small signal modeling of resonant controlled vsc systems · 2013-10-24 · abstract small signal...

Small Signal Modeling of Resonant Controlled VSCSystems

by

Stephen Podrucky

A thesis submitted in conformity with the requirementsfor the degree of Master’s of Applied Science

Graduate Department of Computer and Electrical EngineeringUniversity of Toronto

Copyright c© 2009 by Stephen Podrucky

Abstract

Small Signal Modeling of Resonant Controlled VSC Systems

Stephen Podrucky

Master’s of Applied Science

Graduate Department of Computer and Electrical Engineering

University of Toronto

2009

A major issue with respect to VSC based systems is the propagation of harmonics to DC

side loads due to AC voltage source unbalance. Standard dq-frame control techniques

currently utilized offer little mitigation of these unwanted harmonics. Recently, resonant

controllers have emerged as an alternative to dq-frame controllers for regulation of grid

connected converters for distributed resources. Although these control systems behave

somewhat similar to dq-frame controllers under balanced operating conditions, their be-

haviour under unbalanced operation is superior. Currently, there are no linearized state

space models of resonant controlled VSC systems. This work will develop a linearized

small signal state space model of a VSC system, where resonant current controllers are

used for regulation of the grid currents. It will also investigate the stability of resonant

controlled VSC based systems using eigenvalue analysis for HVDC applications.

ii

Dedication

To my mother, Stephanie Podrucky, who taught me the value of a strong work ethic

and for giving me all the confidence one would need and more. You have been there for

me every step of the way and you have never let me down. This work is a small payment

in return for all the time, effort and love you have invested in me.

iii

Acknowledgements

I would like to thank my supervisor, Prof. Peter W. Lehn, for his guidance and support

during my graduate degree. I would also like to thank my family and Leona for their

many years of support and help throughout my academic career.

I would like to extend a large thanks to Dr. K. Natarajan, who started my journey

into post undergraduate studies. Thank you for all your help, support, guidance, time,

and, most of all, the lessons you have taught me. You have changed my life’s path for the

better, without receiving anything in return, and I cannot thank you enough for doing

so. A special thanks to Prof. Prodic of the University of Toronto, for his great classes,

interesting lectures, and kind disposition.

I would also like to thank Natural Sciences and Engineering Council of Canada

(NSERC) for their financial support for the duration of my thesis work.

iv

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Objective and Scope of Thesis . . . . . . . . . . . . . . . . . . . . . . . . 3

2 VSC Transforms, Equations, and Control 5

2.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Definition of αβ-frame and dq-frame Orientation . . . . . . . . . . . . . . 5

2.3 Voltage Sourced Converter Equations and

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Three Phase Voltage Source . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 AC Line Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.3 Three Phase IGBT Bridge . . . . . . . . . . . . . . . . . . . . . . 13

2.3.4 DC Link and DC Side Load . . . . . . . . . . . . . . . . . . . . . 14

2.4 Voltage Sourced Converter Control Methods . . . . . . . . . . . . . . . . 15

2.4.1 Standard dq-frame PI Control . . . . . . . . . . . . . . . . . . . . 15

2.4.2 αβ-frame Resonant Current Control with DC PI Voltage Control 17

2.4.3 αβ-frame Resonant Current Control with DC Space Vector Control 18

2.5 VSC Control Operation Under Grid Voltage Unbalance . . . . . . . . . . 20

2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

v

3 Linearization of VSC Control and State Space Modeling 23


3.2 Modulator Shifting and Small Signal Modeling . . . . . . . . . . . . . . . 24

3.3 State Space Representation of Linearized VSC Control Loop . . . . . . . 30

3.3.1 State Space Representation of S-domain Transfer

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.2 Combination of Derived State Space Models . . . . . . . . . . . . 35

3.3.3 Validation of Linearized Small Signal Model versus

Non-Linear Large Signal Model . . . . . . . . . . . . . . . . . . . 39

3.3.4 Validation of Large Signal Non-Linear Model versus PSCAD Model 49

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Applications of Linearized State Space VSC System Models 55


4.2 DC Side Impedance of VSC Based Systems . . . . . . . . . . . . . . . . . 55

4.3 Eigenvalue Analysis of VSC Based Systems . . . . . . . . . . . . . . . . . 58

4.3.1 Single VSC System . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 VSC based Back-to-Back HVDC System . . . . . . . . . . . . . . 63

4.3.3 VSC Based HVDC Transmission System . . . . . . . . . . . . . . 69

4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Conclusions 82

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Appendices 84

A State Space Modeling of Controllers 84

B Development of Complete State Space Model for DC SVC 92

vi

C Small Signal State Space Models of VSC Control Schemes 99

C.1 Modulation Index Control: PI DC Voltage

Control with dq-frame PI Current Control . . . . . . . . . . . . . . . . . 99


Control with αβ-frame Resonant Current Control . . . . . . . . . . . . . 102

C.3 VSC Terminal Voltage Control: PI DC Voltage Control with αβ-frame

Resonant Current

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

C.4 VSC Terminal Voltage Control: DCSV Voltage Control with αβ-frame

Resonant Current

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

D Combinations of State Space Systems: Series and Feedback 112

D.1 Series Combination of Two State Space Systems . . . . . . . . . . . . . . 112

D.2 Feedback Combination of Two State Space Systems . . . . . . . . . . . . 113

Bibliography 115

vii

Chapter 1

Introduction

1.1 Background

With the growing popularity of VSC based systems in the fields of power transmission and

distributed power generation, new types of VSC control are emerging. The standard form

of dq-frame PI current control coupled with PI control for DC voltage regulation offers

little mitigation of harmonics during source unbalance. The propagation of harmonic

ripple from a voltage source unbalance is shown in [1]. This poses a definite problem

when these VSC systems are utilized for high voltage DC transmission due to the transfer

of unwanted harmonics from the AC grid to the DC side. Several emerging control

techniques rely on regulation of αβ-frame quantities instead of dq-frame quantities and

identify their potential to limit the unwanted harmonic interactions that result from AC

system unbalance. However, comprehensive linearized models of such systems are not

available in the literature to analytically assess their performance.

1.2 Literature Overview

As previously stated, the problem of harmonic component introduction through source

unbalance has been approached from many angles. Several strategies have been developed

1

Chapter 1. Introduction 2

using a separation of positive and negative sequence components. The separation of these

components require the addition of filtering devices, which add to the latency and the

complexity of control. This strategy is developed in [2], [3], and [4]. This method also

implements a feedforward type of power control to eliminate harmonic ripple on the DC

side of the VSC. This feedforward strategy does not provide the robustness of control as

compared to a standard feedback control systems approach. The implementation of the

feedforward strategy produces non-linear control signals, formed by the product of time-

varying terms. Due to the multiple non-linear terms, and complex signal conditioning

utilized, a linearized model of these VSC systems has not been developed. Without a

linearized model, rigorous linear control design and stability analysis techniques cannot

be performed. Also, a zero steady state value exists only in absence of sensor errors under

the assumption that all system parameters are known with precision. Even sensitivity to

parameter errors, sensor errors, and the like cannot be carried out without a linearized

model.

A solution, which implements feedback control, to the harmonic mitigation issue is

to employ αβ-frame resonant current controllers. The αβ-frame controller, also known

as a stationary frame controller or resonant controller, is presented in [5] and [6]. The

performance of the αβ-frame resonant current controller was compared to the standard

dq-frame PI approach in [7]. The results displayed that the resonant controller can

perform comparably with the dq-frame PI controller, yet the resonant controller does

not require the complicated coordinate transforms to map values to the dq-frame. The

resonant control is further developed in [8]. This work also supports the claim that

the resonant controller’s performance is equivalent to that of the dq-frame PI controller.

The resonant type of control possess the ability to significantly reduce harmonic content

found on the DC side of the converter during source unbalance. In [9] a feedforward

power matrix develops the current references required for the αβ current controller to

eliminate DC voltage ripple caused by source unbalance. Once again, the feedforward


method has the same drawbacks that were previously discussed.

To incorporate a feedback style of control, to mitigate DC ripple, a special DC voltage

controller was developed in [10]. The proposed strategy by Hwang and Lehn creates an

αβ-frame current reference for the resonant controller using a feedback signal from the

DC voltage. This particular method is named DC space vector control or DCSV control.

In the body of this work, focus will be on linearization and analysis of αβ-frame current

control methods, with particular focus on the DCSV control in order to provide insight

into system stability.

The DCSV control developed in [10] has not currently been implemented as a solution

to the mitigation of DC voltage ripple within VSC based HVDC systems. Instead, the

standard dq-frame PI current control is implemented in HVDC transmission work, such

as in [11], [12], [13], and [14]. All of these works deal with bidirectional power flow of

a VSC based HVDC system, but assume (i) back to back HVDC systems and (ii) a

perfectly balanced AC grid voltage.

Linearized models of the VSC systems have been developed in [15] and [16] with

respect to a VSC based HVDC tranmission system with DC cable transmission. In these

works, an eigenvalue analysis is performed to display system stability and to investigate

optimization of controller gains and system parameters. These VSC based HVDC systems

also implement standard dq-frame current control and do not discuss voltage source

unbalance. Currently there are no linearized models of VSC based HVDC systems which

deal with elimination of DC ripple during source unbalance and contain both plant and

controller states.

1.3 Objective and Scope of Thesis

The focus of this thesis will be to develop a linearized model of a VSC system with

resonant αβ-frame current control, both with standard PI DC voltage control and DCSV


voltage control. An introduction to the background theory of VSC modeling will be

provided. The common forms of VSC control will be shown and discussed in order to

highlight the advantages of the αβ-frame current control. Following the introductory

material, the following objectives will be completed:

• Non-LTI terms will be identified with respect to αβ-frame resonant control

• A method for eliminating the time variant modulation blocks is presented, while

other non-linear time invariant blocks are simply linearized

• The linearized small signal model of the DCSV controlled VSC will be validated

against time-domain simulations and its corresponding state space model will be

developed

• Zero steady state error will be proven for the DCSV control under source unbalance

• A stability analysis will be performed for VSC based HVDC systems under bidi-

rectional power flow including back to back HVDC transmission and an HVDC

transmission system with a DC cable interface.

Chapter 2

VSC Transforms, Equations, and

Control

2.1 Chapter Overview

This chapter will present the coordinate transformations used in this study. The time

averaged equations used in the modeling of the voltage sourced converter, or VSC, will

be explored along with any assumptions made. Also, three types of VSC based system

control will be discussed.

2.2 Definition of αβ-frame and dq-frame Orientation

Two crucial elements of voltage sourced converter theory are the dq and αβ transforms.

The αβ-frame is a transform which converts an array of three scalar quantities into a

singular vector quantity expresssed on a two plane system, along with a zero sequence

term. This two plane system has the α-axis analogous to the x-axis and the β-axis

analogous to the y-axis of the generally understood cartesian coordinate system. The final

result of this transformation is a single vector which rotates at the system frequency. The

chosen orientation of the αβ axes, as well as the orientation of the abc frame quantities,

5

Chapter 2. VSC Transforms, Equations, and Control 6

can be viewed below in Fig.2.1.

α

β

°120

°120

°120

a

b

c

Figure 2.1: αβ-Frame Representation

The equation, known as the Clarke transform, in which the αβ-frame vector and the

zero sequence term is achieved from the abc sequence phasors is shown below in (2.1) and

(2.2). The Clarke transformation can be utilized for the transfer of voltages, currents,

and any other set of variables to the αβ-frame.

C =

1 −1

2−1

2

0√

32

−√

32

1√2

1√2

1√2

(2.1)

vαβo =

vα(t)

vβ(t)

vo(t)

=2

3· C ·

va(t)

vb(t)

vc(t)

(2.2)

The next step of the coordinate transform is converting the stationary αβ-frame

into the rotating or dq-frame. The dq transformation applies a negative rotation, at a

particular frequency, to the rotating αβ vector. If this negative rotation is complementary


to the system frequency, the dq-frame will consist of a static vector, i.e. a set of DC terms.

This transformation from a set of three vectors to a set of DC terms simplifies control

and modeling of the voltage sourced converter system. The d-axis is aligned with the

rotating system vector and the q-axis is quadrature to the d-axis, leading it by ninety

degrees. Fig. 2.2, shown below, depicts the assumed orientation of the dq-frame with

respect to the αβ axes.

α

β

Rotating

Vector

dq

0ω+

θ

Figure 2.2: Dq Frame Representation

In Fig. 2.2, the +ω0 depicts the rotation of the vector at system frequency. The

equations listed below, (2.3) and (2.4), display the mathematical transform from the αβ-

frame to the dq-frame in standard representation, as well as in matrix form. It is also

beneficial to note that the zero sequence from the αβ frame is maintained through the

dq-frame transformation.

vdqo = vαβo · e−jωot (2.3)

vdqo =

vd(t)

vq(t)

vo(t)

=

cos(ωot) sin(ωot) 0

−sin(ωot) cos(ωot) 0

0 0 1

∗vα(t)

vβ(t)

vo(t)

(2.4)


2.3 Voltage Sourced Converter Equations and

Assumptions

The minimal setup necessary for operation of a voltage sourced converter, or VSC, is

presented in Fig. 2.3 below. It consists of an AC voltage source, the point of common

coupling (PCC), an AC line reactor, a three phase IGBT bridge, a DC link capacitor and

a DC load.

LR

+−G

v

VSC

i

tv

C

−

+

dcv

capi

dci

−+R

v

PCC

loadi

Figure 2.3: Voltage Sourced Converter with DC Side Load

g1

g4 g6

g3

g2

g5

abc

dcv

Figure 2.4: Three Phase AC to DC IGBT Bridge

2.3.1 Three Phase Voltage Source

The first element is the three-phase voltage source, labeled as Vg in the Fig.2.3 above. It

is made up of a set of balanced or unbalanced voltage vectors. A balanced source will

produce a uniform rotating vector in the αβ-frame with respect to its magnitude and

phase. This means that the αβ-frame equivalent of the abc parameters will trace a circle

that rotates at the AC system frequency in the αβ-frame. An example of this can be


viewed below in Fig. 2.5. A balanced set of abc quantities will also produce a constant

or DC signal in the dq-frame.

α

β

+ωo

Rotating αβ-

frame vector

Figure 2.5: Rotating Vector Representation of balanced abc system

An unbalanced source, meaning a non-uniformity of abc quantities in phase and/or

magnitude, will produce a non-circular rotating vector in the αβ-frame. The αβ-frame

equivalent of the abc quantities will trace an elliptical shape rotating at the AC system

frequency. An example of this unbalancing can be seen below in Fig. 2.6 while rotating

at the AC system frequency. After transforming the unbalanced abc source voltages

into the dq-frame, one is left with a constant or DC term along with sinusoidal terms,

produced by the unbalance, which are called harmonic components.

This phenomena can be explained using positive and negative sequence components.

A positive sequence component is a vector which rotates at the same frequency and in the

direction of the given system’s frequency, +ωo. A negative sequence component is a vector

which rotates at the same frequency, but in the opposite direction of the given system’s

frequency, −ωo. While the source is balanced, only positive sequence components appear

in the αβ-frame. If the source is unbalanced, negative sequence components will appear

along with the positive sequence components. Shown below, from (2.5) to (2.16), is an


α

β

+ωo

Rotating αβ-

frame vector

Figure 2.6: Rotating Vector Representation of an Unbalanced abc System

example displaying how positive and negative sequence voltage and current components

at the AC terminals of a VSC are transferred into average and ripple power to the DC

side of the converter.

S =3

2(vαβ(t)iαβ(t)∗) =

3

2(V p

dqej(ωot+ 6 V pdq)+V n

dqe−j(ωot+ 6 V ndq))(Ipdqe

j(ωot+ 6 Ipdq)+Indqe−j(ωot+6 Indq))∗

(2.5)

S =3

2[V pdqI

pdqe

j( 6 V pdq−6 Ip

dq)+V n

dqIndqe

j(6 V ndq−6 Indq)+V p

dqIndqe

j(2ωot+ 6 V pdq−6 Indq)+V n

dqIpdqe−j(2ωot−6 V ndq+6 I

pdq

)]

(2.6)

Assuming SPWM,

Vdq = vdc(t) ·Mdq (2.7)


S =3

2· vdc(t)[Mp

dqIpdqe

j(6 Mpdq−6 Ip

dq) +Mn

dqIndqe

j(6 Mndq−6 I

ndq) + (2.8)

MpdqI

ndqe

j(2ωot+ 6 Mpdq−6 Indq) +Mn

dqIpdqe−j(2ωot−6 Mn

dq+6 Ip

dq)]

Applying power balance:

Pdc = Re(S) = Po + P2 (2.9)

where, Po = 32· vdc(t)[Mp

dqIpdqcos(6 M

pdq − 6 I

pdq) +Mn

dqIndqcos(6 M

ndq − 6 Indq)] (2.10)

and, P2 = 32· vdc(t)[Mp

dqIndqcos(2ωot+ 6 Mp

dq − 6 Indq) (2.11)

+MndqI

pdqcos(2ωot− 6 Mn

dq + 6 Ipdq)]

Therefore,

idc(t) =Pdc(t)

vdc(t)= Io + I2 (2.12)

where, Io = 32[Mp

dqIpdqcos(6 M

pdq − 6 I

pdq) +Mn

dqIndqcos(6 M

ndq − 6 Indq)] (2.13)

and, I2 = 32[Mp

dqIndqcos(2ωot+ 6 Mp

dq − 6 Indq) (2.14)

+MndqI

pdqcos(2ωot− 6 Mn

dq + 6 Ipdq)]

And,

vdc(t) =1

C

∫idc(t)dt (2.15)


The resulting DC voltage will therefore have the general form of (2.16).

vdc = Vdco + Vdc2 · e−jπ2 (2.16)

where, Vdco is a DC component and Vdc2 is a harmonic ripple component, following

the form of (2.9).

In the equations above the superscripts p and n denote positive and negative sequence

components, while S denotes the apparent power flowing through the VSC system. Also,

the subscript αβ implies the variable is in the αβ-frame while the subscript dq implies the

variables are in the dq-frame. From (2.6) one can note the appearance of two DC terms

and two sinusoidal terms in the dq-frame. By inspection, if the source were balanced,

meaning the negative sequence terms did not appear in the voltage or current, the dq-

frame value would only consist of a DC term. This DC term would be the product of

the positive sequence voltage and current. Equation (2.9) gives the real power through

the VSC in a simplified form by grouping all the DC and sinusoidal terms respectively.

If (2.9) describes the real power on the DC side capacitor then (2.16) will describe the

simplified DC voltage equation during source unbalance.

Following the three phase voltage source is the point of common coupling or PCC. The

PCC refers to the point which all measured signals, such as voltages and currents, take as

a reference for their orientation when performing synchronization and transformations.

2.3.2 AC Line Reactor

The second component of the VSC system is the AC line reactor. The R in Fig. 2.3

represents the non-ideality of the inductor’s resistance. This resistance will be taken into

account in all proceeding calculations. Note, that the source unbalance can be caused by a

variance in inductor value symmetry from phase to phase, rather than an unbalance in the

source voltage from phase to phase. For the reason of simplicity, all source unbalancing


in this study shall be performed by the three phase grid voltage source. The AC line

reactor equations are shown below in (2.17), (2.18), and (2.19), which include the source

voltage, vg, and converter terminal voltage, vt. Note that the subscripts abc, αβo, and

dqo denote the variables are in a given frame of reference.

vgabc = Riabc + Ld(iabc)

dt+ vtabc (2.17)

vgαβo = Riαβo + Ld(iaαβo)

dt+ vtαβo (2.18)

d

dt

id

iq

io

=1

L

−R +ωL 0

−ωL −R 0

0 0 −R

id

iq

io

+1

L

vgd

vgq

vgo

−1

L

vtd

vtq

vto

(2.19)

2.3.3 Three Phase IGBT Bridge

The following element of the VSC system is the three-phase IGBT bridge. For all cases

in this document, sinusoidal pulse width modulation is used to drive the switching of the

IGBT bridge. The IGBT bridge functions as a modulator, producing AC voltages and

currents, when a DC voltage (or current) source is applied to the DC side. In this case,

the system current flow will be from right to left in Fig. 2.3. The IGBT bridge functions

as a demodulator, producing DC voltages and current, in the absence of a DC side source

and presence of an AC side voltage (or current) source. In this case, the system current

flows from left to right in Fig. 2.3. In order to mathematically model the IGBT bridge,

the AC side and DC side powers are equated. It was assumed that the non-idealities

of the IGBT, such as the transistor’s resistances and capacitances between terminals,

were neglected in modeling calculations. Also, the switching harmonics produced by the

high frequency sinusoidal pulse width modulation was disregarded as well. The high

frequency switching of the IGBT bridge is in the order of several thousand hertz. These

two phenomena were assumed negligible for two reasons: (i) to simplify future control


and system modeling equations and (ii) they do not play a large role in lower frequency

system dynamics or steady-state behaviour. These assumptions will be discussed and

validated in the proceeding chapters.

With the assumption of an ideal IGBT bridge the equations, in the αβ and dq frames,

can be expressed by the conversion of power from the AC to DC side. These equations

are shown in (2.20) and (2.21) respectively.

vdcidc =3

2(vtαiα + vtβiβ + vtoio) (2.20)

vdcidc =3

2(vtdid + vtqiq + vtoio) (2.21)

Where, vt describes the AC terminal voltage produced by the IGBT bridge. With

regards to SPWM, the modulation index is the terminal voltage scaled by half the DC

side voltage. The relationship between vt and m and the resulting AC to DC power

conversion equations are listed below.

mαβo =vtαβovdc2

(2.22)

mdqo =vtdqovdc2

(2.23)

idc =3

4(mαiα +mβiβ +moio) (2.24)

idc =3

4(mdid +mqiq +moio) (2.25)

2.3.4 DC Link and DC Side Load

The final components of the minimal VSC setup are the DC link capacitor and the DC

side load. The DC link capacitor equation is shown below in (2.26), which describes

the relationship between the DC side current, voltage and load current. In Fig. 2.3

the DC load is modeled as current source. All DC side loads in further simulations and


validations will be performed as current disturbances. The final applications of this study

will be with respect to VSC based systems, i.e. HVDC transmission. With respect to

VSC based systems applications, the DC side loads of the transmission system will be

currents.

dvdcdt

=1

C(idc − iload) (2.26)

2.4 Voltage Sourced Converter Control Methods

The development of the VSC equations from the preceding section yields the plant trans-

fer function. For the subsequent control discussions the control of the VSC will be

performed utilizing the modulation index, md and mq or mα and mβ, as inputs. The

terminal voltages, vtd and vtq or vtα and vtβ, are also commonly used control variables.

In the case of this work the modulation index control was performed due to its slight

advantage in implementation. The following subsections will describe the VSC control

systems as follows:

1. Dq-frame PI current control with DC PI voltage control

2. αβ-frame resonant current control with DC PI voltage control

3. αβ-frame resonant current control with DC space vector control

2.4.1 Standard dq-frame PI Control

The following diagram, Fig. 2.7, describes the standard dq-frame PI current control of

the VSC. This is the most common form of VSC control.

The dq-frame PI current control multiplied with the − 2Vdc

block produces the dq-

frame modulation indices, mdq. The negative gain in this block is required due to the


dq-frame equivalent current controller dq-frame equivalent AC line reactance dyanmics

di

gdv

ref

qi

ref

di

tdv

tqv

gqv

qi

RLs

1

RLs

1

d

q

s

KK

ii

iP+

dm

2

dcv

s

KK

ii

iP+

qm

2

dcv

Lo

ω oLω

oLω

Lo

ω

dcV

2−

dcV

2−

Figure 2.7: Dq-frame PI Current Control Loop

chosen current reference direction and the 2Vdc

term scales the control signal in order to

cancel the cross-coupling terms from the dq-frame AC line reactor plant.

The cross-coupling terms are before the combination of the current controller and

the −2Vdc

gain block. These terms are utilized in the controller in order to cancel cross-

coupling produced by the oscillatory nature of the dq-frame AC line reactor plant. The

final output of the current controller gives the uncoupled dq-frame modulation indices.

In the AC system plant, the dq-frame terminal voltages, vtd and vtq, are calculated using

a rearrangement of (2.23). This signal is then fed into the system plant transfer function

derived from (2.19). This AC system plant finally produces the dq-frame AC currents.

The following layer of VSC control shows the regulation of the DC side voltage via

current control. The DC voltage controller sets the references for the dq-frame current

controller and can be viewed in Fig. 2.8.

The dq-frame currents and modulation indices produced by the current control loop

are then fed into AC to DC power conversion equation derived in (2.25). The output of

this non-linear function produces the DC side current, idc, which is then fed into (2.26)


s

KK

I

p+

ref

dqi

Inner AC

Current

Control Loop

dq-frame

( )qqdd

imim +4

3ref

dcv

ref

di

ref

qij ⋅

dci

loadi

dcv

dcv

dqm

dqi

Cs

1

Figure 2.8: PI DC voltage control loop

and produces the desired DC voltage. Note, that the inner AC current control loop of

Fig. 2.8 is that of Fig. 2.7 where the dq-frame grid voltage source is an internal input

not shown in Fig. 2.8.

2.4.2 αβ-frame Resonant Current Control with DC PI Voltage

Control

The subsequent figures, Fig. 2.9 and Fig. 2.10, display the αβ-frame current control loop

as well as the modified DC voltage control loop. This form of control is developed in [9].

αβ-frame AC line reactance dyanmicsαβ-frame current controller

ref

iαβ

RLs +

1

ipK

αβi

αβtv

αβgv

αβRv

2

dcv

mαβ

uuuur

2 2iR

o

sK

s ω⋅

+dc

V

2−

Figure 2.9: Resonant αβ-frame Current Control Loop

Once again the appearance of the negative gain in the − 2Vdc

block is required due to

the chosen current reference direction. The final output of the current controller gives

the αβ-frame modulation indices. From viewing Fig. 2.9 it can be noted that both the

α-axis and β-axis control and plant models are combined. Fig. 2.9 is arranged in this

manner to describe the vector nature of its control. This form is analogous to a single


line diagram with respect to a three phase system model.

The resonant αβ-frame current controller provides benefits in the mitigation of un-

wanted harmonic during a source unbalance in the AC system. This is due to the reso-

nance of the controller at frequencies of +ωo and −ωo. Under complete source balancing,

and the appearance of solely positive sequence quantities, the αβ-frame resonant cur-

rent controller will perform the same duty as the dq-frame PI current controller. In the

AC system plant, the αβ-frame terminal voltages, vtα and vtβ, are calculated using a

rearrangement of (2.22). This signal is then fed into the system plant transfer function

derived from (2.18). This AC system plant finally produces the αβ-frame AC currents.

The following layer of VSC control shows the regulation of the DC side voltage via

current control. The DC voltage controller sets the dq-frame current control references

and the modulator converts those references to the correlating αβ-frame values. The DC

voltage controller and the VSC system plants can be viewed in Fig. 2.10.

AC to DC REAL

POWER EQUATION

s

KK

I

p+

ref

dqi

ref

dcv

ref

di

ref

iαβ

ref

qij ⋅

dci

αβi

loadi

dcv

dcv

Modulator

Demodulator

mαβ

uuuur

3( )

4m i m iα α β β+

Cs

1

Inner αβ-frame

AC current

control loop

oj t

eω

Figure 2.10: DC Voltage control loop with αβ-frame Current Control Loop

2.4.3 αβ-frame Resonant Current Control with DC Space Vec-

tor Control

The inner αβ-frame current control loop used for DCSV control is that of Fig. 2.9

and has been discussed in the previous subsection. The DC space vector control, or

DCSV control, has been develop in [10] and experimentally proven to mitigate unwanted

harmonic ripple on the DC side voltage. Coupled with the αβ-frame resonant current


control the DCSV control has the ability to completely eliminate second order harmonic

ripple on the DC side voltage under voltage source unbalance. The DCSV control loop

can be viewed in Fig. 2.11. The DC space vector controller can be viewed in Fig. 2.12.

AC to DC REAL

POWER EQUATIONref

dcv

ref

iαβdc

i

αβi

loadi

dcv

dcv

Demodulator

mαβ

uuuur

3( )


Cs

1

Inner αβ-frame

AC current

control loop

DC Space

Vector Control

Figure 2.11: DC SVC loop with αβ-frame Current Control Loop

ref

dcv

ref

iαβ

dcv

s

KK

I

P+

22)2(

o

R

s

sK

ω+⋅

22)2(

2

o

o

R

s

jK

ω

ω

+

−⋅

To current

control loop

)(DCref

di

)2(ref

di

)2(ref

qij ⋅−

)(DCref

qij ⋅

)(DCref

dqiv

)2(−ref

dqiv

tj oeω

Modulator

)1(+ref

iαβ

v

)1(−ref

iαβ

v

Figure 2.12: DC Space Vector Controller

The DC space vector controller is resonant to second order harmonic ripple at the

frequency of +2ωo. From Fig. 2.12 it is shown how the DCSV controller produces dq-

frame current references for the positive sequence DC value as well as the second order

negative sequence value. The final output of the DC space vector controller are the

positive and negative sequence αβ-frame current control references.


2.5 VSC Control Operation Under Grid Voltage Un-

balance

As discussed in the section 2.3.1, second harmonic components in the dq-frame are created

from the appearance of negative sequence components in the abc-frame quantities. Since

the resonant control can produce negative sequence components as a control signal it has

the ability to reduce the low order harmonic content in the dq-frame. In equation (2.6),

the two DC terms and two harmonic terms at +2ωo and −2ωo are power terms produced

by the source unbalance. If one were to inspect the AC side currents and voltages of the

VSC in the dq-frame, similar harmonic terms would appear at +2ωo and −2ωo. Since

the DC voltage has the form given in (2.16), the d-axis current reference will have the

form given in (2.27). The q-axis current reference will have a similar form. The αβ-frame

current reference will therefore have the form shown in (2.28).

irefd = Ido + Id2cos(2ωot+ φ3) = Ido + (Id22

)[e+j(2ωot+φ3) + e−j(2ωot+φ3)] (2.27)

irefαβ = irefdq ∗ ejωot = Idqo ∗ ejωot + (Idq22

)[e+j(3ωot+φ3) + e−j(ωot+φ3)] (2.28)

By mapping the reference signal into the αβ-frame, as done in equation (2.28), we

clearly see unwanted grid current components being requested during source unbalance.

The resonant αβ-frame current control can eliminate the unwanted−ωo negative sequence

component due to its resonant properties. A clear overview of the αβ-frame resonant

current control advantage during source unbalance, is presented in Fig. 2.13 and Fig.

2.14. Note that both forms of current control utilize a PI DC voltage controller to set

their respective current references.

The standard PI dq-frame current controller will only produce a constant DC signal


LR

+−G

v

VSC

i

tv

C

−

+

dcv

capi

dci

−+R

v

PCC

t

vtd

t

vgd

t

vdc

loadi

Figure 2.13: Dq-frame PI current controller response under source unbalance

LR

+−G

v

VSC

i

tv

C

−

+

dcv

capi

dci

−+R

v

PCC

t

vtd

t

vgd

t

vdc

loadi

Figure 2.14: αβ-frame resonant current controller response under source unbalance

for the d-axis terminal voltage, vtd. With the inability to produce a sinusoidal component

to subtract from the grid voltage’s sinusoidal component, all of the unwanted harmonic

content of the source unbalance is delivered to the DC side voltage. On the other hand,

the αβ-frame resonant current controller can produce sinusoidal components. This will

counteract the source unbalance due to its negative sequence component resonance. The

αβ-frame resonant current controller alone cannot completely eliminate the unwanted

harmonics due to its non-resonance at the positive sequence component at three times the

system frequency. This positive sequence component, +3ωo, can be viewed in (2.28). In

Fig. 2.14 it is shown that the resonant control significantly reduces the harmonic impact

on the DC side voltage. Fig. 2.15 displays the DCSV controller’s ability to produce


LR

+−G

v

VSC

i

tv

C

−

+

dcv

capi

dci

−+R

v

PCC

t

vtd

t

vgd

t

vdc

loadi

Figure 2.15: DC space vector control response under source unbalance

sinusoidal terminal voltage components that completely mitigate harmonic ripple on the

DC side voltage. The positive sequence component, +3ωo, in the αβ-frame is transferred

to a +2ωo component in the dq-frame. The SVC is resonant at this frequency in the

dq-frame and can generate the required harmonic eliminating current references. Please

note that the waveforms in Fig. 2.13, Fig. 2.14, and Fig. 2.15 are not to scale and are

only shown to give a conceptual reference to difference in the dq-frame and αβ-frame

current controllers.

2.6 Chapter Summary

This chapter gave an explanation of the abc frame to αβ-frame transform, as well as

the αβ to dq-frame transform. The assumed orientation of these frames was discussed.

Subsequently, the equations governing the voltage sourced converter were explained along

with any assumptions or neglected terms. Lastly, the current control and DC voltage

control loops were explored for standard dq-frame control as well as resonant αβ-frame

control.

Chapter 3

Linearization of VSC Control and

State Space Modeling


In the previous chapter, the αβ-frame resonant current controller was introduced and

its qualities discussed. A major disadvantage of the resonant current controller is the

appearance of non-linear and time-varying blocks within its control loop. These non-LTI

blocks do not allow the use of standard linear control design techniques to be performed

on this type of control. This drawback withholds the resonant control from being utilized

for the full spectrum of possible applications, particularly high cost applications such as

high voltage DC transmission. In this chapter, the linearization of the αβ-frame current

controller will be performed with DCSV control of the DC voltage. The state space

model will be developed, and the linearized model will be validated using time-domain

simulations. Small signal models of other, more conventional, dq and αβ-frame controlled

systems are provided in the appendix.

23

Chapter 3. Linearization of VSC Control and State Space Modeling 24

3.2 Modulator Shifting and Small Signal Modeling

In order to perform standard control design procedures, such as eigenvalue analysis and

system parametrization, the given control loop must be linearized. In terms of this study

all unwanted terms are non-linear due to the product of two time-varying terms or they

contain sinusoidal components. The strategy towards a linearized model of this control

loop is to eliminate the sinusoidal terms and apply a Taylor series or small signal modeling

to the product of time-varying terms as developed in [17]. In the case of the resonant

current control loop, there are four non-LTI blocks, which are listed below. These terms

can be identified in Fig. 3.1 and Fig. 3.2 shown below.

1. Modulator, ejωot, containing sinusoidal components

2. Demodulating term, e−jωot, containing sinusoidal components

3. Product of time varying terms, mdq ∗ vdc2

4. Product of time varying terms contained within the AC to DC real power equation

ref

dqi

tjoe

ω

)(4

3

qqddimim +

ref

dcv

αβmref

iαβdc

i

αβi

loadi

dcv

dcv

Modulator

Demodulatortj

oeω−

dqm

dqi

Inner AC

Current

Control Loop

αβ-frame

AC to DC REAL POWER

EQUATION

Cs

1DC

SVC

Figure 3.1: DC Voltage Control Loop with Demodulating Term Removed from PowerEquation

The modulator receives the control signal from the DC voltage controller, dq-frame

term, and provides the αβ-frame current control reference. The demodulator term is

embedded in the AC to DC power equation. This is seen in Fig. 3.1 where the power

equation is written in terms of dq-frame quantities. The cancelation of the modula-

tor/demodulator terms can be achieved if the modulating term is shifted through the



ref

iαβ

RLs +

1

ipK

αβi

αβtv

αβgv

αβRv

2

dcv

mαβ

uuuur

2 2iR

o

sK

s ω⋅

+dc

V

2−

Figure 3.2: Resonant αβ-frame Current Control Loop

current control loop and multiplied by the demodulator term. The elimination equation

can be viewed below, where the product of the two terms creates a unity gain.

ejωot ∗ e−jωot = 1 (3.1)

The shifting of the modulating term will produce a dq-frame equivalent of the resonant

αβ-frame current controller. The goal in this case is to nullify the first two non-linear

terms by multiplying them together and therefore avoiding the more mathematically

awkward task of linearizing about sinusoidal time-varying equations.

The following simple example of an exponential term being shifted through a transfer

function can be viewed in Fig. 3.3 and was developed in [18] for VSC system applications.

The shifting of an exponential term through a transfer function can be derived from

Laplace domain exponential shifting theory.

tje 0

ω

τs+1

1

)(1

1

js

αβx

dqx

dqy

αβy

αβy

dqx

tje 0

ω

0

Figure 3.3: Shifting of rotating space vector


Essentially, each pole and zero of a transfer function is shifted by +jωo during the

modulator shift. The transfer function is also transferred from the αβ-frame to the dq-

frame when the modulating term is moved through to the right. Each transfer function

and disturbance in the current control loop is transformed to its dq-frame equivalent

when this shift is performed. The resulting shifts in the αβ-frame current controller can

be viewed in Table I below. These results are in agreement with those of Zmood, Holmes,

and Bode in [19].

Table I: Transformation of αβ-frame Current Controller

αβ-frame Transfer Functions Equivalent dq-frame Transfer Functions

Kip Kip

KiR · ss2+ω2

oKiR · s+jωo

s(s+j2ωo)

= KiR ·[

(s2+2ω2o)

s(s2+4ω2o)− j · ωo

s2+4ω2o

]= CRe(s)− j · CIm(s)

Using the results from Table I, the block diagram of Fig. 3.4 may be constructed

using the following relations:

~vdq = (CRe(s)− jCIm(s))(εd + jεq) (3.2)

~vdq = (εdCRe + εqCIm) + (εqCRe − εdCIm) (3.3)

From Table I it can be observed that this dq-frame equivalent contains a quadrature

or imaginary component. The combination of this added quadrature component of the

dq-frame equivalent current controller with the vector nature of the dq-frame error signal

will yield cross-coupling between the d and q axes. This phenomena can be seen in (3.2)

and (3.3).

The further shifting of the modulator term through the current control loop trans-

forms the grid voltage, AC system plant, and currents into the dq-frame equivalents seen



di

gdv

ref

qi

ref

di

tdv

tqv

gqv

qi

RLs +

1

RLs +

1

dm

2

dcv

2

dcv

qm

oLω

oLω

qε

2 24

o

iR

o

Ks

ω

ω⋅

+

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

iPK

dε

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

2 24

o

iR

o

Ks

ω

ω⋅

+

iPK

dcV

2−

dcV

2−

Figure 3.4: Dq-frame Equivalent of Fig. 3.2

in Fig. 3.4.

Now that the modulating term is shifted through to the right of the current control

loop it can be combined with the demodulating term and both terms are nullified. This

step will remove non-linearities 1 and 2 from the list at the beginning of the chapter, but

non-linearity number 3 is still present within the current control loop. The multiplication

of the two time-varying terms vdc and mdq create this non-linear phenomena and can

therefore be linearized by applying small signal analysis or a Taylor series. The sequence

of linearizing the product of time-varying terms, with respect to the d-axis, can be viewed

below in (3.4) and (3.5). Note that the same procedure applies for the q-axis and the

same result can be achieved by replacing the d in the equations by q.

(Vtd + vtd) = (Md + md)(Vdc2

+vdc2

) (3.4)

vtd = md ·Vdc2

+ vdc ·Md

2(3.5)


In the equations above, the inflection above the variables indicate that they are small

signal perturbation terms and the capital letters denote large signal, steady-state, or

operating point values. In (3.5) the product of two small signal terms are neglected

because the perturbation about the large signal operating point will always be a minute

value. Therefore, the product of the small signal terms will be minuscule. The entire

current control loop is now transformed into its small signal equivalent and can be viewed

in Fig. 3.5 shown below.


di$

gdv$

ref

qi$

ref

di$

tdv$

tqv$

gqv$

qi$

oLω

oLω

RLs +

1

RLs +

1

dε

qε

dm

qm

dcv

2

dcV

2

dM

2 24

o

iR

o

Ks

ω

ω⋅

+

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

iPK

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

2 24

o

iR

o

Ks

ω

ω⋅

+

iPK

dcv

2

dcV

2

qM

dcV

2−

dcV

2−

Figure 3.5: Small Signal Linearized dq-frame Representation of Fig. 3.4

The final component which contains non-linearity due to the product of time-varying

terms is the AC to DC real power conversion equation. The αβ-frame version of this

equation was separated into two blocks. The first is the demodulating term and the

second is the dq-frame equivalent of the AC to DC real power equation. It was shown

in Chapter 2 that the product of an αβ-frame term along with a demodulating or e−jωot

will produce a dq-frame term. This demodulating term was eliminated through its com-

bination with the modulating term earlier in this chapter. To linearize the remaining


product of time-varying terms, the same procedure of small signal modeling or Taylor

series expansion was utilized. This linearization process, taken from [17], can be viewed

in (3.6) and (3.7) shown below.

(Idc + idc) =3

4[(Md + md)(Id + id) + (Mq + mq)(Iq + iq)] (3.6)

idc =3

4(Mdid + Idmd +Mq iq + Iqmq) (3.7)

With the new small signal model of the AC to DC power conversion equation available,

it is possible to construct a complete linear model of the αβ-frame resonant control with

a DCSV voltage controller. This newly linearized DC control loop can be viewed in Fig.

3.6 shown below.

ref

dqi

dci

loadi

dcv

dcv

Linearized

Current

Control Loop

dq-frameCs

1ref

dcv

di

qi

dm

qm

dM

qM

dI

qI

Σ4

3DC

SVC

Figure 3.6: Small Signal Linear Equivalent of Fig. 3.1 with Current Control of Fig. 3.5


3.3 State Space Representation of Linearized VSC

Control Loop

3.3.1 State Space Representation of S-domain Transfer

Functions

State space modeling of the linearized VSC control loop is essential in order to per-

form several linear control design and system development techniques, such as eigenvalue

analysis, finding state participation factors, and system parametrization. The goal of

this study is to develop a mathematical structure of the VSC systems in order to allow

rigorous control and system parameter analysis that, currently, has not been developed.

State space modeling will allow for the easy connection of blocks with multiple inputs

and outputs for multiple VSC systems as well as the interfaces between them. Using

the equations for the VSC system presented in Chapter 2 and the linearized controller

equations developed at the beginning of Chapter 3, a complete set of state space equations

can be developed for the DC voltage control loop of Fig. 3.6. The first step of state

space modeling is to chose system inputs and outputs, as well as the state variables.

The figures below display the linearized DC voltage control loop of Fig. 3.6 and the AC

current control loop of Fig. 3.5 divided into five sections. The sections, as numbered in

Fig. 3.7 and Fig. 3.8, are:

ref

dqi

dci

loadi

dcv

dcv

Linearized

Current

Control Loop

dq-frameCs

1ref

dcv

di

qi

dm

qm

dM

qM

dI

qI

Σ4

3

System #1

System #4 System #5

DC

SVC

Figure 3.7: DC voltage control loop in divided sections



di$

gdv$

ref

qi$

ref

di$

tdv$

tqv$

gqv$

qi$

oLω

oLω

RLs +

1

RLs +

1

dε

qε

dm

qm

dcv

2

dcV

2

dM

2 24

o

iR

o

Ks

ω

ω⋅

+

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

iPK

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

2 24

o

iR

o

Ks

ω

ω⋅

+

iPK

System #2 System #3

dcv

2

dcV

2

qM

dcV

2−

dcV

2−

Figure 3.8: Inner AC current control loop of Fig. 3.7 in divided sections

1. DCSV voltage controller

2. Dq-frame current controller

3. Dq-frame AC line reactance plant combined with terminal voltage equation (3.5)

4. AC to DC real power conversion equation

5. DC Link and DC side load.

These five sections will be transformed from their s-domain transfer function form

into their state space matrix form as shown below. All state space matrix equations will

follow the general state space form listed in (3.8) and (3.9).

x = Ax+Bu (3.8)


y = Cx+Du (3.9)

Where x is the array of state variables, u is the array of system inputs and y is the

array of system outputs. Note that the choice of the system’s states can be found in Ap-

pendix A and the derivations of the A, B, C, and D matrices can be found in Appendix B.


System 1: DC Space Vector Controller (DCSV)

xdc1

xdc2

xdc3

=

0 1 0

0 0 1

0 −(2ωo)2 0

xdc1

xdc2

xdc3

+

0

0

1

u (3.10)

y =

irefd

irefq

=

KI(2ωo)2 0 (KI +KR)

0 −2KRωo 0

xdc1

xdc2

xdc3

+

KP

0

u (3.11)

System 2: Dq-frame Current Controller (Dq-frame equivalent of resonant controller)

ˆx2 =

ˆxd1

ˆxd2

ˆxd3

ˆxq1

ˆxq2

ˆxq3

=

0 1 0 0 0 0

0 0 1 0 0 0

0 −4ω2o 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 −4ω2o 0

xd1

xd2

xd3

xq1

xq2

xq3

+

0 0

0 0

1 0

0 0

0 0

0 1

irefd − id

irefq − iq

(3.12)

y2 =

−md

−mq

=

−4KiRω

2o

Vdc0 −2KiR

Vdc0 −2KiRωo

Vdc0

0 2KiRωoVdc

0 −4KiRω2o

Vdc0 −2KiR

Vdc

xd1

xd2

xd3

xq1

xq2

xq3

+

−2KiPVdc

0

0 −2KiPVdc

irefd − id

irefq − iq

(3.13)


System 3: Dq-frame AC Line Reactance Plant Combined with Terminal Voltage

Equation

ˆx3 =

ˆid

ˆiq

=

−RL

ωo

−ωo −RL

id

iq

+

−Vdc2L

0 1L

0 −Md

2L

0 −Vdc2L

0 1L

−Mq

2L

md

mq

vgd

vgq

vdc

(3.14)

y3 =

id

iq

=

1 0

0 1

id

iq

+

0 0 0 0 0

0 0 0 0 0

md

mq

vgd

vgq

vdc

(3.15)

System 4: AC to DC Real Power Conversion Equation

y4 = idc =[−3KiRω

2oId

Vdc

3KiRωoIq2Vdc

−3KiRId2Vdc

−3KiRω2oIq

Vdc

−3KiRωoId2Vdc

−3KiRIq2Vdc

3Md

43Mq

4

]

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

+

[−3KiP2Vdc

−3KiP2Vdc

] irefd − id

irefq − iq

(3.16)


Note that system 4 contains no state variables. Therefore, (3.16) simply gives the

algebraic input-output relation of this subsystem.

System 5: DC Link and DC Side Load

ˆvdc = [0]vdc +[

1C

−1C

] idc

iload

(3.17)

y5 = vdc = [1]vdc +[

0 0

] idc

iload

(3.18)

3.3.2 Combination of Derived State Space Models

With the five sections transformed into their state space form, the combination of these

sections can be achieved. The goal to perform linear control design techniques, such as

eigenvalue analysis, cannot be performed until the DC voltage control loop for the VSC

system is grouped into a single state space model. This grouping can be achieved by

utilizing a series and feedback block combination.

The series combination of two state space systems can be viewed in the Fig. 3.9

shown below.

State Space

System #1

11111

11111

uDxCy

uBxAx

+=

+=&

State Space

System #2

22222

22222

uDxCy

uBxAx

+=

+=&1

u 21uy =

2y

Figure 3.9: Series formation of two State Space Systems

This series transformation will combine the two state space systems into a single state

space system follow the equations (3.19) and (3.20).


xs =

x1

x2

=

A1 0

B2C1 A2

x1

x2

+

B1

B2D1

u (3.19)

y =[D2C1 C2

] x1

x2

+D2D1u (3.20)

Where xs is the new array of the derivative of state variables, u is the array of system

inputs from the state space systems and y is the output from the second state space

system. The derivation of (3.19) and (3.20) can be viewed in Appendix D.

The formation of two state space system in a feedback arrangement can be viewed in

Fig. 3.10.

State Space

System #1

11111

11111

uDxCy

uBxAx

+=

+=&

State Space

System #2

22222

22222

uDxCy

uBxAx

+=

+=&1

u 21uy =

2yr

Figure 3.10: Feedback Formation of two State Space Systems

This feedback transformation will combine the two state space systems into a singular

state space system. The feedback combination in matrix form is shown in equations (3.21)

and (3.22).

xf =

x1

x2

=

A1 −B1C2

B2C1 A2 −B2D1C2

x1

x2

+

B1

B2D1

r (3.21)

y =[

0 C2

] x1

x2

(3.22)

Where xf is the new array of the derivative of state variables, r is the input reference


from the first state space system and y is the output from the second state space system.

Please note that (3.21) and (3.22) are derived on the basis that the second system is

strictly proper, meaning D2 is zero. The derivation of (3.21) and (3.22) can be viewed in

Appendix D.

Using the series and feedback combination equations developed above, the five sections

of the linearized state space DC voltage control loop can be mathematically fastened

together. The combination of these state space systems are viewable in Appendix III.

The complete series of state space models as a combination for DC voltage control loop,

with resonant current control, can be viewed in (3.23) and (3.24).

ˆ xd

c1

ˆ xd

c2

ˆ xd

c3

ˆ xd1

ˆ xd2

ˆ xd3

ˆ xq1

ˆ xq2

ˆ xq3

ˆ i d ˆ i q ˆ vd

c

=

01

00

00

00

00

00

00

10

00

00

00

00

0−

4ω2 o

00

00

00

00

00

00

00

10

00

00

00

00

00

01

00

00

00

4K

Iω2 o

0K

I+

KR

0−

4ω2 o

00

00

−1

0−

KP

00

00

00

01

00

00

00

00

00

00

10

00

0−

2K

Rω

o0

00

00

−4

ω2 o

00

−1

0

4K

IK

iP

ω2 o

L0

KiP

(KI+

KR

)

L

2K

iR

ω2 o

L0

KiR

L0

KiR

ωo

L0

−R

L−

KiP

L+

ωo

−M

d2

L−

KP

KiP

L

0−

2K

RK

iP

ωo

L0

0−

KiR

ωo

L0

2K

iR

ω2 o

L0

KiR

L−

ωo

−R

L−

KiP

L−

Mq

2L

−6

KiP

KI

ω2 o

Id

Vd

cC

3K

iP

KR

ωo

Iq

Vd

cC

−3

KiP

(KI+

KR

)Id

2V

dc

C

−3

KiR

ω2 o

Id

Vd

cC

3K

iR

ωo

Iq

2V

dc

C

−3

KiR

Id

2V

dc

C

−3

KiR

ω2 o

Iq

Vd

cC

−3

KiR

ωo

Id

2V

dc

C

−3

KiR

Iq

2V

dc

C3 4C

(Md

+2

KiP

Id

Vd

c)

3 4C

(Mq

+2

KiP

Iq

Vd

c)

3K

iP

KP

Id

2V

dc

C

xd

c1

xd

c2

xd

c3

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vd

c

+

00

00

0

00

00

0

10

00

0

00

00

0

00

00

0

KP

00

00

00

00

0

00

00

0

01

00

0K

PK

iP

L0

1 L0

0

0K

iP

L0

1 L0

−3

KP

KiP

Id

2V

dc

C

−3

KiP

Iq

2V

dc

C0

0−

1 C

vr

ef

dc

ir

ef

q vg

d

vg

q

ilo

ad

(3.2

3)(3

.23)


y =[

0 0 0 0 0 0 0 0 0 0 0 1

]

xdc1

xdc2

xdc3

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vdc

+[

0 0 0 0 0

]

vrefdc

irefq

vgd

vgq

iload

(3.24)

3.3.3 Validation of Linearized Small Signal Model versus

Non-Linear Large Signal Model

In order to validate the small signal model of the DC voltage control loop, containing

dq-frame resonant current controller, a series of comparison tests were performed against

the large signal model in MATLAB/SIMULINK. The tests were performed by applying

a small signal perturbation to the large and small signal system models. The operating

point, or large signal values, were then added to the dynamic perturbation of the small

signal model. Once the MATLAB/SIMULINK tests were performed, a model was built

in PSCAD in order to compare with the MATLAB/SIMULINK results. This test was

performed to validate the assumptions made in the system plant modeling of Chapter 2.

The steady state operating point was calculated using the general equations of Chap-


ter 2, and the results and methods are listed below. To begin, a particular grid voltage

is chosen, along with a DC voltage, and DC side load current. The grid voltage is then

transferred to the dq-frame as shown below.

Vgabc =

Vg 6 0

Vg 6 120

Vg 6 240

(3.25)

Vgαβ =2

3· C · Vgabc (3.26)

Vgdq = Vgαβ · e−jωot (3.27)

The following equations will produce the d-axis AC current, Id, for the steady state

operating point. Note that the steady state value of the q-axis AC line current, Iq, is

chosen as zero for the following calculations, but is not required to be zero.

VdcIdc =3

2(VtdId + VtqIq) (3.28)

Vtd = −RId + ωLIq + Vgd (3.29)

Vtq = −RIq − ωLId + Vgq (3.30)

Substituting (3.29) and (3.30) into (3.28) will yield (3.31).

−3

2RI2

d +3

2IdVgd +

3

2(−RI2

q + VgqIq)− VdcIdc = 0 (3.31)

The above equation will produce two roots, one root is a believable value in the

normal range of acceptable currents. This acceptable root is chosen and the remaining


unbelievably large root is disregarded. Using (3.29) and (3.30) equations to produce the

dq-frame VSC terminal voltages, Vtd and Vtq, the d-axis and q-axis modulation indices,

Md and Mq, can be calculated as seen below.

Md =VtdVdc2

(3.32)

Mq =VtqVdc2

(3.33)

Using the equations listed above, the steady operating point of the VSC system was

calculated and tabulated below. Note that the values used are for a VSC system with

high voltage and current ratings as one of the proceeding applications will be for a VSC

based HVDC transmission network.


Table II : Steady State Operating Conditions

System Parameters Variable/Symbol Value

Line to Line Grid Voltage Vgll 83 kV

Converter Parameters Variable/Symbol Value

AC interface inductor L 12.22 mH

AC interface resistance R 0.23 Ω

DC link capacitor C 200 µF

Converter Ratings Variable/Symbol Value

kVA rating Sbase 150 MVA

AC voltage rating Vbase 83 kVl−l

AC current rating Ibase 1.04 kA

DC voltage rating Vdcrated 150 kV

Steady state operating Conditions Variable/Symbol Peak Value

D-axis grid voltage Vgd 67.77 kV

Q-axis grid voltage Vgq 0 V

D-axis modulation index Md 0.90

Q-axis modulation index Mq -0.095

D-axis AC line current Id 1.48 kA

Q-axis AC line current Iq 0 A

DC link current Idc 1 kA

DC link voltage Vdc 150 kV


Three step disturbance tests and two AC grid fault tests were performed, in order

to validate the small signal model developed previously. The dynamic behaviour of the

small signal model was compared graphically to the large signal model during these

disturbances using time-domain simulation performed in MATLAB/SIMULINK. The re-

sponse of the two system models were compared by viewing the DC link voltage, the

d-axis AC line current, and the q-axis AC line current.

Disturbance Comparison Test 1: DC Link Voltage Reference Step Response

A change in the DC link voltage reference, vrefdc , is made to the simulated systems at

the time of one second. The DC voltage reference was increased from 150 kV to 157.5 kV.

All operating conditions were extracted from Table II shown previously. The responses

of the large and small signal systems are given in Fig. 3.11.

From viewing Fig. 3.11, one can note very little deviation between the large and

small signal system responses. The small deviation between the two systems is due to

the slight inaccuracy of the small signal model when the system conditions are moved

from the predetermined steady state operating point. Due to the large overlapping of

the large and small signal system responses, this test validates the small signal model

developed previously.

Disturbance Comparison Test 2: DC Load Current Step Response

A change in the DC load current, iload, is made to the simulated systems at the time

of one second. The DC load current was increased from 1000 A to 1050 A. All operating

conditions were extracted from Table II shown previously. The responses of the large

and small signal systems are given in Fig. 3.12.

From viewing Fig. 3.12, one can note a much smaller deviation between the large

and small signal system responses as compared to Fig 3.11. The DC voltage reference


0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

150

155

160

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to DC Voltage Reference Step

Small Signal Model

Large Signal Model

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

1000

2000

3000

D-a

xis

Curr

ent (A

)

D-axis Current Response to DC Voltage Reference Step

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

-1000

0

1000

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to DC Voltage Reference Step

Figure 3.11: Responses to DC Voltage Reference Step Change

and DC load current disturbances were both a five percent increase from the steady state

operating value, but the nominal DC voltage is in the order of hundreds of thousands

while the DC load current is in the order of thousands. Therefore, the five percent change

in the DC voltage will have a much larger impact on the response of the given small signal

model. Due to the large overlapping of the large and small signal system responses, to

the DC load current step, this test validates the small signal model developed previously.

Disturbance Comparison Test 3: D-axis Grid Voltage Step Response

A change in the d-axis AC grid voltage, vgd, is made to the simulated systems at the

time of one second. The d-axis AC grid voltage was decreased from 67.77 kV to 64.38

kV for fifty milliseconds. After the fifty millisecond period, the d-axis AC grid voltage

is returned to 67.77 kV. All operating conditions were extracted from Table II shown

previously. The responses of the large and small signal systems are given in Fig. 3.13.


0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

149.6

149.8

150

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to DC Load Step

Small Signal Model

Large Signal Model

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

1500

1550

1600

D-a

xis

Curr

ent (A

)

D-axis Current Response to DC Load Step

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

-50

0

50

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to DC Load Step

Figure 3.12: Responses to DC Load Current Step Change

From viewing Fig. 3.13, one can note very little deviation between the large and

small signal system responses. The small deviation between the two systems is due to

the slight inaccuracy of the small signal model when the system conditions are moved

from the predetermined steady state operating point. Due to the large overlapping of

the large and small signal system responses, this test validates the small signal model

developed previously.

AC Grid Fault Comparison Test 1: Distant Three Phase Fault

A three phase fault was applied to the AC grid voltage source of the large and small

signal systems. The fault was applied at one second of the time-domain simulation and

lasted for fifty milliseconds. Following the fault, the AC grid voltage was returned to its

pre-fault value. The pre-fault, fault, and post-fault AC grid conditions can be viewed


0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

149

150

151

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to D-axis AC Grid Votlage Step

Small Signal Model

Large Signal Model

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

1400

1500

1600

1700

D-a

xis

Curr

ent (A

)

D-axis Current Response to D-axis AC Grid Votlage Step

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

-200

-100

0

100

200

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to D-axis AC Grid Votlage Step

Figure 3.13: Responses to D-axis AC Grid Voltage Step Change

in Table III shown below. All operating conditions were extracted from Table II shown


Table III: Grid Voltage Variation due to Three Phase Fault

Va Phasor Vb Phasor Vc Phasor

Pre-fault 47.926 0o kV 47.926 120o kV 47.926 240o kV

Fault 38.346 0o kV 38.346 120o kV 38.346 240o kV

Post-fault 47.926 0o kV 47.926 120o kV 47.926 240o kV

During the three phase fault a significant drop in grid voltage introduces a large

amount of inaccuracy in the linearized small signal model behaviour. Also, one can

note a deviation between the d-axis currents during the fault. Once again, the source of

these inaccuracies in the small signal model is due to the large deviation from the chosen


0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

145

150

155

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to Three Phase Fault

Small Signal Model

Large Signal Model

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

1000

1500

2000

2500

D-a

xis

Curr

ent (A

)

D-axis Current Response to Three Phase Fault

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08-1000

-500

0

500

1000

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to Three Phase Fault

Figure 3.14: Responses to 50 ms Three Phase Fault

steady state operating point. However, when the fault is cleared the recovery dynamics

of the small signal system are highly accurate. Due to the large overlapping of the large

and small signal system responses, during the recovery of the three phase fault, this test

validates the small signal model developed previously.

AC Grid Fault Comparison Test 2: Distant Line to Line Fault

A line to line phase fault was applied to the AC grid voltage source of the large and

small signal systems. The fault was applied at one second of the time-domain simulation

and lasted for fifty milliseconds. The line to line fault creates an asymmetry between the

three phases of the voltage source, thus creating a voltage source unbalance along with

harmonic ripple components. Following the fault, the AC grid voltage was returned to

its pre-fault value. The pre-fault, fault, and post-fault AC grid conditions can be viewed


in Table IV shown below. All operating conditions were extracted from Table II shown


Table IV: Grid Voltage Variation due to Line to Line Phase Fault

Va Phasor Vb Phasor Vc Phasor

Pre-fault 47.926 0o kV 47.926 120o kV 47.926 240o kV

Fault 38.346 0o kV 38.346 124.7o kV 38.346 235.3o kV

Post-fault 47.926 0o kV 47.926 120o kV 47.926 240o kV

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

145

150

155

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to D-axis Source Voltage Step

Small Signal Model

Large Signal Model

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

1000

1500

2000

2500

D-a

xis

Curr

ent (A

)

D-axis Current Response to D-axis Source Voltage Step

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08-1000

-500

0

500

1000

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to D-axis Source Voltage Step

Figure 3.15: Responses to 50 ms Line to Line Fault

During the line to line phase fault a significant drop in grid voltage introduces a

large amount of inaccuracy in the linearized small signal model behaviour. Also, one can

note a deviation between the d-axis currents during the fault. Once again, the source of

these inaccuracies in the small signal model is due to the large deviation from the chosen


steady state operating point. However, when the fault is cleared the recovery dynamics

of the small signal system are highly accurate. From the DC voltage curves of Fig. 3.15,

one can note that second order harmonic ripple, introduced by the source unbalance, is

nearly suppressed by the DCSV controller when the end of the fault is reached. Due to

the large overlapping of the large and small signal system responses, during the recovery

of the three phase fault, this test validates the small signal model developed previously.

3.3.4 Validation of Large Signal Non-Linear Model versus PSCAD

Model

A validation of the large signal model against a PSCAD model is required. A comparison

of the PSCAD model against the large signal model, utilizing time-domain simulation,

will validate the accuracy of the large signal model as well as the assumptions made

during the modeling process. The parameters and values used for the large signal and

PSCAD model correspond to those of Table II. The PSCAD model of the VSC system

uses a three phase IGBT bridge and does not utilize a simplified model of voltage and

current sources. The diagram of the PSCAD VSC system can be viewed in Fig. 3.16.

0.23 [ohm]

0.01222 [H]

0.01222 [H]0.23 [ohm]

0.01222 [H]

0.23 [ohm]

g2

g3

g4

g5

g6

200 [u

F]

R=0

2

6

2

4

2

5

2

3

2

2

g12

1

ia

ib

ic

A

B

C

V_dc BR

K2

ialpharef

ibetaref

B

-

D+

B

-

D+

ialphaf

ibetaf

N(s)

D(s)

Order = 2

*

*

0.16Kip

B

+

F

+

B

+

F

+

0.16Kip

B

+

D-

vtbetaunlim

vga

vgb

vgc

C+

E

+

X2

X

N

D

N/D ArcTan

*

Cos *

Sin

vtd

vtq

vtq

vtd

sin5

sin1

sin3

-1.0

-1.0

sin2

sin4

sin6

sin1

sin3

sin5

-1.0

*

*

*

45

61

23

1

2

3

4

5

6

1

2

3

4

5

6

sin1

sin2

sin4

sin5

sin6

sin3

RSgnOn

RSgnOff

TIME

Dblck

6

6

6

6

L

H

H

ON

OFF

L

(1)

(4)

(5)

(6)

(2)

(3)

g1

g2

g3

g4

g5

g61

2

3

4

5

6

1

2

3

4

5

6

**

53.0

rad2deg

180 by PiModulo

360.0

theta

TrgOn

TrgOff

TrgOn

TrgOff

N(s)

D(s)

Order = 2angle

alpha

beta

theta

d

q

theta

id ia

iq ib

theta ic

dq to abc

*

-1.0

theta

150 [o

hm

]

Kip=2Kir=400

0.0

A

B

C

D

Q

0

ia

ib

ic

ialpha

*

-1.0

ibeta

A

B

C

D

Q

0

vga

vgb

vgc

vgalpha

*

-1.0

vgbeta

ia id

ib iq

ic

theta

abc to dq

ia

ib

ic

theta

id

iq

-1.0

*

Va

Vb

Vc

PLLtheta

vga

vgb

vgc

theta

V_dc

V_dcerror

vtalphaunlim

vtbetaunlim

2nd order lowpass filter

zeta=0.8w=2*pi*1500

N(s)

D(s)

Order = 2V_dc V_dcf

N(s)

D(s)

Order = 2ialpha ialphaf

N(s)

D(s)

Order = 2ibeta ibetaf

ialpharef

ibetaref

Main : Graphs

0.980 0.990 1.000 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 ... ... ...

144.0

146.0

148.0

150.0

152.0

154.0

156.0

158.0

y

V_dc

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

V_dcf

0.00

0.50

1.00

1.50

2.00

2.50

3.00

y

id

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

y

iq

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

ialpha ibeta

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

vtalphaunlim vtbetaunlim

vtd

vtq

Main : Graphs

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 ... ... ...

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

ialphaf ialpha ialpharef

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

ibetaf ibeta ibetaref

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

vtd vtq

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

y

vtdunlim test

test

BR

K3

BRK5

BRK5

BRK5

Figure 3.16: Model of VSC system for PSCAD simulations

The DC side load, contained within the PSCAD system, is chosen to be a resistor

rather than a current source. This was due to simulator limitations. With regards to


the PSCAD simulation software, the time-domain simulation will not run with a current

source as a DC side load. Therefore, a resistor was used in the place of the previously

modeled current source. This alteration to the large signal system model can be viewed

in Fig. 3.17.

AC to DC REAL

POWER EQUATIONref

dqi

ref

dcv

ref

iαβdc

i

αβi

dcv

dcv

Modulator

Demodulator

mαβ

uuuur

3( )


Cs

1Inner AC current

loop : αβ-frame

resonant control

DCloadR

1

dcv

Change in DC

load modeling

tjoe

ω

DC

SVC

Figure 3.17: DC space vector control loop with modification to DC side load

Three tests were performed to validate the large signal mode: a DC voltage reference

step, a DC load step, and a distant three phase fault applied to AC grid voltage source.

Validation Test 1: DC Link Voltage Reference Step Response

A change in the DC link voltage reference, vrefdc , is made to the simulated systems at

the time of one second. The DC voltage reference was increased from 150 kV to 155 kV.

The responses of the large signal system and the PSCAD system are given in Fig. 3.18.

From viewing Fig. 3.18, one can note some deviation between the large signal system

and PSCAD system responses. This deviation between the two systems is due to the

inclusion of second order feedback filters for the DC link voltage signal, d-axis AC line

current, and the q-axis AC line current of the PSCAD system. All the filters are anti-

aliasing filters with high bandwidth to ensure the stable steady state operation of the

PSCAD model, however due to the high gain controllers used, the effect of these filters

is not entirely negligible. These filters were not included in the large signal model of

the VSC model due to the large addition of complexity they would add during small


0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

150

152

154

156

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to DC Voltage Reference Step

PSCAD Model

Large Signal Model

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

1000

2000

3000

D-a

xis

Curr

ent (A

)

D-axis Current Response to DC Voltage Reference Step

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

-1000

0

1000

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to DC Voltage Reference Step

Figure 3.18: Responses to DC Voltage Reference Step Change

signal and state space modeling. The final result of the state space small signal model

would become more convoluted, therefore increasing the difficulty of correlating dominant

system parameters and conditions to system states. Due to close overlapping of the large

signal and PSCAD system responses, this test validates the large signal model developed

previously.

Validation Test 2: DC Load Current Step Response

A change in the DC load current, iload, is made to the simulated systems at the time

of one second. The DC load current was increased from 1000 A to 1200 A. All operating

conditions were extracted from Table II shown previously. The responses of the large

signal system and the PSCAD system are given in Fig. 3.19.

From viewing Fig. 3.19, one can note a similar deviation between the large signal and

PSCAD system responses as in the previous DC link voltage reference step simulation.


0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04148.5

149

149.5

150

150.5

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to DC Load Step

PSCAD Model

Large Signal Model

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04

1400

1600

1800

2000

2200

D-a

xis

Curr

ent (A

)

D-axis Current Response to DC Load Step

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04-500

0

500

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to DC Load Step

Figure 3.19: Responses to DC Load Current Step Change

This deviation is once again caused by the absence of the second order feedback filters in

the large signal system model. This test validates the large signal model developed due

to the accuracy of its response during the DC load change simulation.

Validation Test 3: Distant Three Phase Fault

A three phase fault was applied to the AC grid voltage source of the large and small

signal systems. The fault was applied at one second of the time-domain simulation and

lasted for fifty milliseconds. Following the fifty millisecond fault, the AC grid voltage was

returned to its pre-fault value. The pre-fault, fault, and post-fault AC grid conditions

were taken from Table III. All operating conditions were extracted from Table II shown


During the three phase fault simulation there is high accuracy between the two system

responses for pre-fault, fault, and post-fault dynamics. Once again the slight inaccuracy


0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

145

150

155

DC

Lin

k V

oltage (

kV

)

DC Voltage Response to Three Phase Fault

PSCAD Model

Large Signal Model

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

1000

1500

2000

2500

D-a

xis

Curr

ent (A

)

D-axis Current Response to Three Phase Fault

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08-1000

-500

0

500

1000

Q-a

xis

Curr

ent (A

)

Time (s)

Q-axis Current Response to Three Phase Fault

Figure 3.20: Responses to 50 ms Three Phase Fault

of the large signal response is due to the absence of the second order feedback filters.

Due to the large overlapping of the large signal and PSCAD system responses, during

the three phase fault, this test validates the large signal model developed previously.

In the three phase IGBT bridge section of Chapter 2 an assumption was made that

the switching harmonics of the converter can be neglected in modeling the large signal

VSC system. As seen from Fig. 3.18, Fig. 3.19, and Fig. 3.20, the PSCAD system

response is quite similar to the large signal system response. The PSCAD system only

differs in shape due to the addition of the switching noise to its signal response. From

the above validation results, the assumption to neglect the switching harmonics in the

large signal system modeling is therefore acceptable.


3.4 Chapter Summary

This chapter developed the linearized model of the resonant αβ-frame current control.

The linearized model was transformed into its state space model. This state space model

was validated versus the large signal model using time-domain simulations. The large

signal model was then tested versus a PSCAD model to validate assumptions made in

Chapter 2.

Chapter 4

Applications of Linearized State

Space VSC System Models


This chapter will display some of the possible applications of state space modeling of

the linearized resonant current control. The impedance, from the DC side of the VSC,

will be discussed for two types of control. Also, an eigenvalue analysis will be performed

with respect to several VSC based systems. This investigation will display the necessity

of computing all system eigenvalues and their dependence on the chosen steady state

operating point.

4.2 DC Side Impedance of VSC Based Systems

Developers of electric drives, VSC based HVDC systems or wind turbine systems with

back-to-back converter interface, may be interested in the DC-side input/output impedance

of a converter, as this impedance can be used for investigating DC side interactions. This

application investigates the DC-side output impedance of the VSC when it is operated

as a unity power factor active rectifier.

55

Chapter 4. Applications of Linearized State Space VSC System Models56

The transfer function, Zdc(s), from the DC side load current, iload, to the DC side

voltage, vdc, gives the DC-side output impedance of the converter. This may be found

from the linearized model from:

Zdc(s) =Vdc(s)

Idc(s)= CM(sI − A)−1BM (4.1)

Where, A is taken from the linearized state space model, CM is a modified version of

the C matrix from the linearized state space model, and BM is a modified version of the

B matrix from the linearized state space model.

With respect to a VSC system that implements DC PI voltage control and αβ-frame

resonant current control, the corresponding CM and BM are shown in equations (4.2)

and (4.3).

CM =[

0 0 0 0 0 0 0 0 0 1

](4.2)

BM =[

0 0 0 0 0 0 0 0 0 − 1C

]T(4.3)

Using successive time domain simulations, the small signal impedance, both the mag-

nitude and the corresponding angle, is calculated and plotted in Fig. 4.1 for the DC PI

voltage controlled VSC. The impedance calculation is verified at five discrete frequencies,

as shown by the ’x’ markers in Fig. 4.1. This verification is done utilizing the large signal

model in MATLAB/Simulink.

The linearized results show excellent accuracy up to several hundred hertz.

Next, the DCSV controlled state space model analyzed. With respect to a VSC

system that implements DCSV voltage control and αβ-frame resonant current control,

the corresponding CM and BM are shown in equations (4.4) and (4.5).

CM =[

0 0 0 0 0 0 0 0 0 0 0 1

](4.4)


-200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

10

12

14

Frequency (Hz)

Magnitude o

f Im

pedance

Magnitude of Output DC Impedance

-200 -150 -100 -50 0 50 100 150 200-200

-100

0

100

200

Frequency (Hz)

Phase o

f Im

pedance (

Degre

es)

Phase of Output DC Impedance

Figure 4.1: Small Signal DC impedance of VSC: resonant current control, PI DC voltagecontrol

BM =[

0 0 0 0 0 0 0 0 0 0 0 − 1C

]T(4.5)

Using successive time domain simulations, the small signal impedance, both the mag-

nitude and the corresponding angle, is calculated and plotted in Fig. 4.2 of the DCSV

voltage controlled VSC. The impedance calculation is verified at five discrete frequencies,

as shown by the ’x’ markers in Fig. 4.2.

Once again, the linearized results show excellent accuracy up to several hundred hertz.

With respect to Fig. 4.1 one can note that the magnitude of the output impedance of

the converter is a zero value when the frequency of the DC side signals are zero as well.

This is due to the nature of the PI DC voltage control and its ability to perform perfect

DC voltage regulation. On the other hand, Fig. 4.2 has a zero magnitude at frequencies

of zero, one hundred and twenty, and minus one hundred and twenty hertz. This is due


-200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

10

12

Frequency (Hz)

Magnitude o

f Im

pedance

Magnitude of Output DC Impedance

-200 -150 -100 -50 0 50 100 150 200-200

-100

0

100

200

Frequency (Hz)

Phase o

f Im

pedance (

Degre

es)

Phase of Output DC Impedance

Figure 4.2: Small Signal DC impedance of VSC: resonant current control, DCSV voltagecontrol

to the nature of the DCSV voltage control and its resonance to DC signals and sinusoids

at a frequency of two times the line frequency. The zero value of the DC side impedance,

at one hundred and twenty hertz, correlates to a value of zero for vdc. This proves the

zero steady state error of the DCSV under unbalanced AC voltage source conditions,

causing harmonic ripple on the DC side voltage. While even a standard DC voltage

resonant controller, such as the control strategy of [9], could likely achieve a similar zero

impedance at +/− one hundred and twenty hertz, it would do so at the expense of third

harmonic current injection into the AC grid.

4.3 Eigenvalue Analysis of VSC Based Systems

In this section a series of eigenvalue tests will be performed to determine the VSC system

stability. Three VSC based systems will be analyzed under bilateral power flow in each


case. The three VSC based systems will be the single minimal setup of a VSC, a back

to back VSC based HVDC transmission system, and a VSC based HVDC transmission

system with a DC cable.

4.3.1 Single VSC System

Eigenvalue analysis of the VSC based system will provide the exponents associated with

time-domain solution of the system state variables. A diagram of the system used can

be found in Fig. 4.3.

LR

+−G

v

VSCi

tv

C

−

+

dcv

capi

dci

−+R

v

PCC

loadi

Direction of Positive Power Flow

Direction of Negative Power Flow

Figure 4.3: Minimal setup of single VSC system

Eigenvalue analysis is possible with the development of a linearized small signal model.

Following traditional control and mathematical theory, a complete set of negative eigen-

values denotes the stable operation of the VSC system pertaining to the particular op-

erating condition used in the derivation. One or more positive eigenvalues denote an

instability within the VSC based system pertaining to the given operating condition.

In all following analyses, the eigenvalues were calculated utilizing MATLAB and the

following calculations with the A matrix from the state space model.

Ax = λx (4.6)


Where λ is the matrix of system eigenvalues, A is the matrix associated with the state

space model, and x is the array of state variables.

det(λI − A) = 0 (4.7)

An eigenvalue analysis was performed using the steady state operating conditions of

Table II in Chapter 3. The power flow in this case is into the DC side load and the VSC

is operating as a demodulator. In this case, the power flow will be considered as positive

when the current is flowing from the AC grid into the VSC. The eigenvalues of the VSC

based system can be viewed below in Fig. 4.4 and a magnified view about the imaginary

axis of the same plot can be viewed in Fig. 4.5.

-900 -800 -700 -600 -500 -400 -300 -200 -100 0 100-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Figure 4.4: Eigenvalue plot of VSC based system with positive power flow

With reference to Fig. 4.5, there are two eigenvalues which appear on the imaginary

axis at values of ∓754 radians per second. These two eigenvalues relate to dq-frame

equivalent resonant current controller. In the αβ-frame, the resonant current control has


-25 -20 -15 -10 -5 0 5

-800

-600

-400

-200

0

200

400

600

800

Figure 4.5: Closeup eigenvalue plot of VSC based system with positive power flow

two poles and two eigenvalues at ∓377 radians per seconds. This is due to its resonant

property at the frequencies of ωo and −ωo, where ωo is equal to the product of two times

π and the system frequency of sixty hertz. The two marginally stable eigenvalues of the

dq-frame equivalent resonant current controller do not introduce instability to the VSC

system. This can be concluded due to the stable operation of the same VSC system in

the time-domain simulations of Chapter 3.

The following eigenvalue plots relate to a reversal of the power flow from the previous

system values. This refers to a negative power or current flow within the VSC system.

Table I displays to the steady state operating conditions that coincide to negative power

flow.


Table I: Steady State Operating Conditions for a Negative Power Flow


Line to Line Grid Voltage Vgll 83 kV

Converter Parameters Variable/Symbol Value

AC interface inductor L 12.22 mH

AC interface resistance R 0.23 Ω

DC link capacitor C 200 µF

Converter Ratings Variable/Symbol Value

kVA rating Sbase 150 MVA

AC voltage rating Vbase 83 kVl−l

AC current rating Ibase -1.04 kA

DC voltage rating Vdcrated 150 kV

Steady state operating Conditions Variable/Symbol Peak Value

D-axis grid voltage Vgd 67.77 kV

Q-axis grid voltage Vgq 0 V

D-axis modulation index Md 0.908

Q-axis modulation index Mq 0.090

D-axis AC line current Id -1.48 kA

Q-axis AC line current Iq 0 A

DC link current Idc -1 kA

DC link voltage Vdc 150 kV

The eigenvalue plots displaying the eigenvalues relating to the power flow reversal are

displayed in Fig. 4.6 and Fig. 4.7.


-800 -700 -600 -500 -400 -300 -200 -100 0-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Figure 4.6: Eigenvalue plot of VSC based system with negative power flow

All eigenvalues of Fig. 4.6 and Fig. 4.7 are contained within the real left half plane

and therefore relate to a stable operation of the VSC system under negative power flow.

This result coincides with [20], which states that a VSC has complete bidirectional power

flow.

4.3.2 VSC based Back-to-Back HVDC System

The eigenvalues of the back-to-back HVDC converter system will be analyzed under a

positive and negative power flow. A diagram of the back to back HVDC system is shown

below in Fig. 4.8.

With reference to Fig. 4.8, VSC one operates as the DCSV voltage controller and

VSC two operates as a current controller. The control and system plant model of the

current controlled VSC system can be viewed in Fig. 4.9.

The small signal linearized state space model of the current controlled is developed

is shown in (4.8).

-20 -15 -10 -5 0 5-800

-600

-400

-200

0

200

400

600

Figure 4.7: Closeup eigenvalue plot of VSC based system with negative power flow

VSC 1:

DC

Voltage

Control

VSC 2 :

Current

Control

L1R1

+−1G

v

1i

1tv

R2

2i

2tv

L2 2Gv

−

+

dcv

1dci

2dci



Figure 4.8: VSC based back-to-back HVDC transmission system



di$

gdv$

ref

qi$

ref

di$

tdv$

tqv$

gqv$

qi$

oLω

oLω

RLs +

1

RLs +

1

dε

qε

dm

qm

dcv

2

dcV

2

dM

2 24

o

iR

o

Ks

ω

ω⋅

+

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

iPK

2 2

2 2

( 2 )

( 4 )

o

iR

o

sK

s s

ω

ω

+⋅

+

2 24

o

iR

o

Ks

ω

ω⋅

+

iPK

dcv

2

dcV

2

qM

dcV

2−

dcV

2−

Figure 4.9: Small Signal Current Control Model and AC Line Reactor Plant


x=

01

00

00

00

00

10

00

00

0−

4ω

2 o0

00

0−

10

00

00

10

00

00

00

01

00

00

00

−4ω

2 o0

0−

1

2K

iRω

2 oL

0K

iRL

0K

iRω

oL

0−R L−K

iPL

+ωo

0−K

iRω

oL

02K

iRω

2 oL

0K

iRL

−ωo

−R L−K

iPL

xd1

xd2

xd3

xq1

xq2

xq3

i d i q

+

00

00

0

00

00

0

10

00

0

00

00

0

00

00

0

01

00

0

KiPL

01 L

0−M

d2L

0K

iPL

01 L

−M

q

2L

iref

d iref

q vgd

vgq

vdc

(4

.8)


The state space model of VSC system 1 is that of the DCSV control developed in

Chapter 3. Note, that the DC side capacitor dynamics are attached to the DC voltage

controlled VSC.

Due to the state space modeling of the linearized VSC system, the two state space

equations governing the individual VSC systems can be easily grouped together. Please

note that in system 2 vdc is a system input while, in system 1 vdc is a state variable.

Therefore, a modified matrix Um1 is developed to transfer vdc’s associated coefficients

from the total system’s B matrix to the total system’s A matrix, with respect to system

2. Equation (4.9) shown below, displays the connection matrix for transforming the

individual VSC models into a grouped system model.

x1

x2

=

A1 0

Um1 A2

x1

x2

+

B1 0

0 B2

u1

u2

(4.9)

Where the subscripts one and two denote VSC systems one and two and u1 and u2

are the inputs to VSC systems one and two.

Also where,

Um1 =

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 −Md

2L

0 0 0 0 0 0 0 0 0 0 0 −Mq

2L

(4.10)

Equation (4.11) displays the steady state condition relation between the systems’ DC

side currents.


Idc1 = −Idc2 (4.11)

An eigenvalue analysis was performed to show system stability under bidirectional

power flow. The first analysis will be performed for positive power flow. Positive power

flow, with reference to Fig. 4.8, relates to a current flow direction from left to right.

An eigenvalue analysis for positive flow was performed using the steady state operating

conditions of Table II in Chapter 3 for VSC system one and the steady state operating

conditions of Table I of this chapter for VSC system two. The eigenvalue plot is displayed

in Fig. 4.10.

-1000 -800 -600 -400 -200 0 200-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Figure 4.10: Eigenvalue Plot of Back-to-Back HVDC System with Positive Power Flow

With all eigenvalues contained within the left half plane, the back-to-back HVDC

system is stable under positive power flow.

The second analysis will be performed for negative power flow. Negative power flow,

with reference to Fig. 4.8, relates to a current flow from right to left. An eigenvalue anal-

ysis for negative power flow was performed using the steady state operating conditions

of Table II in Chapter 3 for VSC system two and the steady state operating conditions


-20 -15 -10 -5 0 5-800

-600

-400

-200

0

200

400

600

Figure 4.11: Close up Eigenvalue Plot of Back-to-Back HVDC System with PositivePower Flow

of Table I of this chapter for VSC system one. The eigenvalue plot is displayed in Fig.

4.12.

With all eigenvalues contained with the left half plane, the back-to-back HVDC trans-

mission system is stable under negative power flow. With both eigenvalue plot displaying

stable values the back to back HVDC transmission system is stable for nominal operating

conditions for bidirectional power flow, which validates the findings of [11], [12], [13], and

[14].

4.3.3 VSC Based HVDC Transmission System

The VSC based HVDC transmission system will have the same structure as the back

to back HVDC system but, with the inclusion of a DC cable to transmit power. The

HVDC transmission system is shown in Fig. 4.14. One VSC will act as a DCSV voltage

controller, while the other operates as a current controller.

The VSC model of the DCSV voltage controller will remain the same as in the previous

section. The VSC model of the current controller must be modified slightly to account


-1000 -800 -600 -400 -200 0 200-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Figure 4.12: Eigenvalue Plot of Back-to-Back HVDC System with Negative Power Flow

for the DC cable. The control and plant model of the modified current controlled VSC

can be viewed in Fig. 4.15.

Note that the current controlled VSC model must include the AC to DC power

equation and DC side capacitor dynamics, since the DC sides of the two VSC’s are

now separated by a particular length of DC cable. Thus, the current controlled VSC will

require a DC side capacitor for stable control and the system model will therefore include

the modeling of the power conversion equation and DC side capacitor. The linearized

state space model of the current controlled VSC is displayed in (4.12).

-20 -15 -10 -5 0 5 10-800

-600

-400

-200

0

200

400

600

Figure 4.13: Close up Eigenvalue Plot of Back-to-Back HVDC System with NegativePower Flow

VSC 1:

DC

Voltage

Control

L1R1

+−1G

v

1i

1tv

−

+

dcv

2dci

VSC 2 :

Current

Control

R2

2i

2tv

L22G

v

1dci LDC

RDCLDC

RDC

DCC

2C1

C

−

+

1dcv

−

+

2dcv

T-Model of HVDC Cable



Figure 4.14: VSC based HVDC transmission system with DC cable


ref

dqi

ref

di

ref

qij ˆ⋅

dci

loadi

dcv

dcv

Linearized

Current

Control Loop

dq-frameCs

1

di

qi

dm

qm

dM

qM

dI

qI

Σ4

3

AC to DC Power Equation

Figure 4.15: Small Signal Current Control Model with VSC System Plants


xs1

=

[ x1

x2

] =

01

00

00

00

0

00

10

00

00

0

0−

4ω

2 o0

00

0−

10

0

00

00

10

00

0

00

00

01

00

0

00

00

−4ω

2 o0

0−

10

2K

iRω

2 oL

0K

iRL

0K

iRω

oL

0−R L−K

iPL

+ωo

−M

d2L

0−K

iRω

oL

02K

iRω

2 oL

0K

iRL

−ωo

−R L−K

iPL

−M

q

2L

−3K

iRω

2 oId

Vd

cC

3K

iRω

oIq

2V

dcC

−3K

iRId

2V

dcC

−3K

iRω

2 oIq

Vd

cC

−3K

iRω

oId

2V

dcC

−3K

iRIq

2V

dcC

3 4C

(Md

+2K

iPId

Vd

c)

3 4C

(Mq

+2K

iPIq

Vd

c)

0

xd1

xd2

xd3

xq1

xq2

xq3

i d i q vdc

+(4

.12)

00

00

0

00

00

0

10

00

0

00

00

0

00

00

0

01

00

0

KiPL

01 L

00

0K

iPL

01 L

0

−3K

iPId

2V

dcC

−3K

iPIq

2V

dcC

00

−1C

iref

d iref

q vgd

vgq

i load


The DC cable parameters are taken from [21] and are calculated given a fifty kilometer

length of DC cable. The DC cable parameters can be found in Table II.

Table II: DC Cable Parameters


DC Cable Inductance LDC 8.5mH

DC Cable Resistance RDC 1.2Ω

DC Cable Capacitance CDC 11.5mF

Before an eigenvalue analysis can be performed, the two VSC system models must

be gathered together along with the system model of the HVDC cable. The simplified

model of the DC cable can be viewed in Fig. 4.16, where idc1, idc2, and vdc are system

states and vdc1 and vdc2 are system inputs. The derivation of the DC cable equations can

be viewed in (4.13), (4.14), and (4.15). The state space model of the DC cable can be

viewed in equations (4.16) and (4.17).

−

+

dcv

2dci

1dci LDC

RDC

LDCRDC

DCC

−

+

2dcv

−

+

1dcv

Figure 4.16: T-model of DC cable

d

dtidc1 =

1

LDCvdc1 −

RDC

LDCidc1 −

1

LDCvdc (4.13)

d

dtidc2 =

1

LDCvdc2 −

RDC

LDCidc2 −

1

LDCvdc (4.14)


d

dtvdc =

1

CDCidc1 +

1

CDCidc2 (4.15)

d

dt

idc1

idc2

vdc

=

−RDCLDC

0 − 1LDC

0 −RDCLDC

− 1LDC

1CDC

1CDC

0

idc1

idc2

vdc

+

1

LDC0

0 1LDC

0 0

vdc1

vdc2

(4.16)

y =

idc1

idc2

=

1 0 0

0 1 0

idc1

idc2

vdc

(4.17)

With the formulation of the DC cable’s state space model, a complete state space

model of the two VSC’s along with the DC cable can be achieved. A list of system inputs

and system states can be viewed in Table III for each VSC model as well as the DC cable.


Table III: HVDC System Inputs and State Variables

VSC 1: DC Voltage Controlled VSC 2: Current Controlled DC Cable

Inputs vrefdc irefd vdc1

irefq1 irefq2 vdc2

vgd1 vgd2

vgq1 vgq2

idc1 idc2

States xdc xd4 idc1

xd1 xd5 idc2

xd2 xd6 vdc

xd3 xq4

xq1 xq5

xq2 xq6

xq3 id2

id1 iq2

iq1 vdc2

vdc1

Note the variables in bold font contained within Table III. These emphasized variables

denote the overlap of a particular system’s input or state and another systems input or

state. As in the previous subsection, some overlapping terms arise and appear in the

development of total state space equation required for eigenvalue analysis. This total

state space equation can be viewed in (4.18).

x1

x2

x3

=

A1 0 Um1

0 A2 Um2

Um31 Um32 A3

x1

x2

x3

+

B1 0

0 B2

u1

u2

(4.18)

Where, subscripts 1 relate to the DCSV controlled VSC system 1 model, subscripts


2 relate to the resonant current controlled VSC system 2 model, and the subscripts 3

relate to the DC cable model.

And,

Um1 =

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

− 1C1

0 0

(4.19)

Um2 =

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 − 1C2

0

(4.20)


Um31 =

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1LDC

(4.21)

Um32 =

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1LDC

(4.22)

With the collection of the entire HVDC transmission system A matrix, an eigenvalue

analysis was performed to show system stability under bidirectional power flow. The

first analysis will be performed for positive power flow. Where positive power flow, with

reference to Fig. 4.14, relates to a current flow direction from left to right. An eigenvalue

analysis for positive power flow was performed using the steady state operating conditions

of Table II in Chapter 3 for VSC system one and the steady state operating conditions

of Table I of this chapter for VSC system two. The eigenvalue plot is displayed in Fig.

4.17.

-1000 -800 -600 -400 -200 0 200-1500

-1000

-500

0

500

1000

1500

Figure 4.17: Eigenvalue Plot of HVDC Transmission System with Positive Power Flow


-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5-800

-600

-400

-200

0

200

400

600

800

Figure 4.18: Close up Eigenvalue Plot of HVDC Transmission System with PositivePower Flow

From viewing Fig. 4.17 it is apparent that the system is unstable under the given

operating conditions due to the appearance of two eigenvalues in the right half plane. This

result is significant as all analyses up to this point have been stable under bidirectional

power flow. Therefore, in analysis of VSC based systems it is necessary to evaluate not

only the individual VSC systems, but the entire interconnected system accounting for

the transmission line dynamics. In the previous section, the back-to-back configuration

of the HVDC system was stable due to lack of a DC cable between the VSC’s. In this

case, the effects of the linking terms between states caused by the DC cable introduces

instability.

The second analysis will be performed for negative power flow. Negative power flow,

with reference to Fig. 4.14, relates to a current flow direction from right to left. An eigen-

value analysis for negative power flow was performed using the steady state operating

conditions of Table II in Chapter 3 for VSC system two and the steady state operating

conditions of Table I of this chapter for VSC system one. The eigenvalue plot is displayed

in Fig. 4.19.


-1000 -800 -600 -400 -200 0 200-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Figure 4.19: Eigenvalue Plot of HVDC Transmission System with Negative Power Flow

In this case all eigenvalues are contained within the left half plane, denoting stable

operation of the VSC system under the given operating conditions.

Therefore, when operating an HVDC system with this particular type of control

scheme, the sending end must operate as a current controller and the receiving end must

operate as a DC voltage controller. Thus, for this type of operation, the control duties

of the VSC’s must be swapped when a power flow reversal is required. This result does

not entirely coincide with [20], which states that VSC based HVDC transmission systems

with a DC cable interface have complete bidirectional power flow capabilities. Indeed

the VSC’s allow for bidirectional current flow, but stability of the system must also be

considered. This case study shows that not all VSC based HVDC systems have complete

bidirectional power flow. Therefore, a blanket statement that all VSC systems have

complete directional power flow should not be used.


-50 -40 -30 -20 -10 0-800

-600

-400

-200

0

200

400

600

800

Figure 4.20: Close up Eigenvalue Plot of HVDC Transmission System with NegativePower Flow

4.4 Chapter Summary

This chapter investigated the DC side impedance of the DC PI voltage controlled and the

DCSV voltage controlled VSC systems. Eigenvalue analyses were performed to display

VSC based system stability under a particular set of operating conditions. The eigen-

value analysis of the VSC based HVDC transmission system displayed the importance

of collecting and viewing all system eigenvalues in order to determine complete system

stability.

Chapter 5

Conclusions

5.1 Summary

The objectives of this thesis were to develop state space models for converter systems

that employ emerging αβ-frame control techniques. Modeling of the αβ-frame resonant

current controller was the main focus of this work, along with its complementary DC space

vector controller. A linearized model of the resonant current controller was developed

and validated using time-domain simulations. Also, the large signal time averaged model

of the DCSV DC voltage controller coupled with the αβ-frame resonant current control

was validated versus the same control implementation in PSCAD. These simulations

validated the assumptions made when mathematically modeling the VSC system plants.

The small signal linearized model was then shaped into its state space equivalent. With

the state space model in hand eigenvalue plots and DC output impedance plots were

constructed and discussed. The state space models of several other control strategies

were produced and are viewable in appendix C.

82

Chapter 5. Conclusions 83

5.2 Contributions

The contributions of this work can be summarized as follows:

• With linearized small signal models of several control strategies, developers of power

transmission systems, electric drives, and renewable power production systems can

integrate their respective state space system models into the linearized state space

models of the VSC systems in this work

• For the applications listed above, all standard linear control design techniques can

be performed to better understand system stability, parameterizations, and state

contribution factors

• With respect to the VSC based HVDC transmission system, system stability tests

of Chapter 4, it was shown that the interface between VSC’s must not be overlooked

• The linearized model of the VSC system with DC space vector control will allow

for the design of HVDC transmission systems that can isolate harmonic ripple

disturbances to their respective AC sides with minimal propagation of harmonic

ripple to DC side loads

• The stability of the DCSV controlled VSC based HVDC systems can be investigated

under changes in system parameters and varying steady state operating conditions.

Appendix A

State Space Modeling of Controllers

System 1: PI DC Voltage Controller

irefdεvdc

=Y (s)

U(s)= KP +

KI

s(A.1)

Y (s) = KP · U(s) +KI

s· U(s) = KP · U(s) + Z(s) · U(s) = KP · U(s) + Y (s) (A.2)

Y (s)

U(s)=KI

s(A.3)

˙Y (s) = KI · U(s) (A.4)

y is chosen as state variable xdc, therefore

xdc = [0] · xdc + [KI ] · u (A.5)

Y (s) = KP · U(s) + Y (s) = KP · U(s) +Xdc(s) (A.6)

84

Appendix A. State Space Modeling of Controllers 85

y = irefd = [1] · xdc + [KP ] · u (A.7)

System 2 : Space Vector DC Voltage Control

Real Part

irefdεvdc

=Y1(s)

U(s)= KP +

KI

s+KR ·

s

s2 + (2ωo)2(A.8)

Y1(s)

U(s)=KP s(s

2 + (2ωo)2) +KI(s

2 + (2ωo)2) +KRs

2

s(s2 + (2ωo)2)(A.9)

Y1(s) = KP · U(s) + Z(s) · U(s) = KP · U(s) + Y1(s) (A.10)

Y1(s) = [(KI +KR)s2 +KI(2ωo)

2

s(s2 + (2ωo)2)] · U(s) (A.11)

Q(s)

U(s)=

1

s(s2 + (2ωo)2)=

1

s3 + (2ωo)2s(A.12)

Y1(s)

Q(s)= (KI +KR)s2 +KI(2ωo)

2 (A.13)

From equation (A.12),

q(3) + (2ωo)2q = u (A.14)

Choosing state variables,

xdc1 = q (A.15)


xdc2 = q, xdc1 = xdc2

xdc3 = q, xdc2 = xdc3

xdc3 = −(2ωo)2xdc2 + u (A.16)

Therefore,

xdc1

xdc2

xdc3

=

0 1 0

0 0 1

0 −(2ωo)2 0

xdc1

xdc2

xdc3

+

0

0

1

u (A.17)

Y (s) = KP · U(s) + Y1(s) = KP · U(s) +Q(s)[(KR +KI)s2 +KI(2ωo)

2] (A.18)

y = KPu+ (KI +KR)q +KI(2ωo)2q = (KI +KR)xdc3 +KI(2ωo)

2xdc1 +KPu (A.19)

Therefore,

y =[KI(2ωo)

2 0 (KI +KR)

]xdc1

xdc2

xdc3

+ [KP ]u (A.20)

Imaginary Part

irefqεvdc

=Y2(s)

U(s)= KR ·

−2ωos2 + (2ω2

o)(A.21)


y2 + (2ωo)2y2 = −2KRωou (A.22)

y2 = −(2ωo)2y2 − 2KRωou (A.23)

Choosing state variables,

y2 = x1 (A.24)

x2 = x1 = y2

x2 = y2

Therefore,

x1

x2

=

0 1

−(2ωo)2 0

x1

x2

+

0

−2KRωo

u (A.25)

y2 =[

1 0

] x1

x2

+ [0]u (A.26)

Combining Real and Imaginary Parts,

xdc1

xdc2

xdc3

=

0 1 0

0 0 1

0 −(2ωo)2 0

xdc1

xdc2

xdc3

+

0

0

1

u (A.27)

y =

irefd

irefq

=

KI(2ωo)2 0 (KI +KR)

0 −2KRωo 0

xdc1

xdc2

xdc3

+

KP

0

u (A.28)


System 3: Dq-frame Equivalent of αβ-frame resonant current controller

Real error to Real output state variable definition,

Y11(s)

U1(s)=vdεd

= KiP +KiR ·s2 + 2ω2

o

s2(s2 + 4ω2o)

(A.29)

Y11(s) = [KiP +KiR ·(s2 + 2ω2

o)

s3 + 4ω2os

] · U1(s) (A.30)

Y11(s)

U1(s)= KiR ·

(s2 + 2ω2o)

s3 + 4ω2os

] · U1(s) (A.31)

Q1(s)

U1(s)=

1

s3 + 4ω2os

(A.32)

Y11(s)

Q1(s)= KiR(s2 + 2ω2

o) (A.33)

x1 = q1 (A.34)

x2 = q1, x1 = x2 (A.35)

x3 = x2, x3 = q1 (A.36)

x3 = q(3)1 (A.37)

(A.38)

q(3)1 + (4ω2o) · q1 = u1 (A.39)

q(3)1 = −(4ω2o) · q1 + u1 (A.40)


x1

x2

x3

=

0 1 0

0 0 1

0 −(2ωo)2 0

x1

x2

x3

+

0

0

1

u1 (A.41)

Y11(s) = Y11(s) +KiPU1(s) = KiPU1(s) +Q1(s)[KiR(s2 + 2ω2o)] (A.42)

y11 = q1KiR + q1 · 2KiRω2o +KiPu1 (A.43)

[2KiRω

2o 0 KiR

]x1

x2

x3

+ [1]u1 (A.44)

Imaginary error to Real output state variable definition,

Y12(s)

U2(s)=vdεq

= KiR ·ωo

s2 + 4ω2o

(A.45)

y12 + 4ω2o · y12 = KiRωo · u2 (A.46)

x1 = y12 (A.47)

x2 = x1 = y12

x2 = y12

x1

x2

=

0 1

−(2ωo)2 0

x1

x2

+

0

KiRωo

u2 (A.48)


y12 =[

1 0

] x1

x2

+ [0]u2 (A.49)

Imaginary error to imaginary output state variable definition,

Y21(s)

U2(s)=vqεq

(A.50)

Derivation is same as for Y11(s)U1(s)

.

Real error to imaginary output state variable definition,

Y22(s)

U1(s)= KiR ·

−ωos2 + (2ωo)2

(A.51)

y22 = −(2ωo)2y22 −KiRωo · u2 (A.52)

x1

x2

=

0 1

−(2ωo)2 0

x1

x2

+

0

−KiRωo

u1 (A.53)

y22 =[

1 0

] x1

x2

+ [0]u1 (A.54)

Combining the four sections together will yield the final result shown below,

xd1

xd2

xd3

xq1

xq2

xq3

=

0 1 0 0 0 0

0 0 1 0 0 0

0 −(2ωo)2 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 −(2ωo)2 0

xd1

xd2

xd3

xq1

xq2

xq3

+

0 0

0 0

1 0

0 0

0 0

0 1

u1

u2

(A.55)


y1

y2

=

vd

vq

=

2KiRω2o 0 KiR 0 KiRωo 0

0 −KiRωo 0 2KiRω2o 0 KiR

xd1

xd2

xd3

xq1

xq2

xq3

+

KiP 0

0 KiP

u1

u2

(A.56)

Appendix B

Development of Complete State

Space Model for DC SVC

Development of A, B, C, and D matrices for SVC DC voltage control and dq-frame

equivalent resonant current control.

The first step in completing the state space model of the SVC control loop, is the

combination of the current controller, system 2, and the AC line plant, system 3. This

combination will be performed using the feedback connection of two state space models

outlined in chapter 3.

xf =

x1

x2

=

A1 −B1C2

B2C1 A2 −B2D1C2

x1

x2

+

B1

B2D1

r (B.1)

y =[

0 C2

] x1

x2

(B.2)

Therefore,

92

xf1

=

[ x1

x2

] =

01

00

00

00

00

10

00

00

0−

4ω

2 o0

00

0−

10

00

00

10

00

00

00

01

00

00

00

−4ω

2 o0

0−

1

2K

iRω

2 oL

0K

iRL

0K

iRω

oL

0−R L−K

iPL

+ωo

0−K

iRω

oL

02K

iRω

2 oL

0K

iRL

−ωo

−R L−K

iPL

xd1

xd2

xd3

xq1

xq2

xq3

i d i q

+

00

00

0

00

00

0

10

00

0

00

00

0

00

00

0

01

00

0

KiPL

01 L

0−M

d2L

0K

iPL

01 L

−M

q

2L

iref

d iref

q vgd

vgq

vdc

(B.3

)

Appendix B. Development of Complete State Space Model for DC SVC94

y =

id

iq

=

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

(B.4)

Next the small signal AC to DC real power equation, system 4, is added to the state

space model of the current control loop. The A and B matrices are the same as the

current control loop and the C and D matrices can be viewed below.

y=i dc

=

[ −3K

iRω

2 oId

2

3K

iRω

oIq

4−

3K

iRId

4

−3K

iRω

2 oIq

2−

3K

iRω

oId

4

−3K

iRIq

43 4(M

d+KiPI d

)3 4(M

q+KiPI q

)

] xd1

xd2

xd3

xq1

xq2

xq3

i d i q

+

[ −3K

iPId

4

−3K

iPIq

4−

1

] iref

d iref

q

i load

(B.5

)


The DC link capacitor model, system 5, will now be added using the series combina-

tion of two state space systems and the result can be viewed below,

xs1

=

[ x1

x2

] =

01

00

00

00

0

00

10

00

00

0

0−

4ω

2 o0

00

0−

10

0

00

00

10

00

0

00

00

01

00

0

00

00

−4ω

2 o0

0−

10

2K

iRω

2 oL

0K

iRL

0K

iRω

oL

0−R L−K

iPL

+ωo

−M

d2L

0−K

iRω

oL

02K

iRω

2 oL

0K

iRL

−ωo

−R L−K

iPL

−M

q

2L

−3K

iRω

2 oId

Vd

cC

3K

iRω

oIq

2V

dcC

−3K

iRId

2V

dcC

−3K

iRω

2 oIq

Vd

cC

−3K

iRω

oId

2V

dcC

−3K

iRIq

2V

dcC

3 4C

(Md

+2K

iPId

Vd

c)

3 4C

(Mq

+2K

iPIq

Vd

c)

0

xd1

xd2

xd3

xq1

xq2

xq3

i d i q vdc

+(B

.6)

00

00

0

00

00

0

10

00

0

00

00

0

00

00

0

01

00

0

KiPL

01 L

00

0K

iPL

01 L

0

−3K

iPId

2V

dcC

−3K

iPIq

2V

dcC

00

−1C

iref

d iref

q vgd

vgq

i load


y = vdc =[

0 0 0 0 0 0 0 0 1

]

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vdc

(B.7)

The next step will be to combine the DC space vector control state space model with

the rest of the control loop’s state space model shown above.

The result of this combination and final result of this procedure can be viewed in the

body of chapter 3.

Appendix C

Small Signal State Space Models of

VSC Control Schemes

This chapter will present the state space models of VSC systems using modulation index

control and VSC AC terminal voltage control.


Control with dq-frame PI Current Control

Fig. C.1 and Fig. C.2, shown below, give the current control loop and DC voltage control

loop for the VSC system with modulation index control.

The complete state space model of the DC voltage controlled VSC system is shown

below.

99


di

gdv

ref

qi

ref

di

tdv

tqv

gqv

qi

RLs

1

RLs

1

d

q

s

KK

ii

iP+

dm

2

dcv

s

KK

ii

iP+

qm

2

dcv

Lo

ω oLω

oLω

Lo

ω

dcV

2−

dcV

2−

Figure C.1: Dq-frame PI Current Control Loop

s

KK

I

p+

ref

dqi

Inner AC

Current

Control Loop

dq-frame

( )qqdd

imim +4

3ref

dcv

ref

di

ref

qij ⋅

dci

loadi

dcv

dcv

dqm

dqi

Cs

1

Figure C.2: PI DC voltage control loop

Appendix C. Small Signal State Space Models of VSC Control Schemes101

x=

˙ xdc ˙ xd ˙ xq ˙ i d ˙ i q ˙ v dc

=

00

00

0−KI

Kii

00

−Kii

0−KiiKP

00

00

−Kii

0

KiP L

1 L0

−R L−

KiP L

0−KPKiP

L

00

1 L0

−R L−

KiP L

0

−3KiPI d

2VdcC

−3I d

2VdcC

−3I q

2VdcC

3 4C

(2KiPI d

Vdc−

2ωoLI q

Vdc

+M

d)

3 4C

(2KiPI q

Vdc−

2ωoLI d

Vdc

+M

q)

3KiPKPI d

2VdcC

xdc

xd

xq i d i q v dc

+(C

.1)

KI

00

00

KiiKP

00

00

0Kii

00

0

KPKiP

L0

1 L0

0

0KiP L

01 L

0

−3KiPKPI d

2VdcC

−3KiPI q

2VdcC

00

−1C

vrefdc

irefq v gd

v gq

i load


y = vdc =[

0 0 0 0 0 1

]

xdc

xd

xq

id

iq

vdc

(C.2)


Control with αβ-frame Resonant Current Con-

trol


loop for the VSC system with modulation index control.


ref

iαβ

RLs +

1

ipK

αβi

αβtv

αβgv

αβRv

2

dcv

mαβ

uuuur

2 2iR

o

sK

s ω⋅

+dc

V

2−

Figure C.3: Resonant αβ-frame Current Control Loop


below.

AC to DC REAL

POWER EQUATION

s

KK

I

p+

ref

dqi

ref

dcv

ref

di

ref

iαβ

ref

qij ⋅

dci

αβi

loadi

dcv

dcv

Modulator

Demodulator

mαβ

uuuur

3( )


Cs

1

Inner αβ-frame

AC current

control loop

oj t

eω

Figure C.4: DC Voltage control loop with αβ-frame Current Control Loop


ˆ xdc1

ˆ xd1

ˆ xd2

ˆ xd3

ˆ xq1

ˆ xq2

ˆ xq3

ˆ i d ˆ i q ˆ vdc

=

00

00

00

00

0−KI

00

10

00

00

00

00

01

00

00

00

10

−4ω

2 o0

00

0−

10

−KP

00

00

01

00

00

00

00

00

10

00

00

00

0−

4ω

2 o0

0−

10

KiPV

dc

2L

KiRV

dcω

2 oL

0K

iRV

dc

2L

0K

iRω

oV

dc

2L

0−R L−K

iPV

dc

2L

+ωo

−M

d2L−K

iPK

PV

dc

2L

00

−K

iRω

oV

dc

2L

0K

iRω

2 oV

dc

L0

KiRV

dc

2L

−ω

0−R L−K

iPV

dc

2L

−M

q

2L

−3K

iPId

4C

−3K

iRω

2 oId

2C

3K

iRω

oIq

4C

−3K

iRId

4C

−3K

iRω

2 oIq

2C

−3K

iRω

oId

4C

−3K

iRIq

4C

3M

d4C

+3K

iPId

4C

3M

q

4C

+3K

iPIq

4C

3K

iPK

PId

4C

xdc1

xd1

xd2

xd3

xq1

xq2

xq3

i d i q vdc

+

KI

00

00

00

00

0

00

00

0

KP

00

00

00

00

0

00

00

0

01

00

0

KPK

iPV

dc

2L

01 L

00

0K

iPV

dc

2L

01 L

0

−3K

PK

iPId

4C

−3K

iPIq

4C

00−

1 C

vref

dc

iref

q vgd

vgq

i load

(C

.3)


y =[

0 0 0 0 0 0 0 0 0 1

]

xdc1

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vdc

+[

0 0 0 0 0

]

vrefdc

irefq

vgd

vgq

iload

(C.4)

C.3 VSC Terminal Voltage Control: PI DC Voltage

Control with αβ-frame Resonant Current

Control


loop for the VSC system with VSC AC terminal voltage control.


ref

iαβ

RLs +

1

ipK

αβi

αβtv

αβgv

αβRv

-1αβεr

22

o

iR

s

sK

ω+⋅



s

KK

I

p+

ref

dqi

Inner AC

Current

Control Loopdc

tt

v

iviv ββαα +

2

3

ref

dcv

ref

di

αβtvref

iαβ

ref

qij ⋅

dci

αβi

loadi

sC ⋅

1

dcv

dcv

Modulator

Demodulator

tjoe

ω

Figure C.6: DC Voltage control loop with αβ-frame Current Control Loop


below.

ˆ xdc

ˆ xd1

ˆ xd2

ˆ xd3

ˆ xq1

ˆ xq2

ˆ xq3

ˆ i d ˆ i q ˆ vdc

=

00

00

00

00

0−KI

00

10

00

00

00

00

01

00

00

00

10

−4ω

2 o0

00

0−

10

−KP

00

00

01

00

00

00

00

00

10

00

00

00

0−

4ω

2 o0

0−

10

KiPL

2K

iRω

2 oL

0K

iRL

0K

iRω

oL

0−R L−K

iPL

+ωo

−K

iPK

PL

00

−K

iRω

oL

02K

iRω

2 oL

0K

iRL

−ω

0−R L−K

iPL

0

−3K

iPId

2V

dcC

−3K

iRω

2 oId

Vd

cC

3K

iRω

oIq

2V

dcC

−3K

iRId

2V

dcC

−3K

iRω

2 oIq

Vd

cC

−3K

iRω

oId

2V

dcC

−3K

iRIq

2V

dcC

3K

iPId

2V

dcC

+3V

td

2V

dcC

3K

iPIq

2V

dcC

+3V

tq

2V

dcC

3K

iPK

PId

2V

dcC

−Id

cV

dcC

xdc1

xd

xd2

xd3

xq1

xq2

xq3

i d i q vdc

+

KI

00

00

00

00

0

00

00

0

KP

00

00

00

00

0

00

00

0

01

00

0

KPK

iP

L0

1 L0

0

0K

iPL

01 L

0

−3K

PK

iPId

2V

dcC

−3K

iPIq

2V

dcC

00−

1 C

vref

dc

iref

q vgd

vgq

i load

(C

.5)


y =[

0 0 0 0 0 0 0 0 0 1

]

xdc

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vdc

+[

0 0 0 0 0

]

vrefdc

irefq

vgd

vgq

iload

(C.6)

C.4 VSC Terminal Voltage Control: DCSV Voltage

Control with αβ-frame Resonant Current

Control


loop for the VSC system with VSC AC terminal voltage control.


ref

iαβ

RLs +

1

ipK

αβi

αβtv

αβgv

αβRv

-1αβεr

22

o

iR

s

sK

ω+⋅



ref

dqi

Inner AC

Current

Control Loopdc

tt

v

iviv ββαα +

2

3

ref

dcv

αβtvref

iαβdc

i

αβi

loadi

sC ⋅

1

dcv

dcv

Modulator

Demodulator

tjoe

ω

DCSV

Controller

Figure C.8: DCSV Voltage control loop with αβ-frame Current Control Loop


below.

ˆ xd

c1

ˆ xd

c2

ˆ xd

c3

ˆ xd1

ˆ xd2

ˆ xd3

ˆ xq1

ˆ xq2

ˆ xq3

ˆ i d ˆ i q ˆ vd

c

=

01

00

00

00

00

00

00

10

00

00

00

00

0−

4ω2 o

00

00

00

00

0−

1

00

00

10

00

00

00

00

00

01

00

00

00

4K

Iω2 o

0(K

I+

KR

)0

−4

ω2 o

00

00

−1

0−

KP

00

00

00

01

00

00

00

00

00

00

10

00

0−

2K

Rω

o0

00

00

−4

ω2 o

00

−1

0

4K

IK

iP

ω2 o

L0

KiP

(KI+

KR

)

L

2K

iR

ω2 o

L0

KiR

L0

KiR

ωo

L0

−R

L−

KiP

L+

ωo

−K

PK

iP

L

0−

2K

RK

iP

ωo

L0

0−

KiR

ωo

L0

2K

iR

ω2 o

L0

KiR

L−

ωo

−R

L−

KiP

L0

−6

KiP

KI

ω2 o

Id

Vd

cC

3K

iP

KR

ωo

Iq

Vd

cC

−3

KiP

(KI+

KR

)Id

2V

dc

C

−3

KiR

ω2 o

Id

Vd

cC

3K

iR

ωo

Iq

2V

dc

C

−3

KiR

Id

2V

dc

C

−3

KiR

ω2 o

Iq

Vd

cC

−3

KiR

ωo

Id

2V

dc

C

−3

KiR

Iq

2V

dc

C

3(K

iP

Id+

Vtd)

2V

dc

C

3(K

iP

Iq+

Vtq)

2V

dc

C

3K

iP

KP

Id−

2Id

c2

Vd

cC

xd

c1

xd

c2

xd

c3

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vd

c

+

00

00

0

00

00

0

10

00

0

00

00

0

00

00

0

KP

00

00

00

00

0

00

00

0

01

00

0K

PK

iP

L0

1 L0

0

0K

iP

L0

1 L0

−3

KP

KiP

Id

2V

dc

C

−3

KiP

Iq

2V

dc

C0

0−

1 C

vr

ef

dc

ir

ef

q vg

d

vg

q

ilo

ad

(C

.7)


y =[

0 0 0 0 0 0 0 0 0 0 0 1

]

xdc1

xdc2

xdc3

xd1

xd2

xd3

xq1

xq2

xq3

id

iq

vdc

+[

0 0 0 0 0

]

vrefdc

irefq

vgd

vgq

iload

(C.8)

Appendix D

Combinations of State Space

Systems: Series and Feedback

D.1 Series Combination of Two State Space Systems

x1 = A1x1 +B1u1 (D.1)

y1 = C1x1 +D1u1 (D.2)

x2 = A2x2 +B2u2 (D.3)

Substituting (D.2) into (D.3) where u2 = y1,

x2 = A2x2 +B2(C1x1 +D1u1) (D.4)

x2 = A2x2 +B2C1x1 +B2D1u1 (D.5)

112

Appendix D. Combinations of State Space Systems: Series and Feedback113

y2 = C2x2 +D2u2 (D.6)

Substituting (D.2) into (D.6) where u2 = y1,

y2 = C2x2 +D2(C1x1 +D1u1) (D.7)

y2 = C2x2 +D2C1x1 +D2D1u1 (D.8)

The final result of the series combination is shown below,

xs =

x1

x2

=

A1 0

B2C1 A2

x1

x2

+

B1

B2D1

u (D.9)

y =[D2C1 C2

] x1

x2

+D2D1u (D.10)

D.2 Feedback Combination of Two State Space Sys-

tems

D2 must be zero, assumed strictly proper, for system 2,

y2 = C2x2 (D.11)

u1 = r − y2 = r − C2x2 (D.12)

u2 = y1 = C1x1 +D1u1 = C1x1 +D1(r − C2x2) (D.13)

Appendix D. Combinations of State Space Systems: Series and Feedback114

x1 = A1x1 +B1(r − C2x2) (D.14)

x1 = A1x1 +B1r −B1C2x2 (D.15)

x2 = A2x2 +B2(C1x1 +D1r − C1C2x2) (D.16)

x2 = (A2 −B2D1C2)x2 +B2C1x1 +B2D1r (D.17)

xf =

x1

x2

=

A1 −B1C2

B2C1 A2 −B2D1C2

x1

x2

+

B1

B2D1

r (D.18)

y =[

0 C2

] x1

x2

(D.19)

Bibliography

[1] C. Schauder and H. Mehta, “Vector analysis and control of advanced static var

compensators,” Generation, Transmission and Distribution, IEE Proceedings C, vol.

140, no. 4, pp. 299–306, Jul 1993.

[2] A. Stankovic and T. Lipo, “A novel control method for input output harmonic

elimination of the pwm boost type rectifier under unbalanced operating conditions,”

Power Electronics, IEEE Transactions on, vol. 16, no. 5, pp. 603–611, Sep 2001.

[3] Y. Suh, V. Tijeras, and T. Lipo, “A control method in dq synchronous frame for

pwm boost rectifier under generalized unbalanced operating conditions,” Power Elec-

tronics Specialists Conference, 2002. pesc 02. 2002 IEEE 33rd Annual, vol. 3, pp.

1425–1430 vol.3, 2002.

[4] Y. Suh and T. Lipo, “Control scheme in hybrid synchronous stationary frame for

pwm ac/dc converter under generalized unbalanced operating conditions,” Industry

Applications, IEEE Transactions on, vol. 42, no. 3, pp. 825–835, May-June 2006.

[5] M. Kazmierkowski and M. Dzieniakowski, “Review of current regulation techniques

for three-phase pwm inverters,” in Industrial Electronics, Control and Instrumen-

tation, 1994. IECON ’94., 20th International Conference on, vol. 1, Sep 1994, pp.

567–575 vol.1.

115

Bibliography 116

[6] M. Kazmierkowski and L. Malesani, “Current control techniques for three-phase

voltage-source pwm converters: a survey,” Industrial Electronics, IEEE Transactions

on, vol. 45, no. 5, pp. 691–703, Oct 1998.

[7] M. Kazmierkowski and M. Cichowlas, “Comparison of current control technqiues for

pwm rectifiers,” IEEE International Symposium on Industrial electronics, vol. 4, pp.

1259–1263, Jul 2002.

[8] D. Zmood and D. Holmes, “Stationary frame current regulation of pwm inverters

with zero steady-state error,” Power Electronics, IEEE Transactions on, vol. 18,

no. 3, pp. 814–822, May 2003.

[9] J.-W. Hwang, M. Winkelnkemper, and P. Lehn, “Control of ac-dc-ac converters with

minimized dc link capacitance under grid distortion,” Industrial Electronics, 2006

IEEE International Symposium on, vol. 2, pp. 1217–1222, July 2006.

[10] J.-W. Hwang and P. Lehn, “Dc space vector controller and its application to con-

verter control,” IEEE P.E.S.C. 2008, pp. 830–836, June 2008.

[11] W. Lu and B.-T. Ooi, “Dc overvoltage control during loss of converter in multi-

terminal voltage-source converter-based hvdc (m-vsc-hvdc),” Power Delivery, IEEE

Transactions on, vol. 18, no. 3, pp. 915–920, July 2003.

[12] B. Ooi and X. Wang, “Voltage angle lock loop control of the boost type pwm con-

verter for hvdc application,” Power Electronics, IEEE Transactions on, vol. 5, no. 2,

pp. 229–235, Apr 1990.

[13] B.-T. Ooi and X. Wang, “Boost-type pwm hvdc transmission system,” Power De-

livery, IEEE Transactions on, vol. 6, no. 4, pp. 1557–1563, Oct 1991.

Bibliography 117

[14] W. Lu and B. T. Ooi, “Simultaneous inter-area decoupling and local area damping

by voltage source hvdc,” in Power Engineering Society Winter Meeting, 2001. IEEE,

vol. 3, 2001, pp. 1079–1084 vol.3.

[15] D. Jovcic, L. Lamont, and K. Abbott, “Control system design for vsc transmission,”

Electric Power Systems Research, vol. 77, no. 7, pp. 721–729, 2007.

[16] K. Padiyar and N. Prabhu, “Modelling, control design and analysis of vsc based

hvdc transmission systems,” in Power System Technology, 2004. PowerCon 2004.

2004 International Conference on, vol. 1, Nov. 2004, pp. 774–779 Vol.1.

[17] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, Second

Edition. Springer Science+Business Media, 2001.

[18] C. Sao and P. W. Lehn, “A block diagram approach to reference frame transforma-

tion of converter dynamic models,” in Electrical and Computer Engineering, 2006.

CCECE ’06. Canadian Conference on, May 2006, pp. 2270–2274.

[19] D. Zmood, D. Holmes, and G. Bode, “Frequency-domain analysis of three-phase

linear current regulators,” Industry Applications, IEEE Transactions on, vol. 37,

no. 2, pp. 601–610, Mar/Apr 2001.

[20] A. Reidy and R. Watson, “Comparison of vsc based hvdc and hvac interconnections

to a large offshore wind farm,” in Power Engineering Society General Meeting, 2005.

IEEE, June 2005, pp. 1–8 Vol. 1.

[21] N. B. Negra, J. Todorovic, and T. Ackermann, “Loss evaluation of hvac and hvdc

transmission solutions for large offshore wind farms,” Electric Power Systems Re-

search, vol. 76, no. 11, pp. 916 – 927, 2006.

small signal modeling of resonant controlled vsc systems · 2013-10-24 · abstract small signal...

Documents