Linear Modeling of DFIGs andVSC-HVDC Systems

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IN

DEGREE PROJECT MASTER&APOS;S PROGRAMME, ELECTRIC POWER , SECOND CYCLE

ENGINEERING 120 CREDITS STOCKHOLM SWEDEN 2015,

Linear Modeling of DFIGs and VSC-HVDC Systems

WEIRAN CAO

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING

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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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i | P a g e

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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ii | P a g e

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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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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vi | P a g e

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 𝑉_𝑟𝑑 to 𝑇_𝑒 28

Fig. 4.6 Frequency-response comparison from 𝑉_𝑟𝑞 to 𝑇_𝑒 28

Fig. 4.7 Frequency-response comparison from 𝜔_𝑟 to 𝑇_𝑒 29

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Fig. 4.8 Step response from 𝜔_𝑟 to 𝑇_𝑒 30

Fig. 4.9 Step response from 𝑉_𝑟𝑑 to 𝑇_𝑒 30

Fig. 4.10 Step response from 𝑉_𝑟𝑞 to 𝑇_𝑒 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 ∆𝜃_𝑣 to ∆𝑃 41

Fig. 5.7 Frequency-response comparison from ∆𝜃_𝑣 to ∆𝑈_𝑓 41

Fig. 5.8 Frequency-response comparison from ∆𝑉_𝑟𝑞 to ∆𝑇_𝑒 42

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viii | P a g e List of symbols

𝑃_𝑚 The mechanical power of the wind turbine;

𝜌 The air density;

𝑅 The turbine radius;

𝑉_𝑤 The wind speed;

𝐶_𝑝 The power factor of wind turbine;

𝜆 The tip speed ratio;

𝛽 The blade pitch-angle;

𝑐₁− 𝑐₆ Turbine’s coefficients;

𝜔_𝑡 The rotational speed of wind turbine;

𝐻_𝑡 The inertia constant of the turbine;

𝐾_𝑠ℎ The shaft stiffness;

𝜃_𝑡𝑤 The shaft twist angle;

𝑇_𝑚 The mechanical torque of the wind turbine;

𝐷_𝑠ℎ The shaft damping constant;

𝐻_𝑔 The inertia constant of the generator;

𝜔_𝑟 The generator-rotor speed;

𝑇_𝑒 The electrical torque of the generator;

𝑉_𝑠 The generator-stator voltage;

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ix | P a g e 𝑅_𝑠 The stator resistance;

𝑖_𝑠 The stator current;

𝜓_𝑠 The stator-winding flux;

𝜔_𝑠 The synchrous speed;

𝑅_𝑟 The rotor resistance;

𝑖_𝑟 The rotor current;

𝑠 The generator slip;

𝜓_𝑟 The rotor-winding flux;

𝐿_𝑠 The stator inductance;

𝐿_𝑚 The mutual inductance;

𝐿_𝑟 The rotor inductance;

𝑉_𝑟 The rotor voltage;

𝑉_𝑠𝑑, 𝑉_𝑠𝑞 The stator voltage in d-q frame;

𝑉_𝑟𝑑, 𝑉_𝑟𝑞 The rotor voltage in d-q frame;

𝑖_𝑠𝑑, 𝑖_𝑠𝑞 The stator current in d-q frame;

𝑖_𝑟𝑑, 𝑖_𝑟𝑞 The rotor current in d-q frame;

𝑉_𝑔𝑑1, 𝑉_𝑔𝑞1 The grid-side voltage in d-q frame;

𝑖_𝑔𝑑1, 𝑖_𝑔𝑞1 The grid-side current in d-q frame;

𝑄_𝑠 The stator-reactive power;

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x | P a g e 𝑖_𝑟𝑑^∗ , 𝑖_𝑟𝑞^∗ The current-reference value in d-q frame;

𝜔_𝑟^∗ The rotor-speed reference value;

𝑠₀ The generator slip at steady-state;

𝑖_𝑠𝑑0, 𝑖_𝑠𝑞0 The stator current in d-q frame at steady-state;

𝑖_𝑟𝑑0, 𝑖_𝑟𝑞0 The rotor current in d-q frame at steady-state;

𝜓_𝑠𝑑0, 𝜓_𝑠𝑞0 The stator-winding flux in d-q frame at steady-state;

𝑉_𝑠𝑅, 𝑉_𝑠𝐼 The stator voltage in ac-network R-I frame;

𝑖_𝑠𝑅, 𝑖_𝑠𝐼 The stator current in ac-network R-I frame;

𝑖_𝑔𝑅, 𝑖_𝑔𝐼 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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2 | P a g e 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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3 | P a g e 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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5 | P a g e 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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6 | P a g e 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]:

𝑃_𝑚 =1

2𝜌𝜋𝑅²𝑉_𝑤³𝐶_𝑝 , (3.1)

where 𝑃_𝑚 is the mechanical power; 𝜌 is the air density; 𝑅 is the turbine radius; 𝑉_𝑤 is the wind speed; 𝐶_𝑝 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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8 | P a g e 𝐶_𝑝(𝜆, 𝛽) = 𝑐₁(𝑐₂𝑅

𝜆 − 𝑐₃∙ 𝛽 − 𝑐₄) ∙ 𝑒^−𝑐⁵^𝑅^𝜆+ 𝑐₆∙ 𝜆 𝜆 =𝑅𝜔_𝑡

𝑉_𝑤 ,

(3.2)

where 𝑐₁-𝑐₆ are turbine’s coefficients that depends on the design; 𝜔_𝑡 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. 𝜔_𝑡 is adjusted to keep the tip speed ratio 𝜆 at the level that the power 𝑃_𝑚 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]:

2𝐻_𝑡𝑑𝜔_𝑡

𝑑𝑡 = 𝑇_𝑚− 𝐾_𝑠ℎ𝜃_𝑡𝑤− 𝐷_𝑠ℎ𝑑𝜃_𝑡𝑤 𝑑𝑡 2𝐻_𝑔𝑑𝜔_𝑟

𝑑𝑡 = 𝐾_𝑠ℎ𝜃_𝑡𝑤+ 𝐷_𝑠ℎ𝑑𝜃_𝑡𝑤 𝑑𝑡 − 𝑇_𝑒 𝑑𝜃_𝑡𝑤

𝑑𝑡 = 𝜔_𝑡− 𝜔_𝑟 .

(3.3)

Rewrite the two-mass model into state-space form, with the state variables 𝑥 = [∆𝜔_𝑡 ∆𝜔_𝑟 ∆𝜃_𝑡𝑤]^𝑇, the input variables 𝑢 = [∆𝑇_𝑒 ∆𝑇_𝑚]^𝑇 and the output variable 𝑦 = [∆𝜔_𝑟]. Then it becomes:

𝑥̇ = 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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9 | P a g e 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:

𝑉_𝑠= 𝑅_𝑠𝑖_𝑠+ 𝑗𝜓_𝑠𝜔_𝑠+𝑑𝜓_𝑠 𝑑𝑡 0 = 𝑅_𝑟𝑖_𝑟+ 𝑗𝜓_𝑠𝑠𝜔_𝑠+𝑑𝜓_𝑟

𝑑𝑡 ,

(3.5)

where 𝑠𝜔_𝑠 = 𝜔_𝑠− 𝜔_𝑟, and 𝜓_𝑠 = 𝐿_𝑠𝑖_𝑠+ 𝐿_𝑚𝑖_𝑟, 𝜓_𝑟 = 𝐿_𝑟𝑖_𝑟+ 𝐿_𝑚𝑖_𝑠. 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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10 | P a g e 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]

𝑉_𝑠= 𝑅_𝑠𝑖_𝑠+ 𝑗𝜓_𝑠𝜔_𝑠+𝑑𝜓_𝑠 𝑑𝑡 𝑉_𝑟 = 𝑅_𝑟𝑖_𝑟+ 𝑗𝜓_𝑠𝑠𝜔_𝑠+𝑑𝜓_𝑟

𝑑𝑡

(3.6)

Where 𝑠𝜔_𝑠= 𝜔_𝑠− 𝜔_𝑟, and 𝜓_𝑠= 𝐿_𝑠𝑖_𝑠+ 𝐿_𝑚𝑖_𝑟, 𝜓_𝑟 = 𝐿_𝑟𝑖_𝑟+ 𝐿_𝑚𝑖_𝑠.

Fig. 3.4 Equivalent circuit of a DFIG Rewrite the dynamics in d-q component:

𝑉_𝑠𝑑 = 𝑅_𝑠𝑖_𝑠𝑑− 𝜔_𝑠𝐿_𝑠𝑖_𝑠𝑞− 𝜔_𝑠𝐿_𝑚𝑖_𝑟𝑞+ 𝐿_𝑠𝑑𝑖_𝑠𝑑

𝑑𝑡 + 𝐿_𝑚𝑑𝑖_𝑟𝑑 𝑑𝑡 𝑉_𝑠𝑞 = 𝑅_𝑠𝑖_𝑠𝑞+ 𝜔_𝑠𝐿_𝑠𝑖_𝑠𝑑 + 𝜔_𝑠𝐿_𝑚𝑖_𝑟𝑑+ 𝐿_𝑠𝑑𝑖_𝑠𝑞

𝑑𝑡 + 𝐿_𝑚𝑑𝑖_𝑟𝑞 𝑑𝑡 𝑉_𝑟𝑑 = 𝑅_𝑟𝑖_𝑟𝑑− 𝑠𝜔_𝑠𝐿_𝑚𝑖_𝑠𝑞− 𝑠𝜔_𝑠𝐿_𝑟𝑖_𝑟𝑞+ 𝐿_𝑚𝑑𝑖_𝑠𝑑

𝑑𝑡 + 𝐿_𝑟𝑑𝑖_𝑟𝑑 𝑑𝑡

(3.7)

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11 | P a g e 𝑉_𝑟𝑞 = 𝑅_𝑟𝑖_𝑟𝑞+ 𝑠𝜔_𝑠𝐿_𝑚𝑖_𝑠𝑑+ 𝑠𝜔_𝑠𝐿_𝑟𝑖_𝑟𝑑+ 𝐿_𝑚𝑑𝑖_𝑠𝑞

𝑑𝑡 + 𝐿_𝑟𝑑𝑖_𝑟𝑞 𝑑𝑡 .

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.

𝑉_𝑔𝑑1= 𝑅_𝑔𝑖_𝑔𝑑1− 𝜔_𝑠𝐿_𝑔𝑖_𝑔𝑞1+ 𝐿_𝑔𝑑𝑖_𝑔𝑑1 𝑑𝑡 + 𝑉_𝑠𝑑1 𝑉_𝑔𝑞1= 𝑅_𝑔𝑖_𝑔𝑞1+ 𝜔_𝑠𝐿_𝑔𝑖_𝑔𝑑1+ 𝐿_𝑔𝑑𝑖_𝑔𝑞1

𝑑𝑡 + 𝑉_𝑠𝑞1 .

(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 𝜆_𝑠𝑞= 0 which requires 𝑖_𝑞𝑠 = −^𝐿^𝑚^𝑖^𝑞𝑟

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

𝑇_𝑒= −3 2 𝑝

2𝐿²_𝑚𝑖_𝑚𝑠𝑖_𝑞𝑟⁄ 𝐿_𝑠 (3.9)

𝑄_𝑠= −3

2𝜔_𝑠𝐿²_𝑚𝑖_𝑚𝑠(𝑖_𝑚𝑠− 𝑖_𝑑𝑟) 𝐿⁄ _𝑠 (3.10)

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12 | P a g e 𝑉_𝑟𝑑 = 𝑅_𝑟𝑖_𝑟𝑑+ 𝜎𝐿_𝑟𝑑𝑖_𝑟𝑑

𝑑𝑡 − 𝑠𝜔_𝑠𝐿_𝑟𝑖_𝑟𝑞 (3.11)

𝑉_𝑟𝑞= 𝑅_𝑟𝑖_𝑟𝑞+ 𝜎𝐿_𝑟𝑑𝑖_𝑟𝑞

𝑑𝑡 + 𝑠𝜔_𝑠(𝐿_𝑚²𝑖_𝑚𝑠⁄ + 𝜎𝐿𝐿_𝑠 _𝑟𝑖_𝑟𝑑) , (3.12)

where 𝑖_𝑚𝑠 =^𝑉^𝑠𝑞_𝜔^−𝑅^𝑠^𝑖^𝑠𝑞

𝑠𝐿_𝑚 , 𝜎 = 1 −_𝐿^𝐿^𝑚²

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

(3.9) and (3.10) indicate that the DFIG rotor speed 𝜔_𝑟 can be controlled by regulating the q-axis rotor current components, 𝑖_𝑞𝑟; While the stator reactive power 𝑄_𝑠 can be controlled by regulating the d-axis rotor-current components, 𝑖_𝑑𝑟. So, the reference value 𝑖_𝑟𝑑^∗ and 𝑖_𝑟𝑞^∗ 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 𝑖_𝑞𝑟^∗ for maximum wind power extraction. The speed command 𝜔_𝑟^∗ is determined from the maximum wind power tracking algorithm [3].

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

𝑉_𝑟𝑑= (𝑘_𝑝𝑟+𝑘_𝑖𝑟

𝑠 ) (𝑖_𝑟𝑑^∗ − 𝑖_𝑟𝑑) − 𝑠𝜔_𝑠𝐿_𝑟𝑖_𝑟𝑞 𝑉_𝑟𝑞= (𝑘_𝑝𝑟+𝑘_𝑖𝑟

𝑠 ) (𝑖_𝑟𝑞^∗ − 𝑖_𝑟𝑞) + 𝑠𝜔_𝑠(𝐿_𝑚²𝑖_𝑚𝑠⁄ + 𝜎𝐿𝐿_𝑠 _𝑟𝑖_𝑟𝑑) .

(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

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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 𝑉_𝑠 (𝑉_𝑠= 𝑉_𝑠𝑑, 𝑉_𝑠𝑞 = 0), (3.8) becomes (3.14). Therefore the control block of the grid-side PWM converter is shown in Fig. 3.6.

𝑉_𝑔𝑑1= (𝑘_𝑝𝑔+𝑘_𝑖𝑔

𝑠 ) (𝑖_𝑔𝑑1^∗ − 𝑖_𝑔𝑑1) − 𝜔_𝑠𝐿_𝑔𝑖_𝑔𝑞1+ 𝑉_𝑠 𝑉_𝑔𝑞1= (𝑘_𝑝𝑔+𝑘_𝑖𝑔

𝑠 ) (𝑖_𝑔𝑞1^∗ − 𝑖_𝑔𝑞1) + 𝜔_𝑠𝐿_𝑔𝑖_𝑔𝑑1 .

(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.

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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:

𝑥̇ = 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:

𝐵_𝑢1𝑢₁= 𝑅𝑥 + 𝐿𝑥̇ + 𝐵_𝑢2𝑢₂ , (4.2)

where

𝑥 = [∆𝑖𝑠𝑑 ∆𝑖_𝑠𝑞 ∆𝑖_𝑟𝑑 ∆𝑖_𝑟𝑞 ∆𝑖_𝑔𝑑1 ∆𝑖_𝑔𝑞1]^𝑇

𝑢₁= [∆𝑉_𝑠𝑑 ∆𝑉_𝑠𝑞 ∆𝑉_𝑠𝑑1 ∆𝑉_𝑠𝑞1 ]^𝑇, 𝑢₂= [∆𝜔_𝑟 ∆𝑉_𝑟𝑑 ∆𝑉_𝑟𝑞 ∆𝑉_𝑔𝑑1 ∆𝑉_𝑔𝑞1]^𝑇

𝐵_𝑢1= [

1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0

0 0 0 − 1]

𝐵_𝑢2= [ 0

0 0

0

𝐾₁ −1 0 0 0

𝐾₂ 0 −1 0 0

0 0 0 −1 0 0 0 0 0 −1 ]

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15 | P a g e 𝐿 =

[ 𝐿_𝑠 0

0 𝐿_𝑠 𝐿_𝑚 0

0 𝐿_𝑚 0 0

𝐿_𝑚 0 0 0 0 𝐿_𝑚

𝐿_𝑟 0

0 𝐿_𝑟 0 0

0 0 0 0

0 0

𝐿_𝑔 0 0 𝐿_𝑔]

𝑅 =

[

𝑅_𝑠 −𝜔_𝑠𝐿_𝑠 𝜔_𝑠𝐿_𝑠 𝑅_𝑠

0 −𝜔_𝑠𝐿_𝑚

𝜔_𝑠𝐿_𝑚 0 0 0

0 0 0 −𝑠₀𝜔_𝑠𝐿_𝑚

𝑠₀𝜔_𝑠𝐿_𝑚 0 𝑅_𝑟 −𝑠₀𝜔_𝑠𝐿_𝑟

𝑠₀𝜔_𝑠𝐿_𝑟 𝑅_𝑟 0 0

0 0 0 0

0 0

0 0 0 0

𝑅_𝑔 −𝜔_𝑠𝐿_𝑔 𝜔_𝑠𝐿_𝑔 𝑅_𝑔 ]

,

𝐾₁= 𝐿_𝑟𝑖_𝑟𝑞0+ 𝐿_𝑚𝑖_𝑠𝑞0, 𝐾₂= −𝐿_𝑟𝑖_𝑟𝑑0− 𝐿_𝑚𝑖_𝑠𝑑0, 𝑖_𝑠𝑑0, 𝑖_𝑠𝑞0, 𝑖_𝑟𝑑0, 𝑖_𝑟𝑞0 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:

𝑥̇ = −𝐿⁻¹𝑅𝑥 + 𝐿⁻¹𝐵_𝑢1𝑢₁− 𝐿⁻¹𝐵_𝑢2𝑢₂= 𝐴𝑥 + 𝐵₁𝑢₁+ 𝐵₂𝑢₂ 𝐴 = −𝐿⁻¹𝑅, 𝐵₁= 𝐿⁻¹𝐵_𝑢1, 𝐵₂ = −𝐿⁻¹𝐵_𝑢2 .

(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 = [∆𝑖_𝑠𝑑 ∆𝑖_𝑠𝑞 ∆𝑖_𝑔𝑑1 ∆𝑖_𝑔𝑞1]^𝑇, 𝐶 = [

1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ]

𝐷₁= [0_4×4], 𝐷₂= [0_4×5] .

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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+ ∫ 𝜔₀^𝑡 𝑑𝑡. 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:

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17 | P a g e 𝑥̇ = 𝐴𝑥 + 𝐵𝑢_𝑑𝑞

𝑦_𝑑𝑞= 𝐶𝑥 + 𝐷𝑢_𝑑𝑞

(4.5)

𝑢_𝑑𝑞 = [∆𝑉_𝑠𝑑 ∆𝑉_𝑠𝑞 ∆𝑉_𝑠𝑑1 ∆𝑉_𝑠𝑞1 ∆𝜔_𝑟 ∆𝑉_𝑟𝑑 ∆𝑉_𝑟𝑞 ∆𝑉_𝑔𝑑1 ∆𝑉_𝑔𝑞1]^𝑇, 𝑦_𝑑𝑞 = [∆𝑖_𝑠𝑑 ∆𝑖_𝑠𝑞 ∆𝑖_𝑔𝑑 ∆𝑖_𝑔𝑞]^𝑇 The voltage vectors and current vectors which are connected to the grid need to be transformed. However, ∆𝜔_𝑟 is scalar quantity and ∆𝑉_𝑟𝑑, ∆𝑉_𝑟𝑞, ∆𝑉_𝑔𝑑1 ∆𝑉_𝑔𝑞1 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= 𝑎𝑟𝑐𝑡𝑎𝑛^𝜓^𝑠𝑞0

𝜓_𝑠𝑑0 and the steady state angle of stator voltage 𝜃₁= 𝑎𝑛𝑔𝑙𝑒(𝑉_𝑠0). Therefore we have the relation:

𝑢_𝑑𝑞= 𝑢_𝑅𝐼𝑒^−𝑗𝛿, 𝑦_𝑅𝐼= 𝑦_𝑑𝑞𝑒^𝑗𝛿 (4.6)

In the linearized form:

𝑢_𝑑𝑞= 𝑃_𝐸𝑢_𝑅𝐼+ 𝑃_𝐸1∆𝛿 𝑦_𝑅𝐼 = 𝑃_𝐼𝑦_𝑑𝑞+ 𝑃_𝐼1∆𝛿

(4.7)

Where

𝑃_𝐸= [

cos 𝛿₀ sin 𝛿₀

−sin 𝛿₀ cos 𝛿₀ 0_2×5 cos 𝛿₁ sin 𝛿₁

−sin 𝛿₁ cos 𝛿₁ 0_2×5

0_5×2 𝑒𝑦𝑒(5)]

𝑃_𝐼= [

cos 𝛿₀ −sin 𝛿₀

sin 𝛿₀ cos 𝛿₀ 0_2×2 0_2×2 cos 𝛿₁ −sin 𝛿₁

sin 𝛿₁ cos 𝛿₁ ]

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18 | P a g e 𝑃_𝐸1=

[

−𝑉_𝑠𝑅0sin 𝛿₀+ 𝑉_𝑠𝐼0cos 𝛿₀

−𝑉_𝑠𝑅0cos 𝛿₀− 𝑉_𝑠𝐼0sin 𝛿₀

−𝑉_𝑟𝑅0sin 𝛿₁+ 𝑉_𝑟𝐼0cos 𝛿₁

−𝑉_𝑟𝑅0cos 𝛿₁− 𝑉_𝑟𝐼0sin 𝛿₁ 0

00

00 ]

, 𝑃_𝐼1= [

−𝑖_𝑠𝑑0sin 𝛿₀− 𝑖_𝑠𝑞0cos 𝛿₀

−𝑖_𝑠𝑞0sin 𝛿₀+ 𝑖_𝑠𝑑0cos 𝛿₀

−𝑖_𝑔𝑑0sin 𝛿₁− 𝑖_𝑔𝑞0cos 𝛿₁

−𝑖_𝑔𝑞0sin 𝛿₁+ 𝑖_𝑔𝑑0cos 𝛿₁] .

Substituting equation 4.8 into 4.6, yields

𝑥̇ = 𝐴𝑥 + [𝐵𝑃_𝐸 𝐵𝑃_𝐸1]𝑢_𝑅𝐼 𝑦_𝑅𝐼 = 𝑃_𝐼𝐶𝑥 + [𝑃_𝐼𝐷 𝑃_𝐼𝐷𝑃_𝐸1+ 𝑃_𝐼1]𝑢_𝑅𝐼

(4.8)

𝑢_𝑅𝐼＝[∆𝑉_𝑠𝑅 ∆𝑉_𝑠𝐼 ∆𝜔_𝑟 ∆𝑉_𝑟𝑑 ∆𝑉_𝑟𝑞 ∆𝑉_𝑔𝑑1 ∆𝑉_𝑔𝑞1 ∆𝛿]^𝑇, 𝑦_𝑅𝐼= [∆𝑖𝑠𝑅 ∆𝑖_𝑠𝐼 ∆𝑖_𝑔𝑅 ∆𝑖_𝑔𝐼]^𝑇. As we can see, after the coordinate transformation, 𝑢_𝑅𝐼 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

𝐶_𝑠𝑑∆𝐸_𝑐

𝑑𝑡 = −∆𝑖_𝑚− 𝑗𝜔_𝑠𝐶_𝑠∆𝐸_𝑐 (4.9)

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19 | P a g e

∆𝑉_𝑠= −∆𝐸_𝑐− 𝜔_𝑠𝐿_𝑡𝑚∆𝑖_𝑚− 𝐿_𝑡𝑚𝑑∆𝑖_𝑚 𝑑𝑡 ,

Where 𝑖_𝑚 is the sum of the stator current 𝑖_𝑠 and the grid-side converter current 𝑖_𝑔. Rewrite the network dynamics into state-space form:

𝑥̇_𝑛 = 𝐴_𝑛∙ 𝑥_𝑛+ 𝐵_𝑛∙ 𝑢_𝑛

𝑦_𝑛 = 𝐶_𝑛∙ 𝑥_𝑛+ 𝐷_𝑛1∙ 𝑢_𝑛+ 𝐷_𝑛2𝑑𝑢_𝑛 𝑑𝑡

(4.10)

𝑥_𝑛 = [∆𝐸_𝑐𝑅 ∆𝐸_𝑐𝐼]^𝑇, 𝑢_𝑛= [∆𝑖_𝑚𝑅 ∆𝑖_𝑚𝐼]^𝑇, 𝑦_𝑛 = [∆𝑉_𝑠𝑅 ∆𝑉_𝑠𝐼]^𝑇

𝐴_𝑛= [ 0 𝜔_𝑠

−𝜔_𝑠 0 ] , 𝐵^𝑛 = [

1 𝐶_𝑠 0

0 1

𝐶_𝑠]

, 𝐶_𝑛= [1 00 1]

𝐷_𝑛1= [ 0 𝜔_𝑠𝐿_𝑡𝑚

−𝜔_𝑠𝐿_𝑡𝑚 0 ] , 𝐷^𝑛2= [𝐿_𝑡𝑚 0 0 𝐿_𝑡𝑚]

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:

𝑥̇_𝑚 = 𝐴_𝑚∙ 𝑥_𝑚+ 𝐵_𝑚1∙ 𝑢_𝑚1+ 𝐵_𝑚2∙ 𝑢_𝑚2 𝑦_𝑚 = 𝐶_𝑚∙ 𝑥_𝑚+ 𝐷_𝑚1∙ 𝑢_𝑚1+ 𝐷_𝑚2∙ 𝑢_𝑚2 ,

(4.11)

where 𝑥_𝑚 = [∆𝑖𝑠𝑑 ∆𝑖_𝑠𝑞 ∆𝑖_𝑟𝑑 ∆𝑖_𝑟𝑞 ∆𝑖_𝑔𝑑1 ∆𝑖_𝑔𝑞1]^𝑇 , 𝑦_𝑚= [∆𝑖_𝑚𝑅 ∆𝑖_𝑚𝐼]^𝑇 , 𝑢_𝑚1＝

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20 | P a g e [∆𝑉_𝑠𝑅 ∆𝑉_𝑠𝐼 ]^𝑇, 𝑢_𝑚2＝[∆𝜔_𝑟 ∆𝑉_𝑟𝑑 ∆𝑉_𝑟𝑞 ∆𝑉_𝑔𝑑1 ∆𝑉_𝑔𝑞1 ∆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:

𝑥̇_𝑑 = [𝑁(𝐴_𝑚+ 𝐵_𝑚1𝐷_𝑛1𝐶_𝑚) 𝑁𝐵_𝑚1𝐶_𝑛 𝐵_𝑛𝐶_𝑚 𝐴_𝑛 ] ∙ 𝑥_𝑑 + [𝑁(𝐵_𝑚1𝐷_𝑛1𝐷_𝑚2+ 𝐵_𝑚2)

𝐵_𝑛𝐷_𝑚2 ] ∙ 𝑢_𝑑+ [𝑁𝐵^𝑚1𝐷_𝑛2𝐷_𝑚2 𝑧𝑒𝑟𝑜𝑠 ]𝑑𝑢_𝑑

𝑑𝑡 ,

(4.12)

where 𝑁 = (𝐼 − 𝐵_𝑚1𝐷_𝑛2𝐶_𝑚)⁻¹, 𝑥_𝑑 = [𝑥_𝑚^𝑇 𝑥_𝑛^𝑇]^𝑇, 𝑢_𝑑= 𝑢_𝑚2. 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 ∆𝑖_𝑟𝑑, ∆𝑖_𝑟𝑞, ∆𝑖_𝑔𝑑1,

∆𝑖_𝑔𝑞1, 𝑉_𝑠1, 𝑄_𝑠 and 𝑇_𝑒. The later 3 variable can be linearized as:

𝑉_𝑠1= cos(𝜃₁) 𝑉_𝑠𝑅+ sin(𝜃₁) 𝑉_𝑠𝐼 (4.13)

𝑄_𝑠= −𝑉_𝑠𝑑𝑖_𝑠𝑞+ 𝑉_𝑠𝑞𝑖_𝑠𝑑 → −𝑉_𝑠𝑑0∙ ∆𝑖_𝑠𝑞+ 𝑉_𝑠𝑞0∙ ∆𝑖_𝑠𝑑− 𝑖_𝑠𝑞0∙ ∆𝑉_𝑠𝑑+ 𝑖_𝑠𝑑0∙ ∆𝑉_𝑠𝑞 (4.14)

𝑇_𝑒 = 𝜔_𝑠𝐿_𝑚(𝑖_𝑠𝑞𝑖_𝑟𝑑− 𝑖_𝑠𝑑𝑖_𝑟𝑞)

→ 𝜔_𝑠𝐿_𝑚(𝑖_𝑠𝑞0∙ ∆𝑖_𝑟𝑑+ 𝑖_𝑟𝑑0∙ ∆𝑖_𝑠𝑞− 𝑖_𝑠𝑑0∙ ∆𝑖_𝑟𝑞− 𝑖_𝑟𝑞0∙ ∆𝑖_𝑠𝑑)

(4.15)

So we have 𝑦_𝑑= 𝐶_𝑑∙ 𝑥_𝑑+ 𝐷_𝑑∙ 𝑢_𝑑, which 𝐶_𝑑 and 𝐷_𝑑 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

𝑦_𝑑= [𝐶_𝑑(𝑠𝐼 − 𝐴_𝑑)⁻¹𝐵_𝑑+ 𝐷_𝑑] ∙ 𝑢_𝑑 (4.16)

[𝐶_𝑑(𝑠𝐼 − 𝐴_𝑑)⁻¹𝐵_𝑑+ 𝐷_𝑑] 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

𝑢_𝑑= [∆𝜔𝑟 ∆𝑉_𝑟𝑑 ∆𝑉_𝑟𝑞 ∆𝑉_𝑔𝑑1 ∆𝑉_𝑔𝑞1 ∆𝛿]^𝑇

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21 | P a g e 𝑦_𝑑= [∆𝑖_𝑟𝑑 ∆𝑖_𝑟𝑞 ∆𝑖_𝑔𝑑1 ∆𝑖_𝑔𝑞1 ∆𝑉_𝑠1 ∆𝑄_𝑠 ∆𝑇_𝑒]^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.

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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 ∆𝜔_𝑟 1 ∆𝑖_𝑟𝑑

2 ∆𝑉_𝑟𝑑 2 ∆𝑖_𝑟𝑞

3 ∆𝑉_𝑟𝑞 3 ∆𝑖_𝑔𝑑1

4 ∆𝑉_𝑔𝑑1 4 ∆𝑖_𝑔𝑞1

5 ∆𝑉_𝑔𝑞1 5 ∆𝑉_𝑠1

6 ∆𝛿 6 ∆𝑄_𝑠

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23 | P a g e

7 b1 7 ∆𝑇_𝑒

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:

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24 | P a g e 𝑄 =

[ 2 87 93 1110 124 1413 175 1618 2019 6 1

8

−12

−611

−21

−215

−34 173

−519 217 20 21

−9 100 120 130 140 160 180

−40 00 0 0

0 00 00 00

−150 00 00 00 00 0 0 ]

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 ∆𝑉_𝑟𝑑, 8 stands for the output of ‘b1’, 9 stands for the output of ‘b2’, so the first row means the input of ∆𝑉_𝑟𝑑 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 𝑖_𝑟𝑞^∗ to the electric torque 𝑇_𝑒, these two signals have to be added as input and output accordingly. In this case, we choose ∆𝜔_𝑟,

∆𝑉_𝑟𝑑, ∆𝑉_𝑟𝑞 as the input signals and ∆𝑇_𝑒 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.

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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 ^𝑑𝑢^𝑛

𝑑𝑡 term so that it need to be transformed. The idea is that treat ^𝑑𝑢^𝑛

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 ∆𝑖_𝑚𝑅, ∆𝑖_𝑚𝐼 which is used to connect to the network, but also the signals used for the control part:

∆𝑖_𝑟𝑑, ∆𝑖_𝑟𝑞, ∆𝑖_𝑔𝑑1, ∆𝑖_𝑔𝑞1, ∆𝑉_𝑠1, ∆𝑄_𝑠, ∆𝑇_𝑒;

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

𝑥̇_𝑛 = 𝐴_𝑛∙ 𝑥_𝑛+ 𝐵_𝑛∙ 𝑢_𝑛 𝑦_𝑛= 𝐶_𝑛∙ 𝑥_𝑛+ 𝐷_𝑛∙ 𝑢_𝑛

(4.17)

𝑥_𝑛= [∆𝐸_𝑐𝑅 ∆𝐸_𝑐𝐼]^𝑇, 𝑢_𝑛= [∆𝑖𝑚𝑅 ∆𝑖_𝑚𝐼 𝑑∆𝑖_𝑚𝑅

𝑑𝑡 𝑑∆𝑖_𝑚𝐼 𝑑𝑡 ]

𝑇

, 𝑦_𝑛= [∆𝑉_𝑠𝑅 ∆𝑉_𝑠𝐼]^𝑇

𝐴_𝑛 = [ 0 𝜔_𝑠

−𝜔_𝑠 0 ] , 𝐵^𝑛= [

1

𝐶_𝑠 0 0 0

0 1

𝐶_𝑠 0 0

]

, 𝐶_𝑛= [1 00 1]

𝐷_𝑛= [ 0 𝜔_𝑠𝐿_𝑡𝑚

−𝜔_𝑠𝐿_𝑡𝑚 0

𝐿_𝑡𝑚 0 0 𝐿_𝑡𝑚]

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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.

Linear Modeling of DFIGs andVSC-HVDC Systems

Linear Modeling of DFIGs and VSC-HVDC Systems

Linear Modeling of DFIGs and VSC-HVDC Systems

Weiran Cao

Abstract

Sammanfattning

Keywords

Acknowledgements

Contents

List of Tables

List of Figures

1. Introduction

1.1. Background

1.2. Project Objective and Outline

1.2.1. Objective

1.2.2. Outline

2. Jacobian Transfer Matrix Method

3. Doubly Fed Induction Generator

3.1. DFIG introduction

3.2. Modeling of Wind Turbine

3.3. Turbine-Generator Mechanical Model

3.4. Dynamic Equivalent Circuit

3.4.1. Rotor-side Dynamics

3.4.2. Grid-side Dynamics

3.5. Control strategy for DFIG

3.5.1. Rotor-side Converter

3.5.2. Grid-side converter

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

4.1. Basic state space of a DFIG

4.2. Coordinate transformation of the DFIG state-space

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

4.3.1. Network Model

4.3.2. Option 1

4.3.3. Option 2

4.3.4. Testing for a single DFIG connect to infinite bus