## 3 Proposed Control Scheme of 3-Phase AC/DC PWM Converter Using SOGI-FLL
### Sogi-Fll  
SOGI-FLL is used for detecting input voltage exactly, even if input frequency is varied and due to grid distortion or fault. Configuration and discretization of the SOGI-FLL technique in single-phase grid-connected inverter system is introduced in [22]. In this subsection, not only introducing the structure of SOGI-FLL, but also how SOGI-FLL extracts information of input voltage and tracks frequency variation based on its mathematical explanation and bode plot. If the center frequency \(\omega^{\prime}\) is set to the same frequency as the input signal, the SOGI-based AF outputs \(v^{\prime}\) having the same amplitude and phase as the input signal and \(qv^{\prime}\) delayed only by 90 degrees from the input signal. However, if the frequency of the input signal varies, the amplitudes of \(v^{\prime}\) and \(qv^{\prime}\) is changed. Therefore, it is not appropriate to use SOGI only for power conversion systems that use AC power such as grid-connected systems. In order to apply SOGI to a system where the input frequency is possibly changed, the center frequency must be able to follow the frequency change of the input signal.  
In the FLL method, frequency information is extracted using the phase delayed signal of SOGI \(qv^{\prime}\), which is used as the center frequency \(\omega^{\prime}\) of SOGI. FLL can be operated without the PI controller and trigonometric function calculations that were used in PLL. Figure 10 shows the control block diagram of SOGI-FLL, where \(\omega_{N}\) is the angular frequency of the input voltage in steady state, as used in the PLL.  
Figure 10: Control block diagram of SOGI-frequency locked loop (FLL) for phase synchronization.  
Figure 9: Characteristics of SOGI-based AF various gain \(k\).  
In order to explain the SOCI-based AF with FLL, the error signal \(\varepsilon_{v}\), which is the relation between the output signals \(qv^{\prime}\), \(v^{\prime}\), and \(v\), should be analyzed. The transfer function for the input signal \(v\) and the error signal \(\varepsilon_{v}\) is shown in Equation (8).  
\[E(s)=\frac{\varepsilon_{v}}{v}(s)=\frac{s^{2}+\omega\prime^{2}}{s^{2}+k\omega ^{\prime}s+\omega\prime^{2}} \tag{8}\]  
The transfer function \(E(s)\) is illustrated in the bode plot that is shown in Figure 11.  
As shown in Figure 11, if the frequency of the input signal \(\omega\) is lower than the center frequency \(\omega^{\prime}\), \(qv^{\prime}\), and \(\varepsilon_{v}\) have the same phase. That is, the frequency error variable \(\varepsilon_{f}\) can be calculated using the two signals \(qv^{\prime}\) and \(\varepsilon_{v}\). If \(\omega<\omega^{\prime}\), then \(\varepsilon_{f}\) is a positive value, if \(\omega=\omega^{\prime}\) then \(\varepsilon_{f}=0\) and if \(\omega>\omega^{\prime}\) then \(\varepsilon_{f}\) is a negative value. This means that the negative gain \(-\gamma\) and the integrator expressed in Figure 10 are used to set the DC component of the frequency error variable \(\varepsilon_{f}\) to zero so that the center frequency \(\omega^{\prime}\) tracks the frequency of the input signal \(\omega\).  
To use this SOCI-FLL in 3-phase systems, two SOCIs are used. This is called a double second order generalized integrator (DSOGI). Figure 12 shows a control block diagram of the DSOGI-FLL that is used in the 3-phase AC system.  
Figure 11: Comparison of transfer functions \(E(s)\) and \(Q(s)\).  
Figure 12: 3-phase SOCI-FLL (= DSOGI-FLL).