# Virtual Flux Control Methods for Grid-Forming Converters: A Four-Method Comparison
## 4 HIL Results and Comparison
### Low Voltage Ride-Through  
This test shows the response to a balanced three-phase voltage sag with a depth of 50% and a duration of 250 ms in the grid. Figure 12 presents the instantaneous voltage waveforms during sag for the four control algorithms. As observed, in all cases, the converter withstands voltage sags successfully and maintains stability after recovery, although differences in waveform characteristics can be noted. Both FBC and MPC exhibit resonance effects, distorting the voltage waveform, whereas the DC control introduces noticeable noise in the voltages. In contrast, the VFOC approach maintains voltages more effectively.  
Figure 13 illustrates the \(dq\) components of the virtual flux generated by the converter. The \(d\)-component behaves similarly across all four control strategies, although MPC shows a slightly lower magnitude compared to the others. However, significant differences are observed in the \(q\)-component behavior. Both MPC and FBC introduce noise in this component, which affects the waveform of the voltages and currents injected by the converter. This noise originates from the reactive current limiter. The reason why it impacts MPC and FBC but not DC and VFOC is that these two methods do not directly control the reference flux modulus calculated in the RPC. Instead, FBC regulates the flux produced in the filter inductor, \(\psi_{L}\), while MPC operates based on a future estimation of the virtual flux, \(\psi_{\varphi[k+1]}\).  
Another key observation is that the \(q\)-component of MPC remains shifted (around \(-0.15\) p.u.) compared to the other methods, leading to differences in the power response of the converter during the voltage sag, as will be analyzed later. Lastly, regarding dynamic response, DC proves to be the fastest control algorithm, responding quickly to both the entry and the exit of the voltage sag, whereas VFOC exhibits the slowest recovery once the disturbance clears.  
Figure 12: Instantaneous voltages during a three-phase fault.  
Figures 14 and 15 illustrate the active and reactive power of the converter, along with the output current magnitude, during which both the entry and exit of the voltage sag. All four control strategies behave similarly and according to (8), delivering a reactive current during the sag limited by the reactive current limiter. However, when analyzing the dynamic response at the entry of the fault, VFOC exhibits the most pronounced impact, generating a higher current peak (around 2.5 p.u.) compared to the other control schemes. Once the three-phase fault occurs, DC stabilizes the fastest, whereas MPC takes the longest to settle.  
Additionally, an important distinction is observed in the power delivered by the converter in this scenario. Unlike VFOC, FBC, and DC, the MPC strategy provides slightly less reactive power and instead injects a small amount of active power. This causes a shift in the control angle, leading to a nonzero \(q\)-component of the flux, as evidenced in Figure 13.  
Figure 14: Active and reactive power and current modulus during the voltage fault entry.  
Figure 13: Flux \(dq\) components during a three-phase fault.  
Regarding the exit from the voltage sag, DC is the most affected, producing the highest current peak (around 3 p.u.) at the moment of the fault clearing. On the other hand, as also reflected in the flux components in Figure 13, VFOC exhibits the slowest recovery, taking the longest to return to its pre-fault operating condition. Meanwhile, MPC and FBC demonstrate a nearly identical dynamic response throughout the process.  
Finally, Figure 16 presents the instantaneous converter output currents during the voltage sag, where the reactive current limiter is actively operating. Both MPC and FBC exhibit highly distorted currents, introducing the noise observed in the \(q\)-component of the flux shown in Figure 13.  
Figure 16: Instantaneous currents during a three-phase fault.  
Figure 15: Active and reactive power and current modulus during the voltage fault exit.