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SPICE application in the study of the behaviour of multi-state systems described by Markov models
Microelectron.Reliab.,Vol. 37, No. 4, pp. 609-613, 1997 Copyright 0 1996ElsevierScience Ltd Printed in Great Britain. All rights reserved
Pergamon
0026-2714/97
$17.00+.00
PII: SOO26-2714(%)00077-7
SPICE APPLICATION IN THE STUDY OF THE BEHAVIOUR OF MULTI-STATE SYSTEMS DESCRIBED BY MARKOV MODELS MUHAMMAD TAHER ABUELMA’AT’TIT and ISA SALMAN QAMBARS t King Fahd University of Petroleum and Minerals, Box 203, Dhahran 31261, Saudi Arabia $.University of Bahrain, P.O. Box 32038 Isa Town, Bahrain
(Received for publication 10 April 1996) Abstract-This paper shows the applicability of the SPICE circuit analysis computer program for studying the behaviour of multi-state systems described by Markov models. Two examples are considered and the results are compared with those obtained by the Laplace transform and the flow graph methods. Agreement between the three techniques is excellent. Copyright e 1996 Elsevier Science Ltd.
k,(t) = a,,P,(t)
INTRODUCTION
In a recent
publication, Qambar [l] described procedure for using the graph theory
a
in studying the behaviour of multi-state systems described by Markov models. Although this procedure can accurately predict the steady-state behaviour of a given system, no attempt has been made to investigate the transient response of the system. Moreover, while calculating the steady-state behaviour of four-state and five-state models is simple and can be done manually, it appears that for more sophisticated N-state models the use of a personal computer, to solve the resulting N simultaneous equations, may be inevitable. On the other hand, the SPICE circuit analysis program is a general-purpose circuit simulation program which can be used for d.c., transient and a.c. analysis. Although it was primarily developed for integrated circuit analysis, it is now widely used in many non-integrated circuit applications [2-51. No attempt, however, has been made to exploit SPICE capabilities in studying the behaviour of multi-state systems described by Markov models. Such a study would be useful for obtaining an understanding of how the various elements of a system will interact and how the overall system will perform under the transient and steady-state conditions. The intent of this paper is to show how the capabilities of SPICE can be exploited for predicting the transient and the steady-state behaviour of N-state systems described by Markov models. simple
- a,,P,(t)
+
+ a,,P,(t)
+ .
- aNNPN(t) (lc)
(lb)
and P,(t) = alNPl(t) + a,,P,(t)
where k,,(t) represents the differentiation nth-state probability P.(t)
of the
al,=a,z+a,,+~.‘+a,, 42 = a21 + a23 + ...
+ u2,&
and cl&IN =
apJ1 +
aN2
+
. . . +
uhf,N-l)
where N is the number of states. According to Ref. [l], an appropriate source for the model must be identified and this source-state is recommended to be the equation having the larger number of terms. Then the source equation is replaced by the following equation: 2 P.=l.
(2)
n=1
In the following it will be assumed that eqn (la) is the source equation. After replacing eqn (la) by eqn (2), and rearranging the resulting set of equations, eqns (1 a-c) can be rewritten as PI(t) = 1 -
i
P,(l)
(34
In=2
P2(t) =
f m=l.m#2
%mPrn(~)
+
P2P2U)
(3b)
h~dt).
(3c)
PROPOSED METHOD
PN(t)=
Following the procedure described in Ref. [l], the differential equations for the Markov model of a given system can be expressed as PI(t) = -a, ,P,(t) + a,,P,(t)
+
+ a,,P,(t)
i
ahp,(t) +
l?l=l,lll+N
The electrical equivalent circuit of eqn (3) is shown in Fig. 1. In the circuit, the current is the analog of P,(t), the voltage across the inductor is the analog of
(la) 609
610
M. T. Abuelma’atti
J$n
and I. S. Qambar
n
Fig. 1. Equivalent
circuit
____
n
of eqn (3). R, = 10, n = 1 - N, L, = 1 H, n = 1 - N.
p,(t) and the voltage-controlled current sources (VCCSs) represent the right-hand sides of eqn (3). The polynomial feature of SPICE will be used to represent these VCCSs [6]. In the SPICE input file of the circuit of Fig. 1, the initial conditions of P,,(t), n = 1 - N are realized by the IC option, the transient analysis option TRAN will be used to perform the transient analysis of the circuit of Fig. 1 and the SPICE output file will contain the probabilities P,(t), n = 1 - N as a function of time.
k,(t) = 0005OP,(t) - 0.0041!‘,(t) + 0.0250PJt) P$) = O.O008P,(t) - 0.0250PJt)
(4b) (4c)
&(t) = O.O300P,(t) + O.O250P,(t) - O.O4OOP,(t). (4dI
Following the procedure described in the previous section, eqns (4a-d) can be rewritten as P,(f) = 1 - PJC) - Pj(f) - PJ(f)
(W
Pz(t) = 0.5494505 - 0,5494505P,(f)
Examples
To illustrate the use of the circuit of Fig. 1 and to establish the accuracy of the proposed method, consider the following two examples. Example 1. Four-state model. In this example we consider the four-state model for the rapid start units given in Ref. [7] and described by the following set of differential equations [ 11:
0
- O.O714285P,(t) - 14.285714ek(t).
Fig. 3. SPICE
(5d)
(4a)
circuit
of the four-state
6 7 1.0 -1.0 -1.0 -1.0 6 7 0.5494505 -109.8901 0.0 0.032 -40.0 7 0 0.4285714 -0.4285714
V(4,5)
(5c)
PJt) = 0.4285714 - 0.4285714P,(t)
0
Fig. 2. Equivalent
Rl 1 0 1 R2 2 3 1 R3 4 5 1 R4 6 I 1 L23OlIC=O L3 5 0 1 IC=O L47OlIC=O Cl 0 1 poly(3) 2 3 4 5 (32 0 2 poly(3) 3 0 4 5 G3 0 4 poly(2) 2 3 5 0 G4 0 6 poly(3) 2 3 4 5 .tran 20 1000 UIC .probe .print tran V(l, 0) V(2,3) .end
P3(t) = O.O320P,(t) - 40.0&t)
(5b)
The equivalent cirucit of eqn (5) is shown in Fig. 2 and its SPICE input file is shown in Fig. 3 where the initial conditions of P.(t), n = 2-4 are set to zero. The transient results are shown in Fig. 4 and the
Pt(t) = -O.O350P,(t) + O.O033P,(f) + O.O150P,(f)
u
+ 2.1978021P,(t) - 109.8901&)
model.
-0.5494505 -0.0714285
V(6.7)
input file for the circuit of Fig. 2.
2.1978021 -14.285714
SPICE
611
application reliability Temperature:
Date/Time ran: 03116196 14:31:30
27.0
(F) B: \ISA.DAT
1 .OA
0.6A 0 I (rl) 0
I (r2)
V I (r3)
0.4A
A I (r4)
0.2A
OA OS
0.2Ks
0.6Ks
0.4Ks
0.8Ks
1.OKs
Time Date: March 17, 1996
Page 1
Fig. 4. PROBE
Table
I. Steady-state
probabilities
output
Time: 13:46:23
giving the probabilities
of the four-state
model
of the four-state
model.
Following the procedure described in the previous section, eqns (6a-e) can be rewritten as
Laplace Steady-state
probability
simulation
transform method [I]
Flow graph method [I]
0.1 I51 0.7592 0.0243 0.1015
0.1151 0.7592 0.0243 0.1015
0.11507 0.75914 0.02429 0.10148
SPICE
PI(t) = I - PJt) - P3(t) - P4(t) - P,(t) Pz(t) = 0.5452067 - 0.5452067P,(t)
PI PZ pj p4
- 0.4884143P,(t)
(7a)
- 0.5452067PJt)
- 22.716946+,(t)
(7b)
PJt) = O.O006666P,(t) - 33.333333+,(t)
(7c)
P4(t) = 0.117647 - O.l17647P,(t) + 0.3235294PJt) steady-state
in Table 1 are the results obtained using the flow graph and the Laplace transform methods [l]. From Table 1 it can be seen that the results obtained using SPICE simulation are in excellent agreement with the results obtained using the Laplace transform method. Example 2. The five-state method. In this example we consider the five-state model for the hot-reserve units given in Ref. [7] and described by the following set of differential equations [ 11:
+ 0003P,(t)
(6a)
= O.O24P,(t) - O.O2002P,(t) + O.O025P&)
(6b)
P,(t) = 0.00002P,(t) - 0.030PJt)
(6~)
&f)
= O.O08P,(t) + O.O3OP&) - O.O60P,(t)
P,(t) = 0.025PJt)
- O.O055P,(t).
P,(t) = 4.5454545PJt)
(7d)
- 181.818181~&).
(7e)
The equivalent circuit of eqn (7) is shown in Fig. 5 and its SPICE input file is shown in Fig. 6 where the initial conditions for P,,(t), n = 2-5 are set to zero. The transient results are shown in Fig. 7 and the steady-state results are shown in Table 2. Also shown in Table
2 are the results
obtained
using the flow
Table 2. Steady-state probabilities of the five-state model
PI(t) = -O.O32P,(t) + O.O20P,(t) + o.o35P,(t)
&t)
- O.l17647P,(t) - 14.705882eJt)
results are shown in Table 1. Also shown
(6d) (6e)
Steady-state probability
SPICE simulation
Laplace transform method [l]
Flow graph method [I]
PI p2
0.331500 0.422600 0.000282 0.044340 0.201300
0.331500 0.422600 0.000282 0.044340 0.201300
0.331419 0.422468 0.000281 0.044330 0.201500
p3
P4 PS
612
M. T. Abuelma’atti
Fig 5. Equivalent
Rl 1 0 1 R2 2 3 1 R3 4 5 1 R4 6 7 1 R5 8 9 1 L2301IC=O L35011C=O L47OlIC=O L59OlIC=O Gl 0 1 poly(4) 2 3 4 5 G2 0 2 poly(4) 4 5 6 7 G3 0 4 poly(2) 2 3 5 0 G4 0 6 poly(4) 2 3 4 5 G5 0 8 poly(2) 6 7 9 0 .tran 20 1000 UIC .print tran V(l, 0) V(2,3) .probe .end
and I. S. Qambar
circuit
of the five-state
model
6 7 8 9 1.0 -1.0 - 1.0 - 1.0 - 1.0 8 9 3 0 0.5452067 -0.5452067 -0.5452067 -0.4884143 -22.716946 0.0 0.0006666 -33.333333 8 9 7 0 0.117647 -0.117647 0.3235294 -0.117647 -14.705882 0.0 4.5454545 -181.81818 V(4,5)
V(6,7)
Fig. 6. SPICE
V(8,9)
input file for the circuit
of Fig. 5.
reliability Date/Time run: 03/17/96
Temperature: 27.0
13: lo:40
(A) B: \QAMBAR.DAT
1.OA
0.8A
0.6A
0
I (rl)
0
I (r2)
v
I (r3)
A
I
0
I (r5)
(r4)
0.4A
0.2A
OA OS
0.2Ks
0.6Ks
0.4Ks
0.8Ks
1.OKs
Time
Fig. 7. PROBE
Time: 13:17:55
Page 1
Date: March 17. 1996
output
giving the probabilities
of the four-state
model.
SPICE application graph and the Laplace transform methods. From Table 2 it can be seen the results obtained using SPICE simulation are in excellent agreement with the results obtained using the Laplace transform method.
CONCLUSION From the analysis presented and the examples considered it is evident that the SPICE circuit simulation program is a useful tool for studying the behaviour of multi-state systems described by Markov models. Steady-state and transient solutions for the different probabilities involved are readily available. The procedure for obtaining the electrical equivalent circuit of a given set of linear differential equations is general and can be applied for any number of equations. The choice of a particular equation as the source is not critical and any differential equation can be assigned as the source-state. The use of SPICE is
expected
to
provide
a
better
insight
into
the
613
performance of multi-state Markov models.
systems
described
by
REFERENCES 1. Qambar, I. S., Flow graph development method. Microelectron. Realiab., 1993, 33, 1387-1395. 2. Prigozy, S., Novel applications of SPICE in engineering eduction. IEEE Trans. Educ., 1989, 32, 35-38. 3. Laghari, J. R., Suthar, L., and Cygan, S., SPICE applications in high voltage engineering education. Comput. Educ., 1990, 14, 455-462. 4. Paul, C. R., Analysis of Linear Circuits. McGraw-Hill, New York 1989. 5. Hmurcik, L. V., Hettinger, M., Gottschalck, K. S., and Fitchen, F. C., SPICE applications to an undergraduate electronic program. IEEE Trans. Educ., 1990. 33, 183-189. 6. Kumar, K. B., Novel techniques to solve sets of coupled differential equations with SPICE. IEEE Circuits Devices Magazine, 1991. 7, 11-14. 7. Billinton, R., and Chowdhury, N. A., Operating reserve assessment in interconnected generating systems. IEEE Trans. Power Syst.. 1988, 3, 1479-1487.
Pergamon
0026-2714/97
$17.00+.00
PII: SOO26-2714(%)00077-7
SPICE APPLICATION IN THE STUDY OF THE BEHAVIOUR OF MULTI-STATE SYSTEMS DESCRIBED BY MARKOV MODELS MUHAMMAD TAHER ABUELMA’AT’TIT and ISA SALMAN QAMBARS t King Fahd University of Petroleum and Minerals, Box 203, Dhahran 31261, Saudi Arabia $.University of Bahrain, P.O. Box 32038 Isa Town, Bahrain
(Received for publication 10 April 1996) Abstract-This paper shows the applicability of the SPICE circuit analysis computer program for studying the behaviour of multi-state systems described by Markov models. Two examples are considered and the results are compared with those obtained by the Laplace transform and the flow graph methods. Agreement between the three techniques is excellent. Copyright e 1996 Elsevier Science Ltd.
k,(t) = a,,P,(t)
INTRODUCTION
In a recent
publication, Qambar [l] described procedure for using the graph theory
a
in studying the behaviour of multi-state systems described by Markov models. Although this procedure can accurately predict the steady-state behaviour of a given system, no attempt has been made to investigate the transient response of the system. Moreover, while calculating the steady-state behaviour of four-state and five-state models is simple and can be done manually, it appears that for more sophisticated N-state models the use of a personal computer, to solve the resulting N simultaneous equations, may be inevitable. On the other hand, the SPICE circuit analysis program is a general-purpose circuit simulation program which can be used for d.c., transient and a.c. analysis. Although it was primarily developed for integrated circuit analysis, it is now widely used in many non-integrated circuit applications [2-51. No attempt, however, has been made to exploit SPICE capabilities in studying the behaviour of multi-state systems described by Markov models. Such a study would be useful for obtaining an understanding of how the various elements of a system will interact and how the overall system will perform under the transient and steady-state conditions. The intent of this paper is to show how the capabilities of SPICE can be exploited for predicting the transient and the steady-state behaviour of N-state systems described by Markov models. simple
- a,,P,(t)
+
+ a,,P,(t)
+ .
- aNNPN(t) (lc)
(lb)
and P,(t) = alNPl(t) + a,,P,(t)
where k,,(t) represents the differentiation nth-state probability P.(t)
of the
al,=a,z+a,,+~.‘+a,, 42 = a21 + a23 + ...
+ u2,&
and cl&IN =
apJ1 +
aN2
+
. . . +
uhf,N-l)
where N is the number of states. According to Ref. [l], an appropriate source for the model must be identified and this source-state is recommended to be the equation having the larger number of terms. Then the source equation is replaced by the following equation: 2 P.=l.
(2)
n=1
In the following it will be assumed that eqn (la) is the source equation. After replacing eqn (la) by eqn (2), and rearranging the resulting set of equations, eqns (1 a-c) can be rewritten as PI(t) = 1 -
i
P,(l)
(34
In=2
P2(t) =
f m=l.m#2
%mPrn(~)
+
P2P2U)
(3b)
h~dt).
(3c)
PROPOSED METHOD
PN(t)=
Following the procedure described in Ref. [l], the differential equations for the Markov model of a given system can be expressed as PI(t) = -a, ,P,(t) + a,,P,(t)
+
+ a,,P,(t)
i
ahp,(t) +
l?l=l,lll+N
The electrical equivalent circuit of eqn (3) is shown in Fig. 1. In the circuit, the current is the analog of P,(t), the voltage across the inductor is the analog of
(la) 609
610
M. T. Abuelma’atti
J$n
and I. S. Qambar
n
Fig. 1. Equivalent
circuit
____
n
of eqn (3). R, = 10, n = 1 - N, L, = 1 H, n = 1 - N.
p,(t) and the voltage-controlled current sources (VCCSs) represent the right-hand sides of eqn (3). The polynomial feature of SPICE will be used to represent these VCCSs [6]. In the SPICE input file of the circuit of Fig. 1, the initial conditions of P,,(t), n = 1 - N are realized by the IC option, the transient analysis option TRAN will be used to perform the transient analysis of the circuit of Fig. 1 and the SPICE output file will contain the probabilities P,(t), n = 1 - N as a function of time.
k,(t) = 0005OP,(t) - 0.0041!‘,(t) + 0.0250PJt) P$) = O.O008P,(t) - 0.0250PJt)
(4b) (4c)
&(t) = O.O300P,(t) + O.O250P,(t) - O.O4OOP,(t). (4dI
Following the procedure described in the previous section, eqns (4a-d) can be rewritten as P,(f) = 1 - PJC) - Pj(f) - PJ(f)
(W
Pz(t) = 0.5494505 - 0,5494505P,(f)
Examples
To illustrate the use of the circuit of Fig. 1 and to establish the accuracy of the proposed method, consider the following two examples. Example 1. Four-state model. In this example we consider the four-state model for the rapid start units given in Ref. [7] and described by the following set of differential equations [ 11:
0
- O.O714285P,(t) - 14.285714ek(t).
Fig. 3. SPICE
(5d)
(4a)
circuit
of the four-state
6 7 1.0 -1.0 -1.0 -1.0 6 7 0.5494505 -109.8901 0.0 0.032 -40.0 7 0 0.4285714 -0.4285714
V(4,5)
(5c)
PJt) = 0.4285714 - 0.4285714P,(t)
0
Fig. 2. Equivalent
Rl 1 0 1 R2 2 3 1 R3 4 5 1 R4 6 I 1 L23OlIC=O L3 5 0 1 IC=O L47OlIC=O Cl 0 1 poly(3) 2 3 4 5 (32 0 2 poly(3) 3 0 4 5 G3 0 4 poly(2) 2 3 5 0 G4 0 6 poly(3) 2 3 4 5 .tran 20 1000 UIC .probe .print tran V(l, 0) V(2,3) .end
P3(t) = O.O320P,(t) - 40.0&t)
(5b)
The equivalent cirucit of eqn (5) is shown in Fig. 2 and its SPICE input file is shown in Fig. 3 where the initial conditions of P.(t), n = 2-4 are set to zero. The transient results are shown in Fig. 4 and the
Pt(t) = -O.O350P,(t) + O.O033P,(f) + O.O150P,(f)
u
+ 2.1978021P,(t) - 109.8901&)
model.
-0.5494505 -0.0714285
V(6.7)
input file for the circuit of Fig. 2.
2.1978021 -14.285714
SPICE
611
application reliability Temperature:
Date/Time ran: 03116196 14:31:30
27.0
(F) B: \ISA.DAT
1 .OA
0.6A 0 I (rl) 0
I (r2)
V I (r3)
0.4A
A I (r4)
0.2A
OA OS
0.2Ks
0.6Ks
0.4Ks
0.8Ks
1.OKs
Time Date: March 17, 1996
Page 1
Fig. 4. PROBE
Table
I. Steady-state
probabilities
output
Time: 13:46:23
giving the probabilities
of the four-state
model
of the four-state
model.
Following the procedure described in the previous section, eqns (6a-e) can be rewritten as
Laplace Steady-state
probability
simulation
transform method [I]
Flow graph method [I]
0.1 I51 0.7592 0.0243 0.1015
0.1151 0.7592 0.0243 0.1015
0.11507 0.75914 0.02429 0.10148
SPICE
PI(t) = I - PJt) - P3(t) - P4(t) - P,(t) Pz(t) = 0.5452067 - 0.5452067P,(t)
PI PZ pj p4
- 0.4884143P,(t)
(7a)
- 0.5452067PJt)
- 22.716946+,(t)
(7b)
PJt) = O.O006666P,(t) - 33.333333+,(t)
(7c)
P4(t) = 0.117647 - O.l17647P,(t) + 0.3235294PJt) steady-state
in Table 1 are the results obtained using the flow graph and the Laplace transform methods [l]. From Table 1 it can be seen that the results obtained using SPICE simulation are in excellent agreement with the results obtained using the Laplace transform method. Example 2. The five-state method. In this example we consider the five-state model for the hot-reserve units given in Ref. [7] and described by the following set of differential equations [ 11:
+ 0003P,(t)
(6a)
= O.O24P,(t) - O.O2002P,(t) + O.O025P&)
(6b)
P,(t) = 0.00002P,(t) - 0.030PJt)
(6~)
&f)
= O.O08P,(t) + O.O3OP&) - O.O60P,(t)
P,(t) = 0.025PJt)
- O.O055P,(t).
P,(t) = 4.5454545PJt)
(7d)
- 181.818181~&).
(7e)
The equivalent circuit of eqn (7) is shown in Fig. 5 and its SPICE input file is shown in Fig. 6 where the initial conditions for P,,(t), n = 2-5 are set to zero. The transient results are shown in Fig. 7 and the steady-state results are shown in Table 2. Also shown in Table
2 are the results
obtained
using the flow
Table 2. Steady-state probabilities of the five-state model
PI(t) = -O.O32P,(t) + O.O20P,(t) + o.o35P,(t)
&t)
- O.l17647P,(t) - 14.705882eJt)
results are shown in Table 1. Also shown
(6d) (6e)
Steady-state probability
SPICE simulation
Laplace transform method [l]
Flow graph method [I]
PI p2
0.331500 0.422600 0.000282 0.044340 0.201300
0.331500 0.422600 0.000282 0.044340 0.201300
0.331419 0.422468 0.000281 0.044330 0.201500
p3
P4 PS
612
M. T. Abuelma’atti
Fig 5. Equivalent
Rl 1 0 1 R2 2 3 1 R3 4 5 1 R4 6 7 1 R5 8 9 1 L2301IC=O L35011C=O L47OlIC=O L59OlIC=O Gl 0 1 poly(4) 2 3 4 5 G2 0 2 poly(4) 4 5 6 7 G3 0 4 poly(2) 2 3 5 0 G4 0 6 poly(4) 2 3 4 5 G5 0 8 poly(2) 6 7 9 0 .tran 20 1000 UIC .print tran V(l, 0) V(2,3) .probe .end
and I. S. Qambar
circuit
of the five-state
model
6 7 8 9 1.0 -1.0 - 1.0 - 1.0 - 1.0 8 9 3 0 0.5452067 -0.5452067 -0.5452067 -0.4884143 -22.716946 0.0 0.0006666 -33.333333 8 9 7 0 0.117647 -0.117647 0.3235294 -0.117647 -14.705882 0.0 4.5454545 -181.81818 V(4,5)
V(6,7)
Fig. 6. SPICE
V(8,9)
input file for the circuit
of Fig. 5.
reliability Date/Time run: 03/17/96
Temperature: 27.0
13: lo:40
(A) B: \QAMBAR.DAT
1.OA
0.8A
0.6A
0
I (rl)
0
I (r2)
v
I (r3)
A
I
0
I (r5)
(r4)
0.4A
0.2A
OA OS
0.2Ks
0.6Ks
0.4Ks
0.8Ks
1.OKs
Time
Fig. 7. PROBE
Time: 13:17:55
Page 1
Date: March 17. 1996
output
giving the probabilities
of the four-state
model.
SPICE application graph and the Laplace transform methods. From Table 2 it can be seen the results obtained using SPICE simulation are in excellent agreement with the results obtained using the Laplace transform method.
CONCLUSION From the analysis presented and the examples considered it is evident that the SPICE circuit simulation program is a useful tool for studying the behaviour of multi-state systems described by Markov models. Steady-state and transient solutions for the different probabilities involved are readily available. The procedure for obtaining the electrical equivalent circuit of a given set of linear differential equations is general and can be applied for any number of equations. The choice of a particular equation as the source is not critical and any differential equation can be assigned as the source-state. The use of SPICE is
expected
to
provide
a
better
insight
into
the
613
performance of multi-state Markov models.
systems
described
by
REFERENCES 1. Qambar, I. S., Flow graph development method. Microelectron. Realiab., 1993, 33, 1387-1395. 2. Prigozy, S., Novel applications of SPICE in engineering eduction. IEEE Trans. Educ., 1989, 32, 35-38. 3. Laghari, J. R., Suthar, L., and Cygan, S., SPICE applications in high voltage engineering education. Comput. Educ., 1990, 14, 455-462. 4. Paul, C. R., Analysis of Linear Circuits. McGraw-Hill, New York 1989. 5. Hmurcik, L. V., Hettinger, M., Gottschalck, K. S., and Fitchen, F. C., SPICE applications to an undergraduate electronic program. IEEE Trans. Educ., 1990. 33, 183-189. 6. Kumar, K. B., Novel techniques to solve sets of coupled differential equations with SPICE. IEEE Circuits Devices Magazine, 1991. 7, 11-14. 7. Billinton, R., and Chowdhury, N. A., Operating reserve assessment in interconnected generating systems. IEEE Trans. Power Syst.. 1988, 3, 1479-1487.