5. Phase lock loop

$$\mathbf{G}\_{PLL}(s) = \mathbf{f}(s, w, k) \tag{11}$$

The phase lock loop is required for PI-based controller. So, the structure of PLL should be thoroughly assessed for stability. Proper structure of PLL and control scheme helps to mitigate adverse impact on the system. The purpose of PLL is to synchronise converter with grid. Harmonics in the grid may penetrate to the converter through the PI controller. If control parameters are not properly chosen, this may produce harmonics through circular effect. Figure 5 shows the block diagram of PLL. The detail block diagram is given in Figure 6.

Figure 6 is the traditional second-order generalised integrator–quadrature signal generator (SOGI-QSG) PLL. It can filter out higher-order harmonics, where ui is the input signal; ui <sup>0</sup> and qui <sup>0</sup> are two output signals, which are in quadrature; k is the damping coefficient; and ω<sup>0</sup> is the output angular frequency of PLL. Eqs. (14) and (15) show the transfer function of PLL.

$$G\_1(s) = \frac{u\_i'}{u\_i} \tag{12}$$

In traditional SOGI-QSG PLL, the DC component in the input signal is not suppressed by the PLL. To overcome this problem, a minor modification is made in

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

Transfer functions of modified PLL are given in Eqs. (16)–(18). This structure

<sup>¼</sup> ks<sup>2</sup> s<sup>2</sup> þ kω<sup>0</sup>

s

<sup>¼</sup> <sup>k</sup>ω<sup>0</sup>

ð Þ k þ 1 s<sup>2</sup> þ kω<sup>0</sup>

<sup>¼</sup> <sup>k</sup>ω0<sup>2</sup> ð Þ k þ 1 s<sup>2</sup> þ kω<sup>0</sup>

The bode plot of one of the PLL used in [12] is given here. The bandwidth of the PLL is 33 Hz. It attenuates harmonics of 1 kHz to �30 dB. However, the effect of PLL with other controllers and output filter needs to be investigated for crucial

Different types of passive filters are used in converter-based renewable generation sources. The effectiveness of filter, particularly passive type, primarily depends on the grid strength and variation of grid impedance with time. LCL is the common type of filter used widely. The variation in grid impedance affects the performance

<sup>s</sup> <sup>þ</sup> <sup>ω</sup>0<sup>2</sup> (16)

<sup>s</sup> <sup>þ</sup> ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>ω</sup>0<sup>2</sup> (17)

<sup>s</sup> <sup>þ</sup> ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>ω</sup>0<sup>2</sup> (18)

above PLL. The modified PLL is shown in Figure 7.

G2ðÞ¼ s

<sup>H</sup>2ðÞ¼ <sup>s</sup> qu<sup>0</sup>

F2ðÞ¼ s

i ui

u0 i ui ξu ui

reduces the tracking error of PLL.

Frequency response of simple phase lock loop.

Figure 7.

Figure 8.

Block diagram of modified phase lock loop.

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

stability analysis (Figure 8).

75

6. Damped passive filter topologies

$$\mathbf{G}\_1(s) = \frac{u\_i'}{u\_i} = \left[ (u\_i - u\_i')k - u\_i'^{a'} \slash \*^{a'} \slash \*^{a'} \slash = u\_i' \tag{13}$$

Simplifying above equation gives

$$G\_1(\mathbf{s}) = \frac{u\_i'}{u\_i} = \frac{ko'\mathbf{s}}{\mathbf{s}^2 + ko'\mathbf{s} + o'^2} \tag{14}$$

Similarly, for quadrature output transfer function is

$$H\_1(\mathbf{s}) = \frac{qu\_i^{\prime}}{u\_i} = \frac{ko\prime^2}{\mathfrak{s}^2 + ko\prime\mathfrak{s} + o\prime^2} \tag{15}$$

Figure 5. Phase lock loop.

Figure 6. Block diagram of simple phase lock loop.

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

Figure 7. Block diagram of modified phase lock loop.

5. Phase lock loop

input signal; ui

Figure 5. Phase lock loop.

Figure 6.

74

Block diagram of simple phase lock loop.

GPLLðÞ¼ s f sð Þ , w, k (11)

The phase lock loop is required for PI-based controller. So, the structure of PLL should be thoroughly assessed for stability. Proper structure of PLL and control scheme helps to mitigate adverse impact on the system. The purpose of PLL is to synchronise converter with grid. Harmonics in the grid may penetrate to the converter through the PI controller. If control parameters are not properly chosen, this may produce harmonics through circular effect. Figure 5 shows the block diagram

Figure 6 is the traditional second-order generalised integrator–quadrature signal generator (SOGI-QSG) PLL. It can filter out higher-order harmonics, where ui is the

damping coefficient; and ω<sup>0</sup> is the output angular frequency of PLL. Eqs. (14) and

G1ðÞ¼ s

i <sup>k</sup> � <sup>u</sup><sup>0</sup>

¼ ui � u<sup>0</sup>

u0 i ui

i ui

u0 i ui

∗ <sup>ω</sup><sup>0</sup>

<sup>¼</sup> <sup>k</sup>ω<sup>0</sup>

s<sup>2</sup> þ kω<sup>0</sup>

<sup>¼</sup> <sup>k</sup>ω0<sup>2</sup> s<sup>2</sup> þ kω<sup>0</sup>

i ω0 =s

s

=<sup>s</sup> ¼ u<sup>0</sup>

<sup>0</sup> are two output signals, which are in quadrature; k is the

(12)

<sup>i</sup> (13)

<sup>s</sup> <sup>þ</sup> <sup>ω</sup>0<sup>2</sup> (14)

<sup>s</sup> <sup>þ</sup> <sup>ω</sup>0<sup>2</sup> (15)

of PLL. The detail block diagram is given in Figure 6.

G1ðÞ¼ s

u0 i ui

G1ðÞ¼ s

<sup>H</sup>1ðÞ¼ <sup>s</sup> qu<sup>0</sup>

Similarly, for quadrature output transfer function is

<sup>0</sup> and qui

Wind Solar Hybrid Renewable Energy System

(15) show the transfer function of PLL.

Simplifying above equation gives

Figure 8. Frequency response of simple phase lock loop.

In traditional SOGI-QSG PLL, the DC component in the input signal is not suppressed by the PLL. To overcome this problem, a minor modification is made in above PLL. The modified PLL is shown in Figure 7.

Transfer functions of modified PLL are given in Eqs. (16)–(18). This structure reduces the tracking error of PLL.

$$F\_2(s) = \frac{\xi\_u}{u\_i} = \frac{ks^2}{s^2 + ko's + o'^2} \tag{16}$$

$$G\_2(s) = \frac{u\_i'}{u\_i} = \frac{k\alpha's}{(k+1)s^2 + k\alpha's + (k+1)\alpha'^2} \tag{17}$$

$$H\_2(s) = \frac{qu\_i'}{u\_i} = \frac{k\alpha'^2}{(k+1)s^2 + k\alpha's + (k+1)\alpha'^2} \tag{18}$$

The bode plot of one of the PLL used in [12] is given here. The bandwidth of the PLL is 33 Hz. It attenuates harmonics of 1 kHz to �30 dB. However, the effect of PLL with other controllers and output filter needs to be investigated for crucial stability analysis (Figure 8).

## 6. Damped passive filter topologies

Different types of passive filters are used in converter-based renewable generation sources. The effectiveness of filter, particularly passive type, primarily depends on the grid strength and variation of grid impedance with time. LCL is the common type of filter used widely. The variation in grid impedance affects the performance

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 explained here with RL filter topology.
