2.2. Mathematical model of amplifying the harmonic conditions by reactive power compensation

Switching shunt equivalent impedance in a network containing harmonics, conduce to more or less influence on the harmonic conditions, depending on the character and the value of the impedance.

Switching a shunt capacitor bank in order to improve the power factor in a bus of a distribution network containing harmonics can conduce to the amplification of the harmonic conditions. This is possible only when the harmonic currents are flowing in the network, with the frequencies close to the parallel resonance frequency. The resonance frequency occurs between the capacitance of the capacitor bank and the equivalent inductance of the network.

Figure 1 presents the equivalent schema for the k range harmonic, for the cases before and after the shunt capacitor impedance switching (ZkCB) in the point of common coupling (PCC).

The notations used in Figure 1 are:

Ik is the harmonic current provided by an equivalent fictive current source. It corresponds to all the k range harmonic current sources existing in the network, and it produces the harmonic impedance corresponding to PCC, Zknet, and the harmonic voltage drop Uk.

The harmonic equivalent current source for k range (Ik) does not change its value after connecting the capacitor bank (CB) represented by the equivalent reactance, ZkCB.

In order to determinate the mathematical expressions that will be used, the harmonic impedance "seen" in the PCC before the installation of ZkCB is separated in two components:

Zkload is the transversal impedance of the load existing before the disturbance, usually a load impedance and

Zknet is the equivalent impedance of the rest of the network.

The harmonic voltage of k range in PCC is:

$$
\underline{\mathbf{L}}\_k = \underline{\mathbf{I}}\_k \cdot \underline{\mathbf{Z}}\_k = \underline{\mathbf{I}}\_{klad} \cdot \underline{\mathbf{Z}}\_{klad} \tag{11}
$$

in which: Zk ¼ Zknet ˜ ˜ ˜ Zkload <sup>¼</sup> Zknet�Zkload . ZknetþZkload

Results of the component of the harmonic current distributed through Zkload:

$$
\underline{I}\_{kload} = \underline{d}\_{lond} \cdot \underline{I}\_k \tag{12}
$$

where

$$\underline{\mathbf{d}}\_{\text{kload}} = \frac{\underline{\mathbf{Z}\_k}}{\underline{\mathbf{Z}\_{kload}}} \tag{13}$$

We can also write:

$$\underline{I}\_k = \underline{I}\_{k \text{load}} \cdot \frac{\underline{Z}\_{k \text{load}}}{\underline{Z}\_k} \tag{14}$$

Figure 1. The equivalent circuit: (a) before the capacitor connection; (b) after the capacitor connection.

After the shunt capacitor bank connecting, ZkCB, supposing that the harmonic current sources and the harmonic impedance of the network do not change, we can write:

$$
\underline{\mathbf{L}}\_{k}^{\*} = \underline{\mathbf{I}}\_{k} \cdot \underline{\mathbf{Z}}\_{k}^{\*} = \underline{I}\_{kload}^{\*} \cdot \underline{\mathbf{Z}}\_{kload}^{\*} \tag{15}
$$

Consulting expressions for the harmonic impedance corresponding to PCC after the reactive power compensation (ZkCB switching):

$$\underline{\mathbf{Z}}\_{k}^{\*} = \underline{\mathbf{Z}}\_{k} \left| \underline{\mathbf{Z}}\_{k \text{CB}} = \frac{\underline{\mathbf{Z}}\_{k} \cdot \underline{\mathbf{Z}}\_{k \text{CB}}}{\underline{\mathbf{Z}}\_{k} + \underline{\mathbf{Z}}\_{k \text{CB}}} \right. \tag{16}$$

The total transversal impedance:

$$\underline{\mathbf{Z}}\_{\text{kload}}^{\*} = \underline{\mathbf{Z}}\_{\text{kload}} \left| \underline{\mathbf{Z}}\_{\text{kCB}} = \frac{\underline{\mathbf{Z}}\_{\text{kload}} \cdot \underline{\mathbf{Z}}\_{\text{kCB}}}{\underline{\mathbf{Z}}\_{\text{kload}} + \underline{\mathbf{Z}}\_{\text{kCB}}} \right. \tag{17}$$

From (15), we have the following result:

$$I\_{kload}^\* = I\_k \cdot \frac{\underline{Z\_k^\*}}{\underline{Z\_{kload}^\*}} \tag{18}$$

Using the expression of Ik for the regimen before the reactive power compensation, ZkCB (expression (14)):

$$\underline{I}\_{\text{kload}}^{\*} = \underline{I}\_{\text{kload}} \cdot \frac{\underline{Z}\_{\text{kload}}}{\underline{Z}\_{k}} \cdot \frac{\underline{Z}\_{k}^{\*}}{\underline{Z}\_{\text{kload}}^{\*}} \tag{19}$$

or

$$
\underline{I}\_{kl\alpha d}^\* = \underline{F}\_{l\\_d} \cdot \underline{I}\_{kl\alpha d} \tag{20}
$$

where the notation FIload represents the load current amplification factor, pursuant to the reactive power compensation, ZkCB:

$$\underline{F}\_{kl\_{\text{kud}}} = \frac{\underline{Z}\_{kl\text{kud}}}{\underline{Z}\_{k}} \cdot \frac{\underline{Z}\_{k}^{\*}}{\underline{Z}\_{kl\text{kud}}^{\*}}\tag{21}$$

Using expressions (16) and (17) in (21), we obtain:

$$\underline{F}\_{kl\_{\text{kul}}} = \frac{\underline{\mathbf{Z}\_{k\text{kul}}} + \underline{\mathbf{Z}\_{k\text{C}}}}{\underline{\mathbf{Z}}\_{k} + \underline{\mathbf{Z}\_{k\text{C}}}} \tag{22}$$

So, knowing the impedances Zk and ZkCB from the initial regimen, and the initial load harmonic current (Ikload), we can find the final load current I ∗ kload and also the harmonic voltage in PCC after the capacitor bank connection ZkCB:

MatLab Simulink Modeling for Network-Harmonic Impedance Assessment: Useful Tool to Estimate Harmonics… 11 http://dx.doi.org/10.5772/intechopen.76461

$$
\underline{\mathbf{L}}\_k^\* = \underline{\mathbf{L}}\_{k \text{load}}^\* \cdot \underline{\mathbf{Z}}\_{k \text{load}}^\* \tag{23}
$$

The harmonic voltage amplification factor in PCC is calculated by the following expression:

$$\underline{\mathbf{L}}\_{\mathrm{LI}\_{k}} = \frac{\underline{\mathbf{L}}\_{k}^{\*}}{\underline{\mathbf{L}}\_{k}} = \frac{\underline{\mathbf{Z}}\_{k}^{\*}}{\underline{\mathbf{Z}}\_{k}} = \frac{\frac{\underline{\mathbf{Z}}\_{k} \cdot \underline{\mathbf{Z}}\_{k \text{CB}}}{\underline{\mathbf{Z}}\_{k} + \underline{\mathbf{Z}}\_{\mathrm{CB}}}}{\underline{\mathbf{Z}}\_{k}} = \frac{\underline{\mathbf{Z}}\_{k \text{CB}}}{\underline{\mathbf{Z}}\_{k} + \underline{\mathbf{Z}}\_{k \text{CB}}} \tag{24}$$

and

$$F\_{L\_k} = \frac{|\underline{L}\_k^\*|}{|\underline{L}\_k|} = \frac{|\underline{Z}\_{kCB}|}{|\underline{Z}\_k + \underline{Z}\_{kCB}|} \tag{25}$$

The total harmonic distortion (THD) of the PCC voltage after connecting to the capacitor bank must be lower than the highest value:

$$THD\_{\mathcal{U}^\*} = \frac{\sqrt{\sum\_{k=2}^n \mathcal{U}\_k^{\*2}}}{\mathcal{U}\_1} = \frac{\sqrt{\sum\_{k=2}^n \left(F\_{\mathcal{U}\_k}, \mathcal{U}\_k\right)^2}}{\mathcal{U}\_1} \le THD\_{\mathcal{U}\text{max}}\tag{26}$$

In addition, the voltage in PCC must be lower than the highest value imposed by the capacitor bank manufacturer, usually 10% over the nominal voltage.

Knowing the voltage bus in PCC and the capacitor bank reactance, we can also check if the current trough in the capacitor bank is lower than the highest value imposed by the capacitor bank manufacturer, usually 30% over the nominal current:

$$I\_{\rm CB} = \sqrt{\sum\_{k=1}^{n} I\_{k\rm CB}^2} \le I\_{\rm CB\max} \tag{27}$$
