Achieving linear convergence for differentially private full-decentralized economic dispatch over directed networks

Abstract

Privacy-preserving methods play an indispensable role in protecting agents' cost functions from being disclosed during the dispatch process. In this paper, we aim at resolving Economic Dispatch Problems (EDPs) over multi-agent systems in a fully decentralized and privacy-preserving manner. Based on both row- and column-stochastic weight matrices, we first design a fully decentralized algorithm without any privacy-masking techniques, namely F-Deed, to resolve a class of structured EDPs over directed networks. We further consider an adverse case that potential attackers including both internal honest-but-curious adversaries and external eavesdroppers try to infer or steal the local cost functions of all agents to achieve their malicious goals. To protect agents' local cost functions against differential attacks, a differentially private algorithm, dubbed DPF-Deed, is developed via enhancing F-Deed with local differential privacy (LDP). To attain LDP, DPF-Deed decomposes the gradient tracker of F-Deed into two sub-trackers, with one of them invisible to any other agents, and masks a decomposed gradient with Laplace noise. Under standard assumptions, theoretical analysis validates that DPF-Deed can achieve linear convergence and an explicit trade-off between LDP and convergence accuracy, which are derived by analyzing the contraction relationships among the network consensus error, the optimal gap, and the gradient-tracking error. Theoretical results are validated by case studies for (dynamical) EDPs based on modified IEEE-14 and IEEE-118 bus systems.