## 2 Materials and Methods  
Practical search and optimization problems in real-world production and daily life often evolve over time, necessitating treatment as online optimization problems. These variations manifest in aspects such as objective functions, constraint functions, or variable boundaries. Ideally, solutions should be promptly derived upon problem variations, with temporal parameters corresponding to the iteration counter of optimization algorithms. The NSGA-II (Non-dominated Sorting Genetic Algorithm II), a representative multi-objective evolutionary algorithm (MOEA), is designed to efficiently solve complex multi-objective optimization problems and identify uniformly distributed Pareto optimal solution sets. The NSGA-II algorithm is developed from the NSGA-I (Non-dominated Sorting Genetic Algorithm I). Proposed by [PERSON] and [PERSON] in 1995, NSGA-I is the first generation of multi-objective evolutionary algorithms based on Pareto dominance. Its core idea is to solve multi-objective optimization problems through non-dominated sorting and sharing functions, aiming to generate a uniformly distributed Pareto optimal solution set. Under this algorithm, the population is stratified into multiple non-dominated fronts, where solutions in higher-level fronts obtain higher fitness. Meanwhile, the similarity between solutions is calculated using the sharing radius parameter, and finally, offspring are generated through roulette selection based on fitness. However, this algorithm has three non-negligible draw backs: high computational complexity, parameter sensitivity, and the absence of an elitism preservation mechanism. To address these flaws of NSGA-I, [PERSON]'s team proposed three groundbreaking improvements: fast non-dominated sorting, replacement of the sharing function with crowding distance, and the elitism preservation strategy. These improvements have made NSGA-II a milestone in the field of multi-objective optimization and laid the foundation for dynamic multi-objective optimization. Table 1 presents a comparison of the algorithm before and after improvement. Building upon the traditional NSGA-II workflow, this study implements modifications and enhancements to address the dynamic optimization challenges of relay protection setting coordination in distribution networks following renewable energy integration. A testing mechanism is introduced prior to algorithm execution to detect variable changes in each generation, thereby selecting more appropriate execution strategies. In this research, real-time distribution network parameters--including renewable energy output, load power, line impedance, transformer tap ratios, and historical protection settings--are used as inputs to initialize the parent population. Each individual is encoded as a protection parameter vector. The structure is formulated as \([I_{\text{set1}},I_{\text{set2}},t_{\text{delay}},\Delta t]\); here, \(I_{\text{set1}}\) is the Overcurrent Stage I setting, \(I_{\text{set2}}\) is the Overcurrent Stage II setting, \(t_{\text{delay}}\) is the time delay, and \(\Delta t\) is the time coordination margin. During the testing and decision-making phase, a small number of solutions are randomly selected from the parent population and re-evaluated. If the differences between their objective functions, constraint values, or historical records exceed a threshold, it is determined that significant changes in renewable energy output have occurred. The threshold is defined as \(\theta_{\text{threshold}}=k\cdot\max(\sigma_{\text{wind}},\sigma_{\text{pv}})\), where \(k\) is adjustment coefficient, and \(\sigma_{\text{wind}}\) and \(\sigma_{\text{pv}}\) are the standard deviations of wind power and photovoltaic power outputs, respectively. If the testing mechanism determines that the solution injection strategy (DNSGA-II-A) should be executed, this strategy requires generating uniformly distributed random solutions according to Equation (1) for 20-70%