## Dissertations

### New Renewal theory results on the maintenance of systems with coupled components

5-1-2023

#### Degree Name

Doctor of Philosophy

#### Department

Electrical and Computer Engineering

Kagaris, Dimitri

#### Abstract

Maintenance operations, which are performed in order to bring a system to a certain better condition, play an important role in ensuring availability, reliability, and safety. The best maintenance plan for a system is determined using renewal functions, which predict the frequency of failures of a system or component over a certain period of time.In this thesis, we first study renewals in a system with two components connected in series. Both components can undergo corrective maintenance (i.e., either full replacement/perfect repair or minimal repair) when a failure occurs. When one of the units fails and is correctively maintained, the other one is either preventively replaced (if cost-feasible) or simply left working as is. When a component is minimally repaired or left working as is, its remaining lifetime is reevaluated (``memory effect'') and is taken into consideration in calculating the next renewal. A ``coupled" lifetime that combines the lifetimes of the two serially connected components is proposed to represent the joint lifetime of the system. We develop renewal functions based on the coupled lifetimes and show that they follow the classical or generalized renewal theory depending on whether the components work without memory or not. Approximation formulas for the new renewal functions are also obtained and validated by Monte Carlo simulations for various combinations of distributions, and a comparative cost analysis is conducted.Secondly, we present a novel mathematical framework for computing the number of maintenance cycles of a system component (referred to as ``non-critical'' (NC)) until another reference system component (referred to as ``critical'' (CR)) fails for the first time. Every time the NC component fails, it undergoes corrective maintenance (replacement or minimal repair), provided that the CR component is still in operation. The lifetime of the CR component is assumed to be independent of that of NC and the failure of CR is assumed to mark the end of the cycle counting process. That is, the CR component is never correctively repaired but it can optionally be fully replaced in an opportunistic maintenance fashion every time the NC component fails. We extend traditional renewal theory for various maintenance scenarios for a system composed of one CR and one NC component in order to compute the average number of renewals of NC under the restriction (``bound") imposed by CR. We also develop approximations in closed form for the proposed ``bounded" renewal functions. We validate our formulas by simulations on a variety of component lifetime distributions, including actual lifetime distributions of wind turbine components.