In recent years, subsynchronous resonance (SSR) has frequently occurred in DFIG-connected series-compensated systems. For the analysis and prevention, it is of great importance to achieve wide area monitoring of the incident. This paper presents a Hankel dynamic mode decomposition (DMD) method to identify SSR parameters using synchrophasor data. The basic idea is to employ the DMD technique to explore the subspace of Hankel matrices constructed by synchrophasors. It is analytically demonstrated that the subspace of these Hankel matrices is a combination of fundamental and SSR modes. Therefore, the SSR parameters can be calculated once the modal parameter is extracted. Compared with the existing method, the presented work has better dynamic performances as it requires much less data. Thus, it is more suitable for practical cases in which the SSR characteristics are time-varying. The effectiveness and superiority of the proposed method have been verified by both simulations and field data.

The fast growth and application of doubly-fed induction generators (DFIGs) in series compensated systems have significantly increased the occurrence of subsynchronous resonance (SSR) [

To date, two types of data have been considered for SSR parameter identification (SSRPI). One is the waveform data provided by the fault recorder. This type of data contains the complete information of the oscillation and thus can be easily utilized for SSRPI through various signal-processing algorithms, such as Prony [

Another option is to take advantage of the synchrophasors provided by the wide area monitoring system (WAMS). Currently, phasor measurement units (PMUs) have been widely deployed in transmission networks [

However, the DFT-based methods rely on a long data window to obtain better accuracy for estimating the SSR parameters and assume that these parameters are constant within the window. In practice, the SSR parameters are usually time-varying due to the stochastic nature of wind resources and the volatile operation conditions of the grid [

Within this context, this paper proposes a signal analysis technique based on dynamic mode decomposition (DMD). DMD seeks a linear dynamic operator to best approximate the underlying dynamics of the system. Its performance has been found to be satisfactory in a wide variety of applications, including fluid communities [

In this paper, we implemented the key parameter estimation of SSR by using the DMD method from the eigenvalues of Hankel matrices after the behavior of the synchrophasors under SSR is analyzed. The contributions of this paper include the following: (1) temporal synchrophasors with less data (less than 1 s) are used to construct two Hankel matrices, and the computational efficiency is improved. Note that only a single channel of measurement is required here. (2) The DMD method is performed on two Hankel matrices to estimate the parameters of SSR, and the number of dominant modes is automatically determined rather than predetermined; thus, the dynamic performance of SSR is captured well. (3) The proposed method is performed on simulation and field data, demonstrating the effectiveness of the proposed method.

The remainder of the paper is organized as follows.

This paper focuses on the SSR caused by the interaction between DFIGs and series-compensated systems. For such cases, all wind farms and the network are engaged in one SSR mode [

Commonly, synchrophasors are obtained by applying a discrete Fourier transform (DFT) on

Let

At the

bin (i.e.,

Assuming that

Generally, a series of

By defining

This section first presents a Hankel-DMD method in which two Hankel matrices are constructed to satisfy the requirement of applying DMD. Then, the equations to calculate the frequency, damping and amplitude of the SSR are analytically derived.

One premise to perform DMD is that the rank of the measurement matrix needs to be no less than the number of the dominant modes [

Actually,

With the derivation in the Appendix, the relationship of

where

To reduce the impact of noise, reduced SVD is performed to seek the low-dimensional representation of

By retaining the first

The matrix

Since

The modal parameters are obtained from the eigenvalues of

Once

Finally, the amplitude

The proposed Hankel-DMD method provides a good dynamic performance, as it uses a very short data window for SSRPI. Under noise-free conditions, the proposed method can perform well as long as the dimensions of the constructed

Another parameter that affects the performance of DMD is the selection of the number of dominant modes, i.e.,

The whole procedure of the Hankel-DMD method to identify three key parameters of the SSR component, i.e.,

Construct two Hankel matrices

Perform SVD of

Calculate

Perform eigen-decomposition of

Identify

Identify

This section evaluates the performance of the proposed method using both simulations and field data. Comparative studies with the InpDFT method [

A synthetic SSR current data was constructed as

The parameters of the SSR components, i.e.,

The proposed method applies a sliding window to identify the parameters of SSR, and each window contains

In another test, Gaussian noise was also added to the signal. According to our field data and those reported in the literature [

Furthermore, the mean errors of the three methods are displayed in

SNR (dB) | Time |
Proposed method | InpDFT | |||||||
---|---|---|---|---|---|---|---|---|---|---|

0^{*} |
[2,4) | 0.00 | 0.18 | 3.39 | 0.26 | 52.89 | 1.88 | |||

(4,6) | 0.00 | 0.00 | 2.53 | 0.01 | 13.35 | 2.11 | ||||

40 | [2,4) | 0.00 | 11.74 | 3.40 | 0.26 | 53.47 | 1.87 | |||

(4,6) | 0.00 | 2.30 | 2.63 | 0.01 | 31.94 | 1.75 |

Note: 0^{*} denotes noise-free case.

The proposed method was further tested by simulated SSR data. For this purpose, a series-compensated wind farm system was modeled in MATLAB/Simulink software, as shown in

System voltage | 220 kV |

Equivalent system reactance, |
19.98 |

Transmission line inductance, L | 0.3 H |

Transmission line resistance, R | 5.3 |

Series capacitance, C (20% compensation) | 110 |

Transformer voltages | 220 kV/690 V |

An SSR event is initiated at

After the proposed method was performed on each window with

This subsection investigates the performance of the proposed method using practical SSR incidents that occurred in North China. Two sets of field data at different periods are used.

The estimation results of the first case are shown in

The estimation results of the second case are shown in

This paper presented a Hankel-DMD-based method to identify SSR parameters using synchrophasor data. Through rigorous analytical derivation, it is revealed that SSRPI can be formulated as a DMD problem. By taking advantage of the Hankel matrix, which increases the modes of the subspace, the SSR parameters can be identified using a single channel of synchrophasor data within one second. Its performance has been verified using both simulation and field data. Comparative studies also demonstrate its superiority when compared with state-of-the-art algorithms. Therefore, it is expected that the proposed method can serve as an effective tool for wide area monitoring of SSR parameters.

frequency of subsynchronous component

damping of subsynchronous component

amplitude of subsynchronous component

frequency of fundamental component

amplitude of fundamental component

reporting frequency for PMU

synchrophasor provided in PMU

reported synchrophasor

This work was supported by the

The authors declare that they have no conflicts of interest to report regarding the present study.

Let

From

From

Let

Similar to

Thus, considering

Finally, by extending the relation of the vector in