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

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Abstract

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 (SM). In this way, SM 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.

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Acknowledgements

The authors are grateful to Yelitza Sorrentino for her help in refining details of English language of this article.

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Authors and Affiliations

  1. Dpto. de Ingeniería Eléctrica, Universidad Carlos III de Madrid, Calle Butarque 15, Leganés, 28911, Madrid, Spain

    Elmer Sorrentino

  2. Universidad Simón Bolívar, Caracas, Venezuela

    Elmer Sorrentino

  3. Universidad de Los Andes, Bogotá, Colombia

    Paulo M. De Oliveira-De Jesus

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  1. Elmer Sorrentino
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  2. Paulo M. De Oliveira-De Jesus
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Elmer Sorrentino had the original idea of formulating the coordination of protections as a multi-objective problem, and both authors participated in all the other stages of this research.

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Correspondence to Elmer Sorrentino.

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Appendix: Example of behavior of other speed and selectivity indexes

Appendix: Example of behavior of other speed and selectivity indexes

The assessment tool shown in [49] was fed with the optimal settings obtained for Case C of Fig. 3, in order to compute other speed and selectivity indexes (ν and ξ, respectively). ν is the average speed of the main DOCPi, computed as the average of 1/ψik for 1000 uniformly distributed faults along each line (ψik is the tripping time of DOCPi, considering transient configurations). ξ is a selectivity index, computed as the number of faults with SM ≥ 0.3 s in percentage of 20,000 analyzed faults (corresponding to 20 main backup pairs and 1000 uniformly distributed faults along each transmission line).

Figure 4 shows that the graphs of ξ and ν, as a function of SM and νI, respectively, are monotonically increasing functions. This fact means that the increase in one speed index implies the increase in the other speed index, as well as the increase in one selectivity index implies the increase in the other selectivity index. That is, regardless of individual (and/or subjective) preferences about the selection of these indexes, any of them can be reliably selected as a valid index for the analyzed objectives. Actually, the selectivity index ξ has a saturation level (100%); this fact could be seen as a limitation related to this selectivity index, but this limitation could be easily avoided if the selectivity margin to compute ξ is selected to a convenient greater value (i.e., selecting SM > 0.3 s to compute ξ).

On the other hand, the graphs shown in Fig. 4 depend on the type of curve. This fact implies that the comparison among results with different curve types should be performed carefully, considering that the analysis of the specific index for one objective should probably be complemented with the analysis of other factors. The need for a wide perspective to compare the results with different curve types has been previously shown [25, 49].

Fig. 4
figure 4

Example of behavior of other speed and selectivity indexes in the Pareto frontiers for Case C (ξ in percentage; SM in s; ν and νI in 1/s). IEC EI curve (blue); IEC VI curve (red); IEC NI curve (black)

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Sorrentino, E., De Oliveira-De Jesus, P.M. Coordination of protections as a multi-objective problem, and Pareto frontiers relating speed and selectivity in the particular case of directional overcurrent functions. Electr Eng (2024). https://doi.org/10.1007/s00202-024-02374-z

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