## 2 Proposed Voltage Fluctuation Compensator
### Topology of Voltage Compensator Based on Bipolar AC/AC Chopper  
In order to increase the reliability of the three-phase voltage fluctuation compensation system, the proposed compensator for each phase will consist of two bipolar AC/AC choppers. Each of these choppers is powered from a different line of the power system and these are lines other than the one in which the compensation occurs. The single module of the voltage compensator in phase 1 is shown in Figure 3. The output of each AC/AC chopper (\(MC_{1}\) and \(MC_{2}\)) is connected to the primary side of transformers \(TR_{3}\) and \(TR_{4}\), respectively. The voltage ratio of both transformers is 1:1, and they are interfacing transformers connected in series with the supply voltage.  
Figure 1: Diagram of the bipolar AC/AC controller based on matrix chopper (MC).  
Figure 2: Control characteristics of the bipolar AC/AC controller: (**a**) voltage transmittance, (**b**) phase of voltage transmittance.  
The diagram of the three-phase AC/AC voltage compensator is presented in Figure 4. The considered AC voltage compensator consists of three AC/AC converter modules: AC/AC I, AC/AC II and AC/AC III. Each module is powered by two different phases of the voltage source, e.g., in the AC/AC module in phase 1 (AC/AC I), voltages \(v_{2}\) and \(v_{3}\) are used as the supply. Switches \(W_{1}\), \(W_{2}\) and \(W_{3}\) are used when there is no need to compensate the supply voltage.  
Figure 4: Diagram of the three-phase voltage compensator with a bipolar AC/AC controller.  
Figure 3: Structure of the voltage controller in phase 1 (AC/AC I).  
Considering that the AC/AC converter is supplied by a symmetrical voltage source, the output voltages \(v_{\text{MC1}}\) and \(v_{\text{MC2}}\) (Figure 3) can be described as  
\[v_{\text{MCI}} =v_{2}(2D_{1}-1) \tag{3}\] \[v_{\text{MC2}} =v_{3}(2D_{2}-1) \tag{4}\]  
where \(D_{1}\) and \(D_{2}\)--duty cycle of control signal for \(MC_{1}\) and \(MC_{2}\), respectively.  
The output voltage of the converter AC/AC I (\(v_{\text{C1}}\)) is the sum of the output voltages of \(MC_{1}\) and \(MC_{2}\):  
\[v_{\text{C1}}\ =v_{2}(2D_{1}-1)+v_{3}(2D_{2}-1). \tag{5}\]  
By writing the voltage in the complex form in Equation (5), the following form of the equation for the compensation voltage in phase 1 (\(v_{\text{C1}}\)) was obtained:  
\[V_{\text{C1}}e^{i\varphi_{\text{C1}}}\ =V_{2}e^{-i\frac{2\pi}{3}}(2D_{1}-1)+V_{3}e^{ i\frac{2\pi}{3}}(2D_{2}-1) \tag{6}\]  
where \(V_{\text{C1}}\), \(\varphi_{\text{C1}}\)--amplitude and phase of voltage \(v_{\text{C1}}\), respectively, \(V_{2}\), \(V_{3}\)--amplitude of voltages in phase 2 (\(v_{2}\)) and phase 3 (\(v_{3}\)). The magnitude and phase of the output voltage of each compensator module for the individual phase can be controlled independently. If duty cycles \(D_{1}\) and \(D_{2}\) are equal and \(\leqslant\)0.5, then \(\varphi_{\text{C1}}\) is equal 180\({}^{\circ}\). If duty cycles \(D_{1}\) and \(D_{2}\) are equal and \(>\)0.5, then \(\varphi_{\text{C1}}\) is equal 0\({}^{\circ}\). For different values of \(D_{1}\) and \(D_{2}\), we obtain a compensation voltage \(v_{\text{C1}}\) with adjustable voltage and phase in the range of 0-360\({}^{\circ}\).  
Since the compensating voltage for a given phase of a three-phase system is shaped from the voltages of the other phases, the amplitude range of the compensating voltage varies with the phase shift of the compensating voltage. The calculation of the compensation voltage amplitude, which depends on the phase shift for each phase, will be the subject of the next chapter of this article.