BFO optimized RLS algorithm for power system harmonics estimation

Applied Soft Computing 12 (2012) 1965–1977

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

BFO optimized RLS algorithm for power system harmonics estimation Pravat Kumar Ray, Bidyadhar Subudhi∗ Department of Electrical Engineering, National Institute of Technology, Rourkela 769008, Odisha, India

a r t i c l e

i n f o

Article history: Received 19 October 2010 Received in revised form 18 December 2011 Accepted 4 March 2012 Available online 16 March 2012 Keywords: Harmonic estimation Bacterial Foraging Optimization (BFO) Fuzzy BFO (F-BFO) RLS Genetic Algorithm (GA) Particle Swarm Optimization (PSO) DFT FFT

a b s t r a c t In this paper a new estimation approach combining both Recursive Least Square (RLS) and Bacterial Foraging Optimization (BFO) is developed for accurate estimation of harmonics in distorted power system signals. The proposed RLS–BFO hybrid technique has been employed for estimating the fundamental as well as harmonic components present in power system voltage/current waveforms. The basic foraging strategy is made adaptive by using RLS that sequentially updates the unknown parameters of the signal. Simulation and experimental studies are included justifying the improvement in performance of this new estimation algorithm. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The significant growth of the solid-state power switching devices in recent years has led to a corresponding increase in harmonic levels in power systems. Concurrently, increasing demands upon electrical resources have made the subject of power quality an important issue in modem electric utility operation. Negative effects of harmonic currents and voltages, such as increased I2 R losses and the reduction of the lifespan of sensitive equipment, has prompted the establishment of a number of standards and guidelines regarding acceptable harmonic levels. The interests in harmonic studies include modeling, measurements, mitigation, estimation and a variety of related subjects. Accurate analysis of power system measurements is essential to determine harmonic levels and effectively design mitigating filters. Transformer saturation in a power network produces an increased amount of current harmonics. Consequently, to provide the quality of the delivered power, it is imperative to know the harmonic parameters such as magnitude and phase. This is essential for designing filters for eliminating and reducing the effects of harmonics in a power system. A wide variety of techniques has been investigated [2–10]. Many algorithms are available to evaluate the harmonics of which the Fast Fourier Transform (FFT) is widely used. Other algorithms include,

∗ Corresponding author. E-mail address: [email protected] (Bidyadhar Subudhi). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2012.03.008

recursive Discrete Fourier Transform (DFT) [1], spectral observer and Hartley transform for selecting the range of harmonics. Neural networks have also been extensively studied as a means of harmonic extraction [4]. A popular method for parameter estimation is Least Square (LS) algorithm [5,10,18–20]. The algorithm is very powerful in estimating system parameters. Many researchers for estimating frequency and their deviations in a signal use it. Since variation of frequency is dynamic in nature, the conventional estimation technique assuming that power system voltage waveform is purely sinusoidal and the time between two zero crossing is an indication of frequency, based on constant frequency modeling is significantly affected. The difficulty in estimation of harmonics is due to the fact that harmonic generating loads are dynamic in nature. The harmonics produced have time varying amplitudes. Thus, fast methods of measuring and estimating harmonic signals are required. Search based algorithm such as GA [13,17], PSO [14–16] and BFO [11–13] have been applied to estimate the harmonic components present in power system voltage/current waveforms. BFO rests on a simple principle of the foraging (food searching) behavior of Escherichia coli bacteria in human intestine. The aforementioned problems discussed above have been addressed in this paper using an RLS based Bacterial Foraging Optimization (BFO) hybrid algorithm. In the proposed technique the power system signals with harmonics are estimated using BFO [11,12]. Further RLS is used to update the weights adaptively so that the estimated values converge to the desired values of the signal.

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P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

Fig. 1. Structure of RLS–BFO estimation scheme.

The contributions of the paper includes development of a new combined RLS and BFO approach to power system harmonics estimation that gives improvement in estimation performance. These are achievement of reduced estimation error, reduced processing time in computation and achieving improved performance in presence of inter and sub-harmonic components.

y(k) =

2. Motivation of combining RLS and BFO for harmonics estimation

ydc = Adc − Adc ˛dc kTs

• Evolutionary Computation technique is a population based search algorithm, it works with a population of strings that represent different potential solutions. • It enhances its search capability and the optima can be located more quickly when applied to complex optimization problems. • In this paper, the Bacterial Foraging Optimization scheme is used for estimating the unknown parameters on the basis of minimization of the cost function, which is the sum of squared error of the signal. • The optimized output values of the unknown parameters of BFO are used to initialize the unknown parameter for RLS algorithm. Then they are updated using RLS algorithm. • Amplitude and phase of the fundamental and harmonic components are estimated from the updated output of RLS.

3. Power system harmonic estimation problem Negative effects of harmonics currents and voltages are signal interference, over voltages, data loss and circuit breaker failure, as well as equipment heating, malfunction and damage. Harmonics are responsible for introduction of noise on telephone and data transmission lines, huge increases in computer data loss, excessive heating in transformers and capacitors resulting in shortened life or failure. Rotor heating and pulsating output torque caused by harmonics can result in excessive motor heating and inefficiency. Let us assume the voltage or current waveforms of the known fundamental angular frequency ω as the sum of harmonics of unknown magnitudes and phases. The general form of the waveform is y(t) =

N 

An sin(ωn t + n ) + Adc exp(−˛dc t) + (t)

(1)

n=1

where N is the number of harmonics ωn = n2f0 ; f0 is the fundamental frequency; (t) is the additive noise; Adc exp(− ˛dc t) is the probable decaying term.

The discrete time version of (1) can be represented as N 

An sin(ωn kTs + n ) + Adc exp(−˛dc kTs ) + (k)

(2)

n=1

where Ts is sampling period. Approximating decaying term using first two terms of Taylor series as (3)

Now Eq. (2) becomes y(k) =

N 

An sin(ωn kTs + n ) + Adc − Adc ˛dc kTs + (k)

(4)

n=1

The nonlinearity arises in the model is used to phase of the sinusoids. From the discrete signal of (4), we will estimate the amplitudes and phases of the fundamental and all the harmonics components. 4. Bacterial Foraging Optimization A brief review on the BFO is given here. The survival of species in any natural evolutionary process depends upon their fitness criteria, which relies upon their food searching (foraging) and motile behavior. So a clear understanding and modeling of foraging behavior in any of the evolutionary species, lead to its suitable application in any non-linear system. The E. coli bacteria that are present in our intestines have a foraging strategy governed by four processes, namely, chemo taxis, swarming, reproduction, and elimination and dispersal [11–13]. 4.1. Chemo taxis The characteristics of movement of bacteria in search of food can be defined in two ways, i.e. swimming and tumbling together known as chemo taxis. A bacterium is said to be ‘swimming’ if it moves in a predefined direction, and ‘tumbling’ if moving in an altogether different direction. Depending upon the rotation of the flagella in each bacterium, it decides whether it should go for swimming or for tumbling, in the entire lifetime of the bacterium. 4.2. Swarming For the bacteria to reach at the richest food location (i.e. for the algorithm to converge at the solution point), it is desired that the optimum bacterium till a point of time in the search period should try to attract other bacteria so that together they converge at the

P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

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Fig. 2. Flow chart of the proposed algorithm.

solution point more rapidly. This can be achieved by using a penalty function based upon the relative distances of each bacterium from the fittest bacterium.

This makes the population of bacteria constant in the evolution process. 4.4. Elimination and dispersal

4.3. Reproduction The original set of bacteria, after getting evolved through several chemo tactic stages reach the reproduction stage. Here, the best set of bacteria (chosen out of all the chemo tactic stages) gets divided into two groups. The healthier half replaces the other half of bacteria, which gets eliminated, owing to their poorer foraging abilities.

In the evolution process a sudden unforeseen event can occur, which may drastically alter the smooth process of evolution and cause the elimination of the set of bacteria and/or disperse them to a new environment. Most ironically, instead of disturbing the usual chemo tactic growth of the set of bacteria, this unknown event may place a newer set of bacteria nearer to the food location.

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5. Structure of the proposed RLS–BFO hybrid estimation scheme applied to harmonic estimation Fig. 1 shows the estimation scheme of proposed RLS–BFO combined algorithm. First input signal is fed to BFO algorithm. Unknown parameters (weight vectors before initialization) are optimized using BFO algorithm. Optimized output of BFO is taken as the initial values of weights for RLS algorithm. Then weights are updated using the steps of RLS algorithm. Fundamental as well as harmonic components are estimated from final updated weights of RLS. For estimation amplitudes and phases, (4) can be rewritten as

y(k) =

N 

[An sin(ωn kTs ) cos n + An cos(ωn kTs ) sin n ]

n=1

+ Adc − Adc ˛dc kTs + (k)

(5)

1. Initialization of BFO Parameters • S: total sample number • p: number of parameters to be optimized • Ns : swimming length • Nc : number of chemotactic iterations • Nre : maximum number of reproduction steps • Ned : maximum number of elimination and dispersal events • Ped : probability of elimination and dispersal • The location of each bacterium which is specified by random numbers on [0, 1]. • The value of C(i) assumed constant for simplification of design point of view • The values of dattract , ωattract , hrepellant and ωrepellant 2. Elimination-dispersal loop: l = l + 1 3. Reproduction loop: m = m + 1 4. Chemo taxis loop: n = n + 1 a) For i = 1, 2, . . . , S Compute value of cost function J(i,n,m,l) Ns 

ε2 (t) =

Ns 

2

Signal in parametric form becomes

J=

y(k) = x(k)W (k)

Let Jsw (i, n, m, l) = J(i, n, m, l) + Jcc (xi (n, m, l), P(n, m, l)) (i.e., add on the cell-to-cell attractant effect for swimming behavior), where

t=1

where x(k) = [ sin(ω1 kTS ) W = [ A1 cos(1 )

cos(ω1 kTS )

···

···

A1 sin(1 )

sin(ωN kTS )

An cos(n )

cos(ωN kTS )

An sin(n )

Adc

1 −kTS ] Adc ˛dc ]

T

(6)

Jcc (x, P(n, m, l)) = T

(7) The optimized output of the unknown parameter using BFO algorithm is taken as the initial values of unknown parameter for estimation using RLS. The vector of unknown parameter can be updated as W2 (k)

···

W2N−1 (k)

W2N (k)

W2N+2 (k) ]

W2N+1 (k)

ˆ (k) + K(k + 1)ε(k + 1) ˆ (k + 1) = W W

(8)

Error in measurement is (9)

The gain K is related with covariance of parameter vector T

K(k + 1) = P(k)x(k + 1)[1 + x(k + 1) P(k)x(k + 1)]

−1

(10)

The updated covariance of parameter vector using matrix inversion lemma T

P(k + 1) = [I − K(k + 1)x(k + 1) ]P(k)

(11)

These equations are initialized by taking some initial values for the estimate at instants k, x(k) and P. As the choice of initial covariance matrix is large it is taken P = ˛I where ˛ is a large number and I is a square identity matrix. After updating of unknown parameter vector, amplitudes, phases of the fundamental and nth harmonic parameters and dc decaying parameters are derived as



An =

n = tan

2 + W2n−1

˛dc =

(12)

 W  2n −1

(13)

W2n−1

Adc = W2n+1

W

2n+2

(14)



(15)

W2n+1

Because T

(16) BFO is employed then to optimize the initial weights W given in Eq. (16). The proposed RLS–BFO algorithm is discussed below (Fig. 2).

W = [ A1 cos(1 )

A1 sin(1 )

 S

=

i=1

+

S 

···

An cos(n )

An sin(n )

Adc

Adc ˛dc ]

S 

−dattract exp



i

Jcc (x, xi (n, m, l))

i=1



i=1

T ˆ (k) ε(k + 1) = y(k + 1) − x(k + 1) W

2 W2n

t=1

T

The vector of unknown parameter can be updated as W (k) = [ W1 (k)

[y(t) − yˆ (t)]

 −ωattract

P 

 i 2 (xm − xm )

m=1

hrepellant exp(−ωrepellant

P 

i 2 (xm − xm )

m=1

Jlast = Jsw (i, n, m, l) End of for loop b) For i = 1, 2, . . ., S take the tumbling/swimming decision • Tumble Generate a random vector (i), on [−1, 1] i) Update parameter (i) xi (n + 1, m, l) = xi (n, m, l) + u × C(i)

T (i)(i) This results in an adaptable step size in the direction of tumble for set of solution of parameter i ii) Compute: J(i, n + 1, m, l) Jsw (i, n + 1, m, l) = J(i, n + 1, m, l) + Jcc (xi (n + 1, m, l), P(n + 1, m, l)) • Swim Let nswim = 0; (counter for swim length) While nswim < Ns Let nswim = nswim + 1 If Jsw (i, n + 1, m, l) < Jlast (if doing better) Jlast = Jsw (i, n + 1, m, l) (i) xi (n + 1, m, l) = xi (n, m, l) + u × C(i)

T (i)(i) i Then x (n + 1, m, l) is used to compute the new J(i, n + 1, m, l) Else nswim = Ns This is the end of the while statement.

P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

actual estimated

2.5

c) Go to the next sample (i + 1) if i = / S [i.e. go to b] to process the next sample. d) If min (J) is less than the tolerance limit then break all the loops.

2 1.5

Amplitude in P.U.

5. If J < Nc go to 4, in this case continue chemotaxis since the life of bacteria is not over. 6. Reproduction a) For the given m and l, and for each i = 1, 2, 3, . . ., NC +1 S, let Jhealth = Jsw (i, n, m, l). Sort of parameter in j=1 ascending Jhealth b) Sr = S/2 no. of set parameters with highest Jhealth will be removed and other Sr no. of set of parameters with the best value split 7. If m < Nre go to 3, in this case, specified reproduction steps is not reached, so start the next generation in the chemo tactic loop. 8. Elimination-dispersal For i = 1, 2, . . ., S, with probability Ped , eliminate and disperse each set of parameters 9. Obtain optimized values for Weights (parameters) 10. Employ RLS for final updating of Weights 11. Estimate Amplitudes and phases from updated Weights

1969

1 0.5 0 -0.5 -1 -1.5 5

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Sample No Fig. 4. Actual vs. estimated output of signal using BFO algorithm (20 dB).

act ual est imated

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Amplitude in P.U.

1.5

6. Results and discussions 6.1. Static signal corrupted with random noise and decaying DC component The power system signal corrupted with random noise and decaying DC component is taken. The signal used for the estimation, besides the fundamental frequency, contains higher harmonics of the 3rd, 5th, 7th, 11th and a slowly decaying DC component [4]. This kind of signal is typical in industrial load comprising power electronic converters and arc furnaces [4]. y(t) = 1.5 sin(ωt + 800 ) + 0.5 sin(3ωt + 600 ) + 0.2 sin(5ωt + 450 )

1 0.5 0 -0.5 -1 -1.5 -2 0

5

10

15

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25

30

35

40

Sample No Fig. 5. Actual vs. estimated output of signal using BFO algorithm (10 dB).

+ 0.15 sin(7ωt + 360 ) + 0.1 sin(11ωt + 300 ) + 0.5 exp(−5t) + (t)

(17)

The signal is corrupted by random noise (t) = 0.01 rand(t) having normal distribution with zero mean and unity variance. In

actual estimated

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2

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Amplitude in P.U.

Amplitude in P.U.

2.5

the simulation work for harmonics estimation using BFO based algorithm, we have considered S = 100, p = 12 (without inter and sub-harmonics case), p = 18 (with inter and sub-harmonics case), Nc = 5, Nre = 10, Ned = 10, Ped = 0.1, Ns = 3, C(i) = 0.001, dattract = 0.05, ωattract = 0.3, hrepellant = 0.05 and ωrepellant = 10. The population size

1 0.5 0

actual estimated

1 0.5 0 -0.5

-0.5

-1 -1

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Sample No Fig. 3. Actual vs. estimated output of signal using BFO algorithm (40 dB).

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Sample No Fig. 6. Actual vs. estimated output of signal using RLS–BFO algorithm (40 dB).

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P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

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actual estimated

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Amplitude in P.U.

Amplitude in P.U.

1.5 1 0.5 0

1.2 funda 3rd 5th 7th 11th

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Sample No. 5

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Sample No

Fig. 9. Estimation of amplitude of fundamental and harmonics components of signal using RLS–BFO (40 dB SNR).

Fig. 7. Actual vs. estimated output of signal using RLS–BFO algorithm (20 dB). 1.8 1.6 1.4

Amplitude in P.U.

and the maximum number of generations for PSO and GA are both selected as 100 and 100 respectively. The stop criterion of the algorithm is that the maximum generation is reached. When performing PSO, the acceleration constants c1 and c2 are both set as 0.5; the inertia weight w = 0.9 is adopted. The parameter values for different algorithms are tuned on performing many experiments to get the optimal values. The simulation work is carried out on a PC with a 1.46 GHz CPU and 1 GB RAM. Figs. 3, 4 and 5 show actual vs. estimated signal using BFO algorithm with SNR values of 40, 20 and 10 dB respectively. It is seen that at 40 dB SNR value the estimated value closely matches with the actual value but as SNR value of signal decreases, there is more deviations of estimated value from actual value. Figs. 6, 7 and 8 show actual vs. estimated signal using RLS–BFO algorithm with SNR values of 40, 20 and 10 dB respectively. It is seen that at 40 dB SNR value the estimated value closely matches with the actual value but as SNR value of signal decreases, there is more deviations of estimated value from actual value. Figs. 9, 10 and 11 estimate amplitudes of fundamental as well as harmonics components contained in the signal using RLS–BFO algorithm at SNR of 40, 20 and 10 dB respectively. From these figures, it is seen that more accurate estimation is obtained at SNR of 40 dB.

1.2 1 0.8 funda 3rd 5th 7th 11th

0.6 0.4 0.2 0

0

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Sample No Fig. 10. Estimation of amplitude of fundamental and harmonic components of signal using RLS–BFO (20 dB SNR).

Figs. 12, 13 and 14 show a comparative estimation of phases of fundamental, 3rd, 5th, 7th and 11th harmonics components of signal at 40, 20 and 10 dB SNR respectively using both RLS–BFO algorithms. Figs. 15–19 compare the estimation of fundamental, 3rd, 5th, 7th and 11th harmonics components of signal respectively 1.8

actual estimated

3

1.6

2.5

1.4

Amplitude in P.U.

Amplitude in P.U.

2 1.5 1 0.5 0

1.2 funda 3rd 5th 7th 11th

1 0.8 0.6

-0.5

0.4 -1

0.2

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Sample No Fig. 8. Actual vs. estimated output of signal using RLS–BFO algorithm (10 dB).

Fig. 11. Estimation of amplitude of fundamental and harmonic components of signal using RLS–BFO (10 dB SNR).

P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977 1.7

90 funda 3rd 5th 7th 11th

Phase in degrees

70

bfo rls-bfo

1.65 1.6

Amplitude in P.U.

80

60

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40

1.55 1.5 1.45 1.4

30

1.35 0

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Sample No. 0

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Sample No.

Fig. 15. Comparison of estimation of amplitude of fundamental component of signal.

Fig. 12. Estimation of phase of fundamental and harmonic components of signal using RLS–BFO (40 dB SNR).

0.55 rls-bfo bfo

0.54 0.53 90

Phase in degrees

70

Amplitude in P.U.

funda 3rd 5th 7th 11th

80

60

0.52 0.51 0.5 0.49 0.48 0.47

50

0.46 40

0.45

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Sample No. 30

Fig. 16. Comparison of estimation of amplitude of 3rd harmonics of signal. 20

0

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Sample No. Fig. 13. Estimation of phase of fundamental and harmonic signal using RLS–BFO (20 dB SNR).

90 funda 3rd 5th 7th 11th

80

0.215 rls-bfo bfo

0.21 60

0.205

Amplitude in P.U.

Phase in degrees

70

using both BFO and RLS–BFO algorithms. Comparison shows that RLS–BFO outperforms over BFO in each case of estimation. Figs. 20–24 show a comparative estimation of phases of fundamental, 3rd, 5th, 7th and 11th harmonics components signal respectively using both BFO and RLS–BFO algorithms. RLS–BFO gives more correct estimation compared to BFO in these figures. Fig. 25 shows the comparative estimation of Mean Square Error (MSE) of signal using the two algorithms. From the figure, it is found

50

40

30

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0.195

0.19

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Sample No. 0.18

Fig. 14. Estimation of phase of fundamental and harmonic signal using RLS–BFO (10 dB SNR).

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Sample No. Fig. 17. Comparison of estimation of amplitude of 5th harmonics of signal.

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P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977 0.165

64 bfo rls-bfo

bfo rls-bfo

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0.16

Phase in degrees

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62

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0.145

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Sample No. Fig. 18. Comparison of estimation of amplitude of 7th harmonics of signal. Fig. 21. Comparison of estimation of phase of 3rd harmonics component of signal. 0.11 bfo rls-bfo

0.108

50

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48

0.104

Phase in degrees

Amplitude in P.U.

bfo rls-bfo

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0.094

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Sample No. 41

Fig. 19. Comparison of estimation of amplitude of 11th harmonics of signal.

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Sample No.

that, MSE performance in case of RLS–BFO is comparatively better than BFO algorithm. Table 1 compares the simulation results obtained by the proposed RLS–BFO algorithm with Genetic Algorithm (GA), Particle Swarm Optimization (PSO), BFO [11] and Fuzzy-BFO (F-BFO) [12]. The final harmonics parameters obtained with the proposed approach exhibit the best estimation precision where the largest amplitude deviation is 1.746% occurred at the 11th harmonics

Fig. 22. Comparison of estimation of phase of 5th harmonics component of signal.

estimation and the largest phase angle deviation is 1.4539◦ occurred at the 3rd harmonics estimation. The computational time of estimation using RLS–BFO is the smallest (9.345 s) compared to other four algorithms. 40

bfo rls-bfo

39 81.5 bfo rls-bfo

38

Phase in degrees

Phase in degrees

81 80.5 80 79.5

37 36 35 34

79 33 78.5

32

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Sample No. Fig. 23. Comparison of estimation of phase of 7th harmonics component of signal.

P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

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Table 1 Comparative assessment of harmonics estimation. Methods

Parameter

Fund

3rd

5th

7th

11th

Actual

f (Hz) A (V) ϕ (◦ )

50 1.5 80

150 0.5 60

250 0.2 45

350 0.15 36

550 0.1 30

GA

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

1.48 1.33 80.61 0.61

0.485 3 62.4 2.4

0.18 10 47.03 2.03

0.158 5.33 34.354 1.646

0.0937 6.3 26.7 3.3

15.363

PSO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

1.482 1.2 80.54 0.54

0.488 2.4 62.2 2.2

0.182 9 46.6 1.6

0.1561 4.06 34.621 1.379

0.0948 5.2 27.31 2.69

13.238

BFO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

1.4878 0.8147 80.4732 0.4732

0.5108 2.1631 57.9005 2.0995

0.1945 2.7267 45.8235 0.8235

0.1556 3.7389 34.5606 1.4394

0.1034 3.4202 29.127 0.873

10.931

FBFO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

1.488 0.8 80.42 0.42

0.5103 2.06 58.1 1.9

0.198 1 45.75 0.75

0.1545 3 34.73 1.27

0.1028 2.8 29.358 0.642

10.532

RLS–BFO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

1.4942 0.384 80.3468 0.3468

0.4986 0.2857 58.5461 1.4539

0.2018 0.9021 45.6977 0.6977

0.1526 1.7609 34.8079 1.1921

0.0986 1.746 29.9361 0.0639

9.345

35 bfo rls-bfo

34

Table 2 ANOVA table: amplitudes of the fundamental and harmonics components. Source of variation

Sum of squares

Degrees of Freedom (v)

Mean square

Between samples Within samples Total

6.27 × 10−4 6.7393 6.7399

4 20 24

1.568 × 10−4 0.3369

Phase in degrees

33 32 31 30

Remark: The simulation results of fundamental as well as harmonics components of signal estimation (Figs. 3–24), MSE analysis (Fig. 25) and different data as shown in Table 1, suggests that RLS–BFO outperforms over other four in all cases of estimation of a signal corrupted with random noise and dc offset.

29 28 27 26

0

10

20

30

40

50

60

70

80

90

100

F=

Sample No. Fig. 24. Comparison of estimation of phase of 11th harmonics component of signal. x 10

-4

bfo rls-bfo

Variance between samples 1.568 × 10−4 = = 4.6 × 10−4 0.3369 Variance within samples

The table value of F for v1 = 4 and v2 = 20 at 5% level of significance = 2.8661. The calculated value of F is less than the table value and hence the difference in mean values of the sample is not significant i.e. the samples could have come from the same universe. F=

Phase in degrees

Comp. time (s)

0.4851 = 1.1 × 10−3 436.3212

The table value of F for v1 = 4 and v2 = 20 at 5% level of significance = 2.8661. The calculated value of F is less than the table value and hence the difference in mean values of the sample is not significant i.e. the samples could have come from the same universe. Tables 2 and 3 show the ANOVA tables for the estimated data of fundamental and harmonics components of amplitudes and phases

1

Table 3 ANOVA table: phases of the fundamental and harmonics components.

0

0

10

20

30

40

50

60

70

Sample No. Fig. 25. Comparison of MSE of signal.

80

90

Source of variation

Sum of squares

Degrees of freedom (v)

Mean square

Between samples Within samples Total

1.9405 8726.425 8728.3655

4 20 24

0.4851 436.3212

100

1974

P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977 0.56

4

actual Estimated

3

bfo rls-bfo

0.55 0.54 0.53

Amplitude in P.U.

Amplitude in P.U.

2

1

0

0.52 0.51 0.5 0.49

-1 0.48 -2

0.47 0.46

-3

0

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90

0

100

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60

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100

Sample No.

Sample No.

Fig. 28. Estimation of sub-harmonics having amplitude 0.5 p.u.

Fig. 26. Actual vs. estimated value of signal having sub and inter harmonics using BFO (40 dB SNR).

respectively. As the calculated value of F in both the cases are very much less than their table values, statistical significance of dataset is proved.

78

bfo rls-bfo 77

6.2. Harmonics estimation of signal in presence of inter and sub-harmonics To evaluate the performance of the proposed algorithm in the estimation of a signal in the presence of sub-harmonics and interharmonics, a sub-harmonic and two inter-harmonics components are added to the original signal. The frequency of sub-harmonic is 20 Hz, the amplitude is set to be 0.505 p.u. and the phase is equal to 75◦ . The frequency, amplitude and phase of one of the inter-harmonic is 130 Hz, 0.25 p.u. and 65◦ respectively. The frequency, amplitude and phase of the other inter-harmonic is 180 Hz, 0.35 p.u. and 20◦ respectively. Figs. 26–33 show the estimation of amplitudes and phases of a sub-harmonic and two inter-harmonics. Using RLS–BFO, the estimation is very much perfect with most of the sample converge towards the reference value in each case of estimation. Table 4 compares the simulation results of the signal having two inter harmonics and one sub-harmonic components using

Phase in deg.

76

75

74

73

72

0

10

20

30

40

50

60

70

80

90

100

Sample No. Fig. 29. Estimation of sub-harmonics having phase 75◦ .

0.28 4

bfo rls-bfo

actual Estimated

3

0.27

Amplitude in P.U.

Amplitude in P.U.

2

1

0

0.26

0.25

0.24

-1 0.23 -2

-3

0.22 0

10

20

30

40

50

60

70

80

90

100

Sample No. Fig. 27. Actual vs. estimated value of signal having sub and inter harmonics using RLS–BFO (40 dB SNR).

0

10

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50

60

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80

90

Sample No. Fig. 30. Estimation of inter-harmonics having amplitude 0.25 p.u.

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P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

1975

Table 4 Comparative assessment of estimation performance in presence of inter and sub-harmonics. Methods

Parameter

Sub

Fund

3rd

Inter1

Inter2

5th

7th

Actual

f (Hz) A (V) ϕ (◦ )

20 0.505 75

50 1.5 80

150 0.5 60

180 0.25 65

230 0.35 20

250 0.2 45

350 0.15 36

GA

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

0.532 5.34 73.02 1.98

1.5083 0.553 79.23 0.77

0.472 5.6 57.55 2.45

0.238 4.8 62.41 3.59

0.381 8.85 17.64 2.36

0.215 7.5 48.33 3.33

0.172 14.66 38.78 2.78

0.117 17 32.56 2.56

18.563

PSO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

0.53 4.95 73.51 1.49

1.5049 0.326 79.45 0.55

0.481 3.8 58.12 1.88

0.24 4 63.28 1.72

0.377 7.7 18.23 1.77

0.211 5.5 48.1 3.1

0.165 10 37.109 1.109

0.111 11 31.87 1.87

15.346

BFO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

0.525 3.995 74.48 0.514

1.4788 1.4103 79.8361 0.1639

0.4877 2.4575 61.2316 1.2316

0.2664 6.5574 63.9910 1.0090

0.3729 6.5295 19.6887 0.3113

0.2052 2.5764 47.698 2.6983

0.1464 2.4170 36.7362 0.7462

0.1016 1.5531 29.3928 0.6072

13.833

FBFO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

0.521 3.247 74.61 0.388

1.489 0.733 79.86 0.14

0.489 0.489 61.16 1.16

0.261 4.4 64.33 0.67

0.371 6 19.723 0.277

0.208 4 47.22 2.22

0.1468 2.13 36.658 0.658

0.1019 1.9 30.52 0.52

13.253

RLS–BFO

A (V) Deviation (%) ϕ (◦ ) Deviation (◦ )

0.511 1.190 74.81 0.183

1.5029 0.1952 79.9148 0.0852

0.4921 1.5887 59.076 0.924

0.2581 3.2372 65.3445 0.3445

0.3639 3.9651 19.8677 0.1323

0.2009 0.4541 46.278 1.2783

0.1479 1.4149 36.4473 0.4473

0.1015 1.48 30.0643 0.0643

12.837

68

bfo rls-bfo

67

Phase in deg.

66

65

64

11th

Comp. time (s)

550 0.1 30

proposed RLS–BFO algorithm with four other algorithms. The largest amplitude deviation is 3.9651% occurred at 230 Hz interharmonic estimation and the largest phase angle deviation is 1.2783◦ occurred at 5th harmonics estimation in case of RLS–BFO. The computational time of RLS–BFO is least (12.837 s) as compared to other four algorithms. As a whole RLS–BFO outperforms over other four algorithms in the estimation of fundamental and harmonics components containing sub harmonics and inter harmonics components. 0.8098 × 10−4 = 4.0188 × 10−4 0.2015

63

F=

62

The table value of F for v1 = 4 and v2 = 35 at 5% level of significance = 2.641. The calculated value of F is less than the table value and hence the difference in mean values of the sample is not significant i.e. the samples could have come from the same universe.

61

60

0

10

20

30

40

50

60

70

80

90

100

Sample No.

F=

0.518 = 1.123 × 10−3 461.024

Fig. 31. Estimation of inter-harmonics having phase 65◦ . 22.5

0.38 bfo rls-bfo

0.37

21.5 21

0.36

Phase in deg.

Amplitude in P.U.

bfo rls-bfo

22

0.35

20.5 20 19.5

0.34 19 18.5

0.33

18 0.32

0

10

20

30

40

50

60

70

80

90

Sample No. Fig. 32. Estimation of inter-harmonics having amplitude 0.35 p.u.

100

17.5

0

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20

30

40

50

60

70

80

Sample No. Fig. 33. Estimation of inter-harmonics having phase 20◦ .

90

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P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977

Table 5 ANOVA table: amplitudes of the fundamental and harmonics components (in presence of sub and inter harmonic components). Source of variation

Sum of squares

Degrees of freedom (v)

Mean square

Between samples Within samples Total

3.239 × 10−4 7.05376 7.03408

4 35 39

0.8098 × 10−4 0.2015

Table 6 ANOVA table: phases of the fundamental and harmonics components (with sub and inter harmonics). Source of variation

Sum of squares

Between samples Within samples Total

2.072 16,135.85 16,137.922

Degrees of freedom (v) 4 35 39

Mean square 0.518 461.024

The table value of F for v1 = 4 and v2 = 35 at 5% level of significance = 2.641. The calculated value of F is less than the table value and hence the difference in mean values of the sample is not significant i.e. the samples could have come from the same universe. Tables 5 and 6 show the ANOVA tables for the estimated data of fundamental and harmonics components containing subharmonics and inter-harmonics components of amplitudes and phases respectively. Since the calculated values of F in both the cases are very much less than their table values, statistical significance of dataset is proved. In the simulation studies the performance index ε is estimated by

N ε=

k=1

(y(k) − yˆ (k))

N

k=1

2

y(k)

2

× 100

(18)

The performance index of estimation results of GA, PSO, BFO, FBFO and the proposed RLS–BFO algorithms are given in Table 7. From which it can be seen that RLS–BFO achieves a significant improvements in terms of reducing error for harmonics estimation as compared to other four algorithms. Remark: The simulation results of sub-harmonic and interharmonics components of signal estimation (Figs. 26–33) and different data as shown in Table 4, data shown in Table 7 for

Table 7 Comparison of performance index. SNR

GA

PSO

BFO

F-BFO

RLS–BFO

No noise 40 dB 20 dB 10 dB 0 dB

0.1576 0.1834 1.2037 10.6537 60.2384

0.1236 0.1572 0.9546 7.3645 53.6725

0.1178 0.1381 0.8073 5.2549 45.4871

0.1054 0.1124 0.8021 5.1864 38.6528

0.0870 0.0923 0.7870 4.5482 32.8243

Table 8 Comparison of performance index of real data. Parameter

GA

PSO

BFO

F-BFO

RLS–BFO

ε

11.8351

11.3067

11.0512

11.0113

10.9613

performance index judgment, suggests that RLS–BFO outperforms over other four algorithms in all cases of estimation of a signal containing a sub-harmonic and two inter harmonics components. 7. Experimental studies and results In view of real time application of the algorithm for estimating harmonics in a power system, data is obtained in a laboratory environment on running a DG set on normal working day of the laboratory as per the experimental setup shown in Fig. 34. Specifications of the instruments used are: 1. D–G set: (a) Alternator-3 phase, 50 Hz, Y connected, 415 volt, 1500 rpm, 55.8 A, 40 kVA (b) Diesel Engine – Bore × stroke = 110 × 116, 37.2 kW, 1500 rpm 2. Rheostats: 100 ohm, 5 A (3 in no.) 3. Ammeters: 0–3 A (3 in no.), MI Type 4. Lamp Load: 3 Phase type (had 2; 200 W/215 V bulbs in series in each phase) 5. Digital Storage Oscilloscope: Band Width-200 MHz, Sample rate-2GS/s, Channels-2, Record length-2500 data points, PC Connectivity – USB Port and Open Choice PC Communication software, Probe-P2220 6. PC: 1.46 GHz CPU and 1 GB RAM, Notebook PC

Rheostats

A

A

415 V, 1500 RPM, DG-Set

A

Digital Storage Oscilloscope (200 MHz)

PC (1.46 GHz, 1 GB RAM) with Open Choice PC Communication software

Fig. 34. Experimental setup for online data generation.

Load

P.K. Ray, B. Subudhi / Applied Soft Computing 12 (2012) 1965–1977 20

actual bfo rls-bfo

15

Amplitude in volt.

10

5

0

harmonics components contained in a power system signal contaminated with noise. In the estimation process, the algorithm first applies BFO to estimate the unknown parameters used for determining amplitudes and phases. Then final amplitudes and phases of fundamental and harmonics components are estimated after updating the unknown parameter using RLS algorithm. On comparing this new algorithm with GA, PSO, BFO and F-BFO, it is seen that the proposed RLS–BFO algorithm outperforms over other four in each case of estimation. References

-5

-10

-15

-20

1977

0

100

200

300

400

500

600

Sample No Fig. 35. Estimation of signal from real data.

The waveform is stored in a Digital Storage Oscilloscope (Tektronix Ltd.) across almost 10-ohm resistance (measured using multi-meter) and then through Open Choice PC Communication software, data is acquired to the personal computer. Ammeters are used to ensure that the load is balanced. The used PC had a 1.46 GHz CPU and 1 GB RAM. The sampling time in this case is fixed at 0.4 ms. For the purpose of estimation, the performance index is calculated for all the five algorithms and the results are given in Table 8. In this case RLS–BFO obtains the most accurate estimation result. From Fig. 35, the estimated waveform is close to the real one over the cycle. Hence the obtained results are satisfactory for the application with real data. The performance of the proposed hybrid algorithm is very dependent on the initial choice of maximum and minimum value of unknown parameters taken. In this paper, we have taken maximum and minimum values as 10% deviation from their actual values. By using an optimal choice of parameters, faster convergence to the true value of signal parameter can be achieved. Both the algorithms track the fundamental and harmonic signals very well but the performance of tracking using RLS–BFO is better than BFO. The power system signal is generated in MATLAB platform and the used PC had a 1.46 GHz CPU and 1 GB RAM. The same algorithms can be applied to other areas such as communication channels, telephones and other encrypted signals. 8. Conclusions This paper presents a new algorithm which can accurately estimate the amplitudes and phases of the fundamental as well as

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