## 2 Materials and Methods
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Characteristics** & **NSGA-I** & **NSGA-II** & **Significance of Improvements** \\ \hline Non-dominated Sorting & Ordinary stratification & Quick sorting & 10–100\(\times\) Faster Computational Efficiency \\ \hline Diversity Preservation & Sharing function & Crowding distance & Eliminated Parameter Tuning Burden, 40–90\% Enhanced Distribution Uniformity \\ \hline Elitism Strategy & None & Parent-offspring merging and rank selection & Preserved Historical Optimal Solutions, 30\%+ Accelerated Convergence Speed \\ \hline Selection Mechanism & Roulette wheel & Binary tournament & Reduced Selection Bias, Balanced Exploration and Exploitation \\ \hline Constraint Handling & None & Constraint dominance principle & Direct Support for Engineering Constraints \\ \hline Computational Resource Requirement & High & Low & Capability to Handle Real-World Problems \\ \hline Parameter Dependency & Strong & Weak & Strong Robustness, Reduced Debugging Costs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison table of NSGA-I and NSGA-II algorithms. of the population individuals. These solutions will replace either the worst-performing individuals or randomly selected individuals within the population. \[\mathrm{I_{set}^{new}}=\mathrm{I_{set}^{current}}\cdot U(1-\alpha,1+\alpha) \tag{1}\]  
where \(\alpha=k\cdot\sigma_{\mathrm{wind}}\), where U denotes the uniformly distributed random number generation function, and \(\sigma_{\mathrm{wind}}\) represents the standard deviation of wind power output. If the testing mechanism determines that the Gaussian mutation strategy (DNSGA-II-B) should be executed, this strategy requires performing Gaussian mutation adjustments on the setting values according to Equation (2), which will replace either the individuals with the lowest crowding distance or redundant individuals in the population. \[I_{\mathrm{set}}^{\mathrm{new}}=\mathrm{I_{set}^{current}}\cdot\mathcal{N}(1,\sigma_{\mathrm{mut}}) \tag{2}\]  
where \(\sigma_{\mathrm{mut}}=\eta\cdot\frac{\sigma_{\mathrm{pv}}}{\sqrt{\pi}}\), \(\eta\) is the scaling factor, \(\sigma_{\mathrm{pv}}\) is the standard deviation of renewable energy output, n is the current iteration count, N denotes the normal-distributed random number generation functions. Subsequently, the merged extended population undergoes non-dominated sorting, crowding distance calculation, and elitist preservation based on the composite objective function F to generate a new parent population. The calculation of F is defined by Equation (3). \[F=w_{1}\cdot T_{\mathrm{action}}+w_{2}\cdot(-S)+w_{3}\cdot K_{\mathrm{sens}}+w _{4}\cdot(-R)+w_{5}\cdot E \tag{3}\]  
where \(w_{1}-w_{5}\) are the weight coefficients, \(T_{\mathrm{action}}\) denotes the operation time, S represents selectivity, K sens is the sensitivity coefficient, R signifies reliability, and E corresponds to the misoperation rate. The misoperation rate E is calculated as:  
\[E=\frac{|I_{\mathrm{set}}-I_{\mathrm{opt}}|}{I_{\mathrm{opt}}} \tag{4}\]  
where \(I_{\mathrm{opt}}\) denotes the theoretical optimal setting. When the maximum iteration count is reached or the convergence of the Pareto front change rate (defined as the rate of change in the Pareto front remaining below 1% for five consecutive generations) is achieved, the protection parameter set satisfying the constraint conditions is output to the protection devices. The constraints include: sensitivity \(K_{\mathrm{sens}}>1.5\), selectivity \(t_{\mathrm{upper}}>t_{\mathrm{lower}}+\Delta t\), misoperation rate \(E<1\%\). In terms of computational complexity analysis, the proposed Dynamic Non-dominated Sorting Genetic Algorithm II (DNSGA-II) introduces a scenario-adaptive mechanism based on the conventional NSGA-II framework.