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Load Flow Analysis of
Multi-Converter Transmission Systems
A Thesis
Submitted to the Faculty
of
Drexel University
By
Shaun Mendoza Cruz
in partial fulfillment of the
requirements for the degree
of
Master of Science in Electrical Engineering
June 2014
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Copyright 2014
Shaun Mendoza Cruz. All Rights Reserved
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ACKNOWLEDGEMENTS
I would like to take this opportunity to express the utmost gratitude for my thesis advisor,
Dr. Chikaodinaka Nwankpa for all the assistance, guidance, and advice he has provided throughout
this research endeavor. I also thank him for his mentorship throughout my final years at Drexel as
an undergraduate student.
The Center of Electric Power Engineering (CEPE) has provided me an exceptional
education complemented with excellent hands-on lab experience which I am grateful for. Thank
you to Dr. Karen Miu for introducing the field of power engineering to me. Thank you to Dr.
Thomas Halpin for furthering my fundamental knowledge of power engineering to a graduate level
of thinking. Thank you to Dr. Dagmar Niebur for providing me with exciting alternative power
engineering research opportunities. I would also like to thank Drs. Dagmar Niebur and Thomas
Halpin for serving on my thesis committee. I am appreciative of their feedback and am grateful of
their valuable comments which has been utilized to further this thesis research.
For their financial support, thank you to those at the American Society for Engineering
Education (ASEE), the Science, Mathematics and Research for Transformation (SMART)
Scholarship for Service Program, and the Naval Surface Warfare Center (Carderock Division).
Finally, I would like to thank several people who have supported me on a more personal
basis. To my parents, Marcelino and Maria Cruz, thank you for raising me, supporting me, and
guiding me on all aspects of my life. To my sister, Stacy Cruz, thank you for being a great role
model and loving sibling. To my girlfriend, Yanni Eboras, thank you for being the inspiration for
me to achieve and supporting me even when things looked bleak. Also, thank you to all my
extended family and friends for your love and support. This thesis is dedicated to all of you.
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TABLE OF CONTENTS
LIST OF FIGURES ..................................................................................................................... viii
LIST OF TABLES ......................................................................................................................... xi
1 INTRODUCTION .................................................................................................................. 1
1.1 Overview .......................................................................................................................... 1
1.2 Background ...................................................................................................................... 4
1.2.1 Load Flow Methodologies for Traditional AC Systems ........................................... 4
1.2.2 High Voltage Direct Current (HVDC) Links ............................................................ 6
1.2.3 Load Flow Methodologies for HVDC Transmission Systems ................................. 8
1.3 Motivation ...................................................................................................................... 10
1.4 Problem Statement ......................................................................................................... 11
1.5 Approach ........................................................................................................................ 12
1.6 Organization of Thesis ................................................................................................... 13
2 DEVELOPMENT OF A LOAD FLOW ANALYSIS TOOL .............................................. 15
2.1 Overview ........................................................................................................................ 15
2.2 Sato and Arillagas Method............................................................................................ 15
2.2.1 DC Subroutine ........................................................................................................ 16
2.2.2 Full Load Flow Routine .......................................................................................... 20
2.3 Modifications to Sato and Arillagas Method ................................................................ 21
2.3.1 Simplified DC Load Flow Routine ......................................................................... 22
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2.3.2 Isolated Bus Load Flow for AC Routine ................................................................ 26
2.3.3 Full Double Loop Approach ................................................................................... 28
2.3.4 Example Case: Grainger and Stevensons 4-Bus System ....................................... 31
2.3.5 Ordering of HVDC Link Substitutions ................................................................... 35
3 CONVERGENCE ANALYSIS OF LOAD FLOW FOR DIFFERENT POWER SYSTEMS
38
3.1 Overview ........................................................................................................................ 38
3.2 Grainger and Stevensons 4-Bus System ....................................................................... 39
3.3 IEEE 9-Bus System ........................................................................................................ 44
3.4 IEEE 14-Bus System ...................................................................................................... 49
3.5 IEEE 30-Bus System ...................................................................................................... 53
3.6 IEEE 118-Bus System .................................................................................................... 57
3.7 Summary of Convergence Properties ............................................................................. 62
4 DEVELOPMENT OF AN ENHANCED LOAD FLOW SOLVER .................................... 64
4.1 Overview ........................................................................................................................ 64
4.2 Convergence of Modified Sato and Arillagas Method in Terms of Percent DC
Transmission ............................................................................................................................. 64
4.3 Enhanced Load Flow Solver Algorithm......................................................................... 66
4.4 Results of Enhanced Load Flow Solver ......................................................................... 68
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5 APPLICATION OF ENHANCED LOAD FLOW SOLVER ON VOLTAGE STABILITY
70
5.1 Overview ........................................................................................................................ 70
5.2 HVDC Link Substitutions Based On Transmission Power Magnitude ......................... 71
5.3 Substitution Orderings with Heavily Loaded Bus as First Isolation .............................. 74
6 CONCLUSION ..................................................................................................................... 80
6.1 Summary of Thesis Work .............................................................................................. 80
6.2 Potential Future Work .................................................................................................... 81
LIST OF REFERENCES .............................................................................................................. 84
APPENDIX ................................................................................................................................... 87
APPENDIX A: Additional Information about HVDC Links ................................................... 87
A.1 Calcluation of Vx, Vy, and Ids ...................................................................................... 87
A.2 Reactive Power for HVDC Links .................................................................................. 89
APPENDIX B: Power System Data in MATPOWER Format ................................................. 90
B.1 Data for Grainger and Stevensons 4-Bus System ......................................................... 90
B.2 Data for IEEE 9-Bus System .......................................................................................... 91
B.3 Data for IEEE 14-Bus System ........................................................................................ 93
B.4 Data for IEEE 30-Bus System ........................................................................................ 95
B.5 Data for IEEE 118-Bus System ...................................................................................... 98
APPENDIX C: MATLAB Code ............................................................................................. 107
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C.1 Main Code for Load Flow Analysis Tool .................................................................... 107
C.2 Code to set HVDC Link Parameters ............................................................................ 111
C.3 Code for AC Subroutine ............................................................................................... 114
C.4 Code for DC Subroutine ............................................................................................... 117
C.5 Main Code for Enhanced Load Flow Method .............................................................. 120
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LIST OF FIGURES
Figure 1. Power flow notation at a bus i for active (left) and reactive (right) power ..................... 5
Figure 2. One line diagram for typical HVDC transmission system [10] ...................................... 7
Figure 3. Sato and Arillagas HVDC link model .......................................................................... 17
Figure 4. Flow chart of Sato and Arillagas DC subroutine ......................................................... 19
Figure 5. Flowchart of Sato and Arillagas full load flow routine................................................ 21
Figure 6. Simplified DC subroutine developed in this thesis ....................................................... 26
Figure 7. Power system with included HVDC link (left) and substitution of equivalent power
sources for AC routine (right) ....................................................................................................... 26
Figure 8. Power system with multiple HVDC links tied to one bus (left) resulting in isolated bus
during substitution (right) ............................................................................................................. 27
Figure 9. Modified Sato and Arillagas Full Load Flow Method ................................................. 30
Figure 10. One-line diagram for Grainger and Stevensons 4-bus system ................................... 31
Figure 11. One-line diagram of 4-Bus system with HVDC link connecting bus 3 and 4 ............ 33
Figure 12. Placement of HVDC links (left) resulting in multiple disjoint sub-systems (right) .... 36
Figure 13. Example of minimum spanning tree in which removal of line results in two systems 36
Figure 14. Ordering of HVDC Link Substitutions........................................................................ 37
Figure 15. Time diagram for full load flow routine of 4-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 40
Figure 16. Time plot for AC load flow routine of 4-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 42
Figure 17. Iteration plots for full load flow method and AC load flow routine of 4-bus system using
Newton-Raphson method (a and b) and Gauss Seidel method (c and d)...................................... 43
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Figure 18. IEEE 9-Bus System [22] ............................................................................................. 45
Figure 19. Time diagram for full load flow routine of 9-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 46
Figure 20. Time plot for AC load flow routine of 9-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 47
Figure 21. Iteration plots for full load flow method and AC load flow routine of 9-bus system using
Newton-Raphson method (a and b) and Gauss Seidel method (c and d)...................................... 48
Figure 22. IEEE 14-Bus System [23] ........................................................................................... 49
Figure 23. Time diagram for full load flow routine of 14-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 50
Figure 24. Time plot for AC load flow routine of 14-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 51
Figure 25. Iteration plots for full load flow method and AC load flow routine of 14-bus system
using Newton-Raphson method (a and b) and Gauss Seidel method (c and d) ............................ 52
Figure 26. IEEE 30-Bus System [24] ........................................................................................... 53
Figure 27. Time diagram for full load flow routine of 30-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 54
Figure 28. Time plot for AC load flow routine of 30-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 55
Figure 29. Iteration plots for full load flow method and AC load flow routine of 30-bus system
using Newton-Raphson method (a and b) and Gauss Seidel method (c and d) ............................ 57
Figure 30. IEEE 118-Bus System [25] ......................................................................................... 58
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Figure 31. Time diagram for full load flow routine of 118-bus system using (a) the Newton-
Raphson method and (b) Gauss Seidel method for the AC routine .............................................. 59
Figure 32. Time plot for AC load flow routine of 118-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine ............................................................. 61
Figure 33. Iteration plots for full load flow method and AC load flow routine of 118-bus system
using Newton-Raphson method (a and b) and Gauss Seidel method (c and d) ............................ 62
Figure 34. Comparison of Gauss Seidel method to Newton-Raphson method in regards to faster
convergence time with respect to percentage of DC transmission ............................................... 65
Figure 35. Enhanced Load Flow Solver algorithm. ...................................................................... 67
Figure 36. Generic P-V curve ....................................................................................................... 70
Figure 37. Full load flow convergence times for maximum active power load ........................... 73
Figure 38. Full load flow convergence times for maximum reactive power load ........................ 73
Figure 39. Minimum spanning tree for IEEE 30-Bus system with heavily loaded bus 10 .......... 75
Figure 40. Time partition plot for heavily loaded bus 10 with large active power (a) and large
reactive power (b) ......................................................................................................................... 76
Figure 41. AC time partition plot for heavily loaded bus 10 with large active power (a) and large
reactive power (b) ......................................................................................................................... 76
Figure 42. Minimum spanning tree for IEEE 30-Bus system with heavily loaded bus 30 .......... 78
Figure 43. Time partition plot for heavily loaded bus 10 with large active power (a) and large
reactive power (b) ......................................................................................................................... 79
Figure 44. AC time partition plot for heavily loaded bus 10 with large active power (a) and large
reactive power (b) ......................................................................................................................... 79
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LIST OF TABLES
Table 1. Line Data for Grainger and Stevensons 4-Bus System ................................................. 31
Table 2. Bus Data for Grainger and Stevensons 4-Bus System .................................................. 31
Table 3. Calculated Bus Data from Load Flow Solution of 4-Bus System .................................. 32
Table 4. Calculated Line Flows from Load Flow Solution of 4-Bus System ............................... 32
Table 5. Equivalent Impedances of 4-Bus System ....................................................................... 33
Table 6. HVDC Parameters .......................................................................................................... 34
Table 7. Calculated Bus Voltages for Systems with HVDC Links ............................................. 34
Table 8. Calculated Converter Powers and Line Flows for Systems with HVDC Links ............. 35
Table 9. Convergence Time and Iteration Data for Load Flow of IEEE 14-Bus System ............. 68
Table 10. Stressed System Load Flow for Lines Connected to Bus 10 of IEEE 30-Bus System. 72
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ABSTRACT
Load Flow Analysis of Multi-Converter Transmission Systems
Shaun Mendoza Cruz
Chikaodinaka Nwankpa, Ph.D.
This thesis discusses the development of a load flow methodology capable of solving for
the load flow of power systems inclusive of multiple point-to-point HVDC links. The methodology
is an extension of a previous algorithm developed by Sato and Arillaga. Recent pushes for
renewable power and developments supporting HVDC transmission for short distances will
facilitate the substitution of multiple HVDC links for existing transmission lines. The extended
methodology is utilized as a load flow analysis tool and is applied to various power systems to
determine the effects of multiple HVDC links on the convergence properties of the load flow
methods (i.e. timing and number of iterations). It was found that the percentage of DC transmission
in a system is a good indicator of which embedded AC load flow method should be utilized in the
extended methodology. Specifically at a DC transmission percentage of 90% or higher, the Gauss
Seidel method has speeds comparable to the Newton-Raphson method. Leveraging off this finding,
an Enhanced Load Flow solver is developed, which optimizes for faster convergence with a second
priority to reducing memory requirements. The Enhanced Load Flow solver is a robust
methodology which is easily extended to include other AC load flow methods. This research also
looked into the effects of substituting AC transmission lines with HVDC links on the speed of
convergence for voltage stability studies. It was found that the substitution of HVDC links for lines
connected to a heavily loaded bus for a stressed system yields a lower convergence time for the
voltage stability study.
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1 INTRODUCTION
1.1 Overview
Load flow analysis is a key tool which supports the planning, operation, and control efforts
of power systems. Utilizing numerical methods, the traditional load flow methods (i.e. Gauss
Seidel, Newton-Raphson, Fast Decoupled) all solve for the bus voltages, voltage angles, real
power, and reactive power of AC power systems. The solution then serves as a basis for several
different analyses since it solves for the expected steady state operation of the transmission
network (Ex. voltage stability analysis). However, it should be noted that the initial development
of these load flow methodologies assumed a purely AC transmission system.
It is a well-known fact that there is a push for the use of alternative energy resources as
sources of generation for todays power grid. The green-energy resources of interest (i.e. solar,
wind, geothermal, etc) are sometimes located in remote regions, requiring long-distance
transmission to the main power grid. Alongside the incorporation of these renewable generation
sources therefore comes the incorporation of multiple high voltage direct current (HVDC) links
which have proven to be of economic feasibility for long-distance transmission [1]. In addition,
there has been some recent technological developments to support the use of HVDC links for even
shorter distances, which will further the incorporation of DC components into the current power
system by replacing existing AC transmission lines with HVDC lines [2]. Thus, instead of a pure
AC transmission network, current transmission networks are moving towards an integrated system
of traditional AC transmission and multiple HVDC links.
This said, the load flow solvers initially designed for pure AC transmission needed to be
altered to accommodate the load flow through the converters. It was found that in-depth load flow
calculations for HVDC links required the determination of HVDC output voltages and currents,
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optimum transformer tap position, current settings for constant power control, power transfer at
the converter terminals, etc. This methodology contrasted with AC load flow methods which
mainly require only the bus/generator limits, Y-bus matrix formulation, generator and load data.
Thus, by inspection of the general methodologies, existing load flow methodologies had to be
altered to incorporate a solution methodology for DC transmission alongside retaining the solution
method for AC transmission.
It is clear that a DC sub-routine can be embedded into current AC load flow methods to
accommodate the inclusion of HVDC links. That is, solve for the DC systems separately and
represent the HVDC links as two power sources on the sending/receiving end buses, then perform
a traditional AC load flow analysis. This sub-routine methodology, proposed by Sato and Arrillaga
[3] (and henceforth will be referred as Sato and Arrillagas method throughout this thesis), clearly
impacts the performance metrics of obtaining a steady state solution (i.e. number of convergence
iterations, total convergence time, accuracy, etc). However, it is yet to be determined the relation
between the number of HVDC links and the impacts on performance metrics. Significant impacts
on the performance metrics due to the inclusion of multiple HVDC links into existing power
systems may facilitate an environment in which a certain load flow algorithm is optimal in
comparison to the others.
Another area of interest is the impact of the integration of multiple HVDC links on the
analyses which utilize load flow methods as a basis. In particular, the voltage stability study and
the development of P-V curves for multi-link HVDC transmission systems is of interest. In order
to obtain an accurate representation of the P-V curve for AC transmission networks, full iterative
AC load flow methods (Ex. Newton-Raphson) are needed. In contrast, approximated methods will
result in inaccurate P-V curves, specifically as the point of voltage collapse is approached for
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increased loading. Research has been done to determine the voltage stability at the terminal ends
of the HVDC links integrated into AC networks [4-5]. However little research has be done
regarding the impact of HVDC links on the speed of voltage stability studies. It is highly likely
that a significant impact on the speed would be found due to the nature of the solution methodology
(i.e. including both a DC and AC routine).
This thesis is concerned with several experimental studies on the load flow analysis of
multiple HVDC link transmission systems. In particular, this thesis is concerned with finding a
correlation between the number of HVDC links within a transmission network and the
convergence properties of Sato and Arrillagas method. Leveraging off the findings, the
development of a load flow solver which extends Sato and Arrillagas method to multiple HVDC
link transmission systems is done. The load flow solver optimizes for convergence time and
memory requirements and is called the Enhanced Load Flow solver. In addition, the impact of the
increased amount of DC transmission on the speed of a voltage stability analysis of a multi-link
HVDC transmission system is performed. Specifically, the deliverables are as follows:
1. A summary of the convergence analysis of load flow methods applied to various
benchmark power systems with respect to the inclusion of multiple HVDC links.
2. The development of an Enhanced Load Flow solver for integrated AC/DC systems.
3. A summary of the impact multiple HVDC links have on voltage stability analysis.
The following section provides an overview of load flow analysis, HVDC links, and existing
load flow methodologies for HVDC transmission systems. Section 1.3 discusses the motivation
for exploring the convergence properties of load flow methods applied to HVDC transmission
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systems. Section 1.4 presents the current issues with Sato and Arrillagas method if it is applied to
multi-link HVDC transmission systems. Section 1.5 presents the approach used to characterize the
performance of load flow methodologies on multi-link HVDC systems, how the Enhanced Load
Flow solver was developed, and the effects of HVDC links on voltage stability studies. Finally,
section 1.6 explains the organization of the remaining chapters of the thesis.
1.2 Background
1.2.1 Load Flow Methodologies for Traditional AC Systems
The load flow problem is one of the classical computation problems in power systems
engineering. Load flow studies are utilized as a basis for the development of power systems as
well as the optimal operation of a power system. Because of its importance, a large amount of
work has been done in regards to the formulation of methodologies to solve the load flow problem.
All load flow methodologies are attempts to solve for the voltage magnitude and voltage
phase angle at each bus in a power system. Calculations of the complex power injections at a bus
and power losses on a transmission line can then be performed using the solved complex voltages.
In its basic polar form, the power flow equations are as follows:
= || cos( + )=1 (1)
= || sin( + )=1 (2)
where i is the bus number of the bus injection being calculated (i.e. the from bus), n denotes the
number of a bus in the power system (i.e. the to bus), Yin is the corresponding element in the
Ybus matrix describing the admittance of the power system, in is the phase angle of the
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corresponding element in the Ybus matrix, Vi and Vn are the bus voltages for buses i and n
respectively, i and n are the phase angle of the bus voltages i and n respectively, and Pi and Qi
represent the calculated net real and reactive power entering the network at bus i (respectively)
[6].
The net power entering the network at bus i is equivalent to the generated power, Pgi, minus
the demanded load power, Pdi. We denote the net scheduled power Pi,sched as this difference.
Similar notation is used for reactive power. Figure 1 illustrates the power flow at a bus i.
Figure 1. Power flow notation at a bus i for active (left) and reactive (right) power
Being that Pi and Qi are calculated values obtained from equations (1) and (2) and Pi,sched
and Qi,sched are true values, the goal for any load flow methodology is to find the correct voltage
values such that the following power-balance equations hold true:
0 = , = ( ) (3)
0 = , = ( ) (4)
The set of known and unknown parameters used to further describe a particular load flow
problem for a power system is dependent on the bus types. One bus is considered to be a reference
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bus as is called the slack bus. The remaining buses in the system are categorized into either PQ or
PV buses. PQ buses are load buses in which the complex power at the bus is known, and only the
complex voltage at the bus needs to be found. PV buses are generator buses in which the real
power and voltage magnitude are known, and only the reactive power and bus voltage phase needs
to be found.
For a 2-bus power system, an analytic approach can be used to solve for the complex
voltages. However, the amount of nonlinear equations for larger systems garners the use of
numerical methods. The traditional load flow methodologies of Gauss-Seidel [7], Newton-
Raphson [8], and Fast Decoupled Load Flow [9], are all based on general-purpose numerical
iterative techniques.
1.2.2 High Voltage Direct Current (HVDC) Links
There are several benefits for using direct current transmission as opposed to the traditional
three-phase ac transmission. It has been proven that it is more economically attractive (in
comparison to ac transmission) when a large amount of power needs to be transmitted over a long
distance (typically 300-400 miles). Also, because the transmission is DC, it is possible to connect
two networks operating at different/unsynchronized frequencies. The power flow of DC
transmission is also easier to control, and thus there is an improved stability [10].
Figure 2 shows a typical HVDC transmission system, with details of the power electronics
of one terminal. Using Figure 2 as a guide and proceeding from left to right, we see that terminal
obtains voltage (typically 69-230kV) from the connected ac system. The ac voltage is then filtered
to reduce harmonics generated by the conversion process and the power factor is corrected by
means of a capacitor bank. The voltage is then transformed up to the transmission level via a Y-Y
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and -Y transformer. The stepped up ac voltage is then rectified to a DC voltage which is smoothed
by the inductor and DC filter, and applied to the HVDC transmission line. A similar architecture
is used at the terminal at the other end of the DC transmission line except utilizing inverter
operation and stepping down the voltage to whatever is required by the corresponding ac system.
It should be noted that power flow over the line can be reversed (i.e. bidirectional), depending on
the firing angle used for converter operation as either a rectifier or inverter [10].
Figure 2. One line diagram for typical HVDC transmission system [10]
The actual conversion from AC to DC or DC to AC is achieved by the positive pole 12-
pulse converter and negative pole 12-pulse converter in each terminal. Each pole consists of two
6-pulse line-frequency bridge converters connected via the transformers to yield the 12-pulse
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converter arrangement. The physics behind the 12-pulse line-frequency converters and the
smoothing inductor are the key components which will drive the derivation of power flow
equations though the HVDC link. These equations will be discussed further in Chapter 2, in the
overview of Sato and Arillagas method.
1.2.3 Load Flow Methodologies for HVDC Transmission Systems
After over a century of reliable HVDC transmission [11], there has been vast development
in the load flow methods as applied to integrated AC-DC systems. The initial proposed
methodologies used equivalent-current-source concepts to simulate the HVDC links in an
equivalent AC fashion [12]. Barker and Carr took these equivalent-current-source concepts and
applied it to a full load-flow program which solved both the load flows through the HVDC links
and the remaining AC load flows of the transmission system using the successive overrelaxation
method [13].
One of the main pioneers in the analysis of power networks that included HVDC
transmission, Jos Arrillaga, presented the first investigation on the convergence of the load flow
methods with respect to the influence of the inclusion of a HVDC link into a transmission system
[3]. His paper, Improved load-flow techniques for integrated a.c.-d.c. systems, explained a more
realistic simulation of HVDC links and detailed the incorporation of a DC subroutine into
traditional AC load flow methods (i.e. an extension of Barker and Carrs methodology). It was
concluded that the resulting substitution of a transmission line for an HVDC link resulted in an
increased number of iterations, yielding 30% extra computation (for a 14-bus system). It also
showed that certain load flow methodologies converge faster than others when HVDC links are
incorporated into the transmission system.
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As the AC load flow methodologies were developed, the integrated AC-DC load flow
methods needed to be altered as well. The development of the fast decoupled load flow method
necessitated a representation of HVDC links compatible with the methodology. Arrillaga
accomplished this in his paper Integration of h.v.d.c. links with fast-decoupled load-flow
solutions [14]. The results of the methodology showed retention of the reliability, speed, and
storage advantages of the fast decoupled load flow method. It was also shown that the DC portion
of the methodology had a significant impact on convergence time and storage for small power
systems, such as a 14 bus system. However, as the system grew in size, these impacts grew more
insignificant. In addition, Arrillaga mentioned that multiple HVDC links yielded extra time and
storage proportional to the number of links, but yielded no special problem with respect to
convergence of the methodology.
It should be noted that the methodologies previously explained utilized a sequential
approach the load flow through an HVDC link was solved separately via a DC subroutine,
followed by an AC routine to solve the remaining load flows of the transmission system. Other
methodologies such as [15] also used the sequential approach. However, unified approach
methodologies were also developed such as [16]. The unified approach methods essentially
incorporated the solution of the HVDC links within the equations for the solution of the AC
transmission system (i.e. embedded HVDC equations into AC iteration equation) usually
employing a Newton-Raphson type approach. Unified methods were considered to be less-robust
and thus sequential approaches were preferred.
As the idea of HVDC links being incorporated into the transmission system widened, the
topic of load flow analysis for multi-terminal HVDC links came about. [17-20] explain the
extension of the methodologies to mutli-terminal systems. It should be stated that this thesis is
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concerned not with the question of load flow for multi-terminal systems, but the question of load
flow for multi-link HVDC transmission systems such that each HVDC link contains only one pair
of converters (one inverter and one rectifier), also known as point-to-point transmission.
1.3 Motivation
Load flow analysis is key component of several different applications with respect to power
systems analysis. Being that numerical analysis methodologies are implemented to solve the load
flows, convergence properties of the particular solution methodology applied have always been of
interest. The most ideal method would result in a low iteration count, convergence of quadratic or
higher nature, small number of operations, minimal RAM, and fast overall solution time.
With the introduction of multiple HVDC links into the transmission system, it is clear that
the convergence of current AC-DC load flow methodologies will be impacted. Intuitively, the
convergence will be slower and require additional CPU usage. However, if a convergence analysis
is performed, it may be found that there exists a non-intuitive correlation of convergence properties
to the number of HVDC links in a transmission system. There exists several load flow
methodologies which can be implemented to solve the AC-DC network. Depending on the
convergence characteristics, a certain load flow methodology may be preferred over another.
The increased amount of HVDC links may also generate an impact on the speed of voltage
stability analysis of the transmission network. Due to the nonlinear nature of a stressed (i.e. loading
limit reached) transmission system, full AC iterative methodologies are required for accurate
determination of the current operating point on the P-V curve. Still, the Jacobian matrix used in
majority of load flow algorithms becomes increasingly singular as the maximum loading point is
reached. This results in a higher computation time and larger number of iterations to acquire an
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accurate result for points near the maximum loading point. However, because of the solution
methodology of Sato and Arillagas method, a reduced AC network is solved and the lessened
number of equations can result in a faster convergence time and less iterations for similar accuracy.
In addition to the computational benefits to load flow studies of power systems containing
multiple HVDC links, the main motivation of this thesis is the actual increased inclusion of HVDC
links into existing power networks. HVDC links allow the transfer of more power, an increased
controllability, and the interconnection of two power systems under different operation
characteristics (ex. frequency). Because of the clear utilization benefits and the economic benefits,
the installment of HVDC links will occur in the future. However, there exists very little research
done for multi-link systems. This thesis aims to provide a starting point for power system engineers
to use when exchanging multiple existing transmission lines with HVDC links.
1.4 Problem Statement
Two observations can be made about the implementation of Sato and Arrillagas method
on an integrated AC-DC transmission network. The first and most obvious is that additional time
must be spent in the load flow method to execute the DC subroutine. The second is that after
convergence of the DC subroutine, the substitution of representative power sources on terminal
buses essentially removes a transmission line from the solution of the AC routine. In other words,
a sparser Y-bus matrix is a resultant of the DC subroutine.
Realizing these two observations is key to understanding the impact of utilizing Sato and
Arrillagas method as applied to a multiple HVDC link system. Each HVDC link must be solved
separately by a separate DC subroutine, thus increasing convergence times. However, as the
number of HVDC links increases, the sparseness of the Y-bus matrix is also increased, which
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12
intuitively should decrease the AC routine convergence time. Therefore, to optimize Sato and
Arrillagas method, a correlation between the number of HVDC links to the convergence
properties of the method should be found. This correlation could lead to a decision factor in
choosing what AC routine is best for the particular number of HVDC links in a system.
Another issue which may occur is HVDC link placement within a plant to load flow
convergence. Because several AC transmission lines are being substituted for HVDC links, there
are certain topologies in such that a bus may become isolated in terms of the AC load flow
routine. These isolated buses therefore would cause a convergence issue strictly because a separate
routine to solve for the bus voltages must be performed. In addition to this, whatever components
(i.e. generator, compensator, etc) are connected to the bus will also be removed from the system,
which may also influence the convergence properties.
It is also unknown about the impacts on voltage stability studies with the increased
integration of HVDC links. When operating near the maximum loading point, traditional load flow
methods take high computation time and a large number of iterations to converge to an accurate
result. A decrease in iteration count and convergence time may occur when the same loading point
is analyzed using Sato and Arillagas method, due to the DC subroutine and the reduced AC system
solved by the AC routine.
1.5 Approach
A personal PC with Matlab R2013a (version 8.1.0.604) was used as a platform to code
Sato and Arrillagas method and apply it to various multi-converter AC-DC transmission systems.
MATPOWER 4.1 was used for the embedded AC load flow methods [21].
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13
Transmission lines were substituted for HVDC links in benchmark IEEE power systems.
From analysis of the load flow results for the different configurations, correlations between the
numbers of HVDC links in the transmission system to the convergence properties was determined.
The convergence properties for a 4-bus, 9-bus, 14-bus, 30-bus and 118-bus system were all looked
into. Using the results, the Enhanced Load Flow solver was developed in this thesis. This load
flow solver takes advantage of the benefits of using specific load flow methods, depending on the
structure of the transmission network (i.e. the percentage of DC transmission).
In addition, a system under stressed conditions was created by choosing an operating point
near the maximum loading point of a particular bus. The final bus voltages were determined using
a traditional AC load flow method. Substitution of different transmission lines for HVDC links
operating at the calculated load flows for the stressed system was done, and the Enhanced Load
Flow solver was used to recalculate the load flow. The time for the solver to converge and solve
the integrated AC-DC system was compared to the convergence time of the Newton-Raphson
method applied to an entirely AC system.
1.6 Organization of Thesis
An overview of Sato and Arrillagas method and the modifications made to the methodology
to extend the usage to multi-link HVDC transmission systems and analyze different embedded AC
methods is given in Chapter 2 of this thesis. Chapter 3 provides the results of the convergence
analysis of the modified Sato and Arillagas method applied to different power systems, in an
effort to determine the correlation of the number of HVDC links inclusive in an AC-DC
transmission system to the convergence properties of the methodology. The development of the
Enhanced Load Flow solver which determines the load flows for multi-link HVDC transmission
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14
systems benefiting from the findings of the convergence studies is explained in Chapter 4. An
experimental set up (and results of the experiment) to determine the effects of substituting AC
lines with HVDC links on the speed of voltage stability analysis is presented in Chapter 5. The
conclusion of the thesis as well as recommendations for future work are given in Chapter 6.
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15
2 DEVELOPMENT OF A LOAD FLOW ANALYSIS TOOL
2.1 Overview
A key component to this thesis work is the development of an algorithm capable of solving
the load flow of power systems containing multiple HVDC links. Sato and Arillagas method was
chosen as the benchmark algorithm, being that the algorithm is of a simple and elegant structure
which is easily modified. Also, since Sato and Arillagas method was one of the first developed
for load flow analysis of power systems containing HVDC links, it seemed natural to develop the
original methodology. This chapter explains the modifications made to Sato and Arillagas method
to extend the algorithm to multi-link systems.
It should be noted that the algorithm explained in this section is not to be confused with
the Enhanced Load Flow solver, which will be explained in a later chapter of the thesis. Although
the developments described in this chapter do form a basis for the final load flow solver, the
intended use was for analyzing the effects of including multiple HVDC links in an existing power
system, and hence the claim to be a load flow analysis tool.
2.2 Sato and Arillagas Method
Sato and Arillaga developed a load flow technique which improved the accuracy and
convergence rates of existing load flow methodologies when the power system contained a HVDC
link. In their paper Improved load-flow techniques for integrated a.c.-d.c. systems, they
explained an improved simulation of an HVDC link which incorporated the control concepts used
in HVDC links and optimum tap-change procedures. The improved simulation was also translated
into a DC subroutine and used in conjunction with traditional AC load flow techniques to create a
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16
load flow solver for integrated AC-DC power systems. The load flow methodology is discussed in
further detail in the following sections.
2.2.1 DC Subroutine
Sato and Arillagas method assumes that the HVDC link, depicted in Figure 3, is equipped
with both closed-loop current control and constant margin angle control. Under normal operation,
the inverter end of the link determines the direct voltage and operates on constant margin angle
control, and the rectifier end fixes the operating current by operating on constant current control.
With these assumptions, the following equations can be written for the direct voltages of the
HVDC link:
= 32
cos
3
(5)
= = (6)
= { } (7)
where Vinv is the inverter side DC voltage, Vline is the voltage drop on the transmission line, Vrect
is the rectifier side DC voltage, vinv is the r.m.s phase to phase voltage on the inverter side, is
the extinction angle, Xinv is the commutation reactance of the inverter, Iline is the transmission line
current, Rline is the resistance of the transmission line, Arect is the slope of the constant current
characteristic of the rectifier, and Idsrect is the DC current setting of the rectifier.
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17
Figure 3. Sato and Arillagas HVDC link model
Combining equations (5-7) the equation for the operating current can be found to be:
= 32
cos
3
{1}
(8)
which can be substituted back into equations (5-7) to find the DC voltages of the HVDC link.
In order to find the value of the DC current setting Idsrect, the optimum operating voltage
for the HVDC link needs to be calculated. The operating voltage Vx is found using the rectifier
constant current characteristic and the inverter constant margin angle characteristic, alongside the
power settings of the HVDC link. Using Vx, the current setting Idsrect and the maximum voltage Vy
can be found. The optimum operating voltage is found by minimizing the function
= > 0 (9a)
where =
(9b).
Details of how to find Vx, Vy, and Ids can be found in [3], and are discussed in Appendix A.
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18
Since the purpose of the methodology is to compute the power flows through an HVDC
link, it is necessary to calculate the power transfer at the converter terminals. For each converter,
the power transfer is calculated as follows:
= + tan (10)
= + tan (11)
where tan = ()2+sin 2sin 2{+}
cos 2cos 2{+} (12)
= cos1{
(2)
(32) cos } (13)
such that () = 1, () = 1, is equal to the firing angle or extinction
angle for the rectifier or inverter (respectively).
The converter-transformer, filters, compensators, and other interface elements can be
represented by a network which is easily translated into a matrix of equations to solve for unknown
voltages. A Newton-Raphson approach is used to solve for the unknown voltages and currents in
the network representation. Finally, a tap changer routine is used to update the taps of the
transformers accordingly. These calculations are also detailed in [3], and thus concludes the DC
subroutine of Sato and Arillagas method. Figure 4 shows a flow chart representation of the DC
subroutine.
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19
Figure 4. Flow chart of Sato and Arillagas DC subroutine
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20
2.2.2 Full Load Flow Routine
The full load flow routine depicted in Sato and Arillagas methodology for solving
integrated AC-DC power systems is based off an iterative sequential algorithm. First, the unknown
voltages and phases of the AC system are initialized. Using the initial guess, the DC subroutine
explained in the previous section is used to solve for the load flow through the HVDC link. The
HVDC link in the AC network is then replaced by power sources on the terminal buses equivalent
to the calculated power transfers from equations 10-13 (i.e. a power source of value equal to Srect
is connected to the rectifier end bus terminal, and a power source of value equal to Sinv is connected
to the inverter end bus terminal).
Once the substitution of the HVDC link is made for the equivalent power sources, Sato and
Arillaga allow two different approaches to solve for the bus voltages of the network. The first
approach, called the single loop approach, calls for only one iteration of a selected AC load flow
methodology (Gauss Seidel, Newton-Raphson, or Z-matrix method) to update the bus voltages
before returning to the DC subroutine. The single loop process of one DC subroutine followed by
one AC iteration repeats until convergence is met. The second approach, called a double loop
approach, calls for multiple iterations of the selected AC load flow methodology before returning
to the DC subroutine. The double loop process of one DC subroutine followed by several AC load
flow iterations repeats until system convergence is met. Note that convergence for either approach
is determined by a tolerance on the order less than or equal to 10-5, and is calculated as the
maximum change in voltage magnitude between one full load flow iteration. Figure 5 shows a
flow chart representation of the full load flow routine depicted in Sato and Arillagas methodology.
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21
Figure 5. Flowchart of Sato and Arillagas full load flow routine
2.3 Modifications to Sato and Arillagas Method
A few modifications are now proposed to Sato and Arillagas method in order to increase
the speed of the algorithm, as well as allow it to be used for multi-link HVDC systems. A simplified
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22
DC subroutine was developed to ensure an equivalence of load flow solution between a pure AC
network and an integrated AC-DC network and to reduce computation time. The double loop
approach was also used for the full load flow algorithm, with only the Gauss Seidel and Newton-
Raphson methods used to solve for the network bus voltages. A modified AC routine was also
included to deal with instances where the placement of HVDC links results in an isolation of a
bus, in which traditional AC load flow methods are incapable of calculating the bus voltage. The
following sections discuss the details of the modifications made to Sato and Arillagas method.
2.3.1 Simplified DC Load Flow Routine
For the purposes of this thesis, it is convenient to assume an ideal HVDC link capable
producing the same power flows as the original AC transmission line. By assuring the HVDC link
provides the same power flow, we eliminate non-convergence of the load flow algorithm due to
an infeasible demand by the HVDC link. As is, Sato and Arillagas improved simulation of an
HVDC link returns power flows close to, but not equivalent to, the exact power flows calculated
from a full AC load flow routine on the original AC power system. Thus, a simplified DC load
flow routine was utilized.
To ensure exactness, the original power flows of a pure AC network was first determined
using MATPOWERs netwon raphson function, and stored in memory. For a specific transmission
line, the bus in which the power is flowing from (i.e. the negative line flow) was set as the rectifier
end of the HVDC link. Likewise, the bus in which the power was flowing to (i.e. the positive line
flow) was considered to be the inverter end. The impedance of the line was also stored in memory,
and thus we have now stored Srect, Sinv, and Rline.
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23
Given the initialized bus voltages, the ac voltages on the high side of the converter
transformers can be calculated from the transformer tap positions. Using the number of tap
positions and the regulation range of the transformer, the tap position correlates to a percentage of
the low side voltage, which will be called the operating regulation. That is
=+
(#1)
(1)
100 (14)
where regmin and regmax are the minimum and maximum regulation percentages of the
transformer, and t is the correlated percentage of low side voltage or operating regulation. The
initial tap position of the transformers is set such that the high side and low side voltages are
equivalent (i.e. t = 1). Using the newly calculated t values, the high side voltages can be calculated
as
, = , (15)
, = , (16).
Because we have the expected apparent powers Srect and Sinv, we are able to calculate the
DC current using the expected active powers taken as the real component of Srect and Sinv.
= ||||||/ (17)
Using the calculated line current, the inverter and rectifier DC voltage can be calculated as follows:
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24
= / (18)
= + (19)
From analysis of equation (17), it is clear that if the impedance of the line has a real part
equal to zero, such that the original transmission line has no resistance, a division of zero occurs.
Although in reality all transmission lines contain a resistance, data provided for the benchmark
power systems used in this thesis round down to zero for small numbers. To deal with this issue,
when the impedance values are stored in memory, all elements containing a zero real part are set
to a fictitious value of 0.001 p.u. This assures that the division of zero is avoided. Another issue
that may occur is that the original power flow of the AC transmission line would yield no real line
loss, or in other words, Prect = Pinv, and we obtain a zero line current. A loss of 0.2W is added to
all instances in which this issue would occur.
The power transfer at the converter terminals are calculated using equations (10-13). The
resulting values have a real part equal to the expected Pinv and Prect values. It is assumed that a
synchronous compensator or capacitor bank is included in the HVDC link which is able to balance
the imaginary component of Srect and Sinv to be equivalent to the original load flows. Techniques
to do this are discussed in Appendix A. Note that since the converter powers are forced to be equal
to the original load flow values, recalculating Srect and Sinv seems to be redundant. However, if one
wished to calculate the unknown voltages from the network representation of the converter
interface as explained in the original Sato and Arillaga DC subroutine, the values of the
compensator/capacitor power is needed alongside the power transfers at the converter terminals.
For this thesis, these values are not needed and thus not included in the modified routine.
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25
As a final step, the tap positions of the transformers are updated. The new operating
regulation value is calculated as:
= ( +
3
,
32
cos()||
) 100 (20)
= ( +
3
,
32
cos()||
) 100 (21)
If the new operating regulation value is less than the minimum regulation, the tap position is set to
the lowest tap position as to not exceed the limitations of the transformer. Likewise, if the value is
greater than the maximum regulation, the tap is set to the highest tap position. Otherwise, the tap
is set to the corresponding position which allows the desired regulation value.
The modifications made to the DC subroutine to ensure the one-to-one correspondence
with the original power flows allow for a shorter DC subroutine time. Not only is the complexity
of the equations reduced, but also the optimization process of the operating voltage and the
Newton-Raphson approach to solve for the internal voltages are eliminated, further reducing the
computation time. Figure 6 details the simplified DC subroutine.
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26
Figure 6. Simplified DC subroutine developed in this thesis
2.3.2 Isolated Bus Load Flow for AC Routine
Once the DC subroutine is complete for a particular HVDC link, the AC transmission line is
removed from the power system and power sources equivalent to the calculated power transfers
are placed on the bus terminals, as shown in Figure 7. For a single HVDC link, it is rare that an
issue occurs when an AC load flow method is run to update the bus voltages. All buses are
connected via AC transmission lines and thus all bus voltages can be updated via traditional
methods.
Figure 7. Power system with included HVDC link (left) and substitution of equivalent power
sources for AC routine (right)
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27
However, this thesis is concerned with multiple point-to-point HVDC links throughout an
existing AC network. For a particular arrangement of HVDC links, such that all transmission lines
connected to a particular bus are now HVDC links, the substitution of equivalent power sources
will introduce an isolated bus to the network, as shown in Figure 8. Although the bus is not actually
isolated (as in the HVDC links still connect the bus to the rest of the system), AC load flow
methods are not capable of solving for the isolated bus voltage. This introduces an issue with Sato
and Arillagas method when solving multi-link systems. A simple modification of the AC load
flow methods is able to deal with the issue of isolated buses.
Figure 8. Power system with multiple HVDC links tied to one bus (left) resulting in
isolated bus during substitution (right)
From the load flow solution of the power system prior to the inclusion of HVDC links, the
generation, load, and voltage into a bus are all known values. Using this, we are able to calculate
the equivalent impedance of the ith bus as follows:
, =,
2
,+, (22)
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28
If a bus has no generator and load, such that the denominator of equation (22) would be zero, the
denominator is set to a value of 1 ensuring the calculation of an equivalent impedance. A flag is
also assigned to these buses indicating that there is no attached generation and load. All equivalent
impedances for each bus are stored in memory. Now if an isolation occurs, a separate subroutine
is used to calculate the voltage of the isolated bus as follows:
, = , (,) (23)
where Sinjections are the sum of the power transfer of the converters. Regarding the flagged buses,
Sinjecctions would be equivalent to zero, and thus to ensure equation (23) results in the correct bus
voltage, Sinjections is set to -1.
2.3.3 Full Double Loop Approach
Sato and Arillaga stated in their results that between the two full load flow methodologies,
the double loop approach provided a much faster convergence [3]. This is because the multiple
iterations of AC load flow completed before returning to the DC subroutine updates the bus
voltages such that the DC subroutine is capable of calculating more accurate power transfers at the
converter terminals. Also stated in the paper is that the iteration count is high enough, a converged
AC load flow may occur.
Leveraging of Sato and Arillagas findings, it is clear that full convergence of an AC load
flow method to update the bus voltages should be done before returning to the DC subroutine.
Thus, instead of using a high-iteration count in the double loop approach, it was decided that as
many iterations required for convergence should be completed before returning to the DC
subroutine. This assures that redundant iterations of AC load flow are not performed, as well as
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29
assuring the best approximation of bus voltages is provided before returning to the DC subroutine,
ultimately providing a much faster convergence as explained by Sato and Arillaga. Figure 9 details
the modified Sato and Arillagas method used in this thesis.
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30
Figure 9. Modified Sato and Arillagas Full Load Flow Method
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31
2.3.4 Example Case: Grainger and Stevensons 4-Bus System
Several different power systems are available in MATPOWERs database. The simplest of
the power systems is a 4-Bus system used in Power Systems Analysis by Grainger and
Stevenson [6]. Figure 10 shows the layout of the power system and Tables 1 and 2 list the power
system details, with a bases of 100MVA and 230kV.
Figure 10. One-line diagram for Grainger and Stevensons 4-bus system
Table 1. Line Data for Grainger and Stevensons 4-Bus System
From bus
To bus
G (per unit)
B (per unit)
Shunt Total Charging (Mvar)
Shunt Y/2 (per unit)
1 2 3.815 -19.08 10.25 0.05125 1 3 5.170 -25.85 7.750 0.03875 2 4 5.170 -25.85 7.750 0.03875 3 4 3.024 -15.12 12.75 0.06375
Table 2. Bus Data for Grainger and Stevensons 4-Bus System
Bus Generated
P (MW) Generated Q (Mvar)
Load P (MW)
Load Q
(Mvar) V
(per unit) Bus
Type
1 - - 50 30.99 1 + 0j Slack 2 0 0 170 105.35 1 + 0j PQ 3 0 0 200 123.94 1 + 0j PQ 4 318 - 80 49.58 1 + 0j PV
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32
The first step in the load flow algorithm is to calculate the load flows of the original
unaltered AC power system and store them in memory. This is done using MATPOWERs
Newton-Raphson function. Leveraging off the calculated voltages, the equivalent impedances are
found using equation 22. The results of the load flow are summarized in Tables 3 and 4, and the
calculated equivalent impedances are found in Table 5.
Table 3. Calculated Bus Data from Load Flow Solution of 4-Bus System
Bus Generated
P (MW) Generated Q (Mvar)
Load P (MW)
Load Q (Mvar)
V magnitude (per unit)
V phase (deg)
Bus Type
1 186.81 114.5 50 30.99 1 0 Slack 2 0 0 170 105.35 0.982 -0.976 PQ 3 0 0 200 123.94 0.969 -1.872 PQ 4 318 181.43 80 49.58 1.02 1.523 PV
Table 4. Calculated Line Flows from Load Flow Solution of 4-Bus System
From Bus To Bus P (MW) Q (Mvar)
1 2 38.69 22.3 1 3 98 61.21 2 1 -38 -31.24 2 4 -132 -74.11 3 1 -97 -63.57 3 4 -103 -60.37 4 2 133 74.92 4 3 104.75 56.93
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33
Table 5. Equivalent Impedances of 4-Bus System
Bus Equivalent Impedance (per
unit)
1 0.5325 - 0.3241j 2 -0.4013 + 0.2680j 3 -0.3248 + 0.2319j 4 0.3439 - 0.1673j
Now that the expected power flows and equivalent impedances are stored in memory, a
replacement of one transmission line for an HVDC link is made. In particular, the line connecting
bus 3 and bus 4 is replaced as shown in Figure 11. Parameters for the HVDC link are shown in
Table 6. The parameters were chosen such that it would be a rare occurrence for the HVDC link
to be incapable of operating at the desired power settings. It should be noted that all HVDC links
in this thesis have the same parameters.
Figure 11. One-line diagram of 4-Bus system with HVDC link connecting bus 3 and 4
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34
Table 6. HVDC Parameters
Parameter Inverter (values)
Rectifier (values)
Initial Tap Position 16 16 Number of Taps 31 31
Minimum Regulation (%) 85 85 Maximum Regulation (%) 115 115
Delay/Extinction (degrees) 10 10 Minimum Delay/Extinction
(degrees) 10 10 Amplification Factor (p.u) 53.71 53.71
Commutation Reactance, Xc, (p.u) 0.07275 0.07275
Once the substitution is made, the modified Sato and Arillaga method is now run to obtain
the load flow results of the system using the Newton Rapshon method for the AC routine. Keeping
the current layout of the power system, the next substitution is made for the line connecting bus 2
and bus 4, and a load flow result is obtained with the two links in place. The procedure is continued
for a HVDC link substitution for line connecting bus 1 and bus 3, followed by substitution of the
last remaining line. The results of each of the load flows is detailed in Table 7 and Table 8. As
shown, the calculated voltages and load flows are equivalent to those of the original AC system in
Tables 3 and 4, showing that the modified methodology works as desired.
Table 7. Calculated Bus Voltages for Systems with HVDC Links
Voltage Magnitude at Bus # (p.u) Voltage Angle at Bus # (degrees)
# of HVDC Links 1 2 3 4 1 2 3 4
1 1 1 1 1 0 0 0 0
2 0.982 0.982 0.982 0.982 -0.976 -0.976 -0.976 -0.976
3 0.969 0.969 0.969 0.969 -1.872 -1.872 -1.872 -1.872
4 1.02 1.02 1.02 1.02 1.523 1.523 1.523 1.523
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35
Table 8. Calculated Converter Powers and Line Flows for Systems with HVDC Links
Converter Powers (MW)
1 HVDC Link 2 HVDC Links 3 HVDC Link 4 HVDC Link
Rectifier Bus
Inverter Bus Rectifier Inverter Rectifier Inverter Rectifier Inverter Rectifier Inverter
4 3 104.74 102.91 104.74 102.91 104.74 102.91 104.74 102.91 4 2 - - 133.25 131.53 133.25 131.53 133.25 131.53 1 3 - - - - 98.117 97.086 98.117 97.086 1 2 - - - - - - 38.691 38.465
AC Line Flows
From Bus To Bus P(MW) Q(Mvar) P(MW) Q(Mvar) P(MW) Q(Mvar) P(MW) Q(Mvar)
1 2 38.69 22.29 38.69 22.29 38.69 22.29 - - 2 1 -38.46 -31.23 -38.46 -31.23 -38.46 -31.23 - - 1 3 98.11 61.21 98.11 61.21 - - - - 3 1 -97.08 -63.56 -97.08 -63.56 - - - - 2 4 -131.53 -74.113 - - - - - - 4 2 133.25 74.91 - - - - - -
2.3.5 Ordering of HVDC Link Substitutions
It is clear that the order of substitutions of transmission lines for HVDC links can vary. For
the 4-bus system described in the previous section there are 24 different orderings for the
substitutions. For an N-line system, there are N! different permutations of the orderings. As the
size of the system grows, the number of permutations grows drastically.
It was found that certain orderings caused a non-convergence of the load flow algorithm.
These orderings were such that the substitutions of power sources for the AC routine resulted in
multiple disjoint sub-systems, as depicted in Figure 12. Because there are multiple disjoint sub-
systems, a traditional AC load flow method is required to solve for each sub-system. However, it
is not guaranteed that the load flow of the sub-systems can be solved for. Some sub-systems may
contain no generator buses to serve as the slack bus, while others may just have an inadequate
slack bus capable of supporting the AC load flow method.
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36
Figure 12. Placement of HVDC links (left) resulting in multiple disjoint sub-systems (right)
To deal with this issue, it was decided that all orderings which resulted in disjoint sub-systems
would be avoided by finding the minimum spanning tree of the connected buses using MATLABs
built in functions. The minimum spanning tree of a power system is a subset of all the transmission
lines which uses the minimum amount of lines required to connect all buses in the system, such
that removal of one line in the minimum spanning tree results in two systems, as shown in Figure
13.
Figure 13. Example of minimum spanning tree in which removal of line results in two systems
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37
All remaining lines not contained in the minimum spanning tree are links, and permutations
of the links can be done such that disjoint sub-systems are not formed if a substitution is made for
any ordering of the links. Once all the transmission lines considered to be links are replaced with
HVDC links, the lines furthest from the identified slack node are removed, forming isolated buses.
This assures that no disjoint systems are formed and an adequate slack is provided to support the
AC load flow of the remaining larger system. As such, the ordering of any power system is detailed
in Figure 14. The number of different permutations is now equivalent to M!, where M is the
number of links in the power system.
Figure 14. Ordering of HVDC Link Substitutions
It should be noted that there may exist multiple minimum spanning trees for a particular power
system. However, MATLABs graphminspantree function returns only one solution determined
by Prims algorithm, and this is used as the minimum spanning tree for the ordering of substitutions
of HVDC links.
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38
3 CONVERGENCE ANALYSIS OF LOAD FLOW FOR
DIFFERENT POWER SYSTEMS
3.1 Overview
One of the deliverables of this thesis is the investigation of the influence of multiple HVDC
links on the rate of convergence of different load flow methods. However, it should be mentioned
that the rate of convergence is also influenced by the power system size. Depending on the number
of transmission lines and buses, there may be influences on the number of iterations required for
convergence. Specifically, the total time for the entire load flow, the time devoted to the DC
subroutine, the time for the entire AC load flow method, the time for the isolated bus load flow,
and the time for the connected power system AC load flow (i.e. Gauss Seidel or Newton Raphson)
were measured. The number of full load flow iterations and the number of AC load flow iterations
were also measured. A better understanding of the routines being measured can be obtained by
inspection of Figure 9.
Numerous different power systems can be found in MATPOWERs database. As mentioned
in the previous chapter, Grainger and Stevensons 4-Bus system is the simplest and was used to
test the load flow analysis tool. In addition to the 4-Bus system, the IEEE 9-Bus system, IEEE 14-
Bus system, IEEE 30-Bus system and IEEE 118-Bus system were used as test cases. By using
these systems of varying sizes and substituting lines for HVDC links, similar trends amongst the
systems as well as unique trends (i.e. dependent to the system architectures) could be found
regarding the rate of convergence. Data for the power systems can be found in MATPOWERs
casefiles [21] and are shown in Appendix B. Code written can be found in Appendix C.
Although there are several ways to quantify convergence, this thesis looks into the number
of iterations and the solution time as metrics for the convergence analysis. Plots were generated
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39
using these metrics with respect to the number of HVDC links in a system. Because the program
was run on a personal PC, fluctuations in convergence times occurred between runs of the same
orderings due to non-processor isolation. To overcome this, 500 trials of the same run were
averaged together for systems with only one set ordering (i.e 4-bus and 9-bus). It was also found
that the different orderings due to the permutation of the link transmission lines led to similar
trends with negligible differences, and thus only the first 500 orderings were used and averaged
together for systems with multiple orderings (i.e 14-bus, 30-bus, and 118-bus), also reducing the
error from the non-processor isolation. This chapter details the findings for the various power
systems.
3.2 Grainger and Stevensons 4-Bus System
Grainger and Stevensons 4-Bus system which is depicted in Figure 10 was mainly used as
a test system to ensure the validity of the load flow analysis tool. It is a radial system with two
generators. The system only has four transmission lines to be replaced with HVDC links, and thus
the minimum spanning tree to connect all four buses is three lines. Due to the ordering scheme
explained in section 2.3.5, only one ordering of substitutions for HVDC links is determined: line
connecting buses 3 and 4, then 2 and 4, 1 and 3, and finally 1 and 2. This ordering results in the
second generator bus being isolated after the second HVDC link is placed and the slack bus being
isolated at the 4th HVDC link.
The time it takes to run the load flow method using the Newton-Raphson method and Gauss
Seidel method for the AC routine is shown in Figure 15. As shown, there are three main
components to the partition of the total time: the time for the AC routine (i.e. Newton-Raphson or
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Gauss Seidel), the time for the DC subroutine, and the time for additional computation (i.e.
processing of variables for the program).
Figure 15. Time diagram for full load flow routine of 4-bus system using (a) the Newton-
Raphson method and (b) Gauss Seidel method for the AC routine
With regards to the total time, it is shown that there is an increase of 1.2% in total time for
the Newton-Raphson method when the 1st HVDC link is placed and 1% when the 2nd HVDC link
is placed. The 3rd HVDC link results in a decrease of 6.5% in total time, followed by a dramatic
80% decrease. For the Gauss Seidel method, there is a 5.9% increase in total time for the 1st HVDC
link, followed by a decrease of 21.5%, 7.2%, and 78.8% for the 2nd, 3rd, and 4th HVDC links
respectively. Also, in comparison of the two methods it seems as if the Newton-Raphson approach
is slightly faster than the Gauss Seidel approach as expected.
From inspection of the plots, it is clear that the total time is dominated by the AC routine.
This is confirmed by the dramatic decrease in total time when all the transmission lines are replaced
with HVDC links, and the non-iterative AC load flow routine for isolated buses and the DC
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subroutine are dominant in the algorithm. Both the DC subroutine and additional computation
make (at most) 20% of the total time. However, it should be noted that the spike in DC time which
occurs at 1 HVDC link is a resultant of the creation of DC subroutine variables. Declaring the
variables prior to the load flow routine would eliminate the spike causing the total time to follow
a trend similar to the AC routine.
An interesting phenomena is the spike in the time required for the AC routine once the first
transmission line in the minimum spanning tree is substituted for an HVDC link for the Newton-
Raphson method. For Grainger and Stevensons 4-bus system this occurs at 2 HVDC links since
the minimum spanning tree contains three transmission lines and there are only four total
transmission lines in the system. Looking at Figure 15, it can be calculated that the total increase
in AC time is about 11.6%. In contrast, utilizing the Gauss Seidel method for the AC routine results
in the largest decrease (not including the jump from one remaining AC transmission line to full
HVDC transmission) of approximately 18.8% once the minimum spanning tree is reached. The
point where the minimum spanning tree is reached is also when a generator bus is isolated. Also
noted is the decrease in AC time once more lines in the minimum spanning tree are replaced for
either AC load flow methodology.
The timing partition of the AC routine can be found in Figure 16. It is clear that the total
AC time is dominated by the load flow of the remaining connected power system (i.e. system
buses) up until all buses are isolated, in which the isolated bus routine is the only routine taking
time. Because the isolated bus subroutine is algebraic equations, there is minimal time required to
do the calculations. The extra time deals with the processing of variables for the AC routine.
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Figure 16. Time plot for AC load flow routine of 4-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine
An explanation for the trends described for the system buses portion of the AC routine has
to do with the iterative process of the numerical methods. The number of iterations for the methods
required for convergence is shown in Figure 17. When the first HVDC link is included, the number
of AC iterations remains constant, while the total time for the system bus routine decreases for the
Newton-Raphson method. This is because the admittance matrix is now sparser (due to the removal
of a line) and therefore the jacobian is sparser. MATPOWER is a toolbox for MATLAB which
leverages off sparseness of matricies for faster computation, resulting in the reduction in time.
Once the minimum spanning tree is reached with the second HVDC link, the number of iterations
increases dramatically resulting in an increase in time (i.e. more iterations leads to more
computation time). A decrease in time is again seen at the 3rd HVDC link because of the increased
sparseness of the jacobian.
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Figure 17. Iteration plots for full load flow method and AC load flow routine of 4-bus system
using Newton-Raphson method (a and b) and Gauss Seidel method (c and d)
The Gauss Seidel method utilizes a different approach to load flow which is sensitive to
the amount of non-zero terms in the admittance matrix. More iterations are required for
convergence once 1 HVDC link is included, resulting in the increase in total time for the system
bus routine. When the minimum spanning tree is reached, the number of iterations drops drastically
because the calculation of one bus is removed entirely, and the remaining equations all drop one
term (i.e. terms concerning bus 4). This is why there is a drastic drop in time as well. The third
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HVDC link results in another isolation and thus another removal of an equation, causing the
continued decrease in iteration count and time.
It is shown that the choice of AC routine does not affect the number of full load flow
iterations required for convergence. In fact, it is shown that for the 4-bus system, 2 full load flow
iterations are required for convergence up until the system consists entirely of HVDC links, when
only 1 iteration is required. This is probably due to the full convergence of the AC routine prior to
returning to the DC subroutine. Only 1 iteration is required for a system consisting entirely of
HVDC links because the AC routine for isolated buses utilizes the expected final voltages when
calculating the equivalent impedance.
3.3 IEEE 9-Bus System
The IEEE 9-Bus system shown in Figure 18 is a radial system with three generators and 9
transmission lines. The minimum spanning tree of the system consists of 8 transmission lines to
connect the 9 buses. Thus, due to the ordering scheme described in section 2.3.5, only the following
ordering is considered: 7-8, 2-8, 6-7, 3-6, 8-9, 5-6, 4-9, 4-5, and 1-4. It should be noted that the
first generator bus is isolated after the 2nd HVDC link and the second generator bus after the 4th
HVDC link, with the slack generator isolated at the 9th HVDC link.
In addition, the following plots (and plots for larger systems) should be taken as discretized
data. Although the plots are portrayed as continuous, data was only gathered for integer values of
HVDC links. It was chosen to remove the data markers for aesthetic reasons, and to ensure the rate
of change in convergence was emphasized for increased HVDC link transmission.
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Figure 18. IEEE 9-Bus System [22]
The total time for the load flow method to converge for the 9-bus system follows similar
trends as the 4-bus system for both the Newton-Raphson and Gauss Seidel methods for the AC
routine as shown in Figure 19. The total time remains dominated by the AC routine and follows
the trend strongly. Time allocated for the DC subroutine remains under 2ms although the system
size increases, while the time for the AC routine increased. It is easier to see now that there are
more transmission lines that the time allocated for the DC subroutine has an overall increasing
trend. Again, it is shown that the Newton-Raphson method is superior in required time for
convergence in comparison to the Gauss Seidel method. Also noted is the spike in total time and
AC time for the Newton-Raphson method once the minimum spanning tree is reached at 2 HVDC
links. Unlike the 4-bus system, the Gauss Seidel method does not contain the largest decrease in
total time once the minimum spanning tree is met.
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Figure 19. Time diagram for full load flow routine of 9-bus system using (a) the Newton-
Raphson method and (b) Gauss Seidel method for the AC routine
The partition of the total AC routine time for the 9-bus system is shown in Figure 20. Again
the timing is dominated by the system bus routine with the isolated bus routine taking up negligible
time. It is recognized that there seems to be four different pieces to both AC routine plots once
HVDC links are introduced. The pieces occur at 1 to 2 HVDC links, 2 to 4 HVDC links, 4 to 8
HVDC links and then 8 to 9 HVDC links. This is clearly seen in the plot for the Gauss Seidel
method as the slope for the total AC time subtly changes for these sections from a decrease of
0.00399 s/link, to 0.0066 s/link, to 0.001461 s/link, and finally to 0.006188 s/link. For the Newton-
Raphson method, the slope changes from an increase of 0.001344 s/link, to a decrease of 0.002085
s/link, to a decrease of 0.000733 s/link, to a decrease of 0.00634 s/link. The end point of these
pieces of plot are also representative of the isolation of generator buses at 2 HVDC links, 4 HVDC
links, and 9 HVDC links.
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Figure 20. Time plot for AC load flow routine of 9-bus system using (a) the Newton-Raphson
method and (b) Gauss Seidel method for the AC routine
Further analysis of the iteration plots for the 9-bus system show similar pieces of plot as
seen in Figure 21. The Gauss Seidel method contains the exact same pieces of plot for the number
of AC load flow iterations as compared to the AC time partition. It should be noted however that
the spike which occurred in the 4-bus system when the first HVDC link was introduced is not seen
for the 9-bus system. Instead, there is only a decrease in iteration count as more HVDC links are
included. This leads to a similar trend in total AC time.
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Figure 21. Iteration plots for full load flow method and AC load flow routine of 9-bus system
using Newton-Raphson method (a and b) and Gauss Seidel method (c and d)
The plot for the number of iterations required for convergence of the Newton-Raphson
method shows a different trend than the AC time plot. However, the resulting iteration plot still
coincides with the AC time plot the number of iterations jumps at 2 HVDC links resulting in the
spike in AC time, and increased sparsity of the Jacobian occurs as more HVDC links are included
resulting in the time decrease although the number of iterations remains somewhat constant.
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Also noted from Figure 21 is that the number of full load flow iterations follows a trend
similar to the 4-Bus system. Two full load flow iterations are required for convergence regardless
of the number of links, up until the system consists soley of HVDC links when only one full load
flow iteration is required.
3.4 IEEE 14-Bus System
The IEEE 14-Bus system shown in Figure 22 was used as the test case for Sato and
Arillagas paper [3]. It contains in total 5 generator buses (i.e. slack, one generator, and three
condensors) and 20 transmission lines. The minimum spanning tree is 13 lines in total, and thus a
system with 8 or more HVDC links results in isolations. There are 7 link transmission lines and
thus 7! different orderings. Generator bus isolations occur at 10 HVDC links, 16 HVDC links, 18
HVDC links, and 20 HVDC links.
Figure 22. IEEE 14-Bus System [23]
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The time partition plots for the 14-bus system shown in Figure 23. Although subtle, there
i