Evaluation and Visualization of Multi-Level Contingencies in Power Systems by Anton Lodder A ...

234 Pages·2015·3.24 MB·English

Evaluation and Visualization of Multi-Level Contingencies in Power
Systems
by
Anton Lodder
A thesis submitted in conformity with the requirements
for the degree of Masters of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
(cid:13)c Copyright 2015 by Anton Lodder
Abstract
Evaluation and Visualization of Multi-Level Contingencies in Power Systems
Anton Lodder
Masters of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2015
Contingency analysis is critical to evaluating the operational capacity of power systems and char-
acterizing their vulnerability to component faults. As we look to increase the resilience of networks to
element failure, the instance of multiple contingencies is of growing concern for planners and operators
in identifying weak points in the system. Multi-element contingencies introduce new challenges for how
to reliably and consistently measure the severity of a fault, how to perform contingency analysis on an
expanding range of contingency scenarios in a timely manner, and how to interpret the increasingly
hierarchical data obtained by contingency analysis. This research explores techniques that can be used
to generate, summarize and display the results of multi-element contingency analyses in power systems,
including high-performance computational methods for evaluating contingencies and new visualization
techniques that leverage visual summarization and live interaction to extract valuable insights from the
resulting data.
ii
Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Computations for Contingency Analysis 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Techniques for Evaluating Contingency Scenarios . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Performance Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Voltage Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Continuation Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Formulation of Continuation Power Flow . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Continuation Power Flow Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Choice of Continuation Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Modifications to the Continuation Power Flow . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Adaptive Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Lagrange Polynomial for Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Quantification of Performance Gains in Continuation Power Flow . . . . . . . . . . 20
2.5 Performing Multi-level Contingency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Dealing with Islanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.2 Implementing Participation Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.3 Application of Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Visualizing Multi-Level Contingency Data 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Representing Contingency Data as a Tree Diagram . . . . . . . . . . . . . . . . . . 28
3.2.3 Techniques for Overlaying Quantitative Data on Tree Diagrams . . . . . . . . . . . 30
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3.2.4 Normalization of Data in Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.5 Strengths and Limitations of the Tree Diagram . . . . . . . . . . . . . . . . . . . . 32
3.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Treemap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Representing Contingency Data as a Treemap . . . . . . . . . . . . . . . . . . . . . 34
3.3.3 Normalization of Data in Treemaps. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.4 Different Approaches to Treemap Styling . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.5 Tiling Algorithm for Treemap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.6 Dealing With Quantization Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.7 Use of Colour Coding to Overlay Information Treemaps . . . . . . . . . . . . . . . 43
3.3.8 Strengths and Limitations of Treemaps . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Expanding Content of Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.2 Interactive Discovery of Contingency Details . . . . . . . . . . . . . . . . . . . . . 49
3.4.3 Responsive Highlighting of Diagram Structures . . . . . . . . . . . . . . . . . . . . 49
3.4.4 Cross-referencing to One-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.5 Drill-down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.6 Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Conclusions and Future Work 64
4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 List of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Future Work in Improving Contingency Analysis Techniques . . . . . . . . . . . . 67
4.2.2 Future Work in Improving Visualizations . . . . . . . . . . . . . . . . . . . . . . . 69
Bibliography 70
Appendices 74
A Source Code for Contingency Analysis 75
A.1 Running Contingency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.2 Defining Faults to be Analyzed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.3 Fault Class Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.4 Dealing With Islanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B Source Code for Continuation Power Flow 102
B.1 Continuation Power Flow Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B.2 Prediction step using First Order Approximation . . . . . . . . . . . . . . . . . . . . . . . 137
B.3 Prediction step using Lagrange Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.3.1 Lagrange Polynomial Prediction with λ Continuation . . . . . . . . . . . . . . . . 140
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B.3.2 Lagrange Polynomial prediction with Voltage Continuation . . . . . . . . . . . . . 142
B.3.3 Implementation of Lagrange Polynomial . . . . . . . . . . . . . . . . . . . . . . . . 143
B.4 Correction step of Continuation Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.4.1 Correction Step With λ as Continuation Parameter . . . . . . . . . . . . . . . . . 144
B.4.2 Correction Step With Bus Voltage as Continuation Parameter . . . . . . . . . . . 146
C Source Code for Visualizations 152
C.1 Software Representations of Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.2 Cross-Referencing Elements and Contingencies . . . . . . . . . . . . . . . . . . . . . . . . 178
C.3 Building A Treemap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
C.3.1 Treemap Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
C.3.2 Treemap Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
C.4 Building a Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
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List of Figures
2.1 Example of a power-voltage curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Linear approximation versus Lagrange polynomial interpolation in predictor step of con-
tinuation power flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Comparison of performance of the Lagrange polynomial interpolation scheme for contin-
uation power flow with gradual versus sudden change in step size.. . . . . . . . . . . . . . 19
2.4 Flow chart describing the algorithm for island detection. . . . . . . . . . . . . . . . . . . . 22
2.5 Graphsummarizingperformancegainsresultingfrommodificationstocontinuationpower
flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Road-map describing the relationship between tree diagrams and treemaps. . . . . . . . . 27
3.2 Tree representation of a subset of fault cases for the IEEE 30-bus test system. . . . . . . . 29
3.3 Tree diagram illustrating the use of data overlays. . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Treemap of n−1 contingencies of a set of four elements. . . . . . . . . . . . . . . . . . . . 34
3.5 Treemap of n−1 and n−2 contingencies of a set of four elements. . . . . . . . . . . . . . 35
3.6 Flow chart describing the recursive tiling algorithm for a squarified treemap layout.. . . . 39
3.7 Treemap of n−1 and n−2 contingencies for the IEEE 30 bus test system. . . . . . . . . 45
3.8 Treemap of n−1 and n−2 contingencies for the IEEE 118 bus test system.. . . . . . . . 46
3.9 Treemap demonstrating use of alternating colours to increase visual differentiation. . . . . 47
3.10 Treemap diagram demonstrating break-out of contingency details via mouse click inter-
action for the fault of generator 4 and transformer 2. . . . . . . . . . . . . . . . . . . . . . 50
3.11 Treemap diagram demonstrating break-out of contingency details via mouse click inter-
action for the fault of bus 2 and transformer 2. . . . . . . . . . . . . . . . . . . . . . . . . 51
3.12 Treemap diagram demonstrating break-out of contingency details via mouse click inter-
action for the fault of transformer 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.13 Treemap diagram demonstrating break-out of contingency details via mouse click inter-
action for the fault of generator 4 and bus 6. . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.14 Screen-shots of a tree diagram demonstrating the use of mouse hover interaction. . . . . . 54
3.15 Use of mouse interaction to highlight duplicate contingencies in a treemap. . . . . . . . . 56
3.16 Screen-shotsofatreemapdiagramdemonstratingtheuseofmouseinteractiontoidentify
elements involved in a contingency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.17 Screen-shotsofatreemapdiagramdemonstratingtheuseofmouseinteractiontoidentify
elements involved in a contingency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.18 Treemap diagram showing n−1 through n−3 contingencies involving branch 2 and four
other elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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3.19 Treemap diagram showing n−2 and n−3 contingencies for five elements. . . . . . . . . . 62
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List of Tables
2.1 Performance benchmark comparing execution times for linear approximation versus La-
grange polynomial approximation in the prediction step of continuation power flow. . . . 18
2.2 Performance benchmark comparing execution of continuation power flow with various
modifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Performancebenchmarkcomparingexecutiontimesforcontinuationpowerflowwithand
without the use of multi-processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Comparison of rounding error for elements with large area versus small area in laying out
a treemap diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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List of Code Snippets
1 Pseudo-code describing how to update the step size after completing the prediction and
correction steps of an iteration of continuation power flow. . . . . . . . . . . . . . . . . . . 15
2 Pseudo-code describing how to calculate edge weights for the layout of tree diagrams,
taking into consideration the associated faults. . . . . . . . . . . . . . . . . . . . . . . . . 32
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Chapter 1
Introduction
1