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# Application of Hankel Dynamic Mode Decomposition for Wide Area Monitoring of Subsynchronous Resonance

1 State Grid Hebei Electric Power Research Institute, Shijiazhuang, 050021, China

2 State Grid Hebei Electric Power Company, Shijiazhuang, 050021, China

3 The College of Electrical Engineering, Sichuan University, Chengdu, 610065, China

* Corresponding Author: Xiaomei Yang. Email:

*Energy Engineering* **2023**, *120*(4), 851-867. https://doi.org/10.32604/ee.2023.025383

**Received** 08 July 2022; **Accepted** 09 October 2022; **Issue published** 13 February 2023

## Abstract

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.## Keywords

Nomenclature

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 |

The fast growth and application of doubly-fed induction generators (DFIGs) in series compensated systems have significantly increased the occurrence of subsynchronous resonance (SSR) [1]. Recent SSR events in south Texas of USA [2] and North China [3,4] indicated that the oscillation could be system-wide which involves complex interactions among grid components. It is thus of great importance to provide a wide area monitoring of SSR parameters, including the magnitude, frequency and damping. The data are crucial for replicating SSR events, identifying the sources of SSR [5] and supporting the design of countermeasures, e.g., feedback-linearized sliding mode controller [6], energy-shaping L2-gain controller [7], damping controller [8].

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 [9] and the recursive least square (RLS) method [10]. Additionally, model decomposition-based techniques, such as the Hilbert-Huang transform [11] and variational mode decomposition (VMD) [12], can perform parameter identification decomposition after the time signals are decomposed into multifrequency mode components. Unfortunately, the fault recorder is locally stored, and whether it records the SSR data depends on the triggering mechanism. This deficiency imposes great challenges for wide area monitoring or system-wide analysis [13].

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 [14], making the WAMS a promising platform for SSR monitoring. The main challenge here is that the synchrophasor only captures the fundamental phasors. As a result, the SSR components will appear as the spectral leakage components. Studies have been conducted to address this issue. The work in [15] demonstrated that it is possible to identify the SSR frequency from synchrophasors. Recently, some modal parameter extraction methods, e.g., classic Prony analysis, estimation of signal parameters via the rotational invariance technique (ESPRIT) and the matrix pencil method [16], have been developed to extract the SSO parameters. In addition, the recent study in [17] further proposed a DFT-based correction method to recover the SSR amplitude from spectral leakage components. An interpolated DFT (InpDFT)-based method was also proposed in [13] to achieve better identification accuracy through the consideration of damping parameters.

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 [3]. When a short window is used in DFT-based methods for analyzing nonstationary signals, the spectrum of the SSR component in the synchrophasors will be significantly affected by spectral leakage from the fundamental frequency phasors. In this way, large estimation errors are unavoidable.

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 [18], biomedical fields [19] and power system areas. As an example, the work in [20] successfully applied DMD for spatiotemporal PMU data to monitor low-frequency oscillations.

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. Section 2 analyzes the behavior of synchrophasors under SSR and defines the DMD problem to be solved. The Hankel-DMD method is explained in Section 3 for the identification of SSR components. Section 4 verifies the effectiveness of the proposed method by using simulation data and field data under dynamic and noisy conditions. A comparative study is also conducted to show the superiority of the proposed method.

2 Synchrophasor Model under SSR

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 [3,4,21]. As a result, the current waveform data in the time domain under SSR can be expressed as

where

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

where

Let

with

where

At the

bin (i.e.,

Assuming that

Generally, a series of

with

where

By defining

and

we rewrite (7) as

which denotes that

where

Eq. (12) indicates that the synchrophasors under SSR are a linear combination of four phasors rotating at different frequencies. The index

3 Synchrophasor Model under SSR

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.

3.1 Hankel-Dynamic Mode Decomposition

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 [22,23]. In the case that the measurement matrix is a series of temporal synchrophasors, its rank would be one, which is insufficient for SSRPI [22]. To address this issue, the concept of Hankel matrices is used here. For the

and

where

Actually,

With the derivation in the Appendix, the relationship of

where

where

and

where

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

where

By retaining the first

where

The matrix

Since

where

where

3.2 Calculation of SSR Parameters

The modal parameters are obtained from the eigenvalues of

where

where

Once

with

where

where

Finally, the amplitude

where

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.,

where

where

3.4 The Procedure of SSR Parameter Estimation

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 [13] and classical Prony method are also presented.

A synthetic SSR current data was constructed as (31), where an off-nominal frequency

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

Fig. 2a shows the waveform of

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 [24], the noises in practical PMU data are generally around a signal-to-noise ratio SNR = 45 dB. Thus, the paper considers noise with SNR = 40 dB. The results obtained from the proposed and two comparative methods are shown in Figs. 3b–3d. Since Prony is sensitive to noise, the results of Prony become worse, showing a large deviation from the ground truth under noisy conditions. The proposed method also coincides better with the true values than the InpDFT and Prony methods. Comparatively, the estimation of

Furthermore, the mean errors of the three methods are displayed in Table 1 for two model periods, i.e., [2,4) s and (4,6] s, as shown in Fig. 2a. The mean errors

where

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 Fig. 4a. A sixth-order model of the induction machine is used, with a two-mass drive train model to represent the generator shaft. Figs. 4b and 4c show the control strategies of the grid-side converter (GSC) and the rotor-side converter (RSC), respectively. Table 2 provides the key system parameters, and further details of the simulation model can be found in [25].

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. Figs. 6a and 7a show the waveform data provided by the fault recorder with a sampling rate of 1000 Hz, while the magnitude of the corresponding synchrophasors is shown in Figs. 6b and 7b. It can be seen that the SSR mode varies over time. The possible reasons could be the tripping of wind generators and the change in the wind speed.

The estimation results of the first case are shown in Figs. 6c–6e. Similarly, the InpDFT and Prony methods are considered for comparison, and waveform-based analysis is used as a reference. According to the results, the estimation of the proposed method matches well with the reference value, while the damping results estimated by the two comparative methods deviate from those of the proposed method and waveform analysis.

The estimation results of the second case are shown in Figs. 7c–7e. Different from the first case, the off-nominal condition in the second period is not severe. As a result, the curves of the estimated parameters from the proposed and InpDFT methods match well, whereas the Prony method still cannot achieve satisfactory estimation and cannot effectively capture the variation in the damping and amplitude. The estimated

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.

Funding Statement: This work was supported by the China Key Technology Research on Risk Perception of Sub-Synchronous Oscillation of Grid with Large-Scale New Energy Access SGTYHT/21-JS-223.

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

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Appendix A. Proof of the linear combination

Let

From (12), we obtain

with

and

From (35), we can solve

where

Let

Similar to (35),

where

with

by using (36).

Thus, considering (37) and (40), we can rewrite (39) as

where a linear map

Finally, by extending the relation of the vector in (42) to the matrix, the connection of

where

## Cite This Article

**APA Style**

*Energy Engineering*,

*120*

*(4)*, 851-867. https://doi.org/10.32604/ee.2023.025383

**Vancouver Style**

**IEEE Style**

*Energ. Eng.*, vol. 120, no. 4, pp. 851-867. 2023. https://doi.org/10.32604/ee.2023.025383

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