## 2 Materials and Methods
The computational complexity of the traditional NSGA-II algorithm is given by:  
\[\mathcal{O}(MN^{2}+N^{3}) \tag{5}\]  
where M represents the number of objective functions (in this study, M = 5: operation time, selectivity, sensitivity coefficient, reliability, and maloperation rate); and N denotes the population size (typically ranging from 100 to 200). In the enhanced multi-objective optimization algorithm, the dynamic mechanism introduces additional computational complexity. Specifically, it requires re-evaluation of 10% of the population individuals per generation, contributing the following complexity term:  
\[\mathcal{O}(0.1N\cdot C_{f}) \tag{6}\]  
where \(C_{f}\) represents the cost of a single objective function calculation (power flow calculation for distribution networks takes approximately 0.1-1 ms per time). In addition, different execution schemes will increase different computational complexities. For the DNSGA-II-A scheme, replacing 50% of the individuals requires uniform distribution sampling, and the computational complexity is:  
\[\mathcal{O}(0.5N\cdot D) \tag{7}\]  
where D represents the dimension of decision variables (D = 4 in this study: \(I_{set1}\), \(I_{set2}\), \(t_{delay}\), \(\Delta t\)). For the DNSGA-II-B scheme, which is a mutation operation based on the normal distribution, the computational complexity is:  
\[\mathcal{O}(N\cdot D\cdot\log D) \tag{8}\]  
Synthesizingthe above steps, the complexity per generation in the worst-case scenario (frequent triggering of DNSGA-II-A) is:  
\[\mathcal{O}_{\text{proposed}}=\mathcal{O}(MN^{2}+N^{3}+0.1N\cdot C_{f}+0.5N\cdot D) \tag{9}\]  
Under typical parameters (\(N=100,M=5,D=4,C_{f}\approx 100\) FLOPs), it is calculated that the computational complexity of the traditional NSGA-II algorithm is approximately \(5\times 10^{4}\) FLOPs per generation, while that of the improved algorithm in this paper is about \(5.2\times 10^{4}\) FLOPs per generation, with an increase of approximately 4% in computational complexity. Based on the aforementioned improvements to the NSGA-II algorithm, the workflow of the adaptive overcurrent protection method proposed in this study is illustrated in \[\mathrm{I_{set}^{new}}=\mathrm{I_{set}^{current}}\cdot U(1-\alpha,1+\alpha) \tag{1}\]0 (1) Initialize the population by inputting fundamental parameters from the distribution network, including: real-time renewable power outputs \(P_{wind}(t)\), \(P_{pv}(t)\), load power \(P_{load}(t)\), renewable power fluctuation rates \(\sigma_{\text{wind}}\), line impedance \(Z_{line}\), transformer tap ratio k, and historical protection settings (Overcurrent Stage I setting \(I_{set1}\) and Stage II setting \(I_{set2}\)); (2) Generate the initial parent population P, where each individual is encoded as a protection parameter vector: \([I_{\text{set1}},I_{\text{set2}},t_{\text{delay}},\Delta t]\), with \(t_{\text{delay}}\) denoting the time delay and \(\Delta t\) the time coordination margin. Each encoded individual represents a candidate solution. Following the standard NSGA-II workflow (crossover, mutation), crossover operations are applied to selected parent individuals to produce the offspring population Q. Offspring individuals undergo perturbation, and the parent population P is merged with the offspring population Q to form the extended population R. (3) Determine whether the adaptive overcurrent protection setting optimization triggering conditions are satisfied by detecting if renewable energy power or load fluctuation amplitudes exceed predefined thresholds, which initiates the optimization process. Specifically, at the start of each generation iteration, 10% of parent individuals are randomly selected for re-evaluation of their objective functions and constraint values.