Abstract
The occurrence of low-frequency electromechanical oscillations is a major problem in the effective operation of power systems. The scrutiny of these oscillations provides substantial information about power system stability and security. In this paper, a new method is introduced based on a combination of synchrosqueezed wavelet transform and the stochastic subspace identification (SSI) algorithm to investigate the low-frequency electromechanical oscillations of large-scale power systems. Then, the estimated modes of the power system are used for the design of the power system stabilizer and the flexible alternating current transmission system (FACTS) device. In this optimization problem, the control parameters are set using a hybrid approach composed of the Prony and residual methods and the modified fruit fly optimization algorithm. The proposed mode estimation method and the controller design are simulated in MATLAB using two test case systems, namely IEEE 2-area 4-generator and New England-New York 68-bus 16-generator systems. The simulation results demonstrate the high performance of the proposed method in estimation of local and inter-area modes, and indicate the improvements in oscillation damping and power system stability.
NOWADAYS, low-frequency electromechanical oscillations (LFEOs) with poor or negative damping have become a significant threat to power system stability. It is vital to analyze these oscillations to achieve the stability and security of power system [
There are two categories for the classification of typical data, including ambient and ring-down data. Ambient data are aggregated from a power system under stable conditions without any major disruption, and the perturbation results from load changes [
Several mode estimation methods have been reported. Block processing methods such as the Fourier transform [
A variety of methods have been introduced for controller design. Several control strategies have been proposed for the coordinated design of the power system stabilizer (PSS) and flexible alternating current transmission system (FACTS) device parameters through system models. These models include linear matrix inequality (LMI), H∞, optimal, adaptive, and robust controllers [
In this paper, the proposed SSWT-SSI method estimates the eigenvalues aided by the assembled data of the power system, which is a hybrid of SSWT and SSI. Then, the control parameters are designed using Prony and residual methods. The modified fruit fly optimization algorithm (MFOA) is used to improve the damping and dynamic performances and to meet the constraints of the PSS and FACTS parameters.
The remainder of this paper is organized as follows. Section II provides a brief description of the SSWT and SSI algorithms and some problems concerning the algorithm implementation, mode estimation, and transfer function calculation using the Prony method. In Section III, the proposed design method is used for the explanation of the control parameter tuning with the MFOA. Section IV presents the simulation results on 2-area 4-generator and 68-bus 16-generator IEEE test systems. Finally, conclusions are drawn in Section V.
The wavelet coefficients Ws(a,b) of the studied signal f(t) are acquired using the continuous wavelet transform (CWT):
(1) |
where a is the scale factor, which is inversely related to frequency; and b is the shift factor associated with time. In the small-scale factor, a wavelet indicates time density, which measures the details of the high-frequency signal. In the large-scale factor, on the contrary, a wavelet represents time expansion, which analyzes the approximations of the low-frequency signal. The complex conjugate of the mother wavelet is , which is selected as Morlet wavelet. can be obtained by using the partial derivative of according to instantaneous frequency b [
(2) |
In (2), a, b, and are discretized. is calculated at a discrete ak, and the reconstruction is determined via inverse transformation of :
(3) |
where is the synchrosqueezed transform, which is obtained at the center frequency ; ; and is the Fourier transform of [
The data reconstructed for each mode are placed into a block Hankel matrix based on the SSI method, and the SVD of the weighted projection is calculated. Finally, mode damping and frequency can be obtained from the eigenvalue analysis of the state matrix [
After the estimation of the modes, the Prony algorithm is used to calculate the residue values. If the power system is represented as a transfer function, the function can be approximated by:
(4) |
where is the root of the equation; bj is the equation coefficients of the zeros of the system; and M and N are the numbers of system zeros and poles, respectively. Finally, the inverse z-transform of G(z), , is given by [
(5) |
where Ri is the residue value.
After calculating the eigenvalues, the residue values, and the transfer function, and the controller parameters can be designed. After the controller is added, the transfer function is composed as , where is the transfer function of the controller. The denominator is set to be zero, and since and are very small, the eigenvalue changes can be written as follows:
(6) |
where is the transfer function of the controller; and is the residue pertaining to eigenvalue . Therefore, we can obtain:
(7) |
(8) |
where and are the phases of and , respectively; and is the imaginary part of the complex eigenvalue .
After proper selection of , the controller transfer function can be obtained, and the internal parameters can be calculated in accordance with the structures of the PSSs and unified power flow controller (UPFC) devices [
Because of some limitations in the structures of generators and their exciter systems in recent years, PSS and FACTS have been used simultaneously. These tools are the most important and economical controllers to improve the dynamic performance and small-signal stability of the power system, and they are essential to increase the damping of the local and inter-area modes of the power system. An important point of using FACTS devices is to find their desirable locations and select the best input signal, which is possible through examination of the controllability and observability of the estimated modes [

Fig. 1 Proposed algorithm for signal estimation and controller design.
The purpose of this paper is to improve mode damping by examining the search space of the control parameters. The method used in this research is the MFOA, which is a meta-heuristic algorithm based on the olfactory search process inherent in fruit flies’ foraging behavior. Fruit flies have much stronger senses of smell and vision than other insects, and this helps them find food sources [
In this paper, the PSSs are considered with two inputs. The PSS schematic model and the block diagram of UPFC dynamic model are shown in

Fig. 2 Structure of controllers. (a) Schematic model of PSS with two inputs. (b) UPFC schematic model.
The objective function is defined as follows. If the system is unstable, the function value will increase, and the value of the objective function will be reduced through the improvement of the system damping (the real parts of the eigenvalues should be negative).
(10) |
where is the
The inequality and equality constraints defined for the specification of the optimal parameters can be written using (7) and (8). As a result, the equality constraints are:
(11) |
(12) |
where , Tpss is the transfer function of the PSS, Tupfc is the transfer function of the UPFC, and u is the number of UPFC; p is the number of generators with PSSs; and are the eigenvalues and residual values calculated using the proposed method, respectively; and the eigenvalues are state variables. The inequality constraints are applied to the PSS parameters including T1-T8, Ks1-Ks3, and the parameters of UPFC including , Trv, Trp, and Trq. The importance of the proposed method (SSWT-SSI) concerns the precise estimation of the eigenvalues and residue values , and this method is very accurate compared with the other examined methods, as mentioned above.
In various studies on mode estimation, several types of disturbance have been investigated for the examination of small-signal stability. The favorable disruption would include different operation conditions such as different faults at various locations, e.g., three-phase faults, loss of line, and changes in the exciter reference voltage of generators and load amounts, and the noise behavior of the load. Two types of disturbance are applied to two systems below for investigating the effectiveness of the proposed method, and consequently, the estimation of modes. The first examined system is the 2-area 4-generator system.

Fig. 3 Single-line diagram and active power signal of 2-area 4-generator test system. (a) Single-line diagram. (b) Active power passing through transmission line 101-13.
It is assumed that all the four generators are equipped with PSSs [

Fig. 4 Reconstruction of IMT and time-frequency analysis. (a) Reconstruction of IMT. (b) Time-frequency analysis result.

Fig. 5 Reconstituted signal.
To verify the accuracy and validate the SSWT-SSI method, the estimated eigenvalues are compared with those calculated by the modal, CWT, and SSI methods through the calculation of the signal-to-noise ratio (SNR) in
The SSI method is highly dependent on the selected system order and nonlinear elements, and the CWT method, like the SSWT method, requires human participation when the mode is estimated with the image observation method. However, these problems are resolved aided by the SSWT-SSI method, and the estimation speed increases.
In this paper, we try to apply a disturbance to the power system to demonstrate the effectiveness of the proposed method in handling small-signal stability and controller design. It is noted that the SSWT-SSI method is flexible for studying both stationary and non-stationary signals. Furthermore, given that the Prony and SSI methods alone are not suitable for investigating non-stationary signals, we select some disturbances that exhibit non-stationary behavior. The mode estimation capability of the presented method can thus be observed. Therefore, the selected signal in the 2-area 4-generator test system is stationary, and the signal in the 68-bus 16-generator test system is non-stationary. Moreover, SSI, CWT, and SSWT methods have anti-noise capability.

Fig. 6 Frequency-time analysis of signal obtained by SSWT with noise. (a) Signal at noise level of 10 dB. (b) Time-frequency analysis at noise level of 15 dB with SSWT.
The above analysis demonstrates that SSWT has anti-noise capability, particularly for frequency performance against noise, to a certain degree. However, the other methods such as the Prony and SSI methods are not robust against noise, requiring de-noising, and this problem is resolved aided by the SSWT method.
After the high-accuracy estimation of the eigenvalues, residuals, and transfer function using the proposed method, (11) and (12) are solved by MFOA, and the calculated PSS parameters of generators 1 and 3 are presented in
In

Fig. 7 Comparison of active power of line 101-13 in presence of controllers designed by modal, CWT-Prony, and SSWT-SSI methods and in absence of PSSs.

Fig. 8 Comparison of rotor angle waveforms with controllers designed by different methods.
In this section, the performance of the SSWT-SSI method is studied on a 68-bus 16-generator test system. The single-line diagram of the system is presented in

Fig. 9 Single-line diagram of 68-bus 16-generator system and active power signal. (a) Single-line diagram of 68-bus 16-generator test system. (b) Active power through line connecting buses 17 and 27.
For creating a perturbation, a three-phase fault is applied to the line connected between buses 1 and 27, which is marked in red in

Fig. 10 Reconstruction of IMT and time-frequency spectrum. (a) Reconstruction of IMT. (b) Time-frequency spectrum analysis with SSWT-SSI.
The reconstructed signal is shown in

Fig. 11 Reconstituted signal and error waveform. (a) Reconstituted signal. (b) Signal reconstruction error.
The frequency and damping of the first to third modes and residue values are given in
Note:
To validate the accuracy of the proposed method, its estimated eigenvalues are compared with those obtained by the modal method. The comparison shows that the SSWT-SSI method is more accurate. After the transfer function is identified, the controller parameters are specified using the method of this paper.
The controller parameters are given in
Note: , , , , , and Kiv are the gains of UPFC.

Fig. 12 Comparison of generator speed deviations obtained through application of controller design using different methods. (a) Modal method. (b) CWT-Prony method. (c) SSWT-SSI method.

Fig. 13 Comparison of eigenvalues of 68-bus 16-generator test system before and after application of controllers. (a) Comparison of active power waveforms in two cases with and without controllers. (b) Eigenvalues of system with controllers. (c) Eigenvalues of system without controllers.
The modes are calculated using the SSWT-SSI, CWT-Prony, and modal methods when the controllers are applied to the 68-bus 16-generator test system. The results are shown in
Note:

Fig. 14 Comparison of rotor angle of G1 in four cases without controllers and with PSSs and UPFC as designed by three methods.
In both two disturbances to study the 68-bus 16-generator test system, the active power flow on the line connected between buses 17 and 27 is considered as a window of measured data for analysis. Then, the time-frequency distribution analysis plots by SSWT have been obtained. The frequency and damping of the modes in both two cases are given in
Note:
After the calculation of the eigenvalues and residue values, the control parameters are determined.

Fig. 15 Comparison of active power waveforms of line 17-27 with two different faults. (a) Comparison of active power in two cases with and without controllers when load at bus 1 is lost. (b) Comparison of active power waveforms in two cases with and without controllers when applying 3-phase fault to line 1-2.
It should be noted that this method is suitable for investigating both types of data, including ambient and ringdown data, although the ringdown data measured after a major disturbance are used to study the small-signal stability as presented in this paper.
In this paper, a new method of time-frequency analysis is presented to estimate low-frequency oscillation modes, residue values, and transfer function of power system. The proposed method is a combination of the SSWT, SSI, and Prony methods.
The SSWT method overcomes the problems of energy deviation and mode mixing in WT. Moreover, the method improves the accuracy of identification of the examined oscillations with better anti-noise performance. The combination of SSWT and SSI presents an automatic identification algorithm, thus, the mode prediction occurs before signal reconstruction, and parameter identification becomes effective. The use of the Prony method provides a better estimation of the signal transfer function. For setting the PSSs and UPFC parameters, the equations are obtained from the residual method and solved using MFOA. This optimization algorithm is used to increase the design accuracy, and ultimately, generates parameters that are consistent with the practical limits.
The simulations of two well-known systems, namely the 2-area 4-generator and the 68-bus 16-generator test systems, verify the high performance and efficiency of the SSWT-SSI method. The results also demonstrate a considerable increase in the damping of LFEOs. For comparing the capability of the proposed algorithm and the other methods, the advantages of the hybrid algorithm are briefly presented as follows.
1) A window of data is used as measured signal.
2) The proposed method is robust against noise.
3) The speed and accuracy of calculation for estimation of modes is increased by the proposed automatic method and MFOA.
4) The optimal controller parameters is calculated given the practical ranges.
5) The damping and stability are increased in the power system.
REFERENCES
A. R. Messina, “Advanced data processing and feature extraction,” in Wide-area Monitoring of Interconnected Power Systems, Stevenage: The IET Press, 2015, pp. 63-96. [Baidu Scholar]
J. Turunen, J. Thambirajah, M. Larsson et al., “Comparsion of three electromechanical oscillation damping estimation methods,” IEEE Transactions on Power Systems, vol. 26, no. 4, pp. 2398-2407, Nov. 2011. [Baidu Scholar]
S. You, J. Guo, G. Kou et al., “Oscillation mode identification based on wide-area ambient measurements using multivariate empirical mode decomposition,” Electric Power Systems Research, vol. 134, pp. 158-166, May 2016. [Baidu Scholar]
A. Chakraborttyl and P. P. Khargonekar, “Introduction to wide-area control of power systems,” in Proceedings of American Control Conference, Washington DC, USA, Jun. 2013, pp. 1-13. [Baidu Scholar]
A. Chakrabortty, J. H. Chow, and A. Salazar, “A measurement-based framework for dynamic equivalencing of large power systems using wide-area phasor measurements,” IEEE Transactions on Smart Grid, vol. 2, no. 1, pp. 68-81, Mar. 2011. [Baidu Scholar]
M. Weiss, A. Chakrabortty, and F. Habibi Ashraf, “A wide-area SVC controller design using a dynamic equivalent model of WECC,” in Proceedings of IEEE PES General Meeting, Denver, USA, Jul. 2015, pp. 1-5. [Baidu Scholar]
M. U. Usman and M. O. Faruque, “Applications of synchrophasor technologies in power systems,” Journal of Modern Power Systems and Clean Energy, vol. 7, no. 2, pp. 211-226, Mar. 2019. [Baidu Scholar]
C. Chen, Z. An, X. Dai et al., “Measurement-based solution for low frequency oscillation analysis,” Journal of Modern Power Systems and Clean Energy, vol. 4, no. 3, pp. 406-413, May 2016. [Baidu Scholar]
A. Zamora, V. M. Venkatasubramanian, J. A. Serna et al., “Multi-dimensional ringdown modal analysis by filtering,” Electric Power Systems Research, vol. 143, pp. 748-759, Feb. 2017. [Baidu Scholar]
C. Huang, F. Li, D. Zhou et al., “Data quality issues for synchrophasor applications part I: a review,” Journal of Modern Power Systems and Clean Energy, vol. 4, no. 3, pp. 342-352, Jul. 2016. [Baidu Scholar]
H. Wen, J. Zhang, W. Yao et al., “FFT-based amplitude estimation of power distribution systems signal distorted by harmonics and noise,” IEEE Transactions on Power Systems, vol. 14, no. 4, pp. 1447-1455, Sept. 2018. [Baidu Scholar]
G. Liu, J. Quintero, and V. M. Venkatasubramanian, “Oscillation monitoring system based on wide area synchrophasors,” in Proceedings of Power Systems iREP Symposium–Bulk Power System Dynamics and Control-VII: Revitalizing Operational Reliability, Charleston, USA, Aug. 2007, pp. 1-13. [Baidu Scholar]
K. H. Jin and L. Yilu, “Identification of interarea modes from an effectual impulse response of ringdown frequency data,” Electric Power Systems Research, vol. 144, pp. 96-106, Mar. 2017. [Baidu Scholar]
S. A. N. Sarmadi and V. Venkatasubramanian, “Electromechanical mode estimation using recursive adaptive stochastic subspace identification,” IEEE Transactions on Power Systems, vol. 29, no. 1, pp. 349-358, Sept. 2014. [Baidu Scholar]
M. Yazdanian, A. Mehrizi-sani, and M. Mojiri, “Estimation of electromechanical oscillation parameters using an extended Kalman filter,” IEEE Transactions on Power Systems, vol. 30, no. 6, pp. 2994-3002, Jan. 2015. [Baidu Scholar]
M. Beza and M. Bongiorno, “A modified algorithm for online estimation of low-frequency oscillations in power system,” IEEE Transactions on Power Systems, vol. 31, no, pp. 1703-1714, Jun. 2015. [Baidu Scholar]
L. Cheng, X. Ji, F. Zhang et al., “Wavelet-based data compression for wide-area measurement data of oscillations,” Journal of Modern Power Systems and Clean Energy, vol. 6, no. 6, pp. 1128-1140, Jul. 2018. [Baidu Scholar]
L. Dosiek and J. W. Pierri, “Estimating electromechanical modes and mode shapes using the multichannel ARMAX model,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 1950-1959, Apr. 2013. [Baidu Scholar]
H. Khalid and J. Peng, “Tracking electromechanical oscillations: an enhanced maximum-likelihood based approach,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 1799-1808, May 2015. [Baidu Scholar]
Y. Li, D. Yang, F. Liu et al., “Coordinated design of local PSSs and wide-area damping controller,” in Interconnected Power Systems Wide-area Dynamic Monitoring and Control Applications, Verlag Berlin Heidelberg: Springer, 2015, pp. 103-117. [Baidu Scholar]
I. Daubechies, J. F. Lu, and H. T. Wu, “Synchrosqueezed wavelet transforms: an empirical mode decompositionlike tool,” Applied and Computational Harmonic Analysis, vol. 30, no. 2, pp. 243-261, Mar. 2011. [Baidu Scholar]
M. H. Rafiei and H. Adeli, “A novel unsupervised deep learning model for global and local health condition assessment of structures,” Engineering Structures, vol. 156, pp. 598-607, Feb. 2018. [Baidu Scholar]
M. Yu, B. Wang, X. Chen et al., “Application of synchrosqueezed wavelet transform for extraction of the oscillatory parameters of low frequency oscillation in power systems,” Transactions of China Electrotechnical Society, vol. 32, no. 6, pp. 14-20, Mar. 2017. [Baidu Scholar]
K. Tang, and G. K. Venayagamoorthy, “Damping inter-area oscillations using virtual generator based power system stabilizer,” Electric Power Systems Research, vol. 129, pp. 126-141, Dec. 2015. [Baidu Scholar]
A. Abiee, “An advanced state estimation method using virtual meters,” Journal of Operation and Automation in Power Engineering, vol. 5, no. 1, pp. 11-17, Sept. 2017. [Baidu Scholar]
A. I. Konara, and U. D. Annakkage, “Robust power system stabilizer design using eigenstructure assignment,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 1845-1853, May 2016. [Baidu Scholar]
H. Hasanvand, M. R. Arvan, B. Mozafari et al., “Coordinated design of PSS and TCSC to mitigate interarea oscillations,” International Journal of Electrical Power & Energy Systems, vol. 78, pp. 194-206, Jun. 2016. [Baidu Scholar]
A. Faraji and A. H. Naghshbandy, “A combined approach for power system stabilizer design using continuous wavelet transform and SQP,” International Transactions on Electrical Energy Systems, vol. 29, no. 3, pp. 1-18, Mar. 2019. [Baidu Scholar]
J. Quintero and V. Venkatasubramanian, “SVC compensation on a real-time wide-area control for mitigating small-signal instability in large electric power systems,” in Proceedings of International Conference on Power System Technology, Chongqing, China, Oct. 2006, pp. 1-8. [Baidu Scholar]
B. K. Kumar, S. N. Singh, and S. C. Srivastava, “Placement of FACTS controllers using modal controllability indices to damp out power system oscillations,” IET Generation, Transmission & Distribution, vol. 1, no. 2, pp. 209-217, May 2007. [Baidu Scholar]
A. Asgari and K. G. Firouzjah, “Optimal PMU placement for power system observability considering network expansion and [Baidu Scholar]
contingencies,” IET Generation, Transmission & Distribution, vol. 12, no. 18, pp. 4216-4224, Oct. 2018. [Baidu Scholar]
S. Hajforoosh, S. M. H. Nabavi, and M. A. S. Masoum, “Coordinated aggregated-based particle swarm optimisation algorithm for congestion management in restructured power market by placement and sizing of unified power flow controller,” IET Science, Measurement & Technology, vol. 6, no. 4, pp. 1447-1455, Aug. 2012. [Baidu Scholar]
K. Zhang, Z. Shi, Y. Huang et al., “SVC damping controller design based on novel modified fruit fly optimisation algorithm,” IET Renewable Power Generation, vol. 12, no. 1, pp. 90-97, Jan. 2018. [Baidu Scholar]
Y. Yu, Z. Mi, X. Zheng et al., “Accommodation of curtailed wind power by electric water heaters based on a new hybrid prediction approach,” Journal of Modern Power Systems and Clean Energy, vol. 7, no. 3, pp. 525-537, May 2019. [Baidu Scholar]
Ayda Faraji received the B.S. and M.S. degrees from the University of Kurdistan, Sanandaj, Iran, in 2014 and 2017, respectively, in electrical engineering. She is currently a Ph.D. student at the University of Kurdistan, Sanandaj, Iran. Her main research interests include the dynamic and stability of power systems. [Baidu Scholar]