A Novel Impedance Measurement Technique for
Power Electronic Systems
Peng Xiao, Student Member, IEEE, Ganesh Venayagamoorthy, Senior Member, IEEE,
and Keith Corzine, Senior Member, IEEE
Real-Time Power and Intelligent Systems Laboratory
Electrical and Computer Engineering Department
University of Missouri-Rolla, Rolla, MO 65401, USA
Abstract - When designing and building power systems that
contain power electronic switching sources and loads, system
integrators must consider the frequency-dependent impedance
characteristics at an interface to ensure system stability. Stability
criteria have been developed in terms of source and load
impedance for both dc and ac systems and it is often necessary to
measure system impedance through experiments. Traditional
injection-based impedance measurement techniques require
multiple online tests which lead to many disadvantages. The
impedance identification method proposed in this paper greatly
reduces online test time by modeling the system with recurrent
neural networks. The recurrent networks are trained with
measured signals from the system with only one injection. The
measurement and identification processes for dc and three-phase
ac interfaces are developed. Simulation tests demonstrate the
effectiveness of this new technique.
I. INTRODUCTION
Stability analysis in power electronics based distributed
power systems is a more crucial task than in conventional
power systems due to the nearly ideal control capability of
many modern power converters. The excellent load regulation
capability of a converter is a desirable feature in many
applications, but it also makes the converter a constant-power
load device, which is a potential cause of negative impedance
instability [1].
For small-signal stability analysis, most research focuses on
the impedance/admittance method that involves examining the
Nyquist contour of the product of the source impedance and
load admittance in a dc system [2]. In recent years, based on
the impedance/admittance method, a variety of stability
criteria and design approaches for both dc and ac systems have
been proposed [3-4].
In the design, integration and analysis of distributed power
systems, it is often necessary to obtain the small-signal
impedance/admittance characteristics of an existing power
electronic component or subsystem at a given operating point.
To get the frequency-dependent characteristics by experiment,
periodic voltage or current perturbations are usually injected
to the system while it is under operational power.
Measurements of the perturbed system are then taken and
processed to determine the impedance at a specific frequency.
Several methods have been proposed for impedance
measurement in high-power ac systems, including utilization
of three-phase bridge converters, wound-rotor induction
machines and three-phase chopper circuit [5-6]. An impedance
measurement technique utilizing a line-to-line current
injection chopper circuit was recently proposed [7], which has
a simple structure and is much easier to implement compared
with other methods.
A common problem of these impedance measurement
techniques is that they require injection of perturbation signals
to the system one frequency at a time. To obtain the
impedance characteristics over a wide frequency range for
stability analysis, multiple tests must be repeatedly performed.
During each test, a perturbation signal of a specific frequency
is injected into the system, and the voltages and currents are
measured and recorded. When tests for all frequencies are
finished, the recorded data is processed to calculate the
impedance value at each frequency. The main disadvantages
of this procedure include: (i) It takes a long online time to
complete the injections for all frequencies; (ii) The operating
point of the system may vary during the prolonged test
procedure, which can lead to inconsistency in the measured
system impedance characteristics; (iii) If the impedances at
additional frequencies are needed, new tests must be
performed on the system, which may cause interruption to the
normal operation of the system.
In this paper, a different approach is taken to identify the
impedance characteristics of a dc or three-phase ac system.
Instead of measuring system impedance at one specific
frequency each time, the proposed method requires only one
injection and measurement process. The recorded data is used
not to directly calculate impedances, but to build a model of
the system at the specified operating point by training a
recurrent neural network (RNN). The trained neural network
is then used to obtain the impedance characteristics.
Simulation results show that the proposed method is capable
of accurately identifying impedances of both dc and three-
phase ac systems.
II. IMPEDANCE MEASUREMENT FOR STABILITY ANALYSIS
The analysis of small-signal stability around steady states of
a power electronic system is important for both control design
and component integration. In the design stage, if the
mathematical model of the system is known, it can be used to
extract the impedance characteristics of the system. In
9551-4244-0655-2/07/$20.00©2007 IEEE
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addition, models of different system components can be
connected together to simulate their behaviors under different
operating conditions, and linearization tools are usually
available to determine the state-space matrices of the system.
The situation is different in the component integration stage,
when the hardware components are connected together to
form a system. In this case, the detailed models of the
components are often not available, especially when the
components are designed and manufactured by different
vendors. To evaluate the stability of the integrated system,
measurements and tests are necessary to obtain the impedance
information of each component.
The injection-based impedance measurement techniques
utilize small voltage or current signals to perturb the system
under study, while it is operating in steady state. Various
injection devices have been proposed. For low-power systems,
power amplifiers can be used. For high-power systems,
different configurations of chopper circuits are often used, in
which switching devices are turned on and off to provide a
varying impedance branch that creates the perturbations.
Fig. 1. Impedance Measurement in dc systems.
Fig. 1 shows the shunt injection diagram for dc systems.
The system is divided into two parts, designated as source and
load, although the actual power flow can be either from the
load to the source or from the source to the load. The injection
device is connected at their common interface. In the shunt
injection system, a current signal of a specific frequency is
injected into the system at a steady-state operating point. The
dc voltage at the interface, together with the load and source
currents, are measured. The waveforms of these signals are
recorded. Fourier transform is then used to process these
signals and determine the magnitudes and phase angles of the
components at the injection frequency. The small signal
impedances of the load and source can then be calculated with
� � � �� � � �
� �
� �il
i
il
is
i
is fI
fVfZ
fI
fVfZ � (1)
where fi is the injection frequency, V, Is and Il are complex
numbers obtained from Fourier transform of the dc voltage,
source current, and load current signals. This single injection
test gives the impedance information of the system at a single
frequency fi. To obtain impedances at other frequencies, the
same test procedure is be repeated, each time with a different
injection frequency.
The impedance measurement test for three-phase ac systems
is more complicated. As shown in Fig. 2, the shunt injection
requires a three-phase current source and measurement of nine
signals. Also, the impedance information for the source and
load is represented by a 3 by 3 matrix. For three-phase
balanced systems without neutral wire, reference frame
transformation theory provides a convenient way to study the
impedance characteristics. In the synchronous reference
frame, the impedance and admittance take matrix forms
»
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«
¬
ª
»
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º
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ª
»
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º
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»
¼
º
«
¬
ª
d
q
d
q
d
q
d
q
V
V
I
I
I
I
V
V
YZ (2)
where
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�
dddq
qdqq
dddq
qdqq
YY
YY
ZZ
ZZ
1ZYZ (3)
To determine the four impedance entries in the matrix, two
sets of injection signals are needed at each frequency. This
actually doubles the number of tests needed to identify the
system impedance characteristics over a wide frequency range.
Fig. 2. Impedance Measurement in three-phase ac systems.
III. RNN-BASED IMPEDANCE IDENTIFICATION METHOD
The key point of the proposed method is the modeling of a
dynamic system under study. If a model can be built to
accurately produce the small-signal time-domain responses of
the system to all kinds of inputs, then it also has the ability to
produce the frequency-domain characteristics of the system.
For an existing hardware system, the internal device
parameters are often unavailable, thus it is impractical to build
the model based on knowledge of the device’s internal
structure and control algorithms. Instead, the modeling
process has to rely on measurement of its input and output
signals.
A. Recurrent Neural Network as a Modeling Tool
For dynamic systems, recurrent neural network has been
demonstrated to be an effective modeling tool in many
applications. Unlike the widely-used multilayer feedforward
neural networks that can only establish static mapping
relationship between inputs and outputs, RNNs contain
internal feedback loops and states. The outputs of RNNs are
functions of internal states as well as the inputs, just as they
are in dynamic systems. The feedback mechanism provides a
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memory to the recurrent networks so that they are capable of
modeling systems with internal dynamics. In this study, the
Elman RNN topology is chosen as the modeling tool.
Fig. 3 shows a simplified diagram of a two-layer Elman
recurrent network structure. For a network with l inputs, m
hidden neurons, and n outputs, the hidden layer equations are
� � � � � �¦¦
��
m
j
jjk
l
i
iikk tdwtxwts
1
)2(
1
)1( 1 (4)
where � � � �� �kstd kk sgm (5)
x(t) is the input vector, w(1) is the weight matrix associated
with the inputs and hidden neurons, and w(2) is the weight
matrix associated with the states and hidden neurons.
The outputs of the network are determined by
� � � �¦
m
i
iikk tdwty
1
)3( (6)
where w(3) is the weight matrix associated with the hidden
neurons and the outputs.
Fig. 3. Topology of the Elman recurrent network.
Past research has demonstrated the ability of the RNN to
learn process dynamics and provide efficient forecasts, and it
has found application in many areas such as wind speed and
power forecasting [8], design of a power system stabilizer [9],
induction motor speed estimation[10], and prediction of
elephant migration [11].
B. Modeling with RNN
To model a dynamic system with RNN, the network must
be trained with measured data so that it learns the behaviors of
the system. It should be noted that the purpose of the training
is not to obtain a complete model of the complex nonlinear
power electronic system. Instead, throughout the test, the
system is running at a specific steady-state operating point.
Small variations of voltage or current are added to the system
to create perturbations. The neural network is then used to
model the behavior of the system responding to small signal
inputs.
The measured signals are voltage and current waveforms at
the interface of the source and load. These waveforms are
used as training data for the input and target output of the
RNN. During the training process, input data are fed to the
network to calculate the output, and the internal weight
parameters of the RNN are adjusted based on the output error.
Several RNN training algorithms are available. Both back-
propagation and particle swarm optimization algorithms [12]
are used in this study.
C. Random PWM Signal Injection
Training of the RNN requires measurement data of a
perturbed system, thus injection of perturbation signals is still
necessary in the proposed method. For the shunt injection,
chopper circuits proposed in [7] are used to handle the high
voltage and power of the tested system. Fig. 4 shows the
circuit as being used for line-to-line current injection in a
three-phase ac system. The circuit contains a bi-directional
switch that controls the branch’s impedance, which in turn
causes variations in the branch current. A properly designed
switching pattern can thus introduce a perturbation current
signal into the system. A fixed-frequency fixed-duty-cycle
PWM switching scheme was used in [7] to generate a
perturbation signal of a specific frequency.
Fig. 4. Chopper circuit shown in three-phase system injection.
Fig. 5. Spectrum of a random PWM signal.
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For the RNN to learn the dynamic behavior of the system,
the spectrum of the perturbation signal must cover a wide
frequency range. A random PWM signal with limited
bandwidth is used in this study, which can be generated by a
PWM switching scheme with random duty cycle and random
switching frequency. In each PWM cycle, the switching
frequency is randomly chosen between two bounds, fmin and
fmax, which are determined according to the frequency range of
interest. Fig. 5 shows the spectrum of such a switching signal,
with fmin = 1 kHz and fmax = 5 kHz. It can be seen that the
signal has a relatively even magnitude at frequencies below 3
kHz. At frequencies above 3 kHz, the magnitude decreases
with a slope between 20 dB/decade and 40 dB/decade.
During the period when the system is being perturbed by a
random PWM switching circuit, the voltage and current
signals of the source and load are measured, filtered and
recorded. For a dc system, the recorded data is normalized and
used directly to train the RNN. Either the voltage or the
current signal can be used as the input, and the other signal is
used as the target output. For a three-phase ac system, the
measured signals are first transformed into the synchronous
reference frame so that the fundamental components become
dc signals. After normalization, the data is then used for RNN
training.
The training process of the RNN involves repeatedly
feeding the network with the input data, calculating the
outputs, and comparing the calculated outputs with the target
outputs. The network weights are modified in each epoch to
minimize the error. The training stops when the error is below
a certain threshold value.
A well-trained RNN can produce correct outputs even when
the inputs are different from its training data. It is this
generalization capability of RNNs that makes them suitable
for impedance characteristics extraction. The trained RNN can
be seen as an accurate small-signal model of the system, and
tests can be performed on the RNN instead of on the real
system to obtain the impedance information.
D. Identification Process for dc Systems
For a dc system, to determine the impedance value at a
frequency fi, a sinusoidal signal of frequency fi is fed to the
trained RNN to produce the output. The input and output
signals are then processed with Fourier transform to determine
their magnitudes and phase angles. The impedance/admittance
of the system at fi can be calculated with (1).
E. Identification Process for Three-Phase ac Systems
For a three-phase ac system, if the RNN is trained with
currents as inputs and voltages as outputs, then it is relatively
easier to calculate the impedance matrix. For each frequency fi,
there are four impedance values to be determined and two
steps are needed.
In the first step, a sinusoidal signal of frequency fi is fed to
the trained RNN as iq, while the input signal id is set to a zero
vector. The RNN output voltages vq and vd are then calculated
with (4)-(6). According to (2), two impedance entries can be
determined by
� � � �� � � �
� �
� �iq
id
idq
iq
iq
iqq fI
fVfZ
fI
fV
fZ (7)
where Vq, Vd and Iq are the complex results from Fourier
transform of vq, vd and iq, respectively. The second step is
similar to the first one except that the sinusoidal signal is fed
to the RNN as id, while iq is set to zero. The other two
impedance entries can be determined by
� � � �� � � �
� �
� �id
iq
iqd
id
id
idd fI
fV
fZ
fI
fVfZ (8)
Fig. 6 shows a flowchart of the proposed impedance
measurement procedure for three-phase ac systems. It can be
seen that the online part of the procedure only includes the
injection of the random PWM signal and data measurement,
and the rest of the process only requires offline training and
calculations.
Fig. 6. Flow chart of the proposed impedance measurement procedure for
three-phase ac systems..
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Fig. 7. Test system for dc impedance measurement.
IV. SIMULATION RESULTS
The proposed impedance measurement technique was
verified with simulation results of both dc and three-phase ac
systems.
A. Test Results in dc Systems
A 3.7 kW variable-speed motor drive system is used for the
dc test, and its diagram is shown in Fig. 7. The example
system consists of a three-phase active rectifier, 300 V dc link,
a three-phase inverter, and a 5 hp induction motor. Input and
output filters are used to reduce the PWM switching noises.
The dc link interface of a rectifier-inverter-induction motor
system is used for the dc signal injection. The location of the
injection device is shown in Fig. 7, where the current source
on the dc link represents the chopper circuit as shown in Fig. 4.
To measure the impedance of the subsystem to the right of the
injection device, both vdc and iload are measured and saved. The
frequency range of interest is from 10 Hz to 1 kHz, and the
frequency bounds of the random PWM signal is set to be 400
Hz and 1 kHz. The measured signals are filtered to avoid
aliasing, and sampled at a frequency of 10 kHz. The data is
then normalized to be within the range from -1 to 1. An Elman
recurrent neural network is trained with the voltage data as
input and current data as output. The extracted impedance
characteristics are shown in Fig. 8. The actual impedance
curves are obtained based on the linearized state-space matrix
in a simulation model of the system. As can be seen, a very
close match between the measured and actual values is
achieved.
B. Test Results in ac Systems
The ac test system includes a salient-pole synchronous
generator feeding an R-L load (R = 27.29 :, L = 19.9 mH).
The chopper circuit is connected to the b and c phases of the
generator terminals. The injection and data processing
conditions are similar to those in the dc test, except that the
abc signals are transformed into the synchronous reference
frame before normalization. An Elman RNN is used for the
training, where the inputs are the currents and the outputs are
the voltages.
Fig. 8. Actual and measured impedances of the dc subsystem..
For a symmetric three-phase R-L load, its impedance matrix
in
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