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Frequency Stability of Synchronous Machines and Grid-Forming Power Converters
Ali Tayyebi, Dominic Groß, Adolfo Anta, Friederich Kupzog, Florian Dörfler
TL;DR
The transition to converter-interfaced renewable generation reduces synchronous-machine inertia and creates stability challenges. This paper reviews and evaluates grid-forming controls in an IEEE 9-bus low-inertia case study, finding improved frequency stability but instability risks from current limits and mult time-scale interactions.
Problem
Loss of synchronous-machine inertia and control mechanisms creates stability challenges as power systems transition toward nearly 100% renewable generation.
Method
The paper reviews grid-forming control techniques and evaluates high-fidelity grid-forming converter and synchronous-machine models in the IEEE 9-bus test system.
Results
Grid-forming converters improve frequency-stability metrics versus an all-synchronous-machine baseline, while current saturation and fast converter–slow machine interactions can destabilize some controls.
Takeaways & Limitations
Robust grid-forming design must account for dc dynamics, effective ac current limitation, and differing converter and synchronous-machine time scales.
Takeaways & Limitations
Designing a robust ac current-limitation strategy that responds effectively to load-induced overcurrent and grid faults remains an open research problem.
Abstract
from arXiv · showhide
An inevitable consequence of the global power system transition towards nearly 100% renewable-based generation is the loss of conventional bulk generation by synchronous machines, their inertia, and accompanying frequency and voltage control mechanisms. This gradual transformation of the power system to a low-inertia system leads to critical challenges in maintaining system stability. Novel control techniques for converters, so-called grid-forming strategies, are expected to address these challenges and replicate functionalities that so far have been provided by synchronous machines. This article presents a low-inertia case study that includes synchronous machines and converters controlled under various grid-forming techniques. In this work 1) the positive impact of the grid-forming converters on the frequency stability of synchronous machines is highlighted, 2) a qualitative analysis which provides insights into the frequency stability of the system is presented, 3) we explore the behavior of the grid-forming controls when imposing the converter dc and ac current limitations, 4) the importance of the dc dynamics in grid-forming control design as well as the critical need for an effective ac current limitation scheme are reported, and lastly 5) we analyze how and when the interaction between the fast grid-forming converter and the slow synchronous machine dynamics can contribute to the system instability
I. INTRODUCTION
The transition to converter-interfaced renewable generation reduces synchronous-machine inertia and creates stability challenges. The paper examines grid-forming controls as a way to support frequency stability during the transition, including their limitations and interactions with synchronous-machine dynamics.
- Converter-interfaced renewable generation reduces synchronous-machine inertia and removes accompanying frequency and voltage control mechanisms.
- Grid-following controls respond to frequency measurements with delay, whereas grid-forming converters are envisioned to provide load-sharing, black-start, inertial response, and hierarchical regulation.
- Proposed grid-forming strategies include droop control, virtual synchronous machines, matching control, virtual oscillator control, and dispatchable virtual oscillator control.
- The study explicitly considers dc-link dynamics, dc-source response and limits, ac current limitation, and four grid-forming strategies in EMT simulations of the IEEE 9-bus system.
- Grid-forming converters improve frequency-stability metrics, but current saturation and interactions between fast converters and slow synchronous-machine dynamics can destabilize some controls.
- The article is organized around modeling, control strategies, simulation case studies, qualitative analysis, conclusions, and control-parameter selection.
II. MODEL DESCRIPTION
The study uses a test system containing both power converters and synchronous machines, with the section introducing the models of the individual devices and components.
- The case study models a low-inertia test system containing power converters and synchronous machines.
- The modeling description covers individual devices and system components used in the study.
A. Converter Model
The converter model represents dc-link energy dynamics, dc-source response and saturation, and averaged dc-ac conversion with filtered ac quantities.
- The converter is modeled in αβ-coordinates using an averaged full-bridge switching-stage representation.
- The dc-link model includes capacitance and losses, while the converter filter includes inductance, capacitance, and resistance.
- The model distinguishes switching-node current and voltage from filtered output current and voltage.
- The controllable dc source follows a first-order response characterized by its current reference and time constant.
- The dc-source current is saturated at a maximum value, representing limits of dc-dc converters, storage systems, or renewable generation.
B. Synchronous Machine Model
The synchronous-machine model includes electrical, mechanical, excitation, stabilization, governor, and turbine dynamics within the studied network representations.
- The study uses an eighth-order balanced, symmetrical, three-phase synchronous-machine model with six electrical and two mechanical states.
- The machine equations represent rotor motion, stator flux dynamics, field-winding dynamics, and three damper-winding states.
- Rotor speed, torques, winding fluxes, voltages, currents, and resistances are defined in the machine model.
- Excitation dynamics include an ST1A automatic voltage regulator and a simplified power-system stabilizer addressing the AVR’s destabilizing synchronizing-torque effect.
- Governor and turbine dynamics use proportional speed droop and first-order turbine dynamics, parameterized by governor droop and turbine time constant.
- The broader system representation includes converter-module aggregation and an IEEE 9-bus network with synchronous machines and converter systems.
C. Network Model
The network model uses an EMT simulation of the IEEE 9-bus test system to study transmission-level low-inertia dynamics, including detailed line, transformer, and load models. Each grid-forming converter is represented by an aggregate model matching the synchronous-machine rating.
- The study uses Sim Power Systems for an EMT simulation of the IEEE 9-bus test system.
- Transmission lines are modeled with nominal π sections including RLC dynamics, transformers with three-phase linear models, and loads as constant impedances.
- Each grid-forming converter aggregates 200 commercial modules, with an aggregate rating of 100 MVA equal to the synchronous-machine rating.Each module is rated at 500 kVA.
III. GRID-FORMING CONTROL ARCHITECTURES
The grid-forming architectures control converter dc-side and modulation-stage inputs through reference currents and modulation signals. Their low-level implementation tracks voltage references supplied by a grid-forming reference model.
- Grid-forming controls regulate the converter through a reference current for the dc energy source and a modulation signal for dc-ac conversion.
- The reviewed architecture combines ac voltage control, current limitation and control, and dc-voltage control for two-level voltage-source converters.
- The dc-voltage controller defines the reference dc current while the cascaded control tracks the voltage reference provided by the reference model.
A. Low-Level Cascaded Control Design
The low-level cascaded design uses a reference model and rotating-frame proportional-integral controllers to track the converter voltage reference.
- A reference model provides the dq voltage reference, including its angle and magnitude.
- Cascaded proportional-integral controllers operate in dq-coordinates rotating with the reference angle.
- The modulation signal is determined by controllers that track the voltage reference.
1) AC Voltage Control:
The listed control architecture distinguishes ac-side control, dc-side control, and low-level cascaded control.
- The architecture includes an ac-side control component.
- The architecture includes a dc-side control component.
- The architecture includes low-level cascaded control.
2) AC Current Limitation:
The AC current-limiting scheme scales the reference current when it exceeds the converter limit while preserving its direction, then uses current control to track the limited reference. The paper emphasizes that this simple scheme is useful for comparing grid-forming controls, although robust limitation remains an open design problem.
- The scheme limits the reference current when its magnitude exceeds the predefined AC current limit.
- The limited reference preserves the direction of the original current reference.
- Robust AC current limitation for load-induced overcurrent and grid faults remains an open research problem.Complex strategies also require careful controller tuning.
- The study uses the simple limitation strategy to provide a clear investigation of existing grid-forming control solutions.
- A current PI controller tracks the limited reference for the converter switching-node current.
3) AC Current Control:
The paper implements several grid-forming strategies with cascaded voltage and current control, including droop, VSM, matching control, and dVOC. Their formulations connect converter dynamics to synchronous-machine behavior while retaining distinct treatments of inertia, dc voltage, synchronization, and voltage regulation.
- Droop Control: Droop control trades off active-power and frequency deviations and uses PI voltage control to replicate synchronous-machine AVR behavior.The voltage loop regulates the direct-axis reference while reactive power varies to achieve exact voltage regulation.
- Virtual Synchronous Machine: VSM control adds virtual inertia and damping dynamics, with its dynamics reducing to droop control when Jr/Dp ≈ 0.The VSM voltage is regulated through PI control, and the same voltage, current, and modulation loops are retained.
- Matching Control: Matching control drives converter frequency from dc-link voltage and controls ac power through dc current, structurally matching synchronous-machine equations.The resulting GFC inertia is linked to dc-link capacitance, while dc-side losses and droop map to machine analogues.
- Dispatchable Virtual Oscillator Control: dVOC uses a decentralized oscillator-based law whose stability conditions include transmission-network dynamics and transfer capacity.Its polar-coordinate form exhibits droop-like frequency–active-power behavior near nominal steady state in an inductive network.
- Case Study: The study compares identically tuned paired converter models in the IEEE 9-bus system against an all-synchronous-machine benchmark.The comparison evaluates grid-forming strategies in the presence of synchronous machines using EMT simulation.
C. Impact of Grid-Forming Control on Frequency Metrics
Grid-forming converters improve synchronous-machine frequency metrics across disturbances, but current saturation exposes distinct stability mechanisms and limitations among control strategies.
- C. Impact of Grid-Forming Control on Frequency Metrics: The study normalizes synchronous-machine nadir and RoCoF by disturbance magnitude to compare frequency performance across load increases.Disturbances are applied at node 7 while measurements are taken at the synchronous machine at node 1.
- C. Impact of Grid-Forming Control on Frequency Metrics: Grid-forming converters improve both frequency metrics relative to the all-SM configuration, regardless of the control strategy.Their faster response reduces the remaining power imbalance affecting the synchronous machine.
- C. Impact of Grid-Forming Control on Frequency Metrics: Droop control and dVOC perform similarly, while VSM differs through its inertial term and matching control produces considerably higher RoCoF.The compared strategies respond differently because some regulate ac quantities while matching control also regulates the dc-link voltage.
- C. Impact of Grid-Forming Control on Frequency Metrics: The frequency-metric comparison is sensitive to controller-gain tuning and the RoCoF computation window, whose guideline was derived for systems fully operated by synchronous machines.Faster grid-forming dynamics may require a smaller RoCoF window in low-inertia systems.
- D. Response to a Large Load Disturbance: The large-disturbance study examines dc-current limitation when renewable generation has low headroom or the dc-dc converter stage is undersized.A 0.9 pu load increase is imposed on a 2.25 pu base load, with dc current approaching a ±1.2 pu limit.
- C. Impact of Grid-Forming Control on Frequency Metrics: Converters synchronize with each other quickly and then synchronize more slowly with the synchronous machine after a large load disturbance.This behavior reflects faster converter dynamics than conventional turbine dynamics.
- D. Response to a Large Load Disturbance: Matching control stabilizes the dc voltage despite dc-source saturation by accounting for dc-side dynamics while regulating ac dynamics.The converter can maintain approximately constant ac power when its frequency synchronizes with the synchronous machine and preserves their relative angle.
- D. Response to a Large Load Disturbance: Under prolonged dc-source saturation, droop, VSM, and dVOC discharge the dc-link capacitor because their angle and frequency dynamics are agnostic to dc conditions.An ac-current limitation scheme is identified as a potential remedy for preventing dc-link depletion.
E. Incorporating the AC Current Limitation
AC current limitation alone can destabilize several grid-forming controls under dc-source saturation, while a set-point modification can stabilize them. Stability also depends on the interaction between fast converter synchronization and slower synchronous-machine dynamics.
- Current-limitation behavior: The ac current limiter does not stabilize dc voltage for droop control, VSM, and dVOC under dc-source saturation.The limitation causes integrator windup, loss of ac voltage control, and eventual instability.
- Current-limitation behavior: A set-point modification activated above an ac-current threshold stabilizes dc voltage for droop control, VSM, and dVOC.It steers power injection away from critical limits, although the below-rated threshold changes the post-disturbance operating point.
- Time-scale interaction: A 1 s synchronous-machine turbine delay preserves stability with the standard limitation strategy, whereas the 5 s delay contributes to instability when dc and ac currents saturate.The study therefore identifies different system time scales as important for robust ac current-limitation design.
- Time-scale interaction: Fast converter responses can interact problematically with slow synchronous-machine responses, prolonging dc-source saturation and destabilizing mixed systems.In the reported mixed system, the synchronous machine takes several seconds to reach its increased post-event power injection.
- Frequency stability: Grid-forming converters improve frequency stability metrics relative to an all-synchronous-machine system, with droop, dVOC, and VSM showing greater improvement than matching control.The comparison uses reduced-order analysis and case-study frequency nadir and average RoCoF results.
- Dc-side dynamics: When dc-source current saturates, active power must be controlled to stabilize dc voltage; matching control achieves this through dc-voltage-dependent angle dynamics.The resulting dc-voltage deviation is proportional to frequency deviation, and matching control remains stable in the examined scenario.