Online Mode Shape Estimation using Complex Principal Component Analysis and Clustering, Hallvar Haugdal (NTMU Norway) Mr. Hallvar Haugdal, finished his MSc. in Electrical Engineering at NTNU, Norway in 2016, specializing in numerical field calculations and electrical motor design. After his MSc he worked a Scientific Assistant also at NTNU, dealing with courses on electrical machines, power systems and field calculations. Since January 2018 he is a PhD student on Wide Area Monitoring- and Control Systems.
Online Mode Shape Estimation using Complex Principal Component Analysis and Clustering Hallvar Haugdal
Contents Motivation Background theory (C)PCA Clustering Proposed method Application to Kundur Two Area-system
Contents Motivation Background theory (C)PCA Clustering Proposed method Application to Kundur Two Area-system
Motivation Modal analysis Powerful tool Understanding of system dynamics Stability limits Design of controllers Difficult due to inaccurate modelling Load dependent Propose empirical approach Relying only on Wide Area PMU measurements Correlation Statistical learning
Two-part method Part 1: Provide estimates of modes and mode shapes Moving window ~ 5 10 s length Using correlation, Complex Principal Component Analysis (CPCA) Provide point estimates of modes shapes Part 2: Compute averaged mode shapes Differentiate between noise and modes Clustering points resulting from Part 1 Averaged modes and -shapes computed as centroids of clusters
Contents Motivation Background theory (C)PCA Clustering Proposed method Application to Kundur Two Area-system
Principal Component Analysis Matrix of MM series, NN samples: XX = xx 11 xx 22 xx MM = xx 1 (tt 1 ) xx 1 tt 2 xx 1 tt NN xx 2 tt 1 xx 2 tt 2 xx 2 tt NN xx MM tt 1 xx 3 tt 2 xx MM tt NN Want to transform the correlated series xx 11, xx 22 xx MM into uncorrelated series ss 11, ss 22 ss MM : Eigenvalues (λ ii ) and -vectors (uu ii ): CCuu ii = λ ii uu ii, ii = 1 MM UU = uu 11, uu 22 uu MM UU CCCC = ΛΛ = λλ 1 λλ 2 λλ 1 > λλ 2 > λλ 3 λλ MM λλ MM SS = ss 11 ss 22 ss MM = UU TT XX, ss ii = uu ii TT XX Inversion gives time series from scores XX = UUUU Do this by eigendecomposition of the covariance matrix: CC = 1 1 NN XXXXTT MM xx ii = uu ii,jj ss jj jj=11 Contribution of ss jj in xx ii given by coefficient uu ii,jj
Complex Principal Component Analysis Similar to conventional PCA Additional steps: Empirical Mode Decomposition (EMD) Detrending (Decomposition Intrinsic Mode Functions) Hilbert Transform Complex time series zz ii Complex covariance matrix: CC = 1 NN 1 ZZ ZZ Contribution of ss jj in zz ii given by complex coefficient uu ii,jj Resembles observability mode shapes Slower than PCA Due to EMD and Hilbert Transform Works well with damped exponentials Noisy during steps [3]
Clustering Various methods K-means Gaussian Mixture Models [4] [2]
Contents Motivation Background theory (C)PCA Clustering Proposed method Application to Kundur Two Area-system
Proposed method, Part 1: Generating mode estimates Input: Starting out with time window 1. PCA XX SS (Scores) 2. EMD Detrending 3. Hilbert Transform 4. CPCA SS ZZ (Complex scores) Complex PC Scores resemble modes (ff, ξξ) Complex Coefficients resemble mode shapes Output: Point in 2MM + 1 dimensions pp = ff, Re CC 1, Im CC 1, Re CC 2, Im CC 2 Re CC MM, Im CC MM
Proposed method, Part 2: Averaged mode estimates using Clustering Resulting points from Part 1 are assumed to populate input space more densely close to areas corresponding to modes Input: Matrix of QQ point estimates PP = pp 1 pp 2 pp QQ = ff, Re CC 1, Im CC 1 ff, Re CC 1, Im CC 1 ff, Re CC 1, Im CC 1 Re CC MM, Im CC MM Re CC MM, Im CC MM Re CC MM, Im CC MM Clustering of points (number of clusters unknown) Output: Averaged mode shapes Centroids of clusters
Contents Motivation Background theory (C)PCA Clustering Proposed method Application to Kundur Two Area-system
Application Kundur Two-Area System Simulated data DigSILENT PowerFactory Short Circuits near generators [1]
Application Kundur Two-Area System Part 1 Moving window Point estimates pp 1 pp 2 pp 3
Application Kundur Two-Area System Part 2 Observations Clustering
Application Kundur Two-Area System Part 2 Resulting average mode shapes DigSILENT PowerFactory Modal Analysis
Concluding remarks Appears to give reasonable results Possible to differentiate modes of similar frequency Potential improvement Including damping Clustering Remove noise Frequency estimation Rotation of mode shapes Further thoughts Clustering on «long term» moving window Use parallel windows of different lengths Track modes Adaptive controllers Further testing Simulations with amibent noise Real PMU data
References [1] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. [2] T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning. New York, NY: Springer New York, 2009. [3] J. D. Horel, Complex Principal Component Analysis: Theory and Examples, Journal of Climate and Applied Meteorology, vol. 23, no. 12. pp. 1660 1673, 1984. [4] Yu's Machine Learning Garden, retrieved from http://yulearning.blogspot.com/2014/11/einsteins-most-famous-equation-isemc2.html (2018)