Passerine swarm optimization algorithm for solving optimal reactive power dispatch problem

Received Jan 3, 2020 Revised Feb 11, 2020 Accepted Mar 14, 2020 This paper presents Passerine Swarm Optimization Algorithm (PSOA) for solving optimal reactive power dispatch problem. This algorithm is based on behaviour of social communications of Passerine bird. Basically, Passerine bird has three common behaviours: search behaviour, adherence behaviour and expedition behaviour. Through the shared communications Passerine bird will search for the food and also run away from hunters. By using the Passerine bird communications and behaviour, five basic rules have been created in the PSOA approach to solve the optimal reactive power dispatch problem. Key aspect is to reduce the real power loss and also to keep the variables within the limits. Proposed Passerine Swarm Optimization Algorithm (PSOA) has been tested in standard IEEE 30 bus test system and simulations results reveal about the better performance of the proposed algorithm in reducing the real power loss and enhancing the static voltage stability margin.


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 adjustment of these controls in different levels, the redistribution of the reactive power would minimize transmission losses. This procedure forms an optimal reactive power dispatch problem and it has a major influence on secure and economic operation of power systems. Various mathematical techniques like the gradient method Alsac et al . Lee et al and linear programming mangoli et al [1][2][3][4][5][6][7] have been adopted to solve the optimal reactive power dispatch problem. Both the gradient and Newton methods has the difficulty in handling inequality constraints. If linear programming is applied then the input-output function has to be expressed as a set of linear functions which mostly lead to loss of accuracy. This paper formulates by combining both the real power loss minimization and maximization of static voltage stability margin (SVSM) as the objectives. Global optimization has received extensive research attention, and a great number  [8][9][10][11][12][13]. Basically Passerine bird has three common behaviours: search behaviour, adherence behaviour and expedition behaviour. Through the shared communications Passerine bird will search for the food and also run away from hunters [14][15][16][17][18][19][20]. By using the Passerine bird communications and behaviour, five basic rules have been created in the PSOA approach to solve the optimal reactive power dispatch problem. Key aspect is to reduce the real power loss and also to keep the variables within the limits. Proposed Passerine Swarm Optimization Algorithm (PSOA) has been tested in standard IEEE 30 bus test system and simulations results reveal about the better performance of the proposed algorithm in reducing the real power loss and enhancing the static voltage stability margin.

VOLTAGE STABILITY EVALUATION
odal 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 are given by (1).
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 Where, ξ = right eigenvector matrix of JR η = left eigenvector matrix of JR ∧ = diagonal eigenvalue matrix of JR and From (5) and (8), we have 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, where, Where ξji is the jth element of ξi 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 (14).
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 (15).
Where nl is the number of load busses and Vk is the voltage magnitude at bus k.

System constraints
The following is an objective function that experiences constraints. a. 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.
b. Generator bus voltage (VGi) inequality constraint: c. Load bus voltage (VLi) inequality constraint: d. Switchable reactive power compensations (QCi) inequality constraint: e. Reactive power generation (QGi) inequality constraint: f. Transformers tap setting (Ti) inequality constraint: g. Transmission line flow (SLi) inequality constraint: Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.

PASSERINE BIRD SWARM HYPOTHESIS
The Passerine bird Figure 1 social behavior's can be written as follows: a) Rule 1. Every Passerine bird has choice to alter between the adherence behaviour and search behaviour.
Whether the Passerine bird searches or in observance, it is molded as a stochastic decision. b) Rule 2. While search, each Passerine bird can promptly record and renovate its previous most outstanding experience and the swarms' previous most outstanding experience about food area. This information has been used to discover food. Social information is shared rapidly among the whole swarm. c) Rule 3. During adherence, every Passerine will attempt to move near to the centre of the swarm.
This behaviour can be embroidered by the interference tempted by the rivalry among swarm. The Passerine with the uppermost reserves would be more prone to lie nearer to the centre of the swarm.
Where ∈ [1, . . , ], rand (0, 1) denotes independent uniformly distributed numbers in (0, 1). M and N are two positive numbers, which can be respectively called as cognitive and social accelerated coefficients. , is the best preceding position of the ith passerine and is the most excellent preceding position shared by the swarm. The Rule 1 can be defined as a stochastic decision. If a uniform arbitrary number in (0, 1) is smaller than, ( ∈ (0,1))a constant value, the Passerine would search for food. Otherwise, the passerine would carry on observance.

Adherence behaviour
Rule 3 indicates that passerine would try to move nearthe Centre of the swarm, and they would inevitably contend with each other. Thus, each Passerine cannot directly move towards the Centre of the swarm. This drive can be written as follows: Where ( ≠ 1)is a positive integer, which is illogically chosen between 1 and N. f1 and f2 are two positive constants in [0, 2], denotes the ith passerine best fitness value and sumFit represents the sum of the swarms' best fitness value. 1, which is used to keep away from zero-division error, denotes the jth element of the average position of the whole swarm. When a Passerine move near the Centre of the swarm, it will unavoidably compete with each other. The average fitness value of the swarm is measured by the surrounding swarm when a Passerine move to the Centre of the swarm. Each Passerine always wants to position at the Centre of swarm, the product of F1 and rand (0, 1) should not be more than 1. Here, F2 is used to create the direct effect persuaded by interference when a Passerine move to the Centre of the swarm. If the most outstanding fitness value of a random kth Passerine (ki) is greater than that of the ith Passerine, then F2, f2 which means that the ith may bear Passerine a greater interference than the kth Passerine. The kth Passerine would be move near the centre of the swarm than the ith passerine.

Expedition behaviour
Passerine may fly to other areas due to countless reasons. When the Passerine arrived at an innovative site, they would again search for food. Some Passerine as creators would search for food patches, while other Passerine try to feed from the food patch found by the creators. By the Rule 4 the creators and spongers can be detached from the swarm. The behaviors of the creators and spongers can be written as follows: , Where (0,1) denotes Gaussian distributed arbitrary number with mean zero and standard deviation 1, ∈ [1,2,3, . . , ], ≠ .
( ∈ [0,2])means that the sponger would follow the creator to search for food. We assume that each Passerine fly to alternative place every GH (positive integer) unit interval.

Passerine bird Swarm optimization Algorithm for optimal reactive power dispatch problem
Enter: P: the number of individuals (passerine) bounded in the population, Q: the utmost number of iterations, GH: the rate of repetition of Passerine expedition behaviors', K: the probability of searching for food, M, N, f1, f2, GH: are five constant parameters, = 0 ; Initialize the population Assessment of the N individuals' fitness value, and find the most outstanding solution

SIMULATION RESULTS
The efficiency of the proposed Passerine Swarm Optimization Algorithm (PSOA) 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 (28-27) -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, Table 2, Table 3 & Table 4. 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 107 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 0.2452 to 0.2466, 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.  Table 5. Comparison of real power loss Method Minimum loss (MW) Evolutionary programming [21] 5.0159 Genetic algorithm [22] 4.665 Real coded GA with Lindex as SVSM [23] 4.568 Real coded genetic algorithm [24] 4.5015 Proposed PSOA method 4.2502

CONCLUSION
In this paper, Passerine bird Swarm Optimization (PSOA) algorithm has been successfully implemented to solve optimal reactive power dispatch problem. By using the Passerine bird communications and behaviour, five basic rules have been created in the PSOA approach to solve the optimal reactive power dispatch problem. Key aspect is to reduce the real power loss and also to keep the variables within the limits. Proposed Passerine Swarm Optimization Algorithm (PSOA) has been tested in standard IEEE 30 bus test system and simulations results reveal about the better performance of the proposed algorithm in reducing the real power loss and enhancing the static voltage stability margin.