A Hybrid Method Based on Corrected Kinetic Energy and Statistical Calculation for Real-Time Transient Stability Evaluation

Abstract

This paper proposes an innovative transient stability index (TSI) designed to enhance the real-time assessment of power system stability. The TSI integrates a corrected kinetic energy approach with a modified equal area criterion, offering a novel methodology for evaluating transient stability margins in power systems. Unlike traditional methods, the proposed TSI operates without relying on post-fault data, making it particularly suitable for online applications. A structure-preserving model is utilized to represent the power network, accounting for key factors such as controller behavior during transient events. Additionally, a new statistical classification method is introduced to efficiently determine the individual contribution of generators to the overall system stability. The effectiveness of the proposed approach is validated through comprehensive case studies on IEEE 9-bus and IEEE 39-bus systems. The simulation results confirm that the proposed method provides accurate, real-time insights into the transient stability margins of power systems, demonstrating its practical advantages in both analysis and operation.

1. Introduction

Ensuring the secure and stable functioning of large-scale power systems has long been a great concern, with transient stability assessment (TSA) being a vital method to achieve this aim [1]. Engineers in the system control centers need innovative tools to calculate the transient stability margins in power systems [2]. An accurate and useful security assessment instrument is essential to maintain the desired level of system stability and security, particularly in major disturbances. The evaluation of transient stability is extremely important for two reasons: first to analyze the system’s ability to endure significant disturbance and second to recommend corrective actions [3]. The inherent nonlinear problem of transient stability in large-scale power networks is difficult and time-consuming to solve [4,5]. Thus, TSA can be classified into two primary subcategories: (1) an evaluation, which analyzes current system conditions, and (2) a prediction, which anticipates future stability outcomes [4,6]. The critical clearing time (CCT) for short circuits, which represents the fault tolerance of the power system, is a central step [7,8]. Unfortunately, traditional CCT calculation methods are computationally heavy, demanding extensive simulations. This significant computational cost makes them unsuitable for real-time monitoring and online use.

To assess transient stability, researchers have proposed various methods. These methods include simulating the system’s behavior over time (time–domain simulation), directly assessing stability using mathematical tools (like Lyapunov functions), applying the extended equal area criterion (EEAC), and leveraging pattern recognition techniques. Additionally, researchers have explored combining these approaches for even more robust analysis [9,10,11,12]. While the time–domain approach shines due to its simplicity, flexibility, and accuracy for assessing transient stability during and after outages [3,13], its high computational cost makes it less suitable for real-time monitoring. This is because it relies on extensive offline calculations, limiting its application to pre-event analysis. The Lyapunov approach is another approach used for estimating CCT and transient stability [14,15,16,17]. While the direct method based on Lyapunov functions offers a fast method for TSA, it is not well-suited for real-time and online uses due to several limitations. First, this method requires post-fault information to determine the potential energy function and update the Lyapunov if the system configuration changes amidst a disturbance. Second, considering all of the detailed elements of the power system in the calculations can be computationally expensive [14,18]. Based on the previous discussion, most energy methods based on the Lyapunov function for transient stability assessment, including the based controlling unstable (BCU) method, rely on post-fault calculations to specify the CCT and stability margin of a power system. Consequently, these strategies are not appropriate for real-time uses [14,19]. Additionally, as mentioned by [20,21,22,23], numerous assessment methods employ pattern recognition. Such methodologies utilize heuristic approaches like artificial neural networks, as noted in [4,24,25,26,27,28], and fuzzy neural networks, mentioned in [29,30,31]. Another method employed in transient stability assessment is the support vector machine (SVM) [26,32]. Additionally, researchers have utilized decision trees (DT) [33,34,35] and kernel ridge regression for evaluating transient stability [36,37]. However, these methods often lack generalizability and are dependent on the quality of training data, which can limit their effectiveness in real-time applications. Consequently, they fall short of providing a holistic solution for evaluating the stability of power systems, as outlined in [23]. In [38], an innovative structure for evaluating power system transient stability is developed, incorporating the probabilistic element. This solution calculates the transient stability probability (TSP) by accounting for the stochastic characteristics of system load variations and fault-clearance durations. Nevertheless, the effectiveness of the machine-learning model is closely tied to the quality and representativeness of the training data. Furthermore, the complexity inherent in the algorithm may require substantial computational resources and expertise for successful implementation. To analyze the transient security of power systems for online applications, the DT approach has been employed as an analytical tool, as demonstrated in [39]. This approach makes use of phasor measurement units (PMUs) in wide area measurement systems (WAMS) to assess the transient security of large, interconnected power system networks. However, potential disadvantages might result from the difficulty of putting the ensemble DT classifier into practice and the requirement for careful PMU placement and selection in order to ensure thorough coverage of the power system network. As cited by [40], and as cited by [38], the authors proposed the approach that employs DT, which lies in its ability to swiftly and precisely predict power system operating conditions in real-time online environments, addressing the demand for low complexity and computational time in online security assessment. However, the potential disadvantages may include the complexity involved in developing and implementing the ensemble security predictor, as well as the need for careful selection and extraction of optimal feature sets for dynamic security prediction. Alternative security assessment algorithms designed for online applications evaluate transient stability by leveraging sourced information from the “control supervision” and “data acquisition system/managing energy,” incorporating appropriate delay times [4].

As previously noted, researchers have pursued parallel objectives, characterized into two categories. The first involves methodologies accounting for all intricacies and nonlinearities within the power system, as seen in references [10,41], while the second focuses on real-time evaluation of transient stability, as exemplified by references [1,11,42]. These methods are not suitable for the immediate assessment of transient stability due to the following reasons:

• they entail a significant computational burden;
• they rely on post-fault information to conduct transient stability assessments.

Considering the contemporary focus on TSA in large-scale power systems, the techniques within the primary category are deemed inadequate for conducting real-time analyses on such expansive networks. In contrast, the methodologies outlined in the next group prioritize the real-time evaluation of power systems. Nevertheless, the proponents of such methodologies have opted for network-reduction models to solve the problem, leading to a notable decline in evaluation precision.

The main objective of this research is to develop an innovative TSI that enables a real-time assessment of power system stability, regardless of the severity of the fault. Analyzing system behavior under fault situations is crucial for thorough transient stability analysis, as faults are one of the main disturbances that can cause instability. This approach offers deeper insights into the transient stability limitations of the system. It achieves this by assessing the magnitude and duration of disturbances that the system can endure while maintaining synchronism.

In this regard, the following explanations focus on the key differences and advantages of our method compared to the traditional and modern approaches discussed in the present paper.

Time–Domain Simulation (TDS)

The limitations of TDS are that time–domain simulation is known for its accuracy and flexibility but suffers from significant computational cost, making it unsuitable for real-time applications. TDS requires extensive post-fault data for analysis, and as mentioned in the manuscript, it is often constrained to offline simulations.

The advantage of the proposed method is that our method, unlike TDS, does not rely on post-fault data, which allows for real-time assessment. By utilizing a structure-preserving model and corrected kinetic energy, the proposed method reduces computational load and can provide transient stability margins efficiently during system operation.

Lyapunov-Based Methods

The limitation of Lyapunov methods is that methods based on Lyapunov functions, such as the BCU method, offer fast computation but require post-fault information and detailed system models, which limits their applicability for real-time stability analysis.

The proposed method’s advantage is that the proposed method eliminates the need for potential energy functions and post-fault data, which are essential for Lyapunov-based methods. By focusing on corrected kinetic energy, it achieves faster and more accurate estimates of system stability without requiring recalculations after every disturbance.

Machine-Learning and Pattern Recognition-Based Methods

As to the limitations of ML methods, machine-learning methods, including neural networks, support vector machines, and decision trees, are highly dependent on the quality and representativeness of the training data. These methods also tend to lack generalizability in complex, real-time environments due to potential mismatches between training and real-time operational data [43,44].

The proposed method’s advantage is that our method does not require training data, making it more reliable and robust in real-time environments. The method’s reliance on statistical calculations and corrected kinetic energy ensures accuracy regardless of changes in system conditions.

Extended Equal Area Criterion (EAC)

The limitations of the EAC are that the traditional EAC method is computationally efficient but is limited in its precision, especially in estimating the critical clearing angle (CCA) when considering the nonlinear behavior of large power systems.

The proposed method’s advantage is that we extend the EAC by combining it with corrected kinetic energy to offer more accurate estimates of CCA and real-time stability. The proposed modification enhances the accuracy of critical angle and critical kinetic energy estimations, addressing the shortcomings of traditional EAC.

In summary, the proposed method provides significant advantages over traditional TDS, Lyapunov, and machine-learning approaches. Its real-time capabilities, combined with high accuracy and reduced computational complexity, make it an ideal choice for modern power system transient stability assessment.

1.1. Contributions

In this research, a novel hybrid structure is suggested, aimed at addressing the dual objectives outlined in the aforementioned groups. Moreover, the method is adaptable for use with online monitoring techniques, such as those based on phasor measurement units (PMUs). The proposed method (PM) relies on a modified version of the EAC and corrected kinetic energy of the system. Initially, the adjusted EAC is applied to determine the critical point, which is then refined using the corrected kinetic energy function to achieve higher performance. Additionally, a TSI is suggested for the real-time assessment of the stability margin.

The key innovations are summarized:

• A structure-preserving model for power network modeling is employed to address the limitations of existing methods when considering model intricacies;
• A novel modification to the EAC is introduced, enhancing the accuracy of critical angle point estimation;
• A new method for categorizing generator contributions during fault conditions based on statistical analysis is presented, accounting for fault location effects and identifying participating generators;
• Precise estimates for the system’s critical angles and critical kinetic energy are provided, contingent upon fault location, without the need for post-fault data or potential energy functions;
• Real-time estimation of the transient stability margin for both the system and the generators using kinetic energy exclusively is offered.

Together, these innovations address the real-time challenges that are intrinsic to transient stability assessment. The efficacy of the approach in practical scenarios is showcased by eschewing the reliance on post-fault data and demonstrating swift and accurate assessments during fault clearance.

1.2. Organization

This manuscript is arranged as follows. Section 2 outlines the network-preserving model and its associated mathematical expressions, serving as the network model of the system. Section 3 presents the definition of the problem and the solution methodology in the paper, while Section 4 delves into the simulation results. Ultimately, Section 5 summarizes the main findings of the research.

2. Background (Network-Preserving Model)

A real-time method for assessing transient stability needs to meet two essential requirements. It should be computationally efficient and capable of promptly estimating stability margins without relying on post-fault data. As discussed earlier, a significant drawback of transient stability assessment methods lies in the accuracy of the employed model [45]. Furthermore, when considering real-time applications, determining the minimum point of the potential energy function of the system becomes an additional concern in transient stability assessment. This process often necessitates access to post-fault data, and the minimum point location can vary depending on the fault location. Moreover, the system topology changes prompted by disturbances add complexity, making recalculating the system’s operating point particularly arduous and time intensive. Because network models are not subject to reduction, all node identities remain unchanged. As a response to this issue, the concept of the network-preserving model emerged during the 1980s [46]. This model ensures the preservation of the network’s structure, thereby giving rise to what is known as a “structure-preserving” or “topological” energy function associated with it.

The models presented in the studies [47,48], are utilized with the objective of maintaining the integrity of the network structure. A two-axis generator model with a first-order exciter system is opted for to represent the internal generator bus, as outlined in [49,50].

Where:

$$\begin{array}{l}{P}_{ei}={\displaystyle \frac{1}{{x^{\prime }}_{di}}}{E^{\prime }}_{qi}{V}_i\sin ({\delta }_i-{\theta }_i)\\ +{\displaystyle \frac{1}{{x^{\prime }}_{qi}}}{E^{\prime }}_{di}{V}_i\cos ({\delta }_i-{\theta }_i)\\ +{\displaystyle \frac{{x^{\prime }}_{di}-{x^{\prime }}_{qi}}{2{x^{\prime }}_{di}{x^{\prime }}_{qi}}}{V_i}^2\sin [2({\delta }_i-{\theta }_i)]\end{array}$$

(1)

Furthermore, the simplified dynamics model associated with the exciter, employed in the one-axis generator model, is articulated as follows [50]:

$$T_{vi}{\dot{E}}_{fi}=-E_{fi}-{\mu }_i{k}_i{V}_i\cos ({\delta }_i-{\theta }_i)+l_i$$

(2)

3. Defining the Problem and Solution Methodology

This section outlines the suggested strategy for achieving real-time functionality. To address the limitations associated with the network-reduction model, this paper adopts the network-preserving model to frame the transient stability issue, as detailed in Section 3.1. The process of computing the critical clearing angle and the fundamental concept of the PM will be discussed in the next subsection. After a thorough explanation of the primary issue, the step-by-step solution procedure is detailed. A reliable method for estimating the TSI should have the ability to swiftly and accurately calculate the TSI during both the disturbance and post-disturbance phases at each simulation time step while taking into account all of the complexities of the power system. The initial stage in computing TSI entails identifying the critical clearing angle for all generators within the system. Essentially, this critical angle marks the juncture where the kinetic energy of generators during the fault aligns with the potential energy of generators following the fault. These critical points are denoted as unstable equilibrium points (UEP) of the system [51]. Hence, while a fault happens within the system, the system’s minimum potential energy matches the maximum kinetic energy that each generator is capable of reaching to maintain stability. Additionally, the location and trajectory of the fault affect the UEP within the system. Consequently, the study presents a novel method for estimating the initial UEP by combining a modified EAC with corrected kinetic energy. Finally, an examination of the impacted generators is conducted to account for the influence of fault trajectory and location within the system. Consequently, the PM accurately determines the UEP without reliance on a potential energy function. The computation of the minimum potential energy functions poses significant challenges and consumes considerable time, especially in large power systems with numerous generators. Moreover, potential energy functions necessitate post-fault data for UEP estimation, rendering them unsuitable for real-time and online applications. Algorithm 1 outlines the sequential steps for computing the real-time TSI using the PM. It comprises eight sequential steps as follows:

(1)

Setting the time;

(2)

Reading the real-time load flow data and dynamic data;

(3)

Estimating the initial critical angle using the modified equal area criterion (MEAC) function without accounting for the fault location effect. This involves employing the classical EAC concept to analyze the primary critical angle point, refining it through a stable or unstable case approach, and considering the effects of automatic voltage regulator (AVR) and governors;

(4)

Calculating the derivative gain severely disturbed group (SDG);

(5)

Calculating the load derivative gain less disturbed group (LDG);

(6)

Obtaining accurate estimates of critical kinetic energy;

(7)

Calculating the normalized rotor angle deviation;

(8)

Determining the real-time transient stability index.

Algorithm 1: Calculation procedures of real-time TSI by PM
1. Set: $t+\Delta t\to t$	% Setting the time
2. Input: $L,F,P,\omega ,V,Te,X,x_d^{\prime },\dots$	% Load flow data and dynamic data
3. Modified EAC Function: ${\displaystyle \underset{t_1}{\overset{t_2}{\int }}\delta (t)}dt={\displaystyle \underset{t_3}{\overset{t_4}{\int }}\delta (t)}dt+\Delta KE$	% Estimating initial critical angle
4. Calculating: $SDG={\displaystyle \frac{dX}{dF}}$	% Calculating SDG
5. Calculating: $LDG={\displaystyle \frac{dL}{dF}}$	% Calculating LDG
6. $K{E}_{critical}=K{E}_0+SDG.(F-F_0)+LDG.\Delta L$	% Obtaining accurate estimates of critical kinetic energy
7. $Normalized Deviation={\displaystyle \frac{\Delta {\delta }_i}{{\delta }_{{\max }_i}}}$	% Calculating normalized rotor angle deviation
8. $RTTSI={\displaystyle \frac{1}{2}}{{\displaystyle \sum _{i=1}^n(Normalized Deviatio{n}_i)}}^2$	% Calculating the real-time transient stability index

3.1. Estimation of Critical Clearing Angle

As previously explained, the PM system must determine the critical angle of generators to enable the real-time estimation of the TSI for the system. To accomplish this, the PM employs an innovative practical method, extensively discussed in [52], to estimate the CCA accurately. Figure 1 offers a graphical overview of this method.

As illustrated in Figure 1, the PM unfolds in two stages to estimate the precise CCA.

Stage 1, depicted in the diagram, involves estimating the primary critical angle of generators by adapting the EAC concept. Every generator is represented by an internal voltage (${\rm E}\angle \delta$) that supplies electrical power for the network, which is represented using terminal voltage ($V_g\angle {\theta }_g$). Notably, this phase does not incorporate the impact of AVR and the governor. Thus, the initial CCA is derived using the following equations [52]:

$${\displaystyle \underset{{\delta }_0}{\overset{{\delta }_c}{\int }}{P}_{m}d\delta ={\displaystyle \underset{{\delta }_c}{\overset{{\delta }_{\max }}{\int }}(P_e-P_m)d\delta }}$$

(3)

$$\begin{array}{l}{P}_{ei}={\displaystyle \frac{1}{{x^{\prime }}_{di}}}{E^{\prime }}_{qi}{V}_i\sin ({\delta }_i-{\theta }_{gi})\\ +{\displaystyle \frac{1}{{x^{\prime }}_{qi}}}{E^{\prime }}_{di}{V}_i\cos ({\delta }_i-{\theta }_{gi})\\ +{\displaystyle \frac{{x^{\prime }}_{di}-{x^{\prime }}_{qi}}{2{x^{\prime }}_{di}{x^{\prime }}_{qi}}}{V_i}^2\sin [2({\delta }_i-{\theta }_{gi})]\end{array}$$

(4)

$$\begin{array}{l}{P}_m({\delta }_c-{\delta }_0)=\\ \left[\begin{array}{l}{\displaystyle \frac{1}{{x^{\prime }}_d}}{E^{\prime }}_{q}V\sin ({\delta }_{\max }-\theta )+{\displaystyle \frac{1}{{x^{\prime }}_q}}{E^{\prime }}_{d}V\cos ({\delta }_{\max }-\theta )\\ +{\displaystyle \frac{{x^{\prime }}_d-{x^{\prime }}_q}{2{x^{\prime }}_d{x^{\prime }}_q}}{V}^2\sin [2({\delta }_{\max }-\theta )]\end{array}\right]\\ -\left[\begin{array}{l}{\displaystyle \frac{1}{{x^{\prime }}_d}}{E^{\prime }}_{q}V\sin ({\delta }_c-\theta )+{\displaystyle \frac{1}{{x^{\prime }}_q}}{E^{\prime }}_{d}V\cos ({\delta }_c-\theta )\\ +{\displaystyle \frac{{x^{\prime }}_d-{x^{\prime }}_q}{2{x^{\prime }}_d{x^{\prime }}_q}}{V}^2\sin [2({\delta }_c-\theta )]\end{array}\right]\end{array}$$

(5)

Stage 2, as indicated in Figure 1, involves refining the primary CCA by using a stable or unstable case approach to account for the influence of AVR and governor functions throughout both fault and post-fault periods. Additionally, this stage entails calculating the critical kinetic energy for generators with high accuracy, similar to the approaches mentioned in [52,53]. Consequently, the TSI for each generator within the system can be formulated as shown in Equation (7).

$$S{I}_{generator,i}(t)=\left\{\begin{array}{c}\begin{array}{l}{\displaystyle \frac{K_i^t(t)}{K_i^{crit}}}:duringFaultcondition\\ {\displaystyle \frac{K_i^t(t)}{2K_i^{crit}}}:duringpostfaultcondition\end{array}\\ i=1,2,\mathrm{\dots },n;n:numberofgenerators\end{array}\right.$$

(6)

The preceding discussion has focused on determining the real-time TSI for each generator, irrespective of fault location, through the modified EMC concept detailed in [52,53]. Yet, it is clear that the critical kinetic energy of the system is contingent upon both the fault location and the trajectory. Hence, to address this dependency and accurately estimate the system’s critical kinetic energy, the corrected kinetic energy method is employed.

3.2. Corrected Kinetic Energy Method

As highlighted by [51], not all generators participate in system instability when a disturbance happens. Specifically, it is only the kinetic energy generated by disturbed generators that leads to instability, rather than the total kinetic energy produced within the system. Therefore, considering the involvement of disturbed generators is critical for accurately estimating the TSI in power systems, as it enables the incorporation of fault trajectory effects when determining the critical clearing angle of generators. Following a disturbance like a three-phase short circuit, the system is segregated into two distinct groups of generators: the SDG and the LDG [52,53]. Every group comprises a minimum of two generators, and their identification follows the algorithm outlined next.

The correlation among a pair of objects in a dataset is determined by the feature–feature correlation. Reference [54] proposed a strategy called normalized mean residue similarity (NMRS) to eliminate the redundancy between features. This research proposes an improved normalized mean residue similarity (INMRS) for identifying correlations among generators’ angular velocities. As will be discussed, the INMRS helps to classify the contributions of generators.

The equation of the INMRS between two objects $F_1=(x_{11},x_{12},\dots ,x_{1N})$ and $F_2=(x_{21},x_{22},\dots ,x_{2N})$ is outlined below:

$$INMR{S}_{(F_1,F_2)}=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N{\left|(x_{1i}-x_{1lm(i)})-(x_{2i}-x_{2lm(i)})\right|}^2}}{2\times \max ({{\displaystyle {\sum }_{i=1}^N\left|x_{1i}-x_{1lm(i)}\right|}}^2,{{\displaystyle {\sum }_{i=1}^N\left|x_{2i}-x_{2lm(i)}\right|}}^2)}}$$

(7)

where

$$x_{(k)lm(i)}={\displaystyle \frac{{\left\{x_{(k)1}+x_{(k)2}+\dots +x_{(k)i}\right\}}_{i=1,\dots ,N}}{N}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{x}_{(k)i}}}{N}}$$

(8)

Notably, $F_i\in \left\{\left\{{\omega }_i\right\}\right.,\left\{{\displaystyle \frac{d{\omega }_i}{dt}}\right\}|,\left.i\in n_{GEN}\right\}$ refers to the features under investigation, specifically the angular velocity and acceleration of the generators. $\left[\Lambda \right]$ denotes the matrix of $NMRS$ for ${\omega }_i,i\in n_{GEN}$, and it is explained as follows:

$$\left[\Lambda \right]=\left\{\begin{array}{cc}{\lambda }_{ij}=1& i=j\\ {\lambda }_{ij}=INMRS(\left\{{\omega }_{1i},\dots ,{\omega }_{Ni}\right\},\left\{{\omega }_{1j},\dots ,{\omega }_{Nj}\right\})& i\ne j\end{array}\right.$$

(9)

Similarly, the value of $\left[\Gamma \right]$ is determined as follows:

$$\left[\Gamma \right]=\left\{\begin{array}{cc}{\gamma }_{ij}=1& i=j\\ {\gamma }_{ij}=INMRS\left(\left\{{\displaystyle \frac{d{\omega }_{1i}}{dt}},\dots ,{\displaystyle \frac{d{\omega }_{Ni}}{dt}}\right\},\left\{{\displaystyle \frac{d{\omega }_{1j}}{dt}},\dots ,{\displaystyle \frac{d{\omega }_{Nj}}{dt}}\right\}\right)& i\ne j\end{array}\right.$$

(10)

The implementation procedure is outlined as follows:

• Select $i-th$ generator;
• Determine ${\lambda }_{ij}$ and ${\gamma }_{ij}$ for $j=1\cdots n_{GEN}$;
• If ${\lambda }_{ij}\ge \xi$ and ${\gamma }_{ij}\ge \xi$, then $i-th$ and $j-th$ generators are stated in the SDG of generators. If not, they are considered members of the LDG.

After identifying the generators belonging to the SDG group, the rest of the generators in the system, which are not part of the SDG set, are categorized as members of the LDG category. Subsequently, the corrected critical kinetic energy of the system is determined by these equations:

$$\left\{\begin{array}{l}{M}_{SDG}={\displaystyle \sum _{i=1}^n{M}_i;i\in SDG}\\ M_{LDG}={\displaystyle \sum _{i=1}^n{M}_i;i\in LDG}\\ M_{eq}={\displaystyle \frac{M_{SDG}\times M_{LDG}}{M_{SDG}+M_{LDG}}}\end{array}\right\}$$

(11)

$$\left\{\left.\begin{array}{c}{W}_{SDG}^{crit}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n}M{W}_i^{crit}}}{{\displaystyle \sum _{i=1}^n{M}_i}}}\begin{array}{cc}& ,i\in SDG\end{array}\\ W_{LDG}^{crit}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n}M{W}_i^{crit}}}{{\displaystyle \sum _{i=1}^n{M}_i}}}\begin{array}{cc}& ,i\in LDG\end{array}\end{array}\right\}\right.W_{EQ}^{crit}=W_{SDG}^{crit}-W_{LDG}^{crit}$$

(12)

$$K_{EQ}^{crit}=0.5\times M_{EQ}\times {(W_{EQ}^{crit})}^2$$

(13)

The corrected kinetic energy of the system during the fault duration is denoted as $K_{EQ}^{crit}$. Lastly, the system TSI is estimated using Equation (14).

$$TS{I}_{System}(t)=\left\{\begin{array}{c}{\displaystyle \frac{K_{EQ}^t(t)}{K_{EQ}^{crit}}}\begin{array}{cc}& :Durationoffault\end{array}\\ {\displaystyle \frac{K_{EQ}^t(t)}{2K_{EQ}^{crit}}}\begin{array}{cc}& :Durationofpostfault\end{array}\end{array}\right.$$

(14)

As previously discussed regarding the TSI of all generators, the maximum kinetic energy of the system equals $K_{EQ}^{crit}$ during the fault duration and doubles to $2K_{EQ}^{crit}$ during the post-fault duration. The calculation stages of the system’s TSI are outlined in detail in Algorithm 2.

Algorithm 2: Procedure for calculating system TSI
1. Input data: $L,F,P,\omega ,V,Te,X,x_d^{\prime },\dots$	% Load flow data and dynamic data.
2. $I{C}_{unfaulted}=f(Inputdata)$	% Initial condition of the machine during unfaulted operation.
3.  for $t=0\Rightarrow t+\Delta t\to t$
$E_{critical}={\displaystyle \frac{1}{2}}I.{(\Delta \varpi )}^2$	% Estimating the critical kinetic energy of system generators.
${\theta }_{initial\_critical}={\theta }_0+{\displaystyle \frac{T_{mesh}}{2H}}(1-\sqrt{1-{\displaystyle \frac{4HD}{T_{mesh}^2}}}$	%Estimate initial critical angle of generators using EAC
$E_{\mathrm{initial}\mathrm{critical}}={\displaystyle \frac{1}{2}}I.{\varpi }^2$	% Estimate the initial critical kinetic energy of generators.
Putting fault at terminal of generators
if (Stable)
applying stable case
$E_{\mathrm{critical}\text{\_}precision}={\displaystyle \frac{1}{2}}I.{\varpi }^2$	% Calculate high precision value of critical kinetic energy
else
applying unstable case
end
end
4.   for $t=0\Rightarrow t+\Delta t\to t$
$SDG={\displaystyle \frac{dX}{dF}}$, $LDG={\displaystyle \frac{dL}{dF}}$	%Determine SDG and LSD generators
$K{E}_{corrected}={\displaystyle \sum _{i=1}^n\frac{1}{2}}{J}_i{\varpi }_i^2$
$K{E}_{critical,corrected}=E_{initial}+\Delta KE$
if ($t\in Faultdurationtime$)
$TS{I}_{system}={\displaystyle \frac{K_{EQ}^t}{K_{EQ}^{crit}}}$
else
$TS{I}_{system}={\displaystyle \frac{K_{EQ}^t}{2K_{EQ}^{crit}}}$
end
if ($t=simulationtime$)
Finished the calculation
else
$t+\Delta t\to t$
end
end

4. Simulation Results

To verify the efficacy of the PM, this study uses the IEEE 9-bus and IEEE 39-bus test systems. The simulation of these systems employs the synchronous reference frame, with the assumption, wherein the speed and rotor angle of every generator are evaluated against the reference generator (slack) angle.

4.1. IEEE 9-BUS SYSTEM

The IEEE 9-bus system contains three generators and nine lines, which are shown in Figure 2, as detailed in [55]. The details of IEEE 9-bus are provided in Appendix A. The simulation of the IEEE 9-bus network is conducted with a time step of 0.0001 s, where the rotor angle of generator 1 serves as the reference angle for the study. Figure 3 zooms in on generators 2 and 3 during a three-phase fault at bus 6, occurring at 0.0 s and resolved at 0.4444 s. As depicted in Figure 3, generators 2 and 3 remain stable until the fifth second, after which both of them transition into instability. Consequently, the SM identifies instability after five seconds in this scenario. However, as shown in Figure 4, the PM anticipates the instability of generators prior to fault removal, precisely at 0.332 s, as the system’s TSI reaches one at this time. Indeed, Figure 4 demonstrates the capability of the PM to assess the stability margin of every generator and the entire network during the fault duration, at each iteration of the simulation. As depicted in this figure, the TSI of both generators and the system falls within the range of zero to one. Essentially, the TSI curve illustrates their stability margin when a disturbance arises in the system. In this approach, the TSI approaches one when generators and the system are unstable, while it tends toward zero as the system moves towards stability. Therefore, the system maintains stability in the post-fault duration as long as the TSI remains below the threshold value of one during the fault duration. This demonstrates the high accuracy of the PM in estimating the stability margin of the network for online and real-time applications.

Figure 5 displays the angles of generators 2 and 3 while a three-phase fault happens on bus 6 at 5.0 s and is resolved at 5.30 s. Both generators remain stable after the fault clearance, but they exhibit oscillatory stability. And their stability margins change compared to the fault condition, as illustrated in Figure 6. This figure presents the TSI of the generators and the system throughout each simulation iteration. According to Figure 6, the TSI values for the generators and the system fluctuate between zero and one during the fault period. Consequently, the generators and the system maintain stability after the fault is resolved, as depicted in Figure 5. Essentially, the TSI curve depicts the stability state of the generators and the system during pre-fault, fault, and post-fault durations throughout each simulation iteration.

4.2. IEEE 39-BUS SYSTEM

The diagram illustrating the IEEE 39-bus network, including ten generators and 45 lines, is shown in Figure 7 [55]. The details of IEEE 39-bus are provided in Appendix B. As in the previous section, a three-phase short-circuit fault is initiated near one of the buses to assess the TSI of both the generators and the entire system using the proposed method. The simulation operates with a time step of 0.0001 s, with the rotor angle of generator 2 serving as the reference angle for this investigation.

Figure 8 focuses on the generators of the system during a three-phase fault occurring at bus 24 at 0.0 s and being cleared at 0.2000 s. As illustrated in Figure 8, generator 9 becomes unstable after 3.8 s. Hence, the TD simulation identifies instability in generator 9 at a simulation time of 3.8 s. However, based on Figure 9, the PM forecasts the instability of generator 9 before the fault is cleared, precisely at 0.178 s, as its TSI reaches one at this time. Additionally, as indicated in Figure 6, the remaining generators maintain stability after the fault is cleared, as their TSI values are less than one, as depicted in Figure 9.

Figure 10 displays the TSI of the system during the mentioned fault scenario. As depicted, the stability curve closely approaches one, indicating that the system is nearing the threshold of instability, and the stability margin is minimal for the given fault condition. Essentially, the system remains unstable for a prolonged duration of the fault event, as illustrated in Figure 11b where the system instability persists for a fault duration time of 0.2300 s. Consequently, in Figure 11a, the TSI of the system remains at one throughout the post-fault period.

5. Discussion on Results

As illustrated in the simulation results, the primary contribution of the proposed method is its ability to provide a real-time transient stability assessment (TSA) through a hybrid approach combining a modified equal area criterion (EAC) and a corrected kinetic energy. This offers a significant improvement over the traditional methods that typically fall short in one or more of the following areas: computational efficiency, reliance on post-fault data, or model accuracy. We present the advantages of our approach by comparing it to other methods, both simulated in this work and from the broader literature.

Time–Domain Simulation (TDS)

For the existing methods, time–domain simulations offer accurate TSA by simulating the system’s behavior under various fault conditions. However, they require extensive offline simulations, and real-time application is limited due to high computational costs.

For the proposed method, unlike TDS, which depends on post-fault data and is computationally intensive, our method computes transient stability margins without the need for post-fault data. This reduction in computational overhead makes the proposed method highly suitable for real-time applications. The practical advantage lies in faster decision-making in online monitoring and control scenarios.

Lyapunov-Based Energy Function Methods

For the existing methods, methods like the BCU (boundary of CUEP) utilize potential energy functions and rely heavily on post-fault system configurations. While efficient in theoretical analysis, their real-time application is limited due to the need for recalculating energy functions after disturbances.

For the proposed method, our approach eliminates the need for post-fault data and potential energy functions by focusing on corrected kinetic energy. This simplification ensures that real-time assessment can be conducted without recalculating system configurations, making it faster and more adaptable to real-time applications.

Machine-Learning and Heuristic-Based Methods:

For the existing methods, approaches based on machine learning, including neural networks, support vector machines (SVM), and decision trees have been explored for TSA. However, these methods depend on the availability and quality of training data, and they often suffer from generalizability issues when the system experiences conditions not covered in the training set.

For the proposed method, unlike machine-learning methods, our hybrid approach does not rely on training data or pattern recognition. It instead uses a statistical classification based on the system’s current state, providing a more robust solution for real-time applications. This avoids the unpredictability and potential inaccuracies associated with poorly trained models.

Equal Area Criterion (EAC)

For the existing methods, the traditional EAC methods are limited by their simplified treatment of nonlinear system dynamics, often leading to inaccurate estimations of critical clearing angles and critical kinetic energy.

For the proposed method, we improve upon the EAC by modifying it to account for the corrected kinetic energy and introducing a statistical classification method to determine generator contributions during faults. This leads to more accurate and reliable real-time assessments. By combining EAC with kinetic energy calculations, our method provides better precision in calculating critical clearing angles and stability margins without the computational burden of traditional methods.

Key Contributions

For real-time assessment without post-fault data, unlike many traditional methods, the proposed approach does not require post-fault system data for transient stability analysis, making it uniquely suited for real-time applications.

For the hybrid approach combining EAC and corrected kinetic energy, the integration of a modified EAC and corrected kinetic energy offers higher accuracy in estimating critical angles and kinetic energy, which enhances the transient stability assessment process.

For the efficient classification of generators, by employing a statistical method, generators are categorized into severely disturbed and less-disturbed groups, improving the precision of stability margin estimations under various fault conditions.

For applicability to large-scale systems, the proposed method is tested on IEEE 9-bus and 39-bus systems, demonstrating its effectiveness in both small- and large-scale systems. Its structure-preserving nature allows it to handle complex power network models with high accuracy.

6. Conclusions

This study established a new strategy for accurately estimating the TSI of both the overall system and individual generators, which is suitable for real-time and online uses. The PM employs a hybrid technique that combines corrected kinetic energy and statistical calculations. It is adaptable to various network models, including both network-preserving and network-reduction models, and ensures real-time simulation incorporates all pertinent details of the power system. Through testing on the IEEE 9-bus and IEEE 39-bus test systems, simulation results demonstrate the strategy’s capability to precisely analyze critical clearing angles and critical kinetic energy of system generators. Additionally, it reliably calculates the TSI for the system and all generators throughout each real-time simulation iteration, maintaining accuracy without sacrificing any detail in transient stability assessment. Consequently, this approach proves to be an effective, precise, and straightforward structure with minimal computational overhead, making it suitable for practical deployment in online and real-time uses.