mr. hallvar haugdal, finished his msc. in electrical
TRANSCRIPT
Mr. Hallvar Haugdal, finished his MSc. in Electrical Engineering at NTNU, Norway in 2016, specializing innumerical field calculations and electrical motor design. After his MSc he worked a Scientific Assistant also atNTNU, dealing with courses on electrical machines, power systems and field calculations. Since January 2018he is a PhD student on Wide Area Monitoring- and Control Systems.
โOnline Mode Shape Estimation using Complex Principal Component Analysis and Clusteringโ, Hallvar Haugdal(NTMU Norway)
Online Mode Shape Estimationusing Complex Principal
Component Analysis and Clustering
Hallvar Haugdal
Contents
โข Motivationโข Background theory
โข (C)PCAโข Clustering
โข Proposed methodโข Application to Kundur Two Area-system
Contents
โข Motivationโข Background theory
โข (C)PCAโข Clustering
โข Proposed methodโข Application to Kundur Two Area-system
Motivation
โข Modal analysisโข Powerful toolโข Understanding of system dynamicsโข Stability limitsโข Design of controllersโข Difficult due to inaccurate modellingโข Load dependent
โข Propose empirical approachโข Relying only on Wide Area PMU measurementsโข Correlationโข Statistical learning
Two-part methodPart 1: Provide estimates of modes and mode shapes
โข Moving window ~ 5 โ 10 s lengthโข Using correlation, Complex Principal Component Analysis (CPCA)โข Provide point estimates of modes shapes
Part 2: Compute averaged mode shapesโข Differentiate between noise and modesโข Clustering points resulting from Part 1โข Averaged modes and -shapes computed as centroids of clusters
Contents
โข Motivationโข Background theory
โข (C)PCAโข Clustering
โข Proposed methodโข Application to Kundur Two Area-system
Principal Component Analysisโข Matrix of ๐๐ series, ๐๐ samples:
๐๐ =
๐ฑ๐ฑ๐๐๐ฑ๐ฑ๐๐โฎ๐ฑ๐ฑ๐ด๐ด
=
๐ฅ๐ฅ1(๐ก๐ก1) ๐ฅ๐ฅ1 ๐ก๐ก2๐ฅ๐ฅ2 ๐ก๐ก1 ๐ฅ๐ฅ2 ๐ก๐ก2
โฏ ๐ฅ๐ฅ1 ๐ก๐ก๐๐๐ฅ๐ฅ2 ๐ก๐ก๐๐
โฎ๐ฅ๐ฅ๐๐ ๐ก๐ก1 ๐ฅ๐ฅ3 ๐ก๐ก2
โฑ๐ฅ๐ฅ๐๐ ๐ก๐ก๐๐
โข Want to transform the correlated series ๐ฑ๐ฑ๐๐, ๐ฑ๐ฑ๐๐ โฏ๐ฑ๐ฑ๐ด๐ด into uncorrelatedseries ๐ฌ๐ฌ๐๐, ๐ฌ๐ฌ๐๐ โฏ ๐ฌ๐ฌ๐ด๐ด:
๐๐ =
๐ฌ๐ฌ๐๐๐ฌ๐ฌ๐๐โฎ๐ฌ๐ฌ๐ด๐ด
= ๐๐๐ป๐ป๐๐, ๐๐๐๐ = ๐ฎ๐ฎ๐๐๐ป๐ป๐ฟ๐ฟ
โข Do this by eigendecomposition of the covariance matrix:
๐๐ =1
1 โ ๐๐๐๐๐๐๐๐
โข Eigenvalues (ฮป๐๐) and -vectors (๐ฎ๐ฎ๐๐):
๐๐๐ฎ๐ฎ๐๐ = ฮป๐๐๐ฎ๐ฎ๐๐, ๐๐ = 1 โฆ๐๐๐๐ = ๐ฎ๐ฎ๐๐,๐ฎ๐ฎ๐๐ โฏ๐ฎ๐ฎ๐ด๐ด
โ ๐๐โฒ๐๐๐๐ = ๐ฒ๐ฒ =
๐๐1๐๐2
โฑ๐๐๐๐
๐๐1> ๐๐2 > ๐๐3 โฆ ๐๐๐๐
โข Inversion gives time series from scores
๐๐ = ๐๐๐๐
๐ฑ๐ฑ๐๐ = ๏ฟฝ๐๐=๐๐
๐ด๐ด
๐ข๐ข๐๐,๐๐๐ฌ๐ฌ๐๐
โข Contribution of ๐ฌ๐ฌ๐๐ in ๐ฑ๐ฑ๐๐ given by coefficient ๐ข๐ข๐๐,๐๐
Complex Principal Component Analysisโข Similar to conventional PCAโข Additional steps:
โข Empirical Mode Decomposition (EMD)โข Detrendingโข (Decomposition โ Intrinsic Mode Functions)
โข Hilbert Transformโ Complex time series ๐ณ๐ณ๐๐โข Complex covariance matrix:
๐๐ =1
๐๐ โ 1๐๐โ๐๐
โข Contribution of ๐ฌ๐ฌ๐๐ in ๐ณ๐ณ๐๐ given by complexcoefficient ๐ข๐ข๐๐,๐๐
โข Resembles observability mode shapesโข Slower than PCA
โข Due to EMD and Hilbert Transformโข Works well with damped exponentialsโข Noisy during steps [3]
Contents
โข Motivationโข Background theory
โข (C)PCAโข Clustering
โข Proposed methodโข Application to Kundur Two Area-system
Proposed method, Part 1: Generating mode estimates
โข Input: Starting out with time window1. PCA ๐๐ โ ๐๐ (Scores)2. EMD Detrending3. Hilbert Transform4. CPCA ๐๐ โ ๐๐ (Complex scores)
โข Complex PC Scores resemble modes (๐๐, ๐๐)โข Complex Coefficients resemble mode shapes
โข Output: Point in 2๐๐ + 1 dimensions๐๐ = ๐๐, Re ๐ถ๐ถ1 , Im ๐ถ๐ถ1 , Re ๐ถ๐ถ2 , Im ๐ถ๐ถ2 โฆ Re ๐ถ๐ถ๐๐ , Im ๐ถ๐ถ๐๐
Proposed method, Part 2: Averaged mode estimates using Clusteringโข Resulting points from Part 1 are assumed to populate input space more
densely close to areas corresponding to modesโข Input: Matrix of ๐๐ point estimates
๐๐ =
๐๐1๐๐2โฎ๐๐๐๐
=
๐๐, Re ๐ถ๐ถ1 , Im ๐ถ๐ถ1 โฆ Re ๐ถ๐ถ๐๐ , Im ๐ถ๐ถ๐๐๐๐, Re ๐ถ๐ถ1 , Im ๐ถ๐ถ1 โฆ Re ๐ถ๐ถ๐๐ , Im ๐ถ๐ถ๐๐
โฎ๐๐, Re ๐ถ๐ถ1 , Im ๐ถ๐ถ1 โฆ Re ๐ถ๐ถ๐๐ , Im ๐ถ๐ถ๐๐
โข Clustering of points (number of clusters unknown)โข Output: Averaged mode shapes
โข Centroids of clusters
Contents
โข Motivationโข Background theory
โข (C)PCAโข Clustering
โข Proposed methodโข Application to Kundur Two Area-system
Applicationโข Kundur Two-Area Systemโข Simulated data
โข DigSILENT PowerFactoryโข Short Circuits near generators
[1]
Application
๐๐1 ๐๐2 ๐๐3
Kundur Two-Area System โ Part 1โข Moving windowโข Point estimates
ApplicationKundur Two-Area System โ Part 2โข Resulting
average mode shapes
โข DigSILENTPowerFactoryModal Analysis
Concluding remarksโข Appears to give reasonable results
โข Possible to differentiate modes of similar frequency
โข Potential improvementโข Including dampingโข Clustering
โข Remove noiseโข Frequency estimationโข Rotation of mode shapes
โข Further thoughtsโข Clustering on ยซlong termยป moving windowโข Use parallel windows of different lengthsโข Track modesโข Adaptive controllers
โข Further testingโข Simulations with amibent noiseโข 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, โComplex Principal Component Analysis: Theory and Examples,โ Journal of Climate and Applied Meteorology, vol. 23, no. 12. pp. 1660โ1673, 1984.
[4] Yu's Machine Learning Garden, retrieved from http://yulearning.blogspot.com/2014/11/einsteins-most-famous-equation-is-emc2.html (2018)