Modeling and Control of Power Electronics Based DC Networks
Modeling and Control of Power Electronics Based DC
Networks
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree
Doctor of Philosophy in the Graduate School of The Ohio State
University
By
Luis Herrera,
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2015
Dissertation Committee:
Dr. Jin Wang, Advisor
Dr. Longya Xu
Dr. Mahesh S. Illindala
c Copyright by
Luis Herrera
2015
Abstract
The introduction of power electronic converters provides flexibility in the transformation of electrical energy such as ac-dc and dc-ac. For this reason, dc networks are becoming more popular in applications such as electric aircraft, navy ships, long distance power transmissions, and future microgrids in outer space. These systems are typically composed of multiple sources (ac or dc) and multiple loads (ac or dc) interconnected together through power electronics. In this work, modeling methods aimed at efficient real time simulation, stability analysis, and controller design for dc networks are presented.
Real time simulation is a fundamental tool used in industry to test hardware and/or software without the need of the actual system. For example, Hardware-inthe-Loop (HIL) techniques are commonly used to test a control unit for a vehicle, power converter, etc. The increase in switching frequency of the power electronics
(due to new devices such as SiC/GaN), decreases the time step needed to simulate these systems. For this reason, FPGAs are being commonly used for real time simulation. Nevertheless, the low level programming required introduces challenges in the modeling and implementation. Methods and algorithms are proposed to efficiently model the power devices including ‘on’ characteristics. In addition, a discussion of machine simulation including saturation and its implementation in FPGA are discussed.
ii
The use of power electronics to deliver power to loads (e.g. motors, dc electronic loads, etc.) introduces a negative impedance effect as seen by the network. This effect can potentially cause instability problems in the system. These types of loads are commonly referred in literature as Constant Power Loads (CPLs). To study the stability of dc networks with CPLs, a method is proposed based on Linear Matrix
Inequalities (LMIs) and Semidefinite Programming (SDP). The advantage of the proposed method relies on the use of static feedback without the need of knowing the load power and/or its derivatives.
Lastly, the modeling, simulation, and control of High Voltage DC Networks is presented. In these systems, Modular Multilevel Converters (MMCs) are typically used as an interface between the ac and dc networks. These types of converters can utilize hundreds to thousands of switches. Therefore, the simulation, offline and real time, is a main challenge. In addition, the circulating current through each arm affects the power losses in the power devices and causes larger current and voltage stress on the passive components. In this case, a controller is proposed to eliminate this oscillation and a platform for efficient simulation of MMCs is developed. Real time simulation and HIL validation of the controller are presented.
iii
To
God, my family, and all the inspiring mentors iv
Acknowledgments
My deepest gratitude goes to my PhD advisor, Dr. Jin Wang for all of his support, guidance, and encouragement over my studies. He has allowed me to pursue topics in different areas while at the same time helped me to develop creative and critical thinking. I would not have completed this dissertation without his support.
I would also like to thank Dr. Mahesh S. Illindala and Dr. Longya Xu for serving in all of my exam committees during my PhD studies. Their advice, comments, and support throughout my studies have given me invaluable insight into different details of my research. They are both experts in their respective area and I value their advice sincerely.
I am deeply grateful for the many wonderful and inspiring mentors I have had throughout my education. I very thankful to Mr. Frank Leach, for helping me to begin thinking “outside the box.” Dr. Somsak Sukittanon, who has always been a constant help. Without all of his advice and support I would not have continued on to graduate school. I would also like to thank Dr. Andrea Serrani. His passion for mathematics and control coupled with his amazing lectures have truly shaped my research direction and interests. Lastly, I am indebted to Dr. Wei Zhang, who has given me advice in my research projects and continued to fuel my interest in control through his classes.
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I would also like thank my parents, Fabio Herrera and Amparo Duque, and my brother, Oswaldo Herrera, for all of their support and teachings throughout all of my life. Through many ups and downs, they have always been there for me with their love and support. I would like to thank my parents in law, Enhui Yao and
Yunxia Niu, for all of their support during my PhD. I want to thank my wife, Xiu
Yao, whom I love deeply. She has patiently helped me through many hard times with her unconditional love. Her faith in me has given me confidence whenever I am confronted with difficult situations. Lastly, I want to thank God for all of His blessings and carrying me throughout all of these years.
vi
Vita
May 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B.S. Engineering,
University of Tennessee at Martin
December 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Electrical Engineering,
The Ohio State University
Sept 2010 to present . . . . . . . . . . . . . . . . . . . . . . . . Ph.D. student,
The Ohio State University
Publications
Research Publications
L. Herrera, and J. Wang, “Stability analysis and controller design of dc microgrids with constant power loads,” in Proceedings of 2015 IEEE Applied Power Electronics
Conference and Exposition (APEC) , Charlotte, NC, Mar. 15-19, 2015.
L. Herrera, X. Yao, and J. Wang, “Modeling and circulating current control of MMC,” in Proceedings of 2015 IEEE Applied Power Electronics Conference and Exposition
(APEC) , Charlotte, NC, Mar. 15-19, 2015.
X. Yao, L. Herrera, and J. Wang, “Impact evaluation of series dc arc faults in dc microgrids,” in Proceedings of 2015 IEEE Applied Power Electronics Conference and
Exposition (APEC) , Charlotte, NC, Mar. 15-19, 2015.
L. Herrera, C. Li, X. Yao, and J. Wang, “FPGA based detailed real-time simulation of power converters and electric machines for EV HIL applications,” IEEE Trans.
On Industry Applications , vol. 51, no. 2, pp. 1702-1712, Mar./Apr. 2015.
C. Li, L. Herrera, J. Jia, L. Fu, Y. Huang, A. Isurin, A. Cook, and J. Wang, “Design and implementation of bidirectional isolated cuk converter for low voltage and high current automotive dc source applications,” IEEE Trans. On Vehicular Technology , vol. 63, no. 6, pp. 2567-2577, Jul. 2014.
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X. Yao, L. Herrera, S. Ji, K. Zou, and J. Wang, “Characteristic study and timedomain discrete-wavelet transform based hybrid detection of series dc arc faults,”
IEEE Trans. on Power Electronics , vol. 29, no. 6, pp. 3103-3115, Jun. 2014.
L. Herrera, E. Inoa, F. Guo, J. Wang, and H. Tang, “Small-signal modeling and networked control of a PHEV charging facility,” IEEE Trans. on Industry Applications , vol. 50, no. 2, pp. 1121-1130, Mar. 2014.
F. Guo, L. Herrera, R. Murawski, E. Inoa, C. Wang, P. Beauchamp, E. Ekici, and J.
Wang, “Comprehensive real-time simulations of the smart grid,” IEEE Trans. on
Industry Applications , vol. 49, no. 2, pp. 899-908, Mar./Apr. 2013.
X. Yao, L. Herrera, and J. Wang, “Modulation and control of MMC based multiterminal HVDC,” in Proceedings of 2014 IEEE Energy Conversion Congress and
Exposition (ECCE) , Pittsburgh, PA, Sept. 14-18, 2014.
Fields of Study
Major Field: Electrical and Computer Engineering viii
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modeling and Real Time Simulation of Power Converters . .
Stability Analysis of Converter Based DC Networks . . . . .
High Voltage DC Networks . . . . . . . . . . . . . . . . . .
Organization and Contributions . . . . . . . . . . . . . . . . . . . .
1
2
4
1
2
7
8
Modeling and Simulation of Power Electronic Systems
. . . . . . . . . .
10
Review of Modeling Methods for Power Electronic Systems . . . . .
10
State Space Methods . . . . . . . . . . . . . . . . . . . . . .
10
Modified Nodal Analysis . . . . . . . . . . . . . . . . . . . .
17
Proposed Electrical/Thermal Modeling . . . . . . . . . . . . . . . .
21
Electrical Model . . . . . . . . . . . . . . . . . . . . . . . .
21
Thermal Model . . . . . . . . . . . . . . . . . . . . . . . . .
25
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Modeling of Electric Machines . . . . . . . . . . . . . . . . . . . . .
29 ix
Saturation Modeling . . . . . . . . . . . . . . . . . . . . . .
31
Computational Algorithm . . . . . . . . . . . . . . . . . . .
34
Electro-Thermal Algorithm Validation . . . . . . . . . . . . . . . .
37
Hardware Description . . . . . . . . . . . . . . . . . . . . .
38
Model Description . . . . . . . . . . . . . . . . . . . . . . .
38
Experimental and Real Time Results . . . . . . . . . . . . .
41
Real Time Simulation of an EV Drive
. . . . . . . . . . . . . . . .
42
Model and Saturation Characteristics
. . . . . . . . . . . .
44
Real Time Results . . . . . . . . . . . . . . . . . . . . . . .
45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Stability Analysis of DC Networks
. . . . . . . . . . . . . . . . . . . . .
48
Examples of CPLs . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Stability Analysis of DC Networks . . . . . . . . . . . . . . . . . .
50
Problem Formulation . . . . . . . . . . . . . . . . . . . . . .
50
Region of Attraction Review for 1 CPL
. . . . . . . . . . .
52
Proposed SDP Technique . . . . . . . . . . . . . . . . . . . . . . .
55
Single CPL . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Multiple CPLs . . . . . . . . . . . . . . . . . . . . . . . . .
58
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Feedback Design . . . . . . . . . . . . . . . . . . . . . . . .
64
Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
. . . . . . . . . . . . . . . . . . . . . . . . .
71
. . . . . . . . . . . . . . . . . . . .
71
. . . . . . . . . . . . . . . . . . . . . .
71
Controller Design . . . . . . . . . . . . . . . . . . . . . . . .
74
Efficient Modeling of MMC . . . . . . . . . . . . . . . . . . . . . .
80
Equivalent Circuit - Thevenin Method . . . . . . . . . . . .
81
. . . . . . . . . . . . . . . . . . . . . . .
83
Hardware-in-the-Loop Controller Validation . . . . . . . . . . . . .
86
Platform Design . . . . . . . . . . . . . . . . . . . . . . . .
86
HIL Example . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Real Time Simulation of MMC Based HVDC . . . . . . . . . . . .
90
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . .
98
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98 x
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
Appendices
109
LMI Methods in Stability Analysis . . . . . . . . . . . . . . . . . . . . .
109
A.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . .
109
A.2 Algorithms for Computing Region of Attraction . . . . . . . . . . .
111
HIL Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
B.1 Gain Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
B.2 DSP Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
116 xi
List of Tables
Table Page
Summary of explicit integration methods. . . . . . . . . . . . . . . . .
14
Summary of implicit integration methods.
. . . . . . . . . . . . . . .
15
Norton equivalent elements for MNA. . . . . . . . . . . . . . . . . . .
19
Simulation parameters for circuit and proposed model validation.
. .
23
Power converter electrical parameters. . . . . . . . . . . . . . . . . . .
39
Thermal parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Real time EV drive simulation parameters. . . . . . . . . . . . . . . .
45
Circuit parameters and initial conditions for Fig. 3.2. . . . . . . . . .
52
Parameters for dc microgrid with Q = 2 CPLs. . . . . . . . . . . . . .
61
Parameters for simulation of three phase MMC. . . . . . . . . . . . .
79
Single phase MMC for HIL validation of controller. . . . . . . . . . .
88
HVDC case study simulation parameters (offline). . . . . . . . . . . .
93 xii
List of Figures
Figure Page
Detailed dc microgrid of more electric aircraft, navy ships, etc. . . . .
4
Boost converter with discontinuous mode.
. . . . . . . . . . . . . . .
15
Hybrid automaton for a discontinuous boost converter. . . . . . . . .
16
Results showing the different modes for a boost converter.
. . . . . .
16
Representation of capacitor and inductors by a Norton equivalent circuit. 18
Discretization and MNA formulation for boost converter. . . . . . . .
19
Equivalent methods for modeling power switches. . . . . . . . . . . .
22
Inductor and output current comparison. . . . . . . . . . . . . . . . .
24
Thermal model for an IGBT and diode inside the same module. . . .
25
Parallel modeling of electrical and thermal systems. . . . . . . . . . .
27
2.10 Flowchart of the algorithm to compute the power losses.
. . . . . . .
29
2.11 Simulation Results Showing Output of the Power Loss Computation
. . . . . . . . . . . . . . . . . . . . . . .
30
2.12 Typical saturation characteristics for an induction machine. . . . . . .
33
2.13 High level diagram for FPGA implementation of machine model. . . .
35
− 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 xiii
2.15 Boost converter for real time electro-thermal simulation.
. . . . . . .
37
2.16 Experimental setup for electro-thermal model verification.
. . . . . .
39
2.17 Electro-thermal RT Power Loss Validation.
. . . . . . . . . . . . . .
40
41
42
2.20 Real time results showing the junction temperature of the power devices. 43
2.21 EV traction drive real time HIL example . . . . . . . . . . . . . . . .
43
2.22 HIL results of an EV drive system. . . . . . . . . . . . . . . . . . . .
44
2.23 Saturation characteristics of the machine in Table 2.7. . . . . . . . . .
45
46
Example of regulated loads/CPLs. . . . . . . . . . . . . . . . . . . . .
49
Simplification of a source and power converter load as a CPL. . . . .
50
51
53
Illustration on computing bound α using (3.18) for one CPL. . . . . .
56
Simplified version of the dc microgrid shown in Fig. 1.1 with Q CPLs.
58
64 xiv
65
Robust controller using energy storage to increase robustness margin.
66
3.10 Detailed control diagram for Q = 2 CPLs. . . . . . . . . . . . . . . .
69
3.11 Results of the closed loop system to a short circuit fault in the dc link at t = 0 .
70
MMC single phase structure. . . . . . . . . . . . . . . . . . . . . . . .
72
Equivalent MMC circuit including disturbance.
. . . . . . . . . . . .
74
Equivalent model for circulating current control. . . . . . . . . . . . .
76
Reduced order observer circulating current suppression. . . . . . . . .
79
Implementation of reduced order observer with N = 100 SMs/arm.
The controller is turned on at t = 1 .
1 s. . . . . . . . . . . . . . . . . .
80
Submodule simplification for MMC based on [28]. . . . . . . . . . . .
81
Proposed submodule simplification for real time simulation. . . . . . .
83
Simulation results and comparison of the methods discussed in Section
4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
87
88
4.11 HIL Validation of controller and efficient modeling methods. . . . . .
89
4.12 HIL results for a single phase MMC with parameters in Table 4.2.
Note: the controller is turned on at t = t
. . . . . . . . . . . . . . . .
91
4.13 HVDC case study based on the Lower Churchill project [10] .
. . . .
92 xv
4.14 Output controller in dq frame for Soldier’s Pond Station. Salisbury
93
4.15 Case study I: Salisbury increments its power from 400 MW to 800 MW. 94
4.16 Case study II: Salisbury reverses its power flow to − 400 MW.
. . . .
95
4.17 Real time simulation results of HVDC systems.
Pond whereas R stands for Salisbury. . . . . . . . . . . . . . . . . . .
97
A.1 T-S Fuzzy formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
111 xvi
Chapter 1: Introduction
1.1
Motivation
Significant advances have been achieved in the electrification of the world over the past centuries. One of these, the induction motor in the late 1800s, became a
catalyst for the widespread adoption of an ac electric grid [56]. One of the reasons
for this adoption is the ability to transmit ac power over long distances as opposed to dc; it was impossible to increase the dc voltage levels at the time. Without the help of power electronics, as a discipline/technology, it seemed inevitable that ac systems dominated.
Nowadays, power electronics provide significant flexibility in the manipulation and/or design of ac and dc systems. Devices such as transistors, thyristors, IGBTs, and MOSFETs allow for the seamless conversion of ac to dc and vice-versa with efficiencies as high as 99%. For this reason, hybrid dc/ac systems are becoming more and more attractive. These new types of mixed power networks are being not only applied to the electric grid, but in “small” isolated microgrids for mission specific applications. Some examples include: Plug-in Hybrid Electric Vehicle (PHEV) charging stations, electric car power systems, more electric aircraft, electric navy ships, etc.
1
Recently, the development of new technologies for power devices, namely SiC and GaN, provides several advantages over Si based devices. Some of these include: higher operating temperature, faster switching frequency, etc. As the switching speed of these devices increases, simulation models must become more and more detailed and lower time steps are needed to accurately capture their dynamics. In addition, the bandwidth of the controllers for the devices can also increase. Therefore, modeling/simulation is an important topic nowadays as well as the stability analysis and controller design of power electronics systems. These topics are necessary and fundamental for the safe operation and adoption of these systems.
1.2
Previous Work
1.2.1
Modeling and Real Time Simulation of Power Converters
In order to help in the introduction and adoption of SiC and GaN to the current market, Real Time (RT) simulation plays a fundamental role in the design and testing of these converters/machines drives. Due to the increase in the switching frequency, the time step of the real time model should be much smaller than the converter’s switching time step, i.e.
T s
T sw
. Currently, typical CPU based real time simulation can only achieve a minimum time step of T s
≥ 10 µ s caused by the large bus latencies in a CPU. In order to decrease this time step as much as possi-
ble, Graphical Processing Units (GPU) provide a viable alternative [33]. Although
GPUs have good parallel processing capabilities, they inherit the same drawbacks as
CPUs. Field Programmable Gate Arrays (FPGAs) have excellent parallel processing capabilities and small bus latencies which make them ideal for fast RT simulation.
Its main disadvantage is the low level programming required.
2
In literature, several papers have studied different methods and approaches for modeling power devices/power systems in FPGA. Two well-known methods include
state space modeling, and Modified Nodal Analysis (MNA) [45, 64]. Moreover, a
state machine to model the characteristics of IGBTs and MOSFETs was discussed in
[63]. In [55], the authors studied the modeling of power converters and ac machines in
FPGA for real time simulation. However, the power converter models were developed using MNA and the ac machines were modeled without saturation. Furthermore, most
literature have implemented these models using fixed point operations [63, 55, 49],
while only a few have presented models using single, double, and/or custom floating
Another important factor in power converter modeling and simulation is their thermal behavior. It is required to maintain these devices within allowable operating temperature. For this reason, modeling of the thermal characteristics of power converters are important during the design and troubleshooting stages. Recent papers have dis-
cussed offline electrical and thermal modeling of power converters [19, 29, 36, 46, 59].
In [19], a thermal model of an IGBT is used in order to compute the junction tem-
perature of the devices based on the calculated power losses. In [29], an algorithm
for offline simulations of the electrical and thermal components of power converters
is studied. Lastly, [59] presented a real time implementation of a thermal model
incorporated to an actual controlled full bridge for temperature monitoring.
In EV drives, typical machines for traction drive include permanent magnet syn-
chronous machine (PMSM), induction machines (IM), and reluctance drives [3, 22].
One of the most important factors which influences the dynamic and steady state response of a machine is saturation. Although this topic has been studied extensively
3
for offline simulations [12, 17], it is considerably new in its implementation for FPGA
real time simulation.
1.2.2
Stability Analysis of Converter Based DC Networks
The recent advancement of power electronics presents a wide range of opportunities in the design of microgrids. These dc networks are composed of multiple generation units and ac/dc loads interconnected together through filters and power
converters. A typical structure of these systems is shown in Fig. 1.1 [21, 53, 54].
Figure 1.1: Detailed dc microgrid of more electric aircraft, navy ships, etc.
The controllers of each source or loads in the dc network are designed to track either a voltage, current, or power reference. Due to the advances in power electronics and digital control, these power converters track the desired signal and reject any disturbances with very high accuracy. However, the high bandwidth of the loads introduce a negative impedance effect as seen from the network’s perspective. This
4
effect is very commonly seen in tightly regulated loads such as inverter motor drive
systems [21, 35], electronic loads, and power supplies [14]. Therefore, typical studies
represent these loads as Constant Power Loads (CPL) [5, 37]. The study of CPLs in dc
power networks is fundamental to automotive [21, 35, 54], marine systems [15, 44, 66],
and aircraft applications [50, 51, 70].
The stability analysis of CPLs has encompassed ac systems as well [4]. During
the early stages, priority was given to the small signal behavior of these systems.
Techniques based on linearization [5] and impedance based criteria [24, 72, 13, 77, 42]
have been largely utilized. However, it is still desired to obtain a better technique for the stability analysis during large signal excursions.
An application of global stability analysis is for filter design. The design of the
filters shown in Fig. 1.1 is typically carried out locally. For example, the EMI filter
may be designed based only on its frequency response. However, these filters, as well as the CPLs, will greatly impact the stability and robustness of the dc microgrids.
As an example, a smaller capacitor connected to a CPL will decrease the stability
margin for the system but a bigger capacitor will increase cost and size [53]. It is
therefore important to study how these parameters impact the global behavior of the system. One way to accomplish this goal is to compute a Region of Attraction (ROA) for an operating point. Different methods of computing an estimation on the ROA have been presented in literature:
1. Mixed Lyapunov functions [5]
2. Takagi-Sugeno (T-S) Fuzzy model [53, 54]
5
4. Reverse trajectory tracking [2]
Among these, the methods based on Mixed Lyapunov functions and Genetic algorithms were shown to be highly conservative while the reverse trajectory tracking method is a graphical approach that is difficult to use in a high dimensional system.
Lastly, the T-S Fuzzy model approach shows good results but its computational complexity grows exponentially with respect to the number of CPLs in the microgrid.
Therefore, it is important to design efficient algorithms to compute the region of attraction considering the overall complexity.
While computing a ROA can be useful for the design of the passive components of the dc microgrid, it is also important to consider the possibility of increasing the
robustness of the system. For example, [14] considers the use of passive damping
filters to stabilize the dc network with CPLs. However, this method will increase the overall cost and size of the system. A different approach is to use control strategies to mitigate the effects of the CPLs with the use of energy storage or the dc source. One
control method considers sliding mode and feedback linearization techniques [21, 71].
However, one drawback is the need to use derivatives which in the presence of noise
may not cancel completely the non-linearity [8]. In [50], a control Lyapunov function
is used to generate a feedback law to stabilize the system. The main drawback of this approach is the need to inject a stabilizing current (modifying the load behavior) at all the CPLs. Therefore, this is still an important problem where different approaches can be of great value.
6
1.2.3
High Voltage DC Networks
The electricity consumption has been increasing rapidly all over the world. The possibility of expanding the ac transmission capacity is either limited by the availability of land or by the huge costs from long distance bulk power transmission. As an alternative, High Voltage dc Transmission (HVDC) is becoming a popular solution.
For transmission distances longer than 700 km, the investment cost of HVDC becomes
lower than HVAC [57]. Moreover, HVDC cables do not have the capacitance problem
of AC cables which makes it possible for long distance underwater transmission [11].
Traditional HVDC systems usually consisted of thyristor based stations with three level converters, point-to-point or back-to-back topologies, connected to strong ac systems. In the 1980s, the concept of Multiterminal HVDC (MTDC) has emerged
and has been studied intensively [43, 58]. MTDC was proven to be feasible in theory,
but in practice only three terminals were realized. Commutation failure was expected when the ac system experience transients such as voltage flicking, which then lead
to the complete shut off of that terminal [73]. A Voltage Source Converter (VSC)
based MTDC does not rely on the ac system voltage to accomplish commutation, and therefore can operate at reduced power mode at such situations. This feature is very helpful in connecting remote renewable energy sources or isolated island systems.
For MTDC applications, Modular Multilevel Converters (MMC) offer several advantages over a typical three phase inverter. For example, higher number of voltage levels help to decrease the ac harmonic content and avoid the need of passive filters.
In addition, each switch has a voltage stress of V sw
= V dc
/N , where N is the total number of sub-modules per arm, making its implementation for HVDC applications more feasible.
7
However, without additional modulation strategy or control for a MMC, an os-
cillation occurs in the circulating current of each phase [74]. This oscillation causes
additional stress on each switch and does not contribute to the active power transfer between the dc and ac side. For this reason, several methods have been proposed in recent years to eliminate this oscillation. Among these, there are controllers based
on abc/dq proportional integral (PI) control [74], proportional-resonant (PR) control
[80, 75], repetitive control [82], and energy based controllers [1, 6]. A problem with
PI, PR, or repetitive controllers is choosing the gains to avoid instability. Tuning these controllers is generally accomplished through transfer function and bode plot analysis. Nevertheless, even if the system appears stable through bode plot, a high
gain might diminish the performance and cause instability [82, 75]. This is unwanted
for a larger scale system such as MTDC which requires the controller to be stable under all possible operation situations including contingencies.
1.3
Organization and Contributions
This dissertation makes several contributions aiding in the implementation of dc microgrids with a case study in HVDC systems.
In Chapter 2, a review of numerical integration techniques and their use in the
modeling and simulation of power electronics based systems is presented. In addition, a change in the formulation of the Thevenin equivalent is proposed in order to include the on characteristics of a power device. With this formulation an electrical/thermal algorithm is proposed and implemented in FPGA for real time simulation. Lastly, a numerical algorithm for the simulation of saturation in an induction machine is discussed and the FPGA based simulation of a motor drive is presented. These two
8
types of loads are very common in dc networks and can behave as CPLs discussed in the following chapter.
In Chapter 3, an introduction to the problem of negative impedance is presented
based on a simplified case with one CPL. A method is proposed to compute the region of attraction of the dc network with multiple CPLs based on a Semidefinite
Programming formulation. Considering the use of energy storage, a robust controller design is then introduced. The goal of this controller is to expand the region of attraction by a desired amount while minimizing the battery effort. Simulation results are presented for a case study with two CPLs. The analysis conducted in this chapter has applications in many types of dc networks ranging from low to high voltage.
In Chapter 4, an introduction to HVDC is discussed. Beginning with MMCs,
a re-formulation of the dynamic equation of the circulating current is proposed. A controller based on an internal model of the disturbance is presented and a practical implementation is shown with a Hardware in the Loop (HIL) validation. Lastly, a platform is presented to simulate MMCs with a realistic number of SMs per arm in a HVDC case with both offline and real time simulation.
Lastly, in Chapter 5, a summary of the dissertation and future work which arises
from it are discussed.
9
Chapter 2: Modeling and Simulation of Power Electronic
Systems
In this chapter a review of different methods are introduced to model and simulate power converters and electric machines. A proposed method to efficiently compute the on characteristics of power devices is presented with applications in electrical/thermal modeling of power converters. In addition, the efficient computation of saturation in an induction machine is presented along with an example for an electric drive system.
The modeling techniques presented in this chapter will be used in the remainder of this dissertation.
2.1
Review of Modeling Methods for Power Electronic Systems
2.1.1
State Space Methods
The modeling of power electronics systems falls in the category of hybrid dynam-
ical systems [27]. In this dissertation, a switching dynamic system is considered to
be of the form:
˙ = f q
( t, x, u ) , x (0) = x
0
(2.1) where x ∈
R n , u ∈
R m , t ∈
R ≥ 0
, and q ∈ Q . The set of “modes” Q represents all the feasible vector fields and it is assumed to be finite. For a typical power converter
10
with N > 1 switches, the set Q = 1 , 2 , ..., 2 N , i.e. there are a total of 2 N different combinations of the switches. For power switches, the modes can be a function of the states and time: q = Φ( t, x, σ ) .
(2.2) where σ is the discrete (e.g. ideal switches) inputs. For example, whether an ideal switch is on/off may only be a function of a discrete external input, whereas a diode depends on the states of the systems. In order to obtain a numerical solution for
(2.1), different types of integration techniques are reviewed.
Explicit Integration
For simplicity, assume that we have an autonomous system of the form:
˙ ( t ) = f ( t, x ) , x (0) = x
0
(2.3) and x ∈
R and t ∈
R ≥ 0
. In general, we would like to approximate the solution or trajectory of the system to a certain time step T s
Z
T s dx =
0
Z
T s
0 f ( τ, x ( τ )) dτ (2.4) which simplifies to: x ( T s
) = x (0) +
Z
T s f ( τ, x ( τ )) dτ.
0
(2.5)
Explicit integration approximates the integral term using only the previous state.
One of the most common methods is Forward Euler (FE), where the integral is approximated as:
Z
T s f ( τ, x ( τ )) dτ ≈ T s f (0 , x (0)) .
0
(2.6)
11
Using this method, the numerical solution at time T s is: x ( T s
) ≈ x (0) + T s f (0 , x (0)) .
(2.7)
The local error for FE can be obtained using Taylor’s theorem: x ( T s
) = x x (0) T s
+ ¨ ( ξ )
T
2 s
2
(2.8) where ˙ (0) = f (0 , x (0)) and some ξ ∈ (0 , T s
). This implies that
|| x ( T s
) − ( x (0) + T s f (0 , x (0))) || =
T s
2
|| ¨ ( ξ ) || .
2
(2.9)
The local error is then commonly denoted as O ( T 2 s
) since it is proportional to T 2 s
.
Obtaining an estimate at x ( T s
) then allows us to repeat this procedure to obtain another estimate at time t
2
= 2 T s
. In general: x k +1
= x k
+ T s f ( t k
, x k
) , k = 0 , 1 , ...
(2.10)
Another condition of interests is the global error.
Assuming x ( t ) are the actual values of x at time t , ˜ ( t ) are the Forward Euler approximations, and the following conditions:
∂f ( t, x ( t ))
∂x
≤ L, || ¨ ( t ) || ≤ M, (2.11) for an arbitrary t
, then the global error is defined as [23]:
|| x ( t ) − ˜ ( t ) || ≤
T s
M
2 L e
L ( t − 0)
− 1 .
(2.12)
From (2.12), the global error for Forward Euler is
O ( T s
) which can be a major drawback of using this method. Lastly, the stability of the continuous system does not
12
implies stability of the discrete approximation. For example, assume that we have a one dimensional linear system of the form:
˙ ( t ) = λx, x (0) = x
0
.
(2.13) and λ is real. The exact solution in this case is x ( t ) = e λt x (0). Thus, if λ < 0 this system will decay to 0 as t → ∞ . Applying Forward Euler we obtain: x k +1
= x k
+ T s
λx k
= (1 + T s
λ ) x k
(2.14) which implies: x n
= (1 + T s
λ ) n x
0 for n = 1 , 2 , 3 , ...
(2.15)
We can see (2.15) as a sequence of numbers
{ x n
} which converges to zero as well if
| 1 + λT s
| < 1 ⇒ T s
2
<
| λ | assuming λ < 0 .
(2.16)
In other words, using Forward Euler, the time step must be sufficiently small in order to guarantee the numerical approximation of an asymptotically stable system converges to 0.
A summary of a few explicit integration methods and their errors is shown in
Tab. 2.1. It should be noted that in order to decrease the local and global error the
computations increase as well.
Implicit Integration
Implicit integration methods not only use past approximations of x ( t ), but may use the present values as well. A common example is the Backward Euler (BE)
method. Using (2.3), the approximation of the solution at time
t = T s by BE is given
13
Table 2.1: Summary of explicit integration methods.
Method Procedure
Forward Euler x k +1
= x k
+ T s f ( t k
, x k
)
Midpoint Method x k +1
= x k
+ T s f t k
+
T s
2
, x k
+
T s
2 f ( k, x k
)
Heun’s Method w = 3 f t k
+
2 T s
3
, x k
+
2 T s
3 f ( t k
, x k
) x k +1
= x k
+
T s
4
[ f ( t k
, x k
) + w ]
Runge Kutta 4 w
1
= T s f ( t k
, x k
) w
2
= T s f t k
+
T s
2
, x k
+
1
2 w
1 w
3
= T s f t k
+
T s
2
, x k
+
1
2 w
2 w
4
= T s f ( t k
+ T s
, x k
+ w
3
) x k +1
= x k
+
1
6
( w
1
+ w
2
+ w
3
+ w
4
)
Loc. Error Glob. Error
O ( T 2 s
)
O ( T 3 s
)
O ( T
O ( T s
2 s
)
)
O
O
(
(
T
T
3 s
5 s
)
)
O ( T
O ( T
2 s
4 s
)
) by: x ( T s
) ≈ x (0) + T s f ( T s
, x ( T s
)) .
This sequence converges to 0 if:
1
| 1 − T s
λ |
< 1 ⇒ | 1 − T s
λ | ≥ 1 .
14
(2.17)
The main challenge in solving (2.17) is that the term
f ( T s
, x ( T s
)) is not known. To obtain ˜ ( T s
) for nonlinear systems, an iteration technique such as Newton-Raphson can be employed.
Considering the same linear system as (2.14) and employing BE we obtain:
x k +1
= (1 − T s
λ )
− 1 x k for k = 0 , 1 , ...
(2.18) or x n
= (1 − T s
λ )
− 1 · n x
0
.
(2.19)
(2.20)
Table 2.2: Summary of implicit integration methods.
Method Procedure
Backward Euler x k +1
= x k
+ T s f ( t k +1 k, x k +1
)
Trapezoidal x k +1
= x k
+
T s
2
( f ( t k
, x k
) + f ( t k +1
, x k +1
))
Midpoint Method x k +1
= x k
+ T s f ( t k
+
T s
2
,
1
2
( x k
+ x k +1
))
Loc. Error Glob. Error
O ( T 2 s
)
O ( T 3 s
)
O ( T 3 s
)
O ( T s
O ( T 2 s
O ( T 2 s
)
)
)
This implies that if λ < 0 (asymp. Stability), then the BE numerical approximation also decays as k → ∞ ! This property of the BE method is known as A-stability. A summary of implicit integration methods and its local and global errors is shown in
Boost Converter Example
A hybrid dynamical system can be constructed for a discontinuous boost converter. In this circuit the inductor current is not allowed to be negative due the diode.
The circuit is shown in Fig. 2.1. Let
x
1
, i l
, x
2
, v c
, u
1
, v dc
> 0 be the inductor
Figure 2.1: Boost converter with discontinuous mode.
15
Figure 2.2: Hybrid automaton for a discontinuous boost converter.
Figure 2.3: Results showing the different modes for a boost converter.
current, the capacitor voltage, and the input voltage respectively. Then the hybrid system can be described by the following:
16
• Q = { 1 , 2 , 3 } (set of modes)
• X =
R
2 (state space)
• The vector fields are: f
1
( x, u ) =
1
1 u
− l rc
1 x
2
, (switch on) f
2
( x, u ) =
− 1 l
1 c x x
1
2
−
+
1 rc
1 l u x
2
1
, (switch off-cont. current)
f
3
( x, u ) =
0
− 1 rc x
2
(switch off-disc. current)
The discrete input, σ = s is the switching signal of the IGBT. The transitions from
system was implemented using Forward Euler integration as described in the previous subsections.
2.1.2
Modified Nodal Analysis
Modified Nodal Analysis (MNA) is a very popular method for simulating power
electronics/systems. Nodal analysis was first introduced by H. W. Dommel [18] to
describe a method of finding the time response of electromagnetic transients in arbitrary networks. Other important contributions to the development of MNA were
presented by C. Ho [31] and more recently by J. Mahseredjian [52].
The MNA represents dynamic elements of circuits by a Norton equivalent as shown
Y x
, and current source (also known as history term), j x
, for an inductor/capacitor depends on the type of integration method used. For
17
(a) Inductor and capacitor elements.
(b) Norton equivalent.
Figure 2.4: Representation of capacitor and inductors by a Norton equivalent circuit.
example, beginning with the dynamic equation for an inductor: v l
= L di dt l and using BE
(2.21) i l
( t ) =
T s v l
( t ) + i l
( t − T s
)
L it is possible to see that
Y l
,
T s
L and j l
( t − T s
) , − i l
( t − T s
) .
(2.22)
(2.23)
A summary of the admittances and history terms for inductor/capacitors using BE
and Trapezoidal integrations is shown in Table 2.3.
Using this Norton representation for inductors/capacitors, the numerical integration (using implicit methods) for a certain circuit becomes the solution of a linear system of the form:
Ax k +1
= u k for k = 0 , 1 , ...
(2.24) where the vector x ∈
R n + m contains the voltages at each node ( n nodes) in a circuit as well as certain branch currents (defined by m voltage sources) and u contains the
18
(a) Actual circuit model.
(b) Discretized boost converter model.
Figure 2.5: Discretization and MNA formulation for boost converter.
history and input terms. For linear circuits containing only independent sources, the matrix A is of the following form:
A =
Y B
B T 0
(2.25) where Y ∈
R n × n is the admittance matrix and B ∈
R n × m is a matrix of 1s and 0s depending on the connection of independent voltage sources.
Example for Boost Converter
Consider a typical boost converter as shown in Fig. 2.5a. This passive components
are first replaced by the Norton equivalent form and the switches are modeled by a
Table 2.3: Norton equivalent elements for MNA.
Element Y x
(BE)
L
C
T
L
C
T s s j x
(BE)
− i l
( t − T s
)
C
T s v c
( t − T s
)
Y x
(Trap.)
T
2
2
T s
L
C s j x
(Trap.)
− i l
( t − T s
) +
T s
2 L v l
( t − T s
) i c
( t − T s
) +
2 C v c
( t − T s
)
T s
19
varying resistance (large off resistance/small on resistance). The equivalent network
Using the MNA, we can solve this system as follows:
Y
L
− Y
L
0
0
0
1
− Y
L
Y
L
+ Y rL
− Y rL
0
0
0
j
L
− j
L
0
0
j
C
V dc
0
− Y rL
Y rL
+ Y s 1
+ Y s 2
− Y s 2
0
0
0 0 1
v
1
−
0
Y s 2
0
0
0
0
Y
R
+ Y s 2
+ Y rC
− Y rC
0
− Y rC
Y rC
+ Y
C
0
v
v
3
2
v
4
v
5
0 0 0 i dc
=
(2.26) or in more compact form:
A q x k +1
= u k
.
(2.27)
Notice that the matrix A is time-varying, due to the representation of the switches as a small on resistance/large off resistance. To solve for x it is typically necessary to compute the inverse of A q such that x k +1
= A
− 1 q u k
. However, since A is commonly sparse, an LU decomposition for A is preferred in practice whenever it is not feasible to store all possible A
− 1 (e.g. when the number of switches is very large). Using this
LU decomposition, the solution becomes:
L q
U q x k +1
= u k
⇒ L q y k +1
= u k
U q x k +1
= y k +1
(2.28)
That is, we can solve two sets of linear equations by using forward and backward
20
2.2
Proposed Electrical/Thermal Modeling
This section presents a method to efficiently compute the on characteristics of the switching devices. Based on this approach, an electro-thermal algorithm for parallel modeling of power converters is proposed.
2.2.1
Electrical Model
The method introduced by Pejovic [64] represents a switch as a resistor and current
source in parallel as shown in Fig. 2.6a. By letting the resistor be a constant value,
the current source is changed as follows:
J
N
=
( i k
− G
N v k if s k +1
= 1 if s k +1
= 0
.
(2.29)
Studying the equivalent equations of the circuit in Fig. 2.6a, we can see that the
switch is represented as a small inductor when on and a small capacitor when off.
The value of the capacitance/inductance depends on the type of integration method
used. In [64], Backward Euler was the recommended method due to its decreased
in transient oscillation as opposed to Trapezoidal integration. Using BE, the capacitance/inductance values are:
L = T s
R, C =
T s
R
(2.30)
From (2.30), the following relations can be obtained:
L ∝ R and C ∝ 1 /R . Therefore, there is always a trade-off when choosing R , a smaller R implies a small inductance
but larger capacitance. For small time steps (ns range) it is typically chosen as 1 [64].
Deriving the nodal equations for a complete converter, the system is of the form:
Ax k +1
= u k
⇒ x k +1
= A
− 1 u k
21
(2.31)
(a) (b)
Figure 2.6: Equivalent methods for modeling power switches.
where A, A
− 1 matrices are constant. The current sources for each switch J
N are a component of the input vector u k
.
This method is equivalent to using a ‘Thevenin’ representation shown in Fig. 2.6b
[78]. In this case, the voltage source is represented as:
E
TH k +1
=
(
− R
TH i k v k if s k +1
= 1 if s k +1
= 0
.
(2.32)
Analyzing the on behavior of the Thevenin representation, the equivalent voltage is:
V eq k +1
= E
TH k +1
+ R
TH i k +1
= − R
TH i k
+ R
TH i k +1
(2.33)
We can see that when i k
= i k +1
(i.e.
i is in steady state) V eq
= 0. The on characteristics of a power device are governed by its on resistance r on and forward voltage v f
. In order to include these characteristics into (2.33) as an input to the
system, we can let the new voltage source E
TH be:
E
TH k +1
= − R
TH i k
+ r on i k
+ v f k
(2.34) where it is clear the on steady state is V eq
= r on i + v f
. This proposed representation
22
is particularly useful in the case where r on and v f are a function of the current and device junction temperature. Since they are input to the system, the A
− 1 matrix is still constant. The on parameters, r on
( i, T j
) and v f
( i, T j
), are nonlinear terms and need to be re-computed after every time step. Using conventional MNA, the matrix
A
in (2.31) would have to be re-computed as well as its inverse or
LU matrices at each time step.
Compared to a (hybrid) state space approach, the nonlinear parameters r on and v f also imply the need to recompute the system and input matrices in x k +1
= A j
( x, u ) x k
+ B j
( x, u ) u k for j = 1 , ... , 2 N and N switches during each time step.
To validate the previous method, a circuit simulation using Matlab SimPower
Systems was performed with the parameters shown in Table 2.4. Using SimPower
Systems, the on resistance and forward voltage were incorporated into the switch and diode respectively. The results of this simulation were then compared with the
Table 2.4: Simulation parameters for circuit and proposed model validation.
L r on
T s
1 mH
10 mΩ
1 µ s
V f
F sw r load
1 V
1000 Hz
2 Ω
C
V dc
1 mF
10 V
23
Figure 2.7: Inductor and output current comparison.
proposed method by using (2.35):
R
L
+ r l
E
E
0 i
TH d
TH
J cap
1 0
i
L
− R i
TH
0
0
R
− i
TH
R d
TH
1
1
1 0 − ( G c
0
− 1
+
1 r load
)
i d v sw
v out
=
V in
− E thL
(2.35) where R
L
=
L
T s
, G c
=
C
T s
, J cap
= − G c
V n out
, and E thL
= − R
L i
L k
. Lastly, the voltage input for the IGBT is given by:
E i
TH k +1
=
(
− R i
TH
( i
L k
− i v sw k d k
) + V ce k if s i k +1 if s i k +1
= 1
= 0
(2.36)
24
and for the diode:
E d
TH k +1
=
(
− R d
TH
( i d k
) + V f k v sw k
− v out k if s d k +1 if s d k +1
= 1
= 0
The diode’s on state is decided by:
.
s d k +1
= s d k
( i d k
≥ 0) + ¯ d k
(( v sw k
− v out k
) > V f k
)
(2.37)
(2.38)
Fig. 2.7 shows a comparison between these two results where the green curve repre-
sents the results if r on and v f were to be ignored.
2.2.2
Thermal Model
The thermal behavior of a switch from its junction temperature to the case is typically specified in the data sheet. Since the junction temperature is given by
T j-c
= Z th, j-c
P loss
, (2.39) there exists an analogy between this equation and a regular circuit representation where P loss
↔ I in and T j-c
↔ V j-c
. For this reason, the junction to case impedance
Figure 2.8: Thermal model for an IGBT and diode inside the same module.
25
Z th, j-c
, the case to fin Z th, c-f
, and the fin to ambient Z th, f-a are specified as RC networks.
The complete thermal model for an IGBT and diode inside the same module or
case is presented in Fig. 2.8. Notice that this system can be represented as
˙ ( t ) = A th x ( t ) + B th u ( t ) y ( t ) = C th x ( t ) + D th v ( t ) (2.40) where the matrices A th
, B th
, C th
, and D th are
A th
B th
= Diag
=
1
C i
1
.
..
C
1 i
N
0
.
..
0
1
C c-f
1
C f-a
− 1
, ...,
τ i
1
0
−
τ i
N
1
,
−
τ d
1
1
, ...,
−
τ d
N
1
.
..
0
1
C d
1
.
..
1
C d
N
1
C c-f
1
C f-a
1 · · · 1 0 · · · 0 1 1
,
− 1
R c-f
C c-f
,
R f-a
C f-a
−
1
1
C th
=
0 · · · 0 1 · · · 1 1 1
0 · · · 0 0 · · · 0 1 1
, D th
0 · · · 0 0 · · · 0 0 1
=
1
1
1
(2.41) where τ
α q
= R
α q
C
α q for q ∈ { 1 , ..., N } and α ∈ { i, d } .
The states x = T i
C 1
... T i
CN
T d
C 1
... T d
CN
T
C c-f
T
C f-a
>
, u = P d
P i
>
, v = T a
, and y = T j i T j d T
C
T f
>
. The total power losses for the IGBT and diode, P i and P d respectively, are computed based on the conduction and switching losses of each device. The conduction losses will be taken into account using the approach discussed in the previous subsection. Therefore, for an actual IGBT (superscript i ) and diode
26
(superscript d )
P
α cond
= i
α v f
α
( i
α
, T j
α
) , where α ∈ { i, d } .
(2.42)
These conduction losses are included in the electrical model and the forward voltage will be modified depending on i and T j through each device. To take into account the switching losses in the electrical model will require detailed knowledge of the device.
For example, the leakage inductances of the device affect the transient response at different levels. Since a highly elaborate model of the device increases complexity and overall time step, the switching losses will be calculated using the device datasheets
[29]. The switching energy is usually given for an IGBT as follows:
E i sw
= E i sw − on
+ E i sw − off and the total switching losses can then be computed as:
P i sw
= E i sw
F sw
2.2.3
Algorithm
(2.43)
(2.44)
Motivated by the results presented in the previous subsections, an electro-thermal
algorithm is shown in Figs. 2.9 and 2.10. Note that in Fig. 2.9, an external PWM
Figure 2.9: Parallel modeling of electrical and thermal systems.
27
signal is fed to the power loss block. This signal represents the total time for which the power losses are averaged. Thus if T s-epwm
= T sw
, the program is averaging the power losses over one complete switching cycle. However, if T s-epwm
< T sw then the power losses are updated more frequently and the total losses will have a ripple based on the on current through the device.
The flowchart representing the power loss computation algorithm is composed of three main parts:
1. Switching loss counter C α sw − on
, C α sw-off
.
2. On current addition i α tot
.
3. Forward voltage and power loss computation.
The superscript α = 1 , ..., N implies that each parameter needs to be computed for N switches in a circuit. Although the algorithm is averaging the power losses for a period of T e-pwm
, it runs at the same time step as the electrical circuit model T s
.
For the same boost converter shown previously, this algorithm is applied to compute the power losses and thermal model. The simulation results are shown in Fig.
T e-pwm
= T sw which can be inferred from the power loss waveform.
Lastly, since the conduction losses are taken into account by updating the forward voltage and on resistance in the electrical model, the switching losses should be taken into account as well to achieve power balance.
The power is:
P in
= P out
+ P cond
+ P sw
.
(2.45)
28
Figure 2.10: Flowchart of the algorithm to compute the power losses.
Thus [29] explains the addition of a current source in parallel to the input voltage of
the converter. This current will have a value as:
I sw
=
P sw
V in
(2.46)
and power balance (2.45) can be achieved. The RT implementation and validation of
the results presented will be shown in Section 2.4.
2.3
Modeling of Electric Machines
The development of dynamic models for electric machines has been studied for the past several decades. These models can be categorized by the choice of coordinate
29
Figure 2.11: Simulation Results Showing Output of the Power Loss Computation
Algorithm with T s-epwm
= T sw
.
systems upon which is studied and the choice of state variables (e.g. flux linkage or current). The phase domain model uses three phase coordinates, abc , to represent the state variables. The second uses a combination of phase and quadrature axis variables, for example stator quantities are modeled in phase domain and rotor variables are modeled in dq domain. This method is known as Voltage Behind Reactance. The last method uses dq
coordinates to represent all the variables [76].
For RT implementation, the dq model offers several advantages such as static inductance parameters and a lower order system. For this reason, it is one of the most commonly used machine models. For an induction motor, its dynamics in dq frame are given by: v sr dq
= pλ sr dq
+ Ω λ sr dq
+ R i sr dq
(2.47)
30
where p := d dt
, Ω , R ∈
R
4 × 4 , v sr dq
, λ sr dq
, i sr dq
∈
R
4 . The matrices Ω , R are as follows:
0 − ω 0
Ω =
ω 0
0 0
0
0
0 0 ( ω − ω r
)
R = Diag ( r s
, r s
, ¯ r
, ¯
∗ r
)
0
0
− ( ω − ω r
)
0
(2.48) where ω is the angular speed of the reference frame ( ω = 0 for stationary frame) and ω r is the rotor angular speed. In addition, these matrices assume the vectors
λ sr dq
, i sr dq
, and v sr dq are of the form x sr dq
= x s d
, x s q
, ¯ r d
, ¯ r q
> where x = λ, i, or v .
One of the most influential elements which affect the machine dynamic response
is core saturation. However, the model in (2.47) does not consider saturation in the
mutual inductance. Although several studies of modeling electric machines in FPGA
have been presented in literature, e.g. see [63, 55, 61], very few have modeled these
with saturation characteristics. For this reason, we briefly review what has been done to account to model saturation in FPGA and present implementation of these characteristics using a mathematical derivation.
2.3.1
Saturation Modeling
Saturation of an inductor occurs when its core can no longer store magnetic energy.
For example, a typical B-H curve gives a good representation of saturation as the magnetic field strength ( H ) increases and the flux density ( B ) remains constant. The main effect of saturation in an ac machine occurs in its mutual inductance L m
. The leakage inductances of the stator and rotor remain in air for a significant part of their path and thus saturation does not often occur.
31
FEA Methods
Recent studies focuses on using Finite Element Analysis (FEA) to simulate sat-
uration for a three phase machine, particularly PMSM [32, 20]. The machine is
designed using a FEA software such as JMAG [34] and the saturation characteris-
tics are stored in look-up tables. For FPGA implementation, these tables are then a function of different currents i d
, i q and rotor position θ r
. These tables need to be quite dense and/or include interpolation techniques in order to avoid any large jump discontinuities. This can then be major drawback causing incorrect transient behavior.
Mathematical Method
Typical saturation characteristics for an induction machine is shown in Fig. 2.12.
We assume that the magnetizing flux and current are complex quantities described
in polar coordinates as follows [12]:
λ m
( | λ m
| , µ ) = | λ m
| e jµ i m
( | i m
| , µ ) = | i m
| e jµ
(2.49)
(2.50)
Since there is a one to one relationship between | λ m
| and | i m
| : d | λ m
| dt
= d | λ m
| d | i m
| d | i m
| dt
= L y d | i m
| dt
(2.51) where L y
:= d | λ m
| d | i m
| is defined as the ‘dynamic inductance.’
The following two equations are derived from the derivative with respect to time of λ m
i m
(2.50), and separating it into its real and imaginary parts (denoted
32
Figure 2.12: Typical saturation characteristics for an induction machine.
by subscripts d and q respectively): dλ md dt dλ mq dt
= L dd di md dt
= L qd di md dt
+ L dq di mq dt
+ L qq di mq dt
The new inductance parameters are:
(2.52)
L dd
:= L y cos
2
( µ ) + L m sin
2
( µ )
L qq
:= L m cos
2
( µ ) + L y sin
2
( µ )
L dq
= L qd
:=
L y
− L m
2 sin(2 µ ) .
(2.53) where L m
= | λ m
| / | i m
| (note that it is not constant) and µ = tan
− 1
( i mq
/i md
). The terms L dd and L qq correspond to saturation in their own axis, whereas the terms
L dq
= L qd are cross saturation terms. It is also clear to see that during the linear region
(unsaturated) L y
≡ L m and these equations are identical to the classical unsaturated model of a three phase induction machine.
33
By incorporating (2.52) into (2.47), we can redefine the equations for an induction
machine in a stationary reference frame as: v sr dq
= R i sr dq
+ ΩL r i sr dq
+ L p i sr dq
(2.54)
The matrices L ( | λ m
| , µ ) and L r
( | λ m
| ) vary based on the saturation parameters and are defined as follows:
L =
L ls
+ L dd
L
L
L dq dd dq
L ls
L
+
L
L dq dq qq
L qq
0 0 0
L r
=
0 0 0
0 L m
− L m
0 − ( L lr
0
+ L m
)
L lr
L
L
+ dd dq
L dq
L dd
L dq
L
L qq dq
L lr
+ L qq
0
( L lr
0
+ L m
)
0
(2.55)
Thus (2.54) constitutes the new
dq model of a three phase induction machine including saturation. The same idea is followed to derived saturated models for other types of machines.
2.3.2
Computational Algorithm
The saturation model presented in the previous subsection only requires knowledge
of the saturation curve, such as the one in Fig. 2.12. Therefore, this curve can be
represented as:
| λ m
| ( | i m
| ) =
(
L m
| i m
| for | i m
| < | i mc
| f
˜
( | i m
| ) | i m
| for | i m
| ≥ | i mc
|
(2.56)
L m is the slope of the linear region, and ˜ ( | i m
| ) | i m
| is a (interpolated) function describing the nonlinear saturated region. In order to make sure there is a smooth transition between the unsaturated to saturated region or vice versa, it is required
34
Figure 2.13: High level diagram for FPGA implementation of machine model.
that the function | λ m
| ( | i m
| ) ∈ C 1 , i.e. continuous and with continuous first derivative.
This will ensure that at the point | i mc
| , L m
( | i mc
| ) = L y
( | i mc
| ).
For the FPGA implementation, (2.54) should be transformed into state space form
as: di sr dq dt
= L
− 1
( − R − ΩL r
) i sr dq
+ L
− 1 v sr dq
.
(2.57)
This model is then discretized using typical methods such as Implicit Backward Euler,
Trapezoidal, Runge Kutta, etc. The choice of method depends on the system to be studied. Since the time step used in FPGA is in the ns range, Forward Euler’s method simplifies computation over other integration methods. The percent difference compared with Runge Kutta is < 1 % for a wide range of machine parameters. Thus: i k +1
= ( I + T s
A ) i k
+ T s
B v k
(2.58) for A = L
− 1 ( − R − ΩL r
) and B = L
− 1 .
However, several challenges arise from (2.58): the matrices are time varying, non-
linear, and L
− 1 need to be computed at every time step. A general procedure for
35
the computation of the induction machine system with saturation is shown in Fig.
2.13. There are different options to compute
L
− 1 at every time step. Since it is a symmetrical matrix, a Cholesky decomposition could be used. However, we can see that although this matrix is a function of µ and | I m
| , it is found that the det ( L ) is independent of µ : det ( L ) = d
1
( L m
+ L y
) + d
2
L m
L y
+ d
3
(2.59)
Figure 2.14: Computation of L
− 1 .
where d
1
, d
2
, d
3 are constants defined as d
1
:= L
2 ls
L lr
+ L ls
L
2 lr d
2
:= L
2 ls
+ L
2 lr
+ L ls
L lr d
3
:= L
2 ls
L
2 lr
.
36
(2.60)
It can be seen from (2.59) that det(
L ) = 0 ∀ t . Thus, the matrix L is always non-
singular. Taking advantage of (2.59), the matrix inverse is then computed by calcu-
lating the determinant and the adjoint matrix in parallel , i.e.
L
− 1 = L
∗
/ det ( L ). The
procedure is outlined in Fig. 2.14.
(a) Electro-thermal boost converter.
(b) Equivalent circuit in MNA and Thevenin equivalent.
Figure 2.15: Boost converter for real time electro-thermal simulation.
2.4
Electro-Thermal Algorithm Validation
In order to demonstrate the electro-thermal algorithm discussed in Section 2.2, a
model of a boost converter is built in a real time simulator as shown in Fig. 2.15.
37
2.4.1
Hardware Description
Real Time Platform
An Opal RT based platform is used in order to simulate the described electro-
thermal model in real time [60]. This platform utlilizes a Xilinx Virtex 6 FPGA in
order to model the system. The analog to digital converter (ADC) has a time rate of 2 .
5 µ s, the digital to analog converter (DAC) 1 µ s, and the digital input/output
(DIO) has a refresh rate of 0 .
2 µ s.
Power Electronics Platform
The boost converter is built based on a Powerex PS22A78-E IPM [65].
To accurately measure the power losses, a Yokogawa WT3000 power meter was used
[79]. The dc power supply is a Magna Power TS1500-16.5. The controller is a TI
TMS320F28335 and is used to send the gating signals to the converter. The experi-
mental setup is shown in Fig. 2.16.
2.4.2
Model Description
The main difference between the boost converter in Fig. 2.15a and the one in
Section 2.2 is the introduction of the current source
i sw
The equations for this circuit are similar to (2.35).
The values of the inductor, capacitor, thermal model, etc. Are shown in Tables
2.5 and 2.6. These values are based on the actual power converter parameters. In
order to compute the forward voltages for the IGBT and diode, a surface fit was used
38
based on the device datasheet v f i
T j i
, i on i
= 0 .
8433 + i i on
0 .
02905 + 5 .
818e
− 5
T j i v d f
T j d
, i d on
= 1 .
023 + i d on
0 .
03469 − 5 .
896e
− 5
T j d
(2.61)
Figure 2.16: Experimental setup for electro-thermal model verification.
Table 2.5: Power converter electrical parameters.
L r l
R [40 330] Ω F sw
T s
13 mH
200 ns P in
0.463 Ω C
10800 Hz V in
[600 4500] W d
1.1 mF
[200 325] V
[30 40] %
The same procedure is applied to the switching energy of the IGBT E i sw
. Lastly, the diode switching losses due to its reverse recovery current are assumed to be
39
Table 2.6: Thermal parameters.
R
1
C
1
R
2
C
2
IGBT 0.1568
0.0409
0.3814
0.2474
R
3
0.2318
Diode 0.2546
0.02519
0.6191
0.1524
R c-f
C c-f
R f-a
0.245
0.1405
0.175
0.3763
C f-a
T a
1333 [20 23 .
6] o C
C
3
3.7644
2.3189
Figure 2.17: Electro-thermal RT Power Loss Validation.
constant at P d sw
= 3 W.
The time steps of the boost converter, the power loss algorithm, and the thermal model are T s
= 200 ns.
40
2.4.3
Experimental and Real Time Results
Fig. 2.17 presents a comparison of the RT calculated power losses at different
input current against the actual power converter power losses. We can see that the algorithm closely matches the experimental values.
To test the validity of the electrical circuit, we perform a change in the input voltage from 300 V to 325 V. A comparison of the experimental and real time results
are shown in Fig. 2.18. It can be seen that the real time model follows well the
experimental results.
(a) Experimental resuls (b) Real Time results
Figure 2.18: Comparison of the experimental and real time results by a change in voltage input.
The next step involves validation of the thermal model. The input voltage is changed from V in
= 200 V (1.03 kW) to V in
= 325 V (2.73 kW). In this case, the losses in the IGBT and diode will contribute to rise in the heatsink temperature.
Validating this rise in temperature will also give an indication of the calculation of
41
the IGBT and diode power losses. The results of the experimental and real time
waveforms are shown in Fig. 2.19.
Lastly, in Fig. 2.20, a duty cycle step change is introduced to the system to show
the estimated junction temperatures of the IGBT and diode in real time.
Figure 2.19: Comparison of a change in power to validate algorithm and thermal model.
2.5
Real Time Simulation of an EV Drive
An FPGA model of a three phase induction machine with saturation is modeled
using procedure outlined in Section 2.3. The complete setup is shown in Fig. 2.21.
In order to control the EV drive, an external Wanda 3U target machine running a
Linux operating system is used [60].
42
Figure 2.20: Real time results showing the junction temperature of the power devices.
Figure 2.21: EV traction drive real time HIL example
43
2.5.1
Model and Saturation Characteristics
| i mc
| =
1 .
25 A, the inductances L m
( | i mc
| ) = L y
( | i mc
| ).
(a) System response to step input.
(b) HIL Closed loop speed control
Figure 2.22: HIL results of an EV drive system.
44
2.5.2
Real Time Results
The achieved time step of the EV drive without saturation is T unsat s
= 200 ns and with saturation is T sat s
= 650 ns. The results of the system are shown in Fig. 2.22.
One of the advantages of a fast simulation time step, is the possibility of running
the model at higher switching frequencies. In Fig. 2.22a, the inverter is switching at
F sw
= 24 kHz. In here, rated voltage is applied to the machine without closed loop control.
In Fig. 2.22b, vector control is used to regulate the machine speed and
(a) Mutual Flux (b) Dynamic and static Ind.
Figure 2.23: Saturation characteristics of the machine in Table 2.7.
Table 2.7: Real time EV drive simulation parameters.
R s
L ls
V dc
T unsat s
1 .
97 Ω
200 ns
R r
10 .
23 mH L lr
800 V F sw
T s sat
2 .
82 Ω
8 mH
J
[2 24] kHz P m
650 ns T s-cpu
0 .
11 kgm 2
0 .
598 mH
2
25 µ s
45
Figure 2.24: Results for the machine with input voltage fundamental frequency of f = 30 Hz.
flux. The details of the control can be found in [7]. The impact of the saturation
in the magnetizing flux to the controller is part of future study. The saturation of
the motor is illustrated in Fig. 2.24. It is possible to see that the starting electro-
mechanical torque for the saturated conditions has a lower magnitude than the one for the saturated one. Furthermore, the torque frequency and phase are not affected
by the saturation validating previous results in saturation modeling [12, 38, 17, 39].
2.6
Conclusion
A platform for modeling detailed power converters and electric machines geared towards EV applications was developed. A switch representation including the on
parameters was proposed in Section 2.2. Using this switch representation, an algo-
rithm for electro-thermal modeling of power converters was presented. In addition, this model was validated using a Powerex PS22A78-E IPM.
46
Furthermore, a mathematical derivation was presented to model the saturation and cross-saturation effects in an induction machine. An algorithm to compute L
− 1 was presented to decrease the simulation time step. Lastly, an example was presented to show a comparison of saturated and unsaturated machine conditions. Using the previously discussed methods, a case study of an EV drive using an induction machine was modeled in FPGA at T s sat
= 650 ns and T unsat s
= 200 ns. A comparison of the saturated and unsaturated model was presented showing the importance of modeling saturation.
47
Chapter 3: Stability Analysis of DC Networks
In this chapter, the negative impedance/Constant Power Load (CPL) behavior of power electronics based loads dc networks will be studied in detail. Several examples will be discussed at the beginning of this chapter to illustrate the types of conditions/loads that will behave as CPL. A method based on Semidefinite Programming
(SDP) is then proposed to compute the region of attraction of systems with multiple
CPLs. Lastly, a controller design based on a similar SDP is proposed to increase the robustness of dc networks.
3.1
Examples of CPLs
The study of CPLs has encompassed both ac and dc systems [4]. These types of
loads are regulated by a power converter in order to maintain/deliver constant power to the loads. One of the most common examples of CPL is a regulated electric motor, such as the drive system discussed in Chapter 2. Although it is true that this tightly regulated system behaves as a CPL, the instability problem arises due to the input filters/line parameters connected to the front of the load as will be seen in the next sections. Another type of CPL commonly seen in literature are dc loads with front
filters. These two cases are shown in Fig. 3.1.
48
(a) Example of a regulated electric motor.
(b) Example of regulated dc electronic loads.
(c) Example of back to back HVDC with one station as CPL.
Figure 3.1: Example of regulated loads/CPLs.
49
A more uncommon example arises in the back to back connection of dc converters,
such as for High Voltage DC Transmission (HVDC) shown in Fig. 3.1c. When one
of the converters regulates the power transfer between the two systems, the tightly regulated converter also exhibits negative impedance behavior. Stability issues can also arise due to the line parameters/input filters connected to the front of the inverter
In the subsequent chapters, the following notation will be used. A matrix H ∈
R n × n is said to be positive definite (denoted as H 0), if for any non-zero vector x ∈
R n it satisfies x T Hx > 0 . The set
R
+ is used to denote the set of all positive real numbers greater than 0. The set
N
= { 1 , 2 , ...
} contains the natural numbers.
Lastly, the symbol , is used to denote equality by definition.
3.2
Stability Analysis of DC Networks
3.2.1
Problem Formulation
In order to visualize the problem of negative impedance/constant power loads,
the microgrid shown in Fig. 1.1 is first studied with only 1 CPL and a dc source as
Figure 3.2: Simplification of a source and power converter load as a CPL.
50
Figure 3.3: System response and phase diagram of (3.1) based on two different initial
conditions.
shown in Fig. 3.2. The dynamic equations for this system are given by:
˙
1
= x
2
=
− r x
1
−
L
1
C x
1
−
1
C
1
L x
2
+
P x
2
1
L
V dc
(3.1) where x
1
= i
L and x
2
= v
C are the inductor current and capacitor voltage respectively and r , r dc
+ r
L
. In addition, it is possible to see the nonlinear term P/x
2
. In this study, the load power P is assumed constant.
It is possible to find a maximum value the load can demand by linearizing (3.1)
about the equilibrium point, x o
, ( x
10
, x
20
)
T
, to obtain:
P < min
V dc
,
4 r rCx
2
20
L
, P max
.
(3.2)
The first term is due to finding a real operating point and the second guarantees the
eigenvalues of the Jacobian matrix have negative real part [5]. Due to the nonlinear
51
Table 3.1: Circuit parameters and initial conditions for Fig. 3.2.
r 1 .
1 Ω L
C
P
500 µ F V dc
300 W x
0
Stable i.c.
(14 , − 30) Unstable i.c.
39.5 mH
200 V
(1 .
5 , 198 .
34)
(19 , − 30) nature of the load, a safe operation of these systems is not ensured by guaranteeing local stability. To observe this, a change of coordinates about the operating point, x o
, is performed as ˜ = x − x o
. Under the new coordinate system, (3.1) becomes:
x
1 x
2 r
= −
L x
1
=
1
C x
1
+
−
1
L x
2
P
Cx
20 x
2
˜
2
+ x
20
.
(3.3)
Fig. 3.3 shows the response of the system based on two different initial conditions
with the parameters shown in Table 3.1. It can be seen that only a difference in 5 A in
the initial conditions caused the system from converging to the equilibrium to going unstable. Therefore, it is very important for the applications considered (e.g. aircraft, ships, etc.) to compute a region of attraction based on the system parameters.
3.2.2
Region of Attraction Review for 1 CPL
The method presented in [53] to compute the ROA is discussed briefly in the
remainder of this section to show the importance of the ROA. The system in (3.1) is
formulated as a Takagi Sugeno (T-S) Fuzzy model. For a case with Q ∈
N
CPLs, the
T-S model will contain 2 Q Fuzzy rules and linear systems. Hence, for Q = 1, two linear systems defined as ˙ = A i x, i = 1 , 2 are obtained by approximating the nonlinearity x
2 x
2
∈ [ − x
2 N x
2 N
] , D
δ and
52
(a) Estimated ROA based on different capacitor values.
(b) Estimated ROA based on different resistor values.
(c) Estimated ROA based on different inductor values.
Figure 3.4: Estimated ROAs based on different passive component values deviating
from the nominal points in Table 3.1.
x
2 N
> 0 is a nominal value defining this set. The matrices are computed as:
A
1
=
−
− r
L
1
C
− 1
L f
˜
( x
2 N
)
!
, A
2
=
−
− r
L
1
C
− 1
L f
˜
( − x
2 N
)
!
.
where
(3.4) f
˜
(˜
2
) ,
P 1
Cx
20 x
2
+ x
20
(3.5)
53
Thus, the system with one CPL is stable in the region defined by D
δ if the following
LMI is feasible:
P 0
A
T i
P + P A i
≺ 0 for i = 1 , 2 (3.6)
To compute the ROA, we can iteratively increase D
δ by a small positive number δ > 0
until (3.6) is no longer feasible. This algorithm is described in Alg. 1. More details
on this method are shown in Appendix A.
The estimated domain of attraction is then contained in the set
D = x ∈
R n | | b
T x | < r = x
2 N − prev
, (3.7) where b = (0 , 1)
T
. Therefore, the largest level set of the Lyapunov equation V = x T P x contained in D can be computed by calculating min s.t.
| b T x | = r x
T
P x = r
2 b T P − 1 b
, a (3.8)
Algorithm 1 Algorithm to estimate ROA for Q
10:
11:
12:
13:
7:
8:
9:
4:
5:
6:
1:
2:
3: procedure
TSROA
( A i
, δ )
Set x
2 N − new
= δ
Set x
2 N − prev
Compute A i
= 0 for i = 1 , 2
while
do
Set x
2 N − prev
Enlarge D
δ
Re-compute
=
A i x by x
2 N − new
2 N − new
= x
2 N − prev for i = 1 , 2
+ δ end while
Estimated ROA is computed using x
2 N − prev end procedure
54
and the estimated ROA is Ω a
= x ∈
R
2 | x T P x < a .
As an example, ROAs based
on different capacitor, resistor, and inductor values are shown in Fig. 3.4. It can be
seen that a larger capacitance and resistance increases the robustness of the system whereas the inductance has an opposite effect.
However, there are several drawbacks of this method such as the need to iteratively increase the nonlinear region by a factor δ ∈
R
+ and the need to derive an exponential number of models based on the total CPLs. For a very large number of CPLs ( Q ), it is computational expensive to use this method since at each step the 2 Q matrices
( A i
) need to be re-computed as well as the 2 Q + 1 LMIs.
3.3
Proposed SDP Technique
3.3.1
Single CPL
To overcome the computational drawbacks of the previous methods, (3.1) is ap-
proached from a different perspective. The system in (3.3) is first re-written as a
Lur’e problem [47] of the form:
˙˜
( t ) = A ˜ ( t ) + Dh (˜ ) (3.9) where it is desired for the nonlinearity to satisfy a quadratic bound as: h
T
(˜ ) h (˜ ) ≤ α
2 x
T
H
T
H ˜ where α ∈
R
+
. The matrices A, D, and H 0 are defined as:
A =
− r
L
1
C
− 1
L
0
!
, D =
0
!
, H =
1
C
0 0
!
,
0 1 and the nonlinear term is: h (˜ ) =
P ˜
2 x
20
(˜
2
+ x
20
)
.
55
(3.10)
(3.11)
(3.12)
(a) CPL characteristics for P = 300 W, the red circle shows the operating point.
(b) Nonlinearity and computed bound in new coordinate system.
(c) Intersection points for different load powers.
Figure 3.5: Illustration on computing bound α
The goal is to guarantee asymptotic stability of the system whenever the inequality
(3.10) is satisfied for a certain
α > 0. This goal can be achieved if for an α > 0, it is possible to find a W 0 such that the following holds:
V (˜ ) = ˜
T
W ˜ 0
V
˙
(˜ ) = ˜
T
( A
T
W + W A )˜ + h
T
D
T
W x + ˜
T
W Dh < 0 whenever h
T
(˜ ) h x ) ≤ α
2 x
T
H
T
H ˜
(3.13)
(3.14)
56
Using LMIs, (3.13) and (3.14) can be re-written as:
W 0
A T W + W A W D
!
≺ 0 whenever
D T W 0
α 2 H T H 0
!
0 − I
0
(3.15)
(3.16)
The previous equations can be simplified further using s-procedure and Schur’s com-
Y 0
Y A
T
+
D
T
AY D
− I
Y H
0
T
≺ 0
HY 0 − γI
(3.17) where γ =
1
α 2 and Y = τ W
− 1 for some τ > 0 (s-procedure constant).
A Semidefinite Programming (SDP) problem can be formulated to maximize α to enlarge the region of attraction (equivalent to minimizing γ ). SDP is a specific class of convex optimization problems with linear cost function and LMI constraints
[9]. Therefore, using (3.17), the following SDP is setup to maximize
α and guarantee
asymptotic stability whenever (3.10) holds:
min γ subject to:
Y 0
AY + Y A
D T
HY
T D Y H T
− I 0
0 − γI
≺ 0 (3.18)
A more detailed derivation is presented in Appendix A. Problem (3.18) is derived using
a similar procedure as described in [69]. Using the CVX toolbox [9], a maximum
57
Figure 3.6: Simplified version of the dc microgrid shown in Fig. 1.1 with
Q CPLs.
α = 0 .
0138 was obtained for the parameters in Table 3.1. Note that for the case
with one CPL, the inequality in (3.10) implies
| h ( x ) | ≤ α | ˜
2
| . Thus, it is possible to compute the intersection of | h ( x ) | and α | ˜
2
| to calculate at which value of the capacitor voltage this inequality will be violated (the ROA can then be calculated using this value and W
from (3.18)). A similar ROA is obtained using this procedure
as the ones shown using the previous T-S Fuzzy method. This method and its results
for one CPL are outlined in Fig. 3.5. In addition, Fig. 3.5c shows how the intersection
point (and thus the ROA) decreases as the load power increases.
3.3.2
Multiple CPLs
The proposed method to compute the region of attraction will be expanded to include Q
CPLs and an energy storage connected to the source as shown in Fig. 3.6.
58
To model this large dynamic system, we can decouple it into Q + 1 subsystems. The
CPLs (1 , ..., Q ) are of the form: x j
= A j x j
+ d j h j
+ A js x s
(3.19) for j ∈ J
Q
, { 1 , 2 , ..., Q } and s , Q + 1 represents the source. The matrices
A j
, d j
, and A js are of the following form:
A j
=
− r j
L j
1
C j
− 1
L j
0
!
d j
=
0
1
C j
!
A js
=
0
1
L j
!
.
0 0
The source subsystem can be written as
(3.20) x s
= A s x s
+ b s
V dc
+ b es i es
+
Q
X
A cn x j j =1
(3.21) with matrices
A s
=
− r s
L s
1
C s
− 1
L s
0
!
, b s
=
1
L s
!
,
0
A cn
=
0 0
!
, b es
− 1
C s
0
=
0
− 1
C s
!
.
(3.22)
Assuming a coordinate change about an operating point as ˜ = x − x o and let the energy storage current i es be the control input, the dc microgrid is rewritten as a dynamic system of the form:
˙˜
= A ˜ + Dh + B es i es
+ B s
V dc
(3.23)
59
where the matrices A, D, B es
, and B s are as follows:
A =
A
1
· · · 0 A
1 s
.
..
. ..
..
.
..
.
0 · · · A
Q
A
Qs
A cn
· · · A cn
A s
, D =
d
1
· · · 0
.
..
. ..
..
.
0 · · · d
Q
0 · · · 0
,
B es
=
0
.
..
0
b es
, and B s
=
0
0
.
..
.
b s
(3.24)
The state variables are defined as x = ( x
1
, ..., x
Q
, x s
)
T and the nonlinear term h (˜ ) =
( h
1
, ..., h
Q
)
T where h j x j
) = x j, 20
P j x x j, 2 j, 2
+ x j, 20
)
, j ∈ J
Q
.
(3.25) x j, 2 is used to denote the second state (capacitor voltage) of the j th CPL sub-system and x j, 20 is the corresponding state operating point.
In this case, assume the nonlinear term must satisfy a quadratic bound of the form: h
T
( x ) h ( x ) =
Q
X h
T j h j j =1
≤ α
2 x
T
H
T
H ˜ for a diagonal matrix H 0. Next, consider two cases:
(3.26)
Robustness Index
The parameter α can be seen as a robustness index for the dc microgrid since the larger this number is, the more robust the system is to nonlinear disturbances. As a first case, a matrix H of the following form is considered:
H =
H
1
.
..
0
· · · 0 0
. ..
..
.
.
..
· · · H
Q
0
, H i
0 · · · 0 0
=
0 0
!
0 1
(3.27)
60
Table 3.2: Parameters for dc microgrid with Q = 2 CPLs.
r
1
L
1
C
1
P
1
0 .
8 Ω r
2
40 mH L
2
1 mF C
2
900 W P
2
0.42 Ω r
3
19.6 mH L
3
1.05 mF C
3
600 W V dc
0.4 Ω
17.3 mH
1.05 mF
200 V for i ∈ J
Q
. With the system in (3.23) and
H as shown in the previous equation, we can setup a problem to maximize α
similar to (3.18). This procedure is then applied
to test a case with Q = 2 CPLs (six states). The parameters of a dc microgrid with
two CPLs are shown in Table 3.2. Based on these values, the nominal robustness
index obtained using (3.18) is
α = 0 .
0224.
Algorithm 2 Algorithm to maximize Λ( a ) for arbitrary Q
9:
10:
11:
12:
13:
14:
5:
6:
7:
8:
2:
3:
4:
1: procedure
SDPCPL
( A , D , a , δ )
1 compute F = Φ
2
( a ) set H = F set a prev
= 0
while α ≥ 1 do a prev
= a a = a prev
+ δ re-compute F = Φ
1
2
( a ) set H = F
end while
ROA is given by largest set Π( r ) ⊂ Λ( a prev
) end procedure
61
Region of Attraction
The previous case can be used to compute a maximum bounding parameter α based on a given H . However, it is still important to compute the ROA even for a system with multiple CPLs. From this point on, we let x = ˜ for simplicity. Assuming x ∈
R n , consider a region of the form:
Λ( a ) = { x ∈
R n
| | x j, 2
| ≤ a, a > 0 j ∈ J
Q
} .
(3.28)
Next, assume there exists a diagonal matrix F 0 such that h
T
( x ) h ( x ) ≤ α
2 x
T
F
T
F x, (3.29) for all x ∈ Λ( a ) and a positive constant α ≥
1. If problem (3.18) is feasible with
H = F and α ≥ 1, then the estimated ROA can be computed as the largest set
Π( r ) = x ∈
R n | V ( x ) = x
T
Y
− 1 x ≤ r (3.30) contained in Λ( a
To compute F
, we can exploit the advantages of the nonlinear terms in (3.25)
which can be written of the form: h j
( x j
) = x j, 20
( x
P j j, 2
+ x j, 20
)
· x j, 2
, m j
( x ) x j, 2 for x j, 20
> 0 and j ∈ J
Q
. Thus,
(3.31) h
T
( x ) h ( x ) = x
T
M ( x ) x (3.32) where M ( x ) has diagonal terms consisting of the functions m
2 j
( x ). It is then possible to find a matrix Φ( x ) with diagonal entries φ 2 j
( x ) such that: x
T
M ( x ) x ≤ x
T
Φ( x ) x, ∀ x s.t.
φ j
( x ) > 0 , j ∈ J
Q
.
62
(3.33)
The matrix M ( x ) is of the following form:
M ( x ) =
M
1
.
..
· · · 0
. ..
..
.
0 · · · M
Q
0
..
.
0
, M j
=
0 · · · 0 0
0 0
0 m 2 j
( x )
!
(3.34) where the terms m j
( x
j = 1 , ..., Q . The matrix Φ( x ) is similar
Φ( x ) =
Φ
.
..
1
· · · 0
. ..
..
.
0 · · · Φ
Q
0
..
.
0
0 · · · 0 0
, Φ j
=
0 0
0 φ 2 j
( x )
!
(3.35) where the functions φ j
( x ) are chosen as:
φ j
( x ) =
P j x j, 20
( −| x j, 2
| + x j, 20
)
(3.36) for j = 1 , ..., Q .
Let T = { x ∈
R n | φ j
( x ) > 0 j ∈ J
Q
} . For a positive number a satisfying Λ( a ) ⊂
T , the matrix F = Φ
1
2
( a ) meets the desired requirement: h
T
( x ) h ( x ) ≤ x
T
F
T
F x ∀ x ∈ Λ( a ) .
(3.37)
An algorithm to enlarge the set Λ( a ) and compute the region of attraction is then shown in Alg. 2. An initial value of a should be chosen to satisfy α >
Fig. 3.7 shows several iterations of the proposed algorithm to expand Λ(
a ) based
on the parameters shown in Table 3.1 (one CPL). Furthermore, using this method a
similar region is computed as the ones obtained by calculating the intersection point
shown in Section 3.3.1 and the T-S method.
The advantage of the proposed formulation can be seen for computing the ROA for a dc microgrid multiple CPLs. Using this method only one SDP is needed to be
63
Figure 3.7: Plot showing the expansion of the set Λ( a ) using Alg. 2 for one CPL
using the nominal parameters in Table 3.1.
solved for each iteration, the number of LMIs do not increase exponentially with Q , and only the bounding matrix F is re-computed each iteration. Using Alg. 2 and
the values in Table 3.2 (system with two CPLs), a maximum constraint set Λ(22
.
6) was found for which the ROA can then be computed. Since this network is a six dimensional system, the ROA is illustrated by its intersection with different planes
as shown in Fig. 3.8 (without control).
3.4
Controller Design
3.4.1
Feedback Design
Energy storage can be used to increase the robustness of the system by mitigating the transients which occur due to temporary faults at the dc link or disturbances in
64
(a) Intersection for the estimated ROA with the plane given by x
2
= x
3
= (0 , 0) T .
(b) Intersection for the estimated ROA with the plane given by x
1
= x
3
= (0 , 0) T .
(c) Intersection for the estimated ROA with the plane given by x
1
= x
2
= (0 , 0)
T
.
Figure 3.8: Comparison of the ROAs for 2 CPLs using Alg. 2 with and without proposed controller.
the dc source voltage. Therefore, it is possible to increase the region of attraction using the proper choice of feedback. In this section, we consider the use of a linear controller to achieve this goal by a desired amount while minimizing the norm of the feedback gain.
65
Figure 3.9: Robust controller using energy storage to increase robustness margin.
The method to compute the ROA presented in the last section can be extended to calculate a feedback of the form i es
= K ˜ (using the energy storage input) to increase this region. For example, the nominal robustness index of the network shown in Table
α n
= 0 .
0224. Assuming that we would like to increase this index to α = ¯ n by optimally computing K , we can setup the following SDP problem
66
min bγ + b
L
κ
L
+ b
Y
κ
Y subject to
Y 0
AY + Y A
T
+ B es
L + L
T
B
T es
D
T
HY
− κ
L
I L
T
!
≺ 0 ,
L − I
Y
I κ
I
Y
D Y H
T
−
0
I
I
!
−
0
γI
0
≺ 0
γ −
1
¯ 2
< 0
(3.38) where the feedback matrix can be calculated as: K = LY
− 1 , the constants b, b
L
, and b
Y are used as weighting factors, and the variables κ
L
, κ
Y bound the norm of the feedback matrix as
|| K || ≤ || L || || Y
− 1 || <
√
κ
L
κ
Y
.
(3.39)
Comparing (3.38) to (3.18), the new LMIs in (3.38) can be derived from (3.39) as
follows
⇒
|| L || <
√
κ
L
, ⇔ L
T
L − κ
L
I ≺ 0
− κ
L
I L
T
!
≺ 0 (Schur’s complement)
L − I
(3.40) and
Y I
I κ
Y
I
!
⇒
0 ⇒ κ
|| Y
− 1 || < κ
Y
Y
I − Y
− 1
0
(3.41)
Hence (3.39) is satisfied [69]. The last term,
γ − 1
α 2
< 0, makes sure the desired robustness index, ¯ , can be achieved.
67
It is important to note that the system in (3.23) with the energy storage as an input
does not satisfy the matching conditions described in [69]. These conditions state that
if both the control input ( i es
) and the nonlinear disturbance ( h ( x )) enter the system through the same matrix, e.g.
B , then the system can be robustly stabilized with arbitrarily large index α by a (possibly large) proper choice of feedback K (Theorem
3.5 in [69]). In (3.23), the input and the disturbance enter through different matrices,
B es and D respectively, and Im D 6⊂ Im B es
(if Im D ⊂ Im B es
, then D = B es
C for some matrix C , and the matching conditions would be satisfied). Therefore, the system can only be improved by a limited amount. However, if energy storage or current
source inputs were to be considered in parallel to each CPL, as is the case in [50],
then the matching conditions could be met and the system can be robustly stabilized with arbitrarily large index α
using a simpler version of (3.38), where the norm of the
feedback matrix K is not to be bounded. Nevertheless, for the problem considered in this paper of using only one energy storage, the increase in robustness can be sufficient for even severe fault scenarios.
3.4.2
Case Study
The main structure of the controller is shown in Fig. 3.9 while its implementation
is shown in more detail in Fig. 3.10. Simulations were then conducted to increase
the robustness index of the microgrid in Table 3.2 to
α = 0 .
045 > α n
. The computed
K = (1 .
406 − 0 .
116 0 .
47 0 .
354 0 .
653 0 .
121) (3.42)
To demonstrate the advantages of this controller, assume the system is working with
the nominal parameters shown in Table 3.2 and introduce a fault at the dc link for
68
Figure 3.10: Detailed control diagram for Q = 2 CPLs.
2 ms. This fault causes the system to go unstable without any control, whereas
the response of the system with the proposed controller is shown in Fig. 3.11. It
can be seen that the system with control is able to return to its operating point after the fault. Since the robustness index was increased, the ROA for the system with the proposed controller is also increased. A comparison of the ROAs with and
without control is shown in Fig. 3.8. The solid red curves represent the ROA with
the proposed controller whereas the dashed blue curves shows the ROA without any control. It can be seen that a significant enlargement of the ROA was achieved by increasing the robustness index while at the same time minimizing the norm of the feedback matrix K (amount of current supplied by battery).
3.5
Summary
A new methodology to analyze the stability of dc microgrids composed of multiple
CPLs is proposed. The system was formulated as a Lur’e problem with quadratic
69
(a) Without control.
(b) With control.
Figure 3.11: Results of the closed loop system to a short circuit fault in the dc link at t = 0 .
05 s. The feedback gain K
was computed using (3.38) to increase the robustness
index (hence ROA) to α = 0 .
045.
bounds. A SDP problem was formulated to compute a robustness index and an algorithm using the same SDP problem was presented to calculate an estimate of the domain of attraction. One of the main advantages of this method is that its complexity does not increase exponentially with the number of CPLs. The use of energy storage was studied to mitigate temporary faults such as a short circuit in the dc link or disturbances in the dc source voltage. With a similar SDP formulation, it is shown how to compute the feedback matrix K to achieve a desired robustness index while minimizing the norm of K (proportional to the current supplied by the battery).
70
Chapter 4: High Voltage DC Networks
Multi-terminal High Voltage DC Transmission (MTDC) is used to reduce the power losses over multiple distant regions. However, it is difficult for typical three level inverters to reach the desired dc bus voltage levels, e.g.
> 500 kV, due to the large voltage stress on each switch and switching transients. For this reason,
Modular Multilevel Converters (MMC) have gained popularity in High Voltage DC
Transmission (HVDC) applications and are currently a promising solution. MMC based HVDC will be discussed in this chapter along with its challenges and proposed solutions.
4.1
Modular Multilevel Converters
4.1.1
Dynamic Equations
A typical MMC structure is shown in Fig. 4.1. This case represents a single phase
MMC converter. It contains 2 arms, and N sub-modules per arm. The arm inductors create a current state for which the active power can be computed as:
P dc
= V dc i pj
+ i nj
2
= V dc i circ j
, where i circ
, ( i pj
+ i nj
) / 2 is known as the circulating current and j = a, b, c .
(4.1)
71
Figure 4.1: MMC single phase structure.
The dynamics of the circulating current can be derived by performing KVL around
di circ j dt r o d
+ l o dt
( i pj
+ i nj
) = V dc
− ( v pj
+ v nj
)
= − r o l o i circ j
+
1
2 l o
V dc
−
1
2 l o
( v pj
+ v nj
) .
(4.2)
The output current can be obtained by using KVL around the top and bottom loops: di vj dt
=
− r o i vj l o
+
1 l o
( v nj
− v pj
) −
2 l o v vj
, (4.3) where the control inputs are the top and bottom arm voltages v pj and v nj
. To simplify the control and analysis, these inputs are chosen as: v v pj nj
=
=
V dc
2
V dc
− e j
− v circ j
+ e j
− v circ j
,
2
72
(4.4)
where e j
is the desired output voltage at the middle terminal. Therefore, (4.2) and
di circ j di dt vj dt
= − r o l o i circ j
= − r o l o i vj
+
+
2 l o e j
1 l o v circ j
− l
2 o v vj
.
(4.5)
(4.6)
The advantage of the previous equations is that each state can now independently be controlled by v circ j and e j
.
However, there exists a double fundamental frequency ripple in the arm voltages of the MMC caused by the capacitor charge and discharge in each sub-module. This voltage ripple introduces a disturbance in the circulating current which is not being
In the literature seen thus far, (4.5) has been used for all of the proposed circu-
lating current controllers. For this reason, a detailed analysis of the rejection of the double fundamental frequency (or even higher harmonics) disturbance has not been
performed. Therefore, to include this disturbance in (4.5), it is assumed the arm
voltages v pj
, v nj are not perfect and include a ripple term as follows:
˜ nj
= v nj
− w
1
˜ pj
= v pj
− w
1
.
(4.7)
Based on (4.7), the new equivalent circuit is shown in Fig. 4.2. By KVL around the
outer loop we obtain the following equation: di circ dt
= − r o l o i circ
1
−
2 l o
( v pj
+ v nj
) +
1
2 l o
V dc
+
1 l o w
1
.
Lastly, using (4.4) the new dynamic circulating current equation is derived:
di circ j dt
= − r o l o i circ j
+
1 l o v circ j
+
1 l o w
1
(4.8)
(4.9)
73
Figure 4.2: Equivalent MMC circuit including disturbance.
to include the disturbance. The disturbance term, w
1 does not affect the output cur-
rent dynamics in (4.3). If only the double fundamental frequency is to be considered
in the disturbance, this term can be written as: w
1
= α cos (2 ω f t + φ ) (4.10) where the amplitude α and phase φ are generally not known or measurable.
4.1.2
Controller Design
Using (4.9), a more complete analysis and controller design can be performed.
This system is then viewed from an output regulation / disturbance rejection [25]
perspective. Since (4.10) is sinusoidal with a double frequency component, it can
be assumed to be generated by an autonomous system (exosystem) of the following
74
form:
˙
1
!
= w
2
0
2 ω f
− 2
0
ω f
! w
1
!
w
2
⇔ ˙ = Sw.
(4.11)
Notice the initial condition for w (0) decides the amplitude and phase of the distur-
bance term in (4.10). For convenience, re-write (4.9) as:
˙ = Ax + Bu + P w, y = Cx (4.12) where A, B, C ∈
R
, P ∈
R
1 × 2 , S ∈
R
2 , x = i circ j and w = ( w
1
, w
2
)
T
.
The problem is to design a controller such that lim t →∞ y ( t ) = 0.
The main challenge arises from the fact the disturbance w is not measurable. Therefore, one of the conditions for the solvability of this problem is for the pair:
A P
!
, C 0 ⇔ A e
, C e
0 S
(4.13) to be detectable.
This implies the existence of an observer for ( x, w ) .
For example, assume that the disturbance w can be measured. Then a feedback law of the form:
(4.14) µ ( x, w ) = Kx + Γ w eliminates the disturbance if σ ( A + BK ) ⊂
C
− and B Γ + P = 0.
75
Figure 4.3: Equivalent model for circulating current control.
Full Order Observer
Based on the condition stated in the last section, we considered a dynamic controller of the form:
ξ
˙
0
!
ξ
˙
1
=
"
A P
!
0 S
−
G
0
!
G
1
+
B
!
u
0 u = K Γ
ξ
0
!
ξ
1
C 0
#
ξ
0
!
ξ
1
+
G
0
!
G
1 x
(4.15) or written in a more compact form as:
ξ
˙
= ( A e
− GC e
) ξ + Gx + B e u u = Hξ (4.16) where ξ ∈
R
3
. The gain G is then chosen such that σ ( A e
− GC e
) ⊂
C
− and K so that σ ( A + BK ) ⊂
C
−
. The union of these sets are the closed loop poles of the
76
system. Lastly, the feedback term Γ is calculated by using the regulator equations
[25] or simply canceling the disturbance based on the separation principle [26]:
B Γ + P = 0 (4.17) to ensure lim t →∞ y = 0 by rejecting the disturbance. The structure of this controller
Reduced Order Observer and Gain Design
The dimension of the controller can be further reduced since the variable x , estimated by ξ
0
, is already known/measured. Therefore, we can design a reduced order observer as follows:
ξ
˙˜
= S
˜
+ G x − ( Ax + P
˜
+ Bu )
| {z error
}
(4.18) to estimate w . In this case ˜ ∈
R
2
. To eliminate the derivative term in (4.18), a
simple change of variables v = ˜ − Gx gives us the desired form for the controller: v ˙ = ( S − GP ) v + ( SG − GP G − GA ) x − GBu where σ ( S − GP ) ⊂
C
−
. Lastly, the updated feedback law is of the form:
µ ( x,
˜
) = Kx ξ = ( K + Γ G ) x + Γ v
(4.19)
(4.20) where K and Γ satisfy the previous requirements. To optimize the observer gain G , notice that the error dynamics, ˜ ( t ) = w ( t ) − ξ
˜
( t ), are given by:
˙˜
( t ) = ( S − GP x ( t ) (4.21)
The gain matrix G can be chosen using different techniques such as Kalman gain, pole
placement, etc. [26]. A different method is to consider using LMIs as in the previous
77
chapter. In this case, assume that we would like to find a quadratic Lyapunov function of the form:
V (˜ ) = ˜
T
W ˜ ≥ ˜
T
(4.22) such that
V
˙
(˜ ( t )) = ˜
T
( S
T
W + W S − P
T
G
T
W − W GP )˜ ≤ − α ˜
T
W ˜ (4.23)
The existence of a matrix W and gain G satisfying the previous conditions for some
α ∈
R
> 0 implies the error decays exponentially with rate α , i.e.
|| ˜ ( t ) || 2 < ce
− αt for some positive constant c
[67]. This property is not hard to see after stating the
following fact: x
T
W ˜ ≤ λ max
( W x
T
(4.24) where λ max
( W ) is the largest eigenvalue of the matrix W . Thus:
V
˙
(˜ ( t )) ≤ − αV ( x ) ⇒ V (˜ ( t )) ≤ e
− αt
V (˜ (0))
⇒ ˜
T
˜ ≤ e
− αt
λ max
( W )˜
T
(0)˜ (0) = ce
− αt
(4.25)
(4.26) for c = λ max
( W )˜
T
(0)˜
. Placing conditions (4.22) and (4.23) in LMI form we obtain:
W I, S
T
W + W S − P
T
G
T
W − W GP − αW.
(4.27)
However, assuming that we would like to find G as well, the previous LMI is not linear since W is also unknown. To overcome this problem, let Y , W G . Lastly, a
SDP is formulated: x
T
(0) W ˜ (0)
W I
S
T
W + W S − P
T
Y
T
− Y P − αW
78
(4.28)
Figure 4.4: Reduced order observer circulating current suppression.
where the observer gain is computed G = W
− 1
Y .
The complete diagram of the circulating current suppression controller is shown
controller are presented in Fig. 4.5. It can be seen that the controller completely
Table 4.1: Parameters for simulation of three phase MMC.
V dc
V
LL l o
C sm
1000 kV S
3 φ
400 kV r o
0 .
15 pu N
1 mF f
600 MVA
0 .
02 pu
100
50 Hz
79
(a) Submodule capacitor voltages.
(b) Output voltage and current.
Figure 4.5: Implementation of reduced order observer with N = 100 SMs/arm. The controller is turned on at t = 1 .
1 s.
eliminates the double fundamental frequency oscillation and it does not affect the output current/voltage waveforms.
4.2
Efficient Modeling of MMC
One of the main challenges in the simulation of MMCs is the large number of
SMs N in each arm. This not only increases the size of the states, but also the total possible of matrix combinations to 2 2 N (single phase). Modeling of MMC for
simulation purposes can be defined into four different classes according to [68]:
1.
Detailed IGBT model: most accurate but with large computation time
2.
Equivalent circuit - Thevenin/Nodal analysis: (small R on and large R off
) accurate and no loss of information
3.
Arm switching function: submodule information is lost
80
4.
Average method: simple but major loss of information
Among these methods, the Equivalent circuit - Thevenin analysis is the most
widely adopted one due to its simplicity and accuracy [28]. In this section, we begin
by reviewing the Thevenin method and continue by discussing the proposed approach.
Figure 4.6: Submodule simplification for MMC based on [28].
4.2.1
Equivalent Circuit - Thevenin Method
This method simplifies each sub-module as shown in Fig. 4.6. The capacitor is
first discretized using an implicit form of integration and placed as a resistor and voltage source (Thevenin) of the form v c
( t ) = R c i c
( t ) + V ceq
( t − T s
) (4.29)
81
where T s is the simulation time step and the coefficients using trapezoidal integration are given as:
R c
,
V ceq
( t − T s
) ,
T s
2 C
T s
2 C i c
( t − T s
) + v c
( t − T s
) .
(4.30)
This first step corresponds to Fig. 4.6b. The Thevenin equivalent of a sub-module
can then be obtained as follows
V sm
( t ) = R sm i p
( t ) + V sm
( t − T s
) (4.31) where
R sm
,
V sm
( t − T s
) ,
R
2
( R
1
+ R c
)
R
1
+ R
2
+ R c
,
R
2
R
1
+ R
2
+ R c
V ceq
( t − T s
) , (4.32) and R
1
, R
2 are the switch resistances which vary depending on the switch status
(small on resistance, large off resistance). This representation is shown in Fig. 4.6c.
Since each sub-module is now a Thevenin circuit, for N sub-modules in an arm, the equivalent arm voltage becomes the sum of all the Thevenin resistances and voltages in series:
V arm
( t ) =
N
X
R sm − i
!
i p
( t ) +
N
X
V sm − i
( t − T s
) i =1 i =1
V arm
( t ) , R arm i p
( t ) + V arm
( t − T s
) .
(4.33) where i p
( t ) is the arm current. Lastly, the arm information is converted to a current source in parallel to an admittance (Norton equivalent form) used in EMTP such as:
I
N
( t ) = V arm
/R arm
, and R
N
= 1 /R arm
.
82
(4.34)
Figure 4.7: Proposed submodule simplification for real time simulation.
The main advantage of this method, is that it simplifies N sub-modules in each
arm into an equivalent Thevenin voltage and resistor (4.33). Furthermore, the
i h SM capacitor voltage can be computed by v c − i
( t
i = 1 , ..., N . However, one of the main disadvantages is that the switching status given by R
1 − i
(top) and
R
2 − i
(bottom) needs to be re-defined whenever a switch is on (small resistance) or off (large resistance). The entire system is then placed of the form:
A ( t ) x ( t + T s
) = y ( t ) ⇒ x ( t + T s
) = A
− 1
( t ) y ( t ) , (4.35) where the main MNA matrix A ( t ) needs to be re-computed and inverted during any switch change.
4.2.2
Proposed Method
The proposed method follows a similar logic as the previous one with the main difference that each switch is represented as a small inductor when on and a small
capacitor when off in Thevenin form as shown in Section 2.2.1 (see [30],[78]). This
83
method is illustrated in Fig. 4.7. Its main advantage is that each switch is represented
by a Thevenin network where the resistor is constant :
V sw
( t ) = i sw
( t ) R sw
+ E
T H
( t − T s
) where
E
T H
=
(
− R sw i ( t − T s
) if s ( t ) = 1
V sw
( t − T s
) if s ( t ) = 0
(4.36) and R sw is a constant chosen to minimize the error during transients. Backward
Euler is used as the integration method for the switches and the capacitors. Thus, the switch off capacitor and on inductor can be computed as:
C sw
=
T s
R sw and L sw
= R sw
T s
.
(4.37)
If R sw
= 1 as an example, then C sw
= L sw
= T s
. This introduces a transient behavior which could realistically represent parasitic elements in a submodule. However, to keep realistic values of C sw and L sw
, a very small T s
(ns range) is needed which can be seen as a drawback of this method.
To simplify the SM equation, a similar procedure is followed to compute the equivalent Thevenin resistance and voltage as follows:
V sm − i
( t ) = R sm − i i p
( t ) + V sm − i
( t − T s
) , where in this case
R sm − i
,
R
2 i
( R
1 i
+ R ci
)
R
1 i
+ R
2 i
+ R ci
is constant (in [28] it is recomputed whenever a switch changes) and
R
2 i
V sm − i
( t − T s
) , E
2 i
+
R
1 i
+ R
2 i
+ R ci
( V ci
+ E
1 i
− E
2 i
)
(4.38)
(4.39)
(4.40)
84
for i = 1 , ..., N . The arm equation is then written similar as the previous method
V arm
( t ) =
N
X
R sm − i
!
i p
( t ) +
N
X
V sm − i
( t − T s
) i =1 i =1
V arm
( t ) , R arm i p
( t ) + V arm
( t − T s
) (4.41) where R arm
V arm
using (4.40). If all the sub-modules
are identical to each other, then R arm
= N · R sm
. The arm Thevenin equivalent
representation is then placed into a Norton form similar to (4.34).
One of the main advantages of the proposed method is that the resistance R arm is constant and thus the main MNA matrix A is constant as well. This eliminates the need to re-compute or invert the MNA matrix during each switch change while inducing transients during the switching of the sub-modules. Again, this method requires a small time step T s in order to obtain realistic values of the switch capacitance/inductance.
A case study with a single phase MMC is presented in Fig. 4.8. the time step of
the simulation is T s
= 100 ns. The switch resistances are R sw1
= R sw2
= 0 .
48 (fixed).
The rest of the circuit parameters are V dc
= 140 V , N = 4 , r arm
= 1 mΩ , l arm
=
1 mH , C sm
= 10 mF , r load
= 0 .
9 Ω , l load
= 1 .
3 mH , and F sw
= 1 kHz. The results of
the typical modeling method are presented in Fig. 4.8a, whereas Fig. 4.8b shows the
results using the proposed method. As can be seen, the presented modeling method
achieves similar results as the typical one. Lastly, in Fig. 4.8c the output voltage
using the two methods are shown. It is possible to see from this figure the transients during the switching of the sub-modules due to the representation of the switches as a small inductor/capacitor.
85
(a) Output voltage and current using the
(b) Output voltage and current using proposed method.
(c) Zoomed-in output voltage comparison of two methods.
(d) Closer look at comparison of two methods.
Figure 4.8: Simulation results and comparison of the methods discussed in Section
4.3
Hardware-in-the-Loop Controller Validation
4.3.1
Platform Design
In this section, a Hardware in the Loop (HIL) validation of the reduced order observer will be presented using the simulation methods presented in the previous
86
Figure 4.9: Platform designed for the (real time) simulation of MMC based HVDC with large number of SMs/arm.
section. In order to simulate MMC with large number of SMs using either offline or
real time simulation, a platform has been designed using Matlab as shown in Fig. 4.9.
The steps in Fig. 4.9 are explained as follows:
1. Simpower Sys. Diagram: The desired model is drawn using Matlab’s Simpower
Systems toolbox
2. The netlist for the desired model is generated using the command: power_analyze(filename,‘net’); for the desired model
3. A custom code is then used to convert the netlist to MNA form, keeping track of the switches, inductors, capacitors, and inputs
4. The information is then passed on to a Simulink model where it is solved at
a fixed time step using either of the two methods discussed in Section 4.2 to
simplify the MMC arms
In order to visualize the advantages of the methods for simplifying the MMC arms consider a three phase MMC with N = 100 SMs/arm. The main MNA matrix with
87
(a) Regular MNA matrix.
(b) MNA matrix with efficient methods.
Figure 4.10: Comparison of MNA matrix size based on regular and efficient modeling methods.
× 1222 where as the one with either of the presented methods is 28 × 28.
Table 4.2: Single phase MMC for HIL validation of controller.
V dc l o
1000 kV S
1 φ
0 .
0025 pu N
200 MVA V
LL
63 C sm
400 kV r o
0 .
504 mF f
0 .
0025 pu
50 Hz
4.3.2
HIL Example
Using this developed platform, a single phase MMC was then modeled with the pa-
rameters shown in Table 4.2. A diagram of the setup is shown in Fig. 4.11 along with
88
a picture of the actual hardware implementation. In Fig. 4.11b the real time simula-
tor is an Opal RT Wanda 3U whereas the dsp is a floating point TI TMS320F28335.
(a) High level diagram of the HIL setup for circulating current controller.
(b) Hardware with the real time simulator and TI dsp controller.
Figure 4.11: HIL Validation of controller and efficient modeling methods.
89
The number of SMs/arm should be high in order to realistically represent an MMC with such high dc bus voltage. To decrease the amount of digital outputs required by the dsp to turn on each SM, a six bit binary number/arm was sent to the real time simulator in order to decide the total number of SMs to turn on. For this reason, the maximum number of SMs/arm is 2 6 − 1 = 63. Lastly, the capacitor balancing
algorithm, similar to [74], is performed inside the real time target.
Notice that the reduced order observer in (4.19) is continuous. In order to imple-
ment it in the dsp, the Zero Order Hold (ZOH) was calculated in Matlab first. The
code for the gain optimization and the dsp program are shown in Appendix B. The
results of the simulation are shown in Fig. 4.12. The controller is turned on at time
t = t
1 by creating a flag within Code Composer Studio. The top waveforms (blue and purple) show the first capacitor voltages of the top and bottom arm (there are a total of 126 capacitor voltages), the middle waveform (light blue) shows the circulating current, and the bottom trace (green) shows the ac output current. It can be seen the controller eliminates the double fundamental frequency oscillation without affecting the output currents. Some advantages of this implementation is that the controller does not require any Phase Lock Loop (PLL) to work and it cancels the double oscillation as long as the frequency is relatively close to 2 ω f
.
4.4
Real Time Simulation of MMC Based HVDC
The previous platform is expanded in order to efficiently simulate HVDC/MTDC systems for both offline/real time purposes. To speed up the simulation, the dc network is decoupled using the distributed parameters formulation for the transmission
90
(a)
(b)
Figure 4.12: HIL results for a single phase MMC with parameters in Table 4.2. Note:
the controller is turned on at t = t
1
.
91
line [18]. Each decoupled ac system can then be solved in parallel using different CPU
cores with the platform presented in the previous section.
(a) HVDC case study.
(b) Decoupled system taking into advatanges the transmission lines.
Figure 4.13: HVDC case study based on the Lower Churchill project [10] .
An HVDC case study is shown in Fig. 4.13 based on the Lower Churchill project
[10]. In Fig. 4.13b, it can be seen that Soldiers Pond and Salisbury are computed
in parallel. In this case study, Soldier’s Pond controls the dc bus voltage whereas
Salisbury controls the active power transferred between the two stations. Each station can control the reactive power in their ac areas as well. The main structure of the
output control is shown in Fig. 4.14. The output voltage variables,
e a
, e b
, e c
, are sent to the modulation modules of each MMC arm in order to compute the desired
number of capacitors to be on/off as shown in Fig. 4.11a. More details about the
92
Figure 4.14: Output controller in dq frame for Soldier’s Pond Station. Salisbury
Station is similar with V dc replaced by P and a reference computation instead of P I .
Table 4.3: HVDC case study simulation parameters (offline).
V dc l o
1000 kV S
3 φ
0 .
15 pu N
800 MVA V
LL
100 C sm
400 kV r o
1 .
9 mF f
0 .
02 pu
50 Hz
output controller can be found in [40, 48]. Lastly, each MMC station utilizes the
proposed circulating current control. The parameters for this simulation are shown
Two cases are studied with this simulation and are explained as follows:
• Case 1: Salisbury initially demands 400 MW. At t = 2 s, it increases its power demanded to 800 MW
• Case 2: Salisbury initially demands 400 MW. At t = 2 s, it reverses its power to supply 400 MW
93
The offline simulation results are shown in Figs. 4.15 and 4.16 for Case I and II
respectively. From Fig. 4.15b, the circulating current can be seen to increase very
quickly, as the active power control in Salisbury is very fast. On the other hand, in
Fig. 4.15a, the response of the dc bus voltage controller in Soldiers Pond is slower
which causes oscillation in the circulating current. The same behavior can be seen
in Fig. 4.16, where Salisbury reverses its power flow while Soldiers Pond tries to
maintain a constant dc bus voltage. In addition, the capacitor voltage at the first
SMs of the top and bottom arm (1 st and 101 st in phase A) are shown as well. It is possible to see any of the SMs capacitor voltages.
(a) Waveforms at Soldiers Pond.
(b) Waveforms at Salisbury.
Figure 4.15: Case study I: Salisbury increments its power from 400 MW to 800 MW.
The same case studies are implemented in real time using CPU based real time
simulation from Opal RT [60]. As shown in Fig. 4.13b, the models for Salisbury
and Soldiers Pond are implemented in two CPU cores, exchanging communication in
94
(a) Waveforms at Soldiers Pond.
(b) Waveforms at Salisbury.
Figure 4.16: Case study II: Salisbury reverses its power flow to − 400 MW.
parallel for efficiency. All of the parameters for the simulation are the same as Table
4.3, except the number of SMs per arm is decreased to
N = 50 in order to meet
timing requirements. The results for the two case studies are shown in Fig. 4.17. It
can be seen that the real time simulation waveforms are similar to those of the offline results.
4.5
Summary
The dynamic equations for a MMC are reformulated to include the double fundamental frequency disturbance term. This re-formulation allows the possibility of utilizing a reduced order observer for the elimination of the oscillation. In addition,
LMI and SDP techniques were presented to optimally compute the observer gain.
95
Furthermore, a method was proposed for the simulation of hundreds of SMs/arm inside a MMC which are typical in HVDC systems. A HIL validation of the proposed controller is discussed and the efficient real time simulation of an HVDC system based on the Lower Churchill project was presented.
96
(a) Case I real time results.
(b) Case II real time results.
Figure 4.17: Real time simulation results of HVDC systems.
L stands for Soldiers
Pond whereas R stands for Salisbury.
97
Chapter 5: Conclusion and Future Work
5.1
Summary
The main motivation of this work is to help in the safe adaptation of new technologies and the implementation of dc based networks. By studying the modeling/simulation (Chapter 2), stability analysis (Chapter 3), and a case study for High Voltage
DC system (Chapter 4), several contributions were made along the way towards this goal.
In Chapter 2, a review of typical modeling methods for the simulation of power electronics based networks was presented. A method was proposed to include the on characteristics of power devices without the need to re-compute the main MNA matrix. Taking advantage of this method, an algorithm is proposed and implemented in
FPGA for the electrical/thermal simulation of power circuits. In addition, a detailed model of an electric machine with saturation was developed in FPGA and a HIL case study was presented. This chapter provides a solid foundation for the stability and in-depth case studies in the following chapters.
In Chapter 3, the problem of negative impedance behavior in tightly regulated power converter loads was reviewed. It was shown how the two case studies presented in Chapter 2 behave as a CPL and create stability issues when coupled with
98
a front filter/line parameters. A method based on LMIs and SDPs was proposed to compute an index of robustness and calculate the region of attraction in dc microgrids with multiple CPLs. Using the same approach, a controller was then designed to optimally control energy storage in a system to increase the robustness without the need of increasing the filter capacitance/resistance or decreasing its inductance.
This controller also minimizes the norm of the feedback matrix which in turn reduces the energy supplied by the battery.
In Chapter 4, an in-depth case study for HVDC networks was presented including the typical converters (MMC) in these systems. A re-formulation of the MMC circulating current dynamics was discussed. This new equation is geared towards controller design. Therefore, a reduced order observer was proposed to eliminate the double fundamental frequency disturbance and a method based on SDP was introduced to optimize the gains. Furthermore, an algorithm for the efficient simulation of MMCs with large number of SMs/arm was proposed and a platform was developed for the offline/real time simulation of HVDC systems. The proposed controller was validated with a HIL case study with N = 63 SMs/arm implemented in a TI dsp. Lastly, a case study of an HVDC system was presented utilizing the proposed platform.
5.2
Future Work
With regards to the modeling/simulation aspect of this dissertation, future work aims at considering the temperature effects on the machine model to further complete a detailed model of an EV drive system. In addition, different parallelization strategies are desired for the efficient modeling/simulation of large dc/ac power systems in
99
FPGA. Although the distributed parameters formulation for transmission lines was used to decouple the HVDC system in Chapter 4, it is important to consider different techniques when a transmission line is not present.
The load power was considered constant in the stability analysis presented in
Chapter 3. Future work aims at considering the variation of the load power within a predefined set including the case of reverse/regenerative power (e.g. braking action in motor loads). In addition, decentralized stabilizing control strategies for systems with multiple generators/batteries and multiple CPLs are needed.
With regards to HVDC networks, as the size of the network increases (e.g. more stations are added) power sharing techniques need to be analyzed. Thus, decentralized control strategies are also of interest for the power sharing of these systems.
100
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108
Appendix A: LMI Methods in Stability Analysis
A.1
Mathematical Preliminaries
Notation
The following notation is used in this appendix:
R
,
R k ,
R m × n
R
C
N
>
C
−
S n
0
S n
+
S n
++
,
⊂ ( ⊃ )
M
M
M
T
0
M 0
M ≺ 0
σ ( M )
0
The real numbers, real k vectors, real
Real numbers greater than 0 m × n matrices
The complex numbers
The set of complex numbers with negative real part
The set of natural numbers, i.e.
{ 1 , 2 , ...
}
The set of real symmetric matrices of n × n dimension
The set of all positive semi-definite (symmetric) matrices
The set of all positive definite (symmetric) matrices defined as
Strict subset (superset) of
Transpose of a matrix M
M ∈
R n × n
M = M T is symmetric positive semi-definite and z T M z ≥ 0 ∀ z ∈
R n
The matrix M is symmetric and positive definite
The matrix M is symmetric and negative semi-definite
The matrix M is symmetric and negative definite
The set of eigenvalues of the matrix M
Fundamentals of Linear Matrix Inequalities
A set P is convex if ∀ x, y ∈ P implies αx + (1 − α ) y ∈ P ∀ α ∈ [0 1] .
109
A set P is a cone if λ ≥ 0 x ∈ P ⇒ λx ∈ P
Using the previous two definitions, we can say that the set S n
+
(positive semidefinite symmetric matrices) is a convex cone.
A Linear Matrix Inequality (LMI) has the form:
F ( x ) = F
0
+ m
X x i
F i i =1
0 (A.1) where x i
∈
R and the matrices F i are symmetric. The constraint set { x i
∈
R
| F ( x ) 0 } is then convex due to the convexity of S n
++ and being the inverse image of the affine mapping F ( x ).
LMIs can be used to find Lyapunov functions. For example, given the autonomous linear system ˙ = Ax , we would like to see if it is possible to find a matrix P 0 such that the following holds:
V ( x ) = x
T
P x > 0 ⇒ P 0
V
˙
( x ) =
∂V
∂x
˙ = x
T
A
T
P + P A x < 0 ⇒ A
T
P + P A ≺ 0
Therefore, we encounter an LMI where the variable to solve is P .
A useful tool used for simplifying and/or converting inequalities into LMIs is
Schur’s Complement :
A B
!
B T C
0 ⇔
(
A 0
C − B T A
− 1 B 0
⇔
(
C 0
A − BC
− 1 B T 0 and s-procedure : ∃ α
1
, ..., a k
≥ 0 so that
G
0
G
0
α
1
G
1
+ ...
+ α k
G k
0 whenever G
1 implies:
0 , ..., G k
0
110
A.2
Algorithms for Computing Region of Attraction
Fuzzy Algorithm
The first step in using T-S Fuzzy formulation for finding a Region of Attraction
(RoA), is to re-write the system as:
˙
1
!
= x
2
−
1
C r
L
− 1
L f ( x
2
)
! x
1
!
, where f ( x
2
) = x
2
P 1
Cx
20 x
2
+ x
20
We can then derive membership functions such that:
(A.2) z = f ( x
2
) = f max
M
1
( z ) + f min
M
2
( z ) , M
1
( z ) + M
2
( z ) = 1 (A.3) where f min and f max are computed based on x
2
∈ [ x
2min x
2max
]. This idea is depicted
Figure A.1: T-S Fuzzy formulation
Two rules are now implied:
IF z is in + THEN ˙ = A
1 x, where
IF z is in − THEN ˙
A
1
=
− r
1
L
C
− 1
L f max
!
,
111
A
2
=
− r
1
L
C
− 1
L f min
!
= A
2 x,
(A.4)
To check the stability of this system under arbitrary switching, a quadratic Lyapunov function is used to obtain the following LMI:
P 0
A
T
1
P + A
1
P ≺ 0
A
T
2
P + A
2
P ≺ 0
SDP Formulation
The derivation of (3.18) follows closely from [69] and can be obtained by consid-
ering the following Lur’e problem:
˙ = Ax + Dh ( x ) , h ( x )
T h ( x ) ≤ α
2 x
T
H
T
Hx and using a quadratic Lyapunov function of the form V ( x ) = x T P x . Thus, we want to find a P such that
V ( x ) = x
T
P x > 0
V
˙
( x ) = x
T
( A
T
P + P A ) x + h
T
D
T
P x + x
T
P Dh < 0 whenever h
T h ≤ α
2 x
T
H
T
Hx or in LMI form:
P 0
A T P + P A P D
!
≺ 0 whenever:
D T P 0
α 2 H T H 0
!
0 − I
0 using s-procedure (with constant τ ≥ 0) we can obtain:
P 0
A T P + P A + α 2 τ H T H P D
!
≺ 0
D T P − τ I
112
The challenge is the variables to solve, P, α, τ , make the previous equation non-linear in these variables. Therefore, we can use Schur’s complement again and multiply by
P
− 1 both sides to obtain:
Y 0
Y A T + AY + α 2 Y H T HY D
!
≺ 0
D T − I where Y = τ P
− 1 . Lastly, applying Schur’s complement once again:
Y 0
Y A
T
+
D
T
AY D
− I
Y H
0
T
≺ 0
HY 0 − γI where γ =
1
α 2
113
Appendix B: HIL Code
B.1
Gain Optimization
In order to optimize the gain of the observer G as well as compute the discrete
ZOH model, the following Matlab code is used. This code utilizes the CVX toolbox
clc;clear all;close all;
Ts = 20e-6;
%% Model definitions and initialization
% Parameter definition
R = 1e-3;
L = 1e-3; w = 2*pi*100; % Disturbance frequency
%%% -Model definition Plant- %%%
A = -R/L; B = 1/L; P = [1/L 0];
C = 1; nx = 1; mx = 1;
% dimension
%%% -Model Definition Exosystem- %%%
S = [0 w; -w 0];
Q = [0 0]; % No signal to track nw = 2; px = 1;
% dimension
%% - Controller Design - %%%
% Step 1 Observer Design
F = [S];
114
CQ = [P]; xe0 = [20 10]’; xe0t = xe0’; cvx_begin sdp variable Pmat(nw,nw) symmetric
% To obtain the gain variable Ymat(nw,px) minimize xe0t*Pmat*xe0
Pmat > eye(2);
F’*Pmat + Pmat*F - CQ’*Ymat’ - Ymat*CQ <= -10*Pmat cvx_end
G = inv(Pmat)*Ymat; % Gain matrix after optimization
%% Step 2 Controller design K (stabilization) cvx_begin sdp variable X(nx,nx) symmetric variable Y(mx,nx)
X > eye(nx);
X*A’ + A*X + Y’*B + B*Y < -2000*eye(nx); cvx_end
K = Y*inv(X);
A+B*K;
%% Step 3 Solve for the regulator equations
PImat = 0;
Gamma = -P/B;
Kef = [K+Gamma*G (Gamma)];
Fcl = F-G*CQ;
Gcl = S*G-G*P*G-G*A;
% Obtain the discrete versions of the plant, exosystem, controller exosys = ss(S,[0 0]’,eye(2),[0 0]’); plant = ss(A,[B P],C,0); observer2 = ss(Fcl,[Gcl -G*B],eye(2),[0 0; 0 0]’); exosysdis = c2d(exosys,Ts,’zoh’) plantdis = c2d(plant,Ts,’zoh’) observerdis2 = c2d(observer2,Ts,’zoh’); save(’circ_params.mat’,’Kef’,’observerdis2’);
115
B.2
DSP Implementation
The following code is the interrupt function for the implementation of the controller discussed in Chapter 4 using a TI dsp.
interrupt void adc_isr(void){
// Read the adc data first adcvar.signal1
= ((float)(AdcRegs.ADCRESULT0 >> 4)); adcvar.signal2
= ((float)(AdcRegs.ADCRESULT1 >> 4));
// Convert to actual signals currs.curra = ((adcvar.signal1)*ADCGAIN - 1.5297)*GAIN; currs.currb = ((adcvar.signal2)*ADCGAIN - 1.5)*GAIN;
// Filter the signal currafil.x0 = currs.curra; currbfil.x0 = currs.currb; filtersignal(&currafil); filtersignal(&currbfil); currs.curra = currafil.y0; currs.currb = currbfil.y0; switch(states){
/******************************************/ case START: // 1
// Initialization sequence if(cflags.start_entry == 0){ clearflags(&cflags); cflags.start_entry = 1;
PWMDISABLE;
GpioDataRegs.GPADAT.all = GpioDataRegs.GPADAT.all | (0x0000003F);
} break;
/******************************************/ case OPEN: // 2
// Initialization sequence if(cflags.open_entry == 0){ clearflags(&cflags); cflags.open_entry = 1; input.mi = 230e3*sqrt(2);//50;
}
116
// Regular code
// First update the angle theta omegaupdate(&omega, &ffreq); sincos(omega.thetanew, &omega.sintheta, &omega.costheta);
// Compute the modulation signals input.out = input.mi*omega.sintheta; mod.modtop = (mod.modadd - input.out)*mod.modmult; mod.modbot = (mod.modadd + input.out)*mod.modmult; mod.moddistop = mod.Nsm*mod.modtop; mod.moddisbot = mod.Nsm*mod.modbot;
// Discretize and round the value for the submodules mod.remtop = fmod(mod.moddistop,1); mod.rembot = fmod(mod.moddisbot,1); mod.distop = (Uint16)mod.moddistop; mod.disbot = (Uint16)mod.moddisbot; if(mod.remtop > 0.5001){ mod.distop += 1;
} if(mod.rembot > 0.5001){ mod.disbot += 1;
}
// Bound it up to Nsm (no need to bound below uint) if (mod.distop > (Uint16)mod.Nsm){ mod.distop = (Uint16)mod.Nsm;
} if (mod.disbot > (Uint16)mod.Nsm){ mod.disbot = (Uint16)mod.Nsm;
} mod.distop = ˜mod.distop; mod.disbot = ˜mod.disbot; mod.disbot &= 0x003F;
// Just for visualization purposes mod.moddistop = (float) mod.distop; mod.moddisbot = (float) mod.disbot;
// Shift the bits and combine together to set the register!
mod.distop = mod.distop << 6;
117
mod.total = (mod.distop) | (mod.disbot);
// Assign the inputs to the desired value!
GpioDataRegs.GPADAT.cus = mod.total; break;
/******************************************/ case CLOSED: // 3
// Initialization sequence if(cflags.closed_entry == 0){ clearflags(&cflags); cflags.closed_entry = 1; input.mi = 230e3*sqrt(2);//50;
}
// Regular code
// First update the angle theta omegaupdate(&omega, &ffreq); sincos(omega.thetanew, &omega.sintheta, &omega.costheta);
// Observer computation!
observ.x1_new = observ.a11*observ.x1_prev + observ.a12*observ.
x2_prev + observ.d1*currs.curra + observ.b1*input.circ; observ.x2_new = observ.a21*observ.x1_prev + observ.a22*observ.
x2_prev + observ.d2*currs.curra + observ.b2*input.circ; observ.x1_prev = observ.x1_new; observ.x2_prev = observ.x2_new; input.circ = (observ.k1*currs.curra + observ.k2*observ.x1_new)* observ.on;
// Compute the modulation signals input.out = input.mi*omega.sintheta; mod.modtop = (mod.modadd - input.out - input.circ)*mod.modmult; mod.modbot = (mod.modadd + input.out - input.circ)*mod.modmult; mod.moddistop = mod.Nsm*mod.modtop; mod.moddisbot = mod.Nsm*mod.modbot;
// Discretize and round the value for the submodules mod.remtop = fmod(mod.moddistop,1); mod.rembot = fmod(mod.moddisbot,1);
118
mod.distop = (Uint16)mod.moddistop; mod.disbot = (Uint16)mod.moddisbot; if(mod.remtop > 0.5){ mod.distop += 1;
} if(mod.rembot > 0.5){ mod.disbot += 1;
}
// Lastly bount it up to Nsm if (mod.distop > (Uint16)mod.Nsm){ mod.distop = (Uint16)mod.Nsm;
} if (mod.disbot > (Uint16)mod.Nsm){ mod.disbot = (Uint16)mod.Nsm;
} mod.distop = ˜mod.distop; mod.disbot = ˜mod.disbot; mod.disbot &= 0x003F;
// Just for visualization purposes mod.moddistop = (float) mod.distop; mod.moddisbot = (float) mod.disbot;
// Shift the bits and combine together to set the register!
mod.distop = mod.distop << 6; mod.total = (mod.distop) | (mod.disbot);
// Assign the inputs to the desired value!
GpioDataRegs.GPADAT.cus = mod.total; break;
/******************************************/ case FAULT: // 4
// Initialization sequence if(cflags.fault_entry == 0){ clearflags(&cflags); cflags.fault_entry = 1;
PWMDISABLE;
GpioDataRegs.GPADAT.all = GpioDataRegs.GPADAT.all | (0x0000003F);
}
119
// Regular code break;
/******************************************/ default:
// Initialization sequence
// Regular code states = FAULT; break;
}// end state machine
// Store variables to see them in a graph if(dlogs.sampcounter >= 1){ dlogs.sampcounter = 0; switch(dlogs.choose){ case 0: dlogs.dlogdummy = &currs.curra; break; case 1: dlogs.dlogdummy = &input.out; break; case 2: dlogs.dlogdummy = &mod.modtop; break; case 3: dlogs.dlogdummy = &mod.modbot; break; case 4: dlogs.dlogdummy = &mod.moddistop; break; default: dlogs.dlogdummy = &mod.moddisbot; break;
} storedlogs(&dlogs);
} else{ dlogs.sampcounter += 1;
}
/*EPwm1Regs.CMPA.half.CMPA = (Uint32)mod.modrega;
EPwm2Regs.CMPA.half.CMPA = (Uint32)mod.modregb;
EPwm3Regs.CMPA.half.CMPA = (Uint32)mod.modregc;*/
120
// Reinitialize for next ADC sequence
AdcRegs.ADCTRL2.bit.RST_SEQ1 = 1;
// Reset SEQ1
AdcRegs.ADCST.bit.INT_SEQ1_CLR = 1;
PieCtrlRegs.PIEACK.all = PIEACK_GROUP1;
// Clear INT SEQ1 bit
// Acknowledge interrupt to PIE
} // ** End main ISR
121
