## 2 IMC with Synchronized Decoupled Controller
### Synchronized PI Controllers of IMC  
Clarke and Park transformation matrices are applied to convert AC signals to DC signals in the controlled part. In this case, steady-state errors are minimized by the PI controllers which are based on the following equations:  
\[\mathbf{v}_{d,\textit{ref}}=K_{p}\Big{(}i_{\textit{sd,ref}}-i_{\textit{sd}}\Big{)}+ K_{i}\int(i_{\textit{sd,ref}}-i_{\textit{sd}})\mathbf{dt} \tag{7}\]  
\[\mathbf{v}_{q,\textit{ref}}=K_{p}\Big{(}i_{\textit{sd,ref}}-i_{\textit{sq}}\Big{)}+ K_{i}\int(i_{\textit{sq,ref}}-i_{\textit{sq}})\mathbf{dt} \tag{8}\]  
where \(i_{\textit{sd,ref}}\) and \(i_{\textit{sq,ref}}\) are the reference signals of \(i_{\textit{sd}}\) and \(i_{\textit{sq}}\) in units of Amps., respectively, which are obtained from the measured DC voltage (\(\mathbf{V}_{\textit{dc}}\)) of PWM rectifier, while \(\mathbf{K}_{p}\) and \(\mathbf{K}_{i}\) are the gains of PI controller, and \(i_{\textit{sd}}\) and \(i_{\textit{sq}}\) are the current source in \(\mathbf{dq0}\) frame. The reference voltages (\(\mathbf{v}_{\textit{d,ref}}\) and \(\mathbf{v}_{\textit{q,ref}}\))  
Figure 4: Block configuration phase-locked loop (PLL) combined with Clarke and Park transform.  
Figure 5: Phasor diagram of Clarke and Park transformation.  
in unit of Volt are transformed using inverse Clarke and Park transformation to derive gate switching pulses \(\mathbf{S}_{\mathbf{abc}}\) which controls the operation of PWM rectifier, as shown in Equation (9).  
\[\left[\begin{array}{c}\mathbf{v}_{\mathbf{\alpha},\mathbf{ref}}\\ \mathbf{v}_{\mathbf{\beta},\mathbf{ref}}\end{array}\right]=\left[\begin{array}{cc}\sin \theta&-\cos\theta\\ \cos\theta&\sin\theta\end{array}\right]\left[\begin{array}{c}\mathbf{v}_{\mathbf{d}, \mathbf{ref}}\\ \mathbf{v}_{\mathbf{q},\mathbf{ref}}\end{array}\right] \tag{9}\]  
The DC-link voltage control is used to regulate the active current component (\(\mathbf{i}_{\mathbf{d}}\)) in unit of Amp to obtain active power that flows during the charging process. The value of reactive components (\(\mathbf{i}_{\mathbf{q}}\)) is set to zero in unit of Amp., to achieve unity power factor and to mitigate the THD.  
The proposed approach uses three PI controllers (one voltage controller and two current controllers). The output loop of DC-link voltage (\(\mathbf{V}_{dc}\)) in unit of Volt is regulated by PI voltage controller, the measured \(\mathbf{V}_{dc}\) (V) with its pre-determined reference value \(\mathbf{V}_{dc\_ref}\) are compared to estimate the reference current signal \(\mathbf{i}_{\mathbf{d\_ref}}\). The first current controller is used to regulate the inner loop of \(\mathbf{i}_{\mathbf{d}}\) current component by minimizing the error between \(\mathbf{i}_{\mathbf{d}}\) with \(\mathbf{i}_{\mathbf{d\_ref}}\) in order to estimate the reference voltage signal \(\mathbf{v}_{\mathbf{d\_ref}}\). Likewise, the second PI current controller is used to regulate the second inner loop of \(\mathbf{i}_{\mathbf{q}}\) current component, where its \(\mathbf{i}_{\mathbf{q}}\) is set to zero to estimate the reference voltage signal \(\mathbf{v}_{\mathbf{q\_ref}}\). As illustrated in Figure 6, the characteristic of three PI controllers are given as follow:  
\[\mathbf{i}_{\mathbf{d\_ref}}=\mathbf{K}_{p1}(\mathbf{V}_{dc\_ref}-\mathbf{V}_{dc})+\mathbf{K}_{i1}\int (\mathbf{V}_{dc\_ref}-\mathbf{V}_{dc})\mathbf{d}\mathbf{t} \tag{10}\]  
\[\mathbf{v}_{\mathbf{d\_ref}}=\mathbf{v}_{\mathbf{d}}\ +2\pi\mathbf{f}\mathbf{L}_{s}\ \ \mathbf{i}_{\mathbf{q}}\ -(\mathbf{K}_{p2}(\mathbf{i}_{d \_ref}-\mathbf{i}_{\mathbf{d}})+\mathbf{K}_{i2}\int(\mathbf{i}_{d\_ref}-\mathbf{i}_{\mathbf{d}})\mathbf{d} \mathbf{t}) \tag{11}\]  
\[\mathbf{v}_{\mathbf{q\_ref}}=\mathbf{v}_{\mathbf{q}}\ -2\pi\mathbf{f}\mathbf{L}_{s}\ \ \mathbf{i}_{\mathbf{d}}\ -(\mathbf{K}_{p3}(\mathbf{0}-\mathbf{i}_{\mathbf{q}})+\mathbf{K}_{i3}\int(\mathbf{0}-\mathbf{i}_{\mathbf{q}}) \mathbf{d}\mathbf{t}) \tag{12}\]  
where in PI