Electric Power Systems Research 77 (2007) 1124–1131
Distributed simulation for power system analysis
including shipboard systems
Jian Wu, Noel N. Schulz ∗ , Wenzhong Gao
Department of Electrical and Computer Engineering, Mississippi State University, Box 9571, Mississippi State, MS 39762, United States
Received 8 May 2006; received in revised form 8 August 2006; accepted 15 September 2006
Available online 20 October 2006
Power systems are distributed in nature. Often they can be divided into sections or groups and treated separately. Terrestrial power systems
are divided into separate utilities and are controlled by different regional transmission organization (RTO). Each RTO has detailed data for the
area under its control, but only limited data and boundary measurements of the external network. Additionally, shipboard power systems may be
divided into sections where local information is kept but not distributed to other parts of the system. Thus, performing a comprehensive power
system analysis in such a case is challenging. Also, simulating a large-scale power system with detailed modeling of the components causes a
heavy computational burden.
One possible way of relieving this problem is to decouple the network into subsystems and solve the subsystems respectively, and then combine
the results of the subsystems to get the solution. The way to decouple the network and represent the missing part will affect the result greatly. Also,
due to information distribution in the dispatch or data centers, a problem of doing power system analysis with limited available data arises. The
equivalent for other networks needs to be constructed to analyze the power system.
In this paper, a distributed simulation algorithm is proposed to handle the issues above. A history of distributed simulation is brieﬂy introduced.
A generalized coupling method dealing with natural coupling is proposed. Distributed simulation models are developed and demonstrated in the
virtual test bed (VTB). The models are tested with different network conﬁgurations. The test results are presented and analyzed. The performance
of the distributed simulation is compared with the steady state and time domain simulation results.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Distributed simulation; Shipboard power systems; Transient analysis; Time domain simulation; Resistive companion model; Virtual test bed (VTB)
1. Introduction is installed into the ship power system. While tests with the
actual electric ship hardware are costly and risky, a virtual test
The large-scale terrestrial power systems are composed of environment is more affordable and safer to perform a hardware
several utilities and controlled by different regional transmis- test in the prototype stage. Such hardware-in-the-loop tests can
sion organizations (RTO). Each RTO has detailed parameters be undertaken as distributed simulation with part of the system
for the area under control, but only limited data of the exter- simulated in software and part of the response originating from
nal network. Usually, each RTO has the right to read only the the hardware.
boundary measurements on the tie lines that connect its control Therefore, distributed simulation, which can decouple an
area to others. Thus, performing comprehensive power system entire system into multiple parts, is beneﬁcial to a large-scale
analysis in such case is very difﬁcult. power system and shipboard power systems (SPS) analysis. Dis-
Also, for an all-electric ship to ensure its survivability, a tributed simulation helps provide quick diagnosis of failures in
weakly meshed zonal network is used. In each zone an intelligent SPS and better understanding of the system status. An exten-
controller coordinates the zonal connection. In the development sion of distributed simulation could enable hardware to interact
stage, new equipment needs to be tested before the equipment remotely [1,2].
For the reasons above, ﬁve universities in the US have
teamed up for a Department of Defense Multiple University
∗ Corresponding author. Tel.: +1 662 325 2020; fax: +1 662 325 2298. Research Initiative (MURI) project to develop remote testing
E-mail address: firstname.lastname@example.org (N.N. Schulz). and measurement (RTM) models and procedures to virtually
0378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131 1125
connect power system laboratories over a distributed network. the potential to extend to CORBA and allow for more clients
The MURI project targets setting up a large-scale power sys- and a securer connection. Therefore, these properties provide the
tem laboratory to carry advanced, non-destructive testing and development tools to make an application adjustable within dif-
measurement of power systems . ferent network environments easily and not limit the distributed
simulation algorithm only within VTB.
2. Literature survey In this paper, our research has extended the decoupling sim-
ulation method for dc link to ac systems and explored the
2.1. Distributed simulation decoupling method for a power system simulation. The algo-
rithm’s capability to deal with three-phase coupling will be
Distributed Simulation makes use of the computer network demonstrated with different kinds of networks with different
and computes the overall power system solution through sepa- power sources and power load conﬁgurations. In this paper,
rately and concurrently computing units. One of the most devel- a distributed simulation algorithm is proposed. The models
oped techniques in distributed simulation is parallel processing. dealing with the natural coupling were developed in VTB and
This method requires the information of a whole power system demonstrated with different network conﬁgurations, including
matrix. The advanced techniques divide the computation load a shipboard power system-an 18-bus icebreaker. The distributed
based on the partitions of system matrix [3,4]. This method can simulation performance is analyzed in time domain and steady
greatly improve the computation speed for a large-scale power state. The results, when compared to published work, demon-
system and has been used in simulation software such as the real strate how the new models for distributed simulation expand the
time digital simulator (RTDS). simulation toolbox in VTB.
The other technique in distributed simulation is based on
graphically decoupling. This method is used for dc circuit anal- 3. Decoupling method
ysis such as VLSI circuit analysis and large power electronic
system analysis, where the dc link is selected as the coupling This section describes the algorithm extended from the dc
point [5–7]. This method is attractive when some component coupling method to distributed simulation of power system. The
models are unavailable but a response of the models is avail- problem starts with the entire power system network. Suppose
able. It also conforms to the nature of the power system network, that two sub-networks connect via a tie line as shown in Fig. 1.
which is composed of a number of sub-networks and allows a The key issues for distributed simulation include decoupling
simpliﬁed problem formulation for each sub-network. the circuit and representing the missing subsystem. The choices
of the decoupling point and subsystem model will affect the
2.2. Simulation environment stability and accuracy of the solution.
Using the VI overlap decoupling method described in paper
The virtual test bed (VTB) is a time domain simulation soft- , the whole system can be decoupled into two subsystems
ware package and provides multidisciplinary simulation envi- with the transmission line present in both, as the two circles
ronment including electrical, thermal, chemical and mechanical indicate in Fig. 1.
engineering [8,9]. Its open architecture allows users to develop When solving subsystem A, the subsystem B is treated as the
their own models . Its extension to real-time VTB matches the “missing system.” A stabilizing resistor and a current source in
MURI ﬁnal project goal of the hardware-in-the-loop test. VTB parallel represent the missing subsystem, subsystem B in this
uses the resistive companion form (RCF) to model each compo- case, as shown in Fig. 2(a). The corresponding point in the part-
nent and get a solution through nodal analysis. RCF discretizes ner subsystem B controls their values. Similarly, when solving
the device differential equations and describes the electrical subsystem B, the subsystem A is treated as the “missing system.”
component based on its instantaneous response to its terminals’ A stabilizing resistor and a current source in parallel represent
voltage inputs. The independency allows the models to be devel- the missing subsystem, subsystem A in this case, as shown in
oped separately and interconnect with each other easily. This Fig. 2(b).
technique is widely used in time domain simulation software Fig. 3 shows the general workﬂow of the algorithm. For
such as Pspice, PSIM, and PSCAD [8,9]. detailed implementation, if the inner loop runs once, this
In this work, VTB is selected as the simulation environ- algorithm is called a linear method. For nonlinear methods,
ment. However, the application of the proposed algorithm for
distributed simulation is not limited to VTB. It is applicable
to all time domain simulation software using RCF techniques.
Remote procedure call (RPC)  is selected as the protocol for
communication since it can invoke a function remotely through
a standard interface. The functions interface called by RPC is
deﬁned by the interface deﬁnition language (IDL), which is a
standard language used to describe the interface to a routine or
function. RPC can further migrate to common object request
broker architecture (CORBA), since objects in the CORBA are
deﬁned by an IDL. So, with simpliﬁed programming, RPC has Fig. 1. Whole system without decoupling.
1126 J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131
if the stabilizing resistor is static, this algorithm is called a
non-adjusted stabilizing method. If the stabilizing resistor is
varying, this algorithm is called adjusted stabilizing method.
Here are the detailed steps for solving subsystem A. In this
workﬂow, the superscript, n, indicates the number of the inner
1. At initialization, assume v2 = 0 and i2 = 0.
2. Construct equivalent circuit for subsystem B, set stabilizing
resistor RS A = R0 A , where R0 A is a user deﬁned value.
Set the current source i0 A = (v2 (t − h)/RS A ) + i2 (t − h).
3. Solve for sub-system A and send v0 and i0 to subsystem B.
4. Receive vn and in from subsystem B.
5. Construct equivalent circuit for subsystem B, for non-
adjusted static stabilizing resistance, RS A will keep static;
for adjusted static stabilizing resistance RS A will be adjusted
according to history data. The next section presents a detailed
explanation about RS A selection in A = (vn (t))/(RS A +
6. Solve for subsystem, get vn+1 and in+1 .
Fig. 2. Decoupled system with equivalent circuit.
7. Check convergence of i1 . If convergence, proceed to next time
step. If not convergence, send vn+1 and in+1 to subsystem B
and go step 4.
Similarly, this process goes through subsystem B:
1. At initialization, assume v1 = 0 and i1 = 0.
2. Construct equivalent circuit for subsystem B, set stabilizing
resistor RS B = R0 B , where R0 B is a user deﬁned value.
Set the current source i0 B = (v1 (t − h)/RS B ) + i1 (t − h).
3. Solve for sub-system B and send v0 and i0 to subsystem A.
4. Receive vn and in from subsystem A.
5. Construct equivalent circuit for subsystem A, for non-
adjusted static stabilizing resistance, RS B will keep static;
for adjusted static stabilizing resistance RS B will be adjusted
according to history data. The next section presents a detailed
explanation about RS B selection. in B = (vn (t)/RS B ) +
6. Solve for subsystem B, get vn+1 and in+1 .
7. Check convergence of i2 . If convergence, proceed to next time
step. If no convergence, send vn+1 and in+1 to subsystem A
and go step 4.
At each time step, for high accuracy simulation, the inner loop
executes and stops when the current on the transmission line con-
verges. For low accuracy or future hardware-in-the-loop tests,
the inner loop executes only once (linear method). The outer
loop executes to increase the simulation time until it reaches the
total simulation time.
To test the distributed simulation algorithm, two models
were developed in VTB to collect/send information to a remote
site and receive/reproduce the information at a local site. The
server model sends boundary measurements to other clients, and
the client model receives boundary measurements from other
servers. Fig. 4 shows the client/server’s symbols. The three-
Fig. 3. Workﬂow of distributed simulation. phase client/server models decouple a three-phase system, and
J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131 1127
and in+1 is a function of v1 n and i1 n . Similarly, vn+1 and in+1 is
2 1 1
a function of vn and in . Their relationship is expressed in the Eq.
(1) with the changing portion decoupled from the ﬁxed portion.
⎡ R L − RS A ⎤
i1 (t) ⎢ Req 1 + RS A ⎥
=⎢ ⎣ RS − R S B
i2 (t) 0
Req 2 + RS B
Fig. 4. Server/client models in VTB. in (t)
× + (1)
the single-phase client/server models decouple a single-phase
RL Req 1
4. Numerical analysis J1 = − iL (t) + ieq 1 (t)
RS A + Req 1 RS A + Req 1
This section establishes the mathematical model of the dis- RS Req 2
tributed algorithm through circuit analysis and analyzes the J2 = − iS (t) + ieq 2 (t)
RS B + Req 2 RS B + Req 2
convergence of different methods.
In each time step, after the original system is decoupled, the
two subsystems can be represented in the RCF model, as shown Req 1 RS A
i0 (t) =
1 ieq 1 (t) − i0
in Fig. 5. RS A + Req 1 RS A + Req 1 S A
For subsystem A (Fig. 5(a)), no matter how complex the
known subsystem is, at each time step, it can be simpliﬁed into Req 2 RS B
i0 (t) =
2 ieq 2 (t) − i0
a Norton equivalent and represented by a current source iS and a RS B + Req 2 RS B + Req 2 S B
resistor RS . Following the RCF modeling, the transmission line
is represented by a current source, iT , and a resistor, RT . The v2 (t − h)
missing subsystem is represented by a current source iS A and a i0
S A = + i2 (t − h)
stabilizing resistor RS A . Similarly, for subsystem B (Fig. 5(b)),
no matter how complex the known subsystem is, at each time
v1 (t − h)
step, it can be simpliﬁed into a Norton equivalent and repre- i0
S B = + i1 (t − h)
sented by a current source iL and a resistor RL . Following the RS B
RCF modeling, the transmission line is represented with iT and
vn (t − h)
RT . The missing subsystem A is represented by a current source in
S A = 2
+ in (t − h)
iS B and a stabilizing resistor RS B . RS A
Following the algorithm described in Fig. 3, the relationship
between the voltage and current of each iteration in the inner vn (t − h)
S B = 1
+ in (t − h)
loop can be determined through nodal analysis. According to RS B
the algorithm, vn and in controls the value of iS A , vn and in
2 2 1 1
controls the value of iS B . All other elements, like resistors and 2Lload
current source, are known and ﬁxed at this time step. Thus, vn+1 RL = + Rload
Fig. 5. RCF equivalent circuit for subsystems. (a) Subsystem A and (b) subsystem B.
1128 J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131
With such a stabilizing resistance selection, the solution
RT = + Rtrans line is guaranteed to converge to the true solution, which can be
obtained from nodal analysis of the non-distributed simulation.
1 In Fig. 5(a), after replacing RS A with RL and iS A with iL , and
Req−1 = in Fig. 5(b), replacing RS B with RS and iS B with iS , the true
1/(RS + RT ) − 1/(RS + RT )(RT /RS )(iT (t)/iS (t))
solution can be expressed in the equations below:
1 RL Req 1
Req 2 = i1 = − iL (t) + ieq 1 (t)
1/(RL + RT ) + 1/(RL + RT )(RT /RL )(iT (t)/iL (t)) RL + Req 1 RL + Req 1
1 RS Req 2
iL (t) = − v2 (t − h) i2 = − iS (t) + ieq 2 (t)
(2Lload / h) + Rload RS + Req 2 RS + Req 2
(2Lload / h) − Rload Here, i1 and i2 are of the same magnitude and are the opposite
− i1 (t − h)
(2Lload / h) + Rload of each other.
iT (t) = − v2 (t − h) 5. Test cases and performance analysis
(2Ltrans line / h) + Rtrans line
(2Ltrans line / h) − Rtrans line To demonstrate the distributed simulation algorithm, corre-
− i1 (t − h)
(2Ltrans line / h) + Rtrans line sponding models are developed in VTB using C++ language and
tested with different network conﬁgurations. The models perfor-
RS RT mance is analyzed in time domain and steady state simulation.
ieq 1 (t) = iS (t) − iT (t)
RS + R T RS + R T
5.1. Test case #1
RL RL RT
ieq 2 (t) = = iL (t) + iT (t) The ﬁrst test case is a two-bus system, as shown in Fig. 6,
RL + R T RL + R T RL + R T
and the system speciﬁcation comes from the MURI project as
Here, h is the time step size. All the variables at time t–h are shown in Table 1.
known. For the linear VI coupling model, i0 (t) and i0 (t) are the
1 2 Distributed simulation is carried out on a different connected
solutions for that time step. number of loads with different step sizes. The adjusted sta-
To make the solution stable and converging, the eigen- bilizing resistance algorithm is compared with non-adjusted
value of the iteration matrix must be within the unit circle. stabilizing resistance and the linear algorithm. The results of all
However, the eigenvalues λ, (RL − RS A )/(Req 1 + RS A ) and distributed test cases are compared with a single simulation with
(RS − RS B )/(Req 2 + RS B ), are related to state variables and the same time step to evaluate the performance of distributed
time step size sensitive. To make | | within 1, the best choice is simulation. The mean absolute percentage error (MAPE) mea-
to approximate RS A = RL and RS B = RS . sured its performance in the transmission line current as show
But, as stated in the algorithm, only boundary voltage and in Eq. (6). Table 2 shows the test results.
current measurements are available for the unknown subsys-
|inon−distributed − idistribued |
tem, RL and RS are unknown. To estimate RL and RS from the APE = × 100% (6)
measurement, the following relationship between voltage and inon−distributed
current in iterations is used: 1
MAPE = APE (7)
vn vn−1 N
S = 1
1 = 1
+ in−1 (2) N
RS RS 1
Here, N is the number of sample points in the simulation.
vn vn−1 This comparison shows quicker convergence and higher
+ in =
+ in−1 (3) accuracy for the adjusted stabilizing resistance algorithm. In
RL RL 2
each time step, the adjusted stabilizing resistance method can
Note that in and in are ﬁxed at a speciﬁc time step in Eqs.
S L ﬁnd the solution within tolerance in two iterations, whereas the
(2) and (3). Thus, RL and RS can be calculated through the non-adjusted stabilizing resistance method needs three to four
boundary voltage and current measurement. Make RS A = RL iterations to reach the solution for converged cases.
and RS B = RS , and that creates the following equation for esti-
mation of RS A and RS B : Table 1
System speciﬁcations for test case #1 
vn − vn−1
RS A =− 2 2
(4) Voltage source RMS: 120 V, RS = 0.02
2 − in−1
2 Load #1 R = 12.3 , L = 0.03138 H
Load #2 R = 14.52 , L = 0.031468 H
vn − vn−1 Load #3 R = 14.52 , L = 0.031468 H
RS B =− 1 1
in − in−1
Transmission line R = 0.34 , L = 0.005913 H
J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131 1129
Fig. 6. Two bus system.
MAPE for distributed simulation
Case Step size (s) Linear (%) Non-adjusted stabilizing resistance Adjusted stabilizing resistance (%)
One load 1e-6 0.3312 0.0248% 0.0248
1e-5 2.741 0.0248% 0.0248
1e-4 15.733 Did not converge 0.0248
1e-3 422.338 Did not converge 0.0248
Three load 1e-6 0.322 0.0323% 0.0323
1e-5 3.774 0.0928% 0.0928
Fewer iterations lead to less computation time. The adjusted linear method can have better performance when the correction
stabilizing resistance method always converges and is not time based on the previous calculation is introduced for next step sim-
step sensitive. For the linear method, the MAPE keeps grow- ulation, using a state estimator. The accuracy decreases when
ing exponentially when the time step size increase, because the the number of components in simulation increases for all three
larger the step size reduces accuracy of the discretized RCF methods. This decrease occurs because the VTB solver only has
model for the components in simulation. Without the correction control over the voltage proﬁle, while the currents are calculated
in the minor step for the non-linear method, the error accumu- based on individual device model; thus the truncation error can
lates and ﬁnally makes the simulation results inaccurate. The accumulate.
Fig. 7. 18-Bus ship power system distributed simulation.
1130 J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131
Power ﬂow solution from VTB
Bus no. Voltage magnitude (pu) Voltage angle (degree) Real power Reactive power
generation, P (MW) generation, Q (MVar)
1 1.02 0 5.722 1.292
2 1.02 0 6.329 0.1424
3 1.02 0 6.112 0.1391
4 1.02 0 5.896 0.1335
5 1.0199 −0.0054
6 1.0199 −0.0053
7 1.0197 −0.0136
8 0.9866 −10.5502
9 0.9865 −10.5638
10 1.0198 −0.0111
11 0.9873 −10.6174
12 0.9872 −10.6311
13 1.0196 −0.0167
14 0.9872 −10.4745
15 0.9870 −10.4911
16 1.0197 −0.015
17 0.9851 −10.7343
18 0.9849 −10.7502
5.2. Test case #2 demonstrate the capability of natural level coupling agent model.
Generally, a smaller time step makes the distributed simulation
In this section, a more complicated test case is developed more accurate than it is with non-distributed simulation. With
based on a new benchmark test system of a ship’s distribu- a small enough time step, little difference exists between non-
tion network described in paper . This system is an 18-bus adjusted stabilizing element and adjusted stabilizing element
shipboard power system. It has six polynomial loads and four methods. However, when the time step increases, an inappropri-
generators. This shipboard power system is conﬁgured in two ate stabilizing element will make the simulation diverge. Since
zones. The two zones are weakly coupled through transmission limited information about the external system is available, it
lines 5 and 6. Therefore this line is selected as the decoupling is difﬁcult to guess an appropriate stabilizing element. At this
point. Here single-phase client/server models are used for the point, the advantage of adjusted stabilizing element is obvious.
distributed simulation in VTB, shown in Fig. 7. It can identify the optimal stabilizing element for the system to
Here, the PQ load model is a non-linear model dependent make the simulation converge. This characteristic is especially
on the terminal voltage’s RMS value, as described in detail in useful when the system is nonlinear or there is a time varying
paper . Its performance has been validated. The transformers element.
and lines are represented by an inductor since reference  Also, as noticed from the two-bus test cases for natural cou-
provides only R and L parameters. This test, besides comparing pling model, a single load test has better performance than
the MAPE of current ﬂowing through the tie line (from bus 5 multiple loads. This difference is caused by the VTB solver,
to bus 6), also compares the power ﬂow solution between the where voltage is taken as the state variable and is controlled
distributed simulation and non-distributed simulation. within a given tolerance. The current of each element is com-
After data processing, the MAPE is within 0.5% deviation puted separately. However the truncation error can make the sum
from the non-distributed simulation. Table 3 shows the power of the current mismatch bigger. So the number of branches at the
ﬂow solution using VTB. Compared to the result from the power coupling point causes the simulation to converge with different
ﬂow solution in paper , the voltage magnitude is the same accuracies.
while the voltage angle deviation is within 0.014 degree. This For a steady state test, satisfactory results are achieved for the
error diminishes because in steady state, the error integrated to developed electric ship test case. This demonstrates the natural
zero. The steady state solution in distributed simulation matches coupling model’s capability to work with multiple sources and
the solution in non-distributed simulation . The close voltage multiple loads. Testing the natural coupling model further with
match indicates the simulation conﬁrms power ﬂow analysis and a larger scale power system will be possible when VTB-Pro is
can be used for transient analysis. released, as it can accommodate more nodes and is more stable.
6. Summary 7. Conclusion
This paper compares three methods to represent the external In this paper, a distributed simulation algorithm using the
network, linear, non-adjusted stabilizing element and adjusted generalized VI coupling method is proposed to handle natu-
stabilizing element. Numerical analysis techniques demonstrate ral coupling. With a VI coupling model developed in VTB,
the convergence of the algorithm. Test cases are developed to numerical analysis for convergence and computational stabil-
J. Wu et al. / Electric Power Systems Research 77 (2007) 1124–1131 1131
ity is performed. The adjusted stabilizing resistance method is  G. Cokkinides and B. Beker, VTB Model Developer’s Guide. Vir-
proposed to achieve better convergence and stability. Test cases tual Test Bed Project, Department of Electrical Engineering, Univer-
with different network conﬁgurations are developed. The dis- sity of South Carolina, Columbia, SC, 29208, Available at http://vtb.
engr.sc.edu/modellibrary/modeldev.asp, Mar. 2001.
tributed simulation performance is analyzed in time domain and  W. Gao, E.V. Solodovnik, R.A. Dougal, Symbolically aided model devel-
steady state. The results demonstrate good performance of this opment for an induction machine in Virtual Test Bed, IEEE Trans. Energy
technique. Convers. 19 (1) (2004).
On the other hand, current models for distributed simulation  Remote Procedure Call, http://www.microsoft.com/technet/prodtechnol/
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VTB is progressing to provide a COM server, a more promising tributed simulation, in: Proceedings of North America Power System Con-
feature is coming. Therefore, in the next step in our research we ference, Iowa, October 2005.
will migrate the models from RPC to a COM server and test it  J. Wu, Y. Huang, W. Gao, N.N. Schulz, Power system load modeling in vir-
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Acknowledgments power systems, IEEE Trans. Indus. Appl. 40 (5) (2004) 1183–1190.
This research work has been made possible through the sup- Jian Wu is a Ph.D. candidate in electrical and computer engineering at Mis-
port of the DoD MURI Fund # N00014-04-1-0404 and ONR sissippi State University (MSU). She received B.S.E.E. and M.S.E.E. from
ZheJiang University, China. She has worked with HangZhou Municipal Power
Fund #N00014-02-1-0623. Company. She received another M.S.E.E. from MSU in 2004. Her ﬁelds of
interest include the power system modeling and simulation, common informa-
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tual test bed, in: Proceedings of Power Electronics Specialists Conference, State University. His current research interests include power system model-
Florida, November 2001. ing and simulation, alternative power systems, intelligent system applications in
 S.Y.R. Hui, K.K. Fung, Fast decoupled simulation of large power electronic power engineering, electric and hybrid electric propulsion systems, and electric
systems using new two-port companion link models, IEEE Trans Power machinery and drive.
Electron. 12 (1997) 462–473.