Solving for the Low-Voltage/Large-Angle Power-Flow Solutions by Using the Holomorphic ...

220 Pages·2015·3.79 MB·English

Solving for the Low-Voltage/Large-Angle Power-Flow Solutions by Using the
Holomorphic Embedding Method
by
Yang Feng

A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy

Approved July 2015 by the
Graduate Supervisory Committee:

Daniel J. Tylavsky, Chair
Dieter Armbruster
Keith E. Holbert
Lalitha Sankar

ARIZONA STATE UNIVERSITY
August 2015
ABSTRACT
For a (N+1)-bus power system, possibly 2N solutions exists. One of these solutions
is known as the high-voltage (HV) solution or operable solution. The rest of the solu-
tions are the low-voltage (LV), or large-angle, solutions.
In this report, a recently developed non-iterative algorithm for solving the pow-
er-flow (PF) problem using the holomorphic embedding (HE) method is shown as
being capable of finding the HV solution, while avoiding converging to LV solutions
nearby which is a drawback to all other iterative solutions. The HE method provides a
novel non-iterative procedure to solve the PF problems by eliminating the
non-convergence and initial-estimate dependency issues appeared in the traditional
iterative methods. The detailed implementation of the HE method is discussed in the
report.
While published work focuses mainly on finding the HV PF solution, modified
holomorphically embedded formulations are proposed in this report to find the
LV/large-angle solutions of the PF problem. It is theoretically proven that the pro-
posed method is guaranteed to find a total number of 2N solutions to the PF problem
and if no solution exists, the algorithm is guaranteed to indicate such by the oscilla-
tions in the maximal analytic continuation of the coefficients of the voltage power se-
ries obtained.
After presenting the derivation of the LV/large-angle formulations for both PQ
and PV buses, numerical tests on the five-, seven- and 14-bus systems are conducted
i
to find all the solutions of the system of nonlinear PF equations for those systems us-
ing the proposed HE method.
After completing the derivation of finding all the PF solutions using the HE method,
it is shown that the proposed HE method can also be used to find only the PF solutions
of interest (i.e. type-1 PF solutions with one positive real-part eigenvalue in the
Jacobian matrix), with a proper algorithm developed. The closet unstable equilibrium
point (closest UEP), one of the type-1 UEP’s, can be obtained by the proposed HE
method with limited dynamic models included.
The numerical performance as well as the robustness of the proposed HE method is
investigated and presented by implementing the algorithm on the problematic cases and
large-scale power system.

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ACKNOWLEDGEMENTS
First and foremost, I would like to express my sincere thanks to Dr. Tylavsky, my
advisor for being a commendable source of knowledge and inspiration. From the be-
ginning of my graduate career, he has inculcated a passion for the subject, guided me
through several challenges in research and shaped my writing skills.
I express my sincere thanks to Muthu Kumar Subramanian, a master student
graduated from ASU, Shruti Rao, a doctoral student and Yuting Li, a master student,
currently at ASU. They all are hard-working team members and productive contributor
in developing the algorithms. Besides, Muthu and Shruti have spent much of their
valuable part time to shape my writing skills.
Dr. Antonio Trias, AIA, deserves a special mention for the valuable inputs he gave
to our research group. I would like to extend to my thanks to my graduate committee
Dr. Armbruster, Dr. Holbert and Dr. Sankar for their valuable feedback and suggestions
to my thesis document. I am grateful to the all the faculty members of power engi-
neering for the wonderful learning experience they provided for the last five years since
I joined ASU.
I would like to express my gratitude to the School of Electrical, Computer and
Energy Engineering for providing me the opportunity to work as Teaching Assistant.
Finally, my heart is with my love, Fanjie Lin, my family, especially my mother, and my
friends who have kept me grounded during the entire period of my study.
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TABLE OF CONTENTS
Page
ABSTRACT .................................................................................................................... i
ACKNOWLEDGEMENTS ......................................................................................... iii
TABLE OF CONTENTS .............................................................................................. iv
LIST OF FIGURES ...................................................................................................... ix
LIST OF TABLES ........................................................................................................ xi
NOMENCLATURE .................................................................................................. xiii
CHAPTER
1 INTRODUCTION ............................................................................................... 1
1.1 Power-Flow Problems .............................................................................. 1
1.2 Iterative Methods ..................................................................................... 1
1.3 Holomorphic Embedding ......................................................................... 3
1.3.1 Holomorphic Functions ................................................................. 3
1.3.2 Holomorphic Embedding Method ................................................. 3
1.4 Objectives ................................................................................................ 4
1.5 Organizations ........................................................................................... 5
2 LITERATURE REVIEW .................................................................................... 8
2.1 Iterative Methods ..................................................................................... 8
2.2 All the Solutions for a Power System .................................................... 11
2.3 Improve Convergence for Iterative Method .......................................... 11
iv
CHAPTER Page
2.3.1 Step-Size Adjustment Newton’s Method..................................... 12
2.3.2 Decoupled Newton’s Methods ..................................................... 13
2.3.3 Miscellaneous Load-flow Method ............................................... 14
2.3.4 Non- Iterative Methods ................................................................ 15
2.3.5 Continuation Power Flow Method ............................................... 15
2.4 Finding All the Solutions for Power-Flow Problem .............................. 16
2.4.1 CPF Method ................................................................................. 17
2.4.2 Homotopy Method ....................................................................... 17
2.4.3 Groebner Basis ............................................................................. 18
2.5 Type-1 Algebraic Solutions for Power-Flow Problem .......................... 19
2.5.1 Type-1 PF Solutions .................................................................... 19
2.5.2 The Closest Unstable Equilibrium Point...................................... 21
3 THE HIGH-VOLTAGE SOLUTION USING THE HOLOMORPHIC
EMBEDDING METHOD ................................................................................. 23
3.1 Embedded PBE’s ................................................................................... 23
3.2 Power Series Expansion ......................................................................... 27
3.3 Power Series Coefficients ...................................................................... 29
3.4 Maximal Analytic Continuation ............................................................ 32
3.5 Padé Approximant ................................................................................. 35
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CHAPTER Page
3.5.1 Direct/Matrix Method .................................................................. 35
3.5.2 Viskovatov Method ...................................................................... 38
3.6 Curve Following in HE .......................................................................... 41
4 TWO-BUS LOW-VOLTAGE SOLUTION ...................................................... 45
4.1 Theoretical Derivation ........................................................................... 46
4.1.1 Embedding ................................................................................... 48
4.1.2 Continued Fraction....................................................................... 48
4.1.3 The LV Solution .......................................................................... 49
4.2 Numerical Example for a Two-Bus System .......................................... 59
4.3 Conclusion ............................................................................................. 63
5 MULTI-BUS LOW-VOLTAGE/LARGE-ANGLE FORMULATIONS .......... 64
5.1 Germ ...................................................................................................... 64
5.2 Solution Existence ................................................................................. 65
5.3 Unique Germ-to-Solution Mapping ....................................................... 66
5.4 Finding All Possible Germs for the HE PBE’s ...................................... 66
5.4.1 LV Formulation for PQ Buses and Finding 2NPQ Germs ............. 67
5.4.2 Large-Angle Formulation for PV Buses and Finding 2NPV Germs ..
...................................................................................................... 71
5.4.3 Finding 2N Germs for a Multi-Bus Lossless System ................... 73
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CHAPTER Page
5.4.4 Finding 2N Germs for a Practical Multi-Bus System ................... 75
5.5 The Guarantee to Find All the PF Solutions .......................................... 78
5.6 Discussions ............................................................................................ 79
5.7 Numerical Tests ..................................................................................... 80
5.7.1 Five & Seven-Bus System ........................................................... 80
5.7.2 14-Bus System ........................................................................... 102
6 FINDING THE TYPE-1 POWER-FLOW SOLUTIONS USING THE
PROPOSED HOLOMORPHIC EMBEDDING METHOD ............................ 106
6.1 Type-1 PF Solutions ............................................................................ 106
6.2 The Closest Unstable Equilibrium Point.............................................. 112
6.3 Conclusion ........................................................................................... 115
7 NUMERICAL PERFORMANCE OF THE HOLOMORPHIC EMBEDDING
METHOD ........................................................................................................ 117
7.1 Numerical Performance for Heavily Loaded System .......................... 117
7.1.1 43-Bus System ........................................................................... 118
7.1.2 Multi-Precision Complex (MPC) Application ........................... 125
7.2 Precision Issue for the LV Solution of the 43-bus System .................. 133
7.3 Large Systems ...................................................................................... 136
7.4 Conclusion ........................................................................................... 141
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CHAPTER Page
8 CONCLUSION ................................................................................................ 142
8.1 Summary .............................................................................................. 142
9 REFERENCES ................................................................................................ 144
APPENDIX
A ALL SOLUTIONS FOR IEEE-14 BUS SYSTEM ............................................... 152
B TYPE-1 SOLUTIONS FOR IEEE-118 BUS SYSTEM ........................................ 165

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LIST OF FIGURES
Figure Page
2.1 Convergence Problems for Iterative Methods ....................................................... 28
3.1 Radius of Convergence of Power Series f (s) ........................................................ 51
1
3.2 Regions for f (s) and f (s) Representing 1/(1-s) ..................................................... 52
1 2
3.3 Two-Bus Example with Shunt Reactance.............................................................. 60
3.4 Two-Bus Model: Curve Following in HE ............................................................. 61
4.1 Two-Bus Example without Shunt Reactance ........................................................ 63
4.2 Solution of NR Method from Different Starting Points......................................... 79
4.3 Solution of Proposed LV HE Formulation from Different Starting Points ........... 80
4.4 Solution of HV HE Formulation from Different Starting Points ........................... 80
5.1 Two-Bus System with the PV Bus Model ............................................................. 90
5.2 Five-Bus System .................................................................................................... 98
5.3 HV Solution (1111).............................................................................................. 104
5.4 LV PBE’s Applied to V (0111)........................................................................... 105
1
5.5 LV PBE’s Applied to V & V (0110) ................................................................... 106
1 4
5.6 LV PBE’s Applied to V (1110)........................................................................... 107
4
5.7 LV PBE’s Applied to V &V (0011) .................................................................... 108
1 2
5.8 LV PBE’s Applied to V (1101)........................................................................... 109
3
5.9 LV PBE’s Applied to V &V (1001) .................................................................... 110
2 3
5.10 LV PBE’s Applied to V (1011)......................................................................... 111
2
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