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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

ii

Copyright 2014

Shaun Mendoza Cruz. All Rights Reserved

iii

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

v

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

vi

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

viii

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


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

ix

Figure 18. IEEE 9-Bus System [22] ............................................................................................. 45





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 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 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 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

xi

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

xii

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.

1

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,

2

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

3

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

4

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

5

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

6

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

7

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

8

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.

9

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

10

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

11

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

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].

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

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.

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

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.

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.

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.

19

Figure 4. Flow chart of Sato and Arillagas DC subroutine

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.

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

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.

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:

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.

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.

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)

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)

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

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.

30

Figure 9. Modified Sato and Arillagas Full Load Flow Method

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

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

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

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

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.

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

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.

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

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

40

Gauss Seidel), the time for the DC subroutine, and the time for additional computation (i.e.

processing of variables for the program).


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

41

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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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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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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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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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.

48


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.

49

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

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