## 2 Hybrid Active Power Filter Concept
These topologies are the eight kinds of APFs introduced for the non-linear current compensation so far. The proposed hybrid APF schematic is presented in Figure 3, containing both shunt passive and shunt active power filters with novel control system-based sliding mode control (SMC). Figure 3: \(Cont.\)  
Figure 2: The conventional APFs; (**a**) four-switch three phase shunt APF, (**b**) nine-switch AC/AC converter with a quasi-DC link, (**c**) eight-switch four-leg converter, (**d**) back-to-back two level voltage source converter (VSC), (**e**) three-switch three-capacitor multi-level inverter (MLI), (**f**) six-switch two leg inverter, (**g**) 12-switch APF with feedback harmonic detection, (**h**) eight-switch dual parallel full bridge inverter, Reproduced with permission from [28], Copyright Elsevier, 2017. The state-space equations of the HAPF are mathematically formulated in Equation (1):  
\[ \begin{split} L_{c}\frac{d\left(i_{AFP,d}\right)}{dt}=\omega L_{c}i_{ AFP,d}-v_{Cd}-u_{d}V_{dc}+v_{Ld}\\ L_{c}\frac{d\left(i_{AFP,d}\right)}{dt}=-\omega L_{c}i_{AFP,d}-v_{ Cd}-u_{q}V_{dc}+v_{Lq}\\ C_{c}\frac{d\left(v_{cd}\right)}{dt}=i_{AFP,d}+\omega C_{c}v_{ Cd}\\ C_{c}\frac{d\left(v_{cq}\right)}{dt}=i_{AFP,q}-\omega C_{c}v_{ Cd}\end{split}  \tag{1}\]  
where \(i_{AFP,d}\) and \(i_{AFP,d}\) are the \(dq\) frame compensating currents, the passive filter voltages in the \(dq\) frame are \(v_{Cd}\) and \(v_{Cd}\), respectively, and \(v_{L}\) is the point of common coupling (PCC) voltage. The base equation for DC link capacitance is found in Equation (2). \[C_{dc}\frac{d\left(V_{dc}\right)}{dt}=u_{q}i_{AFP,q}+u_{d}i_{AFP,d} \tag{2}\]  
Therefore, the state-space equations of the above-mentioned relations in normal format are gathered in Equation (3). \[ \begin{split}\frac{d}{dt}X=A(X)+B(X)u\\ Y=C(X)+D(X)u\end{split}  \tag{3}\]  
where the matrixes \(X,\;u,\)\(A(X),\)\(B(X),\)\(C(X),\) and \(D(X)\) are given as follows:  
\[ \begin{split} X=\left[i_{AFP,q}\;,\;i_{AFP,d}\;,\;v_{Cd}\;,\;v_{ Cd}\right]^{T}\\ u=\left[u_{d}\;,\;u_{q}\right]^{T}\end{split}  \tag{4}\]  
Figure 3: HAPF scheme; (**a**) Shunt APF with series in passive filter, (**b**) control diagram. where the matrixes are represented in detail in Equation (5). \[A(X)=\left[ \begin{array}{c}\frac{1}{L_{c}}\Big{(}\omega L_{c}i_{AFP,d}-v_{Cd}-u _{d}V_{dc}+v_{Ld}\Big{)}\\ \frac{1}{L_{c}}\Big{(}-\omega L_{c}i_{AFP,d}-v_{Cq}-u_{q}V_{dc}+v_{Lq}\Big{)}\\ \frac{1}{L_{c}}\Big{(}i_{AFP,d}+\omega C_{c}v_{Cd}\Big{)}\\ \frac{1}{L_{c}}\Big{(}i_{AFP,q}-\omega C_{c}v_{Cq}\Big{)}\\ 0\end{array} \right] \tag{5}\]  
\[ \begin{array}{c}B(X)=\left[\begin{array}{c}\frac{1}{L_{c}}\left(-V_{dc} \right)&0\\ 0&\frac{1}{L_{c}}\left(-V_{dc}\right)\\ 0&0\\ 0&0\\ \frac{1}{L_{c}}\left(i_{cd}\right)&\frac{1}{L_{c}}\Big{(}i_{\alpha}\Big{)} \end{array}\right]\\ C(X)=\left[\begin{array}{c}i_{AFP,d}\\ i_{AFP,d}\end{array}\right]\\ D(X)=0\end{array} \]  
Based on multi-input multi-output (MIMO) control system concepts, around the set point of load feeding, the linearization has been done using the Jacobean matrixes as below:  
\[ \begin{array}{c}A_{L}=\left[\begin{array}{ccc}\frac{\partial A_{1}(X)}{ \partial x_{1}}&\dots&\frac{\partial A_{1}(X)}{\partial x_{n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial A_{n}(X)}{\partial x_{1}}&\dots&\frac{\partial A_{n}(X)}{ \partial x_{n}}\end{array}\right]\\ B_{L}=\left[\begin{array}{c}\frac{\partial B_{1}(X)}{\partial u_{1}}\\ \vdots\\ \frac{\partial B_{n}(X)}{\partial u_{n}}\end{array}\right].\end{array}  \tag{6}\]  
Therefore, the linear system is defined as shown in Equation (7). \[\frac{d}{dt}X=A_{L}X+B_{L}u+H+h \tag{7}\]  
where \(h\) represents the outgoing/external limited attenuations, and \(H\) corresponds to the indeterminate matrix part of the linearized state space model.