Modified Biogeography Optimization Strategy for Optimal Sizing and Performance of Battery Energy Storage System in Microgrid Considering Wind Energy Penetration

Abstract

The nature of renewable energy resources (RERs), such as wind energy, makes them highly unstable, unpredictable, and intermittent. As a result, they must be optimized to reduce costs and emissions, increase reliability, and also to find the optimal size and location for RERs and energy storage systems (ESSs). Microgrids (MG) can be modified using ESSs to gradually reduce traditional energy use. In order to integrate RERs in a financially viable scheme, ESSs should be sized and operated optimally. The paper presents an enhanced biogeography-driven optimization algorithm for optimizing the operations and sizes of battery ESSs (BESSs) taking into account MGs that experience wind energy penetration in a way that migration rates are adaptively adjusted based on habitat suitability indexes and differential perturbations added to migration operators. An optimization problem was applied to a BESS to determine its depth of discharge and lifespan. This paper considers three different scenarios in using simulations and compares them to existing optimization methods for the purpose of demonstrating the effectiveness of the offered scheme. Out of all the case studies examined, the optimized BESS-linked case study was the least expensive. We also show that a BESS must be of an optimum size to function both economically and healthily. For economic and efficient functioning of MGs, it has been shown that finding the optimum size of the ESS is important and potentially extends battery lifespan. The IBBOA obtained a more precise size for BESS’s volume, and the final outcomes are compared in this paper with other methods.

1. Introduction

Recently, battery energy storage systems (BESS) have become increasingly popular in power systems because of their capability of improving power grid reliability and flexibility, providing ride-through capabilities throughout generation losses, performing energy arbitrage, and mitigating instability resulting from renewable energy resources (RERs) such as wind and solar [1,2]. While BESS offers many benefits, it is imperative to plan it optimally, i.e., for the best location and for the best size, since BESS should not be installed on all buses, particularly in a large network [3]. Furthermore, an oversized BESS can increase the price of investment in a utility [4,5]. A complicated non-deterministic polynomial-time problem exists in determining both the location and sizing of BESS together.

Ref. [6] examined an approach utilizing second-order cone programming optimal power flow for determining the best locations and sizes for energy storage systems (ESSs) in electrical grids. According to the outcomes, ESSs are placed at optimum locations to avoid substantial control on generation operations, which saves the energy costs associated with employing large control structures. Ref. [7] used clustering and sensitivity evaluation to specify the optimum locations and volumes of BESSs in radial networks. No matter how many clusters there were, critical buses usually included BESSs. Ref. [8] examined Wind Power Time Series as a basis for optimizing ESS allocation for allocating the ESS according to charge and discharge cycles. Due to the simplicity of derived equations, the suggested time-domain dynamic simulation decreased computational time. Ref. [9] likewise proposed Receding Horizon Control and Benders decomposition algorithms for optimally allocating BESSs in low-voltage networks. By splitting the BESS allocation problem into a master and sub-problems, and solving them sequentially after solving the master problem, the algorithm increased its efficiency in optimizing BESS allocation.

Besides the analytical optimization methods outlined above, meta-heuristics and artificial intelligence have likewise been studied as they do not rely on complex calculations. Furthermore, meta-heuristic algorithms have an advantage in that they can perform global searches over a short period of time [10]. Metaheuristic algorithms may not always lead to optimum solutions, but a well-designed heuristic algorithm can usually provide a solution that is very close to optimum [11]. Ref. [12] used mixed-integer linear programming on the basis of a generated dataset to compute the optimum size of a battery in a microgrid (MG) with respect to battery autonomy. The battery autonomy is predicted using the Support Vector Regression scheme. As the input features’ capabilities are essential and important for battery autonomy prediction, the feature selection methods are used to select the relevant features. Ref. [13] presented and developed a control method and operation scheme for energy storage systems, which are used in hybrid MGs with AC coupling. The suggested method in this paper optimized the indirect control of diesel generators to decrease the working time, raise the efficiency, and increased RER generation in the grid. In order to optimize BESS size and control within wind and solar power systems, refs. [14,15] recommend artificial neural networks. Ref. [15] compares real photovoltaic (PV) production with predicted PV production. Using BESSs with optimum sizes can significantly decrease peak loads and energy costs. Ref. [16] suggested a scheme to specify the optimum size of a BESS and PV in an on-grid MG. Minimizing the cost of energy was chosen as the objective function. The energy management scheme and particle swarm optimization (PSO) algorithm were used to specify the optimum size of the PV and BESS to achieve minimum overall cost. A multi-objective optimal dispatching scheme of distributed power generation in the on-grid MG is proposed in ref. [17], which combines environmental advantages and the operational costs. Simultaneously, the power shifting for the interruptible loads on the customer side is linked to investigating the interruptible loads’ effect on the MG’s operational cost on the basis of the size of battery storage. In this study, an improved biogeography-based optimization algorithm (IBBOA) is used to solve the optimization problem. Ref. [18] optimized ESS size and location using the genetic algorithm (GA) in order to decrease the loss of networks and net present amount of the smart grids and enhance the power output fluctuation. Ref. [19] suggested a scheme to improve the MG’s reliability with integration with an ESS in which the ESS’s size was optimized under wind uncertainties. A probabilistic optimization layout with stochastic programming was applied to this end. Ref. [20] presented a flexible layout to optimize the size, and the BESS was operated in an MG with wind penetration. To solve the optimization problem, the butterfly optimization algorithm has been applied due to its simplicity and speed of implementation, and it can also achieve global optimization variables. The life-cycle and depth of discharge of a BESS are considered in this study. Refs. [21,22] used an optimization algorithm to determine the optimum location and volume of an ESS and optimized the energy hub systems. A whale optimization algorithm [23], a gray wolf algorithm [24], and a bee colony algorithm have been used to optimize BESS allocation in power grids.

An ESS can improve the performance of an MG. Nevertheless, MGs linked to grids receive assistance from networks when RERs fluctuate or when loads change. Additionally, the maintenance and operation costs of a centralized system are higher than those of MGs due to their small sizes. A properly controlled MG system produces more efficient, cleaner, and higher-quality energy compared to a centrally controlled system. As a result, this paper offers a scheme for determining the optimum sizing of different kinds of BESSs and ESSs.

Dan Simon proposed a biogeography-based optimization algorithm (BBOA) in which the biogeography theory serves as the basis for a swarm intelligence optimization algorithm [25]. BBOA has some advantages in convergence performance in comparison to the GA, and PSO algorithms and the capability and performance of BBOA have been investigated and verified by expert scholars via experimental studies. However, the basic BBOA faces the local optimization problem and also suffers from premature convergence. Hence, improvement is essential to improve the performance of the BBOA [17,25].

The life cycle and discharge depth of the BESS were also taken into account when formulating the optimization problem, and an improved BBOA (IBBOA) was used for the problem solution.

Although there is a great deal of discussion of BESS capacity optimization. Gaps remain in improving battery life for high-variability, wind-penetrated MGs (WPMGs), due to the fact that, in these conditions, the battery operates intermittently and requires higher flexibility than it does with other RESs or in normal circumstances. A well- chosen set of case studies was applied to examine the efficacy of wind power on battery performance with the IBBOA, taking into account the depth of discharge (DOD) and the life cycle of the BESS. In addition, the following contributions are offered in this paper:

• Evaluating the effect of using wind power on the DODs and life spans of BESSs;
• Considering the effects of wind power variations on the performance of the BESS in various places and seasons;
• Using the capacity incremental method for sizing the BESS until achieving the optimum size and the effects on the BESS’s DOD;
• Using the IBBOA to solve the optimization problem and proposing an efficient method for integrating BESS into WPMG.

The following is a summary of the remaining parts. Section 2 explains the problem formulation, while Section 3 explains the suggested method and the limitations of optimization. Section 4 presents the IBBOA. Section 5 examines the outcomes and discusses them. Finally, Section 6 concludes the study.

2. Problem Formulation

A model for MG components is provided in the following part, along with the problem formulation. This paper formulates the problem using MG configurations, as shown in Figure 1. The generator bus connects the wind power (the sole RES) and 3 diesel generators (DGs) to the AC bus.

The following are 3 operational case studies based on the purpose of the paper: (i) Scenario-1: BESS non-connected state, in which no BESS has been connected to the MG; (ii) Scenario-2: BESS linked state, powered by a battery of 100 kWh; (iii) Scenario-3: BESS linked state using optimal battery capacity.

2.1. Wind Power Scheme

Using Equation (1) [26], wind turbine power can be calculated. In the wind power scheme, the output power per-hour is related to the wind speed (WS).

$$P_{\omega ,t}=\left\{\begin{array}{c}0\hspace{1em}\hspace{1em}\hspace{1em}{v}_t\le v_{CI} or v_t\ge v_{CO}\\ P_{\omega }^{max}\frac{v_t-v_{CI}}{v_R-v_{CI}}\hspace{1em}\hspace{1em}{v}_{CI}\le v_t\le v_R\\ P_{\omega }^{max}\hspace{1em}\hspace{1em}{v}_R\le v_t\le v_{CO}\end{array}\right.$$

(1)

In Equation (1), $v_t$ shows the speed of the wind during $t$, $v_{CI}$ shows the cut-in WS, $v_{CO}$ indicates the cut-out WS, $v_R$ shows the rated WS, and $P_{\omega }^{max}$ indicates the rated power of the wind farm. Equation (3) provides the overall price of wasted power generated by the wind farm [27].

$$CRF=\frac{i_r{(1+i_r)}^{ly}}{365[{(1+i_r)}^{ly}-1]}$$

(2)

$$C_{\omega }\left(t\right)=\left(\sum _{t=1}^T{P}_{\omega ,t}\right)\times {IC}_{\omega }\times CRF$$

(3)

In Equations (2) and (3), CRF shows the capital recovery factor, calculated using Equation (2), in which $i_r$ shows the interest rate and $l_y$ indicates projected battery lifespan of the wind power farm. ${IC}_{\omega }$ shows the primary wind plant price. T shows the interval time for 24 h.

2.2. Generator Scheme

The cost of the generator $i$ is calculated based on its dispatch power during t, as shown in Equation (4) [28]. This paper assumes the DG as the second resource of power production for the MG. Wind power is the main resource of energy.

$$C_g(t)=\sum _{i=1}^N(a_i{P}_{g,i}^2\left(t\right)+b_i{P}_{g,i}\left(t\right)+c_i)$$

(4)

In Equation (4), $C_g\left(t\right)$ defines the cost of the generator, and $P_{g,i}\left(t\right)$ shows the output power of generator $i$ during $t$. $a_i$, $b_i$, and $c_i$ show the ratio of the $ith$ generator. $N$ indicates the number of generators examined.

2.3. BESS Scheme

BESS costs are calculated according to the battery’s life span [26,27,28]. A battery’s life-span is determined through its DOD. As a result, BESS costs are likewise determined according to DOD. The DOD of the battery $\left({DOD}_b\right(t\left)\right)$, life-span for the determined DOD $\left(l\right(t\left)\right)$, and the battery’s cost $\left(C_b\right)$ are determined through Equations (5)–(7).

$${DOD}_b\left(t\right)=1-{SOC}_b\left(t\right)$$

(5)

$$l\left(t\right)=694\times {\left({DOD}_b\right(t\left)\right)}^{-0.795}$$

(6)

$$C_b(t)=\frac{C_{b,cap}\times P_b\left(t\right)}{E_{sto}\times l\left(t\right)\times {\eta }_b^2}$$

(7)

In Equations (5)–(7), ${DOD}_b\left(t\right)$ is the DOD of the battery. ${SOC}_b\left(t\right)$ shows the state of charge of the battery at time $t$. $C_b\left(t\right)$ defines the cost of the battery. $C_{b,cap}$ is the primary battery capacity price. $E_{sto}$ shows the battery storage capacity. ${\eta }_b$ is the efficiency of the battery. $P_b\left(t\right)$ shows the power available through the BESS during $t$. Here, the unit optimization time $(∆t)$ is one hour.

3. Proposed Layout

This part discusses the suggested size method. Moreover, this part explains the objective function and all limitations.

3.1. Objective Function

Since the individual components of the MG have been discussed in the previous parts, the overall price would be, accordingly, the total of the prices of the wind farm $\left(C_{\omega }\right)$, the generator $\left(C_g\right)$, and the BESS $\left(C_b\right)$. Thus, Equation (8) provides the objective function for optimizing size. A BESS whose size is minimized would have the optimum cost function:

$$J=min\sum _{t=1}^T(C_{\omega }\left(t\right)+C_g\left(t\right)+C_b(t\left)\right)$$

(8)

In Equation (8), $C_{\omega }$, $C_g$, and $C_b$ correspond to the previous definitions, and $T$ represents the overall operational time (24 h in this paper).

3.2. Limitations

There are a number of limitations associated with minimizing the objective function. In order to optimize, the following limitations must be met.

3.2.1. Power Balance Limitation

In the electrical grid, the power balance limitation is the most important limitation. Power supply and demand are maintained by this limitation, as shown in Equation (9).

$$P_{\omega }\left(t\right)+P_b\left(t\right)+\left(\sum _{i=1}^N{P}_g\left(t\right)\right)-P_L\left(t\right)=0$$

(9)

In Equation (9), $P_L\left(t\right)$ shows the power required by the load during $t$.

3.2.2. Generator Limitation

There must always be upper and lower limits on the output power of each generator $i$ during each given time $t$. Equation (10) provides this limitation:

$${P_g\left(i\right)}^{min}\le P_g\left(i,t\right)\le {P_g\left(i\right)}^{max}i=\mathrm{1,2},\dots ,N$$

(10)

3.2.3. BESS Limitation

Equations (11)–(15) express the BESS limitations. In this paper, the negative and positive power convention has been selected for the charging power and discharging power of the battery. Hence, the battery acted as a generator in its discharging mode and acted as a load in its charging mode. A 1-h optimization time interval ($∆t$) is assumed. According to Equation (11), the battery’s power withdrawn should remain under reasonable limitations for the battery all the time. In addition, Equations (12) and (13) express the discharge and charge limitations of the battery. ${\mu }_b$ shows a binary variable stating the operational state of the battery. During discharge, it equals one, while during charging, it equals zero. Additionally, battery energy should always remain under the limitation shown by Equation (14). Equation (15) calculates the charge and discharge power based on the battery’s energy [29,30,31].

$$P_b^{min}\le P_b\left(t\right)\le P_b^{max}$$

(11)

$$0\le P_{b,dch}\left(t\right)\le P_b\times {\mu }_b$$

(12)

$$-P_b * (1-{\mu }_b)\le P_{b,ch}\left(t\right)\le 0$$

(13)

$$E_b^{min}\le E_b\left(t\right)\le E_b^{max}$$

(14)

$$E_b\left(t+1\right)=\left\{\begin{array}{c}{E}_b\left(t\right)-\frac{P_{b,dch}\times ∆t}{{\eta }_{b,dch}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(P_b\left(t\right)>0)\\ E_b\left(t\right)-P_{b,ch}\left(t\right)∆t\times {\eta }_{b,ch}\hspace{1em}\left(P_b\right(t)<0)\end{array}\right.$$

(15)

In Equations (11)–(15), $P_b^{min}$ shows the minimal power and $P_b^{max}$ shows the maximal power available through the BESS during $t$. $E_b^{min}$ shows the minimal energy limit of the BESS, and $E_b^{max}$ shows the maximal energy limit of the BESS during $t$. $P_{b,ch}\left(t\right)$ shows the charge power, and $P_{b,dch}\left(t\right)$ indicates the discharge power during $t$. $P_b\left(t\right)$ shows the battery power, and $E_b\left(t\right)$ shows energy during $t$.

In addition, Equation (15) relates the state of charge (SOC) of the battery at a particular time $t$ to the prior SOC and to the charge efficiency and discharge efficiency.

3.3. Proposed Optimization Layout

As a method of determining the optimum battery capacity, the suggested approach uses the IBBOA to balance power at each hour according to load demand. When the wind power falls below the demand, the algorithm dispatches power from the generators and/or the BESS. It is influenced by battery DOD, thereby affecting operation costs.

The algorithm dispatches the most efficient amount of power from the energy resources and generators according to the price formulas of the power resources and load demand for each hour. A generator could likewise be applied for charging the battery when the BESS price exceeds that of the generator. In this case, the battery’s DOD would be considerable. At the end of each hour, the algorithm calculates the battery SOC and decides what actions to take. A power source planning method, aimed at minimizing the MG’s total price, has been determined for the optimum battery capacity. Batteries ranging from 100 to 300 kWh are used.

4. BBOA

The science of biogeography, founded in the 19th century, examines the spatial and temporal distribution of species [32]. Several mathematical schemes have been proposed for biological community migration, extinction, and distribution by researchers. In 2008, Dan Simon, an American academic, suggested a BBOA based on those mathematical schemes. To find the global optimal solution, a BBOA simulates a species migration process among habitats based on the mathematical scheme of species migration.

4.1. Basic BBOA

In the BBOA, an optimization problem is modeled as a combination of habitats of species in biogeography and suitability index variables (SIV) in the layout, which represent the individual parameters, along with a fitness function (I), which evaluates the quality of a solution set based on habitat suitability indexes. Optimization problems are solved by generating various habitats randomly as an initial solution. By exchanging information about species migration amongst habitats, species variety of habitats has been enhanced, habitat I has been enhanced, and the optimal solution to the problem has been achieved. In the BBOA, there are three main steps: (i) initialization, (ii) migration, and (iii) mutation.

4.1.1. Initialization

There are several factors that are initialized for the BBOA, including the maximum mutation rate $m_{max}$, the maximum rate of emigration $E$, and immigration $I$ and the maximum number of species per habitat $S_{max}$. In order to generate the initial population of habitats, the BBOA uses Equation (16) to generate $N$ habitats, and every habitat is a potential solution with D-dimensional solution parameters. Then, a calculated $I$ is determined for each habitat.

$$x_{ij}=x_{j,min}+rand(x_{j,max}-x_{j,min})$$

(16)

In Equation (16), $x_{ij}$ shows the $jth$ dimensional solution factor of habitat $x_i,i\in [1,2,...,NP],j\in [1,2,...,D]$, and $x_{j,min}$ and $x_{j,max}$ defines the upper and lower restrictions of the $jth$ dimensional parameter, respectively. $rand$ is a number between zero and one that is generated randomly.

4.1.2. Migration

A wide area of the solution space is explored after the BBOA has exchanged data with other habitats with the migration scheme. According to the biogeographic concepts, habitat $I$ directly correlates with species variety, and a habitat with a high $I$ is able to support a greater variety of species. Thus, the number of species $S_i$ and habitat $x_i$ have a mapping relationship. The initial habitat $x_i$ receives a new $i$ after having been reordered from high to low based on the $I$ that belongs to it. Equation (17) calculates how many species $S_i$ have been ranked for $x_i$ after ranking.

$$S_i=S_{max}-i$$

(17)

This formula can be used to calculate the emigration rate ${\mu }_i$ and immigration rate ${\lambda }_i$ of habitat $x_i$:

$$\left\{\begin{array}{c}{\lambda }_i=I(1-\frac{S_i}{S_{max}})\\ {\mu }_i=\frac{E.S_i}{S_{max}}\end{array}\right.$$

(18)

The first step in the migration process is to determine the habitat $x_i$ to be migrated based on the migration rate ${\lambda }_i$. A random number amongst $\left(\mathrm{0,1}\right)$ is produced; if it is lower than ${\lambda }_i$, the habitat $x_i$ will be moved into. Following this, we must determine the habitat through which migration will occur for each dimension of $x_i$. A roulette system determines $x_k$ based on the migration rates (${\mu }_k$) of the residual $N-1$ habitats. Via replacing the solution parameters of habitat $x_i$ with the solution parameters of habitat $x_k$, the solution parameters of habitat $x_i$ are finally replaced.

4.1.3. Mutation

A habitat’s $I$ can change rapidly due to a catastrophic event, and the BBOA simulates such a rapid change using a mutation operation. Equation (19) calculates the probability $P_i$ of species number $S$ in habitat $x_i$ based on the emigration rate ${\mu }_i$ and the immigration rate ${\lambda }_i$.

$$P_i=\left\{\begin{array}{c}-\left({\lambda }_i+{\mu }_i\right)P_i+{\mu }_{i+1}{P}_{i+1},\hspace{1em}\hspace{1em}\hspace{1em}{S}_i=0\\ -\left({\lambda }_i+{\mu }_i\right)P_i+{\lambda }_{i-1}{P}_{i-1}+{\mu }_{i+1}{P}_{i+1}, 1\le S_i\le S_{max}-1\\ -\left({\lambda }_i+{\mu }_i\right)P_i+{\lambda }_{i-1}{P}_{i-1},\hspace{1em}\hspace{1em}\hspace{1em}{S}_i=S_{max}\end{array}\right.$$

(19)

Equation (20) is used to calculate the mutation rate $m_i$ of habitat $x_i$.

$$m_i=m_{max}(1-\frac{P_i}{P_{max}})$$

(20)

In Equation (20), $P_{max}$ defines the maximum probability of entire species in the habitat, and $m_{max}$ defines its maximum mutation factor.

4.2. Improved BBOA

4.2.1. Mechanism of Adaptive Definition of Migration Rate

An individual with wide quality gaps or an uneven distribution can be easily misjudged for their quality level. As a result, either beneficial information from the better individuals cannot be kept or data from the poor individuals who took part early in evolution cannot be retained. Equation (21) indicates that the immigration rate can be adjusted adaptively based on the $I$ value of normalized individuals, according to the above question.

$$\left\{\begin{array}{c}{\lambda }_i=\left(\frac{F_i-F_{min}}{F_{max}-F_{min}}\right)\\ {\mu }_i=1-{\lambda }_i\end{array}\right.$$

(21)

In Equation (21), $F_i$ defines the $I$ of habitat $x_i$. $F_{min}$ and $F_{max}$ show the minimum and maximum amount of $I$ in the existing population. It can be used to solve the problem of assessing individual quality in real time while keeping data about individuals in the existing population in mind to effectively assess the potential immigrants or emigrants.

4.2.2. Dynamic Migration Scheme

A large part of the performance of BBOA is influenced by its migration approach, which is an integral part of its evolution process. This migration approach, in which the SIV of the main solution is directly replaced with the SIV that is to be emigrated, may result in a specific blindness in migration, as well as a poor capacity to identify new solutions, which leads to premature convergence. In order to solve this problem, a new dynamic migration scheme is proposed, utilizing an adaptive migration factor, which was designed in the prior part to adapt the perturbation degree around the differential vector in order to find new solutions. By adjusting the degree of perturbation, adaptive perturbation can be achieved around the SIV which will be emigrated at various evolutionary times. Hence, Equation (22) shows how the migrating procedure is enhanced in terms of its capability to search.

$$x_{ij}=x_{kj}+(r_{min}+{\lambda }_i\times (r_{max}-r_{min}\left)\right)\times (x_{aj}-x_{bj})$$

(22)

In this case, $x_k$ refers to the habitat that needs to be emigrated; the numbers $a$ and $b$ are chosen at random from the range $[1,NP]$; $r_{min}$ and $r_{max}$ define the minimum and maximum disturbance degree, respectively; and $x_{aj}$ and $x_{bj}$ show the $jth$ dimensional parameters of the $ath$ and the $bth$ habitat, respectively.

5. Simulation Outcomes

A low voltage MG with a 150 kW wind turbine, a battery, and 3 DGs, which are shown in Figure 1, are presented to show the proficiency of the suggested approach. The Matlab software (2021b) package is used to solve the optimization problem with IBBOA and simulate the suggested system.

Table 1. Generator variables.

DG	${\mathit{P}}_{\mathit{m}\mathit{i}\mathit{n}} \left(\mathbf{k}\mathbf{W}\right)$	${\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}} \left(\mathbf{k}\mathbf{W}\right)$	${\mathit{a}}_{\mathit{i}} (\mathbf{\$}/\mathbf{k}\mathbf{W})$	${\mathit{b}}_{\mathit{i}} (\mathbf{\$}/\mathbf{k}\mathbf{W})$	${\mathit{c}}_{\mathit{i}} \left(\mathbf{\$}\right)$
${DG}_1$	0	40	1 × 10−4	$4$.38 × 10−4	0.3
${DG}_2$	0	20	1 × 10−4	4.79 × 10−4	0.5
${DG}_3$	0	10	1 × 10−4	4.9 × 10−4	0.4

shows the DGs information. The BESS data such as maintenance and capital costs, interest rate, and lifetime are presented in

Table 2. System variables.

Variable	Primary Capital Cost ($/kWh)	Round-Trip Efficiency (%)	Primary SOC (%)	${\mathit{S}\mathit{O}\mathit{C}}_{\mathit{b}\mathit{a}\mathit{t}}^{\mathit{m}\mathit{a}\mathit{x}}$ (%)	${\mathit{P}}_{\mathit{b}\mathit{a}\mathit{t}}^{\mathit{m}\mathit{a}\mathit{x}}$ (kW)
Value	635	90	80	90	25
Variable	Maintenance cost ($/kWh)/Year	Interest rate (%)	Lifetime (Year)	${\mathit{S}\mathit{O}\mathit{C}}_{\mathit{b}\mathit{a}\mathit{t}}^{\mathit{m}\mathit{i}\mathit{n}}$ (%)	${\mathit{P}}_{\mathit{b}\mathit{a}\mathit{t}}^{\mathit{m}\mathit{i}\mathit{n}}$ (kW)
Value	25	6	3	10	10

.

The algorithm of the suggested method to solve the optimization problem is summarized in Algorithm 1 below.

Algorithm 1: Suggested Algorithm to solve the optimization problem
1.	Defining the system variables, and initializing variables such as queest agents, number of generation units, dimention, cost function.
2.	Specifing the population size, and generating the random population, and computing the cost function based on Equation (8) for the initial population.
3.	Evaluating the habitat individuals, and eliminating the individuals that do not satisfy the constrains.
4.	Sorting the remained habitat individuals in ascendant order and using them as the population.
5.	Updating the parameters ${\mu }_i$, and ${\lambda }_i$ for every habitat $x_i$ in the population (Equation (21)).
6.	Computing the migration according to Equation (22) by using the updated ${\mu }_i$, and ${\lambda }_i$ from previous step.
7.	Specifing the mutation rate $m_i$, determining the species probability $P_i$ for all habitat. Computing the cost function based on Equation (8).
8.	If the iteration reach the maximum, go to next step, if not, go to step 3 for the next iteration.
9.	Print the best answer.

A common residential load profile is shown in Figure 2, along with a profile of wind power applied to evaluate the suggested approach. According to Figure 2, for most of the 24 h, the power demand exceeds the supply of RES power. Aside from the BESS, the generators are used to meet the remaining energy needs. In addition, when the supply of RES exceeds the demand for a few brief times, the extra energy is utilized for charging the BESS. The generators are applied to charge the BESS when it is necessary to ensure a sufficient level of SOC for effective and smooth operation. In this condition, the BESS’s discharge cost rises as the SOC of the BESS depletes from continuous usage. In order to maintain the BESS’ state of health and lengthen its lifespan, it must perform at its optimum SOC.

The following are three operational case studies based on the purpose of the paper: (i) Scenario-1: BESS non-connected state, in which no BESS has been connected to the MG; (ii) Scenario-2: BESS linked state, powered by a battery of 100 kWh; (iii) Scenario-3: BESS linked state using optimal battery capacity. In the following, we will try to find that raising the BESS’s capacity can increase cost or may not decrease the operational costs. For achieving a minimal diurnal operational cost in the MG, an incremental approach is used that focuses on the BESS’s capacity and on its flexible operation. The following three subsections explain the outcomes of the three scenarios examined.

5.1. Scenario-1: BESS Disconnected Status

This case involves the MG being performed sans the BESS. In this case, only generators and the RES provide power. Load shedding cannot occur if the sources are always able to meet the load demand; otherwise, there will be a mismatch. The supply graph for generators and RES, and also the load demands are displayed in Figure 3. In addition, the figure represents the power supply when the generators are running at maximum capacity. In certain hours, especially between 9~11 and 19~21, a lack of installed capacity that is not able to accommodate the demand can be seen.

5.2. Scenario-2: BESS Linked Status, with a Firm Battery Size of 100 kWh

BESS with a constant volume of 100 kWh is connected to the MG to study its impact on its performance. It should be possible to prevent power mismatches at any given point in time with this volume. Figure 4 shows the output power of the BESS, generators, and RES. It was determined that the overall operating cost for a day was USD 287.9, and the scheduling cost for a day was USD 240.3.

During the hours of 8~11, the BESS is charged again. However, while DG output and wind power increased, the two were insufficient to supply the peak demand that occurred at hour 11. Hence, the BESS must discharge some of its capacity to accommodate the demand. Figure 5 shows the DGs’ dispatch in Scenario-2, and it also indicates how each generator is dispatched. The observation clearly indicates that the MG operates quite flexibly.

In the hours 13~16, there is a reduction in BESS power as the demand drops continuously, and the BESS is charging. In addition, the BESS power during hours 18 to 21 is reflected by a significant rise in demand accompanied by a decline in wind power. As a result of this operation, a battery will just charge if demand power is lower than power of wind and/or if the discharging’s cost of the battery is greater than the DGs’ cost. Nonetheless, overstressing the battery might not be good for it. The DOD of the BESS can be seen in Figure 6 when the charging is performed on the BESS. According to this figure, the DOD changes as the BESS switches between charging and discharging. In addition, during peak demand or when the other resources operate at their operational ranges, a unity DOD can be achieved.

5.3. Scenario-3: BESS Connected Status with an Optimum Battery Capacity

The aim of this scenario is to determine the optimum battery volume that will result in a smooth performance that is cost effective for the proposed system. A battery volume optimal for minimizing operational costs, assuring flexibility, and maximizing battery lifespan was sought within different ranges of battery capacities. With a step size of 15 kWh, the battery volumes changes from 100 kWh to 300 kWh. A volume of 150 kWh was determined to be the optimum volume of the battery. In Scenario-3, USD 159.6 and USD 113.2 were derived as total operational costs and scheduling costs for 24 h, respectively. As the MG is operated at optimal capacity, these costs are lower than those calculated in Scenario-2.

Table 3. Comparison of the entire operational costs and scheduling costs in Scenario-3 among various algorithms.

Method	Operational Costs ($)			Scheduling Costs ($)
	Best	Mean	Worst	Best	Mean	Worst
IBBOA	159.6	160.3	161.8	113.2	114.1	115.4
BBOA	198.4	206.7	215.3	161.7	168.5	177.1
PSO	226.8	231.2	235.6	184.9	188.7	195.0
GA	237.1	248.5	262.6	205.3	213.2	228.6

shows the total operational costs and scheduling costs in Scenario-3 for the suggested IBBOA and other algorithms as basic BBOA, PSO, and GA. As can be seen in this table, the suggested method shows better performance, and the obtained costs based on the suggested method are lower than the other methods. Furthermore, in the suggested method, the difference between best cost and lower cost is lower than in other methods, which shows that the performance and efficiency of the suggested method attains the best solution. Figure 7 shows the power output of the BESS, generators, and RES. As can be seen in this figure, load demands were met at all times. Since Generator-1 has the highest power output limit and the lowest cost, it has been given priority among the three generators. This can be seen in Figure 8, which illustrates how each generator dispatches power individually.

Due to the lack of optimization of battery volume in Scenario-2, heavy discharge occurred in the battery to satisfy the power demand. In this case, the DOD remained within the battery’s optimal utilization range, which is shown in Figure 9. Similar to Scenario-2, the BESS was executed hourly. The DOD of the battery supposes a similar sample but is not hiked to the unity. The maximum DOD amount was about 69%, which means the battery will need to be charged in another cycle or else the cost of battery will exceed that of the generators. Battery lifespan increases when the battery operates within the DOD limits.

Figure 10 shows the battery power dispatch for a day when the battery is charged or discharged. If a battery’s power is positive, it is in discharging mode, and if it is negative, it is in charging mode. The maximum charging and discharging power were adjusted to −10 kW, and 25 kW, respectively.

6. Conclusions

A scheme is presented in this study in order to optimize the BESS’s capacity in a WPMG; hence, the IBBOA was used to solve the optimization problem. The entire operational cost of the MG is minimized to specify the optimum capacity of the battery. The study we conducted found that raising the BESS’s capacity causes increased costs and may not decrease operational costs. For achieving a minimal diurnal operational cost in the MG, an incremental approach is used that focuses on the BESS’s capacity and also on its flexible operation. Moreover, this study showed that the BESS must be of an optimum volume to function both economically and healthily. For an economic and efficient function of the MGs, it has been concluded that finding the optimum volume of the ESS is important, as it could potentially extend the battery’s lifespan. The search algorithms could be used to find the BESS’s capacity rather than an incremental method based on a presumed primary capacity for determining the optimum volume. In spite of its computational cost, this scheme results in a more precise size for a BESS’s volume and final outcomes. For future studies, the new and modified control method can be used to control the charging/discharging of the BESS to reach a stable and robust system. Furthermore, the cyber security of the battery optimization and control can be considered an edge layer that transfers its information to the upper layer or to the cloud layer in an IoT-based smart MG.