Offering an extensive multi-objective modeling strategy for optimizing load balancing in regional integrated energy systems

This paper develops an analytical modeling framework for load responsiveness for regional integrated energy systems to capture the coupled dynamics among electricity, thermal, and natural gas networks under dynamic pricing schemes. In contrast to conventional single-carrier demand response models, the proposed load response program includes cross-energy elasticities, inter-temporal coupling, and physical interdependencies among energy carriers using a unified economic-thermodynamic formulation. The behavior of the user is quantified for utility maximization theory and derived closed-form relationships between demand elasticity, price variation, and satisfaction functions. Furthermore, inter-carrier interaction coefficients represent substitution and co-generation effects between electricity and gas consumption. This couples the economic signals and physical system states. The resulting integrated load response model simultaneously yields consistency with both microeconomic rationality and the physics of multi-energy systems. To achieve optimal scheduling under this coupling, a multi-objective optimization framework is defined to minimize total operating cost, environmental impact, and reliability risk. Therefore, a novel entropy- and fuzzy logic-enhanced Multi-Objective Particle Swarm Optimization (EF-MOPSO) algorithm is developed to solve the high-dimensional nonconvex problem. The proposed EF-MOPSO introduces adaptive inertia control, entropy-based diversity metrics, and fuzzy-leader selection to balance convergence speed and solution dispersion. It ensures robust exploration of the Pareto front under stochastic uncertainties. The proposed model applies a time-of-use (TOU) based dynamic pricing structure for electricity and natural gas to stimulate demand-side flexibility. The electricity price is divided into three daily periods: off-peak (01:00–06:00), mid-peak (06:00–10:00 and 14:00–18:00), and peak (10:00–14:00 and 18:00–21:00), with multipliers of 0.75, 1.05, and 1.30, respectively, relative to the base tariff. The natural gas price is set at a baseline of 3.24 CNY/m 3 and becomes responsive in the second scenario under a similar TOU pattern. This pricing mechanism encourages users to shift loads from high-cost peak periods to lower-cost off-peak periods, which improves operational flexibility and enables a 10% reduction in total system cost while preserving supply reliability.