A Set-Valued Variational Analysis Framework for Uncertain Energy Resource Allocation Problems

Abstract

The broad deployment of uncertain intermittent renewables and weather driven fluctuations in load brings uncertainty to energy resource allocation, and single valued stochastic programs calibrated at a single fix nominal probability distribution may not capture this uncertainty. The underlying reason is that the feasibility regions for the dispatch can alter themselves set valued with the uncertainty reality not becoming smaller to a single defined set. In the present work introduce a set-valued variational analysis frame to describe the energy system and network-based dispatch feasibility as a mapping from uncertainty to allocation subsets, and an optimal fair allocation is interpreted as a minimax distributionally robust program for an ambiguity set which includes Wasserstein balls for probability laws to be acceptable. Using the idea of conditional value-of-risk reformulate robust chance constraints with use of generalized subdifferential calculus and develop Fritz-John type optimality conditions for the problem based on normality cones as well as horizon maps for graphical convergence. Generalized Hukuhara and second order proto differentiability for the feasibility map enables precise analysis of optimal solution’s sensitivities based on ambiguity radius, and generalized-Nash equilibrium offers models for problems involving competitions among agents. The study demonstrates convergence properties of the sample average approximate scheme and establish a concentration bound related to cascading allocation failure events. The methodology is validated with numerical case studies illustrating the comparison between an iterative fixed point scheme and a inexact Newton type approach. An estimate of growth for allocation deficiencies according to ambiguity radius is quantitatively verified.