Sensitivity and Covariance in Stochastic Complementarity Problems with an Application to Natural Gas Markets

We provide an efficient method to approximate the covariance between decision variables and uncertain parameters in solutions to a general class of stochastic nonlinear complementarity problems. We also develop a sensitivity metric to quantify the uncertainty propagation in the problem by determining the change in the variance of the output variables due to a change in the variance of an input parameter. The covariance matrix of the solution variables quantifies the uncertainty in the output and pairs correlated variables and parameters. The sensitivity metric helps in identifying the parameters that cause maximum fluctuations in the output. The method developed in this paper optimizes the use of gradients and matrix multiplications which makes it particularly useful for large-scale problems. Having developed this method, we extend the deterministic version of the North American Natural Gas Model (NANGAM) to incorporate effects due to uncertainty in the parameters of the demand function, supply function, tariffs, and investment costs. We then use the sensitivity metrics to identify the parameters that impact the equilibrium the most.