**4. Line impedance emulation control**

### **4.1 Grid side converter control**

**Figure 8** shows the GsC control. It incorporates two control loops. The internal loop controls in the *abc* reference frame the grid currents *ig*(*abc*) and it is based on

**Figure 8.** *Block diagram of GsC control.*

resonant controller. The external loop regulates, via a PI regulator, the voltage at the DC bus *Vdc* and provides grid current reference on *d* axis *igd\** . The grid current reference on the *q* axis *igq\** is selected to have the desired PF. For the *abc* grid current reference components *ig*(*abc*)*\**, they are obtained via the application of Park transformation to *igd\** and *igq\**. In the following, the tuning of the PI and the resonant controller parameters will be detailed.

#### *4.1.1 Tuning of the PI regulator of the voltage at the DC bus*

Based on **Figure 9**, the current *idc* at the output of the GsC is expressed as in Eq. (6). By applying the Laplace transform to Eq. (6), Eq. (7) is obtained.

$$i\_{dc} = i\_c + i\_s = C\frac{dV\_{dc}}{dt} + i\_s \tag{6}$$

$$V\_{dc} = \frac{1}{\text{Cs}}(\dot{\mathbf{i}}\_{dc} - \dot{\mathbf{i}}\_{s}) \tag{7}$$

Then, the form and the dynamics of the response of the DC bus voltage *Vdc* are imposed by setting the natural frequency of the oscillations *ωnc* and a damping coefficient *ξc*. Thus, the gains *Kpdc* and *Kidc* can be obtained based on equations

*Line Impedance Emulator: Modeling, Control Design, Simulation and Experimental Validation*

*Kpdc* <sup>¼</sup> *<sup>C</sup>ω*<sup>2</sup>

The use of the PWM makes it possible to have a fundamental of the voltage *Ui*

Considering **Figure 10**, the closed-loop system transfer function (*Tcig*) is given

*Lgs*<sup>3</sup> þ *Kpig s*<sup>2</sup> þ *Lgω*<sup>2</sup>

For the synthesis of the resonant controller parameters, we consider the pole placement method and more precisely the Naslin criterion [20–21]. The *n* order

> þ *s* <sup>3</sup> *τ*<sup>3</sup> *α*3

From Eq. (14), we deduce the system characteristic polynomial given by

The identification between the system characteristic polynomial *Pig* and the second order Naslin polynomial makes it possible the deduction of resonant con-

*Kpig* ¼ *Lg*

<sup>2</sup> <sup>þ</sup> *Kiig* <sup>þ</sup> *Lgω*<sup>2</sup>

*α*2

single-phase block diagram the grid side regulation loop given by **Figure 10**.

. Thus, based on Eq. (1), we obtain the simplified

<sup>2</sup> <sup>þ</sup> *Kiig <sup>s</sup>* <sup>þ</sup> *Kpigω*<sup>2</sup>

þ *::* … þ *s*

0 *<sup>s</sup>* <sup>þ</sup> *Kpigω*<sup>2</sup>

<sup>¼</sup> *Lg <sup>α</sup>*<sup>2</sup> � <sup>1</sup> *<sup>ω</sup>*<sup>2</sup>

(15)

0

*<sup>n</sup> τ<sup>n</sup> αn n*ð Þ �<sup>1</sup> *<sup>=</sup>*<sup>2</sup>

*<sup>τ</sup>* (17)

<sup>0</sup> (18)

0

<sup>0</sup> (16)

(14)

<sup>0</sup> <sup>þ</sup> *Kiig <sup>s</sup>* <sup>þ</sup> *Kpigω*<sup>2</sup>

*4.1.2 Tuning of the resonant controller of the grid side current*

*<sup>g</sup>* ð Þ�*<sup>s</sup> ig* ð Þ*<sup>s</sup>* <sup>¼</sup> *Kpig <sup>s</sup>*

<sup>2</sup> *τ*<sup>2</sup> *α* 

<sup>3</sup> <sup>þ</sup> *Kpig <sup>s</sup>*

troller parameters *Kpig*, et *Kiig* as shown in Eq. (17) and Eq. (18).

*α*3 *<sup>τ</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> 0 

*Kiig* ¼ *Lg*

*Grid current regulation loop simplified block diagram.*

*\**

*DOI: http://dx.doi.org/10.5772/intechopen.90081*

*ig*ð Þ*s*

polynomial of this criterion is expressed by Eq. (15).

*Pig*ðÞ¼ *s Lg s*

*i* ∗

*PNaslin*ðÞ¼ *s n*<sup>0</sup> 1 þ *sτ* þ *s*

*Kpdc* ¼ 2*Cξcωnc* (12)

*nc* (13)

Eq. (12) and Eq. (13).

equal to its reference *Ui*

*Tcig*ðÞ¼ *s*

by Eq. (14).

Eq. (16).

**Figure 10.**

**137**

Since the current *idc* is instantaneously equal to �*ig* and the current regulation loop time constant is insignificant compared to the one of the DC bus voltage regulation loop, **Figure 9** gives simplified DC bus voltage regulation loop block diagram.

The transfer function of the PI regulator is given by Eq. (8). Based on this equation and neglecting the load current *is*, the closed-loop transfer function of the *Vdc* control is given by Eq. (9).

$$\mathbf{G}\_c(\mathbf{s}) = \frac{i\_{dc}^\*}{\Delta V\_{dc}} = K\_{pdc} + \frac{K\_{idc}}{s} \tag{8}$$

$$\frac{V\_{dc}}{V\_{dc}^\*} = \frac{\frac{K\_{pk}}{C}\mathfrak{s} + \frac{K\_{ik}}{C}}{\mathfrak{s}^2 + \frac{K\_{pk}}{C}\mathfrak{s} + \frac{K\_{ik}}{C}} = \frac{\frac{K\_{pk}}{C}\mathfrak{s} + \frac{K\_{ik}}{C}}{\mathfrak{s}\mathfrak{2} + 2\mathfrak{s}\_\epsilon\alpha\_{nc}\mathfrak{s} + \alpha\_{nc}^2} \tag{9}$$

The transfer function of Eq. (9) is a second-order system whose denominator can be written in the canonical form of a second-order system given by the righthand side of Eq. (9). By identifying the terms of Eq. (9), the obtained transfer function is characterized by a damping ratio *ξ<sup>c</sup>* and a natural frequency of oscillation *ωnc* that satisfy Eq. (10) and Eq. (11).

$$2\xi\_c a\_{nc} = \frac{K\_{pdc}}{C} \tag{10}$$

$$
\rho\_{nc}^2 = \frac{K\_{idc}}{C} \tag{11}
$$

**Figure 9.** *DC bus voltage regulation loop simplified block diagram.*

*Line Impedance Emulator: Modeling, Control Design, Simulation and Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.90081*

Then, the form and the dynamics of the response of the DC bus voltage *Vdc* are imposed by setting the natural frequency of the oscillations *ωnc* and a damping coefficient *ξc*. Thus, the gains *Kpdc* and *Kidc* can be obtained based on equations Eq. (12) and Eq. (13).

$$K\_{pdc} = \mathsf{2C} \xi\_c o\_{nc} \tag{12}$$

$$K\_{pdc} = \mathbb{C}o\_{nc}^2\tag{13}$$

### *4.1.2 Tuning of the resonant controller of the grid side current*

The use of the PWM makes it possible to have a fundamental of the voltage *Ui* equal to its reference *Ui \** . Thus, based on Eq. (1), we obtain the simplified single-phase block diagram the grid side regulation loop given by **Figure 10**.

Considering **Figure 10**, the closed-loop system transfer function (*Tcig*) is given by Eq. (14).

$$T\_{\rm c\dot{g}}(s) = \frac{i\_{\rm g}(s)}{i\_{\rm g}^{\*}(s) - i\_{\rm g}(s)} = \frac{K\_{p\rm g}s^2 + K\_{\rm i\dot{g}}s + K\_{p\rm j\dot{g}}o\_0^2}{L\_{\rm g}s^3 + K\_{p\rm j\dot{g}}s^2 + (L\_{\rm g}o\_0^2 + K\_{\rm i\dot{g}})s + K\_{p\rm j\dot{g}}o\_0^2} \tag{14}$$

For the synthesis of the resonant controller parameters, we consider the pole placement method and more precisely the Naslin criterion [20–21]. The *n* order polynomial of this criterion is expressed by Eq. (15).

$$P\_{Nadin}(s) = n\_0 \left( 1 + s\tau + s^2 \left( \frac{\tau^2}{a} \right) + s^3 \left( \frac{\tau^3}{a^3} \right) + \dots \dots + s^n \left( \frac{\tau^n}{a^{n(n-1)/2}} \right) \right) \tag{15}$$

From Eq. (14), we deduce the system characteristic polynomial given by Eq. (16).

$$P\_{\rm ig}(\mathbf{s}) = L\_{\rm g}\mathbf{s}^3 + K\_{\rm p\dot{g}}\mathbf{s}^2 + \left(K\_{\rm i\dot{g}} + L\_{\rm g}\alpha\_0^2\right)\mathbf{s} + K\_{\rm p\dot{g}}\alpha\_0^2 \tag{16}$$

The identification between the system characteristic polynomial *Pig* and the second order Naslin polynomial makes it possible the deduction of resonant controller parameters *Kpig*, et *Kiig* as shown in Eq. (17) and Eq. (18).

$$K\_{\rm pig} = L\_{\rm g} \frac{a^2}{\tau} \tag{17}$$

$$K\_{\rm i\dot{g}} = L\_{\rm g} \left( \frac{\alpha^3}{\pi^2} - \alpha\_0^2 \right) = L\_{\rm g} (\alpha^2 - 1) \alpha\_0^2 \tag{18}$$

**Figure 10.** *Grid current regulation loop simplified block diagram.*

resonant controller. The external loop regulates, via a PI regulator, the voltage at the

reference on the *q* axis *igq\** is selected to have the desired PF. For the *abc* grid current reference components *ig*(*abc*)*\**, they are obtained via the application of Park transformation to *igd\** and *igq\**. In the following, the tuning of the PI and the resonant

Based on **Figure 9**, the current *idc* at the output of the GsC is expressed as in

Since the current *idc* is instantaneously equal to �*ig* and the current regulation loop time constant is insignificant compared to the one of the DC bus voltage regulation loop, **Figure 9** gives simplified DC bus voltage regulation loop block diagram. The transfer function of the PI regulator is given by Eq. (8). Based on this equation and neglecting the load current *is*, the closed-loop transfer function of the

¼ *Kpdc* þ

*Kidc s*

*s*2 þ 2*ξcωncs* þ *ω*<sup>2</sup>

*Kpdc <sup>C</sup> <sup>s</sup>* <sup>þ</sup> *Kidc C*

*idc* <sup>¼</sup> *ic* <sup>þ</sup> *is* <sup>¼</sup> *<sup>C</sup>dVdc*

Eq. (6). By applying the Laplace transform to Eq. (6), Eq. (7) is obtained.

*Vdc* <sup>¼</sup> <sup>1</sup>

*i* ∗ *dc* Δ*Vdc*

*<sup>C</sup> <sup>s</sup>* <sup>þ</sup> *Kidc C* ¼

The transfer function of Eq. (9) is a second-order system whose denominator can be written in the canonical form of a second-order system given by the righthand side of Eq. (9). By identifying the terms of Eq. (9), the obtained transfer function is characterized by a damping ratio *ξ<sup>c</sup>* and a natural frequency of oscillation

<sup>2</sup>*ξcωnc* <sup>¼</sup> *Kpdc*

*ω*2 *nc* <sup>¼</sup> *Kidc*

*Gc*ðÞ¼ *s*

*Kpdc <sup>C</sup> <sup>s</sup>* <sup>þ</sup> *Kidc C*

*<sup>s</sup>*<sup>2</sup> <sup>þ</sup> *Kpdc*

. The grid current

*dt* <sup>þ</sup> *is* (6)

(8)

(9)

*Cs idc* � *<sup>i</sup>* ð Þ*<sup>s</sup>* (7)

*nc*

*<sup>C</sup>* (10)

*<sup>C</sup>* (11)

DC bus *Vdc* and provides grid current reference on *d* axis *igd\**

*4.1.1 Tuning of the PI regulator of the voltage at the DC bus*

controller parameters will be detailed.

*Numerical Modeling and Computer Simulation*

*Vdc* control is given by Eq. (9).

*Vdc V* <sup>∗</sup> *dc* ¼

*ωnc* that satisfy Eq. (10) and Eq. (11).

*DC bus voltage regulation loop simplified block diagram.*

**Figure 9.**

**136**
