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Paper text:
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. @xcite .
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 @xcite .
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 @xcite .
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 @xcite .
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 @xcite 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 @xmath0 and mode @xmath1 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 @xcite for an application in the nuclear field .
the general model has been introduced in dynamic reliability by dutuit and dufour @xcite .
the objective and originality of this paper is twofold .
first , we propose an optimization procedure for a well - known benchmark in the dynamic reliability literature .
the tank model was first introduced by @xcite where only one continuous variable ( liquid level ) is taken into account , and then in @xcite and @xcite where the second variable ( temperature ) is introduced .
they have tested various monte carlo approaches to simulate the process to compute reliability and safety indices . in @xcite ,
the authors have used the same system to present continuous cell - to - cell mapping markovian approach ( cccmt ) still to simulate the process .
the simulation of the holdup tank example has been and is still widely studied in the literature ( not exhaustive ) @xcite . here
we go one step further and not only propose to simulate the tank process but also we optimize it .
second , even though pdmp s have been recognized as a powerful modeling tool for dynamic reliability problems @xcite , there are very few numerical tools adapted to these processes .
our aim is to further demonstrate the high practical power of the theoretical methodology described in @xcite , by applying it to the tank benchmark . in @xcite ,
the authors have proposed a numerical algorithm to optimize pdmp s and have studied its theoretical properties .
this optimization procedure was first applied to an example of maintenance of a metallic structure subject to corrosion , without closed loop interactions or deterministic jumps .
in addition , the system has only one continuous variable and the cost function is simple and does not depend on time , see @xcite . in this paper , we adapt the numerical procedure proposed in @xcite to the more challenging heated hold - up tank problem with two continuous variables , deterministic jumps when these variables hit some given boundaries and closed loop interactions between continuous and discrete variables .
furthermore , we consider a cost function that depends on both continuous variables as well as on the running time .
the remainder of this paper is organized as follows . in section [ section :
model ] , the dynamics of the heated hold - up tank is presented with more details as well as the framework of pdmp s . in section [ section : opti ] the formulation of the optimal stopping problem for pdmp s and its theoretical solution are briefly recalled and the four main steps of the algorithm are detailed . in section [ section
: res ] the numerical results obtained on the example of the tank are presented and discussed . finally , in section [ section : conclusion ] a conclusion and perspectives are presented .
we are interested in the maintenance of a heated hold - up tank .
the dynamics of the tank can be modeled by a piecewise deterministic markov process ( pdmp ) .
we first describe with more details the dynamics of the tank , then we recall the definition and some basic properties of pdmp s .
the tank model is a well known benchmark in dynamic reliability .
it was first introduced by @xcite where only one continuous variable ( liquid level ) is taken into account , and then in @xcite and @xcite where the second variable ( temperature ) is introduced .
we have kept the values of the parameters defined in those papers .
the system is represented on figure [ fig1 ] .
it consists of a tank containing a fluid whose level is controlled by three components : two inlet pumps ( units 1 and 2 ) and one outlet valve ( unit 3 ) .
a thermal power source heats up the fluid .
the variables of interest are the liquid level @xmath2 , the liquid temperature @xmath3 and the state of the three components and the controller .
each component has four states : on , off , stuck on or stuck off . once a unit is stuck ( either on or off ) it can not change state .
the possible transitions between these four states are given in figure [ fig2 ] .
thus , by a random transition a working unit can only become stuck ( either on or off ) .
the initial state of the components is on for units 1 and 3 and off for unit 2 .    ,
width=226 ]    the intensity of jumps @xmath4 for unit @xmath5 depends on the temperature through the equation @xmath6 with @xmath7 given in eq .
( [ eq : a ] ) , see @xcite @xmath8    function @xmath7 is represented on figure [ fig3 ] and the various parameters come from the literature , see @xcite , and are given in table [ tab1 ] .
the special form of the failure rate @xmath4 as a product of a constant depending on @xmath5 and a function of the temperature allows for all three units to have failure rates with the same dependence on the temperature , but different scaling parameters .
indeed , at the reference temperature of @xmath9 , the mean time to failure of unit @xmath10 is @xmath11 , for unit @xmath12 , it is @xmath13 and for unit @xmath14 it is @xmath15 .
in addition , the shape for function @xmath7 was chosen in the original benchmark so that there is a very high failure rate when the temperature is high .
more specifically , the parameters are chosen such that @xmath7 is lowest ( equal to @xmath10 ) when the temperature is equal to a reference temperature of @xmath9 , it equals @xmath16 when the temperature is @xmath17 and it is highest ( equal to @xmath18 ) when the temperature equals the critical temperature of @xmath19 .
the exponential functions are chosen in order to enable this very high dependence with the temperature .
roughly speaking , the units fail @xmath18 times more often when the temperate is @xmath19 than when it is @xmath9 .     as a function of @xmath3,width=188 ]    in addition
, control laws are used to modify the state of the components to keep the liquid within two acceptable limits : @xmath20 meters and @xmath21 meters .
if the liquid level drops under @xmath22 , the components 1 , 2 , 3 are put respectively in state on , on and off ( provided they are not stuck ) .
if the liquid level rises above @xmath23 , the components are put respectively in the state off , off and on ( provided they are not stuck ) . unlike the classical model presented in @xcite , we also allow the control unit to fail . at each solicitation
, the control may succeed with probability @xmath24 independently from previous successes .
once it has failed , it will never succeed again .
therefore the control unit has two possible states : working @xmath10 or failed @xmath25 .
.parameters for the tank dynamics [ cols="^,^",options="header " , ]
the numerical method described in @xcite has been applied to a well known test case of dynamic reliability to approximate the value function of the optimal stopping problem and an @xmath26-optimal stopping time for a piecewise - deterministic markov process , that is the maintenance date for the tank .
the quantization method proposed can sometimes be costly in computing time , but has a very interesting property : it can be calculated off - line .
moreover it depends only on the dynamics of the model , and not on the cost function chosen , or the actual trajectory of the specific process one wants to monitor .
the calculation of the optimal maintenance time is done in real time , making our procedure applicable in practice .
the optimal maintenance time is updated at the changes of mode and has a conditional threshold form , which allows scheduling maintenance services in advance .
if one only changes the reward function @xmath27 without changing the dynamics of the tank , one just has to run the optimization part of the algorithm , and not the quantization grids .
this can be done in real time . if one wants to change the dynamics of the system , or add some components , one has to rewrite the simulation code for the system , and with this new code re - run the quantization grids , which can be quite long .
however , the general methodology is valid for a wide class of piecewise deterministic markov processes and not at all specific to the tank .
the method has been implemented on the heated hold - up tank .
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 .
the optimization problem under study has no analytic solution .
however , our method is based on a rigorous mathematical construction with proof of convergence .
in addition , simple comparisons between no motoring and our policy also prove its practical validity with a significant improvement of the performance of the system ( the mean performance is increased by @xmath28 and the top events are almost always avoided ) .
our next project is to extend this research in two main directions .
first , we could allow only partial repair of the system .
the problem will then be to find simultaneously the optimal times of maintenance and optimal repair levels .
mathematically , it is an impulse control problem , which complexity exceeds widely that of the optimal stopping .
second , our method requires a perfect observation of the state process at the jump times .
it would be interesting to extend our results to a noisy observation of the process , as often happens in real life .
this work was partially funded by the french national research , anr grant fautocoes , program arpege , anr-09-segi-004 .
the authors also thank pr .
franois dufour for initial discussions on how to define an interesting optimal stopping problem for the test - case of the tank .
this paper was presented at esrel 2012 conference .
we present a numerical method to compute an optimal maintenance date for the test case of the heated hold - up tank . 
 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 rates of the components depends on the temperature , the position of the three components monitors the liquid level in the tank and the liquid level determines the temperature . 
 therefore , this system can be modeled by a hybrid process where the discrete ( components ) and continuous ( level , temperature ) parts interact in a closed loop . 
 we model the system by a piecewise deterministic markov process , propose and implement a numerical method to compute the optimal maintenance date to repair the components before the total failure of the system .