It is always desired for optimizing energy trading to disable manipulation and preserve individual privacy. These two features become increasingly appealing for an energy market where interest parties do not mutually trust each other, such as peer-to-peer energy trading. Traditional centralized or hierarchical optimization schemes are vulnerable to an untrusted coordinator who may dishonestly broadcast coordination results or be curious about individual privacy. Recent blockchain-based optimization schemes resist dishonesty but increase the risk of privacy leakage. This paper proposes a privacy-preserving blockchain-based method to optimize energy trading. In the proposed method, participants submit encrypted bids/offers based on a bid/offer encryption algorithm to preserve their privacy. A privacy-preserving-Byzantine-fault-tolerance-based coordination algorithm is proposed to ensure the correctness of trading results with considering dishonesty. Numerical results in a peer-to-peer energy trading case demonstrate the performance of our method on convergence, resisting dishonesty, preserving privacy, and scalability.

This paper proposes a computationally efficient and accurate method to solve probabilistic power flow (PPF) and its global sensitivity analysis (GSA) problems of power distribution systems. A distinguishing feature of this study is that a closed-form expression of the joint probability distribution function (PDF) of random input and output variables of distribution system power flow is derived based on the Gaussian mixture model (GMM), and then both PPF and GSA problems are calculated analytically. First, we combine kernel density estimation with the density-preserving hierarchical expectation maximization algorithm to obtain the GMM of random input variables, which describes their non-Gaussian distributions and correlations. Second, to characterize the nonlinear mapping relationship from inputs to outputs in power flow calculation, we establish the joint PDF of random input and output variables using the GMM of input variables and a piecewise linear power flow model, and then derive the closed-form PPF and semi-closed-form GSA solutions. Third, we develop a GMM-based decomposition method of global sensitivity indices to identify correlation effects and individual effects of correlated input variables on power flow variations. The proposed method is tested on the IEEE 33-bus and 123-bus distribution systems integrated with photovoltaic power units, wind turbines, and electric vehicle charging stations. Numerical results show that the proposed method is both fast and accurate, and superior over state-of-the-art surrogate model-based PPF and GSA methods.

This article proposes a robust parametric programming (RPP) method to adaptively obtain piecewise linear control functions of photovoltaic (PV) inverters for real-time voltage regulation in distribution systems. First, the voltage regulation problem is designed as a multi-parametric programming problem, which is transformed into single-parametric linear programming (sp-LP) problems for realizing local control of a PV inverter while considering the maximum voltage deviation caused by other PV units. Second, an alternating iterative algorithm is developed to solve the sp-LP problem of each PV unit and obtain the control functions of all the PV units. Third, considering the information on PV power correlation, a two-level parametric programming problem is established to enhance the voltage regulation performance of PV inverters. A geometric method is developed to solve the inner-level problem, and breakpoint reduction with robust interpolation is developed to improve the computational efficiency in solving the outer-level problem. Four distribution systems integrated with PV units are used to validate the effectiveness of the proposed method. Numerical results show that the performance of the proposed PV inverter control strategy is close to that of the ideal optimal solution and superior to existing piecewise linear control strategies.