An Accurate Model for Bifacial Photovoltaic Panels

Abstract

Recently, there has been increasing concerns over bifacial PV (BPV) modules over the conventional monofacial PV (MPV) modules owing to their potential to add extra electrical energy from their rear-side irradiance. However, adding the rear-side irradiance to the front-side irradiance results in the increased nonlinearity of the BPV modules compared to MPV modules. Such nonlinearity makes the conventional methods unable to accurately extract the BPV module parameters. In this context, the precise determination of the BPV module parameters is a crucial issue for establishing energy yield estimations and for the proper planning of BPV installations as well. This paper proposes a new model for the BPV modules based on the MPV modeling, in which a new parameter is added to the MPV model to adjust the value of the model series resistance in order to provide a generic model for BPV modules in both monofacial and bifacial operating regions. Moreover, a new determination method for optimizing BPV model parameters using the recently developed enhanced version of the success-history-based adaptive differential evolution (SHADE) algorithm with linear population size reduction, known as the LSHADE method, is applied. The determination process of the model parameters is adapted using a two-stage optimization scheme to model the full operating range of BPV modules. The accuracy of the obtained parameters using the proposed model is compared with the conventional single-diode and double-diode models of the BPV. The obtained results using the proposed model of the BPV module show the performance superiority and accuracy of the LSHADE method over the existing methods in the literature. Furthermore, the LSHADE method provides the successful and accurate extraction of the global optimized parameters to model MPV and BPV modules. Therefore, the proposed method can provide an accurate model for the whole operating range of BPV that would be beneficial for further studies of their economic and technical feasibility for wide installation plans.

1. Introduction

1.1. General

Recently, with the increased motivation to enhance the participation and involvement of renewable energies in the utility grid, renewable energies have received huge funding and massive investments all over the world [1,2]. More attention has been oriented towards photovoltaic (PV) technology as a promising solution to overcome issues stemming from fossil fuel emissions, to produce green energy and mitigate climate change [3,4]. Significant effort has been dedicated to reducing the overall price of solar power generation technology by improving the efficiency of solar cells or modules using low-cost materials. However, increasing solar cell efficiency cannot be considered a proper solution since these require a more complex and expensive fabrication process which might reduce the gained benefits of improving the solar cell efficiency [5]. Therefore, one of the most promising ways of reduce the cost of solar power generation is boosting the energy yield of PV systems for the same installed capacities, as in the case of using bifacial PV (BPV) modules compared to conventional monofacial PV (MPV) modules [6,7,8,9].

BPV modules have the ability to generate extra electrical power from their rear side as well as their nominal front side [10]. This capability makes BPV systems a more suitable solution to reduce the overall cost of PV-generated power. In addition, BPV can increase the power density of PV systems, which defines the generated power compared to the area of the front side. Several research articles have been developed in the literature to monitor the energy yield of BPV systems compared to MPV modules at different configurations and under different weather conditions [11,12]. Although the efficiency of the BPV modules is higher than the MPV modules, there are not many installations based on BPV modules due to several challenges related to standardization, environmental conditions and the surrounding environment.

Similarly to MPV modules, the electrical power and voltages generated by BPV modules are significantly affected by environmental conditions such as solar irradiance and temperature [13,14,15,16]. Due to the nonlinearity nature of the PV panels, it is difficult to extract the maximum power of PV installations without having an accurate model for BPV modules. Almost all the provided data in datasheets of the BPV modules are obtained indoors and measured under standard test conditions (STC), which are far from the working site conditions. Furthermore, the provided data are not sufficient to be used in the analysis, planning and design process of BPV systems [17]. Therefore, having a more accurate model that can be used to assist the gained energy potential of BPV systems at different site conditions before the installation process is essential issue for the more efficient and economical operation of BPV modules.

1.2. Literature Review

Several articles in the literature have presented the research area of BPV systems. The performance comparisons between the MPV and BPV systems were presented in [18,19,20,21]. In [19], the energy output modeling was presented for BPV systems. BPV systems have led to more energy output and better efficiency in comparison with MPV systems. In [20], an artificial neural network-based output forecasting of BPV modules was presented. Additionally, the impacts of various BPV technologies were presented in [21] with their performance comparisons. On the other hand, the study of the output characteristics of BPV modules and their comparisons with MPV modules have been introduced in [22,23,24]. However, few articles have covered the area of the modeling and parameter identification of BPV modules.

Extracting BPV model parameters is advantageous in terms of many application aspects, including the design of the optimal size of the PV system, calculating the annual energy yield, forecasting the produced energy and studying the system reliability and stability [12,25]. The single-diode model (SDM) and double-diode model (DDM) are the most common models for MPV modules. All of these models have proven their ability to accurately simulate the performance of MPV cells and modules [26,27]. The extraction methods for the optimum parameters of MPV modules possess different accuracy levels, complexity levels and sizes of the required input data [28]. These can mainly be classified into analytical-based methods, numerical-based methods and meta-heuristic optimizer-based methods [29]. In analytical-based methods, mathematical formulas are utilized to identify the various parameters of PV systems [30,31]. However, their accuracy is dependent on the utilized data points and they produce approximate model parameters. However, in numerical-based methods, local searching methods are used. These can generate precise and approximate parameters of PV modules. The nonlinear properties of PV modules make them unreliable parameter determination methods [32]. The SDM and DDM will be covered in this paper for their generality and wide application in this area. However, the proposed method is general and can be extended to other existing models in the literature.

On the other hand, meta-heuristic optimizer-based methods use soft-computing techniques and combine the advantages of numerical and analytical-based methods. The global search schemes are utilized in these techniques to estimate the model parameters [33]. These have proven to have better accuracy levels and generalized applications. Many optimizers have been applied in the literature to define the optimum parameters of various PV modules [34,35,36]. Widely used optimizers include the genetic optimization algorithm (GA), the approximation and correction optimization technique (ACT) [37], artificial bee colony optimizers (ABCs), particle swarm optimizers (PSOs), the bacterial foraging optimizer algorithm (BFO), the differential evolution-based algorithms (DEAM), the evaporation rate-based WCA optimizer (ER-WCA) [38], the harmony searching optimizers (HS), the flower pollination optimizer algorithm (FPA), Nelder–Mead MFO methods (NMSOLMFO) [39], simulated annealing optimizer algorithm (SA) [33,36,40,41] and the improved JAYA optimizers (IJAYA) [42].

Another method based on the modified search and rescue optimizer algorithm (mSAR) has been presented in Houssein et al. [43]. The statistical analysis showed its superiority over existing optimizers. The bald-eagle search optimizer (BES) algorithm was introduced with the modified triple-diode model (TDM) to identify the parameters of perovskite PV cells in [44]. Moreover, the adaptive fractional order Archimedes optimizer algorithm (A-FAOA) was proposed in [45] with the SDM and DDM of PV panels. Another novel marine predator optimizer algorithm (MPA) was applied with TDM and various PV technologies in [2]. In [46], the stochastic fractal-search optimizer (SFS) was employed with the SDM and DDM. However, the gradient-based optimizer algorithm (GBO) and the improved GBO algorithms were proposed in [47] for the SDM and DDM. To properly model BPV modules, continuous improvements in the existing algorithms are required to precisely identify the different parameters of PV models.

Actually, BPV modules employ simultaneous power generation from both the front and rear PV sides. This, in turn, makes the conventional SDM and DDM less accurate models due to the nonlinear operation of BPV modules. Currently, almost all BPV module manufacturers show the module characteristics by conducing the linear addition of the efficiency/power of the front, which is achieved as a MPV module, and the rear side under a specific irradiance condition [48]. However, the forward summation of the efficiencies of the front and rear sides is not accurate due the fact that the efficiency/power does not vary linearly with the irradiance. Therefore, the nonlinearity characteristics of the BPV module should be considered when performing BPV models.

In the literature, several attempts to measure the current–voltage curves for BPV modules have been presented. However, almost all of these attempts have not considered the different combinations of the front- and rear-side illuminations. In [24], the shading effects on the power performance of the BPV were investigated. The generated power from the BPV was modeled with a consideration of the module’s shaded area and temperature in addition to the operation of the bypass diode. The results were obtained using a constant rear-irradiation level. In [25], a single-side illumination method was presented to measure the performance of BPV instead of using double-side illumination. The results show consistence between the short-circuit current ($I_{SC}$) and the open-circuit voltage ($V_{OC}$), whereas the maximum power ($P_{max}$) of the double-side illumination differed little with a distinct irradiation combination on the front and rear sides. Moreover, in [49], an improved approach for enhancing the accuracy of a single-diode model based on five parameters was introduced. The series resistance and shunt resistance were modeled using two physical equations rather than the established expression. However, the other parameters were defined according to previously used expressions. In [50], a hybrid optimization method based on the differential evolution and the artificial bee colony intelligence algorithms (nDEBCO) was proposed for determining the model parameters of MPV and BPV modules. However, this method cannot represent the full operating range of BPV modules.

1.3. Article Contributions

Based on the aforementioned literature review, improved methods are essential for the precise modeling of BPV characteristics. Since existing models in the literature lack the modeling of a nonlinear region of BPV systems, they fail to model the whole operating regions of BPV systems. The main contributions in this paper can be summarized as follows:

• An improved modeling method of BPV modules is proposed in this paper. The proposed method adapts the module’s series resistance to effectively describe the performance of BPV modules in the full operating range whilst considering the nonlinear properties of BPV systems. Thence, this paper introduces new modeling for BPV modules, which has seen little coverage in the literature.
• A generalized method is proposed to determine the unified model parameters to describe the BPV for the whole operating front-side and back-side irradiance levels. The proposed method represents a two-step optimization method for modeling BPV systems. Compared to existing methods in the literature, the proposed method has high precision for modeling both MPV operation and BPV operation.
• An enhanced application of the powerful LSHADE optimizer is introduced for determining the optimum model parameters of BPV using SDM and DDM. The LSHADE method is advantageous at adapting the internal parameters of the algorithm based on the acquired knowledge along evolutionary processes with the continuous reduction of populations.

What remains of this paper is organized as follows: Section 2 provides the SDM and DDM of PV modules. Section 3 introduces the problem formulation of this research. The proposed modified method for modeling BPV modules is presented in Section 4. The operating principle and main mathematical representation LSHADE optimizer are introduced in Section 5. The obtained results are shown in Section 6. The performance comparisons of the LSHADE method with other optimization algorithms are provided in Section 7. Finally, paper conclusions are provided in Section 8.

2. SDM and DDM for PV Modules

The accurate modeling of BPV modules is very important for precisely predicting and analyzing their performance. The SDM and DDM represent the widely employed models in the literature for PV modules. The SDM represents a reduced calculation model due to using the lower number of model parameters. The DDM can provide improved accuracy for PV modeling, however, it requires a higher number of model parameters. The two models are employed in this paper to validate the proposed BPV modeling method.

2.1. The SDM-Based Modeling

Figure 1 shows the equivalent circuit for the employed SDM of PV modules. The model includes a current source to model photoelectric current ($I_{ph}$) which is based on the ambient operating conditions of PV systems. This reflects the physical effects modeling of the existing PN junction of PV modules. It is place in paralleled to the diode D and the shunt resistance ($R_{sh}$) that model the leakage current loss in PV modeling. Furthermore, a series resistance ($R_s$) is connected to model the voltage drop of the PV output. This denotes the ohmic power loss in addition to the utilized metal contact material resistivity. The output PV current is expressed as follows [5]:

$$I_{PV}=I_{ph}-I_d-I_p$$

(1)

where $I_d$ and $I_p$ are the diode and $R_{sh}$ currents, respectively. Using the Shockley equation, it can express the $I_d$ current as follows [2]:

$$I_d=I_0[exp(\frac{V_{PV}+I_{PV}{R}_s}{a{V}_t})-1]$$

(2)

where $I_0$ is the reverse diode saturation current and $V_{PV}$ is PV model outputted voltage. Moreover, a denotes the ideality factor of the diode and $V_t$ is thermal voltage which is obtained as follows [5]:

$$V_t=\frac{N_{s}KT}{q}$$

(3)

where $N_s$ is the number of series cells in the PV module, K is the Boltzman constant $(K=1.3806503\times {10}^{-23}J/K)$, q is the electron charge $(q=1.60217646\times {10}^{-19}Coulombs)$, and T is the junction temperature of the module (in Kelvins). The current through $R_{sh}$ can be expressed as follows:

$$I_p=\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}$$

(4)

By substituting from (2) and (4) in (1), the output PV current $I_{PV}$ can be expressed as follows [34]:

$$I_{PV}=I_{ph}-I_0[exp(\frac{V_{PV}+I_{PV}{R}_s}{a{V}_t})-1]-\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}$$

(5)

From (5), it can be seen that the SDM requires the determination of five different parameters ($Para{m}_{SDM}$ includes $[I_{ph},I_0,a,R_s,R_{sh}]$). The obtained five parameters are used afterwards to model the I-V characteristics of BPV modules.

2.2. The DDM-Based Modeling

Figure 2 shows the equivalent circuit of DDM of BPV modules. Compared to SDM, the DDM includes two parallel diodes with an $I_{ph}$ source. The additional diode is beneficial as it better describes the PN junction physical effects. The diodes $D_1$ and $D_2$ represent the diffusion and recombination currents of PN junction. The DDM provides a more precise and accurate physical model than SDM. The output PV current $I_{PV}$ in DDM is expressed as follows [34]:

$$I_{PV}=I_{ph}-I_{d1}-I_{d2}-I_p$$

(6)

where $I_{d1}$ and $I_{d2}$ are the currents passing through $D_1$ and $D_2$, respectively. The currents $I_{d1}$ and $I_{d2}$ are expressed as follows [5]:

$$I_{d1}=I_{01}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_1{V}_t})-1]$$

(7)

$$I_{d2}=I_{02}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_2{V}_t})-1]$$

(8)

where $I_{01}$ and $I_{02}$ are the reverse diode saturation currents in $D_1$ and $D_2$, respectively. However, $a_1$ and $a_2$ are the ideality factor parameters for $D_1$ and $D_2$, respectively. $I_p$ is the current through $R_{sh}$. It is calculated using (4) as in SDM. By using (7), (8) and (4) in (6), the output PV current $I_{PV}$ is expressed as follows [5]:

$$I_{PV}=I_{ph}-I_{01}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_1{V}_t})-1]-I_{02}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_2{V}_t})-1]-\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}$$

(9)

It is clear from (9) that DDM requires the determination of seven different parameters ($Para{m}_{DDM}$ includes $[I_{ph},I_{01},I_{02},a_1,a_2,R_s,R_{sh}]$). The obtained seven parameters are used afterwards for modeling the I-V characteristics of BPV modules.

3. Formulation of the Optimization Problem

3.1. For SDM Model

It has become clear that the SDM requires the determination of five different parameters in the model ($Para{m}_{SDM}$ includes $[I_{ph},I_0,a,R_s,R_{sh}]$). These are determined using an analytical, numerical and meta-heuristic optimizer-based method. In this paper, a new application of the LSHADE optimizer is proposed to determine the optimum model parameters. The objective function is responsible for the accurate selection of the parameters to model the BPV module. The selected parameter set has to preserve the minimum error between the actual data points and the estimated data points. The function for representing the error between the actual data points and the estimated data points can be expressed as follows:

$$f(V_{PV},I_{PV},Para{m}_{SDM})=I_{ph}-I_0[exp(\frac{V_{PV}+I_{PV}{R}_s}{a{V}_t})-1]-\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}-I_{PV}$$

(10)

In the proposed method, the root-mean square-error (RMSE) is utilized for expressing the difference in the error between all the points as follows:

$$RMSE\left(X\right)=\sqrt{\frac{1}{N}\sum _{k=1}^{N}f{(V_{PV},I_{PV},Para{m}_{SDM})}^2}$$

(11)

where N is number of actual data points from the datasheet. The proposed method tracks the minimum value of RMSE to define the best-fit parameters.

3.2. For DDM Model

From another side, the DDM requires the modeling of BPV modules using seven parameters ($Para{m}_{DDM}$ includes $[I_{ph},I_{01},I_{02},a_1,a_2,R_s,R_{sh}]$). The function for representing the error in DDM between the actual data points and the estimated data points can be expressed as follows:

$$\begin{array}{c}\hfill f(V_{PV},I_{PV},Para{m}_{DDM})=I_{ph}-I_{01}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_1{V}_t})-1]-I_{02}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_2{V}_t})-1]\\ \hfill -\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}-I_{PV}\end{array}$$

(12)

The RMSE is calculated for DDM as follows:

$$RMSE\left(X\right)=\sqrt{\frac{1}{N}\sum _{k=1}^{N}f{(V_{PV},I_{PV},Para{m}_{SDM/DDM})}^2}$$

(13)

4. The Proposed Modified Method for BPV Modeling

4.1. For SDM Model

The aforementioned methodology can perfectly determine the five parameters of the SDM ($Para{m}_{SDM}$ = $[I_{ph},I_0,a,R_s,R_{sh}]$) for the MPV operating region. However, this model cannot determine a generic model for modeling the MPV in addition to BPV operating regions. In this paper, a modified method is proposed for the determination of the parameters for BPV systems as follows:

Step 1: The LSHADE optimization algorithm is run using the datasheet parameters for the MPV operating region as in the conventional PV model parameter extraction methods at a solar irradiance of 1000 W/m${}^2$. This step determines the five parameters for the MPV operation using (10) and (11) as follows:

$$RMSE\left(X\right)=\sqrt{\frac{1}{N}\sum _{k=1}^N(I_{ph}-I_0[exp(\frac{V_{PV}+I_{PV}{R}_s}{a{V}_t})-1]}-\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}-I_{PV}{)}^2$$

(14)

After minimizing this function using the LSHADE algorithm, five parameters were obtained for the PV model, ($Para{m}_{SDM}$ = $[I_{ph},I_0,a,R_s,R_{sh}]$).

Step 2: The LSHADE algorithm is run another time but with the remaining data of the BPV module. In this step, a modified model is used with additional parameters to the conventional five parameters of the SDM of PV systems. The objective function and the mathematical representation of the BPV modeling using the SDM can be represented as follows:

$$\begin{array}{c}{f}_{mod}(V_{PV},I_{PV},Para{m}_{SDM,mod})=(1+B_{Isc})I_{ph}-I_0[exp(\frac{V_{PV}+I_{PV}(R_s+R_{add,SDM}{B}_{Isc})}{a{V}_t})-1]\\ -\frac{V_{PV}+I_{PV}(R_s+R_{add,SDM}{B}_{Isc})}{R_{sh}}-I_{PV}\end{array}$$

(15)

where $B_{Isc}$ represents the bifaciality factor of the short circuit current and it is equal to the ratio between the short circuit current at BPV to the short circuit current at the MPV operation. In addition, another resistance $R_{add,SDM}$ is added in series with $R_s$ to compensate for the nonlinearity of the BPV systems. However, the RMSE function can be expressed as follows:

$$RMSE\left(X\right)=\sqrt{\frac{1}{N}\sum _{k=1}^N{f}_{mod}{(V_{PV},I_{PV},Para{m}_{SDM,mod})}^2}$$

(16)

In the optimization of this step, only the determination of the adjusted value of the resistance $R_{add,SDM}$ is performed. Thence, the total number of determined parameters for the model will be six at the end of this step. As such, the parameters $[I_{ph},I_0,a,R_s,R_{sh}]$ are determined using the datasheet data of 1000 W/m${}^2$ in step 1. On the other hand, the parameter $R_{add,SDM}$ is determined by step 2 using the datasheet data of the bifacial region.

Figure 3 shows the new proposed generic model for the SDM representation of BPV systems. It can be seen that the SDM of BPV systems has six different parameters to model the PV module in both the MPV and BPV regions. The first five parameters are determined based on the linear MPV region at 1000 W/m${}^2$. The modified value of the series resistance $R_{add,SDM}$ is determined using the nonlinear region of the BPV datasheet. Hence, the LSHADE optimizer is run in two steps with different curve data in each step.

4.2. For DDM Model

In the same way, a two-stage optimization process is performed to obtain the model parameters of the DDM for the BPV modules. The proposed modified method can be represented using the following steps:

Step 1: The seven parameters of the DDM (with $Para{m}_{DDM}$ includes [$I_{ph}$, $I_{01}$, $I_{02}$, $a_1$, $a_2$, $R_s$, $R_{sh}$]) are determined using the LSHADE optimization method and the datasheet curves of the MPV region at 1000 W/m${}^2$. Thus,

$$\begin{array}{c}RMSE\left(X\right)=\sqrt{\frac{1}{N}{\sum }_{k=1}^N(I_{ph}-I_{01}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_1{V}_t})-1]-I_{02}[exp(\frac{V_{PV}+I_{PV}{R}_s}{a_2{V}_t})-1]}\\ -\frac{V_{PV}+I_{PV}{R}_s}{R_{sh}}-I_{PV}{)}^2\end{array}$$

(17)

Step 2: In this step, the LSHADE algorithm is run using the seven parameters determined in Step 1. Meanwhile, the extra added resistance $R_{add,DDM}$ is determined using the data from the BPV curves with both front- and rear-side irradiance data. The objective function of this step can be represented as follows:

$$\begin{gathered} \begin{array}{c}{f}_{mod}(V_{PV},I_{PV},Para{m}_{DDM,mod})=(1+B_{Isc})\end{array}{I}_{ph}-I_{01}[exp(\frac{V_{PV}+I_{PV}(R_s+R_{add,DDM}{B}_{Isc})}{a_1{V}_t})-1] \\ -I_{02}[exp(\frac{V_{PV}+I_{PV}(R_s+R_{add,DDM}{B}_{Isc})}{a_2{V}_t})-1]-\frac{V_{PV}+I_{PV}(R_s+R_{add,DDM}{B}_{Isc})}{R_{sh}}-I_{PV} \end{gathered}$$

(18)

The RMSE function for the LSHADE algorithm can be expressed as follows:

$$RMSE\left(X\right)=\sqrt{\frac{1}{N}\sum _{k=1}^N{f}_{mod}{(V_{PV},I_{PV},Para{m}_{DDM,mod})}^2}$$

(19)

Figure 4 shows the proposed modified version of DDM for BPV systems. It can be seen that the modification of the current source in addition to the extra added resistance form a generic modeling of both the MPV and BPV operating regions. Thence, the proposed method can determine the appropriate model for the BPV modules in both monofacial and bifacial operating regions. The determination process is valid for various representations of PV modules, while a two-step optimization is performed for the parameter determination process. An additional parameter that represents the incremental value of the series resistance is added for the process to model the nonentities of the BPV modules.

5. The Proposed LSHADE Method

The differential evaluation (DE) was originally suggested by Storn et al. [51] for solving different problems. It is reliant on some parameters, such as the scaling factor and the crossover rate. The algorithm adjusts the scaling factor and the crossover rate based on the chronological data of such parameters [52]. Based on an ensemble of two sinusoidal adaptation approaches, Awad et al. suggested the LSHADE–EpSin algorithm [53]. The best solution can be achieved with a high convergence speed. The steps of LSHADE–EpSin are outlined as follows [54]:

5.1. Initialization Step

The initialization of every element h for every variable k is made by adjusting its values within its upper $x_k^U$ and lower $x_k^L$ limits arbitrarily as follows [53]:

$$\begin{array}{c}{x}_{kh}^{\left(0\right)}=x_h^L+ran{d}_{kh}[0,1]\times (x_h^U-x_h^L)\hfill \end{array}$$

(20)

where $k=\{1,2,3,\dots ,NP\}$, $h=\{1,2,3,\dots ,N_d\}$, $N_d$ denote the number of variables.

5.2. Mutation Step

For every generation u, the mutation stage generates a donor vector $v_k^{\left(u\right)}$ based on the ‘current-to-best/1’ mutation approach as follows [53]:

$$\begin{array}{c}{v}_k^u=x^{\left(u\right)}+S{F}_k^{\left(u\right)}\times (x_{pbest}^{\left(u\right)}-x_x^{\left(u\right)})+S{F}_k^{\left(u\right)}\times (x_{t_1^k}^{\left(u\right)}-x_{t_2^k}^{\left(u\right)})\hfill \end{array}$$

(21)

where $x_{pbest}^{\left(u\right)}$ is the best individual in the current generation u. $S{F}_k^{\left(u\right)}$ is a scaling factor.

5.3. Adaptation of Parameter Step

This step contains two parts: the first part of the total number of iterations $\left(u^{max}\right)$ and the second part for the remainder of these iterations. For the first part, $SF$ is modified by applying an ensemble of two sinusoidal methodologies: the non-adaptive and adaptive sinusoidal methods. One of them is selected arbitrarily at a time for $SF$ adjustment as follows [54]:

$$\begin{array}{c}S{F}_k^{\left(u\right)}=0.5\times (sin\left(2\pi fre{q}_k\right)\times u+\pi )\times \left(\frac{u}{u^{max}}\right)+1),\forall u\le \frac{u^{max}}{2}\hfill \end{array}$$

(22)

$$\begin{array}{c}fre{q}_k=randc(\mu fre{q}_{rk},0.1)\hfill \end{array}$$

(23)

where $freq$ is fixed, $fre{q}_k$ is adaptive. Furthermore, these are modified corresponding to their previously stored knowledge, and thus [53]:

$$\begin{array}{c}S{F}_k^{\left(u\right)}=randc(\mu S{F}_{rk},0.1),\forall u>\frac{u^{max}}{2}\hfill \end{array}$$

(24)

$$\begin{array}{c}C{R}_k^{\left(u\right)}=randn(\mu C{R}_{rk},0.1),\forall u>\frac{u^{max}}{2}\hfill \end{array}$$

(25)

where the mean of the successful values of both $SF$ and $CR$, which are stored in M from previous generations, are $\mu S{F}_{rk}$ and $\mu C{R}_{rk}$, respectively.

Similarly to the original LSHADE, if the values generated for both SF and CR were sufficiently successful to obtain a vector better than $x_k^{\left(u\right)}$, those values are stored in $S_{SF}$ and $S_{CR}$, respectively, and the index that controls the position of the next best value is denoted as q, which is incremented by 1 after storing the new better $SF$ and $CR$ values. Moreover, $\mu S{F}_{rk}$ and $\mu C{R}_{rk}$ are updated using the weighted Lehmer mean $\left(mea{n}_L\right)$ as follows [54]:

$$\begin{array}{c}\mu S{F}_{rq}^{(u+1)}=mea{n}_L\left(S_{SF}\right)\hfill \end{array}$$

(26)

$$\begin{array}{c}\mu C{R}_{rq}^{(u+1)}=mea{n}_L\left(S_{CR}\right)\hfill \end{array}$$

(27)

5.4. Crossover

A vector $Y_k^{\left(u\right)}$ is established by combining $x_{kh}^{\left(u\right)}$ and $v_{kh}^{\left(u\right)}$ elements based on the crossover probability of the assignment. The crossover assignment probability is controlled by $C{R}_k^{\left(u\right)}$, which is calculated in the previous phase. The elements of $Y_k^{\left(u\right)}$ are combined as follows [54]:

$$Y_k^{\left(u\right)}=\left\{\begin{array}{cc}{v}_{kh}^{\left(u\right)},\hfill & h=h_{rand}\hfill \\ v_{kh}^{\left(u\right)},\hfill & ran{d}_{kh}[0,1]\le C{R}_k^{\left(u\right)}\hfill \\ x_{kh}^{\left(u\right)},\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$

(28)

where $h_{rand}$ is randomly chosen from D.

5.5. Population Reduction Stage

A linear population size strategy is implemented in LSHADE–EpSin to effectively select the best population size as follows [53]:

$$\begin{array}{c}N{P}^{(u+1)}=round\{(\frac{N{P}_{min}-N{P}_{initial}}{N{F}_{max}})\times NF+N{P}_{initial}\}\hfill \end{array}$$

(29)

where $N{P}_{initial}$ denotes the initial population size $NF$ and $N{F}_{max}$ are the present number of objective function evaluations and their maximum value, respectively. $N{P}_{min}$ is set to four because of the requirements requested by the mutation strategy implemented in LSHADE. At the $u^{th}$ generation, if $N{P}^{(u+1)}<N{P}^{\left(u\right)}$, $N{P}^{(u+1)}-N{P}^{\left(u\right)}$ individuals with the worst objective function values are eliminated from the population. The parameters of the LSHADE algorithm were set according to the original LSHADE parameters as in [54]. Figure 5 and Figure 6 summarized the optimization process and pseudo-code of the LSHADE optimization algorithm for the proposed method.

6. Results and Discussion

The aforementioned LSHADE optimizer algorithm was employed for the determination process of the SDM and DDM parameters in both the MPV and BPV operation. The characteristics of the LSHADE algorithm provide the proper adaptation of the algorithm’s internal parameters based on the acquired knowledge along evolutionary processes with a continuous reduction in populations. Hence, it was selected in this paper to determine the optimal parameters of BPV modules. Moreover, the datasheet parameters of the Silfab’s BPV 360 ultra-high-efficiency PV module (SLG-X 360 Bifacial) with 72 Cell [48] was selected as a case study. The SLG-X 360 BPV are based on premium N-Type BPV cells with up to 21.5% of front efficiency (wherein 23.2% is the efficiency of module and up to 30% is the contribution of the back side).

Table 1. Specifications of considered solar PV modules [48].

Electrical	STC at	STC at Front + Irradiance % on Back Side				NOCT at
Specifications	Front	15%	20%	25%	30%	Front
$Pmp$ (W)	360.0	405.9	421.2	436.5	451.9	274.5
$Imp$ (A)	8.9	9.95	10.33	10.68	11.04	6.8
$Vmp$ (V)	40.3	39.44	39.46	39.52	39.53	40.3
$Isc$ (A)	9.7	10.57	10.98	11.38	11.77	7.7
$Voc$ (V)	47.	5 47.81	47.86	47.90	47.98	47.0
Efficiency	18.46%	20.24%	21.60%	22.38%	23.17%	17.6%

shows the main datasheet data of the SLG-X 360 BPV modules [48]. The table includes the datasheet data at STC with only front-side irradiance, as well as with different percentages of rear-side irradiance.

6.1. Results for SDM of BPV Module

The I-V curves for the SLG-X 360 bifacial at 1000 W/m${}^2$ front irradiation with 0% rear, and at 1000 W/m${}^2$ front irradiation with 30% rear are shown in Figure 7. It can be seen that the 30% rear-side irradiation increases the generated power from the BPV module compared to the case without rear irradiance. This in turn makes the BPV modules advantageous in the applications with a large amount of required energy output from PV systems. It can also be seen that the short circuit current with 30% rear-side irradiation is higher than the case without rear irradiance.

Firstly, the LSHADE algorithm is run to determine the SDM parameters at the monofacial case with 1000 W/m${}^2$ front irradiance and 0% rear irradiance. The five model parameters are shown in

Table 2. Boundaries and the obtained optimal SDM parameters for the SLG-X 360 bifacial module.

Parameter	Boundary		Optimal Parameters (DDM) Conventional			Proposed
	Min.	Max.	MPV Only	BPV Only	Scaling MPV	Method
$I_{ph}$ (A)	0.0	12	9.2464	11.5113	9.2464 $\times (1+B_{Isc})$	9.2464 $\times (1+B_{Isc})$
$I_o$ (A)	$1.0\times {10}^{-10}$	$1.0\times {10}^{-4}$	$3.0146\times {10}^{-7}$	$1.2347\times {10}^{-7}$	$3.0146\times {10}^{-7}$	$3.0146\times {10}^{-7}$
a	0.0	3.0	1.2280	1.1664	1.2280	1.2280
$R_{sh}\left(\mathsf{\Omega }\right)$	0.0	5000	661.2200	4705.3	661.2200	661.2200
$R_s\left(\mathsf{\Omega }\right)$	0.0	1.0	0.01	0.0754	0.01	0.01
$R_{add,SDM}\left(\mathsf{\Omega }\right)$	0.0	1.0	—	—	—	0.39
$RMSE$			0.0321	0.0345	0.4519	0.0538
$MAE$			0.0236	0.0259	0.3125	0.0435
$R^2$			0.9999	0.9999	0.9858	0.9998

. Figure 8 shows the estimated model using the proposed LSHADE method compared to the original data from the datasheet of the BPV module. It can be seen that the proposed LSHADE method has very good fit to the original data from the datasheet curves. The RMSE value of this case is equal to 0.0321, whereas the mean absolute error (MAE) is equal to 0.0236. The R-square ($R^2$) value for this case is 0.9999, which confirms the excellent performance of the LSHADE algorithm to determine the SDM parameters.

The autocorrelation test is also made to provide a statistical measure for the accurateness of the modeling method. The autocorrelation test provides the degree of correlations of the same variable between two different successive timing intervals. The output values of the test range between −1 and +1. In the case of autocorrelation values between −1 and 0, it represents the negative autocorrelation case. However, in the case of autocorrelation values between +1 and 0, it represents the positive autocorrelation case. Figure 9 shows the autocorrelation test of the proposed LSHADE method at an SDM with 1000 W/m${}^2$ front irradiance and 0% rear irradiance. It can be seen that the proposed model possesses reduced values of the autocorrelation test in both the positive and negative autocorrelation regions.

The comparison of the estimated model using the proposed LSHADE method with the original data from the datasheet of the BPV module at the monofacial case with 1000 W/m${}^2$ front irradiance and 30% rear irradiance is shown in Figure 10. The proposed LSHADE method can effectively represent the curve of BPV with the existing rear irradiation. The values of the RMSE, MAE and $R^2$ are 0.0345, 0.0259 and 0.9999, respectively. The values of error measures show that the proposed LSHADE method can effectively determine the SDM parameters with very good accuracy. Moreover, the autocorrelation test of the proposed LSHADE method at SDM with 1000 W/m${}^2$ front irradiance and 30% rear irradiance is shown in Figure 11. The results of the autocorrelation test show that the proposed LSHADE method can effectively represent the curves of BPV, regardless of the rear irradiance data.

In this part, a check of the generality of the obtained models in Figure 8 is made to represent the BPV in the bifacial operating region of the studied BPV module. Figure 12 shows the estimated model using the scaled irradiance value with the original data at 1000 W/m${}^2$ front irradiance and 30% rear irradiance. The nonlinearity of the BPV performance is reflected as a deviation of the estimated curve compared to the original curve. The RMSE value is 0.4519, the MAE value is 0.3125 and the $R^2$ value is 0.9858. The values show the deviation of the estimated model versus the original datasheet data. Additionally, the result of the autocorrelation test for this case is shown in Figure 13. It can be seen that most of the values are large and with a positive autocorrelation type. It is necessary to make an improved model to reduce the values of the autocorrelation test with reduced deviation between the estimated curve and the original curve.

In this part, the performance of the proposed improved model of BPV using the description in Section 4.1 is tested. Figure 14 shows the obtained estimated curve using the proposed method and the original curve at 1000 W/m${}^2$ front irradiance and 30% rear irradiance.

The RMSE value under the proposed method is 0.0538 compared to 0.4519 of the conventional method. In addition, the MAE value is 0.0435 under the proposed method compared to 0.3125 under the conventional method. The $R^2$ value is 0.9998 under the proposed method, whereas its value is 0.9858 under the conventional method. This value shows the improvements and superiority of the proposed method to model the BPV modules. It can be also seen from Figure 14 that the improved modeling of the BPV is obtained using the proposed method. This in turn verifies the capability of the proposed method to model the BPV in both the MPV and BPV operating regions. The proposed two-stage optimization approach can compensate the deviations that result from the nonlinearity property of BPV modules at the bifacial region. Moreover, the autocorrelation test for the proposed method is shown in Figure 15. It can be seen that the proposed method achieves improved autocorrelation results compared to the previously obtained results in Figure 13. This in turn proves the feasibility of the proposed method to improve the generalized SDM modeling of BPV modules.

6.2. Results for DDM of BPV Module

On the other hand, the proposed LSHADE method has been tested with DDM to model the BPV systems.

Table 3. Boundaries and the obtained optimal DDM parameters for the SLG-X 360 bifacial module.

Parameter	Boundary		Optimal Parameters (DDM) Conventional			Proposed
	Min.	Max.	MPV Only	BPV Only	Scaling MPV	Method
$I_{ph}$ (A)	0.0	12	9.2379	11.5232	9.2379 $\times (1+B_{Isc})$	9.2379 $\times (1+B_{Isc})$
$I_{o1}$ (A)	$1.0\times {10}^{-10}$	$1.0\times {10}^{-4}$	$8.596\times {10}^{-5}$	$1.3917\times {10}^{-7}$	$8.596\times {10}^{-5}$	$8.596\times {10}^{-5}$
$I_{o2}$ (A)	$1.0\times {10}^{-10}$	$1.0\times {10}^{-4}$	$1.2728\times {10}^{-7}$	$2.6899\times {10}^{-7}$	$1.2728\times {10}^{-7}$	$1.2728\times {10}^{-7}$
$a_1$	0.0	3.0	2.3665	2.4942	2.3665	2.3665
$a_2$	0.0	3.0	1.1738	1.2178	1.1738	1.1738
$R_{sh}\left(\mathsf{\Omega }\right)$	0.0	5000	4488.9748	4348.9552	4488.9748	4488.9748
$R_s\left(\mathsf{\Omega }\right)$	0.0	1.0	0.01	0.0629	0.01	0.01
$R_{add,SDM}\left(\mathsf{\Omega }\right)$	0.0	1.0	—	—	—	0.41
$RMSE$			0.0279	0.0427	0.4621	0.0530
$MAE$			0.0191	0.0323	0.3143	0.0396
$R^2$			0.9999	0.9999	0.9852	0.9998

summarizes the boundaries and the obtained optimal model parameters for DDM. The estimated curve based on the proposed LSHADE method versus the original curve from the manufacturer at 1000 W/m${}^2$ front irradiance and 0% rear irradiance is shown in Figure 16. In this case, the values of the RMSE, MAE and $R^2$ are 0.0279, 0.019 and 0.9999, respectively. The proposed LSHADE algorithm can perfectly determine the optimal model parameters of BPV systems using DDM. The value of the RMSE for DDM is 0.0279 compared to 0.0321 for SDM. Furthermore, the value of MAE using DDM is lower than that using SDM. However, the corresponding autocorrelation test results are shown in Figure 17. It can be seen that an improved matching of both curves is obtained using the DDM with the proposed LSHADE method. This can also be seen from the small values in the autocorrelation test in both the positive and negative sides. Moreover, the results show the effectiveness of the proposed LSHADE method with both the SDM and DDM of PV systems.

Additionally, Figure 18 shows the obtained estimated curve of DDM and the original datasheet curve at 1000 W/m${}^2$ front irradiance and 30% rear irradiance. The LSHADE method can also determine the optimal parameters of the DDM in the bifacial PV operating region. The proposed method achieves an RMSE value in this case which is equal to 0.0427, the value of MAE is 0.0323 and the value of $R^2$ is 0.9999. The obtained results show the effective fitting of the estimated and the original curves with the existing rear-side irradiance. Furthermore, the obtained autocorrelation test is shown in Figure 19 at 1000 W/m${}^2$ front irradiance and 30% rear irradiance. It can be seen that the values of the autocorrelation test are small and have positive and negative values.

Figure 20 shows the representation of the BPV module with 30% rear irradiance using the obtained parameters in the case without rear irradiance while scaling the solar irradiance. The estimated values under this case are 0.4621 for RMSE, 0.3143 for MAE and 0.9852 for $R^2$. It has become clear that the model fails in modeling both of the operating regions using only seven of the obtained parameters. This in turn verifies the failure of the generality of the model in this step. The result of the autocorrelation test is shown in Figure 21 for this case. The values of the autocorrelation are high and most of them are in the positive autocorrelation type. The results show that the classical modeling cannot provide a general representation of the BPV for the full operating range.

The proposed modified modeling of BPV using DDM has been tested using the LSHADE method. Figure 22 compares the estimated curve using the proposed modeling method with the original curve at 1000 W/m${}^2$ front irradiance and 30% rear irradiance. The LSHADE was utilized to determine the eight parameters using two subsequent steps. It can be seen from the two curves that there is improved modeling of the BPV in the bifacial region using the obtained model parameters of the monofacial region. Thence, a more general model to describe the full operating range of BPV systems is presented in this paper using a modified process of the conventional model determination methods. The proposed modeling method achieves an RMSE value of 0.0530 compared to 0.4519 under the conventional method. In addition, the proposed method achieves an MAE value of 0.0396 compared to 0.3125 under the conventional method. The value of $R^2$ in this case is equal to 0.9998. Additionally, the autocorrelation test was performed with the obtained model using the proposed method as shown in Figure 23. It has become clear that the obtained autocorrelation values have been reduced using the proposed method. This in turn confirms the generality and the accuracy of the proposed method.

7. Performance Comparisons of LSHADE Method

Table 4. Comparison of different optimizers for BPV modeling at 1000 W/m${}^2$ front irradiance and 0% rear irradiance.

Run No.	LSHADE	SCA	AEO	ASO	SMA	MPA
1	0.031301782	0.347265266	0.029924963	0.166870224	0.039802333	0.045475419
2	0.045984225	0.24030415	0.046179282	0.063442297	0.040931443	0.050654598
3	0.029802696	0.466750656	0.051108826	0.191040182	0.053404346	0.0467553
4	0.035614495	0.296088206	0.041756257	0.250520982	0.029934247	0.030048178
5	0.032802254	1.057749553	0.041832899	0.180936404	0.119548975	0.049326078
6	0.029511653	0.391723026	0.037515532	0.075012915	0.057295955	0.061429031
7	0.032339831	0.381258343	0.040422073	0.098276595	0.045873362	0.030357461
8	0.036047852	0.750971209	0.033204699	0.125610159	0.06117019	0.062945774
9	0.074731026	1.20277875	0.052478552	0.145481508	0.049403854	0.046329233
10	0.035976252	0.224309244	0.038183257	0.154148555	0.140792614	0.082601218
11	0.029629578	0.740312706	0.032829873	0.234284739	0.05388134	0.04635604
12	0.033848511	0.531043584	0.032043539	0.147601301	0.056233513	0.036072274
13	0.03603033	0.449658834	0.03040842	0.222619863	0.084209319	0.100400849
14	0.031229036	0.470646191	0.029435805	0.109929286	0.201742478	0.091687324
15	0.02933224	0.672743035	0.042047279	0.050210287	0.085406166	0.04548336
16	0.032110862	0.904973735	0.040845861	0.194559707	0.088988915	0.041790774
17	0.030198193	0.26228278	0.040616418	0.198487163	0.075866397	0.114174565
18	0.029618632	0.346358068	0.040736633	0.100026198	0.056941617	0.04408729
19	0.032846288	0.518007716	0.032301602	0.149009044	0.065389952	0.037144361
20	0.029487126	0.382742771	0.032709624	0.109345045	0.196806016	0.054565684
21	0.031792066	0.340719384	0.032855151	0.120429719	0.041650539	0.036442765
22	0.029675494	0.428000437	0.033792674	0.104403584	0.043391312	0.055121835
23	0.029587059	0.271264182	0.04967191	0.111079526	0.055018014	0.029865091
24	0.031958269	0.474475342	0.031990669	0.207917859	0.031257339	0.033222618
25	0.046578843	0.371585699	0.050241951	0.070425603	0.102030862	0.097472745
26	0.038340429	0.622467131	0.034101394	0.197425253	0.091590727	0.075567015
27	0.038326347	0.314203704	0.035322589	0.165007498	0.071699543	0.086275356
28	0.037019875	0.686659217	0.039293975	0.175400184	0.161269555	0.050266469
29	0.033515796	0.472381055	0.04029131	0.12853282	0.076445682	0.042383453
30	0.037207863	0.462041639	0.029658382	0.078324976	0.033063702	0.063550457
AVG	0.035081497	0.50272552	0.038126713	0.144211983	0.077034677	0.056261754
MEDIAN	0.032571043	0.455850236	0.037849394	0.146541405	0.059233072	0.048040689
STD	0.008713375	0.237788143	0.006768953	0.053741309	0.045578528	0.022960587
Best	0.02933224	0.224309244	0.029435805	0.050210287	0.029934247	0.029865091
Worst	0.074731026	1.20277875	0.052478552	0.250520982	0.201742478	0.114174565

and

Table 5. Comparison of different optimizers for BPV modeling at 1000 W/m${}^2$ front irradiance and 30% rear irradiance.

Run No.	LSHADE	SCA	AEO	ASO	SMA	MPA
1	0.034189644	1.15038179	0.070227853	0.212619162	0.17526472	0.054383022
2	0.075608676	0.607142681	0.041205081	0.182964487	0.059772625	0.048403619
3	0.029375189	0.79847023	0.032489498	0.119831767	0.089395072	0.056169344
4	0.029972137	0.587178015	0.031385305	0.194920938	0.034786902	0.078303381
5	0.057890571	0.671252247	0.052218187	0.201460221	0.129323623	0.032842401
6	0.029431209	0.639823702	0.038841115	0.149651521	0.092778928	0.049020376
7	0.031359129	0.41687846	0.051115721	0.161138388	0.138624157	0.045564405
8	0.029713905	0.53164077	0.050408847	0.164832292	0.05143026	0.054441371
9	0.03484947	0.269608415	0.03421572	0.150982301	0.062980588	0.052784859
10	0.031794089	0.29843143	0.053907787	0.201369713	0.045888967	0.113319899
11	0.036144978	1.100389227	0.044837591	0.237964732	0.080216062	0.029601339
12	0.029367499	0.260918309	0.031696511	0.11353111	0.07596078	0.032960304
13	0.029733773	0.704985325	0.031034332	0.17357378	0.065408403	0.041994491
14	0.031157119	0.575123037	0.036432051	0.157664742	0.120467385	0.103452354
15	0.046433024	0.255027002	0.030314219	0.166697265	0.052410896	0.042363482
16	0.035369477	0.37682641	0.051371581	0.246406846	0.037315972	0.029573451
17	0.030676923	0.134579089	0.03224564	0.181094703	0.129906454	0.060407736
18	0.02937632	0.275787438	0.036128159	0.123221235	0.163800274	0.030474554
19	0.029986625	0.240822575	0.030722509	0.28706597	0.049410134	0.068679264
20	0.031393463	0.854545038	0.03141997	0.136451342	0.050175771	0.048891979
21	0.048517246	0.242747409	0.037811702	0.103428411	0.045823967	0.067398488
22	0.029736597	0.367581708	0.035198521	0.156284979	0.10065641	0.036609147
23	0.032714922	0.346105366	0.032915964	0.069557276	0.112741654	0.03991321
24	0.03416113	0.403895356	0.029616908	0.136677946	0.041276835	0.073532893
25	0.02942883	0.215784161	0.032114661	0.18441361	0.052199929	0.054687175
26	0.029549189	0.560195627	0.032038952	0.075486615	0.046135944	0.029596169
27	0.042657314	0.235168368	0.032072312	0.27318809	0.039768249	0.029394282
28	0.035201235	0.412166883	0.030664794	0.351240761	0.072347623	0.033064002
29	0.036647765	0.23171018	0.052483907	0.134934404	0.04206238	0.042623468
30	0.029437371	0.168715711	0.036771123	0.218715367	0.045259652	0.038191538
AVG	0.035395827	0.464462732	0.038796884	0.175578999	0.076786354	0.0506214
MEDIAN	0.031376296	0.390360883	0.03470712	0.165764778	0.061376606	0.046984012
STD	0.010124565	0.26402316	0.009970382	0.061606304	0.039743572	0.02086518
Best	0.029367499	0.134579089	0.029616908	0.069557276	0.034786902	0.029394282
Worst	0.075608676	1.15038179	0.070227853	0.351240761	0.17526472	0.113319899

were added to investigate the performance of the proposed LSHADE technique compared with some other common techniques such as the sine cosine algorithm (SCA), the artificial ecosystem-based optimization (AEO), anarchic society optimization (ASO), slime mould algorithm (SMA) and the marine predators algorithm (MPA). The optimizers were run 30 times to compare the addressed optimizers. The two comparisons corresponded to the dataset curves at 1000 W/m${}^2$ front irradiance with 0% rear irradiance and with 30% rear irradiance, respectively.

Table 4 shows the performance of the addressed optimization techniques for BPV modeling at 1000 W/m${}^2$ front irradiance and 0% rear irradiance with SDM. It can be seen that the LSHADE technique achieved the best solution compared to the others; as LSHADE achieves 0.02933224 compared to 0.029435 for AEO, 0.02993424 for SMA. Moreover, LSHADE possesses the minimum average and the median for the solutions compared to the other techniques. On the other hand, Table 5 shows the comparison of different optimizers for BPV modeling at 1000 W/m${}^2$ front irradiance and 30% rear irradiance with SDM. It is also clear that LSHADE has achieved the best solutions, minimum average and median compared to other techniques. Therefore, Table 4 and Table 5 verified the superiority of LSHADE results compared to the other techniques.

8. Conclusions

An improved generalized modeling of bifacial PV modules was presented in this paper. In addition, a new application of the recently developed LSHADE algorithm was presented for the optimum determination of the parameters for bifacial PV systems. The main findings of the paper are as follows:

• The proposed method is advantageous in terms of model accuracy and precisely modeling both the MPV and BPV.
• In addition, the proposed method is general and can be applied to the various single- and double-diode models of bifacial PV systems.
• The proposed method achieves the RMSE values of 0.0538 and 0.0530 compared to 0.4519 and 0.4621 under the conventional method for SDM and DDM, respectively.
• The proposed method has MAE values of 0.0435 and 0.0396 compared to 0.3125 and 0.3143 under the conventional method for SDM and DDM, respectively.
• The obtained results show the effectiveness and the accuracy of the proposed method with a reduced error in the obtained representations for both the SDM and DDM of BPV modules.

The future extension of this work includes the comparison of the performance of the LSHADE optimizer in BPV applications with other existing optimization methods in the literature. The study of different BPV modules will be included in future research. Moreover, future research can cover other modeling methods of rear-side effects rather than the proposed added series resistance.