Distributed online optimal power flow for distribution system

Highlights

• A distributed online OPF algorithm for distribution systems is proposed.

• It formulates the second-order OPF approximations in a distributed way.

• It only needs local measurements and boundary information for each area.

• It can continuously drive the distribution operation towards given OPF targets.

• The algorithm can cope with various setups for the OPF models.

Abstract

This paper proposes a distributed online OPF algorithm for distribution systems with high integration of electronic-interfaced distributed energy resources (DERs). The algorithm can coordinate the update of DER set-points on fast time scales only depending on local measurements and boundary information for each distribution area. Comparing with the existing distributed online algorithms that require some special setups (e.g., without considering the constraints on currents), the proposed algorithm can cope with various setups for the objective functions and constraints. The algorithm is based on solving the Taylor approximations of OPF problems by a distributed solution. In this solution, we design a strategy to formulate the second-order Taylor information in a distributed way by decomposing the sensitivities of the OPF targets among different areas; moreover, we design a method to simplify the constraints on DER outputs to accelerate the solution. The validity of the algorithm is verified by case studies, which show that it can continuously drive DER outputs towards the time-varying OPF targets composed of power losses and operation costs of DERs while satisfying the constraints on voltages and currents.

Introduction

Future distribution systems are expected to integrate large numbers of electronic-interfaced distributed energy resources (DERs), such as photovoltaics (PVs) and battery energy storage systems (BESSs) [1], [2], [3]. They introduce fast control capacities, and provide the potential to coordinate the operation of distribution systems more accurately. However, most of the optimal power flow (OPF) algorithms, such as the classical reduced gradient method [4] and the Newton method [5] and some distributed OPF algorithms [6], [7], [8], [9], are offline. The offline OPF algorithms require to iterate on all variables several times in cyberspace until getting an optimal solution that can be applied to the electrical systems. This solving process is time-consuming and unable to fully utilize the fast control capacities of DERs, as well as to respond to fluctuations of outputs of renewable energy sources (RESs) [10]. Future distribution systems require online OPF algorithms working on fast timescales.

Some recent efforts have focused on the online OPF algorithms. The online algorithms utilize the measurements to iterate only on the controllable variables, and simultaneously apply the intermediate iterates to update DER set-points and drive DER outputs towards given OPF targets in feedback interaction with the electrical systems [10]. Ref. [11] proposed a gradient-based online OPF algorithm for radial distribution grids and established sufficient conditions for its convergence to a global optimum. Ref. [12] proposed an OPF seeking controller for grids with arbitrary structure, and introduced semi-definite programming relaxations to achieve global convergence. Ref. [13] designed an adaptive feedback controller introducing a modeling approach in terms of manifold optimization. Ref. [14] proposed a model-free online OPF algorithm based on extremum seeking methods, where the gradients are estimated in procedures of injecting sinusoidal probing signals into electrical grids. Ref. [15] designed an OPF seeking controller based on alternating direction method of multipliers. Ref. [16] proposed an OPF-pursuit controller and analytically established its OPF-target tracking capabilities in terms of operation changes and measurement errors. Ref. [17] proposed a quasi-Newton-based online OPF algorithm, where the solution is accelerated by adaptively shrinking the size of Jacobian matrixes according to the binding constraints in real time. Ref. [18] proposed an incentive-based solution to track time-varying targets of maximizing social-welfare. Ref. [19] designed a gradient-based controller and studied its convergence when the set-points of DERs are discrete. Ref. [20] designed a stochastic dual algorithm for solving online OPF problems considering the multi-period constraints. However, the above online solutions [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] are all based on a centralized framework [10], where a center is needed to collect all the parameter and measurement data throughout the considered system to formulate the sensitivities related to given OPF targets. This framework faces the problem of single point of failure and therefore ‘limited reliability’ due to centralized data collections [21], [22], [23], [24].

To overcome the shortage, some works [25], [26], [27], [28], [29], [30], [31], [32], [33], [34] proposed the distributed online algorithm only depending on the local and/or neighbors’ information. Refs. [25], [26], [27], [28], [29], [30], [31] deduced the sensitivities used to update DER set-points into a distributed fashion, and thus developed the online algorithm into a distributed solution; however, the deductions of these methods [25], [26], [27], [28], [29], [30], [31] were based on OPF models without considering the current constraints, and moreover, the consideration of power losses in [27], [28], [31] needs an assumption of homogeneity of the X/R ratios across the considered distribution grids, and [29], [30] can not consider the hard constraints on voltages. Refs. [32], [33] proposed the online algorithms for frequency control, but they were based on DC power flow models without considering voltage regulation and power losses minimization. Ref. [34] proposed a heuristic-based algorithm for minimizing power losses, but the consideration of the other objectives (e.g., minimization of the operation costs of DERs) needs further improvements for the heuristic strategy.

To this end, this paper proposes a novel distributed online OPF algorithm. Comparisons between the proposed algorithm and the others [25], [26], [27], [28], [29], [30], [31], [32], [33], [34] are summarized in Table 1. One can see that the proposed algorithm has more extensive application scopes and can cope with various setups for the objective functions and constraints. The algorithm is based on solving the Taylor approximations of OPF problems by a distributed solution. In this solution, we design a strategy to formulate the second-order Taylor information in a distributed way by decomposing the sensitivities of the OPF targets among different areas; moreover, we design a method to simplify the constraints on DER outputs to accelerate the solution. Case studies show that it can continuously drive DER outputs towards time-varying OPF targets while satisfying the constraints on voltages and currents.

Overall, the main contribution of this paper is three-fold:

• We proposed a distributed online OPF algorithm only depending on local measurements and boundary information for each distribution area. Comparing with the existing distributed online solutions, the proposed algorithm has more extensive application scopes and can cope with various setups for the objective functions and constraints.

• We design a strategy to formulate the Jacobian and Hessian of OPF problems for distribution systems in a distributed way. This is useful for the optimization and analysis of distribution systems where the information is unable to be processed centrally.

• We design a strategy to simplify the coupling constraints on DER outputs. It can accelerate the solution and is significant for achieving close-followed OPF tracking capacities.

The rest of this paper is organized as follows. Section 2 describes a time-varying OPF problem for multi-area distribution systems. Section 3 presents the distributed algorithm to solve the OPF problem. Section 4 presents the case studies and the test results. Section 5 concludes this paper. The main symbols used in the paper and their meanings are listed in Table 2.