A Branch and Bound algorithm to solve nonconvex MINLP problems via novel division strategy: An electric power system case study

This paper presents two Branch and Bound algorithms (B&B) for solving mixed-integer nonlinear programming (MINLP) problems with nonconvex search space. The main advantage of the proposed algorithms, comparing with the commonly used B&B algorithms, is using an innovative way of variables' separation and subproblems' division while, if necessary, one more variable is used in the separation process. This approach allows circumventing the probable difficulties caused by nonlinearity and nonconvexity. This paper aims at addressing the following issues of how to: 1) deal with nonlinear programming problems, 2) detect the infeasibility of the resulted NLP problems, and 3) deal with the nonconvexity of the problem. In order to show the applicability, the proposed algorithms are applied to one of the most complicated problems in power system, the long-term static transmission expansion planning, which is modeled as an MINLP problem. Several case studies such as Garver 6-bus, IEEE 24-bus, South Brazilian 46-bus, Bolivian 57-bus, and the Colombian 93-bus are conducted to reveal the effectiveness and shortcoming of the proposed algorithms. Results show that the proposed algorithms can find the best-known solutions for most of the aforementioned systems with a significant reduction in the number of subproblems.

I. Introduction

The Branch and Bound (B&B) algorithm was proposed in early 1960 by Land and Doig [1], and was used to solve the Traveling Salesman problem [2], [3]. Because of its capability in solving practical problems, which have finite but several feasible solutions, it has been widely used in operational-based researchers in different areas. This methodology is a well-distinguished and powerful tool to solve mixed-integer linear programming (MILP) problems.