A memory and update strategy-based social group optimization for unknown parameter identification of photovoltaic modules

PV module model

As shown in Fig. 1, the PV module model comprises several diodes configured in series and/or parallel. The output current I is formulated by Eq. (1) (Abd El-Mageed et al., 2023).

(1) $$I=I_{pd}{N}_p-I_o{N}_p\left[\exp \left(\frac{Q\cdot (V{N}_p+I{R}_s{N}_s)}{a{N}_s{N}_{p}kT}\right)-1\right]-\frac{V{N}_p+E{R}_s{N}_s}{R_{sh}{N}_s}$$where $I_{ph}$ denotes the photo-generated current, $N_P$ represents the count of shunt resistors, $I_o$ indicates the diode reverse saturation current, V signifies the output voltage, $R_s$ refers to the series resistance, $R_{sh}$ is the shunt resistance, $a$ is the diode’s ideal factor, $N_s$ is the count of series resistors, $k=1.380503\times {10}^{-23}J/K$ is the Boltzmann constant, T is the temperature, $Q=1.60217646\times {10}^{-19}C$ is the electron charge.

Single diode model

As shown in Fig. 2, when both $N_s$ and $N_P$ are set to 1 in the PV module model, the single-diode model is derived. The expression for I can be seen in Eq. (2) (Naraharisetti, Devarapalli & Bathina, 2024).

(2) $$I=I_{ph}-I_{sh}-I_d=I_{ph}-I_o\left[\exp \left(\frac{Q(V+I{R}_s)}{akT}\right)-1\right]-\frac{V+I{R}_s}{R_{sh}}$$where $I_d$ denotes the diode current, $I_{sh}$ represents the shunt resistor current. Derived from Eq. (2), five unknown parameters can be identified: $I_{ph}$, $I_o$, $R_s$, $R_{sh}$ and $a$.

Optimization objective

RMSE serves as the optimization objective function, as it more effectively reflects the dispersion degree between actual data and simulated data (Abd El-Mageed et al., 2024). The definition of RMSE is given by Eq. (3).

(3) $$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}(X)=\sqrt{\frac{1}{L}\sum _{i=1}^L{\left(I_i-{\hat{I}}_i\right)}^2}$$where $X=\left[I_{ph},I_o,R_s,R_{sh},a\right]$ denotes the vector of unknown parameters. L represents the amount of measured I–V data that the PV cell manufacturer supplied. ${\hat{I}}_i$ is the simulated current obtained by MUS-SGO. A smaller RMSE signifies less discrepancy between simulated and measured currents, thereby indicating greater accuracy in the identification of unknown parameters.

Improving phase

In the improving phase, each individual seeks to expand their knowledge by learning from the best individual ( $X_{best}$) in the population. The update rule for $i$ is shown in Eq. (4).

(4) $$X_i^{new}=C\cdot X_i+r\cdot \left(X_{best}-X_i\right).$$

If $X_i^{new}$ has a better fitness than $X_i$, accept $X_i^{new}$; otherwise, keep $X_i$. Where C is the self-reflection factor ( $C=0.2$), $r\sim U(0,1)$.

Acquiring phase

In the acquiring phase, each individual gains knowledge not only from the best individual, but also from randomly selected peers. Another individual $j$ ( $j\ne i$) is randomly selected, and the update rule is given by Eq. (5).

(5) $$\{\begin{array}{cc}{X}_i^{new}=X_i+r_1\cdot \left(X_i-X_j\right)+r_2\cdot \left(X_{best}-X_i\right),\hfill & f\left(X_i\right)<f\left(X_j\right)\hfill \\ X_i^{new}=X_i+r_1\cdot \left(X_j-X_i\right)+r_2\cdot \left(X_{best}-X_i\right),\hfill & f\left(X_j\right)<f\left(X_i\right)\hfill \end{array}$$where $r_1,r_2\sim U(0,1)$. If $X_i^{new}$ has a better fitness than $X_i$, $X_i^{new}$ is accepted.

Population initialization phase

In the population initialization phase, randomly generate the decision variables of each individual in the population within the domain of each dimension, as shown in Eq. (14).

(14) $$X_{i,j}=l{b}_j+r\cdot (u{b}_j-l{b}_j),i=1,2,\dots ,NP,j=1,2,\dots ,D$$where $ub$ and $lb$ represent the upper and lower bounds, NP is the population size, D is the problem dimension, $r\sim U(0,1)$.

Generate historical memory repository

To fully utilize historical information, the historical memory repository (M) is defined as an ordered set for storing historical optimal solutions and corresponding objective function values, is shown in Eq. (15).

(15) $$M=\{(X_1,f_1),(X_2,f_2),\dots ,(X_m,f_m)\}$$where $m$ denotes the memory capacity and follows the general configuration rule $m=\lfloor 0.1\times NP\rfloor$. This rule balances two key goals. On the one hand, it allows M to accumulate sufficient historical high-quality information to guide the algorithm towards promising regions and thereby accelerate convergence. On the other hand, it avoids storing too many similar solutions, which would otherwise reduce population diversity.

The M adopts an elite replacement strategy based on the objective function value. When M is not full, newly generated solutions are directly inserted into M. Once M is full, the worst solution in M is replaced only if the new solution is better. This strategy improves the overall quality of M and strengthens the guidance provided by historical information during the search. In addition, to enable differentiated use of memory solutions, the $i$-th solution in M is assigned a weight, and the dynamic memory weight $w_i$ is computed as Eq. (16).

(16) $$w_i={\lambda }^{m-i}$$where $i$ denotes the index of the solution ( $1\le i\le m$) and $\lambda$ is the forgetting factor (set to $\lambda =0.9$ in this work). The factor $\lambda$ controls the rate of weight decay, so that more recent memory solutions receive higher weights and are used more frequently. This design prioritizes recent information and enhances the ability of MUS-SGO to explore new search directions.

Improving phase

In the improving phase, each individual actively learns from the best individual ( $X_{best}$) in the current population based on their own current experience, thereby achieving self-improvement. The updated rule is shown in Eq. (17).

(17) $$X_i^{new}=C\cdot X_i+r\cdot \left(X_{best}-X_i\right)$$where C is self-reflection factor (set to $C=0.2$ in this work), $r\sim U(0,1)$.

After completing the update, if $X_i^{new}$ has a better fitness than $X_i$, replace $X_i$, and update M according to the elite replacement strategy to accumulate high-quality search information.

Acquiring phase

In the acquiring phase, each individual performs knowledge transfer and experiential learning through multidimensional interactions. Each individual actively exchanges information with the current best individual and also randomly selects other individuals in the population to exchange knowledge. When an individual finds that another individual holds superior knowledge, it adopts this information to improve its own state. On this basis, a dynamic memory-guided strategy is introduced, which allows individuals to combine the weighted high-quality information stored in M to further correct and enhance their positions.

Randomly chosen an individual $j$ ( $j\ne i$) from the population, and chosen the best memory solution $M^{\ast }$ from M. The update rule is shown in Eq. (18).

(18) $$\{\begin{array}{cc}{X}_i^{new}=X_i+r_1\cdot (X_i-X_j)+w_i\cdot r_2\cdot (M^{\ast }-X_i)+(1-w_i)\cdot r_2\cdot (X_{best}-X_i),\hfill & f(X_i)<f(X_j)\hfill \\ X_i^{new}=X_i+r_1\cdot (X_j-X_i)+w_i\cdot r_2\cdot (M^{\ast }-X_i)+(1-w_i)\cdot r_2\cdot (X_{best}-X_i),\hfill & f(X_j)<f(X_i)\hfill \end{array}$$where $w$ denotes the dynamic memory weight, $r_1,r_2\sim U(0,1)$.

After completing the update, if $X_i^{new}$ outperforms $X_i$, accept $X_i$, and update M.

Adaptive population update strategy

To dynamically maintain population diversity and accelerate convergence, an adaptive population update strategy is introduced. This strategy uses an exponential decay mechanism to eliminate low-fitness individuals and introduce new individuals in different proportions at different stages of the algorithm. In the early stage, the strategy favors global exploration and encourages the population to explore a broad solution space. In the later stage, it gradually shifts the search toward local exploitation around already identified high-quality solutions. Population update operations are triggered when the condition in Eq. (19) is satisfied.

(19) $$\mathrm{m}\mathrm{o}\mathrm{d}(T_{iter},t)=0$$where $T_{iter}$ is the current number of iterations, $t$ is the fixed elimination interval (set to $t=10$ in this work). That is, the population update is performed after every 10 iterations.

During each update, the fitness of the current population individuals is sorted from highest to lowest, and the worst-performing individuals are selected for elimination. The specific number of individuals ( $n$) to be eliminated is dynamically controlled by an exponential decay mechanism, as shown in Eqs. (20), (21).

(20) $$n=max(1,\lfloor R(t)\cdot NP\rfloor )$$

(21) $$\{\begin{array}{c}R(t)=R_0\cdot e^{-\alpha p}\hfill \\ p=\frac{NFE}{Max\mathrm{\_}NFE}\hfill \end{array}$$where $R(t)$ denotes the current elimination ratio, $R_0$ is the initial elimination ratio (set to $R_0=0.3$), and $\alpha$ is the attenuation coefficient (set to $\alpha =0.5$). $Max\mathrm{\_}NFE$ is the maximum number of function evaluations and NFE is the current number of evaluations. The operator $\lfloor \cdot \rfloor$ denotes the floor function and ensures that at least one individual is eliminated in each update. The motivation for and selection of these hyperparameters are further discussed in the “Experimental Setting” subsection.

The eliminated individuals are replaced by newly generated ones, which are produced using the same strategy as in the initialization phase. After evaluation, these new individuals are also used to update the historical memory repository M, so as to maintain the integrity and timeliness of the stored information.

Results of poly-crystalline KC200GT at distinct irradiance

Table 4 presents the parameters of the poly-crystalline KC200GT identified by different algorithms at various irradiance levels when $T={25}^{\circ }\mathrm{C}$. From the parameter identification results, it can be seen that the RMSE values of MUS-SGO are 0.0014, 0.0014, 0.0012, 0.0016 and 0.0015 at irradiance levels of $G=200$, $400$, $600$, $800$ and $\mathrm{1,000}\mathrm{W}/{\mathrm{m}}^2$, respectively, which are significantly lower than those of the other algorithms. It can also be observed that $I_{ph}$ increases as irradiance increases. In contrast, the remaining parameters such as $R_s$ and $R_{sh}$ vary only slightly with changes in irradiance, indicating that these parameters are less affected by irradiance variations.

To validate the accuracy of parameter identification by MUS-SGO, the simulated current values are calculated using Eq. (1). Table 5 reports the coefficients of determination ( $r^2$) between the simulated and measured currents. As shown in Table 5, MUS-SGO achieves consistently high $r^2$ values. For clearer illustration, Fig. 4 depicts the fitting curves of simulated and measured currents, and it can be seen that the simulated currents obtained by MUS-SGO closely match the measured data. These results verify the high accuracy of MUS-SGO in identifying the parameters of the poly-crystalline KC200GT under varying irradiance levels.

To further validate the superiority of MUS-SGO, Table 6 presents the statistical results derived from 30 independent runs, including the average (Mean), minimum (Min), maximum (Max), standard deviation (Std), and the Wilcoxon signed ranks test at a significance level of $\alpha =0.05$. In the Wilcoxon test, $R^{+}$ and $R^{-}$ denote the sums of positive and negative ranks, corresponding to the cases where MUS-SGO performs better or worse than its comparator, respectively; A p-value smaller than 0.05 indicates that MUS-SGO outperforms the corresponding algorithm in a statistically significant sense, which is marked with an asterisk (*).

Across all tested irradiance levels, MUS-SGO consistently outperforms the other algorithms in every statistical metric. Its Mean, Min, and Max RMSE values are identical at each irradiance level, indicating exceptional stability, whereas the other algorithms exhibit markedly larger fluctuations in RMSE across all metrics. In addition, in the Wilcoxon signed-ranks test the sum of positive ranks $R^{+}$ for MUS-SGO is much larger than the sum of negative ranks $R^{-}$, showing that its performance gains over the comparators are statistically significant. Taken together, these results confirm that MUS-SGO is superior to the competing algorithms in both accuracy and stability for identifying the parameters of the poly-crystalline KC200GT under varying irradiance.

Results of poly-crystalline KC200GT at distinct temperature

Table 7 presents the parameter identification results of different algorithms for the poly-crystalline KC200GT at varying temperatures with $G=\mathrm{1,000}\mathrm{W}/{\mathrm{m}}^2$. At $T={25}^{\circ }\mathrm{C}$, MUS-SGO achieves the smallest RMSE of 0.0015. The RMSE values of the other algorithms are markedly higher: 0.0390 for EJADE, 0.0572 for SATLBO, 0.0701 for TLABC, 0.0418 for PGJAYA, 0.0636 for L-SHADE, and 0.0789 for ATLDE. At $T={50}^{\circ }\mathrm{C}$, MUS-SGO again attains the minimum RMSE of 0.0027. At $T={75}^{\circ }\mathrm{C}$, MUS-SGO still achieves the smallest RMSE value of 0.0044. These results indicate that, across all three temperatures, MUS-SGO consistently demonstrates superior accuracy in identifying the parameters of the polycrystalline KC200GT compared with the other algorithms.

In addition, Table 8 reports the $r^2$ values of the different algorithms. At $T={75}^{\circ }\mathrm{C}$, MUS-SGO, PGJAYA, and EJADE achieve relatively high $r^2$ values, while at $T={25}^{\circ }\mathrm{C}$ and $T={50}^{\circ }\mathrm{C}$, MUS-SGO attains the highest $r^2$ among all algorithms. Figure 5 depicts the fitting curves between the measured and simulated currents generated by MUS-SGO for the poly-crystalline KC200GT module at various temperatures. It can be clearly observed that the simulated currents match the measured currents very well, which further demonstrates that the parameter values identified by MUS-SGO at different temperatures are highly accurate.

Table 9 presents the statistical RMSE results of the different algorithms at various temperatures. From these metrics, MUS-SGO outperforms the other algorithms across all temperatures, and the Wilcoxon signed ranks test yields $p$-values below 0.05 for all pairwise comparisons with MUS-SGO, indicating that its advantages are statistically significant. In contrast, the other algorithms exhibit larger variability in their RMSE values. This consistently superior performance of MUS-SGO in terms of RMSE confirms its advantage in accuracy over competing algorithms for parameter identification of the poly-crystalline KC200GT under varying temperatures.

From an error-analysis perspective, the very small RMSE values and $r^2$ values close to 1.0 reported in Tables 4–9, together with the almost perfect overlap between the measured and simulated I–V curves in Figs. 4, 5, indicate that the residuals between measured and simulated currents are globally small and do not show a clear systematic bias along the I–V curve. In addition, by comparing the parameter estimates obtained by different algorithms in Tables 4–9, it can be observed that lower RMSE values are generally accompanied by more consistent estimates of $a$ and $R_s$, whereas larger variations in $R_{sh}$ have a comparatively weaker impact on the objective values. This suggests that, for the poly-crystalline KC200GT module, the overall residuals are more sensitive to $a$ and $R_s$ than to $R_{sh}$.

Results of mono-crystalline SM55 at distinct irradiance

Table 10 summarizes the parameter values of the mono-crystalline SM55 identified by different algorithms at various irradiance levels when $T={25}^{\circ }\mathrm{C}$. For MUS-SGO, the corresponding RMSE values are 0.0003 at $G=200\mathrm{W}/{\mathrm{m}}^2$, 0.0007 at $G=400\mathrm{W}/{\mathrm{m}}^2$, 0.0008 at $G=600\mathrm{W}/{\mathrm{m}}^2$, 0.0006 at $G=800\mathrm{W}/{\mathrm{m}}^2$, and 0.0011 at $G=\mathrm{1,000}\mathrm{W}/{\mathrm{m}}^2$. These results indicate that, among all compared algorithms, only MUS-SGO consistently provides the smallest RMSE across all irradiance levels.

Table 11 reports the $r^2$ values of all algorithms, and Fig. 6 shows the fitting curves obtained by MUS-SGO. As seen from Table 11, MUS-SGO generally attains the highest $r^2$ values at all irradiance levels. In addition, Fig. 6 indicates that the simulated currents generated by MUS-SGO closely match the measured currents. These results confirm that MUS-SGO can accurately identify the parameters of the mono-crystalline SM55 under different irradiance levels.

The statistical results obtained by all algorithms for the mono-crystalline SM55 at different irradiance levels are summarized in Table 12. As can be seen, MUS-SGO performs best in terms of the minimum, mean, and maximum RMSE, as well as the standard deviation. The Wilcoxon signed ranks test also rejects the null hypothesis of equal performance in favour of MUS-SGO at all irradiance levels. It is worth noting that the standard deviation of MUS-SGO is much smaller than those of the other algorithms, indicating more stable optimization performance.