Harmonics and Interharmonics Estimation of DFIG based Standalone Wind Power Plant

Sanjay Agrawal, Student Member, IEEE,

Electrical Engineering Department MNNIT, Allahabad, India

Sanjay.ucer2009@gmail.com

S. R. Mohanty, Senior Member, IEEE,

Electrical Engineering Department MNNIT, Allahabad, India

soumya@mnnit.ac.in

Vineeta Agarwal, Senior Member, IEEE,

Electrical Engineering Department MNNIT, Allahabad, India

vineeta@mnnit.ac.in

Abstract—Detection of harmonics and inter-harmonics produced by wind power plant is very challenging task for the modern power system researchers. This paper presents estimation of harmonics produced by wind generator under variable wind speed by sliding window ESPRIT and sliding window Root-MUSIC algorithm. The series of simulation results demonstrate advantages of the sliding window ESPRIT over the sliding window Root MUSIC in estimation of harmonics and inter harmonics generated by standalone doubly fed Induction Generator.

Keywords—Sliding window ESPRIT; sliding window Root MUSIC;harmonics; inter-harmonics; Doubly fed induction generator(DFIG).

I. INTRODUCTION Modern variable speed wind turbines utilizes voltage

source converter based on high power semiconductor devices like SCRs, IGBT, GTOs etc. These devices injects harmonics and inter-harmonics (non integer multiple of fundamental) into the system. These harmonics and inter-harmonics frequency and magnitude are not constant over the time but it is dependent upon the wind speed and load conditions. In this context variable nature of harmonics and inter harmonics production may lead to resonance, control malfunction, protection equipment failure of nearby electrical equipments. That’s why accurate estimation of harmonics and inter-harmonics is essential in order to avoid these unwanted situations.

Various techniques [1] have been developed by several researchers in the past few decades. Many of these are based on the Fast Fourier Transform. FFT suffers from many disadvantages like poor spectral resolution, highly sensitive to power system frequency variations and require a minimum number of samples. IEC Standard 61000-4-7 [2] also utilizes FFT for harmonics and inter-harmonics estimation and uses grouping methodology. Due to grouping methodology, it can’t able give exact inter harmonic frequency component present in the signal. Short time Fourier transform (STFT) suffers from leakage and piket-fence effect when power system frequency varies or signal contains inter harmonic component and also have poor frequency resolution. Discrete wavelet transform [3] is a non parametric technique gives spectrum in the terms of frequency bands. DWT can also be used for harmonics

estimation but it require additional tool for extracting frequency from these bands. Its accuracy is highly dependent upon the mother wavelet and resolution levels.

Artificial intelligent based harmonics estimation techniques [4]–[5], and Kalman filters [6], [7] have been found in applications where the prior knowledge of harmonics frequency available. These methods give poor accuracy when an unknown frequency component appears signal.

Almost all the non parametric techniques suffer from frequency resolution. In past few years, parametric techniques have been applied widely in harmonics and inter harmonics estimation. These techniques give better frequency resolution than the nonparametric techniques and its accuracy not depends upon the number of samples and the knowledge of fundamental frequency. prony [8] is also used for estimation of power system harmonics[9]-[10] but it is highly prone to noise. Multiple signal classification (MUSIC)[11], Estimation of signal parameters by rotational invariance (ESPRIT)[12], are the parametric techniques which are capable to detect closely spaced frequency components but these techniques can determine the frequency of steady state signal. Sliding window ESPRIT [ 13] is used to determine the inter harmonics of time varying signal which gives efficient results in harmonics and inter harmonics estimation.

In the present study, sliding window ESPRIT and sliding window root music is implemented to estimate harmonics and inter harmonics component produced by DFIG.

The present paper is organized as follows. In Section II, origin of harmonics and inter harmonics component by DFIG is explained. Theory and principle of harmonics estimation by ESPRIT and Root MUSIC is explained in the section III and IV respectively. Sliding window Root MUSIC and sliding window ESPRIT concepts explained in the section V. Section VI gives description about the DFIG model taken for signal generation. Simulated results and discussion on the performance of estimation methods are presented in the section VII. Finally conclusions are drawn in the section VIII.

II. HARMONICS GENERATION IN STATOR OF DFIG In standalone DFIG, Harmonics and Inter Harmonics are

generated in stator side mainly due to following reasons.

978-1-4799-2399-1/14/$31.00 ©2014 IEEE 2601

A. MMF Space Harmonics Due to Machine design limitation, mechanical distribution

of rotor and stator winding creates air flux which is not perfectly sinusoidal in nature. Harmonics generated by this flux is referred to as MMF space harmonics. Inter harmonics frequency due this flux is given by following equation [14]

s =| 6 (1- ) 1| (1)kspacef k s f±

where k =0, 1, 2, 3 ………….

s f is synchronous or stator frequency

s is the actual slip of the induction generator

As stated in [15] (1) is valid if both stator and rotor supplies are balanced. If this condition is not fulfilled, many other frequency components are also produced.

From the (1) it is clear that these frequencies are dependent upon the slip. Value of slip depends upon the wind speed. That’s why, space harmonics frequency is changed as wind speed changes.

B. Slot Harmonics These harmonics generated by variations in reluctance due

to the slots. Slot harmonics is given by following equation.[14]

( )2 1 1 (2)slot sSf f s

P⎛ ⎞= − ±⎜ ⎟⎝ ⎠

Where S is the number of slots Slot harmonics frequency also varies with wind speed.

C. Harmonics and inter harmonics produced by RSC

In DFIG, rotor side converter (RSC) injects voltage of variable magnitude and frequency in rotor circuit. Frequency and magnitude of this injection depends upon the wind turbine speed. MMF produced rotor voltage injection rotates in the same direction when rotor speed is below synchronous speed circuit. When rotor speed is above the synchronous speed, it rotates in opposite direction to the rotor. This is an inherent property of DFIG to make net flux linkage and stator current/voltage frequency at synchronous frequency.

(3)rs injectedff f= ±

where, injectedf is the injected rotor voltage frequency.

rf is the rotor frequency. In (3) +ve sign is used when rotor speed is below the

synchronous speed and − ve sign is used when rotor speed is above the synchronous speed.

6k-1 harmonics components of rotor injected voltage have different phase difference (0,240,120) from fundamental rotor injected voltage. These rotor harmonics causes harmonics in stator voltage and current of order

6 1 (4)hsub r kf f f −= ∓

In (4) − ve sign is used when rotor speed is below the synchronous speed and +ve sign is used when rotor speed is above the synchronous speed.

6k+1 harmonics components have same phase difference as the fundamental rotor injected voltage. These rotor harmonics causes harmonics in stator voltage and current of order

6 1 (5)hsub r kf f f −= ±

In (5) +ve sign is used when rotor speed is below the synchronous speed and − ve sign is used when rotor speed is above the synchronous speed.

III. ROOT MULTIPLE SIGNAL CLASSIFICATION (ROOT MUSIC)

MUSIC [1], [11] is subspace based parametric technique. It is based on the eigenvalue decomposition of the correlation matrix and its partitioning into signal and noise subspaces. It utilizes noise subspace for frequency estimation.

Harmonics polluted power supply signal is expressed as.

1y( ) cos(2 ) ( ) (6)k

K

kk

n n fa w nπ=

= +∑

Where ka is amplitude, kf is harmonics frequency, K is the number harmonics frequency components present in the signal and ( )w n is white noise with zero mean. MUSIC employs complex exponential model to estimate frequencies and magnitude of harmonics in signal. Equation (6) can be expressed as

2 2

1 1y( ) ( ) (7)k k

K Kj n j n

k kk

f f

kn A e A e w nπ π−

= =

= + +∑ ∑

Where kA the complex is the complex amplitude and can be related to ka as

( 8 )2k

kaA =

Correlation matrix yR of M × M, obtained from available data samples of y(n) of length 1L N M= + − is given by (9).

1 (9 )HyR Y Y

N=

where (.)H is Hermitian operator Y is data matrix of size N×M and is described as

2602

(0) (1) ( 1)

y(1) (2) ( )

( 1) ( ) ( 2)

(10)

y y y M

y y M

y N y N y N M

Y

−

− + −

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

…

This correlation matrix is divided into two orthogonal subspaces signal subspace and noise subspace. Signal subspace has 2K eigenvectors corresponding to the 2K largest eigenvalues. 2K corresponds to no. of harmonics frequency component of interest. Remaining M−2K eigenvectors corresponds to noise subspace. Noise subspace is used to compute pseudo spectrum in (11). Peaks of pseudo spectrum correspond to those of the dominant frequency components.

2

2 1

1( e ) ( 1 1 )jwM U S IC M

Hi

i K

PN

= +

=ϒ∑

where, Hϒ represents complex conjugate transpose of signal eigenvector, iN represent noise space eigenvector. Amplitude of harmonics component is obtained by solving [16](12)

1A = (V ) V ( 1 2 )H HV X−

Where V is defined as

1 2 2

1 2 2

1 2 2

2 2 2

2*2 2*2 2*2

( 1)*2 ( 1)*2 ( 1)*2

1 1 1

(13)

K

K

K

j f j f j f

j f j f j f

j L f j L f j L f

e e ee e e

e e e

π π π

π π π

π π π− − −

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

…

And X is defined as

[ ](0 ) (1) ( 2 ) (L 1) (1 4 )TX x x x x= −

IV. ESTIMATION OF SIGNAL PARAMETERS VIA ROTATIONAL INVARIANCE TECHNIQUE (ESPRIT)

ESPRIT [1], [12] is also subspace based parametric technique. It is also employs complex exponential model to determine the signal parameters. However, It uses signal subspace for parameters calculation rather than the noise subspace. This employs rotational property between staggered subspaces to determine the frequency parameter. Once the frequency is calculated other parameter magnitude is determined by using (12).

Basic ESPRIT algorithm utilizes same procedure as in MUSIC to decompose correlation matrix in to subspace matrix. After that it uses signal subspace and determines the two shifted matrix from signal subspace matrix ϒ . The basic

ESPRIT employs following steps to determine the signal parameters.

1) It adopted similar procedure to determine signal subspace as described in section III.

2) After that it uses two selection matrixes 1V and 2V to determine two shifted matrix 1ϒ and 2ϒ from ϒ using (16) for i=1, 2.

1 1[I 0 ] (15(a))M dV −=

2 1[0 I ] (15(b))d MV −=

* (16)i iVϒ = ϒ

Where 1IM − an identity matrix of order M−1 and d is is the distance between two sub matrices.

3) Applying shift invariance property, 1ϒ and 2ϒ can be related using a matrix ψ , whose eigen values represents exponential terms 2 kj fe π where k varies 1, 2 , 3, ... 2K.

11 1 1 2( ) (1 7 )H Hψ −= ϒ ϒ ϒ ϒ

4) The frequency components are obtained from the eigen values of ψ by using (18)

(1 8 )2

kkf ψλ

π∠

=

V. SLIDING WINDOW ESPRIT AND SLIDING WINDOW ROOT MUSIC

Both the ESPRIT and Root MUSIC are able to determine accurate frequency when the signal is under steady state which means the signal parameter like frequency and magnitude of harmonics component is not time varying in nature. It is clear from the section II that harmonics magnitude and frequency introduced by DFIG is not constant. In the present section, Sliding window concept shown in Fig. 2 is discussed. In sliding window ESPRIT [13] and Root MUSIC a small window is taken which moves over the time with overlapping previous. By using this sliding window concept both the methods are able to determine frequency of a time varying signal. In implementing With the Sliding window concept it is assumed that signal is stationary within this window.

Fig. 2 Sliding window methodology

2603

VI. MODEL OF STANDALONE DFIG Model of Standalone DFIG is shown in Fig.1is developed

in MATLAB/Simulink. In this Model DFIG is connected to three phase star connected balanced load. Each phase have 11Ω resistive load. In rotor circuit voltage is injected by using DC/AC converter. When Rotor speed is below the synchronous speed, it acts like inverter and supplies power to rotor circuit. It acts as rectifier when rotor speed is above the synchronous speed and stores power in to battery. Different switching techniques are available converter like space vector pulse width modulation (SVPWM), pulse width modulation (PWM) and six pulse switching etc. Out of these techniques PWM technique is widely adopted because of its simplicity in control and reduced switching losses.

In present study, six pulse switching technique is used to inject rotor voltage. In six pulse switching techniques, triplen harmonics never comes in to output voltage and only 6k±1 order harmonics present in output voltage. Due to these harmonics output voltage waveform is quasi-sine wave.

Fig. 1 Standalone DFIG Wind Power Plant

5 HP 4 pole slip ring induction motor is used to model DFIG. Parameters of motor is taken from the [17] given in Table I. Whole the model is simulated in the MATLAB/SIMULINK environment. As per the discussion in section II, It is clear that harmonics produced in the stator due to rotor voltage harmonics is depends upon the rotor speed or wind speed. To check the accuracy of parametric techniques under variable wind speed 10 seconds stator current is recorded and rotor rotation frequency is changed from 1200 RPM or 40 Hz to 1140 RPM or 38 Hz at the instant 3s and then it changes to 40 Hz at the instant 7s. Rotor side converter injects 20 Hz and 22 Hz voltage in to rotor circuit for the rotor speed 40 Hz and 38 Hz respectively.

TABLE I. MACHINE PARAMETER OF 5 HP DFIGS

S.No. Parameters Values 1 ( )sR Ω 0.32

2 ( )lsL mH 1.19

3 ( )M mH 39.46 4 ( )lrL mH 1.34

5 r ( )r Ω 0.36

6 Turn ratio a 1.38

VII. SIMULATION RESULTS AND DISCUSSIONS First the performance of the parametric techniques

described in section III, IV are V are tested in the synthetic signal and after that technique described in section V is applied to determine the harmonics in stator current of DFIG due to the injected rotor voltage is determined. First of all description of frequency component present in the synthetic signal is shown in the Table II to test the performance of two algorithms. Synthetic signal is sampled by sampling frequency 10 kHz.

TABLE II. DISCRIPTION OF SYNTHETIC SIGNAL

Frequency (Hz)

Magnitude

420 5 300 3 250 4 200 2.5 95 3.5 70 6 60 50

First 2500 samples of synthetic signal are taken and both Root MUSIC and ESPRIT is applied for determining the frequency components. Results of estimation by Root MUSIC and ESPRIT are shown in Fig. 3(a) and Fig. 3(b) respectively.

Fig. 3(a) Harmonics parameter estimation by Root MUSIC

Fig. 3(b) Harmonics parameter estimation by ESPRIT

0 500 1000 1500 2000 2500-80

-60

-40

-20

0

20

40

60

80

No of Samples

Mag

nitu

de

Estimated Signal by Root Music

TrueEstimated

0 500 1000 1500 2000 2500-80

-60

-40

-20

0

20

40

60

80

No. of Samples

Mag

nitu

de

Estimated Signal by ESPRIT

TrueEstimated

2604

It is clearly seen from the above results harmonics parameters obtained from ESPRIT is likely converging to true value as compared to Root MUSIC.

For testing the accuracy of sliding window ESPRIT and sliding window Root MUSIC, synthetic signal described in Table II of 10s duration are taken. Both the methods are using 0.25 seconds sliding window with 80% overlapping to determine the harmonics parameters. Results of estimation by Sliding window root music and sliding window ESPRIT is shown in Fig 4 (a) and 4(b) respectively.

Fig. 4(a) Harmonics parameter estimation by Sliding window Root MUSIC

Fig. 4(b) Harmonics parameter estimation by Sliding window ESPRIT.

It is clear from the results that both the methods are estimating the harmonics parameters as much as close true value. From results shown in the Fig 4(a) and Fig. 4(b), it is difficult to analyze which method performs better. That’s why average estimation error over 10 seconds duration is also calculated by (19) and show in Table III.

1 (1 9 )

P

ii

a vg

EE

P==∑

where P is the total no of sliding windows. iE is the estimation error in single window.

avgE is the average estimation error. From the average estimation error, it is clear that signal

parameter estimation by sliding window ESPRIT is much better than the sliding window root MUSIC. Average

computation time taken by sliding window ESPRIT is much less as compared to sliding window Root MUSIC. This feature of ESPRIT makes it more powerful than the Root MUSIC.

TABLE III. AVERAGE ESTIMATION ERROR BY SLIDING WINDOW ESPRIT AND SLIDING WINDOW ROOT MUSIC

Methods Root MUSIC ESPRIT TrueFrequency

TrueMagnitude

Average Frequency Estimation Error

Average Magnitude Estimation Error

Average Frequency Estimation Error

Average Magnitude Estimation Error

420 5 -1.1983*10^

-05

1.368*10^-03

2.5068*10^-11

8.3933*10^-08

300 3 1.7281*10^-04

2.8219*10^-03

5.4247*10^-9

1.7685*10^-07

250 4 8.0386*10^-04

2.869*10^-03

2.8660*10^-8

6.0471*10^-07

200 2.5 9.2950*10^-04

6.843*10^-03

1.6424*10^-7

4.4591*10^-07

95 3.5 0.04183 0.04065 9.8040*10^-6

4.6736*10^-06

70 6 0.1740 0.03257 3.5826*10^-5

2.8808*10^-06

60 50 0.04160 0.0224 2.1908*10^-6

1.4093*10^-06

Computation Time

0.1187 s 0.0795 s

Further the above analysis is carried out for the estimation of harmonics for standalone wind generation system. If injected rotor voltage having harmonics, it causes harmonics and inter harmonics in stator current and voltage. This phenomenon is clearly explained in the section II. Harmonics component due to this is also depends upon wind speed means harmonics components frequency is varies as the wind speed varies. That’s why, to check the accuracy and performances of methods described in V. Stator current of model shown in section VI is captured by simulating the model in MATLAB. In this 10s duration, rotor speed varied from 1200 RPM to 1140 RPM at the instant 3 seconds and rotor regain speed of 1200 RPM at the instant of 7seconds. Stator current waveform is shown in Fig. 5a and Fig. 5b. It is clear that magnitude and frequency of harmonics is also varies as wind speed varies. Frequency and magnitude of estimated by Sliding Window Root MUSIC and Sliding window ESPRIT is shown in the Fig. 6(a) and Fig 6(b). From this result it is clear that both methods are capable of detecting frequency and magnitude accurately but Sliding window Root MUSIC takes more computational time than the sliding window ESPRIT. Average computational time taken by Sliding window Root MUSIC and sliding window ESPRIT is 1.96 and 0.35 seconds respectively.

Fig. 5a Stator current of Phase A of duration 1.05 Seconds to 1.2 Seconds

1.05 1.1 1.15 1.2-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (Seconds)

Am

plitu

de (A)

2605

Fig. 5b Stator current of Phase A of duration 4.05 Seconds to 4.2 Seconds

Fig. 6(a)Harmonics parameter estimation by Sliding window Root MUSIC

Fig. 6(b) Harmonics parameter estimation by Sliding window ESPRIT

VIII. CONCLUSION The work in this paper has introduced a study on the estimation of harmonics and inters harmonics in DFIG using sliding window ESPRIT and sliding window Root MUSIC under variable wind speed conditions. Results obtained in the section VII clearly shows that estimation by sliding window

ESPRIT is more accurate and computational efficient than the sliding window Root MUSIC in the context of qualitative and quantitative analysis.

REFERENCES

[1] H. J. Math and I. Y.-H. Gu, Signal Processing of Power Quality Disturbances.Hoboken, NJ: Wiley, 2006.

[2] IEC Std. 61000-4-7: General guide on harmonics and interharmonics measurements and instrumentation, for power supply system and equipment connected thereto,2009.

[3] S.G.Mallat, “A theory for multiresolution signal decomposition wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. vol. 11 pp. 674–693, 1989.

[4] P. K. Dash, D. P. Swain, A. C. Liew, and S. Rahman, “An adaptive linear combiner for on-line tracking of power system harmonics,” IEEE Trans.Power Syst., vol. 11, no. 4, pp. 1730–1735, Nov. 1996.

[5] G. W. Chang, C.-I Chen, and Q. W. Liang, “A two-stage ADALINE for harmonics and interharmonics measurement,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 2220–2228, Jun. 2009.

[6] S. H. Jo, S. E. Son, S. Lee, and J. W. Park, “Kalman-filter-based multilevel analysis to estimate electric load composition,” IEEE Trans. Ind. Electron., vol. 59, no. 11, pp. 4263–4271, Nov. 2012.

[7] J. A. R. Macias and A. G. Exposito, “Self-tuning of Kalman filters for harmonic computation,” IEEE Trans. Power Del., vol. 21, no. 1, pp. 501– 504, Jan. 2006.

[8] G.R.B. Prony, Essai experimantal et analytique, etc, Journal de L’Ecole Poly- Technique 124–76.

[9] F.F. Costa, A.J.M. Cardoso, “Harmonic and interharmonic identification based on improved Prony’s method,” Ann. Conf. IEEE Ind. Electr. pp. 1047–1052, 2006.

[10] Z. Hu, J. Guo, M. Yu, Z. Du, C. Wang, “The studies on power system harmonic analysis based on extended Prony method,” Int. Conf. Power Syst. Technol, pp.1–8, 2006.

[11] Petre Stoica, Anders Eriksson, “MUSIC estimation of real-valued sine-wave frequencies,” Signal Processing, vol. 42, pp. 139- 146, 1995.

[12] R. Roy and T. Kailath, “ESPRIT - Estimation of signal parameters via rotational invariance techniques”, IEEE Trans. Acoust. Speech Signal Processing., vol. 37.

[13] I. Y.-H. Gu and M. H. J. Bollen, “Estimating interharmonics by usingsliding-window ESPRIT,” IEEE Trans. Power Del., vol. 23, no. 1, pp.13–23,Jan.2008.

[14] M. Lindholm and T. W. Rasmussen,“Harmonic analysis of doubly-fed induction generators,” in Proc. 5th Int. Conf. Power Electronics and Drive System , vol. 2, pp. 837–841 Nov. 2003.

[15] S. Williamson and S. Djurovic, “Origins of stator current spectra with winding faults and excitation asymmetries,” in Proc. IEEE Int. Electric Machines and Drives Conf. (IEMDC’09), pp. 563–570, May 2009.

[16] Waleed K. A. Najy, H. H. Zeineldin, Ali H. Kasem Alaboudy, and Wei Lee Woon, “A bayesian passive islanding detection method for inverter-based distributed generation using ESPRIT,” IEEE Trans. Power Del., vol. 26, no. 4, OCT 2011.

[17] Lingling Fan, Subbaraya Yuvarajan, and Rajesh Kavasseri, “Harmonic Analysis of a DFIG for a Wind Energy Conversion System,”IEEE Trans. ON Energy Conversion, vol.. 25, no. 1, MARCH 2010.

4.05 4.1 4.15 4.2-5

-4

-3

-2

-1

0

1

2

3

4

5

Time(Seconds)

Am

plitu

de(A

)

2606

Powered by TCPDF (www.tcpdf.org)