## 4 Limitations of AIDMs When Applied with the Presence of GSCI
### Active Method-Impedance Measurement Type  
The aim of any IM active method is to determine the impedance measurement at the PCC. In order to achieve this objective, [PERSON]'s law can be applied, then the voltage and current values of the PCC are needed. However, the proportionality between voltage and current is only maintained if there are not any active sources participating. In this sense, to decouple the impedance value from any voltage source in the system, the voltage and current information should be obtained at a different frequency from the grid-rated one. Thus, a little disturbance is added to produce a non-characteristic harmonic current in the grid. The same harmonic voltage value is obtained from the PCC. The key challenge is \begin{table}
\begin{tabular}{l l r r} \hline \hline  & Output phase inductance \(L_{1}\) & 250 & \(\upmu\)H \\ LCLfilter & Transformer leakage inductance \(L_{2}\) & 70 & \(\upmu\)H \\  & AC capacitor C & 350 & \(\upmu\)F \\  & Switching frequency & 8 & kHz \\ Droop & Proportional constant for \(P\), \(Q\) & 0.00001 & \\ loop & Integral constant for \(Q\) & 0.003 & \\ Voltage & PRproportional constant & 0.07 & \\ loop & PR integral constant & 0.07 & \\ Current & PR proportional constant & 0.75 & \\ loop & PR integral constant & 3.93 & \\ PLL loop & Settling time & 5 & ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hardware and control parameters. Figure 7: SFS detection times applied to CC-VSI and VC-VSI. (**a**) Operating with a grid supply inverter (GSI) (CC-VSI). (**b**) Operating with a grid support and constitution inverter (GSCI) (VC-VSI). to select a suitable frequency that makes the detection feasible without exceeding the maximum total harmonic distortion level. Analyzing the transfer function of the PCC impedance, the asymptotic magnitude Bode diagram for the grid-connected and grid-disconnected operation modes has been parametrized. Figure 8 represents the transfer function where the open loop block is the resonant load impedance and the feedback block is the grid equivalent series \(RL\) (resistive-inductive) model admittance, both represented also in Figure 1.  
The following section introduces the detection frequency bandwidth (DFBW) as a valid mechanism to explore the mentioned desired suitable frequency making IM valid and optimal. The DFBW for IM Applicability  
This paper presents a new analysis tool called DFBW that allows identifying the applicability limits of IM anti-islanding methods. The transfer function represented in Figure 8 is analyzed in detail to define the introduced concept of DFBW. On the one hand, for the grid-connected case, the impedance Bode diagram has three characteristic angular frequencies:  
* The resonant frequency between the grid resistance and the resonant load inductance \(\omega_{1}=R_{G}/L_{RL}\). * The resonant frequency between the grid resistance and inductance \(\omega_{2}=R_{G}/L_{G}\). * The resonant frequency between the grid inductance and the resonant load capacitance \(\omega_{4}=1/\sqrt{L_{G}\cdot\mathbb{C}_{RL}}\). On the other hand, for the grid-disconnected case, the Bode diagram presents a resonance between the resonant load inductance and capacitance at \(\omega_{3}=1/\sqrt{L_{RL}\cdot\mathbb{C}_{RL}}\). If the different asymptotes of the transfer function depicted in Figure 8 are drawn, considering \(\omega_{1}\)-\(\omega_{4}\), the area between the grid and the local load impedance Bode diagram allows defining a DFBW, as shown in Figure 9. This is a bandwidth that goes from the resonance between the grid resistance and the local load inductance \(\omega_{1}\) to the resonance between the grid inductance and the local load capacitance \(\omega_{4}\), and it can be expressed as:  
\[DFBW=\frac{L_{RL}}{R_{G}\cdot\sqrt{L_{G}\cdot\mathbb{C}_{RL}}} \tag{9}\]  
where \(R_{G}\) and \(L_{G}\) are the resistance and the inductance of the grid. Nevertheless, harmonic injection is usually used in an IM.