1
Dimensioning of a Current Source Inverter for the Feed-in of Electrical
Energy from Fuel Cells to the Mains
M. Mohr1), M. Bierhoff2), F.W. Fuchs3)
Christian-Albrechts-Universität zu Kiel
Chair for Power Electronics and Electrical Drives
Kaiserstraße 2, 24143 Kiel, Germany
Fax.: +49 431 880 6103; URL: http://www.tf.uni-kiel.de/etech/LEA
1)
mam@tf.uni-kiel.de, Tel. +49 431 8806104 2)mib@tf.uni-kiel.de, Tel. +49 431 8806104
3)
fwf@tf.uni-kiel.de, Tel. +49 431 8806100
Abstract—Fuel cells deliver dc current which demands a
dc/ac converter between the fuel cell and the mains to feed in
electrical energy. An inverter of the current source type makes it
possible to convert the energy supplied by a fuel cell into the
mains without using an auxiliary dc/dc-converter. The current
source inverter’s efficiency depends on the input dc voltage of the
inverter, so the rating of the fuel cell voltage influences the
overall efficiency of the system.
After a short introduction of the fuel cell’s electrical
behaviour, the current source inverter and the impact of its LCFilter on the inverter’s input voltage are described. The
dimensioning process dependent on the power range of the fuel
cell and the current source inverter is shown. In doing so, the
characteristics of the fuel cell and the inverter are taken into
account.
A laboratory setup of a current source inverter has been built
and a control scheme is presented. Simulation results of the
general system behaviour are shown.
Index Terms— current source inverter, dc-ac power
conversion, fuel cell, renewable energy
F
A current source inverter, which combines several
advantages is proposed as an interesting alternative for the
feed-in of electrical energy from fuel cells. As far as the
authors know only one previous paper covers this subject [2],
dealing with the comparison of current source inverter losses
to losses of other topologies. More papers deal with the
current source inverter in general, as drive converter, for
example [4], [5].
In this paper the dimensioning of the current source inverter
in relation to the fuel cell dimensioning is analysed.
Semiconductor losses are calculated and a control method for
the inverter is presented. Simulation results show the general
system behaviour including its control. A laboratory setup has
been built up and taken into operation.
The paper is organised as follows: In section II the system
components, fuel cell and current source inverter, are
introduced. Section III contains the analysis of the combined
inverter and fuel cell dimensioning. Section IV gives an
introduction to the inverter laboratory setup and the simulation
results.
I. INTRODUCTION
UEL cells convert chemical energy from fuels, especially
hydrogen, directly into electricity. They are more efficient
than other energy conversion systems which are based on a
thermodynamic process [1]. Due to their high efficiency, fuel
cells are nowadays an alternative to systems like gas turbines.
In future, fuel cells are ment to reconvert hydrogen, which
will be produced from renewable energy systems like solar
cells or wind power plants, back into electrical energy.
If a fuel cell feeds electrical energy into the mains, its dc
current has to be converted into a three-phase sinusoidal ac
current. For these purposes an inverter has to be inserted
between the fuel cell and the mains. In addition to the
conversion of dc current into ac current, the inverter has to
adapt the variable voltage of the fuel cell to the voltage
amplitude of the mains. Apart from the request for high
efficiency, the inverter has to draw a well smoothed dc current
from the fuel cell for a high lifetime of the cell. In addition it
has to deliver low current harmonics to the mains in respect to
the standards. Usually, a dc/dc converter and a voltage source
inverter are used in series to reach the requirements for this
kind of application [2], [3].
II. SYSTEM COMPONENTS
A. Fuel cell characteristics
Fuel cells provide a variable dc current at a varying fuel cell
voltage which depends on the current. The maximum possible
fuel cell current depends on the amount of combustibles
Vfc [V]
Arbeitspunkte
operating
points
op
AP11
op 22
AP
J
Fig. 1. Characteristic curves of a PEM fuel cell at different fuel supply
conditions, operated with air [1]. Left curve: cell suplied with few
combustibles; right curve: cell supplied with nominal amount of
combustibles. Operating point op 1 at partial load (at left curve),
operating point op 2 at nominal load (at right curve).
2
which the fuel cell is supplied with. Figure 1 shows two
characteristic curves of a fuel cell at different supply
conditions (like in [1]). If the fuel cell operates at partial load
the compustible supply is reduced. The aim is to tune in to an
operating point in which maximum available power is drawn
from the fuel cell. In the following this operating point will be
termed as “relative maximum power point”.
The fuel cell voltages typically specified in fig. 1 refer to
one single fuel cell, the current density refers to a certain area.
The current, delivered by the fuel cell, depends on the cell
area, higher system voltages can be obtained by connecting
several fuel cells in series (so called “stacks“).
Operation at higher current densities in the decreasing part
of the characteristic curve yields on the one hand to a bad
efficiency of the fuel cell, in addition it may damage the fuelcell [6].
The no load voltage of the fuel cell is higher than the fuel
cell voltage in load condition, see fig. 1. In the following an
operation with an inverter maximum input voltage lower than
the no load fuel cell voltage is proposed. In this case the
system’s operating point cannot be approached in a controlled
way. The fuel cell voltage will jump from the no load voltage
to the voltage at the operating point, at the time when the cell
is connected with the inverter. This switching-in is permissible
if there is sufficient fuel at the fuel cell so that the current
density does not exceed its critical value, not even in dynamic
operation.
If the load is disconnected, the voltage rises up to the no
load voltage within a short time [7]. The voltage steps can be
smoothed by connecting capacitors in parallell to the fuel cell.
As the time constant of the fuel cell is much longer than the
time constant of the inverter, it is necessary to operate the
inverter’s current set value synchronously to the fuel cell.
B. Current source inverter, general function
The inverter has to adapt the varying dc fuel cell voltage to
the nearly constant ac voltage of the mains. If a voltage source
inverter is used, the dc link voltage has to be greater than the
maximum value of the phase to phase voltage of the mains
( Vˆline ∆ = 2 Vline ∆ ). This can be achieved by using a dc/dc
converter, like a boost-converter, for example [2], [3]. An
interesting alternative solution is the use of a current source
inverter. It increases the voltage towards the mains, so the fuel
cell voltage can be lower than the mains voltage.
Fig. 2 shows the topology of a current source inverter. At
the mains side the inverter is equipped with a LC-Filter. The
inductance Lline combines the line inductance and the attached
filter inductance, resistor R characterises the ohmic losses in
the filter inductances and the ohmic line resistance. Because
of the dc link inductor Ld a low dc current ripple can be
accomplished. This yields to a longer lifetime of the fuel cell
[6]. Furthermore, a current source inverter has a low harmonic
content in the line current. Ideally the inverter provides a unity
power factor in the mains, however a variation of the power
factor from 0,9 < cos(φ) < 1 for example is desirable.
The switches of the current source inverter have to be
Ld
Iinv
C
Lline
Iline
Vline
VC
Id
Fig. 2. Current source inverter topology
reverse blocking. If IGBTs are used for the current source
inverter, at present the reverse blocking capability can only be
achieved with diodes connected in series to the IGBTs [5].
This yields to relatively high semiconductor conduction
losses.
C. Current source inverter, voltage control
The current source inverter increases the voltages towards
the mains, but at a given mains-voltage there is a maximum dc
voltage the inverter can operate with. So the inverter’s input
voltage has to be limited when it exceeds a maximum value,
otherwise the dc current increases and the inverter gets
beyond control.
The inverter is controlled by a space vector modulation
which causes a nearly sinusoidal mains current. The switching
states in which the dc current is connected directly to the
mains side are designated as active switching states. A so
called passive switching state is present if the inductance Ld is
directly connected to the fuel cell (thus the dc side of the
inverter is shortened).
Presuming a sinusoidal inverter current iinv the modulation
index [5]
M =
iinv
Id
; 0 < M ≤1
(1)
terms the ratio between the norm of the current space vector
and the dc current. The norm of the current space vector
corresponds to the amplitude of the line current.
Due to the power equilibrium at the dc and the ac side of
the inverter, the average dc voltage can be adjusted varying
the modulation index, assuming a constant line voltage.
At a modulation index of M<1 (boost operation), passive
switching states as described above, are used more frequently,
leading to a worse efficiency factor of the inverter. Hence the
aim is to operate the current source inverter at the highest
possible modulation index. This means the same as operating
the inverter at the smallest possible difference between dc
voltage and maximum possible rectified voltage. For practical
use, a safety factor has to be included.
In the following the inverter’s maximum input dc voltage
Vd , max is derived, the influences of the ac filter components
are thereby included.
At the highest possible modulation index of M = 1 the
amplitude of the line current corresponds to the dc current Id.
3
Thus the root mean square value of the inverter’s ac current is:
I
I inv = d
(2)
2
The power equilibrium between the ac side and the dc side
of the inverter
PAC = 3 VC ⋅ I inv ⋅ cos(ϕ ) = Vd ⋅ I d = PDC
yields with (2) and with the capacitor voltage
 ω L I line


VC = (Vline + R I line ) ⋅ arctan
(
+
)
V
R
I
line 
 line
(3)
(4)
at a modulation index of M = 1 to the maximum value of the
inverter’s input dc voltage as follows:
 ω L I line

3
 ⋅ cos(ϕ ) (5)
⋅ (Vline + R I line ) ⋅ arctan
Vd ,max =
2
 (Vline + R I line ) 
Thus, at a constant modulation index, the maximum dc
voltage Vd, max depends mainly on the line voltage and the
phase angle φ between inverter current and inverter voltage
(which is similar to the capacitor voltage).
The current source inverter has a LCR filter at the mains
side. Assuming unity power factor towards the mains, the
filter’s reactive power has to be supplied by the inverter itself.
So the maximum dc voltage depends on the value of the filter
capacitors C, the line inductances Lline and the line resistors R.
Figure 3 shows the phasor diagram of the single-phase
equivalent circuit of the ac filter. The angle between the
capacitor voltage VC and the inverter current iinv can be
regarded as the inverter’s phase angle φ as used in (3).
It depends on the value of the capacitor’s reactive current and
the voltage drop across the line inductances Lline. In Figure 3,
unity power factor to the mains is assumed, so the angle
between line current Iline and line voltage Vline equals zero.
Including the reactive current of the filter capacitors and the
voltage drop across the line inductances, the maximum dc
voltage at a modulation index of M = 1, figured out from
fundamental component analysis, is as follows (capacitors in
delta connection):
 ω L I line

3

(Vline + R I line ) ⋅ arctan
Vd ,max =
2
 (Vline + R I line ) 

 ω C 3 (Vline + R I line ) 


 − arctan ω L I line

⋅ cosarctan



I line

 (Vline + R I line ) 


(6)
Higher dc voltages as derived in equation (6) are possible
using a modulation index greater than one. But this will result
in increasing current harmonics and is therefore not
recommended.
III. RATING OF THE COMPLETE FUEL CELL INVERTER SYSTEM
A. Design scheme
The design process of the fuel cell inverter system has to
fulfil many requirements. The fuel cell is the most expensive
component of the whole system. During operation it is
required, that the combustibles achieve their maximum
utilisation. So the most important aim is to operate the system
at the fuel cell’s relative maximum power.
The inverter has to operate with the highest possible
efficiency to apply the provided power to the mains. In
addition the power rating of the current source inverter has to
be as small as possible to keep down costs. Additional
conditions like an adjustable power factor or the ability of the
system to operate in a certain partial load range affects the
design of the inverter and the rating of its components.
Figure 4 shows the dimensioning process of the fuel cell
inverter system starting from the given mains voltage and the
required power of the system. Thereupon the voltage rating of
the fuell cell has to be set, taking the current source inverter’s
constraints into account (eq. (6)).
ωLIline
RIline
given mains voltage
voltage rating of the fuel cell
Vline
VC
required mains power/
required power range
current rating of the fuel cell
ϕ
Iinv
ωCVC
power rating of the inverter
Iline
Fig 3. Phasor diagram of
the one phase equivalent
circuit diagram of the
mains-side filter. Mains
power factor equals 1.
dc-filter
Fig. 4. Design scheme
ac-filter
semiconductors
4
Then the current rating of the fuel cell, determined by the
recommended power, has to be set. The fuel cell current
involves the rating of the current source inverter and its
components. The separate dimensioning steps are presented in
the following sections.
B. Fuel cell stack voltage and inverter power rating
The rated dc side voltage of the current source inverter
should be as close as possible to the maximum inverter dc side
voltage as mentioned above. The higher the dc input voltage
the lower the input current of the current source inverter is at a
certain load, so the semiconductor’s current capability can be
chosen to lower values at higher input voltages.
In the following a special way of operation of a fuel cell
inverter system with reduced inverter power rating is
developed.
Figure 5 shows characteristic curves of three supposed fuel
cell stacks with a different number of cells corresponding to
different stack voltages. The maximum input dc voltage of the
current source inverter assuming sinusoidal line currents is
Vd, max.
If a fuel cell stack with n single fuel cells (dashed line) is
chosen, the inverter can operate at a minimum partial load
(operating point op 1). The current source inverter is able to
operate at varying input voltages. Therefore it can operate at
lower input voltages which correspond with higher loads (as
in operating point op 2 for example). If the fuel cell stack
consists of n+m single cells (solid line), the inverter cannot
operate at partial load because the inverter input voltage
equals the maximum input voltage already at nominal load
(operating point op 3).
Due to the fact that the semiconductor losses of the inverter
depend only on the dc current as mentioned below, the losses
at operating point op 3 are equal to the losses at operating
point op 2. But at operating point op 2 the input power is
smaller (area 1 + area 2) than the input power at operating
point op 3 (area 1 + area 2 + area 3) because of the lower
voltage at op 2.
For operation at full load range the no load voltage of the
fuel cell stack has to be the same as the inverter’s maximum
input voltage Vd, max (dotted line). In this case at the inverter’s
maximum current Iinv, max the system operates at op 4 (fig 5).
The power in this operating point is then represented by
area 1.
As the rated power of the inverter depends on its maximal
dc current, we can see, that the total system power can be
increased, using fuel cell stacks with higher voltages, whereas
the rated inverter power does not increase. The drawback is
thereby, that we cannot operate the system over the entire
power range.
The characteristic curves in figure 6 clarify this topic. In
this diagram fuel cell current Ifc and fuel cell voltage Vfc of
different fuel cell stacks (different numbers of fuel cells
resulting in different stack-voltages) are plotted against the
fuel cell power. Vd, max labels the maximum possible inverter
input voltage, Pmax labels the maximum output power of the
fuel cell stack.
As we can see the fuel cell stack with lower voltage (Vstack 2)
allows the inverter to operate at a wider power range whereas
the inverter’s power range at the fuel cell stack with higher
output voltage (Vstack 1) is more constricted. If the fuel cell no
load voltage equals the maximum inverter dc voltage (Vstack 3)
the inverter can operate at the full load range.
In the diagram (fig. 6. ) the fuel cell currents corresponding
to the characteristic voltage vs. power curves of the two
exemplary fuel cell stacks are shown.
Due to the fact that the inverter losses PV are proportional to
the inverter current Id [8] it can be seen that the efficiency
P − PV
η=
(7)
P
for a given inverter is higher for higher fuel cell stack voltages
at a certain power P than for fuel cells with less voltage.
Vfc
Ifc
Vfc
n fuel cells at nominal load
n fuel cells at partial load
n+m fuel cells at nominal load
Vstack 2
Vstack 1
Vd,max
Vstack 3
Vd,max
op 1
op 3
area 3
Istack 2
op 2
Istack 1
area 2
op 4
0
area 1
0
Pmax P
load range at higher
fuel cell voltage
Iinv, max
Fig 5. Characteristic curves of fuel cell stacks with different numbers
of fuel cells. Including inverter input voltage limit
Id
load range at lower fuel cell voltage
full load range, fuel cell no-load voltage = max. inverter dc voltage Vd,max
Fig 6. Fuel cell voltage and dc current versus system power of different
fuel cell stacks, including inverter input voltage limit
5
Therefore, the inverter’s efficiency at nominal load is lower if
the system is designed to operate down to a lower partial load
compared to a system optimised for nominal load.
Furthermore, the inverter’s rated power as well as its costs
increase proportional to the input dc current. Table 1 gives an
overview of rated inverter power and achieveable power range
dependent on inverter input voltage at the given maximum
fuel cell power Pmax. In general one can see, that by reducing
the control range of the power, for a given maximum
powerflow, the inverter’s nominal power rating can be
reduced and its efficiency can be increased.
TABLE 1
RATED INVERTER POWER AND POWER RANGE AT DIFFERENT SYSTEM
LAYOUTS.
maximum power Pmax at
inverter input voltage
Vd, max
0.8 · Vd, max
0.66 · Vd, max
rated inverter
power
100 %
125 %
150 %
power range,
based on Pmax*
0%
45 %
90 %
*
The power range is dependent on the ohmic losses of the fuel cell. Values
based on fuel cell data from [Ledjeff-Hey].
At partial load the efficiency factor of the inverter increases
lightly due to the fact that the fuel cell voltage increases, too.
In addition, the ohmic losses in the fuel cell are smaller at
lower current rates, so the total efficiency of fuel cell and
current source inverter increases at partial load operation.
C. Rating of the current source inverter’s filter elements
The value of the dc filter (inductance Ld) affects the ripple
of the dc current. In [9] the equations for the design of the
filter components are derived as follows. Equation (8)
specifies the value of Ld dependent on the switching
frequency fS and the permitted current fluctuation ∆Id, max.
2 Vline
Ld =
4 f S ⋅ ∆ I d , max
(8)
The value of the filter capacitor C affects on the one hand
the voltage fluctuation at the filter capacitors C, caused by the
pulsed capacitor currents. On the other hand the value of the
capacitor affects the maximum dc voltage at the inverter, as
shown in section II B.
Equation (9) [9] specifies the value of the capacitor C
dependent on the switching frequency fS and the permitted
voltage fluctuation at the capacitors, ∆VC, max.
1
C=
⋅ Id
(9)
6 f S ⋅ ∆VC , max
Thus, the minimal value of the filter capacitors is
determined by the desired voltage fluctuation. As shown in the
vector diagram (Fig. 3) and derived in equation (6) it has to be
taken into account that a high capacitance results in a
decreasing input voltage of the inverter. Although the inductor
counteracts the reactive component of the capacitors, the
capacitance has a major impact on the reduction of the
maximum dc-voltage. Particularly at partial load, the
inductor’s reactive component decreases (function of line
current Iline) wheras the capacitor’s reactive component
persists (function of line voltage Vline).
D. Semiconductor losses
The semiconductor losses, switching losses PS and
conduction losses PC, of the current source inverter are
calculated according to (10) and (11) [8].
I
Vˆ
3
PS = f S ⋅ ⋅ (EOn, IGBT + EOff , IGBT + E Off , Diode ) ⋅ d ⋅ N (10)
π
I ref Vref
PC = 2 I d { VCE (I d ) + VF (I d )}
(11)
At constant line to line voltage Vline∆, neglecting the voltage
oscillation at the filter capacitors, the semiconductor losses
depend only on the dc link current Id. The chosen operating
point has a line to line voltage of Vline∆ = 400 V, a dc link
current of Id = 40 A and a switching frequency of fS = 5 kHz.
The datasheet values of the IGBTs and diodes, assembled in
the laboratory inverter described below are used. The applied
modulation method is fully symmetrical modulation (FSM) as
described in [10].
The switching losses are calculated to PS =53 W and the
conduction loses are calculated to PC = 280 W. It can be seen
that the conduction losses have a larger amount than the
switching losses, as described in section II B.
The total inverter powerflow for the choosen operating
point at a modulation index of M = 1 is Pinv = 19.6 kW.
Including the total losses of 333 W as calculated above, this
yields to a theoretical efficiency factor of the inverter of
η inv = 0.983 without considering filter losses.
IV. SIMULATION RESULTS AND FUEL CELL INVERTER
LABORATORY SETUP
A. Control System, Simulation results
For the control of the dc side current and the phase shift of
the mains current, a cascaded feed back control loop was
chosen working with the corresponding coordinates of a
rotating space vector dq-plane, see also [4]. As such, the dc
side current control loop contains two inner loops for the
capacitor voltage and the mains current, respectively. Thus the
innermost control loop directly affects the capacitor voltage
which in turn influences the current that is driven through the
filter inductors. As the d-component of the mains current is
proportional to the active power fed into the mains (proper
angle adjustment of the rotating coordinate system provided)
it would be addressed by the outermost loop controlling the dc
link current. To the set value of the mains current’s qcomponent a value proportional to the desired reactive power
has to be assigned, thus controlling the power factor. Cross
couplings of the two components being controlled are
compensated by decoupling elements.
The design of the dc link current controller’s parameters is
difficult, because of the non-linearity of the plant as the gain
varies with the dc link current itself. The fuel cell’s no load
6
voltage VB, which can be interpreted as a disturbance value,
may be compensated by a feed forward term added to the
controller’s set value to realise better step responses. The nonlinearity of the control path is indicated by the coefficient VS
which is determined by (12), Vline, d is the d-component of the
line voltage.
3 Vline, d
VS = ⋅
(12)
2 Id
With this the control parameters can be determined by
applying the symmetrical optimum [11] with the restriction
that the dc link current’s control path has to be linearised.
Based on these results, simulations have been carried out with
Simplorer®.
As an approximation, the fuel cell has been modelled by a
constant voltage source with a current dependent internal
resistance to achieve the voltage-current characteristic of a
fuel cell stack. Furthermore the capacitor voltage’s actual
value has been filtered by a low pass first order with a cut off
frequency equal to the switching frequency (5 kHz).
In fig. 7 the simulation results of a step response of the dclink current (set value Id* = 40 A) are shown. It can be seen
that the rise time of the dc current is quite fast, the dc-current
achieves its set value within less than a quarter of a line
period.
100
A, V
Id*
Id
Vline
50
0
Id*, Id,
Vline/5, iline
-50
iline
-100
0
12.5
25
67.5
ms
50
t
Fig. 7. Simulation results of a step response of the dc-link current
C = 16 µF in delta connection
Ld = 20 mH
Lline = 3 mH
Vline ∆ = 400 V
set value Id* = 40 A
B. Inverter laboratory setup
For investigations at practical operation, a current source
inverter laboratory setup with a maximum output power of
20 kVA has been built. Fig. 8 shows the schematical diagram
of the developed inverter including controller and protection
circuits.
The IGBT S7 in conjunction with the free wheeling diode
D1 will shorten the dc link inductor Ld in case of overvoltages
in the power element to dissipate the energy which is stored in
the inductor Ld. This protection circuit is described in [12]. In
this case, the switch S8 separates the fuel cell from the
inverter. In addition the diode D1 protects the fuel cell against
Ld
C
S8
Fuel
cell
Vfc
S7
D1
overvoltage
Lline
Vline
Current
Source
Inverter
Space vector
modulation
VC
protection
*
IDC
IDC
Controller
Iline
Fig. 8. Fuel cell inverter system
possibly arising negative voltages caused by a faulty
modulation for example.
The inverter is controlled by a space vector modulation,
generated in a C167 microcontroller, while the system control
is implemented in a DSP-system.
Fig. 9 shows a picture of the power circuit, the protection
circuits, measurement devices and space vector modulator of
the current source inverter.
The value of the dc inductor Ldc can be set from to 5 to
30 mH, the ac filter inductances are set to 3 mH. The
capacitors are delta connected and have a value of 16 µF. At
nominal load this filter dimensioning yield to a phase angle
φ = 0° between capacitor voltage and inverter current.
The power circuit consists of six diodes SEMIKRON®
SKKD 60 F17 with six IGBTs eupec® BSM55 GB120DLC in
series. The IGBTs and the filter capacitors are linked to each
other with copper plates to maintain low leakage inductances.
The maximum current capability (diode: IFAV = 58 A, IGBT:
IC, nom = 55 A) and withstand voltage (diode: VRRM = 1700 V,
IGBT: VCES = 1200 V) of this semiconductors are relatively
high to prevent damages during test operation. Further
investigations should show if especially the semiconductor’s
withstand voltage can be reduced.
Fig. 9. Power circuit, measurement device and space vector modulator
of the laboaratory setup
7
V. CONCLUSION
In this paper, the dimensioning of a fuel cell inverter system
has been analysed. A current source inverter renders some
advantages as for example limited component effort, well
smoothed fuel cell current and mains current with low
harmonics. In contrast to that there is a drawback of the
inverter in conjunction with a fuel cell system. If the inverter
is designed for the full load range, the inverter’s power losses
are comparatively high at full load operation. In this paper a
design with limited load range is developed. Thereby the
efficiency can be increased and the rated inverter power can
be reduced. The rating of the current source inverter is
derived, in doing so the influence of the filter elements on the
design is included. The performance of the inverter is
analysed by means of numerical simulation. A laboratory
setup with a nominal load of 20 kVA at a line to line voltage
of 400 V has been built to verify the above mentioned results
in practical use.
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