Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Online Tracking of Local Damping in Power Systems with High Proportion of Renewable Energy Sources Under Ambient Data  PDF

  • Zhenglong Sun
  • Zewei Li
  • Hao Yang
  • Lixin Wang
  • Bo Wang
  • Chao Pan
  • Cheng Liu
  • Guowei Cai
Key Laboratory of Modern Power System Simulation and Control & Renewable Energy Technology, Ministry of Education, Northeast Electric Power University, Jilin, China

Updated:2025-03-26

DOI:10.35833/MPCE.2024.000169

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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.

I. Introduction

THE utilization of wind, solar, and other renewable energy sources in power systems is increasing [

1], [2]. This shift is accompanied by the emergence of features characterized by low inertia and damping. Consequently, power systems are becoming more vulnerable to low-frequency oscillations [3]-[5], which could compromise the secure operation.

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 [

6]-[8]. Thus, the online tracking and assessment of local damping and identification of sources with negative damping are crucial for curtailing oscillations and enhancing the power system stability.

The eigenvalue analysis is a classic method in small-disturbance stability analysis [

9], [10]. The eigenvalue analysis can be employed to select critical sources and install them in power system stabilizers in multi-machine systems [11]. In addition, this method can be used to assess the stability of electronic-based power systems [12]. The dimensionality increases as the power system scale expands, leading to the curse of dimensionality during the computation of eigenvalues. The curse of dimensionality poses challenges for efficient online analysis. The component of the generator electromagnetic torque, known as damping torque, is an essential factor in studying the dynamic stability of power systems [13]. The principle of weakened system damping in weakly interconnected systems caused by reduced line impedance can be explained using the damping torque method [14]. An analysis of the effects of different generator rotor damping structures on the damping torque coefficients can provide a theoretical foundation for enhancing the power system stability by altering the damping structures [15]. However, the damping torque of generators in multi-machine systems is highly complex, and the online computation of torque coefficients is challenging.

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 [

16], [17]. In addition, phasor measurement data support power system stability analysis. A calculation method for network energy flow based on WAMS data is proposed in [18] to address the challenge to constructing an energy function within a system. This method operates independently of the energy function and facilitates energy flow computation using the phasor measurement data. The correlation between the energy consumption of generator and its damping torque in a single-machine infinite-bus system is explored in [19], highlighting the close relationship between system damping and energy dissipation. Furthermore, the energy flow analysis method, which involves signal reconstruction and decomposition, can be employed to investigate low-frequency oscillations in power systems. The consistency between the energy flow and modal analysis methods in multi-system scenarios is demonstrated through derivation in [20]. Finally, [21] derives the damping torque coefficient of multi-machine power systems, defines the energy attenuation coefficient, and rigorously establishes the essential equivalence between the damping torque and energy flow analysis methods.

As the energy flow analysis method aligns with system damping, an online assessment of damping can be achieved using this method [

22]. By integrating the energy flow theory with the damping representation in elasticity, an online assessment of local damping during oscillation becomes feasible. Reference [23] proposes a method for identifying oscillation sources by analyzing the energy sources and flows within the network. In [24], the dissipation energy curve is computed using eigenvectors, and the energy dissipation factor (EDF) is defined. Reference [25] separates the average dissipation power across the oscillation modes of varying frequencies. It employs eigenvectors to calculate the energy dissipation coefficient specific to a single mode of generator. In addition, [26] introduces an energy-based method that utilizes phasor measurement data to trace poorly damped natural and forced oscillation sources in power systems. However, the assessment of local damping requires filtering the phasor measurement data. Bandwidth selection significantly affects the evaluation results, and real-time applications are hindered by the substantial amount of data required for assessment.

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.

II. Expression of Dissipation Energy in Frequency Domain

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.

A. Ambient Response of Power Systems

The differential algebraic equations of a power system can be expressed as:

x˙=f(x,y)0=g(x,y,l) (1)

where x is the state variable of the system; y is the algebraic variable; l is the load fluctuation variable; and f() and g() 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:

I˙=-K(l-leq)+δξ (2)

where leq is the load value at the equilibrium point; K is the load response rate; δ is the noise intensity; and ξ is an ambient fluctuation following a Gaussian distribution.

Linearizing (1) and (2) results in:

Δx˙Δl˙=fx-fygy-1gx   -fygy-1gl         0                  -K     ΔxΔl+δ0Inlξ (3)

where gx, gy, and gl are the Jacobian matrices corresponding to variables x, y, and l, respectively; fx and fy are the Jacobian matrices corresponding to variables x and y, respectively; Δx is the change in state variable x; Δl is the change in load relative to the equilibrium point; and Inl is the identity matrix.

Let A=fx-fygy-1gx   -fygy-1gl         0                  -K     , z=ΔxΔl, and B=0Inl. Equation (3) can then be represented as:

z˙=Az+δBξ (4)

Given that matrix A contains the characteristic information of electromechanical oscillations under load fluctuations, the eigenvalues λk=σk+jωk (k=1,2,...,n) corresponding to n electromechanical oscillation modes can be calculated. Therefore, the time-domain analytical solution of the system state variables can be expressed as:

z(t)=k=1nvkukTz0eσktsin(ωkt+φk)+δBξ (5)

where uk and vk are the left and right eigenvectors corresponding to eigenvalue λk, respectively; z(0) is the initial value of the state variables; and φk 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 equation (6) shows that the ambient response of the power system contains rich modal information. Let Ck=vkukTz(0)eσkt and ε=δBξ. Substituting these into (5) yields:

z(t)=k=1nCksin(ωkt+φk)+ε (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:

ΔXi(t)=k=1nCXksin(ωkt+φXk)+εnoise,Xi (7)

where Xi can be Pi, Qi, Ui, or θi, and Pi, Qi, Ui, and θi are the active power, reactive power, voltage amplitude, and voltage phase angle of generator port i, respectively; and εnoise,Xi is the random component of the measured noise composition in Xi.

B. Theory of Energy Flow

The energy flowing from node i to branch Lij in a power system can be expressed as:

Wi=Im(Iij*dUi)=Pijdθi+Qijd(lnUi) (8)

where Iij 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 Im() 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:

Wi=WiO+WiD (9)
WiO=Pij,sdΔθi+Qij,sd(ΔlnUi)WiD=ΔPijdΔθi+ΔQijd(ΔlnUi)=WiD1+WiD2 (10)

where Δ is the variation in each electrical quantity; Pij,s and Qij,s are the steady-state values of active power and reactive power, respectively; WiO is the oscillation component representing the transient energy of the power system; WiD is the continuously changing component over time, known as the dissipation energy; and WiD1 and WiD2 are the integral terms containing the variation in active power and the variation in reactive power, respectively. A positive WiD 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 WiD, 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 WiD1 and WiD2 are expressed in (11) and (12), respectively.

WiD1=a=b=1nMaPθ(t2-t1)+a=1,abnb=1nNabPθ+ε1 (11)
WiD2a=b=1nMaQU(t2-t1)+a=1,abnb=1nNabQU+ε2 (12)
WiD=WiD1+WiD2a=b=1nMa(t2-t1)+a=1nb=1,abnNab+ε1+ε2  (13)

where a and b are the modes; MaPθ is the damping term for mode a calculated using the incremental variables ΔPi(t) and Δθi(t); NabPθ is the harmonic and coupling term for modes a and b computed using the incremental variables ΔPi(t) and Δθi(t); ε1 is the ambient component in ΔPi(t) and Δθi(t) composed of measurement noise; MaQU is the damping term for mode a calculated using the incremental variables ΔQi(t) and ΔUi(t); NabQU is the harmonic and coupling term for modes a and b computed using the incremental variables ΔQi(t) and ΔUi(t); ε2 is the ambient component in ΔQi(t) and ΔUi(t) composed of measurement noise, and the derivation is presented in Supplementary Material A; Ma=MaPθ+MaQU; and Nab=NabPθ+NabQU.

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.

III. Assessment of Local Damping in Frequency Domain Using EDF

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.

A. Parseval’s Theorem

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 f(t) as an example. The following equation is given:

-+f(t)2dt=12π-+F(jω)2dω (14)

where F(jω) is obtained from f(t) through Fourier transform.

For two signals fi(t) and fj(t), in line with (14), the derivation process is given as:

-+fi(t)fj(t)dt=-+fj(t)12π-+Fi(ω)ejωtdωdt=12πFi(ω)-+fj(t)ejωtdtdω=12π-+Fi(ω)Fj*(ω)dω (15)

where Fi(ω) and Fj(ω) are the Fourier transforms of signals fi(t) and fj(t), respectively. The integrated product of the two time-domain signals can be transformed into the frequency domain for analysis by applying (15).

B. Local Damping Assessment Index in Frequency Domain Based on Dissipation Energy

Analysis of the generator ports via (10) allows to express the dissipation energy at generator port i as:

WGiD=ΔPi(t)dΔθi(t)+ΔQi(t)d(ΔlnUi(t))=ΔPi(t)dΔθi(t)dtdt+ΔQi(t)d(ΔlnUi(t))dtdt (16)

Let A(t)=ΔPi(t), B(t)=d(Δθi(t))/dt, C(t)=ΔQi(t), and D(t)=d(lnUi(t))/dt. In this case, the expression for WGiD is:

WGiD=A(t)B(t)dt+C(t)D(t)dt (17)

Fourier transform is then performed on the aforementioned variables. Using A(t) as an example, the transformation process can be expressed as:

A(jω)=-+A(t)e-jωtdt=-+A(t)cos(ωt)dt--+A(t)jsin(ωt)dt=RA(ω)-jXA(ω) (18)

where RA(ω) and XA(ω) are the real and imaginary parts of A(t) 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 ω.

Similarly, the remaining variables can be expressed as:

B(jω)=RB(ω)-jXB(ω)C(jω)=RC(ω)-jXC(ω)D(jω)=RD(ω)-jXD(ω) (19)

Equations (14), (18), and (19) can be substituted into (17) to obtain:

WGiD=A(t)B(t)dt+C(t)D(t)dt=12π-+A(jω)B(jω)*dω+-+C(jω)D(jω)*dω (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:

RA(ω)jXB(ω)dω=0RC(ω)jXD(ω)dω=0RB(ω)jXA(ω)dω=0RD(ω)jXC(ω)dω=0 (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:

WGiD=A(t)B(t)dt+C(t)D(t)dt=12π(RA(ω)RB(ω)+XA(ω)XB(ω)+RC(ω)RD(ω)+XC(ω)XD(ω))dω=ξ(ω)dω (22)

where ξ(ω) 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:

WGiD=ξ(ω)dω=ω1-α1lω1+α1uξ(ω)dω+ω2-α2lω2+α2uξ(ω)dω+...+ωk-αklωk+αkuξ(ω)dω (23)
WGiD=WGiD(f1)+WGiD(f2)+...+WGiD(fk)=DGi(f1)+DGi(f2)+...+DGi(fk) (24)

where ωk is the angular frequency of the system mode k; αku and αkl 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 (ωk=2πfk); (fk-αku,fk+αkl) is the entire integration range of the energy spectral function; and WGiD(fk) 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, DGi(fk) is defined as the EDF for mode k, which corresponds to generator i. Figure 1 illustrates the division of the theoretical integration intervals corresponding to different modes.

Fig. 1  Division of theoretical integration intervals.

Based on (23), the GEDF is defined as:

GDGi(fd)=ωd-βdωd+βd(RA(ω)RB(ω)+XA(ω)XB(ω)+RC(ω)RD(ω)+XC(ω)XD(ω))dω=ωd-βdωd+βdξ(ω)dω (25)

where GDGi(fd) is the GEDF; and (ωd-βd,ωd+βd) is the integration interval of the dissipation energy in frequency domain corresponding to the dominant oscillation mode d, and βd 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. Figure 2 shows the selection process of the integration intervals.

Fig. 2  Selection process of integration intervals.

According to (25), GDGi(fd) is the dissipation energy corresponding to mode d. Under dominant oscillation mode d, a large value of GDGi(fd) for a source indicates high energy dissipation, which suggests that the source contributes abundant damping to the power system. Positive and negative values of GDGi(fd) indicate a positive or negative contribution to the damping, respectively.

C. Relationship Between Spectral Function and Eigenvalues of Dissipation Energy

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 WiD1. The terms ΔPi(t) and Δθi'(t) can be expressed as:

ΔPi(t)=k=1nvpkupkP(0)eσktsin(ωkt+φk)+εnoise,P (26)
Δθi'(t)=k=1nvθ'kuθ'kθ0(σkeσktsin(ωkt+φk)+ωkeσktcos(ωkt+φk))+εnoise,θ' (27)

where upk and vpk are the values of the left and right eigenvectors corresponding to the active power variation for mode k, respectively; and uθ'k and vθ'k are the values of the left and right eigenvectors corresponding to the phase angle derivative variation for mode k, respectively.

Performing Fourier transform on ΔPi(t) yields:

ΔPi(jω)=k=1nvpkupkTP(0)eσktsin(ωkt+φk)+εnoise,P=k=1nvpkupkTP0eσktejωktejφk-e-jωkte-jφk2j+εP (28)

where εP is the Fourier transform of the measurement noise in variable ΔPi(t).

Given that σk 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 t>0 in ΔPi(t). 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).

fP(t)=eσktejωktejφk-e-jωkte-jφk2j (29)
(fP(t))=ωkcosφk+(σk+jω)sinφk(σk+jω)2+ωk2 (30)
ΔPi(jω)=k=1nvpkupkTP0ωkcosφk+(σk+jω)sinφk(σk+jω)2+ωk2+εP (31)

Similarly, applying Fourier transform to variable Δθi'(t) results in:

Δθi'(jω)=k=1nvθ'kuθ'kTθ(0)3σkωk+2jωωkcosφk(σk+jω)2+ωk2+(σk2+jσkω-2ωk2)sinφk(σk+jω)2+ωk2+εθ' (32)

where εθ' is the Fourier transform of the measurement noise in Δθi'(t).

Therefore, when (31) is multiplied by (32), WGiD1 in (10) can be expressed as:

WGiD1=-+A(jω)B(jω)*2πdω=-+ΔPi(jω)Δθi'(jω)*2πdω=-+F(σk,ωk,φk,ω)dω+εPθ' (33)
F(σk,ωk,φk,ω)=k=1nvpkupkTvθ'kuθ'kTP0θ0ωkcosφk+(σk+jω)sinφk2π(σk+jω)2+ωk23σkωk+2jωωkcosφk(σk+jω)2+ωk2+(σk2+jσkω-2ωk2)sinφk(σk+jω)2+ωk2* (34)

where the term obtained by multiplying the measured noise εP or εθ' is expressed as εPθ'.

Equation (33) indicates that for a system with n oscillation modes, the energy dissipation in the frequency domain can be represented using the real and imaginary parts of the eigenvalues. The relationship between these parameters can be derived via (34).

Similarly, WGiD2 in (10) can be expressed by function L involving σk, ωk, φk, and ω as:

WGiD2=12π-+C(jω)D(jω)*dω=-+L(σk,ωk,φk,ω)dω+εQU (35)

where εQU 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:

WGiD=-+F(σk,ωk,φk,ω)+L(σk,ωk,φk,ω)dω+εPθ'+εQU  (36)

where the original functions εPθ' and εQU 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, εPθ' and εQU can be disregarded.

Equation (36) can then be rewritten as:

WGiD=(F(σk,ωk,φk,ω)+L(σk,ωk,φk,ω))dω=ξ(ω)dω (37)

Equation (37) shows that a certain functional relationship exists between the spectral function of the dissipation energy at the generator port and the real and imaginary parts of the eigenvalues derived from the analysis. The spectral function in each source also contains modal information regarding the power system. This phenomenon indicates that this paper successfully uses the energy spectral function as an indicator and reveals the feasibility of using various online tracking methods to assess the damping in each source.

IV. Assessment of Local Damping in Frequency Domain Under Ambient Data

A. Identification of Dominant Oscillation Mode Based on Dissipation Energy Spectrum Analysis (DESA)

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:

ξi(ω)=Rξi(ω)+jXξi(ω)=ξi(ω)arctanXξi(ω)Rξi(ω) (38)

where Rξi(ω) and Xξi(ω) 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.

ξi(ω)max=ξi(2πfd)    0.2 Hzfd2.5 Hz (39)

Second, the spectral function value of fd is extracted from ξi(ω) 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.

B. Online Tracking of Local Damping

Figure 3 shows a schematic of the online tracking of local damping under ambient data. The main steps of this method are described as follows.

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 ΔPi(t), Δθi(t), ΔQi(t), and ΔUi(t) 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 ξ(ω) 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.

C. Discussion

1) Assumptions

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.

2) Window Selection

Under simulations conducted in DIGSILENT software, a time step of 0.01 s is used. Accordingly, the sampling interval Ts is 0.01 s, and the sampling frequency of Fs=1/Ts=100 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, df=0.01 Hz and N=Fs/df=10000. In the calculation of GEDF, βd=0.1 Hz, and the integration interval is (fd-0.1 Hz,fd+0.1 Hz).

3) Advantages

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.

V. Case Studies

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.

A. Four-machine 11-bus Power System

Figure 4 illustrates the four-machine 11-bus power system with wind power sources. The original active power output of generator G2 in this system is 700 MW. In this paper, the active power output of G2 is adjusted to 100 MW. In addition, a wind farm consisting of 120 wind turbines is connected to bus 6 through a transformer with an active power output of 5 MW for each wind turbine. The dominant oscillation mode of the initial power system is the weak damping mode, as indicated by the modal analysis results, which reveals a damping ratio of 2.33% for the dominant oscillation mode of the power system.

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, where Load1 and Load2 are two loads in the four-machine 11-bus power system.

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. Table I shows the comparison of modal analysis results.

TABLE I  Comparison of Modal Analysis Results of Four-machine 11-bus Power System
Analysis methodModal 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. 4) and comparing the trends of the damping ratio and sum of GEDFs. The original length of L67 in the system is 10 km, and the length is increased or decreased in increments of 0.5 km to observe the trends in the GEDF indices. Figure 6 presents the results.

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. Figure 6 shows that as the line length increases from 8.5 km to 11 km, the damping ratio of the dominant oscillation mode monotonically decreases, which is in accordance with actual physical laws. Similarly, the sum of GEDFs decreases monotonically with the increase of line length, and the trend is consistent with that of the damping ratio. This preliminary analysis demonstrates the effectiveness of the GEDF indices.

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.

TABLE II  Local Damping Assessment Result Without Virtual Inertia for Wind Turbine
Generator or loadGEDFGenerator or loadGEDF
G1 1.83×10-4 Gwind -1.97×10-5
G2 2.29×10-4 Load1 -1.45×10-4
G3 -8.04×10-5 Load2 -2.08×10-4
G4 -4.13×10-5
TABLE Ⅲ  Local Damping Assessment Result with a Virtual Inertia Coefficient KVD=0.5 for Wind Turbine
Generator or loadGEDFGenerator or loadGEDF
G1 2.01×10-5 Gwind 1.41×10-4
G2 3.14×10-5 Load1 -1.74×10-4
G3 7.45×10-7 Load2 -1.68×10-5
G4 -8.70×10-7
TABLE Ⅳ  Local Damping Assessment Result with a Virtual Inertia Coefficient KVD=1 for Wind Turbine
Generator or loadGEDFGenerator or loadGEDF
G1 5.21×10-6 Gwind 1.89×10-4
G2 2.26×10-5 Load1 -1.89×10-4
G3 -4.76×10-7 Load2 -2.28×10-5
G4 -2.86×10-6
TABLE Ⅴ  Comparison Results of Sum of GEDFs and Damping Ratio in Different Scenarios
ScenarioSum of GEDFsDamping ratio (%)
Without virtual inertia control 9.99×10-7 1.28
With virtual inertia control (KVD=0.5) 1.61×10-6 2.25
With virtual inertia control (KVD=1) 2.00×10-6 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.

B. Actual Power System in Northwest China

An actual power system in Northwest China, as depicted in Fig. 7, is studied in this subsection. The power system includes wind power sources, photovoltaics (PVs), and high-voltage direct current components.

Fig. 7  Actual power system in Northwest China.

The DESA method is employed to extract the modal information of the power system. Figure 8 illustrates the amplitude of the energy spectral function, where G represents the generator, and the subscripts show the corresponding type (e.g., wind denotes the wind power generator represented by A-F; and PV denotes the PV generator represented by 1-6). The amplitude shows two peaks in the low-frequency interval corresponding to the frequencies of 1.51 Hz and 2.68 Hz. Given that the frequency interval for low-frequency oscillations should be between 0.2 Hz and 2.5 Hz, the dominant oscillation mode frequency of the system is determined to be 1.51 Hz.

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. The sources in Areas 1 and 2 exhibit negative and positive phases, respectively. The phases of the wind and PV sources are zero. The oscillation areas of the power system show that the wind and PV sources do not participate in the oscillation.

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.

TABLE Ⅵ  GEDF Values for Synchronous Generators in Area 1
GeneratorGEDFGeneratorGEDF
GA1 2.65×10-5 GA4 1.72×10-5
GA2 2.65×10-5 GA5 1.26×10-5
GA3 2.65×10-5
TABLE Ⅶ  GEDF Values for Wind Turbines in Area 1
GeneratorGEDFGeneratorGEDF
GwindA 9.53×10-7 GwindD 9.05×10-7
GwindB 9.54×10-7 GwindE 9.83×10-7
GwindC 9.06×10-7 GwindF 9.52×10-7
TABLE Ⅷ  GEDF Values for Synchronous Generators in Area 2
GeneratorGEDFGeneratorGEDF
GB1 -8.73×10-6 GB5 -1.32×10-5
GB2 -1.41×10-5 GB61 -1.29×10-5
GB3 -1.18×10-5 GB62 -1.29×10-5
GB4 -1.17×10-5
TABLE Ⅸ  GEDF Values for PV Generators in Area 2
GeneratorGEDFGeneratorGEDF
GPV1 1.19×10-9 GPV4 9.48×10-10
GPV2 1.19×10-9 GPV5 8.81×10-10
GPV3 1.16×10-9 GPV6 1.57×10-9

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. The lengths of transmission lines AC13 and AC14 are modified (red lines in Fig. 7) to simulate the effect. The initial lengths of both lines are 512 km, and adjustments are made with 10 km intervals by either increasing or decreasing their lengths.

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 11 and 12 present the simulation results.

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 [

27].

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 Table X.

TABLE X  Comparison of Modal Analysis Results of Actual Power System
ScenarioSum of GEDFsDamping ratio (%)
Without WADC 6.13×10-5 3.57
With WADC 2.09×10-4 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.

VI. Conclusion

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.

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