# Identification of Mode Shapes Based on Ambient Signals and the IA-VMD Method
## 1 Introduction  
In recent years, with the development of electric power technology and the scale of power grids, power exchanges across regions and over long distances have become more common. However, this outcome brings many new security problems to power grids, among which the low-frequency oscillation (LFO) of the power system is a very important problem [1, 2]. If the power grid oscillation is not detected and suppressed in time, it may have a serious impact on the power system. At present, the analysis methods for LFOs of power systems mainly include offline analysis methods based on mathematical models of power systems [3, 4] and signal analysis methods based on measured signals. The former method has problems such as the uncertainty of the operation mode of the power grid and the curse of dimensionality in the calculation process; these problems make it difficult to apply this method to the online monitoring of LFOs. Compared with the former method, the latter method can better adapt to the operation mode of the power grid. The acquisition of measured signals is not restricted by the size of the power system, which can reduce the difficulty of oscillation analyses. Additionally, by identifying and analyzing the system output signal, the system characteristics contained in the signal can be obtained. In recent years, a wide area measurement system (WAMS) has been widely applied to power systems, which provides the foundation for signal analysis methods. Therefore, the signal analysis method, based on measured signals, has become the main analysis method of power system LFOs [5]. There are many small fluctuation signals, called ambient signals (similar to environmental noise), in the normal operation of the power grid [6]. Ambient signals are generallycaused by normal electrical operation during the operation of a power grid, such as load switching and line parameter adjustment. These signals widely exist in measurement signals, contain rich characteristic information of system operation, and can be used for early warning of oscillation. However, ambient signals are easily obscured by environmental noise, so how to extract the characteristic information of system operation from these signals is a key problem [7]. At present, there are many methods used to analyze LFO, such as discrete Fourier transform (DFT) [8], empirical mode decomposition (EMD) [9], wavelet-based methods [10], Hilbert-Huang transform (HHT) [11], estimation of signal parameters via rotational invariance technique (ESPRIT) [12; 13; 14], matrix pencil [15] and the Prony algorithm [16]. In [13], a method combining ESPRIT with the exact model order (EMO) was proposed to analyze LFO. In addition, in [15], the basis pursuit denoising method, combined with a tuneable Q-factor wavelet transform, was proposed to increase the signal-to-noise ratio, and then an improved version of the matrix pencil algorithm was used to identify the parameters of LFO. The Prony algorithm, the most widely used method in many oscillation mode parameter extraction algorithms, can fit the actual LFO signals by a linear combination of multiorder exponential functions. The oscillation characteristic parameters of the oscillation signal can be obtained directly by using the Prony algorithm. However, the Prony algorithm is sensitive to noise, so it cannot identify the ambient noise signals of the power grid directly [17; 18]. Therefore, it is necessary to perform modal decomposition on the measured signals so as to obtain modal parameters by the Prony algorithm in the future. At present, there are multiple modal decomposition methods, such as the Kalman filter, wavelet transform, EMD, eigensystem realization algorithm (ERA), and variational mode decomposition (VMD). By combining the above methods with the Prony algorithm, the interference in the original signal can be removed and the identification accuracy of the Prony algorithm can be greatly improved [19; 20; 21].