Data gaps and degraded space-time resolution for modal decomposition: A compensator approach

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Highlights

  • Compensation of data gaps and degraded resolution of wide-area measurements.

  • The multi-component modeling of virtually-designed data records.

  • The derivation of the empirical coherence to the area classification.

Abstract

High-resolution empirical mapping methods of dynamic oscillation responses for large power systems have recently been considered to treat the data gaps and its space-time compensation. Although virtually-designed data of signals are not a real condition for monitoring systems, these can be implemented to study objectively the moving features and dominant paths of inter-area oscillation modes. This paper demonstrates the application of the space-time interpolation of wide-area measurements to compensate data gaps and degraded space-time resolution for modal decomposition. Contributions of empirical mode coherence for splitting and clustering data in density factors are also presented to the classification area. The method involves the space-time high-resolution interpolation of empirical mode components for splitting closely-related oscillation modes at intrinsic timescales, preserving their multivariate waveform features. Therefore, high-resolution empirical viewing maps of remotely-sensed dynamic responses, with degraded space-time resolutions, are derived to study the observance and dominant paths of wide-area oscillation modes. The proposed approach is numerically applied to the IEEE 16-generator 68-bus test system for efficiently defining empirical viewing maps of inter-area oscillation modes.

Introduction

In sensor-based monitoring networks for large power systems, it is often unattainable to collect observations in time or space with sufficient regularity of sampling rate. This drawback is mainly caused by the relatively high cost of synchronized phasor measurement technologies or due to geo-structural and technical aspects during its design. Nevertheless, it is possible to identify and prioritize both major areas for which data records are currently needed to define and classify modal oscillation sources, and general approaches by which datasets in high-resolution arrays are generated. It is largely known that most analysis methods by data-based modal decomposition, e.g. the proposed in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] require measurements that are regularly sampled in time and space. Underlying issues associated with numerical conditioning [7], [11], [13], spatial resolution and orientation [2], [7], [8], [9], collection and sampling rate [4], [12], missing and scaling data [1], [3], [13], coherence and clustering data [1], [2], [5], [8], [9], data fusion [6], [10], have been reported. Efforts are usually made for recording measurements as evenly as possible, but for a number of reasons, station spacing and irregular time steps during the data collection process are often inherently problematic. This leads almost invariably to unwanted gaps in the dataset. Furthermore, the difficulty induced by the data gaps or holes is still not completely solved in studies of empirical mode observance and multi-path wave propagation which are being treated in this paper. An option to overcome this drawback is generally necessary to use an interpolation procedure to create numerical approximations of the required regular data series as part of the signal processing. This conditioning data process allows treating the modal observance, sparsity, and dispersion as a problem of closely-related oscillation modes and neighborhoods in empirical decomposition algorithms. In [14], a self-organizing neural network for nonlinear mapping of datasets by curvilinear component analysis (CCA) is developed. The mainly purpose of CCA is to give a revealing representation of data in low dimension, preparing a basis for further clustering and classification. A novel representation of datasets in 3-D mapping is given in [15]. The method based on k-means clustering is used to obtain the cluster centers of high dimensional datasets, and get the correspondence coordinates in 3-D space, with the projection along the center direction. In [16], an algorithm for the analysis of multivariate data is presented, where the inherent data structure is approximately preserved under the dimensional mapping of vectors. Several mapping techniques for exploratory pattern analysis are described in [17], where it is compared to various mappings and classical clustering techniques, whose objective is to find clusters in the data. Distribution maps of frequency propagation speeds in large-scale power networks using a bicubic interpolation method are presented in [7]. This approach is structured from model-based measurements in order to derive distributed viewing maps of frequency dynamics on a two-dimensional regular grid. However, new contributions to treat the data gaps and degraded space-time resolutions in studies of empirical observance of modal oscillation sources using wide-area measurements may also be introduced. The space-time interpolation methods of data records designed through high-resolution numerical models for decomposition in empirical mode components, provide a practical manner to study and split closely-related oscillation modes in distributed dynamics of large power networks. Arrays of datasets with irregular resolution in its space-time structure can also be integrated to develop a new empirical analysis algorithm with multivariate space-time attributes. Thus, diverse datasets collected with irregular sampling rates can be integrated in modal oscillation analysis connecting different sources of devices and technologies to the data acquisition and sampling. Therefore, in this paper, an alternative framework to the conventional interpolation methods of wide-area measurements, as presented in [18], [19], [20], [21], exploiting the properties of space-time modal decomposition by empirical orthogonal functions (EOFs) analysis and intrinsic mode functions (IMFs) is introduced and discussed in detail. Our proposal involves the space-time high-resolution interpolation of empirical components to split closely-spaced oscillation modes for IMFs, preserving their multivariate waveform features objectively. The method overcomes the drawbacks of modal independence and spatial geometry in virtually-designed data using the intrinsic linear combination of timescales as empirical functions [22]. The paper demonstrates the application of the space-time interpolation of modal components to compensate data gaps and degraded space-time resolution of wide-area measurements. The proposed method has two major contributions, firstly, discussing data gaps, and secondly, proposing approaches to filling the gaps. The presented technique yields a numerical estimation of the data that lie in the gaps and permits to fill them in an empirical manner by virtually-designed records, preserving the multivariate waveform features of sensed signals. Therefore, high-resolution empirical viewing maps to inter-area oscillation modes are structured as new arrays of databases of equable space-time dimension preserving their modal moving features objectively. Numerical results show that the obtained spectra and eigenfunctions are closely-related to those obtained from remotely-sensed datasets. The proposed approach is numerically applied to the IEEE 16-generator 68-bus test system for efficiently defining empirical viewing maps of inter-area oscillation modes.

The main contributions of this research are:

  • The application of the space-time interpolation of empirical mode components to compensate data gaps and degraded resolution of wide-area measurements.

  • The multi-component modeling of virtually-designed data records with regular sampling intervals to define datasets of equable space-time dimension.

  • The derivation of the empirical mode coherence to the area classification for splitting and clustering data in density factors, employing sensor-based monitoring networks.

The paper is organized as follows. An analytical framework to the estimations of empirical orthogonal functions from spatially-sensed signals is presented in Section 2. Section 3 reports the proposed approach to derive the high-resolution empirical viewing maps and empirical coherence for splitting and clustering data using the empirical mode decomposition and density factors. Test cases and the main results are discussed in Section 4. Conclusions are given in Section 5. Finally, detailed descriptions on analytic tools used in the proposed approach are relegated to an appendix.

Section snippets

EOFs analysis: overview for wave variance

Let us consider the study of a non-dispersive plane wave expressed by [9]:uj(t)=ω[α(ω)cos(kxjωt)+β(ω)sin(kxjωt)]which is propagated at a phase speed c with wavenumber k=ω/c, crossing an array of sensors spatially distributed at positions xj, with arbitrary angular frequency ω in time t. Then, (1) can be rewritten as:uj(t)=ω[aj(ω)cos(ωt)+bj(ω)sin(ωt)]withaj(ω)=α(ω)cos(kxj)+β(ω)sin(kxj)bj(ω)=α(ω)cos(kxj)β(ω)sin(kxj).For this analysis, it is assumed that uj(t) is a white and band-limited

High-resolution empirical components

Let us assume that X(x,t)Rm×n, with nm, is a two-dimensional data array structured through a sensor-based monitoring network. The spatial location is denoted by variable xj, with j=1,2,...,n whereas th, to h=1,2,...,m is the instants of time. Hence, using the complex EOFs analysis, the modal decomposition of X(x,t) is obtained as a linear combination of L proper orthogonal modes with form:X(x,t)real{q=1Laq(t)φqH(x)},(Lm).Following the theoretical development of (4)–(10), the optimal

Two-area four-machine system

The two-area four-machine system is shown in Fig. 2, consisting of two identical areas, interconnected by two long transmission lines with network parameters given in [36]. At the base-case condition, each generator is represented by the classical model. This modeling includes only two state variables per generator, rotor angle, and rotor speed deviation since the inter-area modes are mainly associated with these two state variables. The loads are considered as constant impedance and the

Conclusion

This paper demonstrates the application of the space-time interpolation of wide-area measurements to compensate data gaps and degraded space-time resolution for the linear combination of band-limited components in intrinsic timescales. The proposed approach is derived for using the intrinsic mode functions (IMFs) decomposition of modal components estimated for empirical orthogonal functions (EOFs) analysis of datasets. Furthermore, the method employing the linear interpolation of magnitude and

CRediT authorship contribution statement

P. Esquivel: Conceptualization, Investigation, Methodology. E.N. Reyes: Methodology, Validation. F. Ornelas-Tellez: Visualization, Investigation. L. Garza: Supervision. Carlos E. Castañeda: Writing - original draft, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This paper was supported in part by the National Council of Science and Technology (CONACYT-México) under Cátedras program, and by Academic and Multidisciplinary Unit Reynosa-RODHE of the Autonomous University of Tamaulipas México, under project no. 1429.

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