Comparative study of optimization methods for optimal coordination of directional overcurrent relays with distributed generators

Due to the growing penetration of distributed generators (DGs), that are based on renewable energy, into the distribution network, it is necessary to address the coordination of directional overcurrent relays (DOCR) in the presence of these generators. This problem has been solved by many metaheuristic optimization techniques to obtain the optimal relay parameters and to have an optimal coordination of the protection relays by considering the coordination constraints. In this article, a comparative study of the optimization techniques proposed in the literature addresses the optimal coordination problem using digital DOCRs with standard properties according to IEC60-255. For this purpose, the three most efficient and robust optimization techniques, which are particle swarm optimization (PSO), genetic algorithm (GA) and differential evolution (DE), are considered. Simulations were performed using MATLAB R2021a by applying the optimization methods to an interconnected 9-bus and 15-bus power distribution systems. The obtained simulation results show that, in case of distributed generation, the best optimization method to solve the relay protection coordination problem is the differential evolution DE.


INTRODUCTION
The main role of protection relays is to detect and eliminate faults as quickly as possible by transmitting an opening command to the related circuit breaker. This circuit breaker isolates the faulty part of the network to ensure that the electrical equipment is not affected by the fault current [1]. The directional overcurrent relay (DOCR) is the most widely used type of relay in the coordination of protection relays due to their simplicity of application and their technical and economic characteristics [2]. The coordination of DOCRs protection has been considered a necessity for distribution networks, as it quickly isolates the faulty area, keeps the system safe and overcomes current faults so that the relays are reliable, flexible and selective [3]. In a properly coordinated system, the main relay must first function on overcurrent faults within a predefined time. After this predetermined time, known as the coordination time interval (CTI), the emergency relay must operate to isolate the default if the main one failed to trip [4]. Relay coordination is usually based on the evaluation of both fault currents and power flow. To optimize relay coordination, two important parameters are considered; relay settings which include the time dial setting (TDS) and the plug The coordination constraint should be satisfied for all primary/backup relay pairs (P/B). This constraint is indicated in (4). The CTI value depends on the type of relay (digital or electromechanical) and it varies between 0.2 and 0.5 s. The parameters tp and tb are respectively the running time of the main and emergency relays [23]. The reliability constraint is presented in (5), the relay must operate within a time margin, it must respond in a minimum time tmin and it must not exceed a maximum time tmax, the relay operating time generally varies between 0.1 and 4 s [21].
The sensitivity constraints are presented in (6) and (7). The parameters TDS and PS must respect the minimum values TDSmin and PSmin and the maximum values TDSmax and PSmax. The limits of TDS are generally 0.1 and 1.1 s [24]. The limits of PS are calculated using (8) and (9), where ILmax is the maximal load current and IFmin is the minimal fault current [5].

DOCRs coordination with DGs
Due to its various advantages, the integration of distributed generators into power grids has become more widespread in the global energy sector, but they also have negative impacts on the distribution network parameters in terms of load current and fault current level, which increases according to the capacity and location of DGs relative to the fault [10], [25]. Therefore, power grid protection systems may be affected. The type of DG and the characteristics of the distribution network have a significant effect on the coordination of the protection [11]. The consequences related to the connection of distributed generators to the network are nuisance tripping, blinding of protections and loss of coordination of the protection relays. This can lead to a decrease in system reliability and an increase in corrective maintenance costs [26]. Therefore, it is necessary to optimize the DOCRs protection relay parameters according to the new system configuration.

OPTIMIZATION METHODS FOR THE OPTIMAL DOCRs COORDINATION
This section presents the three most efficient and robust optimization methods used to solving DOCRs coordination, which are PSO, GA and DE. These algorithms have a randomly generalized initial population, in order to obtain the best solution by reaching the optimal point in the search space. x is the variable vector presented in (10); D is the dimensions of each element of the population, which is the number of variables and N is the population size.

Particle swarm optimization
Particle swarm optimization is an optimization approach based on the social behavior of birds and school fish, combined with the swarm intelligence. Individuals can perform extremely complex tasks when interacting with each other because each individual has little or no wisdom [20]. Each particle is initialized randomly with its velocity vj and its position xj. At each step, each particle moves in the D-dimensional search space according to three criteria: its best score (Pbset), the best score of all particles (Gbest) and random factors rand1 and rand2. In (11) and (12)  each iteration k. Where c1 is the personal learning coefficient, c2 is the global learning coefficient and w is the inertia weight.

Genetic algorithm
Darwin's natural selection theory was based on the GA to find optimal solutions that should be best suited to the objective function of the problem taking into account the constraints. At each iteration, the genes of each individual, which are the decision variables, undergo genetic operations (selection, crossover, mutation, and elitism) to generate new individuals better at solving the problem. In this algorithm, the individual is estimated and receives a score referring to its competence to execute the objective function and the constraints. The selection process consists in choosing in the middle of the randomly generated population a series of individuals, this process is totally random and does not favour choice within the population. In the crossing process, the two best individuals obtained during the selection process will be chosen as parents. The fundamental role of the crossover process is the exchanging of genetic information in order to increase the genetic variety between the population individuals. The process of mutation inserts diversity in the population, it allows the creation of new genetic traits that are not present in any previous generation individual, which ensures the best research in the resolution space of the system. The crossover factor (CF) represents the probabilities that pairs of chromosomes will produce offspring and the mutation factor (MF) represents the probabilities of a change in status of a chromosome [7].

Differential evolution
The DE algorithm represents a simple and efficient evolutionary algorithm based on natural gene selection. The DE algorithm has been shown to be faster than other evolutionary algorithms since it involves less mathematical operations and execution time [13]. An initial population is first randomly generated. For each population element, a mutant vector is created using the (13), with a1, a2 and a3 ϵ {1, 2,…, N} are three mutually different random indices and F is the mutation factor that regulates the differential variation amplification 23 kk a , j a , j . A crossover is inserted to increase the variety of perturbed parameter vectors to obtain the test vector. Crossover execution on the test solution is performed using the crossover rate (CR) and the random index randk where randk equals randi(D), as expressed in (14) [27], the Selection of the trial solution is made using system at (15), with TFi is the trial fitness. ,

RESULTS AND DISCUSION
To ensure coordination of DOCRs in distribution networks with integrated DGs, the optimization methods described in the previous section are applied to interconnected 9-bus and 15-bus distribution systems with digital protection relays and standard characteristics. The PS limits are calculated using (8) and (9), the TDS boundaries are 0.1 and 1.1 s and The CTI value for both networks is 0.2 s. The relay operating time limits are 0.1 and 4 s. Comparative study of optimization methods for optimal coordination of … (Zineb El Idrissi) 213 and 24 protection relays (R1, R2, . . ., R24). They have 44 pairs of P/B relays between them. The CTR is 500/1 for all relays. Table 1 gives the optimal adjustments of the 24 protection relays, which are PS and TDS, obtained by PSO, GA and DE optimization methods, while Table 2 shows the values for the running time of the main and the emergency relays tp and tb, as well as the coordination time interval corresponding to 44 P/B relay combinations for this optimization approach. The last two rows of Table 1 show the objective function (OF) and the time of convergence for each method. The objective function is the total of the running times of all main and emergency relays with the obtained optimal settings. The CTI between the main and emergency relay running times of the 44 pairs of P/B relays for the different methods is illustrated in Figure 2.

Distribution system 15 bus
The 15-bus interconnected distribution system presents an example of a high DG penetration distribution system. Six 15 MVA generators with 15% synchronous reactance are connected to buses 1, 3, 4, 6, 13 and 15. Therefore, this system is composed of 42 directional overcurrent relays and 82 pairs of main/emergency relays with 84 decision variables including 42 variables for TDS and 42 variables for PS. More details about this system are provided in [28], [29]. This information contains the fault current values, the current transformation ratio of the relays as well as the P/B pairs of relays. Tables 3 and 4 show respectively the optimal relay setting of the 42 protection relays, the primary relay running time tp and the emergency relay running time tb as well as the CTI values corresponding to 82 P/B relay combinations, for the the proposed optimization methods. Figure 3 illustrates the plot of the CTI values. The last two rows of Table 3 show the objective function and convergence time for each method. It is observed from Table 3 that the methods applied to this network give optimal results that respect the sensitivity constraint given that the TDS and PS parameters are within the previously mentioned limits. However, it is also observed that the value of OF is the smallest for DE compared to PSO and GA as well as the convergence time is shorter for this method. From Table 4, it can be seen that the running times of all main and emergency relays are greater than 0.1 s and less than 4 s and the coordination constraint is greater than or equal to 0.2 s for all primary and emergency relay combinations for the studied methods, which means that the reliability and coordination constraints are well respected. Table 4 and Figure 3 show that the CTI values for DE have the smallest values, for the 44 P/B relay pairs, compared to the other two methods. In Figure 3, it can also be observed that the CTI values obtained by DE vary between 0 and 1, while the CTI values of PSO and GA methods belong to the interval [0,2], then it can be concluded that the best optimization technique for solving the relay protection coordination problem is differential evolution DE.

CONCLUSION
This paper proposes three different optimization methods, PSO, GA and DE dealing with the problem of coordination of directional overcurrent relays. These techniques are applied on two distribution networks with 9 and 15 buses integrating distributed generators in order to determine the most efficient 218 method to solve this problem with the integration of DGs, the objective function and the time of convergence obtained by each method are compared between them. The comparative analysis shows that the differential evolution gives optimal values of the objective function and a shorter convergence time compared to the other methods for both distribution networks. Even more, the CTI values obtained by DE are found to be the most optimal, which explains the choice of DE as the method that offers the most satisfactory results among the methods investigated in this work. Therefore, DE can be regarded as the most efficient method to reach the best solution respecting the constraint of coordination between relays in the presence of DGs.