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Design-Oriented Transient Stability Analysis of PLL-Synchronized Voltage-Source Converters

Heng Wu, Xiongfei Wang

arXiv:1811.08926v2eess.SY

TL;DR

PLL-synchronized VSCs lack physical synchronization laws and can lose synchronism during large grid disturbances, particularly through VSC-grid interaction. The paper uses phase portraits to analyze first- and second-order PLL stability, quantifies damping effects, and proposes an adaptive PLL. It finds that the adaptive design preserves transient stability and phase-tracking accuracy, with simulations and experiments validating the analysis.

  • Problem

    Transient stability of PLL-synchronized VSCs during large grid disturbances is insufficiently characterized, especially when converter current alters the synchronizing PCC voltage through grid impedance.

  • Method

    The paper uses phase-portrait analysis of a nonlinear PLL model including VSC-grid interaction, supported by time-domain simulations and experimental tests.

  • Results

    The adaptive PLL is the only tested design reported as remaining synchronized in the severe-fault experiment, while simulations show both the higher-damping SRF-PLL and adaptive PLL remaining synchronized for a 0.14 p.u. voltage fault.

  • Takeaways & Limitations

    Switching between the SRF-PLL in steady state and first-order PLL during fault-occurring and fault-clearing transients preserves transient stability and phase-tracking accuracy when equilibrium points exist.

Abstract

from arXiv · show

Differing from synchronous generators, there are lack of physical laws governing the synchronization dynamics of voltage-source converters (VSCs). The widely used phase-locked loop (PLL) plays a critical role in maintaining the synchronism of current-controlled VSCs, whose dynamics are highly affected by the power exchange between VSCs and the grid. This paper presents a design-oriented analysis on the transient stability of PLL-synchronized VSCs, i.e., the synchronization stability of VSCs under large disturbances, by employing the phase portrait approach. Insights into the stabilizing effects of the first- and second-order PLLs are provided with the quantitative analysis. It is revealed that simply increasing the damping ratio of the second-order PLL may fail to stabilize VSCs during severe grid faults, while the first-order PLL can always guarantee the transient stability of VSCs when equilibrium operation points exist. An adaptive PLL that switches between the second-order and the first-order PLL during the fault-occurring/-clearing transient is proposed for preserving both the transient stability and the phase tracking accuracy. Time-domain simulations and experimental tests, considering both the grid fault and the fault recovery, are performed, and the obtained results validate the theoretical findings.

I. INTRODUCTION

The paper addresses the limited transient-stability analysis of PLL-synchronized VSCs during large grid disturbances, where VSC-grid interaction can drive PLL loss of synchronization. It develops a design-oriented analysis and proposes an adaptive PLL to preserve stability and phase tracking.

  • VSC control dynamics create instability phenomena including harmonic instability and loss of synchronization under changing grid conditions.
  • PLL dynamics introduce negative damping within their bandwidth and can destabilize VSCs under weak-grid conditions.
  • Transient synchronization stability during large grid disturbances remains comparatively underexplored despite PLL loss of synchronization contributing to a photovoltaic plant trip.
  • Existing adaptive PLL studies primarily consider PLL performance while overlooking VSC-grid interaction through voltage drops across grid impedance.
  • Freezing the PLL can avoid instability but prevents correct grid-phase detection and can violate reactive-current grid-code requirements.
  • The paper quantifies PLL damping requirements, shows when second-order PLL stabilization fails, and proposes switching to a first-order PLL during fault transients.

II. GRID-CONNECTED VOLTAGE SOURCE CONVERTERS

The paper models a grid-connected VSC as a controlled current source whose phase is regulated by an SRF-PLL, then incorporates line-impedance effects into the PLL dynamics. Equilibrium existence depends on grid voltage, impedance, and injected currents during faults.

  • During normal operation, current references are selected from the dc-link-voltage and reactive-power loops; during faults, reactive current follows grid-code requirements and active current avoids overcurrent.
  • Fast inner-loop dynamics are neglected because the PLL bandwidth is much lower, leaving PLL dynamics dominant during faults when outer loops are deactivated.
  • The SRF-PLL tracks grid phase by regulating the q-axis PCC voltage with a PI controller.
  • The PLL model includes PCC voltage contributions from the grid-connection-point voltage and line-impedance voltage drop caused by converter current.
  • The resulting second-order nonlinear phase-swing model characterizes PLL behavior while accounting for VSC-grid interaction.
  • Stable operation requires an equilibrium with vPCCq = 0, and equilibrium existence depends on injected currents, grid impedance, and fault voltage magnitude.

III. DESIGN-ORIENTED TRANSIENT STABILITY ANALYSIS

The analysis characterizes VSC transient stability through equilibrium-point structure and PLL phase portraits, showing how grid faults and PLL parameters govern loss of synchronism.

  • LOS mechanism: Severe faults can eliminate equilibrium points, making VSC synchronism loss inevitable under the specified current-injection condition.When −ImaxRline < −Vgcpfault, the voltage-angle curve has no equilibrium points.
  • LOS mechanism: With two equilibrium points, the stable equilibrium point c and unstable equilibrium point e determine whether the post-fault trajectory remains synchronized.The system has two points when −ImaxRline > −Vgcpfault and one point at equality.
  • LOS mechanism: Crossing the unstable equilibrium point can cause loss of synchronism, whereas trajectories returning to the stable equilibrium point recover synchronization after oscillation.The outcome depends on whether the PLL output frequency has recovered to the grid frequency before reaching the unstable equilibrium point.
  • PLL parameter effects: Shorter PLL settling time worsens transient stability when frequency-dependent line reactance and active-current injection create positive feedback.The smaller settling time implies a larger Kp and increased positive-feedback loop gain; pure resistance or zero active current removes this effect.
  • PLL parameter effects: Increasing the PLL damping ratio reduces phase overshoot and enhances transient stability in systems with two equilibrium points.The phase portrait is stable when it converges to the post-fault stable equilibrium point and unstable when it diverges.
  • Critical damping ratio: The required critical damping ratio increases with |vzq|/|Vgcpfault|, but no damping ratio can ensure convergence when the ratio equals 1.At this ratio, the PLL has a single equilibrium point and δ cannot converge regardless of the adopted damping ratio.

IV. ADAPTIVE PLL FOR THE TRANSIENT STABILITY ENHANCEMENT

The proposed adaptive PLL switches between the SRF-PLL and first-order PLL during grid transients to combine transient stability with phase-tracking accuracy. Its switching logic uses PLL-detected ROCOF, with design thresholds and filtering selected for stability and noise robustness.

  • First-order PLL: Setting Ki=0 converts the SRF-PLL into a first-order PLL that guarantees transient stability when equilibrium points exist.The first-order PLL avoids the loss-of-synchronization problem under this condition, but its phase-tracking limitation motivates adaptive switching.
  • Adaptive switching: The adaptive PLL operates as an SRF-PLL during steady state and switches to the first-order PLL only during fault-occurring and fault-clearing transients.Switching back to the SRF-PLL restores the steady-state phase-tracking behavior after the transient.
  • Adaptive switching: ROCOF detection changes the integral gain Ki between PLL modes, using the detected |dωPLL/dt| and a low-pass filter to attenuate high-frequency noise.The switching thresholds ROCOFPLL1 and ROCOFPLL2 determine transitions between the two modes.
  • Threshold selection: ROCOFPLL1 is selected between 2.5 Hz/s and 8.8 Hz/s, with ROCOFPLL1=5 Hz/s used in simulations and experiments.The upper boundary follows the stated parameter selection, while the lower boundary is tied to avoiding erroneous steady-state switching and a 2.5 Hz/s withstand capability.
  • Threshold selection: ROCOFPLL2=0.5 Hz/s is adopted to improve robustness against practical noise after the system reaches the new equilibrium point.In theory, |dωPLL/dt| converges to zero at equilibrium.
  • Limitations and implementation: A grid frequency deviation during the fault causes an inevitable steady-state phase-tracking error in first-order mode, although switching back to the SRF-PLL can compensate it.A 200 ms low-pass-filter time constant is selected to balance switching dynamics and robustness against noise; accurate reactive current injection can be maintained during most of the fault period.

V. SIMULATION AND EXPERIMENTAL RESULTS

Simulations and experiments validate the phase-portrait findings across severe symmetrical and asymmetrical faults, fault recovery, and distorted grid voltages. The adaptive PLL preserves synchronization while maintaining accurate phase tracking under conditions where fixed PLLs or freezing the PLL can fail.

  • Symmetrical faults: At Vgcp=0.14 p.u., ζ=0.5 becomes unstable, whereas ζ=1.5 and the adaptive PLL remain synchronized.The ζ=0.5 PLL saturates at 45 Hz, causing diverging δ; the results agree with the phase-portrait analysis.
  • Adaptive PLL: The adaptive PLL switches Ki to zero only during fault-occurring and fault-clearing transients, then restores SRF-PLL operation for accurate phase tracking.This switching provides transient stability, accurate phase tracking during the fault, and seamless transfer between operating modes.
  • Comparison with PLL freezing: Freezing the PLL preserves stability but causes incorrect phase detection, producing a 55º phase difference and violating the grid code.With the PLL active, the phase difference is 90º, indicating pure reactive current injection when loss of synchronization does not occur.
  • Symmetrical faults: At Vgcp=0.10 p.u., only the adaptive PLL remains synchronized when a single equilibrium point exists during the fault.The adaptive PLL also maintains purely reactive current injection, while freezing the PLL produces a 64º phase difference and fails to inject rated reactive current.
  • Asymmetrical faults: For asymmetrical faults, the adaptive PLL alone remains stable at the severe 0.10 p.u. positive-sequence-voltage condition.The simulations support applying the analysis to asymmetrical faults with a pre-filtered PLL and confirm further stabilization from first-order PLL operation.
  • Distorted grids: Under grid voltage containing 5% fifth-order and 8% seventh-order harmonics, the adaptive PLL retains switching logic and phase-detection accuracy.Experiments under the same harmonic distortion report satisfactory adaptive-PLL performance, corroborating the simulation.
  • Experimental validation: Experimental tests confirm that only the adaptive PLL remains synchronized under a 0.10 p.u. symmetrical fault.The result is attributed to automatically switching Ki to zero during fault-occurring and fault-clearing transients.

VI. CONCLUSION

The conclusion summarizes a phase-portrait analysis of PLL effects on VSC transient stability and validates the findings through simulations and experiments. It identifies when damping-ratio increases help, when first-order PLLs avoid loss of synchronization, and how adaptive switching combines stability with phase-tracking accuracy.

  • Main findings: Increasing the SRF-PLL damping ratio enhances transient stability when two equilibrium points exist during the fault.The critical damping ratio is identified from the voltage ratio |vzq|/|Vgcpfault|.
  • Main findings: When only one equilibrium point exists during the fault, SRF-PLL loss of synchronization is inevitable.
  • Main findings: The first-order PLL avoids loss of synchronization whenever equilibrium points exist but incurs steady-state phase-tracking error under steady grid-frequency drift.
  • Adaptive PLL: The adaptive PLL uses SRF-PLL operation in steady state and first-order PLL operation during fault-occurring and fault-clearing transients.It guarantees both transient stability and phase-tracking accuracy during the grid fault.
  • Validation: Time-domain simulations and experimental tests confirm all reported theoretical findings.
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