1999

Figure 1. One-line diagram of the AC power distribution with closed-loop torque control of the induction machine consisting of

30 state variables and 31 parameters.

Sensitivity Analysis and Low-Dimensional Stochastic Modeling of Shipboard Integrated

Power Systems P. Prempraneerach*, F.S. Hover*, M.S. Triantafyllou*, C. Chryssostomidis*,+ and G.E. Karniadakis*,**

* Massachusetts Institute of Technology/Department of Mechanical Engineering, Cambridge, MA, USA ** Brown University/Division of Applied Mathematics, Providence, RI, USA

+ Corresponding author

Abstract— For design purposes, we use stochastic modeling and sensitivity analysis to identify the most important parameters of the AC power distribution in the Integrated Power System (IPS) model. The main AC subsystem of the IPS consists of a 59 kW synchronous generator and a motor drive of 50 hp induction motor. First, we perform sensitivity analysis treating all 31 parameters of the full model as stochastic variables. Second, we construct two low-dimensional stochastic models (with 15 and 8 parameters) and compare the multi-rate dynamics of the reduced models to the full model. For parameter variations larger than 50%, the low-dimensional models still predict accurately the mean values but underpredict the variance of most state variables.

I. INTRODUCTION

In the All-Electric Ship (AES) architecture, the power sharing between the propulsion units and the high-power equipment, especially under heavy propulsion demand and casualty conditions, has recently been identified as an important issue [2], [3], [12], [15]. In a new AES configuration, there is an increasing demand for electric power for ship system automation, electrical weaponry, electric propulsion, and ship service distribution. About 70% to 90% of power from the generator units in the fully integrated power system (IPS) is consumed in the propulsion systems [2]. Thus, when a large power demand is imposed on the electrical bus during a mission or life-critical situation, the power distribution must be optimally modified to yield the most efficient power usage and to maintain continuity of service [15].

Many machines and electrical components form the power generation and propulsion drive in the AES. As a result, sensitivity analysis that can identify the influential and interactive parameters from a large number of parameters is needed for the IPS designer to improve

performances of the integrated system and to prevent a cascaded failure. The sensitivity analysis, based on the One-At-a-Time stochastic variation [7], [8], has been shown to be able to identify and prioritize the important parameters of an AC subsection in the entire IPS [4]. In addition, to further accelerate large-scale simulation or even guide experimental studies, a reduction of parametric space can be accomplished by fixing the less important parameters at their nominal values.

In this paper, we first describe the modeling of the IPS and subsequently introduce our sensitivity analysis approach, based on stochastic perturbations of the system. We then present the application of the sensitivity analysis to three cases of parameter-space reduction using the Quasi Monte Carlo technique [10].

II. SYSTEM MODELING

In this study, the integrated power system is based on the model developed by the Office of Naval Research (ONR) [4]. The main AC subsystem of the IPS model consists of a 59 kW, 3-phase Synchronous Machine (SM) as generator and a close-loop drive of 50 hp, 3-phase Induction Machine (IM) as propulsor. This simulation model was validated by experimental results obtained on the Naval Combat Survivability at Purdue University testbed [12], [13]. A power converter and a constant-slip current control system are incorporated into the propulsion system to control the torque of the induction machine. According to [5], the constant-slip current control is composed of an inner loop with a current feedback from IM and an outer loop with IM's shaft angular velocity feedback for controlling motor torque and speed, respectively. The detailed derivation of all machine equations can be found in [5], [8] and [9]. The model of AC power distribution and propulsion systems is composed of 30 state variables (see Table II) and 31 parameters (see Table III). The configuration of IPS considered here is shown in Figure 1.

III. SENSITIVITY ANALYSIS

Global sensitivity analysis [11], [14] is an effective tool in identifying the most significant parameters of the system, particularly in large-scale systems with many parameters, i.e., the electric ship IPS [8], [9]. First, we

978-1-4244-1668-4/08/$25.00 ©2008 IEEE

2000

Figure 2. Morris Method for p = 4 or = 2/3. Shown are two randomized trajectories in three-dimensional input space within

[0,1].

Figure 3. Monte Carlo Sampling Method: With N = 9, the random direction and initial condition in each direction of a three-dimensional input space within [0,1] range are used for computing the j

iEE for i =1,2,3 with a random-directional fixed step .

briefly introduce the concept of a gradient-based sensitivity analysis technique. For a system with dparameters ( ix ) and n outputs ( jy ), the elementary

effect of the i input on the j output ( jiEE ) can be

defined as the One-At-a-Time (OAT) approximated gradient, where only the ix parameter is perturbed from its nominal value by and the other 1d parameters are fixed. It is defined by:

tytxxxytEE jdijj

i

,,,,,1 (1)

Morris [6] proposed to construct a p -level grid structure in the parametric space and to generate randomized trajectories from random initial conditions on this grid for r trials such that the mean of j

iEE from a large number of trials can accurately identify the influence of the i input on the j output. In the original Morris method, input parameter range is assumes to be a uniform distribution only. The larger the number of inputs compared to the number of outputs, the more efficient the Morris method becomes. By setting 12 pp and using even value of p (where 2p ), the randomized trajectories will equally distribute in the d -dimensional parametric space. The detailed technique on constructing randomized trajectories can be found in [6]. An example of two trajectories on a 4-level grid structure in three-dimension space is shown in Figure 2.

Our gradient-based sensitivity analysis technique is not restricted on the grid structure, and as a result the sensitivity index converges faster for the same computational cost [8]. All ix are assumed to fluctuate within a specified range, ba, . The perturbation magnitude, , is fixed at 2ba , similarly to the Morris method with very large p , because the OAT perturbation will cover the parametric space with fewer trials. The direction of in the gradient computation is random such that the OAT perturbation is equally

distributed in the parametric space. Using the Monte Carlo (MC) or Quasi Monte Carlo (QMC) sampling method [8] to generate N random values of ix in the d -dimensional space, the mean of j

iEE with respect to N , jiEEE

referred to as sensitivity index, can be computed from Nrandomized gradient values of i th parameters, as shown in Figure 3. The larger the j

iEEE becomes, the more influence ix has on jy . Moreover, this technique is suitable for the metamodels since the convergence rate of the Monte Carlo is independent of the input or parameter dimension.

IV. RESULTS

In IPS simulation, we assume that the generator initially operates at its rated speed near steady-state conditions. Then, at 0.35 seconds the torque-control propulsion drive of the induction motor running at the synchronous speed is suddenly connected to the RC bus. The torque command to the controller is kept constant at 2 N-m. First, all 31 parameters are assumed to be uniform random variables,

, whose values vary within from their nominal values, 0x , (i.e., 10xx ). Performing sensitivity analysis using the QMC sampling method on this system yields j

iEEE for 311 ,,i and 301 ,,j as a function of time. To present a comprehensible measure of the sensitivity over time, we compute the average sensitivity index of each parameter to different states or outputs as the follows :

22 tEEEES jiji,, . Figure 4

shows the average sensitivity index of the AC power distribution system with the propulsion drive for

910 .,t second when all parameters vary within %30of their nominal values. According to Figure 4, we can easily identify the 15 most sensitive parameters to all state variables of SM, IM, RC bus, and power converter, as shown in Table I. Both generator and motor are sensitive to their mutual flux reactance between stator and rotor windings ( [c,4-6 and 9-10] and [d,1-3] for generator, [o,20-23] for motor ), which physically govern the interaction of stator and rotor windings in these machines.

2001

Figure 4. For 910 .,t second, the jiES ,,2

plot with

%30 variations using the QMC Sampling method with

1500N . (See Tables II and III for the variables on both axes.)

Figure 5. Electrical transients: For 60350 .,.t and %20variation, the sensitivity reduction to 15 and 8 most sensitive

parameters of AC power distribution and torque-control propulsion drive: Mean (Left column) and Variance (Right column).

Furthermore, the induction motor is highly sensitive to the rotor inertia ([r,20-23 and 25]) and load coefficient ([s,20-23 and 25]), determining the motor mechanical time constant. Capacitive and inductive elements of the RC bus and harmonic filter have strong influences on the generator field-winding voltage and harmonic filter current ([t,7 and 9 and 14-16] for RC bus, [v and x,7 and 9 and 14-16] for harmonic filter). These elements have direct effects on the electrical time constant of the synchronous machine and RC bus.

Because of the multi-rate dynamical states in this system, we must consider both the fast and slow transient

dynamics of the electrical and mechanical states, respectively. To demonstrate the advantage of the parametric-space reduction using sensitivity analysis with the same number of evaluations, stochastic simulations are performed for three perturbation levels, = [0.1, 0.2, 0.5], for comparison purposes. The mean and variance responses from the stochastic analysis with 31, 15, and 8 random parameters are plotted overlaying each other in Figures 5 and 6 (for = 0.2) up to 0.6 and 1.9 seconds, respectively. Stochastic responses of the stator q-axis flux linkage per second ( qs ) of SM, stator q-axis flux linkage per second ( qs ) of IM, IM rotor speed ( r ), and rectifier output voltage ( outV ) of RC bus from the 15- and 8-parameter models are very similar to those with the full 31-parameter as random variables. The only difference is in the variance magnitude in the 15-parameter case because the SM state variables are also sensitive to mqXand

1lkqX in addition to those 15 parameters in Table I. When we further reduce the parametric space in the stochastic simulations to 8 random variables, the stochastic solutions are also similar to the previous case except that the variance of SM qs and of outV is much lower. The influence on these two state variables confirms the sensitivity of SM qs to mqX , 1lkqX ([d and j,1]) and

of outV to mdX ([c,27]), as shown in Figure 4.

There are small differences in the stochastic responses of these four states with reduced- and full-parameter variations for = 0.1 and 0.2. Even though, the variance responses with = 0.1 and 0.2 have similar characteristics in fast- and slow-transient regimes, the

magnitude of variance responses with = 0.2 are much larger than those with = 0.1, especially in the mechanical-transients regime. Because of the strong nonlinearities in power systems, the response variation does not increase proportionally when the perturbation increases from 10 to 20 percent.

When the percent variation of parameters increases to 50 percent, the stochastic responses in electrical-transient regime (as shown in Figure 7) and in mechanical-transient regime (as shown in Figure 8) with 8-dimensional stochastic modeling underpredict the mean responses slightly, but the variance responses largely deviate from those obtained from stochastic simulations with full-parameter perturbation. This confirms the strong nonlinearities and interactions in the AC power distribution and propulsion systems of the IPS, expressed

TABLE I. THREE DIFFERENT STOCHASTIC MODELS CORRESPONDING TO 31, 15 AND

8 UNCERTAIN PARAMETERS. WHERE SM: SYNCHRONOUS MACHINE,IM: INDUCTION MACHINE, BUS: RC BUS AND HARMONIC FILTER,

RECT: RECTIFIER, AND DCF: DC LINK FILTER.

Component 31 Parameters 15 Parameters 8 Parameters

SM2211 lkqkqlkqkq

lkdkdlfdfd

mqmdlss

XrXr

XrXrXXXr

,,,

,,,,,,,,

lfdfd

mdls

XrXX,,

lsX

IMloadlrr

mlss

JXr

XXr

,,,

,,,''

22

222loadrm JrX ,,, '22

loadJ ,

BUS fff LRCRC ,,,, ff LCC ,, fCC,

RECT dcdcdcc CLrL ,,, dcdc CL , dcC

DCF LLL CRL ,, LL CL , LL CL ,

2002

Figure 6. Mechanical transients: For 91350 .,.t and %20variation, the sensitivity reduction to 15 and 8 most sensitive

parameters of AC power distribution and torque-control propulsion drive: Mean (Left column) and Variance (Right column).

Figure 7. Electrical transients: For 60350 .,.t and %50variation, the sensitivity reduction to 15 and 8 most sensitive

parameters of AC power distribution and torque-control propulsion drive: Mean (Left column) and Variance (Right column).

Figure 8. Mechanical transients: For 91350 .,.t and %50variation, the sensitivity reduction to 15 and 8 most sensitive

parameters of AC power distribution and torque-control propulsion drive: Mean (Left column) and Variance (Right column).

via the magnitude of the variances in our sensitivity analysis.

In terms of the computational cost on an Intel Pentium 4 3.0GHz processor, the reduction of stochastic variables from 31 to 15 and from 15 to 8 can save the computational time of the stochastic technique by about 15 and 4 hours, respectively.

V. CONCLUSION

Stochastic sensitivity analysis can identify the most important parameters in a large-scale system with numerous parameters and hence can accelerate substantially simulations of integrated power systems. This, in turn, can be very useful in validation and other experimental studies as only a small subset of parameters need to be considered for testing, thus saving human time and resources. Specifically, in this paper we demonstrated that we can reduce the number of parameters by a fourfold with no significant adverse effect in predicting the dynamics of the system even for large perturbations of 50

percent. In particular, the mean responses predicted by the two reduced systems were in close agreement with the full-parameter system although the nonlinear interactions were consistently underestimated by the reduced models. The results of this paper represent a first step towards

constructing robust low-dimensional descriptions for integrated power systems governed by multi-rate dynamics.

ACKNOWLEDGMENT

This work is supported by the Office of Naval Research (N00014-02-1-0623 ESRD Consortium, Also N00014-07-1-0846) and Sea Grant (NA060AR4170019 NOAA/DOC).

REFERENCES

[1] F. Campolongo, J. Cariboni, A. Saltelli, and W. Schoutens, “Enhancing The Morris Method,” Los Alamos National Laboratory, http://library.lanl.gov, 2005.

[2] D.H. Clayton, S.D. Sudhoff, G.F. Grater, “Electric Ship Drive and Power System,” Conf. Rec. 2000 24th Int. Power Modulation Symp., 2000, pp. 85-88.

[3] A. Chaboki, L. Thurmond, G. Grater, D. Bauer, F. Beach, and D. Clayton, “Integration of electromagnetic rail gun into future electric warship,” High Powered Weapons Systems for Electric Ship Conference, Dec. 2004.

[4] PC Krause and Associates, “Power System Control Development,” Final Report, Contract F33615-99-D-2974 for NSF/ONR Partnership in Electric Power Network Efficiency and Security (EPNES), March 2003.

[5] P.C. Krause, O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery and Drive Systems, 2nd e.d., IEEE Press and Wiley-Interscience, NY, 2002.

[6] M.D. Morris, “Factorial Sampling Plans for Preliminary Computational Experiments,” Technometrics, vol. 33, no. 2, pp. 161-174, May 1991.

[7] P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, G.E. Karniadakis, “Uncertainty Quantification in Simulations of Power Systems: Multi-Element Polynomial Chaos methods,” submitted to International Journal of Electrical Power and Energy Systems,May 2007.

2003

[8] P. Prempraneerach, F.S. Hover, M.S. Triantafyllou, T.J. McCoy, C. Chryssostomidis, G.E. Karniadakis, “Sensitivity Analysis of the Shipboard Integrated Power System,” Naval Engineering Journal,vol. 119, no. 2, 2007.

[9] P. Prempraneerach, Uncertainty Analysis in a Shipboard Integrated Power System using Multi-Element Polynomial Chaos,Ph.D. thesis, Massachusetts Institute of Technology, 2007.

[10] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, 2nd e.d., Cambridge University Press, 1992.

[11] A. Saltelli, K. Chan, M. Scott, Sensitivity Analysis, John Wiley \& Sons Ltd, 2000.

[12] S.D. Sudhoff, B.T. Kuhn, E. Zivi, D.E. Delisle, and D. Clayton, “Impact of Pulsed Power Loads on Naval Power and Propulsion Systems,” Electric Ship Research and Development Consortium

(ESRDC), Recent Papers, nerc.aticorp.org/esrdc_recent_papers.html.

[13] S.D. Sudhoff, S. Pekarek, B. Kuhn, S. Glover, J. Sauer, D. Delisle, “Naval Combat Survivability Testbeds for Investigation of Issues in Shipboard Power Electronics Based Power and Propulsion Systems,” Power Engineering Society Summer Meeting, IEEE,vol. 1, pp. 347 - 350, July 2002.

[14] A. Saltelli, S. Tarantola, F. Campolongo, M. Ratto, Sensitivity Analysis In Practice: A Guide to Assessing Scientific Models, John Wiley & Sons Ltd, 2004.

[15] E. Zivi, T.J. McCoy, “Control of a Shipboard Integrated Power System,” Proceedings of the 33th Annual Conference on Information Sciences and Systems, March 1999.

TABLE II. STATE VARIABLES OF THE SYNCHRONOUS GENERATOR/EXCITER (1-10),

THE INDUCTION MACHINES (20-25), THE POWER CONVERTER (26-30),AND THE RC BUS (11-19) WITH VBASE = 450 V.

1 eqs q-axis stator flux linkage

2ekq1 first q-axis damper winding flux linkage

3ekq2 second q-axis damper winding flux linkage

4 eds d-axis stator flux linkage

5efd d-axis field winding flux linkage

6 ekd d-axis damper winding flux linkage

7 es0 0-axis stator flux linkage

8 power angle

9'xfde exciter voltage

10 VF exciter state

11 anV a-phase bus voltage

12 bnV b-phase bus voltage

13 cnV c-phase bus voltage

14 aI a-phase harmonic filter current

15 bI b-phase harmonic filter current

16 cI c-phase harmonic filter current

17 faV a-phase harmonic filter voltage

18 fbV b-phase harmonic filter voltage

19 fcV c-phase harmonic filter voltage

20 eqs q-axis stator flux linkage

21 eqr q-axis rotor flux linkage

22 eds d-axis stator flux linkage

23 edr d-axis stator flux linkage

24 es0 0-axis rotor flux linkage

25 r rotor angular speed

26 outI rectifier average DC current over one switching cycle

27 outV DC voltage across rectifier's capacitor

28 invV DC voltage across DC-link filter's capacitor

29 invI DC current across DC-link filter's inductor

30 *s

optimum slip speed of constant-slip current controller

TABLE III. PARAMETERS OF THE SYNCHRONOUS GENERATOR (A-L), THE INDUCTION MACHINES (M-S), THE POWER CONVERTER (Y-EE), AND THE RC BUS (T-

X) WITH VBASE = 450 V.

a sr stator resistance

b lsX stator flux leakage reactance

c mdX d-axis mutual flux reactance

d mqX q-axis mutual flux reactance

e fdr rotor field winding resistance

f lfdX field winding's flux leakage reactance

g kdr d-axis rotor damper winding resistance

h lkdX d-axis damper winding's flux leakage reactance

i 1kqr first q-axis rotor damper winding resistance

j 1lkqX first q-axis damper winding flux leakage reactance

k 2kqr second q-axis rotor damper winding resistance

l 2lkqX second q-axis damper winding flux leakage reactance

m 2sr stator resistance

n 2lsX stator flux leakage reactance

o 2mX mutual flux reactance

p '2lrX rotor flux leakage reactance

q '2rr rotor resistance

r J rotor inertia

s load torque load coefficient

t C RC bus capacitance

u R RC bus resistance

v fC harmonic filter capacitance

w fR harmonic filter resistance

x fL harmonic filter inductance

y cL inductance of voltage source

z dcr resistance of rectifier

aa dcL inductance of rectifier

bb dcC capacitance of rectifier

cc LL inductance of DC-link filter

dd LR resistance of DC-link filter

ee LC capacitance of DC-link filter