Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation

Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx

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Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation Santosh Kumar Singh a,⇑, Deepika Kumari a, Nilotpal Sinha b, Arup Kumar Goswami c, Nidul Sinha c a

Electrical and Electronics Engineering Department, Rajeev Gandhi Memorial College of Engineering and Technology, Nandyal, Andhra Pradesh, India Electrical and Computer Engineering Department, National Cheng Kung University, Tainan City, Taiwan c Electrical Engineering Department, National Institute of Technology, Silchar, Assam, India b

a r t i c l e

i n f o

Article history: Received 5 September 2016 Revised 23 December 2016 Accepted 24 January 2017 Available online xxxx Keywords: Gravity Search Algorithm Recursive Least Square Harmonic pollution Power quality

a b s t r a c t This paper presents a new hybrid method based on Gravity Search Algorithm (GSA) and Recursive Least Square (RLS), known as GSA-RLS, to solve the harmonic estimation problems in the case of time varying power signals in presence of different noises. GSA is based on the Newton’s law of gravity and mass interactions. In the proposed method, the searcher agents are a collection of masses that interact with each other using Newton’s laws of gravity and motion. The basic GSA algorithm strategy is combined with RLS algorithm sequentially in an adaptive way to update the unknown parameters (weights) of the harmonic signal. Simulation and practical validation are made with the experimentation of the proposed algorithm with real time data obtained from a heavy paper industry. A comparative performance of the proposed algorithm is evaluated with other recently reported algorithms like, Differential Evolution (DE), Particle Swarm Optimization (PSO), Bacteria Foraging Optimization (BFO), Fuzzy-BFO (F-BFO) hybridized with Least Square (LS) and BFO hybridized with RLS algorithm, which reveals that the proposed GSA-RLS algorithm is the best in terms of accuracy, convergence and computational time. Ó 2017 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Harmonic estimation is the prime step in mitigating the most prominent problems of power quality in the power system. The various negative impacts of harmonic pollution in electrical network include – increase of I2R losses for whole power network, various signal interferences, over and under voltages, loss of information and data, failure of circuit breaker operation, equipment heating, malfunction in the devices including damage, motor heating, pulsating and varying output torque, inefficiency and reduction in longitivity of the electrical appliances. The various impacts of harmonics have prompted to establish a number of suitable standards and proper guidelines regarding acceptable harmonic levels [1,2]. Accurate estimation and analysis of power system harmonic is essential to determine harmonic levels for effectively designing filters for mitigation of harmonics by way of removing the harmonic levels from the signal [2]. So from this point of view, it is highly imperative to know the quality of power delivered along

⇑ Corresponding author. E-mail address: [email protected] (S.K. Singh).

with harmonic parameters, such as amplitude and phase[2–5]. These two parameters are highly essential for designing the suitable filters for the elimination and reduction of harmonics and their effects in power system. The harmonic signals produced in the power network are dynamic in nature. This nature of the harmonic signal calls for some fast methods of measuring and estimating harmonic signals [1,2]. Some works are reported in the literature to address this problem wherein various approaches have been proposed to estimate the parameters of harmonics [1,2]. The Fast Fourier Transform (FFT) is considered as a suitable approach for stationary signal, but it suffers from loss of accuracy under time varying frequency conditions and posses picket and fence problems. The International Electrotechnical Commission (IEC) standard drafts has specified signal processing recommendations and definitions for harmonic, sub-harmonic and inter-harmonic measurement [1–3]. So far the hybrid approaches are concerned, significant contributions are reported in the literature under Refs. [6–10]. All these approaches are based on integrating both digital signal processing and soft computing techniques, namely, Genetic Algorithm-Least Square (GA-LS) [6], Fuzzy Bacteria foraging-least square (FBFOLS) [7], Particle swarm optimization with passive congregationleast square (PSOPC-LS) [8] and Artificial bee colony-least square

Peer review under responsibility of Karabuk University. http://dx.doi.org/10.1016/j.jestch.2017.01.006 2215-0986/Ó 2017 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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Nomenclature

vt

H

xh

f1 hh

distorted voltage signal harmonic order harmonic angular frequency fundamental frequency phase of the harmonic signal

(ABC-LS) [9]. In these hybrid approaches, the prime objectives of combining both least square and soft computing algorithms are to improve the convergence and accuracy of the harmonic estimation method. All of the heuristic algorithms are based on population search algorithms, and work with a population of strings representing different potential solutions [9]. Therefore, each of them has implicit parallelism that enhances their search capability and the optimum can be located more quickly when they are applied to complex optimization problems [10]. In this category of hybrid approaches, first the attempt has been made to optimize the phase of the harmonic components of the power signal by using metaheuristic algorithms and then the conventional recursive least squares are applied to get the amplitude of the harmonic signal [10]. Such hybrid algorithms have shown encouraging performances in solving harmonic estimation problems essentially because the actual models of voltage and current signals are nonlinear in phase and linear in amplitude [10]. Another hybrid approach [10] based on bacteria foraging optimization (BFO) is reported in which the phases of the fundamental and harmonic components are estimated using BFO, whereas, the conventional recursive least square (RLS) technique is used for estimating the amplitude of these components. Gravitational Search Algorithm (GSA) is a recently reported algorithm that has been inspired by the Newtonian’s law of gravity and motion. Since its introduction in 2009, GSA has undergone a lot of changes to the algorithm itself and has been applied in various applications [11]. At present, there are various variants of GSA such as Chaotic GSA incorporated to use sequences generated from chaotic systems to substitute random numbers for different parameters of GSA where it is necessary to make a randombased choice and another variant of chaotic is incorporated to utilize the chaotic search as a local search procedure of GSA. Oppositional GSA is developed to enhance and improve the original version of GSA. The original GSA is chosen as a parent algorithm and opposition-based ideas are embedded in it with an intention to exhibit accelerated convergence profile [12–15]. In GSA algorithm the different agents are considered as the objects and their performance is measured by their masses. All these objects attract each other by the gravity force and this force causes a global movement of all objects towards the objects with heavier masses [12,13]. Hence, masses cooperate using a direct form of communication through gravitational force. The heavy masses, which correspond to good solutions, move more slowly than lighter ones, which in turn guarantee the exploitation step of the algorithm [12,13]. In GSA, each mass (agent) has four specifications namely position, inertial mass, active gravitational mass, and passive gravitational mass [11]. The different positions of the masses correspond to a solution of the problem, and its gravitational and inertial masses are determined using a fitness function. Basically, each mass presents a solution, and the algorithm is navigated by properly adjusting the gravitational and inertia masses [11]. As the time passes, it is expected that the masses will be attracted by the heaviest mass. This heaviest mass will be the optimum solution in the search space [12,13]. GSA is considered as one

bdc

et

Hk hk T Samp

DC decaying term additive noise observation vector vector of unknown parameters (weight) sampling time

of the newest Evolutionary Algorithm (EA) and it has already proved itself as a very competent optimization technique in comparison to other developed heuristic optimization techniques [12,13] in terms of faster convergence, higher capability to escape from local optima and better quality of solutions. The estimation of harmonics is carried out in two phases; first the phases of the fundamental and other harmonic components are to be obtained as optimal weights using a nonlinear optimization algorithm and then their amplitudes are to be computed using recursive least square algorithm. The non-linearity arises due to the phases of the harmonic signals. To add further the time varying dynamic signal in presence of noise make this problem more non-linear and likely to have more local optima which makes it an appropriate field of application of heuristic search algorithms. Also, the estimation of amplitudes is more of linear one calling for conventional recursive least square algorithms. Hence, an urge is felt to consider the GSA algorithm for estimating the phases of the fundamental and other harmonic components, while their amplitudes are estimated by using the RLS algorithm to enhance the convergence as well as the accuracy of the estimation algorithm. Motivation: Every population based algorithm uses two search processes: exploration for more of global best and exploitation mostly for local best. This algorithm uses exploration capability at the beginning to escape local optimum problems followed by more exploitation at later generations. A time function, named as K best particle/agent, is used to attract other particles. The performance of GSA is improved by controlling exploration and exploitation. The value of K best function decreases with time linearly and at last only one agent will be there with heavy mass that represents the global solution [12,15–17]. In the view of the above following are the main objectives of the present work. (a) Maiden application of Gravity Search Algorithm (GSA) based Recursive Least Square (GSA-RLS) algorithm is proposed for estimating amplitudes and phases of the fundamental, harmonics, inter and sub harmonics in presence of various noises in power system signal. (b) To evaluate the comparative performance of the proposed algorithm as compared to other hybrid algorithms like GALS [6,10], PSOPC-LS [8,10], BFO [7,10], F-BFO-LS [7,10], BFO-RLS [10] in finding the best harmonic estimator. (c) To evaluate the performance of the algorithms for accurately estimating harmonic parameters on the data obtained from a real time industrial data setup for finding the best and appropriate method for harmonic estimation.

2. Gravitational Search Algorithm (GSA) The GSA algorithm can be considered as an isolated system of masses based on the Newtonian laws of gravitation and motion [11,12].

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1. The law of gravity: Each particle of the universe attracts every other particle and the gravitational force between them is directly proportional to the product of their masses and inversely proportional to the square of the distance (R) between them [11]. 2. Law of motion: The current velocity of any mass is equal to the sum of the fraction of its previous velocity and the variation in the velocity. Variation in the velocity or acceleration of any mass is equal to the force acted on the system divided by mass of inertia [11]. Now, consider a system with N agents (masses). The position of the ith agent is defined by:

X i ¼ ðx1i ; . . . xdi ; . . . xni Þ for i ¼ 1; 2; . . . ; N

ð1Þ

where xdi represents the position of ith agent in the dth dimension. At a specific time ‘t’, it is defined that the force acting on masses ‘i’ from mass ‘j’ as following:

F dij ðtÞ ¼ GðtÞ

M pi ðtÞxM aj ðtÞ d ðxj ðtÞ  xdi ðtÞÞ Rij ðtÞ þ e

ð2Þ

where Maj is the active gravitational mass related to agent j, M pi is the passive gravitational mass related to agent i, GðtÞ is gravitational constant at time t, e is a small constant, and Rij ðtÞ is the Euclidian distance between two agents i and j:

Rij ðtÞ ¼ kX i ðtÞ; X j ðtÞk2

ð3Þ

For stochastic characteristic of GSA algorithm, the total force that acts on agent i in a dimension (d) be a randomly weighted sum of dth components of the forces exerted from other agents are supposed which is given by (4).

F di ðtÞ ¼

N X

randj F dij ðtÞ

ð4Þ

j¼1;j–i

where randj is a uniform random number in the interval [0, 1]. Hence, by the law of motion, the acceleration of the agent i at time t, and in direction dth , ati ðtÞ is given as follows [11,12]:

adi ðtÞ ¼

F di ðtÞ

ð5Þ

Mii ðtÞ

where Mii is the inertial mass of ith agent. Then the next velocity of an agent is considered as a fraction of its current velocity added to its acceleration. Therefore, its position and its velocity could be calculated as follows [11,12]:

v di ðt þ 1Þ ¼ randi  v di ðtÞ þ adi ðtÞ

ð6Þ

xdi ðt þ 1Þ ¼ xdi ðtÞ þ v di ðt þ 1Þ

ð7Þ

where randi is a uniform random variable in the interval [0, 1]. The use of this random number is to give a randomized characteristic to the search. The gravitational constant (G) is initialized at the beginning and will be reduced with time to control the search accuracy where G is a function of the initial value G0 and time t [11,12]:

GðtÞ ¼ GðG0 ; tÞ

ð8Þ

The gravitational and inertial masses are updated by the fitness evaluation function. In this case, a heavier mass is supposed to have more efficient agent. Whereas the gravitational and inertial masses are updated by the following equations [11,12]:

M ai ¼ M pi ¼ M ii ¼ M i M i ðtÞ ¼

i ¼ 1; 2; . . . ; N

fit i ðtÞ  worstðtÞ bestðtÞ  worstðtÞ

ð9Þ

mi ðtÞ Mi ðtÞ ¼ PN j¼1 mj ðtÞ

3

ð11Þ

where fit i ðtÞ represent the fitness value of the agent i at time t, worstðtÞ and bestðtÞ are defined as follows for a minimization problem [11]:

bestðtÞ ¼ min fitj ðtÞ

ð12Þ

worstðtÞ ¼ max fitj ðtÞ

ð13Þ

j2f1;...;Ng

j2f1;...;Ng

To improve the performance of GSA by controlling exploration and exploitation only the K best agents will attract the others. K best is a function of time, with the initial value K 0 at the beginning and decreasing with time. In such a way, at the beginning, all agents apply the force, and as time passes K best are decreased linearly and at the end there will be just one agent applying force to the others [11]. Therefore, (4) could be modified as:

X

F di ðtÞ ¼

randj F dij ðtÞ

ð14Þ

j2Kbest;j–i

where K best is the set of first K agents with the best fitness value and biggest mass. The readers may be referred to [11] and [12] for more details of GSA algorithm. 3. Harmonic estimation problem The nonlinearity that arises in the sinusoidal model is due to the phase of the harmonic signal. In this paper, Gravity Search Algorithm (GSA) algorithm is used for the optimizing the unknown parameters (weights) of the harmonic signal for the estimation and the optimized weights are used by the RLS algorithm for estimating the amplitudes of the harmonic signal in an adaptive way [10,18–20]. 3.1. Mathematical modeling for harmonic estimation A distorted voltage signal can be modeled as the sum of higher order harmonics of unknown magnitudes and phases and can be represented as (15).

vt ¼

H X

v h Sinðxh t þ hh Þ þ v dc expðbdc tÞ þ et

ð15Þ

h¼1

where xh ¼ h  2pf 1 h ¼ 1; 2; . . . ; H So, the discrete time version of (15) can be represented as (16).

vk ¼

H X

v h Sinðxh kT Samp þ hh Þ þ v dc expðbdc kT Samp Þ þ ek

ð16Þ

h¼1

Now, using the first two terms of the Taylor series and also neglecting higher order terms, the decayed part of the signal can be approximated as-

v dc expðbdc kT Samp Þ ¼ v dc  v dc bdc kT Samp

ð17Þ

Substituting (17) in (16), (17) becomes

vk ¼

H X

v h Sinðxh kT Samp þ hh Þ þ v dc  v dc bdc kT Samp þ ek

ð18Þ

h¼1

So, this signal can be written in the form as (19)

vk ¼

H X ½v h sinðxh kT Samp Þ cos hh þ v h cosðxh kT Samp Þsinhh h¼1

ð10Þ

þ v dc  v dc bdc kT Samp þ ek 

ð19Þ

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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Hence, for the purpose of estimation of the harmonic parameters, this signal can be written in the parametric form as-

v k ¼ H k hk

ð20Þ

where Hk ¼ ½sinðx1 kT Samp Þcosðx1 kT Samp Þ    sinðxh kT Samp Þcosðxh kT Samp Þ    1  kT Samp 

T

ð21Þ

The overall vector of unknown parameters are-

hk ¼ ½h1k h2k    hð2h1Þk h2hk hð2hþ1Þk hð2hþ2Þk T

ð22Þ

h ¼ ½v 1 cos h1 v 1 sin h1    v h cos hh v h sin hn    v dc v dc bdc 

T

ð23Þ

Finally the objective function J for optimizing the vectors of unknown parameters (weights) due to the errors for harmonic signal estimation may be written as (24)K X J ¼ min e2k ðkÞ

!

K X 2 ¼ min ½v k  vd kest 

k¼1

! ð24Þ

k¼1

where vd kest represents the estimated output of the harmonic signal. This signal contains the vectors of unknown parameters (weights) which are optimized using GSA algorithm. 3.2. GSA-RLS approach for harmonic estimation problem The detailed procedure to apply GSA-RLS approach for solving harmonic estimation problem is discussed in this section. The basic control variables of the harmonic estimation problems are the weights of the unknown parameter of the harmonic signal. All these weights constitute the individual position of several masses that represent a complete solution set. The position of any agent i, consisting of N number of masses, may be defined as

Xi ¼

ðx1i ; x2i ; . . . ; xdi ; . . . ; xni Þ;

i ¼ 1; 2; 3; . . . ; N

cx11 ; x21 ; . . . ; xd1 ; . . . ; xn1

6 x1 ; x2 ; . . . ; x d ; . . . ; x d 6 2 2 2 2 6 6 : X¼6 6 x1 ; x 2 ; . . . ; x d ; . . . ; x n 6 i i i i 6 4 :

ð26Þ

x1N ; x2N ; . . . ; xdN ; . . . ; xnN The different steps to solve Harmonic Estimation problem using GSA-RLS approach is reported belowStep 1: Initialize GSA parameters like initial value of gravitational constant (G0 ), acceleration (a), and maximum iteration. Step 2: The initial positions of the control variables (weights of the unknown parameters) of each agent should be randomly selected while satisfying different harmonics present in the signal based on several numbers of agents ðNÞ depending upon the generated agent size. Each set in the agent matrix X represents a potential solution of the harmonic estimation problem. Step 3: Calculate the fitness value for each member of the agent matrix X. In GSA algorithm the fitness values of agents are represented by their masses. In this paper, different weights of the unknown parameters of the harmonic signals are considered. Their fitness values are calculated with the help of (9)–(14). Step 4: Update GðtÞ, bestðtÞ, worstðtÞ and M i ðtÞ for each set of agents. Step 5: Check excursion of newly generated elements of each population beyond their respective operating limits.

ð27Þ

Error in measurement is given by (28).

ekþ1 ¼ v kþ1  ½Hkþ1 T hd kest

ð28Þ

The gain K is related with covariance of parameter vector is represented as (29).

K kþ1 ¼ Pk Hkþ1 ½1 þ ½Hkþ1 T Pk Hkþ1 

1

ð29Þ

The updated covariance using matrix inversion lemma for the parameter vector is represented as (30).

Pkþ1 ¼ ½I  K kþ1 ½Hkþ1 T Pk

ð30Þ

The various parameters of these equations are initialized by considering some values for the estimate at different instants K, Hk and Pk . After the update of the vector of unknown parameters (weights) using the GSA-RLS approach, the amplitudes and phases of the funth

damental and h harmonic parameters along with dc decaying parameters can be computed using expressions (31)-(34) as follows:

vh ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh22h þ h22h1 Þ 

hh ¼ tan1

3 7 7 7 7 7 7 7 7 5

b h ðkþ1Þest ¼ b h kest þ K kþ1 ekþ1

ð25Þ

Several agents together form the agent matrix X

2

Step 6: If required, re-compute the agent matrix X, which have been generated after steps 4 and 5 for some population sets of the modified agent matrix X. Step 7: Compute the fitness value for each newly generated set of agent matrix X. Step 8: Repeat steps 4–7, for a predefined number of iterations. Step 9: The optimized output of the unknown parameter (weights) using GSA algorithm is taken as the input values for estimation using RLS. The vector of unknown parameter can be updated as (27)-(30).

h2h h2h1

ð31Þ

 ð32Þ

v dc ¼ h2hþ1

ð33Þ

  h2hþ2 bdc ¼ h2hþ1

ð34Þ

Because, h ¼ ½v 1 cos h1 v 1 sin h1 . . . v h cos hh v h sin hh . . . v dc v dc bdc  GSA is employed then to optimize the initial weights h given in (23) (see Figs. 1 and 2). T

4. Simulation results and discussion For the purpose of experimentation and to investigate the performance of the proposed algorithm, a synthetic signal is generated as a test signal, which is the same as the one used in [10,20]. This type of signal is typically nonlinear in nature and present in industrial loads, which are comprised of power electronic devices, variable frequency drives (VFD’s) and arc furnaces [10,20].

v t ¼ 1:5 sinð2  pi  f 1  t þ 80 Þ þ 0:5 sinð2  pi  f 3  t þ 60 Þ þ 0:2 sinð2  pi  f 5  t þ 45 Þ þ 0:15 sinð2  pi  f 7  t þ 36 Þ þ 0:1 sinð2  pi  f 11  t þ 30 Þ þ 0:5  expð5tÞ þ et

ð35Þ

The synthetic test signal is sampled at 64 samples per cycle from a 50-Hz voltage waveform after satisfying the Nyquist sampling criterion. The sampling frequency of the harmonic signal is considered to be at 2 kHz. The stochastic GSA-RLS approach is run for several times and then the mean value is reported. The obtained results

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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Fig. 1. Scheme of proposed GSA-RLS approach.

Fig. 2. Flow chart of proposed GSA-RLS approach.

are used to reconstruct the harmonic signal and then the estimated harmonic signal is compared with the original harmonic signal. The proposed approach is evaluated under several non-noisy and noisy situations, respectively. The Gaussian noises are added in different signal-to-noise ratios (SNRs) as 10, 20 and 40 dB respectively in the simulation study. Therefore, three signals are generated by adding these noises to the original harmonic signal. Under each noisy condition, including the original synthetic signal, the three signals are processed separately by the approach. The aforementioned signal is also corrupted by 1% of Gaussian random noise (et ) with zero mean and

unity variance. All the amplitudes given are in per unit (p.u) values. The proposed GSA-RLS approach is implemented in MATLAB2009a in a PC with 2.2 GHz Intel CPU and 2 GB RAM and the results are obtained based on 50 Hz nominal and fundamental frequency signal. To successfully implement the GSA for solution of the harmonic estimation problem, the optimum settings of different parameters are required to be determined. Different trials runs have been performed for the entire test systems. Based on the trials, the following input parameters are found to be the best for the optimal performance of the proposed algorithm: a = 20, Maximum No. of

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Iteration = 100, N = No. of agents = 50 Initial value of K 0 = N and its final value decreases to 2. The stop criterion of the algorithms is when the maximal generation is reached. Figs. 3 and 4 represent the actual vs. estimated output signal using GSA-RLS algorithm with SNR values of 10 dB and 40 dB respectively. It can be observed that at 40 dB SNR the estimated value nearly matches with the actual value, but as the SNR of the signal decreases as in the case of Fig. 4, there is more deviations of the estimated value from the actual value of the harmonic signal. Figs. 5 and 6 represent the estimated amplitude plot of the whole harmonic signal including fundamental, 3rd, 5th, 7th and 11th harmonics at 10 dB and 40 dB SNRs. Fig. 7 represents the estimated phase output plot of the whole harmonic signal including fundamental, 3rd, 5th, 7th and 11th harmonic signal with 40 dB SNR using GSA-RLS algorithm. It can be concluded that at 40 dB SNR the estimated value nearly matches with the actual value as is evident from the figures. Figs. 8 and 9 represent the estimated amplitude and estimated phase plot of the whole harmonic signal including fundamental, 3rd, 5th, 7th and 11th harmonics at 20 dB SNRs using GSA-RLS algorithm. It can be observed that at 20 dB SNR value the estimated value slightly deviated from the actual value and the performance of estimation is also degraded as compared to 40 dB SNRs as is evident from the figures. Figs. 10 and 11 represent the comparative performance of amplitude MSE and the phase MSE plot of the whole harmonic

signal including fundamental, 3rd, 5th, 7th and 11th harmonics at 10 dB and 20 dB SNRs using proposed GSA-RLS algorithm with recently reported BFO-RLS algorithm [10]. It can be observed that at 10 dB and 20 dB SNRs the performance of the proposed GSARLS is better than BFO-RLS in terms of accuracy and convergence. To investigate the performance of the proposed GSA-RLS algorithm in the estimation of harmonic signal in presence of interharmonics two inter-harmonics components are generated. The frequency, amplitude and phase of the inter-harmonic-1 are 130 Hz, 0.25 p.u. and 65° respectively. And the frequency, amplitude and phase of the other inter-harmonic-2 are 180 Hz, 0.35 p. u. and 20° respectively. Fig. 12 shows the amplitude estimation plot of inter-2 harmonic at 40 dB SNR. The estimation achieved with the proposed GSA-RLS is much more accurate with most of the samples converging towards the reference value in each case of inter harmonic amplitude estimation. Further, to investigate the performance of the proposed GSARLS approach for sub harmonic signals, a signal consisting of 20 Hz frequency, 0.5 p.u. amplitude and 75° is considered. Fig. 13 shows the amplitude estimation plot of the sub harmonic signal with 40 dB SNR. In this case too the estimated sub harmonic signal using GSA-RLS is more accurate and most of the samples have converged towards the reference value in each case of sub harmonic signal compared to other algorithms (see Fig. 14). Tables 1 and 2 report about the simulation results obtained with the proposed GSA-RLS algorithm along with recently reported GA-LS [6,10], PSOPC-LS [8,10], BFO [7,10], F-BFO-LS [7,10], and

Fig. 3. Actual vs. estimated output plot of the overall signal containing harmonics at 10 dB SNR.

Fig. 4. Actual vs. estimated output plot of the overall signal containing harmonics at 40 dB SNR.

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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Fig. 5. Estimated amplitude output plot of the overall signal containing harmonics at 10 dB SNR.

Fig. 6. Estimated amplitude output plot of the overall signal containing harmonics at 40 dB SNR.

Fig. 7. Estimated phase output plot of the overall signal containing harmonics at 40 dB SNR.

Fig. 8. Estimated amplitude output plot of the overall signal containing harmonics at 20 dB SNR.

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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Fig. 9. Estimated phase output plot of the overall signal containing harmonics at 20 dB SNR.

Fig. 10. Amplitude MSE plot of the overall signal containing harmonics at 10 dB SNR.

Fig. 11. Phase MSE plot of the overall signal containing harmonics at 20 dB SNR.

Fig. 12. Estimated amplitude plot of the inter-2 harmonic signal at 40 dB SNR.

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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Fig. 13. Estimated amplitude plot of the sub harmonic signal at 40 dB SNR.

BFO-RLS [10] algorithms. The final harmonic parameters obtained with the proposed approach exhibit the best estimation where the largest amplitude deviation is 1.45355% occurred at the inter-1 harmonics estimation and the largest phase angle deviation is 1.1055% occurred at the 5th harmonics estimation. The computational estimation time using GSA-RLS approach is the smallest (6.1575 s) as compared to other five algorithms. In the simulation studies the performance index n is estimated by (36)-

PH n¼

Fig. 14. Experimental setup for online real time voltage data recording.

v

v

d k¼1 ½ð k  kest Þ PH 2 k¼1 k

v

2



 100

ð36Þ

The performance index of estimation results of GA-LS, PSOPC-LS, BFO-LS, F-BFO-LS, BFO-RLS and the proposed GSA-RLS algorithms are given in Table 3. It can be observed from the table that the

Table 1 Comparative assessment of harmonic estimation using different approaches including GSA-RLS. Algorithm

Parameters

Fund

3rd

5th

7th

11th

Comp. time (s)

Actual

Frequency Amp (V) Phase (deg) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%)

50 1.5 80 1.48 1.33 80.61 0.61 1.482 1.2 80.54 0.54 1.4878 0.8147 80.4732 0.4732 1.488 0.8 80.42 0.42 1.4942 0.384 80.3468 0.3468 1.4956 0.2905 79.7960 0.2550

150 0.5 60 0.485 3.0 62.4 2.4 0.488 2.4 62.2 2.2 0.5108 2.1631 57.9005 2.0995 0.5103 2.06 58.1 1.9 0.4986 0.2857 58.5461 1.4539 0.5003 0.07450 59.6865 0.5225

250 0.2 45 0.18 10.0 47.03 2.03 0.182 9.0 46.6 1.6 0.1945 2.7267 45.8235 0.8235 0.198 1.0 45.75 0.75 0.2018 0.9021 45.6977 0.6977 0.2007 0.35501 45.4734 1.0525

350 0.15 36 0.158 5.33 34.354 1.646 0.1561 4.06 34.621 1.379 0.1556 3.7389 34.5606 1.4394 0.1545 3.0 34.73 1.27 0.1526 1.7609 34.8079 1.1921 0.1497 0.1505 36.0912 0.2535

550 0.1 30 0.0937 6.3 26.7 3.3 0.0948 5.2 27.31 2.69 0.1034 3.4202 29.127 0.873 0.1028 2.8 29.358 0.642 0.0986 1.746 29.9361 0.0639 0.0999 0.07502 30.0121 0.0405

–

GA-LS [6,10]

PSOPC-LS [8,10]

BFO [10]

F-BFO-LS [7,10]

BFO-RLS [10]

GSA-RLS

15.363

13.238

10.931

10.532

9.345

5.6545

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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S.K. Singh et al. / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx

Table 2 Comparative assessment of harmonic estimation including sub and inter harmonics. Algorithm

Parameters

Sub

Fund

3rd

Inter-1

Inter-2

5th

7th

11th

Comp. time (s)

Actual

Frequency Amp (V) Phase (deg) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%) Amp (V) Error (%) Phase Error (%)

20 0.505 75 0.532 5.34 73.02 1.98 0.53 4.95 73.51 1.49 0.525 3.995 74.48 0.514 0.521 3.247 74.61 0.388 0.511 1.190 74.81 0.183 0.494 1.1065 74.943 0.0755

50 1.5 80 1.5083 0.553 79.23 0.77 1.5049 0.326 79.45 0.55 1.4788 1.4103 79.8361 0.1639 1.489 0.733 79.86 0.14 1.5029 0.1952 79.9148 0.0852 1.4985 0.0945 79.9588 0.0515

150 0.5 60 0.472 5.6 57.55 2.45 0.481 3.8 58.12 1.88 0.4877 2.4575 61.2316 1.2316 0.489 0.489 61.16 1.16 0.4921 1.5887 59.076 0.924 0.5002 0.05525 59.6068 0.6552

180 0.25 65 0.238 4.8 62.41 3.59 0.24 4 63.28 1.72 0.2664 6.5574 63.9910 1.0090 0.261 4.4 64.33 0.67 0.2581 3.2372 65.3445 0.3445 0.2029 1.45355 65.0146 0.22552

230 0.35 20 0.381 8.85 17.64 2.36 0.377 7.7 18.23 1.77 0.3729 6.5295 19.6887 0.3113 0.371 6.0 19.723 0.277 0.3639 3.9651 19.8677 0.1323 0.3497 0.06575 19.9792 0.10355

250 0.2 45 0.215 7.5 48.33 3.33 0.211 5.5 48.1 3.1 0.2052 2.5764 47.698 2.6983 0.208 4.0 47.22 2.22 0.2009 0.4541 46.278 1.2783 0.2007 0.3552 45.4974 1.1055

350 0.15 36 0.172 14.66 38.78 2.78 0.165 10 37.109 1.109 0.1464 2.4170 36.7362 0.7462 0.1468 2.13 36.658 0.658 0.1479 1.4149 36.4473 0.4473 0.1497 0.7556 36.1107 0.30752

550 0.1 30 0.117 17 32.56 2.56 0.111 11.0 31.87 1.87 0.1016 1.5531 29.3928 0.6072 0.1019 1.9 30.52 0.52 0.1015 1.48 30.0643 0.0643 0.0999 0.09000 30.0097 0.03255

–

GA-LS [6,10]

PSOPC-LS [8,10]

BFO[10]

F-BFO-LS [7,10]

BFO-RLS [10]

GSA-RLS

Table 3 Comparison of performance index (n).

18.563

15.346

13.833

13.253

12.837

6.1575

Table 4 Comparison of the performance index (n) for real time data.

SNR

No noise

10 dB

20 dB

40 dB

Algorithm

Amplitude

Phase

Computational time (s)

GA-LS [6,10] PSOPC-LS [8,10] BFO [10] F-BFO-LS [7,10] BFO-RLS [10] GSA-RLS

0.1576 0.1236 0.1178 0.1054 0.0870 0.05575

10.6537 7.3645 5.2549 5.1864 4.5482 3.6525

1.2037 0.9546 0.8073 0.8021 0.7870 0.5475

0.1834 0.1572 0.1381 0.1124 0.0923 0.0652

GA-LS PSOPC-LS BFO F-BFO-LS BFO-RLS GSA-RLS

15.7458 14.2595 12.1545 11.2575 9.1475 6.7545

14.3575 12.5456 10.5514 9.5255 8.2548 6.3556

15.4595 13.4555 11.7545 10.6554 9.5576 7.1525

proposed GSA-RLS approach achieves a significant improvements in terms of reducing estimation error for harmonic signal as compared to other five algorithms.

5. Real time validity of the proposed GSA-RLS approach To investigate the performance of the proposed GSA-RLS algorithm under real time environment for estimating harmonics in power system, a voltage signal data is captured and recorded across a Variable Frequency Drive (VFD) panel, which is used for controlling the speed and torque of the induction motor at Hindustan Paper Corporation Limited (HPCL), a heavy paper industry

located at Panchgram in Cachar district of Assam, India through an experimental setup depicted in Fig. 10 with power quality analyzer. The distorted voltage signal data is acquired through USB connecting port of a Power Quality Analyzer (PQA) and sent to the laptop for analysis through the proposed GSA-RLS algorithm. Specifications of the instrument used are: 1. Laptop (Maker HP):1.5 GHz, 2 GB RAM, Intel Pentium3 Processor 2. Power Quality Analyzer (Maker Fluke):  True RMS Voltage (AC/DC): 5-1250 V  True RMS Current (AC/DC): 5-5000 Å

Fig. 15. Real time estimated voltage plot of the harmonic signal at 40 dB SNR.

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006

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 Frequency Range: 40 Hz to 15.9 kHz 3. Variable Frequency Drive (AC) Make- Siemens  Model No-MICROMASTER-440  Rating-250 kW/750 RPM 4. 3-Phase Induction Motor Make- Siemens  Rating-260 kW, 458 A, 50 Hz, 988 RPM, 415 V The real time distorted voltage data across the VFD panel is recorded while the motor was in running condition. The several data are recorded and then the amplitude of the voltage waveform is estimated using GSA-RLS approach. For all the data, voltage amplitude is obtained and the estimated results are compared. It is evident that the proposed GSA-RLS approach is found to be the best amongst all in terms of accuracy and faster convergence. Hence, the results obtained with real time data from a real time system validate the theoretical results obtained. As per IEC 61000-4-30 for computing power quality parameters 10 cycles in a 50 Hz system, which is 200 ms windowing at a sample time of 0.4 ms has been used for the experiment. The estimation performance of the proposed GSA-RLS algorithm on real time data is depicted in Fig. 15. Whereas, the performance of all the algorithms under realtime environment is reported in Table 4. It is evident from the table and the figure that the proposed algorithm is the best amongst all in terms accuracy and convergence. 6. Conclusion A new hybrid algorithm approach, called GSA-RLS algorithm has been developed and applied for the first time in which the GSA algorithm is used for estimating the phases and conventional RLS algorithm is applied to estimate the amplitudes of a time varying fundamental signal, its harmonics, sub harmonics and inter harmonics corrupted with different noises. The comparison of the performance of the proposed algorithm as compared to other recently reported five algorithms, namely, GA-LS, PSOPC-LS, BFO, F-BFO-LS and BFO-RLS, demonstrate that the proposed algorithm is the best in terms of accuracy and convergence time in estimating harmonic, sub-harmonic and inter harmonics. In addition, the real time experimentation at HPCL has further validated the superior performance of the proposed algorithm as compared to other five hybrid algorithms.

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Science and Technology (DST), Govt. of India, for the financial help provided for completing this work. In addition, the authors thank Dr. M Santhi Ramudu, Chairman of RGM group of Institutions for providing the research facilities at RGMCET Nandyal, AP, India. References [1] C.I. Chen, Y.C. Chen, Comparative study of harmonic and inter harmonic estimation methods for stationary and time-varying signals, IEEE Trans. Ind. Electron. 61 (1) (2014). 0278–0046. [2] H. Akagi, New trends in active filters for power conditioning, IEEE Trans. Ind. Appl. 32 (6) (1996) 1312–1322. [3] Testing and Measurement Techniques—General Guide on Harmonics and Inter harmonics Measurements and Instrumentation, for Power Supply Systems and Equipment Connected Thereto, IEC Std. 61000-4-7, 2009. [4] J. Barros, E. Perez, Automatic detection and analysis of voltage events in power systems, IEEE Trans. Instrum. Meas. 55 (5) (2006) 1487–1493. [5] T.A. George, D. Bones, Harmonic power flow determination using the Fast Fourier Transform, IEEE Trans. Power Delivery 6 (1991) 530–535. [6] M. Bettayeb, U. Qidwai, A hybrid least squares-ga-based algorithm for harmonic estimation, IEEE Trans. Power Delivery 18 (2) (2003) 377–382. [7] S. Mishra, A hybrid least square-fuzzy bacterial foraging strategy for harmonic estimation, IEEE Trans. Evol. Comput. 16 (2005) 61–73. [8] S. He, Q.H. Wu, J.Y. Wen, J.R. Saunders, R.C. Paton, A particle swarm optimizer with passive congregation, Bio System 78 (1–3) (2004) 135–147. [9] S. Biswas, A. Chatterjee, S.K. Goswami, An artificial bee colony-least square algorithm for solving harmonic estimation problems, Appl. Soft Comput. (2013) 2343–2355. [10] P.K. Ray, B. Subudhi, BFO optimized RLS algorithm for power system harmonics estimation, Appl. Soft Comput. 12 (8) (2012) 1965–1977. [11] E. Rashedi, H. Nezamabadi-pour, Saeid Saryazdi, GSA: a gravitational search algorithm, Elsevier Inf. Sci. 179 (2009) 2232–2248. [12] M. Crepinsek, S. Liu, M. Mernik, Exploration and exploitation in evolutionary algorithms: a survey, ACM Comput. Survey 45 (3) (2013). article 35. [13] B. Schutz, Gravity from the Ground Up, Cambridge University Press, 2003. [14] S. Gao, C. Vairappan, Y. Wang, Q. Cao, Z. Tang, Gravitational search algorithm combined with chaos for unconstrained numerical optimization, Appl. Math. Comput. 231 (2014) 48–62. [15] B. Shaw, V. Mukherjee, S. Ghoshal, A novel opposition based gravitational search algorithm for combined economic and emission dispatch problems of power systems, Int. J. Electr. Power Energy Syst. 35 (1) (2012) 21–33. [16] Y. Cai, J. Wang, Differential evolution with hybrid linkage crossover, Inf. Sci. 320 (2015) 244–287. [17] A. Bhattacharya, P.K. Roy, Solution of multi-objective optimal power flow using gravitational search algorithm, IET Gener. Transm. Distrib. 6 (2012) 751– 763. [18] S.K. Singh, A.K. Goswami, N. Sinha, Harmonic parameter estimation of a power signal using FT-RLS algorithm, in: IEEE, ICHQP-2014, May 2014. [19] S.K. Singh, A.K. Goswami, N. Sinha, Power system harmonic parameter estimation using BRLS algorithm, IJEPES-Elsevier J. 67 (2014) 1–10. [20] S.K. Singh, N. Sinha, A.K. Goswami, N. Sinha, Optimal estimation of power system harmonics using a hybrid firefly algorithm based least square method, Soft Comput (SOCO) J. Springer, http://dx.doi.org/10.1007/s00500-015-1877-0.

Acknowledgements This work was supported by SERB Project No: SR/FTP/ETA12/2011. The authors would like to acknowledge Department of

Please cite this article in press as: S.K. Singh et al., Gravity Search Algorithm hybridized Recursive Least Square method for power system harmonic estimation, Eng. Sci. Tech., Int. J. (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.006