Reliability study of two engineering models using LU decomposition

Reliability Engineering and System Safety 64 (1999) 359–364

Reliability study of two engineering models using LU decomposition I.S. Qamber Department of Electrical Engineering, University of Bahrain, PO Box 33831, Isa Town, Bahrain Received 6 July 1997; accepted 4 September 1998

Abstract This paper presents two different methods — LU decomposition and Runge–Kutta — for reliability engineering assessment. These methods are used to calculate the steady-state probabilities and frequencies of two different engineering models. The effect of the different methods is shown on the simple numerical examples by comparing the steady-state probabilities and frequencies of both models. The paper concludes that the LU decomposition method is useful to practicing engineers and students of reliability concepts. Also, the final results for both methods is almost the same. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Power plant model; Steady-state probability; Steady-state frequency; Markov model; LUD; Runge–Kutta methods

Nomenclature

l ij fi i LUD n Pi SSP SSF

transition rate from state i to state j steady-state frequency of encountering state i represents the i-th state Lower–upper decomposition number of states steady-state probability of state i steady-state probability steady-state frequency

1. Introduction The steady-state probability and frequency can be calculated by different techniques such as developed graph method [1,2], the solution of simultaneous equations [3,4] and the frequency-balance approach [4,5]. In this paper, the LU decomposition [6] is used to calculate the steady-state probabilities and frequencies. This method — to my knowledge and from the literature review [7–12] — has not been used to calculate the steady-state probabilities and frequencies. In Ref. [1], the author demonstrates the use of graph theory to find the steady-state behavior of multi-state systems described by Markov models. Two applications are demonstrated and compared with results obtained by the Laplace transform technique. The developed method was also compared with another method which uses graph theory to calculate the steady-state probabilities. Compared with the developed method [1,2], the LU decomposition

method is faster and easier. Moreover, while a number of steps are eliminated from the calculation, the obtained accuracy of methods is almost the same. In Ref. [2], the authors use the Markov process to determine both the steady-state and transient general formulas for an arbitrary three-state model using the Laplace Transforms method. The calculations were carried-out on four models. Then, the flow-graph method developed is also applied on the four models to find the steady-state probabilities. These steady-state probabilities obtained by the flow-graph method are then compared with the results obtained using the Laplace Transform method when the steady-state limit has been reached. The accuracy obtained in both cases are almost the same. In Ref. [3], the authors describes a general purpose graphical approach for obtaining steady-state availability and frequency expressions from a flow graph based on the Markov model. In Ref. [3], self-loops and a number of sources and sinks are selected for their proposed approach. In the present paper the LU decomposition is adopted to find the steady-state probabilities and frequencies. However, in Refs.[1] and [2] no attempt has been made to calculate the steady-state frequencies. For large sparse system of linear equations, a parallel solution is presented. The solution is based on a parallel LU decomposition technique on a shared multiprocessor memory. The author in Ref. [7] worked on the switching between parallel and single pivoting steps to assure numerical stability. In the same paper [7] the algorithms, their implementation and performance of the solution methods, on actual multiprocessor are presented and compared. The authors in Ref. [8] deal with sparse matrix

0951-8320/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0951-832 0(98)00082-9

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Fig. 1. The seven-state model state-space diagram.

factorization algorithms for general problems. These problems are typically characterized by irregular memory access patterns that limit their performance on parallelvector supercomputers. One advantage is the repetitive structure taken in the matrix. The performance is compared with the classical multifrontal method. The authors in their paper [9] for many cases applied a preconditioning technique and it shows grid-independent convergence. This technique requires only an ordering of the unknowns based on the different levels of multigrid and an incomplete LU decomposition based on a drop tolerance. The authors in Ref. [10] present a new approach for solving the magnetic field integral equation for the current induced on an infinite perfectly conducting rough surface. By splitting the propagator matrix into contributions from the left and the right of the point of observation, a second Table 1 Sixteen-state model State

Unit #1

Unit #2

Unit #3

Unit #4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

On On On On On On On On Off Off Off Off Off Off Off Off

On On On On Off Off Off Off On On On On Off Off Off Off

On On Off Off On On Off Off On On Off Off On On Off Off

On Off On Off On Off On Off On Off On Off On Off On Off

kind integral equation can be formed with a new born term and a new kernel. The unknown currents can be determined more rapidly and significantly less storage requirements than conventional LU decomposition. The author, in Ref. [11], deals with an electromagnetics problem. In order to solve these problems, the magnitude of the problem is demonstrated by looking at the time required today to solve large complex dense linear systems of various sizes utilizing current LU decomposition techniques. The authors, in their paper [12], present a parallel processing algorithm to solve efficiently the linear network equations for power flow calculation on a transputer based multiprocessor system. In the proposed approach LU decomposition and forward/backward substitution are executed by parallel processing. The computation time for a parallel algorithm was measured and compared with that obtained by the sequential algorithm. The Runge–Kutta technique is well documented in [6,13]. This method is useful for determining the steadystate and transient probabilities of any system. Also, LU decomposition is very useful for determining the steadystate probabilities and frequencies. The purpose of this paper is to present LU decomposition (LUD) method [6] for obtaining the steady-state probabilities and frequencies by means of two different engineering examples. The results will be compared with the results obtainable using the Runge–Kutta method.

2. The proposed method The proposed method — LUD — is well documented in [6]. In the present paper, the LUD is applied to calculate the steady-state probabilities and frequencies after some modifications. One of these modifications is by replacing one of

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Fig. 2. The system block diagram and the estimated repair and failure times.

the model equations by X Pi ˆ 1:0 ᭙ i ˆ 1; 2; 3; …; n

…1†

to satisfy the condition that the summation of the probabilities must be equal to one. Then, the steady-state frequencies are calculated using the following formula: X  lij Pi ᭙ j ˆ 1; 2; 3; …; n and j 苷 i …2† fi ˆ This method, going through the literature, it seems that nobody has used it for this type of reliability engineering application. 3. Numerical examples In the present section, two models will be considered (seven- and 16-state models). The seven-state model (Fig. 1) [14] represents three-unit system. The sixteen-state model (Table 1) [15] represents a combination of four generation units for part of one of the power stations in Bahrain. The LUD and the Runge–Kutta methods are applied on both models. The results are given below:

Example 1. As a first example consider the seven-state model. This model is taken from a paper written by Barlow and Hunter [14]. This system consisting of two transmitters, A and B, and power supply C providing a high voltage with a modulator and modulation programmer. The two transmitters are run from the same power supply C. If one of the two transmitters fails, the other one can perform the system task satisfactorily but with a decrease in expected time. The system block diagram and the estimated repair and failure times of each unit are presented in Fig. 2. The system is considered as being in one of the seven possible states; these are as follows: S1 S2 S3

All three units are working properly. Both B and C are working properly, but A fails. Both A and B are working properly, but C fails.

S4 S5 S6 S7

Both A and C are working properly, but B fails. B is working properly, but A and C have failed. C is working properly, but A and B have failed. A is working properly, but B and C have failed.

The four states S3, S5, S6 and S7 represent the system failures when the system shut down, requiring repair. In the case of failure of power supply unit, the system will fail as a whole. Similarly, if both transmitters fail, the system serves no purpose and operation is discontinued until it is repaired. For the above two reasons the state represented by failures of all three units can be ignored. The state-space diagram is given in Fig. 1. The system differential equations are written as follows: dP1 …t†=dt ˆ ⫺0:0283P1 …t† ⫹ 0:5P2 …t† ⫹ 0:2P3 …t† ⫹ 0:5P4 …t† …3† dP2 …t†=dt ˆ 0:0133P1 …t† ⫺ 0:525P2 …t† ⫹ 0:2P5 …t† ⫹ 0:5P6 …t† …4† dP3 …t†=dt ˆ 0:005P1 …t† ⫺ 0:2P3 …t†

…5†

dP4 …t†=dt ˆ 0:01P1 …t† ⫺ 0:525P4 …t† ⫹ 0:5P6 …t† ⫹ 0:2P7 …t† …6† dP5 …t†=dt ˆ 0:005P2 …t† ⫺ 0:2P5 …t†

…7†

dP6 …t†=dt ˆ 0:02P2 …t† ⫹ 0:02P4 …t† ⫺ P6 …t†

…8†

dP7 …t†=dt ˆ 0:005P4 …t† ⫺ 0:2P7 …t†

…9†

(i) Using the LU decomposition: Following the procedure explained in the Refs. [6–12] with the needed modifications, where the model can be represented by the equations given in the present section: The first equation of the seven-state model can be

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Table 2 Seven-state model results State

LUD SSP

LUD SSF

Runge–Kutta SSP

Runge–Kutta SSF

% Error SSP

% Error SSF

1 2 3 4 5 6 7

9.31E ⫺ 01 2.47E ⫺ 02 2.33E ⫺ 02 1.87E ⫺ 02 6.16E ⫺ 04 8.68E ⫺ 04 4.69E ⫺ 04

0.026357533 0.0129444 0.004656806 0.009841356 0.00012328 0.000868029 9.37272E ⫺ 04

0.93136 0.02466 0.02328 0.01875 0.00062 0.00087 0.00047

0.026357488 0.0129465 0.004656 0.00984375 0.000124 0.00087 0.000094

1.72E ⫺ 03 ⫺ 0.016223232 0.017308 ⫺ 0.024325916 ⫺ 0.584003704 ⫺ 0.227089239 ⫺ 0.291057452

0.000171791 ⫺ 0.016223232 0.02 ⫺ 0.024325916 ⫺ 0.584003704 ⫺ 0.227089239 ⫺ 0.291057452

dP3 …t†=dt ˆ 0:000343P1 …t† ⫺ 1:43064P3 …t† ⫹ 0:137615P4 …t†

replaced by: P1 ⫹ P2 ⫹ P3 ⫹ P4 ⫹ P5 ⫹ P6 ⫹ P7 ˆ 1:0

…10†

Then, following the other steps of the proposed LU decomposition to find the steady-state probabilities and frequencies, one can see that the steady-state probabilities and frequencies are as shown in Table 2. (ii) Using the Runge–Kutta method: Following the Runge–Kutta method explained in [6,13], the steady-state probabilities and frequencies are obtained for the seven-state model and are given in Table 2. Table 2 shows the results of both the LU decomposition and Runge–Kutta methods. From Table 2 it is easy to conclude that the percentage error obtained is very small or zero.

⫹ 0:02184P7 …t† ⫹ 0:113314P11 …t†

…13†

dP4 …t†=dt ˆ 0:000343P2 …t† ⫹ 0:00115P3 …t† ⫺ 1:567105P4 …t† ⫹ 0:024184P8 …t† ⫹ 0:11314P12 …t†

…14†

dP5 …t†=dt ˆ 0:000346P1 …t† ⫺ 0:02625P5 …t† ⫹ 0:137615P6 …t† ⫹ 1:428571P7 …t† ⫹ 0:11314P13 …t†

…15†

dP6 …t†=dt ˆ 0:000346P2 …t† ⫹ 0:00115P5 …t† ⫺ 0:162715P6 …t† ⫹ 1:428571P8 …t† ⫹ 0:11314P14 …t†

…16†

dP7 …t†=dt ˆ 0:000346P3 …t† ⫹ 0:0003430P5 …t† ⫺ 1:452134P7 …t† ⫹ 0:137615P8 …t† ⫹ 0:11314P15 …t† …17† dP8 …t†=dt ˆ 0:000346P4 …t† ⫹ 0:000343P6 …t† ⫹ 0:00115P7 …t† Example 2. The 16-state model is represented as a second example, where both the LUD and the Runge–Kutta are used to determine the steady-state probabilities and frequencies. The sixteen-state model represents four generators (real example). These generators are part of one of Bahrain power stations. The combination of the four units under operation and failing conditions are shown in Table 1. The following equations can be written to represents the system:

…11†

⫹ 1:428571P4 …t† ⫹ 0:024184P6 …t† ⫹ 0:113314P10 …t†

⫹ 0:137615P10 …t† ⫹ 1:428571P11 …t† ⫹ 0:024184P13 …t† …19† dP10 …t†=dt ˆ 0:000573P2 …t† ⫹ 0:00115P9 …t† ⫺ 0:251618P10 …t† …20†

dP11 …t†=dt ˆ 0:000573P3 …t† ⫹ 0:000343P9 …t† ⫺ 1:543381P11 …t†

⫹ 0:137615P12 …t† ⫹ 0:024184P15 …t†

dP2 …t†=dt ˆ 0:00115P1 …t† ⫺ 0:138877P2 …t†

…18†

dP9 …t†=dt ˆ 0:000573P1 …t† ⫺ 0:115153P9 …t†

⫹ 1:428571P12 …t† ⫹ 0:024184P14 …t†

dP1 …t†=dt ˆ ⫺0:02412P1 …t† ⫹ 0:137615P2 …t† ⫹ 1:428571P3 …t† ⫹ 0:024184P5 …t† ⫹ 0:1131P9 …t†

⫺ 1:590943P8 …t† ⫹ 0:11314P16 …t†

…21†

dP12 …t†=dt ˆ 0:000573P4 …t† ⫹ :000343P10 …t† ⫹ :00115P11 …t† …12†

⫺ 1:679846P12 …t† ⫹ 0:024184P16 …t†

…22†

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Table 3 Sixteen-state model results State

LUD SSP

LUD SSF

Runge–Kutta SSP

Runge–Kutta SSF

% Error SSP

% Error SSF

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.9725712 0.0081274 0.0002335 1.951E ⫺ 06 0.0139149 0.0001163 3.346E ⫺ 06 2.792E ⫺ 08 0.004918 4.11E ⫺ 05 1.181E ⫺ 06 9.868E ⫺ 09 7.036E ⫺ 05 5.88E ⫺ 07 1.69E ⫺ 08 1.412E ⫺ 10

0.0234584 0.0011287 0.0003341 3.058E ⫺ 06 0.0003653 1.892E ⫺ 05 4.859E ⫺ 06 4.442E ⫺ 08 0.0005663 1.034E ⫺ 05 1.822E ⫺ 06 1.658E ⫺ 08 9.78E ⫺ 06 1.62E ⫺ 07 2.648E ⫺ 08 2.405E ⫺ 10

0.972582527 0.008127547 0.000233517 1.9514E ⫺ 06 0.013914832 0.000116282 0.000003341 2.79E ⫺ 08 0.004918085 4.10988E ⫺ 05 1.1808E ⫺ 06 9.9E ⫺ 09 7.03635E ⫺ 05 0.000000588 1.69E ⫺ 08 1E ⫺ 10

0.0234587 0.0011287 0.0003341 3.058E ⫺ 06 0.0003653 1.892E ⫺ 05 4.852E ⫺ 06 4.439E ⫺ 08 0.0005663 1.034E ⫺ 05 1.822E ⫺ 06 1.663E ⫺ 08 9.78E ⫺ 06 1.62E ⫺ 07 2.649E ⫺ 08 1.704E ⫺ 10

⫺ 0.001164614 ⫺ 0.001373127 ⫺ 0.003425991 ⫺ 0.000307473 0.000273089 ⫺ 0.000257995 0.159965192 0.082548289 ⫺ 0.000823498 ⫺ 0.00104627 0.001778424 ⫺ 0.327147574 0.000284237 0.000646254 ⫺ 0.021898285 29.16964154

⫺ 0.001164614 ⫺ 0.001373127 ⫺ 0.003425991 ⫺ 0.000307473 0.000273089 ⫺ 0.000257995 0.159965192 0.082548289 ⫺ 0.000823498 ⫺ 0.00104627 0.001778424 ⫺ 0.327147574 0.000284237 0.000646254 ⫺ 0.021898285 29.16964154

dP13 …t†=dt ˆ 0:000573P5 …t† ⫹ 0:000346P9 …t† ⫺ 0:138991P13 …t† ⫹ 0:137615P14 …t† ⫹ 1:428571P15 …t†

…23†

dP14 …t†=dt ˆ 0:000573P6 …t† ⫹ 0:000346P10 …t† ⫹ 0:00115P13 …t† ⫺ 0:275456P14 …t† ⫹ 1:428571P16 …t†

the steady-state probabilities and frequencies are obtained for the sixteen-state model are shown in Table 3. One can see that the values of the steady-state probabilities and frequencies obtained, for the sixteen-state model, using the Runge–Kutta method are almost similar to that obtained using the LUD. Table 3 shows the results. From Table 3 it is easy to conclude that the percentage error obtained is very small or almost zero.

…24† 4. Conclusion

dP15 …t†=dt ˆ 0:000573P7 …t† ⫹ 0:000346P11 …t† ⫹ 0:000343P13 …t† ⫺ 1:567219P15 …t† ⫹ 0:137615P16 …t† …25† dP16 …t†=dt ˆ 0:000573P8 …t† ⫹ 0:000346P12 …t† ⫹ 0:000343P14 …t†⫹0:00115P15 …t† ⫺ 1:70368P16 …t†

…26†

(i) Using the LUD: By applying the method, explained in [6–12] with the needed modifications on the sixteen-state model, and replacing the first equation of the model by P1 ⫹ P2 ⫹ P3 ⫹ P4 ⫹ P5 ⫹ P6 ⫹ P7 ⫹ P8 ⫹ P9 ⫹ P10 ⫹ P11 ⫹ P12 ⫹ P13 ⫹ P14 ⫹ P15 ⫹ P16 ˆ 1:0

…27†

to satisfy the condition that the summation of the probabilities must be equal to one. The steady-state probabilities and frequencies are as shown in Table 3. (ii) Using the Runge–Kutta method: following the Runge–Kutta method explained in [6,13],

The present paper presents LU decomposition method to calculate the steady-state probabilities and frequencies of two different engineering models described by Markov systems. Both models give accurate results using the LU decomposition method. The Runge–Kutta approach is applied on both models for comparison with the results obtained by LUD. The results obtained by LUD approach shows an excellent accuracy with a very small percentage error compared with the Runge–Kutta method which has a high accuracy. It can be conclude that LUD can be used as a tool for the calculations of the steady-state probabilities and frequencies for any engineering model. This method is being successfully applied in practical communication and power systems models.

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