Turbulent Flow of Water-Based Optimization Using New Objective Function for Parameter Extraction of Six Photovoltaic Models

With the development of new energy power systems, the estimation of the parameters of photovoltaic (PV) models has become increasingly important. Weather changes are random; therefore, the changes in the PV output power are periodic and nonlinear. Traditional power prediction methods are based on linearity, and relying only on a time series is not feasible. Consequently, metaheuristic algorithms have received considerable attention to extract the parameters of solar cell models and achieve excellent performance. In this study, the Turbulent Flow of Water-based Optimization (TFWO) is used to estimate the parameters of three traditional solar cell models, namely, Single-Diode Solar Cell Model (SDSCM), Double-Diode Solar Cell Model (DDSCM), and Three-Diode Solar Cell Model (TDSCM), in addition to three modified solar cell models, namely, modified SDSCM (MSDSCM), modified DDSCM (MDDSCM), and modified TDSCM (MTDSCM). Moreover, a polynomial equation of five degrees for the sum of squared errors (PE5DSSE) between the measured and calculated currents was used as a new objective function for extracting the parameters of the solar cell models. The proposed objective function delivered improved prediction accuracy than common objective functions. Experimental results revealed the effectiveness of TFWO compared with six counterparts, namely, “Tunicate Swarm Algorithm (TSA), Grey wolf optimizer (GWO), modified particle swarm optimization (MPSO), Cuckoo Search algorithm (CSA), Moth flame optimizer (MFO) and Teaching Learning based optimization algorithm (TLBO),) for all the traditional and modified solar cell models based on the optimal parameters extracted using best PE5DSSE values.


INDEX TERMS
Human activities release excess carbon dioxide and other global warming gases into the atmosphere. Such gases act like a cover and trap heat in the atmosphere, resulting in significant and destructive impacts, including frequent storms and dry spells, rise in sea levels, and termination of animals [1].
Most renewable energy sources emit small to no global warming gases. Renewable energy obtained from biomass can produce a large number of global warming gases depending on the assets and whether the biomass is economically sourced and collected. Expanding the renewable energy supply can allow us to supplant carbon-intensive vitality sources and altogether decrease global warming emissions [2]. Solar energy is renewable. It is abundantly available, free, does not require transportation, and does not contaminate the environment. Sun-powered vitality facilitates a modern way of life for humankind and considers environment and ecological life into vitality preservation to diminish environmental contamination. It is the most common renewable source after wind energy [3]. Moreover, it is a promising vitality source that has expanded to incorporate numerous applications, including sun-powered water warming, sun-powered heating of buildings, sun-powered refining, sun-oriented pumping, sun-powered heaters, sun-oriented cooking, sun-powered electric control, sun-powered warm-control generation, and sun-powered greenhouses [4]. Solar energy is converted into a valuable power source using a photovoltaic (PV) model based on system based on the building block of the solar cell that converts light into electricity directly [5]. Solar PV innovation is considered a significant renewable energy source worldwide [6]. Recently, considerable financial enhancements within the PV control industry have been achieved, thus enabling a clean future for this innovation. PV can influence humans from all walks of life, such as mortgage holder, agriculturist, planner, architect, or electricity user. PV was first used in space programs. Currently, PV frameworks are used to produce power-topump water, illuminate the night sky, actuate switches, charge batteries, and supply electric utility lattice, among others.
An essential aspect of improving the efficiency of PV systems is determining an optimal design for the system. Therefore, a reliable and optimal PV system model must be established to improve the operating characteristics of the overall PV system [7]. The modeling process can be divided into two independent stages. The first stage involves the preparation of the mathematical model of the PV system, and the second stage involves the accurate parameter estimation technique for precise PV cell modeling and the analysis of PV system characteristics. However, the high nonlinearity of the output I-V curve makes the optimal design for the system very difficult. Many PV solar cell models describe the nonlinear performance of the solar PV system, such as The Single Diod PV (SDPV), a Double Diode PV (DDPV), and a Triple-Diode PV (TDPV) [8] The SDPV model is the basic PV solar cell model and is the most popularly used model owing to its simplicity, smaller number of unidentified parameters, and higher accuracy. Therefore, the SDPV model exhibits a straightforward structure with fast dynamic behavior. The DDPV model achieves more accurate modeling of the PV panels than the SDPV model; further, the DDPV model shows losses in the P-N junction's quasineutral and space-charge regions [9].
The DDPV model has seven unknown parameters that must be estimated; therefore, it simulated as a mathematical model. In a study by Kaur et al. [10], the three-diode model is presented as a model with ten parameters that increase the exactness of the estimation process and make the PV cell model suitable for manufacturing applications. The TDPVis considered the best PV solar cell model, although its design is complicated. The consistent modeling of the PV system is a challenging task that typically depends on the formulation of the mathematical simulation model and the exact estimation of the unknown parameters [11].
Different approaches have been used to fine-tune the unknown parameters (maximum ten parameters) of the PV model. In recent literature, the SDPV, DDPV, and TDPV models of the PV modules have been widely investigated because of their unknown parameters determined using analytical techniques and metaheuristic (MH) optimization approaches [12].
Analytical techniques have been used to estimate the PV parameters using different selected points. The analytical approach is based on the derivation of mathematical equations that necessarily provide simple and rapid identification and calculation of the PV parameters. In analytical approaches, the main points of the I-V characteristic curves were used, namely, the point of the short circuit current, open-circuit voltage, and the maximum power. Despite the simplicity and short calculation time, the accuracy of analytical approaches can decrease if one or more critical points of the I-V characteristics are incorrectly determined [13]. Moreover, the analytical approach does not reflect the real operating conditions. Several analytical techniques have been reported in the literature [14].
The Lambert method is a strategy for predicting the obscure parameter values of the one-and two-diode modes of the sun-powered PV cells. This approach is less accurate than the numerical approaches. Numerical calculation strategies generally involve nonlinear calculations, such as the Newton-Raphson method [15], Nelder-Mead simplex strategy [16], conductivity strategy [17], and Levenberg-Marquardt (LM) calculation, to distinguish the parameters of the reenactment models of the PV framework [18]. In this setting, multiple numerical procedures are presented in a study [?]. In any case, notwithstanding the precision of the explanatory approaches, numerous obscure parameters within the numerical strategies complicate the extraction procedure.
A comprehensive survey on MH calculations and related variations [20]- [22] has been performed on the PV cell parameters. These calculations are primarily classified into four categories: biology-, physics-, sociology-, and mathematics-based calculations.
The primary category of MH calculations for PV cell modeling and parameter estimation is biology-based calculations, such as genetic algorithm (GA) and differential evolution (DE). In the same setting, the adaptive GA method can yield higher computation efficiency of parameter VOLUME 9, 2021 estimation [23] than traditional GA methods. In the same context, the improved versions of DE [24],have been proposed to enhance the convergence speed and global search quality, such as artificial bee swarm optimization [25], artificial bee colony algorithm, teaching-learning-based artificial bee colony [26], whale optimization algorithm (WOA) [27], improved WOA [28], and chaotic WOA [29]. In addition to modified WOA, improved antlion optimizer [30], and biogeography-based optimization (BBO) [31], the BBO-M strategy incorporates the mutation strategy of DE into the original migration of BBO [31], to effectively enhance the exploitation capability and overcome the shortcomings of the conventional BBO, which can easily determine an optimum when the PV cell parameter identification is applied using the cuckoo search (CS) [32]. A hybrid version of CS, called biogeography-based heterogeneous CS algorithm, is proposed [33] to improve the accuracy and reliability of various algorithms, such as original CS, bird-mating optimization (BMO) [34], simplified BMO [35], Flower pollination algorithm (FPA) [?], hybrid bee pollinator FPA [36], Grey wolf optimization (GWO) [37], Bacterial foraging algorithm [38], Artificial immune system [39], and Salp swarm algorithm.
Simulated results show the significant superiority of MH algorithms for modeling model the PV cells based on the category of biology-based algorithms. Hence, such strategies can be considered robust and effective tools to solve the identification problem of PV cells.
The second category of MH calculations for PV cell modeling and parameter estimation is the physics-based calculation. This category incorporates calculations that are used for parameter identification, such as particle swarm optimization (PSO) (and its improved models [40],), parallel chaos optimization algorithm (PCOA), modified PCOA (MPCOA) [41], simulated annealing (SA) algorithm [42], hybrid method (LM+SA) [17], firework algorithm [43], wind-driven optimization [44], evaporation rate-based water cycle algorithm (an improved version of water cycle algorithm (WCA)), and improved Lozi map-based chaotic optimization algorithm [45]. All these physics-based algorithms have been used for identifying the parameters of PV cells.
The third category of MH algorithms PV cell modeling and parameter estimation is the sociology-based algorithms. This category includes algorithms that are used for parameter identification, such as harmony search (HS) algorithm, grouping-based global HS [46], (an improved variant of HS), teaching-learning-based optimization (TLBO) algorithm [47], improved and simplified TLBO (STLBO) algorithm [44], imperialist competitive algorithm (ICA) [44] and multiple learning backtracking search algorithm. Compared with GA and PSO, ICA has a higher convergence speed and accuracy and more substantial convergence stability, particularly for low-dimensional optimizations. Based on backtracking search algorithm [48], all sociology-based algorithms and approaches have been used for PV cell parameter identification for SDM and DDM.
The fourth category of MH calculations for PV cell modeling and parameter estimation is the mathematics-based calculation. This category incorporates calculations used for parameter estimation, such as pattern search algorithm [49], shuffled complex evolution algorithm [50] Jaya algorithm, and modified Jaya algorithm.
In summary, the major contributions of this study are as follows: • The polynomial equation of five degrees for the sum of squared errors (PE5DSSE) between the measured and calculated currents is used as a new objective function for extracting parameters of the solar cell models.
• New MH algorithms and turbulent flow of water-based optimization (TFWO) are used for identifying the solar cell parameters.
• The performance of the proposed algorithm is compared with those of other algorithms, such as tunicate swarm algorithm (TSA), GWO, modified particle swarm optimization (MPSO), CS algorithm (CSA), moth flame optimizer (MFO), and TLBO algorithm. • The modified and traditional solar cell models are compared based on the new PE5DSSE value.
• The characteristic curves of SDSCM, DDSCM, TDSCM, MSDSCM, MDDSCM, and MTDSCM are simulated based on the optimal parameters extracted using the best PE5DSSE value from the TFWO algorithm. The remainder of this paper organization is as follows. The analysis of photovoltaic models is presented in Section II. The new objective function is discussed in Section III. An overview of TFWO is shown in Section IV. The experimental results and discussion are presented in Section V. The conclusion of this paper is summarized in Section VI.

II. ANALYSIS OF PHOTOVOLTAIC MODELS
The traditional PV models, such as SDSCM, DDSCM, and TDSCM, and modified PV models (MSDSCM, MDDSCM, and MTDSCM) are analyzed in this section.
A. SINGLE DIODE SOLAR CELL MODEL (SDSCM) Figure 1 shows the equivalent circuit of SDSCM. Based on this circuit, the current generated from SDSCM is determined using the following equation:  where I ; is the current output from SDSCM, I ph is the photo generated current, the shunt current is I sh , I d1 is the diode current, R sh is the shunt resistance, R s is the series resistance, n 1 is the diode ideality factor, K is Boltzmann's constant, q is the charge of electron, T c is the cell temperature.
B. MODIFIED SINGLE DIODE SOLAR CELL MODEL (MSDSCM) Figure 2 shows the equivalent circuit of MSDSCM. Based on this circuit, the current generated from MSDSCM is determined using the following equation: The losses in the quasi-neutral region is expressed by adding the modified series resistance; R sm .
C. DOUBLE DIODE SOLAR CELL MODEL (DDSCM) Figure 3 shows the equivalent circuit of DDSCM. Based on this circuit, the current generated from DDSCM is determined using the following equation: where I d2 is the current in the second diode, n 2 is the ideality factor of the second diode.    Figure 4 shows the equivalent circuit of MDDSCM. Based on this circuit, the current generated from MDDSCM is determined using the following equation.
The losses in the space charge region is expressed by adding the modified series resistance; R sm in the second diode. Figure 5 shows the equivalent circuit of TDSCM. Based on this circuit; the current generated from TDSCM is determined using the following equation:

E. THREE DIODE SOLAR CELL MODEL (TDSCM
where I d3 is the current in the third diode, n 3 is the ideality factor of the third diode.   Figure 6 shows the equivalent circuit of MTDSCM. Based on this circuit, the current generated from MTDSCM is determined using the following equation.
The losses in the defect region is expressed by adding the modified series resistance; R sm in the third diode.

III. OBJECTIVE FUNCTION
To measure the consistency between the measured and simulated data, the PV models (SDSCM, MSDSCM, DDSCM, MDDSCM, TDSCM, and MTDSCM) were evaluated using a new objective function for estimating the parameters of each model. The new objective function is PE5DSSE between the measured and simulated current data. The mathematical equations for PE5DSSE are expressed as follows: (12) where I exp , is the measured current, N is the number data set and X is the variables required to estimate. The decision variable vector for SDSCM is: n. R s and R sh ).
The decision variable vector for DDSCM is: n. R s . R sh . I o2 and n 2 ).
The decision variable vector for TDSCM is: o3 and n 3 ).
The decision variable vector for MSDSCM is X = (I ph . I o1 . n 1 . R s . R sh and R sm ).

IV. TURBULENT FLOW OF WATER-BASED OPTIMIZATION (TFWO)
TFWO [57], is a recent MH algorithm. It is inspired by the whirlpool phenomenon created in a turbulent flow of water. A whirlpool moves in a circular motion along a tight route. The center of the whirlpool is a hole that sucks the particles around it toward the middle and then draws the particles inside the vortex. In the TFWO algorithm, the population is divided into NWh groups and the best member of each group is set in the center of the whirlpool.

a: FORMATION AND EFFECTS OF WHIRLPOOLS
The algorithm divides the initial population (x 0 ) and comprising N p members set equally between the N Wh whirlpool. Moreover, by applying a centripetal force on each whirlpool (Wh), and plunging them into its well, the positions of objects in a specific set (X ) were integrated with its central position. Then, each whirlpool jth (with their local position on Wh j , integrating the X i object position with itself; this implies that ((X i = Wh j ). If this integration is not performed, some deviations ( X i ) will occur for other (Wh) whirlpools owing to the distance (Wh − Wh j ) between them and their objective values (f ()). Therefore, the ith new position of the object will be equal toX i new = Wh j − X i . Moreover, around their whirlpool's center and approach, the motion of the objects (X ) is restricted by the special angle (δ), which changes at each iteration as follows: The angle X i has been calculated depending on the distance of the whirlpools from all the objects with the least and most weights based on Eqs. 14 and 15 respectively. Then, the particle's position is updated from Eq. (16).
where the δ i is the ith object's angle and the whirlpools with the minimum and maximum values of t are Wh f and Wh W , respectively.

A. THE MATHEMATICAL MODEL
This subsection presents an overview of the mathematical steps for the TFWO algorithm as follows: 1) Updating object's position phase: The updating of object's position is summarized in the following two steps: Step 1: From Newton's first law of motion, although the centripetal force FE i pulls the moving objects toward their whirlpool, FEi occasionally overcomes the centripetal force of the whirlpool; therefore, the object randomly moves to a new position. Then, the centrifugal force moves them away from the corresponding center. Moreover, the centrifugal force and action use Eq. (17) and Eq. (18) respectively. Additionally, the mathematical model of the centrifugal force phase is summarized in Step 3.

) Interactions between the whirlpools phase:
The effects of whirlpools on the objects have been modeled. Every whirlpool tends to unite its own position with that of the considered whirlpool, which is similar to the effects of a whirlpool on the surrounding objects. Therefore, the minimum amount of nearest whirlpool is calculated using Eq. (19) based on its objective function. To update the whirlpool position, Eqs. (20) and (21) are defined as follows.
Wh new j = Wh f − Wh j (21) where, value of the j th whirlpool hole's angle is acts by δ.
To summarize the above phenomenon, we use Steps 4 and 5, which illustrate the relation between the whirlpool interactions: Step 4: Step 5:

) The strongest member phase:
Among the new members obtained for the whirlpool, the strongest member is selected for the next iteration according to the least value of the objective function compared with its corresponding whirlpool. To summarize this phase, Step 6 illustrates the selected new strongest whirlpool.

RESULTS OF SOLAR CELL MODELS
The identified parameters of the traditional and improved solar cell models are established in this section using the R.T.C France solar cell. The proposed TFWO algorithm uses the estimated parameters. Several algorithms, such VOLUME 9, 2021  as TSA [51], Grey wolf optimizer (GWO) [52], modified particle swarm optimization (MPSO) algorithm [53], Cuckoo Search algorithm (CSA) [54], Moth flame optimizer (MFO) [55] and Teaching Learning based optimization algorithm (TLBO) [56], are compared with the proposed TFWO algorithm. The algorithms used in this study extract the solar cell parameters of each model based on the new objective function. TFWO and all compared algorithms were evaluated using 30 independent runs (with 1000 iterations in each run) and 30 search agents.

A. RESULTS OF THE TRADITIONAL SOLAR CELL MODELS
The results for the traditional solar cell models, namely, SDSCM, DDSCM, and TDSCM, are discussed in this subsection. The estimated parameters of these models are based on the new objective function. PE5DSSE in terms of the proposed TFWO algorithm and other compared algorithms is discussed. The I-V and P-V curves of the R.T.C France solar cell for SDSCM, DDSCM, and TDSCM are illustrated using the best PE5DSSE value for the proposed TFWO algorithm.

1) SDSCM RESULTS
The parameters extracted from the seven algorithms for SDSCM explain in table: 2. Based on this data the best value of PE5DSSE is 2.5278E-05, that is achieved by the TFWO algorithm, the TLBO algorithm achieve the second best PE5DSSE (2.5308E-05), then CSA, MFO, GWO, TSA and MPSO respectively. Figure 7 explains the I-V and P-V curves for SDSCM at the best value of PE5DSSE from the proposed TFWO. Figure 8 explains the absolute error for current and power curves for SDSCM at the best value of PE5DSSE from the proposed TFWO. Based on these figures; the maximum absolute error for current is 0.00250741232809032, the maximum absolute error for power is 0.00146257361097508.

2) DDSCM RESULTS
The parameters extracted from the seven algorithms for DDSCM explain in table 3. Based on this data the best value of PE5DSSE is 2.51E-05, that is achieved by the TFWO algorithm, the TLBO algorithm achieve the second best PE5DSSE (2.52E-05) then MFO, CSA, GWO, TSA and MPSO respectively. Figure 9 explains the I-V and P-V curves for DDSCM at the best value of PE5DSSE from the proposed TFWO. Figure 10 explains the absolute error for current and power curves for DDSCM at the best value of PE5DSSE from the proposed TFWO. Based on these figures; the maximum absolute error for current is 0.00255209505168791, the maximum absolute error for power is 0.00148863704364956.

3) TDSCM RESULTS
The parameters extracted from the seven algorithms for TDSCM explain in table 4. Based on this data the best value of PE5DSSE is 2.51E-05, that is achieved by the TFWO algorithm, the TLBO algorithm achieve the second best PE5DSSE then MFO, CSA, GWO, TLBO and MPSO respectively. Figure 11 explains the I-V and P-V curves for TDSCM at the best value of PE5DSSE from the proposed TFWO.     proposed TFWO algorithm and other compared algorithms is discussed. The I-V and P-V curves for the R.T.C France solar cell are illustrated for SDSCM, DDSCM, and TDSCM using the best PE5DSSE value by employing the proposed TFWO algorithm.

1) MSDSCM RESULTS
The parameters extracted from the seven algorithms for MSDSCM are presented in Table 5. Based on these data, the best PE5DSSE value is 2.5278E-05, which is achieved using the proposed TFWO algorithm. The TLBO algorithm achieves the second-best PE5DSSE value (2.5298E-05),  followed by CSA, MFO, GWO, TSA, and MPSO, in the given order. Figure 13 presents the I-V and P-V curves for SDSCM using the best PE5DSSE value by employing the proposed TFWO algorithm. Figure 14 shows the absolute error of the current and power curves for SDSCM using the best PE5DSSE value by employing the proposed TFWO algorithm. Based on these figures, the maximum absolute errors of the current and power curves are 0.00250741122972475 and 0.00146257297029845, respectively.

2) MDDSCM RESULTS
The parameters extracted from the seven algorithms for DDSCM explain in table 6. Based on this data the best value of PE5DSSE is 2.51E-05, that is achieved by the TFWO algorithm, the TLBO algorithm achieve the second best PE5DSSE (2.522E-05) then MFO, CSA, GWO, TSA and MPSO respectively. Figure 15 explains the I-V and P-V curves for DDSCM at the best value of PE5DSSE from the proposed TFWO. Figure 16 presents the absolute error of the current and power curves for DDSCM using the best PE5DSSE value by employing the proposed TFWO algorithm. Based on these figures, the maximum absolute errors of the current and power curves are 0.00254956327080924 and 0.00148716025586303, respectively.

3) MTDSCM RESULTS
The parameters extracted from the seven algorithms for MTDSCM explain in table7. Based on these data, the best VOLUME 9, 2021   PE5DSSE value is 2.509E-05, which is achieved using the TFWO algorithm. The TLBO algorithm achieves the second-best PE5DSSE value, followed by MFO, CSA, GWO, TLBO, and MPSO, in the given order. Figure 17 shows the I-V and P-V curves for TDSCM using the best PE5DSSE value by employing the proposed TFWO algorithm. Figure 18 shows the absolute error of the current and power curves for TDSCM using the best PE5DSSE value by employing the proposed TFWO algorithm. Based In this section, we compared the performance of the modified and traditional solar cell models. The accuracy and    reliability of each model are compared. The accuracy of the model is measured using the best PE5DSSE value.
The reliability of the model is measured using the standard  Table 8, those for DDSCM and MDDSCM are shown in Table 9, and those for SDSCM and MSDSCM are presented in Table 10. The results provided in Tables 7, 8, and 9 show that MSDSCM is more accurate and reliable than SDSCM. Furthermore, MDDSCM and MTDSCM are more reliable and accurate than DDSCM and TDSCM, respectively. The proposed TFWO algorithm  achieves the optimal results for the minimum, mean, maximum, and standard deviation values using PE5DSSE compared with other algorithms.
We performed statistical analysis for 30 independent runs for all algorithms. The robustness curves for TFWO, TSA, GWO, CSA, MFO, MPSO, and TLBO are presented in

VI. CONCLUSION AND FUTURE WORK
The PV systems are becoming one of the most popular renewable energy technologies for generating electric power.
Establishing an accurate PV model that emulates the system behavior under different environmental conditions is essential. The challenge of PV cell parameter estimation has garnered the attention of researchers and industrialists and gained immense momentum for the development of PV models. The accuracy of the PV model depends on its identified parameters that are mainly based on the executed optimization technique and employed objective function. Therefore, we use PE5DSSE between the measured and calculated currents. PE5DSSE is used as a new objective function for extracting the parameters of the solar cell models. Herein, we present an efficient MH method, called TFWO, for estimating the parameters of PVs in solar cell systems. Regarding the proposed objective function, the experimental and comparative results show that TFWO with high stability achieves more precise and accurate parameters than other competitor algorithms. The main conclusions of this study are listed: • We propose a new objective function based on polynomial equations to extract the parameters of the solar cell models.
• This function yields the most accurate, precise, and consistent solutions.
• The modified solar cell models, namely, MSDSCM, MDDSCM, and MTDSCM, are more accurate and reliable than the traditional solar cell models, namely, SDSCM, DDSCM, and TDSCM.
• TFWO shows more flexibility and effectiveness performance than the other algorithms.
• The good fit confirms the superior reliability and stability of TFWO.
• The superiority of the TFWO algorithm in comparison with other competitor algorithms is confirmed in terms of the experimental dataset fitting accuracy, convergence rate, stability, and consistency of the results. the future, we aim to improve the TFWO and other MH algorithms for enhanced renewable energy and power systems. Moreover, we will consider the output power prediction of multiple energy systems under the integrated energy system. We also aim to perform research on the optimal control of energy internet.