A coherent generator group identification algorithm under extreme conditions

0 Introduction

The development of an advanced society is based on the reliability and affordability of energy distribution[1].With the rapid increase in power demand, modern power systems have become increasingly complex and interconnected, leading to a heightened risk of cascading accidents in extreme situations.Moreover, the extensive integration of renewable-energy-based distributed generators has led to a significant reduction in system inertia[2,3],thereby aggravating the severity of the stability problem.In recent years, extreme disasters have resulted in numerous large-scale power outages and significant economic losses [4,5].Network attacks targeting cyberphysical power systems, such as replay, denial-of-service,and false data injection attacks[6],may also lead to severe blackout incidents.Controlled islanding is considered an effective method for maintaining maximum stability and safety of power systems.The principle is that when a disturbance occurs, generators can be divided into two or more groups according to their coherency, serving as power sources in islands to maintain the electricity supply.As a crucial component of the controlled islanding technology, fast and accurate coherency detection must first be performed to prevent the system from losing stability after islanding, especially under extreme conditions.

So far, the proposed coherency detection methods can be classified into two general categories: measurementbased and model-based methods [7].The key objective of measurement-based methods is to analyze the similarity of the rotor angle data between different generators collected by the phasor measurement unit (PMU) and wide area measurement system (WAMS).Among the relevant studies, Senroy et al.introduced a method utilizing the Hilbert-Huang transform to capture the generator coherency by analyzing instantaneous phase variations between oscillations [8].Jonsson et al.developed an approach for identifying coherent generators by applying a fast Fourier transform to wide-generator-area speed measurement data[9].Avdakovic´ et al.proposed a technique that uses the wavelet phase difference to analyze the signal of generator movement, finding the coherent groups [10].Anaparthi et al.presented an innovative solution using principal component analysis that reduces the dimensionality of data to identify coherent generator groups [11].Susuki et al.presented a novel modal analysis framework for multimachine power systems based on the Koopman operator,introducing Koopman modes that extend linear oscillatory modes to nonlinear dynamics,and providing a method for coherency identification in swing dynamics [12].Lin et al.proposed a coherency detection algorithm for power systems with high renewable penetration utilizing WAMS data in combination with kernel principal component analysis and affinity propagation clustering techniques[13].These measurement-based methods rely on the real rotor-angle data of the generators over a period of time.However, when a large-scale cascading event occurs,obtaining coherent generator information is an urgent necessity, which means that acquiring rotor angle data over a period of time is not feasible.In addition,measurement-based methods require substantial computing resources and are therefore unsuitable for online applications.

The fundamental concept of the model-based method is to analyze the linearized model of generators to obtain coherency information.Reference [14] provides a method that utilizes the singular value decomposition of the reachability Grammian of the system model.Reference[15] presents a Krylov subspace-based approach for power system model reduction, highlighting its utility in the identification of coherent generator groups.Among the model-based methods, the slow coherency method is one of the most extensively used.An approach based on slow coherency was proposed in [16], with a detailed algorithm and its application in large-scale power systems described in [17]. [18,19] developed the slow coherency method based on the singular perturbation theory and the multi-time scale characteristics of power systems.A method for accurately detecting slow coherent generator groups by effectively minimizing generic normalized cuts was presented in [20], along with guidance on choosing the number of groups.Reference [21] offered a novel approach to assess the impact of inertia changes on generator coherency using the matrix perturbation theory.Considering the impact of integrating doubly fed induction generators into power systems, [22] presented a mathematical framework to analyze the alteration of the weighted Laplacian matrix and the slow eigenspace of generators, which can potentially change the coherent areas.When the slow coherency method is used, the system must be linearized at the stable equilibrium point(SEP),thereby transforming the coherent generator group identification problem into a classification problem of the state matrix eigenvectors.Owing to their lower computational complexity, slow coherency-based methods are well-suited for online applications and can accommodate more scenarios.

Generally, traditional slow coherency methods are based on two assumptions:

1) The coherent generator grouping result is independent of the disturbance magnitude.It means the coherency can be determined by the system linearized at the SEP.

2) The coherent generator grouping result is independent of the generator model details.This implies that the impact of the excitation and turbine-governor systems can be neglected.

However, the magnitude of disturbances in power systems is uncertain under extreme conditions, which means that Assumption 1) is not valid.Therefore, most current slow coherency methods are unsuitable for identifying coherent generator groups under extreme conditions.This fact considerably diminishes the practicality of the method.

To overcome this limitation, we propose an algorithm that can adapt to disturbances of various magnitudes for coherent generator identification under extreme conditions.The contributions of this work are as follows: 1) A method is proposed to distinguish the magnitude of disturbances ranging from small disturbances to unstable scenarios.2) An algorithm is extended to identify coherent groups under large disturbances based on the slow coherency method.3)An efficient and effective algorithm is proposed for the identification of coherent groups.

1 Concepts and preliminaries

1.1 Slow coherency method

In power systems, some generators exhibit similar dynamical response characteristics to certain disturbances.This phenomenon is known as the generator coherency.That is,over a time interval t ∈[0,τ], differences in power angle between two generators i and j remain within a certain range ε:

where ε is a positive value,δi is the power angle for generator i, and max represents the maximum function that determines the largest value of its argument over the time interval [0, τ].

The second-order model, which ignores the excitation and turbine-governor systems of synchronous generators,is widely used when conducting slow coherency analysis[23]:

where,for generator i,δi is the power angle,ωi is the rotor speed,Ti is the inertia time constant,Pmi is the mechanical power,Pei is the electromagnetic power,and ω0 is the synchronous velocity.

The electromagnetic power can be expressed as

where n is the number of generators in the system;Ei is the voltage behind the transient reactance of generator i; Bij and Gij are the real and imaginary parts of the admittance matrix reduced to the internal voltage nodes of the generators.

The linearized system can be formulated as

where Δδ is the deviation of the state variable δ from the equilibrium point (EP), M = diag[T1/ω0, T2/ω0,..., Tn/ω0] is the diagonal matrix of inertia constant, and K is the matrix of synchronous power coefficient:

Let state matrix A denote M-1K,whose eigenvalues and eigenvectors are denoted as λi and ui, for i = 1...n.Each eigenvalue of A represents a time domain mode of Δδ, in which the positive eigenvalue represents a divergent mode,the zero eigenvalue represents a linear mode, and the negative eigenvalue represents an oscillation mode with frequency The elements in the row vector of the eigenvector matrix U = [u1,u2,...,un] represent the coeffi-cients of each mode.

Two deviations of power angle Δδi and Δδj are similar if the corresponding row vectors in U share a high degree of similarity, which implies that generator i and generator j are approximately coherent.For slow coherency methods,the slowest r mode is considered,that is,the vectors in the eigenvector matrix Ur = [u1,u2,...,ur] that correspond to the r smallest absolute eigenvalues are taken into account.The detailed steps of the slow coherency algorithm are described in [24].

1.2 Boundary of stability region based controlling unstable equilibrium point (BCU) method

In power systems,two main methods are used for transient stability analysis: time-domain simulation and direct methods [25].Time-domain simulation methods integrate dynamical equations to capture the transient characteristics of power systems under fault conditions.This method provides accurate trajectory and transient stability estimation information.However, its high computational cost and time consumption,particularly for large-scale systems,limit its real-time application.For the direct method, the earliest approach was based on the Lyapunov stability theory,which assesses the stability from the perspective of the energy function[26]by comparing the transient energy Vcl at the fault-clearing time with the critical energy.Compared with time-domain simulation methods, the advantages of direct methods include lower computational complexity and the capability to provide stability indices.Based on the perspective of the energy function,Kakimoto et al.proposed the potential energy boundary surface method (PEBS).The main objective was to determine the exit point of the fault-on trajectory on the potential energy surface, thereby estimating the critical stability energy of the system.Building on the theoretical analysis of the direct method, Chiang et al.proposed the BCU method.The BCU method serves not only as a direct method for transient stability analysis but also for searching the CUEP,which utilizes the controlling unstable equilibrium point (CUEP) of a reduced system to identify the CUEP of the original system [27].The relevant theoretical foundations and definitions are as follows.

For a nonlinear system described by a set of differential equations

the EPs are the state that satisfies the equation ˙x=0.Typically, a nonlinear dynamical system may have multiple EPs.The type of an EP is determined by the eigenvalue of Jacobian matrix J at the EP.If all the eigenvalues have negative real part, the EP is referred to as SEP.If J has exactly k eigenvalues with positive real part, the EP is referred to as a type-k unstable equilibrium point (UEP).If J has no eigenvalue with zero real part, then the EP is referred to as a hyperbolic equilibrium point.

Let denote a hyperbolic equilibrium point of system(6).The stable manifold and unstable manifold of ^x can be expressed as follows:

As shown in Fig.1, a stable manifold is a collection of trajectories that converge to^x, whereas an unstable manifold is a collection of trajectories that diverge from .

For systems that are not globally asymptotically stable,the stability region A（xs） of xs can be defined as follows:

where xs denotes an SEP of the system.Let ∂A(xs) denote the boundary of the stability region.

For systems that satisfy the three assumptions in [28],let xi denote the UEP on the stability boundary of xs,then∂A(xs) can be defined as

where E represents the set of all the UEPs of the system.This theorem implies that the stability boundary of xs is composed of the stable manifolds of the UEPs on the stable boundary.

The CUEP concept was first proposed in the 1970s.CUEP xc is defined as a UEP whose stable manifold contains the exit point of the fault-on trajectory xf (t) [27], as shown in Fig.2.The fault-on trajectory refers to the trajectory in which a sustained fault occurs at the SEPof pre-fault system.The exit point xep is defined as the point at which the fault-on trajectory intersects the stability region.For a system with an energy function,if the stability region of the post-fault SEP contains the pre-fault SEP,the CUEP exists and is unique [29].

Fig.1 Trajectories near the stable and unstable manifold.

Fig.2 Diagram of the CUEP.

It can be proven that the CUEP is a type-I UEP on the stability boundary, whose unstable manifold converges to the SEP of the post-fault system xs[29],as shown in Fig.3.

In direct methods,it is necessary to transform model(2)into a center of inertia (COI) reference frame [30] as follows:

Fig.3 Unstable manifold of the CUEP.

At the core of the direct method is the energy function.Athay et al.proposed a practical energy function based on the COI reference frame [31].

where Cij =EiEjBij and Dij =EiEjGij.In (11), the first term represents the kinetic energy of the system, denoted as V k（ω～）, and the last three terms represent the potential energy, denoted as V p（θ）.Specifically, the fourth term is an integral of trajectory from the SEP to the current state,which is unknown if a time-domain simulation is not performed.However, the time-domain simulation is an exact avoidance in direct methods.Therefore, the following linear trajectory approximation is proposed:

Based on the above theory, a reduced system is introduced in the BCU method.

which is related to the original system(10)in the following manner under the assumption of a small transfer conductance [32]:

Thus,the problem of searching for a CUEP in the original system can be transformed into finding a CUEP in the reduced system.The specific implementation steps for the BCU can be found in [27].

Based on the characterization of nonlinear dynamical systems, the BCU method can determine the location of the CUEP and accurately estimate the system stability.In addition,this method exhibits high speed and low computational complexity owing to the use of a reduced system, which indicates its potential for online applications.

2 Algorithm for identification of coherent generator groups under extreme conditions

2.1 Classification of disturbance magnitudes

The analysis presented in the slow coherency method is based on linearized system (4).Typically, the widely used slow coherency method locates the linearization point at the SEP based on the small-perturbation assumption.This approach is generally effective under standard operational conditions, where disturbances are relatively minor.However, the magnitude of the disturbance varies significantly for different fault cases in power systems.Under extreme conditions, such as those resulting from military strikes or severe natural disasters, the disturbances inflicted on power systems can be substantial.In these scenarios, the small perturbation assumption is no longer valid.According to nonlinear dynamical system theory, the state variables deviate from the SEP, potentially moving towards the UEP under large disturbances.In this case,the system should be linearized at the UEP.

Therefore, the magnitude of the disturbance results in different requirements for the system’s operational state and control strategies.In this section, an algorithm for identifying the magnitude of the disturbance is proposed,which is able to distinguish the system’s states ranging from small disturbances to unstable scenarios, providing decision support for coherent grouping.

In a disturbed system,the trajectory can be divided into three phases: pre-fault, fault-on, and post-fault trajectory.For a system operating in a stable state, the pre-fault trajectory is exactly that of the SEP.When a fault occurs,the trajectory moves away from the SEP of the pre-fault system.Once the fault is cleared, the evolution of the trajectory depends on whether the state is within the stability region of the post-fault system.Specifically, if the state at the time of fault clearance is within the stability region of the post-fault system,the trajectory approaches the SEP of the post-fault system, and the system remains stable.Conversely, if the state is outside the stability region, the trajectory does not converge to the SEP, resulting in an unstable situation.

The magnitude of the disturbance determines the distance between the initial state of the disturbed trajectory and the SEP.From another perspective, as the magnitude of the disturbance increases, the initial state of the disturbed trajectory moves closer to the UEP (which is exactly the CUEP).Therefore,the magnitude of the disturbance determines the dynamical characteristics of the trajectory.In other words, the magnitude of the disturbance can be inferred by analyzing the properties of the trajectories.

In a system described by differential equations,the EP is defined as the point at ‖ ˙x‖=0.At other locations in the state space, the Euclidean norm of the state-variable derivatives is greater than zero.Because the state variables change continuously, it can be concluded that in the area near the EPs, the value of ‖ ˙x‖ decreases on trajectories approaching the EPs and increases on trajectories moving away from the EPs[33].For example,for a trajectory that linearly connects two EPs, xe1 and xe2, the ‖ ˙x‖ value changes as shown in Fig.4.

Thus,for a certain state x in the state space,by analyzing the change trend of ‖ ˙x‖along its trajectory, the position of x can be determined.In addition,the magnitude of the disturbance can be inferred.

1) Small disturbance scenario

Under normal conditions, the analysis is conducted based on small disturbance scenarios.That is, from the occurrence to clearance of a fault, it is assumed that the state remains near the SEP.

As shown in Fig.5, is the SEP of the pre-fault system, xs and xc is the SEP and the CUEP of the post-fault system,W s（xc）and W u（xc）are its stable and unstable manifold, respectively.The solid line represents the fault-on trajectory and the dashed line represents the post-fault trajectory.In this case, the disturbed trajectory does not move far from SEP and eventually converges to xs; therefore, the value of decreases monotonically.Alternatively, if exhibits such a trend of change, we can infer that the current state may be near SEP, and the disturbance magnitude may be small.

2) Large disturbance scenario

Fig.4 Change of along a trajectory linearly connecting two EPs.

Fig.5 Trajectory in the small disturbance scenario.

Fig.6 Trajectory in the large disturbance scenario.

If the disturbance is large and the disturbed trajectory approaches the stability boundary, two possible cases arise, represented by trajectories c and d in Fig.6.First,the two post-fault trajectories are attracted by the CUEP and move along its stable manifold W s（xc）.When reaching the neighboring area of xc,the trajectories move along the unstable manifold W u（xc）.Trajectory d proceeds along the unstable manifold away from xs,whereas c moves towards xs.

In this scenario, because the Euclidean norm of the state variable derivatives is nearly zero in the neighboring area of xc, the value of first decreases and then increases.Alternatively, if exhibits such a trend of change, we can infer that the current state is near the CUEP, and the disturbance magnitude is large.

3) Intermediate scenario

As shown in Fig.7,in the intermediate scenario,trajectory b is already at a certain distance from the SEP of the pre-fault system but is not attracted by the CUEP.It finally converges to the SEP of the post-fault system.Therefore, the value of first increases and then decreases.If exhibits such a pattern of change, we can infer that the current state may be in an intermediate position.

4) Unstable scenario

As shown in Fig.8,if the trajectory significantly crosses the stability boundary at the time of fault clearance, the system state moves towards another EP along the unstable manifold.Thus,the value ofexhibits a monotonically decreasing trend.Similar to the previous analysis,the variation in along the intermediate trajectory e first increases and then decreases.Therefore, if exhibits such a trend of change, we can infer that the current state may be outside the stability region,indicating that the system could be unstable.

These four scenarios describe possible trajectories in the system after experiencing various degrees of disturbance.By analyzing the change trends in , it is possible to distinguish the position of the current state and the magnitude of the disturbance.

However, the same trend in may represent different trajectories.That is, the change trends in of trajectories a and f, c and d, and b and e are identical.The difference between these pairs of trajectories is whether the trajectory is within the stability region.Therefore, a method to determine the positional relationship between a state and the stability region should be proposed.

Fig.7 Trajectory in the intermediate scenario.

Fig.8 Trajectory in the unstable scenario.

2.2 Distinction of specific trajectories

The analysis in the previous section is also applicable to the reduced system (13).To simplify the judgment using the reduced system, for a certain state in the original system, one needs to project it onto the reduced system and integrate it to obtain a trajectory.The position of the state in the reduced system can be inferred by assessing the changing trend of the Euclidean norm of the state variable derivatives along the trajectory.Accordingly, the position of the state in the original system and the magnitude of the disturbance can be determined.Therefore, the following analysis focuses on the reduced system.

According to the previous analysis,the criterion for distinguishing trajectories with the same trend in the Euclidean norm of the state variables lies in whether they are within the stability region.Chaing et al.[32]demonstrated that the stability region of a reduced system is exactly the-PEBS of the original system,which is perpendicular to the equipotential surface {θ|V p（θ）=c}.This theorem indicates that, along the normal direction of the PEBS, there is a local maximum of the potential energy Vp on the PEBS.That is,the state with the local maximum potential energy in the state space around the SEP constitutes the PEBS, which is equivalent to the stability boundary of the reduced system.

Therefore,for a given state on a trajectory,whether it is within the stability region can be determined by judging whether the potential energy of the states on the trajectory has a local maximum.

The criteria for distinguishing between the six types of trajectories are summarized in Table 1.

2.3 Identification of coherent group

In this subsection, a grouping method based on state position is proposed.For different scenarios, differentcoherency detection methods are used to adapt the characteristics of disturbances with varying magnitudes.

Table 1 Trajectory type classification and coherency detection method.

Trajectory type Variation of Euclidean norm of state variable derivatives Value of potential energy before the state Magnitude of the disturbance Coherent group identification method a monotonically decreaseno local maximumsmall disturbanceslow coherency method b first increase and then decrease no local maximumintermediate disturbance slow coherency methodV CUEP-V x||＞V SEP-V x||coherency detection algorithm under large disturbance|V CUEP-V x |＜V SEP-V x||c first decrease and then increase no local maximumlarge disturbance coherency detection algorithm under large disturbance d first decrease and then increase exist a local maximumlarge disturbance coherency detection algorithm under large disturbance e first increase and then decrease exist a local maximumunstable conditioncluster algorithm f monotonically decreaseexist a local maximumunstable conditioncluster algorithm

First, in the small disturbance case, the trajectory is near the SEP.Therefore, the system should be linearized at the SEP.The traditional slow coherency method can be applied to identify the coherent groups.

Second,if the disturbance is large,the system should be linearized at the CUEP to analyze its coherent properties.However, the slow coherency method is not practical because the CUEP is a type-I UEP, and the eigenvalues of A are not all negative.Based on the traditional slow coherency method, a coherent grouping algorithm for large disturbances can be developed as follows:

1) Linearize the system at the CUEP and calculate the state matrix A.

2) Calculate the eigenvalues and eigenvectors of A.Specifically, there is one positive eigenvalue and the rest are non-positive.Let denote by λ1 the positive eigenvalue and u1 its corresponding normalized eigenvector.

3) Use a cluster algorithm (such as k-means) to cluster u1 into two categories, and the corresponding generator is the coherent grouping result.

As indicated by the previous analysis, the positive eigenvalue λ1 represents a divergent mode.This implies that the modes corresponding to the other eigenvalues can be neglected.As the elements in u1 represent the coefficients of each generator for mode λ1, u1 can serve as the basis of the cluster.Consequently,the cluster is performed only on u1.By executing this algorithm, coherent groups can be identified under large disturbances.

Third, in the case of an intermediate disturbance, that is, b-type trajectory, the state is neither near the SEP nor near the CUEP.Linearizing the system at the SEP or CUEP depends on the distance between the assessed state and equilibrium points.A naive concept is to evaluate the distance by calculating Euclidean distance.That is, compare the Euclidean distance between the state and SEP or CUEP, and choose the one with a shorter distance as the linearization position.This approach is simple to compute and is effective in certain cases.However,it simplifies the trajectory between the two states as a straight line and does not consider the dynamical characteristics of the system.This study proposes a more practical method for evaluating the distance between two states from an energyfunction perspective:

1) Use the BCU method to find the CUEP.

2) Calculate the energy function of the CUEP, current state, and SEP.These energy function values are denoted as VCUEP, Vx, and VSEP.

3) If the energy function of the current state is closer to the energy function of the CUEP, that is,|V CUEP-V x |＜|V SEP-V x|, linearize the system at the CUEP and apply the coherency detection algorithm under large disturbance proposed in the previousdiscussion.Conversely,if|V CUEP-V x |＞|V SEP-V x|, linearize the system at the SEP and use the slow coherency method to recognize the coherent groups.

Finally, in the unstable scenario, because the state far exceeds the stability boundary, the difference in power angle between the coherent groups is significant.Grouping can be achieved by simply applying cluster algorithms(such as k-means) to the power angle data of the generators.

The coherent group identification methods for all the scenarios are listed in Table 1.

In practical power systems, PMUs can collect dynamic real-time data [34], thereby providing accurate state variable information.At each sampling step of the PMU,the required information includes the voltage,phase angle,rotor angle, angular velocity, active power, and reactive power of the buses in the system.Based on the above analysis and methods, the complete algorithm for identifying coherent groups under extreme conditions is as follows.

1) For a disturbed system,use the PMU to collect state information, and calculate the corresponding potential energy function of the current state.

2) Integrate the reduced system

with the projection of the current state on the reduced system as the initial condition.Judge if there is a local minimum of ‖ ˙θ‖ along the trajectory.

3) If there exists a local minimum denoted as θ*,which means that the trajectory is type-c or type-d,calculate the CUEP.Solve the equation using the Newton-Raphson method with θ* as the initial value:

where superscript PF represents the value of the corresponding variable in the post-fault system.The solution^θ is the CUEP of the reduced system, and （^θ,0） is the CUEP of the original system.Linearize the original system at（^θ,0）,use the coherency detection algorithm under large disturbance to obtain the grouping result, and the algorithm ends.If there is no local minimum, determine whether there is a local maximum of the potential energy before the current state.

4) If a local maximum of potential energy exists,which means that the trajectory is type-e or type-f,perform k-means on the power-angle data to distinguish the coherent groups,and the algorithm ends.If the local maximum does not exist, use the BCU algorithm to determine CUEP xCUEP.

5) If the value of ‖ ˙θ‖ along the trajectory in (1)decreases monotonically, which means that the trajectory is type-a, linearize the system at xSEP, use the slow coherency method to obtain the grouping result, and the algorithm ends.If the value of ‖ ˙θ‖first increases and then decreases, which means that the trajectory is type-b,calculate the energy function at the current state, SEP, and CUEP.

6) If |V CUEP-V x |＜|V SEP-V x|,linearize the system at xCUEP, use the coherency detection algorithm under large disturbance, and the algorithm ends.If|V CUEP -V x |＞|V SEP -V x |, linearize the system at xSEP, use the slow coherency method for grouping,and the algorithm ends.

A flowchart of the algorithm is shown in Fig.9.It should be emphasized that the algorithm should be executed at every sampling step of the PMU to provide real-time coherent group results.To balance efficiency and effectiveness, the PMU sampling interval can be set to approximately 100 ms.

3 Case studies

In this section,the proposed generator coherent grouping algorithm is validated using the IEEE 39-bus system model, and the node indices are shown in Fig.10.Power System Analysis Toolbox(PSAT)and MATLAB are used as analysis tools in this study.

3.1 System trajectories under different disturbance magnitudes

The fault is chosen to be a perfect three-phase short circuit on bus 22 applied at Tf=0 s.Let Tc denote the clearance time.The larger Tc is,the longer the disturbance lasts,and the greater the disturbance magnitude is.According to the previous analysis, as the disturbance magnitude increases, the trajectories after fault clearance will appear in the order of types a, b, c, d, e, f.

Fig.9 Flowchart of the coherent generator group identification algorithm under extreme conditions.

Fig.10 IEEE 39-bus system.

The simulation results with different Tc values are shown in Fig.11 and are consistent with the theoretical analysis: Type-a and type-f trajectories (shown in the first subplot of Fig.11) monotonically decrease.Type-c and type-d trajectories(second subplot of Fig.11)first decrease and then increase.Type-b and type-e trajectories (third subplot of Fig.11) first increase and then decrease.

The distinction between the two trajectories in the same subplot lies on whether the trajectory reaches the local maximum of potential energy.The value of the potential energy function for this case is shown in Fig.12.The results show that the local maximum appears at 0.514 s;that is,the trajectories can be considered within the stability region before 0.514 s and categorized as type a/b/c,whereas they can be regarded as outside the stability region after 0.514 s and categorized as type d/e/f.

3.2 Dynamic change of trajectory type

Once the fault is cleared,the state of the original system changes dynamically.Simultaneously, the trajectory type continues to vary.In contrast to the previous sections which consider the projection of the fault clearance state as the integral initial state in the reduced system,the analysis in this section focuses on the projection of state ts after fault clearance.That is, after the fault is cleared, the projection of the state that evolves in the state space for ts is used as the initial state for integration in the reduced system to determine the trajectory type.Applying a perfect three-phase short circuit on bus 22 at Tf = 0 s, and with the clearance time set to Tc = 0.38 s, the simulation result with different ts values is shown in Fig.13.

Fig.11.Changes in‖ ˙θ‖after fault clearance under different disturbance magnitudes.

Fig.12 Change of potential energy.

The results show that,as ts increases,the type of trajectory changes in the order of a,b,d,e,f.This indicates that when the fault is cleared, the projection of the state in the original system onto the reduced system is near the SEP,which causes the trajectory to exhibit the characteristics of type-a.As ts increases, the state of the original system continues to change because of inertia, resulting in the appearance of the b-type trajectory.The state of the original system continues to change, causing the projection state in the reduced system to cross the stability boundary.Therefore, the trajectory evolves into type-d.Similarly, as ts increases further, type-e and type-f trajectories appear.

The experimental results indicate that, as the initial state in the reduced system changes after the fault is cleared, the type of trajectory may also change; hence,the coherent grouping result may change as well.Therefore,in practical applications,the proposed coherent algorithm should be executed at every sampling step to address the changes in grouping results over time.This is also consistent with the actual phenomenon:the power angle under unstable conditions exhibits a divergent mode and the differences in the power angle among coherent groups increase over time, causing the coherency to vary.

Fig.13.Changes in of different initial states.

3.3 Comparison between the proposed method and slow coherency method

Fig.14 Power angle curves of the generators under the short circuit fault.

In small disturbance scenarios, the proposed algorithm is consistent with the traditional slow coherency method.Consider the large disturbance scenario: A perfect threephase short circuit on bus 22 occurs at Tf = 0 s, the fault is cleared at Tc = 0.38 s, and ts is set to be 0.6 s.This scenario simulates a system that experiences a short circuit caused by a graphite bomb strike or other similar events.The simulation of the generator power angle is presented in Fig.14, which indicates that two coherent groups exist in the system: {G35, G36} and {G30, G31, G32, G33,G34, G37, G38, G39}.

For the short-circuit fault described in the preceding paragraph, the traditional slow coherency method linearizes the system at the SEP.The eigenvalues of the linearized state matrix A are 0, -0.0938, -0.1018,-0.1358, -0.1590, -0.1623, -0.2210, -0.2463, -0.2525,and-0.3315.The grouping of eigenvectors and the results are listed in Table 2.

For the same fault, according to the algorithm proposed in this paper,the type of trajectory is c.The coordinate of CUEP is(0,0,0,0,0,0,0,0,0,0,0.8894,-0.3897,-0.4196, 0.1620, 0.2402, 1.8532, 2.322, -0.4901, -0.2085,-0.9808).When linearizing the system at CUEP, the eigenvalues of A are calculated to be 0.1268, 0, -0.0052,-0.0368, -0.0732, -0.1122, -0.1242, -0.1319, -0.1985,-0.2553.As mentioned previously, the grouping basis is the eigenvector corresponding to the positive eigenvalue.The grouping results are shown in Table 3.

Under extreme conditions, power system may also experience node failures cause by military strike or othersimilar events.To validate the effectiveness of the algorithm under such conditions, bus 9 and bus 12 are simulated to fail.The simulation result of the generator power angle is shown in Fig.15,from which it can be confirmed that two coherent groups exist after the failure occurs: {G31, G32} and {G30, G33, G34, G35, G36,G37, G38, G39}.

Table 2 Grouping eigenvectors and the grouping results of the slow coherency method.

Generatorsu1u2u3Grouping result G30-0.1150-0.0904-0.0839group 1 G31-0.0405-0.54750.3041group 2 G32-0.0194-0.46210.1790group 2 G330.27540.14780.0027group 2 G340.62550.41030.5103group 1 G350.16170.0439-0.5529group 1 G360.15740.0430-0.5211group 1 G37-0.1600-0.0390-0.0810group 1 G38-0.66210.50650.1553group 3 G39-0.0631-0.1631-0.0106group 2

Table 3 Grouping result of coherency detection algorithm under large disturbance in the short circuit conditions.

Generatorsu1Grouping result G300.1025group 1 G310.0599group 1 G320.0672group 1 G33-0.0001group 1 G34-0.0202group 1 G35-0.6412group 2 G36-0.7405group 2 G370.0800group 1 G380.0376group 1 G390.1171group 1

Fig.15 Power angle curves of the generators under the node failure fault.

Table 4 Grouping result of coherency detection algorithm under large disturbance in the node failure conditions.

Generatorsu1Grouping result G300.2072group 1 G310.6810group 2 G320.3991group 2 G330.2249group 1 G340.2131group 1 G350.2236group 1 G360.2254group 1 G370.2149group 1 G380.2106group 1 G390.2162group 1

Similar to the calculation process of the short circuit fault, the identification results of the proposed algorithm 0.6 s after the fault are shown in Table 4.In comparison,the slow coherency method still produces the same results as before in this case due to its inability to account for large disturbances.

Comparing traditional slow coherency method with the algorithm proposed in this paper, the grouping results are summarized in Table 5 and Table 6.

The simulation results demonstrate that traditional slow coherency method is unable to distinguish the coherent groups under large disturbance correctly, whereas the algorithm proposed in this paper can accurately identify the coherency.This proves the effectiveness and superiority of the proposed algorithm.In addition,the proposed algorithm has low computational complexity due to the use of the reduced system model, which improves its applicability.

4 Conclusion

This paper proposed an algorithm to identify the coherency of generators under extreme conditions.Specifically,an approach for determining the types of system trajectory and identification of varying disturbance magnitudes is proposed based on the properties of nonlinear dynamical systems.A coherent grouping approach for large disturbances and unstable conditions is also explored based on the traditional slow coherency algorithm.Combining these two approaches, a coherent generator group identification algorithm under extreme conditions is proposed, which is effective for different magnitudes of disturbance.The validation results on the IEEE 39-bus system indicate that compared with traditional slow coherency method, the method proposed in this paper is able to identify the coherent generator groups correctly under large disturbances.The research results can provide novel strategies andtechnical support for the safe operation of power systems under extreme conditions.

Table 5 Grouping result comparison of the slow coherency method and the proposed algorithm under the short circuit fault.

Grouping algorithmSlow coherency methodAlgorithm proposed in this paperCoherent groups result from simulation curves Group1G30, G34, G35, G36, G37G35, G36G35, G36 Group2G31, G32, G33, G39G30, G31, G32, G33, G34, G37, G38, G39G30, G31, G32, G33, G34, G37, G38, G39 Group3G38nonenone

Table 6 Grouping result comparison of the slow coherency method and the proposed algorithm under the node failure fault.

Grouping algorithmSlow coherency methodAlgorithm proposed in this paperCoherent groups result from simulation curves Group1G30, G34, G35, G36, G37G31, G32G31, G32 Group2G31, G32, G33, G39G30, G33, G34, G35, G36, G37, G38, G39G30, G33, G34, G35, G36, G37, G38, G39 Group3G38nonenone

CRediT authorship contribution statement

Yizhe Zhu: Writing - original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Yulin Chen:Writing - review & editing, Validation, Resources,Methodology, Investigation, Funding acquisition, Formal analysis.Li Li: Writing - review & editing, Supervision,Project administration, Investigation, Funding acquisition.Donglian QI:Writing-review&editing,Supervision,Project administration, Investigation, Funding acquisition.Jinhua Huang:Formal analysis,Supervision.Xudong Song: Funding acquisition, Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No: 52477133), Science and Technology Project of China Southern Power Grid(Grant No.GDKJXM20231178(036100KC23110012),GDKJXM20240389(030000KC24040053)) and Sanya Yazhou Bay Science and Technology City (Grant No:SKJC-JYRC-2024-66).