Amit Kumar and Mangey Ram

Department of Mathematics, Graphic Era University, Dehradun 248002, India

Performance of Marine Power Plant Given Generator, Main and Distribution Switchboard Failures

Amit Kumar and Mangey Ram*

Department of Mathematics, Graphic Era University, Dehradun 248002, India

Power generation is one of the most essential functions of any plant for continuous functioning without any interruption. A marine power plant (MPP) is the same section. In the present paper, the authors have tried to find the various reliability characteristics of an MPP. A marine power plant which is a composition of two generators and in which one of them is located at the stern and another at the bow, both associated to the main switchboard (MSB). The distribution switchboards (DSB) receive power from the MSB through cables and their respective junctions. Given that arrangement, a working based transition state diagram has been generated. With the help of the Markov process, a number of intro-differential equations are formed and solved by Laplace transform. Various reliability characteristics are calculated and discussed with the help of graphs.

marine engineering; marine power plant (MPP); Markov process; main switchboards (MSB); distribution switchboards (DSB); sensitivity analysis; reliability theory

1 Introduction1

Power supply without any interruption is the major necessity of every engineering/industrial plant. For continuous power supply, it is essential to identify exactly and accurately that which failures affect the availability/ reliability of the system.

In this area, a lot of research has been done to improve the reliability and production efficiency of various industrial plants (Ram, 2013). Arora and Kumar (1997) discussed the availability of a steam generating system, with three subsystems and a power generation system, consisting of four subsystems arranged in series. Kumaret al. (2006) developed a method for analyzing the fuzzy system reliability for a mixed configuration system using interval valued trapezoidal vague sets. Then with the help of that method analyzed fuzzy reliability of a marine power plant. Sanninoet al. (2006) discussed the availability of large offshore wind parks for three different topologies of collection grid and investigated sensitivity and mean time to failure. Ben Elghaliet al. (2007) reviewed marine tidal power fundamental concepts then reported on issues regarding electric generator topologies associated with tidal turbines. Etiet al. (2007) deliberated an applications of failure mode effect analysis, failure mode effects, criticality analysis, feedback, support system and risk analysis, in order to reduce the frequency of failures and maintenance costs.

As the failure of any component of a system leads to the failure of the system or degrades the system, fault diagnosis is important (Bhushan and Rengaswamy, 2002; Ram and Manglik, 2014) for the successful operation of any industry/system and for increasing reliability (Huet al., 2009). In comparison to other production plants, like hydro power plants, Tavneret al. (2006) described an investigation of the reliability of generators and convertors, based on failure data collected in Germany and Denmark, and recommended how the designers and operators of wind turbines can increase their reliability by the choice of design concept and the operating regime. Yeet al. (2001) discussed reliability analysis of a hydro power station. Hosseiniet al. (2005) proposed the technique to calculate the annual energy for a small hydro plant, for this, they developed two programs: one using Excel software and other Matlab software. With the help of these programs, the authors analyzed the most important economic indices of a small hydro power plant using sensitivity analysis methods and calculated the reliability indices for number of units for small hydro power plants using the Monte Carlo method. Kumar and Ram (2013) discussed the coal handling unit of a thermal power plant, and found some most important reliability measures for the same, using a probabilistic approach. Wang and Zhang (2007) discussed a repairable complex system, with the assumption that a repaired unit cannot work with the same efficiency as a new one, and determined an optimal replacement policy for the system in order to minimize long run expected cost.

2 System description

In the present paper, the authors have developed a mathematical model to find the reliability characteristics of an MPP. An MPP is a system which is responsible for power generation in a ship. The MPP consists of two generators, one at the stern and the other at the bow, two main switchboards (MSB) and one distribution switchboard(DSB). The two MSBs are interconnected by a cable (Srinath, 2008). The DSB gets power from the MSB for further distribution and the MSB gets power from the generators. The configuration and transition diagrams of the MPP are shown in Figs. 1 and 2, respectively.

Some assumptions are as follows:

1) Initially, the MPP system is working with full efficiency.

2) The MPP can also work at reduced capacity (i.e. in degraded state).

3) At every instant, sufficient repair facilities are available.

4) Failure rate of each unit is taken to be a constant.

5) No two units will fail simultaneously.

Fig. 1 Configuration of MPP

Fig. 2 State transition diagram of MPP

3 Mathematical formulation and solution of the problem

The set of intro-differential equations which governs the present mathematical model (on the basis of state transition diagram) is as follows:

Boundary conditions:

and all other state probabilities are zero att=0.

Taking Laplace transform from the Eqs. (1)-(6), one gets

Initial condition:

Solving the above set of equations with the help of initial and boundary conditions, the operative transition state probabilities are given as:

where:

From the transition state diagram, the probability that the system is up (i.e. in good or degraded state) and down (failed state) state at any time is given as:

4 Assessment of various reliability measures

4.1 Availability assessment

For computing the availability of MPP, setting the values of different failure rates asλG=0.003,λM=0.011,λD=0.111 (Kumaret al., 2006) and repair rates asμG=μM=μD=μMD=μGMD=μGM=μGD=1 (Ram and Kumar, 2014; Manglik and Ram, 2015) in Eq. (26) then taking the inverse Laplace transform, one can gets the availability of the system as:

Now varying time unittin Eq. (28), one gets Table 1 and corresponding Fig. 3, as given below, for availability of MPP.

Table 1 Availability of MPP

Fig. 3 Availability vs. time

4.2 Reliability assessment

For computing the reliability of the MPP setting the values of different failure rates asλG=0.003,λM=0.011,λD=0.111 (Kumaret al., 2006) repair rates asμG=μM=μD=μMD=μGMD=μGM=μGD=0 (Ram and Kumar, 2015) in Eq. (26) then taking the inverse Laplace transform, one gets the reliability of the MPP as:

Now vary the time unittfrom 0 to 15 in Eq. (29), one gets Table 2 and corresponding Fig. 4, as given below, for reliability of MPP.

Table 2 Reliability of MPP

Fig. 4 Reliability vs. time

4.3 Mean time to failure assessment

For computing the MTTF of MPP, taking all repairs equal to zero for exponential distribution in Eq. (26) and taking asstends to zero, one can obtain the MTTF of MPP as:

Table 3 MTTF of MPP

Fig. 5 MTTF of MPP

Setting failure rates asλG=0.003,λM=0.011,λD=0.111 (Kumaret al., 2006) and varying failure rates from 0.01 to 0.10 (with a time interval of 0.01) one by one in Eq. (30), we get the MTTF of the MPP as tabulated in Table 3 and Fig. 5.

4.4 Sensitivity assessment

4.4.1 Sensitivity with respect to MTTF

Table 4 Sensitivity of MTTF

Fig. 6 Sensitivity of MTTF

4.4.2 Sensitivity with respect to reliability

Table 5 Sensitivity of reliability

Fig. 7 Sensitivity of reliability

4.5 Expected profit from MPP

The profit function (Ram and Kumar, 2014) for the MPP in time interval [0,t) is given as:

Using Eq. (28) in Eq. (31), one obtains the profit function as given below:

Setting revenueK1=1 and service cost asK2=0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 respectively then varying time scaletin Eq. (32), one gets Table 6 and correspondingly Fig. 8 forexpected profit of an MPP as follows.

Table 6 Expected profit of MPP

Fig. 8 Expected profit of MPP

5 Results and discussion

In the present paper, the authors have developed a mathematical model to find the reliability characteristics of an MPP. The graph between availability and time shown in Fig. 3 reveals that the availability of an MPP decreases smoothly as time passes. The reliability of an MPP with respect to the time scale t is shown in Fig. 4. It can be seen from the figure that the reliability of an MPP decreases faster than availability as time passes. This reflects the importance of repair policy. Fig. 5 shows the graph between MTTF of an MPP and variation in failure rates. From the graph it is observed that the MTTF of a marine power plant decreases with respect to all types of failure rates. It is highest with failure rates of the distribution switchboard i.e. MTTF of an MPP is greatly affected by the failure rate of distribution switchboards. The sensitivity of MTTF for an MPP is shown in Fig. 6. It shows that the MTTF is equally sensitive to the failure rates of generators and main switchboards. Also, it is very sensitive to the failure rate of the distribution switchboard. From this one can say that we have to focus more on the failure rate of distribution switchboards to enhance reliability of an MPP. Fig. 7 shows the sensitivity analysis of reliability of an MPP with respect to different failure rates, and shows the reliability of an MPP is most sensitive to the failure rate of the distribution switchboards. Further, in order to make the system reliability less sensitive, one has to control the failure rate of distribution switchboards. Keeping the revenue per unit time fixed at 1 and varying service cost as 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6, Fig. 8 is obtained. It is very clear from the graph that the profit decreases as the service cost increases with the passage of time. So, in order to optimize the profit function one has to control service cost as well as various failure rates of an MPP.

6 Conclusion

On the basis of the calculation and discussion above for marine power plants, it is noticeable that the system is highly sensitive to failure rate of distribution switchboards. So in order to make marine power plants more reliable, one has to especially control failure rate of the distribution switchboards. Also, we see that it is equally sensitive to failure rates of generators and main switchboards. The difference between the graph of availability and reliability shows the importance of repairs. Hence it is observed that good maintenance is the major requirement for the successful functioning of any system. It is asserted that the findings of this paper are highly beneficial for the management of marine power plants.

Nomenclatures

The following notations have been used throughout the MPP model:

The revenue/service cost from the MPP.

t/sTime scale variable in years/Laplace transform variable.

Appendix A

At any timet, if the system is in stateSi, then the probability of the system to be in that state is defined as: Probability that the system is in stateSiat timetand remains there in interval （t,t+Δt） or/and if it is in some other state at timetthen it should transit to the stateSiin the interval （t,t+Δt） provided transition exist between the states and Δt→0.

Accordingly the equations are interpreted as: the probability of the system to be in stateS0in the interval（t,t+Δt） is given by

Appendix B

Boundary conditions of the system are obtained corresponding to transitions between the states where transition from a state with and without elapsed repair time exists, with elapsed repair timesxand 0. Hence we have the following boundary conditions:

Boundary conditions

and all other state probabilities are zero att= 0.

Appendix C

System description:

S0The MPP is working with full efficiency.

S1The MPP is operational in the degraded state with one failed generator.

S2The MPP is operational in the degraded state with one failed generator and one failed MSB.

S3The MPP is operational in the degraded state with one failed MSB.

S4The MPP has failed due to the failure of both the generators.

S5The MPP has failed due to the failure of DSB.

S6The MPP has failed due to the failure of one MSB and DSB.

S7The MPP has failed due to the failure of both the MSB.

S8The MPP has failed due to the failure of one generator, one MSB and complete failure of DSB.

S9The MPP has failed due to the failure of both generators and one MSB.

S10The MPP has failed due to the failure of one generator and both the MSB.

S11The MPP has failed due to the failure of one generators and DSB.

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10.1007/s11804-015-1335-5

1671-9433(2015)04-0450-09

Received date: 2015-05-26.

Accepted date: 2015-10-03.

*Corresponding author Email: drmrswami@yahoo.com

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015