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Low‐loss image‐based compression for synchrophasor measurements

2020; Institution of Engineering and Technology; Volume: 14; Issue: 20 Linguagem: Inglês

10.1049/iet-gtd.2020.0551

1751-8695

Sowmya Acharya, Christopher L. DeMarco,

Advanced Electrical Measurement Techniques

IET Generation, Transmission & DistributionVolume 14, Issue 20 p. 4571-4579 Research ArticleFree Access Low-loss image-based compression for synchrophasor measurements Sowmya Acharya, Corresponding Author Sowmya Acharya sacharya2@wisc.edu orcid.org/0000-0003-2283-1318 Department of Electrical and Computer Engineering, University of Wisconsin, Madison, 53706 USASearch for more papers by this authorChristopher L. DeMarco, Christopher L. DeMarco Department of Electrical and Computer Engineering, University of Wisconsin, Madison, 53706 USASearch for more papers by this author Sowmya Acharya, Corresponding Author Sowmya Acharya sacharya2@wisc.edu orcid.org/0000-0003-2283-1318 Department of Electrical and Computer Engineering, University of Wisconsin, Madison, 53706 USASearch for more papers by this authorChristopher L. DeMarco, Christopher L. DeMarco Department of Electrical and Computer Engineering, University of Wisconsin, Madison, 53706 USASearch for more papers by this author First published: 04 September 2020 https://doi.org/10.1049/iet-gtd.2020.0551AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Deployments of high-sampling rate synchronised phasor measurement units (PMUs) are growing rapidly throughout the world, and with the advent of microPMUs, spreading from bulk transmission through distribution systems. The growing volume of PMU data presents challenges in its communication and storage, motivating consideration of compression algorithms. This study presents a novel lossy compression algorithm that exploits particular characteristics of power system measurements to improve the compression. Concepts successfully applied in image compression are tailored to the spatio-temporal correlations induced between electrical quantities via their network interconnections. The quality of the resulting compression is judged on the balance of storage space savings versus the accuracy of data reconstruction. In representative real-world and transient simulation datasets, the technique developed can provide storage compression in the range of 40:1 when different physical quantities are compressed together. The compression ratios can be in the range of 90:1 for voltage magnitudes and 190:1 for frequency when the measurements are compressed separately. The high-compression ratios are achieved while maintaining low-loss (high-accuracy) reconstruction. 1 Introduction Phasor measurement unit (PMU) measurements have become an integral part of power system operations. Data from PMUs are employed for state estimation, security assessment, wide-area control, and a host of other applications in protection and automation. More than 2500 PMUs are deployed across North America today. The raw data generated and transmitted by these devices is obviously voluminous and growing. North American Electric Reliability Corporation mandates measurement data be retained for ten calendar days in its raw format, and to be available for up to three years [1]. These requirements make clear the value of methods to efficiently communicate and archive large volumes of PMU data. A range of lossless (guaranteed exact reconstruction) and lossy (reconstruction error allowed) compression algorithms have been proposed for PMU data. Most lossless methods treat the input data as generic files and apply variants of existing data compression algorithms. For example, Olivo et al. [2] evaluated the use of off-the-shelf algorithms such as 7-zip, rar, and zip. Similarly, Top and Breneman [3] evaluated the effectiveness of gzip, 7-zip, bzip2, and szip for compressing PMU measurements. An approach termed as slack reference encoding (SRE) is proposed in [4] as a pre-processing step before employing file compression algorithms. SRE represents the first effort to exploit correlations in PMU measurement data to improve compression performance. The algorithm is applied only to voltage angle measurements, with reported compression ratios (CRs) ranging up to 14:1. Lossy compression algorithms reported in the literature typically use some variant of singular value decomposition (SVD) [5-7]. The work in [8] clusters correlated PMU measurements before compressing with a method based on principal component analysis (PCA). A method that combines PCA and discrete cosine transform (DCT) for real-time compression of PMU data is proposed in [9]. Although these approaches offer high CRs, the associated reconstruction errors and computational costs are relatively high. Wavelet-based approaches are typically used to compress disturbance data [10-12]. The survey in [13] discusses various approaches for compressing the disturbance data from smart grids. Ning et al. [14] proposed an algorithm based on discrete wavelet transform (DWT) with an average CR of 5.4:1. The multi-scale PCA compression algorithm in [15] combines DWT and PCA to achieve a 19.02:1 CR for voltage magnitudes and a 14.81:1 CR for frequency measurements. The lossy compression algorithm in [16] is shown to perform better than wavelet-based approaches. CRs up to 20:1 are achieved with low-reconstruction errors. The authors of [16] also note that measurement noise adversely affects compression. Perhaps the highest achieved CR of 31:1 appears in [17], which proposes a three-stage lossy compression approach: PCA in the first stage, DWT or DCT in the second stage, and a standard lossless file compression algorithm in the third stage. Although our paper also uses PCA, DCT, and lossless compression, the way in which these techniques are employed is distinctly different from that of [17]. In the approach proposed here, PCA is performed only once on a small subset of the data to obtain a fixed transformation matrix, which can be applied to the remaining data. DCT is applied to each block of the data. In comparison, the algorithm proposed in [17] requires performing repeated PCA on each data matrix, for every 10 s of time series data. The DCT coefficients are obtained separately for each singular vector and not the data matrix itself. The three-stage algorithm achieves high CR values. However, an effective compression by lossy algorithms must consider both CR and reconstruction error. The errors reported in [17] are for normalised measurements where the normalisation is dependent on the statistical features of the input data. Therefore, a consistent 'apples-to-apples' comparison of performance with the algorithm presented in this paper is not fully possible. However, we have made a sincere attempt to implement the approach in [17] and compare it with the proposed algorithm. Image compression algorithms such as the widely used JPEG achieve compression by exploiting features specific to images [18]. These features include similarity of nearby pixels and smooth variation of intensity across the image. In many ways, arrays constructed of chronological power system measurements have properties analogous to an image, or a stream of image frames. Measurements from network 'near neighbours' will show high-spatial correlations in the PMU data array. Measurements also display a high-temporal correlation. This paper extends the preliminary work reported in [19] and outlines a PMU data compression algorithm based on JPEG-like compression. The contributions are twofold. First, a lossy compression algorithm that leverages the spatio-temporal correlations in power system measurement data is proposed. This algorithm is then further enhanced by the use of an orthogonal transformation, computed from empirical covariance over a short starting time interval. The overall algorithm achieves CRs in the range of 40:1, with the average reconstruction error below 0.5%. When measurements are compressed separately, even higher CRs are achieved with low-reconstruction errors. The paper is organised as follows: Section 2 describes the data matrix and the spatio-temporal correlations present in the measurement. Section 3 introduces the algorithm and discusses its steps in detail. Numerical results are presented in Section 4, followed by concluding remarks. 2 Correlations in synchrophasor measurements PMU measurements from multiple locations in the same power system can be arranged as a matrix for simplifying its storage and manipulation. The matrix representation reveals the spatio-temporal correlations inherent in the measurement data and allows one to think of the data points as analogous to pixels in an image. This section explains how this concept can be used for developing the compression algorithm. 2.1 Measurement data matrix Measurements arriving from different locations are time-aligned at the phasor data concentrator with the help of global positioning system-based time tags. A set of m time-aligned PMU measurements for a given time interval can be stored in a data matrix as (1) where each is a vector of n time samples corresponding to the measurement i (2) The value of n depends on the reporting rate of the PMU and the period under consideration. The matrix is referred to as data matrix in the rest of the paper. may contain measurements of voltage magnitudes, voltage angles, line currents flows, line power flows, frequency etc. Measurements of like physical quantities are arranged in adjacent columns. 2.2 Spatio-temporal correlations Transmission lines interconnect various parts of the bulk power system, typically with complex meshed topologies. This interconnection induces couplings between electrical quantities such as voltages, currents, frequencies, and power flows. The coupling manifests as correlations in the temporal evolution of these quantities. During steady-state operation, the measured quantities do not vary much or change only gradually over time. It can be said that the measurements are temporally correlated. The spatial correlation is more observable when there are events causing transients. During such events, the measured quantities at electrically-close locations exhibit similar patterns. The spatio-temporal correlations suggest treatment of the data matrix as a continuous-tone digital image. While the matrix representation of an image is composed of intensity or colour values of the pixels, the individual elements of correspond to the value of a specific measurement at a specific time. Fig. 1 shows image representations of a data matrix consisting entirely of reactive power measurements from real-world PMUs. Fig. 1a shows normalised measurement magnitudes corresponding to one minute and Fig. 1b shows a zoomed-in version showing the magnitude variations in the first 16 samples. Fig. 1Open in figure viewerPowerPoint Image representation of reactive power measurements(a) 3600 rows of , (b) First 16 rows of Inspection of Fig. 1 shows that the intensity variations along each column of are indeed gradual, owing to the strong temporal correlations in measurement. Sudden changes in intensity along the rows indicate that there is considerable variation of magnitude between measurements from different locations. Fig. 2 shows the image representation of a data matrix with voltage magnitude (), active power (P), reactive power (Q), and frequency (f) measurements. It may be noted that variations along rows are more pronounced in power flow measurements (as compared to voltage and frequency measurements). The correlation matrix depicted in Fig. 2b confirms this observation, where the strong correlations within and f measurements are evident. Fig. 2Open in figure viewerPowerPoint Image representation of a matrix with different measurements and the corresponding correlation matrix(a) matrix, (b) Correlation matrix Most image compression algorithms exploit the correlations in the intensity and colour values to obtain high CRs. Efficient compression of PMU data can similarly be achieved by taking advantage of the inherent correlations. 3 Compression algorithm The proposed algorithm is inspired by the popular image compression standard JPEG [20]. The steps involved in the approach are summarised in Fig. 3. The preparation of the data matrix is carried out as explained in Section 2.1. The remaining steps of the algorithm can be classified into two major processes: data conditioning and JPEG-like compression. The following sections explain the proposed compression algorithm in detail. While the data conditioning steps are optional, the numerical results indicate that data conditioning has a significant impact on the performance of the compression algorithm. Fig. 3Open in figure viewerPowerPoint Flowchart of the proposed compression algorithm 3.1 Data conditioning Measurement noise can significantly affect the performance of the compression algorithm outlined in Section 3. The spatial and temporal correlations present in the measurements may be masked by the noise, thereby reducing the efficiency of the linear transformation and DCT-based decorrelation. The pre-processing approaches presented in this paper tackle two specific issues observed in raw PMU data. In the first scenario, noise is present along with valid measurement data. With sufficient knowledge about the nature of the actual measurement and the type of noise, the noise can be eliminated by designing a suitable filter. The second scenario is one in which some measurements are completely dominated by noise content. This can arise due to faulty equipment or because the magnitude of the actual measurement is considerably lower than the magnitude of the noise. 3.1.1 Filtering noise using low-pass filter PMU measurements are subject to disturbances from events in the power system. Fig. 4 shows the frequency ranges of different phenomena that can affect measurements. High-frequency non-sinusoidal events are removed in the PMU itself by an anti-aliasing filter, which limits the bandwidth of the input signal. The events that can be faithfully captured by the PMU are typical of fairly low frequency, such as power swings. Inspecting the spectral properties of real PMU measurements confirms that the noise content is of significantly higher frequency than the phenomena of interest. A simple low-pass filter with an appropriately designed cut-off frequency can be helpful in effectively retrieving meaningful information from the raw data. Fig. 4Open in figure viewerPowerPoint Spectral ranges of power phenomena. Source: [21] The filter chosen here is implemented as a 30th-order Hamming window-based low-pass filter with a suitable cut-off frequency (). The value of is chosen such that the information about important system dynamics is preserved while the noise is eliminated. In particular datasets, the authors have observed long intervals over which it is possible to choose to be as low as without significantly altering the information. However, for the results presented in this paper, is chosen as , consistent with the classification in Fig. 4. Fig. 5 compares a typical voltage measurement with noise to its filtered version. The spectral content of these signals is quantified using periodograms. Inspection of the filtered signal and its periodogram confirms that the filtering has not altered the signal content of interest. Fig. 5Open in figure viewerPowerPoint Original and filtered signals and their periodograms(a) Original, (b) Filtered 3.1.2 Removing noise-only measurements When a measured quantity has magnitude near zero, the noise can dominate the signal of interest. An example is active and reactive power measurements from a PMU monitoring a line carrying negligible power. In such cases, the filtering approach described above cannot satisfactorily remove the noise content. Tests performed using real PMU data show that the presence of such measurements adversely affects the CR and reconstruction accuracy. Since the magnitude of the measurement is zero (or near zero) for the period of interest, the measurement can be removed from the data matrix before any conditioning is performed. If a small number of such highly noisy measurements are judged worth archiving, they may be separated from the compression set and stored directly with modest impact on overall achieved compression. A long-standing approach for quantifying noise in signals examines the autocorrelation in the noisy signal. Autocorrelation of a random process quantifies the correlation between its values at different points in time as a function of the time lag. For a discrete signal x of finite length N, the autocorrelation coefficient can be defined as a function of the lag as (3) where and are the sample mean and sample variance of x, respectively. The autocorrelation coefficients have the following important properties: The autocorrelation function in (3) is an even function. The maximum value of occurs at . If x is non-periodic and has zero mean, then . The last property can be exploited to quantify the noise content in signals. Observe that the rate at which decays as increases with higher noise content in the signal. This suggests defining the half-life as follows: (4) Fig. 6 shows an active power measurement of 10 min duration and its autocorrelation coefficients. For this signal, the half-life can be identified as 297.95 s (corresponding to ). Fig. 7 shows an example of how the half-life can be used to identify measurements with high-noise content. The magnitude of the measurement is nearly zero for the entire window considered. The autocorrelation coefficients reduce to small values very quickly, yielding a half-life of 0.05 s (corresponding to ). This significant separation in the half-life of signals can be used to identify and remove measurements with high-noise content. Fig. 6Open in figure viewerPowerPoint Autocorrelation of a typical measurement Fig. 7Open in figure viewerPowerPoint Autocorrelation of a noisy measurement A possible concern regarding the approach described in this section is of ensuring that signals with electro-mechanical oscillations of interest are not being wrongly classified as noise. As an initial effort to address this concern, dynamic data generated by simulating various transient events such as series and shunt faults and load drops have been examined, and the half-life of their autocorrelations computed. In all of the examples examined, the decay of observed was always much slower than noisy signals. The analysis reinforced confidence in the ability of the half-life test to discriminate between signals with high-noise content and measurements corresponding to transients with large excursions in signal magnitude. Studies using these simulated measurements are not included here due to space constraints. 3.1.3 Normalising measurement magnitudes When mixing PMU measurements of physically different quantities, their relative scaling strongly influences the success of compression. This is analogous to experience in image compression, where sharp changes in intensity compress poorly and create undesirable effects. A simple approach is to normalise the magnitudes of different quantities, equilibrating magnitudes along a row of . Voltage magnitude measurements are scaled with the respective rated voltage levels of the buses at which those measurements are obtained. For example, measurements from a 230 kV bus are scaled with 230 kV line-to-line voltage as the base voltage. Similarly, the base value for real and reactive powers is chosen based on the system size. A typical base value for measurements from the bulk transmission system is 100 MVA. The nominal system frequency is chosen as the base for frequency measurements. Angle measurements in degrees have a wide range and may span the full range of PMU reporting. Converting the angles to radians scales these measurements to a numeric range comparable in magnitude to other per-unitised measurements. PMUs typically measure angles modulo , but some data sources may report angles 'unwound' over a larger range. Here, values are projected to the range before they are converted to radians. 3.2 JPEG-like compression The JPEG image compression algorithm leverages the correlations in the image pixels to achieve compression. The following sections explain how different processes in the JPEG compression algorithm have been adapted for compression of PMU data. 3.2.1 Orthogonal transformation In JPEG compression, the performance of the algorithm is improved when the red–green–blue values are transformed into YCbCr colour space. This transformation is introduced to reflect the manner in which the human eye perceives intensity and colour information from an image. Similarly, the characteristics of power system quantities can be exploited to improve the compression of PMU data by suitably transforming the data before applying DCT. We capture these characteristics with the help of an empirical covariance matrix, computed from a limited time interval of measurements. The pair-wise covariances between different measurements in are used to form the symmetric covariance matrix (5) where is computed empirically as (6) with and as the means of the measurement vectors and , respectively. The samples used in covariance calculation can be a modest-sized subset of the data being compressed (typically a window composed of the starting samples). Alternatively, historical measurements from the system under consideration may be used to obtain . Singular value decomposition of gives (7) where is a diagonal matrix of the singular values. Matrix serves as the transformation matrix for the measured data (8) Similar to the YCbCr transformation, the proposed transformation is linear. This transformation has the effect of decorrelating the data matrix , as illustrated in Fig. 8. The bright areas are indicators of high correlations between measurements. The decorrelating effect of the covariance-based pre-processing is evident in Fig. 8b. Fig. 8Open in figure viewerPowerPoint Intensity maps of the correlation matrix of the original and processed PMU data. Darker regions correspond to smaller correlations(a) Original data, , (b) Transformed data, To compress PMU data, the transformation matrix might be generated by considering the samples corresponding to the early hours of a day and then employed as a fixed transformation throughout the remaining hours of the day. This approach allows for storing the transformation matrix a priori and avoids the need to recompute the transformation matrix for each data matrix. A variant of this approach is used in the numerical results of Section 4. 3.2.2 Discrete cosine transform JPEG uses DCT as the decorrelating transform. The DCT concentrates the energy of input data in a small number of transformed coefficients, allowing coefficients with negligible magnitude to be discarded. The general formula for the 2D DCT of an matrix with data points can be written as (9) where and . and are defined as (10) (11) The 2D DCT defined in (9) can be written in the form of a transformation matrix and applied to to obtain the matrix of transformed coefficients as (12) DCT is applied individually to square blocks of size in traditional JPEG compression. The proposed algorithm does not limit the transformation to square blocks. Compression performance for PMU data can be improved by applying the transformation to rectangular blocks of . When cannot be evenly divided into blocks of the chosen size, its dimensions are made compatible by suitably padding the rows (columns) by repeating the values in the last row (column). 3.2.3 Quantisation and storage After applying DCT as per (12), each block in F will have a few elements with relatively large magnitudes, which contain the low-frequency information that one desires to retain. These large coefficients appear in the top-left corner of the block of DCT coefficients. The coefficients with smaller magnitudes are discarded through quantisation to achieve lossy compression. A user-selected parameter determines the threshold below which coefficients are rounded off to zero. A user has the option of 'customising' the choice of based on the anticipated operating conditions (steady-state or transient) and the amount of compression desired, but the good performance was observed throughout even with a fixed . For easy implementation of data reconstruction, we choose to represent the quantised coefficients in matrix form, , by retaining both zero and non-zero elements in their respective locations. Practically, only the non-zero DCT coefficients need to be stored using a suitable sparse storage scheme [22]. The compressed representation is encoded using a lossless dictionary-compression method before saving to storage. In this paper, Lempel–Ziv–Markov chain Algorithm (LZMA) [23] is used. 3.3 Data reconstruction The quantised DCT coefficients, the transformation matrix U, the base values used in scaling, and the dimensions of the blocks used are required to reconstruct the data matrix. The steps involved in reconstructing the data from the compressed representation are summarised as follows: If lossless compression is employed to compress further, decompress the compressed file to obtain . Apply inverse DCT to each block of . Invert the covariance-based transformation to obtain the conditioned representation of original data. Appropriately scale the elements (with the chosen base values) to the actual magnitudes to get the reconstructed data matrix . 3.4 Performance metrics Compression performance is evaluated by quantifying space savings and the accuracy of reconstruction. Space savings are captured using the CR defined simply as (13) Larger values of CR imply higher space savings. The matrix is stored on the physical storage as a comma-separated file, whereas the DCT coefficients are stored as a compressed archive using the LZMA file compression. The compressed representation includes and the measurements required to generate the transformation matrix used in pre-processing. Relevant metadata such as block sizes used in DCT and the base values used for magnitude normalisation are also included in the compressed representation. The accuracy of data reconstruction can be quantified using the difference between original and reconstructed data points. The element-wise differences of and the reconstructed matrix are obtained as (14) The matrix is then used to define the following error metrics: Average relative percentage error (ARPE) Since measurements of different quantities are compressed together, one may wish to judge the severity of the error by comparing with the average or nominal values of the measurements. Based on the type of measurement and its average values, the significance of the error may vary. For example, an error of in an active power measurement with a mean value of may be considered negligible. However, an error of magnitude in frequency measurements with nominal frequency is significant. Relative percentage error (RPE) is defined for each measurement to capture this aspect of the proposed approach: (15) The results report the average RPE across all measurements (16) Root mean square error (RMSE) This is a standard metric that captures the average error across all data points. RMSE is calculated using the elements of as (17) 4 Numerical results The proposed compression algorithm is tested on real PMU data obtained from a North American power system. The PMUs are located at different geographical locations within the network. The data contains 95 measurements comprising voltage magnitudes (from 11 different PMUs), frequencies (from 11 different PMUs), active and reactive power flows (from 37 and 36 lines, respectively). All the measurements are given in actual magnitudes and contain 60 sample points per second. The transients in the measurements correspond to excursions in the wind generation present in the network. CEII concerns prohibit the release of further details about the data. For the purpose of reporting the results, five 10-min windows exhibiting transients of different characteristics are chosen from the dataset. These 10-min windows of measurements are referred to as Dataset A–Dataset E in the rest of the paper. The voltage magnitude and active power waveforms depicted in Fig. 9, in fact, correspond to measurements available in Dataset A. Fig. 9Open in figure viewerPowerPoint Representation of voltage magnitude and active power; see Fig. 10 for an enlarged view Fig. 10Open in figure viewerPowerPoint Enlarged view of original and reconstructed voltage magnitude and active power measurements The numerical results presented highlight the effects of covariance-based pre-processing transformation, choice of block sizes, and data conditioning on compression performance. 4.1 Effect of the covariance-based pre-processing Table 1 showcases the effect of the covariance-based pre-processing transformation by comparing it with unprocessed compression. In the unprocessed variant, DCT is applied to filtered and normalised without employing the orthogonal transformation. For the pre-processed variant, the transformation matrix is generated using the