**5.2 Testing kalman filter algorithm**

#### **5.2.1 Description of the load**

The proposed algorithm is tested on an actual recorded data to obtain the damped subharmonics which contaminated the three phase current waveforms of a dynamic load. The load is a variable frequency drive controlling a 3000 HP induction motor connected to an oil pipe line compressor. The solid state drive is of 12 pulses designed with harmonic filter. The data given is the three phase currents at different motor speed, and is given in per unit. The three phase currents are given in Figure 42. This figure shows high harmonics in each phase current as well as sub-harmonics. It is clear that the currents have variable magnitudes from one cycle to another (non-stationary waveforms).

#### **5.2.2 Sub-harmonic estimation**

After the harmonic contents of the waveforms had been estimated, the waveform was reconstructed to get the error in this estimation. Figure 71 gives the real current and the reconstructed current for phase A as well as the error in this estimation. It has been found

Electric Power Systems Harmonics - Identification and Measurements 45

Once the sub-harmonic parameters are estimated, the total reconstructed current can be obtained by adding the harmonic contents to the sub-harmonic contents. Figure 45 gives the

Fig. 44. The phase angle of the 30 Hz component.

total resultant error which now is very small, less than 3%.

Fig. 45. the final error in the estimate using KF algorithm.

that the error has a maximum value of about 10%. The error signal is analyzed again to find if there are any sub-harmonics in this signal. The Kalman filtering algorithm is used here to find the amplitude and the phase angle of each sub-harmonic frequency. It was found that the signal has sub-harmonic frequencies of 15 and 30 Hz. The sub-harmonic amplitudes are given in Figure 43 while the phase angle of the 30 Hz component is given in Figure 44. The sub-harmonic magnitudes were found to be time varying, without any exponential decay, as seen clearly in Figure 43.

Fig. 42. Actual and reconstructed current for phase *A*

Fig. 43. The sub-harmonic amplitudes.

that the error has a maximum value of about 10%. The error signal is analyzed again to find if there are any sub-harmonics in this signal. The Kalman filtering algorithm is used here to find the amplitude and the phase angle of each sub-harmonic frequency. It was found that the signal has sub-harmonic frequencies of 15 and 30 Hz. The sub-harmonic amplitudes are given in Figure 43 while the phase angle of the 30 Hz component is given in Figure 44. The sub-harmonic magnitudes were found to be time varying, without any exponential decay, as

seen clearly in Figure 43.

Fig. 42. Actual and reconstructed current for phase *A*

Fig. 43. The sub-harmonic amplitudes.

Fig. 44. The phase angle of the 30 Hz component.

Once the sub-harmonic parameters are estimated, the total reconstructed current can be obtained by adding the harmonic contents to the sub-harmonic contents. Figure 45 gives the total resultant error which now is very small, less than 3%.

Fig. 45. the final error in the estimate using KF algorithm.

Electric Power Systems Harmonics - Identification and Measurements 47

Fig. 47. Sub-harmonic amplitudes using the WLAF algorithm.

Fig. 48. Phase angle of the 30 Hz component using the WLAVF algorithm.

#### **5.2.3 Remarks**


#### **5.3 SUb-harmonics indentification with DLAV algorithm (Soliman and Christensen algorithm)**

In this section, the application of discrete dynamic least absolute value algorithm for identification and measurement of sub-harmonics is discussed. The model used with Kalman filter algorithm and explained earlier, will be used in this section, a comparison with Kalman filter is offered at the end of this section. No needs to report here, the dynamic equations for the DLAV filter, since they are already given in the previous chapters, in the place, where we need them. The steps used with Kalman filter in the previous section, will be followed here. Hence, we discuss the testing of the algorithm.

As we said earlier, the algorithm first estimates the harmonics that contaminated one of the phases current waveforms, say phase *A*; in this estimation, we assumed a large number of harmonics. The reconstructed waveform and the error for this estimation, which is the difference between the actual recorded data and the reconstructed waveform, are then obtained. Figure 46 gives the real current and the reconstructed current for phase *A*, as well as the resultant error. The maximum error in this estimation was found to be about 13%. This error signal is then analyzed to identify the sub-harmonic parameters. Figure 47 gives the sub-harmonic amplitudes for sub-harmonic frequencies for 15 and 30 Hz, while Fig. 48 gives the phase angle estimate for the 30 Hz sub-harmonic. Note that in the sub-harmonic estimation process we assume that the frequencies of these sub-harmonics are known in advance, and hence the matrix *H* can easily be formulated in an off-line mode.

Fig. 46. Actual (full curve) and reconstructed (dotted curve) current for phase *A* using the WLAVF algorithm.

Kalman filter algorithm is implemented, in this section, for identification of sub-

By identifying the harmonics and sub-harmonics of the signal under investigation, the

In this section, the application of discrete dynamic least absolute value algorithm for identification and measurement of sub-harmonics is discussed. The model used with Kalman filter algorithm and explained earlier, will be used in this section, a comparison with Kalman filter is offered at the end of this section. No needs to report here, the dynamic equations for the DLAV filter, since they are already given in the previous chapters, in the place, where we need them. The steps used with Kalman filter in the previous section, will

As we said earlier, the algorithm first estimates the harmonics that contaminated one of the phases current waveforms, say phase *A*; in this estimation, we assumed a large number of harmonics. The reconstructed waveform and the error for this estimation, which is the difference between the actual recorded data and the reconstructed waveform, are then obtained. Figure 46 gives the real current and the reconstructed current for phase *A*, as well as the resultant error. The maximum error in this estimation was found to be about 13%. This error signal is then analyzed to identify the sub-harmonic parameters. Figure 47 gives the sub-harmonic amplitudes for sub-harmonic frequencies for 15 and 30 Hz, while Fig. 48 gives the phase angle estimate for the 30 Hz sub-harmonic. Note that in the sub-harmonic estimation process we assume that the frequencies of these sub-harmonics are known in

advance, and hence the matrix *H* can easily be formulated in an off-line mode.

Fig. 46. Actual (full curve) and reconstructed (dotted curve) current for phase *A* using the

**5.3 SUb-harmonics indentification with DLAV algorithm (Soliman and Christensen** 

harmonics parameters that contaminated the power system signals.

total error in the reconstructed waveform is reduced greatly.

be followed here. Hence, we discuss the testing of the algorithm.

**5.2.3 Remarks** 

**algorithm)** 

WLAVF algorithm.

Fig. 47. Sub-harmonic amplitudes using the WLAF algorithm.

Fig. 48. Phase angle of the 30 Hz component using the WLAVF algorithm.

Electric Power Systems Harmonics - Identification and Measurements 49

1. The estimate obtained via the WLAVF algorithm is damped more than that obtained via the KF algorithm. This is probably due to the fact that the WLAVF gain is more

Park's transformation is well known in the analysis of electric machines, where the three rotating phases abc are transferred to three equivalent stationary dq0 phases (d-q reference frame). This section presents the application of Park's transformation in identifying and measuring power system harmonics. The technique does not need a harmonics model, as well as number of harmonics expected to be in the voltage or current signal. The algorithm uses the digitized samples of the three phases of voltage or current to identify and measure the harmonics content in their signals. Sampling frequency is tied to the harmonic in question to verify the sampling theorem. The identification process is very simple and easy

In the following steps we assume that m samples of the three phase currents or voltage are available at the preselected sampling frequency that satisfying the sampling theorem. i.e. the sampling frequency will change according to the order of harmonic in question, for example if we like to identify the 9th harmonics in the signal. In this case the sampling frequency

damped and reaches a steady state faster than the KF gain, as shown in Fig. 50. 2. The overall error in the estimate was found to be very close in both cases, with a maximum value of about 3%. The overall error for both cases is given in Fig. 51. 3. Both algorithms were found to act similarly when the effects of the data window size,

sampling frequency and the number of harmonics were studied

Fig. 51. Final errors in the estimation using the two filters.

**6. Park's transformation** 

**6.1 Identification processes** 

must be greater than 2\*50\*90=900 Hz and so on.

to apply.

Fig. 49. Final error in the estimate using the WLAVF algorithm.

Finally, the total error is found by subtracting the combination of the harmonic and subharmonic contents, the total reconstructed waveform, from the actual waveform. This error is given in Fig. 49. It is clear from this Figure that the final error is very small, with a maximum value of about 3%.

#### **5.4 Comparison between DLAV and KF algorithm**

The proposed WLAVF algorithm was compared with the well-known linear KF algorithm. It can be shown that if there is no gross error contaminating the data, both filters produce very close results. However, some points may be mentioned here.

Fig. 50. Comparison between the filter gains for component *x*1.

Finally, the total error is found by subtracting the combination of the harmonic and subharmonic contents, the total reconstructed waveform, from the actual waveform. This error is given in Fig. 49. It is clear from this Figure that the final error is very small, with a

The proposed WLAVF algorithm was compared with the well-known linear KF algorithm. It can be shown that if there is no gross error contaminating the data, both filters produce

Fig. 49. Final error in the estimate using the WLAVF algorithm.

very close results. However, some points may be mentioned here.

Fig. 50. Comparison between the filter gains for component *x*1.

**5.4 Comparison between DLAV and KF algorithm** 

maximum value of about 3%.

Fig. 51. Final errors in the estimation using the two filters.


### **6. Park's transformation**

Park's transformation is well known in the analysis of electric machines, where the three rotating phases abc are transferred to three equivalent stationary dq0 phases (d-q reference frame). This section presents the application of Park's transformation in identifying and measuring power system harmonics. The technique does not need a harmonics model, as well as number of harmonics expected to be in the voltage or current signal. The algorithm uses the digitized samples of the three phases of voltage or current to identify and measure the harmonics content in their signals. Sampling frequency is tied to the harmonic in question to verify the sampling theorem. The identification process is very simple and easy to apply.

#### **6.1 Identification processes**

In the following steps we assume that m samples of the three phase currents or voltage are available at the preselected sampling frequency that satisfying the sampling theorem. i.e. the sampling frequency will change according to the order of harmonic in question, for example if we like to identify the 9th harmonics in the signal. In this case the sampling frequency must be greater than 2\*50\*90=900 Hz and so on.

Electric Power Systems Harmonics - Identification and Measurements 51

Then, the resulting samples represent the samples of the harmonic components in each

1. Decide what the order of harmonic you would like to identify, and then adjust the sampling frequency to satisfy the sampling theory. Obtain m digital samples of harmonics polluted three-phase voltage or current samples, sampled at the specified sampling frequency Fs. Or you can obtain these m samples at one sampling frequency that satisfies the sampling theorem and cover the entire range of harmonic frequency you expect to be in the voltage or current signals. Simply choose the sampling frequency to be greater than double the highest frequency you expect in the signal 2. Calculate the matrices, given in equations (55) and (57) at m samples and the order of harmonics you identify. Here, we assume that the signal frequency is constant and

3. Multiplying the samples of the three-phase signal by the transformation matrix given

4. Remove the dc offset from the original samples; simply by subtracting the average of the new samples generated in step 2 using equation (56) from the original samples. The

6. Subtract these samples from the original samples; we obtain m samples for the

7. Use the least error squares algorithm explained in the preceding section to estimate the amplitude and phase angle of the component. If the harmonics are balanced in the three phases, the identified component will be the positive sequence for the 1st, 4th, 7th,etc and no negative or zero sequence components. Also, it will be the negative sequence for the 2nd, 5th , 8th etc component, and will be the zero sequence for the 3rd 6th, 9th etc components. But if the expected harmonics in the three phases are not balanced go to

8. Replace by - in the transformation matrix of equation (55) and the inverse transformation matrix of equation (57). Repeat steps 1 to 7 to obtain the negative

v ( ) cos( ) <sup>a</sup> *am <sup>a</sup> t V nt*

where Vam is the amplitude of harmonic component n in phase a, is the fundamental frequency and a its phase angle measured with respect to certain reference. Using the

> v ( ) cos sin <sup>a</sup> *a a t x nt y nt*

> > cos *a am a x V*

 

(58)

(59)

(60)

**6.2 Measurement of magnitude and phase angle of harmonic component**  Assume that the harmonic component of the phase a voltage signal is presented as:

trigonometric identity, equation (58) can be written as:

generated samples in this step are the samples of the ac samples of dqo signal. 5. Multiplying the resulting samples of step 3 by the inverse matrix given by equation (57). The resulting samples are the samples of harmonics that contaminate the three

phase of the three phases. The following are the identification steps.

equal the nominal frequency 50 or 60 Hz.

harmonic component in question

phase signals except for the fundamental components.

by equation (57)

step 8.

where we define

sequence components.

The forward transformation matrix at harmonic order n; n=1,2,.., N, N is the total expected harmonics in the signal, resulting from the multiplication of the modulating matrix to the signal and the - transformation matrix is given as (dqo transformation or Park`s transformation)

$$\mathbf{P} = \begin{bmatrix} \sin not & \cos not & 0 \\ \cos not & \sin not & 0 \\ 0 & 0 & 1 \end{bmatrix} \mathbf{x}$$

$$\sqrt{\frac{2}{3}} \quad \begin{bmatrix} 1 & -0.5 & -0.5 \\ 0 & \frac{\sqrt{3}}{2} & \frac{-\sqrt{3}}{2} \\ 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{bmatrix} \sin not & \sin(not + 120n) & \sin(not + 240n) \\ \cos not & \cos(not + 120n) & \cos(not + 240n) \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \tag{55}$$

The matrix of equation (69) can be computed off line if the frequencies of the voltage or current signal as well as the order of harmonic to be identified are known in advance as well as the sampling frequency and the number of samples used. If this matrix is multiplied digitally by the samples of the three-phase voltage and current signals sampled at the same sampling frequency of matrix (55), a new set of three -phase samples are obtained, we call this set a dq0 set (reference frame). This set of new three phase samples contains the ac component of the three-phase voltage or current signals as well as the dc offset. The dc off set components can be calculated as;

$$
\mathcal{V}\_{\text{d}}(\text{dc}) = -\frac{1}{m} \sum\_{i=1}^{m} \langle V\_{i} \rangle\_{i}
$$

$$
\mathcal{V}\_{\text{q}}(\text{dc}) = -\frac{1}{m} \sum\_{i=1}^{m} \langle Vq \rangle\_{i} \tag{56}
$$

$$
\mathcal{V}\_{\text{O}}(\text{dc}) = -\frac{1}{m} \sum\_{i=1}^{m} \langle V\_{i} \rangle\_{i}
$$

If these dc components are eliminated from the new pqo set, a new ac harmonic set is produced. We call this set as Vd(ac), Vq(ac) and V0(ac). If we multiply this set by the inverse of the matrix of equation (56), which is given as:

$$\mathbf{P}^{\cdot 1} = \sqrt{\frac{2}{3}} \begin{bmatrix} \sin \alpha t & \cos \alpha t & \frac{1}{\sqrt{2}} \\\\ \sin (\alpha t + 240\mathbf{n}) & \cos (\alpha t + 240\mathbf{n}) & \frac{1}{\sqrt{2}} \\\\ \sin (\alpha t + 120\mathbf{n}) & \cos (\alpha t + 120\mathbf{n}) & \frac{1}{\sqrt{2}} \end{bmatrix} \tag{57}$$

The forward transformation matrix at harmonic order n; n=1,2,.., N, N is the total expected harmonics in the signal, resulting from the multiplication of the modulating matrix to the signal and the - transformation matrix is given as (dqo transformation or Park`s

*n*

The matrix of equation (69) can be computed off line if the frequencies of the voltage or current signal as well as the order of harmonic to be identified are known in advance as well as the sampling frequency and the number of samples used. If this matrix is multiplied digitally by the samples of the three-phase voltage and current signals sampled at the same sampling frequency of matrix (55), a new set of three -phase samples are obtained, we call this set a dq0 set (reference frame). This set of new three phase samples contains the ac component of the three-phase voltage or current signals as well as the dc offset. The dc off

Vd(dc)=

VO(dc)=

sin t cosn t 0 cos t sin t 0 0 01

1 <sup>1</sup> ( ) *m*

1 <sup>1</sup> ( ) *m*

1 <sup>1</sup> ( ) *m*

<sup>1</sup> sin cosn

 

   

<sup>1</sup> sin(n 240n) cos(n 240n) <sup>2</sup>

*t t*

*t t*

*n t t*

<sup>1</sup> sin(n 120n) cos(n 120n) <sup>2</sup>

*i V*

*m* 

If these dc components are eliminated from the new pqo set, a new ac harmonic set is produced. We call this set as Vd(ac), Vq(ac) and V0(ac). If we multiply this set by the inverse

*i*

*m*

*i V*

*m* 

*d i*

*i*

*o i*

*Vq*

 

  x

sin sin( 120 ) sin( 240 ) cos cos( 120 ) cos( 240 ) 11 1 22 2

*nt nt n nt n nt nt n nt n*

 

 

(56)

2

(57)

(55)

*n n*

P =

<sup>=</sup> <sup>2</sup> 3

 

set components can be calculated as;

Vq(dc)=

of the matrix of equation (56), which is given as:

P -1 =

2 3

transformation)

2 3 Then, the resulting samples represent the samples of the harmonic components in each phase of the three phases. The following are the identification steps.


#### **6.2 Measurement of magnitude and phase angle of harmonic component**

Assume that the harmonic component of the phase a voltage signal is presented as:

$$\mathbf{v}\_{\text{a}}(t) = V\_{am}\cos(n\alpha t + \phi\_{\text{a}}) \tag{58}$$

where Vam is the amplitude of harmonic component n in phase a, is the fundamental frequency and a its phase angle measured with respect to certain reference. Using the trigonometric identity, equation (58) can be written as:

$$\mathbf{v\_a(t)} = \mathbf{x\_a}\cos n\mathbf{t} + \mathbf{y\_a}\sin n\mathbf{t}\tag{59}$$

where we define

$$\propto\_a = V\_{aw}\cos\phi\_a\tag{60}$$

$$\propto\_a = V\_{an} \sin \phi\_a \tag{61}$$

Electric Power Systems Harmonics - Identification and Measurements 53

Examining this table reveals that the proposed transformation is succeeded in estimating the harmonics content of a balanced three phase system. Furthermore, there is no need to model each harmonic component as was done earlier in the literature. Another test is conducted in this section, where we assume that the harmonics in the three phases are unbalanced. In this

<sup>0</sup> ( ) sin( 30 ) 0.25sin(3 ) 0.1sin(5 ) 0.05sin(7 ) *<sup>a</sup> vt t*

0 0 <sup>0</sup> ( ) 0.9sin( 150 ) 0.2sin(3 ) 0.15sin(5 120 ) 0.03sin(7 120 ) *<sup>b</sup> vt t*

0 0 <sup>0</sup> ( ) 0.8sin( 90 ) 0.15sin(3 ) 0.12sin(5 120 ) 0.04sin(7 120 ) *<sup>c</sup> vt t*

The sampling frequency used in this case is 1000Hz, using 50 samples. Table 4 gives the results obtained for the positive sequence of each harmonics component including the

Harmonic 1st harmonic 3rd harmonic 5th harmonic 7th harmonic Phase V V V V

A 0.9012 -29.9 0.2495 179.91 0.124 0.110 0.0301 0.441 B 0.8986 -149.97 0.2495 179.93 0.123 119.85 0.0298 -120.0 C 0.900 89.91 0.2495 179.9 0.123 -119.96 0.0301 119.58

Examining this table reveals that the proposed transformation is produced a good estimate in such unbalanced harmonics for magnitude and phase angle of each harmonics

We present in this section an algorithm to identifying and measuring harmonics components in a power system for quality analysis. The main features of the proposed

 It filters out the dc components of the voltage or current signal under consideration. The proposed algorithm avoids the draw backs of the previous algorithms, published

It uses samples of the three-phase signals that gives better view to the system status,

It has the ability to identify a large number of harmonics, since it does not need a

The only drawback, like other algorithms, if there is a frequency drift, it produces inaccurate estimate for the components under study. Thus a frequency estimation algorithm is needed in this case. Also, we assume that the amplitude and phase angles of each harmonic

Table 4. Estimated positive sequence for each harmonics component

component. In this case the components for the phases are balanced.

It needs no model for the harmonic components in question.

earlier in the literature, such as FFT, DFT, etc

mathematical model for harmonic components.

component are time independent, steady state harmonics identification.

especially in the fault conditions.

*t t t*

*t t t*

*t t t*

 

> 

 

test, we assume that the three phase voltages are as follows;

fundamental component.

**6.4 Remarks** 

algorithm are:

As stated earlier in step 5 m samples are available for a harmonic component of phase a, sampled at a preselected rate, then equation (73) can be written as:

$$\mathbf{Z} = \mathbf{A}\theta + \mathbf{z} \tag{62}$$

Where **Z** is mx1 samples of the voltage of any of the three phases, A is mx2 matrix of measurement and can be calculated off line if the sampling frequencies as well as the signal frequency are known in advance. The elements of this matrix are;

1 2 *a t n ta t n t* ( ) cos , ( ) sin ; is a 2x1 parameters vector to be estimated and is mx1 error vector due to the filtering process to be minimized. The solution to equation (62) based on least error squares is

$$\boldsymbol{\theta}^{\prime} = \left[\boldsymbol{A}^{\top}\boldsymbol{A}\right]^{-1}\boldsymbol{A}^{\top}\boldsymbol{Z} \tag{63}$$

Having identified the parameters vector \* the magnitude and phase angle of the voltage of phase a can be calculated as follows:

$$V\_{av} = \left[\text{x}^2 + \text{y}^2\right]^{\frac{1}{2}}\tag{64}$$

$$\phi\_a = \tan^{-1} \frac{y}{x} \tag{65}$$

#### **6.3 Testing the algorithm using simulated data**

The proposed algorithm is tested using a highly harmonic contaminated signal for the threephase voltage as:

$$w\_s(t) = \sin(\alpha t - \Im 0^0) + 0.25 \sin(3\alpha t) + 0.1 \sin(5\alpha t) + 0.05 \sin(7\alpha t)$$

The harmonics in other two phases are displaced backward and forward from phase a by 120o and equal in magnitudes, balanced harmonics contamination.

The sampling frequency is chosen to be *Fs =4. \* fo \* n*, *fo = 50* Hz, where n is the order of harmonic to be identified, n = 1,..,..,N, N is the largest order of harmonics to be expected in the waveform. In this example N=8. A number of sample equals 50 is chosen to estimate the parameters of each harmonic components. Table 3 gives the results obtained when n take the values of 1,3,5,7 for the three phases.


Table 3. The estimated harmonic in each phase, sampling frequency=1000 Hz and the number of samples=50

Examining this table reveals that the proposed transformation is succeeded in estimating the harmonics content of a balanced three phase system. Furthermore, there is no need to model each harmonic component as was done earlier in the literature. Another test is conducted in this section, where we assume that the harmonics in the three phases are unbalanced. In this test, we assume that the three phase voltages are as follows;

$$v\_a(t) = \sin(\alpha t - 30^\circ) + 0.25\sin(3\alpha t) + 0.1\sin(5\alpha t) + 0.05\sin(7\alpha t)$$

$$v\_b(t) = 0.9\sin(\alpha t - 150^0) + 0.2\sin(3\alpha t) + 0.15\sin(5\alpha t + 120^0) + 0.03\sin(7\alpha t - 120^0)$$

$$\sigma\_{\varepsilon}(t) = 0.8\sin(\alpha t + 90^{\circ}) + 0.15\sin(3\alpha t) + 0.12\sin(5\alpha t - 120^{\circ}) + 0.04\sin(7\alpha t + 120^{\circ})$$

The sampling frequency used in this case is 1000Hz, using 50 samples. Table 4 gives the results obtained for the positive sequence of each harmonics component including the fundamental component.


Table 4. Estimated positive sequence for each harmonics component

Examining this table reveals that the proposed transformation is produced a good estimate in such unbalanced harmonics for magnitude and phase angle of each harmonics component. In this case the components for the phases are balanced.

## **6.4 Remarks**

52 Power Quality Harmonics Analysis and Real Measurements Data

*x V a am a* sin

As stated earlier in step 5 m samples are available for a harmonic component of phase a,

 Z=A+ (62) Where **Z** is mx1 samples of the voltage of any of the three phases, A is mx2 matrix of measurement and can be calculated off line if the sampling frequencies as well as the signal

error vector due to the filtering process to be minimized. The solution to equation (62)

 *AA AZ* 

1

1 *<sup>a</sup>* tan *<sup>y</sup>*

The proposed algorithm is tested using a highly harmonic contaminated signal for the three-

<sup>0</sup> ( ) sin( 30 ) 0.25sin(3 ) 0.1sin(5 ) 0.05sin(7 ) *<sup>a</sup> vt t*

The harmonics in other two phases are displaced backward and forward from phase a by

The sampling frequency is chosen to be *Fs =4. \* fo \* n*, *fo = 50* Hz, where n is the order of harmonic to be identified, n = 1,..,..,N, N is the largest order of harmonics to be expected in the waveform. In this example N=8. A number of sample equals 50 is chosen to estimate the parameters of each harmonic components. Table 3 gives the results obtained when n take

Harmonic 1st harmonic 3rd harmonic 5th harmonic 7th harmonic Phase V V V V A 1.0 -30. 0.2497 179.95 0.1 0.0 0.0501 0.200 B 1.0 -150 0.2496 179.95 0.1 119.83 0.04876 -120.01 C 1.0 89.9 0.2496 179.95 0.0997 -119.95 0.0501 119.8

Table 3. The estimated harmonic in each phase, sampling frequency=1000 Hz and the

*x*

*t t t*

sampled at a preselected rate, then equation (73) can be written as:

frequency are known in advance. The elements of this matrix are;

 ; 

<sup>1</sup> \* *T T*

**6.3 Testing the algorithm using simulated data** 

the values of 1,3,5,7 for the three phases.

120o and equal in magnitudes, balanced harmonics contamination.

Having identified the parameters vector \*

phase a can be calculated as follows:

1 2 *a t n ta t n t* ( ) cos , ( ) sin 

based on least error squares is

phase voltage as:

number of samples=50

(61)

is a 2x1 parameters vector to be estimated and is mx1

(63)

the magnitude and phase angle of the voltage of

2 2 <sup>2</sup> *V xy am* (64)

(65)

 

> We present in this section an algorithm to identifying and measuring harmonics components in a power system for quality analysis. The main features of the proposed algorithm are:


The only drawback, like other algorithms, if there is a frequency drift, it produces inaccurate estimate for the components under study. Thus a frequency estimation algorithm is needed in this case. Also, we assume that the amplitude and phase angles of each harmonic component are time independent, steady state harmonics identification.

Electric Power Systems Harmonics - Identification and Measurements 55

Fuzzy numbers can be though of as crisp sets with moving boundaries with the following

 *[a, b] + [c, d] = [a+c , b+d]* 

 *[a, b] - [c, d] = [a-d , b-c]* 

 *[a, b] \* [c, d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]* 

*[a, b] / [c, d] = [min(a/c, a/d, b/c, b/d), max(a/c, a/d, b/c, b/d)]* (70)

A voltage or current waveform in a power system beside the fundamental one can be contaminated with noise and transient harmonics. For simplicity and without loss of

where *v1(t)* contains harmonics to be identified, and *v2(t)* contains other harmonics and

1 00

*vt V n t V n t*

*N*

*n*

( ) [ cos sin( ) sin cos( )]

 

*<sup>n</sup>* are the amplitude and phase angle of the *nth* harmonic, respectively. *N* is

*n n n n*

the number of harmonics to be identified in the waveform. Using trigonometric identity *v1(t)*

1 12 1 2 1

*vt Ax Ax* 

1 () [ ] *N*

Eq.(74) is a linear model with coefficients *A0, An1, An2, n=1, 2, …, N*. The model can be treated as a fuzzy model with fuzzy parameters each has a symmetric triangular membership

( ) ( ) [( ) ( ) ]

*v t p c p c x p c x*

*n vt A A x A x* 

function characterized by a central and spread values as described by Eq.(68).

1

*n*

*N*

0 0 1 11 1 12

() [ ]

*n n n n*

*ot)* n=1, 2, …, N

0 12 1 2

*<sup>n</sup>* n=1, 2, …, N

*n n n n*

*n nn n nn*

(75)

*v(t) = v1(t) + v2(t)* (71)

(73)

(74)

(72)

In the next section, waveform harmonics will be modeled as a linear fuzzy model.

four basic arithmetic operations [9]:

**7.2 Modeling of harmonics** *as a fuzzy model*

generality consider a non-sinusoidal waveform given by

transient that will not be identified. Consider *v1(t*) as Fourier series:

( ) sin( )

*n n*

 

1 0 1

*vt V n t*

*N*

*n N*

1

*n*

Where *Vn* and

can be written as:

Where *xn1 = sin(n*

*An1 = Vn cos*

Now v(t) can be written as:

Where A0 is effective (rms) value of v2(t).

*ot), xn2 = cos(n*

*n , An2 = Vn sin*
