Multi-Stage Quality-Diversity and Gradient-Assisted Memetic Optimization for Strongly Constrained Continuous Multi-Reservoir Scheduling

This study addresses a modified continuous multi-reservoir scheduling problem characterized by a high-dimensional continuous decision space, strong time-varying storage constraints, strict terminal storage closure requirements, and a highly nonconvex composite objective. To solve this challenging problem, a multi-stage collaborative memetic algorithm based on CVT-MAP-Elites and clustering gradient (MCMA-CCG) is proposed. The framework consists of three tightly coupled stages: an exploration stage based on CVT-MAP-Elites to preserve diverse high-potential elites, a clustering stage using DBSCAN with a customized noise-retention strategy, and a refinement stage that combines DE with gradient-enhanced SLSQP to perform accurate exploitation. Under a unified experimental setting, MCMA-CCG was evaluated against several representative optimization algorithms, including DE, GA, SAPHTLR, HBMO, MFA, and MBWOHHO, over 30 independent runs. The updated results show that MCMA-CCG consistently achieves the best overall performance in both the four-cycle and five-cycle reservoir scheduling scenarios while also exhibiting superior empirical runtime and feasibility behavior. In the four-cycle case, it attained a best value of 6.08 × 103, an average of 5.97 × 103, and a standard deviation of 4.62 × 101; meanwhile, it produced feasible solutions in 26 of 30 runs, achieved a mean feasibility distance of 1.20 × 10−3, and required only 54.91 s on average under the 30,000-function-evaluation budget. In the more challenging five-cycle case, it attained a best value of 7.60 × 103, an average of 7.44 × 103, and a standard deviation of 7.55 × 101; it still generated feasible solutions in 19 of 30 runs, with a mean feasibility distance of 4.84 × 10−3 and an average runtime of 93.80 s under the 50,000-function-evaluation budget. By contrast, all baseline algorithms produced no fully feasible runs under the same feasibility criterion and generally required longer wall-clock time. Ablation studies further demonstrate that the superior performance of MCMA-CCG does not arise from any single module, but from the effective synergy among quality-diversity exploration, cluster-guided seed extraction, and gradient-assisted local refinement. These results confirm both the numerical superiority and the physical interpretability of the proposed framework for complex continuous multi-reservoir scheduling problems.