IN DEGREE PROJECT MASTER'S PROGRAMME, ELECTRIC POWER, SECOND CYCLEENGINEERING 120 CREDITS

, STOCKHOLM SWEDEN 2015

Linear Modeling of DFIGs andVSC-HVDC Systems

WEIRAN CAO

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING

Linear Modeling of DFIGs and VSC-HVDC Systems

Weiran Cao

School of Electrical Engineering

Royal Institute of Technology

Examiner: Hans-Peter Nee

Comissioned by ABB Corporate Research Center in Västerås, Sweden

Supervisor: Lidong Zhang, Pinaki Mitra

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Abstract

Recently, with growing application of wind power, the system based on the doubly fed

induction generator (DFIG) has become the one of the most popular concepts. The

problem of connecting to the grid is also gradually revealed. As an effective solution to

connect offshore wind farm, VSC-HVDC line is the most suitable choice for stability

reasons. However, there are possibilities that the converter of a VSC-HVDC link can

adversely interact with the wind turbine and generate poorly damped sub-synchronous

oscillations. Therefore, this master thesis will derive the linear model of a single DFIG as

well as the linear model of several DFIGs connecting to a VSC-HVDC link. For the

linearization method, the Jacobian transfer matrix modeling method will be explained

and adopted. The frequency response and time-domain response comparison between

the linear model and the identical system in PSCAD will be presented for validation.

Sammanfattning

Nyligen, med ökande tillämpning av vindkraft, det system som bygger på den dubbelt

matad induktion generator (DFIG) har blivit en av de mest populära begrepp. Problemet

med att ansluta till nätet är också gradvis avslöjas. Som en effektiv lösning för att ansluta

vindkraftpark är VSC -HVDC linje det lämpligaste valet av stabilitetsskäl. Det finns dock

möjligheter att omvandlaren en VSC-HVDC länk negativt kan interagera med

vindturbinen och genererar dåligt dämpade under synkron svängningar. Därför kommer

detta examensarbete härleda den linjära modellen av en enda DFIG liksom den linjära

modellen av flera DFIGs ansluter till en VSC-HVDC -länk. För arise metoden kommer

Jacobian transfer matrix modelleringsmetodförklaras och antas. Jämförelse mellan den

linjära modellen och identiskt system i PSCAD frekvensgången och tidsdomänensvar

kommer att presenteras för godkännande.

Keywords

Wind farm, DFIG, VSC-HVDC link, sub-synchronous oscillations, linear modeling,

Jacobian transfer matrix, frequency response, PSCAD

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Acknowledgements

First of all, I would like to express my sincere gratitude to my supervisors at ABB, Dr. Lidong Zhang

and Dr. Pinaki Mitra, for their instructive advices and help in model building and testing.

Secondly, I’m also indebted to my Examiner at KTH, Professor Hans-Peter Nee, who has put his

considerable time and support into the completion of this thesis.

Last but not the least, I’d like to thank my friends, teachers and colleagues. Without their help

and encouragement, it would be much harder for me to finish my thesis and this paper.

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Contents

1. Introduction ............................................................................................................. 1

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

1.2. Project Objective and Outline .............................................................................................. 2

1.2.1. Objective .................................................................................................................................. 2

1.2.2. Outline ...................................................................................................................................... 3

2. Jacobian Transfer Matrix Method .............................................................................. 4

3. Doubly Fed Induction Generator ................................................................................ 7

3.1. DFIG introduction ..................................................................................................................... 7

3.2. Modeling of Wind Turbine .................................................................................................... 7

3.3. Turbine-Generator Mechanical Model ............................................................................. 8

3.4. Dynamic Equivalent Circuit .................................................................................................. 9

3.4.1. Rotor-side Dynamics .......................................................................................................... 9

3.4.2. Grid-side Dynamics .......................................................................................................... 11

3.5. Control strategy for DFIG ................................................................................................... 11

3.5.1. Rotor-side Converter ....................................................................................................... 11

3.5.2. Grid-side converter .......................................................................................................... 13

4. Modeling of a Single DFIG Connected to an Infinite Bus ........................................... 14

4.1. Basic state space of a DFIG................................................................................................. 14

4.2. Coordinate transformation of the DFIG state-space ............................................... 16

4.3. Combine DFIG Dynamics with the Network Dynamics and the Control

Strategy .................................................................................................................................................... 18

4.3.1. Network Model .................................................................................................................. 18

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4.3.2. Option 1 ................................................................................................................................ 19

4.3.3. Option 2 ................................................................................................................................ 25

4.3.4. Testing for a single DFIG connect to infinite bus ................................................. 26

5. Modeling of DFIGs Connecting to a VSC-HVDC converter .......................................... 32

5.1. Modeling of a VSC-HVDC converter ............................................................................... 32

5.2. Connecting 2 DFIGs to the VSC-HVDC converter ...................................................... 34

5.2.1. Network Model .................................................................................................................. 35

5.2.2. Jacobian Transfer Matrix J(s) of the System ........................................................... 37

5.2.3. Testing for two DFIGs and a VSC-HVDC System ................................................... 40

6. Conclusions ............................................................................................................ 42

7. Future Work ........................................................................................................... 43

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List of Tables

Table 1 The sequence of inputs and outputs of system in Fig. 4.2 22

Table 2 Parameters of the DFIG 27

Table 3 Rated value of the VSC-HVDC and the infinite bus 35

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List of Figures

Fig. 2.1 An AC-network connected to various input devices 4

Fig. 2.2 First modeling option 5

Fig. 2.3 Second modeling option 6

Fig. 3.1 Typical configuration of a DFIG 7

Fig. 3.2 Two-mass model block diagram 9

Fig. 3.3 Equivalent circuit of a regular induction machine 10

Fig. 3.4 Equivalent circuit of a DFIG 10

Fig. 3.5 Rotor-side PWM control block 12

Fig. 3.6 Grid-side PWM control block 13

Fig. 3.7 Coordinate transformation to the R-I frame 16

Fig. 4.1 Single DFIG connected to an infinite bus 18

Fig. 4.2 Complete linear model of the DFIG model using option 1 22

Fig. 4.3 Complete linear model of the DFIG model using option 2 26

Fig. 4.4 Single DFIG connected to an infinite bus in PSCAD 27

Fig. 4.5 Frequency-response comparison from 𝑉𝑟𝑑 to 𝑇𝑒 28

Fig. 4.6 Frequency-response comparison from 𝑉𝑟𝑞 to 𝑇𝑒 28

Fig. 4.7 Frequency-response comparison from 𝜔𝑟 to 𝑇𝑒 29

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Fig. 4.8 Step response from 𝜔𝑟 to 𝑇𝑒 30

Fig. 4.9 Step response from 𝑉𝑟𝑑 to 𝑇𝑒 30

Fig. 4.10 Step response from 𝑉𝑟𝑞 to 𝑇𝑒 31

Fig. 5.1 Main circuit of the VSC-HVDC converter 32

Fig. 5.2 Control block diagram of a VSC-HVDC converter using

Power-synchronization control

33

Fig. 5.3 The system diagram of two DFIGs connecting to a VSC-HVDC converter 34

Fig. 5.4 Simplified linear model of the system in Fig. 5.3 35

Fig. 5.5 Two DFIGs and a VSC-HVDC system in PSCAD 40

Fig. 5.6 Frequency-response comparison from ∆𝜃𝑣 to ∆𝑃 41

Fig. 5.7 Frequency-response comparison from ∆𝜃𝑣 to ∆𝑈𝑓 41

Fig. 5.8 Frequency-response comparison from ∆𝑉𝑟𝑞 to ∆𝑇𝑒 42

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List of symbols

𝑃𝑚 The mechanical power of the wind turbine;

𝜌 The air density;

𝑅 The turbine radius;

𝑉𝑤 The wind speed;

𝐶𝑝 The power factor of wind turbine;

𝜆 The tip speed ratio;

𝛽 The blade pitch-angle;

𝑐1 − 𝑐6 Turbine’s coefficients;

𝜔𝑡 The rotational speed of wind turbine;

𝐻𝑡 The inertia constant of the turbine;

𝐾𝑠ℎ The shaft stiffness;

𝜃𝑡𝑤 The shaft twist angle;

𝑇𝑚 The mechanical torque of the wind turbine;

𝐷𝑠ℎ The shaft damping constant;

𝐻𝑔 The inertia constant of the generator;

𝜔𝑟 The generator-rotor speed;

𝑇𝑒 The electrical torque of the generator;

𝑉𝑠 The generator-stator voltage;

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𝑅𝑠 The stator resistance;

𝑖𝑠 The stator current;

𝜓𝑠 The stator-winding flux;

𝜔𝑠 The synchrous speed;

𝑅𝑟 The rotor resistance;

𝑖𝑟 The rotor current;

𝑠 The generator slip;

𝜓𝑟 The rotor-winding flux;

𝐿𝑠 The stator inductance;

𝐿𝑚 The mutual inductance;

𝐿𝑟 The rotor inductance;

𝑉𝑟 The rotor voltage;

𝑉𝑠𝑑 , 𝑉𝑠𝑞 The stator voltage in d-q frame;

𝑉𝑟𝑑 , 𝑉𝑟𝑞 The rotor voltage in d-q frame;

𝑖𝑠𝑑 , 𝑖𝑠𝑞 The stator current in d-q frame;

𝑖𝑟𝑑 , 𝑖𝑟𝑞 The rotor current in d-q frame;

𝑉𝑔𝑑1, 𝑉𝑔𝑞1 The grid-side voltage in d-q frame;

𝑖𝑔𝑑1, 𝑖𝑔𝑞1 The grid-side current in d-q frame;

𝑄𝑠 The stator-reactive power;

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𝑖𝑟𝑑∗ , 𝑖𝑟𝑞

∗ The current-reference value in d-q frame;

𝜔𝑟∗ The rotor-speed reference value;

𝑠0 The generator slip at steady-state;

𝑖𝑠𝑑0, 𝑖𝑠𝑞0 The stator current in d-q frame at steady-state;

𝑖𝑟𝑑0, 𝑖𝑟𝑞0 The rotor current in d-q frame at steady-state;

𝜓𝑠𝑑0, 𝜓𝑠𝑞0 The stator-winding flux in d-q frame at steady-state;

𝑉𝑠𝑅 , 𝑉𝑠𝐼 The stator voltage in ac-network R-I frame;

𝑖𝑠𝑅 , 𝑖𝑠𝐼 The stator current in ac-network R-I frame;

𝑖𝑔𝑅 , 𝑖𝑔𝐼 The grid-side current in ac-network R-I frame.

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1. Introduction

1.1. Background

In recent years, high-voltage direct-current (HVDC) based long-distance power

transmission is gaining immense importance throughout the world. Now, more than 145

projects using HVDC are in operation worldwide. The European Union is also vigorously

promoting the European electricity transmission system development, which is mainly

based on DC technology, designed to facilitate large-scale sustainable power generation

in remote areas for transmission to centers of consumption [1].

Due to the lower losses in DC cables, HVDC technology has become more popular than

HVAC technology especially for long distance transmission. There are mainly two types

of HVDC technology available today. One is based on line-commutated converters (LCCs)

and the other is based on voltage-source converters (VSCs). Among these, VSC-HVDC

system, apart from addressing conventional network issues such as bulk power

transmission, asynchronous network interconnections, back-to-back ac system

connection, and voltage/ stability support etc., is particularly suitable for integration of

large-scale renewable energy sources with the grid [2].

On the other hand, with growing concerns about environmental pollution and a possible

energy shortage, wind energy has been considered as one of the solutions. Ever since the

first large grid connected wind farm appeared in California (U.S.) in 1980s, wind power

generation has been undergoing a significant development. With developing techniques,

reducing costs and low environmental impact, wind energy will definitely play a major

role in the world’s energy future [3].

Today, most wind turbines above 1 MW are using variable-speed technique. Amongst

many variable-speed concepts, the system based on the doubly fed induction generator

(DFIG) has become the most popular and effective one [4]. The reason is that the power

converter for DFIG only deals with rotor power, therefore, the converter rating can be

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kept fairly low, around 20 percent of the total machine power [3]. Thus the costs and

losses of the converter will be really small. Another feature is that the DFIG is able to

control the reactive power which is similar as a synchronous generator. There are

several different control strategies for DFIGs. Among them, the voltage-vector regulation

of the rotor in the stator-flux oriented reference frame is one of the most effective

method [5]. Therefore it will be used as the control strategy for DFIGs in this thesis.

VSC-HVDC line is the most suitable choice among all the HVDC technologies. So it is

useful and necessary to analyze the interaction between offshore wind farm and VSC-

HVDC links. In such a case, the converter of a VSC-HVDC link can adversely interact with

the wind turbine and generate poorly damped sub-synchronous oscillations [6].

Power-synchronization control has been demonstrated to SSCI can be very effective for

offshore wind integration by VSC-HVDC system. Therefore, this thesis will only consider

power synchronization control as the strategy for the HVDC converters. So far, the

effectiveness of power synchronization control for offshore wind integration was

established only through digital simulation results and no analysis was involved. It is

therefore important to derive the linear model to develop better insight. For the

mathematical modeling, a so-called Jacobian transfer matrix approach has been shown

to be capable of reflecting the frequency response characteristics. The objective of this

thesis is therefore to utilize transfer matrix formulation to understand and analyze the

interaction between wind farms and HVDC converters.

1.2. Project Objective and Outline

1.2.1. Objective

The objectives of the thesis are:

1. Develop an aggregated mathematical model of a single DFIG with back to back PWM

converter;

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2. Develop mathematical model of a VSC-HVDC link with power-synchronization control;

3. Integrate the DFIG model with the VSC-HVDC model and obtain the transfer matrix of

the combined system;

4. Carry out rigorous frequency domain analysis of the wind integrated HVDC system

and study the interaction of the converter controls especially in the sub-synchronous

range;

5. Based on the analysis, provide possible recommendations for control design of the

VSC-HVDC stations while integrating large offshore wind farms.

1.2.2. Outline

This thesis is conducted by both theoretical derivation and physical validation. Chapter 2

will introduce the Jacobian transfer matrix modeling concept; In Chapter 3, the model of

a single DFIG, which includes the electric equivalent model, the control strategy and

mechanical transient, will be proposed. Chapter 4 will present the linear model of a

single DFIG connected to an infinite bus, the linear model will be validated by comparing

the frequency response from identical system in PSCAD; Chapter 5 will present the

linear model of a VSC-HVDC converter. Besides, the system of two DFIGs and a

VSC-HVDC converter connected to an infinite bus will be linearized. The linear model

will be validated by the frequency-response comparison. In Chapter 6 and 7, the

conclusions and future works will be discussed.

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2. Jacobian Transfer Matrix Method

Jacobian transfer matrix is a new method for power system modeling. The idea of this

method came from the Jacobian matrix. Originally, the Jacobian matrix was proposed to

solve power-flow iteration using Newton-Raphson algorithm. It was found that the

singularity of the Jacobian matrix is closely related to the voltage stability issues. For

instance, when the Jacobian matrix is singular, the operation point will be identical to

the critical points on the P-V curve [7]. Based on the Jacobian matrix, a modal analysis

technique was developed for analyzing small-signal stability. However, either voltage

stability or small signal stability is a dynamic issue, while the Jacobian matrix is a static

matrix that can only reflect the power-flow deviation at a certain operating point. This

can be understood that the Jacobian matrix can describe the power system dynamic

behavior when the frequency range is “quasi-static”. So in order to fulfill the requirement

of dynamic response analysis, the Jacobian matrix needs to be improved so that it can be

valid in the whole frequency range. This is how the idea of Jacobian transfer matrix came

from.

Fig. 2.1 An AC-network connected to various input devices

The main idea of Jacobian transfer matrix modeling is that the power system can be

treated as one multi-inputs multi-outputs feedback-control system. As shown in Fig. 2.1,

all the components in a power system can be divided into two groups:

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1 Passive network: including transmission lines, transformers, line inductors, shunt

capacitors, RLC loads and so on.

2 Input device: including any power component in the system which has a feedback

property, such as synchronous generators, induction motors, HVDC line and so on.

For the passive network, it includes the inductance and capacitance. So based on the

Kirchhoff’s law and the characteristic of inductance and capacitance, the dynamic

equations of the network can be derived. Then rewrite into the state-space form, with

the inject current vectors of each input device as input signals and the voltage vector of

each input device as output signals.

Fig. 2.2 First modeling option

For input devices, each device has electrical part and controller part. There are two

modeling options:

First one which is also proposed in [7], only model the electrical part of input devices

into state-space form, with voltage vector as input signals and output current vectors as

output signals, which is reciprocal to network state-space. And then combine the

state-space of electrical part and aforementioned network state-space into a new

state-space called the Jacobian transfer matrix. The input and output signals are

determined by the controller of input devices. For instance, if a VSC-HVDC line is using

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active-power control to adjust the voltage angle, then the active power of the HVDC line

should be included as one of the output signal and the voltage angle should be one of the

input signal of the Jacobian transfer matrix. The Jacobian transfer matrix has been

derived, the controller part can be added as outer feedback loop. This concept can be

explained in Fig. 2.2.

Fig. 2.3 Second modeling option

The second option is to model each input device as an individual state-space. This means

each input device will become a state-space including both the electrical and controller

part. And then combine network state-space and each input device’s state-space.

Therefore the final state-space for the whole system is derived as shown in Fig. 2.3. In

principle, this option is the same as the first one, the difference is that this option

strengthens the modular idea, each input device will be modeled separately so that it

simplifies the procedure of modeling in MATLAB for some cases.

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3. Doubly Fed Induction Generator

3.1. DFIG introduction

Fig. 3.1 shows a typical configuration of a DFIG. The rotor winding connects to the grid

through a back to back PWM converter. The grid-side converter is to keep the voltage of

the DC link constant while the rotor-side converter is to control the rotor speed and the

reactive power through the stator. With such a structure, DFIGs can keep the stator

voltage at constant magnitude and frequency when wind speed varies. Besides, due to

the back-to-back converter only deals with the rotor power, so the converter rating can

be kept fairly small which saves the total costs.

Fig. 3.1 Typical configuration of a DFIG

3.2. Modeling of Wind Turbine

The mechanical energy capture of a wind turbine is given by (3.1) [8] [13]:

𝑃𝑚 =1

2𝜌𝜋𝑅2𝑉𝑤

3𝐶𝑝 , (3.1)

where 𝑃𝑚 is the mechanical power; 𝜌 is the air density; 𝑅 is the turbine radius; 𝑉𝑤 is

the wind speed; 𝐶𝑝 is the power factor which is related to the tip speed ratio 𝜆 and

blade pitch-angle 𝛽 given by (3.2) [8] [13]:

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𝐶𝑝(𝜆, 𝛽) = 𝑐1 (𝑐2𝑅

𝜆− 𝑐3 ∙ 𝛽 − 𝑐4) ∙ 𝑒

−𝑐5𝑅𝜆 + 𝑐6 ∙ 𝜆

𝜆 =𝑅𝜔𝑡

𝑉𝑤 ,

(3.2)

where 𝑐1-𝑐6 are turbine’s coefficients that depends on the design; 𝜔𝑡 is the rotational

speed of wind turbine.

If the wind speed is below the rated value, the wind turbine operates in the

variable-speed mode and the pitch-angle 𝛽 is kept at minimum limit. 𝜔𝑡 is adjusted to

keep the tip speed ratio 𝜆 at the level that the power 𝑃𝑚 is maximized; if the wind

speed is above the rated value, then the pitch-angle 𝛽 will be adjusted to reduce the

mechanical power extracted from wind.

3.3. Turbine-Generator Mechanical Model

The turbine’s mechanical dynamics is usually represented by a two-mass model for the

combination of the turbine’s low speed shaft and generator’s high speed shaft coupled

by the gear box. The two-mass model is given by (3.3) [9] [10]:

2𝐻𝑡

𝑑𝜔𝑡

𝑑𝑡= 𝑇𝑚 − 𝐾𝑠ℎ𝜃𝑡𝑤 − 𝐷𝑠ℎ

𝑑𝜃𝑡𝑤

𝑑𝑡

2𝐻𝑔

𝑑𝜔𝑟

𝑑𝑡= 𝐾𝑠ℎ𝜃𝑡𝑤 + 𝐷𝑠ℎ

𝑑𝜃𝑡𝑤

𝑑𝑡− 𝑇𝑒

𝑑𝜃𝑡𝑤

𝑑𝑡= 𝜔𝑡 − 𝜔𝑟 .

(3.3)

Rewrite the two-mass model into state-space form, with the state variables 𝑥 =

[∆𝜔𝑡 ∆𝜔𝑟 ∆𝜃𝑡𝑤]𝑇, the input variables 𝑢 = [∆𝑇𝑒 ∆𝑇𝑚]𝑇 and the output variable 𝑦 =

[∆𝜔𝑟]. Then it becomes:

�̇� = 𝐴 ∙ 𝑥 + 𝐵𝑢

𝑦 = 𝐶 ∙ 𝑥 .

(3.4)

The block diagram of the two-mass model can be expressed in Fig. 3.2.

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Fig. 3.2 Two-mass model block diagram

3.4. Dynamic Equivalent Circuit

As shown in Fig. 3.1, the dynamic equation of a DFIG can be divided into two parts, the

rotor-side dynamics and the grid-side dynamics. The rotor-side dynamics refer to the

machine’s dynamics and grid-side dynamics refer to the dynamic equation on the link

between the grid-side converter and the ac-network.

3.4.1. Rotor-side Dynamics

Fig. 3.3 shows the equivalent circuit of a regular induction machine. There is no rotor

winding. The dynamic equation of this circuit can be written as:

𝑉𝑠 = 𝑅𝑠𝑖𝑠 + 𝑗𝜓𝑠𝜔𝑠 +𝑑𝜓𝑠

𝑑𝑡

0 = 𝑅𝑟𝑖𝑟 + 𝑗𝜓𝑠𝑠𝜔𝑠 +𝑑𝜓𝑟

𝑑𝑡 ,

(3.5)

where 𝑠𝜔𝑠 = 𝜔𝑠 − 𝜔𝑟 , and 𝜓𝑠 = 𝐿𝑠𝑖𝑠 + 𝐿𝑚𝑖𝑟, 𝜓𝑟 = 𝐿𝑟𝑖𝑟 + 𝐿𝑚𝑖𝑠. The mutual reluctance

is neglected. It can be seen that the left side of the second equation is zero which

represents no inserted voltage in the rotor circuit.

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Fig. 3.3 Equivalent circuit of a regular induction machine

Similar to the regular induction machines, the difference of DFIG is that there is an

equivalent voltage injection on the rotor winding as shown in Fig. 3.4. Then the

dynamics of DFIG can be written as: [5] [9] [11]

𝑉𝑠 = 𝑅𝑠𝑖𝑠 + 𝑗𝜓𝑠𝜔𝑠 +𝑑𝜓𝑠

𝑑𝑡

𝑉𝑟 = 𝑅𝑟𝑖𝑟 + 𝑗𝜓𝑠𝑠𝜔𝑠 +𝑑𝜓𝑟

𝑑𝑡

(3.6)

Where 𝑠𝜔𝑠 = 𝜔𝑠 − 𝜔𝑟, and 𝜓𝑠 = 𝐿𝑠𝑖𝑠 + 𝐿𝑚𝑖𝑟 , 𝜓𝑟 = 𝐿𝑟𝑖𝑟 + 𝐿𝑚𝑖𝑠.

Fig. 3.4 Equivalent circuit of a DFIG

Rewrite the dynamics in d-q component:

𝑉𝑠𝑑 = 𝑅𝑠𝑖𝑠𝑑 − 𝜔𝑠𝐿𝑠𝑖𝑠𝑞 − 𝜔𝑠𝐿𝑚𝑖𝑟𝑞 + 𝐿𝑠

𝑑𝑖𝑠𝑑𝑑𝑡

+ 𝐿𝑚

𝑑𝑖𝑟𝑑𝑑𝑡

𝑉𝑠𝑞 = 𝑅𝑠𝑖𝑠𝑞 + 𝜔𝑠𝐿𝑠𝑖𝑠𝑑 + 𝜔𝑠𝐿𝑚𝑖𝑟𝑑 + 𝐿𝑠

𝑑𝑖𝑠𝑞

𝑑𝑡+ 𝐿𝑚

𝑑𝑖𝑟𝑞

𝑑𝑡

𝑉𝑟𝑑 = 𝑅𝑟𝑖𝑟𝑑 − 𝑠𝜔𝑠𝐿𝑚𝑖𝑠𝑞 − 𝑠𝜔𝑠𝐿𝑟𝑖𝑟𝑞 + 𝐿𝑚

𝑑𝑖𝑠𝑑𝑑𝑡

+ 𝐿𝑟

𝑑𝑖𝑟𝑑𝑑𝑡

(3.7)

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𝑉𝑟𝑞 = 𝑅𝑟𝑖𝑟𝑞 + 𝑠𝜔𝑠𝐿𝑚𝑖𝑠𝑑 + 𝑠𝜔𝑠𝐿𝑟𝑖𝑟𝑑 + 𝐿𝑚

𝑑𝑖𝑠𝑞𝑑𝑡

+ 𝐿𝑟

𝑑𝑖𝑟𝑞𝑑𝑡

.

3.4.2. Grid-side Dynamics

As shown in Fig. 3.1, the grid-side PWM converter connects to the AC-network through a

transformer which can be seen as a reactance. So the dynamic equations can be

expressed as equation 3.8 [5]. Due to the different control strategy for the grid-side

PWM and rotor-side PWM, so the d-q axis for (3.8) is different to (3.7). This will be

explained in the later section.

𝑉𝑔𝑑1 = 𝑅𝑔𝑖𝑔𝑑1 − 𝜔𝑠𝐿𝑔𝑖𝑔𝑞1 + 𝐿𝑔

𝑑𝑖𝑔𝑑1

𝑑𝑡+ 𝑉𝑠𝑑1

𝑉𝑔𝑞1 = 𝑅𝑔𝑖𝑔𝑞1 + 𝜔𝑠𝐿𝑔𝑖𝑔𝑑1 + 𝐿𝑔

𝑑𝑖𝑔𝑞1

𝑑𝑡+ 𝑉𝑠𝑞1 .

(3.8)

3.5. Control strategy for DFIG

3.5.1. Rotor-side Converter

The objective of rotor-side converter control is as follows [5] [12]:

1. Regulating the DFIG rotor speed for maximum wind power capture;

2. Maintaining the DFIG stator output voltage-frequency constant;

3. Controlling the DFIG reactive power.

It has been shown that, these objectives are commonly achieved by rotor-current

regulation in the stator-flux oriented reference frame. This means 𝜆𝑠𝑞 = 0 which

requires 𝑖𝑞𝑠 = −𝐿𝑚𝑖𝑞𝑟

𝐿𝑠. With this relation, the following expression can be derived:

𝑇𝑒 = −3

2

𝑝

2𝐿𝑚2 𝑖𝑚𝑠𝑖𝑞𝑟 𝐿𝑠⁄ (3.9)

𝑄𝑠 = −3

2𝜔𝑠𝐿𝑚

2 𝑖𝑚𝑠(𝑖𝑚𝑠 − 𝑖𝑑𝑟) 𝐿𝑠⁄ (3.10)

12 | P a g e

𝑉𝑟𝑑 = 𝑅𝑟𝑖𝑟𝑑 + 𝜎𝐿𝑟

𝑑𝑖𝑟𝑑𝑑𝑡

− 𝑠𝜔𝑠𝐿𝑟𝑖𝑟𝑞 (3.11)

𝑉𝑟𝑞 = 𝑅𝑟𝑖𝑟𝑞 + 𝜎𝐿𝑟

𝑑𝑖𝑟𝑞𝑑𝑡

+ 𝑠𝜔𝑠(𝐿𝑚2𝑖𝑚𝑠 𝐿𝑠⁄ + 𝜎𝐿𝑟𝑖𝑟𝑑) , (3.12)

where 𝑖𝑚𝑠 =𝑉𝑠𝑞−𝑅𝑠𝑖𝑠𝑞

𝜔𝑠𝐿𝑚, 𝜎 = 1 −

𝐿𝑚2

𝐿𝑠𝐿𝑟, 𝑝 is the number of poles of the induction machine.

(3.9) and (3.10) indicate that the DFIG rotor speed 𝜔𝑟 can be controlled by regulating

the q-axis rotor current components, 𝑖𝑞𝑟; While the stator reactive power 𝑄𝑠 can be

controlled by regulating the d-axis rotor-current components, 𝑖𝑑𝑟. So, the reference

value 𝑖𝑟𝑑∗ and 𝑖𝑟𝑞

∗ will be determined directly from the stator reactive power error and

DFIG rotor-speed error. Here PI-type speed controller that generates the reference value

𝑖𝑞𝑟∗ for maximum wind power extraction. The speed command 𝜔𝑟

∗ is determined from

the maximum wind power tracking algorithm [3].

(3.11) and (3.12) can be expressed as:

𝑉𝑟𝑑 = (𝑘𝑝𝑟 +𝑘𝑖𝑟

𝑠) (𝑖𝑟𝑑

∗ − 𝑖𝑟𝑑) − 𝑠𝜔𝑠𝐿𝑟𝑖𝑟𝑞

𝑉𝑟𝑞 = (𝑘𝑝𝑟 +𝑘𝑖𝑟

𝑠) (𝑖𝑟𝑞

∗ − 𝑖𝑟𝑞) + 𝑠𝜔𝑠(𝐿𝑚2𝑖𝑚𝑠 𝐿𝑠⁄ + 𝜎𝐿𝑟𝑖𝑟𝑑) .

(3.13)

PI-type controllers are also applied to regulate the current to the reference value.

Therefore the control block of the rotor-side PWM converter is shown in Fig. 3.5.

Fig. 3.5 Rotor-side PWM control block

13 | P a g e

3.5.2. Grid-side converter

The objective of the grid-side converter is to keep the dc-link voltage constant

regardless of the magnitude and direction of the rotor power. This can be achieved by

voltage regulation in stator voltage-reference frame [5] [12]. In the synchronously

rotating reference frame with the d-axis aligned to the grid-voltage vector 𝑉𝑠 (𝑉𝑠 =

𝑉𝑠𝑑 , 𝑉𝑠𝑞 = 0), (3.8) becomes (3.14). Therefore the control block of the grid-side PWM

converter is shown in Fig. 3.6.

𝑉𝑔𝑑1 = (𝑘𝑝𝑔 +𝑘𝑖𝑔

𝑠) (𝑖𝑔𝑑1

∗ − 𝑖𝑔𝑑1) − 𝜔𝑠𝐿𝑔𝑖𝑔𝑞1 + 𝑉𝑠

𝑉𝑔𝑞1 = (𝑘𝑝𝑔 +𝑘𝑖𝑔

𝑠) (𝑖𝑔𝑞1

∗ − 𝑖𝑔𝑞1) + 𝜔𝑠𝐿𝑔𝑖𝑔𝑑1 .

(3.14)

Fig. 3.6 Grid-side PWM control block

As it is shown, the d-q reference frame of each PWM converter is different, so the

subscript d and q in Fig. 3.5 and 3.6 is also different.

14 | P a g e

4. Modeling of a Single DFIG Connected to

an Infinite Bus

4.1. Basic state space of a DFIG

From chapter 3, the dynamic equations for both side have been derived as shown in (3.7)

and (3.8). The subscript 1 indicates the stator voltage reference frame in grid-side PWM

control. The linear time-invariant system can be expressed as:

�̇� = 𝐴 ∙ 𝑥 + 𝐵 ∙ 𝑢

𝑦 = 𝐶 ∙ 𝑥 + 𝐷 ∙ 𝑢 ,

(4.1)

where 𝑥 is state variable, 𝑢 is input variable, 𝑦 is output variable, 𝐴, 𝐵, 𝐶, 𝐷 are

matrix that determines the property of the system. So the target is to represent the DFIG

in this form.

According to the Jacobian Transfer Matrix method, the DFIG should be modeled with the

grid voltage as input 1 and other input signals as input 2. The output should include the

current injected to the grid and other signals depend on the control strategy. So linearize

the dynamic equations in the following form:

𝐵𝑢1𝑢1 = 𝑅𝑥 + 𝐿�̇� + 𝐵𝑢2𝑢2 , (4.2)

where

𝑥 = [∆𝑖𝑠𝑑 ∆𝑖𝑠𝑞 ∆𝑖𝑟𝑑 ∆𝑖𝑟𝑞 ∆𝑖𝑔𝑑1 ∆𝑖𝑔𝑞1]𝑇

𝑢1 = [∆𝑉𝑠𝑑 ∆𝑉𝑠𝑞 ∆𝑉𝑠𝑑1 ∆𝑉𝑠𝑞1 ]𝑇, 𝑢2 = [∆𝜔𝑟 ∆𝑉𝑟𝑑 ∆𝑉𝑟𝑞 ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1]

𝑇

𝐵𝑢1 =

[ 1 0 0 00 1 0 00 0 0 00 0 0 00 0 − 1 00 0 0 − 1]

𝐵𝑢2 =

[

00

00

00

00

00

𝐾1 −1 0 0 0𝐾2 0 −1 0 00 0 0 −1 00 0 0 0 −1 ]

15 | P a g e

𝐿 =

[ 𝐿𝑠 00 𝐿𝑠

𝐿𝑚 00 𝐿𝑚

0 00 0

𝐿𝑚 00 𝐿𝑚

𝐿𝑟 00 𝐿𝑟

0 00 0

0 00 0

0 00 0

𝐿𝑔 0

0 𝐿𝑔]

𝑅 =

[

𝑅𝑠 −𝜔𝑠𝐿𝑠

𝜔𝑠𝐿𝑠 𝑅𝑠

0 −𝜔𝑠𝐿𝑚

𝜔𝑠𝐿𝑚 00 00 0

0 −𝑠0𝜔𝑠𝐿𝑚

𝑠0𝜔𝑠𝐿𝑚 0𝑅𝑟 −𝑠0𝜔𝑠𝐿𝑟

𝑠0𝜔𝑠𝐿𝑟 𝑅𝑟

0 00 0

0 00 0

0 00 0

𝑅𝑔 −𝜔𝑠𝐿𝑔

𝜔𝑠𝐿𝑔 𝑅𝑔 ]

,

𝐾1 = 𝐿𝑟𝑖𝑟𝑞0 + 𝐿𝑚𝑖𝑠𝑞0, 𝐾2 = −𝐿𝑟𝑖𝑟𝑑0 − 𝐿𝑚𝑖𝑠𝑑0, 𝑖𝑠𝑑0, 𝑖𝑠𝑞0, 𝑖𝑟𝑑0, 𝑖𝑟𝑞0 are stator and rotor

current vectors in d-q reference frame in steady state; 𝑠0 is the generator slip in steady

state.

The input signals have been divided into two parts, 𝑢1 and 𝑢2. 𝑢1 is the input signal

from ac-network; 𝑢2 is the input signal from the mechanical block and the control

block. Thus:

�̇� = −𝐿−1𝑅𝑥 + 𝐿−1𝐵𝑢1𝑢1 − 𝐿−1𝐵𝑢2𝑢2 = 𝐴𝑥 + 𝐵1𝑢1 + 𝐵2𝑢2

𝐴 = −𝐿−1𝑅, 𝐵1 = 𝐿−1𝐵𝑢1, 𝐵2 = −𝐿−1𝐵𝑢2 .

(4.3)

For the output, according to the Jacobian transfer matrix modelling technique, the

currents injected to the grid should be considered as output signals. Thus the stator and

grid-side converter currents should be regarded as output signals:

𝑦 = 𝐶𝑥 + 𝐷1𝑢1 + 𝐷2𝑢2 (4.4)

𝑦 = [∆𝑖𝑠𝑑 ∆𝑖𝑠𝑞 ∆𝑖𝑔𝑑1 ∆𝑖𝑔𝑞1]𝑇, 𝐶 = [

1 0 0 0 0 00 1 0 0 0 00 0 0 0 1 00 0 0 0 0 1

]

𝐷1 = [04×4], 𝐷2 = [04×5] .

16 | P a g e

4.2. Coordinate transformation of the DFIG state-space

When using Jacobian transfer matrix modeling method, all the electrical machines are

built in its own d-q rotating reference frame while the ac-network has its own R-I

reference frame. The angle between these two reference frame can be expressed as:

𝜃 = 𝜃0 + ∫ 𝜔𝑡

0𝑑𝑡. For the synchronous generator, itself determines the d-q reference

frame so 𝜃0 ≠ 0 and the state-space of synchronous generator has to be transformed

into AC-network R-I reference frame; For a normal induction machine, the d-q reference

frame can be chosen directly as the R-I frame, which means 𝜃0 = 0, so there is no need

for coordinate transformation [7]. However, as it is shown in the previous section, for a

DFIG, the rotor-side PWM control strategy requires the state-space of DFIG be built in

stator-flux oriented reference frame and the grid-side PWM control requires the d-axis

aligned with the grid voltage vector. Therefore, the DFIG state-space has to be

transformed to the AC-network R-I reference frame. The concept of the coordinate

transformation can be described by Fig. 3.7. As shown in this Fig., the coordinate

transformation only deals with the input and output variables, the state variables are

still in the d-q reference frame.

Fig. 3.7 Coordinate transformation to the R-I frame

The DFIG state-space in the d-q reference frame can be shown as:

17 | P a g e

�̇� = 𝐴𝑥 + 𝐵𝑢𝑑𝑞

𝑦𝑑𝑞 = 𝐶𝑥 + 𝐷𝑢𝑑𝑞

(4.5)

𝑢𝑑𝑞 = [∆𝑉𝑠𝑑 ∆𝑉𝑠𝑞 ∆𝑉𝑠𝑑1 ∆𝑉𝑠𝑞1 ∆𝜔𝑟 ∆𝑉𝑟𝑑 ∆𝑉𝑟𝑞 ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1]𝑇, 𝑦𝑑𝑞 = [∆𝑖𝑠𝑑 ∆𝑖𝑠𝑞 ∆𝑖𝑔𝑑 ∆𝑖𝑔𝑞]

𝑇

The voltage vectors and current vectors which are connected to the grid need to be

transformed. However, ∆𝜔𝑟 is scalar quantity and ∆𝑉𝑟𝑑 , ∆𝑉𝑟𝑞 , ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1 are

controlled directly in d-q reference frame, so these signals do not need to transform. The

steady state angle of the stator flux is 𝜃0 = 𝑎𝑟𝑐𝑡𝑎𝑛𝜓𝑠𝑞0

𝜓𝑠𝑑0 and the steady state angle of

stator voltage 𝜃1 = 𝑎𝑛𝑔𝑙𝑒(𝑉𝑠0). Therefore we have the relation:

𝑢𝑑𝑞 = 𝑢𝑅𝐼𝑒−𝑗𝛿 , 𝑦𝑅𝐼 = 𝑦𝑑𝑞𝑒

𝑗𝛿 (4.6)

In the linearized form:

𝑢𝑑𝑞 = 𝑃𝐸𝑢𝑅𝐼 + 𝑃𝐸1∆𝛿

𝑦𝑅𝐼 = 𝑃𝐼𝑦𝑑𝑞 + 𝑃𝐼1∆𝛿

(4.7)

Where

𝑃𝐸 =

[ cos 𝛿0 sin 𝛿0

−sin 𝛿0 cos 𝛿002×5

cos 𝛿1 sin 𝛿1

−sin 𝛿1 cos 𝛿102×5

05×2 𝑒𝑦𝑒(5)]

𝑃𝐼 = [

cos𝛿0 −sin 𝛿0

sin𝛿0 cos 𝛿002×2

02×2cos 𝛿1 −sin𝛿1

sin 𝛿1 cos𝛿1

]

18 | P a g e

𝑃𝐸1 =

[ −𝑉𝑠𝑅0 sin𝛿0 + 𝑉𝑠𝐼0 cos𝛿0

−𝑉𝑠𝑅0 cos 𝛿0 − 𝑉𝑠𝐼0 sin𝛿0

−𝑉𝑟𝑅0 sin𝛿1 + 𝑉𝑟𝐼0 cos𝛿1

−𝑉𝑟𝑅0 cos 𝛿1 − 𝑉𝑟𝐼0 sin𝛿1

00000 ]

, 𝑃𝐼1 =

[ −𝑖𝑠𝑑0 sin 𝛿0 − 𝑖𝑠𝑞0 cos 𝛿0

−𝑖𝑠𝑞0 sin𝛿0 + 𝑖𝑠𝑑0 cos 𝛿0

−𝑖𝑔𝑑0 sin 𝛿1 − 𝑖𝑔𝑞0 cos 𝛿1

−𝑖𝑔𝑞0 sin𝛿1 + 𝑖𝑔𝑑0 cos 𝛿1]

.

Substituting equation 4.8 into 4.6, yields

�̇� = 𝐴𝑥 + [𝐵𝑃𝐸 𝐵𝑃𝐸1]𝑢𝑅𝐼

𝑦𝑅𝐼 = 𝑃𝐼𝐶𝑥 + [𝑃𝐼𝐷 𝑃𝐼𝐷𝑃𝐸1 + 𝑃𝐼1]𝑢𝑅𝐼

(4.8)

𝑢𝑅𝐼＝[∆𝑉𝑠𝑅 ∆𝑉𝑠𝐼 ∆𝜔𝑟 ∆𝑉𝑟𝑑 ∆𝑉𝑟𝑞 ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1 ∆𝛿]𝑇, 𝑦𝑅𝐼 = [∆𝑖𝑠𝑅 ∆𝑖𝑠𝐼 ∆𝑖𝑔𝑅 ∆𝑖𝑔𝐼]

𝑇.

As we can see, after the coordinate transformation, 𝑢𝑅𝐼 has one additional input

variable ∆𝛿, which is connected to the rotor transfer function.

4.3. Combine DFIG Dynamics with the Network

Dynamics and the Control Strategy

4.3.1. Network Model

The system shown in Fig. 3.1 where a single DFIG is connected to the infinite bus

through a transformer will be used as example. So the network in Fig. 3.1 can be

simplified to Fig. 4.1. The dynamic equation of this network can be expressed by:

Fig. 4.1 Single DFIG connected to an infinite bus

𝐶𝑠

𝑑∆𝐸𝑐

𝑑𝑡= −∆𝑖𝑚 − 𝑗𝜔𝑠𝐶𝑠∆𝐸𝑐 (4.9)

19 | P a g e

∆𝑉𝑠 = −∆𝐸𝑐 − 𝜔𝑠𝐿𝑡𝑚∆𝑖𝑚 − 𝐿𝑡𝑚

𝑑∆𝑖𝑚𝑑𝑡

,

Where 𝑖𝑚 is the sum of the stator current 𝑖𝑠 and the grid-side converter current 𝑖𝑔.

Rewrite the network dynamics into state-space form:

�̇�𝑛 = 𝐴𝑛 ∙ 𝑥𝑛 + 𝐵𝑛 ∙ 𝑢𝑛

𝑦𝑛 = 𝐶𝑛 ∙ 𝑥𝑛 + 𝐷𝑛1 ∙ 𝑢𝑛 + 𝐷𝑛2

𝑑𝑢𝑛

𝑑𝑡

(4.10)

𝑥𝑛 = [∆𝐸𝑐𝑅 ∆𝐸𝑐𝐼]𝑇 , 𝑢𝑛 = [∆𝑖𝑚𝑅 ∆𝑖𝑚𝐼]

𝑇 , 𝑦𝑛 = [∆𝑉𝑠𝑅 ∆𝑉𝑠𝐼]𝑇

𝐴𝑛 = [0 𝜔𝑠

−𝜔𝑠 0] , 𝐵𝑛 =

[ 1

𝐶𝑠0

01

𝐶𝑠]

, 𝐶𝑛 = [1 00 1

]

𝐷𝑛1 = [0 𝜔𝑠𝐿𝑡𝑚

−𝜔𝑠𝐿𝑡𝑚 0] , 𝐷𝑛2 = [

𝐿𝑡𝑚 00 𝐿𝑡𝑚

]

For now, the linearized dynamic system of the DFIG without any control has been

obtained. According to the concepts in Fig. 2.2 and 2.3, there are two options for

continuing the modelling: the first option is to connect the DFIG dynamics to the

network dynamics so that the dynamic system of all electrical components is derived,

and then combine the system with the control strategy proposed in chapter 3.3; the

second option is to equip the DFIG dynamics with the control strategy first, and then

connect it to the network. Both options will be explained in detail and tested in the

system shown in Fig. 4.1.

4.3.2. Option 1

For Option 1, the DFIG dynamics in (4.8) can be written into such form:

�̇�𝑚 = 𝐴𝑚 ∙ 𝑥𝑚 + 𝐵𝑚1 ∙ 𝑢𝑚1 + 𝐵𝑚2 ∙ 𝑢𝑚2

𝑦𝑚 = 𝐶𝑚 ∙ 𝑥𝑚 + 𝐷𝑚1 ∙ 𝑢𝑚1 + 𝐷𝑚2 ∙ 𝑢𝑚2 ,

(4.11)

where 𝑥𝑚 = [∆𝑖𝑠𝑑 ∆𝑖𝑠𝑞 ∆𝑖𝑟𝑑 ∆𝑖𝑟𝑞 ∆𝑖𝑔𝑑1 ∆𝑖𝑔𝑞1]𝑇

, 𝑦𝑚 = [∆𝑖𝑚𝑅 ∆𝑖𝑚𝐼]𝑇 , 𝑢𝑚1＝

20 | P a g e

[∆𝑉𝑠𝑅 ∆𝑉𝑠𝐼 ]𝑇, 𝑢𝑚2＝[∆𝜔𝑟 ∆𝑉𝑟𝑑 ∆𝑉𝑟𝑞 ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1 ∆𝛿]

𝑇.

As explained in [7], the input and output of the DFIG’s model in (4.11) and the network

state-space in (4.10) are reciprocal. Therefore, the state-space model of the combined

electrical systems can be solved as:

�̇�𝑑 = [𝑁(𝐴𝑚 + 𝐵𝑚1𝐷𝑛1𝐶𝑚) 𝑁𝐵𝑚1𝐶𝑛

𝐵𝑛𝐶𝑚 𝐴𝑛] ∙ 𝑥𝑑

+[𝑁(𝐵𝑚1𝐷𝑛1𝐷𝑚2 + 𝐵𝑚2)

𝐵𝑛𝐷𝑚2] ∙ 𝑢𝑑 + [

𝑁𝐵𝑚1𝐷𝑛2𝐷𝑚2

𝑧𝑒𝑟𝑜𝑠]𝑑𝑢𝑑

𝑑𝑡 ,

(4.12)

where 𝑁 = (𝐼 − 𝐵𝑚1𝐷𝑛2𝐶𝑚)−1, 𝑥𝑑 = [𝑥𝑚𝑇 𝑥𝑛

𝑇]𝑇, 𝑢𝑑 = 𝑢𝑚2. Now the output variables

can be chosen randomly. However, due to the requirement on the control strategy, the

signals which are inputs of the control part should be the output of the combined system.

So in this case, the output variables of the system should include ∆𝑖𝑟𝑑, ∆𝑖𝑟𝑞 , ∆𝑖𝑔𝑑1,

∆𝑖𝑔𝑞1, 𝑉𝑠1, 𝑄𝑠 and 𝑇𝑒 . The later 3 variable can be linearized as:

𝑉𝑠1 = cos(𝜃1) 𝑉𝑠𝑅 + sin(𝜃1) 𝑉𝑠𝐼 (4.13)

𝑄𝑠 = −𝑉𝑠𝑑𝑖𝑠𝑞 + 𝑉𝑠𝑞𝑖𝑠𝑑 → −𝑉𝑠𝑑0 ∙ ∆𝑖𝑠𝑞 + 𝑉𝑠𝑞0 ∙ ∆𝑖𝑠𝑑 − 𝑖𝑠𝑞0 ∙ ∆𝑉𝑠𝑑 + 𝑖𝑠𝑑0 ∙ ∆𝑉𝑠𝑞 (4.14)

𝑇𝑒 = 𝜔𝑠𝐿𝑚(𝑖𝑠𝑞𝑖𝑟𝑑 − 𝑖𝑠𝑑𝑖𝑟𝑞)

→ 𝜔𝑠𝐿𝑚(𝑖𝑠𝑞0 ∙ ∆𝑖𝑟𝑑 + 𝑖𝑟𝑑0 ∙ ∆𝑖𝑠𝑞 − 𝑖𝑠𝑑0 ∙ ∆𝑖𝑟𝑞 − 𝑖𝑟𝑞0 ∙ ∆𝑖𝑠𝑑)

(4.15)

So we have 𝑦𝑑 = 𝐶𝑑 ∙ 𝑥𝑑 + 𝐷𝑑 ∙ 𝑢𝑑, which 𝐶𝑑 and 𝐷𝑑 can be determined according to

the relation in (4.13)- (4.15). The state-space representation can be further written in

input-output transfer matrix form

𝑦𝑑 = [𝐶𝑑(𝑠𝐼 − 𝐴𝑑)−1𝐵𝑑 + 𝐷𝑑] ∙ 𝑢𝑑 (4.16)

[𝐶𝑑(𝑠𝐼 − 𝐴𝑑)−1𝐵𝑑 + 𝐷𝑑] is the Jacobian transfer matrix 𝐽(𝑠) which is the linear

description of the electrical part of the system. That is, 𝐽(𝑠) is a 7 × 6 transfer matrix

which has

𝑢𝑑 = [∆𝜔𝑟 ∆𝑉𝑟𝑑 ∆𝑉𝑟𝑞 ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1 ∆𝛿]𝑇

21 | P a g e

𝑦𝑑 = [∆𝑖𝑟𝑑 ∆𝑖𝑟𝑞 ∆𝑖𝑔𝑑1 ∆𝑖𝑔𝑞1 ∆𝑉𝑠1 ∆𝑄𝑠 ∆𝑇𝑒]𝑇

The next step is to connect the control loop which has been proposed in Section 3.5 to

the Jacobian transfer matrix 𝐽(𝑠). This step can be achieved in MATLAB by using

“connect” function, first draw the diagram which includes the Jacobian transfer matrix

and each control block; then number them in sequence and use “append” function to

combine Jacobian transfer matrix 𝐽(𝑠) with all blocks in the same sequence; next the

sequence of all the inputs and outputs is derived; according to the input and output

sequence and the diagram, write the matrix to define how the system is interconnected;

last define which signal is input and output.

As shown in Fig. 4.2, the rotor and grid-side PWM controls the voltage vectors in each

reference frame; due to the dynamic model is dealing with the impact on small change of

each signal, so the reference value of each signal can be neglected; two-mass model is

adopted to represent the turbine’s mechanical behavior.

The red number from “b1” to “b14” defines the sequence of the connection, so in

MATLAB, the function should be ”append (J(s), b1, b2, …, b14)”. Therefore, the sequence

of inputs and outputs can also be obtained as shown in table 1.

22 | P a g e

Fig. 4.2 Complete linear model of the DFIG model using option 1

Table 1 The sequence of inputs and outputs of system in Fig. 4.2

Input Output

1 ∆𝜔𝑟 1 ∆𝑖𝑟𝑑

2 ∆𝑉𝑟𝑑 2 ∆𝑖𝑟𝑞

3 ∆𝑉𝑟𝑞 3 ∆𝑖𝑔𝑑1

4 ∆𝑉𝑔𝑑1 4 ∆𝑖𝑔𝑞1

5 ∆𝑉𝑔𝑞1 5 ∆𝑉𝑠1

6 ∆𝛿 6 ∆𝑄𝑠

23 | P a g e

7 b1 7 ∆𝑇𝑒

8 b2 8 b1

9 b3 9 b2

10 b4 10 b3

11 b5 11 b4

12 b6 12 b5

13 b7 13 b6

14 b8 14 b7

15 b9 15 b8

16 b10 16 b9

17 b11 17 b10

18 b12 18 b11

19 b13 19 b12

20 b14 20 b13

21 b14 21 b14

Then according to Fig. 4.2 and Table 1, the connection matrix can be determined as

shown in Q matrix:

24 | P a g e

𝑄 =

[ 28793111012414135171618201961

82

−1−6111

−2−2154

−317319−57212021

−9010012013014016180

−400000

00000000

−150000000000 ]

The first column stands for the input number, the later 3 columns are outputs which

connect to the input. Each row defines a connection. Take the first row as example: 2

stands for ∆𝑉𝑟𝑑, 8 stands for the output of ‘b1’, 9 stands for the output of ‘b2’, so the first

row means the input of ∆𝑉𝑟𝑑 is the output of ‘b1’ minus the output of ‘b2’.

The next step it to define inputs and outputs for the new combined system. The inputs

and outputs can be defined as needed, to check the frequency response from any part of

the control loop to any output signals. For instance, if we want to check the frequency

response from q-axis rotor reference current 𝑖𝑟𝑞∗ to the electric torque 𝑇𝑒 , these two

signals have to be added as input and output accordingly. In this case, we choose ∆𝜔𝑟,

∆𝑉𝑟𝑑, ∆𝑉𝑟𝑞 as the input signals and ∆𝑇𝑒 as the output signal. So the input and output

matrix become:

𝑖𝑛𝑝𝑢𝑡𝑠 = [1 2 3]

𝑜𝑢𝑡𝑝𝑢𝑡𝑠 = [7]

Connect the Jacobian matrix 𝐽(𝑠), Q matrix, input and output matrix by

𝑐𝑜𝑛𝑛𝑒𝑐𝑡(𝐽(𝑠), 𝑄, 𝑖𝑛𝑝𝑢𝑡𝑠, 𝑜𝑢𝑡𝑝𝑢𝑡𝑠)

Then we derive the whole system’s 3 × 1 transfer matrix.

25 | P a g e

4.3.3. Option 2

Most procedure of option 2 is similar to option 1, the only difference is the way that used

for connecting network dynamics and DFIG dynamics. For option 1, the connecting

method is derived from manual mathematical derivation from []. However, it is difficult

to achieve manually for each case, and the debug work is also inconvenient. Therefore,

same as DFIG model, the network model can be also treated as a module. So the

connecting process can be achieved by using the ‘𝑐𝑜𝑛𝑛𝑒𝑐𝑡’ function in option 1. The only

problem for this concept is that, for the input of network model, there is 𝑑𝑢𝑛

𝑑𝑡 term so

that it need to be transformed. The idea is that treat 𝑑𝑢𝑛

𝑑𝑡 as a new input, as shown in Fig.

4.3, and add a transfer function 𝑠 in 𝑠-domain to achieve the derivative of the signal.

Since the linear model is dealing with small change, so it is reliable to achieve derivative

by multiplying 𝑠.

According to this idea, both DFIG dynamics from and network dynamics need to be

changed slightly:

For DFIG, the outputs should include not only the current vectors ∆𝑖𝑚𝑅 , ∆𝑖𝑚𝐼 which is

used to connect to the network, but also the signals used for the control part:

∆𝑖𝑟𝑑 , ∆𝑖𝑟𝑞 , ∆𝑖𝑔𝑑1, ∆𝑖𝑔𝑞1, ∆𝑉𝑠1, ∆𝑄𝑠, ∆𝑇𝑒;

For network dynamics in Fig. 10, the new form becomes:

�̇�𝑛 = 𝐴𝑛 ∙ 𝑥𝑛 + 𝐵𝑛 ∙ 𝑢𝑛

𝑦𝑛 = 𝐶𝑛 ∙ 𝑥𝑛 + 𝐷𝑛 ∙ 𝑢𝑛

(4.17)

𝑥𝑛 = [∆𝐸𝑐𝑅 ∆𝐸𝑐𝐼]𝑇 , 𝑢𝑛 = [∆𝑖𝑚𝑅 ∆𝑖𝑚𝐼

𝑑∆𝑖𝑚𝑅

𝑑𝑡 𝑑∆𝑖𝑚𝐼

𝑑𝑡]𝑇

, 𝑦𝑛 = [∆𝑉𝑠𝑅 ∆𝑉𝑠𝐼]𝑇

𝐴𝑛 = [0 𝜔𝑠

−𝜔𝑠 0] , 𝐵𝑛 =

[ 1

𝐶𝑠0 0 0

01

𝐶𝑠0 0

]

, 𝐶𝑛 = [1 00 1

]

𝐷𝑛 = [0 𝜔𝑠𝐿𝑡𝑚

−𝜔𝑠𝐿𝑡𝑚 0

𝐿𝑡𝑚 00 𝐿𝑡𝑚

]

26 | P a g e

Which is a 4 × 2 transfer matrix.

Then apply the same connecting method as option1, combine the DFIG model, network

model, control blocks for PWM control and two-mass model.

Fig. 4.3 Complete linear model of the DFIG model using option 2

Option 2 is more clear and simple to follow. However, the process is more complicated,

usually the diagram consists too many blocks. In the following section, only the results of

option 1 will be presented.

4.3.4. Testing for a single DFIG connect to infinite bus

To demonstrate the dynamic performance of the DFIG, the system in Fig. 4.1 is simulated

with PSCAD. In Fig. 4.4, it shows the single DFIG connecting to the infinite bus in PSCAD,

the rating of the DFIG is listed in Table 2. The DFIG Converters and controls page

contains the electrical circuits and the control part as shown in Section 3.3. The DFIG

will start at constant speed mode at first 0.5 s and then switch to the torque control

mode. The most efficient way to validate the linear model in MATLAB, is to compare the

frequency-response curve and the step-response curve. For frequency-response, we

apply the frequency scan module in PSCAD. This module will inject a variable frequency

signal at the input signal, and detect the magnitude response and the phase shift at the

output signal.

27 | P a g e

However, there are several tips when using the frequency scan module in PSCAD:

1. The PWM converter should switch to the average model which use three

controllable voltage sources to control three phase voltage. In principle this is

same as the PWM converter, but the PWM strategy will disable the frequency

signal injection.

2. There should not be any close loop in the system when apply the frequency scan.

The feedback loop will amplify the amplitude which results in an inaccurate

response. So we should run this model first, and derive the value of controlled

signals at steady-state, for instance 𝑉𝑟𝑑 and 𝑉𝑟𝑞. Next set these signals constant

values and switch the DFIG to the constant speed mode, and turn on the

frequency scan module and then the frequency response curve can be obtained.

Fig. 4.4 Single DFIG connected to an infinite bus in PSCAD

Table 2 Parameters of the DFIG

Rated Power 1 MVA Stator Resistance 0.0111 pu

Rated Voltage (L-L) 0.69 kV Wound Rotor Resistance 0.0108 pu

Frequency 60 Hz Magnetizing Inductance 4.7 pu

Stator/ Rotor Ratio 1 Stator Leakage Inductance 0.1487 pu

Angular Moment of Inertia 0.85 s Rotor Leakage Inductance 0.1366 pu

Fig. 4.5-4.7 show the 3 frequency responses from the linear model in Section 4.3.2,

overlapped with the frequency-scanning results from PSCAD, with 𝑉𝑟𝑑 , 𝑉𝑟𝑞 and 𝜔𝑟 as

28 | P a g e

the input signal, electric torque 𝑇𝑒 as the output signal. The red dashed line is from

PSCAD, the black solid line is for using option 1 from MATLAB. Due to limitations of the

applied frequency-scanning technique in PSCAD, only the results with frequencies

higher than 1 Hz are shown.

Fig. 4.5 Frequency-response comparison from 𝑉𝑟𝑑 to 𝑇𝑒

Fig. 4.6 Frequency-response comparison from 𝑉𝑟𝑞 to 𝑇𝑒

29 | P a g e

Fig. 4.7 Frequency-response comparison from 𝜔𝑟 to 𝑇𝑒

As it is shown in the figures above, the linear model gives identical results with

frequency-scanning results in every case. Due to the inaccurate response detection in the

lower frequency in PSCAD, there are some differences when the frequency is lower than

2 Hz. As it has shown, the identical results prove the accuracy of the linear model in

MATLAB.

Fig. 4.8- 4.10 shows the time-domain comparison between the linear model in MATLAB

and simulation in PSCAD. The step responses from 𝜔𝑟, 𝑉𝑟𝑑 and 𝑉𝑟𝑞 to output 𝑇𝑒 are

shown accordingly. The red dashed line stands for the results from PSCAD, the black

solid line is derived from ‘step’ function in MATLAB. One thing should be noted that the

magnitude of red dashed line from PSCAD has been adjusted in order to compare with

the results from MATLAB.

30 | P a g e

Fig. 4.8 Step response from 𝜔𝑟 to 𝑇𝑒

Fig. 4.9 Step response from 𝑉𝑟𝑑 to 𝑇𝑒

31 | P a g e

Fig. 4.10 Step response from 𝑉𝑟𝑞 to 𝑇𝑒

32 | P a g e

5. Modeling of DFIGs Connecting to a

VSC-HVDC converter

5.1. Modeling of a VSC-HVDC converter

The modeling of the VSC-HVDC converter based on Jacobian transfer matrix is proposed

and validated in [7]. The detailed derivation will not be presented. Fig. 5.1 shows the

main circuit of the VSC-HVDC converter. The dynamic equations of the VSC-HVDC

converter can be expressed as:

Fig. 5.1 Main circuit of the VSC-HVDC converter

𝐿𝑐

𝑑𝑖𝑐𝑑𝑡

= (𝑉0 + ∆𝑉)𝑒𝑗𝜃𝑣 − 𝐻𝐻𝑃(𝑠)𝑖𝑐 − 𝑢𝑓 − 𝑅𝑐𝑖𝑐 − 𝑗𝜔𝑠𝐿𝑐𝑖𝑐

𝐶𝑓

𝑑𝑢𝑓

𝑑𝑡= 𝑖𝑐 − 𝑖𝑣 − 𝑗𝜔𝑠𝐶𝑓𝑢𝑓

𝐿𝑣

𝑑𝑖𝑣𝑑𝑡

= 𝑢𝑓 − 𝑒𝑣 − 𝑗𝜔𝑠𝐿𝑣𝑖𝑣

(5.1)

𝐻𝐻𝑃(𝑠) is the Laplace transform variable, it has been shown in [7], 𝐻𝐻𝑃(𝑠) can be

eliminated by introducing new state variables. As a result, the dynamic equation of the

VSC-HVDC converter can be written in state-space form:

𝑑𝑥𝑣

𝑑𝑡= 𝐴𝑣𝑥𝑣 + 𝐵𝑣1𝑢𝑣1 + 𝐵𝑣2𝑢𝑣2

𝑦𝑣 = 𝐶𝑣𝑥𝑣

(5.2)

33 | P a g e

where

𝐴𝑣 =

[ −𝛼𝑣 −

𝑅𝑐 + 𝑘𝑣

𝐿𝑐−𝜔𝑠

−𝛼𝑣𝑅𝑐

𝐿𝑐−𝜔𝑠𝛼𝑣

1

𝐶𝑓

000

−𝜔𝑠

−𝛼𝑣 −𝑅𝑐 + 𝑘𝑣

𝐿𝑐𝜔𝑠𝛼𝑣

−𝛼𝑣𝑅𝑐

𝐿𝑐

01

𝐶𝑓

00

10000000

01000000

−1

𝐿𝑐

0

−𝛼𝑣

𝐿𝑐

00

−𝜔𝑠

1

𝐿𝑣

0

0

−1

𝐿𝑐

0

−𝛼𝑣

𝐿𝑐𝜔𝑠

001

𝐿𝑣

0000

−1

𝐶𝑓

00

−𝜔𝑠

00000

−1

𝐶𝑓

𝜔𝑠

0

]

𝐵𝑣1 =

[

02×6

−1

𝐿𝑣0

0 −1

𝐿𝑣] 𝑇

𝐵𝑣2 =

[ −

𝑉0𝑠𝑖𝑛𝜃𝑣0

𝐿𝑐

𝑉0𝑐𝑜𝑠𝜃𝑣0

𝐿𝑐

𝑉0𝑐𝑜𝑠𝜃𝑣0

𝐿𝑐

𝑉0𝑠𝑖𝑛𝜃𝑣0

𝐿𝑐

−𝛼𝑣𝑉0𝑠𝑖𝑛𝜃𝑣0

𝐿𝑐

𝛼𝑣𝑉0𝑐𝑜𝑠𝜃𝑣0

𝐿𝑐

𝛼𝑣𝑉0𝑐𝑜𝑠𝜃𝑣0

𝐿𝑐

𝛼𝑣𝑉0𝑠𝑖𝑛𝜃𝑣0

𝐿𝑐

02×4

] 𝑇

𝐶𝑣 = [02×61 00 1

] ∙𝑆𝑁𝑣

𝑆𝑁𝑛

And the inputs, outputs and state variables are

𝑢𝑣1 = [∆𝑒𝑣𝑑 ∆𝑒𝑣𝑞]𝑇 , 𝑢𝑣2 = [∆𝜃𝑣

∆𝑉

𝑉0]𝑇

, 𝑦𝑣 = [∆𝑖𝑣𝑑 ∆𝑖𝑣𝑞]𝑇

𝑥𝑣 = [∆𝑖𝑐𝑑 ∆𝑖𝑐𝑞 ∆𝜌𝑐𝑑 ∆𝜌𝑐𝑞 ∆𝑢𝑓𝑑 ∆𝑢𝑓𝑞 ∆𝑖𝑣𝑑 ∆𝑖𝑣𝑞]𝑇

(5.3)

The input 𝑢𝑣1 is the voltage-vector connected to the AC-network; The input 𝑢𝑣2 is

control signals based on the power-synchronization control. For the VSC-HVDC

converter, the d-q reference frame can be chosen as same as AC-network R-I frame. Thus,

there is no need for the coordinate transformation. 𝑆𝑁𝑣 and 𝑆𝑁𝑛 are rated power of

the VSC-HVDC converter and the AC-network. The reason of C matrix multiplying 𝑆𝑁𝑣

𝑆𝑁𝑛 is

the different rated power between the network and the VSC-HVDC converter. When the

VSC-HVDC converter connects to the AC-network through a transformer, the voltages do

not need to transform while the currents need to be transformed depending on the rated

power. This current-base transformation is necessary for all the electrical machines

34 | P a g e

connecting to the AC-network when the base value is different. In the later section,

DFIGs will also equip this transformation.

As explained in [7], ∆𝜃𝑣 is connected to the active-power controller, while ∆𝑉

𝑉0 is

connected to the alternating-voltage controller. The control block diagram of a

VSC-HVDC converter using Power-synchronization control can be shown in Fig. 5.2. 𝑘𝑝,

𝑘𝑢, 𝑘𝑣, 𝛼𝑣, 𝐾𝑓 , 𝑇𝑓 and 𝑇𝑚 are controller parameters of the VSC-HVDC converter. The

frequency-controller loop is only using when the system does not have frequency

regulation. For instance, if there is an island system which does not connect to a large

AC-network, then the VSC-HVDC converter needs to keep the frequency constant, thus,

the frequency control should be adopted instead of constant power control. It should

also be noted that the ’Load compensation’ in Fig. 5.2 is in the nonlinear form, it requires

linearization if their effects are taking into consideration.

Fig. 5.2 Control block diagram of a VSC-HVDC converter using Power-synchronization

control

5.2. Connecting 2 DFIGs to the VSC-HVDC converter

The system’s diagram can be shown in Fig. 5.3. 2 DFIGs and one VSC-HVDC converter

connects in parallel to an infinite bus. The parameters of DFIGs is the same as shown in

35 | P a g e

Table 2. The parameters of the VSC-HVDC converter and the infinite bus are shown in

Table 3.

Fig. 5.3 the system diagram of two DFIGs connecting to a VSC-HVDC converter

Table 3 Rated value of the VSC-HVDC and the infinite bus

VSC-HVDC Infinite bus

Rated Power 350 MVA 100 MVA

Rated Voltage (L-L) 195 kV 195 kV

5.2.1. Network Model

The dynamic equations of the AC-network in Fig. 5.3 can be expressed as

𝐶𝑠

𝑑𝑈𝑐

𝑑𝑡= 𝑖𝑚1 + 𝑖𝑚2 + 𝑖𝑣 − 𝑖𝑓 − 𝑗𝜔𝑠𝐶𝑠𝑈𝑐

𝐿𝑓

𝑑𝑖𝑓

𝑑𝑡= 𝑈𝑐 − 1 − 𝑅𝑓𝑖𝑓 − 𝑗𝜔𝑠𝐿𝑓𝑖𝑓

𝑉𝑠1 = 𝑈𝑐 + 𝑗𝜔𝑠𝐿𝑡𝑚1𝑖𝑚1 + 𝐿𝑡𝑚1

𝑑𝑖𝑚1

𝑑𝑡

𝑉𝑠2 = 𝑈𝑐 + 𝑗𝜔𝑠𝐿𝑡𝑚2𝑖𝑚2 + 𝐿𝑡𝑚2

𝑑𝑖𝑚2

𝑑𝑡

(5.4)

36 | P a g e

𝑉𝑣 = 𝑈𝑐 + 𝑗𝜔𝑠𝐿𝑣𝑖𝑣 + 𝐿𝑣

𝑑𝑖𝑣𝑑𝑡

In order to eliminate the complex component j, (5.4) can be written in AC-network R-I

reference frame:

𝐶𝑠

𝑑𝑈𝑐𝑅

𝑑𝑡= 𝑖𝑚1𝑅 + 𝑖𝑚2𝑅 + 𝑖𝑣𝑅 − 𝑖𝑓𝑅 + 𝜔𝑠𝐶𝑠𝑈𝑐𝐼

𝐶𝑠

𝑑𝑈𝑐𝐼

𝑑𝑡= 𝑖𝑚1𝐼 + 𝑖𝑚2𝐼 + 𝑖𝑣𝐼 − 𝑖𝑓𝐼 − 𝜔𝑠𝐶𝑠𝑈𝑐𝑅

𝐿𝑓

𝑑𝑖𝑓𝑅

𝑑𝑡= 𝑈𝑐𝑅 − 1 − 𝑅𝑓𝑖𝑓𝑅 + 𝜔𝑠𝐿𝑓𝑖𝑓𝐼

𝐿𝑓

𝑑𝑖𝑓𝐼

𝑑𝑡= 𝑈𝑐𝐼 − 1 − 𝑅𝑓𝑖𝑓𝐼 − 𝜔𝑠𝐿𝑓𝑖𝑓𝑅

𝑉𝑠1𝑅 = 𝑈𝑐𝑅 − 𝜔𝑠𝐿𝑡𝑚1𝑖𝑚1𝐼 + 𝐿𝑡𝑚1

𝑑𝑖𝑚1𝑅

𝑑𝑡

𝑉𝑠1𝐼 = 𝑈𝑐𝐼 + 𝜔𝑠𝐿𝑡𝑚1𝑖𝑚1𝑅 + 𝐿𝑡𝑚1

𝑑𝑖𝑚1𝐼

𝑑𝑡

𝑉𝑠2𝑅 = 𝑈𝑐𝑅 − 𝜔𝑠𝐿𝑡𝑚2𝑖𝑚2𝐼 + 𝐿𝑡𝑚2

𝑑𝑖𝑚2𝑅

𝑑𝑡

𝑉𝑠2𝐼 = 𝑈𝑐𝐼 + 𝜔𝑠𝐿𝑡𝑚2𝑖𝑚2𝑅 + 𝐿𝑡𝑚2

𝑑𝑖𝑚2𝐼

𝑑𝑡

𝑉𝑣𝑅 = 𝑈𝑐𝑅 − 𝜔𝑠𝐿𝑣𝑖𝑣𝐼 + 𝐿𝑣

𝑑𝑖𝑣𝑅

𝑑𝑡

𝑉𝑣𝐼 = 𝑈𝑐𝐼 + 𝜔𝑠𝐿𝑣𝑖𝑣𝑅 + 𝐿𝑣

𝑑𝑖𝑣𝐼

𝑑𝑡

(5.5)

According to the Jacobian transfer matrix method, by linearization and writing in

component form, (5.5) can be expressed in state-space form

�̇�𝑛 = 𝐴𝑛 ∙ 𝑥𝑛 + 𝐵𝑛 ∙ 𝑢𝑛

𝑦𝑛 = 𝐶𝑛 ∙ 𝑥𝑛 + 𝐷𝑛1 ∙ 𝑢𝑛 + 𝐷𝑛2

𝑑𝑢𝑛

𝑑𝑡

(5.6)

where

37 | P a g e

𝐴𝑛 =

[

0 𝜔𝑠

−𝜔𝑠 0−1 𝐶𝑠⁄ 0

0 −1 𝐶𝑠⁄

−1 𝐿𝑓⁄ 0

0 −1 𝐿𝑓⁄

−𝑅𝑓 𝐿𝑓⁄ 𝜔𝑠

−𝜔𝑠 −𝑅𝑓 𝐿𝑓⁄ ]

𝐵𝑛 = [

1 𝐶𝑠⁄ 0

0 1 𝐶𝑠⁄1 𝐶𝑠⁄ 0

0 1 𝐶𝑠⁄1 𝐶𝑠⁄ 0

0 1 𝐶𝑠⁄02×6

]

𝐶𝑛 =

[ 1 00 11 00 11 00 1

06×2

]

𝐷𝑛1 =

[

0 −𝜔𝑠𝐿𝑡𝑚1

𝜔𝑠𝐿𝑡𝑚1 00 00 0

0 00 0

0 00 0

0 −𝜔𝑠𝐿𝑡𝑚2

𝜔𝑠𝐿𝑡𝑚2 00 00 0

0 00 0

0 00 0

0 −𝜔𝑠𝐿𝑣

𝜔𝑠𝐿𝑣 0 ]

𝐷𝑛2 =

[ 𝐿𝑡𝑚1 0

0 𝐿𝑡𝑚1

0 00 0

0 00 0

0 00 0

𝐿𝑡𝑚2 00 𝐿𝑡𝑚2

0 00 0

0 00 0

0 00 0

𝐿𝑣 00 𝐿𝑣]

With the state variables: 𝑥𝑛 = [∆𝑈𝑐𝑅 ∆𝑈𝑐𝐼 ∆𝑖𝑓𝑅 ∆𝑖𝑓𝐼]𝑇

, the input variables: 𝑢𝑛 =

[∆𝑖𝑚1𝑅 ∆𝑖𝑚1𝐼 ∆𝑖𝑚2𝑅 ∆𝑖𝑚2𝐼 ∆𝑖𝑣𝑅 ∆𝑖𝑣𝐼]𝑇 and the output variables: 𝑦𝑛 =

[∆𝑉𝑠1𝑅 ∆𝑉𝑠1𝐼 ∆𝑉𝑠2𝑅 ∆𝑉𝑠2𝐼 ∆𝑉𝑣𝑅 ∆𝑉𝑣𝐼]𝑇.

5.2.2. Jacobian Transfer Matrix J(s) of the System

According to the (4.11), the linear model of ‘DFIG1’ can be expressed as:

�̇�𝑚1 = 𝐴𝑚1 ∙ 𝑥𝑚1 + 𝐵𝑚11 ∙ 𝑢𝑚11 + 𝐵𝑚12 ∙ 𝑢𝑚12

𝑦𝑚1 = 𝐶𝑚1 ∙ 𝑥𝑚1 + 𝐷𝑚11 ∙ 𝑢𝑚11 + 𝐷𝑚12 ∙ 𝑢𝑚12

(5.7)

For ‘DFIG2’:

�̇�𝑚2 = 𝐴𝑚2 ∙ 𝑥𝑚2 + 𝐵𝑚21 ∙ 𝑢𝑚21 + 𝐵𝑚22 ∙ 𝑢𝑚22 (5.8)

38 | P a g e

𝑦𝑚2 = 𝐶𝑚2 ∙ 𝑥𝑚2 + 𝐷𝑚21 ∙ 𝑢𝑚21 + 𝐷𝑚22 ∙ 𝑢𝑚22

As mentioned in Section 5.1, the DFIGs also need to equip the current-base

transformation. So in this case, 𝐶𝑚1 = 𝐶𝑚2 = 𝐶𝑚 ∙𝑆𝑁𝑚

𝑆𝑁𝑛, where 𝐶𝑚 is from (4.11), 𝑆𝑁𝑚

and 𝑆𝑁𝑛 are the base power of the DFIG and the VSC-HVDC converter respectively.

Similar to the procedure in Section 4.3.2, the first step is to derive the Jacobian transfer

matrix 𝐽(𝑠) of the combined electrical system based on the model of the DFIG in (5.7)

and (5.8), the VSC-HVDC model in (5.2) and the AC-network model in (5.6). Since there

are more than one electric component connected to the AC-network, the DFIGs and the

VSC-HVDC need to be lumped into one state-space model as []:

�̇�𝑧 = 𝐴𝑧 ∙ 𝑥𝑧 + 𝐵𝑧1 ∙ 𝑢𝑧1 + 𝐵𝑧2 ∙ 𝑢𝑧2

𝑦𝑧 = 𝐶𝑧 ∙ 𝑥𝑧 + 𝐷𝑧2 ∙ 𝑢𝑧2

(5.9)

where 𝑥𝑧 = [𝑥𝑚1𝑇 𝑥𝑚2

𝑇 𝑥𝑣𝑇]𝑇 , 𝑢𝑧1 = [𝑢𝑚11

𝑇 𝑢𝑚21𝑇 𝑢𝑣1

𝑇 ]𝑇, 𝑢𝑧2 = [𝑢𝑚12𝑇 𝑥𝑚22

𝑇 𝑥𝑣2𝑇 ]𝑇 ,

𝑦𝑧 = [𝑦𝑚1𝑇 𝑦𝑚2

𝑇 𝑦𝑣𝑇]𝑇 . 𝐴𝑧 , 𝐵𝑧1 , 𝐵𝑧2 , 𝐶𝑧 , 𝐷𝑧2 are the block diagonal matrices

composed of the corresponding input-device matrices in (), () and (), i.e.,

𝐴𝑧 = [

𝐴𝑚1

𝐴𝑚2

𝐴𝑣

] , 𝐵𝑧1 = [

𝐵𝑚11

𝐵𝑚21

𝐵𝑣1

] , 𝐵𝑧2 = [

𝐵𝑚12

𝐵𝑚22

𝐵𝑣2

]

𝐶𝑧 = [

𝐶𝑚1

𝐶𝑚2

𝐶𝑣

] , 𝐷𝑧2 = [𝐷𝑚12

𝐷𝑚22

𝑧𝑒𝑟𝑜𝑠

].

(5.10)

The reason of a non-zero 𝐷𝑧2 is the coordinate transformation of DFIGs, the

current-vectors ∆𝑖𝑚1 and ∆𝑖𝑚2 are related to the rotor dynamic ∆𝛿 . For the

VSC-HVDC converter, since there is no coordinate transformation, the current-vector ∆𝑖𝑣

is independent on the rotor dynamic ∆𝛿.

The input and output of the state-space model in (5.9) and AC-network model (5.2) are

reciprocal. Then the state-space model of the combined electrical systems can be solved

as:

39 | P a g e

�̇�𝑑 = [𝑁(𝐴𝑧 + 𝐵𝑧1𝐷𝑛1𝐶𝑧) 𝑁𝐵𝑧1𝐶𝑛

𝐵𝑛𝐶𝑧 𝐴𝑛] ∙ 𝑥𝑑

+[𝑁(𝐵𝑧1𝐷𝑛1𝐷𝑧2 + 𝐵𝑧2)

𝐵𝑛𝐷𝑧2] ∙ 𝑢𝑑 + [

𝑁𝐵𝑧1𝐷𝑛2𝐷𝑧2

𝑧𝑒𝑟𝑜𝑠]𝑑𝑢𝑑

𝑑𝑡

(5.11)

where 𝑁 = (𝐼 − 𝐵𝑧1𝐷𝑛2𝐶𝑧)−1, 𝑥𝑑 = [𝑥𝑧

𝑇 𝑥𝑛𝑇]𝑇 , 𝑢𝑑 = 𝑢𝑧2. For outputs, same principle

as in Section 4.3.2: the signals which are inputs of the control part should be the output

of the combined system. Therefore, for each DFIG, ∆𝑖𝑟𝑑, ∆𝑖𝑟𝑞 , ∆𝑖𝑔𝑑1, ∆𝑖𝑔𝑞1, 𝑉𝑠1, 𝑄𝑠

and 𝑇𝑒 should be output signals; for the VSC-HVDC converter, the active power P and

the voltage 𝑈𝑓 should also be included as output signals. Accordingly, by having all the

output variables expressed by state variables and input signals, the output of the

electrical system in Fig. 6.3 can be expressed by

𝑦𝑑 = 𝐶𝑑 ∙ 𝑥𝑑 + 𝐷𝑑 ∙ 𝑢𝑑 (5.12)

The input-output transfer matrix form can be written as

𝑦𝑑 = [𝐶𝑑(𝑠𝐼 − 𝐴𝑑)−1𝐵𝑑 + 𝐷𝑑] ∙ 𝑢𝑑 (5.13)

[𝐶𝑑(𝑠𝐼 − 𝐴𝑑)−1𝐵𝑑 + 𝐷𝑑] is the Jacobian transfer matrix 𝐽(𝑠) which is the linear

description of the electrical part of the island system. That is, 𝐽(𝑠) is a 16 × 14

transfer matrix which has

𝑢𝑑 = [∆𝜔𝑟1 ∆𝑉𝑟𝑑1 ∆𝑉𝑟𝑞1 ∆𝑉𝑔𝑑1 ∆𝑉𝑔𝑞1 ∆𝛿1 ∆𝜔𝑟2 ∆𝑉𝑟𝑑2 ∆𝑉𝑟𝑞2 ∆𝑉𝑔𝑑2 ∆𝑉𝑔𝑞2 ∆𝛿2 ∆𝜃𝑣 ∆𝑉

𝑉0

]𝑇

𝑦𝑑 = [∆𝑖𝑟𝑑1 ∆𝑖𝑟𝑞1 ∆𝑖𝑔𝑑1 ∆𝑖𝑔𝑞1 ∆𝑉𝑠1 ∆𝑄𝑠1 ∆𝑇𝑒1 ∆𝑖𝑟𝑑2 ∆𝑖𝑟𝑞2 ∆𝑖𝑔𝑑2 ∆𝑖𝑔𝑞2 ∆𝑉𝑠2 ∆𝑄𝑠2 ∆𝑇𝑒2 ∆𝑃 ∆𝑈𝑓]𝑇

(5.14)

The next step is to connect control loop to the Jacobian transfer matrix 𝐽(𝑠). Fig. 5.4 is

the simplified linear model of the Jacobian transfer matrix 𝐽(𝑠) combined with control

blocks. In Fig. 5.4, the ‘VSC control blocks’ refers to the control blocks in Fig. 5.2; ‘DFIG1

Control Blocks’ and ‘DFIG2 Control Blocks’ are the same as the outer control loops of a

single DFIG in Fig. 4.2. Due to the complexity of the detailed model, only the simplified

diagram is presented to show the concept. This step is accomplished by using ‘connect’

function in MATLAB, same as Section 4.3.2.

40 | P a g e

Then the linear model of the system in Fig. 5.3 is derived, the input and output signal can

be chosen depending on which signals are interested for the frequency response

Fig. 5.4 Simplified linear model of the system in Fig. 5.3

5.2.3. Testing for two DFIGs and a VSC-HVDC System

In this section, the identical system in Fig 5.3 is simulated with PSCAD. Fig. 5.5 shows the

two DFIGs and a VSC HVDC converter connected to an infinite bus in PSCAD. In order to

validate the Jacobian transfer matrix 𝐽(𝑠), so the frequency scan should be derived

when the feedback control is turned off. The specific process in PSCAD is as follows: first,

run the system when all the control blocks operates normally; second, when the system

reaches steady state, decrease the gain of each feedback loop as much as possible, as

long as the system remaining stable; then turn on the frequency scan module and derive

the frequency response.

41 | P a g e

Fig. 5.5 Two DFIGs and a VSC-HVDC system in PSCAD

Fig. 5.6-5.8 are the frequency-response comparison from ∆𝜃𝑣 to ∆𝑃, ∆𝜃𝑣 to ∆𝑈𝑓 and

∆𝑉𝑟𝑞 to ∆𝑇𝑒 respectively. The black solid line is the bode plot from the linear model in

MATLAB. The red dashed line is the frequency response from PSCAD. The upper plot of

each response is the amplitude response and the lower one is the phase shift.

Fig 5.6 Frequency-response comparison from ∆𝜃𝑣 to ∆𝑃

Fig 5.7 Frequency-response comparison from ∆𝜃𝑣 to ∆𝑈𝑓

42 | P a g e

Fig. 5.8 Frequency-response comparison from ∆𝑉𝑟𝑞 to ∆𝑇𝑒

As it is shown in the figures above, the linear model gives identical results with

frequency-scanning results which proves the accuracy of the linear model.

6. Conclusions

Doubly Fed Induction Generators show many advantages compared to other variable

speed concept. This thesis mainly focuses on the linear modeling of DFIGs connected

system.

The Jacobian transfer matrix modeling is adopted as the linearization method and

summarized in details. The power system is viewed as a multivariable feedback-control

system. Two modeling options are illustrated and tested. It has been shown that the

Jacobian transfer matrix modeling method is a suitable choice to model a large power

system.

The Doubly Fed Induction Generator as the main concern of this thesis, is introduced

comprehensively. Based on Jacobian transfer matrix, the linear model of a DFIG is

derived, including the dynamics of the DFIG, the dynamics of wind turbine and the

control strategy of both rotor-side and grid-side PWM converter.

43 | P a g e

Two systems are studied and simulated in PSCAD. The first system is a single DFIG

connected to an infinite bus. The second system is two DFIGs and a VSC-HVDC converter

connected to an infinite bus. By comparing the results from linear model and PSCAD, it

has already shown that the linear model has the same frequency response for any input/

output signal. Therefore the linear modeling method for DFIGs connected system is

accurate and the linear model can reflect the dynamic performance of the non-linear

system.

7. Future Work

The following is a list of possible future work:

1. Analyze the equivalent model for multiple DFIGs. One wind farm usually

contains plenty of DFIGs, and the reality is each DFIG will have an influence on

each other, i.e. there is interaction within the wind farm. Only increasing the

rating of a DFIG is not an accurate representation when analyzing multiple

DFIGs system. Also it is not feasible to build all the DFIGs into simulation

software. Therefore it is necessary to find the equivalent representation of

multiple DFIGs which can capture the dynamic behavior of the wind farm;

2. Carry out frequency domain analysis of wind integrated HVDC system and study

the interaction of the converter controls especially in the sub-synchronous

range;

3. Based on the analysis, provide possible recommendations for control design of

the VSC-HVDC stations while integrating large offshore wind farms.

44 | P a g e

1 | P a g e

Reference

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University, 2001.

[2] Dechao Kong, “Advanced HVDC Systems for Renewable Energy Integration and Power

Transmission: Modeling and Control for Power System Transient Stability”, Ph.D dissertation,

University of Birmingham, 2013.

[3] Yazhou Lei, Alan Mullane, Gordon Lightbody and Robert Yacamini, “Modeling of the Wind

Turbine With a Doubly Fed Induction Generator for Grid Integration Studies”, IEEE Trans. Energy

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[4] Jesu ́s Lo ́pez, Eugenio Gub ́ıa, Pablo Sanchis, Xavier Roboam and Luis Marroyo, “Wind

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[8] Lingling Fan, Rajesh Kavasseri, Zhixin Lee Miao and Chanxia Zhu, “Modeling of DFIG-Based

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[11] R. Pena, J. C. Clare and G. M. Asher, “A doubly fed induction generator using back-to-back

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2 | P a g e

[12] R. Pena, J. C. Clare and G. M. Asher, “Doubly fed induction generator using back-to-back

PWM converters and its application to variable-speed wind-energy generation”.

[13] Yin Chin Choo, A. P. Agalgaonkar, K. M. Muttaqi, S. Perera and M. Negnevitsky,

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TRITA EE 2015:68

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