Improving Stability Margins with Grid-Forming Damper Winding Emulation

TL;DR

This work tackles the challenge of ensuring small-signal frequency stability in bulk power systems that feature line dynamics and heterogeneous bus dynamics. It introduces a reduced-order damper winding model for synchronous machines and a PD damper winding emulation control for grid-forming converters, together with a framework to certify stability in networks with line dynamics. The manuscript demonstrates that the damper-winding model can increase stability margins for both generators and condensers, and that PD emulation can outperform conventional droop control in heterogeneous networks, as verified by EMT simulations. The results offer a physically intuitive, practically implementable path to enhance grid-forming converter stability in future grids with higher penetration of converters and complex line dynamics.

Abstract

This work presents (i) a framework for certifying small-signal frequency stability of a power system with line dynamics and heterogeneous bus dynamics, (ii) a novel reduced-order model of damper windings in synchronous machines, and (iii) a proportional-derivative (PD) damper winding emulation control for voltage-source converters (VSCs). Damper windings have long been understood to improve the frequency synchronization between machines. However, the dynamics of the damper windings are complex, making them difficult to analyze and directly emulate in the control of VSCs. This paper derives a reduced-order model of the damper windings as a PD term that allows grid-forming controls for VSCs to emulate their effect on frequency dynamics. Next, a framework for certifying small-signal frequency stability of a network with heterogeneous bus dynamics is developed that extends prior results by incorporating line dynamics. Finally, we analytically demonstrate that PD damper winding emulation can improve the stability of grid-forming converter controls. These results are validated with electromagnetic-transient (EMT) simulation.

Figures (12)

• Figure 1: Model of the $d$-axis synchronous machine circuit, where $L_{\ell}$ denotes the stator leakage inductance, $L_{\text{a}d}$ denotes the $d$-axis magnetizing inductance, $L_{\text{D}d}$ and $R_{\text{D}d}$ denote the $d$-axis damper winding inductance and resistance, and $L_{\text{f}}$ and $\psi_{\text{f}}$ denote the field winding inductance and flux linkage. The $q$-axis circuit is modeled by replacing the field winding with an open circuit.
• Figure 2: The small-signal system model, using the synchronous generator model \ref{['eq:sg_tf']} and line model \ref{['eq:powerflow_vec']} is compared to an EMT simulation of the IEEE 9-bus model after a load step of $0.5$ p.u. at Bus 7. Incorporating the damper winding model allows the small-signal model to capture the frequency synchronization between the machines (dark lines in (c)), which a swing equation model without damper windings (faint lines in (c)) cannot capture.
• Figure 3: Block diagram of the normalized small-signal power system model. The normalized bus dynamics are represented by $G'(s)$ and the normalized power flow model is represented by $\frac{\mu(s)}{s}L'$.
• Figure 4: The interoperability regions in Condition \ref{['cond:interpretable']} are shown on the complex plane. Region 1 is shaded in blue, Region 2 in red, and Region 3 in green.
• Figure 5: The Bode plot of $\frac{1}{\omega}\mu(j\omega)g(j\omega)$ is shown for a 60 Hz system where $\rho=0.1$ and $g(j\omega) = g_{\text{SG}}(j\omega)$ with $H=3.7$, $T_{\text{G}} = 3$, and $k_g = 20$. The transfer function is plotted with no damper winding model (i.e. $\xi_{\text{SM}} = 0$) and with $\xi_{\text{SM}} = 0.0131$. The crossover frequency $\omega^{\text{SG}}_{\text{c}}$ for each value of $\xi_{\text{SM}}$ is indicated on the phase plot, and $\Delta M$ on the magnitude plot depicts the increased stability margin for $\xi_{\text{SM}} = 0.0131$.

Theorems & Definitions (3)

• Theorem 1: Frequency stability
• Remark 1
• proof