## 3 Control Strategy
### Shapf  
The prime motive role of SHAPF is to reduce the THD of the supply current thereby improving the power factor (PF) by injecting the appropriate current. In addition, it maintains a constant voltage across the DC-Link. The proposed system (i) performs abc-\(\alpha\beta\), \(\alpha\beta\)-dq, dq-\(\alpha\beta\), and \(\alpha\beta\)-abc domain conversions; and (ii) NFIHC is used for regulating voltage across DC-Link capacitor and reducing harmonics in the current waveform. The NFIHC compares the actual DC capacitor voltage \(V_{dc}\) with \(V^{ref}_{dc}\) and injects the error into the axis. The schematic diagram of the NFIHC is given in Figure 6. STF is used to generate SYP and for the splitting of FC and HC of the current. The domain transformations of STF, the design, and control part of the NFIHC are given below:  
The SHAPF is controlled by the d-q theory. According to it, load currents are transferred to \(\alpha\)-\(\beta\)-0 coordinates using [PERSON]'s transformation by Equation (20). \[ \begin{bmatrix}i_{l\_a}\\ i_{l\_\beta}\\ i_{l\_0}\end{bmatrix} =\sqrt{\frac{2}{3}} \begin{bmatrix}1&-\frac{1}{2}&-\frac{ 1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{bmatrix}\begin{bmatrix}i_{l\_a}\\ i_{l\_b}\\ i_{l\_c}\end{bmatrix}  \tag{20}\]  
By the implementation of the Laplace transformation, the STF separates the HC from FC by using the Equation (21). \[ \begin{bmatrix}i_{l\_a}(s)\\ i_{l\_\beta}(s)\end{bmatrix} =\frac{k_{2}}{s} \begin{bmatrix}i_{l\_a}(s)-i_{l\_a }(s)\\ i_{l\_\beta}(s)+i_{l\_\beta}(s)\end{bmatrix} +\frac{2\pi f_{\_2}}{s} \begin{bmatrix} i_{l\_a}(s)\\ -i_{l\_a}(s)\end{bmatrix}  \tag{21}\]  
Here \(k_{2}\) is the constant gain whose value is chosen as 20 and \(f_{\_2}\) is the cut-off frequency rating equal to system frequency 50 Hz. The HC in \(\alpha\)-\(\beta\)-0 coordinates are obtained by Equation (22). \[ \begin{array}{c}i_{l\_a}^{\prime\prime}=i_{l\_a}-i_{l\_a}\\ i_{l\_\beta}^{\prime\prime}=i_{l\_\beta}-i_{l\_\beta}\end{array}  \tag{22}\]  
In Equation (23), the Clarke transformation is applied to transform the supply voltage from \(abc\) to \(\alpha\)-\(\beta\)-0 domain. \[ \begin{bmatrix}V_{S\_a}\\ V_{S\_b}\\ V_{S\_0}\end{bmatrix} =\sqrt{\frac{2}{3}} \begin{bmatrix}1&-\frac{1}{2}&-\frac {1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{bmatrix}\begin{bmatrix}V_{S\_a}\\ V_{S\_b}\\ V_{S\_c}\end{bmatrix}  \tag{23}\]  
Figure 6: NFIHC for Shunt Converter. The SYP \(\sin(wt)\) and \(\cos(wt)\) are obtained from Equation (24), which eliminates conventional PLL. \[ \begin{bmatrix}\sin(wt)\\ \cos(wt)\end{bmatrix} =\frac{1}{\sqrt{\left(V{t_{S_{-}\!a}}\right)^{2}+\left(V{t_ {S_{-}\!\beta}}\right)^{2}}} \begin{bmatrix}V{t_{S_{-}\!a}}\\ -V{t_{S_{-}\!\beta}}\end{bmatrix}  \tag{24}\]  
The HC obtained in Equation (22) and the SYP obtained from Equation (24) are used to obtain HC in the d-frame. \[i^{\prime\prime}_{l\_d}=i^{\prime\prime}_{l\_a}\sin(wt)-i^{\prime\prime}_{l\_a }\cos(wt) \tag{25}\]  
The performance of SHAPF depends on both the reference current generation and DC-Link voltage. Therefore, it is important to maintain a constant DC-Link voltage. Whenever the load varies, the active power flow in the SHAPF varies, which in turn varies the DC-Link voltage. Therefore, to maintain the constant DC-Link voltage, the active power in SHAPF must be regulated. To achieve it, the active power flow in the SHAPF was made equal to the switching loss of the filter. The proposed NFIHC injects a suitable error signal \(\Delta i_{dc}\) in the d axis from the difference between the actual and reference DC-Link voltages. Therefore, the proper selection and design of the controller play a key role in DC-Link voltage regulation. The grid reference current in the d-axis is given by Equation (26). \[i^{ref}_{l\_d}=i^{\prime\prime}_{l\_d}-\Delta i_{dc} \tag{26}\]  
The reference \(d\)-\(q\) load currents are converted into \(\alpha\)-\(\beta\) domain by using Equation (27).