An efficient Equilibrium Optimizer for parameters identification of photovoltaic modules

Single diode model

In the SDM, five unknown parameters must be obtained. These parameters are the photon current (I_pv), reverse saturation current of the diode (I_o), the ideality factor (a), and the series and parallel resistances (R_s and R_p). Figure 2 presents the SDM of a PV cell with N_C series-connected cells.

Therefore, the I−V characteristics of the SDM of the PV are represented as:

(4) $$I=I_{PV}-I_O\left[exp{\displaystyle \frac{q(V+R_{S}I)}{aK{N}_{C}T}-1}\right]-{\displaystyle \frac{V+R_{S}I}{R_S}}$$

At the point of the open circuit, we find that (V = V_OC) and (I = 0) (opinion); thereafter, from Eq. (4), we obtain;

(5) $$0=I_{PV}-I_O[exp({\displaystyle \frac{q{V}_{OC}}{aK{N}_{C}T})-1}]-{\displaystyle \frac{V_{OC}}{R_P}}$$

(6) $$\text{Thus},I_{PV}=I_O[exp({\displaystyle \frac{q{V}_{OC}}{aK{N}_{C}T})-1}]-{\displaystyle \frac{V_{OC}}{R_P}}$$

At the point of the short circuit (I = I_SC) and (V = 0) (opinion); thereafter, from Eq. (4) we obtain;

(7) $$I_{SC}=I_{PV}-I_O[exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{aK{N}_{C}T}}\right)-1]-{\displaystyle \frac{R_S{I}_{SC}}{R_P}}$$

(8) $$Therefore,I_{PV}=I_{SC}+I_O[exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{aK{N}_{C}T}}\right)-1]-{\displaystyle \frac{R_S{I}_{SC}}{R_P}}$$

From Eqs. (6) and (8) we have;

(9) $$I_O={\displaystyle \frac{I_{SC}+{\displaystyle \frac{R_S{I}_{SC}}{R_P}-{\displaystyle \frac{V_{OC}}{R_P}}}}{exp\left({\displaystyle \frac{q{V}_{OC}}{aK{N}_{C}T}}\right)-exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{aK{N}_{C}T}}\right)}}$$

Substituting Eq. (9) into Eq. (6) yields;

(10) $$I_{PV}={\displaystyle \frac{(I_{SC}+{\displaystyle \frac{R_s{I}_{SC}}{R_P}-{\displaystyle \frac{V_{OC}}{R_P}}})\left[exp\left({\displaystyle \frac{q{V}_{OC}}{aK{N}_{C}T}-1}\right)\right]}{exp\left({\displaystyle \frac{q{V}_{OC}}{aK{N}_{C}T}}\right)-exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{aK{N}_{C}T}}\right)}}$$

At the maximum power point (V = V_MPP) and (I = I_MPP) (say), from Eq. (4) we have;

(11) $$I_{MPP}=I_{PV}-I_O\left[exp{\displaystyle \frac{q(V_{MPP}+R_S{I}_{MPP})}{aK{N}_{C}T}-1}\right]-{\displaystyle \frac{V_{MPP}+R_s{I}_{MPP}}{R_P}}$$

Double-diode model

In the DDM in Fig. 3, seven unknown parameters must be obtained. These parameters are the photon current (I_pv), reverse saturation current of the first diode (I_O1), reverse saturation current of second diode (I_O2) , ideality factors of the first and second diodes (a₁) and a₂, and the PV cell series and parallel resistances (R_S and R_P). Therefore, the I-V characteristics of the DDM are as follows:

(12) $$I=I_{PV}-I_{D1}-I_{D2}-{\displaystyle \frac{V+R_{s}I}{R_P}}$$

(13) $$i.e.,I=I_{PV}-I_{O1}\left[exp{\displaystyle \frac{q(V+R_{S}I)}{a_{1}K{N}_{C}T}-1}\right]-I_{O2}\left[exp{\displaystyle \frac{q(V+R_{S}I)}{a_{2}K{N}_{C}T}-1}\right]-{\displaystyle \frac{V+R_{S}I}{R_P}}$$

At the point of the open circuit, I = 0 and V = V_OC; thus, from Eq. (13) we obtain;

(14) $$0=I_{PV}-I_{O1}\left[exp{\displaystyle \frac{q(V+R_{S}I)}{a_{1}K{N}_{C}T}-1}\right]-I_{O2}\left[exp{\displaystyle \frac{q(V+R_{S}I)}{a_{2}K{N}_{C}T}-1}\right]-{\displaystyle \frac{V_{OC}}{R_S}}$$

(15) $$Therefore,I_{PV}=O1\left[exp{\displaystyle \frac{q(V+R_{S}I)}{a_{1}K{N}_{C}T}-1}\right]-I_{O2}\left[exp{\displaystyle \frac{q(V+R_{S}I)}{a_{2}K{N}_{C}T}-1}\right]-{\displaystyle \frac{V_{OC}}{R_S}}$$

At the point of the short circuit, V = 0 and I = I_SC; thus, from Eq. (13) we obtain;

(16) $$I_{SC}=I_{PV}-I_{O1}[exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{1}k{N}_{C}T}}\right)-1]-I_{O2}[exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{2}k{N}_{C}T}}\right)-1]-{\displaystyle \frac{R_S{I}_{SC}}{R_P}}$$

(17) $$\begin{array}{l}Therefore,I_{PV}=I-SC+I_{O1}\left[exp\left(\frac{q{R}_S{I}_{SC}}{a_{1}k{N}_{C}T}\right)-1\right]\\ \begin{array}{ll}& +I_{O2}\left[exp\left(\frac{q{R}_S{I}_{SC}}{a_{2}k{N}_{C}T}\right)-1\right]+\frac{R_S{I}_{SC}}{R_P}\end{array}\end{array}$$

From Eqs. (15) and (17) we have;

(18) $$I_{O2}={\displaystyle \frac{I_{SC}+{\displaystyle \frac{R_S{I}_{SC}}{R_P}-{\displaystyle \frac{V_{OC}}{R_P}-I_{O1}[exp\left({\displaystyle \frac{q{V}_{OC}}{a_{1}k{N}_{C}T}}\right)-exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{1}k{N}_{C}T}}\right)]}}}{exp\left({\displaystyle \frac{q{V}_{OC}}{a_{2}k{N}_{C}T}}\right)-exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{2}k{N}_{C}T}}\right)}}$$

Substituting Eq. (18) into Eq. (15) yields;

(19) $$I_{PV}=I_{O1}[exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{1}k{N}_{C}T}}\right)-1]+{\displaystyle \frac{I_{SC}+{\displaystyle \frac{R_S{I}_{SC}}{R_P}-{\displaystyle \frac{V_{OC}}{R_P}-I_{O1}[exp\left({\displaystyle \frac{q{V}_{OC}}{a_{1}k{N}_{C}T}}\right)-exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{1}k{N}_{C}T}}\right)]}}}{[exp({\displaystyle \frac{q{V}_{OC}}{a_{2}k{N}_{C}T})-exp\left({\displaystyle \frac{q{R}_S{I}_{SC}}{a_{2}k{N}_{C}T}}\right)}][exp({\displaystyle \frac{q{V}_{OC}}{a_{2}k{N}_{C}T})-1{]}^{-}1}}+{\displaystyle \frac{V_{OC}}{R_P}}}$$

At the maximum power point, V = V_MPP and I = I_MPP (say); then, from Eq. (13) we obtain;

(20) $$I_{MPP}=I_{PV}-I_{O1}\left[exp{\displaystyle \frac{q(V_{MPP}+R_S{I}_{MPP})}{a_{1}k{N}_{C}T}-1}\right]-I_{O2}\left[exp{\displaystyle \frac{q(V_{MPP}+R_S{I}_{MPP})}{a_{2}k{N}_{C}T}-1}\right]-{\displaystyle \frac{V_{MPP}+R_S{I}_{MPP}}{R_P}}$$

Initialization and evaluation of functions

Like many metaheuristic algorithms, the EO uses the initial population for beginning the optimization process. The processing in this method is divided into two phases: the first phase is the “teacher phase,” which means learning from the teacher, and the second phase is the “learner phase,” which means learning by interacting with other learners. These phases are consequently iterated to produce better results until convergence is achieved. In EO, each particle (solution) with its concentration (position) acts as a search agent. Some solutions and their dimensions experience mutation, defined by a function, and selected by a parameter such as mutation probability and percentage The initial concentration is structured according to the particle number and dimensions, with normal initialization appearing as follows:

(21) $$C_i^{initial}=C_{min}+ran{d}_i(C_{max}-C_{min})i=1,2,.....,n$$

$C_i^{initial}$ is the initial condensation of the ith particle, C_max and C_min are the maximum and minimum of the dimensions, respectively, and rand_i is in the domain [0, 1] so that n is the particle number representing a population. Particles are examined and thereafter classified as candidates.

Equilibrium pool and candidates C_eq

The equilibrium case takes into consideration the last case of convergence of the algorithm, which should be the global optimum. At the beginning of the optimization algorithm, the equilibrium state is unknown, and it is only determined by the candidate that the equilibrium data creates a research model for particles. Depending on various factors, these candidates are identified, four particles specific to the optimization, and the condensation of others particle is the arithmetic mean of those. The number of candidates is random and depends on the nature of the problem. For example, the GWO uses the three best candidates (β, γ and α wolves) to change place with other wolves. More than four candidates can positively impact these five particles and are used as candidates for balancing to build a vector that is the equilibrium set in Eq. (22):

(22) $${\overrightarrow{C}}_{eq,pool}=\{{\overrightarrow{C}}_{eq(1)},{\overrightarrow{C}}_{eq(2)},{\overrightarrow{C}}_{eq(3)},{\overrightarrow{C}}_{eq(ave)}\}$$

Every particle at every repetition changes its condensation to select randomly from the candidate’s selection with the same prospect.

Exponential term (F)

The following exponential expression (F) contributes to the rule of changing condensation. A precise introduction of this expression leads to the balance of exploration and exploitation. The fluctuation range can change over time in an actual dominance volume, and should be a vector in the domain of [0, 1] as in Eq. (23):

(23) $$\overrightarrow{F}=e^{-\overrightarrow{\lambda }(t-t_o)}$$

Thus, time, t, known as the task of iteration (Iter), reduces the repetition number of Eq. (24):

(24) $$t=(1-{\displaystyle \frac{Iter}{Ma{x}_{-}iter}{)}^{(a_2{\displaystyle \frac{Iter}{Ma{x}_{-}iter})}}}$$where, Max_iter and Iter represent the maximum number of iterations and the current, respectively, and a2 is a constant utilized to administer the capacity of use. To warranty concourse and improve the exploration and exploitation capacity of the EO, this is given as;

(25) $${\overrightarrow{t}}_o={\displaystyle \frac{1}{\overrightarrow{\lambda }}\ln (-a_{1}sign(\overrightarrow{r}-0.5)[1-e^{-\overrightarrow{\lambda }t}])+t}$$

Here, a₁ is a constant that controls the exploration capacity, while a₂ is a constant that controls the exploitation ability. Furthermore, r is a vector in the range [0–1] selected randomly. Sign(r − 0.5) impacts the direction of exploration and exploitation. a₁ and a₂ are two and one, respectively, for all problems. The derived form of Eq. (23) with the substitution Eq. (25) is represented in Eq. (26) as follows:

(26) $$\overrightarrow{F}=a_{1}sign(\overrightarrow{r}-0.5)[e^{-\overrightarrow{\lambda }t}-1]$$

Generation rate (G)

G is another key expression of the proposed technique for providing the precise solution by enhancing exploitation. For example, a multipurpose model that characterizes G as the top-order exponential decomposition operation is as follows:

(27) $$\overrightarrow{G}={\overrightarrow{G}}_o{e}^{-\overrightarrow{K}(t-t_o)}$$where, G_o is the initial value and k represents the attenuation constant. For restricting the random variables, suppose that K = λ, and utilize the formerly derived exponential expression. The conclusive set of G is as follows:

(28) $$\overrightarrow{G}={\overrightarrow{G}}_o{e}^{-\overrightarrow{\lambda }(t-t_o)}={\overrightarrow{G}}_o\overrightarrow{F}$$

(29) $$where:{\overrightarrow{G}}_o=\overrightarrow{G}CP({\overrightarrow{C}}_{eq}-\overrightarrow{\lambda }\overrightarrow{C})$$

(30) $$\overrightarrow{G}CP={\{}_0^{0.5r_1}{}_{r_2<GP}^{r_2\ge GP}$$Here, GCP is the control parameter for the generation rate that contains the potential of the generation expression assistance for the update procedure. In addition, r₁ and r₂ are random vectors in [0, 1]. Finally, the EO update rule is as in Eq. (31):

(31) $$\overrightarrow{C}={\overrightarrow{C}}_{eq}+(\overrightarrow{C}-{\overrightarrow{C}}_{eq}).\overrightarrow{F}+{\displaystyle \frac{\overrightarrow{G}}{\overrightarrow{\lambda }V}(1-\overrightarrow{F})}$$where, F is presented in Eq. (26), and V is a unit. To summarize all the aforementioned steps, a framework for EO is drawn in Fig. 4. It displays a conceptual sketch of the cooperation of all candidates for balance in a set of sample particles and how they sequentially affect the concentration update in the proposed algorithm. Since the topological positions of the equilibrium candidates are different in the initial iteration, and the exponential term generates large random numbers, this step-by-step update process helps particles that cover the entire area of the research. In the latter case, the opposite scenario occurs iterations when candidates circle the sweet stop with similar attitudes. To this sometimes the exponential term generates small random numbers, which helps in refining the solutions. Providing smaller step sizes. This concept can also be extended to higher dimensions such as hyperspace in which the concentration is updated with the movement of the particle in n-dimensional space.

The EO algorithm has been tested for three known engineering design problems. A simple method of constraint management, the static penalty, is used here to punish the objective function with high score when constraints are violated at any level from their defined limits. The penalty coefficient must be large enough to correctly punish the objective function under the conditions of equality/inequality. Note that all technical test problems ran with the same number of iterations (500) and particles (30) according to the math functions described above.

Opposition based learning (OBL)

The basic concept of Opposition-Based Learning (OBL) was originally introduced by Tizhoosh (2005). The basic aim of this algorithm is optimization, to find the best candidate solution, the simultaneous consideration of an estimate and its corresponding opposite estimate which is closer to the global optimum. This has been achieved in a very short time and it has been used in various areas of soft computing. Opposition-based Learning (OBL), which has been used for improving convergence of the metaheuristic techniques for finding the global solution of the optimization problem. OBL theory has been applied to solve many real-world problems such as optimization and estimation of photovoltaic parameters. In general, any metaheuristic algorithm begins with creating an initial population as an attempt for finding the optimum solution(s), these initial solutions are generated randomly or based on a specific search range.

Numerical data

This study examined three types of SCs: polycrystalline, thin-film, and monocrystalline. Exemplary data supplied by the industries are displayed in Table 3. This table illustrates which current and voltage data at the three main points are explicitly specified at standard test conditions, such as irradiance of 1,000 w/m2 and temperature of $25$ °C. Additional details of the parameters measured by the aforementioned equations are listed in Table 4.

Results of the single diode model

Every group of the optimal variables was estimated by several runs led to zero error at the three main I–V points. There were three types of SCs: Polycrystalline cell Kyocera KC200GT, monocrystalline cell Shell SQ85, and thin film ST40. The run of each type is presented in this subsection. Table 5 presents the optimal parameters of the SDM for the polycrystalline cell Kyocera KC200GT using the IEO. Table 5 also presents the accuracy of the IEO in generating the PV parameters. The results of the best run of the monocrystalline PV cell (Shell SQ85) are presented in Table 6. Table 7 presents the results obtained by the run of the thin-film PV cell (Shell ST40) using the IEO.

Tables 5–7 reveal a small difference in the values of series resistance R_S for both multicrystalline and monocrystalline. The I–V curve is strongly affected by the series resistance. According to the results, lower values of Rs lead the I–V characteristics to move farther from the axis with respect to the maximum point, while higher values of Rs lead the I–V characteristics to move closer to the axis as in Fig. 5. This tendency occurs naturally with the IEO parameter values.

The difference between I–V characteristics can be attributed not only to different values of R_S and R_P, but also to ideality factor of the diode, a. It is demonstrated that the values of all variables change significantly. However, every optimal solution can arrive at zero error for the three major points. All prior studies displayed one I–V characteristic that passed across the three main points. However, a PV cell characteristic was observed that is not normal when the designer concentrated on the three major points. For each cell, there is an operating point on the characteristic in which the performance is greatest. These points are referred as the major points. The algorithm arrived at the global optimal solution after 0.10 s with an error of zero value. This algorithm took less time to run compared to other metaheuristic algorithms.

A scatter plot of the optimal values (30 points) of these three parameters is provided in Figs. 6–8. It is showing that the value of each parameter changes significantly. This demonstrates that under the same parameter values $I_{PV},I_O,a,R_S\mathrm{a}\mathrm{n}\mathrm{d}{R}_P$, the IEO algorithm has the highest efficiency for the experimental I–V data for every module.

As displayed in the previous figures, every curve passes through the three known points. A plot of the optimal solution is presented in Fig. 9. A difference in (a) is present approximately between 0.2 and 1.7, while that of (R_p) is approximately between 80 and 175 ohms.

Results of the double diode model

Using the (same parameter values) I_ph, I_o1, I_o2, a₁, a₂, R_S, and R_P, IEO is more consistent with the experimental I–V data of all types of modules. The estimation of the parameters of SCs based on the DDM has high precision, and the optimization of parameters is implemented for the DDM of different types. A total of five parameters are optimized for the DDM (a₁, a₂, R_S, R_P, I_O1), while two sustainable parameters are calculated (IO2, IPV) for their relation with the optimized parameters. The optimized parameters in various runs of the IEO algorithm for the DDM polycrystalline, KC200GT cell are listed in Table 8, SQ85 monocrystalline dish are listed in Table 9 and thin cells, shell ST40 are displayed in Table 10. For the SDM, the optimized parameters resulted in zero error at the three main points. The 30 I–V characteristics resulting from various combinations of the variables are presented in Fig. 9. A plot of the optimal solution is presented in Fig. 10. All the curves passed through the open circuit, short circuit, and maximum power points. To easily analyze the optimized variables, standardized parameter values are presented in a parallel coordinate graph in Fig. 11. The value of R_P differs widely. However, I_O1 ranges from approximately 0.01 to 0.9p.u. The final value of the diode saturation current is very low, with an average of only $\mu A$

To summarize, regardless of the type of the PV cell or the PV parameter extraction model used, we cannot obtain one I–V characteristic with details of short circuits, maximum power, and open circuit points. A total of three point optimizations can reveal various characteristics in several tests. ‘Simulation Results and Analysis’ confirms the effectiveness of the IEO algorithm in estimating the parameters of the SC, taking into account the SDM and DDM. In addition, the SDM used three types of PV modules to estimate the parameters of the PV model.

The single diode model

This subsection presents a comparison between the solutions generated by the IEO algorithm and other algorithms that have been applied under the same conditions. The optimal error values obtained by the IEO was zero, as presented in Table 11. The IEO achieves the best results among all the six algorithms tested, followed by SCA, while the GA achieves the poorest results.

The double diode model

This subsection provides a comparison between the solutions generated by the IEO algorithm and other algorithms for the DDM. Table 12 indicates that the best error values obtained by the IEO were zero. The IEO thus achieved the best results of the six algorithms tested, followed by SCA. Whereas, GA achieved the poorest results.

To sum up the obtained results, Tables 11 and 12 indicate that PSO produces better results than GA. In addition, GWO generates fewer errors than PSO, GA, WOA, SCA, and MVO. However, the IEO results are superior to those of all other approaches. Fig. 12 presents a comparison between the optimal results for the SDM of the KC200GT cell using different methods.