Fuzzy Availability Analysis of a Marine Power Plant

Available online at www.sciencedirect.com

ScienceDirect Materials Today: Proceedings 5 (2018) 25195–25202

www.materialstoday.com/proceedings

IConAMMA_2017

Fuzzy Availability Analysis of a Marine Power Plant Ashish Kumar*, Monika Sainia *, a Department of Mathematics & Statistics, Manipal University Jaipur, Jaipur-303007, Rajasthan, India

Abstract The main objective of the present study is to analyze the availability of a marine power plant using fuzzy approach of system reliability. The marine power plant is a complex system having five subsystems. Due to lack of sufficient probabilistic information, one cannot relies on the conventional reliability of probability model. Here a mathematical model is formulated using mnemonic rule and Chapmen - Kolmogorov differential equations are developed. The differential equations are solved by Runge–Kutta method of order four. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advances in Materials and Manufacturing Applications [IConAMMA 2017]. Keywords: Marine Power Plant; Markov process; Runge–Kutta method; Fuzzy availability.

1. Introduction In the present era of science and technology, the market value of any product/item is measured by its reliability. Reliability is the most important characteristic of the products/systems. The role of reliability is visualized in our daily life from switching on the bulb to the use of car, mobile. In all these jobs, the customer expects that these equipment perform their intended function properly with full efficiency. But sometimes, we found that system/product does not able to provide the service according to expectation and work with reduced capacity. But, in the approach of conventional reliability most of the systems are assumed in binary states, i.e., operative or failed. But, with the advancement of technology all the industrial, manufacturing, home appliances and medical equipment’s becomes more and more complex. And, in the reliability analysis of these equipment’s the conventional

* Corresponding author. Tel.:0141-399-9100; fax: 0141-399-9100. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advances in Materials and Manufacturing Applications [IConAMMA 2017].

25196

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

reliability approach seems unrealistic due to lack of sufficient probability information. To overcome such situations, many researchers developed several techniques to measure the reliability and performance of the systems. Event tree, fault tree analysis, reliability block diagram, Petri nets, Markovian approach, regenerative point technique and semi- Markovian technique. Zadeh [1] gives the initial ideas and concepts of fuzzy theory and it changed the complete context of reliability theory, because it can easily handle all the states between a normal working and total failed state. This approach is named as profust reliability. Though, we cannot completely ignore the conventional reliability but the fuzzy approach also be considered in parallel. A new methodology for finding reliability characteristics using fault tree and fuzzy set theory has been developed by Singer [2]. Here all repair and failure times are represented by triangular fuzzy numbers. Cheng and Mo [3] developed a method for fuzzy reliability analysis by confidence interval. Fuzzy number arithmetic operations have been used in the development of a new method for fuzzy reliability evaluation by Chen [4]. The physical interpretation of fuzzy reliability as probability has been suggested by Cai [5]. Biswas and Sarkar [6] discussed the effect of various imperfect repairs on the system’s availability. A new methodology by using Petri nets has been developed by Knezevic and Odoom [7]. A new method based on vague sets for analyzing the fuzzy system reliability has been presented by Chen [8]. Kumar et al. [9] obtained fuzzy reliability of a marine power plant using interval valued vague sets. Kumar & Kumar [10] analyzed the fuzzy availability of a biscuit manufacturing plant. Garg and Sharma [11] carried out the behavioral analysis of a synthesis unit in fertilizer plant. Kumar and Ram [12] derived the expressions for performance of a marine power plant with various component failure. Aggarwal et al.[13]used fuzzy reliability approach in the mathematical modeling of a crystallization system in a sugar plant. The literature discussed above shows that most of the techniques used to obtain the conventional reliability of complex systems involves complex numerical computations and widely studied by the researchers. So, in the present study an advance technique, Runge-Kutta fourth –order method is used for obtaining the fuzzy availability of a marine power plant. The required data is randomly generated. And the fuzzy availability measure are obtained and shown by graphs. 2. Material and Method 2.1. Profust Reliability Cai [5] defined the profust reliability as n n R(t 0 ,t 0 +t)=P{TSFdoesn't occur during theinterval(t 0 ,t 0 +T)}=1-   μ T (mij )P{mij occurs during (t 0 ,t 0 +T)} i=1 j=1 SF

Where mij is confined to be the transition from state

Si

to state S j without passing via any intermediate state.

2.2. Coverage Factor Coverage factor is the probability of successful reconfiguration operation of a fault-tolerant system. It is denoted by ‘c’ and if its value lies between 0 and 1.When coverage factor is less than 1, it is called imperfect coverage.

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

25197

2.3. Fuzzy Availability Kumar and Kumar [10] stated a fuzzy stochastic semi- Markov model {(Sn, Tn), n € N} having ‘n’ states with transition time. Let U = {S1, S2,…,Sn} represent the population of discourse. On this population, we define a fuzzy success state S, S= {(Si , S (Si )); i  1, 2,...n} and a fuzzy failure state F, F  {(Si , F (Si )); i  1, 2,...n} , where S (Si ) and kF (Si ) are trapezoidal fuzzy numbers, respectively. The fuzzy availability of the system is  S ( Si ) Pi (t ) , where k denotes the operative states. defined as: A(t ) 

 i 1

2.4. Markov Process If the state of the system is probability based, then the model is a Markov probability model. The fundamental assumption in Markov process is that, the probability Pij , depends entirely on states Si and S j , and is independent of all previous states except the last one state Si . 2.5. Marine Power Plant as a Complex System In a marine, for its continuous functionality power is essentially required that is generated using a marine power plant. It is a complex system which consists two generators located at stern and bow, one distributed switch board and two main switch boards interconnected by a cable. The power flows generators to main switch board and then to distributed switch board in a series configuration. The configuration is shown in the figure 1.1 as follows: Generator

MSB

DSB

Generator

MSB

Fig.1: Marine Power Plant Configuration

2.6. Assumptions    

All failure and repair rates are constant. All random variables are statistically independent. Repairs are perfect. No simultaneous failure of the subsystems.

2.7. Notations

i i

: Denotes the repair rate of the subsystems generators, MSB and DSB respectively : Denotes the failure rate of the subsystems generators, MSB and DSB respectively C : Coverage Factor th Pj (t ) : The probability that at time t system remain in j state P1 , P2 , P3 & P4 : Fuzzy availability of the system at 1, 2, 3 & 4 state respectively.

25198

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

3. System State Description

S1(GMD) : S2 (GMD) : S3 (GMD) : S4 (GMD) :

System is in operative state as generator, main switch board and distributed switch board available and system working with full capacity. System is in operative state as generator, distributed switch board and standby main switch board available and system work with reduced capacity. System is in operative state as standby generator, distributed switch board and main switch board available and system work with reduced capacity. System is in operative state as standby generator, distributed switch board and standby main switch board available and system work with reduced capacity.

S5 (GMD) :

System is in failed state as both original and standby generators failed.

S6 (GMD) :

System is in failed state as both original and standby main switch boards failed.

S7 (GMd ) :

System is in failed state as distributed switch board failed.

S8 (GMd ) :

System is in failed state as distributed switch board failed.

S9 (GMD) :

System is in failed state as both original and standby generators failed.

S10 (GMd ) :

System is in failed state as distributed switch board failed.

S11 (GMD) :

System is in failed state as both original and standby main switch board failed.

S12 (GMd ) :

System is in failed state as distributed switch board failed.

4. Mathematical Modeling of the System: The mathematical modeling of the marine power plant is carried out using birth- death process and corresponding Chapman-Kolmogorov differential equations are developed. The equations of fuzzy availability for marine power plant based on system states are as follows:

P1 (t  t )  [1  1c  2c  3 (1  c)]P1 (t )  2 P2 (t )  1P3 (t )  3 P12 (t ) dP1  [ 1c   2c  3 (1  c)]P1 (t )  2 P2 (t )  1P3 (t )  3 P12 (t ) dt dP2  [ 2  5c  3 (1  c)   4 (1  c)]P2 (t )  3 P10 (t )  4 P11 (t )  5 P4 (t )   2cP1 (t ) dt dP3  [ 1  4c  3 (1  c)  5 (1  c)]P3 (t )  3 P8 (t )  2 P9 (t )  4 P4 (t )  1cP1 (t ) dt dP4  [ 4  5  3 (1  c)   4 (1  c)  5 (1  c)]P4 (t )  5 P5 (t )  3 P7 (t ) dt  4 P6 (t )  5cP2 (t )   4cP3 (t ) dP5  5 P5 (t )  5 (1  c) P4 (t ) dt dP6  4 P6 (t )   4 (1  c) P4 (t ) dt

(1) (2) (3) (4) (5) (6)

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

dP7  3 P7 (t )  3 (1  c) P4 (t ) dt dP8   3 P8 (t )  3 (1  c) P3 (t ) dt dP9  2 P9 (t )  5 (1  c) P3 (t ) dt dP10  3 P10 (t )  3 (1  c) P2 (t ) dt dP11  4 P11 (t )   4 (1  c) P2 (t ) dt dP12  3 P12 (t )  3 (1  c) P1 (t ) dt

25199

(7) (8) (9) (10) (11) (12)

With initial conditions:

1, if j  1 Pj (0)   0, if j  1

(13)

This system of differential equations (1) to (12) along with initial condition (13) has been solved using Runge-Kutta fourth –order method. The numerical computations were derived by using the assumption in consideration that the repair and failure rates of the original and standby subsystems were different. The expression for fuzzy availability of the marine power plant is as follows:

5 5 2 AF  P1  P2  P3  P4 6 6 3

(14)

5. Performance Analysis: Here, using equation (14) the fuzzy availability of marine power plant is obtained and the effect of various failure rates, repair rates and coverage factor on the fuzzy availability is shown in graphs 2-12. Effect of failure rates of the generator on the fuzzy availability of marine power plant are shown in graphs 2,3 , and 11 for varying values of the parameters as follows: 1  0.0002, 0.9 5  0.005, 0.6 for fixed values of repair rates 1  0.5 & 5  0.65 and different values of coverage factor as c=0, 0.5, 1. The repair and failure rates of other subsystems were kept fixed as follows: 2  0.0002 & 2  0.31 , 3  0.0021 & 3  0.4 , 4  0.0001 & 4  0.25 . The fuzzy availability decrease with respect to time for all values of c. For c=0, there is no effect of the variation in failure rate on the fuzzy availability. For non -zero values of c, the fuzzy availability decrease sharply as failure rate increased for original ( 1 ) and standby ( 5 ) subsystems. As, the value of coverage factor increase the availability of the system also increase.

Fig. 2: Fuzzy Availability vs. Time

Fig. 3: Fuzzy Availability vs. Time

25200

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

Effect of repair rates of the generator on the fuzzy availability of marine power plant are shown in graphs 2, 4, and 12 for varying values of the parameters as follows:

1  0.5,1.2 & 5  0.65,1.4 for

fixed values of failure

1  0.0002 5  0.005 and different values of coverage factor as c=0, 0.5, 1. The repair and failure rates of other subsystems were kept fixed as follows: 2  0.0002 & 2  0.31 , 3  0.0021 & 3  0.4 , 4  0.0001 & 4  0.25 . The fuzzy availability decrease with respect to time for all values of c. For c=0, there rates

is no effect of the variation in repair rate on the fuzzy availability. For non -zero values of c, the fuzzy availability increase rapidly as we increase the value of repair rates for original ( 1 ) and standby ( 5 ) subsystems. As, the value of coverage factor increase the availability of the system also increase.

Fig. 4: Fuzzy Availability vs. Time

Fig. 5: Fuzzy Availability vs. Time

Effect of failure rates of the main switch board on the fuzzy availability of marine power plant are shown in graphs 2, 5, and 9 for varying values of the parameters as follows: of repair rates failure

rates

2  0.31 & 4  0.25 and of

other

subsystems

2  0.0002, 0.7 4  0.0001, 0.6 for fixed values

different values of coverage factor as c=0, 0.5, 1. The repair and

were

kept

fixed

as

follows:

1  0.0002 & 1  0.5 ,

3  0.0021 & 3  0.4 , 5  0.005 & 5  0.65 . The fuzzy availability decrease with respect to time for all values of c. For c=0, there is no effect of the variation in failure rate on the fuzzy availability. For non -zero

values of c, the fuzzy availability decrease sharply as failure rate increased for original ( 2 ) and standby ( 4 ) subsystems. As, the value of coverage factor increase the availability of the system also increase.

Fig. 6: Fuzzy Availability vs. Time

Fig. 7: Fuzzy Availability vs. Time

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

25201

Effect of repair rates of the main switch board on the fuzzy availability of marine power plant are shown in graphs 2, 6, and 10 for varying values of the parameters as follows: 2  0.31,0.9 & 4  0.25,0.9 for fixed values of failure rates 4  0.0001& 2  0.0002 and different values of coverage factor as c=0, 0.5, 1. The repair and failure rates of other subsystems were kept fixed as follows: 1  0.0002 & 1  0.5 , 3  0.0021 & 3  0.4 , 5  0.005 & 5  0.65 . The fuzzy availability decrease with respect to time for all values of c. For c=0, there is no effect of the variation in repair rate on the fuzzy availability. For non -zero values of c, the fuzzy availability increase rapidly as we increase the value of repair rates for original ( 2 ) and standby ( 5 ) subsystems. As, the value of coverage factor increase the availability of the system also increase.

Fig. 8: Fuzzy Availability vs. Time

Fig. 9: Fuzzy Availability vs. Time

Effect of failure rates of the distributed switch board on the fuzzy availability of marine power plant are shown in graphs 2, and 7 for varying values of the parameters as follows:

3  0.4

3  0.0021, 0.8 for fixed values of repair rates

and different values of coverage factor as c=0, 0.5, 1. The repair and failure rates of other subsystems

1  0.0002 & 1  0.5 , 2  0.0002 & 2  0.31 , 4  0.0001& 4  0.25 5  0.005 & 5  0.65 . The fuzzy availability decrease with respect to time for

were

kept

fixed

as

follows:

all values of c. For c=0, there is no effect of the variation in failure rate on the fuzzy availability. For non -zero values of c, the fuzzy availability decrease sharply as failure rate ( 3 ) increased for subsystems. As, the value of coverage factor increase the availability of the system also increase.

Fig. 10: Fuzzy Availability vs. Time

Fig. 11: Fuzzy Availability vs. Time

Effect of repair rates of the distributed switch board on the fuzzy availability of marine power plant are shown in graphs 2, and 8 for varying values of the parameters as follows:

3  0.4,1.3

for fixed values of failure rate

25202

Ashish Kumar et al./ Materials Today: Proceedings 5 (2018) 25195–25202

3  0.0021and different values of coverage factor as c=0, 0.5, 1. The repair and failure rates of other subsystems 1  0.0002 & 1  0.5 , 2  0.0002 & 2  0.31 , were kept fixed as follows: 4  0.0001& 4  0.25 5  0.005 & 5  0.65 . The fuzzy availability decrease with respect to time for all values of c. For c=0, there is no effect of the variation in repair rate on the fuzzy availability. For non -zero values of c, the fuzzy availability increase rapidly as we increase the value of repair rate ( 3 ) of original subsystem. As, the value of coverage factor increase the availability of the system also increase.

Fig. 12: Fuzzy Availability vs. Time

6. Conclusion The detailed study of fuzzy availability of marine power plant helps in increasing the power production in a marine power plant. The effect of coverage factor, fuzziness, failure rates and repair rates of all the subsystems are shown in graphs 2-12. As a comparative study, we conclude that distributed switch board has a prominent role in the variation of the fuzzy availability of the marine power plant. So, by applying proper, repair and maintenance facility for distributed switch board we can improve the reliability of a marine power plant. 7. References [1] L.A. Zadeh, Fuzzy Sets, Information and control, 8 (1965) 338-353. [2] D. Singer, A Fuzzy Set Approach to Fault Tree and Reliability Analysis, Fuzzy sets and systems, 34 (1990) 145-155. [3] C. H. Cheng, D. L. Mon, Fuzzy System Reliability Analysis by Interval of Confidence, Fuzzy Sets and Systems, 56 (1993) 29-35. [4] K.Y. Cai, Introduction to Fuzzy Reliability, Kluwer Academic Publishers, Norwell, 1996. [5] S.M. Chen, Fuzzy System Reliability Analysis using Fuzzy Number Arithmetic Operations, Fuzzy sets and systems, 64 (1994) 31-38. [6] A. Biswas, J. Sarkar, Availability of a System Maintained through Several Imperfect Repairs before a Replacement or a Perfect Repair. Statistics & Probability Letters, 50 (2000) 105-114. [7] J. Knezevic, E.R. Odoom, Reliability Modelling of Repairable Systems using Petri Nets and Fuzzy Lambda–Tau Methodology, Reliability Engineering & System Safety, 73 (2001) 1-17. [8] S.M. Chen, Analyzing Fuzzy System Reliability using Vague Set Theory, International Journal of Applied Science and Engineering, 1(2003) 82-88. [9] A. Kumar , S. P. Yadav, S. Kumar, Fuzzy Reliability of a Marine Power Plant using Interval Valued Vague Sets, International Journal of Applied Science and Engineering, 4 (2006) 71–82. [10] K. Kumar, P. Kumar, Fuzzy Availability Modeling and Analysis of Biscuit Manufacturing Plant: a Case Study, International Journal of System Assurance Engineering and Management, 2 (2011) 193-204. [11] H. Garg, S. P. Sharma, Behavior Analysis of Synthesis Unit in Fertilizer Plant, International Journal of Quality & Reliability Management, 29 (2012) 217-232. [12] A. Kumar, M. Ram, Performance of Marine Power Plant given Generator, Main and Distribution Switchboard Failures, Journal of Marine Science and Application, 14 (2015) 450-458. [13] A. K. Aggarwal, S. Kumar, V. Singh, Mathematical Modeling and Fuzzy Availability Analysis for Serial Processes in the Crystallization System of a Sugar Plant, Journal of Industrial Engineering International, (2016) 1-12.