A Novel Approach Based on Crow Search Algorithm for Solving Reactive Power Dispatch Problem

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Introduction
Optimal reactive power dispatch (ORPD) is one of the strategic problems in which inappropriate management can compromise the security and the reliability of the power systems [1,2]. ORPD represents a specific part of the more general OPF problem. It can be divided into two parts: the real and reactive power dispatch problems. The real power dispatch problem aims to minimize the total cost of real power generated by the different production plants [3,4]. On the other hand, the reactive power dispatch controls the power system stability, power quality and power losses. The main objectives of ORPD are to minimize real power losses and to increase voltage stability by enhancing the load bus voltage deviations while simultaneously satisfying a certain set of desired operating and security constraints [5][6][7]. Therefore, the ORPD is formulated as a complex optimization problem involving a nonlinear objective function with continuous and discrete control variables subject to nonlinear constraints [8][9][10]. The controlled variables of the ORPD problem are generator bus voltages, transformer tap-settings, shunt capacitors and output of static reactive power compensators. These variables must be set simultaneously to minimize power losses and to improve the voltage profile by satisfying the equality and inequality constraints [11][12][13].
confirm the performance of the proposed approach. Consequently, significant technical and economic merits are satisfied, which prove the robustness and the effectiveness of the proposed CSA approach, thus providing an alternative solution to the ORPD problem.
The paper is organized as follows: Section 2 presents the ORPD problem formulation. Section 3 introduces the crow search algorithm. Section 4 derives the proposed CSA for the ORPD problem. Simulation and comparison results are presented in Section 5. In section 6, a sensitivity analysis to test the differences among CSA, PSO, WOA, and ALO is presented. The main findings and future work directions are summarized in Section 7.

Problem Formulation
The ORPD problem is formulated as a nonlinear and non-convex optimization problem. The objective function is to minimize the power losses of the transmission system, taking into account the equality and inequality constraints [12,13]. Mathematically: where F(x, u) represents the objective function of power losses, g(x, u) is the equality constraints, h(x, u) is the inequality constraints, and x and u represent the vectors of control variables and state variables, respectively.

Control Variables
The control variables vector x consists of load bus voltages, transmission line loading and reactive power output of generators as follows: where V L is the load bus voltages, NPQ is the number of load buses, Q G is the reactive power of ith generator bus, NG denotes the number of generators, S L represents the transmission line loadings, and NTL denotes the number of transmission lines.

State Variables
The state variables vector u includes discrete and continuous variables: voltages of PV bus (continuous variables), switching shunt capacitor banks (discrete variables), and transformer tap settings (discrete variables). It is expressed by: where V G represents the generator voltages, Q C is the reactive power of ith load bus, NC denotes the number of shunt VAR compensators, T is the transformer tap settings, and NT denotes the number of regulating transformers.

Constraints
The ORPD problem is subject to satisfy the required equality and inequality constraints [12].

Equality Constraints
The equality constraints represent the power balance of load flow equations. The difference between power generated and power demand is equal to power loss as follows:

Inequality Constraints
The inequality constraints are based on the operating limits that satisfy the normal operation of the system. The inequality constraints of control and state variables are given by: • Generator voltages limits: • Tap positions of transformers limits: • Amount of reactive compensation limits: • Load bus voltage constraints limits: • Reactive power outputs limits of generators: • The transmission lines loading are limited by upper values:

The Crow Search Algorithm (CSA)
Crow search algorithm (CSA) is a novel meta-heuristic optimization method recently proposed by Askarzadeh [26] to solve constrained engineering problems. This algorithm is modeled related to the intelligent behavior of the crows while saving their excess food in hiding places and retrieving it when the food is needed. Crows can remember the faces of other crows and warn each other when an unfriendly crow approaches as they have large brains relative to their bodies. In addition, they can use communication tools in sophisticated ways and remember the places where food is hidden for up to several months. Therefore, the main principles of crows is that they monitor the food locations of other birds and steal when there are no birds in that place. Moreover, if the crow has committed thievery, it will take extra precautions such as moving hiding places to avoid becoming a victim in the future.
Let us assume that d is a dimensional environment and N is the number of crows. The position of crows i at iteration iter is given by: where i = 1, 2, · · · , N and iter = 1, 2, · · · , iter max . iter max is the maximum number of iterations. Crows move in the environment and search for better sources of food (hiding places). Each crow has a memory capacity that can memorize the position of its food hiding places. One of the main activities of the crow is to follow crow j to get close to the hiding place of the food and hunt it. Therefore, two major cases may occur in CSA:

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Case 1: Crow j does not know that crow i is following it. Therefore, crow i will approach the hiding place of crow j and change its position to a new one as follows: where r i is a random number with uniform distribution (between 0 and 1), f l i,iter indicates the flight length of crow i at iteration iter (see Figure 1), and m j,iter represents the best visited location by crow j at iteration iter.

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Case 2: Crow j detects that it is followed by crow i. Thus, to protect its cache from being pilfered, crow j will change the flight pass to mislead followers by moving to another position in the search space.
Therefore, the position of crow can be expressed as follows: where AP j,iter indicates the awareness probability of each crow.

ORPD Problem Using Crow Search Algorithm
We propose the CSA approach to solve the ORPD problem. The main objective is to find the optimal solution of control variables to minimize the objective function while satisfying all the constraints imposed by the power system. The implementation of the CSA for the ORPD problem is achieved through by the following steps:

1: Initialization of algorithm parameters and constraints
Initialize the number of crows N, the maximum number of iterations iter max , flight length f l and awareness probability AP. Determine the decision variables and constraints. •

Step 2: Initialize position and memory of crows
Generate N crows randomly in the d-dimensional search space. Each crow presents an appropriate solution to the problem. Initially, crows have no experience; it is supposed that they have hidden their food in the initial positions.

Step 3: Evaluate fitness function
For each crow, the quality of its position is computed by adjusting the values of the control variable into the objective function. •

Step 4: Generate new position
The crow i generates a new position as follows: It selects randomly one of the other crows and follows it to find the position of the hidden food. Thus, the new position of crow i is given by Equation (14). This procedure is repeated for all crows.

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Step 5: Check the feasibility of new positions The feasibility of the new position of each crow is examined. For the ORPD problem, all variables are checked for any violation for certain limits. If the new position of a crow is valid, the crow updates its position. Otherwise, it stays in its current position and does not move to the new position that was created. •

Step 6: Evaluate fitness function of new positions
For each crow, the fitness function value of the new position is computed.

Step 7: Update memory
If the evaluation of the fitness function value of each crow is better than the memorized fitness function value, the crows update its memory as follows: where f (.) indicates the objective function value.

Step 8: Check stop criterion
Repeat Steps 4-7 until iter max is reached. The solution of the ORPD problem is the best position of the memory corresponding to the best objective function value. Figure 2 illustrates the flowchart of the proposed algorithm in order to solve the ORPD problem.

Application and Results
The proposed approach based on CSA for solving the ORPD problem was applied to the standard IEEE-14 and the IEEE-30 bus benchmark power systems as well as the Tunisian power system. It was implemented using MATLAB Platform 2016a on a Windows 10 Intel(R) Core(TM) i3-5010U CPU @2.10 GHz 6 GB RAM. For all test systems, the base power was 100 MW. The results obtained with CSA were compared with three other meta-heuristic algorithms: Particle Swarm Optimization (PSO), Ant Lion Optimization (ALO), and Whale Optimization Algorithm (WOA).The features of all test systems and the limits of control variables are given in Tables 1 and 2.

IEEE 14-Bus Test Power System
This test system includes five generators at the buses 1, 2, 3, 6 and 8, three tap-changing transformers located at Lines 4-7, 4-9 and 5-6 and two shunt reactive compensators connected to buses 9 and 14. All data of this test system are specified in [34]. CSA, PSO, ALO, and WOA were applied to minimize real power loss as the main objective function. Table 3 presents the results of the optimal settings using the four implemented algorithms. The optimal result was given by the proposed CSA where the active power loss is 12.2307 MW (8.68%) compared to PSO 12.5881 MW (6.00%), ALO 12.6266 MW (5.72%), and WOA 12.4245 MW (7.23%). In addition, we noticed that the CSA is the fastest (946.0803 s) compared to PSO (958.9098 s), ALO (946.4107 s), and WOA (1029.973 s). Obviously, the computing time of CSA is less than the other algorithms. Figure 3 represents the performance loss convergence characteristics for the corresponding approaches: CSA is in red, PSO is in blue, WOA is in green and ALO is in black. Moreover, the CSA algorithm has a great ability to locate the optimal solutions and effectively deal the constraints of the optimization problem.

IEEE 30-Bus Test Power System
In this case study, the IEEE 30-bus test system includes six generators connected to buses 1, 2, 5, 8, 11 and 13. Four tap-changing transformers are located at lines 6-9, 4-12, 9-12 and 27-28. Nine shunt reactive power sources are connected to buses 10, 12, 15, 17, 20, 21, 23, 24 and 29. The system data are given in. To evaluate the efficacy and the superiority of the proposed algorithm to minimize the power losses of this test system, we tested PSO, ALO, WOA, and CSA. Table 4 summarizes the results of the optimal settings using CSA, PSO, ALO, and WOA. The performance real power loss convergence characteristics of the all mentioned approaches are shown in Figure 4. We note that the minimum

A Real Case Study: Tunisian 86-Bus Power System
For the examination of the appropriateness of the proposed algorithm on the large-scale power system, we considered a large real power system: Tunisian 86−bus power system. It includes 22 generators, 120 lines, 21 tap-changing transformers, and 15 shunt reactive power sources (see Figure 5). Table 5 introduces the results of the optimal settings using the swarm approaches for this case. The optimal solution for the ORPD problem in Tunisian power system is given by the proposed CSA where the obtained minimum active power loss P loss = 161.  Figure 6 shows the performance loss convergence characteristics of the swarm approaches. We observed that the proposed CSA has good convergence characteristics compared to PSO, ALO and WOA. It is substantial to note that the CSA algorithm can solve the ORPD even for a large-scale power system.

Sensitivity Analysis
Sensitivity analysis (SA) is one of the methods that help to make decisions regarding with more than a solution to a problem. It provides a decent idea about how sensitive is the optimum solution chosen to any changes in the input values of one or more parameters. ANOVA is a model independent sensitivity analysis method that evaluates whether there is any statistically significant association between an output and one or more inputs [35]. Therefore, ANOVA uses the F-statistic ratio to determine whether a significant difference exists among mean responses for main effects or interactions between factors. The relative magnitude of F-values can be used to rank the factors in the sensitivity analysis [36]. The higher is the F-value, the more sensitive the response variable is to the factor. Therefore, factors with higher F-values are given higher rankings. The p-value indicates that differences between column means are significant. One-way ANOVA was used to compare the means between the results obtained by the proposed CSA algorithm and the other developed PSO, WOA, and ALO techniques.
Tables 6-8 summarize the statistical analysis of the proposed and the conventional methods for IEEE-14 and IEEE-30 bus benchmark power systems and the real Tunisian power system, respectively. Note that CSA always provides the competing performance over the other algorithms in terms of accuracy (Mean) and robustness (standard deviation). Table 9 depicts the results of the one-way ANOVA test obtained for the different systems. We observed that the results of p-value are much lower than 0.01 for all tests, which confirm that CSA is statistically significant from ALO, WOA, and PSO.

Conclusions
We addressed the problem of optimal reactive power dispatch (ORPD) using a recent meta-heuristic technique, called Crow Search Algorithm (CSA). The ORPD problem is formulated as a nonlinear constrained optimization problem to minimize power losses and enhance the voltage stability. The main advantages of the proposed CSA relay on the high ability to solve complex optimization problems and faster convergence. The proposed CSA has been successfully implemented and tested on benchmark test bus systems which are IEEE-14 and the IEEE-30 as well as on the real and large-scale Tunisian 86-bus system to solve the ORPD problem. Based on sensitivity analysis test, the proposed CSA has a superior performance (for IEEE-14 bus p < 0.0006, for IEEE-30 bus p < 0.006, and for Tunisian 86-bus p < 0.0000001) compared to ALO, WOA, and PSO. It provides the optimal and faster solution to minimize the real power loss. In our future work, we will improve the performance of CSA to minimize the number of iterations which leads to minimizing the convergence time and suggest this approach as a powerful solution for the Tunisian Company of Electricity and Gas.
Author Contributions: All the authors have contributed to this work including the production of the idea, the analysis and interpretation of the results and finally the writing and reviewing of the paper.
Funding: This research received no external funding.