7.1.2 Type II filter

In Type II LCL filter (Figure 19), the damping can be achieved by simply adding a parallel combination of resistor and inductor in series with capacitor C. The effect of inductor is investigated using Nyquist plot. The value of Rf = 10 Ohm and Lf is varied from 50 to 500 μH. The effect of increase in Lf observed using Nyquist plot is given in Figure 20.

Impedance of Type II LCL filter with damping inductor and resistance is given by

$$Z\_o = \frac{a\_4s^4 + a\_3s^3 + a\_2s^2 + a\_1s^1 + a\_0s^0}{b\_3s^3 + b\_2s^2 + b\_1s^1 + b\_0s^0} \tag{33}$$

b<sup>3</sup> ¼ CLf L<sup>2</sup> þ Lg

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

b<sup>2</sup> ¼ CLfRf þ CLfRg þ CRf L<sup>2</sup> þ Lg

b<sup>1</sup> ¼ Lf þ CRfRg b<sup>0</sup> ¼ Rf

From the bode graph, it is observed that the gain margin is 70–100 dB for various values of Lf. Similarly, the phase margin is 270°. As the value of Lf is

In Type III LCL filter, the damping can be achieved by simply adding a parallel

combination of resistor and inductor in series with capacitor C. The effect of

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

a<sup>4</sup> ¼ CCfL<sup>1</sup> Lg þ L<sup>2</sup>

a<sup>2</sup> ¼ L1C þ Cf L<sup>2</sup> þ Lg

b<sup>3</sup> ¼ CCf L<sup>2</sup> þ Lg

Impedance of Type III LCL filter with damping inductor and resistance is

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

a<sup>3</sup> ¼ L1CCfRg þ CL1CfRf þ CfLf L<sup>2</sup> þ Lg

a<sup>1</sup> ¼ L1Cf þ CfRgRf þ L<sup>2</sup> þ Lg

a<sup>0</sup> ¼ Rg

b<sup>2</sup> ¼ CCfRg þ CCfRf b<sup>1</sup> ¼ Cf b<sup>0</sup> ¼ 0

Impedance shows two resonance points: first is parallel resonance and second is

series resonance (Figure 22). At first resonance point, the output impedance increases, and at series resistance it is at the minimum value. The phase margin of filter is around 210°. This ensures the stability of filter. The resonant frequency will shift with change in grid resistance, so the damping effect is difficult to predict.

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

<sup>þ</sup> CL1LfCf

Rf <sup>þ</sup> CfLfRg

<sup>þ</sup> CLfCf

<sup>1</sup> <sup>þ</sup> <sup>a</sup>0<sup>s</sup> 0 <sup>b</sup>3s<sup>3</sup> <sup>þ</sup> <sup>b</sup>2s<sup>2</sup> <sup>þ</sup> <sup>b</sup>1s<sup>1</sup> <sup>þ</sup> <sup>b</sup>0s<sup>0</sup> (34)

increased, there will be reduction in the gain margin.

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

inductor is investigated using Nyquist plot (Figure 21).

Zo <sup>¼</sup> <sup>a</sup>4<sup>s</sup>

7.1.3 Type III filter

given by

where

Figure 21. Type III LCL filter.

83

where

$$a\_4 = \mathrm{CL}\_1 \mathrm{L}\_f \left(\mathrm{L}\_\mathrm{g} + \mathrm{L}\_2\right)$$

$$a\_3 = \mathrm{L}\_1 \mathrm{CL}\_f \mathrm{R}\_f + \mathrm{CL}\_1 \mathrm{L}\_f \mathrm{R}\_\mathrm{g} + \mathrm{CR}\_f \left(\mathrm{L}\_2 + \mathrm{L}\_\mathrm{g}\right) + \mathrm{CR}\_f \mathrm{L}\_f \left(\mathrm{L}\_2 + \mathrm{L}\_\mathrm{g}\right)$$

$$a\_2 = \mathrm{L}\_1 \mathrm{L}\_f + \mathrm{CL}\_1 \mathrm{R}\_f \mathrm{R}\_\mathrm{g} + \mathrm{CL}\_f \mathrm{R}\_f \mathrm{R}\_\mathrm{g} + \mathrm{L}\_f \left(\mathrm{L}\_2 + \mathrm{L}\_\mathrm{g}\right)$$

$$a\_1 = \mathrm{L}\_1 \mathrm{R}\_f + \mathrm{L}\_f \mathrm{R}\_\mathrm{g} + \mathrm{R}\_f \left(\mathrm{L}\_2 + \mathrm{L}\_\mathrm{g}\right)$$

$$a\_0 = \mathrm{R}\_f \mathrm{R}\_\mathrm{g}$$

Figure 20. Nyquist plot of Type II LCL filter.

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

$$b\_3 = CL\_f \left(L\_2 + L\_\mathfrak{g}\right)$$

$$b\_2 = CL\_f R\_f + CL\_f R\_\mathfrak{g} + CR\_f \left(L\_2 + L\_\mathfrak{g}\right)$$

$$b\_1 = L\_f + CR\_f R\_\mathfrak{g}$$

$$b\_0 = R\_f$$

From the bode graph, it is observed that the gain margin is 70–100 dB for various values of Lf. Similarly, the phase margin is 270°. As the value of Lf is increased, there will be reduction in the gain margin.

#### 7.1.3 Type III filter

The gain margin is �40 dB and phase margin is around 120°. So, as per the Nyquist

In Type II LCL filter (Figure 19), the damping can be achieved by simply adding a parallel combination of resistor and inductor in series with capacitor C. The effect of inductor is investigated using Nyquist plot. The value of Rf = 10 Ohm and Lf is varied from 50 to 500 μH. The effect of increase in Lf observed using Nyquist plot is

Impedance of Type II LCL filter with damping inductor and resistance is

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

a<sup>4</sup> ¼ CL1Lf Lg þ L<sup>2</sup>

a<sup>2</sup> ¼ L1Lf þ CL1RfRg þ CLfRfRg þ Lf L<sup>2</sup> þ Lg

a<sup>1</sup> ¼ L1Rf þ LfRg þ Rf L<sup>2</sup> þ Lg

a<sup>0</sup> ¼ RfRg

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

<sup>1</sup> <sup>þ</sup> <sup>a</sup>0<sup>s</sup> 0 <sup>b</sup>3s<sup>3</sup> <sup>þ</sup> <sup>b</sup>2s<sup>2</sup> <sup>þ</sup> <sup>b</sup>1s<sup>1</sup> <sup>þ</sup> <sup>b</sup>0s<sup>0</sup> (33)

<sup>þ</sup> CRfLf <sup>L</sup><sup>2</sup> <sup>þ</sup> Lg

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

a<sup>3</sup> ¼ L1CLfRf þ CL1LfRg þ CRf L<sup>2</sup> þ Lg

Zo <sup>¼</sup> <sup>a</sup>4<sup>s</sup>

criterion, the filter response is very stable.

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7.1.2 Type II filter

given in Figure 20.

given by

where

Figure 19. Type II LCL filter.

Figure 20.

82

Nyquist plot of Type II LCL filter.

In Type III LCL filter, the damping can be achieved by simply adding a parallel combination of resistor and inductor in series with capacitor C. The effect of inductor is investigated using Nyquist plot (Figure 21).

Impedance of Type III LCL filter with damping inductor and resistance is given by

$$Z\_o = \frac{a\_4s^4 + a\_3s^3 + a\_2s^2 + a\_1s^1 + a\_0s^0}{b\_3s^3 + b\_2s^2 + b\_1s^1 + b\_0s^0} \tag{34}$$

where

$$a\_4 = \mathbf{C} \mathbf{C}\_f \mathbf{L}\_1 \left(\mathbf{L}\_\mathbf{g} + \mathbf{L}\_2\right) + \mathbf{C} \mathbf{L}\_1 \mathbf{L}\_f \mathbf{C}\_f$$

$$a\_3 = \mathbf{L}\_1 \mathbf{C} \mathbf{C}\_f \mathbf{R}\_\mathbf{g} + \mathbf{C} \mathbf{L}\_1 \mathbf{C}\_f \mathbf{R}\_f + \mathbf{C}\_f \mathbf{L}\_f \left(\mathbf{L}\_2 + \mathbf{L}\_\mathbf{g}\right)$$

$$a\_2 = \mathbf{L}\_1 \mathbf{C} + \mathbf{C}\_f \left(\mathbf{L}\_2 + \mathbf{L}\_\mathbf{g}\right) \mathbf{R}\_f + \mathbf{C}\_f \mathbf{L}\_f \mathbf{R}\_\mathbf{g}$$

$$a\_1 = \mathbf{L}\_1 \mathbf{C}\_f + \mathbf{C}\_f \mathbf{R}\_\mathbf{g} \mathbf{R}\_f + \left(\mathbf{L}\_2 + \mathbf{L}\_\mathbf{g}\right)$$

$$a\_0 = \mathbf{R}\_\mathbf{g}$$

$$b\_3 = \mathbf{C} \mathbf{C}\_f \left(\mathbf{L}\_2 + \mathbf{L}\_\mathbf{g}\right) + \mathbf{C} \mathbf{L}\_f \mathbf{C}\_f$$

$$b\_2 = \mathbf{C} \mathbf{C}\_f \mathbf{R}\_\mathbf{g} + \mathbf{C} \mathbf{C}\_f \mathbf{R}\_f$$

$$b\_1 = \mathbf{C}\_f$$

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

Impedance shows two resonance points: first is parallel resonance and second is series resonance (Figure 22). At first resonance point, the output impedance increases, and at series resistance it is at the minimum value. The phase margin of filter is around 210°. This ensures the stability of filter. The resonant frequency will shift with change in grid resistance, so the damping effect is difficult to predict.

Figure 21. Type III LCL filter.

Figure 22. Nyquist plot of Type III LCL filter.

#### 7.1.4 Type IV filter

In Type IV LCL filter, the damping can be achieved by simply adding a parallel combination of resistor and capacitor Cf. The effect of resistor value is investigated using Nyquist plot (Figure 23).

Impedance of Type IV LCL filter with damping resistor Rf in series with capacitor Cf is given by

$$Z\_o = \frac{a\_4s^4 + a\_3s^3 + a\_2s^2 + a\_1s^1 + a\_0s^0}{b\_3s^3 + b\_2s^2 + b\_1s^1 + b\_0s^0} \tag{35}$$

b<sup>2</sup> ¼ CCfRg þ CCfRf b<sup>1</sup> ¼ Cf þ C b<sup>0</sup> ¼ 0

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

The response of Type IV filter is similar to the tuned filter. The notch in frequency response of impedance is observed at the resonant frequency (Figure 24). This notch can be damped by putting higher value of resistor in series with the capacitor Cf. The phase of impedance sharply changes from �90 to 90°, which means the nature of impedance turns from capacitive to inductive. The ratio of Cf/C decides the damping effectiveness. The larger the Cf/C ratio, the larger will be the damping. Also, with increasing value of Cf/C ratio, the loss in resistor Rf is also

In Type V LCL filter, the damping can be achieved by simply adding a parallel combination of resistor and inductor with capacitor C. The effect of inductor is

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

<sup>þ</sup> LfCfRfRg

<sup>þ</sup> LfRg

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

<sup>b</sup>4s<sup>4</sup> <sup>þ</sup> <sup>b</sup>3s<sup>3</sup> <sup>þ</sup> <sup>b</sup>2s<sup>2</sup> <sup>þ</sup> <sup>b</sup>1s<sup>1</sup> <sup>þ</sup> <sup>b</sup>0s<sup>0</sup> (36)

<sup>þ</sup> CCfL1LfRg

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

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

a<sup>5</sup> ¼ CCfL1Lf Lg þ L<sup>2</sup>

a<sup>3</sup> ¼ CLfL<sup>1</sup> þ CCfL1Rf þ CfLfRf Lg þ L<sup>2</sup>

a<sup>0</sup> ¼ RfRg

increases as more and more current tends to flow in it [16–18].

<sup>5</sup> <sup>þ</sup> <sup>a</sup>4<sup>s</sup>

a<sup>4</sup> ¼ CCfL1LfRf þ CCfL1Rf Lg þ L<sup>2</sup>

a<sup>2</sup> ¼ L1CRf þ Lf L<sup>2</sup> þ Lg

a<sup>1</sup> ¼ L<sup>1</sup> þ Rf L<sup>2</sup> þ Lg

investigated using Nyquist plot (Figure 25). Impedance of Type V LCL filter is given by

Zo <sup>¼</sup> <sup>a</sup>5<sup>s</sup>

7.1.5 Type V filter

Figure 24.

Nyquist plot of Type IV LCL filter.

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

where

85

where

$$a\_4 = \mathbf{CC}\_f L\_1 (L\_\xi + L\_2)$$

$$a\_3 = \mathbf{CC}\_f L\_1 R\_f + \mathbf{CC}\_f R\_\xi L\_1$$

$$a\_2 = L\_1 \mathbf{C}\_f + L\_1 \mathbf{C} + C\_f R\_f \left(L\_2 + L\_\xi\right) + \mathbf{C}\_f \left(L\_2 + L\_\xi\right)$$

$$a\_1 = C\_f R\_f R\_\xi$$

$$a\_0 = R\_\xi$$

$$b\_3 = \mathbf{CC}\_f \left(L\_2 + L\_\xi\right)$$

$$\begin{array}{c|c|c|c|c|c} \hline \text{1} & \text{1} & \text{1} & \text{1} & \text{1} & \text{1} & \text{1} \\ \hline \text{1} & \text{1} & \text{1} & \text{1} & \text{1} & \text{1} \\ \hline \text{1} & \text{1} & \text{1} & \text{1} & \text{1} & \text{1} \\ \hline \end{array}$$

Figure 23. Type IV LCL filter.

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

Figure 24. Nyquist plot of Type IV LCL filter.

$$b\_2 = \mathbf{C} \mathbf{C}\_f \mathbf{R}\_g + \mathbf{C} \mathbf{C}\_f \mathbf{R}\_f$$

$$b\_1 = \mathbf{C}\_f + \mathbf{C}$$

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

The response of Type IV filter is similar to the tuned filter. The notch in frequency response of impedance is observed at the resonant frequency (Figure 24). This notch can be damped by putting higher value of resistor in series with the capacitor Cf. The phase of impedance sharply changes from �90 to 90°, which means the nature of impedance turns from capacitive to inductive. The ratio of Cf/C decides the damping effectiveness. The larger the Cf/C ratio, the larger will be the damping. Also, with increasing value of Cf/C ratio, the loss in resistor Rf is also increases as more and more current tends to flow in it [16–18].

#### 7.1.5 Type V filter

7.1.4 Type IV filter

Nyquist plot of Type III LCL filter.

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Figure 22.

itor Cf is given by

where

Figure 23. Type IV LCL filter.

84

using Nyquist plot (Figure 23).

Zo <sup>¼</sup> <sup>a</sup>4<sup>s</sup>

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

a<sup>2</sup> ¼ L1Cf þ L1C þ CfRf L<sup>2</sup> þ Lg

In Type IV LCL filter, the damping can be achieved by simply adding a parallel combination of resistor and capacitor Cf. The effect of resistor value is investigated

Impedance of Type IV LCL filter with damping resistor Rf in series with capac-

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

a<sup>4</sup> ¼ CCfL<sup>1</sup> Lg þ L<sup>2</sup>

a<sup>3</sup> ¼ CCfL1Rf þ CCfRgL<sup>1</sup>

a<sup>1</sup> ¼ CfRfRg a<sup>0</sup> ¼ Rg b<sup>3</sup> ¼ CCf L<sup>2</sup> þ Lg

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

<sup>1</sup> <sup>þ</sup> <sup>a</sup>0<sup>s</sup> 0 <sup>b</sup>3s<sup>3</sup> <sup>þ</sup> <sup>b</sup>2s<sup>2</sup> <sup>þ</sup> <sup>b</sup>1s<sup>1</sup> <sup>þ</sup> <sup>b</sup>0s<sup>0</sup> (35)

<sup>þ</sup> Cf <sup>L</sup><sup>2</sup> <sup>þ</sup> Lg

In Type V LCL filter, the damping can be achieved by simply adding a parallel combination of resistor and inductor with capacitor C. The effect of inductor is investigated using Nyquist plot (Figure 25).

Impedance of Type V LCL filter is given by

$$Z\_o = \frac{a\_5s^5 + a\_4s^4 + a\_3s^3 + a\_2s^2 + a\_1s^1 + a\_0s^0}{b\_4s^4 + b\_3s^3 + b\_2s^2 + b\_1s^1 + b\_0s^0} \tag{36}$$

where

$$a\_5 = \mathbf{C} \mathbf{C}\_f \mathbf{L}\_1 \mathbf{L}\_f \left(\mathbf{L}\_\mathbf{g} + \mathbf{L}\_2\right)$$

$$a\_4 = \mathbf{C} \mathbf{C}\_f \mathbf{L}\_1 \mathbf{L}\_f \mathbf{R}\_f + \mathbf{C} \mathbf{C}\_f \mathbf{L}\_1 \mathbf{R}\_f \left(\mathbf{L}\_\mathbf{g} + \mathbf{L}\_2\right) + \mathbf{C} \mathbf{C}\_f \mathbf{L}\_1 \mathbf{L}\_f \mathbf{R}\_g$$

$$a\_3 = \mathbf{C} \mathbf{L}\_f \mathbf{L}\_1 + \mathbf{C} \mathbf{C}\_f \mathbf{L}\_1 \mathbf{R}\_f + \mathbf{C}\_f \mathbf{L}\_f \mathbf{R}\_f \left(\mathbf{L}\_\mathbf{g} + \mathbf{L}\_2\right)$$

$$a\_2 = \mathbf{L}\_1 \mathbf{C} \mathbf{R}\_f + \mathbf{L}\_f \left(\mathbf{L}\_2 + \mathbf{L}\_\mathbf{g}\right) + \mathbf{L}\_f \mathbf{C}\_f \mathbf{R}\_f \mathbf{R}\_g$$

$$a\_1 = \mathbf{L}\_1 + \mathbf{R}\_f \left(\mathbf{L}\_2 + \mathbf{L}\_\mathbf{g}\right) + \mathbf{L}\_f \mathbf{R}\_\mathbf{g}$$

$$a\_0 = \mathbf{R}\_f \mathbf{R}\_\mathbf{g}$$

The inverter output characteristics depend on several factors like parameter of LCL filter, grid impedance, type of controller and control parameters. By selecting the appropriate parameter, the inverter can be operated with resistive output impedance, inductive output impedance and capacitive output impedance. Most of the inverters are operated with inductive (L-type) output impedance. This inverter is known as the L-type inverter. Here it is explained how an inverter impedance can be made capacitive (C-type inverter) with selection of virtual

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

The output impedance of inverter is controlled by the virtual impedance. The single-line diagram of inverter with control loop is shown in Figures 27 and 28. The control is implemented with current loop and voltage loop, which gives good tracking behaviour and good output voltage. The feedback of current is taken from branch between the two capacitors. The first capacitor has a value of βC, and the second capacitor has a value of (1-β)C. Thus, the overall value of capacitor is C. The

Cs Lð Þþ <sup>s</sup> <sup>þ</sup> <sup>r</sup> Csð Þ <sup>1</sup> � <sup>β</sup> KPWMð Þþ GI <sup>þ</sup> GV GIGUKPWM <sup>þ</sup> <sup>1</sup> (37)

CsβKPWM

Cs (39)

(38)

The value of virtual impedance GV decides the overall impedance characteristics

GV <sup>þ</sup> GI <sup>¼</sup> GIGUKPWM <sup>þ</sup> <sup>1</sup>

<sup>Z</sup><sup>0</sup> <sup>≈</sup> <sup>1</sup>

output impedance of system after adding virtual branch GV is given by

Zo <sup>¼</sup> ð Þ GI <sup>þ</sup> GV KPWM <sup>þ</sup> Ls <sup>þ</sup> <sup>r</sup>

of output impedance. If GV is selected as per Eq. (38),

current control loop.

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

Figure 27.

Figure 28.

87

Block diagram for active damping.

Control diagram of active damping of filter.

Figure 25. Type V LCL filter.

$$b\_4 = L\_f \mathbf{C} \mathbf{C}\_f \left( L\_2 + L\_\mathbf{g} \right)$$

$$b\_3 = L\_f \mathbf{C} \mathbf{C}\_f \mathbf{R}\_f + L\_f \mathbf{C} \mathbf{C}\_f \mathbf{R}\_\mathbf{g} + \mathbf{C}\_f \mathbf{R}\_f \left( L\_\mathbf{g} + L\_2 \right)$$

$$b\_2 = L\_f \mathbf{C} + \mathbf{C} \mathbf{C}\_f \mathbf{R}\_f$$

$$b\_1 = R\_f \mathbf{C}$$

$$b\_0 = \mathbf{1}$$

Type V filter response shows two resonance frequency (Figure 26). The first resonance creates high voltage distortion, while the second resonance point gives rise to current distortion. If converter generates harmonics equal to this resonance, then there will be high voltage distortion and current distortion. However, with lower value of resistance, damping can be achieved. An additional inductor Lf can reduce the resistive loss.

#### 7.2 Active damping of filter

Grid-connected converters may not function stably under the harmonic resonance condition. Harmonic resonance occurs when the converter impedance and grid impedance becomes equal in magnitude and 180° out of phase. The converter is generally connected to grid through filters, which is mostly LCL type. So, the parameters of LCL filters play an important role in keeping the successful functioning of converter. Also, the control structure of the converter shapes its output impedance. So, the control parameter should be selected such that it keeps the converter in the safe zone at all frequency. This is explained further here with analysis.

Figure 26. Nyquist plot of Type V LCL filter.

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

The inverter output characteristics depend on several factors like parameter of LCL filter, grid impedance, type of controller and control parameters. By selecting the appropriate parameter, the inverter can be operated with resistive output impedance, inductive output impedance and capacitive output impedance. Most of the inverters are operated with inductive (L-type) output impedance. This inverter is known as the L-type inverter. Here it is explained how an inverter impedance can be made capacitive (C-type inverter) with selection of virtual current control loop.

The output impedance of inverter is controlled by the virtual impedance. The single-line diagram of inverter with control loop is shown in Figures 27 and 28. The control is implemented with current loop and voltage loop, which gives good tracking behaviour and good output voltage. The feedback of current is taken from branch between the two capacitors. The first capacitor has a value of βC, and the second capacitor has a value of (1-β)C. Thus, the overall value of capacitor is C. The output impedance of system after adding virtual branch GV is given by

$$Z\_o = \frac{(G\_I + G\_V)K\_{PWM} + L\_s + r}{\text{Cs}(L\_s + r) + \text{Cs}(1 - \beta)K\_{PWM}(G\_I + G\_V) + G\_I G\_U K\_{PWM} + 1} \tag{37}$$

The value of virtual impedance GV decides the overall impedance characteristics of output impedance. If GV is selected as per Eq. (38),

$$\mathbf{G}\_V + \mathbf{G}\_I = \frac{\mathbf{G}\_I \mathbf{G}\_U \mathbf{K}\_{P\text{WM}} + \mathbf{1}}{\mathbf{G}\beta \mathbf{K}\_{P\text{WM}}} \tag{38}$$

$$Z\_0 \approx \frac{1}{\text{Cs}}\tag{39}$$

Figure 27. Block diagram for active damping.

Figure 28. Control diagram of active damping of filter.

b<sup>4</sup> ¼ LfCCf L<sup>2</sup> þ Lg

b<sup>3</sup> ¼ LfCCfRf þ LfCCfRg þ CfRf Lg þ L<sup>2</sup>

b<sup>2</sup> ¼ LfC þ CCfRf b<sup>1</sup> ¼ RfC b<sup>0</sup> ¼ 1

Type V filter response shows two resonance frequency (Figure 26). The first resonance creates high voltage distortion, while the second resonance point gives rise to current distortion. If converter generates harmonics equal to this resonance, then there will be high voltage distortion and current distortion. However, with lower value of resistance, damping can be achieved. An additional inductor Lf can

Grid-connected converters may not function stably under the harmonic resonance condition. Harmonic resonance occurs when the converter impedance and grid impedance becomes equal in magnitude and 180° out of phase. The converter is generally connected to grid through filters, which is mostly LCL type. So, the parameters of LCL filters play an important role in keeping the successful functioning of converter. Also, the control structure of the converter shapes its output impedance. So, the control parameter should be selected such that it keeps the converter in the

safe zone at all frequency. This is explained further here with analysis.

reduce the resistive loss.

Figure 25. Type V LCL filter.

Figure 26.

86

Nyquist plot of Type V LCL filter.

7.2 Active damping of filter

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With the above value of GV, the output impedance becomes capacitive. So, by selection of proper control loop and control parameter, the shaping of inverter output impedance can be effectively done. Also, the desired value of damping can be achieved. This method is known as active damping method. Like this, the inverter can be made either L-type or R-type.
