A Comprehensive Review and Application of Metaheuristics in Solving the Optimal Parameter Identification Problems

Abstract

For many electrical systems, such as renewable energy sources, their internal parameters are exposed to degradation due to the operating conditions. Since the model’s accuracy is required for establishing proper control and management plans, identifying their parameters is a critical and prominent task. Various techniques have been developed to identify these parameters. However, metaheuristic algorithms have received much attention for their use in tackling a wide range of optimization issues relating to parameter extraction. This work provides an exhaustive literature review on solving parameter extraction utilizing recently developed metaheuristic algorithms. This paper includes newly published articles in each studied context and its discussion. It aims to approve the applicability of these algorithms and make understanding their deployment easier. However, there are not any exact optimization algorithms that can offer a satisfactory performance to all optimization issues, especially for problems that have large search space dimensions. As a result, metaheuristic algorithms capable of searching very large spaces of possible solutions have been thoroughly investigated in the literature review. Furthermore, depending on their behavior, metaheuristic algorithms have been divided into four types. These types and their details are included in this paper. Then, the basics of the identification process are presented and discussed. Fuel cells, electrochemical batteries, and photovoltaic panel parameters identification are investigated and analyzed.

1. Introduction

With the rapid depletion of different fossil-fuel-based energy supplies, such as coal, oil, and natural gas, and the air environment being severely polluted in recent decades, an alternative energy supply has become an urgent and vital subject that has piqued widespread interest. As a result, the extraction and usage of renewable energy will undoubtedly play an important part in future growth, with solar energy serving as one of the most promising options [1].

Solar energy may produce electricity or thermal energy without emitting pollutants, which is critical for environmental issues. Nevertheless, there are still significant problems with their practical deployment, such as poor photoelectric conversion efficiency and a lack of precision in photovoltaic (PV) cell modeling. Accurate PV cell modeling is essential for understanding and forecasting the particular properties of PV systems [2]. Due to nonlinear PV features and its enormous dependence on radiation level and operating temperature, the research on PV panel model development remains an open subject. In this context, a number of PV models have been created and published, including the single diode model (SDM) [3], the double diode model (DDM) [4], the triple diode model (TDM) [5], the improved single diode model (ISDM) [6], SDM with a parasitic capacitor [7], the improved two diode model (IDDM) [8], the modified double diode model (MDDM) [4], the diffusion-based model [9] and multi-diode model [10]. On the other hand, the model’s accuracy varies depending on its predicted parameters. Regretfully, it is challenging to define fixed numbers for these parameters using manufacturers’ datasheets since they vary with time. As a result, specific model parameters are required to develop an accurate and trustworthy PV model.

On the other hand, the generated renewable power is submitted to weather conditions that may lead to power fluctuations. In addition, sunlight is unavailable at night, which limits the generation during these specified times. To resolve these problems, energy storage systems such as batteries are required [11]. There are several types of electrochemical batteries; lead acid is the most common, but the lithium-ion type is becoming more commonly used due to its significant advantages [12]. However, the lifetime of each battery is related to its physical features, which provides a high nonlinearity to its mode [13]. The lifespan of a battery cannot be prolonged by reducing power consumption at a certain stage. Instead, it can be extended by how the power is utilized. Furthermore, persistent high-current pulling reduces residual battery capacity [14]. As a result, a battery management system (BMS) is necessary to guarantee that batteries operate safely, reliably, and efficiently. The BMS’ tasks include guaranteeing a battery’s safe operation, over-temperature prevention, managing the charging/discharging phases, and calculating the state of charge (SoC) based on the measured current, voltage, and temperature [15]. However, performing these tasks and estimating these states depends on the battery model. Therefore, an accurate battery model is required, and its accuracy is related to the accuracy of the parameters identification. Furthermore, parameter identification is critical in PV system modeling, performance assessment and optimization, and real-time control [16]. Since the importance of parameter identification has grown significantly, a wide range of studies has been conducted to find realistic and practical solutions to such difficulties.

In such applications, the batteries cannot store all the produced renewable energy. Fuel cell systems offer an alternative solution to store this energy and other benefits. These systems are mainly composed of electrolyzers and fuel cells [17,18]. The electrolyzer splits the water into oxygen and hydrogen using generated renewable power. Fuel cells transform hydrogen energy into electricity with a controlled flow of electrons based on electrochemical reactions between a fuel (the hydrogen) and an oxidant. The fuel cells usually have an electrical efficiency of >50% in the case of regular cycles and >70% in the case of hybrid cycles [19]. Additionally, the pollution created by fuel cells is almost nonexistent [18], and the carbonic emissions per unit of energy produced are decreased using renewable fuels. Fuel cells are split into four types [20]: Alkaline FC (AFC) [21], Molten carbonate FC (MCFC) [22], Proton electrolyte membrane FC (PEMFC) [23], Direct Methanol FC (DMFC) [24], Solid oxide FC (SOFC) [21], and Phosphoric acid FC (PAFC) [25]. The mathematical modeling of fuel cells is one of the most significant problems associated with their technological development. Modeling can reveal more information about how this device works, providing a simulation tool that helps in understanding their performance and improving it [26]. Accurately assessing the parameters is one of the modeling process’ most significant issues [27]. The multi-physics system primarily causes this challenge, and its operating circumstances directly impact its parameters [28] due to its ability to accurately reproduce the behavior of the fuel cell under various operating conditions employing the polarization curve. The semi-empirical equations-based mathematical models have received the most attention among all the currently used modeling approaches [29]. The present problem with these models is that the precise parameters are not readily available. To produce reliable results, accurately identifying the model’s parameters is crucial.

A variety of techniques, including artificial neural networks and adaptive filter approaches, have been used for the parameter extraction and modeling of PV panels, batteries, and fuel cells. Recently, numerous research has focused on using metaheuristic optimization algorithms (MOAs) for parameter identification. This is because of the scalability and the parallel computing capabilities for identifying the model’s linear and nonlinear parameters. In addition, their capacity for exploration and finding intriguing domains in the specified search space at a certain moment makes them an excellent solution. Metaheuristic optimization algorithms provide optimum or sub-optimal results. They need the objective function and constraints to solve linear, nonlinear, and nonconvex problems [23]. According to the reported study by A. Tzanetos and G. Dounias [30], the number of published works that presents newly created metaheuristic optimization algorithm is rising. The increased interest in them can explain this. This attracted the authors to review their applicability in the context of optimal parameter extraction strategies for PVs, FCs, and Li-ion batteries.

To further accelerate the optimal PV cell parameters identification tendency, a number of metaheuristic algorithms have been used. RTC France solar cell is one of the most popular PV cells. Its model has been widely used to approve the performance of the optimizers. The parameters of three models (SDM, DDM, and TDM) have been identified for this type using the Artificial Hummingbird Algorithm (AHA) [31]. A similar study has been proposed in [32] using Atomic Orbital Search (AOS), and another one in [33] by using the Marine Predators Optimizer (MPA). Kyocera KC200GT PV also recieved increased interest in the parameters identification context, such as Northern Goshawk Optimization (NGO) [34], Gorilla Troops Optimizer (GTO) [35], Transient Search Optimization (TSO) [36], and Coyote Optimization Algorithm (COA) [37]. On the other hand, papers that present battery parameter extraction strategies are more frequently published. The Genetic Algorithm (GA) has been widely used to extract the parameters of many types of Li-ion batteries, such as SPM [38,39], simplified SPM [40], and P2D [39,41,42]. Identifying the PEMFC is also an attractive topic to the academic community. BCS 500 W, NedStack PS6, 250 W stack, and SR-12 500 W are the most commonly used to extract the parameters. More details about these and the other systems (PV and Li-ion batteries) will be presented in this paper.

The identification problem can be constructed as an optimization problem, where the optimization variables are the unknown parameters, and the objective function is based on the model and the measured system output. This study comprehensively reviews the metaheuristic optimization algorithms used in the parameter estimation of PV, battery, and fuel cell models. This paper first presented the model of each system and the parameters to be identified. Then, three well-known metaheuristic optimization algorithms are presented to illustrate the optimization process of each one. Then, a set of recent metaheuristic optimization algorithms used to extract its parameters is listed. The reported results are discussed, and the optimizer with the best performance is indicated. The following are this paper’s significant contributions:

• A review of the photovoltaic models, including the SDM, DDM, and TDM;
• A review of the lead-acid and lithium-ion battery models;
• A review of the electrochemical modeling of the PEMFC;
• A comprehensive review of the metaheuristic optimization algorithms’ implementation in each system’s parameters extraction.

The rest of the paper can be organized according to Figure 1.

2. Photovoltaic Models

Developing a PV cell model is necessary for studying the characteristics of solar PV output. Only an exact fitting of PV cell output current–voltage (I-V) and power–voltage (P-V) curves can accurately analyze and predict PV system performance, which is heavily reliant on adequately identifying the needed DC parameters from the PV cell model. This section will provide a quick overview of some frequently used and typical PV models, such as SDM, DDM, and TDM. Their basic architecture is similar: an ideal constant current source (i_ph), a series resistor (R_s), and a shunt resistor (R_sh), with the primary difference being the number of parallel diodes. The architecture of each type is presented in Figure 2, and the equation that expresses the output current can be provided as

$$\begin{array}{c}{i}_c=i_{ph}-i_d-i_{ohm}\\ \hspace{1em}\hspace{1em}=i_{ph}-i_d-\frac{V_{cell}+i{R}_s}{R_{sh}}\end{array}$$

(1)

where i_d is the diode current and i_ohm is the ohmic losses current.

2.1. Single Diode Model (SDM) [43]

It is the simplest model that consists of a photocurrent source, a diode representing the semiconductor’s losses due to the optical and recombination, and series and shunt resistances accounting for leakage losses. However, they suffer from low accuracy with significant irregularity in replicating the characteristics of the I-V curve, especially for the partial shading conditions. The diode current is denoted as

$$\begin{array}{l}{i}_d=i_0\left(e^{q\frac{V_{cell}+i{R}_s}{a{V}_t}}-1\right)\\ V_t=\frac{N_{s}KT}{q}\end{array}$$

(2)

where q = 1.6 × 10⁻¹⁹, N_s expresses the number of cells connected in series, K is the Boltzmann constant (=1.38 × 10⁻²³ J/K), i_d, i₀, R_s, R_sh, and α are the unknown parameters.

2.2. Double Diode Model (DDM) [44]

The presence of an extra diode is the only distinction between SDD and DDM. Compared to the SDD, introducing a second diode improves model accuracy, especially at low solar irradiation. The second diode works with the first to reflect the recombination losses in the depletion layer under weak solar irradiation. The diode current can be expressed as

$$i_d=i_{01}\left(e^{q\frac{V_{cell}+i{R}_s}{a_1{V}_t}}-1\right)+i_{02}\left(e^{q\frac{V_{cell}+i{R}_s}{a_2{V}_t}}-1\right)$$

(3a)

where i_d, i₀₁, i₀₂, R_s, R_sh, α₁ and α₂ are the unknown parameters.

2.3. Trible Diode Model (TDM) [45]

This type of PV model includes an additional diode, which can express the various elements of PV cells with better curve-fitting accuracy. However, the additional diode increases the model complexity, which makes its hardware implementation challenging. The diode current equation can be expressed as follows:

$$i_d=i_{01}\left(e^{q\frac{V_{cell}+i{R}_s}{a_1{V}_t}}-1\right)+i_{02}\left(e^{q\frac{V_{cell}+i{R}_s}{a_2{V}_t}}-1\right)+i_{03}\left(e^{q\frac{V_{cell}+i{R}_s}{a_3{V}_t}}-1\right)$$

(3b)

where i_d, i₀₁, i₀₂, i₀₃, R_s, R_sh, α₁, α₂ and α₃ are the unknown parameters.

The classification of these modes is presented in Figure 3.

3. Lithium-Ion Battery Modeling

There are a number of studies concerning lithium-ion battery modeling in the literature. These models can be constructed based on specific physical, electrical, and thermal factors or a combination of these [46]. Concerning the electric models, they attempt to imitate the electric variables’ behavior, such as the voltage, the current, and the SoC. Thermal models, on the other hand, try to mimic the temperature distribution on the battery cell in one, two, or three dimensions. Concerning the aging effect, the related models are designed to simulate battery degradation, manifested as capacity fading and an increase in internal impedance. The literature review divides the battery models into four classes [47]: empirical, equivalent circuit (ECM), electrochemical, and data-driven models. The battery parameters can be extracted based on the measured data and the considered model. After contracting the model, the current and/or the SoC data will be used to simulate it and generate the voltage. The generated voltage will be compared with the measured one, and the error between them will be used to generate the fitness value for the optimization algorithm. Based on the fitness value, the optimizer updates the candidate solutions.

3.1. Empirical Models [48]

These are streamlined electrochemical models. They use reduced-order polynomials or mathematical expressions to reflect the principal nonlinear features of a battery. In this model, the output voltage is empirically expressed as a function of the SOC and current. The Shepherd model [49], Unnewehr universal model [50], and Nernst model [51] are the most common ones. These models have the following generalized equation [52]:

$$V_{batt}=E_0-R\cdot i_{batt}-\frac{a_1}{SoC}-a_2\cdot SoC+a_3\cdot \ln (SoC)+a_4\cdot \ln (1-SoC)$$

(4)

where E₀ is the OCV, R expresses the internal resistance, and a_1,2,3,4 are model parameters.

3.2. Equivalent Circuit Models [53]

ECM consists of a SOC-related voltage source, an internal ohmic resistor, and resistance–capacitance (RC) pairs that may represent the inputs (current, temperature, and SOC) and the output (voltage) relationship. The resistor represents self-discharge. The diffusion process in the electrolyte, porous electrode charge transfer and double-layer effect in the electrode is represented by the RC pairs with various time constants. The commonly used models in this category include the Rint model [54], Thevenin model [55], and General Nonlinear (GNL) [56]. The architecture of the models is presented in Figure 4.

3.3. Electrochemical Models [57]

For the objective of representing the internal processes in the battery, electrochemical models are constructed in accordance with the charge transfer process, including the electrochemical kinetics. They are constructed as nonlinear partial differential equations based on several laws, including Ohm’s law, Faraday’s first law, Butler–Volmer equation, and Fick’s law of diffusion. There are two main types of these models: the single particle model (SPM) and the pseudo-2D model (P2DM). P2DM is built based on the porous electrode and the concentrated solution theories [58]. The porous electrode’s structure expands the surface area in a way that effectively aids the electrochemical processes. The porous construction has the advantage of allowing the active material to touch the electrolyte sufficiently. P2DMs view the electrode’s active component as a sphere with uniformly sized and differently sized particles. SPM is a P2DM simplification that treats the electrode as a single particle [59]. The reactions in the electrode are the same for various particles if the concentration of the liquid phase is considered constant, as well as the electrode voltage. As a result, their electrochemical responses may be regarded as a single spherical one. SPM makes it considerably easier to describe the motion of lithium ions within a solid particle compared to P2DM.

3.4. Data-Driven Models [60,61]

Because of the rapid growth of data mining algorithms in the artificial intelligence (AI) field, the link between the battery variables may be easily created based on preliminary data. After collecting sufficient training samples, a data-driven model is constructed using the AI algorithm’s training process. This model automatically updates the input (the current, temperature, and SOC) and the output (voltage) link. A radial basis function neural network (RBFNN), support vector machine (SVM), and extreme learning machine (ELM) are the most commonly used methods for these models.

The classification of the lithium-ion battery models is presented in Figure 5.

4. Proton Membrane Exchange Fuel Cell Modeling

Based on its polarization curve, the PEMFC output voltage can be modeled as follows [26]:

$$V_{FC}=V_{Nernest}-V_{Act}-V_{Ohm}-V_{Con}$$

(5)

where V_Act is the activation voltage drop, V_Con is the concentration voltage drop, V_Ohm is the ohmic losses voltage, and V_Nernest is the Nernst voltage. The Nernst voltage can be calculated for temperature values <100 °C as follows:

$$V_{Nernest}=1.229-0.85\times {10}^{-3}(T-298.15)+4.3085\times {10}^{-5}\times T\left(\ln ({\mathrm{P}}_{{\mathrm{H}}_2})+\frac{\ln ({\mathrm{P}}_{{\mathrm{O}}_2})}{2}\right)$$

(6)

where P_H2 and P_O2 are the partial pressures of both hydrogen and oxygen, which can be calculated as follows:

$$P_{H_2}=\frac{R_{ha}\times P_{H_{2}O}}{2}\left[\frac{1}{\frac{R_{ha}\times P_{H_{2}O}}{P_a}\times e^{\frac{1.635\left(i/A\right)}{T^{1.334}}}}-1\right]$$

(7)

$$P_{O_2}=R_{hc}\times P_{H_{2}O}\left[\frac{1}{\frac{R_{hc}\times P_{H_{2}O}}{P_c}\times e^{\frac{4.192\left(i/A\right)}{T^{1.334}}}}-1\right]$$

(8)

where P_c and P_a express the inlet pressures in the cathode and the anode (atm), R_hc and R_ha are the vapor humilities in the cathode and anode, i is the generated current (A), A is the electrode area (cm²), P_H2O is the water vapor saturation pressure(atm).

The mathematical expression of the activation losses can be formulated as

$$\begin{array}{l}{V}_{Act}=-\left({\zeta }_1+{\zeta }_{2}T+{\zeta }_{3}T\ln ({\mathrm{C}}_{{\mathrm{O}}_{\partial }})+{\zeta }_{4}T\ln (i)\right)\\ C_{O_2}=\frac{P_{O_2}}{5.08\times {10}^6}{e}^{{}^{\left(\frac{498}{T}\right)}}\end{array}$$

(9)

where ξ_1,2,3,4 express the semi-empirical coefficients of the polarization phase; C_O2 describes the concentration of the oxygen (O₂) at the cathode’ surface (mol.cm⁻³).

The mathematical expression of the concentration loses can be formulated as

$$V_{Cons}=-\beta \ln (1-\frac{i}{i_{\lim }})$$

(10)

where β symbolizes the diffusion constant and i_lim symbolize the limiting current value.

The mathematical expression of the ohmic losses can be formulated as

$$V_{Ohm}=i(R_m+R_c)$$

(11)

where R_c denotes the resistance of the connectors, and R_m represents the ohmic membrane resistance. R_m can be calculated as follows:

$$\begin{array}{l}{R}_m=\raisebox{1ex}{${\rho }_m\cdot l$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.\\ {\rho }_m=\frac{181.6\left[1+0.03\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.+0.062\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$303$}\right.{\left(\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.\right)}^{2.5}\right]}{\left[\lambda -0.634-3\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$A_m$}\right.\right]e^{4.18\frac{T-303}{T}}}\end{array}$$

(12)

where A_m is the membrane’s surface (cm²), l represents the membrane’s thickness (cm), and λ is the membrane material water content’s constant.

5. Parameters Extraction Using Metaheuristic Optimization Algorithm

5.1. Brief Review on Metaheuristic Optimization Algorithms

“Meta” and “heuristic” are Greek terms that mean upper level or beyond for the meta term, and to find, to know, to lead an investigation, or to discover for the heuristic term [62]. They are strategies created to find (sub-)optimum solutions at a low computing effort without ensuring feasibility or optimality [30]. Most of these algorithms imitate biological or physical processes and have a stochastic behavior. Metaheuristic algorithms have been classified according to five metrics [63]:

• Inspiration sources: nature-inspired and non-nature inspired;
• The number of parallel computing solutions: population-based and single-point search;
• Objective function nature: dynamic and static objective function;
• Neighborhood structures: single and various neighborhood structures.
• Memory: memory usage and memory-less methods.

Figure 6 summarizes this classification.

5.2. Deployment of the Metaheuristic Optimization Algorithms in the Identification Process

The identification algorithm provides the model parameters based on the input data, the used model, and the implemented objective function. The optimizer generates a set of random parameters limited in the search space. These parameters are used to initialize the model, and its output will be compared to the used data. Based on this error, the objective function is calculated, and the candidate solutions will be updated. As illustrated in Figure 7, the system parameters can be extracted based on the measured data and the considered model. After establishing the model, input data required by the model will be used to generate the estimated output data. These generated data will be compared with the measured one, and the error between them will be used to generate the fitness value for the optimization algorithm. The optimizer updates the candidate solutions and steps to the next iteration based on the fitness value.

Several assessment criteria are used to generate the objective functions required by the algorithms. The algorithm can successfully achieve the intended outcomes in extracting various parameters using the appropriate criteria.

Table 1. Criteria equations and characteristics.

Criteria	Equation	Characteristics
Root mean square error (RMSE)	$f(x)=\sqrt{\frac{1}{M}{\displaystyle {\sum }_{k=1}^M{(x_{\mathrm{data}}(k)-x_{\mathrm{model}}(k))}^2}}$	Square value
Normalized RMSE (NRMSE)	$f(x)=\frac{\sqrt{\frac{1}{M}{\displaystyle {\sum }_{k=1}^M{(x_{\mathrm{data}}(k)-x_{\mathrm{model}}(k))}^2}}}{\sqrt{\frac{1}{M}{\displaystyle {\sum }_{k=1}^M{(x_{\mathrm{data}}(k))}^2}}}$	Square value
Mean absolute error (MAE)	$f(x)=\frac{1}{M}{\displaystyle {\sum }_{k=1}^M\left|x_{\mathrm{data}}(k)-x_{\mathrm{model}}(k)\right|}$	Absolute value
Relative error (RE)	$f(x)=\frac{\left|x_{\mathrm{data}}(k)-x_{\mathrm{model}}(k)\right|}{\left|x_{\mathrm{data}}(k)\right|}$	Absolute value
Mean relative error (MRE)	$f(x)=\frac{1}{M}{\displaystyle {\sum }_{k=1}^M\frac{\left|x_{\mathrm{data}}(k)-x_{\mathrm{model}}(k)\right|}{\left|x_{\mathrm{data}}(k)\right|}}$	Absolute value
Sum square error (SSE)	$f(x)={\displaystyle {\sum }_{k=1}^M{(x_{\mathrm{data}}(k)-x_{\mathrm{model}}(k))}^2}$	Square value

summarizes the typical equations and characteristics of each criterion. Using absolute value in the computation prevents negative numbers; however, formulae involving squares can yield more exact answers.

5.3. Presentation of Some Metaheuristic Optimization Algorithms

5.3.1. Salp Swarm Algorithm

A meta-heuristic technique called SSA looks for the best solutions to a given issue within a constrained search area [64]. The search begins at arbitrary locations. The individuals (the salps) will chain together and move in the direction of the best solution. Leaders and followers are the two categories that make up this chain. The leaders pursue the target position rapidly. Each follower will evolve his position in line with the location of his prior agent as they move seamlessly. The leaders can update their positions (P_L) at the iteration (t) as follows:

$$\begin{array}{l}{P}_L(t)=\left\{\begin{array}{ccc}\begin{array}{l}{P}_T(t)+c_1((\mathrm{ub}-\mathrm{lb})\cdot r_1+\mathrm{lb})\\ P_T(t)-c_1((\mathrm{ub}-\mathrm{lb})\cdot r_1+\mathrm{lb})\end{array}& \begin{array}{l}if\\ if\end{array}& \begin{array}{l}{r}_2<0.5\\ r_2>0.5\end{array}\end{array}\right.\\ c_1=2e^{-{(\raisebox{1ex}{$4t$}\!\left/ \!\raisebox{-1ex}{$T_{\max }$}\right.)}^2}\end{array}$$

(13)

where P_T is the target position, r₁ and r₂ are random numbers in [0, 1], and T_max is the max number of iterations.

The ith follower can update their positions (P_F ⁱ) at the iteration (t) as follows:

$$P_F{}^i(t)=0.5(P_F{}^i(t-1)+P_F{}^{i-1}(t))$$

(14)

The evolution of this algorithm can be presented in Figure 8.

5.3.2. Marine Predator Algorithm

A metaheuristic algorithm called MPA mimics the behavior of aquatic predators in search of prey [65]. The following are the main steps of this algorithm:

-

Phase 1(if t < T_max/3): based on Brownian motion, the prey updates its position (P_Prey) as follows:

$$\begin{array}{c}step(t)=R_B\otimes \left(P_{\mathrm{Elite}}(t)-R_B\otimes P_{\mathrm{Prey}}(t)\right)\\ P_{\mathrm{Prey}}(t+1)=P_{\mathrm{Prey}}(t)+0.5\cdot r\otimes step(t)\end{array}$$

(15)

where P_Elite is a set of best positions, r is a random number in [0, 1], R_B is the Brownian motion’s normal distribution vector. The notation ⨂ expresses the entry wise multiplications.

-

Phase 2 (if t > T_max/3 and if t < 2T_max/3): while the prey utilizes Levy motion, the predator uses Brownian motion. If n < N_pop/2, the updating equation is expressed as follows:

$$\begin{array}{c}step(t)=R_L\otimes \left(P_{\mathrm{Elite}}(t)-R_L\otimes P_{\mathrm{Prey}}(t)\right)\\ P_{\mathrm{Prey}}(t+1)=P_{\mathrm{Prey}}(t)+0.5\cdot r\otimes step(t)\end{array}$$

(16)

where N_pop is the population size, R_L is the Levy motion’s normal distribution vector. If n > N_pop/2, the updating equation is expressed as follows:

$$\begin{array}{l}step(t)=R_L\otimes \left(R_L\otimes P_{\mathrm{Elite}}(t)-P_{\mathrm{Prey}}(t)\right)\\ P_{\mathrm{Prey}}(t+1)=P_{\mathrm{Elite}}(t)+0.5\cdot c(t)\otimes step(t)\\ c(t)=[1-(t/T_{\max }){]}^{\frac{2t}{T_{\max }}}\end{array}$$

(17)

-

Phase 3 (t > 2T_max/3): the predator travels utilizing Levy throughout this phase, and the mathematical model is stated as follows:

$$\begin{array}{c}step(t)=R_L\otimes \left(P_{\mathrm{Elite}}(t)-R_L\otimes P_{\mathrm{Prey}}(t)\right)\\ P_{\mathrm{Prey}}(t+1)=P_{\mathrm{Prey}}(t)+0.5\cdot r\otimes P_{\mathrm{Prey}}(t)\end{array}$$

(18)

The main steps of this algorithm can be presented in Figure 9.

5.3.3. Bald Eagle Search Algorithm

The BES algorithm, which replicates the movement and hunting tactics, is given in [66]. The BES algorithm is divided into three stages:

-

Select phase: the eagle discovers the search space and determines the area with the best food availability. This phase can be expressed as follows:

$$P(t+1)=P_{\mathrm{prey}}(t)+\alpha \cdot r\cdot \left(P_m-P(t)\right)$$

(19)

where P_prey represents the prey position, α is a constant in [1.5, 2], r represents a random in [0, 1]. P_m is the mean value of all the current positions.

-

Search phase: to speed up its investigation, the eagle moves in different directions inside a spiral zone while it seeks prey. This phase can be modeled as follows:

$$\begin{array}{l}{P}^i(t+1)=P^i(t)+y^i.(P^i(t)-P^{i-1}(t))+x^i.(P^i(t)-P_m)\\ x^i=\frac{r{x}^i}{\max (\left|rx\right|)};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r{x}^i=r^i\cdot \sin ({\theta }^i)\\ y^i=\frac{r{y}^i}{\max (\left|ry\right|)};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r{y}^i=r^i\cdot cos({\theta }^i)\\ {\theta }^i={\beta }_1\cdot \pi \cdot r;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}^i={\theta }^i\cdot R\cdot r\end{array}$$

(20)

where β₁ is a constant in [5, 10], R is a constant in [0.5, 2], and r is random in [0, 1].

-

Swoop phase: the eagle attacked the target from the best position achieved in the previous phases. This phase can be represented as follows:

$$\begin{array}{l}P(t+1)=r.P_{prey}+x{1}^i.(P^i(t)-r_1.P_{mean})+y{1}^i.(P^i(t)-r_2.P_{prey})\\ x{1}^i=\frac{r{x}^i}{\max (\left|rx\right|)};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}r{x}^i=r^i.\sinh ({\theta }^i)\\ y{1}^i=\frac{r{y}^i}{\max (\left|ry\right|)};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}ry(i)=r^i.cosh({\theta }^i)\\ {\theta }^i={\beta }_2.\pi .r;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}^i={\theta }^i\end{array}$$

(21)

where r is a random number in [0, 1], r_1,2 are random numbers in [1, 2], β₂ is a constant in [5, 10].

The main steps of this algorithm can be presented in Figure 10.

5.4. Photovoltaic Parameters Extraction

Any PV system requires a PV model for simulation analysis, design optimization, and problem diagnostics. Furthermore, the model’s capacity to express precise I-V characteristics under all conditions (solar irradiance and temperature) is critical. However, the precision of the unknown parameters’ determination completely determines how well the I-V curve is simulated. Manufacturers only supply experimental I-V curves under STC (1000 W/m² and 25 °C), which the identification algorithms will use. The manufacturer data of the commonly investigated PV panels and modules are shown in

Table 2. The most used cells/modules for PV parameter identification.

Cell/Module	Type	Pmp (W)	Vmp (V)	Imp (A)	Voc (V)	Isc (A)	kv (V/°C)	ki (A/°C)	Ns
Sanyo HIT215 [67]	Mono-crystalline	215	42	5.13	51.6	5.61	−0.143	1.96 × 10−3	72
KC200 GT [67]	Poly-crystalline	200	26.3	7.61	32.9	8.21	−0.123	3.2 × 10−3	54
ST40 PV [67]	Thin-film	39.9	16.9	2.36	23.3	2.68	−0.1	3.5 × 10−4	42
RTC France solar cell57 mm [31]	Poly-crystalline	0.3101	0.4507	0.688	0.5728	0.7603	NA	0.035	1
SM55 [68]	Mono-crystalline	55	17.4	3.15	21.7	3.45	NA	0.04	36
S75 [69]	Poly-crystalline	74.976	17.6	4.26	21.6	4.7	−0.0076	2 × 10−3	36
ST40 [69]	Thin-film	40	16.6	2.41	23.3	2.68	−0.01	0.35 × 10−3	42
Photowatt PWP201 [70]	Poly-crystalline	11.315	12.64	0.912	16.778	1.03	NA	NA	36
Canadian Solar CS6K-280M [34]	Mono-crystalline	280	31.5	8.89	38.5	9.34	NA	NA	60
PVM752 GaAs	Thin-film	0.075	0.8053	0.0937	0.9926	0.0999	NA	NA	1

.

Table 3. Recent MOAs used to extract the PV parameters.

Ref	Author & Year	MOA	Cell/Module	Model	Criterion	Best Results
[31]	M. Navarro et al. 2023	Artificial Hummingbird Algorithm (AHA)	RTC France solar cell	SDM DDM TDM	RMSE	9.860 × 10−4 6.835 × 10−4 9.855 × 10−4
[34]	M. El-Dabah et al. 2023	Northern Goshawk Optimization (NGO)	Photowatt PWP-201 Kyocera KC200GT Canadian Solar CS6K-M	TDM	Customized	1.346 × 10−5 9.4174 × 10−5 9.4174 × 10−5
[32]	F. Ali et al. 2023	Atomic Orbital Search (AOS)	RTC France solar cell	SDM DDM TDM	RMSE	7.752 × 10−4 7.606 × 10−4 7.950 × 10−4
[32,71]	F. Ali et al. 2023 A. Beşkirli, I. Dağ 2023	Atomic Orbital Search (AOS) Tree Seed Algorithm (TSA)	PVM752 GaAs	SDM DDM TDM	RMSE RMSE	1.618 × 10−4 1.780 × 10−3 3.904 × 10−4
			STM6-40/36 module	SDM		2.655 × 10−3
[72]	A. M. Shaheen et al. 2022	Supply Demand Optimizer (SDO)	PVM752 GaAs	TDM	RMSE	1.249 × 10−3
[35]	A. Ginidi et al. 2021	Gorilla Troops Optimizer (GTO)	Kyocera KC200GT PV	SDM DDM	RMSE	6.367 × 10−4 9.482 × 10−5
[35,73]	A. Ginidi et al. 2021 M. El-Dabah et al. 2022	Gorilla Troops Optimizer (GTO) Runge Kutta optimizer (RKO)	STM6-40/36 PV	SDM DDM	RMSE RMSE	1.333 × 10−17 1.730 × 10−3
			RTC France solar cell Photowatt PWP-201	DDM		9.829 × 10−4 3.139 × 10−3
[33]	A. Bayoumi et al.2021	Marine Predators Optimizer (MPA)	RTC France solar cell	SDM DDM TDM	RMSE	8.438 × 10−4 7.590 × 10−4 7.561 × 10−4
[33,74]	A. Bayoumi et al.2021 N. F. Nicaire et al. 2021	Marine Predators Optimizer (MPA) Bald Eagle Search Algorithm (BES)	Q6-1380witharea	SDM DDM TDM	RMSE RMSE	1.610 × 10−5 1.460 × 10−5 1.420 × 10−5
			RTC France solar cell	SDM DDM		9.860 × 10−4 9.824 × 10−4
[74]	N. F. Nicaire et al. 2021	Bald Eagle Search Algorithm (BES)	Photowatt-PWP201	SDM	RMSE SAE	2.425 × 10−3
			STM6-40/36	SDM		1.729 × 10−3
			STP6-120/36	SDM		1.678 × 10−3
[36]	MH. Qais et al. 2020	Transient Search Optimization (TSO)	Kyocera KC200GT PV MSX-60 CS6K280M	TDM		0 0 1.740 × 10−13
[75]	A. Abbassi et al. 2020	Modified Salp Swarm Algorithm (mSSA)	TITAN12-50 solar panel	DDM	MSE	3.602 × 10−5
[37]	A. Diab et al. 2020	Coyote Optimization Algorithm (COA)	RTC France solar cell	SDM DDM TDM	RMSE	7.754 × 10−4 7.468 × 10−4 7.597 × 10−4
[37]	A. Diab et al. 2020	Coyote Optimization Algorithm (COA)	Photowatt-PWP201	SDM DDM TDM	RMSE	2.949 × 10−3 2.404 × 10−3 2.406 × 10−3
			SM55	SDM DDM TDM		3.837 × 10−3 3.541 × 10−3 4.403 × 10−3
			ST40	SDM DDM TDM		43.944 × 10−3 34.562 × 10−3 34.562 × 10−3
			Kyocera KC200GT PV	SDM DDM TDM		30.185 × 10−3 31.742 × 10−3 30.326 × 10−3

presents the most recent MOAs applied in PV parameters extraction published in Scopus. Figure 11 illustrates their graphical distributions. This table includes the references, the used MOA, cell/module type, the model type, the used criterion, and the best obtained results.

From this table, the parameter extraction of PV panels has been carried out for various commercialized types. Concerning the RTC France solar cell, the best result for the SDM has been provided by the Atomic Orbital Search (AOS) [32] with a fitness value of 7.752 × 10⁻⁴. The DDM’s best result is 6.835 × 10⁻⁴, provided by the Artificial Hummingbird Algorithm (AHA) [31]. The TDM best result is 7.950 × 10⁻⁴ provided by the Atomic Orbital Search (AOS) [32]. This may confirm that no algorithm can give the best results for all types of problems. The Kyocera KC200GT model also has been identified in several papers. The base value of its SDM has been provided by Gorilla Troops Optimizer (GTO) [35] with a final fitness of 6.367 × 10⁻⁴. The same optimizer also offers the best result of the DDM with an ultimate fitness of 9.482 × 10⁻⁵. The TDM best result is recorded by the Transient Search Optimization (TSO) [36], where the final SAE is near zero. Concerning the Photowatt PWP-201 model, the best SDM result is offered by the Bald Eagle Search Algorithm (BES) [74] with an ultimate fitness of 2.425 × 10⁻³. The DDM best result is 2.404 × 10⁻³ provided by Coyote Optimization Algorithm (COA) [37]. The best result of the TDM is provided by the Northern Goshawk Optimization (NGO) [34] with a final fitness of 1.346 × 10⁻⁵.

As shown in this table, the deployment of the MOAs for extracting the PV parameters for the three models (SDM, DDM, and TDM) has become more attractive and more frequently published in the last two years. This approves the ability of the MOAs to solve this problem efficiently.

5.5. Lithium-Ion Battery Parameters Extraction

The battery identification strategy deals mainly with the model parameters. Hence, the identification strategy is built based on the selected model.

Table 4. Recent MOAs used to extract the Li-ion battery parameters.

Ref	Author & Year	MOA	Model
[76]	R. El-Sehiemy et al. 2022	Supply–demand algorithm (SDA)	2RC-ECM
[77]	R. Rizk-Allah et al. 2022	Manta Ray Foraging Optimizer (MRFO)	Tremblay
[78]	Y. Hao et al. 2022	An improved coyote optimization algorithm (ICOA)	FO-ECM
[79]	T. Pan et al. 2022	Whale optimization algorithm (WOA)	P2D
[80]	R. El-Sehiemy et al. 2022	Enhanced sunflower optimization algorithm (ESOA)	RC-ECM
[81]	J. Hou et al. 2022	Chaotic quantum sparrow search algorithm (CQSSA)	FO-ECM
[82]	S. Ferahtia et al. 2022	Modified Bald Eagle Search (mBES)	Shepherd
[83]	A. Fatgi et al. 2022	Bald Eagle Search (BES)	Shepherd
[84]	S. Ferahtia et al. 2022	Salp Swarm Algorithm (SSA)	Shepherd
[85]	E. Houssein et al. 2022	Modified Coot algorithm (mCOOT)	Shepherd
[86]	S. Ferahtia et al. 2022	Artificial eco-system optimization (AEO)	Shepherd
[87]	A. Shaheen et al. 2021	Equilibrium Optimizer (EO)	3RC-ECM
[88]	M. Elmarghichi et al. 2021	Sunflower optimization algorithm (SOA)	RC-ECM
[89]	S. Zhou et al. 2021	Adaptive particle swarm optimization (APSO)	Thevenin 2RC-ECM FO-ECM
[90]	W. Shuai et al. 2020	Differential evolution (DE)	Modified Thevenin
[91]	X.Lai et al. 2019	Particle swarm optimization (PSO)	PNGV
[38]	H. Pang et al. 2019	Genetic Algorithm (GA)	SPM
[40]	L. Chen 2019	Genetic Algorithm (GA)	Simplified SPM
[39]	Y. Qi et al. 2017	Genetic Algorithm (GA)	SPM P2D
[92]	M. Rahman et al. 2016	Particle swarm optimization (PSO)	SPM
[41]	A. Jokar et al. 2016	Genetic Algorithms (GA)	P2D
[42]	J. Li et al. 2016	Genetic Algorithms (GA)	P2D
[93]	C. Forman et al. 2012	Genetic Algorithm (GA)	Doyle–Fuller–Newman

presents the most recent MOAs applied in Li-ion battery parameters extraction according to the Scopus database. This table includes the references, the used MOA, and the model type. Figure 12 presents their yearly distribution.

As shown in this table, the utilization of the MOAs for extracting the lithium-ion battery parameters for all models (equivalent circuit, empirical, and electrochemical) has become more attractive for academics in the last ten years. The empirical models such as Shepherd are noticed as the most used ones to validate the electrical model of the battery. The RC electrical circuit models are also used to achieve the same objective. The electrochemical models such as the SPM and P2D are identified to determine the whole state of the battery and to predict its health and remaining helpful capacity more accurately. From Figure 12, most of the cited papers are recent and published in 2022.

Using MOAs for other types of electrochemical batteries, such as lead acid, is also becoming more frequent [94,95]. The battery management system (BMS) improves battery operation and extends the lifecycle. It is recommended to optimize it using these optimization algorithms.

5.6. Proton Exchange Membrane Fuel Cell Parameters Extraction

Arbitrary solutions that fall inside the search space restrictions are initialized and sent to the model as candidate solutions. The model’s output will then be compared with the collected data after being simulated using these settings. By minimizing the objective function generated based on the errors between the output voltage of each PEMFC stack and the voltage estimated by the model, the fitness function definition, on which all the algorithms are compared, aims to extract the steady-state model parameters. The optimizer will then choose the top solutions, and the subsequent iteration will provide a new set of modified solutions. Up to the final iteration, this procedure will be repeated. The parameters of the most studied PEMFC types are presented in

Table 5. Specifications of the most studied PEM fuel cells for parameter identification [96].

Cell/Module	A (cm2)	l (μm)	${\mathit{P}}_{\mathit{H}2}^{\ast }\left(\mathbf{atm}\right)$	${\mathit{P}}_{\mathit{O}2}^{\ast }\left(\mathbf{atm}\right)$	T (K)	Imax (mA/cm2)	Pout (W)	n
NedStack PS6	240	178	0.5	1.0	343	1200	6000	65
BCS 500W	64	178	1.0	0.2	333	469	500	32
AVISTA SR-12 500 W	64	178	1.47628	0.2095	323	672	500	32
250 W stack	27	127	1.5	1.5	328.15	860	250	24
Temasek 1 kW PEMFC	150	51	0.5	0.5	323	1500	1000	20
Horizon H-12 stack	8.1	0	0.4	0.55	328.15	246.9	100	13
Horizon 500-W PEMFC	52	25	0.55	1	323	446	500	36
Ballard 5 kW Mark V FC	50.6	178	1	1	343	1500		35
Ballard 1.2-kW Nexa	50	400	5	5	333.15	1200		47

where the symbol * means the rated pressure value.

Table 6. Recent MOAs used to extract the PV parameters.

Ref	Author & Year	MOA	FC Module	Criterion	Best Results
[97]	M. Abd Elaziz et al. 2023	Gorilla Troops Optimizer (GTO)	BCS 500 W NedStack PS6 250 W stack	SSE	0.0118 0.3378 1.38
[98]	R. Hegazy et al. 2022	Bald Eagle Search (BES)	BCS 500 W NedStack PS6	SSE	2.07974 0.01136
[99]	R. Hegazy et al. 2022	Gradient-based Optimizer (GBO)	250 W stack	RMSE SSE	0.00684 0.0557
			BCS 500 W	RMSE SSE	0.00234 0.01129
			SR-12 500 W	RMSE SSE	0.05546 0.49883
[100]	E. Houssein et al. 2021	modified artificial electric field algorithm (mAEFA)	NedStack PS6	RMSE SSE	2.07974 0.13164
			SR-12 500 W	RMSE SSE	0.56067 0.05637
[96]	M. Özdemir. 2021	Chaos embedded particle swarm optimization (CEPSO)	250 W Stack	RMSE ASE MASE	0.6112 19.834 0.6402
			BCS-500 W	RMSE ASE SSE	0.01151 2.22845 0.01219
			Nedstack PS6	RMSE ASE SSE	2.680 649.41 2.18067
[101]	M. Abdel-Basset et al. 2021	Improved Heap-based Optimizer (IHBO)	BCS 500 W NedStack PS6 H-12 stack SR-12 500 W	SSE	0.01170 2.14570 0.11802 0.00014
[102]	R. Rizk-Allah et al. 2020	Improved Artificial Eeco-system Optimizer (AEO)	NedStack PS6 BCS 500 W 250 W stack	SSE	2.14590 0.01160 1.1510
[103]	J. Jiang et al. 2020	Sine Tree-Seed Algorithm (STSA)	NedStack PS6	SSE	2.14576
[104]	M. Sultan et al. 2020	Improved salp swarm algorithm (ISSA)	BCS 500 W SR-12 500 W 250 W stack Temasek-1 kW	SSE	0.01160 0.79157 0.64340 0.79268
[105]	A.Diab et al. 2020	Political optimizer (PO)	BCS 500 W SR-12 500 W 250 W stack	SSE	0.01155 1.05662 0.64421
[105,106]	A. Diab et al. 2020 M. Fawzi et al. 2019	Marin predator algorithm (MPA)	BCS 500 W SR-12 500 W 250 W stack	SSE SSE	0.01155 1.05662 0.59405
		Neural network optimizer (NNO)	Ballard Mark V 5 kW		0.85361
[106,107]	M. Fawzi et al. 2019 A. El-Fergany. 2018	Neural network optimizer (NNO) Salp swarm algorithm (SSA)	BCS 500 W BCS stack Nedstack PS6	SSE SSE	0.011698 2.14487
			NedStack PS6 BCS 500 W		2.18067 0.01219

shows the recent Scopus papers that explain the utilization of the MOAs in PEMFC parameters extraction. This table includes the references, the used MOA, FC module, the used criterion, and the best obtained results.

Extracting the parameters of the BCS 500 W types has been widely investigated in the literature review. The gradient-based optimizer (GBO) provides the best result with an RMSE of 0.00234 [99]. Concerning the NedStack PS6, the best result has been reported by the Bald Eagle Search (BES) with a final SSE of 0.01136 [98]. The best result for the 250 W stack is 0.00684, provided by the gradient-based optimizer (GBO) [99]. The best result for the SR-12 500 W is provided by the improved heap-based optimizer (IHBO) with an ultimate SSE of 0.00014 [101].

The provided information in this table approves the benefit of deployment of the MOAs for extracting the PEMFC parameters. This topic has been more frequently published in high-impact factor journals in the last few years. This confirms their contribution to solving this problem efficiently. In addition, these algorithms can be used to extract the parameters of other FC types, such as solid oxide FCs (SOFCs). The promising results when developing these algorithms for these applications can lead to enhancing the operation for a longer lifecycle and better efficiency. It is recommended to optimize their operation using these algorithms.

6. Future Research Directions

As mentioned in the previous sections, metaheuristic optimization algorithms have been increasingly used in the parameter extraction of PV cells, Li-ion batteries, and PEMFCs. These algorithms are used to extract the parameters of other systems, such as motors. The recent algorithms may provide better performance in such applications or may not. According to the no free lunch (NFL) theory, no optimizer can provide consistent and superior performance for all optimization problems. This encourages academics to develop more recent optimization algorithms that may provide better performance. The revolution in artificial intelligence may be used by coupling the metaheuristic optimization algorithms with the learning features of the AI. This may significantly increase their performance.

7. Conclusions

This paper has contributed to the discussion on using a metaheuristic optimization algorithm to solve parameter extraction problems related to photovoltaic generators, lithium-ion batteries, and PEM fuel cells. Starting with the problematic definition mainly imposed by the degradation phenomenon will certainly change the model parameters. Hence, the model’s accuracy of each system should be updated. A brief review of the models of the photovoltaic generators, the lithium-ion batteries, and the PEM fuel cells has been provided in this study. The principles of the metaheuristic optimization algorithms have also been presented. The deployment manner of these algorithms within the problem is also provided and discussed. A summary of recently published papers in the Scopus database for each system has been provided, starting with published papers that identify the single, the double, and the terrible diode models. Then, a set of new papers identifying the lithium-ion models, including the empirical, the equivalent circuits, and the electrochemical models, were listed. Finally, the recently published papers that introduced the employment of the metaheuristic algorithm in extracting the parameters of a PEMFC were regrouped. From this report, it can be noticed that the deployment of these optimization algorithms is becoming more frequently used in parameter extraction. For the case of PV parameters extraction, the identification error for all models, including the TDM that has a higher complexity, has been reduced for various commercialized PV panels. Various metaheuristic optimization algorithms have been deployed to extract the parameters of the Li-ion battery for different types of models. The parameters of different types and prototypes of PEMFC are successfully extracted using various metaheuristic optimization algorithms. Most of these works have been published recently, which is linked to the recent surge in interest in these algorithms for this kind of application. To conclude, the main purpose of this paper is to investigate the effect of the metaheuristic optimization algorithm on the parameter extraction of several vital energy systems. Their contribution has been reported, and they are expected to be employed in many other applications related to power systems.