Coordination of protections as a multi-objective problem, and Pareto frontiers relating speed and selectivity in the particular case of directional overcurrent functions

This article shows the formal formulation of the coordination of protections as the multi-objective problem of simultaneous maximization of desired features of the protective system, which is a key a contribution of this article. From this perspective, the Pareto frontiers to correlate speed and selectivity in the Optimal Coordination of Directional Over-Current Protections (OC-DOCP) are explicitly shown, assuming that the maximum sensitivity is kept (i.e., only speed, selectivity and sensitivity as desired features are considered for the sake of simplicity). The proposed method to compute these Pareto frontiers is based on the obtaining of solutions for the traditional problem of OC-DOCP, formulated as the maximization of speed subject to selectivity constraints, using different selectivity margins (S_M). In this way, S_M simply becomes the selectivity index to be maximized, which complements the maximization of the speed index from the perspective of the multi-objective formulation. This method can be easily applied to many different formulations of the OC-DOCP. The Pareto frontiers are herein shown for a power system taken as an example, using the simplest formulation of the OC-DOCP as well as two formulations that consider the transient configurations due to sequential tripping at both line ends. The solutions for three different cases of standardized IEC curves are shown in the numerical examples. It is herein explicitly shown that an increase in speed implies a decrease of selectivity for the optimal solutions in the Pareto frontier, regardless of the applied OC-DOCP formulation, as theoretically expected. The presentation of Pareto frontiers relating speed and selectivity indexes, as two desired features of the protection system, as well as the method to obtain them in the case of OC-DOCP, are other contributions of this article.