The Branch-and-Bound Algorithm in Optimizing Mathematical Programming Models to Achieve Power Grid Observability

Abstract

Phasor Measurement Units (PMUs) are the backbone of smart grids that are able to measure power system observability in real-time. The deployment of synchronized sensors in power networks opens up the advantage of real-time monitoring of the network state. An optimal number of PMUs must be installed to ensure system observability. For that reason, an objective function is minimized, reflecting the cost of PMU installation around the power grid. As a result, a minimization model is declared where the objective function is defined over an adequate number of constraints on a binary decision variable domain. To achieve maximum network observability, there is a need to find the best number of PMUs and put them in appropriate locations around the power grid. Hence, maximization models are declared in a decision-making way to obtain optimality satisfying a guaranteed stopping and optimality criteria. The best performance metrics are achieved using binary integer, semi-definite, and binary polynomial models to encounter the optimal number of PMUs with suitable PMU positioning sites. All optimization models are implemented with powerful optimization solvers in MATLAB to obtain the global solution point.

Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

“…Hence, it is still significant to optimize/minimize the number of PMUs placed within the power grid transmission system [36][37][38]41,42,51]. The optimization output gives an appropriate number of PMUs that is sufficient to monitor the power system [37,38,41,51].…”

“…Hence, it is still significant to optimize/minimize the number of PMUs placed within the power grid transmission system [36][37][38]41,42,51]. The optimization output gives an appropriate number of PMUs that is sufficient to monitor the power system [37,38,41,51]. In practice, PMU manufacturers yield a few types of PMU, with distinct functions and varying channel capacities [56,[59][60][61][63][64][65][66][67].…”

“…Mathematical algorithms are utilized in the OPP problem solving [31][32][33][34][35][36][37][38][39][40][41][42], whereas heuristic algorithms are also implemented to handle the integer program's inequality to examine the solution, thereby yielding heuristically sound knowledge [39,[43][44][45][46][47][48][49][50][51][52][53][54][55].…”

“…Many research works on OPP rely on a cumulative single objective function that is minimized subject to a constraint function defined on specific decision bounds [32][33][34][35][36][37][38][39][40][41]. On the contrary, a two-product objective minimization model is proposed in [42,47,49,51] to find those optima points with a maximum system observability index declared in [33,37,38,41,42,51].…”

“…The optimization problem in [37,38,[42][43][44][45]47,49,51] considers obtaining more than one objective product. This multi-objective function consists of two conflicting products for the consideration of those solutions satisfying complete and maximum observability at the same time.…”