Abstract
As the proportion of renewable energy sources continues to increase, the local damping contributions of sources in power system decrease, posing a challenge to the power system stability. Therefore, online tracking of the damping contributions of each source is crucial for the prevention of low-frequency oscillations. This paper proposes an online tracking method of local damping under ambient data. The proposed method is based on dissipation energy spectrum analysis (DESA) and the energy dissipation factor (EDF). First, the feasibility of using frequency-domain analysis for the dissipation energy of generator is analyzed. The frequency spectral function of dissipation energy of generator is then derived by integrating with Parseval’s theorem, and the EDF is defined. Second, the generator energy dissipation factor (GEDF) for the dominant oscillation mode frequency is established. The modal information of the dominant oscillation in the power system is obtained through DESA. The relationship between the frequency spectral function and eigenvalues is also established. Finally, an online tracking method of local damping is proposed based on DESA and GEDF. The effectiveness of the proposed method is validated through simulations on a four-machine 11-bus power system and an actual power system in Northwest China.
THE utilization of wind, solar, and other renewable energy sources in power systems is increasing [
Damping is a crucial aspect in the study of oscillations. Sources with positive damping dissipate energy during oscillations, thereby aiding the attenuation of oscillations. By contrast, sources with negative damping emit energy during oscillations, thereby exacerbating the oscillations. Alterations in the structural configuration of the power system can cause some sources to exhibit negative damping, which triggers oscillations [
The eigenvalue analysis is a classic method in small-disturbance stability analysis [
With the implementation of wide-area measurement systems (WAMSs) in power systems, the obtained phasor measurement data play an essential role in real-time state detection [
As the energy flow analysis method aligns with system damping, an online assessment of damping can be achieved using this method [
To overcome the limitations of previous research works, this paper integrates Parseval’s theorem with dissipation energy, derives the expression of dissipation energy in frequency domain, and defines a spectral function. The relationship between the spectral function and the eigenvalue is elucidated. Then, through spectral and phase analysis of the derived spectral function, an online tracking method of local damping under ambient data is proposed.
The contributions of this paper are as follows.
1) Based on Parseval’s theorem, the frequency spectral function and expression of dissipation energy in frequency domain are derived. An EDF is established based on the modal coupling.
2) The generator energy dissipation factor (GEDF) is defined by analyzing and calculating the amplitude and phase of the spectral function of dissipation energy. The relationship between the spectral function and eigenvalues is analyzed.
3) An online tracking method of local damping in frequency domain under ambient data is proposed, where simulation results verify the effectiveness of the proposed method.
The remainder of this paper is organized as follows. Section II describes the expression of dissipation energy in the frequency domain. Section III introduces the assessment of local damping in frequency domain using EDF. Section IV introduces the assessment of local damping in frequency domain under ambient data. Section V validates the proposed method through case studies. Section VI concludes this paper.
The feasibility of assessing system damping in the frequency domain can be illustrated through the ambient response of power systems, dissipation energy, and time-frequency transformations of signals.
The differential algebraic equations of a power system can be expressed as:
(1) |
where x is the state variable of the system; y is the algebraic variable; l is the load fluctuation variable; and and are the continuous functions.
Based on the assumption that the load fluctuations follow an Ornstein-Uhlenbeck distribution, the dynamic model of the load can be expressed as:
(2) |
where leq is the load value at the equilibrium point; K is the load response rate; is the noise intensity; and is an ambient fluctuation following a Gaussian distribution.
Linearizing (1) and (2) results in:
(3) |
where , , and are the Jacobian matrices corresponding to variables x, y, and l, respectively; and are the Jacobian matrices corresponding to variables x and y, respectively; is the change in state variable x; is the change in load relative to the equilibrium point; and Inl is the identity matrix.
Let , , and .
(4) |
Given that matrix A contains the characteristic information of electromechanical oscillations under load fluctuations, the eigenvalues corresponding to n electromechanical oscillation modes can be calculated. Therefore, the time-domain analytical solution of the system state variables can be expressed as:
(5) |
where uk and vk are the left and right eigenvectors corresponding to eigenvalue , respectively; z(0) is the initial value of the state variables; and is the initial phase of the sinusoidal oscillation for mode k.
The mathematical representation of the ambient response of the power system in (5) consists of two parts: an oscillation component, and a stochastic component that contains the measurement noise. During normal operation, the power system is subjected to small disturbances uniformly distributed in the spectrum carrying a specific amount of energy. Therefore, when the frequency of environmental excitation is close to that of a specific mode within the power system, oscillation modes are triggered. The following
(6) |
Thus, during the operation of the power system under ambient data, the variations in the active power, reactive power, voltage magnitude, and voltage phase angle at generator port i can be expressed as:
(7) |
where Xi can be , , , or , and , , , and are the active power, reactive power, voltage amplitude, and voltage phase angle of generator port i, respectively; and is the random component of the measured noise composition in Xi.
The energy flowing from node i to branch Lij in a power system can be expressed as:
(8) |
where is the current flowing on branch Lij; Pij and Qij are the active power and reactive power flowing from node i into branch Lij, respectively; and represents the imaginary part. Here, the term “energy” refers to an energy function derived from Lyapunov functions, which are commonly used to describe the energetic states of systems or objects.
The electrical quantities previously referenced can be presented using steady-state and incremental values as:
(9) |
(10) |
where is the variation in each electrical quantity; and are the steady-state values of active power and reactive power, respectively; is the oscillation component representing the transient energy of the power system; is the continuously changing component over time, known as the dissipation energy; and and are the integral terms containing the variation in active power and the variation in reactive power, respectively. A positive indicates energy consumption that contributes to positive damping, whereas a negative value indicates negative damping. Given the consistency between the dissipation energy and damping torque, the power system damping can be assessed by analyzing the dissipation energy.
The dissipation energy calculated based on the response data from different generator ports includes contributions from various modal components because the system response to ambient data encompasses multiple modal components. Therefore, the measurement data from generator port i can be used to calculate the dissipation energy , and these measurement data enable an analysis of how each mode manifests in dissipating energy. The measurement data from generator port i are substituted into the formula, and the results for and are expressed in (11) and (12), respectively.
(11) |
(12) |
(13) |
where a and b are the modes; is the damping term for mode a calculated using the incremental variables and ; is the harmonic and coupling term for modes a and b computed using the incremental variables and ; is the ambient component in and composed of measurement noise; is the damping term for mode a calculated using the incremental variables and ; is the harmonic and coupling term for modes a and b computed using the incremental variables and ; is the ambient component in and composed of measurement noise, and the derivation is presented in Supplementary Material A; ; and .
According to (13), between time t1 and t2, the dissipation energy at generator port i is primarily divided into the following three components.
1) The first component represents the monotonic damping term that changes over time. This component accumulates contributions from the damping terms of various modes.
2) The second component represents the time-varying periodic terms including harmonic terms for different modes and coupling terms between various modes.
3) The third component is the ambient term formed by measurement noise.
Time-domain methods ignore the possibility that the frequencies of the harmonic and coupling terms in the second component may match the frequency of the dominant oscillation mode. This oversight can reduce the precision of assessment. Local damping assessment in the frequency domain can solve the aforementioned problem and obtain accurate results. In addition, the sum of the damping terms of the various modes in (13) collectively constitutes the damping of the generator port, indicating the feasibility of assessing local damping in frequency domain.
Section II presents qualitative analysis of local damping of the power system, which demonstrates the feasibility of assessing the local damping in frequency domain. The specific generator modes of interest are quantitatively analyzed based on Parseval’s theorem. This section presents the expression of dissipation energy under ambient data and introduces the EDF.
Parseval’s theorem states that the energy contained in a signal equals the sum of the energies of its components within a complete orthogonal function set. The total energy of a signal in time domain is equal to the total energy of the signal in frequency domain. Here, the term “energy” refers to signal energy, specifically the total energy of signals over a defined period. Use signal as an example. The following equation is given:
(14) |
where is obtained from through Fourier transform.
For two signals and , in line with (14), the derivation process is given as:
(15) |
where and are the Fourier transforms of signals and , respectively. The integrated product of the two time-domain signals can be transformed into the frequency domain for analysis by applying (15).
Analysis of the generator ports via (10) allows to express the dissipation energy at generator port i as:
(16) |
Let , , , and . In this case, the expression for is:
(17) |
Fourier transform is then performed on the aforementioned variables. Using A(t) as an example, the transformation process can be expressed as:
(18) |
where and are the real and imaginary parts of after Fourier transformation, respectively, and the same applies below.
The results of the Fourier transform reveal that the real and imaginary parts are even and odd functions, respectively, with respect to .
Similarly, the remaining variables can be expressed as:
(19) |
Equations (
(20) |
Given that the real parts of the complex numbers obtained from the Fourier transform are all even functions, whereas the imaginary parts are all odd functions, which satisfy the following equations:
(21) |
The expression of dissipation energy in frequency domain at the generator port can be obtained by substituting (21) into (20) and simplifying the result to obtain:
(22) |
where is the energy spectral function of the dissipation energy. The value of the dissipation energy is equal to the integral of its energy spectral function over the entire frequency domain.
Given that the dissipation energy is composed of the superposition of various modes in the power system, in the analysis of low-frequency oscillation problems, the focus is on the low-frequency modes, particularly the interval oscillation modes that significantly affect the power system stability. Therefore, the dominant oscillation modes must be filtered from the dissipation energy of various modes, and the damping contribution of the source must be assessed. If the frequency of the dissipation energy is divided into multiple frequency intervals with the modal frequency as the center, the dissipation energy at generator port i can be expressed as:
(23) |
(24) |
where is the angular frequency of the system mode k; and are the upper and lower limits of the integration interval, respectively, which are used to calculate the dissipation energy corresponding to mode k; fk is the oscillation frequency corresponding to mode k (); is the entire integration range of the energy spectral function; and is the dissipation energy corresponding to mode k. The integration intervals for the dissipation energy of each mode are connected successively without any overlap. Given that the dissipation energy in the generator is consistent with the damping torque, is defined as the EDF for mode k, which corresponds to generator i.

Fig. 1 Division of theoretical integration intervals.
Based on (23), the GEDF is defined as:
(25) |
where is the GEDF; and is the integration interval of the dissipation energy in frequency domain corresponding to the dominant oscillation mode d, and is the optimal integration interval used to calculate the GEDF (generally set to be 0.1 Hz). Although some power systems may have two dominant oscillation modes with similar (distant) frequencies that lead to overlapping (or vacant) parts in the selected integration interval, this phenomenon does not affect the accuracy or effectiveness of the GEDF.

Fig. 2 Selection process of integration intervals.
According to (25), is the dissipation energy corresponding to mode d. Under dominant oscillation mode d, a large value of for a source indicates high energy dissipation, which suggests that the source contributes abundant damping to the power system. Positive and negative values of indicate a positive or negative contribution to the damping, respectively.
The time-domain response of a power system under ambient conditions can be represented by the analysis solution given in (5). The relationship between the eigenvalues and energy spectral function is exemplified by using . The terms and can be expressed as:
(26) |
(27) |
where and are the values of the left and right eigenvectors corresponding to the active power variation for mode k, respectively; and and are the values of the left and right eigenvectors corresponding to the phase angle derivative variation for mode k, respectively.
Performing Fourier transform on yields:
(28) |
where is the Fourier transform of the measurement noise in variable .
Given that represents the real part of the eigenvalue and that only modes with positive damping are considered, the real part of the eigenvalues is negative for in . Therefore, after the linearity and frequency shift properties of the Fourier transform are leveraged, as shown in (29) and (30), (28) can be transformed into (30).
(29) |
(30) |
(31) |
Similarly, applying Fourier transform to variable results in:
(32) |
where is the Fourier transform of the measurement noise in .
Therefore, when (31) is multiplied by (32), in (10) can be expressed as:
(33) |
(34) |
where the term obtained by multiplying the measured noise or is expressed as .
Similarly, in (10) can be expressed by function L involving , , , and as:
(35) |
where is the term multiplied by the noise component.
Accordingly, the dissipation energy in the frequency domain can be represented by (36), which includes the eigenvalue:
(36) |
where the original functions and are the noise terms, and their Fourier transforms are obtained by multiplying each element in the vector by the impulse function. In the spectral analysis without considering the fundamental frequency component, and can be disregarded.
(37) |
As the GEDF described in Section III tracks the damping contribution of each source under ambient data, the dominant oscillation modes of the power system must be identified. In this paper, the modal identification is achieved primarily by applying the DESA method, and the frequency of the dominant oscillation mode and the oscillation area of the power system under that mode can be obtained.
The spectral function of dissipation energy for generator port i can be expressed as:
(38) |
where and are the real and imaginary parts of the spectral function, respectively. The spectral function can also be presented in terms of the magnitude and phase angle.
First, the amplitude spectrum of the spectral function is analyzed, and then the point with the maximum amplitude in the low-frequency oscillation band is identified. The frequency corresponding to this point (fd) is the dominant oscillation frequency.
(39) |
Second, the spectral function value of fd is extracted from of each generator, and its phase is calculated. Given that the sources participating in the dominant oscillation mode have a phase difference of 180° during the oscillation, the phases of the various sources can be categorized into three types based on their direction and magnitude. The first and second types are the sources with positive and negative phases, respectively, and these sources participate in the dominant oscillation mode. The third type has zero phase, indicating that the sources do not participate in the oscillation.
Identifying the oscillation frequency and area of the dominant oscillation mode could facilitate the development of strategies to enhance the local damping after assessment.

Fig. 3 Online tracking of local damping under ambient data.
Step 1: measurement data are obtained from each generator port under ambient data using a phasor measurement unit. The obtained data include active power, reactive power, voltage magnitude, and voltage phase angle.
Step 2: the response data from Step 1 are preprocessed, and variations in , , , and are calculated for the measurement data of each generator port.
Step 3: initial calculations are performed based on the aforementioned variations. The logarithm of the voltage amplitude is obtained, and the derivative of the voltage phase is calculated with respect to time to obtain A(t), B(t), C(t), and D(t). Fourier transform is then conducted on these variables. The energy spectral function is computed, and the DESA method is employed to determine fd and the partition of generators.
Step 4: based on fd and the energy spectral function obtained in Step 3 for each source, a suitable integration interval is used to calculate the local damping assessment index GEDF.
Step 5: the damping contributions of each source are estimated based on the GEDF, and the nature of the contribution is determined by the sign of the GEDF.
Several theoretical assumptions are made in this paper, which may affect the accuracy of local damping assessment. The first assumption is that each source in the power system is equipped with a phasor measurement unit at its generator port, which enables real-time measurements of electrical quantities. The second assumption is that when the local damping is being tracked online, the system operates normally under ambient data. The proposed method is inapplicable to damping assessment during oscillations caused by disturbances in the power system. The third assumption is that during the local damping assessment, the values of some frequency components in the spectral function of the dissipation energy are small and located between two closely spaced modes. It is difficult to precisely determine the modes to which they belong. However, since their effects on the local damping assessment are minimal, they can be ignored.
Under simulations conducted in DIGSILENT software, a time step of 0.01 s is used. Accordingly, the sampling interval is 0.01 s, and the sampling frequency of Hz. The window selection is crucial for assessing the local damping. In power systems, a small window allows for rapid online acquisition of the damping characteristics, thus facilitating timely adjustments to the unstable operating conditions of the power system. However, for an accurate local damping assessment, the minimum frequency spacing df of the spectral function is determined by the ratio of the sampling frequency Fs to the window length N. Therefore, a large window results in small df, leading to an accurate local damping assessment. Given that the proposed method primarily addresses the low-frequency oscillation modes of the system, a df of less than 0.01 Hz is sufficient. Therefore, Hz and . In the calculation of GEDF, Hz, and the integration interval is .
First, when a local damping assessment of the power system is conducted using a spectral function based on the dissipation energy, the coupling between modes is considered. In addition, the integration interval is small, leading to the precise local damping assessment results closely aligned with real-world scenarios. Second, the energy spectral function derived in this paper enables modal partitioning when conducting the local damping assessment. Finally, the proposed method requires minimal data and uses a short window length, making it suitable for online applications.
The proposed online tracking method of local damping is validated through simulation using a four-machine 11-bus power system with renewable energy sources and an actual power system in Northwest China. The aim of simulation is to demonstrate the effectiveness of DESA and GEDF.

Fig. 4 Four-machine 11-bus power system with wind power sources.
The energy spectral function of the dissipation energy for each source in the power system is calculated using the measurement data. The dominant oscillation mode is then obtained using DESA. Analysis of the amplitude of the spectral function reveals that the dominant oscillation mode of the power system has a frequency of 0.6 Hz, as shown in

Fig. 5 Amplitude of spectral function of four-machine 11-bus power system.
Two peaks are observed near the frequency of 0.6 Hz, but they belong to the same mode. Therefore, the frequency corresponding to the maximum peak is used as the frequency for this mode. The phase of the spectral function is then analyzed to determine the phase value corresponding to a frequency of 0.6 Hz. The results indicate that G1, G2, and the new source belong to the same area, whereas G3 and G4 form another area.
A modal analysis is conducted with wind power source Gwind.
Analysis method | Modal frequency (Hz) | Oscillation area division |
---|---|---|
Eigenvalue analysis | 0.61 | (G1, G2), (G3, G4, Gwind) |
DESA | 0.60 | (G1, G2), (G3, G4, Gwind) |
The modal analysis reveals that the dominant oscillation frequency is 0.61 Hz. The results for the oscillation area division are consistent with the results of DESA. This demonstrates the effectiveness of the DESA.
The damping ratio is currently considered as an assessment index. Given that damping is the sum of the contributions from the local damping of each generator and load in the power system, this paper validates the effectiveness of the local damping index GEDF by modifying the length of line L67 (red line in

Fig. 6 Trends of damping ratio and sum of GEDFs of four-machine 11-bus power system.
The damping ratio of the power system is used to measure the stability of the corresponding modes.
Unlike traditional synchronous generators, traditional doubly-fed asynchronous wind turbines lack inertial response or primary frequency regulation capabilities due to the decoupling between their rotational speed and grid frequency, which affects the frequency regulation capabilities of power systems. Currently, virtual inertia control is integrated into wind turbine. However, the integration of virtual inertia control introduces coupling between the wind turbine and existing synchronous generators, causing significant changes in the electromechanical oscillation characteristics of the power system.
This paper analyzes and compares the presence of virtual inertia control in wind turbines as well as different droop control coefficient settings to demonstrate the effectiveness of the proposed method. Tables II-V present the results.
Generator or load | GEDF | Generator or load | GEDF |
---|---|---|---|
G1 |
1.83×1 | Gwind |
-1.97×1 |
G2 |
2.29×1 | Load1 |
-1.45×1 |
G3 |
-8.04×1 | Load2 |
-2.08×1 |
G4 |
-4.13×1 |
Generator or load | GEDF | Generator or load | GEDF |
---|---|---|---|
G1 |
2.01×1 | Gwind |
1.41×1 |
G2 |
3.14×1 | Load1 |
-1.74×1 |
G3 |
7.45×1 | Load2 |
-1.68×1 |
G4 |
-8.70×1 |
Generator or load | GEDF | Generator or load | GEDF |
---|---|---|---|
G1 |
5.21×1 | Gwind |
1.89×1 |
G2 |
2.26×1 | Load1 |
-1.89×1 |
G3 |
-4.76×1 | Load2 |
-2.28×1 |
G4 |
-2.86×1 |
Scenario | Sum of GEDFs | Damping ratio (%) |
---|---|---|
Without virtual inertia control |
9.99×1 | 1.28 |
With virtual inertia control (KVD=0.5) |
1.61×1 | 2.25 |
With virtual inertia control (KVD=1) |
2.00×1 | 2.33 |
The results show that the GEDF of the wind turbine is negative in the absence of virtual inertial control. Introducing virtual inertial control in a wind turbine changes the GEDF to a positive value, demonstrating that virtual inertial control tightens the coupling between the wind farm and the original power system, thereby substantially boosting the damping contribution of wind farm to the power system. In addition, an increase in the droop control coefficient KVD amplifies the positive damping contribution from the wind turbine.
A comparison of the relationship between the sum of GEDFs and damping ratio of the power system across various scenarios clearly reveals that the integration of virtual inertia control into wind farms enhances the overall damping characteristics of the power system. In addition, the consistent trends in the changes of the sum of GEDFs and damping ratio underscore the effectiveness of the proposed method.
An actual power system in Northwest China, as depicted in

Fig. 7 Actual power system in Northwest China.
The DESA method is employed to extract the modal information of the power system.

Fig. 8 Amplitude of energy spectrum function of actual power system.
The phase of each source at 1.51 Hz following the analysis of energy phase spectrum is obtained, as shown in

Fig. 9 Phase of energy spectrum function of actual power system.
Tables VI-IX present the results of the local damping assessment for the actual power system.
Generator | GEDF | Generator | GEDF |
---|---|---|---|
GA1 |
2.65×1 | GA4 |
1.72×1 |
GA2 |
2.65×1 | GA5 |
1.26×1 |
GA3 |
2.65×1 |
Generator | GEDF | Generator | GEDF |
---|---|---|---|
GwindA |
9.53×1 | GwindD |
9.05×1 |
GwindB |
9.54×1 | GwindE |
9.83×1 |
GwindC |
9.06×1 | GwindF |
9.52×1 |
Generator | GEDF | Generator | GEDF |
---|---|---|---|
GB1 |
-8.73×1 | GB5 |
-1.32×1 |
GB2 |
-1.41×1 | GB61 |
-1.29×1 |
GB3 |
-1.18×1 | GB62 |
-1.29×1 |
GB4 |
-1.17×1 |
Generator | GEDF | Generator | GEDF |
---|---|---|---|
GPV1 |
1.19×1 | GPV4 |
9.48×1 |
GPV2 |
1.19×1 | GPV5 |
8.81×1 |
GPV3 |
1.16×1 | GPV6 |
1.57×1 |
The effectiveness of the proposed method in the actual power system is also validated. The lengths of certain transmission lines in the power system are adjusted, and the trends of the system damping ratio and sum of GEDFs are compared, as shown in

Fig. 10 Trends of damping ratio and sum of GEDFs of actual power system.
The results indicate that the damping ratio and sum of GEDFs decrease gradually with the increase of line length. The consistent trends in their variations confirm the effectiveness of the proposed method.
Load fluctuations with different signal-to-noise ratios (SNRs) are introduced into the power system to illustrate the stability of the proposed method. The relative magnitudes of the GEDF indices for various generators under different noise levels and the nature of their contributions to damping are analyzed. SNRs of 34, 35, and 36 are considered. Figures

Fig. 11 GEDFs of synchronous generators under different SNRs.

Fig. 12 GEDFs of renewable energy generators under different SNRs. (a) Wind power generator. (b) PV generator.
The simulation results indicate that under different SNRs, the relative magnitudes of GEDFs of various generators in the power system and the nature of their damping contributions remain consistent. The results suggest the stability of the proposed method in conducting online tracking of local damping of individual generators. The renewable energy generators in the power system contribute minimally to the overall damping, whereas the synchronous generators play a major role.
To further demonstrate the effectiveness of the proposed method in power systems with renewable energy sources, this paper compares the local damping assessment results without wide-area damping controllers (WADCs) with those with WADCs.
In the scenarios with WADCs, a compact design based on a power system stabilizer (PSS) is used, which adjusts the relative residual index by targeting specific dominant oscillation modes and by enhancing the control loop selection strategy. The PSS-based WADC is as straightforward as the classical PSS in which only two lead-lag phase compensation sources are incorporated [
The local damping indices of the power system with and without WADCs are assessed separately. The changes in the sum of GEDFs and damping ratio in both scenarios are compared, as presented in
Scenario | Sum of GEDFs | Damping ratio (%) |
---|---|---|
Without WADC |
6.13×1 | 3.57 |
With WADC |
2.09×1 | 4.01 |
Theoretically, the addition of a WADC to the power system should enhance its stability, which is confirmed by simulation results. The results also indicate that the damping ratio with WADC is higher than that without WADC. In addition, the sum of GEDFs is greater in the system with WADC than that without it. These findings are aligned with those of theoretical analysis and further validate the effectiveness of the proposed method.
This paper proposes an online tracking method of local damping in power systems with high proportion of renewable energy sources under ambient data. The expression of dissipation energy in frequency domain at the generator ports is derived, which enables the spectral functions for the dissipation energy in each generator to be obtained. The dominant oscillation modes of the power system are identified by analyzing the amplitude and phase spectra of the spectral functions. The EDFs for each generator are then obtained by integration with the selected frequency interval. The simulation results demonstrate the effectiveness of the proposed method. The relationship between the eigenvalues and spectral functions is also analyzed. The indicators remain stable under varying ambient disturbances and accurately reflect the relationship between the damping contributions of different generators and their relative magnitudes. Notably, the proposed method requires the system to be in a normal operation state, and small disturbances may render the operation data unsuitable for the online tracking of local damping.
Future research work should consider the introduction of a decision-making step during the data acquisition process to extend the proposed method to a broad range of scenarios for the online tracking of local damping in power systems.
References
T. Wang, M. Jin, Y. Li et al., “Adaptive damping control scheme for wind grid-connected power systems with virtual inertia control,” IEEE Transactions on Power Systems, vol. 37, no. 5, pp. 3902-3912, Sept. 2022. [Baidu Scholar]
S. Bu, W. Du, and H. Wang, “Investigation on probabilistic small-signal stability of power systems as affected by offshore wind generation,” IEEE Transactions on Power Systems, vol. 30, no. 5, pp. 2479-2486, Sept. 2015. [Baidu Scholar]
S. Bu, W. Du, H. Wang et al., “Probabilistic analysis of small-signal stability of large-scale power systems as affected by penetration of wind generation,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 762-770, May 2012. [Baidu Scholar]
B. Wang, D. Yang, and G. Cai, “Review of research on power system inertia related issues in the context of high penetration of renewable power generation,” Power System Technology, vol. 44, no. 8, pp. 2998-3007, Aug. 2020. [Baidu Scholar]
Q. Wu, Y. Huang, Y. Zhou et al., “Damping characteristics of synchronous motor-generator pair system for integration of new energy,” Automation of Electric Power Systems, vol. 43, no. 15, pp. 80-86, Aug. 2019. [Baidu Scholar]
L. Chen, Y. Min, Y. Chen et al., “Relationship between oscillation energy analysis and eigenvalue analysis and assessment of generator damping,” Automation of Electric Power Systems, vol. 37, no. 19, pp. 33-40, Oct. 2013. [Baidu Scholar]
Z. Sun, Q. Jiang, J. Wang et al., “Evaluation method of local damping of electromechanical oscillation of power systems containing wind turbines,” High Voltage Engineering, vol. 47, no. 10, pp. 3452-3466, Oct. 2021. [Baidu Scholar]
L. Chen, Y. Chen, Y. Min et al., “Low frequency oscillation analysis and oscillation source location based on oscillation energy, Part two: method for oscillation source location and case studies,” Automation of Electric Power Systems, vol. 36, no. 4, pp. 1-5, Feb. 2012. [Baidu Scholar]
P. Kundur, “Small signal stability,” in Power System Stability and Control. New York: McGraw-Hill, 1994. [Baidu Scholar]
S. Chou, X. Wang, and F. Blaabjerg, “Frequency-domain modal analysis for power-electronic-based power systems,” IEEE Transactions on Power Electronics, vol. 36, no. 5, pp. 4910-4914, May 2021. [Baidu Scholar]
V. Aijarapu, “Reducibility and eigenvalue sensitivity for identifying critical generators in multimachine power systems,” IEEE Transactions on Power Systems, vol. 5, no. 3, pp. 712-719, Aug. 1990. [Baidu Scholar]
M. Amin and M. Molinas, “Small-signal stability assessment of power electronics based power systems: a discussion of impedance- and eigenvalue-based methods,” IEEE Transactions on Industry Applications, vol. 53, no. 5, pp. 5014-5030, Sept.-Oct. 2017. [Baidu Scholar]
H. Ghasemi and C. Canizares, “Damping torque estimation and oscillatory stability margin prediction,” in Proceedings of 2006 IEEE PES General Meeting, Montreal, Canada, Jun. 2006, pp. 667-674. [Baidu Scholar]
Y. Yu and P. Li, “The impact of weak internection of bulk power grids to damping and dynamic stability of power systems,” Proceedings of CSEE, vol. 25, no. 11, pp. 6-11, Jun. 2005. [Baidu Scholar]
G. Xu, J. Zhou, Z. Li et al., “Damping torque coefficient affected by different rotor damping structures of turbo-generator during large disturbance,” in Proceedings of 2019 IEEE Energy Conversion Congress and Exposition (ECCE), Baltimore, USA, Oct. 2019, pp. 1729-1734. [Baidu Scholar]
Q. Dong, J. Sun, Q. Wu et al., “A method for filtering low frequency disturbance in PMU data before coordinated usage in SCADA,” IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 2810-2816, Jul. 2017. [Baidu Scholar]
Z. Sun, J. Wang, C. Pan et al., “Low-frequency oscillation area location method based on branch mode oscillation energy,” Proceedings of CSEE, vol. 43, no. 17, pp. 6589-6601, Sept. 2023. [Baidu Scholar]
L. Chen, Y. Min, and W. Hu, “An energy-based method for location of power system oscillation source,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 828-836, May 2013. [Baidu Scholar]
L. Chen, Y. Min, and W. Hu, “Low frequency oscillation analysis and oscillation source location based on oscillation energy, Part one: mathematical foundation and energy flow computation,” Automation of Electric Power Systems, vol. 36, no. 3, pp. 22-27, Feb. 2012. [Baidu Scholar]
Y. Hu, S. Bu, S. Yi et al., “A novel energy flow analysis and its connection with modal analysis for investigating electromechanical oscillations in multi-machine power systems,” IEEE Transactions on Power Systems, vol. 37, no. 2, pp. 1139-1150, Mar. 2022. [Baidu Scholar]
Y. Hu, S. Bu, X. Zhang et al., “Connection between damping torque analysis and energy flow analysis in damping performance evaluation for electromechanical oscillations in power systems,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 1, pp. 19-28, Jan. 2022. [Baidu Scholar]
Z. Sun, G. Cai, D. Yang et al., “A method for the evaluation of generator damping during low-frequency oscillations,” IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 109-119, Jan. 2019. [Baidu Scholar]
L. Chen, Y. Chen, Y. Min et al., “Low frequency oscillation analysis and oscillation source location based on oscillation energy, Part two: method for oscillation source location and case studies,” Automation of Electric Power Systems, vol. 36, no. 4, pp. 1-5, Feb. 2012. [Baidu Scholar]
L. Chen, S. Ming, Y. Min et al., “Online monitoring of generator damping using dissipation energy flow computed from ambient data,” IET Generation, Transmission & Distribution, vol.11, no. 18, pp. 4430-4435, Dec. 2017. [Baidu Scholar]
L. Chen, Y. Min, Y. Chen et al., “Evaluation of generator damping using oscillation energy dissipation and the connection with modal analysis,” IEEE Transactions on Power Systems, vol. 29, no. 3, pp. 1393-1402, May 2014. [Baidu Scholar]
S. Maslennikov, B. Wang, and E. Litvinov, “Dissipating energy flow method for locating the source of sustained oscillations,” International Journal of Electrical Power & Energy Systems, vol. 88, pp. 55-62, Jun. 2017. [Baidu Scholar]
J. Qi, Q. Wu, Y. Zhang et al., “Unified residue method for design of compact wide-area damping controller based on power system stabilizer,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 2, pp. 367-376, Mar. 2020. [Baidu Scholar]