Abstract
This paper presents the solution of directional overcurrent relay (DOCR) problems using Simulated Annealing based Symbiotic Organism Search (SASOS). The objective function of the problem is to minimize the sum of the operating times of all primary relays. The DOCR problem is nonlinear and highly constrained with two types of decision variables, namely, the time dial settings (TDS) and plug setting (PS). In this paper, three models of the problem are considered, the IEEE 3-bus, 4-bus, and 6-bus, respectively. We have applied SASOS to solve the problem and the obtained results are compared with other algorithms available in the literature.
1. Introduction
Due to rapidly growing power systems, the stability and security issues are highly important for the power system researchers [1–3]. The protection systems are mainly used to detect and clear faults as fast and selective as possible [4–6]. The protection relays are used for detecting the faults in the system and to detach the faulty parts from the system in real time. Proper coordination of relays is essential to maintain the appropriate operation of the overall protection system. There are various types of relays with different operating principles. An example of these relays, which are used as a good technical tool for the protection of power systems, is directional overcurrent relay [7–9]. Such a relay is divided into two units, that is, the instantaneous unit and the time overcurrent unit. The parameters to be defined in the overcurrent unit are the time dial settings (TDS) and the plug settings (PS). Computers have made the huge calculations in DOCR problems in power systems easy [10, 11]. Different optimization algorithms are used to solve the relays coordination problems in which the objective function is to minimize activity time of all main relays. The constraints of this optimization problem are considered in the second layer of relay, which should respond, if the main layer of relay fails to operate on nearby fault. This depends on the variables, TDS, PS, and the minimized working time of relay. There is a nonlinear relationship between the operating time of overcurrent relays, TDS, and PS.
In electrical engineering, the power system engineering has the longest history among all other areas. Ever since, different numerical optimization techniques have been applied to power systems engineering and played an important role [12]. Optimization problems are usually nonlinear, which have nonlinear objective functions and constraints [13, 14].
Nowadays, researchers use different optimization algorithms to find the optimal solutions for the problems of relays settings and coordination. Examples of these optimization algorithms are Evolutionary Algorithm (EA) [15], Differential Evolution (DE) [16], Modified Differential Evolution (MDE) [17], Self-Adaptive Differential Evolutionary (SADE) [18], Particle Swarm Optimization (PSO) [19], Modified Particle Swarm Optimizer [20, 21], Evolutionary Particle Swarm Optimization (EPSO) [22], Box-Muller Harmony Search (BMHS) [23], Zero-One Integer Programming (ZOIP) approach [24, 25], Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [26], Seeker Algorithm [27], Chaotic Differential Evolution Algorithm (CDEA) [28], Adaptive Differential Evolution [29], Artificial Bee Colony (ABC) [30], Firefly Optimization Algorithm (FOA) [31], Modified Swarm Firefly Algorithm (MSFA) [32], and Biogeography Based Optimization (BBO) [33]. BFOA has been applied to obtain the optimal location and size of multiple distributed generators (DG) [34], optimal placement and sizing of DG [35], power system harmonics estimation [36], distribution systems reconfiguration for loss minimization [37], minimum load balancing index for distribution system [38], power system stabilizer for the suppression of oscillations [39], and optimum economic load dispatch [40].
During the last few years, PSO algorithm has been applied for Optimal Power Flow (OPF) control in power systems [41], OPF problem with FACTS devices [42], economic dispatch problems [43], optimal sizing and placement of DG [44], optimal location and sizing of static synchronous series compensator [45], and optimized controller design of energy storage devices [45, 46].
This paper proposes the use of a hybrid optimization technique—namely, Simulated Annealing based Symbiotic Organism Search (SASOS) [1, 47]—to find the optimal solutions for the relays settings. This algorithm is applied to different models of the DOCR problems such as the IEEE 3-bus, 4-bus, and 6-bus models. To check the efficiency of the proposed algorithm for the three cases, we have minimized the total activity time for each relay.
2. Problem Formulation
There are two important settings in each overcurrent relay for its satisfactory operations. The time dial settings (TDS) represent the activation time of each relay and the relay operation is decided by the plug settings (PS). The plug settings (PS) depend on the maximum load current and fault current due to short circuit. The main factors which control the total operating time of the relay are TDS and PS, and the fault current is represented by [15, 22], wherewhere denotes the fault current at the current transformer (CT) initial terminal and is the primary rating of CT. Constants , , and are assigned values 0.14, 0.02, and 1.0, respectively, and that is according to IEEE standards [26].
The current, seen by the relay, denoted by , is equal to the ratio between and , which is a nonlinear equation:
2.1. Objective Function
In coordination studies [22], the main objective is to minimize the total time taken in operation of primary relays for clearing a fault. The objective function takes the following form:wherewhere is the relay operation time to clear a near-end fault while is its operation time in case of a far end fault. and represent the relays fixed at the ends of the front line.
2.2. Constraints
The objective function is bound to the three constraints related to , , and .
Equation (5) represents the bound constraints on TDS: and are the lower and upper bounds for TDS, whose values are given by 0.05 and 1.1, respectively, while varies from 1 to .
The second constraint is PS of the relay that takes the following form: and are the lower and upper bounds of PS, whose values are given by 1.25 and 1.50, respectively, while varies from 1 to .
The third constraint is related to the fault current and pickup current. The operating time of relay depends on the type of relay. According to [24, 33], the operating time of relay is defined by and are the lower and upper bounds for relay functioning time, whose values are adopted as 0.05 and 1, respectively.
During the optimization procedure, the time taken by primary relays to coordinate with the backup relays is constrained as inwhere CTI represents the specified coordination time.
and represent the working time of primary and backup relays, which can be obtained using the following equations:
2.3. The Standard IEEE System of 3-Bus
There are six overcurrent relays in this model. According to the number of relays, the value assigned to each of and is 6 and the number of decision variables is 12, and this means to and the variables to . Figure 1 shows the 3-bus model.
Mathematically the objective function can be written aswhereThe values of constants , , , and are shown in Table 6.
Constraints imposed on the 3-bus system are as follows.
Limits on Variables . , where varies from 1 to 6.
Limits on Variables . , where varies from 1 to 6.
Another constraint limits each term of objective between 0.05 and 1.
Selectivity constraints on working time of backup relay and the primary relay are given in the following relation:where CTI takes the value 0.3. Here,The values of the constants for and of 3-bus model are given in Table 7.
2.4. The Standard IEEE System of 4-Bus
In the IEEE 4-bus model, there are 8 overcurrent relays. Accordingly, the value of and is 8, which is twice the number of transmission lines and the number of decision variables is 16. The 4-bus model is shown in Figure 2. CTI takes the value for this model as 0.3. Total selectivity constraints are 9.
Moreover, the values of constants for the 4-bus model are given in Tables 8 and 9.
2.5. The Standard IEEE System of 6-Bus
In this section, the IEEE 6-bus model is elaborated. Here, value of and is 14. Hence, there are 28 variables in this problem, namely, to and to . The 6-bus model is shown in Figure 3. CTI takes the value 0.2 for this model. There are 48 selectivity constraints in this problem. Based on the observation of [48], we have relaxed 10 constraints. The values of constants , , , and for Model 3 are given in Tables 10 and 11. A summary of all components involved in models of IEEE 3-bus, 4-bus, and 6-bus is presented in Table 5.
3. Hybridization of Simulated Annealing (SA) and Symbiotic Organism Search (SOS)
The hybrid algorithm is named as SASOS [47]. In SASOS algorithm, the new solutions are created using the three phases of SOS algorithm by shifting the earlier solutions towards randomly picked solutions from the system [49]. The convergence rate of SOS algorithm is slow due to its strategy for locating the global optima. If the updated solution is better than previous solution, the convergence will be faster. Therefore, to further improve the convergence rate, Simulated Annealing technique is employed in the mutualism and commensalism phase of the SOS algorithm. The parasitism stage is left unchanged [47]. A pseudo code for SASOS algorithm is given in Algorithm 1. The SA technique accepts poor neighbour solutions along with the current best solutions by implementing certain probability. In each generation, probability of selecting the bad solutions is higher at the initial stages but decreases at the later stages of the search. This probability of either selecting poor solutions or good ones is calculated as inwhere and are the fitness values of th solution and current best solutions, respectively, and here is the control parameter. Equation (15) is used to reduce the temperature in Simulated Annealing:
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4. Experimental Settings and Discussion
4.1. Experimental Settings
The parameters involved in SASOS are ecosize, initial temperature , final temperature , and cooling rate . The ecosize is taken as 50 for all the cases. The value of initial temperature is taken as 1, while the final temperature is and cooling rate is taken as .
For a fair comparison, a total of 30 simulations were performed for each of the three cases and the current best solution through the simulation was recorded as the current best solution.
4.2. Discussion
Performance of the SASOS algorithm is analyzed based on the best objective values (see Figures 5, 7, and 9) and total function evaluations taken by algorithms (see Figures 4, 6, and 8). The outcomes of SASOS are highlighted and compared with the values obtained by different versions of Differential Evolution and other algorithms available in the literature [17]. The results for 3-bus, 4-bus, and 6-bus systems are given in Tables 1, 2, and 3, respectively.
The best solutions obtained by SASOS and other versions of DE for IEEE 3-bus model in terms of best decision features, minimum objective function value, and number of function evaluations taken to complete each simulation are given in Table 1 and Figures 4 and 5. Here, it is evident that, in terms of objective function value, DE gave the worse objective value as and all the other versions of DE algorithm gave almost similar values. On the other hand, SASOS gave us an objective value which is much better than other algorithms. However, if we compare the number of function evaluations (NFE), then the performances of DE and its variants are worse as compared to SASOS algorithm as it took only NFE to complete the simulations. Thus, SASOS is significantly better than all the other algorithms under consideration.
The simulation results of IEEE 4-bus model are given in Table 2 and Figures 6 and 7, respectively. It is observed for the IEEE 4-bus model that DE algorithm and all the variants of DE algorithm performed in a similar manner in terms of objective function values with MDE-4 giving a slightly better value than other variant algorithms of DE. But SASOS outperformed the algorithms under consideration in terms of the number of function evaluations (NFE). The better performance was shown by SASOS, which took NFE to converge to a solution .
Furthermore, the results for IEEE 6-bus model as in Table 3 and Figures 8 and 9, in terms of best objective function value, are again slightly different to each other with MDE-4 and MDE-5 giving slightly improved solutions than other variant algorithms of DE. On the other hand, SASOS gave us an objective value which is much better than other results in the table. However, in terms of NFE, the worst convergence rate was shown by DE while SASOS took NFE to get the best solution presented in the table.
Summarily, it can be observed from Tables 1, 2, and 3 that SASOS provides better solutions to the given problems compared to DE. Also, the number of function evaluations (NFE) taken by SASOS algorithm is considerably reduced in comparison to DE algorithms. In Table 4, the objective function values obtained by SASOS are compared with the values obtained by other algorithms such as DE, RST2, SOMA, and SOMGA for all the three models, as reported in [50], LX-POL and LX-PM [51], and MDE algorithms [17]. It can be observed from Table 4 that SASOS provided better results for all the three systems among all results quoted in [17], in terms of best objective values.
5. Conclusion
We present a conclusion of this research as follows:(i)The problems of optimal coordination of directional overcurrent relays are highly nonlinear, NP-hard, and highly constrained in nature.(ii)Dealing with the DOCR problems, use of efficient metaheuristic is needed.(iii)SASOS is implemented to solve the problem of DOCRs for standard IEEE 3-, 4-, and 6-bus systems.(iv)The outcome of our simulations shows that SASOS can efficiently minimize all the three models of the problem.(v)The efficiency of SASOS can be observed from the minimum function evaluations required by the algorithm to reach the optimum as compared to other standard algorithms.(vi)In future, one can extend this work to solve problems of higher buses and complex power systems. Moreover, extensive statistical analysis and parameters tuning can further highlight and improve the efficiency of SASOS.
Nomenclature
| : | Constants according to IEEE standard |
| : | Cooling rate in Simulated Annealing |
| CT: | Current transformer |
| : | Primary rating of current transformer |
| CTI: | Coordination time interval |
| DE: | Differential Evolution |
| DOCR: | Directional overcurrent relays |
| : | Objective function |
| : | Fitness value at the ith solution |
| : | Current best solution |
| : | Fault current at the current transformer initial terminal |
| : | The current, seen by relay |
| MDE: | Modified Differential Evolution |
| : | Number of relays responding for close end fault |
| : | Number of relays responding for far-bus fault |
| PS: | Plug settings |
| SA: | Simulated Annealing |
| SOS: | Symbiotic Organism Search |
| : | Operating time of relay |
| : | Operating time of backup relay |
| : | Final temperature |
| : | Initial temperature |
| : | The relay operation time to clear near end fault |
| : | The relay operation time in case of far end fault |
| : | Operating time of primary relay |
| TDS: | Time dial settings |
| NFE: | Net function evaluations. |
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.