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Generic Wind Turbine Generator Model Comparison basedon Optimal Parameter Fitting

by

Zhen Dai

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

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

Copyright c© 2014 by Zhen Dai

Abstract

Generic Wind Turbine Generator Model Comparison based on Optimal Parameter

Fitting

Zhen Dai

Master of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

2014

Parameter fitting will facilitate model validation of the generic dynamic model for type-3

WTGs. In this thesis, a test system including a single 1.5 MW DFIG has been built and

tested in the PSCAD/EMTDC environment for dynamic responses. The data generated

during these tests have been used as measurements for the parameter fitting which is

carried out using the unscented Kalman filter. Two variations of the generic type-3

WTG model (the detailed model and the simplified model) have been compared and

used for parameter estimation. The detailed model is able to capture the dynamics

caused by the converter and thus has been used for parameter fitting when inputs are

from a fault scenario. On the other hand, the simplified model works well for parameter

fitting when a wind speed disturbance is of interest. Given measurements from PSCAD,

the estimated parameters using both models are indeed improvements compared to the

original belief of the parameters in terms of prediction error.

ii

Acknowledgements

First and foremost I wish to express my thanks to Professor Zeb Tate for his invaluable

guidance, advice and patience.

I also want to express my gratitude to my thesis committee members: Professor Peter

Lehn, Professor Reza Iravani and Professor Jason Anderson, for their insights and sug-

gestions.

I would like to thank the members of Energy Systems Group at University of Toronto

for their selfless help.

Finally, I’d like to thank my grandmother and my parents.

iii

Contents

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Parameter Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Generic Models of Type-3 Wind Turbine Generators 8

2.1 An Introduction To The Type-3 Wind

Turbine Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Generic Wind Turbine Generator (WTG) Models . . . . . . . . . 9

2.1.2 Type-3 Generic WTG Models . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Power Flow in a DFIG . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Aerodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Aerodynamic Model with Cp Curves - Two-Dimensional Model . . 17

2.2.2 Aerodynamic Model without Cp Curves - Linear Model . . . . . . 19

2.2.3 Comparison of the Two Models . . . . . . . . . . . . . . . . . . . 22

2.3 The Turbine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

iv

2.4 The Pitch Control Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Generator and Converter Model . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.1 Generator and Converter Model . . . . . . . . . . . . . . . . . . . 30

2.5.2 Converter Controller . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.3 The Scope of the Models . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Test System and Model Performance . . . . . . . . . . . . . . . . . . . . 40

2.6.1 Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.2 Model Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Type-3 WTG Generic Models for Parameter Fitting 45

3.1 Type-3 WTG Detailed Transient Stability Model M1 . . . . . . . . . . . 46

3.2 Reduced-order Type-3 WTG Transient Stability Model M2 . . . . . . . . 52

3.3 Discrete Type-3 WTG Generic Models . . . . . . . . . . . . . . . . . . . 55

3.4 Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1 Equilibrium and initialization . . . . . . . . . . . . . . . . . . . . 59

3.4.2 Model test results . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 System Identifiability 72

5 UKF Parameter Estimation 78

5.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 UKF Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.1 Annealing Method . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.2 Forgetting-Factor Method . . . . . . . . . . . . . . . . . . . . . . 85

5.2.3 Robbins-Monro Method . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.4 Parameter Estimation with ReBEL toolkit . . . . . . . . . . . . . 86

5.3 Parameter Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Estimation results and analysis . . . . . . . . . . . . . . . . . . . 89

5.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

v

6 Conclusion 103

6.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Appendices 106

A The Torque Control Model 107

B System Parameters 109

Bibliography 112

vi

List of Tables

2.1 Notations of the Aerodynamic Model [26] . . . . . . . . . . . . . . . . . . 17

2.2 Cp curve coefficients αi,j [26] . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Parameters of the Turbine Model [39] . . . . . . . . . . . . . . . . . . . . 25

2.4 Parameters of the Pitch Control Model [39] . . . . . . . . . . . . . . . . . 27

2.5 Base Quantities Used in The Converter Controllers . . . . . . . . . . . . 36

2.6 Parameters of the RSC Converter Control Model [39] . . . . . . . . . . . 36

3.1 Variable Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Parameter Notation and Value . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Prediction Error between Models (both compared to M0, Case 1) . . . . 63

5.1 UKF parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Estimation Results M1 Fault Scenario . . . . . . . . . . . . . . . . . . . 92

5.3 Estimation Results M1 Fault Scenario Compared to M0 . . . . . . . . . 93

5.4 Estimation Results M2 Wind Speed Jump . . . . . . . . . . . . . . . . . 96

5.5 Estimation Results M2 Wind Speed Jump Compared to M0 . . . . . . . 97

B.1 Transmission Line Parameters . . . . . . . . . . . . . . . . . . . . . . . . 109

B.2 Parameters of the Induction Machine . . . . . . . . . . . . . . . . . . . . 110

B.3 Parameters of the Step-up Transformer . . . . . . . . . . . . . . . . . . . 110

B.4 Parameters of the Transformer (stator-converter) . . . . . . . . . . . . . 111

vii

List of Figures

1.1 Parameter Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Type-3 WTG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Power Flow in the DFIG (Motor Convention) . . . . . . . . . . . . . . . 15

2.3 Power Coefficient Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Simplified Aerodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Pitch Angle vs. Wind Speed . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Single-mass Turbine Model . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Pitch Control Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8 Stator-flux Model for the DFIG . . . . . . . . . . . . . . . . . . . . . . . 32

2.9 Grid-Side Converter Control Model in PSCAD (M0) . . . . . . . . . . . 33

2.10 Torque Control Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.11 Turbine Speed Setpoints (pu) vs. Real Power (pu) . . . . . . . . . . . . . 38

2.12 Rotor-side converter control in PSCAD (M0) . . . . . . . . . . . . . . . 39

2.13 Test System with Single Type-3 WTG . . . . . . . . . . . . . . . . . . . 41

2.14 Real Power Under Different Wind Speeds: Results of Two Models . . . . 43

2.15 Dynamic Response During Fault . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Pelec − ωref curve - data fitting results . . . . . . . . . . . . . . . . . . . 50

3.2 System Response M1, Wind Speed Change . . . . . . . . . . . . . . . . . 64

3.3 System Response Comparison Using M1, M2, noise free . . . . . . . . . 65

viii

3.4 Comparison of System Response using Case 1 . . . . . . . . . . . . . . . 66

3.5 Comparison of System Response using Case 2 . . . . . . . . . . . . . . . 68

3.6 System Response During 20% Voltage Drop, Model M1 Tg = 100 . . . . 69

3.7 System Response During 20% Voltage Drop, Model M1 . . . . . . . . . . 70

3.8 Ip and Ipcmd during 20% voltage drop, model M1 . . . . . . . . . . . . . 71

3.9 Ip and Ipcmd during 20% voltage drop, model M1 from 29.6 sec to 31.4 sec 71

ix

Abbreviations

CIGRE Council on Large Electric Systems

DFIG Doubly Fed Induction Generator

EKF Extended Kalman Filter

EPRI Electrical Power Research Institute

IEC International Electrotechnical Commission

L-G Line-to-Ground

L-L Line-to-Line

LVRT Low-Voltage Ride-Through

ML Maximum Likelihood

MAP Maximum A Posterior

MMSE Minimum-Mean Squared Error

MSE Mean Squared Error

NDE Nonlinear Differential Equation

NERC North American Electric Reliability Corporation

ODE Ordinary Differential Equation

PCC Point of Common Coupling

PWM Pulse-Width Modulation

RLS Recursive Least-Squares

RSC Rotor-Side Converter (of DFIG)

WECC Western Electricity Coordinating Council

WTG Wind Turbine Generator

UKF Unscented Kalman Filter

x

Chapter 1

Introduction

1.1 Background and Motivation

Dynamic models of wind turbine generators are vital since wind generation is gaining in-

creasing importance throughout the world. Much research has been done in order to find

both accurate and simple dynamic models of wind plants and their components. One im-

portant characteristic of such models should be “generality”. Namely the models should

be general enough to represent a type of wind turbine generator regardless of its man-

ufacturer. Undoubtedly introducing such generic models will facilitate the development

of wind energy technology since exchanging research results becomes possible without

disclosing specific manufacturer’s data. Many organizations (e.g. WECC, IEC, CIGRE,

etc.) have separate working groups and task forces for developing generic models of wind

turbine generators. They are also dedicated to standardize the generic models. So far the

generic models have been validated by comparing to the real-world measurements from

different products. However, the model parameters may vary depending on the vendor

or rating of the units. It will be very helpful if the parameters can be identified using

device measurements like active and reactive power at the wind turbine unit. Based on

the estimation results, different model structures can be compared and chosen.

1

Chapter 1. Introduction 2

There have been studies in terms of parameter estimation of DFIGs [4, 35]. However

the focus of these papers is not the complete generic dynamic model. For example, they

normally exclude the pitch control. So far not much research has been done in terms

of parameter fitting using the complete generic dynamic model. EPRI has developed a

software tool to facilitate model validation for type-3 and type-4 WTG [9]. It will opti-

mize the user’s initial guess of parameters given terminal voltage (of the WTG), active

and reactive power measurements. Despite the limitation of the software, no information

from the parameter estimation is given since the algorithm is not disclosed. Therefore the

goal of this thesis is to compare different type-3 WTG (also known as the DFIG configu-

ration) model structures based on parameter estimation results. Three model structures

are involved in the thesis, the time-domain model M0 and two transient stability models

M1 and M2. M1 is a detailed transient stability model and M2 is a simplified version

based on M1. First the sophisticated time-domain model M0 will be built in PSCAD to

generate sample data for the purpose of identification of transient stability models M1

and M2.

The following paragraphs will briefly introduce the source of the models, the parameter

estimation procedure and how to evaluate the models, also known as model structure,

algorithm and criterion of identification.

1.1.1 Model structure

The development of generic WTG models is the work of many organizations and man-

ufacturers in collaboration. WECC has been the leading force on North America [38].

So far two versions of four major types of generic WTG models have been purposed (in

literature, the two versions are sometimes called two phases). The first phase of models

have been implemented in several software tools including PSS/E. Recently, the second

phase has been prepared by EPRI [31, 32]. Among the four types of WTGs, type-3 is


the dominant model installed in the market [3]. Type-3 WTG (the variable speed wind

turbine generator with partial-scale converter) is well known as the DFIG (doubly fed

induction generator) configuration. The advantages of type-3 WTGs includes separate

control over active and reactive power, ability to be magnetized from the rotor circuit,

capability of delivering reactive power to the stator/grid side and low cost due to the

small size of the converter [3]. This thesis is based on the first phase of the type-3 WTG

generic models purposed by WECC.

Given some assumptions, the type-3 WTG model can be described by nonlinear differ-

ential equations which can be easily implemented for the purpose of parameter identifi-

cation. The main assumption made in this thesis is that no limiters are active. The test

system in the thesis consists of a single 1.5 MW DFIG unit (instead of a lumped model

for wind plant) therefore the supervisory control at the plant level is not considered. It

is also worth mentioning that in the generic model the DC link and related protections

are not modelled. Thus they are not considered in our transient stability models either.

1.1.2 Algorithm

As mentioned earlier, although the generic models of type-3 WTGs have been widely

accepted, the parameters vary depending on the vendors. In search for the best param-

eters given certain criterion (which will be introduced in the next paragraph), various

parameter estimation techniques are available. However, a crucial question before appli-

cation of the identification procedure is determine if the system is identifiable. Given the

particular structure of the type-3 WTG model, we are able to gain some insights about

the identifiability of its subsystems.

There are two commonly used estimation techniques for nonlinear estimation, the ex-

tended Kalman filter (EKF) and the unscented Kalman filter (UKF). The EKF uses

the first-order linearization of the model whereas the UKF uses the original nonlinear


functions of the model. Therefore, the EKF can only work when the Jacobian exists

and it only works well when the model is not highly nonlinear. By contrast, the UKF is

accurate when highly nonlinear model is involved with comparable computation cost to

the EKF. In this thesis, the UKF is used for parameter estimation.

1.1.3 Criterion

The criterion, model structure and the algorithm can not be separated. The criterion can

be expressed as a cost function. Normally the cost function is the expected squared error

between the desired measurements and the estimates (also known as prediction error).

Since the WTG unit can be viewed as a controlled current source, the prediction error

of the injected current will be used for evaluating the models.

In this thesis, the measurements are generated using PSCAD simulation. The test system

in this environment is more detailed compared the transient stability models used for

parameter estimation. For example, the converter is implemented in PSCAD and the

detailed aerodynamic model has been used. Given the measurements, the two transient

stability models M1 and M2 will be tested for parameter estimation in MATLAB with

the help of ReBEL toolkit [2]. The estimated parameters will be evaluated in terms of

prediction error of the injected current.

By comparing the measurements from the time-domain model M0 and predictions given

by M1 and M2 with estimated parameters, we are able to determine which candidate

model structure should be chosen.

1.2 Parameter Estimation Procedure

The parameter estimation procedure used in the thesis can be summarized in Fig. 1.1

and illustrated as follows:


1. Generate inputs and states.

For parameter estimation, inputs and states are needed. Inputs u can be generated

using M0. However, u can also be given by M1 and M2. This is what is normally

done when only one model structure is available or when the structure is believed

to be correct and sufficient.

Then the inputs are passed to numerical integration process using model Mi (i can

be 1 or 2 in our case). Then the states will be calculated based on the original

guess of the parameter θA1.

2. Then we pass the measurements to the UKF and get estimate θB.

3. We evaluate the estimates by comparingMi(θA),Mi(θB) toM0(θA) with respect

to the prediction error. Based on the results, candidate model structures can be

compared and chosen.

Note that this procedure can be extended to evaluate multiple candidate model struc-

tures, i.e. i in Mi can be any integral greater than 1.

1.3 Objectives

The objectives of this thesis are:

1. Implement the detailed test system M0 which consists of a type-3 WTG unit

in PSCAD/EMTDC environment. Test the system under different circumstances

(wind speeds, faults). And record the signals as measurements.

2. Develop models M1,M2 for parameter estimation.

3. Use the measurements from the PSCAD and estimate the parameters with the

detailed transient stability model M1 and the simplified model M2.

4. Evaluate the estimated parameters by comparing prediction errors.


Figure 1.1: Parameter Estimation Procedure


1.4 Outline

In the first part of the thesis, the type-3 WTG generic model purposed by WECC will

be discussed. The whole model consists of five parts: the aerodynamic model (Sec-

tion 2.2), the turbine model (Section 2.3), the pitch control model (Section 2.4), the gen-

erator/converter model and the converter controller model (Section 2.5). Three model

structures will be discussed: M0, M1 and M2. M0 is the one implemented in the

PSCAD/EMTDC environment and it will serve as the “benchmark”. This system was

tested under different wind speeds and fault scenarios and the results are presented in

Section 2.6. M1 and M2 are dynamic models represented by differential equations and

were used for parameter estimation.

The second part of the thesis focuses on parameter estimation. Chapter 3 gives the

model descriptions of M1 and M2. Then both models will be discretized so that the

UKF, or any other recursive methods, can be applied. The UKF algorithm for param-

eter estimation is covered in Chapter 5, along with a short introduction of the software

toolkit ReBEL [2]. In addition, the parameter estimation results and analysis are also

summarized in this chapter. Lastly, the conclusion and future work are provided in

Chapter 6.

Chapter 2

Generic Models of Type-3 Wind

Turbine Generators

8

Chapter 2. Generic Models of Type-3 Wind Turbine Generators 9

2.1 An Introduction To The Type-3 Wind

Turbine Generator

2.1.1 Generic Wind Turbine Generator (WTG) Models

An imminent need for dynamic models of wind power plants and their components has

arisen as a result of the rapid growth of wind generation. Most dynamic models have

been developed by manufacturers and they usually require the users to follow certain non-

disclosure agreements. Generic WTG models, on the other hand, are non-proprietary by

nature. Therefore such generic models enable exchanging data between system operators,

manufacturers and researchers. Many working groups from different organizations are

dedicated to develop and validate generic WTG models. The International Electrotech-

nical Commission (IEC) and the Western Electricity Coordinating Council (WECC) are

two leading organizations on this topic.

The IEC Technical Committee 88 Working Group 27 started the standardization work:

the IEC 61400 - 27 since 2009. So far, the first committee draft for the dynamic wind

turbine model has been presented. However, the first printed edition is still being pre-

pared. The standard for wind plants is not expected to be available until 2015.

Meanwhile, the Wind Generation Modeling Group (WGMG) of the WECC initiated the

development of its generic WTG models for four major types [38]. The first generation

of the generic WTG models have been developed and implemented in major commercial

simulation tools, for example the GE PSLF and the Siemens PTI PSS/E. There are

other detailed reports (technical brochure, special reports, subcontract reports) available

prepared by the Council on Large Electric Systems (CIGRE) [6], the Electrical Power

Research Institute (EPRI), the National Renewable Energy Laboratory (NREL) and the

North American Electric Reliability Corporation (NERC) [28]. These groups and many

industry cooperations (such as GE, ABB, Siemens, etc.) have been working in collab-


oration on the development, the improvement and the validation of the WECC generic

WTG models. Also, in parallel with the work of WECC WGMG, the Working Group

on Dynamic Performance of Wind Power Generation under the IEEE Power System Dy-

namic Performance Committee is also focusing on the generic dynamic modeling of the

wind plants. A joint status report published in 2011 by the WECC and the IEEE [8]

summarizes the first generation of the generic WTG models, their development and

specifications briefly. Recently, the second generation of the type-3 and the type-4 WTG

models have been proposed through the WECC Renewable Energy Modeling Task Force

(REMTF) in reports prepared by the EPRI [31, 32]. Compared to the first generation,

the second phase of modeling focuses on improving and completing the control strategies.

However, the main structure of all parts remain the same. In order to form a relatively

simpler model for parameter fitting, the first generation of the type-3 WTG model is

used in this thesis.

2.1.2 Type-3 Generic WTG Models

Each WTG unit in the market can be classified as one of four basic types based on their

respective topologies and the grid interfaces.

- Type 1: Fixed-speed induction generators

- Type 2: Variable-slip induction generators with variable rotor resistance

- Type 3: Doubly-fed induction generators

- Type 4: Full-converter variable-speed generators

Between these four types, the type-3 and the type-4 units are most commonly sold and

installed. In this thesis, we focus on the type-3 WTG model which is characterized by

the doubly fed induction generator.


A Doubly Fed Induction Generator (DFIG) or a Doubly Fed Asynchronous Generator

(DFAG) system consists of two major parts: a wound rotor induction machine and a

power electronic converter. The rotor is fed at slip frequency from the back-to-back in-

verter to the grid thereby providing a wide speed range of ±20%−±30%. In all, the

application of power electronic control in the type-3 WTGs increases the generator op-

eration speed range compared to the type-1 and the type-2 WTGs. The converter size

is smaller (20% to 30% of the generator rated power) thus more budget-friendly and

efficient compared to the type-4 [3].

The first generic type-3 WTG model prepared by WECC consists of four major com-

ponents: the generator/converter model, the converter control model, the pitch control

model and the wind turbine model. During the second phase of modeling, an updated

and slightly detailed model connectivity has been presented [31]. Specifically, the con-

verter control model can be replaced by the active (or torque) control model and the

reactive control model. Meanwhile the wind turbine model can be represented by the

aerodynamic model and the drive-train (or shaft-dynamic) model. The updated model

is better in terms of clarity, especially when users are not interested in implementing

all control blocks or intending to make changes to a certain block without altering the

connectivity. Given this update, Fig. 2.1 shows the block diagrams of the type-3 WTG

generic model in details. Note that in Fig. 2.1 blocks circled by dashed lines mean they

are not included. For clarity, here is a list of the models which will be discussed in details

in subsequent sections:

1. Aerodynamic model

2. Drivetrain model: one-mass model

3. Pitch control model

4. Electrical (converter) controls: torque control model, reactive control model


5. Generator/converter model

Nevertheless, essentially the model components are the same despite the connectivity or

name preferences in different references.

Though the converter/generator model is described and included in the generic model, it

will be unnecessary to implement it in the PSCAD/EMTDC environment. Since in order

to implement a working test system, the converter and the generator can be modeled in

details using the PSCAD/EMTDC components. Also similarly, the one-mass turbine

model has been included in the generator model. Considering the amount of overlapping

models, we will not list the same ones used for PSCAD simulation. From now on, if the

model is only used for PSCAD simulation and not included in the generic WTG model,

the section title will be followed by M0. Snapshots from PSCAD will also be provided.

2.1.3 Power Flow in a DFIG

The type-3 WTG is also known as the DFIG configuration. Because of the converter

configuration, the power flow in a type-3 WTG is conceptually different compared to

the other types. With the four-quadrant converter connected to the rotor, power can be

extracted from or injected to the rotor. In other words, a DFIG can operate as a gener-

ator under supersynchronous conditions or subsynchronous conditions. The relationship

between the rotor speed and power is showed in Fig. 2.2. Pr is the power delivered by

the rotor and Ps is the power delivered to the stator. Hence the total supply power

(neglecting the losses) yields Pg = Ps − Pr = Ps(1− s). Here the directions are chosen

to follow the motor convention (motoring positive) in accordance with [25]. Since in the

PSCAD/EMTDC the motor convention is also used by default for the induction machine,

the torque of the DFIG will be negative during the simulation. However, the directions

of the power are not necessarily the same as defined in Fig. 2.2. It is more convenient

with the directions in Fig. 2.13 thus Pgen will be positive and meanwhile Pm will fulfill

the request of the PSCAD/EMTDC.


Fig

ure

2.1:

Typ

e-3

WT

GM

odel


It is possible that the rotor speed drops below the synchronous speed if the wind speed

is relatively low. For the type-1 WTG, the induction machine will act as a motor as

soon as the slip becomes positive. However, for a type-3 WTG, the rotor speed can vary

within the range ±20%−±30%.


Fig

ure

2.2:

Pow

erF

low

inth

eD

FIG

(Mot

orC

onve

nti

on)


2.2 Aerodynamic Model

The aerodynamic model purposes to calculate the turbine mechanical power based on the

wind speed Vwind, the blade pitch angle θ and the turbine rotor speed ωt. The relationship

is defined as:

Pmech =1

2ρAVwind

3Cp(λ, θ) (2.1)

λ =VtipVwind

= KbωtVwind

(2.2)

where Cp is the power coefficient and a function of the blade pitch angle θ and the tip-

speed ratio λ. λ is defined as the ratio of the rotor blade tip speed to the wind speed in

Eqn. 2.2. A is the cross-sectional area or the swept area, which is either given directly

in the data sheets of the wind turbines or computed approximately using πR2. R is the

radius of the turbine i.e. the blade length.

Eqn. 2.1 indicates the complexity of the aerodynamic model highly depends on the rep-

resentation of the power coefficient Cp. Three representations of the Cp curves have been

presented in [33] by Price and Sanchez-Gasca. As far as the transient stability study is

concerned, the two simplified models (the two-dimensional model and the linear model)

have been validated in comparison with the most detailed three-dimensional model (the

three dimensions are λ, θ and Cp) [33]. As the paper mentioned, such detailed and ac-

curate three-dimensional representation is not required for transient stability analyses.

Also it’s unpractical to assume this model is available to system analysts. Therefore only

the two simplified models will be described in subsequent paragraphs.

For clarity, Table 2.1 shows the notations for the variables and the constants used in

the aerodynamic model. Some variable, for example the pitch angle θ, will also show

up later in other models. The notations of these variables remain consistent throughout.

Note that the unit of Pmech can either be W or pu. Conventionally the per-unit system

is preferred in order to facilitate analysis of the whole system. The per-unit system for

the machine (in our case, induction generator) and the network are well established.


Table 2.1: Notations of the Aerodynamic Model [26]

Notation Value Unit Description

θ - degree Blade Pitch Angle

λ - - Tip-speed Ratio

Cp - - Power Coefficient

ω - pu Generator Rotor Speed

ωt - pu Turbine Rotor Speed

Vwind - m/s Wind Speed

Pmech - W or pu Turbine Mechanical Power

A πR2 m2 Swept Area

Kb 56.6 - Constant

ρ 1.225 kg/m3 Air Density

R 35.5 m Radius of Turbine

However, it should be pointed out that the aerodynamic model (Eqn. 2.1) is described

in SI forms, in which case Pmech should be the actual value of the turbine mechanical

power in W. Alternatively, for a given WTG unit, the mechanical power in per unit can

be computed directly provided proper scaling of the constant A or ρA. For example, a

typical value of 1/2ρA for the 1.5 MW GE WTG is 0.00159. In this case, Pmech is in per

unit since the rated power has been considered in the constant.

2.2.1 Aerodynamic Model with Cp Curves - Two-Dimensional

Model

A well-known set of Cp curves is presented in [26, 33] for the GE Energy’s 1.5 MW and 3.6

MW wind turbines initially. The curves can be reproduced using a polynomial approx-


Figure 2.3: Rotor Power Coefficient Cp vs. TSR λ Using Polynomial Approximation

imation in Eqn. 2.3. The coefficients used are listed in Table. 2.2 [26]. Thus the curves

were implemented in the PSCAD/EMTDC environment using a FORTRAN-based user

defined component. See Fig. 2.3 for the approximation results. The accuracy of the

polynomial approximation has been validated since satisfactory results in terms of the

power curve have been achieved in Section 2.6.2. In other words, this model is able to

provide correct Cp given the right θ and λ values. Therefore, the mechanical power can

be regulated correctly using this power coefficient.

Note that there is more than one way to represent the coefficient curves mathematically.

Besides the polynomial approximation, there have been other attempts to represent the

Cp involving exponential functions [3][13]. However these functions were not developed

for the type-3 WTG originally. Thus the variation can be quite large between different

models. Therefore the curves should be changed accordingly if a different model of WTGs

is of interest.


Cp(λ, θ) =4∑i=0

4∑j=0

αi,jθiλj (2.3)

Although the polynomial approximation is simpler compared to the original three-dimensional

power coefficient curves, there are 25 parameters which need to be determined from the

perspective of parameter estimation (Table. 2.2). The model will be simplified consider-

ably if the Cp representation can be replaced with a linear function.

2.2.2 Aerodynamic Model without Cp Curves - Linear Model

Although the two-dimensional model is much simpler compared to the actual turbine

model in reality, it can be further simplified assuming constant wind speed. The simpli-

fied model is a linear approximation based on the GE’s WTG parameters initially. By

using this linear model, the calculation of the power coefficient in Eqn. 2.3 can be avoided

altogether. Due to the substantial variations between manufacturers, the parameter val-

ues proposed in [33] can’t serve for all the turbine models in the market. Nevertheless,

the simplified linear aerodynamic model has been accepted as part of the generic type-3

WTG model in the WECC report and implemented in commercial softwares (for exam-

ple, the PSS/E).

The model is based on a linear approximation of the relation between the rate of power

with respect of the pitch angle (i.e. dP/dθ) and the pitch angle θ. In other words, it’s

based on the linear relation below (see Table 2.3 for Kaero value):

dP

dθ= −Kaeroθ

which leads to Eqn. 2.4.

Pmech = Pm0 −Kaeroθ(θ − θ0) (2.4)


Table 2.2: Cp curve coefficients αi,j [26]

i j αi,j

0 0 -4.1909E-01

0 1 2.1808E-01

0 2 -1.2406E-02

0 3 -1.3365E-04

0 4 1.1524E-05

1 0 -6.7606E-02

1 1 6.0405E-02

1 2 -1.3934E-02

1 3 1.0683E-03

1 4 -2.3895E-05

2 0 1.5727E-02

2 1 -1.0996E-02

2 2 2.1495E-03

2 3 -1.4855E-04

2 4 2.7937E-06

3 0 -8.6018E-04

3 1 5.7051E-04

3 2 -1.0479E-04

3 3 5.9924E-06

3 4 -8.9194E-08

4 0 1.4787E-05

4 1 -9.4839E-06

4 2 1.6167E-06

4 3 -7.1535E-08

4 4 4.9686E-10


where Kaero is the aerodynamic gain; Pm0 and θ0 are the initial values of the mechanical

power and the pitch angle respectively. Note that in the linear aerodynamic model, Pmech

is now a function of θ rather than three variables ( θ, Vwind and ωt). This is the because

of the constant wind speed assumption. However, the linear aerodynamic model requires

proper initialization of the power and the pitch angle. The initialization is based on the

investigation in [33].

There are two parameters to be determined: Pm0 is the mechanical power initial value

and θ0 the pitch angle initial value. When the pitch angle θ equals to θ0, the mechanical

power yields Pmech = Pm0. Normally Pm0 = Pelec which can be determined by the power

flow condition. For example, in PSS/E after solving the steady state load flow of a

system, Pm0 will be initialized. If:

1. Pm0 < 1.0 pu, then θ0 = 0, Vwind is initialized based on Eqn. 2.5

2. Pm0 = 1.0 pu, and Vwind,user > 5.75Pm0 + 5.60 = 11.35 (Eqn. 2.5), then θ0 is calcu-

lated using Eqn. 2.6.

3. Pm0 = 1.0 pu, and Vwind,user ≤ 5.75Pm0 + 5.60 = 11.35 (Eqn. 2.5), then θ0 = 0.

Vwind = 5.75Pmech + 5.60 (2.5)

θ = 1.46Vwind − 11.0 (2.6)

Here are some comments about the initialization steps:

First, this linear aerodynamic model is developed so that it can be implemented easily

as a part of the dynamic model. This initialization has to rely on the load flow solution

of the system.

Secondly, the linear aerodynamic model has its limitations. It is based on the linear

relation dPdθ

= −Kaeroθ which suggests it is only applicable when θ is changing. Namely,

the linear relation is only true when the pitch control is active. This corresponds to

the situation Pm0 = 1 pu. When θ remains at its minimum, there are many possible


Figure 2.4: Simplified Aerodynamic Model

Pm0 and they are all less than 1 pu. In this case, dθ = 0 but apparently the real power

may change. Therefore, θ is initialized using Eqn. 2.6 only when Pm0 = 1 pu. Otherwise

θ0 = θmin = 0.

Thirdly, it is not very obvious why the wind speed Vwind,user is involved. As the conse-

quence of the limitation of the linear approximation mentioned above, generally speaking

when the wind speed is less than the rated speed, the pitch angle will remain at its min-

imum in order to get maximum power out of the wind. However the question is when

the wind speed is lower than what value, θ should be set to zero. This threshold can be

computed by substituting the critical Pm0 (i.e. 1 pu) into Eqn. 2.5. Setting the threshold

is crucial because in the region around the threshold which corresponding to the neigh-

bourhood near the rated wind speed, the linear approximation becomes less and less

accurate as the wind speed decreases. If the threshold is not set properly, the operating

point will shift far away from what it should be provided by the 2-D aerodynamic model.

2.2.3 Comparison of the Two Models

In order to compare the two models, a relationship between the wind speed and the pitch

angle has been plotted in Fig. 2.5. The 2-D aerodynamic model was implemented in

PSCAD and tested under various wind speeds. After the system reaches steady state,

the pitch angle is recorded and used to generate Fig. 2.5. The details about the test


system is described in Section 2.6.1. The discrete dots are results from the system with

the polynomial representation of the aerodynamic model. The solid line is from the linear

model using Eqn. 2.6. The figure shows that the model works as expected by comparison

with results in [33].

The result indicates the pitch angle from the linear fitting is larger than its counterparts

based on the coefficient curves when the wind speed exceeds 14 m/s. This is a tradeoff

considering the approximation accuracy near the rated wind speed. There is an obvious

outlier at 12 m/s and this is due to the limitation of the linear model. As mentioned

earlier, the linear approximation does not work very well when the wind speed is around

the rated value. As a result, we have the initialization steps above. For more discussion

see Section 2.6.2. In all, the simplified linear model works well regarding the overall

accuracy and the simplicity.


Figure 2.5: Pitch Angle vs. Wind Speed

Discrete Dots from 2-D Model and Solid Line is from Linear Model Eqn. 2.6


Table 2.3: Parameters of the Turbine Model [39]

Parameter Value Unit Description

Kaero 0.007 - Aerodynamic Gain Factor

H 4.94 s Equivalent Total Inertia Constant

D 0 pu Shaft Damping Factor

2.3 The Turbine Model

The turbine model, also called rotor mechanical model in some literature, describes the

conversion from mechanical power to electrical power. It computes the generator rotor

speed based on the mechanical power given by the aerodynamic/wind model and the

electrical power given by the generator model. Despite a variety of existed drive train

models (from the detailed six-mass model to the lumped one-mass model), the WECC

presented the one-mass turbine model and the two-mass turbine model. In the two-

mass model, the angular velocity of the turbine differs from the angular velocity of the

generator rotor since the inertia of the turbine and the generator are considered as two

separate parts. Only the one-mass model is chosen in this paper.

In the PSCAD/EMTDC, the multi-mass turbine model is available by adding a separate

component which supports up to the six-mass model. The one-mass turbine model (see

Fig. 2.6)can be implemented directly using the wound induction machine block. The

parameter value H used in Eqn. 2.7 is listed in Table 2.3.

dω

dt=

1

2Hω(Pmech − Pelec) (2.7)

where ω is the angular speed of the generator as well as the turbine angular speed due

to the choice of the lumped model.


Figure 2.6: Single-mass Turbine Model

2.4 The Pitch Control Model

The pitch control model allows the blades to be pitched so that the maximum mechanical

power (1.0 pu if the wind speed exceeds the rated value) can be drawn from the wind.

Figure 2.7 shows the pitch control model consists of two PI controllers driven by the

rotor speed error ωerr and the real power error respectively. Sometimes, the former is re-

ferred to as pitch controller alone while the latter is referred to as the pitch compensator.

Sometimes, the compensator is not included [3]. The blade pitch angle is both amplitude

limited and rate limited. Because the pitch angle can not be changed abruptly due to

the size of the rotor blades and the expensive blade drives. The rate depends on the size

of the wind turbine. The maximum rate of change in the pitch angle is said to be in the

order of 10 degree per second according to [36]. A time constant Tpi is associated with

the translation of the pitch angle to the mechanical output [8]. The parameter values are

in Table 2.4.

Even though the relationship between the pitch angle and the mechanical power is not

linear as indicated in Eqn. 2.1 (Pmech = 12ρAVwind

3Cp(λ, θ)), in general as the wind speed

increases the pitch angle would also increase accordingly. The upper limit of the pitch

angle is 27 degrees nevertheless it might vary depending on the wind turbine model. In

addition, generally speaking before the pitch angle hits its upper limit, the wind speed


Table 2.4: Parameters of the Pitch Control Model [39]

Parameter Value Description

Kpp 150 Proportional gain in pitch controller

Kip 25 Integrator gain in pitch controller

Kpc 3 Proportional gain in compensator

Kic 30 Integrator gain in compensator

Tpi 0.3 Blade response time constant

PImax 27 Pitch angle upper Limit (degree)

PImin 0 Pitch angle lower limit (degree)

PIrate 10 Pitch angle rate limit

has past the cut-off speed. Thus the upper limit of the pitch angle is rarely a problem.

In contrast, when the wind speed is lower than the rated speed (13 m/s in this case), the

pitch angle decreases and remains at its minimum value in order to maximize the power

that can be extracted from the wind.

dxpdt

= Kip(ω − ωref ) (2.8)

dxcdt

= Kic(Pord − Pcmd) (2.9)

dθ

dt=

1

Tpi(θcmd − θ) (2.10)

θcmd = xp + xc +Kpp(ω − ωref ) +Kpc(Pord − Pcmd) (2.11)

xp in Eqn. 2.8 represents the integral of ωerr in the first integrator (the pitch control

integrator, Figure 2.7). And xc represents the integration result of the pitch compen-

sation integrator. Hence the unregulated pitch angle θcmd is the summation of the two

aforementioned integrals and the results of the proportional controllers.

Note that in the generic models, the integrators are supposed to be anti-windup or

non-windup integrators which means when the pitch angle is at its lower limit, if ωerr is


negative, then the integrator should be blocked. Similarly the integrator will be blocked

when ωerr > 0 and θ reaches its upper limit. This mechanism aims to preventing the

integrated result from going further down while the pitch angle remains zero degree.

Without the anti-windup logic, the pitch angle can still remain at its limit. However, it

might take much longer for xc and xp to recover from the limit.

In the PSCAD simulation, all limiters can be easily applied but the anti-windup logic

is not considered. To begin with, according to [15], the anti-windup logic may lead

to trajectory deadlock. In short, it is possible there exist situations when xc and θ

lingers at the edge of block/unblock indefinitely and infinitely. The author proposed to

solve the problem by introducing hysteresis. Although this will solve the deadlock, more

states as well as switching behaviours will be involved as a result. Such phenomena are

much undesirable for parameter estimation since the system continuity assumption will

be compromised which will lead to much complicated model description and potential

numerical problems. Such systems with discrete events can be classified as hybrid dy-

namical systems [37]. An alternative to work around this problem without implementing

the anti-wind up logic is setting up a threshold directly with respect to the wind speed.

For example, despite the limiters, when the wind speed is lower than 11.35 m/s (the

rated speed is 13 m/s), θ is set to be zero. This is helpful when the anti-windup blocks

are not implemented otherwise it may take longer for the angle to settle as the angle is

approaching the limit. In addition, the threshold value 11.35 m/s comes from the linear

aerodynamic model, in other words, only when the wind speed exceeds the threshold,

the pitch control model will take over and start to regulate the pitch angle.

In summary, in this thesis, the anti-windup limited logic will not be discussed. During

the PSCAD simulation, all limiters including rate of change limiters were considered.

And the threshold mentioned in this section was implemented. But later as we extract

the model for parameter fitting, we assume no limits are hit. Otherwise, for each limiter,

there would be at least one more state. In addition, the anti-windup block in the torque


Figure 2.7: Pitch Controller

control isn’t considered.

Pcmd is the real power reference which can be provided by plant controller or external

model. In short, through the pitch compensator, the real power will be driven to its

reference Pcmd by changing the pitch angle. Note that Pcmd can not exceed the maximum

power which can be extracted from the wind. In general, if there isn’t plant controller

and we want the maximum power, Pcmd can be set according to the power/wind speed

relation which is available in all WTG product data sheet. The power curve of our system

is given later in Fig. 2.14.


2.5 Generator and Converter Model

2.5.1 Generator and Converter Model

In the PSCAD/EMTDC environment, the generator and the converter can be modelled

in great details, whereas in the generic WTG model (as indicated in Fig. 2.1), the gen-

erator/converter is modelled as one block which also serves as the interface with the

grid. The generator/converter model takes in two current inputs Iqcmd and Ipcmd and

gives Pelec and Qelec or injected current components Ip and Iq as outputs. A full gener-

ator/converter model consists of three logic/management blocks: the low voltage power

logic block, the high voltage reactive current management block and the low voltage

active current management block [26][31]. However, high voltage and low voltage sce-

narios are not considered in this thesis since our interests are mainly scenarios when

the terminal voltage is near the nominal value. Moreover, adding these logic blocks will

compromise the desirable continuity and differentiability of the models thus will make

parameter fitting extremely difficult. Therefore, the generator/control model is:

dIpdt

=1

Tg(Ipcmd − Ip) (2.12)

Pelec = VtermIp (2.13)

dIqdt

=1

Tg(Iqcmd − Iq) (2.14)

Qelec = IqVterm (2.15)

The lags in converter action are characterized by a small time constant Tg = 0.02s. In

other words, the complicate electronic control is replaced by two low-pass filters approx-

imately [7]. Thus the current “commands” Iqcmd and Ipcmd become the actual current

components Iq and Ip after the “converter”. Tg is also the smallest time constant in the

whole model.


2.5.2 Converter Controller

As mentioned above, having a partial-scale converter is a key feature of the type-3 WTG.

The rating of the converter is about 20% to 30% of the generator rating depending on

the operating slip range. The control of the AC-DC-AC converter consists of two parts:

the grid-side control and the rotor-side control. Both are outlined in this subsection.

There are several different methods to implement the converter control system. Two-

dimensional frames, i.e. the αβ-frame and the dq-frame, are most commonly used. By

controlling two subsystems, designing controllers for the converters becomes easier. Es-

pecially when the dq-frame is employed, the commands needed to be tracked will be DC

signals thus simple PI controllers are sufficient to provide a satisfactory performance.

Such control methods are presented frequently in literatures, and one of many resources

is the book by Yazdani and Iravani [40]. In addition, there are two different choices: the

reference frame aligned with the stator flux or the stator voltage. Here the stator-flux

orientated frame will be used.

The Stator Flux Model (M0)

As is known, in order to accomplish the abc-to-dq transformation or the αβ-to-dq trans-

formation, a time-variant angle is tracked all the time. Depending on the choice of

reference frame orientation, there are two options: one is the stator flux orientation, the

other one is the grid voltage orientation (also known as grid flux orientation). In the

stator flux orientation, the d axis is aligned with the stator flux. In contrast, in the

grid voltage orientation, the q axis is aligned with the stator (grid) voltage. The flux

orientated frame requires a stator flux model (flux observer) while the voltage orientated

frame requires a PLL model in order to obtain the angle of the fundamental.

In the type-3 WTG control, both models are implemented in the PSCAD/EMTDC. The

angle θs which is needed to accomplish the abc-to-dq transformation on the grid-side is

given by computing the angle of the rotor voltage in the αβ-frame. Meanwhile, the angle


Figure 2.8: Stator-flux Model for the DFIG

ρ for the rotor-side is given by the stator flux observer.

~λs = λsejθ(t)

θ(t) = ρ(t) + θr(t)

where ~λs represents the stator flux space phasor and θr is the rotor angle. After the

transformation using ρ, the rotor side current can be presented as ird + jirq. In addition,

irq is proportional to the machine electrical torque in steady state (to be specific, when

the amplitude of the stator flux is constant). Thus the torque as well as the real power

output can be controlled through the torque control model which is discussed in the

subsequent section.

Grid-Side Converter Controller (M0)

The grid-side controller is responsible for maintaining the dc link voltage as well as the

stator side reactive power. The configuration is readily available in literature. Fig. 2.9

shows the main part of the grid-side converter control, where Ed and phi are the stator

voltage magnitude and angle (α− β frame). Thus Vabcref is the reference for generating

PWM gating signals.


Figure 2.9: Grid-Side Converter Control Model in PSCAD (M0)


Rotor-Side Converter Controller

In the dq-frame, controlling over the torque and the reactive power output can be achieved

by assigning the q-axis and d-axis rotor currents independently. This is also one of the

advantages of type-3 WTG.The torque control model and the reactive power control

model are the major two parts of the rotor-side converter controller. In Fig. 2.1, there

are two blocks: Q control and P control, which follow the voltage regulation/power fac-

tor control block and the torque control block respectively. These two blocks can take

in either external signals or outputs from the previous blocks as commands (Qcmd and

Pcmd). Then the current commands for Ip and Iq will be computed and passed to gener-

ate gating signals for the converter. In other words, the torque control and the voltage

regulation/power factor control block are optional. Here the torque control model has

been implemented. Again, together with the P control block, they are called the real

power control model. Sometimes, it is merely called the torque control model.

The naming and scope of the blocks can be confusing given the various options in lit-

erature. However, essentially there are only two parts, one for the active power control

corresponding to the q-axis rotor current in this case. The other one is the reactive power

control corresponding to the d-axis rotor current. In the active power control part, the

current reference is computed based on a relationship involving the machine torque. In

contrast, the reactive power control part is a bit simpler. In this thesis, the d-axis current

reference is simply computed using a PI controller driven by the reactive power error.

1. The WTG Real Power Control Model (Torque Control Model)

In steady state, the electrical torque of the induction machine is proportional to

the q-axis rotor current (stator flux orientation). Therefore, the torque can be

controlled by assigning irq,ref properly. Fig. 2.10 shows the structure of the torque

control model. The PI controller acts on the rotor speed error and gives the torque

reference, which is translated to the q-axis current reference in the end. Further-

more, a nonlinear relationship extracted from [39] has been used to determine the


turbine speed reference ωref based on the real power Pelec. A piecewise linear ap-

proximation is implemented in the PSCAD/EMTDC environment as showed in

Fig. 2.11. In addition, the intermediate signals ωref and Pord will be used in the

pitch control model mentioned in Section 2.4. See Table 2.6 for parameter values.

The torque controller presented is based on per-unit values. The base quantities are

as listed in Table 2.5. More discussion about the torque control model is included

in Appendix A. Note that the torque control model used in the generic WTG

model is based on the model used in real world however their configurations are

not the same. The most important difference is the output of the torque control

Ipcmd is not the same as the d-axis component of the rotor current reference used

to generate gating signals for the converter. In the generic model, Ipcmd will be

passed to the generator/converter model. After some limiters, Ipcmd will become

Ip. Then, through the grid interface, the injected current to the grid from the WTG

can be computed (along with Iq). In other words, Ip is a component of the injected

current of the WTG. Meanwhile Irqcmd is the true reference used to control the real

power. Fig. 2.12 shows the torque control used in PSCAD. In short, the torque

control in PSCAD doesn’t include Pord. The reason Pord is considered as a part of

the torque control in the generic model is so that it can be used as a replacement

of Pelec. However, there is no such need to include in Pord in the torque control in

PSCAD since Pelec is available directly. In addition, in order to decrease the time

the system takes from start-up to steady state, proper initial values should be used

in PSCAD.

dTωdt

= Kitrq(ω − ωref ) (2.16)

dPorddt

=1

Tpc(ω(Tω +Kptrq(ω − ωref ))− Pord) (2.17)


Table 2.5: Base Quantities Used in The Converter Controllers

Expression Value Description

Pbase 1.5 MVA The rated power of the DFIG

Vbase 0.5634 kV Amplitude of the nominalline-to-neutral voltage

Ibase 1.0248 kA Amplitude of the nominal line current

ωbase 60 Hz System frequency

Table 2.6: Parameters of the RSC Converter Control Model [39]

Parameter Value Unit Description

Kptrq 3 - Torque control proportional gain

Kitrq 0.6 - Torque control integrator gain

Tpc 0.05 s Power control lag

Tsp 5 s Speed reference lag

Ipmax 1.1 - Current reference upper limit

Pmax 1.12 pu Power reference upper limit

Pmin 0.1 pu Power reference lower limit

Kqvp 0.25 - Reactive power control proportional gain

Kqvi 2 - Reactive power control integrator gain


Fig

ure

2.10

:T

orque

Con

trol

Model


Figure 2.11: Turbine Speed Setpoints (pu) vs. Real Power (pu)

2. The WTG Reactive Power Control Model

Similar to the real power control, in steady state, the reactive power of the WTG

can be controlled by the d-axis rotor current. Therefore the command for the d-

axis rotor current can be calculated given the desired reactive power Qcmd. As

mentioned before, the voltage regulation and the power factor control block can be

ignored by directly assigning the reactive power command. In our case, Qcmd is

held to be constant. This is also referred as “fixed” reactive power control. In other

cases, it can also be computed by a separate model constantly. For instance, if a

certain power factor expected, the reactive power command can be computed based

on the real power. Fig. 2.12 shows both the real and reactive control implemented

in PSCAD. The parameter values are included in Table 2.6.

dxqdt

= Kqvi(Qcmd −Qelec) (2.18)

Iqcmd = xq +Kqvp(Qcmd −Qelec) (2.19)


Figure 2.12: Rotor-side converter control in PSCAD (M0)

2.5.3 The Scope of the Models

So far all major components of the time-domain model and the transient stability model

which will be used in the thesis have been introduced. However, it might be helpful to

remind the readers of the assumptions we made on the models.

1. There are no anti-windup logic implemented and all limiters in the transient sta-

bility models are assumed to be inactive.

2. Low-voltage ride through requirement is not considered. In other words, also fol-

lowing the previous assumption, the low voltage and high voltage logic in the gen-

erator/converter model is not considered.

3. Historically, the DC-link and grid-side converter are not considered in the transient

stability models for the current two generations of WECC models. Therefore,

these two parts and related features (e.g. crow-bar model) are not considered in

the transient stability models in this thesis.


2.6 Test System and Model Performance

2.6.1 Test System

PSCAD provides its users a very simple example of a wind farm system consisting of a

single 2 MW DFIG unit. However, the system is not complete since the pitch control

model and the aerodynamic model are missing. In addition, the example system uses

several simplified models, for instance, the active power control model. Our test system,

on the other hand, includes all major parts of generic type-3 WTG model.

In the test sytsem, the generator (rated power 1.5 MVA), is connected to a 20 kV infi-

nite bus (ideal voltage source) behind a 0.69 kV/ 20 kV transformer and the Thevenin

equivalent of the grid. The back-to-back converter including its control system has been

implemented in the converter module. The “wind power” module takes in the wind

speed, the power coefficient Cp and the machine rotor speed as inputs. After computa-

tion, it gives out the mechanical torque to the generator. This module won’t be used if

the simplified aerodynamic model is employed. The parameter values of the system are

listed in Appendix B. Fault can be introduced by adding impedance at between the grid

and the step-up transformer.

2.6.2 Model Performance

To maintain a smooth transition between the speed control mode and the torque con-

trol mode, all initial values such as the rotor speed, real and reactive power commands

should be set accordingly to match those in steady state. Furthermore, if the simplified

aerodynamic model is used, the initialization of this part is also necessary. At 0.5 s, the

control mode is switched.


Figure 2.13: Test System with Single Type-3 WTG

Start-Up Settings

Incorrect initialization will lead to fairly long settling time since the rate of the pitch

angle is regulated. The shaft speed normally takes several seconds to settle down. The

simulation duration should be set to 20 s to 30 s before any change made (for example

changing the wind speed or introducing a fault).

Simulation Results under Different Wind Speeds

The system is tested under different wind speeds ranging from the cut-in speed (6 m/s)

to the cut-off speed (20 m/s). A relationship between the wind speed and the real power

output has be established based on the test results. The values are recorded after the

system reaches steady state under a constant wind speed. As mentioned in Section 2.2.1

and Section 2.2.2, there are two ways of modelling the aerodynamics. Fig. 2.14 shows the

power curve (red/solid line) involving Cp computation. By contrast the power curve of

the system with the simplified linear aerodynamic model is also included (blue/dash line).


This well-known curve (from the 2-D aerodynamic model) shows the relation between

the wind speed and active power of the WTG which is available in wind turbine product

brochure.

The linear relationship in Eqn. 2.5 is an approximation of the power curve when the

wind speed is less than 11.35 m/s. In other words, Eqn. 2.5 is a simplified version of the

rising part of the curve. The result is in accordance with [33]. However, due to the sharp

change of the slop when Vwind is between 11 m/s and 12 m/s, the linear model is less

satisfactory in this particular range given the consideration of the overall approximation

accuracy. Thus if the linear aerodynamic model is used, one should make sure θ0 is

properly initialized around this region. When the wind speed is higher than the rated

speed, the mechanical power is determined by the pitch angle error directly (see Eqn. 2.4).

In all, the linear model works well as expected.

Dynamic Response of the System

To test the dynamic response of the WTG system, a voltage dip of 20% at the WTG

terminal is designed to happen at 30s under the rated wind speed condition. The fault

voltage is introduced by adding fault branch between the grid and the transformer. The

duration of the voltage drop is 18 cycles (i.e. 0.3 s). The dynamic responses are indicated

in Fig. 2.15 including the real and reactive power, the terminal voltage and the q-axis

rotor current reference.

The real power drops about 13% due to the fault and increased back to around 1.5 MW

after the fault is cleared. Meanwhile, an increase of the reactive has been observed during

the fault. The q-axis rotor current jumps up due to the voltage sag as an effort to increase

the real power. Also during the fault, the pitch angle goes down trying to increase Cp

in order to increase the real power. The results agree with [33] in general. Note that

since the two-mass turbine model is used in [33], the turbine shaft speed oscillates and

takes longer to settle down compared to the generator shaft speed. As a result, the


Figure 2.14: Real Power Under Different Wind Speeds: Results of Two Models

Solid/Red Line for the Two-Dimensional Model; Dash/Blue Line for the Linear Model


(a) Real Power During Fault (b) Reactive Power During Fault

(c) Voltage During Fault (d) Q-axis Current Reference During Fault

Figure 2.15: Dynamic Response During Fault

dynamic response when the turbine shaft speed oscillates violently (i.e. when the fault

just happens) can be different from the response with the one-mass turbine model.

Again, the PSCAD-type model is sophisticated time-domain model which is capable of

capturing all kinds of system dynamics. The simulation results will be used to evaluate

the dynamic models which will be covered in the subsequent chapters.

Chapter 3

Type-3 WTG Generic Models for

Parameter Fitting

In the previous chapters, the type-3 WTG generic models were reviewed and a test sys-

tem was implemented in PSCAD. Historically, this is the initial point of WTG generic

dynamic model development. The PSCAD-type models are sophisticated three-phase

time-domain models with detailed representation of the electronic components. In con-

trast, the dynamic models or generic transient stability models are phasor models which

focus merely on transient stability. So one should bear in mind, the system responses

using PSCAD-type model and the generic dynamic model are not necessarily the same

even during transient stability studies.

This chapter provides two different models represented by nonlinear differential equa-

tions. Both are simplified models based on the type-3 WTG generic model. In general,

we assume no limits are active and there are no integrators with anti-windup logic (see

previous chapters for details). So far we have been using the word model loosely from

the perspective of system identification. Before we move forward to model description,

it is worth mentioning that in the field of system identification, models like PSCAD-type

model and the dynamic model are not just different models. They are different models

45

Chapter 3. Type-3 WTG Generic Models for Parameter Fitting 46

from different model structures, i.e. they are fundamentally different in structures since

the PSCAD model contains more parameters. In other words, models are considered

different even from the same structure, as long as their parameters are different. For

simplicity, we will not begin to distinguish model and model structure. Instead, the

PSCAD-type model structure is represented by M0, the first generic model (structure)

we are about to introduce is M1 and the second is M2. Mi(θ?) represent a specific

model from structure Mi, i = 1, 2 and it is parameterized by θ?. Meanwhile, the PSCAD

model M0(θ?,θext? ) is not only parameterized by θ?, it is also parameterized by extra

parameters θext? (e.g. converter transformer leakage, transmission line impedance). Also,

M2 is obtained by reducing the order of model M1 by fixing a parameter. In this case

M2 is said to be contained in M1.

After the model introduction, the method for discretizing the system is discussed briefly.

Simple tests of the systems will be presented before we move to parameter identification.

3.1 Type-3 WTG Detailed Transient Stability Model

M1

The type-3 WTG model is summarized by the nonlinear differential equations below.

Again, this model consists of the linear aerodynamic model, single-mass turbine model,

the pitch control model, the torque control model, the reactive control model and the

generator/converter model. This ninth-order differential equations (Eqn. 3.1) can be used

as a simplified version of a WTG including the interface with the grid. In this sense,

the whole WTG model is a block which will determine how much current to be injected

into the grid based on the terminal voltage, initial power flow condition and the wind

speed. Eqn. 3.2 shows the possible output functions of the model. Note that they are

not states since the nine differential equations are capable of representing the full model.


In addition, not all the output functions are necessary.

dTωdt

= Kitrq(ω − f(VtermIp)) (3.1a)

dPorddt

=1

Tpc(ω(Tω +Kptrq(ω − f(VtermIp)))− Pord) (3.1b)

dIpdt

=1

Tg(PordVterm

− Ip) (3.1c)

dxqdt

= Kqvi(Qcmd − IqVterm) (3.1d)

dIqdt

=1

Tg(xq +Kqvp(Qcmd − IqVterm)− Iq) (3.1e)

dxpdt

= Kip(ω − f(VtermIp)) (3.1f)

dxcdt

= Kic(Pord − Pcmd) (3.1g)

dθ

dt=

1

Tpi(xp + xc +Kpp(ω − f(VtermIp)) +Kpc(Pord − Pcmd)− θ) (3.1h)

dω

dt=

1

2Hω(Pm0 −Kaeroθ(θ − θ0)− VtermIp) (3.1i)

ωref = f(Pelec) (3.2a)

Pelec = VtermIp (3.2b)

Ipcmd =PordVterm

(3.2c)

Qelec = IqVterm (3.2d)

Iqcmd = xq +Kqvp(Qcmd −Qelec) (3.2e)

θcmd = xp + xc +Kpp(ω − f(Pelec)) +Kpc(Pord − Pcmd) (3.2f)

Pmech = Pm0 −Kaeroθ(θ − θ0) (3.2g)

There are five inputs in this model:

1. Vterm: Amplitude of the terminal/stator voltage

2. Pm0: Initial value of the mechanical power, required by the simplified aerodynamic

model and provided by load flow result Pp.f..


3. θ0: Initial value of the pitch angle, required by the simplified aerodynamic model;

It is also a function of the wind speed Vwind

Pm0 = Pp.f.

θ0 = θmin, if Pm0 < 1;

= 1.46Vwind − 11, if Pm0 = 1;

4. Pcmd: Real power command. If Vwind ≥ 13m/s, Pcmd = 1 pu.

Otherwise Pcmd = (Vwind − 5.6)/5.75 (Eqn. 2.5). As mentioned earlier, this is not

the only way to assign Pcmd.

5. Qcmd: Reactive power command (some fixed value).

In order to arrive the 9-th order model, we assume that there are no active limiters.

Otherwise, more states have to be added in order to track the signals before and after

the limiter.

Note that Eqn. 3.2a represents the relation between the electrical power and the rotor

speed reference. In the PSCAD simulation, it’s expressed by a piecewise-linear function

whereas in the MATLAB, it’s better approximated by polynomial functions based on the

same data. According to [7] and [8], a second-order polynomial function is used. The

coefficients were computed by parameter fitting using the typical Pelec − ωref curve given

in [8] which is also used in the PSCAD simulation. The choice of using second order poly-

nomial function is based on [7]. In order to get better approximation, Eq. 3.3 was used

only when 0.15pu ≤ Pelec ≤ 0.75pu. For Pelec ≥ 0.75, ωref = 1.2pu while Pelec ≤ 0.15,

ωref = 0.689pu. Fig. 3.1 shows the data fitting results.

ωref = −0.791319P 2elec + 1.526046Pelec + 0.4918893 (3.3)

In order to get the standard form as in Eqn. 3.4, the states and parameters can be


Table 3.1: Variable Notation

Variable Notation Meaning

Tω x1 integrator result, torque control

Pord x2 real power order, torque control

ωref x3 rotor speed reference

Ipcmd x4 Ip current command

Ip x5 component of injected current controlling P

Pelec x6 electrical power of the DFIG

xq x7 integrator result, reactive power control

Iqcmd x8 Iq current command

Iq x9 component of injected current controlling Q

Qelec x10 reactive power of the DFIG

xp x11 integrator result, pitch control

xc x12 integrator result, pitch compensator

θcmd x13 pitch angle command

θ x14 pitch angle

Pmech x15 mechanical power

ω x16 rotor speed

Vterm u1 terminal voltage

Pm0 u2 initial value of mechanical power

θ0 u3 initial value of pitch angle

Pcmd u4 real power command

Qcmd u5 reactive power command


Figure 3.1: Pelec − ωref curve - data fitting results

replaced by their counterparts in Table 3.1 and Table 3.2 respectively. Thus we can write

the system in standard state-space form in Eqn. 3.5. In the standard form, u is the input

vector and u = [Vterm, Pm0, θ0, Pcmd, Qcmd]ᵀ. Meanwhile y represents the observations in

the system. Later this chapter, we will review the method for discretizing the system so

that it can be used for recursive methods.

x = F(u,x,θ) (3.4a)

y = H(x,u,θ) (3.4b)


Table 3.2: Parameter Notation and Value

Parameter Value Notation Meaning

Kitrq 0.6 p1 integrator gain, torque control

Kptrq 3 p2 proportional gain, torque control

1Tpc

20 p3 time constant, torque control

1Tg

50 p4 time constant, converter/generator control

Kqvi 2 p5 integrator gain, Q control

Kqvp 0.25 p6 proportional gain, Q control

Kip 25 p7 integrator gain, pitch control

Kic 30 p8 integrator gain, pitch compensator

Kpp 150 p9 proportional gain, pitch control

Kpc 3 p10 proportional gain, pitch compensator

1Tpi

3.33 p11 time constant, pitch control

Kaero 0.007 p12 Gain, aerodynamic model

12H

0.101215 p13 inertia constant


dx1dt

= p1(x16 − f(u1x5)) (3.5a)

dx2dt

= p3(x16(x1 + p2(x16 − f(u1x5))− x2) (3.5b)

dx5dt

= p4(x2u1− x5) (3.5c)

dx7dt

= p5(u5 − x9u1) (3.5d)

dx9dt

= p4(x7 + p6(u4 − x10)− x9) (3.5e)

dx11dt

= p7(x16 − f(u1x5)) (3.5f)

dx12dt

= p8(x2 − u4) (3.5g)

dx14dt

= p11(x11 + x12 + p9(x16 − x3) + p10(x2 − u4)− x14) (3.5h)

dx16dt

= p131

x16(u2 − p12x14(x14 − u3)− u1x5) (3.5i)

x3 = f(x6) (3.6a)

x4 =x2u1

(3.6b)

x6 = u1x5 (3.6c)

x8 = x7 + p6(u4 − x10) (3.6d)

x10 = x9u1 (3.6e)

x13 = x11 + x12 + p9(x16 − x3) + p10(x2 − u4) (3.6f)

x15 = u2 − p12x14(x14 − u3) (3.6g)

3.2 Reduced-order Type-3 WTG Transient Stability

Model M2

As noted earlier, despite the fact that model M1 is already a simplified version compared

to what was used in the PSCAD/EMTDC simulation, there are still 9 states and 13 pa-

rameters in the system. In this section, we seek for a further simplified model description


by ignoring the smallest time constant (i.e. the time constant Tg representing the lags in

converter actions).

The time constants in the system vary from milliseconds to seconds indicating a (near)

stiff system. Historically, a common model reduction technique is to neglect the dynam-

ics corresponding to the fast response components of the system. This technique has

been used in applications of many fields, including power system. According to [19],

small time constant T in a subsystem T x = f(x) can be neglected (i.e. set as zero) given

some conditions (e.g. the subsystem is stable). See [19] for details about stability of the

system. Then by solving the equation f(x) = 0, the related x can be determined thus

the order of the whole system is reduced. Therefore in our system, after neglecting Tg,

the derivatives of Ip and Iq becomes zero. As a result, the dynamics with regard to the

converter lag is neglected. By solving for the roots, we have Ipcmd = Ip and Iqcmd = Iq.

The further simplified model M2 becomes:

dTωdt

= Kitrq(ω − f(Pord)) (3.7a)

dPorddt

=1

Tpc(ω(Tω +Kptrq(ω − f(Pord)))− Pord) (3.7b)

dxqdt

= Kqvi(Qcmd −xq +KqvpQcmd

1 +KqvpVtermVterm) (3.7c)

dxpdt

= Kip(ω − f(Pord)) (3.7d)

dxcdt

= Kic(Pord − Pcmd) (3.7e)

dθ

dt=

1

Tpi(xp + xc +Kpp(ω − f(Pord)) +Kpc(Pord − Pcmd)− θ) (3.7f)

dω

dt=

1

2Hω(Pm0 −Kaeroθ(θ − θ0)− Pord) (3.7g)


ωref = f(Pord) (3.8a)

Ip =PordVterm

(3.8b)

Iq =xq +KqvpQcmd

1 +KqvpVterm(3.8c)

Qelec = IqVterm (3.8d)

θcmd = xp + xc +Kpp(ω − f(Pord)) +Kpc(Pord − Pcmd) (3.8e)

Pmech = Pm0 −Kaeroθ(θ − θ0) (3.8f)

Note that by replacing Ipcmd by Ip, we have Pelec = VtermIp = Pord. Namely in this case

the electrical power of the WTG is the same as the state Pord. This agrees with the work

by Hiskens in [15], in which the author used the assumption Pelec can be replaced by

Pord when no limits are hit. Note that his assumption is true under the circumstances

that the generator/converter model is ignored. In our case, the conclusion Pelec = Pord is

based on the assumptions there is no active limiters and Tg is neglected. Now there are 7

differentiable equations (Eqn. 3.7) characterized by 12 parameters. The output functions

are changed accordingly (Eqn. 3.8). A further simplified model might be possible if

the reactive power controller isn’t the main focus. There are alternatives for simplified

models [3]. For example, instead of having a pitch controller and a pitch compensator,

the pitch control model can be represented by only the pitch controller which is driven


by the rotor speed error.

dx1dt

= p1(x13 − f(x2)) (3.9a)

dx2dt

= p3(x13(x1 + p2(x13 − f(x2)))− x2) (3.9b)

dx5dt

= p4(u5 −x5 + p5u51 + p5u1

u1) (3.9c)

dx8dt

= p6(x13 − f(x2)) (3.9d)

dx9dt

= p7(x2 − u4) (3.9e)

dx11dt

= p10(x8 + x9 + p8(x13 − x3) + p9(x2 − u4)− x11) (3.9f)

dx13dt

= p121

x13(x12 − x2) (3.9g)

x3 = f(x2) (3.10a)

x4 =x2u1

(3.10b)

x6 =x5 + p5u51 + p5u1

(3.10c)

x7 = x6u1 (3.10d)

x10 = x8 + x9 + p8(x13 − x3) + p9(x2 − u4) (3.10e)

x12 = u2 − p11x11(x11 − u3) (3.10f)

3.3 Discrete Type-3 WTG Generic Models

The two models for type-3 generic WTG M1 and M2 are represented in terms of (first-

order ordinary) differential equations. To integrate the system numerically, the system

needs to be expressed as difference equations. In other words, the model needs to be

discretized first. The numerical methods for solving ordinary differential equations are

fundamental in almost all fields in engineering. Numerical integration is also the core

technology of computer simulation [20] and the first step in digital control [10].


In general, there are explicit integration methods and implicit integration methods. Some

famous and relatively simple methods, for example forward Euler and Runge-Kutta, are

explicit methods. The implicit methods show great advantage of dealing with stiff sys-

tems. These methods (e.g. implicit trapezoidal, Gear, Adams and Rosenbrock methods)

are much more robust. However, they also require more time for computation. In some

cases, the only choice for solving a stiff system is using implicit methods. One way to

identify a stiff system is to compare the time responses of different components of the

system. Roughly speaking if the time responses vary by several order of magnitude, the

system is considered stiff [10]. The ratio of the stiffness can be measured by the ratio

of corresponding eigenvalues (which are inversely proportional to the time constants in-

volved) [30].

The power system is generally considered as a stiff system. Details of system represen-

tations for the purpose of stability studies associated with various time constants were

presented in [21]. According to [25], there are four types of dynamic phenomena of the

basic power system: wave phenomena, electromagnetic phenomena, electromechanical

phenomena and thermodynamic phenomena. For the type-3 WTG system in question,

wave phenomena are normally not our interests. In [30], time constants are classified

into three categories accordingly (small, medium and large time constants). Specifically,

for wind turbines, the stator flux dynamics and power electronics are considered to have

small time constants. Meanwhile rotor flux dynamics, mechanical and aerodynamic com-

ponents, consist of medium time constants. Meanwhile large time constants are involved

in wind transport phenomena.

By reviewing the models in the previous sections, the largest time constant can be found

in the shaft dynamics (turbine inertia 2H = 9.88 s) while the smallest time constant is

Tg = 0.02 s inside the converter model.

The integration time step of the simulation can be set to be very small which would

mitigate the impact caused by explicit methods. The integration time step using the


explicit methods is required to be less than the smallest time constant of the model. In

our case, if the complete model is used, the integration time step must be less than 0.02

seconds (Tg). First we used one tenth of Tg i.e. 0.002 seconds as the time step. The

integration results are satisfying. If the simplified model is used then the time step must

be less than 0.05 seconds (Tpc in torque control).

In addition, for a nonlinear system as the wind turbine generator system is, the compu-

tation time can be very long with the implicit methods applied. To begin with, every

recursive step involving solving all nonlinear equations in the system. Given the time

span, the computation time can be quite long. Besides, during the parameter estimation

process, in some algorithm (e.g. the unscented Kalman transformation) at each step, the

computation time will multiply a factor L which is the number of sigma points in the

UKF. The exact value depends on the model, in general it’s several times of the number

of parameters.

The improved Euler method and the (implicit) trapezoidal method have been widely used

in power system simulation software tools. For example, the trapezoidal rule is used for

numerical integration in PSCAD by default. Meanwhile the improved Euler method (for

short-term stability) and the implicit trapezoidal method (for long-term stability) are

used by PSS/E. Also PSS/E supports the generic WTG model, and studies have been

done using the improved Euler method [30].The improved Euler method is an explicit in-

tegration method involving two steps. For an ODE of the form x = f(x, t) the improved

Euler method is in Eqn. 3.11.

x(t+ ∆t) = x(t) + k1∆t (3.11a)

x(t+ ∆t) = x(t) +k1 + k2

2∆t (3.11b)


where

k1 = f(x(t), t) (3.11c)

k2 = f(x(t+ ∆t), t+ ∆t) (3.11d)

To validate our choice of the improved Euler method, a simple experiment can be used

with the help of MATLAB. Using the full system model description in Sec. 3.1, given the

same initial values, solutions provided by the improved Euler method and the Matlab

ODE initial value problem solvers for stiff systems (e.g. ode23t, ode23tb and ode13s)

have been compared. In particular, ode23t follows the trapezoidal rule. The results show

that when the sampling time (or integration time step) is small enough, the results using

the improved Euler method is considered to be satisfying given the trade-off between

the accuracy and the computation time. In summary, the improved Euler method was

implemented.

3.4 Model Testing

Before we proceed to the parameter estimation part, the models should be tested first.

The tests are simple: we start from an equilibrium of the system and solve the differential

equations by integration. Given the initial states and inputs corresponding to some

disturbances over a period, the two models are used to predict the dynamic responses

under the disturbances. By comparing the dynamic responses from these two models

M1 and M2 and the PSCAD simulation results (M0), the models can be evaluated. We

consider two types of disturbances: wind speed change and terminal voltage drops due

to fault. First, we need to find an equilibrium (steady-state case).


3.4.1 Equilibrium and initialization

The equilibrium points of the system in Eqn. 3.5 and 3.9 should be associated with

steady-state operating points. Here are some points worth mentioning:

1. There is more than one solution to the same steady state operating point, i.e. the

equilibrium is not unique.

2. Not all solutions which yield equilibria are possible in the real world. In order to get

identical equilibria, the external system must be consistent. However, the external

system is not considered in M1 or M2.

3. Finding an equilibrium is relatively easy for our system.

For a state-space model, when it reaches the system equilibrium, all the derivatives should

be zero. In our case, all the differential equations should be set to zero for solving the

states. Despite the high order of the whole system, every equation is easy to solve. More-

over, all errors between the reference and the state are zero, such as ωref − ω, Pord − Pelec,

Qcmd −Qelec. Pm0 is the initial value for the turbine model. Pcmd is the electrical power

reference provided by user or upper level control logic. They are independent variables.

Here we only consider the steady-state situations when Pm0 − Pcmd = 0. This indicate

that in the generated power is the same as the required power. Other possibilities do exist

when the limits are active and the anti-windup logic is implemented (discussed in [14]).

However, these possibilities are not the main focus.

Here we provide a simple example for initialization when the wind speed Vwind is 13 m/s.

Based on the relation θ0 = 1.46Vwind − 11, θ0 = 7.98. Under the rated wind speed or

higher, Pcmd should be 1 pu. If the wind speed is lower than the rated speed, Pcmd can

be set according to the wind speed versus generator power curve. In steady state, Tω

is close to the torque. The torque can be computed by T = Pelec/ω. The rotor speed

ω = ωref = 1.2 when the wind speed is 13 m/s (referred to the power-speed set point


curve in the torque control). Ip = Ipcmd = Pelec/Vterm and Iq = Iqcmd = xq = Qelec/Vterm.

Though θ = θ0 = 7.98, there is no information about what xp and xc should be except

for the relation xp + xc = θ. This is another reason why the equilibrium is not unique

under the same operating points. In addition, these are the integrating results of the

pitch controller and the pitch compensator, they are by no means available as measure-

ments in real world. Fortunately, for initialization their values are not as important as

other states. Ideally they should be non-negative values if the anti-windup logic is im-

plemented. As long as the summation is θ, an equilibrium can be guaranteed.

3.4.2 Model test results

Two scenarios will be discussed in this section. The system will start from steady state

when the wind speed is 13 m/s. The first case (Case 1) is the wind speed jumps up to

15 m/s. The second case (Case 2) is the terminal voltage drops from 1 pu to around 0.8

pu while the wind speed remains 13 m/s all the time. Extra noise can be added to the

system to represent process noise as well as observation noise. The first case will be used

for parameter estimation later.

Case 1 wind speed jump

Under constant wind speed, the system remains at its equilibrium. After the wind speed

jumps up to 15 m/s and then remains constant, the system should be able to settle down

at a new equilibrium. Since after the disturbance, the wind speed is still greater than the

rated value, the generated active power should be 1 pu. But during the transient period,

after the wind speed increases abruptly, the power coming from the wind increases.

Since the pitch angle can not follow the wind speed closely, it might take the pitch

control several seconds (depending on how much the wind speed changes) to reach the


new steady state value. Thus during the same time, the real power will increases and

decreases back to 1 pu. In addition, the rotor speed will also give similar dynamic

response. All graphs showed here agree with this analysis. We will compare the results

from M1 and M2; from M0, M1 and M2.

Fig. 3.2 shows the prediction results without adding any noises using M1. The dynamic

responses under the same condition using M1 and the simplified model M2 (without Tg)

are presented in Fig. 3.3. The result shows that the two models are almost identical in

terms of wind speed disturbances. The time interval is 0.002 seconds in the three figures

and the total simulation time is 10 seconds. A quantitative measure can be chosen to

compare these two models. Since the dynamic model is essentially connected to the grid

as a current source, the error of Ip and Iq over time are compared. errip and erriq are the

cumulative error of Ip and Iq respectively. The cumulative total current error is errI . It

can be computed by first integrating the error of Ip and Iq over time and then calculating

errI =√errip2 + erriq2. The integration is done using the trapezoidal rule.

errip =

√∫(ip,i(t)− ip,0(t))2dt (3.12a)

erriq =

√∫(iq,i(t)− iq,0(t))2dt (3.12b)

errI =√errip2 + erriq2 (3.12c)

, where i can be 1 or 2 depending on which model structure is being evaluated. Note

that there is not much point calculating the discrepancy in current between M1 and

M2 using the “clear” simulation data except we can see how much the discrepancy is

compared to the steady state value. A third model should be brought in and serve as

the benchmark. In our case, the PSCAD simulation results will be used for this purpose.

Fig. 3.4a and Fig. 3.4b show the comparison of M0 and both models respectively under

the wind speed jump condition.

If we compare Pelec from the three model structures in Fig. 3.4a and Fig. 3.4b, it is


clear that Pelec from PSCAD simulation has the highest magnitude of high frequency

oscillation. This is because time-domain simulation is able to capture all dynamics. For

example, those caused by the converter switching. By comparing Pelec from M1 and M2,

we can see that the result given by M1 has high frequency oscillation while the result

from M2 is very smooth. This agrees with the difference in model structure. As a result

of neglecting Tg, M2 lose all converter dynamic. Thus Ipcmd and Ip are identical. In fact

there is no Ipcmd in M2. By contrast, Ipcmd and Ip from M1 are different (see Fig. 3.9).

Ip lags Ipcmd due to the existence of Tg.

The prediction error of the current is in Table. 3.3. The results are based on simulation

results mentioned above. The integration time step is 0.0001 s. The entire simulation in

PSCAD lasts 100 s. However these two graphs are from a small window from 24.7 s to

about 35 s. At 25 s the wind speed increases to 15 m/s from 13 m/s. The results also

quantitatively show that there is almost no difference between M1 and M2 under this

type of disturbance.

Note that the pitch angle from the PSCAD simulation is different from the one given

by M1 and M2. This is because in PSCAD simulation, the 2-D aerodynamic model

was used while M1 and M2 used the linear approximated aerodynamic model. Thus the

exact values of the pitch angle are different. Note that this is not just about an inaccurate

parameter in the linear aerodynamic model, i.e. Kaero. This discrepancy comes from the

limitation of the linear model. As we mentioned before, the linear model can only be

used when the wind speed is higher than the rated speed. The linear approximation is

not satisfying near the rated wind speed. However, the exact value of the pitch angle

is far less important compared to states like Pelec and Qelec. The discrepancy in steady

state value is acceptable.

The dynamic responses after the wind speed jump is similar but not exactly the same

due to the fundamental differences in model structures. As we mentioned earlier in the

chapter,M0(θ?,θext? ) is not only parameterized by θ? (which is shared withM1 andM2),


Table 3.3: Prediction Error between Models (both compared to M0, Case 1)

Model errI errip erriq

M1 0.047389435 0.047386859 4.94E-04

M2 0.047270903 0.047268319 4.94E-04

it is also parameterized by extra parameters θext? . Therefore, the dynamic response can

only be very close but not the same. In order to get nearly identical response, the rest

of the system (e.g. transmission line, transformer) have to be included in the MATLAB

simulation. As a result, those parameters will have to be considered for parameter fitting.

Passing hybrid dynamical system (characterized by differential equations and algebraic

equations) through the parameter estimation algorithm is not trivial. However, this is

our further interests. By neglecting the algebraic constraints from the rest of the system,

the results with M1 and M2 can be solved by integration starting from the steady state

operating points.

Case 2 fault

In Case 2, a 20% voltage drop occurs at the WTG terminal for 0.5 seconds from 30 s

to 30.5 s. As mentioned in Section 2.6.2, the three-phase voltage drop is introduced by

adding fault branch between the grid and the step-up transformer. See the DFIG example

provided by Manitoba HVDC for model settings [1]. The time interval (channel output

time interval in PSCAD and the integration time interval in MATLAB) in this part of

simulation is 100 µs. The simulation step size in the PSCAD is 50 µs. The whole time

horizon is 2 seconds in Fig 3.5. Note that the system hasn’t reaches its post-disturbance

equilibrium yet. However, it’s for better comparison of the system response during the

fault. See Fig. 3.7 for 15 s simulation results. Fig. 3.8 shows the Ipmd and Ip using M1

over the same period. During the fault, the current Ip jumps up in order to maintain the


Figure 3.2: System Response M1, Wind Speed Change


Figure 3.3: System Response Comparison Using M1, M2, noise free


Figure 3.4: Comparison of System Response using Case 1

(a) system response Case 1, M0 vs M1

[blue-M1, green-M0]

(b) system response Case 1, M0 vs M2

[blue-M2, green-M0]


real power.

The most significant difference between the two models is the system response of the

simplified model M2 is very smooth, particularly with Pelec. At the beginning and the

end of the fault, Pelec curve computed by the complete model M1 shows small notches.

By contrast, no such dynamic response is captured using the simplified model. In other

words, this kind of dynamic response is due to the converter time constant Tg. Also the

simulation results from the PSCAD shows the shape of the notches is not exactly the

same as those from the complete model. This is because, in the PSCAD simulation, the

converter is fully modelled while in the model M1 it is replaced by time constant Tg.

The peak of the “notches” is determined by the value of Tg. The smaller Tg is, in general

the smaller the peak is. When Tg = 0.01 rather than 0.02 seconds, the system response

is Fig. 3.6. When Tg is zero, the notches disappear and this part of dynamic is lost. This

is the case with the simplified model M1.

Inevitably due to the model structures, the system responses during fault are significantly

different. Again, this is because the rest of the test system in M0 are not modeled in

M1 and M2. However, this does not mean M1 and M2 are not capable of delivering the

same dynamic responses. For example, after the fault is cleared the real power Pelec given

by PSCAD simulation is very close to those given by M1 and M2. This only indicates

inputs during the fault are not very ideal for our parameter estimation approach. But

for the time period when the fault just happens or is cleared, the inputs may still be

informative enough for parameter identification.


Figure 3.5: Comparison of System Response using Case 2

(a) system response during 20% voltage drop,

M1

[blue-M1, green-M0]

(b) system response during 20% voltage drop,

M2

[blue-M2, green-M0]


Figure 3.6: System Response During 20% Voltage Drop, Model M1 Tg = 100

[blue-M1, green-M0]


Figure 3.7: System Response During 20% Voltage Drop, Model M1

[blue-M1, green-M0]


Figure 3.8: Ip and Ipcmd during 20% voltage drop, model M1

[blue-Ipcmd, green-Ip]

Figure 3.9: Ip and Ipcmd during 20% voltage drop, model M1 from 29.6 sec to 31.4 sec

[blue-Ipcmd, green-Ip]

Chapter 4

System Identifiability

Identifiability is the central concept in identification problems (see [23] for rigorous def-

inition). The rigorous definition is very general and difficult to put to use in practice.

Loosely speaking, before identification procedure, we would like to know whether the

procedure will yield a unique θ?. It raises two questions:

1. whether the data set (inputs and outputs) is informative enough to distinguish

between different models.

2. whether the resulting model (parameterized by θ?) is equal to the true system

Namely, if the data set is indeed informative enough, we can get some θ? from the

identification procedure, then is this θ? truly is the one. Does there exist a different

θ which will yield same outputs under the same conditions?

A model structure can be globally identifiable, locally identifiable and non-identifiable.

However, determining whether a structure is identifiable or not is in general very diffi-

cult. Fortunately for a specific type of the model there is a well developed method to

answer these questions. If a model can be written in standard continuous state-space

form (first-order differential equations) and all equations including observation functions

are polynomial or rational functions, the identifiability of the model structure can be de-

termined based on differential algebra techniques. A software tool DAISY [5] is available

72

Chapter 4. System Identifiability 73

for this purpose.

[5] gives a simpler definition of priori global identifiability with respect to input-output

experiments. The underlying assumption of this definition is still the same model struc-

ture. Given arbitrary parameters θ1 and θ2, θ1 6= θ2 and model M(θ1) and M(θ2) are

initialized at the same state; does there exist an input function u for which the two

models gives out different outputs? If there is at least one such input, then M(θ1) and

M(θ2) can be distinguished. Model structure M is said to be a priori globally identifi-

able from input-output data. And if no matter driven by what input function, the two

models will always give the same output under the same initialization condition, the two

parameters are indistinguishable from input-output data. In addition, if the number of

indistinguishable parameters are finite, then the structure is called locally identifiable

from input-output data. And if the number is infinite, then the structure is unidentifi-

able. Note that strictly globally identifiable requires the condition is met for all possible

parameters θ. Similarly an unidentifiable structure means no parameters can be iden-

tified over all possible values. The definition is very demanding thus difficult to justify.

The point mentioning these definitions is to clarify the following facts.

1. Even if a structure is globally identifiable, there is no guarantee that all parameters

can always be identified from a specific input. Since globally identifiability requires

only one input for which different outputs are expected. Similarly, if the structure

is locally identifiable, it might still be satisfying enough. The model structure is

often formed with physical parameters (in contrary to black-box model structure),

therefore the parameters normally have a relatively narrow range.

2. Practically, one might not have the luxury to test all inputs. For example, one

parameter might be identifiable only during severe faults in power system. This

is still inspiring since it might suggest different model structures can be used for

different inputs. In other words, for different interests, more experiments can be


done so that different model structures can be chosen accordingly.

3. Based on the demanding definition of global identifiability, multiple tests are re-

quired. This is particularly true for those algorithm where a random parameter

vector is picked for each test, for example the algorithm used in DAISY.

4. If the initial states are known, they can be integrated into identification problem.

And under this specific initial states, the identifiability property of the structure

might change (e.g. from local identifiable to global identifiable). However this

requires the model structure to be algebraically observable.

DAISY implements a differential algebra algorithm to perform parameter identifiability

analysis for dynamic models described by polynomial or rational equations. It is based on

an interactive program REDUCE. The detail of the algorithm is in [5]. Here we provide

a short description of how the algorithm works. The goal is to see if the parameters can

be identified based on the input and output data. First we can find a set of equations

which only involves 1) parameters 2) input and output variables through elimination.

This set is called the characteristic set of the model structure. Then we extract the

coefficients of these equations, and the coefficients are functions of only parameters.

For example one coefficient is θ21θ2 if we can find a, b such that θ21θ2 − a2b = 0 then we

have a = θ1, b = θ2. Similarly more parameters and more coefficient functions can be

added and computed accordingly. Instead of symbolic calculation, in this algorithm, a

random numerical vector will be used, e.g. [a, b] = [1, 2]. Now the problem becomes

solving θ21θ2 = 1 ∗ 2. Apparently if this is the only equation we have, then infinite many

solutions exist and the model structure is unidentifiable. If there is only one solution,

then the model structure is identifiable around this particular parameter vector value.

And in order to test the globally identifiability, this procedure should be repeated many

times until confident enough or user has found the results of interest. And if there are

finite number of solutions, the structure is locally identifiable.


Given the fact that the WTG dynamic model involves rational equations, DAISY can be

used for identifiability analysis. However, a series of assumptions have to be made. The

model structures used for testing identifiability is based on M2.

1. ωref is constant rather than a function of Pelec.

The reason is the relation between Pelec and ωref is highly nonlinear over all possible

Pelec while ωref remains constant when the wind speed exceeds the rated value.

Since we will only focus on the situations when the wind speed is greater than the

rated value during parameter fitting, in this identifiability test, ωref is replaced by

a constant (1.2 pu).

2. Considering the fact the the reactive power control and the active power control

are separate, the reactive control model can be tested separately.

3. The generator/converter and the interface to grid are ignored.

4. All switch operations and limits are neglected.

Under these assumptions, the model becomes a six-order system including pitch control,

torque control, aerodynamic model and drivetrain model. The model consists of three

inputs, ten parameters. The three inputs are the real power on the grid side, commands

for the pitch angle and the generated real power respectively. The real power reference,

as mentioned in the pitch control model, can be computed in different ways. So is the

pitch control command.

The results from DAISY indicate that when three outputs (i.e. The pitch angle, the real

power command and the shaft speed) are available, the system is globally identifiable.

However, the pitch angle and the shaft speed are normally unavailable to utility compa-

nies, which suggests they should be treated merely as internal states rather than outputs.

When the outputs are the rotor speed and the real power, the system is still globally

identifiable at 10 random parameter vectors. However, if we reduce the system to only


one output Pord DAISY fails to give any conclusive results as the identification process

couldn’t finish. According to the software introduction, this may happen due to the

computation complexity. Therefore we tried to reduce the system further by ignoring

corresponding dynamics and replacing states by inputs, thereby reducing the order of

the system. The minimum subset including Pord we can get consists of the torque con-

trol, the aerodynamic model and the drivetrain model. There are only three states Tω,

Pord and ω. When only Pord is the output, DAISY still failed to terminate. Since the

prescribed order of the system might impact the computation time, we tried change the

order in which the equations are given. DAISY still couldn’t finish the identification pro-

cess. Naturally adding more parameters will not help the identification test. In addition,

the system is algebraic unobservable which means adding initial conditions will not help

the identification process either.

Unfortunately we cannot get conclusive results in terms of the whole model structures

M1 and M2. Since even the 3-rd order subsystem contained in these model structures

is too complex and demanding for our computer. But we are still able to provide some

results:

1. The reactive control part is separate from the rest of model given no current limit

is active. Under such circumstances, the three parameters involved Kqvp, Kqvi

and Tg can be globally identified provided inputs Vterm, Qcmd and output Qelec.

This indicates the reactive control part can also be considered separately during

identification procedure. Also the structure of the reactive control part is simple

and typical: a PI controller driven by known error signal and the output passes

through a low pass filter. This means if the output and the error signal is given, the

proportional gain, the integration time constant and the time constant associated

with lag action can be uniquely identified.

2. In order to identify the parameters, if the rotor speed and pitch angle are available

then most of the parameters are identifiable (since we didn’t use the whole model


structure of M2). However this does not necessarily make identification problem

easier. Without these measurements, the structure may still be identifiable. This

only means there is not more efficient differential algebraic method that can quickly

solve the problem.

Chapter 5

UKF Parameter Estimation

Parameter estimation is a process to determine the “best” parameters of known model

structures given some observations and a criterion. A general introduction of parameter

estimation can be found in [24, 23]. There are many methods which may fall within

different categories based on estimation problems. In general, a parameter estimation

method is a mapping from a batch of data (observations and inputs) ZN to a proper

estimated parameter vector θN within a model set DM : ZN → θN ∈ DM [23]. In this

section, we will briefly review the general terms used in parameter estimation methods

then introduce our choice: the unscented Kalman filter.

As noted earlier, our system is described as nonlinear differential equations. For a re-

cursive method, the system should be discrete in time. In the previous chapter we have

reviewed the method of discretizing the models. Now we begin with the general state

space representation of a discrete-time nonlinear system. This is also the form used in

state estimation.

process function: xk+1 = F(xk,uk,vk,θk) (5.1a)

observation function: yk = H(xk,uk,nk,θk) (5.1b)

where k = 1, 2, . . . , N . xk represents the state of the system, yk is the observed measure-

ment and uk is the input. vk is the process noise and nk is the observation noise. The

78

Chapter 5. UKF Parameter Estimation 79

nonlinear function F and H are parameterized by vector θ.

The prediction error given θ? is :

ε(k,θ?) = y(k)− y(k|θ?) (5.2)

In order to evaluate the models, a criterion must be set. The criterion is usually a

scalar-valued function (e.g. norm function) of the prediction error ε(k, θN). Or from the

perspective of statistical inference, the criterion also can be a likelihood function of the

given data (or a function involving the likelihood function). Apparently these functions

are parameterized by θ. Therefore by minimizing the prediction error or maximizing the

likelihood, an estimate of the parameter θ, can be determined. For example, the famous

least-squares criterion is often used as a way to evaluate the prediction error. The second

approach contains different estimation methods, such as the maximum likelihood (ML)

estimation and the Bayesian estimation. In the Bayesian estimation, the “best” estimate

(also called the mode) is referred to as the maximum a posterior (MAP) estimate [29].

The most important features of the Bayesian estimation compared to the ML estimation

include: the parameters are treated as random variables; the prior knowledge of the pa-

rameters can be considered in providing parameter estimates.

Among these techniques, the Kalman filter family has many candidates for solving dif-

ferent estimation problems. Broadly speaking, Kalman filtering is closely related to

Bayesian estimation (the parameter or the state is considered to be Gaussian random

variable). In addition, Kalman filtering is a recursive method which means at each dis-

crete moment, the updates only depend on previous estimate and the new input. In this

family, the extended Kalman filter is famous for solving nonlinear estimation problems

through a linearization procedure. It is an extension of the Kalman filter thus is referred

to as the extended Kalman filter (EKF) [16]. An alternative method is the unscented

Kalman filter (UKF). First proposed by Julier et al. [18] and further developed by Wan

and van der Merwe [11], the UKF has been widely accepted for its many advantages.

Although both are used for nonlinear parameter estimation, there are significant dif-


ferences between the EKF and the UKF. Firstly, in general, the goal of a estimation

procedure is to estimate a state vector. In this sense, the unknown parameter vector can

be viewed as a state vector. The state is assumed to be a Gaussian random variable.

When the new state or observation is updated, the state distribution should be propa-

gated through the nonlinear functions. In the EKF, the first-order linearization of the

functions will be used. In contrast, in the UKF, the original nonlinear system is used.

Secondly, the UKF is able to capture the first two moments of the posterior distribution

correctly to the second order without explicitly computing Jacobian or Hessian matrix.

Its computation complexity is comparable to the EKF. [11]

Due to the linearization, there are many limitations with the EKF. For example, it is

difficult to implement and tune. It’s only reliable when the system is near linear (more

explanation see [17],[22]). In addition, it’s only useful when the Jacobian exists. In

other words, the EKF can not be used if the system contains discontinuities. Therefore,

the UKF algorithm was used in this thesis. The algorithm discussed in the subsequent

chapter is based on the work of Wan and van der Merwe in [11]. Also, Wan and van

der Merwe are the main developers of ReBEL (Recursive Bayesian Estimation Library)

which is a MATLAB toolkit that facilitating sequential Bayesian inference in general

dynamic Bayesian networks (DBNs). The software is available online through [2]. The

UKF algorithm was implemented in this toolkit and was used for parameter estimation

in this thesis.

In this chapter, the first section will present the UKF algorithm; then the structure of

the algorithm implemented using the ReBEL will be introduced; the results of parameter

estimation and the evaluations will be discussed last.


5.1 Parameter Estimation

According to [11], estimation problems may be divided into three types: state estimation,

parameter estimation and dual estimation. The main focus is the application of parame-

ter estimation (sometimes also known as system identification or machine learning) of the

system described in Chapter 3 using the UKF. The standard form of discrete nonlinear

model for parameter estimation is:

θk+1 = θk + wk (5.3a)

dk = G(xk,θk) + ek (5.3b)

In this system, xk represents the input, yk is output of the nonlinear function yk = G(xk,θk)

and dk is the desired output (measured output). Be careful that despite having the same

notations, xk and yk are not the same vectors in the two forms. The standard system

representation may vary but they can be converted in order to fit different estimation

application. In other words, the implementation of the UKF is the essentially the same

nonetheless. Here we present one way to convert a system having the state estimation

form above into the form suited for the parameter estimation.

In order to form the new state-space representation in Eqn. 5.3a, xk should be an aug-

mented vector including the original system states xk and the exogenous inputs uk. wk

is the process noise in parameter estimation and the distribution of which can not be

determined accurately but can be replaced by an educated guess. ek is the error between

the computed output and the desired output (sometimes also referred to as the error of

the machine). ek is an augmented vector including both vk and nk. Namely, θk is the

new state (equivalent to xk in Eqn. 5.1a); xk is the new input (same as original uk),

rk is equivalent to vk while ek is equivalent to nk. Therefore, Eqn. 5.1 Eqn. 5.3 essen-

tially are the same. The UKF algorithm for parameter estimation will be introduced next.


5.2 UKF Parameter Estimation

Based on the system representation in Eqn. 5.3a and 5.3b, the algorithm for UKF pa-

rameter estimation is in Table 5.1. Where:

E(wkwᵀk) = Rw

k

E(ekeᵀk) = Re

k

γ =√L+ λ

λ = α2(L+ κ)− L

Wm,0 =λ

λ+ L

Wc,0 = Wm,0 + 1− α2 − β

Wm,0 = Wc,0 =1

λ+ L, i = 1, 2, . . . , 2L.

L = dim(d) + dim(w) + dim(e)

For the UKF, three parameters need to be determined: α, β and κ. α is a small positive

value (1 in our case) which determines the spread of the sigma points around the expected

parameter θ. κ is usually 3−L [18], and β is used to incorporate prior knowledge of the

distribution of the parameter. Assuming the parameter follows Gaussian distribution,

β = 2 is optimal [11].

Although it’s not obvious in parameter estimation, the noise v and n in the system

Eqn. 5.1 are considered in parameter estimation. θ, v and n will form an augmented

vector and they will be propagated through the unscented transformation at the same

time. If d (measurements in the parameter estimation form) and y (measurements in

the state estimation form) are the same, the e can be n. However, its distribution will

not necessarily be updated. For simplicity, the covariance of e is assumed to a constant.

As mentioned before, the unscented Kalman filter is one method based on the Bayesian

estimation. In Bayesian estimation, we assume the parameter is a Gaussian random


Table 5.1: UKF parameter estimation

Initialize with: (5.4a)

θ0 = E[θ] (5.4b)

P−θ0 = E[(θ − θ0)(θ − θ0)ᵀ] (5.4c)

Time update: (5.4d)

θ−k = θk−1 (5.4e)

P−θk = Pθk−1+ Rw

k−1 (5.4f)

Θk|k−1 = [θ−k , θ−k + γ

√P−θk , θ

−k − γ

√P−θk ] (5.4g)

Dk|k−1 = G(xk,Θk|k−1) (5.4h)

dk =2L∑i=0

Wm,iDi,k|k−1 (5.4i)

Measurement update: (5.4j)

Pdkdk =2L∑i=0

Wc,i(Di,k|k−1 − dk)(Di,k|k−1 − dk)ᵀ + Rek (5.4k)

Pθkdk =2L∑i=0

Wc,i(Θi,k|k−1 − θ−k )(Di,k|k−1 − dk)ᵀ (5.4l)

Kalman Gain: (5.4m)

Kk = PθkdkP−1dkdk

(5.4n)

θk = θ−k + Kk(dk − dk) (5.4o)

Pθk = P−θk −KkPdkdkKᵀk (5.4p)


variable. We initialize using E[θ] which is our guess of the parameter vector. P−θ0 is the

covariance matrix which reflects our belief on the guess. This is also the key point of

Bayesian estimation because it allows the use of prior knowledge of the distribution. As

a random variable, the parameter vector θk follows a stationary process driven by the

process noise w. At the end of each iteration, Rwk (the process-noise covariance) and Re

k

(the measurement-noise covariance) will be updated.

Though it is called noise, w is in fact the change in θ at each step since θk+1 = θk + wk.

It is also considered as a Gaussian random variable with zero mean. In other words,

by assigning Rwk we set a range what the next parameter could be. According to [11],

roughly speaking the larger Rwk is, the more quickly the old data is discarded therefore

Rwk affects the convergence speed. Apparently, if the estimate of parameter converges, w

should be very small and close to zero eventually which suggests the estimated parameter

is stationary around a point. Moreover, from the optimization perspective, Rek can be

considered to be the weight matrix for specifying MSE costs:

J(θ) =N∑k=1

[dk −G(xk,θ)]ᵀ(Rek)−1[dk −G(xk,θ)] (5.5)

The question is how to update Rwk and Re

k. There are different methods to determine

Rwk and Re

k. Three methods are listed below and included in ReBEL toolkit. There

isn’t conclusive results indicating any of the three methods is superior to the rest. The

study of trade-offs between different approaches is still ongoing. Different models and

estimation purposes may lead to different choices. In general, the annealing method is

able to achieve good minimum mean-squared error and it’s very simple. From now on,

the annealing method would be the default choice unless stated otherwise.

1. “annealing” method Rwk = αRw

k−1, where α ⊂ (0, 1]

2. “forgetting factor” method Rwk = (λRLS

−1 − 1)Pθk−1, where λRLS ⊂ (0, 1]


3. Robbins-Monro method

Rwk = (1− γrm)Rw

k−1 + γrmKk[dk − dk][dk − dk]ᵀ(Kk)

ᵀ

5.2.1 Annealing Method

The annealing method is the simplest among the three approaches. To start with, ar-

bitrarily set Rwk to be a diagonal matrix; then Rw

k will be “annealed” (by multiplying

the scaler α after each iteration) towards zero. According to [11], the annealing method

can achieve good final performance; however it requires a greater prior knowledge of the

noise levels. As the “confidence” of the initial guess might vary over different parameters,

Rwk is initialized with fixed values however each entry is not necessarily the same. The

annealing method doesn’t explicitly use any other knowledge e.g. the Kalman gain or

the parameter covariance Pθk .

5.2.2 Forgetting-Factor Method

As the subscript suggests, λRLS is initially defined as the exponential weighting factor or

the forgetting factor in the recursive least-squares (RLS) method [12]. The RLS method

is considered to be a special case of the Kalman filter. And a detailed derivation is

included in [27]. In short, by updating Rwk in this way, Pθk = λ−1RLSPθk−1

.

The choice of the forgetting factor depends on the number of measurements considered

to be relevant for the current properties of the system. In our case, λ is a constant and

slightly less than 1 (which is also the most common choice for systems without abrupt

changes). Ljung discusses the choice of λ in [23] in general for recursive estimation

methods, RLS included. And typical choices are in the range between 0.98 and 0.995.

Basically, the smaller λ is, the more the filter will reply on the recent measurements.

When λ is 1, the weight for all measurements would be the same. A quantitative analysis

can be achieved by defining the memory time constant T0 = 1/(1− λ). This method is

appropriate for online learning process.


5.2.3 Robbins-Monro Method

Robbins and Monro suggested a scheme for estimating the innovations, thereby solving

problems for stochastic approximation recursively. The innovation equation above is one

implementation of this scheme. A general introduction of the scheme can be found in [24].

The Robbins-Monro method provides the fastest rate of absolute convergence and lowest

final minimum mean-squared error (MMSE) values according to investigation in [11].

The gain sequence γ(t) is defined as a time-varying sequence in stochastic approxima-

tion algorithms. In our case, γ(t) = γrm at all t. The gain sequence has been proved to

yield a relationship with the forgetting factor sequence, which is discussed fully in [24].

When γrm is taken as a constant, γrm = 1− λrm where λrm is the corresponding forget-

ting factor. Therefore γrm is normally a small positive number based on the choice of

λrm. Notice that the choice of λrm and λRLS should be in the same range. However, the

forgetting-factor method and the Robbins-Monro method are not the same algorithm.

5.2.4 Parameter Estimation with ReBEL toolkit

The UKF function in the ReBEL toolkit is applicable to state estimation, parameter es-

timation and dual estimation. The toolkit allows the user to define a model description

file in the form described in Eqn. 5.1 and it will generate new form in Eqn. 5.3. For

example, we can define the WTG model M1 and its 9 states, 2 outputs (Pelec and Qelec)

and 5 inputs in one model description file. But the states in parameter estimation are

in fact the parameters. With ReBEL, we can still define the models in their standard

form and the new model for parameter estimation can be automatically generated. The

model interface in ReBEL has to be defined using the equation Eqn. 5.3a and 5.3b first.

In addition, four approaches for parameter estimation are provided. The difference be-

tween approaches stems from the functions used for propagation. The new measurement-

update equation Eqn. 5.3b can include transition functions F (time-update equations,


Eqn. 5.1a) and/or observation functions H (measurement-update equations, Eqn. 5.1b).

Therefore in order to get better estimation results, if possible both F and H should be

included during parameter estimation.

The goal of parameter estimation is to estimate the parameters assuming noisy xk and

dk are known over a certain time period. It can not estimate the states and the parame-

ters simultaneously regardless of the type of filters. That’s the task for dual estimation.

However, even the noisy states are not always available. For example, the integration

results xc and xp will never be available as measurements. In order to get the states,

the actual parameters are used to to solve the NDEs through integration first. As a

result, the states can be generated. In addition, due to the complexity of notation in this

section, from now on to the end of this section, vectork,vector = x,y,w,d, e,u,v,n is

replaced by their non-bold counterparts.

1. both

In this mode, both the transition function F and the observation function H will

be used during estimation. As mentioned above, the “external” state xk and the

external input uk now form the new inputs. The new state are the parameter vector

θk.

θk+1 = θk + wk (5.6)

dk = G(xk, θk) + ek

= H(xk, u2,k, nk)

= H(F (xk−1, u1,k−1, vk−1)|θk , u2,k, nk

) (5.7)

where Eqn. 5.1a and Eqn. 5.1b are substituted. ek could be assigned directly as an

augmented vector based on vk and nk; meanwhile, the value of wk depends on the


method used to update Rwk .

u2,k =[xᵀk−1, u

ᵀ1,k−1, u

ᵀ2,k

]ᵀ=

xk−1

u1,k

u2,k

=

xk−1

u(1)2,k

u(2)2,k

ek = [vᵀk, n

ᵀk]

ᵀ =

e(1)ke(2)k

Since u2,k is the actual second input(the first input vector u1,k is assumed to be

zero due to the stationary transition process), Eqn. 5.7 can be rewritten as below:

dk = G(xk, θk) + ek

= H(xk, u2,k, nk)

= H(F (xk−1, u

12,k, vk−1)

∣∣θk, u

(2)2,k, nk

)However, the notations would become less readable. We’d rather stick to the previ-

ous notations. Just remember that x, u1, u2 should be included in the second actual

input vector u2.

2. both-p

The difference between this model and ”both” is that external states are considered

to be another source of observations as well as inputs. dk =[d(1)k

ᵀ, d

(2)k

ᵀ]ᵀd(1)k = F (xk−1, u1,k, vk−1)|θk

d(2)k = H(xk, u2,k, nk)

Here u2,k =[xᵀk−1, u

ᵀ1,k−1, x

ᵀk, u

ᵀ2,k

]ᵀ3. f-function

The name f-function implies that observation equation will be excluded and the


new observation would be external states xk.

dk = F (xk−1, u1,k, vk−1)|θk

4. h-function

Only the observation function will be used.

dk = H(xk, u2,k, nk)|θk

5.3 Parameter Estimation Results

5.3.1 Estimation results and analysis

Both models have been used for parameter estimation using self-generated and PSCAD

generated inputs. Recall that the identification procedure normally involves different

types of inputs. Following our preliminary model testing results in the previous chapter,

two types of dynamic scenarios are considered in this parameter estimation part: the

wind speed jump and the voltage drop due to fault.

The parameter estimation results in this section will show that the UKF algorithm is

able to identify the parameters of interests successfully according to different dynamic

scenarios. In all, the full model M1 is works well for parameter estimation purpose when

the inputs come from the fault scenario. By contrast, the simplified model M2 is sufficient

for parameter identification when there is a wind speed change. In this section, we will

also go through the difficulties we have encountered so far with the UKF and give some

analysis.


parameter estimation results - model M1

Model M1 was used first to proceed parameter estimation when the inputs are from the

wind speed disturbance. Unfortunately, the estimation process ran into numerical prob-

lems. One reason is because of the limited scope of the test cases. Again, all these cases

are about the wind speed change (from 13 m/s to 15 m/s). The wind speed change is slow

dynamic, the time constant is in seconds. Fig. 3.3 shows the system responses are almost

identical using the two different models. As a result, even assuming full observability,

i.e. when all states are assumed to be known, Tg may not be identified even the noise

level is very low. The simplified model is in fact the same as the complete model with

Tg = 0, p4 = 1/Tg =∞. Given the almost identical measurements, p4 can vary from 50

to infinity. Thus even the slightest noise will cause numerical problems. p4 is unidentified

under such circumstances.

The second reason is the system is stiff and thus the problem is ill-conditioned. If we

look inside the UKF algorithm during the estimation, the Kalman gain corresponding to

p4 has shown to be extremely large after several iterations while the rest remains around

1e-10 level (the exact number depends on the initial guess, noise and UKF parameters).

In this case, the Cholesky factorization method used in the ReBEL toolkit isn’t accurate

enough to facilitate the UKF estimation procedure using MATLAB. Later the topic will

be summarized in the numerical challenges section.

This is also one of the motivations for the simplified model M2. Note that this doesn’t

mean the model structure M1 is unidentifiable. It only means the models can not be

distinguished using these inputs. If there is at least one input will yield different outputs

using different parameters, the system is still globally identifiable. Only when there isn’t

any input which will yield different outputs, the system is unidentifiable.

Since the algorithm can not successfully estimate the parameters given the inputs corre-

sponding to the wind speed change, a different disturbance in which the impact of Tg is

obvious should be used. Naturally a fault scenario (voltage dip 20% due to the fault in


our case) should be used. Table 5.2 shows the estimates result given inputs from fault

scenario. The measurements are given by numerical integration using M1 with the actual

values listed in the table. The initial guess of 1Tg

is 100 while the actual value used for

integration is 50. The initial guess of the rest of the parameters is the actual parameters

plus a uniformly distributed random variable in the interval of [−0.1%, 0.1%] ∗ θ. The

algorithm successfully estimates 1Tg

should be 50 despite its far-off initial guess in only a

few interations. And the ratio of variance to actual value of this parameter is 3.42E-25,

the lowest of all. This result is expected since when the voltage starts to drop the fast

dynamics caused by electronics will mask the slow dynamics. As a result, Tg should be

estimated with great confidence since it is the most crucial parameter in this case. And

indeed it can be identified correctly even from a very inaccurate initial guess.

Table 5.3 shows the results when the PSCAD measurements are directly used as inputs

and desired observations for the UKF. In this case, we don’t know the actual value of Tg

because the PSCAD model has the entire converter while the dynamic model replace the

converter by the time constant Tg. However we do have an initial guess of 1Tg

which is 50.

This is also what we believe to be the actual value. Then we pass the states generated by

numerical integration and the PSCAD measurements to the UKF. The estimated 1Tg

is

70.40140516. Then in order to evaluate the estimates and the original parameters (those

we believe to be the actual parameters), the cumulative error of Ip and Iq: errip and erriq

and the total error are calculated (Eqn. 3.12). The reason the prediction error of injected

current is chosen is because the dynamic WTG model is in fact a controlled current

source by the terminal voltage, the wind speed and power commands. In addition, for

DFIG the real and reactive power controls are separately controlled by Ip and Iq. This

choice of prediction error in a way evaluates the predicted observations compared to the

desired observations.

From the table, we can see that the estimated parameter is able to reduce the total error

by decreasing errip. In other words, using the estimated error, the real power during the


Table 5.2: Estimation Results M1 Fault Scenario

Initial guess of 1Tg

= 100, Pθ0 = 0.01. Final PwN= 1e− 8.

parameter estimate variance actual value variance/θ

Kitrq 0.600620846 1.16E-12 0.6 1.93E-12

Kptrq 2.997675737 2.90E-11 3 9.66E-12

1Tpc

19.97100203 1.29E-09 20 6.44E-11

1Tg

50 1.71E-23 50 3.42E-25

Kqvi 2.001800492 1.29E-11 2 6.44E-12

Kqvp 0.249912427 2.01E-13 0.25 8.05E-13

Kip 25.03286881 2.01E-09 25 8.05E-11

Kic 29.96844648 2.90E-09 30 9.66E-11

Kpp 149.9361362 7.25E-08 150 4.83E-10

Kpc 2.999537581 2.90E-11 3 9.66E-12

1Tpi

3.330074241 3.58E-11 3.333 1.07E-11

Kaero 0.006989634 1.58E-16 0.007 2.25E-14

12H

0.101205177 3.30E-14 0.10215 3.23E-13


Table 5.3: Estimation Results M1 Fault Scenario Compared to M0

Initial guess of 1Tg

= 50, final PwN= 1e− 8

parameter estimates variance variance/θ

Kitrq 0.600110392 1.11E-12 1.86E-12

Kptrq 2.996345516 2.78E-11 9.28E-12

1Tpc

20.01913126 1.24E-09 6.19E-11

1Tg

70.40140516 1.09E-23 *

Kqvi 2.001833353 1.24E-11 6.19E-12

Kqvp 0.249635297 1.93E-13 7.73E-13

Kip 24.96244536 1.93E-09 7.73E-11

Kic 29.99223722 2.78E-09 9.28E-11

Kpp 149.8774847 6.96E-08 4.64E-10

Kpc 2.996086977 2.78E-11 9.28E-12

1Tpi

3.322690155 3.44E-11 1.03E-11

Kaero 0.006999046 1.52E-16 2.17E-14

12H

0.101154213 3.17E-14 3.10E-13

- errI errip erriq

estimates 0.006845545 0.00198905 0.006550204

original 0.007534706 0.003723791 0.006550204


transients is closer to the PSCAD simulation result compared to the real power we get

from our original belief. Note that the cumulative error is computed using the trape-

zoidal rule over a very short period just after the voltage starts to drop and before the

real power starts to recover from the fault. As the consequence of the different model

structures (M0 and M1), the dynamic responses are very different except when the fault

just starts and get cleared. In other words, we can only use about several milliseconds

worth measurements. If we use more than that, due to the structural difference, the

estimated parameters are likely to oscillate and blow up eventually. But if we use too

few, then the risk that the measurement number is insufficient exists. Thus the UKF

might not be able to give satisfying estimates or even converge. One solution to this is

using high sampling rate to get as many data during the fault as possible especially when

the fault just starts (or ends). In summary, the estimates using M1 work well when the

fault scenario is used.

parameter estimation results - simplified model M2

The results in this part are given by the simplified model M2 using the self-generated

inputs and the PSCAD generated inputs. Both the process noise v and the observa-

tion noise n follow normal distribution (zero mean). The variance of v is 1e-5 while the

variance of n is 2e-5. The variance for all states (or all observations) is assumed to be

the same. In addition, the process noise for all states (or the observation noise for all

observations) are assumed to be independent.

Table 5.4 show the estimated parameters using the UKF given self-generated inputs (i.e.

the validation of the UKF). The column “variance/θ” provides how big the variance is,

therefore we can know how reliable the estimates are. The differences between the two

tables are the initial guess of Pθ0 . The initial guess of the parameter in both cases,

is the actual parameter plus a uniformly distributed random variable in the interval of

[−0.1%, 0.1%] ∗ θ.


In comparison, the results in Table 5.4 shows that when the initial guess of the parameter

covariance is accurate enough, given the same amount of measurements, the final variance

can be comparably smaller. However, when the prior knowledge impact the estimation

results very much it might indicate that the parameters can not be easily identified in

this situation.

Table 5.5 shows the results using input from the PSCAD. The observations in the param-

eter estimation are Pelec and Qelec. From optimization perspective, the algorithm will try

to find the best parameter that minimizes the squared error between the desired obser-

vations and the predicted observations. In order to evaluate how the estimate work, we

still use the error errip, erriq and the cumulative total current error errI . The cumulative

errors are computed by integrating over the entire simulation period.

errI decreases using the estimated parameters compared to the original parameters. It

indicates the model does improve by using the estimated parameters. The absolute value

of errI is of less importance since it’s related to the simulation time and how well the

simplified model works. Note that in order to evaluate the parameters better, no noise

was added to generate the predictions in Table 5.5. In other words, the predicted errI

will not change between runs if the parameter is fixed. However, for parameter estima-

tion, it is allowed to assume the measurements are noisy.

Despite the fact that the initial value of the parameter is a random variable which is

close to actual value, the initial guess of the variance of the distribution is quite small.

This requires great prior knowledge of the parameters.

Annealing method is not the fastest method in terms of convergence speed. However,

when dealing with problematic estimation, it might provide insights of the numerical

problems. On the other hand, the Robbins-Monro is believed to have the lowest mini-

mum mean squared error in general. However, when dealing with problematic estimation

especially with many parameters and states, it is very difficult for the user to identify

the problem since w can vary a lot between steps and it’s different for all parameters.


Table 5.4: Estimation Results M2 Wind Speed Jump

Initial Pθ0 = 0.001, final PwN= 1e− 8, N = 2000.

parameter estimate variance actual value variance/θ

Kitrq 0.599898341 1.88E-29 0.6 3.13E-29

Kptrq 3.000000628 8.28E-27 3 2.76E-27

1Tpc

19.99992825 2.24E-23 20 1.12E-24

Kqvi 2.000151045 2.32E-27 5 4.63E-28

Kqvp 0.249999578 8.54E-31 0.25 3.41E-30

Kip 24.99997254 5.66E-23 25 2.26E-24

Kic 29.999895 1.17E-22 30 3.91E-24

Kpp 149.9998831 7.33E-20 150 4.89E-22

Kpc 2.999895848 1.17E-26 3 3.91E-27

1Tpi

3.33291903 1.79E-26 3.33 5.37E-27

Kaero 0.006999186 2.17E-32 0.007 3.10E-30

12H

0.101250722 1.52E-32 0.1012 1.51E-31


Table 5.5: Estimation Results M2 Wind Speed Jump Compared to M0

Initial Pθ0 = 0.001, final PwN= 1e− 8, N = 5000

parameter estimates variance variance/θ

Kitrq 0.599844212 1.00E-16 1.67E-16

Kptrq 2.999683143 5.64E-17 1.88E-17

1Tpc

20.03503501 1.00E-16 5.00E-18

Kqvi 2.001455958 1.00E-16 2.00E-17

Kqvp 0.250029161 7.20E-17 2.88E-16

Kip 24.99708497 1.00E-16 4.00E-18

Kic 29.9991479 1.00E-16 3.33E-18

Kpp 149.998246 1.00E-16 6.67E-19

Kpc 2.999853706 1.00E-16 3.33E-17

1Tpi

3.334797074 1.00E-16 3.00E-17

Kaero 0.006577771 7.17E-17 1.02E-14

12H

0.101935636 9.99E-17 9.87E-16

- errI errip erriq

estimates 0.099730117 0.08638564 0.049835905

actual 0.099937006 0.086624326 0.049836044


Convergence

The predominant evidence of convergence is that Pθ decreases to a small value around

zero at the end of estimation. For absolute convergence, the Kalman gain will also goes

to zero. Pθ is the covariance computed after the estimation, i.e. posterior covariance.

The mean is roughly the latest estimate from the last innovation where all data over

the whole period has been used. Therefore, Pθ is not necessarily very close to zero. For

example if the parameter is large and this is no way to scale it down, then the acceptable

covariance should also be large. More specifically, if the ratio between the biggest pa-

rameter and the smallest parameter is very large, the ratio of the acceptable covariance

between the two parameters should also be high. Numerically, Pθ should be comparable

to a small percent of the corresponding parameter value. If this is true, then we can draw

the conclusion that the estimates converge to a small region and we will also know from

Pθ the range of the parameter.

If Pθ is much bigger compared to the parameter value, the algorithm fails to converge.

There might be several explanations: One is such “best” parameter doesn’t exist in this

particular case. This might suggest that the parameter is unidentifiable under the cur-

rent input. This may also occur when measurements from a different model structure

are used. If the structures are different, then during the parameter estimation, it is pos-

sible the structure used for parameter estimation can not fully predict a similar response

compared to the measurements no matter what parameter values are chosen. It is also

possible the “best” parameters estimated vary a lot and lose its original physical meaning.

In other words, the structural difference will cause potential numerical problems even the

same parameters are used to produce the measurements. Another reason of divergence

might be the filter is not well tuned. Even though the UKF is much better compared to

the EKF in dealing with nonlinear models, it doesn’t mean any arbitrarily chosen initial

points and UKF parameters (α, β, κ for generating sigma points) will guarantee absolute

convergence. Also the parameters used for updating Pw will impact the convergence


speed (e.g. annealing factor in the annealing method). In addition, the prior knowledge

noise level might also cause numerical problems.

Numerical challenges in the UKF

In Table 5.1, the algorithm will be executed for N times. N is the length of the discrete

time horizon. At each iteration, the new input will be included into the estimation along

with all previous data. In the UKF, there are several numerical challenges which might

prevent the estimation from convergence or even completing. Note that when numerical

ill-conditioning exists, the problem might be amplified through innovations.

• Cholesky decomposition, Eqn. 5.4g

The Cholesky decomposition requires the covariance matrix to be positive semi-

definite. Although the Cholesky factorization is indeed generally much better com-

pared to using the covariance matrix directly with respect to numerical condition-

ing, it is possible the covariance matrix becomes non-positive semidefinite during

the parameter estimation procedure. As a result, the Cholesky factorization can

not be done. Note that there are two parts of covariance, the first part is the

parameter covariance while the second part is the noise covariance. During the

update, the Cholesky decomposition will be calculated for both parts.

This leads to a trade-off, as the covariance is small, numerical problems tend to

happen. If the noise covariance is small, which suggests the estimated parameter

falls into a small neighbourhood, it is possible the estimate is in fact not very good

and the parameter covariance is big. However, the whole estimation can be com-

pleted because for all methods of updating the noise covariance, the covariance will

be kept constant after it passes a threshold e.g. 1e-8. As a result, the estimate

will be stuck at the region and can not escape. Such cases will happen particularly


when the “annealing” method is used for updating the noise covariance.

Given the fact above, Rw should be large enough. However, if the noise covariance

is big, which means the parameter can vary a lot after the update, it is possible the

next step is too aggressive thus the innovation error blows up and consequently the

parameter covariance and the noise covariance become non-positive semi-definite.

This will happen particularly with Robbins-Monro method since it uses innovation

error as well as the Kalman gain to update the noise.

• Kalman gain, Eqn. 5.4n

In order to calculate the Kalman gain Kk = PθkdkP−1dkdk

, the inverse of Pdkdk has

to be computed. This can be a problem when Pdkdk and Pθkdk are very close to

zero. Inaccurate Kalman gain will lead to ill numerical conditioning.

Here we provide an example where Cholesky factorization does not work well thus ill-

conditioned Kalman gain will appear. Given the fault scenario, as we mentioned before,

the system response just after the fault is mainly determined by the converter time

constant Tg. Loosely speaking, Tg is the dominant parameter in terms of this type of

disturbance. And by varying Tg, the prediction of Pelec will change a lot. The UKF

algorithm or any sigma point Kalman filter propagates a distribution of the parameter

through the nonlinear model at each innovation. Thus the innovation error associates

with Tg and Pelec will be very large (possibly 1e14 times larger). This will lead to a huge

entry of the Kalman gain correspondingly. Thus after several innovation, Pθk becomes

non semi-positive definite and the estimation terminates.

The solution for this is to use QR decomposition instead of Cholesky factorization. QR

decomposition has been proved to have better numerical properties especially for ill-

conditioned or stiff problems. Cholesky factorization method may lead to substantially

less accurate solutions than methods based on the QR decomposition [34]. Since our

model M1 is stiff, the Cholesky factorization will directly lead to divergence of the UKF

no matter which scenario is used. The QR decomposition on the other hand, works well


for M1 given fault scenario. However, due to the reason we mentioned earlier (it is very

difficult to identify Tg given just wind speed jumps), even with QR decomposition, the

problem is ill-conditioned with M1. In contrast, when we neglect Tg, the model M2

becomes less stiff compared to M1. Therefore the UKF is able to give out reasonable

estimates without running into numerical problems.

5.3.2 Summary

In this chapter, the UKF parameter estimation algorithm has been reviewed. And the

parameter estimation results given two types of transient events were given. One is the

wind speed disturbance and the other one is voltage drop due to system fault. The same

scenarios were used to test the two models described in the previous chapter: the full

modelM1 and the simplified model M2 without converter lag time constant. The cumu-

lative current error over the simulation period was used to quantitatively evaluate the

models/parameters.

Both models show satisfying results in terms of the active and reactive power outputs

of the WTG. However, both models are less accurate with respect to the pitch angle

and the rotor speed especially during the transients. This is the result of the linear

approximation of the power coefficient curves in the aerodynamic model. The power

coefficient curves take the rotor speed into account all the time. Meanwhile, in the linear

aerodynamic model, the rotor speed is implicit. Therefore, it’s understandable that the

detailed aerodynamic model with Cp curves settles down more quickly. However, due

to the fact that the detailed aerodynamic model leads to smaller change in pitch angle,

this also means the active power will be larger during the transits as the result of pitch

control. And higher active power contributes to higher rotor speed, which partly explains

the discrepancy of the rotor speed during the same period.

Furthermore, compared to M1 and M2, M0 is the time-domain detailed model which

consists of many more parameters and constraints. Thus the results won’t be the same.


In other words, these discrepancies in system responses are partially due to the different

model structures.

Due to the fact that the converter dynamics can not be observed under the disturbance of

the wind speed jump, parameter Tg in M1 can not be identified using this type of inputs

no matter if the better factorization method in the UKF is used. Due to the stiffness of

M1 and the nature of the disturbance, even the better factorization - QR decomposition

can not help the estimation process.

The measurements from the fault scenario was then used for parameter identification

using M1. With the QR decomposition method, the UKF was able to give out satis-

fying estimates. Especially Tg can be identified with great confidence even with very

inaccurate initial guess with self-generated inputs. When given PSCAD measurements,

the estimated parameters will reduce the prediction error of currents compared to the

original parameters.

Inspired by the first experiment, M2 was used to estimate parameters except for Tg given

wind speed disturbance instead. The simplified model (neglecting Tg) is less stiff there-

fore the Cholesky factorization method is able to give reasonable estimates as well. When

given PSCAD measurements, the current error has been improved using the estimated

parameters compared to the original ones using the simplified model.

In addition, if the estimation procedure becomes ill-conditioned due to Tg, one way to

facilitate the identification is comparing prediction using different guesses during faults

to PSCAD simulation results. Thus Tg can be narrowed down to a relatively narrow

region which may help the estimation procedure.

Chapter 6

Conclusion

6.1 Summary and Conclusions

In this thesis, the type-3 generic WTG model purposed by WECC has been reviewed in

details and two simplified models have been proposed for the purpose of parameter fit-

ting. Both models have been verified in comparison with the detailed model. All models

include major parts of the generic model: the aerodynamic model (Section 2.2), the tur-

bine model (Section 2.3), the pitch control model (Section 2.4), the generator/converter

model and the converter controller models (Section 2.5).

The detailed time-domain model M0 has been implemented in the PSCAD environment

and it provides measurements for parameter fitting. The test system consists of one 1.5

MVA DFIG, its converter, a transformer between the converter and the stator, a trans-

former between the WTG and the grid. And the grid is represented by an ideal voltage

source behind the grid Thevenin equivalent.

The full model M1 ignores most of the limiters in the detailed model. And its aero-

dynamic model is a linear approximation of its counterpart in the detailed model. The

further simplified model M2 neglects the converter lag time constant.

The simulation results indicate that M1 and M2 yield almost identical outputs given the

103

Chapter 6. Conclusion 104

same inputs and parameters when the focus is the transient stability under wind speed

change. And thus M2 should be used for parameter estimation if the real power is the

only concern. M1 has been proved to be stiff and the estimation becomes ill-conditioned

given such inputs despite the choice of factorization methods.

When a voltage drop happens due to fault, M1 is able to capture the fast dynamic caused

by the electronics. Meanwhile, M2 can not accurately predict the system response under

such circumstances for neglecting the associated time constant Tg. Thus M1 is used for

the model in parameter estimation. The results show that Tg can be successfully identi-

fied with very small covariance.

The parameter estimation is carried out by the UKF in this thesis. It has many ad-

vantages in terms of estimation involving nonlinear systems. First and foremost, it does

not require the system to be smooth. And it gives out the posterior distribution of

the parameters. And by tuning the UKF parameters, users can get information about

the convergence. It is very helpful for complicated system of which the identifiability is

uncertain. And through the UKF, one can arbitrarily choose which parameters to be es-

timated. In short, the UKF provides freedom in terms of nonlinear parameter estimation

compared to other parameter identification tools. Our results show that when the initial

guess is close to the actual parameters, using the PSCAD measurements as inputs, the

UKF is able to improve the model in terms of chosen criterion. Specifically, using the

estimated parameters for M1 and M2, the prediction error of the injected current will

decrease with respect to the measurements in M0.

6.2 Contributions

The main contributions of this work are:

1. Implementation of full WECC generic model in PSCAD.

Chapter 6. Conclusion 105

2. UKF based parameter estimation for type-3 WTG (both full and simplified model).

The results show that the full model works well in terms of parameter estimation

when a fault scenario is used. And the simplified model works well when the wind

speed changes. The estimated parameters indeed reduce the error compared to the

time-domain model results.

3. Insights about using the UKF for nonlinear stiff system. Through the presentation

of the results, we also provide 1) Evaluation of UKF convergence property when

different methods for updating process covariance are used; 2) Factorization method

and their impact on the dynamic WTG models.

4. Identifiability analysis of the type-3 WTG based on differential algebraic algorithm.

6.3 Future Work

Not much research has been done on parameter estimation of the generic WTG models.

The work in this thesis is a start. The future work may include:

1. Low-voltage ride through (LVRT) requirement is an important feature for DFIG to

fulfill grid code. However it is not considered in the stability models in this thesis.

In addition, in our case, only one single DFIG (1.5 MW) has been implemented in

the simulation. The future work may include adding the LVRT and the supervisory

control of the wind plant to the model.

2. Implement state estimator to directly estimate states based on measurements from

the time-domain model. In other words, implement dual/joint estimation instead

of just parameters estimation.

3. Consider the whole system (not just WTG) during parameter estimation process.

This involves in solving differential-algebraic system.

Appendices

106

Appendix A

The Torque Control Model

A detailed discussion of the torque control model is outlined in this appendix. Unsur-

prisingly the torque controller presented is based on per-unit values. Since the torque

control is part of the rotor-side converter, the base values should be consistent with the

machine base values. That is the voltage base for the RSC control is the peak value of

rated phase voltage while the current base is the peak value of rated line current. For

the convenience of analysis, the rotor-side values are often referred to the stator side.

Eqn. A.1 shows the relationship between the electrical torque and the q-axis rotor cur-

rent of a DFIG using actual values (the negative torque suggests the motor convention).

Vs is the amplitude of the stator voltage (i.e. the terminal voltage of the PCC). Irq is

the q-axis rotor current (referred to the stator side). And ω0 represents the frequency

of the voltage which is 60 Hz since the stator is connected to a three-phase balanced

voltage source in the test system with symmetrical components. Eqn. A.2 is a good

approximation based on the fact that the magnetizing inductance is much greater than

the stator resistance (normally several hundred times larger). Hence Eqn. A.3 is achieved

using the base quantities listed in Table 2.5. Therefore Irq,ref can be computed when the

torque reference is known. Note that though the per-unit equation is derived based upon

the assumption that the rotor current has been referred to the stator side, nonetheless

107

Appendix A. The Torque Control Model 108

Irq,ref,pu will be the same value on the rotor side. By multiplying the rotor speed, the

per-unit electrical power yields Eqn. A.4.

In the PSCAD/EMTDC simulation, the actual values must be used in terms of control-

ling the converter. Irq,ref,pu will be translated to the corresponding actual value using

the peak value of the rated rotor current (without referring).

Telec = −3

2

1

σs + 1λsirq (A.1)

Telec≈−3

2

Vs(σs + 1)ω0

irq (A.2)

Telec,pu = − Vs,pu(σs + 1)

irq,pu (A.3)

Pelec,pu = −Vs,puωr,pu(σs + 1)

irq,pu (A.4)

where

σs =LsLm

Ls is the stator leakage inductance and Lm is the mutual inductance. In general, σs

is quite small, for example when σs = 0.0408 if Lm = 4.2 pu and Ls = 0.1714 pu, which

means σs can be neglected. As a result, the per-unit electrical power is presented in

Eqn. A.4. If ωr,puirq,pu is treated as a new control variable irq,pu,new, then we get the

torque control model, where in steady state Pord ≈ Pelec,pu.

Appendix B

System Parameters

The system configuration is in Fig 2.13. And the tables below show all the parameter

values.

Table B.1: Transmission Line Parameters

Parameter Value Per-unit Value

resistance 2.5 Ω 0.009375

inductance 0.04 H 0.05655

109

Appendix B. System Parameters 110

Table B.2: Parameters of the Induction Machine

Parameter Value

rated power 1.5 MVA

rated voltage (L-L) 0.69 kV

frequency 60 Hz

stator/rotor turns ratio 0.379

stator resistance 0.0071 pu

rotor resistance 0.005 pu

magnetizing inductance 4.2 pu

stator leakage inductance 0.1714 pu

rotor leakage inductance 0.1563 pu

Table B.3: Parameters of the Step-up Transformer

Parameter Value

rated power 1.5 MVA

rated voltage (L-L) 0.69 kV/20 kV

frequency 60 Hz

connection Y-Y

no load losses 0.001 pu

copper losses 0.002 pu

positive sequence leakage reactance 0.06 pu

Appendix B. System Parameters 111

Table B.4: Parameters of the Transformer (stator-converter)

Parameter Value

rated power 0.3 MVA

rated voltage (L-L) 0.6 kV/0.5 kV

frequency 60 Hz

connection Y-Y

positive sequence leakage reactance 0.1 pu

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