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Placing Grid-Forming Converters to Enhance Small Signal Stability of PLL-Integrated Power Systems

Chaoran Yang, Linbin Huang, Huanhai Xin, Ping Ju

arXiv:2007.03997v2eess.SY

TL;DR

The paper addresses the unresolved interaction between grid-forming and PLL-based converters in multi-converter systems, where PLL-based converters can become unstable under weak-grid conditions. Using matrix perturbation theory and network-based stability analysis, it shows that grid-forming placement increases effective grid strength and improves PLL-induced small-signal stability, while simulations validate the analysis and placement strategy.

  • Problem

    It remains unclear how grid-forming converters interact with PLL-based converters and how their placement affects or optimizes small-signal stability in multi-converter systems.

  • Method

    The paper derives a multi-converter small-signal model, applies matrix perturbation theory, and optimizes placement through the smallest eigenvalue of the weighted and Kron-reduced network Laplacian.

  • Results

    Grid-forming placement is equivalent to increasing grid strength characterized by gSCR, improving PLL-induced small-signal stability; simulations validate the analysis on test systems.

  • Takeaways & Limitations

    The results provide a network-based way to understand mixed grid-forming and PLL-based systems and select grid-forming locations for stability improvement.

Abstract

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The modern power grid features the high penetration of power converters, which widely employ a phase-locked loop (PLL) for grid synchronization. However, it has been pointed out that PLL can give rise to small-signal instabilities under weak grid conditions. This problem can be potentially resolved by operating the converters in grid-forming mode, namely, without using a PLL. Nonetheless, it has not been theoretically revealed how the placement of grid-forming converters enhances the small-signal stability of power systems integrated with large-scale PLL-based converters. This paper aims at filling this gap. Based on matrix perturbation theory, we explicitly demonstrate that the placement of grid-forming converters is equivalent to increasing the power grid strength and thus improving the small-signal stability of PLL-based converters. Furthermore, we investigate the optimal locations to place grid-forming converters by increasing the smallest eigenvalue of the weighted and Kron-reduced Laplacian matrix of the power network. The analysis in this paper is validated through high-fidelity simulation studies on a modified two-area test system and a modified 39-bus test system. This paper potentially lays the foundation for understanding the interaction between PLL-based (i.e., grid-following) converters and grid-forming converters, and coordinating their placements in future converter-dominated power systems.

I. INTRODUCTION

PLL-based converters dominate modern converter-integrated power systems but can suffer small-signal instability under weak-grid conditions. The paper analyzes how grid-forming converter placement affects this stability and develops an approach for choosing locations.

  • Motivation: PLL-based converters use phase-locked loops for grid synchronization and are widely deployed in renewable, HVDC, and energy-storage interfaces.They are also called grid-following converters because the PLL tracks grid angle and frequency.
  • Motivation: PLL-induced small-signal instability can arise under weak-grid conditions, potentially causing oscillations, converter trips, and economic loss.The instability is dominated by PLL dynamics, while oscillation frequency and stability margin also depend on other loop designs.
  • Motivation: Grid-forming converters do not require a PLL for synchronization and are robust to varying power-grid strength, making mixed converter populations practically relevant.Existing installations make replacing every PLL-based converter unrealistic, motivating systems containing both control types.
  • Research gap: The paper addresses how grid-forming and PLL-based converters interact, whether placement improves PLL-induced stability, and how locations can be optimized.These questions concern multi-converter systems rather than only single-converter infinite-bus settings.
  • Contributions: Using matrix perturbation theory, the paper shows that grid-forming placement is equivalent to changing grid strength characterized by gSCR and formulates an optimization for placement.The contributions include theoretical stability analysis, placement-dependent improvement, and a tractable optimization problem.

II. MULTI-CONVERTER SYSTEM MODELING

The paper models grid-forming and PLL-based converters through admittances and combines them into a closed-loop multi-converter system coupled by the power network.

  • System modeling: The modeling framework introduces admittance models for PLL-based converters and grid-forming converters, with VSM used as the prototypical grid-forming control.The models are combined to represent their interaction through the network.
  • System modeling: A grid-connected converter with an LCL filter can operate in either PLL-based or grid-forming mode.The corresponding control diagrams are presented in Fig. 1.

A. Admittance Modeling of PLL-based Converters

The PLL-based converter model represents synchronization, power control, filtering, and converter dynamics in a global dq-frame, yielding a 2×2 admittance matrix after linearization.

  • Reference frames: The converter’s three-phase voltages and currents are represented as space vectors in the controller’s dq-frame and transformed to the global dq-frame.The global frame uses angular frequency ωg = ω0 = 100πrad/s.
  • Converter dynamics: The LCL dynamics and current loop use converter, grid, and filter parameters together with PI-regulator transfer functions.The voltage-feedforward filter removes high-frequency components from voltage feed-forward signals.
  • Power control: Power control generates active- and reactive-power references that contribute to the current reference used by the converter.The relevant quantities include P_ref, Q_ref, P_E, and Q_E.
  • PLL synchronization: A PLL synchronizes the converter with the grid and determines the angle and angular frequency of the controller’s dq-frame.Its PI regulator defines the PLL dynamics used in the synchronization model.
  • Admittance model: Linearizing and combining the converter equations produces the PLL-based converter admittance model, represented by a 2×2 admittance matrix.The detailed expression of YPLL(s) is referenced separately in the paper.

B. Admittance Modeling of Grid-Forming Converters

The grid-forming converter model generates its current reference through voltage control and synchronizes by emulating a swing equation rather than using a PLL.

  • Voltage control: Unlike the PLL-based controller, the grid-forming current reference comes from a voltage loop with voltage-reference, capacitor-current, and current-feed-forward terms.The current reference is Iref = PIVC(s)(Vref − V) + jωCFV + kF Ig.
  • Voltage control: The voltage reference is Vref = 1 + j0, and kF denotes the current feed-forward coefficient.The voltage reference can also be supplied by a reactive-power control loop if needed.
  • Synchronization: Grid synchronization is achieved by emulating the swing equation, which determines the controller dq-frame angle and frequency.This replaces the PLL-based synchronization mechanism.
  • Admittance model: Combining the linearized converter, filter, voltage-loop, and swing-equation relations yields the grid-forming converter admittance model.The paper treats VSMs as the representative grid-forming control.
  • Admittance model: The grid-forming admittance generally has large magnitudes because voltage control is designed to provide limited output impedance.This property depends on appropriate control design.

C. Closed-Loop Dynamics of Multi-Converter Systems

The closed-loop model separates converter dynamics from network dynamics and couples them through the power network. Kron reduction represents the network seen by the converters, while block-diagonal converter models enable multi-converter analysis.

  • System configuration: The system contains PLL-based converters, grid-forming converters, interior nodes, and an infinite bus.Interior nodes are eliminated through Kron reduction, and the infinite bus is treated as grounded in small-signal modeling.
  • Network model: The network topology is modeled using transmission-line dynamics and a grounded Laplacian matrix.The grounded Laplacian is partitioned before Kron reduction eliminates the interior nodes.
  • Converter model: The converter model stacks injected currents and voltages and represents PLL-based and grid-forming dynamics in separate blocks.The capacity-ratio matrix scales converter quantities for per-unit calculations, while small power-angle differences are neglected.
  • Closed-loop model: The network and converter transfer-function models are combined to obtain the closed-loop dynamics of the multi-converter system.The network dynamics are derived from the Kron-reduced Laplacian, and the converter block-diagonal structure facilitates subsequent analysis.

III. IMPACTS OF GRID-FORMING CONVERTERS

This section analyzes how grid-forming converters affect PLL-based multi-converter systems using matrix perturbation theory and eigenvalue-based grid-strength measures. It establishes that grid-forming placement increases the effective grid strength and improves PLL-induced small-signal stability, with location determining the improvement.

  • Analysis setup: The analysis considers one grid-forming converter first and extends the result to multiple converters by repeating the same procedure.The single-converter case is set as m = 1 for analytical simplicity.
  • Perturbation analysis: Matrix perturbation theory shows that grid-forming converters interact with PLL-based converters through the network and alter the dominant poles.The interaction is represented through the frequency-dependent perturbation associated with the grid-forming converter.
  • Baseline stability measure: The dominant poles of PLL-based multi-converter systems are determined by the smallest eigenvalue λ1 of S−1Qred, the generalized short-circuit ratio.This eigenvalue reflects network connectivity and grid strength; larger λ1 indicates a larger stability margin.
  • Effective grid strength: Changing a converter from PLL-based to grid-forming control is equivalent to changing the effective grid strength from λ1 to λ′1.When the grid-forming converter has sufficiently large capacity, the perturbation term can be small in the control-effect frequency range.
  • Placement effect: The smallest eigenvalue increases after deleting the row and column associated with a grid-forming converter, yielding λ1 < λ′1.The same eigenvalue-deletion procedure extends the analysis to multiple grid-forming converters and their placement choices.
  • Stability conclusion: Changing any PLL-based converter to grid-forming control improves the PLL-induced small-signal stability of the multi-converter system.The result follows from the relationship between grid strength, the effective eigenvalue, and stability margin.

IV. OPTIMAL PLACEMENT OF GRID-FORMING CONVERTERS

The paper formulates grid-forming converter placement as selecting nodes that maximize the smallest eigenvalue of a weighted, Kron-reduced Laplacian. A greedy procedure approximates this combinatorial optimization more efficiently than enumeration.

  • Optimization formulation: Grid-forming locations are selected by maximizing the smallest eigenvalue after deleting the corresponding rows and columns from the weighted Laplacian.The objective represents equivalent enhancement of grid strength and system stability.
  • Greedy solution: Enumeration solves the placement problem for small systems, but its computational burden becomes unacceptable as converter scale increases.The paper therefore introduces a greedy suboptimal method for larger systems.
  • Greedy solution: The greedy method iteratively deletes one selected node and resolves the smallest-eigenvalue problem for the remaining network.The process repeats for q placements, updating L[i+1] from L[i].
  • Node selection: The first selected node can be interpreted as the farthest node from the grounded infinite bus, whose connection most increases network connectivity.This interpretation links the optimization to the smallest eigenvalue and network connectivity.
  • Node selection: Participation factors of the smallest eigenvalue identify a computationally efficient suboptimal node by selecting the node with the largest factor.This requires one eigendecomposition rather than repeated eigenvalue calculations.

V. SIMULATION RESULTS

The simulation section evaluates the proposed placement analysis and algorithm through detailed simulation results.

  • Simulation objective: Simulation studies assess the validity of the analysis and the effectiveness of the proposed algorithm for placing grid-forming converters.The evaluation uses the simulation results described in the subsequent case studies.

A. Case studies on a four-converter test system

The four-converter case study compares candidate placements using eigenvalue analysis and time-domain responses. Placement at Converters 3 and 4 gives the strongest reported stability improvement and matches the greedy solution.

  • Test system: The two-area system contains four converter nodes, interior nodes eliminated by Kron reduction, and an infinite bus treated as the grounded node.Equal converter capacities make S an identity matrix, so L equals Qred.
  • Model validation: The Bode analysis finds |∆λ(s)| around 0.01 from 1Hz to 200Hz, supporting its omission in PLL-induced instability analysis.This approximation underpins the subsequent placement results.
  • Optimal placement: For q = 2, exhaustive evaluation identifies Converters 3 and 4 as the optimal locations for increasing the PLL-induced stability margin.The candidate combinations are summarized in Table I.
  • Greedy placement: The greedy procedure selects α1 = 3 and α2 = 4, fully matching the optimal solution obtained by direct optimization.The first and second selections are supported by the corresponding eigenvalue and participation-factor comparisons.
  • Limitation: The greedy algorithm can significantly improve stability but is not guaranteed to produce the globally optimal placement.Developing rigorously optimal and computationally efficient solutions is left for future work.
  • Stability comparison: Damping ratio increases from very low in Case 1 to about 0.05 with grid-forming control at Converters 1 and 2, and about 0.4 at Converters 3 and 4.The eigenvalue results align with the time-domain responses under the overload event.

B. Case studies on a nine-converter test system

The nine-converter study evaluates optimal and suboptimal grid-forming placements in a modified 39-bus network through eigenvalue-based grid-strength analysis and overload-response simulations.

  • Test system: The nine converters are interconnected through a 39-bus network connected to an infinite bus at Bus 39.
  • Placement optimization: λ1 = 3.31 is the smallest eigenvalue of the network’s Kron-reduced Laplacian matrix before placement optimization.
  • Placement optimization: The globally optimal placement of two grid-forming converters is I = {1, 4}.
  • Placement optimization: The suboptimal placement {8, 9} increases the smallest eigenvalue to 12.03.
  • Dynamic responses: Under an overload event, both placements produce superior damping performance compared with all converters using PLL-based control.The overload occurs at 0.2 s and is cleared after 0.02 s.

VI. CONCLUSIONS AND DISCUSSIONS

The paper shows theoretically and through high-fidelity simulations that properly placed grid-forming converters strengthen the effective network and improve PLL-induced small-signal stability.

  • Conclusions: The placement of grid-forming converters is equivalent to enhancing power grid strength measured by the generalized short-circuit ratio.The gSCR is characterized by the smallest eigenvalue of the weighted and Kron-reduced Laplacian matrix.
  • Validation: High-fidelity two-area and 39-bus simulations verify that proper grid-forming placement can significantly improve small-signal stability in PLL-integrated systems.
  • Conclusions: Optimal locations can be determined by maximizing the relevant smallest eigenvalue through a computationally efficient optimization approach.This avoids directly optimizing the damping ratio of the dominant poles, which would require the full system state matrix.
  • Future directions: Future work includes placement decisions considering both small-signal and frequency stability, as well as stability-constrained network planning.

APPENDIX A SYSTEM PARAMETERS

Appendix A contains Table VI, which presents parameters for the two-area test system.

  • System parameters: Table VI is titled “Parameters of the Two-Area Test System.”
  • System parameters: The table belongs to the appendix’s system-parameter material.
  • System parameters: The supplied passage identifies the table but does not include its parameter values.
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