Static Synchronous Stability Analysis of Synchronous Condensers Based on the Simplified Heffron–Phillips Model

Abstract

To address insufficient dynamic reactive power support during large-scale new energy grid connection, synchronous condensers are widely used in centralized new energy delivery. As a special rotating electrical machine, their operational stability is critical to new energy power stations’ safe operation. Targeting practical application scenarios (synchronous condenser power delivery and new energy grid-connected systems with synchronous condensers), this paper establishes a simplified Heffron–Phillips model for their static stability analysis by integrating their actual operating characteristics into the traditional model. Specific electromagnetic torque component expressions are derived to reflect static stability. Mechanistically, it reveals the correlation between excitation system gain and torque, their impact on static synchronous stability, and obtains critical gain parameters for positive damping. Influencing factors are determined, and essential differences in additional torque characteristics between synchronous condensers and generators are clarified. Simulation models of the two systems verify the conclusions, providing theoretical support for engineering applications and a reference for practical parameter setting.

1. Introduction

1.1. Motivation

Over the past decade, increasing wind power penetration and other intermittent renewable sources have compromised grid stability [1,2]. To enhance system stability, it is necessary to deploy dynamic reactive power compensation devices [3]. Among these compensation devices, synchronous condensers outperform traditional power electronic equipment in terms of transient voltage support and short-circuit capacity supply [4,5]. In particular, distributed synchronous condensers are characterized by small capacity and short electrical distance to new energy units. They can be deployed in a layered manner to alleviate the voltage-induced tripping problem of new energy units and improve transmission capacity [6]. With their excellent inertia support and voltage regulation capabilities, synchronous condensers have become core equipment in new energy power systems [7,8].

In new energy power stations deploying synchronous condensers, their transient stability is intrinsically linked to the stable operation of the system. This stability is influenced by multiple factors, including fault types and LVRT depth. Liu et al. conducted systematic research into stability control schemes tailored to this specific scenario [9,10]. Poulose and Kim analyzed methods for enhancing the transient stability of new energy power stations [11], while Nair et al. formulated control strategies to suppress transient instability in synchronous condensers [12].

As another critical issue, the instability mechanism of static power angle stability has not been fully elucidated. Decomposing electromagnetic torque into synchronous torque and damping torque is a core method for small-signal stability analysis [13,14]. Ghimire et al. investigated the impact of synchronous condensers on small disturbance stability and short-circuit ratio enhancement under different converter control strategies for offshore wind farm scenarios [15]. Zhu et al. proposed an improved complex torque coefficient method, providing an effective tool for sub-synchronous resonance analysis in multi-machine systems [16]. Wang et al. (2025) established an optimized configuration model for synchronous condensers in multi-DC systems [17]. Wang et al. (2021) validated the suppression effect of synchronous condensers on sub- and super-synchronous oscillations in wind farms based on an impedance model [18].

Although existing studies have illustrated the effectiveness of synchronous condensers in suppressing power oscillations and other disturbances, research on their static stability characteristics under emerging new energy scenarios remains insufficient. Mechanically, torque characteristics directly reflect the static performance of synchronous condensers. This research gap indicates that in-depth analysis of torque characteristics is essential for improving the stable operation capability of power systems.

1.2. Contribution

To address the deficiencies in existing research regarding the stability mechanism and electromagnetic torque characteristics of synchronous condensers under new application scenarios, this paper makes the following contributions:

(1) Within the practical engineering application scenarios of synchronous condensers in modern power grids, this paper derives the parameters of the simplified Heffron–Phillips model for synchronous condensers under reasonable assumptions applicable to new energy integration.

This simplified model features a concise formulation, and its parameters can characterize the operating behaviors of synchronous condensers under both power output scenarios and new energy grid integration scenarios with synchronous condensers incorporated. It provides a novel methodological tool for synchronous stability analysis, oscillation damping assessment and small-signal analysis of power systems equipped with synchronous condensers, which enables direct investigation into the influences of key synchronous condenser parameters—the excitation system gain—on system synchronous stability.

(2) It clarifies the unique characteristics of the excitation system gain and additional torque of synchronous condensers, and establishes their correlations with operating conditions, rotor oscillation frequency, and excitation system gain. It is revealed that the torque characteristics and excitation system gain laws of synchronous condensers are opposite to those of generators.

(3) It builds typical power system simulation models, namely the synchronous condenser power delivery system and the new energy grid-connected system with synchronous condensers, to verify the correctness of theoretical analysis conclusions in the actual operation scenarios of current synchronous condensers, providing a reliable theoretical basis for engineering applications of synchronous condensers.

The core innovations of this study are reflected in three aspects. First, targeting the current application scenarios of synchronous condensers and combining practical operating conditions, the traditional Heffron–Phillips model is simplified, and a simplified model with clear physical significance and strong applicability is proposed. Second, this paper systematically reveals the unique dynamic characteristics of synchronous condensers that distinguish them from conventional generators, improving and enriching the theoretical system for dynamic analysis of synchronous condensers. Third, since the stability analysis of synchronous machines based on the Heffron–Phillips model relies on damping torque and synchronizing torque, the simplified Heffron–Phillips model proposed in this paper provides a novel methodological tool for synchronous stability analysis, oscillation suppression and small-signal analysis of power systems integrated with synchronous condensers, realizing methodological innovation in the modeling and analysis of synchronous condensers.

Compared with existing studies on synchronous stability, oscillation suppression and small-signal analysis, this paper focuses on the modeling operating conditions adapted to synchronous condensers. The established simplified model is more suitable for practical engineering scenarios, filling the methodological gap in the simplified modeling of synchronous condensers. Meanwhile, it also lays a theoretical foundation for the safe and stable operation of new energy grid-connected power systems.

2. Simplified Heffron–Phillips Model

2.1. Dynamic Equations of Synchronous Condenser

The Heffron–Phillips model is suitable for the static stability analysis of generators in single-machine infinite bus systems. Using this model, the electromagnetic torque generated by the generator excitation system can be resolved into synchronous torque components and damping torque components, enabling static stability analysis of the system based on physical meanings.

The Heffron–Phillips model is suitable for the dynamic analysis of synchronous machines under small disturbances and fixed operating points in a single-machine infinite-bus system, while it cannot accurately characterize the dynamic behaviors under strong nonlinearity and severe transient processes. Therefore, this model is only applicable to the theoretical analysis of small-disturbance stability.

The research system, shown in Figure 1, is simplified from an actual system.

The synchronous condenser adopts a third-order model, and a first-order differential equation of the fast excitation system is taken into account, resulting in a system of equations describing the dynamic process of the synchronous condenser, totaling four equations.

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\delta }{\mathrm{d}t}}=\omega {\omega }_0\\ {\displaystyle \frac{\mathrm{d}\omega }{\mathrm{d}t}}={\displaystyle \frac{1}{T_{\mathrm{j}}}}\left(P_{\mathrm{m}}-P_{\mathrm{e}}\right)\\ {\displaystyle \frac{\mathrm{d}{E}_{\mathrm{q}}^{\prime }}{\mathrm{d}t}}={\displaystyle \frac{1}{T_{\mathrm{d}0}^{\prime }}}\left(E_{\mathrm{fd}}-E_{\mathrm{q}}\right)\\ {\displaystyle \frac{\mathrm{d}{E}_{\mathrm{fd}}}{\mathrm{d}t}}={\displaystyle \frac{1}{T_{\mathrm{e}}}}\left(-E_{\mathrm{fd}}-K_{\mathrm{e}}{U}_{\mathrm{G}}\right)\end{array}\right.$$

(1)

where $\delta$ is the power angle of the synchronous condenser, $\omega$ is the angular velocity of the synchronous condenser, $T_{\mathrm{j}}$ is the synchronous condenser’s inertia time constant, P_m is the mechanical power of the synchronous condenser, P_e is the electromagnetic power, $E_{\mathrm{q}}^{\prime }$ is the transient electromotive force, $T_{\mathrm{d}0}^{\prime }$ is the direct-axis open-circuit transient time constant, $E_{\mathrm{fd}}$ is the excitation voltage, $E_{\mathrm{q}}$ is the no-load electromotive force, $T_{\mathrm{e}}$ is the excitation system time constant, $K_{\mathrm{e}}$ is the excitation system gain, and $U_{\mathrm{G}}$ is the terminal voltage of the synchronous condenser.

We linearize the equations to derive the system’s state equations. We further transform the equations into a component sum form for electromagnetic power, terminal voltage, and transient electromotive force—namely the Heffron–Phillips model. The model coefficients $K_1\text{–}{K}_6$ are essentially sensitivity coefficients of operating parameters to state parameters.

For electromagnetic power, it contains component $\Delta P_{\mathrm{e}1}$ proportional to the power angle and component $\Delta P_{\mathrm{e}2}$ proportional to the transient electromotive force. The primary focus is the partial derivative with respect to the power angle, which has the significance of a static stability criterion. Therefore, $\Delta P_{\mathrm{e}2}$ is substituted with a form proportional to the power angle. To simplify the analysis, $T_{\mathrm{e}}=0$ is adopted in the process, as shown in Equation (2).

$$\Delta P_{\mathrm{e}2}=-{\displaystyle \frac{K_2{K}_3\left(K_{\mathrm{e}}{K}_5+K_4\right)}{1+K_{\mathrm{e}}{K}_3{K}_6+s{K}_3{T}_{\mathrm{d}0}^{\prime }}}\Delta \delta$$

(2)

Among them, $s$ is the solution to the system state equation, corresponding to the rotor oscillation frequency. We substitute this into $s=j{\omega }_{\mathrm{d}}$ and separate the real and imaginary parts to obtain Equation (3).

$$\Delta P_{\mathrm{e}2}=\left(\Delta K_{\mathrm{S}}+\mathrm{j}\Delta K_{\mathrm{D}}\right)\Delta \delta =\Delta K_{\mathrm{S}}\Delta \delta +\Delta K_{\mathrm{D}}\Delta \omega$$

(3)

The real part is $\Delta K_{\mathrm{S}}\Delta \delta$ and the imaginary part is $\Delta K_{\mathrm{D}}\Delta \delta$. From a physical perspective, they correspond to the ability to maintain synchronization and the ability to suppress rotor oscillations, respectively. These two coefficients are the additional synchronous torque coefficient and the additional damping torque coefficient, distinguished from the synchronous torque coefficient $K_1$ and the inherent damping coefficient $D$. Their expressions are given in Equation (4).

$$\left\{\begin{array}{c}\Delta K_{\mathrm{S}}=-{\displaystyle \frac{K_2{K}_3\left(K_4+K_{\mathrm{e}}{K}_5\right)\left(1+K_{\mathrm{e}}{K}_3{K}_6\right)}{{\left(1+K_{\mathrm{e}}{K}_3{K}_6\right)}^2+{\left({\omega }_{\mathrm{d}}{K}_3{T}_{\mathrm{d}0}^{\prime }\right)}^2}}\hfill \\ \Delta K_{\mathrm{D}}={\displaystyle \frac{K_2{K}_3\left(K_4+K_{\mathrm{e}}{K}_5\right){\omega }_{\mathrm{d}}{K}_3{T}_{\mathrm{d}0}^{\prime }}{{\left(1+K_{\mathrm{e}}{K}_3{K}_6\right)}^2+{\left({\omega }_{\mathrm{d}}{K}_3{T}_{\mathrm{d}0}^{\prime }\right)}^2}}\hfill \end{array}\right.$$

(4)

The phasor relationship between the torque components is shown in Figure 2.

In Figure 2, the red arrow represents the increment of electromagnetic torque; the blue arrows denote the two components of the electromagnetic torque increment. Among them, the component with subscript 1 stands for the increment generated by the mechanical link, and the component with subscript 2 stands for the increment generated by the electromagnetic link. The electromagnetic torque component produced by the electromagnetic link can be further decomposed into the damping torque and synchronizing torque, which are indicated by the orange arrows.

Using torque to judge the static synchronous stability of a synchronous condenser, the condition for no self-excited oscillation is $K_{\mathrm{D}}>0$, and the condition for no aperiodic power angle instability is $K_{\mathrm{S}}>0$.

2.2. Model Simplification

A synchronous condenser is defined as a synchronous generator operating without a prime mover. During steady-state operation, it does not output active power, and its power angle is very small, approximately near zero. Based on the characteristics of actual operating conditions, two simplification conditions are proposed:

• The power angle is assumed to be zero, so $\sin \delta =\delta$ and $\cos \delta =1$ is satisfied;
• Based on simplification of condition (1), the operating variables directly related to the model coefficients are further simplified: $U_{\mathrm{Gd}}=0$, $U_{\mathrm{Gq}}=U_{\mathrm{G}}$, and $I_{\mathrm{q}}=0$.

The expression of the simplified model coefficients is shown in Equation (5).

$$\left\{\begin{array}{c}{K}_1={\displaystyle \frac{E_{\mathrm{Q}}U\left(x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }+r\delta \right)}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\hfill \\ K_2={\displaystyle \frac{E_{\mathrm{Q}}r}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\hfill \\ K_3={\displaystyle \frac{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}{x}_{\mathrm{q}\mathsf{\Sigma }}}}\hfill \\ K_4={\displaystyle \frac{U\left(x_{\mathrm{d}}-x_{\mathrm{d}}^{\prime }\right)\left(x_{\mathrm{q}\mathsf{\Sigma }}\delta -r\right)}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\hfill \\ K_5={\displaystyle \frac{U{x}_{\mathrm{d}}^{\prime }\left(r-x_{\mathrm{q}\mathsf{\Sigma }}\delta \right)}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\hfill \\ K_6={\displaystyle \frac{r^2+x_{\mathrm{e}}{x}_{\mathrm{q}\mathsf{\Sigma }}}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\hfill \end{array}\right.$$

(5)

where $r$ is the resistance of the system, $x_{\mathrm{d}}^{\prime }$ is the d-axis transient reactance, $x_{\mathrm{d}}$ is the d-axis synchronous reactance, $x_{\mathrm{q}}$ is the q-axis synchronous reactance, $x_{\mathrm{e}}$ is the interconnecting reactance between the synchronous condenser terminal and the infinite bus system, and $U$ is the infinite system voltage.

After simplification, not only is coefficient $K_3$ a constant, but coefficient $K_6$ is also a constant. For the sum terms involving the power angle (e.g., $x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }+r\delta$ in $K_1$), since the power angle is approximated to zero, $x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }+r\delta =x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }$, $x_{\mathrm{q}\mathsf{\Sigma }}\delta -r=-r$, and $r-x_{\mathrm{q}\mathsf{\Sigma }}\delta =r$ can be approximately considered. Therefore, the expressions of $K_1$, $K_4$, and $K_5$ can be further simplified to

$$\left\{\begin{array}{c}{K}_1={\displaystyle \frac{E_{\mathrm{Q}}U{x}_{\mathrm{d}\mathsf{\Sigma }}^{\prime }}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\\ K_4=-{\displaystyle \frac{U\left(x_{\mathrm{d}}-x_{\mathrm{d}}^{\prime }\right)r}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\\ K_5={\displaystyle \frac{U{x}_{\mathrm{d}}^{\prime }r}{r^2+x_{\mathrm{d}\mathsf{\Sigma }}^{\prime }{x}_{\mathrm{q}\mathsf{\Sigma }}}}\end{array}\right..$$

(6)

With the infinite bus voltage kept constant, an increase in the synchronous condenser’s reactive power output leads to a reduction in the power angle across its operating range, the current amplitude increases, and $E_{\mathrm{Q}}$ increases. Thus, $K_1$ and $K_2$ increase with the growth of reactive power output. Since $E_{\mathrm{Q}}$ is relatively large, $K_1$ is also large, which can maintain a certain level of synchronizing stability. The remaining model coefficients are approximated as constant values after simplification. As the amplitudes of all operating variables are positive, all model coefficients except $K_4$ are greater than zero.

2.3. Scope of Application of Model

The limitation of the simplified model lies in that its applicable scope is determined by the underlying assumptions. The assumption is that the excitation system time constant $T_{\mathrm{e}}=0$ corresponds to the scenario with a fast-response excitation system. The assumption of a near-zero power angle, as well as its derived conditions such as $I_{\mathrm{q}}=0$ and $U_{\mathrm{Gd}}=0$, is satisfied for all operating conditions when the synchronous condenser is connected as a single unit to the sending-end system.

For new energy integration scenarios, in renewable energy delivery systems equipped with synchronous condensers, if the point of common coupling (PCC) of new energy units is selected as the reference node, the power angle of synchronous condensers—which only output reactive power during steady-state operation—remains approximately zero. Consequently, all simplification conditions adopted for the traditional Heffron–Phillips model in the above context are still satisfied, and the subsequent analysis conclusions regarding the static stability mechanism of synchronous condensers are equally applicable.

3. Synchronous Condenser Torque Characteristic Analysis

3.1. Critical Value of Excitation System Gain

For the expression of the additional torque coefficient, the sign (positive or negative) of the item $K_4+K_{\mathrm{e}}{K}_5$ determines the signs of the two torque coefficients. When $K_4+K_{\mathrm{e}}{K}_5>0$ is satisfied, the additional synchronous torque coefficient is negative and the additional damping torque coefficient is positive; the opposite is true when $K_4+K_{\mathrm{e}}{K}_5<0$ is satisfied. Synchronous condensers have a large synchronous torque. When the signs are opposite between the additional synchronous torque and additional damping torque, the condenser should be configured with a positive additional damping torque and a negative additional synchronous torque to ensure its static stability.

This analysis demonstrates that $K_5$ is greater than zero and $K_4$ is less than zero. $K_4+K_{\mathrm{e}}{K}_5$ is a linear function of the excitation system gain $K_{\mathrm{e}}$ with a negative intercept and a positive slope, and there exists a minimum value of the magnification $K_{\mathrm{emin}}$ that satisfies $K_4+K_{\mathrm{emin}}{K}_5=0$; at this point, the damping torque of the synchronous condenser is zero, operating at the critical value. Substituting the equation into Equation (6), the expression for $K_{\mathrm{emin}}$ is obtained.

$$K_{\mathrm{emin}}=-{\displaystyle \frac{K_4}{K_5}}={\displaystyle \frac{x_{\mathrm{d}}-x_{\mathrm{d}}^{\prime }}{x_{\mathrm{d}}^{\prime }}}$$

(7)

For a system with fixed parameters, $K_{\mathrm{emin}}$ is a fixed value independent of the operating state. Only when the excitation system gain exceeds $K_{\mathrm{emin}}$ can the synchronous condenser achieve positive damping, thus avoiding self-sustained oscillation under small disturbances.

By applying the Routh–Hurwitz criterion to perform stability analysis on the state equation, another criterion that constrains the excitation system gain is derived, as shown in Equation (8).

$${\displaystyle \frac{K_1}{K_3}}-K_2{K}_4+K_{\mathrm{e}}\left(K_1{K}_6-K_2{K}_5\right)>0$$

(8)

The inequality-constrained excitation system gain should be greater than a certain critical value. However, for the synchronous condenser, since both terms $K_1/K_3-K_2{K}_4$ and $K_1{K}_6-K_2{K}_5$ are greater than zero, the critical value obtained from Equation (9) is less than zero. Meanwhile, $K_{\mathrm{e}}$ is always greater than zero, so the determined feasible range of $K_{\mathrm{e}}$ can always be satisfied. Therefore, it no longer acts as a constraint on the maximum value of $K_{\mathrm{e}}$.

Synchronous generators supplying both active and reactive power have a relatively large power angle at heavy load, leading to the condition $K_5<0$. However, its power angle is large enough to satisfy $K_4>0$; this is exactly the opposite of the model coefficient characteristics of the synchronous condenser analyzed earlier. For synchronous generators, $K_4+K_{\mathrm{e}}{K}_5$ is a linear function with a positive intercept and a negative slope, which requires the excitation system gain to be less than a certain value to ensure the damping coefficient is greater than zero. If the excitation system gain is too large, a Power System Stabilizer (PSS) needs to be installed for phase compensation. The difference in their basic operating conditions determines the difference in parameter tuning of the excitation system gain.

3.2. Analysis of Influencing Factors

Since insufficient damping is more prominent in synchronous condensers, the damping torque is taken as the main research object, and the variation trend of the synchronous torque is opposite to that of the damping torque. Substituting Equations (5) and (6) into Equation (4), the specific expressions of the additional synchronous torque and additional damping torque are obtained.

$$\left\{\begin{array}{l}\Delta K_{\mathrm{S}}=-{\displaystyle \frac{E_{\mathrm{Q}}r\left(N_3{K}_{\mathrm{e}}+N_2\right)Ur\left[\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}\right]}{N_1\left[{\left(N_3{K}_{\mathrm{e}}+N_2\right)}^2+T_{\mathrm{d}0}^{\prime 2}{\omega }_{\mathrm{d}}^2{N}_1^2\right]}}\\ \Delta K_{\mathrm{D}}={\displaystyle \frac{E_{\mathrm{Q}}{r}^2\left[\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}\right]U{T}_{\mathrm{d}0}^{\prime }{\omega }_{\mathrm{d}}}{{\left[N_2+K_{\mathrm{e}}\left(r^2+x_{\mathrm{q}\mathsf{\Sigma }}{x}_{\mathrm{e}}\right)\right]}^2+T_{\mathrm{d}0}^{\prime 2}{\omega }_{\mathrm{d}}^2 N_1^2}}\end{array}\right.$$

(9)

In the expression, $N_1=r^2+x_{\mathrm{q}\mathsf{\Sigma }}{x}_{\mathrm{d}\mathsf{\Sigma }}^{\prime }$ , $N_2=r^2+x_{\mathrm{q}\mathsf{\Sigma }}{x}_{\mathrm{d}\mathsf{\Sigma }}$ and $N_3=r^2+x_{\mathrm{q}\mathsf{\Sigma }}{x}_{\mathrm{e}}$. As can be observed from the expression, the additional damping torque is related to operating conditions, system parameters, the excitation system gain, time constants, and the rotor oscillation frequency.

3.2.1. Excitation System Gain

The term $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}$ is a deformation of the expression of $K_{\mathrm{emin}}$ and it serves to determine the sign of the additional damping torque. When $K_{\mathrm{e}}>K_{\mathrm{emin}}$, both the numerator and denominator of the expression are positive, so the additional damping torque is positive; the opposite holds when $K_{\mathrm{e}}<K_{\mathrm{emin}}$. As the excitation system gain increases, the numerator of the rate of change exceeds the denominator, leading to an increase in the additional damping torque and a decrease in the additional synchronizing torque.

3.2.2. System Resistance

When $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}>0$, as the system resistance $r$ increases, considering actual operation, the per-unit value of the resistance is less than 1, and $r^2$ is also smaller than both $x_{\mathrm{q}\mathsf{\Sigma }}{x}_{\mathrm{d}\mathsf{\Sigma }}^{\prime }$ and $x_{\mathrm{q}\mathsf{\Sigma }}{x}_{\mathrm{d}\mathsf{\Sigma }}$. The order of magnitude of the numerator is slightly smaller than that of the denominator, but its rate of change is much larger than the denominator’s. Therefore, the additional damping torque increases with increasing resistance, whereas the additional synchronizing torque decreases. When $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}<0$, the trend of change is the opposite.

3.2.3. Reactive Power Output

In the expression, only $E_{\mathrm{Q}}$ is associated with the reactive power output and the power angle of the synchronous condenser. When the reactive power output increases, the power angle becomes larger, and $E_{\mathrm{Q}}$ also increases.

When $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}>0$, as the reactive power output of the condenser increases, the additional damping torque rises while the additional synchronous torque decreases. When $\left(K_{\mathrm{e}}+1 \right){x^{\prime }}_{\mathrm{d}}-x_{\mathrm{d}}<0$, the variation trend is the opposite.

3.2.4. Rotor Oscillation Frequency

For the additional synchronous torque coefficient, the rotor oscillation frequency ${\omega }_{\mathrm{d}}$ is only related to the denominator of the expression; thus, the additional synchronous torque coefficient varies monotonically with the change in ${\omega }_{\mathrm{d}}$. When $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}>0$, the additional synchronous torque takes a negative value and increases as ${\omega }_{\mathrm{d}}$ increases; when $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}<0$, it takes a positive value and decreases as ${\omega }_{\mathrm{d}}$ increases.

For the additional damping torque, ${\omega }_{\mathrm{d}}$ affects both the numerator and denominator of the expression simultaneously. Its variation is not monotonic, and there exists a maximum value.

3.2.5. D-Axis Open-Circuit Transient Time Constant

Consistent with the rotor oscillation frequency, for the additional synchronous torque, the time constant $T_{\mathrm{d}0}^{\prime }$ is only related to the denominator; for the additional damping torque, $T_{\mathrm{d}0}^{\prime }$ is related to both the numerator and the denominator. Thus, when $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}>0$, as the time constant increases, the additional synchronous torque takes a negative value, and its absolute value decreases with the increase in the time constant; the additional damping torque grows and subsequently declines. When $\left(K_{\mathrm{e}}+1 \right)x_{\mathrm{d}}^{\prime }-x_{\mathrm{d}}<0$, the variation trend is the opposite.

Based on the above analysis, the influencing factors of the additional torque coefficients and their variation laws are shown in

Table 1. Influencing factors and variation laws of additional torque coefficient.

Influencing Factor	Trend of Additional Damping Torque		Trend of Additional Synchronizing Torque
	${\mathit{K}}_{\mathbf{e}}>{\mathit{K}}_{\mathbf{emin}}$	${\mathit{K}}_{\mathbf{e}}<{\mathit{K}}_{\mathbf{emin}}$	${\mathit{K}}_{\mathbf{e}}>{\mathit{K}}_{\mathbf{emin}}$	${\mathit{K}}_{\mathbf{e}}<{\mathit{K}}_{\mathbf{emin}}$
Increased reactive power output	Increases	Decreases	Decreases	Increases
Increased excitation system gain	Increases		Decreases
Increased rotor oscillation frequency	First increases then decreases (positive value)	First decreases then increases (negative value)	Increases (negative value)	Decreases (positive value)
Increased system resistance	Increases	Decreases	Decreases	Increases

.

From the torque variation rate induced by increasing system resistance at a gain of 20, it can be seen that simultaneously increasing system resistance and gain significantly improves the torque, with the average variation rate being considerably higher than in other cases. By contrast, the influence of operating condition variations on the additional torque is weaker than that of system resistance, gain, and oscillation frequency, resulting in a relatively small increase in the average variation rate.

The critical excitation system gain $K_{\mathrm{emin}}$ derived above is not a constant value, but a threshold that varies with the parameters of the synchronous condenser. From an engineering perspective, $K_{\mathrm{emin}}$ serves as the boundary between negative damping torque and positive damping torque, which can provide a clear reference for the practical tuning of the excitation system gain. It also plays a critical role in both the steady-state stability and transient stability of systems containing synchronous condensers.

Specifically, the setting value of $K_{\mathrm{e}}$ should be higher than $K_{\mathrm{emin}}$ to ensure that the synchronous condenser maintains positive damping and suppresses low-frequency oscillations. Meanwhile, an excessively high $K_{\mathrm{e}}$ should be avoided, as it will reduce the overall system stability margin, trigger high-frequency oscillations, and amplify measurement noise. Therefore, the practical tuning value of $K_{\mathrm{e}}$ needs to strike a balance between the demand for sufficient positive damping and the constraints of hardware components in real engineering design.

4. Case Study

The example system adopts the configuration shown in Figure 1. The parameters of the synchronous condenser are configured as follows: rated capacity of 50 Mvar, $x_{\mathrm{d}}=x_{\mathrm{q}}=$0.88 p.u., $x_{\mathrm{d}}^{\prime }=$0.1163 p.u., $T_{\mathrm{j}}=$5.569 s, and $T_{\mathrm{d}0}^{\prime }=$7.634 s. According to the analysis presented in the previous section, the corresponding minimum value of the excitation system gain $K_{\mathrm{emin}}=6.6$ can be calculated.

The excitation system gain is set to 5 and 20. The infinite bus voltage is used as the reference phasor. We assumed a rotor oscillation frequency ${\omega }_{\mathrm{d}}$ = 10 rad/s and reactive power output of 10 Mvar. Parameters such as the K coefficient and additional torque coefficient of the synchronous condenser are listed in

Table 2. Operating parameters of synchronous condenser under steady operating conditions.

Synchronous Condenser Operating Parameters	Calculated Values of Operating Parameters
Power Angle/deg	−2.95
$K_1$	0.44
$K_2$	0.25
$K_3$	0.42
$K_4$	−0.30
$K_5$	0.05
$K_6$	0.79
Torque	Ke = 5	Ke = 20
Additional Synchronous Torque/p.u.	1.91 × 10−5	−4.49 × 10−4
Additional Damping Torque/p.u.	−2.31 × 10−4	1.90 × 10−3

.

As can be seen from Table 2, when $K_{\mathrm{e}}$ = 20, it is greater than the critical excitation system gains for generating positive damping torque. At this time, the additional synchronous torque is less than zero but far smaller than the synchronous torque $K_1$; thus, the impact of weakening the synchronization capability can be approximately neglected. Meanwhile, the additional damping torque is greater than zero, which enhances the damping performance of the synchronous condenser. Whereas when $K_{\mathrm{e}}<K_{\mathrm{emin}}$, the opposite is true: the additional damping torque is negative, and the additional synchronous torque is positive. The contribution to the synchronization capability is small, and the negative damping will cause self-excited oscillation in the system, leading to an unstable state.

To analyze the relationships between operating conditions, excitation system gain, rotor oscillation frequency, and additional torque coefficient, three operating conditions are selected within the reactive power output range that meets the voltage level constraints:

Operating Condition 1:The synchronous condenser absorbs 5 Mvar;

Operating Condition 2:The synchronous condenser neither absorbs nor injects reactive power;

Operating Condition 3:The synchronous condenser injects 5 Mvar.

The active power output is zero under all three operating conditions.

4.1. Influencing Factors

4.1.1. Reactive Power Output of Synchronous Condenser

When the active power output of the synchronous condenser is zero and the reactive power output ranges from −20 Mvar to 50 Mvar, the comprehensive excitation system gain is set to 5 and 20, respectively, with the rotor oscillation frequency assumed constant at 10 rad/s. The variation curves of the additional torque as a function of reactive power output are shown in Figure 3.

For the target system under study, its parameters are fixed, and neither $K_4$ nor $K_5$ changes with variations in operating conditions.

The larger the excitation system gain, the greater the rate of torque change and the absolute value of the torque, resulting in stronger positive damping. Furthermore, the order of magnitude of the additional synchronous torque is much smaller than that of the synchronous torque coefficient $K_1$, so the effect of the additional synchronous torque can be approximately neglected.

The reactive power output has a more significant impact on the damping characteristics, and increasing the excitation system gain can further enhance the influence of the damping torque. In contrast, the synchronous characteristics are relatively limitedly affected by both factors.

4.1.2. Excitation System Gain

Keeping the system parameters and rotor oscillation frequency constant, while varying the excitation system gain from 0 to 50, the variation curves of the additional torque with the excitation system gain under three operating conditions are shown in Figure 4.

As the excitation system gain increases, the additional synchronous torque decreases monotonically, while the additional damping torque increases monotonically. When comparing the variation characteristics across the three operating conditions, the higher the level of reactive power generated, the larger the variation range of the torque and the higher the rate of change.

4.1.3. Rotor Oscillation Frequency

Keeping the system parameters unchanged, under the same operating condition, the variation curves of the additional torque with the oscillation frequency—when the rotor oscillation frequency is varied within the low-frequency oscillation range (between 0.1 Hz and 2.5 Hz) and the excitation system gains are 5 and 20—are shown in Figure 5.

From Figure 5, it is evident that the variation trend of the additional damping torque is not monotonic. When the excitation system gain is less than $K_{\mathrm{emin}}$, it decreases first and then increases; when it is greater than $K_{\mathrm{emin}}$, it increases first and then decreases. In contrast, the variation in the additional synchronous torque is monotonic: it decreases when the excitation system gain is less than $K_{\mathrm{emin}}$ and increases when it is greater than $K_{\mathrm{emin}}$.

4.1.4. System Resistance

Keeping the rotor oscillation frequency constant and under the same operating condition, when the system resistance value is varied within the range of (0 p.u., 1 p.u.) and the excitation system gains are set to 5 and 20, respectively; the curves of the additional torque versus the system resistance are shown in Figure 6.

Based on Figure 6, the effect of increased resistance on the additional torque is similar to that on the reactive power output.

Simultaneously increasing the system’s resistance and excitation system gain is of great significance for enhancing the damping torque, with the average rate of change being much higher than that in other cases.

In summary, within the normal variation range of each variable, the degree of influence of operating conditions on the magnitude of the additional torque is less significant than that of system resistance, excitation system gain, and oscillation frequency, and the increasing amplitude of the average rate of change is relatively small.

4.2. Frequency Domain Analysis

The coefficient K represents the first-order partial derivatives of operating variables with respect to state variables. Using the coefficient K derived in the preceding section, the system state matrix is constructed, and the corresponding eigenvalues are calculated. For each complex eigenvalue ${\lambda }_{\mathrm{i}}={\sigma }_{\mathrm{i}}+j{\omega }_{\mathrm{i}}$, the damping ratio is determined as follows:

$${\zeta }_{\mathrm{i}}=-{\displaystyle \frac{{\sigma }_{\mathrm{i}}}{\sqrt{{\sigma }_{\mathrm{i}}^2+{\omega }_{\mathrm{i}}^2}}}$$

(10)

The eigenvalues, undamped oscillation frequencies, and damping ratios calculated under four different comprehensive excitation system gains are presented in

Table 3. Eigenvalues of system state equations under different excitation system gains.

Excitation System Gain	Eigenvalue		Undamped Oscillation Frequency/(Rad/s)	Damping Ratio
	Conjugate Complex Roots	Real Roots
2	0.008 ± j7.028	−0.534	7.028	−1.726 × 10−5
5	0.003 ± j7.028	−0.834	7.028	−5.895 × 10−6
6.6	0.000 ± j7.028	−0.992	7.028	0
20	−0.020 ± j7.021	−2.339	7.024	4.820 × 10−5

.

In Table 3, j represents the imaginary unit of complex numbers, and j² = −1. It is adopted in electrical engineering to distinguish it from the current symbol i.

As the excitation system gain increases, both a pair of conjugate complex roots ${\lambda }_{1,2}$ and the real root ${\lambda }_3$ of the system state equation move toward the left half-plane of the complex plane. Consequently, the system gradually tends to stabilize, with its stability degree being progressively enhanced.

In response to an increase in the excitation system gain, the damping ratio increases, changing from negative to positive. When the excitation system gain is less than the critical value of 6.6, the damping ratio is negative, indicating that the system is in a negative damping mode. Correspondingly, the damping torque coefficient is also negative, and the system will experience oscillatory instability.

When the excitation system gain is 6.6, the damping ratio is exactly zero, which corresponds to the case of $K_4+K_{\mathrm{e}}{K}_5=0$ derived earlier. At this point, both the damping torque and the synchronous torque are zero.

4.3. Simulation Verification

To verify the effects of factors such as the comprehensive excitation system gain of the synchronous condenser (derived from the mechanism analysis in this paper) on the additional torque coefficient and static stability of the synchronous condenser, the system shown in Figure 1 is built in the PSASP (Power System Analysis Software Package) Version 7 for simulation studies. The parameter settings are shown in

Table 4. Simulation parameters.

Component	Parameter	Value
Synchronous Condenser	Rated power	1
	d-axis synchronous reactance	0.88
	q-axis synchronous reactance	0.88
	d-axis transient reactance	0.13
	d-axis open-circuit transient time constant	8 s
	Inertia time constant	6 s
	Damping coefficient	0
Wind Farm	Rated power	500 MW
System	Base power	100 MVA
	Wind farm step-up transformer reactance	0.03
	Synchronous condenser step-up transformer reactance	0.12
	35/220 kV step-up transformer reactance	0.03
	35 kV line impedance	0.5 + j0.3
	220 kV line reactance	0.05
	Receiving-end system voltage	1.0

.

In Table 4, the parameters following j represent the reactance magnitude in the impedance.

A 1% step disturbance in the synchronous condenser terminal voltage is introduced into the system at 1 s. Parameters such as the excitation system gain, operating conditions, and system resistance of the synchronous condenser were systematically modified. The variation curves of the synchronous condenser’s power angle and angular velocity over time during the disturbance are analyzed, thereby deriving the variation law of the synchronous condenser’s damping torque under small disturbance conditions. As a typical small-signal disturbance in power system stability analysis, it can effectively verify the damping torque characteristics and static stability of the synchronous condenser. The small amplitude ensures that the system operates within the linearized range required by the theoretical analysis.

4.3.1. Excitation System Gain

We analyze the influence of the excitation system gain on the system’s damping characteristics. We set the excitation system gain of the synchronous condenser to 2, 20, and 50, with other parameters unchanged. Figure 7 presents the time-domain variation curves of the power angle and angular velocity of the synchronous condenser.

The critical value of the excitation system gain $K_{\mathrm{emin}}$ remains 6.6, as calculated earlier. It can be seen from the figure that when the excitation system gain is 2 (less than the critical value), the excitation system provides negative damping for the synchronous condenser, and the power angle exhibits increasing-amplitude oscillation over time. When the excitation system gain exceeds the critical value, the damping intensity is sufficient to maintain the stability of the power angle, and the power angle curve shows decreasing-amplitude oscillation—with a larger excitation system gain corresponding to a smaller power angle oscillation amplitude.

For the angular velocity, when the excitation system gain is less than the critical value, it undergoes increasing-amplitude oscillation, and the oscillation axis during the oscillation process is lower than the original steady-state value, indicating a loss of static stability. In contrast, when the excitation system gain is greater than the critical value, the angular velocity exhibits decreasing-amplitude oscillation; the larger the excitation system gain, the more quickly it converges to stability.

4.3.2. Reactive Power Output of Synchronous Condenser

We analyze the influence of the synchronous condenser’s reactive power output on the system’s damping characteristics. By setting the reactive power output to −10 Mvar, 0, and 10 Mvar, the variation curves of the condenser’s power angle and angular velocity with time are shown in Figure 8.

4.3.3. System Resistance

We analyze the influence of system resistance on the system’s damping characteristics. We set the resistance to 0.3 p.u., 0.5 p.u., and 1 p.u.; the variation curves of the power angle and angular velocity of the synchronous condenser with time are shown in Figure 9.

As the resistance increases, the power consumption increases accordingly. When a small disturbance occurs, the oscillation amplitude grows with the increase in resistance. However, the existence of resistance also enhances the damping torque, enabling the power angle and angular velocity to more quickly exit the oscillatory state and tend to stabilize, as shown in Figure 9.

4.3.4. Renewable Energy Grid Integration Scenario

In order to verify the applicability of the conclusions of this paper in new energy grid-connected scenarios, a simulation system as shown in Figure 10 is established in this section, with all parameters consistent with those listed in Table 4. Based on the original topology consisting of a synchronous condenser, conventional generating units and an infinite power grid, a wind farm is connected at the point of common coupling. The wind farm adopts a constant power control mode and only outputs active power during steady-state operation.

To reflect practical engineering operation, two typical operating conditions of the wind farm are selected, including the low-power condition (0.2 P_N) and the medium-power condition (0.5 P_N), where P_N denotes the rated active power of the wind farm. Under each operating condition, parameters such as the line resistance and reactive power output of the synchronous condenser remain unchanged, while only the excitation system gain K_e is adjusted. Three typical values of excitation system gain (2, 20 and 50) are set for comparative analysis to investigate the influence of excitation system gain on the damping characteristics of the synchronous condenser.

The small disturbance adopted in this section is consistent with that in the previous cases. The time-domain curves of power angle and angular velocity of the synchronous condenser under the three excitation system gains for low- and medium-output conditions are presented in Figure 11 and Figure 12, respectively.

Figure 11 and Figure 12 present the dynamic responses of the power angle and angular velocity of the synchronous condenser in the wind farm under different excitation system gains K_e, in low and medium wind power output scenarios, respectively. The simulation results show that as the excitation system gain K_e increases from 2 to 50, the oscillation decay rate of both the power angle and angular velocity of the synchronous condenser is significantly accelerated, indicating that increasing K_e can effectively improve the damping torque of the synchronous condenser, thereby suppressing power angle oscillations.

5. Conclusions

Based on theoretical assumptions, this paper proceeds from mechanism analysis to simulation validation in a stepwise manner, drawing the following two conclusions:

• The opposite variation characteristics of the Heffron–Phillips model parameters between synchronous condensers and generators lead to opposite trends of the additional torque coefficient with varying excitation system gain.
• If the excitation system gain of a synchronous condenser is lower than the critical value defined by system parameters, the condenser will suffer periodic instability due to insufficient damping. Thus, to avoid such instability risks, the excitation system gain of synchronous condensers should be set at a relatively high level.

Finally, simulation experiments based on two typical systems, including the synchronous condenser power delivery system and the new energy grid-connected system with synchronous condensers, verify the validity of the theoretical analysis. This conclusion is highly significant for mitigating synchronous condenser oscillation risks and ensuring the stable operation of new energy power grids through the optimal tuning of excitation system parameters. On this basis, further research can focus on the optimization of excitation control strategies for synchronous condensers under complex new energy grid operating conditions and explore the engineering implementation scheme of parameter tuning, so as to promote the practical application of synchronous condensers in large-scale new energy power systems.