modeling of DGs and BESS. Section 3 introduces the optimization model which consists of the objective function and the PSO algorithm. Section 4 provides the simulation results and result discussions. Finally, Section 5 concludes the paper.

2. DGs and BESS Models

In this section, the mathematical models of PV, WT and BESS used in the proposed optimization problem are briefly explained. A small industrial load is used for the case study in which PV and WT power generation systems are installed. The battery energy storage systems are used for power demand periods where the DGs are unable to supply the load for only some periods. Hence, BESS is small in size, and costs are reduced accordingly. However, the proper size of a BESS affects its longevity and maintenance or replacement costs. This model can be applied to the larger systems.

2.1. PV Model

The amount of PV power depends on the solar intensity and temperature and is given by [16],

$$P_{PV}=P_{STC}·\frac{G_c}{G_{STC}}·[1+k\left(T_c\right)-T_{STC}],$$

(1)

where $P_{PV}$ is the output power; $P_{STC}$ is the rated output power at standard conditions (the solar irradiance $G_{STC}$ of 1000 W/m${}^2$ and the temperature $T_{STC}$ of 25 ${}^{\circ }$C; $G_C$ is the solar irradiance during operation; k is the discrete-time index and $T_c$ is the temperature during operation. The installed capacity of the PV system in this paper is 65 kW${}_{\mathrm{p}}$.

2.2. Wind Turbine Model

The amount of generated WT power depends on the wind speed [19]. The output power of the WT system is expressed by,

$$P_{wt}=\left\{\begin{array}{cc}0,\hfill & v\le v_{ci}\mathrm{OR}v\ge v_{co},\hfill \\ P_{rated-wt}\frac{(v-v_{ci})}{(v_r-v_{ci})},\hfill & v_{ci}\le v\ge v_r,\hfill \\ P_{rated-wt},\hfill & v_r\le v\ge v_{co},\hfill \end{array}\right.$$

(2)

where $P_{WT}$ is the output power of the wind turbine; v is the wind speed and $P_{rated-wt}$ is the rated output power of WT. The capacity of the WT system is selected as 40 kW${}_{\mathrm{p}}$.

2.3. BESS Model

In this paper, a lead-acid battery is used for the calculation of the BESS cost because it is more cost-effective and safer compared to Li-ion battery. Although price of the Li-ion battery is continuing to decrease, it is still expensive in Thailand. In Thailand, the batteries widely used for energy storage in PV power generation systems are lead-acid batteries. In order to simulate the operation of the BESS, mathematical models for calculating the charge and discharge parameters and State of Charge (SOC) of the BESS are required. Batteries are bidirectional energy sources, meaning they can store energy in a manner comparable to an electrical load. In contrast, they provide energy as an energy source. The battery energy balance equation and its constraint are,

$$\begin{array}{cc}\hfill E_B(t+\mathsf{\Delta }t)& =E_B\left(t\right)+(P_{B,ch}\left(t\right)·{\eta }_{B,ch}-P_{B,d}\left(t\right)/{\eta }_{B,d})\mathsf{\Delta }t,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill E_B^{min}& \le E_B\left(t\right)\le E_B^{max},\hfill \end{array}$$

(4)

where $E_B$ is the energy of the BESS at time t; $P_{B,ch},P_{B,d}$ are the charging and discharging powers of BESS; ${\eta }_{B,ch},{\eta }_{B,d}$ are the charging and discharging efficiency of BESS respectively.

The maximum permissible output power of the BESS is limited as,

$$-P_{B,ch}^{max}\le P_b\left(t\right)\le B_{B,d}^{max},$$

(5)

where $P_{B,ch}^{max},P_{B,d}^{max}$ are the maximum charging and discharging powers of BESS.

The battery SOC and DOD can be obtained as,

$$\begin{array}{cc}\hfill SOC\left(t\right)& =\frac{E_B\left(t\right)}{E_{B,rated}},0\le SOC\left(t\right)\le 1,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill DOD& =1-SOC,\hfill \end{array}$$

(7)

where SOC is the battery state of charge; $E_B$ is the energy of BESS at time t; $E_{B,reted}$ is the rated energy and $DOD$ is the battery depth of discharge.

The $SOC=$ 1 or 0 means that the battery is fully charged or completely discharged respectively. The SOC at time t can be expressed as,

$$SOC(t+\mathsf{\Delta }t)=SOC\left(t\right)·(1-\beta )+\frac{E_B(t+\mathsf{\Delta }t)}{E_{B,rated}},$$

(8)

where $\beta$ is the battery’s self-discharge rate.

The battery SOC should be maintained within the specific ranges that depend on the type of the battery,

$$SO{C}^{min}\le SOC\left(t\right)\le SO{C}^{max},$$

(9)

where $SO{C}^{min},SO{C}^{max}$ are the minimum and maximum limits of battery SOC.

2.4. BESS Capacity Model

BESS is used to maintain the microgrid’s power balance. Thus, the required power of a BESS is the power mismatch between the load and the power generation. The balance between load and generation can be achieved through,

$$P_{B,required}\left(t\right)=P_L\left(t\right)-P_{WT}\left(t\right)-P_{PV}\left(t\right),$$

(10)

where $P_{B,required}$ is the BESS’ required power; $P_L$ is the load demand; $P_{WT}$ is the output power of WT and $P_{PV}$ is the output power of PV.

Taking into account the charging and discharging efficiencies, the BESS power and its rated power can be rewritten as follows: [20],

$$\begin{array}{cc}\hfill P_B\left(t\right)& =\left\{\begin{array}{cc}{P}_{B,required}\left(t\right)/{\eta }_{B,d},\hfill & P_{B,required}\left(t\right)>0,\hfill \\ P_{B,required}\left(t\right)\times {\eta }_{B,ch},\hfill & P_{B,required}\left(t\right)<0,\hfill \\ 0,\hfill & P_{B,required}\left(t\right)=0,\hfill \end{array}\right.\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill P_{B,rated}& =\max \left(\right|P_B\left(t\right)\left|\right),\hfill \end{array}$$

(12)

where $P_{B,rated}$ is the BESS’s rated power.

The required energy from BESS can be calculated as,

$$E_{B,required}=\max \left(E_B\left(t\right)\right)-\min \left(E_B\left(t\right)\right).$$

(13)

The SOC of BESS should be kept within a specific range. Therefore, the rated energy considering DOD limits is expressed by

$$E_{B,rated}=\frac{E_{B,required}}{DO{D}^{max}-DO{D}^{min}},$$

(14)

where $DO{D}^{max},DO{D}^{min}$ are the DOD limits of the BESS.

In order to reduce the SOC level of the BESS, oversizing of the BESS is considered in this paper, as discussed in the next section. The oversized BESS does not have the same lifetime as an optimal size BESS [21,22]. BESS’s non-optimal size also increases investment costs. An adjusting factor is used to obtain the BESS’ oversized energy as,

$$E_{B,Orated}=E_{B,rated}\times q_B,$$

(15)

where $E_{B,Orated}$ is the BESS’s oversized energy, $E_{B,rated}$ is the BESS’ rated energy and $q_B$ is the adjusting factor.

The adjusting factor $q_B$ is equal to or greater than 1. As the size of the battery can affect the DOD range, it follows that a battery with a higher capacity will last longer under identical conditions. However, the cost of installation will rise. Consequently, the parameter that must be optimized in this paper is $q_B$. The optimal values will result in longer battery life and a reduction in total system costs.

2.5. BESS Lifetime Estimation

In this paper, a weighted Wh throughput method is used to calculate the lifetime of a lead-acid battery. This method has been employed by many researchers [15,23], as it can roughly estimate the battery lifetime by using the SOC as the most significant factor for battery lifetime. The battery has an overall throughput calculated from the manufacturer’s datasheet, which usually provides the relationship between the Cycles to Failure (CTF) and the different DODs as shown in Figure 1. In this study, the employed battery is the Sun Xtender PVX-2580L model (12 V, 3.096 kWh) [24].

The energy throughput corresponding to a specific DOD is expressed as,

$$E_{throughput}\left(DOD\right)=2\times E_{B,rated}\times DOD\times CTF,$$

(16)

where $E_{throughput}$ is the energy throughput corresponding to a specific DOD and $CTF$ is the cycles to failure.

Because the DOD varies within the range of DOD limits, the energy throughput varies with the DOD, as shown in Figure 2. By considering Equation (16) and the data shown in the Figure 1, the relationship between energy throughput and DOD in the Figure 2 can be obtained. The nonlinear curve in the Figure 2 is caused by the nonlinear function in the Figure which is the relationship between the Cycles to Failure (CTF) and different DODs. The data from the Figure 1 was obtained from the datasheet of the lead-acid battery in [24]. The overall energy throughput is the average energy throughput between the DOD limits and can be determined as,

$$E_{throughput,avg}=avg{\left\{E_{throughput}\left(DOD\right)\right\}}_{DOD\left(min\right)}^{DOD\left(max\right)}.$$

(17)

The cumulative energy throughput $E_{Wsoc}$ multiplied by the SOC weighting factor $W_{SOC}$ during period $\mathsf{\Delta }$t is given as,

$$E_{Wsoc}=W_{SOC}\times \left|P_B\left(t\right)\right|\times \mathsf{\Delta }t.$$

(18)

The relationship between SOC and the weighting factor $W_{SOC}$ is shown in Figure 3. For example, using 1 Wh of battery at 0.2 p.u. of SOC is equivalent to 1.3 Wh of the cumulative energy throughput. On the other hand, using 1 Wh at 1 p.u. of SOC is equal to 0.55 Wh of the cumulative throughput. Based on the Figure 3 that shows the relationship between SOC and SOC weighting factor and on Equations (18) and (19), battery life loss mainly depends on SOC and SOC weighting factor ($SOC$ and $W_{SOC}$). If the SOC of the battery is kept lower, the battery life loss will be affected (see Equation (19)). Thus, the battery should operate at a high SOC or within its upper SOC limits to extend its lifetime. It has to be noted that this method ignores many influencing factors for simplicity, such as temperature, because temperature can be controlled. Therefore, the battery life loss can be calculated as,

$$L_{life}=\frac{E_{Wsoc}}{E_{throughput,avg}},$$

(19)

where $L_{life}$ is the battery life loss and $E_{throughput,avg}$ is the average energy throughput.

The battery is considered to reach the end of life when the total cumulative energy throughput is equal to the overall energy throughput.

The following exponential function can be used to express the relationship between the number of cycles and the depth of discharge,

$$\begin{array}{cc}\hfill N_{CTF}& =f\left(DOD\right)=a\times e^{b\times DO{D}_B}+c\times e^{d\times DO{D}_B},\hfill \end{array}$$

(20)

where $a,b,c,d$ are constants.

The depreciation ($D_B$) of BESS can then be calculated as,

$$\begin{array}{cc}\hfill D_B& =\sum _{i=1}^m\frac{N_{B,cycle}\left(DO{D}_i\right)}{N_{CTF}\left(DO{D}_i\right)},\hfill \end{array}$$

(21)

where $N_{B,cycle}$ is the number of cycles according to the depth of discharge.

Therefore, the BESS lifetime $L_B$ can be obtained as,

$$L_B=\frac{1}{D_{B,1year}}\times years,$$

(22)

where $D_{B,1year}$ is the depreciation of the battery in one year.

Hence, the total number of BESS replacements $N{R}_B$ throughout the project is given by,

$$N{R}_B=\frac{L_p}{L_B}-1.$$

(23)

where $L_p$ is the project life.

3. Optimization Model

The optimization problem is introduced in this section, and the PSO algorithm used to solve it is briefly described.

3.1. Objective Function

The cost of the microgrid is represented by the concept of the Net Present Value ($NPV$). The total cost includes the initial cost, fixed operation and maintenance costs, and replacement costs. Therefore, the objective function used to minimize the $NPV$ of the total cost can be defined as:

$$\mathrm{Min}\left(NPV{C}_{total}\right)=min(C_{initial}+NPV{C}_{om}+NPV{C}_{re}),$$

(24)

where $NPV{C}_{total}$ represents the net present value of the total cost; $NPV{C}_{om}$ is the net present value of the fixed operation and maintenance costs; $NPV{C}_{re}$ is the net present value of the replacement cost and $C_{initial}$ is the initial cost.

The initial cost, the net present values of the replacement cost, and the fixed operation and maintenance costs respectively are given as,

$$\begin{array}{cc}\hfill C_{initial}& =(k_{BE,initial}\times E_{B,Orated})+(k_{BP,initial}\times P_{B,rated}),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill NPV{C}_{re}& =\sum _{n=1}^{N{R}_B}\frac{C_{initial}}{{(1+d)}^{N{R}_B\times L_B}},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill NPV{C}_{om}& =k_{B,om}\times E_{B,Orated}\times USW,\hfill \end{array}$$

(27)

where $k_{BE,initial}$ is the BESS initial cost per kWh ($/kWh); $E_{B,Orated}$ is the BESS’ oversized energy; $k_{BP,initial}$ is the BESS initial cost per kW ($/kW); $k_{B,om}$ is the operation and maintenance costs set to 5% of the initial cost ($/kWh/year); d is the discount rate (5%); $L_B$ is the BESS lifetime and $USW$ is the uniform series present worth factor.

The uniform series present worth factor is given as,

$$USW=\frac{{(1+d)}^{L_p}-1}{L_p{(1+d)}^{L_p}},$$

(28)

where d is the discount rate.

Figure 4 depicts the procedure flowchart for the BESS capacity determination. It starts by obtaining the input power of WT, PV, and load, and then calculating the rated power and energy capacity of the battery. Then, it estimates the BESS lifetime using the BESS model and obtains the objective function’s value. If $NPV{C}_{total}$ is minimal, the calculation ends. If not, $q_B$ is adjusted by incremental increases of 0.001 between 1 and 5 and the procedure is repeated.

3.2. PSO Algorithm

Particle swarm optimization (PSO) is an intelligent optimization algorithm introduced by J. Kennedy and R. Eberhart [25]. It has been widely used in many works owing to its effectiveness and simplicity. The basics of PSO can be found in [26,27,28]. The reason for choosing the PSO algorithm is that it has a simple structure with few parameters and provides a quick solution. It creates particle swarms, which simulate a system or structure of a group of organisms that group together for some specific purpose in each organism. The PSO chooses each i-th particle initial position and velocity at random. Then the i-th particle position and velocity are updated until the termination conditions are met. The particle’s position and velocity of the particle i are expressed by,

$$\begin{array}{cc}\hfill x_i(k+1)& =x_i\left(k\right)+v_i(k+1),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill v_i(k+1)& =w\times v_i\left(k\right)+c_1{r}_{1i}(p_{i,best}-x_i\left(k\right)+c_{2i}{r}_{2i}(g_{i,best}-x_{i}k)),\hfill \end{array}$$

(30)

where $x_i$ is the position of particle i; $v_i$ is the velocity of the particle i; w is the inertia factor; $c_1,c_2$ are the acceleration factors; $r_1,r_2$ are the random number [0–1]; $p_{best}$ is the best particle and $g_{best}$ is the best global solution.

Both acceleration factors ($c_1,c_2$) are commonly set to 2 [29]. The inertia factor which linearly decreases in each iteration $ite$ is calculated as,

$$w_{ite}=w_{max}-ite\times \frac{w_{max}-w_{min}}{it{e}_{max}},$$

(31)

where $w_{max},w_{min}$ are the weights of the maximum and minimum speeds and $it{e}_{max}$ is the maximum iteration.

This paper applies the PSO algorithm to determine the optimal installation size of the BESS based on the battery size multiplier and the cost of the energy storage system throughout the project as a measure of fitness or sufficiency. Each particle in the herd has the battery size adjuster ($q_B$) as a variable and is assigned to represent the answer point. The initialization parameters for the PSO are the number of particles ($n=$ 50), the maximum number of iterations ($it{e}_{max}=$ 100), the range of velocity weighted values ($w_{max}/w_{min}=$ 0.9/0.4), and the learning weighting value ($c_1,c_2=$ 2). The optimization toolbox in MATLAB is used to solve the PSO algorithm [30]. The effectiveness of the proposed strategy was verified using MATLAB running on MacBook Air (Early 2015), 1.6 GHz Dual-Core Intel Core i5, RAM 8 GB 1600 MHz DDR3. The verification findings for the suggested optimization approach are presented in the following section.

4. Results

In this study, a standalone microgrid with an industrial load, a WT, a PV and a BESS illustrated in Figure 5 is considered. The daily load demand, PV and WT generations with a 5-min resolution shown in Figure 6, tasken from [31], are used to verify the effectiveness of the proposed method. The industrial load has a peak power of 100 kW at 12:30 p.m. The PV generates power between 7:00 a.m. and 5:00 p.m. with a maximum value of 63 kW at noon. The power generated by wind turbines fluctuates around 35 kW throughout the day. The required power from the BESS is obtained based on the mismatch between the load demand and the total power generation. In Figure 7, the BESS’ discharging power is positive and the charging power is negative. The initial SOC is set at 50%. The standalone microgrid project life is set to 20 years. The case parameters used in the optimization are given in

Table 1. Case study parameters.

Parameter	Variable	Unit	Value
Project life	$L_p$	years	20
BESS Calendar life	$L_B$	years	10
BESS SOC limits	$SO{C}_{min}$-$SO{C}_{max}$	%	20–80
Charge/discharge efficiency	${\eta }_{B,ch}$$/{\eta }_{B,d}$	%	90/90
Initial cost per energy [32,33,34]	$k_{BE,initial}$	$/kWh	183.86
Initial cost per power	$k_{BP,initial}$	$/kW	183.86
Operation & maintenance cost	$k_{B,om}$	$/kWh/year	9.19
Discount rate [35]	d	%	5

.

The rated energy of BESS $E_{B,rated} \mathrm{is}$ 82 kWh ($q_B=$ 1). For $q_B=$ 1, the BESS lifetime is 1.2 years, and the $NPV{C}_{total}$ throughout the project is US$204,436. When $q_B=$ 1, the SOC value is in the range of 20% to 80%. As the $q_B$ is increases, the battery size is increasing, and the deviation of the battery SOC from 50% is decreasing. However, the increased battery capacity results in higher total costs. Thus, the PSO optimization method is applied to find the optimal value of $q_B$. The BESS SOCs for $q_B=$ 1, 1.761, and 5 are illustrated in Figure 8.

Figure 9 shows a stepped pattern of the relationship between $N{R}_B$ and $q_B$. As the $q_B$ increases, the number of battery replacements will decrease. As the size of the battery increases, the SOH and the battery lifetime increase as well. The relationship between $N{R}_B$ and $L_B$ is shown as the purple line (saw-tooth pattern) and is used to obtain the $NPV{C}_{re}$ using Equation (26).

Using the proposed method, the obtained value of US$200,653 was the lowest $NPV{C}_{total}$ at the $q_B$ of 1.761. This corresponds to the optimal capacity of the BESS of 144.4 kWh. The effects of $q_B$ on the $C_{initial}$, the $NPV{C}_{om}$ and the $NPV{C}_{re}$ are depicted in Figure 10. As the $q_B$ increases, the size of the battery and the cost increase, and the number of replacements decreases. If the battery size is small, the cost of battery replacement is low, but the cost of replacing the battery is high. As the $q_B$ varied from 1 to 5, in steps of 0.001, the $C_{initial}$ increased from US$18,419 to US$78,724, and the $NPV{C}_{om}$ increased from US$23.48 to US$117.43. In contrast, the $C_{re}$ has downward trends because the $q_B$ can extend the BESS lifetime and the BESS replacement was reduced.

The effects of $q_B$ on the BESS lifetime and the number of BESS replacements are shown in Figure 11. The number of BESS replacements will decrease step by step with respect to the DOD and the size of the battery in relation to the $q_B$ factor. However, when calculating the BESS replacement cost in the low $q_B$ range, the replacement cost is low, and when $q_B$ is high, the cost of replacing the battery increases. The minimum saw-tooth pattern range is the point where the number of BESS replacements is reduced. The $q_B$ has a positive effect on the BESS lifetime. The BESS lifetime also increased from 1.2 years to 5.44 years when the $q_B$ increased from 1 to 5. The number of BESS replacements decreased because the number of replacements remains equivalent even when $q_B$ is adjusted slightly, such as when $N{R}_B=$ 8 and $q_B=$ 2.0 and 2.2.

The $q_B$ has an effect on the $NPV{C}_{total}$ of the microgrid, as shown in Figure 12. This curve is similar to a saw-tooth line due to the nonlinear characteristic of the replacement cost. When $q_B=$ 1.761 and $q_B=$ 2.0 are compared, the $NPV{C}_{total}$ of the $q_B=$ 1.761 is the lowest. Even though the $C_{initial}$ and $C_{om}$ for $q_B=$ 1.761 are greater than those for $q_B=$ 2.0, the $C_{re}$ is lower because it can reduce the $N{R}_B$ from 9 to 8. This shows how the $q_B$ affects the $NPV{C}_{total}$. The BESS lifetime and cost analysis versus some of the $q_B$ values are given in

Table 2. Simulation results for some of the $q_B$ values.

${\mathit{q}}_{\mathit{B}}$	${\mathit{L}}_{\mathit{B}}$	${\mathit{NR}}_{\mathit{B}}$	${\mathit{C}}_{\mathit{initial}}$ ($)	${\mathit{NPVC}}_{\mathit{om}}$ ($)	${\mathit{NPVC}}_{\mathit{re}}$ ($)	${\mathit{NPVC}}_{\mathit{total}}$ ($)
1.0	1.20	16	18,419	23.48	185,992	204,436
1.5	1.72	11	25,958	35.23	178,562	204,557
1.761	2.00	9	29,896	41.36	170,468	200,653
2.0	2.25	8	33,497	46.97	168.626	202,172
2.2	2.46	8	36,513	51.67	176,524	213,088
3.0	3.31	6	48,573	70.46	171,838	220,482
4.0	4.38	4	63,649	93.94	153,539	217,284
5.0	5.44	3	78,724	117.43	142,152	220,994

. Battery lifetime estimates show that increasing the size of the battery by the $q_B$ factor increases battery life because it reduces the number of cycles at a high DOD value.

5. Conclusions

This paper proposed a capacity optimization method for a BESS in a standalone microgrid while taking the BESS’ lifetime into account. The BESS’ capacity influenced the initial cost, operation and maintenance costs, and replacement cost. The case study demonstrated the efficacy of the proposed method. According to the PSO algorithm results, the optimal capacity of the BESS ($q_B$ = 1.761, $E_{B,rated}$ = 144.4 kWh, and $NPV{C}_{total}$ = US$200,653) has the lowest NPV of the total cost. According to the simulation results, the capacity and lifetime of the BESS were the major factors influencing the total cost of the system. Based on the results, the overall cost was reduced slightly. This is due to the small size of the test system, which caused the battery system to be small as well. However, if the proposed method is applied to a larger system, the total cost can be greatly reduced. Future research will concentrate on scaling up the proposed method and comparing it to other current battery technologies.