Calculating Global Minimum Points to Binary Polynomial Optimization Problem: Optimizing the Optimal PMU Localization Problem as a Case-Study

Abstract

State estimation (SE) is an algorithmic function of an energy management system (EMS). SE provides an actual-time monitoring and control of modern electrical power grids. State Estimation can be worked with sufficiency using Phasor Measurement Units optimally placed within a power grid. This paper concerns the implementation of proper algorithms embedded in optimization solvers to the optimal PMU localization problem solving globally. The optimization model is formulated as a 0 - 1 nonlinear minimization problem. The problem is transformed to a polyhedron using linearization methods and B&B tree. In this model, we use a linear cost function under polynomial constraints and binary restrictions on the design variables in a symbolic format. This mathematical model is programmed in the YALMIP environment which is fully compatible with MATLAB. The 0 - 1 Nonlinear Programming (NLP) model is suitable for getting concisely global optimal solutions. The optimal solution is given by a wrapped optimization engine including a local optimizer routine performing together with a mixed-Integer-Linear Programming routine. The solution is achieved within a zero-gap precisely encountered during the iterative process. This tolerance criterion is a necessity for a successful implementation of the B&B tree because it ensures global optimality with an acceptance relative gap. The minimization model is implemented in a YALMIP code fully compatible with MATLAB in two stages. Initially, an objective function with one term is minimized to discover a number of sensors for wide-area monitoring, control and state estimator applications. Then, an extra product is considered in the objective to suffice maximum reliability for observing the network buses. The numerical minimization models are applied to standard power networks in the direction to be solved globally.

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.…”