# A Two-Grid Interline Dynamic Voltage Restorer Based on Two Three-Phase Input Matrix Converters
## 3 TI-IDVR System Analysis  
For the analysis of the proposed TI-IDVR shown in Figure 2, we assume ideal conditions and that the feeders and loads have the same nominal characteristics. It is also considered that a voltage sag arises in feeder 1. Under these considerations, it is assumed that the series converter (CSI 1) operates in compensation mode and the shunt converter (TTI-MC) operates in power flow control mode. Figure 3a shows the relationship of the voltage vectors or voltage phasors in this case. Taking into consideration Figures 2 and 3a, the following equation can be written:  
\[\overset{\rightarrow}{V}_{L1,rated}=\overset{\rightarrow}{V}_{F1,sag}+ \overset{\rightarrow}{V}_{comp,sag} \tag{1}\]  
where, \(\overset{\rightarrow}{V}_{L1,rated}\), \(\overset{\rightarrow}{V}_{F1,sag}\) and \(\overset{\rightarrow}{V}_{comp,sag}\) are the voltage vectors of load voltage 1, feeder 1 voltage (with sag), and compensation voltage injected by the TI-IDVR, respectively. On the other hand, neglecting the voltage drop in the CSI output filter and the injection transformer, the absolute value of the compensation voltage \(\left|\overset{\rightarrow}{V}_{comp,sg}\right|\) is related to the voltage in the virtual dc-link (\(V_{dc}\)), the modulation index of the CSI (\(M\)), and the transformation ratio of the injection transformer (\(a\), ratio of the converter side to the network side) in the following way:  
\[\left|\overset{\rightarrow}{V}_{comp,sg}\right|=aMV_{dc} \tag{2}\]  
The per unit value of the voltage sag (\(V_{sqg,pu}\)) based on the load voltage nominal value (\(V_{base}=\left|V_{L1,ratal}\right|\)) is defined as follows:  
\[V_{sqg,pu}=\frac{\left|V_{L1,ratal}\right|-\left|V_{F1,sqg}\right|}{\left|V_{L1,rated}\right|}V_{sqg,pu}=1-\left|V_{F1,sqg,pu}\right| \tag{3}\]  
Now, for clarity, assuming that the voltages \(\overset{\rightarrow}{V}_{L1,rated}\) and \(\overset{\rightarrow}{V}_{F1,sqg}\) are in phase, Equation (1) can be written as follows:  
\[\left|V_{L1,rated}\right|=\left|V_{F1,sqg}\right|+\left|V_{comp,sqg}\right| \tag{4}\]  
The purpose of the TI-IDVR is to maintain the load voltage at its nominal value ( \(\left|V_{L1}\right|\) = \(\left|V_{L1,rated}\right|\)), in this way, using Equation (4), it can be written as:  
\[1=\left|V_{F1,sqg,pu}\right|+\left|V_{comp,sqg,pu}\right| \tag{5}\]  
Combining Equations (2), (3) and (5) we can write:  
\[V_{sqg,pu}=aMV_{dc,pu} \tag{6}\]  
The maximum value of a voltage sag that can be compensated by the CSI of the TI-IDVR can be obtained considering that the maximum value of the modulation index in Equation (6) is 1 (\(M\) = 1). In this way, we can write:  
\[V_{sqgmax,pu}=aV_{dc,pu}\]This equation indicates that the maximum value of a voltage sag (\(V_{sgmax,pu}\)) that can be fully compensated by a conventional DVR is directly related to the voltage present on the dc-link and to the transformation ratio of the injection transformer. Theoretically, Equation (7) states that a conventional DVR can cope with any voltage sag value. However, there are some inherent limitations to the DVR configuration or topology. For a DVR with topology but without a power storage system connected on the feeder side (in this case feeder 1) and without a capacitor on the dc-link, the voltage on the dc-link can be estimated by \(V_{dc,pu}=\left|V_{F1,pu}\right|\); then, Equation (7) can be written as:  
\[V_{sgmax,pu}=a\left|V_{F1,pu}\right| \tag{8}\]  
This equation indicates that for a DVR in this topology, the maximum value of a voltage sag that can be fully compensated by the DVR is limited by the voltage present in the voltage supply. That is, if we suppose that \(a=1\), Equation (8) tells us that the DVR under this configuration can compensate for, at most, a voltage sag of 0.5 pu. On the other hand, the proposed TI-IDVR is formed by the merge of two DVRs without energy storage systems connected on the side of the power supply and without a capacitor in the dc-link.