# A method to extract instantaneous features of low frequency oscillation based on trajectory section eigenvalues

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## Abstract

Affected by the nonlinear time-varying factors due to fault scenarios, protection relaying, and control measures, the dynamic behaviors of a power system may be significantly different from the results of previous methods. In order to analyze the oscillation characteristics of complex power systems more accurately and suppress the low frequency oscillation more effectively, this paper improves the trajectory section eigenvalue method. Firstly, the time response of a system is obtained by numerical simulation in a given fault scenario. Secondly, the algebraic variables are substituted to the differential equations along the trajectory. Thus, the original time-varying differential-algebraic equations are approximated by a set of linear ordinary differential equations, which can be updated along the trajectory. On this basis, this paper proposes a method to extract instantaneous features of the oscillation from the micro perspective. The non-equilibrium points with strong nonlinearity or critical eigenmodes are identified by the proposed method. The simulation test results of the IEEE 3-machine 9-bus system and the New England system illustrate the validity of the proposed method.

## Keywords

Low frequency oscillation Nonlinear factor Time-varying factor Trajectory section eigenvalue Instantaneous feature## 1 Introduction

In recent years, the low frequency oscillation (LFO) has become increasingly intricate, with the enlargement of power grids [1], the integration of renewable energy and the development of high voltage direct-current transmission [2, 3]. The universality and accuracy of previous analysis approaches for small disturbance stability can hardly satisfy the requirements of modern power systems. Influenced by the essential nonlinear factors and time-varying factors due to fault scenarios, protection relaying, and control measures, the results obtained by traditional methods may be significantly different with the dynamic behaviors of a power system [4]. Hence, it is urgent to put forward an LFO analysis method that can describe the dynamic behaviors of a power system more accurately with a complete consideration of the nonlinear time-varying factors.

Generally, LFOs can be divided into three forms based on the causes: ① steady increase in generator rotor angle due to lack of synchronizing torque; ② rotor oscillation with increasing amplitude due to lack of sufficient damping torque [5]; ③ sustained oscillation due to periodic disturbance resulting from failures or malfunctioning equipment [6]. The eigen-analysis method is most widely used in practice. It describes the dynamic system behaviors near the equilibrium point by a linear time-invariant system. However, the nonlinearity of a power system may lead to bifurcation and blow-up of solutions [7, 8]. Reference [9] addresses the nonlinear singularity phenomenon by Hopf bifurcation theory, which demonstrates that the dynamic behaviors of a simple power system could be different from the results of eigen-analysis. However, limited by the heavy computation in large-scale systems, this method is still under investigation as well in methodological aspects as in concrete applications. In order to consider the effects of nonlinear factors, the normal form theory [10] and the modal series method [11] are proposed. More detailed information of eigenmodes is extracted by solving the high-order approximate equations. One major drawback of these approaches is that the solutions are obtained at the equilibrium point, neglecting time-varying factors. For the sake of considering the effects of random factors, probability models are used to improve the universality of traditional methods [12]. All of these methods reviewed thus far, however, suffer from the limitation of disturbance scenarios. As a result, some critical factors of a power system such as the non-differentiable components and control measures are ignored [13].

To entirely analyze the influence of nonlinear time-varying factors on power system oscillation, it is quite necessary to extract information from trajectories. Some signal analysis techniques are used to identify the oscillation modals. Common approaches like time-frequency transformations take the disturbed trajectory as research object. The frequency features are identified by decoupling frequency components, and the damping features are extracted by differentiating mode amplitudes with respect to time. As a classical method, the Prony analysis replaces the trajectory by a set of exponential components [14]. This method can describe the dynamic behaviors in an interval approximately, though the influences of time-varying factors are averaged. Based on wide-area measurement signals and the proper orthogonal decomposition (POD) method, [15] uses dynamic mode decoupling (DMD) method to achieve oscillation modes decoupling and features extracting. Nevertheless, these methods can hardly describe the instantaneous characteristics, which are time-varying. The empirical mode decoupling (EMD) method extracts instantaneous features by decomposing the original signal into locally symmetric components on the basis of envelope characteristics and the Hilbert transform [16]. However, the precision of EMD method is strongly affected by the frequencies of adjacent components. To extract instantaneous features more accurately, window Prony method [17], and window fast Fourier transformation (ridges) method [18] use slide windows to divide the entire time to a set of overlapping periods along the trajectory. The feature sequence is composed of the modals of all windows. Despite of that, the accuracy of the methods is limited by the width of the window. Moreover, the truncated error caused by the edges on both sides of the signal is inevitable. To improve the adaptive ability of the time-frequency transformation approaches, wavelet (ridge) method is introduced to assist analyzing LFOs [19]. Researches show that the method can identify the modals of an actual trajectory with noises effectively [20], although the choice of wavelet basis relies on empirical parameters. Reference [21] approximates the observed process as a linear system for more information of models. The stability margins are obtained as posterior probabilities that the poles of the estimated system are unstable.

A considerable number of researches have been done to achieve the extraction of instantaneous features by signal analysis techniques during the last decades. To be objective, these approaches are instrumental in identifying modals online. However, all these mentioned methods suffer from a serious limitation. The essence of signal analysis techniques is identification of the average characteristics in a time window. The time scale of the oscillation analysis is limited by the resolution of the methods. On the other hand, the major goal of modal identification of LFO is to suppress the oscillation. To locate critical generators or parameters, the signal analysis techniques have to be unified with the models at the equilibrium point, which can hardly reflect the real dynamic behaviors of a nonlinear time-varying system.

Attempts to resolve this dilemma have resulted in the invention of trajectory section eigenvalue (TSE) theory, which extends the concept of eigenvalue from the equilibrium point to other non-equilibrium points of the trajectory. The pioneer work can be traced to [22]. Reference [23] attempts to use TSE method to analyze transient stability. The method is further developed to estimate the energy of a system before the visual far end point or the visual dynamic saddle point [24]. Reference [25] concludes that the global stability of linear time-varying systems cannot be determined by TSEs. The major reason is that the global dynamics of time-varying systems are affected by the kinetic energy at non-equilibrium points, which could not be considered by TSE method. However, [26] has proved that the effects of the kinetic energy can be ignored when analyzing the instantaneous characteristics in one integration step. Meanwhile, the mathematical and physical foundations of TSE method have been given in this literature.

The contributions of this paper are threefold. Firstly, it improves the mathematics of TSE method to give a better explanation of the physical meaning of the eigenvalues at non-equilibrium points. Based on the improved theory, this paper proposes a method to extract the instantaneous features of LFOs. Secondly, the paper proposes the criterion to identify the non-equilibrium points with strong nonlinearity, and illustrates the mechanism of nonlinear oscillations in power system. Thirdly, it proposes the criterion to identify the dangerous non-equilibrium points and the critical eigenmodes. Compared with previous methods, the critical generators and parameters identified at above non-equilibrium points can achieve better inhibition effects.

## 2 Mathematics of trajectory section eigenvalues

### 2.1 Piecewise-linearized system models and solution in time domain

*n*first-order nonlinear differential-algebraic equations (DAEs) as the following form (the derivatives of the state variables are not explicit functions of time) [5]:

*is referred as the state vector, and its entries \({x}_{i}\) as the state variables, such as rotor angles and angular velocities of the rotors. The column vector*

**X***is referred as the algebraic vector, and its entries \({y}_{{i}}\) as the algebraic variables, such as node voltages and node current injections. The solutions of*

**Y***and*

**X***in the time domain can be obtained by numerical simulation in given fault scenarios. By substituting*

**Y***into the differential equations \({\dot{\varvec{X}}} = \varvec{f} (\varvec{X}, \varvec{Y} )\) and linearizing the nonlinear components of the functions along the trajectory, the DAEs are replaced by a set of linear ordinary differential equations (ODEs). For an illustrative purpose, the linear ODE at \({t}_{{k}}\) is taken as an example to introduce TSE method.*

**Y***; \(\varvec{B_{k}} = \varvec{f}\left( {\varvec{X} ({t}_{{k}} ), \varvec{Y} ({t}_{{k}} )} \right)\) is the initial values of the right side of the differential equation at \({t}_{{k}}\). Supposing that*

**f**

**U**_{k}is the right modal matrix of

**A**_{k}. New state vector is defined by \(\varvec{Z}_{k} (\Delta {t} )= \varvec{U}_{k}^{ - 1}\Delta \varvec{X}_{k} (\Delta {t} )\). Both sides of (2) are left-multiplied by \(\varvec{U}_{k}^{ - 1}\):

**A**_{k}at

*t*

_{k}. Two procedure parameter matrices

**D**_{k}and

**C**_{k}are defined as follows:

*d*

_{k,i}is the

*i*

^{th}element of

**D**_{k}; \({v}_{{{k,ij}}}\) is the element in the

*i*

^{th}row and the

*j*

^{th}column of \(\varvec{U}_{{k} }^{ - 1}\);

*f*

_{i}is the

*i*

^{th}element of

**B**_{k}. Both sides of (4) are left-multiplied by

**U**_{k}:

Obviously, (7) describes the dynamic behaviors of the changes of state variables in \([t_{{k} } ,t_{{{k} + 1}} )\). The values of \(\varvec{U}_{{k} } \varvec{C}_{{k} }\) indicate the degrees of eigenmodes excitation [27].

### 2.2 Physical meaning of trajectory section eigenvalues

**Z**_{k}(Δ

*t*) are zero, we have \(\varvec{Z}_{{k} } (0 )=\varvec{ C}_{{k} } \varvec{E}_{{k} } (0 )-\varvec{\varLambda}_{{k} }^{ - 1} \varvec{U}_{{k} }^{ - 1} \varvec{B}_{{k} } \varvec = {\mathbf{0}}\). Equation (7) can be rearranged to:

**X**_{k,new}(

*t*) satisfies:

**C**_{k}and

**Λ**_{k}are both diagonal matrices, and the expression in (11) must satisfy \(\varvec{U}_{{k} } \varvec{C}_{{k} }\varvec{\varLambda}_{{k} } (\varvec{U}_{{k} } \varvec{C}_{{k} } )^{ - 1} =\varvec{ U}_{{k} }\varvec{\varLambda}_{{k} } \varvec{U}_{{k} }^{ - 1} = \varvec{A}_{{k} }\). Equation (11) can be rearranged as:

Thus, the original system without regard to unbalanced power is approximated by a linear system in \([t_{k} ,t_{k + 1} )\). Although **X**_{k,new}(*t*) may be different from **X**_{k}(*t*), the changes of their state vectors are the same in \([t_{k} ,t_{k + 1} )\). The section eigenvalues \(\left( {{\lambda }_{{{k} , 1}} {,\lambda }_{{{k,} 2}} , \cdots {,\lambda }_{{{k,n}}} } \right)\) describe the dynamic behavior of the imaginary system in \([t_{k} ,t_{k + 1} )\).

Besides, the parameter matrices **U**_{k} and **C**_{k} are updated along the trajectory according to the results of numerical simulation. The effect of the unbalanced power is reflected in following time sections. Furthermore, the accumulate error is avoided by updating the parameters of (7) and (12) along the trajectory. Consequently, the effects of nonlinear factors and time-varying factors are considered by the section state matrix changing with time.

## 3 Extraction method of instantaneous features of LFO

To extract the instantaneous features of LFO, the eigenvalues of the section state matrix are calculated. In general, an *n*-order matrix has *n* eigenvalues, which may consist of couples of complex eigenmodes and real eigenmodes. A couple of complex eigenmodes represent a sinusoidal component, while a real eigenmode represents an exponential component. In this paper \({f}_{{{k} ,i,ins}}\) is used to represent the imaginary part of the *i*^{th} eigenvalue at *t*_{k}, divided by a constant 2π. \({\sigma }_{{{k} ,i,ins}}\) is used to represent the real part of the *i*^{th} eigenvalue at *t*_{k}. As for a complex root, \({f}_{{{k} ,i,ins}}\) represents the instantaneous frequency characteristic of the *i*^{th} eigenmode at *t*_{k}. \({\sigma }_{{{k} ,i,ins}}\) describes the trend of the amplitude of the *i*^{th} eigenvalue at *t*_{k}. As for a real root, \({f}_{{{k} ,i,ins}}\) is zero. \({\sigma }_{{{k} ,i,ins}}\) represents the rate of a trend change within one integration step.

### 3.1 Identification of points with strong nonlinearity

In an oscillation, a couple of complex eigenmodes may reversibly transform into two real eigenmodes occasionally. A one-machine infinite bus (OMIB) system is used to illuminate the mechanism of the variation of TSEs.

*V*represents the voltage of the infinite bus.

*P*

_{e}is the input power. The summation of reactance is denoted as

*X*

_{Σ}. \({E'}\) and

*δ*are referred as the module and angle of the transient electromotance after the transient reactance of the generator, respectively. The classical model is used to represent the generator. The dynamic equations of the generator can be found in Appendix A (A1). When a disturbance occurs, the TSEs can be obtained along the trajectory:

*D*is the damping torque coefficient of the generator; \({K}_{{k}} =(E'V{ \cos \delta }_{k} )/{X}_{{\varSigma }}\) represents the slope of the electromagnetic power-rotor angle curve at

*t*

_{k}, geometrically; \({\delta }_{{k}}\) is the value of the rotor angle at \({t}_{{k}}\) with a value range of \(\left( {0,{\pi }} \right]\). As can be expected, if \({\delta }_{{k}} \in \left( {0,{ \arccos }\left( {{D}^{2} {X}_{{\varSigma }}/(4{ME'V} )} \right)} \right)\), \({\lambda }_{{{k,}1,2}}\) would be a couple of complex roots. With the increase of \({\delta }_{{k}}\), the imaginary parts of \({\lambda }_{{{k,}1,2}}\) decrease. If \({\delta }_{{k}} \in \left( {{ \arccos }\left( {{D}^{2} {X}_{{\varSigma }} {/} (4{ME'V} )} \right),\uppi} \right]\), \({\lambda }_{{{k,}1,2}}\) would be two different real roots. With the increase of \({\delta }_{{k}}\), one of them decreases, and the other increases. Therefore, the non-equilibrium points with strong nonlinearity can be identified briefly below: ① aggregating the complex eigenmodes at each time section; ② comparing the numbers of the complex eigenmodes at non-equilibrium points to the equilibrium point; ③ those non-equilibrium points are signed, the number of which is inconsistent with the equilibrium point.

### 3.2 Identification of critical points

The TSEs of this two-machine system after being disturbed can be of two forms: ① a couple of complex roots, a real root (can be zero), and a zero root; ② three real roots (can be zero) and a zero root. Equation (12) suggests that the positive signs of the real parts of TSEs represent the injection of dangerous energy. Thus, the critical non-equilibrium points are identified by the signs of the real parts of the eigenvalues along the trajectory.

In the neighborhood of the equilibrium point, the TSEs are consistent with the eigenvalues at the equilibrium point. However, power system is an essential nonlinear dynamic system. Once the operating point of the system runs away from the equilibrium point, the dynamic behaviors may be different with the eigenvalues at the equilibrium point.

In a linear system, the accelerated power gradually increases with the decreasing relative angular velocity from DCP to FEP. However, affected by the nonlinearity of the *P*_{e}–*δ* curve, the accelerated power of a power system is lower than the linear system before the FEP. As a result, the damping characteristics near FEPs would be worse than the neighborhood of the equilibrium point. The closer the FEP is to the DSP, the lower the stability margin of the system suffers, and the trajectories of the real parts of TSEs change correspondingly. Figure 4a, b depicts the evolution of \({\sigma }_{{{k} ,i,ins}}\) in convergent and divergent oscillation, respectively. The values of \({\sigma }_{{{k} ,i,ins}}\) have obviously positive correlation with the damping characteristics.

Therefore, the trajectories of the real parts of TSEs indicate the instantaneous damping characteristics. Importance should be attached to the critical non-equilibrium points with positive real parts of TSEs.

To suppress power system oscillation, it is necessary to optimize the parameters of critical generators or controllers, which can be located by the eigenmodes with positive real parts. The participation factor and the electromechanical mode correlation ratio can also be extended from the equilibrium point to non-equilibrium points. Section 4 demonstrates the effectiveness of the instantaneous features in locating critical generator and guiding the optimization.

In addition, the relative positions of eigenvalues between adjacent time sections are random, which limits the application of TSE theory in identifying critical eigenmodes. For an illustrative purpose, assuming that the eigenvalues at \({t}_{{a}}\) and \({t}_{{b}}\) are \({\lambda }_{{{a},1}} {,\lambda }_{{{a,}2}} , \cdots {,\lambda }_{{{a,n}}}\) and \({\lambda }_{{{b,}1}} {,\lambda }_{{{b,}2}} , \cdots {,\lambda }_{{{b,n}}}\), respectively. \({\lambda }_{{{a,}1}}\) may relate to \({\lambda }_{{{b,}2}}\) or even \({\lambda }_{{{b,n}}}\). The eigen-polynomial is a continuous function of the parameters of state matrix, and the zero points of the eigen-polynomial are continuous functions of the parameters of eigen-polynomial. Consequently, the eigenvalues of a matrix are continuous functions of the parameters of the matrix. The TSEs between adjacent time sections can match with the similarity of their values.

### 3.3 Process of instantaneous features extraction

*Step* 1: the time-domain responses of state variables are obtained by numerical simulation or phase measurement units.

*Step* 2: the algebraic variables are substituted to the differential equations along the trajectory and the section state matrices are obtained by linearizing the nonlinear components of the differential equations. In this paper, the size of analysis step is equal to the integration step.

*Step* 3: the TSEs are calculated from the state matrix and the instantaneous features are extracted along the trajectory.

*Step* 4: the eigenvalues between adjacent time sections are matched.

*Step* 5: the points with strong nonlinearity are identified by corresponding criterion. *R*^{n} and *I*^{n} represent the fields of real number and imaginary number, respectively.

*Step* 6: critical points are identified by corresponding criterion.

## 4 Case study

### 4.1 IEEE 3-machine 9-bus system

In this section, a 3-machine 9-bus system is used to illustrate the limitation of the traditional eigen-analysis method.

Eigenvalues of IEEE 3-machine 9-bus system at equilibrium point

Number | Real part | Imaginary part | Cluster of generators |
---|---|---|---|

1 | − 0.1042 | 14.4300 | {G1, G2} vs. {G3} |

2 | − 0.1042 | 14.4300 | {G1, G2} vs. {G3} |

3 | − 0.0256 | 8.7149 | {G2, G3} vs. {G1} |

4 | − 0.0256 | 8.7149 | {G2, G3} vs. {G1} |

5 | − 0.2482 | 0.6203 | {G2} vs. {G1, G3} |

6 | − 0.2482 | 0.6203 | {G2} vs. {G1, G3} |

7 | − 1.1960 | 0.0763 | {G1, G2} vs. {G3} |

8 | − 1.1960 | 0.0763 | {G1, G2} vs. {G3} |

9 | − 4.0447 | 0.0000 | – |

10 | − 19.1085 | 0.0000 | – |

11 | − 19.3779 | 0.0000 | – |

12 | − 19.3979 | 0.0000 | – |

13 | − 5.7904 | 0.0000 | – |

14 | − 0.7561 | 0.0000 | – |

15 | − 0.2343 | 0.0000 | – |

16 | 0.0000 | 0.0000 | – |

17 | − 0.0203 | 0.0000 | – |

18 | − 0.0149 | 0.0000 | – |

On the other side, the most dangerous eigenmode identified by the traditional eigen-analysis method should be the 3^{rd} and 4^{th} eigenvalues with the minimum real parts. Moreover, the corresponding cluster of generators is {G2, G3} vs. {G1}.

The oscillation characteristics of this system are analyzed in different fault scenarios. The numerical integration is used to assess the validity of the proposed method.

#### 4.1.1 Scenario 1: weak nonlinear time-varying scenario

To demonstrate that the instantaneous features of the oscillation are consistent with the traditional eigen-analysis method in weak nonlinear time-varying scenarios, the instantaneous features are obtained as follows.

#### 4.1.2 Scenario 2: strong nonlinear time-varying scenario

The inconsistency between the trajectories of rotor angles and the eigenvalues at the equilibrium point reveals the limitation of traditional eigen-analysis method in strong nonlinear time-varying scenarios. To illustrate the mechanism of the oscillation from the micro point of view, the instantaneous features are obtained as follows.

Therefore, the characteristics of instantaneous features can be summarized as follows: ① in weak nonlinear time-varying scenarios, the operating point of the system runs near the equilibrium point, and the instantaneous features are similar to the traditional eigen-analysis method; ② in strong nonlinear time-varying scenario, the operating point of the system runs far away from the equilibrium point periodically, and the instantaneous features may be totally different from the traditional eigen-analysis method.

### 4.2 IEEE 10-machine 39-bus New England system

In the view of the traditional eigen-analysis method, the eigenmodes with real parts greater than zero are dangerous. The eigenmodes with real parts approximate to zero are defined as weak damping eigenmodes. For the sake of suppressing LFO, these eigenmodes should be controlled. However, Section 4.1 has revealed that the traditional eigen-analysis method cannot describe the dynamic behaviors accurately. Moreover, the actually dangerous or weak damping eigenmodes may be ignored.

The New England system is used to illuminate the superiority of the proposed method in locating critical eigenmodes.

Before the 3^{rd} second, the amplitudes of the rotor angles are decreasing and the dominant oscillation mode is {G39} to the set of the other generators. After the 3^{rd} second, the amplitudes of the rotor angles are increasing and finally the system loses stability. The critical mode is {G34} to the set of the other generators.

#### 4.2.1 Real parts of trajectory section eigenvalues

#### 4.2.2 Location of critical generator

Analyses of eigenmodes at typical time sections from intervals 1 to 6

Interval number | Typical time section (s) | Highest participating generator | Participation factor | Electromechanical mode correlation ratio |
---|---|---|---|---|

1 | 1.00 | G39 | 0.1397 | 89.5934 |

2 | 2.20 | G39 | 0.8408 | 9.3670 |

3 | 3.50 | G34 | 0.5048 | 1102.7740 |

4 | 5.20 | G34 | 0.4824 | 709.6118 |

5 | 5.80 | G34 | 0.4781 | 717.3367 |

6 | 6.20 | G34 | 0.4774 | 623.5736 |

In intervals 1 and 2, *λ*_{29} and *λ*_{30} are the most dominant eigenmodes with the highest participation factors and the critical generator is G39. In intervals 3 to 6, the new eigenmodes are the most dominant eigenmodes with the highest participation factor, and the critical generator is G34. The descriptions of the instantaneous features are the same as the actual trajectories of the rotor angles, as shown in Fig. 11. Considering that the electromechanical mode correlation ratios of these eigenmodes are much larger than 1, the state variables participating in the eigenmodes are mainly rotor angles and angular velocities.

## 5 Conclusion

Based on TSEs, this paper proposes a method to extract instantaneous oscillation features from the micro perspective. To better illustrate the mechanism of power system oscillation and effectively suppress the LFO, the criteria for non-equilibrium points with strong nonlinearity and critical eigenmodes are proposed. At non-equilibrium points with strong nonlinearity, the dynamic behaviors may be totally different from the eigenvalues at the equilibrium point. The critical points rarely appear in weak nonlinear time-varying scenarios. Even though the oscillation is convergent, the real parts of TSEs may be positive at some non-equilibrium points with strong nonlinearity. The instability of oscillations may be caused by the accumulation effects of the critical non-equilibrium points, which massively appear in strong nonlinear time-varying scenarios. Moreover, simulation results clearly indicate the enhanced accuracy of critical generators location.

## 6 Discussion

Over the last 3 decades, the rotor angle stability has been characterized individually into two subcategories: small-disturbance stability and transient stability. As a result, it is impossible to analyze the relationship between oscillation stability and synchronous stability within the current framework.

- 1)
It is difficult to distinguish small disturbance and large disturbance. In a negative damping oscillation defined by the traditional method, a complex dynamic process is inevitable before synchronous instability. Thus, this study begins with the aim of studying the mechanism and characteristics of the evolution of LFO, no matter the disturbance is large or small. Although TSE method cannot be used to analyze the global stability of a power system, simulation test results indicate a strong correlation between the oscillation characteristics and the proposed features.

- 2)
In order to study the stability of a power system, a unified framework has been proposed by [29] to unify the analysis approaches of the oscillation stability and the transient stability in terms of energy. The improved TSE method and the achievements of this paper contribute to the unified framework. This paper is a tentative study on unifying small disturbance analysis and large disturbance analysis through TSE method.

- 3)
The eigenvalues can only describe the dynamic behaviors in the neighboring region of one point, whether equilibrium point or non-equilibrium point it is. To analyze the oscillation characteristics in strong nonlinear time-varying scenarios, the state matrix and the variables are updated along the trajectory, according to the numerical results.

Finally, the proposed method would be used in engineering only when the traditional eigen-analysis cannot describe the dynamic behaviors accurately, or detailed analysis is required in some complex scenarios. Thus, compared with the traditional eigen-analysis, the increasing computation is acceptable. The quantity of information and computation can be balanced by determining the step size more intelligently. Moreover, mature methods have been used to reduce the dimension curse when calculating eigenvalues at the equilibrium point. The theorem of the inverse operator can be used to determine the circle where the eigenvalues lie. The radius of the circle can be estimated by Gerschgorin theory. Many results have been made in this field [30]. These achievements can also be used to estimate the distribution of the eigenvalues at the non-equilibrium points.

## Notes

### Acknowledgements

This work was supported by Science and Technology Program of State Grid Corporation of China (Theoretical Basis, Algorithm and Application of Trajectory Eigenvalue Method).

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