REAL POWER LOSS MINIMIZATION AND MAXIMIZATION OF STATIC VOLTAGE STABILITY MARGIN BY HYBRIDIZED ALGORITHM

This paper presents a new Hybridized Algorithm (HA) for solving the multi-objective reactive power dispatch problem. Inspired by Genetic Algorithm (GA), Particle Swarm Optimization (PSO) & the Bat Algorithm (BA), the HA was designed to retain some advantages of each method to improve the exploration and exploitation of the search. Scrutinizing PSO and BA reveals some differences, in that BA rejects the historical experience of each individual’s own position but admits an improved personal solution with some probability. We will adjust some of the updating mechanisms of BA and add a mutation method in order to try to solve reactive power problem more accurately. Proposed (HA) algorithm has been tested on standard IEEE 30 bus test system and simulation results shows clearly about the good performance of the proposed algorithm.


Introduction
Optimal reactive power dispatch problem is subject to number of uncertainties and at least in the best case to uncertainty parameters given in the demand and about the availability equivalent amount of shunt reactive power compensators. Optimal reactive power dispatch plays a major role for the operation of power systems, and it should be carried out in a proper manner, such that system reliability is not got affected. The main objective of the optimal reactive power dispatch is to maintain the level of voltage and reactive power flow within the specified limits under various operating conditions and network configurations. By utilizing a number of control tools such as switching of shunt reactive power sources, changing generator voltages or by adjusting transformer tap-settings the reactive power dispatch can be done. By doing optimal (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 (5) Where, ξ = right eigenvector matrix of JR η = left eigenvector matrix of JR ∧ = diagonal eigenvalue matrix of JR and J R −1 = ξ˄ −1 η (6) From (5) and (8), we have ∆V = ξ˄ −1 η∆Q 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,

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. Minimize VD = ∑ |V k − 1.0| nl k=1 (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: G ij cos θ ij +B ij sin θ ij ] = 0, i = 1,2 … . , nb 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.

Genetic Algorithm with Laplace Crossover Operator
The crossover operator is a scheme for producing genetic information from parents; it combines the characters of two parents to form two off-springs, with the possibility that good chromosomes may evaluate better ones. The crossover operator is not regularly imposed to all pairs of parent solution the intermediate generation. An incidental choice is made, where the possibility of crossover being applied depends on probability determined by a crossover rate, known as crossover probability. The crossover operator is most significant part in GAs. It combines portion of good solution to construct new favorable solution. Information involved in one solution mixed with information involved in another solution and the rising solution will either have good quality fitness or stay alive to commutate this information again. If generated two off-springs are the same then crossover operator show strong heritability [21,22]. Crossover operators play key role in genetic algorithm which combines the characteristic of existing solutions and generate new solutions. The optimization problems depend upon the data they used so they are classified in to two categories. One is based on real data set and another one is based on binary or discrete data set. Crossover operator also considered as binary crossover operators and real coded crossover operators. Two particles distribute their positional information in the search space and a new particle is formed. The particle, is known as laplacian particle, replaces the nastiest performing particle in the swarm. Using this fresh operator, this paper introduces two algorithms namely Laplace Crossover PSO with inertia weight (LXPSO-W) and Laplace Crossover PSO with constriction factor (LXPSO-C) [23]. A. H. Wright suggests a genetic algorithm that uses real parameter vectors as chromosomes, real parameters as genes, and real numbers as alleles [24].Linear crossover [23,24] is one of the most primitive operator in real coded crossover it develops three solutions from two parents and the best two off-springs substitute parents. Let ( 1 (1, ) , 2 (1, ) , . . , (1, ) ) and ( 1 (2, ) , 2 (2, ) , . . , (2, ) ) are two parent solutions of dimension n at generation t. Linear crossover develops three offspring from these parents as shown in Eq. (24, 25 and 26) and best two offspring being chosen as off-springs.

Particle Swarm Optimization (PSO)
PSO [25][26][27][28] is a population based optimization tool, where the system is initialized with a population of random particles and the algorithm searches for optima by updating generations. Suppose that the search space is D-dimensional. The position of the i-th particle can be represented by a D-dimensional vector = ( 1 , 2 , . . , ) and the velocity of this particle is = ( 1 , 2 , . . , ).The best previously visited position of the i-th particle is represented by = ( 1 , 2 , . . , ) and the global best position of the swarm found so far is denoted by = ( 1 , 2 , . . , ). The fitness of each particle can be evaluated through putting its position into a designated objective function. The particle's velocity and its new position are updated as follows: Where ∈ {1,2, . . , }, ∈ {1,2, . . , } N is the population size, the superscript t denotes the iteration number, is the inertia weight, r 1 and r 2 are two random values in the range [0, 1], c 1 and c 2 are the cognitive and social scaling parameters which are positive constants.
These both equations are used to update the velocity and position of a particle in the exploration space .The equation (27) is used to balance the search abilities of the particle in the search space. The equation (28) uses the velocity obtained in first equation to get the new position of the particle. Crossover is a Genetic operator which is used after selection in Genetic Algorithm to get the new children using two or more than two parent .It is used to get the healthier solution than current solution. There are various improved version of crossover available to get the value of new-fangled species. Intermingling crossover is also an improved operator which is used to get the new healthier child by using current parent. This operator is applied in PSO to optimize the multi-dimensional function and upsurge the probing capability of the PSO, So that Particle Swarm Optimization optimizes the functions efficiently and did not jammed in the local optima.

Bat Algorithm
Bat algorithm has been developed by Xin-She Yang in 2010 [29]. Bats use sonar echoes to identify and evade obstacles. They use time delay from emanation to replication and utilize it for navigation. They classically emit short loud, sound impulse and the rate of pulse is usually 10 to20 times per second. Bats are in-bound to frequencies about 20,500 kHz. By execution [31], Pulse rate can be simply determined from range 0 to 1, where 0 means there is no emanation and by 1, bats are emitting maximum [30], By utilizing above behavior new bat algorithm can be formulated. Yang [29] used three generalized rules for bat algorithm: a) All bats use echolocation to sense distance, and they also guess the difference between prey and background barriers in some magical way. b) Bats fly arbitrarily with velocity ϑ i at position x i with a fixed frequency f min , varying wavelength λ and loudness A 0 to search for prey. They can automatically adjust the wavelength of their emitted pulses and adjust the rate of pulse emission r ∈ [0; 1], depending on the proximity of their target. c) Although the loudness can vary in many ways, we assume that the loudness varies from a large (positive) A 0 to a minimum constant valueA min .

Original Bat Algorithm
Where U (0; 1) is a uniform distribution.
An arbitrary walk with direct exploitation is used for local exploration that modifies the existing best solution according to equation: Where ϵ is the scaling factor, and A i (t) the loudness. The local exploration is launched with the proximity depending on the pulse rate ri and the new solutions accepted with some proximity depending on parameter. In natural bats, where the rate of pulse emission ri increases and the loudness Ai decreases when a bat finds a prey. The above characteristics can be written by the following equations:

Hybridized algorithm
The projected Hybridized Algorithm with greedy strategy resembles the outline of the Genetic Algorithm, which can be described as follows: initialization, evaluation, selection, crossover, and mutation. A difference from the Genetic Algorithm is that there is no distinct selection mechanism in the HA, because each individual will generate its offspring by recombination with the global best individual and it does not require an operator to select an individual to evolve. In addition, local search is also employed to increase the algorithm's exploitation capability. Even though Bat Algorithm can solve some tough problems and converge quickly, it frequently cannot evade converging to a local optimum. Scrutinizing PSO and BA reveals some differences, in that BA rejects the historical experience of each individual's own position but admits an improved personal solution with some probability. We will adjust some of the updating mechanisms of BA and add a mutation method in order to try to solve reactive power problem more accurately.

Mutation
The drive of mutation is to upsurge the diversity of the population and avert them trapping into a local optimum, particularly in the later iterations. So, the probability of mutation will be made low at the beginning and higher later. We set the mutation probability (MP) as follows, Where K is a limiting parameter which can be a constant or a variable, max is the maximum number of generations, and is the current generation.
The mutation formula is given as follows, is the solution of an individual after crossover, ∈ [−1, 1] is a uniform random number, and T is a vector which determines the scope of mutation.

Local Search
As we know, most of global optimization algorithms have outstanding competence in exploration but are feeble at exploitation. To augment this capability, particularly in the later iterations, we will expect the algorithm to be able to locate the global best rapidly with local search, once it has found the right neighbourhood. The probability of local search will be maintained low in early iterations and elevated later in the search process. The probability of local search will follow the same distribution as mutation (34).
The following formula used for local search, Where * −1 is the best individual of the current population, ∈ [−1, 1] is a uniform random number, and S is a vector which determines the search scope of the random walk, formulated in the variable space.

Simulation Results
The efficiency of the proposed HA method 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  -are with the tap setting transformers. The lower voltage magnitude limits at all buses are 0.95 p.u. and the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all the PQ buses and the reference bus. The simulation results have been presented in Tables 1, 2, 3 &4. And in the 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 0.2469 to 0.2478, 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.    [32] 5.0159 Genetic algorithm [33] 4.665 Real coded GA with Lindex as SVSM [34] 4.568 Real coded genetic algorithm [35] 4.5015 Proposed HA method 4.2989

Conclusion
In this paper, proposed HA has been successfully implemented to solve optimal reactive power dispatch (ORPD) problem. The main advantages of HA when applied to the ORPD problem is optimization of different type of objective function, i.e real coded of both continuous and discrete control variables, and without difficulty in handling nonlinear constraints. Proposed HA algorithm has been tested on the IEEE 30-bus system. Simulation Results clearly show the good performance of the proposed algorithm in reducing the real power loss and enhancing the voltage stability.