On the Uniqueness of the Optimal Flow Distribution LP Solution in an Electrical Network

In [1], a mathematical model for minimizing the power moment in an electrical network was developed, taking into account transmission line capacities, balance relations at the main network nodes, and load levels of energy nodes. The model is formulated as a mixed-integer linear programming problem, where the integer variables are binary and determine one of the possible directions of power transmission along the network lines. This paper presents a mathematical model of the same problem in the form of a linear programming (LP) problem and shows that the optimal power flows along the network edges are transmitted in only one direction, as required for power transmission in electrical energy systems. An algorithm for verifying the uniqueness of the solution to the resulting LP problem is proposed and tested on a sample case. It is based on comparing the solutions of two quadratic programming problems with separable strictly convex quadratic objective functions, each of which has a unique solution. The paper is organized as follows. In Section 1, the LP problem for flow distribution with respect to the minimum power moment in the electrical network is formulated. Section 2 investigates the properties of the LP problem related to power flow transmission in the network. Section 3 describes a quadratic programming problem and, based on it, justifies a criterion for determining the uniqueness of the LP solution. Section 4 presents the results of computational experiments for a model electrical network performed using the Gurobi solver. The developed mathematical model can be used for balancing regional power systems under conditions of de-energization of main nodes by setting the corresponding node power values to zero. The proposed approach can also be applied to modeling a wide class of network systems, such as flow control problems in energy, transportation, and logistics networks.