Optimal Sizing of Modular Multilevel Converters
Thèse
Amin Zabihinejad
Doctorat en génie électrique
Philosophiae Doctor (Ph. D.)
Québec, Canada
© Amin Zabihinejad , 2017
Optimal Sizing of Modular Multilevel Converters
Thèse
Amin Zabihinejad
Sous la direction de:
Philippe Viarouge, directeur de recherche
iii
iii. Résumé
L’électronique de puissance a pénétré depuis quelques décennies les applications à forte
puissance dans de nombreux domaines de l’industrie électrique. Au-delà de l’apparition des
technologies d’interrupteur à forte puissance commutable en moyenne tension, ces
applications imposaient également des avancées dans le domaine des topologies de
convertisseurs statiques : les principaux défis à affronter concernaient l’atteinte de niveaux
de tension compatibles avec le domaine de puissance des applications, l’augmentation de la
fréquence de commutation apparente en sortie afin d’augmenter la bande passante de la
commande, de réduire la taille des éléments de filtrage et de limiter les harmoniques de
courant injectés dans le réseau d’alimentation. Les topologies de convertisseurs modulaires
multiniveaux (MMC) sont issues de cette problématique de recherche : elles permettent grâce
à l’association de cellules de commutation d’atteindre des niveaux de tension exploitables en
grande puissance avec les technologies d’interrupteurs existantes, de limiter les fréquences
et les pertes de commutation des interrupteurs élémentaires tout en maîtrisant la distorsion
harmonique totale (THD). La modularité, la redondance, les degrés de liberté et les
fonctionnalités des MMC leur permettent aussi d’augmenter la tolérance aux défauts. Ils
pénètrent à présent une large gamme d'applications comme le transport à courant continu en
haute tension (HVDC), les systèmes d'énergie renouvelable, les entraînements à vitesse
variables de grande puissance, la traction ferroviaire et maritime ainsi que des applications
spécifiques très contraignantes en matière de performance dynamique comme les systèmes
d’alimentation des électro-aimants dans les accélérateurs de particules.
Les topologies MMC sont composées de cellules de commutation élémentaires utilisant des
interrupteurs électroniques tels que le Thyristor à Commande Intégrée (IGCT) standard ou
les dernières génération d’IGBT. Les convertisseurs MMC ont fait l’objet de nombreux
travaux de recherche et de développement en ce qui concerne les topologies, la modélisation
et le calcul du fonctionnement en régime permanent et transitoire, le calcul des pertes, le
contenu harmonique des grandeurs électriques et les systèmes de commande et de régulation.
Par contre le dimensionnement de ces structures est rarement abordé dans les travaux publiés.
Comme la grande majorité des topologies de convertisseurs statiques, les convertisseurs
iv
MMC sont composés non seulement d’interrupteurs mais aussi d’organes de stockage
d’énergie de type composants diélectriques (condensateurs) et magnétiques (inductances,
coupleurs) qui sont essentiels pour assurer la conversion des grandeurs électriques en entrée
et en sortie. Ces composants ont une forte influence sur la taille, le volume et le rendement
des convertisseurs et le dimensionnement optimal de ces derniers résulte souvent de
compromis entre la taille des composants passifs, la fréquence et la puissance commutable
par les interrupteurs élémentaires.
Le travail de recherche présenté dans ce mémoire concerne le développement d’une
méthodologie de dimensionnement optimal et global des MMC intégrant les composants
actifs et passifs, respectant les contraintes des spécifications de l’application et maximisant
certains objectifs de performance. Cette méthodologie est utilisée pour analyser divers
compromis entre le rendement global du convertisseur et sa masse, voire son volume. Ces
divers scénarios peuvent être également traduits en termes de coût si l’utilisateur dispose du
prix des composants disponibles. Diverses solutions concurrentes mettant en œuvre un
nombre de cellules spécifique adaptées à des interrupteurs de caractéristiques différentes en
termes de calibre de tension, de courant et de pertes associés peuvent ainsi être comparées
sur la base de spécifications d’entrée-sortie identiques. La méthodologie est appliquée au
dimensionnement d’un convertisseur MMC utilisé comme étage d’entrée (« Active Front-
end » : AFE) d’une alimentation d’électro-aimant pulsée de grande puissance.
Dans une première partie, une méthode de calcul rapide, précise et générique du régime
permanent du convertisseur MMC est développée. Elle présente la particularité de prendre
en compte la fréquence de commutation contrairement aux approches conventionnelles
utilisant la modélisation en valeurs moyennes. Cet outil se révèle très utile dans l’évaluation
du contenu harmonique qui est contraint par les spécifications, il constitue le cœur de
l’environnement de conception du convertisseur.
Contrairement aux convertisseurs conventionnels, il existe des courants de circulation dans
les convertisseurs MMC qui les rendent complexe à analyser. Les inductances de limitation
incorporées dans les bras de la topologie sont généralement volumineux et pénalisants en
termes de volume et de masse. Il est courant d’utiliser des inductances couplées afin de
réduire l'ondulation , la THD et la masse. Dans le travail présenté, un circuit équivalent des
inductances couplée tenant compte de l'effet de saturation est développé et intégré à
v
l’environnement. L’utilisation d’inductances couplée augmente la complexité de l'analyse du
fonctionnement et la précision de leur méthode de dimensionnement est critique pour
l’optimisation globale du convertisseur. Un modèle analytique de dimensionnement de ces
composants a été développé et intégré dans l’environnement ainsi qu’un modèle de
complexité supérieure qui utilise le calcul des champs par éléments finis.
La méthodologie de conception optimale et globale proposée utilise une procédure
d’optimisation non linéaire avec contraintes qui pilote l’outil de calcul de régime permanent,
le modèles de dimensionnements à plusieurs niveaux de complexité des composants passifs
ainsi que d’autres modules permettant de quantifier les régimes de défaut. Pour pallier à la
précision réduite des modèles analytiques, une approche d'optimisation hybride est
également implantée dans l’environnement. Dans la boucle d'optimisation hybride, le modèle
de dimensionnement des inductances peut être corrigé par le modèle de complexité
supérieure qui utilise le calcul des champs. On obtient ainsi un meilleure compromis entre la
précision de la solution optimale et le temps de convergence de la méthode itérative
d’optimisation globale.
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iv. Abstract
In the last decades, power electronics has penetrated high power applications in many areas
of the electrical industry. After the emergence of high-voltage semiconductor switch
technologies these applications also required advances in the field of static converter
topologies: The main challenges were to achieve voltage levels compatible with the
application power domain, to increase the apparent switching frequency at the output, to
increase the control bandwidth, to reduce the size of the elements of filtering and of limiting
the current harmonics injected into the supply network. The topologies of multi-level
modular converters (MMC) are based on this research problem: they enable the use of
switching cells to achieve high power levels that can be used with existing switch
technologies, frequencies and switching losses of the elementary switches while controlling
the total harmonic distortion (THD). Modularity, redundancy, degrees of freedom and MMC
functionality also allow them to increase fault tolerance. They now penetrated a wide range
of applications, such as high-voltage DC (HVDC), renewable energy systems, high-speed
variable speed drives, rail and marine traction, and very specific applications in terms of
dynamic performance such as electromagnet power systems in particle accelerators.
MMC topologies are composed of elementary switching cells using electronic switches such
as the standard Integrated Control Thyristor (IGCT) or the latest generation of IGBTs. MMC
converters have been the subject of extensive research and development work on topologies,
modeling, and calculation of steady-state and transient operation, loss calculation, the
harmonic content of electrical quantities and systems control and regulation functions. On
the other hand, the dimensioning methodology of these structures is rarely addressed in the
published works.
Like most static converter topologies, MMC converters are composed not only of switches
but also passive components of energy storage devices (capacitors) and magnetic (inductors,
couplers) that are essential to ensure the conversion of the input and output electrical
quantities. These components have a strong influence on the size, the volume and the
efficiency of the converters and the optimal dimensioning of the latter often result from a
vii
compromise between the size of the passive components, the frequency and the power
switchable by the elementary switches.
The research presented in this thesis concerns the development of an optimal and
comprehensive design methodology for MMCs integrating active and passive components,
respecting the constraints of the application specifications and maximizing certain
performance objectives. This methodology is used to analyze the various trade-off between
the overall efficiency of the converter and its mass, or even its volume. These various
scenarios can also be translated into cost if the user has the price of the available components.
Various competing solutions using a specific number of cells adapted to switches with
different characteristics in terms of voltage, current, and associated losses can thus be
compared on the basis of identical input-output specifications. The methodology is applied
to the dimensioning of an MMC converter used as an active front-end (AFE) input of a high-
power pulsed solenoid power supply.
In the first part, a fast, precise and generic method for calculating the steady-state model of
MMC converter is developed. It has the particularity of taking into account the switching
frequency as opposed to conventional approaches using modeling in mean values. This tool
is very useful in evaluating the harmonic content that is constrained by the specifications, it
is the heart of the design environment of the converter.
Unlike conventional converters, there are circulation currents in MMC converter structure
that make it complex to analyze. The inductors which are used in the arms of the topology
are generally bulky and expensive in terms of volume and mass. It is common to use coupled
inductors to reduce ripple, THD, and mass. In the presented work, an equivalent circuit of
coupled inductances considering the saturation effect is developed and integrated. The use of
coupled inductors increases the complexity of the analysis and the precision of its sizing
method is critical for the overall optimization of the converter. An analytical model for the
dimensioning of these components has been developed and integrated as well as a higher
complexity model which uses the finite element method calculation.
The proposed optimal and global design methodology uses a nonlinear optimization
procedure with constraints that drive the steady-state computing tool, multi-level design
models of passive component complexity, and other modules to quantify the fault state. To
compensate the low precision of the analytical models, a hybrid optimization approach is
viii
also implemented. In the hybrid optimization loop, the inductance-sizing model can be
corrected by the higher complexity model that uses finite element computation. A better
compromise is thus obtained between the precision of the optimal results and convergence
time of the iterative global optimization method.
ix
List of Contents
iii. Résumé ...................................................................................................... iii
iv. Abstract .................................................................................................... vi
List of Contents ............................................................................................................... ix
List of Tables xv
List of Figures ............................................................................................................... xvi
List of Symbols .............................................................................................................. xxi
CHAPTER I .................................................................................................................... 1
1 Introduction to Multilevel converters ........................................................... 1
1.1. Introduction ..................................................................................................... 1
1.2. Relevant State of the Art and Problem Description........................................... 4
1.3. History and MMC Definition ........................................................................... 7
1.4. Description of Multilevel structures ................................................................. 9
1.4.1. Neutral Point Clamped (NPC) ..................................................................... 9
1.4.2. Flying capacitor......................................................................................... 10
1.4.3. Cascaded Multilevel Converters ................................................................ 12
1.5. Applications and Industrial Relevance ............................................................17
1.5.1. Multilevel converters and renewable energy .............................................. 19
1.5.2. Multilevel converters and HVDC and FACT systems ................................ 21
1.5.3. Multilevel converters and Marine propulsion ............................................. 24
1.5.4. Traction motor drive .................................................................................. 24
1.5.5. Losses analysis of multilevel converters .................................................... 25
1.6. Main Objective: Converter sizing methodology applied to MMC structures of
static converters ............................................................................................................26
1.7. Secondary Objectives......................................................................................27
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1.7.1. Specify the global optimization approach for Optimization of MMC static
converters ................................................................................................................. 27
1.7.2. Applying the converter sizing method to an industrial application ............. 30
1.7.3. Finding the optimal solution for the MMC AFE converter ......................... 32
1.7.4. Using the Converter sizing method for other Applications ......................... 33
1.8. Conclusion ......................................................................................................34
CHAPTER II...................................................................................................................35
2 Design of MMC converter based on the load specification ........................................35
2.1. Introduction ....................................................................................................35
2.2. Calculation of MMC converter variables.........................................................36
2.2.1. Converter and sub-module topology .......................................................... 36
2.2.2. Converter inputs and outputs variables ...................................................... 38
2.2.3. Semiconductor sizing in steady state ......................................................... 39
2.2.4. Passive components sizing in steady state .................................................. 41
2.3. Investigation of adjustable parameters of multilevel converter ........................43
2.3.1. Converter topology .................................................................................... 43
2.3.2. Number of sub-modules per arm ............................................................... 43
2.3.3. Passive component values ......................................................................... 44
2.3.4. IGBT selection .......................................................................................... 45
2.4. Investigation of converter performance and limitations ...................................45
2.4.1. Converter Losses and efficiency ................................................................ 45
2.4.2. Power quality and harmonic Investigation ................................................. 46
2.4.3. Converter volume and mass....................................................................... 47
2.5. Comprehensive analysis of MMC converter ....................................................47
2.5.1. Circuit Analysis......................................................................................... 48
2.5.2. Electromagnetic Analysis .......................................................................... 49
2.5.3. Thermal Analysis ...................................................................................... 50
2.6. MMC dimensioning Analysis .........................................................................51
xi
2.6.1. Capacitor dimensioning analysis ............................................................... 51
2.6.2. Inductance dimensioning analysis ............................................................. 52
2.7. MMC Multiphase Analysis .............................................................................53
2.7.1. Global analysis using analytical model ...................................................... 53
2.7.2. Global analysis using modified analytical model ....................................... 55
2.8. Optimal sizing of MMC converter ..................................................................56
2.9. Conclusion ......................................................................................................60
CHAPTER III .................................................................................................................61
3 Circuit Model of Modular Multilevel AFE..............................................61
3.1. Introduction ....................................................................................................61
3.2. Steady-State Average Model of Modular Multilevel Active-Front-End Converter
62
3.2.1. Sub-module circuit analysis ....................................................................... 62
3.2.2. Single phase average model parameters ..................................................... 63
3.2.3. Average switching function ....................................................................... 64
3.2.4. Circulating current and capacitor voltage ripple estimation ........................ 66
3.2.5. Advantages and disadvantages of steady-state average model of MMC Active-
Front-End ................................................................................................................. 68
3.3. Time-domain steady-state model of Modular Multilevel Active-Front-End .....69
3.3.1. Sub-module switching function ................................................................. 69
3.3.2. Time-domain state equations ..................................................................... 69
3.3.3. Proposed time-domain model .................................................................... 70
3.4. Steady-State Model Verification using Simulink .............................................72
3.5. Conclusion ......................................................................................................74
CHAPTER IV .................................................................................................................76
4 Electromagnetic and Dimensioning Analysis of Passive Components ...76
4.1. Introduction ....................................................................................................76
4.2. Capacitor Dimensioning Analysis ...................................................................76
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4.2.1. Capacitor Mass Function ........................................................................... 77
4.2.2. Transient Equivalent Model of Capacitor .................................................. 78
4.3. Dimensioning Analysis of Inductor .................................................................80
4.3.1. Core Topologies ........................................................................................ 80
4.3.2. Magnetic Equivalent Circuit (type 1) ......................................................... 81
4.3.3. Magnetic Equivalent Circuit (type 2) ......................................................... 82
4.3.4. Inductance and Resistance Estimation ....................................................... 84
4.3.5. Volume and Mass Function ....................................................................... 86
4.4. Inductor Thermal Analysis ..............................................................................86
4.4.1. Inductor Losses ......................................................................................... 86
4.4.2. Thermal Model of Inductor ....................................................................... 92
4.5. Investigation the effect of core saturation ........................................................94
4.5.1. Finding the Mathematical Core Magnetizing Function .............................. 95
4.5.2. Inductance circuit Equation Considering Core Saturation .......................... 95
4.6. Finite Element Analysis of Coupled Inductors ................................................96
4.6.1. Magneto-static Analysis using Finite Element Method .............................. 96
4.6.2. Correction of Analytical Model using Finite Element Method ................... 98
4.7. Conclusion .................................................................................................... 100
CHAPTER V ................................................................................................................. 102
5 Converter Analysis in the Fault Condition ........................................... 102
5.1. Introduction .................................................................................................. 102
5.2. Investigation of Standard Defects in MMC Converter ................................... 102
5.2.1. DC Link Fault ......................................................................................... 102
5.2.2. Sub-module Fault .................................................................................... 103
5.2.3. Inductance Fault ...................................................................................... 104
5.3. Close Loop Control of MMC converter using Simulink ................................ 104
5.4. Investigation of Converter Performance in Defect Condition ........................ 105
xiii
5.5. Combination of Time-Domain Steady-State Model and Faults in the unit package
..................................................................................................................... 109
5.5.1. Sub-module faults investigation .............................................................. 109
5.5.2. Proposed global optimization considering fault analysis .......................... 111
5.6. Conclusion .................................................................................................... 111
CHAPTER VI ............................................................................................................... 113
6 Optimal Design of Modular Multilevel Converter ................................ 113
6.1. Introduction .................................................................................................. 113
6.2. Optimization algorithm using numerical solver ............................................. 114
6.3. Load Specification of the MMC Active Front End converter application ....... 116
6.4. Constraints Calculation ................................................................................. 117
6.4.1. Sub-module capacitor voltage ripple........................................................ 117
6.4.2. THD ........................................................................................................ 117
6.4.3. Semiconductor Losses ............................................................................. 117
6.4.4. Inductor Losses ....................................................................................... 118
6.5. Goal function ................................................................................................ 118
6.6. MMC Optimization using analytical circuit model ........................................ 119
6.6.1. Optimization algorithm ........................................................................... 119
6.7. Optimal design of modular multilevel converter using dimensioning model .. 121
6.7.1. Global Mass Minimization Algorithm ..................................................... 121
6.8. Hybrid Optimization Model using 2-D FEM ................................................. 123
6.8.1. Hybrid Global Optimization Algorithm ................................................... 124
6.8.2. Hybrid Global Optimization Algorithm considering fault margin ............ 125
6.9. Conclusion .................................................................................................... 126
CHAPTER VII .............................................................................................................. 128
7 Investigation of Optimization Results ................................................... 128
7.1. Introduction .................................................................................................. 128
xiv
7.2. High power IGBT specifications ................................................................... 129
7.3. Optimization results using proposed time-domain circuit model.................... 130
7.4. Mass Minimization of Modular Multilevel Converter ................................... 134
7.4.1. Selection of inductor core topology ......................................................... 136
7.4.2. Optimization using 3.3KV/1500A IGBT ................................................. 137
7.4.3. Optimization using 6.5KV/750A IGBT ................................................... 140
7.5. Optimization Results using Hybrid Analytical Model ................................... 143
7.5.1. Optimization results using 3.3KV/1500A IGBT ...................................... 144
7.5.2. Optimization results using 6.5 kV/750 A IGBT ....................................... 148
7.6. Parameter sensitivity analysis ....................................................................... 150
7.6.1. Sensitivity analysis of maximum Temperature Rise ................................. 151
7.6.2. Investigation the effect of maximum Flux Density on Converter Mass .... 153
7.6.3. Investigation the effect of maximum THD on Converter Mass ................ 154
7.6.4. Investigation of the effect of Capacitor Voltage Ripple on Converter Mass ...
................................................................................................................ 156
7.6.5. Sensitivity analysis converter mass against Fault margin ......................... 157
7.7. Conclusion .................................................................................................... 159
CHAPTER VIII ............................................................................................................ 160
8 Conclusion and Future Researches ....................................................... 160
8.1. Conclusion ............................................................................................... 160
8.2. Future Researches .................................................................................... 163
REFERENCES ............................................................................................................... 165
xv
List of Tables
Table 1.1: Some of the today’s MMC projects ..................................................................18
Table 1.2: ABB SVC Light for electrical transmission grids ..............................................23
Table 3.1 The MMC converter parameters and operating point .........................................72
Table 4.1: Calculated coefficients using fitting algorithm ..................................................78
Table 4.2: Rac/Rdc of copper conductor with 13.21mm diameter in 50Hz, 100Hz and 200Hz
using three estimation methods ...................................................................91
Table 4.3: Calculation of conductor copper losses using the estimated AC resistances and
error calculation compared to utilization of DC resistance ..........................92
Table 5.1: Simulation parameters of MMC converter in simpower .................................. 106
Table 5.2: Normal operation of a half-bridge sub-module................................................ 109
Table 5.3 Investigation of Open-Circuit Fault in T1 ........................................................ 110
Table 5.4 Investigation of Open-Circuit Fault in T2 ........................................................ 110
Table 5.5 Investigation of Short-Circuit Faults in T1 or T2 .............................................. 111
Table 6.1: Load specification of MMC Active Front End converter application............... 116
Table 6.2 The main optimization constraints ................................................................... 120
Table 6.3: List of optimization variables ......................................................................... 122
Table 6.4: List of main constraints in optimization algorithm .......................................... 123
Table 7.1 Technical specifications of high power IGBTs ................................................. 130
Table 7.2: Comparison of total mass of two different inductor core topologies ................ 137
Table 7.3: Converter parameter and thermal coefficient .................................................. 151
xvi
List of Figures
Figure 1.1: Multilevel neutral point clamped converter....................................................... 9
Figure 1.2: Multilevel flying capacitor topology ................................................................11
Figure 1.3: Multilevel cascaded converter with elementary converters ..............................12
Figure 1.4: Typical configuration of sub-module (Commutation Cell) ...............................13
Figure 1.5: Single-leg multilevel converter ........................................................................14
Figure 1.6: Multi-leg multilevel converter .........................................................................15
Figure 1.7: Single-leg two levels converter with interleaving inductors .............................16
Figure 1.8: MMC converter with interleaving inductors ....................................................17
Figure 1.9: Single-line schematic diagram of the proposed multilevel converter-based wind
energy conversion system [28] ...................................................................20
Figure 1.10: Multilevel inverter topology proposed for solar energy systems [32] .............21
Figure 1.11: The modular cascaded multilevel converter used by Siemens Company for
HVDC application [33] ..............................................................................22
Figure 1.12: Three-phase five-level structure of a diode-clamped multilevel converter [36]
...................................................................................................................23
Figure 1.13: Multilevel converter in electric marine propulsion system [39] .....................24
Figure 1.14: 6-level diode-clamped back to back converter for traction motor drive [15] ...25
Figure 1.15: Optimization and verification loop ................................................................28
Figure 1.16: Optimization and verification loop ................................................................30
Figure 1.17: Multi-megawatt power supply of the PS Booster ...........................................31
Figure 1.18: Electromagnet Current pulsed Cycle delivered by the Multilevel H-bridge
converter ....................................................................................................31
Figure 1.19: Corresponding capacitor voltage cycle in the DC bus ....................................31
Figure 1.20: Example of High Power Converter in CERN complex...................................33
Figure 2.1 Modular multilevel active front-end converter ..................................................37
Figure 2.2 a)half-bridge sub-module configuration b)full-bridge sub-module configuration
...................................................................................................................37
Figure 2.3 Inputs and output variables of active front-end converter ..................................38
xvii
Figure 2.4 Thermal dissipation circuit of IGBT .................................................................41
Figure 2.5 MMC circuit model and input/output variables .................................................48
Figure 2.6 Analytical electromagnetic model of arm inductance and its input/output .........49
Figure 2.7 Proposed model for electromagnetic analysis includes analytical model and finite
element analysis .........................................................................................50
Figure 2.8 The analytical thermal model of MMC converter .............................................51
Figure 2.9 The magnetic core topology .............................................................................52
Figure 2.10 Global analysis plan of MMC converter using analytical models ....................54
Figure 2.11 Global analysis plan of MMC converter using finite element correction loop .55
Figure 2.12 First proposed optimization plan of MMC converter.......................................57
Figure 2.13 Second proposed optimization plan using analytical inductor model ...............58
Figure 2.14 Third optimization plan includes the correction loop using FEM ....................59
Figure 3.1: MMC-based topology of the Multi-megawatt power supply of Fig.1.17 ..........62
Figure 3.2: half-bridge sub-module topology .....................................................................63
Figure 3.3: Single phase average equivalent circuit ...........................................................64
Figure 3.4: initializing diagram using average steady-state model .....................................71
Figure 3.5: Diagram of Time-domain analytical model .....................................................72
Figure 3.6: The sub-module capacitor current using analytical model and Simulink ..........73
Figure 3.7: The sub-module capacitor voltage using analytical model and Simulink ..........73
Figure 3.8: The lower and upper arm current using analytical model and Simulink ...........74
Figure 3.9: The input line current using analytical model and Simulink .............................74
Figure 4.1 Cylindrical capacitor of VISHAY Company designed for power electronic
applications ................................................................................................77
Figure 4.2: Capacitor weight versus the capacitance and maximum voltage value .............78
Figure 4.3 The simple model of high power capacitor .......................................................79
Figure 4.4 The modified high power capacitor model ........................................................80
Figure 4.5: The proposed core topologies for independent and dependent mutual inductances
...................................................................................................................81
Figure 4.6 The equivalent magnetic circuit of inductance (type 1) .....................................81
Figure 4.7: The inductor core topology and sizing parameters ...........................................83
Figure 4.8 Magnetic flux lines in the core and leakage flux lines .......................................85
xviii
Figure 4.9: AC resistance of round copper conductor versus conductor diameter using exact
equation .....................................................................................................89
Figure 4.10: AC resistance of round copper conductor versus conductor diameter using
simplified equation .....................................................................................89
Figure 4.11 AC resistance of round copper conductor versus conductor diameter using IEC
standard equation .......................................................................................90
Figure 4.12: Finite element analysis of skin effect .............................................................91
Figure 4.13: Inductor equivalent thermal circuit ................................................................92
Figure 4.14: The B-H curve of iron sheet core ...................................................................94
Figure 4.15: Finite element analysis of coupled arm inductance ........................................97
Figure 4.16: The flowchart of the proposed correction approach ..................................... 100
Figure 5.1: The DC link fault and currents path in the converter ...................................... 103
Figure 5.2: Various kind of sub-module faults ................................................................. 104
Figure 5.3 Close loop control diagram of MMC converter............................................... 105
Figure 5.4: DC Link voltage variation via various converter faults .................................. 106
Figure 5.5: Line current variation via various converter faults ......................................... 107
Figure 5.6: Upper inductor current variation via various converter fault .......................... 108
Figure 5.7: Magnetic flux density of inductor core via various converter fault ................. 108
Figure 6.1: Implementation of Global Optimization Algorithm with Microsoft Excel ...... 115
Figure 6.2: Optimization flowchart of MMC converter.................................................... 120
Figure 6.3: The proposed global optimization algorithm using analytical model .............. 121
Figure 6.4: The proposed hybrid optimization algorithm ................................................. 124
Figure 6.5 Global optimization algorithm considering fault margin ................................. 125
Figure 6.6 Hybrid optimization algorithm considering the fault margin ........................... 126
Figure 7.1: Electric energy stored in the capacitors versus the number of sub-module per arm
................................................................................................................. 131
Figure 7.2: Magnetic energy stored in the inductors versus the number of sub-module per
arm ........................................................................................................... 131
Figure 7.3: Total energy stored in the converter versus the number of sub-module per arm
................................................................................................................. 132
Figure 7.4: Total converter efficiency versus the number of sub-module per arm ............ 132
xix
Figure 7.5: Optimal switching frequency versus the number of sub-modules per arm ...... 133
Figure 7.6: Sub-module capacitor ripple versus capacitor energy and sub-module number
................................................................................................................. 133
Figure 7.7: Contour of capacitor ripple versus capacitor energy and sub-module number 134
Figure 7.8: THD and total efficiency versus coupling factor ............................................ 134
Figure 7.9: MMC topology using 3.3 kV/1500 A IGBT .................................................. 135
Figure 7.10: MMC topology using 6.5 kV/750 A IGBT .................................................. 136
Figure 7.11: Optimal arm inductance value versus number of sub-modules per arm ........ 138
Figure 7.12: Optimal sub-module capacitor value versus number of sub-modules per arm
................................................................................................................. 138
Figure 7.13: Optimal total inductor mass versus number of sub-modules per arm ............ 139
Figure 7.14: Optimal total capacitor mass versus number of sub-modules per arm .......... 139
Figure 7.15: Optimal converter mass versus number of sub-modules per arm .................. 140
Figure 7.16 The optimal arm inductance versus the number of sub-modules per arm ....... 141
Figure 7.17 Optimal value of the sub-module capacitor versus the number of sub-modules
per arm ..................................................................................................... 141
Figure 7.18 Total inductor mass versus the number of sub-modules per arm ................... 142
Figure 7.19 Total capacitor mass versus the number of sub-modules per arm .................. 142
Figure 7.20 Total converter mass versus the number of sub-modules per arm .................. 143
Figure 7.21: The optimal arm inductance value versus the sub-modules per arm ............. 144
Figure 7.22: The optimal sub-module capacitor value versus the sub-modules per arm ... 145
Figure 7.23: The optimal arm inductance mass versus the sub-modules per arm .............. 146
Figure 7.24: The optimal capacitor mass versus the sub-modules per arm ....................... 146
Figure 7.25: The optimal converter mass versus the sub-modules per arm ....................... 147
Figure 7.26: Total converter efficiency versus the sub-modules per arm .......................... 147
Figure 7.27 Total inductor mass versus the number of sub-modules per arm ................... 148
Figure 7.28 The total sub-module capacitor mass versus the number of sub-module per arm
................................................................................................................. 149
Figure 7.29 Total converter mass versus the number of sub-modules per arm .................. 150
Figure 7.30 Total converter efficiency versus the number of sub-modules per arm .......... 150
Figure 7.31: Optimal inductor mass versus maximum temperature rise ........................... 152
xx
Figure 7.32: Total converter mass versus the maximum temperature rise ........................ 152
Figure 7.33: The contour of optimal inductor mass versus the sub-modules number and the
maximum flux density .............................................................................. 153
Figure 7.34 The discontinuity value versus number of sub-modules per arm ................... 154
Figure 7.35 The THD value of input current versus the arm inductance value ................. 155
Figure 7.36 The total inductor mass versus the arm inductance value .............................. 156
Figure 7.37: The total converter mass versus the capacitor ripple .................................... 157
Figure 7.38 Total converter mass sensitivity against fault margin for 3.3KV IGBT ......... 158
Figure 7.39 Total converter mass sensitivity against fault margin for 6.5KV IGBT ......... 158
xxi
List of Symbols
𝐴𝑐 Core section of magnetic circuit [𝑚2]
𝐴𝑐𝑢 Wire section area [𝑚2]
𝐴𝑒 The effective value of core section of magnetic circuit [𝑚2]
𝐴𝑤𝑇 Total copper section of each winding [𝑚2]
𝑎 Core window width [𝑚]
𝑏 Core window height [𝑚]
𝐵 Magnetic flux density [𝑇]
𝐵ℎ1𝑚𝑎𝑥 Maximum flux density of first harmonic [𝑇]
𝐵ℎ2𝑚𝑎𝑥 Maximum flux density of second harmonic [𝑇]
𝐵𝑚𝑎𝑥 Maximum magnetic flux density [𝑇]
𝐵𝑙𝑖𝑛𝑒𝑎𝑟 Linear flux density of the core [𝑇]
𝐵𝑛𝑜𝑙𝑖𝑛𝑒𝑎𝑟 Flux density in the saturation region [𝑇]
𝐵𝑠𝑎𝑡 Saturation flux density [𝑇]
𝑐 Inductor core depth [𝑚]
𝐶 Capacitor value [𝐹]
𝐶𝑎𝑟𝑚 Arm capacitor value [𝐹]
𝐶𝑠𝑚 Sub-module capacitor value [𝐹]
𝐶1 The capacitance between the anode and cathode of the
capacitor
[𝐹]
𝐶2 Correction capacitor [𝐹]
𝑑 Core Width [𝑚]
𝑑𝑐 Conductor diameter [𝑚]
𝐸𝑐𝑚𝑎𝑥 Maximum energy stored in capacitors [𝐽]
𝐸𝑐𝑎𝑝 Electrical energy of capacitors [𝐽]
𝐸𝑐𝑜𝑛𝑣 Total energy stored in converter [𝐽]
𝐸𝑖𝑛𝑑 Magnetic energy stored in the inductors [𝐽]
𝐸𝑜𝑛𝑖 Turn-on energy losses of IGBT [𝐽]
𝐸𝑜𝑛𝑑 Turn-off energy losses of diode [𝐽]
xxii
𝐸𝑜𝑓𝑓𝑖 Turn-off energy losses of IGBT [𝐽]
𝐸𝑜𝑓𝑓𝑑 Turn-off energy losses of diode [𝐽]
𝐸𝑢 Unit capacitor voltage [𝑉]
𝐸𝑃 Energy Power ration
𝑓𝑠 AC source frequency [𝐻𝑧]
𝑓𝑠𝑤 Switching frequency [𝐻𝑧]
𝐹𝑇 Fault time constant [𝑠𝑒𝑐]
ℎ Even-order harmonics
ℎ𝑓𝑒 Heat transfer coefficient of iron core
ℎ𝑐𝑢 Heat transfer coefficient of copper
ℎ𝑓𝑒−𝑐𝑢 Heat transfer coefficient between core and copper
𝐻 Magnetic field value [𝐴/𝑚]
𝐻𝑙𝑖𝑛𝑒𝑎𝑟 Linear magnetic field [𝐴/𝑚]
𝐻𝑠𝑎𝑡 Saturation magnetic field [𝐴/𝑚]
𝐼𝑎 AC phase current of phase a (peak) [𝐴]
𝐼1 The current of first winding [𝐴]
𝐼2 The current of second winding [𝐴]
𝑖𝑎𝑏𝑐 Instantaneous three phase AC line currents [𝐴]
𝐼𝑎𝑢 Upper arm current of phase a (peak) [𝐴]
𝐼𝑎𝑙 Lower arm current of phase a (peak) [𝐴]
𝑖𝑐𝑒 Collector-emitter current of IGBT [𝐴]
𝐼𝑐𝑢 Capacitor current of upper sub-modules (peak) [𝐴]
𝐼𝑐𝑢𝑖 Capacitor current of ith upper sub-modules (peak) [𝐴]
𝐼𝑐𝑙 Capacitor current of lower sub-module (peak) [𝐴]
𝐼𝑐𝑙𝑖 Capacitor current of ith lower sub-module (peak) [𝐴]
𝐼𝑐𝑖𝑟𝑐 Circulation current (peak) [𝐴]
𝐼𝑐𝑖𝑟𝑐ℎ2 Second harmonic of circulation current (peak) [𝐴]
𝐼𝑑𝑐 DC link current (peak) [𝐴]
𝑖𝑑𝑐 Instantaneous DC link current [𝐴]
𝐼𝑐𝑢ℎ1 Main component of capacitor current (peak) [𝐴]
xxiii
𝐼𝑐𝑙ℎ1 Main component of capacitor current (peak) [𝐴]
𝐼𝑐𝑖𝑟𝑐𝑑𝑐 DC component of circulation current (peak) [𝐴]
𝐼𝑓 Fault current (peak) [𝐴]
𝑖𝐹𝐷 Diode forward current [𝐴]
𝐼𝐿 Inductor current (peak) [𝐴]
𝐼𝑛𝑓 No fault current [𝐴]
𝑖𝑜𝑢𝑡 Instantaneous load current [𝐴]
𝐼𝑎𝑟𝑚𝑟𝑚𝑠 Effective value of arm current [𝐴]
𝐼𝑟𝑚𝑠 Effective value of first winding current [𝐴]
𝐼𝑟𝑚𝑠ℎ1 Effective value of first harmonic of winding current [𝐴]
𝐼𝑟𝑚𝑠ℎ2 Effective value of second harmonic of winding current [𝐴]
𝐼𝑟𝑚𝑠ℎ4 Effective value of fourth harmonic of winding current [𝐴]
𝐼𝑠1 Main switch current [𝐴]
𝐼𝑠2 Bypass switch current [𝐴]
𝐼𝑠𝑚 Submodule current [𝐴]
𝐼𝑢 Upper arm current [𝐴]
𝐽 Current density of inductor [𝐴/𝑚2]
𝐾𝑚𝑢 Estimated Coupling factor
𝑘𝑤 Filling factor of inductor winding
𝐾ℎ Hysteresis losses coefficient
𝐾𝑒 Eddy current losses coefficient
𝐾𝑐 Correction coefficient of analytical model
𝐾𝑠 AC resistance coefficient
𝐾𝑠𝑎𝑡 Core saturation coefficient
𝑙𝑐 The magnetic length of the core considering air gap [𝑚]
𝑙𝑚 The magnetic length of the core [𝑚]
𝑙𝑒 Effective value of magnetic circuit length [𝑚]
𝑙𝑔 Air gap length [𝑚]
𝑙𝑔1 Left and right leg air gap length [𝑚]
𝑙𝑔2 Center-leg air gap [𝑚]
xxiv
𝑙𝑖𝑠𝑜 Isolation thickness [𝑚]
𝐿 Inductance value [𝐻]
𝐿11 Self-inductance value [𝐻]
𝐿12 Mutual inductance value [𝐻]
𝐿21 Mutual inductance value [𝐻]
𝐿𝑎𝑛 Inductance value from analytical model [𝐻]
𝐿𝑓𝑒𝑚 Inductance value from FEM calculation [𝐻]
𝐿𝑐 Inductance of capacitor [𝐻]
𝑀 Mutual inductance [𝐻]
𝑀𝑖𝑛𝑑 Inductance total mass [𝐾𝑔]
𝑀𝑐𝑎𝑝 Capacitor mass [𝐾𝑔]
𝑀𝑐𝑎𝑝−𝑇 Capacitor total mass [𝐾𝑔]
𝑀𝑤𝑖𝑛𝑑𝑖𝑛𝑔 Copper mass [𝐾𝑔]
𝑀𝑐𝑜𝑟𝑒 Core mass [𝐾𝑔]
𝑀𝐼𝐺𝐵𝑇 IGBT mass [𝐾𝑔]
𝑀𝐼𝐺𝐵𝑇−𝑇 Total IGBT mass [𝐾𝑔]
MLT Mean length per turn [𝑚]
𝑁 Number of parallel arms
𝑛 Inductor turn number
𝑛1 First winding turn number
𝑚 Number of sub-modules per arm
𝑃𝑎𝑐 Active power of AC side [𝑊]
𝑃𝑑𝑐 Power of DC side [𝑊]
𝑃𝑐𝑢 Copper losses [𝑊]
𝑃𝑐𝑢𝑇 Total inductor copper losses [𝑊]
𝑃𝑒 Eddy current losses [𝑊]
𝑃ℎ Hysteresis losses [𝑊]
𝑃𝐼𝐺𝐵𝑇−𝑇 Total IGBT Losses [𝑊]
𝑃𝑐 Total conduction losses of semiconductor switch [𝑊]
𝑃𝑐𝑖𝑇 Total conduction losses of IGBT [𝑊]
xxv
𝑃𝑐𝑖 Conduction losses of IGBT [𝑊]
𝑃𝑐𝑑𝑇 Total conduction losses of diode [𝑊]
𝑃𝑐𝑑 Conduction losses of diode [𝑊]
𝑃𝑠𝑤 Total switching losses of semiconductor switch [𝑊]
𝑃𝑠𝑤𝑖 Switching losses of IGBT [𝑊]
𝑃𝑠𝑤𝑑 Switching losses of diode [𝑊]
𝑞 Conductor constant
𝑅1 Left side core reluctance of the core [𝐻−1]
𝑅2 Right side core reluctance of the core [𝐻−1]
𝑅3 Center core reluctance of the core [𝐻−1]
𝑅4 Right side core reluctance of the core [𝐻−1]
𝑅5 Left side core reluctance of the core [𝐻−1]
𝑅𝑎 Capacitor equivalent resistance [𝑜ℎ𝑚]
𝑅𝑎𝑐 AC resistance of the winding [𝑜ℎ𝑚]
𝑅𝑎𝑐1 AC resistance of the conductor in the main frequency [𝑜ℎ𝑚]
𝑅𝑎𝑐2 AC resistance of the conductor in the second harmonic [𝑜ℎ𝑚]
𝑅𝑎𝑐4 AC resistance of the conductor in the fourth harmonic [𝑜ℎ𝑚]
𝑅𝑎𝑔 Air gap reluctance [𝐻−1]
𝑅𝑙𝑔1 Center leg air gap reluctance [𝐻−1]
𝑅𝑙𝑔2 Side leg air gap reluctance [𝐻−1]
𝑅𝑏 Modeling resistance [𝑜ℎ𝑚]
𝑅𝑐 Dielectric leakage resistance [𝑜ℎ𝑚]
𝑅𝑐𝑜𝑟𝑒 Magnetic core reluctance [𝐻−1]
𝑅𝐿1 Inductor winding resistance [𝑜ℎ𝑚]
𝑅𝑇 Total magnetic reluctance [𝐻−1]
𝑅2𝑎 Compensation resistance [𝑜ℎ𝑚]
𝑅2𝑐 Compensation resistance [𝑜ℎ𝑚]
𝑅𝑐𝑢−𝑎𝑖𝑟 Thermal resistivity of copper [𝑚2℃/𝑊]
𝑅𝑑𝑐 DC resistance of the winding [𝑜ℎ𝑚]
𝑅𝑓𝑒−𝑎𝑖𝑟 Thermal resistivity of core [𝑚2℃/𝑊]
xxvi
𝑅𝑓𝑒−𝑐𝑢 Thermal resistivity of core and copper [𝑚2℃/𝑊]
𝑅𝑗𝑐 Junction-case thermal resistance [𝑚2℃/𝑊]
𝑅𝑐𝑠 Case-Sink thermal resistance [𝑚2℃/𝑊]
𝑅𝑠𝑎 Sink-air thermal resistance [𝑚2℃/𝑊]
𝑟 Conductor radius [𝑚]
𝑆𝑎𝑙 Switching function of lower arm
𝑆𝑎𝑢 Switching function of upper arm
𝑆𝑐𝑢 Copper external area [𝑚2]
𝑆𝑓𝑒 Core external area [𝑚2]
𝑆𝑓𝑒−𝑐𝑢 The area between the core and copper [𝑚2]
𝑆𝑚 Modulation index
𝑆𝑛 Nominal power [𝑉𝐴]
𝑆𝑢𝑖 Switching function of ith submodule of upper arm
𝑆𝑙𝑖 Switching function of ith submodule of lower arm
𝑇𝑎 Ambient temperature [℃]
𝑇𝑐𝑢 Copper temperature [℃]
𝑇𝑐𝑢𝑚𝑎𝑥 Maximum copper temperature [℃]
𝑇𝑓𝑒 Core temperature [℃]
𝑇𝐻𝐷 Total harmonic distortion [%]
𝑣𝑎𝑏𝑐 Instantaneous three phase AC voltage [𝑉]
𝑉𝑐 Submodule capacitor voltage [𝑉]
𝑣𝑐𝑒 Collector-emitter voltage of IGBT [𝑉]
𝑉𝑐𝑢 Upper side submodule capacitor voltage [𝑉]
𝑉𝑐𝑢𝑖 Upper side ith submodule capacitor voltage [𝑉]
𝑉𝑐𝑙 Lower side submodule capacitor voltage [𝑉]
𝑉𝑐𝑙𝑖 Lower side ith submodule capacitor voltage [𝑉]
𝑉𝑐𝑜𝑟𝑒 Total core volume [𝑚3]
𝑣𝐹 Forward conduction voltage of diode [𝑉]
𝑣𝑑 D-axis AC voltage [𝑉]
𝑉𝑑𝑐 DC link voltage (mean value) [𝑉]
xxvii
𝑣𝑑𝑐 Instantaneous DC link voltage [𝑉]
𝑉𝑙 Lower arm voltage [𝑉]
𝑉𝐿 Inductance voltage [𝑉]
𝑉𝐿−𝐿,𝑟𝑚𝑠 Line to line rms voltage [𝑉]
𝑉𝑢𝑇 Total upper submodule voltage [𝑉]
𝑉𝐿𝑢 Upper arm inductance voltage [𝑉]
𝑉𝐿𝑙 Lower arm inductance voltage [𝑉]
𝑉𝑙𝑖 Voltage of ith submodule in lower arm [𝑉]
𝑉𝑙𝑠 Lower side submodule voltage [𝑉]
𝑉𝐼𝐺𝐵𝑇 IGBT voltage value [𝑉]
𝑉𝑜 Terminal voltage [𝑉]
𝑉𝑜𝑢𝑡 Out put converter voltage [𝑉]
𝑣𝑞 Q-axis voltage of AC side [𝑉]
𝑉𝑠𝑚 Submodule terminal voltage [𝑉]
𝑉𝑢 Upper arm voltage [𝑉]
𝑉𝑢𝑇 Total upper submodule voltage [𝑉]
𝑉𝑢𝑖 Voltage of ith submodule in upper arm [𝑉]
𝑉𝑢𝑠 Upper side submodule voltage [𝑉]
𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 Total copper volume [𝑚3]
𝑥𝑠 AC resistance IEEE constant
𝑦𝑠 AC resistance IEEE constant
𝜔1 Circular frequency of AC source [𝑟𝑎𝑑/𝑠𝑒𝑐]
𝜃1 Phase of the main component of arm current [𝑟𝑎𝑑]
𝜃2 Phase of second harmonic of arm current [𝑟𝑎𝑑]
Δ𝑉𝑐𝑢ℎ1 Main component of upper sub-module capacitor voltage [𝑉]
Δ𝑉𝑐𝑙ℎ1 Main component of lower sub-module capacitor voltage [𝑉]
Δ𝑉𝑐𝑢ℎ2 Second harmonic of upper sub-module capacitor voltage [𝑉]
Δ𝑉𝑐𝑙ℎ2 Second harmonic of lower sub-module capacitor voltage [𝑉]
Δ𝑉𝑐𝑢ℎ3 Third harmonic of upper sub-module capacitor voltage [𝑉]
Δ𝑉𝑐𝑙ℎ3 Third harmonic of lower sub-module capacitor voltage [𝑉]
xxviii
Δ𝑉𝑎𝑟𝑚 Arm voltage ripple [𝑉]
Δ𝑉𝑎𝑟𝑚𝑑𝑐 DC component of arm voltage [𝑉]
Δ𝑉𝑎𝑟𝑚ℎ2 Second harmonic of arm voltage [𝑉]
Δ𝑉𝑎𝑟𝑚ℎ4 Fourth harmonic of arm voltage [𝑉]
Δ𝑡 Fault period [𝑠𝑒𝑐]
Δ𝑇 Inductor temprature rise [℃]
𝜇0 Vacuum permeability [𝐻/𝑚]
𝜇𝑟 Core permeability
𝜂 Converter efficiency [%]
𝜙 Core flux [𝑊𝑏]
𝜙1 Core flux of first winding [𝑊𝑏]
𝜙11 First winding flux of first winding current [𝑊𝑏]
𝜙12 First winding flux of second winding current [𝑊𝑏]
𝜙21 Second winding flux of first winding current [𝑊𝑏]
𝜌𝑐𝑢 Electrical resistivity [𝛺.𝑚]
𝜌0 Copper electrical resistivity constant at 25℃ [𝛺.𝑚]
𝐷𝑐𝑜𝑟𝑒 Iron volumetric mass [𝑘𝑔/𝑚3]
𝐷𝑤𝑖𝑛𝑑𝑖𝑛𝑔 Copper volumetric mass [𝑘𝑔/𝑚3]
𝛼 Hysteresis losses field constant
휀25 Copper resistivity coefficient at 25℃
𝛽 Eddy current losses frequency constant
𝛾 Eddy current losses field constant
𝜎 Copper conductivity [𝛺.𝑚]−1
𝜆 Thermal conduction factor of the isolation
𝛿 Air gap correction coefficient
1
CHAPTER I
1 Introduction to Multilevel converters
1.1. Introduction
Recently, multilevel converters have emerged as an interesting solution in the power
industry. The general structure of the multilevel converter is to synthesize a sinusoidal
voltage from several levels of voltages, typically obtained from capacitor voltage sources.
For several years, multilevel voltage source converters allow working at the high voltage
level and producing a quasi-sinusoidal voltage waveform. Classical multilevel topologies
such as NPC and Flying Capacitor VSIs were introduced twenty years ago, and are widely
used in Medium Power applications such as traction drives nowadays. In the scope of High
Voltage AC/DC converters, the Modular Multilevel Converter (MMC), proposed ten years
ago, by Professor R. Marquardt from the University of Munich (Germany), appeared
particularly interesting for HVDC transmissions. On the base of the MMC principle, this
thesis considers different topologies of elementary cells, which make the High Voltage
AC/DC converter more flexible to achieve different voltage and current levels.
Trends in power semiconductor technology indicate a trade off in the selection of power
devices in terms of switching frequency and voltage sustaining capability. New multi-level
high-power converter topologies have been proposed using a hybrid approach involving
integrated gate commutated thyristors (IGCTs) or gate turned off thyristors (GTOs) and
insulated gate bipolar transistors (IGBTs) operation in synergism. The new multilevel power
conversion concept combines the flexibility of the frequency converter with the robustness
of the industrial active neutral point clamped converter (ANPC) to generate multilevel
voltages.
In recent years, the industry has begun to demand higher power equipment, which now
reaches the megawatt level. Controlled ac drives in the megawatt range are usually connected
2
to the medium-voltage network. Today, it is hard to connect a single power semiconductor
switch directly to medium voltage grids (2.3, 3.3, 4.16, or 6.9 kV). For these reasons, a new
family of multilevel converters has emerged as the solution for working at higher voltage
levels [1].
Multilevel converters are one of the best solutions in order to use in the high voltage
applications. Increasing the demands for high power converters in order to connect to the
grid for renewable energy systems, HVDC and other applications made multi-level
converters topology more suitable than two-level PWM rectifier [2].
Using multilevel converters is increasing more and more, especially in high power industries
of electric power conversion. Multilevel converters are customized for a wide range of
applications, such as extruders, compressors, conveyors, crushers, pumps, grinding mills,
fans, rolling mills, blast furnace blowers, gas turbine starters, mixers, mine hoists, reactive
power compensation, marine propulsion, high-voltage direct-current (HVDC) transmission,
hydro pumped storage, wind energy conversion, and railway traction [2].
Multilevel converters provide us a lot of challenges and offer a wide range of possibilities.
Researchers are trying to further improve them in the fields of efficiency, reliability, power
density, simplicity, and cost of multilevel converters and its dimension.
Recently, the interleaved converters with coupled-inductors have been widely used in
medium- to low-power applications, mainly to increase the output current, while the current
ripple in power devices reduces. Using the interleave technique; the size of converter’s
passive components (inductances and capacitances) and the harmonic content of output
voltage are considerably decreased with respect to the classical approach.
There are many publications in the field of MMC converters. In the past decade, due to
increasing the demands of MMC converters, researchers investigated the multilevel
converter from a different point of view. These converters have been analyzed in order to use
in the various applications such as HVDC, electric vehicle, renewable energy system and etc.
also, new modulation approaches and modern control strategies have been proposed to
increase the power quality and decrease the harmonics.
One of the interesting subjects is to estimate the losses of multilevel converters. Due to its
complex topology, researchers proposed various losses models based on its topology and
switch type. According to use the multilevel converter in high power application, its final
3
dimension is very big and bulky. Also, utilization of different components such as the
semiconductor switch, coupled inductor, heat sinks, capacitors, bus bars and other
accessories makes it very difficult to implement. The analysis which is used to find the best
component value of the multilevel converter in order to achieve the minimum dimension is
called “converter sizing analysis.” Therefore, the dimensioning analysis will be an important
part of the design procedure.
In literature, there are few publications about the global dimensioning methodology of MMC
converters, especially in high power application. It seems there are two points, which affect
the lack of researches. First, academic researchers do not have access to the information of
high power applications. Secondly, dimensioning design data is a part of confidential
engineering documents of the companies and therefore, highly unlikely to be published
publicly.
The main objective of this dissertation is to propose and implement of the converter sizing
of the static converters applied to MMC structures. Providing a platform, which analyzes and
executes the global optimal dimensioning algorithm, utilizing the specifications of a MMC
industrial converter application, verification of algorithm parameters, finding the optimal
solution for the industrial application and extending the algorithm for other applications are
the secondary objectives.
This dissertation introduces the converter sizing methodology in order to optimize the MMC
converters. Chapter 1 presents a definition for MMC converter and explains the history of
modular multilevel converters. Then it introduces different topologies of MMC converter
and investigates their advantages and disadvantages. In addition, the important industrial
application of MMC converters with an industrial example is presented.
In chapter 2, the main Ph.D. objectives are explained. The converter sizing model is defined
to implement to the high power MMC converters. Also, the validation tool in order to verify
our methodology is introduced.
In chapter 3, the average steady-state model and its advantages and weakness are presented.
Then, to enhance the model performance, the time-domain steady-state model is introduced
and investigated. Finally, in order to validate the proposed model, the outputs are compared
with the outputs of a Simulink model.
4
In chapter 4, the methodology of MMC converter optimization is investigated and clarified.
The plan of the optimization loop is proposed and detail sections of the global optimization
loop are introduced and explained. Also, the software which must be used to integrate the
optimization loop is identified and the validation method with the specifications of the MMC
converter industrial application is investigated.
The dimensioning analysis of converter components is presented in chapter 5. Using an
analytical approach, the mathematical mass function of capacitor and inductor are extracted
and added to the optimization algorithm. Then, an optimization is done to minimize the total
converter mass regarding the mass functions.
In order to increase the analytical model accuracy especially in the case of the magnetic
model, a novel hybrid optimization model is presented in chapter 6. It consists a combination
of analytical model and finite element approach which intensely increases the model
accuracy while the optimization time does not increase so much. The mass minimization is
repeated using new optimization algorithm and the result is discussed.
In chapter 7, the optimal converter is investigated and analyzed in the defect condition in
order to minimize the converter damage in fault condition. This study is done using two
different methods. The first one is to send the optimal parameters to Simulink and make a
co-simulation in the defect condition using Matlab/Simulink. The second one is to add the
defect analysis to the time domain model to calculate the extra constraints while the
optimization is running.
Finally, in chapter 8, the future works in the field of the high power modular multilevel
converter is investigated. The optimization results and the proposed approach is investigated
and summarized.
1.2. Relevant State of the Art and Problem Description
In response to the demands of high-power systems and to supply the requirements of the
industrial processes powered by large electric ac drive systems, two different solutions are
proposed by power electronics researchers:
5
• Conventional two-level voltage/current source topologies comprising high-
voltage/current-rated power switches based on developing and immature high-
voltage semiconductor technology (currently 8 kV and 6 kA);
• Multilevel power converters covering a power range from several MW to tens of
MW based on matured semiconductor technology of medium-voltage/current rated
power switches (currently 1.2 kV up to 6.6 kV) [3].
Although the first approach leads to the simplicity of power and control circuit, two-level
power converters suffer from major disadvantages of augmented price of newer high-power
semiconductors and power quality concerns, specifically as going higher in the power ranges.
In turn, multilevel power converters bring many technical advantages such as extended power
range due to the capability of the multilevel topologies to handle the voltage and power in
the range of several kV and MW utilizing reliable medium voltage insulated gate bipolar
transistors (IGBTs), improved harmonic content of the switched output voltage, and hence
increased power quality, increased reliance on power converter operation owing to possible
fault-tolerant feature, lowered electromagnetic interference and upgraded electromagnetic
compatibility, lowered switching losses, enhanced efficiency, and reduced amount of output
filter, etc. [3-7].
Nowadays, three commercial topologies of multilevel voltage-source inverters were
introduced as classical topologies: neutral point clamped (NPC) [8], cascaded H-bridge
(CHB) [9], and flying capacitors (FCS) [10]. Among these inverter topologies, cascaded
multilevel inverter reaches the higher output voltage and power levels (13.8 kV, 30 MVA)
and the higher reliability due to its modular structure.
In the field of high power application, multilevel converters with a high number of levels
seem to be the most suitable types, because of the need for series connection of
semiconductors in combination with low voltage distortion on the line side. There are many
other important aspects have to be taken into account for these applications. MMC converter
is the technique of using standard commutation cells in series and parallel compositions in
order to achieve higher voltage and current with conventional semiconductor switches.
The main technical and economic aspects of the development of multilevel converters are
[11]:
6
1. Modular realization
• Scalable to different power voltage levels
• Independent of the fast development of semiconductor switches
• Developing power devices
• Expandable to any number of voltage steps
• Dynamic division of voltage to the power devices
2. Multilevel waveform
• low total harmonic distortion
• use of approved devices
• redundant operation
3. Investment and life cycle cost
• standard components
• modular construction
4. High availability
5. Failure management
Since 50 years ago that multilevel conversion was introduced, several multilevel converters
are used to increase power and reduce THD. Because of the limited current capability of the
cables and semiconductor devices, high power systems need a type of converters which able
to operate with more than a thousand hundred volts.
The multilevel converters provide us the possibility of working at a high-voltage and high
current level, with a better efficiency and power quality. In high power systems, the current
and voltage ratings can easily go beyond the range the existing semiconductor switches.
Multicell interleaved converters which are connected in parallel or series is an interesting
solution for high power application. However, extra measures should be taken for equal
sharing of the current or voltage among the parallel or series devices. In multilevel structures,
due to the interleaved modulation technique, it is possible to achieve a series of advantages
[12], such as:
• Quasi-sinusoidal AC voltage waveform
• Low harmonic impact
• Reduced costs for the filtering elements
7
• Possible direct connection to the MV grid
• Reduction of semiconductor losses due to a very low single-switching frequency per
device
Multilevel converters include an array of power semiconductors and capacitor voltage
sources, the output of which generate voltages with stepped waveforms. The commutation
of the switches permits the addition of the capacitor voltages, which reach a high voltage at
the output, while the power semiconductors need to be able to withstand only partial of the
total voltage.
For completeness and a better understanding of the advances in multilevel technology, it is
necessary to review the classic multilevel converter topologies. However, in order to focus
the content of this thesis on the most recent advances and ongoing research lines, well-
established topologies will only be briefly introduced and referred to existing literature.
1.3. History and MMC Definition
History of multilevel inverters began in 1975 with Baker and Bannister. This first patent
described a converter topology capable of producing a multilevel voltage by connecting
single phase inverter in series.
The multilevel converter has been developed to compensate the shortcomings in solid-state
switching device ratings and technology so that they can be applied to high-voltage electrical
systems. The special topology of multilevel voltage converters allows them to obtain high
voltages with low harmonics without the use of transformers [11].
Since last two decades, the demand for medium and high voltage power converters has grown
to provide medium and high voltage output with the low harmonic rate. The application of
these converters is HVDC links, static VAR compensators, traction motor variable speed
drives and active filtering. Multilevel power converters have been introduced and presented
as a solution in high voltage and medium voltage applications. MMC provides a cost-
effective solution in the medium and high voltage energy management market. A multilevel
converter has several advantages over a conventional two-level converter that uses high
switching frequency Pulse Width Modulation (PWM). The attractive advantages of a
multilevel converter can be briefly summarized as follows:
8
Harmonic distortion: Multilevel converter is based on energy conversion using small
voltage steps, their output waveforms are close to the sinusoidal wave. Hence, it contains less
harmonic distortion.
Electromagnetic compatibility: Multilevel converters not only can generate the output
voltage with very low distortion but also can reduce the dv/dt stresses; therefore,
Electromagnetic Compatibility (EMC) problems can be reduced.
Common-Mode (CM) voltage: Multilevel converters produce smaller CM voltage;
therefore, the stress in the bearings of a motor connected to a multilevel motor drive can be
reduced. Furthermore, CM voltage can be eliminated by using advanced modulation
strategies [13].
Input current: The input current of multilevel converters has very low distortion.
Switching frequency: Multilevel converters operates at both fundamental switching
frequency as well as high switching frequency PWM [1, 11].
Multilevel converters have some disadvantages. One of the most important disadvantages is
the greater number of power semiconductor switches needed. Although lower voltage rated
switches can be utilized in a multilevel converter, each switch requires a related gate drive
circuit. This may cause the overall system to be more expensive and complex.
The number of multilevel converter topologies has been introduced during the last two
decades. The researches have concentrated on novel converter topologies and unique
modulation schemes. Moreover, three different major multilevel converter structures have
been reported in the literature: cascaded H-bridges converter with separate dc sources, diode
clamped (neutral clamped), and flying capacitors (capacitor clamped). Furthermore,
abundant modulation techniques and control schemes have been developed for multilevel
converters such as sinusoidal pulse width modulation (SPWM), selective harmonic
elimination (SHE-PWM), space vector modulation (SVM), and others. In addition, many
multilevel converter applications focus on industrial medium-voltage motor drives [1, 14,
15], utility interface for renewable energy systems [16], flexible AC transmission system
(FACTS) [17], and traction drive systems [18].
9
1.4. Description of Multilevel structures
In the following, classic topologies will be referred to those that have extensively been
analyzed and documented and have been commercialized and used in practical applications
for more than a decade. The more important configurations of multilevel converters are:
1. Neutral Point Clamped (NPC)
2. Flying capacitor
3. Cascaded Multilevel Converters
1.4.1. Neutral Point Clamped (NPC)
The most commonly used multilevel topology is the diode clamped inverter, in which the
diode is used as the clamping device to connect the dc bus voltage in order to take steps in
the output voltage. In the natural point clamped topology, the diodes are the key difference
between the two-level inverter and the three-level inverter. In Figure 1.1, a single-phase
three-level and four levels version is shown, but it is possible to increase the number of level
and legs (phase). In the three levels topology, using two diodes, it is possible to convert the
voltage to half the level of the dc-bus voltage. In general, the voltage across each capacitor
for an N level diode-clamped inverter at steady state is1
dcV
N .
Figure 1.1: Multilevel neutral point clamped converter
10
In general, for an N level diode-clamped inverter, for each leg 2(N-1) switching devices, (N-
1)(N-2) clamping diodes and (N-1) dc link capacitors are required. Increasing the number of
levels leads to increase the number of diodes and the number of switching devices and makes
the system impracticable to implement. In the PWM converters, the main constraint will be
the reverse recovery of the clamping diode.
The component which characterizes this topology is the diode necessary to clamp the
switching voltage to the half level of the DC bus, which is split into three levels by two series
of connected bulk capacitors. In this topology, the middle point is also called the neutral
point. By increasing the number of levels, the voltage which the diodes have to sustain rises.
For a specific diode rating voltage, more devices are necessary to withstand the whole
voltage. Therefore, if the number of voltage levels that the system can impose is N, 2(N-1)
diodes are necessary. For high-DC voltages, the system becomes less convenient due to the
huge number of diodes.
1.4.2. Flying capacitor
The Flying Capacitor is another multilevel topology, which is suitable for high-power
applications. This topology is composed of the series connection of capacitor clamped
switching cells. Figure 1.2 shows the topology of the multilevel flying capacitor.
The flying capacitor topology has some advantages when compared to the diode-clamped
inverter. It does not need to clamping diodes. In addition, the flying capacitor inverter has
switching redundancy within the phase, which is used to balance the flying capacitors so that
only one dc source is needed.
The flying capacitor and diode clamped inverter have the same problem in implementation.
A large number of bulk capacitors must be used and install. The voltage rating of each
capacitor must be the same as the main power switch rating, an N level converter will require
a total of (𝑁 − 1)(𝑁 − 2)/2 clamping capacitors per phase in addition to (𝑁 − 1) main dc
bus capacitors.
11
Figure 1.2: Multilevel flying capacitor topology
This topology also has other disadvantages which have limited its utilization. First of all, it
needs the converter initialization. Before the converter can be modulated by any modulation
scheme, the capacitors need to be charged with the required voltage level as the initial
voltage. This complicates the modulation process and becomes a hindrance to the operation
of the converter. The capacitor voltages must also be regulated under normal operation in a
similar way to the capacitors of a diode clamped converter. Another problem of this topology
is the rating of the capacitors. The capacitors have large fractions of the dc bus voltage across
them.
The two topologies analyzed present a better reduction in the harmonics. Despite the
improvements which they are able to reach, these kinds of multilevel converters present a
series of limitations. For this reason, they did not succeed in these HV-application demands.
Also, there are some problems which limit the utilization of this topology in industrial
applications.
1. Unwanted EMI disturbances generated by a very high slope (di/dt) of the arm
currents
2. The DC bulk capacitor stores a huge quantity of energy which leads to damages
under faulty conditions.
12
3. The stored energy of the concentrated DC capacitor at the DC-Bus results in
extremely high surge currents and subsequent damage if short circuits at the DC-
Bus cannot be excluded.
4. Harmonics on the AC current must always be suppressed.
1.4.3. Cascaded Multilevel Converters
These structures are characterized by a series connection of elementary converters that are
normally identical. Each cell corresponds to a voltage level according to the particular
modulation technique. It is possible to achieve the desired voltage waveform according to
the imposed reference. Figure 1.3 shows the cascaded structure with elementary converter
modules.
Figure 1.3: Multilevel cascaded converter with elementary converters
The cascaded structures ensure the modularity of the system by ensuring series industrial
production. Due to modularity, they do not present upper DC voltage limits. In fact, it is
possible to add more series cells to sustain the desired voltage. The converter is a composition
of series-connected elementary cells. This converter offers the possibility to regulate the
active and reactive power independently. Each phase is composed of two groups of
elementary cells (1…N and N+1…2N), called branches. Each branch conducts the half-phase
current. The advantages of using Cascaded Multilevel Converters are:
• Each arm conducts half current and in continuous conduction mode.
13
• Arm inductances contribute to limit faulty conditions.
• The bulk capacitor is not necessary because there are two terminal cells.
• Each capacitor cell voltage can be controlled very slowly with respect to the current
regulator.
• The DC link voltage can be controlled by the converter.
The typical structure of an MMC and the configuration of a Sub-Module (SM) are shown in
figure 1.4. This sub-module is known as general commutation cell. Each sub-module is a
simple chopper cell which is composed of two IGBT switches (T1 and T2), two anti-parallel
diodes (D1 and D2) and a capacitor C. The commutation cell limits the maximum switch
voltage. So, it is possible to achieve high voltage by a series connection of commutation cells
(sub-modules).
Icu T1
T2
Iu
C +Vsm
-
+Vc
-
D1
D2
Figure 1.4: Typical configuration of sub-module (Commutation Cell)
14
SM1
SM2
SM1
SM2
L1
L1
Vac
E
E
-Vdc
+Vdc
Figure 1.5: Single-leg multilevel converter
The configuration with T1 and T2 both ON should not be considered because it creates a
short circuit across the capacitor. Also, the configuration with T1 and T2 both OFF is not
useful as it produces different output voltages depending on the current direction.
In an MMC the number of steps of the output voltage is related to the number of series
connected sub-modules. In order to show how the voltage levels are generated, in the
following, reference is made to the simple three level MMC configuration shown in figure
1.5.
In this case, in order to get the positive output, +E, the two upper SMs 1 and 2 are bypassed.
Accordingly, for the negative output, - E, the two lower SMs 3 and 4 are bypassed. The zero
state can be obtained through two possible switch configurations. The first one is when the
two SMs in the middle of a leg (2 and 3) are bypassed, and the second one is when the end
SMs of a leg (1 and 4) are bypassed. It has to be noted that the current flow through the SMs
that are not bypassed determining the charging or discharging of the capacitors depending on
the current direction. Therefore, in order to keep the capacitor voltages balanced, both zero
15
states must be used alternatively. The principle of operation can be extended to any multi-
level configuration.
In this type of inverter, the only states that have no redundant configurations are two states
that generate the maximum positive and negative voltages, +E and –E. For generating the
other levels, in general, there are several possible switching configurations that can be
selected in order to keep the capacitor voltages balanced. In MMC of Figure 1.6, the
switching sequence is controlled so that at each instant only N SMs are in the on-state. As an
example, if at a given instant in the upper arm SMs from 2 to N are in the on-state, in the
lower arm only one SM will be in on-state. It is clear that there are several possible switching
configurations. Equal voltage sharing among the capacitor of each arm can be achieved by
control of bypassed SMs during each sampling.
In the applications that we need to pass more than 10 kA, the nominal IGBT current is not
sufficient to pass the high value of current. The parallel arm topology was emerged to solve
this problem. Figure 1.6 shows the multilevel converter with parallel arms. The switching
function of the parallel arms will be the same. The arm inductances limit the circulation
current between the parallel arms.
+Vdc
L1
L2
E
E
VacL1
L2
-Vdc
...
Figure 1.6: Multi-leg multilevel converter
Another constraint of converters in high power application is their capability to suffer the
high value of current. The parallelization of the commutation cells by coupled inductors is
16
an effective approach in order to obtain a high current output with low current switches [19].
The interleaving technique which is the dual of the series connection of commutation cells
to achieve high voltage consists of connecting converters in parallel, with synchronized and
complementary operation, connected to the same load and with the same power source.
Figure 1.7 shows a single-leg two-level converter with interleaving inductors. Interleaved
converters can be classified in two ways: without magnetic coupling and with magnetic
coupling [20].
-V
+V
SM1
SM1'
-V
+V
SM2
SM2'
-V
+V
SMn
SMn'
Vout
Figure 1.7: Single-leg two levels converter with interleaving inductors
The converters are using an output filtering structure with magnetic coupling. These filtering
structures are usually employed to minimize the mass of the converters in a significant way.
These filters are normally sized to work with a constant number of phases. To optimize the
magnetic components, it is necessary to take into account some physical constraint, as the
iron sections of the magnetic circuits and saturation problems. These problems are
challenging, especially when the number of parallel cells is one of the variables [21].
The multicell interleaving converter with small inductance has proved to be desirable for
voltage regulator modules (VRMs) with low voltages, high currents, and fast transients.
Integrated magnetic components are used to reduce the size of the converter and improve the
efficiency. Researchers mentioned that with the proper design, coupling inductors can
improve both the steady-state and dynamic performances of VRMs with easier
manufacturing [22].
Interleaved power converters are used in many different conversion systems involving
various topologies (series or parallel) and related to different fields or loads. Researchers deal
17
with interleaved parallel commutation cells using coupling transformers with a possibly high
number of cells.
In high power applications, a composition of series and parallel connection of commutation
cells which provides the possibility of working in the high voltage and high current. This
configuration is known as a multilevel multicell converter (MMC). Figure 1.8 shows an
MMC converter which is consisted of a composition of the series and parallel commutation
cells.
-V
+V
SM1
SMn
SM1'
SMn'
SM1
SMn
Vout
-V
+V
SM1'
SMn'
-V
+V
SM1
SMn
SM1'
SMn'
Figure 1.8: MMC converter with interleaving inductors
1.5. Applications and Industrial Relevance
Researchers presented various topologies with different characteristics. Multilevel converters
make possible to connect to the medium and high voltage grids. In addition, they can connect
to the separated DC sources such as photovoltaic cells and battery package. Multilevel
converters increase the number of switching vectors, therefore they create the possibility to
use modern switching approach and design the progressive controller. There are many
18
research papers, which classified the modern control of multilevel converters. Recently,
multilevel converters are introducing as a good choice in the case of renewable energy
systems. Because of its scalable technology, it is possible to work in very high power
application. Todays, multilevel converters are the best choice in HVDC and FACT projects.
Another application which is very interesting in literature is electric vehicles. The modular
structure of multilevel converters is ideal to connect to the battery and photovoltaic sources.
MMC has received great attention because of its modular structure, which gives many
benefits as a multilevel converter. Lately, MMC technology has been used in large HVDC
transmissions, and hence well suited for high voltage structures. It is therefore natural to
include MMC in studies regarding high voltage installations. Mentionable projects are the
new cable connection between Norway and Denmark (Skagerrak 4 [18]), Trans Bay Cable
in San Fransisco [23], German offshore wind projects (HelWin, BorWin, DolWin), and the
cable connection between Sweden and Lithuania [24]. Table 1.1 demonstrates some of the
latest projects that include MMC technology. Both Siemens and ABB have developed HVDC
concepts which use some sort of modular multilevel converter topologies. The two concepts
are called HVDC Plus (Siemens) and HVDC Light (ABB).
Table 1.1: Some of the today’s MMC projects
Site Contractor Power (MW)
Trans-Bay San Francisco (2010) Siemens
400
Skagerrak 4 (2014) ABB
700
DolWin1 Germany (2013) ABB
800
DolWin2 Germany (2015) ABB
900
BorWin2 Germany (2013) Siemens
800
NordBalt Swe-Lit (2013) ABB
700
Losses analysis has been a popular subject in the multilevel converter literature. Researchers
presented some solutions to reduce the semiconductor losses. One of the most important
solutions is the multicell converters, which employs the coupled inductors. Some researchers
tried to optimize the converter parameters like ripple, efficiency and etc.
Many researchers tried to introduced and develop the multilevel topologies such as diode-
clamped inverter (neutral-point-clamped), capacitor-clamped (flying capacitor), and
19
cascaded multicell with separate dc sources. In the recent years, new multilevel topologies
like asymmetric hybrid cells and soft-switched multilevel converters have been presented [1,
25].
Also, multilevel converters have been compared with two-level converters in simulations to
investigate the advantages of using multilevel converters [25]. The symmetrical and
asymmetrical multilevel inverter has been classified by researchers. Both types are very
effective and efficient for improving the quality of the inverter output voltage [26].
On the other hand, researchers presented the weakness of these topologies. One of the major
limitations of multilevel converters is the voltage unbalancing between different levels. The
voltage balancing techniques between different levels normally involve voltage clamping or
capacitor charge control. There are several ways to provide the voltage balance in multilevel
converters [13, 25].
Due to the special topology of multilevel converters, various switching strategies and control
principles have been proposed and developed in the literature and advantages and
disadvantages were classified [25, 26].
The important section of research in the field of the multilevel converter is modular
converters which are suitable for very high voltage applications, especially network interties
in power generation and transmission [11].
The Modular Multilevel Converter (MMC) was represented as a scalable technology making
high voltage and power capability possible. The mathematical model of MMC was presented
with the aim to develop the model converter control system [27].
1.5.1. Multilevel converters and renewable energy
The general function of the multilevel inverter is to synthesize the desired AC voltage from
several levels of DC voltages. For this reason, multilevel converters are ideal for connecting
either in series or in parallel an AC grid with renewable energy sources such as photovoltaics
or fuel cells or with energy storage devices.
Researchers introduced multilevel converters to control the frequency and voltage output of
renewable energy because of its fast response and autonomous control [16]. In addition, they
20
represented the multilevel converters as a significant tool to control the real and reactive
power flow from a utility connected to the renewable energy source.
The rotor of a doubly-fed induction generator (DFIG) driven by a wind turbine needs rotor
excitation. In a variable-speed wind energy conversion system (WECS), the mechanical
frequency of the generator varies, and to keep the stator voltage and frequency constant, the
rotor voltage and its frequency have to be varied. Thus, the system requires a power
conversion unit to supply the rotor with a variable frequency voltage that keeps the stator
frequency constant irrespective of the wind speed. Also, in the case of permanent magnet
generators, a high power multilevel converter is used to convert the output AC voltage of the
generator to a fixed frequency voltage to supply the grid [28]. Figure 1.9 shows the single-
line diagram of multilevel converter proposed for wind energy conversion system [29].
Figure 1.9: Single-line schematic diagram of the proposed multilevel converter-based wind energy
conversion system [28]
Researchers are interested in the structure of Cascaded H-bridge DC-AC in order to use with
photovoltaic cells. In this structure, each module needs to connect to a DC source. Thus,
modular multilevel converters can be an ideal topology to connect to the PV panels [30]. The
multilevel converter is realized using a multicell topology where the total AC output of the
system is formed by series connection of several full-bridge converter stages. The dc links of
full bridges are supplied by individual DC-DC isolation stages which are arranged in parallel
concerning the DC input of the total system [31]. Figure 1.10 shows the multilevel topology
21
using full bridge commutation cells which are utilized to make a connection between solar
panels and single phase grid [32].
Figure 1.10: Multilevel inverter topology proposed for solar energy systems [32]
1.5.2. Multilevel converters and HVDC and FACT systems
Over the twenty-first century, HVDC transmissions will be a key point in green electric
energy development [33]. Modular multilevel converter (MMC) was proposed by many
researchers for high voltage AC/DC power conversion applications, such as HVDC and
FACTS [34]. Multilevel voltage source converters allow working at a high voltage level and
draw a quasi-sinusoidal voltage waveform. Classical multilevel topologies such as NPC and
Flying Capacitor VSIs were introduced twenty years ago, and are nowadays widely used in
Medium Power applications such as traction drives. In the scope of High Voltage AC/DC
converters, the Modular Multilevel Converter (MMC) appeared particularly interesting for
HVDC transmissions. On the base of the MMC principle, modular topologies of elementary
cells make the High Voltage AC/DC converter more flexible and easily suitable for different
voltage and current levels [33, 34]. The first modular multilevel converter for HVDC
application went operational in 2010. Figure 1.11 shows the cascaded multilevel topology
used for HVDC application by Siemens Company.
22
Figure 1.11: The modular cascaded multilevel converter used by Siemens Company for HVDC
application [33]
The advantages of multilevel topologies were proven in practice: efficiency comparable to
thyristor-based HVDC plants, very low space consumption due to the absence of filters and,
most important for the customer, a very good reliability. The same success could be achieved
for Static VAR Compensators (SVC) based on the modular multilevel concept. Both products
will certainly play a vital role in meeting the future challenges imposed upon the transmission
grids, like the integration of large amounts of renewable energy sources or increasing the
power transmission capability of the existing grids [35]. In the case of FACT applications,
various types of converter topologies are proposed. The most popular topology in this
application is neutral point clamped converters. Figure 1.12 shows a five level NPC
converter. In [36], a five-level FCMC is selected and designed for a 6.6 kV STATCOM.
23
Figure 1.12: Three-phase five-level structure of a diode-clamped multilevel converter [36]
Recently, ABB Company introduced two generations of SVC light for electrical transmission
grids [37]. In the first generation, ABB utilized 3-level NPC topology, while in the new
generation multi-level chain link (full bridge commutation cells) topology was used.
Table 1.2: ABB SVC Light for electrical transmission grids
First Generation Next Generation
Introduced 1997 2014
VSC tech. 3-level NPC Multi-level chain link
Converter voltage ≤ 35 kV ≤ 69 kV
Power range per block +/- 90 MVAr +/- 360 MVAr
Losses Medium Low Losses Medium Low Losses Medium Low
Active filtering Yes, up to 9th harm. Yes, up to 9th harm.
Need for filters Yes, high-pass No (depending on design)
DC capacitor Common Distributed
IGBT ABB StakPak, 2.5 kV, 1600 A ABB StakPak, 4.5 kV, 1800 A
24
1.5.3. Multilevel converters and Marine propulsion
Marine propulsion systems are undergoing rapid development with a significant thrust
towards the use of electric propulsion and the integration of auxiliary and propulsion power
systems. Multilevel converters are very popular for high power AC drives for large
electric/hybrid vessel. The marine propulsion drive systems usually are in the range of
medium power. The NPC converter is proposed in order to improve the system reliability
and increase its expansion flexibility [38]. The multilevel inverter can generate a high-quality
output voltage with less switching frequency and losses since only the small power cells of
the inverter operate at high switching rate. In the case of marine propulsion, increasing the
quality of power generated by multilevel inverter causes to less torque variation and smooth
propulsion. Furthermore, in this application, converter efficiency, reliability and power
quality are important. Figure 1.13 shows the NPC configuration in the electric propulsion
system [39].
Figure 1.13: Multilevel converter in electric marine propulsion system [39]
1.5.4. Traction motor drive
In the rapid development process of a high-speed electrified railway, power quality problems
in the traction power grid have become increasingly deteriorative. Increasing the usage of
electric railways in transportation systems augments the demands of high power converters.
Therefore, the power quality prolusion related to the high-power converters has increased
more and more [4]. Power traction and load characteristics of electrified railway give rise to
lots of power quality problems, such as negative-sequence current (NSC), harmonic and
25
voltage fluctuation. These will not only deteriorate power quality in traction power grid but
also threaten the safe and economical operation of power system [40-42]. The multilevel
converters were proposed as an appropriate solution to enhance the power quality of electric
railways. The main advantage of this kind of topology is that it can generate almost perfect
current or voltage waveforms because it is modulated by amplitude instead of pulse-width.
That means that the pulsating torque generated by harmonics can be eliminated, and power
losses into the machine due to harmonic currents can also be eliminated [43]. Another
advantage of this kind of drive is that the switching frequency and power rating of the
semiconductors are reduced considerably [15]. Figure 1.14 shows a 6-level diode-clamped
back to back converter that was proposed for traction drive application.
Figure 1.14: 6-level diode-clamped back to back converter for traction motor drive [15]
1.5.5. Losses analysis of multilevel converters
Some applications require light, efficient and reliable converters. Multilevel converters have
some potential advantages due to their lower output harmonic distortion and also the lower
device voltage rating requirements. Multilevel converters were compared to conventional
converters in terms of power losses and harmonic distortion of the output waveforms. Also,
the effects of modulation strategy which plays a significant role in the switching loss
distribution were investigated by some researchers [44].
Some researchers compared different structures of the cascaded multilevel converter with
IGBT technologies with the intention of minimizing power loss. The total power losses in
26
the IGBTs and diodes of each cell in the chain were estimated [45]. The hard-switching
transients of the power semiconductors at high commutation voltage cause high switching
losses and a poor harmonic spectrum which produces additional losses in the machine [46].
1.6. Main Objective: Converter sizing methodology applied to
MMC structures of static converters
The main objective of this research is to define and develop a converter sizing analysis
algorithm in order to optimize the dimension of power stack of MMC structures of static
converters. Converter sizing or converter dimensioning is an approach, which is used to find
the optimal dimension of a power electronic converter with respect to the inputs and outputs
specifications to achieve the best possible performance. Generally, the converter-sizing
procedure consists of several steps.
• Finding suitable multilevel topology
• Number of converter levels
• Switch sizing based on the specification and circuit model
• Passive component sizing includes capacitor, inductors
The first step is to propose an appropriate converter topology. With the certain source and
load specifications, it is possible to utilize different topologies. After choosing topology,
converter sizing algorithm tries to find the size of semiconductor switches, dimension of
electromagnetic components such as coupled inductors and specification of passive
components such as the capacitors, cables, fuses and etc. Furthermore, in the case of modular
MMC converters, converter sizing algorithm calculates the optimal number of parallel and
series commutation cells in order to achieve the maximum efficiency. Due to a limited
number of power IGBTs and converter topologies, the power semiconductor switches and
the converter topologies will be considered as discrete variables in the optimization
procedure. Therefore, the converter-sizing algorithm must be done for each power switch
and converter topology in order to find the best performance. The most important constraints
are:
• Capacitor voltage ripple
27
• Maximum switch losses
• Circulation current
• THD in AC side
• Efficiency
In this project, it is considered that the converter topology is determined. The converter sizing
procedure is used to determine the switch size, the dimension of coupled inductors, the
specification of passive components and number of series and parallel commutation cells.
Because of the limited number of IGBTs in the field of high power application, the
optimization algorithm will repeat for each IGBT to find the best performance. In these
project, we will focus on the MMC converter industrial application and its load specifications
and the sizing algorithm will calculate the optimal dimensions of power components.
Converter sizing analysis is a comprehensive analysis approach which consists of different
sub-methods from related knowledge. It is composed of circuit analysis, electromagnetic and
thermal analysis, mechanical analysis and manufacturing rules. Converter sizing analysis of
a multilevel multicell converter is a subject, which has rarely been studied, especially in high
power applications. Generally, the dimensioning analysis data are the confidential documents
of the companies. This may be the reason why this area has not been studied well in the
literature. Furthermore, it is impossible to verify the dimensioning analysis of the high-power
applications by university laboratories.
1.7. Secondary Objectives
1.7.1. Specify the global optimization approach for Optimization of MMC
static converters
In this thesis, we will propose an optimization model for MMC converters in order to
minimize its volume and mass in high power applications. This model consists of circuit
analysis, magnetic and thermal analysis and optimization program which work together to
find the best dimension.
Calculation of converter dimension is dependent on the total efficiency, total dimension and
manufacturing cost. The switch losses, inductor losses, inductor dimension, total efficiency,
28
and dimension are not independent variables. Therefore, an optimization loop must be
employed in order to find the number of levels, arm and inductor value, which obtain minimal
losses and dimension. Additionally, in each optimization loop, we have an internal
verification loop in order to achieve the precise result. The converter sizing of multilevel
converters is composed of several analysis methods, which need to work together.
• Electrical analysis
• Magnetic analysis
• Thermal analysis
• Dimensioning analysis
• Cost analysis
The converter-sizing algorithm of the multilevel converter has two main optimization cores.
The first core optimizes the semiconductor switches rating and the number of cells. It works
based on the circuit analytical model. There is an internal verification loop, which transfers
the circuit data to a circuit analysis software in order to verify the core results. The second
core finds the optimal dimension of the passive components (coupled inductors and
capacitors) which is offered by the first core. The analysis in this core is more complicates
than the first one. It receives the circuit data of inductors and capacitors and tries to optimize
the inductor losses, capacitor ripple and the dimension.
Figure 1.15: Optimization and verification loop
Dim
ensio
n V
ariab
les
Circuit Analysis
Electromagnetic Analysis
Thermal Analysis
Dimensioning Analysis
Op
timal
Dim
ensio
ns
Solver
Constraints Goal Function
29
Figure 1.15 shows the main parts of the optimization loop which is used in the converter-
dimensioning algorithm. The inductor and capacitor value and the dimensions are the outputs
of the second core. The passive components variables must return to the first core in order to
complete the global optimization loop. The design of the inductors is done in the second core.
Thus, the output of this core must be verified by an advanced software. In order to verify the
inductor analysis output, two internal verification loops have been used. The electromagnetic
verification loop is composed of a finite element electromagnetic analysis which receives the
data from the core and corrects the design parameters to achieve the response. With the same
approach, a thermal verification loop is used to verify the thermal design.
Optimal dimensioning is an art more than to be a science. It makes a connection between
classical analysis approach and industrial manufacturing methods. In this dissertation, a
converter sizing methodology will be proposed in order to find the optimal dimension of
passive components that lead to minimize the material consumption and cost.
Dimensioning analysis needs to collect some technical and experimental data of high power
semiconductors, heat sink systems, isolation materials and inductor core. This information
must be formulized in order to use in the optimization algorithm.
The optimization core generally uses the analytical analysis of the system. It decreases the
solution time, which is very important in the complex issues. Analytical approaches have
some errors in their results in comparison to the actual data. In order to eliminate the error,
an internal verification loop will be connected to each optimization loop. Each verification
loop consists of a software, which corrects the error coefficient until the result of analytical
approach is fitted to the simulation results. The analytical analysis consists of circuit analysis,
electromagnetic analysis, and thermal analysis. Thus, we need to connect to three software
in order to verify the circuit data, electromagnetic design, and thermal analysis. Figure 1.16
shows the proposed converter-sizing algorithm with verification loop applied to MMC static
converters.
30
fswNL
abcdlgJ
Converter Optimization
Core
InductorOptimization
Core
L,M
Pcu,Pcore
2D-3D FEM Magnetic
Verification
Initial Value
2D-3D FEM Thermal
Verification
Circuit Verification
Dimensioning
data
Figure 1.16: Optimization and verification loop
1.7.2. Applying the converter sizing method to an industrial application
Using multi-level converters will lead to the lower voltage stress across semiconductor
devices, lower switching frequency and less harmonic distortion in AC side. Additionally,
regarding design the appropriate controller, multi-level converters provide more adjustable
states.
The simplified power diagram of the specific MMC industrial application is shown in Figure
1.17. This multi-megawatt power supply is used in the PS Booster, a circular particle
accelerator of the CERN accelerator complex (European Organisation for Nuclear Research
in Geneva Switzerland) [75]. An Active AC/DC Front End (AFE) converter is supplying
electrical power to a DC/DC H-bridge Multilevel converter that is feeding the accelerator
electromagnets ring.
The capacitor bank in the DC bus is oversized in order to be used as a storage capacitor to
exchange the energy with the electromagnets during their current pulsed operation. The rated
electromagnet current pulsed cycle and the corresponding capacitor voltage waveform are
presented in Fig 1.17 & 1.18 respectively [75]. The multilevel H-bridge converter provides
the current of the load which is adjusted with a current controlled controller. The current of
the load changes from 300 A to 6000 A in less than 0.3 second. The extreme change of load
31
current affects the DC link voltage and drops it. In this condition, the conventional controller
does not have satisfactory performance. Also, the controller must set the reactive power to
zero to decrease the mis-effect of the converter on the grid, especially in the low switching
frequencies.
Figure 1.17: Multi-megawatt power supply of the PS Booster
Figure 1.18: Electromagnet Current pulsed Cycle delivered by the Multilevel H-bridge converter
Figure 1.19: Corresponding capacitor voltage cycle in the DC bus
Filtercircuit
Multilevel Rectifier
Transformer
MultilevelH-Bridge
Converter
Load
(18M
W)
Gri
d
18KV, 50 Hz 18KV:2KV
AFEActive
Front End
MultilevelActive
rectifier
DC BusStorage
Capacitor
32
The initial topological structure of the Active Front End converter (AFE) presented in [75]
was a three phase Neutral Point Clamped (NPC) converter. In this document, the replacement
of this NPC based AFE by a MMC is investigated as an application example of the proposed
design methodology.
Due to the variation of output current, it is necessary to do a precise investigation in order to
find the optimal size of the active and passive component. In addition, the DC link voltage
oscillation is another problem, which affects the performance of the power module and
controller. This will be possible using a converter sizing analysis, which considers all rules,
constraints, and limitations.
In high power applications, the role of switch losses will be very important. It plays a
significant role in order to increase the power of the converters. Decreasing the switching
frequency is one of the prevalent tasks to decrease the switch losses. This limitation forces
us to choose a low switching frequency for this application. The value of switching frequency
depends on the maximum losses of IGBT, which is the summation of conduction and
switching losses.
Due to high power rating, it is not possible to implement the converters with the classical
approach. There is not any semiconductor switch, which endures in this voltage and current
rating. Therefore, we need to use series and parallel configurations. Voltage limitation of the
semiconductor switches will be resolved using multilevel topologies whereas its current
limitation will remain. In the literature, the multi-leg (multi-phase) interleaved converter was
proposed for this problem. Therefore, in this application, we need to use a composition of
multilevel multicell topology to achieve our desired outputs. The power quality, final
dimension and implementation cost are the main constraints in this design approach. The
number of switches per arm must be determined. It is dependent on the DC bus voltage and
switches characteristics.
1.7.3. Finding the optimal solution for the MMC AFE converter
Result verification is an important part of high power application research. Due to the
difficulty of proving the validity of the result, researchers that work in the academic
laboratories are not interested in working in the field of dimensioning analysis of high power
33
converters. It is impossible to manufacture the high-power converters in the university
laboratory. Therefore, result validation is usually an important obstacle against the
development of academic research in the field of high power converters.
Figure 1.20: Example of High Power Converter in CERN complex
Fortunately, it was possible to have access to the specifications and technical information of
the industrial application presented in [75]. Figure 1.20 shows one of high power converter
cubicles on the CERN complex.
After result verification of our converter-sizing algorithm with load specification of the
MMC AFE converter application of Fig. , we need to propose an optimal solution for the
next generation of this application. The optimal dimensioning method ensures us to achieve
the best performance with minimum material and manufacturing cost. In this optimization,
we consider that the converter topology was already chosen and the optimization algorithm
tries to find the switch specification, the number of levels, the number of parallel cells, cables,
capacitor specifications, and inductor dimensions.
1.7.4. Using the Converter sizing method for other Applications
The next goal of this research is to extend our methodology in order to design and upgrade
other high power converters applications. It will be possible to extend our algorithm to these
converters with small variations.
34
1.8. Conclusion
To outline this section, there are many types of research about multilevel multicell converters.
Increasing the demands of high power converters causes to increase the number of researches
in this field. Developing the multilevel multicell topology, control system, losses reduction,
harmonic distortion and enhancement of transient behavior is the most important field of
research.
In high power applications, one of the important issues is to minimize the final volume and
the mass of the converter in order to decrease the manufacturing cost. Also, the consumer
expects to achieve the best possible desired outputs in return.
In this dissertation, a systematic approach will be proposed in order to achieve the converter
desired outputs with minimum mass, volume, and price. This type of analysis is called the
converter sizing analysis. The converter sizing analysis is an optimization between switch
and cable specifications, number of levels, number of parallel cells, capacitors size, and
inductor dimension with respect to different constraints in order to minimize the final mass
or volume of MMC converter. The details and methodology of this analyzing method will be
clarified in the next chapters.
35
CHAPTER II
2 Design of MMC converter based on the load
specification
2.1. Introduction
This chapter explains the design procedure of an active-front-end MMC converter. The
design procedure is an algorithm, which leads to determining the components value and size
regarding the nominal operating point. Unlike the conventional converters, modular
multilevel converters have many parameters to determine. The selection of converter
topology, sub-module capacitor, electromagnetic components and number of sub-modules
are known as the most important issues in the design procedure
The selection of converter topology depends on the converter application and the technical
and manufacturing constraints. The nominal voltage and current, power quality and total
efficiency are some of the parameters, which are important to choose the topology. In
addition, the sub-module topology is chosen dependent on the application.
The passive components such as capacitor and inductor are the main parts of converter
topology. There is an internal interaction between the components value and circuit variables
that made it very complex to investigate. The circuit analysis approach, which estimates the
circuit variables in transient and steady state condition.
The arm inductance is one of the important components of MMC converter. The arm
inductance should be investigated not only in term of circuit evaluation but also in term of
electromagnetic analysis. Hence, the MMC converter needs an extra analysis in comparison
with conventional converters.
36
The maximum voltage of IGBT is the most important factor that affects the number of sub-
modules. In high voltage application, increasing the number of a series sub-module is a
solution to endure the voltage. In addition, increasing the number of sub-modules leads to
higher power quality of converter. On the other side, the increasing of sub-module augments
the converter complexity and price.
In this chapter, the design procedure of MMC converter is explained. The converter and sub-
module topology and the input/output variables of the converter are investigated. A sizing
procedure is introduced to design the passive components and IGBTs in steady state and fault
condition. The adjustable parameters are introduced that should be design based on the
nominal. In order to calculate the converter efficiency, THD, and volume, a simple approach
has been presented. Finally, the advanced analysis tools are presented in order to investigate
the converter in term of circuit, electromagnetic and thermal analysis. In addition, utilizing
the analysis tools, a global optimization algorithm has been introduced to find the best
converter performance using the converter model and analysis tools.
2.2. Calculation of MMC converter variables
2.2.1. Converter and sub-module topology
MMC converters have a modular structure and each converter arm is composed of several
sub-modules in series. This modularity provides the possibility to employ the MMC
converter in a wide range of voltage from medium voltage to very high voltage applications.
Also, each converter arm includes an inductor that reduces the harmonics and the circulation
current between the arms. Figure 2.1 represents the topology of a modular multilevel active
front-end converter with parallel arms. The parallel arms topology is used to provide the
possibility to pass high current values, which are higher than IGBT nominal currents. In the
parallel topology, the voltages are the same while the currents are divided by the number of
parallel arms. Hence, in this chapter, to simplify the equations, the single leg topology is
investigated. There are various sub-module topologies that are used in the MMC converters.
The half-bridge and full bridge are the most important sub-module that is employed in high
power application. Figure 2.2 shows the circuit configuration of half-bridge and full bridge
sub-modules. The full-bridge sub-module includes four switches and one bypass capacitor
while half-bridge sub-module includes two switches and on bypass capacitor. The full bridge
37
configuration generates more voltage steps in combination with other sub-modules. Hence,
it provides better power quality and lower harmonics. In addition, the input current is divided
between two arms and it is possible to employ the switch with lower power. On the other
side, full bridge configuration needs more semiconductor switches and more drivers that
made it bulky and expensive.
SM1
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Figure 2.1 Modular multilevel active front-end converter
+
Vc
- +
Vsm
-
Ic
S1 D1
S2 D2
S1 D1
S2 D2
S3D3
S4D4
+
Vc
-
Ic
- Vsm +
a) b)
Figure 2.2 a)half-bridge sub-module configuration b)full-bridge sub-module configuration
38
2.2.2. Converter inputs and outputs variables
If an active front-end converter is considered as a black box, in the steady-state condition,
the active power relation in the AC side and DC side are written as below:
𝑃𝑎𝑐 = 𝑣𝑎𝑖𝑎 + 𝑣𝑏𝑖𝑏 + 𝑣𝑐𝑖𝑐 (2.1)
𝑃𝑑𝑐 = 𝑖𝑑𝑐𝑣𝑑𝑐 (2.2)
It was assumed that the harmonics are negligible. Figure 2.3 represents the input and output
parameters of a three-phase active front-end converter regardless of the converter topology.
c
ia
ib
ic
idc
vdc
va
C
iout
R L
vb R L
vc R L
a
b
c
n
Figure 2.3 Inputs and output variables of active front-end converter
The circuit equation of input and output circuit are written as below:
𝑣𝑎𝑏𝑐 = (𝑅 + 𝐿𝑝)𝑖𝑎𝑏𝑐𝑠 + 𝑣𝑎𝑏𝑐𝑛 (2.3)
𝐶𝑝𝑣𝑑𝑐 = 𝑖𝑑𝑐 − 𝑖𝑜𝑢𝑡 (2.4)
Where 𝑝 =𝑑
𝑑𝑡 and 𝑣𝑎𝑏𝑐𝑛 is the phase voltages between the converter terminals (abc) and
neutral point (n) , 𝑖𝑎𝑏𝑐𝑠 are three phase line currents, 𝑅, 𝐿 are line resistance and inductance
respectively..
Regarding the switching index, the relation between the input and output variables for the
fundamental frequency is defined in the DQ axis as below:
𝑣𝑞 = 𝑅𝑖𝑞𝑠 +𝜔𝑒𝐿𝑖𝑑𝑠 +𝑚𝑞
2𝑣𝑑𝑐
(2.5)
𝑣𝑑 = 𝑅𝑖𝑑𝑠 −𝜔𝑒𝐿𝑖𝑞𝑠 +𝑚𝑑
2𝑣𝑑𝑐
(2.6)
3
4(𝑚𝑞𝑖𝑞𝑠 +𝑚𝑑𝑖𝑑𝑠) = 𝑖𝑑𝑐
(2.7)
39
Where 𝑚𝑞 and 𝑚𝑑 are the average switching index on q and d axis, respectively and
𝑖𝑑𝑠𝑎𝑛𝑑 𝑖𝑞𝑠 are d and q axis currents. Also, the active and reactive power on AC side.
𝑃𝑖𝑛 =3
2(𝑉𝑞𝑠𝑖𝑞𝑠 + 𝑉𝑑𝑠𝑖𝑑𝑠)
(2.8)
𝑄𝑖𝑛 =3
2(𝑉𝑞𝑠𝑖𝑑𝑠 − 𝑉𝑑𝑠𝑖𝑞𝑠)
(2.9)
2.2.3. Semiconductor sizing in steady state
The semiconductor switch is chosen regarding its nominal voltage and current and its losses
function. In the half-bridge configuration, the capacitor voltage is the maximum voltage that
should be endured by IGBTs. The capacitor voltage is a DC part with a ripple. The IGBT
voltage must be greater than 𝑉𝑐𝑚𝑎𝑥.
𝑉𝑐𝑚𝑎𝑥 =
𝑉𝑑𝑐𝑚+𝑅𝑖𝑝𝑝𝑙𝑒
2
(2.9)
where 𝑅𝑖𝑝𝑝𝑙𝑒 is the difference between maximum and minimum voltage of the capacitor.
The current of IGBTs in half bridge configuration depends on the switching function and
load specification. The arm current, 𝐼𝑎𝑟𝑚 is composed of phase current and circulating
current.
𝐼𝑎𝑟𝑚 = 𝐼𝑐𝑖𝑟𝑐 ±𝐼𝑎2
(2.10)
Where 𝐼𝑎is the instantaneous value of line current and 𝐼𝑐𝑖𝑟𝑐 is the the instantaneous value of
circulation current in arm. The arm current is divided between two IGBT regarding to the
modulation index. The current of sub-module switches 𝐼𝑠1, 𝐼𝑠2 are computed as below
𝐼𝑠1 = Sm. 𝐼𝑎𝑟𝑚 (2.11)
𝐼𝑠2 = (1 − 𝑆𝑚)𝐼𝑎𝑟𝑚 (2.12)
Where 𝑆𝑚 is the modulation Index of the sub-module. In the next sections, the precise
calculation of the capacitor voltage and IGBT current is presented and clarified.
The IGBT losses are the most important parameter in order to select the appropriate switch.
High power IGBT are manufactured by companies based on specific applications to provide
a balance between conduction losses and switching losses. The switching and conduction
losses of IGBTs depends on the device structure. Fundamental and early device structures
40
include symmetric blocking IGBTs and asymmetric blocking IGBTs. Symmetric structures,
also called “reverse blocking,” have inherent forward and reverse blocking capabilities,
which make them well suited for AC applications such as matrix (AC-to-AC) converters or
three-level inverters. Asymmetric structures maintain only forward blocking capability and
offer a lower on-state voltage drop than symmetric IGBTs. This makes them ideal for DC
applications like a variable speed motor control, where an anti-parallel diode is used across
the device allowing operation in only the first quadrant of i-v characteristics.
Faster IGBTs that achieve a higher PWM frequency will reduce ripple current and the
required filters can be made smaller because the output waveform is closer to the desired
waveform. Comparing the two device families to determine suitability for different
applications is based on several key parameters: conduction losses (𝑉𝐶𝐸𝑆𝐴𝑇 (for IGBT), 𝑉𝐹
(for diode)) and switching losses (ETS (for IGBT), 𝑄𝑅𝑅(for diode). In general, the RC-DF
devices [47] have at least 50 percent lower switching loss (ETS (mJ)) values when compared
to the lower frequency devices. The RC-D devices [47] stand out in terms of lower
𝑉𝐶𝐸𝑆𝐴𝑇 values and 𝑉𝐹 values which are the thermal or conduction losses, at the expense of
the higher switching losses [48].
Each IGBT switch includes an IGBT and a freewheel diode. The conduction losses of the
switch are the summation of IGBT and diode conduction losses. The conduction losses
depend on the conduction resistance, saturation voltage and the current of IGBT and
freewheel diode.
𝑃𝑐𝑖 = 𝑣𝑐𝑒𝑖𝑐𝑒 + 𝑟𝑐𝑖𝑐𝑒2 (2.13)
𝑃𝑐𝑑 = 𝑣𝐹𝑖𝐹𝐷 + 𝑟𝑑𝑖𝐹𝐷2 (2.14)
𝑃𝑐 = 𝑃𝑐𝑖 + 𝑃𝑐𝑑 (2.15)
where 𝑖𝑐𝑒and 𝑖𝐹𝐷 are the effective current of IGBT and reverse diode, respectively. 𝑟𝑐 and 𝑟𝑑
are resistance of IGBT and diode in conduction cycle. The switching losses depends on the
switching frequency 𝑓𝑠𝑤 and commutation energy of IGBT.
𝑃𝑠𝑤𝑖 = (𝐸𝑜𝑛𝑖 + 𝐸𝑜𝑓𝑓𝑖)𝑓𝑠𝑤 (2.16)
𝑃𝑠𝑤𝑑 = (𝐸𝑜𝑛𝑑 + 𝐸𝑜𝑓𝑓𝑑)𝑓𝑠𝑤 ≈ 𝐸𝑜𝑛𝑑𝑓𝑠𝑤 (2.17)
Where 𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 are the energy losses in one switching period. Figure 2.4 shows the
thermal dissipation circuit of IGBT. The maximum temperature of junction is determined by
manufacturer. The junction temperature is calculated as below:
41
(𝑇𝑗 − 𝑇𝑎) =𝑃𝑐 + 𝑃𝑠𝑤
𝑅𝑗𝑐 + 𝑅𝑐𝑠 + 𝑅𝑠𝑎
(2.18)
Where 𝑇𝑗 is the junction temperature, 𝑇𝑎 is the ambient temperature, 𝑅𝑗𝑐 is the thermal
resistance of junction-case, 𝑅𝑐𝑠 is the thermal resistance of case-sink and 𝑅𝑠𝑎 is the thermal
resistance of sink-air.
Rjc Rcs Rsa
PcPsw Ta
Tj Tc Tr
Figure 2.4 Thermal dissipation circuit of IGBT
2.2.4. Passive components sizing in steady state
A. Selection of sub-module capacitor
In MMC converters, the electric energy is stored in sub-module capacitors. The maximum
energy stored in capacitors ECmax is determined by the rated converter power Sn and the
energy-power ratio(𝐸𝑃) [49].
𝐸𝑃 =𝐸𝑐𝑚𝑎𝑥𝑆𝑛
(2.18)
The energy-power ratio varies from EP = 10 J/kVA to 50 J/kVA and depends on the converter
application. Lower values mean a reduction of the converter cost while it leads to higher
voltage ripples in the DC-link circuit.
The design procedure of an active front-end converter starts by selection of two main
converter parameters. It is the rated converter power 𝑆𝑛 and RMS value of the line-to-line
voltage 𝑉𝐿−𝐿,𝑟𝑚𝑠 at the ac side of the converter or the voltage 𝑉𝑑𝑐at the dc side of the
converter. Assuming that in the MMC there are no redundant sub-modules, the relation
between AC side and dc side voltages is given as
𝑉𝐿−𝐿,𝑟𝑚𝑠 =√3𝑆𝑚2
𝑉𝑑𝑐
√2
(2.19)
where the modulation index 𝑆𝑚 can be changed from 0 up to 2/√3.
42
The maximum energy stored in sub-module capacitors of the three-phase MMC consisting
of 6m sub-modules is given by:
𝐸𝑐𝑚𝑎𝑥 = 6𝑚𝐶𝑠𝑚2(𝑉𝑑𝑐𝑚)2
= 3𝐶𝑎𝑟𝑚𝑉𝑑𝑐2
(2.20)
Where 𝐶𝑎𝑟𝑚 = 𝐶𝑠𝑚/𝑚 . Hence the arm capacitance 𝐶𝑎𝑟𝑚 can be calculated using the energy-
power ratio EP.
𝐶𝑎𝑟𝑚 =𝐸𝑐𝑚𝑎𝑥3𝑉𝑑𝑐
2 = 𝐸𝑃𝑆𝑛3𝑉𝑑𝑐
2 (2.21)
The value of 𝐶𝑎𝑟𝑚 could not increase so much due to the ratio of 𝑆𝑛/𝑉𝑑𝑐2 , which affects the
dimension of current, limits the rated current of the converter, load and other components.
The maximum energy-stored in capacitors (𝐸𝑐𝑚𝑎𝑥) and the energy-power ratio (𝐸𝑃) are the
parameters that are important to determine the sub-module capacitor.
B. Arm inductance selection
In the structure of MMC converter, the role of the arm inductors 𝐿𝑎𝑟𝑚 in the MMC is to
suppress any high-frequency components of the arm currents caused by differences in upper
and lower arm voltages. The voltage difference between upper and lower arms exists due to
the different switching function of upper and lower switches.
The arm inductances value Larm have been chosen regarding a number of parameters. The
exact value of the arm inductance depends on the sub-module capacitor voltage 𝑉𝑑𝑐/𝑚, the
modulation technique, the switching frequency and an additional controller optionally used
for suppressing the circulating current.
It is assumed that the open loop sinusoidal PWM is used as the modulation technique and the
circulating current is not suppressed by any other control methods. It means that the
circulating current has to be suppressed only by the proper selection of the arm inductance
Larm. The arm inductance must be calculated in term of avoiding resonances that occur in the
circulating current for the given arm capacitance 𝐶𝑎𝑟𝑚.
𝐿𝑎𝑟𝑚 =1
𝐶𝑎𝑟𝑚𝜔22(ℎ2 − 1) + 𝑆𝑚
2 ℎ2
8ℎ2(ℎ2 − 1)
(2.22)
Where ℎ is the existing even-order harmonics (ℎ = 2, 4,…).
The proper selection procedure should be restricted to the particular values of harmonic order
and modulation indices, which are possible for the converter application. In a situation when
43
the modulation index 𝑆𝑚 is limited during the converter operation to values much closer to
the maximum value 𝑆𝑚 = 1. The best value concerning the RMS value of the circulating
current can be obtained by performing a simulation of the MMC averaged model.
2.3. Investigation of adjustable parameters of multilevel
converter
The design of multilevel converters depends on lots of parameters that should be considered
to achieve a high-performance converter.
2.3.1. Converter topology
Multilevel topologies provide different properties in term of efficiency, harmonic and losses,
complexity, volume, and mass. Neutral point clamped converter, flying capacitor and
modular multilevel converter are the most important topologies of multilevel converters. The
selection of topology depends on the load specification and its application. In the medium
power applications, the neutral point clamped converter is used because of its performance
and reliability. By increasing the converter voltage where the number of levels must be
augmented to endure high voltage, the modular converter is the best choice for high voltage
applications. Because of modular structure, there is no limit to achieving high voltage values.
Also, MMC converters provide the possibility of harmonic cancellation using their internal
passive filter.
2.3.2. Number of sub-modules per arm
The number of sub-module per arm is the most important parameter which separates
conventional and multilevel converters. By increasing the converter voltage level, there is no
semiconductor that suffers the converter voltage. The converter voltage is divided between
the sub-modules. Hence, the maximum voltage of IGBT must be greater than the DC link
voltage divided by the number of sub-module per arm 𝑚.
𝑉𝐼𝐺𝐵𝑇 >𝑉𝑑𝑐𝑚
(2.23)
Increasing the number of sub-modules leads to reduce the sub-module voltage and switch
voltage. The switching and conduction losses of IGBT depend on the voltage [50, 51] and it
44
could be reduced by increasing the number of series sub-modules per arm. In result,
according to the IGBT voltage limit, increasing the number of sub-modules can affect and
reduce the switching losses.
The number of sub-modules per arm can effectively decrease the converter harmonic and
enhance the power quality of the system. In high power application, regarding the high
voltage and current values, the switching losses of semiconductor switches is high. Hence,
to avoid high switching losses and respect the thermal limit of semiconductors, a low
switching frequency is utilized. Semiconductor losses are composed of conduction and
switching losses. Increasing the switching frequency leads to increase the switching losses.
In high power application, due to the high value of voltage and current, the low switching
frequency is employed to avoid high switching losses. On the other hand, utilizing low
switching frequency leads to diminishing the converter power quality [52, 53]. Multilevel
converters provide the possibility to work with the low switching frequency without
decreasing the power quality. The effect low switching frequency could be compensated by
increasing the number of commutation cells [52, 53]. The frequency that emerges in the
converter output is the switching frequency multiplied by the number of series sub-modules
per arm. Hence, in the case of high voltage converters where a high number of sub-modules
is used, the low switching frequency is employed to avoid high switching losses while high
power quality is achieved.
2.3.3. Passive component values
MMC converters have passive components (sub-module capacitors and arm inductors) in
their topology. The value of circulation current depends on the capacitor and inductor value.
Therefore, the passive component value affects the converter variable and in result converter
performance. The main role of sub-module capacitor is to balance the sub-module voltage
and divide the DC link voltage between sub-modules. Changing the current direction of the
capacitor leads to vary the capacitor voltage and generate the voltage ripple, which could
make unbalance voltage condition. Increasing the sub-module capacitor value directly
reduces the voltage ripple.
In MMC topology, arm inductance is used to control and diminish the arm circulation current
due to the imbalance voltage between upper and lower arms. Also, it works as a passive filter
45
at the converter input that eliminates or reduces the current harmonics. The arm inductance
directly affects the THD value and IGBT losses by minimizing the circulation currently.
Also, in the fault condition, the arm inductance limits the fault current.
Unlike the conventional converters, passive components play an important role in the
structure of MMC converters. They play not only the role of the passive filter but also the
fundamental role in converter circuit performance. Selection of proper value for the passive
components depends on the converter specifications, constraints, and its application. The
passive component values will be computed using a comprehensive circuit analysis model.
2.3.4. IGBT selection
The selection of IGBT specification is another parameter that could be determined.
Unfortunately, in term of high power application, the number of IGBTs that are introduced
by the manufacturer is not so much. Hence. The IGBT selection is a discrete parameter and
there are not lots choices for it. The design procedure of MMC converter could be repeated
for the available IGBTs in order to find the best specification.
2.4. Investigation of converter performance and limitations
The term of converter performance is generally defined regarding its application. The
converter performance could be defined as total efficiency, converter power quality, total
losses, converter volume or mass and etc.
2.4.1. Converter Losses and efficiency
MMC converter losses include IGBT losses and inductor losses. The IGBT losses are
composed of switching and conduction losses. Each sub-module includes two IGBT: main
IGBT and bypass IGBT. The conduction losses of each IGBT includes the collector losses
and diode losses. Also, the IGBT switching losses include the IGBT and diode switching
losses. The total IGBT losses are calculated as below:
{𝑃𝑐𝑖𝑇 = 6𝑚(𝑃𝑐𝑖1 + 𝑃𝑐𝑖2)𝑃𝑐𝑑𝑇 = 6𝑚(𝑃𝑐𝑑1 + 𝑃𝑐𝑑2)
(2.24)
{𝑃𝑠𝑤𝑖 = 12𝑚𝑓𝑠𝑤(𝐸𝑜𝑛𝑖1 + 𝐸𝑜𝑓𝑓𝑖1 + 𝐸𝑜𝑛𝑖2 + 𝐸𝑜𝑓𝑓𝑖2)
𝑃𝑠𝑤𝑑 = 12𝑚𝑓𝑠𝑤(𝐸𝑜𝑛𝑑1 + 𝐸𝑜𝑛𝑑2)
(2.25)
46
𝑃𝐼𝐺𝐵𝑇−𝑇 = 𝑃𝑐𝑖𝑇 + 𝑃𝑐𝑑𝑇 + 𝑃𝑠𝑤𝑖 + 𝑃𝑠𝑤𝑑 (2.26)
The total IGBT losses depends on the switching frequency (𝑓𝑠𝑤) and number of sub-modules
per arm (𝑚).
The inductor losses are composed of copper losses of the winding and core losses. The copper
losses depend on the inductor current and current frequency. In the first step, the effect of
frequency is neglected. The total copper losses are calculated as below:
𝑃𝑐𝑢𝑇 = 6 𝑅𝐿𝐼𝑎𝑟𝑚𝑟𝑚𝑠2 (2.27)
The core losses include hysteresis and eddy current losses. The core losses depend on the
frequency and magnetic flux density of the core. The total hysteresis and eddy current losses
are calculated as below:
𝑃ℎ = 6𝐾ℎ𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝐵ℎ1.𝑚𝑎𝑥α + 2𝑓𝑠𝐵ℎ2.𝑚𝑎𝑥
α ) (2.28)
𝑃𝑒 = 6𝐾𝑒𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝛽𝐵ℎ1.𝑚𝑎𝑥γ
+ (2𝑓𝑠)β𝐵ℎ2.𝑚𝑎𝑥
γ) (2.29)
where 𝛼 is the hysteresis losses constant and 𝛽 𝑎𝑛𝑑 𝛾 are eddy current losses constants,
which are dependent on material magnetic properties. The converter efficiency is known as
most important criterion that shows the converter performance. The total converter efficiency
is:
η =𝑃𝑑𝑐
𝑃𝑑𝑐 + 𝑃𝐼𝐺𝐵𝑇−𝑇 + 𝑃𝑐𝑢𝑇 + 𝑃ℎ + 𝑃𝑒× 100
(2.30)
2.4.2. Power quality and harmonic Investigation
One of the most important advantages of MMC converter is to eliminate the current
harmonics that are injected into the grid by the power electronic converter. MMC converters
utilize low switching frequency to minimize the switching losses. The multilevel structures
transport and shift the voltage to high-frequency spectrum.
There are two different harmonic frequencies in MMC structures. The double-frequency
component of single-phase ac power is emerged in the phase and compose the low-order
harmonics of MMC converter. On the other hand, the current harmonic generation due to the
switching frequency that was shifted to high spectrum frequency is the high-frequency part
of MMC harmonics. Low frequencies are more expensive in terms of filters because the
components of filters are large and costly. The arm inductances should be designed in order
47
to limit the low-order harmonics which can affect the converter losses. In term of high
frequency, researchers proposed several methods to control and diminish the high-frequency
harmonics [54].
2.4.3. Converter volume and mass
MMC converter is composed of multiple sub-modules that are series connected to sustain the
high voltage values. By increasing the converter voltage, the total converter volume and size
will be an important issue due to the high number of components. Hence, the converter mass
estimation is a criterion in converter design and optimization.
The converter mass depends on the number of sub-modules. Each sub-module consists of
two IGBTs and one capacitor. The total mass of IGBTs (𝑀𝐼𝐺𝐵𝑇) and sub-module capacitors
(𝑀𝑐𝑎𝑝) is calculated as below:
𝑀𝐼𝐺𝐵𝑇−𝑇 = 12𝑚𝑀𝐼𝐺𝐵𝑇 (2.31)
𝑀𝑐𝑎𝑝−𝑇 = 6𝑚𝑀𝑐𝑎𝑝 (2.32)
The value of IGBT and capacitor mass are written in the datasheet. Unlike the IGBT and
capacitor, the number of arm inductances is not dependent on the number of sub-module per
arm. The inductor mass is composed of core mass and winding mass. The total inductance
mass (𝑀𝑖𝑛𝑑) is written as below:
𝑀𝑖𝑛𝑑 = 6(𝑉𝑐𝑜𝑟𝑒𝐷𝑐𝑜𝑟𝑒 + 𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔𝐷winding) (2.33)
2.5. Comprehensive analysis of MMC converter
Due to the complexity of MMC structures, a comprehensive analysis includes the circuit,
electromagnetic and thermal model must be done to achieve the reliable results. The circuit
model determines the electrical quantity of MMC circuit regarding the load specification,
IGBT model, and switching function. The electromagnetic model estimates the magnetic
variables of inductors regarding the core parameters and nominal values. The thermal model
investigates the heat distribution which is produced via converter losses in the inductor and
semiconductors.
48
2.5.1. Circuit Analysis
The converter circuit model provides the quantity of all electrical variables versus the
operating load values, component value, and switching function. In the case of MMC
converter, there is a circular interaction between the passive component values and the
electrical quantity of the converter that made it complex to investigate. Figure 2.5 shows the
input/output variables of MMC circuit model of an active front-end converter. Various circuit
models were proposed by the researchers to explain the circuit behavior of MMC converter
and estimate the important electrical variables that are important to determine the converter
performance.
In the literature, various circuit models have been proposed for MMC converter. The MMC
circuit model based on the average switching function are the most popular models [55-58].
The average model neglects the effect of switching frequency. Hence, it is not accurate
enough for harmonic study [59-61]. In this research, a time-domain MMC circuit model is
presented and developed to estimate the ripple and THD value.
MMC Circuit
Model and
Fault Analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch losses
Spe
cifi
ctio
ns
Co
nve
rte
r to
po
logy
N.
.
.
Pas
sive
co
mp
on
en
t v
alu
e
M
IGB
T
Spe
cifi
cati
on
s Vmax
Imax
Psw
Inputs
Stead
y-state Variab
les
Stead
y-state Pe
rform
ance
Inductor losses
Efficiency
Energy stored
Outputs
Figure 2.5 MMC circuit model and input/output variables
49
2.5.2. Electromagnetic Analysis
Unlike the conventional converters, the electromagnetic components are the part of MMC
converter topology. The electromagnetic components a circuit model which is used in circuit
analysis and a magnetic model that calculates the magnetic parameters of the electromagnetic
core.
The electromagnetic model of inductor investigates the magnetic behavior of inductor core
and the winding. The electromagnetic model estimates the inductance, resistance, magnetic
core flux, losses, volume and mass regarding the core size and winding parameters. The
magnetic analysis is a complex analysis which is generally done using numerical solution
approaches. The simplest way to analyze a magnetic component is to use the analytical
model. The analytical model explains the magnetic behavior of an electromagnetic system
using simple mathematical equations. The analytical model accuracy is not high but, it is an
appropriate tool in the initial design procedure. Figure 2.6 shows the input/output of the
analytical model of arm inductance which calculates the inductance value regarding the core
and winding parameters.
Dimension model
of arm
inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cifi
ctio
ns
Co
re t
op
olo
gy
IL2
Win
din
g
n1
n2
Inputs
Electro
magn
etic V
ariable
s
Pe
rform
ance
Outputs
η%
Figure 2.6 Analytical electromagnetic model of arm inductance and its input/output
The major weakness of the analytical model is the low model accuracy especially in the case
of electromagnetic analysis. Various numerical solution approach is used to analyze the
electromagnetic structures. Finite element method (FEM) is the most popular approach which
is used to solve the electromagnetic issues.
50
Finite element method provides high accuracy results in comparison with the analytical
model. Finite element method is a time-consuming analysis method which is not suitable for
the initial step of design. In this research, an innovative model is presented that is a
combination of analytical model and finite element method to provide the accuracy and
analysis speed at the same time. Figure 2.7 shows the proposed model topology which utilizes
the analytical model and finite element analysis at the same time.
FEM
softwareLfem=Larm
Correction
coefficient
no
Finish
Dimension model
of arm
inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cifi
ctio
ns
Co
re t
op
olo
gy
IL2
Win
din
g
n1
n2
InputsE
lectrom
agne
tic Variab
les
Pe
rform
ance
Outputs
η%
yes
Figure 2.7 Proposed model for electromagnetic analysis includes analytical model and finite
element analysis
2.5.3. Thermal Analysis
The thermal losses which are generated in the components increase the components
temperature and might leads to destroying them. The IGBTs and arm inductances are major
heat sources in the converter topology. In the case of IGBT, the thermal limitation and its
dissipation function are described in the technical datasheet. The cooling system is designed
based on the technical data and the IGBT losses.
51
In the case of an inductor, the thermal losses are generated in the winding and the magnetic
core. Hence, the thermal analysis should be done to estimate the maximum temperature rise.
Figure 2.8 shows the proposed thermal model based on the analytical approach. The model
output is the temperature rise while the inputs are the calculated losses and cooling system
specifications.
Thermal Model
Pcu
Pcore
Psw
IGBT
cooling
Inductor
cooling
Co
nve
rte
r lo
sses
Co
olin
g
InputsTe
mp
erature rise
ΔTIGBT
ΔTcore
ΔTcu
Outputs
Figure 2.8 The analytical thermal model of MMC converter
2.6. MMC dimensioning Analysis
The last analysis which should be done to determine the MMC converter performance is
dimensioning or sizing analysis. The circuit value of passive components does not represent
the component dimension. The passive component dimension depends on lots of technical
and manufacturing and thermal parameters that must be considered to achieve the real size.
Hence, a physical dimensioning analysis is employed by designers to estimate the size of
components. The total converter size which is composed of capacitor, inductor, and
semiconductor switches and cooling system can be considered as converter performance.
2.6.1. Capacitor dimensioning analysis
The capacitor is an industrial component which is manufactured by the industrial companies
base on the standard tables. Actually, regarding the product of each manufacturer, there is a
limited number of capacitors, especially in high voltage applications. Hence, the capacitor
value must be sized regarding the available products.
Power electronic capacitors are designed regarding their applications [62]. Power electronic
capacitors are generally manufactured based on the Polypropylene/Polyester film and
52
Metallized polypropylene film technologies in order to minimize the internal inductance and
eliminate the high-frequency noises. The capacitor container is a robust rectangular or
cylinder case.
The volume or mass of capacitor depends on the capacitor type (technology), capacitance
value and capacitor voltage. Investigation of industrial products shows that in low voltage,
the capacitor volume strongly depends on the capacitance. In high voltage, increasing the
capacitor voltage intensely affect the capacitor mass.
2.6.2. Inductance dimensioning analysis
Unlike the capacitors, inductors that are used in power electronic are not pre-designed
products. Power electronic designers calculate the technical parameters of the inductor to
make an order to the manufacturing companies.
The magnetic core and winding are the most important parts of the inductor. The magnetic
cores are designed with various shapes and topologies. The most popular and simplest
topology which is used to design inductor is shown in figure 2.9. The core volume is
calculated as below:
𝑉𝑐𝑜𝑟𝑒 = 𝑐[(𝑎 + 2𝑑)(𝑏 + 2𝑑) − 𝑎𝑏] (2.34)
𝑀𝑐𝑜𝑟𝑒 = 𝑉𝑐𝑜𝑟𝑒𝐷𝑐𝑜𝑟𝑒 (2.35)
d
a
b
d
d
g
c
Figure 2.9 The magnetic core topology
53
The core parameters should be designed with respect to the magnetic parameters such as
maximum flux density and core losses and the thermal constraints.
The inductor winding is another part of an inductor that affects the inductor mass. The
average length of one turn is calculated as below:
MLT = 2𝑐 + 4𝑎 + 2𝑑 (2.36)
The total copper volume and mass are estimated as below:
𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑛.𝐼𝐿𝐽. MLT
(2.37)
𝑀𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔𝐷𝑤𝑖𝑛𝑑𝑖𝑛𝑔 (2.38)
where 𝐷𝑤𝑖𝑛𝑑𝑖𝑛𝑔 is the density of the winding’s material.
2.7. MMC Multiphase Analysis
As it was mentioned, MMC converter is composed of various components that need several
analysis models to design. The design of electrical quantity must be done at the same time
that passive components are designed. This kind of analysis which investigates all technical
parameters in a unique design procedure is known as global analysis approach. In the global
analysis, all models should be executed in the same algorithm. The models interact together
to design all MMC variables considering all technical or manufacturing constraints.
2.7.1. Global analysis using analytical model
Analysis of an MMC converter is composed of circuit, electromagnetic and thermal analysis.
Each model includes several inputs/outputs. Some inputs of the models come from the
outputs of other models. For example, the circuit model provides the inductance current value
for the electromagnetic model, while electromagnetic model determines the self and mutual
inductances which are utilized in circuit model calculation.
The circuit and electromagnetic models estimate the switch losses and inductor losses
respectively. The calculated losses are the inputs of the thermal model which estimates the
maximum temperature rise in the semiconductors and inductors.
54
MMC Circuit
Model and
Fault analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch losses
Spe
cifi
ctio
ns
Co
nve
rte
r to
po
logy
N
.
.
.
Pas
sive
co
mp
on
en
t v
alu
e
M
IGB
T
Spe
cifi
cati
on
s Vmax
Imax
Psw
Inputs
Stead
y-state Variab
les
Stead
y-state Pe
rform
ance
Inductor losses
Efficiency
Energy stored
Outputs
Dimension model
of arm
inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cifi
ctio
ns
Co
re t
op
olo
gy
IL2
Win
din
g
n1
n2
Electro
magn
etic V
ariable
s
Pe
rform
ance
η%
Thermal Model
Pcu
Pcore
Psw
IGBT
cooling
Inductor
cooling
Co
nve
rte
r lo
sses
Co
olin
g
Tem
peratu
re rise
ΔTIGBT
ΔTcore
ΔTcu
Figure 2.10 Global analysis plan of MMC converter using analytical models
Figure 2.10 shows the plan of global analysis approach which represents the models,
inputs/outputs variables and interconnections. The circuit model gets the nominal operating
point, passive components values and switching function to calculate all electrical quantity
of the converter. The electromagnetic model gets the magnetic core size and parameters to
55
calculate the self and mutual inductance that is used by circuit model. Finally, the thermal
model estimates the temperature rise based on the calculated losses and cooling system
parameters.
2.7.2. Global analysis using modified analytical model
In order to compensate the low accuracy of an analytical model of the inductor, a correction
loop has been added to the plan to correct the model parameters regarding the finite element
results.
MMC Circuit
Model and
Fault analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch losses
Spe
cifi
ctio
ns
Co
nve
rte
r to
po
logy
N
.
.
.
Pas
sive
co
mp
on
en
t v
alu
e
M
IGB
T
Spe
cifi
cati
on
s Vmax
Imax
Psw
Inputs
Stead
y-state Variab
les
Stead
y-state Pe
rform
ance
Inductor losses
Efficiency
Energy stored
Outputs
Dimension model
of arm
inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cifi
ctio
ns
Co
re t
op
olo
gy
IL2
Win
din
g
n1
n2
Electro
magn
etic V
ariable
s
Pe
rform
ance
η%
Thermal Model
Pcu
Pcore
Psw
IGBT
cooling
Inductor
cooling
Co
nve
rte
r lo
sses
Co
olin
g
Tem
peratu
re rise
ΔTIGBT
ΔTcore
ΔTcu
FEM
softwareLfem=Larm
Correction
coefficient
no
Finish
yes
Figure 2.11 Global analysis plan of MMC converter using finite element correction loop
56
Figure 2.11 shows the proposed plan of global analysis of MMC converter that employs the
finite element correction loop. This plan provides more accurate results compared to the
previous plan.
2.8. Optimal sizing of MMC converter
By increasing the demands of high power converters in medium and high power applications,
MMC converters were changing to complex, bulky and expensive structures. At this time,
mass minimization was emerged to reduce and optimize the volume of passive components
such as capacitors and inductors. In the case of MMC converter, the mass minimization
algorithm is dependent on the circuit operation, electromagnetic and thermal functionalities.
The passive components make the converter very bulky especially by increasing the number
of sub-modules in high voltage application. The most important goal of converter
optimization is to minimize the converter volume by optimal selection of passive components
regarding the technical and manufacturing constraints.
In this dissertation, three optimization plans are presented and investigated. The first plan is
to minimize the passive component value regardless of its mechanical volume and size. In
this plan, the inductance value, sub-module capacitor value, number of sub-modules per arm
and switching frequency are optimization variables that should be calculated by the solver.
The capacitor voltage ripple, total harmonic distortion, and maximum temperature rise are
the most important constraints. In this plan, the mechanical volume of passive components
is neglected. Hence, it is important to define a criterion which has relation to the mechanical
volume. The energy stored in the components is the best criterion to estimate the mechanical
volume. The electric energy stored in the capacitors and magnetic energy stored in the
inductors are chosen as goal function which must be minimized to achieve the optimal design.
Figure 2.12 shows the first plan of MMC optimization using energy stored in the converter.
57
MMC Circuit
Model and
Fault analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch lossesSp
eci
fict
ion
s
Co
nve
rte
r to
po
logy
N
.
.
.P
assi
ve
com
po
ne
nt
val
ue
M
IGB
T
Spe
cifi
cati
on
s Vmax
Imax
Psw
InputsStea
dy-state V
ariable
sStea
dy-state P
erfo
rman
ce
Inductor losses
Efficiency
Energy stored
Outputs
Non-linear
Solver
Total mass function
Capacitor energy
Csm
Const
rain
ts
Init
ial V
alues
Op
tim
izat
ion
par
am
eter
s
Inductor energy
Larm,M
Figure 2.12 First proposed optimization plan of MMC converter
The design of magnetic components is complex and depends on lots of parameters. It is not
possible to investigate and design an MMC converter without considering the inductor
topology and core and winding parameters. In the second plan, the analytical model of arm
inductances and capacitor volume estimation block are added to the optimization plan. The
core size and winding parameters are added to the optimization variables vector. Also, some
new constraints concerning the magnetic flux in the core are added to the constraints vector.
The most important change of the second plan in comparison with the first one is the goal
function. Utilizing the analytical model of inductor and capacitor mass function, the total
converter mass is chosen as main goal function. The optimization solver tries to minimize
the total converter mass by adjusting the optimization variables. Figure 2.13 shows the
second plan of optimization that employs the analytical model of inductors.
58
abcdg
Non-linear
Solver
Init
ial
Val
ues
Op
tim
izat
ion
par
am
eters
Total mass function
Capacitor mass
Csm
Co
nst
rain
ts
MMC Circuit
Model and
Fault analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch losses
Spe
cifi
ctio
ns
Co
nve
rte
r to
po
logy
N
.
.
.
Pas
sive
co
mp
on
en
t v
alu
eM
IGB
T
Spe
cifi
cati
on
s Vmax
Imax
Psw
Inputs
Stead
y-state Variab
les
Stead
y-state Pe
rform
ance
Inductor losses
Efficiency
Energy stored
Outputs
Dimension model
of arm
inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cifi
ctio
ns
Co
re t
op
olo
gy
IL2
Win
din
g
n1
n2
Electro
magn
etic V
ariable
s
Pe
rform
ance
η%
Thermal Model
Pcu
Pcore
Psw
IGBT
cooling
Inductor
cooling
Co
nve
rte
r lo
sses
Co
olin
g
Tem
peratu
re rise
ΔTIGBT
ΔTcore
ΔTcu
Figure 2.13 Second proposed optimization plan using analytical inductor model
59
MMC Circuit
Model and
Fault analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch losses
Spe
cifi
ctio
ns
Co
nve
rte
r to
po
logy
N.
.
.
Pas
sive
co
mp
on
en
t v
alu
e
M
IGB
T
Spe
cifi
cati
on
s Vmax
Imax
Psw
Inputs
Stead
y-state Variab
les
Stead
y-state Pe
rform
ance
Inductor losses
Efficiency
Energy stored
Outputs
Dimension model
of arm
inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cifi
ctio
ns
Co
re t
op
olo
gy
IL2
Win
din
g
n1
n2
Electro
magn
etic V
ariable
s
Pe
rform
ance
η%
Thermal Model
Pcu
Pcore
Psw
IGBT
cooling
Inductor
cooling
Co
nve
rte
r lo
sses
Co
olin
g
Tem
peratu
re rise
ΔTIGBT
ΔTcore
ΔTcu
FEM
softwareLfem=Larm
Correction
coefficient
no
Finish
yes
Init
ial
Val
ues
Op
tim
izat
ion
par
am
eters
Non-linear
Solver
Total mass function
Capacitor mass
Csm
Perf
orm
ance f
un
ctio
n,
dim
en
sio
n a
nd
co
nst
rain
ts
Figure 2.14 Third optimization plan includes the correction loop using FEM
The accuracy of inductor analytical model is not sufficient enough to be sure about the
optimization result especially in the case of inductor parameters. Hence, the analytical model
accuracy must be evaluated using a precise approach such as finite element method. In this
plan, a correction loop is designed to modify the analytical model parameters in each solution
step using finite element results. This approach increases the model accuracy while the
optimization time does not increase so much. Figure 2.14 shows the third plan of optimization
60
which includes the correction loop using finite element method. The analytical model
parameters are modified in each cycle regarding the FEM results.
2.9. Conclusion
In this chapter, the initial design of an MMC converter using the nominal load specifications
was presented. The IGBT selection, sub-module capacitor value and arm inductance sizing
were the most important point in MMC design procedure. Regarding the variety of
components in MMC topology, several analyses should be done to verify the converter
performance in terms of circuit functionality, magnetic performance, and thermal dissipation.
In order to achieve the reliable results, all analyses should be integrated into a unique model
as global analysis model which includes all technical variables and analysis tools.
Due to the high number of variables that should be determined and the number of technical
constraints, the optimization idea has been introduced and investigated. Three optimization
plans with a different degree of complexity were presented and studied. The inputs/outputs,
constraints, interconnections and goal function were introduced. Also, the method of
integration with a non-linear solver was represented.
In the next chapters, the analysis tools that were introduced in this chapter will be investigated
in detail. Also, the optimization plans will be employed to minimize the MMC converter
volume and mass with respect to the technical and manufacturing constraints. The
optimization result will be compared and discussed to find the accuracy of different models.
61
CHAPTER III
3 Circuit Model of Modular Multilevel AFE
3.1. Introduction
The average steady-state model of the converter is an analytical model, which widely used
in power electronic applications. Regardless to the conventional converters, there is a
circulating current in the MMC converter arms. The value of circulating current is dependent
on the operating point and component values. It makes a circular interaction between the
circuit variables such as arm current, capacitor voltage ripple, and component values. This
circular interaction made the MMC converter complex to analyze. Hence, the average steady-
state model of Modular Multilevel is a significant tool in order to estimate the parameter
values in a steady-state condition. The average steady-state model of Modular Multilevel
AFE is explained in this chapter. The circuit model investigates the multicell (series sub-
modules) and multi-leg (parallel arms). In order to simplify the analysis, the single leg
topology is investigated. In the multi-leg topology, the voltages are the same while the
currents are divided by the number of parallel arms.
62
Figure 3.1: MMC-based topology of the Multi-megawatt power supply of Fig.1.17
Figure 3.1 shows the MMC-based alternative topology of the multi-megawatt power supply
presented in Figure 1.17 and [75]. The design of the MMC Active Front End converter on
the left of Figure 3.1 will be investigated using the methodology proposed in this document
only.
3.2. Steady-State Average Model of Modular Multilevel Active-
Front-End Converter
The most popular circuit model of the converter which is employed by power electronic
designer is the average model. The switching function of IGBTs is a discrete function. The
average model uses the average switching function in the circuit model calculation. The
average model provides the converter electrical quantities in steady-state. It represents a
suitable approximation of converter variables in a steady-state condition. The accuracy of the
average model increases by increasing the switching frequency.
3.2.1. Sub-module circuit analysis
Figure 3.2 shows the half-bridge sub-module topology and the important sub-module
variables. Each converter arm is composed of the number of the series sub-module. Each
sub-module consists of two IGBTs and one capacitor as shown in figure 3.2. The sub-module
Transformer
Multilevel Multicell H-Bridge Converter
Load
(18M
W)
Gri
d
18KV, 50 Hz 18KV:2KV
SM1
SMN
SM1
SMN
..
SM1
SMN
SM1
SMN
..
SM1
SMN
SM1
SMN
..
Phase A
Phase B
Phase C
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
SM1
SMN
.
.
.
.
.
.
Multilevel Multicell Three-Phase Rectifier
SM1
SMN
SM1
SMN
..
SM1
SMN
SM1
SMN
..
SM1
SMN
SM1
SMN
..
Phase A
Phase B
Phase C
SM1
SMN
SM1
SMN
..
SM1
SMN
SM1
SMN
..
SM1
SMN
SM1
SMN
..
Phase A Phase B Phase C
63
current is divided between two IGBTs dependent on the switching function. According to the
KCL, the IGBTs currents are calculated as below:
𝐼𝑠𝑚(𝑡) = 𝐼𝑠1(𝑡) + 𝐼𝑠2(𝑡) (3.1)
𝐼𝑠1(𝑡) = −𝐼𝑐𝑢(𝑡) (3.2)
Icu S1
S2
Ism
C +Vsm
-
+Vc
-
D1
D2
Is2
Is1
Figure 3.2: half-bridge sub-module topology
3.2.2. Single phase average model parameters
Figure 3.3 shows the single-phase average model of modular multilevel AFE and the
important variables which should be calculated to solve the average model. Each arm is
modeled by the total sub-module voltage and arm inductance which could be coupled or not.
According to the KCL:
𝐼𝑎𝑢 = 𝐼𝑎 + 𝐼𝑎𝑙 (3.1)
{𝐼𝑐𝑖𝑟𝑐 = 𝐼𝑎𝑢 −
𝐼𝑎2
𝐼𝑐𝑖𝑟𝑐 = 𝐼𝑎𝑙 +𝐼𝑎2
(3.2)
64
Figure 3.3: Single phase average equivalent circuit
In order to calculate the precise values, it is important to estimate the circulating current. The
existing literature has proven that the most significant component of the circulating current
is the second-order harmonic [23].
𝐼𝑐𝑖𝑟𝑐 =𝐼𝑑𝑐3+ 𝐼𝑐𝑖𝑟𝑐
ℎ2 cos (2𝜔1𝑡 + 𝜃2) (3.3)
Where, 𝐼𝑑𝑐 is the current of DC side, 𝐼𝑐𝑖𝑟𝑐ℎ2 is the peak value of second order circulation
current. Thus, the upper arm current for a three-phase converter is defined as below:
{𝐼𝑎𝑢(𝑡) =
𝐼𝑑𝑐3+ 𝐼𝑐𝑖𝑟𝑐
ℎ2 𝑐𝑜𝑠(2𝜔1 + 𝜃2) +𝐼𝑎2𝑐𝑜𝑠(𝜔1𝑡 + 𝜃1)
𝐼𝑎𝑙(𝑡) =𝐼𝑑𝑐3+ 𝐼𝑐𝑖𝑟𝑐
ℎ2 𝑐𝑜𝑠 (2𝜔1 + 𝜃2) −𝐼𝑎2𝑐𝑜𝑠(𝜔1𝑡 + 𝜃1)
(3.4)
The upper and lower arm currents consist of DC source current divided by three, phase
current divided by two and the circulating current.
3.2.3. Average switching function
The internal sub-module variables such as current and voltage are dependent on the switching
function. In order to reduce the harmonics, the sinusoidal PWM switching is used. Generally,
Vus
.
.
R
L
R
L
M
Vls
Vdc/2
Vdc/2
Ia
Idc
Iau
Ial
Icirc
65
the switching function is not a pure sinusoidal waveform. If the number of sub-modules is
high enough or the switching frequency is high enough, the harmonic components in the
switching function can be ignored. Hence, the average switching waveform will be a
sinusoidal waveform. The upper and lower arm sub-module switching function will be
written as below:
{𝑆𝑎𝑢 =
1
2(1 − 𝑆𝑚𝐶𝑜𝑠 𝜔1𝑡)
𝑆𝑎𝑙 =1
2(1 + 𝑆𝑚𝐶𝑜𝑠 𝜔1𝑡)
(3.5)
where 𝑆𝑚 is the modulation index.
Considering to the half-bridge configuration, the current passes through the capacitor when
the lower switch is ON and the upper is off. The capacitor current waveform is calculated as
below:
{𝐼𝑐𝑢(𝑡) = 𝑆𝑎𝑢(𝑡)𝐼𝑎𝑢(𝑡)
𝐼𝑐𝑙(𝑡) = 𝑆𝑎𝑙(𝑡)𝐼𝑎𝑙(𝑡)
(3.6)
The sub-module capacitor current waveform consists of DC component, the main frequency
component, and second-order and third-order harmonics:
𝐼𝑐𝑢𝑑𝑐(𝑡) = 𝐼𝑐𝑙
𝑑𝑐(𝑡) =1
6𝐼𝑐𝑖𝑟𝑐𝑑𝑐 −
1
8𝑆𝑚𝐼𝑎𝑐𝑜𝑠𝜃1
(3.7)
𝐼𝑐𝑢ℎ1(𝑡) = −𝐼𝑐𝑙
ℎ1(𝑡)
= −1
6𝑆𝑚𝐼𝑐𝑖𝑟𝑐
𝑑𝑐 cos(𝜔1𝑡) +𝐼𝑎4cos(𝜔1𝑡 + 𝜃1)
−𝐼𝑐𝑖𝑟𝑐2𝑆𝑚𝑐𝑜𝑠(𝜔1𝑡 + 𝜃2)
(3.8)
𝐼𝑐𝑢ℎ2(𝑡) = 𝐼𝑐𝑙
ℎ2(𝑡) =𝐼𝑐𝑖𝑟𝑐2cos(2𝜔1𝑡 + 𝜃2) −
𝐼𝑎8𝑆𝑚𝑐𝑜𝑠(2𝜔1𝑡 + 𝜃1)
(3.9)
𝐼𝑐𝑢ℎ3(𝑡) = −𝐼𝑐𝑙
ℎ3(𝑡) = −𝐼𝑐𝑖𝑟𝑐4𝑆𝑚𝑐𝑜𝑠(3𝜔1𝑡 + 𝜃2)
(3.10)
The DC part of capacitor current must be zero in a steady-state condition. So,
66
𝜃1 = 𝑐𝑜𝑠−1(8𝐼𝑑𝑐6𝑆𝑚𝐼𝑎
) (3.11)
3.2.4. Circulating current and capacitor voltage ripple estimation
To calculate the capacitor voltage, it is sufficient to multiply the capacitor current by the
capacitor impedance. By eliminating the DC section of the result, the capacitor voltage ripple
is determined. The capacitor voltage ripple consists of main frequency, second-order and
third-order harmonics. By adding three terms of ripple, the total capacitor voltage ripple is
calculated for each sub-module. The importance of capacitor voltage ripple comes from the
limitation of semiconductor rating. Finding the maximum capacitor voltage is a criterion to
select the semiconductor switch. On the other hand, to find the precise losses function, the
capacitor voltage ripple plays an important role [21], [22].
∆𝑉𝑐𝑢ℎ1(𝑡) = −∆𝑉𝑐𝑙
ℎ1(𝑡)
=−𝑆𝑚𝐼𝑐𝑖𝑟𝑐
𝑑𝑐
6𝜔1𝐶𝑠𝑚sin(𝜔1𝑡) +
𝐼𝑎4𝜔1𝐶𝑠𝑚
sin(𝜔1𝑡 + 𝜃1)
−𝑆𝑚𝐼𝑐𝑖𝑟𝑐2𝜔1𝐶𝑠𝑚
sin (𝜔1𝑡 + 𝜃2)
(3.12)
∆𝑉𝑐𝑢ℎ2(𝑡) = ∆𝑉𝑐𝑙
ℎ2(𝑡)
=𝐼𝑐𝑖𝑟𝑐
4𝜔1𝐶𝑠𝑚sin(2𝜔1𝑡 + 𝜃2) −
𝐼𝑎𝑆𝑚16𝜔1𝐶𝑠𝑚
sin(2𝜔1𝑡 + 𝜃1)
(3.13)
∆𝑉𝑐𝑢ℎ3(𝑡) = −∆𝑉𝑐𝑙
ℎ3(𝑡) =−𝐼𝑐𝑖𝑟𝑐𝑆𝑚12𝜔1𝐶𝑠𝑚
sin (3𝜔1𝑡 + 𝜃2) (3.14)
Where ∆𝑉𝑐𝑢ℎ1, ∆𝑉𝑐𝑢
ℎ2, ∆𝑉𝑐𝑢ℎ3 are first, second and third voltage harmonics of capacitor. It should
be noted that the DC part of sub-module capacitors in steady state condition is 𝑉𝑑𝑐
𝑚, where 𝑚
is the number of series sub-modules per arm.
The phase voltage ripple is the summation of all sub-module capacitors voltage ripple
multiplied by the switching function. The phase voltage ripple is composed of the pair
harmonics and the odd harmonics were eliminated. The output voltage ripple consists of DC
component, second-order and fourth order harmonics [24].
67
∆𝑉𝑎𝑟𝑚 = [𝑚
2−𝑚
2𝑆𝑚𝑐𝑜𝑠(𝜔1𝑡)] (∆𝑉𝑐𝑢
ℎ1 + ∆𝑉𝑐𝑢ℎ2 + ∆𝑉𝑐𝑢
ℎ3)
+ [𝑚
2+𝑚
2𝑆𝑚𝑐𝑜𝑠(𝜔1𝑡)] (∆𝑉𝑐𝑙
ℎ1 + ∆𝑉𝑐𝑙ℎ2 + ∆𝑉𝑐𝑙
ℎ3)
(3.15)
The voltage ripple function is a valuable analytical function which is important to estimate
the grid-side THD value. This function is used to estimate the THD constraint in the
optimization procedure or it could be an appropriate criterion to design the external filter.
∆𝑉𝑎𝑟𝑚𝑑𝑐 = −
𝑚𝑆𝑀𝐼𝑎8𝜔1𝐶𝑠𝑚
𝑠𝑖𝑛𝜃1 +𝑚𝑆𝑚
2 𝐼𝑐𝑖𝑟𝑐4𝜔1𝐶𝑠𝑚
𝑠𝑖𝑛𝜃2 (3.16)
∆𝑉𝑎𝑟𝑚ℎ2 =
𝑚𝑆𝑚2 𝐼𝑐𝑖𝑟𝑐
𝑑𝑐
12𝜔1𝐶𝑠𝑚sin(2𝜔1𝑡) −
3𝑚𝑆𝑚𝐼𝑎16𝜔1𝐶𝑠𝑚
sin(2𝜔1𝑡 + 𝜃1)
+𝑚𝐼𝑐𝑖𝑟𝑐(1 +
76 𝑆𝑚
2 )
4𝜔1𝐶𝑠𝑚sin (2𝜔1𝑡 + 𝜃2)
(3.17)
∆𝑉𝑎𝑟𝑚ℎ4 =
𝑚𝑆𝑚2 𝐼𝑐𝑖𝑟𝑐
24𝜔1𝐶𝑠𝑚sin (4𝜔1𝑡 + 𝜃2)
(3.18)
It is noted that the circulating current consists of second-order harmonic. Therefore, to
estimate the circulating current (𝐼𝑐𝑖𝑟𝑐), the second-order harmonic of the voltage is utilized:
𝐼𝑐𝑖𝑟𝑐 cos(2𝜔1𝑡 + 𝜃2) =∆𝑉𝑎𝑟𝑚
ℎ2
−𝑗2𝜔12(𝐿 −𝑀)=
∆𝑉𝑎𝑟𝑚ℎ2
−𝑗4𝜔1(𝐿 − 𝑀)
(3.20)
According to Eq.(3.17), the circulating current 𝐼𝑐𝑖𝑟𝑐 is calculated as below:
𝐼𝑐𝑖𝑟𝑐 cos(2𝜔1𝑡 + 𝜃2) (1 −𝑚(1 +
76𝑆𝑚
2 )
16𝜔12(𝐿 + 𝑀)𝐶𝑠𝑚
)
=𝑚𝑆𝑚
2 𝐼𝑐𝑖𝑟𝑐𝑑𝑐
96𝜔12(𝐿 + 𝑀)𝐶𝑠𝑚
cos(2𝜔1𝑡)
−3𝑚𝑆𝑚𝐼𝑎
64𝜔12(𝐿 + 𝑀)𝐶𝑠𝑚
cos(2𝜔1𝑡 + 𝜃1)
(3.21)
By solving the above equation, the magnitude and phase of the circulating current are
obtained as below:
𝐼𝑐𝑖𝑟𝑐 =√(𝐵𝑐𝑜𝑠𝜃1 + 𝐶)2 + (𝐵𝑠𝑖𝑛𝜃1)2
1 − 𝐴
(3.22)
< 𝜃2 = atan (𝐵𝑠𝑖𝑛𝜃1
𝐵𝑐𝑜𝑠𝜃1 + 𝐶)
(3.23)
Where
68
{
𝐴 =
𝑚(1 +76 𝑆𝑚
2 )
16𝜔12(𝐿 −𝑀)𝐶𝑠𝑚
𝐵 =−3𝑚𝑆𝑚𝐼𝑎
64𝜔12(𝐿 −𝑀)𝐶𝑠𝑚
𝐶 =𝑚𝑆𝑚
2 𝐼𝑐𝑖𝑟𝑐𝑑𝑐
96𝜔12(𝐿 −𝑀)𝐶𝑠𝑚
(3.24)
According to Eq.(3.22), there is a point of discontinuity. On the other word, the circulating
current goes to infinitive value if 𝐴 = 1. Hence, a new constraint is added to avoid the
instability. The capacitor and arm inductance values should be adjusted as below to avoid the
instability.
𝐶𝑠𝑚(𝐿 −𝑀) ≠𝑚(1 +
76𝑆𝑚
2 )
8𝜔12
(3.25)
3.2.5. Advantages and disadvantages of steady-state average model of
MMC Active-Front-End
Because of the circular interaction, MMC converter analysis is complicated behavior.
Generally, the numerical solvers are used to simulate and analyze the MMC converters. This
is a complex and time-consuming analysis approach which is not suitable for component
selection and optimization. The average model provides the significant information of the
converter which could be very helpful for power electronic designers. Also, it could be an
appropriate model to use optimization loop. On the other hand, the average steady-state
model neglects the switching frequency effect which is important especially in high power
application which utilizes low switching frequency. The most important weakness of the
average model is in the harmonic calculation.
69
3.3. Time-domain steady-state model of Modular Multilevel
Active-Front-End
The average model neglects the effect of switching function and switching frequency. High
power converters use low switching frequency. Hence, neglecting the switching frequency
leads to diminishing the model accuracy. The time-domain steady-state model is presented
in order to resolve the average model weakness and increase the accuracy. It estimates the
waveform of the important parameters considering the real switching function. Using time-
domain model, it is possible to calculate the more accurate harmonic and ripple values.
3.3.1. Sub-module switching function
The time -domain steady-state model employs real sub-module switching function which is
a discrete function. The switching function could be written as below:
1 2
1 2
1 , , ,( )
0 , , 0, 0
sm cu u cu
sm cu
S ON S OFF V V I IS t
S OFF S ON V I
(3.26)
The sinusoidal PWM (SPWM) is utilized to reduce the harmonic. By comparing a sinusoidal
waveform with a high-frequency sawtooth waveform, the switching function is generated.
3.3.2. Time-domain state equations
In order to analyze the converter in time-domain, the state equations must be extracted. Using
the state equations, it is possible to calculate all converter circuit parameters in the time
domain. The first step is to calculate the sub-module capacitor current from the arm current.
{𝐼𝑐𝑢𝑖 = 𝑆𝑢𝑖(𝑡)𝐼𝑎𝑢(𝑡)
𝐼𝑐𝑙𝑖 = 𝑆𝑙𝑖(𝑡)𝐼𝑎𝑙(𝑡)
(3.27)
In each sub-module, the capacitor current and capacitor voltage could be calculated using
switching function (𝑆𝑢 , 𝑆𝑙).
70
{
𝑉𝑐𝑢𝑖 =1
𝐶𝑠𝑚∫𝑆𝑢𝑖(𝑡)𝐼𝑎𝑢(𝑡)
𝑉𝑐𝑙𝑖 =1
𝐶𝑠𝑚∫𝑆𝑙𝑖(𝑡)𝐼𝑎𝑙(𝑡)
(3.28)
Also, using the capacitor sub-modules and the switching function, it is possible to determine
the sub-module terminal voltage. Also, the total sub-module voltage is the summation of all
sub-modules voltage in each arm.
{𝑉𝑢𝑖(𝑡) = (1 − 𝑆𝑢𝑖(𝑡))𝑉𝑐𝑢𝑖(𝑡)
𝑉𝑙𝑖(𝑡) = (1 − 𝑆𝑙𝑖(𝑡))𝑉𝑐𝑙𝑖(𝑡)
(3.29)
{
𝑉𝑢𝑇(𝑡) = ∑𝑉𝑢𝑖(𝑡)
𝑁
𝑖=1
𝑉𝑙𝑇(𝑡) = ∑𝑉𝑙𝑖(𝑡)
𝑁
𝑖=1
(3.30)
According to KVL, the arm inductance voltage is calculated versus total sub-module voltage
and phase voltage of the source. The upper and lower arm inductance are calculated versus
the inductance voltages (𝑉𝐿) and inductance (𝐿) and coupling factor (𝑀). The calculated
currents are utilized in the next step of the numerical solution approach.
{𝑉𝐿𝑢(𝑡) =
𝑉𝑑𝑐2− 𝑉𝑎(𝑡) − 𝑉𝑢𝑇(𝑡) + 𝑉𝑛
𝑉𝐿𝑙(𝑡) =𝑉𝑑𝑐2− 𝑉𝑙𝑇(𝑡) + 𝑉𝑎(𝑡) − 𝑉𝑛
(3.31)
{
𝐼𝑢(𝑡) =
𝐿2
𝐿2 +𝑀2∫𝑉𝐿𝑢(𝑡)𝑑𝑡 +
𝑀2
𝐿2 +𝑀2∫𝑉𝐿𝑙(𝑡)𝑑𝑡
𝐼𝑙(𝑡) =𝐿2
𝐿2 +𝑀2∫𝑉𝐿𝑙(𝑡)𝑑𝑡 +
𝑀2
𝐿2 +𝑀2∫𝑉𝐿𝑢(𝑡)𝑑𝑡
(3.32)
By elimination of 𝐼𝑐𝑖𝑟𝑐, the input current of phase A is calculated,
𝐼𝑎(𝑡) = 𝐼𝑢(𝑡) − 𝐼𝑙(𝑡) (3.33)
3.3.3. Proposed time-domain model
Utilizing the state equations, a time domain model will be developed in order to use as an
analytical model. In order to simulate the converter in a steady-state condition, the state
71
variables should be initialized. There is a possibility to initialize the state variables using
average steady-state model. Figure 3.4 shows the diagram using an average steady-state
model which is used to initialize the state variables.
Average Switching Functions
Average Switching Functions
IaIa
IcircIcirc
+
±
Iau,Ial
11+
- ×
× Submodule terminal voltages
Submodule terminal voltages
Capacitor Currents
Capacitor Voltages
Capacitor Voltages
Total upper sub. VoltagesTotal lower sub. Voltages
Total upper sub. VoltagesTotal lower sub. Voltages
Upper inductor voltageLower inductor voltage
Upper inductor voltageLower inductor voltage
Figure 3.4: initializing diagram using average steady-state model
The initial value of sub-module capacitors and arm inductances are estimated using average
steady-state model. This is a useful tool to eliminate the transient part of the simulation.
Figure 3.5 shows the complete analytical model which consists of the average model to
initialize and the state space model to simulate. The most important advantages of the
proposed model are:
1- Fast initialization
2- Using real SPWM switching function
3- Accurate steady-state simulation using state variables
4- Harmonic investigation
5- Investigation of voltage and current ripple
6- Accurate semiconductor losses calculation
72
Figure 3.5: Diagram of Time-domain analytical model
3.4. Steady-State Model Verification using Simulink
An important section of introducing a model is to validate and verify the model accuracy.
The proposed model is implemented using visual basic programming code in Excel. The
Microsoft Excel provides an appropriate graphic space which simplifies the modeling and
parameter adjusting. The Excel cells support mathematical functions. Also, it connects to the
visual basic code sheets. In order to validate the outputs of the proposed model, the MMC
converter is simulated using Simulink/Matlab.
Table 3.1 The MMC converter parameters and operating point
Parameter Value Parameter Value
Nominal Power 2.5 MW Number of module per arm 6
AC Line Voltage 2000 V Number of parallel arm 1
DC Link Voltage 5000 V Sub-module capacitor 11.1 mF
Power Factor 1 Arm inductance 4.22 mH
Nominal Frequency 50 Hz Switching frequency 755 Hz
Load SpecificationPout, Vac,Iac,cosɸ,Vdc,...
InitilizationInductance, Capacitor, Modulation Index,...
Estimation of required parametersCirculation current, line current, dc current
Calculation of submodule Capacitor current and Voltage
𝑉𝑐𝑢𝑡𝑜𝑡𝑎𝑙 =∑𝑉𝑐𝑢 𝑖
𝑚
𝑖=1
𝑉𝑐𝑙𝑡𝑜𝑡𝑎𝑙 =∑𝑉𝑐𝑙 𝑖
𝑚
𝑖=1
Calculation of arm currents
𝐼𝑢𝐼𝑙
= 𝐿 −1 𝑉𝑢 𝑡𝑜𝑡𝑎𝑙𝑉𝑙 𝑡𝑜𝑡𝑎𝑙
Calculation of line Current, THD and capacitor voltage
ripple
𝐼𝑙𝑖𝑛𝑒, THD, ∆𝑉𝑐
Calculation of Losses, Efficiency and Energy
Stored in the converter components
𝑆𝑎𝑢𝑆𝑎𝑙𝑓𝑠𝑤
73
Table 3.1 shows the MMC converter parameters and the operating point which is used in the
simulation. The simulation has been done using Simulink/Matlab and Excel/VB
programming. Then the output waveforms are compared. The sub-module capacitor currents
simulated by Excel vb code and Simulink model have been compared in figure 3.6. The open
loop controller using SPWM switching function has used.
a) Visual basic code
b) Simulink/Matlab
Figure 3.6: The sub-module capacitor current using analytical model and Simulink
Figure 3.7 shows the sub-module capacitor voltage waveforms which have obtained from the
analytical model and Simulink model. The comparison proves the analytical model accuracy.
Figures 3.8 and 3.9 show the upper/lower arm currents and the input line current respectively.
Also, the THD calculation proves the analytical model accuracy again. Using Simulink, the
THD value of the line current is about 0.7% while the analytical model estimates 0.64%.
a) Visual Basic code
b) Simulink/Matlab
Figure 3.7: The sub-module capacitor voltage using analytical model and Simulink
74
a) Visual basic code
b) Simulink/Matlab
Figure 3.8: The lower and upper arm current using analytical model and Simulink
a) Visual basic code
b) Simulink/Matlab
Figure 3.9: The input line current using analytical model and Simulink
3.5. Conclusion
In this chapter, a circuit model of MMC converter was investigated and developed. The
average model was presented as most popular converter model. In high power application,
the low switching frequency is used to avoid high switch losses. The accuracy of average
switching function is suitable for high switching frequencies. Therefore, the accuracy of the
average model is not sufficient enough in MMC converters, especially in low switching
frequency. In the high power applications, the low switching frequency is utilized in order to
minimize the switching losses. To enhance the model accuracy, a time-domain circuit model
was introduced which employs the real switching function. The proposed model provides
75
high accuracy results especially in term of harmonic analysis. The accuracy of the proposed
model was evaluated using Simulink/MATLAB.
In the next chapter, the electromagnetic analytical model of arm inductances will be
presented. The dimensioning analysis of passive components will be investigated and
formulated.
76
CHAPTER IV
4 Electromagnetic and Dimensioning Analysis of
Passive Components
4.1. Introduction
In chapter 4, the electromagnetic arm inductance and dimensioning analysis of the passive
components are presented. In this chapter, the mass function of the main components is
extracted in order to achieve the total converter mass function. The mass function of IGBTs
and their cooling system could be extracted from the datasheet. The mass function of the
capacitor is related to the capacitance, capacitor nominal voltage, and the capacitor type.
Regarding the products of a manufacturer, an analytical equation is fitted to estimate the
capacitor mass value. In term of arm inductance, the dimensioning model will be a part of
the electromagnetic analysis model.
In this chapter, the analytical electromagnetic model of arm inductances is reviewed and
investigated based on the electrical variables of the MMC converter. The electromagnetic
model should calculate the magnetic quantities such as inductance, magnetic flux, core and
winding losses regarding the core size and winding parameters.
4.2. Capacitor Dimensioning Analysis
In this section, the mass function of power capacitor is extracted. Also, the losses function of
the capacitor is calculated.
77
4.2.1. Capacitor Mass Function
Capacitors are industrial components which are designed and manufactured in the standard
packages and sizes. Power electronic designer generally utilizes the standard capacitors in
their applications. Therefore, the mass function must be extracted based on the product
datasheets. In this research, the cylindrical capacitors of VISHAY Company are chosen.
Figure 4.1 Cylindrical capacitor of VISHAY Company designed for power electronic applications
Figure 4.1 shows the cylindrical capacitor of VISHAY Company designed for power
electronic applications. The nominal voltage of capacitors is between 880 V and 2200 V.
Also, the capacitances are from 30 μF to 1000 μF.
The most important variables which affect the capacitor mass are the capacitance and
maximum voltage. Figure 4.2 shows the contour of capacitor mass versus the capacitance
and maximum voltage according to the product's datasheet. A second-order linear function
has been fitted to the points which are obtained from company datasheet. Table 4.1 shows
the function coefficients which are calculated using curve fitting algorithm.
𝑊𝑒𝑖𝑔ℎ𝑡 = 𝑝00 + 𝑝10𝑉𝑑𝑐 + 𝑝01𝐶 + 𝑝20𝑉𝑑𝑐2 + 𝑝02𝐶
2 + 𝑝11𝑉𝑑𝑐𝐶 (4.1)
78
Figure 4.2: Capacitor weight versus the capacitance and maximum voltage value
Table 4.1: Calculated coefficients using fitting algorithm
Function coefficient Value
𝑝00 1.005
𝑝10 -0.00119
𝑝01 -3550
𝑝20 3.87E-07
𝑝02 43200
𝑝11 5.27
4.2.2. Transient Equivalent Model of Capacitor
Various electrical models have been represented to simulate the electrical and thermal
behavior of the capacitor. In this research, two different models are used with different
complexity.
79
Model 1: the simplest model that was presented for an electrical model of the capacitor is
shown in figure 4.3. The capacitance 𝐶1 represents total capacitance between the anode and
the cathode terminals. Resistance 𝑅𝑎 includes different terms: terminal resistance, tabs
resistance, foils resistance, resistance of the impregnated electrolyte paper, dielectric
resistance, and tunnel-electrolyte resistance. 𝑅𝑐 represents the leakage current that depends
on the quality of the dielectric material. Inductance 𝐿𝑐 is especially dominated by the loop
area from the terminals and tabs outside of the active winding. Resistance 𝑅𝑏 is essential to
have in a physical representation of the model [63, 64]. The Laplace formulation
corresponding to this model is:
𝑍1(𝑠)
=(𝑅𝑎 + 𝑅𝑏)𝑅𝑐𝐿c𝐶1𝑠
2 + (𝑅𝑎𝑅𝑏𝑅𝑐𝐶1 + 𝐿c𝑅𝑐 + 𝐿c𝑅𝑏 + 𝐿c𝑅𝑎)𝑠 + (𝑅𝑎 + 𝑅𝑐)𝑅𝑐𝑅𝑐𝐿c𝐶1𝑠2 + (𝑅𝑏𝑅𝑐𝐶1 + 𝐿c)𝑠 + 𝑅𝑏
(4.2)
Rc
C1 Lc
Rb
Ra
Figure 4.3 The simple model of high power capacitor
Model 2: the simple model is not accurate especially by changing the capacitor temperature.
In order to enhance the model accuracy throughout the temperature range, a modified
capacitor model has been proposed [63, 64]. Figure 4.4 represents the modified capacitor
model that provides more accurate results in a wide range of temperature.
80
Rc
C1 Lc
Rb
RaC2
R2a
R2c
Figure 4.4 The modified high power capacitor model
The element values are identified at the maximum temperature of the component for which
the impedance is weakest (85℃). The elements (𝐶2, 𝑅2𝑎 , 𝑅2𝑐), are added in series to this
model in order to take into account the influence of the reduction of temperature and the
shape of the curve Z versus frequency.
The Laplace function corresponding to the modified capacitor model is:
𝑍2(𝑠) = 𝑍1(𝑠) + 𝑍02(𝑠) (4.3)
Where
𝑍02(𝑠) =𝐶2𝑅2𝑎 + 𝑅2𝑐
𝐶2(𝑅2𝑎 + 𝑅2𝑐)𝑠 + 1
(4.4)
4.3. Dimensioning Analysis of Inductor
The dimension analysis is done to provide the mass function of arm inductance. This analysis
is based on the analytical model of the inductor.
4.3.1. Core Topologies
Various core topologies which are suitable for MMC converter application are investigated.
Each topology provides some advantages and some disadvantages. In the case of coupled
inductor, two basic core topologies are utilized. Figure 4.5 shows two core topologies that
bring suitable technical and manufacturing properties.
81
a) Independent mutual inductance (type 1) b) Dependent mutual inductance (type 2)
Figure 4.5: The proposed core topologies for independent and dependent mutual inductances
The first topology provides the possibility to determine the coupling factor 𝑀 independently,
while the final mass will be higher because of the center yoke. In the second topology, the
coupling factor is not an independent variable. The coupling factor will be so close to one
regarding to the leakage flux. Instead, it has lower mass in compare with the first one. In this
application, the second topology is chosen to minimize the total converter mass.
4.3.2. Magnetic Equivalent Circuit (type 1)
The magnetic equivalent circuit provides important magnetic variables of inductance core.
The first step is to calculate the core reluctances. Figure 4.6 shows the equivalent magnetic
circuit of inductance with the core type 1.
N1I1 N2I2
Rag1 Rag1Rag2
R1 R2
R3
R5R4
Φ1 Φ2
Figure 4.6 The equivalent magnetic circuit of inductance (type 1)
d
b
a
d/2
lg1lg2lg1
a
b
lg
d
82
If 𝑅1 = 𝑅2and 𝑅4 = 𝑅5 then, the total reluctance value of each winding based on the
geometry in figure 4.5.a is calculated as below:
𝑅𝑇 = (𝑅1 + 𝑅𝑎𝑔1 + 𝑅4) + ((𝑅3 + 𝑅𝑎𝑔2)‖(𝑅2 + 𝑅5 + 𝑅𝑎𝑔1)) (4.5)
where
𝑅1 = 𝑅2 =2(𝑎 +
32𝑑 + 𝑏)
𝜇0𝜇𝑟 𝑐 𝑑
(4.6)
𝑅3 =𝑏 − 𝑙𝑔2𝜇0𝜇𝑟 𝑐 𝑑
(4.7)
𝑅4 = 𝑅5 =2(𝑎 + 𝑑)
𝜇0𝜇𝑟 𝑐 𝑑
(4.8)
𝑅𝑎𝑔1 =2𝑙𝑔1𝜇0 𝑐 𝑑
(4.9)
𝑅𝑎𝑔2 =2𝑙𝑔2𝜇0𝑐 𝑑
(4.10)
To calculate the magnetic flux that passes through the windings,
𝛷1 =𝑁1𝐼1𝑅𝑇
(4.11)
Φ2 =𝑁2𝐼2𝑅𝑇
(4.12)
Φ12 = Φ21 =Φ1(𝑅3+𝑅𝑙𝑔2)
𝑅2 + 𝑅5 + 𝑅𝑙𝑔1
(4.13)
Also, the magnetic flux density is calculated as below:
𝐵 =𝜙1 + 𝜙12
𝐴𝑐=(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑁1𝐼1 + (𝑅3+𝑅𝑙𝑔2)𝑁1𝐼1
𝑐. 𝑑(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑅𝑇/2
(4.14)
4.3.3. Magnetic Equivalent Circuit (type 2)
The magnetic circuit analysis is done to calculate the important parameters such as flux, flux
density, reluctance and etc. Figure 4.7 shows the core topology and parameters of the core.
The core reluctance is calculated as below:
𝑅𝑐 =𝑙𝑐
𝜇0𝜇𝑟𝐴𝑐
(4.15)
83
where 𝑙𝑐 is the magnetic core length, 𝐴𝑐 is the cross-sectional area of the core, 𝜇0 is the
vacuum permeability and 𝜇𝑟 is the relative magnetic permeability of the material. The
magnetic core length and cross-sectional area are computed as below:
𝑙𝑐 = 2𝑎 + 2𝑏 + 4𝑑 (4.16)
𝐴𝑐 = 𝑐𝑑 (4.17)
where 𝑎 is the width of the core window, 𝑏 is the height of the core window, 𝑐 is the core
depth and 𝑑 is the core width. The total reluctance is the summation of core reluctance and
airgap reluctance. The airgap reluctance and total reluctance are calculated as below:
𝑅𝑎𝑔 =2𝑙𝑔𝜇0𝐴𝑐
(4.18)
𝑅𝑇 = 𝑅𝑐 + 𝑅𝑎𝑔 =𝑙𝑐 + 2𝜇𝑟𝑙𝑔𝜇0𝜇𝑟𝐴𝑐
(4.19)
where 𝑙𝑔 is the airgap length.
ab
lg
d
c
lc
Figure 4.7: The inductor core topology and sizing parameters
The magnetic flux which passes through the core and the magnetic flux density is calculated
as below:
𝜙 =𝑛1𝐼1 + 𝑛2𝐼2
𝑅𝑇
(4.20)
𝐵 =𝜙
𝐴𝑐=𝜇0𝜇𝑟(𝑛1𝐼1 + 𝑛2𝐼2)
(𝑙𝑐 + 2𝜇𝑟𝑙𝑔)
(4.21)
84
4.3.4. Inductance and Resistance Estimation
According to the magnetic flux value that was calculated in the previous section, the
inductance value and the mutual inductance are:
𝐿11 =𝜙1𝐼1=𝑛12
𝑅𝑇
(4.22)
𝐿12 = 𝐿21 =𝜙11𝐼2
(4.23)
The winding resistance value is calculated as below:
𝑅𝑑𝑐 =𝜌𝑐𝑢𝐽
𝐼𝑟𝑚𝑠1(2𝑎 + 2𝑑 + 2𝑐)𝑛1
(4.24)
where 𝜌𝑐𝑢 is the copper resistivity, 𝐽 is the current density.
In the analytical model of the inductor, the most complex parameter to compute is the mutual
inductance. In the case of center column topology, the magnetic flux which generated by the
winding is divided between two magnetic paths. It helps to estimate the coupling factor. In
the centerless core topology, the magnetic flux that passes through the first and second
windings are the same. Hence, the self-inductance and the mutual inductance are equal. On
the other words, the coupling factor is equal to one. In reality, there is some flux leakage in
the inductor structure and the coupling factor will be always less than one. The flux leakage
which is dependent on the material, shape, and size of the electromagnetic core. To estimate
the coupling factor in for this application, a simulation has done using electromagnetic
analysis software. Figure 4.8 shows the magnetic flux lines and leakage flux lines. In this
simulation, 91% percent of the flux generated by first winding passes through the second
winding. Hence, in this model, the mutual inductance is a function of self-inductance (𝐿12 =
𝐾𝑚𝑢𝐿11), in this research 𝐾𝑚𝑢 = 0.9 that could be estimated using electromagnetic analysis
software.
85
Figure 4.8 Magnetic flux lines in the core and leakage flux lines
The most important constraints of the inductor are the maximum flux density and the core
window area. Each magnetic material has a maximum flux density to avoid the saturation.
The core flux density must be lower than these values in the case of utilizing center column
and centerless core topology, respectively.
{
𝐵𝑚𝑎𝑥 ≥
(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑛1𝐼1 + (𝑅3+𝑅𝑙𝑔2)𝑛1𝐼1𝑐. 𝑑(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑅𝑇
𝐵𝑚𝑎𝑥 ≥𝑛(𝐼1 + 𝐼2)𝜇0𝜇𝑟𝑙𝑐 + 2𝜇𝑟𝑙𝑔
(4.25)
The core window area and total winding cross-section are calculated as below:
𝐶𝑜𝑟𝑒 𝑤𝑖𝑛𝑑𝑜𝑤 𝑎𝑟𝑒𝑎 = 𝑎𝑏 (4.26)
𝐴𝑤𝑇 = 𝑛𝐼𝑟𝑚𝑠𝐽
(4.27)
Also, the core window area should be greater than the total winding cross-section. In the first
type topology, each winding is placed in a window, while in the second topology, the core
window must consist of two windings.
{
𝑘𝑤 ∗ 𝑎 ∗ 𝑏 >𝑛 ∗ 𝐼𝑟𝑚𝑠
𝐽
𝑘𝑤 ∗ 𝑎 ∗ 𝑏 >2 ∗ 𝑛 ∗ 𝐼𝑟𝑚𝑠
𝐽
(4.28)
where 𝑘𝑤 is the filling factor of the winding.
86
4.3.5. Volume and Mass Function
Regarding the analytical model, a mass function is proposed to estimate the total inductor
mass dependent on the core and winding size. The inductor core volume is calculated for two
different core topologies, respectively as below:
{𝑉𝑐𝑜𝑟𝑒 = 𝑐(2𝑎 + 3𝑑)(2𝑑 + 𝑏) − 2𝑎𝑏 − (𝑙𝑔2 − 𝑙𝑔1)𝑐𝑑
𝑉𝑐𝑜𝑟𝑒 = 𝑐 ((𝑎 + 2𝑑)(𝑏 − 𝑙𝑔 + 2𝑑) − 𝑎𝑏)
(4.29)
The copper volume is dependent on the winding turn number and wire cross-section.
Regarding the inductor topology, the copper volume for each winding is:
{
𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑛(4𝑎 + 2𝑐 + 2𝑑)𝐴𝑟𝑚𝑠𝐽
𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑛(4𝑎 + 2𝑐 + 𝑑)𝐴𝑟𝑚𝑠𝐽
(4.30)
Also, considering the volumetric mass density, the total inductor mass is obtained:
𝑀𝑖𝑛𝑑 = 𝐷𝑐𝑜𝑟𝑒𝑉𝑐𝑜𝑟𝑒 + Dwinding𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 (4.31)
4.4. Inductor Thermal Analysis
Thermal analysis of inductor is important to calculate the final inductor size. The inductor
loss consists of core losses and winding losses as heat source increase the inductor
temperature. In addition to the magnetic analysis, a thermal analysis should be done to
determine the thermal dissipation and maximum temperature rise in the inductor material.
4.4.1. Inductor Losses
The inductor losses are divided to core losses and copper losses. The core loss consists of
hysteresis and eddy current losses. Based on the literature, the hysteresis and eddy current
losses are calculated as below:
𝑃ℎ = 𝐾ℎ𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝐵ℎ1.𝑚𝑎𝑥α + 2𝑓𝑠𝐵ℎ2.𝑚𝑎𝑥
α ) (4.32)
𝑃𝑒 = 𝐾𝑒𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝛽𝐵ℎ1.𝑚𝑎𝑥γ
+ (2𝑓𝑠)β𝐵ℎ2.𝑚𝑎𝑥
γ) (4.33)
where 𝐾ℎ is the hysteresis constant, 𝐾𝑒 is the eddy current constant.
87
The electric resistivity of copper augments by increasing its temperature. Hence, the copper
losses will increase. The copper resistivity versus copper temperature is written as below:
𝜌𝑐𝑢 = 𝜌0(1 + 휀25(𝑇𝑐𝑢 − 25)) (4.34)
where 𝜌0 is the copper resistivity in 25℃. On the other hand, the copper losses and in results
the copper temperature depends on the electric resistivity. It makes a circular interaction
which makes it complex to compute. Regarding to our goal which is to achieve the minimum
inductor mass, the copper temperature that is used to compute the copper resistivity is
considered as maximum copper temperature in order to compute the worst-case copper
losses. Therefore, Eq.(4.24) is rewritten as below:
𝜌𝑐𝑢 = 𝜌0(1 + 휀25(𝑇𝑐𝑢𝑚𝑎𝑥 − 25)) (4.35)
where 𝑇𝑐𝑢𝑚𝑎𝑥 is the maximum copper temperature which can be determined regarding to
some technical limitations such as isolation class. The copper losses of the winding depend
on the winding resistance and effective current of the winding. According to the skin effect
of the conductors, the copper resistance depends on the current frequency. The copper
resistance will augment by increasing the frequency of the current. The copper losses can be
calculated regarding to the Superposition Principle in the linear circuit. The inductor current
is consisted of DC current, main component (50 Hz), second-order component (100 Hz) and
fourth-order component (200 Hz). Hence, the copper loss for each winding is calculated as
below:
𝑃𝑐𝑢 = 𝑅𝑑𝑐𝐼𝑑𝑐2 + 𝑅𝑎𝑐1𝐼𝑟𝑚𝑠ℎ1
2 + 𝑅𝑎𝑐2𝐼𝑟𝑚𝑠ℎ22 + 𝑅𝑎𝑐4𝐼𝑟𝑚𝑠ℎ4
2 (4.36)
The winding DC resistance is calculated as below:
𝑅𝑑𝑐 =𝑛 𝜌0(1 + 휀25(𝑇𝑐𝑢𝑚𝑎𝑥 − 25)).MLT
𝐴𝑐𝑢
(4.37)
where 𝑀𝐿𝑇 is length of winding and 𝐴𝑐𝑢 is the conductor cross section.
In order to calculate the AC resistance, several equations have been presented in the literature
[65-68]. The most popular equation which calculates the AC resistance of round wires based
on the zero-Kelvin function is [67]:
𝑅𝑎𝑐 =√2
𝜋𝑑𝑐𝜎𝛿
𝑏𝑒𝑟(𝑞)𝑏𝑒𝑖′(𝑞) − 𝑏𝑒𝑖(𝑞)𝑏𝑒𝑟′(𝑞)
[𝑏𝑒𝑟′(𝑞)]2 + [𝑏𝑒𝑖′(𝑞)]2
(4.38)
88
where 𝑑𝑐is the conductor diameter, 𝑞 =𝑑𝑐
√2𝜎 , 𝜎 is the copper conductivity, ber and bei being,
respectively, the real and imaginary parts of the zero-order Kelvin functions of first kind, and
ber’ and bei’ their derivatives. The real and imaginary parts of the ith-order Kelvin functions
of first kind and their derivatives are as follow [69],
𝑏𝑒𝑟(𝑖, 𝑞) = (𝑞
2)𝑖
∑cos [(
3𝑖4 +
𝐾2)𝜋
]
𝐾! (𝑖 + 𝐾)!𝐾≥0
(𝑞2
4)
𝐾
(4.39)
𝑏𝑒𝑖(𝑖, 𝑞) = (𝑞
2)𝑖
∑sin [(
3𝑖4 +
𝐾2)𝜋
]
𝐾! (𝑖 + 𝐾)!𝐾≥0
(𝑞2
4)
𝐾
(4.40)
𝑏𝑒𝑟′(𝑖, 𝑞) =𝑏𝑒𝑟(𝑖 + 1, 𝑞) + 𝑏𝑒𝑟(𝑖 + 1, 𝑞)
√2
+ (𝑖
𝑞) . 𝑏𝑒𝑟(𝑖, 𝑞)
(4.41)
𝑏𝑒𝑖′(𝑖, 𝑞) =𝑏𝑒𝑖(𝑖 + 1, 𝑞) − 𝑏𝑒𝑖(𝑖 + 1, 𝑞)
√2+ (
𝑖
𝑞) . 𝑏𝑒𝑖(𝑖, 𝑞)
(4.42)
Regarding the complexity of this equation, it is not suitable for the optimization loop. Hence,
in the literature, some approximate formulas derived from Eq.4.38 such as below [68]:
𝑅𝑎𝑐 =1
𝜋𝜎𝛿 (1 − 𝑒−𝑟𝛿) [2𝑟 − 𝜎 (1 − 𝑒−
r𝜎)]
(4.43)
where r is the conductor radius.
Also, the international standard (IEC 60287-1-1) [66] provides an effective and accurate
approach to estimate the AC resistance of a solid round wire in the air. Using IEC standard,
the AC resistance is calculated as below:
𝑅𝑎𝑐 = 𝑅𝑑𝑐(1 + 𝑦𝑠) [Ω/m] (4.44)
The skin effect factor 𝑦𝑠 is calculated as,
𝑦𝑠 =𝑥𝑠4
192 + 0.8𝑥𝑠4
(4.45)
where,
89
𝑥𝑠4 = (
8𝜋𝑓𝐾𝑠𝑅𝑑𝑐107
)2
(4.46)
and 𝐾𝑠 = 1 in the case of a solid round conductor. However, according to [66], to obtain
accurate results is only applicable when xs ≤ 2.8.
Figure 4.9: AC resistance of round copper conductor versus conductor diameter using exact
equation
Figure 4.10: AC resistance of round copper conductor versus conductor diameter using simplified
equation
0.9
1.1
1.3
1.5
1.7
1.9
2.1
9 11 13 15 17 19 21 23 25 27 29 31 33 35
Rac/R
dc
Copper diameter (mm)
Copper resistivity versus current frequency (Kelvin function)
f=50Hz
f=100
f=200Hz
0.9
1.1
1.3
1.5
1.7
1.9
2.1
9 11 13 15 17 19 21 23 25 27 29 31 33 35
Rac/R
dc
Copper diameter (mm)
Copper resistivity versus current frequency (Simplified)
f=50Hz
f=100
f=200Hz
90
Figures 4.9, 4.10 and 4.11 represent the variation of AC resistance of copper conductor versus
the conductor diameter calculated using Kelvin Function, simplified equation, and IEC
standard equation, respectively.
Figure 4.11 AC resistance of round copper conductor versus conductor diameter using IEC
standard equation
In order to verify the analytical equation that was proposed to estimate the AC resistance, a
finite element analysis has been done. Figure 4.12 shows the finite element simulation of skin
effect. Figure 4.12.and 4.12.b show the magnetic flux density and current density in wire
cross-section respectively. The wire diameter is 13.1mm, the current is 528A RMS and the
frequency is 50Hz
0.9
1.1
1.3
1.5
1.7
1.9
2.1
9 11 13 15 17 19 21 23 25 27 29 31 33 35
Rac/R
dc
Copper diameter (mm)
Copper resistivity versus current frequency (IEC standard)
f=50Hz
f=100
f=200Hz
91
a) Magnetic flux density in wire section
b) current density in the wire section
Figure 4.12: Finite element analysis of skin effect
Table 4.2 shows the values of 𝑅𝑎𝑐/𝑅𝑑𝑐 of a round copper conductor with the diameter of
13.21 mm which were calculated using three methods for 50Hz, 100Hz and 200Hz.
Table 4.2: 𝑅𝑎𝑐/𝑅𝑑𝑐 of copper conductor with 13.21mm diameter in 50Hz, 100Hz and 200Hz using
three estimation methods
Frequency (Hz) Kelvin Function Simplified IEC Standard FEM
0 1.00 1.00 1.00 1.00
50 1.002 1.062 1.002 1.002
100 1.009 1.113 1.009 1.0091
200 1.037 1.202 1.037 1.0371
Also, regarding the current value of each frequency, the total copper losses are computed
using Eq.4.36. Table 4.3 shows the copper losses in each frequency and the error compared
to utilizing the same resistance in AC current. The IEC standard equation is not very complex
to implement, Hence, it will be appropriate to employ in the analytical model. It provides
suitable accuracy compared to the simplified equation.
92
Table 4.3: Calculation of conductor copper losses using the estimated AC resistances and error
calculation compared to utilization of DC resistance
Frequency
(Hz)
Current (A)
Kelvin Function
Copper
losses(W)
Simplified
Copper
losses(W)
IEC Standard
Copper
losses(W)
FEM
Copper losses
(W)
0Hz, 166A 198.27 198.27 198.27 198.27
50Hz, 528A 2001.56 2121.86 2001.56 2001.55
100Hz, 476A 1632.97 1801.445 1632.97 1632.97
200Hz, 211A 331.22 384.22 331.56 331.25
Error 8.4% 15% 8.42% 8.4%
4.4.2. Thermal Model of Inductor
The thermal model simulates the thermal dissipation and the temperature rise in a different
node of the system. The temperature rise is a function of time and it depends on the thermal
capacity of the materials and the heat source value. It was assumed that the thermal
distribution in the materials is homogenous and the internal thermal resistivity is neglected.
Pfe Pcu
Ta
Rfe-air
Rfe-cu Rcu-airTfe
Tcu
Figure 4.13: Inductor equivalent thermal circuit
Figure 4.13 shows the thermal equivalent circuit of the inductor. The core loss is divided to
the vertical and horizontal parts regarding the volume.
93
The temperature values of the core and copper are calculated as below:
𝑇𝑓𝑒 − 𝑇𝑎 =𝑅𝑓𝑒−𝑎𝑖𝑟(𝑅𝑓𝑒−𝑐𝑢 + 𝑅𝑐𝑢−𝑎𝑖𝑟)𝑃𝑓𝑒 + 𝑅𝑐𝑢−𝑎𝑖𝑟𝑃𝑐𝑢
𝑅𝑓𝑒−𝑎𝑖𝑟 + 𝑅𝑓𝑒−𝑐𝑢 + 𝑅𝑐𝑢−𝑎𝑖𝑟
(4.47)
𝑇𝑐𝑢 − 𝑇𝑎 =𝑅𝑓𝑒−𝑎𝑖𝑟𝑅𝑐𝑢−𝑎𝑖𝑟𝑃𝑓𝑒 + (𝑅𝑓𝑒−𝑎𝑖𝑟 + 𝑅𝑓𝑒−𝑐𝑢)𝑅𝑐𝑢−𝑎𝑖𝑟𝑃𝑐𝑢
𝑅𝑓𝑒−𝑎𝑖𝑟 + 𝑅𝑓𝑒−𝑐𝑢 + 𝑅𝑐𝑢−𝑎𝑖𝑟
(4.48)
where 𝑅𝑓𝑒−𝑎𝑖𝑟 , 𝑅𝑓𝑒−𝑐𝑢 , 𝑅𝑐𝑢−𝑎𝑖𝑟 are the thermal resistivity between core-air, core-winding and
winding-air respectively.
The thermal resistance between the materials are calculated as below:
𝑅𝑓𝑒−𝑎𝑖𝑟 =1
ℎ𝑓𝑒𝑆𝑓𝑒
(4.49)
𝑅𝑐𝑢−𝑎𝑖𝑟 =1
ℎ𝑐𝑢𝑆𝑐𝑢
(4.50)
where 𝑆𝑓𝑒 , 𝑆𝑐𝑢 are the contact surface between two materials; core-air, and copper-air
respectively. There is an isolation space between the winding and the core. Hence, the
isolation is consisted of isolation materials and immobile air. Hence, the thermal conduction
between the winding and the core is weak. The thermal resistivity between the winding and
the core is calculated as below:
𝑅𝑓𝑒−𝑐𝑢 =1
𝜆
𝑙𝑖𝑠𝑜𝑆𝑐𝑢
(4.51)
where 𝜆 is the thermal conduction factor of isolation and 𝑙𝑖𝑠𝑜 is the isolation thickness.
𝑆𝑓𝑒 = 4𝑑(2𝑑 + 𝑎 + 𝑐) + 2𝑐(2𝑑 + 𝑎) (4.52)
𝑆𝑐𝑢 =4𝑑𝐴𝑐𝑢𝑏
𝑐 + 4𝑑𝑏 (4.53)
Also, ℎ𝑓𝑒 , ℎ𝑐𝑢 are the dissipation factors; core-air and copper-air. The value of dissipation
factors is dependent on the cooling system specifications. In the case of natural air cooling
system, the value of dissipation factors 15 W/m^2/°C. Utilizing the ventilation air cooling
system, the dissipation factor can be augmented four times and it can be increased to eight
times using water cooled system.
94
4.5. Investigation the effect of core saturation
Power electronic designers often consider that inductance core has linear characteristic and
neglect the nonlinearity and saturation region. Hence, dependent on the maximum flux
density in the core, a nonlinearity will emerge in inductance function.
Figure 4.14: The B-H curve of iron sheet core
Figure 4.14 shows a specific B-H curve of the iron sheet which is utilized in inductor
production. The B-H curve is usually linearized to use in the mathematical calculation. Power
electronic engineers prefer to keep the flux density in the linear part while increasing the flux
density can be lead to reduce the inductor size. On the other hand, in the transient state or
fault condition, a converter might enter to the nonlinear region. Therefore, performance
analysis of the converter with a nonlinear inductor will be important. In order to add the effect
of nonlinearity to the model, the relation between voltage and current must be calculated.
95
4.5.1. Finding the Mathematical Core Magnetizing Function
If the inductance core is linear or the magnetic flux density is kept in the linear range, 𝑉𝐿 =
𝑛𝑑𝜙
𝑑𝑡 where 𝜙 =
𝑛𝐼𝐿
𝑅𝑇, 𝑅𝑇 =
𝑙𝑒
𝜇0𝜇𝑟𝐴𝑐. Therefore, in the case of linear inductance, the inductance
voltage will be 𝑉𝐿 = 𝐿𝑑𝐼𝐿
𝑑𝑡. In the case of non-linear core inductance, to find the relation
between the voltage and current of the inductor, we must pass several steps. In the first step,
the magnetic field (𝐻), must be computed.
𝐻 =𝑛𝐼𝐿𝑙𝑒
(4.54)
𝐻𝑙𝑖𝑛𝑒𝑎𝑟 =𝐵𝑙𝑖𝑛𝑒𝑎𝑟𝜇0𝜇1
(4.55)
𝐻𝑠𝑎𝑡 =𝐵𝑠𝑎𝑡𝜇0𝜇𝑟
(4.56)
where 𝑙𝑒 is the effective magnetic length, 𝑛 number of turn, 𝐵𝑙𝑖𝑛𝑒𝑎𝑟 maximum linear magnetic
flux density and 𝐵𝑠𝑎𝑡 is the saturation flux density. To calculate the magnetic flux density, a
hyperbolic function provides the best fitting factor to show nonlinear effect of the core.
𝐾𝑠𝑎𝑡 = (tanh−1(
𝐵𝑠𝑎𝑡𝐵𝑙𝑖𝑛𝑒𝑎𝑟
))/𝐻𝑙𝑖𝑛𝑒𝑎𝑟 (4.57)
𝐵 = 𝐵𝑠𝑎𝑡tanh (𝐾𝑠𝑎𝑡𝐻) (4.58)
4.5.2. Inductance circuit Equation Considering Core Saturation
Using the nonlinear flux function, the inductor voltage function will be as below:
𝑉𝐿 = 𝑛𝑑𝜙
𝑑𝑡
=𝑛2𝑘𝑠𝑎𝑡𝐴𝑒𝐵𝑠𝑎𝑡
𝑙𝑒sech2 (
𝐾𝑠𝑎𝑡𝑛𝐼𝐿𝑙𝑒
)𝑑𝐼𝐿𝑑𝑡
(4.59)
we can easily extend this approach for coupled inductors. In the case of coupled inductors,
inductor voltage will be calculated as below:
96
𝑉𝐿 =𝑛1𝑑𝜙𝑇𝑑𝑡
= 𝑛1𝑑
𝑑𝑡(𝐵𝑠𝑎𝑡𝐴𝑒 tanh (
𝐾𝑠𝑎𝑡𝑛1𝐼1𝑙𝑒1
+⋯+𝐾𝑠𝑎𝑡𝐾𝑛𝑛1𝐼1
𝑙𝑒𝑛))
= (𝑛12𝐾𝑠𝑎𝑡𝐴𝑒𝑙𝑒1
𝑑𝐼1𝑑𝑡
+⋯+𝐾𝑁𝑛1
2𝐾𝑠𝑎𝑡𝐴𝑒𝑙𝑒𝑛
𝑑𝐼𝑛𝑑𝑡) sech2(
𝐾𝑠𝑎𝑡𝑛1𝐼1𝑙𝑒1
+⋯+𝐾𝑛𝐾𝑠𝑎𝑡𝑛1𝐼𝑛
𝑙𝑒𝑛)
(4.60)
In order to add the saturation effect, it is sufficient to modify the flux density relation as
below:
𝐵𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 = 𝐵𝑠𝑎𝑡 tanh−1(
𝐾𝑠𝑎𝑡𝑛(𝐼1 + 𝐼2)
𝑙𝑚 + 2𝜇𝑟𝑙𝑔)
(4.61)
4.6. Finite Element Analysis of Coupled Inductors
The finite element method (FEM) is a numerical technique for solving problems which are
described by partial differential equations or can be formulated as functional minimization.
A domain of interest is represented as an assembly of finite elements. Approximating
functions in finite elements are determined in terms of nodal values of a physical field which
is sought. A continuous physical problem is transformed into a discretized finite element
problem with unknown nodal values. For a linear problem, a system of linear algebraic
equations should be solved. Values inside finite elements can be recovered using nodal values
[70].
The finite element approach could be employed in 2-D or 3D space. 2-D finite element has
fewer nodes to solve in compare with the 3-D finite element approach. Therefore, it is faster
and needs less computer memory to solve. In the case of symmetrical volume on the z axis,
2-D finite element approach could be an effective solution to accelerate the simulation time
and decrease the required computer memory.
4.6.1. Magneto-static Analysis using Finite Element Method
In this section, the finite element analysis using a software is explained. The finite element
software calculates the precise magnetic field density, self-inductance, and the magnetic
97
coupling factor. Also, the copper and magnetic losses are calculated by the software. There
is a number of electromagnetic analysis software which use the finite element method to
solve the partial differential equations. The main sections of a finite element software are:
1- Geometry
2- Generate mesh
3- Solve equations
4- Show result
The finite element software is complex and expensive. In this project, a free version of FEM
software is used to analyze the coupled inductor. This version is an open source version and
free for the academic researches. The most advantages of this software are the possibility to
execute the netlist code. The netlist code consists of geometry parameters, material
parameters, and mesh constraints are written as an LUA file. In this project, the notepad++
is used to write the LUA file. The LUA file is executed by FEM software. This is a useful
possibility to make a connection between the EXCEL and FEM software. Figure 4.15 shows
the magnetic flux lines and magnetic flux density of the inductor.
a) Magnetic flux lines
b) Magnetic flux density
Figure 4.15: Finite element analysis of coupled arm inductance
98
4.6.2. Correction of Analytical Model using Finite Element Method
The analytical model of coupled inductor provides an appropriate platform to integrate the
optimization loop. The accuracy of the analytical model will reduce by increasing the length
of the inductor air gap. Hence, in most of the operating points, the analytical model accuracy
is not sufficient enough for the optimization algorithm. The precise value should be
calculated using a finite element method. On the other hand, utilizing the finite element
software in the optimization loop made it very slow and some time difficult to converge. To
find the global optimal point, solver chooses the initial values from a different region to avoid
falling in a local point. If the finite element model is employed directly in the optimization
loop, it takes so much time to find the global points. In this research, an innovative solution
has been proposed to resolve the time and accuracy issues at the same time.
The FEM analysis results are employed in order to verify and correct the analytical model
parameters. Utilization the FEM software in the optimization loop, intensely increase the
optimization time and makes it complex to converge. On the other hand, the analytical model
does not provide accurate results at all operating points. In this research, a hybrid
optimization algorithm has been proposed and developed to modify the analytical model with
FEM software in order to decrease the optimization time and increase the model preciseness.
2-D finite element method is employed to analyze the inductance and modify the analytical
model. A Hybrid optimization model consisted of the analytical model and finite element
approach is employed to increase the model accuracy without increasing the optimization
time.
The air gap length is known as the most important variable which affects the accuracy of the
analytical model. Therefore, the inductance relation is defined for the analytical model and
FEM analysis as below:
𝐿𝑎𝑛 =𝑛2𝜇0𝜇1𝐴𝑐𝑙𝑚 + 2𝜇𝑟𝑙𝑔
(4.62)
𝐿𝑓𝑒𝑚 =𝑛2𝜇0𝜇1𝐴𝑐
𝑙𝑚 + 2𝐾𝑐𝜇𝑟𝑙𝑔
(4.63)
If we consider that 𝛿 =𝜇𝑟𝑙𝑔
𝑙𝑚, then the correction factor 𝐾𝑐 and model error 𝐸 will be:
99
𝐾𝑐 =𝐿𝑎𝑛(1 + 𝛿) − 𝐿𝑓𝑒𝑚
𝛿𝐿𝑓𝑒𝑚
(4.64)
𝐸 =𝐿𝑓𝑒𝑚 − 𝐿𝑎𝐿𝑓𝑒𝑚
× 100 (4.65)
Figure 4.16 shows the flowchart of the proposed correction loop, which was employed to
combine the analytical model and finite element method. It consists of two separated loops;
the optimization loop and correction loop. The analytical model is utilized to solve and find
the optimal values while the correction loop is used to modify the analytical model and
eliminate the error. In this method, the optimization loop will be repeated for more than 1000
times while the correction loop is called for 3 or 4 times. Using multi-start algorithm, we can
find the global optimal point and avoid falling in local optimal points. This is exactly the
point, which makes difficult to use the finite element model in optimization. The solver
chooses different initial points to find the global optimal point. If we use the finite element
in optimization without suitable initial values, it may go to the region without an optimal
point that leads to missing the optimization time. In this approach, the initial evaluation has
done using analytical model, then the variables are sent to finite element software to validate.
This innovative method effectively decreases the optimization time and increases the
converging probability.
100
FEM
software
Dimension model of
arm inductances
IL1
µr
Larm
Jabcdg
M
Bmax
RL
Pcu
Pcore
Vind
Mind
Spe
cific
tio
ns
Co
re t
op
olo
gy
IL2
Win
din
gn1
n2
Electromagne
tic Variab
les
Perfo
rmance
Outputs
η%
yes
Error calculation
Calculate correction factor
Figure 4.16: The flowchart of the proposed correction approach
4.7. Conclusion
In this chapter, the dimension analysis of MMC passive components is presented. The mass
equation of sub-module capacitor depends on the capacitance value and nominal capacitor
voltage. The capacitor mass equation is determined regarding product datasheet of the
manufacturer. The dimension analysis of the arm inductances is dependent on
electromagnetic analysis.
The analytical electromagnetic model of the arm inductances is investigated and developed
regarding MMC specifications. Utilizing the analytical model, the inductance value, mutual
inductance, magnetic flux, core and winding losses are calculated. Also, the electromagnetic
model of coupled inductor and its circuit model were investigated.
101
Finally, the finite element analysis of the arm inductance was presented as an accurate
approach to analyzing the magnetic components. The dimensioning variables are sent to
finite element software to generate the model geometry. The inductance model is supplied
using a current source to compute the electrical and magnetic parameters. Utilizing the finite
element approach is complex and time-consuming. In this chapter, a hybrid model was
presented to correct the analytical model parameters and enhance its accuracy. In the next
chapter, the converter analysis in the fault condition is presented.
102
CHAPTER V
5 Converter Analysis in the Fault Condition
5.1. Introduction
The steady-state model guarantees the appropriate converter functionality in the nominal
operating point. Regarding the nonlinearity of the magnetic core, the functionality of MMC
converter should be studied in the fault condition. In the final design, a margin should be
considered to minimize the components damage in defect condition. In this chapter, the
standard MMC converter faults are introduced and investigated. Also, two different methods
are proposed to analyze the defect condition. The first one is to use a Simulink model in order
to simulate the defect condition and measure the circuit parameters. The second one is to add
the analytical model of the fault condition to the proposed time-domain model. In order to
investigate the fault condition, the standard faults are designed and analyzed using state
equations. The results could be added to the constraints vector.
5.2. Investigation of Standard Defects in MMC Converter
In this section, the standard defect type which is happened in MMC converter is investigated.
Based on the literature, three kinds of converter fault have been introduced and investigated.
The faults might happen in the sub-modules, arm inductance or at the converter output. The
most important faults are DC link fault, sub-module short circuit or open circuit and arm
inductance fault [71-73].
5.2.1. DC Link Fault
The first kind of faults that were studied in the literature is the DC bus fault [72]. The DC
bus dependent on its application might be a short or a long DC line. The short circuit might
happen on the different points of the DC line. Therefore, the fault’s impedance is different.
103
The most important effect of DC link fault is to affect the DC line voltage and increase the
arm currents. Figure 5.1 shows the current paths in an MMC converter when a DC link fault
happens. DC link fault strongly affects the arm currents. Hence, it might lead to destroy the
semiconductors and saturate the arm inductances.
Figure 5.1: The DC link fault and currents path in the converter
5.2.2. Sub-module Fault
There are two different types of the switching device fault in a sub-module: open-circuit fault
which appears due to lifting of the bonding wires in a switch module caused by over-
temperature or aging and usually does not cause additional serious damage to the system if
the protection system functions well; short-circuit faults are caused by wrong gating signals,
overvoltage, or high temperature and could cause additional damage to other components in
the circuit, so the short circuit fault should be treated fast and carefully. Generally, the
hardware overcurrent protection which stops the operation of the system is the most common
solution for short-circuit faults [17]. However, for MMC, the overcurrent appears in the
faulty module rather than flowing through the whole arm or phase which means it is the faulty
module that needs to be bypassed by the action of overcurrent protection devices and the
system operation can continue without stopping. The failure configurations of the open-
circuit and short-circuit fault in a sub-module concerning on the failure of the switching
devices T1, T2 are shown in Figure 5.2.
104
D1
C
D2
Vc
IcT1
T2
Ism
Vsm
a) Short-circuit of
bypass IGBT
D1
C
D2
T1
T2
Vsm
IsmVc
Ic
b) Short-circuit of the
main IGBT
D1
C
D2
T1
T2
Vsm
IsmVc
Ic
c) Open circuit of
sub-module IGBTs
Figure 5.2: Various kind of sub-module faults
5.2.3. Inductance Fault
There are two types of inductor fault that are investigated in literature; short-circuit and open-
circuit. Instantaneous changes in inductor parameters are unphysical. Therefore, when the
Inductor enters the faulted state, short-circuit and open-circuit voltages transition to their
faulted values over a period of time, according to the following formula:
𝐼𝐿 = 𝐼𝑓 − (𝐼𝑓 − 𝐼𝑛𝑓)sech (Δ𝑡
𝐹𝑇)
(5.1)
For short-circuit types of faults, the conductance of the short-circuit path also changes
according to a 𝒔𝒆𝒄𝒉(𝚫𝒕/𝑭𝑻) function from a small value (representing an open-circuit path)
to a large value. where Δ𝑡 is the time since the onset of the fault condition and 𝐹𝑇 is the time
constant associated with the fault transition.
5.3. Close Loop Control of MMC converter using Simulink
In power converters, there is a current control loop which limits the maximum current in the
fault condition. The inductance of non-ideal core inductors is reduced by increasing the
current. In the fault condition where the current is massively increased, there are serious
concerns about the inductors functionality. The inductance reduction intensely affects the
converters circuit performance and may lead to component failure. In this situation, the
recommissioning is usually so expensive and time-consuming. Hence, it is important to
105
verify the component safety in the fault condition. There are various types of faults that
should be investigated in MMC converter.
The performance of optimal inductor must be evaluated in the fault condition in order to
make sure about the converter and inductance stability in a fault condition. A close loop
controller has employed to control the DC link voltage while the dc current is changing.
Figure 5.3 shows the close loop control which is utilized to control the MMC converter.
MMC Converter
+
Vdc
-
Grid
Vdc*
Vdc
PID ×
Kv
Vabc,Iabc
Vabc
abc/dq0
abc/dq0
Iabc
Park Trans.
Park Trans.
Id
Iq
Vd
Vq
PID
PID
dq0/abc
1
0.5
fsw/m
(m-1)fsw/m
.
.
.
Submodule pulses
+
+
++
-
-
-
+
Figure 5.3 Close loop control diagram of MMC converter
5.4. Investigation of Converter Performance in Defect Condition
Using Simulink, the performance of MMC converter and its components is evaluated in
defect condition. Figure 5.4 and 5.5 show the DC link voltage and AC line current of MMC
converter in a complete converter cycle. The faults happen at t=0.5. The normal condition
was compared to the faults condition.
Figures 4.9, 4.10 and 4.11 represent the variation of AC resistance of copper conductor versus
the conductor diameter calculated using Kelvin Function, simplified equation, and IEC
standard equation, respectively.
106
Table 5.1: Simulation parameters of MMC converter in simpower
Parameters Value
Number of sub-modules per arm (m) 3
Sub-module Capacitor value 3.5 𝜇𝐹
Inductance value 2.83 𝑚𝐻
Coupling factor 0.9
DC voltage 5000 V
AC voltage L-L 2000 V
Figure 5.4: DC Link voltage variation via various converter faults
107
Figure 5.5: Line current variation via various converter faults
In the normal condition, inductor current does not pass 2500A in the full load. The sub-
module short circuit or DC link fault cause to jump the current to 5000 A. Therefore, the
inductor core might enter to the saturation region. It leads to increase the circulation current
and damage the IGBTs and inductor. In order to verify the core saturation, the maximum flux
density in the core should be evaluated. Figure 5.7 shows the magnetic flux density of the
core in the normal and fault condition. It should be noted that in the coupled inductor
topology, the magnetic flux density is affected by upper and lower arm currents at the same
time. In the normal condition, the maximum flux density is about 0.9T, while in the fault
condition it goes to 1T which is not a critical condition for the electromagnetic core.
Therefore, the optimized inductor parameters could pass the fault condition.
108
Figure 5.6: Upper inductor current variation via various converter fault
Figure 5.7: Magnetic flux density of inductor core via various converter fault
109
5.5. Combination of Time-Domain Steady-State Model and
Faults in the unit package
In order to investigate the MMC converter under a fault condition, it is necessary to review
the sub-module operation in the fault condition. First of all, the normal operation of half-
bridge sub-module must be investigated. Table 5.1 shows the normal operation of half-bridge
sub-module in four sub-module statuses.
5.5.1. Sub-module faults investigation
Normal operation: In normal operation, as listed in Table 5.2, when the arm current 𝑖𝑎𝑟𝑚 is
positive, if T1 is turned on, T2 is turned off in the sub-module module which means 𝑆 = 1,
the current flows through D1 and C, the capacitor is charged; otherwise, T1 is turned off, T2 is
turned on, the current will go through T2 and the capacitor voltage maintains stable; when
𝑖𝑎𝑟𝑚 is negative, T1 is turned off, T2 is turned on, the current flows through D2; oppositely,
𝑆 = 0, the current will go through T1 and C, the capacitor is discharged.
Table 5.2: Normal operation of a half-bridge sub-module
St
No.
Current Status Gate Current goes
through
capacitor Capacitor
voltage
1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Charged Increased
2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 T2 Bypassed Stable
3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 T1 and C Discharged Decreased
4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 Bypassed Stable
Open-circuit fault in T1: as shown in Table 5.3, the sub-module operates as normal when the
arm current 𝑖𝑎𝑟𝑚 > 0, the arm current still goes through D1 and C to charge the capacitor
when the gating signal 𝑆 = 1 and the arm current flows through T2 to bypass the capacitor
when the gating signal 𝑆 = 0; when 𝑖𝑎𝑟𝑚 < 0, the module is in normal operation when 𝑆 =
0, the arm current flows through D2 and the capacitor voltage is stable; however, when the
110
gating signal 𝑆 = 1, the arm current will be forced to go through D2 instead of T1 and C in
the normal condition;
Table 5.3 Investigation of Open-Circuit Fault in T1
St
No.
Current Status Gate Current goes
through
capacitor Capacitor
voltage
1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Charged Increased
2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 T2 Bypassed Stable
3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 D2 Bypassed Stable
4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 Bypassed Stable
Open-circuit fault in T2: the open-circuit fault is shown in Table 5.4, the sub-module
operates as normal when the arm current 𝑖𝑎𝑟𝑚 > 0 and 𝑆 = 1, if T1 is turned off, T2 is turned
on, the arm current is forced to go through D1 and C to charge the capacitor instead of T2 to
bypass the capacitor; when the arm current 𝑖𝑎𝑟𝑚 < 0, the module is in normal operation;
Table 5.4 Investigation of Open-Circuit Fault in T2
St
No.
Current Status Gate Current goes
through
capacitor Capacitor
voltage
1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Charged Increased
2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 D1 and C Charged Increased
3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 T1 and C Discharged Decreased
4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 Bypassed Stable
Short-circuit fault in T1 or T2 : as shown in Table 5.5, when the short-circuit fault happens
in T1 (T2), the sub-module operates as normal if the corresponding IGBT T1 (T2) is turned on
and the complementary IGBT T2 (T1) is turned off; when the complementary IGBT T2 (T1) is
turned on, the capacitor discharged through the capacitor discharging loop which is formed
by the short-circuited T1 (T2), the complementary IGBT T2 (T1) and the capacitor C. Due to
the small-time constant of the capacitor discharging loop, the capacitor discharged very
quickly which leads to the rapid declines of capacitor voltage and the large short-circuit
111
current in the faulty module. Generally, the faulty module is bypassed and the arm current
goes through the switch used to do overcurrent protection. However, with MMC topology,
the arm current will go through D1 to charge C when the arm current is positive and go
through D2 to discharge C with negative arm current. The capacitor voltage of the faulty
module changes from zero to a small value compared to the normal capacitor voltages.
Table 5.5 Investigation of Short-Circuit Faults in T1 or T2
St
No.
Current Status Gate Current goes
through
capacitor Capacitor
voltage
1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Bypassed Increased
2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 D1 and C Bypassed Increased
3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 D2 and C Bypassed Decreased
4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 and C Bypassed Decreased
It can be seen that various module performances are caused by different faults which mean
that the different faults can be identified by analyzing the performance of the system.
Moreover, the system performance under fault conditions can be limited by changing the
gating signals which makes the below fault identification possible.
5.5.2. Proposed global optimization considering fault analysis
Another approach is to make a combination of transient model and fault analysis. In this
method, while the transient model is calculating the value of the normal operation, the fault
model calculates the important parameters in the fault condition. The value of the converter
parameters in the fault condition is considered as constraints of the optimization algorithm.
5.6. Conclusion
Power electronic designers design the converters base on the nominal operating point and
usually neglect the fault condition. In high power application, the fault occurrence could lead
to costly damages. Hence, in high power application, the fault condition must be noted in
design procedure in order to minimize the damages and reduce the cost of fault.
112
In this chapter, various kinds of the faults in MMC structure was studied and their effects on
the sub-module voltage and arm current were investigated. To simulate the fault condition, a
close loop control system was proposed and explained. The Saturable model of coupled
inductances which was presented in the previous chapter was employed in the converter
model. The fault condition was simulated using Simulink/MATLAB to investigate the
important converter variables in the fault condition. One of the most important variables in
the fault condition is the magnetic flux density of the arm inductance core. The magnetic flux
density is the criterion to determine the saturation condition of inductance core.
In the next chapter, various optimization scenarios with different complexity will be proposed
and developed in order to minimize the final volume and mass of MMC converter. The results
of different optimization algorithm are compared and discussed to achieve the best
optimization approach.
113
CHAPTER VI
6 Optimal Design of Modular Multilevel Converter
6.1. Introduction
In this chapter, different optimization scenarios of MMC converter are proposed and
developed. The major optimization goal is to minimize the converter size with respect to the
technical and manufacturing constraints. Each optimization algorithm consists of a
mathematical model, optimization variables, constraints, goal function and the solver. In term
of MMC converter, the converter model is divided to circuit model, electromagnetic model,
thermal model, and dimensioning model.
In this research, various optimization algorithms with different degree of complexity have
been proposed. The first optimization algorithm employs the proposed time-domain circuit
model which was explained in chapter 3 as converter model and neglects the structure of the
arm inductance and capacitor size. It investigates the inductor and capacitor as circuit
components. This algorithm is fast and provides a suitable guideline to get closer to the
optimal point. The weakness of this approach is to neglect the electromagnetic characteristics
of the inductances and its dimension.
The arm inductance is the bulkiest part of MMC converter which strangely affects the total
converter mass. In chapter 4, the electromagnetic and thermal model of the model of arm
inductances was presented. Utilizing the proposed circuit model, electromagnetic model, and
thermal model, a new optimization algorithm has proposed and developed. Unlike the
previous algorithm, the core and winding parameters of arm inductance are considered as
part of optimization variables. Utilizing the analytical dimensioning model of inductor and
capacitor, the total converter mass is calculated as goal function.
114
The electromagnetic model of inductors is not accurate enough especially in terms of using
airgap in the core topology. In chapter 4, finite element method was introduced as an effective
tool to analyze the electromagnetic components. A hybrid correction loop was proposed to
modify the analytical model parameters and enhance the model accuracy. The third
optimization algorithm employs the proposed hybrid correction loop to enhance the accuracy
of optimization results. Finally, results of various optimization algorithm are discussed and
investigated.
6.2. Optimization algorithm using numerical solver
Numerical solvers are the mathematical tools which are used in lots of engineering domain
such as mathematics, engineering, science, business, and economics. One of the interesting
application of the numerical solvers is to use in the optimization problems. In the
optimization issues, an engineering issue is formulated using mathematical approaches called
analytical model to represent the situation. There are lots of software which were designed
to implement the optimization issues. Microsoft excel provides an appropriate environment
and functions to implement the optimization problems.
Microsoft excel 2013 supports three solvers, Linear, nonlinear and evolutionary (Genetic
algorithm based). Using Microsoft excel leads to visualize the optimization loop and will be
a very suitable platform to implement the problem. Also, we have a possibility to make a
connection between Microsoft Excel and MATLAB software in order to communicate and
use their software facilities. Using the link between Microsoft Excel and MATLAB, we are
able to use the optimization toolbox of the MATLAB in the Excel environment. Therefore,
we will have a chance to utilize the modern optimization function and all Matlab functions.
It will be an attractive option in future researches. Figure 6.1 shows the image of excel sheet
which is consist of different parts of the optimization problem.
Optimization toolbox of MATLAB is another solution to implement an optimization
problem. MATLAB software is a professional and powerful mathematical package which
consists of recent numerical functions. Some researchers, use Microsoft excel to implement
the analytical model, afterward the model is sent to the MATLAB optimization toolbox in
115
order to solve and optimize the problem. An optimization loop generally consists of the
following components:
1- Optimization variables: Optimization variables are independent variables which are
chosen to optimize. The decisions of the problem are the variables which must be
changed by the numerical solver to find the optimal point.
2- The objective of the problem is expressed as a mathematical expression based on
decision variables. The objective may be maximizing or minimizing by changing the
decision variables.
3- Constraints: The limitations or requirements of the problem are expressed as
inequalities or equations in decision variables. The constraints might be a simple value
or a mathematical function which depends on lots of parameters.
Figure 6.1 shows the proposed global optimization algorithm for modular multilevel
converters. The Excel cells are connected together via mathematical equations.
Figure 6.1: Implementation of Global Optimization Algorithm with Microsoft Excel
Microsoft excel represented three solver functions in order to solve the different types of
models; linear solver, nonlinear solver and evolutionary which employs genetic algorithm.
The linear solver is used to solve the models which consist of linear objective function and
116
linear constraints in decision variables, it is called linear programming model. The nonlinear
solver is used to solve the nonlinear programming model consists of a nonlinear objective
function and nonlinear constraints. The advantage of Microsoft Excel is the possibility of
visualization of the model, constraints and goal function.
6.3. Load Specification of the MMC Active Front End converter
application
The next step is to determine the load specification based on the application. In order to
investigate the application using the proposed optimization algorithm, it is necessary to use
the load specifications of the converter which must be studied. In the application, there is no
nominal point and the output power changes from zero to 30MW in each cycle. Hence, the
average power should be considered as a nominal point. The average converter power is
2.5MW. Table 6.1 shows the converter load specifications.
Table 6.1: Load specification of MMC Active Front End converter application
Converter Parameter Value
Nominal Power 2.5 MW
AC Line Voltage 2000 V
DC Link Voltage 5000 V
Power Factor 0.8
Nominal Frequency 50 Hz
THD < 2%
Efficiency > 95%
ΔT < 50 ℃
Capacitor voltage ripple < 20%
117
6.4. Constraints Calculation
One of the important parts of optimization loop is to choose and calculate the appropriate
constraints. The optimization finds the optimal values concerning the technical and
manufacturing constraints. The main constraints related to the application and method which
is used to calculate is presented as below:
6.4.1. Sub-module capacitor voltage ripple
The DC link voltage is divided between the series sub-modules. The DC part of sub-module
capacitor voltage is 𝑉𝑑𝑐
𝑚 in steady-state condition. The voltage ripple increases the maximum
capacitor voltage and converter harmonic and reduces the converter stability. Therefore, it
should be limited in the specific band. The sub-module capacitor voltage must not be greater
than the IGBT voltage. The capacitor voltage ripple is considered as a percentage of dc part.
The IGBT voltage is chosen based on the maximum capacitor voltage, generally with 50%
safety margin.
6.4.2. THD
Total harmonic distortion (THD) is an important parameter which shows the converter power
quality. Reducing the THD depends on lots of parameters such as switching frequency, a
switching method, the number of sub-module per arm, arm inductance, input filter and
capacitor value. High power converters generally use low switching frequency to minimize
the IGBT losses. Hence, reducing the THD will lead to increase the passive component
values. On the other word, there is a trade-off between the THD and MMC component value
especially arm inductance.
6.4.3. Semiconductor Losses
According to the IGBT datasheet, each IGBT can dissipate a maximum thermal loss which
depends on the cooling system. Therefore, the IGBT loss will be an important constraint
which guarantees the semiconductor safety. On the other hand, IGBT loss intensely affects
the converter efficiency. In some applications, the IGBT loss must be limited to achieve
higher converter efficiency. To calculate the IGBT loss, the technical data of power IGBT is
118
necessary. The number of power switches which are appropriate for our application is
limited. Therefore, the optimization has been done with two IGBTs that their specifications
are closer to our application. The technical data of ABB IGBT (5SNA 1500E330305) was
utilized to calculate the precise IGBT losses value. The nominal value of voltage and current
of this switch is 3.3 kV and 1500 A respectively.
6.4.4. Inductor Losses
The inductor loss is should be limited regarding the thermal dissipation function. There is a
maximum allowable temperature rise which must be considered to avoid the inductor fault.
On the other hand, it affects the total converter efficiency and reduces the converter
performance.
6.5. Goal function
In this thesis, the main goal function is to minimize the total converter volume and mass.
Mass optimization needs the component mathematical mass function to put in the
optimization loop. It needs a dimensioning analysis in order to obtain all mass functions. The
dimensioning analysis will be done in the next chapters. Therefore, the mass function should
be replaced by a circuit criterion which could explain the mass function.
The energy stored in the capacitor and inductor are suitable functions in order to estimate the
capacitor and inductor mass. In each converter arm, there are 𝑚 number of series sub-
modules, while there is on inductor per arm. The total electric and magnetic energy stored in
capacitors and arm inductance are calculated as below:
𝐸𝑐𝑎𝑝 = 3𝑚𝐶𝑠𝑚𝑉𝑐2 (6.1)
𝐸𝑖𝑛𝑑 =3
2𝐿(𝐼𝑢
2 + 𝐼𝑙2) + 3𝑀𝐼𝑢𝐼𝑙
(6.2)
Finally, the total converter energy stored could be calculated as goal function.
𝐸𝑐𝑜𝑛𝑣 = 𝐸𝑐𝑎𝑝 + 𝐸𝑖𝑛𝑑 (6.3)
119
6.6. MMC Optimization using analytical circuit model
The first optimization algorithm utilizes the proposed time-domain circuit model to
determine the optimal value of capacitor and inductor value of MMC converter. In addition
to the passive components value, the switching frequency and the number of sub-modules
per arm are considered as optimization variables. In this algorithm, the magnetic parameters
of inductors are neglected. Hence, an energy criterion is chosen to estimate the size of
inductors and capacitors. Also, the total harmonic distortion, total converter efficiency, and
sub-module capacitor ripple are the most important constraints of this optimization
algorithm.
6.6.1. Optimization algorithm
Figure 6.2 shows the optimization algorithm based on the analytical circuit model which is
proposed for MMC converter. The most important role of this algorithm is to find the optimal
circuit value of the sub-module capacitor and arm inductance. The optimization variables are
sub-module capacitor 𝐶𝑠𝑚, arm inductance 𝐿, 𝑀, switching frequency 𝑓𝑠𝑤 and modulation
index. The circuit analytical model of MMC regarding to the load specifications and some
constant values organize the main core of the algorithm. With the same reason, the main
constraints are the circuit constraints. The constraints calculation is done to estimate the
important limitation such as ripple, THD, losses and DC bus voltage and finally the goal
function is calculated. The circuit model does not obtain the size or dimension of any
components. Therefore, it is necessary to define some criteria which are proportional to the
components dimension. The capacitors and inductors store the energy in their structure. This
energy is proportionally depends on it size. Hence, the best parameter is to use the maximum
energy stored in the capacitor and inductor to estimate the size and dimension. The final part
of optimization algorithm is the solver. The solver is a calculation engine which solves the
nonlinear equation using advanced numerical approach. The nonlinear solver searches to find
the optimal values to minimize the total energy stored in the converter. Table 6.2 shows the
optimization constraints.
120
MMC Circuit
Model and
Fault analysis
Vabc
Idc
S(t)
fsw m
Csm
Larm
ωs
Iabc
Is1,Is2
Icirc
VCsm
Voltage ripple
THD
IL
Ifault
Switch losses
Spe
cifi
ctio
ns
Co
nve
rte
r to
po
logy
N
.
.
.
Pas
sive
co
mp
on
en
t v
alu
e
MIG
BT
Sp
eci
fica
tio
ns Vmax
Imax
Psw
Inputs
Stead
y-state Variab
les
Stead
y-state Pe
rform
ance
Inductor losses
Efficiency
Energy stored
Outputs
Non-linear
Solver
Total mass function
Capacitor energy
Csm
Co
nst
rain
ts
Vo
ltag
e ri
pp
le,
TH
D,e
ffic
ien
cy,
swit
ch l
oss
es
Init
ial
Val
ues
Op
tim
izat
ion
par
amet
ers
(L,M
,Csm
,Sm
,fsw
)
Inductor energy
Larm,M
Figure 6.2: Optimization flowchart of MMC converter
Table 6.2 The main optimization constraints
Constraints Value
DC bus voltage 5000 V
Efficiency >95%
THD <2%
Capacitor voltage ripple 0.2 ∗
𝑉𝑑𝑐𝑚
Number of series sub-modules 3< 𝑚 < 8
Number of parallel branches 1 < 𝑁 < 3
121
6.7. Optimal design of modular multilevel converter using
dimensioning model
The mass function of capacitor and inductors are added to the optimization algorithm and the
optimization is done to minimize the total converter mass. The optimization results should
be discussed and investigated.
6.7.1. Global Mass Minimization Algorithm
By increasing the demands of medium and high power converters, MMC converters were
changing to complex, bulky and expensive structures. In this time, mass minimization was
emerged to reduce and optimize the volume of the main components such as capacitors and
inductors. In the case of MMC converter, the mass minimization algorithm is dependent on
the circuit operation, electromagnetic and thermal functionalities.
Figure 6.3: The proposed global optimization algorithm using analytical model
122
Accordingly, the global optimization algorithm has been proposed to consider all technical
issues and constraints. Figure 6.3 shows the proposed mass minimization algorithm includes
circuit, electromagnetic and thermal analytical model. The mass minimization algorithm
consists of four main sections; mathematical model, goal function, constraints and numerical
solver.
Given that in MMC topology, the circuit initial values are dependent on the component
values, an extra part as initializing section has been added to the algorithm. The initializing
section employs the steady-state model to find the circuit initial values. It should be
considered that the converter does not stay in a constant operating point. Hence, after
initialization, the transient model will be started to calculate the state variables. In the same
time, the electromagnetic model of coupled inductors is employed in order to estimate the
inductor parameters and core size. Also, thermal model investigates the thermal distribution
in the various converter components and thermal exchange with the cooling system.
Table 6.3: List of optimization variables
Variables definition
𝐶𝑠𝑚 Sub-module capacitor
𝑓𝑠𝑤 Switching Frequency
A Core window width
B Core window height
C Core depth
D Core Width
G Air gap
J Current density
n Turn Number
The constraints are the important part of optimization loop to determine the technical and
manufacturing limits of MMC converter. The constraints values are determined by
mathematical equations and compared to model outputs. If the outputs do not satisfy the
constraints, the solver chooses other values for optimization variables. If the outputs are
123
placed in the allowed region, the goal function is calculated using optimization variable. This
subroutine will repeat until the optimization loop finds the global optimal point of goal
function.
Table 6.4: List of main constraints in optimization algorithm
Constraints Value
Total efficiency >95%
Capacitor voltage ripple < 0.2𝑉𝑑𝑐/𝑚
THD < 2%
𝐵𝑚𝑎𝑥 0.9T
Core window area > 𝐴𝑐𝑢−𝑡𝑜𝑡𝑎𝑙
Δ𝑇 < 60
Utilizing the circuit and electromagnetic model, the switch and inductor losses are calculated
in the transient state. Therefore, the total converter efficiency was calculated as a constraint
in the optimization algorithm. The optimization must determine the sub-module capacitor,
inductor value, and size. Table 6.3 shows the optimization variables which have to optimize
by the solver. Table 6.4 shows the constraints values in this optimization.
6.8. Hybrid Optimization Model using 2-D FEM
Utilizing the air gap leads to reduce the analytical model accuracy. Finite element analysis is
used to increase the analysis accuracy. Utilization of finite element model in optimization
procedure increases the time of convergence. It will be a trade-off between the optimization
time and result accuracy. The hybrid optimization model is proposed and developed to
increase the result accuracy without increasing the optimization time.
124
6.8.1. Hybrid Global Optimization Algorithm
A hybrid optimization algorithm is presented using a combination of the analytical model
and finite element method. The most important advantage of this method is to achieve high
accuracy and low convergence time in comparison with other optimization algorithms.
By a combination of global optimization algorithm and the proposed correction loop, a hybrid
optimization algorithm is presented which provides the advantages of conventional global
optimization algorithm while the results accuracy has been enhanced.
Figure 6.4: The proposed hybrid optimization algorithm
Figure 6.4 shows the proposed hybrid optimization algorithm which provides the optimal
converter size. After an optimization cycle, outputs are sent to the software to analyze using
finite element method. The software results are compared to the analytical results and the
model will modify to minimize the error. This subroutine is repeated until achieving the same
125
results by analytical model and software. The optimization algorithm estimates the optimal
value of these variables to achieve the minimum size.
6.8.2. Hybrid Global Optimization Algorithm considering fault margin
Figure 6.5 shows the global optimization of MMC converter considering fault margin. The
outputs of the fault calculation block are sent to the constraint block. These new constraints
increase the converter margin against the fault condition.
Steady State Model
Nominal Specifications
InitializingSet Initial Values
MMC circuit Analytical Model
Optimization Variables
Electromagnetic Analytical Model of coupled Inductors
Thermal ModelOutputs satisfy
the constraints?
Circuit Constraints
Electromagnetic Constraints
Thermal Constraints
Goal function(Converter or inductor Mass Minimization)
Nonlinear Solver
Optimal Values
Initializing
Analytical Model Constraints
Numerical Solver
No
yesGoal Function
IGBT open circuit
IGBT short circuit
Inductor short circuit
Inductor open circuit
Fau
lt c
alc
ula
tio
n
Fault condition model
Figure 6.5 Global optimization algorithm considering fault margin
Figure 6.6 shows the proposed hybrid optimization algorithm considering the fault margin.
This algorithm is more complex and slower in comparison to the previous algorithm. The
main optimization loop and auxiliary correction loop affect the outputs of the fault model at
the same time. Utilizing this mode, the output of optimization is calculated regarding actual
126
core specifications such as nonlinear permeability and saturation effect. The fault margins
will minimize the converter damage in the failure time.
Steady State Model
Nominal Specifications
InitializingSet Initial Values
MMC circuit Analytical Model
Optimization Variables
Electromagnetic Analytical Model of coupled Inductors
Thermal Model
Outputs satisfy the
constraints?
Circuit Constraints
Electromagnetic Constraints
Thermal Constraints
Goal function(Converter or inductor Mass Minimization)
Nonlinear Solver
Optimal Values
Initializing
Analytical Model Constraints
Numerical Solver
No
yes
Goal Function
Sending inductor size to FEM software
FEM Analysis
La=Lfem
Generate model correction factor
No
Verified Optimal Values
Yes
IGBT open circuit
IGBT short circuit
Inductor short circuit
Inductor open circuit
Fau
lt c
alc
ula
tio
n
Fault condition model
Figure 6.6 Hybrid optimization algorithm considering the fault margin
6.9. Conclusion
In this chapter, the proposed optimization plans for MMC converter were introduced and
developed. In the first section, the general parts of an optimization loop are introduced. Each
optimization loop is composed of optimization variables, mathematical model, constraints,
goal function and solver. Depending on the application, the structure of optimization
algorithm should be changed.
In this project, the optimization algorithm has chosen with different complexity levels. The
first algorithm employed the converter circuit model to find the optimal circuit value of
passive components. In this algorithm, the mechanical dimensioning of components was
neglected and the design process should be done regarding the circuit values. The goal
127
function is the total energy stored in the converter which is the summation of electrical energy
stored in the capacitors and the magnetic energy stored in the inductors. The most important
advantages of this approach are the simplicity and fast converging that could be used to find
the best optimal bond of variables. The optimization variables are the sub-module capacitor,
arm inductance, switching frequency and the modulation index.
The second algorithm employed the circuit model, electromagnetic model and thermal model
in the unique shell in combination with the dimensioning model in order to minimize the total
converter volume and mass. Unlike the first algorithm, this algorithm not only calculates the
circuit values but also the dimension parameters of the passive components. Because the high
number of optimization variables and constraints, the complexity and convergence time of
optimization algorithm is higher in comparison with the first algorithm.
The analytical model of arm inductance does not provide high precision results. The accuracy
of the analytical model varies by changing the air gap value. In order to correct the model
parameters and enhance its accuracy, a novel correction loop was presented and utilized in
the optimization loop. In this approach, the analytical model parameters are modified
regarding the finite element analysis results. This approach enhances the model accuracy,
while the optimization time does not increase so much.
Finally, the proposed hybrid optimization algorithm was combined with the fault condition
model of MMC converter. The fault analysis adds a margin to the variables in order to suffer
the fault condition in a small-time period. This margin leads to minimize the converter
damage if a fault happens.
In the next chapter, the optimization results using the proposed optimization algorithms are
presented and discussed. Two power IGBT is used in the optimization in order to investigate
the switch influence.
128
CHAPTER VII
7 Investigation of Optimization Results
7.1. Introduction
In the previous chapter, several optimization plans with the aim of minimization volume of
MMC converter were explained in details and developed regarding the MMC Active Front
End converter application. The optimization plans were presented with a different level of
complexity. The algorithms with lower complexity have a higher probability to converge.
The result of each algorithm are used by another as a guide to initialize the optimization
variables. This approach increases the probability of finding a solution and converging of
optimization loop.
The first optimization plan works based on the converter circuit model. The proposed circuit
model calculates the time-domain waveform of each electrical variables in a sinusoidal cycle.
In this plan, the mechanical dimensioning of components was neglected. The optimization
algorithm just tries to find the optimal circuit value of passive components. The goal function
is the total energy stored in the converter that is the summation of electric energy stored in
the capacitors and the magnetic energy stored in the inductors. The major constraints are the
capacitor voltage ripple, AC current THD, and the total converter efficiency.
The first optimization plan neglects the component dimensions and the magnetic parameters.
In the second plan, the analytical electromagnetic model of arm inductances and the
dimensioning model of passive components were added to the optimization algorithm. The
electromagnetic model estimates the magnetic variables regarding the core size and winding
parameters. In the optimization variables, the inductance value was replaced by the core and
winding parameters of the inductor. Also, the new constraints concerning the magnetic
analysis and inductor manufacturing were added to the constraint vector. Unlike the first plan
129
that a criterion was assumed as goal function, in this optimization plan, the dimensioning
model estimates the total converter volume and size using the dimensioning model.
The analytical electromagnetic model of inductor does not provide precise results compare
to the real data. Hence, to enhance the model accuracy, a composition of the analytical model
and finite element method was employed. In this approach, an extra loop was added to the
optimization algorithm. In each iteration, the inductor parameters are sent to the finite
element software and the results are used to modify the analytical model parameters to
enhance the model accuracy. This approach increases the model accuracy while the
optimization time does not increase very much.
As it was mentioned in chapter 2, because of the limit number of IGBT switches in high
power application, the type of IGBT could not be considered as an optimization variable. In
this research, regarding the application, two high power IGBTs with different specifications
were selected to use. The optimization is repeated for each IGBT and the results were
recorded for each of them in order to find the best switch.
In this chapter, the optimization results are investigated and discussed to the complexity of
algorithms. The results of simpler algorithms can be used as the initial values of more
complex algorithms. This solution increases the convergence probability of optimization.
Finally, the sensitivity of converter mass versus several parameters are studied and discussed.
7.2. High power IGBT specifications
As it was mentioned in chapter 2, in high power application the number of IGBTs is not
unlimited. Regarding to the nominal values and load specifications, there are a limited
number of IGBTs to choose. Hence, the IGBT is a discrete variable, which could not be
considered as optimization variables. The best solution is to do optimization using a specific
IGBT and then repeat it with the other ones.
In this dissertation, two high power IGBTs with different specifications from ABB Company
were employed to design the MMC converter [50, 74]. Table 7.1 shows some specifications
of selected IGBTs.
130
Table 7.1 Technical specifications of high power IGBTs
Parameters 5SNA1500E330305
5SNA 0750G650300
Collector-emitter voltage 3300 V 6500 V
DC collector current 1500 A 750 A
Peak collector current 3000 A 1500 A
Total power dissipation 14700 W 9500 W
Junction operating temperature -50 ℃ to 150 ℃ -50 ℃ to 125 ℃
Short circuit current 6400 A 3400 A
IGBT thermal resistance junction to case 0.0085 K/W 0.011 K/W
Diode thermal resistance junction to case 0.017 K/W 0.021 K/W
IGBT thermal resistance case to heatsink 0.009 K/W 0.009 K/W
Diode thermal resistance case to heatsink 0.018 K/W 0.018 K/W
Dimensions 190×140×38 mm 190×140×48
Turn-on switching energy (𝐸𝑜𝑛) 2150 mJ @125 ℃ 6400 mJ @125 ℃
Turn-off switching energy (𝐸𝑜𝑓𝑓) 2800 mJ @125 ℃ 5300 mJ @125 ℃
IGBT forward resistance 1.5 mΩ 3 mΩ
Diode forward resistance 0.75 mΩ 2.2 mΩ
On state collector/emitter voltage 0.95 V 1.55 V
On state diode voltage 1.125 V 1.9 V
7.3. Optimization results using proposed time-domain circuit
model
In this section, the optimization result using circuit model is investigated. The optimization
has been done for two high power IGBTs that were introduced in the previous section. The
optimization procedure was repeated for a different number of sub-modules and parallel
arms. The number of sub-modules per arm was changed between 3 and 8 and number of
131
parallel arms was changed between 1 and 3. The optimized capacitor energy stored is shown
in figure 7.1. The capacitor energy stored is the criterion to estimate the total capacitor size
which should be installed. The total capacitor size is almost constant and is independent of
the number of sub-modules and parallel arms. It seems that the capacitor size is dependent
on the voltage ripple, which will be explained in the next section.
Figure 7.1: Electric energy stored in the capacitors versus the number of sub-module per arm
Figure 7.2: Magnetic energy stored in the inductors versus the number of sub-module per arm
Figure 7.2 shows the magnetic energy stored in the inductors which represent the total size
of the inductors. The magnetic energy stored is decreased by increasing the number of sub-
modules. The reduction rate of magnetic energy increases in higher sub-module numbers.
Also, figure 7.3 shows the total energy stored in the converter which is the summation of
electric and magnetic energies.
3 4 5 6 7 8
N=1 103585 100184 105540 102949 106808 99888
N=2 98158 104942 105160 105185 107066 105528
N=3 100318 108120 106945 102918 105662 102333
80000
90000
100000
110000
120000
Ener
gy S
tore
d in
Cap
acit
ors
(J)
Number of Series Modules
3 4 5 6 7 8
N=1 3719 1351 1321 1152 797 799
N=2 3348 1482 1215 902 843 676
N=3 3520 1765 1243 1107 811 830
0
2000
4000
Ener
gy s
tore
d in
Co
up
led
ind
uct
ors
(J
)
Number of Series Modules
132
Figure 7.3: Total energy stored in the converter versus the number of sub-module per arm
Figure 7.4: Total converter efficiency versus the number of sub-module per arm
Figure 7.4 shows the total converter efficiency versus the number of series sub-modules. The
total converter efficiency is reduced by increasing the number of sub-modules and parallel
arms. By increasing the number of modules, optimization procedure decreases the switching
frequency to reduce the losses and keep the efficiency higher than 95%. Therefore, by
increasing the number of sub-modules per arm, it is possible to reduce the total converter size
respecting to the efficiency and THD constraints. Figure 7.5 shows the optimal switching
frequency. Increasing the switching frequency leads to reduce the total harmonic distortion
and enhance the converter power quality. It leads to smaller passive filter size and hence the
smaller converter volume. On the other hand, increasing the switching frequency augments
the IGBT switching losses and reduces the converter efficiency. In the constraint vector, the
3 4 5 6 7 8
N=1 107304 101535 106861 104101 107605 108007
N=2 101507 106425 106375 106088 107910 106205
N=3 103839 109885 108189 104025 106474 103164
80000
90000
100000
110000
120000
Tota
l En
ergy
Sto
red
in C
on
vert
er
(J)
Number of Series Modules
3 4 5 6 7 8
N=1 96.99 96.72 96.51 95.93 95.44 95.36
N=2 96.67 96.33 95.4 95.2 95.24 95.25
N=3 96.2 95.45 95.16 95.07 95.06 95.29
94
94.5
95
95.5
96
96.5
97
97.5
Tota
l Eff
icie
ncy
(%)
Number of Series Modules
133
total converter efficiency is kept higher than 95%. Therefore, in order to compensate the
switching losses, the switching frequency must be reduced.
Figure 7.5: Optimal switching frequency versus the number of sub-modules per arm
Figure 7.6 presents a better recognition of the effect of capacitor voltage variation on the
capacitor voltage ripple. The contour of capacitor voltage ripple versus capacitor energy
stored and a number of sub-module is shown in figure 7.7 while the arm inductance is 2.3
mH and the switching frequency is 1000 Hz. It shows that the rate of ripple reduction is
decreased by increasing the capacitor energy stored. It means that to achieve lower ripple,
more capacitor volume is needed. Also, it is possible to reduce the ripple by increasing the
number of sub-modules.
Figure 7.6: Sub-module capacitor ripple versus capacitor energy and sub-module number
3 4 5 6 7 8
N=1 1000 994 947 947 947 828
N=3 1000 948 814 696 598 496
N=2 996 949 947 884 650 647
0
200
400
600
800
1000
1200
Swit
chin
g Fr
equ
ency
(H
z)
Number of Series Modules
134
Figure 7.8 shows the effect of magnetic coupling between the upper and lower arm
inductances on the line current THD and total efficiency. It shows that the negative coupling
factor enhances the converter efficiency while reduces the power quality. On the other side,
the positive coupling factor leads to lower THD and efficiency.
Figure 7.7: Contour of capacitor ripple versus capacitor energy and sub-module number
a) THD versus coupling
b) Efficiency versus coupling
Figure 7.8: THD and total efficiency versus coupling factor
7.4. Mass Minimization of Modular Multilevel Converter
To minimize the MMC converter mass, the dimensioning model of the passive component
must be added to the optimization algorithm. In addition, in the case of arm inductance, the
electromagnetic model of arm inductance should be considered. The electromagnetic model
135
provides the magnetic parameters versus the core and winding size and parameters. Utilizing
these two models, it makes possible to estimate the component mass and change the goal
function to total converter mass.
This optimization has done utilizing two different high power IGBTs from ABB
semiconductor products. The IGBT specifications and technical data were used in the
optimization algorithm to estimate the losses. Two IGBTs with different nominal voltage and
current are employed in this research. The first one has 3.3 kV and 1500 A. regarding the
nominal value of the converter, the IGBT current is suitable but its voltage is less than the
maximum sub-module voltage. Regarding the DC link voltage and the capacitor voltage
value, at least three series sub-modules must be used to endure the converter and sub-module
voltages. The second IGBT has 6.5 kV and 750 A. The nominal current of this IGBT is not
sufficient enough to endure the converter arm current. Hence, at least two parallel branches
must be employed. In term of IGBT voltage, two series sub-modules are sufficient to work
safely. Figures 7.9 and 7.10 show the proposed MMC topologies in the case of using different
IGBTs.
SM1
SMN
SM1
SMN
.
.Ia
R
L
R
L
M
Iau
Vu1
VuN
Vl1
VlN
Vcu1
VcuN
Vcl1
VclN
SM1
SMN
SM1
SMN
.
.Ib
R
L
R
L
M
Ibu
Vu1
VuN
Vl1
VlN
Ial
SM1
SMN
SM1
SMN
.
.Ic
R
L
R
L
M
Icu
Vu1
VuN
Vl1
VlN
Idc
Vdc/2
Vdc/2
Figure 7.9: MMC topology using 3.3 kV/1500 A IGBT
136
SM1
SMN
SM1
SMN
.
.Ia
R
L
R
L
M
Iau
Vu1
VuN
Vl1
VlN
Vcu1
VcuN
Vcl1
VclN
SM1
SMN
SM1
SMN
.
.Ib
R
L
R
L
M
Ibu
Vu1
VuN
Vl1
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Ial
SM1
SMN
SM1
SMN
.
.Ic
R
L
R
L
M
Icu
Vu1
VuN
Vl1
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Idc
Vdc/2
Vdc/2
SM1
SMN
SM1
SMN
.
.Ia
R
L
R
L
Iau
Vu1
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Vcu1
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SM1
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.
.Ib
R
L
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SM1
SMN
SM1
SMN
.
.Ib
R
L
R
L
Ibu
Vu1
VuN
Vl1
VlN
Figure 7.10: MMC topology using 6.5 kV/750 A IGBT
7.4.1. Selection of inductor core topology
In chapter 4, two inductor core topologies were introduced and their dimensioning models
were extracted. In the first topology, there is the possibility to adjust the coupling factor,
while in the second one, the coupling is not an independent variable and it depends on the
core parameters. Using the second topology, the coupling factor will be close to the unit
coupling factor.
The circuit analysis of MMC converter shows that the high coupling factor enhances the
converter performance in terms of ripple, THD, and efficiency. Hence, in this application,
the coupling factor must be kept at the maximum to optimize the performance. In order to
maximize the coupling factor in the center leg topology, the middle air gap must be increased.
The best coupling will be obtained by elimination of the center leg. Therefore, we will
achieve the centerless topology.
The main optimization goal in this research is to minimize the total converter mass. Hence,
the inductor mass as the important part of the converter must be minimized. Utilizing the
dimensioning model proposed in chapter 4, the inductor mass using two proposed topologies
was calculated in the same circuit specifications. Table 7.2 shows the total inductor mass of
center leg and centerless topologies. The results show that with the same specifications, the
137
centerless topology provides lower total mass. Hence, the optimization will continue using
centerless topology.
In the case of multi-leg topology, the separated inductor core for each leg is used. For
example in the case of two parallel legs, two inductor cores per phase were utilized.
Table 7.2: Comparison of total mass of two different inductor core topologies
Inductance
(mH)
Coupling
factor
Mass -type 1
(Kg)
Mass-type2 (Kg)
1 mH 0.92 354 420
2mH 0.91 630 725
3mH 0.91 1025 1180
7.4.2. Optimization using 3.3KV/1500A IGBT
The optimization algorithm solves the mathematical equations of the analytical model to find
the best value of the optimization variables considering technical and manufacturing
constraints. The number of sub-modules per arm is not considered as an optimization
variable. Hence, the optimization has been repeated for the different sub-modules number.
Figure 7.11 and 7.12 show the optimal arm inductance and sub-module capacitor values,
respectively. The arm inductance value decreases by increasing the number of sub-modules.
By increasing the number of the sub-module, the current harmonic reduces, therefore a
smaller arm inductance is required. On the other hand, the sub-module capacitor value
augments by increasing the number of sub-modules. Connecting the capacitor in series leads
to the smaller equivalent capacitor. Therefore, to achieve the same voltage ripple the
capacitance should be augmented. It is clear that utilizing the coupled inductor strongly
affects the arm inductance and capacitor value and decreases required passive components.
138
Figure 7.11: Optimal arm inductance value versus number of sub-modules per arm
Figure 7.12: Optimal sub-module capacitor value versus number of sub-modules per arm
Figure 7.13 shows the arm inductance mass that is computed according to the analytical
model. The inductor mass decreases by increasing the sub-module number, especially in the
lower sub-module numbers. Also, in the case of using coupled inductor, the inductor mass
strongly decreases in comparison to the uncoupled inductor.
Figure 7.14 shows the total sub-module capacitor mass that is calculated regarding the
capacitor mass function that was obtained from the fitting algorithm. The capacitor mass
depends on the capacitance and the capacitor voltage. In this application, according to the
load specifications, the capacitor mass increases on the small sub-module numbers. In this
state, the capacitance augments and therefore the capacitor mass increases. Then, in the
139
higher sub-module numbers, the capacitor mass decreases. By increasing the sub-module
number, the capacitor voltage reduced. It leads to reduce the capacitor mass according to the
capacitor mass function. The interesting point in this figure is the effect of coupled inductor
on capacitor mass. The total capacitor mass in the state of using coupled inductor is lower in
comparison to the state of using uncoupled inductors. On the other word, utilizing the coupled
inductors, it is possible to reduce the inductance and capacitor mass at the same time.
Figure 7.13: Optimal total inductor mass versus number of sub-modules per arm
Figure 7.14: Optimal total capacitor mass versus number of sub-modules per arm
140
Figure 7.15 shows the total converter mass which consists of inductance, capacitor and
semiconductor mass. The total converter mass decreases by increasing the sub-modules
number. Also, it is clear that utilizing the coupled inductor strongly reduces the total
converter mass.
Figure 7.15: Optimal converter mass versus number of sub-modules per arm
7.4.3. Optimization using 6.5KV/750A IGBT
The next optimization has done using 6.5KV/750A IGBT. Regarding the IGBT
specifications, the MMC converter must have at least two parallel branches in each arm.
Hence, the arm current is divided by two. The IGBTs current and inductor current are almost
divided by two. It affects the capacitor and inductor values. Also, the losses parameters of
this IGBT is different which affects the switch losses and converter efficiency.
Regarding IGBT voltage, the optimization is started with two sub-modules per arm and two
parallel branches in each arm. The main constraints are the capacitor voltage ripple less than
20%, the THD current less than 2% and Δ𝑇 less than 60℃. Figure 7.16 shows the optimal
arm inductance value versus the number of series sub-modules per arm in the case of using
coupled and non-coupled inductors. The arm inductance value strongly reduces by increasing
the number of sub-modules. Also, the reduction rate of inductance versus the sub-module
number decreases in high sub-module numbers. In comparison with the results of pervious
IGBT, the arm inductance value is higher. All parts of arm current are divided by two except
141
the second order part of circulation current. It leads to bigger arm inductance value to
eliminate the harmonics and provide the same converter power quality.
Figure 7.16 The optimal arm inductance versus the number of sub-modules per arm
Figure 7.17 shows the optimal sub-module capacitor values versus the number of sub-
modules per arm. The capacitor value augments by increasing the sub-module numbers. The
increase rate of the capacitor is almost linear while utilizing of coupled inductance decreases
the capacitor value and enhances the converter performance. Due to decreasing the sub-
module current, the capacitor ripple will decrease. Hence, the sub-module capacitor will be
lower compared to the previous IGBT.
Figure 7.17 Optimal value of the sub-module capacitor versus the number of sub-modules per arm
142
The investigation of circuit value of the components is not sufficient to explain the
component mass. Figure 7.18 shows the total inductor mass versus the number of sub-
modules per arm in the case of coupled and non-coupled inductor. Increasing the number of
sub-modules per arm leads to decrease the inductor mass. Also, utilization of coupled
inductor provides lower inductor mass.
Figure 7.18 Total inductor mass versus the number of sub-modules per arm
Figure 7.19 Total capacitor mass versus the number of sub-modules per arm
Figure 7.19 shows the total sub-module capacitor mass versus the number of sub-modules
per arm. The capacitor mass depends on the capacitance and capacitor voltage. As it was
1159
612
435359
314202 213
191 171
1470
962
650
511 467405 426 408 404
0
200
400
600
800
1000
1200
1400
1600
2 3 4 5 6 7 8 9 10
Ma
ss (
Kg
)
Number of submodules per arm
Total Inductor mass
coupled
uncoupled
379401
505439
384
298
201150
140
401
480
591 591545
508
380
280235
0
100
200
300
400
500
600
700
2 3 4 5 6 7 8 9 10
Ma
ss (
Kg
)
Number of submodules per arm
Total Capacitor mass
coupled
uncoupled
143
shown in figure 7.19, the variation of capacitor mass is not linear. The capacitance value and
capacitor voltage do not increase in the same direction. Hence, first, the capacitor mass
increases due to increase the capacitance and then it reduces due to voltage reduction.
The total converter mass includes the inductor mass, the sub-module capacitor mass, and
IGBTs. Figure 7.20 shows the total converter mass versus the number of sub-modules per
arm. The curve of total converter mass is a descending curve versus the number of sub-
modules per arm. The total converter mass is higher compared to the previous optimization.
Figure 7.20 Total converter mass versus the number of sub-modules per arm
7.5. Optimization Results using Hybrid Analytical Model
In chapter 6, the proposed hybrid analytical model of arm inductance and the proposed
optimization algorithm using hybrid model was introduced and investigated. The hybrid
model is an innovative approach to enhance the model accuracy while the optimization time
does not increase very much. In this section, the optimization results using hybrid
optimization model are explained and discussed. Also, the effect of some important
constraints such as temperature rise, flux density, THD and voltage ripple on converter mass
are investigated.
1592
10651003
943 872703
607 602 601
1929
1528
1301 1247 1178 1116970 949 929
0
500
1000
1500
2000
2500
2 3 4 5 6 7 8 9 10
Ma
ss (
Kg
)
Number of submodules per arm
Total Converter mass
coupled
uncoupled
144
7.5.1. Optimization results using 3.3 kV/1500 A IGBT
The optimization results using the proposed hybrid algorithm are shown in Figures 7.21 to
7.26. The nominal values and constraints are the same as previous analysis. Figure 7.21 and
7.22 show the optimal inductance and capacitor values in states of utilizing coupled and
uncoupled inductors versus a number of series sub-modules per arm respectively. In both of
them, the use of coupled inductors reduces the required inductance and capacitor values
which may lead to lower converter cost. By increasing the number of sub-modules, the
inductance value reduces while capacitor value is increasing. Also, it should be considered
that the reduction rate of inductance is more in lower sub-module numbers while the rate of
increase of capacitor value will be higher by rising the number of series sub-modules. It
means the midpoints could be the desired points.
Figure 7.21: The optimal arm inductance value versus the sub-modules per arm
4.8
3.062.62
2.191.83
1.35 1.2 1.05
2.56
0.806 0.801 0.659 0.496 0.49 0.46 0.45
0
1
2
3
4
5
6
3 4 5 6 7 8 9 10
Ind
uct
ance
valu
e (m
H)
Number of sub-modules per arm
Arm inductance value
uncoupled
coupled
145
Figure 7.22: The optimal sub-module capacitor value versus the sub-modules per arm
The electromagnetic model of inductor provides a mathematical relation between the
inductance value and its physical size. The hybrid optimization algorithm uses an analytical
model to calculate and finite element analysis to verify and modify the model and finally, it
estimates the physical size of the optimal inductor. In the case of a capacitor, the final mass
depends on the type of capacitor and its technical data which provides by manufacturing
company. Utilizing the capacitor value and maximum voltage, it is possible to find a
mathematical function which estimates the capacitor mass dependent on the capacitance and
voltage.
Figure 7.23 and 7.24 show the estimated inductor and capacitor mass versus the number of
sub-modules per arm which were calculated using hybrid optimization algorithm. The curve
of inductor mass is descending the same as the inductor value in figure 7.21. Also, the
inductor mass reduces in presence of coupled inductors. Figure 7.23 shows the capacitor
mass which is estimated based on capacitor datasheet. The curve of capacitor mass is
different from the curve of capacitor value in figure 7.22. First, the capacitor voltage is high
and capacitance is low. As a result, capacitor mass is low. By increasing the number of the
sub-module, the voltage reduces while capacitance increases. The variation of capacitor mass
depends on the mass function that is provided by the manufacturer. In this case, it leads to
increase the capacitor mass. In continue, the capacitor value increases while the capacitor
4.546.07
7.75
9.53
11.5
14.1
16.718.2
3.134.2
5.656.65
7.828.89
10.111.25
0
4
8
12
16
20
3 4 5 6 7 8 9 10
cap
acit
or
valu
e (
mF)
Number of sub-modules per arm
Sub-module capacitor value
uncoupled
coupled
146
voltage effectively reduces by increasing the number of sub-modules. Therefore, the effect
of voltage overcomes the capacitor value and the capacitor mass decreases.
Figure 7.23: The optimal arm inductance mass versus the sub-modules per arm
Figure 7.24: The optimal capacitor mass versus the sub-modules per arm
1090
699603
520453
396 385 374
726
227 192 159 144 133 124 118
0
200
400
600
800
1000
1200
3 4 5 6 7 8 9 10
Mas
s (K
g)
Number of sub-modules per arm
Total inductor mass
uncoupled
coupled
518
579605 590
539
480
410 388354
393427
384
318
213172
145
0
100
200
300
400
500
600
700
3 4 5 6 7 8 9 10
Mas
s (K
g)
Number of sub-modules per arm
Total capacitor mass
uncoupled
coupled
147
Figure 7.25: The optimal converter mass versus the sub-modules per arm
Figure 7.26: Total converter efficiency versus the sub-modules per arm
According to the figure 7.23 and 7.24, by increasing the number of the sub-module, it is
possible to reduce the total converter mass. Figure 7.25 shows the total converter mass which
includes capacitor, inductor and IGBT mass. It is clear that the converter mass is reduced by
increasing the number of sub-module per arm. Also, utilizing coupled inductor intensely
affects and reduces the converter mass. It should be noted that the total converter efficiency
1652
1336 12811197
1094992 926 907
1124
678 692 630 564462 427 408
0
400
800
1200
1600
2000
3 4 5 6 7 8 9 10
Mas
s (K
g)
Number of sub-modules per arm
Total converter mass
uncoupled
coupled
97.68
97.05
96.4
95.73
95.04 95.11 95 95.01
97.897.44
96.74
96.19
95.59
95.01 95 95
93.5
94
94.5
95
95.5
96
96.5
97
97.5
98
98.5
3 4 5 6 7 8 9 10
Effi
cien
cy (%
)
Number of sub-modules per arm
Total converter efficiency
uncoupled
coupled
148
is reduced too. Figure 7.26 shows the total converter efficiency in case of using coupled and
uncoupled inductors. Total converter efficiency is dependent on IGBT losses and inductor
losses. The switch losses increase and as a result, the efficiency is reduced by increasing the
number of sub-modules and IGBTs. However, the total efficiency could be mentioned as a
constraint to find the minimum converter mass. In this research, the total efficiency should
be kept greater than 95%.
7.5.2. Optimization results using 6.5 kV/750 A IGBT
This section represents the optimization result using hybrid optimization algorithm and
second type IGBT (6.5 kV/750 A). Utilizing the 6.5 kV IGBT, it is possible to decrease the
number of sub-modules to two sub-modules. Due to the nominal current of the converter, at
least two parallel branches must be employed in each arm.
Figure 7.27 Total inductor mass versus the number of sub-modules per arm
Figure 7.27 shows the total inductor mass versus the number of sub-modules per arm in term
of using coupled and non-coupled inductors. The value of total inductor mass is lower
compared to the optimization using the analytical model. The correction loop strongly
enhances the accuracy of the analytical model. Also, it should be considered that the coupled
inductor intensely reduces the total inductor mass.
1095
565
335295
242177 185 165 162
1320
862
598
445372
315 335 303 285
0
200
400
600
800
1000
1200
1400
2 3 4 5 6 7 8 9 10
Mas
s (K
g)
Number of sub-modules per arm
Total Inductor mass
coupled
uncoupled
149
Figure 7.28 shows the total sub-module capacitor mass versus the number of sub-modules
per arm that was computed using hybrid optimization algorithm. Unlike the inductor mass,
the capacitor mass does not change very much compared to the optimization results using
analytical model.
Figure 7.28 The total sub-module capacitor mass versus the number of sub-module per arm
Figure 7.29 shows the total converter mass that is the summation of inductor mass, capacitor
mass, and IGBT mass. The total converter mass is significantly higher in a low number of
series sub-modules than the same condition using another IGBT. By increasing the number
of sub-modules, the first IGBT provides lower converter mass. Also, it is clear that utilizing
of coupled inductors effectively reduce the converter size and enhance the converter
performance.
Figure 7.30 shows the total converter efficiency versus the number of sub-modules per arm
in terms of using coupled and non-coupled inductors. By increasing the number of sub-
modules, the switch losses will increase and the total converter efficiency is reduced. Hence,
the optimization algorithm decreases the switching frequency to keep the efficiency more
than 95%.
350
398
500
421382
304
195158 142
392
485
598 589543 528
372
286242
0
100
200
300
400
500
600
700
2 3 4 5 6 7 8 9 10
Mas
s (K
g)
Number of sub-modules per arm
Total Capacitor mass
coupled
uncoupled
150
Figure 7.29 Total converter mass versus the number of sub-modules per arm
Figure 7.30 Total converter efficiency versus the number of sub-modules per arm
7.6. Parameter sensitivity analysis
Sensitivity analysis is a type of study that investigates and computes the goal function
sensitivity versus the different design parameters. This study provides very interesting results
15031050
951861
798 684612 584 594
1770
14341312
11791089 1046
939850 817
0
200
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600
800
1000
1200
1400
1600
1800
2000
2 3 4 5 6 7 8 9 10
Mas
s (K
g)
Number of sub-modules per arm
Total Converter mass
coupled
uncoupled
97.88
97.12
96.22
95.3995 95 95 95 95
97.84 97.24
96.51
95.74
95 95 95 95 95
93.5
94
94.5
95
95.5
96
96.5
97
97.5
98
98.5
2 3 4 5 6 7 8 9 10
Effi
cien
cy (%
)
Number of sub-modules per arm
Total converter efficiency
uncoupled
coupled
151
that could use in the design process. In the case of MMC converter, especially in high power
applications, designers are confronted by many parameters and variables which make
complex to select the appropriate parameters to design. The sensitivity study presents the
sensitivity of goal function for one parameter while others are constant.
In this section, the sensitivity of total converter mass function against some important
parameters such as maximum temperature rise, maximum core flux density, maximum THD,
maximum capacitor voltage ripple, fault margin and inductor core material are investigated
and discussed.
7.6.1. Sensitivity analysis of maximum Temperature Rise
In the single objective optimization algorithm, there is a goal function and number of
constraints. In this research, the main goal function is to minimize the total converter mass.
Other functions such as temperature rise, capacitor voltage ripple, and THD are considered
as constraints. Therefore, the objective sensitivity versus constraints variation should be
investigated. Table 7.3 shows the converter parameters that are used in optimization.
Table 7.3: Converter parameter and thermal coefficient
Parameters Value Unit
Number of sub-modules 3-8
DC Link Voltage 5000 V
Capacitor voltage ripple 0.2𝑉𝑑𝑐/𝑚 V
Copper thermal exchange 80 𝑊/𝑚2𝐾
Iron thermal exchange 80 𝑊/𝑚2𝐾
Ambient temperature 40 ℃
Modulation Index 65 %
One of the most important constraints which intensely affects the converter mass is the
maximum temperature rise. The optimization has been done by changing the maximum
temperature rise as a constraint in the different sub-module number per arm. Figure 7.31
152
shows the optimal inductor value versus the maximum temperature rise. Increasing the
maximum temperature rise intensely reduces the inductor mass.
Figure 7.31: Optimal inductor mass versus maximum temperature rise
Also, the total converter mass versus the maximum temperature rise is shown in figure 7.32.
The optimal mass converter is reduced by increasing the maximum temperature rise. The
high value of the capacitor mass in comparison with the inductor mass leads to move the
results versus the sub-module numbers.
Figure 7.32: Total converter mass versus the maximum temperature rise
153
7.6.2. Investigation the effect of maximum Flux Density on Converter
Mass
One of the most important constraints of the arm inductance is the maximum core flux
density. The maximum flux density is dependent on the material and its B-H characteristic.
Regarding the saturation effect, power electronic designers try to keep the operating point in
the linear region. The silicon electrical steel which is used to design the electrical machine
and transformer is in the linear region until 0.9T and in the saturation region until 1.8T.
Increasing the maximum flux density of the core might lead to reducing the inductor mass
while it might lead to increase the core losses. Figure 7.33 shows the contour of optimal
inductor mass versus the sub-module number and the maximum flux density of the core. It
is clear that the inductor mass is reducing while the maximum flux density is increased.
Therefore, it proves the effect of maximum flux density in reducing the arm inductance mass.
Figure 7.33: The contour of optimal inductor mass versus the sub-modules number and the
maximum flux density
154
7.6.3. Investigation the effect of maximum THD on Converter Mass
Finding the relation between the component mass and the converter THD is dependent on
lots of parameters such as passive component values, circulating current, switching
frequency, the number of sub-module per arm, switching index, switch characteristics and
etc. Hence, extracting the relation between THD and converter mass will be more complex
than other constraints. Regarding the circulating current formula (Eq. 3.25) which was
investigated in chapter 3, shows that there is a point of discontinuity that could lead to
augment the circulating current. The circulating current is composed of the second harmonic
which increases the THD value. The point of discontinuity is dependent on the switching
index, capacitor value, self-inductance and mutual inductance and the number of sub-module
per arm. Figure 7.34 shows the point of discontinuity for a different number of sub-modules.
The value of arm inductance and sub-module capacitor must be chosen to avoid this point.
Figure 7.34 The discontinuity value versus number of sub-modules per arm
Figure 7.35 shows the THD value of three phase current versus the arm inductance value and
number of sub-module per arm. It is clear that the THD value reduces by increasing the arm
inductance. Also, the THD value is smaller in the higher sub-modules per arm. But it does
not sufficient to be sure about the performance of the converter. The inductor mass should
be investigated to avoid the point of discontinuity.
0.00E+00
4.00E-06
8.00E-06
1.20E-05
1.60E-05
2.00E-05
3 4 5 6 7 8 9 10
(L-M
)Csm
The number of sub-modules per arm
The point of discontinuty (A=1)
155
Figure 7.35 The THD value of input current versus the arm inductance value
Figure 7.36 shows the total inductor mass versus the arm inductance and sub-module number
per arm. Theoretically, the inductor mass should be reduced by decreasing the arm inductance
value if they work at the same operating point. But, in the case of MMC converter, the
circulating current value could affect the inductor mass. If 𝐴 ≫ 1 𝑜𝑟 𝐴 ≪ 1 then the inductor
mass is dependent on the inductance value. The results show that the inductor mass reduces
by decreasing the arm inductance except one for point. At 𝑚 = 3 when the arm inductance
becomes smaller than 1.5 𝑚𝐻 the inductor mass augments. This is the effect of discontinuity
which strongly increases the circulating current.
0
2
4
6
8
10
12
1 1.5 2 2.5 3 3.5 4 4.5
THD
(%
)
Arm inductance value (mH)
THD of input current versus inductance value
m=3 m=4
m=5 m=6
m=7 m=8
156
Figure 7.36 The total inductor mass versus the arm inductance value
7.6.4. Investigation of the effect of Capacitor Voltage Ripple on Converter
Mass
The constraint that strongly affects the capacitor mass is the voltage ripple of the sub-module
capacitors. To enhance the converter power quality and the capacitor safety, the capacitor
ripple should be minimized. The voltage ripple reduction augments the capacitor mass
regarding the capacitor mass function. Figure 7.37 shows the optimal total capacitor mass
versus the voltage ripple percentage of sub-module capacitors. The capacitor mass
augmentation is not linear. The capacitor mass and the final converter price intensely
increases to achieve the low voltage ripple rates. Hence, the constraints must be selected
carefully to achieve a suitable balance between the converter quality and the final price.
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5
ind
uct
ance
Mas
s (k
g)
Arm inductance (mH)
The Inductor mass versus arm inductance value
m=3 m=4 m=5
m=6 m=7 m=8
157
Figure 7.37: The total converter mass versus the capacitor ripple
7.6.5. Sensitivity analysis converter mass against Fault margin
In chapter 5, the fault margin of MMC converter and the proposed optimization algorithm
that considers the fault margin was investigated and explained. The fault margin guarantees
the functionality of the components in a transient fault condition. The fault margin is defined
based on a percentage of nominal converter values. Various types of fault affect different
variables such as capacitor voltage, inductor current and IGBT’s voltage/current. The fault
model computes all variables in a fault condition and it computes the converter mass
regarding the fault values and fault margin.
Utilizing the proposed optimization model with fault margin block, the total converter mass
is computed versus the number of sub-modules per arm and different fault margin value. The
optimization has done using two different IGBTs. Figure 7.38 shows the total converter mass
using 3.3 kV IGBT versus the number of sub-modules per arm and different fault margins.
Figure 7.39 shows the same results for 6.5 kV IGBT. The results show that the converter
mass sensitivity is higher in a lower number of sub-modules. Increasing the number of sub-
modules reduces the converter mass sensitivity to the fault condition. Hence, to increase the
converter stability and reliability, the number of sub-modules should be increased.
158
Figure 7.38 Total converter mass sensitivity against fault margin for 3.3KV IGBT
Figure 7.39 Total converter mass sensitivity against fault margin for 6.5KV IGBT
159
7.7. Conclusion
In the previous chapter, several global optimization algorithms were proposed and explained in order
to minimize the MMC converter volume and mass. The optimization algorithms were presented with
a different level of complexity. The first optimization model employs the time-domain circuit model
to compute the circuit value of passive components and minimize the total energy stored in the
converter. In the second optimization algorithm, the electromagnetic model of arm inductance and
dimensioning model of passive components were added to the optimization algorithm. The final
optimization algorithm has an internal correction loop using finite element approach in order to
enhance the electromagnetic model accuracy. Also, the fault margin calculation block was added to
the optimization algorithm to compute the components parameters based on the fault condition.
In this chapter, the optimization results of MMC converter using variously proposed optimization
algorithm were presented, investigated and discussed. The optimization has done using two
high power IGBT with different nominal values. The optimization results show that by
increasing the number of sub-modules per arm, the total inductor mass is reduced. The total
capacitor mass depends on the capacitor mass function that is provided by the manufacturer.
The total converter mass which is consisted of inductor mass, capacitor mass and IGBTs is
reduced by increasing the sub-module number per arm.
Also, the optimization has done in terms of utilizing coupled and non-coupled inductor. The
results prove that utilization of coupled inductors strongly increases the MMC performance
and reduce the inductor and converter mass. Another interesting point is the effect of coupled
inductors on the sub-module capacitor value. The lower sub-module capacitor is needed
when the coupled inductors are employed. Finally, a parameter sensitivity analysis has done
to determine the effects of several parameters on the total converter mass.
160
CHAPTER VIII
8 Conclusion and Future Researches
8.1. Conclusion
In the field of electrical energy conversion, Modular Multilevel Converter (MMC) have
emerged in the recent years as an attractive solution for high/medium power and voltage
applications. These topologies of power electronics converters can now be used up to
900MW and higher in AC/DC & DC/AC applications like High-Voltage Direct Current
(HVDC) light transmissions systems, Flexible Alternating Current Transmissions Systems
(FACTS), hydro pumped storage, wind energy conversion, marine propulsion and railway
traction drives.
The MMC topologies are composed of elementary commutation cells using standard
Integrated Gate-Commutated Thyristor (IGCT) or insulated gate bipolar transistors (IGBT)
associated with capacitors. They are easily scalable to be adapted to different voltage and
current levels. Their modularity can be used to improve the redundancy and the fault-tolerant
operation, and to implement the multilevel control concept to provide a specified AC
waveform quality with a reduced size of filtering magnetic components.
The main objective of this research project is to propose a generic and versatile sizing
methodology of MMC converters based on a global optimization approach constrained by
the specifications of each application. With such an approach, it is possible to a fixed set of
input-output specifications of the application to compare the performances of different MMC
161
topologies in terms of converter volume and weight including passive (magnetic &
capacitive) components, cell cooling and insulation system, the number of cells and switches,
global efficiency and power quality. The methodology will be validated on different
applications including a high pulsed power supply for particle accelerator electromagnets
with an active front-end.
In this dissertation, a systematic optimization approach has been proposed and developed in
order to minimize high power modular multilevel converters considering the technical,
thermal and manufacturing constraints. Capacitors volume, arm inductances size and a
number of sub-modules per arm are the most important parameters which affect the final
converter mass. In the case of variable load converters, the steady-state model is not
appropriate to analyze the converter. Hence, the transient converter model should be
employed to analyze MMC circuit. The steady-state model is utilized in the initializing step.
The analytical model of the converter and coupled inductors are utilized as a part of
optimization algorithm to find the capacitor value, arm inductance value, and size and a
number of sub-modules per arm. The electromagnetic core has saturation characteristic in
reality. The electromagnetic analytical model accuracy is not sufficient enough in
comparison with the real data to present the saturation effect. On the other side, utilization of
complex model such as finite element model in the optimization loop is time-consuming. In
this research, a novel hybrid optimization algorithm by a combination of analytical model
and finite element model has been proposed to investigate the saturation effect of
electromagnetic core and enhance the model preciseness. The proposed hybrid optimization
algorithm was employed to optimize the high power Active-Front-End converter for the
application. The result proves that couple inductors reduce the converter mass and improve
the total performance. Also, according to the possibility of core saturation, the performance
of MMC converter has been evaluated in a fault condition.
Chapter 1 presented a brief history of the modular multilevel converter. Some of the
important researches on the field of MMC converter has presented and studied. In the
literature, MMC converters were investigated from different points of view. Literature
investigation proved that the global optimization of MMC converter is a subject that was
rarely investigated in the previous works. Then the most important types and topologies were
introduced and their advantages and limitations were discussed. The structure of neutral point
162
converter, flying capacitor, and modular converter were investigated. Also, MMC converters
were sorted base on its applications. Finally, the goals of this dissertation were explained and
the methodology and approach which should be employed were introduced.
In chapter 2, the systematic design procedure of MMC converter was presented and reviewed.
First of all, the important parameters and variables of MMC converter were determined. The
adjustable MMC parameters and variables were discussed and explained. Various analysis
tools that are needed to investigate the MMC structure are introduced. The conventional
approach which is used to design and size the MMC components were investigated. Finally,
several integrated analysis models with different level of complexity which consisted of
circuit model, electromagnetic model, thermal model, and dimensioning model were
proposed and presented in order to global analysis of MMC converter.
Chapter 3 presented a modified circuit model of MMC converter which is suitable to estimate
time-dependent variables such as ripple and THD. The conventional average model of MMC
converter neglects the effect of switching frequency and uses the average switching function
to calculate the steady-state values. A time-domain circuit model was proposed and
developed in order to calculate the precise value of time-dependent parameters such as
capacitor voltage ripple, circulating current and total harmonic distortion. The accuracy of
the proposed model was verified using Simulink/Matlab.
The circuit model analyzes the arm inductances as circuit component and neglects the
magnetic and dimensioning variables. Chapter 4 presented the analytical electromagnetic
model of arm inductances. The electromagnetic model estimates the magnetic variables such
as magnetic flux, inductance value, reluctance and magnetic losses regarding the core and
winding parameters. In addition, the magnetic and circuit model of saturable inductance in
the case of coupled and non-coupled inductors. Utilizing the analytical models, the
dimensioning model of inductor and capacitors were extracted. Also, the thermal model of
arm inductances and semiconductor switches were presented in chapter 4. Finally, the finite
element analysis method was presented in order to achieve more precise results in arm
inductance analysis.
In chapter 5, the MMC converter in the fault condition was investigated. In high power, it is
very important to control and protect the system against the faults in order to minimize the
damage and cost. Hence, the converter should be designed with a fault margin to increase the
163
component capability to endure in the fault condition for a short time. In chapter 5, various
standard MMC converter faults were presented and investigated. Also, the behavior of
different converter variables in the fault condition was studied. Finally, a close loop control
system was proposed and simulated to control the converter in the fault condition.
In chapter 6, the most important parts of an optimization algorithm were presented and
investigated. Various strategies of MMC converter optimization were proposed and
developed. The proposed optimization algorithms were sorted regarding its complexity level.
The first optimization algorithm employs the time-domain circuit model to minimize the total
energy stored in the converter. It neglects the dimension of passive components. The second
optimization model utilizes the circuit model in combination with the electromagnetic model
of arm inductances and dimensioning model of passive components. This optimization
algorithm minimizes the total converter mass and searches to find the optimal size of
capacitor and inductor. The last optimization model utilizes an internal correction loop using
finite element method in order to correct the analytical model parameters based on the finite
element analysis results. It is an innovative approach which leads to enhance the model
accuracy while the optimization time does not increase very much.
In chapter 7, the optimization results of the various proposed optimization algorithms were
presented, investigated and discussed. The total inductor mass, capacitor mass, and converter
mass were calculated versus the number of sub-modules per arm in terms of using coupled
and non-coupled inductors. The THD of AC current, capacitor voltage ripple, and total
converter efficiency were considered as the constraints. Finally, the sensitivity of total
converter mass versus the variation of several variables was investigated.
8.2. Future Researches
The climate change and global warming increase the tendency to employ the renewable
energies such as the wind and solar generation. In the last decade, demands too high power
electronic converters have intensely augmented in order to use in renewable generation,
electric railways, HVDC and other power applications. Emerging the modular multilevel
converters increased the tendency to employ high power converters which provide the high
164
global efficiency and power quality. MMC converter provides high power quality using low
switching frequency that leads to low switching losses and high converter efficiency.
Regarding the high number of components, MMC converters have two weakness in
comparison to the conventional converters; high volume of passive components and high
possibility to enter to fault condition due to the number of components. By increasing the
converter voltage, the number of sub-modules an as result, the number of components will
intensely increase. It leads to increase the size of the converter and made it so expensive. On
the other hand, increasing the number of components augments the chance of fault and
malfunction in the converter. It reduces the reliability of the converter.
As a future activity, it is recommended to investigate the fault tolerant optimization that
considers a margin for each component to resist under the fault condition. Regarding the
modularity of MMC converter, it will lead to minimizing the converter maintenance and
repairing cost. Also, according to the new information, the appropriate protection setting
could be chosen for each section.
165
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