Coupling Short-Circuit Ratio Calculation Method Based on the Source Network Load Correlation Thevenin Equivalent

Abstract

The short-circuit ratio is a key index used to measure the voltage support capability of new energy grid-connected systems, which contradicts its accuracy and practicability. The current short-circuit ratio index (SCR-U) does not consider the influence of the coupling impedance between power supply and load on the short-circuit ratio in the calculation process. This paper derives the coupled short-circuit ratio (SCR-O) based on SCR-U and the coupled equivalent impedance. The coupled critical short-circuit ratio (CSCR-O) is then numerically obtained based on the maximum transmission power, and the extreme value of the critical short-circuit ratio is calculated to provide a numerical basis to determine the accuracy of SCR-U and SCR-O. Finally, simulations demonstrate the superiority of the coupling short circuit to the short-circuit ratio index. It can provide more accurate results and reflect the system’s real-time voltage stability.

1. Introduction

With the increase in the proportion of new energy in power systems, issues related to voltage stability and system security have become increasingly prominent, posing critical challenges that must be addressed. As an essential indicator for evaluating the voltage support capability of new energy grid-connected systems, the short-circuit ratio (SCR) has been widely discussed in related research fields [1,2,3]. The value of SCR directly reflects the system’s ability to preserve the voltage levels after integrating new energy, providing a quantitative basis for the voltage support strength under different operating conditions, thereby ensuring the reliability and efficient operation of new energy grid-connected systems [4].

However, traditional SCR indicators and their extended forms still have certain limitations in new energy grid-connected systems. The multi-infeed short-circuit ratio (MISCR) was proposed to evaluate the relative strength of AC systems in multi-infeed DC systems [5,6] using the impedance and influence factor methods equivalent to the calculation methods [7,8]. In [9], the multi-infeed interaction factor (MIIF) was proposed to characterize the interaction between different direct current (DC) systems and derive the multi-infeed effective short-circuit ratio (MIESCR) based on MISCR, while its applicability remains unclear. In [10], the multi-infeed equivalent effective short-circuit ratio (MEESCR) and its critical value (MCEESCR) were proposed by equivalently transforming multi-infeed systems into single-infeed ones considering the impact of the DC current variations on the voltage and power transfer at infeed points. The application prerequisites for MIESCR and effective short-circuit ratio (ESCR) were also discussed.

Additionally, although the multi-station short-circuit ratio (MRSCR) [11] considers the impact of multiple new energy stations, it is challenging to accurately and quickly calculate the mutual influence between stations. The generalized short-circuit ratio (GSCR) decouples multi-infeed systems using modal analysis [12,13] to improve the analysis accuracy, while its computational complexity limits its practical application in engineering. Weighted short-circuit ratio [14] and composite short-circuit ratio [15] treat thermal and new energies for estimation [16]. However, due to their insufficient accuracies [17], the mutual influence between infeed branches cannot be fully reflected. In [18], constant impedance loads were considered when deriving equivalent impedance without fully considering the impact of equivalent node current variations on the generator and other load node currents, thereby changing the equivalent impedance.

To address the above issues, this paper proposes a novel and practical coupling short-circuit ratio (SCR-O) indicator using the coupling critical short-circuit ratio (CSCR-O) as a reference to locate the system’s SCR and effectively evaluate the voltage support strength. Specifically, the main contributions of this paper are as follows: (1) A new node admittance matrix is constructed based on the source–grid–load-correlated Thevenin equivalent, and the SCR-O indicator is proposed based on voltage calculation. SCR-O has stronger practicality than MRSCR and can evaluate the system’s voltage support strength in real time. (2) CSCR-O is established based on maximum power transfer and coupling equivalent impedance, which is more accurate than the traditional CSCR [19] and can serve as a reference to measure SCR-O. Simulation results demonstrate the superiority of the proposed coupling short-circuit ratio indicator to traditional methods in terms of applicability and accuracy, more accurately reflecting the system’s voltage stability.

2. Short-Circuit Ratio Indicator Based on Voltage Calculation

Voltage source converters (VSCs) with current control as the core can effectively simulate current source characteristics, while external connection conditions can significantly affect output voltage stability. This characteristic makes VSCs valuable in new energy grid-connected systems, particularly in wind farms and photovoltaic power stations. However, voltage fluctuations considerably influence system stability. Voltage stability issues involve the dynamic characteristics of power systems, including the dynamic responses of generators, loads, reactive power compensation devices, and power electronic devices. In practical engineering, static analysis methods based on power flow calculations remain significantly valuable for rapid approximate analysis due to dynamic analysis’s computational complexity and time cost [20]. Static analysis methods simplify system models and provide rapid preliminary assessments of system voltage stability, offering important references for subsequent dynamic analysis and control strategy design. Therefore, combining the advantages of static and dynamic analyses to develop efficient and accurate voltage stability assessment methods for new energy grid-connected systems is crucial.

2.1. Equivalent Analysis Model

Figure 1 shows the simplified structure of the Thevenin equivalent AC system with new energy integration, and its equivalent analysis model is shown in Figure 2.

2.2. Definition and Analysis of Short-Circuit Ratio Based on Voltage Calculation

In the AC infeed system structure shown in Figure 1, nodes of $1~\mathrm{m}$ is the new energy integration point, and node $\mathrm{m}+1~\mathrm{n}$ is the synchronous machine node. The node voltage equation is as follows:

$$U_{\mathrm{ac}}=\left[\begin{array}{l}{\dot{Z}}_{1m+1}{\dot{I}}_{ac,m+1}+\dots +{\dot{Z}}_{1n}{\dot{I}}_{ac,n}\\ {\dot{Z}}_{nm+1}{\dot{I}}_{ac,m+1}+\dots +{\dot{Z}}_{nn}{\dot{I}}_{ac,n}\end{array}\right]$$

(1)

In the $Z_{ij}$ matrix, $Z_{ij}$ describes the electrical distance between nodes; $I_{ac,i}$ is the node current injected by the synchronous machine with new energy open-circuited.

When the new energy integration node and the synchronous machine node are far apart in the circuit, the synchronous machine node has a negligible impact on voltage fluctuations. In (1), node $1~m$ injects current, and node $m+1~n$ is open-circuited. The simplified node voltage equation can be used when considering the impact of new energy integration on the grid, and the voltage change can be described as follows:

$$\Delta {\dot{U}}_i={\dot{Z}}_{ii}{\dot{I}}_{E,i}+{\displaystyle \sum _{j\ne i}{\dot{Z}}_{ij}{\dot{I}}_{E,j}}$$

(2)

The ratio of the nominal voltage at the new energy integration point to the voltage variation caused by new energy integration is as follows:

$${\displaystyle \frac{U_N}{\Delta U_i}}={\displaystyle \frac{U_N}{\left|{\dot{Z}}_{ii}{\dot{I}}_{E,j}+{\displaystyle \sum _{j\ne i}{\dot{Z}}_{ij}{\dot{I}}_{E,j}}\right|}}$$

(3)

According to (3), one can obtain the following:

$$SCR-U_i={\displaystyle \frac{\left|U_N{\dot{E}}_{eq,i}\right|}{\left|\Delta U_i{U}_i\right|}}$$

(4)

where $SCR-U_i$ represents the voltage support strength at the integration point $i$.

To further understand and apply the short-circuit ratio, the basic definition and characteristics of SCR were employed to propose two new short-circuit ratio indicators [19]: the capacity-based short-circuit ratio (SCR-S) and the voltage-based short-circuit ratio (SCR-U). These indicators can be employed for voltage stability analysis in new energy grid-connected systems.

Regarding Equation (4), SCR-U has the following advantages:

(a) Traditional short-circuit ratio indicators can only be effectively employed for simple single-infeed systems and cannot accurately reflect complex voltage stability issues caused by large-scale new energy integration. In contrast, the improved SCR-U metric can be properly employed for modern new energy integration scenarios while comprehensively considering the dynamic mutual influence between multiple converter stations, thus providing a more thorough and reliable evaluation of the system’s overall voltage support capability under varying operating conditions.

(b) Both SCR-S and SCR-U exhibit fundamentally consistent performance when evaluating the voltage support strength of modern new energy grid-connected systems under various operating conditions. However, SCR-U demonstrates superior practicality by featuring a more concise mathematical expression and simplified computational requirements. More importantly, all required input parameters for SCR-U can be directly obtained through real-time measurement systems, enabling continuous dynamic evaluation of the system’s voltage support strength. This advanced capability provides power system operators with timely, accurate, and reliable decision-making references for stability assessment and control actions, particularly during critical grid transitions or contingency scenarios.

Nevertheless, the voltage calculation-based SCR-U methodology still exhibits fundamental limitations in properly handling Thevenin equivalent impedance estimation, particularly in complex grid scenarios. Specifically, SCR-U’s formulation does not fully account for the dynamic coupling effects between multiple distributed power sources and variable loads, which significantly distorts the equivalent impedance calculation in Equation (2) during actual operation. This inherent deficiency causes non-negligible errors in SCR-U calculations, especially in weak grids with high renewable penetration, making it increasingly difficult to accurately reflect the system’s true operating conditions during both steady-state and transient scenarios. The impedance estimation inaccuracies become particularly pronounced during rapid generation or load variations, where traditional SCR-U fails to capture the instantaneous network characteristics.

Therefore, this paper conducts an in-depth theoretical and computational investigation into advanced SCR indicators that properly account for the impact of multi-source coupling impedance dynamics. We propose an improved SCR calculation methodology incorporating both static network characteristics and dynamic interaction factors, which significantly enhances the indicator’s computational accuracy and practical applicability across different operating scenarios. This novel approach establishes a more comprehensive and reliable theoretical framework for precise voltage stability analysis in modern power systems with high penetration of renewable energy integration, particularly addressing the challenges posed by converter-interfaced generation and complex grid interactions.

3. Short-Circuit Ratio Calculation Considering Load Coupling Relationship

First, the Thevenin equivalent impedance used in SCR-U is analyzed. Then, the source–grid–load-correlated Thevenin equivalent is employed to accurately calculate the system equivalent impedance, which is applied to the short-circuit ratio calculation. Finally, a coupling SCR calculation method is proposed based on SCR-U.

3.1. Analysis of Equivalent Coupling Impedance

In the voltage calculation-based equivalent circuit, the voltage equation for node $i$ is as follows:

$${\dot{V}}_{Li}={\dot{E}}_{Thev,i}-{\dot{Z}}_{Thev,i}{\dot{I}}_{Li}$$

(5)

where ${\dot{E}}_{Thev,i}$ is the equivalent Thevenin voltage on equivalent node $i$, and ${\dot{Z}}_{Thev,i}$ is its equivalent impedance. ${\dot{E}}_{Thev,i}$ and ${\dot{Z}}_{Thev,i}$ describe the effect of the system on the voltage of node 1, using ${\dot{E}}_{Thev,i}$ and ${\dot{Z}}_{Thev,i}$, indirectly affecting the voltage support strength at node $i$.

Traditional Thevenin equivalence cannot reflect the intrinsic change mechanism of ${\dot{E}}_{Thev,i}$ and ${\dot{Z}}_{Thev,i}$. In addition, it is necessary to reflect the system equivalent impedance in real time. Thus, providing adequate data support for voltage-based SCR indicators is challenging.

Thevenin equivalence is a commonly used method for simplifying system analysis in power systems. Traditional Thevenin equivalent voltage is usually independent of changes in equivalent node current, while the system equivalent impedance is considered constant. However, this traditional equivalence is applicable in the case of source–grid–load (power source, grid, and load) correlation. The source–grid–load-correlated Thevenin equivalence considers this variation and divides the system impedance into the following three components:

Load equivalent impedance:

$${\dot{Z}}_{Leq,i}={\displaystyle \frac{1}{{\dot{I}}_{Li}}}{\displaystyle \sum _{j=1}^n{\dot{Z}}_{LLij}\left({\dot{I}}_{Lj}-{\dot{I}}_{Lj,i=0}\right)}$$

(6)

Generator equivalent impedance:

$${\dot{Z}}_{Geq,i}={\displaystyle \frac{1}{{\dot{I}}_{Li}}}{\displaystyle \sum _{j=1}^n{\dot{Z}}_{LGij}\left({\dot{I}}_{Gj}-{\dot{I}}_{Gj,i=0}\right)}$$

(7)

Inductive equivalent impedance:

$${\dot{Z}}_{LGeq,i}={\displaystyle \frac{1}{{\dot{I}}_{Li}}}{\displaystyle \sum _{j=1}^n\left[\left({\dot{Z}}_{LGij}-{\dot{Z}}_{LLij}\right)\left({\dot{I}}_{Lj}-{\dot{I}}_{Lj,i=0}\right)\right]}$$

(8)

Finally, the simplified form of the coupling equivalent impedance can be defined as follows:

$${\dot{Z}}_{Thev,i}={\dot{Z}}_{Leq,i}+{\dot{Z}}_{Geq,i}+{\dot{Z}}_{LGeq,i}$$

(9)

3.2. Short-Circuit Ratio Indicator Based on Coupling Impedance Calculation

The ratio of the nominal voltage of the new energy grid-connected system to the voltage variation caused by new energy integration can be obtained as follows:

$${\displaystyle \frac{U_N}{\left|\Delta U_i\right|}}={\displaystyle \frac{U_N}{\left|{\dot{Z}}_{Thev,i}{\dot{I}}_{E,i}+{\displaystyle \sum _{j\ne i}{\dot{Z}}_{ij}{\dot{I}}_{E,j}}\right|}}$$

(10)

From (10), we have the following:

$${\displaystyle \frac{\left|U_N{\dot{E}}_{eq,i}\right|}{\left|\Delta U_i{U}_i\right|}}={\displaystyle \frac{\left|U_N{\dot{E}}_{eq,i}/{\dot{Z}}_{Thev,i}\right|}{\left|{\dot{S}}_i+{\displaystyle \sum _{j\ne i}\frac{{\dot{Z}}_{ij}^{\ast }{U}_i}{{\dot{Z}}_{Thev,i}^{\ast }{U}_j{\dot{S}}_j}}\right|}}$$

(11)

The coupling SCR indicator based on voltage and coupling equivalent impedance is defined as follows:

$$SCR-O_i={\displaystyle \frac{\left|U_N{\dot{E}}_{eq,i}\right|}{\left|\Delta U_i{U}_i\right|}}$$

(12)

where $SCR-O$ represents the voltage support capability at node $i$.

In modern power system analysis considering the complex source–grid–load relationship, the Thevenin equivalent impedance approach has become increasingly important for studying networks with high penetration of new energy integration. Besides these fundamental considerations, the precise mathematical relationship between the SCR metric and the system’s equivalent impedance is carefully employed to derive the more advanced coupled short-circuit ratio (CSCR) indicator, which utilizes detailed voltage calculation methods. Compared with traditional SCRs that were developed for conventional power systems, the innovative CSCR formulation explicitly considers both the integration of intermittent new energy sources and the crucial impact of coupling equivalent impedance between distributed power sources and variable loads on the overall SCR value. When compared with alternative approaches like SCR-U and SCR-S, the optimized SCR (SCR-O) variant offers distinct advantages through its more straightforward mathematical form and closed-form analytical expression while maintaining higher computational accuracy. Since the CSCR methodology inherently requires real-time system parameters for its calculations, this advanced method naturally supports continuous real-time evaluation and can accurately reflect the actual real-time system’s dynamic operating status under varying conditions.

3.3. Impact of Coupling Impedance on Short-Circuit Ratio

Traditional Thevenin’s theorem mainly focuses on the equivalence of port characteristics while ignoring the mutual influence between active components inside the circuit, particularly in modern power electronic-dominated systems. However, these mutual influences significantly affect the actual circuits’ performance and stability, especially in weak grid conditions with high renewable penetration, and simple equivalent circuits often cannot fully reflect these internal dynamic interactions. Therefore, traditional Thevenin’s theorem cannot be directly employed in complex power systems without substantial modifications to account for multi-source coupling effects and nonlinear behaviors.

In the equivalent network shown in Figure 3, the system’s effect on each node’s voltage can be divided into equivalent voltage, equivalent resistance, and load. The following three factors mainly limit the configuration of voltage and resistance at these nodes: the system’s voltage support framework, active power output mode, and load attributes. These factors jointly determine node voltage distribution characteristics and the system’s stability.

The terminal network configuration, related parameters, and the voltage characteristics of each node determine the power system’s stability. These factors further affect the mathematical characteristics of network parameters. When the system’s voltage support structure changes, such as when the generator of the power station reaches maximum capacity and can no longer provide stable support or when the grid’s structural layout is adjusted, Formulas (6)–(8) can quantitatively analyze the impact of these variations on the equivalent impedance value at the equivalent node. Accordingly, this paper optimizes the equivalent impedance in Formula (12) to propose a more accurate SCR calculation method and further derives the CSCR indicator (SCR-O). This improved method considers the coupling relationship between power sources and loads while accurately reflecting the system’s voltage support capability in actual operation, providing a more reliable theoretical basis for stability analysis of new energy grid-connected systems.

4. Critical Coupling SCR Calculation for New Energy Grid-Connected Systems

The critical short-circuit ratio (CSCR) fundamentally defines the power system’s critical stable operating state and represents another essential manifestation of the SCR precisely at the moment of imminent system collapse, serving as a key stability boundary indicator. This crucial metric specifically takes the theoretical maximum power transfer capacity under steady-state conditions as its primary static voltage stability evaluation criterion, incorporating both network impedance characteristics and generation capability limits.

Critical CSCR Based on Maximum Active Power

The fault current capacity provided by the nodes in the grid-connected system can be obtained as follows:

$${\dot{S}}_{ac,i}={\displaystyle \frac{U_N{\dot{E}}_{eq,i}}{Z_{Thev,i}}}$$

(13)

Meanwhile, based on the system shown in Figure 3, the power flow at the grid connection point can be described as follows:

$$\dot{S}=P+jQ=\dot{U}{\dot{I}}^{\ast }$$

(14)

$$\left\{\begin{array}{l}P={\displaystyle \frac{U{\dot{E}}_{Thev,i}{X}_{Thev,i}\sin \theta }{X_{Thev,i}^2}}\\ Q={\displaystyle \frac{-U{\dot{E}}_{Thev,i}{X}_{Thev,i}\sin \theta }{X_{Thev,i}^2}}\end{array}\right.$$

(15)

where ${\dot{E}}_{Thev,i}$ is the equivalent potential; $\dot{U}$ is the voltage at the grid connection point; $\theta$ is the voltage phase angle; $\dot{I}$ is the current; $X_{Thev,i}$ is the reactance of the equivalent impedance; and $\dot{S}$, $P$, and $Q$ are the apparent, active, and reactive powers, respectively.

The voltage equation has a unique solution when the system becomes critically stable. In this state, the following equations calculate the maximum power transfer $P_{\max }$ and the critical voltage at the grid connection point $U_c$:

$$P_{\max }={\displaystyle \frac{{\dot{E}}_{Thev,i}{Z}_{Thev,i}\sqrt{{\dot{E}}_{Thev,i}^2+4Q_{Thev,i}{X}_{Thev,i}}}{2X^2}}$$

(16)

$$U_c=\sqrt{{\displaystyle \frac{E^2+2QX}{2}}}$$

(17)

In summary, the system’s coupling critical SCR under a constant system’s reactive power level can be obtained as follows:

$$CSCR-O={\displaystyle \frac{S_{i,ac}}{S_{\max }}}$$

(18)

The ratio obtained at this time is the extreme value of the critical SCR, reflecting the system’s most vulnerable and unstable state.

5. Simulation Verification

5.1. System Case

This paper takes a provincial power grid as the research object, focusing on the performance of different SCRs in the critical stable state. Figure 4 shows the topology structure used in the study, and the system structure is simplified using the Thevenin equivalent method, as presented in Figure 3.

To verify the accuracy of the SCR-U and SCR-O indicators, the following experiment is designed: a 30% increase in load is applied at bus 3. This state is maintained for 1 s to simulate an instantaneous fault condition. The performance differences between SCR-U and SCR-O in system voltage stability evaluation are compared and analyzed by observing the static voltage fluctuation at bus 4. This experimental scheme can effectively reflect the system’s voltage response characteristics during the transient process, providing a reliable experimental basis to evaluate the applicability and accuracy of the two SCR indicators.

5.2. Case Analysis

Under normal stable system operating conditions, there exists a clearly defined and measurable margin between the actual operational SCR values and their corresponding critical thresholds, quantitatively indicating that the power network maintains a sufficient safety reserve capacity. However, the system rapidly transitions into an unstable operating regime when the effective SCR deteriorates and falls below this predetermined critical stability value threshold.

Taking a provincial power grid as an example, due to voltage support capability limitations, the maximum grid-connected scale of new energy is 1875 MW when the critical SCR is below 1.98. Based on this scenario, this paper studies the impact of a new energy grid-connected scale on SCR-U and SCR-O and compares the evaluation effects of the two using the critical SCR.

The CSCR-U value in this case is calculated as 1.98. Figure 5 shows the simulation results when SCR-U is in the range of 1.98 to 1.94. In addition,

Table 1. System stability under different SCR-U values.

SCR-U	Stability
1.98	Stable
1.96	Stable
1.94	Instability

describes the system’s stability under different SCR values, further verifying the differences between SCR-U and SCR-O in system stability evaluation.

According to Equation (18), the CSCR-O in this case is 1.93. Figure 6 shows the simulation results when the SCR-O is between 1.96 and 1.92.

Table 2. System stability under different SCR-O values.

SCR-O	Stability
1.96	Stable
1.94	Stable
1.92	Instability

describes the system’s stability under different SCR-O values.

The analysis of the calculated data from Figure 4 and Figure 5 and Table 1 and Table 2 indicates that the CSCR-U value at the moment of system collapse is 1.98, while the corresponding SCR-U value at that moment is 1.94. This indicates that the obtained SCR-U is more conservative than the critical SCR, potentially with some margin of deviation. In contrast, at the moment of system collapse, the SCR-O value is calculated as 1.92, while the corresponding CSCR-O value is 1.93, closer to the SCR-O value than the CSCR-U. This result demonstrates that SCR-O can more accurately reflect the system’s actual state at the moment of collapse, indicating higher precision and reliability.

6. Conclusions

This study proposed an improved SCR index and established a calculation method based on the critical value of the CSCR. The critical SCR was employed as a benchmark to evaluate the voltage support capability of new energy grid-connected systems through the SCR index. The main contributions are as follows:

(1) A novel SCR evaluation method was designed by integrating voltage calculation and coupling impedance. Although its form is similar to traditional SCR calculations, introducing a coupling impedance factor significantly enhances computational accuracy and real-time performance. In addition, it more precisely reflects the voltage characteristics at various nodes in the power system.

(2) A critical CSCR model based on maximum power transfer theory was established, providing a theoretical foundation to evaluate the reliability of the new SCR index. The study first validated the conservative deviation of the voltage-based SCR index using the traditional critical SCR. The comparative analysis using the critical CSCR confirmed the compatibility of the improved CSCR index with the system’s actual operating conditions.

The proposed method optimized the traditional SCR calculation framework to address the adaptability limitations of traditional indices in new energy grid-connected scenarios, providing a more accurate quantitative tool for real-time assessment of system voltage stability.