Towards fairness and low-carbon operation of energy hub: A multi-objective optimization framework with energy–carbon bidirectional coupling and heterogeneous uncertainties

The energy hub plays a growingly vital role in improving energy utilization efficiency and reducing carbon emissions. However, current low-carbon economic optimization methods for energy hub suffer from the following key limitations: overlook the bidirectional coupling of energy and carbon emission flow, neglect fairness concerns due to geographical location, and inadequately address the multiple heterogeneous uncertainties. To address these challenges, a multi-objective coordinated optimization framework is developed for energy hub under bidirectional energy–carbon coupling and heterogeneous uncertainties. Firstly, an energy hub model embedded with carbon emission flow constraints is constructed, enabling simultaneous tracking and bidirectional coupling of energy and carbon flows. Secondly, a hybrid stochastic programming–distributionally robust optimization method is developed to capture the distinct characteristics of uncertainty for energy hub nodes' carbon intensity and renewable generation. Specifically, a temporally-spatially correlated stochastic programming model is employed to address the uncertainty of carbon intensity, while the moment–metric based distributionally robust optimization method is adopted for handling the uncertainty of renewable generation. Furthermore, a tri-objective optimization model is proposed to minimize energy hub operating costs, the aggregated carbon intensity on the load side, and the carbon intensity inequality among loads based on the Theil index. To obtain Pareto-optimal solutions, an ε-constraint method combined with multiple reformulation strategies is employed. An energy-carbon–fairness balance index is proposed to identify the most balanced Pareto solution. Numerical experiments demonstrate that incorporating spatiotemporal correlation in carbon intensity uncertainty modeling improves carbon performance by approximately 7.9% and fairness by approximately 10%. For renewable generation uncertainty management, the distributionally robust optimization achieves 3.5% cost reduction compared to robust optimization while maintaining reliability. The multi-objective optimization achieves 19.4% fairness improvement compared to economic-only optimization. The balanced solution identified by the proposed index delivers substantial carbon reduction (51.57%) and fairness improvement (40.40%) with modest cost increase (18.81%). The framework offers new theoretical insights and practical guidance for the economic, low-carbon, and fairness-aware operation of energy hubs.