Approximately Adaptive Distributionally Robust Optimization for Energy and Reserve Dispatch

This paper proposes two novel paradigms of approximately adaptive distributionally robust optimization (AADRO) for the energy and reserve dispatch with wind uncertainty. The piecewise linear policy-based AADRO (PLP-AADRO) approximates the adaptive optimization-based recourse decision as piecewise affine adjustment, while the piecewise value function-based AADRO (PVF-AADRO) approximates the quadratic recourse problem as piecewise linear recourse problems. Moreover, an equal probability principle is developed to achieve a high-quality segmentation of the wind power ambiguity set. Consequently, the distributionally robust quadratic cost constraint can be decomposed into decoupled piecewise constraints, allowing the dispatch problem to be formulated as a less-iterative or even non-iterative program. The two-stage AADROs with polyhedron supported uncertainties are first recast precisely as tractable forms with semidefinite constraints, by employing duality theory and S-lemma. Then, the distributionally robust cost constraint in PVF-AADRO is handled by dual vertex generation, and the bilinear terms in both AADROs are addressed by alternating optimization. Numerical simulations verify the efficiency of AADROs in approximating the strict adaptive distributionally robust optimization, and their adaptability in different cases is discussed.