## 3 Regulation Range of Compensation Voltage Amplitude
### Pulse Duty Factors Calculation  
Duty cycles \(D_{1}\) and \(D_{2}\) (5) are determined by the law of sines, which relates the lengths of the sides of any triangle to the sines of its angles. In each switching sequence, the ratio of the lengths of the sides of the triangle and the sines of its angle formed from the basis vectors (\(\pm\upsilon_{1}\), \(\pm\upsilon_{2}\), \(\pm\upsilon_{3}\)) and the reference voltage vector \(V_{ref}\) are determined, as shown in Figure 10. Then, the coefficients \(d_{1}\) and \(d_{2}\) for symmetrical supply voltages for sector I and phase 1 are calculated from the following relations:  
\[d_{1}=\frac{2}{\sqrt{3}}\frac{\left|V_{1\;ref}\right|}{\left|V_{2}\right|} \sin(60^{\circ}-\varphi) \tag{11}\]  
\[d_{2}=\frac{2}{\sqrt{3}}\frac{\left|V_{1\;ref}\right|}{\left|V_{3}\right|} \sin(60^{\circ}+\varphi) \tag{12}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sector** & **Sector Range** & **Output Voltage Vector** & **Value of \(\varphi\) in Each Sector** \\ \hline I & \(0^{\circ}\) to \(60^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(60^{\circ}+\varphi)}\) & \(0^{\circ}\) to \(60^{\circ}\) \\ II & \(60^{\circ}\) to \(120^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(60^{\circ}+\varphi)}\) & \(0^{\circ}\) to \(60^{\circ}\) \\ III & \(120^{\circ}\) to \(150^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(60^{\circ}-\varphi)}\) & \(0^{\circ}\) to \(30^{\circ}\) \\ IV & \(150^{\circ}\) to \(180^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(30^{\circ}+\varphi)}\) & \(0^{\circ}\) to \(30^{\circ}\) \\ V & \(180^{\circ}\) to \(240^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(60^{\circ}+\varphi)}\) & \(0^{\circ}\) to \(60^{\circ}\) \\ VI & \(240^{\circ}\) to \(300^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(60^{\circ}+\varphi)}\) & \(0^{\circ}\) to \(60^{\circ}\) \\ VII & \(300^{\circ}\) to \(330^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{5}\;V_{\
u}}{2\sin(60^{\circ}-\varphi)}\) & \(0^{\circ}\) to \(30^{\circ}\) \\ VIII & \(330^{\circ}\) to \(360^{\circ}\) & \(\left|V_{ref}\right|=\frac{\sqrt{3}\;V_{\
u}}{2\sin(30^{\circ}+\varphi)}\) & \(0^{\circ}\) to \(30^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The obtained equations for calculation of the amplitude of the output voltage vector for AC/AC III. Figure 9: Voltage phasor diagram for AC/AC III, (Figure 4), where I—VIII are the number of sectors. Figure 12 shows the situation of a three-phase voltage dip in the LV distribution line. At time \(t=0.2\) s, there is a three-phase voltage sag from 0.95 to 0.7 of the nominal value (\(V_{m}=325\) V--nominal voltage amplitude, \(V_{m\ sag}=0.7\)\(V_{m}\)--voltage amplitude during the sag). The compensator generated three compensation voltages \(v_{c1}\), \(v_{c2}\) and \(v_{c3}\), which, when added to the mains voltages \(v_{1}\), \(v_{2}\) and \(v_{3}\) gave the resultant output voltages \(v_{L1}\), \(v_{L2}\) and \(v_{L3}\) with an amplitude equal to the set value of 325 V. The \(D_{1}\) and \(D_{2}\) coefficients controlling the individual AC/AC controllers are equal, as shown in Figure 12.  
A similar situation is shown in Figure 13, but for a three-phase symmetrical overvoltage, where the voltage value in time \(t=0.2\) s changed from 0.95 \(V_{m}\) to 1.3 \(V_{m}\) (\(V_{m\ over}=422\) V--voltage amplitude during the overvoltage). Also, the values of coefficients \(D_{1}\) and \(D_{2}\) are equal, and when the overvoltage is less than 0.5, then the phase shift of the voltage is \(\pi\) (Figure 2), i.e., the sum of mains voltages and compensating voltages results in the lowering of the main voltage-overvoltage compensation. The compensation of slow-changing voltage fluctuations, which means the voltage amplitude is sometimes higher and sometimes lower than the nominal value \(V_{m}\), is shown in Figure 14. On the basis of the voltage measurement, the change in the coefficients of the starting fillers \(D_{1}\) and \(D_{2}\) is calculated.