2006-01-3050
Modeling and Simulation of an Electric Warship Integrated
Engineering Plant
A.M. Cramer, R.R. Chan, S.D. Sudhoff
Purdue University
Y. Lee, M.R. Surprenant, N.S. Tyler, E.L. Zivi
U.S. Naval Academy
R.A. Youngs
Anteon Corporation
Copyright © 2006 SAE International
ABSTRACT
A layered approach to the simulation of dynamically
interdependent systems is presented. In particular, the
approach is applied to the integrated engineering plant
of a notional all-electric warship. The models and
parameters of the notional ship are presented herein.
This approach is used to study disruptions to the
integrated engineering plant caused by anti-ship
missiles. Example simulation results establish the
effectiveness of this approach in examining the
propagation of faults and cascading failures throughout a
dynamically interdependent system of systems.
INTRODUCTION
Effective design of advanced, integrated systems often
requires consideration of the dynamic interdependence
of the constituent subsystems. Although shipboard
machinery systems have traditionally been designed
independently, the integrated engineering plant (IEP) of
an all-electric warship includes tightly coupled energy
and fluid transport systems. This plant includes electrical
generation, distribution, and propulsion; freshwater and
seawater based thermal management; and distributed
controls. The modeling and simulation of a notional
electric warship is described herein.
In the past, it has often been the case that, for example,
the electrical and thermal systems are represented
independently; however, they are strongly interrelated.
The power electronics converters in the system require
cooling to operate, but the pumps that circulate the
cooling fluids require electric power. By developing an
integrated approach for the simulation of this
interconnected system, it is possible to investigate the
effects of this interdependency.
The simulation approach presented in this paper is a
layered approach wherein each of the subsystems is
represented in a separate layer with well-defined inputs
and outputs from other layers. The various layers
encompass different aspects of the system’s overall
behavior. These layers include a spatial layer (used to
determine effects of incoming weapon detonations), an
automation layer, an ac medium voltage high power
distribution layer, a zonal dc power distribution layer, a
seawater cooling network layer, and a freshwater cooling
network layer. The layered approach is flexible enough
to incorporate future layers as more aspects of the
plant’s behavior are modeled. Models describing the
components of each layer and their parameters are
included. These models are based on a physical test
site—the land based Naval Combat Survivability
Testbed located at Purdue University [1].
The goal is to produce concise, computationally efficient
models which capture the complex dynamical
interdependence of the disparate subsystems.
Moreover, a model interconnection scheme allows
subsystem models to be considered independently or as
an integrated, dynamically interdependent composite.
Once the subsystem models have been formulated, the
use of the model to study disruptions from anti-ship
missiles on the IEP is considered.
These example simulation results establish the
effectiveness of this approach to examine the
propagation of faults and cascading of failures
throughout a dynamically interdependent system of
systems.
SYSTEM OVERVIEW
Electric warship IEPs provide services to the ship which
are vital to completing the ship’s mission. These
services include electrical power, mobility, and thermal
management. In the notional ship considered herein, the
IEP includes ac electric power networks, a zonal dc
electric power distribution network, a seawater cooling
network, and freshwater cooling loops.
The IEP considered in this work has two ac networks
and a zonal dc network with three zones. One of the ac
networks is located forward and the other aft. This
geographical separation decreases the likelihood that
both networks would be damaged during an explosion.
The structure of the electric system is shown in Figure 1.
In the figure, each ac network is supplied with
mechanical energy from a prime mover (PM) which
represents a gas turbine. The PM controls the speed at
which the shaft of the corresponding synchronous
machine (SM) rotates, controlling the electrical
frequency of the ac that is generated by the SM. A
brushless exciter/voltage regulator (BE/VR) regulates
the output voltage of each SM. Each SM is connected to
an ac bus which feeds two loads, a propulsion drive
(PD) which drives the propulsion motor and a power
supply (PS) that delivers power to the zonal distribution
network. The power supplies from each of the ac
networks supply a primary dc bus (PDCB) on each side
of the ship. A converter module (CM) is connected to
each PDCB in each zone. Opposite CMs are connected
to a zonal dc bus (ZDCB) which supplies power to the
inverter module (IM) in each zone. The IM provides ac
power to the ship service loads located in that zone.
Figure 2. Seawater Network Overview
Figure 3. Freshwater Loop System Overview
The control scheme used herein has been called
anarchist because it only relies on local supervisory
controllers to determine when to operate each device.
There is no communication between devices. Each
device is operated when the local measurements
suggest that conditions are viable to operate. The overall
layout of the notional ship is shown in Figure 4.
Figure 1. Electric System Overview
Shipboard electrical components are cooled by the
seawater network depicted in Figure 2. In this figure,
numbers enclosed in squares indicate seawater nodes
(SWNs), numbers enclosed in circles indicate seawater
valves (SWVs), and unenclosed numbers indicate
seawater branches (SWBs). In each zone of the ship a
seawater pump (SWP) provides pressure to the
network. Component heat exchangers (CHXs) are used
to cool larger loads (particularly SMs and PDs) directly
from the seawater network, while freshwater loop
systems (FWLSs) are used to cool the smaller loads.
The FWLSs are shown in Figure 3. Each component is
cooled by a freshwater loop (FWL). The FWLs come
together to be cooled by a fluid heat exchanger (FHX)
that is cooled by the seawater network.
Figure 4. Layout of Notional Ship
MODEL STRUCTURE
The simulation approach presented herein is a layered
approach wherein each of the subsystems described
above is represented in a separate layer with welldefined inputs and outputs from the other layers. The
various layers encompass different aspects of the
system’s overall behavior. These layers include the
spatial, automation, ac, dc, seawater, and thermal
layers, but the layered approach is flexible enough to
incorporate future layers as more aspects of the plant’s
behavior are modeled.
The different layers of the ship are shown in Figure 5.
The spatial layer describes the geographic location of
each of the components of the IEP. This description and
information about a potential weapon event are used to
determine which components would survive that event.
The automation layer contains models of the supervisory
controllers that govern the operation of IEP components.
This layer combines information from other layers to
determine when a given device can operate. This
includes checking whether the device has survived the
weapon event. The ac layer contains models of the two
ac networks in the system, which interacts extensively
with the dc layer which models the zonal distribution
network. The seawater layer models the pressure and
flow of seawater through the seawater network, while
the thermal layer models the heat flow through heat
exchangers and freshwater cooling loops.
Before the models are presented, some notational
issues will be discussed. Heaviside notation is used
throughout this paper. Thus the time derivative of a
variable x is denoted px . The dot product operator is
written as ⋅,⋅ , and all norms, ⋅ , are Euclidean norms,
i.e. ⋅ 2 . The operator := represents assignment rather
than algebraic equality. The function
bound(⋅,⋅,⋅) is
x>u
⎧u
⎪
defined such that bound(l , u, x ) = ⎨ x l ≤ x ≤ u . Finally, I n
⎪x
x<l
⎩
is taken to be the n × n identity matrix.
SPATIAL LAYER
The spatial layer represents the ship components as
geometric objects. With this representation, it is possible
to determine the effect of a spatially located disturbance
on the overall performance of the IEP. In particular, it is
possible to determine which components will be
damaged by the detonation of an anti-ship missile.
A Cartesian coordinate system is defined as shown in
Figure 6. The origin is located at the ship’s baseline and
lines up with the ship’s forward perpendicular. The
positive x-axis is oriented to the aft, the positive y-axis is
oriented to the starboard, and the positive z-axis is
oriented upwards.
Figure 5. Modeling Approach
A single component will often exist in multiple layers
simultaneously, each representation capturing a different
aspect of the component’s behavior. Consider a CM in
the zonal distribution network. A traditional electrical
model of a CM exists in the dc layer, but a rectangular
prism represents the physical location of the CM in the
spatial layer. A supervisory controller model that governs
the operation of the CM exists in the automation layer.
Finally, the power electronics in the CM are cooled by a
FWL in the thermal layer.
As illustrated with the CM, the behavior of a single
component is divided across logical boundaries. This
division helps to manage the complexity involved in the
modeling and simulation of such an interdependent
system. Additionally, the division can improve the
efficiency of the simulation by allowing loosely coupled
layers to be simulated using different integration
algorithms and time steps. In particular, the faster
electrical dynamics are simulated separately from the
slower thermal dynamic behavior.
NOTATION
Figure 6. Cartesian Coordinate System
An explosion is represented as a sphere with radius rd
centered at d = ( xd , yd , z d ) . Any component whose
geometric representation in the spatial layer intersects
the sphere is considered destroyed by the missile. The
primary function of the models in the spatial layer is to
test for this intersection and determine the hit status, h ,
of the component.
POINT
Very small components can be represented
geometrically as a single point in three space. In this
context, very small is defined in relation to the radius of
the explosion, rd . An example of such a component is
an SWV.
To determine whether the point is hit, the distance from
the point, c = (xc , yc , zc ) , to the center of the explosion is
calculated using
d = c −d .
(1)
Then the hit status of the component is determined by
comparing this distance to the radius of the explosion.
Thus,
h = (d ≤ rd ) .
(2)
POLYGONAL PATH
Most interconnections, whether they are signals, power
connections, or pipes for fluid transport, can be
represented in a geometric sense as a polygonal path
consisting of a finite sequence of line segments where
the first endpoint of one segment is the second endpoint
of the previous segment.
To determine whether the path is hit, each of the ns
total segments should be considered separately. The
endpoints of the s th segment are c s = (xc, s , yc, s , z c, s ) and
c s + 1 = (xc , s +1 , yc, s +1 , z c, s +1 ) . If c s = c s + 1 then this segment
is actually a point. If s > 1 then the point was the second
endpoint of the previous segment and has already been
tested for intersection with the explosion. If s = 1 then
the point should be checked for intersection using (1)
and (2).
In the general case that c s ≠ c s +1 then any point on the
line connecting c s and c s + 1 can be written as
c = αc s + (1 − α )c s + 1 . If α ∈ [0,1] then the point c is
actually on the line segment. To determine the closest
point on the line to the center of the explosion
F = c−d
2
2 is minimized with respect to α . The value
of α that minimizes F is given by
α=
d − c s+1 , c s − c s+1
c s − c s+1
.
(3)
Projecting α onto [0,1] yields the closest point on the
line segment, c , to the explosion center. Then (1) and
(2) can be used to check this point for intersection.
As soon as any point on the path has been found to
intersect the explosion sphere it is not necessary to
consider remaining segments.
component. An example of such a component is a PD.
The bounding box would contain the actual propulsion
motor in addition to the drive itself. The prism is defined
by its centroid, c = ( xc , yc , zc ) and by its length, width,
and height, Δ = (Δx, Δy , Δz ) .
To determine whether the prism is hit, one can project
the explosion center into the prism so that
c−
Δ ≤p≤c+ Δ
2
2
where p is the projected version of d , and
inequalities in (4) are taken to hold elementwise.
example, the x-component of p is given
x p = bound( xc − Δx 2 , xc + Δx 2 , xd ) . Then p is
(4)
the
For
by
the
closest point in the prism to d . The distance from the
explosion center to the closest point in the prism is given
by
d = p −d .
(5)
Finally, (2) can be used to check for intersection.
AUTOMATION LAYER
The automation layer contains models of all of the
supervisory controllers in the system. It has
representations of the local protective controllers that
ensure that components only operate when it is safe and
possible. Recall that the control scheme used herein is
anarchist and only relies on these local controllers to
determine when to operate each device. However, if a
hierarchal control system were implemented it would be
represented in this layer; although any required
communication links would require a communications
layer. That addition would be straightforward because of
the flexible nature of the layered modeling approach.
The automation layer also contains models of the
actuators that are used to reconfigure the system. These
actuators serve a vital role in the ship’s ability to recover
from damage situations.
Each of the local supervisory controllers determines
when it is possible to activate and when it is necessary
to deactivate the associated device. Each constructs an
activate and a deactivate signal, α and β , respectively.
Then, on any change in α or β , the operation of the
device is updated in accordance with
o := α + β o .
(6)
RECTANGULAR PRISM
Most components can be represented in three
dimensional space as rectangular prisms. Even if a
component is not strictly a rectangular prism, it is
possible to find a bounding box that contains the
Throughout the descriptions below, o * represents the
commanded operation status of the device, and h
represents the hit status of the device which is
determined in the spatial layer. Also, x y ,a ,min and x y ,a ,max
represent the minimum and maximum values of the
variable x y for which the device is allowed to activate.
Similarly, x y ,o ,min and x y,o ,max represent the minimum
and maximum values for which the device is allowed to
operate. In general, x y,o ,min < x y ,a ,min < x y ,a ,max < x y ,o ,max .
SYNCHRONOUS MACHINE OPERATION
The SM supervisory controller determines when a SM
can operate. The SM is permitted to operate when it is
commanded to, it has not been damaged, the rotor
speed is in an acceptable range, and it has not
overheated. Thus the activate and deactivate signals are
given by
α = o*h (ω rm ≥ ω rm,a ,min )(ω rm ≤ ω rm ,a ,max )oh
(7)
where Thx is the cold plate temperature of the CHX that
is cooling the PD power electronics.
POWER SUPPLY OPERATION
The PS supervisory controller determines when a PS
can operate. The PS is permitted to operate when it is
commanded to, it has not been damaged, its input
voltage is in an acceptable range, and its temperature is
reasonable. Thus the activate and deactivate signals are
given by
α = o* h (vin ≥ vin,a ,min )(vin ≤ vin,a ,max ) ⋅
and
β = o * + h + (vin < vin,o ,min ) + (vin > vin,o ,max ) +
and
(Thx > Thx,o,max )
β = o * + h + (ω rm < ω rm,o,min ) + (ω rm > ω rm,o ,max ) + oh , (8)
respectively, where ωrm is the mechanical rotor speed,
and oh is the overheat status which is governed by
oh := oh + (Thx > Thx,o ,max )
(9)
where Thx is the cold plate temperature of the CHX that
is cooling the SM.
PROPULSION DRIVE OPERATION
The PD supervisory controller determines when a PD
can operate. The PD is permitted to operate when it is
commanded to, it has not been damaged, its input and
dc link capacitor voltages are in acceptable ranges, and
it has not overheated. Thus the activate and deactivate
signals are given by
α = o* h (vin ≥ vin,a ,min )(vin ≤ vin,a ,max ) ⋅
(10)
(vc ≥ vc,a,min )oh
and
β = o * + h + (vin < vin,o ,min ) + (vin > vin,o ,max ) +
(vc < vc,o,min ) + oh
,
(11)
,
(14)
respectively, where vin is the peak input voltage and Thx
is the cold plate temperature of the CHX that is cooling
the PS.
CONVERTER/INVERTER MODULE OPERATION
The CM/IM supervisory controller determines when a
CM or an IM can operate. The CM or IM is permitted to
operate when it is commanded to, it has not been
damaged, its input voltage is in an acceptable range,
and its temperature is reasonable. Thus the activate and
deactivate signals are given by
α = o* h (vin ≥ vin,a ,min )(vin ≤ vin,a ,max ) ⋅
(15)
(Thx ≤ Thx,a,max )
and
β = o * + h + (vin < vin ,o,min ) + (vin > vin ,o ,max ) +
(Thx > Thx,o,max )
,
(16)
respectively, where vin is the dc input voltage and Thx is
the cold plate temperature of the CHX that is cooling the
device.
LOAD OPERATION
respectively, where vin is the peak input voltage, v c is
the dc link capacitor voltage (described in the propulsion
drive model below), and oh is the overheat status which
is governed by
oh := oh + (Thx > Thx,o ,max )
(13)
(Thx ≤ Thx,a,max )
(12)
The load operation controller determines when an
electrical load (SWP or FWL) can operate. The load is
permitted to operate when it is commanded to, it and its
line to the source IM have not been damaged, and the
source IM is operating. Thus the activate and deactivate
signals are given by
α = o * h oin
(17)
and
3] in each network is taken to be that of the rotor
reference frame of the corresponding SM.
β = o * + h + oin ,
(18)
respectively, where h is the hit status of both the load
and the line that connects the load to the IM that
supplies the load’s power. Also, oin is the operation
status of that IM.
SEAWATER VALVE
Two types of SWVs are considered for securing the
seawater network after battle damage has occurred.
Type 1 SWVs are initially open, and, if a certain flow rate
is exceeded, they close and cannot be reopened. Type 2
SWVs are somewhat more sophisticated. After a flow
rate has been exceeded they allow a specified volume to
flow before turning off. Ideally, this results in a better
choice in valve closures. In the future, additional valve
types may be added, which, for example, may limit flow
to a specified value rather than closing completely.
The model of a type 1 SWV is as follows. The status of
a valve is denoted c and is initially true meaning that the
valve is open and allows fluid to flow. Thereafter, it is
governed by
c := c ( q ≤ qT ) + h
(19)
where q is the flow rate through the SWV, h is the hit
status of the valve, and qT is the threshold flow rate,
above which the SWV closes.
The model of a type 2 SWV is as follows. First, the
excess flow is defined as
⎧
⎪ 0
qe = ⎨
⎪
⎩ q − qT
q ≤ qT
q > qT
(20)
To interconnect the different models in the ac layer, it is
assumed that the ac bus model inputs are currents and
outputs are voltages. The models that are connected to
buses (SMs, PDs, and PSs) input voltage and output
current. Since the ac bus calculates the voltage
algebraically, the other models must produce the current
without an algebraic dependence on the voltage.
AC BUS
An ac bus is represented as a fictitious resistance r fict
to ground. The model of this bus depends on the hit
status of the bus. The hit status h is taken as the logical
AND of the hit statuses of all the polygonal paths
composing the ac network. The q-axis and d-axis
voltages of the n th bus can be found using
⎧⎪− r i e
v eqd,n = ⎨ fict qd,n
⎪⎩
0
h
h
(23)
where i eqd,n is a vector of the sum of the q-axis and daxis currents flowing out of the n th bus.
PRIME MOVER
The PM model represents the source of mechanical
power for the SM. This source would typically be a gas
turbine, but the specific details are abstracted by the
model. The PM also regulates the rotational speed of the
shaft.
The PM model is described by the block diagram shown
in Figure 7. The parameters of this model are given in
the appendix.
where q and qT are defined above. The excess volume
is then defined as
pVe = qe .
(21)
The status is governed by
c := c(Ve ≤ VT ) + h
(22)
where VT is the threshold excess volume.
AC LAYER
The ac layer includes models of the generation plants
(PM, SM, and BE/VR), the PDs, the PSs, and the ac
buses. In the system studied herein, there are two
separate ac networks, each including one SM. The
position of the synchronous reference frame [2, Chapter
Figure 7. Prime Mover Model
SYNCHRONOUS MACHINE
The SM model is based on [3]. The interesting features
of this model are the incorporation of magnetic
saturation and an arbitrary rotor network representation.
The model parameters as well as the method of
obtaining the parameters are set forth in [4].
The SM model is well documented in [3], and as such
will not be discussed in detail here. A modification was
made to the model to support the machine shutting
down (either by choice or necessity). When the machine
shuts down, it disconnects from the system. The PM
continues to spin, and the BE/VR continues to regulate
the terminal voltage of the machine. However, when
shut down the machine no longer interacts with the rest
of the network. In order to determine the terminal
voltage required by the VR, a fictitious resistance r fict is
added behind the circuit breaker to compute the terminal
voltage. This is illustrated in Figure 8. In this figure, the
e
internal values ( v̂ eqds and î qds
) are interfaced to the
model in [4], but the values are determined using
e
⎧
⎪ v qds
vˆ eqds = ⎨
ˆe
⎪
⎩− r fict i qds
o
o
(24)
BRUSHLESS EXCITER/VOLTAGE REGULATOR
The excitation system model encompasses both the BE
which is based on [5] and the VR. The BE model
incorporates multiple rectifier modes, but hysteresis is
neglected by representing the magnetizing flux linkage in
the d-axis as an affine function of the d-axis magnetizing
current. The assumed exciter parameters as well as the
method of obtaining these parameters are set forth in
[6].
The stationary field winding of the exciter machine is
driven by the field drive circuit shown in Figure 9. This
circuit attempts to drive the field current i fde to the
commanded field current i *fde .
and
⎧
⎪ˆi e
i eqds = ⎨ qds
⎪
⎩ 0
o
o
(25)
where o is the operation status of the SM determined in
the automation layer. In this way, the terminal voltage
can be calculated using
Figure 9. Field Drive
Vt =
v̂ eqds
.
(26)
To model the action of this circuit an effective dc voltage
is found using
v dc, pow
⎧
v rect = ⎨
⎩v dc, pow − K fix iifde
i fde ≥ 0
i fde < 0
.
(29)
where vdc, pow is the voltage into the field drive circuit and
K fix is a large positive constant. If i fde < 0 then vrect
becomes very large, which tends to increase i fde . A
negative value of i fde is physically impossible because
Figure 8. Synchronous Machine Modification
To determine the power loss of the SM the output power
is calculated using
Pout = −
3 e
v qds , i eqds .
2
⎛1
⎝η
⎞
− 1⎟⎟ Pout .
⎠
circuit can produce are computed using
(27)
Then the losses are computed using an assumed
efficiency η ,
Ploss = ⎜⎜
of the topology of the circuit in Figure 9. The upper and
lower limits on the field voltage v fde that the field drive
(28)
v fde,max = vrect − vswitch − rswitch i fde
(30)
v fde,min = −vrect − vdiode − rdiodei fde
(31)
and
where v switch and rswitch are the voltage drop and
resistance of the transistors in Figure 9, and vdiode and
rdiode are the voltage drop and resistance of the diodes.
The current error is defined as
ierror = i *fde − i fde .
(32)
The exciter field voltage is then calculated using
(
(
⎧ bound 0, v fde,max , v fde ,max ierror Δ I
v fde = ⎨
⎩bound v fde ,min ,0,− v fde ,minierror Δ I
)
)
ierror > 0
ierror ≤ 0
(33)
It should be noted that the introduction of this filter is
artificial. It is added to solve the simulation problem of
interfacing different models together.
The open circuit rectifier voltage is given by
where Δ I is the hysteresis band of the controller. The
current command in (32) is determined using the VR
model shown in Figure 10.
Vr =
3 6E
(36)
π
where E is the input RMS voltage. The dc link variables
are governed by
pil =
Vr cos α − (rdc + 3Lcωe π )il − vc
Ldc + 2 Lc
(37)
and
pvc =
Figure 10. Voltage Regulator
The field drive and VR parameters are described in the
appendix.
Cdc
.
(38)
Finally, the current into the constant power load, i p , is
given by
PROPULSION DRIVE
The PD model consists of an uncontrolled rectifier model
connected to a constant power load model as shown in
Figure 11.
il − i p
⎧⎪P * vc
ip = ⎨
⎪⎩ 0
o
o
.
(39)
Since the current i l flows through the rectifier, it must
be nonnegative. Also, since the voltage vc is the voltage
across an electrolytic capacitor, it must also be
nonnegative.
It should be noted that there is an error in [2] regarding
the expressions for the commutation components of the
ac currents into the rectifier. The correct expressions for
(11.3-53) and (11.3-54) are
g
iqg
,com =
Figure 11. Propulsion Drive Model
3
(34)
Second, in order to interface the rectifier with the ac
system it is desirable that the ac currents are state
variables, which is not the case in [2]. To facilitate the
interconnection of the rectifier to the ac system, the ac
e
) are filtered
currents from the model described in [2] ( î qd
as shown in
i eqd =
1
ˆi e .
qd
τ filter s + 1
(35)
π
2E
π l cω g
The rectifier model is fundamentally the same as that
described in [2, Chapter 11]. Nevertheless, there are a
few details that should be discussed. First, since the
rectifier is uncontrolled then there is no firing delay so
α =0.
2 3
⎡
⎛
⎣
⎝
id ⎢sin⎜ u + α −
5π
5π ⎞
⎛
⎟ − sin⎜ α −
6
6 ⎠
⎝
⎞⎤
⎟⎥ +
⎠⎦
cos α [cos (u + α ) − cos α ] +
(40)
1 3 2E
[cos 2α − cos(2α + 2u )]
4 π l cω g
and
g
idg
, com =
3
2 3
2E
π l cω g
π
⎡
⎛
⎣
⎝
id ⎢− cos ⎜ u + α −
5π ⎞
5π
⎛
⎟ + cos ⎜ α −
6 ⎠
6
⎝
cos α [sin (u + α ) − sin α ] +
⎞⎤
⎟⎥ +
⎠⎦
. (41)
1 3 2E
[sin 2α − sin(2α + 2u )] − 3 2 E 1 u
4 π l cω g
π l cω g 2
Finally, the power into the rectifier is computed using
Pin =
3 e e
v qd , i qd ,
2
(42)
and assuming an efficiency of η the power loss is
computed using
Ploss = (1 − η )Pin .
(43)
POWER SUPPLY
Recall that both of these dc variables are required to be
nonnegative. The power loss is computed using (42) and
(43).
The PS control model is depicted in Figures 13–16
below. The purpose of the control module is to provide
the firing angle α to the rectifier. The slew rate limited
and short circuit protected commanded output voltage is
determined according to Figure 13.
The PS is shown in Figure 12. The PS is similar to the
PD model presented above. There are two differences.
In the PS, a transformer is situated between the ac
system and the rectifier. Also, the rectifier consists of
controlled thyristors instead of the uncontrolled diodes in
the propulsion drive. Thus there is control logic to
determine the firing angle.
Figure 13. Commanded Output Voltage
The commanded voltage is achieved through the use of
two control signals. The PI control signal is determined
using Figure 14. The D control signal is determined
using Figure 15.
Figure 12. Power Supply Model
Because of the transformer, the voltages and currents in
the model in [2] need to be scaled by the turns ratio TR .
That is the RMS voltage used by the model should be
E = TR ⋅ Eˆ
(44)
where Ê is the RMS voltage of the bus. The q-axis and
d-axis currents into the transformer are then
ˆg
i qdg
g
(45)
i qdg
=
TR
Figure 14. PI Control Signal
ˆg
is a vector of the q-axis and d-axis currents
where i qdg
into the rectifier. Again, the filter in (35) is used to
interface the PS model to the bus model. The open
circuit rectifier voltage Vr is again defined by (36). The
dc variables are governed by
pil =
Vr cos α − (rdc + 3Lcωe π )il − vout
Ldc + 2 Lc
(46)
and
pvout =
il + iout
.
Cdc
(47)
Figure 15. D Control Signal
Finally, the firing angle α is determined using Figure 16.
Figure 16. Combined Control Signal
DC LAYER
The dc layer has models of the PDCBs, CMs, ZDCBs,
and IMs. These components form the backbone of the
zonal electric distribution system. This zonal distribution
architecture is fundamental to the plant’s ability to
reliably provide power in the event of battle damage.
Figure 18. Zonal Tie Line Model
In the Norton equivalent circuit formulation of the
network, each zone will be represented using the bus
equivalent shown in Figure 19. The algorithm below
calculates the Norton equivalent circuit parameters ( jc ,
g c , and g z ).
PRIMARY DC BUS
The zonal distribution architecture consists of two
PDCBs. These buses, on the port and starboard sides of
the ship, conduct dc from the PSs to the CMs. The
model of the bus contains not only the bus itself, but
also the tie lines that connect components (PSs and
CMs) to the bus.
The PDCB model is purely resistive, requiring only the
solution of a linear system to determine the solution.
Difficulty arises because switch configuration can cause
the voltage at certain nodes to be undefined. An
algorithm to determine the voltages and currents is set
forth. This algorithm uses an aggregate Norton
equivalent circuit to represent the components
connected to each zone.
Figure 19. Bus Equivalent
Step 0 – Initialization. Set Norton equivalent
conductance and current vectors to zero ( g c := 0 and
j c := 0 ).
Step 1 – Process component tie lines. For each
c ∈ {1, , nc } modify the Norton equivalent vectors as
follows
…
The model of the c th component tie line (of nc total
component tie lines) is shown in Figure 17. In this figure,
lc,c represents the location (in terms of zone number) of
the c th component. Thus, v z ,l
c ,c
⎧1 rc,c
⎪
+ ⎨2 rc,c
⎪ 0
⎩
cc,c hc,c
⎧⎪v r
jc ,l := jc,l + ⎨ c,c c,c
⎪⎩ 0
cc,c hc,c
g c,l := g c,l
c ,c
c ,c
is the zonal voltage of
the zone to which the c th component is connected.
(48)
cc,c hc,c
cc , c
and
c ,c
c ,c
cc,c + hc,c
.
Step 2 – Process zonal tie lines. For
z ∈ {1, , n z − 1} calculate the conductance as
…
Figure 17. Component Tie Line Model
The model of the zonal tie line connecting the z th zone
to the z + 1 th zone is shown in Figure 18. There are n z
total zones.
⎧1 rz , z
g z,z = ⎨
⎩ 0
c z,z
c z,z
.
(49)
each
(50)
Step 3 – Build system matrix. At
this
point,
an
equivalent electrical network has been found. From
Figure 19 it follows that the network equations may then
be expressed as
⎡ g c,1 + g z ,1
⎢
− g z ,1
⎢
⎢
0
⎢
⎢
⎢
0
⎣
− g z ,1
g c , 2 + g z ,1 + g z, 2
− g z ,2
0
− g z ,2
0
− g z, n
⎤ ⎡ v z ,1 ⎤
⎥
⎥⎢
v
⎥ ⎢ z, 2 ⎥
⎥
⎥⎢
0
⎥
⎥⎢
− g z, n z −1 ⎥ ⎢v z , n z −1 ⎥
g c, n z + g z , nz −1 ⎥⎦ ⎢⎣ v z, n z ⎥⎦
0
z
−1
=
⎡ jc,1 ⎤
⎥
⎢
j
⎢ c, 2 ⎥
⎥
⎢
⎥
⎢
⎢ jc, n z −1 ⎥
⎥
⎢ j
c
,
n
z
⎦
⎣
. (51)
Step 4 – Adjust for damage and isolated buses. If hz , z ,
indicating that the z th zone is hit, then the z th row is
set to zero except for the diagonal element which is set
to one, and jc, z is replaced with zero. Also, each block
Figure 20. Converter Module
diagonal matrix must be invertible. A block diagonal
matrix that is not invertible represents a section of the
bus that is not connected to anything, causing the
voltage in this section to be undefined. The voltage in
this section can be arbitrarily defined to be zero. To
implement this definition, the same modification that is
performed for a damaged zone can be applied to one of
the zones in the unconnected section.
Step 5 – Solve system. A solver for tridiagonal systems
is described in [7, Chapter 2].
Step 6 – Solve for currents. The circuits depicted in
Figures 17 and 18 can be used to solve for the
component and zonal tie line currents.
Figure 21. Reduced Order Converter Module Model
ZONAL DC BUS
The ZDCB brings current from the port and starboard
CMs through ORing diodes into the IM. In this way, the
IM can function using power from either or both CMs,
thus ensuring continuity of service.
CONVERTER MODULE
The ZDCB model is shown in Figure 22. In this figure,
v p and i p represent the output voltage and current of
CMs are dc/dc converters that serve to isolate the PDCB
from damage inside the zone. Additionally, they
coordinate the load sharing between the two PDCBs.
the CM connected to the port side of the bus, and h p
The CM is shown in Figure 20. Since the inductor
current is tightly regulated in this hysteresis modulated
CM, it is appropriate to use the reduced order model
shown in Figure 21. In this figure, the inductor current is
given by
(
(
) )
*
⎧⎪bound 0, ilimit , vout
− vout d
il = ⎨
⎪⎩
0
o
o
o
o
.
and the power loss is given by (43).
Table 1. Determination of i p
ip =
hp
(53)
hl
v p ≥ vd
(54)
ip =
v p < vd
hp
hl
v p > vl + vd
v p ≤ vl + v d
Recall that the state variables associated with this circuit
are constrained to be nonnegative because the
capacitors are electrolytic. Finally, the input power is
given by
Pin = vin iin ,
1.
(52)
*
is the
where ilimit is the current limit of the CM, vout
commanded output voltage, d is the droop of the CM,
and o is the operation status of the CM determined in
the automation layer. The current i p is determined by
⎧v i v
i p = ⎨ out l in
⎩ 0
represents the hit status (determined in the spatial layer)
of the portion of the bus that connects to this port CM.
Similarly, the s subscript relates to the CM on the
starboard side, and the l subscript refers to the load,
the IM connected to the bus. A careful examination of
this circuit shows that i p can be calculated using Table
(
2v p
rp
2 v p − vd
)
rp
ip = 0
ip =
v p − vl − vd
rp
ip = 0
The current ilp represents the contribution to the load
current coming from the port side and can be calculated
using Table 2.
SEAWATER LAYER
Table 2. Determination of ilp
ilp = −
hl
hl
hp
vl < −v d
v p > vl + vd
2v l
rp
ilp = 0
vl ≥ −v d
hp
The seawater layer contains models of the components
of the seawater network. These components, including
SWBs and SWPs, are essential for removing heat from
the ship.
ilp = −
ilp =
v p ≤ vl + v d
SEAWATER NODE
2(vl + vd )
rp
An SWN is a location at which the pressure will be
determined. Figure 24 illustrates an SWN. Therein PN ,n
v p − vl − vd
denotes the pressure at the n th node and G Nlk is a
leakage conductance to ground. The flow into this
leakage conductance is the node flow q N , n . Since a
rp
ilp = 0
common value of leakage conductance is used for all
nodes, it is a parameter of the seawater solver (SWS).
The ground node is designated as node 0. The physical
aspects of the node are represented within the SWS.
Similar relationships exist for the starboard currents.
Figure 24. Seawater Node
SEAWATER BRANCH
The SWB model represents a pipe. The behavior of the
pipe model is described below, but much of this behavior
is contained within the SWS.
Figure 22. Zonal DC Bus Model
INVERTER MODULE
The IM is a dc/ac converter that consumes power from
the zonal distribution system and provides ac to zonal
loads.
The IM model is an input capacitor in parallel with a
constant power load. This constant power load
represents the power delivered to the zonal loads by the
IM. The model is shown in Figure 23. In the figure the
current into the constant power load is given by
⎧
⎪P * vin
ip = ⎨
⎪
⎩ 0
The basic structure of this model is depicted in Figure
25. Therein, as part of the first portion of a subscript, α
and β denote pipe ends. In the second part of the
subscript, α and β represent node numbers
associated with the α and β node ends, and b
represents the branch number. Hence, PN ,α and PN , β
are node pressures at nodes α and β , and q B,b is the
branch flow through branch b . The resistances R vα ,b
and R vβ ,b are associated with the two possible valves on
o
o
.
The power loss is calculated using (54) and (43).
(55)
the α and β side of branch b , respectively. These
valve resistances change depending upon whether the
valve is open or closed according to
⎧0
Rvx,b = ⎨
⎩RC
cvx,b
cvx,b
(56)
where x ∈ {α , β } , cvx,b is the valve status (calculated in
Figure 23. Inverter Module Model
the automation layer) of the x valve of the b th branch,
and RC is a large resistance indicating that the valve is
closed. The logical variable hB, b is the hit status. During
a branch fault, the nodes are isolated from each other
and the pipe is assumed to be faulted at both ends. The
pipe conductance to ground is G Bflt . The nonlinear
resistance R B,b represents the pipe resistance, and the
PP, p (q) =
∑
∑
constant pressure drop PB 0,b is associated with a
physical rise in the pipe ( PB0,b is positive if node β is
higher than node α ).
α P , np
⎛
N
q
⎜
aP,np
⎜
q Pb , p
n=1
⎜
PP0, p 1 +
β P , mp
⎜
M
q
⎜
1 + bP,mp
⎜
qPb , p
m=1
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(60)
where qPb , p is the base pump flow of the p th pump. In
(60), the nominal pump pressure PP 0, p is a variable
which is assumed to be governed by
PP 0, p =
⎧ PPss , p
1
⎨
τ P, p s + 1 ⎩ 0
o
o
.
(61)
In (61), o is the operation status of the pump, PPss , p is
the steady state value of PP 0, p , and
τ P, p
is the pump
time constant.
Figure 25. Seawater Branch
Assuming that a fault is not present, the pressure drop
from node α to node β may be expressed
(
)
PN ,α − PN ,β = q B,b Rvα ,b + Rvβ ,b + RB,b (q B,b ) + PB0,b (57)
Note that numbers of terms in the summations in the
numerator, N , and the denominator, M , of (60) have
been fixed at 2 and 3, respectively—these values have
been found to give an excellent representation of sample
data.
SEAWATER SOLVER
where
(58)
The fundamental function of the SWS is to solve for the
node pressures, as well as the node, branch, and pump
flows.
In (58), q B 0,b is the base flow for the b th branch. RB 0,b
Upon combining the flow relationships for the nodes,
branches, and pumps, a system of equations of the form
q ⎞⎟
⎟
⎜q
⎝ B 0 ,b ⎠
⎛
RB, b (q ) = RB0, b + RBq ,b ⎜
m B ,b
.
is the linear resistance, RBq ,b is the flow dependent
resistance, and mB,b is the exponent for resistance
calculation. This possibly nonlinear resistance can be
used to model turbulent flow in the seawater network.
SEAWATER PUMP
The SWP provides pressure to the seawater network.
Much of the behavior of the SWP is represented in the
SWS.
The relationship between the pump pressure and flow
for the SWP is given by
(
)
PN ,n = PP , p q P, p + RP, p q P, p
where
(
PN , n
PP, p q P, p
)
denotes pressure,
qP , p
(59)
A(q S )PN = b(q S )
(62)
q S = g(PN )
(63)
can be formed, where PN is the vector of node
pressures, q S is a vector of seawater flows, A (q S ) is a
square matrix whose elements are a function of q S , and
b(q S ) is a vector whose elements are a function of q S ,
and g is a function of PN .
In way of further definitions, q S is the concatenation of
the node flow vector q N , branch flow vector, q B , and
pump flow vector q P . In particular,
denotes flow,
is the flow and state dependent pressure in
the p th pump, and RP, p is a constant resistance term
for the p th pump.
The flow dependent pressure term is assumed to have
the form
qS =
⎡q N ⎤
⎢
q ⎥.
⎢ B⎥
⎢
⎣q P ⎥
⎦
(64)
An algorithm to solve (62) and (63) based on the GaussSeidel Method is now set forth. Let ( PNk , q Sk ) denote an
estimated solution of (62)-(63). The goal will be to
determine an improved solution ( PNk +1 , q Sk +1 ).
Step 0 – Initialization step. The iteration count is set to
zero ( k := 0 ) and, if the no other estimate of the flow is
available, q 0S := 0 .
( )
q Sk
q Sk
( )
q Sk
, A
and b
are
Step 1 – Build step. Using
determined. A detailed procedure to accomplish this will
be set forth below.
~
q S := q
S
The algorithm set forth is quite effective. However, the
calculation of A(⋅) , b(⋅) , and g(⋅) has yet to be
discussed. It can be shown that A(⋅) and b(⋅) may be
constructed using the following algorithm:
Step 0 – Initialization. The matrices are set to zero,
A := 0 and b := 0 .
Step 1 – Process node list.
A := A + G Nlk I N
Step 2 – Linear solution step. The linear system
( )
( )
A q Sk PNk +1 = b q Sk
(72)
(65)
is solved for PNk +1 . An efficient linear system solver is
presented in [7, Chapter 2]. Then
n
Step 2 – Process branch list. For each b ∈ {1,
where N b is the number of branches:
(73)
…, N }
b
Step 2a: Let α = nBα ,b and β = nBβ , b denote the two
nodes to which branch b is connected.
( )
~ := g P k +1
q
S
N
(66)
~ is a candidate for the updated flow vector.
where q
S
Step 2b: If hB, b , which is to say the branch is faulted,
then the following updates are performed:
Step 3 – Convergence test. Let ε and δ be small
positive constants which will determine the convergence
requirement. If
q~S ,i − qSk ,i ≤ ε q~S ,i + q Sk ,i
A αα := Aαα +
1
Rvα , b +1 GBflt
(74)
A ββ := Aββ +
1
Rvβ ,b +1 GBflt
(75)
(67)
Step 2c: If hB,b , which is to say the branch is not faulted,
then the value of the nonlinear resistance is saved using
or
q~S ,i − qSk ,i ≤ δ
{…
is satisfied ∀i ∈ 1, , N n + N b + N p
}
then it is assumed
that the convergence has been obtained and the
algorithm proceeds to Step 4. If the iteration count is
less than the maximum allowed count k max , an updated
flow vector is calculated as
q Sk +1
:= γq Sk
+ (1 − γ )~
qS
(69)
where γ is a convergence factor. In addition, the
iteration count is updated:
k := k + 1
(70)
Then execution continues at Step 2. If the iteration count
is equal to the maximum allowed count execution
terminates and the algorithm fails.
Step 4 – Convergence. If the convergence has been
obtained, then
PN :=
PNk +1
( )
~
RB,b := RB, b q Bk , b
(68)
(71)
( )
where R B,b q Bk ,b
(76)
may be computed from (58). Then if
the branch is active (all branches are assumed to be
active if they have not been deactivated below) the total
branch conductance is computed:
Gbranch :=
Rvα ,b
1
~
+ Rvβ ,b + RB,b
(77)
Next the following updates are performed:
Aαα := Aαα + Gbranch
(78)
Aββ := Aββ + Gbranch
(79)
Aαβ := Aαβ − Gbranch
(80)
Aβα := Aβα − Gbranch
(81)
bα := bα + PB 0,b Gbranch
(82)
bβ := bβ − PB0, b Gbranch
(83)
Terms involving α are not updated if α = 0 , and the
same is true for β .
{ …N p}
branch is activated. Branch deactivation prevents water
from flowing into the system from the ground node.
{ …, N p }:
Step 3 – Process pump list. For each p ∈ 1,
Step 3 – Process pump list. For each p ∈ 1,
Step 3a: Let n = nPn , p denote the node to which the
where N p is the number of pumps:
pump is connected.
Step 3a: Let n = nPn , p denote the node to which the
pump is connected.
Step 3b: The pump pressure is saved using
( )
~
PP, p := PP, p q Pk , p
(84)
( )
where PP, p qPk , p may be computed from (60). Then the
following updates are performed:
A nn := A nn +
bn := bn +
1
RP , p
(85)
~
PP, p
(86)
RP , p
~ can be calculated
The updated flow vector candidate q
S
using the following algorithm.
Step 3b: The pump flow is calculated using:
~
PNk ,+n1 − PP, p
~
qP, p :=
RP , p
(90)
THERMAL LAYER
The thermal layer contains models of the CHXs, FHXs,
and FWLs in the system. Whereas the seawater layer
was primarily concerned with fluid behavior, the thermal
layer is responsible for capturing the thermal
characteristics of the system.
COMPONENT HEAT EXCHANGER
Power component heat dissipation is removed through
conduction from the power components to a “cold plate”
heat sink. This cold plate is then cooled via convective
heat transfer to a cooling fluid as illustrated in the Figure
26.
Step 1 – Process node list.
~ := G P k +1
q
Nlk N
N
Step 2 – Process branch list. For each b ∈ {1,
(87)
…N }:
b
Step 2a: Let α = nBα ,b and β = nBβ ,b denote the two
nodes to which branch b is connected.
Step 2b: If hB, b , which is to say the branch is faulted,
then:
Figure 26. Cold Plate Heat Exchanger
q~B, b := 0
(88)
Step 2c: If hB,b , which is to say the branch is not faulted,
then:
PNk ,+α1 − PNk ,+β1 − PB0, b
q~B, b :=
~
Rvα ,b + Rvβ , b + RB ,b
(89)
If α = 0 and q~B ,b > 0 then q~B ,b := 0 and the branch is
deactivated. Similarly, if β = 0 and q~B ,b < 0 then
q~ := 0 and the branch is deactivated. Otherwise the
B ,b
Typically the cooling fluid is seawater, freshwater, or
chilled water. The heat exchanger is assumed to be well
insulated and the fluid flow is assumed to be one
dimensional “plug” flow.
The net heat flow into the heat exchanger cold plate is
the difference between the heat flow from the power
components and the heat flow removed by the cooling
fluid. This net heat flow determines the cold plate rate of
temperature change
pThx =
Qin − Qhx
mhx chx
(91)
where mhx and chx are the mass and the specific heat
of the cold plate, Qin is the heat into the cold plate from
the component, and Qhx is the heat out of the cold plate
to the cooling fluid.
Let wcf and ccf denote the mass flow rate and the
specific heat of the cooling fluid, and let A and h
denote the cold plate cooling fluid contact area and heat
transfer coefficient. If 2wcf ccf ≥ Ah then the rate at
which heat can be removed from the cold plate is
governed by
Qhx =
2wcf ccf Ah
2wcf ccf + Ah
where Tci is the inlet
temperature is given by
Tco = Tci +
(Thx − Tci )
temperature.
Qhx
.
wcf ccf
(92)
The
outlet
(93)
If 2wcf ccf < Ah then the cooling fluid heat removal
capacity is saturated and the rate at which heat can be
removed from the cold plate is governed by
Qhx = wcf ccf (Thx − Tci ) ,
(94)
and the outlet temperature is equal to the cold plate
temperature,
Tco = Thx .
(95)
FLUID HEAT EXCHANGER
The fluid heat exchanger cools the freshwater cooling
loop fluid using seawater which is then discharged
overboard. Once again, the heat exchanger is assumed
to be well insulated and the fluid flow is assumed to be
one dimensional “plug” flow. Heat transfer from the
freshwater to the seawater occurs through the
freshwater tube wall as depicted in Figure 27. The
following model requires the practical assumption that
the freshwater inlet temperature is always at least as
warm as the seawater inlet temperature. The tube wall is
assumed to be thin so that the energy stored in the tube
wall can be neglected. As is the case in actual shipboard
application, the heat exchanger has a counter flow
configuration meaning that the freshwater and seawater
flow in opposite directions [8]. Neglecting the spatial
component of the transient fluid thermodynamics along
the interior of the heat exchanger produces lumped
parameter fluid models.
Figure 27. Fluid Heat Exchanger
The heat flow through the heat exchanger cools the
incoming fresh water by heating the counter flowing
seawater. This heat flow is assumed to be a linear
function of the Log Mean Temperature Difference
(LMTD) between the freshwater and seawater [9-10]
Qhx = Asf hsf ΔTLMTD
(96)
where Asf and hsf are the heat exchanger contact area
and heat transfer coefficient, and ΔTLMTD is the LMTD.
However, for fresh water cooling loops, the lumped
parameter LMTD formulation does not behave well
during transient conditions or during saturation.
Saturation occurs when the temperature based
predicted heat transfer rate exceeds the heat removal
capacity of the seawater flow rate. Let T fi and T fo
denote the inlet and outlet temperatures of the
freshwater, and let Tsi and Tso denote the inlet and
outlet temperatures of the seawater. To avoid problems
with the LMTD formulation, the LMTD is approximated
by the average temperature difference
ΔTLMTD ≈ ΔTavg =
ΔTi + ΔTo
2
(97)
where ΔTi = T fi − Tso and ΔTo = T fo − Tsi . In [8], it is
shown that the average temperature formulation
introduces a 5% percent error if ΔTi and ΔTo differ by a
factor of two. Extensive testing indicates that errors of
this magnitude are negligible compared to the problems
created by the introducing saturation and transient
behavior into the LMTD formulation.
Let w f , c f , and m f denote the mass flow rate, specific
heat, and mass of the freshwater in the heat exchanger.
Also, let ws , c s , and ms denote the mass flow rate,
specific heat, and mass of the seawater in the heat
exchanger. When unsaturated, the heat flow through the
heat exchanger is given by
Qhx = Asf hsf ΔTavg
(98)
and the derivative of the freshwater outlet temperature is
given by
pT fo =
(
)
w f c f T fi − T fo − Qhx
mf cf
(
.
(99)
)
However, if Qhx > ws cs T fi − Tsi + ms cs pT fo then the heat
exchanger is saturated and (98) and (99) are replaced
with
pT fo =
(
)
(
w f c f T fi − T fo − ws c s T fi − Tsi
m f c f + ms c s
)
(100)
The flow rate is governed by
wcf =
⎧⎪wcf*
⎨
τ pump s + 1 ⎪⎩ 0
1
o
o
(103)
where wcf* is the nominal mass flow rate of the cooling
fluid, o is the operation status of the pump determined
in the automation layer, and τ pump is the pump time
constant. The transport lags are calculated using a
“plug” flow assumption based on the fluid transit time.
The transit time calculation assumes one dimensional
incompressible fluid flow through a connecting piping leg
of characteristic diameter and length.
t lag , x =
A pipe, x l pipe, x
qf
(104)
where x ∈ {in, out} , A pipe, x is the pipe cross-sectional
and
(
)
Qhx = ws c s T fi − Tsi + ms c s pT fo .
(101)
For both saturated and unsaturated conditions, the outlet
seawater temperature is governed by
pTso =
Qhx − ws c s (Tso − Tsi )
.
ms c s
(102)
FRESHWATER LOOP
The freshwater cooling loop circulates freshwater
between the “cold plate” component heat exchanger and
the fluid heat exchanger. The component heat
exchanger removes waste heat from heat producing
devices and the freshwater to seawater exchanger
transfers the waste heat from the freshwater loop to
seawater which is then discharged overboard.
The power component heat exchanger defined in an
earlier section is combined with a circulating pump, a
supply pipe and a return pipe to represent one
freshwater cooling fluid loop. As illustrated in Figure 28,
incoming cooling fluid goes through a circulating pump
and then travels through the supply pipe to the
component heat exchanger. The heated fluid then
travels through the return pipe to be cooled by the fluid
heat exchanger and returned to the circulating pump.
Figure 28. Component Heat Exchanger with Circulating
Pump and Piping
area, l pipe, x is the pipe length, and q f = wcf ρ is the
volume flow rate where ρ is the density of the
freshwater. For zero flow rate, the transport lag
becomes infinite which creates an unbounded numerical
memory storage problem. To avoid this problem, the
transport lag is limited to a maximum value, t lag ,max , and
if A pipe, x l pipe, x > q f t lag ,max then t lag , x = t lag ,max .
EXAMPLE SIMULATION
The primary benefit of the layered modeling approach is
its ability to capture the dynamic interdependence of the
subsystems. To illustrate this, a time domain simulation
of the notional IEP was conducted for the following
situation. The system had been running for 15 minutes
under full load, with both PDs consuming 37 kW and
each IM providing 5 kW to ship service loads. Within
fifteen minutes the IEP had reached steady-state
operation. Then an anti-ship missile with an explosion
radius of 2.00 m was detonated at the point (100.00, 4.18, 3.38) m as shown in Figure 29. This explosion
destroyed SWB 20. The simulation continued to predict
system response from the explosion instant for an
additional 45 minutes.
Figure 29. Missile Detonation Event
The response of the system starting at 800 s (100 s
before the explosion) is shown in Figure 30. Therein,
vbus is the line-to-line rms voltage of the two ac buses.
The dotted and solid traces correspond to the forward
and aft ac buses. The variable iout , sm is the rms output
current of the SMs. Again, the dotted and solid traces
correspond to the forward (SM 1) and aft (SM 2)
machines.
The power supply output voltages and currents are
designated vout , ps and iout , ps . In each case, the dotted
trace is the forward power supply (PS 1) and the solid
trace the aft power supply (PS 2). The output current
( iout , cm ) of the Zone 2 CMs is shown next (CM 2—
dotted; CM 5—solid). The forward most (IM 1—dotted)
and aft most (IM 3—solid) inverter module input voltages
are labeled vin , im .
Seawater pump flow rates for SWP 1 in Zone 1 (dotted)
and SWP 3 in Zone 3 (solid) are qP . The cold plate
temperature of the CHXs cooling the SMs is designated
Thx, chx (CHX 1—dotted; CHX 4—solid). Freshwater flow
rates wcf and cold plate temperatures Thx, fwl of the
FWLs cooling IMs 1 and 3 are also shown (FWL 4—
dotted; FWL 11—solid).
The scenario that unfolded as a result of the weapons
impact, which is illustrated in Figure 30, is as follows. At
900 s, SWB 20 was destroyed, causing SWVs 3, 4, 7, 8,
and 14 to close in order to isolate this fault in Zone 3
from the rest of the seawater network. As can be seen,
the flow rate in SWP 3 goes up because SWB 20 is
leaking. The remainder of the seawater network settled
to a new equilibrium point. In Zone 3, the damaged
piping was no longer capable of pumping cooling fluid to
the thermal loads in the zone. In particular, the
temperature of both CHX 4 and FWL 11 rise following
the weapon detonation. At approximately 1016 s (116 s
after the initial event), SM 2 overheats and is forced to
shut down. This causes PD 2 and PS 2 to shut down.
The PS 2 shutdown causes CM 4, CM 5, and CM 6 to
shut down which can be seen in the input voltage to IM
1. The input voltages drop due to the increased output
current of CMs 1 and 3 caused by the loss of CMs 4 and
6. Also, the output current of CM 2 and CM 5 were
shared prior to this point, but at 1016 s the current that
CM 5 was providing had to shift to CM 2. The output
currents of both SM 1 and PS 1 increase to cover the
load that was previously shared with SM 2 and PS 2.
Figure 30. System Response to Event
At 3363 s (2347 s after SM 2 shutdown), IM 3
overheats. This causes the pumps in FWLs 8, 9, 10, and
11, and SWP 3 to shut down, as seen in the pump flow
rates. The shutdown of IM3 causes the input voltage of
IM 3 to rise as the CM3 output current decreases. Also,
the output currents of SM 1 and PS 1 drop due to the
decreased load in the dc system. At this point, the
system stabilizes in its new configuration. The following
devices have been shut down: SM 2, PD 2, PS 2, CMs
4, 5, and 6, IM 3, SWP 3, and FWLs 8, 9, 10, and 11.
CONCLUSION
The example simulation illustrates that the layered
approach is an effective technique for capturing the
interdependent behavior of an IEP. In particular, the
cascading failures resulting from a weapon detonation
were studied. Future work will be aimed toward using
this approach to identify and investigate worst case
scenarios for missile impact points. This will be done by
identifying suitable metrics for system operability and
dependability,
and
then
applying
optimization
techniques.
ACKNOWLEDGMENTS
This work was sponsored by the Office of Naval
Research under contract numbers N00014-04-1-0351
and N00014-02-1-0623.
REFERENCES
1. S.D. Sudhoff, S. Pekarek, B. Kuhn, S. Glover, J.
Sauer, and D. Delisle, “Naval combat survivability
testbeds for investigation of issues in shipboard
power electronics based power and propulsion
systems,” in 2002 IEEE Power Engineering Society
Summer Meeting, pp. 347-350, 2002.
2. P.C. Krause, O. Wasynczuk, and S.D. Sudhoff,
Analysis of Electric Machinery and Drive Systems,
2nd ed., John Wiley and Sons/IEEE Press, New
York, 2002.
3. D.C. Aliprantis, S.D. Sudhoff, and B.T. Kuhn, “A
synchronous machine model with saturation and
arbitrary rotor network representation,” IEEE Trans.
Energy Conversion, vol. 20, no. 3, pp. 585-594,
Sept. 2005.
4. D.C. Aliprantis, S.D. Sudhoff, and B.T. Kuhn,
“Experimental characterization procedure for a
synchronous machine model with saturation and
arbitrary rotor network representation,” IEEE Trans.
Energy Conversion, vol. 20, no. 3, pp. 595-603,
Sept. 2005.
5. D.C. Aliprantis, S.D. Sudhoff, and B.T. Kuhn, “A
brushless exciter model incorporating multiple
rectifier modes and Preisach’s hysteresis theory,”
IEEE Trans. Energy Conversion, to be published.
6. D.C. Aliprantis, S.D. Sudhoff, and B.T. Kuhn,
“Genetic algorithm-based parameter identification of
a hysteretic brushless exciter model,” IEEE Trans.
Energy Conversion, to be published.
7. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and
B.P. Flannery, Numerical Recipes in C: The Art of
Scientific Computing, 2nd ed., Cambridge University
Press, Cambridge, 1992.
8. J.G. Hawley, P.F. Wiggins, P.G. Vining, K.W.
Lindler, Thermodynamics of Marine Engineering
Systems, 3rd ed., Kendall/Hunt Publishing
Company, Dubuque, 2000.
9. M.
Horsley,
Engineering
Thermodynamics,
Chapman & Hall, London, 1993.
10. S. Kakaç, H. Liu, Heat Exchangers: Selection,
Rating and, Thermal Design, CRC Press, Boca
Raton, 1998.
APPENDIX
The dimensions of various types of components are
given in Table 3. Table 4 shows the locations of the
centroids of the rectangular prisms representing specific
components of the notional ship. The vertices of each of
the polygonal paths in the spatial layer are listed in Table
5.
Table 3. Component Dimensions
Δy (m)
Δx (m)
Δz
Type
SM
10.24
4.1
PD
7.17
4.1
PS
3.81
1.83
CM
1.83
1.22
IM
1.83
1.22
SWP
3.05
1.52
FHX
3.05
1.52
(m)
4.1
4.1
2.13
1.91
1.91
1.52
1.52
Table 4. Rectangular Prism Centroids
xc (m)
yc (m)
zc (m)
Component
SM 1
44.5
-3.66
4.11
SM 2
92.96
-0.61
5.79
PD 1
82.91
-4.08
4.72
PD 2
82.91
4.08
4.72
PS 1
41.45
0
2.46
PS 2
91.82
0
5.56
CM 1
39.84
-4.57
10.86
CM 2
64.6
-4.57
10.86
CM 3
111.03
-4.57
10.86
CM 4
39.84
4.57
3.04
CM 5
64.6
4.57
3.04
CM 6
111.03
4.57
3.04
IM 1
36.12
0
13.72
IM 2
63.81
0
10.86
IM 3
109.65
0
10.86
SWP 1
50.29
0
2.47
SWP 2
77.11
0
2.47
SWP 3
88.39
0
2.47
FHX 1
39.82
-0.11
2.46
FHX 2
64.3
0.08
3.04
FHX 3
105.89
0
3.04
Path
SM 1 to
PD 1
SM 1 to
PS 1
SM 2 to
PD 2
SM 2 to
PS 2
PDCB 1
Zone 1
PDCB 1
Zone 2
PDCB 1
Zone 3
Table 5. Polygonal Path Vertices
Vertices (m)
(44.5, -3.66, 4.11), (44.5, -3.66, 4.72),
(44.5, -4.08, 4.72), (82.91, -4.08, 4.72)
(44.5, -3.66, 4.11), (44.5, -3.66, 2.46),
(44.5, 0, 2.46), (41.45, 0, 2.46)
(92.96, -0.61, 5.79), (92.96, -0.61, 4.72),
(92.96, 4.08, 4.72), (82.91, 4.08, 4.72)
(92.96, -0.61, 5.79), (92.96, -0.61, 5.56),
(92.96, 0, 5.56), (91.82, 0, 5.56)
(39.84, -7.32, 5.49), (50.6, -7.32, 5.49)
(50.6, -7.32, 5.49), (88.09, -7.32, 5.49)
(88.09, -7.32, 5.49), (111.03, -7.32, 5.49)
PDCB 2
Zone 1
PDCB 2
Zone 2
PDCB 2
Zone 3
PS 1 to
PDCB 1
CM 1 to
PDCB 1
CM 2 to
PDCB 1
CM 3 to
PDCB 1
PS 2 to
PDCB 2
CM 4 to
PDCB 2
CM 5 to
PDCB 2
CM 6 to
PDCB 2
CM 1 to
ZDCB 1
CM 4 to
ZDCB 1
IM 1 to
ZDCB 1
CM 2 to
ZDCB 2
CM 5 to
ZDCB 2
IM 2 to
ZDCB 2
CM 3 to
ZDCB 3
CM 6 to
ZDCB 3
IM 3 to
ZDCB 3
IM 1 to
SWP 1
IM 2 to
SWP 2
IM 3 to
SWP 3
IM 1 to
FHX 1
IM 2 to
FHX 2
IM 3 to
FHX 3
SWB 1
SWB 2
SWB 3
SWB 4
SWB 5
(39.84, 7.32, 5.49), (50.6, 7.32, 5.49)
(50.6, 7.32, 5.49), (88.09, 7.32, 5.49)
(88.09, 7.32, 5.49), (111.03, 7.32, 5.49)
(41.45, 0, 2.46), (41.45, 0, 5.49),
(41.45, -7.32, 5.49)
(39.84, -4.57, 10.86), (39.84, -4.57, 5.49),
(39.84, -7.32, 5.49)
(64.6, -4.57, 10.86), (64.6, -4.57, 5.49),
(64.6, -7.32, 5.49)
(111.03, -4.57, 10.86), (111.03, -4.57, 5.49),
(111.03, -7.32, 5.49)
(91.82, 0, 5.56), (91.82, 0, 5.49),
(91.82, 7.32, 5.49)
(39.84, 4.57, 3.04), (39.84, 4.57, 5.49),
(39.84, 7.32, 5.49)
(64.6, 4.57, 3.04), (64.6, 4.57, 5.49),
(64.6, 7.32, 5.49)
(111.03, 4.57, 3.04), (111.03, 4.57, 5.49),
(111.03, 7.32, 5.49)
(39.84, -4.57, 10.86), (39.84, -4.57, 9.21),
(39.84, 0, 9.21), (38.6, 0, 9.21)
(39.84, 4.57, 3.04), (39.84, 4.57, 9.21),
(39.84, 0, 9.21), (38.6, 0, 9.21)
(36.12, 0, 13.72), (36.12, 0, 9.21),
(36.12, 0, 9.21), (38.6, 0, 9.21)
(64.6, -4.57, 10.86), (64.6, -4.57, 8.25),
(64.6, 0, 8.25), (64.34, 0, 8.25)
(64.6, 4.57, 3.04), (64.6, 4.57, 8.25),
(64.6, 0, 8.25), (64.34, 0, 8.25)
(63.81, 0, 10.86), (63.81, 0, 8.25),
(63.81, 0, 8.25), (64.34, 0, 8.25)
(111.03, -4.57, 10.86), (111.03, -4.57, 8.25),
(111.03, 0, 8.25), (110.57, 0, 8.25)
(111.03, 4.57, 3.04), (111.03, 4.57, 8.25),
(111.03, 0, 8.25), (110.57, 0, 8.25)
(109.65, 0, 10.86), (109.65, 0, 8.25),
(109.65, 0, 8.25), (110.57, 0, 8.25)
(36.12, 0, 13.72), (36.12, 0, 2.47),
(36.12, 0, 2.47), (50.29, 0, 2.47)
(63.81, 0, 10.86), (63.81, 0, 2.47),
(63.81, 0, 2.47), (77.11, 0, 2.47)
(109.65, 0, 10.86), (109.65, 0, 2.47),
(109.65, 0, 2.47), (88.39, 0, 2.47)
(36.12, 0, 13.72), (36.12, 0, 2.46),
(36.12, -0.11, 2.46), (39.82, -0.11, 2.46)
(63.81, 0, 10.86), (63.81, 0, 3.04),
(63.81, 0.08, 3.04), (64.3, 0.08, 3.04)
(109.65, 0, 10.86), (109.65, 0, 3.04),
(109.65, 0, 3.04), (105.89, 0, 3.04)
(38.4, 0, 2.47), (38.4, 4.57, 2.47),
(50.29, 4.57, 2.47)
(50.29, 4.57, 2.47), (65.23, 4.57, 2.47)
(65.23, 4.57, 2.47), (77.11, 4.57, 2.47)
(77.11, 4.57, 2.47), (88.39, 4.57, 2.47)
(88.39, 4.57, 2.47), (106.68, 4.57, 2.47),
(106.68, 0, 2.47)
SWB 6
SWB 7
SWB 8
SWB 9
SWB 10
SWB 11
SWB 12
SWB 13
SWB 14
SWB 15
SWB 16
SWB 17
SWB 18
SWB 19
SWB 20
SWB 21
SWB 22
SWB L1
SWB L2
SWB L3
SWB L5
SWB L6
SWB L8
SWB L9
FHX 1 to
PS 1
FHX 1 to
CM 1
FHX 1 to
CM 4
FHX 1 to
IM 1
FHX 2 to
CM 2
FHX 2 to
CM 5
FHX 2 to
IM 2
FHX 3 to
PS 2
FHX 3 to
CM 3
FHX 3 to
CM 6
FHX 3 to
IM 3
(38.4, 0, 2.47), (38.4, -3.66, 2.47)
(50.29, 4.57, 2.47), (50.29, 0, 2.47)
(65.23, 4.57, 2.47), (65.23, 0.3, 2.47)
(77.11, 4.57, 2.47), (77.11, 0, 2.47)
(88.39, 4.57, 2.47), (88.39, 0, 2.47)
(106.68, 0, 2.47), (106.68, -0.61, 2.47)
(50.29, -4.57, 2.47), (50.29, 0, 2.47)
(65.23, -4.57, 2.47), (65.23, 0.3, 2.47)
(77.11, -4.57, 2.47), (77.11, 0, 2.47)
(88.39, -4.57, 2.47), (88.39, 0, 2.47)
(38.4, -3.66, 2.47), (38.4, -4.57, 2.47),
(50.29, -4.57, 2.47)
(50.29, -4.57, 2.47), (65.23, -4.57, 2.47)
(65.23, -4.57, 2.47), (77.11, -4.57, 2.47)
(77.11, -4.57, 2.47), (88.39, -4.57, 2.47)
(88.39, -4.57, 2.47), (106.68, -4.57, 2.47),
(106.68, -0.61, 2.47)
(38.4, 0, 2.47), (38.4, 0, 2.46),
(38.4, -0.11, 2.46), (39.82, -0.11, 2.46)
(65.23, 4.57, 2.47), (65.23, 4.57, 4.72),
(65.23, 4.08, 4.72), (82.91, 4.08, 4.72)
(65.23, 0.3, 2.47), (65.23, 0.3, 3.04),
(65.23, 0.08, 3.04), (64.3, 0.08, 3.04)
(65.23, -4.57, 2.47), (65.23, -4.57, 4.72),
(65.23, -4.08, 4.72), (82.91, -4.08, 4.72)
(106.68, 0, 2.47), (106.68, 0, 3.04),
(106.68, 0, 3.04), (105.89, 0, 3.04)
(106.68, -0.61, 2.47), (106.68, -0.61, 5.79),
(106.68, -0.61, 5.79), (92.96, -0.61, 5.79)
(38.4, -3.66, 2.47), (38.4, -3.66, 4.11),
(38.4, -3.66, 4.11), (44.5, -3.66, 4.11)
(39.82, -0.11, 2.46), (39.82, -0.11, 2.46),
(39.82, 0, 2.46), (41.45, 0, 2.46)
(39.82, -0.11, 2.46), (39.82, -0.11, 10.86),
(39.82, -4.57, 10.86), (39.84, -4.57, 10.86)
(39.82, -0.11, 2.46), (39.82, -0.11, 3.04),
(39.82, 4.57, 3.04), (39.84, 4.57, 3.04)
(39.82, -0.11, 2.46), (39.82, -0.11, 13.72),
(39.82, 0, 13.72), (36.12, 0, 13.72)
(64.3, 0.08, 3.04), (64.3, 0.08, 10.86),
(64.3, -4.57, 10.86), (64.6, -4.57, 10.86)
(64.3, 0.08, 3.04), (64.3, 0.08, 3.04),
(64.3, 4.57, 3.04), (64.6, 4.57, 3.04)
(64.3, 0.08, 3.04), (64.3, 0.08, 10.86),
(64.3, 0, 10.86), (63.81, 0, 10.86)
(105.89, 0, 3.04), (105.89, 0, 5.56),
(105.89, 0, 5.56), (91.82, 0, 5.56)
(105.89, 0, 3.04), (105.89, 0, 10.86),
(105.89, -4.57, 10.86), (111.03, -4.57, 10.86)
(105.89, 0, 3.04), (105.89, 0, 3.04),
(105.89, 4.57, 3.04), (111.03, 4.57, 3.04)
(105.89, 0, 3.04), (105.89, 0, 10.86),
(105.89, 0, 10.86), (109.65, 0, 10.86)
(105.89, 0, 3.04), (105.89, 0, 3.04),
(105.89, 4.57, 3.04), (111.03, 4.57, 3.04)
(105.89, 0, 3.04), (105.89, 0, 10.86),
(105.89, 0, 10.86), (109.65, 0, 10.86)
The SM operation parameters are shown in Table 6. The
PD operation parameters are given in Table 7. Table 8
lists the PS operation parameters. Tables 9 and 10 show
the CM and the IM operation parameters, respectively.
Table 6. Synchronous Machine Operation Parameters
ω rm,a ,min
ω rm,o,max
178.5 rad/s
208.5 rad/s
ω rm,a ,max
Thx,o ,max
198.5 rad/s
365 K
ω rm ,o,min
168.5 rad/s
Table 7. Propulsion Drive Operation Parameters
vin ,a ,min
vin ,o ,max
400 V
600 V
Table 12. Prime Mover Parameters
τ lag ,δ
188.5 rad/s
3.51 ms
2
k
326 N∙m∙s2/rad2
8.4026 kg∙m
ω
*
ω rm
J
20.94 rad/s2
*
pω max
τ srl
δ max
kδ
τ lead ,δ
49.8 ms
402 rad
91.5 N∙m/rad
τω
803 ms
ω err ,thr
10 rad/s
Tmin
Tmax
-600 N∙m
600 N∙m
255 ms
vin ,a ,max
500 V
vc,o ,min
400 V
The SM machine parameters are described in [4], but
two parameters not included in [4] will be described
here. The parameter values are r fict = 531.5254 Ω and
vc, a , min
500 V
Thx,o ,max
365 K
η = 94.56% .
vin ,o ,min
300 V
Table 8. Power Supply Operation Parameters
vin ,a ,min
vin ,o ,min
400 V
300 V
vin ,a ,max
Thx,a ,max
500 V
vin ,o ,max
600 V
325 K
Thx,o ,max
365 K
Table 9. Converter Module Operation Parameters
vin ,a ,min
vin ,o ,min
450 V
400 V
vin ,a ,max
550 V
vin ,o ,max
600 V
Thx,a ,max
325 K
Thx,o ,max
365 K
Table 10. Inverter Module Operation Parameters
vin ,a ,min
vin ,o ,min
370 V
320 V
vin ,a ,max
470 V
vin ,o ,max
520 V
Thx,a ,max
325 K
Thx,o ,max
365 K
The BE parameters are described in [6]. The VR and
field drive parameters are given in Table 13. The
constant term in the d-axis magnetizing flux linkage
equation is 0.0105 V∙s.
Table 13. Voltage Regulator and Field Drive Parameters
vll* ,rms
560 V,l-l,rms vdc, pow
172 V
τf
10 ms
rswitch
0.402 τv
200 ms
vswitch
0V
k pv
0.0245 A/V
rdiode
0.07 Ω
3A
vdiode
1.37 V
0.018 A
K fix
i fde,max
ΔI
The PD parameters are given in Table 14.
Table 14. Propulsion Drive Parameters
τ filter
Ldc
2.858 mH
10 ms
The SWV locations are given in Table 11. All of the
SWVs in the notional plant are type 1.
Table 11. Seawater Valve Parameters
y (m)
qT (m3/s)
x (m)
z (m)
Valve
1
50.29
4.57
2.47
0.15
2
65.23
4.57
2.47
0.15
3
77.11
4.57
2.47
0.15
4
88.39
4.57
2.47
0.15
5
50.29
-4.57
2.47
0.15
6
65.23
-4.57
2.47
0.15
7
77.11
-4.57
2.47
0.15
8
88.39
-4.57
2.47
0.15
9
50.29
0
2.47
0.3
10
50.29
0
2.47
0.3
11
77.11
0
2.47
0.3
12
77.11
0
2.47
0.3
13
88.39
0
2.47
0.3
14
88.39
0
2.47
0.3
For the ac buses, r fict = 531.5254 Ω .
The PM model parameters are given in Table 12.
4
5×10 V/A
Lc
rdc
0.2 mH
0.0556 Cdc
η
1.988 mF
95.82%
The PS parameters are given in Table 15.
*
vout
τ filter
Table 15. Power Supply Parameters
*
500 V
pvmax
2500 V/s
10 ms
kv
2.657
TR
Lc
0.79
τv
100 ms
0.662 mH
kip
0.00112 V/A
rdc
Ldc
0.075 11.3 mH
4.4×10-5 V∙s/A
597 V
Cdc
4.55 mF
kid
Vr 0
reff
I sc
55 A
Leff
12.624 mH
I thr
35 A
Cα ,min
-0.9
0.1 ms
Cα ,max
1.0
τ srl
pv*min
η
0.3133 94.89%
-2 MV/s
The two PDCBs are similarly constructed. They each
consist of three zones. Adjacent zones have a zonal tie
line resistance of 0.1 , and component tie lines have a
resistance of 0.1 .
19
20
L1
L2
L3
L4
L5
L6
L7
The CM parameters are given in Table 16. The ZDCB
parameters are given in Table 17. The IM parameters
are given in Table 18.
Table 16. Converter Module Parameters
*
d
0.84 V/A
vout
420 V
Cin
ilimit
449 F
20 A
η
Cout
447 F
96.97%
GNlk
GBflt
Table 17. Zonal DC Bus Parameters
rp
vd
1.2 V
0.05 rs
RC
0.05 ε
Table 18. Inverter Module Parameters
η
Cin
590 F
95.90%
All three SWPs have the same parameters, given in
Table 19.
Table 19. Seawater Pump Parameters
a P,1
β P ,1
704.72
4.92
a P ,2
529.48
β P, 2
1.59
α P ,1
0.84
β P ,3
0.000153
α P, 2
1.01
qPb
0.31 m3/s
bP ,1
611.34
RP
352.234 kPa∙s/m3
bP ,2
411.45
PPss
60.556 kPa
bP ,3
469.27
τP
5s
All the SWBs exhibit turbulent flow so RB 0 = 1 Pa ⋅ s/m 3 ,
q B 0 = 1 m 3 /s , and mB = 0.75 . The branches are shown in
Table 20. The SWS parameters are in Table 21.
Table 20. Seawater Branch Parameters
3
RBq (kPa∙s/m )
β
α
PB 0 (kPa)
Branch
1
1
3
0
62.869
2
3
6
0
57.101
3
6
9
0
45.375
4
9
12
0
43.084
5
12
15
0
87.313
6
1
2
0
13.979
7
3
5
0
17.455
8
6
8
0
16.309
9
9
11
0
17.455
10
12
14
0
17.455
11
15
16
0
2.33
12
4
5
0
17.455
13
7
8
0
18.601
14
10
11
0
17.455
15
13
14
0
17.455
16
2
4
0
48.889
17
4
7
0
57.063
18
7
10
0
45.375
10
13
1
6
8
7
15
16
2
13
16
0
0
0
0
0
0
0
0
0
0.1
22.624
5.72
22.624
5.72
33.33
16.507
43.084
84.983
99650.369
4129.37
466623.529
4141.387
91111.427
807.009
1280.271
Table 21. Seawater Solver Parameters
δ
1×10-10 m3/s/kPa
1×10-9 m3/s
3
1 m /s/kPa
10
3
1×10 kPa∙s/m
1×10-3
kmax
γ
2000
0.5
The losses used in the thermal models are scaled
versions of the electrical model losses. The SM and PD
losses are scaled by 1000 and the PS, CM, and IM
losses are scaled by 250. The parameters of the CHXs
that cool SMs are given in Table 22. The PD CHX
parameters are given in Table 23.
Table 22. Synchronous Machine Component Heat
Exchanger Parameters
2
mhx
127.22 m
A
14327 kg
chx
h
400 J/K/kg
4800 W/K/m2
ccf
3998 J/K/kg
Table 23. Propulsion Drive Component Heat
Exchanger Parameters
2
mhx
6813 kg
60.5 m
A
chx
h
400 J/K/kg
4800 W/K/m2
ccf
3998 J/K/kg
There are three sets of FWL parameters. In each set,
A pipe,out = Apipe,in . The parameters for an FWL cooling a
PS are given in Table 24. The parameters for an FWL
cooling a CM are given in Table 25. The parameters for
an FWL cooling an IM are given in Table 26. Table 27
shows the lengths of the pipes for each FWL, with
l pipe,out = l pipe,in .
Table 24. Power Supply Freshwater Loop Parameters
mhx
h
380.1 kg
4800 W/K/m2
chx
896 J/K/kg
wcf*
8.691 kg/s
ccf
4180 J/K/kg
A pipe,in
45.6 cm
A
7.569 m
2
τ pump
2
5s
7
8
9
10
11
Table 25. Converter Module Freshwater Loop
Parameters
mhx
h
73.5 kg
4800 W/K/m2
chx
896 J/K/kg
wcf*
1.679 kg/s
ccf
4180 J/K/kg
A pipe,in
45.6 cm
A
1.463 m
τ pump
2
Table 28 shows the parameters of the FHXs in Zones 1
and 3. Table 29 shows the parameters of the FHX in
Zone 2.
2
5s
Table 26. Inverter Module Freshwater Loop Parameters
mhx
h
100.6 kg
4800 W/K/m2
chx
896 J/K/kg
wcf*
2.30 kg/s
ccf
4180 J/K/kg
A pipe,in
45.6 cm
A
2.003 m
2
τ pump
Table 27. Freshwater Loop Lengths
l pipe,in (m)
Loop
1
1.75
2
5.28
3
12.88
4
15.07
5
4.95
6
12.77
8.39
16.59
9.72
17.53
11.58
2
5s
Table 28. Zones 1 and 3 Fluid Heat Exchanger
Parameters
2
Asf
mf
23.65 m
446 kg
2
hsf
cs
3998 J/K/kg
2536.6 W/K/m
cf
4180 J/K/kg
ms
358 kg
Table 29. Zone 2 Fluid Heat Exchanger
Parameters
2
Asf
mf
9.32 m
175.9 kg
2
hsf
cs
3998 J/K/kg
2536.6 W/K/m
cf
4180 J/K/kg
ms
141.2 kg