Copyright

by

Quan Huy Nguyen

2016

The Thesis Committee for Quan Huy Nguyen

Certifies that this is the approved version of the following thesis:

Power Flow Solution for Multi-Frequency AC Power

Systems

APPROVED BY

SUPERVISING COMMITTEE:

Surya Santoso, Supervisor

Ross Baldick

Power Flow Solution for Multi-Frequency AC Power

Systems

by

Quan Huy Nguyen, B.E.

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN ENGINEERING

THE UNIVERSITY OF TEXAS AT AUSTIN

May 2016

This work is dedicated to my parents and my sister.

Acknowledgments

I would like to express my sincere gratitude to my advisor, Dr. San-

toso, for his continuous support and valuable guidance, which assisted me in

completing this work. I would also like to thank Dr. Baldick for reading this

thesis.

I gratefully acknowledge Tuan Ngo for insightful discussions and pieces

of advice about the problem. I also thank Pisitpol Chirapongsananurak and

Kyung Woo Min for their help in typesetting this manuscript.

v

Power Flow Solution for Multi-Frequency AC Power

Systems

Quan Huy Nguyen, M.S.E.

The University of Texas at Austin, 2016

Supervisor: Surya Santoso

This work proposes the enhancement of existing power flow solutions for

application in multi-frequency AC power systems. Such power flow solutions

can be obtained using existing methods, provided that impedances of lines

operating at different frequencies be reflected to their equivalent impedances

at the fundamental power frequency. The study first presents a mathematical

proof of equivalent parameters in terms of power flow of a transmission line

operated at a low frequency and the fundamental frequency. It is then val-

idated by analyzing the power flow solutions of a power system modeled in

PSCAD/EMTDC when it is operated at multi-frequency and at conventional

60 Hz conditions. An application of a low frequency AC transmission line is

illustrated to demonstrate its superiority in terms of power carrying capac-

ity over a typical fundamental frequency AC transmission line. Finally, the

power flow problem in a more general power system, which is a multi-frequency

AC and multi-terminal HVDC system, is also completely solved by using two

different approaches.

vi

Table of Contents

Acknowledgments v

Abstract vi

List of Tables ix

List of Figures x

Chapter 1. Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 2. Low Frequency AC Transmission: State Of The Art 6

2.1 Characteristics of Low Frequency AC Transmission . . . . . . 6

2.2 LFAC in Comparison to Conventional HVAC and HVDC Systems 8

Chapter 3. Equivalent AC Transmission Line Parameters atDifferent Frequencies 13

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Equivalent Parameters of an AC Transmission Line Operated atDifferent Frequencies . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 4. VSC Converter Station in Multi-Frequency AC PowerSystems 20

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Structure of a VSC Converter Station . . . . . . . . . . . . . . 21

4.3 Back-to-back Converter . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Control Configuration and Controller Design . . . . . . . . . . 26

4.5 Pulse Width Modulation for Three-Phase VSC Converters . . 31

4.6 Simulation of a Back-to-back Converter in PSCAD/EMTDC . 33

vii

4.6.1 Functional Blocks . . . . . . . . . . . . . . . . . . . . . 33

4.6.2 Converter Models . . . . . . . . . . . . . . . . . . . . . 39

Chapter 5. Power Flow Simulation in HVAC and Multi-FrequencySystems 43

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Power Flow in a Conventional HVAC System . . . . . . . . . . 44

5.3 Validation of the Equivalent Parameters of a Transmission Lineat Different Frequencies . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Application of Low Frequency Transmission . . . . . . . . . . . 52

Chapter 6. Power Flow in Multi-Frequency AC and HVDC Sys-tems 57

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 VSC Converter Stations in Multi-Terminal HVDC Systems . . 60

6.2.1 Equivalent Model of VSC Converter Station . . . . . . . 60

6.2.2 Converter Operating Modes . . . . . . . . . . . . . . . . 62

6.3 Sequential Method . . . . . . . . . . . . . . . . . . . . . . . . 64

6.3.1 Power Flow in Multi-Frequency AC System . . . . . . . 65

6.3.2 Converter Calculation . . . . . . . . . . . . . . . . . . . 66

6.3.3 Power Flow in DC system . . . . . . . . . . . . . . . . . 67

6.3.4 DC Slack Bus Iteration . . . . . . . . . . . . . . . . . . 68

6.4 Unified Method . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.5 Simulation of Power Flow in a Multi-Frequency AC - Multi-Terminal HVDC System . . . . . . . . . . . . . . . . . . . . . 75

Chapter 7. Conclusion and Future Work 83

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Bibliography 86

viii

List of Tables

2.1 Comparison between HVAC, HVDC, and LFAC technologies. . 11

5.1 Impedances and Admittances of Transmission Lines (SystemBases are 100 MVA and 345 kV). . . . . . . . . . . . . . . . . 45

5.2 Power Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Voltages and Powers in Conventional Working Condition (Sys-tem Bases are 100 MVA and 345 kV). . . . . . . . . . . . . . . 48

5.4 Voltages and Powers in Multi-Frequency Working Condition(System Bases are 100 MVA and 345 kV). . . . . . . . . . . . 52

5.5 Voltages and Powers in a Multi-Frequency System with Im-proved Power Capability in Line 1-2 (System Bases are 100MVA and 345 kV). . . . . . . . . . . . . . . . . . . . . . . . . 55

6.1 Per-unit converter loss coefficients. . . . . . . . . . . . . . . . 64

6.2 Converter Data (System Bases are 100 MVA and 345 kV AC). 77

6.3 DC Line Resistances (System Bases are 100 MW and 120 kVDC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Power Flow Solution of the Multi-frequency AC-MTDC Systemfrom MATLAB and PSCAD/EMTDC (System Bases are 100MVA, 345 kV AC, and 120 kV DC). . . . . . . . . . . . . . . 79

ix

List of Figures

2.1 The investment-cost comparison of the three transmission tech-nologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Short-line model operated at different frequencies. . . . . . . . 15

3.2 Equivalent π model of a long transmission line. . . . . . . . . 18

4.1 VSC converter station. . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Back-to-back converter and its control blocks . . . . . . . . . . 23

4.3 A typical control block diagram for a back-to-back converter. . 27

4.4 DC-link and current control loops . . . . . . . . . . . . . . . . 29

4.5 Implementation of the SVPWM in a three-phase three-leg VSC. 32

4.6 Rectifier and inverter blocks. . . . . . . . . . . . . . . . . . . . 33

4.7 Angle detection block. . . . . . . . . . . . . . . . . . . . . . . 34

4.8 abc-to-dq block. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.9 SVPWM block. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.10 PWM block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.11 DC controller block. . . . . . . . . . . . . . . . . . . . . . . . 38

4.12 Simulation model of a rectifier in PSCAD/EMTDC. . . . . . . 39

4.13 Simulation model of an inverter in PSCAD/EMTDC. . . . . . 40

4.14 Simulation model of a back-to-back converter in PSCAD/EMTDC. 41

5.1 5-bus test system (System bases are 100 MVA and 345 kV, Loadpower unit is MVA). . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 Line power of a 5-bus test system operated at 60 Hz (Systembases are 100 MVA and 345 kV, Power unit are MVA, MW, andMvar). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Multi-frequency system with line 1-2 operated at 10 Hz (Systembases are 100 MVA and 345 kV, Load power unit is MVA). . . 49

5.4 Multi-frequency system with line power equal to that of theequivalent HVAC system (System bases are 100 MVA and 345kV, Power unit are MVA, MW, and Mvar). . . . . . . . . . . . 51

x

5.5 Multi-frequency system with improved power carrying capabil-ity in line 1-2 (System bases are 100 MVA and 345 kV, Powerunit are MVA, MW, and Mvar). . . . . . . . . . . . . . . . . . 54

6.1 A VSC converter station . . . . . . . . . . . . . . . . . . . . . 61

6.2 Flow chart of the two iteration loop at the slack converter. . . 69

6.3 Flow chart of the sequential method. . . . . . . . . . . . . . . 71

6.4 Flow chart of the unified method. . . . . . . . . . . . . . . . . 74

6.5 A multi-frequency AC - multi-terminal HVDC system and inputdistributed power (System bases are 100 MVA, 345 kV AC, and120 kV DC; Power unit are MVA, MW, and Mvar). . . . . . . 76

6.6 A multi-frequency AC - multi-terminal HVDC system and theinput distributed power (System bases are 100 MVA, 345 kVAC, and 120 kV DC; Power unit are MVA, MW, and Mvar). . 81

xi

Chapter 1

Introduction

1.1 Motivation

Recently, the increasing electricity demand leads to an ever-growing

interest in the improvement of transmission capacity. Conventional transmis-

sion methods are either high voltage AC (HVAC), operated at 50 Hz or 60 Hz,

or high voltage DC (HVDC) to transfer bulk power efficiently and reliably.

HVAC systems are designed to operate at a high voltage level with an aim

to reduce losses and increase bulk power transfer. This method is limited,

however, by the constraints of installed overhead lines such as power carrying

capability through a transmission line. On the contrary, HVDC systems are

able to handle a large amount of power on transmission lines by utilizing DC

current instead of AC current. HVDC systems have no limitation in trans-

mission line length for power transfer, yet they reduce the efficiency due to

switching losses and require a high initial cost for converter stations and spe-

cialized protection devices [1]. Accordingly, HVDC systems are only beneficial

for long transmission distances. Besides, HVDC systems are typically point-

to-point connections, which impedes the possibility of utilizing multi-terminal

DC grids [2].

1

Another feasible solution for bulk power transfer is to employ low fre-

quency AC (LFAC) transmission lines. By operating the systems at a fre-

quency that is lower than 60 Hz, the transmission line series reactance can be

reduced, thus extending power carrying capacity. LFAC, thus, not only in-

herits natural advantages of the conventional HVAC systems, such as meshed

network and distant protection using AC based circuit breakers, but also im-

proves the system’s power-transfer capability close to that in HVDC systems.

These valuable characteristics leave multi-frequency systems consisting of both

short 60 Hz transmission lines and LFAC transmission lines (up to 200 km for

cable), a promising solution for bulk power transfer with high reliability and

flexibility [3], [4].

For the analysis of future multi-frequency AC systems, besides inter-

ests in electromagnetic and electromechanical phenomena, another important

issue is steady-state performance. Insight in steady-state performance can be

obtained by solving power flow in such multi-frequency AC systems, which is

objective of this work. A power flow study is a steady-state analysis whose

target is to determine the bus voltages, line currents, and real and reactive

power flows in a system under a given load condition. The purpose of power

flow studies is to plan ahead and account for various hypothetical situations.

For example, if a transmission line is be taken off for maintenance, power flow

study needs to determine if the remaining lines in the system are able to han-

dle the required loads without exceeding rated values such as bus voltage or

line thermal limits.

2

Based on the existing power flow algorithms for conventional 60 Hz AC

transmission systems, this thesis provides a more generalized method to obtain

power flow solution for multi-frequency AC systems and for multi-frequency

AC and HVDC systems by taking into account the properties in terms of

power flow of LFAC transmission lines. The theoretical advantage in terms of

power carrying capacity of low frequency transmission over the conventional

transmission is also validated.

1.2 The Thesis Outline

Chapter 1, an introductory chapter, provides an overview of the two

conventional transmission methods, which are high voltage alternating current

(HVAC) and high voltage direct current (HVDC). Their characteristics, as well

as their advantages and disadvantages, are discussed in detail. This chapter

also contains a brief introduction about the content and contribution of the

thesis.

Chapter 2 reviews recent studies about LFAC transmission, which is

an alternative technology of the two conventional transmission methods. The

chapter focuses on the advantages of LFAC and provides insights in why multi-

frequency systems consisting of both 60 Hz and LFAC transmission lines are

promising solutions for bulk power transfer with high reliability and flexibility.

Chapter 3 presents a mathematical derivation of equivalent parameters

in terms of power flow of a transmission line operated at different frequencies.

Based on that relation, the parameters of LFAC transmission lines in a multi-

3

frequency system can be precisely converted into those of the equivalent line

operated at 60 Hz. Therefore, the conventional load flow methods for AC

systems are still applicable in multi-frequency power systems. The benefit in

terms of high power carrying capacity when using LFAC transmission lines is

discussed as well.

Chapter 4 describes the modeling and control of a back-to-back con-

verter to simulate a multi-frequency power system that will be used to vali-

date the proposed power flow solution. The study includes control strategies,

converters dynamic model, control configuration, controller design, and space

vector pulse width modulation (SVPWM) technique. The converter model,

control blocks, and simulation result in PSCAD/EMTDC are then discussed.

Chapter 5 analyzes the power flow solution of a multi-frequency system

from MATLAB and time-domain simulation performed in PSCAD/EMTD to

verify the theoretical concept developed in Chapter 3. An application of LFAC

in a multi-frequency system is also discussed to demonstrate its superiority in

terms of power carrying capacity over a typical fundamental frequency AC

transmission line.

Chapter 6 addresses a case in which a multi-terminal VSC HVDC

(MTDC) system is integrated into a multi-frequency AC system. By using

a simplified converter loss model, approximate converter losses are taken into

account. The power flow problem in such a hybrid AC/DC system is then

solved in MATLAB by two different approaches, the sequential method and

the unified method. Both methods are then discussed in terms of complexity

4

and accuracy.

The thesis is finalized in Chapter 7, which summarizes major points

and addresses future work.

5

Chapter 2

Low Frequency AC Transmission: State Of

The Art

2.1 Characteristics of Low Frequency AC Transmission

The concept of LFAC transmission for power systems was originally

proposed in [5]. It is widely known that in order to raise transmission capability

in a stability limited system, we can either increase the voltage level or decrease

the reactance of the transmission lines. With the current development in

material science, the maximum voltage level is limited at 800 kV AC; thus,

power transfer capacity can be boosted further by reducing the reactance of

transmission lines. This is the main motivation to use low frequencies to

transmit electrical power in LFAC systems.

Potential configurations for LFAC transmission in one application, which

is wind power generation, are introduced with consideration of optimal voltage

level, cost, and power carrying capability in [6], [7]. The power from wind tur-

bine generations can be directly generated at 20 Hz and then collected at the

point of common coupling (PCC), or it can be converted into DC power and

then collected at PCC before being converted into 20 Hz AC power for trans-

mission. The former configuration eliminates the necessity of converters at the

6

generation side while the latter configuration overcomes the problem when the

wind turbine systems operate at different speeds and voltage levels [7].

To convert frequency from the fundamental value to another lower one,

different types of converter need to be evaluated. Take cycloconverters, line

commutated or self-commutated converters having back-to-back configuration

at the DC terminal for examples. Most of related research focus on using

cycloconverter as a method to change the frequency due to its lower cost

and more transfered power compared to back-to-back voltage-source converter

(VSC) [8]. The main disadvantage of a cycloconverter is that its operating

frequency is fixed at 1/3 of the grid frequency. In other words, the operating

frequency of cycloconverter is 20 Hz when the grid frequency is 60 Hz or 16.67

Hz when the grid frequency is 50 Hz. At this frequency, however, the generated

voltage waveform contains harmonics at the output side. In Chapter 4, a back-

to-back VSC converter is analyzed as an alternative topology for frequency

converter.

In [9], the availability for a LFAC system to use existing equipment

in the market such as submarine cables, gas insulated switch gears, power

transformer, shunt reactors, or protective AC circuit breakers is discussed and

compared with those in conventional systems. The possibility to operate a gen-

erator at a low frequency is studied as well. Expanding on LFAC transmission

lines, [5] addresses the difficulty of establishing new overhead transmission line

due to electric magnetic field and spoiling the land-scape. The author focuses

on discussing potential advantages of LFAC transmission using cross-linked

7

polyethylene (XLPE) cable. XLPE cable has significant advantages over oil

filled (OF) cable in terms of environmental compatibility such as the impossi-

bility of oil leakage and better electrical properties such as low dielectric loss,

stronger fire resistance, and easier handling and maintenance [10]. Experi-

mental results reports that the space charge accumulation in the dielectrics of

normal XLPE cable is found when the frequency is lower than 0.1 Hz, but it

is neutralized when the frequency is higher than 1 Hz [11]. This characteris-

tic makes XLPE a suitable option for LFAC transmission with the operating

frequency higher than 0.1 Hz.

2.2 LFAC in Comparison to Conventional HVAC andHVDC Systems

The comparison between LFAC systems over conventional 60 Hz trans-

mission systems and HVDC systems are also deeply investigated in [5] - [4].

Compared to conventional AC transmission systems, LFAC systems are su-

perior in terms of power carrying capacity. In addition, the charging current

induced by cable susceptance decreases as the frequency becomes low, which

extends the maximum transmissible length in LFAC systems [5]. Furthermore,

the voltage drop is inversely proportional to the square of voltage and propor-

tional to the reactance of the transmission line; therefore, LFAC systems also

can reduce the voltage drop across the transmission lines [12]. At low fre-

quency, the thermal rating under the same environmental conditions is even

higher since the resistance of the cables or overhead lines is lower due to lower

8

skin effect [9], [13]. However, operating at low frequencies means that the size

of transformers and shunt reactors in LFAC systems is larger than that in the

conventional HVAC systems. Also, the fault clearing time of circuit breakers

in LFAC systems is prolonged in accordance with the low system frequency [5].

With regard to HVDC systems, as mentioned above, the first advan-

tage of LFAC systems is that LFAC transmission systems can use existing

equipment such as cables, standard transformers with tap changers, or circuit

breaker while HVDC systems must use other special types of XLPE cables or

circuit breakers. XLPE cables cannot be used directly in HVDC systems be-

cause of the phenomenon of so-called space charge, which likely malfunctions

to the consistency of the electrical field in the insulator dielectrics and causes

a breakdown [5]. The second advantage of LFAC systems over HVDC systems

is that while HVDC systems are currently point-to-point connections, LFAC

systems have the natural property of multi-terminal connection [2]. However,

the disadvantage of LFAC system compared to HVDC systems is that LFAC

systems are confronted with the problem related to charging current in case of

cable or large voltage drop due to series reactance in case of overhead trans-

mission line. In HVDC systems, that concern does not exist, which makes

the transmission length in HVDC systems unlimited. Besides, the initial cost

for building converter stations is gradually compensated after a certain break-

even distance by the lower conductance losses, smaller right-of-way, and less

number of conductors with lower insulation level, which all establish HVDC

as the best solution up to this time for long transmission [2], [14].

9

The characteristics of conventional AC transmission, HVDC transmis-

sion, and LFAC transmission systems are summarized in Table 2.1. Regarding

the investment cost of the three technologies, Fig. 2.1 shows that LFAC sys-

tems are less expensive than both HVAC and HVDC systems for a range of

distance of transmission line [15].

Figure 2.1: The investment-cost comparison of the three transmission tech-nologies.

10

Table 2.1: Comparison between HVAC, HVDC, and LFAC technologies.

11

In conclusion, this chapter reviews the state of art of LFAC transmission

systems. Main points are the motivation of LFAC transmission, its potential

connection configurations, the topologies of frequency converter, and the abil-

ity to apply the existing equipment from the conventional HVAC systems in

LFAC systems. The advantages and disadvantages of LFAC systems compared

to conventional HVAC systems and HVDC systems are mentioned as well. In

the next chapter, the equivalent parameters of a transmission line operated at

different frequencies will be derived, facilitating the process of solving power

flow problem in multi-frequency AC systems.

12

Chapter 3

Equivalent AC Transmission Line Parameters

at Different Frequencies

3.1 Introduction

Given a transmission line, which is desired to be operated at a low

frequency, it is expected that it transfers more power than it does when it is

operated at the fundamental frequency. However, it is necessary to have a good

estimation about that power carrying capacity before deciding the operating

points for the associated converters. Unfortunately, in a multi-frequency power

system, it is not possible to directly determine that power capacity level of low

frequency lines when there are different operating frequencies at the same time.

An alternative approach is to convert those transmission lines operated at low

frequencies in a multi-frequency system into their equivalent lines operated at

the fundamental frequency in terms of transferred power, which needs to be

preserved. The main contribution of this chapter is to discover the relation

between parameters of a transmission line that works at different frequencies

but transfers the same amount of active and reactive power at these operating

frequencies. Based on that relation, the parameters of any LFAC transmission

line in a multi-frequency system can be converted into those of its equivalent

lines operated at the fundamental 60 Hz without changing the power flow.

13

Therefore, the power flow in the original multi-frequency system can be deter-

mined from the solution of the equivalent system operated at the fundamental

frequency, which is easily obtained by applying existing algorithms.

3.2 Equivalent Parameters of an AC Transmission LineOperated at Different Frequencies

For the sake of simplicity, the analysis in this section starts with the

consideration of a short-length model of a transmission line [16], as shown in

Fig. 3.1(a) and Fig. 3.1(b). Parameters VS, VR, R, L denote the sending-end

voltage, receiving-end voltage, resistor, and inductor of the transmission line,

respectively. The subscript “s” stands for parameters at a low frequency ωs

while “e” stands for parameters at the fundamental ωe = 60 Hz, respectively.

Since the purpose is to find the equivalent parameters in terms of power

flow, the real and reactive power of the line operated at these frequencies are

considered to be exactly analogous. The voltage magnitude ratios between the

sending ends and receiving ends in two cases are assumed to be equal to each

other, i.e. (VSeVSs

)2

=

(VReVRs

)2

= a (3.1)

and the frequency ratio is defined as:

s =ωsωe

(3.2)

The power of the transmission line at the sending ends and receiving

14

(a)

(b)

(c)

Figure 3.1: Short-line model operated at different frequencies.

15

ends in two cases are given by:

SSs = VSsI∗s = VSs∠0

VSs∠0− VRs∠−δsRs − jωsLs

(3.3)

SRs = VRsI∗s = VRs∠δs

VSs∠0− VRs∠−δsRs − jωsLs

(3.4)

SSe = VSeI∗e = VSe∠0

VSe∠0− VRe∠−δeRe − jωeLe

(3.5)

SRe = VReI∗e = VRe∠δe

VSe∠0− VRe∠−δeRe − jωeLe

(3.6)

where δs and δe are the power angles at the receiving ends, which can be

negative or positive depending on the direction of power flow. Since the power

is preserved, i.e. SSs = SSe and SRs = SRe, the following equations are inferred:

VSsVSs − VRs∠−δsRs − jωsLs

= VSeVSe − VRe∠−δeRe − jωeLe

(3.7)

VRs∠δsVSs − VRs∠−δsRs − jωsLs

= VRe∠δeVSe − VRe∠−δeRe − jωeLe

(3.8)

By manipulating both sides of each equation, these equalities become:

VSs(VSs − VRscosδs)Rs − VRssinδsωsLs + j[VRssinδsRs + (VSs − VRscosδs)ωsLs]

R2s + ω2

sL2s

= VSe(VSe − VRecosδe)Re − VResinδeωeLe + j[VResinδeRe + (VSe − VRecosδe)ωeLe]

R2e + ω2

eL2e

(3.9)

VRs(Rscosδs − ωsLssinδs)VSs −RsVRs + j[(Rssinδs + ωsLscosδs)VSs − ωsLsVRs]

R2s + ω2

sL2s

= VRe(Recosδe − ωeLesinδe)VSe −ReVRe + j[(Resinδe + ωeLecosδe)VSe − ωeLeVRe]

R2e + ω2

eL2e

(3.10)

16

By equating the real parts and imaginary parts of these two equations,

the following result can be obtained:

VSs(VSs − VRscosδs)Rs − VRssinδsωsLs

R2s + ω2

sL2s

= VSe(VSe − VRecosδe)Re − VResinδeωeLe

R2e + ω2

eL2e

(3.11)

VSsVRssinδsRs + (VS − VRscosδs)ωsLs

R2s + ω2

sL2s

= VSeVResinδeRe + (VSe − VRecosδe)ωeLe

R2e + ω2

eL2e

(3.12)

VRs(Rscosδs − ωsLssinδs)VSs −RsVRs

R2s + ω2

sL2s

= VRe(Recosδe − ωeLesinδe)VSe −ReVRe

R2e + ω2

eL2e

(3.13)

VRs(Rssinδs + ωsLscosδs)VSs − ωsLsVRs

R2s + ω2

sL2s

= VRe(Resinδe + ωeLecosδe)VSe − ωeLeVRe

R2e + ω2

eL2e

(3.14)

The solution of the above four equations with four variables is:

Re = aRs

Le = asLs (3.15)

δe = δs

The equivalent circuit for a LFAC transmission line referring to 60 Hz

is represented in more details in Fig. 3.1(c), in which the voltage source VSs

and VRs are the same as those of the original LFAC line. An ideal transformer,

which has turn ratio 1/√a, represents voltage conversion. The resistor and

inductor are scaled by the voltage ratio and the product of voltage ratio and

frequency ratio, respectively. Notice that when a = 1, then from (3.15), the

17

following conclusion is obvious:

Re = Rs

Le = sLs (3.16)

δe = δs

which means that the equivalent line can be simply obtained by keeping the

resistance constant while scaling the inductance by the frequency ratio s.

Figure 3.2: Equivalent π model of a long transmission line.

Now, let’s move on to a more general case. Fig. 3.2 shows the equivalent

π model of a long transmission line [16], which is assumed to be operated at

a low frequency and at 60 Hz with matching amounts of transferred power.

If the sending-end voltages are similar and the receiving-end voltages are the

same as well, i.e. a = 1, applying a completely similar manipulation to this

model leads to the following relations between transmission line parameters in

the two working conditions:

1) The resistors and the phase angles at the sending ends and receiving ends

are equal to each other.

2) The capacitors and series inductors are scaled by the frequency ratio.

18

It is also worthy to estimate the active power at the sending end of the

line operated at the low frequency ωs in (3.3) and (3.11):

PSs = VSs(VSs − VRscosδs)Rs − VRssinδsωsLs

R2s + ω2

sL2s

(3.17)

Since the AC transmission line is dominated by the reactance, the ac-

tive power is, therefore, proportional to the reciprocal of reactance. Thus, the

power carrying capacity will be improved significantly by reducing the reac-

tance, which can be resulted from operating the system at low frequencies.

This property is the most important benefit of LFAC transmission over the

conventional 60 Hz transmission.

The theoretical work in this chapter, which includes the derivation of

equivalent of an AC transmission lines operated at different frequencies and

the benefit in terms of higher power carrying capacity of LFAC, will be verified

in Chapter 5 by analyzing power flow in a real power system.

19

Chapter 4

VSC Converter Station in Multi-Frequency

AC Power Systems

4.1 Introduction

The purpose of this chapter is to describe the modeling of a voltage

source converter (VSC) to simulate a multi-frequency power system that will

be used to validate the proposed power flow solution described in Chapter 3.

In a multi-frequency AC power system, the function of VSC converter stations

is to convert the fundamental frequency, which is 60 Hz, to the desirable val-

ues. Cycloconverters are mentioned as a solution for converting the operating

frequency [5] - [13]. However, cycloconverters are not able to help the system

achieve an operating frequency lower than one-third of the grid frequency.

Since the optimal frequency for LFAC transmission, which has not been ex-

actly evaluated yet, might go below this value, other converter topologies must

be carefully investigated.

In this chapter, the typical structure of a VSC converter station is first

addressed Section 4.2. Section 4.3 elaborates on the model of a back-to-back

converter, which can be used in multi-frequency power systems and its com-

ponents, based on the structure of the aforementioned VSC converter. Section

20

4.4 of this chapter discusses the control strategy for back-to-back converters

while the pulse width modulation (PWM) technique for VSC converters are

treated in Section 4.5. Simulation blocks in PSCAD/EMTDC to evaluate the

performance of back-to-back converter and its control strategy are given in

Section 4.6.

4.2 Structure of a VSC Converter Station

Fig. 4.1 shows the typical structure of a VSC converter station. Its

main components are a transformer, phase reactors, capacitors, and switching

valves [17], [32].

Figure 4.1: VSC converter station.

The function of the transformer is to reduce the rated voltage to a

value that is compatible with the normal working condition of the compo-

nents of the station, especially switching valves. The transformer is normally

equipped with a tap changer to regulate the voltage within a certain range.

The transformer can be represented by its leakage impedance.

The phase reactor is responsible for controlling the current flowing be-

21

tween the AC grid and the converter, which also means that it can control

active and reactive power. In addition, phase reactor is part of the low pass

filter that prevents high frequency harmonics created from switching process

to enter the transformer.

The filter incorporating with the phase reactor act like a low-pass filter,

which lets the fundamental-frequency and low-frequency signals pass by and

significantly attenuate high frequency signals. By doing so, the transformer is

not exposed to high frequency stress created by the switching process.

Another important element is switching valves, which are connected in

series in order to be able to switch voltages that might be higher than the rated

voltage of a particular one. Currently, the most widely used switching device

for VSC HVDC systems is the Insulated Gate Bipolar Transistor (IGBT).

The capacitor at the DC side of the converter station acts as energy

storage. The DC voltage is kept within a narrow band by charging or dis-

charging the capacitors.

4.3 Back-to-back Converter

Fig. 4.2 shows a back-to-back converter station model and its control

blocks used in multi-frequency AC power systems. This power converter is

structured by two identical VSC converters sharing a common DC-link ca-

pacitor. The capacitor decouples the operation of these two converters. The

converters are connected to AC grids via an equivalent inductor Ls, which

22

Figure 4.2: Back-to-back converter and its control blocks

represents the transformer and filter inductor, and a resistor Rs which takes

into account all the losses in inductor and switching stages [17]. In Fig. 4.2,

vabc1, vabc2, vsabc1 and vsabc2 stand for the three-phase AC voltage at grid sides

and converter side of the two converters, iabc1 and iabc2 stand for the measured

three-phase currents flowing between AC grids and converters, Vdc and V ∗dc

stand for the measured and reference values of DC-link voltage. P ∗1 , Q∗1, P∗2 ,

and Q∗2 represent the reference active and reactive powers of the two AC buses

with power flowing into converter defined as positive.

Back-to-back converters have significant advantages such as near sinu-

soidal current waveforms and bidirectional power flow operation [18]. Depend-

ing on the direction of active power flow, one converter operates as a rectifier

while the other one works as an inverter. Moreover, in back-to-back convert-

ers, active and reactive powers are independently controlled, resulting from

23

the fact that each converter has two degrees of control freedom. One degree

regulates reactive power while the other one is used for active power or DC-

link voltage control. For reactive power regulation, each converter controls its

reactive power separately from the other. For active power regulation, one

converter has to be responsible for keeping the DC-link voltage constant, and

due to the consistent flow of active power, its active power must be equal to

the controlled active-power value of the other converter if all the losses in Rs

are negligible [17].

In order to design controllers for back-to-back converters, its mathe-

matical model must be obtained. As previously mentioned, the controls of

these two rectifier and inverter are independent from each other. Without loss

of generality, only the model and control of converter V SC1 is specified here.

The voltage equation at the interconnection between AC grid and converter is

given by: va1vb1vc1

= Rs

ia1ib1ic1

+ Lsd

dt

ia1ib1ic1

+

vsa1vsb1vsc1

(4.1)

The next step is to convert this equation, which is in abc frame, into the

stationary frame αβ by using the amplitude-invariant Clarke transformation.

This transformation is defined by [19]:abc

=

0 1

−12

√32

−12−√32

[αβ

](4.2)

Therefore, in the stationary frame αβ, the voltage equation (4.1) be-

24

comes: [vα1vβ1

]= Rs

[iα1iβ1

]+ Ls

d

dt

[iα1iβ1

]+

[vsα1vsβ1

](4.3)

where vα1, vβ1, vsα1, and vsβ1 denotes the α-β components of voltages at the

grid side and at the converter side; iα1 and iβ1 signify the d-q components of

measured current flowing between the grid and converter VSC1. For conve-

nience of controller design, this equation still needs to be converted from the

stationary αβ frame into the synchronous frame dq by using Park transforma-

tion. The transformation is defined by [19]:[αβ

]=

[cosθ −sinθsinθ cosθ

] [dq

](4.4)

where θ = ω1t is the angle between the stationary frame αβ and the syn-

chronous frame dq. The equivalent form in terms of vector of equation (4.4)

is:

•αβ = •dqejθ (4.5)

Therefore, in the synchronous frame dq, the voltage equation (4.3) becomes:

vdq1ejθ = Rs.idq1e

jθ + Lsd

dt(idq1e

jθ) + vsdq1ejθ (4.6)

vdq1ejθ = Rsidq1e

jθ + Ls

(d

dt(idq1) + jω1idq1

)ejθ + vsdq1e

jθ (4.7)

vdq1 = Rsidq1 + Lsd

dt(idq1) + jω1Lsidq1 + vsdq1 (4.8)

25

[vd1vq1

]= Rs

[id1iq1

]+ Ls

d

dt

[id1iq1

]+ ω1Ls

[−iq1id1

]+

[vsd1vsq1

](4.9)

where ω1 is the grid frequency at the grid side of converter VSC1; vd1, vq1,

vsd1, and vsq1 denotes the d-q components of voltages at the grid side and at

the converter side; id1 and iq1 signify the d-q components of measured current

flowing between the grid and converter VSC1.

In addition, the active and reactive power injected from the AC grid to

the converter is given by [19]:[P1

Q1

]=

3

2

[vd1 vq1vq1 −vd1

] [id1iq1

](4.10)

In synchronous d-q frame, it is important to note that the d-axis compo-

nent of grid-side voltage vd1 is constant while the q-axis component of grid-side

voltage vq1 is zero. Thus, the equation of the power injected from AC grid to

the converter can be simplified further as the following equation:[P1

Q1

]=

3

2

[vd1id1−vd1iq1

](4.11)

The active and reactive power, therefore, will be proportional to id and

iq, respectively.

4.4 Control Configuration and Controller Design

The decoupled control of a rectifier/inverter, as shown in Fig. 4.3, has

a nested-loop structure consisting of a fast inner current loop and a slow outer

control loop that generates d− and q−axis current references i∗d1 and i∗q1 to the

26

current controllers [17], [20]. Speaking of bandwidth, the outer voltage control

loop should have at least a decade less in bandwidth compared to the inner

current control loop.

Figure 4.3: A typical control block diagram for a back-to-back converter.

For the current control loop, it is obvious to see from equation (4.9)

that id1 and iq1 or the ac system active and reactive power can be regulated

using v′

sd1 and v′sq1, respectively. The output current to input voltage transfer

function is given by:

G1(s) =id1(s)

v′sd1(s)

=iq1(s)

v′sq1(s)

=1

Lss+Rs

(4.12)

where:

v∗sd1 = −v′

sd1 + (ω1Liq1 + vd1) (4.13)

v∗sq1 = −v′

sq1 − (ω1Lid1) (4.14)

27

Since the above transfer function is of first order type, a PI controller is enough.

The PI controller used for the current loop is defined as:

GPI(s) = Kp +Ki

s(4.15)

Therefore, the open-loop transfer function of the current loop is given

by:

GI(s) = (Kp +Ki

s)

1

Lss+Rs

(4.16)

or:

GI(s) =Kp(s+ Ki

Kp)

s

1Ls

s+ Rs

Ls

(4.17)

Assume ωI is the bandwidth of the current loop [21], the standard form

of a closed-loop transfer function with the bandwidth equal to ωI is given

by [22]:

Gclosed−loop1(s) =1

1 + sωI

(4.18)

and the corresponding open-loop transfer function is of the form:

GI(s) =ωIs

(4.19)

By equating (4.17) and (4.19), the coefficients of the PI controller for

the current loop are obtained as below:

Kp = ωILs (4.20)

Ki = ωIRs (4.21)

28

The next step is to design a controller for the outer DC-voltage loop. In

order to maintain the balance between the active power supplied or absorbed

by the BTB converter, it is necessary to keep the DC-link voltage at a constant

value. For the DC-link control loop, start with the capacitor energy equation:

CVdcdVdcdt

= P1 − Pload =3

2vd1id1 − Pload (4.22)

Assume that CVdc is constant, the output voltage to input current

transfer function is obtained:

G2(s) =Vdc(s)

id1(s)=

3vd12VdcCs

(4.23)

For DC-link voltage control loop, an IP controller is used to achieve

the closed-loop transfer function with the standard second-order form:

Gclosed−loop2(s) =

3vd1Ki

2VdcC

3vd1Ki

2VdcC+ 3vd1Kp

2VdcCs+ s2

(4.24)

The detail of the DC-link control structure is shown in Fig. 4.4.

Figure 4.4: DC-link and current control loops

Assume ωv and ζ are the bandwidth and damping factor of the DC-

link voltage control loop, from (4.24) the coefficients of the IP controller are

29

determined as:

Kp = 4ζωvCVdc/3vd1 (4.25)

Ki = 2ω2vCVdc/3vd1 (4.26)

30

4.5 Pulse Width Modulation for Three-Phase VSC Con-verters

A large variety of methods for PWM has been studied; among them,

sinusoidal PWM (SPWM) and space vector PWM (SVPWM) are the most

popular ones and have been widely applied in industry. Between these two

methods, SVPWM has some important advantages such as lower harmonic

currents and higher modulation index [23]. However, the implementation of the

conventional SVPWM is difficult to apply in practice due to the complicated

process, which includes sector identification, effective-time determination, and

switching-time calculation. In [23], [24], a new technique is introduced, and

it proves a great simplification in implementation compared with the con-

ventional method. The output voltage is now directly synthesized from the

reference phase voltages and the off-set voltage without sector identification.

Specifically, a common offset voltage Vsn defined by (4.30) is added into the

phase voltages Vas, Vbs, and Vcs to obtain the pole voltages Van, Vbn, and Vcn:

Van = Vas + Vsn (4.27)

Vbn = Vbs + Vsn (4.28)

Vcn = Vcs + Vsn (4.29)

where

Vsn = −max(Vas, Vbs, Vcs) +min(Vas, Vbs, Vcs)

2(4.30)

The switching states of IGBTs are obtained by the comparison between

31

each of the pole voltages Van, Vbn, and Vcn with a triangular carrier wave, and

the optimum switching sequence is obtained, as shown in Fig. 4.5.

Figure 4.5: Implementation of the SVPWM in a three-phase three-leg VSC.

32

4.6 Simulation of a Back-to-back Converter in PSCAD/EMTDC

4.6.1 Functional Blocks

Rectifier and inverter blocks: each block contains six IGBTs as

in Fig. 4.6(b) and requires PWM gate signals for upper switches S1, S3, S5

to operate. The signals for lower switches S2, S4, S6 are the inverse of the

corresponding signals for upper switches.

(a)

(b)

Figure 4.6: Rectifier and inverter blocks.

33

Angle detection block: The inputs of this block are three-phase

voltage while the output of this block is the electrical angular speed of the

generator or the grid.

(a)

(b)

Figure 4.7: Angle detection block.

34

abc-to-dq block: This block converts the voltage or current from the

stationary frame into those in the synchronous frame. It needs the theta angle

from the angle detection block to proceed. Fig. 4.8(b) shows one example of

voltage conversion result in the synchronous d− q frame.

(a)

(b)

Figure 4.8: abc-to-dq block.

35

SVPWM block: This block models the current controller for the

rectifier/inverter and it generates the reference voltage signals to compare

with the PWM signals.

(a) (b)

Figure 4.9: SVPWM block.

The inputs are the reference currents (Idref , Iqref ) and the feedback

currents (Id, Iq). This block needs information of electrical angular speed

we, electrical angle, the magnitude of voltage in synchronous frame (Vd, Vq).

In addition, the filter inductance and its resistance (Ls, Rs) are considered as

inputs. To configure the speed of current controller, the bandwidth can be

changed. The default value of current controller bandwidth is 500 rad/sec. As

shown in Fig. 4.9(b), there are two PWM methods for current controller: the

36

space vector PWM (SVPWM) and the sinusoidal PWM (SPWM).

PWM generation block: This block has function in generating

PWM signals for the upper switches (S1, S3, S5) of rectifier or inverter. The

voltage references (Varef , Vbref , Vcref ) from the SVPWM block are compared

with the carrier signal.

(a)

(b)

Figure 4.10: PWM block.

37

Vdc controller block: This block includes the controller of the DC-link

voltage. Based on the reference V ∗dc and the feedback value Vdc, the reference

current i∗q is calculated. This block is configured by changing the output power,

the DC-link capacitor, the bandwidth, and the peak voltage. Fig. 4.11(c)

shows the matched DC-link voltage when the reference value is 1.2 kV DC.

(a) (b)

(c)

Figure 4.11: DC controller block.

38

4.6.2 Converter Models

Fig. 4.12 and Fig. 4.13 shows the simulation models of a rectifier

and an inverter, respectively, in PSCAD/EMTDC. The models include all the

functional blocks described in the previous section.

Figure 4.12: Simulation model of a rectifier in PSCAD/EMTDC.

39

Figure 4.13: Simulation model of an inverter in PSCAD/EMTDC.

40

Finally, Fig. 4.14 shows the model of a back-to-back converter in

PSCAD/EMTDC, which is simply a combination of the models of a rectifier

and an inverter. This model will be used to simulate the frequency conversion

in a multi-frequency AC system in Chapter 5 and Chapter 6.

Figure 4.14: Simulation model of a back-to-back converter inPSCAD/EMTDC.

41

In conclusion, this chapter focuses on the typical structure of a VSC

converter station and the model of a back-to-back converter, which can be used

in multi-frequency power systems and its components. Another important

target of this chapter is the control strategy for back-to-back converters and

the pulse width modulation (PWM) technique for VSC converters. Simulation

blocks and models of a rectifier, an inverter, and a back-to-back converter in

PSCAD/EMTDC are also given in detail.

42

Chapter 5

Power Flow Simulation in HVAC and

Multi-Frequency Systems

5.1 Introduction

This chapter focuses on the validation of the power flow solution pro-

posed in Chapter 3 using a power system operated at the conventional 60 Hz

and at multi-frequency conditions. At first, the power flow problem of the

system when it is operated at the fundamental 60 Hz is solved using Newton-

Raphson method. Next, one transmission line of the system is selected to be

operated at a low frequency. The parameters of this line are modified by the

way described in Chapter 3. The new multi-frequency system is then sim-

ulated and obtained in PSCAD/EMTDC. The two power flow solutions are

expected to be the same if the losses in converter stations are considered to

be negligible.

In addition, this chapter also analyzes one application of LFAC tech-

nology to prove its superiority in terms of power carrying capacity over the

system operated at the typical fundamental frequency.

Publication: Quan Nguyen, Tuan Ngo, Surya Santoso, “Power Flow Solution forMulti-Frequency AC Power Systems,” accepted to 2016 IEEE Transmission and Dis-tribution Conference and Exposition.

43

5.2 Power Flow in a Conventional HVAC System

Figure 5.1: 5-bus test system (System bases are 100 MVA and 345 kV, Loadpower unit is MVA).

Fig. 5.1 shows a 5-bus AC power system [25], which is used to verify

the proposed power flow solution in multi-frequency operating condition. Bus

1 is chosen to be the slack bus of the system, and its voltage magnitude and

phase angle are known and are used as reference for other buses. Bus 2 is

a PV bus or generator bus, where its voltage magnitude is kept constant by

adjusting the field current of a synchronous generator and its active power is

assigned according to economic dispatch. Bus 3, bus 4, and bus 5 are PQ

buses, where their voltages are unknown but their active and reactive power

are already specified. The power and transmission line data can be found in

Table 5.1 and Table 5.2.

44

Table 5.1: Impedances and Admittances of Transmission Lines (System Basesare 100 MVA and 345 kV).

Line Impedance Admittance

(pu) (pu)

1-2 0.02 + j0.06 j0.06

1-3 0.08 + j0.24 j0.05

2-3 0.06 + j0.18 j0.04

2-4 0.06 + j0.18 j0.04

2-5 0.04 + j0.12 j0.03

3-4 0.01 + j0.03 j0.02

4-5 0.08 + j0.24 j0.05

Table 5.2: Power Data.

Generation Load

Bus Type Pgen Qgen Pload Qload

(MW) (Mvar) (MW) (Mvar)

1 slack - - 0 0

2 PV 40 0 20 10

3 PQ 0 0 45 15

4 PQ 0 0 40 5

5 PQ 0 0 60 10

45

The transmission line data is used to construct the admittance ma-

trix of the system, and the power flow solution is obtained by solving highly

non-linear power balance equations [16]. These equations need to be solved

by iterative methods such as Newton-Raphson or Gauss-Seidel instead of an-

alytical methods. In this thesis, all sets of non-linear equations are solved by

Newton-Raphson method because of its good convergence speed and ease in

implementation. The standard procedure is given as follows:

1) Construct the bus admittance matrix for the power system.

2) Make an initial guess for the unknown bus voltages (both magnitude and

phase angle).

3) Substitute the values of bus voltages into the power equations, and deter-

mine the power mismatch compared to the specified power values.

4) Update the estimated voltages based on Newton-Raphson algorithm.

5) Repeat the above process until the deviations from the solution are within

an acceptable range.

In order to validate the result from implementing Newton-Raphson

method in MATLAB, the entire system is modeled in PSCAD/EMTDC. Bus

1 and bus 2 are modeled as voltage sources, and the active power at bus 2 is

controlled to be constant. Bus 3, bus 4, and bus 5 are modeled as constant PQ

loads despite the variations of their voltages. Transmission lines are model as

equivalent π model [16], and their actual values of series inductors, resistors,

and shunt capacitors are calculated based on the data given in Table 5.1 and

the chosen system bases.

46

Fig. 5.2 shows the line power while the bus voltages and power solved

by MATLAB and indicated from PSCAD/EMTDC model. Bus voltages and

line powers are summarized in Table 5.3. From the table, it is apparent that

the results match with each other. They will be continually used to evaluate

the power flow in the multi-frequency working condition, which is the subject

of the following sections.

Figure 5.2: Line power of a 5-bus test system operated at 60 Hz (System basesare 100 MVA and 345 kV, Power unit are MVA, MW, and Mvar).

47

Table 5.3: Voltages and Powers in Conventional Working Condition (SystemBases are 100 MVA and 345 kV).

Conventional HVAC (MATLAB) Conventional HVAC (PSCAD)

Bus Type U θ Pgen Qgen U θ Pgen Qgen

(pu) (deg) (MW) (Mvar) (pu) (deg) (MW) (Mvar)

1 slack 1.0600 0.00 131.12 90.82 1.0600 0.00 131.12 90.83

2 PV 1.0000 -2.06 20.00 -71.59 1.0000 -2.06 20.02 -71.60

3 PQ 0.9872 -4.64 -45.00 -15.00 0.9872 -4.64 -45.01 -15.00

4 PQ 0.9841 -4.96 -40.00 -5.00 0.9841 -4.96 -40.01 -5.00

5 PQ 0.9717 -5.76 -60.00 -10.00 0.9717 -5.76 -60.01 -9.99

5.3 Validation of the Equivalent Parameters of a Trans-mission Line at Different Frequencies

The system described above now includes a transmission line operated

at a different frequency. Specifically, the transmission line between bus 1 and

bus 2 is operated at 10 Hz instead of 60 Hz. This variation in frequency is

achieved by employing two back-to-back converters located at the sending end

and receiving end of the line, as shown in Fig. 5.3.

The equivalent parameters in terms of power flow of the line 1-2 at

60 Hz and 10 Hz, elaborated in Chapter 3, is now verified. The entire data

about power at all buses and transmission lines when the system is operated at

multi-frequency case is the same with those in the conventional case; the only

difference is in the line between bus 1 and bus 2, where its shunt capacitors and

series inductor are scaled by the inverse of the frequency ratio. Specifically,

the values of the shunt capacitors and series inductor are now equal to six

48

Figure 5.3: Multi-frequency system with line 1-2 operated at 10 Hz (Systembases are 100 MVA and 345 kV, Load power unit is MVA).

49

times the values of the shunt capacitors and series inductor when the line

is operated at 60 Hz. However, since the frequency is now one-sixth of the

fundamental frequency, which is 60 Hz, the impedance and the admittance

are now still equal to (0.02 + j0.06) pu and (j0.06) pu, respectively, similar to

what is shown in Table 5.1.

As mentioned in Section 3.1 of Chapter 3, it is not possible to directly

solve power flow in this multi-frequency system by applying numerical methods

such as Newton-Raphson method. The power flow in such system can only be

solved by simulation. The PSCAD/EMTDC model of back-to-back converter

described in Chapter 4 now is used to convert the operating frequency of line

1-2 from 60 Hz to 10 Hz. Also, as discussed in Section 3.1, the power flows

at the two ends of line 1-2 from the solution solved in the Section 5.2 are

used as the setting points for the powers of these two back-to-back converters

in the PSCAD/EMTDC model, which are (89.4 + j74) MVA and (86.9 +

j72.9) MVA as shown in Fig. 5.4. In practice, if these converters are desired

to be operated at different values, they can be represented by constant PQ

loads, and the power flow is obtained immediately by applying the existing

algorithms for conventional 60 Hz AC systems. Fig. 5.4 indicates the line

power from PSCAD/EMTDC simulation while the bus voltages and powers

are given in Table 5.4.

By comparing these results with those in Fig. 5.2 and Table 5.3, a close

resemblance in most values of the conventional and multi-frequency systems

is clearly seen. There is only about 0.1 MW (0.06 percent) increase in active

50

power generated at the slack bus in the multi-frequency case, which could

be attributed to the losses of transformers, reactor, and switching stages ac-

companying with the utilization of back-to-back converters. These analogous

results confirm the conclusion about equivalent transmission line parameters

in terms of power flow of a line operated at various frequencies. Therefore,

any low frequency line in a given multi-frequency system can be converted

into their equivalent lines operated at the fundamental frequency; and solving

power flow in this equivalent system gives us an estimation about the power

carrying capacity of low frequency lines in the original system.

Figure 5.4: Multi-frequency system with line power equal to that of the equiv-alent HVAC system (System bases are 100 MVA and 345 kV, Power unit areMVA, MW, and Mvar).

51

Table 5.4: Voltages and Powers in Multi-Frequency Working Condition (Sys-tem Bases are 100 MVA and 345 kV).

Multi-frequency (PSCAD)

Bus Type U θ Pgen Qgen

(pu) (deg) (MW) (Mvar)

1 slack 1.0600 0.00 131.20 90.83

2 PV 1.0000 -2.06 20.01 -71.61

3 PQ 0.9872 -4.64 -45.01 -15.00

4 PQ 0.9841 -4.96 -40.01 -5.00

5 PQ 0.9717 -5.77 -60.01 -9.99

5.4 Application of Low Frequency Transmission

Based on the conclusion in Section 5.3, an application of LFAC trans-

mission in a multi-frequency power system is now discussed to illustrate the ad-

vantage related to improved power carrying capacity of applying LFAC trans-

mission technique. Specifically, the transmission line between bus 1 and bus 2

of the original system in Section 5.2 is operated at 10 Hz instead of 60 Hz. The

values of the shunt capacitors and series inductor of this line are the same as

those in the original multi-frequency AC circuit. Since the operating frequency

of line 1-2 now is only 10 Hz, its impedance is only (0.02 + j0.01) pu, and its

admittance is only (j0.01) pu. These values are smaller than those when the

entire system is operated at the fundamental frequency, as shown in Table 5.1,

which leads to the assumption that the transferred power carrying capacity

and operating power set points of back-to-back converters increase.

52

Based on the conclusion in Section 5.3, the power flow in this multi-

frequency system now can be obtained indirectly in MATLAB by solving power

flow in its equivalent system operated at 60 Hz with Newton-Raphson method.

Note that when doing so, the value of series inductor and shunt capacitors are

scaled to be one-sixth less than the original values; however, at 60 Hz, the

impedance and admittance of line 1-2 are still equal to (0.02 + j0.01) pu and

(j0.01) pu, respectively. Also, power flow in the multi-frequency system can

be obtained from PSCAD/EMTDC simulation, as described in Section 5.3,

with the only change in the actual values of the series inductor and shunt

capacitors of line 1-2. Again, it is worth to note that setting points of the

two back-to-back converters associated with line 1-2 are chosen based on the

MATLAB solution, which are (333.6 - j24.5) MVA and (313.5 - j33.4) MVA

as shown in Fig. 5.5.

Fig. 5.5 indicates the line power while the bus voltages and powers

are given in Table 5.5. Compared with the power flow solution in Section 5.2

and Section 5.3, there is a significant increase in active power flow, which is

from (89.4 - 86.9) MW in the previous cases to (333.6 - 313.5) MW, from bus

1 to bus 2. There is also a negligible difference (0.27 MW) in active power

generated at bus 1 between the MATLAB and PSCAD/EMTDC solutions.

These mismatches result from the losses associated with transformers, filters,

and converter switching. The voltages at all 5 buses and power flows in other

transmission lines remain constant. Thus, if the thermal capacity is neglected

for simplification, it is clearly seen that one transmission line in a system

53

can deliver more power when it is operated at a low frequency instead of at

conventional 60 Hz.

Figure 5.5: Multi-frequency system with improved power carrying capabilityin line 1-2 (System bases are 100 MVA and 345 kV, Power unit are MVA,MW, and Mvar).

54

Table 5.5: Voltages and Powers in a Multi-Frequency System with ImprovedPower Capability in Line 1-2 (System Bases are 100 MVA and 345 kV).

Multi-frequency (MATLAB) Multi-frequency (PSCAD)

Bus Type U θ Pgen Qgen U θ Pgen Qgen

(pu) (deg) (MW) (Mvar) (pu) (deg) (MW) (Mvar)

1 slack 1.0600 0.00 375.18 -7.67 1.0600 0.00 375.45 -7.71

2 PV 1.0000 -2.06 -206.66 34.69 1.0000 -2.07 -206.69 34.75

3 PQ 0.9872 -4.64 -45.00 -15.00 0.9872 -4.65 -45.00 -15.01

4 PQ 0.9841 -4.96 -40.00 -5.00 0.9841 -4.97 -40.00 -5.00

5 PQ 0.9717 -5.76 -60.00 -10.00 0.9717 -5.77 -60.00 -10.01

55

In conclusion, this chapter verifies the precision of the equivalent pa-

rameters in terms of power flow of a transmission line operated at different

frequencies. The power flow of a multi-frequency AC system in general and

the power flow of its low frequency lines in particular now can be estimated

by converting the parameters of lines operated at low frequencies into those

of lines operated only at the fundamental frequency before applying exist-

ing power flow algorithms. Also, this chapter proves the superiority in terms

of power carrying capacity of a multi-frequency AC system over the system

operated at the typical fundamental frequency.

56

Chapter 6

Power Flow in Multi-Frequency AC and

HVDC Systems

6.1 Introduction

Recently, power generation based on renewable energy sources is ex-

panding rapidly as a solution for the resources depletion and air pollution

caused by conventional energy sources such as fossil fuel. As mentioned be-

fore, wind power is the leading renewable source of energy. Large-scale offshore

wind farms with capacity exceeding hundreds of megawatts become an inter-

national trend, and they have been planed and constructed [26], [27]. These

large-scale wind farms are connected to the main power grid via either high

voltage alternating current (HVAC) or high voltage direct current (HVDC)

transmission systems. HVAC transmission systems are widespread with a his-

tory over 100 years. However, with the cutting-edge developments in power

electronics and advanced controls, a new era in HVDC technology has com-

menced. The most important advantage of HVDC systems is that they are able

to transmit a large amount of power without any limitation in transmission

line length [2].

Currently, most of the installed HVDC systems are of the point-to-point

57

connection type using line-commutated current-source converter (CSC) tech-

nology. However, the recent appearance of forced-commutated voltage-source

converter (VSC) technology draws interest in establishing multi-terminal HVDC

networks due to their smaller footprint and improved controllability [28]. Thus,

multi-terminal HVDC would be a great solution in terms of both technical

and economical aspects for meshed transmission networks between distant lo-

cations [28], [29].

Debates on using DC transmission or AC transmission might never

cease since each method has its own advantages and disadvantages. Therefore,

suggestions have been made that their combination, which is a multi-frequency

AC and multi-terminal HVDC transmission system, would be a promising

solution for bulk power transmission in the near future [30] - [31].

For the analysis of the future systems that consist of multi-frequency

AC and multi-terminal HVDC transmission lines, besides interests in elec-

tromagnetic and electromechanical phenomena, another important issue is

steady-state performance. Insight in steady-state performance can be gained

by using an efficient power flow algorithm. An algorithm is mainly evaluated

based on convergence speed, implementation complexity, and computational

calculation. The purpose of a power flow algorithm is to find a correct steady-

state operating point of the power system as fast as possible, even when the

system is involved with several thousand buses and hundreds of machines. In

the past, numerous attempts have been made to find the power flow solution

for AC - CSC HVDC systems. However, since the operating principles and

58

control strategy of VSC converters differ from those of CSC converters, new ef-

forts have to be made to deal with the power flow problem in AC - VSC HVDC

systems. Reviewing very recent literature in this area shows that two different

approaches have emerged, which are the sequential approach and the unified

approach [32] - [35]. The aim of this chapter is to solve power flow problem in

multi-frequency AC and multi-terminal HVDC systems using what has been

done in the previous chapters and applying both of these two approaches with

a modification to reduce computational effort.

Publication: Quan Nguyen, Tuan Ngo, Surya Santoso, “Power Flow Solution forMulti-Frequency AC and Multi-Terminal HVDC Power Systems,” accepted to 2016IEEE Power and Energy Society General Meeting Conference.

59

6.2 VSC Converter Stations in Multi-Terminal HVDCSystems

6.2.1 Equivalent Model of VSC Converter Station

Basic understanding about the structure of a typical VSC converter

station is already mentioned in Chapter 4. In the previous chapters, VSC con-

verters are only mentioned as a way to convert frequency from one frequency

value to another one, and the object of power flow is still homogeneous AC

power systems. In this chapter, the object is extended with the integration of

multi-terminal HVDC systems into multi-frequency AC systems. Therefore,

it is necessary to construct the equivalent model of VSC converter stations to

facilitate the calculation and solve power flow problem in multi-frequency AC

and multi-terminal HVDC systems.

The typical components of a VSC converter station, as mentioned in

Chapter 4, are shown in Fig. 6.1 (a) while its equivalent model is shown in

Fig. 6.1 (b). The impedance in the equivalent model includes both the phase

reactor and the transformer. In the equivalent model, the shunt capacitor is

not taken into account for calculating simplification.

In Fig. 6.1 (a), Us∠θs, Uc∠θc, and Zc denote the AC bus voltage, DC

bus voltage, and equivalent impedance of the converter station. Ps and Qs

are the active and reactive powers injected into the AC grid; Pc and Qc are

the active and reactive powers injected into converter, and Pdc is the active

power injected into DC grid from the corresponding converter. Representing

the equivalent admittance of the converter station Yc = 1Zc

= Gc + jBc, the

60

(a) Typical structure of a VSC converter station

(b) Equivalent model of a VSC converterstation

Figure 6.1: A VSC converter station

active and reactive powers injected into the AC grid are calculated as below:

Ss = UsI∗s = Us

(Us − Uc)∗

Z∗c= Us(Us − Uc)

∗(Gc − jBc) (6.1)

Ps = −U2sGc + UsUc[Gccos(θs − θc) +Bcsin(θs − θc)] (6.2)

Qs = U2sBc + UsUc[Gcsin(θs − θc)−Bccos(θs − θc)] (6.3)

The active and reactive powers at the converter end are given by:

Sc = UcI∗c = Uc

(Uc − Us)∗

Z∗c= Uc(Uc − Us)

∗(Gc − jBc) (6.4)

Pc = U2cGc − UsUc[Gccos(θs − θc)−Bcsin(θs − θc)] (6.5)

Qc = U2cBc + UsUc[Gcsin(θs − θc) +Bccos(θs − θc)] (6.6)

61

6.2.2 Converter Operating Modes

Contrary to CSC converter, which constantly consumes reactive power

due to the firing delay and commutation process, any VSC converter is able to

independently control the active and reactive power at the AC grid side [37].

Regarding the active power control, it can be modeled in two different ways

[32], [33]:

1) Constant Ps: the active power injected into AC grid from converter side is

kept constant.

2) Constant Udc: the converter adapts the injected active power to obtain a

constant DC bus voltage Udc.

In a multi-terminal configuration, all except one VSC converter control

their active power injection into or withdrawn from the AC grid (constant

P-control). Only one converter, which is called “DC slack converter,” adopts

its active power injection to control the corresponding DC bus voltage Udc.

In addition, the reactive power control can also be represented in two

ways:

1) Constant Q: the reactive power injected into the AC system from the

converter side is kept constant.

2) Constant Us: the injected reactive power can varies in order to maintain

constant corresponding AC bus voltage magnitude.

Therefore, converter stations are either in PQ- or PV-control, with an

exception made for the DC slack converter. The actual value of the active

62

power injected into the AC grid from this slack converter is not known prior

to the power flow, as this quantity depends on the losses in the DC network

as well as in the converter station.

With regard to DC grid, as mentioned above, the slack converter con-

trols its DC bus voltage and adopts active power to compensate for the DC

line losses. The remaining converters operate in active power control mode.

Compared with power flow in AC systems, several considerations should be

taken into account when dealing with power flow problem in DC systems:

- Effort to deal with reactive power Q or voltage angle θ is not necessary.

- Instead of three types of bus as in AC systems (slack bus, PV bus,

and PQ bus), there are only two types of DC bus, which are slack bus and

active power bus.

- The nodal admittance matrix is only composed by resistive part (no

imaginary part).

It is important to notice that in the converter model mentioned here,

the loss of converter stations is taken into account. That loss is obtained by

using a generalized loss formula, in which the value of the loss is quadratically

dependent on the converter current magnitude Ic [36]:

Ploss = a+ bIc + cI2c (6.7)

where the converter current magnitude is given by:

Ic =

√P 2c +Q2

c√3 ∗ Uc

(6.8)

63

and the coefficients a, b, and c is given in Table 6.1, which is corresponding to

the system bases equal to 600 kW and 300 kV [36]:

Table 6.1: Per-unit converter loss coefficients.

Converter Mode a b c

rectifier 11.033x10−3 3.464x10−3 4.400x10−3

inverter 11.033x10−3 3.464x10−3 6.667x10−3

6.3 Sequential Method

The first approach to solve the power flow problem in a multi-frequency

AC and multi-terminal HVDC system is a sequential approach [32], [33]. In a

sequential approach, the power flow problem is solved separately and sequen-

tially: start with AC system’s power flow, then calculate converter stations’

variables, and finally solve DC system’s power flow. The advantage of this

algorithm is that the conventional implementation scheme used to solve AC

systems can be applied to DC system with minor modifications based on the

aforementioned considerations in Section 6.2.2.

After solving power flow for DC system, it is necessary to have an extra

loop to calculate the slack converter voltage Uc since its actual value depends

on the active power injected into the AC grid from slack converter, which is

not known beforehand. Apart from that loop, there must be an extra outer

loop to calculate exactly the value for active power Ps at the slack converter

station. Therefore, five different loops, in which the two last loops only relate

64

to the slack converter, are presented in sequence in the following sections.

6.3.1 Power Flow in Multi-Frequency AC System

In Chapter 2, the equivalent parameters are derived in terms of power

flow of an AC transmission line operated at various frequencies. By scaling

the series inductor and shunt capacitors of the LFAC transmission lines by

the frequency ratio, a multi-frequency AC power system can be converted into

an equivalent conventional AC power system operated at the fundamental

frequency without changing power flow in any branch. Therefore, in this

chapter, we assume that the multi-frequency AC system is already converted

into its equivalent conventional AC system in terms of preserving power flow.

Assuming this AC system has m buses, the power flow equations for

bus i can be written as [16]:

Pi = Ui

m∑k=1

Uk[Gikcos(θi − θk) +Biksin(θi − θk)] (6.9)

Qi = Ui

m∑k=1

Uk[Giksin(θi − θk)−Bikcos(θi − θk)] (6.10)

The above non-linear set of power flow equations for all AC buses can

be solved using the Newton-Raphson method. Its matrix form is given by:[∂Pi

∂θi

∂Pi

∂Ui∂Qi

∂θi

∂Qi

∂Ui

][∆θi∆Ui

]=

[∆Pi∆Qi

](6.11)

where the power mismatches are calculated as:

∆Pi = Pi − (Pgen,i − Pload,i + Ps,i)

∆Qi = Qi − (Qgen,i −Qload,i +Qs,i) (6.12)

65

Since at the slack converter station the injected active power Ps,slack is

not known yet, an initial guess must be made. The injected active power at

the slack converter station is assumed to be the negative sum of active powers

injected into AC grid from other converter stations.

P(0)s,slack = −

m∑i=1,i 6=slack

Ps,i (6.13)

From the second iteration, the result of Ps,slackbus from the previous

iteration is used as an estimate of the injected power into AC grid at the slack

converter station.

6.3.2 Converter Calculation

Once the power injections into AC grid from converter side Ps and Qs

as well as the bus voltages Us of all AC buses are known as a result of the

AC power flow, the voltages Uc at the converter side can be found by first

calculating the converter currents Ic.

Ic,i =Ps,i − jQs,i

U∗s,i(6.14)

The converter voltage at each station can now be calculated from the

corresponding AC bus voltage and voltage drop at that converter station:

Uc,i = Us,i − Ic,iZi (6.15)

Also, after the current flowing between converter and the AC grid is

known, the loss at each converter station can be obtained from (6.7). There-

fore, the actual active powers injected into DC gird from all converters except

66

from slack converter into DC grid is now known:

Pdc−inj,i = −Pc,i − Ploss,i = −real(Uc,iI∗c,i)− Ploss,i (6.16)

6.3.3 Power Flow in DC system

A DC system with n buses uses the injected power Pdc,i calculated

above as input to solve power flow. Section 6.2.1 mentions that the power flow

problem in a standalone DC system is similar to the power flow problem in AC

system; in fact, it is a simplified case of the power flow problem in AC systems.

After converter calculation, all the actual injected active power at all buses

except the slack converter is already known. The power at slack converter can

be assumed to be an initial value, which is zero in here. On the other hand,

for a DC grid with n buses, the power at each bus can be given as follows:

Pdc,i = kDCtype ∗ UiIi = kDCtypeUi

n∑k=1

UkYdc(i, k) (6.17)

where kDCtype is a coefficient depending on the type of HVDC (kDCtype = 1

for monopolar HVDC and kDCtype = 2 for bipolar HVDC). The DC mismatch

equations are given by:

∆Pdc,i = Pdc,i − Pdc−inj,i (6.18)

The Newton Raphson method continues to be applied. The matrix

form of non-linear active power equations at n DC buses is given by:

[∂P∂U

] [∆U

]=[∆P]

(6.19)

67

After the power flow problem in the DC system is solved, all injected

active powers, including the one at the slack converter, are known. Only

this injected active power at the slack converter is used for the next step to

figure out the actual value of the slack converter voltage Uc,slack and the active

power injected into the AC grid at the slack converter station. It is also worth

remembering that there is a relationship between the DC bus voltage Udc and

Uc, which depends on the unknown modulation index used in switching process

of the slack converter station. However, since this work only focuses on power

flow, that relationship is not discussed in detail.

6.3.4 DC Slack Bus Iteration

The purpose of this iteration is to re-calculate the actual value of the

slack converter voltage Uc,slack and the active power injected into the AC grid

at the slack converter station, which is then compared with the value estimated

at the beginning of the current iteration loop. The flow chart of the iteration

at the slack converter is represented in Fig. 6.2.

After calculating the power flowing in the DC grid, the active power at

the slack converter side Pc,slack can be calculated from the power of the DC

bus connected to the slack converter and from the power loss given in (6.7),

which is already calculated in the above converter calculation step:

P(k)c,slack = −P (k)

dc,slack − P(k)loss,slack (6.20)

The variable we need to determine is the active power Ps,slack injected into

the AC grid from the DC slack bus converter, which depends on the unknown

68

Figure 6.2: Flow chart of the two iteration loop at the slack converter.

69

current Ic,slack flowing in that converter station. In order to determine that

current, we need to calculate both the magnitude and phase angle of the con-

verter voltage Uc,slack. These two variables are determined by solving the two

nonlinear equations (6.2) and (6.3). In these equations, Pc,slack are accessible

from (6.20) while Qs,slack, Us,slack, and θs,slack are kept constant after solving

AC system’s power flow due to the strong relation between reactive power and

voltage magnitude. The Newton Raphson method is continually used with an

initial flat start of voltage, and the non-linear set of equations is given by:[∂Pc,slack

∂θc,slack

∂Pc,slack

∂Uc,slack∂Qs,slack

∂θc,slack

∂Qs,slack

∂Uc,slack

][∆θc,slack∆Uc,slack

]=

[∆Pc,slack∆Qs,slack

](6.21)

Once the converter voltage Vc,slack is determined, the converter current

is calculated from (6.8), and then the power loss is achieved from (6.7). The

active power Pc,slack is then calculated again using (6.20). Therefore, an extra

loop is required to get this value of Pc,slack matching with the one we got

right after the DC system’s power flow is found. Once Pc,slack converges, the

active power injected into AC grid from the slack converter station Ps,slack is

updated by (6.1). Now, similar to the aforementioned case about Pc,slack, an

extra outer iteration is needed to get Ps,slack converge. In other words, if the

difference between two consecutive values of Ps,slack is sufficiently small, the

loop is terminated. If it is not the case, the updated value of Ps,slack is used as

the input for the next outermost iteration loop, which starts with AC system’s

power flow.

The flow chart of the sequential method is shown in Fig. 6.3.

70

Figure 6.3: Flow chart of the sequential method.

71

6.4 Unified Method

In contrast to the sequential method, in which AC power flow and

DC power flow are separately solved in sequence, in a unified method, the

AC system and the DC system are treated as a unit, and their power flow is

solved simultaneously [34], [35]. Although the unified method gives raise to

a modification in power flow scheme, its implementation is less complicated

compared to what is done in the sequential method since no extra loop is

needed.

The mismatch equations now contain both AC and DC groups, and

the variables are also constructed by both the AC system’s and the DC sys-

tem’s variables. In [34], [35], the total loss in the slack converter station is

represented by an additional explicit variable. However, in this thesis, a mod-

ification is made. The total loss in the slack converter station is represented

implicitly through the AC system’s and the DC system’s variables. The mis-

match equation related to the losses in the slack converter station, therefore,

is not needed, and the Jacobian matrix is also reduced by one. The sets of

mismatch equations and variables are defined as follows:

F (XAC, XDC) = 0 (6.22)

where XAC = [UTAC , θ

TAC ] is a vector containing voltage magnitudes of PQ

buses and phase angles of the non-slack bus in AC system, respectively, and

XDC = [UTDC ] is a vector containing the voltages of non-slack buses in DC

systems.

72

Similar to other algorithms, the unified algorithm starts with an initial

guess, typically a flat guess, for the variables. The next step is to calculate the

power injected into AC buses by using (6.9) and (6.10). The power mismatch

equations set of AC system is then calculated using (6.12) and (6.13). For the

DC system, the powers at each DC bus Pdc,i are calculated from (6.17). Alter-

natively, since AC system and DC system are connected through converters,

it is possible to get the injected power Pdc−inj,i into DC bus i by applying the

same process of converter calculation presented in the sequential method. The

converter current is given by (6.14), then the converter voltage is calculated by

(6.15). The power injected into DC is calculated from (6.16) in which the loss

of the corresponding converter is given in (6.7). The power mismatch equation

in DC system, therefore, can be given by (6.18).

The solution for the system is obtained by solving mismatch equations

(6.12) and (6.18) simultaneously. The Jacobian matrix of the unified method

has be to modified by integrating both AC and DC equations. Its general

non-linear form is shown as below: ∂Pac

∂θac∂Pac

∂Uac0

∂Qac

∂θac

∂Qac

∂Uac0

0 0 ∂Pdc

∂Udc

∆θac∆Uac∆Udc

=

∆Pac∆Qac

∆Pdc

(6.23)

The flow chart of the unified method is illustrated in Fig. 6.4.

73

Figure 6.4: Flow chart of the unified method.

74

6.5 Simulation of Power Flow in a Multi-Frequency AC- Multi-Terminal HVDC System

In this section, the power flow for a multi-frequency AC and multi-

terminal HVDC system is solved. The 5-bus multi-frequency AC system in

Chapter 5 is incorporated with a 3-bus HVDC system as shown in Fig. 6.5.

Similar to Chapter 5, the transmission line from bus 1 to bus 2 is operated at

10 Hz instead of the fundamental 60 Hz. The load power and AC transmission

line data are the same as specified in Section 2.4 of Chapter 5.

The 3-terminal monopolar HVDC system is also embedded into it

through three converters connected to bus 1, 4, and 5. The slack converter is

chosen to be the one connected to bus 1, which also controls the voltage level

of the entire DC grid. The DC voltage at bus 1 is kept constant at 1 pu or 120

kV, and the reactive power injected into the AC grid from the slack converter

is already known while the injected active power is only known after solving

power flow. The active and reactive powers injected from the AC grid at bus 4

and 5, as shown in Fig. 6.5(b), are also known before power flow is solved since

the corresponding converters connected to these buses independently control

active and reactive powers. The load power and AC transmission line data are

the same as specified in Chapter 5.

For the DC grid, the resistance of DC transmission line is given in per

unit with respect to the DC system base. The power base of DC system is

chosen to be equal to the power base of the AC system, which is 100 MVA.

The voltage base of the AC system and the DC system are 345 kV AC and

75

(a)

(b)

Figure 6.5: A multi-frequency AC - multi-terminal HVDC system and inputdistributed power (System bases are 100 MVA, 345 kV AC, and 120 kV DC;Power unit are MVA, MW, and Mvar).

76

120 kV DC, respectively. The data for converter stations is given in Table 6.2

while the data for the DC transmission line is given in Table 6.3 [38].

Table 6.2: Converter Data (System Bases are 100 MVA and 345 kV AC).

Converter Control Mode P (into AC) Q (into AC) Impedance

(MW) (Mvar) (pu)

1 slack - -0.30 0.0007 + j0.0397

4 PQ 0.35 0.05 0.0007 + j0.0397

5 PQ 0.45 0.10 0.0007 + j0.0397

Table 6.3: DC Line Resistances (System Bases are 100 MW and 120 kV DC).

Bus to Bus Resistance

(pu)

1 4 0.052

1 5 0.073

4 5 0.052

In order to apply the sequential and unified algorithms described in the

previous sections, it is necessary to first convert the line between bus 1 and 2,

which is operated at 10 Hz, into its equivalent line operated at the fundamental

frequency. According to Chapter 3, it is done by scaling the series inductor

and shunt capacitors by the inverse of the frequency ratio. Specifically, the

equivalent line operating at the fundamental 60 Hz has the series inductor and

shunt capacitors equal to one-sixth of those in the original line operating at

10 Hz. However, since the fundamental frequency is six times the original low

77

frequency, the impedance and admittance of the equivalent line are still (0.02

+ j0.01) pu and (j0.01) pu, respectively. The sequential and unified algorithms

are then implemented in MATLAB with a flat start for all variables and with

the tolerance equal to 1e-5.

The system is then modeled in PSCAD/EMTDC to verify the power

flow solution obtained from MATLAB. Table 6.4 shows the AC bus voltages,

powers injected into AC grid, DC bus voltages, and DC bus powers from

MATLAB and PSCAD/EMTDC. By convention, power is chosen to be pos-

itive when it flows towards AC buses. First, it is noticed that the power

flow solutions using the sequential method and unified method are exactly the

same with each other. Second, power flow results from MATLAB and from

PSCAD/EMTDC are also found to be closely matched with each other, which

confirms the conclusion about equivalent transmission line parameters in con-

ventional 60 Hz and multi-frequency working conditions as well as the validity

of the pre-described methods when solving power flow for multi-frequency AC-

MTDC systems.

78

Table 6.4: Power Flow Solution of the Multi-frequency AC-MTDC Systemfrom MATLAB and PSCAD/EMTDC (System Bases are 100 MVA, 345 kVAC, and 120 kV DC).

MATLAB (using the sequential method and unified method)

Bus Type AC Voltage AC Bus Injection DC Voltage DC Power

AC Converter Uc θc Ps Qs Udc Pdc

grid mode (pu) (deg) (MW) (Mvar) (pu) (deg)

1 slack slack 1.0600 0.00 -81.40 -30.00 1.0000 -79.70

2 PV - 1.0000 -2.23 - - - -

3 PQ - 1.0007 -2.09 - - - -

4 PQ PQ 1.0009 -1.96 30.00 5.00 0.9780 31.28

5 PQ PQ 0.9993 -1.56 45.00 10.00 0.9727 46.42

PSCAD/EMTDC

Bus Type AC Voltage AC Bus Injection DC Voltage DC Power

AC Converter Uc θc Ps Qs Udc Pdc

grid mode (pu) (deg) (MW) (Mvar) (pu) (deg)

1 slack slack 1.0600 0.00 -80.63 -30.00 1.0000 -79.09

2 PV - 1.0000 -2.22 - - - -

3 PQ - 1.0010 -2.08 - - - -

4 PQ PQ 1.0010 -1.97 30.00 5.00 0.9782 31.19

5 PQ PQ 0.9992 -1.54 45.00 10.00 0.9729 45.92

79

Fig. 6.6 shows the line power of the multi-frequency AC-MTDC system.

As expected, the transferred power through transmission line between bus 1

and 2, which is operated at 10 Hz, significantly increases. The load at bus

2, thus, can be set to be 240 MW instead of just 20 MW without violating

the voltage limit requirement. In addition, by embedding an MTDC system

into an AC system, the AC transmission lines do not have to transfer a large

amount of power. Therefore, the line currents and line losses are reduced,

which improves the voltage profiles at PQ buses of the AC grid compared

with those shown in Table 5.3 in the conventional AC system.

80

(a)

(b)

Figure 6.6: A multi-frequency AC - multi-terminal HVDC system and theinput distributed power (System bases are 100 MVA, 345 kV AC, and 120 kVDC; Power unit are MVA, MW, and Mvar).

81

From the above results and during the implementation time, the fol-

lowing is concluded:

1) Both the sequential and unified algorithms converge to the same solution.

2) The advantage of the sequential method is that it can be implemented as an

extension of the existing AC power flow scheme, whereas the unified method

requires certain modification both in the scheme and Jacobian matrix. How-

ever, the unified method is much more easier to implement in MATLAB.

3) The reduction in the number of mismatch equations in the unified method,

and in turn its lower computation complexity in that method, do not reduce

the convergence speed compared to the conventional unified method in [34].

82

Chapter 7

Conclusion and Future Work

7.1 Conclusion

In this thesis, state of the art LFAC systems are discussed in com-

parison to conventional HVAC systems and HVDC systems to show the great

potential that LFAC systems can bring to the future power transmission. With

the ultimate goal of solving power flow in multi-frequency power systems, the

derivation of equivalent parameters in terms of power flow of a transmission

line operated at different frequencies is first proved. The result shows that a

transmission line operated at a low frequency can be converted into an equiv-

alent transmission line operated at the fundamental frequency by keeping the

resistance constant and scaling the series reactor and shunt capacitors by the

inverse of the frequency ratio. The model of a back-to-back converter as well

as their control configuration are also described, which is applied later to verify

the proposed previous power flow solution method. The converter model and

control blocks are built in PSCAD/EMTDC software, which can be effectively

utilized for design and analysis of any multi-frequency AC power systems. A

5-bus multi-frequency AC power system is then modeled in PSCAD/EMTDC

in conventional 60 Hz and in multi-frequency AC. The simulation results ver-

ify the proposed method to solve power flow solution in multi-frequency power

83

systems. In addition, the potential advantages in term of higher power carry-

ing capacity when using low frequency technology instead of the conventional

method are illustrated, consolidating LFAC technology as a promising solution

for bulk power transmission. The thesis also discusses the application of two

existing approaches to solve the power flow problem in multi-frequency AC and

multi-terminal HVDC systems. The aforementioned 5-bus multi-frequency AC

system is now integrated with a 3-bus HVDC system, and the power flow solu-

tion from its PSCAD/EMTDC simulation model validates the precision of the

two approaches. It is revealed from the solution that by implementing such

multi-frequency AC and multi-terminal HVDC systems, not only the trans-

ferred power increases significantly but the bus voltage profiles also improve.

7.2 Future Work

The next stage of this work could be investigate scenarios that could

be beneficial from utilizing or cooperating LFAC technology. For example, the

removal of offshore converter stations and the associated cost reduction would

greatly motivate the application of LFAC transmission for offshore wind power,

either standalone or in cooperation with HVDC systems or conventional AC

systems. The design considerations, system modeling, and study of operational

characteristic of such systems will be further addressed. For instance, with

regard to system protection point of view in multi-frequency AC systems, one

challenging issue for those transmission lines operated at low frequencies is

that circuit breakers associated with these lines have to wait longer for the AC

84

current to cross the zero point. It means that the fault clearing time of these

circuit breakers is prolonged, leading to more serious impacts on the entire

system. One possible solution to reduce the clearing time is to convert the

current operating frequency of the line where the fault happens back to the

conventional 60 Hz as soon as possible (within a cycle). By doing so, it takes

the circuit breakers less time to approach the zero crossing point and interrupt

the fault current. Therefore, the control of back-to-back converters needs to

be improved to quickly detect the fault current and perform the frequency

conversion.

Another issue requiring attention to is at the DC side of a multi-

frequency AC and multi-terminal HVDC system. As mentioned in Chapter 6,

wind energy and solar energy are the most rapidly developing ones from all

renewable resources, and their penetration into the power grids is expected

to increase exponentially in the near future. However, when an offshore wind

farm or a large-scale solar PV is penetrated into the DC grid, to what extend

this integration does not cause damage in the AC-DC connection, converter

control, voltage stability, or system protection needs to be investigated.

85

Bibliography

[1] P. Bresesti, W. L. Kling, R. L. Hendriks, and R. Vailati, “HVDC connection

of offshore wind farms to the transmission system,” IEEE Trans. Energy

Convers., vol. 22, no. 1, pp. 3743, Mar. 2007.

[2] J. Kreusel (ABB), “High Voltage Direct Current (HVDC) Power Trans-

mission,” Workshop “Transporting Tens of Gigawatt of Green Power to the

Market”, Potsdam, May 12, 2011.

[3] J. Ruddy, R. Meere, T. ODonnell, “Low Frequency AC transmission for off-

shore wind power: A review,” Renewable and Sustainable Energy Reviews,

Volume 56, April 2016, Pages 7586.

[4] J. Ruddy, R. Meere, T. ODonnell, “A Comparison of VSC-HVDC with

Low Frequency AC for Offshore Wind Farm Design and Interconnection,”

12th Deep Sea Offshore Wind RnD Conference, EERA Deep Wind 2015,

Volume 80, 2015, Pages 185192.

[5] T. Funaki, K. Matsuura, “Feasibility of the low frequency AC transmis-

sion,” IEEE Proc. Elect. Power App., NO.3, pp 231-240, 1996.

[6] PSERC,“Low Frequency Transmission,” PSERC Publication, 2012. Avail-

able: http://www.pserc.wisc.edu/documents/publications/reports/

2012reports/MeliopoulosPSERCReportS-42Oct2012ExSum.pdf.

86

[7] Y. Cho, G. Cokkinides, A. Meliopoulos, “LFAC - Transmission systems for

remote wind farms using a three-phase, six-pulse cycloconverter,” in Power

Electronics and Machines in Wind Applications (PEMWA), 2012 IEEE, pp.

17, July 2012.

[8] H. Chen, M. Johnson, D. Aliprantis, “Low-frequency AC transmission for

offshore wind power,” IEEE Transactions on Power Delivery, vol. 28, pp.

22362244, Oct. 2013.

[9] W. Fischer, R. Braun, I. Erlich, “Low frequency high voltage offshore grid

for transmission of renewable power,” in PES Innovative Smart Grid Tech-

nologies Europe, 2012 IEEE, pp. 16, Oct. 2012.

[10] H. E. Orton, R. Sarnm, “Worldwide Underground Transmission Cable

Practices,” IEEE Trans on PD, V01.12, No.2, pp.533-541, 1997.

[11] Takada et.al, “New Direct Observation Technique for Electric Charge Be-

havior in Insulating Materials and Its Application to Power Cables,” Proc.

CIGRE session 1998, 15-303, 1998.

[12] W. Xifan, C. Chengjun, Z. Zhichao, “Experiment on fractional frequency

transmission system,” IEEE Transactions on Power Systems, vol. 21, pp.

372377, Feb. 2006.

[13] T. Ngo, M. Lwin, S. Santoso, “Steady-State Analysis and Performance

of Low Frequency AC Transmission Lines,” IEEE Transactions on Power

Systems, 2015.

87

[14] L. Tang, “HVDC Technologies and ABB Experience, OE Workshop Ap-

plications for High-Voltage Direct Current Transmission Technologies,” ABB,

April 22, 2013.

[15] N. Qin, S. You, Z. Xu, V. Akhmatov, “Offshore wind farm connection with

low frequency ac transmission technology,” Power Energy Society General

Meeting, 2009.

[16] J. Glover, M. Sarma, and T. Overbye, “Power System Analysis and De-

sign,” Cengage Learning, 2011.

[17] S. Li, T.A Haskew, Ling Xu, “Control of HVDC Light System Using

Conventional and Direct Current Vector Control Approaches,” IEEE Trans.

Power Electronics, vol. 25, no. 12, pp. 31063118, 2010.

[18] V. Pedro, G. D. Marques, “DC voltage control and stability analysis of

PWM-voltage-type reversible rectifiers,” IEEE Transactions on Industrial

Electronics, 45:263-273, 1998.

[19] C. M. Ong, “Dynamic simulations of Electric Machinery using Matlab/Simulink,”

Prentice Hall Ptr, 1998.

[20] A. Timbus, M. Liserre, R. Teodorescu, P. Rodriguez, and F. Blaabjerg,

“Evaluation of current controllers for distributed power generation systems,”

IEEE Trans. Power Electron., vol. 24, pp. 654664, Mar. 2009.

[21] F. Golnaraghi, B. C. Kuo, “Automatic Control System,” 9th edition, Wi-

ley, 2010.

88

[22] R. Erickson, D.Maksimovic, “Fundamentals of power electronics,” 2nd

edition, Kluwer Academic Publishers, 2004.

[23] J. H. Kim, S. K. Sul, “A Carrier-Based PWM Method for Three-Phase

Four-Leg Voltage Source Converters,” IEEE Transaction on Power Electron-

ics, VOL. 19, NO. 1, January 2004.

[24] D. Woong, J. S. Kim, S. K. Sul, “Unified Voltage Modulation Technique

for Real-Time Three-Phase Power Conversion,” IEEE Transaction on In-

dustry Application, VOL. 34, NO. 2, March/April 1998.

[25] S. G. W. and A. H. El-Abaid, “Computer Methods in Power System Anal-

ysis,” Kogakusha, Japan: McGraw-Hill, 1968.

[26] A. Purvins, A. Zubaryeva, M. Llorente, E. Tzimas, A. Mercier, “Chal-

lenges and options for a large wind power uptake by the European electricity

system,” Appl. Energy, vol. 88, no. 5, pp. 14611469, 2011.

[27] EWEA, “Large scale intergration of wind energy in the European power

supply: analysis, issues, and recommendations,”, 2005 report. Available:

http://www.uwig.org/eweastudy/051215_grid_report.pdf.

[28] R. T. Pinto, P. Bauer, S. F. Rodrigues, E. J. Wiggelinkhuizen, J. Pierik,

and B. Ferreira, “A novel distributed direct-voltage control strategy for grid

integration of offshore wind energy systems through MTDC network,” IEEE

Trans. Ind. Electron., vol. 60, pp. 24292441, Jun. 2013.

89

[29] S. Rodrigues, R. T. Pinto, P. Bauer, J. Pierik, “Optimal Power Flow Con-

trol of VSC-Based Multiterminal DC Network for Offshore Wind Integration

in the North Sea,” IEEE Journal of Emerging and Selected Topics in Power

Electronics, vol. 1, no. 4, Dec. 2013.

[30] D. V. Hertema, M. Ghandhari, “Multi-terminal VSC HVDC for the Eu-

ropean supergrid: Obstacles,” Renew. sustain. Energy Rev., 14 (9) (2010),

pp. 31563163.

[31] P. Wang, L. Goel, X. Liu, F. H. Choo, “Harmonizing AC and DC: A

Hybrid AC/DC Future Grid Solution,” IEEE Power Energy Mag., vol. 11,

no. 3, pp. 7683, May 2013.

[32] J. Beerten, S. Cole, and R. Belmans, “Generalized dynamic VSC MTDC

model for power system stability studies,” IEEE Trans. Power System, vol.

25, no. 3, pp. 16551662, Aug. 2010.

[33] J. Beerten, S. Cole, and R. Belmans, “A sequential AC/DC power flow

algorithm for networks containing multi-terminal VSC HVDC systems,” in

Proc. IEEE PES GM 10, Minneapolis, MN, Jul. 2529, 2010.

[34] M. Baradar, M. Ghandhari, “A Multi-Option Unified Power Flow Ap-

proach for Hybrid AC/DC Grids Incorporating Multi-Terminal VSC-HVDC,”

Power Systems, IEEE Transactions on August 2013.

[35] M. Baradar, M. Ghandhari, D. Van Hertem, “The modeling multi-terminal

VSC-HVDC in power flow calculation using unified methodology,” Innova-

90

tive Smart Grid Technologies (ISGT Europe), 2011 2nd IEEE PES Interna-

tional Conference and Exhibition on 5-7 Dec. 2011.

[36] M. R. S. C. G. Daelemans, K. Srivastava, and R. Belmans, “Minimization

of steady-state losses in meshed networks using VSC HVDC,” in Proc. IEEE

PES General Meeting 09, Jul. 2009.

[37] R. Rudervall, J.P. Charpentier, R. Sharma, “High Voltage Direct Current

(HVDC) Transmission Systems Technology Review Paper,” Presented at

Energy Week 2000, Washington, D.C, USA, March 7-8, 2000.

[38] P. Haugland, “Its time to connect - technical description of HVDC Light

technology,” ABB, Tech. Rep., Mar. 2008.

91