load flow analysis of multi-converter transmission systems · pdf file ·...

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 Arillaga’s Method............................................................................................ 15

2.2.1 DC Subroutine ........................................................................................................ 16

2.2.2 Full Load Flow Routine .......................................................................................... 20

2.3 Modifications to Sato and Arillaga’s 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 Stevenson’s 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 Stevenson’s 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 Arillaga’s 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 Stevenson’s 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 Arillaga’s HVDC link model .......................................................................... 17

Figure 4. Flow chart of Sato and Arillaga’s DC subroutine ......................................................... 19

Figure 5. Flowchart of Sato and Arillaga’s 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 Arillaga’s Full Load Flow Method ................................................. 30

Figure 10. One-line diagram for Grainger and Stevenson’s 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


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 Stevenson’s 4-Bus System ................................................. 31

Table 2. Bus Data for Grainger and Stevenson’s 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.

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 today’s 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 Arrillaga’s 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 Arrillaga’s method. Leveraging off the findings, the

development of a load flow solver which extends Sato and Arrillaga’s 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 Arrillaga’s 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:

𝑃𝑖 = ∑ |𝑌𝑖𝑛𝑉𝑖𝑉𝑛| cos(𝜃𝑖𝑛 + 𝛿𝑛 − 𝛿𝑖)𝑁𝑛=1 (1)

𝑄𝑖 = − ∑ |𝑌𝑖𝑛𝑉𝑖𝑉𝑛| sin(𝜃𝑖𝑛 + 𝛿𝑛 − 𝛿𝑖)𝑁𝑛=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, θin 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, δi and δn 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 = 𝑃𝑖 − 𝑃𝑖,𝑠𝑐ℎ𝑒𝑑 = 𝑃𝑖 − (𝑃𝑔𝑖 − 𝑃𝑑𝑖) (3)

0 = 𝑄𝑖 − 𝑄𝑖,𝑠𝑐ℎ𝑒𝑑 = 𝑄𝑖 − (𝑄𝑔𝑖 − 𝑄𝑑𝑖) (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

7

and Δ-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 Arillaga’s 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 Carré 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 Carré’s 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 Arillaga’s 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 Arrillaga’s 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

Arrillaga’s 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

Arrillaga’s 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 Arillaga’s 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 Arrillaga’s 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 Arrillaga’s 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 Arillaga’s 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 Arillaga’s 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 Arillaga’s 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 Arillaga’s 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 Arillaga’s 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 Arillaga’s 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:

𝑉𝑖𝑛𝑣 = 3√2

𝜋 𝑣𝑖𝑛𝑣 cos 𝛾 −

3

𝜋𝑋𝑖𝑛𝑣𝐼𝑙𝑖𝑛𝑒 (5)

𝑉𝑙𝑖𝑛𝑒 = 𝑉𝑟𝑒𝑐𝑡 − 𝑉𝑖𝑛𝑣 = 𝑅𝑙𝑖𝑛𝑒𝐼𝑙𝑖𝑛𝑒 (6)

𝑉𝑟𝑒𝑐𝑡 = −𝐴𝑟𝑒𝑐𝑡{𝐼𝑙𝑖𝑛𝑒 − 𝐼𝑑𝑠𝑟𝑒𝑐𝑡} (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, γ 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 Arillaga’s HVDC link model

Combining equations (5-7) the equation for the operating current can be found to be:

𝐼𝑙𝑖𝑛𝑒 = 3√2

𝜋 𝑣𝑖𝑛𝑣 cos 𝛾−𝐼𝑑𝑠𝑟𝑒𝑐𝑡𝐴𝑟𝑒𝑐𝑡

3

𝜋𝑋𝑖𝑛𝑣−{𝑅𝑙𝑖𝑛𝑒−1}𝐴𝑟𝑒𝑐𝑡

(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

𝑢 = 𝑉𝑦 − 𝑉𝑥 > 0 (9a)

where 𝑉𝑦 =𝐾

𝑉𝑥 (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:

𝑆𝑟𝑒𝑐𝑡 = 𝑉𝑟𝑒𝑐𝑡𝐼𝑙𝑖𝑛𝑒 + 𝑗𝑉𝑟𝑒𝑐𝑡𝐼𝑙𝑖𝑛𝑒 tan 𝜙𝑟𝑒𝑐𝑡 (10)

𝑆𝑖𝑛𝑣 = 𝑉𝑖𝑛𝑣𝐼𝑙𝑖𝑛𝑒 + 𝑗𝑉𝑖𝑛𝑣𝐼𝑙𝑖𝑛𝑒 tan 𝜙𝑖𝑛𝑣 (11)

where tan 𝜙𝑘 = 𝑠𝑔𝑛(𝑘)2𝑢𝑘+sin 2𝜃𝑘−sin 2{𝜃𝑘+𝑢𝑘}

cos 2𝜃𝑘−cos 2{𝜃𝑘+𝑢𝑘} (12)

𝑢𝑘 = cos−1{(2𝜋)𝑉𝑘

(3√2)𝑣𝑘− cos 𝜃𝑘} − 𝜃𝑘 (13)

such that 𝑠𝑔𝑛(𝑖𝑛𝑣) = −1, 𝑠𝑔𝑛 (𝑟𝑒𝑐𝑡) = 1, 𝑎𝑛𝑑 𝜃𝑘 is equal to the firing angle α or extinction

angle γ 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 Arillaga’s method. Figure 4 shows a flow chart representation of the DC

subroutine.

19

Figure 4. Flow chart of Sato and Arillaga’s DC subroutine

20

2.2.2 Full Load Flow Routine

The full load flow routine depicted in Sato and Arillaga’s 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 Arillaga’s methodology.

21

Figure 5. Flowchart of Sato and Arillaga’s full load flow routine

2.3 Modifications to Sato and Arillaga’s Method

A few modifications are now proposed to Sato and Arillaga’s 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 Arillaga’s 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 Arillaga’s 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 MATPOWER’s 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

𝑡 =𝑟𝑒𝑔𝑚𝑖𝑛+

𝑟𝑒𝑔𝑚𝑎𝑥−𝑟𝑒𝑔𝑚𝑖𝑛(#𝑜𝑓𝑡𝑎𝑝𝑠−1)

∗(𝑡𝑎𝑝𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛−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

𝑣𝑖𝑛𝑣,ℎ𝑖𝑔ℎ = 𝑣𝑖𝑛𝑣,𝑙𝑜𝑤 ∗ 𝑡𝑖𝑛𝑣 (15)

𝑣𝑟𝑒𝑐,ℎ𝑖𝑔ℎ = 𝑣𝑟𝑒𝑐,𝑙𝑜𝑤 ∗ 𝑡𝑟𝑒𝑐 (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.

𝐼𝑙𝑖𝑛𝑒 = √||𝑃𝑟𝑒𝑐𝑡|−|𝑃𝑖𝑛𝑣||/𝑅𝑙𝑖𝑛𝑒 (17)

Using the calculated line current, the inverter and rectifier DC voltage can be calculated as follows:

24

𝑉𝑖𝑛𝑣 = 𝑃𝑖𝑛𝑣/𝐼𝑙𝑖𝑛𝑒 (18)

𝑉𝑟𝑒𝑐𝑡 = 𝑉𝑖𝑛𝑣 + 𝑅𝑙𝑖𝑛𝑒 ∗ 𝐼𝑙𝑖𝑛𝑒 (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:

𝑡𝑖𝑛𝑣 = 𝑐𝑒𝑖𝑙 ( 𝑉𝑖𝑛𝑣+

3

𝜋𝑥𝑐,𝑖𝑛𝑣𝐼𝑙𝑖𝑛𝑒

3√2

𝜋cos(𝛾)∗|𝑣𝑖𝑛𝑣|

) 𝑥 100 (20)

𝑡𝑟𝑒𝑐 = 𝑐𝑒𝑖𝑙 ( 𝑉𝑟𝑒𝑐+

3

𝜋𝑥𝑐,𝑟𝑒𝑐𝐼𝑙𝑖𝑛𝑒

3√2

𝜋cos(𝛼)∗|𝑣𝑟𝑒𝑐|

) 𝑥 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 Arillaga’s 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 i’th bus as follows:

𝑍𝑏𝑢𝑠,𝑖 =𝑉𝑏𝑢𝑠,𝑖

2

𝑆𝑔𝑒𝑛,𝑖+𝑆𝑙𝑜𝑎𝑑,𝑖 (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:

𝑉𝑏𝑢𝑠,𝑖 = √𝑍𝑏𝑢𝑠,𝑖 ∗ (−𝑆𝑖𝑛𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠,𝑖) (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 Arillaga’s 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 Arillaga’s method used in this thesis.

30

Figure 9. Modified Sato and Arillaga’s Full Load Flow Method

31

2.3.4 Example Case: Grainger and Stevenson’s 4-Bus System

Several different power systems are available in MATPOWER’s database. The simplest of

the power systems is a 4-Bus system used in “Power System’s 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 Stevenson’s 4-bus system

Table 1. Line Data for Grainger and Stevenson’s 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 Stevenson’s 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 MATPOWER’s

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 MATLAB’s

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, MATLAB’s graphminspantree function returns only one solution determined

by Prim’s 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 MATPOWER’s database. As mentioned

in the previous chapter, Grainger and Stevenson’s 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 MATPOWER’s

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 Stevenson’s 4-Bus System

Grainger and Stevenson’s 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 Stevenson’s 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.

42


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.

43


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

44

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.

45

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.

46



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.

47



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



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.


The IEEE 14-Bus system shown in Figure 22 was used as the test case for Sato and

Arillaga’s 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.


50

The time partition plots for the 14-bus system shown in Figure 23. Although subtle, there

is a slight increase in time (4.1%) for the Newton-Raphson plots and a decrease in time (23%) in

the Gauss Seidel plots of total time once the minimum spanning tree is reached. The DC time for

both methods still contains the spike in time once the first link is introduced. For both the AC time

and DC time, it is shown that the time essentially doubles once the 14th HVDC link is put into

place. This has to do with the iteration counts which will be explained later.



Plots of the AC time partition for the 14-bus system are shown in Figure 24. The Gauss

Seidel method plot highlights some key events that occur from substitution of for HVDC links. A

general increase in time occurs prior to the minimum spanning tree being met at the 8th HVDC

link. The first generator bus is isolated at the 10th link which results in a change of slope. Subtle

changes in slope also occur at 16 and 18 HVDC links, which result in the other generator buses

being isolated. For the Newton-Raphson method, an increase in time occurs at the 8th HVDC link,

51

and subtle changes in slope are seen at the points where generation buses are isolated. More

apparent changes in slope are not seen as it was in the 9-bus and 4-bus system for either AC

method, which may indicate that the isolation of a generator bus in which the generation is

produced by a condenser contributes less influence on the convergence time in comparison to a

generator.



A key point to make is the difference in simulation time of the modified algorithm in

comparison to Sato and Arillaga’s original method [3]. Sato and Arillaga utilized the 14-bus

system and replaced only the line connecting bus 4 and 5 with an HVDC link. This resulted in a

30% increase in simulation time, which Sato and Arillaga state occurs for the substitution of any

line with an HVDC link. For this analysis, the line connecting bus 4 and 5 is replaced as the 5th

HVDC link. For the Gauss Seidel method, this resulted in an increase of 9.5% in time while the

Newton-Raphson resulted in a 0.31% increase in time. Also, the substitution of one HVDC link

52

results in a decrease in total time of 5.5% for the Gauss Seidel method and an increase in total time

of 3.7% for the Newton-Raphson method. Clearly the modified algorithm results in a faster

simulation time.

The iteration plots seen in Figure 25 explain why the time doubles at the 14th HVDC link.

As shown, the number of full-load flow iterations jumps to 2, in order for the methodology to

converge, resulting in the increase in time. Also shown is the repeated occurrence that once the

minimum spanning tree is met at 8 HVDC links, more iterations are required for convergence for

the Newton-Raphson method, and less iterations for the Gauss Seidel method



53


The IEEE 30-Bus system shown in Figure 26 is a standard for both load flow and voltage

stability studies. It contains 41 transmission lines and 6 generators. At 13 HVDC links, the

minimum spanning tree for the system is reached and bus isolations occur. Generator buses are

isolated at 15, 19, 24, 27, and 41 HVDC links (with the slack and a generator being removed at 41

links).


Figure 27 shows the time partition plots for the 30-Bus system. Looking at the plot for the

Newton-Raphson method, the convergence patterns seen in the previous power systems are

confirmed. Again, the total time follows a trend similar to the AC time. Total DC time increases

54

linearly as more links are introduced. The drop in DC time at the last link is due to the drop in full

loadflow iteration count as will be discussed shortly. An interesting take away from the Newton-

Raphson plot is the curvature of the total time. It seems to increase up until a certain point

(approximately 24 links) and then start to decrease. From inspection of the AC and DC times, it

seems as if the hypothesis that although the timing for the DC subroutine increases with the number

of links, the AC routine is able to take advantage of reduced equations and matricies as the buses

are isolated resulting in an overall decrease in total time.



Regarding the Gauss Seidel method, it should be noted that the algorithm did not converge

for 4 HVDC links to 14 HVDC links. This is expected to occur for larger power systems, as the

Gauss Seidel method is less reliable than the Newton-Raphson method when it comes to

convergence to a solution. Interestingly, the method is able to converge once an isolation of the

first generator occurs at 15 HVDC links. Overall, a decreasing trend in total time is seen. It is also

55

shown that the total time taken is considerably higher in comparison to the Newton-Raphson

method approach, but retains competitive timing once the system contains a high percentage of

HVDC links.

From inspection of the AC time partition plots for the 30-bus system in Figure 28, it is

shown that the methods retain the spikes and changes in slopes caused by the isolation of a

generation bus. At 13 HVDC links, there is a clear increase in the timing required for the isolated

buses (seen easily for the Newton-Raphson method) as well as an increase in time required for the

Newton-Raphson method to converge. No conclusions can be derived for the Gauss Seidel method

regarding the minimum spanning tree due to the non-convergence of the methodology.



It is also interesting to see the first instance in where the total time taken for the system

buses is larger than the time allotted for extra computation. At 13 HVDC links, the timing for the

extra computation begins to decrease somewhat linearly, indicating a direct relationship to the

56

number of remaining connected buses in the power system. Because of the system size, it is easier

to see the relationship in comparison to the previous smaller power systems.

The iteration plots for the 30-bus system are seen in Figure 29. As shown, the number of

full load flow iterations is consistent at 2 up until the last HVDC link when it drops to 1 iteration.

This explains the drop in total time as well as the drop in DC time as previously discussed. Also

shown is the spike in AC load flow iterations for the Newton-Raphson approach at 13 HVDC

links, which again corresponds to the first isolation from the minimum spanning tree of the system.

Comparing Figure 28 and 29, it is clear that the number of iterations resulted in a jump in AC time

at 13 HVDC links. However, because the number of iterations is consistent for the full load flow

and AC load flow portions (for the Newton-Raphson method), there is further support for the

hypothesis that the equations describing the remaining connected network result in the trends

shown for the total time. In other words, the curvature of the plots is not affected by the number

of iterations and thus is strictly based on the math.

57




The IEEE 118-Bus system shown in Figure 30 is a large power system which pushes the

limits of load flow solvers. For instance, for the base case starting from flat start, MATPOWER

solves the system using the Newton-Raphson approach in 30ms, which is 50% more time than

required for the 30-bus system. MATPOWER also is unable to solve for the system using the

Gauss Seidel approach (i.e. non-convergence).

58


The 118-Bus system contains 186 transmission lines and 54 generator buses. 35 of the 54

generators supply only reactive power. Also unique to this system in comparison to the previous

systems are multiple transmission lines connecting the same buses (ex. two transmission lines

connecting buses 42 and 49). It should be noted that the ordering scheme described in section 2.3.5

was not coded to take into account multiple transmission lines connecting the same buses. Instead,

the ordering scheme removes the lines consecutively and considers both lines to be a part of the

minimum spanning tree. Thus, there are 119 minimum spanning tree branches plus 2 “extra”

59

transmission lines sizing the minimum spanning tree at 121 lines. This means there are 65 links

and therefore isolations that occur starting at 66 HVDC links.

Only 150 orderings were used as data for the 118-bus system due to the length of time

required for the program to calculate and save the data. All orderings provided similar results and

thus it seemed plausible to gather data for 150 instead of the 500. Results for the timing partition

of the full load flow method of the 118-bus system are shown in Figure 31.



As shown in Figure 31, the total time follows a trend similar to the AC routine. Because

there are so many link transmission lines (i.e. 65 in total) it is easier to see a decrease in AC routine

time and an increase in DC time as more transmission lines are removed for the Newton-Raphson

plot. It is also shown that at 66 HVDC links the spike in time which occurs due to the beginning

of bus isolations is present for the Newton-Raphson method. A drop of 32% in total time (and a

drop in all other timings) is seen at 141 HVDC links and is a result of iteration count as will be

60

discussed later. Also noticed is the curvature of the plot – a general increase in time is seen up until

about 134 HVDC links where the total time begins to decrease. This is similar to the findings in

the 30-bus system.

The Gauss Seidel method did not converge for 0 to 163 HVDC links. Convergence is

achieved at 164 HVDC links where the system has reduced considerably in size (only 21 AC

transmission lines remain). A downward trend is seen in the total time for the Gauss Seidel method

because of the previously mentioned reduction in terms for the load flow equations. The timing

for the Gauss Seidel method does not become competitive until the end, however closer analysis

shows that the Newton-Raphson method still converges faster.

A further breakdown of the AC timing for the 118-bus system is shown in Figure 32.

Shown again is the linear decrease in extra computation time for the AC routine with regards to

the Newton-Raphson method. Also shown is the negligible time allotted for the isolated bus routine

in comparison to the time required for the system buses calculations. Although it seems as if the

Gauss Seidel method requires less extra computation time, attention should be drawn to the scale

of the timing – both methods are somewhat equivalent for extra and system bus time. From

inspection of the plots, it seems as if the dynamics regarding the system size dominates any

dynamics regarding the isolation of a generator bus and thus is not needed to be discussed further.

61



Figure 33 details the iteration plots for the 118-bus system. As previously mentioned, the

drop in time at 141 links is a result of full load flow iterations. Also, the spike in time at the

minimum spanning tree is also a result of the increased amount of AC iterations for the Newton-

Raphson method.

62



3.7 Summary of Convergence Properties

It was found that the placement of multiple HVDC links directly effects the convergence

time of the load flow method. For instance, placement of HVDC links which result in disjoint

power systems will most likely lead to non-convergence. Also, an increase in convergence time is

seen once the first bus is isolated from the rest of the system due to placement of the links (i.e. an

isolation from the minimum spanning tree of a power system). If an isolation of generator bus

occurs, there will be a change in the convergence time. For larger systems, it is also seen that

63

although the convergence time increases as more isolations occur, there exists a point where the

timing peaks, and the decreased amount of connected buses starts to downtrend in required time

for convergence. Also, the total time is greatly influenced by the AC load flow methodology.

64

4 DEVELOPMENT OF AN ENHANCED LOAD FLOW

SOLVER

4.1 Overview

This chapter deals with leveraging off the findings from the previous test cases to develop

an enhanced load flow solver capable of determining the load flows of a multi-link power system.

In particular, the load flow solver looks at the percentage of DC transmission in the system as a

determining factor of what load flow solver to use for the AC routine. The enhanced load flow

solver is then applied to the 14 bus system and the results are compared to the previous load flow

method.

Only the Newton-Raphson and Gauss Seidel methods were looked into in this thesis. It

should be noted that this method is robust in the sense that if the convergence properties (i.e. time,

computation, and iteration) of another method are determined as in Chapter 2 and 3, it can be

integrated into the enhanced load flow solver easily. Only a comparison needs to be made between

the load flow methods to determine which methodology is optimal for the certain percentage of

DC transmission.

4.2 Convergence of Modified Sato and Arillaga’s Method in Terms of

Percent DC Transmission

From inspection of the convergence plots in Chapter 3, it is clear that the Newton-Raphson

method out performs the Gauss Seidel method in terms of minimizing convergence time for the

full load flow method and reducing iteration count for majority of the case studies. A bar graph

depiction of the comparison of the two methodologies for convergence time with respect to

65

percentage of DC transmission is shown in Figure 34. Percentage of DC transmission was

calculated as the number of HVDC links divided by the total amount of transmission lines (Ex. 1

HVDC link for Granger and Stevenson’s 4-bus system yields 25%). The size of the bar determines

how many times the certain methodology had faster full load flow convergence.

Figure 34. Comparison of Gauss Seidel method to Newton-Raphson method in regards to faster

convergence time with respect to percentage of DC transmission

As shown in Figure 34, the Newton-Raphson method out performs the Gauss Seidel

method in all percentages of DC transmission. However, it should be noted that the Gauss Seidel

method becomes competitive as the percentage of DC transmission increases. Specifically, it is

shown that the Gauss Seidel method converges faster than the Newton-Raphson method 41.53%

of the time when at least 90% of the transmission is via HVDC links. During these instances,

convergence is on average 1.1ms faster, with a maximum of being 186ms faster.

0 10 20 30 40 50 60 70 80 90 1000

500

1000

1500

2000

2500

%DC Transmission

Num

ber

of

Ord

eri

ng

s

Frequency of Faster Convergence Time of Load Flow

Newton Raphson

Gauss Seidel

66

It should be stated that the Gauss Seidel is known to require less memory for storage in

comparison to the Newton-Raphson method [6]. This is mainly because the Jacobian does not need

to be recalculated and stored every iteration for the Gauss Seidel method. Because memory storage

is of consideration for load flow methods, a 41.53% opportunity to save memory storage while

obtaining faster convergence is considered to be a good enough reason to choose the Gauss Seidel

method over the Newton-Raphson method for power systems containing over 90% DC

transmission.

4.3 Enhanced Load Flow Solver Algorithm

The algorithm for the enhanced load flow solver is actually a simple modification of the already

modified Sato and Arillaga’s method explained in Chapter 2. Prior to calculating the load flow of

the system, the number of HVDC links and the total number of transmission lines in are used to

determine the percentage of DC transmission. Leveraging off that, the optimum AC methodology

for the AC routine is chosen to minimize both the convergence time and memory requirements,

with a priority given to convergence time. As previously stated, at over 90% DC transmission, the

Gauss Seidel method becomes competitive with the Newton-Raphson method with regards to

speed of convergence. Thus, only one if statement is added which states if %DC transmission is

greater than or equal to 90%, use the Gauss Seidel method, else use the Newton-Raphson method.

The algorithm is depicted in Figure 35. The algorithm was coded in MATLAB and can be found

in Appendix C.

67

Figure 35. Enhanced Load Flow Solver algorithm.

68

4.4 Results of Enhanced Load Flow Solver

The IEEE 14-Bus power system was used as the test system for the Enhanced Load Flow

solver. It was chosen because it is the medium sized system used in this thesis as well as the fact

that the inclusion of every HVDC link increases the total percentage of DC transmission by 5%.

The Enhanced Load Flow solver was applied to the first 500 orderings of the 14-bus system. Again,

the results are all similar and were averaged together to reduce error from non-processor isolation.

The convergence time and iteration count for the load flow method using only Gauss Seidel, the

load flow method using only Newton-Raphson, and for the Enhanced Load Flow is detailed in

Table 9.

Table 9. Convergence Time and Iteration Data for Load Flow of IEEE 14-Bus System

Number of AC Iterations Full Convergence Time (s)

Number of Links %DC

Gauss Seidel

Newton-Raphson

Enhanced Load Flow

Gauss Seidel

Newton-Raphson

Enhanced Load Flow

0 0 103 2 2 0.0273 0.0080 0.0082 1 5 93 2 2 0.0258 0.0083 0.0081 2 10 110 2 2 0.0288 0.0071 0.0071 3 15 115 2 2 0.0294 0.0074 0.0072 4 20 122 2 2 0.0310 0.0074 0.0073 5 25 136 2 2 0.0339 0.0075 0.0071 6 30 152 2 2 0.0370 0.0071 0.0072 7 35 166 2 2 0.0398 0.0072 0.0073 8 40 134 3 3 0.0323 0.0075 0.0074 9 45 70 3 3 0.0195 0.0076 0.0076

10 50 70 3 3 0.0185 0.0076 0.0078 11 55 45 3 3 0.0144 0.0076 0.0077 12 60 33 3 3 0.0121 0.0077 0.0079 13 65 30 3 3 0.0115 0.0078 0.0080 14 70 32 4 4 0.0180 0.0140 0.0147 15 75 17 4 4 0.0156 0.0142 0.0142 16 80 17 4 4 0.0159 0.0142 0.0141 17 85 12 4 4 0.0150 0.0141 0.0147 18 90 9 4 9 0.0147 0.0144 0.0144 19 95 8 3 8 0.0140 0.0139 0.0143 20 100 1 1 1 0.0043 0.0042 0.0040

69

As shown in Table 9, the algorithm can be confirmed as working correctly by inspection

of the iteration count. The Enhanced Load Flow has the same iteration count as the Newton-

Raphson method (highlighted in green) up until the %DC reaches 90%, where the iteration count

matches the Gauss Seidel method (highlighted in yellow). Convergence time is not equivalent

because of the error due to the non-processor isolation being included in the averaging of the 500

orderings. However, it can be seen that the Enhanced load flow solver has an average time similar

to the Newton-Raphson method up until 18 HVDC links when it again follows the Gauss Seidel

pattern. From inspection of the last three rows of Table 9 (i.e. when the %DC is at least 90%), it

can be seen how competitive the timings are between the Newton-Raphson and Gauss Seidel

methods. Although at 95% DC transmission the Enhanced Load Flow gives a convergence time

higher than the Newton-Raphson and Gauss Seidel method, the difference in timing is on the order

of microseconds and is attributed yet again to the non-processor isolation.

Although memory requirements are not directly recorded, the time per iterate can be

calculated by a simple division using the data in Table 9. It can be shown that the Gauss Seidel

method and Enhanced Load Flow solver provide a smaller time per iterate which is indicative of

the memory requirements. Since the total time is competitive and the time per iterate is smaller

(indicating a smaller memory requirement), it is confirmed that the Enhanced Load Flow solver

optimizes for both memory and time at these high DC percentages while prioritizing the

convergence time at lower DC percentages.

70

5 APPLICATION OF ENHANCED LOAD FLOW SOLVER ON

VOLTAGE STABILITY

5.1 Overview

Load flow solvers are used in several different applications in power system engineering.

One of the main concerns of power system engineers is the security and stability of a power system

to ensure reliable power is delivered and blackouts do not occur. As the system load is increased,

there begins to be a stress on the generators which begin to reach their limitations. Buses also begin

to reach a maximum loading point in which the voltage begins to collapse. This point is nicknamed

the “tip of the nose curve” as depicted in Figure 36.

Figure 36. Generic P-V curve

Load flow solvers are used to generate the P-V curve of a particular bus of a power system.

A load flow is run to calculate the bus voltages as the load is increased. However, as the maximum

loadability point is approached, it becomes increasingly more difficult for load flow solvers to

71

achieve convergence. In fact, the Newton-Raphson method is unable to converge for points near

the tip of the nose curve due to the increased singularity of the Jacobian. For closest point near the

maximum loadabilty point in which the Newton-Raphson method converges, the time for

convergence is considerably large.

This chapter explores the influence of using the Enhanced Load Flow solver for voltage

stability studies. The hypothesis is that because substitutions for AC transmission lines are made

with equivalent power sources, a reduced power system is solved for leading to a decrease in

convergence time. In addition, the isolation of the heavily loaded bus from the power system may

result in less of a convergence time increase when the minimum spanning tree is reached. Also

looked into is the influence of selecting particular lines for HVDC substitution with regards to the

amount of power being transmitted over the line to the heavily loaded bus. In particular, the voltage

stability of the IEEE 30-Bus system was looked into.

5.2 HVDC Link Substitutions Based On Transmission Power Magnitude

To create a stressed system, the load at bus 10 for the IEEE-30 Bus system was increased

until convergence was not achieved using MATPOWER’s built in Newton-Raphson method. Bus

10 is normally loaded at 5.8 + 2.0j MVA. It was found that increasing the active power of the load

at bus 10 beyond 165MW (while keeping the reactive power at the original 2 MVar) resulted in

non-convergence and thus was utilized as the test case scenario. Similarly, increasing the reactive

power of the load beyond 170MVar (while keep the active power at the original 5.8 MW) resulted

in non-convergence. The calculated load flows of the transmission lines connected to bus 10 are

depicted in Table 10. MATPOWER’s Newton-Raphson method converged in 0.07s and 0.08s for

the maximum active power and maximum reactive power loads (respectively).

72

Table 10. Stressed System Load Flow for Lines Connected to Bus 10 of IEEE 30-Bus System

Bus 10 Power = 165 + 2j MVA Bus 10 Power = 5.8 + 170j MVA

Bus Number From Bus Injection To Bus Injection

From Bus Injection To Bus Injection

From Bus

To Bus

P (MW)

Q (Mvar)

P (MW)

Q (Mvar)

P (MW)

Q (Mvar)

P (MW)

Q (Mvar)

6 10 38.75 -1 -38.75 11.48 0.25 32.16 -0.25 -24.6 9 10 67.81 -13.79 -67.81 20.09 0.43 47.59 -0.43 -43.04

10 20 -9.12 2.02 9.21 -1.81 3.64 -13.37 -3.25 14.27 10 17 -22.3 7.38 22.48 -6.87 1.31 -18.85 -1.07 19.48 10 21 -15.34 -26.08 15.66 26.82 -4.68 -43.06 5.94 46 10 22 -11.68 -16.88 12.02 17.6 -5.39 -27.08 6.59 29.64

Utilizing the results in Table 10, the HVDC link parameters are set for the lines connected

to bus 10. Six different orderings were looked into: substitutions for HVDC links delivering the

least to most active power, least to most reactive power, least to most apparent power, and the

opposite orderings (i.e. most to least). As with the convergence studies performed in Chapter 3 of

the thesis, reduction of error due to non-processor isolation was reduced by the averaging of 500

runs of each ordering. Figures 37 and 38 show plots of the resulting full load flow convergence

times for the different orderings.

73

Figure 37. Full load flow convergence times for maximum active power load

Figure 38. Full load flow convergence times for maximum reactive power load

74

From inspection of Figures 37 and 38, it is clearly shown that the average convergence time

for the load flow of a system with HVDC links connected to the heavily loaded bus is considerably

less than a full Newton-Raphson load flow method. In fact, up to a 75% decrease in time can be

achieved by substitutions of HVDC links. An explanation of this derives from the findings in

Chapter 3 in which the substitution of lines not connected to the minimum spanning tree of the

system normally result in a decrease in AC convergence time for the Newton-Raphson method.

Because the percent DC transmission is less than 90% for the power system, the Newton-Raphson

method is the chosen methodology for the AC routine by the Enhanced Load Flow solver, and thus

follows a similar pattern. Mathematically speaking, the decrease in time is due to the increased

sparsity of the Y-Bus matrix alongside the unneeded calculation of the injected power flows from

the HVDC link (since the HVDC link is set to deliver the original power injection calculated).

It is also shown that the ordering of the substitutions of HVDC links does not impact the

convergence times greatly. As shown in Figures 37 and 38, all orderings of substitutions lie around

the same convergence times. Minor fluctuations and differences can be attributed to the error

caused by the non-processor isolation. Removal of the line delivering the largest amount of active

power, reactive power, or apparent power does not create a larger decrease in convergence time.

Substitutions at which majority of the delivered active power, reactive power, or apparent power

are now done by DC transmission also does not create a drastic change in convergence time.

5.3 Substitution Orderings with Heavily Loaded Bus as First Isolation

Utilizing the same stressed scenario with bus 10 as the heavily loaded bus (described in the

previous section) a HVDC substitution ordering was made with bus 10 as the furthest bus from the

slack bus in the minimum spanning tree. In doing so, the heavily loaded bus is the first bus to be

75

isolated from the system. Figure 39 depicts the minimum spanning tree used for this ordering, and

was created from sheer inspection of the IEEE 30-Bus system.

Figure 39. Minimum spanning tree for IEEE 30-Bus system with heavily loaded bus 10

The convergence plots for the total time partition and AC time partition are shown in

Figures 40 and 41. From inspection of the plots, it is seen that regardless of how bus 10 is brought

to a heavy loading point (i.e. reactive or active power increase) the plots follow a similar trend.

Also, all trends previously mentioned in Chapter 3 for the non-stressed systems are retained.

Because the Enhanced Load Flow method is used, a spike in convergence time is seen at around

76

37 links when 90% of the transmission is via HVDC links and the Gauss Seidel method is used.

Since the system is stressed, the Gauss Seidel method takes a longer time to converge.


reactive power (b)

Figure 41. AC time partition plot for heavily loaded bus 10 – with large active power (a) and

large reactive power (b)

77

In comparison to the plots made for the base case load flow study (Figures 27 and 28) one

major difference is seen. Once the minimum spanning tree is reached at 13 HVDC links, the

convergence time starts to increase until a certain point is reached (approximately 24 HVDC links)

where the decreased system size results in a decrease in convergence time. This occurs for both

the stressed system and base case. However, the difference in total convergence time between the

24th (approximate peak) and 13th HVDC link (when minimum spanning tree is reached) for the

base case is 3ms while the stressed case yields 2.25ms, a 25% decrease in time when the system

is stressed. With respect to the AC time devoted to the system buses calculation, the stressed case

yields a difference in time of 2.075ms while the base case yields 3.028ms, a 31.47% decrease in

time when the system is stressed. This confirms the hypothesis that the isolation of the heavily

loaded bus results in a smaller increase in convergence time when the minimum spanning tree is

reached.

To further confirm the hypothesis, a second scenario where bus 30 is heavily loaded was

looked into. Figure 42 shows the minimum spanning tree used for this ordering. By increasing the

load to 82.5 + 1.9j MVAR (highest allowable active power) or 10.6 + 62j MVAR (highest

allowable reactive power), the system is considered to be stressed.

78

Figure 42. Minimum spanning tree for IEEE 30-Bus system with heavily loaded bus 30

Figures 43 and 44 show the convergence plots for the stressed system with a heavy load

on bus 30. In comparison to the base case plots, the peak of the curvature resulting from the

isolation of buses is again lower. Specifically, the stressed system yields a difference of total time

(between links 24 and 13) of 2.57ms which is approximately 15% less than the base case. For the

AC time devoted to the system buses, the stressed system yields a time difference of 2.24ms which

is approximately 26% less than the base case.

79


reactive power (b)

Figure 44. AC time partition plot for heavily loaded bus 10 – with large active power (a) and

large reactive power (b)

80

6 CONCLUSION

6.1 Summary of Thesis Work

A load flow analysis tool was developed which determines the effects of the inclusion of

multiple HVDC links on the convergence properties of particular load flow methods. This tool

utilizes a simplified DC subroutine based off Sato and Arillaga’s method and maximizes the

efficiencies explained by Sato and Arillaga to ensure only the influence of multiple links on the

AC routine is determined.

Using the load flow analysis tool, convergence patterns were determined from analysis of

HVDC link substitutions on Grainger and Stevenson’s 4-bus system and the IEEE 9-bus, 14-bus,

30-bus, and 118-bus systems. It was found that the placement of HVDC links was a key factor in

the convergence time. For instance, if multiple HVDC links were placed such that two disjoint

power systems were formed – the load flow methods would not converge to a solution. Also, when

enough transmission lines are replaced such that a bus isolation has to occur, a spike in

convergence time is usually present. The isolation of generator buses usually effects convergence

time in some way as well, although only when the generator supplies active power. In addition,

for large systems it is shown that the convergence time peaks at a certain amount of DC

transmission and begins to decrease. This pattern is not a factor of iteration count and thus is

concluded to be a resultant of solution methodology (i.e. less terms and sparser matricies leading

to faster convergence).

Overall, the Newton-Raphson method is superior in comparison to the Gauss Seidel method

when calculating the load flows. However, when the power system has 90% of transmission done

via HVDC links, the Gauss Seidel method becomes competitive in time with the Newton-Raphson

method. Leveraging off this finding, an Enhance Load Flow solver was developed to optimize for

81

speed, but also reduce memory requirements once a system architecture has enough DC

transmission to allow for Gauss Seidel to be utilized. It was shown that the Enhanced Load Flow

solver was faster and produced a lower time-per-iterate value in comparison to Sato and Arillaga’s

method utilizing only one load flow method.

Finally, the application of the Enhanced Load Flow solver on voltage stability was looked

into. It was found that the substitution of HVDC links for lines connected to the heavily loaded

bus results in a large decrease in convergence time for the load flow method in comparison to a

traditional Newton-Raphson approach. Also, it was shown that there is minimal variation in the

convergence time depending on the selection of transmission line substitution (Ex. the line

delivering the most active power vs the line delivering the least). The isolation of the heavily

loaded bus also results in a less steep curvature of the trend seen when the minimum spanning tree

is met.

6.2 Potential Future Work

There are several avenues to extend the work presented in this thesis. One of the most

obvious is the inclusion of additional load flow methods within the AC routine of the Enhanced

Load Flow solver. An analysis similar to what was described in Chapter 2 can be performed to

compare the convergence properties of the other load flow methodologies to see at what percentage

of DC transmission would the methods would be more beneficial. The inclusion of these additional

load flow methods would increase the optimality of the algorithm. It is recommended that the work

be performed using an isolated processor to ensure background noise does not introduce error into

the results.

82

The Enhanced Load Flow solver could also be modified to utilize the full Sato and Arillaga

DC subroutine, instead of the simplified version described in this thesis. Including this portion

would increase the amount of time required for the DC subroutine, but also provide more realistic

results since controllability of the links is included as well as in depth calculations of the load flows

for the terminal interfaces. The introduction of competition between multiple HVDC links

(connected to the same bus) to control the bus voltage would also become a factor for the

methodology.

An analysis could be performed utilizing a different ordering scheme. Although it was found

that all orderings retained similar trends for a particular power system, the basis of the orderings

all used the same minimum spanning tree. The minimum spanning tree utilized for each power

system was constant due to MATLAB’s functionality, although there exists several different

permutations of spanning trees. It was shown in Chapter 5 that different minimum spanning trees

for stressed systems yielded slight variations in trends, however was attributed more to the

isolation of the heavily loaded bus. Utilizing a different minimum spanning tree for the orderings

may lead to new insights on the convergence properties for load flow of multi-link HVDC systems.

The substitution of HVDC links which lead to multiple sub-systems resulted in non-

convergence of the load flow method. However, this was not looked into much detail as for an N-

bus system containing M transmission lines, there exists M! permutations of orderings. It was

assumed that all instances of orderings in which multiple sub-systems occurred resulted in non-

convergence, due to the alarming amount of non-convergences which occurred due to it. However,

not all instances were looked at. An avenue of research could be the determination of what size of

sub-system results in the non-convergence (whether it be bus size, amount of load, generation to

load ratio, etc).

83

Finally it is most important to leverage off this thesis work to determine the impacts of

HVDC link placement in existing power networks. It was shown that load flow convergence is

effected by the placement of the links, in particular because of the isolation of buses and the

creation of sub-systems. The impacts of replacing transmission lines which carry large amounts of

power with HVDC links, as well as the impacts of replacing transmission lines which have high

variation in transmitted power (depending on the forecasted load) with HVDC links should be

looked into. Load flow only calculates the steady-state voltages and power flows – taking this

thesis work and extending it to the effects on dynamic changes is a good next step.

84

LIST OF REFERENCES

[1] R. T. H. Alden and F. A. Qureshy, “HVDC – A review of applications and systems analysis

methods”, Volume: 9, Issue 4, Page 139-145, Canadian Electrical Engineering Journal,

October 1984.

[2] V. Akhmatov, M. Callavik, C. M. Franck, S.E. Rye, T. Ahndorf, M. K. Bucher, H. Muller, F.

Schettler, and R. Wiget, “Technical Guidelines and Prestandarization Work for First HVDC

Grids”, Volume 29, Issue 1, Page 327-335. IEEE Transactions on Power Delivery, October

2013.

[3] H. Sato and J. Arrillaga, “Improved load-flow techniques for integrated a.c.-d.c. systems”,

Volume 116, Issue 4, Page 525-532. Proceedings of the Institution of Electrical Engineers,

April 1969.

[4] M. H. Nguyen, T. K. Saha, and M. Eghbal, “A comparative study of voltage stability for long

distance HVAC and HVDC interconnections”, Page 1-8, 2010 IEEE Power and Energy

Society General Meeting, July 2010.

[5] A. E. Hammad and W. Kuhn, “A Computation Algorithm for Assessing Voltage Stability at

AC/DC Interconnections”, Volume: PER-6, Issue: 2, Page 45-46, IEEE Power Engineering

Review, February 1986.

[6] J. Grainger and W. Stevenson, Power System Analysis, McGraw-Hill Inc, United States of

America, 1994.

[7] J.B. Ward and H.W. Hale, “Digital computer solution of power-flow problems”, Volume 75,

No 3, Page 398-404, Power Apparatus and systems, Part III. Transactions of the American

Institute of Electrical Engineers, January 1956.

[8] W.Tinney and C.Hart, “Power flow solution by newton’s method”, Volume: PAS-86, No 11,

Page 1449-1460, IEEE Transactions on Power Apparatus and Systems, November 1967.

[9] B. Stott and O. Alsac, “Fast decoupled load flow”, Volume: PAS-93, No.3, Page 859-869,

IEEE Transactions on Power Apparatus and Systems, May 1974.

85

[10] N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics: Converters,

Applications and Design, John Wiley & Sons Inc, United States of America, 2003.

[11] J. Arrillaga, High Voltage Direct Current Transmission, The Institution of Electrical

Engineers, London, United Kingdom, 1998.

[12] E. Uhlmann, “Representation of an h.v.d.c. link in a network analyser’, Cigre, Paris, Paper

404, 1960.

[13] I.E. Barker, B.A. Carré, “Load flow calculations for systems containing h.v.d.c. links”, IEE

Conference, Publ 22, Pages 115-118, 1966.

[14] J. Arrillaga and P. Bodger, “Integration of h.v.d.c. links with fast-decoupled load-flow

solutions”, Volume 124, Issue 5, Page 463-468. Proceedings of the Institution of Electrical

Engineers, May 1977.

[15] H.A. Sanghavi and S.K. Banerjee, “Load flow analysis of integrated AC-DC power

systems”, Page 746-751. TENCON 89’, Fourth IEEE Region 10 International Conference,

November 1989.

[16] D.A. Braunagel, L.A. Kraft, and J.L Whysong, “Inclusion of DC converter and transmission

equations directly in a Newton power flow”, Volume: 95, Issue: 1, IEEE Transactions on

Power Apparatus and Systems, January 1976.

[17] J. Reeve, G. Fahny, and B. Stott, “Versatile load flow method for multiterminal HVDC

systems”, Volume 96, Issue 3, Page 925-933, IEEE Transactions on Power Apparatus and

Systems, May 1977.

[18] H. Fudeh and C.M. Ong, “A Simple and Efficient AC-DC Load-Flow Method for

Multiterminal DC Systems”, Volume: PAS-100, Issue:11, Page 4389-4396, IEEE

Transactions on Power Apparatus and Systems, November 1981.

[19] C.M. Ong and A. Hamzei-nejad, “A General-Purpose Multiterminal DC Load-Flow”,

Volume: PAS-100, Issue:7, Page 3166-3174, IEEE Transactions on Power Apparatus and

Systems, July 1981.

86

[20] M.M. El-Marsafawy and R.M. Mathur, “A New Fast Technique for Load-Flow Solution of

Integrated Multi-Terminal DC/AC Systems”, Volume: PAS-99, Issue:1, Page 246-255, IEEE

Transactions on Power Apparatus and Systems, January 1980.

[21] http://www.pserc.cornell.edu/matpower/

[22] http://www.emeraldinsight.com/content_images/fig/1740250406025.png

[23] http://power.elec.kitami-it.ac.jp/ueda/demo/WebPF/14bus.jpg

[24] http://fglongatt.org/OLD/TEST%20SYSTEMS/IEEE_30/IEEE_30bus.png

[25] http://fglongatt.org/OLD/TEST%20SYSTEMS/IEEE_118/IEEE_118bus_2.JPG

http://www.pserc.cornell.edu/matpower/

87

APPENDIX

APPENDIX A: Additional Information about HVDC Links

A.1 Calcluation of Vx, Vy, and Ids

The following are a series of calculations assuming an optimum tap with constant-power

control. The equations are detailed in Sato and Arillaga’s paper [3] and are restated here for

convenience.

The operating voltage Vx is the intersection of the rectifier constant current characteristic

and invertor constant margin angle characteristic. It is obtained from the power setting as follows:

𝑉𝑥 = −𝐴𝑟𝑒𝑐𝑡(𝐼𝑥 − 𝐼𝑑𝑠𝑟𝑒𝑐𝑡) (A1)

𝑉𝑥 = 𝑣𝑖𝑛𝑣 cos 𝛾 −3

𝜋𝑋𝑖𝑛𝑣𝐼𝑥 + 𝑅𝑑𝑐𝐼𝑥 (A2).

By eliminating Ix from equations (A1) and (A2), the operating voltage Vx can be found by

following the following equation:

𝑉𝑥(𝐾2𝑋𝑖𝑛𝑣 − 𝑅𝑙𝑖𝑛𝑒 − 𝐴𝑟𝑒𝑐𝑡) + 𝐴𝑟𝑒𝑐𝑡𝑣𝑖𝑛𝑣 cos(𝛾) −𝐴𝑟𝑒𝑐𝑡3

𝜋𝑋𝑖𝑛𝑣 − 𝑅𝑙𝑖𝑛𝑒𝐼𝑑𝑠𝑟𝑒𝑐𝑡 = 0 (A3)

The following relation holds between the current and power settings:

𝑃𝑑𝑠 = 𝑉𝑥𝐼𝑑𝑠𝑟𝑒𝑐𝑡 (A4)

Substituting equation (A4) into (A3):

88

𝑉𝑥2(𝐾2𝑋𝑖𝑛𝑣 − 𝑅𝑙𝑖𝑛𝑒 − 𝐴𝑟𝑒𝑐𝑡) + 𝐴𝑟𝑒𝑐𝑡𝑣𝑖𝑛𝑣 cos(𝛾)𝑉𝑥 −𝐴𝑟𝑒𝑐𝑡(

3

𝜋𝑋𝑖𝑛𝑣 − 𝑅𝑙𝑖𝑛𝑒)(𝑃𝑑𝑠) = 0 (A5)

which can be solved for Vx using the quadratic formula.

Once the operating voltage Vx is found, the operating current Ix can be found by

substitution back into equation (A2).

Ids is the current setting and is a function of the power settings and operating voltage. That

is:

𝐼𝑑𝑠𝑟𝑒𝑐𝑡 = 𝑃𝑑𝑠/𝑉𝑥 (A6)

Vy, the maximum obtainable voltage, can be found using the following equation:

𝑉𝑦 = 𝐴𝑟𝑒𝑐𝑡(𝐴𝑖𝑛𝑣𝑃𝑑𝑚 + 𝑅𝑙𝑖𝑛𝑒𝑃𝑑𝑠)/(𝑉𝑥(𝐴𝑟𝑒𝑐𝑡 + 𝐴𝑖𝑛𝑣 + 𝑅𝑙𝑖𝑛𝑒)) (A7).

The optimum operating voltage can then be found using the constrained optimization equation

depicted in equations (9a) and (9b).

89

A.2 Reactive Power for HVDC Links

The converter terminals of an HVDC link absorb a large amount of reactive power. The

reactive power demand increases with increasing power transfer level. A large percentage of this

reactive power draw is provided by the harmonic filters. In fact, for a 60-Hz fundamental frequency

power system, the per-phase reactive power supplied by the filters is approximated by the

following equation [10]:

𝑄𝑓 = 377𝐶𝑓𝑉𝑠2 (A8)

where Qf is the reactive power supplied by the filter, Cf is the capacitance of the filter, and Vs is

the rms phase voltage applied across the filters.

The capacitors utilized for the AC-side filtering of the harmonics are chosen such that the

reactive power supplied never exceeds the reactive power demand of the converters when

operating under minimum power transmission. This assures over voltage does not occur. In order

to compensate for higher reactive power demand by the converters at higher power transmission

levels, additional capacitors are put in place parallel to the AC side filter. These capacitors are

switched on by a means of mechanical contactors if additional reactive power is demanded by the

converter terminals.

90

APPENDIX B: Power System Data in MATPOWER Format

B.1 Data for Grainger and Stevenson’s 4-Bus System

%% MATPOWER Case Format : Version 2 mpc.version = '2';

%%----- Power Flow Data -----%% %% system MVA base mpc.baseMVA = 100;

%% bus data % bus_i type Pd Qd Gs Bs area Vm Va baseKV zone Vmax

Vmin mpc.bus = [ 1 3 50 30.99 0 0 1 1 0 230 1 1.1 0.9; 2 1 170 105.35 0 0 1 1 0 230 1 1.1 0.9; 3 1 200 123.94 0 0 1 1 0 230 1 1.1 0.9; 4 2 80 49.58 0 0 1 1 0 230 1 1.1 0.9; ];

%% generator data % bus Pg Qg Qmax Qmin Vg mBase status Pmax Pmin Pc1 Pc2

Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 4 318 0 100 -100 1.02 100 1 318 0 0 0 0 0 0 0 0

0 0 0 0; 1 0 0 100 -100 1 100 1 0 0 0 0 0 0 0 0 0 0

0 0 0; ];

%% branch data % fbus tbus r x b rateA rateB rateC ratio angle

status angmin angmax mpc.branch = [ 1 2 0.01008 0.0504 0.1025 250 250 250 0 0 1 -360 360; 1 3 0.00744 0.0372 0.0775 250 250 250 0 0 1 -360 360; 2 4 0.00744 0.0372 0.0775 250 250 250 0 0 1 -360 360; 3 4 0.01272 0.0636 0.1275 250 250 250 0 0 1 -360 360; ];

91

B.2 Data for IEEE 9-Bus System




Vmin mpc.bus = [ 1 3 0 0 0 0 1 1 0 345 1 1.1 0.9; 2 2 0 0 0 0 1 1 0 345 1 1.1 0.9; 3 2 0 0 0 0 1 1 0 345 1 1.1 0.9; 4 1 0 0 0 0 1 1 0 345 1 1.1 0.9; 5 1 90 30 0 0 1 1 0 345 1 1.1 0.9; 6 1 0 0 0 0 1 1 0 345 1 1.1 0.9; 7 1 100 35 0 0 1 1 0 345 1 1.1 0.9; 8 1 0 0 0 0 1 1 0 345 1 1.1 0.9; 9 1 125 50 0 0 1 1 0 345 1 1.1 0.9; ];


Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 1 0 0 300 -300 1 100 1 250 10 0 0 0 0 0 0 0 0

0 0 0; 2 163 0 300 -300 1 100 1 300 10 0 0 0 0 0 0 0 0

0 0 0; 3 85 0 300 -300 1 100 1 270 10 0 0 0 0 0 0 0 0

0 0 0; ];


status angmin angmax mpc.branch = [ 1 4 0 0.0576 0 250 250 250 0 0 1 -360 360; 4 5 0.017 0.092 0.158 250 250 250 0 0 1 -360 360; 5 6 0.039 0.17 0.358 150 150 150 0 0 1 -360 360; 3 6 0 0.0586 0 300 300 300 0 0 1 -360 360; 6 7 0.0119 0.1008 0.209 150 150 150 0 0 1 -360 360; 7 8 0.0085 0.072 0.149 250 250 250 0 0 1 -360 360; 8 2 0 0.0625 0 250 250 250 0 0 1 -360 360; 8 9 0.032 0.161 0.306 250 250 250 0 0 1 -360 360; 9 4 0.01 0.085 0.176 250 250 250 0 0 1 -360 360; ];

%%----- OPF Data -----%% %% area data % area refbus mpc.areas = [

92

1 5; ];

%% generator cost data % 1 startup shutdown n x1 y1 ... xn yn % 2 startup shutdown n c(n-1) ... c0 mpc.gencost = [ 2 1500 0 3 0.11 5 150; 2 2000 0 3 0.085 1.2 600; 2 3000 0 3 0.1225 1 335; ];

93





Vmin mpc.bus = [ 1 3 0 0 0 0 1 1.06 0 0 1 1.06 0.94; 2 2 21.7 12.7 0 0 1 1.045 -4.98 0 1 1.06 0.94; 3 2 94.2 19 0 0 1 1.01 -12.72 0 1 1.06 0.94; 4 1 47.8 -3.9 0 0 1 1.019 -10.33 0 1 1.06 0.94; 5 1 7.6 1.6 0 0 1 1.02 -8.78 0 1 1.06 0.94; 6 2 11.2 7.5 0 0 1 1.07 -14.22 0 1 1.06 0.94; 7 1 0 0 0 0 1 1.062 -13.37 0 1 1.06 0.94; 8 2 0 0 0 0 1 1.09 -13.36 0 1 1.06 0.94; 9 1 29.5 16.6 0 19 1 1.056 -14.94 0 1 1.06 0.94; 10 1 9 5.8 0 0 1 1.051 -15.1 0 1 1.06 0.94; 11 1 3.5 1.8 0 0 1 1.057 -14.79 0 1 1.06 0.94; 12 1 6.1 1.6 0 0 1 1.055 -15.07 0 1 1.06 0.94; 13 1 13.5 5.8 0 0 1 1.05 -15.16 0 1 1.06 0.94; 14 1 14.9 5 0 0 1 1.036 -16.04 0 1 1.06 0.94; ];


Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 1 232.4 -16.9 10 0 1.06 100 1 332.4 0 0 0 0 0 0

0 0 0 0 0 0; 2 40 42.4 50 -40 1.045 100 1 140 0 0 0 0 0 0 0 0

0 0 0 0; 3 0 23.4 40 0 1.01 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 6 0 12.2 24 -6 1.07 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 8 0 17.4 24 -6 1.09 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; ];


status angmin angmax mpc.branch = [ 1 2 0.01938 0.05917 0.0528 9900 0 0 0 0 1 -360 360; 1 5 0.05403 0.22304 0.0492 9900 0 0 0 0 1 -360 360; 2 3 0.04699 0.19797 0.0438 9900 0 0 0 0 1 -360 360; 2 4 0.05811 0.17632 0.034 9900 0 0 0 0 1 -360 360; 2 5 0.05695 0.17388 0.0346 9900 0 0 0 0 1 -360 360; 3 4 0.06701 0.17103 0.0128 9900 0 0 0 0 1 -360 360;

94

4 5 0.01335 0.04211 0 9900 0 0 0 0 1 -360 360; 4 7 0 0.20912 0 9900 0 0 0.978 0 1 -360 360; 4 9 0 0.55618 0 9900 0 0 0.969 0 1 -360 360; 5 6 0 0.25202 0 9900 0 0 0.932 0 1 -360 360; 6 11 0.09498 0.1989 0 9900 0 0 0 0 1 -360 360; 6 12 0.12291 0.25581 0 9900 0 0 0 0 1 -360 360; 6 13 0.06615 0.13027 0 9900 0 0 0 0 1 -360 360; 7 8 0 0.17615 0 9900 0 0 0 0 1 -360 360; 7 9 0 0.11001 0 9900 0 0 0 0 1 -360 360; 9 10 0.03181 0.0845 0 9900 0 0 0 0 1 -360 360; 9 14 0.12711 0.27038 0 9900 0 0 0 0 1 -360 360; 10 11 0.08205 0.19207 0 9900 0 0 0 0 1 -360 360; 12 13 0.22092 0.19988 0 9900 0 0 0 0 1 -360 360; 13 14 0.17093 0.34802 0 9900 0 0 0 0 1 -360 360; ];

%%----- OPF Data -----%% %% generator cost data % 1 startup shutdown n x1 y1 ... xn yn % 2 startup shutdown n c(n-1) ... c0 mpc.gencost = [ 2 0 0 3 0.0430293 20 0; 2 0 0 3 0.25 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; ];

95





Vmin mpc.bus = [ 1 3 0 0 0 0 1 1 0 135 1 1.05 0.95; 2 2 21.7 12.7 0 0 1 1 0 135 1 1.1 0.95; 3 1 2.4 1.2 0 0 1 1 0 135 1 1.05 0.95; 4 1 7.6 1.6 0 0 1 1 0 135 1 1.05 0.95; 5 1 0 0 0 0.19 1 1 0 135 1 1.05 0.95; 6 1 0 0 0 0 1 1 0 135 1 1.05 0.95; 7 1 22.8 10.9 0 0 1 1 0 135 1 1.05 0.95; 8 1 30 30 0 0 1 1 0 135 1 1.05 0.95; 9 1 0 0 0 0 1 1 0 135 1 1.05 0.95; 10 1 5.8 2 0 0 3 1 0 135 1 1.05 0.95; 11 1 0 0 0 0 1 1 0 135 1 1.05 0.95; 12 1 11.2 7.5 0 0 2 1 0 135 1 1.05 0.95; 13 2 0 0 0 0 2 1 0 135 1 1.1 0.95; 14 1 6.2 1.6 0 0 2 1 0 135 1 1.05 0.95; 15 1 8.2 2.5 0 0 2 1 0 135 1 1.05 0.95; 16 1 3.5 1.8 0 0 2 1 0 135 1 1.05 0.95; 17 1 9 5.8 0 0 2 1 0 135 1 1.05 0.95; 18 1 3.2 0.9 0 0 2 1 0 135 1 1.05 0.95; 19 1 9.5 3.4 0 0 2 1 0 135 1 1.05 0.95; 20 1 2.2 0.7 0 0 2 1 0 135 1 1.05 0.95; 21 1 17.5 11.2 0 0 3 1 0 135 1 1.05 0.95; 22 2 0 0 0 0 3 1 0 135 1 1.1 0.95; 23 2 3.2 1.6 0 0 2 1 0 135 1 1.1 0.95; 24 1 8.7 6.7 0 0.04 3 1 0 135 1 1.05 0.95; 25 1 0 0 0 0 3 1 0 135 1 1.05 0.95; 26 1 3.5 2.3 0 0 3 1 0 135 1 1.05 0.95; 27 2 0 0 0 0 3 1 0 135 1 1.1 0.95; 28 1 0 0 0 0 1 1 0 135 1 1.05 0.95; 29 1 2.4 0.9 0 0 3 1 0 135 1 1.05 0.95; 30 1 10.6 1.9 0 0 3 1 0 135 1 1.05 0.95; ];


Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 1 23.54 0 150 -20 1 100 1 80 0 0 0 0 0 0 0 0 0

0 0 0; 2 60.97 0 60 -20 1 100 1 80 0 0 0 0 0 0 0 0 0

0 0 0; 22 21.59 0 62.5 -15 1 100 1 50 0 0 0 0 0 0 0 0

0 0 0 0;

96

27 26.91 0 48.7 -15 1 100 1 55 0 0 0 0 0 0 0 0

0 0 0 0; 23 19.2 0 40 -10 1 100 1 30 0 0 0 0 0 0 0 0 0

0 0 0; 13 37 0 44.7 -15 1 100 1 40 0 0 0 0 0 0 0 0 0

0 0 0; ];


status angmin angmax mpc.branch = [ 1 2 0.02 0.06 0.03 130 130 130 0 0 1 -360 360; 1 3 0.05 0.19 0.02 130 130 130 0 0 1 -360 360; 2 4 0.06 0.17 0.02 65 65 65 0 0 1 -360 360; 3 4 0.01 0.04 0 130 130 130 0 0 1 -360 360; 2 5 0.05 0.2 0.02 130 130 130 0 0 1 -360 360; 2 6 0.06 0.18 0.02 65 65 65 0 0 1 -360 360; 4 6 0.01 0.04 0 90 90 90 0 0 1 -360 360; 5 7 0.05 0.12 0.01 70 70 70 0 0 1 -360 360; 6 7 0.03 0.08 0.01 130 130 130 0 0 1 -360 360; 6 8 0.01 0.04 0 32 32 32 0 0 1 -360 360; 6 9 0 0.21 0 65 65 65 0 0 1 -360 360; 6 10 0 0.56 0 32 32 32 0 0 1 -360 360; 9 11 0 0.21 0 65 65 65 0 0 1 -360 360; 9 10 0 0.11 0 65 65 65 0 0 1 -360 360; 4 12 0 0.26 0 65 65 65 0 0 1 -360 360; 12 13 0 0.14 0 65 65 65 0 0 1 -360 360; 12 14 0.12 0.26 0 32 32 32 0 0 1 -360 360; 12 15 0.07 0.13 0 32 32 32 0 0 1 -360 360; 12 16 0.09 0.2 0 32 32 32 0 0 1 -360 360; 14 15 0.22 0.2 0 16 16 16 0 0 1 -360 360; 16 17 0.08 0.19 0 16 16 16 0 0 1 -360 360; 15 18 0.11 0.22 0 16 16 16 0 0 1 -360 360; 18 19 0.06 0.13 0 16 16 16 0 0 1 -360 360; 19 20 0.03 0.07 0 32 32 32 0 0 1 -360 360; 10 20 0.09 0.21 0 32 32 32 0 0 1 -360 360; 10 17 0.03 0.08 0 32 32 32 0 0 1 -360 360; 10 21 0.03 0.07 0 32 32 32 0 0 1 -360 360; 10 22 0.07 0.15 0 32 32 32 0 0 1 -360 360; 21 22 0.01 0.02 0 32 32 32 0 0 1 -360 360; 15 23 0.1 0.2 0 16 16 16 0 0 1 -360 360; 22 24 0.12 0.18 0 16 16 16 0 0 1 -360 360; 23 24 0.13 0.27 0 16 16 16 0 0 1 -360 360; 24 25 0.19 0.33 0 16 16 16 0 0 1 -360 360; 25 26 0.25 0.38 0 16 16 16 0 0 1 -360 360; 25 27 0.11 0.21 0 16 16 16 0 0 1 -360 360; 28 27 0 0.4 0 65 65 65 0 0 1 -360 360; 27 29 0.22 0.42 0 16 16 16 0 0 1 -360 360; 27 30 0.32 0.6 0 16 16 16 0 0 1 -360 360; 29 30 0.24 0.45 0 16 16 16 0 0 1 -360 360; 8 28 0.06 0.2 0.02 32 32 32 0 0 1 -360 360; 6 28 0.02 0.06 0.01 32 32 32 0 0 1 -360 360; ];

%%----- OPF Data -----%%

97

%% area data % area refbus mpc.areas = [ 1 8; 2 23; 3 26; ];

%% generator cost data % 1 startup shutdown n x1 y1 ... xn yn % 2 startup shutdown n c(n-1) ... c0 mpc.gencost = [ 2 0 0 3 0.02 2 0; 2 0 0 3 0.0175 1.75 0; 2 0 0 3 0.0625 1 0; 2 0 0 3 0.00834 3.25 0; 2 0 0 3 0.025 3 0; 2 0 0 3 0.025 3 0; ];

98





Vmin mpc.bus = [ 1 2 51 27 0 0 1 0.955 10.67 138 1 1.06 0.94; 2 1 20 9 0 0 1 0.971 11.22 138 1 1.06 0.94; 3 1 39 10 0 0 1 0.968 11.56 138 1 1.06 0.94; 4 2 39 12 0 0 1 0.998 15.28 138 1 1.06 0.94; 5 1 0 0 0 -40 1 1.002 15.73 138 1 1.06 0.94; 6 2 52 22 0 0 1 0.99 13 138 1 1.06 0.94; 7 1 19 2 0 0 1 0.989 12.56 138 1 1.06 0.94; 8 2 28 0 0 0 1 1.015 20.77 345 1 1.06 0.94; 9 1 0 0 0 0 1 1.043 28.02 345 1 1.06 0.94; 10 2 0 0 0 0 1 1.05 35.61 345 1 1.06 0.94; 11 1 70 23 0 0 1 0.985 12.72 138 1 1.06 0.94; 12 2 47 10 0 0 1 0.99 12.2 138 1 1.06 0.94; 13 1 34 16 0 0 1 0.968 11.35 138 1 1.06 0.94; 14 1 14 1 0 0 1 0.984 11.5 138 1 1.06 0.94; 15 2 90 30 0 0 1 0.97 11.23 138 1 1.06 0.94; 16 1 25 10 0 0 1 0.984 11.91 138 1 1.06 0.94; 17 1 11 3 0 0 1 0.995 13.74 138 1 1.06 0.94; 18 2 60 34 0 0 1 0.973 11.53 138 1 1.06 0.94; 19 2 45 25 0 0 1 0.963 11.05 138 1 1.06 0.94; 20 1 18 3 0 0 1 0.958 11.93 138 1 1.06 0.94; 21 1 14 8 0 0 1 0.959 13.52 138 1 1.06 0.94; 22 1 10 5 0 0 1 0.97 16.08 138 1 1.06 0.94; 23 1 7 3 0 0 1 1 21 138 1 1.06 0.94; 24 2 13 0 0 0 1 0.992 20.89 138 1 1.06 0.94; 25 2 0 0 0 0 1 1.05 27.93 138 1 1.06 0.94; 26 2 0 0 0 0 1 1.015 29.71 345 1 1.06 0.94; 27 2 71 13 0 0 1 0.968 15.35 138 1 1.06 0.94; 28 1 17 7 0 0 1 0.962 13.62 138 1 1.06 0.94; 29 1 24 4 0 0 1 0.963 12.63 138 1 1.06 0.94; 30 1 0 0 0 0 1 0.968 18.79 345 1 1.06 0.94; 31 2 43 27 0 0 1 0.967 12.75 138 1 1.06 0.94; 32 2 59 23 0 0 1 0.964 14.8 138 1 1.06 0.94; 33 1 23 9 0 0 1 0.972 10.63 138 1 1.06 0.94; 34 2 59 26 0 14 1 0.986 11.3 138 1 1.06 0.94; 35 1 33 9 0 0 1 0.981 10.87 138 1 1.06 0.94; 36 2 31 17 0 0 1 0.98 10.87 138 1 1.06 0.94; 37 1 0 0 0 -25 1 0.992 11.77 138 1 1.06 0.94; 38 1 0 0 0 0 1 0.962 16.91 345 1 1.06 0.94; 39 1 27 11 0 0 1 0.97 8.41 138 1 1.06 0.94; 40 2 66 23 0 0 1 0.97 7.35 138 1 1.06 0.94; 41 1 37 10 0 0 1 0.967 6.92 138 1 1.06 0.94; 42 2 96 23 0 0 1 0.985 8.53 138 1 1.06 0.94; 43 1 18 7 0 0 1 0.978 11.28 138 1 1.06 0.94;

99

44 1 16 8 0 10 1 0.985 13.82 138 1 1.06 0.94; 45 1 53 22 0 10 1 0.987 15.67 138 1 1.06 0.94; 46 2 28 10 0 10 1 1.005 18.49 138 1 1.06 0.94; 47 1 34 0 0 0 1 1.017 20.73 138 1 1.06 0.94; 48 1 20 11 0 15 1 1.021 19.93 138 1 1.06 0.94; 49 2 87 30 0 0 1 1.025 20.94 138 1 1.06 0.94; 50 1 17 4 0 0 1 1.001 18.9 138 1 1.06 0.94; 51 1 17 8 0 0 1 0.967 16.28 138 1 1.06 0.94; 52 1 18 5 0 0 1 0.957 15.32 138 1 1.06 0.94; 53 1 23 11 0 0 1 0.946 14.35 138 1 1.06 0.94; 54 2 113 32 0 0 1 0.955 15.26 138 1 1.06 0.94; 55 2 63 22 0 0 1 0.952 14.97 138 1 1.06 0.94; 56 2 84 18 0 0 1 0.954 15.16 138 1 1.06 0.94; 57 1 12 3 0 0 1 0.971 16.36 138 1 1.06 0.94; 58 1 12 3 0 0 1 0.959 15.51 138 1 1.06 0.94; 59 2 277 113 0 0 1 0.985 19.37 138 1 1.06 0.94; 60 1 78 3 0 0 1 0.993 23.15 138 1 1.06 0.94; 61 2 0 0 0 0 1 0.995 24.04 138 1 1.06 0.94; 62 2 77 14 0 0 1 0.998 23.43 138 1 1.06 0.94; 63 1 0 0 0 0 1 0.969 22.75 345 1 1.06 0.94; 64 1 0 0 0 0 1 0.984 24.52 345 1 1.06 0.94; 65 2 0 0 0 0 1 1.005 27.65 345 1 1.06 0.94; 66 2 39 18 0 0 1 1.05 27.48 138 1 1.06 0.94; 67 1 28 7 0 0 1 1.02 24.84 138 1 1.06 0.94; 68 1 0 0 0 0 1 1.003 27.55 345 1 1.06 0.94; 69 3 0 0 0 0 1 1.035 30 138 1 1.06 0.94; 70 2 66 20 0 0 1 0.984 22.58 138 1 1.06 0.94; 71 1 0 0 0 0 1 0.987 22.15 138 1 1.06 0.94; 72 2 12 0 0 0 1 0.98 20.98 138 1 1.06 0.94; 73 2 6 0 0 0 1 0.991 21.94 138 1 1.06 0.94; 74 2 68 27 0 12 1 0.958 21.64 138 1 1.06 0.94; 75 1 47 11 0 0 1 0.967 22.91 138 1 1.06 0.94; 76 2 68 36 0 0 1 0.943 21.77 138 1 1.06 0.94; 77 2 61 28 0 0 1 1.006 26.72 138 1 1.06 0.94; 78 1 71 26 0 0 1 1.003 26.42 138 1 1.06 0.94; 79 1 39 32 0 20 1 1.009 26.72 138 1 1.06 0.94; 80 2 130 26 0 0 1 1.04 28.96 138 1 1.06 0.94; 81 1 0 0 0 0 1 0.997 28.1 345 1 1.06 0.94; 82 1 54 27 0 20 1 0.989 27.24 138 1 1.06 0.94; 83 1 20 10 0 10 1 0.985 28.42 138 1 1.06 0.94; 84 1 11 7 0 0 1 0.98 30.95 138 1 1.06 0.94; 85 2 24 15 0 0 1 0.985 32.51 138 1 1.06 0.94; 86 1 21 10 0 0 1 0.987 31.14 138 1 1.06 0.94; 87 2 0 0 0 0 1 1.015 31.4 161 1 1.06 0.94; 88 1 48 10 0 0 1 0.987 35.64 138 1 1.06 0.94; 89 2 0 0 0 0 1 1.005 39.69 138 1 1.06 0.94; 90 2 163 42 0 0 1 0.985 33.29 138 1 1.06 0.94; 91 2 10 0 0 0 1 0.98 33.31 138 1 1.06 0.94; 92 2 65 10 0 0 1 0.993 33.8 138 1 1.06 0.94; 93 1 12 7 0 0 1 0.987 30.79 138 1 1.06 0.94; 94 1 30 16 0 0 1 0.991 28.64 138 1 1.06 0.94; 95 1 42 31 0 0 1 0.981 27.67 138 1 1.06 0.94; 96 1 38 15 0 0 1 0.993 27.51 138 1 1.06 0.94; 97 1 15 9 0 0 1 1.011 27.88 138 1 1.06 0.94; 98 1 34 8 0 0 1 1.024 27.4 138 1 1.06 0.94; 99 2 42 0 0 0 1 1.01 27.04 138 1 1.06 0.94; 100 2 37 18 0 0 1 1.017 28.03 138 1 1.06 0.94;

100

101 1 22 15 0 0 1 0.993 29.61 138 1 1.06 0.94; 102 1 5 3 0 0 1 0.991 32.3 138 1 1.06 0.94; 103 2 23 16 0 0 1 1.001 24.44 138 1 1.06 0.94; 104 2 38 25 0 0 1 0.971 21.69 138 1 1.06 0.94; 105 2 31 26 0 20 1 0.965 20.57 138 1 1.06 0.94; 106 1 43 16 0 0 1 0.962 20.32 138 1 1.06 0.94; 107 2 50 12 0 6 1 0.952 17.53 138 1 1.06 0.94; 108 1 2 1 0 0 1 0.967 19.38 138 1 1.06 0.94; 109 1 8 3 0 0 1 0.967 18.93 138 1 1.06 0.94; 110 2 39 30 0 6 1 0.973 18.09 138 1 1.06 0.94; 111 2 0 0 0 0 1 0.98 19.74 138 1 1.06 0.94; 112 2 68 13 0 0 1 0.975 14.99 138 1 1.06 0.94; 113 2 6 0 0 0 1 0.993 13.74 138 1 1.06 0.94; 114 1 8 3 0 0 1 0.96 14.46 138 1 1.06 0.94; 115 1 22 7 0 0 1 0.96 14.46 138 1 1.06 0.94; 116 2 184 0 0 0 1 1.005 27.12 138 1 1.06 0.94; 117 1 20 8 0 0 1 0.974 10.67 138 1 1.06 0.94; 118 1 33 15 0 0 1 0.949 21.92 138 1 1.06 0.94; ];


Qc1min Qc1max Qc2min Qc2max ramp_agc ramp_10 ramp_30 ramp_q apf mpc.gen = [ 1 0 0 15 -5 0.955 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 4 0 0 300 -300 0.998 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 6 0 0 50 -13 0.99 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 8 0 0 300 -300 1.015 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 10 450 0 200 -147 1.05 100 1 550 0 0 0 0 0 0 0 0

0 0 0 0; 12 85 0 120 -35 0.99 100 1 185 0 0 0 0 0 0 0 0 0

0 0 0; 15 0 0 30 -10 0.97 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 18 0 0 50 -16 0.973 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 19 0 0 24 -8 0.962 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 24 0 0 300 -300 0.992 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 25 220 0 140 -47 1.05 100 1 320 0 0 0 0 0 0 0 0 0

0 0 0; 26 314 0 1000 -1000 1.015 100 1 414 0 0 0 0 0 0 0

0 0 0 0 0; 27 0 0 300 -300 0.968 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 31 7 0 300 -300 0.967 100 1 107 0 0 0 0 0 0 0 0

0 0 0 0; 32 0 0 42 -14 0.963 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 34 0 0 24 -8 0.984 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0;

101

36 0 0 24 -8 0.98 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 40 0 0 300 -300 0.97 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 42 0 0 300 -300 0.985 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 46 19 0 100 -100 1.005 100 1 119 0 0 0 0 0 0 0 0

0 0 0 0; 49 204 0 210 -85 1.025 100 1 304 0 0 0 0 0 0 0 0 0

0 0 0; 54 48 0 300 -300 0.955 100 1 148 0 0 0 0 0 0 0 0

0 0 0 0; 55 0 0 23 -8 0.952 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 56 0 0 15 -8 0.954 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 59 155 0 180 -60 0.985 100 1 255 0 0 0 0 0 0 0 0 0

0 0 0; 61 160 0 300 -100 0.995 100 1 260 0 0 0 0 0 0 0 0

0 0 0 0; 62 0 0 20 -20 0.998 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 65 391 0 200 -67 1.005 100 1 491 0 0 0 0 0 0 0 0 0

0 0 0; 66 392 0 200 -67 1.05 100 1 492 0 0 0 0 0 0 0 0 0

0 0 0; 69 516.4 0 300 -300 1.035 100 1 805.2 0 0 0 0 0 0

0 0 0 0 0 0; 70 0 0 32 -10 0.984 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 72 0 0 100 -100 0.98 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 73 0 0 100 -100 0.991 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 74 0 0 9 -6 0.958 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 76 0 0 23 -8 0.943 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 77 0 0 70 -20 1.006 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 80 477 0 280 -165 1.04 100 1 577 0 0 0 0 0 0 0 0

0 0 0 0; 85 0 0 23 -8 0.985 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 87 4 0 1000 -100 1.015 100 1 104 0 0 0 0 0 0 0

0 0 0 0 0; 89 607 0 300 -210 1.005 100 1 707 0 0 0 0 0 0 0 0

0 0 0 0; 90 0 0 300 -300 0.985 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 91 0 0 100 -100 0.98 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 92 0 0 9 -3 0.99 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 99 0 0 100 -100 1.01 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0;

102

100 252 0 155 -50 1.017 100 1 352 0 0 0 0 0 0 0 0 0

0 0 0; 103 40 0 40 -15 1.01 100 1 140 0 0 0 0 0 0 0 0 0

0 0 0; 104 0 0 23 -8 0.971 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 105 0 0 23 -8 0.965 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 107 0 0 200 -200 0.952 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 110 0 0 23 -8 0.973 100 1 100 0 0 0 0 0 0 0 0 0

0 0 0; 111 36 0 1000 -100 0.98 100 1 136 0 0 0 0 0 0 0

0 0 0 0 0; 112 0 0 1000 -100 0.975 100 1 100 0 0 0 0 0 0 0

0 0 0 0 0; 113 0 0 200 -100 0.993 100 1 100 0 0 0 0 0 0 0 0

0 0 0 0; 116 0 0 1000 -1000 1.005 100 1 100 0 0 0 0 0 0 0

0 0 0 0 0; ];


status angmin angmax mpc.branch = [ 1 2 0.0303 0.0999 0.0254 9900 0 0 0 0 1 -360 360; 1 3 0.0129 0.0424 0.01082 9900 0 0 0 0 1 -360 360; 4 5 0.00176 0.00798 0.0021 9900 0 0 0 0 1 -360 360; 3 5 0.0241 0.108 0.0284 9900 0 0 0 0 1 -360 360; 5 6 0.0119 0.054 0.01426 9900 0 0 0 0 1 -360 360; 6 7 0.00459 0.0208 0.0055 9900 0 0 0 0 1 -360 360; 8 9 0.00244 0.0305 1.162 9900 0 0 0 0 1 -360 360; 8 5 0 0.0267 0 9900 0 0 0.985 0 1 -360 360; 9 10 0.00258 0.0322 1.23 9900 0 0 0 0 1 -360 360; 4 11 0.0209 0.0688 0.01748 9900 0 0 0 0 1 -360 360; 5 11 0.0203 0.0682 0.01738 9900 0 0 0 0 1 -360 360; 11 12 0.00595 0.0196 0.00502 9900 0 0 0 0 1 -360 360; 2 12 0.0187 0.0616 0.01572 9900 0 0 0 0 1 -360 360; 3 12 0.0484 0.16 0.0406 9900 0 0 0 0 1 -360 360; 7 12 0.00862 0.034 0.00874 9900 0 0 0 0 1 -360 360; 11 13 0.02225 0.0731 0.01876 9900 0 0 0 0 1 -360 360; 12 14 0.0215 0.0707 0.01816 9900 0 0 0 0 1 -360 360; 13 15 0.0744 0.2444 0.06268 9900 0 0 0 0 1 -360 360; 14 15 0.0595 0.195 0.0502 9900 0 0 0 0 1 -360 360; 12 16 0.0212 0.0834 0.0214 9900 0 0 0 0 1 -360 360; 15 17 0.0132 0.0437 0.0444 9900 0 0 0 0 1 -360 360; 16 17 0.0454 0.1801 0.0466 9900 0 0 0 0 1 -360 360; 17 18 0.0123 0.0505 0.01298 9900 0 0 0 0 1 -360 360; 18 19 0.01119 0.0493 0.01142 9900 0 0 0 0 1 -360 360; 19 20 0.0252 0.117 0.0298 9900 0 0 0 0 1 -360 360; 15 19 0.012 0.0394 0.0101 9900 0 0 0 0 1 -360 360; 20 21 0.0183 0.0849 0.0216 9900 0 0 0 0 1 -360 360; 21 22 0.0209 0.097 0.0246 9900 0 0 0 0 1 -360 360; 22 23 0.0342 0.159 0.0404 9900 0 0 0 0 1 -360 360; 23 24 0.0135 0.0492 0.0498 9900 0 0 0 0 1 -360 360; 23 25 0.0156 0.08 0.0864 9900 0 0 0 0 1 -360 360;

103

26 25 0 0.0382 0 9900 0 0 0.96 0 1 -360 360; 25 27 0.0318 0.163 0.1764 9900 0 0 0 0 1 -360 360; 27 28 0.01913 0.0855 0.0216 9900 0 0 0 0 1 -360 360; 28 29 0.0237 0.0943 0.0238 9900 0 0 0 0 1 -360 360; 30 17 0 0.0388 0 9900 0 0 0.96 0 1 -360 360; 8 30 0.00431 0.0504 0.514 9900 0 0 0 0 1 -360 360; 26 30 0.00799 0.086 0.908 9900 0 0 0 0 1 -360 360; 17 31 0.0474 0.1563 0.0399 9900 0 0 0 0 1 -360 360; 29 31 0.0108 0.0331 0.0083 9900 0 0 0 0 1 -360 360; 23 32 0.0317 0.1153 0.1173 9900 0 0 0 0 1 -360 360; 31 32 0.0298 0.0985 0.0251 9900 0 0 0 0 1 -360 360; 27 32 0.0229 0.0755 0.01926 9900 0 0 0 0 1 -360 360; 15 33 0.038 0.1244 0.03194 9900 0 0 0 0 1 -360 360; 19 34 0.0752 0.247 0.0632 9900 0 0 0 0 1 -360 360; 35 36 0.00224 0.0102 0.00268 9900 0 0 0 0 1 -360 360; 35 37 0.011 0.0497 0.01318 9900 0 0 0 0 1 -360 360; 33 37 0.0415 0.142 0.0366 9900 0 0 0 0 1 -360 360; 34 36 0.00871 0.0268 0.00568 9900 0 0 0 0 1 -360 360; 34 37 0.00256 0.0094 0.00984 9900 0 0 0 0 1 -360 360; 38 37 0 0.0375 0 9900 0 0 0.935 0 1 -360 360; 37 39 0.0321 0.106 0.027 9900 0 0 0 0 1 -360 360; 37 40 0.0593 0.168 0.042 9900 0 0 0 0 1 -360 360; 30 38 0.00464 0.054 0.422 9900 0 0 0 0 1 -360 360; 39 40 0.0184 0.0605 0.01552 9900 0 0 0 0 1 -360 360; 40 41 0.0145 0.0487 0.01222 9900 0 0 0 0 1 -360 360; 40 42 0.0555 0.183 0.0466 9900 0 0 0 0 1 -360 360; 41 42 0.041 0.135 0.0344 9900 0 0 0 0 1 -360 360; 43 44 0.0608 0.2454 0.06068 9900 0 0 0 0 1 -360 360; 34 43 0.0413 0.1681 0.04226 9900 0 0 0 0 1 -360 360; 44 45 0.0224 0.0901 0.0224 9900 0 0 0 0 1 -360 360; 45 46 0.04 0.1356 0.0332 9900 0 0 0 0 1 -360 360; 46 47 0.038 0.127 0.0316 9900 0 0 0 0 1 -360 360; 46 48 0.0601 0.189 0.0472 9900 0 0 0 0 1 -360 360; 47 49 0.0191 0.0625 0.01604 9900 0 0 0 0 1 -360 360; 42 49 0.0715 0.323 0.086 9900 0 0 0 0 1 -360 360; 42 49 0.0715 0.323 0.086 9900 0 0 0 0 1 -360 360; 45 49 0.0684 0.186 0.0444 9900 0 0 0 0 1 -360 360; 48 49 0.0179 0.0505 0.01258 9900 0 0 0 0 1 -360 360; 49 50 0.0267 0.0752 0.01874 9900 0 0 0 0 1 -360 360; 49 51 0.0486 0.137 0.0342 9900 0 0 0 0 1 -360 360; 51 52 0.0203 0.0588 0.01396 9900 0 0 0 0 1 -360 360; 52 53 0.0405 0.1635 0.04058 9900 0 0 0 0 1 -360 360; 53 54 0.0263 0.122 0.031 9900 0 0 0 0 1 -360 360; 49 54 0.073 0.289 0.0738 9900 0 0 0 0 1 -360 360; 49 54 0.0869 0.291 0.073 9900 0 0 0 0 1 -360 360; 54 55 0.0169 0.0707 0.0202 9900 0 0 0 0 1 -360 360; 54 56 0.00275 0.00955 0.00732 9900 0 0 0 0 1 -360 360; 55 56 0.00488 0.0151 0.00374 9900 0 0 0 0 1 -360 360; 56 57 0.0343 0.0966 0.0242 9900 0 0 0 0 1 -360 360; 50 57 0.0474 0.134 0.0332 9900 0 0 0 0 1 -360 360; 56 58 0.0343 0.0966 0.0242 9900 0 0 0 0 1 -360 360; 51 58 0.0255 0.0719 0.01788 9900 0 0 0 0 1 -360 360; 54 59 0.0503 0.2293 0.0598 9900 0 0 0 0 1 -360 360; 56 59 0.0825 0.251 0.0569 9900 0 0 0 0 1 -360 360; 56 59 0.0803 0.239 0.0536 9900 0 0 0 0 1 -360 360; 55 59 0.04739 0.2158 0.05646 9900 0 0 0 0 1 -360 360; 59 60 0.0317 0.145 0.0376 9900 0 0 0 0 1 -360 360;

104

59 61 0.0328 0.15 0.0388 9900 0 0 0 0 1 -360 360; 60 61 0.00264 0.0135 0.01456 9900 0 0 0 0 1 -360 360; 60 62 0.0123 0.0561 0.01468 9900 0 0 0 0 1 -360 360; 61 62 0.00824 0.0376 0.0098 9900 0 0 0 0 1 -360 360; 63 59 0 0.0386 0 9900 0 0 0.96 0 1 -360 360; 63 64 0.00172 0.02 0.216 9900 0 0 0 0 1 -360 360; 64 61 0 0.0268 0 9900 0 0 0.985 0 1 -360 360; 38 65 0.00901 0.0986 1.046 9900 0 0 0 0 1 -360 360; 64 65 0.00269 0.0302 0.38 9900 0 0 0 0 1 -360 360; 49 66 0.018 0.0919 0.0248 9900 0 0 0 0 1 -360 360; 49 66 0.018 0.0919 0.0248 9900 0 0 0 0 1 -360 360; 62 66 0.0482 0.218 0.0578 9900 0 0 0 0 1 -360 360; 62 67 0.0258 0.117 0.031 9900 0 0 0 0 1 -360 360; 65 66 0 0.037 0 9900 0 0 0.935 0 1 -360 360; 66 67 0.0224 0.1015 0.02682 9900 0 0 0 0 1 -360 360; 65 68 0.00138 0.016 0.638 9900 0 0 0 0 1 -360 360; 47 69 0.0844 0.2778 0.07092 9900 0 0 0 0 1 -360 360; 49 69 0.0985 0.324 0.0828 9900 0 0 0 0 1 -360 360; 68 69 0 0.037 0 9900 0 0 0.935 0 1 -360 360; 69 70 0.03 0.127 0.122 9900 0 0 0 0 1 -360 360; 24 70 0.00221 0.4115 0.10198 9900 0 0 0 0 1 -360 360; 70 71 0.00882 0.0355 0.00878 9900 0 0 0 0 1 -360 360; 24 72 0.0488 0.196 0.0488 9900 0 0 0 0 1 -360 360; 71 72 0.0446 0.18 0.04444 9900 0 0 0 0 1 -360 360; 71 73 0.00866 0.0454 0.01178 9900 0 0 0 0 1 -360 360; 70 74 0.0401 0.1323 0.03368 9900 0 0 0 0 1 -360 360; 70 75 0.0428 0.141 0.036 9900 0 0 0 0 1 -360 360; 69 75 0.0405 0.122 0.124 9900 0 0 0 0 1 -360 360; 74 75 0.0123 0.0406 0.01034 9900 0 0 0 0 1 -360 360; 76 77 0.0444 0.148 0.0368 9900 0 0 0 0 1 -360 360; 69 77 0.0309 0.101 0.1038 9900 0 0 0 0 1 -360 360; 75 77 0.0601 0.1999 0.04978 9900 0 0 0 0 1 -360 360; 77 78 0.00376 0.0124 0.01264 9900 0 0 0 0 1 -360 360; 78 79 0.00546 0.0244 0.00648 9900 0 0 0 0 1 -360 360; 77 80 0.017 0.0485 0.0472 9900 0 0 0 0 1 -360 360; 77 80 0.0294 0.105 0.0228 9900 0 0 0 0 1 -360 360; 79 80 0.0156 0.0704 0.0187 9900 0 0 0 0 1 -360 360; 68 81 0.00175 0.0202 0.808 9900 0 0 0 0 1 -360 360; 81 80 0 0.037 0 9900 0 0 0.935 0 1 -360 360; 77 82 0.0298 0.0853 0.08174 9900 0 0 0 0 1 -360 360; 82 83 0.0112 0.03665 0.03796 9900 0 0 0 0 1 -360 360; 83 84 0.0625 0.132 0.0258 9900 0 0 0 0 1 -360 360; 83 85 0.043 0.148 0.0348 9900 0 0 0 0 1 -360 360; 84 85 0.0302 0.0641 0.01234 9900 0 0 0 0 1 -360 360; 85 86 0.035 0.123 0.0276 9900 0 0 0 0 1 -360 360; 86 87 0.02828 0.2074 0.0445 9900 0 0 0 0 1 -360 360; 85 88 0.02 0.102 0.0276 9900 0 0 0 0 1 -360 360; 85 89 0.0239 0.173 0.047 9900 0 0 0 0 1 -360 360; 88 89 0.0139 0.0712 0.01934 9900 0 0 0 0 1 -360 360; 89 90 0.0518 0.188 0.0528 9900 0 0 0 0 1 -360 360; 89 90 0.0238 0.0997 0.106 9900 0 0 0 0 1 -360 360; 90 91 0.0254 0.0836 0.0214 9900 0 0 0 0 1 -360 360; 89 92 0.0099 0.0505 0.0548 9900 0 0 0 0 1 -360 360; 89 92 0.0393 0.1581 0.0414 9900 0 0 0 0 1 -360 360; 91 92 0.0387 0.1272 0.03268 9900 0 0 0 0 1 -360 360; 92 93 0.0258 0.0848 0.0218 9900 0 0 0 0 1 -360 360; 92 94 0.0481 0.158 0.0406 9900 0 0 0 0 1 -360 360;

105

93 94 0.0223 0.0732 0.01876 9900 0 0 0 0 1 -360 360; 94 95 0.0132 0.0434 0.0111 9900 0 0 0 0 1 -360 360; 80 96 0.0356 0.182 0.0494 9900 0 0 0 0 1 -360 360; 82 96 0.0162 0.053 0.0544 9900 0 0 0 0 1 -360 360; 94 96 0.0269 0.0869 0.023 9900 0 0 0 0 1 -360 360; 80 97 0.0183 0.0934 0.0254 9900 0 0 0 0 1 -360 360; 80 98 0.0238 0.108 0.0286 9900 0 0 0 0 1 -360 360; 80 99 0.0454 0.206 0.0546 9900 0 0 0 0 1 -360 360; 92 100 0.0648 0.295 0.0472 9900 0 0 0 0 1 -360 360; 94 100 0.0178 0.058 0.0604 9900 0 0 0 0 1 -360 360; 95 96 0.0171 0.0547 0.01474 9900 0 0 0 0 1 -360 360; 96 97 0.0173 0.0885 0.024 9900 0 0 0 0 1 -360 360; 98 100 0.0397 0.179 0.0476 9900 0 0 0 0 1 -360 360; 99 100 0.018 0.0813 0.0216 9900 0 0 0 0 1 -360 360; 100 101 0.0277 0.1262 0.0328 9900 0 0 0 0 1 -360 360; 92 102 0.0123 0.0559 0.01464 9900 0 0 0 0 1 -360 360; 101 102 0.0246 0.112 0.0294 9900 0 0 0 0 1 -360 360; 100 103 0.016 0.0525 0.0536 9900 0 0 0 0 1 -360 360; 100 104 0.0451 0.204 0.0541 9900 0 0 0 0 1 -360 360; 103 104 0.0466 0.1584 0.0407 9900 0 0 0 0 1 -360 360; 103 105 0.0535 0.1625 0.0408 9900 0 0 0 0 1 -360 360; 100 106 0.0605 0.229 0.062 9900 0 0 0 0 1 -360 360; 104 105 0.00994 0.0378 0.00986 9900 0 0 0 0 1 -360 360; 105 106 0.014 0.0547 0.01434 9900 0 0 0 0 1 -360 360; 105 107 0.053 0.183 0.0472 9900 0 0 0 0 1 -360 360; 105 108 0.0261 0.0703 0.01844 9900 0 0 0 0 1 -360 360; 106 107 0.053 0.183 0.0472 9900 0 0 0 0 1 -360 360; 108 109 0.0105 0.0288 0.0076 9900 0 0 0 0 1 -360 360; 103 110 0.03906 0.1813 0.0461 9900 0 0 0 0 1 -360 360; 109 110 0.0278 0.0762 0.0202 9900 0 0 0 0 1 -360 360; 110 111 0.022 0.0755 0.02 9900 0 0 0 0 1 -360 360; 110 112 0.0247 0.064 0.062 9900 0 0 0 0 1 -360 360; 17 113 0.00913 0.0301 0.00768 9900 0 0 0 0 1 -360 360; 32 113 0.0615 0.203 0.0518 9900 0 0 0 0 1 -360 360; 32 114 0.0135 0.0612 0.01628 9900 0 0 0 0 1 -360 360; 27 115 0.0164 0.0741 0.01972 9900 0 0 0 0 1 -360 360; 114 115 0.0023 0.0104 0.00276 9900 0 0 0 0 1 -360 360; 68 116 0.00034 0.00405 0.164 9900 0 0 0 0 1 -360 360; 12 117 0.0329 0.14 0.0358 9900 0 0 0 0 1 -360 360; 75 118 0.0145 0.0481 0.01198 9900 0 0 0 0 1 -360 360; 76 118 0.0164 0.0544 0.01356 9900 0 0 0 0 1 -360 360; ];

%%----- OPF Data -----%% %% generator cost data % 1 startup shutdown n x1 y1 ... xn yn % 2 startup shutdown n c(n-1) ... c0 mpc.gencost = [ 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.0222222 20 0; 2 0 0 3 0.117647 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0;

106

2 0 0 3 0.01 40 0; 2 0 0 3 0.0454545 20 0; 2 0 0 3 0.0318471 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 1.42857 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.526316 20 0; 2 0 0 3 0.0490196 20 0; 2 0 0 3 0.208333 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.0645161 20 0; 2 0 0 3 0.0625 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.0255754 20 0; 2 0 0 3 0.0255102 20 0; 2 0 0 3 0.0193648 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.0209644 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 2.5 20 0; 2 0 0 3 0.0164745 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.0396825 20 0; 2 0 0 3 0.25 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.277778 20 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; 2 0 0 3 0.01 40 0; ];

107

APPENDIX C: MATLAB Code

C.1 Main Code for Load Flow Analysis Tool

%for repeat=1:1:100 clear all; clc; perm_num=1; perm_max=10000000; while perm_num <= perm_max for method = 1:2:1 clearvars -EXCEPT method perm_num perm_max %% Convergence Test % This file is used to test the convergence properties of several different % AC loadflow methodologies in a modified Sato and Arrillaga's method with % respect to the number of HVDC links in a system. %clear all; clc; close all;

%% Set up for Load flow %% casename = 'case4gs'; casedata = loadcase(casename); % load the casedata numbranch = size(casedata.branch,1); % how many branches linkdata = setlinkdata(casedata);% set HVDC link parameters convergence_times = []; % data matrix to store convergence times full_iterations = []; %datamatrix to store convergence interations AC_iterations = []; %datamatrix to store convergence interations orderings = makeorders(casedata); if strcmp(casename,'case4gs') || strcmp(casename,'case9') neworderings=[]; for i=1:1:500 neworderings = [orderings;neworderings]; end orderings = neworderings; end %perm_max = floor(length(orderings(:,1))*0.01); perm_max = 1; %perm_max = length(orderings(:,1)); order = orderings(perm_num,:); %order = numbranch:-1:1; %order of replacing links type = method; if type == 3 type = method+1; end; %1 = nr, 2=fd, 4 = gs

global ACtime_iso ACtime_sys etime failures = []; AC_time = []; DC_time = []; AC_time_iso =[]; AC_time_sys = []; AC_time_extra = [];

%% Run Modified Sato and Arrillaga's Method for #HVDC Links = 0...numbranch numbranch=length(order) for numlinks = 0:1:numbranch numlinks

108

success=1; total = 0; ac_t_total = 0; dc_t_total = 0; ACtime_sys = 0; ACtime_iso = 0; etime = 0; if numlinks > 0 [ACdata,DCdata]=buildDC(casedata,numlinks,order,linkdata); else ACdata = casedata; DCdata = casedata; end tol = 20; i = 0;

t_start = tic; reset = [ACdata.bus(:,3) ACdata.bus(:,4)]; store = [real(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180)) imag(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180))]; while tol > 10^-5 && success == 1 if numlinks > 0 dc_t_start = tic; [ACdata,DCdata] = runsimphvdcpf(ACdata,DCdata); dc_t_total = dc_t_total + toc(dc_t_start); end ac_t_start = tic; [ACdata,success,iterations] = runACpfv2(ACdata,DCdata,type); ac_t_total = ac_t_total + toc(ac_t_start); total = total + iterations; DCdata.bus(:,2) = ACdata.bus(:,8); DCdata.bus(:,3) = ACdata.bus(:,9); new = [real(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180)) imag(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180))]; tol = max(abs(store - new).^2); store = new; ACdata.bus(:,3)=reset(:,1); ACdata.bus(:,4)=reset(:,2); if success == 0 failures=[failures numlinks]; end i=i+1; end store convergence_times = [convergence_times toc(t_start)]; %store convergence

times AC_iterations = [AC_iterations total]; full_iterations = [full_iterations i]; %store convergence iterations AC_time = [AC_time ac_t_total]; DC_time = [DC_time dc_t_total]; AC_time_iso =[AC_time_iso ACtime_iso]; AC_time_sys =[AC_time_sys ACtime_sys]; AC_time_extra=[AC_time_extra etime]; end %% Delete Failed Trials xdata = 0:1:numbranch; if ~isempty(failures)

109

failures = sort(failures,2); for j = length(failures):-1:1 convergence_times(failures(j)+1) = []; AC_iterations(failures(j)+1)=[]; full_iterations(failures(j)+1)=[]; AC_time(failures(j)+1)=[]; DC_time(failures(j)+1)=[]; AC_time_iso(failures(j)+1)=[]; AC_time_sys(failures(j)+1)=[]; AC_time_extra(failures(j)+1)=[]; xdata(failures(j)+1)=[]; end end

%% Create Plots

figure(1) hold on plot(xdata, convergence_times,'r-',... xdata, AC_time,'b-',... xdata, DC_time,'g-',... xdata, convergence_times-AC_time-DC_time,'k-') if method ==1 title(strcat(casename,': Time Partition for Newton Raphson Method')) elseif method == 2 title(strcat(casename,': Time Partition for FDLF Method')) else title(strcat(casename,': Time Partition for Gauss Seidel Method')) end xlabel('Number of Links') ylabel('time (s)') axis([0 numbranch 0 (max(convergence_times)*1.2)]) legend('Total Time','AC Routine','DC Routine','Additional Computation') hold off

figure(2) hold on plot(xdata, AC_time,'b',... xdata, AC_time_iso,'r',... xdata, AC_time_sys,'g',... xdata, AC_time_extra,'k') if method ==1 title(strcat(casename,': AC Time Partition for Newton Raphson Method')) elseif method == 2 title(strcat(casename,': AC Time Partition for FDLF Method')) else title(strcat(casename,': AC Time Partition for Gauss Seidel Method')) end xlabel('Number of Links') ylabel('time (s)') axis([0 numbranch 0 (max(convergence_times)*1.2)]) legend('AC Total','Isolated Buses','System Buses','Extra') hold off

110

figure(3) hold on plot(xdata, full_iterations, 'r-') if method ==1 title(strcat(casename,': # Full Loadflow Iterations for Newton Raphson

Method')) elseif method == 2 title(strcat(casename,': # Full Loadflow Iterations for FDLF Method')) else title(strcat(casename,': # Full Loadflow Iterations for Gauss Seidel

Method')) end xlabel('Number of Links') ylabel('# of Iterations') axis([0 numbranch 0 (max(full_iterations)*1.2)]) hold off

figure(4) hold on plot(xdata, AC_iterations, 'r-') if method ==1 title(strcat(casename,': # AC Loadflow Iterations for Newton Raphson

Method')) elseif method == 2 title(strcat(casename,': # AC Loadflow Iterations for FDLF Method')) else title(strcat(casename,': # AC Loadflow Iterations for Gauss Seidel

Method')) end xlabel('Number of Links') ylabel('# of Iterations') axis([0 numbranch 0 (max(AC_iterations)*1.2)]) hold off

%}

%% Save if method ==1 filename = strcat(casename,'_NR_',int2str(perm_num)); elseif method == 2 filename = strcat(casename,'_FDLF_',int2str(perm_num)); else filename = strcat(casename,'_GS_',int2str(perm_num)); end save(filename) end perm_num=perm_num+1; end %end

111

C.2 Code to set HVDC Link Parameters

function linkdata = setlinkdata(casedata)

%bus format: % 1 busnum % 2 voltage % 3 angle % 4 Stotal % 5 Impedance % 6 flag for 0=gen-load

linkdata.bus = [casedata.bus(:,1), casedata.bus(:,8), casedata.bus(:,9)... zeros(size(casedata.bus(:,1)))];

%branch format: % 1 Rec_bus % 2 Inv_bus % 3 Rec_bus_P % 4 Rec_bus_Q % 5 Inv_bus_P % 6 Inv_bus_Q % 7 RLine

casedata=runpf(casedata); linkdata.branch = [casedata.branch(:,1), casedata.branch(:,2), ... casedata.branch(:,14), casedata.branch(:,15), casedata.branch(:,16),... casedata.branch(:,17), casedata.branch(:,3)/4];

for i=1:1:length(linkdata.branch(:,1)) %sort rec and inv buses if linkdata.branch(i,3) < 0 dummy = linkdata.branch(i,1); linkdata.branch(i,1)=linkdata.branch(i,2); linkdata.branch(i,2)=dummy; dummy = linkdata.branch(i,3); linkdata.branch(i,3)=linkdata.branch(i,5); linkdata.branch(i,5)=dummy; dummy = linkdata.branch(i,4); linkdata.branch(i,4)=linkdata.branch(i,6); linkdata.branch(i,6)=dummy; end if linkdata.branch(i,7) == 0 linkdata.branch(i,7) = 0.001; %% if AC line has R=0, change to

ficticious 0.01; linkdata.branch(i,5) = linkdata.branch(i,5)+0.2; %also add 0.2W to

simulate Ploss end end

%Create "impedance" for buses voltages = casedata.bus(:,8).*exp(sqrt(-1)*casedata.bus(:,9)*pi/180); powers = zeros(size(linkdata.bus(:,1))); flag = zeros(size(powers)); %{ for i=1:1:length(linkdata.branch(:,1))

112

powers(linkdata.branch(i,1)) = powers(linkdata.branch(i,1)) - ...

%subtract rectifier power (linkdata.branch(i,3)+sqrt(-1)*linkdata.branch(i,4)); powers(linkdata.branch(i,2)) = powers(linkdata.branch(i,2)) - ... %add

invertor power (linkdata.branch(i,5)+sqrt(-1)*linkdata.branch(i,6)); end %} powers = powers - (casedata.bus(:,3) + sqrt(-1)*casedata.bus(:,4)); for i = 1:1:length(casedata.gen(:,1)) powers(casedata.gen(i,1)) =

powers(casedata.gen(i,1))+(casedata.gen(i,2)+sqrt(-1)*casedata.gen(i,3)); end powers=powers/100; for iter = 1:1:length(powers) if powers(iter) == 0 powers(iter)=1; flag(iter)=1; end end impedance = (voltages.^2)./(powers) linkdata.bus = [linkdata.bus impedance flag];

%interface (xfmr, filter, etc) format: % 1 tapr % 2 tapi % 3 numtapr % 4 numtapi % 5 reg%min % 6 reg%max % 7 xp % 8 xs % 9 xt % 10 xc % 11 zf

dummy = ones(size(linkdata.branch(:,1))); linkdata.interface = [16*dummy, 16*dummy, 31*dummy, 31*dummy,... 85*dummy, 115*dummy, 0.077* dummy, 0.049*dummy,... 0.00225*dummy, 0.07275*dummy,... (0.000728+sqrt(-1)*0.4902)*dummy];

%converter control format: % 1 delay % 2 extinction % 3 mindelay % 4 minextinct % 5 Arec % 6 Ainv]

linkdata.control = [10*dummy, 10*dummy, 10*dummy, 10*dummy, 53.71*dummy,

53.71*dummy];

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end

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C.3 Code for AC Subroutine

function [ACdata,success,total] = runACpfv2(ACdata,DCdata,type) global ACtime_iso ACtime_sys etime ACtime_iso = ACtime_iso; ACtime_sys = ACtime_sys; etime = etime;

extratiming_start= tic;

AC = mpoption('PF_ALG', type); success=1; failure = []; total = 0; ybus = makeYbus(ACdata); [S,C] = graphconncomp(ybus,'Directed',false); % S is the number of disjoint % systems, C is what bus belongs to what % system numbuses = histc(C,unique(C)); %count of the #buses in each system

ACdata.branch = [ACdata.branch zeros(length(ACdata.branch(:,1)),4)]; if isfield(ACdata,'areas') ACdata=rmfield(ACdata,'areas'); end if isfield(ACdata,'gencost') ACdata=rmfield(ACdata,'gencost'); end

currentAC = ACdata; busnum = []; volmag =[]; ref =[]; flag = [];

%this block of code finds all isolated buses for j = length(C):-1:1 if numbuses(C(j)) == 1 busnum = [busnum; currentAC.bus(j,1)]; %{ if currentAC.bus(j,2) == 2 || currentAC.bus(j,2) == 3 volmag = [volmag; [currentAC.bus(j,1) length(busnum)]]; if currentAC.bus(j,2)==3 ref = [currentAC.bus(j,1) length(busnum)]; end end %} end end %%%% etime= etime+toc(extratiming_start);

%%%%%%% This block of code calculates the isolated bus voltages isostart=tic; if ~isempty(busnum)

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% power = -currentAC.bus(:,3) - sqrt(-1)*currentAC.bus(:,4); % power(currentAC.gen(:,1)) = power(currentAC.gen(:,1)) + ... % currentAC.gen(:,2)+ sqrt(-1)*currentAC.gen(:,3); % power = -power/100;

%{ for i=1:1:length(DCdata.branch(:,1)) power(DCdata.branch(i,1)) = power(DCdata.branch(i,1)) - ... (DCdata.branch(i,3)+sqrt(-1)*DCdata.branch(i,4)); power(DCdata.branch(i,2)) = power(DCdata.branch(i,2)) - ... (DCdata.branch(i,5)+sqrt(-1)*DCdata.branch(i,6)); end %} %power = power/100 %} %DCdata.bus(busnum,4) %voltage = sqrt(power(busnum).*DCdata.bus(busnum,5)); power = DCdata.bus(busnum,4); power(DCdata.bus(busnum,6)==1)=-100; %voltage = sqrt(-DCdata.bus(busnum,4)/100.*DCdata.bus(busnum,5)); voltage = sqrt(-power/100.*DCdata.bus(busnum,5));

%{ if ~isempty(volmag) voltage(volmag(:,2)) = currentAC.bus(volmag(:,1),8);%keep PV and Ref

vol mags end if ~isempty(ref) voltage(ref(:,2)) = currentAC.bus(ref(:,1),9)*pi/180;%keep ref angle end %} ACdata.bus(busnum,8) = abs(voltage); ACdata.bus(busnum,9) = angle(voltage)*180/pi; total = total + 1; end ACtime_iso = ACtime_iso + toc(isostart);

sys_start = tic;

% This block of code calculates the voltages for the remaining buses for i = 1:1:S if numbuses(i)>1 for j = length(C):-1:1 if C(j) ~= i for k = length(currentAC.gen(:,1)):-1:1 if currentAC.gen(k,1) == j currentAC.gen(k,:) = []; end end for k = length(currentAC.branch(:,1)):-1:1 if currentAC.branch(k,1) == j || ... currentAC.branch(k,2) == j currentAC.branch(k,:)=[]; end end currentAC.bus(j,:)=[];

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end end [updateAC,f,iterations] = runpf(currentAC, AC); failure = [failure f-1]; total = total + iterations; currentAC = ACdata; if ~isempty(updateAC.gen) for j = 1:1:length(updateAC.gen(:,1)) for k = 1:1:length(currentAC.gen(:,1)) if currentAC.gen(k,1) == updateAC.gen(j,1) currentAC.gen(k,:) = updateAC.gen(j,:); end end end end for j = 1:1:length(updateAC.bus(:,1)) for k = 1:1:length(currentAC.bus(:,1)) if currentAC.bus(k,1) == updateAC.bus(j,1) currentAC.bus(k,:) = updateAC.bus(j,:); end end end for j = 1:1:length(updateAC.branch(:,1)) for k = 1:1:length(currentAC.branch(:,1)) if currentAC.branch(k,1) == updateAC.branch(j,1) && ... currentAC.branch(k,2) == updateAC.branch(j,2) currentAC.branch(k,:) = updateAC.branch(j,:); end end end end end ACtime_sys = ACtime_sys + toc(sys_start);

extratiming_start=tic;

results = [currentAC.branch(:,1), currentAC.branch(:,2), ... currentAC.branch(:,14), currentAC.branch(:,15), ... currentAC.branch(:,16), currentAC.branch(:,17), ... abs((abs(currentAC.branch(:,14))-abs(currentAC.branch(:,16)))), ... abs((abs(currentAC.branch(:,15))-abs(currentAC.branch(:,17)))) ];

currentAC.results=results; printacpf(results); ACdata = currentAC; if sum(failure,2)< 0 success = 0; end

etime= etime+toc(extratiming_start);

end

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C.4 Code for DC Subroutine

function [ACdata,DCdata] = runsimphvdcpf(ACdata,DCdata) %for simplicity, set variables to names:

busnum=DCdata.bus(:,1); voltage=DCdata.bus(:,2); angle=DCdata.bus(:,3); Stotal=DCdata.bus(:,4);

Rec_bus=DCdata.branch(:,1); Inv_bus=DCdata.branch(:,2); Rec_bus_P=abs(DCdata.branch(:,3)/100); Rec_bus_Q=DCdata.branch(:,4)/100; Inv_bus_P=abs(DCdata.branch(:,5)/100); Inv_bus_Q=DCdata.branch(:,6)/100; RLine=DCdata.branch(:,7);

tapr = DCdata.interface(:,1); tapi = DCdata.interface(:,2); numtapr = DCdata.interface(:,3); numtapi = DCdata.interface(:,4); regmin = DCdata.interface(:,5); regmax = DCdata.interface(:,6); xp = DCdata.interface(:,7); xs = DCdata.interface(:,8); xt = DCdata.interface(:,8); xc = DCdata.interface(:,10); zf = DCdata.interface(:,11);

delay = DCdata.control(:,1)*pi/180; extinction = DCdata.control(:,2)*pi/180; mindelay = DCdata.control(:,3)*pi/180; minextinct = DCdata.control(:,4)*pi/180; Arec = DCdata.control(:,5); Ainv= DCdata.control(:,6);

%Calculate current tap position

tr = (85+(regmax-regmin)./(numtapr-1).*(tapr-1))/100; ti = (85+(regmax-regmin)./(numtapi-1).*(tapi-1))/100;

%Set AC voltages (primary and secondary) vr = []; vi = []; for i = 1:1:length(Rec_bus) vr = [vr; voltage(Rec_bus(i)).*exp(sqrt(-1)*angle(Rec_bus(i))*pi/180)]; vi = [vi; voltage(Inv_bus(i)).*exp(sqrt(-1)*angle(Inv_bus(i))*pi/180)]; end

vr_sec = abs(vr.*tr); vi_sec = abs(vi.*ti);

% Calculate converter DC voltages and current

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Id = sqrt((Rec_bus_P - Inv_bus_P)./RLine); Vd_inv = Inv_bus_P./Id; Vd_rec = Vd_inv + RLine.*Id;

%Calculate Power Transfer (Eq 14-16. Arrillaga)

urec = acos(2*pi*Vd_rec./(3*sqrt(2)*vr_sec)-cos(delay))-delay; uinv = acos(2*pi*Vd_inv./(3*sqrt(2)*vi_sec)-cos(extinction))-extinction;

tanrec = (2*urec+sin(2*delay)-sin(2*(delay+urec)))./... (cos(2*delay)-cos(2*(delay+urec))); taninv =-(2*uinv+sin(2*extinction)-sin(2*(extinction+uinv)))./... (cos(2*extinction)-cos(2*(extinction+uinv)));

Srec = Vd_rec.*Id + sqrt(-1)*Vd_rec.*Id.*tanrec; Sinv = Vd_inv.*Id + sqrt(-1)*Vd_inv.*Id.*taninv;

%[real(Srec) Rec_bus_P; real(Sinv) Inv_bus_P]

Srec(real(Srec) ~= Rec_bus_P) = Rec_bus_P(real(Srec) ~= Rec_bus_P); Sinv(real(Sinv) ~= Inv_bus_P) = Inv_bus_P(real(Sinv) ~= Inv_bus_P);

%Calculate new taps

tr = ceil((Vd_rec+3/pi*xc.*Id)./(3*sqrt(2)/pi.*cos(delay).*abs(vr)))*100; ti =

ceil((Vd_inv+3/pi*xc.*Id)./(3*sqrt(2)/pi.*cos(extinction).*abs(vi)))*100;

for n = 1:1:length(Rec_bus)

if tr(n) < regmin(n) tapr(n)=1; elseif tr(n) > regmax(n) tapr(n)=numtapr(n); else tapr(n)=(tr(n)-regmin(n)+1); end

if ti(n) < regmin(n) tapi(n)=1; elseif ti(n) > regmax(n) tapi(n)=numtapr(n); else tapi(n)=(ti(n)-regmin(n)+1); end end

%Update Svalues in column 4 of DCdata.bus Stotal = zeros(size(Stotal));

for n = 1:1:length(Rec_bus)

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Stotal(Rec_bus(n)) = Stotal(Rec_bus(n)) - real(Srec(n))*100 -... sqrt(-1)*Rec_bus_Q(n)*100; Stotal(Inv_bus(n)) = Stotal(Inv_bus(n)) + real(Sinv(n))*100 -... sqrt(-1)*Inv_bus_Q(n)*100; end

DCdata.bus(:,4) = Stotal;

%Update DCdata taps

DCdata.interface(:,1) = tapr; DCdata.interface(:,2) = tapi;

%Update ACdata Svalues ACdata.bus(:,3)=ACdata.bus(:,3)-real(Stotal); ACdata.bus(:,4)=ACdata.bus(:,4)-imag(Stotal); ACdata.bus(:,3); ACdata.bus(:,4); %Print Data results = [Rec_bus Inv_bus real(Srec)*100 real(Sinv)*100 100*(real(Srec)-

real(Sinv))]; DCdata.results = results; printhvdcpf(results);

end

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C.5 Main Code for Enhanced Load Flow Method

%for repeat=1:1:100 clear all; clc; perm_num=1; perm_max=10000000; while perm_num <= perm_max

clearvars -EXCEPT method perm_num perm_max %% Convergence Test % This file is used to test the convergence properties of several different % AC loadflow methodologies in a modified Sato and Arrillaga's method with % respect to the number of HVDC links in a system. %clear all; clc; close all;

%% Set up for Load flow %% casename = 'case14'; casedata = loadcase(casename); % load the casedata numbranch = size(casedata.branch,1); % how many branches linkdata = setlinkdata(casedata);% set HVDC link parameters convergence_times = []; % data matrix to store convergence times full_iterations = []; %datamatrix to store convergence interations AC_iterations = []; %datamatrix to store convergence interations orderings = makeorders(casedata); if strcmp(casename,'case4gs') || strcmp(casename,'case9') neworderings=[]; for i=1:1:500 neworderings = [orderings;neworderings]; end orderings = neworderings; end perm_max = 500; order = orderings(perm_num,:);

global ACtime_iso ACtime_sys etime failures = []; AC_time = []; DC_time = []; AC_time_iso =[]; AC_time_sys = []; AC_time_extra = [];

%% Run Modified Sato and Arrillaga's Method for #HVDC Links = 0...numbranch numbranch=length(order) for numlinks = 0:1:numbranch numlinks success=1; total = 0; ac_t_total = 0; dc_t_total = 0; ACtime_sys = 0; ACtime_iso = 0; etime = 0;

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percentDC = numlinks/numbranch*100; if percentDC >= 90 type = 4; %% gs else type = 1; %% nr end

if numlinks > 0 [ACdata,DCdata]=buildDC(casedata,numlinks,order,linkdata); else ACdata = casedata; DCdata = casedata; end tol = 20; i = 0;

t_start = tic; reset = [ACdata.bus(:,3) ACdata.bus(:,4)]; store = [real(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180)) imag(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180))]; while tol > 10^-5 && success == 1 if numlinks > 0 dc_t_start = tic; [ACdata,DCdata] = runsimphvdcpf(ACdata,DCdata); dc_t_total = dc_t_total + toc(dc_t_start); end ac_t_start = tic; [ACdata,success,iterations] = runACpfv2(ACdata,DCdata,type); ac_t_total = ac_t_total + toc(ac_t_start); total = total + iterations; DCdata.bus(:,2) = ACdata.bus(:,8); DCdata.bus(:,3) = ACdata.bus(:,9); new = [real(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180)) imag(ACdata.bus(:,8).*exp(sqrt(-1)*ACdata.bus(:,9)*pi/180))]; tol = max(abs(store - new).^2); store = new; ACdata.bus(:,3)=reset(:,1); ACdata.bus(:,4)=reset(:,2); if success == 0 failures=[failures numlinks]; end i=i+1; end store convergence_times = [convergence_times toc(t_start)]; %store convergence

times AC_iterations = [AC_iterations total]; full_iterations = [full_iterations i]; %store convergence iterations AC_time = [AC_time ac_t_total]; DC_time = [DC_time dc_t_total]; AC_time_iso =[AC_time_iso ACtime_iso]; AC_time_sys =[AC_time_sys ACtime_sys]; AC_time_extra=[AC_time_extra etime]; end %% Delete Failed Trials

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xdata = 0:1:numbranch; if ~isempty(failures) failures = sort(failures,2); for j = length(failures):-1:1 convergence_times(failures(j)+1) = []; AC_iterations(failures(j)+1)=[]; full_iterations(failures(j)+1)=[]; AC_time(failures(j)+1)=[]; DC_time(failures(j)+1)=[]; AC_time_iso(failures(j)+1)=[]; AC_time_sys(failures(j)+1)=[]; AC_time_extra(failures(j)+1)=[]; xdata(failures(j)+1)=[]; end end

%% Save filename = strcat(casename,'_ELF_',int2str(perm_num)); save(filename)

perm_num=perm_num+1; end %end

load flow analysis of multi-converter transmission systems · pdf file ·...

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