Equal Increment Rate Optimization for Power System Stochastic Optimal Power Flow with Carbon Footprint Quantification and Optimal Carbon Trading

High shares of wind and solar power make generation outputs strongly random. Traditional stochastic optimal power flow (SOPF) also lacks power–carbon coordination. Low-carbon constraints are often disconnected from operational scheduling. To address these issues, this paper proposes an SOPF method that includes an optimal carbon-trading mechanism. First, Monte Carlo simulation and K-means clustering generate typical renewable-generation scenarios. Each scenario is assigned a probability. Scenario reduction preserves the statistical features of uncertainty while lowering computational cost. Second, a full-link carbon-footprint index system is built for the generation, grid, and load sides. It covers carbon-emission measurement, efficiency constraints, and low-carbon constraints. It describes the spatial distribution and time-series transmission of carbon flow. Third, a carbon-trading scheme-selection framework is proposed. It combines a Nash equilibrium game with AISM hierarchical topology analysis. It balances the revenues of generators, the grid, and users, and it follows the hierarchical transmission of carbon indicators. The optimal scheme is selected and the system carbon-reduction benchmark is set. Finally, a multi-objective model is built. It minimizes generation and transmission cost, minimizes carbon-emission cost, and maximizes the carbon-reduction contribution rate of renewable consumption. The equal incremental rate criterion is used for iteration. Case studies on an improved IEEE 33-bus network verify the method. The method links carbon-trading rules with grid scheduling. It improves the accuracy and engineering value of low-carbon regulation.