Abstract
An ambient modal framework for inertia estimation using synchrophasor data is proposed in this letter. Specifically, an analytical formulation is developed for the estimation of inertia based on the frequency and damping ratio modes extracted from ambient data. An advantage of the proposed framework is that it can rely on synchronized ambient data under non-disturbed conditions for online estimation and tracking of inertia. Ultimately, numerical simulation studies and physical experiments demonstrate the feasibility of the proposed approach.
THE increasing penetration rate of renewable power generation units has been a key factor for the concept of power system inertia estimation to garner significant attention [
The common deployment of a phasor measurement unit (PMU) in the power system enables the inertia to be estimated from the measurement data. The strong correlation between the frequency change rate at the disturbance moment and the inertia, i.e., the swing equation without the damping coefficient, is utilized to estimate the inertia by means of the preset sudden power shock and measured frequency deviation [
The main contributions of this letter are as follows:
1) According to the power system dynamic equations under ambient excitation, a mathematical relationship between the inertia and ambient modal is established.
2) An analytical formulation for the estimation of inertia is developed.
3) Numerical simulations and physical experiments are conducted to test and validate the proposed framework.
Natural excitation exists in a real power system. The typical causes of this excitation are the electrical loads, which vary randomly by nature. Under natural excitation circumstances, the oscillatory behavior of a generator can be described by the transfer function shown in
(1) |

Fig. 1 Dynamic behavior of a generator under natural excitation.
where is the rotor angle deviation; TJ is the inertia constant; D is the damping coefficient; Ks is the synchronizing power coefficient; and F(t) is the natural excitation of the system. In
Let fn () and be the natural frequency and damping ratio, respectively. We can then obtain:
(2) |
Based on the random vibration theory [
(3) |
where h(t) is the unit pulse response of the linear system described in (2). The power spectral density of h(t) can be described as:
(4) |
Suppose that the natural excitation F(t) is approximately a stationary Gaussian process, and that the power spectral density of F(t) is constant, i.e., .
Furthermore, the autocorrelation power spectral density of the response (t) can be obtained as follows:
(5) |
Therefore, when (as the damping ratio is much smaller than 1), there exist two stress peaks in the autocorrelation power spectral density of the ambient response, as shown in
The inertia is strongly coupled with the electromechanical dynamic behavior of a power system, which affects not only the transient response but also the electromechanical oscillation by the oscillation frequency and damping ratio . According to the modal analysis theory [
(6) |
(7) |
where Pe0 is the steady-state electrical power; is the rated rotor speed; and is the steady-state rotor angle.
Based on (6) and (7), the inertia constant expressed by the oscillation modes is derived as:
(8) |
where is the steady-state coefficient.
As the steady-state variables Pe0 and can be measured or calculated directly by a PMU, the inertia can be estimated once the oscillation modes are extracted from the synchronized ambient data.
In this letter, numerical simulations of a single-generator infinite bus system are conducted to demonstrate the performance of the proposed method using the Power System Toolbox (PST) [
The extracted electromechanical oscillation frequency and damping ratio are shown in

Fig. 2 Electromechanical modes extracted by SSI.
In the proposed inertia estimation method, the electrical power and rotor angle should be determined using the measured data, while the modes are to be extracted. To avoid errors caused by random fluctuations, the mean values of the measured power and rotor angle are used in each simulation. Meanwhile, two calculation approaches are considered. The first approach, denoted by C-1, is to calculate the inertia based on the extracted modes and steady-state variables for each simulation so that the results contain 500 inertia estimation results. The second approach, denoted by C-2, is to calculate the inertia based on the means of the extracted modes and steady-state variables in 500 simulations to obtain only one inertia estimation result.
The inertia estimations obtained via C-1 are shown in

Fig. 3 Inertia estimation results obtained via C-1. (a) Inertia estimation results. (b) Counting results.
The inertia obtained via the C-2 approach is 2.855 s, which is close to the real value. The inertia estimations obtained via the C-1 and C-2 approaches are both close to the real value, which indicates the accuracy of the proposed method.
To verify the effectiveness of the proposed method further, a physical experiment is performed and the system parameters are given in Appendix A. The system configuration is shown in

Fig. 4 System configuration of physical experiment.

Fig. 5 PMU-measured electrical power signal.
As presented in
In this letter, an ambient modal based framework is proposed for power system inertia estimation using synchronized data. The results of a numerical test system simulation and a physical experiment show that the proposed method achieves high computation efficiency and exhibits robust performance under ambient excitation conditions. These advantages render the proposed method an appropriate and promising approach for effective inertia estimation in online rolling applications.
Appendix
The system parameters for the physical experiment are as follows: the rated capacity kVA, the rated voltage V, the line reactance p.u., the synchronous reactance p.u., the transient reactance p.u., the sub-transient reactance p.u., the inertia constant s, and the open circuit time constants s, s.
REFERENCES
Z. Wu, W. Gao, T. Gao et al., “State-of-the-art review on frequency response of wind power plants in power systems,” Journal of Modern Power Systems and Clean Energy, vol. 6, no. 1, pp. 1-16, Jan. 2018. [Baidu Scholar]
D. P. Chassin, Z. Huang, M. K. Donnelly et al., “Estimation of WECC system inertia using observed frequency transients,” IEEE Transactions on Power Systems, vol. 20, no. 2, pp. 1190-1192, May 2005. [Baidu Scholar]
G. Cai, B. Wang, D. Yang et al., “Inertia estimation based on observed electromechanical oscillation response for power systems,” IEEE Transactions on Power Systems, vol. 34, no. 6, pp. 4291-4299, Nov. 2019. [Baidu Scholar]
A. Gorbunov, A. Dymarsky, and J. Bialek, “Estimation of parameters of a dynamic generator model from modal PMU measurements,” IEEE Transactions on Power Systems, vol. 35, no. 1, pp. 53-62, Jan. 2020. [Baidu Scholar]
M. Paz and Y. H. Kim. Random Vibration: Theory and Computation. Berlin: Springer, 2019. [Baidu Scholar]
L. Dosiek, N. Zhou, J. W. Pierre et al., “Mode shape estimation algorithms under ambient conditions: a comparative review,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 779-787, May 2013. [Baidu Scholar]
P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1992. [Baidu Scholar]
J. H. Chow and K. W. Cheung, “A toolbox for power system dynamics and control engineering education and research,” IEEE Transactions on Power Systems, vol. 7, no. 4, pp. 1559-1564, Nov. 1992. [Baidu Scholar]