Pricing Uncertainties With General Parametric Distributions in Power Systems

With the increasing uncertainty brought by renewable energy sources, it is crucial to price uncertainty and quantify the impact on operation costs. For certain types of uncertainty, such as those following jointly Gaussian distributions, the price of uncertainty has been defined. However, the definition of prices for uncertainties following more general parametric forms of distributions is not clear. To fill the gap, this article proposes a uniform price definition of uncertainties with general parametric distributions and a practical calculation method. First, the price definition for uncertainty is proposed based on the marginal pricing principle, which can be derived from the Lagrangian function of the chance-constrained optimal power flow (CCOPF) problem. Next, the scenario-based method is used to analytically transform CCOPF into a deterministic form. The prices of uncertainties in each scenario can be calculated from the Lagrangian function of the scenario-based model. Then, the relationship between the distribution parameters and the uncertainties in scenarios is revealed. We compute the prices of distribution parameters through the prices of uncertainties in multiple scenarios and their relationship. Finally, this article demonstrates the entire calculation procedure using the correlated non-Gaussian uncertainties as typical examples. The performance of proposed method is verified in the PJM 5-bus, IEEE 30-bus, and 118-bus test systems. It shows that the proposed method can accurately compute the price of uncertainty, ensuring nonnegative market surplus, and keeping the constraint violation probability within the tolerance.