Optimal Maintenance Decisions for Hydraulic Structures under Isotropic Deterioration by Jan M. van Noortwijk (Summary of Ph.D. Thesis)

To protect the Dutch lowlands against flooding, a flood defence system has been constructed in which The Netherlands is subdivided into fifty-three dyke-ring areas. These areas are surrounded by dyke rings consisting of dunes, dykes, special water-retaining structures (e.g. the Eastern-Scheldt storm-surge barrier), and higher ground. Each component of a dyke ring has to fulfill certain requirements in the areas of flood protection, environment, recreation, shipping access, road connection, transport, agriculture, fishery, and landscape. As soon as a component deteriorates to such a degree that it fails to meet its main requirements, it should be maintained; preferably against minimal costs. The Ph.D. thesis is devoted to determining cost-optimal maintenance decisions for hydraulic structures subject to deterioration.

Maintenance is defined as a combination of actions carried out to restore a hydraulic structure to, or "renew" it to, its desired condition. In hydraulic engineering, expensive condition-based preventive maintenance, i.e. maintenance based on inspecting or monitoring a structure's condition, is mainly applied. In The Netherlands, the attention is shifting from building structures to maintaining structures and the use of maintenance optimisation models is therefore of considerable interest.

There are two phases of a structure's life cycle in which it is economic to apply maintenance optimisation techniques: the design phase and the use phase. In the design phase, one might obtain an optimum balance between the initial costs of building and the future costs of maintenance and failure (being the area of life cycle costing). In the use phase, one might minimise the costs of inspection, repair, replacement, and failure. A large number of papers on maintenance optimisation models, mainly focussing on the mathematical aspects, have been published. Unfortunately, since the use of these models is restricted to situations in which abundant data is available, only a few of them have been applied.

In hydraulic engineering, a distinction can often be made between a structure's resistance (e.g. the crest-level of a dyke) and its design stress (e.g. the maximal water level to be withstood). A failure may then be defined as the event in which - due to deterioration - the resistance drops below the stress. Since deterioration is uncertain, it can best be regarded as a stochastic process.

Even though it is common to model a deterioration process mathematically as a so-called 'Brownian motion with drift' (a stochastic process with stationary independent decrements and increments having a normal distribution), the 'Brownian motion' is inadequate in describing the deterioration of hydraulic structures. To illustrate, a dyke whose height is subject to a Brownian deterioration can, according to the model, spontaneously rise up, which cannot occur in practice. Furthermore, in most applications there is only information available in terms of a probability distribution (uncertainty distribution) of the average rate of deterioration.

In order that a stochastic deterioration process has the desired properties, we consider it as a so-called 'generalised gamma process'. A gamma process is a stochastic process with independent non-negative increments (e.g. the increments of crest-level decline of a dyke) having a gamma distribution with known (certain) average rate. A generalised gamma process is then defined as a so-called 'mixture' of gamma processes, where the mixture represents the uncertainty in the unknown average rate. In addition to the classical characterisation of gamma processes in terms of compound Poisson processes, the thesis presents two new mathematical characterisations of generalised gamma processes: (i) in terms of conditional probability distributions (given a cumulative amount of deterioration which serves as a summarising, sufficient, statistic for the unknown average rate) and (ii) in terms of isotropic probability distributions (an l_p-istropic probability distribution can be written as a function of the l_p-norm).

A useful property of the generalised gamma process is that various probabilistic properties, such as the probability of exceedence of a failure level per unit time, can be expressed in explicit form when the average rate of deterioration is given. In mathematical terms, this means that we can always find units of time of equal length for which the joint probability density function of the increments of deterioration can be written as a mixture of exponential probability densities. This mixture represents the uncertainty in the unknown average rate of deterioration. Since the probability density function of any finite sequence of increments can then be written as a function of the sum of the increments (i.e. the l_1-norm of the increments), the infinite sequence of increments is said to be l_1-isotropic or l_1-norm symmetric. Due to the exchangeability of the l_1-isotropic increments of deterioration, the expected cumulative amount of deterioration is linear in time.

To make optimal maintenance decisions while explicitly taking account of the uncertainty in the average rate of deterioration, statistical decision theory can be used. A decision-maker can then best choose a maintenance decision whose expected monetary loss (in terms of the expected costs of maintenance and failure) is minimal; such a decision is called an optimal decision. The expected loss is determined with respect to the probability distribution of the average rate of deterioration, which can be updated with (new) observations using Bayes' theorem.

The maintenance of hydraulic structures can best be modelled as a so-called 'renewal process', where the renewals are the maintenance actions restoring a structure to its desired condition. After each renewal we start, in a statistical sense, all over again. Since the planned lifetime (including maintenance) of the Dutch dyke rings is essentially unbounded, maintenance decisions can best be compared over an unbounded time-horizon. There are basically three cost-based criteria that can serve as loss functions:

  1. the expected average costs per unit time (which are determined by averaging the costs over an unbounded horizon);
  2. the expected discounted costs over an unbounded horizon (which are determined by summing the present discounted values of the costs over an unbounded horizon); and
  3. the expected equivalent average costs per unit time (which are determined by averaging the discounted costs).
These cost-based criteria can be computed using renewal theory (the discrete renewal theorem). The notion of equivalent average costs relates the notions of average costs and discounted costs. Although in the literature most attention has been focussed on the criterion of average costs, the cost-based criteria of discounted costs and equivalent average costs are most suitable for optimally balancing the initial building costs against the future maintenance costs.

On the basis of generalised gamma processes, tailor-made models have been built and implemented to enable optimal maintenance decisions to be determined for four characteristic components of a dyke ring:

The models for sand nourishment and dyke heightening are examples of maintenance optimisation in the design phase, whereas the two inspection models are examples of maintenance optimisation in the use phase.

In the last part of the thesis, two decision models have been presented which are not directly based on deterioration processes: one model for evaluating and comparing decisions that reduce flood damage along the Meuse river (by using l_1-isotropy and discounting) and one model for optimising maintenance when the uncertainty in failure probabilities can be expressed in terms of a Dirichlet distribution (this model is useful when both resistance and stress are stochastic).

Although the decision models in the thesis have primarily been developed for the maintenance of beaches, dykes, berm breakwaters, and the Eastern-Scheldt barrier, they can also be applied to other hydraulic structures and other engineering systems for solving many decision problems in maintenance optimisation and life cycle costing.

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