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Stability of Droop-Controlled Low-Frequency Transmission Lines
Rafael Castillo-Sierra, Giri Venkataramanan, Dionisio Ramirez
TL;DR
LFAC converter control has received limited stability study, particularly beyond the Following-Follower Strategy. This paper uses eigenvalue-based small-signal analysis of a Droop-Controlled LFAC line and finds that dynamics depend on the converters’ combined droop gains, with the critical stability limit determined by line and operating parameters.
Problem
Stability of power-converter control in LFAC lines has been little studied, with prior studies using only the Following-Follower Strategy.
Method
The paper applies eigenvalue analysis to the small-signal model of a Droop-Controlled LFAC transmission line.
Results
The LFAC line dynamic is governed by the sum of the AC/AC converters’ droop gains, and a symbolic expression predicts the marginal-stability Global Droop Gain.
Takeaways & Limitations
The stability limit depends on the line’s length, R/X ratio, operating frequency, and voltage.
Takeaways & Limitations
The final paper will provide further details on system modeling and the mathematical derivation of the global critical-gain expression.
Abstract
from arXiv · showhide
Low-Frequency AC (LFAC) transmission systems employing power converters are being considered for a varity of applications. This work studies the small-signal stability of an LFAC transmission line controlled by the Droop Control Strategy. Eigenvalue Analysis is used to determine how the controller droop gains, operating frequency, transmission line parameters, and the operating point affect system stability. The results show that the overall system's dynamic is governed by the sum of the droop gains of the AC/AC converters. Analytical results that give insights on how the different system's parameters affect the critical stability point are presented. The results indicate that system stability is affected by the line's length, the line's R/X ratio, operating frequency and voltage.
1 Introduction
LFAC reduces transmission frequency through AC/AC converters, and this paper addresses the limited stability literature by analyzing a Droop-Controlled LFAC line. The analysis examines how line parameters, operating conditions, and converter droop gains affect stability.
- Motivation and system concept: LFAC is considered for long-distance transmission and offshore wind-farm interconnection because lower frequency mitigates voltage drops and transient-stability constraints.The stated motivation is that transmission corridors can consequently be used up to the conductor’s thermal limit.
- Motivation and system concept: LFAC uses AC/AC converters to interface low-frequency transmission lines or sections with standard-frequency systems.The reduced-frequency network can use control schemes including Following-Follower, Droop Control, or Virtual Synchronous Machine strategies.
- Research gap and contribution: Very few studies have examined stability in power-converter-controlled LFAC lines, and prior studies used only the Following-Follower Strategy.This work therefore focuses on small-signal stability under Droop Control.
- Research gap and contribution: The analysis identifies how line parameters, operating-point variables, and the two converters’ power-frequency droop gains affect point-to-point LFAC system stability.These are the principal parameter categories examined in the work.
2 Control Architecture Of The LFAC Branch
Each LFAC converter combines grid-side and low-frequency-side control loops, while an outer power loop implements Droop Control for voltage and frequency regulation. The architecture also includes inner current and voltage loops and a grid-side DC-voltage loop.
- Controller structure: Each frequency converter has grid-side and low-frequency-side controllers that regulate converter and LFAC-network variables.The grid-side controller stabilizes stored energy, while the low-frequency-side controller forms the network at the desired voltage and frequency.
- Grid-side control: The grid-side controller uses cascaded inner dq-axis current loops and an outer DC-voltage loop that modifies the d-axis current command.The inner loops are based on power commands.
- Low-frequency-side control: The low-frequency-side controller uses inner dq-axis current and terminal-voltage loops, with an outer loop controlling power-flow quantities.The outer power loop is the subject of the paper’s stability analysis.
- Droop control: Droop Control implements the outer power loop for LFAC voltage and frequency regulation and allows the converters to share power.The frequency and voltage commands are adjusted using active- and reactive-power errors with gains m_p and m_q.
3 Dynamic Modeling and Linearization of The LFAC System
The LFAC low-frequency side is modeled with an r-L transmission-line branch and converter power-frequency droop loops, then linearized for eigenvalue-based stability analysis. Under the stated assumptions, stability behavior is governed by the combined droop gains, with marginal stability occurring at a fixed sum in the case studied.
- Model assumptions and structure: The low-frequency side model represents the transmission line as an r-L branch between converter terminal voltages.The reduced-order model uses v1 and v2 as the converter voltages at the LF-side terminals.
- Model assumptions and structure: The model assumes constant converter voltage magnitudes, zero reactive-power droop gains, stiff inner DC dynamics, and omitted measurement filters.Measurement filters are explicitly excluded and deferred for future consideration.
- Linearization and stability analysis: The linearized transmission-line and transmission-angle dynamics are expressed in state-space form and assessed through the characteristic equation’s eigenvalues.The characteristic equation uses s as the Laplace operator, I as the identity matrix, and A as the state matrix.
- Droop-gain stability effects: Three roots arise: one remains real and negative, while the other two are complex conjugates that migrate rightward as mp1 increases.The pole-migration study varies mp1 from 0.0 to 20mrad/s/MW for mp2 values of 0.00, 0.99, and 1.98mrad/s/MW.
- Droop-gain stability effects: The marginally stable operating point occurs when mp1 + mp2 equals 1.98mrad/s/MW in the studied case.This result motivates defining the Global Droop Gain as mpg = mp1 + mp2.
- Droop-gain stability effects: The Global Droop Gain’s role is explained by the characteristic equation coefficient, which contains the combined droop gains when converter voltage magnitudes are approximately equal.The voltage-equality approximation follows from transmission-system voltage-regulation limits.
5 Critical Global Droop Gain
The Critical Global Droop Gain is the global gain that makes the system marginally stable, and it depends on line parameters, operating frequency, and voltage. The analysis identifies how line length, R/X ratio, frequency, and voltage affect this stability limit.
- The marginally stable operating point occurs when mpg reaches the critical value determined by the transmission line and operating conditions.
- Longer transmission lines have higher stability regions than shorter lines.
- The Critical Global Droop Gain, mpg(critical), is the global gain that makes the system marginally stable.
- Inductive transmission lines have lower mpg(critical) values than lines with more resistive components because ωl/ω0 measures the line’s R/X ratio.
- When ω0 >> ωl, mpg(critical) is directly proportional to the operating frequency.
- Higher transmission voltage negatively affects the system’s stability range because the critical gain depends on the inverse square of voltage.
6 Conclusions and Future Work
The paper presents an eigenvalue-based small-signal stability analysis of a Droop-Controlled LFAC transmission line. It finds that dynamics depend on the converters’ summed droop gains and that the allowable gain is constrained by line and operating parameters.
- The work presents a small-signal stability analysis of a low-frequency AC transmission line controlled by Droop Control.
- Eigenvalue Analysis identifies the sum of the converters’ droop gains as governing the LFAC line dynamic.
- A symbolic expression predicts the Global Droop Gain that makes the system marginally stable.
- The stability limit depends on line length, line R/X ratio, operating frequency, and voltage.
- The system imposes constraints on how much droop gain is allowed in the converters’ controllers.
- The final paper is expected to add detailed system modeling, mathematical derivation, reactive power-voltage loop effects, and simulation and experimental verification.