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With the increasing deployment of distributed energy resources (DERs), dispatching DERs subject to operational constraints in distribution networks draws much attention. One challenge is the non-convexities in 1) system-wide AC power flow constraints and 2) the individual complementarity constraint of energy storage. To resolve this challenge, this paper studies an AC-feasible linear model in distribution networks with energy storage, including its formulation, analysis, and applications. First, an AC-feasible linear model is formulated as a set of linear constraints on controllable DERs and uncontrollable power demand by 1) converting the non-convex system-wide constraints into linear constraints based on the Brouwer's fixed-point theorem and the second-order Taylor expansion, and 2) replacing the non-convex individual complementarity constraint of energy storage with one properly designed linear constraint. Furthermore, to analyze the power demand level at which the proposed linear model can provide a solution, this paper proposes an examination-based projection method under the Monte Carlo framework to handle projections of thousands of dimensions from linear constraints over time periods. Finally, the potential applications of our AC-feasible linear model are discussed. Numerical experiments are conducted in the IEEE 33-bus and 136-bus test systems to demonstrate the effectiveness of the proposed methods.

To determine the stability of a power system integrated with numerous wind power generators, considering the stochastic fluctuation of wind power output, this paper mainly studies the stability criterion of power systems under stochastic continuous disturbances (SCDs). The dynamic performance of a power system under SCDs is described through stochastic differential equations (SDEs) based on nonlinear state equations. Under the framework of SDEs, a criterion for determining the mean square stability of a power system under SCDs is proposed, and it is proved that the proposed stability criterion is a necessary and sufficient condition for determining the mean square stability. In addition, the bi-level nonlinear semidefinite programming (NLSDP) problem in the stability criterion is transformed into a single-level NLSDP problem by using the improved Karush–Kuhn–Tucker condition of the inner-layer problem, and an extended primal-dual interior point algorithm is proposed to solve the single-level NLSDP problem. In addition, the optimal objective value calculated from the proposed stability criterion is used to quantify the stability degree of the system under SCDs. Finally, case studies on the modified IEEE-39 and 118 bus systems, and an actual provincial power grid demonstrate the correctness and effectiveness of the proposed stability criterion and solution algorithm.