# Virtual Flux Control Methods for Grid-Forming Converters: A Four-Method Comparison
## 1 Introduction
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**GFM Approach** & **Class** & **Definition** & **Advantages / Disadvantages** \\ \hline \multirow{8}{*}{Methodology} & Droop control & & Simple implementation \\  & [14; 15; 16; 17; 18] & Mimics the behavior of a SG governor & � Doses not provide inertial response \\ \cline{2-4}  & Synchronous machine-based controllers [19; 20; 21; 22; 23; 24] & Mimics the behavior of a SG & ✓ Emulates inertial response and providing damping capabilities \\  & & & ✗ Output power can be more oscillating \\ \cline{2-4}  & Nonlinear control algorithms & Emulates the dynamics of a weakly nonlinear oscillator using a single dead-zone oscillator & ✗ Does not provide inertial emulation or fault-ride through capability \\ \hline \multirow{8}{*}{Voltage Vector Control} & Field-Oriented Control & Decouples control of both the converter’s voltage and frequency using two PI controllers & ✓ Good dynamics and waveforms Fixed switching frequency (PWM) \\  & Flatness theory & Ensures full control over all system variables by imposing a desired trajectory on flat outputs & ✓ Easy implementation Fixed switching frequency (PWM) \\ \cline{2-4}  & Direct control & Regulates a key variable of the converter without relying on conventional internal control loops & ✓ Fast dynamic response and high robustness Variable switching frequency \\ \cline{2-4}  & Model predictive control & Uses a mathematical model to predict the converter’s future behavior and determine the optimal control action & ✓ Fast transient performance, inclusion of nonlinearities and constraints Variable switching frequency (can be addressed through the cost function) \\ \hline \multirow{8}{*}{Current Limitation} & Switch to GFL control & Uses a backup PLL & ✗ May cause instabilities \\  & [49; 50] & States the reference currents of the internal current control & ✗ Can adversely affect the control angle, potentially compromising the converter’s stability \\ \cline{1-1} \cline{2-4}  & Virtual impedances & Increase the converter’s output impedance & ✓ Balances current limitation and stability \\ \cline{1-1} \cline{2-4}  & [54; 55; 56] & & ✗ Requires complex design \\ \cline{1-1} \cline{2-4}  & Virtual-flux-based control & Does not use current control loops & ✓ Easy design and implementation, limiting current in a robust way \\ \cline{1-1} \cline{2-4}  & Switch to GFL control & & ✗ May cause instabilities \\ \cline{1-1} \cline{2-4}  & [49; 50] & Uses a backup PLL & ✗ May cause instabilities \\ \cline{1-1} \cline{2-4}  & [51; 52; 53] & Saturates the reference currents of the internal current control & ✗ Can adversely affect the control angle, potentially compromising the converter’s stability \\ \cline{1-1} \cline{2-4}  & Virtual impedances & & ✗ Balances current limitation and stability \\ \cline{1-1} \cline{2-4}  & [54; 55; 56] & Increase the converter’s output impedance & ✗ Requires complex design \\ \cline{1-1} \cline{2-4}  & Virtual-flux-based control & & ✗ Easy design and implementation, limiting current in a robust way \\ \cline{1-1} \cline{2-4}  & [57; 58] & & ✗ \\ \hline \hline \end{tabular} _Nomenclature:_✓ for advantages and ✗ for disadvantages.  
\end{table} 
Table 1: Summary of GFM control methods and current limitation strategies. This paper proposes, as its main contributions, three new control schemes based on the virtual flux variable. However, unlike the vector control approach presented in [57], the new schemes are based on flatness theory, DC, and MPC. Their performance as GFM is evaluated and compared following the requirements set by the National Energy System Operator (NESO) [59; 60], using the results obtained on a Hardware-in-the-Loop (HIL) platform. This study is structured as follows. Section II describes the implemented system and how the virtual flux variable is measured. Section III presents the four control algorithms to be compared.