ThesisPDF Available

Methods for Identification of Instantaneous Frequencies for Application in Isolated Microgrids

Authors:

Abstract and Figures

Several cases with time varying frequencies have been reported in isolated electrical systems such as stand-alone microgrids and marine vessel power systems. This thesis studies the use of several types of Kalman filters (KF), Hilbert-Huang Transform (HHT) and the proposed method of merging empirical mode decomposition (EMD) and KF for the purpose of tracking instantaneous values of voltage- and current waveforms in isolated microgrids with the aforementioned challenges. Both synthetic signals and real measurements from a marine vessel power system were used to validate the methods. The algorithms and methods were implemented in Matlab and Simulink. In varying degrees, the methods did all prove to be viable options for tracking of the fundamental frequency on the marine vessel. The proposed method turned out to be particularly powerful to decompose multicomponent signals consisting of several time-varying monocomponents, and track their instantaneous amplitude and frequency.
Content may be subject to copyright.
Methods for Identification of
Instantaneous Frequencies for
Application in Isolated Microgrid
Haakon Jondal Helle
Master of Science in Cybernetics and Robotics
Supervisor: Marta Maria Cabrera Molinas, ITK
Co-supervisor: Mohammad Amin, ITK
Department of Engineering Cybernetics
Submission date: June 2017
Norwegian University of Science and Technology
Problem Description
Name: Haakon Jondal Helle
Faculty: Faculty of Information Technology and Electrical Engineering
Department: Department of Engineering Cybernetics
Study program: Cybernetics and robotics (Master, 5 years)
Main profile: Control of smart grids and renewable energy
Start date: 9 January 2017
Due date: 5 June 2017
Supervisor: Professor Marta Molinas
Title: Methods for Identification of Instantaneous Frequencies for Application in Isolated
Microgrids
Work Description:
Find a suitable model for the extended Kalman filter and unscented Kalman filter for
estimating ”instantaneous frequencies” or time-varying frequencies in multicompo-
nent signals.
Implement an Extended Kalman filter and an unscented Kalman filter using this
model, in Matlab and Simulink.
Assess the strengths and limitations of this model when identifying instantaneous
frequencies in synthetic signals.
Implement Hilbert Huang Transform (empirical mode decomposition and Hilbert
Transform) in Matlab and Simulink.
Propose the method of merging empirical mode decomposition and Kalman fil-
tering. Suggest structures for single-phase and three-phase systems. Validate the
merged tool using synthetic signals.
Assess the aforementioned methods with real voltage measurements from a marine
vessel power system. Implement a phase-locked loop to serve as reference for the
tracking of the fundamental frequency.
i
ii
Abstract
THE introduction of new methods for production and distribution of electrical energy
has increased the attention related to problems with power quality and the presence
of time-varying frequencies. It has been reported several cases with such problems in iso-
lated electrical systems such as isolated microgrids for incorporation of renewable energy
sources and marine vessel power systems. The sources and loads in such systems are usu-
ally interfaced with power electronic equipment, meaning that there is low or no inertia.
The low inertia and the stochastic nature of the generation and loads results in systems
that are prone to nonlinear distortions and variations in the fundamental frequency. The
hitherto used measurement- and monitoring equipment have mostly been based on average
value calculation. The aforementioned problems in isolated electrical systems have made
the need of measurement of instantaneous values instead of average values apparent, in
order to have monitoring- and control systems with satisfying performance and accuracy.
This thesis studies the use of several types of Kalman filters (KF), Hilbert-Huang Trans-
form (HHT) and the proposed method of merging empirical mode decomposition (EMD)
and KF for the purpose of tracking instantaneous values of voltage- and current waveforms
in isolated microgrids with the aforementioned challenges. Both synthetic signals and real
measurements from a marine vessel power system were used to validate the methods. The
algorithms and methods were implemented in Matlab and Simulink.
In varying degrees, the methods did all prove to be viable options for tracking of the funda-
mental frequency on the marine vessel. The proposed method turned out to be particularly
powerful to decompose multicomponent signals consisting of several time-varying mono-
components, and track their instantaneous amplitude and frequency.
iii
iv
Sammendrag
INNFØRINGEN av nye metoder for produksjon og fordeling av elektrisk energi har økt
oppmerksomheten rundt problemer med strømkvalitet og tidsvarierende frekvenser.
Det har blitt rapportert flere hendelser med slike problemer i isolerte kraftsystem som for
eksempel isolerte microgrids for inkorporering av fornybare energikilder og kraftsystem
ombord p˚
a marinfartøy med elektrisk fremdrift. Kilder og laster i slike system er van-
ligvis knyttet sammen gjennom kraftelektronisk utstyr, noe som fører til lavt treghetsmo-
ment. Det lave treghetsmomentet, i tillegg til kraftproduksjonens og lastenes stokastiske
natur, resulterer i system som er utsatt for ulineære forvrengninger og variasjoner i grunn-
frekvensen. Det hittil benyttede m˚
ale- og overv˚
akingsutstyret har for det meste vært basert
p˚
a gjennom- snittsverdi beregninger. De tidligere nevnte problemene i isolerte kraftsys-
tem har gjort det tydelig at m˚
alinger heller burde baseres p˚
a momentanverdier, slik at
overv˚
akings- og kontrollsystemer opprettholder tilfredsstillende ytelse og nøyaktighet.
Denne masteroppgaven studerer bruken av forskjellige Kalman-filtre (KF), Hilbert-Huang
Transform (HHT) og den foresl˚
atte metoden for sammensl˚
aing av empirical mode decom-
position (EMD) og KF for følging av momentantverdier i spennings- og strøm bølgeformer
i isolerte microgrids med de nevnte utfordringene. B ˚
ade syntetiske signaler og ekte spen-
ningsm˚
alinger fra et kraftystem p˚
a marinfartøy ble brukt til ˚
a validere de forskjellige meto-
dene. Algoritmene og metodene ble implementert i Matlab og Simulink.
I varierende grad, viste de foresl˚
atte metodene seg ˚
a være gode metoder for følging av
grunnfrekvensen om bord p˚
a marinfartøyet. Den foresl˚
atte metoden viste seg ˚
a være spe-
sielt kraftig for ˚
a dekomponere signal bestende av flere tidsvarierende monokomponenter,
og estimere deres momentane amplitude og frekvens.
v
vi
Preface
THIS is the master’s thesis to conclude the Master of Science degree in Cybernetics
and Robotics at the Norwegian University of Science and Technology. The work
was carried out at the Department of Engineering Cybernetics during spring 2017.
I would like to express my sincere gratitude to my supervisor, professor Marta Molinas
for giving me the opportunity to work with such an interesting topic. The completion of
the thesis would not have been possible without her tireless guidance and her remarkable
insight on the topic. Secondly, I would like to thank Vijay Venu Vadlamudi. His great
lectures aroused my interest for the world of electric power engineering.
I am grateful for the mail conversations with Manuel Duarte Ortigueira and Raul Rato for
increasing my understanding of their version of the EMD.
I would also like to thank Tomasz Tarasiuk from Gdynia Maritime Institute for providing
the measurements from the marine vessel power system.
Trondheim, 5 June 2017
Haakon Jondal Helle
vii
viii
Table of Contents
Problem Description i
Abstract iii
Sammendrag v
Preface vii
Table of Contents ix
List of Tables xiii
List of Figures xv
Abbreviations xvii
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Objective and Scope of Work . . . . . . . . . . . . . . . . . . . . 2
1.3 MainContributions............................. 2
1.4 StructureoftheReport ........................... 2
2 State-of-the-art in Methods for Frequency Identification in Microgrids 5
2.1 Introduction to Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Fundamental Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Time- and Phasor Domain . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 RMS, Effective Value . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Harmonics and Total Harmonic Distortion . . . . . . . . . . . . . 8
2.2.4 FourierAnalysis .......................... 8
2.2.5 Clarke Transformation . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.6 Symmetrical Component Theory - Fortescue’s Theorem . . . . . 10
2.3 Frequency Identification Methods . . . . . . . . . . . . . . . . . . . . . 12
ix
2.3.1 Kalman Filter and Extended Kalman Filter . . . . . . . . . . . . 12
2.3.2 The Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . 14
2.3.3 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.4 Hilbert-Huang Transform . . . . . . . . . . . . . . . . . . . . . 19
2.3.5 Phase-Locked Loop . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Summary of Previous Work 25
3.1 The Kalman filter and the Adaptive Kalman filter . . . . . . . . . . . . . 25
3.2 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Harmonics- and Frequency Tracking Using Kalman Filters 31
4.1 Tracking of Three-Phase Harmonics Based on Linear Kalman Filter . . . 31
4.2 Tracking of Fundamental Frequency Based on Extended- and Unscented
KalmanFilter ................................ 35
4.3 Simulations ................................. 37
4.3.1 Estimation of the Phase Angle . . . . . . . . . . . . . . . . . . . 37
4.3.2 Tracking of Time-Varying- Amplitude and Fundamental Frequency 38
5 Merging Empirical Mode Decompositon and Kalman Filtering 41
5.1 Single-PhaseSystems............................ 41
5.2 Three-PhaseSystems ............................ 42
5.3 Merging Empirical Mode Decompositon and Kalman Filtering - A Vali-
dationStudy................................. 44
5.3.1 Experiment1 ............................ 44
5.3.2 Experiment2 ............................ 47
6 Assessment of Methods for Tracking of Time-Varying Frequencies in real data
from a Marine Vessel Power System 51
6.1 3.33secondanalysis ............................ 53
6.2 60secondanalysis ............................. 55
6.3 Interpreting the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7 Conclusion and Future Work 63
Bibliography 65
Appendices 71
A Simulink Models 73
A.1 Tracking in Three-Phase Test System . . . . . . . . . . . . . . . . . . . . 73
A.2 Tracking in Marine Vessel Power System . . . . . . . . . . . . . . . . . 75
B Matlab Code 77
B.1 Voltage Source in the Three-Phase Test System . . . . . . . . . . . . . . 77
B.2 Empirical Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . 78
B.3 Calculation of Instantaneous Amplitude and Frequency . . . . . . . . . . 80
B.4 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
x
B.5 Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.6 From Estimated States to Estimated Magnitude, Phase Angle and Frequency 83
B.7 ParkTransformation ............................ 84
xi
xii
List of Tables
3.1 Parameters for project experiment 1. . . . . . . . . . . . . . . . . . . . . 27
3.2 Parameters for project experiment 2. . . . . . . . . . . . . . . . . . . . . 29
5.1 Parameters and tuning in experiment 1. . . . . . . . . . . . . . . . . . . . 45
5.2 Parameters and tuning in experiment 2. . . . . . . . . . . . . . . . . . . . 47
6.1 UKF tunings for tracking on the marine vessel. . . . . . . . . . . . . . . 52
xiii
xiv
List of Figures
2.1 An example microgrid with DC bus and the possibility to connect to the
maingrid. .................................. 6
2.2 Three-phase voltages in the time domain. Phase voltages plotted with solid
lines and line voltages plotted with dotted lines. . . . . . . . . . . . . . . 7
2.3 Three-phase voltages shown in a phasor diagram. . . . . . . . . . . . . . 7
2.4 The Clarke transformation shown graphically. . . . . . . . . . . . . . . . 10
2.5 Unbalanced three phase voltages. . . . . . . . . . . . . . . . . . . . . . . 11
2.6 The sequence components of the unbalanced voltages. . . . . . . . . . . 12
2.7 TheKFalgorithm. ............................. 14
2.8 Comparison between the HT and FFT. . . . . . . . . . . . . . . . . . . . 19
2.9 Signal with spline interpolations and mean. . . . . . . . . . . . . . . . . 21
2.10StepsoftheEMD............................... 22
2.11 Instantaneous amplitude and frequency of the three first IMFs. . . . . . . 23
2.12 PLL structure and step response. . . . . . . . . . . . . . . . . . . . . . . 24
3.1 The regular KF and the AKF compared. . . . . . . . . . . . . . . . . . . 26
3.2 Comparison between the KF and the AKF. . . . . . . . . . . . . . . . . . 27
3.3 The analytical impedance of the MMC compared with impedance obtained
by small-signal perturbation. . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 The analytical impedance of the MMC compared with impedance obtained
by KF and FFT for selected harmonics. . . . . . . . . . . . . . . . . . . 28
3.5 Simulations of the EKF. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 A small section of the three-phase voltages. . . . . . . . . . . . . . . . . 37
4.2 Simulations of the EKF, no noise. . . . . . . . . . . . . . . . . . . . . . 38
4.3 Simulations of the EKF, with noise. . . . . . . . . . . . . . . . . . . . . 38
4.4 Three-phase voltages and frequency. . . . . . . . . . . . . . . . . . . . . 39
4.5 Tracking of voltages with time-varying amplitude and frequency with fv=
2Hz, fm= 1 H z, fc= 10 H z. ...................... 39
xv
4.6 Tracking of voltages with time-varying amplitude and frequency with fv=
10 Hz, fm= 5 H z, fc= 100 H z...................... 40
4.7 Tracking of voltages time-varying amplitude and frequency with fv=
2Hz, fm= 1 H z, fc= 3 H z........................ 40
5.1 Merging of EMD and KF. Single-phase structure. . . . . . . . . . . . . . 42
5.2 Merging of EMD and KF. Three-phase structures. . . . . . . . . . . . . . 43
5.3 The IMFs and residue of the space vector. . . . . . . . . . . . . . . . . . 45
5.4 Experiment1results. ............................ 46
5.5 The IMFs and residue of the space vector. . . . . . . . . . . . . . . . . . 48
5.6 Experiment2results. ............................ 49
6.1 Line voltages and FFT plot. . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 IMFs of the 3.33 seconds space vector. . . . . . . . . . . . . . . . . . . . 53
6.3 Results using the different methods. . . . . . . . . . . . . . . . . . . . . 54
6.4 IMFs of the 60 seconds space vector. . . . . . . . . . . . . . . . . . . . . 55
6.5 Results using the different methods. . . . . . . . . . . . . . . . . . . . . 56
6.6 The instantaneous amplitude and frequency obtained by HHT for the first
andsecondIMFs............................... 57
6.7 The instantaneous amplitude and frequency obtained by the merged EMD
and UKF for the two first IMFs. . . . . . . . . . . . . . . . . . . . . . . 57
6.8 The three first IMFs of the space vector, and the space vector it self, plotted
for three fundamental periods. . . . . . . . . . . . . . . . . . . . . . . . 58
6.9 Comparison between HHT and merged EMD and UKF . . . . . . . . . . 59
6.10SinettingtoIMF1. ............................ 60
6.11 Sine wave fitted to the instantaneous amplitude and frequency obtained by
HHT. .................................... 61
6.12 Sine wave fitted to the instantaneous frequency obtained by HHT. . . . . 62
A.1 Implementation of the three-phase test system in Simulink. . . . . . . . . 73
A.2 Implementation of three-phase voltage source in Simulink. . . . . . . . . 74
A.3 Implementation of the extended Kalman filter in the test system. . . . . . 74
A.4 Implementation of the unscented Kalman filter in the test system. . . . . . 75
A.5 Implementation of the unscented Kalman filter for tracking on the marine
vessel..................................... 75
A.6 Implementation of the different tracking methods on the marine vessel. . . 76
A.7 Implementation of the PLL. . . . . . . . . . . . . . . . . . . . . . . . . . 76
xvi
Abbreviations
AC Alternating current
AKF Adaptive Kalman filter
DC Direct current
DFT Discrete Fourier transform
EKF Extended Kalman filter
EMD Empirical mode decomposition
FFT Fast Fourier transform
FIR Finite impulse response
HHT Hilbert-Huang transform
HT Hilbert transform
IMF Intrinsic mode function
PV Photovoltaics
KF Kalman filter
MAF Moving average filter
MMC Modular multilevel converter
PI Proportional-Integral
PLL Phase-locked loop
RMS Root mean square
THD Total harmonic distortion
UKF Unscented Kalman filter
xvii
xviii
Chapter 1
Introduction
1.1 Background and Motivation
DUE to a large amount of rotating masses in the electric generators and loads, classi-
cal electric power systems have been characterized by excellent power quality and
frequency with very small drifts away from the fundamental frequency, usually at 50 Hz
or 60 Hz. With the increasing amount of power electronics equipment in the grids, such
as rectifiers, inverters, adjustable speed drives and the like, harmonics and nonlinear dis-
tortions are increasingly injected. This makes real-time measurement and estimation of
essential quantities such as frequency and magnitude a challenging, but necessary task.
With no connections to a strong-/stiff grid, stand-alone power systems such as isolated
microgrids (e.g photovoltaics (PV) microgrids or marine vessel power systems) are prone
to problems with severe frequency deviations. The low inertia in such systems requires
properly implemented controllers in order to maintain satisfying operation. In addition to
the harmonic pollution, advanced inverter control may produce time-varying oscillations
in conditions with varying load demand [1]. The role of these invertes are to serve as an
interface between the generation source and the loads. ”Static” power electronics equip-
ment such as the inverter is a means of inverting DC power to AC power in the cases where
the generation source is not rotating, e.g solar panels, fuel cells or battery banks supplying
DC power [1]. Hitherto, data acquisition- and measurement systems have generally been
based on average value calculations and harmonics that are integer multiples of a con-
stant fundamental frequency [2, 3]. After an extensive study of IEC Standard 61000-4-7
measuring methods, [4] states that these methods do not produce accurate results in envi-
ronments with time-varying angular frequency. Keeping the aforementioned problems of
nonlinearities and time-varying quantities in mind, measurements and estimation in iso-
lated microgrids should rather be based on instantaneous amplitude and frequency rather
than the usage of average values [1,5]. With improved data acquisition- and measurement
tools, the supervisory control systems in isolated microgrids may perform better actions,
and earlier hidden distortions may be revealed.
1
Chapter 1. Introduction
1.2 Thesis Objective and Scope of Work
THE objective of this thesis is to review current literature on specific frequency iden-
tification methods, and assess their viability in microgrids with time-varying funda-
mental frequency and nonlinear distortions. This is obtained by running simulations using
synthetic signals and real voltage measurements from a marine vessel power system.
Fundamental theory for microgrids and the methods used in this thesis are carried out in a
thorough manner. The extended Kalman filter (EKF) and unscented Kalman filter (UKF)
algorithms, and their models are derived. Also a phase-locked loop (PLL), EMD, Hilbert
transform (HT) and the proposed method of merging EMD and UKF are outlined. These
methods are implemented in Matlab and Simulink for assessment of their potential for
tracking of instantaneous amplitude and frequency of voltage- and current waveforms in
isolated microgrids, and for verification of the derived models used by the EKF and UKF.
It is also studied how some of the methods are able to break down multicomponent signals
into monocomponents, from which instantaneous amplitude and frequency with physical
meaning can be studied. Synthetic signals and real voltage measurements from a marine
vessel power system during sea voyage will be used for this purpose.
1.3 Main Contributions
THE main contributions of this thesis are:
An extensive literature review of papers and books relevant to the objective is ren-
dered in an easy-to-read and compact thesis.
A proposed method for better phase angle estimation in noisy environments.
Study of the nonlinear KF model limitations, regarding tracking of signals with
time-varying amplitude and frequency.
The merging of EMD and KF is in this thesis proposed by the supervisor and author,
for the first time to the authors knowledge. Single-phase and three-phase topologies
are developed. This new method is tested with synthetic signals and real voltage
measurements, showing promising results.
1.4 Structure of the Report
THE thesis is outlined as follows:
Chapter 1: Introduction - This chapter presents the background and motivation for
the work in this thesis. It also contains the thesis objective, scope of work, main contribu-
tions and the structure of the report.
Chapter 2: State-of-the-art in Methods for Frequency Identification in Microgrids
- This chapter introduces the concept of microgrids, fundamental definitions and impor-
tant frequency identification methods. It also contains literature review and state-of-the-art
within the field.
2
1.4 Structure of the Report
Chapter 3: Summary of Previous Work - This chapter summarizes the most important
results from the specialization project carried out autumn 2016.
Chapter 4: Harmonics- and Frequency Tracking using Kalman Filters - Linear and
nonlinear mathematical models for the different KFs are developed. Simulations are con-
ducted to confirm the models and to study their feasibility.
Chapter 5: Merging Empirical Mode Decomposition and Kalman Filtering - The pro-
posed method of combining EMD and KF is outlined, and validated with simulations.
Chapter 6: Assessment of Methods for Tracking of Time-Varying Frequencies in a
Marine Vessel Power System - The feasibility of the different frequency identification
methods outlined in this thesis are assessed by using real voltage measurements from a
marine vessel power system during sea voyage in rough sea conditions.
Chapter 7: Conclusion & Future Work - Conclusion of the thesis and recommended
work for future papers and master theses.
Appendix A: Simulink Models - Some of the Simulink models used in this thesis are
included in this appendix.
Appendix B: Matlab Code - Some of the scripts and functions used in Matlab and
Simulink are listed here.
3
Chapter 1. Introduction
4
Chapter 2
State-of-the-art in Methods for
Frequency Identification in
Microgrids
2.1 Introduction to Microgrids
ACCORDING to the U.S Department of Energy a microgrid is ”a group of intercon-
nected loads and distributed energy resources with clearly defined electrical bound-
aries that acts as a single controllable entity with respect to the grid and can connect and
disconnect from the grid to enable it to operate in both grid connected or island modes”
[6]. The International Energy Agency (IEA) estimates that to achieve the goal of universal
access to electrical energy, 70% of rural areas lacking electricity will have to connect to
mini-grid or off-grid solutions [7]. Often microgrids are intended as a means of incorpo-
rating distributed and renewable energy sources such as solar PV, wind power, small hydro
and more. Microgrids are also used for military applications, offshore and maritime appli-
cations. Figure 2.1, inspired by [8], shows an example DC microgrid with a PV array and
battery bank supplying DC and AC loads. The microgrid is able to connect to and discon-
nect from the main grid. This point is called the point of common coupling (PCC). When
operating in island mode the available capacity from the energy storage and PV array must
exceed the power drawn from critical loads. Should there be any surplus energy in the mi-
crogrid, it may be sent to the main grid or be stored in the battery bank. The AC/DC block
is a rectifier/inverter, depending on the direction of the power flow. The DC/DC blocks are
mainly buck or boost converters, depending on whether the voltage levels should decrease
or increase. The DC/DC block connected to the battery bank is a so-called bi-directional
converter, having the ability to increase or decrease the voltage levels depending on the
direction of the power flow.
5
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
Though having many advantages, there are several challenges related to the microgrids. In
isolated microgrids the amount of rotating mass is usually low, i.e low inertia. This can
often lead to severe frequency deviations if the controllers are not properly implemented
[9].
Figure 2.1: An example microgrid with DC bus and the possibility to connect to the main grid.
2.2 Fundamental Definitions
INthe following subsections several fundamental terms and definitions used in the the-
sis are explained, containing basic three-phase system and circuit analysis theory, har-
monics, Fourier analysis, different transformations and more.
2.2.1 Time- and Phasor Domain
Balanced three-phase voltages are shown in the time- and phasor domain in Figure 2.2 and
Figure 2.3. The three phases are denoted a,band c, and can be represented as in equation
(2.1). The three phases are displaced by 120or 2π
3radians in balanced systems, and has
the frequency fn, usually 50 Hz or 60 H z in most power systems. Currents will have the
same waveforms as the voltages.
va(t) = ˆ
Vacos(2πfnt)(2.1a)
vb(t) = ˆ
Vbcos(2πfnt120)(2.1b)
vc(t) = ˆ
Vccos(2πfnt+ 120)(2.1c)
Line voltages, or line-line voltages may be expressed as vab(t) = va(t)vb(t)and so on.
Phase- and line voltages are often denoted as vφand vl. The following relations applies:
vl=3vφ30(2.2)
Three-phase quantities may also be represented in the phasor domain:
Va=ˆ
Vaej0=ˆ
Va0(2.3a)
6
2.2 Fundamental Definitions
Vb=ˆ
Vbej120=ˆ
Vb120(2.3b)
Vc=ˆ
Vcej120=ˆ
Vc120(2.3c)
where the the angular frequency ωn= 2πfnis assumed to be known. In fact each phasor
is multiplied with the term ent, but is usually omitted as it is implied.
Vl=1.732·Vφ
-Vφ
-Vl
120°
30°
Vφ
VaVbVc
Vab Vbc Vca
Figure 2.2: Three-phase voltages in the time domain. Phase voltages plotted with solid lines and
line voltages plotted with dotted lines.
Re
Im
Vca Vab
Va
Vb
Vbc
Vc
Figure 2.3: Three-phase voltages shown in a phasor diagram.
7
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
2.2.2 RMS, Effective Value
The term root mean square (RMS) is commonly used in electrical power engineering.
Vrms is the effective value of v(t), it can be regarded as the DC-value that will give the
same dissipated power over an resistance as v(t)does on average [10]. The instantaneous
power over an resistance can be expressed as:
p(t) = v(t)i(t) = v(t)2
R(2.4)
The RMS value of any voltage can be found by an integral, as in (2.5).
V2
RMS
R=1
R
1
TZT
0
v(t)2dt (2.5)
giving
VRMS =s1
TZT
0
v(t)2dt (2.6)
For sinusoids, it can be shown that ˆ
V=2VRM S .
2.2.3 Harmonics and Total Harmonic Distortion
Total harmonic distortion (THD) is a measure of harmonic distortion, and is defined as the
ratio between the sum of the power of the harmonics and the power of the fundamental
frequency. Harmonic components are often introduced by power electronics equipment,
such as rectifiers and inverters, and has frequencies with an integer times the fundamental
frequency. Let a distorted voltage be defined as:
vs(t) = vs1(t) + X
h6=1
vsh(t),(2.7)
where vs1(t)is the fundamental component and vsh(t)are the harmonics. In terms of
RMS, the THD of the voltage defined in equation (2.7) can be found as follows [10]:
%T H Dv= 100 ·pV2
sV2
s1
Vs1
= 100 ·v
u
u
tX
h6=1 Vsh
Vs12
(2.8)
2.2.4 Fourier Analysis
Fourier analysis of electrical voltage and current waveforms is hitherto the most widely
used method in instrumentation for electrical applications such as smart meters, spectrum
analyser, phasor measurement units and many more. With the presence of time-varying
quantities the frequency based methods may suffer from the leakage and picket-fence ef-
fects [11]. Leakage refers to the spreading of energy from one frequency to the adjacent
frequencies. The picket-fence effect occurs if the analysed signal contains harmonics that
are not an integer times the fundamental.
8
2.2 Fundamental Definitions
According to IEEE 519 and IEC 61000-4-7, harmonic measurements using Fourier series
should be done in the following manner [2,3, 12]:
x(t) = a0+
X
h=1
chsin( h
Nωnt+φh)(2.9a)
ch=qa2
h+b2
h(2.9b)
φh= tan1ah
bh,if bh0(2.9c)
φh=π+ tan1ah
bh,if bh<0(2.9d)
and
ah=2
TwZTw
0
x(t) cos h
Nωnt+φhdt (2.10a)
bh=2
TwZTw
0
x(t) sin h
Nωnt+φhdt (2.10b)
a0=1
TwZTw
0
x(t)dt (2.10c)
where wnis the fundamental frequency, his the harmonic order, Twis the duration of
the window and Nis the number of fundamental periods within the window width. As
the measurements obtained by data acquisition- and measurement systems are in discrete
time, the equations above is not used exactly as they are given. The aforementioned trans-
forms DFT and FFT are in that case used, where the only difference between the two is
that the latter is a more efficient implementation. The FFT takes advantage of symmetries,
reducing the number of computing operations needed from O(N2)(DFT) to O(Nlog N)
(FFT) [13].
2.2.5 Clarke Transformation
The Clarke transformation, also known as αβγ-, or αβ0transformation, is a means of
simplifying three-phase quantities by projecting them onto a stationary reference frame
denoted αand β[14]. This transformation is conceptually similar to the Park-, or dq0
transformation, where the difference is that the Park transformation projects three-phase
quantities onto a rotating reference frame. The transformation from abc reference frame
to the stationary reference frame, αand β, can be obtained by:
Xαβ0=T Xabc ,(2.11)
and inversely from αβ0to abc:
Xabc =T1Xαβ0.(2.12)
9
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
The transformation matrices in equation (2.11) and (2.12) are defined as in (2.13) and
(2.14).
T=2
3
11
21
2
03
23
2
1
2
1
2
1
2
(2.13)
T1=
1 0 1
1
2
3
21
1
23
21
(2.14)
The transformation is shown in Figure 2.4, where balanced three phase quantities are trans-
formed into the αand βaxis. Since the system is balanced, the γor 0component equals
zero.
Re
Im
Clarke Transform
a
α
β
b
c
Figure 2.4: The Clarke transformation shown graphically.
2.2.6 Symmetrical Component Theory - Fortescue’s Theorem
The fundamentals for symmetrical component theory was first presented by Fortescue in
[15], which is a paper that is by many seen upon as one of the most important papers
in electric power engineering. Fortescue proposed that any set of N unbalanced phasors
could be expressed as the sum of N symmetrical sets of balanced phasors. This does of
course apply to three phase system, where the three symmetrical sets of balanced phasors
are called positive-, negative- and zero sequence components. The positive sequence com-
ponents is a set of three balanced vectors, with equal magnitude, same phase sequence and
displaced by 120with respect to each other. The same properties applies to the negative
sequence components except that the phase sequence is opposite. The zero sequence com-
ponents is a set of three balanced vectors with equal magnitudes and equal phase angles,
i.e in phase.
10
2.2 Fundamental Definitions
Any set of unbalanced voltages or currents in the abc reference frame, Xabc, can be trans-
formed into balanced positive, neqative and zero sequence components, X012.X012 =
A1Xabc, where A1is the transformation matrix:
A1=1
3
1 1 1
1a a2
1a2a
,(2.15)
where a=ej120. Similarly the sequence components can be transformed back to the
original unbalanced quantities. Xabc.Xabc =AX012 where Ais the transformation
matrix:
A=
1 1 1
1a2a
1a a2
(2.16)
Figure 2.5 shows an unbalanced three-phase system, whereas Figure 2.6 shows the se-
quence components.
Re
Im
Vabc
Vc
Va
Vb
Figure 2.5: Unbalanced three phase voltages.
11
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
Re
Im
Positive sequence
Re
Im
Negative sequence
Re
Im
Zero sequence
Vc1
Va1
Vb1
Vb2
Va2
Vc2
Va0=b0 =V c0
Figure 2.6: The sequence components of the unbalanced voltages.
2.3 Frequency Identification Methods
THE task of estimating frequency and other parameters in isolated microgrids is a task
of grave importance, for many different reasons. The methods mentioned in section
1.2 will here be outlined. Rich literature on these methods is available, and also for other
methods that will not be covered in this thesis. Some other widely used methods for
on-line fundamental frequency estimation (and other parameters) are: adaptive Prony’s
method [16], least squares methods [17, 18], adaptive notch filtering [19, 20] and Newton
type algorithms [21, 22].
2.3.1 Kalman Filter and Extended Kalman Filter
For time-domain- and model based methods, KF is a widely used tool for a variety of
power system applications. The use of KF for electrical engineering purposes has been
around since the early 80’s [23,24], and has been of interest in applications and research
ever since. Several great approaches have been suggested for both balanced and unbal-
anced power system, under the assumption of both stationary- and non-stationary fre-
quency in [11, 25–29]. Another example of usage is state estimation in electrical drives
[30].
The KF has been around for over 50 years, and is still alive and well and widely used in
many applications. In 1960 R. E. Kalman found a recursive solution for the Wiener filter
problem, the solution was called KF and was first presented in [31]. The theory in this
section is based on [32].
It is assumed that the following discrete state-space model can be used to estimate the
random process:
xk+1 =Akxk+wk(2.17a)
yk=Ckxk+vk(2.17b)
12
2.3 Frequency Identification Methods
xkand ykare the process state- and measurement vector at time tk.Akis the system
matrix relating xkto xk+1, assuming there is no noise. Matrix Ckgives the noiseless
relationship between the state- and measurement vector at time tk.wkand vkare the
model error and measurement error, assumed to be white sequences with zero mean, and
normal distributed covariances wk∼ N(0, Qk)and vk∼ N(0, Rk). Furthermore the
covariance matrices Qkand Rkare uncorrelated, and are uncorrelated at different time
instants, as stated in (2.18).
E[wkwT
i] = (Qk, i =k
0=, i 6=k,(2.18a)
E[vkvT
i] = (Rk, i =k
0=, i 6=k,(2.18b)
E[wkvT
i] = 0,k, i (2.18c)
To start the KF algorithm, an initial estimate ˆx
0and its error covariance P
0is needed.
The ”hat” denotes that this is an estimate, and the superscript ”-” denotes that this is the a
priori estimate, meaning this is the best estimate before taking the measurement at time tk
into consideration. Furthermore the estimation error is defined as:
e
k=xkˆx
k.(2.19)
Assuming the estimation error in (2.19) has zero mean, the error covariance matrix is given
by (2.20).
P
k=E[e
ke
k
T] = E[(xkˆx
k)(xkˆx
k)T](2.20)
We now want to improve our a priori estimate, ˆx
k, by using the measurement yk. The a
posteriori estimate ˆxkcan be found by:
ˆxk= ˆx
k+Kk(ykCkˆx
k).(2.21)
Kkis the yet to be determined Kalman gain, with the purpose of giving an optimal updated
estimate through properly weighting the measurement residual ykˆykand adding it to
the a priori estimate. The a posteriori estimate results in the following error covariance
matrix:
Pk=E[ekeT
k] = E[(xkˆxk)(xkˆxk)T].(2.22)
The purpose of the Kalman gain Kk, and the KF itself, is to minimize the individual terms
along the major diagonal of Pk. It can be shown that (2.23) minimizes the mean-square
estimation error.
Kk=P
kCT
k(CkP
kCT
k+Rk)1(2.23)
For optimal gain conditions the error covariance matrix for the updated estimate is given
as:
Pk= (IKkCk)P
k(2.24)
The a priori estimate for the next time step, tk+1, can be found by simply projecting ˆxk
ahead via the system matrix Ak, as given in (2.25).
ˆx
k+1 =Akˆxk(2.25)
13
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
The error covariance matrix for the next time step is given by:
P
k+1 =E[e
k+1eT
k+1] = AkPkAT
k+Qk(2.26)
For the case of systems with highly nonlinear characteristics , the KF will normally behave
badly and the EKF may be a better candidate. Assuming the state-space model is on the
form:
˙x=f(x, t) + w(2.27a)
y=h(x, t) + v(2.27b)
To obtain the state transition matrix, Ak, and measurement matrix, Ck, at time step k
equation (2.28) and (2.29) are used.
Ak=∂f
∂x ˆxk
(2.28)
Ck=∂h
∂x ˆxk
(2.29)
Though the above equations may seem abstract at first, the goal of the KF is quite sim-
ple. It is simply a computer algorithm that processes discrete measurements into optimal
estimates. Figure 2.7 shows the KF loop visually, and summarizes equations (2.21)-(2.26).
Figure 2.7: The KF algorithm.
2.3.2 The Unscented Kalman Filter
For nonlinear systems, the EKF is the most widely used state estimation algorithm. It has
low complexity and is easy to implement. Though the EKF possess many advantages,
such as simple tuning, it also falls short when compared to other nonlinear state estimation
algorithms, mostly due to linearization errors and the need of calculating derivatives [33]
[34]. The theory in this part is based on [33] and [35].
Assume we have a n-dimensional state-space model as given in (2.30):
xk+1 =f(xk, tk) + wk(2.30a)
14
2.3 Frequency Identification Methods
yk=h(hk, tk) + vk(2.30b)
wk(0, Qk)(2.30c)
vk(0, Rk)(2.30d)
The UKF is initalized:
ˆx0=E[x0](2.31a)
P0= [ekeT
k] = E[(x0ˆx0)(x0ˆx0)T](2.31b)
As opposed to the EKF which uses linearized models to propagate from time step k1
to k, the UKF picks a minimal set of sigma points around the mean, which is further
propagated through the nonlinear functions, meaning it is derivative free. This is known
as the unscented transformation, which results in a new mean and covariance estimate.
The unscented transformation is based on the idea that it is easy to perform nonlinear
transformations on single points. By using Cholesky factorization to find matrix square
roots, i.e nP TnP =nP , the sigma points are defined as follows:
χ(i)
k1= ˆxk1+ ˜x(i), i = 1, ..., 2n(2.32a)
˜x(i)=pnPk1T
i, i = 1, ..., n (2.32b)
˜x(n+i)=pnPk1T
i, i = 1, ..., n (2.32c)
The sigma points are then propagated through the nonlinear function f(·), making the
vector χ(i)
k:
χ(i)
k=f(χ(i)
k1, uk, tk)(2.33)
The elements of χ(i)
kare then combined, serving as the a priori estimate at time k.
ˆx
k=1
2n
2n
X
i=1
χ(i)
k(2.34)
The a priori error covariance is defined as in (2.35), where it should be noted that Qk1is
added to take model error into account.
P
k=1
2n
2n
X
i=1
(χ(i)
kˆx
k)(χ(i)
kˆx
k)T+Qk1(2.35)
The equations defined above are usually denoted as the time update equations. The mea-
surement update equations are yet to be defined. Equation (2.36) defines new sigma points
using the current best estimates, in fact ˆx
kand P
k. To save computational effort one may
omit the calculation of new sigma points, if one is willing to sacrifice accuracy, that is.
χ(i)
k= ˆx
k+ ˜x(i), i = 1, ..., 2n(2.36a)
15
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
˜x(i)=qnP
kT
i
, i = 1, ..., n (2.36b)
˜x(n+i)=qnP
kT
i
, i = 1, ..., n (2.36c)
The nonlinear measurement function h(·)is now used to transform the sigma points into
vectors of predicted measurements, γ(i)
k.
γ(i)
k=h(χ(i)
k, tk)(2.37)
The vectors of predicted measurements are then combined to obtain the predicted mea-
surement ˆyk, as shown in (2.38).
ˆyk=1
2n
2n
X
i=1
γ(i)
k(2.38)
Equation (2.39) defines the covariance of the predicted measurement.
Py=1
2n
2n
X
i=1
(γ(i)
kˆyk)(γ(i)
kˆyk)T+Rk(2.39)
The cross covariance between ˆx
kand ˆykis defined as:
Pxy =1
2n
2n
X
i=1
(χ(i)
kˆx
k)(γ(i)
kˆyk)T(2.40)
The a posteriori estimates can now be found as in equation (2.41)
Kk=PxyP1
y(2.41a)
ˆxk= ˆx
k+Kk(ykˆyk)(2.41b)
Pk=P
kKkPyKT
k(2.41c)
Using 2n+ 1 Sigma Points
[35] suggests different scaling parameters and weights for combining the sigma points into
estimates. Also an additional sigma point is added, i.e 2n+ 1 sigma points. Compared to
equation (2.32) the a priori sigma points vectors χ(i)
kare now defined as:
χ(0)
k1= ˆxk1, i = 0 (2.42a)
χ(i)
k1= ˆxk1+ ˜x(i), i = 1, ..., 2n(2.42b)
˜x(i)=p(n+λ)Pk1T
i, i = 1, ..., n (2.42c)
16
2.3 Frequency Identification Methods
˜x(n+i)=p(n+λ)Pk1T
i, i = 1, ..., n (2.42d)
χ(i)
k1will be on the form:
χ(i)
k1=ˆxk1ˆxk1+p(n+λ)Pk1T
ˆxk1p(n+λ)Pk1T(2.43)
The scaling parameter λis defined as in equation (2.44):
λ=α2(n+κ)n, (2.44)
where αdetermines the spread of the sigma points around the mean ¯x= ˆxk1, usually
set to a small positive value, e.g 1e-3. κis a secondary scaling parameter, usually set to 0.
Then a set of mean and covariance weights are defined as in equation (2.45):
ω(m)
0=λ
n+λ(2.45a)
ω(c)
0=λ
n+λ+ (1 α2+β)(2.45b)
ω(m)
i=ω(c)
i=1
2(n+λ), i = 1, ..., 2n(2.45c)
βis used to incorporate prior knowledge about the probabilistic distribution, e.g β= 2 for
Gaussian distribution. The different state, measurement and covariance estimates are now
found as:
ˆx
k=
2n
X
i=0
ω(i)
mχ(i)
k(2.46a)
ˆyk=
2n
X
i=0
ω(i)
mγ(i)
k(2.46b)
P
k=
2n
X
i=0
ω(i)
c(χ(i)
kˆx
k)(χ(i)
kˆx
k)T+Qk1(2.46c)
Py=
2n
X
i=0
ω(i)
c(γ(i)
kˆyk)(γ(i)
kˆyk)T+Rk(2.46d)
Pxy =
2n
X
i=0
ω(i)
c(χ(i)
kˆx
k)(γ(i)
kˆyk)T(2.46e)
The above changes are also applied to the a posteriori estimates. The observant reader
may notice that with λ= 0, equations (2.42)-(2.46) reduces to the equations from the
preceding subsection.
17
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
Adaptive update of model- and measurement error covariances
[34] proposes an adaptive update of the model- and measurement error covariance matri-
ces, in order to better respond to rapid fluctuations and varying measurement noise. Let
Zk= ˆxkˆx
k=Kk(ykˆyk) = ψ1kψ2kT(2.47)
The iteratively updated model error covariance matrix is then defined as follows:
Qk=1
2(ψ2
1k+ψ2
2k)×I2×2(2.48)
It can be seen that a large error in one of the states will affect the whole model. The
adaptive measurement noise covariance matrix is given as:
Rk=ξRk1+ (1 ξ)|ek||ek1|(2.49)
where ξis the forgetting factor between 0 and 1.
2.3.3 Hilbert Transform
The HT is an important tool in signal processing. It is a tool for projecting real signals
onto the imaginary axis, which further makes it possible to obtain instantaneous amplitude
and frequency. Let the HT of the function x(t)be defined as in [36]:
y(t) = H[x(t)] = 1
πp.v Z
−∞
x(τ)
tτdτ, (2.50)
where p.v indicates the Cauchy principal value. The original signal can now be expressed
in an exponential form as shown equation (2.51).
z(t) = x(t) + jy(t) = a(t)e(t),(2.51)
where
a(t) = px2(t) + y2(t)(2.52a)
θ(t) = tan1y(t)
x(t)(2.52b)
a(t)and θ(t)are the instantaneous amplitude and phase respectively. Knowing the instan-
taneous phase, the instantaneous frequency f(t)can then be found:
f(t) = 1
2π
(t)
dt (2.53)
The notions of instantaneous amplitude and frequency for general signals are not well-
defined [5, 37]. For a perfect sinusoid, the instantaneous frequency will be f=1
T. Figure
2.8 compares the HT and FFT applied to a signal with varying amplitude and frequency,
given as:
x(t) = (cos(2π3t),0t < 2.5
0.5·cos(2π6t),2.5t10 (2.54)
18
2.3 Frequency Identification Methods
It can be seen that the Fourier transform struggles when analysing signals with time-
varying components, and is representing the signal energy as a leakage around 3Hz and
6Hz, whereas the instantaneous amplitude and frequency found from the HT quickly set-
tles to the correct values.
012345678910
Time [s]
-1
0
1Original signal
Hilbert transformed
012345678910
Time [s]
0
0.5
1
1.5
Amplitude
012345678910
Time [s]
2
4
6
Frequency [Hz]
(a) HT (b) FFT
Figure 2.8: Comparison between the HT and FFT.
2.3.4 Hilbert-Huang Transform
The HHT, in fact a NASA designated name, combines EMD and the aforementioned HT,
and is well suited for analysis of non-stationary signals. The use of HHT has proven
useful for obtaining instantaneous amplitude, frequency and power in power systems and
isolated microgrids [1, 5, 38, 39]. As will be outlined in the next subsection, the EMD
decomposes signals into monocomponents/IMFs. From the IMFs it is possible to obtain
instantaneous amplitudes and frequencies, as the monocomponents often can be regarded
as near periodic and near sinusoidal [5,37]. The amount of IMFs greatly varies depending
on the signal at hand. Knowing the IMFs of a signal one may reveal important information,
e.g oscillatory modes. EMD is originally an offline method, but there are also examples of
on-line EMD implementations [40–42]. To realize the HHT for real-time applications, a
FIR filter can be used to estimate the HT, as in [43–45].
Empirical Mode Decomposition - The Sifting Process
The EMD and sifting process aims to extract IMF from non-stationary and nonlinear data
in a systematic manner. [46] states that an IMF must satisfy the following conditions:
19
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
The number of extrema and the number of zero crossings must either be equal or
differ at most by one
At any point, the average value of the envelopes defined by the local maxima and by
the local minima is zero.
The EMD starts by finding all the local maxima and minima of the signal x(t). The
extrema points are connected by cubic spline interpolation lines as shown in Figure 2.9.
The average of the upper and lower spline envelopes is computed and here denoted as m1.
The mean is subtracted from the original signal [46]:
h1=x(t)m1(2.55)
Further, the extrema points of h1are identified and again connected by cubic spline in-
terpolation lines. The new mean is defined as m11. This so called sifting procedure is
repeated k times until h1kis an IMF or the standard deviation, SD, computed from two
consecutive sifting is below a certain value, typically 0.2 or 0.3 [46] .
SD =
T
X
t=0 "|(h1(k1)(t)h1k(t)|2
h2
1(k1)(t)#(2.56)
h1k=h1(k1) m1k(2.57)
The first IMF is designated as
c1=h1k,(2.58)
and subtracted from the original signal, resulting in the residue r1.
r1=x(t)c1(2.59)
The steps mentioned above are repeated until the residue is a monotonic function, or so
small that it is less than a predefined value of substantial consequence. From the resulting
IMFs and the last residue the signal x(t)can be represented as [46]:
x(t) =
n
X
i=1
ci+rn(2.60)
To illustrate, the signal x(t) = cos(2π50t)+0.5 cos(2π250t)+0.3 cos(2π750t)is passed
through the HHT. Figure 2.10 clearly shows the steps of the sifting process for the three
first IMFs, while Figure 2.11 shows the instantaneous amplitudes and frequency obtained
from the three first IMFs. As can be seen, the highest frequency components are extracted
first.
20
2.3 Frequency Identification Methods
1.975 1.98 1.985 1.99 1.995 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Signal
Maxima
Minima
Cubic spline interpolation
Cubic spline interpolation
Mean
Figure 2.9: Signal with spline interpolations and mean.
21
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
Figure 2.10: Steps of the EMD.
22
2.3 Frequency Identification Methods
Figure 2.11: Instantaneous amplitude and frequency of the three first IMFs.
2.3.5 Phase-Locked Loop
The PLL is the state-of-the-art method for extracting the phase angle of grid voltages
[47, 48]. Many versions of the PLL are implemented in the dq synchronous reference
frame, as can be seen in Figure 2.12a, which is based on [49]. The estimated d- and q-axis
voltages are low-pass filtered with the cut off frequency ωLP . The inverse tangent of the
23
Chapter 2. State-of-the-art in Methods for Frequency Identification in Microgrids
filtered voltages is used as input to the Proportional-Integral (PI) controller [49], giving:
ωP LL =θv(kp·1 + Tis
Tis) + ωg,(2.61)
where ωgis the nominal grid angular frequency. A typical step response is showed in
Figure 2.12b. The Park-/dq transform is given as [50]:
vd
vq=r2
3cos(θ) cos(θ2π
3) cos(θ+2π
3)
sin(θ)sin(θ2π
3)sin(θ+2π
3)
va
vb
vc
(2.62)
g
PLL
1
s
b
v
c
v
abc
dq
PLL
d
v
q
v
LP
LP
s
LP
LP
s
atan2
v
Low-pass
filtering
PI-controller
a
v
1i
pi
Ts
kTs
(a) Example of a PLL.
0 0.5 1 1.5
Time [s]
49
49.5
50
50.5
51
51.5
52
Frequency [Hz]
(b) Step response of the PLL.
Figure 2.12: PLL structure and step response.
24
Chapter 3
Summary of Previous Work
THIS thesis is a continuation of the specialization project carried out autumn 2016 at
the Department of Engineering Cybernetics [51]. The specialization project mainly
involved the development and testing of different types of Kalman filters for harmonics-
and frequency tracking in electric power systems. This chapter includes the main results
and findings of the specialization project.
3.1 The Kalman filter and the Adaptive Kalman filter
THE mathematical models for the KF, the adaptive Kalman filter (AKF) and the three-
phase voltages subject to harmonic pollution are given in section 4.1. The KF and
AKF were subjected to the experiment as given in table 3.1
As can be seen from Figure 3.1, the AKF outperformed the KF. This was to be expected as
the AKF was designed to quickly handle fluctuations. Figure 3.2 shows the error of the KF
and AKF, and also how the tuning of the AKF impacted the error. The error was defined
as in equation 3.1.
M agnitude errordB = 10 ·logsX
ihphn
(|Vi,ref |−|Vi|)2(3.1a)
Angle errordB = 10 ·logsX
ihphn
(Vi,ref Vi)2(3.1b)
25
Chapter 3. Summary of Previous Work
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time [s]
-500
-400
-300
-200
-100
0
100
200
300
400
500
Magnitude [V]
Phase a
Phase b
Phase c
(a) Three-phase voltages in the abc frame
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
50
100
Magnitude [V]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time [s]
-20
0
20
40
60
80
Angle [°]
(b) KF
(c) AKF
Figure 3.1: The regular KF and the AKF compared.
26
3.1 The Kalman filter and the Adaptive Kalman filter
Project experiment 1
Parameter Value
Simulation time 0.2 s
Sample time, Ts106s
Measurement noise, v=v1=v2, turn on 0.08 s
Noise mean, E[v] 0
Noise variance, var(v) = E[v2] 1
Positive sequence harmonics, hp[1,7,13]
Negative sequence harmonics, hn[5,11,17]
|Vi,ref |,0st < 0.06 s[100,80,50]
|Vi,ref |,0.06 st0.2s[50,40,25]
Vi,ref ,0st < 0.12 s[0,45,60]
Vi,ref ,0.12 st0.2s[0,22.5,30]
ihp
|Vj,ref |,0st < 0.06 s[90,70,40]
|Vj,ref |,0.06 st0.2s[45,35,20]
Vj,ref ,0st < 0.12 s[20,55,70]
Vj,ref ,0.12 st0.2s[10,27.5,35]
jhn
Table 3.1: Parameters for project experiment 1.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-50
-25
0
25
50
Magnitude error [dB]
KF
AKF
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-20
0
20
40
Angle error [dB]
KF
AKF
(a) KF and AKF compared.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-40
-20
0
20
Magnitude error [dB]
R=5·I2
R=25·I2
R=0.5·I2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
time [s]
-20
-10
0
10
20
Angle error [dB]
R=5·I2
R=25·I2
R=0.5·I2
(b) Different tunings of the AKF compared.
Figure 3.2: Comparison between the KF and the AKF.
Figure 3.3 shows the modular multilevel converter (MMC) impedance found by small-
signal perturbation in a MMC-diode bridge system. This was obtained by injecting shunt
currents of 0.01 pu between the MMC and the diode bridge. Figure 3.4 compares the
analytical impedance, and impedance obtained by FFT and KF at specified harmonics
present in the system.
27
Chapter 3. Summary of Previous Work
100101102103
Frequency [Hz]
100
101
102
103
104
Magnitude [ ]
MMC impedance
FFT
Analytical
100101102103
Frequency [Hz]
-360
-180
0
180
360
Angle [°]
FFT
Analytical
Figure 3.3: The analytical impedance of the MMC compared with impedance obtained by small-
signal perturbation.
100101102103
Frequency [Hz]
100
101
102
103
104
Magnitude [ ]
MMC impedance
KF
Analytical
FFT
100101102103
Frequency [Hz]
-360
-180
0
180
360
Angle [°]
KF
Analytical
FFT
Figure 3.4: The analytical impedance of the MMC compared with impedance obtained by KF and
FFT for selected harmonics.
28
3.2 The Extended Kalman Filter
3.2 The Extended Kalman Filter
THE equations for the model used by the EKF can be found in section 4.2. Figure 3.5
shows how the EKF was able to track the voltage magnitude and frequency as given
in table 3.2. At this point the author had had no luck tracking the phase angle. It was found
that the performance of the EKF was linked to the ratio between the diagonal elements of
the model error covariance matrix Q, defined as λ=q1,1
q2,2.
Project experiment 2
Parameter Value
Simulation time 2 s
Sample time, Ts105s
Measurement noise, v, turn on 0 s
Noise mean, E[v] 0
Noise variance, var(v) = E[v2] 0.1
Positive sequence harmonics, hp[1]
Negative sequence harmonics, hn
|Vi,ref |,0st < 0.2s[100]
|Vi,ref |,0.2st2s[50]
Vi,ref ,0st2s[45]
ihp
fn,0st < 0.8s50 Hz
fn,0.8st2s51 Hz
Table 3.2: Parameters for project experiment 2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s]
-150
-100
-50
0
50
100
150
Magnitude [V]
Phase a
Phase b
Phase c
(a) Three-phase voltages in the abc frame
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
40
60
80
100
Magnitude [V]
EKF Estimate
Reference
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s]
-20
0
20
40
60
80
Magnitude [V]
EKF Estimate
Reference
(b) Magnitude and phase angle
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time [s]
49
49.5
50
50.5
51
51.5
52
Frequency [Hz]
Reference
λ=1· 10-12
λ=1· 10-13
λ=1· 10-14
(c) Frequency
Figure 3.5: Simulations of the EKF.
29
Chapter 3. Summary of Previous Work
30
Chapter 4
Harmonics- and Frequency
Tracking Using Kalman Filters
INthis chapter two Kalman filter models will be developed, one for tracking of harmon-
ics and one for tracking of time-varying fundamental frequency. The models will be
validated by simulations.
The first model is based on [26,52], and [25] for the adaptive approach. Here it is assumed
that the angular frequency ωis constant, and that the system is balanced. In addition a
nonlinear model suitable for the EKF and UKF, as in [27] and [34] is developed. This
model can be used to include tracking of the fundamental frequency.
4.1 Tracking of Three-Phase Harmonics Based on Linear
Kalman Filter
FOR now, ω(t) = ωnis assumed to be known and constant. It is also assumed that
the system is balanced, hence Va=Vb=Vc. Equation (4.1) represents three-phase
voltages in the abc frame with amplitudes Va, Vb, Vc, phase angles φa, φb, φc, and a
known angular frequency ωn. The angular frequency is given by ωn= 2πfn, where fnin
this case is the fundamental frequency at 50 Hz. The model to be developed also applies
for three-phase currents in the abc frame.
va(t) = Vacos(ωnt+φa)(4.1a)
vb(t) = Vbcos(ωnt+φb)(4.1b)
vc(t) = Vccos(ωnt+φc)(4.1c)
Let φb=φa2π
3and φc=φa+2π
3, i.e the phases are aligned in the positive sequence. As
explained in section 2.2.6, any unbalanced systems can be transformed into three sets of
balanced phasors. The positive-, negative- and zero sequence will from now on be denoted
as p, n, 0, respectively.
31
Chapter 3. Summary of Previous Work
va(t) = va,p(t) + va,n (t) + va,0(t)(4.2a)
vb(t) = vb,p(t) + vb,n (t) + vb,0(t)(4.2b)
vc(t) = vc,p(t) + vc,n (t) + vc,0(t)(4.2c)
Furthermore we define Vp=[Va,p,Vb,p ,Vc,p]T,Vn=[Va,n ,Vb,n ,Vc,n]Tand V0=[Va,0,Vb,0,Vc,0]T
as in [26]. Equation (4.1) can be rearranged as in equation (4.3).
va(t)
vb(t)
vc(t)
=Vp
cos(ωnt+φp)
cos(ωnt+φp2π
3)
cos(ωnt+φp+2π
3)
+Vn
cos(ωnt+φn)
cos(ωnt+φn+2π
3)
cos(ωnt+φn2π
3)
+V0
cos(φ0)
cos(φ0)
cos(φ0)
,
(4.3)
where φp, φn, φ0are the phase angles for each sequence. Further the voltages in the abc
frame are transformed into the αβ0frame using the Clarke transform, where the transfor-
mation matrix Tis given in (4.4) and (4.5).
T=2
3
11
21
2
03
23
2
1
2
1
2
1
2
(4.4)
By assuming we have a balanced system, the ”0”-component is omitted and we get:
T=2
311
21
2
03
23
2(4.5)
Multiplying (4.5) with every part of (4.3) yields:
vαβ (t) = T vabc(t) = T Vpcos(ωnt+φp)
sin(ωnt+φp)+T Vncos(ωnt+φn)
sin(ωnt+φn)(4.6)
To obtain a more convenient structure, the trigonometric theorem of addition and subtrac-
tion, as in (4.7) is used.
sin(a±b) = sin(a) cos(b)±cos(a) sin(b)(4.7a)
cos(a±b) = cos(a) cos(b)sin(a) sin(b)(4.7b)
Applying (4.7) to (4.6):
vα(t)
vβ(t)=cos(ωnt)sin(ωnt)
sin(ωnt) cos(ωnt)Vpcos(φp)
Vpsin(φp)+cos(ωnt)sin(ωnt)
sin(ωnt)cos(ωnt)Vncos(φn)
Vnsin(φn)
(4.8)
In (4.9) the model is restated as a discrete state-space model fitting the KF, where itakes
the values of the positive harmonic set hpand negative harmonic set hn,xi,1and xi,2are
the αand βcomponents at harmonic i,ωnis the grid angular frequency, Tsis the sampling
period and subscript ”k” denotes the time instant. It is also assumed random walk for the
states.
32
3.2 The Extended Kalman Filter
xi,1
xi,2k+1
=1 0
0 1xi,1
xi,2k
+wi,1
wi,2k
(4.9a)
yk=vα
vβk
=X
ihpcos(nkTs)sin(nkTs)
sin(nkTs) cos(nkTs)xi,1
xi,2k
+X
ihncos(nkTs)sin(nkTs)
sin(nkTs) cos(nkTs)xi,1
xi,2k
+vi,1
vi,2k
(4.9b)
Let the number of harmonics in the set hpand hnbe npand nn, and furthermore the
total number of harmonics N=np+nn. The total number of states will be 2N. The
model error covariance matrix, Qk, will be a 2N×2Nmatrix. The measurement noise
covariance matrix, Rkwill be a 2×2matrix. The system matrix Ak, and measurement
matrix Ckwill be matrices with dimensions 2N×2Nand 2×2Nrespectively. The
amplitude and phase at harmonic iis found by:
|Vi|=qx2
i,1+x2
i,2(4.10a)
φi=tan1xi,2
xi,1(4.10b)
The self-tuning AKF algorithm from [25] is adopted and slightly modified to include sev-
eral harmonics. The AKF is implemented so that the model error covariance matrix is
adaptively updated, to handle fast fluctuations in the studied signal. The model error ˆwk
can be estimated as:
ˆwk= ˆxkˆx
k= ˆx
k+Kk(ykCkˆx
k)ˆx
k
=Kk(ykCkˆx
k).(4.11)
Inspired by the algorithm in [25], the diagonal terms of the model error covariance matrix
takes the value of the average of the sum of ( ˆw2
i,1+ ˆw2
i,2)for every harmonic iin the set
hphn, as in (4.12) and (4.13):
qk=1
2NX
ihphn
( ˆw2
i,1+ ˆw2
i,2)k(4.12)
Qk=qkI(4.13)
The AKF algorithm is given in Algorithm 1.
33
Chapter 3. Summary of Previous Work
Algorithm 1 Adaptive Kalman filter algorithm for time instant k= 0,1, ...
1: q
k=Qk(1,1)
2: for i= 1 to Nmax iter do
3: Pk=AkP
kAT
k+Qk
4: Kk=PkCT
k(CkPkCT
k+Rk)1
5: ˆxk= ˆx
k+Kk(ykCkˆx
k)
6: ˆwk=Kk(ykCkˆx
k)
7: qk=1
2NPi( ˆw2
i,1+ ˆw2
i,2)k, i hphn
8: Qk=qkI
9: if |qkqq
k|<  then
10: break
11: end if
12: q
k=qk
13: end for
14: ˆx
k+1 =Akˆxk
15: P
k+1 = (IKkCk)Pk
16: Qk+1 =Qk
34