Energy-Efficient Optimization of Homogeneous and Heterogeneous Parallel Pumping Systems Using an Improved Snake Optimizer with Stage-Wise Search Strategies

Under varying demand, pump states, load allocation, speed settings, and supply pressure are strongly coupled in parallel pumping systems. This study develops a steady-state energy-efficiency optimization framework for homogeneous and heterogeneous three-branch systems by integrating continuous pump-performance modeling, hard hydraulic-feasibility handling, mixed-variable optimization, and demand-dependent pressure regulation. An improved snake optimizer (ISO) is constructed through targeted strategies for population initialization, global exploration, and local exploitation. Flow–head and flow–power models established from LVR5-5 and CDL3-50 experimental data achieved a total-power MAPE of 2.36% and R2 = 0.9972 over 40 implemented steady-state schemes. Deterministic constrained solutions for all effective pump-state combinations were used as references for the pumping-system optimization. Under the same budget of 21,000 function evaluations, ISO, SO, PSO, DE, and GA completed 1500 independent runs; ISO achieved the best overall search performance, with all 300 runs reaching the 0.1% neighborhood of the reference solution and the lowest mean NAUC and final gaps. Using ISO for both pressure-control modes, variable-pressure operation reduced mean power consumption by 10.92% and 7.51% for the homogeneous and heterogeneous systems, respectively, relative to constant-pressure operation under the same minimum service-pressure requirement. The results demonstrate that combining high-quality mixed-variable optimization with demand-dependent pressure regulation can effectively improve the steady-state energy efficiency of parallel pumping systems.