A novel frequency tracking method based on complex adaptive linear neural network state vector in power systems

Electric Power Systems Research 79 (2009) 1216–1225

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A novel frequency tracking method based on complex adaptive linear neural network state vector in power systems I. Sadinezhad, M. Joorabian ∗ Electrical Engineering Department, Shahid Chamran University, Ahvaz 61355, Iran

a r t i c l e

i n f o

Article history: Received 30 January 2008 Received in revised form 28 October 2008 Accepted 7 March 2009 Keywords: Frequency Tracking Complex state observer Adaptive Linear Neural Networks (ADALINE)

a b s t r a c t This paper presents the application of a Complex Adaptive Linear Neural Network (CADALINE) in tracking the fundamental power system frequency. In this method, by using stationary-axes Park transformation in addition to producing a complex input measurement, the decaying DC offset is eliminated. As the proposed method uses first-order differentiator to estimate frequency changes, a Hamming filter is used to smoothen the response and cancel high-frequency noises. The most distinguishing features of the proposed method are the reduction in the size of observation state vector required by a simple Adaptive Linear Neural Network (ADALINE) and increase in the accuracy and convergence speed under transient conditions. This paper concludes with the presentation of the representative results obtained in numerical simulation and simulation in PSCAD/EMTDC software as well as in practical study. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In some digital application systems, power system frequency tracking is an important task. Accurate power frequency estimation is a necessity to check the state of health of power index, and a guarantee for accurate quantitative measurement of power parameters such as voltages, currents, active power, and reactive power, in multi-function power meters under steady states [1]. Akke [2] in his paper specified three criteria that a frequency tracking method should satisfy in this application: (i) fast speed of convergence, (ii) accuracy of frequency estimation, and (iii) robustness to noise. Many well-proven techniques such as zero-crossing technique [3,4], level-crossing technique [5], least squares error technique [6–8], Newton method [9], Kalman filter [10–14], Fourier transform [15–21] and wavelet transform [22–24] have been used for power harmonic frequency estimation which brings in many benefits in the fields of measurement, instrumentation, control and monitoring of power systems. Besides, a comprehensive analysis of Discrete Fourier Transform (DFT) error is given in [25], including the cases of synchronous sampling and error rises when sampling frequency does not synchronize with signal frequency. In [26], a frequency tracking method based on Linear Estimation of Phase (LEP) has been introduced. Also, a processing unit for symmetrical components and harmonic estimation based on an Adaptive Linear

∗ Corresponding author. Tel.: +98 916 118 3017; fax: +98 611 3360017. E-mail address: [email protected] (M. Joorabian). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.03.001

Combiner has been introduced in [27]. In this paper, the ADALINE structure is used to estimate the fundamental frequency of power system based on the principle of Decomposing of Single Phase into Orthogonal Component (DSPOC), introduced by Moore [18,28]. The main drawback of this method is that it needs at least three cycles of sample feeding to converge. Besides, as Taylor series expansion is used to estimate decaying DC offset, in dynamic changes the sloop of the Taylor expansion may not match with the real sloop of the decaying DC offset. In this paper, we propose the use of a new complex ADALINE (CADALINE) structure, using complex state observer to estimate fundamental frequency of the power system. In this novel approach, the accuracy and transient response are improved. Moreover, the proposed method reduces parameters to be estimated to half of the number that a simple ADALINE uses. To produce the input vectors for CADALINE and deal with the decaying DC offset, stationary-axes Park transformation is used. Before a discrete differentiator, which uses the normalized output of the CADALINE, a Hamming window is used as a middle-filter. The performance of the proposed method is compared with Kalman filter and DFT approaches. This paper is divided as follows: ADALINE structure for tracking fundamental frequency is described in Section 2 and complex ADALINE structure for tracking fundamental frequency is presented in Section 3. Kalman filter and DFT approaches are reviewed in Section 4. Finally, the results obtained in testing the proposed algorithm, which are represented in Section 5, confirm improved measurement performance in the presence of abrupt conditions, during dynamic changes and transient conditions in power systems.

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Nomenclature sampling frequency fs f0 fundamental frequency Ns = fs /f0 sampling rate W(k) weighting vector X(k) time-varying observation matrix N total number of harmonics Av amplitude of decaying DC offset  time constant of decaying DC offset ˛ ADALINE constant learning parameter e(k) tracking error for ADALINE H(i) ith coefficient of the FIR Hamming window coefficients N0 sampling rate given by N0 = (fs /f0 ) f1 frequency deviation yd (t) D-axis Park components Q-axis Park components yq (t) [T] orthogonal matrix for Park transformation  (KTs ) complex harmonic estimation vector for CADALINE Z(KTs ) complex observation matrix for CADALINE − → e (KTs ) complex error for CADALINE A1 (KTs ) complex phasorial expression of center frequency in the D–Q frame An1 (KTs ) complex normalized state rotating vector in the D–Q frame xˆ k estimate of the state vector xk in Kalman approach P observer gain matrix in Kalman approach

2. ADALINE structure to track fundamental frequency Assume that the waveform of voltage of power system comprises unknown fundamental, harmonic, and decaying DC offset components, as V (t) =

N 

Vl sin (lω0 t + ˚1 ) + Av e(−t/)

(1)

l=1

Fig. 1. ADALINE structure to track fundamental frequency.

where V1 and ˚1 are the amplitude and phase of the fundamental frequency, respectively. Av and  are the amplitude and time constant of decaying DC offset, respectively; N is the total number of harmonics; and ω0 is the fundamental angular frequency in (rad/s). Time-discrete expression of (1) is V (k) =

N 





Vl sin l + ˚1 + Av e(−/ω0 )

where  = 2k/Ns and Ns is sampling rate given by Ns = fs /f0 , in which fs is sampling frequency and f0 is fundamental frequency of power system. By using the triangular equality:



 

 

sin ˛ + ˇ = sin (˛) cos ˇ + sin ˇ cos (˛)

(3)

Eq. (2) can be rewritten as V (k) =

N 

Vl sin(˚l ) cos(l) + Vl cos(˚l ) sin(l) + Av e(−/ω0 ) (4)

l=1

Rearranging the above equation in the matrix form, we obtain: T

V (k) = V X (k)



V =

(2)

l=1



each iteration to be tracked. These variables for phase voltages are shown as follows: V1 sin (˚1 ) V1 cos (˚1 ) V2 sin (˚2 ) V2 cos (˚2 ) Av ... VN sin (˚N ) VN cos (˚N ) Av − 

 X(k) =

 

where V(k) represents the measurement at each sampling, X(k) is the time varying observation matrix, and  V (k) is the parameter at

 

(6)

 

cos  sin  cos 2 sin 2      ... cos N sin N 1 − ω0

According to Fig. 1, at kth iteration the input vector X(k) is multiplied by the weighting vector W (k) = [w1 (k) w2 (k) · · ·wP (k)], and then these weighted inputs are summed to produce the linear output y(k) = W(k) × X(k)T . In order for the ADALINE output to precisely mimic the desired value d(k), the weight vector is adjusted utilizing an adaptation rule that is mainly based on Least Mean Square (LMS) algorithm. This rule is also known as Widrow–Hoff Delta Rule [27] and given by W (k + 1) = W (k) +

(5)

 



˛ e (k) X (k) T

X (k) X(k)

(7)

where ˛ is the constant learning parameter and e (k) = y (k) − d (k) is the error. When perfect learning is attained, the error is reduced to T near zero and the desired output becomes equal to d (k) = W0 X(k) ,

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where W0 is the weight vector after the complete algorithm convergence. Thus, the neural model exactly predicts the incoming signal. To track harmonic components of a voltage signal with ADALINE, the variables  V (k) and V(k) are simply assigned to W(k) and d(k), respectively, with P = 2N + 2. After that error converges to zero, the weight vector yields the Fourier coefficients of power signal as

 W0 =

V1 sin (˚1 ) V1 cos (˚1 ) V2 sin (˚2 ) V2 cos (˚2 ) Av ... VN sin (˚N ) VN cos (˚N ) Av − 

 (8)

and voltage amplitude and phase of Nth harmonic are VN = ˚V,N



W02 (2N − 1) + W02 (2N) W 0 (2N − 1) = tan−1 W0 (2N)

(9) Fig. 2. Impulse response of the Hamming window with 20 Hz cut frequency.

Voltage amplitude and phase of fundamental frequency extracted by (9) are V1 =



W02 (1) + W02 (2) W0 (1) tan−1

˚V,1 =





(10)

W0 (2)

N 

 

 

Il sin (˚l ) cos l + Il cos (˚l ) sin l + Ai e(−/w0  ) (11)

in which variables  I (k) and I(k) are simply assigned to W(k) and d(k), respectively, with P = 2N + 2 [27]. I =

I1 sin (˚1 ) I1 cos (˚1 ) I2 sin (˚2 ) I2 cos (˚2 ) A ... IN sin (˚N ) IN cos (˚N ) Ai − i 



˚I,N  = tan−1 (W0 (2N − 1) /W0 (2N)) W02 (2N − 1) + W02 (2N)

˚I,1

(14)

(15)

that can be rewritten as s(KTs ) = x1 sin(2f0 KTs ) + x2 cos(2f0 KTs )

f1 (KTs ) =

•

1 y1 (KTs )y2 (KTs ) − y2 (KTs )y1 (KTs ) × 2 (y2 (KTs ) + y2 (KTs ))

(18)

•

y1 (KTs ) − y1 (KTs − Ts ) Ts • y2 (KTs ) − y2 (KTs − Ts ) y2 (KTs ) = Ts •

(13)

To track frequency, a center frequency is assumed to be the real value. It would be the operational frequency of the power system usually 50 Hz or 60 Hz. Under situations that the base power frequency changes, the Kth sample of fundamental voltage or current signal is modeled by [29] s(KTs ) = A sin(2fx KTs + )

x2 ((K − i + 1)) × H(i)

where the cut frequency for a low pass Hamming window is 20 Hz and the length of filter is 40. Fig. 2 shows the Impulse Response of this Hamming window. H(i) is the ith coefficient of the FIR Hamming window coefficients and N0 is the sampling rate given by N0 = (fs /f0 ). By using DSOPC principle, f1 is obtained as

•



W02 (1) + W02 (2) W 0 (1) = tan−1 W0 (2)

y2 (KTs ) =

(17)

where y1 and y2 are first-order discrete derivatives defined as

Current amplitude and phase of fundamental frequency are achieved by I1 =

N0 

•

(12)

Current amplitude and phase of Nth harmonic are as follows:

IN =

x1 ((K − i + 1)) × H(i)

i=1

l=1



N0 

y1 (KTs ) =

i=1

With the same approach by sampling current signal, discrete expression of current is I (k) =

obtained as

(16)

where x1 is the in-phase component, x2 is the quadrature phase component, f0 is the center frequency (60 Hz), f1 is the frequency deviation and Ts is the sampling interval (1/fs ). Before calculating the frequency deviation (f1 ), x1 and x2 pass through an FIR Hamming window and parameters y1 and y2 are

y1 (KTs ) =

(19)

Finally, the real value of fundamental frequency (fx ) is calculated by adding the frequency deviation to the assumed center frequency as fx = f0 + f1

(20)

3. Complex ADALINE structure to track fundamental frequency The proposed Complex ADALINE (CADALINE) structure is based on the Widrow–Hoff Delta Rule, explained earlier. The improvement in ADALINE structure is made by introducing a complex observation vector. This approach reduces the number of weight updates, and so, the number of parameters to be estimated. To produce a complex vector measurement and remove DC offset, the use of the stationary-axes Park transformation is proposed. Stationaryaxes Park transformation is widely employed to study the behavior of rotating electrical machines in transient conditions [30]. However, it can be considered a more general and powerful tool to study the behavior of three-phase systems. This transformation applied to the signals ya (t), yb (t) and yc (t) (voltages and currents) of a threephase system leads to the Park components yd (t), yq (t) and y0 (t)

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defined as



yd yq y0





= [T ]

ya yb yc

1219

 (21)

where [T] is the orthogonal matrix:

⎡

2 3

1 6

− ⎢ ⎢

⎢ 1 T =⎢ 0 ⎢ 2 ⎢

⎣ 1 1 3

3

⎤

−

1

6

⎥ ⎥ 1⎥ ⎥ − ⎥ 2

⎥ 1 ⎦

(22)

3

In the D–Q frame, it is then possible to define the Park vector as a complex quantity as y = yd + jyq

(23)

This vector is used as a desired value. The complex observation matrix Z is introduced by Z(KTs ) = [ejω0 KTs ej2ω0 KTs . . . ejNω0 KTs ]T

(24)

where ω0 is the center angular frequency (rad/s), defined as ω0 = 2f0 . The complex harmonic vector to be tracked at Kth sample is  (KTs ), defined as  (KTs ) = [A1 (KTs ) A2 (KTs ) . . . AN (KTs )]T

(25)

where A1 (KTs) is the complex phasorial expression of center frequency in the D–Q frame. Weight update, according to LMS rule is → e (KTs − Ts )  (KTs ) =  (KTs − Ts ) + ˛ · −

z(KTs − Ts ) z T (KTs − Ts )z(KTs − Ts ) (26)

− → e (KTs ) is the complex error obtained as follows: Ys (KTs ) =  T (KTs )Z(KTs ) − → e (KTs ) = Ys (KTs ) − y(KTs )

(27)

N0

Ah1 (KTs ) =

A1 ((K − i + 1)Ts ) × H(i)

Therefore, complex normalized rotating state vector An1 (KTs ) is obtained as An1 (KTs ) =

(28)

Ah1 (KTs ) abs (Ah1 (KTs ))

(29)

where abs(x) stands for absolute value of x. Therefore, by using the first-order discrete differentiator, f1 is obtained as f1 (KTs ) =

where Ys (KTs) is the complex estimation of real values of y(KTs) in D–Q frame. It should be noted that under conditions where power system operates with the nominal frequency, A1 (KTs ) is a constant vector, which does not rotate with respect to the time in complex frame. When the base frequency changes, A1 (KTs ) becomes a rotating vector. It is the result of the fact that when the base frequency changes, A1 (KTs ) components appear as modulated signals and their carrier is the occurred frequency-drift. Thus, the rate of this rotation is the key element to track frequency deviation from the center frequency. The frequency deviation (f1 ) is achieved by normalizing and differentiating A1 (KTs ). For the types of power swing events studied here, it has been found that the non-fundamental components cannot be characterized as harmonics. A middle-filter is, therefore, required so that the signal is dominated by the fundamental component. The middle-filter, used here, is the FIR Hamming type filter as has been used in [18]. It has been used in Section 2. A1 (KTs ) passes through an FIR Hamming window and Ah1 (KTs ) is obtained as



Fig. 3. Complex ADALINE structure to track fundamental frequency.



1 j2An1 (KTs )

 An (KT ) − An (KT − T ) s s s 1 1 Ts

It can be seen that observation matrix size and the parameters to be estimated have been reduced to (N) elements in comparison with the simple ADALINE that uses (2N + 2) elements. After all, the most important aspect of the proposed technique is that decaying DC offset has been eliminated by applying stationary-axes Park transformation. Furthermore, owing to the fact that data from three phases are combined, the convergence speed is considerably improved. Fig. 3 shows the Complex ADALINE structure to track fundamental frequency. 4. Review of Kalman and DFT approaches 4.1. Kalman filter to track fundamental frequency Kalman filter has also been used to track fundamental frequency in power systems [10,11]. Consider the following deterministic state-variable equation for a periodic signal having harmonic components up to nth order with samples zk , at time tk , (2n + 1) samples per period [31,32]. xk+1 = Fxk zk = Qxk where (2n + 1)-dimensional state vector xk is as follows:

i=1

Then, Ah1 (KTs ) should be normalized to produce the rotating operator (e(j2f1 KTs ) ). e(j2f1 KTs ) stands for a normal rotating vector in which its amplitude is unity and f1 is the frequency deviation.

(30)

xk (2i − 1) = real component of the ith harmonic phasor; xk (2i) = imaginary component of the ith harmonic phasor, xk (2n + 1) = DC component,

(31)

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where the ith element of xk is represented by xk (i), and F is

⎡ 



f 1 ⎢ 0 ⎢ ⎢ . . F =⎢ ⎢ .

⎢ ⎣

0  · · · ···

f 2 .. .

0 0

0

···



0 .. .

··· f n ··· 0

0 0

0

⎤

0⎥ ⎥ .. ⎥ .⎥ ⎥ .. ⎥ .⎦ 1



(32)

where = ω0 Ts , ω0 is the fundamental supply frequency in (rad/s) and Ts is the sampling interval in seconds.

 

f i



=

      − sin i  , i = 1, 2, . . . , n

cos i sin i

cos i

(33) Fig. 4. Tracked frequency (Hz).

and Q is a (1 × (2n + 1)) matrix, which gives the connection between the measurement (zk ) and the state vector (xk ). The sampled value of the signal is considered to be the sum of the real components of the harmonic phasors and the DC component. Therefore, the matrix Q is given by Q = [1, 0, 1, 0, . . . , 1, 0, 1]



Herein, a disturbance is simulated at time 0.3 s. Three-phase non-sinusoidal unbalanced signals, including decaying DC offset up to third harmony, are produced as

=



xk2 (2i − 1) + xk2 (2i) 2

,

i = 1, 2, . . . , n

(35)

The problem of estimating the present state of the signal model (Eq. (31)) from output measurements zk involves the design of standard state observers [33]. The observer state can be represented by



xˆ k+1 = F xˆ k + P zk − Q xˆ k



(36)

where xˆ k denotes the estimate of the state vector xk and P is the observer gain matrix. The primary objective in choosing P is to obtain a stable observer, which is achieved by assigning the eigenvalues of the matrix (F − PQ) within the unit circle. The locations of the eigenvalues determine, among other things, the transient response of the observer. For the purpose of frequency tracking, the speed of response and tracking ability are of particular importance. After studying various choices, the following case is considered. P = [0.248, 0.0513, 0.173, 0.046, 0.0674, 0.0434, 0.00916, 0.0236, − 0.000415, 0.113, 0.0802]T

(37)

To estimate fundamental frequency, the approach is based on DSPOC, which has been described in Eqs. (15)–(20) is used. 4.2. DFT filter to track fundamental frequency Under situation of frequency change, the Kth sample of fundamental voltage or current signal is described as denoted in Eq. (15). By using a DFT dynamic window, parameters x1 and x2 in Eq. (16) can be achieved as follows: x1 (KTs ) = ( x2 (KTs ) = (

5.1. Simulated signals

(34)

The harmonic components hi (RMS) are given by h2i

and Section 5.3 includes practical measurement of a real fault incidence in Fars province, Iran.

Ns −1





Nk=0 s −1





2  K ) s(KTs ) sin(2 ) Ns Ns

2  K ) s(KTs ) cos(2 ) Ns Ns

⎧ V = 220 sin(ω0 t) ⎪ ⎨ A 2 VB = 220 sin(ω0 t −

)

3 ⎪ ⎩ V = 220 sin(ω t + 2 ) 0 C

0 ≤ t ≤ 0.3

3

After disturbance at 0.3 s, signals are

⎧ 40 sin(3ωx t) + 400e(−10t) ⎨ VA = 400 sin(ωx t) +2

) + 60 sin(3ωx t) + 800e(−10t) 3 2 ) + 20 sin(3ωx t) = 800 sin(ωx t + 3

VB = 800 sin(ωx t −

⎩V

C

0.3 ≤ t ≤ 0.6

(40)

where ω0 is the base frequency and ωx is the actual frequency after disturbance. 5.1.1. Case In this case, a 1-Hz frequency deviation occurs and tracked frequency using CADALINE, ADALINE, Kalman, and DFT approaches is revealed in Fig. 4; three-phase signals are shown in Fig. 5. Estimation error percentage according to samples fed to each algorithm after frequency drift is shown in Fig. 6 and second two and half cycles including 200 samples fed to all algorithms is magnified. It can be seen that CADALINE converges to the real value after first 116 samples, less than two and half power cycles, with error of −0.4%; and reaches a perfect estimation after having more few samples. Other estimation methods are too far from real value. DFT, ADALINE and Kalman respectively need 120, 200 and 360 samples to reach an estimation with less than one percent error. It should be

(38)

k=0

The fundamental frequency tracking process includes the same approach that has been expressed in Eqs. (15)–(20). 5. Simulation results Simulation examples include the following three categories: numerical simulations are represented in Section 5.1 for two cases, simulation in PSCAD/EMTDC software is represented in Section 5.2,

(39)

Fig. 5. Three-phase signals.

I. Sadinezhad, M. Joorabian / Electric Power Systems Research 79 (2009) 1216–1225

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Fig. 6. Estimation error percentage according to samples fed to each algorithm after frequency drift.

considered that for 2.4 kHz sampling frequency and power system frequency of 60 Hz, each power cycle includes 40 samples. The complex normalized rotating state vector An1 (KTs ) with respect to time and in D-Q frame is shown in Fig. 7. It has been seen that for 1-Hz frequency deviation (f1 = 1 Hz), CADALINE has the best convergence response in terms of speed and over/under shoot. ADALINE method convergence speed is half that in the CADALINE and shows a really high overshoot. Besides, Kalman approach shows the biggest error, at the first seven power system cycles it converges to 61.7 Hz instead of 61 Hz and its computational burden is considerably higher than in other methods. In this case, presence of a long-lasting decaying DC offset affects the DFT performance. Consequently, its convergence speed and overshoot is not as improved as CADALINE. As can

Fig. 7. Complex normalized rotating state vector (An1 ).

be seen in Fig. 7, An1 (KTs ) starts rotation simultaneously when the frequency changes at time 0.3 s. 5.1.2. Case In this case, a three-phase balanced voltage is simulated numerically. The only change applied is a step-by-step 1-Hz change in fundamental frequency to study the steady-state response of the

Table 1 Samples needed to estimate with 1% error for 50–70 frequency range. fx (Hz)

70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50

Approaches CADALINE

KALMAN

ADALINE

DFT

95 97 93 90 95 97 92 93 98 96 92 83 81 88 98 97 96 90 96 96 105

360 421 358 384 385 305 361 328 430 360 385 234 281 313 216 377 336 331 290 374 405

202 188 186 187 178 138 211 193 206 231 220 155 181 197 178 192 206 195 190 184 113

111 114 118 114 114 139 114 116 115 116 112 97 116 117 123 117 122 114 108 120 112

Table 2 Marvdasht substation capacities. No.

Substation name

1

Marvdasht 230/66 (kV) Marvdasht City Mojtama Kenare Sahl Abad Dinarloo Seydan Arsanjan

2

Fig. 8. Average convergence time (cycles) to track static frequency changes.

3 4 5 6 7 8

Feeder no. tag

3-Phase short circuit capacity (MVA)

1-Phase short circuit capacity (MVA)

–

1460

1184

602

896

640

601 607 603 604 608 605

631 1005 751 203 381 145

423 718 500 121 237 84

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Fig. 9. A three-machine connected system simulated in PSCAD/EMTDC software.

Fig. 10. Tracked frequency (Hz).

Fig. 11. Phase-A voltage (kV).

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1223

5.2. Simulation in PSCAD/EMTDC software

Fig. 12. Complex normalized rotating state vector (An1 ).

proposed method when the power system operates under/over frequency conditions. The three-phase signals are

⎧ V = 220 sin(ωx t) ⎪ ⎨ A 2

In this case, a three-machine system controlled by governors is simulated in PSCAD/EMTDC software, shown in Fig. 9. Information of the simulated system is given in Appendix A. A three-phase fault occurs at 1 s. Real frequency changes, estimations by use of ADALINE, CADALINE, and Kalman approaches are shown in Fig. 10. Instead of DFT method, the Frequency Measurement Module (FMM) performance, existing in PSCAD Library, is compared with the presented methods. Phase-A voltage signal is shown in Fig. 11. The complex normalized rotating state vector (An1 (KTs )) is shown in Fig. 12. The best transient response and accuracy belongs to ADALINE and CADALINE, but CADALINE has faster response with considerable lower overshoot, as can be seen in Fig. 10. Kalman approach has a suitable response in this case, but its error and overshoot in estimating frequency is bigger than that in CADALINE. The PSCAD FMM shows drastic fluctuations in comparison with other methods proposed and reviewed here. 5.3. Practical Study

VB = 220 sin(ωx t −

) 3 ⎪ ⎩ V = 220 sin(ω t + 2 ) x C 3

(41)

where ωx = 2fx , and values of fx are shown in Table 1. The range of frequency that has been studied here is 60–70 Hz. Results are revealed in Table 1 and average convergence time is shown in Fig. 8 for CADALINE, ADALINE, Kalman filter, and DFT approaches. The results from this section can give an insight into the number of samples that each algorithm needs to converge to a reasonable estimation. According to the fact that each power cycle is equivalent to 40 samples, number of samples that is needed for each algorithm to have estimation with less than one percent error is represented in Table 1.

In this case, a practical example is represented. Voltage signal measurements are applied from the Marvdasht power station in Fars province, Iran. The recorder sampling frequency fs is 6.39 kHz and fundamental frequency of power system is 50 Hz. A phase-C-toground (C–G) fault occurred on March 4, 2006. The fault location is 46.557 km from Arsanjan substation. Main information on the Marvdasht 230/66 kV station and other substation supplied by this station is given in Tables 2 and 3. Fig. 13 shows the performance of CADALINE, ADALINE, Kaman, and DFT approaches. Besides, phaseC voltage and residual voltage are revealed in Fig. 14(A) and (B), respectively. Complex normalized rotating state vector (An1 ) is shown in Fig. 15.

Table 3 Marvdasht substation three-phase and single-phase short circuit capacities and impedances (|Z|). No.

Substation name

Feeder no. tag

3-Phase short circuit current (kA)

1-Phase short circuit current (kA)

|Z| ( )

1 2 3 4 5 6 7 8

Marvdasht 230/66 (kV) Marvdasht City Mojtama Kenare Sahl Abad Dinarloo Seydan Arsanjan

– 602 601 607 603 604 608 605

12.77169 7.837967 5.519818 8.79147 6.569546 1.775789 3.332886 1.268421

10.35731 5.598548 3.70029 6.280871 4.373866 1.058475 2.073212 0.734809

2.983562 4.861607 6.903328 4.334328 5.800266 21.45813 11.43307 30.04138

Fig. 13. Tracked frequency (Hz), Section 5.3.

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3. Load characteristics: Load active power: 190 MW Load nominal line-to-line voltage: 13.8 kV

References

Fig. 14. (A) Phase-C voltage and (B) residual voltage, Section 5.3.

Fig. 15. Complex normalized rotating state vector (An1 ), Section 5.3.

6. Conclusion This paper proposes an adaptive approach for frequency estimation in electrical power systems by introducing a novel complex ADALINE structure. The proposed technique is based on tracking and analyzing a complex rotation state vector in D–Q frame that appears when a frequency-drift occurs. This method improves the convergence speed both in steady states and dynamic disturbances, including a change in base frequency of power system. Besides, the proposed method reduces the size of the state observer vector that has been used by simple ADALINE structure in other references. The numerical and simulation examples have verified that the technique proposed is far more robust and accurate in estimating the instantaneous frequency under various conditions compared with methods that have been reviewed in this paper. Appendix A. Multi-machine system information simulated in PSCAD/EMTDC software 1. Basic data of all generators are Number of machines: 3 Rated RMS line-to-neutral voltage: 7.967 kV Rated RMS line current: 5.02 kA Base angular frequency: 376.991118 rad/s Inertia constant: 3.117 s Mechanical friction and windage: 0.04 p.u. Neutral series resistance: 1.0E5 p.u. Neutral series reactance: 0 p.u. Iron loss resistance: 300.0 p.u. 2. Fault characteristics: Fault inductance: 0.00014 H Fault resistance: 0.001

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I. Sadinezhad, M. Joorabian / Electric Power Systems Research 79 (2009) 1216–1225 [29] D.W.P. Thomas, M.S. Woolfson, Evaluation of frequency tracking methods, IEEE Trans. Power Delivery 16 (July (3)) (2001) 367–371. [30] G.J. Retter, Matrix and Space-Phasor Theory of Electrical Machines, Akademiai Kiado, Budapest, Rumania, 1987. [31] G.H. Hostetter, Recursive discrete Fourier transformation, IEEE Trans. ASSP-28 (1980) 184–190. [32] R.R. Bitmead, A.C. Tsoi, P.J. Parker, A Kalman filtering approach to short-time Fourier analysis, IEEE Trans. ASSP-34 (1986) 1493–1501. [33] T. Kailath, Linear Systems, Prentice-Hall, NJ, 1980. I. Sadinezhad was born in Ahvaz, Iran, on April 20, 1982. He received M.Sc. (Electricalfirst class Hons.) in Electrical Engineering from Shahied Chamran University, Ahvaz, Iran, in 2007. He received B.Sc. (Electrical-first class Hons.) in Electrical Engineering from Iran University of Science and Technology (IUST), Tehran, Iran, in 2004. His research interests are power system protection and monitoring, power quality, signal

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processing, and power electronics. He is currently pursuing his Ph.D. in Electrical Engineering at The University of Sydney, Sydney, Australia. M. Joorabian was born in Shooshtar, Iran, on April 29, 1961. He graduated in B.E.E. from University of New Haven, CT, USA, in 1983. He received M.Sc. in Electrical Power Engineering from Rensselaer Polytechnic Institute, NY, USA, in 1985, and received Ph.D. in Electrical Engineering from University of Bath, Bath, UK, in 1996. Since 1985, he has been with the Department of Electrical Engineering, Shahid Chamran University, as Senior Lecturer, since 1996 as assistant professor, and since 2004 as associate professor. He was the manager of national project entitled “Design and implementation of intelligent accurate fault locator for electrical transmission lines” affiliated to Khozestan Water and Power Authority (K.W.P.A). His research interests are power system modelling, power quality, power system protection, renewable energy, and applications of intelligence technique in power systems.