## 3 The Hysteresis Current Controller  
The implementation of the traditional constant bandwidth variable frequency HCC (CBVF-HCC) involves comparing the reference current to the inverter current within the hysteresis band, as seen in Figure 2. The aim is to obtain an error current, \(e_{r}(t)\), as minimum as possible within a hysteresis bandwidth and generate an appropriate gating signal. Considering a sinusoidal supply voltage with no harmonics, the inverter voltage, \(V_{f}\), is dependent on the appropriate switching scheme of the inverter switches as well as the modulation index. Under linear modulation mode and considering the current direction in Figure 2, \(V_{f}\) can be generalized as:  
\[V_{f}=(S_{1}-S_{4})\times\frac{V_{dc}}{2} \tag{8}\]  
For \(\text{Vc}=\frac{V_{dc}}{2}\)  
\[V_{f}= \begin{cases}V_{C}\ for\ S_{1}=1\\ -V_{C}\ for\ S_{4}=1\end{cases}  \tag{9}\]  
Figure 2: Per phase equivalent of Figure 1 using CBVF\(-\)HCC. \(v_{s(t)}\): supply voltage; \(i_{f}(t)\): inverter current; \(i^{*}_{f}(t)\): reference current; \(\epsilon\): hysteresis band current; \(e_{r}(t)\): error current; \(S_{1},S_{4}\) power inverter switches. Again, from Figure 2, the following expressions can be obtained for the rate of change in the inverter current and that of the reference current, respectively. \[\frac{di_{f}(t)}{dt}=\frac{1}{L}\Big{(}v_{f}-v_{s}(t)\Big{)} \tag{10}\]  
\[\frac{di_{f}^{*}(t)}{dt}=\frac{1}{L}\Big{(}v_{f}^{*}-v_{s}(t)\Big{)} \tag{11}\]  
The error current, \(e_{r}(t)\), is calculated for each phase of the VSC and obtained from Figure 3, where the inverter current tracks the reference current, within the hysteresis bandwidth, to generate gating pulses for the VSC. \[e_{r}(t)= \begin{bmatrix}e_{ra}\\ e_{rb}\\ e_{rc}\end{bmatrix} = \begin{bmatrix}i_{f}^{*}\\ i_{fp}^{*}\\ i_{fc}^{*}\end{bmatrix} - \begin{bmatrix}i_{f}a\\ i_{f}b\\ i_{fc}\end{bmatrix}  \tag{12}\]  
\[\frac{\Delta e_{r}}{\Delta t}=\frac{2\varepsilon}{t_{1}}=\frac{di_{f}^{*}(t)} {dt}-\frac{di_{f}(t)}{dt} \tag{13}\]  
From Figure 3, the rate of change in the error current can be deduced, for the rising and falling periods, as Equation (14) and Equation (15), respectively. From these, a relation between the hysteresis bandwidth and the switching frequency can be obtained by substituting Equations (14) and (15) into Equation (13). \[\frac{di_{f}^{+}(t)}{dt}=\frac{1}{L}\Big{(}v_{f}-v_{s}(t)\Big{)} \tag{14}\]  
\[\frac{di_{f}^{-}(t)}{dt}=-\frac{1}{L}\Big{(}v_{f}+v_{s}(t)\Big{)} \tag{15}\]  
\[\left\{ \begin{array}{l}2\varepsilon=\frac{di_{f}^{*}(t)}{dt}\times t_{1}- \frac{di_{f}^{+}(t)}{dt}\times t_{1}\\ -2\varepsilon=\frac{di_{f}^{*}(t)}{dt}\times t_{2}-\frac{di_{f}^{*}(t)}{dt} \times t_{2}\end{array} \right. \tag{16}\]  
Equation (16) is the traditional relation for obtaining the hysteresis current error, \(\varepsilon\). From the expression, a fixed band is obtained, but variable and uncontrollable switching frequencies will be observed. This results in several challenges in using the HCC controller including difficulties in harmonic filter design due to the uncontrollable nature of the  
Figure 3: Hysteresis current control bandwidth gate pulses. \(i_{rr}^{*}\): Reference current; \(i_{Hinv}^{+}\): Rising inverter current; \(i_{Hinv}^{-}\): Falling inverter current; \(\Delta i/\)HB: Hysteresis current bandwidth; \(S_{1}/S_{4}\): VSC switch 1 and 4; \(V_{out}\): VSC output voltage; \(V_{dc}\): VSC Capacitor voltage. switching frequency. This article resolves this issue by presenting a novel equation that expresses the maximum instantaneous switching frequency with respect to the error band current. The basic condition for the successful implementation of this controller is for the rate of change in the reference current to be smaller than that of the inverter current shown in Equation (16).