## 2 Nyquist Analysis for Ideal FPR Controller on Voltage Applications
### Initial System Considerations  
The purpose of grid-forming inverters is to generate a stable and robust voltage. Figure 1 shows the used schematic for a voltage-controlled voltage source converter (VC-VSC). An inductive (\(L\))-capacitive (\(C\)) output coupling filter interfaces the inverter with the AC side. The VC-VSC control is based on two nested control loops. The inner loop regulates the current (\(i_{l}\)) through the inductance \(L\), and the outer loop controls the output capacitor's voltage (\(v_{c}\)) using \(i_{l}\). The capacitor includes an equivalent series resistance (ESR), (\(r_{C}\)). This \(r_{C}\) only considers the parasitic resistance of the capacitor to allow a more realistic model. Thus, the capacitor voltage is  
\[V_{c}(s)=\frac{Cr_{C}s+1}{Cs}I_{l}(s). \tag{1}\]  
As an assumption, the inner current control loop is considered pre-tuned, with enough bandwidth and adequate time response to avoid cross-coupling effects [30]. Thus, the inner control loop is considered as a unitary gain in the following analysis. The FPR formulation considered in this paper follows [23]  
\[G_{\text{FPR}}(s)=k_{\text{p}}+k_{\text{i}}\frac{\omega_{0}s^{\alpha}}{s^{2}+ \omega_{0}^{2}}. \tag{2}\]  
In (2), the \(\alpha\) value is the fractional-order value \(\{\alpha\in(0,2];\alpha\in\mathbb{R}\}\), \(\omega_{0}\) is the resonant angular speed, and \(k_{\text{p}}\) and \(k_{\text{i}}\) are the proportional and integral gains, respectively. The \(\alpha\) value is not considered negative as the studied behaviour is moved from a fractional derivative response to an integral one. In the same direction, values higher than two involve non-causal controllers. Note that when \(\alpha\) is one, (2) results in a classical ideal PR controller. Considering (1) and (2), the open-loop transfer function is  
\[G_{\text{ol}}(s)=\frac{Cr_{\text{c}}s+1}{Cs}\Bigg{(}k_{p}+k_{i}\frac{\omega_{ 0}s^{\alpha}}{s^{2}+\omega_{0}^{2}}\Bigg{)}. \tag{3}\]  
Consequently, the closed-loop transfer function results  
\[G_{\text{cl}}(s)=\frac{V_{\text{c}}(s)}{V_{\text{c}}^{*}(s)}=\frac{\frac{Cr_{ \text{c}}s+1}{Cs}\bigg{(}k_{p}+k_{i}\frac{\omega_{0}s^{\alpha}}{s^{2}+\omega_ {0}^{2}}\bigg{)}}{1+\frac{Cr_{\text{c}}s+1}{Cs}\bigg{(}k_{p}+k_{i}\frac{ \omega_{0}s^{\alpha}}{s^{2}+\omega_{0}^{2}}\bigg{)}} \tag{4}\]  
where the super-index \(*\) designs to the reference signal. Once (3) and (4) are obtained, it would be interesting to gain an initial insight into the effect of the \(\alpha\) parameter when applied to the voltage control of the capacitor \(C\). Thus, for computations, and without loss of generality, it will be considered a real system with specific parameters as a case study. Table 2 summarises the base system parameters. Figure 2 and Figure 3 show the Bode diagram for \(G_{\text{ol}}(s)\) and \(G_{\text{cl}}(s)\), respectively. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Descriptor** & **Value** & **Unit** \\ \hline \(C\) & Capacitance & 75 & \(\mu\)F \\ \(L\) & Inductance & 492 & \(\mu\)H \\ \(r_{\text{C}}\) & Capacitor ESR & 8 & \(m\Omega\) \\ \(r_{l}\) & Inductor ESR & 50 & \(m\Omega\) \\ \(k_{\text{p}}\) & Proportional gain & 0.0494 & \\ \(k_{\text{i}}\) & Integral gain & 0.0363 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Base plant and controller parameters. Figure 1: Voltage-controlled voltage source converter (VC-VSC) schematic with inductive (\(L\))–capacitive (\(C\)) filter. Analysing Figure 2a it can be observed that \(\alpha\) values above 1 contribute to an \(G_{\text{ol}}(s)\) with higher gains outside \(\omega_{0}\). This effect is translated to the \(G_{\text{cl}}(s)\) behaviour extending the bandwidth, refer to Figure 3a. On the contrary, \(\alpha\) values below 1 result in increased selectivity. Furthermore, Figure 2b shows that at frequencies above \(\omega_{0}\), it reduces faster its phase. This is translated into a \(G_{\text{cl}}(s)\) with the worst tracking capability near \(\omega_{0}\), but with the same phase delay for higher frequencies.