Water cycle algorithm for optimal overcurrent relays coordination in electric power systems

Abstract

The coordination of overcurrent relays in interconnected mesh systems with many sources can be formulated as an optimization problem. Different conventional and heuristic algorithm-based optimization procedures have been presented to deal with this nonlinear highly constrained optimization problem. This paper presents an attempt to apply water cycle algorithm (WCA) in order to optimally deal with this coordination problem. The design variables contain the time dial, pickup current, and type of inverse characteristic of each relay. The viability of the proposed method is compared to other competing methods for different interconnected mesh systems including distributed generation units such as the 15-bus system. For obtaining a realistic study, the proposed WCA method is tested in solving the coordination problem for a detailed IEEE 30-bus system, which involves 111 industrial commercial relays type SEPAM-2000 and 333 design variables within the search space along with 726 inequality constraints. The IEEE 30-bus system is modeled using the Electrical Transient Analyzer Program. The strength of the WCA based on methodology is extensively confirmed using the simulation results and comprehensive comparisons.

Introduction

Coordination of the protective devices aims at obtaining the primary protective devices used to clear the fault in an expedite way for each fault location, and if the primary device fails, another backup protective device will operate (Saleh et al. 2015). These protection systems play an important role to ensure the security and operation of power networks (Papaspiliotopoulos et al. 2017). The protection schemes based on directional overcurrent relays (DOCRs) are considered as secondary or backup protection of transmission systems, cooperating with distance relays, and firstly or primary protection of distribution or sub-transmission systems (Mahari and Seyedi 2013). The high penetration of the distributed generation systems into the electric grids and smart grids can help in reconfiguration of the distribution networks to meshed sub-transmission systems (Camacho-Gómez et al. 2018; El-Fergany 2015, 2016a, b; Quadri et al. 2018). A method based on syntactic pattern recognition has been presented (Pavlatos et al. 2015). If the fault happens in these interconnected mesh systems, a bidirectional fault current will flow, which may result in a failure of overcurrent relays coordination. Therefore, DOCRs are implemented in these systems (Saleh et al. 2015). The main target of optimal coordination of overcurrent relays is to find the time dial \( \left( {T_{\text{D}} } \right) \), pickup current \( \left( {I_{\text{P}} } \right) \), and type of inverse curve of each relay taking into account the coordination and bounds constraints. The whole operating times of the primary relays are minimized to achieve the target. Therefore, the coordination problem can be dealt as an optimization problem.

Several optimization methods have been applied to solve the coordination problem. In Urdaneta et al. (1999), Chattopadhyay et al. (1996), and Saberi and Amraee (2017), the linear programming (LP) method is proposed to deal with the problem in a linear form. In this method, \( T_{\text{D}} \) of the relays is the control variable and \( I_{\text{P}} \) has a fixed value in acceptable ranges. Moreover, the nonlinear programming (NLP) and mixed integer NLP (MINLP) were successfully used to solve this problem, where \( T_{\text{D}} \) and \( I_{\text{P}} \) are considered as continuous design variables. This selection agrees with the settings of modern microprocessor-based protective relays. Although the NLP method exhibits better solutions to the coordination problem, it has some demerits such as its convergence to a local minima, its complexity, and its dependence on the initial values of the design variables (Albasri et al. 2015). The interior point method is presented for minimizing the operating times of primary \( \left( {{\text{OT}}_{\text{P}} } \right) \) and backup \( \left( {{\text{OT}}_{\text{PB}} } \right) \) relays simultaneously to realize the optimum settings of relays in Alam et al. (2016). Moreover, a novel method based on network splitting has been proposed with the aim of reducing the calculation time in Azari and Akhbari (2015).

The computational evolutionary algorithms such as genetic algorithm (GA) (Adelnia et al. 2015; Thakur and Kumar 2016), particle swarm optimization (Maancer et al. 2015; Srivastava et al. 2016), gravitational search algorithm (Srivastava et al. 2016), differential evolution (Moirangthem et al. 2013), firefly algorithm (Tjahjono et al. 2017), seeker algorithm (SA) in Amraee (2012), artificial bee colony algorithm (Singh et al. 2013), and biogeography-based optimization algorithm (Albasri et al. 2015) have been utilized to effectively solve this optimization problem. In addition, the succeeding recent articles reported in the literature (El-Fergany 2016a, b; Bouchekara et al. 2016; Shih et al. 2015, 2017; Saha et al. 2016; Huchel and Zeineldin 2016; Radosavljević and Jevtić 2016) applied further heuristic interesting algorithms to explore their features for solving optimal relay coordination problems. Among these promising algorithms are flower pollination algorithm (FPA) (El-Fergany 2016a, b), backtracking search algorithm 9 (Bouchekara et al. 2016), ant colony optimizer (Shih et al. 2015), symbiotic organism search algorithm (Saha et al. 2016) and many more (Huchel and Zeineldin 2016; Shih et al. 2017; Radosavljević and Jevtić 2016; Costa et al. 2017; El-Fergany and Hasanien 2017).

From the above literature survey, it can be emphasized that little works are done to deal with a large-scale power systems to solve coordination problem such as the IEEE 30-bus network. In addition, it has been proved that there are some differences among these evolutionary algorithms involving the convergence speed, computational time, and procedures complexity. The tremendous development of these algorithms indicates the authors’ impetus to apply the proposed water cycle algorithm (WCA) to solve several engineering problems. This algorithm represents one of the newest metaheuristic algorithms, and it is motivated by the water life cycle in nature. The WCA demonstrates flow of rivers and imitates their behaviors to finally reach the seas. The WCA points out these rivers are establishing and travel to the seas. The proposed algorithm was proposed by Hadi Eskander in Eskandar et al. (2012) and Sadollah et al. (2015). It possesses several merits as compared with other optimization algorithms like its high velocity of convergence plus the minimum number of variables that are needed to fine-tune. The WCA was utilized in solving several optimization electrical engineering problems in industry (Elhameed and El-Fergany 2016, 2017; Hadjaissa et al. 2018; Hasanien and Matar 2018; Hasanien 2018).

In this paper, WCA is utilized optimally in tackling the coordination problem of DOCRs in several interconnected mesh systems. The design variables of the optimization problem include TD, IP, and curve type characteristic of each relay. The viability of the proposed WCA-based coordination method is compared with other optimization methods for different interconnected mesh systems such as the 15-bus system. For obtaining a realistic study, the proposed WCA method is tested to solve the coordination problem for a detailed IEEE 30-bus system, which involves a large number of design variables within the search space. This represents a significant challenge to WCA to deal with a large complex coordination problem. The IEEE 30-bus system is modeled using the Electrical Transient Analyzer Program (ETAP). The load current of the relays and near-end fault currents of the \( P \) and \( {\text{PB}} \) relays are investigated. Moreover, real industrial protective relays are implemented through this study. The cogency of the proposed WCA-based method is extensively confirmed using the numerical results, which are carried out using MATLAB platform.

Problem definition and mathematical formulation

Based on the IEC standard, the time–current characteristics (TCCs) of overcurrent relays can be described, in general, by the following equation:

$$ {\text{OT}} = \frac{{T_{\text{D}} \cdot \alpha }}{{\left( {\frac{{I_{\text{F}} }}{{I_{\text{P}} }}} \right)^{n} - 1}} $$
(1)

where OT defines the relay tripping time (s) and \( I_{\text{F}} \) defines the fault current (A). α and n are fixed values, which depend on the type of relay characteristics. Generally, these characteristics are long inverse (LI), extremely inverse (EI), very inverse (VI), and normally inverse (NI). The values of α and n are illustrated in (Mahari and Seyedi 2013; El-Fergany and Hasanien 2017).

The coordination of overcurrent relays can be expressed as a nonlinear optimization problem, which possesses an objective function and set of constraints (El-Fergany 2016a, b; Shih et al. 2015). The ultimate target of this formulation is to minimize the total operating time of main relays for faults occurrence at their primary protection area. It is worthy emphasizing that the \( {\text{OT}}_{\text{P}} \) and \( {\text{OT}}_{\text{BP}} \) are strongly interrelated together by the coordination time interval (\( {\text{CTI}} \)). Thus, the objective function (OF) can be written as depicted in (2).

$$ {\text{OF}} = {\text{minimize}}\left( {\mathop \sum \limits_{i = 1}^{N} {\text{OT}}_{{{\text{P}},i}} } \right) $$
(2)

where \( {\text{OT}}_{{{\text{P}},i}} \) is the operating time of ith primary relay for a fault taking place close to that relay and \( N \) equals the number of main relays along the system under study.

This objective function is subjected to set of constraints, which can be defined in the subsequent sections:

Relay setting constraints

The relay setting constraints consist of the boundary constraints of \( T_{\text{D}} \) and \( I_{\text{P}} \), which can be represented as depicted in (3) and (4), respectively. On the other hand, the type of relay characteristics is selected as defined in (5).

$$ T_{{{\text{D}}i}}^{ \hbox{min} } \le T_{{{\text{D}}i}} \le T_{{{\text{D}}i}}^{ \hbox{max} } \quad \forall i \in N $$
(3)
$$ I_{{{\text{P}}i}}^{ \hbox{min} } \le I_{{{\text{P}}i}} \le I_{{{\text{P}}i}}^{ \hbox{max} } \quad \forall i \in N $$
(4)
$$ {\text{Relay}}\;{\text{curve}}_{i} \in \left\{ {\text{NI, VI, EI, LI}} \right\} \quad \forall i \in N $$
(5)

where \( T_{{{\text{D}}i}} \) is the time dial of ith relay and \( T_{{{\text{D}}i}}^{ \hbox{min} } \) and \( T_{{{\text{D}}i}}^{ \hbox{max} } \) are the minimum and maximum time dials of ith relay, respectively. \( I_{{{\text{P}}i}} \) is the pickup current of ith relay, and \( I_{{{\text{P}}i}}^{ \hbox{min} } \) and \( I_{{{\text{P}}i}}^{ \hbox{max} } \) are the minimum and maximum pickup currents of ith relay. It is worthy to emphasize that \( I_{{{\text{P}}i}} \) should be greater than the maximum load current (\( I_{\text{L}} \)) to enable the full capacity operation of the ith equipment.

Coordination constraints

The coordination constraints refer to adjustments that should be done between the \( {\text{OT}}_{\text{P}} \) and \( {\text{OT}}_{\text{PB}} \) relays. The \( {\text{OT}}_{\text{P}} \) is less than the \( {\text{OT}}_{\text{PB}} \) by the predefined \( {\text{CTI}} \) to ensure the good selectivity of the protection scheme. This coordination can be described by the formula shown in (6).

$$ {\text{OT}}_{{{\text{BP}},k}} \ge {\text{OT}}_{{{\text{P}},k}} + {\text{CTI}}_{k} \quad \forall k \in {\text{RP}} $$
(6)

where \( {\text{OT}}_{{{\text{BP}},k}} \) and \( {\text{OT}}_{{{\text{P}},k}} \) are the operating times of backup and main relays of kth relay pair, respectively, and \( {\text{RP}} \) defines the entire predefined relay pairs.

Minimum OT

To ensure the practical limitation of the relay minimum operating:

$$ {\text{OT}}_{{{\text{P}},i}} \ge T_{ \hbox{min} } \quad \forall i \in N $$
(7)

where \( T_{ \hbox{min} } \) is the minimum operating time of the relay along the network.

It is important to disclose that the design control variables stated in (3)–(5) are self-constrained by WCA. However, in an attempt to respect the inequality constraints specified in (6) and (7), a penalty function is superimposed to the objective function stated in (2) to produce feasible solutions within acceptable predefined limits. To implement such penalties (\( {\text{Pen}} \)), the formula depicted in (8) is used.

$$ {\text{Pen}} = \psi_{T} \mathop \sum \limits_{i = 1}^{N} \left| {{ \hbox{min} }\left( {{\text{OT}}_{{{\text{P}},i}} - T_{ \hbox{min} } , 0} \right)} \right| + \psi_{\text{CTI}} \mathop \sum \limits_{K = 1}^{\text{RP}} \left| {{ \hbox{min} }\left( {\left\{ {{\text{OT}}_{{{\text{BP}},k}} - {\text{OT}}_{{{\text{P}},k}} } \right\} - {\text{CTI}}_{k} , 0} \right)} \right| $$
(8)

where \( \psi_{T} \) and \( \psi_{\text{CTI}} \) are constants with large values.

Water cycle algorithm

In our life, rivers are formed by fall of rains/snows on mountains/hills. The river flows from the mountains. All branches of dropped water are combined together to create a river, which can flow from mountains into the sea. In addition, the water of rivers evaporates according to the environmental conditions into the atmosphere to create clouds in the sky. In the winter season, the steam is condensed and it comes back to earth forming rains. Actually, these statements represent the water cycle in nature.

The proposed algorithm as all metaheuristic algorithms starts with random populations (raindrops) through the search field. In this algorithm, the best candidate solution can be selected as a sea. The better raindrops can be chosen as rivers, and their rest are selected as streams. The raindrop can be mathematically formulated as follows (Eskandar et al. 2012; Sadollah et al. 2015):

$$ {\text{Raindrop}} = \left[ {x_{1} , x_{2} ,x_{3} , \ldots ,x_{D} } \right] $$
(9)

where D represents control variables number and x is the control variable value. In general, these raindrops can be demonstrated as a matrix Np× D, and NP represents the population number, as follows:

$$ {\text{Population}} = \left[ {x_{ij} } \right]_{{N_{\text{p}} \times D}} \quad \forall i \in N_{\text{p}} , \forall j \in D $$
(10)

In the WCA, each population or raindrop is assessed by the fitness function, which can be formulated as follows:

$$ C_{i} = f\left( {x_{1}^{i} , x_{2}^{i} , \ldots ,x_{D}^{i} } \right) \quad \forall i \in N_{\text{p}} $$
(11)

In the starting process of the WCA, \( N_{\text{p}} \) is specified. A number of the best populations \( N_{\text{sr}} \), which possess lower values, are chosen as number of rivers \( N_{\text{r}} \) plus one sea. The rest of these raindrops represent the streams number, which may form rivers or go to the sea. The formulation of these statements is as follows (Sadollah et al. 2015):

$$ N_{\text{sr}} = N_{\text{r}} + 1 $$
(12)
$$ N_{\text{streams}} = N_{\text{p}} - N_{\text{sr}} $$
(13)

To assign streams to certain rivers/sea based on the intensity flow, (14) is used for this purpose.

$$ {\text{NS}}_{n} = {\text{round}}\left\{ {\left| {\frac{{C_{n} }}{{\mathop \sum \nolimits_{i = 1}^{{N_{\text{sr}} }} C_{i} }}} \right| \times N_{\text{Streams}} } \right\}, \quad \forall n \in N_{\text{sr}} $$
(14)

The randomly chosen distance between the stream and river is as follows:

$$ X \in \left( {0, C \times d} \right), C > 1 $$
(15)

where C ϵ [1, 2] and its best value is 2 and d is the current distance between the stream and river. The updated position of them can be formulated as follows:

$$ X_{\text{stream}}^{i + 1} = X_{\text{stream}}^{i} + {\text{rand}} \times C \times \left( {X_{\text{river}}^{i} - X_{\text{stream}}^{i} } \right) $$
(16)
$$ X_{\text{river}}^{i + 1} = X_{\text{river}}^{i} + {\text{rand}} \times C \times \left( {X_{\text{sea}}^{i} - X_{\text{river}}^{i} } \right) $$
(17)

where rand is a random number ϵ [0, 1]. If the stream solution is lower than that of river, their positions will be replaced. Moreover, if the river solution is lower than that of the sea, their positions will be replaced.

In the proposed algorithm, the sea water is evaporated and the rivers/streams flow to the sea. The following criterion is applied to perform this evaporation condition:

$$ \begin{aligned} & if\; X_{\text{Sea}}^{i} - X_{\text{River}}^{i} < d_{ \hbox{max} } \quad \forall i \in \left( {N_{\text{sr}} - 1} \right) \\ & {\text{Carry }}\;{\text{out }}\;{\text{evaporation }}\;{\text{and}}\; {\text{raining }}\;{\text{process}} \\ & {\text{End}} \\ \end{aligned} $$

where \( d_{ \hbox{max} } \) is a small value near to zero and controls the search of the optimal solution. \( d_{ \hbox{max} } \) is updated as follows:

$$ d_{ \hbox{max} }^{i + 1} = d_{ \hbox{max} }^{i} - \frac{{d_{ \hbox{max} }^{i} }}{{{\text{Max\_iter}}}} $$
(18)

Max_iter represents the iterations number.

The raining process is implemented after finishing the evaporation process. In the raining process, new streams can be formed in several locations, as follows:

$$ X_{\text{Stream}}^{\text{new}} = {\text{LB}} + {\text{rand}} \times \left( {{\text{UB}} - {\text{LB}}} \right) $$
(19)

where \( {\text{LB}} \) and \( {\text{UB}} \) are lower and upper boundaries. The streams that go to the sea and help in flowing to the best solution are modified as pointed out in the following equation:

$$ X_{\text{Stream}}^{\text{new}} = X_{\text{sea}} + \sqrt \mu \cdot {\text{rand}}\left( {1, D} \right) $$
(20)

where µ is a small value, which suitably equals 0.1 such that the search closes to the sea area.

A detailed flowchart of the proposed WCA is demonstrated in Fig. 1.

Fig. 1
figure1

Flowchart of WCA

Simulation results and discussion

In this study, the proposed WCA is utilized to deal with this adapted coordination problem of DOCRs and non-DOCRs for two meshed power systems such as the 15-bus and detailed IEEE 30-bus systems. The demonstrated results are extensively carried out using MATLAB environment. Both \( T_{\text{D}} \) and \( I_{\text{P}} \) are considered continuous design variables, which agree with recent digital microprocessor-based relays, and the TCC of the relay is a discrete design variable. The typical values of \( \psi_{T} \) and \( \psi_{\text{CTI}} \) are 100,000 and 10,000, respectively for the two test cases. The simulation results are investigated in the form of different scenarios as follows:

The 15-bus transmission system

The one-line diagram of the 15-bus transmission network is illustrated in Fig. 2. The data of the system are given in Amraee (2012) and El-Fergany (2016a, b). \( T_{{{\text{D}}i}}^{ \hbox{min} } \) and \( T_{{{\text{D}}i}}^{ \hbox{max} } \) are selected as 0.01 and 1s, and \( I_{{{\text{P}}i}}^{ \hbox{min} } \) and \( I_{{{\text{P}}i}}^{ \hbox{max} } \) are equal to 100% and 150% of \( I_{\text{L}} \), respectively. For this test case, the optimal parameters of the proposed algorithm involve \( N_{\text{P}} \) = 100, \( N_{\text{r}} \) = 8, \( d_{ \hbox{max} } \) = 1e−7, and \( {\text{Max}}\_{\text{iter}} \) = 2000. The parameters are established by the trial-and-error procedure, which is a well-known procedure that is applied to adjust algorithm parameters. Two cases are considered to the system under study: Case (1) represents the coordination problem with variable TCC of the relays and case (2) cares with fixed NI TCCs. For obtaining a realistic study, the lowest operating time of the relays equals to 50 ms. The optimal values of design variables using the proposed WCA are listed in Table 1. The objective function convergence is indicated in Fig. 3a, b. It can be noted here that the convergence of fitness function is fast and smooth for both cases. The total operating time of the main relays in the entire system for both cases records 3.45 and 10.89 s, respectively. Tables 2 and 3 point out the \( {\text{OT}}_{\text{P}} \), \( {\text{OT}}_{\text{BP}} \), and computed CTI for these cases, respectively. It is worth noting that no violations are found in the constraints and the CTI is within the defined range.

Fig. 2
figure2

One-line diagram of the 15-bus transmission network

Table 1 Optimal/best values of the design variables for the 15-bus network with \( T_{ \hbox{min} } = 50\, {\text{ms}} \)
Fig. 3
figure3

Fitness value convergence of the 15-bus transmission system with \( T_{ \hbox{min} } = 50\, {\text{ms}} \): a Case (1), b Case (2)

Table 2 Operating times of relays with their associated CTIs for case (1) of the 15-bus system with \( T_{ \hbox{min} } = 50\, {\text{ms}} \)
Table 3 Operating times of relays with their associated CTIs for case (2) of the 15-bus system with \( T_{ \hbox{min} } = 50\, {\text{ms}} \)

In addition, the effectiveness of the proposed WCA method is verified by making a fair comparison among its results and those obtained from other optimization methods. In this comparison, the minimum operating time of the relay is chosen 0.01 s for case (1) in order to compare the results of WCA with that obtained using other optimization methods in the literature under the same conditions. Table 4 illustrates the detailed comparison of these results. It can be realized that the total operating times of the primary relays using WCA method are superior to those by using other optimization methods such as SA (Amraee 2012), FPA (El-Fergany 2016a, b), and the stochastic fractal search algorithm (SFSA) (El-Fergany and Hasanien 2017). The authors would like to highlight that FPA (El-Fergany 2016a, b) and SFSA (El-Fergany and Hasanien 2017) have employed different objective functions to achieve the overcurrent coordination and the comparisons are made with them for the sum of \( {\text{OT}}_{\text{P}} \) along the network irrespective of the objective function used. Therefore, WCA method is successfully capable of dealing with the formulated problem of the DOCRs.

Table 4 Comparisons among the results of WCA method and other optimization methods for the 15-bus system

The standard IEEE 30-bus system

This scenario aims to test the usefulness of the WCA method when dealing with a complex system and try to solve the coordination problem of such system. Accordingly, the IEEE 30-bus standard network is selected for this case study, and its detailed standard data are collected as indicated in https://www.ee.washington.edu/research/pstca/pf30/pg_tca30bus.htm. The IEEE 30-bus system consists of 6 synchronous generating units having 6 non-DOCRs, 41 lines having 82 DOCRs, two synchronous condensers with two non-DOCRs, and 21 loads having 21 non-DOCRs, as shown in Fig. 4. Therefore, the system has a total number of 111 relays to be coordinated, which represents a significant challenge to the proposed methodology and reveals a good realistic study.

Fig. 4
figure4

One-line diagram of the IEEE 30-bus standard network

The proposed system is modeled using ETAP (http://www.etap.com) to estimate \( I_{\text{L}} \) and to calculate maximum near-fault currents \( \left( {I_{\text{F}} } \right) \) for various relay pairs. The values of \( I_{\text{L}} \) and \( I_{\text{F}} \) for relays and various relay pairs are enumerated in Tables 5 and 6, respectively. \( T_{\text{D}} \), \( I_{\text{P}} \), and the TCC of each relay are the design variables. Therefore, this coordination problem possesses 333 design mixing continuous and discrete variables within the search space along with 393 inequality constraints (282 for CTI’s and 111 for \( T_{ \hbox{min} } \)). As a result of this formulation, a total of 726 inequality constraints should be respected [Among them, 333 are self-constrained by WCA and others are achieved by the proposed penalties as indicated in (8)]. For obtaining a realistic study, an industrial overcurrent relay of Schneider Electric (Sepam-2000) is selected for this purpose (http://mt.schneider-electric.be/OP_MAIN/Sepam/3140750UK.pdf). Therefore, various IEC curves are incorporated in addition to DOCRs and non-DOCRs. For this relay, \( T_{{{\text{D}}i}}^{ \hbox{min} } \) and \( T_{{{\text{D}}i}}^{ \hbox{max} } \) are chosen as 0.1 and 12.5 s and \( I_{{{\text{P}}i}}^{ \hbox{min} } \) and \( I_{{{\text{P}}i}}^{ \hbox{max} } \) are equal to 120% and 200% of \( I_{\text{L}} \), respectively. The range of CTI is from 150 to 600 ms. The adopted WCA parameters are \( N_{\text{P}} \) = 500, \( N_{\text{r}} \) = 20, \( d_{ \hbox{max} } \) = 1e−8, and Max_iter = 5000. The proposed WCA method is applied to solve this coordination problem. The best values of relays’ settings using WCA method are pointed out in Table 7. The best/optimal value of the total operating time of primary relays along the network is 29.08 s. Figure 5a, b represents the realized operating times of primary relays and the obtained CTI values for various 282 relay pairs. The reader can obviously see that the minimum of operating time of any relay along the network is 50 ms as indicated in Fig. 5a. Additionally, it can be seen from Fig. 5b that there are five violations out of total experiments of 282 for various relay pairs. This violation represents 1.77%, resulting in a successful coordination of overcurrent relays by 98.23%, which may be acceptable for a large complex interconnected power system.

Table 5 Load currents of the relays for the IEEE 30-bus system
Table 6 Relay pairs fault currents for the IEEE 30-bus system
Table 7 Best settings of relays along the IEEE 30-bus system
Fig. 5
figure5

Operating times of various relay pairs. a OTP, b CTI

In addition, Table 8 depicts samples of the operating times of the \( {\text{P}} \) and \( {\text{PB}} \) relays and CTI with special focus to those violating the CTI predefined limits. For a fair comparison, the optimization results using the proposed WCA are compared with those by using the GA for this case study. The GA code is executed via MATLAB software. The optimal characteristics of GA include: population size is 500, crossover fraction equals 0.85, mutation fraction is 0.15, the number of iterations is 5000, the fitness scale function is rank, and the selection function is uniform. The GA results reveal 12 violations out of total experiments of 282 for the entire CTI’s. This violation indicates a ratio of 4.25% and results in a successful coordination of overcurrent relays by 95.75%. These results depict the superiority of WCA to another heuristic algorithm (i.e., GA) for realizing the optimal settings of overcurrent relays in a large complex power system. This superiority of WCA reflects its high convergence speed and its proper design.

Table 8 Samples of operating times with relevant CTIs for the IEEE 30-bus system

Performance category of WCA evaluations

The performance test of the proposed WCA for solving the coordination problem is implemented on 64-bit Windows 10 PC of Intel(R) Core™ i7-4510U CPU@2.00 GHz, 12 GB-RAM, 2.6 GHz. WCA code is performed 50 and 30 independent runs for the test cases of 15- and 30-bus networks in addition to the best results achieved. It can be noted that the average elapsed times per each run (2000 and 5000 iterations) intended for those test cases are 17 s and 830 s with 3 control variables per each relay, respectively. The t test with 1% confidence interval is performed to examine the decision of the evidence against the ‘null hypothesis,’ and the results are ‘statistically significant.’ The obtained values of objective function for these cases are divided into two groups with equal variances, and the p values are computed. The p values are 0.006138 and 0.008107 for the 15-bus and the 30-bus study cases, respectively. The reader can see that the p values are lesser than 0.01, which indicates no evidence against the ‘null hypothesis.’

In addition to the abovementioned, the performance category stated in Beiranvand et al. (2017) is performed to examine the reliability and robustness of the WCA for the studied cases. Among these factors are the number of function evaluations (NFEs), CPU time, success rate, number of constraint violations, and computational accuracy along with traditional statistical performance measures. Table 9 lists these various performance factors for WCA evaluations. It may be useful to report that the parallel processing is not used for any further possible future comparisons.

Table 9 Various performance factors for WCA evaluations

In addition to the above, performance indices which indicate without doubt the effectiveness of WCA in realizing the best settings of overcurrent relays along the networks under study are carried out. It may be interesting to the readers to see examples for the TCC plots for the optimized settings obtained from some relay pairs. In order to avoid a lengthy article, few of these TCC plots are reported as a representative samples. By passing the obtained optimized settings of overcurrent relays with various inverse characteristics as defined in Table 1 of case 1 concerning the 15-bus network to the STAR package of ETAP 16.00 (http://www.etap.com), various TCCs can be generated. Then, and in particular, Fig. 6a, b shows the TCC plots of relay pairs of relays (1 and 6) and relays (2 and 4). It can be seen that the coordination of such mentioned pairs is indicated clearly without any intersection of the curves. In addition, the maximum corresponding three-phase fault currents are shown in their relevant TCC plots for easy reference.

Fig. 6
figure6

TCC plots for the relay pairs

Conclusion

A new attempt to apply the WCA to deal with the overcurrent relays coordination problem in electric power systems is presented. The adapted problem is mathematically formulated to define the best settings of overcurrent relays along the network under study. The viability of WCA method is compared with other conventional and evolutionary challenging methods for two interconnected networks such as the 15-bus and 30-bus systems. The numerical results showed that WCA method has efficiently succeeded to solve the coordination problem of such systems. Moreover, the WCA is successfully examined to deal with a large complex (i.e., detailed IEEE-30 bus) system with the purpose of achieving a realistic study comprising real commercial type of relays. It can be confirmed from the obtained results that WCA based on methodology is superior to other challenging optimization methods for dealing with the coordination problem. Moreover, the WCA-based relay coordination tool is flexible and adequate for dealing with a large industrial real power system, and the percentage of successful coordination is within acceptable ranges.

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El-Fergany, A.A., Hasanien, H.M. Water cycle algorithm for optimal overcurrent relays coordination in electric power systems. Soft Comput 23, 12761–12778 (2019). https://doi.org/10.1007/s00500-019-03826-6

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Keywords

  • Overcurrent relays
  • Optimization methods
  • Power system relay coordination and relay pairs
  • Distributed generations