# Comprehensive model of a synchronous generator for single-node studies

Guillermo Bautista, Miguel García, Javier Marcos, Íñigo de la Parra, Luis Marroyo

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Guillermo Bautista, Miguel García, Javier Marcos, Íñigo de la Parra, Luis Marroyo. Comprehensive model of a synchronous generator for single-node studies. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0194 . hal-05101140

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Distributed under a Creative Commons Attribution 4.0 International License
# Comprehensive model of a synchronous
generator for single-node studies

Guillermo Bautista, Miguel García, Javier Marcos, Íñigo de la Parra, Luis Marroyo

Department of Electrical, Electronic and Communications Engineering

Institute of Smart Cities (ISC)

Public University of Navarre (UPNA)

Campus de Arrosadia, 31006 Pamplona, Spain

guillermo.bautista@unavarra.es

## ACKNOWLEDGMENT

This investigation is supported by the Spanish State Research Agency (AEI) under grant PID2022-142179OB-I00, funded by MCIN/AEI/10.13039/501100011033 and FSE+.

*Index Terms*—Generator, Grid integration, Low-Inertia Grid, State-Space.

**Abstract**—As power electronics increasingly replace Synchronous Generators (SGs), the focus on system frequency stability has grown. To study frequency dynamics during disturbances, time-domain models, such as the simple single-node model, are used. The commonly used single-node models lack sufficient representation of electric power dynamics, simplifying converter-interfaced generation (CIG) behavior as reduced system inertia. However, control strategies, such as Power System Stabilizers (PSS) on SGs or Power Oscillation Dampers (POD) on CIG, can modify the system-wide frequency response. This paper presents a comprehensive state-space model of a SG for single-node studies, offering flexibility to analyze electrical interactions involving the Automatic Voltage Regulator (AVR) and PSS, while facilitating the inclusion of power electronics. Additionally, the model provides a mathematical framework for understanding the relationships between system parameters through state-space representation and could be used to explore other critical interactions.

## I. INTRODUCTION

In the early stages of renewable energy generation, transmission system operators (TSOs) paid little attention to converter-interfaced generation (CIG). Grid codes often required these converters to inject power into the grid when available. However, they were often designed to disconnect or remain inactive during contingencies.

Voltage and frequency control tasks were always left to the synchronous generators (SGs). In such systems, a strong relationship exists between the inertia of rotating masses and frequency stability, defined as the ability of the power system to maintain frequency within an acceptable range following a load-generation imbalance caused by a severe disturbance [1]–[3]. For this reason, the increasingly widespread presence of CIG, at the expense of SGs, has raised concerns among TSOs about the system’s available inertia [4]. A reduction in the system inertia might cause the grid frequency to drop too low or too fast after a severe disturbance. In such a situation, low-frequency or Rate of Change of Frequency (RoCoF) relays will trip generators and load shedding protocols might activate. To avoid such outcomes, frequency control strategies must be in place to meet system requirements. Although current regulations do not fully address frequency control for CIG, several strategies have been proposed to enhance frequency control, such as reducing the response time of power reserves [5].

Additional control strategies are being increasingly included in CIG and SGs, such as Power Oscillation Dampers controllers (POD) or Power System Stabilizers (PSS), which can also modify the system-wide frequency response. For these reasons, although frequency stability has been extensively studied in the literature [6], [7], the relevance of this issue has persisted due to the continuous evolution of the electrical grid. In this context, the proper modelling of system-wide frequency response remains an open research question in the field, posing a major challenge for TSOs.

TSOs often analyze system-wide frequency response through time-domain simulations, utilizing either detailed dynamic models or simplified single-node approaches. Detailed dynamic simulations are employed by the North American Electric Reliability Corporation
(NERC) [8], and Transpower [9]. These models entail high complexity. In this approach, the system equivalent frequency can be calculated as the MVA-weighted average speed of synchronous machines, since measuring the frequency at a single node after a disturbance can lead to potential confusion and inaccuracies. The single-node approach is a much simpler model and is based on aggregating all SGs, non-synchronous generators, and loads. This approach is used by TSOs in Australia (AEMO) [10], the Nordic synchronous area [11], and Europe (ENTSOE) [4]. Some authors have also proposed a single-node model for the UK system in order to identify the future requirements for primary and secondary response to maintain frequency control [12]. Preparing and tuning time-domain simulations remains a relevant and challenging task [4].

One disadvantage of single-node models proposed in the literature is that they neglect the effects on voltage, angle deviations, and changes in line flows. These models only consider the influence of CIG in the system-wide frequency response through a reduction in inertia, as the electrical dynamics are not modeled. To study the future frequency behavior of the grid, it will likely be necessary to account for certain controls that influence voltage and may impact frequency, such as PODs or PSSs.

This paper proposes a simple state-space model of a SG for single-node studies that includes both the Automatic Voltage Regulator (AVR) and PSS to account for voltage effects within the system. Adding more degrees of freedom, the model enhances physical insights into system behavior. Additionally, it offers flexibility by allowing integration of other models to simulate effects of CIG or other load dynamics. The state-space representation results in a linearized model, which simplifies analysis and makes it more comprehensive than those created in commercially available simulation tools, providing a mathematical framework for understanding the relationships between system parameters.

## II. TITLE OF THE FIRST PARAGRAPH

## III. SYSTEM MODEL

### A. Swing equation

In a system composed of SGs, system-wide frequency response is an important parameter as it captures the mismatch between generation and consumption. This imbalance is described by the swing equation, expressed in per-unit and state-space form in Ec. 1, and is formulated in terms of deviations $\Delta$, as is common in small-signal models. However, in this paper, a linearized model is used to approximate the system's response to an

imbalance between generation and demand. The smaller the disturbance, the more accurately the linearized model represents the actual system behavior.

$$ \Delta\omega_r = \frac{1}{2Hs + D}(\Delta P_m - \Delta P_e) \quad (1) $$

Where $\Delta\omega_r$ is the per-unit deviation in system-wide frequency. $\Delta P_m - \Delta P_e$ represents the imbalance between generation and demand. D is the damping coefficient (pu torque/pu speed deviation) and H is the constant of inertia of the aggregated SGs (MWs/MVA), expressed as Ec. 2, where $H_i$ is the constant of inertia of an individual machine and $S_{B,i}$ its rated power.

$$ H = \frac{\sum_{i=1}^{n} H_i S_{B,i}}{\sum_{i=1}^{n} S_{B,i}} \quad (2) $$

### B. Assessing the basic single machine infinite bus model

To fully comprehend the single-node model, it is helpful to recapitulate some concepts covered in the single machine infinite bus model (SMIB) [13], represented in Fig. 1. This model is based on three fundamental state variables (Ec. 3 - Ec. 5) and on the 1.0 model [14], which includes the effects of one rotor winding in the d-axis and none in the q-axis, thereby accounting for the field winding and zero damper windings.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Circuit representation of a SMIB system.</figcaption></figure>

$$ s\Delta\omega_r = \frac{1}{2H}(\Delta P_m - \Delta P_e - D\Delta\omega_r) \quad (3) $$

$$ s\Delta\delta = \omega_0\Delta\omega_r \quad (4) $$

$$ s\Delta\psi_{fd} = \frac{\omega_0 R_{fd}}{L_{ad}} \Delta E_{fd} - \omega_0 R_{fd} \Delta i_{fd} \quad (5) $$

$\Delta\delta$ (rad) is the rotor angle and $\omega_0$ (rad/s) is the nominal angular velocity. Generation torque $\Delta P_m$, Electric demand $\Delta P_e$, rotor current $\Delta i_{fd}$ and rotor terminal voltage $\Delta E_{fd}$ are not state variables. Therefore, it is necessary to express these variables in terms of the state variables to develop the state-space model. This
expression forms the core of the model. Other dynamics of little interest for these studies will not be included in the model.

First, the dynamic of $\Delta P_e$ in rotor angle stability studies is described with the per-unit equations of the electrical machine [6], [14]. the effects of speed variation and flux derivatives are neglected.

Furthermore, it is known that electric torque, i.e., generation torque, depends on currents and total flux linkages. The currents in dq components are expressed in terms of state variables. These currents are obtained from the stator terminal voltage equations (Ec. 6 – Ec. 9) by eliminating $e_d$ and $e_q$. A list of nomenclature is in Appendix A.

$$e_d = -R_s i_d - \psi_q = -R_s i_d + (L_{ls} i_q - \psi_{aq}) \quad (6)$$

$$e_q = -R_s i_q + \psi_d = -R_s i_q - (L_{ls} i_d - \psi_{ad}) \quad (7)$$

$$e_d = R_E i_d - \omega_r L_E i_q + E_{bd} \quad (8)$$

$$e_q = R_E i_q + \omega_r L_E i_d + E_{bq} \quad (9)$$

Additionally, by linearizing the currents, equations Ec. 10 – Ec. 11 are obtained. Constants $m_1$, $m_2$, $n_1$ and $n_2$ depend on the total impedance (i.e., the aggregation of generator and infinite bus reactance) and initial conditions of the system. A detailed derivation of Ec. 10 – Ec. 11 is covered in [6].

$$\Delta i_d = m_1 \Delta \delta + m_2 \Delta \psi_{fd} \quad (10)$$

$$\Delta i_q = n_1 \Delta \delta + n_2 \Delta \psi_{fd} \quad (11)$$

Total flux linkages also depend on dq currents Ec. 12 – Ec. 13.

$$\Delta \psi_{ad} = \frac{1}{\frac{1}{L_{ad}} + \frac{1}{L_{fd}}} (-\Delta i_d + \frac{\Delta \psi_{fd}}{L_{fd}}) \quad (12)$$

$$\Delta \psi_{aq} = -L_{aq} \Delta i_q \quad (13)$$

By replacing Ec. 10 – Ec. 13 in Ec. 14, the torque is found to be a function only of state variables and can be substituted into the equation of motion (Ec. 15).

$$\Delta P_e = \psi_{ad0} \Delta i_q + i_{q0} \Delta \psi_{ad} - \psi_{aq0} \Delta i_d - i_{d0} \Delta \psi_{aq} \quad (14)$$

$$\Delta P_e = K_1 \Delta \delta + K_2 \Delta \psi_{fd} \quad (15)$$

Next, $\Delta i_{fd}$ is obtained by substituting Ec. 12 in Ec. 16.

$$\Delta i_{fd} = \frac{\Delta \psi_{fd} - \Delta \psi_{ad}}{L_{fd}} \quad (16)$$

Following this, the terminal voltage is expressed in Ec. 17.

$$E_{t0} \Delta E_t = (e_{d0} + \Delta e_d)^2 + (e_{q0} + \Delta e_q)^2 \quad (17)$$

By linearizing Ec. 6 and Ec. 7 and replacing Ec. 10 – Ec. 13 the terminal voltage is reduced to Ec. 18.

$$\Delta E_t = K_5 \Delta \delta + K_6 \Delta \psi_{fd} \quad (18)$$

$\Delta E_t$ is the input of the AVR and $\Delta E_{fd}$ is its output. The transfer function of the AVR is represented by a gain and a first-order low-pass block with time constant $T_a$. The PSS changes the voltage reference of the AVR and its input signal is the frequency deviation. These are ST1C and PSS1A type models, respectively [15]. Thus, the electric dynamics are derived with constants $K_1$ to $K_6$. Fig. 2 shows the block diagram representation of the SMIB system, and Fig. 3 shows the state-space form. The dot symbol (·) denotes the derivatives of the state variables with respect to time, and $\Delta Y$ indicates the flow area through the valve/gate.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Block diagram representation of a SMIB system.</figcaption></figure>

This model often neglects the influence of $\Delta P_m$ since it is primarily used in small-signal studies. To include it, from Ec. 3, $\Delta P_m$ is given as an output of the governor and turbine model, e.g., a unit with reheat steam technology.

The key factor of the model is that the electrical behavior ultimately depends on how the currents on the dq axis are modeled.
$$
\begin{bmatrix} \Delta\dot{\omega}_r \\ \Delta\dot{\delta} \\ \Delta\dot{\psi}_{fd} \\ \Delta\dot{v}_1 \\ \Delta\dot{v}_2 \\ \Delta\dot{v}_s \\ \Delta\dot{Y} \\ \Delta\dot{P}_m \end{bmatrix} = \begin{bmatrix} -\frac{D}{2H} & -\frac{K_1}{2H} & -\frac{K_2}{2H} & 0 & 0 & 0 & 0 & \frac{1}{2H} \\ \omega_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{\omega_0 R_{fd}}{L_{fd}} m_1 L'_{md} & -\frac{\omega_0 R_{fd}}{L_{fd}} \left(1 - \frac{L'_{md}}{L_{fd}} + m_2 L'_{md} \right) & -\frac{\omega_0 R_{fd}}{L_{mdu}} K_a & 0 & \frac{\omega_0 R_{fd}}{L_{mdu}} K_a & 0 & 0 \\ 0 & \frac{K_5}{T_r} & \frac{K_6}{T_r} & -\frac{1}{T_r} & 0 & 0 & 0 & 0 \\ -\frac{K_{STAB}D}{2H} & -\frac{K_{STAB}K_1}{2H} & -\frac{K_{STAB}K_2}{2H} & 0 & -\frac{1}{T_W} & 0 & 0 & \frac{K_{STAB}}{2H} \\ -\frac{T_1}{T_2} \frac{K_{STAB}D}{2H} & -\frac{T_1}{T_2} \frac{K_{STAB}K_1}{2H} & -\frac{T_1}{T_2} \frac{K_{STAB}K_2}{2H} & 0 & -\frac{T_1}{T_2} \frac{1}{T_W} + \frac{1}{T_2} & -\frac{1}{T_2} & 0 & \frac{T_1}{T_2} \frac{K_{STAB}}{2H} \\ -\frac{1}{RT_G} & 0 & 0 & 0 & 0 & 0 & -\frac{1}{T_G} & \frac{1}{T_{CH}} - \frac{1}{T_{CH}} \\ 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{T_G} & \frac{1}{T_{CH}} \end{bmatrix} \begin{bmatrix} \Delta\omega_r \\ \Delta\delta \\ \Delta\psi_{fd} \\ \Delta v_1 \\ \Delta v_2 \\ \Delta v_s \\ \Delta Y \\ \Delta P_m \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1/T_G \\ 0 \end{bmatrix} \Delta P_{m,ref}
$$

<caption>Fig. 3. State-space representation of a SMIB system.</caption>

### C. Single-node model

Without an infinite bus, the concept of power angle loses its meaning, and deviation of speed becomes a state variable. The model proposed in this paper is depicted in Fig. 4 and is also based on the elimination of stator terminal voltages in terms of dq components as in Ec. 6 - Ec. 9. The primary difference from the SMIB system is the consideration of the speed or frequency variation in Ec. 6 - Ec. 7 (stator equations) and represent the voltage drop across a resistance instead of the grid equations (Ec. 8 - Ec. 9):

<figure><img src="image_4.png" /><figcaption>Fig. 4. Circuit representation of a single-node system.</figcaption></figure>

$$e_d = -R_s i_d - \omega_r \psi_q = -R_s i_d + \omega_r (L_{ls} i_q - \psi_{aq}) \quad (19)$$

$$e_q = -R_s i_q + \omega_r \psi_d = -R_s i_q - \omega_r (L_{ls} i_q - \psi_{ad}) \quad (20)$$

$$e_d = R_E i_d \quad (21)$$

$$e_q = R_E i_q \quad (22)$$

Hence, by eliminating $e_d$ and $e_q$ in Ec. 19 - Ec. 22 we obtain Ec. 23 - Ec. 24 which indicate the expressions of the currents in dq axis.

$$i_d = \frac{\omega_r^2 L_{Tq}}{R_T^2 + \omega_r^2 L_{Tq} L_{Td}} \frac{L'_{ad}}{L_{fd}} \psi_{fd} \quad (23)$$

$$i_q = \frac{\omega_r R_T}{R_T^2 + \omega_r^2 L_{Tq} L_{Td}} \frac{L'_{ad}}{L_{fd}} \psi_{fd} \quad (24)$$

Where $R_T = R_s + R_E$, $L'_{ad} = L_{ad} // L_{fd}$, $L_{Td} = L'_{ad} + L_{ls} + L_E$ and $L_{Tq} = L_{aq} + L_{ls} + L_E$. Next, from Ec. 23 - Ec. 24, we find the partial derivatives of the state variables, that is, the speed variation $\omega_r$ and the flux linkage $\psi_{fd}$. These are the state variables we fundamentally care about, nevertheless we can include the influence of the load by adding the partial derivative of the load resistance $R_{load}$ as stated in Ec. 27 for constant $m_{3\omega}$.

$$\Delta i_d = m_{1\omega} \Delta \omega_r + m_{2\omega} \Delta \psi_{fd} + m_{3\omega} \Delta R_{Load} \quad (25)$$

$$\Delta i_q = n_{1\omega} \Delta \omega_r + n_{2\omega} \Delta \psi_{fd} + n_{3\omega} \Delta R_{Load} \quad (26)$$

$$m_{3\omega} = \frac{\partial i_d}{\partial R} = - \frac{2 R_T \omega_{r0}^2 L q}{(R_T^2 + \omega_{r0}^2 L_q L_{Td})^2} \frac{L'_{ad}}{L_{fd}} \psi_{fd0} \quad (27)$$

The current equations are once again replaced in Ec. 12 - Ec. 14 is repeated to create the model (Fig. 5). The linearized equations of dq currents are Ec. 25 - Ec. 26. See Appendix B for constants $m_{1\omega} - m_{3\omega}$ and $n_{1\omega} - n_{3\omega}$.

Hence, by substituting Ec. 25 and Ec. 26 into Ec. 12-Ec. 14 and using the linearized forms of Ec. 19-Ec. 20, we observe that the influence of another state variable on the dq axis currents can be seen as a direct disturbance on the generation torque, terminal voltage, and field voltage. See appendix B for the expressions of generation the torque and terminal voltage.

The state-space form is depicted in Fig. 6 and it differs from Fig. 3 by considering the load resistance $R_{load}$ as a disturbance instead of $\Delta P_m$ and by not including the influence of rotor angle on the constants in Appendix B. This distinction also applies to the block diagram representation. Furthermore, Fig. 7 summarizes the process of obtaining the model.
<figure><img src="image_5.png" /><figcaption>Fig. 5. Block diagram representation of a single-node system.</figcaption></figure>

## IV. VALIDATION OF DE MODEL BY SIMULATION

To validate the model, a time-domain simulation was carried out in the MATLAB/Simulink environment with the Simscape/Electrical/Specialized Power Systems library blocks. System parameters are specified in Table I.

<table><caption>TABLE I
SYSTEM PARAMETERS</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td><strong>Generator</strong></td><td></td></tr><tr><td>Nominal power S<sub>b</sub></td><td>555 MVA</td></tr><tr><td>Nominal voltage U</td><td>24 kV</td></tr><tr><td>Inductance d-axis L<sub>d</sub></td><td>1.81 p.u.</td></tr><tr><td>Inductance q-axis L<sub>q</sub></td><td>1.76 p.u.</td></tr><tr><td>Stator leakage inductance L<sub>fs</sub></td><td>0.16 p.u.</td></tr><tr><td>Rotor inductance L<sub>fd</sub></td><td>0.153 p.u.</td></tr><tr><td>Stator resistance R<sub>s</sub></td><td>0.003 p.u.</td></tr><tr><td>Rotor resistance R<sub>fd</sub></td><td>5.9782e-04 p.u.</td></tr><tr><td>Inertia constant H</td><td>4 s</td></tr><tr><td><strong>Load</strong></td><td></td></tr><tr><td>Load resistance R<sub>Load1</sub></td><td>1.2 Ω</td></tr><tr><td>Load resistance R<sub>Load2</sub></td><td>10.8 Ω</td></tr><tr><td>Damping coefficient D</td><td>1.5</td></tr><tr><td><strong>Governor & turbine</strong></td><td></td></tr><tr><td>Droop R</td><td>5%</td></tr><tr><td>Steam chest time constant T<sub>CH</sub></td><td>0.3 s</td></tr><tr><td>Main servo time constant T<sub>G</sub></td><td>0.2 s</td></tr><tr><td><strong>AVR</strong></td><td></td></tr><tr><td>Regulator gain K<sub>a</sub></td><td>90 p.u.</td></tr><tr><td>Regulator time constant T<sub>r</sub></td><td>0.02 s</td></tr><tr><td><strong>PSS</strong></td><td></td></tr><tr><td>Stabilizer gain K<sub>STAB</sub></td><td>9.5 p.u.</td></tr><tr><td>Washout filter time constant T<sub>W</sub></td><td>1.4 s</td></tr><tr><td>Numerator time constant for phase compensation T<sub>1</sub></td><td>0.154 s</td></tr><tr><td>Denominator time constant for phase compensation T<sub>2</sub></td><td>0.033 s</td></tr></tbody></table>

The grid nominal frequency, $f_0$, is 60 Hz, and the machine has one pair of poles, resulting in a synchronous speed $\Omega_m$ of 3600 rpm. The dynamic response of the state-space model is compared to that of the MATLAB/Simulink model at the nominal frequency, with an initial power injection of $P = 0.86$ p.u. to load 1 ($R_{Load1}$). The graphs in Fig. 8 - Fig. 9 represent the response of the system (frequency and generator active power) without PSS when a step of the load is introduced ($R_{Load1} = 1.2 \Omega$ to $R_{Load1}/R_{Load2} = 1.08 \Omega$). The state-space model is also compared to a classical single-node model, with an electric torque disturbance equivalent to the load disturbance. It is observed that when the AVR has a fast response, there is almost no voltage deviation, which makes its behavior similar to a classic single-node model that ignores voltage effects.

The graphs in Fig. 10 - Fig. 12 represent the response of the system (frequency, active power and generator terminal voltage) with PSS when the same step of load is introduced. The developed state-space model demonstrates the same dynamic response as the Simulink model. Fig. 8 - Fig. 12 show that ignoring the effect of voltage control dynamics in frequency response analysis can lead to errors. Specifically, Fig. 10 reveals that excluding the effect of PSS on voltage tends to result in a higher nadir (i.e., the lowest point on the graph). For single-node models, this can bias parameter selection since they do not fully capture how voltage and frequency response interact. It is important to note that in our analysis, the impact of the AVR and PSS has been intentionally exaggerated to illustrate this effect.

## V. CONCLUSION

This paper presents a comprehensive state-space model of a synchronous generator for single-node studies, incorporating an AVR and a PSS to simulate voltage effects. The model introduces additional degrees of freedom compared to commonly used single-node models, offering a flexible framework for analyzing interactions without relying solely on time-domain simulations. Validation of the model through comparison with a MATLAB/Simulink simulation demonstrated that the developed state-space model accurately replicates the dynamic response of more complex models created with commercially available simulation tools. Future research will focus on integrating power electronic generation models to properly analyze the frequency response of modern power systems.
$$
\begin{bmatrix}
\Delta \dot{\omega}_r \\
\Delta \dot{\psi}_{fd} \\
\Delta \dot{v}_1 \\
\Delta \dot{v}_2 \\
\Delta \dot{v}_s \\
\Delta \dot{Y} \\
\Delta \dot{P}_m
\end{bmatrix}
=
\begin{bmatrix}
-\frac{D'}{2H} & -\frac{K_{2\omega}}{2H} & 0 & 0 & 0 & 0 & \frac{1}{2H} \\
-\frac{\omega_0 R_{fd}}{L_{fd}} m_1 \omega L'_{md} & -\frac{\omega_0 R_{fd}}{L_{fd}} \left(1 - \frac{L'_{md}}{L_{fd}} + m_2 \omega L'_{md}\right) & -\frac{\omega_0 R_{fd}}{L_{md}} K_a & 0 & \frac{\omega_0 R_{fd}}{L_{md}} K_a & 0 & 0 \\
-\frac{K_{5\omega}}{T_r} & -\frac{K_{6\omega}}{T_r} & 0 & -\frac{1}{T_r} & 0 & 0 & 0 \\
-\frac{K_{STAB} D}{2H} & -\frac{K_{STAB} K_2}{2H} & 0 & -\frac{1}{T_W} & 0 & 0 & \frac{K_{STAB}}{2H} \\
-\frac{T_1}{T_2} \frac{K_{STAB} D}{2H} & -\frac{T_1}{T_2} \frac{K_{STAB} K_2}{2H} & 0 & -\frac{T_1}{T_2} \frac{1}{T_W} + \frac{1}{T_2} & -\frac{1}{T_2} & 0 & \frac{T_1}{T_2} \frac{K_{STAB}}{2H} \\
-\frac{1}{RT_G} & 0 & 0 & 0 & 0 & -\frac{1}{T_G} & 0 \\
0 & 0 & 0 & 0 & 0 & -\frac{1}{T_{CH}} & 0
\end{bmatrix}
\begin{bmatrix}
\Delta \omega_r \\
\Delta \psi_{fd} \\
\Delta v_1 \\
\Delta v_2 \\
\Delta v_s \\
\Delta Y \\
\Delta P_m
\end{bmatrix}
+
\begin{bmatrix}
-\frac{K_R}{2H} \\
-\frac{\omega_0 R_{fd}}{L_{fd}} m_1 L'_{md} \\
\frac{K_{6\omega}}{T_r} \\
-\frac{K_{STAB} K_R}{T_1} \\
-\frac{K_{STAB} K_R}{T_2} \\
0 \\
0
\end{bmatrix}
\Delta R_{Load}
$$

<figure><img src="image_6.png" /><figcaption>Fig. 7. Method of creating the state-space single-node model.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 9. Response to a step change in demand: Active power generation.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 8. Response to a step change in demand: System-wide frequency</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 10. Response to a step change in demand: System-wide frequency</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 11. Response to a step change in demand: Active power generation.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 12. Response to a step change in demand: Voltage deviation.</figcaption></figure>

# APPENDIX

## A. Nomenclature

*e_d*: d-axis voltage.

*e_q*: q-axis voltage.

*E_t*: terminal voltage.

*EFD*: field voltage in nonreciprocal per-unit system.

*E_b*: infinite bus voltage.

*ψ_d*: d-axis flux linkages.

*ψ_ad*: d-axis mutual flux linkage.

*ψ_q*: q-axis flux linkage.

*ψFD*: field flux linkage.

*L_d*: d-axis synchronous inductance.

*L_q*: q-axis synchronous inductance.

*L_ad*: d-axis mutual inductance.

*L'_ad*: d-axis transient mutual inductance.

*L_1s*: leakage inductance.

*L_E*: system inductance.

*R_E*: system resistance.

*R_s*: stator resistance.

*RFD*: field resistance.

*i_d*: d-axis current.

*i_q*: q-axis current.

*iFD*: field current.

*δ*: angle between q-axis and the infinite bus voltage.

*δ_i*: angle between q-axis and SG terminal voltage.

*ω_0*: nominal angular velocity

*ω_r*: system-wide frequency

*s*: laplace variable.

*H*: inertia constant.

*D*: damping coefficient.

*P_m*: generation torque.

*P_e*: demand torque.

## B. Equations

$$R_T = R_E + R_a \quad (B28)$$

$$L'_{ad} = \frac{L_{fd}L_{ad}}{L_{ad} + L_{fd}} \quad (B29)$$

$$L_{Td} = L_E + L'_{ad} + L_{ls} \quad (B30)$$

$$L_{Tq} = L_E + L_{aq} + L_{ls} \quad (B31)$$

$$D_{\omega} = R_{T}^{2} + \omega_{0}^{2} L_{Tq} L_{Td} \quad (B32)$$

$$m_{1\omega} = \frac{2\omega_0 L_{Tq} L'_{ad} \psi_{fd0}}{D_{\omega} L_{fd}} - \frac{2\omega_0 L_{Tq} L_{Td} \left( \frac{\omega_0^2 L_{Tq} L'_{ad} \psi_{fd0}}{D_{\omega}^2} \right)}{D_{\omega}^2 L_{fd}} \quad (B33)$$

$$m_{2\omega} = \frac{\omega_0^2 L_{Tq} L'_{ad}}{D_{\omega} L_{fd}} \quad (B34)$$

$$m_{3\omega} = -\frac{2R_T}{D_{\omega}^2} \left( \omega_0^2 L_{Tq} L'_{ad} \frac{\psi_{fd0}}{L_{fd}} \right) \quad (B35)$$

$$n_{1\omega} = -\frac{2\omega_0^2 L_{Tq} L_{Td}}{D_{\omega}^2} R_T L'_{ad} \frac{\psi_{fd0}}{L_{fd}} + \frac{R_T}{D_{\omega}} L'_{ad} \frac{\psi_{fd0}}{L_{fd}} \quad (B36)$$
$$n_{2\omega} = \frac{\omega_0 R_T L'_{ad}}{D_{\omega} L_{fd}} \qquad (B37)$$

$$n_{3\omega} = \frac{\omega_0 L'_{ad} \psi_{fd0}}{D_{\omega} L_{fd}} \left(1 - \frac{2R_T}{D}\right) \qquad (B38)$$

$$D' = n_{1\omega}(\psi_{ad0} + L_q i_{d0}) - m_{1\omega}(\psi_{aq0} + L'_{ad} i_{q0}) + D \qquad (B39)$$

$$K_{2\omega} = n_{2\omega}(\psi_{ad0} + L_q i_{d0}) - m_{2\omega}(\psi_{aq0} + L'_{ad} i_{q0}) + \frac{L'_{ad}}{L_{fd}} i_{q0} \qquad (B40)$$

$$K_R = n_{3\omega}(\psi_{ad0} + L_q i_{d0}) - m_{3\omega}(\psi_{aq0} + L'_{ad} i_{q0}) \qquad (B41)$$

$$K_{5\omega} = \frac{e_{d0}}{E_{t0}}(-R_s m_{1\omega} + \omega_0 L_{ls} n_{1\omega} + \omega_0 L_q n_{1\omega} + L_{ls} i_{q0} - \psi_{q0}) + \frac{e_{q0}}{E_{t0}}(-R_s n_{1\omega} - \omega_0 L'_{ad} m_{1\omega} - L_{ls} i_{d0} + \psi_{ad0}) \qquad (B42)$$

$$K_{6\omega} = \frac{e_{d0}}{E_{t0}}(-R_s m_{2\omega} + \omega_0 L_{ls} n_{2\omega} + \omega_0 L_q n_{2\omega}) + \frac{e_{q0}}{E_{t0}}(-R_s n_{2\omega} - \omega_0 L_{ls} m_{2\omega}) + \omega_0 L'_{ad} \left(\frac{1}{L_{fd}} - m_{2\omega}\right) \qquad (B43)$$

$$K_{7\omega} = \frac{e_{d0}}{E_{t0}}(-R_s m_{3\omega} + \omega_0 L_{ls} n_{3\omega} + \omega_0 L_q n_{3\omega}) + \frac{e_{q0}}{E_{t0}}(-R_s n_{3\omega} - \omega_0 L_{ls} m_{3\omega} - \omega_0 L'_{ad} m_{3\omega}) \qquad (B44)$$

$$K_{8\omega} = - \frac{m_{3\omega} L'_{ad} L_{ad}}{L_{fd}} \qquad (B45)$$

[3] R. Marconato and A. Berizzi, Small-Disturbance Angle Stability and Electromechanical Oscillation Damping. John Wiley & Sons, Ltd, 2013.

[4] G. Chown, J. Wright, R. van Heerden, and M. Coker, "System inertia and rate of change of frequency (rocof) with increasing non-synchronous renewable energy penetration," 11 2017.

[5] A. Dixon, Modern Aspects of Power System Frequency Stability and Control. Elsevier Science, 2019. [Online]. Available: https://books.google.es/books?id=9niWDwAAQBAJ

[6] P. Kundur, N. Balu, and M. Lauby, Power System Stability and Control, ser. EPRI power system engineering series. McGraw-Hill, 1994. [Online]. Available: https://books.google.es/books?id=0fPGngEACAAJ

[7] P. Anderson, A. Fouad, I. of Electrical, and E. Engineers, Power System Control and Stability, ser. IEEE Press power engineering series Power system control and stability. Wiley, 2003. [Online]. Available: https://books.google.es/books?id=8-xSAAAAMAAJ

[8] J. Tan, Y. Zhang, S. You, Y. Liu, and Y. Liu, "Frequency response study of u.s. western interconnection under extra-high photovoltaic generation penetrations," in 2018 IEEE Power & Energy Society General Meeting (PESGM), 2018, pp. 1-5.

[9] A. Prince-Pike, D. I. Wilson, I. Ilieva, A. Li, and m. Phethean, "Maintaining new zealand's electrical reserve management tool," in 2012 10th International Power & Energy Conference (IPEC), 2012, pp. 273-278.

[10] "Wem procedure: Essential system service quantities," Australian Energy Market Operator (AEMO), "Sydney, AU", Tech. Rep. "0.1", 2022.

[11] E. Agneholm, A. S., M. Kuivaniemi, and P. Ruokolainen, "Fcr-d design of requirements - phase 2," European Network of Transmission System Operators (ENTSO-E), "Brussels, BELG", Tech. Rep. 1, 2019.

[12] W. Murrell, L. Ran, and J. Wang, "Modelling uk power system frequency response with increasing wind penetration," in 2014 IEEE Innovative Smart Grid Technologies - Asia (ISGT ASIA), 2014, pp. 1-6.

[13] W. G. Heffron and R. A. Phillips, "Effect of a modern amplityne voltage regulator on underexcited operation of large turbine generators [includes discussion]," Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, vol. 71, no. 3, pp. 692-697, 1952.

[14] "Ieee guide for synchronous generator modeling practices and parameter verification with applications in power system stability analyses," IEEE Std 1110-2019 (Revision of IEEE Std 1110-2002), pp. 1-92, 2020.

[15] "Ieee recommended practice for excitation system models for power system stability studies," IEEE Std 421.5-2016 (Revision of IEEE Std 421.5-2005), pp. 1-207, 2016.

REFERENCES

[1] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. Van Cutsem, and V. Vittal, "Definition and classification of power system stability iee/cigre joint task force on stability terms and definitions," IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 1387-1401, 2004.

[2] N. Hatziargyriou, J. Milanovic, C. Rahmann, V. Ajjarapu, C. Canizares, I. Erlich, D. Hill, I. Hiskens, I. Kamwa, B. Pal, P. Pourbeik, J. Sanchez-Gasca, A. Stankovic, T. Van Cutsem, V. Vittal, and C. Vournas, "Definition and classification of power system stability - revisited & extended," IEEE Transactions on Power Systems, vol. 36, no. 4, pp. 3271-3281, 2021.