**Abbreviations**

The following abbreviations are used in this manuscript:


## **Appendix A**

The subscript 0 indicates the value around which the linearization is performed.

## LCL filter matrices:

⎣

*ALCL* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −ω*<sup>b</sup> Ri* + *Rf Li* ω*r*0ω*<sup>b</sup>* ω*<sup>b</sup> Rf Li* <sup>0</sup> <sup>−</sup>ω*<sup>b</sup> Li* 0 −ω*r*0ω*<sup>b</sup>* −ω*<sup>b</sup> Ri* + *Rf Li* 0 ω*<sup>b</sup> Rf Li* <sup>0</sup> <sup>−</sup>ω*<sup>b</sup> Li* ω*<sup>b</sup> Rf Lf g* + *Lg* 0 −ω*<sup>b</sup> Rf g* + *Rg* + *Rf Lf g* + *Lg* ω*r*0ω*<sup>b</sup>* ω*<sup>b</sup> Lf g* + *Lg* 0 0 ω*<sup>b</sup> Rf Lf g* + *Lg* −ω*r*0ω*<sup>b</sup>* −ω*<sup>b</sup> Rf g* + *Rg* + *Rf Lf g* + *Lg* <sup>0</sup> <sup>ω</sup>*<sup>b</sup> Lf g* <sup>+</sup> *Lg* <sup>ω</sup>*<sup>b</sup> Cf* <sup>0</sup> <sup>−</sup>ω*<sup>b</sup> Cf* 0 0 ω*r*0ω*<sup>b</sup>* <sup>0</sup> <sup>ω</sup>*<sup>b</sup> Cf* <sup>0</sup> <sup>−</sup>ω*<sup>b</sup> Cf* −ω*r*0ω*<sup>b</sup>* 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ *BLCL* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ω*<sup>b</sup> Li* 00 0 ω*<sup>b</sup> I q i*0 <sup>0</sup> <sup>ω</sup>*<sup>b</sup> Li* 0 0 −ω*<sup>b</sup> <sup>I</sup><sup>d</sup> i*0 0 0 <sup>−</sup> <sup>ω</sup>*<sup>b</sup> Lf g* + *Lg* 0 ω*<sup>b</sup> I q g*0 00 0 <sup>−</sup> <sup>ω</sup>*<sup>b</sup> Lf g* + *Lg* −ω*<sup>b</sup> <sup>I</sup><sup>d</sup> g*0 00 0 0 <sup>ω</sup>*bV<sup>q</sup>* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

*c*0

*c*0

⎦

00 0 0 −ω*bV<sup>d</sup>*

(A1)

$$\mathbf{C}\_{\text{LCL}} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ R\_f & 0 & -R\_f & 0 & 1 & 0 \\ 0 & R\_f & 0 & -R\_f & 0 & 1 \\ \frac{L\_g R\_f}{L\_{f\text{x}} + L\_{\text{x}}} & 0 & \frac{R\_g L\_{f\text{y}} + R\_{f\text{z}} L\_{\text{y}} - L\_{\text{x}} R\_f}{L\_{f\text{x}} + L\_{\text{x}}} & 0 & \frac{L\_{\text{g}}}{L\_{f\text{x}} + L\_{\text{g}}} & 0 \\ 0 & \frac{L\_g R\_f}{L\_{f\text{x}} + L\_{\text{g}}} & 0 & \frac{R\_g L\_{f\text{z}} + R\_{f\text{z}} L\_{\text{g}} - L\_{\text{g}} R\_f}{L\_{f\text{x}} + L\_{\text{g}}} & 0 & \frac{L\_g}{L\_{f\text{x}} + L\_{\text{g}}} \end{bmatrix}$$

$$D\_{LCL} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{L\_{f\_{\mathcal{S}}}}{L\_{f\_{\mathcal{S}}} + L\_{\mathcal{S}}} & 0 & 0 \\ 0 & 0 & 0 & \frac{L\_{f\_{\mathcal{S}}}}{L\_{f\_{\mathcal{S}}} + L\_{\mathcal{S}}} & 0 \end{bmatrix}$$

*Energies* **2020**, *13*, 4824

Current controller matrices:

$$\begin{array}{rcl} A\_{\text{Inv}} &=& \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{2}{T\_d} & 0 \\ 0 & \frac{4}{T\_d} & 0 & -\frac{2}{T\_d} \end{bmatrix} \\\\ B\_{\text{Inv}} &=& \begin{bmatrix} k\_i & 0 & -k\_i & 0 \\ 0 & k\_i & 0 & -k\_i \\ k\_p \frac{4}{T\_d} & 0 & -k\_p \frac{4}{T\_d} & \omega\_{r0} L\_i \frac{4}{T\_d} \\ 0 & k\_p \frac{4}{T\_d} & -\omega\_{r0} L\_i \frac{4}{T\_d} & -k\_p \frac{4}{T\_d} \end{bmatrix} \\\\ C\_{\text{Inv}} &=& \begin{bmatrix} -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \\\\ D\_{\text{Inv}} &=& \begin{bmatrix} -k\_p & 0 & k\_p & -\omega\_{r0} L\_i \\ 0 & -k\_p & \omega\_{r0} L\_i & k\_p \end{bmatrix} \end{array} \tag{A2}$$

S-VSC electromagnetic part matrices:

$$\begin{array}{rcl} A\_{Elt} &=& \begin{bmatrix} -\omega\_b \frac{R\_v}{L\_d'} & \omega\_{r0} w\_b & 0\\ -\omega\_{r0} \omega\_b & -\omega\_b \frac{R\_v}{L\_q'} & \omega\_b \frac{R\_v}{L\_q'}\\ 0 & \frac{L\_{rq}}{\tau\_{rq} 0 L\_q'} & -\frac{1 + L\_{rq}/L\_q'}{\tau\_{rq0}} \end{bmatrix} \\\\ B\_{Elt} &=& \begin{bmatrix} \omega\_b & 0 & 0 & \omega\_b \Delta\_{q0} & \omega\_b \frac{R\_v}{L\_q'}\\ 0 & \omega\_b & -\omega\_b \Delta\_{40} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \\\\ C\_{Elt} &=& \begin{bmatrix} -\frac{1}{L\_d'} & 0 & 0\\ 0 & -\frac{1}{L\_q'} & \frac{1}{L\_q'} \end{bmatrix} \\\\ D\_{Elt} &=& \begin{bmatrix} 0 & 0 & 0 & \frac{1}{L\_d'}\\ 0 & 0 & 0 & 0 \end{bmatrix} \end{array} \tag{A3}$$

S-VSC power loops matrices:

$$\begin{array}{rcl}\text{A power} &=& \begin{bmatrix} 0 & 0 & 0 \\ \omega\_b & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\\\ B\_{\text{Power}} &=& \begin{bmatrix} 0 & 0 & -\frac{L\_{\text{x00}}}{2H} & -\frac{L\_{\text{x0}}}{2H} & -\frac{V\_{g00}}{2H} & -\frac{V\_{g00}}{2H} & 0 \\ 0 & 0 & 0 & 0 & 0 & -\omega\_b \\ 0 & 0 & k\frac{L\_{\text{x0}}}{V\_{g0}} & -k\epsilon\frac{L\_{\text{x0}}}{V\_{g0}} & -k\epsilon\frac{V\_{g00}}{V\_{g0}} & k\epsilon\frac{V\_{g00}}{V\_{g0}} & 0 \end{bmatrix} \\\\ \mathbf{C}\_{\text{Power}} &=& \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\\\ D\_{\text{Power}} &=& \begin{bmatrix} 0 & 0 & I\_{\text{x00}} & I\_{\text{x00}} & V\_{g00} & V\_{g00} & 0 \\ 0 & 0 & -I\_{\text{x00}} & I\_{\text{x00}} & V\_{g00} & -V\_{g00} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{array} \tag{A4}$$

Power to current reference calculation:

*DRef* =

$$\begin{bmatrix} \frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_0^\*(\mathbf{z}\_{g00}^2 - \mathbf{v}\_{g00}^2) - 2Q\_0^\* v\_{g00} v\_{g00}}{V\_{g0}^4} & \frac{Q\_0^\*(\mathbf{v}\_{g00}^2 - \mathbf{v}\_{g00}^2) - 2Q\_0^\* v\_{g00} v\_{g00}}{V\_{g0}^4} \\ \frac{v\_{g00}}{V\_{g0}^2} & -\frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & -\frac{v\_{g00}}{V\_{g0}^2} & \frac{v\_{g00}}{V\_{g0}^2} & -\frac{v\_{g00}}{V\_{g0}^2} & -\frac{2P\_0^\* v\_{g00} v\_{g00} + Q\_0^\*(v\_{g00}^2 - v\_{g00}^2)}{V\_{g0}^4} & \frac{V\_0^\*(v\_{g00}^2 - v\_{g00}^2) + 2Q\_0^\* v\_{g00} v\_{g00}}{V\_{g0}^4} \end{bmatrix} \tag{A5}$$

Grid perturbation matrix:

$$D\_{Grid} = \begin{bmatrix} \mathfrak{e}\_{\mathcal{S}\mathcal{A}} & \frac{\mathfrak{e}\_{\mathcal{S}\mathcal{A}}}{E\_{\mathcal{S}\mathcal{0}}} & -\mathfrak{e}\_{\mathcal{S}\mathcal{A}} \\ -\mathfrak{e}\_{\mathcal{S}\mathcal{A}0} & \frac{\mathfrak{e}\_{\mathcal{S}\mathcal{A}}}{E\_{\mathcal{S}\mathcal{0}}} & \mathfrak{e}\_{\mathcal{S}\mathcal{A}0} \end{bmatrix} \tag{A6}$$

Connection matrices. Only the non-zero elements are given:

$$\begin{aligned} T\_{xy}(1,7)&=1 & T\_{xy}(2,8)=1 & T\_{xy}(3,18)=1 & T\_{xy}(4,19)=1 & T\_{xy}(5,13)=1\\ T\_{xy}(6,16)&=1 & T\_{xy}(7,17)=1 & T\_{xy}(8,1)=1 & T\_{xy}(9,2)=1 & T\_{xy}(10,3)=1\\ T\_{xy}(11,4)=1 & T\_{xy}(12,13)=1 & T\_{xy}(13,15)=1 & T\_{xy}(14,4)=1 & T\_{xy}(15,4)=1\\ T\_{xy}(16,9)=1 & T\_{xy}(17,10)=1 & T\_{xy}(21,11)=1 & T\_{xy}(22,12)=1 & T\_{xy}(23,3)=1\\ T\_{xy}(24,4)=1 & T\_{xy}(25,14)=1 &\\ T\_{m2}(24,4)=1 & T\_{m}(18,1)=1 & T\_{m}(20,2)=1 & T\_{m}(26,4)=1 & T\_{m}(27,5)=1\\ T\_{yy}(1,1)=\frac{3}{2}T\_{y^{6}}^{4} & T\_{xy}(1,3)=\frac{3}{2}T\_{y^{6}}^{4} & T\_{y}(1,2)=\frac{3}{2}T\_{y^{6}}^{4} & T\_{y}(1,4)=\frac{3}{2}T\_{y^{6}}^{4}\\ T\_{y^{6}\_{6}(19)} &= \frac{3}{2}T\_{y^{6}}^{4} & T\_{y}(2,4)=\frac{3}{2}T\_{y^{6}}^{4} & T\_{y}(2,1)=-\frac{3}{2}T\_{y^{6}}^{4} & T\_{y}(2,3)=-\frac{3}{2}T\_{y^{6}}^{4}\\ T\_{xy}(3,1)=1 & T\_{xy}(4,1)=1 & T\_{xy}(3,3)=1 & T\_{xy}(6,4)=1 \end{aligned}\tag{A7}$$

#### **References**


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