Parameter Estimation of Photovoltaic Cell/Modules Using Bonobo Optimizer

Abstract

In this paper, a new application of Bonobo (BO) metaheuristic optimizer is presented for PV parameter extraction. Its processes depict a reproductive approach and the social conduct of Bonobos. The BO algorithm is employed to extract the parameters of both the single diode and double diode model. The good performance of the BO is experimentally investigated on three commercial PV modules (STM6-40 and STP6-120/36) and an R.T.C. France silicon solar cell under various operating circumstances. The algorithm is easy to implement with less computational time. BO is extensively compared to other state of the art algorithms, manta ray foraging optimization (MRFO), artificial bee colony (ABO), particle swarm optimization (PSO), flower pollination algorithm (FPA), and supply-demand-based optimization (SDO) algorithms. Throughout the 50 runs, the BO algorithm has the best performance in terms of minimal simulation time for the R.T.C. France silicon, STM6-40/36 and STP6-120/36 modules. The fitness results obtained through root mean square (RMSE), standard deviation (SD), and consistency of solution demonstrate the robustness of BO.

1. Introduction

The concern over increasing energy costs, losses in the contemporary energy system and the greenhouse gas effect have shifted the world focus towards renewable energy resources [1]. Among the sources of renewable energy, solar energy (the PV system) is the fastest growing source of renewable energy in the world [2]. Unlike conventional power sources, the output power of the PV system depends on solar irradiation. Consequently, the increasing capacity of PV system installations poses a major challenge to power system operation and planning. Furthermore, PV system manufacturers only provide data relating to standard test conditions (STC), which are short circuit current, open circuit voltage, maximum power, current and voltage at maximum power point at irradiance, temperature and mass spectral distribution of 1000 W/m², 25 °C and 1.5 air mass, respectively. However, this does not reflect the punitive environmental conditions in which PV systems are deployed for utilization [3]. As a result of this, the optimal extraction of PV model parameters is very important to analyze the dynamic characteristics of a PV module under different environmental conditions [4]. The accurate knowledge and characterization of PV modules are centred on the model parameters extracted, and this is important for several practical applications [5]. This will enhance the performance of the PV modules, fault analysis, quality control of the solar cells and, likewise, the maximum power point tracking [6]. Several models have been established to describe the input and output characteristics of the PV cells, the most popular is lumped parameter equivalent electrical circuit model: the single diode model (SDM) and the double diode model (DDM) [7]. However, the transcendental nature and existence of exponential terms in the model equations of the SDM and DDM make the optimum parameter extraction and efficient PV system simulation very challenging and complex by using basic functions [8,9,10]. In the light of these considerations, the challenges have attracted the attention of several researchers over the past years, and various computational procedures have been recommended.

Quite a number of methods are available for extracting parameters of both the SDM and DDM from the current-voltage (I–V) curves or datasheet [6,7,11]. The application of parameter extraction techniques that provide solutions based on empirical relations among several parameters or major points of PV characteristics is regarded as an analytical method [12]. Generally, analytical methods estimate PV cell parameters in one iteration. These methods are centred on deriving an observation and experimental relationship by solving a collection of equations that is obtained from the I–V (Current-Voltage) curve of the PV cell under the STC conditions [13,14]. The STC specifications can easily be obtained from the datasheet; these make analytical methods naturally handy. In spite of the benefits, analytical methods are prone to noise because of minimal selected points used in the calculation and PV degradation over time [15,16]. Furthermore, because of the non-linear model of the equations, and to ease computations, some assumptions are made which lead to inaccurate solutions [13,17]. Although these methods are befitting for the SDM, their aptness is yet to be substantiated for the DDM [3].

Conversely, numerical methods obtain the parameters by reducing the margin between simulated and experimental data points in multiple iterations. The numerical methods are known to be more dependable than the analytical methods [18]. Numerical methods can be divided into two categories: heuristic methods and deterministic methods. The latter relies on gradient information with initial constraint condition. As a result, they get stuck in a local search easily, which leads to erroneous and undependable solutions. In addition, the increment in decision parameters tends to reduce the accuracy of the optimizer [19,20]. Examples of these types of techniques are iterative curve-fitting [21], the Runge-Kutta method [22], Levenberg-Marquardt algorithm [23] and Newton-Raphson method [24]. To overcome the challenges associated with deterministic methods, several heuristic optimization approaches have been presented for the parameter extractions. Examples of these methods are genetic algorithms [25,26], particle swarm optimization [27], cuckoo search optimization [3], adaptive differential evolution algorithm [7], flower pollination algorithm [28], simulated annealing [20], harmony search [29], cat swarm optimization [30], coyote optimization algorithm [31], supply-demand-based optimization [32], improved chaotic whale optimization algorithm [10], firefly algorithm [33], differential evolution [34], symbiotic organisms search algorithm [35] and artificial bee colony [36]. Unlike deterministic methods, heuristic methods are notable for their global search operation and their efficacy towards handling non-linear functions without the requirements for gradient information. Furthermore, owing to their non-derivative nature, initial conditions are not required for the computation. Heuristic methods are robust, simple and easy to implement [31,32].

The performance and selection of any optimization methods are evaluated based on the speed of convergence, precision and ease of execution. Although the highlighted heuristic methods above have proved to be hopeful for parameter extraction, some heuristic algorithms contain control parameters, and the incorrect settings could lead to untimely and slow convergence [37,38]. Therefore, since accurate model parameters have not been established, any improvement in the fitting solution is highly indispensable for the model utilization. Moreover, considering the popular no free lunch theorem [39], no sole algorithm is appropriate to solve all optimization challenges. Therefore, the pursuit for different optimization algorithms that can effectively ascertain accurate parameters of PV cells is a cogent area of research. As a result, a Bonobo optimizer (BO) [40] for the parameter extraction of PV cells is presented in this study. The algorithm is selected because of its ease of implementation and fast computation time. Its process depicts a reproductive approach and the social conduct of Bonobos. The algorithm adopts the fission–fusion search approach of Bonobos, by splitting into smaller groups and later uniting for proficient optimization. The performance of the BO algorithm is extensively compared to MRFO [41,42], ABO [36], PSO [27], FPA [28] and SDO [32].

In terms of accuracy, convergence rate and dependability, experimental data reveals that BO outperforms the most recent well-established algorithms (e.g., MRFO, ABO, PSO, FPA and SDO). The following are the primary contributions of this paper:

• For the first time, a simple and time-saving metaheuristic algorithm called Bonobo optimizer (BO) has been presented to extract the electrical parameters of PV modules models based on several series of experiments.
• By combining different mating techniques found in bonobo culture, the suggested BO achieves a balance between exploration and exploitation.
• Throughout the search process, BO keeps the solution diverse, preventing premature convergence.
• BO avoids population stagnation by exploring diverse parts of a multidimensional search space while being guided by a group of bonobos with different mating strategies.
• The proposed BO is tested on different PV models datasets, and experimental results show that BO has better or competitive performance in comparison with other state of the art algorithms.

The rest of the paper is organized as follows: Section 2 describes the mathematical model of a PV cell; Section 3 presents the BO optimization strategy; Section 4 presents the objective function; Section 5 explains the simulation results; Section 6 provides the conclusion.

2. Mathematical Model of PV Module

A mathematical model can be employed to accurately depict the output characteristics of the PV model. This will portray the physical processes that occurs in the cell of the module. The most popular utilized model is the single diode and double diode PV cell model [17]. The parameter extraction of the PV cell is achieved by assuming the cells to be identical under similar operating circumstances. This criterion is used to devise the objective functions for the model description.

2.1. Single Diode Model (SDM)

Figure 1 shows the circuit model of a single diode PV cell. Using Kirchhoff’s Current Law (KCL), the output current, $I$, of the model is calculated as [43]

$$I=I_{ph}-I_d-I_{sh}$$

(1)

where $I_{ph}$ is the photocurrent,$I_d$ is the diode current and $I_{sh}$ represents the shunt resistor current [32].

The Shockley’s diode equation, $I_d,$ and the shunt resistor current can be represented as

$$I_d=I_o\left(e^{\frac{V_L+R_{s}I}{n{V}_t}}-1\right)$$

(2)

$$I_{sh}=\frac{V_L+R_{s}I}{R_{sh}}$$

(3)

where $I_o$ is the saturation current, $n$ is the diode ideal factor, $V_L$ is the output voltage, $R_{sh}$ and $R_s$ are the shunt and series resistance, respectively,$V_t=\frac{kT}{q}$ is the PV cell thermal voltage. $k$ denotes the Boltzmann constant, $T$ is the temperature of the $p-n$ junction in kelvin $\left(K\right)$,$q$ is the electron charge.

By substituting Equations (2) and (3) into Equation (1), the output current of the PV cell module is represented as

$$I=I_{ph}-I_o\left(e^{\frac{V_L+R_{s}I}{n{V}_t}}-1\right)-\frac{V_L+R_{s}I}{R_{sh}}$$

(4)

It can be seen from Equation (4) that the single diode model has five parameters ($I_{ph},I_o,R_{sh},R_s,n$) that are required to be extracted.

2.2. Double Diode Model (DDM)

The ease of implementation and accuracy account for the utilization of the SDM. However, the DDM accounts for the effects of recombination current loss in the depletion region. This allows for a detailed model of the DDM and, therefore, advances the operation in some applications; specifically, in thin films at low irradiance [17,44]. Figure 2 depicts the model for the DDM with a second diode connected in parallel with the first. From KCL, the output current from $I$ is calculated as

$$I=I_{ph}-I_{d1}-I_{d2}-I_{sh}$$

(5)

where $I_{d1}$ and $I_{d2}$ are the first and second diode’s current, respectively. From Shockley’s diode equations, the two currents are represented as

$$\begin{gathered} I_{d1}=I_o\left(e^{\frac{V_L+R_{s}I}{n_1{V}_t}}-1\right) \\ {I}_{d2}=I_o\left(e^{\frac{V_L+R_{s}I}{n_2{V}_t}}-1\right) \end{gathered}$$

(6)

By incorporating Equations (6) and (3) in Equation (5), the output current of the DDM is represented as

$$I=I_{ph}-I_{o1}\left(e^{\frac{V_L+R_{s}I}{n_1{V}_t}}-1\right)-I_{o2}\left(e^{\frac{V_L+R_{s}I}{n_2{V}_t}}-1\right)-\frac{V_L+R_{s}I}{R_{sh}}$$

(7)

where $I_{o1} \mathrm{and} I_{o2}$ are the saturation currents. $n_1$ and $n_2$ are the diode ideality factors of the respective diodes. It is evident from Equation (7) that the DDM has seven unknown parameters ($I_{ph}, I_o,R_{sh},R_s,I_{o1},I_{o2}, n_1,n_2$).

2.3. PV Module Model

By considering $N_s$ by $N_p$ solar cells with various connection in series and or parallel, the output current, $I$, for the SDM and DDM can be represented as Equations (8) and (9), respectively

$$I=I_{ph}{N}_p-I_o{N}_p\left(e^{\frac{V_L+R_{s}I\left(\frac{N_s}{N_p}\right)}{n{V}_t}}-1\right)-\frac{V_L+R_{s}I\left(\frac{N_s}{N_p}\right)}{R_{sh}\left(\frac{N_s}{N_p}\right)}$$

(8)

$$I=I_{ph}{N}_p-I_{o1}{N}_p\left(e^{\frac{V_L+R_{s}I\left(\frac{N_s}{N_p}\right)}{n_1{V}_t}}-1\right)-I_{o2}{N}_p\left(e^{\frac{V_L+R_{s}I\left(\frac{N_s}{N_p}\right)}{n_2{V}_t}}-1\right)-\frac{V_L+R_{s}I\left(\frac{N_s}{N_p}\right)}{R_{sh}\left(\frac{N_s}{N_p}\right)}$$

(9)

3. Bonobo Optimizer

Bonobo Optimizer (BO) is a new metaheuristic algorithm which is influenced by the reproductive approach and social conduct of Bonobos. It is a population-based algorithm developed by Amit Kumar Das et al. [40]. The algorithm adopts the fission-fusion search approach of Bonobos for proficient optimization. Bonobos partition into smaller groups, which are referred to as fission, for locating foods and later are reunited (fusion) at night for sleeping (refer to Figure 3). This distinct method was utilized in the algorithm to make the searching process more effective. Similarly to other heuristics, BO is also an algorithm based on population. An individual solution in the population is named a bonobo. The most dominant bonobo with the best rank hierarchy is called the alpha bonobo (αbonobo). Additionally, Bonobos advance through phase probability, which is either population diversity or selection pressure and through positive (PP) and negative (NP) phases, as shown in Figure 4. The counts of a successive number of iterations of PP and NP are called positive phase counts (ppc) and negative phase counts (npc), respectively. In order to reach the global optimum solution, the algorithm explores the natural behaviours and strategies of Bonobos. Bonobo utilizes four different mating strategies to create new Bonobos, they are: promiscuous, restrictive, consortship and extra-group mating [40].

The mating strategies will change depending on the phase condition (positive (PP) or negative (NP)). The PP physically depicts the bonobo community when there is sufficient food, protection from neighboring communities, mating success and genetic variety among bonobos. The likelihood of the first two kinds of mating, i.e., promiscuous and restricted mating, will be greater during this period. Promiscuous mating makes an oestrus female available to both the alpha bonobo (i.e., the highest-ranking male in a group) and other lower-ranking males. However, in the event of restricted mating, only males of higher status may join. The NP, which denotes an unpleasant condition in the society, increases the likelihood of consortship mating and extra-group mating. A couple is isolated from their original community and spends their time together in a kind of mating called consortship. They rejoin their community after a few days or weeks. However, in the instance of extra-group mating, a female bonobo is observed mating with males from different communities. Additionally, the likelihood of extra-group mating is quite low in comparison to the possibility of mating with another group. These physical events are artificially reproduced in the suggested BO by the use of mathematics for optimization. The flowchart of the proposed algorithm is shown in Figure 5.

3.1. Promiscuous and Restrictive Mating Strategies

The phase probability parameter (p_p) determines the mating approach of the Bonobo. At the initial stage, the value of p_p is designated as 0.5, which is updated at every iteration. A new Bonobo is formed if a random number, r, generated as a value between 0 and 1, is found to be either less than or equal to p_p as described in the following equation

$$n_{-}{b}_j=b_j^i+r_1{s}^{\alpha }({\alpha }_j^b-b_j^i)+(1-r_1)s^{s}flag(b_j^i-b_j^p)$$

(10)

where b is bonobo, $n_{-}{b}_j$ and ${\alpha }_j^b$ are the $j^{th}$ variables of the new offspring and α^bonobo, respectively. $j$ as a value that varies between 1 and d number of variables. $b_j^i and b_j^p$ represent variable values of $i^{th}$ and $p^{th}$ bonobos, respectively. $r_1$ represents a value within the range from 0 to 1. The $s^{\alpha }$ and $s^s$ are sharing coefficients for α^bonobo and $p^{th}$ bonobos, respectively. The parameter: $flag$ exists between −1 and 1. Promiscuous mating occurs when the best solution of $i^{th}$ bonobo yields a better result than $p^{th}$ bonobos. In this situation, the flag is allotted 1. Alternatively, it is referred to as restrictive mating. In this regard, the $flag$ and α^bonobo are assigned −1.

3.2. Consortship and Extra-Group Mating Strategies

These kinds of matings occur if phase $p_p$ is lesser than the random number, $r$. Furthermore, if $r_2$ is equivalent to or less than the probability of extra-group $(p_{xgm})$, this will lead to updating of the solution via extra-group mating.

$$n_{-}{b}_j=\left\{\begin{array}{cc}{b}_j^i+e^{\left(r_3^2+r_3-2r_3^{-1}\right)}\left(Var\text{\_}{\mathit{max}}_j-b_j^i\right)\hfill & \hfill {\alpha }_j^{\mathrm{bonobo}}\ge b_j^i\\ b_j^i-e^{\left(-r_4^2+2r_4-2r_4^{-1}\right)}\left(b_j^i-Var\text{\_}{\mathit{min}}_j\right)\hfill & \hfill r_3\le p_d\\ \left.b_j^i-e^{\left(r_3^2+r_3-2r_3^{-1}\right.}\right)\left(b_j^i-Var\text{\_}{\mathit{min}}_j\right)\hfill & \hfill {\alpha }_j^{\mathrm{bonobo}}\le b_j^i\\ b_j^i+e^{\left(-r_4^2+2r_4-2r_4^{-1}\right)}\left(Var\text{\_}{\mathit{max}}_j-b_j^i\right)\hfill & \hfill r_3\ge p_d\end{array}\right.$$

(11)

The $p_d$ is initialized with 0.5 with a gradual updating base on the nature of the evolution. The $p_d$ optimizes the searching process for the most promising result. $Var\text{\_}mi{n}_j$ and $Var\text{\_}ma{x}_j$ represent the lower and upper boundaries of the j^th-variable, respectively.

In other cases, where the value of $r_2$ is found to be greater than that of $p_{xgm}$, a new offspring is created using the consortship mating strategy as follows

$$n_{\text{\_}}{b}_j=\{\begin{array}{cc}{n}_{-}{b}_j+e^{r_5}flag(1+r_1)(b_j^i-b_j^p)\hfill & \hfill r_6\le p_d\\ b_j^p\hfill & \hfill otherwise\end{array}$$

(12)

where $r_1, r_2, r_3, r_4, r_5$ are random numbers between 0 and 1.

4. Objective Function

The implementation of parameter extraction requires the defining of a suitable algorithm for the objective function. Considering the fact that the best collection of solutions should produce the nearest fit to measured data, therefore, the best solution is obtained by minimizing the difference between the measured data and the calculated data. The most popular method to minimize the difference between the measured and calculated data is the root mean square error [6,10,11,31].

$$\min h(x)=RMSE(x)=\sqrt{\frac{1}{N}{\displaystyle \sum }_{k=1}^N{(I_{cal}^k(x)-I_{meas.}^k)}^2}$$

(13)

where $I_{cal}$ is the calculated current value, $I_{meas}$ is the measured current value and $N$ is the number of measured data.

5. Results and Discussion

BO, MRFO, ABO, PSO, FPA and SDO are utilized in determining parameters of the SDM and DDM. The parameters of models are obtained using the objection function in Equation (10). A notable standard solar data cell, R.T.C. France solar cell at 33 °C, which is extensively reported in the literature is selected as the benchmark for comparing and testing the algorithms [45]. The performance of the algorithms is further evaluated from the practical I-V curve of the single diode model of STM6-40 with 36 series cells at 51 °C and the single diode model of STP6-120 with 36 series cells at 55 °C. These datasets are obtained from [6,46]. To achieve a rational comparison, the three algorithms are compared under the same parameters as shown in

Table 1. Parameter boundary conditions.

Parameters	R.T.C France Solar Cell		STM6-40/36 Module		STP6-120/36 Module
	Upper Bound	Lower Bound	Upper Bound	Lower Bound	Upper Bound	Lower Bound
$I_{ph(A)}$	1	0	2	0	8	0
$R_s(\Omega )$	0.5	0	0.36	0	0.36	0
$R_{sh}(\Omega )$	100	0	1000	0	1500	0
$n,n_1,n_2$	2	1	60	1	50	1
$I_o,I_{o1},I_{o2}(\mu A)$	1	0	50	0	50	0

, 500 iterations, 100 population size through 50 independent runs. The search ranges are determined based on boundaries conditions reported in [6]. The simulations were implemented on an Intel Core i5-8250U [email protected] GHz, 16 GB under Window 1064bit with MATLAB 2020b.

5.1. Results for R.T.C France Single Diode Model (SDM) and Double Diode Model (DDM)

Since the accurate parameters are yet to be established, a meaningful way of establishing extracted parameters is the comparison of their fitness function (FF). As described in Equation (10), RMSE is adopted to evaluate the fitness function. Out of the 50 runs, the run with the minimum fitness function for each algorithm is selected. Figure 6 shows the fitness function with respect to the number of runs for the R.T.C. France SDM and DDM solar cell at 33 °C. From

Table 2. Standard Deviation, Mean, FF minimum value, FF maximum value.

	Double Diode Model (DDM)				Single Diode Model (SDM)
	Standard Deviation	Mean	Fitness Minimum Value	Fitness Maximum Value	Standard Deviation	Mean	Fitness Minimum Value	Fitness Maximum Value
BO	0.00000295	0.00098519	0.00098248	0.00100052	0.00000295	0.00098519	0.00098249	0.00100052
SDO	0.00017	0.00111	0.00098	0.00176	0.000171	0.001113	0.000984	0.001764
MRFO	0.00021562	0.00123364	0.00098452	0.00171098	0.0006	0.001234	0.000985	0.001711
ABO	0.000239804	0.00148831	0.00107507	0.0020606	0.00019812	0.001464191	0.00110559	0.001857987
PSO	0.000391466	0.00144547	0.000983104	0.00247564	0.00037916	0.001409098	0.00098637	0.002448049
FPA	0.000712527	0.00329423	0.001742104	0.00520968	0.00021841	0.001608247	0.00123105	0.002185476

, BO exhibits the lowest standard deviation, minimum FF, and lowest maximum FF for both the SDM and DDM model. It is evident from the results that BO achieved optimum parameter extraction with minimal oscillations as compared to MRFO, ABO, PSO, FPA and SDO.

In order to substantiate values of the extracted parameters as shown in

Table 3. R.T.C France SSD optimal extracted parameters for BO, MRFO, SDO, ABO, PSO and FPA.

	BO	MRFO	SDO	ABO	PSO	FPA
Run	49	1	23	36	49	42
$I_{ph}$	0.7607755305	0.7607571621	0.7607754965	0.7611001744	0.7607827949	0.7595131346
$I_o$	0.0000003230	0.0000003229	0.0000003230	0.0000004017	0.0000003186	0.0000003532
$R_s$	0.0363770927	0.0363815063	0.0363770838	0.0353805457	0.0364316575	0.0362325539
$R_{sh}$	53.718522699	53.780846761	53.718861398	55.283210547	53.342564009	68.976842040
$n$	1.4811835905	1.4811678529	1.4811838910	1.5035323016	1.4797978438	1.4901822291
Elapsed time	43.07508	70.60042	72.76333	112.22356	78.26339	82.48219

and

Table 4. R.T.C France DDM optimal extracted parameters for BO, MRFO, SDO, ABO, PSO and FPA.

	BO	MRFO	SDO	ABO	PSO	FPA
Run	18	24	44	9	41	25
$I_{ph}$	0.7607810893	0.7607691048	0.7607743683	0.7607204717	0.760798850	0.7602945481
$I_{o1}$	0.0000002258	0.0000002333	0.0000002277	0.0000000000	0.0000006947	0.0000005723
$R_s$	0.0367410499	0.0366017040	0.0366398681	0.0355916525	0.0367859727	0.0341192527
$R_{sh}$	55.487603064	54.994252147	54.834589923	58.378735135	54.7912151386	97.089233631
$n_1$	1.4509691134	1.4556162769	1.4535325349	2.0000000000	2.0000000000	1.5406252960
$I_{o2}$	0.0000007504	0.0000003357	0.0000003670	0.0000003859	0.0000002279	0.0000000000
$n_2$	2.0000000000	1.8237018689	1.8320837572	1.4992684406	1.4514809290	1.3501730772
Elapsed time	39.74895	67.52181	62.96226	249.60160	60.90088	81.40994

the values are substituted in Equation (4) to obtain the I-V and P-V curve of Figure 7 and Figure 8 for both the SDM and DDM. The results demonstrate the ability of the BO algorithm to fit well with the experimental data.

Table 5. R.TC France SSD experimental and calculated for BO, SDO, MRFO, ABO, PSO and FPA.

Experimental Data		Calculated I(A) for SDM
V(V)	I(A)	BO	SDO	MRFO	ABO	PSO	FPA
−0.2057	0.764	0.7644892	0.7644892	0.7645101	0.7648214	0.7646393	0.7624956
−0.1291	0.762	0.7626024	0.7626024	0.7626083	0.7629463	0.7626811	0.7609835
−0.0588	0.7605	0.7613363	0.7613363	0.761331	0.7616757	0.7613644	0.7599652
0.0057	0.7605	0.7601733	0.7601733	0.7601581	0.7605091	0.7601555	0.7590301
0.0646	0.76	0.7591076	0.7591076	0.7590842	0.7594413	0.7590495	0.758174
0.1185	0.759	0.7581214	0.7581214	0.7580921	0.7584553	0.7580297	0.7573823
0.1678	0.757	0.7571879	0.7571879	0.7571564	0.7575256	0.7570721	0.7566319
0.2132	0.757	0.7562436	0.7562436	0.7562151	0.7565888	0.7561173	0.7558641
0.2545	0.7555	0.7551766	0.7551766	0.755158	0.7555303	0.7550583	0.7549673
0.2924	0.754	0.7537203	0.7537203	0.7537193	0.7540747	0.7536324	0.7536744
0.3269	0.7505	0.7513882	0.7513882	0.7514109	0.7517181	0.7513525	0.7514881
0.3585	0.7465	0.7472689	0.7472689	0.747317	0.7475238	0.7473	0.7474783
0.3873	0.7385	0.7398967	0.7398967	0.7399649	0.7400034	0.7399934	0.7401517
0.4137	0.728	0.7269549	0.7269549	0.72703	0.7268356	0.7270954	0.7271653
0.4373	0.7065	0.7059596	0.7059596	0.7060247	0.7055864	0.7061055	0.7060315
0.459	0.6755	0.6729697	0.6729697	0.6730124	0.6724117	0.673084	0.6728371
0.4784	0.632	0.6259944	0.6259944	0.6260168	0.6254587	0.6260643	0.6256682
0.496	0.573	0.5627924	0.5627924	0.5628134	0.5625816	0.5628435	0.5623725
0.5119	0.499	0.4840904	0.4840904	0.484137	0.4844714	0.4841741	0.4837394
0.5265	0.413	0.3902978	0.3902978	0.3903938	0.3914176	0.3904696	0.3901971
0.5398	0.3165	0.2862293	0.2862293	0.2863767	0.2879671	0.2865062	0.2864963
0.5736	−0.01	−0.0593831	−0.0593831	−0.0591741	−0.0574303	−0.0589092	−0.0578659
0.5833	−0.123	−0.1799648	−0.1799648	−0.1797837	−0.1785751	−0.1795084	−0.1781157
0.59	−0.21	−0.2360522	−0.2360522	−0.2361043	−0.21	−0.21	−0.21

and

Table 6. R.TC France DDM experimental and calculated for BO, SDO, MRFO, ABO, PSO and FPA.

Experimental Data		Calculated I(A) for DDM
V(V)	I(A)	BO	SDO	MRFO	ABO	PSO	FPA
−0.2057	0.764	0.7644892	0.7645262	0.7645101	0.7642444	0.764554	0.7624138
−0.1291	0.762	0.7626024	0.7626184	0.7626083	0.7624663	0.7626426	0.7613569
−0.0588	0.7605	0.7613363	0.7613374	0.761331	0.7612631	0.7613604	0.7606329
0.0057	0.7605	0.7601733	0.760161	0.7601581	0.7601582	0.7601827	0.7599679
0.0646	0.76	0.7591076	0.7590839	0.7590842	0.7591469	0.7591038	0.7593574
0.1185	0.759	0.7581214	0.7580888	0.7580921	0.7582129	0.7581058	0.7587878
0.1678	0.757	0.7571879	0.75715	0.7571564	0.7573312	0.7571621	0.758233
0.2132	0.757	0.7562436	0.7562054	0.7562151	0.7564399	0.7562098	0.7576276
0.2545	0.7555	0.7551766	0.7551447	0.755158	0.7554254	0.755138	0.7568411
0.2924	0.754	0.7537203	0.753702	0.7537193	0.754015	0.753681	0.7555791
0.3269	0.7505	0.7513882	0.7513897	0.7514109	0.7517069	0.7513537	0.7533049
0.3585	0.7465	0.7472689	0.7472927	0.747317	0.7475667	0.7472454	0.7490456
0.3873	0.7385	0.7398967	0.7399393	0.7399649	0.7401047	0.7398904	0.7412843
0.4137	0.728	0.7269549	0.727006	0.72703	0.7269937	0.7269694	0.7276999
0.4373	0.7065	0.7059596	0.7060049	0.7060247	0.7057837	0.7059932	0.7059313
0.459	0.6755	0.6729697	0.6729978	0.6730124	0.6726099	0.6730143	0.6722487
0.4784	0.632	0.6259944	0.6260046	0.6260168	0.6255994	0.6260354	0.6249768
0.496	0.573	0.5627924	0.562798	0.5628134	0.5625993	0.5628132	0.5620882
0.5398	0.3165	0.2862293	0.286325	0.2863767	0.2874774	0.2861564	0.2887384
0.5521	0.212	0.1729364	0.17307	0.1731305	0.1745112	0.1728487	0.1762278
0.5633	0.1035	0.0581082	0.0582546	0.0583128	0.0596348	0.0580264	0.0613355
0.5736	−0.01	−0.0593831	−0.0592274	−0.059174	−0.058024	−0.0594505	−0.0565282
0.5833	−0.123	−0.1799648	−0.1798224	−0.179784	−0.17908	−0.1800015	−0.1781746
0.59	−0.21	−0.2360522	−0.2360795	−0.236104	−0.2370736	−0.2360138	−0.2385659

show the experimental data and the calculated data for the I-V curve. Furthermore, BO utilized the lowest time in extracting the parameters for both the SDM and DDM, as shown in Figure 9, throughout the 50 runs.

5.2. Results for STM6-40/36 and STP6-120/36

Table 7. Standard Deviation, Mean, FF minimum value, FF maximum value for STM6-40/36.

Factor	BO	SDO	MRFO	ABO	PSO	FPA
Standard Deviation	0.000661	0.000414	0.00023	0.005806	0.072792	0.008792
Mean	0.001459	0.003109	0.00268	0.020241	0.024412	0.021274
FF Minimum value	0.001459	0.001797	0.00223	0.004632	0.002637	0.006056
FF Maximum Value	0.003459	0.004241	0.00323	0.02981	0.310757	0.044158

and

Table 8. Standard Deviation, Mean, FF minimum value, FF maximum value for STP6-120/36.

Factor	BO	SDO	MRFO	ABO	PSO	FPA
Standard Deviation	0.00065	0.000406	0.000262	0.000662	0.510067	0.051422
Mean	0.002082	0.003253	0.002779	0.075556	0.32932	0.12537
FF Minimum value	0.001416	0.002144	0.001946	0.066829	0.050038	0.060189
FF Maximum Value	0.003291	0.004131	0.003222	0.098554	1.41092	0.27506

provide information of the calculated standard deviation (SD), mean values and FF for both STM6-40/36 and STP6-120/36. From Figure 10, it is obvious that BO yields the smallest values of FF for both STM6-40/36 and STP6-120/36. STM6-40/36 for both MRFO and SDO show minimal SD and mean values but failed to achieve the smallest FF, as depicted in Table 7. Their smaller SD and mean are as a result of the close range of their FF, as shown in Figure 10. BO performs better than all the algorithms for STP6-120/36 in terms of SD, mean and FF. This demonstrates the heftiness of the BO algorithm for parameter extraction in comparison to other algorithms.

The extracted results of the unknown parameters for both STM6-40/36 and STP6-120/36 are shown in

Table 9. STM6-40/36 optimal extracted parameters for BO, MRFO, SDO, ABO, PSO and FPA.

	BO	SDO	MRFO	ABO	PSO	FPA
Run	5	3	14	33	35	38
$I_{ph}$	1.663905	1.662921	1.66344	1.657659	1.661664	1.653499
$I_o$	1.74 × 10−6	2.01 × 10−6	3.06 × 10−6	7.41 × 10−6	4.01 × 10−6	5.40 × 10−6
$R_s$	0.004274	0.003889	0.002295	0	0.001432	0.002095
$R_{sh}$	15.92829	17.3181	18.29602	62.21173	22.08826	995.2841
$n$	1.520303	1.536179	1.585073	1.697509	1.617942	1.655169
Elapsed time	35.4192	84.92486	70.82105	109.9931	56.22952	79.7934

and

Table 10. STP6-120/36 optimal extracted parameters for BO, MRFO, SDO, ABO, PSO and FPA.

	BO	SDO	MRFO	ABO	PSO	FPA
Run	39	2	1	28	23	17
$I_{ph}$	7.469177	7.464503	7.473341	7.534997	7.47682	7.505643
$I_o$	3.41 × 10−6	3.85× 10−6	4.02 × 10−6	4.12 × 10−5	6.62 × 10−5	2.27 × 10−5
$R_s$	0.004298	0.004239	0.004211	0.002747	0.003932	0.003185
$R_{sh}$	43.80992	995.2117	45.17533	1500	1500	549.1424
$n$	1.292478	1.303346	1.307213	1.556064	1.353659	1.485287
Elapsed time	23.52998	45.85164	48.23066	36.85526	25.1744	18.64645

, respectively. These results are selected from the runs that produce the minimum FF for each algorithm out of the 50 independent runs. These values are substituted in Equation (4) to obtain the calculated current value as shown in

Table 11. STM6-40/36 experimental and calculated results for BO, MRFO, SDO, ABO, PSO and FPA.

Experimental Data		Calculated I(A) for STM6-40/36
V(V)	I(A)	BO	MRFO	SDO	ABO	PSO	FPA
0	1.663	1.663905	1.66292117	1.66344	1.65765939	1.66166384	1.65349868
0.118	1.663	1.663252	1.66235792	1.6630516	1.65760617	1.66140721	1.65349106
2.237	1.661	1.65955	1.65895166	1.6598245	1.65664049	1.65873025	1.65341586
5.434	1.653	1.653913	1.65375291	1.6548838	1.65506168	1.65460749	1.65319671
7.26	1.65	1.650565	1.65063729	1.6518961	1.65390403	1.6520646	1.652842
9.68	1.645	1.645428	1.64571343	1.6470932	1.65119355	1.64776789	1.65127889
11.59	1.64	1.63923	1.63938726	1.6408486	1.64590616	1.64174778	1.64702935
12.6	1.636	1.633709	1.6334902	1.635081	1.64014042	1.6359733	1.64187252
13.37	1.629	1.627278	1.62646658	1.6283112	1.63301323	1.62911169	1.63522206
14.09	1.619	1.618294	1.61651927	1.6188745	1.62288851	1.61950807	1.62551932
14.88	1.597	1.60301	1.59943252	1.6029574	1.60577071	1.60331591	1.60875888
15.59	1.581	1.581461	1.57518899	1.5807788	1.5821133	1.58082591	1.585209
16.4	1.542	1.541952	1.53055692	1.5407276	1.54009977	1.5404388	1.54276731
16.71	1.524	1.520983	1.50683545	1.5195506	1.51800725	1.51911183	1.52043796
16.98	1.5	1.498904	1.48183471	1.4974554	1.49523214	1.4969484	1.49723574
17.13	1.485	1.485064	1.46616111	1.48359	1.48094194	1.48303494	1.48271859
17.32	1.465	1.46531	1.44377622	1.4639751	1.46095055	1.46342751	1.46224496
17.91	1.388	1.385657	1.35343767	1.3858464	1.38265855	1.38575453	1.38131947
19.08	1.118	1.104342	1.03414803	1.1150816	1.11923948	1.11894108	1.10626389
21.02	0	0	0	0	0	0	0

and

Table 12. STP6-120/36 experimental and calculated results for BO, MRFO, SDO, ABO, PSO and FPA.

Experimental Data		Calculated I(A) for STP6-120/36
V(V)	I(A)	BO	MRFO	SDO	ABO	PSO	FPA
0	7.48	7.46918	7.4645	7.4733	7.53499667	7.47682032	7.50564293
9.06	7.45	7.45468	7.45583	7.4586	7.51474383	7.46641706	7.48911528
9.74	7.42	7.44885	7.4502	7.4525	7.50398719	7.4599007	7.47999877
10.32	7.44	7.44105	7.44252	7.4444	7.49036269	7.45115383	7.46824519
11.17	7.41	7.42162	7.42309	7.4243	7.45883025	7.42946015	7.44045427
11.81	7.38	7.39646	7.39775	7.3984	7.42121315	7.40175011	7.40658704
12.36	7.37	7.36294	7.36393	7.3641	7.37442691	7.36534759	7.36373638
12.74	7.34	7.3305	7.33121	7.3311	7.33153747	7.33056871	7.32394556
13.16	7.29	7.28208	7.28239	7.2819	7.27052867	7.27924173	7.26668027
13.59	7.23	7.2144	7.21423	7.2133	7.18942674	7.20839009	7.18965044
14.17	7.1	7.08109	7.08023	7.0789	7.03923158	7.07100522	7.04490598
14.58	6.97	6.95052	6.94924	6.9477	6.89883003	6.93812038	6.90818887
14.93	6.83	6.8054	6.80384	6.8022	6.74850564	6.7918385	6.7605129
15.39	6.58	6.54725	6.54573	6.5442	6.49339037	6.53475866	6.50705075
15.71	6.36	6.33087	6.32944	6.328	6.28040097	6.31954286	6.29532996
16.08	6	6.00198	6.00135	6.0005	5.97129385	5.9962751	5.98459042
16.34	5.75	5.75098	5.75059	5.7499	5.72762172	5.74728621	5.74139715
16.76	5.27	5.17707	5.17938	5.1801	5.21451528	5.19032851	5.21929979
16.9	5.27	5.08653	5.08656	5.0863	5.08001074	5.08657396	5.09490568
17.1	4.79	4.72634	4.72912	4.7302	4.77295366	4.74256746	4.77991903
17.25	4.56	4.52339	4.5251	4.5257	4.5562867	4.53400965	4.56675918
17.41	4.29	4.22203	4.22533	4.2267	4.28399121	4.24178211	4.29118311
17.65	3.83	3.70965	3.71469	3.7169	3.81595553	3.74023422	3.81627385
19.21	0	0	0	0	0	0	0

. The obtained results in Table 9 and Table 10 were further utilized to compare the fitness of the I-V and P-V curve of STM6-40/36 and STP6-120/36 with the experimental data, as described in Figure 11 and Figure 12. From the figures, it is evident that the calculated parameter by BO fits well with the experimental parameters. Figure 13 shows the simulation time for the 50 runs. The figures clearly reveal that the BO algorithm has the best performance in terms of the minimal simulation time for STM6-40/36 but not for STP6-120/36 throughout the 50 runs. FPA achieved the smallest simulation time for STP6-120/36 but without achieving the minimal FF. This is an indication of premature convergence by the FPA algorithm.

5.3. Accuracy and Consistency of the Algorithms

In the same way as other types of metaheuristics algorithm, BO involves a stochastic model in its computational analysis. As a result, each run of the algorithm will not yield the same result. This creates an enormous challenge to determine the quality of the solution. Hence, it is imperative to investigate the consistency of the solution. By running the algorithms for 50 runs, a unique distinct pattern is obtained for the algorithms, which implies the parameters always unite at a resolute location, as shown in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19. The standard deviation of the BO algorithm computed denotes a good level of consistency, as shown in

Table 13. R.T.C France SDM Standard Deviation.

SDM
Parameters	BO	MRFO	SDO	ABO	PSO	FPA
$I_{ph}$	9.85 × 10−6	0.0003141	5.87 × 10−5	0.00093	0.000238	0.001006
$I_o$	2.27 × 10−13	4.08 × 10−8	1.76 × 10−8	1.09 × 10−7	1.85 × 10−7	1.1 × 10−7
$R_s$	2.79 × 10−9	0.00043925	0.0001992	0.001099	0.001643	0.001081
$R_{sh}$	1.91 × 10−5	6.36523412	1.7285297	16.78214	17.04909	14.90616
$n$	7.08 × 10−8	0.01127051	0.0050644	0.027712	0.041237	0.024983

and

Table 14. R.T.C France DDM Standard Deviation.

DDM
Parameters	BO	MRFO	SDO	ABO	PSO	FPA
$I_{ph}$	1.96 × 10−5	0.00054151	0.0002694	0.000887	0.000272	0.002332
$I_{o1}$	2.09 × 10−7	1.79 × 10−7	1.77 × 10−7	2.17 × 10−7	2.74 × 10−7	2.85 × 10−7
$R_s$	0.00018987	0.00084782	0.0008177	0.001387	0.001433	0.00241
$R_{sh}$	0.72765192	12.003639	9.0188604	17.09158	16.21457	21.68888
$n_1$	0.26510014	0.18173549	0.1764134	0.191918	0.215425	0.194329
$I_{o2}$	2.13 × 10−7	1.94 × 10−7	1.69 × 10−7	2.93 × 10−7	2.66 × 10−7	3.21 × 10−7
$n_2$	0.2749555	0.1292507	0.1592397	0.207477	0.209314	0.191603

, for both the R.T.C France SDM and DDM.

Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 show the consistency patterns of the six algorithms for both STM6-40/36 and STP6-120/361. From the figures, it can be seen that BO achieves the highest level of consistency compared to other algorithms. The high deviation of BO as described in

Table 15. Standard Deviation for STM6-40/36 parameters.

Parameters	STM6-40/36
	BO	MRFO	SDO	ABO	PSO	FPA
$I_{ph}$	0.00107878	0.0017	0.00074	0.010495	0.048065	0.013852
$I_o$	6.86 × 10−6	1.00 × 10−6	5.10 × 10−7	9.03 × 10−6	1.44 × 10−5	1.09 × 10−5
$R_s$	0.00158919	0.0008	0.00046	0.000606	0.000205	0.001121
$R_{sh}$	3.99490299	6.866	1.77258	412.1749	356.5776	328.7578
$n$	0.09479771	0.0296	0.01547	0.085978	12.48113	0.103517

and

Table 16. Standard Deviation for STP6-120/361 parameters.

Parameters	STP6-120/361
	BO	MRFO	SDO	ABO	PSO	FPA
$I_{ph}$	0.165	0.0102	0.0049253	0.03305	0.199091	0.102149
$I_o$	2.00 × 10−5	2.00 × 10−6	1.26 × 10−6	3.35 × 10−6	1.94 × 10−5	1.39 × 10−5
$R_s$	0.0011	0.0002	0.0001111	0.000228	0.001609	0.000891
$R_{sh}$	612.42	398.42	394.43889	597.3546	649.5019	471.9173
$n$	0.1638	0.0295	0.01839	0.010267	17.5496	0.073633

is as a result of the high level of differences between its consistent level and deviation point.

6. Conclusions

This paper presents the new application of the BO algorithm for PV parameter extraction. The performance of the algorithm is compared with MRFO, ABO, PSO, FPA and SDO which are reported in the literature for PV parameter extraction. It is evident from the results that BO achieved the best precision. With respect to the fitness function, the BO achieved the lowest fitness function using the RMSE. The shortest time of the BO algorithm in extracting the parameters of the PV system also demonstrates its robustness and efficacy. Furthermore, the consistency of the BO algorithm also strengthened its resolve for PV parameter extraction compared with the five other algorithms. While the BO optimizer discussed in this paper has been shown to be rather effective, it may not be as effective in multi-objective issues. Other objectives that impact the on performance of PV systems should be included in future research. It would also be interesting to use BO to address other optimization problems, such as optimal sizing and siting of distributed generation, optimal load flow and optimal load dispatch.