## 2 System Description and Cancellation Problem
### Injected Signal Cancellation  
By using the proposed IDM in [16] in the multi-PV system, cancellation can occur when the perturbation signals of PV1 and PV2 are in the opposite positions, as shown in Figure 3.  
* The perturbation signal factors at normal condition in normal case are \(n\) = 0.9-1.1. * The perturbation signal factors at normal condition in the cancellation case are \(n\) = 1.1-0.9 for PV1 and \(n\) = 0.9-1.1 for PV2. Figure 4 describes in detail on how the injected signal cancellation can occur. Figure 3: Perturbation signals in cancellation signal scenario. Figure 2: System model under analysis. The cancellation problem is explained below:  
Based on the equation in [16], after islanding, the DC-link voltage is shown in Equation (1):  
\[v_{dc}(t)=[v(0)-N_{k}Ri_{pv}(t)]e^{-\frac{1}{RC}t}+N_{k}Ri_{pv}(t) \tag{1}\]  
and  
\[N_{k}=1-n_{k}D \tag{2}\]  
where:  
* \(v_{dc}(t)\) is the DC-link voltage at time \(t\). * \(v(0)\) is the initial DC-link voltage. * \(i_{pv}(t)\) is the photovoltaic current at time \(t\). * \(N_{k}\) is the output DC/DC converter current perturbation signal. * \(n_{k}\) is the perturbation factor generated by the islanding program. * \(D\) is the duty cycle. * \(R\) is the DC load (resistor). * \(C\) is the capacitor. The DC-link voltage at perturbation \(N_{1}\) from PV1 is as follows:  
\[v_{dc1}(t)=N_{1}Ri_{pv}(t)+[v(0)-N_{1}Ri_{pv}(t)]e^{-\frac{1}{RC}t} \tag{3}\]  
where: \(v_{dc1}(t)\) is the DC-link voltage at time \(t\) when perturbation signal 1 is injected and, \(N_{1}\) is the perturbation signal of PV1. The DC-link voltage at perturbation \(N_{2}\) from PV2 is shown in Equation (4):  
\[v_{dc2}(t)=N_{2}Ri_{pv}(t)+[v(0)-N_{2}Ri_{pv}(t)]e^{-\frac{1}{RC}t} \tag{4}\]  
Figure 4: Controller perturbation signal circuit (\(V_{dc}\) is the measurement DC bus voltage; \(V_{pv1}\) and \(V_{pv2}\) are the PV1 and PV2 voltages, respectively; \(I_{pv1}\) and \(I_{pv2}\) are the PV1 and PV2 currents, respectively; \(n_{1}\) and \(n_{2}\) are the perturbation factors of IDM programs 1 and 2, respectively; \(D_{1}\) and \(D_{2}\) are the duty cycles of boost converter 1 and 2, respectively). where: \(v_{dc2}(t)\) is the DC-link voltage at time \(t\) when perturbation signal 2 is injected. The DC-link voltage without injecting the perturbation signal is as follows:  
\[v_{dc0}(t)=N_{0}Ri_{pv}(t)+[v(0)-N_{0}Ri_{pv}(t)]e^{-\frac{1}{RC}t} \tag{5}\]  
where: \(v_{dc0}(t)\) is the DC-link voltage without perturbation signal at time \(t\) after islanding occurs. The fluctuation of DC-link voltage caused by the perturbation signal from PV1 is shown in Equation (6):  
\[v_{dc1}(t)-v_{dc0}(t)=\{N_{1}Ri_{pv}(t)+[v(0)-N_{1}Ri_{pv}(t)]e^{-\frac{1}{RC} t}\}-\{N_{0}Ri_{pv}(t)+[v(0)-N_{0}Ri_{pv}(t)]e^{-\frac{1}{RC}t}\} \tag{6}\]  
The Equation (6) is simplified  
Figure 3:0  
Equation (7) becomes  
Figure 3:1  
or  
Figure 3:2  
The fluctuation of DC-link voltage caused by the perturbation signal from PV2 is shown in Equation (10):  
Figure 3:3  
The Equation (10) is simplified  
Figure 3:4  
Equation (11) becomes  
Figure 3:5  
or  
Figure 3:6  
Based on Equations (9) and (13), the injected signal cancellation occurs when:  
Figure 3:7  
Figure 3:8  
If \(n_{0}=1\) (without perturbation signal), \(n_{1}=1.1\), and \(n_{2}=0.9\), then Equation (15) is satisfied. Consequently, the fluctuation of DC-link voltage is cancelled because the perturbation signals from PV1 and PV2 have the same value but opposite directions. Thus, the DC-link voltage fluctuations have been cancelled. The simulation result verifies the problem.