Estimating burst probability of water pipelines with a competing hazard model

Water-supply pipelines, which form important components of the infrastructure of cities, require huge annual maintenance budgets. Consequently, the establishment of optimal regimes for maintaining water pipeline systems has become a major issue for water-utility managers throughout the world. In the management of infrastructure assets, optimal maintenance strategies are frequently based on lifecycle-cost analysis, which is dependent on the deterioration model (Kobayashi et al. 2010).

In the field of water-supply systems, many studies have been conducted to assess the condition of pipeline systems and to predict their deterioration process. Shamir & Howard (1979) and Marvin (1996) assumed that breaks in pipelines increase exponentially with their age, and they obtained break-prediction models by using regression analysis. Clark et al. (1982) reported a method for estimating the expected failure time of pipelines, whereas Shinstine (1999) examined the relationship between pipeline breaks and the diameters of pipes. Because water pipeline systems are usually buried underground, the monitoring and inspection of such systems is difficult and it is hard to accumulate adequate observational data for use as a basis for deterioration forecasting analysis. Because of the difficulties in observing deteriorations of pipelines directly, we decided to predict the deterioration of pipelines by examining failures caused by the deterioration process.

Marks (1985), Constantine & Darroch (1995), and Park (2004) used proportional hazards models, based on the failure-prediction model proposed by Cox (1972), to predict the risk of a pipeline break. Many probabilistic models that use various probability functions have been developed with the average annual number of pipe breaks on the pipeline systems as an indicator of the structural state and the times to failure between pipe breaks considered as random variables (Le Gat & Eisenbeis 2000; Mailhot et al. 2000, 2003; Pelletier et al. 2003). These models have overcome challenges that observation data typically show properties, right censored observations (Eisenbeis et al. 1999; Mailhot et al. 2000), left truncation (Mailhot et al. 2000) and selective survival bias (Scheidegger et al. 2013). By setting the deterioration state as a binary condition, ‘failure’ or ‘normal operation’, it is possible to predict the service life of a pipeline by using a conventional hazard model. There are numerous reports of studies in which this type of deterioration prediction method has been applied to other types of system. Aoki et al. (2005) proposed a method in which a Weibull hazard model is used to predict the lifetime of tunnel lighting equipment. Tanaka et al. (2010) similarly used Weibull hazard models to predict the deterioration of pipelines.

In general, the major cause of interruption of water pipeline systems is deterioration of the pipes. In the conventional models for the prediction of pipeline deterioration prediction, the type of pipe failure is not classified, and all failures are considered as a single type of failure. In a real pipeline system, however, pipe failures caused by deterioration appear in various forms. We therefore classified pipeline failures as ‘B-burst’, which occur in the pipe body, or ‘C-burst’, which occur in pipe-connection parts. The lifetime of a given part is defined as the period from its installation to burst, and in this study it is assumed that the burst is regarded as major damage and the damaged pipeline is replaced immediately. The Weibull deterioration-hazard model is used to address the lifetime of each pipeline, and takes into account the nature of the competition between several types of failure by using a competing deterioration-hazard model. The deterioration of the pipeline is predicted by developing a competing deterioration-hazard model that considers competition between C-burst and B-burst. The proposed competing deterioration-hazard model allows us to determine the probability density of bursts in the pipe body and connection.

The competing hazard model assumes that competing causes of failure are independent of one another and that the incidence of each cause of failure can be analyzed from lifetime data. Such methods have been used in many fields, including medicine, economics, and engineering. The competing hazard model is widely used in accelerated lifetime testing (ALT) to estimate the lifetime distribution of components. Nelson (1990) discussed an analysis of typical competing hazard models for constant stress ALT data. Kim & Bai (2002) reported a competing hazard model that considered only two competing causes of failure by using ALT data.

Because the pipeline systems are underground, system administrators face difficulties due to insufficient amounts of observation data. Thus, the insufficiency of data interrupts the practical application of the statistical model. In order to overcome this problem, in this study, the competing deterioration-hazard model is estimated by a Bayesian technique based on the Metropolis–Hasting method (M-H method), a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult.

Competing hazards appear in cases in which two or more events can occur. The main idea of a competing hazard model is that the occurrence of an event of interest has to be taken into account while considering the occurrence of competing events. In pipeline systems, the case in which a pipeline is replaced because of a B-burst can be considered. However, there might be the events that could lead to replacement of the pipeline, such as C-bursts. It is therefore possible to assume that a C-burst is a competing event, if we assume that a B-burst is the event of interest, because a B-burst interrupts the occurrence of a C-burst. To introduce the competing hazard among the pipe failure types, the major damage which requires pipe replacement is considered. In this study we focus on the B-burst and C-burst.

Herein, we classify the state of a pipeline as being one of two condition levels: a ‘healthy condition’ and a ‘burst’ resulting from B-burst or C-burst. It is assumed that the burst denotes a state in which major damage is found and replacement is required immediately. On the other hand, the healthy condition reflects not only a normal operation but a condition where no major damage is found. In addition, it is also assumed that the records of past repaired incidents, leakages or breaks are not considered as burst because these incidents would not be major damage.

The pipeline system must be discretized into pipe segments (Mailhot et al. 2000). A pipe segment is defined as the same as a pipeline is a series of pipes with relatively homogeneous characteristics such as pipe diameter and type of material and installation period.

In the MLE method, huge amounts of data are required to secure precision but it is not always possible to accumulate a great number of data, especially in the pipeline systems. Pipeline system administrators face difficulties of insufficient amounts of observation data because pipeline systems are underground. The Bayesian estimation method can provide estimation results by fusing prior information, such as human experience and expert knowledge, with an insufficient amount of observation data (Kobayashi et al. 2012). In addition, the Bayesian estimation method is easy in comparison with the MLE method because in the Bayesian estimation method it is not required to derive the Jacobian and Hessian matrices. Furthermore, in the high dimensional nonlinear equation problem, the equation has multiple local optimum solutions. In this case, a poor choice of starting point in the MLE method can cause converging to a local optimum that is not the global optimum, or failure to converge entirely. The competing deterioration-hazard model is a high dimensional nonlinear expression of parameter and the optimization problem may have a large number of solutions including complex valued solutions. Thus, in this case, using the Bayesian estimation method instead of the maximum likelihood estimation method can solve the high dimensional nonlinear multinomial expression. In this section, we present a methodology for estimating the unknown parameter vector of the competing deterioration-hazard model by means of a Bayesian estimation method using observed data.

The M-H method is used to perform sampling from an empirical distribution that is similar to and accordingly obtains samples from the original distribution (Kobayashi & Kaito 2012). Furthermore, a random walk is used to improve the efficiency of sampling. The M-H method is described below.

Step 1. Initial establishment

The initial value of parameters ⁠, the number of iterations for parameter sampling ⁠, and the burn-in period are established. In addition the stride of the random walk is set.

Step 2. Sample extraction for estimation of the parameter

When the number of simulations is ⁠, the parameter estimation is generated as described in Steps 2-1 to 2-3.

Step 2-1

Step 3. Final judgment of the algorithm

Step 2 is repeated until the number of samplings reaches N.

The samples are then accumulated except for those that were generated during the burn-in period. If the number of samples N is sufficiently large, the parameters estimated by using the above algorithm will converge on the estimated value of the posterior distribution. Geweke test statistics (Geweke 1992) are used to test whether the sampling process of the M-H method reaches a steady state and the number of samplings N is appropriate or not.

To analyze the deterioration of a real pipeline, we focused on the water distribution system of S city in South Korea. The pipe material, ductile cast iron pipe (DCIP), is regarded as the target for this study. The whole data of DCIPs comprise approximately 26,500 pipelines, 850 km in length, with an average age of around 13 years. Inspection data were obtained from historical records for pipe bursts in S city during the 9-year period 2001–2009. During this period, 1,405 cases of pipe replacement caused by B- and C-burst were recorded. Here, in this study, it is assumed that the replaced pipelines had major damage and its condition state is classified as burst. On the other hand, the historical records of past repair are not considered as burst because a repair is not associated with major damage. Table 1 shows the basic information of the data used in this study.

The inspection data contain information on whether or not pipe burst occurred and the type of burst for each damaged pipeline. In this study, the type of burst is classified as either a B-burst in a pipe body or a C-burst in pipe connections. Accidents that occurred in other subcomponents, such as valves, rubber packings, and so on, are neglected. On the other hand, the pipe diameter and length are used as characteristic information that affects pipe burst. On the basis of this information, the duration of survival before burst of a pipe is expressed by using the Weibull hazard model, and the competing deterioration-hazard model is used to consider the competition between C-bursts and B-bursts in the pipeline. The model is then estimated by using the Bayesian estimation method.

In order to improve the precision of estimation, the Bayesian updating rule (Kobayashi et al. 2012) is used. We created three different data groups (⁠⁠, ⁠, ⁠) which are extracted based on original data set. Here, the subscript numbers denote the number of extracted data. The estimation is performed in the order of the small size of the data and the estimation results (the mean, variance and covariance) are used as prior information of the next estimation using the Bayesian updating rule.

To conduct the M-H method, the number of iterations required to reach a steady state (the burn-in period) was set to and the number of iterations for parameter sampling was set to ⁠. The 10,000 burn-in samples were omitted and the remaining 10,000 parameter samples were used to carry out the estimation.

Table 2 shows the results of the Bayesian estimation of competing deterioration-hazard models for each of the databases ⁠, ⁠, and original data set. The estimations obtained by the M-H method show the probability distribution of the parameters. In Table 2 the values estimated by the Bayesian estimation method are the sample average of parameters, and the values in parentheses refer to 95% credible intervals. All the credible intervals of estimated parameters do not contain zero. Because all the 95% confidence intervals do not contain zero, the estimated values will be significant at the 5% level (Wu & Hamada 2009). As shown in Table 2, as the amount of observation data increases the credible intervals become narrower. The absolute value of the Geweke test statistics shown in italic type are all less than 1.96, so the convergent hypothesis cannot be dismissed at a significance level of 5%.

Table 2

Results of estimation of parameters for the competing deterioration-hazard model

Parameters .	D2000 .	D5000 .	D10000 .	Original data set .
	−11.681 (−12.874, −10.617)	−9.955 (−10.717, −9.293)	−9.504 (−9.963, −9.055)	−9.594 (−9.941, −9.277)
	0.053	0.086	0.036	0.069
	−3.284 (−5.175, −1.165)	−2.929 (−4.369, −1.534)	−0.763 (−1.615, 0.047)	−0.994 (−1.595, −0.406)
	0.134	0.035	0.060	0.011
	6.344 (1.065, 10.737)	5.194 (2.239, 7.885)	2.755 (1.412, 3.800)	2.657 (1.941, 3.284)
	0.042	0.129	0.124	0.038
	−12.461 (−14.264, −10.842)	−10.332 (−11.358, −9.374)	−10.166 (−10.790, −9.644)	−10.094 (−10.429, −9.768)
	0.139	0.200	0.067	0.226
	−2.489 (−5.115, 0.287)	−3.100 (−5.468, −0.822)	−1.228 (−2.422, −0.189)	−1.884 (−2.863, −1.022)
	0.391	0.354	0.197	0.112
	4.510 (−0.797, 7.570)	8.126 (5.808, 10.720)	3.396 (1.966, 4.409)	3.106 (2.442,3.764)
	0.303	0.036	0.145	0.001
	2.954 (2.600, 3.398)	2.379 (2.178, 2.601)	2.199 (2.051, 2.338)	2.256 (2.161, 2.360)
	0.090	0.120	0.032	0.070
	3.001 (2.528, 3.537)	2.370 (2.078, 2.630)	2.306 (2.147, 2.492)	2.338 (2.237, 2.449)
	0.083	0.105	0.003	0.188

Parameters .	D2000 .	D5000 .	D10000 .	Original data set .
	−11.681 (−12.874, −10.617)	−9.955 (−10.717, −9.293)	−9.504 (−9.963, −9.055)	−9.594 (−9.941, −9.277)
	0.053	0.086	0.036	0.069
	−3.284 (−5.175, −1.165)	−2.929 (−4.369, −1.534)	−0.763 (−1.615, 0.047)	−0.994 (−1.595, −0.406)
	0.134	0.035	0.060	0.011
	6.344 (1.065, 10.737)	5.194 (2.239, 7.885)	2.755 (1.412, 3.800)	2.657 (1.941, 3.284)
	0.042	0.129	0.124	0.038
	−12.461 (−14.264, −10.842)	−10.332 (−11.358, −9.374)	−10.166 (−10.790, −9.644)	−10.094 (−10.429, −9.768)
	0.139	0.200	0.067	0.226
	−2.489 (−5.115, 0.287)	−3.100 (−5.468, −0.822)	−1.228 (−2.422, −0.189)	−1.884 (−2.863, −1.022)
	0.391	0.354	0.197	0.112
	4.510 (−0.797, 7.570)	8.126 (5.808, 10.720)	3.396 (1.966, 4.409)	3.106 (2.442,3.764)
	0.303	0.036	0.145	0.001
	2.954 (2.600, 3.398)	2.379 (2.178, 2.601)	2.199 (2.051, 2.338)	2.256 (2.161, 2.360)
	0.090	0.120	0.032	0.070
	3.001 (2.528, 3.537)	2.370 (2.078, 2.630)	2.306 (2.147, 2.492)	2.338 (2.237, 2.449)
	0.083	0.105	0.003	0.188

Values in (·) show 95% credible intervals and values shown in italic type in each row are the Geweke statistical test.

View Large

Figures 2 and 3 also show that the survival probabilities for both C-burst and B-burst decrease over time and that the survival probability for C-burst decreases more rapidly than that for B-burst. In other words, in a ductile cast-iron pipe, bursts in pipe connections (C-bursts) occur at a higher rate than bursts in the pipe body (B-bursts).

A pipe deterioration model is important for the asset management of pipeline systems. Pipe failures caused by deterioration appear in various forms. Thus, a deterioration forecasting model which considers failure types enables the establishment of an efficient rehabilitation strategy. We have developed a competing deterioration-hazard model that considers competition among several types of burst in pipeline systems and the proposed model allows us to determine the probability of burst for each type of burst. The competing deterioration-hazard model is estimated using Bayesian estimation method.

The empirical study was carried out by using an inspection data set of a real pipeline system. In the empirical study, because of the absence of detailed substantive knowledge, the competing deterioration-hazard model is estimated with non-informative prior distribution and the Bayesian updating rule is used to improve the precision of estimation. The results show that the more estimation results are updated, the more precise estimation results can be obtained. The estimation results obtained from the database show almost the same results obtained from the original data set. This result indicates that the Bayesian updating rule improves the efficiency of model estimation and data acquisition. In addition, in this study, although we used non-informative prior distribution because of the absence of detailed substantive knowledge, if we can accumulate prior information, the proposed method would be a good way forward.

According to the results of the occurrence probability prediction of C- and B-burst obtained by competing deterioration-hazard model, more care is necessary for the pipe connection because the probability of pipe burst in a pipe connection (C-burst) is higher than that in pipe bodies (B-burst). In addition, the results show that the conventional Weibull deterioration-hazard model, which does not consider competing properties, overestimates pipe burst rates. The bias which arises between the competing deterioration-hazard model and the conventional Weibull deterioration-hazard model comes from the feature of competing hazard model that considers the occurrence of a competing event when the probability of the event of interest is sought. Even though the prediction accuracy of the competing deterioration-hazard model is slightly high in comparison with the conventional Weibull deterioration-hazard model, it is noteworthy that there is the potential for improvement. The proposed competing deterioration-hazard model can be improved further if we could overcome some problems such as left truncated data and high percentage of right censored data, which were not considered in this study. In addition, it is required that much more observed data set and empirical studies are accumulated.

In this study, we classified the pipe burst type into B-burst which occurred in the pipe body and C-burst which occurred in the pipe connection. As shown in the results, we were able to see that the C-burst and B-burst had different deterioration rates. Because the choice of a maintenance and repair method will depend on the type of burst, the competing deterioration-hazard model enables us to establish an optimum maintenance strategy for the pipeline system. In addition, we believe that our new model can be extended to other items of infrastructure and will contribute to advancing asset management.

Our proposed model has not discussed the following points, which are considered for a future extension of our study:

• (1) In this paper, most of the failures that typically affect real pipeline systems (i.e. pipe breaks, leakages, etc.) are disregarded. To establish optimal maintenance strategy, it is important to consider the repairs due to breaks or leakages.

• (2) The limited and missing information, left-truncated or survival selection, which are often embedded in observed data, have not been mentioned.

• (3) Supposedly, considering competing hazards would be more relevant if more than two competing hazards exist. This could be explored by using synthetic data and considering different amounts of competing failure types.