## 2 The MC and Supraharmonics
\[v_{s(t)}=V_{m}\sin\omega t \tag{1}\]  
\[i_{S}(t)=i_{L}(t)=i_{0}+i_{1}\sin(\omega t+\theta_{1})+\sum_{n=57.9.11...}^{ \infty}i_{n}\sin(n\omega t+\theta_{n}) \tag{2}\]  
\[P_{L(t)}=i_{1}V_{m}sin^{2}\omega tcos\theta_{1}+i_{1}V_{m}sin\omega tcos\omega t sin \theta_{1}+\sum_{n=3,57.-}^{\infty}V_{m}sin\omega ti_{n}\sin(\omega nt+\theta_ {n}) \tag{3}\]  
where  
\(v_{s(t)}\): supply voltage;  
\(V_{m}\): supply voltage amplitude;  
\(i_{S}(t)\): supply current;  
\(i_{L}(t)\): load current;  
\(i_{0}\): DC load current component;  
\(i_{1}\): first harmonic component;  
\(\omega\): supply frequency (50/60) Hz;  
n: harmonic number;  
\(P_{L(t)}\): load power. Active power filter design always begins with the extraction of the reference current from the load [30; 31; 32]. The instantaneous active and reactive power theory (P-Q theory) proposed by the authors in [33] is the most used technique for reference current generation. The principle is to extract the compensational error current from three-phase voltage and current measurements expressed in their alpha/beta domain using a transformation matrix, T, as seen in Equation (4). Ignoring the zero sequence currents, the instantaneous active, p(t), and reactive q(t), powers can be computed with Equation (5). Finally, decomposing p(t) and q(t) into direct and oscillatory components, the reference currents can be calculated from the oscillatory component of the active power, \(\widetilde{P}\), and the total reactive power, q, for a partial compensation. The compensational current extracted in Equation (7) is of same amplitude but 180\({}^{\circ}\) out of phase with the harmonic load current [34]. \[T=\sqrt{\frac{2}{3}} \begin{bmatrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{1}&-\frac{\sqrt{3}}{1}\\ \frac{1}{2}&\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}  \tag{4}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**References** & **Method** & **Conclusion** \\ \hline
[1] & Supraharmonics in electric vehicles & Frequency beating, tripping of residual devices \\
[3;6;17] & Measurement and standardizations & Reduces life expectancy of electrical equipment \\
[16] & Supraharmonics in multiple VSCs & Supraharmonics are distributed integer multiple of the switching frequency of VSC \\
[4;5] & Supraharmonics in power electronic loads and mode of compensation & Power converters are sources of supraharmonics. \\  & compensation & Zone-phase compensation \\
[18] & Modelling of EMI from supraharmonic devices & It is possible to model the effects of supraharmonics \\  & Power quality issues with MC and possible solution & Lyapunov control strategy, APF, Shunt and series power \\  & with active power filters (APF) & filters \\
[24; 25; 26; 27; 29] & Supraharmonics in MC & Large switching states of MC causes supraharmonics \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Similar works review. \[ \begin{bmatrix}P(t)\\ q(t)\end{bmatrix} = \begin{bmatrix}v_{a}(t)&v_{\beta}(t)\\ -v_{\beta}(t)&v_{a}(t)\end{bmatrix}\begin{bmatrix}i^{*}_{ca}(t)\\ i^{*}_{c\beta}(t)\end{bmatrix}  \tag{5}\]  
\[ \begin{bmatrix}i^{*}_{ca}(t)\\ i^{*}_{c\beta}(t)\end{bmatrix} = \begin{bmatrix}v_{a}(t)&v_{\beta}(t)\\ -v_{\beta}(t)&v_{a}(t)\end{bmatrix}\begin{bmatrix}-\bar{P}(t)\\ -q(t)\end{bmatrix}  \tag{6}\]  
\[ \begin{bmatrix}i^{*}_{ca}\\ i^{*}_{cb}\\ i^{*}_{cc}\end{bmatrix} =T^{-1} \begin{bmatrix}i^{*}_{ca}(t)\\ i^{*}_{c\beta}(t)\end{bmatrix}  \tag{7}\]  
The obtained reference currents are compared with the inverter current, in a closed-loop system, under an appropriate control mechanism to produce gating signals for the VSC. Triangular carrier pulse width modulated (PWM) control and the hysteresis current control strategies are the most applied control techniques for active filters [31]. HCCs are most applied due to their stability, easy implementation, and faster dynamic responses. Irrespectively, HCC controllers are most applied in low-frequency applications, and not very much is known about their responses in higher frequency applications.