# Power System Optimization Methods: Convex Relaxation and Benders Decomposition

2020

Dissertation

Ph.D.

## Degree Name

Doctor of Philosophy (Ph.D.)

## Degree Granting Department

Electrical Engineering

## Major Professor

Lingling Fan, Ph.D.

## Committee Member

Zhixin Miao, Ph.D.

## Committee Member

Nasir Ghani, Ph.D.

Tapas Das, Ph.D.

Bo Zeng, Ph.D.

## Keywords

alternating current optimal power flow, model predict control, relaxation exactness, security constrained optimal power flow

## Abstract

Power system optimization methods are wildly used to solve power system problems. Engineers adopt different methods to keep the reliability and efficiency of the power system operation, planning and control. This dissertation focuses on the application and implementation of two optimization methods: Convex relaxation and Benders' decomposition.

The first part of the dissertation focuses on the application of convex relaxation to solve Alternating Current Optimal Power Flow (ACOPF) problems. In the completed work, a 3-node cycle based sparse convex relaxation is proposed to solve ACOPF problems. This method adds virtual lines in minimal chordless cycles to decompose each of them into 3-node cycles. By enforcing the submatrices related to 3-node cycles Positive Semi-Definite (PSD), the resulting convex relaxation has a tight gap. For the majority of the test instances, the resulting gap is as tight as that of a semi-definite programming (SDP) relaxation, yet the computing efficiency is much higher. Furthermore, to achieve the exactness of the convex relaxation, two algorithms are designed to decrease the relaxation gap. The first method is based on the convex iteration technique. It could help the proposed convex relaxation to achieve the exactness by enforcing all submatrices corresponding lines and virtual lines rank-1. The second method is based on the nonlinear programming formulation of ACOPF with the PSD matrix as the decision variable. In this method, the rank-1 PSD matrix constraint is reformulated to equality constraints: all 2 $\times$ 2 minors of the PSD matrix are zeros. The graph decomposition-based approach is implemented to reduce the computation burden.

In the second part of the work, the application of Benders' decomposition is investigated through two problems. The first problem is the Model Predict Control (MPC) problem for Modular Multilevel Converter (MMC). The objective of the MPC is to determine the best switching sequences for the submodules in the MMC to track the phase current references for multiple time horizons. The MPC is formulated as a nonlinear mixed-integer programming (MIP) problem with the on/off status of submodules as binary decision variables and MMC dynamic states such as phase currents, circulating currents and submodule capacitor voltages as continuous decision variables. With a large number of submodules and a large number of time horizons, the dimension of the nonlinear MIP problem is difficult to handle. Our contribution is to formulate this problem and solve this problem using Benders' decomposition. In the second problem, an efficient Benders' decomposition strategy is designed to solve the Security Constrained DCOPF (DC-SCOPF) with generator response constraints. The major difficulties to solve such SCOPF are the large number of contingencies and non-convexity of the generator response constraints. In this work, Benders' decomposition strategies were investigated to seek an efficient computing. We formulate the generator response constraints via bilinear expressing, and adopt Benders' decomposition to decompose the problem into a master problem with multiple sub-problems, each associated with a contingency. Based on the case study results, the proposed method has faster computing speed compares with the traditional big-M based mixed-integer linear programming method.

This dissertation has led to three journal papers and one conference paper.

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