Analytical approach for the optimal design of
combined energy storage devices
in ship power system
E. Fasano*, D. Lauria*, F. Mottola**, R. Rizzo*
*
**
University of Naples Federico II, via Claudio 21, Naples, (Italy)
University of Naples Parthenope, Centro Direzionale di Napoli Isola C4, Naples, (Italy)
modern paradigm of integrated power system allows
itself the naval ship survivability due to the intrinsic
ability in adapting the electrical distribution system,
exploiting the redundancy of the structure: the zonal
survivability, regarded as the ability to supply loads in
zones not affected by faults, can be realized in a feasible
way [3], [4]. Furthermore it permits a reduction of the
prime movers and a noticeable improvement of their
efficiency and then the same efficiency of the propulsion
systems can be significantly enhanced. At this purpose, it
has to be noted that the integrated power system could
allow fixed pitched propellers in place of the traditionally
employed controlled pitched propellers. Medium-voltage
direct current (MVDC) power systems, conceived as a
promising candidate to implement the integrated power
system on ships, have been deeply investigated in the last
decade since it allows to realize many remarkable
advantages over alternating current (AC) distribution
systems [4]. MVDC systems show a high level of
flexibility due to the massive penetration of the
innovative components as power electronic devices
capable of performing better and better performances in
terms of reliability and power quality. Silicon carbide
power electronics in particular is a promising evolving
technology for reducing dramatically the size and the
weight of the capacitive and magnetic devices, due to
high switching speeds. The intrinsic ability in
withstanding high voltages allows lowering the number
of series modules requested for MVDC technology.
Another potential advantage of MVDC system relies
in the inherent ability in counteracting the fault currents,
even if this topic requires also deeper investigations. One
of the interesting advantages of MVDC is the inherent
ability in integrating energy storage systems for
improving fuel efficiency while maintaining the desired
level of reliability. Energy storage systems have been
extensively investigated, since they play a crucial role in
all the sectors of the electrical systems, thanks to their
versatility and to optimal performances obtainable
through the employment of electronically-coupled
devices [5]. The optimal sizing of these devices is not a
trivial task and many proposals are available in the
relevant literature aimed to determine the optimal size.
Focusing on the DC applications, a new analytical
methodology for sizing energy storage systems has been
Abstract— In the last years a great interest has been paid
in the relevant literature to the benefits deriving from the
use of direct current for ship power systems. This is in
compliance with the most recent concept about all-electricship and smart electrical distribution for achieving better
performances by exploiting as far as possible the
potentiality of integrated power systems. In this context the
storage devices play a crucial role in improving the
reliability, the efficiency and the dynamic performances of
the whole electric system. Moreover, compared to
conventional ship propulsion plants, this approach has a
significant impact on ship design (particularly when
combined to azimuthing thruster or podded thrust units) as
regards comfortable (noise and vibrations), flexibility (for
general plan), consumption and environment protection.
The variety of pulsed and intermittent loads also requires a
proper choice of the storage devices and of the techniques
for the optimal system design. Focusing on DC ship power
systems, in this paper two different methodologies for the
design of combined storage systems constituted by batteries
and supercapacitors are proposed. Based on some typical
loads, some relevant case studies are analyzed and
discussed.
Index Terms-- Energy storage, design optimization, allelectric-ship.
I. INTRODUCTION
The worldwide challenge to environmental friendly
systems, with consequent more stringent requirements for
emissions, has pulled the researchers to investigate more
efficient solutions for the large variety of electrical
applications. The marine sector is universally recognized
as one of the most significant energy consumer needed
for ensuring the electrical propulsion demand and the
other appliances. This is confirmed also by the recent
advances of IEEE Standards [1]. The revolutionary
concept of all-electric allows improving the design and
the management of the ship power systems through the
widespread integration of all of the electrical and
electronic subsystems. The rationale behind this solution
is a power station able to supply all loads [2]. Two
primary aspects have to be highlighted in the design
philosophy. The first refers to the fact that the overall
electrical load consequent to the high-power combat
system and the gradual electrification of auxiliary
equipment is critically increased. The second one is
related to the more stringent requirements in terms of
survivability and continuity of the electrical service. The
978-1-5090-4682-9/17/$31.00 ©2017 IEEE
761
recently presented in the literature [6]. In the present
paper an effort is paid at the aim of extending the design
procedure proposed in [6] to ship power system, where
the network exhibits various independent generation
buses. The original contribution of this paper is to discuss
the proper application of the design procedure to the case
of onboard ship applications and, particularly, to the
variety of loads typically supplied by the ship power
systems. At this regard, two different applications of the
sizing procedure are discussed:
- the storage system is used to make more efficient
the use of the vessel diesel generators;
- the storage system is used to provide the onboard
ship power system to face with severe pulse loads.
Based on these applications, the use of batteries and
supercapacitors are discussed and an original analytical
procedure for sizing the supercapacitors is also proposed.
The rest of the paper is organized as follows. In
Section II, possible architectures of the power distribution
of all-electric-ships are proposed and discussed. By
focusing on different kind of load profiles, two different
analytical optimization are solved for the optimal sizing
of storage devices in Section III; based on the derived
optimal storage capacity, the features of battery and
supercapacitors are derived in Section IV. In Section V a
discussion on the use of the analytical approach is
proposes with reference to some relevant case studies.
Finally, our conclusions are drawn in Section VI.
levels of security and reliability of the onboard ship
power system while guaranteeing cheaper operation,
basically relies in the optimal integration of storage
devices with the other system components and their
portioning in the frame of the zonal loads system. At this
aim, both control and sizing of storage devices must be
properly tailored with respect to different typologies of
loads and generators. Two goals are expected: to prevent
undesired operative condition (e.g., undesired peak power
request) and to allow cheaper operation (e.g., reduction of
fuel consumption).
Generator
Generator
storage
storage
Zonal
loads
Zonal
loads
Zonal
loads
Zonal
loads
M
M
Propulsion
Propulsion
Pulsed
loads
Fig. 1. DC Ring-bus microgrid for ship power system.
II. MVDC ONBOARD SHIP POWER DISTRIBUTION
In this framework this paper focuses on the optimal
design of the storage system by proposing an analytical
optimal sizing of storage system aimed at guarantee an
efficient operation of the grid. More in depth the optimal
storage sizing is exploited considering the objective to
level the power production of the on board generators
against pulsed loads’ profiles; the second approach is
devoted to the classical network loss minimization.
In the power systems of all electric ships the use of DC
allows better integrating power sources and loads able to
make more efficient and cheaper energy management on
the ships [7]. The integration of loads and resources can
be further improved by including storage devices in the
system. This is even more apparent in the presence of
critical loads which require high level of secure power
supply. At this regard, the ring-bus based microgrid
architecture has been evidenced as DC architecture for
ship application (e.g., [3], [8]) which better fits the needs
of more reliability and safety of the on board electrical
distribution networks. The ring bus architecture is
characterized by different zones that can be
independently monitored and controlled. As shown in
Fig. 1, different loads and resources can be connected at
the ring-bus, such as generators, energy storage devices,
propulsion, navigation apparatus and instruments, pulsed
loads (e.g., weapons systems) and zonal load centers
(lighting, refrigeration, etc.). The ring-bus architecture
implies the possibility to guarantee higher levels of
survival in case of single points failures by avoiding fatal
interruptions in the network operation. This is reached by
partitioning the ship distribution network in properly
identified zones that can be supplied according to
different paths selected through a proper combination of
switches operating modes.
The ability of ring-bus architecture to improve the
III. ENERGY STORAGE OPTIMAL CURRENT PROFILE
In this section the current profiles requested to the
storage devices (i) to lowering the power output of the on
board generators and (ii) to minimize the network power
losses are analytically derived. The current profiles are
derived by solving two optimization practices formulated
in terms of minimization problems combined with
isoperimetric constraints. The objective functions and
constraints are properly tailored to face with the
peculiarities of ship power systems. The derived current
profiles will be then used for the optimal design of the
storage systems, as discussed next section, with reference
to the case of supercapacitors.
The first formulation is based upon the assumption of
negligible resistances, since the design objective requires
the lowering as far as possible of the current supplied by
the generators, while taking into account the cost of the
storage components.
762
The second formulation is based on the same approach
proposed in [6]. Based on the peculiarities of ship power
systems, the optimal current profiles of the storage
devices in the onboard MVDC network is investigated.
The purpose is the identification of the analytical
expression of the current profile able to minimize
network losses over a specified time period while
accounting both the costs of losses and storage devices.
with k 2' = Esto k 2 . By simple manipulations the following
solution is obtained:
A. Lowering current drawn from the power generators
k'
J sto (t ) + 2 sign J sto (t * ) =
2k1
ng
J sto (t ) =
j =1
ng
³
¦
³E
sto J sto (t ) dt
ng
J al =
³E
sto
J sto (t ) dt (1)
³¦J
j (ξ ) dξ
0 j =1
T ng
³¦J
j (ξ ) dξ
0 j =1
Jal(A)
2000
0
-2000
0
0
8
t(h)
16
24
200
Jsto(A)
(2)
0
-2000
-200
-100
0 (A) 100
Jsto
200
-100
0 (A) 100
Jsto
200
200
0
-200
0
8
t(h)
16
24
0
-200
-200
Fig. 2. Graphical interpretation of the solution of (5).
In Section IV, details on a possible methodology for
the optimal sizing of supercapacitors based on the desired
current profile (5) is discussed.
B. Minimizing losses in the Onboard Ship Power
Network
The MVDC onboard ship network’s line resistances
are considered and the aim of the storage sizing problem
is devoted to the network losses minimization. In order to
formulate the considered problem in mathematical terms,
let us consider the following classification for the MVDC
busses:
nG generation busses;
operations of the generators, k 2 is related to the per unit
cost of the energy charged and discharged during T by
the storage devices. The isoperimetric constraint (2)
allows obtaining a null net energy exchanged by the
storage device during T .
The assumption of zero resistances implies that the
voltage at the storage device’s terminal and generators’
terminal is equal Esto = E gen . According to the theory of
n L load busses;
n S storage busses.
)
Thus, nL + nG + nS = n , with n the number of internal
busses of the network. The busses are considered in the
following order:
{1, …, nG , nG +1, … nG +nL , nL +nG +1, …, n} .
The loads are described in terms of their power
profiles, assumed to be repetitive signals over a certain
time period T and independent of the respective bus
voltages. The network can then be described in a compact
way as (for the sake of simplicity, the time dependence is
not evidenced):
the calculus of variations, the following system to be
solved is derived from the minimization problem (1)(ޤ2):
2
º
·
k 2'
¸
I j (t ) +
J sto (t ) + λ J sto (t ) »» = 0
¸
2
j =1
¹
»¼
(3)
ng
¦
T
sto (t ) dt
1
T
2000
is the current of the jth generator (j=1,…ng) at time t ,
E sto is the voltage at the storage’s terminal, k1 and k 2
are the weights of the two terms of the objective function
(1): k1 is related to the additional cost due to variable
³J
¦
T ng
The solution of this equation can be graphically
interpreted as shown in Fig. 2.
where T is the considered time period, J sto (t ) is the
current of the storage device at time t (for the sake of
clearness, only one storage device is considered), J j (t )
ª
d « §¨
k1 J sto (t ) −
dI c « ¨
«¬ ©
¦
J j (t * ) −
j =1
0
(
1
J j (t ) −
T
j =1
*
T
= E sto J sto (t ) dt = 0
³
ng
The solution of this equation can be graphically
interpreted as shown in Fig. 2, where it has been assumed
T
0
0 j =1
k 2'
signJ sto (t ) (4)
2k1
(5)
s.t.
T
³¦
J j (ξ )dξ −
*
Jal(A)
2
·
k
J j (t ) ¸ dt + 2
¸
2
j =1
¹
T ng
Jsto(A)
§
min k1 ¨ J sto (t ) −
¨
0©
1
T
By rewriting (4) for an assigned t = t * as:
Two hypotheses are made in this subsection: (i) the
cost of storage system are expressed in terms of unit
storage capacity of the single charge/discharge cycle; and
(ii) network power losses are neglected. By keeping this
last on mind, the whole system can be regarded as a
common bus to which generators, loads and storage
devices are connected. The objective function is given in
terms of a weighted sum of the cost to be sustained for
purchasing the storage device and the squared deviation
of the currents of storage devices and generators. In
mathematical terms, the problem can then be formulated
as follows:
T
¦
J j (t ) −
=0
0
763
T
ªJ G º
ªEº
« J » = G «V » ,
¬ ¼
¬ ¼
min J T RJ dt
(6)
³
where E is the vector of voltages at the generators
busses, J G is the vector of the injected currents at the
generator buses, G is the network’s conductance matrix,
V and J are the vectors of voltages and injected
currents, respectively, at the remaining busses of the
network. By partitioning
G , the equivalent
representation of (6) can be derived:
ªJ G º ªG 00 G 0E º ª E º
»« »
« J » = «G
¬ ¼ ¬ E0 G EE ¼ ¬V¼
s.t.
T
³ diag(E
P = diag(J)V
(7)
(8)
d
T
T
° dJ (J RJ + Ȝ J sto ) = 0
° sto
® T
° diag(Esto )J sto dt = 0
°
¯ 0
³
(9)
Ploss = E J G + V J
d
T
T
° dJ (J RJ + Ȝ J sto ) = 0
° sto
T
®
°
J sto dt = 0
°
0
¯
(10)
By taking into account (7), the right hand side of (10) can
³
be rewritten as
−1
−1
ETG 00E + ETG 0EG EE
J − ETG 0EG EE
G E0E +
−1
−1
J TG EE
J − ETG TE0G EE
J
(11)
Since G 0E = G TE0 , (11) reads:
E G 00 E − E
T
T
−1
−1
G 0EG EE
G E0 E + J TG EE
J
(12)
generators busses are the same:
(13)
(19)
(20)
Since J load , that is the vector of injected currents at
the load busses, does not dependent on J sto , the previous
expression becomes:
(21)
where I is the identity matrix. By transposing (21), a
suitable equivalent formulation is derived:
(14)
ª J* º
2[0 I ]R J + Ȝ = 2[0 I ]R «
»+Ȝ=
«¬ J sto »¼
ª J* º
2[R 1 R 2 ]«
»+Ȝ=0
¬« J sto ¼»
thus implying that
−1
Ploss = J T G EE
J
dJ
+ ȜT = 0
dJ sto
ª0º
2J T R « » + Ȝ T = 0
¬I ¼
with 1 is the vector of all ones, it can be easily deduced
that:
−1
E TG 00 E − E TG 0EG EE
G E0 E = 0
(18)
From the matrix differentiation theory, it follows that the
solution of the system (19) can be derived by solving the
equation:
2J TR
It is worth to note that, in case the voltages at all
E = E1 ,
(17)
Under the hypothesis of constant voltage at the storages’
busses, the equation system can be rewritten as:
By observing that network power losses can be
obtained as:
T
) J sto dt = 0 ,
−1
with R = G EE
, E sto and J sto the n S -vector of voltages
and injected currents, respectively, at the storage busses.
In the case hypothesis (13) is not satisfied, further power
losses have to be considered as consequence of
circulating currents due to the different voltages at the
generators’ terminals.
The isoperimetric constraint (17) allows obtaining a
null net energy exchanged by the storage device during
the considered time T (e.g., the day). The solution of this
isoperimetric problem is provided by the solution of the
system:
with:
0
0º
ª J nG +1
« 0
J nG +2 0 »»
diag(J ) = «
« 0»
»
«
0
0 Jn ¼
¬ 0
sto
0
where the matrix G EE is clearly not singular.
The vector of the injected powers at all the busses P
can be derived from J and V as:
T
(16)
0
(15)
Under this hypothesis, the optimization problem that is
aimed to minimize power losses, can then be formalized
as follows:
(22)
where J * = J load . The elements of R 1 and R 2 can be
764
easily derived as:
R 1i , j = R(nL +i , j )
i = 1,..., nS ,
j = 1,..., nL
(23)
R 2i , j = R(nL +i ,nL + j ) i = 1,..., nS ,
j = 1,..., nS
(24)
DC ship network
Jsto
Thus (22) becomes:
2R1 J* + 2R 2 J sto + Ȝ = 0
Vsto
(25)
DC
Once the Lagrange multipliers are evaluated, the optimal
storage current profiles which realizes the minimization
of the network power losses can finally be derived as:
DC
a)
T
·
§1
J sto = R −21R1 ¨
J*dt − J* ¸
¸
¨T
¹
© 0
³
Vsc
Once having determined J sto , the actual voltages at
the load and storage busses can be easily derived.
Furthermore, the estimation of injected currents at the
generator busses are given by:
(
)
−1
J G = G 00 − G 0EG EE
G E0 E +
−1
G 0EG EE
ª
§ T
·º
«J * , R 2−1 R 1 ¨ 1 J *dt − J * ¸»
¨T
¸»
«
© 0
¹¼
¬
DC ship network
T
³
(27)
Vsto
Jsto
DC
Based on the required current profiles at the storage
busses, the size of the storage capacity can be easily
derived based on the considered technology. Details on
economic implications can be found in [6], where the
problem of finding the capacity of the storage device is
discussed.
AC
b)
AC
DC
supercap
IV. SUPERCAPACITOR OPTIMAL DESIGN
dVsc
dt
ESS
Vsc
In this section, the optimal sizing of supercapacitors in
DC networks is discussed. The procedure is able to select
the size of a suparcapacitor able to guarantee the current
profile evaluated by (26). In Fig. 3, two possible
configurations of a supercapacitor is showed: to connect
the storage device to the network an interfacing DC/DC
converter (Fig. 3.a) or AC/DC converters coupled with a
transformer (Fig. 3.b) can be used.
In both case of Fig. 3, it is assumed that the
supercapacitor operates at a specific voltage (Vsc) which
is different from the voltage value of the bus where the
ESS is connected (Vsto). The optimal sizing of the
supercapacitor consists in evaluating the value of the
capacitance which is able to provide the desired current
profile:
J sup = C (Vsc )
ESS
supercap
(26)
Fig. 3 Possible configurations of the connection of supercapacitors to
the DC network.
A widely accepted model of the capacitance is based
on a simplified first order equivalent circuit which
includes a constant term (C0 ) and a term linearly variable
with the voltage (C1 ) [9], [10], which reads:
J sup = (C 0 + 2C1Vsc )
dVsc
dt
(29)
Under the hypothesis of ideal conversion, the
following relationship applies:
(28)
(C0 + 2C1Vsc ) dVsc
where J sup is the supercapacitor’s current, Vsc is the
dt
supercapacitor’s voltage and C (Vsc ) is the capacitance
value; the latter depending on the value assumed by the
supercapactor’s voltage.
which can be rewritten as:
=
Vsto J sto
Vsc
1
2
C0 dVsc 2 + C1dVsc 3 = Vsc J sto dt
2
3
(30)
(31)
By integrating (31), the values of the terms of
765
capacitance and voltage can be expressed in terms of the
energy stored in the supercapacitor at the generic time t.
In particular, two given times can be selected during the
time period T: the first refers to the maximum energy
stored in the supercapacitor; the latter refers to the
minimum energy stored in the supercapacitor:
(
)
(
)
C0 2
2
V0 − Vsc2 ,min + C1 V03 − Vsc3 , min =
2
3
³
(
)
(
)
³
3
0
where V0 is the initial value of the voltage, Vsc, min is the
)
)
(
)
(34)
(
)
(
)
(35)
C0 2
1 C0
V0 − Vsc2 , max +
V03 − Vsc3 , max = K min
2
6 Vsc , rated
t
K max = max Vsto J stodt
t
0
t
K min = min Vsto J sto dt
t
³
(V
2
0
− Vsc2 , max
)
2
K min
1 1
+
V03 − Vsc3 , max
3 Vsc , max
(
)
(39)
0
V. DISCUSSION
Eqs. (34) and (35) consist in a third order system where
the values of C 0 and V0 are unknown. In practical cases,
it is assumed that the maximum value of the voltage is
the rated voltage of the supercapacitor and the minimum
voltage is that corresponding to a specified percentage of
the rated voltage, being its value corresponding to the
specified maximum value of the depth of discharge. As is
well known, this type of equations has a real solution. In
this case, the solutions can be derived from a third order
algebraic equation obtained by the ratio of (34) and (35):
The solution of the design problem has been obtained
by considering the cost coefficients as variable
parameters at the aim to make it usable by the designers,
who may choice the most proper value of size for the
storage . This allows also to perform a sensitivity analysis
which could be also employed to derive off-line control
trajectories which can be successively employed for realtime control, for which suitable stability margins and
robustness are required. The sensitivity analysis is
particularly simple due to the linearity of the obtained
solution: it is expected a behavior close to the absolute
optimum referring to the rated parameters.
The randomness of the various parameters related to
the load uncertainties can be characterized by describing
them by means of probabilistic density functions. At this
regard, it has to be noted that the assumption of
multivariate Gaussian distribution is widely performed in
the technical literature. Then, it can be observed that, as a
consequence of the linearity of the solution, a Gaussian
multivariate distribution results for the storage current
profiles. Therefore, a risk-based design can be afforded in
a correct way, by ensuring the required robustness against
1 § 3 27 3
§ 2 9 2
· 1
·
Vsc , rated ¸
¨V0 − Vsc ,rated ¸ +
¨V0 −
25
125
©
¹ 3 Vsc ,rated ©
¹
= ξ (36)
1
1
2
2
3
3
V0 − Vsc, rated +
V0 − Vsc, rated
3 Vsc, rated
(
(38)
3
Finally, the value of the rated value of the capacitance
is derived as C = 3 / 2C0 .
Based on the capacitance value, the number of
supercapacitors’ elements can be found, as well as the
size of the interfacing converter and/or transformer
depends on the voltage levels of both supercapacitor and
network bus. Moreover, a correct sizing of an ESS should
imply the consideration of the uncertainties in managing
both loads and generators, the economic aspects related to
the cost sustained for the installation and operation of
ESSs and the benefits obtained in terms of losses
reduction. However, this is out of the scope of this paper.
where
³
2
Vsc3 ,rated §
162 ·
¸.
¨ 4ξ −
(1 − ξ ) ©
125 ¹
C0 =
to the discharging stage. In most of the practical cases,
the suitable approximation C1 = C0 / 4Vsc, max [11] is
(
2
3
Once V0 is evaluated, C 0 can be derived as:
voltage corresponding to the charging stage of the
supercapacitor and Vsc, max is the voltage corresponding
considered. Thus, (32) and (33) read:
C0 2
1 C0
V03 − Vsc3 , min = K max
V0 − Vsc2 , min +
2
6 Vsc , rated
3
where
α = 3V sc , rated
β=
(
(37)
β·
β · §α ·
α
§ −α
§ −α
− ¸− ¨
− ¸ −¨ ¸ −
¨
27
2
27
2
9
3
©
¹
©
¹ © ¹
3
(33)
t
min Vsto J sto dt t ∈ [0, T ]
t
2
§ −α3 β · §α2 ·
§ −α3 β ·
− ¸ −¨ ¸ +
V0 = 3 ¨
− ¸+ ¨
2¹
2¹ © 9 ¹
© 27
© 27
0
C0 2
2
V0 − Vsc2 , max + C1 V03 − Vsc3 , max =
2
3
1
162 ·
§
Vsc3 ,rated ¨ 4ξ −
¸=0
125 ¹
(1 − ξ )
©
whose solution is:
(32)
t
max Vsto J sto dt t ∈ [0, T ]
t
V03 + 3Vsc,rated V02 +
)
(
)
where Vsc,max = Vsc,rated , Vsc, min = 3 / 5 Vsc,rated (i.e., it is
assumed that the maximum voltage is assumed to be the
rated voltage and its minimum value is assumed 60% of
the rated value) and ξ = K max / K min . Eq. (36) depends
only on the unknown variable V0 and can be rewritten in
the following suitable expression:
766
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the loads uncertainties. This is particularly important in
case of ship applications where complexities related to
the survivability in case of system failure have to be
handled through effective and accurate approaches.
VI. CONCLUSION
The complexity of the on board power networks of
modern ship requires the use of new paradigms of the
planning and control of loads and generators. At this aim,
in this paper efforts have been made in order to discuss
on the usefulness of analytical approaches for the optimal
sizing of storage devices used in the context of the all
electric ship. Based on some requirements particularly
debated in literature, which are aimed at reducing power
losses and lowering the power supplied by the on board
generator, two different optimization problems are
formulated and analytically solved. Some details and
hypotheses are discussed and the implications analyzed
with reference to the use of different storage
technologies. With reference to the supercapacitors,
which can be used in the case of severe pulsed loads, an
analytical sizing approach for the capacity determination
is also detailed. Future research efforts will be devoted at
applying these analytical methods to real case study
where the use of methods based on numerical solution is
discouraged by the complexities of the system.
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