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 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 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 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 well-defined 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.

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

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

defined such that ( )⎪⎩

⎪⎨

⎧

<≤≤

>=

lxx

uxlx

uxu

xul ,,bound . Finally, nI

is taken to be the nn× 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 6. Cartesian Coordinate System

An explosion is represented as a sphere with radius dr

centered at ( )ddd zyx ,,=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, dr . An example of such a component is an SWV.

To determine whether the point is hit, the distance from the point, ( )ccc zyx ,,=c , to the center of the explosion is calculated using

dc −=d . (1)

Then the hit status of the component is determined by comparing this distance to the radius of the explosion. Thus,

( )drdh ≤= . (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 sn total segments should be considered separately. The endpoints of the s th segment are ( )scscsc zyx ,,, ,,=sc and

( )1,1,1, ,, ++++ = scscsc zyx1sc . If 1ss cc += then this segment

is actually a point. If 1>s then the point was the second endpoint of the previous segment and has already been tested for intersection with the explosion. If 1=s then the point should be checked for intersection using (1) and (2).

In the general case that 1ss cc +≠ then any point on the

line connecting sc and 1sc + can be written as

( ) 1ss ccc +−+= αα 1 . If [ ]1,0∈α then the point c is actually on the line segment. To determine the closest point on the line to the center of the explosion

22

dc −=F is minimized with respect to α . The value

of α that minimizes F is given by

1ss

1ss1s

cc

cccd

+

++

−−−

=,

α . (3)

Projecting α onto [ ]1,0 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.

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

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, ( )ccc zyx ,,=c and by its length, width,

and height, ( )zyx ΔΔΔ= ,,Δ .

To determine whether the prism is hit, one can project the explosion center into the prism so that

22

Δcp

Δc +≤≤− (4)

where p is the projected version of d , and the inequalities in (4) are taken to hold elementwise. For example, the x-component of p is given by

( )dccp xxxxxx ,2,2bound Δ+Δ−= . Then p is the

closest point in the prism to d . The distance from the explosion center to the closest point in the prism is given by

dp −=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

oo βα +=: . (6)

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, minayx ,, and maxayx ,,

represent the minimum and maximum values of the variable yx for which the device is allowed to activate.

Similarly, minoyx ,, and maxoyx ,, represent the minimum

and maximum values for which the device is allowed to operate. In general, maxoymaxayminayminoy xxxx ,,,,,,,, <<< .

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

( )( ) hmaxarmrmminarmrm oho ,,,,* ωωωωα ≤≥= (7)

and

( ) ( ) hmaxormrmminormrm oho +>+<++= ,,,,* ωωωωβ , (8)

respectively, where rmω is the mechanical rotor speed,

and ho is the overheat status which is governed by

( )maxohxhxhh TToo ,,: >+= (9)

where hxT 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

( )( )

( ) hminacc

maxaininminainin

ovv

vvvvho

,,

,,,,*

≥⋅≤≥=α (10)

and

( ) ( )

( ) hminocc

maxoininminoinin

ovv

vvvvho

+<+>+<++=

,,

,,,,*β

, (11)

respectively, where inv is the peak input voltage, cv is the dc link capacitor voltage (described in the propulsion drive model below), and ho is the overheat status which is governed by

( )maxohxhxhh TToo ,,: >+= (12)

where hxT 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

( )( )

( )maxahxhx

maxaininminainin

TT

vvvvho

,,

,,,,*

≤⋅≤≥=α (13)

and

( ) ( )

( )maxohxhx

maxoininminoinin

TT

vvvvho

,,

,,,,*

>+>+<++=β

, (14)

respectively, where inv is the peak input voltage and hxT 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

( )( )

( )maxahxhx

maxaininminainin

TT

vvvvho

,,

,,,,*

≤⋅≤≥=α (15)

and

( ) ( )

( )maxohxhx

maxoininminoinin

TT

vvvvho

,,

,,,,*

>+>+<++=β

, (16)

respectively, where inv is the dc input voltage and hxT is the cold plate temperature of the CHX that is cooling the device.

LOAD OPERATION

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

inoho*=α (17)

and

inoho ++= *β , (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, ino 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

( ) hqqcc T +≤=: (19)

where q is the flow rate through the SWV, h is the hit

status of the valve, and Tq 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

⎪⎩

⎪⎨⎧

>−≤

=TT

Te qqqq

qqq

0 (20)

where q and Tq are defined above. The excess volume is then defined as

ee qpV = . (21)

The status is governed by

( ) hVVcc Te +≤=: (22)

where TV 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

3] in each network is taken to be that of the rotor reference frame of the corresponding SM.

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 fictr

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

⎪⎩

⎪⎨⎧−=

h

hrfict

0

eqd,ne

qd,ni

v (23)

where enqd,i is a vector of the sum of the q-axis and d-

axis 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.

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 fictr is

added behind the circuit breaker to compute the terminal voltage. This is illustrated in Figure 8. In this figure, the

internal values ( eqdsv and e

qdsi ) are interfaced to the

model in [4], but the values are determined using

⎪⎩

⎪⎨⎧

−=

or

o

ficteqds

eqdse

qdsi

vv ˆˆ (24)

and

⎪⎩

⎪⎨⎧

=o

o

0

ˆeqdse

qdsi

i (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

eqdsv=tV . (26)

Figure 8. Synchronous Machine Modification

To determine the power loss of the SM the output power is calculated using

eqds

eqds iv ,

2

3−=outP . (27)

Then the losses are computed using an assumed efficiency η ,

outloss PP ⎟⎟⎠

⎞⎜⎜⎝

⎛−= 1

1η

. (28)

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 fdei to the

commanded field current *fdei .

Figure 9. Field Drive

To model the action of this circuit an effective dc voltage is found using

⎩⎨⎧

<−≥

=0

0

,

,

fdeifdefixpowdc

fdepowdcrect iiKv

ivv . (29)

where powdcv , is the voltage into the field drive circuit and

fixK is a large positive constant. If 0<fdei then rectv

becomes very large, which tends to increase fdei . A

negative value of fdei is physically impossible because

of the topology of the circuit in Figure 9. The upper and lower limits on the field voltage fdev that the field drive

circuit can produce are computed using

fdeswitchswitchrectmaxfde irvvv −−=, (30)

and

fdediodedioderectminfde irvvv −−−=, (31)

where switchv and switchr are the voltage drop and

resistance of the transistors in Figure 9, and diodev and

dioder are the voltage drop and resistance of the diodes. The current error is defined as

fdefdeerror iii −= * . (32)

The exciter field voltage is then calculated using

( )( )⎩

⎨⎧

≤Δ−>Δ

=0,0,bound

0,,0bound

,,

,,

errorIerrorminfdeminfde

errorIerrormaxfdemaxfdefde iivv

iivvv (33)

where IΔ is the hysteresis band of the controller. The current command in (32) is determined using the VR model shown in Figure 10.

Figure 10. Voltage Regulator

The field drive and VR parameters are described in the appendix.

PROPULSION DRIVE

The PD model consists of an uncontrolled rectifier model connected to a constant power load model as shown in Figure 11.

Figure 11. Propulsion Drive Model

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

0=α . (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

currents from the model described in [2] ( eqdi ) are filtered

as shown in

eqd

eqd ii ˆ

1

1

+=

sfilterτ. (35)

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

π

EVr

63= (36)

where E is the input RMS voltage. The dc link variables are governed by

( )

cdc

clecdcrl LL

viLrVpi

2

3cos

+−+−= πωα

(37)

and

dc

plc C

iipv

−= . (38)

Finally, the current into the constant power load, pi , is

given by

⎪⎩

⎪⎨⎧

=o

ovPi cp

0

*

. (39)

Since the current li flows through the rectifier, it must

be nonnegative. Also, since the voltage cv 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

( )[ ]

( )[ ]ul

E

ul

E

uii

gc

gc

dg

comqg

22cos2cos23

4

1

coscoscos23

6

5sin

6

5sin

32,

+−

+−+

+⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −−⎟⎠

⎞⎜⎝

⎛ −+=

ααωπ

αααωπ

παπαπ

(40)

and

( )[ ]

( )[ ] ul

Eu

l

E

ul

E

uii

gcgc

gc

dg

comdg

2

12322sin2sin

23

4

1

sinsincos23

6

5cos

6

5cos

32,

ωπαα

ωπ

αααωπ

παπαπ

−+−

+−+

+⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −+⎟⎠

⎞⎜⎝

⎛ −+−=

. (41)

Finally, the power into the rectifier is computed using

eqd

eqd iv ,

2

3=inP , (42)

and assuming an efficiency of η the power loss is computed using

( ) inloss PP η−= 1 . (43)

POWER SUPPLY

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 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

ETRE ˆ⋅= (44)

where E is the RMS voltage of the bus. The q-axis and d-axis currents into the transformer are then

TR

gqdgg

qdg

ii

ˆ= (45)

where gqdgi is a vector of the q-axis and d-axis currents

into the rectifier. Again, the filter in (35) is used to interface the PS model to the bus model. The open circuit rectifier voltage rV is again defined by (36). The dc variables are governed by

( )

cdc

outlecdcrl LL

viLrVpi

2

3cos

+−+−= πωα

(46)

and

dc

outlout C

iipv

+= . (47)

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.

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 14. PI Control Signal

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.

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.

The model of the c th component tie line (of cn total component tie lines) is shown in Figure 17. In this figure,

ccl , represents the location (in terms of zone number) of

the c th component. Thus, cclzv

,, is the zonal voltage of

the zone to which the c th component is connected.

Figure 17. Component Tie Line Model

The model of the zonal tie line connecting the z th zone to the 1+z th zone is shown in Figure 18. There are zn total zones.

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 ( cj ,

cg , and zg ).

Figure 19. Bus Equivalent

Step 0 – Initialization. Set Norton equivalent conductance and current vectors to zero ( 0:=cg and

0:=cj ).

Step 1 – Process component tie lines. For each { }cnc ,,1…∈ modify the Norton equivalent vectors as

follows

⎪⎩

⎪⎨

⎧

+=

cc

cccccc

cccccc

lclc

c

hcr

hcr

ggcccc

,

,,,

,,,

,,

0

2

1

:,,

(48)

and

⎪⎩

⎪⎨⎧

++=

cccc

cccccccclclc hc

hcrvjj

cccc

,,

,,,,,, 0

:,,

. (49)

Step 2 – Process zonal tie lines. For each { }1,,1 −∈ znz … calculate the conductance as

⎩⎨⎧

=zz

zzzzzz c

crg

,

,,, 0

1. (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

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−

−−++−

−+

−−

−−

−

z

z

z

z

zzz

z

nc

nc

c

c

nz

nz

z

z

nzncnz

nz

z

zzzcz

zzc

j

j

j

j

v

v

v

v

ggg

g

g

ggggg

ggg

,

1,

2,

1,

,

1,

2,

1,

1,,1,

1,

2,

2,2,1,2,1,

1,1,1,

00

00

00

��

�

����

��

��

� . (51)

Step 4 – Adjust for damage and isolated buses. If zzh , ,

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 zcj , is replaced with zero. Also, each block

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.

CONVERTER MODULE

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 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

( )( )

⎪⎩

⎪⎨⎧ −=

o

odvvii outoutlimitl

0

,,0bound *

(52)

where limiti is the current limit of the CM, *outv is the

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 pi is determined by

⎩⎨⎧

=o

ovivi inloutp 0

. (53)

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

ininin ivP = , (54)

and the power loss is given by (43).

Figure 20. Converter Module

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.

The ZDCB model is shown in Figure 22. In this figure,

pv and pi represent the output voltage and current of

the CM connected to the port side of the bus, and ph

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 pi can be calculated using Table

1.

Table 1. Determination of pi

ph p

pp r

vi

2=

dp vv ≥ ( )

p

dpp r

vvi

−=

2

lh

dp vv < 0=pi

dlp vvv +> p

dlpp r

vvvi

−−=

ph

lh

dlp vvv +≤ 0=pi

The current lpi represents the contribution to the load

current coming from the port side and can be calculated using Table 2.

Table 2. Determination of lpi

lh p

llp r

vi

2−=

dl vv −≥ 0=lpi

ph dl vv −<

( )p

dllp r

vvi

+−= 2

dlp vvv +> p

dlplp r

vvvi

−−=

lh

ph

dlp vvv +≤ 0=lpi

Similar relationships exist for the starboard currents.

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

⎪⎩

⎪⎨⎧

=o

ovPi inp

0

*

. (55)

The power loss is calculated using (54) and (43).

Figure 23. Inverter Module Model

SEAWATER LAYER

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.

SEAWATER NODE

An SWN is a location at which the pressure will be determined. Figure 24 illustrates an SWN. Therein nNP ,

denotes the pressure at the n th node and NlkG is a leakage conductance to ground. The flow into this leakage conductance is the node flow nNq , . Since a

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.

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.

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, α,NP and β,NP

are node pressures at nodes α and β , and bBq , is the

branch flow through branch b . The resistances bvR ,α

and bvR ,β are associated with the two possible valves on

the α and β side of branch b , respectively. These valve resistances change depending upon whether the valve is open or closed according to

⎩⎨⎧

=bvxC

bvxbvx cR

cR

,

,,

0 (56)

where { }βα ,∈x , bvxc , is the valve status (calculated in

the automation layer) of the x valve of the b th branch, and CR is a large resistance indicating that the valve is

closed. The logical variable bBh , 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 BfltG . The nonlinear

resistance bBR , represents the pipe resistance, and the

constant pressure drop bBP ,0 is associated with a

physical rise in the pipe ( bBP ,0 is positive if node β is

higher than node α ).

Figure 25. Seawater Branch

Assuming that a fault is not present, the pressure drop from node α to node β may be expressed

( ) bBbBbBbvbvbBNN PqRRRqPP ,0,,,,,,, )( +++=− βαβα (57)

where

( )bBm

bBbBqbBbB q

qRRqR

,

,0,,0, ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+= . (58)

In (58), bBq ,0 is the base flow for the b th branch. bBR ,0

is the linear resistance, bBqR , is the flow dependent

resistance, and bBm , 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

( ) pPpPpPpPnN qRqPP ,,,,, += (59)

where nNP , denotes pressure, pPq , denotes flow,

( )pPpP qP ,, is the flow and state dependent pressure in

the p th pump, and pPR , is a constant resistance term

for the p th pump.

The flow dependent pressure term is assumed to have the form

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+=

∑

∑

=

=

M

m

β

pPbmpP

N

n

α

pPbnpP

pPpPmpP

npP

q

qb

q

qa

PqP

1 ,,

1 ,,

,0,,

,

1

1)( (60)

where pPbq , is the base pump flow of the p th pump. In

(60), the nominal pump pressure pPP ,0 is a variable

which is assumed to be governed by

⎩⎨⎧

+=

o

oP

sP pPss

pPpP 01

1 ,

,,0 τ

. (61)

In (61), o is the operation status of the pump, pPssP , is

the steady state value of pPP ,0 , and pPτ , is the pump

time constant.

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

The fundamental function of the SWS is to solve for the node pressures, as well as the node, branch, and pump flows.

Upon combining the flow relationships for the nodes, branches, and pumps, a system of equations of the form

( ) ( )SNS qbPqA = (62)

( )NS Pgq = (63)

can be formed, where NP is the vector of node

pressures, Sq is a vector of seawater flows, ( )SqA is a

square matrix whose elements are a function of Sq , and

( )Sqb is a vector whose elements are a function of Sq ,

and g is a function of NP .

In way of further definitions, Sq is the concatenation of

the node flow vector Nq , branch flow vector, Bq , and

pump flow vector Pq . In particular,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

P

B

N

S

q

q

q

q . (64)

An algorithm to solve (62) and (63) based on the Gauss-Seidel Method is now set forth. Let ( k

NP , kSq ) denote an

estimated solution of (62)-(63). The goal will be to determine an improved solution ( 1+k

NP , 1+kSq ).

Step 0 – Initialization step. The iteration count is set to zero ( 0:=k ) and, if the no other estimate of the flow is

available, 0qS =:0 .

Step 1 – Build step. Using kSq , ( )k

SqA and ( )kSqb are

determined. A detailed procedure to accomplish this will be set forth below.

Step 2 – Linear solution step. The linear system

( ) ( )kkkSNS qbPqA =+1 (65)

is solved for 1+kNP . An efficient linear system solver is

presented in [7, Chapter 2]. Then

( )1:~ += kNS Pgq (66)

where Sq~ is a candidate for the updated flow vector.

Step 3 – Convergence test. Let ε and δ be small positive constants which will determine the convergence requirement. If

kiSiS

kiSiS qqqq ,,,,

~~ +≤− ε (67)

or

δ≤− kiSiS qq ,,

~ (68)

is satisfied { }pbn NNNi ++∈∀ ,,1… 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 maxk , an updated flow vector is calculated as

( ) SSS qqq ~1:1 γγ −+=+ kk (69)

where γ is a convergence factor. In addition, the iteration count is updated:

1: += kk (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

1: += kNN PP (71)

SS qq ~:= (72)

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, 0A =: and 0b =: .

Step 1 – Process node list.

nNIAA NlkG+=: (73)

Step 2 – Process branch list. For each { }bNb ,,1…∈

where bN is the number of branches:

Step 2a: Let bBn ,αα = and bBn ,ββ = denote the two

nodes to which branch b is connected.

Step 2b: If bBh , , which is to say the branch is faulted,

then the following updates are performed:

Bfltbv GR

AA 11

,: ++=

ααααα (74)

Bfltbv GR

AA 11

,: ++=

βββββ (75)

Step 2c: If bBh , , which is to say the branch is not faulted,

then the value of the nonlinear resistance is saved using

( )kbBbBbB qRR ,,, :

~ = (76)

where ( )kbBbB qR ,, 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:

bBbvbv

branchRRR

G,,,

~1

:++

=βα

(77)

Next the following updates are performed:

branchGAA += αααα : (78)

branchGAA += ββββ : (79)

branchGAA −= αβαβ : (80)

branchGAA −= βαβα : (81)

branchbB GPbb ,0: += αα (82)

branchbB GPbb ,0: −= ββ (83)

Terms involving α are not updated if 0=α , and the same is true for β .

Step 3 – Process pump list. For each { }pNp …,1∈

where pN is the number of pumps:

Step 3a: Let pPnnn ,= denote the node to which the

pump is connected.

Step 3b: The pump pressure is saved using

( )kpPpPpP qPP ,,, :

~ = (84)

where ( )kpPpP qP ,, may be computed from (60). Then the

following updates are performed:

pP

nnnn RAA

,

1: += (85)

pP

pPnn R

Pbb

,

,~

: += (86)

The updated flow vector candidate Sq~ can be calculated using the following algorithm.

Step 1 – Process node list.

1kNN Pq += NlkG:~ (87)

Step 2 – Process branch list. For each { }bNb …,1∈ :

Step 2a: Let bBn ,αα = and bBn ,ββ = denote the two

nodes to which branch b is connected.

Step 2b: If bBh , , which is to say the branch is faulted,

then:

0:~, =bBq (88)

Step 2c: If bBh , , which is to say the branch is not faulted,

then:

bBbvbv

bBkN

kN

bBRRR

PPPq

,,,

,01

,1

,, ~:~

++−−

=++

βα

βα (89)

If 0=α and 0~, >bBq then 0:~

, =bBq and the branch is

deactivated. Similarly, if 0=β and 0~, <bBq then

0:~, =bBq and the branch is deactivated. Otherwise the

branch is activated. Branch deactivation prevents water from flowing into the system from the ground node.

Step 3 – Process pump list. For each { }pNp ,,1…∈ :

Step 3a: Let pPnnn ,= denote the node to which the

pump is connected.

Step 3b: The pump flow is calculated using:

pP

pPk

nNpP R

PPq

,

,1

,,

~

:~ −=

+

(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.

Figure 26. Cold Plate Heat Exchanger

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

hxhx

hxinhx cm

QQpT

−= (91)

where hxm and hxc are the mass and the specific heat

of the cold plate, inQ is the heat into the cold plate from

the component, and hxQ is the heat out of the cold plate to the cooling fluid.

Let cfw and cfc 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 Ahcw cfcf ≥2 then the rate at

which heat can be removed from the cold plate is governed by

( )cihxcfcf

cfcfhx TT

Ahcw

AhcwQ −

+=

2

2 (92)

where ciT is the inlet temperature. The outlet temperature is given by

cfcf

hxcico cw

QTT += . (93)

If Ahcw cfcf <2 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

( )cihxcfcfhx TTcwQ −= , (94)

and the outlet temperature is equal to the cold plate temperature,

hxco TT = . (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]

LMTDsfsfhx ThAQ Δ= (96)

where sfA and sfh are the heat exchanger contact area

and heat transfer coefficient, and LMTDTΔ 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 fiT and foT

denote the inlet and outlet temperatures of the freshwater, and let siT and soT 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

2

oiavgLMTD

TTTT

Δ+Δ=Δ≈Δ (97)

where sofii TTT −=Δ and sifoo TTT −=Δ . In [8], it is

shown that the average temperature formulation introduces a 5% percent error if iTΔ and oTΔ 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 fw , fc , and fm denote the mass flow rate, specific

heat, and mass of the freshwater in the heat exchanger. Also, let sw , sc , and sm 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

avgsfsfhx ThAQ Δ= (98)

and the derivative of the freshwater outlet temperature is given by

( )

ff

hxfofifffo cm

QTTcwpT

−−= . (99)

However, if ( ) fosssifisshx pTcmTTcwQ +−> then the heat

exchanger is saturated and (98) and (99) are replaced with

( ) ( )

ssff

sifissfofifffo cmcm

TTcwTTcwpT

+−−−

= (100)

and

( ) fosssifisshx pTcmTTcwQ +−= . (101)

For both saturated and unsaturated conditions, the outlet seawater temperature is governed by

( )ss

sisosshxso cm

TTcwQpT

−−= . (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

The flow rate is governed by

⎪⎩

⎪⎨⎧

+=

o

ow

sw cf

pumpcf

01

1 *

τ (103)

where *cfw is the nominal mass flow rate of the cooling

fluid, o is the operation status of the pump determined in the automation layer, and pumpτ 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.

f

xpipexpipexlag q

lAt

,,, = (104)

where { }outinx ,∈ , xpipeA , is the pipe cross-sectional

area, xpipel , is the pipe length, and ρcff wq = 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, maxlagt , , and

if maxlagfxpipexpipe tqlA ,,, > then maxlagxlag tt ,, = .

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,

busv 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 smouti , 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 psoutv , and psouti , . 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 ( cmouti , ) 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 iminv , .

Seawater pump flow rates for SWP 1 in Zone 1 (dotted) and SWP 3 in Zone 3 (solid) are Pq . The cold plate temperature of the CHXs cooling the SMs is designated

chxhxT , (CHX 1—dotted; CHX 4—solid). Freshwater flow

rates cfw and cold plate temperatures fwlhxT , 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.

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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 Type xΔ (m) yΔ (m) zΔ (m) SM 10.24 4.1 4.1 PD 7.17 4.1 4.1 PS 3.81 1.83 2.13 CM 1.83 1.22 1.91 IM 1.83 1.22 1.91

SWP 3.05 1.52 1.52 FHX 3.05 1.52 1.52

Table 4. Rectangular Prism Centroids

Component cx (m) cy (m) cz (m)

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

Table 5. Polygonal Path Vertices

Path Vertices (m) SM 1 to

PD 1 (44.5, -3.66, 4.11), (44.5, -3.66, 4.72), (44.5, -4.08, 4.72), (82.91, -4.08, 4.72)

SM 1 to PS 1

(44.5, -3.66, 4.11), (44.5, -3.66, 2.46), (44.5, 0, 2.46), (41.45, 0, 2.46)

SM 2 to PD 2

(92.96, -0.61, 5.79), (92.96, -0.61, 4.72), (92.96, 4.08, 4.72), (82.91, 4.08, 4.72)

SM 2 to PS 2

(92.96, -0.61, 5.79), (92.96, -0.61, 5.56), (92.96, 0, 5.56), (91.82, 0, 5.56)

PDCB 1 Zone 1 (39.84, -7.32, 5.49), (50.6, -7.32, 5.49)

PDCB 1 Zone 2 (50.6, -7.32, 5.49), (88.09, -7.32, 5.49)

PDCB 1 Zone 3 (88.09, -7.32, 5.49), (111.03, -7.32, 5.49)

PDCB 2 Zone 1 (39.84, 7.32, 5.49), (50.6, 7.32, 5.49)

PDCB 2 Zone 2 (50.6, 7.32, 5.49), (88.09, 7.32, 5.49)

PDCB 2 Zone 3 (88.09, 7.32, 5.49), (111.03, 7.32, 5.49)

PS 1 to PDCB 1

(41.45, 0, 2.46), (41.45, 0, 5.49), (41.45, -7.32, 5.49)

CM 1 to PDCB 1

(39.84, -4.57, 10.86), (39.84, -4.57, 5.49), (39.84, -7.32, 5.49)

CM 2 to PDCB 1

(64.6, -4.57, 10.86), (64.6, -4.57, 5.49), (64.6, -7.32, 5.49)

CM 3 to PDCB 1

(111.03, -4.57, 10.86), (111.03, -4.57, 5.49), (111.03, -7.32, 5.49)

PS 2 to PDCB 2

(91.82, 0, 5.56), (91.82, 0, 5.49), (91.82, 7.32, 5.49)

CM 4 to PDCB 2

(39.84, 4.57, 3.04), (39.84, 4.57, 5.49), (39.84, 7.32, 5.49)

CM 5 to PDCB 2

(64.6, 4.57, 3.04), (64.6, 4.57, 5.49), (64.6, 7.32, 5.49)

CM 6 to PDCB 2

(111.03, 4.57, 3.04), (111.03, 4.57, 5.49), (111.03, 7.32, 5.49)

CM 1 to ZDCB 1

(39.84, -4.57, 10.86), (39.84, -4.57, 9.21), (39.84, 0, 9.21), (38.6, 0, 9.21)

CM 4 to ZDCB 1

(39.84, 4.57, 3.04), (39.84, 4.57, 9.21), (39.84, 0, 9.21), (38.6, 0, 9.21)

IM 1 to ZDCB 1

(36.12, 0, 13.72), (36.12, 0, 9.21), (36.12, 0, 9.21), (38.6, 0, 9.21)

CM 2 to ZDCB 2

(64.6, -4.57, 10.86), (64.6, -4.57, 8.25), (64.6, 0, 8.25), (64.34, 0, 8.25)

CM 5 to ZDCB 2

(64.6, 4.57, 3.04), (64.6, 4.57, 8.25), (64.6, 0, 8.25), (64.34, 0, 8.25)

IM 2 to ZDCB 2

(63.81, 0, 10.86), (63.81, 0, 8.25), (63.81, 0, 8.25), (64.34, 0, 8.25)

CM 3 to ZDCB 3

(111.03, -4.57, 10.86), (111.03, -4.57, 8.25), (111.03, 0, 8.25), (110.57, 0, 8.25)

CM 6 to ZDCB 3

(111.03, 4.57, 3.04), (111.03, 4.57, 8.25), (111.03, 0, 8.25), (110.57, 0, 8.25)

IM 3 to ZDCB 3

(109.65, 0, 10.86), (109.65, 0, 8.25), (109.65, 0, 8.25), (110.57, 0, 8.25)

IM 1 to SWP 1

(36.12, 0, 13.72), (36.12, 0, 2.47), (36.12, 0, 2.47), (50.29, 0, 2.47)

IM 2 to SWP 2

(63.81, 0, 10.86), (63.81, 0, 2.47), (63.81, 0, 2.47), (77.11, 0, 2.47)

IM 3 to SWP 3

(109.65, 0, 10.86), (109.65, 0, 2.47), (109.65, 0, 2.47), (88.39, 0, 2.47)

IM 1 to FHX 1

(36.12, 0, 13.72), (36.12, 0, 2.46), (36.12, -0.11, 2.46), (39.82, -0.11, 2.46)

IM 2 to FHX 2

(63.81, 0, 10.86), (63.81, 0, 3.04), (63.81, 0.08, 3.04), (64.3, 0.08, 3.04)

IM 3 to FHX 3

(109.65, 0, 10.86), (109.65, 0, 3.04), (109.65, 0, 3.04), (105.89, 0, 3.04)

SWB 1 (38.4, 0, 2.47), (38.4, 4.57, 2.47), (50.29, 4.57, 2.47)

SWB 2 (50.29, 4.57, 2.47), (65.23, 4.57, 2.47) SWB 3 (65.23, 4.57, 2.47), (77.11, 4.57, 2.47) SWB 4 (77.11, 4.57, 2.47), (88.39, 4.57, 2.47)

SWB 5 (88.39, 4.57, 2.47), (106.68, 4.57, 2.47), (106.68, 0, 2.47)

SWB 6 (38.4, 0, 2.47), (38.4, -3.66, 2.47) SWB 7 (50.29, 4.57, 2.47), (50.29, 0, 2.47) SWB 8 (65.23, 4.57, 2.47), (65.23, 0.3, 2.47) SWB 9 (77.11, 4.57, 2.47), (77.11, 0, 2.47)

SWB 10 (88.39, 4.57, 2.47), (88.39, 0, 2.47) SWB 11 (106.68, 0, 2.47), (106.68, -0.61, 2.47) SWB 12 (50.29, -4.57, 2.47), (50.29, 0, 2.47) SWB 13 (65.23, -4.57, 2.47), (65.23, 0.3, 2.47) SWB 14 (77.11, -4.57, 2.47), (77.11, 0, 2.47) SWB 15 (88.39, -4.57, 2.47), (88.39, 0, 2.47)

SWB 16 (38.4, -3.66, 2.47), (38.4, -4.57, 2.47), (50.29, -4.57, 2.47)

SWB 17 (50.29, -4.57, 2.47), (65.23, -4.57, 2.47) SWB 18 (65.23, -4.57, 2.47), (77.11, -4.57, 2.47) SWB 19 (77.11, -4.57, 2.47), (88.39, -4.57, 2.47)

SWB 20 (88.39, -4.57, 2.47), (106.68, -4.57, 2.47), (106.68, -0.61, 2.47)

SWB 21 (38.4, 0, 2.47), (38.4, 0, 2.46), (38.4, -0.11, 2.46), (39.82, -0.11, 2.46)

SWB 22 (65.23, 4.57, 2.47), (65.23, 4.57, 4.72), (65.23, 4.08, 4.72), (82.91, 4.08, 4.72)

SWB L1 (65.23, 0.3, 2.47), (65.23, 0.3, 3.04), (65.23, 0.08, 3.04), (64.3, 0.08, 3.04)

SWB L2 (65.23, -4.57, 2.47), (65.23, -4.57, 4.72), (65.23, -4.08, 4.72), (82.91, -4.08, 4.72)

SWB L3 (106.68, 0, 2.47), (106.68, 0, 3.04), (106.68, 0, 3.04), (105.89, 0, 3.04)

SWB L5 (106.68, -0.61, 2.47), (106.68, -0.61, 5.79), (106.68, -0.61, 5.79), (92.96, -0.61, 5.79)

SWB L6 (38.4, -3.66, 2.47), (38.4, -3.66, 4.11), (38.4, -3.66, 4.11), (44.5, -3.66, 4.11)

SWB L8 (39.82, -0.11, 2.46), (39.82, -0.11, 2.46), (39.82, 0, 2.46), (41.45, 0, 2.46)

SWB L9 (39.82, -0.11, 2.46), (39.82, -0.11, 10.86), (39.82, -4.57, 10.86), (39.84, -4.57, 10.86)

FHX 1 to PS 1

(39.82, -0.11, 2.46), (39.82, -0.11, 3.04), (39.82, 4.57, 3.04), (39.84, 4.57, 3.04)

FHX 1 to CM 1

(39.82, -0.11, 2.46), (39.82, -0.11, 13.72), (39.82, 0, 13.72), (36.12, 0, 13.72)

FHX 1 to CM 4

(64.3, 0.08, 3.04), (64.3, 0.08, 10.86), (64.3, -4.57, 10.86), (64.6, -4.57, 10.86)

FHX 1 to IM 1

(64.3, 0.08, 3.04), (64.3, 0.08, 3.04), (64.3, 4.57, 3.04), (64.6, 4.57, 3.04)

FHX 2 to CM 2

(64.3, 0.08, 3.04), (64.3, 0.08, 10.86), (64.3, 0, 10.86), (63.81, 0, 10.86)

FHX 2 to CM 5

(105.89, 0, 3.04), (105.89, 0, 5.56), (105.89, 0, 5.56), (91.82, 0, 5.56)

FHX 2 to IM 2

(105.89, 0, 3.04), (105.89, 0, 10.86), (105.89, -4.57, 10.86), (111.03, -4.57, 10.86)

FHX 3 to PS 2

(105.89, 0, 3.04), (105.89, 0, 3.04), (105.89, 4.57, 3.04), (111.03, 4.57, 3.04)

FHX 3 to CM 3

(105.89, 0, 3.04), (105.89, 0, 10.86), (105.89, 0, 10.86), (109.65, 0, 10.86)

FHX 3 to CM 6

(105.89, 0, 3.04), (105.89, 0, 3.04), (105.89, 4.57, 3.04), (111.03, 4.57, 3.04)

FHX 3 to IM 3

(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

minarm ,,ω 178.5 rad/s maxorm ,,ω 208.5 rad/s

maxarm ,,ω 198.5 rad/s maxohxT ,, 365 K

minorm ,,ω 168.5 rad/s

Table 7. Propulsion Drive Operation Parameters

minainv ,, 400 V maxoinv ,, 600 V

maxainv ,, 500 V minocv ,, 400 V

minacv ,, 500 V maxohxT ,, 365 K

minoinv ,, 300 V

Table 8. Power Supply Operation Parameters

minainv ,, 400 V minoinv ,, 300 V

maxainv ,, 500 V maxoinv ,, 600 V

maxahxT ,, 325 K maxohxT ,, 365 K

Table 9. Converter Module Operation Parameters

minainv ,, 450 V minoinv ,, 400 V

maxainv ,, 550 V maxoinv ,, 600 V

maxahxT ,, 325 K maxohxT ,, 365 K

Table 10. Inverter Module Operation Parameters

minainv ,, 370 V minoinv ,, 320 V

maxainv ,, 470 V maxoinv ,, 520 V

maxahxT ,, 325 K maxohxT ,, 365 K The SWV locations are given in Table 11. All of the SWVs in the notional plant are type 1.

Table 11. Seawater Valve Parameters Valve x (m) y (m) z (m) Tq (m3/s)

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, Ω= 5254.531fictr .

The PM model parameters are given in Table 12.

Table 12. Prime Mover Parameters *rmω 188.5 rad/s δτ ,lag 3.51 ms

J 8.4026 kg∙m2 ωk 326 N∙m∙s2/rad2

*maxpω 20.94 rad/s2

ωτ 803 ms

srlτ 49.8 ms threrr ,ω 10 rad/s

maxδ 402 rad minT -600 N∙m

δk 91.5 N∙m/rad maxT 600 N∙m

δτ ,lead 255 ms The SM machine parameters are described in [4], but two parameters not included in [4] will be described here. The parameter values are Ω= 5254.531fictr and

%56.94=η .

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 *

,rmsllv 560 V,l-l,rms powdcv , 172 V

fτ 10 ms switchr 0.402 �

vτ 200 ms switchv 0 V

pvk 0.0245 A/V dioder 0.07 Ω

maxfdei , 3 A diodev 1.37 V

IΔ 0.018 A fixK 5×104 V/A The PD parameters are given in Table 14.

Table 14. Propulsion Drive Parameters

filterτ 10 ms dcL 2.858 mH

cL 0.2 mH dcC 1.988 mF

dcr 0.0556 � η 95.82% The PS parameters are given in Table 15.

Table 15. Power Supply Parameters *outv 500 V *

maxpv 2500 V/s

filterτ 10 ms vk 2.657

TR 0.79 vτ 100 ms

cL 0.662 mH ipk 0.00112 V/A

dcr 0.075 � idk 4.4×10-5 V∙s/A

dcL 11.3 mH 0rV 597 V

dcC 4.55 mF effr 0.3133 �

scI 55 A effL 12.624 mH

thrI 35 A minC ,α -0.9

srlτ 0.1 ms maxC ,α 1.0 *minpv -2 MV/s η 94.89%

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 �.

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 *outv 420 V d 0.84 V/A

inC 449 �F limiti 20 A

outC 447 �F η 96.97%

Table 17. Zonal DC Bus Parameters

pr 0.05 � dv 1.2 V

sr 0.05 �

Table 18. Inverter Module Parameters

inC 590 �F η 95.90% All three SWPs have the same parameters, given in Table 19.

Table 19. Seawater Pump Parameters

1,Pa 704.72 1,Pβ 4.92

2,Pa 529.48 2,Pβ 1.59

1,Pα 0.84 3,Pβ 0.000153

2,Pα 1.01 Pbq 0.31 m3/s

1,Pb 611.34 PR 352.234 kPa∙s/m3

2,Pb 411.45 PssP 60.556 kPa

3,Pb 469.27 Pτ 5 s All the SWBs exhibit turbulent flow so 3

0 s/mPa 1 ⋅=BR ,

/sm 1 30 =Bq , and 75.0=Bm . The branches are shown in

Table 20. The SWS parameters are in Table 21.

Table 20. Seawater Branch Parameters

Branch α β 0BP (kPa) BqR (kPa∙s/m3)

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

19 10 13 0 43.084 20 13 16 0 84.983 L1 1 0 0.1 99650.369 L2 6 0 22.624 4129.37 L3 8 0 5.72 466623.529 L4 7 0 22.624 4141.387 L5 15 0 5.72 91111.427 L6 16 0 33.33 807.009 L7 2 0 16.507 1280.271

Table 21. Seawater Solver Parameters

NlkG 1×10-10 m3/s/kPa δ 1×10-9 m3/s

BfltG 1 m3/s/kPa maxk 2000

CR 1×1010 kPa∙s/m3 γ 0.5 ε 1×10-3

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

hxm 14327 kg A 127.22 m2

hxc 400 J/K/kg h 4800 W/K/m2

cfc 3998 J/K/kg

Table 23. Propulsion Drive Component Heat

Exchanger Parameters

hxm 6813 kg A 60.5 m2

hxc 400 J/K/kg h 4800 W/K/m2

cfc 3998 J/K/kg

There are three sets of FWL parameters. In each set,

inpipeoutpipe AA ,, = . 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

inpipeoutpipe ll ,, = .

Table 24. Power Supply Freshwater Loop Parameters

hxm 380.1 kg h 4800 W/K/m2

hxc 896 J/K/kg *cfw 8.691 kg/s

cfc 4180 J/K/kg inpipeA , 45.6 cm2

A 7.569 m2 pumpτ 5 s

Table 25. Converter Module Freshwater Loop

Parameters

hxm 73.5 kg h 4800 W/K/m2

hxc 896 J/K/kg *cfw 1.679 kg/s

cfc 4180 J/K/kg inpipeA , 45.6 cm2

A 1.463 m2 pumpτ 5 s

Table 26. Inverter Module Freshwater Loop Parameters

hxm 100.6 kg h 4800 W/K/m2

hxc 896 J/K/kg *cfw 2.30 kg/s

cfc 4180 J/K/kg inpipeA , 45.6 cm2

A 2.003 m2 pumpτ 5 s

Table 27. Freshwater Loop Lengths

Loop inpipel , (m)

1 1.75 2 5.28 3 12.88 4 15.07 5 4.95 6 12.77

7 8.39 8 16.59 9 9.72

10 17.53 11 11.58

Table 28 shows the parameters of the FHXs in Zones 1 and 3. Table 29 shows the parameters of the FHX in Zone 2.

Table 28. Zones 1 and 3 Fluid Heat Exchanger Parameters

sfA 23.65 m2 fm 446 kg

sfh 2536.6 W/K/m2 sc 3998 J/K/kg

fc 4180 J/K/kg sm 358 kg

Table 29. Zone 2 Fluid Heat Exchanger Parameters

sfA 9.32 m2 fm 175.9 kg

sfh 2536.6 W/K/m2 sc 3998 J/K/kg

fc 4180 J/K/kg sm 141.2 kg