6.1 R-L filter

Applying KVL at the converter output gives

$$V\_{\varepsilon} - V\_{\text{POC}} = I\_{\text{f\%}} \left( \mathbf{R}\_{f\varepsilon} + \mathbf{S} \mathbf{L}\_{f\varepsilon} \right) \tag{19}$$

Converter output is controlled by output current

$$\mathbf{V}\_c = f\left(\mathbf{I}\_{\hat{\mathbb{f}}\hat{\mathbb{g}}}\right) \tag{20}$$

With simple PI controller, the output becomes

$$V\_c = \left(k\_P + \frac{k\_i}{s}\right) \left(I\_{f\text{\%}}\right) \tag{21}$$

Putting Eq. (20) into Eq. (18)

$$\left(\mathbf{k}\_P + \frac{\mathbf{k}\_i}{\mathbf{s}}\right) \left(\mathbf{I}\_{\mathbf{f}\mathbf{g}}\right) - \mathbf{V}\_{\rm POC} = \mathbf{I}\_{\mathbf{f}\mathbf{g}} \left(\mathbf{R}\_{\mathbf{f}\mathbf{c}} + \mathbf{S} \mathbf{L}\_{\mathbf{f}\mathbf{c}}\right) \tag{22}$$

$$I\left(\left(k\_P + \frac{k\_i}{s}\right)I\_{\text{f\text{g}}}\right) - I\_{\text{f\text{g}}}\left(R\_{\text{f\text{c}}} + \text{SL}\_{\text{f\text{c}}}\right) = V\_{\text{POC}}\tag{23}$$

$$Z\_o = \frac{\left(\mathcal{S}^2 L\_{fc} + \mathcal{S}\left(R\_{fc} - k\_p\right) - k\_i\right)}{\mathcal{S}} \tag{24}$$

$$Z\_o = \frac{(\mathbb{S} + a)(\mathbb{S} + \beta)}{\mathbb{S}} \text{ (considering } a < \beta\text{)}\tag{25}$$

6.2 L-C filter

Figure 11.

Figure 10.

Frequency response of impedance with RL filter without PI controller.

DOI: http://dx.doi.org/10.5772/intechopen.89167

Frequency response of impedance with RL filter with PI controller.

(Figure 12).

Figure 12. L-C filter.

77

LC filter is used in converters for industrial applications like variable frequency drive (VFD) and uninterrupted power supply (UPS). It is simple in construction and relatively less costly. The analysis of LC filter with PI controller is given here

Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design

As per the standard practice, the value of L is selected such that its impedance at

fundamental frequency should not drop by more than 3% of rated voltage. The capacitive reactance offers 1/5th of the fundamental inductive reactance at switching frequency of converter (around 3–4 kHz) to absorb harmonics effectively. Based on these criteria, LC filter is widely designed. The frequency response

of LC filter with and without PI controller is given in Figures 13 and 14,

Equation (25) shows that there are two zeros and one pole. First, the output impedance decreases at �20 dB per decade up to first zero at α. At α, the impedance response becomes flat, and at β the impedance starts increasing at 20 dB per decade (Figure 9). So, the response of integrated filter becomes similar to that of series resonance filter. The selection of α and β depends on the parameter selection of Lfc, Rfc, kP and ki. Bode plot for RL filter without and with PI current controller is given in Figures 10 and 11, respectively.

Figure 9. Grid-connected inverter with filter.

Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design DOI: http://dx.doi.org/10.5772/intechopen.89167

Figure 10. Frequency response of impedance with RL filter without PI controller.

Figure 11. Frequency response of impedance with RL filter with PI controller.

### 6.2 L-C filter

of LCL filter. Hence, the design of LCL filter is a trade-off between robustness and damping of resonance. The effective impedance with simple PI controller is

Vc � VPOC ¼ Ifg Rfc þ SLfc

Vc ¼ f Ifg

ki s 

� VPOC <sup>¼</sup> Ifg Rfc <sup>þ</sup> SLfc

� Ifg Rfc þ SLfc

 � ki 

Lfc þ S Rfc � kp

Equation (25) shows that there are two zeros and one pole. First, the output impedance decreases at �20 dB per decade up to first zero at α. At α, the impedance response becomes flat, and at β the impedance starts increasing at 20 dB per decade (Figure 9). So, the response of integrated filter becomes similar to that of series resonance filter. The selection of α and β depends on the parameter selection of Lfc, Rfc, kP and ki. Bode plot for RL filter without and with PI current controller is given

Ifg

Vc ¼ kP þ

Ifg

Ifg

(19)

(20)

(21)

(22)

<sup>¼</sup> VPOC (23)

<sup>S</sup> (24)

<sup>S</sup> ð Þ considering <sup>α</sup><sup>&</sup>lt; <sup>β</sup> (25)

explained here with RL filter topology.

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Putting Eq. (20) into Eq. (18)

in Figures 10 and 11, respectively.

Figure 9.

76

Grid-connected inverter with filter.

Applying KVL at the converter output gives

Converter output is controlled by output current

With simple PI controller, the output becomes

kP þ ki s 

> kP þ ki s

Zo <sup>¼</sup> <sup>S</sup><sup>2</sup>

Zo <sup>¼</sup> ð Þ <sup>S</sup> <sup>þ</sup> <sup>α</sup> ð Þ <sup>S</sup> <sup>þ</sup> <sup>β</sup>

6.1 R-L filter

LC filter is used in converters for industrial applications like variable frequency drive (VFD) and uninterrupted power supply (UPS). It is simple in construction and relatively less costly. The analysis of LC filter with PI controller is given here (Figure 12).

As per the standard practice, the value of L is selected such that its impedance at fundamental frequency should not drop by more than 3% of rated voltage. The capacitive reactance offers 1/5th of the fundamental inductive reactance at switching frequency of converter (around 3–4 kHz) to absorb harmonics effectively. Based on these criteria, LC filter is widely designed. The frequency response of LC filter with and without PI controller is given in Figures 13 and 14,

Figure 12. L-C filter.

The converter output voltage Vc is a function of the grid current Ig. Then

Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design

Vc ¼ kp þ

Putting this in Eq. (26) and simplifying further give

<sup>Z</sup><sup>0</sup> <sup>¼</sup> Vg �Ig <sup>¼</sup> <sup>a</sup>4<sup>s</sup>

DOI: http://dx.doi.org/10.5772/intechopen.89167

where

Figure 16.

79

Frequency response of LCL filter with PI controller.

Figure 15. L-C-L filter.

Vc ¼ f Ig

<sup>4</sup> <sup>þ</sup> <sup>a</sup>3<sup>s</sup>

a<sup>4</sup> ¼ LgCf kp a<sup>3</sup> ¼ LgCf ki þ RCfLgkp a<sup>2</sup> ¼ RLgCf ki þ Lckp � Cf a<sup>1</sup> ¼ Lcki a<sup>0</sup> ¼ 0 b<sup>3</sup> ¼ LcCf kp b<sup>2</sup> ¼ LcCf ki

ki s

<sup>3</sup> <sup>þ</sup> <sup>a</sup>2<sup>s</sup>

b3s<sup>3</sup> þ b2s<sup>2</sup> þ b1s<sup>1</sup> þ b<sup>0</sup>

<sup>2</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

(27)

Ig (28)

<sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>0</sup>

(29)

Figure 13. Frequency response of LC filter without PI controller.

Figure 14. Frequency response of LC filter with PI controller.

respectively. It is clear from the difference in bode plot that PI controller changes the frequency response.

#### 6.3 L-C-L filter

Initially LC filter was used for converter applications, but grid-connected inverter has unique requirements that LC filter may not provide. Properly designed LCL filter may overcome the drawbacks of LC filter (Figure 15).

Applying KVL,

$$V\_c = \left(\frac{\left(V\_{\text{g}} + L\_{\text{g}}sI\_{\text{g}}\right)}{\frac{1}{sC\_{\text{f}}} + R} + I\_{\text{g}}\right)sL\_c \tag{26}$$

Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design DOI: http://dx.doi.org/10.5772/intechopen.89167

Figure 15. L-C-L filter.

The converter output voltage Vc is a function of the grid current Ig. Then

$$\mathbf{V}\_{\mathfrak{c}} = f\left(\mathbf{I}\_{\mathfrak{g}}\right) \tag{27}$$

$$V\_c = \left(k\_p + \frac{k\_i}{s}\right)I\_{\rm g} \tag{28}$$

Putting this in Eq. (26) and simplifying further give

$$Z\_0 = \frac{V\_{\rm g}}{(-I\_{\rm g})} = \frac{a\_4s^4 + a\_3s^3 + a\_2s^2 + a\_1s^1 + a\_0}{b\_3s^3 + b\_2s^2 + b\_1s^1 + b\_0} \tag{29}$$

where

$$a\_4 = L\_\text{g} \mathbf{C}\_f k\_p$$

$$a\_3 = L\_\text{g} \mathbf{C}\_f k\_i + R \mathbf{C}\_f L\_\text{g} k\_p$$

$$a\_2 = R L\_\text{g} \mathbf{C}\_f k\_i + L\_\text{c} k\_p - \mathbf{C}\_f$$

$$a\_1 = L\_\text{c} k\_i$$

$$a\_0 = \mathbf{0}$$

$$b\_3 = L\_\text{c} \mathbf{C}\_f k\_p$$

$$b\_2 = L\_\text{c} \mathbf{C}\_f k\_i$$

Figure 16. Frequency response of LCL filter with PI controller.

respectively. It is clear from the difference in bode plot that PI controller changes

Initially LC filter was used for converter applications, but grid-connected inverter has unique requirements that LC filter may not provide. Properly designed

> � � 1 sCf

!

<sup>þ</sup> <sup>R</sup> <sup>þ</sup> Ig

sLc (26)

LCL filter may overcome the drawbacks of LC filter (Figure 15).

Vc <sup>¼</sup> Vg <sup>þ</sup> LgsIg

the frequency response.

Frequency response of LC filter with PI controller.

Applying KVL,

6.3 L-C-L filter

78

Figure 14.

Figure 13.

Frequency response of LC filter without PI controller.

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$$b\_1 = \mathbf{0}$$

$$b\_0 = \mathbf{0}$$

Figure 16 shows the frequency response of LCL filter together with PI controller. It is clear from Eq. (29) and from Figure 16 that the PI controller increases the order of filter, so the frequency response of passive filter gets changed by the controller action.
