A Data-Driven Optimization Method Considering Data Correlations for Optimal Power Flow Under Uncertainty

Abstract

This paper introduces a data-driven optimization (DDO) method based on novel strategic sampling (SS) considering data correlations for multiperiod optimal power flow (OPF) considering energy storage devices under uncertainty (OPF-ESDUU) of uncertain renewable energy and power loads (UREPL). This DDO method depends only on the uncertainty samples to yield an optimal solution that satisfies a specific confidence level, which is effective because of two resounding learning algorithms: Bayesian hierarchical modeling (BHM) and determinantal point process (DPP). Considering both the local bus information and spatial correlations over all buses, BHM learns the convex approximation of AC power flow (CAACPF) more accurately than the existing learning methods, converting the originally non-convex OPF-ESDUU to a convex optimization problem. DPP considers the correlations between samples to find a small set of significant samples by measuring the relative weight of each sample using the random matrix theory, significantly decreasing the data samples required by the existing SS. The experimental analysis in IEEE test cases shows that after considering data correlations, 1) BHM learns CAACPF better with 13-90% accuracy improvement on average, compared with the existing learning methods, and 2) the proposed DDO performs more efficiently than the existing DDO as DPP-based SS boosts the sampling efficiency by 50% at least. INDEX TERMSBayesian hierarchical modeling, determinantal point process, power flow, strategic sampling. NOMENCLATURE A. SETS AND INDICES n Number of buses (nodes) in the system. 𝑖, 𝑗 Index for the buses, 𝑖, 𝑗 =1, 2, …, n. 𝑇 ′ Index set of the time periods. T Superscript to denote the transpose operation. 𝑡 Index for any time-period, 𝑡 ∈ 𝑇′. 𝑔 Superscript to denote generators. 𝑛𝑒𝑡 Superscript to denote the net power injection. 𝑀 Scenario (sample) set. 𝑠 Superscript for the 𝑠-th sample, 𝑠 ∈ 𝑀. 𝑖 ′ 𝑗 ′ Index for the i'-th and j'-th samples. 𝒫 A point process on a finite set 𝒴. 𝒜 Any subset of 𝒴. B. PARAMETERS 𝜔 𝑝𝑖 (𝜔 𝑞𝑖 ) Participation factors of the active (reactive) power of the generator or energy storage. 𝑃 𝑗,𝑡(𝑓𝑟𝑡) 𝑛𝑒𝑡 Forecast values of the net active power injection of UREPL. 𝑄 𝑗,𝑡(𝑓𝑟𝑡) 𝑛𝑒𝑡 Forecast values of the net reactive power injection of UREPL. ∆𝑝 𝑗 , ∆𝑞 𝑗 Corresponding forecast errors of net active and reactive power injections. 𝛼 Probability level. 𝜖 Violation probability level. 𝛽 Confidence level. 𝑑 Number of decision variables. 𝑁 𝑢 , 𝑁 3 Minimum sample size (MSS) considered in strategic sampling-based DDO methods. 𝑁 1 , 𝑁 2 MSSs by the conventional scenario optimization (SO)-based methods. 𝑁 𝑑𝑝𝑝 MSS by DPP-based strategic sampling.