# A Two-Grid Interline Dynamic Voltage Restorer Based on Two Three-Phase Input Matrix Converters
## 3 TI-IDVR System Analysis
The merge of the two DVRs is done by replacing the shunt converters of both DVRs for the TTI-MC as shown in Figure 2. Taking the reasoning used to obtain Equation (2), we can say that the voltage on the dc-link (\(V_{dc}\)) is directly related to the voltages of the two feeders through the TTI-MC. Now, neglecting the voltage drop across the TTI-MC input filters, the voltage value on the dc-link relates to the feeder voltages (\(V_{F1}\) y \(V_{F2}\)) and the modulation index of the TTI-MC (\(M_{TTI-MC}\)) as follows:  
\[ \begin{array}{c}V_{dc}=M_{TTI-MC}(\left|V_{F1}\right|+\left|V_{F2}\right|)\\ V_{dc,pu}=M_{TTI-MC}(\left|V_{F1,pu}\right|+\left|V_{F2,pu}\right|)\end{array}  \tag{9}\]  
The maximum voltage value on the virtual dc-link (\(V_{dcmax}\)) that the TTI-MC can provide can be obtained by considering that the maximum value of the TTI-MC modulation index in Equation (9) is 1 (\(M_{TTI-MC}=1\)). In this way, we can write:  
\[V_{dcmax,pu}=\left|V_{F1,pu}\right|+\left|V_{F2,pu}\right| \tag{10}\]  
This equation indicates that in the absence of disturbances, the maximum voltage value on the virtual dc-link can be equal to 2 pu, \(V_{dcmax,pu}=2\) pu, and in the presence of a voltage sag in feeder 1, \(V_{dcmax,pu}>1\) pu. If \(V_{dcmax,pu}=V_{dc,pu}\) in Equation (7) we obtain the following expression:  
\[V_{sgmax,pu}=aV_{dcmax,pu} \tag{11}\]  
This equation indicates that for the proposed TI-IDVR, the maximum value of a voltage sag that can be fully compensated for in feeder 1 by the TI-IDVR is limited by the maximum voltage present on the virtual dc-link. Now, assuming a unit conversion ratio in the injection transformer (\(a=1\)) and substituting Equation (10) into Equation (11), we obtain:  
\[V_{sgmax,pu}=\left|V_{F1,pu}\right|+\left|V_{F2,pu}\right| \tag{12}\]  
Considering ideal conditions and that \(V_{F1,rated}=V_{F2,rated}\), this expression clearly shows that the maximum value of a voltage sag that can be fully compensated in feeder 1 by the TI-IDVR depends directly on the sum of the voltages of the two feeders. Theoretically, Equation (12) indicates an increase in the compensation range. Therefore, the proposed TI-IDVR has the ability to compensate for voltage sags greater than 0.5 pu. For example, suppose an extreme case where feeder 1 suffers a voltage interruption and the voltages on the feeders are \(V_{F1 Int,pu}=0.05\) pu and \(V_{F2,pu}=1\) pu. Now, substituting the values of the voltages in pu of the feeders in Equation (12), we have \(V_{sgmax,pu}=1.05\) pu. Theoretically this value is outside the range of a voltage interruption. Therefore, the voltage interruption in feeder 1 can be compensated for by the CSI1 of the TI-IDVR. It should be noted that for this configuration of the DVRs, it is assumed that the power supplies are strong or robust enough to withstand a higher current demand when compensating for voltage sags. Now, suppose the value of the feeders is different and \(\left|V_{F1,rated}\right|>\left|V_{F2,rated}\right|\). If we consider that a sag appears in feeder 2, Equation (12) can be written as follows:  
\[V_{F2,sagmax}=\left|V_{F1,rated}\right|+\left|V_{F2,sag}\right| \tag{13}\]  
This expression tells us that a sag of any magnitude that occurs in feeder 2 can be compensated for, including voltage interruption. This is because \(\left|V_{F1,rated}\right|>\left|V_{F2,rated}\right|\). On the other hand, if the sag appears in feeder 1, the magnitude of feeder 2 imposes limits on the value of the sag to compensate. Considering Figures 2 and 3a it can be written as:  
\[\left|V_{L1,rated}\right|=\left|2V_{F1,sag}\right|+\left|V_{F2,rated}\right| \tag{14}\]  
\[\left|2V_{F1,sag}\right|=\left|V_{L1,rated}\right|-\left|V_{F2,rated}\right| \tag{15}\]  
since \(\left|V_{F1,rated}\right|>\left|V_{F2,rated}\right|\), we can write:  
\[\left|V_{L1,rated}\right|=k\left|V_{F2,rated}\right|\text{, \ \ }k>1 \tag{16}\]  
where k can be defined as the relationship between the feeders.