# Inner-Loop Controllers for Grid-Forming Converters
### Virtual Impedance  
Virtual inductors have been used in inverter-based distributed energy resources to emulate the stator output impedance of conventional synchronous generators [6]. This is because high-frequency PWM (Pulse-Width Modulation) inverters typically require a small output filter to attenuate high-frequency harmonics and do not exhibit significant impedance at the nominal grid frequency. As a result, the feeder impedance becomes crucial for reactive power sharing among parallel DERs. To address this limitation, virtual impedance can be employed, allowing the inverter's output impedance to be made resistive, capacitive, or inductive, as desired. Since legacy power systems operate with inductive feeders, which have a strong relationship between active power frequency and reactive power voltage, it is reasonable to expect a trend toward the adoption of predominantly inductive virtual impedance. Furthermore, using virtual impedance in inverter-based DERs reduces the parameter sensitivity of microgrids to feeder impedance variations. Different virtual impedance implementations can be characterized, including through the Thevenin equivalent (or Theorem--Figure 4) or the Norton equivalent (Figure 5) and the relation between them [8]. Figure 4: Thevenin equivalent for virtual impedance implementation for a grid-forming converter. Recent research has explored enhancements to inner-loop controllers for grid-forming inverters by leveraging virtual impedance and admittance concepts. In the context of current limiting, research on grid-forming inverters emphasizes the use of virtual impedance [7]. Advanced strategies, such as adaptive transient virtual impedance control [22], have been proposed for both balanced [23] and unbalanced faults [24]. Similarly, the emulation of inertia and grid impedance as a means of assessing and improving the stability of grid-forming converters has been explored, emphasizing the critical interaction between inverters and the grid [25]. On the other hand, virtual admittance has been used for designing recent grid-forming converters' inner controllers aimed at current limiting [26]. The virtual resistance is adjusted to limit instantaneous overcurrent, while the virtual inductance is increased during the short circuit to maintain sinusoidal short-circuit currents [27]. An enhanced design that incorporates hybrid synchronization and virtual admittance loops to optimize system performance and robustness has also been introduced [28]. These advancements underscore the significance of virtual impedance and admittance-based approaches in addressing the challenges of stability and fault management in power grids dominated by renewable energy sources. Nevertheless, no direct comparison is made between virtual impedance and admittance. The integration of machine learning techniques into controller design for power electronic converters has garnered significant attention due to their rapid feedforward capabilities, compatibility with black-box models, and proficiency in function approximation. To address the stability challenges arising from the operational scenario-dependent small-signal dynamics of grid-following converters, the Easy Transfer Reinforcement Learning (ETRL) method, as presented in [29], leverages transfer learning and deep reinforcement learning to adapt a multi-objective controller designed for one converter to others with varying system parameters. Furthermore, a Physics-Informed Deep Transfer Reinforcement Learning (PIDTRL) approach for power balance control and triple phase shift modulation in Input-Series Output-Parallel Dual-Active Bridge (ISOP-DAB) converter-based auxiliary power modules for electric aircraft is proposed in [30]. This approach demonstrates the capability to adaptively determine optimal modulation variable combinations in stochastic and uncertain environments, circumventing the need for precise model information.