ANTELOPE ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM

In this paper, Antelope Algorithm (AA) is proposed for solving optimal reactive power dispatch problem. A population of candidate solution move toward as a herd of Antelope out a sequence of jumps through the exploration space in order to find the most outstanding solution. The main idea of this algorithm is fairly different from the population based algorithms, as the individual solutions are stirred collectively in a herd-like approach. Projected Antelope Algorithm (AA) algorithm has been tested in standard IEEE 30 bus test system and simulation results show clearly about the superior performance of the projected algorithm in reducing the real power loss.


Introduction
Optimal reactive power dispatch (ORPD) problem is a multi-objective optimization problem that diminishes the real power loss and bus voltage deviation. Various mathematical techniques like the gradient method [1][2], Newton method [3] and linear programming [4][5][6][7] have been adopted to solve the optimal reactive power dispatch problem. Both the gradient and Newton methods has the complexity in managing inequality constraints. If linear programming is applied then the input-output function has to be uttered as a set of linear functions which mostly lead to loss of accurateness. The problem of voltage stability and collapse play a major role in power system planning and operation [8]. Global optimization has received extensive research awareness, and a great number of methods have been applied to solve this problem. Evolutionary algorithms such as genetic algorithm have been already proposed to solve the reactive power flow problem [9,10].Evolutionary algorithm is a heuristic approach used for minimization problems by utilizing nonlinear and non-differentiable continuous space functions. In [11], Genetic algorithm has been used to solve optimal reactive power flow problem. In [12], Hybrid differential evolution algorithm is proposed to improve the voltage stability index. In [13] Biogeography Based algorithm is projected to solve the reactive power dispatch problem. In [14], a fuzzy based Http://www.granthaalayah.com ©International Journal of Research -GRANTHAALAYAH [192] method is used to solve the optimal reactive power scheduling method. In [15], an improved evolutionary programming is used to solve the optimal reactive power dispatch problem. In [16], the optimal reactive power flow problem is solved by integrating a genetic algorithm with a nonlinear interior point method. In [17], a pattern algorithm is used to solve ac-dc optimal reactive power flow model with the generator capability limits. In [18], proposes a two-step approach to evaluate Reactive power reserves with respect to operating constraints and voltage stability. In [19], a programming based proposed approach used to solve the optimal reactive power dispatch problem. In [20], presents a probabilistic algorithm for optimal reactive power provision in hybrid electricity markets with uncertain loads. This paper proposes Antelope algorithm (AA) is used to solve the optimal reactive power dispatch problem. AA imitates the behavior of a herd [21,22] of antelopes that jump their way through a solution space to find the optimal point. Antelope algorithm tries to find an optimal point by iteratively altering a population of candidate solutions. Yet it does not depend on swarm intelligence, but rather on herd-like behavior with innermost assessment. Projected Antelope Algorithm (AA) has been evaluated in standard IEEE 30 bus test system & the simulation results shows that the projected approach outperforms all reported standard algorithms in reducing the real power loss.

Modal Analysis for Voltage Stability Evaluation
Modal analysis is one among best methods for voltage stability enhancement in power systems. The steady state system power flow equations are given by.
[ (4) J R is called the reduced Jacobian matrix of the system.

Modes of Voltage Instability
Voltage Stability characteristics of the system have been identified by computing the Eigen values and Eigen vectors. Let J R = ξ˄η Where ξi is the ith column right eigenvector and η the ith row left eigenvector of JR. λi is the ith Eigen value of JR.
The ith modal reactive power variation is, The corresponding ith modal voltage variation is If | λi | =0 then the ith modal voltage will collapse . In (10), let ΔQ = ek where ek has all its elements zero except the kth one being 1. Then,

Problem Formulation
The objectives of the reactive power dispatch problem is to minimize the system real power loss and maximize the static voltage stability margins (SVSM).

Minimization of Real Power Loss
Minimization of the real power loss (Ploss) in transmission lines is mathematically stated as follows.
Where n is the number of transmission lines, gk is the conductance of branch k, Vi and Vj are voltage magnitude at bus i and bus j, and θij is the voltage angle difference between bus i and bus j.

Minimization of Voltage Deviation
Minimization of the voltage deviation magnitudes (VD) at load buses is mathematically stated as follows.
(15) Where nl is the number of load busses and Vk is the voltage magnitude at bus k.

System Constraints
Objective functions are subjected to these constraints shown below. Load flow equality constraints: where, nb is the number of buses, PG and QG are the real and reactive power of the generator, PD and QD are the real and reactive load of the generator, and Gij and Bij are the mutual conductance and susceptance between bus i and bus j.

Antelope Algorithm
The deeds of the Antelope is jumping around their habitat in explore of the locating best food. And an Antelope finding itself in any point in space can give it a real-valued mark, indicating food quality shown in Fig .1. ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ be an D-dimensional vector containing the leader's position, A be a P×D matrix whose first row is ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ and whose remaining rows contain the other Antelope's positions, ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ be a P-dimensional vector containing the cost-ranking of each of the Antelope's, ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ be an D-dimensional vector giving the direction of the leader's jump, j length be a scalar value representing the leader's jump's length, scatter be a scalar value controlling how close to the leader the Antelope's will be scattered, + and − be scalar values used to automatically lengthen or shorten the jumps, m length be a scalar value representing the minimal allowed jump length. Antelope Algorithm (AA) for solving optimal reactive power dispatch problem a) Initialization of parameters i) Initialize the leader to a random point in the search space.
ii) Update the other Antelope's positions by randomly placing them in a region around the leader, the size of which is controlled by parameter scatter. iii) Evaluate the cost function at each of the P points. iv) Rank the costs and store the result in ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . v) Store the lowest cost and the associated position if it is lower than the stored best. vi) Compute the jump direction according to formula (25). vii) Update the jump-length variable: if the new jump is made a direction opposite to that of the last jump (if +1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ < 0) then multiply j length by − , else multiply it by + . If this makes j length smaller than m length, set it to m length. vii) Update the scatter parameter.
The algorithm fundamentally takes three parameters: the population size P, the scatter-range scatter, and the minimal jump-length m length. P is the number of cost-function evaluations in each iteration and the concentration of exploitation of each visited area. The scatter range controls "how local" the search is at each iteration: a lower scatter value will let the random evaluations occur in a narrower region around the leader. The minimum jump-length is used to avoid convergence to local optima. Parameters + and − are directly inspired by those of Riedmiller and Braun's RPROP algorithm [23] for the training of feed forward neural networks. The idea, interpret the original article, is that two successive jumps in conflicting directions Http://www.granthaalayah.com ©International Journal of Research -GRANTHAALAYAH [197] designate that the last jump was too long and the algorithm has jumped over a local minimum; jump-length is then decreased by factor − . Otherwise, jump-length is slightly increased by factor + in order to increase speed of convergence in shallow regions. Although our setting is quite different, we used the original values of both parameters, + = 0.49 and − = 1.1. We chose to use the rank of costs instead of costs to compute the jump direction. This makes the algorithm invariant to any increasing transformation of the cost function. Antelope's were uniformly distributed inside a hyper parallel piped centred on the leader, according to the following formula: Where a ij denotes element (i, j) of matrix A, ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the leader-vector's j-th element, scatter is a scalar value chosen in [0,1], ub j and lb j are respectively the upper and lower bounds of dimension j, and r ij is a random value uniformly drawn from [0,1]. A better choice might be to generate normal deviates from ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ .

Simulation Results
The efficiency of the proposed Antelope Algorithm (AA) for solving the multi-objective reactive power dispatch problem is demonstrated by testing it on standard IEEE-30 bus system. The IEEE-30 bus system has 6 generator buses, 24 load buses and 41 transmission lines of which four branches are (6-9), (6-10) , (4)(5)(6)(7)(8)(9)(10)(11)(12) and ( Table 5 shows the proposed algorithm powerfully reduces the real power losses when compared to other given algorithms. The optimal values of the control variables along with the minimum loss obtained are given in Table 1. Corresponding to this control variable setting, it was found that there are no limit violations in any of the state variables. Optimal Reactive Power Dispatch problem together with voltage stability constraint problem was handled in this case as a multi-objective optimization problem where both power loss and maximum voltage stability margin of the system were optimized simultaneously. Table 2 indicates the optimal values of these control variables. Also it is found that there are no limit violations of the state variables. It indicates the voltage stability index has increased from 0.2478 to 0.2489, an advance in the system voltage stability. To determine the voltage security of the system, contingency analysis was conducted using the control variable setting obtained in case 1 and case 2. The Eigen values equivalents to the four critical contingencies are given in Table 3. From this result it is observed that the Eigen value has been improved considerably for all contingencies in the second case.    Minimum loss (MW) Evolutionary programming [24] 5.0159 Genetic algorithm [25] 4.665 Real coded GA with Lindex as SVSM [26] 4.568 Real coded genetic algorithm [27] 4.5015 Proposed AA method 4.2958

Conclusion
Antelope Algorithm (AA) has been effectively applied for solving Optimal Reactive Power Dispatch problem. The Antelope Algorithm (AA) based Optimal Reactive Power Dispatch has been successfully tested in standard IEEE 30 bus system. Performance comparisons with wellknown population-based algorithms give advance results. Antelope Algorithm (AA) succeeded in plummeting real power loss, when compare to other reported standard algorithms. The simulation results presented in preceding section prove the capability of Antelope Algorithm (AA) approach to arrive at near global optimal solution.