# Hybrid Fuzzy-DDPG Approach for Efficient MPPT in Partially Shaded Photovoltaic Panels
## 2 Materials and Methods
### Reinforcement Learning Fuzzy-DDPG
\begin{table}
\begin{tabular}{c c c} \hline
**Method** & **\(\boldsymbol{\eta}\) (Efficiency)** & **\(\boldsymbol{\sigma}\) (Oscillations)** \\ \hline fuzzy-DDPG & \(0.95\pm 0.02\) & \(0.01\pm 0.005\) \\ \hline P\_O & \(0.85\pm 0.05\) & \(0.05\pm 0.01\) \\ \hline IC & \(0.88\pm 0.03\) & \(0.03\pm 0.007\) \\ \hline \end{tabular}
\end{table} 
Table 4: Approximate statistics of generated data for MPPT methods. * Sample Mean \[\overline{x}_{m}=\frac{1}{n}\sum_{k=1}^{n}x_{m,k}\] (42) where \(n=270\) is the total number of observations per method. Although the notation \(x_{m,k}\) might be interpreted as a multivariate observation, in our analysis, each \(x_{m,k}\) represents a unidimensional measurement for method \(m\) at the \(k\)-th trial, where \(n\) is the total number of observations per method
* Sample Standard Deviation \[s_{m}=\sqrt{\frac{1}{n-1}\sum_{k=1}^{n}(x_{m,k}-\overline{x}_{m})^{2}}\] (43) where \(n\) is the number of observations for method \(m\), \(x_{m,k}\) represents the \(k\)th univariate measurement, and \(\overline{x}_{m}\) is the sample mean of the method \(m\). * 95% Confidence Interval \[\text{CI}_{95\%}(\overline{x}_{m})=\overline{x}_{m}\pm 1.96\ \frac{s_{m}}{\sqrt{n}}\] (44) Such intervals measure the precision of the estimated mean. Notably, if the CIs of two methods do not overlap, we infer a significant difference in means at the 5% level. Given that each method was evaluated using \(n=270\) independent trials, we invoke the Central Limit Theorem (CLT), which states that the sampling distribution of the sample mean approximates a normal distribution for sufficiently large \(n\), regardless of the underlying distribution. Consequently, the standard normal distribution was used to compute the 95% confidence intervals. The critical value \(z=1.96\) corresponds to the two-tailed 95% confidence level under the standard normal distribution. While Student's \(t\)-distribution is more appropriate for small sample sizes (\(n<30\)), for \(n=270\), the difference between \(z\) and the \(t\) critical value is negligible. Therefore, the use of \(z=1.96\) is statistically justified in this context. The boxplots in Figure 3 are constructed for each method to illustrate the distribution of \(\eta\) and \(\sigma\). The interquartile range (IQR) is encompassed by the box, the horizontal line denotes the median, and outliers beyond \(1.5\times\text{IQR}\) are marked with a \(+\). This visual layout provides an immediate sense of central tendency and variability, highlighting that Fuzzy-DDPG typically clusters near higher efficiency with lower spread in oscillations compared to the other two methods. The results summarized in Table 4 align with the statistical distributions observed in the boxplot comparisons shown in Figure 3. The proposed method, Fuzzy-DDPG, demonstrates the highest efficiency (\(\eta\sim 0.95\pm 0.02\)) with minimal oscillations (\(\sigma\sim 0.01\pm 0.005\)), as evident from its tightly grouped boxplot distributions in both subplots. In contrast, P_0 exhibits significantly wider variation in efficiency (\(\eta\sim 0.85\pm 0.05\)) and higher oscillations (\(\sigma\sim 0.05\pm 0.01\)), as indicated by the broader box and whisker ranges. Similarly, IC achieves intermediate performance with \(\eta\sim 0.88\pm 0.03\) and \(\sigma\sim 0.03\pm 0.007\), reflecting a compromise between efficiency and stability. The non-overlapping CIs between Fuzzy-DDPG and the other two methods confirm that the observed differences in both efficiency and oscillation are statistically significant at the 5% level. The following set of simulations demonstrates the performance of the Fuzzy-DDPG algorithm in comparison to conventional methods, including Perturbation and Observation (P&O) and Incremental Conductance (IC), under various irradiance and temperature scenarios designed to reflect realistic and challenging operating conditions for two PV panels in series. * Case 1: Uniform Irradiance with Different Temperatures.