Section: New Results

Predictive maintenance for the heated hold-up tank

Participants : Benoîte de Saporta, François Dufour, Huilong Zhang.

A complex system is inherently sensitive to failures of its components. One must therefore determine maintenance policies in order to maintain an acceptable operating condition. Optimizing the maintenance is a very important problem in the analysis of complex systems. It determines when it is best that maintenance tasks should be performed on the system in order to optimize a cost function: either maximize a performance function or conversely minimize a loss function. Moreover, this optimization must take into account the random nature of failures and random evolution and dynamics of the system.

The example considered here is the maintenance of the heated hold-up tank, a well know test case for dynamic reliability, see e.g. [75] , [89] , [90] , [94] . The system consists of a tank containing a fluid whose level is controlled by three components: two inlet pumps and one outlet valve. A thermal power source heats up the fluid. The failure rate of the components depends on the temperature, the position of the three components monitors the liquid level in the tank, and in turn, the liquid level determines the temperature. The main characteristic of this system is that it can be modeled by a stochastic hybrid process, where the discrete and continuous parts interact in a closed loop. As a consequence, simulating this process and computing related reliability indices has been a challenge for the dynamic reliability community. To our best knowledge, optimization of maintenance policies for the heated hold-up tank has not been addressed yet in the literature.

The only maintenance operation considered here is the complete replacement of all the failed components and the system restarts in its initial equilibrium state. Partial repairs are not allowed. Mathematically, this problem of preventive maintenance corresponds to a stochastic optimal stopping problem as explained by example in the book of Aven and Jensen [68] . It is a difficult problem because of the closed loop interactions between the state of the components and the liquid level and temperature. A classical approach consists in using condition-based maintenance (CBM) to act on the system based on its current state and before its failure. One can for example calculate the remaining useful life (RUL) of the system and the preventive replacement is carried out when the deterioration level exceeds a certain threshold or enters in a certain state [96] , [80] . Our approach also takes into account the current state of the process, but our decision rule is not based on damage accumulation nor does it correspond to hitting some threshold. Instead, it involves a performance function that reflects that the longer the system is in a functioning state the better.

The dynamics of the heated hold-up tank can be modeled by a piecewise deterministic Markov process (PDMP), see [94] . Therefore, our maintenance problem boils down to an optimal stopping problem for PDMP's. PDMP's are a class of stochastic hybrid processes that has been introduced by Davis [73] in the 80's. These processes have two components: a Euclidean component that represents the physical system (e.g. temperature, pressure, ...) and a discrete component that describes its regime of operation and/or its environment. Starting from a state $x$ and mode $m$ at the initial time, the process follows a deterministic trajectory given by the laws of physics until a jump time that can be either random (e.g. it corresponds to a component failure or a change of environment) or deterministic (when a magnitude reaches a certain physical threshold, for example the pressure reaches a critical value that triggers a valve). The process restarts from a new state and a new mode of operation, and so on. This defines a Markov process. Such processes can naturally take into account the dynamic and uncertain aspects of the evolution of the system. A subclass of these processes has been introduced by Devooght [75] for an application in the nuclear field. The general model has been introduced in dynamic reliability by Dutuit and Dufour [79] .

As illustrated above, it is crucial to have an efficient numerical tool to compute the optimal maintenance time in practical cases. To this aim, a general numerical approach was developed in [8] . It was first applied to an example of maintenance of a metallic structure subject to corrosion, without closed loop interactions or deterministic jumps, and with a simple cost function that did not depend on time, see [23] . The objective of the present paper is to further demonstrate the high practical power of the theoretical methodology described in [8] , by applying it to the more challenging heated hold-up tank problem. The cost function chosen here is also more complex as it takes into account both continuous components as well as the running time. More precisely, we propose to compute the optimal cost as well as a quasi-optimal stopping rule, which is the date when the maintenance should be performed. As a by-product of our procedure, the distribution of the optimal maintenance dates is also obtained, as well as the distributions of the liquid level and temperature at the chosen maintenance date.

This work is submitted for publication [66] and presented in an international conference [32] .