Voltage Flicker Parameters Estimation UsingShuffled Frog Leaping Algorithm (SFLA) andImperialistic Competitive Algorithm (ICA)

Measurement of magnitude and frequency ofthe voltage flicker is very important for monitoring and controlling voltage flicker efficiently to improve the network power quality. This paper presents two new methods for measurement of flicker signal parameters using Shuffled Frog Leaping Algorithm (SFLA) and Imperialist Competitive Algorithm (ICA). This paper estimates fundamental voltage and flicker magnitudes and frequencies with proposed methods. The goal is to minimize the error of the estimated magnitudes and frequencies via a designed fitness function. At first we introduce voltage flicker and its measuring techniques. Then voltage flicker model is analyzed. At the next part a review of SFLA and ICA is presented these methods are applied to a test voltage signal and the results are given and analyzed.


INTRODUCTION
Voltage flicker is "the impression of unsteadiness of visual sensation induced by a light stimulus whose luminance or spectral distribution fluctuates with time".Voltage fluctuations are a systematic variations or a series of random changes in the voltage envelope.Cyclic flicker is repeatedly occurring over perhaps an extended period and is caused by periodic voltage fluctuations due to the operation of loads such as arc furnaces.Random changes in the voltage magnitude (noncyclic flicker) refer to occasional voltage fluctuations due to operations such as the starting of large motors.Exhaustion of human eyes, malfunction of electronic controllers and protection devices, reducing the life span of the electronic, incandescent, and cathode ray tubes are in the result of voltage flicker.Thus it is necessary to measure voltage flicker parameters in the power grid and evaluate them for designing suitable control devices to eliminate voltage flicker.Up to now, many techniques have been presented for digital analysis of voltage flicker parameters.Some of these used techniques are listed as follows: Techniques based on fast Fourier transform (FFT) [1], Kalman Filtering technique [2], Wavelet transforms technique [5], Least Absolute Value state estimation technique (LAV), [4].
These conventional techniques have some Deficiencies.Main disadvantage of applying either FFT or FFT prune techniques is the assumption of stationary signals.The application of such algorithms on non-stationary signals may lead to inaccurate results.Kalman filtering technique is a difficult method because of its large mathematical burden and needs to accurate adjustment for its parameters.The wavelet transform method involves computational complexity and also it has difficult process of choosing the candidate wavelet.The main disadvantage of the LAV state estimation technique is the assumption of knowing the flicker frequency in advance, which is not a realistic assumption.Moreover, it has slow convergence, which makes this algorithm not attractive for on-line tracking and implementation.This paper presents two new methods for measuring flicker signals magnitude and frequency.For this aim, the methods based on SFLA and ICA optimization techniques are used.The goal is to minimize the error in estimated parameters.The proposed methods are tested with a flicker signal and the ability of these methods to measurement of flicker magnitude and frequency is verified.At the end these two methods are compared.

VOLTAGE FLICKER MODEL
Mathematically, voltage flicker signal is shown as follows [1,3]: where Where () Zt is the measurement N*1 matrix.This matrix is given by measurement of digital signal samples with a suitable sampling frequency.
( , ) F t x is the N*1 information matrix given by Eq. ( 1) and is the function of time t and unknown variables x .
() et is the N*1 error matrix associated with the estimation of unknowns x .The error at each time step is calculated by: is the measurement voltage at time t that is given by measurement instruments.

Vt i actual
is the voltage that calculates from Eq. ( 1) by estimating the unknowns x with SFLA and ICA algorithms.In the problem of estimating the voltage flicker parameters, unknowns x are fundamental voltage amplitude, flicker amplitude, frequency and phase angle.The goal is to minimize the error between actual voltage and calculated voltage (e ) i by accurately estimating the optimum value of unknowns of the voltage flicker equation.

A. SFLA algorithm [9]
A mimetic meta-heuristic called the shuffled frog-leaping algorithm (SFLA) has been developed for solving combinatorial optimization problems.The algorithm contains elements of local search and global information exchange.At first an initial population of N frogs is created randomly.Then the frogs are sorted in a descending order according to their fitness.After that, the frogs divided into " m " memeplexes in such a way that the first frog goes to the first memeplex, the second frog S e p t e m b e r 2 2 , 2 0 1 5 goes to the second memeplex, the m th frog goes to the m thmemeplex and the ( 1 m  )th frog goes back to the first memeplex.This algorithm continues until N th frog.The SFLA performs simultaneously an independent local search in each memeplex using a particle swarm optimization like method.Within each memeplex, the frogs with the best and worst fitness are identified.During each memeplex evolution, the worst frog leaps toward the best frog.The position of the worst frog in each memeplex is updated as follows: Where " r " is a random number between 0 and 1,

B. ICA algorithm [10]
Imperialist competitive algorithm is an evolutionary algorithm for the optimization problems.ICA starts with an initial population..All the colonies are divided among the imperialists based on their power which is inversely proportional to colonies costs.Afterwards these colonies start moving toward their relevant imperialist country by x units.
x is a random variable with uniform (or any proper distribution).Thus x is as follows:   x ~U 0, β*d (6 Where β is a number greater than 1 and d is the distance between colony and imperialist.The direction of the movement is the vector from colony to the imperialist The imperialistic competition will gradually result in an increase in the power of powerful empires and a decrease in the power of weaker ones.Weak empires will lose their power and ultimately they will collapse.The movement of colonies toward their relevant imperialists along with competition among empires and also the collapse mechanism will hopefully cause all the countries to converge to a state in which there exist just one empire in the world and all the other countries are colonies of that empire.In this ideal new world, colonies have the same position and power as the imperialist.The best country in this world is the optimum response.The flowchart of ICA is shown in figure (2).Total cost of an empire computes as follows: Where .. TC n is the total cost of n th empire and  is a positive number which is considered to be less than 1..In this paper the value of 0.1 for  is used.To start the competition, first, we find the possession probability of each empire based on its total power.The normalized total cost is simply obtained by E.q (8): Where .. TC n and N.T.C n are respectively total cost and normalized total cost of n th empire.The possession probability of each empire is given by: To divide the mentioned colonies among empires based on the possession probability of them, we form the vector P as follows:

C. Fitness Function
Fitness function is responsible for evaluation of the solution at each step.For reaching to the minimum error () e i , we should consider a suitable fitness function for these optimization algorithms.The fitness function that is used in this paper is given in Eq. (8): Where e i is calculated by Eq. ( 3) and N is the number of samples.
For reaching to optimum estimates of flicker parameters, we should minimize SS with SFLA and ICA methods.

A. Case study one:
We consider a sample voltage flicker model according to Eq. ( 14) to test proposed methods: Competitive Algorithm results is better than Shuffled Frog Leaping Algorithm, as the flicker parameters estimation error in ICA technique is about zero (based on the number of samples and sampling frequency and the number of imperialists and colonies, it may have slightly error but it is less than 0.5% ).

CONCLUSION
Two new methods for measurement of flicker signal parameters (magnitudes and frequencies) are proposed in this paper.These proposed approaches estimates fundamental voltage amplitude and flicker amplitude and frequency with high accuracy.The proposed algorithms are tested with two sample voltage flicker signals and the results are presented.Also these two methods are compared and the effect of the number of samples and sampling frequency has analyzed.The results verify high efficiency of Shuffled Frog Leaping Algorithm (SFLA) and Imperialist Competitive Algorithm (ICA) in estimating of voltage flicker parameters.
of the frogs with best and worst fitness respectively and max D is the maximum allowed change of frog's position in one jump.The flowchart of SFLA is showed in figure (1).

Table 1 . Results of flicker parameters estimation with SFLA and ICA B. Case study two:
The number of samples is 100 and sampling frequency is considered to be 150Hz.SFLA and ICA are applied to this sample voltage signal.The results are given in table (1): (15)The number of samples is 50 and sampling frequency is 200Hz.Proposed methods are applied to this sample voltage signal.The results are given in table (2):

Table 2 .Results of flicker parameters estimation with SFLA and ICA As
it is seen from tables (1) and (2), proposed methods estimates unknown parameters of voltage flicker with error less than 1%.With the same number of samples, Shuffled Frog Leaping Algorithm estimates fundamental voltage amplitude and flicker amplitude and frequency in about 6 0 seconds, but Imperialist Competitive Algorithm estimates those parameters in about 7 seconds.So Imperialist Competitive Algorithm convergence to the optimum parameters very faster than Shuffled Frog Leaping Algorithm.These two optimization methods have excellent results for the flicker parameters estimation problem when the number of samples is more than 20 andsampling frequency be more than 100Hz.With increasing the number of samples and sampling frequency, SFLA and ICA gives best results for this problem.Imperialist 8 S e p t e m b e r 2 2 , 2 0 1 5