Detection of System Disturbances Using Sparsely Placed Phasor Measurements Ali Abur Department of Electrical and Computer Engineering Northeastern University, Boston abur@ece.neu.edu CÁTEDRA ENDESA DE LA UNIVERSIDAD DE SEVILLA Escuela Técnica Superior de Ingeniería de Sevilla 29 de mayo de 2015
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Talk Outline Phasor Measurement Units (PMUs) External Network Model Error Identification Fault Detection and Location Closing remarks
Phasor measurement units (PMU) Synchronized Phasor Measurements and Their Applications by A.G. Phadke and J.S. Thorp Invented by Professors Phadke and Thorp in 1988. They calculate the real-time phasor measurements synchronized to an absolute time reference provided by the Global Positioning System. PMUs facilitate direct measurement of phase angle differences between remote bus voltages in a power grid.
Phasor Measurement Units (PMU) Phasor Data Concentrators (PDC) [*] PMU 2 PMU 1 Antenna CT GPS CLOCK PT PMU 3 PDC CORPORATE PDC REGIONAL PDC PMU 4 PMU K DATA STORAGE & APPLICATIONS DATA STORAGE & APPLICATIONS [*] IEEE PSRC Working Group C37 Report
Reference phasor +ϴ 1 --ϴ 2 --ϴ 3 +ϴ 1 --ϴ 2 Reference Reference --ϴ 3
Measurements provided by PMUs I phasors PMU V phasor ALL 3-PHASES ARE TYPICALLY MEASURED BUT ONLY POSITIVE SEQUENCE COMPONENTS ARE REPORTED V A V B VC = [T] V 0 V + V V + = 1 3 [V A+ V B + 2 V C ] = e jjπ 3
Solving for the Unknown Sparse Vector Underdetermined set of equations: C X = [b] P x N N x 1 = P x 1 X is known to be k sparse, i.e. k out of N entries are known to be significantly larger than the remaining (N-k) entries.
Two Examples Line Outage Detection Few equations, large number of candidate lines, one of which is disconnected. Fault detection and location Few equations, large number of candidate branches, one of which is faulted. Further complication: unknown fault location along the suspect branch.
Line Outage Detection Problem: Detect and identify topology changes (line switching) in the external system by monitoring the internal system. Assume limited access to external system PMUs. EXTERNAL SYSTEM [2] INTERNAL SYSTEM [1] REAL-TIME INTERNAL SYSTEM STATE REAL-TIME INTERNAL TOPOLOGY BASE CASE EXTERNAL STATE BASE CASE EXTERNAL TOPOLOGY
Problem Formulation BEFORE AFTER AFTER
Problem Formulation EXTERNAL SYSTEM [2] INTERNAL SYSTEM [1]
Problem Formulation EXTERNAL SYSTEM [2]?? INTERNAL SYSTEM [1]
Linear Approximation: Problem Formulation Partitioned into internal 1 and external 2 networks:
Line Switching Modeled by Bus Injections LINE IS OPEN k Pkm m P km - P km k m P km = 0
Line Switching Modeled by Bus Injections Bθ 0 = P 0 ( B + B) θ 1 = P 0 Bθ 1 = P 0 P Sparse Bus Injection Vector Pre-outage topology Post-outage state of the system k k b m -b [ 0 0 p 0 0 p 0 ] T P = 0 ΔB= m -b b
Linearized Problem Formulation
Linearized Problem Formulation J Internal Only Buses Boundary Buses
Linearized Problem Formulation
Conversion to Binary Variables
Optimization Problem min j u j + v j s. t. J = Λ. p. X + u v X j = 2 j e bb = u j ii e bb > 0 e bb = v j ii e bb < 0 v j = u j = 0 ii e bb = 0
Incorporating Losses ρ = min (u j + v j ) j s. t. J = Λ. p. A + Λ. q. B + u v A j = 2 j B j = 1 j j, A j B j j, u j + v j < p > 0 q > 0 p 10
Example: 118-Bus System / Two Zones
Example: 118-Bus System / Two Zones 24 72 Internal System 38 42 70 65 External System 45 49 46
Identified External Line Outages [DC Approximation, No Load Changes] Line outage P (Pre-outage flow on removed line) Solution found for X -1 at bus +1 at bus 59-63 7.608 59 63 77-80 3.003 77 80 60-61 6.810 60 61 68-116 0.948 116 68 96-82 0.332 82 96 100-101 0.532 100 101 103-104 0.562 104 103 80-98 1.631 98 80
Lossy Case Simulation Results [Different Line Outages in the External System] Line outage p Solution found for A q e +1@ bus -1@ bus 77-80 3.534 80 77 0.066 @ 77 0.132 59-63 5.101 59 63 0.248 @ 59 0.191 60-61 7.216 61 60 0.012 @61 0.055 94-95 1.389 94 95 0.021 @94 0.075
Lossy Case Simulation Results [Simultaneous topology and load/generation changes] Line outage p Solution found for A q e +1@ bus -1@ bus 77-80 3.608 80 77 0.026@ 77 0.474 59-63 5.497 63 59 0.239 @ 59 0.379 60-61 6.979 61 60 0.031 @61 0.251
Fault location problem Fault occurs, typically along a transmission line, less frequently right at a bus / substation Determine the faulted line or line segment Determine the distance between one of the line terminals to the point of fault Branch l Distance x
Fault location methods Power frequency / impedance based methods Record fault transients at line terminals Filter HF signals, estimate impedance to fault Traveling wave based methods High frequency sampling ( > 20KHz ) Capture wave front arrival instants using synchronized sensors AI / pattern recognition / machine learning based methods
Motivation for this work +ΔV k V k V m +ΔV m I f V k +ΔV k I k I m V m +ΔV m
Virtual injections and fault location I 1 + I 2 = I 3 I 2 I 1 = Z 13 Z 23
Fault Detection and Location Can line faults be detected and located using a sparse set of phasor voltage measurements? Can all types of faults in three-phase systems be identified? How do we ensure that there are sufficient measurements to detect and identify faults on any line in the power grid?
Faulted Line 2-5 AB-G Fault 10-bus Example
Network Partitioning U Bus without Measurement M Measured Bus
Estimation Model TI u = RHS
Complex to Real Conversion [ T ][ I ] = [ RHS] = [ I Y V ] u m red m [ X][ β ] = [ y]
Least Angle Regression and Shrinkage (LARS) [*] [*] R. Tibshirani, Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society. Series B (Methodological), pp. 267 288, 1996. B. Efron, T. Hastie, I. Johnstone, R. Tibshirani et al., Least angle regression, The Annals of statistics, vol. 32, no. 2, pp. 407 499, 2004.
Uniqueness of Solution Define bus groups (column indices) for line terminals: g 1, g 2...g L Then the uniqueness condition will be: rank{ T } = full for i = 1, 2,, L g i That is: columns of each g i must be linearly independent.
Measurement Placement Flowchart
Positive Sequence Model 118 buses 41 voltage phasor measurements 131 fault cases required the use of LARS T I = [(100 + j200) (300 + j600)] pu.. Results Summary for Noise-Free Cases Ave. Running Time Ave. No. of Iterations Ave. Error 0.0134 s 3.4242 s 5.76 e-14
Positive Sequence Model
Three Phase Model 30-bus three-phase system All five types of faults (SLG, LL, DLG, 3P, 3PG) are simulated 16 voltage phasor measurements 23 line faults of all 5 types required the use of LARS
Three-Phase Model
Remarks and Conclusions PMUs can be used to track network changes (line outages / faults) without the need for full network observability. Measurement noise remains an issue which requires further work. Work is currently being extended to detection of faults in T-circuits.
Acknowledgements National Science Foundation/DoE NSF/ERC CURENT Thank You Collaborators: Any Questions? Guangyu Feng, Graduate student Roozbeh Emami, Ph.D. 2011