Optimum Coordination of Directional Overcurrent Relays Using Modified Adaptive Teaching Learning Based Optimization Algorithm

Abstract

Relay coordination problem is highly constrained optimization problem. Heuristic techniques are often used to solve optimization problem. These techniques have a drawback of converging to a non-optimum solution due to the wide range of design variables. On the other hand, initial solution becomes difficult to find with shorter range of design variables. This paper presents modified adaptive teaching learning based optimization algorithm to overcome this drawback of conventional heuristic techniques. The coordination problem is formulated as a constrained non-linear optimization problem to determine the optimum solution for the time multiplier setting (TMS) and plug setting (PS) of DOCRs. Initial solution for TMS is heuristically obtained with the commonly chosen widest range for TMS values. The upper bound of TMS range then substituted by the maximum TMS value in the first initial solution. The new upper limit is obviously lower than the earlier one. Next phase of optimization is carried out with the new range of TMS for the pre-determined iterations of teacher phase. Consequent to the completion of the teacher phase, new upper bound is obtained from the available solution and optimization is carried out for the pre-determined iterations of learner phase. This process is repeated to get the optimum solution. Fixed range for PS is used to obtain the selectivity. Such a strategy of iteratively updating the upper bound of TMS range shows remarkable improvement over the techniques which employ fixed TMS range. This algorithm is tested on different networks and has been found more effective. Four case studies have been presented here to show the effectiveness of the proposed algorithm. The impact of distributed generation (DG) and application of superconducting fault current limiter to mitigate DG impact is presented in case study—III.

Introduction

Shunt faults in a power system give rise to sudden built up of current. This magnitude of fault current can be utilized for the indication of fault existence. The over-current protection is provided using directional DOCRs for distribution system. These relays are also used as secondary protection of the transmission system. In distribution feeders, they play a more prominent role and there it may be the only protection provided. A relay must trip for a fault under its primary zone of protection. Only if, the primary relay fails to operate, the back-up relay should takeover tripping. If backup relays are not well coordinated, the relay may get mal-operated. Therefore, relay coordination is a major concern of power system protection. Each relay in the power system must be coordinated with the other relays in the power system [1, 2]. Several optimization techniques are proposed for optimum coordination of DOCRs [3–17]. The optimum settings for TMS and PS are obtained using different algorithms proposed by the researchers. In some cases, pickup currents are determined based on experience and only the value of TMS is optimized using linear programming techniques. Several non-linear programming (NLP) methods are used to optimize both TMS and PS. However, NLP methods are complex as well as time-consuming. To avoid the complexity of the NLP methods, the DOCR coordination problem is commonly formulated as a linear programming problem (LPP). Various LPP techniques are presented by the researchers for DOCRs coordination [3, 4]. In [5, 6], optimum coordination is achieved by considering different network topologies. Some heuristic-based optimization algorithms such as genetic algorithm (GA) [7, 8], TLBO [9, 10] are used to find the optimum relay settings. In [10], authors have presented the impact of TMS range on the optimum solution. The relay coordination problem was solved in [11, 12], using a hybrid GA considering the effects of the different network topologies. GA is used to find the initial solution with less iteration, and final optimum solution is obtained using LP [11] or NLP method [12]. Informative Differential Evolution (IDE) [13], and Seeker Algorithm [14] are used to find optimum relay settings. The application of Modified Differential Evolution Algorithms [15], Opposition based Chaotic Differential Evolution Algorithm [16], Particle Swarm Optimization (PSO) [17] and Modified Particle Swarm Optimization (MPSO) [18] algorithms are also presented by the researchers to find optimum settings for TMS and PS. The use of dual setting relays is presented in [19] for the optimum coordination of DOCRs. An adaptive protection scheme is presented in [20] to mitigate impact of distributed generation. A case study is presented to determine the size of fault current limiter (FCL) to restore relay coordination [21]. Hybrid protection scheme using adaptive relaying and small size FCL is presented in [22]. The algorithm step and application of stochastic disturbance factor to bring the search to global minimum by escaping from local minima is explained in [23]. The run time of algorithm with respect to change in population size is presented in [24]. Different heuristic-based computational algorithms are available in the literature to solve constrained nonlinear optimization problem. One of such algorithm based on conventional classroom Teaching-Learning process is presented by R. Venkata Rao and Savsani in 2011 [25–27].

The relay coordination problem is generally formulated as constrained non-linear programming problem (NLP) to minimize the sum of operating time of primary relays [9, 11–16]. In this paper, the objective function is defined to minimize the sum of operating time of all primary and backup relays to avoid the delayed operation of backup relays. MATLBO algorithm is proposed to determine optimum values of TMS and PS. Four case studies are presented to illustrate the proposed algorithm.

Problem Formulation

The overcurrent relay has two decision variables, time multiplier setting (TMS) and plug setting (PS). The operating time of relay is a function of TMS, PS and current seen by relay. The operating time of relay is given by Eq. (1) [1–5, 8–16].

$$\begin{aligned} t_{op} =\frac{\alpha *{\textit{TMS}}}{\left( {\frac{If}{{\textit{PS}}}} \right) ^{\beta }-\gamma } \end{aligned}$$

(1)

where ‘*’ represents the scalar multiplication. $\alpha$, $\beta$, $\gamma$, are the constants representing the overcurrent relay characteristic in a mathematical form. It is assumed that inverse-definite minimum time (IDMT) type OCRs are used. $\alpha$, $\beta$ and $\gamma$ constants for normal IDMT characteristic are considered as 0.14, 0.02 and 1.0 respectively as per IEEE standards.

Optimal Relay Coordination Problem

Optimum relay coordination problem can be formulated as constrained non-linear optimization problem and solved by different optimization methods. Some researchers have defined the objective function as to minimize the sum of operating time of all primary relays for their near end faults [11–14]. The objective function is also defined as to minimize the sum of operating time of all primary relays for their near end and far end faults [9, 15, 16]. In these techniques the backup relay operating time is not optimized. This may lead to a delayed operation of backup relays. To overcome this difficulty the objective function is formulated to minimize the sum of operating time of all primary and backup relays. This can be stated as-

$$\begin{aligned} \hbox {minimize Z}_\mathrm{k} =\mathop \sum \limits _{\mathrm{i}=1}^\mathrm{N} \hbox {t}_{\mathrm{i},\hbox {k}} +\hbox {}\mathop \sum \limits _{\mathrm{j}=1}^\mathrm{N} \hbox {t}_{\mathrm{j},\hbox {k}} \end{aligned}$$

(2)

where $\hbox {Z}_{\mathrm{k} }$ is the objective function in zone k, $\hbox {t}_{\mathrm{i,k}}$ is the operating time of $\hbox {i}\mathrm{th}$ primary relay for its near end fault in zone—k, $\hbox {t}_{\mathrm{j,k}}$ is the operating time of $\hbox {j}{\mathrm{th}}$ backup relay for its far end fault in zone—k and N is the total number of directional over-current relays.

Depending upon relay characteristics and primary/backup relationship the above optimization problem has following constraints.

Relay Setting

Each relay has TMS and PS settings. PS limit has chosen based on the maximum load current and the minimum fault current seen by the relay, and the available relay setting. The TMS limits are based on the available relay current-time characteristics. This can be mathematically stated as,

$$\begin{aligned} \left. {{\begin{array}{l} {\hbox {PS}_{\mathrm{imin}} \le \hbox {PS}_\mathrm{i} \le \hbox {PS}_{\mathrm{imax}} } \\ {\hbox {TMS}_{\mathrm{imin}} \le \hbox {TMS}_\mathrm{i} \le \hbox {TMS}_{\mathrm{imax}} } \\ \end{array} }\hbox {}} \right\} \end{aligned}$$

(3)

Bounds on Relay Operating Time

Relay needs certain minimum amount of time to operate. Also, a relay should not be allowed to take too long time to operate. This can be mathematically stated as

$$\begin{aligned} \hbox {t}_{\mathrm{imin}} \le \hbox {t}_{\mathrm{i}} \le \hbox {t}_{\mathrm{imax}} \end{aligned}$$

(4)

where $\hbox {t}_{\mathrm{imin}}$ is the minimum operating time of the relay for the fault at any point in the zone k and $\hbox {t}_{\mathrm{imax}}$ is the maximum operating time of the relay for the fault at any point in the zone k.

Backup: Primary Relays Coordination Time Interval

Fault is sensed by both primary as well as secondary relay simultaneously. To avoid mal-operation, the backup relay should take over the tripping action only if primary relay fails to operate. If $\hbox {R}_{\mathrm{i}}$ is the primary relay for fault at k, and $\hbox {R}_{\mathrm{j}}$ is backup relay for the same fault, then the coordination constraint can be stated as

$$\begin{aligned} \hbox {t}_{\mathrm{j},\hbox {k}} -\hbox {t}_{\mathrm{i},\hbox {k}} \ge \Delta \hbox {t} \end{aligned}$$

(5)

where $\hbox {t}_{\mathrm{i,k}}$ is the operating time of the primary relay $\hbox {R}_{\mathrm{i}}$, for the fault in zone k, $\hbox {t}_{\mathrm{j,k}}$ is the operating time of the backup relay $\hbox {R}_{\mathrm{j}}$, for the same fault in zone k and $\Delta \hbox {t}$ is the co-ordination time interval (CTI).

Teaching Learning Based Optimization Algorithm

Teaching learning is a process where every student tries to learn something from the teacher as well as from other students to improve the performance. Inspiring from traditional teaching-learning phenomenon of the classroom, R. Venkata Rao and Savsani proposed an algorithm known as teaching-learning based optimization algorithm (TLBO) [25–27]. This is related to the effect of influence of a teacher on the output of students (learners) in a class. The algorithm simulates two basic modes of the learning through classroom teaching (known as teacher phase) and interacting with the other students (known as learner phase). Like any other search algorithm, TLBO is a population-based algorithm where the population is represented by a group of students (i.e. learners) and the design variables are represented by the different subjects offered to the learners. The possible solution of the problem is represented by the grades obtained by a learner in each subject. The solution in the entire population which represents the minimum value of the objective function is considered as the teacher. At the first step, the TLBO randomly generates initial population ‘$\hbox {P}_{\mathrm{initial}}$’ of ‘n’ solutions, where ‘n’ denotes the size of the population. Each solution Xk, where k = 1, 2, ..., n is a ‘m’ dimensional vector where ‘m’ is the number of design variables. After initialization, the population of the solutions is repeated for predefined iterations (for i = 1, 2, ..., g) of the teacher phase and learner phase. The teacher phase and learner phase of the TLBO algorithm is explained below.

Teacher Phase

In this phase of the algorithm, the students (i.e. learners) increase their knowledge through the teacher. During this phase, a teacher delivers knowledge among the learners and tries to increase the mean result of the class. Suppose there is ‘m’ number of subjects offered to ‘n’ number of students. At any teaching-learning iteration i, let us consider Mj,i is the mean result of the student in a given subject ‘j’. Since a teacher is having more knowledge of that subject, the best solution in the entire population is considered as a teacher in the algorithm. Let $\hbox {Xd}_{\mathrm{j,i}}$, ($\mathrm{d} \in \mathrm{k}$) be the grades of the best student and f(Xd) is the result of the best student with all the subjects, who is identified as a teacher for that cycle. Teacher will give maximum input to increase the result of the whole class, but the knowledge gained by the students will depend upon the quality of teaching delivered by a teacher as well as the quality of the students. The difference between the grade of the teacher and mean grade of the learners in each subject is expressed as,

$$\begin{aligned} \hbox {Diff}\_\hbox {Mean}_{\mathrm{j,i}} =\hbox { rand}_\mathrm{i} *\hbox { }\left( {\hbox {Xd}_{\mathrm{j,i}} -\hbox {TF}*\hbox {M}_{\mathrm{j,i}} } \right) \end{aligned}$$

(6)

where $\hbox {rand}_{\mathrm{i}}$ is a random number in the range [0, 1], $\hbox {Xd}_{\mathrm{j,i }}$ is the grade of the teacher in the subject—j and $\hbox {T}_{\mathrm{F} }$ is the teaching factor which decides the value of the mean to be changed.

The value of $\hbox {T}_{\mathrm{F}}$ can be either 1 or 2 and decided randomly as,

$$\begin{aligned} T_F =\hbox {round}\left[ {1+\hbox {rand}_\mathrm{i}} \right] \end{aligned}$$

(7)

The value of TF is randomly decided by an algorithm. Based on the $\hbox {Diff}\_\hbox {Mean}_{\mathrm{j,i}}$ the existing solution ‘k’ is updated in the teacher phase according to the following expression.

$$\begin{aligned} \hbox {Xnew}_{\mathrm{kj,i}} =\hbox {X}_{\mathrm{kj,i}} +\hbox {Diff}\_\hbox {Mean}_{\mathrm{j,i}} \end{aligned}$$

(8)

where $\hbox {Xnew}_{\mathrm{kj,i}}$ is the updated value of $\hbox {X}_{\mathrm{kj,i}}$.

Substituting Eq. (6) in Eq. (8), we have

$$\begin{aligned} \hbox {Xnew}_{\mathrm{kj,i}} =\hbox {X}_{\mathrm{kj,i}} +\hbox {rand}_\mathrm{i} *\left( {\hbox {Xd}_{\mathrm{j,i}} -\hbox {TF}*\hbox {M}_{\mathrm{j,i}} } \right) \end{aligned}$$

(9)

The term $\hbox {rand}_{\mathrm{i}}$ is the stochastic step of the algorithm while the term $\hbox {TF}*\hbox {M}_{\mathrm{j,i}}$ enables the algorithm to escape from the local minima. The algorithm accepts $\hbox {Xnew}_{\mathrm{kj,i}}$ if it gives a better function value otherwise keeps the previous solution. All the accepted grades (i.e. design variables) are maintained at the end of the teacher phase which becomes the input to the learner phase (Fig. 1).

Fig. 1

Flow Chart of TLBO algorithm [25, 27]

Learner Phase

A learner in a class gets its input in two different ways. This phase of the algorithm simulates the learning of the students (i.e. learners) through interaction through the group of learners. The students can also increase their knowledge by discussing with the other students. A learner will learn new information if the other learners have more knowledge than him or her. The learning phenomenon of this phase is expressed below. The algorithm randomly selects two learners p and q such that $\hbox {f}(\hbox {X}^{\mathrm{P}}) \ne \hbox {f}(\hbox {X}^{\mathrm{q}})$. $\hbox {f}(\hbox {X}^{\mathrm{P}})$ and $\hbox {f}(\hbox {X}^{\mathrm{q}})$ are the updated result of the learners p and q considering grades of all the subjects at the end of teacher phase.

$$\begin{aligned} \hbox {Xnew}_{\mathrm{j},\hbox {i}}^\mathrm{p} =\left\{ \begin{array}{ll} \mathrm{X}_\mathrm{j},\mathrm{i}^\mathrm{p} +\mathrm{rand}_{\left( \mathrm{i} \right) }^\mathrm{p} *\left( \mathrm{X}_\mathrm{j},\mathrm{i}^\mathrm{p} -\mathrm{X}_\mathrm{j},\mathrm{i}^\mathrm{q} \right) &{}\quad \mathrm{if}~\hbox {f}(\mathrm{X}^\mathrm{p})<^\mathrm{i}\left( \mathrm{X}^\mathrm{q} \right) \\ \mathrm{X}_\mathrm{j},\mathrm{i}^\mathrm{p} +\mathrm{rand}_{\left( \mathrm{i} \right) }^\mathrm{p} *\left( \mathrm{X}_\mathrm{j}, \mathrm{i}^\mathrm{q} -\mathrm{X}_\mathrm{j},\mathrm{i}^\mathrm{p} \right) &{}\quad \mathrm{otherwise} \\ \end{array}\right\} \end{aligned}$$

(10)

$\hbox {Xnew}^{\mathrm{P}}$ is the updated value of $\hbox {X}^{\mathrm{P}}$. The algorithm then accepts $\hbox {Xnew}^{\mathrm{P}}$ if it gives a better function value.

Algorithm Termination

The algorithm is terminated after completion of pre-determined iterations. The final set of learners represents the best value of decision variables.

Comparison of TLBO with other Optimization Techniques

Teaching Learning Based Optimization (TLBO) is a population-based optimization algorithm like Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC) algorithms. These algorithms randomly generate a group of possible solutions to find the optimum solution. Many optimization algorithms have the optimization parameters. These parameters affect the performance of the optimization algorithm. GA has the parameters like the mutation rate, crossover probability and selection method. Similarly, PSO is having optimization parameters like learning factors, weight factors and the maximum value of velocity. Unlike these optimization techniques, TLBO does not have any parameters to be tuned. This makes the implementation of TLBO algorithm much simpler than the other optimization algorithms. In TLBO the existing solutions from the population are updated using best solution of the iteration as on PSO. TLBO does not divide the population into different groups like ABC. TLBO uses two different phases, the ‘teacher phase’ and the ‘learner phase’. TLBO algorithm has two different phases, namely, the ‘teacher phase’ and the ‘learner phase’ like crossover and mutation in GA, employed, onlooker and scout bees in ABC.

Proposed Modified Adaptive TLBO algorithm

To protect the power system, relay must operate for minimum fault current and relay should not operate for maximum load current, which is achieved with proper relay settings. The minimum current to start the relay operation is set with the help of PS and TMS decides the total operating time of the relay.

Relay coordination problem becomes more complicated with an increase in relay numbers. The number of constraints increases with an increase in primary/backup relay pairs. The heuristic base optimization techniques are used to solve the optimization problem. These techniques randomly generate a set of possible solutions (population) to get a feasible initial solution. This solution is converted to the optimum solution using an iterative process. During the iterative process, the solution for the objective function is improved by modifying the possible solutions of design variables. The new solution is accepted if it gives a better solution than the previous one. With a wide range of TMS, the number of worst solutions is generated in the population. This leads to a non-optimum solution.

Fig. 2

Possible areas for solution of TMS and PS

Figure 2 shows the three areas, A, B, and C with fixed range of PS and different range of TMS. The range of TMS is considered 0.05 onward to satisfy the constraints of minimum operating time. With the maximum range of TMS (i.e. 0.05 to 1.1), the solution is available in all the three areas, but the solution available from the area B and C contains the higher value of TMS. This increases the operating time of the relay. Similarly for medium range of TMS (0.05 to 0.8), the solution is available in the area A and B, where area B gives a non-optimum solution. With minimum range of TMS, the solution is available from area A, which gives the optimum solution with a smaller value of TMS. The population is generated with a wide range of design variables (TMS) as it is very difficult to get feasible initial solution with a small range of design variables. After some iteration, number of worst solutions (possible solutions for TMS with high value) will remain in the population. Due to this, the optimization method may converge to the solution that may not be optimum.

In this paper, MATLBO algorithm is proposed to overcome this difficulty. This algorithm has two stages. In stage one initial solution is generated and in stage two the objective function is optimized to obtain the optimum solution with a new population for TMS. Figure 3 shows the flowchart of proposed MATLBO algorithm.

Stage: I

This stage of the algorithm finds feasible initial solution with a wide range of TMS. The feasible initial solution is passed to next stage—II.

Stage: II

This stage simulates the optimization problem to find the optimum value of the objective function. This stage generates new population for TMS. The upper bound of TMS range (UB_TMS) is replaced with a maximum value of TMS obtained from the initial solution (UB_TMS = MAX_TMS). Then objective function is optimized for pre-defined iterations of teacher phase. After completing the teacher phase the solution is optimized for pre-defined iterations of a learner phase with new population using MAX_TMS of a teacher phase. New solution will be available after completing total iterations of the learner phase. This solution is compared with the initial solution. If the new solution is better than the initial solution then, the new solution is treated as the initial solution and the process is repeated to get the optimum solution.

Algorithm Termination

The algorithm is terminated after completion of pre-determined iterations/cycles of teacher and learner phase. The final set of learners represents the best value of decision variables.

Fig. 3

Flow chart of modified adaptive TLBO algorithm

Fig. 4

IEEE 6-bus test system [9, 15, 16]

Table 1 TMS and PS for Illustration—I

Impact of Distributed Generation (DG) on Protection Coordination

The presence of DG in distribution system changes the fault current level. This leads to loss of original relay coordination. The original relay coordination is restored by disconnecting all DGs during the fault conditions. This will lead to the loss of DG power as well as it will create resynchronization problems for connecting DGs after clearing the fault. A fault current limiter (FCL) can effectively used in series with DGs to limit their fault currents. The resistive type of fault current limiter is more effective as compared to other type. The size of FCL depends upon the level of DG injection. FCL offers the impedance only during fault conditions and zero impedance is offered during steady state. The unique characteristics equation of resistive type superconducting fault current limiter (SFCL) can be expressed by Eq. (11)

$$\begin{aligned} R_{SFCL} \left( t \right) =R_m \left( {1-exp^{\left( {-t/T_{sc} } \right) }} \right) \end{aligned}$$

(11)

where $\hbox {R}_{\mathrm{m}}$ is the maximum resistance of the SFCL in the normal state.

Table 2 Primary/backup operating time of relays and CTI for Illustration—I

Fig. 5

Convergence curve of MATLBO algorithm (Illustration—I)

$\hbox {T}_{\mathrm{sc}}$ is the time constant of transition from the superconducting state to the normal state, which is assumed to be 1ms [21]. The FCL has some limitations such as the size of FCL increases with the increase in DG level. This increases the cost of FCL and also leads to increase in backup relay time. The other option is to replace all existing relays with microprocessor based digital relays and communication systems for adaptive relaying. However, this option is economically expensive and also increases the complications in control systems.

To overcome these difficulties, a small size of FCL is used in series with DG to restore the settings of far end relays. The near end relays are replaced by digital relays to restore the settings using adaptive relaying, thus making use of advantages of both the techniques (and at the same time overcoming the drawbacks of the above techniques). The application of hybrid protection scheme using adaptive relaying and RSFCL is presented in Illustration—III [22].

Implementation of Proposed Algorithm

MATLAB program is developed to find optimum settings for TMS and PS using MATLBO algorithm. The relay coordination problem discussed in the proposed research is suitable for off-line planning stage. The use of evolutionary type methods in real time monitoring and fault management in the power grid requires a significant decrease of the algorithms’ cycle-time and is still an open research topic.

Fig. 6

Convergence curve of MATLBO and TLBO algorithms (first 2500 iterations)

The parameters of MATLBO algorithm are tabulated in Appendix 1. The proposed algorithm was successfully tested for various systems, out of which four are presented in this paper. In these illustrations, all the relays are considered as numerical relays with normal IDMT characteristics with $\alpha$, $\beta$, and $\gamma$ constants as 0.14, 0.02 and 1.0 respectively. A total of 50 runs for each test systems are conducted and the best solution throughout the run is recorded as a global optimum solution.

Illustration: I

The proposed algorithm is tested on the IEEE 6-bus system shown in Fig. 3 reported in [9, 15, 16]. This network consists of 6 buses and 7 lines and 14 relays. The optimization problem is formulated as constrained nonlinear optimization problem. There are total 28 variables exists in the optimization problem. The problem has 14 constraints because of minimum operating time and 38 constraints due to coordination criteria. The initial range for TMS is considered 0.05 to 1.1 [15] (Fig. 4).

Fig. 7

Comparison of operating time of primary relays (Illustration—I)

Fig. 8

Comparison of operating time of backup relays (Illustration—I)

The CTI value is considered as 0.3 s for the fair comparison with the previous work presented in [9]. The primary/backup relationship of relay pairs and fault current data is taken from [9, 15]. Table 1 represents the optimum solution obtained using TLBO [9] and MATLBO algorithm.

Table 1 shows that the optimum solution obtained using MATLBO algorithm is better than the solution obtained using TLBO algorithm. The primary/backup relay operating time and CTI value associated with primary/backup relay pair is tabulated in Table 2. This table shows that the MATLBO algorithm satisfies all the constraints of minimum operating time and CTI. The convergence curve of MATLBO algorithm is presented in Fig. 5. This shows that the MATLBO algorithm converges to its global optimum solution within 2200 iterations.

The comparison of Convergence curve of MATLBO and TLBO algorithm is presented in Fig. 6. This figure shows shows that the convergence curve of MATLBO and TLBO algorithm is almost same for first 500 iterations. The objective function value after 500 iterations is 62.4599s with the maximum value of TMS as 0.6431. After 500 iterations, the MATLBO algorithm generates the new population with upper bound on TMS value as 0.6431. This gives remarkable improvement in objective function value due to replacement of worst solutions with better solutions for TMS as explained in “Proposed Modified Adaptive TLBO algorithm” section. Figure 6 also shows the improvement in objective function value with new population of TMS for the first four cycles of MATLBO algorithm. The MATLBO algorithm converges to the global optimum solution with less iterations as compared to TLBO algorithm.

The comparison of primary relay operating time, backup relay operating time and CTI is represented in Figs. 7, 8 and 9 respectively. These figures show that the MATLBO algorithm gives better performance as compared to TLBO algorithm. The simulation time with respect to the population size, using the MATLAB R2012a on Personal Computer CPU Core i3 4010U 1.70 GHz. Processor with 4 GB DDR3 RAM is presented in Fig. 10 and tabulated in Appendix 2.

Illustration: II

The proposed algorithm is tested on the 8-bus system shown in Fig. 11 reported in [11, 14]. This network consists of 8 buses, 7 lines, 2 transformers and 2 generators. The optimization problem is formulated as constrained nonlinear optimization problem. There are total 28 variables exists in the optimization problem. The problem has 14 constraints because of minimum operating time and 20 constraints due to coordination criteria. The initial range for TMS is considered 0.05 to 1.1 [11]. In this illustration, for the fair comparison CTI is taken as 0.3 s as in [11, 14]. The primary/backup relationship of relay pairs and fault current data is given in Table 3 [11, 14]. Table 4 presents the optimum solution obtained using TLBO, MATLBO and other optimization method used in [11, 14], this table shows that the optimum solution obtained using MATLBO algorithm is better than other optimization algorithms. The primary/backup relay operating time and CTI value associated with primary/backup relay pair is tabulated in Table 5. This table shows that the MATLBO algorithm satisfies all the constraints of minimum operating time and CTI. The convergence curve of MATLBO algorithm is presented in Fig. 12. This shows that the MATLBO algorithm converges to its global optimum solution within 4350 iterations but there is no significant improvement after 3100 iterations.

Fig. 9

Comparison of coordination time interval (CTI) (Illustration—I)

Fig. 10

Comparison of coordination time interval (CTI) (Illustration—I)

Fig. 11

8-Bus meshed distribution system [11, 14]

Table 3 Fault current data for Illustration—II [11, 14]

Table 4 TMS and PS for Illustration—II

Table 5 Primary/backup operating time of relays and CTI for Illustration—II

The comparison of Convergence curve of MATLBO and TLBO algorithm is presented in Fig. 13. Figure 13 shows that the convergence curve of MATLBO and TLBO algorithm is almost same for first 500 iterations. The objective function value after 500 iterations is 92.3822s with the maximum value of TMS as 0.8074. After 500 iterations, the MATLBO algorithm generates the new population with upper bound on TMS value as 0.8074. This gives remarkable improvement in objective function value due to replacement of worst solutions with better solutions for TMS as explained in “Proposed Modified Adaptive TLBO algorithm” section. Figure 12 also shows the improvement in objective function value with new population of TMS for the first four cycles of MATLBO algorithm. The MATLBO algorithm converges to the global optimum solution with less iterations as compared to TLBO algorithm.

Fig. 12

Convergence curve of MATLBO algorithm (Illustration—II)

Fig. 13

Convergence curve of MATLBO and TLBO algorithms (first 2500 iterations)

The comparison of primary relay operating time, backup relay operating time and CTI is represented in Figs. 14, 15 and 16 respectively. These figures show that the MATLBO algorithm gives better performance as compared to TLBO algorithm.

Illustration: III

A proposed method is applied to 9-bus interconnected distribution system [12, 13]. A single line diagram is shown in Fig. 17. In [12, 13] faults are generated at the middle of each line. As the line impedance plays an important role, the magnitude of fault current seen by relay will be more for its near end faults as compared to the fault at the middle of the line. This may lead to mal-operation of DOCRs. The fault analysis for the near end faults is carried out using power world simulator software. The primary/backup relationship of relay pairs and nearend fault current data is given in Table 6. The minimum operating time of each relay as well as CTI is considered as 0.2s as in [12, 13].

Fig. 14

Comparison of operating time of primary relays (Illustration—II)

Fig. 15

Comparison of operating time of backup relays (Illustration—II)

Fig. 16

Comparison of Coordination Time Interval (CTI) (Illustration—II)

The optimization problem is formulated as constrained non-linear optimization problem. There are total 48 variables exists in the optimization problem. The problem has 24 constraints because of minimum operating time and 32 constraints due to coordination criteria. The MATLBO algorithm is used to solve relay coordination problem to minimize total operating time of primary and backup relays. The optimum solution with proposed algorithm is present in Table 7. The optimum solution with proposed algorithm is 41.9041 s while the solution obtained with IDE and TLBO algorithm is 59.6741 s [13] and 82.9012 s respectively. The primary/backup relay operating time and CTI value associated with primary/backup relay pair is tabulated in Table 8. This table shows that the MATLBO algorithm satisfies all the constraints of minimum operating time and CTI. The convergence curve of MATLBO algorithm is presented in Fig. 18.

Fig. 17

9-Bus meshed distribution system [12, 13]

Table 6 Near end fault current data for Illustration—III

Table 7 TMS and PS for Illustration—III

Table 8 Primary/backup operating time of relays and CTI for Illustration—III

Fig. 18

Convergence curve of MATLBO algorithm (Illustration—III)

Fig. 19

Comparison of operating time of primary relays (Illustration—III)

Fig. 20

Comparison of operating time of backup relays (Illustration—III)

Fig. 21

Comparison of Coordination Time Interval (CTI) (Illustration—III)

Fig. 22

Improvement in CTI with FCL size (Illustration—III)

Fig. 23

15-Bus meshed distribution system [14, 15]

Table 9 Fault current data for Illustration—IV [14]

The MATLBO algorithm generates the new population for TMS after each cycle. This shows remarkable improvement in objective function value due to replacement of worst solutions with better solutions for TMS as explained in “Proposed Modified Adaptive TLBO algorithm” section. The comparison of primary relay operating time, backup relay operating time and CTI is represented in Figs. 19, 20 and 21 respectively. These figures show that the MATLBO algorithm gives better performance as compared to TLBO algorithm.

Table 10 TMS and PS for Illustration—IV

To observe impact of DG a 2 MVA distributed generated is connected at bus 3. The change in fault current data with DG and with different size fault current limiter is presented in Table 6. The results show that to restore fault current level the size of FCL increases. This increases the cost of FCL. This table also shows that with small size FCL the fault current level and protection coordination get restored for the relays other than the near end relays of DG. So to restore protection coordination the near end relays (i.e. Relay No. 8 and 9) can be replaced by digital relays with adaptive settings. This method presents a use of FCL to restore protection coordination of far end using FCL and near end relays using Adaptive Relaying. The improvement in CTI with increase in FCL size is presented in Fig. 22.

Illustration: IV

The proposed MATLBO algorithm is implemented in a 15-bus test network presented in [14]. This case is a highly distributed generation (DG) penetrated distribution network as shown in Fig. 23. Each generator has a synchronous reactance of 15 % with 15 MVA and 20-kV ratings. The external grid has 200-MVA short-circuit capacity. The test case has 42 relays and 82 backup-primary pairs [14].

Fig. 24

Comparison of operating time of primary relays (Illustration—IV)

Fig. 25

Comparison of operating time of backup relays (Illustration—IV)

Near end fault, current data is given in Table 9. The MATLBO algorithm is used to solve relay coordination problem to minimize total operating time of primary relays. The optimum solution with proposed algorithm is 52.5039 s while the solution obtained with MINLP, and seeker algorithm is 75.3655 s [14], and 66.8062 s [14] respectively. The optimum values of TMS and PS are presented in Table 10. Table 10 represents that the solution obtained using MATLBO algorithm satisfies all the constraints. The comparison of primary and backup relay operating time is presented in Figs. 24 and 25 respectively.

Conclusion

MATLBO algorithm to determine the optimum values of TMS and PS of DOCRs is presented in this paper. In this algorithm an advance set of possible design variables (population) is generated with a maximum value of TMS available from the earlier solution. This increases the probability of better feasible solutions. Such a strategy of iteratively updating the upper bound of TMS range shows remarkable improvement over the techniques which employ fixed TMS range. The results show that the proposed MATLBO algorithm overcomes the weakness of TLBO algorithm and capable to find superior TMS and PS settings as compared to previously proposed optimization algorithms in the literature.

Appendices

Appendix 1

Parameters of MATLBO algorithm	Illustration—I	Illustration—II	Illustration—III	Illustration—IV
No. of design Variables	28	28	48	84
Population size	50	50	100	100
No. of iterations	500	500	500	500
No. of cycles	10	10	20	50
Computational time	47 s	52 s	27 min	53 min

Appendix 2

Parameters of MATLBO algorithm	Illustration—I
No. of design variables	28	28	28	28	28	28
Population size	50	100	150	200	250	300
No. of iterations	500	500	500	500	500	500
No. of cycles	10	10	10	10	10	10
Computational time	47 sec	82 sec	127 sec	160 sec	176 sec	218 sec
Objective function (s)	27.0556	28.3245	27.1276	27.1193	28.6874	27.9468