HEADER 1
Adaptive Bus-Impedance-Damping Control
of Multi-Converter System Applying
Bidirectional Converters
Roosa-Maria Sallinen, Student Member, IEEE, Tomi Roinila, Member, IEEE
Abstract—Modern dc-power-distribution systems utilizing en-
ergy storages are often dependent on the operation of bidirec-
tional power-electronics converters. Such distribution systems
typically consist of several feedback-controlled converters prone
to experience stability issues due to cross-effects among the
different converters. Studies have presented adaptive control-
based techniques to prevent such stability issues, but most studies
have not fully considered their implementation on a bidirectional
converter. The system dynamics may vary significantly depending
on the operating point and particularly the direction of the
bidirectional power flow. Therefore, specific care should be taken
in the design of the adaptive stabilizing control to guarantee
that the system’s regular operation is not impeded when the
stabilization is implemented on a bidirectional converter. This
paper proposes a procedure to implement an adaptive stabiliz-
ing control method on a bidirectional converter with minimal
changes to the regular controller. We add an adaptive resonance
term to the bidirectional converter’s voltage controller that
enhances stability and damping around the identified resonance
frequency without impeding the converter’s regular operation.
The resonance term is adjusted periodically based on online
impedance measurements and the chosen design criteria. As a
result, the controller can dampen resonances and prevent adverse
impedance-based interaction. Experimental measurements based
on a multi-converter setup demonstrate the effectiveness of the
proposed methods.
Index Terms—adaptive stabilization, dc power systems,
resonance-based controller, bidirectional converter, multi-
converter system
I. INTRODUCTION
BATTERY energy storage systems (BESSs) play an in-creasingly important role in many power-distribution sys-
tems, such as dc microgrids [2], electric ships [3], and electric
aircraft [4]. The operation of these systems typically relies on
a bidirectional power-electronics converter, which enables the
bidirectional power flow and controls the charge and discharge
processes of the energy storage. For such modern power-
distribution systems, a dual active bridge (DAB) converter
has gained prominence due to its flexible power flow control,
zero voltage-switching, high efficiency, galvanic isolation, and
modular structure [5]–[7].
Typically, the bidirectional converter is a part of a more
complicated multi-converter system in which several convert-
ers are connected to a common dc bus, as shown in Fig. 1.
A preliminary version of the present paper was presented at the IEEE
Workshop on Control and Modeling for Power Electronics (COMPEL) 2020,
see [1].
R.-M. Sallinen and T. Roinila are with the Faculty of Information Tech-
nology and Communication Sciences, Tampere University, 33101 Tampere,
Finland (e-mail: roosa.sallinen@tuni.fi, tomi.roinila@tuni.fi).
v
bus
i
batt
Bidirectional converter
Z
o2
Z
o1
Z
bus
DC
DC
Z
on
Z
o3
Converter 2
Converter 3
Converter n
Fig. 1. Multi-converter system with a bidirectional converter.
Such a multi-converter system may experience performance
degradation due to impedance interactions between different
converters even though the converters may operate well in a
standalone mode. The controller of each converter tracks either
a voltage, current, or power reference. The high control band-
width of the load converters introduces a negative incremental
impedance at the point of coupling with the dc bus. These load
converters act as constant power loads (CPLs). The negative
incremental impedance of CPLs has a destabilizing effect on
the system [8] and is the main reason for interaction dynamics
in dc multi-converter systems [9], [10].
One way of avoiding impedance-based system performance
degradation caused by the CPLs is to add passive/active circuit
components to the existing system [9], [11]. However, the
addition of new components can increase the cost and size
of the overall system and slow down the voltage response
[12]. Therefore, a more attractive solution is to modify the
feedback loops of the individual converters based on system
stability assessment [13]–[15].
Considering the stability assessment of a multi-converter
system, traditional methods may not be effective with sys-
tems including bidirectional converters and varying control
structures. Such methods are typically based on minor-loop
gain (MLG), which is the ratio between the source subsystem
output impedance and the load subsystem input impedance.
Examples of such stability assessment methods include the
Middlebrook criterion [16] and gain-margin and phase-margin
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
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HEADER 2
criterion [17]. These MLG-based methods are not directly
applicable to systems with bidirectional power flow because
they require the subsystems and system grouping to have
specific definitions and formulas [18]. A refined approach for
analyzing the stability of a multi-converter system is to apply
the MLG by characterizing the converters based on their role
in contributing to the current and voltage control rather than
assigning them as load and source converters [19]. However,
because the control roles may change based on the application
operating modes, this method may not be straightforward for
all multi-converter systems.
Recent studies have presented passivity-based stability anal-
ysis [18], which provides an alternative approach for the
stability assessment. This analysis is based on the system
bus impedance, rather than the multi-converter system in
MLG. The bus impedance represents the total impedance
of all the bus-connected subsystems. As the passivity-based
method only requires bus-impedance identification, the method
is independent of the power-flow directions, the converter
operating modes, and the system grouping. This makes the
method especially suitable for the stability assessment of
multi-converter systems with bidirectional power flow.
The stability enhancement of CPL-affected multi-converter
systems on bidirectional converters is still an important prob-
lem where different approaches can be useful. Most stabilizing
control designs in the literature focus on either the load- or the
source-side converters rather than on bidirectional converters;
These methods reshape either the output admittance of the
CPL [10], [12], [20] or the output admittance of a source
converter [21]–[24]. Thus, these methods may not directly
apply when implementing a stabilizing controller on a bidirec-
tional converter. Stabilizing control designs for bidirectional
converters were presented in [9], [25]–[27]. However, these
methods might not be suitable for all applications, as they
utilize specific control methods (e.g., semidefinite program-
ming [9]) or are mainly focused on droop-controlled inverters
[25]–[27]. Moreover, multi-converter systems can benefit the
most from adaptive, nonparametric stabilizing methods that do
not require detailed information about the system variables.
Such a method was used in [1], where a bus impedance-based
stabilizing controller was extended to bidirectional converters,
but no experimental results were used to validate the study.
This paper proposes a general, adaptive, nonparametric
stabilizing control design method for bidirectional converters
in multi-converter systems. This method assesses the multi-
converter system stability through bus-impedance identifica-
tion, and then optimizes the stabilizing controller to dampen
resonances within the allowable frequency range. The bus-
impedance identification utilizes the existing converters of the
system, and employs measurements of the bus-side voltage
and currents of each converter. Based on the identification,
the stabilizing controller alters the bidirectional converter
impedance to provide the desired level of damping for the
identified bus impedance in both load- and source-operation
modes. Since the stabilizing controller adapts to transitions in
the bus impedance, the variations in the operating modes do
not degrade the performance. However, owing to the presented
design criteria, the stabilizing controller only functions in such
a way that the resonance is within the allowable frequency
range. This is important because the converter dynamics,
such as the current control bandwidth and poles/zeros, limit
the available stabilizing controller bandwidth. Therefore, the
stabilizing controller does not interfere with the regular con-
troller operation in a degrading manner. The controller scheme
is validated with experimental results on a multi-converter
system with a DAB converter and two inverters.
The remainder of this paper is organized as follows. Sec-
tion II presents the bus-impedance-based stability and perfor-
mance analysis of multi-converter systems. Section III uses
this performance assessment to facilitate a resonance-damping
control method for adaptive bus-voltage-damping stabilization
and describes the identification of bus impedance and the
resonance-damping control parameters, as well as the effect
of the resonance-damping control on the voltage controller. In
Section IV, experimental results validate the effectiveness of
the resonance controller on a bidirectional converter operating
both as a load and as a source. Conclusions are drawn in
Section V.
II. BUS IMPEDANCE IN STABILITY ANALYSIS OF
MULTI-CONVERTER SYSTEMS
Consider the multi-converter system in Fig. 1; for a multi-
converter system of N bus-connected converters, the system’s
bus impedance (i.e., single-port impedance) Zbus can be given
as a parallel connection of the bus-connected impedances, i.e.
[28]
Zbus(s) =
1
Z−1o1 + Z
−1
o2 + ...+ Z
−1
oN
, (1)
where N impedances are identified at the bus-side of the
corresponding subsystem (denoted by subscript o in Fig. 1),
and the positive signs for the currents correspond to the
direction into the converter from the common dc bus. The
interconnected multi-converter system can be shown to be
passive if the following requirements are met [28].
1) Zbus(s) does not have right half plane (RHP) poles
2) Re{Zbus(jω)} ≥ 0,∀ω > 0
Passivity is a sufficient but not a necessary condition for
stability. Additional concepts are required to assess other
performance metrics (e.g., the level of damping), such as the
allowable impedance region (AIR) introduced in [29]. Whereas
passivity limits the bus impedance to the RHP, the AIR is
defined as a semicircle within the RHP and its radius relates
to the chosen attenuation level.
In the case of adverse impedance-based interaction, the bus
impedance is typically characterized by a single prominent
resonance peak [29]. In such a case, the bus impedance can
be expressed as
Zbus(jω) = Zo−bus
sωo
s2 + sωo/Qbus + ω
2
o
, (2)
where Zo−bus is the characteristic impedance, ωo is the
resonance frequency, and the quality factor Qbus specifies the
level of damping. Thus, the bus impedance has a real value at
the resonance frequency, Zbus(jωo) = Zo−busQbus. In other
words, this value depicts the bus impedance peak magnitude.
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
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To achieve a chosen attenuation, the AIR demands that the
bus impedance achieves a specified quality factor, Qmax.
Therefore, the AIR can be defined in the complex plane as a
semicircle with a chosen radius of Zo−busQmax. In addition,
to simplify the performance analysis, the AIR condition can be
normalized by dividing the bus impedance and the AIR radius
by the characteristic impedance. This normalization results
in a straightforward expression for the AIR with a radius of
Qmax. Accordingly, the chosen attenuation is achieved if the
normalized bus impedance Zbus−N(jω) remains within the
specified AIR.
Based on (1), the bus impedance peaks when the parallel
sum of the impedances is zero, i.e., the impedances are of
similar magnitude but an opposite phase. Concerning the
individual converters, the shape of their impedance is strongly
affected by their control structure. In the case of current
controlling converters, their impedance can be described as a
CPL since they have a negative incremental impedance within
their feedback control loop bandwidth, i.e., the impedance
magnitude is resistive with a −180o phase. In contrast, the
impedance of voltage-controlling converters has a relatively
small magnitude except for a resonance peak around the
voltage control cross-over frequency. Likewise, droop control
has the same characteristics as voltage control except for
very low frequencies, in which the impedance magnitude is
affected by the droop coefficient [30]. When converters of
these different types are used together, the resonance peak
in the voltage-controlling converter’s impedance may cause
the parallel sum of the converter impedances to have equal
magnitudes at some frequency. If the impedances’ phase differ-
ence is around −180o at that same frequency, the denominator
in (1) becomes very small, and the bus impedance exhibits
significant resonance. This phenomenon is a typical example
of adverse impedance-based interactions of multi-converter
systems. Fig. 2 shows an example of a typical bus impedance
based on (2) in which resonance occurs at 75 Hz.
Utilizing a virtual impedance is a straightforward way to
prevent a resonance caused by impedance-based interactions.
Essentially, the resonance in the bus impedance is caused by
a resonant pole that occurs close to the voltage control cross-
over frequency. The resonance typically occurs at a frequency
close to the voltage control loop bandwidth. Thus, a virtual
impedance in the voltage controlling converter can be designed
to dampen the resonance. Even though the resonant behavior
could also be smoothed with, for example, an additional
capacitor, an adaptive control-oriented solution can offer a
more optimized solution.
III. IMPEDANCE-BASED ADAPTIVE STABILIZING
CONTROL OF BIDIRECTIONAL CONVERTERS
In a multi-converter system, one of the converter impedances
can be reshaped in such a way that the bus impedance
achieves a chosen attenuation level. When performed adap-
tively, impedance reshaping can prevent impedance-based in-
teractions from degrading the multi-converter system perfor-
mance. Instead of using additional hardware, impedance re-
shaping can be carried out by modifying the converter control.
10 1 10 2 10 3
-20
0
20
40
M
ag
ni
tu
de
(d
B)
10 1 10 2 10 3
Frequency (Hz)
-135
-90
-45
0
45
90
135
Ph
as
e
(de
g)
Fig. 2. Frequency response of the bus impedance as given in (2) with
Zo−bus = 9, ωo = 477 rad/s, and Qbus = 6.5.
The control can be designed to offer additional damping to
the converter’s closed-loop impedance, thus affecting the bus
impedance. The range of this damping should be around the
identified resonance frequency. For such a stabilizing control
design, the following design criteria are required:
• The added virtual impedance itself has a damping level
of a chosen quality factor Qd
• The resulting normalized bus impedance Zbus−N(jω)
remains within an AIR specified by a chosen quality
factor Qmax
• At the resonance frequency, the resulting normalized bus
impedance Zbus−N(jωo) is limited to remain within an
AIR specified by a chosen quality factor Qmax − Km,
where Km is an additional margin; The quality factor
of the resulting normalized bus impedance Zbus−N(jωo)
corresponds to Qmax −Km at the resonance frequency
Typical ranges for these parameters are Qmax = 0.7...1,
Km = 0...1, and Qd = 0.7...1. One method of fulfilling the
stabilizing control design criteria is to add a specific damping
term to the voltage controlling converter’s voltage control loop.
More specifically, a resonance gain (R-gain) can be added in
parallel with the regular voltage controller, given as [31]
GR =
2Krωrs
s2 + 2ωrs+ ω2o
, (3)
where Kr determines the damping at the resonance frequency
and ωr is the resonance bandwidth. Fig. 3 shows the controller
block diagram. The converter operates under cascaded control;
the outer voltage loop provides a reference to the inner current
loop.
Sufficient values for the R-gain parameters Kr, ωr, and
ωo can be determined adaptively with online bus impedance
identification. First, the resonance frequency ωo can be directly
determined from the bus impedance identification. Second, the
following design criteria are derived in [24]:
Kr =
Qd
Zo−damp
;ωr =
ωo
2Qd
, (4)
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
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HEADER 4
PWM
ibattvbus
R
vbus
ref
PI PI
Update ω , ω ando r rK
Fig. 3. Block diagram of the converter controller with a resonance term.
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-1
-0.5
0
0.5
1
Re
Im
Fig. 4. Normalized bus impedance without R-gain (blue line) and with R-
gain (black line), normalized R-gain (green line), and AIR boundary (red line)
with Qd = 0.7, Qmax = 1, Km = 0.5. Bus impedance as given in (2) with
Zo−bus = 9, ωo = 477 rad/s, and Qbus = 6.5.
where
Zo−damp = Zo−bus
QdQbus (Qmax −Km)
Qbus − (Qmax −Km) . (5)
The adaptive R-gain improves the damping of the bus
impedance around the resonance frequency, which improves
the multi-converter system’s performance and stability. Fig. 4
presents an illustrative example of an R-gain’s effect on
the normalized bus impedance. The addition of the R-gain
decreases the normalized bus impedance quality factor from
6 (out of scale) to within the AIR. Due to the chosen R-
gain, the normalized bus impedance attenuation corresponds
to Qmax − Km at the resonance frequency. In addition, Fig.
4 shows the normalized R-gain (i.e., R-gain multiplied by
Zo−damp). The virtual impedance itself has damping corre-
sponding to the chosen quality factor Qd. Accordingly, all the
design criteria have been achieved. Note that if the resonance
originates from two CPLs that are coupled through identical
resonance frequencies, the resulting virtual impedance cannot
prevent the resonance but other methods are required [21].
An R-gain-based stabilizing control is demonstrated in [24]
for a source buck converter. In the case of bidirectional con-
verters, the fundamental idea behind the control method is not
affected as the bus impedance derivation is independent of the
power-flow direction. However, for bidirectional converters,
the converter dynamics may change profoundly depending
on the operating mode and the power-flow direction. Thus,
the stabilizing R-gain control may disturb the regular control
Battery
emulator
LV side HV side
ϕ
vac-hv
iL Chv
Ltot
iload
vbus
ibatt
n nlv hv:
0 rad
vac-lv
Clv
Fig. 5. DAB converter with a battery emulator.
performance and stability if it affects the voltage control loop
in a degrading manner. The dynamic changes caused by the
change in the operating mode and the power flow direction
require further consideration for the successful implementation
of the stabilizing controller without regular voltage control
degradation or loss of stability.
A. R-Gain Effect on Voltage Control
The nominal voltage control (e.g., PI-based) is typically
designed based on the desired phase-margin φm and crossover
frequency fc. In this work, the stabilizing controller is im-
plemented on a bidirectional DAB dc-dc converter shown in
Fig. 5, but the procedure can be utilized in any bidirectional
converter topology. The regular current and voltage controllers
are based on PI control; the voltage controller is given by
Gv−PI(s) = Kp−v + Ki−v/s. With the R-gain, the voltage
controller becomes Gv−PIR(s) = Gv−PI(s) +GR(s).
Fig. 6 shows an example of the R-gain effect on the voltage
controller gain. The voltage controller gain is presented with
and without the R-gain. The R-gain is also shown. The chosen
R-gain parameters are Kr = 0.5, ωr = 2π80 rad/s, and
ωo = 2π80 rad/s; the PI-gain parameters are Kp−v = 0.55
and Ki−v = 704. As Fig. 6 shows, the R-gain only affects the
area around the chosen frequency.
The R-gain effect on the original voltage (PI) controller
is small but not necessarily negligible. Since the R-gain
increases the voltage control gain within the chosen frequency
range, its effect on the phase margin and crossover frequency
should be considered to guarantee that the R-gain does not
degrade the regular controller performance and stability. The
converter dynamics may change profoundly depending on the
operating mode and the power flow direction, which should
be taken into account in the stabilizing control design and
its implementation. For example, if the R-gain bandwidth
is too wide, the voltage control crossover frequency may
increase beyond the limits of stable operation, which are
restricted both by the inner control loop (i.e., the control
loops should be decoupled in a cascaded controller) and
possible zeros and poles in the converter dynamics. This can
be especially important with bidirectional converters, as the
system dynamics, specifically, the system poles and zeros, may
change profoundly depending on the power direction (e.g.,
buck vs. boost converter) and thus more easily lead to degraded
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
HEADER 5
10 1 10 2 10 3 10 4
-40
-20
0
20
M
ag
ni
tu
de
(d
B)
10 1 10 2 10 3 10 4
Frequency (Hz)
-135
-90
-45
0
45
90
Ph
as
e
(de
g)
Fig. 6. Frequency response of the voltage controller without (GPI) and with
damping (GPI-R) and the damping R-gain (GR).
control performance. Therefore, further design criteria and
implementation methods are required to guarantee the proper
operation of the stabilizing controller. As a general guideline,
the resonance bandwidth should be limited to
ωr−max <
2πfminc−innerloop
10
<
2πfsw
10
(6)
and
ωr−max <
2πfrhp−zero
2
, (7)
where fsw is the converter switching frequency, fc−innerloop
is the inner (current) control loop cross-over frequency, and
frhp−zero is the frequency of a possible right-half-plane zero
in the control-to-output voltage transfer function. This transfer
function can be given as
Gcv =
Lin
1 + Lin
Gco−o
Gci−o
, (8)
where Lin is the inner loop gain and Gco−o and Gci−o
are the control-to-output voltage and control-to-input current
open-loop transfer functions, respectively. Note that since the
converter’s internal dynamics change based on the operating
point and the power direction, these restrictions should be
considered in the R-gain stabilizing control design of a bidi-
rectional converter for both power directions. Following these
guidelines, the controller can dampen possible resonances in
the bus impedance without regular voltage control degradation
or loss of stability.
B. Bus Impedance Identification
An adaptive stabilizing controller is desirable in a multi-
converter system with varying operating states and conditions.
Fortunately, bus impedance identification provides a straight-
forward method for adjusting the stabilizing control variables
because the impedance can be conveniently measured using
the converter input and output currents/voltages [32].
Based on (1), the bus-impedance identification requires
information about all the interconnected terminal impedances.
One method of obtaining the bus impedance is to utilize
broadband excitations such as pseudorandom binary sequences
(PRBS). While injecting these binary sequences into the con-
verter controllers, the impedances can be identified from the
resulting currents and bus voltage with Fourier techniques [28].
The PRBS perturbations are particularly suitable for the identi-
fication of power systems, as they have only two signal levels
and they have a low crest factor, which means high signal
energy in relation to the signal amplitude in the time domain
[33]. Note that the PRBS signal’s time-domain amplitude and
frequency-domain spectrum must be carefully designed for the
system under study to guarantee that the system currents and
voltage stay within allowable limits and that the perturbations
do not excessively degrade the power quality [34].
The impedance identification process should be as fast as
possible to enable efficient use of the adaptive stabilizing
control so that the system damping can be optimized and
a possible distortion can be damped before reaching over-
voltage or over-current conditions. For example, unnecessarily
low frequencies can be excluded from the PRBS to accelerate
the identification process, i.e., frequencies much lower than
the voltage control crossover frequency, as specified in (6) and
(7). In addition, orthogonal binary sequences can be applied
to speed up further the bus impedance identification process
and enable simultaneous impedance measurements rather than
measuring the required impedances sequentially [35], [36].
One advantage of the impedance-based method is its black
box feature; specific knowledge of the system parameters and
properties is not needed [37]. Similarly, one of the drawbacks
of the impedance-based stability assessment is that it cannot
necessarily point the original causes of the resonance without
further analysis. Nevertheless, the method offers sufficient
information for adapting the virtual impedance according to
changes in the multi-converter system so that a possible
resonance can be dampened regardless of its root cause.
IV. EXPERIMENTAL RESULTS
The proposed method is validated experimentally using a
dc multi-converter system consisting of custom-built power
converters. The built system and its specifications are shown
in Fig. 7, in which a DAB converter is connected to two
inverters. The topology of the DAB converter is shown in
Fig. 5 and the inverter topologies are typical three-phase, two-
level inverters; Their parameters are given in Table I. The DAB
converter and Inverter #2 are bidirectional and operate either
as a load or as a source depending on the chosen operating
point. The DAB converter and Inverter #1 operate under two-
loop PI control and the Inverter #2 controls the grid current
with PI control. Regular voltage and current measurements
are marked in Fig. 7 with black arrows and the orange arrows
relate to the current measurements required for the stabilizing
R-gain controller. The DAB controller and the Inverter #2
controller are implemented using rapid prototyping controllers
by Imperix, whereas the Inverter #1 controller is implemented
on dSPACE platform. All the converters are standalone stable,
so degradation in the system performance originates from the
interactions between the single converters.
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
HEADER 6
vbusibatt
DAB
Zo3
Zo1
Zbus
Zo2
control pulsesmeas.
Control
Platform
Identification
and monitoring
ibus-1
ibus-3
ibus-2
DC
DC
C
C
Inverter #2
DC
AC
iL
Inverter #1
DC
AC
vload L, i
Rload
L,Cf
L,Cf
L2
C
Fig. 7. Laboratory setup for the experimental tests; a multi-converter system consisting of two three-phase inverters and one DAB converter with a battery
emulator.
TABLE I
CONVERTER PARAMETERS FOR THE DAB CONVERTER AND THE INVERTERS #1 AND #2 IN FIG. 7.
Parameters Values (DAB / inv #1 / inv #2) Description
Vbus 400 V bus voltage
Vin 200 V (dc) / 120 Vrms (ac) / 120 Vrms (ac) source- or load-side voltage
Rload 25 Ω resistive load of Inverter #1 (star-connection)
fsw 50 kHz / 8 kHz / 20 kHz switching frequency
C 1.5 mF / 1.5 mF / 1.95 mF bus-side capacitance
Cf none / 25 µF / 10 µF inverter ac-side filter capacitance
Ltot 300 µH DAB total inductance
L none / 2.2 mH / 2.5 mH inverter ac-side filter inductance
L2 none / none / 0.6 mH inverter grid-side filter inductance
fc−c 1 kHz / 500 Hz / 450 Hz current loop cross-over frequency
φm−c 65o / 65o / 60o current loop phase margin
fc−v 10 Hz / 6 Hz / none voltage loop cross-over frequency
φm−v 55o / 60o / none voltage loop phase margin
0 0.2 0.4 0.6 0.8 1 1.2
-1
0
1
Se
qu
en
ce
1
1 2 3 4
0 0.2 0.4 0.6 0.8 1 1.2
-1
0
1
Se
qu
en
ce
2
1 2
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
-1
0
1
Se
qu
en
ce
3
1
(a)
10 0 10 1 10 2 10 3 10 4
Frequency (Hz)
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
En
er
gy
(d
B)
(b)
Fig. 8. MLBS signals used for perturbation for frequency response measurements: (a) in time domain and (b) their energy spectrum.
The perturbation sequences are implemented in all three
control platforms. Three orthogonal MLBS sequences are
used: Sequence 1 for DAB converter, Sequence 2 for In-
verter #1, and Sequence 3 for Inverter #2. Sequence 1 is
of length N = 29 − 1. Since the sequences are orthogonal,
the length of Sequence 2 is 2N , and the length of Sequence
3 is 4N . Fig. 8 shows the three sequences in both the
time and frequency domain. The sequences are generated at
fgen = 2 kHz, which provides an 800 Hz bandwidth for the
measured frequency responses. The frequency range of interest
(i.e., around the voltage control cross-over frequency) is well
within the chosen energy spectrum. As demonstrated in Fig. 8,
Sequence 3 has a period length of 1.022 s, during which the
other sequences are repeated periodically. The actual injection
amplitudes are selected such that their values are 5-7 percent
of the nominal voltage/current reference values.
The designed perturbations were simultaneously placed on
top of the controller reference voltages and/or currents of
each converter. The first perturbation was applied with three
periods, the second with six periods, and the third with 12
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
HEADER 7
M
ea
su
re
m
en
ts
Id
en
ti
fi
ca
ti
o
n
P
ro
ce
ss
Add perturbationa small
in the control signals
Measure resulting vbus
and converter currents
Derive (jω)Zbus
Identify:
ω , | (jω )|, | (jω )|o bus o bus 1Z Z
Determine ω andr rK
Update ω , ω ando r rK
Design criteria:
ωQ Q Kd, max, m, r-max
Fig. 9. Flowchart representing the adaptive resonance-damping algorithm
with the chosen design criteria.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
394
396
398
400
402
404
406
V
ol
ta
ge
(V
)
Fig. 10. Low-pass filtered (500 Hz) bus voltage without (blue) and with
(orange) damping R-gain while discharging the battery.
periods (because each perturbation length is twice compared to
the previous one). The resulting bus voltage and output current
of each converter were then measured using a sampling rate
of 50 kHz. The measurements were averaged over the applied
periods and Fourier transform was used to obtain the output
impedances of each converter. The bus impedance was then
computed based on (1). After the identification process, the
R-gain was added to the DAB controller and its parameters
were assigned based on the identified bus impedance and (4)
with Qd = 0.7, Qmax = 1, and Km = 0.4.
The experiments were conducted at two different operating
points: the DAB converter was either feeding the dc bus with
500 W while discharging the battery, or consuming 350 W
while charging the battery. This change in the operating point
was achieved by changing the Inverter #2 from feeding the
grid with 150 W to feeding the dc bus with 700 W. Fig. 9
outlines the bus impedance identification and the controller
update process. First, the discharging mode was considered.
10 1 10 2
-20
-10
0
10
20
30
40
50
M
ag
ni
tu
de
(d
B)
10 1 10 2
Frequency (Hz)
-270
-180
-90
0
90
180
Ph
as
e
(de
g)
Fig. 11. The identified impedances without damping R-gain while discharging
the battery.
The blue line in Fig. 10 shows the bus voltage when the
battery is discharging and no R-gain is applied. The minimum
(394.0 V) and maximum (405.09 V) values are also marked,
meaning a voltage deviation of 2.77 percent. Fig. 11 shows
the identified impedances, and the resulting bus impedance
is shown in Fig. 12 with a blue line. Without the R-gain,
the bus impedance has a magnitude of 37 dB at 21 Hz,
highlighted with a marker in Fig. 12. To improve stability,
the proposed PI-R controller is activated so that an enhanced
damping is achieved around the chosen frequency. Following
the aforementioned design procedure and the identified bus
impedance, the values of Kr = 0.17 and ωr = 18 Hz are
obtained for the R-gain. The resulting bus impedance with the
added R-gain is shown in Fig. 12 with a red line. The R-gain
increases the bus impedance damping around the identified
resonance frequency (21 Hz), lowering it to 11 dB and thus
improving the system damping. The resulting bus voltage is
shown with the orange line in Fig. 10, and the minimum
(395.4 V) and maximum (404.9 V) values are shown resulting
in 2.38 percent voltage deviation. Since the voltage variation
is lower, the added damping shows an improvement compared
to the case without the R-gain.
Next, a similar study is performed while charging the
battery. The voltages are comparable to Fig. 10. The identified
bus impedance without the R-gain is shown in Fig. 12 with
a purple line: without the R-gain, the bus impedance has a
magnitude of 23 dB at 13 Hz, highlighted with a marker in
the figure. As before, the proposed PI-R controller is activated
(now with Kr = 0.25, ωr = 7 Hz) and the resulting bus
impedance is shown in Fig. 12 with the yellow line. The R-gain
increases the bus impedance damping around 13 Hz, lowering
it to 12 dB.
In both the discharging and charging experiments, the damp-
ing R-gain controller enhanced the system stability and damp-
ing around the identified resonance frequency. The R-gain con-
trollers optimized the identified bus impedances, which were
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
HEADER 8
10 1 10 2
-10
0
10
20
30
40
50
60
70
80
M
ag
ni
tu
de
(d
B)
10 1 10 2
Frequency (Hz)
-180
-90
0
90
180
Ph
as
e
(de
g)
Fig. 12. The identified bus impedances with and without damping R-gain
while discharging or charging the battery. Marker highlights the identified
resonance.
0 0.5 1 1.5 2 2.5 3 3.5
Re
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Im
Fig. 13. The Nyquist contour of the normalized bus impedance with and
without damping R-gain. AIR boundary indicated with the red line. Marker
highlights the identified resonance.
well-stabilized in both charging mode and discharging mode.
Fig. 13 presents the corresponding normalized bus impedances
in a complex plane. The normalized bus impedances with the
R-gain have more damping and are well-confined within the
AIR boundary. The Nyquist contour of the R-gain affected
normalized bus impedances intersects with the real axis around
the requested magnitude, indicating the achievement of the
desired damping level. With the R-gain, the system damping
is within the desired limits. The experimental results confirm
the effectiveness of the proposed adaptive stabilization method
in both power directions.
V. CONCLUSION
This paper has implemented adaptive virtual-impedance-
based stabilizing control on a bidirectional dc-dc converter.
With the stabilizing resonance-damping control and the pro-
vided design guidelines, the bidirectional converter can op-
erate as a virtual impedance that dampens resonances in
the bus impedance when required without deteriorating the
regular control operation. As a result, the stabilizing con-
trol can prevent adverse impedance-based interactions within
a multi-converter system and optimize system stability and
performance. The stabilizing controller is tuned adaptively
based on an online bus impedance identification that estimates
the system’s stability and performance. The broadband-based
identification method is well-suited for adaptive control due
to its short measurement cycle. Therefore, the stabilizing
control is suitable for multi-converter systems with changing
operating states and conditions, such as bidirectional power
flow. The analysis and experiments show that the bidirectional
converter can efficiently optimize the damping in the bus
impedance in both power directions. As a result, the multi-
converter system performance is maintained regardless of
changes (e.g., in connections or operating modes) and without
hardware updates or re-tuning of the regular (such as PI)
controller. The resonance-damping control method’s simplicity
(including bus impedance-based stability assessment and lack
of more excessive computations) and basic requirements (such
as voltage and current measurements at the bus-side of each
converter) mean it can be efficiently embedded in parallel with
a regular controller without disturbing its usual operation.
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content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
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Roosa-Maria Sallinen (S’18) received the
B.Sc. (Tech.) and M.Sc. (Tech.) degrees in electrical
engineering from the Tampere University of
Technology, Tampere, Finland, in 2015 and 2017,
respectively.
Since 2017, she has been pursuing the
Ph.D. degree at the Faculty of Information
Technology and Communication Sciences, Tampere
University, Tampere, Finland. In 2022, she joined
GE Grid Solutions, Tampere, Finland, where she
is currently a Lead Control Engineer. Her main
research interests include impedance-based interactions in power electronic
systems and adaptive stabilization.
Tomi Roinila (M’10) received the M.Sc.(Tech.) and
Dr.Tech. degrees in automation and control engi-
neering from Tampere University of Technology,
Tampere, Finland, in 2006 and 2010, respectively.
He is currently an Associate Professor in Tampere
University, Finland.
His main research interests include modeling and
control of grid-connected power-electronics systems,
analysis of energy-storage systems, and modeling of
multi-converter systems.
This article has been accepted for publication in IEEE Journal of Emerging and Selected Topics in Power Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JESTPE.2022.3213724
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/