Optimal Risk-Managed Dispatch of Multi-Terminal High-Voltage Direct Current Systems Integrating Renewable Energy and Battery Storage Through Mixed-Integer Convex Chance-Constrained Programming

This paper proposes a stochastic dispatch framework for multi-terminal high-voltage direct current (MT-HVDC) systems that explicitly accounts for uncertainty in photovoltaic (PV) generation and electrical demand while preserving computational tractability. The economic–environmental dispatch problem is formulated as a mixed-integer second-order cone programming (MI-SOCP) model, where the SOCP relaxation provides a convex representation of the network constraints, and the mixed-integer component captures the discrete charging/discharging states of battery energy storage systems (BESS). This formulation ensures that, for any fixed set of binary decisions, the remaining problem reduces to a standard convex SOCP, enabling efficient solution via branch-and-bound methods with tight continuous relaxations. Uncertainty is incorporated through a chance-constrained optimization (CCP) approach, where forecast errors are modeled using bounded truncated distributions and reformulated into deterministic convex constraints via quantile-based approximations, yielding a risk-aware dispatch strategy that avoids optimistic bias. Numerical studies on an 11-bus MT-HVDC test system demonstrate that accounting for uncertainty increases total operating costs by up to 31.87% and CO2 emissions by up to 37.41% when both PV and demand uncertainties are considered simultaneously at a confidence level of 0.9, compared to the deterministic solution. Power demand uncertainty has a substantially greater impact than PV generation uncertainty, leading to cost increases of 28.09% versus 3.85% at the highest confidence level. Validation on a modified IEEE 24-bus MT-HVDC system confirms the scalability and computational efficiency of the proposed approach, achieving global optimality in a pure solver time of 1.34 s with a maximum relative relaxation gap of 5.81×10−9, demonstrating numerical exactness and suitability for day-ahead scheduling. The results also highlight the critical role of BESSs in providing operational flexibility, with storage strategies differing significantly under uncertainty during early hours while converging to deterministic behavior later. The findings reveal a clear trade-off between economic performance and operational reliability as the confidence level increases, confirming the effectiveness of the proposed approach for integrating renewables and storage in modern HVDC grids, while also identifying important limitations regarding independence assumptions and scalability to larger systems.