Probabilistic OPF using linear fuzzy relation

Optimal Power Flow (OPF) is a very important tool for planning and analysis of power systems. In the recent times, uncertain renewable energy is being integrated into power systems in a large scale. Appropriate modeling of renewables in OPF requires using stochastic models. Using stochastic models of renewables in OPF is numerically and algorithmically challenging due to the complexity of stochastic models and nonlinear nature of bus power balance equations. Hitherto, Monte Carlo Simulation technique and Cumulant technique have been proposed, but they are not computationally viable for large systems. In this paper, we propose the use of linear fuzzy relation technique to relate stochastic models of dependent variables of OPF formulation in terms of control variables that include power output of renewables. This fuzzy relation uses Hessian matrix of the LaGrangian of the OPF formulation at optimal solution point. The method is tested on a 6-bus system and results are reported. One can intuitively see that this method can be easily extended to larger systems.