**4.2 EUT side converter control**

The control based on resonant controller for the EsC is depicted on **Figure 11**. This control includes an external and an internal loops. The external one controls the voltages through the filter capacitor *Vc*(*a,b,c*). The internal one controls the inverter side current *i1*(*a,b,c*) and generates then the inverter voltages references *Vi*(*a,b,c*). For the external loop, a resonant controller is adopted. For the internal loop, the resonant controller is replaced by a constant gain (G) in order to ensure a faster loop than the external one. In the following, the tuning of the resonant controller parameters will be detailed and discussed in order to ensure good control performances.

### *4.2.1 Tuning of the resonant controller of the voltage through the LCL filter capacitor*

For reasons of simplification, it is assumed that the internal loop of the current is faster than the external loop of the voltage. Thus, we can approximate it equal to the unity by associating the PWM function. Consequently, the block diagram of the voltage regulation loop is given by **Figure 12**.

Hence, the closed loop system transfer function (*Tc*) is given by Eq. (19).

$$T\_c(s) = \frac{V\_c}{V\_c^\*} = \frac{a\_{2c}s^2 + a\_{1c}s + a\_{0c}}{C\_f s^3 + a\_{2c}s^2 + \left(C\_f w\_0^2 + a\_{1c}\right)s + a\_{0c}}\tag{19}$$

The method chosen for the computation of the resonant controller parameters is based on the generalized stability criterion [22]. In this case, the *n* order polynomial

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

On the other hand, based on Eq. (19), the system characteristic polynomial *Pc* is

<sup>2</sup> <sup>þ</sup> *Cfω*<sup>2</sup>

The identification of *Pc* and second order generalized stability criterion polyno-

*<sup>c</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *i* � � � *Cfω*<sup>2</sup>

*<sup>c</sup>* <sup>þ</sup> *rcω*<sup>2</sup> *i* � �

ð Þ *s* þ *r* þ *jω<sup>i</sup>* ð Þ *s* þ *r* � *jω<sup>i</sup>* ½ �

<sup>0</sup> þ *a*1*<sup>c</sup>*

0

where *<sup>τ</sup><sup>c</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>1</sup>

� �*<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*0*<sup>c</sup>* (21)

(20)

(22)

*<sup>G</sup>* (23)

*i*¼1

<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*cs*

*a*2*<sup>c</sup>* ¼ 3*rcλ<sup>c</sup> <sup>a</sup>*1*<sup>c</sup>* <sup>¼</sup> *<sup>λ</sup><sup>c</sup>* <sup>3</sup>*r*<sup>2</sup>

*<sup>a</sup>*0*<sup>c</sup>* <sup>¼</sup> *<sup>λ</sup><sup>c</sup> <sup>r</sup>*<sup>3</sup>

*Avec λ<sup>c</sup>* ¼ *Cf*

The simplified internal current regulation loop block diagram is given by

Hence, the transfer function of the closed-loop system *Ti*1(*s*) is given by

*<sup>G</sup> <sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>1</sup>

*G* is chosen so that the real part of the inverse of the closed-loop time constant (1/*τc*) is greater than the stability margin chosen for the synthesis of the voltage external loop in order to ensure that the internal loop is faster than the external one.

In this section, two methods of the line impedance emulator algorithm synthesis

are presented: the trigonometric functions-based algorithm and the voltage

1 þ *τcs*

mial allows the deduction of the resonant controller parameters as shown in

*PGSC*ðÞ¼ *<sup>s</sup> <sup>λ</sup>*ð Þ *<sup>s</sup>* <sup>þ</sup> *<sup>r</sup>* <sup>Y</sup>*<sup>n</sup>*

f g *λ*,*r*,*ω<sup>i</sup>* ∈ ℜ; *i*, *n* ∈ *N*

*Pc*ðÞ¼ *s Cf s*

8 >>><

>>>:

*i*1ð Þ*s i* ∗

<sup>1</sup> ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>1</sup> *L*

*4.2.2 Tuning of the gain of the current i1*

*Ti*1ðÞ¼ *s*

**5. Line impedance emulation algorithms**

drop-based algorithm.

**139**

is expressed as in Eq. (20).

*Current regulation loop simplified block diagram.*

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

given by Eq. (21).

Eq. (22).

**Figure 13.**

**Figure 13**.

Eq. (23).

**Figure 11.** *Block diagram of the EsC control.*

**Figure 12.** *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*

**Figure 13.** *Current regulation loop simplified block diagram.*

**4.2 EUT side converter control**

*Numerical Modeling and Computer Simulation*

voltage regulation loop is given by **Figure 12**.

*Tc*ðÞ¼ *s*

*Vc V* <sup>∗</sup> *c*

performances.

**Figure 11.**

**Figure 12.**

**138**

*Block diagram of the EsC control.*

*Voltage regulation loop simplified block diagram.*

The control based on resonant controller for the EsC is depicted on **Figure 11**. This control includes an external and an internal loops. The external one controls the voltages through the filter capacitor *Vc*(*a,b,c*). The internal one controls the inverter side current *i1*(*a,b,c*) and generates then the inverter voltages references *Vi*(*a,b,c*). For the external loop, a resonant controller is adopted. For the internal loop, the resonant controller is replaced by a constant gain (G) in order to ensure a faster loop than the external one. In the following, the tuning of the resonant controller parameters will be detailed and discussed in order to ensure good control

*4.2.1 Tuning of the resonant controller of the voltage through the LCL filter capacitor*

Hence, the closed loop system transfer function (*Tc*) is given by Eq. (19).

*Cf <sup>s</sup>*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*cs*<sup>2</sup> <sup>þ</sup> *Cfω*<sup>2</sup>

<sup>¼</sup> *<sup>a</sup>*2*cs*

For reasons of simplification, it is assumed that the internal loop of the current is faster than the external loop of the voltage. Thus, we can approximate it equal to the unity by associating the PWM function. Consequently, the block diagram of the

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*cs* <sup>þ</sup> *<sup>a</sup>*0*<sup>c</sup>*

<sup>0</sup> þ *a*1*<sup>c</sup> <sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*0*<sup>c</sup>* (19)

The method chosen for the computation of the resonant controller parameters is based on the generalized stability criterion [22]. In this case, the *n* order polynomial is expressed as in Eq. (20).

$$\begin{aligned} P\_{GSC}(s) &= \lambda(s+r) \prod\_{i=1}^{n} [(s+r+jo\_i)(s+r-jo\_i)] \\ \{\lambda, r, o\_i \in \mathfrak{R}; \ i, n \in N\} \end{aligned} \tag{20}$$

On the other hand, based on Eq. (19), the system characteristic polynomial *Pc* is given by Eq. (21).

$$P\_{\mathfrak{c}}(\mathfrak{s}) = \mathbf{C}\_{f}\mathfrak{s}^{3} + \mathfrak{a}\_{2\mathfrak{s}}\mathfrak{s}^{2} + \left(\mathbf{C}\_{f}\mathfrak{o}\_{0}^{2} + \mathfrak{a}\_{1\mathfrak{c}}\right)\mathfrak{s} + \mathfrak{a}\_{0\mathfrak{c}}\tag{21}$$

The identification of *Pc* and second order generalized stability criterion polynomial allows the deduction of the resonant controller parameters as shown in Eq. (22).

$$\begin{cases} a\_{2c} = 3r\_c \lambda\_c \\ a\_{1c} = \lambda\_c \left( 3r\_c^2 + o\_i^2 \right) - \mathbf{C}\_f o\_0^2 \\ a\_{0c} = \lambda\_c \left( r\_c^3 + r\_c o\_i^2 \right) \\ \text{Ave } \lambda\_c = \mathbf{C}\_f \end{cases} \tag{22}$$
