Virtual Flux Observer-Based Synchronization

• Virtual flux observer-based synchronization is a method for reconstructing internal electromagnetic flux from voltage and current measurements to derive synchronization variables.
• It employs advanced observer design methodologies like KRE, VIM, and full-state virtual-flux observers to achieve fast adaptation and robust performance under varying conditions.
• Experimental validations show rapid transient response, reduced overshoot, and enhanced stability for sensorless motor control and grid-connected converter applications.

Virtual flux observer-based synchronization refers to a family of control and estimation methodologies for power electronic systems and AC machines in which the system’s synchronization variables (typically angle, speed, or frequency) are derived through dynamic observation of a “virtual” electromagnetic flux variable. This virtual flux, reconstructed from measurable quantities such as voltages and currents, serves as an internal reference for controlling machine position, speed, or synchronizing power converters to a grid. Virtual flux observer strategies address sensorless machine operation, grid-forming and grid-following converter synchronization, and robust control in uncertain or weak grid conditions (Yi et al., 2022, Stanojev et al., 2021, Gao et al., 23 Jan 2026).

1. Theoretical Principles and Mathematical Foundations

Virtual flux observer-based synchronization is predicated on reconstructing the underlying electromagnetic flux dynamics of the system using voltage and current measurements. In AC machines (e.g., IPMSM, induction machine), the stator or rotor flux can be recovered or estimated using

$$\dot\lambda = -R\,i + v,\quad\lambda\in\mathbb{R}^2$$

where $\lambda$ is stator flux, $i$ stator current, $v$ stator voltage, and $R$ the resistance (Yi et al., 2022). For grid-connected converters, the virtual flux is often given by

$$\psi = L\,i + \psi_g,\quad \psi_g = (\omega_g J)^{-1}\,u_g$$

where $L$ is the series inductance and $J$ the $90^\circ$ rotation matrix (Gao et al., 23 Jan 2026).

The angle of the virtual flux vector can be used to extract synchronization variables such as electrical angle or frequency, for example:

$$\theta = \arctan\left(\frac{\varphi_2}{\varphi_1}\right)$$

for IPMSM active flux $\varphi$ (Yi et al., 2022).

In converter-grid applications, dynamic observer models reconstruct the virtual flux in the controller’s reference frame. Observer correction is applied using feedback from the estimated error,

$$\dot{\hat{\psi}} = -\omega_c J \hat{\psi} + u_c + K_o\,e$$

with $e$ the flux estimation error and $K_o$ the observer gain (Gao et al., 23 Jan 2026).

2. Observer Design Methodologies

Contemporary design for virtual flux observers encompasses both classical and advanced observer architectures.

• Kreisselmeier Regression Extension (KRE) Observer: For the IPMSM, a globally exponentially stable (GES) flux observer is synthesized using a virtual invariant manifold. Linear-time-invariant (LTI) filters process measured currents and voltages, yielding a linear regression in the unknown active flux, augmented by auxiliary estimator states ($Q$, $Y$, $\xi$), which constrain the observer to a forward-invariant manifold. The observer update law incorporates a dynamic correction via $E = -\gamma Y$, enabling high adaptation gains and fast transients without stability loss (Yi et al., 2022).
• Virtual Induction Machine (VIM)-Based Synchronizer: For grid-following voltage source converters, a VIM model is constructed, capturing the essential slip and torque dynamics of a real induction machine. Flux estimation is performed by integrating electrical and mechanical dynamics in the dq-frame, with virtual rotor-flux alignment, and the slip frequency is reconstructed using measured currents and a first-order filter. The synthesized synchronous speed is used in place of a conventional PLL (Stanojev et al., 2021).
• Full-State Virtual-Flux Observer (Grid-Forming Converter): The estimator uses the converter’s internal model, measurement-based feedback, and pole-placement design. A PI regulator operating on the flux error vector shapes the synchronization and flux estimation dynamics with high precision, allowing robust and decoupled frequency/angle tracking (Gao et al., 23 Jan 2026).

3. Synchronization Control Architectures

Virtual flux observer outputs are integrated into control architectures for both sensorless machine drives and power converter synchronization.

• Sensorless AC Motor Control: The observer-derived angle and speed estimates (e.g., $\hat{\theta}$, $\hat{\omega}$ for IPMSM) are used as feedback in vector-control or field-oriented control (FOC) schemes, enabling closed-loop operation without direct position sensors. The block flow for the KRE-based observer proceeds from measurement, filtering, dynamic regression, flux estimation, disturbance rejection, angle extraction, and integration into the FOC loop (Yi et al., 2022).
• Grid-Connected Converter Synchronization:
  • Grid-Forming (GFM) Control: The virtual flux observer feeds into a load-angle controller, forming a second-order synchronization subsystem decoupled from voltage regulation. Desired power references are mapped to load-angle setpoints; observer gains are chosen for arbitrarily precise bandwidth and damping via pole placement. The system operates as a stiff voltage source with model-based synchronization (Gao et al., 23 Jan 2026).
  • Grid-Following (GFL) Virtual IM Synchronizer: The VIM observer supplies a dynamically-estimated synchronization signal, replacing the Phase-Locked Loop (PLL) and providing grid-friendly features such as self-synchronization, oscillation damping, and standalone capability (Stanojev et al., 2021).

4. Stability Analysis and Performance Metrics

• IPMSM Flux Observer: Global exponential convergence is established using a quadratic Lyapunov function for the stacked observer error state, provided the regressor excitation, model parameters, and observer gains are properly bounded. Unlike earlier designs, the KRE observer’s stability is not restricted by low adaptation gain; high gains yield faster transients without compromising global stability (Yi et al., 2022).
• GFM Converter Virtual-Flux Observer: Linearization of the cascaded observer–synchronization–voltage plant yields analytic decoupling of flux and synchronization eigenmodes. Closed-loop pole locations remain robust to large variations in grid strength, with all dominant poles in the left half-plane across SCR variation (Gao et al., 23 Jan 2026).
• VIM Synchronization: Replacing the PLL with VIM augments damping of low-frequency modes, increases the small-signal stability region with respect to converter penetration and weak grids, and reduces the maximum instantaneous rate of change of frequency (RoCoF). No explicit Lyapunov function is found for the full system, but Jacobian eigenvalue placement suffices for stability assurance (Stanojev et al., 2021).

Performance summary for the KRE IPMSM flux observer (R=2.5 Ω, L_d=L_q=7.82 mH, ψ_m=0.10 Wb):

Mode	γ	Sim Settling Time	Exp Settling Time	Overshoot
Proposed	1	≈30 ms	≈40 ms	<5 %
Proposed	5	≈10 ms	≈15 ms	<3 %
[Ort21]	1	≈40 ms	≈60 ms	<8 %
[Ort21]	5	diverges/slows	large error	>10 %

The proposed observer achieves order-of-magnitude improvements in transient settling and accuracy at high adaptation gains (Yi et al., 2022).

5. Practical Implementation and Gain Selection

Key parameter guidelines are provided for stable and performant observer-based synchronization:

• Filter pole $\alpha$ (in LTI filters): Set as large as possible for rapid signal processing, but below a maximum critical value to preserve Lyapunov-based stability (experimentally, $\alpha_{\max} \approx 200\pi$) (Yi et al., 2022).
• Dynamic regressor weighting $a$: Determines memory depth of the dynamic regressor; typical recommended values $a \in [5\pi,50\pi]$.
• Adaptation gain $\gamma$: Large $\gamma$ directly accelerates error convergence. KRE architecture allows high $\gamma$ without destabilization (Yi et al., 2022).
• VIM Parameters: Begin from physical IM parameters; adjust virtual inertia and damping for desired system equivalents. Carefully tune slip derivative gain to avoid high-frequency instability (Stanojev et al., 2021).
• GFM Observer Gains: Set $K_o$ for decoupled flux dynamics and $k_p$, $k_i$ for desired synchronization damping and bandwidth. Match pole placement to anticipated SCR and converter bandwidth (Gao et al., 23 Jan 2026).

Limitations include loss of persistent excitation in very low speed regimes (requiring signal injection), parameter mismatch sensitivity, and digital control bandwidth constraints.

6. Experimental and Simulation Validation

• GFM Converter (20 kVA, 10 kHz DSP): Across grid impedance variation, active-power step tracking is achieved in ~20 ms with negligible voltage deviations. Under grid frequency ramps and large setpoint changes, the observer maintains synchronism and droop-like power control. Weak grid (L=1 pu) operation remains stable with nominal performance preservation (Gao et al., 23 Jan 2026).
• Grid-Following VIM Synchronizer: Demonstrated to lock onto grid frequency within 0.5 s (starting from $±0.1$ Hz error), remain synchronized through three-phase faults, and retain internal reference under grid loss (islanding). As converter penetration increases in large-scale networks (IEEE-39 bus), VIM enables ~85% converter replacement before instability—substantially higher than with PLL (Stanojev et al., 2021).
• IPMSM Sensorless Observer: Under 1000 rpm operation and large initial errors (90° angle, 200% flux), full synchronization is reached in ≤15 ms with <3% overshoot at high gain, in both simulation and hardware (Yi et al., 2022).

7. Comparative Advantages, Limitations, and Open Directions

Advantages:

• Eliminates the need for direct position sensors or external PLLs by reconstructing synchronization states from internal dynamical models and real-time measurements.
• Robust to variations in grid strength and machine parameters; enables grid-forming and grid-friendly converter operation.
• Loop dynamics and transient response are tunable via internal model-based gain selection and pole-placement.

Limitations and Open Questions:

• Parameter accuracy is essential; significant mismatches in inductance or voltage produce steady-state errors, addressable with adaptive or disturbance observer extensions (Gao et al., 23 Jan 2026).
• Operation on unbalanced or harmonic-rich grids is not addressed by current implementations, motivating future work generalizing the observer to multifrequency or negative-sequence flux estimation.
• Multi-converter synchronization, microgrid coordination, and digital implementation effects (sampling, quantization, latency) represent key domains for ongoing research (Gao et al., 23 Jan 2026).

Virtual flux observer-based synchronization unifies physical-model observer concepts from AC machine control with contemporary grid-connected converter architectures. Recent advances in invariant manifold-based observer design and integration with robust control law synthesis demonstrate globally exponential convergence, high dynamic performance, and wide-area applicability in experimental studies (Yi et al., 2022, Stanojev et al., 2021, Gao et al., 23 Jan 2026).