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Electronic Theses, Treatises and Dissertations
The Graduate School
2014
Nonlinear Modeling of DC Constant Power
Loads with Frequency Domain Volterra
Kernels
Jesse Paul Leonard
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FLORIDA STATE UNIVERSITY
COLLEGE OF ENGINEERING
NONLINEAR MODELING OF DC CONSTANT POWER LOADS
WITH FREQUENCY DOMAIN VOLTERRA KERNELS
By
JESSE LEONARD
A Dissertation submitted to the
Department of Electrical and Computer Engineering
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Fall Semester, 2014
Jesse Leonard defended this dissertation on August 12, 2014.
The members of the supervisory committee were:
Chris S. Edrington
Professor Directing Dissertation
Juan Ordonez
University Representative
Hui Li
Committee Member
Rodney Roberts
Committee Member
The Graduate School has verified and approved the above-named committee members, and certifies
that the dissertation has been approved in accordance with university requirements.
ii
ACKNOWLEDGMENTS
I would like to thank my parents for pushing me to excel in my education since I can remember and
for all their support. I also thank Elizabeth for her support in finishing this multi-year project.
My advisor, Dr. Chris Edrington, has supported my studies since the final semesters of my
bachelor’s degree and helped with the idea to pursue the Volterra Series for power systems and
power electronics applications. I would also like to acknowledge my past and present colleagues
in the Energy Conversion and Integration group at CAPS. The kernel measurement experiments
would not have been possible without the years of team effort. The Florida State University Center
for Advanced Power Systems faculty and staff have been a huge help in learning new techniques for
research in modeling and simulation and Hardware-in-the-Loop (HIL) testing. I am also grateful to
my friends at GE’s Global Research Center in Niskayuna, NY.
iii
TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
1.1 Distributed Power Systems
1.2 Impedance Measurement . .
1.3 Motivation . . . . . . . . .
1.4 Contributions . . . . . . . .
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2 Linear Systems Theory for DC Constant Power
2.1 Resistive Loads . . . . . . . . . . . . . . . . . . .
2.2 Constant Power Loads . . . . . . . . . . . . . . .
2.3 Taylor Series and Negative Resistance . . . . . .
2.4 Inclusion of Input Filter and Stability Analysis .
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Loads
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ix
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1
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15
3 Impedance Measurement of Constant Power Loads
3.1 LCR Meters . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Series Voltage Injection . . . . . . . . . . . . . . . . . . .
3.3 Design of High Bandwidth DC Supply . . . . . . . . . . .
3.4 Experiment Automation with High BW DC Supply . . . .
3.5 Measurements of RLC Circuit . . . . . . . . . . . . . . . .
3.6 Measurements of DC Constant Power Load . . . . . . . .
3.7 Limitations of Linear Methods for Constant Power Loads
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16
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4 Nonlinear System Modeling with Volterra Series
4.1 Second Order Taylor Series Expansion . . . . . . .
4.2 Second Order Dynamic Response . . . . . . . . . .
4.3 Introduction to Volterra Series . . . . . . . . . . .
4.4 Time Domain Volterra Kernels . . . . . . . . . . .
4.5 Frequency Domain Volterra Kernels . . . . . . . .
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43
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48
49
5 Volterra Kernel Measurements of Constant Power Loads
5.1 Fundamentals of Frequency Domain Kernel Measurement .
5.2 Single Frequency Perturbation . . . . . . . . . . . . . . . .
5.3 Additional Experiment Automation Features . . . . . . . .
5.4 First Two Frequency Perturbation Test . . . . . . . . . . .
5.5 Use of Prime Numbers for Perturbation Frequencies . . . .
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51
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iv
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6 Discussion and Conclusion
6.1 Volterra Series and DC Constant Power Loads . . . . . . . . . . . . . . . . . . . . .
6.2 Interpretation of Second Order Kernel . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
63
64
64
Appendix
A Kernel Measurement Automation Supplemental
66
A.1 Automation Script in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B Scalability to Higher Power
69
B.1 Higher Power Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
v
LIST OF TABLES
3.1
H-bridge Inverter Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2
High Bandwidth DC Supply Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3
Measurements on LC Output Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4
RLC Circuit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5
NHR 9200/4960 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
vi
LIST OF FIGURES
2.1
Resistor circuit.
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6
2.2
Curves for circuit in Fig. 2.1 with R = 10 Ω. . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Resistor circuit with regulated power converter. . . . . . . . . . . . . . . . . . . . . . .
8
2.4
Curves for circuit in Fig. 2.3 with VR regulated to 12 VDC. . . . . . . . . . . . . . . .
8
2.5
Block diagram of SISO system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.6
I-V curve for Vicor 600 W Max Converter. . . . . . . . . . . . . . . . . . . . . . . . .
11
2.7
Equivalent circuit of Eq. 2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8
Simulation of Fig. 2.7 circuit approximation of 380 VDC, 600 W CPL.
2.9
Taylor Series approximations for 1 kVDC and 6 kVDC, 1 MW loads. . . . . . . . . . . 13
2.10
Bode plot of negative impedance model for 380 VDC, 600 W load. . . . . . . . . . . . 14
2.11
First order Volterra kernel for negative admittance model. . . . . . . . . . . . . . . . . 15
3.1
Two methods for impedance measurements of DC constant power loads. . . . . . . . . 17
3.2
Block diagram for series injection apparatus. . . . . . . . . . . . . . . . . . . . . . . . 19
3.3
Photographs of series injection apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4
Oscilloscope capture of test at 48 VDC bias and 10 V perturbation. . . . . . . . . . .
3.5
Circuit diagram for high bandwidth DC supply.
3.6
Hardware implementation of high bandwidth DC supply. . . . . . . . . . . . . . . . . 24
3.7
Testing of interleaved DSP PWM with Saleae Logic16. . . . . . . . . . . . . . . . . . . 24
3.8
Logic analyzer capture of interleaved PWM.
3.9
Oscilloscope capture of interleaved inductor currents, 50% duty, no load.
3.10
Oscilloscope capture of interleaved inductor currents, 50% duty, 2 kW load. . . . . . . 26
3.11
Input voltage to output voltage gain of LC filter. . . . . . . . . . . . . . . . . . . . . . 26
3.12
Oscilloscope capture of 5 Hz perturbation, 200 VDC bias, 2 kW on BK8522. . . . . . 29
3.13
Oscilloscope capture of 1 Hz perturbation, 380 VDC bias, 3 kW on NHR 9200. . . . . 30
vii
. . . . . . . . 12
21
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. . . . . . . . . . . . . . . . . . . . . . . 25
. . . . . . . 25
3.14
Photograph of NHR 9200, RLC circuit, and automation equipment. . . . . . . . . . . 33
3.15
Flow chart of experiment automation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.16
RLC circuit under test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.17
Samples from AC sweep for RLC circuit, time and frequency domain. . . . . . . . . . 37
3.18
Bode plot of AC sweep on RLC circuit showing impedance. . . . . . . . . . . . . . . . 38
3.19
Samples from AC sweep for DC CPL, time and frequency domain. . . . . . . . . . . .
3.20
Bode plot of admittance, Y , from AC sweep on NHR 9200 at 380 VDC, 3 kW. . . . . 42
4.1
I-V curve for Vicor 600 W Max Converter with second order approximation. . . . . . 45
4.2
Convergence problem for Taylor Series expansion for CPL curve. . . . . . . . . . . . . 46
4.3
Vicor converter power calculation for first and second order approximations. . . . . . 46
5.1
H1 (f ) measured with single frequency sweep. . . . . . . . . . . . . . . . . . . . . . . . 54
5.2
Magnitude and phase of H2 (f1 , f2 ) kernel in R3 showing f1 = f2 plane. . . . . . . . . 55
5.3
H2 (f1 , f2 ) in f1 = f2 plane measured with single frequency sweep. . . . . . . . . . . . 56
5.4
Perturbation frequency selection for first experiment.
5.5
Two frequency sweep measuring H2 (f1 , f2 ) kernel measurement, 50-500 Hz. . . . . . . 58
5.6
Prime number two frequency sweep for H2 (f1 , f2 ), 50-1000 Hz. . . . . . . . . . . . . .
5.7
Symmetrical H2 (f1 , f2 ) Volterra kernel, 50-1000 Hz. . . . . . . . . . . . . . . . . . . . 62
viii
41
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61
ABSTRACT
Power system designers have more creative flexibility than ever before due to improvements in
power electronics technology. The invention of the silicon (Si) insulated gate bipolar transistor
(IGBT) in the 1980s was a major improvement over commonly used Si MOSFETs, for higher power
applications, and thyristors, because they provided faster switching capabilities. New developments
in silicon carbide (SiC) semiconductors are causing a similarly disruptive effect to the industry
because of higher possible switching frequencies than Si IGBTs and the ability to create 10 kV
devices with switching frequencies beyond 20 kHz. Higher breakdown voltage and faster switching
enable converter designs with higher power densities (watts per cubic meter) that interface with
higher voltage systems. These two factors along with decreasing costs of Si IGBTs and low voltage
SiC MOSFETs make the increased use of power converters throughout a distributed power system
possible.
Power converters with a regulated output draw a constant input power from a distribution
system. While constant power loads have a nonlinear relationship between input voltage and load
current, linear systems theory historically dominated their analysis. The negative admittance model
is often used with input filter parameters to create linear models of constant power loads suitable
for small-signal stability analysis. However, systems with limited generation capacity and large
constant power loads are susceptible to large-signal instability. Therefore, system stability analysis
must include nonlinear models of system components to form an analytical, large-signal stability
metric. We used the Volterra Series to model nonlinear responses of constant power loads through
Volterra kernel measurement. A switch-mode power converter was designed to synthesize large-signal
perturbations to measure frequency domain Volterra kernels of 380 VDC loads up to 5 kW. We
measured the first and second order kernels of a 3 kW, 380 VDC constant power load from 0.1 Hz
to 1000 Hz and verified significance of the second order kernel.
ix
CHAPTER 1
INTRODUCTION
1.1
Distributed Power Systems
Distributed power systems connect spatially separate electrical power sources to electrical loads.
Most designs use more than one voltage level, usually lower voltages at the source and load ends
and higher voltages for long distance interconnects to reduce I 2 R losses in cabling. Before power
electronic converters were available, all voltage conversions were made using passive transformers.
Power electronic converters provide the freedom to scale voltage levels, change between AC and
DC, and produce regulated output voltages. Sensitive loads like microprocessors require tightly
regulated voltage supplies which can be achieved with an adjacent point-of-load converter.
The International Space Station (ISS) power system is very unique due its physical location
and the variety of electrical loads it supplies. During sunlight hours, solar panels provide power to
the loads and charge batteries that discharge during dark hours. The complete system was split
up into modules which were individually designed by the various countries involved in the project.
Interface requirements for the input and output ports of each module are based on source and load
impedance although this approach only provided calculations for small-signal stability; large-signal
stability was analyzed with computer simulations and hardware testing [1] [2]. The station has
maintained operation since its first components launched in 1998.
Datacenter power systems must be self-sufficient in the event of a power outage. Diesel generators
and fast responding uninterruptable power supplies (UPS) produce back-up power until grid power
is restored. The internal distribution system has protective features to prevent cascading failures if
an electrical fault occurs. Circuit breakers isolate faults and redundant electrical paths allow the
remaining components to stay online. Tier 4 datacenters are designed for 99.995 % availability, or
less than 0.4 hours of downtime per year and are able to recover from one worst-case event with no
impact to critical loads [3]. Modern datacenters consume megawatts of electrical power for servers
and cooling. Electrical component suppliers collaborate with the Emerge Alliance [4] to expand
a 380 VDC voltage standard to increase overall system efficiency from microchips to the grid by
1
reducing the number of power conversions and increasing efficiency at each stage with new power
electronics technology. Several demonstration systems have been constructed to collect experimental
results on efficiency gains and feasibility including Lawrence Berkeley National Laboratory [5], Duke
Energy [6], and many others [7]. Gearey and co-authors also outlined the commercially available
components necessary to create high reliability 380 VDC systems from hot plug connectors capable
of extinguishing DC arcs to DC circuit breakers.
Distributed power systems on naval vessels have high reliability requirements too. The Office
of Naval Research started the Electric Ship Research and Development Consortium (ESRDC) in
2002 to direct university research efforts toward new developments in shipboard power systems.
Consortium members contributed to IEEE Standards 1709-2010 with guidelines for medium-voltage
DC distribution systems in ship applications [8]. Research has focused on destroyer class ships:
DDG-51 with ship service gas turbines to produce 6 MW of electrical power and four LM2500
gas turbines for mechanical propulsion [9], USS Zumwalt (DDG-1000) with all 78 MW of gas
turbine capacity available to a 4160 VAC network to power electric propulsion motors and a high
survivability Integrated Fight Through Power (IFTP) system [9], and a hybrid electric drive system
for DDG-51 to add electric generators to the LM2500 gearbox to increase electric power capacity
[10]. New mission systems like higher power radar, lasers, and electromagnetic rail guns have high
input power slew rates which require energy storage buffering to interface with comparatively low
slew rate gas turbines. Shipboard distributed power systems must also be fault tolerant to survive
in the harsh and sometimes hostile sea environment. These distributed power systems maintain
operation even under large-signal disturbances including faults and component failures. The design
process to achieve this requires analysis with suitable models that predict system behavior during
such events.
1.2
Impedance Measurement
Impedance measurements on power system components are useful to test and tweak analytical
models thereby improving the validity of small-signal stability analysis prior to energizing a complete
system. These models are used with the Nyquist criteria to determine the stability of a system
described by Middlebrook [11]. Impedance measurement was initially restricted to DC systems, [2],
but Belkhayat at Purdue University [12] showed that AC systems can be analyzed with the use of
2
D-Q transformation and continued research on AC system impedance measurement in collaboration
with MS&T in Rolla, MO [13] and Virginia Tech [14] [15].
These measurements rely on the mathematics of linear systems theory to form impedance models
even though constant power loads are nonlinear systems. Harmonic currents produced by diode
rectifiers are one nonlinear response considered in impedance measurement literature [16] [17] but
all constant power loads, both DC and AC, have nonlinear responses that appear during impedance
measurement simply due to the constant input power characteristic. These nonlinear responses of
constant power loads are not discussed in the literature but can be excited by large-signal system
disturbances. We extend impedance measurement methods to analyze these nonlinear responses
with the accompanying nonlinear systems theory.
1.3
Motivation
Electrical loads in the above systems utilize power converters to interface with distributed
power systems. These converters produce regulated outputs to the respective sensitive loads
which in turn yield a constant power input from the distribution system. Impedance models,
developed analytically or from impedance measurements, can predict system stability to small-signal
perturbations. Response to large-signal perturbation are primarily analyzed by simulation since
linear models significantly deviate from real behavior in low voltage, fault conditions.
A new model is needed to model nonlinear responses of constant power loads in low input voltage
conditions which occur during faults. Nonlinear systems theory is necessary but should also allow
for experimental model verification and identification much like impedance measurement for linear
systems theory.
1.4
Contributions
We use the Volterra Series with frequency domain kernels to model nonlinear responses of
constant power loads and measure frequency domain kernels by modifying impedance measurement
methods to build upon previously developed equipment. The contributions are as follows:
(1) Explanation of harmonic and intermodulation responses to input voltage perturbations for
constant power loads using Volterra Series
3
(2) Converter design for Volterra kernel measurement of 380 VDC loads up to 5 kW, scalable
to higher power levels.
(3) Experimental results of Volterra kernel measurement for a 3 kW DC constant power load.
4
CHAPTER 2
LINEAR SYSTEMS THEORY FOR DC CONSTANT
POWER LOADS
A discussion of constant power loads, CPLs, where they appear in physical systems, and their
importance to modern power systems is beneficial due to the prevalence of power electronic converters
in everyday devices. Resistive loads are considered first along with familiar systems that can be
modeled this way. Next, power electronic loads are introduced and why they can be considered
constant power loads. Prior art in modeling the dynamics of constant power loads using linear
systems theory is presented along with impedance measurement techniques. Hardware apparatus is
presented to perform impedance measurement for DC power electronic loads. Limitations of linear
systems theory methods are also discussed especially for loads that have multiple operating points
such as a varying intensity light and also the case when input voltage has large perturbations from
ideal.
2.1
Resistive Loads
Traditionally, electrical loads are represented by resistors that dissipate heat, converting electrical
energy into thermal energy. First, consider the incandescent light bulb. When a voltage is applied to
the wire filament within the bulb, current begins to flow which heats the filament causing emission
of light. A more complicated model of the bulb includes the change in resistance as a function of
temperature but in general the bulb behaves like a resistor. If the input voltage decreases then
current through the filament will also decrease, dimming the bulb, and if input voltage increases
the current will increase causing the bulb to brighten. This effect can be seen during a brownout
condition, or voltage sag, on the electric utility causing a dim condition for incandescent bulbs. A
10 Ω resistor circuit shown in Fig. 2.1 has an I-V curve given in Fig. 2.2a with input voltage, Vin ,
as the independent variable and current, Iload , the dependent variable defined by Ohm’s Law in
Eq. 2.1. The slope of the line is the inverse of the resistance, in Ohms. This models the decrease
5
Iload
Vin
R
Figure 2.1: Resistor circuit.
in current for a low voltage condition and increased current for high voltage condition. Power
dissipated by the resistor is given in Eq. 2.2, the corresponding plot is shown in 2.2b.
Vin
R
(2.1)
2
Vin
2
= Iload
R
R
(2.2)
Iload =
Pload =
It is evident from Fig. 2.2b there is a drastic change in power dissipated by the resistor for
changes in input voltage. This is a very important concept when designing components that will be
supplied by unregulated sources with large deviations from nominal voltage. Heating elements and
incandescent bulb filaments can deform or melt if overheated. This is why it is important that the
electric utility maintain standards for voltage regulation.
Some other common resistive loads include toasters and toaster ovens, automatic coffee makers,
dryer heating elements, heating elements in water heaters, space heaters and stove top heating
elements. Some of these utilize Nichrome (Nickel-Chromium alloy) wires which heat up without
melting allowing heat transfer to warm up a room or cook food.
2.2
Constant Power Loads
What is meant by the term constant power load? In this context it refers to a load that can
maintain a constant input power regardless of the input voltage supplied. This contrasts the resistor
described above which has a load power that deviates from nominal when input voltage is perturbed.
Another way a load can be considered constant power is that it draws the same power all the
time i.e. it can be described time invariant. This definition can be used in conjunction with the
6
70
2.5
60
Load Power (W)
Load Current (A)
2
1.5
1
50
40
30
20
0.5
10
0
0
5
10
15
Input Voltage (V)
20
0
25
(a) Current vs. Input Voltage
0
5
10
15
Input Voltage (V)
20
25
(b) Power vs. Input Voltage
Figure 2.2: Curves for circuit in Fig. 2.1 with R = 10 Ω.
previous one to describe a load that draws a constant power at all times and is resilient to input
voltage perturbations. Consider a modern LED lamp as an example to contrast the two definitions.
An LED bulb has a power converter to regulate power to the bulb so it will draw a constant input
power even for perturbations on the input - when the light is on it draws rated power. Another
LED lamp has a dimmer knob so the internal power converter can increase or decrease light output
power. This second case has constant input power at each operating point (resilient to input voltage
perturbations) but may not meet the time invariant requirement due to dimming capability.
The discussion focuses on the load response to perturbations on the input voltage away from
nominal rather than specifically maintaining the same load power through time, although this aspect
will also be considered in Section 3.7.
Consider the same resistor discussed above but with an added specification that it must be
supplied by a constant voltage regardless of the system to which it is connected. In this case a
power converter can be added to the input stage that regulates the resistor voltage as is shown in
Fig. 2.3. Ideally this converter is 100% efficient and provides the resistor with a perfectly regulated
voltage, VR , for any input voltage, Vin .
Suppose this 10 Ω resistor is designed to be connected to a 12 VDC system in order to dissipate
14.4 W of power. The power converter must then provide the resistor with a regulated 12 VDC to
7
Iload
IR
Power
Converter VR
Vin
R
Figure 2.3: Resistor circuit with regulated power converter.
15
15.5
10
Load Power (W)
Input Current (A)
15
5
14.5
14
13.5
0
0
5
15
10
Input Voltage (V)
20
13
25
0
(a) Input Current vs. Input Voltage
5
10
15
Input Voltage (V)
20
25
(b) Resistor Power vs. Input Voltage
Figure 2.4: Curves for circuit in Fig. 2.3 with VR regulated to 12 VDC.
maintain output power regardless of deviations on the supply voltage, Vin . The load current, Iload ,
can therefore be calculated by Eq. 2.3 where Pnominal equals 14.4 W. Plots of the input current and
resistor dissipated power versus input voltage are given in Fig. 2.4a and 2.4b respectively.
Iload =
Pnominal
Vin
(2.3)
Some assumptions have already been made to create these plots. First is the 100% efficiency of
the power converter which is impossible - there will always be losses. Second is the possibility of
maintaining a constant power to the resistor even at an input voltage of zero volts, Fig. 2.4b. Real
power electronics converters either include a low voltage lockout feature which disables operation
8
Vin
System
Iload
Figure 2.5: Block diagram of SISO system.
below a certain input voltage or a protective fuse is included which protects the internal components
from over current destruction - Fig. 2.4a shows input current increasing as input voltage approaches
zero which would quickly destroy all the converter components depending on the oversizing design
specification. Also, power converters have finite bandwidth so quick changes on the input voltage
cause non-constant instantaneous input power but in the steady-state condition the curves in Fig.
2.4 are still valid.
Many common, household devices fall in this category of using power electronics to interface
sensitive electrical devices to sources without tight regulation. Laptop and desktop computer power
supplies, battery chargers that provide a regulated 5 VDC supply for phones, tablets, etc. all use
power electronics to provide regulated output voltage and can be considered constant power loads.
The prevalence of these devices raises the question of how these loads affect the source compared
to resistive loads and whether the overall system stability can be compromised. Answering these
questions requires mathematical models be developed and used in the design and analysis of new
and existing power systems. Extensive research in this area has already been performed using linear
systems theory. The mathematics used to model these load characteristics with linear systems
concepts is presented next.
2.3
Taylor Series and Negative Resistance
Reconsider the constant power load curve described by Eq. 2.3 and shown in Fig. 2.4a. This
represents a nonlinear system with input Vin and output Iload where Pnominal is considered a constant.
A block diagram of this single input single output (SISO) system is shown in Fig. 2.5. Suppose a
linear model of the power converter and resistor circuit, Fig. 2.3, is desired in order to combine it
with a linear model of the source to determine system performance. A Taylor Series can be used to
find the linear approximation for the constant power load at a nominal operating point.
9
y=
iload = Pnominal
∞
X
1
x
(2.4)
f n (vnom )
n=0
(vin − vnom )n
n!
(2.5)
Starting with the function in Eq. 2.4 the Taylor Series expansion about the nominal voltage,
vnom is given in Eq. 2.5 where f n (vnom ) denotes the nth derivative of the function evaluated at
vnom . Nominal power is moved in front of the summation for simplicity. This can be expanded
to calculate the first two terms of the series shown in Eq. 2.6. This is the linear approximation
through truncation of terms beyond n=1.
1
0
iload = Pnominal f 0 (vnom ) (vin −v0!nom ) + f 1 (vnom ) (vin −v1!nom ) + . . .
= Pnominal
≈ Pnominal
1
vnom
1
vnom
≈ 2 Pnominal
vnom −
−
−
1
2
vnom
(vin − vnom ) + · · · − · · · + . . .
vin
2
vnom
+
1
vnom
(2.6)
Pnominal
vin
2
vnom
The remaining two terms after linearization, Eq. 2.9, reveal the load current, iload , can be
approximated by summing twice the nominal current, Eq. 2.7, and a negative admittance term, Eq.
2.8, multiplied by the input voltage. This negative admittance term is simply the inverse of the
nominal resistance but has serious ramifications to system behavior as will be discussed in the next
section.
inominal =
Pnominal
vnom
(2.7)
Pnomianl
1
− =− 2
r
vnom
(2.8)
1
iload ≈ 2inominal − vin
r
(2.9)
A Vicor Maxi V37A48x600Bx is used as a case example. This 600 W converter is designed for
380 VDC nominal input voltage, outputs a regulated 48 VDC and offers a wide operating range
10
10
Nominal
CPL curve
Taylor Series n=0,1
iload (A)
8
Under Voltage
Lockout
6
Over Voltage
Lockout
4
2
0
0
50
100
150
200 250
Vin (V)
300
350
400
450
Figure 2.6: I-V curve for Vicor 600 W Max Converter.
from 250 VDC to 425 VDC as shown by the under voltage and over voltage lockout lines in Fig. 2.6.
A plot of the linear approximation, Eq. 2.9, is also shown. Visual inspection suggests the linearized
model approximates the constant power load curve well near the nominal operating point, especially
since this is far away from the knee of the curve. The linear approximation for this converter is also
well suited due to the under voltage and over voltage lockout functions. This means in the event
the input voltage has very large perturbations, outside the 250-425 VDC range, the converter will
turn off until the source has recovered. These lockout functions represent sharp discontinuities in
the I-V curve and are difficult to model without the use of piecewise models.
An equivalent circuit of Eq. 2.9 is shown in Fig. 2.7. This is used to show the time-domain
response of the circuit to perturbations on the input voltage. Simulation data from PLECS (Piecewise
Linear Electronics Circuit Simulator from Plexim GmbH) environment is shown in Fig. 2.8.
Fig. 2.8a shows a step change for input voltage from 380 VDC to 360 VDC. This decrease
in input voltage causes an increase in the load current but the resulting load power is less than
nominal. Recall the linear approximation of the CPL curve; the linear approximation will always
under estimate the load current if the input voltage is not exactly nominal, in which case the
approximation for load power matches. This is due to the CPL curve being convex for input voltage
greater than zero. Also notice that the step change is instantaneous due to the model having infinite
11
iload
A
V
-r
vin
2 i nominal
Figure 2.7: Equivalent circuit of Eq. 2.9.
400
Vin (V)
Vin (V)
400
380
360
0.05
0.15
360
0
1.7
0.2
1.6
1.5
0
0.05
0.1
0.15
1.5
0
600
598
0
0.05
0.1
Time (s)
0.15
0.1
0.15
0.2
0.05
0.1
0.15
0.2
0.05
0.1
Time (s)
0.15
0.2
600
599.5
0
0.2
0.05
1.6
0.2
Pload (W)
Pload (W)
0.1
Iload (A)
Iload (A)
0
1.7
380
(a) Input voltage step perturbation
(b) Input voltage sine perturbation
Figure 2.8: Simulation of Fig. 2.7 circuit approximation of 380 VDC, 600 W CPL.
bandwidth, no input filters or models of finite bandwidth feedback controllers that exist on the real
converter.
This behavior is also evident in Fig. 2.8b where a 10 V peak sinusoid perturbation is added to a
380 VDC bias. Notice the 180 degree phase shift between the load current and input voltage. This
example also shows the load power never rising above rated even with a higher than nominal input
voltage which has the same cause as mentioned before.
It was briefly mentioned that the linear model approximates the constant power load curve well
when the nominal operating point is far from the knee of the curve. Practically this means that
the constant power load must have nominal current and voltage that aren’t close to each other.
12
7000
7000
CPL curve
Nominal - 1 kV
Taylor Series n=0,1
6000
5000
I load(A)
I load(A)
5000
4000
3000
4000
3000
2000
2000
1000
1000
0
0
CPL curve
Nominal - 6 kV
Taylor Series n=0,1
6000
1000
2000
3000 4000
Vin (V)
5000
6000
0
0
7000
(a) 1000 VDC, 1 MW Load
1000
2000
3000 4000
Vin (V)
5000
6000
7000
(b) 6000 VDC, 1 MW Load
Figure 2.9: Taylor Series approximations for 1 kVDC and 6 kVDC, 1 MW loads.
Comparison of two 1 MW loads that have different nominal DC input voltage is shown in Fig. 2.9.
It is apparent the linear approximation is better for a 1 MW load connected to a 6 kVDC system as
opposed to a 1 MW load connected to a 1 kVDC system. Further discussion of this problem will be
considered in later sections.
The terms negative admittance, negative resistance and negative impedance have been used
several times already without a thorough explanation. There is no physical resistance with a
negative value. This is simply a result of the mathematics for approximating a constant power load
- the process of deriving the negative admittance term, useful for input to output characteristics
of systems like Fig. 2.5, in the Taylor Series expansion and the corresponding negative resistance
useful for equivalent circuits like Fig. 2.7. There are two other forms of the negative admittance, or
impedance, that need to be presented before moving on.
The first is a frequency domain representation. A small-signal approximation is made to Eq. 2.9
to fit into a frequency domain expression. This makes the constant current source open circuit,zeroing
out the constant term of Eq. 2.9. The result is given in Eq. 2.10 where s denotes the Laplace
Transform variable. A Bode plot of this transfer function for the 600 W regulated converter load
is shown in Fig. 2.10. The phase plot of the impedance makes it easy to see why the current and
voltage are 180 degrees out of phase for sinusoidal perturbations like the one in Fig. 2.8b.
13
242
Z (abs)
241.5
241
240.5
240
Phase (deg)
239.5
181
180.5
180
179.5
179
100
Frequency (Hz)
10−1
101
102
Figure 2.10: Bode plot of negative impedance model for 380 VDC, 600 W load.
z(s) =
vin (s)
= −r
iload (s)
(2.10)
The second is the time-domain impulse response model used in the convolution integral for LTI
systems shown in Eq. 2.11 with impulse response h(τ ) given in Eq. 2.12, plotted in Fig. 2.11.
iload =
Z ∞
0
h(τ )vin (t − τ )dτ
1
h(τ ) = − δ(τ )
r
(2.11)
(2.12)
It should be noted that both of these are small-signal approximations which eliminate the
constant current source term, the zero order term from the Taylor Series expansion. If the zero order
term is not included in the time-domain simulation circuit the load will appear to be supplying
power to the source since current supplied to the load will be negative. The constant current source
was included for the time-domain simulations shown in Fig. 2.8a and 2.8b. The zero order Taylor
Series term can simply be added to the convolution integral, Eq. 2.13 but the transfer function
requires additional manipulation. Eq. 2.13 will be revisited in Chapter 4.
iload = 2inominal +
Z ∞
0
14
h(τ )vin (t − τ )dτ
(2.13)
h(τ)
τ
( 1r )
Figure 2.11: First order Volterra kernel for negative admittance model.
2.4
Inclusion of Input Filter and Stability Analysis
Input filters must accompany power converters due to the switch-mode operation of semiconductor
devices used. Analysis of input filters and their effects on power converters was the focus of
Middlebrook’s 1976 paper [11], presented at the IEEE Industrial Applications Society meeting. This
paper is referenced by the majority of modern publications that deal with impedance measurement,
input filter design, and stability analysis of distributed power systems that include power electronics.
Although it is unavailable in the online IEEE Xplore database, a scanned copy is available through
Washington State University Pullman interlibrary loan.
As stated by Middlebrook, the primary functions of the input filter is to reduce current ripple
drawn from a distribution system although the input filter alters the transfer function of the
converter and should therefore be designed simultaneously with the converter control loop if possible.
The major points of the paper revolve around different input filter networks that have low Q-factors
at resonant frequencies but also have high overall power efficiency. This is a difficult design task
since damping elements that lower Q-factor introduce additional losses to the filter.
Increasingly elaborate input filter designs that lower Q-factor are described by Middlebrook and
create more complicated Bode plots for input port impedance. In order to verify the component values
along with any additional parasitics impedance measurement is beneficial for system identification
of unknown load as well as model verification to analyze system stability prior to installation.
Impedance measurement for DC constant power loads is discussed in the next chapter.
15
CHAPTER 3
IMPEDANCE MEASUREMENT OF CONSTANT
POWER LOADS
The previous section motivates the necessity of knowing the impedance looking into the terminals
of a load to have an analytical expression for stability. Just like any engineered system, if the load
design process is known start to finish then a model can likely be made based on control loop
parameters, input filter parameters, and load power. However, many times a converter is purchased,
rather than designed from the ground up, to interface with load to speed up the design process
without all the internal component details and parameters available. Or a model needs to be verified
by testing a hardware implementation. These examples lead to a system identification approach
and in this case is referred to as impedance measurement.
Impedance measurement was instrumental in debugging problems with first implementations of
DC distributed power systems for the international space station. This system had some unique
requirements to accommodate several voltage standards for each country’s equipment as well as
finding ways of connecting them all together for increased redundancy. Research for impedance
measurement techniques has followed trends in distributed power system design from PCB level
distributed power systems such as motherboards with regulated point-of-load (POL) converters,
to more spatially distributed systems. [2] More recently a trend toward impedance measurements
for AC power systems in the D-Q coordinate frame has risen for several reasons. [12] [13] [14]
[15] The first is the push for smart-grid concepts in conventional AC systems. The second is
new semiconductor technology allowing for impedance measurement of higher AC voltage systems,
such as 10 kV SiC MOSFETS [18] to perform impedances measurements on 4.16 kV AC systems.
Renewed interest in DC distribution for 380 VDC in data centers and telecommunications [7] [4],
medium voltage DC (MVDC) for shipboard [8] and land-based power systems, and multi-terminal
high voltage DC (HVDC) may show a trend in research focus back to DC, or at least an equal
distribution between MVAC and MVDC papers. Impedance measurement for HVDC will lag due
to the measurement equipment requirement to handle the large voltage isolation requirements.
16
Iload
PIU
DC
Supply
Vin
Constant
Power
Load
(a) Series voltage injection method.
Iload
High BW
DC
Supply
Vin
Constant
Power
Load
(b) High bandwidth DC supply method.
Figure 3.1: Two methods for impedance measurements of DC constant power loads.
At the core, impedance measurement for DC constant power loads provides an input voltage Vin
with DC bias at the nominal DC voltage as well as a superimposed perturbation signal in order to
measure the gain and phase of the load response found in the Iload signal. The gain and phase data
can then be placed into a Bode plot to visually analyze the load frequency response followed by
Nyquist contour analysis or system identification and model verification to analyze each component
parameter such as ESR of an electrolytic capacitor, ESR of an inductor, inductance and capacitance.
Implementing this DC bias with superimposed perturbation is not trivial and several considerations
must be made prior to performing tests.
A series voltage injection is the most common method for performing impedance measurements
of loads, Fig. 3.1a for both DC [19] and AC systems [14] [15]. Commercial products for DC systems
are available from Venable [20]. This allows the bulk power consumed by the load to be provided
by common power supplies with a fixed voltage at low cost and can be repurposed for other lab
tests. The high bandwidth components specific to impedance measurement only have to be rated
for the line current which can reduce total cost.
The series perturbation injection unit must be isolated since the internal voltage of the perturbation injection unit is raised to the nominal DC voltage potential above the negative rail of the
17
main DC supply. The Venable equipment and others have made these considerations to provide
safe testing scenarios for devices and operators as well as compensating for ground loops causing
erroneous measurements.
An alternative method is possible using a high effective switching frequency switch-mode
converter which provides both the DC bias and the superimposed AC perturbation, Fig. 3.1b. This
requires that the converter components be rated to supply the load with its rated power along
with perturbations. At first glance this may sound like a more expensive alternative since the
requirements of the converter include both the bulk DC power as well as high bandwidth capabilities.
However, common three phase AC converter topologies can be repurposed with minimal modification
which reduces this perceived cost. There were a few complications with the series voltage injection
as well which solidified the decision to use the high bandwidth supply route, Fig. 3.1b.
3.1
LCR Meters
One may ask why a simple LCR meter cannot be used for this process. LCR meters, such as an
Agilent U1733C handheld LCR meter [21], are designed to measure passive devices and specifically
mention in the users manual to discharge any stored energy in components prior to testing. Even if
this were not the case, the internal components are not rated for the line currents of high power
loads. Also, low cost LCR meters like the U1733C only perturb the device at a few frequencies and
a more thorough frequency sweep is necessary here.
3.2
Series Voltage Injection
A prototype for series voltage injection was constructed by repurposing available lab components
due to limited budget for specialized hardware. This is shown in Fig. 3.2 and 3.3. An NHResearch
9200 cabinet with 4960 module provides a 380 VDC output while an H-bridge inverter, fed by
a Magna Power Electronics DC supply, is used for series perturbation injection. Control for the
H-bridge inverter was implemented on a dSPACE 1103 using the PWM module of the internal slave
DSP - this made a 50 kHz switching frequency possible. A BK Precision 8522 DC electronic load
was used in constant power mode as the device under test. Further details of the H-bridge inverter
are given in Table 3.1.
18
208 V
three-phase
wall outlet
Magna Power
Electronics XR +
Series II
200V/20 A DC
Supply
(2) 380uF
Film
capacitors
150 uH
2uF
IGBT modules with
CONCEPT Scale 2 Drivers
High slew rate
option
Isolated output
up to
1 kVDC
+
20 Vpk
sine
-
Powerex CM200DX-24S
-
Sinusoidal modulation index, open loop
Conductance
Loop
208 V
three-phase
wall outlet
+
NHResearch
9200-4960
600 V/40 A
DC Supply
+
chassis
gnd 380 VDC
-
Figure 3.2: Block diagram for series injection apparatus.
+
+
-
-
20 Vpk
sine
+
+
380 VDC
+
-
-
chassis ground
Figure 3.3: Photographs of series injection apparatus.
19
+ BK Precision
8522 DC
Electronics
Load 500 V/
120 A/
2.4 kW
-
Table 3.1: H-bridge Inverter Parameters
Component
IGBT half-bridge
IGBT driver
CDC
Lf
Cf
Controller
fsw
finterrupt
Value
1200 V, 200 A
SCALE-2
380 µF
150 µH
2 µF
dSPACE
50 kHz
20 kHz
Part Number
Powerex CM200DX-24S
CONCEPT 2SP0115T
Kemet C4CDEFPQ6380A8TK
Magnetics Inc. C058906A2
CDE SCD205K122A3Z25-F
DS1103
Several problems arose during testing. Most importantly was the shutdown procedures causing
over-voltage trips on the Magna Power Electronics power supply due to its 200 VDC rating versus
the 380 VDC being supplied by the NHR 9200. This could likely have been prevented by placing a
blocking diode between the Magna Power Electronics supply and the H-bridge DC link capacitors.
Another issue was the unacceptable voltage waveform at the BK Precision 8522 terminals as
shown in Fig. 3.4. Voltage across the H-bridge inverter output is shown in the middle of the screen,
20 V/division, approximately 10 V peak sinusoidal perturbation. Voltage across the load is also
centered in the middle of the screen but with 50 V/division. There are a few possible reasons for
the apparent noise in this waveform. The conduction path was not originally considered to avoid
stray inductance; the large conduction loop is emphasized in Fig. 3.2. Another cause was likely the
slow internal constant power control loop of the BK 8522 load, unknown at the time, but showed
up again in later tests using a different perturbation injection method, Section 3.6.
The last and possibly most important issue with this perturbation implementation was the automation capability of the controller, the dSPACE 1103. There are automation scripting capabilities
within dSPACE ControlDesk but effort in this direction did not benefit the rest of the research
group working with TI TMS320F28335 DSP controllers. The H-bridge shown in Fig. 3.3 was part of
a three phase inverter designed by the group for use with the TI DSP. Rather than pursue scripting
in dSPACE, this method was abandoned in favor of the TI DSP. This allow efforts to cross over
into other group tasks utilizing the DSP, especially in the communication capabilities development
for sending and receiving data from the DSP. The problems with the series perturbation injection
led to its abandonment in favor of a high bandwidth DC supply design described in Section 3.3.
20
Load current
Load voltage
Inverter
output
voltage
Power at
load (V*I)
Figure 3.4: Oscilloscope capture of test at 48 VDC bias and 10 V perturbation.
3.3
Design of High Bandwidth DC Supply
The design criteria for the high bandwidth DC supply consisted of several diverse factors. The
end goal was to perform Volterra kernel measurements for DC constant power loads, described in
Chapters 4 and 5, but many of the requirements for DC load impedance measurements overlap.
Several previous impedance measurement procedures call for perturbations from 0.1 Hz to a few
kHz [14] [15] [2] although there have been measurement devices in the MHz range for lower power
loads with high switching frequencies where measurements in this range are useful.[19] [20] The
second design goal was to utilize as much preexisting equipment as possible with minimal and
non-permanent modification. A readily available back-to-back arrangement of two two-level three
phase bridge IGBT converters was used as a starting point. The next requirement was the ability
to make tests on 380 VDC equipment to complement the 380 VDC standard. [4] Lastly was the
ease of experiment automation. This was difficult to plan for as there are a multitude of ways to
write a script to automate the necessary sequence of events but some are much easier than others.
A suitable approach was discovered with minimal cost and also allowed for cross development of
code to other concurrent projects.
21
As mentioned above, impedance measurement for DC loads requires the load be provided with
an input voltage signal with DC bias at the nominal input voltage as well as a superimposed
perturbation for response measurement. Since the goal here was simply measurements for loads it
was unnecessary to include flexibility to perform both series voltage injection and shunt current
injection (useful for measuring source impedance) with the same equipment. This simplified the
design requirements and after prior trouble with series voltage injection, a single source approach
was taken to synthesize both the DC bias and the perturbation. A single source eliminates the
extra requirements of a secondary isolated series perturbation injection and makes for simple
interconnections to reduce stray inductance between the measurement equipment and the load
under test.
The previously developed back-to-back two-level converter used in [22] [23] was modified to
the topology in Fig. 3.5 to meet these requirements, photographs of the original back-to-back
configuration are shown in Fig. 3.6. Each converter uses the same three phase bridge building
block made up of three half-bridge IGBT modules, heatsink, DC link capacitors and DC bus bar
PCB described in Table 3.2. A three-phase interleaved DC-DC buck converter can be made from
simple rewiring of a three-phase DC/AC inverter by tying the bottom of the output capacitor to the
negative rail of the DC link, disconnecting the B and C phase output capacitors, and connecting
the three output inductors together as shown in Fig. 3.5. Capacitor, CDC , is implemented with two
sets of capacitors since the two three phase IGBT bridges are spatially separated on two different
sliders of a 19" rack. Placing DC link capacitors close to the IGBTs and using a bus bar, or similar
geometry PCB, is important to minimize undesirable stray inductance between the bridge and
capacitors. Inductors for the interleaved converter were hand wound using two Magnetics Inc. cores
in series to implement each. Two twisted, parallel 18 AWG enamel coated conductors make up the
windings.
Simple modifications to the inverter PWM settings in the DSP are well documented by Texas
Instruments ePWM User Guide for the F28335 [24] Section 3.8 Controlling a 3-Phase Interleaved
DC/DC Converter. The PWM carriers, or time-based counters as they are called in the TI User
Guide, for the interleaved converter are phase shifted 120 degrees apart. This corresponds to phase
shifted inductor currents allowing for an overall higher effective switching frequency which pushes
switching ripple and harmonics further away from the desired perturbation frequency range. The
22
1A
208
VLL
208
VLL
2A
3A
4A
5A
6A
Lr
Li
CDC
Ci
1B
2B
3B
4B
5B
6B
Figure 3.5: Circuit diagram for high bandwidth DC supply.
internal DSP PWM carriers are not accessible to measure so a sample test with a fixed 50 % duty
cycle was setup to measure the phase shift between carriers for each half-bridge prior to testing with
real power flow. All twelve PWM signals were probed using the Logic16 USB logic analyzer from
Saleae as shown in Fig. 3.7. Only the A channel signals from ePWM modules 1-6 are shown in Fig.
3.8 for ease of visualization but the B channels are simply the complements of A channels with a 1
µs deadtime between the two on states. The channel number and letter designations correspond to
those shown in Fig. 3.5. Modules 1-3 correspond to the rectifier pulses which are left unmodified
from their original settings and module 4, phase A of the interleaved converter also shared the same
settings. Interleaved phases B, module 5, and phase C, module 6 each have 120 degree and 240
degree phase shift shown in Fig. 3.8.
Following this test the rectifier was reverted back to its original settings with a DC link voltage
reference of 400 V across CDC ; the interleaved converter PWM remained at a fixed 50% duty cycle.
The three inductor currents of the interleaved converter are shown in Fig. 3.9 and Fig. 3.10 for
no load condition and a 2 kW load condition respectively. The 50% duty cycle yields a 200 VDC
output just like a conventional buck converter topology given a 400 V input. Inductor currents
clearly show the interleaving of 120 and 240 degree phase shifts for phases B and C. The DC bias in
the inductor currents also corresponds to the output current to the load evident in the near zero
offset for no load and approximately 3.26 A offset in each inductor current for the 2 kW load test.
The LC filter at the interleaved converter output has a large Q factor and can cause problems
if not considered prior to perturbing a load at the resonant frequency of the filter. Nominal filter
parameters, shown in Table 3.2, were not chosen specifically for perturbation studies but carried
over from the original DC/AC inverter designed primarily for 60 Hz operation. The three phase
23
(a) Three phase boost rectifier
(b) Three phase interleaved DC-DC
Figure 3.6: Hardware implementation of high bandwidth DC supply.
Figure 3.7: Testing of interleaved DSP PWM with Saleae Logic16.
24
6A
5A
4A
3A
2A
1A
1
0
1
0
1
0
1
0
1
0
1
0
0
100
Time (µs)
200
Figure 3.8: Logic analyzer capture of interleaved PWM.
No load
Vout = 200V DC
Interleaved Inductor currents a, b, c
Figure 3.9: Oscilloscope capture of interleaved inductor currents, 50% duty, no load.
25
2 kW BK load
Vout = 200V DC
Interleaved Inductor currents a, b, c
Magnitude (abs)
40
35
30
25
20
15
10
5
0
0
−45
−90
−135
−180
2
10
4
3
2
1
0
0
Phase (deg)
Phase (deg)
Magnitude (abs)
Figure 3.10: Oscilloscope capture of interleaved inductor currents, 50% duty, 2 kW load.
3
10
Frequency (Hz)
−45
−90
−135
−180
2
10
4
10
(a) DC resistance
3
10
Frequency (Hz)
4
10
(b) Equivalent series resistance at 10 kHz
Figure 3.11: Input voltage to output voltage gain of LC filter.
26
Table 3.2: High Bandwidth DC Supply Parameters
Component
IGBT half-bridge
IGBT driver
CDC rectifier
CDC interleaved
Lr
Li
Ci
Controller
fsw
finterrupt
Value
1200 V, 200 A
SCALE-2
(2 parallel) 100 µF / 1200 V
(2 series) 380 µF / 400 V
1.2 mH
830 µH
110 µF
TI DSP
10 kHz
10 kHz
Part Number
Powerex CM200DX-24S
CONCEPT 2SP0115T
CDE 944U101K122ACM
Kemet C4CDEFPQ6380A8TK
MTE RL-03503
Magnetics Inc. C058906A2
Epcos B32798G3556K
TMS320F28335
filter configuration produced a resonant frequency at 530 Hz approximately. Reconfiguring the LC
filter for interleaved topology parallels the inductors as the three half-bridges can be considered as
three paralleled sources. Two of the original capacitors are disconnected as well. Assuming matched
inductances, the three parallel inductors and single output capacitor have a resonant frequency of
912 Hz, given in Eq. 3.1.
1
fresonant =
2π
q
830µH
3 110µF
= 912Hz
(3.1)
Since the inductors do not have appreciable resistance there is little damping at this frequency
so the interleaved converter duty cycle needs to be attenuated when perturbing at frequencies near
912 Hz. Each inductor was measured for inductance and equivalent series resistance at DC and at
10 kHz using an Agilent U1733C LCR meter; measurements are given in Table 3.3. The input to
output voltage transfer function for LC filter with winding resistance included is given in Eq. 3.2.
VCi (s)
=
Vbridge (s)
1
Li avg
2
3 Ci s
+
Ri avg
3 Ci s
+1
(3.2)
Bode plots are shown in Fig. 3.11 using the average of the three measured DC resistance, DCR,
and equivalent series resistance, ESR, at 10 kHz values. Even though a small 18 AWG wire was
used to wind the inductors there is still a skin effect which raises the effective resistance by almost a
factor of ten. This actually helps the damping problem at higher perturbation frequencies, lowering
the Q factor at the resonant frequency easily seen by the much lower peak in Fig. 3.11b. Further
27
hardware modifications of the filter was undesirable since the equipment was only temporarily
being used for this experiment, otherwise a more elaborate filter could have been implemented with
additional branches to reduce the Q factor and raise the cutoff frequency beyond 1000 Hz, the
maximum desired perturbation frequency.
Table 3.3: Measurements on LC Output Filter
Component
Li1
Li2
Li3
Ri1
Ri2
Ri3
Ci
Value
815 µH
816 µH
810 µH
0.147 Ω @ DC, 1.250 Ω @ 10 kHz
0.136 Ω @ DC, 1.234 Ω @ 10 kHz
0.137 Ω @ DC, 1.206 Ω @ 10 kHz
108.88 µF
An open loop controller was used with DC link voltage compensation. This produced acceptable
waveforms with perturbations at the desired frequencies and minimal loading effects. Voltage on
capacitor CDC was measured for feed forward control, scaling the reference voltage to produce a 0
to 1 duty cycle for the DSP PWM comparator, Eq. 3.3. This compensated for any DC link ripple
produced by the rectifier, preventing any components of the rectifier input frequency, 60 Hz, or
harmonics of 60 Hz from passing through to the output of the interleaved converter.
duty cycle =
V output ref erence
VDC
(3.3)
A proportional, integral, resonant (PIR) controller with VCi feedback could be used if the device
under test starts to distort the perturbation waveforms. The integral term provides a zero steady
state error for the DC bias portion of the reference and resonant terms, equal to the number of
simultaneous perturbation frequencies, yields a zero steady state error for the sinusoidal components
of the voltage reference. However, the phase reversal at the LC filter resonant frequency causes
some problems with maintaining an output voltage in phase with the digitally synthesized reference
inside the DSP. An additional compensation term for the LC filter transfer function can assist with
this problem. Additional inductor current feedback is also possible to ensure balancing and can be
integrated into a full state feedback controller. The perturbation voltage frequency spectrum did
not show significant distortion from loading effects so closing the loop was not deemed necessary.
28
Figure 3.12: Oscilloscope capture of 5 Hz perturbation, 200 VDC bias, 2 kW on BK8522.
The BK Precision 8522 load was used again as the first device under test to ensure the developed
high bandwidth DC supply was producing a suitable perturbation voltage. Unfortunately this only
revealed further limitations of the 8522 as a representative DC constant power load. Fig 3.12 shows
the results of a 25 V peak, 5 Hz perturbation with 200 V bias fed into the BK 8522 set to 2 kW in
constant power mode. The load current response is far from sinusoidal and resembles a series of
constant current steps. Details of the 8522 input filter and internal controls are not available but it
appears the constant power mode uses a current control loop with references calculated by updating
the measured input voltage. This loop is unable to track the 5 Hz perturbation to the input voltage
so an alternative load was sought out that better represented DC constant power loads likely found
in distributed power systems.
An NHResearch 9200 cabinet with 4960 module was available for testing. This arrangement
is a bidirectional DC supply designed for dynamic charge/discharge testing of batteries such as
start/stop features and regenerative breaking in new automobiles. It also has some built in functions
for emulating battery terminals for testing power converters that interface batteries to electric power
systems. The converter inside is therefore capable of fast changes between charge and discharge
modes to coincide with these automotive applications. Manufacturer supplied user interface via a
29
i load 1.5 A/div
v in 100 V/div
0V, 0A
Figure 3.13: Oscilloscope capture of 1 Hz perturbation, 380 VDC bias, 3 kW on NHR 9200.
touch screen also allows for a constant power charge and discharge mode. In this case the constant
power discharge was used to represent a DC constant power load - the charge and discharge are
always in reference to the device under test so a charge setting supplies the DUT power and discharge
mode acts as a load to the DUT. Some successful preliminary tests were performed which yielded
sinusoidal responses in the load current. Modifications were made to the high bandwidth DC supply
to synthesize a 380 VDC bias followed by additional tests on the NHR 9200 unit.
In order to synthesize a 380 VDC bias the DC link voltage across CDC must be higher than
380 plus the peak of the perturbation signal. The DC link capacitors used in the active rectifier
power stack are rated for 1200 V and the interleaved power stack uses two 400 V rated capacitors
in series which permits a higher DC voltage at the output of the rectifier. A DC link reference of
470 VDC was chosen after considering the voltage sensors utilized, LEM LV20-P, are limited to 500
V maximum and over voltage lockout for representative 380 VDC converters is approximately 425
VDC. The boost rectifier was capable of producing this DC link voltage by changing the voltage
reference in the DSP code but no further hardware changes were required.
The first test for perturbations with 380 VDC bias is shown in Fig. 3.13 for a 30 V peak, 1 Hz
perturbation using the NHR 9200 set for 3 kW constant power load. Low perturbation frequencies
30
show the frequency response of the load near DC, a 0.1 Hz perturbation is shown later, and the
characteristic 180 degree phase shift between the voltage perturbation and the load current response.
This is the expected behavior for constant power loads and helps to show why the negative resistance
model is so useful. Increasing the perturbation frequency allows measurement of the input filter
impedance and any associated resonance. The DSP was hard coded for the test in Fig. 3.13 which
is not the best way to perform a sweep so an automation method was developed to implement
a sweep across multiple perturbation frequencies. Additional details for the NHR 9200 and 4960
module are given later along with experimental results from AC sweep tests.
3.4
Experiment Automation with High BW DC Supply
Automating impedance measurement experiments is useful for saving time as well as helping to
organize data collection. The sequence of necessary events differs depending on the perturbation
signal used to measure impedance of a constant power load. A decision must also be made as to the
amount of data to be collected either simply the frequency domain data necessary for production of
a Bode plot or if time domain data should also be stored for documentation plots.
The most basic AC sweep consists of a single AC signal at the desired perturbation frequency
superimposed on the nominal DC bias. This signal is held long enough to measure the magnitude
and phase of the current response, possibly storing the time-domain data as well if so desired. The
perturbation frequency is then increased and the process repeated until enough data points are
recorded to produce a smooth Bode plot. The time needed to perform this AC sweep depends on
the minimum perturbation frequency and the number of steady state cycles chosen for analysis. For
example, if 10 cycles are selected for analysis at a minimum 0.1 Hz this amounts to 100 seconds for
the first perturbation. Higher perturbation frequencies require less time to arrive at 10 cycles but the
entire process can add up to several minutes. This method was used since an automation algorithm
was utilized, load conditions could be held constant, and thermal issues were not a problem for the
load or perturbation source for test lasting several minutes.
Alternative perturbation signals apply several simultaneous perturbations such as a multi-sine
perturbation or a chirp signal that has a wide range of frequency content.[15] These signals yield a
current response at multiple frequencies allowing measurement of multiple data points simultaneously.
This reduces the total time needed for measurement which is very beneficial if the load is time
31
varying since complete perturbation must be performed while the load is at a consistent power
level. There are some important implications for using these signals with constant power loads, a
nonlinear system, that will be considered later.
The TI TMS320F28335 controller is the heart of the high bandwidth DC supply so communication
of the desired perturbation frequency and magnitude was necessary. In addition, voltage and current
waveforms had to be measured and a capture triggered to correspond to each perturbation frequency.
Several factors were considered on the best way to control and automate the entire experiment process.
Finally a laptop running Matlab was chosen to run an automation script with communication
interfaces to the TI DSP over CAN bus using a Kvaser Leaf Light V2 USB-CAN cable and
communication with a Yokogawa DLM2024 oscilloscope over USB. Scripting in .m code allowed
for storing time domain waveforms in .mat files along with documentation of the perturbation
frequencies critical for organizing data for post processing. Further details of the automation
script and communication protocols for interfacing Matlab with the Kvaser cable and Yokogawa
oscilloscope can be found in Appendix A. A photograph of all the equipment used is shown in Fig.
3.14.
A flow chart of the experiment automation is shown in Fig. 3.15. The rectifier and interleaved
converter cabinet contactor is energized manually then the .m script is run which first initializes the
desired perturbation frequencies into the Matlab workspace. Communication is initialized with the
oscilloscope and CAN bus cable followed by PWM enable signals for the rectifier to produce the
intermediate DC link voltage for the interleaved converter. The interleaved converter is initialized
inside the DSP to have a 0 V, 0 Hz perturbation so when its PWM pulses are enable it produces a
380 VDC output. Next the perturbation amplitude and frequency are sent to the DSP over CAN and
the oscilloscope time/division is set to coincide with the perturbation frequency. This time/division
decreases as perturbation frequency increases and the oscilloscope automatically adjusts the sample
rate for a constant acquisition length, reducing the overall data stored on the laptop, but maintains
capture capability of approximately 10 cycles for each perturbation frequency. A wait statement
is used to allow the system to reach steady state and 10 of these steady state cycles are captured.
The data from the oscilloscope is then transferred to the laptop over USB and saved to a .mat file.
Perturbation frequency is advanced and the cycle continues and if completed then the interleaved
32
Interleaved
DC-DC
NHR
9200/4960
TI DSP
Rectifier
Laptop
Kvaser
USB-CAN
Yokogawa
DLM 2024
Figure 3.14: Photograph of NHR 9200, RLC circuit, and automation equipment.
33
Begin
Setup desired test
frequencies
Initialize Kvaser USB-CAN
Initialize scope USB
Enable rectifier
Enable interleaved
Communicate
To DSP: amplitude, frequency
To scope: Time/div
Wait for 10 cycles
Advance to next
desired frequency
Fetch scope data
Re-arm scope
Save data to .mat
Done?
End
Figure 3.15: Flow chart of experiment automation.
34
L
RC
C
R
Figure 3.16: RLC circuit under test.
converter PWM is disabled, followed by the rectifier PWM and finally the clean up phase to close
the two communication channels and clear any workspace variables left over.
A post processing script reads each .mat file produced during the perturbation sweep and performs
a Fourier transform to analyze the magnitude and phase of voltage and current at the perturbation
frequency and any other frequencies of interest. This data can then be placed into a Bode plot for
viewing the frequency response of the constant power load to input voltage perturbations.
3.5
Measurements of RLC Circuit
An RLC circuit, shown in Fig. 3.16 with parameters in Table 3.4, was tested using a single
frequency perturbation sweep. The equivalent series resistance, RC , for the electrolytic capacitor is
also shown. Initially, 380 VDC was used for the DC bias in the reference voltage but inrush current
into the circuit caused an over current trip on one of the cabinet circuit breakers so a 100 V bias was
used instead with a 20 V peak perturbation. Fig. 3.17 shows three sets of time domain waveforms
for the input voltage and load current and corresponding Fourier spectra of the load current. Note
that the average value of the time domain waveform was removed prior to Fourier analysis since the
DC response was not of interest, only the response to the perturbation. Perturbing at 0.1 Hz, Fig.
3.17a and 3.17b did not produce a measurable result since the response is dominated by the 132 Ω
resistor, only a 15 mA peak response is expected for a 20 V perturbation. Perturbations at 500 Hz
and 1000 Hz are shown in Fig. 3.17c and 3.17d and Fig. 3.17e and Fig. 3.17f respectively. The
35
load current Fourier spectra for 500 Hz and 1000 Hz spectra show that there is only a significant
response at the perturbation frequency, as expected for a linear circuit.
The perturbation magnitude was not exactly 20 V since an open loop control was implemented on
the DSP. However, attenuation of the duty cycle for the perturbation frequency was pre-calculated
on the laptop in the automation script before transmission over CAN bus. Compensation for the
gain near the resonant frequency using the transfer function model of the interleaved LC filter was
not implemented until measurements for the second order kernel in Chapter 5.
The magnitude and phase of the circuit admittance is calculated using measurements of the
voltage and current at the perturbation frequency, a, extracted from the Fourier spectra. Impedance
inverse is calculated in Eq. 3.4, 3.5, and 3.6 since this is the last time impedance will be used, all
calculations in following sections will use admittance as this corresponds to the input output SISO
block diagram in Fig. 2.5. Magnitude and phase of the impedance for each perturbation from 0.1 Hz
to 1000 Hz is shown in a Bode plot, Fig. 3.18, along with the theoretical values based on the Table
3.4 values and Eq. 3.7. The error at lower frequencies is due to the small load current response
being difficult to measure but higher frequencies show good agreement with the theoretical values.
This test helped to verify the functionality of the developed high bandwidth DC supply and
the automation script with communication interfaces over CAN bus to the DSP and with the
oscilloscope over USB. The time domain waveforms in Fig. 3.17 show how the time/division setting
on the oscilloscope was scaled as perturbation frequency increase. These figures show the complete
capture without truncation.
1
Iload (a)
=
Z(a)
Vin (a)
Iload (a)
1
= abs
|Z(a)|
Vin (a)
(3.4)
(3.5)
Iload (a)
1
= angle
∠Z(a)
Vin (a)
(3.6)
Z(s)theoretical =
(Rc LC + RLC)s2 + Rc RCs + Ls + R
(Rc C + RC)s + 1
36
(3.7)
130
4
120
3
110
2
100
1
90
0
80
−1
0.2
|I load| (A)
Load Current (A)
Input Voltage (V)
0.15
0.1
0.05
70
0
20
60
40
Time (s)
80
0
−2
100
0
10
120
5
0
80
60
0.005
0.01
Time (s)
0.015
1
4
−5
0
0.8
5
|I load| (A)
140
100
0.4
0.6
Frequency (Hz)
(b) 0.1 Hz perturbation
Load Current (A)
Input Voltage (V)
(a) 0.1 Hz perturbation
0.2
3
2
1
0
−10
0.02
0
(c) 500 Hz perturbation
1000
2000
3000
Frequency (Hz)
4000
5000
(d) 500 Hz perturbation
200
20
150
10
12
0
50
8
|I load| (A)
100
Load Current (A)
Input Voltage (V)
10
6
4
−10
2
0
0
0.002
0.004
0.006
Time (s)
0.008
0
−20
0.01
(e) 1000 Hz perturbation
0
2000
4000
6000
Frequency (Hz)
8000
10000
(f) 1000 Hz perturbation
Figure 3.17: Samples from AC sweep for RLC circuit, time and frequency domain.
37
Table 3.4: RLC Circuit Parameters
Component
R
L
Rc
C
Value
132 Ω
1 mH
0.06 Ω
1.4 mF
Part Number
(3)TE500B22RJ (3)TE200B22RJ
Hammond 195C30
Mallory CGS142T450W5C
Mallory CGS142T450W5C
|Z| (abs)
150
Experimental
Theoretical
100
50
0
10−2
10−1
100
101
102
103
10−1
101
100
Frequency (Hz)
102
103
Phase (deg)
100
0
−100
10−2
Figure 3.18: Bode plot of AC sweep on RLC circuit showing impedance.
3.6
Measurements of DC Constant Power Load
The NHR 9200-4960 was tested next using the automation script to measure admittance from
0.1 Hz to 1000 Hz. The 4960 module is housed in the 9200 cabinet which also holds an isolation
transformer, additional protection, and a touch screen user interface. Voltage, current, sourcing
power and sinking power limits for the 4960 unit are given in Table 3.5. Discharge mode was used
with minimum voltage set to 200 V. This setting was very helpful during the experiment sequence.
After placing the 4960 into this mode, the perturbation source cabinet contactor was closed with a
push button followed by execution of the automation script. This yielded a no load condition until
the interleaved converter output reached 200 V then the unit started drawing current to reach a 3
kW constant power load. The 3 kW setting was chosen since it is not the maximum rating of the
38
high bandwidth DC supply but produced a measurable load current. At the end of the automation
script the interleaved converter PWM is disabled and its output voltage drops to zero, placing the
4960 back into under voltage lockout.
Table 3.5: NHR 9200/4960 Parameters
Parameter
Voltage
Current
Sourcing power
Sinking power
Value
0-600 VDC
0-40 ADC
0-8 kW
0-12 kW
Oscilloscope data captures for 0.1 Hz, 500 Hz, and 1000 Hz perturbations are shown in Fig.
3.19. Time domain plots show the input voltage and load current on the same plot and the Fourier
spectrum of the load current is plotted beside them for each frequency. Again, the average value of
each time domain waveforms was removed prior to Fourier analysis to only analyze the response to
the perturbations, not the response at DC. The characteristic 180 degree phase shift is easily seen in
the 0.1 Hz perturbation much like the waveform observed in initial tests of the 4960 module. Tests
at low frequencies show this phase response very clearly since this perturbation is so close to DC;
any input filter inductors approximate to short circuit and capacitors approximate to open circuit.
This means the load current response is not significantly affected by any input filter stage within
the module. The Fourier spectrum of the load current shows a response at 0.1 Hz as well as 0.2
Hz, the second harmonic of the perturbation frequency. Perturbations at 500 Hz do not show the
180 degree phase shift like the 0.1 Hz perturbation, likely due to 4960 input filter effects. The load
current Fourier spectra shows responses at 1000 Hz, the second harmonic, and a small response at
1500 Hz, the third harmonic. At 1000 Hz there is also a response at the second harmonic but higher
order responses are not visibly apparent.
The gain and phase of the admittance, Y , were calculated at each frequency using Eq. 3.8, 3.9,
and 3.10. Results from these calculations are shown in a Bode plot of admittance in Fig. 3.20.
Y (a) =
Iload (a)
Vin (a)
Iload (a)
|Y (a)| = abs
Vin (a)
39
(3.8)
(3.9)
∠Y (a) = angle
Iload (a)
Vin (a)
(3.10)
The internal parameters of the 4960 unit are not available but some analysis can be made to
compare the results to the ideal negative admittance model which has a theoretically value calculated
in Eq. 3.11 in siemens. The measured gain at 0.1 Hz was 0.02123 which is a 2.07% error. Phase of
the admittance at 0.1 Hz matches with the theoretical phase of a constant power load at DC as well.
Y (0)theoretical =
3kW
Pnominal
=
= 0.0208S
2
380V 2
Vnominal
(3.11)
The additional response at harmonics of the perturbation frequency cannot be accounted for
with linear systems theory. It should be noted that these are not a result of the internal transistor
switching like ripple, these additional responses are present due to the internal closed loop controller
tasked to maintain a constant input power even under input voltage perturbations. The 4960
module is slightly different than other power converters since it has controls to directly control
the input power to remain constant whereas distributed power system converters have regulated
outputs which indirectly causes a constant input power. Further discussion of harmonic responses
in load currents from constant power loads are discussed in the following chapters with a nonlinear
modeling method to mathematically explain their existence.
3.7
Limitations of Linear Methods for Constant Power Loads
It is clear that constant power loads have a nonlinear I-V characteristic. A linear approximation
leads to a negative resistance model which can then be used to design converter input filters
analytically. Panizza [25] recommends using the under voltage lockout limit for input filter design
as this is worst case scenario. However, this linear approximation is only suitable for small-signal
perturbations on the input voltage. In many cases this still does not present a major problem since
modern digitally controlled converters include under voltage and over voltage lockout features for
protection. In the case of high power loads this can present an issue since a trip event of a large
load is a large-signal perturbation on the system.
A solution to this problem is presented in the next chapter by means of nonlinear modeling by
Volterra Series to extend impedance measurement techniques to nonlinear systems.
40
10
0.7
420
9.5
0.6
400
9
380
8.5
360
8
340
7.5
320
0
20
60
40
Time (s)
80
0.5
|I load| (A)
Load Current (A)
Input Voltage (V)
440
0.4
0.3
0.2
0.1
0
7
100
0
(a) 0.1 Hz perturbation
0.2
0.4
0.6
Frequency (Hz)
0.8
1
(b) 0.1 Hz perturbation
460
12
440
11
420
10
400
9
380
8
360
7
340
6
2.5
|I load| (A)
Load Current (A)
Input Voltage (V)
2
1.5
1
0.5
320
0
0.005
0.01
Time (s)
0.015
0
5
0.02
0
420
12
3
410
11
2.5
400
10
390
9
380
8
370
7
360
6
0.5
5
0.01
0
350
0
0.002
0.004
0.006
Time (s)
0.008
2000
3000
Frequency (Hz)
4000
5000
(d) 500 Hz perturbation
Load Current (A)
2
|I load| (A)
Input Voltage (V)
(c) 500 Hz perturbation
1000
1.5
1
(e) 1000 Hz perturbation
0
2000
4000
6000
Frequency (Hz)
8000
(f) 1000 Hz perturbation
Figure 3.19: Samples from AC sweep for DC CPL, time and frequency domain.
41
10000
Gain (abs)
0.2
0.1
0
10−1
100
101
102
103
100
101
Frequency (Hz)
102
103
Phase (deg)
−150
−200
−250
−300
10−1
Figure 3.20: Bode plot of admittance, Y , from AC sweep on NHR 9200 at 380 VDC, 3 kW.
The second major limitation to modeling a constant power load with a frequency response,
admittance, or impedance is the lacking ability to model loads that change power with time. For
a DC load at steady state modeled with linear systems theory, the load current is function of the
input voltage and the gain at 0 Hz. In order to model loads that change with time, a family of
frequency responses must be made. This requires designers to decide in which case to design passive
components (as these are more difficult to switch in and out compared to digital control code).
Again, it is likely the worst case should be selected in the event that all time varying loads happen
to all operate at maximum power simultaneously. This problem of developing a single model to
characterize a load with multiple input power operating points is not solved here and still presents
a major issue in mathematically modeling modern electrical loads utilizing power converters.
42
CHAPTER 4
NONLINEAR SYSTEM MODELING WITH
VOLTERRA SERIES
The Volterra Series is a nonlinear system modeling method that in its time domain form utilizes
convolution integrals similar to the one associated with LTI systems but with extensions to nonlinear
systems. It has been described as a Taylor Series with memory [26] - in this context, it allows
modeling of finite bandwidth nonlinear systems.
Many previous works utilize Gaussian white noise for perturbation signals to identify the system.
Signals used for linear system identification are often insufficient to excite all the higher order
responses necessary for thoroughly identifying the system. This approach is useful if time-domain
kernels are desired, similar to an impulse response for LTI systems denoted as h(τ ) in a convolution
integral. However, others have developed alternative measurement methods in the frequency domain
which do not require a Gaussian white noise signal which is difficult to synthesize in practice,
especially at high power levels. These frequency domain measurements are actually very similar to
the perturbation signals used for impedance measurement allowing reuse of previously developed
perturbation equipment.
Once a Volterra model is developed, it is able to model the static nonlinearity described by
higher order terms (greater than one) in the Taylor Series expansions for the constant power load
I-V curve as well as dynamics which appear as additional responses in the load current at harmonics
and intermodulations of the perturbation frequency. These harmonic currents were visible in the
load current frequency spectra for the NHR 9200 impedance measurement tests shown in Fig. 3.19.
First the Taylor Series expansion for the constant power load I-V curve is revisited to include
the second order term. Next the Volterra Series is introduced along with relevant prior work to
extend linear systems theory for impedance measurement to nonlinear systems. A brief overview of
time domain kernels is given before moving on to frequency domain kernels and how they relate
back to frequency response and Bode plots for linear systems. This leads to Chapter 5 which covers
measuring Volterra kernels in the frequency domain using equipment developed in Chapter 2.
43
4.1
Second Order Taylor Series Expansion
Reconsider the Taylor Series expansion used to develop the negative resistance model for constant
power loads and how even though this model was restricted to static situations, only capable of
modeling memoryless, infinite bandwidth systems, it characterized the behavior of the constant
power load I-V curve close to the nominal operating point. The first order approximation has
been used extensively under the name negative impedance or negative admittance to design input
filters and feedback controllers for small-signal stability as discussed in Section 2.4. However, linear
approximations begin to break down for large perturbations to the input voltage. Designers can use
the worst case input voltage, the under voltage lockout limit, to make a conservative input filter
design even with the shortcomings of the small-signal model. The second order Taylor Series is used
to create a nonlinear static model followed by discussion of how to add memory, or finite bandwidth
to the model similar to adding dynamics to the first order static model but for nonlinear dynamics.
The Taylor Series expansion with addition of the second order term is given in Eq. 4.1 and 4.2.
By substituting in Eq. 2.7 and 2.8 the final load current is given in Eq. 4.3 with nominal current
and nominal resistance used to derive the final first order approximation for load current in Eq.
2.9. The third term of the approximation does not have an equivalent circuit element such as the
constant current source or resistance like the first two terms.
iload ≈ Pnominal
2
X
f n (vnom )
n=0
(vin − vnom )n
n!
0
1
iload ≈ Pnominal f 0 (vnom ) (vin −v0!nom ) + f 1 (vnom ) (vin −v1!nom ) + f 2 (vnom ) (vin −vnom)
2!
≈ Pnominal
≈ Pnominal
1
vnom
1
vnom
−
−
1
2
vnom
vin
2
vnom
(vin − vnom ) +
+
1
vnom
Pnominal
≈ 3 Pnominal
vin +
vnom − 3 v 2
nom
+
2
vin
3
vnom
2
3
vnom
−
(vin −vnom )2
2
2vin
2
vnom
+
1
vnom
(4.1)
2
(4.2)
Pnominal 2
vin
3
vnom
Pnominal 2
1
vin
iload ≈ 3inominal − 3 vin +
3
r
vnom
(4.3)
The previously discussed 600 W Vicor converter with regulated output for 380 VDC systems is
used again to compare the ideal CPL I-V curve and the second order Taylor Series approximation.
44
10
Nominal
CPL curve
Taylor Series n=0,1
Taylor Series n=0,1,2
Iload (A)
8
Under Voltage
Lockout
6
Over Voltage
Lockout
4
2
0
0
100
200
Vin (V)
300
400
Figure 4.1: I-V curve for Vicor 600 W Max Converter with second order approximation.
A plot of the two curves is shown in Fig. 4.1 using the same axis limits from the previous plot in Fig.
2.6. The second order approximation appears to be a better approximation for the ideal CPL curve,
especially between the under voltage and over voltage lockouts. Taylor Series approximation breaks
down beyond the nominal operating point due to divergence as input voltage approaches infinity.
This means that even though increasing the Series order may improve curve fitness at lower
than nominal input voltage, the end behavior diverges. Fig. 4.1 is plotted again with higher x-axis
limits in Fig. 4.2 to show this behavior. The limit at twice the nominal input voltage is oscillatory
between the nominal current and zero with each increasing term - the values of the Taylor Series
approximations for n = 0, 1 and n = 0, 1, 2, 3 are equivalent double the input voltage. This may
not be a problem for the model as over voltage conditions quickly destroy semiconductor devices if
protection mechanisms are not employed. The linear model also shows zero load current at double
the input voltage so end behavior for this model has serious problems as well. As long as under
voltage and over voltage lockouts accompany the model into the simulation environment this should
not be a problem.
Power calculations for the two approximations are shown in Fig. 4.3. This shows more clearly
than Fig. 4.1 that a second order model predicts a higher input power than a first order approximation
but also predicts input power increasing beyond the nominal voltage.
45
10
Nominal
CPL curve
Taylor Series n=0,1
Taylor Series n=0,1,2
Iload (A)
8
6
4
2
0
0
200
400
Vin (V)
600
800
Figure 4.2: Convergence problem for Taylor Series expansion for CPL curve.
700
600
Pload (W)
500
400
300
Nominal
CPL curve
Taylor Series n=0,1
Taylor Series n=0,1,2
200
100
0
0
200
400
Vin (V)
600
800
Figure 4.3: Vicor converter power calculation for first and second order approximations.
46
The figures shown here make it clear that increasing the order of the Taylor Series are best
suited to improvements in the voltage range below nominal. Above nominal voltage the expansion
has problems due to the unstable converge at two times nominal input voltage, oscillating between
zero amps and nominal current with each increase in order. However, this does not mean the effort
is hopeless - there is still room to be made in dynamic modeling of these higher order responses.
4.2
Second Order Dynamic Response
In Section 2.4 the constant power load was shown to have finite bandwidth due to the input
filter and controller bandwidth. A linear dynamic model was applied to characterize the frequency
response that can be visually seen in a Bode plot related to the first order term of the Taylor Series.
What about an equivalent frequency domain construct for the second order Taylor Series term to
characterize the harmonics seen under large-signal perturbations? Linear systems theory is unable
to model response current at the second harmonic of an input voltage perturbation as observed
in the impedance measurement tests for the CPL. A nonlinear modeling method is required to
accomplish this. There are many methods for nonlinear modeling although none have caught on
in power electronics analysis likely due to the lack of connection back to previously developed
impedance measurement methods. A method that can leverage all the work in linear impedance
measurement is beneficial but must be able to explain harmonics under large-signal perturbations
and match the I-V curve of the constant power load better than a negative resistance. This model
can serve analysis for systems that require broader lockout limits especially in the lower voltage
region. The Volterra Series provides this extension of impedance measurement to nonlinear systems
analysis and allows for modeling of harmonic current responses to input voltage perturbations.
4.3
Introduction to Volterra Series
The Volterra Series has a long history starting with its creator, Vito Volterra (1860-1940), then
passing through students, colleagues, and collaborators to current researchers in several fields from
engineering to biology. A short history of relevant milestone achievements in the theory and how
they can be traced back to Volterra is presented.
Volterra extended the well-known convolution integral for linear systems to nonlinear systems.
A series of his lectures was compiled into a book which was translated to English and published by
47
Blackie and Son in 1930 then reissued by Dover publishing in 1959 and 2005 [27]. Paul Lévy, one
of Volterra’s students, became known for the Lévy process related to Brownian motion [28] [29].
The Volterra Series was passed to Norbert Wiener who continued the research of Lévy in Brownian
motion with the Wiener process and Wiener theory that furthered Volterra theory. Wiener’s
research in nonlinear systems was originally classified due to wartime importance in the 1940’s with
applications to radar but was later declassified in 1946 [30]. In 1949, Wiener sent some of his findings
back to colleagues at the Research Laboratory of Electronics at MIT to conduct experiments to
accompany his theoretical work, this correspondence is recorded by Martin Schetzen in [26]. Wiener
also published a book on his work with nonlinear systems in 1958 [31]. Yuk-Wing Lee, a student
of Wiener, and Martin Schetzen, a student of Lee, later developed the Lee-Schetzen method for
measuring Volterra kernels.
Schetzen published his work in a book in 1980 [32] and an invited paper in the Proceedings
of IEEE [26]. Multiple contributors published papers and books on the Volterra Series in the
1980’s. Rugh published a book in ’81 on nonlinear system modeling using Volterra Series [33].
Leon O. Chua has several publications contributing to the Volterra Series [34] [35] and advised
several students who wrote their thesis and dissertations on the topic. Stephen Boyd, advised by
Chua, studied the Volterra Series at University of California, Berekeley in the early 1980’s [36] [37].
Chua and Liao wrote two follow-up papers to Boyd’s on measuring Volterra kernels [38] [39] after
Boyd graduated. Tymerski followed Boyd in a 1991 IEEE Transactions on Power Electronics paper
regarding analytical formulation of control-to-output Volterra kernels for a DC-DC boost converter.
Hai-Wen Chen presents an exhaustive bibliography in his paper published in the Proceedings of
the IEEE [40] which references all three of Chua’s papers on measuring Volterra kernels.
More recent publications include a book by Ogunfunmi [41] published in 2007 that uses least mean
squares methods for measuring discrete time-domain Volterra kernels useful for digital computations.
Fard, Karrari and Malik used Volterra series to model a synchronous generator with field voltage as
input and outputs of terminal voltage and active output power [42].
4.4
Time Domain Volterra Kernels
The first four terms of the series are given in Eq. 4.4 starting with the zero order term, h0 and
the general form in Eq. 4.5. Compared to the Taylor Series, the Volterra Series can model systems
48
with outputs that are a function of previous inputs whereas the Taylor Series can only model systems
with outputs as a function of present inputs, synonymous with memoryless and infinite bandwidth
as discussed previously. This helps to explain why Schetzen described the Volterra Series as a Taylor
Series with memory. [26]
y(t) = h0 +
R∞
h1 (τ1 )u(t − τ1 )dτ1
0
+
R∞R∞
+
R∞R∞R∞
0
0
0
0
h2 (τ1 , τ2 )u(t − τ1 )u(t − τ2 )dτ1 dτ2
(4.4)
0
h3 (τ1 , τ2 , τ3 )u(t − τ1 )u(t − τ2 )u(t − τ3 )dτ1 dτ2 dτ3
+ ...
y(t) = h0 +
∞ Z ∞
X
n=1 0
...
Z ∞
0
hn (τ1 , ..., τn )u(t − τ1 )...u(t − τn )dτ1 ...dτn
(4.5)
The first order term of the Volterra Series is the LTI convolution integral which characterizes
linear responses of a system. Impulse response h1 (τ1 ) is the same as an impulse response for a
normal LTI system but is called the first order Volterra kernel.
The zero order term, h0 , can be used to model systems that have a non-zero output even with a
system input of zero. Physical systems rarely have this property so the Volterra Series is sometimes
written without this term. The negative admittance model that includes the constant current source
for large signal modeling has both zero order and first order terms as shown above in Eq. 2.13.
However, the existence of a load current for a zero input voltage condition is not realistic and is
simply a mathematical construct to yield a correct result at an input voltage near nominal.
The second order term represents a two-dimensional convolution between the input, u, and itself
across a two-dimensional impulse response h2 (τ1 , τ2 ). Including the second order Volterra kernels
enables modeling of second harmonic responses to system perturbation frequencies, third order
kernels can model third harmonic responses and so on. The focus here is on the second order kernel
for constant power loads although higher order kernels may still be useful for constant power loads
if deemed necessary in the future.
4.5
Frequency Domain Volterra Kernels
Although the majority of previous work with Volterra Series focused on the time domain
kernels there is also ability to use frequency domain kernels. This approach makes more sense for
49
extending impedance measurement methods for use with Volterra Series. Chua and his students
made significant contributions to Volterra kernel measurement in the frequency domain. [34] [35]
[36] [37] The first order kernel in the frequency domain can be viewed using a Bode plot since it
has the same form as the transfer function of an LTI system with a magnitude and phase at each
frequency. The second order Volterra kernel can be viewed as two surfaces for the magnitude and
phase.
50
CHAPTER 5
VOLTERRA KERNEL MEASUREMENTS OF
CONSTANT POWER LOADS
5.1
Fundamentals of Frequency Domain Kernel Measurement
The Volterra kernel measurement process is a multidisciplinary in itself. Understanding the
Volterra Series and characteristics of the kernels is not sufficient to deal with all the nuances of
kernel measurement.
Stephen Boyd, advised by Leon O. Chua, contributed to Volterra kernel measurement methodology by incorporating new technologies of the time and insistence on developing practical strategies
for empirical measurements.[36] When this paper was published, Boyd had only seen attempts at
actually measuring kernels in the biology area and attempted to expand measurement efforts to
more electrical systems. He proposed that the use of modern (for 1983) D/A and A/D equipment
to synthesize waveforms to allow perturbation at specific frequencies enabling a better measurement
process than using white noise and correlation methods. Using these methods, frequency domain
Volterra kernels can be measured by perturbing a nonlinear system with specific frequencies and
measuring the response at harmonic and intermodulation of the perturbation frequencies. Improvements in signal synthesis and processing have been made since the 1980s that further broaden the
possibilities for kernel measurements on constant power loads.
Boyd measured the kernels of an electro-acoustic transducer made up of a compression driver and
radial horn. Perturbation signals were produced using an 8085 computer and responses measured
with an HP3582A spectrum analyzer. He compares two types of perturbation signals the first being
a two frequency sweep where two frequencies, a and b, are produced simultaneously to perturb
the system and responses at possible second order frequencies are measured: 2a, 2b, and a ± b
corresponding to points on the second order kernel H(a, a), H(b, b), H(a, b), and H(a, −b) for a > b.
Responses at two times the perturbation frequency are referred to as harmonics and responses at
the sum and difference of the input perturbation are called intermodulation frequencies. There are
not too many requirements for selecting perturbation frequencies a and b although it is best to
51
avoid overlapping of intermodulation frequencies and harmonics. If this happens then the measured
response can potentially be erroneously higher since responses from 2a or 2b are occurring at the
same frequency as a + b or a − b. This can easily be avoided by choosing perturbation frequencies
a and b so that neither are multiples of the other and the sum and difference do not align with
multiples either.
A two frequency sweep is the minimum frequency content needed to measure points on the second
order kernel but signals with more than two frequencies can also be used to speed up measurements
by measuring multiple points on the kernel simultaneously. The primary contribution in [36] is a
quick kernel measurement method to provide the user with a fast analysis of the kernel to tweak
input levels prior to beginning the length multitone process. This technique assists in selection of
perturbation frequencies that helps avoid overlapping responses and produces several responses on
the surface that aren’t on the f1 = f2 plane. Additional features, attributed to D. J. Newman, are
presented to select optimal phase for each perturbation frequency cosine in order to minimize the
overall amplitude of the signal to pack in as much frequency content as possible without saturation.
Chua wrote parts two and three on Volterra kernel measurement [38] [39] as a continuation of
the first kernel measurement paper [36]. The nth order response is calculated using Eq. 5.1 in
the same form as it is shown in [38]. This is applicable if there is only one combination of input
frequencies ωi1 , ωi2 , . . . , ωin to that particular frequency component of the system output, y. The
value of the nth order kernel at coordinate (jωi1 , jωi1 , . . . , jωin ) is the system response at frequency
Pn
k=1 ωik
as a complex number divided by product of the input perturbations as complex numbers,
ai1 ai2 . . . ain . Chua published additional papers using frequency domain kernels and multi-tone
perturbation signals. [43]
Yn j
Hn (jωi1 , jωi1 , . . . , jωin ) =
n
P
k=1
ωik
ai1 ai2 . . . ain
(5.1)
More modern publications on kernel measurement topics include Evans [44] [45] [46] whose
dissertation [47] applied the Volterra Series to gas turbine modeling. These contributions focus
on perturbation frequency selection for designing a signal to measure several points on the kernel
without overlap, similar to Boyd but with further additions.
A perturbation source is necessary to do any sort of kernel measurement on a hardware device.
A linear amplifier, with transistors operating in their linear region, is suitable for perturbing low
52
power systems but switch mode amplifiers, with transistors operating in saturation or cutoff, can
produce perturbations are higher power levels. Major improvements in semiconductors for switch
mode amplifiers have been made since the 1980s, notably the IGBT and more recently SiC based
devices, that allow synthesis of perturbation signals with low distortion for impedance measurements
as well as Volterra kernel measurements. The high bandwidth DC supply described in Chapter 3 is
reused here for Volterra kernel measurements.
5.2
Single Frequency Perturbation
A single frequency sweep was used in Chapter 3 to measure the impedance of a NHR 9200-4960
in constant power load mode at 3 kW. The experiment automation script stored input voltage and
load current data at each frequency and spectral analysis was performed during post processing.
Magnitude and phase of voltage and current at the perturbation frequency were used to calculate the
input voltage to load current response according to Eq. 5.2 - Vin (a) and Iload (a) are the magnitude
and phase in complex number notation for each signal at the perturbation frequency a. Gain
and phase are calculated with Eq. 5.3 and 5.4 respectively, showing Matlab functions syntax for
magnitude and angle of complex numbers, which can then be viewed in a Bode plot, 3.20. This is
the first order frequency domain Volterra kernel H1 (f1 ) where f1 is the complementary frequency
domain variable in hertz as τ1 is for the time domain Volterra kernel.
H1 (a) =
Iload (a)
Vin (a)
Iload (a)
|H1 (a)| = abs
Vin (a)
(5.2)
(5.3)
Iload (a)
∠H1 (a) = angle
Vin (a)
(5.4)
The time domain data gathered for each data point in Fig. 5.1 can be reanalyzed to measure
the response at the second harmonic of the perturbation frequency. These data points lie in the
f1 = f2 plane of the second order Volterra kernel, H2 (f1 , f2 ), shown in Fig. 5.2. In order to calculate
the gain and phase to place into the second order kernel, the single frequency perturbation should
be thought of as two cosines with the same frequency and phase each contributing one half the
53
Gain (abs)
0.2
0.1
0
−1
10
0
1
10
10
2
3
10
10
Phase (deg)
200
150
100
50
−1
10
0
10
1
10
Frequency (Hz)
2
3
10
10
Figure 5.1: H1 (f ) measured with single frequency sweep.
perturbation magnitude. The input voltage magnitude and phase is measured at the perturbation
frequency but the load current response is measured at twice the perturbation frequency as shown
in Eq. 5.5, 5.6, and 5.7. Gain and phase calculations for each perturbation frequency are shown in
Fig. 5.3 which is a planar slice of the second order kernel in the f1 = f2 plane described above and
highlighted in Fig. 5.2.
H2 (a, a) =
Iload (2a)
0.5Vin (a)0.5Vin (a)
Iload (2a)
0.5Vin (a)0.5Vin (a)
(5.5)
(5.6)
Iload (2a)
∠H2 (a, a) = angle
0.5Vin (a)0.5Vin (a)
(5.7)
|H2 (a, a)| = abs
It appears that the magnitude of the second order kernel in this plane may be insignificant but
recall that the responses corresponding to this kernel are a function of the input squared so even a
small gain can produce a significant output. The time domain waveforms and frequency analysis in
Fig. 3.19 show the second harmonic of the perturbation frequency in the load current signal analysis.
Also recall that the load current responses are occurring at twice the frequency as the corresponding
54
|H(f2,f3)|
∠ H(f ,f )
2 3
1
100
0.5
0
-100
0
2
0
10
f3(Hz)
2
10
0
0
10
10
f3(Hz)
f2(Hz)
10
0
10
f2(Hz)
Figure 5.2: Magnitude and phase of H2 (f1 , f2 ) kernel in R3 showing f1 = f2 plane.
value on the x-axis. This may explain the large change in phase at approximately 500 Hz in Fig. 5.3
from -175 degrees to 156 degrees, or approximately -30 degree phase shift if 156 degrees is viewed as
-204 degrees. The single frequency sweep only tested up to 1000 Hz and did not show any significant
effects from an input filter at this frequency, Fig. 5.1, but the second order response for a 500 Hz
perturbation is right at 1000 Hz. Higher single frequency perturbations beyond 1000 Hz is necessary
to investigate further. Opening the case to analyze all of the passive filter components was not an
option at the time of testing. If this behavior in the phase of Fig. 5.3 is due to a passive input filter,
and assuming it is not a nonlinear effect such as an inductor saturating since there is not sharp
change in the magnitude, the first order kernel will also be affected since this is part of the linear
response.
5.3
Additional Experiment Automation Features
A two frequency scan is required to measure the remaining points on the second order kernel
surface that are not in the f1 = f2 plane. The experiment automation script was modified to
accommodate a two frequency perturbation sweep. Additional CAN bus message mailboxes were
configured on the DSP to receive a second perturbation frequency and magnitude. Procedures for
capturing data from the Yokogawa oscilloscope were also modified due to perturbations with a low
frequency and higher frequency simultaneously. Single frequency perturbation sweeps allow the
55
−3
Gain (abs)
1.5
x 10
1
0.5
X: 0.1
Y: 0.0002423
0 −1
10
0
1
10
10
2
10
3
10
Phase (deg)
200
0
−200 −1
10
0
10
1
10
Frequency (Hz)
2
10
3
10
Figure 5.3: H2 (f1 , f2 ) in f1 = f2 plane measured with single frequency sweep.
capture window to be scaled as frequency increases through the sweep using the time per division
setting. The time per division setting for two frequency perturbation sweep was held constant at
0.01 seconds per division and an acquisition length of 12.5 kilosamples for each of the two channels.
These were chosen for perturbing at a minimum of 50 Hz and a maximum 1000 Hz since there are
ten time divisions, 0.1 seconds total in this case, per oscilloscope capture. This means five cycles of a
50 Hz signal can be captured with 100 cycles of a 1000 Hz signal simultaneously if necessary without
sampling problems due to 125 kHz sampling. The Yokogawa DLM2024 has a 1.25 megasample
maximum acquisition length if a lower minimum perturbation frequency has to coincide with 1000
Hz. Larger acquisition lengths increase post processing time and can fill mass storage on a PC
quickly so taking a larger sample set that necessary is not desirable.
Another feature was added to the automation script to compensate for the LC output filter
resonance at 912 Hz. The DSP was controlled in open loop so an estimation of the LC filter gain
at each perturbation frequency was calculated and used to scale an ideal voltage perturbation
magnitude. The single frequency sweep was performed in stages for sections of the frequency range
so perturbation amplitudes were scaled along the way but automated calculations based on the LC
filter parameters allowed amplitudes to be scaled on the fly.
Closed loop control on the the DSP is still possible in the future but a purpose built LC output
filter should be employed with a cutoff frequency higher than 1000 Hz or at least a lower Q-factor.
56
H2(f2,f3)
1
0.5
0
500
500
400
400
300
300
200
200
100
f3(Hz)
100
0 0
f2(Hz)
Figure 5.4: Perturbation frequency selection for first experiment.
5.4
First Two Frequency Perturbation Test
A two frequency sweep was performed to measure additional points on the second order kernel
that do not lie in the f1 = f2 plane. This first test measured from 50 Hz to 500 Hz in increments
of 50 Hz on the NHR 9200 at 3 kW setting, identical load settings as in previous tests. Due to
symmetry of self kernels only measurement of half the kernel surface was necessary. Perturbation
frequencies were cycled by choosing the lower triangle of quadrant I, or the f1 > f2 region. Fig. 5.4
shows the perturbation frequency pairs for a and b used during the sweep with a star in the z = 0
plane, which is also the (f1 , f2 ) coordinates on the kernel surface corresponding to a load current
response at f1 + f2 Hz. Consider the two perturbations at frequencies a and b to produce a voltage
reference signal given in Eq. 5.8 for a 15 V peak sinusoid for each perturbation frequency a and b.
Vref erence = 380 + 15cos(2πat) + 15cos(2πbt)V
(5.8)
The value of the second order Volterra kernel can be calculated by first measuring the magnitude
and phase of the two input perturbations, V (a) and V (b), as complex numbers and the load current
response at frequency a + b or I(a + b). This is shown in Eq. 5.9, 5.10, and 5.11. Values of a and b
were swept from 50 Hz to 500 Hz in increments of 50 - only half of the space, a > b, was actually
measured. Results of the magnitude and phase responses are shown in Fig. 5.5.
H2 (a, b) =
Iload (a + b)
Vin (a)Vin (b)
57
(5.9)
|H2(f2,f3)|
−4
∠ H2(f2,f3) (deg)
x 10
100
3
−4
x 10
50
2.5
3
0
2
2
100
1.5
1
0
500
300
200
f3 (Hz)
0
100
300
−200
0.5
400
200
f2 (Hz)
−50
400
−100
1
400
400
500
0
200
200
f3 (Hz)
0
100
f2 (Hz)
−100
−150
Figure 5.5: Two frequency sweep measuring H2 (f1 , f2 ) kernel measurement, 50-500 Hz.
Iload (a + b)
|H2 (a, b)| = abs
Vin (a)Vin (b)
∠H2 (a, b) = angle
(5.10)
Iload (a + b)
Vin (a)Vin (b)
(5.11)
The phase data from the two frequency sweep in Fig. 5.5 appears to match the trend seen in
the single frequency sweep data from Fig. 5.3 approaching approximately -160 degrees at 500 Hz.
Data points in Fig. 5.5 do not fall in the f1 = f2 plane but these can be measured simultaneously
by analyzing the second harmonics of the two perturbation frequencies. Each of these points is
measured more than once since frequency b is held constant for several samples as a is incremented.
Once half the surface is measured the data points can be mirrored across the f1 = f2 plane to yield
a completed kernel surface.
This first sweep was done to test new features in the two frequency sweep so a limited frequency
range was used. A sweep from approximately 50 Hz to 1000 Hz is shown in the next section with
some modifications to perturbation frequency selection.
58
5.5
Use of Prime Numbers for Perturbation Frequencies
Recall the possibility of a nonlinear system with significant second order kernel to produce
responses at intermodulation frequencies when perturbed with two input perturbation frequencies.
These responses can occur at 2a, 2b, a + b, and a − b where a and b are the perturbation frequencies.
This does not take into account the potential for a significant third order kernel which could add
even more responses to the output at 3a, 3b, 2a + b, 2a − b, and a + 2b. If a and b are not chosen
with this in mind it is possible for overlap to occur between 2a, 2b, and a + b.
Selecting prime numbers as perturbation frequencies helps to avoid responses from orders greater
than one from overlapping on the frequencies to be measured. Since a prime number is not a
multiple of any other number this eliminates the possibility of having a harmonic response at the
perturbation frequency. Intermodulations will not occur at the perturbation frequency either if only
a two frequency perturbation is given. Prior research in the literature considers signal selection
for perturbation signals with more than two signals which can produce a multitude of responses
especially if a system has significant Volterra kernels beyond the second and third order, which
expands the possibility of having harmonics and intermodulation responses at the frequencies with
which one wishes to make measurements.
A set of prime numbers was selected for perturbation frequencies in the range of 50 Hz to 1000
Hz approximately, given as a vector in Eq. 5.12. The two tone sweep was produced similar to the
one shown in Fig. 5.4 only with the new set of prime numbers.
P rime N umbers = [53 73 113 139 181 227 257 293 331 353 389
433 479 521 563 593661 727 757 809 839 883
919 941 971 1009]
(5.12)
Additional considerations needed to be made when measuring the magnitude and phase of
Vin and Iload since the frequencies in question did not have an integer number of periods in the
oscilloscope capture window. Multiple FFTs were performed for each data set, truncating the length
of data points to achieve as close to an integer multiple of periods of each perturbation frequency
on both the input voltage and load current signals. This yielded accurate frequency bins very close
to the harmonic and intermodulation frequencies as well as the original perturbation frequencies.
Fig. 5.6 shows the magnitude and phase of the second order kernel H2 (f1 , f2 ) calculated using
Eq. 5.10 and 5.11. The measured magnitude values near the f1 = f2 plane follows a similar trend as
59
the measured values from the single frequency perturbation in Fig. 5.3 with a maximum near 400
Hz then decreasing followed by another increase near 1000 Hz. The phase surface near the f1 = f2
plane is also similar to the phase measurement results from the single frequency perturbations, near
zero degrees at 50 Hz decreasing to -170 degree approximately near 500 Hz with a sharp increase to
150 degrees. This behavior carries over to the rest of the surface.
This second order kernel is symmetric across the f1 = f2 plane since it is a self kernel of an
SISO system. The data points are mirrored across this plane to plot the full second order Volterra
kernel, H2 (f1 , f2 ) in Fig. 5.7. Note that no new information is given in this plot, simply a mirroring
of data points in order to view the complete surface in octant I.
60
−4
|H2(f1,f2)|
x 10
5
−4
x 10
6
4.5
4
4
2
3.5
0
3
3
10
2.5
2
1.5
1
2
3
10
10
f2 (Hz)
0.5
2
10
f1 (Hz)
∠ H2(f1,f2) (deg)
150
200
100
0
50
−200
3
0
10
−50
−100
3
10
2
−150
10
f2 (Hz)
2
10
f1 (Hz)
Figure 5.6: Prime number two frequency sweep for H2 (f1 , f2 ), 50-1000 Hz.
61
−4
|H2(f1,f2)|
x 10
5
−4
x 10
6
4.5
4
4
3.5
2
3
2.5
0
3
10
2
1.5
1
3
2
10
10
f2 (Hz)
0.5
2
10
f1 (Hz)
∠ H2(f1,f2) (deg)
150
100
200
50
0
0
−200
3
10
−50
3
10
2
10
f2 (Hz)
−100
−150
2
10
f1 (Hz)
Figure 5.7: Symmetrical H2 (f1 , f2 ) Volterra kernel, 50-1000 Hz.
62
CHAPTER 6
DISCUSSION AND CONCLUSION
6.1
Volterra Series and DC Constant Power Loads
Measurements from distributed power system components allow system designers to refine
analytical models and predict stability problems before energizing a complete system. Previous
modeling techniques rely on linear models to measure linear responses of constant power loads to
input voltage perturbations. These responses occur at the same frequency as the perturbation.
However, constant power loads have a nonlinear relationship between input voltage and load current
which produces load current responses at harmonic and intermodulation frequencies of input voltage
perturbations. Linear systems theory is unable to account for these responses so a nonlinear model is
necessary to characterize these extra responses. We use the Volterra Series to model these nonlinear
responses in the frequency domain through Volterra kernel measurement, an extension of impedance
measurement.
We developed a measurement unit for 380 VDC loads up to 5 kW was developed through simple
modifications of a common three phase power converter topology. We used interleaved modulation
to achieve an effective switching frequency of 30 kHz which is sufficient for impedance and Volterra
kernel measurements up to 1000 Hz.
We also identified limitations of Taylor Series models for constant power loads. The Taylor
Series expansion is inadequate for modeling the constant power load curve beyond the nominal
operating voltage. While it may be tempting to assume that increasing the order of the series will
always yield a better model, in this case the series diverges beyond the nominal operating voltage.
This is only problematic if the designer wants to raise the over voltage trip limit. This is not as
common as broadening under voltage operation since this complements system wide protective
actions for fault recovery.
The possibility of higher order responses from constant power loads must be taken into account
to avoid erroneous measurements, even if measurement goals do not include Volterra kernels.
Development of a perturbation signals for impedance measurement on constant power loads,
63
corresponding to a linear model, must include checks to ensure that possible higher order responses
do not overlap with the linear frequency responses to be measured. If this is not taken into
account, error will be introduced by contributions from higher order responses that overlap the
linear response frequencies. Many researchers attempt to develop perturbation signals that have
both large frequency content for measuring multiple data points simultaneously, either for linear
impedance or for Volterra kernels, by carefully selecting frequencies so possible response frequencies
do not overlap. We used a simple two frequency perturbation sweep to avoid many of these problems
with the trade off of increased measurement time. The device under test was controlled by the user
to have a fixed operating power point, which may not be the case of testing a converter operating
within a power system in situ. Thermal problems did not arise from running the equipment for an
extended time but this could easily be a problem for loads which are not designed for continuous
operation over several minutes.
6.2
Interpretation of Second Order Kernel
Constant power loads have significant kernels above the first order due to the nonlinear constant
input power characteristic resulting from regulated outputs. If a converter closed loop controller
was disabled and a fixed duty cycle remained, the overall behavior at the input terminals would
appear much like a resistor that draws more current with an increase in input voltage. In this case
the kernels above the first order would all be zero. The significance of higher order kernels, above
the first order, can therefore be interpretted as the ability of a converter to maintain constant input
power even under input voltage perturbations.
6.3
Future Work
Further research with Volterra Series for distributed power systems can provide additional
nonlinear models for source and corresponding analysis using both source and load models.
Impedance measurements of electrical sources use a shunt current injection between a source and
load. Kernel measurements can use similar equipment with modifications to perturbation signals
and signal processing of measurements.
64
System stability analysis using Volterra models has not been studied for distributed power
systems. This may be possible once Volterra kernel models of sources and loads are identified but
additional mathematics is necessary similar to the impedance ratio for linear models.
Finally, Volterra kernel measurements can also be applied to AC systems in the D-Q reference
frame. Constant power load integration problems are just as prevalent in AC systems as in DC
systems, even more so due to use of AC in terrestrial power systems.
65
APPENDIX A
KERNEL MEASUREMENT AUTOMATION
SUPPLEMENTAL
A.1
Automation Script in Matlab
% Test 3 of two-tone perturbation script
% adding feature to scale the amplitude based on the gain
% of the perturber LC filter having resonance near 912 Hz.
% Model of perturber LC filter
L=830e-6;
C=110e-6;
myfilt=tf([1],[L/3*C 0 1]);
primnum = [53 73 113 139 181 227 257 293 331 353 389 433 479 521 ...
563 593 661 727 757 809 839 883 919 941 971 1009];
a=1;
for i=1:length(primnum)
for j=1:length(primnum)
if i==j || i>j %only measure half triangle for H2
mymultitone(a,1:2)=[primnum(i) primnum(j)];
a=a+1;
end
end
end
clear a i j
%Initialize CAN and start
canch1 = canChannel(’Kvaser’,’Leaf Light 1’,1);
configBusSpeed(canch1,100000);
start(canch1)
%CAN message objects
afe_enable = canMessage(268435456, true,1);
inv_enable = canMessage(268435457, true,1);
test_frequency1 = canMessage(101,true,4);
test_amplitude1 = canMessage(102,true,4);
66
test_frequency2 = canMessage(103,true,4);
test_amplitude2 = canMessage(104,true,4);
ret = mexDLComStart(7, ’91N819620’); %91N819620 is DLM2024 serial no.
ret = mexDLSend(’ACQUIRE:RLENGTH 12500’);
ret = mexDLSend([’TIMEBASE:TDIV’,’ ’,num2str(0.01)]);
%enable AFE
pack(afe_enable,uint8(1),0,8,’LittleEndian’);
transmit(canch1,afe_enable)
%wait 1 seconds
pause(1)
%enable INV
pack(inv_enable,uint8(1),0,8,’LittleEndian’);
transmit(canch1,inv_enable)
pause(3)
for i=106:length(mymultitone)
freq1 = mymultitone(i,1)
freq2 = mymultitone(i,2)
% Evaluate the amplitudes needed for 15V AC signals based
% on gain of the perturber LC filter
amplitude1 = 15/abs(evalfr(myfilt,j*2*pi*freq1));
if freq1==freq2
amplitude2 = 0;
else
amplitude2 = 15/abs(evalfr(myfilt,j*2*pi*freq2));
end
% Send new amplitudes to DSP over CAN, if frequencies the same
% i.e. single frequency perturbation, make amplitude 2 = 0
pack(test_amplitude1,single(amplitude1),0,32,’LittleEndian’); %set amplitude
transmit(canch1,test_amplitude1)
pack(test_amplitude2,single(amplitude2),0,32,’LittleEndian’); %set amplitude
transmit(canch1,test_amplitude2)
% Send new frequencies to DSP over CAN
%ret = mexDLSend([’TIMEBASE:TDIV’,’ ’,num2str(tdivvec(i))]);
pack(test_frequency1,single(freq1),0,32,’LittleEndian’);
pack(test_frequency2,single(freq2),0,32,’LittleEndian’);
67
transmit(canch1,test_frequency1);
transmit(canch1,test_frequency2);
pause(1)
%pause(1)
%pause(3)
[ret, waveData,Time] = mexDLGetWave;
ret = mexDLSend(’:START’);
%save(strcat(num2str(fvec(i)),’.mat’),’waveData’,’Time’)
save(strcat(’run’,num2str(i)),’freq1’,’freq2’,’waveData’,’Time’)
pause(1)
%plot(Time(:,1),waveData(:,1),Time(:,1),waveData(:,2),’r’)
%print(myfig,’-dpng’,num2str(fvec(i)))
%pause(2)
end
%disable INV
pack(inv_enable,uint8(0),0,8,’LittleEndian’);
transmit(canch1,inv_enable)
%wait 3 seconds
pause(3)
%disable AFE
pack(afe_enable,uint8(0),0,8,’LittleEndian’);
transmit(canch1,afe_enable)
%clean up
stop(canch1); %stop CAN
mexDLComEnd; %stop Oscilloscope comm.
68
APPENDIX B
SCALABILITY TO HIGHER POWER
B.1
Higher Power Testing
The current interleaved topology using 1200 V IGBTs is easily capable of testing 380 VDC
loads for the 380 V ecosystem initiatives of the EMerge Alliance and accompanying contributors.
The power stack used here is capable of much higher power but is limited by the upstream sources
notable the 7 kVA transformer and 35 kVA lab feed. A direct connection to the lab 208 V feed is
possible, bypassing the input transformer, which would then be limited by lab circuit breakers. If
both of these were modified then the next piece needing upgrade is the IGBT cooling either with a
larger air cooled heatsink or a water cooled cold plate such as one specifically made for the IGBTs
used from Microcool Division of Wolverine Tube.
Additional increased power capability is possible by adding additional interleaved phases which
also has the benefit of higher effective switching frequency if the corresponding changing to the
PWM carriers is performed. In the current configuration the F28335 is maxed out on PWM outputs
so a different configuration is necessary, perhaps moving the rectifier controls to a different controller
which then would allow a 6 phase interleaved converter driven from a single F28335.
Testing higher voltage loads may require a different topology. The interleaved topology used
here with 1700 V IGBTs would be capable of a 1000 VDC input since some derating of the IGBTs
is necessary for safe operating area (SOA). A 1000 VDC feed to the interleaved converter would
provide capability to tests DC loads for systems up to 900 V approximately or even 1000 VDC
if the feed is increased to a slightly higher value although this is pushing the limit. A different
approach would use a three-level topology allowing an input of 1500-2000 VDC which could then
produce safe outputs at 1000 VDC. This voltage class has been used aboard the DDG 1000 ship,
some photovoltaic plants, and also has industry support such as DC circuit breakers from ABB in
the Tmax molded circuit breaker line.
69
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73
BIOGRAPHICAL SKETCH
Jesse Leonard was born in Sarasota, FL. After graduating from Sarasota High School in 2006, he
began his undergraduate studies at Florida State University in electrical engineering. In 2010 he
graduated Summa Cum Laude from FSU with a Bachelor of Science in electrical engineering and
began his doctoral studies at FSU. He has conducted research with his advisor Dr. Chris S. Edrington
at the FSU Center for Advanced Power Systems since 2009 in the areas of power electronics, digital
control with embedded systems, and real-time simulation with power and controller Hardware-inthe-Loop (HIL) techniques using RTDS and dSPACE. In the summer of 2013 Jesse was an R&D
Intern at GE Global Research Niskayuna, NY in the Electric Propulsion Systems laboratory. Jesse
has been a student member of IEEE since 2009.
74