*2.2. ES-2 Model*

Although the CL and the NCL can be loads of resistive, capacitive or inductive type, the CL and the NCL are here taken to be of the purely resistive type to make the ES modeling simpler. The state-space equations of ES are

$$\begin{cases} \dot{\mathbf{x}} = A\mathbf{x} + Bu\\ y = \mathbf{C}\mathbf{x} \end{cases} \tag{1}$$

 .*x* = *Ax* + *Bu y* = *Cx* .*x* = *Ax* + *Bu y* = *Cx* .*x* = *Ax* + *Bu y* = *Cx* where, *x* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *iL vES iG* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *u* = *vGvi* , *y* = [*vs*]*x* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *iL vES iG* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *u* = *vGvi* , *y* = [*vs*]*x* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *iL vES iG* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *u* = *vGvi* , *y* = [*vs*]*x* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *iL vES iG* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *u* = *vGvi* , *y* = [*vs*], *A* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 − 1*Lf* 0 1*Cf* − 1 *Cf*(*<sup>R</sup>*2+*R*3) *R*2 *Cf*(*<sup>R</sup>*2+*R*3) 0 − *R*2 *<sup>L</sup>*1(*<sup>R</sup>*2+*R*3) −*R*1*R*2+*R*1*R*3+*R*2*R*<sup>3</sup> *<sup>L</sup>*1(*<sup>R</sup>*2+*R*3) ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦*A* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 − 1*Lf* 0 1*Cf* − 1 *Cf*(*<sup>R</sup>*2+*R*3) *R*2 *Cf*(*<sup>R</sup>*2+*R*3) 0 − *R*2 *<sup>L</sup>*1(*<sup>R</sup>*2+*R*3) −*R*1*R*2+*R*1*R*3+*R*2*R*<sup>3</sup> *<sup>L</sup>*1(*<sup>R</sup>*2+*R*3) ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦*A* =

*Energies* **2019**, *12*, 3511

$$\begin{bmatrix} 0 & -\frac{1}{L\_{f}} & 0\\ \frac{1}{\mathcal{C}\_{f}} & -\frac{1}{\mathcal{C}\_{f}(R\_{2}+R\_{3})} & \frac{R\_{2}}{\mathcal{C}\_{f}(R\_{2}+R\_{3})}\\ 0 & -\frac{R\_{2}}{L\_{1}(R\_{2}+R\_{3})} & -\frac{R\_{1}R\_{2}+R\_{1}R\_{3}+R\_{2}R\_{3}}{L\_{1}(R\_{2}+R\_{3})}\\ \text{and } \mathcal{C} = \begin{bmatrix} 0 & -\frac{1}{L\_{f}} & 0\\ \frac{1}{\mathcal{C}\_{f}} & -\frac{1}{\mathcal{C}\_{f}(R\_{2}+R\_{3})} & \frac{R\_{2}}{\mathcal{C}\_{f}(R\_{2}+R\_{3})}\\ 0 & -\frac{R\_{2}}{L\_{1}(R\_{2}+R\_{3})} & -\frac{R\_{1}R\_{2}+R\_{1}R\_{3}+R\_{2}R\_{3}}{L\_{1}(R\_{2}+R\_{3})} \end{bmatrix}, B = \begin{bmatrix} 0 & \frac{1}{L\_{1}}\\ 0 & 0\\ \frac{1}{L\_{1}} & 0 \end{bmatrix}, \\\ \text{and } \mathcal{C} = \begin{bmatrix} 0 & -\frac{1}{L\_{1}} & 0\\ 0 & \frac{R\_{2}}{L\_{1}} & 0\\ \frac{1}{L\_{2}} & 0 \end{bmatrix}$$

The block scheme of the ES, derived from the above equations, is drawn in Figure 3. The voltage *vi* is equal to *d*(*t*)*Vdc*, where *d*(*t*) is the modulation signal. In modeling the ES, it is assumed that (i) the VSI modulation frequency is much higher than the fundamental frequency, (ii) the VSI dynamics are negligible, and (iii) the pair LC at the VSI output makes almost sinusoidal the quantities in the downstream circuit. It is also assumed that the amplitude of the DC voltage source is constant.

**Figure 3.** ES-2 block scheme.
