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Modelling and Energy Management for DC Microgrid
Systems
by
Kyle Everett Muehlegg
A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science
Graduate Department of Electrical and Computer Engineering University of Toronto
c© Copyright 2017 by Kyle Everett Muehlegg
Abstract
Modelling and Energy Management for DC Microgrid Systems
Kyle Everett Muehlegg
Master of Applied Science
Graduate Department of Electrical and Computer Engineering University of Toronto
2017
Greenhouse Gas (GHG) emissions and climate change has generated a need for renew-
able energy sources. DC microgrids require less complex power conversion and commu-
nication equipment, making it a promising candidate for renewable energy integration.
This thesis investigates modelling techniques to demonstrate modularity and scalabil-
ity of DC microgrid systems. Specifically, a flexible state-space modelling technique is
developed to accurately represent a complete DC microgrid system to investigate the
effects additional energy storage media and generation sources have on stability. An
autonomous energy management scheme is proposed to further DC microgrid robustness
and reliability. The goal of the thesis is to further prove that DC microgrids can operate
as an alternative to the traditional AC grid infrastructure.
ii
Acknowledgements
I have been learning and developing as a student at U of T for 6 years now, and this
thesis represents the end of that amazing journey. I feel confident in my abilities thanks
to my professor, my friends and my family.
First and foremost, I would like to thank my supervisor, Professor Peter W. Lehn,
for being an exceptional mentor with his guidance, knowledge and insight that made
my success possible. His leadership has allowed me to achieve the goals that I always
dreamed of.
Secondly, I would like to thank NSERC for their financial support to sponsor my
research.
Thirdly, I would like to thank Professor Aleksander Prodic for the recommendation
to pursue Masters research. Although I did not originally intend to, I am glad I chose
this option and am eternally grateful for his initial push.
Fourthly, I would like to thank my fellow graduate students and post-doctoral fel-
lows, Ruoyun Shi, Amrit Singh, Sepehr Semsar, Mike Ranjram, Sebastian Rivera, Rafael
Oliveira and Caniggia Diniz for always being willing to help with any questions I had
and creating everlasting friendships.
Finally, I would like to thank my mother, Marilyn Muehlegg, my father, Peter Mueh-
legg, and sister, Danielle Muehlegg, for their everlasting love and support. I especially
would like to thank my father, who passed away during my final year of my undergradu-
ate degree. His support during my education was beyond what I could ever ask for and
I dedicate this thesis as a reminder of the success I achieved in life thanks to him.
iii
Contents
Acknowledgements iii
List of Figures vii
List of Tables xi
1 Introduction 1
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Converter Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Energy Management: Droop Control in Low Voltage AC Microgrids 4
1.1.3 Turbine-Governor Control & Automatic Generation Control . . . 4
1.1.4 Power Sharing in DC Systems . . . . . . . . . . . . . . . . . . . . 6
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 High-Level DC Microgrid Layout . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 DC Microgrid System Modelling 10
2.1 Component State Space Models . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Battery ESS Model . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Solar Converter State-Space Model . . . . . . . . . . . . . . . . . 22
2.1.3 VSC State-Space Model . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.4 Line State-Space Model . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Connecting State-Space Models . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Line Inductance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 State-Space Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 State-Space Model Verification . . . . . . . . . . . . . . . . . . . 39
2.4.3 Eigenvalue and Participation Factor Verification . . . . . . . . . . 40
iv
2.4.4 Eigenvalue and Participation Factor Analysis . . . . . . . . . . . . 45
2.5 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Scalability Analysis 48
3.1 Scalable State-Space Model . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 Scalable Battery Model . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 Scalable Solar Model . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Model Verification: Simulation Results . . . . . . . . . . . . . . . . . . . 52
3.2.1 Battery Scaling Simulation . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Solar Scaling Simulation . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Scalability Analysis: Eigenvalue Movement . . . . . . . . . . . . . . . . . 56
3.3.1 Battery Model Eigenvalue Movement . . . . . . . . . . . . . . . . 56
3.3.2 Solar Model Eigenvalue Movement . . . . . . . . . . . . . . . . . 56
3.4 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Autonomous Energy Management Method 59
4.1 Energy Management Justification . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 SOC Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 Overcharge Protection (OCP) . . . . . . . . . . . . . . . . . . . . 60
4.1.3 Load Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Proposed Energy Management Scheme . . . . . . . . . . . . . . . . . . . 60
4.2.1 Droop Curve Per-Unitisation . . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Droop Curve Adjustment . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Test Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2 Test Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Conclusion 77
5.1 Summary of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Bibliography 80
Appendices 83
v
A Voltage Source Converter Design 84
A.1 Topology Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.2 LCL Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.3.1 αβ0-Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.3.2 dq0-Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A.4 Theoretical Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.4.1 Reference Signal Generation . . . . . . . . . . . . . . . . . . . . . 90
A.5 Detailed Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.6 VSC Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.6.1 Delay Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.6.2 Delay Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B LCL Filter Design Methodology 100
B.0.1 Step 1: Capacitor Sizing . . . . . . . . . . . . . . . . . . . . . . . 102
B.0.2 Step 2: Inductor Sizing . . . . . . . . . . . . . . . . . . . . . . . . 103
C Transfer Function Approximation Validation 108
D Controller Values 110
E Solar Current Distribution Calculation 112
vi
List of Figures
1.1 Proposed DC/DC Converter for DC Microgrid . . . . . . . . . . . . . . . 3
1.2 Droop Curve Characteristic for AC Microgrid Systems, for P/f, Q/V
Droop [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Mechanical Power vs. Frequency for Two Turbines w/ Different Charac-
teristics [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 System Bus Voltage vs. Output Current for DC Microgrid [14] . . . . . . 6
1.5 General DC Microgrid Diagram . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Schematic for the BESS . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Average Model of MBESC . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Battery Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 New State Introduced by PI Controller . . . . . . . . . . . . . . . . . . . 17
2.5 Battery Control Scheme with Droop Control . . . . . . . . . . . . . . . . 20
2.6 Battery Control Scheme with Limiters Included . . . . . . . . . . . . . . 22
2.7 Schematic for the Solar Converter . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Average Model for the Solar Converter . . . . . . . . . . . . . . . . . . . 23
2.9 Solar Converter Control Scheme . . . . . . . . . . . . . . . . . . . . . . . 26
2.10 Solar Control Scheme with Limiters Included . . . . . . . . . . . . . . . . 28
2.11 VSC Simplified Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.12 Schematic of Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.13 High-Level Diagram of the Micro-Grid System . . . . . . . . . . . . . . . 32
2.14 Input and Output Definitions for Component State Space Models . . . . 33
2.15 Connections for the System State-Space Model . . . . . . . . . . . . . . . 34
2.16 Line Inductance Model [19] . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.17 PSCAD vs. MATLAB Step Response: Bus Voltage (Vbus) and Battery
Output Current (ibattout ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.18 PSCAD vs. MATLAB Step Response: Solar Converter Inductor Current
(iL) and Input Capacitor Sum Voltage (V∑) . . . . . . . . . . . . . . . . 40
vii
2.19 Participation Factor of the Battery, Line and VSC Model States . . . . . 42
2.20 Participation Factor of the Solar Model States . . . . . . . . . . . . . . . 42
2.21 1A Ivsc Step - Bus Voltage (Vbus) and Battery Output Current (ibattout ) . . . 43
2.22 1A Is Step - Solar Inductor Current (iL) and Input Capacitor Sum Voltage
(v∑) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.23 Participation Factor of the Battery Converter, Line and VSC Model States,
Reorganized into Sum and Difference Eigenvalues . . . . . . . . . . . . . 46
2.24 Participation Factor of the Solar Converter States, Reorganized into Sum
and Difference Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.25 Participation Factor of the Battery Converter, Line and VSC States, with
Poorly-Chosen Controller Values . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Schematic of Scaled Battery Model . . . . . . . . . . . . . . . . . . . . . 49
3.2 Schematic of Scaled Solar Model . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 (Two Battery Converters, One Solar Converter) PSCAD vs. MATLAB
Step Response: 1A Step in the VSC Load Demand (ivsc). Top to Bottom:
Bus Voltage (vbus), Output Current (iout) . . . . . . . . . . . . . . . . . . 53
3.4 (Three Battery Converters, One Solar Converter) PSCAD vs. MATLAB
Step Response: 1A Step the VSC Load Demand (ivsc). Top to Bottom:
Bus Voltage (vbus), Output Current (iout) . . . . . . . . . . . . . . . . . . 54
3.5 (Two Solar Converters, One Battery Converter) PSCAD vs. MATLAB
Step Response: 1A Step in is. Top to Bottom: Inductor Current (iL),
Input Capacitor Sum Voltage (v∑), Output Current (iout) . . . . . . . . 55
3.6 (Three Solar Converters, One Battery Converter) PSCAD vs. MATLAB
Step Response: 1A Step in is. Top to Bottom: Inductor Current (iL),
Input Capacitor Sum Voltage (v∑), Output Current (iout) . . . . . . . . 55
3.7 Eigenvalue Movement: Battery Scaling (1-300 Battery Converters, Single
Solar Converter). Solar Converter Eigenvalues Removed . . . . . . . . . . 57
3.8 Eigenvalue Movement: Solar Scaling (1-300 Solar Converters, Single Bat-
tery Converter). Battery Converter Eigenvalues Removed . . . . . . . . . 57
4.1 General Control Scheme for the Energy Management Method . . . . . . 61
4.2 Droop Curve Adjustment for Energy Management Scheme . . . . . . . . 62
4.3 Implemented Droop Adjustment Curve . . . . . . . . . . . . . . . . . . . 63
4.4 Droop Curves for SOC = 0%, 80%, 100% . . . . . . . . . . . . . . . . . . 64
4.5 Aggregate Droop Curve of DC Micro-Grid w/ Three BESS at SOC = 0%,
80%, 100%, where Iaggbase =∑n
k=1 Ibase,k . . . . . . . . . . . . . . . . . . . . 65
viii
4.6 Aggregate Droop Curve of DC Micro-Grid w/ Three BESS at SOC = 0%,
80%, 100% and the Renewable Source Limiter, where Iaggbase =∑n
k=1 Ibase,k 66
4.7 DC Micro-Grid System to Validate Energy Management System . . . . . 68
4.8 Energy Management Simulation Results; Top to Bottom: SOC of Bat-
tery 1 & 2, Bus Voltage, Total Renewable Generation Source and Battery
Output Current, Output Current for Battery Converter 1 & 2 . . . . . . 70
4.9 Test Scenario 1: Entire Simulation. The bus voltage and output current
from the simulation results (Fig. 4.8) are overlapped to see how the oper-
ating point changes due to the proposed energy management scheme. . . 71
4.10 Test Scenario 1: Interval 1. During this interval, SOC balancing is occur-
ring and the SOC is increasing, resulting in the droop curve rising. . . . . 72
4.11 Test Scenario 1: Interval 2. During this interval, The SOC is increasing,
but the OCP is limiting the renewable source current so the SOC does not
exceed 100%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.12 Test Scenario 1: Beginning of Interval 3. A 50A (0.625pu) load is intro-
duced, which lowers the bus voltage and the source limiter opens. . . . . 73
4.13 Test Scenario 1: Interval 3. During this interval, the renewable source
limiter is progressively opening, which reduces the output current required
by the BESS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.14 Test Scenario 1: Interval 4. During this interval, the renewable source is
suppliying its maximum permissive power and the BESS SOC is reduced,
which lowers the droop curve. . . . . . . . . . . . . . . . . . . . . . . . . 74
4.15 Energy Management Test Scenario 2 Simulation Results: Output Current
for Individual BESSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.1 Schematic for the VSC with an LCL Filter. The components ”L1”, ”L2”
and ”C” are the same size in each phase. . . . . . . . . . . . . . . . . . . 85
A.2 Transformation of abc-Frame (Blue) to αβ0-Frame (Red) . . . . . . . . . 86
A.3 Transformation of αβ0-Frame (Blue) to dq0-Frame (Red) . . . . . . . . . 88
A.4 Current Controller Block Diagram . . . . . . . . . . . . . . . . . . . . . . 90
A.5 Detailed Control Diagram for the VSC . . . . . . . . . . . . . . . . . . . 91
A.6 Root Locus Plot for the VSC System . . . . . . . . . . . . . . . . . . . . 92
A.7 Controller Delay Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.8 VSC Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.9 VSC Simulation - Zoomed-In Grid Current . . . . . . . . . . . . . . . . . 98
A.10 VSC Simulation - Zoomed-In DQ-Frame Current Tracking . . . . . . . . 99
ix
B.1 Harmonics Produced from VSC vs. Switching Frequency [24] . . . . . . . 100
B.2 LCL Filter Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.3 Energy Requirement vs. ”k” . . . . . . . . . . . . . . . . . . . . . . . . . 106
B.4 L1 Largest Harmonic vs. ”k” . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.1 Bode Plot: Exact TF vs. Approximated TF . . . . . . . . . . . . . . . . 109
E.1 Is Current Distribution Schematic . . . . . . . . . . . . . . . . . . . . . . 112
x
List of Tables
1.1 Comparison of Droop Concepts for the Low Voltage Level [11] . . . . . . 5
2.1 Line Inductance Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Line Inductance Calculations . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 System Parameters: DC Micro-Grid . . . . . . . . . . . . . . . . . . . . . 38
2.4 Battery Module Electrical Parameters . . . . . . . . . . . . . . . . . . . . 38
2.5 BESS On-Board Component Sizes . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Solar Module Electrical Parameters . . . . . . . . . . . . . . . . . . . . . 39
2.7 Line and VSC Component Values . . . . . . . . . . . . . . . . . . . . . . 39
4.1 System Per-Unit Base Values . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Droop Curve Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Individual BESS Output Current for V refbus = 1 [pu] . . . . . . . . . . . . 65
4.4 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Initial Conditions for Simulation . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Initial SOC for Test Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Load Steps for Test Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . 75
A.1 LCL Filter Component Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.2 Comparison of αβ-Frame and dq-Frame . . . . . . . . . . . . . . . . . . . 89
A.3 VSC System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.4 VSC Simulation: Power Demands vs. Time . . . . . . . . . . . . . . . . 96
B.1 Grid Rating Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.2 System Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.3 LCL Filter Component Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 105
D.1 Control Parameters: Battery Converter . . . . . . . . . . . . . . . . . . . 110
D.2 Control Parameters: Solar Converter . . . . . . . . . . . . . . . . . . . . 111
xi
Chapter 1
Introduction
Climate change and the reduction of Greenhouse Gases (GHG) has become a growing
societal and environmental concern. According to the Energy Information Administra-
tion (EIA), traditional energy production methods (i.e. fossil fuels) represent 39.8% of
the GHG emissions in North America [1], making it one of the leading causes of climate
change. This has resulted in a global push to fund research into clean, renewable energy
sources and implementing environmental policies [2, 3]. As of 2015, renewable energy
sources represent 19.2% of the global energy consumption (including hydro-power and
biomass) and this number is rapidly growing [4]. Globally, the goal is to have renewable
generation represent 100% of the global energy consumption or to have a 80% reduction
in GHG production by 2050 [4]. This represents a desire to better utilize renewable
energy sources in future years to meet consumer demands while reducing GHG.
One of the main concerns with renewable sources (i.e. wind, solar) is their intermit-
tency. Natural, unavoidable events like daily variance in irradiance, clouds and variable
wind speeds result in a large power variance throughout the day. This is undesirable
since excess power must be sold or dumped while times of insufficient power production
may not meet the demands of the system. This can be mitigated by utilizing energy
storage media. Specifically, battery energy storage (BES) is the most commonly used
method within urban and suburban areas [5,6]. For grid-scale applications, BES systems
require high power and energy storage ratings (∼ MW/MWh). Existing technology with
these ratings typically operates on a 480 V AC bus and require complex infrastructures
to manage [7]. Therefore, decreasing the power ratings of the BES while maintaining
a similar bus voltage can allow for similar conversion techniques while reducing system
complexity. This led to the development and research into microgrid systems.
Both AC and DC microgrid systems are being developed. However, there are potential
benefits to using DC microgrids as opposed to AC. For example, PV arrays and BES
1
technology output DC power. Therefore, utilizing a DC micro-grid reduces complexity
and cost of power conversion equipment. Also, DC micro-grids require no frequency
tracking or reactive power management. The challenge of maintaining both voltage and
frequency regulation is replaced with the singular challenge of only maintaining voltage
regulation. Furthermore, the elimination of reactive power inherently reduces current
levels within the microgrid system and eliminates associated costs. Finally, according
to studies, DC/DC conversion offers better semiconductor utilization [8]. Therefore, the
focus of this thesis is on DC microgrids.
1.1 Literature Review
The purpose of this literature review is to develop an understanding of the DC/DC
converters used during this thesis and existing energy management methods.
1.1.1 Converter Topology
DC microgrids require DC/DC conversion for a variety of applications, ranging from
Battery Energy Storage Systems (BESS) to solar PV. Common DC/DC converters like
buck, boost and buck/boost can be utilized. However, their limitations (eg. voltage
operating range, efficiency, component size requirements) can increase design costs. These
can be optimized by utilizing a novel converter to handle all DC/DC conversion. Ranjram
and Rivera provide analysis on a converter topology that meet these requirements [9,10].
The topology is illustrated in Fig. 1.1.
The peak efficiency of this converter is 99.4%. Ranjram also notes that the decou-
pling capacitors (Ca & Cb) provide current harmonic cancellation, which reduces filtering
requirements and, consequently, component sizes. Based on the semiconductor devices
selected, the converter can also provide bidirectional or unidirectional power flow. For
example, if all four switches are MOSFETs/IGBTs, then the converter provides bidirec-
tional power flow. However, if S1a and Sb2 are replaced with diodes, then the converter
provides unidirectional power flow.
This converter has two input configurations. Firstly, the inputs can be connected to
V1 and V2. The conversion ratio of the converter in this configuration is provided by
(1.1).
Vo = d1V1 + d2V2 (1.1)
Under the assumption that d1 = d2 = d, then the conversion ratio is redefined as
2
Ca
S1a
C1
C2 Cb
S1b
S2a
S2b
L
V1
V2
V3 Vo
IoI1
I2 IL
d1
d2
Figure 1.1: Proposed DC/DC Converter for DC Microgrid
(1.2), similar to a buck converter. Therefore, this configuration is defined as the “buck
configuration.”
Vo = d(V1 + V2) (1.2)
In the buck configuration, Ranjram and Rivera note that the rated power is high
under medium to high duty cycles due to high switch and inductor utilization. However,
much like the traditional buck converter, it requires Vin > Vout, or more specifically,
(V1 +V2) > Vo. Also, this configuration does not provide input fault blocking. Therefore,
Ranjram and Rivera recommend this configuration for a BESS due to the small voltage
operating range.
The second configuration option is to connected the input to V3. The conversion ratio
of the converter in this configuration (for d1 = d2 = d) is provided by (1.3), which is
similar to a buck/boost converter. Therefore, this configuration is defined as the ”buck-
boost configuration.”
Vo =d
1− dV3 (1.3)
In the buck-boost configuration, the voltage operating range is increased, but rated
power and efficiency are lower than the buck configuration due to lower switch and
inductor utilization. Additionally, the buck-boost configuration offers bi-directional fault
blocking. Ranjram and Rivera recommend this configuration for solar PV since (i) it
3
does not excessively restrict the range of solar PV voltages and (ii) it can extinguish
fault currents that could occur in case of a fault within the solar array.
1.1.2 Energy Management: Droop Control in Low Voltage AC
Microgrids
Droop control is a common method for energy management in AC microgrids since data
can be communicated by signals that are locally measurable. In [11], Engler notes that,
if microgrid inverters set their instantaneous active and reactive power, then droop can
be utilized to provide voltage and frequency control. Specifically, Engler relates active
and reactive power to inverter output frequency and voltage and compares both pairings
(P/f, Q/V) and (P/V, Q/f). This is visually explained in Fig. 1.2.
Figure 1.2: Droop Curve Characteristic for AC Microgrid Systems, for P/f, Q/V Droop[11]
Engler observed that, for low-voltage grids, ”conventional droop” (P/f, Q/V) can
provide active power dispatch and is compatible with generators and HV-level systems.
”Opposite droop” (P/V, Q/f) is capable of providing direct voltage control for low-
voltage grids. These are outlined in Table 1.1. Engler therefore concludes, based on the
objectives of the system, droop can be utilized to control different parameters.
1.1.3 Turbine-Governor Control & Automatic Generation Con-
trol
Turbine generators power and frequency have a similar relationship to that of the droop
characteristic. Specially, they experience a linear frequency change that is related to
4
Table 1.1: Comparison of Droop Concepts for the Low Voltage Level [11]
Conventional Droop Opposite Droop
Compatible with HV-level yes noCompatible with generators yes no
Direct voltage control no yesActive power dispatch yes no
the system load and its own rating. Given an external power reference demand, the
relationship between mechanical power and generator frequency is provided by (1.4) [12].
Note that R is a constant that is based on the turbine parameters.
Pm = Pref −1
Rf (1.4)
If two generators are interconnected to supply a load and their characteristics are
different, a power imbalance is introduced. This is due to the interconnection of the two
turbines forcing the frequency to match. Since the power reference (Pref ) is externally
defined, it can be altered to change the total power provided to the load while maintaining
a desired system frequency. This is illustrated in Fig. 1.3.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
1.06
Turbine Mechanical Output Power [pu]
Fre
quen
cy [p
u]
Pref
= 1.05
Pref
= 1.025
Desired Frequency
Figure 1.3: Mechanical Power vs. Frequency for Two Turbines w/ Different Character-istics [12]
To maintain the desired frequency, Kundur proposes utilizing an integral control to
5
adjust Pref [13]. This is provided by (1.5).
Pref =KI
s(M w) (1.5)
Where:
M w = wref − wmeas (1.6)
This is also commonly referred to as frequency restoration since it maintains a specific
frequency irrespective of load. Since the curve produced by the turbine matches a typical
droop characteristic, this concept has the potential to be extended to DC microgrid
systems.
1.1.4 Power Sharing in DC Systems
Akagi utilized droop control to provide power sharing in a DC microgrid [14]. Akagi’s
system consisted of a battery energy storage system (BESS) and a grid-tied inverter to
reliably provide power to the microgrid. In his paper, Akagi proposes a piecewise linear
function to relate the microgrid’s bus voltage to the output current of each supply. This
is illustrated in Fig. 1.4.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.94
0.96
0.98
1
1.02
1.04
1.06
Output Current [pu]
Bus
Vol
tage
Ref
eren
ce [p
u]
Energy Storage UnitAC Inverter Unit
Figure 1.4: System Bus Voltage vs. Output Current for DC Microgrid [14]
6
At low current demands, the energy storage unit supplies more power than the inverter
to increase microgrid independence. At high current demands, however, the inverter
begins to supply more power since the energy storage unit is approaching its rated limit.
In conclusion, Akagi defines the droop characteristic with different slopes to alter power
demand from individual units.
1.2 Motivation
One of the significant unknowns about DC microgrids is the robustness and reliability of
such systems as the number of connected sources increase. In most studies, system power
levels are between 1 - 30 kW [15–18], with those studies conducting tests at a single power
rating. This presents an opportunity to investigate the modularity of DC microgrids.
Specifically, developing a modular DC microgrid model that can accommodate a varying
number of components and power ratings would demonstrate the DC microgrid’s ability
to operate at more commercially viable power ranges (kW - MW).
As the DC microgrid system expands and additional BESS and generation sources are
added, managing the energy in the system becomes critical. Firstly, BESSs have a maxi-
mum charge capacity before the cell experiences physical damage. Secondly, asymmetry
between BESSs can result in diverging State-Of-Charges (SOCs). If one BESS depletes
before the others, the system power rating is reduced, which defines a non-robust system.
Therefore, designing an autonomous energy management method that is applicable to
DC microgrid systems with variable component numbers and power ratings compliments
system modularity.
1.3 High-Level DC Microgrid Layout
An AC grid typically has multiple components connected to it, including:
• Battery Energy Storage Systems (BESSs)
• Generation Sources (eg. solar, diesel generator)
• Connections to existing AC grids
• Loads (AC and DC)
Therefore, DC microgrids will comprise of similar components and models to conduct
analysis, comparable to existing AC grids. A general DC microgrid diagram is provided
in Fig. 1.5.
7
ACDC
ACDCDiesel
Generator
Grid
DCDC
Solar
DCDC
B2B1 B3
L12 L23
DC or AC
Loads
Battery
Figure 1.5: General DC Microgrid Diagram
1.4 Thesis Objectives
The objectives of this thesis are to develop a modular state-space modelling method to
accurately represent a complete DC microgrid system, then develop an energy manage-
ment scheme to increase its robustness.
These objectives can be subdivided into the following components:
1. Develop open-loop component state-space models for the BESS and solar PV uti-
lizing the proposed multi-port converter topology.
2. Add controllers into state-space models of BESS and solar PV to create modular
mathematical building blocks for eigenvalue analysis.
3. Expand the BESS and solar PV models that accommodate multiple converters on
the same bus to investigate eigenvalue movement and stability as the number of
converters increase.
4. Develop an autonomous energy management scheme that provides BESS state-
of-charge (SOC) balancing, overcharge protection (OCP) and load shedding to
increase DC microgrid robustness.
8
The goal is to demonstrate that DC microgrids are capable of operating over a sig-
nificant power range and the plausibility of future additions without risk of stability
concerns.
1.5 Thesis Outline
The contents of this thesis are divided into five chapters, including the introduction. The
following chapters are outlined:
Chapter 2 presents the DC microgrid component modelling, connection method,
single BESS/source eigenvalue analysis and model verification via comparison with com-
prehensive PSCAD/EMTDC simulation results.
Chapter 3 presents the scalable version of the component state-space models, up-
dated model verification via simulation result comparison and studying how the eigen-
values move as additional BESS / generation sources are added.
Chapter 4 presents the proposed energy management scheme and verification via
PSCAD/EMTDC simulation results.
Chapter 5 provides concluding remarks and future work.
9
Chapter 2
DC Microgrid System Modelling
Since DC microgrid research is in the early stages, modular modelling methods are lack-
ing. Therefore, developing a model of DC microgrid systems and its individual compo-
nents will enable further research. The purpose of this chapter is to create these models.
This includes models for the battery energy storage system (BESS), photovoltaic (PV)
system, VSC and interconnecting lines. With the models, investigation into system dy-
namics and how each component contributes to these dynamics are conducted.
For this chapter, theoretical state-space models are produced for each component.
Secondly, the method to connect the component models to represent a complete DC
microgrid is summarized. Thirdly, a DC microgrid comprising of a single BESS, single
PV converter, a grid-tied VSC and loading is tested via MATLAB to investigate the
dynamic response of the system. Finally, PSCAD simulations of the DC microgrid system
are conducted to validate the models and connection method.
2.1 Component State Space Models
In order to thoroughly investigate the dynamics of a microgrid system, a complete state-
space model must be created for each component within the microgrid. These models
are used to understand the dynamics of each state, the coupling between states in the
system and, in later chapters, used to investigate the scalability of microgrids.
2.1.1 Battery ESS Model
The implemented BESS topology is the dual-input converter introduced in Section 1.1.1.
The battery is represented by a voltage source with a line impedance (Lb). More advanced
models, such as those representing state-of-charge, are unnecessary for short time-scale
10
Lb
CIN
Lb
CIN
COUT COUT
S1
L
S’1
S2
S’2
V1
V2
Va Vb
IOUTVbus
Ib1
Ib2
IL
Vb1
Vb2
Simplified Load/Source Model
Figure 2.1: Schematic for the BESS
dynamic studies, while exclusion of battery internal resistance is purposefully neglected
since battery loss mechanisms should not be relied on to provide dumping of system
dynamics. The system generation/load that the BESS experiences is modelled as a
current source. The BESS is illustrated in Figure 2.1.
By defining d1 as the duty cycle that controls S1 and d2 as the duty cycle that controls
S2, the output steady-state voltage is defined in equation 2.1.
Vbus = d1V1 + d2V2 (2.1)
Under the assumption that the batteries are balanced, V1 = V2 = Vin. Therefore,
equation 2.1 becomes 2.2, which means the BESS converter operates as a quasi-buck
converter, subject to the constraint Vbus ≤ 2Vin.
Vbus = d1Vin + d2Vin = (d1 + d2)Vin (2.2)
The inductor (L) passes current through all semiconductor switches and is what
results in power flow between the input and output terminals.
The first step in creating the state-space model is to develop the differential equations
for the system of interest. To create the differential equations, the converter average
model is utilized. This is depicted in Figure 2.2. A major benefit of this dual-input
converter is its ability to reduce filtering requirements. This occurs, in part through
cancellation of voltage ripple across coupling input/output capacitor networks. While
11
this benefits performance and cost, it introduces additional model complexity.
Open-Loop State Space Model
CIN
CIN
COUT COUTL
V1
Va Vb Vbus
d1V1
d2V2
d1iL
d2iL
IL
V2
Lb
Ib1Vb1
Lb
Ib2Vb2
Iout
Figure 2.2: Average Model of MBESC
Using Figure 2.2, the differential equations for the BESS model is given in equations
2.3 through 2.9.
Ld
dt〈iL〉 = 〈d1〉〈v1〉+ (〈d2〉 − 1)〈v2〉 − 〈va〉 − (2RON +RL)〈iL〉 (2.3)
CINd
dt〈v1〉 = (1− k)〈ib1〉+ k〈ib2〉+ (−(1− k)〈d1〉 − k〈d2〉+
1
2)〈iL〉 −
1
2〈iOUT 〉 (2.4)
CINd
dt〈v2〉 = k〈ib1〉+ (1− k)〈ib2〉+ (−k〈d1〉 − (1− k)〈d2〉+
1
2)〈iL〉 −
1
2〈iOUT 〉 (2.5)
COUTd
dt〈va〉 = k〈ib1〉 − k〈ib2〉+ (−k〈d1〉+ k〈d2〉+
1
2)〈iL〉 −
1
2〈iOUT 〉 (2.6)
COUTd
dt〈vb〉 = −k〈ib1〉+ k〈ib2〉+ (k〈d1〉 − k〈d2〉+
1
2)〈iL〉 −
1
2〈iOUT 〉 (2.7)
Lbd
dt〈ib1〉 = 〈vb1〉 −RLb〈ib1〉 − 〈v1〉 (2.8)
12
Lbd
dt〈ib2〉 = 〈vb1〉 −RLb〈ib2〉 − 〈v2〉 (2.9)
Where:
k =1
2(
1
1 + CIN
COUT
) (2.10)
Differential equations 2.3 through 2.9 are calculated directly through KCL and KVL.
However, equation 2.2 shows that only the sum duty cycle would normally affect the
net bus voltage, vbus This suggests a sum-difference controller structure. Therefore, the
differential equations are transformed into their sum and difference form. This is done
by defining the states as shown in 2.11 through 2.15.[v∑vM
]=
[1 1
1 −1
][v1
v2
](2.11)
[vout∑voutM
]=
[1 1
1 −1
][va
vb
](2.12)
[ib
∑ibM
]=
[1 1
1 −1
][ib1
ib2
](2.13)
[d∑dM
]=
[1 1
1 −1
][d1
d2
](2.14)
[vb∑vbM
]=
[1 1
1 −1
][vb1
vb2
](2.15)
vbus =v∑ + vout∑
2(2.16)
Using differential equations 2.3 through 2.9 and definitions 2.11 through 2.15, the
sum-difference differential equations all given in 2.17 through 2.22.
d
dt〈vbus〉 = (
1− 〈d∑〉2CIN
+1
2COUT)〈iL〉+
1
2CIN〈i∑〉 − (
1
2CIN+
1
2COUT)〈iOUT 〉 (2.17)
Ld
dt〈iL〉 = −〈vbus〉 − (2RON +RL)〈iL〉+
〈d∑〉2〈v∑〉+
〈dM〉2〈vM〉 (2.18)
13
CINd
dt〈v∑〉 = (1− 〈d∑〉)〈iL〉+ 〈i∑〉 − 〈iOUT 〉 (2.19)
CINd
dt〈vM〉 = −(1− 2k)〈dM〉〈iL〉+ (1− 2k)〈iM〉 (2.20)
Lbd
dt〈i∑〉 = −〈v∑〉 −RLb〈i∑〉+ 〈vb∑〉 (2.21)
Lbd
dt〈iM〉 = −〈vM〉 −RLb〈iM〉+ 〈vbM〉 (2.22)
Now that the differential equations are in the form that the controller will utilize,
state-space models can be produced for the battery converter. State-space equations
take on the form given in 2.23.
x = Ax+Bu
y = Cx+Du(2.23)
Observe that equations 2.17 through 2.22 are non-linear equations. Therefore, to
create the state-space models, the system must be linearised using the Jacobian. The
calculation of the Jacobian is defined in equation 2.24.
J =
df1dx1
df1dx2
· · · df1dxn
df2dx1
df2dx2
· · · df2dxn
......
. . ....
dfndx1
dfndx2
· · · dfndxn
∣∣∣∣∣x=x;u=u
(2.24)
Where:
fx - The differential equation that is being linearised
xx - The different states/inputs in the linearised model
x - The equilibrium point of the states
u - The equilibrium point of the inputs
The Jacobian was used to create the A and B matrices in equation 2.23. For the
battery model, there are six (6) states (vbus, iL, v∑, vM, i∑, iM). There are also five (5)
inputs (d∑, dM, iout, vb∑, vbM). Using differential equations 2.17 through 2.22 and the
Jacobian, the open-loop A and B matrix in the state-space model are given by equations
2.25 through 2.28.
14
AOLbatt =
01− ¯d∑2CIN
+ 12COUT
0 0 12CIN
0
− 1L
−2RON+RL
L
¯d∑2L
dM2L
0 0
01− ¯d∑CIN
0 0 1CIN
0
0 − (1−2k)dMCIN
0 0 0 1−2kCIN
0 0 − 1Lb
0 −RLb
Lb0
0 0 0 − 1Lb
0 −RLb
Lb
(2.25)
BOLbatt =
− iL2CIN
0 −( 12CIN
+ 12COUT
) 0 0¯VC∑
2L
¯VCM2L
0 0 0
− iLCIN
0 − 1CIN
0 0
0 − (1−2k)iLCIN
0 0 0
0 0 0 − 1Lb
0
0 0 0 0 − 1Lb
(2.26)
xOLbatt =
vbus
iL
v∑vM
i∑iM
(2.27)
uOLbatt =
d∑dM
iout
vb∑
vbM
(2.28)
Equations 2.25 through 2.28 represent the system’s natural response. Therefore, the
next step is to model the controller and add it to the open-loop state space model.
Battery Control Scheme
As mentioned in Section 2.1.1, the battery model uses a sum-difference controller. In
other words, the controller outputs two signals: d∑ and dM. The controller then uses
both values to calculate the duty cycle of each battery (d1 and d2) using equation 2.14.
The goal of utilizing a sum-difference controller is to decouple states as much as pos-
sible. As mentioned in the beginning of this subsection, the input and output capacitors
introduces modelling complexity where multiple states are coupled. Decoupling can po-
15
tentially occur if d∑ and dM are regulated as opposed to d1 and d2. By substituting d∑and dM into equation 2.2, it takes on the following form:
Vbus = (d1 + d2)Vin = d∑Vin (2.29)
Equation 2.29 shows that the bus voltage, vbus, is dependent on d∑ and independent
of dM. Therefore, a sum-difference control structure can reduce coupling between states.
Furthermore, with an appropriately tuned controller, further states can be decoupled in
a manner that is beneficial for scalability.
The proposed control scheme is illustrated in Figure 2.3. The purpose of the sum
controller is to regulate the bus voltage (vbus). The sum controller is regulated by a
nested Proportional-Integral (PI) control loop in a conventional fashion. The outer loop
measures vbus and compares it with an external reference voltage, which goes through a
(PI) controller and outputs a reference inductor current (irefL ). This reference is compared
to the measured iL, which goes through another PI controller and outputs d∑. As
mentioned earlier, the inductor current directly flows through the semiconductor switches,
meaning that the control scheme must prevent excessive over-current to protect the
switches.
The purpose of the difference controller is to balance the battery voltages by balancing
converter terminal voltages v1 and v2. This ensures that one battery does not completely
deplete, which would force a single battery to maintain the bus voltage. This is done
by measuring the difference between the two battery voltages and varying dM. Changing
dM simultaneously increases the power drawn from the battery with higher voltage and
decreases the power draw from the battery with the lower voltage, which slowly balances
the voltage. The difference controller is regulated with a single PI controller. The battery
voltage difference is measured and compared with the reference voltage difference (vrefM ),
which goes through the PI controller and outputs dM. Both d∑ and dM are used to
calculate the individual duty cycles (d1 and d2), which control the switches.
Battery Closed-Loop State Space Model
With the controller defined, the next step is to integrate it into the open-loop state space
model. Each PI controller introduces a new state into the system due to the integral
term. Figure 2.4 helps illustrate this.
By defining the new state, Ue, which is the error term of the controller, the new state
(Ue) is defined by equation 2.30.
16
PI+
-
PIVΔ
+
-
VΔ
iL
Upper
Lower
V2
V1
d2
d1
+
-
+
+Inductor Current
Control
Difference Control
VbusiL
PI+
-
Vbus
Bus Voltage Control
Sum Controliout
d∑
2
dΔ
2
iLrefVbus
ref
ref
Figure 2.3: Battery Control Scheme
Uref+
-
U Ue
+
+
y
New State
Kp
Ki
Ue
1s
Figure 2.4: New State Introduced by PI Controller
y = Kp(Uref − U) +KiUe (2.30)
For the nested sum controller, the output (d∑) can be defined by equation 2.31.
d∑ = Kp1(irefL − iL) +Ki1ieL (2.31)
From there, the inductor’s reference current (irefL ) can be defined by equation 2.32.
irefL = Kp2(vrefbus − vbus) +Ki2vebus (2.32)
Finally, for the difference controller, the output (dM) can be defined in equation 2.33.
17
dM = Kp3(vrefM − vM) +Ki3veM (2.33)
By directly substituting equations 2.31 through 2.33 into the battery state model, the
new closed-loop A and B matrices are as shown in equations 2.34 through 2.37.
ACLbatt =
Kp1Kp2¯iL
2CIN−
Kp1Ki2¯iL
2CIN
1− ¯d∑+Kp1¯iL
2CIN+ 1
2COUT−Ki1
¯iL2CIN
0 0 0 12CIN
0
−1 0 0 0 0 0 0 0 0
−2+Kp1Kp2
¯VC∑
2L
Kp1Ki2¯VC∑
2L−
2(2RON+RL)+Kp1¯VC∑
2L
Ki1¯VC∑
2L
¯d∑2L
dM−Kp3¯VCM
2LKi3
¯VCM2L
0 0
−Kp2 Ki2 −1 0 0 0 0 0 0
Kp1Kp2¯iL
CIN−
Kp1Ki2¯iL
CIN
1− ¯d∑+Kp1¯iL
CIN−Ki1
¯iLCIN
0 0 0 1CIN
0
0 0 − (1−2k)dMCIN
0 0Kp3
¯iLCIN
−Ki3¯iL
CIN0 1−2k
CIN0 0 0 0 0 −1 0 0 0
0 0 0 0 − 1Lb
0 0 −RLbLb
0
0 0 0 0 0 − 1Lb
0 0 −RLbLb
(2.34)
BCLbatt =
−Kp1Kp2 iL2CIN
0 −( 12CIN
+ 12COUT
) 0 0
1 0 0 0 0Kp1Kp2
¯VC∑
2L
Kp3¯VCM
2L0 0 0
Kp2 0 0 0 0
−Kp1Kp2 iLCIN
0 − 1CIN
0 0
0 −Kp3(1−2k)iLCIN
0 0 0
0 1 0 0 0
0 0 0 − 1Lb
0
0 0 0 0 − 1Lb
(2.35)
xCLbatt =
vbus
vebusiL
ieLv∑vM
veM
i∑iM
(2.36)
18
uCLbatt =
vrefbus
vrefM
iout
vb∑
vbM
(2.37)
This state-space model can accurately model a single battery converter’s dynamics.
However, since demonstrating scalability is a key aspect to investigate, the model must
be prepared to consider the dynamics when the controller includes additional features
used with multiple battery modules (eg. SOC Balancing).
Droop Control
Droop Control is a method for implementing various energy management methods, as
Akagi demonstrated in Section 1.1.4 [14]. It involves measuring the output current of the
module and adjusting the reference bus voltage to increase/decrease the power demand
from that module, which is detailed in Chapter 4. The formula is given in equation 2.38.
vrefbus = vnombus −Kdiout (2.38)
Droop control also improves the transient response since the reference voltage changes
in the same direction as the initial voltage transient, resulting in a smaller error. It is
important to note that this assumes Kd > 0. Droop control is added to the closed-loop
state space model by directly substituting equation 2.38 into equations 2.34 through 2.37.
This is expressed by equations 2.39 through 2.42. The new control diagram that includes
droop control is provided in Fig. 2.5.
ACLbatt =
Kp1Kp2¯iL
2CIN−
Kp1Ki2¯iL
2CIN
1− ¯d∑+Kp1¯iL
2CIN+ 1
2COUT−Ki1
¯iL2CIN
0 0 0 12CIN
0
−1 0 0 0 0 0 0 0 0
−2+Kp1Kp2
¯VC∑
2L
Kp1Ki2¯VC∑
2L−
2(2RON+RL)+Kp1¯VC∑
2L
Ki1¯VC∑
2L
¯d∑2L
dM−Kp3¯VCM
2LKi3VCM
2L0 0
−Kp2 Ki2 −1 0 0 0 0 0 0
Kp1Kp2¯iL
CIN−
Kp1Ki2¯iL
CIN
1− ¯d∑+Kp1¯iL
CIN−Ki1
¯iLCIN
0 0 0 1CIN
0
0 0 − (1−2k)dMCIN
0 0Kp3
¯iLCIN
−Ki3¯iL
CIN0 1−2k
CIN0 0 0 0 0 −1 0 0 0
0 0 0 0 − 1Lb
0 0 −RLbLb
0
0 0 0 0 0 − 1Lb
0 0 −RLbLb
(2.39)
19
PI+
-
PIVΔ
+
-
VΔ
iL
Upper
Lower
V2
V1
d2
d1
+
-
+
+Inductor Current
Control
Difference Control
VbusiL
PI+
-
Vbus
Bus Voltage Control
Sum Controliout
-
Kd
io
+
Droop Control
d∑
2
dΔ
2
iLref
Vonom
Vbusref
ref
Figure 2.5: Battery Control Scheme with Droop Control
BCLbatt =
−Kp1Kp2 iL2CIN
0 −( 12CIN
+ 12COUT
− KdKp1Kp2 iL2CIN
) 0 0
1 0 −Kd 0 0Kp1Kp2
¯VC∑
2L
Kp3¯VCM
2L−KdKp1Kp2
¯VC∑
2L0 0
Kp2 0 −KdKp2 0 0
−Kp1Kp2 iLCIN
0 −1−KdKp1Kp2 iLCIN
0 0
0 −Kp3(1−2k)iLCIN
0 0 0
0 1 0 0 0
0 0 0 − 1Lb
0
0 0 0 0 − 1Lb
(2.40)
20
xCLbatt =
vbus
vebusiL
ieLv∑vM
veM
i∑iM
(2.41)
uCLbatt =
vnombus
vrefM
iout
vb∑
vbM
(2.42)
Equations 2.39 through 2.42 will be used to fully model the battery dynamics, in-
cluding when it is connected to the remainder of the system.
Practical Control Limiters
Equations 2.39 through 2.42 represent the battery model that is used for all analysis
for the remainder of the thesis. However, it is important to note that, for practical
applications, non-linear features (eg. limiters) are utilized in the BESS control scheme.
The control diagram with the limiters included is provided by Fig. 2.6.
For the sum controller, two limiters are present. The first limiter is for irefL to prevent
the inductor current (iL) from exceeding its rated value. The second limiter is for d∑,
since each duty cycle (d1 and d2) cannot exceed one.
For the difference controller, one limiter is present, which is for dM. The limit changes
based on d∑ to prevent both d1 and d2 from exceeding one. The calculation for these
limits are defined by equations 2.43 and 2.44.
dmaxM = min(d∑, dmax∑ − d∑) (2.43)
dminM = −min(d∑, dmax∑ − d∑) (2.44)
21
PI+
-
PIVΔ
+
-
VΔ
iL
Upper
Lower
V2
V1
d2
d1
+
-
+
+Inductor Current
Control
Difference Control
VbusiL
PI+
-
Vbus
Bus Voltage Control
Sum Control iout
-
Kd
io
+
Droop Control
d∑
2
dΔ
2
iLref
Vonom
Vbusref
ref
d∑max
d∑min
dΔmax
dΔmin
iLmax
iLmin
Figure 2.6: Battery Control Scheme with Limiters Included
2.1.2 Solar Converter State-Space Model
This subsection will define the state-space model that will be used to represent the so-
lar converter. The converter topology used is still the one discussed in Section 1.1.1.
However, since the converter is connected to solar panels, it must be restricted to uni-
directional power flow. This is done by either turning switches S′1 and S
′2 off. Then,
by utilizing the anti-parallel diode in the active switch, the current can flow out of the
converter while restricting current into the converter. Alternatively, both switches can
be replaced with diodes. The converter operates in the quasi-buck/boost configuration
as shown in Figure 2.7.
Assuming the duty cycles are equal (d1 = d2 = d), the output voltage is defined by
equation 2.45, which is identical to the buck-boost converter.
Vbus = Vsd
1− d(2.45)
The solar panel is represented as a current source for purposes of state-space mod-
elling1. The output is modelled as a voltage source with a line inductance, which rep-
resents the DC microgrid bus and the line connection. Typically, the external networks
are represented as either voltage or current source inputs to the models as this allows
1For the energy management method in Chapter 4, a complete solar model that includes irradianceand temperature is utilized.
22
CIN
CIN
COUT COUT
S1
L
S’1
S2
S’2
V1
V2
Va Vb
VcVbus
IL
Is
Lx
Vs
Simplified Load/Source Model
Figure 2.7: Schematic for the Solar Converter
future integration of the component models into a larger system. However, for the solar
converter model, neither of these options are suitable. If the external network is modelled
as a current source, then the inductor current is determined by the current source repre-
senting the solar panel and the external network and, consequently, cannot be controlled.
If the external network is represented as a voltage source, then the voltage across the
input capacitors and the solar panel (v1 + vs + v2) is fixed and, consequently, cannot be
controlled. Instead, the line inductance is included in the solar model to overcome these
limitations. Therefore, the average model is represented by Figure 2.8.
CIN
CIN
COUT COUTL
V1
Va Vb Vbus
d1V1
d2V2
d1iL
d2iL
IL
V2
Is
Lx
Vc
Vs
Figure 2.8: Average Model for the Solar Converter
23
The differential equations for this configuration are provided in equations 2.46 through
2.49.
Ld
dt〈iL〉 = −(2RON +RL)〈iL〉+
〈d∑〉 − 1
2〈v∑〉+
dM2vM −
1
2〈vout∑ 〉 (2.46)
CINd
dt〈v∑〉 = (1− 〈d∑〉)〈iL〉+ 〈i∑〉 − 〈iout〉+ 〈is〉 (2.47)
CINd
dt〈vM〉 = −(1− 2k)〈dM〉〈iL〉 (2.48)
COUTd
dt〈vout∑ 〉 = 〈iL〉 − 〈iout〉 − 〈is〉 (2.49)
Lxd
dt〈iout〉 =
1
2〈v∑〉+
1
2〈vout∑ 〉 −Rx〈iout〉 − 〈Vc〉 (2.50)
For the solar model, there are five (5) states (iL, v∑, vM, vout∑ , iout). There are also
four (4) inputs (d∑,dM,vc,is). Using equations 2.46 through 2.50 and the Jacobian, the
open-loop A and B matrix in the state-space model are given by equations 2.51 through
2.54.
AOLsolar =
−2RON+RL
L
¯d∑−1
2LdM2L− 1
2L0
1− ¯d∑CIN
0 0 0 − 1CIN
− (1−2k)dMCIN
0 0 0 01
COUT0 0 0 − 1
COUT
0 12Lx
0 12Lx
−Rx
Lx
(2.51)
BOLsolar =
V∑2L
VM2L
0 0
− iLCIN
0 0 1CIN
0 − (1−2k)iLCIN
0 0
0 0 0 − 1COUT
0 0 − 1Lx
0
(2.52)
xOLsolar =
iL
v∑vM
vout∑iout
(2.53)
24
uOLsolar =
d∑dM
vc
is
(2.54)
Equations 2.51 through 2.54 represent the natural response of the solar converter.
The next section will focus on the control scheme for this converter.
Solar Control Scheme
Similar to the battery converter, the solar converter utilizes a sum-difference controller.
However, the goal of the controller is not to regulate the bus voltage. Instead, the solar
converter must provide current regulation and PV voltage regulation, as required to allow
implementation of PV peak power tracking.
The purpose of the sum controller is to regulate the panel voltage by controlling
the inductor current. First, it is important to note that the panel voltage (vs) can be
controlled by regulating the input capacitor sum voltage (v∑ = v1 +v2). This is enforced
by the KVL equation of the solar converter that is provided in equation 2.55.
vs = v∑ − vbus (2.55)
Therefore, under the assumption that the bus voltage (vbus) is externally maintained,
by controlling v∑, vs is regulated. Since the BESS is responsible for bus voltage regula-
tion, this assumption is valid.
The solar converter control scheme is illustrated in Figure 2.9. The reference sum
voltage (vref∑ ) is determined by using Maximum Power Point Tracking (MPPT), which is
then controlled by a nested sum controller that is similar to the battery control scheme.
However, since v∑ is being regulated, an increase in vref∑ should result in a decrease of
the inductor current (iL). Therefore, the sum voltage controller input is the difference
between v∑ and vref∑ . This difference then goes through a PI controller to determine
irefL . The current reference, in turn, is compared to the measured iL, which goes through
another PI controller and outputs d∑.
The purpose of the difference controller is to ensure the input capacitor voltages are
equal; in other words, v1 = v2. This input configuration does ensure this in the ideal
case. However, in practice, a symmetrical circuit with identical component values is not
possible. Therefore, this controller deals with this non-ideality and ensures v1 and v2
do not diverge. This is necessary since harmonic cancellation requires symmetry for this
25
topology.
PI+
-
PIVΔ+
-
VΔ
iL
Upper
Lowerd2
d1+
-
+
+Inductor Current
Control
Difference Control
VoiL
PI+
-
V∑
Input Voltage Control
Sum Controlio
Reference Signal Generation
Vs+-
isMPPT
Vs
is+
Vo
+
dΔ
2
d∑
2
V∑ref iL
refVsref
ref
Figure 2.9: Solar Converter Control Scheme
Solar Closed-Loop State Space Model
The closed-loop state-space model is generated using an identical process to section 2.1.1.
First, define the input d∑ as provided in equation 2.56.
d∑ = Kp1(irefL − iL) +Ki1ieL (2.56)
From there, the inductor’s reference current (irefL ) can be defined by equation 2.57.
irefL = Kp2(v∑ − vref∑ ) +Ki2ve∑ (2.57)
Finally, for the difference controller, the output (dM) can be defined by equation 2.58.
dM = Kp3(vM − vrefM ) +Ki3veM (2.58)
By combining the open-loop state-space equation provided by 2.51 through 2.54 and
equations 2.56 through 2.58, the closed-loop state-space model for the solar converter is
generated and provided by equations 2.59 through 2.62.
26
ACLsolar =
−2(2RON+RL)+Kp1
¯V∑2L
Ki1¯V∑
2L
¯d∑−1+Kp1Kp2¯V∑
2L
Kp1Ki2¯V∑
2L
dM+Kp3VM2L
Ki3VM2L
− 12L
0
−1 0 Kp2 Ki2 0 0 0 01− ¯d∑+Kp1
¯iLCIN
−Ki1¯iL
CIN−
Kp1Kp2¯iL
CIN−
Kp1Ki2¯iL
CIN0 0 0 − 1
CIN0 0 1 0 0 0 0 0
− (1−2k)dMCIN
0 0 0 −(1−2k)Kp3
¯iLCIN
− (1−2k)Ki3¯iL
CIN0 0
0 0 0 0 1 0 0 01
COUT0 0 0 0 0 0 − 1
COUT
0 0 12Lx
0 0 0 12Lx
−RxLx
(2.59)
BCLsolar =
−Kp1Kp2V∑2L
−Kp3VM2L
0 0
−Kp2 0 0 0Kp1Kp2 iLCIN
0 0 1CIN
−1 0 0 0
0 Kp3(1−2k)iLCIN
0 0
0 −1 0 0
0 0 0 − 1COUT
0 0 − 1Lx
0
(2.60)
xCLsolar =
iL
ieLv∑ve∑vM
veM
vout∑iout
(2.61)
uCLsolar =
vref∑vrefM
vc
is
(2.62)
Equations 2.59 through 2.62 will be used to fully model the solar converter dynamics.
Practical Control Limiters
For the same justification as the battery converter, the solar converter has practical
control limiters that are not included in the state-space model. These are outlined in
Fig. 2.10.
27
PI+
-
PIVΔ+
-
VΔ
iL
Upper
Lowerd2
d1+
-
+
+Inductor Current
Control
Difference Control
VoiL
PI+
-
V∑
Input Voltage Control
Sum Control io
Reference Signal Generation
Vs+-
isMPPT
Vs
is+
Vo
+
dΔ
2
d∑
2
V∑ref iL
refVsref
ref
iLmax
iLmin
d∑max
d∑min
dΔmax
dΔmin
Figure 2.10: Solar Control Scheme with Limiters Included
2.1.3 VSC State-Space Model
This subsection defines the state-space model that is used to represent the VSC. Rep-
resenting the complete VSC introduces dynamics that are not relevant to DC microgrid
research. Therefore, from the perspective of DC microgrid dynamics, this model can be
simplified while still offering an accurate representation of the VSC. The simplified model
is illustrated in Figure 2.11. For the PSCAD energy management simulations outlined
in Chapter 4, a full VSC model was utilized, which is designed as outlined in Appendix
A.
CdcVC
idc
ivsc
Figure 2.11: VSC Simplified Schematic
Where:
28
ivsc - The DC current demand from the VSC
idc - The DC current before the DC link capacitor (ibattout + isolarout )
vc - The voltage across the DC link capacitor
Cdc - The DC link capacitor for the VSC (See Figure A.1)
The differential equation for the simplified VSC is provided in equation 2.63.
Cdcd
dt〈vc〉 = 〈iout〉 − 〈ivsc〉 (2.63)
Since this is a linear differential equation, this can be directly transformed into the
state space model. For this VSC model, there is one state (vc). There are also two inputs
(idc, ivsc). This model is described in equation 2.64 through 2.67.
AV SC =[0]
(2.64)
BV SC =[
1Cdc
− 1Cdc
](2.65)
xV SC =[vC
](2.66)
uV SC =
[idc
ivsc
](2.67)
This state-space model will be used to represent the VSC when connecting the state-
space models.
2.1.4 Line State-Space Model
This subsection defines the state-space model for the line that connects all the compo-
nents in the system. In other words, it will model the inductance introduced from the
line connections. The schematic for this model is given in Figure 2.12. Note that this is
the equivalent to the short-line model that is utilized in AC power systems. According
to Glover, a for line lengths below 80 km, the short-line model is an acceptable approx-
imation [12]. DC microgrids focus on connecting local generation sources, BESSs and
loads, which means long lines (∼ km) are not necessary. Therefore, the approximation
is suitable for DC microgrid applications.
29
Lxiout
Vbus VC
Rx
Figure 2.12: Schematic of Line Model
Where:
vbus - The DC bus voltage at the converter output terminals
vC - The DC bus voltage at the VSC input terminal
iout - The current through inductor Lx
Lx - The line inductance
The differential equation for the line model is proved in equation 2.68.
Lxd
dt〈iout〉 = 〈vbus〉 − 〈vc〉 −Rx〈iout〉 (2.68)
Just like the VSC, the line differential equations are linear. Therefore, a direct trans-
formation into the state-space model can be done. There is one state (iout). There are
also two inputs (vbus, vc). This model is described in equations 2.69 through 2.72.
AL =[−Rx
Lx
](2.69)
BL =[
1Lx− 1Lx
](2.70)
xL =[iout
](2.71)
uL =
[vbus
vC
](2.72)
This state-space model will be used to represent the lines when connecting the state-
space models.
30
2.2 Connecting State-Space Models
Now that state-space models have been made for all the key components, this section
will focus on how to mathematically connect these models to accurately represent the
complete DC microgrid system. This section assumes one battery converter, one solar
converter, one VSC and one line. Chapter 3 will discuss the addition of multiple battery
and solar converters.
Section 2.1 only discussed the calculation of the A and B matrix of the state-space
models. This was advantageous because defining the outputs of the state-space model
(C and D) is easier when all states and inputs of the components are predefined. For the
sake of simplicity, the closed-loop battery model state-space matrices (Equations 2.39
through 2.42) are referred to as Ab, Bb, Cb and Db for the remainder of this section and
and the solar model matrices (Equations 2.59 through 2.62) are referred to as As, Bs, Cs
and Ds.
To connect the state-space models, the inputs of all models must be provided either
externally or from the output of another model. Therefore, choosing the outputs of each
model appropriately will simplify the mathematical connection process. The battery
model is connected to the line model then to the VSC. The solar model connects directly
to the VSC since the line inductance (Lx) is included in the model. A high-level visual
of this is provided in Figure 2.13.
From Figure 2.13, the output of the battery model as vbus, the output of the solar and
line model is iout and the output of the VSC model as vc. In other words, the outputs
should be defined as shown in equation 2.73 through 2.80.
ybatt = vbus (2.73)
Cb =[1 0 0 0 0 0 0 0 0
]Db =
[0 0 0 0 0
](2.74)
ys = iout (2.75)
Cs =[0 0 0 0 0 0 0 1
]Ds =
[0 0 0 0
](2.76)
yL = iout (2.77)
CL =[1]DL =
[0 0
](2.78)
yvsc = vC (2.79)
31
Battery Converter
Line VSC
Vb1
Vb2
Vbus Vc iVSC
idciout
Solar Converter
Is
batt
ioutsolar
Linesolar
batt
Ab Bb Cb Db
AL BL CL DL
Avsc Bvsc Cvsc Dvsc
As Bs Cs Ds
Vbus
Figure 2.13: High-Level Diagram of the Micro-Grid System
Cvsc =[1]Dvsc =
[0 0
](2.80)
Now that the a full state-space model has been produced for every component, the
goal is connect the models together to create a state-space model for the micro-grid
system. The state space inputs/outputs are illustrated in Figure 2.14.
Using these outputs, the third input for the battery model (iout) is defined as the
output of the line model. Secondly, the first input of the line model (vbus) is defined as
the output of the battery model and the second input of the line model (vC) is defined
as the output of VSC model. Thirdly, the third input of the solar model is defined as
the output of the VSC model. Finally, the input of the VSC model (idc) is defined as the
sum of line and solar model outputs (ibattout + isolarout ). This is illustrated in Figure 2.15.
Note that the connections highlighted in red represent the inputs that are defined
externally, while the connections in black are determined by the outputs of the models.
Mathematically, each state space model is defined by equations 2.81 through 2.84.
32
Battery Model
Ab Bb Cb Db
iout
Vb∑
Vb∆
Vbus
Line Model
AL BL CL DL
VC
iout
VSC Model
Avsc Bvsc Cvsc Dvsc
idc
ivsc
VC
VbusSolar
Model
As Bs Cs Ds
Vc
is
iout
Vonom
V∆ref
V∑ref
V∆ref
Figure 2.14: Input and Output Definitions for Component State Space Models
xb = Abxb +Bbub
yb = Cbxb +Dbub(2.81)
xs = Asxs +Bsus
ys = Csxs +Dsus(2.82)
xL = ALxL +BLuL
yL = CLxL +DLuL(2.83)
˙xvsc = Avscxvsc +Bvscuvsc
yvsc = Cvscxvsc +Dvscuvsc(2.84)
Using the connections outlined in Figure 2.15, the inputs are defined as equation 2.85.
ub(3) = iout = yL
us(3) = vc = yvsc
uL(1) = vbus = yb
uL(2) = vc = yvsc
uvsc(1) = ibattout + isolarout = yL + ys
(2.85)
33
Battery Model
Ab Bb Cb Db
iout
Vb∑
Vb∆
Vbus
Line Model
AL BL CL DL
VC
VSC Model
AVSC BVSC CVSC DVSC
iout
iVSCVC
Solar Model
As Bs Cs Ds
Vc
is
iout
V∑ref
V∆ref
idc
Vonom
V∆ref
++
iout
Figure 2.15: Connections for the System State-Space Model
An important thing to note is that there are multiple feedback loops in this model.
This is problematic because it can potentially introduce an algebraic loop. To prevent
this, there are assumptions that must hold for all cases. The assumptions are provided
in equation 2.86.
Db ·DL = 0
DL ·Dvsc = 0
Ds ·Dvsc = 0
(2.86)
Assuming the equations in 2.86 are true, then the micro-grid state-space equation is
provided in 2.87 through 2.92.
Asys =
Ab 0 Bb(:, 3)CL(1, :) 0
0 As 0 Bs(:, 3)Cvsc
BL(:, 1)Cb 0 AL + BL(:, 1)Db(3)CL(1, :) + BL(:, 2)Dvsc(1)CL(1, :) BL(:, 2)Cvsc
Bvsc(1)DL(1, :)Cb Bv(:, 1)Cs Bvsc(:, 1)CL(1, :) Avsc + Bvsc(:, 1)DL(:, 2)Cvsc
(2.87)
Bsys =
Bb(:, 1) Bb(:, 2) 0 0 0 Bb(:, 4) Bb(:, 5) 0
0 0 Bs(:, 1) Bs(:, 2) 0 0 0 Bs(:, 4)
BL(:, 1)Db(:, 1) BL(:, 1)Db(:, 2) 0 0 BL(:, 1)Db(:, 3) + BL(:, 2)Dvsc(:, 2) BL(:, 1)Db(:, 4) BL(:, 1)Db(:, 5) 0
0 0 0 0 Bvsc(:, 2) 0 0 0
(2.88)
34
Csys =
Cb 0 Db(:, 3)CL(1, :) 0
0 Cs 0 Ds(:, 3)Cvsc
DL(:, 1)Cb 0 CL DL(:, 2)Cvsc
0 Dvsc(1, :)Cs Dvsc(:, 1)CL(1, :) Cvsc
(2.89)
Dsys =
Db(:, 1) Db(:, 2) 0 0 0 Db(:, 4) Db(:, 5) 0
0 0 Ds(:, 1) Ds(:, 2) 0 0 0 Ds(:, 4)
0 0 0 0 0 0 0 0
0 0 0 0 Dvsc(:, 2) 0 0 0
(2.90)
xsys =
xb
xs
xL
xvsc
=
vbus
vebusiL
ieLvC∑vCM
veCM
i∑iM
iL
ieLv∑ve∑vM
veM
vout∑isolarout
ibattout
vc
(2.91)
35
usys =
vnomo
vrefM
vref∑vrefM
iV SC
vb∑vbM
is
(2.92)
This form is now capable of modelling the DC microgrid system. In Section 2.4,
specific controller and component values are assigned, which will be used to demonstrate
the dynamics of the system due to various inputs.
2.3 Line Inductance Calculation
There is a wired connection between the batteries and the converter and amongst con-
verters in the DC microgrid. These connections introduce inductance in the line that will
affect the dynamics of the system. Therefore, an accurate model of the the inductance
must be made.
The connections are best modelled as two wires in parallel with no ground plane.
This is best illustrated in Figure 2.16.
Figure 2.16: Line Inductance Model [19]
Where:D - Diameter of the wire
S - Distance between wires (centre-to-centre)
` - Length of the wire
The mathematical formula to calculate the self inductance of a parallel wire can be
simplified with a few assumptions.
1. The distance between the wires (S) is constant for the entire length.
36
2. The length of both parallel wires (`) are identical.
3. There are no external Electromagnetic Interference (EMI) that are affecting the
parallel wires
With these assumptions, the formula is given by equation 2.93 [19].
Lwires ≈µ0µrπ
(cosh(S
D))−1` (2.93)
It is important to note that variables S and D can be in any units of measurement as
long they match. However, ` must be in meters to ensure proper unit cancellation.
For micro-grids, these lengths can vary greatly. Therefore, when doing the dynamic
analysis, an extensive range of was tested. These are detailed in table 2.1.
Table 2.1: Line Inductance Ranges
Unit Range
D 5.19 - 25.4 mm (4 AWG - 1000 kcmil)S 15.24 - 60.96 mm (0.5 ft - 2.0 ft)` 3.048 - 304.8 m (10 ft - 1000 ft)
Using equation 2.93 and the ranges provided in table 2.1, the inductance per unit
length and inductance over the range of lengths are given in table 2.2.
Table 2.2: Line Inductance Calculations
Cable Diameter Wire Spacing Inductance Per Unit Inductance Range
[mm (AWG or kcmil)] [mm] [µHm
] [µH]
5.19 (4) 15.24 1.63 5 - 50060.96 2.18 6.7 - 670
7.35 (1) 15.24 1.49 4.5 - 45060.96 2.05 6.2 - 620
25.4 (1000) 15.24 0.97 3- 30060.96 1.54 4.7 - 470
These values will be used when modelling any wired connections between different
components in the micro-grid system.
2.4 State-Space Model Results
This section demonstrates accurate modelling of the state-space models generated in the
previous sections. This is conducted by comparing the MATLAB state-space simula-
tions to the PSCAD/EMTDC results. The verified MATLAB model is then utilized
37
to calculate eigenvalues and participation factor, which is used for analysis and develop
conclusions about the DC microgrid system.
2.4.1 System Parameters
The purpose of this subsection is highlight the parameters for each component and pro-
vide a brief explanation of the chosen values. The general system parameters are provided
in Table 2.3.
Table 2.3: System Parameters: DC Micro-Grid
Parameter Symbol Value
Nominal Bus Voltage V nombus 380 V
The battery and solar module for this analysis uses the electrical parameters provided
in Table 2.4 and 2.6 respectively. The component sizes for both modules are provided in
Table 2.5.
Table 2.4: Battery Module Electrical Parameters
Parameter Symbol Value
Power Rating Prated 15.6 kWNominal Battery Voltage Vbat1, Vbat2 240 V
Nominal Output Bus Voltage V nomo 380 V
Rated Inductor Current iratedL 40 ASwitching Frequency fsw 20 kHz
Table 2.5 defines the component sizes used for the on-board converter components.
The range of line inductance, Lb, is defined in Section 2.3.
Table 2.5: BESS On-Board Component Sizes
Parameter Symbol Size
On-Board Inductor L 215 µHRL 3.88 mΩ
Input Capacitors CIN 60 µFDecoupling Capacitors COUT 30 µF
MOSFETs RON 25 mΩ
For proof of concept, the line inductance (See Fig. 2.12) and VSC DC link capacitance
(See Fig. 2.11) are provided in Table 2.7.
38
Table 2.6: Solar Module Electrical Parameters
Parameter Symbol Value
Power Rating Prated 8 kWNominal Output Bus Voltage V nom
bus 380 VRated Inductor Current iratedOUT 40 A
Switching Frequency fsw 20 kHz
Table 2.7: Line and VSC Component Values
Parameter Symbol Value
Line Inductance Lx 50 µHVSC Capacitance Cdc 5 mF
2.4.2 State-Space Model Verification
This section validates the state-space model of the system by creating the system in
PSCAD and comparing the transient responses. One battery converter, one solar con-
verter, one VSC and one line model are all included in the PSCAD model. The PSCAD
model also includes all controller delays to ensure bandwidth requirements are met. Ad-
ditionally, the PSCAD model utilizes the switching model of the converters to further
demonstrate accuracy.
Firstly, the plots in Figure 2.21 are verified to validate the battery converter, line and
VSC models. This is provided in Figure 2.17, which shows the output current (iOUT ) and
the bus voltage (Vbus). Both the MATLAB and PSCAD model match, which confirms
the accuracy of the model used for the battery converter, line and VSC.
Secondly, the plots in Figure 2.22 are verified to validate the solar converter model.
This is provided in Figure 2.18, which shows the inductor current (iL) and the input
capacitor sum voltage (V∑). Both the MATLAB and PSCAD model match, which
confirms the accuracy of the model used for the solar converter.
39
0 0.02 0.04 0.06 0.08 0.1 0.12−1
−0.8
−0.6
−0.4
−0.2
0
Vol
tage
(V
)
ivsc
Step − Vbus
PSCAD
ivsc
Step − Vbus
MATLAB
0 0.02 0.04 0.06 0.08 0.1 0.120
0.5
1
1.5
Time (s)
Cur
rent
(A
)
ivsc
Step − ioutbatt PSCAD
ivsc
Step − ioutbatt MATLAB
Figure 2.17: PSCAD vs. MATLAB Step Response: Bus Voltage (Vbus) and BatteryOutput Current (ibattout )
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
1
2
3
Cur
rent
(A
)
iS Step − i
L PSCAD
iS Step − i
L MATLAB
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
5
10
Time (s)
Vol
tage
(V
)
iS Step − V
sum PSCAD
iS Step − V
sum MATLAB
Figure 2.18: PSCAD vs. MATLAB Step Response: Solar Converter Inductor Current(iL) and Input Capacitor Sum Voltage (V∑)
2.4.3 Eigenvalue and Participation Factor Verification
This section uses the state-space model provided by equations 2.87 through 2.92 to sim-
ulate the step responses of the system. Recall that a complex eigenvalue is defined by
40
equation 2.94.
λ = α + jβ (2.94)
By expressing equation 2.94 as an exponential, it takes on the form of equation 2.95.
eλt = eα+jβ = eα(cos(βt) + jsin(βt)) (2.95)
Therefore, α represents the exponential decay, meaning it determines the time con-
stant of the state when an input is changed (in 1s). Also, this means that β represents
the resonant frequency of the state when an input is changed (in rads
). This is used to
approximately determine transient times of the states.
Using the state-space model provided by equations 2.87 through 2.92 and the values
defined in Section 2.4.1, the eigenvalues of the micro-grid system can be calculated using
MATLAB. However, this will only provide the eigenvalue numbers with no details about
their association to each state. This can be resolved by using the Participation Factor [13]
from Kundur’s “Power System Stability” textbook.
Conceptually, Participation Factor (PF) combines the left and right eigenvectors of
a state-space model to “measure the association between the state variables and the
modes” [13]. Mathematically, Kundur defines the participation factor by equation 2.96.
pfi =
p1i
p2i
...
pni
=
φ1iψ1i
φ2iψ2i
...
φniψni
(2.96)
Where:
pfki - The participation factor of the kth state on the ith eigenvalue
φki - The kth entry of the right eigenvector φi
ψki - The kth entry of the left eigenvector ψi
Using equation 2.96, the participation factor of the system are provided in Figure
2.19 & 2.20. Due the large number of states, the system states are divided into two plots.
The battery, line and VSC states and associated eigenvalues are provided in Fig. 2.19,
while the solar converter states and associated eigenvalues are provided in Fig. 2.20.
Firstly, the battery model is verified. Remember that the sum controller regulates the
bus voltage (Vbus), which is a nested control loop that regulates Vbus and iL. Therefore,
one can approximately determine the time constant of the step response by using the
41
Vbus Vbus_e iL iL_e Vsum Vdelta Vdelta_e isum idelta iout Vc0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Micro−Grid States
Par
ticip
atio
n F
acto
r
λ1=−586+8.27e+04i
λ2=−586−8.27e+04i
λ3=−1.69e+03+1.94e+04i
λ4=−1.69e+03−1.94e+04i
λ5=−9.86e+03+9.26e+03i
λ6=−9.86e+03−9.26e+03i
λ7=−82.5+240i
λ8=−82.5−240i
λ9=−4.93e+03+6.66e+04i
λ10
=−4.93e+03−6.66e+04i
λ11
=−0.000716
Figure 2.19: Participation Factor of the Battery, Line and VSC Model States
iL iLe Vsum Vsume Vdelta Vdeltae Vsum−out iout0
0.5
1
1.5
2
2.5
3
Micro−Grid States
Par
ticip
atio
n F
acto
r
λ1=−376+1.58e+04i
λ2=−376−1.58e+04i
λ3=−1.16e+04
λ4=−7.57e+03
λ5=−794
λ6=−169
λ7=−277+315i
λ8=−277−315i
Figure 2.20: Participation Factor of the Solar Model States
42
eigenvalue associated to the regulated terms that has the smallest real component. For
the sum controller, these are Vbus, Vebus, iL and ieL. Based on Figure 2.19, the smallest
real component is associated to V ebus. Therefore, the sum controller response will contain
a transient with a time constant (τ) that is approximately 182.5
= 12ms. Note that τ
represents the time taken to reach 63.2% of the final value. The settling time of a step
response is approximated as 5τ . Therefore, the settling time of the this system’s step
response should be approximately 60ms.
Secondly, the solar converter model is verified. The sum controller for the solar
converter regulates V∑ and iL. From Fig. 2.20, the smallest real component is associated
to V e∑. Therefore, the sum controller time constant (τ) is 1169
= 6ms. Therefore, the
settling time should be approximately 30ms.
0 0.02 0.04 0.06 0.08 0.1 0.12−1
−0.8
−0.6
−0.4
−0.2
0
X: 0.1187Y: −0.475
Vol
tage
(V
)
ivsc
Step − Vbus
0 0.02 0.04 0.06 0.08 0.1 0.120
0.5
1
1.5
Time (s)
Cur
rent
(A
)
ivsc
Step − ioutbatt
Figure 2.21: 1A Ivsc Step - Bus Voltage (Vbus) and Battery Output Current (ibattout )
43
To validate the eigenvalue analysis, the next step is to plot the step responses of
the DC microgrid components. Figure 2.21 depicts the response of the microgrid DC
bus voltage (vbus) and battery output current (ibattout ) due to a step in the VSC current
(ivsc). The settling time is approximately 60ms, which matches the estimation calculated
from the eigenvalues. Additionally, a 1A current step produced a 0.475V drop in the
bus voltage. This is due to the droop control, whose virtual resistance (Kd) is 0.475Ω.
Therefore, the battery converter, line and VSC eigenvalues and participation factors align
with the model step responses.
0 0.01 0.02 0.03 0.04 0.050
1
2
3
X: 0.04626Y: 2.124
Cur
rent
(A
)
iS Step − i
L
0 0.01 0.02 0.03 0.04 0.050
5
10
Time (s)
Cur
rent
(A
)
iS Step − V
sum
Figure 2.22: 1A Is Step - Solar Inductor Current (iL) and Input Capacitor Sum Voltage(v∑)
44
Figure 2.22 depicts the response of the solar converter’s inductor current (iL) and
input capacitor sum voltage (v∑) due to a step in the solar panel current (is). The
settling time is approximately 30ms, which matches the estimation calculated from the
eigenvalues. Additionally, the inductor current demonstrates the steady-state relation-
ship between the panel current and the inductor current is valid (IL = Isd
). Remember
that, for the buck/boost configuration, the conversion ratio is defined by equation 2.97.
Vbus =d
1− dVs (2.97)
When Vs = 435V and Vbus = 380V , d = 0.466. Therefore, IL = 2.145A, which aligns
with Figure 2.22. Therefore, the solar converter eigenvalues and participation factors
align with the model step responses.
2.4.4 Eigenvalue and Participation Factor Analysis
With the eigenvalue and participation factor calculations verified, the results can be
utilized to make conclusions about the DC microgrid model developed in this chapter.
Figure 2.23 shows the participation factor of the battery converter, line and VSC
eigenvalues, while Fig. 2.24 shows the participation factor of the solar converter. They
are reorganized to show the eigenvalues associated with the sum and difference controllers
separately. Section 2.1.1 discussed that the sum and difference control scheme was utilized
to decouple their states. Both figures show that the sum and difference eigenvalues are
completely decoupled. This implies that a mismatch in the battery voltage does not
affect the DC system, which verifies the utilization of this control scheme.
Furthermore, Fig. 2.23 shows that the DC microgrid bus voltage (Vbus) and output
current (iout) are approximately decoupled from the battery converter states (eg. iL, ieL,
V∑, i∑) for the controller values outlined in Appendix D. This is significant because Vbus
and iout are states that represent important parameters in the DC microgrid (eg. system
bus voltage, output currents). Decoupling these from the converter states reduces the
impact converters have on the DC microgrid system. An example participation factor
plot for poorly chosen controller values is provided in Fig. 2.25, which demonstrates the
significance of controller design on state decoupling.
In contrast with the battery converter, the participation factor plot for the solar
converter in Fig. 2.24 show significant coupling between states. This implies that the
topology configuration and control scheme utilized by the solar converter can have sig-
nificant influence on the DC microgrid (eg. Vbus =V∑+V out∑
2).
45
Vbus Vbus_e iL iL_e Vsum isum iout Vc Vdelta Vdelta_e idelta0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Micro−Grid States
Par
ticip
atio
n F
acto
r
Participation Factor of the Micro−Grid Model
λ1=−586+8.27e+04i
λ2=−586−8.27e+04i
λ3=−1.69e+03+1.94e+04i
λ4=−1.69e+03−1.94e+04i
λ5=−9.86e+03+9.26e+03i
λ6=−9.86e+03−9.26e+03i
λ7=−82.5+240i
λ8=−82.5−240i
λ9=−4.93e+03+6.66e+04i
λ10
=−4.93e+03−6.66e+04i
λ11
=−0.000716
Sum ControllerEigenvalues
Diff. ControllerEigenvalues
Figure 2.23: Participation Factor of the Battery Converter, Line and VSC Model States,Reorganized into Sum and Difference Eigenvalues
iL iLe Vsum Vsume Vsum−out iout Vdelta Vdeltae0
0.5
1
1.5
2
2.5
3
Micro−Grid States
Par
ticip
atio
n F
acto
r
λ1=−376+1.58e+04i
λ2=−376−1.58e+04i
λ3=−1.16e+04
λ4=−7.57e+03
λ5=−794
λ6=−169
λ7=−277+315i
λ8=−277−315i
Sum ControllerEigenvalues
Diff. ControllerEigenvalues
Figure 2.24: Participation Factor of the Solar Converter States, Reorganized into Sumand Difference Eigenvalues
46
Vbus Vbus_e iL iL_e Vsum isum iout Vc Vdelta Vdelta_e idelta0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Micro−Grid States
Par
ticip
atio
n F
acto
r
Participation Factor of the Micro−Grid Model
λ1=−710+8.27e+04i
λ2=−710−8.27e+04i
λ3=−2.98e+03+2.11e+04i
λ4=−2.98e+03−2.11e+04i
λ5=−2.98e+03+1.49e+04i
λ6=−2.98e+03−1.49e+04i
λ7=−81.1+243i
λ8=−81.1−243i
λ9=−4.93e+03+6.66e+04i
λ10
=−4.93e+03−6.66e+04i
λ11
=−0.000716
Figure 2.25: Participation Factor of the Battery Converter, Line and VSC States, withPoorly-Chosen Controller Values
2.5 Chapter Conclusion
This chapter provided and validated a state-space model for the BESs, solar PV, line and
VSC. Additionally, a method to mathematically connect these models are provided and
validated. The connection method is formulated such that it allows modularity, as will be
exploited in Chapter 3. By equating the inputs and outputs in different configurations,
a variety of connection combinations can be modelled.
As outlined in Section 2.4.4, the sum and difference states are completely decoupled.
For the battery converter, the difference controller is utilized to balance the state-of-
charge (SOC) of both connected batteries, which implies that an SOC mismatch has no
effect on the dynamics and stability of the DC microgrid.
Furthermore, Section 2.4.4 outlines the significance of controller design and the cou-
pling between states. Specifically, appropriately-chosen controller values can reduce state
coupling, which reduces the impact converters have on the DC microgrid and can poten-
tially offer improved system scalability, which is further discussed in Chapter 3.
47
Chapter 3
Scalability Analysis
Chapter 2 created a model that accurately represents any small DC microgrid. Using this
model, an investigation into system scalability is conducted. In this chapter, scaling is
defined as increasing the system capacity by the addition of multiple BESSs or generation
sources. Scalability is defined as the ability to demonstrate system stability as the number
of BESSs or generation sources is scaled. Demonstrating scalability for a wide range of
power ratings develops understanding of the effect future component installations have
on the system. Additionally, this demonstrates that DC microgrids are not restricted to
a narrow power range.
The first objective is to expand the existing component models for a variable number
of BESS and solar converters. This model is then verified by comparing the MATLAB
responses to PSCAD simulations, much like Chapter 2. Once the model is confirmed, an
investigation into the eigenvalue’s movements with the addition of multiple converters in
conducted. If the eigenvalues remain in left-half plane (LHP) as the number of converters
is increased, then the system is considered scalable. To keep the problem tractable,
scalability of the BESS and PV systems are exclusively examined, while the microgrid
network remains (i.e. no addition of new buses).
3.1 Scalable State-Space Model
The goal of this section is to expand the BESS and solar component models to include
multiple converters. This allows the system connection method developed in Section 2.2
to be utilized for the scalable model as well.
48
3.1.1 Scalable Battery Model
The battery component model expansion is illustrated in Fig. 3.1. All battery converters
are assumed to be co-located at the same bus. This has an effect on the way currents
distribute. A portion of current from one converter can go into the adjacent converters.
This changes the differential equations noticeably, and can lead to internal instability
amongst the battery converters or between the multiple battery converters and the solar
converter or inverter.
CIN
CIN
COUT COUTL
V11
Va1 Vb1 Vbus
d11V11
d21V21
d11iL1
d21iL1
IL1
V21
Lb
Ib11Vb11
Lb
Ib21Vb21
Iout
CIN
CIN
COUT COUTL
V1n
Van Vbn
d1nV1n
d2nV2n
d1niLn
d2niLn
ILn
V2n
Lb
Ib1nVb1n
Lb
Ib2nVb2n
n
Figure 3.1: Schematic of Scaled Battery Model
The first change is the definition of the bus voltage (vbus). The original definition was
(2.16). For the scalable case, the bus voltage is influenced by all converters. Therefore,
the bus voltage is redefined as (3.1).
49
vbus =
∑ni=1(v∑ i + vout∑
i)
2n(3.1)
The second change is the definition of the constant ”k,” which was originally defined
by (2.10). This definition assumed that only the four capacitors in the converter were
present. With the scalable model, it is redefined as (3.2).
k =Ceq
Cin + Ceq(3.2)
Where:
Ceq =(2n− 1) CinCout
Cin+CoutCout
(2n− 1) CinCout
Cin+Cout+ Cout
(3.3)
These new definitions were integrated into (2.3) - (2.9), which is provided by (3.4) -
(3.9) (Note: jε[1, · · · , n]∧j 6= i).
ddt〈vbus〉 =
∑ni=1[(
1n−K〈d∑ i〉2nCIN
+ (1n−2k
′ 〈d∑ i〉2nCIN
)(n− 1) +1n
+(k−k′ )〈d∑ i〉2nCOUT
+ (1n−2k
′ 〈d∑ i〉2nCOUT
)(n− 1))〈iLi〉+( K
2nCIN+ ( 2k
′
2nCIN+ 2k
′
2nCOUT))〈i∑ i〉]− ( 1
2nCIN+ 1
2nCOUT)〈iOUT 〉
(3.4)
Ld
dt〈iLi〉 = −〈vbus〉 − (2RON +RL)〈iLi〉+
〈d∑ i〉2〈v∑ i〉+
〈dMi〉2〈vMi〉 (3.5)
CINddt〈v∑ i〉 = ( 1
n−K〈d∑ i〉)〈iLi〉+K〈i∑ i〉 − 1
n〈iOUT 〉
+∑n
j=1[( 1n− 2k
′〈d∑ j〉)〈iLj〉+ 2k′〈i∑ j〉]
(3.6)
CINd
dt〈vMi〉 = −Kinv〈dMi〉〈iLi〉+Kinv〈iMi〉 (3.7)
Lbd
dt〈i∑ i〉 = −〈v∑ i〉 −RLb〈i∑ i〉+ 〈vb∑ i〉 (3.8)
Lbd
dt〈iMi〉 = −〈vMi〉 −RLb〈iMi〉+ 〈vbMi〉 (3.9)
Where:
k′=
k
2n− 1(3.10)
50
K = k′+ 1− k (3.11)
Kinv = 1− k − k′ (3.12)
Equations 3.4 - 3.9 now represent the open-loop differential equations for the scalable
battery model. By linearising them and substituting (2.31) - (2.33) and 2.38, the closed-
loop scalable battery model is created, much like the technique in Chapter 2. For the
remainder of the chapter, this is defined by (3.13).
˙xbS = AbSxbS +BbSubS
ybS = CbSxbS +DbSubS(3.13)
3.1.2 Scalable Solar Model
The battery component model expansion is illustrated in Fig. 3.2. Like the battery
model, the current distributions are different than the single converter case and the
output terminals are all connected.
The input current (Is) results in a current distribution that is not encapsulated by
equations 3.2 or 3.10 - 3.12. Therefore, a new variable is defined to depict the distribution
of Is for the scalable model (M1, M2 & Mx). Appendix E defines these variables. The
differential equations for the scalable solar model are defined by (3.14) - (3.18).
Ld
dt〈iLi〉 = −(2RON +RL)〈iLi〉+
〈d∑ i〉 − 1
2〈v∑ i〉+
〈dMi〉2〈vMi〉 −
1
2〈vout∑
i〉 (3.14)
CINddt〈v∑ i〉 = ( 1
m−K〈d∑ i〉)〈iLi〉 − 1
m〈iOUT 〉+M1〈isi〉
+∑m
j=1[( 1m− 2k
′〈d∑ j〉)〈iLj〉+ Mx
m〈isj〉]
(3.15)
CINd
dt〈vMi〉 = −Kinv〈dMi〉〈iLi〉+Kinv〈iMi〉 (3.16)
CINddt〈vout∑
i〉 = ( 1m− (k − k′)〈d∑ i〉)〈iLi〉 − 1
m〈iOUT 〉+M2〈isi〉
+∑m
j=1[( 1m− 2k
′〈d∑ j〉)〈iLj〉+ Mx
m〈isj〉]
(3.17)
Lxd
dt〈iout〉 =
1
2〈v∑ 1〉+
1
2〈vout∑
1〉 −Rx〈iout〉 − 〈vc〉 (3.18)
Equations 3.14 - 3.18 now represent the open-loop differential equations for the scal-
51
CIN
CIN
COUT COUTL
V11
Va1 Vb1 Vbus
d11V11
d21V21
d11iL1
d21iL1
IL1
V21
Is1
Lx
Vc
Vs1
CIN
CIN
COUT COUTL
V1m
Vam Vbm
d1mV1m
d2mV2m
d1miLm
d2miLm
ILm
V2m
Ism Vsm
m
Figure 3.2: Schematic of Scaled Solar Model
able solar model. By linearising them and substituting (2.56) - (2.58), the closed-loop
scalable solar model is created. For the remainder of the chapter, this is defined by (3.19).
˙xsS = AsSxsS +BsSusS
ysS = CsSxsS +DsSusS(3.19)
3.2 Model Verification: Simulation Results
Before scalability can be investigated, the model must be verified. This is conducted using
the same method utilized in Chapter 2. Simulations were conducted in PSCAD with
multiple solar and battery converters and compared with the step responses produced
in MATLAB. The state-space connection method is identical to Section 2.2. Only the
52
battery and solar model are updated based on the number of converters, which are
outlined in each results section.
3.2.1 Battery Scaling Simulation
Much like Chapter 2, the battery model is verified by providing the bus voltage (Vbus)
and the converter output current’s (iout) response to a step in the VSC current (ivsc).
Two Battery Converters
For this section, there are two battery converters, one solar converter, one line model and
one VSC model. The simulation results are provided in Fig. 3.3.
0 0.02 0.04 0.06 0.08 0.1 0.12−0.8
−0.6
−0.4
−0.2
0
Vol
tage
(V
)
ivsc
Step − Vbus
PSCAD
ivsc
Step − Vbus
MATLAB
0 0.02 0.04 0.06 0.08 0.1 0.120
0.5
1
1.5
Time (s)
Cur
rent
(A
)
ivsc
Step − ioutbatt PSCAD
ivsc
Step − ioutbatt MATLAB
Figure 3.3: (Two Battery Converters, One Solar Converter) PSCAD vs. MATLAB StepResponse: 1A Step in the VSC Load Demand (ivsc). Top to Bottom: Bus Voltage (vbus),Output Current (iout)
Three Battery Converters
For this section, there are three battery converters, one solar converter, one line model
and one VSC model. The simulation results are provided in Fig. 3.4.
For both cases, the step responses are identical, which validates the scaled battery
converter model.
53
0 0.02 0.04 0.06 0.08 0.1 0.12−0.8
−0.6
−0.4
−0.2
0
Vol
tage
(V
)
ivsc
Step − Vbus
PSCAD
ivsc
Step − Vbus
MATLAB
0 0.02 0.04 0.06 0.08 0.1 0.120
0.5
1
1.5
Time (s)
Cur
rent
(A
)
ivsc
Step − ioutbatt PSCAD
ivsc
Step − ioutbatt MATLAB
Figure 3.4: (Three Battery Converters, One Solar Converter) PSCAD vs. MATLABStep Response: 1A Step the VSC Load Demand (ivsc). Top to Bottom: Bus Voltage(vbus), Output Current (iout)
3.2.2 Solar Scaling Simulation
Much like Chapter 2, the solar model is verified by providing the inductor current (iL),
the input capacitor sum voltage (V∑) and the converter output current’s (iout) response
to a step in one of the solar panel’s current (is).
Two Solar Converters
For this section, there are two solar converters, one battery converter, one line model and
one VSC model. The simulation results are provided in Fig. 3.3.
Three Solar Converters
For this section, there are three solar converters, one battery converter, one line model
and one VSC model. The simulation results are provided in Fig. 3.4.
For both cases, the step responses are identical, which validates the scaled solar
converter model.
54
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
1
2
3
Cur
rent
(A
)
iS Step − i
L PSCAD i
S Step − i
L MATLAB
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
5
10
Time (s)
Cur
rent
(A
)
iS Step − V
sum PSCAD i
S Step − V
sum MATLAB
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
1
2
Time (s)
Cur
rent
(A
)
iS Step − i
out PSCAD i
S Step − i
out MATLAB
Figure 3.5: (Two Solar Converters, One Battery Converter) PSCAD vs. MATLAB StepResponse: 1A Step in is. Top to Bottom: Inductor Current (iL), Input Capacitor SumVoltage (v∑), Output Current (iout)
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
1
2
3
Cur
rent
(A
)
iS Step − i
L PSCAD i
S Step − i
L MATLAB
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
5
10
Time (s)
Cur
rent
(A
)
iS Step − V
sum PSCAD i
S Step − V
sum MATLAB
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
1
2
Time (s)
Cur
rent
(A
)
iS Step − i
out PSCAD i
S Step − i
out MATLAB
Figure 3.6: (Three Solar Converters, One Battery Converter) PSCAD vs. MATLABStep Response: 1A Step in is. Top to Bottom: Inductor Current (iL), Input CapacitorSum Voltage (v∑), Output Current (iout)
55
3.3 Scalability Analysis: Eigenvalue Movement
With model verifications complete, the system can add numerous converters, which in-
creases the power rating. As mentioned at the beginning of the chapter, if the system
is stable when an additional converter is added, then it is considered scalable. However,
adding multiple converters in a EMTDC simulation is tedious and results in excessive
run-times. Therefore, system stability is investigated by calculating the eigenvalues. If
the eigenvalues remain in the LHP, stability/scalability is confirmed.
Both the battery and solar models were scaled separately, so that their eigenvalue
movements can be studied individually. Converters are continuously added into the state-
space model until instability occurs or the system exceeds MATLAB’s computational
limits (∼300 Converters).
After each converter is added, the eigenvalues are calculated and plotted onto the
real/imaginary axis, with the single converter eigenvalues bolded to show the unscaled
location. This helps visualize the direction that the eigenvalues are moving as the sys-
tem expands. Note that, like the participation factor discussion in Section 2.4.3, the
eigenvalues are separated due to the numerous number of states.
3.3.1 Battery Model Eigenvalue Movement
The eigenvalue plot for the scaled battery model is provided by Fig. 3.7. This includes
the scaled battery model, line and VSC states.
The eigenvalues experience a large movement when adding the first converter. The
movement then slows substantially as the number of converters grow. When scaled to 300
converters, the eigenvalues are still in the LHP, which implies scalability. As mentioned
in Table 2.4, a single battery converter is capable of supplying 15.6 kW. Therefore, the
BESS is confirmed to be scalable up to 4,680 kW.
3.3.2 Solar Model Eigenvalue Movement
The eigenvalue plot for the scaled solar model is provided by Fig. 3.8. This includes the
scaled solar states.
When scaled to 300 converters, the eigenvalues are still in the LHP, which implies
scalability. As mentioned in Table 2.6, a single solar converter is capable of supplying 8
kW. Therefore, the solar is confirmed to be scalable up to 2,400 kW.
56
Figure 3.7: Eigenvalue Movement: Battery Scaling (1-300 Battery Converters, SingleSolar Converter). Solar Converter Eigenvalues Removed
Figure 3.8: Eigenvalue Movement: Solar Scaling (1-300 Solar Converters, Single BatteryConverter). Battery Converter Eigenvalues Removed
3.4 Chapter Conclusions
This chapter expanded the battery and solar component state-space models to include
multiple converters whose output terminals are connected to the same bus, which means
57
all DC microgrid configurations can be modelled and investigated for further analysis.
Furthermore, the expanded model was utilized and eigenvalues were calculated with
each additional converter to investigate their movement. Both the BESS and solar PV
demonstrated exceptional scalability, reaching 300 converters without demonstrating in-
stability. Figures 3.7 & 3.8 show that the eigenvalues movement rate is reduced after
each additional converter, meaning this DC microgrid system is further scalable. These
results propose that, with an appropriately designed converter topology, control structure
and controller values, DC microgrids are applicable for a significant power range (kW -
MW) with an acceptable influence on stability as additional battery and solar models
are integrated into the system.
58
Chapter 4
Autonomous Energy Management
Method
In Chapter 3, system scalability was investigated. The concept of scalability, however,
is incomplete without designing a method that can manage such a large system with
multiple loads and sources. Specifically, the goal is to design an autonomous energy
management method that adapts to the proposed modular system. This will include
designing the following:
1. Battery state-of-charge (SOC) balancing.
2. Battery overcharge protection (OCP).
3. Load shedding.
Firstly, the energy management requirements will be defined. Secondly, the design
procedure and how it meets the requirements is discussed. Finally, the design will be
verified via PSCAD simulation software.
4.1 Energy Management Justification
4.1.1 SOC Balancing
During operation, the SOC of the battery energy storage systems (BESS) can vary. This
is undesirable because it may result in one of the BESSs to deplete prematurely, which
reduces the the rated power of the microgrid system. SOC balancing also reduces the
likelihood of batteries becoming overcharged or undercharged, which significantly reduces
59
deterioration [21]. Therefore, adding a method to provide SOC balancing allows rated
system capacity for longer periods of time and increases BESS operational lifetime.
4.1.2 Overcharge Protection (OCP)
In the event that the BESSs are approaching a fully charged state, there is a potential
risk that the system will attempt to overcharge the batteries. Typical batteries, including
lithium-ion, can suffer from characteristic degradation, short life cycle, overheating or
exploding when they are in the state of overcharge [22]. Therefore, preventing OCP is
required for reliability of the DC microgrid system.
4.1.3 Load Shedding
This system has a rated power it is capable of providing to all loads in the system.
However, loads can potentially demand power that exceeds the system’s capabilities.
In the event that this occurs, the bus voltage will slowly collapse until the bus can no
longer maintain functionality. Therefore, the energy management system must include a
method to detect and provide a solution to deactivate loads when this occurs.
4.2 Proposed Energy Management Scheme
This section outlines the proposed energy management method that provides the require-
ments discussed, which are:
1. Battery Energy Storage System (BESS) State-of-Charge (SOC) Balancing
2. Battery Overcharge Protection (OCP)
3. Load Shedding When Demand Exceeds System Ratings
The general control scheme for the energy management scheme is provided in Fig.
4.1.
Conceptually, the proposed scheme utilizes the droop characteristic set by the BESS.
Specifically, the V-I curve of the droop characteristic’s y-intercept becomes a function of
SOC. The slope of the droop characteristic will remain unchanged to prevent eigenvalues
from changing based on the SOC. This is illustrated in Fig. 4.2. It is important to note
that each BESS defines its droop curve based the SOC of the battery connected to it.
The main advantage of this design is that there is no direct communication required
between converters to implement energy management amongst the batteries. Instead,
60
+
-
IL
DC/DC Conv.
Controller+
-
Vbus
-
Kd
Iout
+IL
VbusrefVbus
nom
Imax
Imax
+
-
ControllerDuty Cycle
Controller
Circuit
Duty Cycle
VbusIout
Vbatt
ref
DC/DC Conv. Internal Current Control
Figure 4.1: General Control Scheme for the Energy Management Method
each converter measures the bus voltage to interpret the state of the system. This allows
the local controllers to adjust without the need to wait for information from a system
supervisory control. Therefore, this method offers a cost-effective method to implement
an indirect-communication energy management that responds quickly and has reduced
effect on stability.
4.2.1 Droop Curve Per-Unitisation
To discuss the details of the energy management scheme, it is beneficial to define the
droop curve of each converter on a normalized per-unit basis. The droop curve is defined
by (4.1).
Vref = Vnom −KdIout (4.1)
Both the voltage and current can be defined in per-unit by (4.2) and (4.3).
Vpu =V
Vbase(4.2)
ipu =I
Ibase(4.3)
By substituting (4.2) and (4.3) into (4.1), the per-unit equation of the droop curve is
described by (4.4).
61
Vbus
Iout
Vnom = f(SOC)
Imax+Imax
-
Figure 4.2: Droop Curve Adjustment for Energy Management Scheme
V purefVbase = V pu
nomVbase −Kdipuoutibase (4.4)
From here, the goal is to define the droop curve’s slope in per-unit. This can be
done by dividing both sides of (4.4) by Vbase. In doing so, the droop curve slope (Kpud ) is
defined by (4.5).
Kpud = Kd
IbaseVbase
(4.5)
Therefore, the new per-unit droop characteristic is defined by (4.6). Each converter
has its own unique base values, Vbase and Ibase, which are determined by the power rating
of the associated converter. For the sake of simplicity, this chapter assumes that each
converter has the same base voltage and current.
V puref = V pu
nom −Kpud I
puout (4.6)
For the example system, the base values for the converter are given in Table 4.1.
Table 4.1: System Per-Unit Base Values
Parameter Symbol Value
Base Voltage Vbase 380 VBase Current Ibase,k 40 A
62
4.2.2 Droop Curve Adjustment
This subsection will discuss how the droop curve is adjusted to provide state-of-charge
(SOC) balancing of the BESS, overcharge protection (OCP) and load shedding. For
ease of understanding, the adjustment curve is provided first, then how it provides each
requirement in the energy management system is discussed individually.
The nominal voltage of the droop curve (V punom) is a non-linear adjustment based on
the SOC of the local BESS. this is illustrated by Fig. 4.3. In this example, for SOC >
80%, the nominal voltage rises considerably quicker than at lower SOCs and the non-
linear adjustment happens to be piecewise linear. This is beneficial for OCP, which is
explained in more detail later in the section. In general, the shape of the non-linear
adjustment curve would depend on battery characteristics, amongst other variables.
0 10 20 30 40 50 60 70 80 90 100−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
SOC [%]
Nom
inal
Vol
tage
Offs
et [p
u]
SOC Curve
Figure 4.3: Implemented Droop Adjustment Curve
The droop characteristic used in the example system are outlined in Table 4.2. Note
that the 5% droop slope is based off the NERC BAL-001-TRE-1 standard [23], which is
used for AC power systems. Using these values, the droop curve for various SOCs take
on the form shown in Fig. 4.4. The significant SOC values are highlighted (0%, 80%,
100%).
As expected, there is a significant rise between SOC = 80% and SOC = 100%. Sup-
pose that these three curves represented three unique BESS in the system. Assuming no
line losses, all BESSs have the same output voltage during steady-state operation. This
phenomenon results in SOC balancing. To further illustrate this, Fig. 4.4 is expanded on
63
Table 4.2: Droop Curve Values
Parameter Symbol Per-Unit Value
Nominal Voltage V punom 1
Droop Slope Kpud 0.05
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
SOC1= 100%SOC2= 80%SOC3= 0%
Figure 4.4: Droop Curves for SOC = 0%, 80%, 100%
by adding the aggregate droop curve. This aggregate curve represents how the microgrid
bus voltage varies based on the output current of all BESSs. Intuitively, the aggregate
curve is piecewise linear since the current limiter does not allow individual converters
to exceed its rated output. This is provided by Fig. 4.5. Since the aggregate is for all
converters, the current base for that curve is altered. The aggregate curve base is defined
in (4.7).
Iaggbase =n∑k=1
Ibase,k (4.7)
The aggregate curve in Fig. 4.5 states that, for a no load scenario, V refbus = 1[pu].
Based on the curve, the output current for each converter is highlighted in Table 4.3.
Since there is no load on the system, the output current of BESS 1 is providing rated
power to BESS 3, while BESS 2 outputs no current, resulting in no net output current
(no load). This continues until SOC1 = SOC2 = SOC3. At that point, all BESS droop
curves are identical and SOC balancing is achieved.
There are two important ramifications to observe at this point:
64
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
AggregateSOC1= 100%SOC2= 80%SOC3= 0%
Figure 4.5: Aggregate Droop Curve of DC Micro-Grid w/ Three BESS at SOC = 0%,80%, 100%, where Iaggbase =
∑nk=1 Ibase,k
Table 4.3: Individual BESS Output Current for V refbus = 1 [pu]
Battery No. SOC Value [%] Base Current Output Current @ No Load [pu]
1 100 ibase,i 12 80 ibase,i 03 0 ibase,i -1
1. At high negative currents (-1 to -0.7 pu), the bus voltage exceeds 1.1 [pu], which is
here taken as the maximum permissible bus voltage.
2. At high SOCs, the BESS has the ability to receive rated current.
These introduce a risk, since the system is rated to handle a maximum bus voltage
here of 1.1 [pu]. Additionally, since the BESS can still receive rated power at high SOC,
the batteries could potentially be overcharged and permanently damaged. At high SOC,
many batteries cannot be charged with a constant current. Therefore, an overcharge
protection (OCP) method is needed to ensure the safety of the BESS.
This protection can be provided via a generation source limiter. By limiting the
maximum power (or current) the source can provide, the power (or current) that the
BESS will receive is indirectly limited. The goal of the OCP scheme is to steadily reduce
the maximum output power of the sources until all battery’s SOCs are at 100%. Once
this is achieved, the generation sources halt power production until a load demand occurs.
65
This is illustrated by Fig. 4.6. It is important to note that the base for the renewable
source limiter is the same base as the aggregate BESS curve, which implies that the
generation source and BESSs are rated for equal power. That is not necessary for OCP
and is chosen to simplify the explanation. To implement OCP, the generation source
measures the bus voltage and reduces its own maximum current based on the red dashed
line. At V refbus = 1.1 pu, that maximum current is set to zero. Therefore, the system
cannot enter the grey region.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
Out of Bounds RegionAggregateRenewable Source LimiterSOC1= 100%SOC2= 80%SOC3= 0%
Figure 4.6: Aggregate Droop Curve of DC Micro-Grid w/ Three BESS at SOC = 0%,80%, 100% and the Renewable Source Limiter, where Iaggbase =
∑nk=1 Ibase,k
By observing the magenta line, an observation can be made that, if all the BESS are
below SOC = 80%, then the renewable source’s maximum permissible power (or current)
will always remain at its rated value. On the other hand, if any of the BESS exceed 80%,
then maximum permissible power can potentially decrease depending on the system load.
Therefore, the region between 1.05 < V refbus < 1.1 can be defined as the OCP region.
Another observation is that, if all the BESS are fully charged and a load is intro-
duced, the maximum power that the renewable generation source can provide is increased.
Therefore, if the BESSs are discharging, the renewable generation source attempts to pro-
vide a portion of the load demand depending on the loading conditions of the system.
The final requirement is system load shedding. If the loads demand power that
exceeds the BESS and renewable generation source’s capabilities, the system bus voltage
collapses. However, the proposed energy management scheme is capable of preventing
this. The acceptable bus voltage operating range is here assumed to be 0.9 < Vbus < 1.1.
66
Therefore, if a load occurs beyond the system’s capabilities, the bus voltage reduces below
0.9 pu. When this occurs for a time scale that exceeds normal transient times, selected
loads are tripped off-line based on a priority basis, consistent with standard practice in
AC power systems, until the load is under the system’s power rating. Using this method,
load shedding can be achieved. Therefore, this energy management method is capable of
providing all requirements mentioned at the beginning of the chapter.
67
4.3 Simulation Results
4.3.1 Test Scenario 1
To validate the proposed energy management scheme, the system was simulated using
PSCADTM . The system consists of two BESSs, two solar PV arrays and one VSC, which
is depicted in Fig. 4.7. Line impedances are other details are included in the model, but
not depicted in Fig. 4.7 to enhance clarity.
Load
STORAGE
___
___- - -
___- - -___- - -
___- - -
___- - -
Vbus
___- - -
~___- - -
~GRID
VSC
Ivsc
ILOADIB1 IB2IS2
SOLAR
___- - -
___- - -
IS1
Figure 4.7: DC Micro-Grid System to Validate Energy Management System
The system parameters for this system are provided in Table 4.4 and uses the base
values in Table 4.1. The initial conditions for the system are outlined in Table 4.5.
It should be noted that a disproportionately small BESS capacity, Q, is employed for
simulations to depict change/discharge behaviour over an accelerated time frame.
68
Table 4.4: System Parameters
BESSParameter Symbol ValueRated Power P rated
batt 15.2kWRated Output Current Iratedbatt 40A
Nominal Capacity Q 0.02AhDroop Nominal Voltage Vnom 380V
Renewable Generation Source (PV)Parameter Symbol ValueRated Power P rated
solar 7.3kWRated Output Current Iratedsolar 19.2A
Table 4.5: Initial Conditions for Simulation
BESSParameter Symbol Value
SOC of Battery 1 SOCB1 80%SOC of Battery 2 SOCB2 75%
Renewable Generation Source (PV)Parameter Symbol ValueIrradiance G 900 W
m2
Temperature Tcell 20oC
69
1 2 3 4 5 6 7 840
60
80
100
SO
C [%
]
1 2 3 4 5 6 7 8360
380
400
420
Bus
Vol
tage
[V]
1 2 3 4 5 6 7 8−50
0
50
Cur
rent
[A]
Time (s)
1 2 3 4 5 6 7 8−40
−20
0
20
Cur
rent
[A]
Time (s)
SOCB1
SOCB2
Vbus
Ioutsolar
Ioutbatt
IoutB1
IoutB2
Int. 1 Int. 2 Int. 3 Int. 4
Figure 4.8: Energy Management Simulation Results; Top to Bottom: SOC of Battery1 & 2, Bus Voltage, Total Renewable Generation Source and Battery Output Current,Output Current for Battery Converter 1 & 2
The simulation results are provided in Fig. 4.82. During interval 1, SOC balancing
occurs to correct the 5% charge difference between both BESSs. During interval 2,
the overcharge protection (OCP) is activated because the bus voltage at the specific
generation current crossed the source limiter line in Fig. 4.6. At the beginning of interval
3, a 50A load is introduced, which reduces the bus voltage and the renewable source
limiter steadily open ups. During interval 4, the renewable source is outputting its
maximum permissible power and the BESSs begin to discharge, which reduces the bus
voltage. To understand the role of the energy management scheme in this simulation,
Fig. 4.9 shows the simulation bus voltage and output current added onto the droop curve
plots. Each interval is described by Fig. 4.10 through 4.14. Figure 4.8 demonstrates the
proposed energy management method meets the requirements discussed in the beginning
of the chapter (SOC balancing, OCP). Therefore, the combination of vertically adjusting
2The generation source used in this simulation is the converter topology and control scheme discussedin Chapter 2. Therefore, the limited current is the inductor current, which has a non-linear relationshipwith the output current. Therefore, the renewable source limiting curve is non-linear in practice and isshown linearly for simplification purposes.
70
the droop characteristic based on SOC together with limiting the renewable generation
source current offers an effective energy management scheme.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Bus
Vol
tage
[pu]
Out of Bounds RegionAggregateRenewable Source LimiterSimulation Results (See Fig. 4.8)SOC1= 75%SOC2= 80%
Figure 4.9: Test Scenario 1: Entire Simulation. The bus voltage and output current fromthe simulation results (Fig. 4.8) are overlapped to see how the operating point changesdue to the proposed energy management scheme.
71
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
Out of Bounds RegionAggregateRenewable Source LimiterSimulation Results (See Fig. 4.8)SOC1= 75%SOC2= 80%
Figure 4.10: Test Scenario 1: Interval 1. During this interval, SOC balancing is occurringand the SOC is increasing, resulting in the droop curve rising.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
Out of Bounds RegionAggregateRenewable Source LimiterSimulation Results (See Fig. 4.8)SOC1= 88%SOC2= 88%
Figure 4.11: Test Scenario 1: Interval 2. During this interval, The SOC is increasing,but the OCP is limiting the renewable source current so the SOC does not exceed 100%.
72
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
Out of Bounds RegionAggregateRenewable Source LimiterSimulation Results (See Fig. 4.8)SOC1= 99%SOC2= 99%
Figure 4.12: Test Scenario 1: Beginning of Interval 3. A 50A (0.625pu) load is introduced,which lowers the bus voltage and the source limiter opens.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
Out of Bounds RegionAggregateRenewable Source LimiterSimulation Results (See Fig. 4.8)SOC1= 99%SOC2= 99%
Figure 4.13: Test Scenario 1: Interval 3. During this interval, the renewable sourcelimiter is progressively opening, which reduces the output current required by the BESS.
73
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Output Current [pu]
Vol
tage
Ref
eren
ce [p
u]
Out of Bounds RegionAggregateRenewable Source LimiterSimulation Results (See Fig. 4.8)SOC1= 94%SOC2= 94%
Figure 4.14: Test Scenario 1: Interval 4. During this interval, the renewable source issuppliying its maximum permissive power and the BESS SOC is reduced, which lowersthe droop curve.
74
4.3.2 Test Scenario 2
To further demonstrate the energy management method, a scenario was created to
demonstrate the current distribution of the BESS with different SOCs. The initial SOCs
are outlined in Table 4.6 and the load demand during each time interval is highlighted
in Table 4.7. This demonstrates that the BESS with a higher SOC provides more power
than the BESS with a lower SOC. This occurs until the BESS with higher SOC cannot
supply more power and the BESS with a lower SOC compensates instead. This demon-
strates the energy management method’s ability to prioritize BESS power demands to
provide SOC balancing.
Table 4.6: Initial SOC for Test Scenario 2
Parameter Symbol Value
Initial SOC: BESS 1 SOCB1 90%Initial SOC: BESS 2 SOCB2 50%
Table 4.7: Load Steps for Test Scenario 2
Time [s] Load Demand [A] Load Demand [pu] (80A base)
t < 0.4s 0 00.4s < t < 0.8s 35 0.43750.8s < t < 1.2s 60 0.75
t > 1.2s 70 0.875
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3−30
−20
−10
0
10
20
30
40
50
BE
SS
Out
put C
urre
nt (
A)
Time (s)
IoutB1 I
outB2
Figure 4.15: Energy Management Test Scenario 2 Simulation Results: Output Currentfor Individual BESSs
75
4.4 Chapter Conclusion
The proposed energy management scheme provides SOC balancing, OCP and load shed-
ding for a DC microgrid system using indirect communication between local controllers,
which increases response time and reduces communication infrastructure costs. The com-
bined SOC balancing / OCP scheme yields a net microgrid bus voltage that decrease in
a piecewise linear manner with DC microgrid loading. Keeping the virtual resistance
(Kd) unchanged during operation reduces eigenvalue movement, which implies that the
energy management method has minimal influence on system stability. Therefore, the en-
ergy management method is applicable to all DC microgrid systems, regardless of power
capacity, number of BESSs, generation sources or loads.
76
Chapter 5
Conclusion
5.1 Summary of Work
This thesis has developed a method to accurately model individual components and
complete DC microgrid systems. The method to connect individual models is flexible
and, therefore, allows multiple configurations to be analysed based on the physical layout
of the system. The modelling technique was verified by utilizing eigenvalue analysis and
comparison with simulation results.
With the modelling technique developed, individual component models were ex-
panded to accommodate multiple converters within a single model and verified by the
same technique as the base model. With the expanded component models, BESS and
solar PV models were scaled to investigate eigenvalue movement to determine stability
and, consequently, scalability. The results confirmed scalability up to 300 converters and
the trend of the eigenvalue movements implying further possible scaling.
To further demonstrate modularity and scalability, the proposed autonomous energy
management method can provide SOC balancing, overcharge protection and load shed-
ding with a reduced effect on stability with minimal communication equipment. There-
fore, the method improves robustness and reliability of DC microgrids with reduced
complexity, which compliments the scalability analysis.
The results of this thesis demonstrate that DC microgrids have the potential to be
a scalable, modular solution that integrates with renewable energy sources and common
BESS technology. The sizeable power scaling capabilities and fast-response autonomous
energy management methods demonstrate the potential to develop a robust distributed
grid infrastructure.
77
5.2 Impact
As mentioned in Chapter 1, DC microgrid studies focus their research on a single power
rating with a set number of BESSs and generation sources. This thesis has investigated
and analysed DC microgrids over a broad range of power capacities. The implications of
these results is that, with an appropriately designed converter topology, control structure
and controller values, DC microgrids are applicable to multiple scenarios that require
varying power ranges (kW - MW).
Furthermore, the modelling technique provided is structured to allow a variety of
connection configurations with numerous BESSs, sources and loads. With this mod-
ular modelling technique, additional research can be conducted to investigate system
dynamics and DC microgrid design with minimal additional effort. Therefore, this thesis
proposes a novel modelling framework that can encapsulate all potential DC microgrid
infrastructures.
Finally, the proposed autonomous energy management method is low-complexity de-
sign that is applicable to all DC microgrid applications, irrespective of component selec-
tion/numbers and power ratings. If the power provided by the source can be limited and
BESSs are present, it is applicable. Since this aligns with the general model discussed
in Chapter 1, this method has the potential to be the benchmark energy management
technique for DC microgrid systems.
5.3 Future Work
Future work can apply the conclusion of thesis to validate other system configurations,
converter topologies and control schemes. By developing individual component mod-
els, the state-space connection method can be intuitively expanded to other generation
sources (eg. wind, diesel generators) with minimal additional work.
Additionally, as the participation factor for the solar converter in Fig. 2.20 demon-
strated, the current control structure and controller values results in significant coupling.
This coupling influences stability and scalability, which presents an opportunity to im-
prove on the control design of the solar converter.
Furthermore, the energy management system could potentially be expanded to pro-
vide additional system-level communication that is common in grid infrastructures. For
example, by measuring the bus voltage, an ”aggregate SOC” could be calculated and
utilized by a monitoring/dispatch operator to do high-level power system optimization,
load scheduling and economic dispatch.
78
Finally, by conducting experimental work on the findings of this thesis, scalability
and energy management methods could be further investigated to account for practical
limitations (eg. sensor errors, component asymmetry, BESS degradation). With these
additional research topics, DC microgrids could further prove to be a reliable alternative
to traditional grid infrastructures.
79
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82
Appendices
83
Appendix A
Voltage Source Converter Design
The purpose of this chapter is to outline the design process used for the Voltage Source
Converter (VSC) that connects the microgrid’s DC bus to the grid. This is significant
because, as mentioned earlier, microgrid research is increasing because it provides storage
for existing grid infrastructures. Therefore, defining a VSC design methodology enables
utility companies to utilize microgrids in future projects.
The goal of this VSC is to use the power available within the microgrid to transmit
real and reactive power to the grid based on its demand. The circuit topology is dis-
cussed, followed by the design of the LCL filter. Next, the current control scheme is
discussed, including the justification for the mathematical frame of the controller, the
theoretical control scheme, the reference signal generation calculations and the detailed
control scheme. Finally, simulation results will be provided to validate the design.
A.1 Topology Overview
The implemented topology is a two-level VSC, which is one of the most common DC/AC
topologies in the industry. An LCL power filter is used between the VSC and the grid.
This topology is advantageous because of its exceptional controllability and filtering
capabilities. Furthermore, to prevent the possibility of zero-sequence current flowing
into the grid, an ungrounded Wye / grounded Wye transformer is located between the
filter and the grid. Figure A.1 illustrates the schematic of the implemented topology.
By defining the line-to-neutral output voltage of the VSC to be Vt, the transfer
functions for the current I1 and I2 with respects to Vt are given in equation A.1a and
A.1b respectively.
84
ma
m'a
mb
m'b
mc
m'c
CdcVdc
L1
C
L2
Grid
I1 I2VtVs
Figure A.1: Schematic for the VSC with an LCL Filter. The components ”L1”, ”L2”and ”C” are the same size in each phase.
I1
Vt=
L2Cs2 + 1
L1L2Cs3 + (L1 + L2)s(A.1a)
I2
Vt=
1
L1L2Cs3 + (L1 + L2)s(A.1b)
Observe that, for equation A.1b, there is no complex conjugate zeroes in the transfer
function. This is significant from a control perspective. Without the complex conjugate
zeroes, the system is mathematically uncontrollable. However, in equation A.1a, there
are existing complex conjugate poles that make the system mathematically controllable.
Therefore, the controller measures I1 to ensure controllability. This means that the plant
model for the controller will be equation A.1a.
A.2 LCL Filter Design
Table A.1 gives the individual component sizes of the LCL filter (See Appendix B for
Design Methodology).
Table A.1: LCL Filter Component Sizes
Component Size
C 4.91 µFL1 1.6 mHL2 0.81 mH
For both equation A.1a and A.1b, there is a complex conjugate pole that is given by
equation A.2
85
Figure A.2: Transformation of abc-Frame (Blue) to αβ0-Frame (Red)
w =1√L1L2CL1+L2
(A.2)
Using the values from A.1, the resonant frequency is w = 19,460 rads
(3.1 kHz). As
mentioned in Appendix B, when calculating the inductor sizes, an approximation can be
made about the transfer function assuming the switching frequency is greater than the
resonant frequency. Therefore, this converter has a switching frequency of 37,700 rads
(6
kHz). Based on the results from Appendix C, it is clear that the approximation is valid.
A.3 Controller Design
With the completion of the LCL filter design, a complete plant model for the controller is
defined. Firstly, a decision of which mathematical frame is most suitable for the controller
must be decided. The two available options are the dq0-frame or the αβ0-frame.
A.3.1 αβ0-Frame
Conceptually, the αβ0-frame is a projection of the three-phase signals onto a stationary
two-phase reference frame. Specifically, the calculation from the abc-frame to the αβ0-
frame is given by equation A.3. This transformation is visually explained in Figure A.2.
Uαβ0 =2
3
1 −12−1
2
0√
32−√
32
12
12
12
Uabc (A.3)
Under balanced, steady-state conditions, this transformation produces a vector of
constant magnitude rotating at an identical frequency to the signals generated in the
86
abc-frame. Assuming the zero-sequence component equates to zero, the αβ-frame can be
defined as a single complex equation (A.4).
Uαβ = Uα + jUβ (A.4)
From a control aspect, this is beneficial because it only requires the control of two
signals (Uα and Uβ), which reduces the number of controllers and computation time
required compared to the three signals in the abc-frame. the αβ-frame, the current
signals are completely decoupled. In other words, a change in Iα has no effect on Iβ
and vice-versa. This is beneficial because it allows for the control the individual α and
β signals separately without any decoupling calculations added in the feedback system.
However, based on the internal model principle, if the goal is to track a signal of a
particular frequency, the transfer function of the controller must have infinite gain at
that frequency. In the situation of the αβ-frame and this system, the goal is to track
a 60 Hz sinusodial signal. The general Laplace form of a sinusoidal signal is given in
equation A.5.
H(s) =N(s)
s2 + w2o
(A.5)
Where:N(s) - The Numerator Polynomial Based on the Specific Signal
wo - The frequency (in rads
) that must be tracked
Based on the internal model principle, this signifies that the controller must take on
form given in equation A.6.
C(s) =K(s+ a)(s+ b)
s2 + w20
(A.6)
Equation A.6 is also known as a Proportional Resonant (PR) controller. This in-
troduces a challenge in the controller design. Specifically, the complex conjugate pole
that is required in the controller creates a system that has a limited range of control
parameter (K,a,b) that are stable. In other words, creating a controller that is stable
with acceptable performance is more difficult compared to a typical Proportional Integral
(PI) controller.
A.3.2 dq0-Frame
Conceptually, the dq0-frame is simply applying a rotational shift to the provide abc-
frame or αβ0-frame. Specifically, equation A.7 and A.8 are the transformations used.
87
Figure A.3: Transformation of αβ0-Frame (Blue) to dq0-Frame (Red)
Figure A.3 depicts this transformation.
Udq0 =
√2
3
cos(wt) cos(wt− 2π3
) cos(wt− 4π3
)
sin(wt) sin(wt− 2π3
) sin(wt− 4π3
)√
22
√2
2
√2
2
Uabc (A.7)
Udq0 =
cos(wt) sin(wt) 0
−sin(wt) cos(wt) 0
0 0 1
Uαβ0 (A.8)
For this system and a rotational shift of 60 Hz, the dq0-frame transformation produces
a DC signal out of the measured 60 Hz signal. From a control aspect, this is advantageous;
for this system, a controller in the dq0-frame will track a DC signal. By the internal model
principle, this means that the controller transfer function must be capable of tracking DC
quantities. In the Laplace domain, a constant-valued signal can be defined by equation
A.9.
C(s) =1
s(A.9)
By the internal model principle, the control function must take on the form given in
equation A.10.
C(s) =K(s+ a)
s(A.10)
88
Equation A.10 is also known as a Proportional Integral (PI) controller. This is benefi-
cial from a control aspect, since a real pole at w = 0 (DC) does not influence the stability
of the system as heavily compared to a PR controller. This means that there is a larger
range of values (K,a) that can produce a stable system with acceptable performances,
making control significantly easier. However, the main issue of the dq0-frame is coupling
between the two components. Especially due to the nature of the LCL filter, the coupling
between the d-component and the q-component are complex. This can be overcome, but
it involves a complex feedback system, which can be costly and increase computation
time. Table A.2 summarizes the benefits and consequences of both frames.
Table A.2: Comparison of αβ-Frame and dq-Frame
Pros Cons
αβ Frame • Decoupled Components • Limited Control Parameters• Simple Feedback System
dq Frame • More Control Parameters • Coupled Components• Complex Feedback System
Based on these comparisons and the requirements of the system, the chosen frame is
the αβ0-frame.
A.4 Theoretical Control Scheme
As mentioned earlier, the plant model will be the transfer function provided in equation
A.1a. Therefore, the PR controller’s input will be the error between the measured and
reference current signal, and the output will be the line-to-neutral output voltage of the
VSC.
Even though the current, I1, is measured before the LCL filter, it is not necessary to
add a signal filter before directing it to the controller. A PR controller will track a 60 Hz
signal, even with the additional high-frequency noise from the measured signal. This will
introduce some harmonics in the output signal of the controller, but the LCL filter will
eliminate these and ensure a clean signal is transmitted to the grid. The elimination of
a signal filter is beneficial in creating a stable controller, since it reduces the order of the
closed-loop system. Figure A.4 illustrates the conceptual block diagram for the control
scheme, which will be used to design the controller.
89
PRVtαβ+
-
Plant (Eqn. 2.1a)
iαβ iαβref
Figure A.4: Current Controller Block Diagram
A.4.1 Reference Signal Generation
The reference current signal is generated based on the real and reactive power demands
of the grid. Therefore, a mathematical model must be created to convert the real and
reactive power requirements into a reference αβ0-frame current. The calculation is con-
ducted by utilizing the definition of complex power. Assuming there is no zero-sequence
voltage/current, we can define the voltage and current signals as equation A.11 and A.12
respectively.
Vdq = Vd + jVq (A.11)
Idq = Id + jIq (A.12)
The definition of complex three-phase power is defined by equation A.13.
S , P + jQ =3
2V I∗ (A.13)
By substituting equation A.11 and A.12 into A.13, the real and reactive equations
are given in equation A.14 and A.15 respectively.
P =3
2(VdId + VqIq) (A.14)
Q =3
2(VqId − VdIq) (A.15)
By assuming that Vq = 0 and re-arranging to solve for Id and Iq, the following for-
mulations are given by equation A.16 and A.17 respectively.
90
Id =2P
3Vd(A.16)
Iq = − 2Q
3Vd(A.17)
Taking the inverse of equation A.3 and using equations A.16 and A.17, the reference
Iα and Iβ signals become equations A.18 and A.19.
Iα = Idcos(wot)− Iqsin(wot) (A.18)
Iβ = Idsin(wot) + Iqcos(wot) (A.19)
A.5 Detailed Control Scheme
Although the theoretical control scheme is used to design the controller, implementing
the closed-loop control scheme requires additional conversions and calculations to use in
practice. Figure A.5 illustrates the detailed control scheme that is implemented.
PR+
-
i1αβ
+
+
VSC Output Current Control
X
Reference Signal Generation
_2_ 3VsdPref
_-2_ 3VsdQref
ejwt
Vsαβ
Vtαβ
Vdc
VSC
LCL FilterVt
+
-
i1
~GRID
Vs
mαβ
idref
idqref
iqref
iαβref
Vdc
_2_
Figure A.5: Detailed Control Diagram for the VSC
As mentioned earlier, the current signals, I1, are used in the control scheme. All
three current signals are measured, then converted into the αβ0-frame using equation
A.3. This measured quantity is then compared to the reference signal, which is generated
using the methods discussed in the previous section. The PR controller then tracks this
reference signal. Using the theoretical block diagram illustrated in Figure A.4 and the
MATLAB/SISOTOOL application, the proposed controller has a transfer function given
in equation A.20.
91
−12000−10000 −8000 −6000 −4000 −2000 0 2000 4000−8
−6
−4
−2
0
2
4
6
8x 10
4
Real Axis (seconds−1)
Imag
inar
y A
xis
(sec
onds
−1 )
Figure A.6: Root Locus Plot for the VSC System
C(s) =12.1(s2 + 950s+ 250, 000)
s2 + w2o
(A.20)
In order to demonstrate stability, the Root Locus plot (Figure A.6) for this system is
provided.
The output signal of the PR controller is then summated with the αβ0-frame grid
voltage quantities, which acts as a feed-forward loop. This is beneficial for the system
because, during start-up, the controller will initially have a signal for the converter output
voltage (Vt) that is equal to the grid voltage (Vs), which results in zero current flow
between the VSC and the grid. Without the feed-forward system, the initial converter
output voltage would be zero, resulting in a substantial current flow. The large initial
current could result in component damage. Also, the feed-forward system improves the
dynamic performance of the system, since the required signal and the measured signal
would be closer initially, which reduces the required time to reach steady-state conditions.
Furthermore, in practice, a controller will have some steady-state error that will result
in a higher percentage of Total Harmonic Distortion (THD). However, the grid voltage
produced has a negligible THD, meaning that the feed-forward system will produce an
output signal that has lower THD.
Once the PR controller output and the feed-forward system are summated, the gener-
ated αβ-frame signals are converted back to the abc-frame using the inverse of equation
92
A.3. This gives the required converter line-to-neutral output voltage in the abc-frame.
The final goal is to generate the signal to determine the switching intervals for the VSC
transistors (ma, mb, mc). For a full-bridge inverter, these signals are generated using
equation A.21.
mx(t) =Vtx(t)Vdc2
(A.21)
Where:mx(t) - Modulation Index (ε[-1,1])
Vtx(t) - VSC Line-To-Neutral Output Voltage
Vdc - DC Voltage at the Input Bus
Once the modulation index is calculated, it is passed through a comparator that
compares the modulation index to a unit sawtooth signal with a controllable switching
frequency. For this system, as mentioned earlier, a 6 kHz switching frequency is used.
Using this control scheme, the VSC is capable of generating signals that will provide
specified real and reactive power to the grid with minimal current THD.
A.6 VSC Simulation
The VSC was validated by utilizing Power System Computer Aided Design (PSCAD).
The simulated system used the parameters provided in Table A.3. In order to maintain
results that will match the experimental results, a delay was added between controller
measurements and the output duty cycles. This is meant to represent the calculation
time required by the physical controller device. This includes controller and PWM delays.
Table A.3: VSC System Parameters
Component Label Value
Grid Voltage (Line-to-Line RMS) Vg 208 VRated Complex Power (Three-Phase) S3φ 6 kVA
DC Voltage Vdc 380 VSwitching Frequency fsw 6 kHz
A.6.1 Delay Calculation
Many controllers specify a certain time required to do all necessary calculations and
measurements, which correlate to controller bandwidths. Specifically, during a single
93
Figure A.7: Controller Delay Breakdown
sample period, the analogue measurement signal must be converted to a digital signal
that the controller can utilize, read the value, then use that value to do all necessary
control calculations. Once these are completed, the controller must prepare to send the
data to the PWM. The remainder of the sample time is used as free time for the controller
to do other small tasks. Once that time passes, the controller then releases this data to
the PWM. This is explained visually in Fig. A.7. This means that there is a delay of a
single sample period that the controller introduces. Traditionally, the controller samples
at the peaks of the switching signal. Therefore, the controller delay is half the switching
period (Equation A.22).
Ts =1
2∗ Tsw (A.22)
In addition to the controller delay, there is also the delay between the PWM signal
being released and the PWM physically changing the state of the switches. This can be
calculated by sending a sinusoidal signal to the PWM and then calculate the delay using
Fourier analysis. On average, this delay will be a quarter of the switching frequency
(Equation A.23).
TPWM =1
4∗ Tsw (A.23)
Adding the two delays, the total signal delay (Td) is given in Equation A.24.
Td =3
4∗ Tsw (A.24)
As mentioned in Table A.3, the switching frequency is 6 kHz, meaning the switching
period (Tsw) is 166.67 µs. Therefore, using equation A.24, the total delay (Td) is 125 µs.
94
A.6.2 Delay Modelling
Now that the time delay is defined, a model is produced that will accurately simulate
this delay in PSCAD. Theoretically, a delay can be modelled using equation A.25.
y(t) = u(t− τ) = u(t− Td) (A.25)
By using a Laplace transformation, equation A.25 can be expressed by equation A.26.
Y (s) = e−sTd (A.26)
This form is not ideal because transfer functions are generally written in the form
G(s) = N(s)/D(s), where N(s) and D(s) are polynomial functions. This is overcome by
utilizing the Taylor Series expansion, which is expressed in equation A.27.
eax = 1 +(ax)1
1!+
(ax)2
2!+
(ax)3
3!+
(ax)4
4!+ ... =
∞∑n=0
(ax)n
n!(A.27)
Using equation A.26 and A.27, the delay function in the Laplace domain can be
described as equation A.28.
e−sTd =1
esTd=
1
1 + (Tds)1
1!+ (Tds)2
2!+ (Tds)3
3!+ (Tds)4
4!+ ...
(A.28)
This can be simplified by assuming that the first two terms of the Taylor Series
expansion are the dominant terms. This approximation is valid if Td << 1. For this
system, Td = 125 µs, which meets this assumption. Therefore, the expression simplifies
to equation A.29.
H(s) = e−sTd =1
1 + Tds(A.29)
Equation A.29 will be used to represent the delay signal in the PSCAD model.
A.6.3 Simulation Results
As mentioned earlier, real and reactive power demands are determined by the grid. There-
fore, PSCAD will simulate various steps in real and reactive power demands. The PR
controller uses the transfer function given in equation A.20 and the delay transfer function
uses equation A.29. For this simulation, the power steps are outlined in Table A.4.
The first plot shows the grid current in the dq0-frame compared to the reference.
The second plot shows a single phase of the grid voltage. The third plot shows the grid
95
Table A.4: VSC Simulation: Power Demands vs. Time
0s < t < 0.4s 0.4s < t < 0.7s t > 0.7s
Real Power (kW) 0 6 3Equivalent Id (A) 0 16.6 8.3
Reactive Power (kVAR) 0 0 6Equivalent Iq (A) 0 0 -8.3
current (I2 from Fig. A.1) of a single phase. The final plot shows the THD (in %). The
results are shown in Figure A.8.
From Figure A.8, it is clear that the Id tracks Irefd . From 0.4s < t < 0.7s, the measured
THD is approximately 0.9%, which confirms that the accuracy of the LCL filter design.
Note that from 0 < t < 0.4s, the current demand is 0 A, which results in a very large
THD measurement.
Figure A.9 is a zoomed-in snapshot of the grid current. From this picture, it is clear
that THD is minimal. A second quantity to confirm is that the reference Id and Iq
matches the required values for the grid power demands. Figure A.10 shows a zoomed-in
version of the dq0-frame grid currents. Figure A.10 shows that Idq tracks Irefdq accurately.
96
0.3 0.4 0.5 0.6 0.7 0.8 0.9−20
0
20
40C
urre
nt (
A)
Idref I
d Iqref I
q
0.3 0.4 0.5 0.6 0.7 0.8 0.9−1
0
1
Dut
y C
ycle
m
a
0.3 0.4 0.5 0.6 0.7 0.8 0.9
−20
0
20
Cur
rent
(A
)
IL2
0.3 0.4 0.5 0.6 0.7 0.8 0.90
1
2
TH
D (
%)
Time (s)
THD
IL2
Figure A.8: VSC Simulation Results
97
0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74−200
−100
0
100
200
Vol
tage
(V
)
0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74−30
−20
−10
0
10
20
30
Cur
rent
(A
)
Time (s)
Vsa
IL2a
Figure A.9: VSC Simulation - Zoomed-In Grid Current
98
0.3 0.4 0.5 0.6 0.7 0.8 0.9−20
−10
0
10
20
30
40
Cur
rent
(A
)
Idref
Id
Iqref
Iq
Figure A.10: VSC Simulation - Zoomed-In DQ-Frame Current Tracking
99
Appendix B
LCL Filter Design Methodology
The typical three-phase Voltage Source Converter (VSC) topologies introduce unaccept-
able harmonics that are related to the switching frequency and the DC link voltage.
Figure B.1 provides the numerical values of the VSC output voltage harmonics.
Figure B.1: Harmonics Produced from VSC vs. Switching Frequency [24]
mf =fswf0
,ma =2√
2√3
VllrmsVdc
(B.1)
Where:fsw - Switching Frequency of the Converter
fo - Grid Switching Frequency (60Hz)
Vllrms - Line-To-Line RMS Output Voltage of the VSC
Vdc - DC Link Voltage
100
For this application at no load (grid current demand is zero), Vdc = 380V and Vllrms
= 208V. therefore, ma = 0.89. At full load, ma approached 1.0, which makes it a realistic
worst-case scenario for harmonics. This is worrisome because, for mf ± 2 and 2mf ± 1,
the harmonic values are 19.2% and 11.2% of the DC link voltage respectively. These must
be filtered out, or else the transmitted power will not meet the grid THD requirements.
Therefore, an LCL filter (illustrated in Figure B.2) is used to filter the harmonics to
acceptable levels.
Vgrid
L1 L2
CVα
I1I2
IC VC
Figure B.2: LCL Filter Schematic
Where Vα represents the voltage output from the VSC.
The requirements of the LCL filter design are as followed:
1. The amount of current through the capacitor (Ic) must be below 1% of the funda-
mental output current.
2. The largest harmonic output current (I2h) must be less than 1% of the fundamental
output current at rated current.
3. Must aim to optimize the component sizes to minimize energy stored within them.
In addition to these requirements, typically L1 > L2 because it reduces the current
ripple in I1, which reduces the current rating requirement of the inductor.
The first step for designing this filter is to get transfer functions for the LCL filter.
The three important transfer functions are B.2, B.3 and B.4.
101
I1
Vα(s) (B.2)
I2
Vα(s) (B.3)
ICVα
(s) (B.4)
Transfer functions B.2 and B.3 will be used to size the components and determine the
harmonic current ripples. This will help minimize the energy stored within the inductors,
since the energy stored within an inductor takes on the form of equation B.5.
EL =1
2LI2 (B.5)
Transfer function B.4 will be used to determine the current through the capacitor,
which will used to confirm that the current through it is below 1%.
Using frequency-domain circuit analysis, the required transfer functions were calcu-
lated and provided in Equations B.6, B.7 and B.8.
I1
Vα=
L2Cs2 + 1
L1L2Cs3 + (L1 + L2)s(B.6)
I2
Vα=
1
L1L2Cs3 + (L1 + L2)s(B.7)
ICVα
=L2Cs
2
L1L2Cs3 + (L1 + L2)s(B.8)
For these transfer functions, two poles exist:
w = 0;w =1√
L1L2
L1+L2C
(B.9)
B.0.1 Step 1: Capacitor Sizing
To choose the size, an assumption is made that the voltage across the capacitor is ap-
proximately equivalent to the grid voltage (VC u Vgrid). This assumption is acceptable
because the inductor impedance is small due to the low grid frequency (60 Hz). This
implies that the current through the capacitor can be represented using Ohm’s Law
(Equation B.10).
102
%fundIrated1
woC= VgridLN
(B.10)
C =%fundIratedwoVgridLN
(B.11)
Where:%fund - Percentage of fundamental current allowed to flow through the capacitor
Irated - Rated grid current
wo - Grid frequency in rad/s (2π*60)
VgridLN- RMS Line-to-Neutral Grid Voltage
B.0.2 Step 2: Inductor Sizing
To determine the size of the inductors, the transfer function equations B.6, B.7 and
Design Requirement 2 are used. The goal is to have the magnitude of equation B.7 be
less than 1% for a given VSC voltage and switching frequency. This is generally defined
in equation B.12.
%harmonicI2rated
Vα(f = fsw), A (B.12)
According to Table B.1, for ma = 1, the largest (line-to-neutral) harmonic is Vα =0.195√
3*Vdc at mf ± 2. Therefore, the required magnitude of equation B.7 at the switching
frequency is described in equation B.13.
%harmonicI2rated0.195√
3Vdc
, A (B.13)
Equation B.13 is used with transfer function B.7 to calculate the minimum product of
the inductances (L1L2). The method to determine the magnitude of a transfer function
is given in equation B.14.
‖H‖2 = HH∗ (B.14)
Equation B.14 does not result in an explicit solution for the inductance product.
However, using an approximation, this problem can be resolved. For a switching fre-
quency greater than the filter’s resonant frequency, the s3 term in the transfer function
dominates, meaning the transfer function can be approximated as equation B.15.
A , ‖ I2
Vα‖ ≈ 1
w3L1L2C(B.15)
103
The switching frequency can be controlled, so it can be ensured that this requirement
is met when designing the filter. Using this assumption, the L1L2 product is given by
equation B.16.
L1L2 =1
w3AC(B.16)
Now that the product for L1 and L2 is defined, any L1, L2 combination will satisfy
Design Requirement 2. However, due to the nature of the LCL filters, a smaller L1 will
result in a larger current ripple across it, resulting in a larger energy capacity requirement.
Since the energy is related by equation B.5, there is an optimal value that minimizes
the stored energy. To start, define inductor L2 and L1 as equations B.17 and B.18
respectively.
L2 ,
√L1L2
k(B.17)
L1 , kL2 (B.18)
From here, the goal is to determine the value of k that best satisfies Design Require-
ment 3. Using a combination of Table B.1 and transfer functions B.6 and B.7, it is
possible to determine the peak current seen by both inductors. Then, using equation
B.5, it is possible to calculate the energy requirement and L1 ripple current for different
values of k to determine the optimal value. This solution is not unique. To get a unique
result, specific ratings and requirements must be defined. For the application investigated
in this thesis, the ratings and requirements are given in Table B.1 and B.2.
Table B.1: Grid Rating Values
Grid Ratings (RMS) Variable Definition Value
Line-to-Line Grid Voltage [V] VgridLL208
DC Voltage [V] Vdc 380Grid Frequency [Hz] fo 60
Three-Phase Complex Power Rating [kVA] S3φ 6Grid Line Current Rating [A] Irated 16.6
VSC Switching Frequency [Hz] fsw 6,000
Using these system parameters, plots B.3 and B.4 were produced to show the inductor
energy requirements, L1 harmonic current ripple magnitude and resonant frequencies for
different values of k.
Based on these plots and system requirements, the ideal value of k = 1.97. The
104
Table B.2: System Requirements
Grid Ratings (RMS) Variable Definition Value
Max Fundamental-Frequency Current Through the Cap. %fund 1%Max Harmonic Current Leaving Inductor L2 %harmonic 1%
Largest Harmonic Magntidue Through L1 Iripple 1.5 A
resulting L1, L2 and C values are listed in Table B.3.
Table B.3: LCL Filter Component Sizes
Component Size
C 4.91 µFL1 1.6 mHL2 0.81 mH
105
Figure B.3: Energy Requirement vs. ”k”
106
Figure B.4: L1 Largest Harmonic vs. ”k”
107
Appendix C
Transfer Function Approximation
Validation
The approximation made on transfer function B.7 to get an explicit solution for the L1L2
product in Appendix A is based off the assumption that, at a switching frequency greater
than the resonant frequency of the LCL filter, the gains for the exact transfer function
and the approximated one are approximately equal. Therefore, this must be validated in
order to ensure accuracy in the design.
To validate the approximation, a bode plot for the exact and approximated transfer
function is produced and the gains will be compared at the switching frequency of the
VSC. The exact transfer function is given by equation C.1.
I2
Vα=
1
L1L2Cs3 + (L1 + L2)s(C.1)
The approximated transfer function is given by equation C.2.
I2
Vα=
1
L1L2Cs3(C.2)
The bode plot for both these transfer functions are provided in Figure C.1.
Visually, it is clear that, at a switching frequency of 6 kHz (37,700 rads
), the exact
and approximated transfer function are almost identical. More specifically, at the switch-
ing frequency, the gains for the exact and approximated transfer functions are given in
equation and respectively.
‖ I2
Vα‖exact = 0.00362 = 0.362% (C.3)
108
Figure C.1: Bode Plot: Exact TF vs. Approximated TF
‖ I2
Vα‖approx = 0.00270 = 0.270% (C.4)
Since the gains are very similar at the switching frequency, this confirms that the
approximated transfer function is a valid approximation that can be used for the design.
109
Appendix D
Controller Values
Table D.1: Control Parameters: Battery Converter
Sum Control: Inductor Current (iL)Symbol ValueKp 0.0205Ki 179
Sum Control: Bus Voltage (Vbus)Symbol ValueKp 0.0565Ki 343
Sum Control: Droop Control (V refbus )
Symbol ValueKd 0.475
Difference Control: Input Cap. Difference Voltage (VM)Symbol ValueKp 0.025Ki 0.0063
110
Table D.2: Control Parameters: Solar Converter
Sum Control: Inductor Current (iL)Symbol ValueKp 0.0103Ki 22
Sum Control: Input Cap. Sum Voltage (V∑)Symbol ValueKp 0.2436Ki 27
Difference Control: Input Cap. Difference Voltage (VM)Symbol ValueKp 0.0111Ki 0.352
111
Appendix E
Solar Current Distribution
Calculation
The purpose of this appendix is to define the current distribution that occurs due to
the solar panel current in the scalable model outlined in Subsection 3.1.2. The current
distribution schematic is illustrated in Figure E.1, where:
Cx = 2CINCOUTCIN + COUT
(E.1)
CIN
CIN
COUT COUT(n-1)Cx
IS
I1
I2
Ia
Ib
Ix
Figure E.1: Is Current Distribution Schematic
112
To determine the variables M1, M2 and Mx mentioned in Subsection 3.1.2, KCL and
KVL equations were made. These are provided below:
I1 + Ia + Ix = 0 (E.2)
Is − I1 + Ib = 0 (E.3)
Is − I2 + Ia = 0 (E.4)
I2
CIN+
IaCOUT
− Ix(n− 1)Cx
= 0 (E.5)
I1
CIN− I2
CIN− IaCOUT
+Ib
COUT= 0 (E.6)
The matrix form of these equations are provided by Equation E.7.1 0 1 0 1
−1 0 0 1 0
0 −1 1 0 0
0 1CIN
1COUT
0 1(n−1)Cx
1CIN
− 1CIN
− 1COUT
1COUT
0
.I1
I2
Ia
Ib
Ix
=
0
−1
−1
0
0
(E.7)
Once I1 through Ix are solved, M1, M2 and Mx are defined by Equations E.8 through
E.10.
M1 = I1 + I2 (E.8)
M2 = Ia + Ib (E.9)
Mx = Ix (E.10)
113
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