**5.1 Modeling of the system sub-harmonics**

For Kalman filter application, equation (28) is the measurement equation, and we recall it here as

$$
\underline{Z}(t) = \underline{H}(t)\underline{\theta}(t) + \underline{\alpha}(t) \tag{28}
$$

If the voltage is sampled at a pre-selected rate, its samples would be obtained at equal time intervals, say *t* seconds. Then equation (26) can be written at stage *k*, *k* = 1, 2, …, *k*, where *K* is the total number of intervals, *K* = [window size in seconds/*t*] = [window size in seconds sampling frequency (Hz)].

$$z(k\Delta t) = h\_{11}(k\Delta t)\mathbf{x}\_1(k) + h\_{12}(k\Delta t)\mathbf{x}\_2(k) + \dots + h\_{16}(k\Delta t)\mathbf{x}\_6(k)\tag{50}$$

If there are m samples, equation (8.64) turns out to be a set of equations. Each equation defines the system at a certain time (*kt*).

$$\mathbf{z}\_{\mathbf{1}}\left(k\Delta t\right) = \mathbf{H}\_{\mathbf{j}}\left(k\Delta t\right)\theta\left(k\right) + w\_{\mathbf{j}}\left(k\right); \mathbf{i} = \mathbf{1}, \mathbf{2}, \dots, m\tag{51}$$

This equation can be written in vector form as:

$$z\left(k\Delta t\right) = H\left(k\Delta t\right)\theta\left(k\right) + w\left(k\right)\tag{52}$$

where

*z*(*k*) is *m* 1 measurement vector taken over the window size


Equation (52) describes the measurement system equation at time *kt*. The state space variable equation for this model may be expressed as

$$
\begin{vmatrix}
\mathbf{x}\_1(k+1) \\
\mathbf{x}\_2(k+1) \\
\mathbf{x}\_3(k+1) \\
\cdots \\
\cdots \\
\mathbf{x}\_{\mu}(k+1) \\
\end{vmatrix} = \begin{vmatrix}
\mathbf{x}\_1(k) \\
\mathbf{x}\_2(k) \\
\mathbf{x}\_3(k) \\
\cdots \\
\cdots \\
\mathbf{x}\_{\mu}(k) \\
\end{vmatrix} + w(k) \tag{53}
$$

Equation (67) can be rewritten in vector form as:

$$
\theta(k+1) = \phi(k)\theta(k) + w(k)\tag{54}
$$

where

42 Power Quality Harmonics Analysis and Real Measurements Data

It has been pointed out in the simulated results that the harmonic filter is sensitive to the deviations of frequency of the fundamental component. An algorithm to measure the power

As we said in the beginning of this chapter, the off-on switching of the power electronics equipment in power system control may produce damped transients of high and/or low frequency on the voltage and/or current waveforms. Equation (20) gives the model for such voltage waveform. The first term in this equation presents the damping inter-harmonics model, while the second term presents the harmonics that contaminated the voltage waveform including the fundamental. In this section, we explain the application of the linear dynamic Kalman filtering algorithm for measuring and identifying these interharmonics. As we said before, the identification process is split into two sub-problems. In the first problem, the harmonic contents of the waveform are identified. Once the harmonic contents of the waveform are identified, the reconstructed waveform can be obtained and the error in the waveform, which is the difference between the actual and the reconstructed waveform, can be obtained. In the second problem, this error is analyzed to identify the sub-

Finally, the final error is obtained by subtracting the combination of the harmonic and the sub-harmonic contents, the total reconstructed, from the actual waveform. It has been

For Kalman filter application, equation (28) is the measurement equation, and we recall it

*Zt Ht t t* 

If the voltage is sampled at a pre-selected rate, its samples would be obtained at equal time intervals, say *t* seconds. Then equation (26) can be written at stage *k*, *k* = 1, 2, …, *k*, where *K* is the total number of intervals, *K* = [window size in seconds/*t*] = [window size in seconds

If there are m samples, equation (8.64) turns out to be a set of equations. Each equation

*zkt Hkt k wk* 

 <sup>1</sup> *z kt H kt k w k i i* 

 

11 1 12 2 16 6 *zkt h ktx k h ktx k h ktx k* (50)

(28)

; 1,2, , *i m* (51)

(52)

shown that by identifying these sub-harmonics, the final error is reduced greatly.

**5. Power system sub-harmonics (interharmonics); dynamic case** 

system frequency should precede the harmonics filter.

**5.1 Modeling of the system sub-harmonics** 

sampling frequency (Hz)].

defines the system at a certain time (*kt*).

This equation can be written in vector form as:

*z*(*k*) is *m* 1 measurement vector taken over the window size

harmonics.

here as

where

(*k*) is *n n* state transition matrix and it is an identity matrix

*w*(*k*) is *n* 1 plant noise vector

Together equation (52) and (54) form the system dynamic model. It is worthwhile to state here that in this state space representation the time reference was chosen as a rotating time reference which caused the state transition matrix to be the identity matrix and the *H* matrix to be a time varying matrix.

Having estimated the parameter vector, the amplitude, damping constant, and the phase angle can be determined using equations (30) to (32), at any step
