Optimal time scheduling for carrying out minor maintenance on a steam turbine

Abstract

Almost all the major equipment of ships or industry today undergo regular maintenance on its components and sub-components. In most of the cases it is observed that there are generally two kinds of maintenance in terms of times to repair and its effect on the availability of the main equipment; minor maintenance and a major maintenance or overhaul. The minor maintenance are generally carried at a component or a sub-component level, either at fixed intervals or based on the component condition. The minor maintenance however, requires that the equipment is brought down from operation for a shorter duration and put back in operation immediately after completion of the minor maintenance. The major maintenance on the other hand is carried out preferably in a workshop, where the equipment is completely stripped down to undertake maintenance which otherwise would not have been possible in its original location (for e.g. inside a ship). The time required for carrying out major maintenance is therefore substantial when compared to the minor maintenance period. The paper puts forward a method for scheduling minor maintenance of a major equipment—a steam turbine of a ship, within the time frame of the major maintenance interval based upon the deterioration of the components. The wear processes of the important components of the turbine are assumed to be time variant gamma processes and allow for continuous monitoring of the wear. Since turbines are a complex mix of components, sub-components and other ancillary systems, most of which are easily replaceable in miniscule of time periods, the paper considers only the most important wearing components which play a direct part in deterioration of performance of the turbine. All other random failures, which based on the past experience are only a very few in numbers and have very little impact on the availability of the turbine, have been ignored. MLE and a Gibbs sampling method has been used to estimate the parameters of distribution of gamma wear processes for the components of turbine.

Appendix

1.1 Maximum likelihood function for gamma variables

Let there be “n” observations of the subject component whose parameters of gamma wear are to be known. Let these observations be carried out at different times t₁, t₂…t_n, then if the shape parameter is function of time such that α = λt^ζ and scale parameter is β; then probability of observation of exactly these “n” values of variables can be given by

$$ {\text{L}} = \prod\limits_{i = 1}^{i = n} {\frac{{\beta^{{\lambda \left( {t_{i}^{\zeta } - t_{i}^{\zeta } - 1} \right)}} }}{{\Upgamma \left( {\lambda \left[ {t_{i}^{\zeta } - t_{i}^{\zeta } - 1} \right]} \right)}}}\,dw_{i}^{{\lambda \left( {t_{i}^{\zeta } - t_{i}^{\zeta } - 1} \right)^{ - 1} }} e^{{ - \beta dw_{i} }} \;{\text{where}}\;dw_{i} = {\text{wear}}\;{\text{between}}\;{\text{intervals}} $$

Taking logarithm of maximum likelihood we have

$$ {\text{Log}} \, \left( {\text{L}} \right) = \sum\limits_{i = 1}^{i = n} {\frac{{\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log (\beta )}}{{\log \Upgamma \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)}}} + \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right) - 1} \right)\log (\delta_{i} ) - \beta dw_{i} $$

Taking partial derivatives for wrt β, λ and ζ we will have

$$ \frac{\partial (\log L)}{\partial \beta } \,= \, 0 \, = \, \sum\limits_{i = 1}^{n} {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\frac{1}{\beta }} \, - \, dw_{i} \hfill \\ \therefore \;\lambda t_{n} \, =\, W_{n} \beta \frac{\partial (\log L)}{\partial \lambda } \,= \,0 = \,\sum\limits_{i = 1}^{n} {\left( {t_{i}^{b} - t_{i - 1}^{b} } \right)\log \beta \,- \,\frac{{\log \, \left( {\Upgamma \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)} \right)}}{\partial \lambda }} + \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log \, dw_{i} = \sum\limits_{i = 1}^{n} {\left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log \beta \, - \, \frac{{\log (\Upgamma \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)}}{{\partial \left[ {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right]}}} \frac{{\partial \left[ {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right]}}{\partial \lambda } + \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log dw_{i} = \sum\limits_{i = 1}^{n} {\left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log \beta - \frac{{\log \left( {\Upgamma \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)} \right)}}{{\partial \left[ {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right]}}} \left[ {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right] + \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log dw_{i} $$

(16)

$$ \begin{gathered} \therefore \sum\limits_{i = 1}^{n} {\left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \left[ {\log dw_{i} - \frac{{\log \left( {\Upgamma \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)} \right)}}{{\partial \left[ {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right]}}} \right] = - \sum\limits_{i = 1}^{n} {\left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)\log \beta } \hfill \\ \therefore \,t_{n}^{\zeta } \log \beta = \sum\limits_{i = 1}^{n} {\left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \left[ {\frac{{\log (\left( {\Upgamma \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)} \right)}}{{\partial \left[ {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right]}} - \log dw_{i} } \right] \hfill \\ \end{gathered} $$

(17)

$$ \begin{aligned} \frac{\partial (\log L)}{\partial \zeta } \, & = \, \sum\limits_{i \,= \,1}^{n} {\lambda \log \beta \left[ {\log t_{i} t_{i}^{\zeta } \, - \, \log t_{i - 1} t_{i - 1}^{\zeta } } \right]}\\ & \quad - \, \frac{{\partial \left( {\log \Upgamma (\lambda (t_{i}^{\zeta } \, - \, t_{i - 1}^{\zeta } } \right)}}{\partial \zeta }\\ & \quad + \lambda \log dw_{i} \left[ {\log t_{i} t_{i}^{\zeta } - \log t_{i - 1} t_{i - 1}^{\zeta } } \right] \\ & =\, \sum\limits_{i \,= \,1}^{n} {\lambda \log \beta \left[ {\log t_{i} t_{i}^{\zeta } - \log t_{i - 1} t_{i - 1}^{\zeta } } \right]} \\ & \quad - \frac{{\partial \left( {\log \Upgamma (\lambda (t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)}}{{\partial \left( {\lambda \left( {t_{i}^{\zeta } - t_{i - 1}^{\zeta } } \right)} \right)}}\lambda \left[ {\log t_{i} t_{i}^{\zeta } - \log t_{i - 1} t_{i - 1}^{\zeta } } \right] \\ & \quad + \lambda \log dw_{i} \left[ {\log t_{i} t_{i}^{\zeta } - \log t_{i - 1} t_{i - 1}^{\zeta } } \right]l \\ \end{aligned} $$

(18)

$\therefore \, {\text{where}}\quad \frac{\partial (\log \Upgamma (A)}{\partial (A)}\quad {\text{in}}\;{\text{above}}\;{\text{equations}}\;{\text{is}}\;{\text{a}}\;{\text{digamma}}\;{\text{function}}$ Nicolai et al. (2007).