**7. Fuzzy harmonic components identification**

In this section, we present a fuzzy Kalman filter to identify the fuzzy parameters of a general non-sinusoidal voltage or current waveform. The waveform is expressed as a Fourier series of sines and cosines terms that contain a fundamental harmonic and other harmonics to be measured. The rest of the series is considered as additive noise and unmeasured distortion. The noise is filtered out and the unmeasured distortion contributes to the fuzziness of the measured parameters. The problem is formulated as one of linear fuzzy problems. The nth harmonic component to be identified, in the waveform, is expressed as a linear equation: *An1 sin(n0t) + An2 cos(n0t)*. The *An1* and *An2* are fuzzy parameters that are used to determine the fuzzy values of the amplitude and phase of the nth harmonic. Each fuzzy parameter belongs to a symmetrical triangular membership function with a middle and spread values. For example *An1 = (pn1, cn1),* where *pn1* is the center and *cn1* is the spread. Kalman filtering is used to identify fuzzy parameters pn1, cn1, pn2, and cn2 for each harmonic required to be identified.

An overview of the necessary linear fuzzy model and harmonic waveform modeling is presented in the next section.

#### **7.1 Fuzzy function and fuzzy linear modeling**

The fuzzy sets were first introduced by Zadeh [20]. Modeling fuzzy linear systems has been addressed in [8,9]. In this section an overview of fuzzy linear models is presented. A fuzzy linear model is given by:

$$\mathbf{Y} = f(\mathbf{x}) = \underline{A}\_0 + \underline{A}\_1 \ge \underline{x}\_1 + \underline{A}\_2 \ge \underline{x}\_2 + \dots + \underline{A}\_n \ge \underline{x}\_n \tag{66}$$

where Y is the dependent fuzzy variable (output), {x1, x2, …, xn} set of crisp (not fuzzy) independent variables, and {A0, A1, …, An} is a set of symmetric fuzzy numbers. The membership function of Ai is symmetrical triangular defined by center and spread values, pi and ci, respectively and can be expressed as

$$\mu\_{A\_i} \left( a\_i \right) = \begin{cases} 1 - \frac{|p\_i - a\_i|}{c\_i} & p\_i - c\_i \le a\_i \le p\_i + c\_i \\\\ 0 & \text{otherwise} \end{cases} \tag{67}$$

Therefore, the function Y can be expressed as:

$$\mathbf{Y} = \mathbf{f}(\mathbf{x}) = \begin{pmatrix} \mathbf{p}\_{0\prime} \ c\_{0} \end{pmatrix} + \begin{pmatrix} \mathbf{p}\_{1\prime} \ c\_{1} \end{pmatrix} \mathbf{x}\_{1} + \dots + \begin{pmatrix} \mathbf{p}\_{n\prime} \ c\_{n} \end{pmatrix} \mathbf{x}\_{n} \tag{68}$$

Where *Ai = (pi, ci)* and the membership function of *Y* is given by:

$$\mu\_Y(y) = \begin{cases} \mathbf{1} - \frac{\left| y - \sum\_{i=1}^n p\_i \mathbf{x}\_i \right|}{\sum\_{i=1}^n c\_i \left| \mathbf{x}\_i \right|} & \mathbf{x}\_i \neq \mathbf{0} \\\\ \mathbf{1} & \mathbf{x}\_i = \mathbf{0}, \quad y = \mathbf{0} \\\ \mathbf{0} & \mathbf{x}\_i = \mathbf{0}, \quad y \neq \mathbf{0} \end{cases} \tag{69}$$

In this section, we present a fuzzy Kalman filter to identify the fuzzy parameters of a general non-sinusoidal voltage or current waveform. The waveform is expressed as a Fourier series of sines and cosines terms that contain a fundamental harmonic and other harmonics to be measured. The rest of the series is considered as additive noise and unmeasured distortion. The noise is filtered out and the unmeasured distortion contributes to the fuzziness of the measured parameters. The problem is formulated as one of linear fuzzy problems. The nth harmonic component to be identified, in the waveform, is expressed as a linear equation: *An1*

fuzzy values of the amplitude and phase of the nth harmonic. Each fuzzy parameter belongs to a symmetrical triangular membership function with a middle and spread values. For example *An1 = (pn1, cn1),* where *pn1* is the center and *cn1* is the spread. Kalman filtering is used to identify

An overview of the necessary linear fuzzy model and harmonic waveform modeling is

The fuzzy sets were first introduced by Zadeh [20]. Modeling fuzzy linear systems has been addressed in [8,9]. In this section an overview of fuzzy linear models is presented. A fuzzy

where Y is the dependent fuzzy variable (output), {x1, x2, …, xn} set of crisp (not fuzzy) independent variables, and {A0, A1, …, An} is a set of symmetric fuzzy numbers. The membership function of Ai is symmetrical triangular defined by center and spread values, pi

*p a pc apc*

1

*c x*

*n i i i*

*y px*

1 0

 

*n i i i*

1 0, 0 0 0, 0

*i i*

1

*ii i ii*

*otherwise*

Y = f(x)= (p0, c0) + (p1, c1) x1 + … + (pn, cn) xn (68)

*x*

(69)

*x y x y*

fuzzy parameters pn1, cn1, pn2, and cn2 for each harmonic required to be identified.

*0t)*. The *An1* and *An2* are fuzzy parameters that are used to determine the

*Y= f(x) = A0 + A1 x1 + A2 x2 + … + An xn* (66)

(67)

**7. Fuzzy harmonic components identification** 

*sin(n*

*0t) + An2 cos(n*

presented in the next section.

linear model is given by:

**7.1 Fuzzy function and fuzzy linear modeling** 

and ci, respectively and can be expressed as

*a*

*i*

Therefore, the function Y can be expressed as:

*A i*

1

Where *Ai = (pi, ci)* and the membership function of *Y* is given by:

*y*

*Y i*

 

 

0

*i i*

*i*

*c*

Fuzzy numbers can be though of as crisp sets with moving boundaries with the following four basic arithmetic operations [9]:

$$[a, b] + \quad [c, d] = [a+c, b+d]$$

$$[a, b] - \quad [c, d] = [a-d, b-c]$$

$$[a, b] \text{ \* } \quad [c, d] = \{\min(ac, ad, bc, bd), \max(ac, ad, bc, bd)\}$$

$$[a, b] / \quad \quad [c, d] = \{\min(a/c, a/d, b/c, b/d), \max(a/c, a/d, b/c, b/d)\}\tag{70}$$

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

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

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 generality consider a non-sinusoidal waveform given by

$$
\psi v(t) \equiv \upsilon\_1(t) + \upsilon\_2(t) \tag{71}
$$

where *v1(t)* contains harmonics to be identified, and *v2(t)* contains other harmonics and transient that will not be identified. Consider *v1(t*) as Fourier series:

$$\begin{aligned} \upsilon\_1(t) &= \sum\_{n=1}^{N} V\_n \sin(n\alpha\_0 t + \varphi\_n) \\ \upsilon\_1(t) &= \sum\_{n=1}^{N} [V\_n \cos \varphi\_n \sin(n\alpha\_0 t) + V\_n \sin \varphi\_n \cos(n\alpha\_0 t)] \end{aligned} \tag{72}$$

Where *Vn* and *<sup>n</sup>* are the amplitude and phase angle of the *nth* harmonic, respectively. *N* is the number of harmonics to be identified in the waveform. Using trigonometric identity *v1(t)* can be written as:

$$\boldsymbol{\upsilon}\_{1}(t) = \sum\_{n=1}^{N} [\underline{\underline{A}}\_{n1} \boldsymbol{\chi}\_{n1} + \underline{\underline{A}}\_{n2} \boldsymbol{\chi}\_{n2}] \tag{73}$$

$$\begin{aligned} \text{Where} \quad \mathbf{x}\_{n1} &= \sin(n \, a\_b t), \, \mathbf{x}\_{n2} = \cos(n \, a\_b t) & \quad \text{n=1, 2, ..., N} \\ \underline{A}\_{n1} &= V\_n \cos \eta\_n \, \underline{A}\_{n2} = V\_n \sin \eta\_n \quad \quad \text{n=1, 2, ..., N} \\ \text{Now } \mathbf{v}(t) \text{ can be written as:} \end{aligned}$$

$$w(t) = \underline{A}\_0 + \sum\_{n=1}^{N} [\underline{A}\_{n1}\underline{x}\_{n1} + \underline{A}\_{n2}\underline{x}\_{n2}] \tag{74}$$

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

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

$$\mathbf{v}(t) = (p\_0 + c\_0) + \sum\_{n=1}^{N} [(p\_{n1} + c\_{n1})\mathbf{x}\_{n1} + (p\_{n1} + c\_{n1})\mathbf{x}\_{n2}] \tag{75}$$

Electric Power Systems Harmonics - Identification and Measurements 57

1 1

*p p pp u u uu*

 

 

 

 

tan ( ) tan ( / ) tan ( ) tan ( / ) *n nn n n nn n*

1 1

The detailed derivation of Kalman filtering can be found in [23, 24]. In this section, only the necessary equation for the development of the basic recursive discrete Kalman filter will be

*x(k +1) = A(k) x(k) + w(k)* 

Zero means: *E[w(k)] = E[v(k)] = 0.* (84)

No time correlation: *E[w(i) wT(j)] = E[v(i) vT(j)] = 0, for i = j.* (85)

*E[w(k) wT(k)] = Q1* 

where *Q1* and *Q2* are positive semi-definite and positive definite matrices, respectively. The basic discrete-time Kalman filter algorithm given by the following set of recursive equations. Given as priori estimates of the state vector *x^(0) = x^0* and its error covariance matrix, *P(0)=* 

Kalman gain: *K(k) = [A(k) P(k) CT(k)] [C(k) P(k) CT(k) + Q2]-1* (87)

*x^(k+1) = A(k) x^(k) + K(k) [z(k) – C(k)x^(k)]* (88)

An intelligent choice of the priori estimate of the state **x**^0 and its covariance error **P**<sup>0</sup> enhances the convergence characteristics of the Kalman filter. Few samples of the output waveform **z**(k) can be used to get a weighted least squares as an initial values for **x^**0 and **P**0:

 *x^0 = [HT Q2-1 H]-1 HT Q2-1 z0* 

*P(k+1) = [A(k) – K(k) C(k)] p(k) [A(k) – K(k) C(k)]T + K(k) Q2 KT(k)* (89)

*n nn*

*cup*

 *A(k) is n x n time varying state transition matrix.* 

The noise vectors *w(k)* and *v(k)* are uncorrected white noises that have:

 

**7.5 Fuzzy modeling for Kalman filter algorithm** 

addressed. Given the discrete state equations:

where *x(k) is n x 1 system states.* 

 *w(k) is n x 1 system error. v(k) is m x 1 measurement error.* 

Known covariance matrices (noise levels):

*P0*, set *k=0* then recursively computer:

New state estimate:

Error Covariance update:

 *z(k) is m x 1 vector measurement. C(k) is m x n time varying output matrix.* 

**7.5.1 The basic Kalman filter** 

tan 2 1

tan 2 1

*z(k) = C(k) x(k) + v(k)* (83)

*E[v(k) vT(k)] = Q2* (86)

(82)

In the next section, Kalman filtering technique is used to identify the fuzzy parameters. Once the fuzzy parameters are identified then fuzzy values of amplitude and phase angle of each harmonic can be calculated using mathematical operations on fuzzy numbers. If crisp values of the amplitudes and phase angles of the harmonics are required, the defuzzefication is used. The fuzziness in the parameters gives the possible extreme variation that the parameter can take. This variation is due to the distortion in the waveform because of contamination with harmonic components, *v2(t),* that have not been identified. If all harmonics are identified, *v2(t)=0*, then the spread values would be zeros and identified parameters would be crisp rather than fuzzy ones.

Having identified the fuzzy parameters, the nth harmonic amplitude and phase can be calculated as:

$$\begin{aligned} \boldsymbol{\upsilon}\_{\boldsymbol{u}}^{2} &= \underline{\underline{A}}\_{\boldsymbol{u}1}^{2} + \underline{\underline{A}}\_{\boldsymbol{u}2}^{2} \\ \tan \boldsymbol{\varrho}\_{\boldsymbol{n}} &= \underline{\underline{A}}\_{\boldsymbol{n}2}^{2} + \underline{\underline{A}}\_{\boldsymbol{n}1}^{2} \end{aligned} \tag{76}$$

The parameters in Eq. (76) are fuzzy numbers, the mathematical operations defined in Eq. (70) are employed to obtain fuzzy values of the amplitude and phase angle.

#### **7.3 Fuzzy amplitude calculation:**

Writing amplitude Eq.(76) in fuzzy form:

$$\sigma\_n^2 = (p\_{\upsilon\_n^{2,\prime}} c\_{\upsilon\_n^2}) = (p\_{n1\prime} c\_{n1})(p\_{n1\prime} c\_{n1}) + (p\_{n2\prime} c\_{n2})(p\_{n2\prime} c\_{n2}) \tag{77}$$

To perform the above arithmetic operations, the fuzzy numbers are converted to crisp sets of the form *[pi -ci, pi+c*i]. Since symmetric membership functions are assumed, for simplicity, only one half of the set is considered, *[pi , pi+ci].* Denoting the upper boundary of the set *pi+ci* by ui, the fuzzy numbers are represented by sets of the form *[pi, ui]* where *ui > pi*. Accordingly,

$$\mathbb{E}\left[\boldsymbol{\sigma}\_{\boldsymbol{u}\_{n}^{2}}^{2}\boldsymbol{\mu}\_{\boldsymbol{v}\_{n}^{2}}^{2}\right] = [\boldsymbol{p}\_{n1}, \boldsymbol{\mu}\_{n1}][\boldsymbol{p}\_{n1'}\boldsymbol{\mu}\_{n1}] + [\boldsymbol{p}\_{n2'}, \boldsymbol{\mu}\_{n2}][\boldsymbol{p}\_{n2'}\boldsymbol{\mu}\_{n2}] \tag{78}$$

Then the center and spread values of the amplitude of the nth harmonic are computed as follows:

$$\begin{aligned} p\_{\nu\_{\boldsymbol{v}}} &= \sqrt{p\_{\boldsymbol{v}\_{\boldsymbol{n}}^{2}}} = \sqrt{p\_{\boldsymbol{n}1}^{2} + p\_{\boldsymbol{n}2}^{2}} \\ \mu\_{\boldsymbol{v}\_{\boldsymbol{n}}} &= \sqrt{\mu\_{\boldsymbol{v}\_{\boldsymbol{n}}^{2}}} = \sqrt{\mu\_{\boldsymbol{n}1}^{2} + \mu\_{\boldsymbol{n}2}^{2}} \\ \mathcal{c}\_{\boldsymbol{v}\_{\boldsymbol{n}}} &= \mu\_{\boldsymbol{v}\_{\boldsymbol{n}}} - p\_{\boldsymbol{v}\_{\boldsymbol{n}}} \end{aligned} \tag{79}$$

#### **7.4 Fuzzy phase angle calculation**

Writing phase angle Eq.(79) in fuzzy form:

$$\tan \varphi\_n = (p\_{\tan \circ n}, \mathcal{c}\_{\tan \circ n}) = (p\_{n2}, \mathcal{c}\_{n2}) \div (p\_{n1}, \mathcal{c}\_{n1}) \tag{80}$$

Converting fuzzy numbers to sets:

$$\tan \varphi\_n = [\boldsymbol{p}\_{\tan \phi n}, \boldsymbol{\mu}\_{\tan \phi n}] = [\boldsymbol{p}\_{n2}, \boldsymbol{\mu}\_{n2}] \div [\boldsymbol{p}\_{n1}, \boldsymbol{c}\_{n1}] \tag{81}$$

then the central and spread values of the phase angle is given by:

$$\begin{aligned} p\_{\phi n} &= \tan^{-1}(p\_{\tan \phi n}) = \tan^{-1}(p\_{n2} \; / \; p\_{n1}) \\ \mu\_{\phi n} &= \tan^{-1}(\mu\_{\tan \phi n}) = \tan^{-1}(\mu\_{n2} \; / \; \mu\_{n1}) \\ \mathcal{L}\_{\phi n} &= \mu\_{\phi n} - p\_{\phi n} \end{aligned} \tag{82}$$

#### **7.5 Fuzzy modeling for Kalman filter algorithm 7.5.1 The basic Kalman filter**

56 Power Quality Harmonics Analysis and Real Measurements Data

In the next section, Kalman filtering technique is used to identify the fuzzy parameters. Once the fuzzy parameters are identified then fuzzy values of amplitude and phase angle of each harmonic can be calculated using mathematical operations on fuzzy numbers. If crisp values of the amplitudes and phase angles of the harmonics are required, the defuzzefication is used. The fuzziness in the parameters gives the possible extreme variation that the parameter can take. This variation is due to the distortion in the waveform because of contamination with harmonic components, *v2(t),* that have not been identified. If all harmonics are identified, *v2(t)=0*, then the spread values would be zeros and identified

Having identified the fuzzy parameters, the nth harmonic amplitude and phase can be

2 2 2 1 2 2 2 2 1 tan *n n n n n n*

11 11 22 22 ( , ) ( , )( , ) ( , )( , ) *n n n n v v <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> v pc pc pc pc pc* (77)

11 11 22 22 [ , ] [ , ][ , ] [ , ][ , ] *n n n n v v <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> v pu pu pu pu pu* (78)

(76)

(79)

(80)

(81)

*vA A*

 *A A* 

The parameters in Eq. (76) are fuzzy numbers, the mathematical operations defined in Eq.

To perform the above arithmetic operations, the fuzzy numbers are converted to crisp sets of the form *[pi -ci, pi+c*i]. Since symmetric membership functions are assumed, for simplicity, only one half of the set is considered, *[pi , pi+ci].* Denoting the upper boundary of the set *pi+ci* by ui, the fuzzy numbers are represented by sets of the form *[pi, ui]* where *ui > pi*. Accordingly,

Then the center and spread values of the amplitude of the nth harmonic are computed as

2

*v n v n*

*p p pp*

*u u uu*

*v n v n*

tan tan 22 11 tan ( , ) ( , ) ( , ) *n n n nn nn p c pc pc*

tan tan 2 2 11 tan [ , ] [ , ] [ , ] *n n n n n nn p u pu pc*

 

 

*n n*

*n n n nn*

*v vv*

*cup*

2

(70) are employed to obtain fuzzy values of the amplitude and phase angle.

2 2

2 2 2 22

then the central and spread values of the phase angle is given by:

parameters would be crisp rather than fuzzy ones.

**7.3 Fuzzy amplitude calculation:**  Writing amplitude Eq.(76) in fuzzy form:

**7.4 Fuzzy phase angle calculation**  Writing phase angle Eq.(79) in fuzzy form:

Converting fuzzy numbers to sets:

2

calculated as:

follows:

The detailed derivation of Kalman filtering can be found in [23, 24]. In this section, only the necessary equation for the development of the basic recursive discrete Kalman filter will be addressed. Given the discrete state equations:

$$\mathbf{x}(k+1) = A(k)\mathbf{x}(k) + w(k)$$

$$z(k) = \mathbf{C}(k)\mathbf{x}(k) + \upsilon(k)\tag{83}$$
where  $\mathbf{x}(k)$  is 
$$\begin{array}{ccccc} \text{where} & \mathbf{x}(k) & \text{is} & \text{n x 1} & \text{system states.}\\ A(k) & \text{is} & \text{n x } n & \text{time varying state transition} & \text{matrix.}\\ z(k) & \text{is} & \text{n x 1} & \text{vector measurement.}\\ \mathbf{C}(k) & \text{is} & \text{n x 1} & \text{time varying output matrix.}\\ w(k) & \text{is} & \text{n x 1} & \text{system error.}\\ \upsilon(k) & \text{is} & \text{n x 1} & \text{measure.} \end{array}$$

The noise vectors *w(k)* and *v(k)* are uncorrected white noises that have:

$$\text{Zero means:}\tag{87} = E[w(k)] = E[v(k)] = 0.\tag{84}$$

$$\text{No time correlation:} \qquad E[w(\mathbf{i}) \; w^{\text{T}}(\mathbf{j})] = E[v(\mathbf{i}) \; v^{\text{T}}(\mathbf{j})] = 0, \quad \text{for } \mathbf{i} = \mathbf{j}. \tag{85}$$

Known covariance matrices (noise levels):

$$E\left[\mathfrak{w}(k)\,\,\mathfrak{w}^{\mathsf{T}}(k)\right] = \mathbf{Q}\_{\mathsf{I}}$$

$$E\left[\mathfrak{w}(k)\,\,\,\,\mathfrak{w}^{\mathsf{T}}(k)\right] = \mathbf{Q}\_{\mathsf{I}}\tag{86}$$

where *Q1* and *Q2* are positive semi-definite and positive definite matrices, respectively. The basic discrete-time Kalman filter algorithm given by the following set of recursive equations. Given as priori estimates of the state vector *x^(0) = x^0* and its error covariance matrix, *P(0)= P0*, set *k=0* then recursively computer:

$$\text{Kalman gain:} \qquad \qquad \text{K(k)} = \left[ \text{A(k)} \text{ P(k)} \text{ C\text{(k)}} \right] \left[ \text{C(k)} \text{ P(k)} \text{ C\text{(k)}} + Q\_2 \right] \text{1} \tag{87}$$

New state estimate:

$$\mathbf{x}^{\circ}(\mathbf{k}+\mathbf{1}) = \mathbf{A}(\mathbf{k})\,\mathbf{x}^{\circ}(\mathbf{k}) + \mathbf{K}(\mathbf{k})\,\left[\mathbf{z}(\mathbf{k}) - \mathbf{C}(\mathbf{k})\mathbf{x}^{\circ}(\mathbf{k})\right] \tag{88}$$

Error Covariance update:

$$P(\text{k}+1) = \left[\text{A}(\text{k}) - \text{K}(\text{k}) \cdot \text{C}(\text{k})\right] p(\text{k}) \left[\text{A}(\text{k}) - \text{K}(\text{k}) \cdot \text{C}(\text{k})\right]^{\text{T}} + \text{K}(\text{k}) \cdot \text{Q}\_{2} \text{ K}^{\text{T}}(\text{k}) \tag{89}$$

An intelligent choice of the priori estimate of the state **x**^0 and its covariance error **P**<sup>0</sup> enhances the convergence characteristics of the Kalman filter. Few samples of the output waveform **z**(k) can be used to get a weighted least squares as an initial values for **x^**0 and **P**0:

$$\mathfrak{x}^{\square}{}\_{0} = \{H^{\mathsf{T}} \: \mathbb{Q} \mathbb{Z}^{\mathsf{-1}} H\}^{\mathsf{-1}} H^{\mathsf{T}} \: \mathbb{Q} \mathbb{z}^{\mathsf{1}} \: \ zo$$

Electric Power Systems Harmonics - Identification and Measurements 59

Where *xn1, xn2, n=1, 2, …, N* are defined in Eq.(73) with sampling at time instant tTk, T is

The first row of *C(k)* is used to identify the center of the fuzzy parameters, while the second row is used to identify the spread parameters. The third raw is used to identify the magnitude of the error produced by the unidentified harmonics and noise. The observation vector *z(k)* consists of three values. *v(k)* is the value of the measured waveform signal at

2 2

11 22

11 22

*nn nn*

() [ () () () (]

*k c kx k c kx k*

The *xn1* and *xn2* are the *v1(t)* harmonics and they are well defined at time instant k, but cn1 and cn2 are the measurement error components in the direction of the nth harmonic of v1(t).

> ( ) ( )cos( )( ) ( ) ( )sin( )( ) *n peak n n peak n ck e k k ck e k k*

Where epeak is the peak error defined in Eq.(93), sinn and sinn are defined in Eq(87). Since the peak error depends on the measured samples, its mean square is estimated as a separate parameter. It is p0(k), the last parameter in the state vector. Similarly, cosn and sinn are unknown parameters that are estimated in the state vector. epeak, cosn and sinn are

1/2

cos ( ) ( 1) /[ ( 1) ( 1)] cos ( ) ( 1) /[ ( 1) ( 1)]

To verify the effectiveness of the proposed harmonic fuzzy parameter identification

*k pk pk pk k pk pk pk*

*nn n n nn n n*

1 12

 

2 12

2 2

2 2

*nn nn*

() () () () ()

The estimated error in Eq. (93) is computed using the estimated central values of the harmonics of v1(t) (Ref. Eq. (85)). The reason for estimating the square of the error rather than the error its self is due to the intrinsic nature of the Kalman filter of filtering out any

*vk p kx k p kx k*

*(k)* depends on *v(k)* and the state vector at time instant *k-1*.

(94)

*(k),* which is the measured spread of the identified

(95)

(96)

(97)

( ) [ ( ) ( )] *k e vk vk* (93)

the signal period and k = 1, 2, … .

*k)* and 

*(k),* it is defined as the square of the error:

*N*

1

*n*

harmonics,v1(t) . It can be thought of as *v2(t)* modeled as *v1(t)* harmonics.

*N*

1

*n*

1 2 0

( ) [2 ( 1)]

*e k pk*

*peak*

approach, simulation examples are given below.

sampling instant *k.,* 

Start with

They are defined below.

zero means noise.

The second entry of *z(k)* is

They are computed as follows:

computed as follows:

**7.5.3 Simulation results** 

$$P\_0 \, \, = \, \{ \mathbf{H}^\mathrm{T} \, \mathbf{Q} \mathbf{z}^\mathrm{1} \, \mathbf{H} \}^{\mathrm{I} \, \mathrm{I}} \, \tag{90}$$

where z0 is (m m1) x 1 vector of m1 measured samples. H is (m m1) x n matrix.

$$z\_0 = \begin{bmatrix} z(1) \\ z(2) \\ \vdots \\ z(m\_1) \end{bmatrix} \qquad and \qquad H = \begin{bmatrix} \mathcal{C}(1) \\ \mathcal{C}(2) \\ \vdots \\ \mathcal{C}(m\_1) \end{bmatrix} \tag{91}$$

#### **7.5.2 Fuzzy harmonic estimation dynamic model**

In this sub-section the harmonic waveform is modeled as a time varying discrete dynamic system suited for Kalman filtering. The dynamic system of Eq.(83) is used with the following definitions:


The observation equation **z**(k)=**C**(k) **x**(k) has the following form:

$$
\begin{bmatrix} v(k) \\ \delta(k) \\ \eta(k) \end{bmatrix} = \begin{bmatrix} x\_{11} & x\_{12} & \cdots & x\_{N1} & x\_{N2} & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & \cdots & 0 & 0 & x\_{11} & x\_{12} & \cdots & x\_{N1} & x\_{N2} & 0 \\ 0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p\_{11} \\ p\_{12} \\ \vdots \\ p\_{N1} \\ p\_{N2} \\ \vdots \\ p\_{N2} \\ p\_{N3} \\ \vdots \\ p\_{N3} \\ p\_{N4} \end{bmatrix} \tag{92}
$$

1 1

( ) ( )

*z m C m*

In this sub-section the harmonic waveform is modeled as a time varying discrete dynamic system suited for Kalman filtering. The dynamic system of Eq.(83) is used with the

3. **Q1** and **Q2** values are based on some knowledge of the actual characteristics of the process and measurement noises, respectively. **Q1** and **Q2** are chosen to be identity matrices for this simulation, **Q1** would be assigned better value if more knowledge were

5. Two parameters (center and spread) per harmonic to be identified. That mounts to 2N parameters. The last parameter is reserved for the magnitude of the error resulted from

6. **C**(k) is *3x(2N+1*) time varying measure matrix, which relates the measured signal to the

7. The observation vector, *z(k),* is 3x(2N+1) time varying vector, depends on the signal

0 0 0 00 ( ) 11 12 1 2 <sup>2</sup> () 0 0 0 0 <sup>0</sup> 11 12 1 2 <sup>11</sup> ( ) 0 0 0 0 0 0 0 01 <sup>12</sup>

*xx x x <sup>p</sup> v k N N <sup>N</sup> k xx x x c*

 

*k c*

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

*z C z C*

*z and H*

where z0 is (m m1) x 1 vector of m1 measured samples.

0

1. The state transition matrix, **A**(k), is a constant identity matrix. 2. The error covariance matrices, **Q1** and **Q2**, are constant matrices.

4. The state vector, **x**(k), consists of *2N+1* fuzzy parameters.

the unidentified harmonics and noise. (Refer to Eq. (92)).

The observation equation **z**(k)=**C**(k) **x**(k) has the following form:

**7.5.2 Fuzzy harmonic estimation dynamic model** 

obtained on the sensor accuracy.

state vector. (Refer to Eq. (106)).

measurement. (Refer to Eq. (92)).

H is (m m1) x n matrix.

following definitions:

*P0 = [HT Q2-1 H]-1* (90)

(91)

11 12

 

*p p*

*p N*

*N N*

1

(92)

1 2 0

 

*c N c N p*

Where *xn1, xn2, n=1, 2, …, N* are defined in Eq.(73) with sampling at time instant tTk, T is the signal period and k = 1, 2, … .

The first row of *C(k)* is used to identify the center of the fuzzy parameters, while the second row is used to identify the spread parameters. The third raw is used to identify the magnitude of the error produced by the unidentified harmonics and noise. The observation vector *z(k)* consists of three values. *v(k)* is the value of the measured waveform signal at sampling instant *k., k)* and *(k)* depends on *v(k)* and the state vector at time instant *k-1*. They are defined below.

Start with *(k),* it is defined as the square of the error:

$$\text{tr}(k) = e^2 = [\upsilon(k) - \hat{\upsilon}(k)]^2 \tag{93}$$

$$
\hat{w}(k) = \sum\_{n=1}^{N} \left[ p\_{n1}(k)\mathbf{x}\_{n1}(k) + p\_{n2}(k)\mathbf{x}\_{n2}(k) \right] \tag{94}
$$

The estimated error in Eq. (93) is computed using the estimated central values of the harmonics of v1(t) (Ref. Eq. (85)). The reason for estimating the square of the error rather than the error its self is due to the intrinsic nature of the Kalman filter of filtering out any zero means noise.

The second entry of *z(k)* is *(k),* which is the measured spread of the identified harmonics,v1(t) . It can be thought of as *v2(t)* modeled as *v1(t)* harmonics.

$$\mathcal{S}(k) = \sum\_{n=1}^{N} \left[ \mathbf{c}\_{n1}(k)\mathbf{x}\_{n1}(k) + \mathbf{c}\_{n2}(k)\mathbf{x}\_{n2}(k) \right] \tag{95}$$

The *xn1* and *xn2* are the *v1(t)* harmonics and they are well defined at time instant k, but cn1 and cn2 are the measurement error components in the direction of the nth harmonic of v1(t). They are computed as follows:

$$\begin{aligned} \varepsilon\_{n1}(k) &= \varepsilon\_{\text{peak}}(k) \cos(\varphi\_n)(k) \\ \varepsilon\_{n2}(k) &= \varepsilon\_{\text{peak}}(k) \sin(\varphi\_n)(k) \end{aligned} \tag{96}$$

Where epeak is the peak error defined in Eq.(93), sinn and sinn are defined in Eq(87). Since the peak error depends on the measured samples, its mean square is estimated as a separate parameter. It is p0(k), the last parameter in the state vector. Similarly, cosn and sinn are unknown parameters that are estimated in the state vector. epeak, cosn and sinn are computed as follows:

$$\begin{aligned} \boldsymbol{e}\_{p\boldsymbol{n}k}(k) &= \left[2\boldsymbol{p}\_0(k-1)\right]^{1/2} \\ \cos\boldsymbol{\rho}\_n(k) &= \boldsymbol{p}\_{n1}(k-1) \left/\left[p\_{n1}^2(k-1) + p\_{n2}^2(k-1)\right]\right. \\ \left.\cos\boldsymbol{\rho}\_n(k) &= \boldsymbol{p}\_{n2}(k-1) \left/\left[p\_{n1}^2(k-1) + p\_{n2}^2(k-1)\right]\right. \end{aligned} \tag{97}$$

#### **7.5.3 Simulation results**

To verify the effectiveness of the proposed harmonic fuzzy parameter identification approach, simulation examples are given below.

Electric Power Systems Harmonics - Identification and Measurements 61

Fig. 52. First Harmonic Centre Paramaters.

Fig. 53. First Harmonic spread parameters.

Fig. 54. Mauserd waveform and estimated central of the first harmonic.

#### **7.5.4 One harmonic identification**

As a first example consider identification of one harmonic only, N=1. Consider a voltage waveform that consists of two harmonics, one fundamental at 50Hz and a sub-harmonic at 150Hz which is considered as undesired distortion contaminating the first harmonic.

$$\begin{aligned} v(t) &= 1.414 \sin(100 \pi t + \pi / 6) \\ &+ 0.3 \sin(300 \pi t + \pi / 5) \end{aligned} \tag{98}$$

For parameter estimation using Kalman filter, the voltage signal is sampled at frequency 1250Hz and used as measurement samples. Converting Eq (88) to discrete time, t 0.08k, where k is the sampling time, and using the notation of Eq. (71), *v1(k)* and *v2(k)* are defined as:

$$\begin{aligned} v\_1(k) &= 1.414\sin(0.08\pi k + \pi/6) \\ v\_2(k) &= 0.30\sin(0.24\pi k + \pi/5) \end{aligned} \tag{99}$$

using the notation of Eq.(88), the time fuzzy model is given by:

*v(k) = Ao + A11 x11(k) + A12 x12(k)* 

where x11(k) = sin(0.08k), x12(k) = cos(0.08k) and the parameters to be identified are: Ao=(po, 0), A11=(p11, c11) and A12=(p12, c12). The observation equation, z(k) = **C**(k) **x**(k), becomes:

$$z(k) = \begin{bmatrix} v \\ \delta \\ \sigma \\ \eta \end{bmatrix} = \begin{bmatrix} \mathbf{x}\_{11} & \mathbf{x}\_{12} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{x}\_{11} & \mathbf{x}\_{12} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix} \begin{bmatrix} p\_{11} \\ p\_{12} \\ c\_{11} \\ c\_{12} \\ p\_{\alpha} \end{bmatrix} \tag{100}$$

The argument (k) of all variables in Eq.(100) has been omitted for simplicity of notation. With initial state vector **x**(0)=[1 1 1 1 1]T the following estimated parameters are obtained:

*A0* = (0.052, 0.0) *A11*= (1.223, 0.330) *A12*= (0.710, 0.219)

Computing the amplitude and phase:

*V1*= (1.414, 0.395)

*<sup>1</sup>*= (0.166, 0.014)

Figures 52 and 53 show the convergence of the center and spread of the first harmonic parameters, respectively.

Figure (54) shows the measured *v(t)* and estimated (crisp) first harmonic, while Figure (55) illustrates the estimated fuzziness of *v(t)* by reconstructing waveforms of the form.

$$
\pi\_{pc}(t) \mathbf{\bar{r}} = (p\_{11} \pm \mathbf{x}\_{11}) \times\_{11} + (p\_{12} \pm \mathbf{x}\_{12}) \times\_{12} \tag{101}
$$

Fig. 52. First Harmonic Centre Paramaters.

As a first example consider identification of one harmonic only, N=1. Consider a voltage waveform that consists of two harmonics, one fundamental at 50Hz and a sub-harmonic at

> ( ) 1.414sin(100 /6) 0.3sin(300 /5)

For parameter estimation using Kalman filter, the voltage signal is sampled at frequency 1250Hz and used as measurement samples. Converting Eq (88) to discrete time, t 0.08k, where k is the sampling time, and using the notation of Eq. (71), *v1(k)* and *v2(k)* are defined

> ( ) 1.414sin(0.08 /6) ( ) 0.30sin(0.24 /5)

*v(k) = Ao + A11 x11(k) + A12 x12(k)*  where x11(k) = sin(0.08k), x12(k) = cos(0.08k) and the parameters to be identified are: Ao=(po, 0), A11=(p11, c11) and A12=(p12, c12). The observation equation, z(k) = **C**(k) **x**(k),

() 0 0 0

*z k x x c*

The argument (k) of all variables in Eq.(100) has been omitted for simplicity of notation. With initial state vector **x**(0)=[1 1 1 1 1]T the following estimated parameters are obtained:

Figures 52 and 53 show the convergence of the center and spread of the first harmonic

Figure (54) shows the measured *v(t)* and estimated (crisp) first harmonic, while Figure (55)

*c11) x11 + (p12*

illustrates the estimated fuzziness of *v(t)* by reconstructing waveforms of the form.

*vpc(t)= (p11*

 

 

11 12 12

0 0 0 01

*v xx p*

0 00

11 12 11

*v k k v k k*

*t* 

(98)

(99)

11

*p*

12 0

(100)

*c12) x12* (101)

*c p*

 

150Hz which is considered as undesired distortion contaminating the first harmonic.

*v t t*

1 2

using the notation of Eq.(88), the time fuzzy model is given by:

**7.5.4 One harmonic identification** 

as:

becomes:

*A0* = (0.052, 0.0) *A11*= (1.223, 0.330) *A12*= (0.710, 0.219)

*V1*= (1.414, 0.395)

*<sup>1</sup>*= (0.166, 0.014)

parameters, respectively.

Computing the amplitude and phase:

Fig. 53. First Harmonic spread parameters.

Fig. 54. Mauserd waveform and estimated central of the first harmonic.

Electric Power Systems Harmonics - Identification and Measurements 63

Then, for estimating the first two harmonics and using Eq.(71) *v1(k)* and *v2(k)* are obtained as

( ) 1.414sin(0.08 0.16667 ) <sup>1</sup> 1.0sin(0.16 0.26667 ) ( ) 0.3sin(0.24 0.2 ) <sup>2</sup>

*k*

*v k k*

*v k k*

 

0.1sin(0.32 0.35 )

*v(k) = Ao + A11 x11(k) + A12 x12(k) + A21 x21(k) + A22 x22(k)* 

Therefore, there are nine parameters to be estimated and their estimated values are found to

*k), x21(k)=sin(0.16*

*k*

 (103)

*k).*

 

 

*k), x22(k)=cos(0.16*

 

follows:

And the linear fuzzy model is given by:

Computing the amplitude and phase:

Fig. 57. Efect of removing 2nd Harmonic

*k), x12(k)=cos(0.08*

Figures (57-59) show the crisp and fuzzy variations of v(t).

Where *x11(k)=sin(0.08*

*Ao* = (0.058, 0.0) *A11*= (1.224, 0.330) *A12*= (0.707, 0.219) *A21*= (0.669, 0.267) *A22*= (0.743, 0.307)

*V1*= (1.414, 0.395)

*<sup>1</sup>*= (0.166, 0.014) *V2*= (1.00, 0.406)

*<sup>2</sup>*= (0.266, 0.005)

be:

Fig. 55. 1 st Harmonic with its fuzzy variations.

Figure (87) shows *v(t)* together with maximum and minimum possible variation (fuzzy) v(t) can take. It can be observed that the measured v(t) is within the estimated fuzziness and that the extreme fuzzy variations is shaped up according to the measured v(t).

Fig. 56. Mauserd waveform and ist maximum and minimum fuzzy variation

#### **7.5.5 Two harmonics identification**

Next, consider identifying four harmonics at 50Hz, 100Hz, 150 Hz and 200Hz.The voltage waveform is given in Eq.(102).

$$\begin{aligned} v(t) &= 1.414\sin(100\pi t + 0.16667\pi) \\ &+ 1.0\sin(200\pi t + 0.26667\pi) \\ &+ 0.3\sin(300\pi t + 0.2\pi) \\ &+ 0.1\sin(400\pi t + 0.35\pi) \end{aligned} \tag{102}$$

Then, for estimating the first two harmonics and using Eq.(71) *v1(k)* and *v2(k)* are obtained as follows:

$$\begin{aligned} v\_1(k) &= 1.414\sin(0.08\pi k + 0.16667\pi) \\ &+ 1.0\sin(0.16\pi k + 0.26667\pi) \\ v\_2(k) &= 0.3\sin(0.24\pi k + 0.2\pi) \\ &+ 0.1\sin(0.32\pi k + 0.35\pi) \end{aligned} \tag{103}$$

And the linear fuzzy model is given by:

62 Power Quality Harmonics Analysis and Real Measurements Data

Figure (87) shows *v(t)* together with maximum and minimum possible variation (fuzzy) v(t) can take. It can be observed that the measured v(t) is within the estimated fuzziness and that

the extreme fuzzy variations is shaped up according to the measured v(t).

Fig. 56. Mauserd waveform and ist maximum and minimum fuzzy variation

*v t t*

 

Next, consider identifying four harmonics at 50Hz, 100Hz, 150 Hz and 200Hz.The voltage

( ) 1.414sin(100 0.16667 ) 1.0sin(200 0.26667 ) 0.3sin(300 0.2 ) 0.1sin(400 0.35 )

*t*

 

*t t*

 

(102)

 

> 

 

Fig. 55. 1 st Harmonic with its fuzzy variations.

**7.5.5 Two harmonics identification** 

waveform is given in Eq.(102).

$$\boldsymbol{\upsilon}(\mathbf{k}) = \underline{\mathbf{A}}\_{\boldsymbol{\upsilon}} + \underline{\mathbf{A}}\_{11} \,\boldsymbol{\upsilon}\_{11}(\mathbf{k}) + \underline{\mathbf{A}}\_{12} \,\boldsymbol{\upsilon}\_{12}(\mathbf{k}) + \underline{\mathbf{A}}\_{21} \,\boldsymbol{\upsilon}\_{21}(\mathbf{k}) + \underline{\mathbf{A}}\_{22} \,\boldsymbol{\upsilon}\_{22}(\mathbf{k})$$

Where *x11(k)=sin(0.08k), x12(k)=cos(0.08k), x21(k)=sin(0.16k), x22(k)=cos(0.16k).* Therefore, there are nine parameters to be estimated and their estimated values are found to be:


Fig. 57. Efect of removing 2nd Harmonic

Electric Power Systems Harmonics - Identification and Measurements 65

S.A. Soliman, G.S. Christensen, D.H. Kelly and K.M. El-Naggar, "A State Estimation

M.S. Saddev and M. Nagpal, "A Recursive Least Error Squares Algorithm for Power System

S.A. Soliman, K. El-Naggar and A. Al-Kandari, "Kalman Filtering Algorithm for Low

E.A. Abu Al-Feilat, I. El-Amin and M. Bettayeb, "Power System Harmonic Estimation: A Comparative Study," Electric Power Systems Research, Vol. 29, pp. 91-97, 1991. A.A. Girgis, W.B. Chang and E.B. Markram, "A Digital Recursive Measurement Scheme for

H.M. Beides and G.T. Heydt, "Dynamic State Estimation of Power System Harmonics Using

H. Ma and A.A. Girgis, "Identification and Tracking of Harmonic Sources in a Power

V.M.M. Saiz and J. Barros Gaudalupe, "Application of Kalman Filtering for Continuous

S.A. Soliman and M.E. El-Hawary, "New Dynamic Filter Based on Least Absolute Value

S.A. Soliman, K. El-Naggar, and A. Al-Kandari, "Kalman Filtering Based Algorithm for Low

A. Al-Kandari, S.A. Soliman and K. El-Naggar, "Digital Dynamic Identification of Power

A. A. Girgis and J. Qiu, Measurement of the parameters of slowly time varying high frequency transients, IEEE Trans. On Inst. And Meas., 38(6) (1989) 1057-1062. A. A. Girgis, M. C. Clapp, E. B. Makram, J. Qiu, J. G. Dalton and R. C. Satoe, Measurement

A. A. Girgis, W. Chang, and E. B. Makram, Analysis of high-impedance fault generated

S. A. Soliman, K. El-Naggar, and A. Al-Kandari, Kalman filtering based algorithm for low

S. A. Soliman and M. E. El-Hawary, Application of Kalaman filtering for online estimation

Generation, Trans. Distribution., Vol. 142, No. 1, pp. 37-44, 1005.

industrial load, IEEE Trans. Power Delivery, 5(1) (1990) 427-434

Electrical Power System Research Jr., Vol. 19, pp. 195-206, 1990.

Energy Systems, Vol. 17, No. 1, pp. 38-43, 1998.

No. 3, pp. 1008-1015, 1991.

Vol. 6, No. 3, pp. 1153-1160, 1991.

Distrib. Vol. 14, No. 1, pp. 13-20, 1998.

Research, Vol. 28, pp. 99-104, 1993.

Energy Systems, Vol. 17, No. 1, pp. 38-43, 1998.

Power and Energy Systems 17(1), (1997) 38-42.

Research 38 (1997) 113-123.

1663-1670, 1991.

1659-1665, 1998.

(1990).

Algorithm for Identification and Measurement of Power System Harmonics,"

Relaying and Measurement Applications," IEEE Trans. on Power Delivery, Vol. 6,

Frequency Power Systems Sub-harmonics Identification," Int. Jr. of Power and

On-Line Tracking of Power System Harmonics," IEEE Trans. on Power Delivery,

Kalman Filter Methodology," IEEE Trans. on Power Delivery, Vol. 6, No. 4, pp.

System Using a Kalman Filter," IEEE Trans. on Power Delivery, Vol. 11, No. 3, pp.

Real-Time Traching of Power System Harmonics," IEE Proc.-Gener. Transm.

Algorithm for On-Line Tracking of Power System Harmonics," IEE Proc.-

Frequency Power Systems Sub-harmonics Identification," Int. Jr. of Power and

System Sub-harmonics Based on Least Absolute Value," Electric Power Systems

and characterization of harmonic and high frequency distortion for a large

signals using a Kalaman filtering approach, IEE Trans. On Power Delivery, 5(4)

frequency power systems sub-harmonics identification, International Journal of

of symmetrical components for power system protection, Electric Power Systems

Fig. 58. Second Harmonic with its fuzzy variation

Fig. 59. Measurd waveform with its fuzzy variations

#### **7.5.6 Conclusion and remarks**

In this paper, the harmonics of a non-sinusoidal waveform is identified. The approach is based on fuzzy Kalman filtering. The basic idea is to identify fuzzy parameters rather than crisp parameters. The waveform is written as a linear model with fuzzy parameters from which the amplitude and phase of the harmonics are measured. Kalman filter is used to identify the fuzzy parameters. Each fuzzy parameter belongs to a triangular symmetric membership function consisting of center and spread values. Obtaining fuzzy parameters rather than crisp ones yields all possible extreme variations the parameters can take. This is useful in designing filters to filter out undesired harmonics that cause distortion.

#### **8. References**


In this paper, the harmonics of a non-sinusoidal waveform is identified. The approach is based on fuzzy Kalman filtering. The basic idea is to identify fuzzy parameters rather than crisp parameters. The waveform is written as a linear model with fuzzy parameters from which the amplitude and phase of the harmonics are measured. Kalman filter is used to identify the fuzzy parameters. Each fuzzy parameter belongs to a triangular symmetric membership function consisting of center and spread values. Obtaining fuzzy parameters rather than crisp ones yields all possible extreme variations the parameters can take. This is

J. Arrillaga, D.A. Bradley and P.S. Bodger, "Power System Harmonics," John Wiley & Sons,

IEEE Working Group on Power System Harmonics, "Power System Harmonics: An

Overview," IEEE Trans. on Power Apparatus and Systems, Vol. PAS-102, No. 8, pp.

useful in designing filters to filter out undesired harmonics that cause distortion.

Fig. 58. Second Harmonic with its fuzzy variation

Fig. 59. Measurd waveform with its fuzzy variations

**7.5.6 Conclusion and remarks** 

New York, 1985.

2455-2460, August 1983.

**8. References** 


Electric Power Systems Harmonics - Identification and Measurements 67

P. J. Moore, J. H. Allmeling and A. T. Johns, " Frequency Relaying Based on Instantaneous

T. Lobos and J. Rezmer, "Real -Time Determination of Power System Frequency", IEEE

T. S. Sidhu, and M.S. Sachdev, "An Iterative Techniques for Fast and Accurate Measurement

J. Szafran, and W.,"Power System Frequency Estimation", IEE Proc.-Genre. Trans., Distrib.,

T. S. Sidhu, " Accurate Measurement of Power System Frequency Using a Digital Signal

P. K. Dash, A. K. Pradhan, and G. Panda, " Frequency Estimation of Distorted Power System

S.A. Soliman, H. K Temraz and M. E. El-Hawary, "Estimation of Power System Voltage and

M. E. El-Hawary, "Electric Power Applications of Fuzzy Systems", IEEE Press, Piscataway,

Quanming Zhang, Huijin Liu, Hongkun Chen, Qionglin Li, and Zhenhuan Zhang," A

Walid A. Omran, Hamdy S. K. El-Goharey, Mehrdad Kazerani, and M. M. A. Salama,"

Ekrem Gursoy*,* and Dagmar Niebur*,"* Harmonic Load Identification Using Complex

Jing Yong*,* Liang Chen, and Shuangyan Chen," Modeling of Home Appliances for Power

Elcio F. de Arruda*,* Nelson Kagan*,* and Paulo F. Ribeiro*,"* Harmonic Distortion State

Mohsen Mojiri, Masoud Karimi-Ghartemani and Alireza Bakhshai*,"* Processing of

TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 2, APRIL 2010 Cong-Hui Huang, Chia-Hung Lin, and Chao-Lin Kuo,' Chaos Synchronization-Based

TRANSACTIONS ON POWER DELIVERY, VOL. 26, NO. 2, APRIL 2011

pp.1737-1742, 1996

pp.109-115, 1998.

NJ, 1998.

2009

OCTOBER 2008.

Vol. 145, No. 5, pp.578-582, 1998.

Vol. 48, No. 1 , pp.75-81, 1999.

Shams University, March 2000.

Delivery, Vol. 14, No. 3, pp.761- 766,1999.

DELIVERY, VOL. 24, NO. 1, JANUARY 2009

DELIVERY, VOL. 25, NO. 4, OCTOBER 2010

DELIVERY, VOL. 25, NO. 2, APRIL 2010

Frequency Measurement", IEEE Transaction on Power Delivery, Vol. 11, No. 4,

Transaction on Instrumentation and Measurement, Vol. 46, No. 4, pp.877-881, 1998.

of Power System Frequency", IEEE Transaction on Power Delivery, Vol. 13, No. 1,

Processing Technique", IEEE Transaction on Instrumentation and Measurement,

Signals Using Extended Complex Kalman Filter", IEEE Transaction on Power

Frequency Using the Three-Phase Voltage Measurements andTransformation", Proceeding of Middle East Power System Conference, MEPCON`2000, Cairo, Ain

Precise and Adaptive Algorithm for Interharmonics Measurement Based on Iterative DFT", IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4,

Identification and Measurement of Harmonic Pollution for Radial and Nonradial Systems", IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY

Independent Component Analysis *",*IEEE TRANSACTIONS ON POWER

Distribution System Harmonic Analysis", IEEE TRANSACTIONS ON POWER

Estimation Using an Evolutionary Strategy", IEEE TRANSACTIONS ON POWER

Harmonics and Interharmonics Using an Adaptive Notch Filter", IEEE

Detector for Power-Quality Disturbances Classification in a Power System" IEEE


S.A. Soliman, I. Helal, and A. M. Al-Kandari, Fuzzy linear regression for measurement of

L.A. Zadeh, Fuzzy sets as a basis for theory of possibility, Fussy Sets and Systems, Vol. 1, pp

H. Tanaka, S. Vejima, K. Asai, Linear regression analysis with fussy model, IEEE Trans. On

R. G. Brown, Introduction to random signal analysis and Kalman filtering, New York: John

G. F. Franklin, J. D. Powel and M. L. Workman, Digital control of dynamic system, 2nd

S. K. Tso and W. L. Chan, "Frequency and Harmonic Evaluation Using Non-Linear Least

M. M. Begovic, P. M. Djuric S. Dunlap and A. G. Phadke, "Frequency Tracking in Power

S. A. Soliman, G. S. Christensen, and K. M. El-Naggar, "A New Approximate Least

S. A. Soliman and G. S. Christensen, "Estimating of Steady State Voltage and Frequency of

P. K. Dash, D. P. Swain, A. C. Liew, and S. Rahman, "An Adaptive Linear Combiner for on-

A. Cavallini and G. C. Montanari, "A Deterministic/Stochastic Framework for Power System Harmonics Modeling", IEEE Transaction on Power Systems, Vol.114, 1996. S. Osowski, "Neural Network for Estimation of Harmonic Components in a Power System",

S. Osowski, "SVD Technique for Estimation of Harmonic Components in a Power System: a

S. A. Soliman , G. S. Christensen, D.H. Kelly, and K. M. El-Naggar, " Least Absolute Value

P. J. Moore, R. D. Carranza and A. T. Johns, " Model System Tests on a New Numeric

Relaying", Elect. Machines & Power Systems, Vol. 20, pp. 569-592, 1992. S. A. Soliman, G. S. Christensen, D. H. Kelly, and K. M. El-Naggar, "Dynamic Tracking of

Squares Techniques" Jr. of Electrical and Electronic Engineers., Australia , Vol. 14,

network in the Presence of Harmonics" IEEE Trans. on Power Delivery, Vol. 8, No.

Absolute Value Based on Dynamic Filtering for on-line Power System Frequency

the Steady State Power System Magnitude and Frequency Using Linear Kalman Filter: a Variable Frequency Model", Elect. Machines & Power Systems, Vol. 20, pp.

Power Systems from Digitized Bus Voltage Samples", Elect. Machines & Power

line Tracking of Power System Harmonics", IEEE Trans. on Power Systems,

Statistical Approach", IEE Proceedings, Gen. Trans. & Distrb., Vol. 141, No.5,

Based on Linear Programming Algorithm for Measurement of Power System Frequency from a Distorted Bus Voltage Signal", Elect. Machines & Power Systems,

Method of Power System Frequency Measurement" IEEE Transactions on Power

System, Man, and Cybernetics, Vol. 12, No. 6, pp 903-907, 1982. Timothy J. Ross, Fuzzy logic with engineering applications, McGraw Hill, 1995.

99-105.

3-28, 1978.

Wiley and Sons, 1983.

No. 2, pp. 124-132, 1994.

2, pp. 480-486, 1993.

593-611, 1992.

pp.473-479, 1994.

edition, Addison Wesley, 1990.

Systems, Vol. 19, pp. 555-576, 1991.

IEE Proceeding-C, Vol.139, No.2, pp.129-135, 1992.

Vol.114, pp. 1730-1735, 1996.

Vol. 20, No. 6, pp. 549-568, 1992.

Delivery, Vol. 11, No. 2, pp.696-701, 1996.

harmonic components in a power system, Electric Power System Research 50 (1999)


**2** 

*Canada* 

**On the Reliability of Real Measurement** 

Power quality assessment is a power engineering field that is first and foremost driven by real data measurements. All the power quality assessment applications rely on results from real data processing. Take as an example the art of harmonic filter design, which is an engineering field notoriously known for relying on simulation-based planning; in this technical assessment, data recordings are indirectly used for finding the frequency response (or *R-X* plots) of the system impedance that is/are in turn used to determine the filters'

With so much reliance on the acquired data, the quality of such has become a very sensitive issue in power quality. An imperative action is to always employ high-resolution recording equipment in any instance of power quality analysis. Nevertheless, high-resolution equipment does not guarantee data usefulness because the measured data may be inherently of very low energy in a variety of ways. Therefore, to investigate such cases and to propose methods to identify useful data were the motivations for this research. This chapter proposes methods for data selection to be used in two applications where the reliability issue is crucial: the power system impedance estimation and the interharmonic

Network impedance is power system parameter of great importance, and its accurate estimation is essential for power system analysis at fundamental and harmonic frequencies. This parameter is deemed of being of great importance for a variety of power system applications, such as evaluating the system short-circuit capacity, or defining the customer harmonic limits (Kimbark, 1971)-(IEEE Std. 519-1992). Several methods have been proposed to measure the network harmonic impedance and are available in literature. In this chapter, the transient-based approach is used to demonstrate the data selection methods. In the transient-based approach, the network impedance is conventionally calculated by using

, *Z h Vh Ih eq* (1)

**1. Introduction** 

source determination.

(Robert & Deflandre, 1997)

tuning frequencies (Kimbark, 1971).

**1.1 The network harmonic impedance estimation** 

**Data for Assessing Power** 

 **Quality Disturbances** 

Alexandre Brandao Nassif *Hydro One Inc.,Toronto, ON,* 

