Virtual Induction Machine Synchronizer

• Virtual Induction Machine synchronizer is a control strategy that emulates induction machines to provide self-synchronization and improved damping in low-inertia grids.
• It infers grid frequency directly from converter voltage and current measurements, eliminating the need for a Phase-Locked Loop while enhancing system stability.
• Simulation results indicate that VIM-based synchronization supports higher converter penetration and superior transient performance compared to conventional PLL-based methods.

A Virtual Induction Machine (VIM)-based synchronizer is a control and synchronization strategy for grid-following Voltage Source Converters (VSCs) in electric power systems, primarily targeting operations in low-inertia grids. The VIM approach emulates the dynamic properties of a physical induction machine, notably self-synchronization, oscillation damping, and standalone capability, using only converter output voltage and current measurements. By directly inferring grid frequency, the VIM excises the need for a traditional Phase-Locked Loop (PLL) synchronizer and thereby enhances small- and large-signal stability, while retaining conventional outer and inner converter control architectures (Stanojev et al., 2021).

1. Induction Machine Emulation: Mathematical Foundations

The VIM methodology is found on a rigorous emulation of the dq-frame model of a squirrel-cage induction machine, rotating at unknown $\omega_s$. The core stator and rotor equations are: $$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$

$$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$

$$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$

where $R_s$, $R_r$ are stator/rotor resistances, $L_s$, $L_r$, $L_m$ are inductances, all $d, q$ refer to the synchronous reference frame, and $$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$0 (slip).

Adopting a standard field-oriented alignment ($$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$1), $$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$2 and $$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$3 are algebraically eliminated. The slip, rotor dynamics, and electromagnetic torque equations simplify and support Laplace-domain transfer function derivations for robust controller synthesis. These relationships are central to the VIM structure (Stanojev et al., 2021).

2. Grid Frequency Recovery Without a PLL

The VIM synchronizer reconstructs the grid's synchronous speed $$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$4 by analogizing the converter’s filter-side voltage and current measurements ($$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$5, $$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$6) to an induction machine stator. The estimation proceeds as:

• Slip estimation: Based on the measured stator currents,

$$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$7

• Rotor speed (“swing” equation):

$$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$8

• Synchronous speed:

$$v_s^d = R_s i_s^d + \dot\psi_s^d - \omega_s \psi_s^q,\qquad v_s^q = R_s i_s^q + \dot\psi_s^q + \omega_s \psi_s^d$$9

where $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$0 is an initialization, $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$1 the emulated inertia, $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$2 the damping constant, and $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$3 the converter power.

By integrating $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$4 (with $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$5 the base frequency in radians/sec), the approach generates angle and speed references $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$6 needed for transformation and current-regulation.

3. Index-1 Differential-Algebraic System Representation

The VIM synchronizer is mathematically compacted into an index-1 differential-algebraic equation (DAE) system, which is structurally suitable for small-signal and eigenvalue stability analyses:

Differential states include: $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$7 Algebraic constraints incorporate current derivatives and frequency saturations: $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$8 The state and algebraic partitioning enables systematic linearization, well-posedness, and integration with other converter or network models (Stanojev et al., 2021).

4. Role within Converter Control Architectures

The VIM synchronizer is slotted as a direct replacement for the PLL in conventional grid-following VSC designs and is agnostic to the outer-loop structure:

• Outer (system-level) loop: Computes current setpoints via $$0 = R_r i_r^d + \dot\psi_r^d - \omega_\nu \psi_r^q,\qquad 0 = R_r i_r^q + \dot\psi_r^q + \omega_\nu \psi_r^d$$9–$$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$0 and $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$1–$$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$2 droop controllers, based on $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$3 provided by the VIM.
• Inner (device-level) loop: Cascaded current PI control (grid-following) or voltage+current PI (grid-forming), unchanged.
• Synchronization block: VIM derives state variables and angle from $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$4, as opposed to the PLL which uses only $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$5 and a PI on $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$6.

This modularity preserves existing controller infrastructure while delivering improved synchronization dynamics (Stanojev et al., 2021).

5. Stability Enhancement and Performance Studies

Linearization and eigenanalysis of the combined converter-network DAE with the VIM synchronizer underpin several key findings:

• The VIM synchronizer supports a larger droop-gain stability region than the PLL, approaching grid-forming converter performance (see Fig. 7 in (Stanojev et al., 2021)).
• Under weakening short-circuit ratio (SCR), VIM–VSCs remain stable even in very low-inertia or "very weak grids," whereas PLL–VSCs have a minimum SCR requirement of ≈1 p.u. (see Fig. 10).
• In multi-converter penetration studies, the maximum VSC share before instability rises from ≈70% (PLL) to ≈77–78% (VIM), nearly matching grid-forming limits (78.5%, see Fig. 9).
• Electromagnetic transient (EMT) simulation shows the VIM–VSC enhances damping after load- or generation-disturbances, yielding improved frequency nadir and RoCoF metrics.

These results demonstrate that VIM-based synchronization offers significantly improved small- and large-signal stability margins compared to PLL-based synchronization in low-inertia, high-penetration network settings (Stanojev et al., 2021).

6. Representative Simulation Results

Multiple EMT case studies confirm the operational and dynamic robustness of the VIM synchronizer:

Scenario	VIM–VSC Behavior	Comparison with PLL–VSC
Start-up and synchronization	Synchronizes within ≈0.5 s, automatic rotor alignment	Standard transients, no lock loss
Set-point tracking	Clean performance on 20% power/5% voltage steps	Outer/inner loops unchanged
Fault ride-through	Stable under 150 ms three-phase short-circuit	PLL–VSC may lose synchronization
Islanding	Maintains autonomous operation after grid loss	PLL–VSC loses lock
Frequency disturbances	Positive damping, nadir/RoCoF improvement	PLL–VSC less effective

Sensitivity analysis reveals that the initialization $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$7 has minimal impact, and the VIM exhibits robustness under moderate parameter uncertainties and a variety of network events (Stanojev et al., 2021).

7. Tuning Guidelines and Practical Recommendations

Initial VIM parameter selection is guided by physical induction machine designs for $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$8, $$\begin{aligned} \psi_s^d &= L_s i_s^d + L_m i_r^d, \qquad \psi_s^q = L_s i_s^q + L_m i_r^q \ \psi_r^d &= L_r i_r^d + L_m i_s^d, \qquad \psi_r^q = L_r i_r^q + L_m i_s^q \end{aligned}$$9, $R_s$0, $R_s$1, and $R_s$2. Key recommendations include:

• Set slip estimator’s proportional gain as $R_s$3; the derivative component $R_s$4 requires careful tuning (e.g., Ziegler–Nichols, typically $R_s$5) to prevent overshoot.
• Explore the $R_s$6 parameter space to avoid “holes” of instability; shift towards domains with guaranteed damping.
• Implement appropriate saturation on slip estimation ($R_s$7), consistent with expected slip ranges such as ±0.5 Hz, to guard against measurement noise and transients.
• Validate VIM performance in EMT or hardware-in-the-loop environments, with special attention to input measurement latencies and operation under unbalanced or distorted voltages.

By following these guidelines, practitioners can achieve the desired trade-off of improved system stability, disturbance rejection, and seamless integration into existing VSC control structures (Stanojev et al., 2021).