Introduction:

Water distribution networks in arid regions face persistent challenges including pressure standard maintenance, infrastructure cost reduction, and water loss management. Iran, with annual precipitation of 240–250 mm, confronts severe water scarcity, making efficient network management critically important. Leakage — an unavoidable phenomenon in distribution systems — is directly governed by network pressure, rendering pressure management a primary strategy for loss reduction (Lambert, 2001). Recent advances in multi-objective evolutionary algorithms have expanded optimization capabilities for water distribution networks. While NSGA-II and MOEA/D have been extensively applied in hydraulic engineering, AGE-MOEA — developed by Panichella (2022) based on non-Euclidean Pareto front geometry approximation — has not previously been evaluated in water distribution optimization contexts, as confirmed through systematic searches of Scopus and Web of Science databases. Furthermore, existing studies predominantly assess Pareto front quality metrics while neglecting the operational implementability of optimization outputs, particularly the stability of pressure-reducing valve (PRV) hourly scheduling patterns under real operating conditions. This study presents a comprehensive two-phase framework integrating hydraulic simulation (WaterGEMS/EPANET) and evolutionary optimization (Python/DEAP/WNTR) to evaluate NSGA-II, MOEA/D, and AGE-MOEA across benchmark networks (Hanoi, Balerma) and the real Poldokhtar western zone network, assessing both Pareto front quality and practical operational feasibility

Materials and Methods:

This study developed an integrated framework combining hydraulic simulation and multi-objective evolutionary optimization across three water distribution networks of increasing complexity: the Hanoi benchmark (34 pipes, 32 nodes, single gravity reservoir), the Balerma irrigation network (454 pipes, 443 nodes, four reservoirs), and the real Poldokhtar western zone network (261 pipes, 206 consumption nodes, one 5,000 m³ gravity reservoir, two pressure-reducing valves). Hydraulic models were initially constructed in WaterGEMS and converted to EPANET 2.2 standard format, with pressure-driven analysis (PDA) implemented through the WNTR library in Python, where nodal demand follows q_i^actual=q_i^demand⋅((P_i-P_i^min)/(P_i^req-P_i^min ))^0.5for P_i^min≤P_i≤P_i^req. The Poldokhtar model was calibrated against field measurements of reservoir outflow and PRV discharge rates, achieving acceptable agreement between simulated and observed values; full nodal pressure calibration was constrained by limited field sensor deployment. Three multi-objective evolutionary algorithms — NSGA-II (dominance-based), MOEA/D (decomposition-based), and AGE-MOEA (geometry approximation-based) — were implemented using the DEAP library with identical computational budgets of 200,000 function evaluations (population N=200, generations=1,000), SBX crossover (pc=0.9, ηc=20), polynomial mutation (pm=1/n, ηm=20), and 30 independent runs with random seeds 1–30. For Hanoi and Balerma, decision variables comprised all pipe diameters selected from discrete commercial sizes, with objectives of minimizing total network cost (Z₁) and pressure deviation from permissible ranges. For Poldokhtar, 53 decision variables encompassed hourly scheduling coefficients of two PRVs (48 variables) and diameters of five selected pipes (5 variables), with three objectives: minimizing total hydraulic penalty from pressure and velocity violations (Z₁), minimizing excess pressure (Z₂), and minimizing the sum of selected pipe diameters (Z₃). Algorithm performance was evaluated using Hypervolume (HV) with reference point r=1.1×fmax and Inverted Generational Distance (IGD) computed against a reference front constructed from all non-dominated solutions across 30 runs of all three algorithms, where higher HV and lower IGD indicate superior performance.

Results and Discussions:

Performance evaluation across three networks revealed distinct algorithmic behaviors reflecting fundamental differences in search mechanisms. On the Hanoi network, NSGA-II achieved the only fully feasible solution with zero pressure deviation at a network cost of $77,551, demonstrating superior capability in navigating tight hydraulic constraints, while AGE-MOEA recorded the highest overall Pareto front quality metrics (HV=4.7, IGD=1.87) and MOEA/D performed weakest (HV=3.6, IGD=3.75) due to its inherent sensitivity to objective scale differences — the Tchebycheff scalarization function becomes dominated by the larger-magnitude cost objective, disrupting convergence toward feasible pressure-compliant solutions. On the large-scale Balerma network (454 pipes, 10⁴⁵⁴ search space), AGE-MOEA maintained its quantitative superiority (HV=0.85, IGD=0.019), followed closely by NSGA-II (HV=0.82, IGD=0.024), while MOEA/D exhibited significant diversity loss (HV=0.61, IGD=0.041), confirming that decomposition-based approaches struggle with high-dimensional design spaces where neighborhood assumptions break down. The Poldokhtar three-objective case produced the most operationally informative results. Pareto front comparison (Z₁: total hydraulic penalty vs. Z₂: excess pressure) revealed that AGE-MOEA dominated NSGA-II across the majority of the objective space in the Z₁ range of 5,750–6,000, delivering lower excess pressure for equivalent hydraulic penalty values, while both algorithms provided substantially wider solution diversity than MOEA/D, which concentrated exclusively in the high-penalty low-excess-pressure region (Z₁>6,050, Z₂≈28,800–29,600) — a computationally significant weakness for multi-objective optimization. Hypervolume convergence analysis (Figure 15) revealed an important distinction: MOEA/D achieved the highest final cumulative HV (4,603,078) compared to NSGA-II (4,089,945) and AGE-MOEA (3,369,721), which does not contradict its weaker Pareto front coverage but reflects its deep convergence within a narrow objective region — a fundamentally different metric capturing exploration history rather than final solution quality. PRV hourly scheduling analysis exposed a critical operational dimension absent from purely quantitative assessments: despite their Pareto front superiority, NSGA-II and AGE-MOEA generated highly oscillatory valve settings fluctuating between 18–23 m within single-hour intervals, with quantitative indicators confirming significantly higher variance, directional change frequency, and oscillation amplitude compared to MOEA/D, which produced conservative near-constant patterns with only minor adjustments — most notably a smooth transition at hour 7 for PRV-1. This operational contrast, supported by the equal-weight MCDM compromise solution extracted from each algorithm's Pareto front, demonstrates that Pareto front quality metrics alone provide an incomplete basis for algorithm selection in real operational contexts: AGE-MOEA is recommended for large-scale design optimization where Pareto coverage is paramount, NSGA-II excels when strictly feasible solutions under tight hydraulic constraints are required, and MOEA/D — despite its quantitative limitations — produces the most operationally stable PRV scheduling patterns suitable for direct field implementation, highlighting a fundamental trade-off between mathematical optimality and engineering practicability that warrants explicit consideration in future water network optimization frameworks.

Conclusion:

This study presented a comprehensive evaluation of NSGA-II, MOEA/D, and AGE-MOEA for water distribution network optimization through an integrated WaterGEMS-EPANET-Python framework. AGE-MOEA demonstrated superior Pareto front coverage in large-scale networks, while NSGA-II proved most effective for constraint-intensive design problems requiring fully feasible solutions. Despite weaker Pareto front diversity, MOEA/D generated operationally implementable PRV scheduling patterns with minimal oscillations, revealing a fundamental trade-off between algorithmic optimality and practical deployability. Algorithm selection should therefore be governed by decision-maker priorities: AGE-MOEA for Pareto quality, NSGA-II for constraint satisfaction, and MOEA/D for operational stability. Quantified leakage reduction requires dedicated pressure-leakage modeling in future research.