AbstractUK railway drainage systems are facing increasing challenges due to poor completeness of the asset inventory, long asset life cycles, more intense use of the UK railway system, and a changing climate. It is therefore important for drainage managers to acquire a better understanding of the current and future condition of the drainage assets for which they are responsible. This study presents a Markov model for simulating the potential future service condition of various classes of UK railway drainage assets based on observed historical changes in asset condition. Linear regression analysis was performed on distinct asset groups and the influence of the characteristics of asset construction material, size, shape, and location on the rate of the degradation process was quantified. These results were incorporated with the continuous time Markov chain model to improve the accuracy of the degradation rate prediction for several drainage asset classes. The model is illustrated on a case study of the Network Rail drainage assets showing the minimum number of samples required to make a reliable estimation of the service condition degradation process.IntroductionNetwork Rail (NR) is the owner of the vast majority of railway infrastructure, including associated drainage systems, in England, Scotland, and Wales. The rail network is divided into nine strategic geographical routes (pre-2020), each responsible for its own day-to-day asset management decisions, with standards, assurance, and support systems provided by a technical authority from a central strategic department. Each consecutive 5-year period is referred to as a control period (CP) for NR; a strategic business plan is agreed at the beginning of a control period stating goals and objectives for the period. For drainage, asset management plans are created with the aim of developing strategies to prevent increase in risks to passengers, workers, and members of the public due to drainage asset failure, while minimizing whole life, whole system costs.There is an ever-increasing recognition that effective and reliable drainage systems can significantly enhance the operational performance of the entire railway system (Drainage Asset Policy, unpublished report, 2017). Inadequate hydraulic capacity in railway drainage systems can cause unexpected trackside flooding, which can lead to temporary speed restrictions or temporary closures of railway lines. In the past 5 years, there were on average 450 flooding events per year, which caused 0.3 million hours of delay each year, leading to a compensation costs to Network Rail at an average of £17 million per year. Such cost is made up solely of payments to impacted train operation companies and does not include the cost of replacing damaged assets.Adequate control of water is also crucial to the management and maintenance of other railway infrastructure, such as tracks, track beds, earthworks, and signaling (known as parent assets). This is because water can play a role in many failure mechanisms that affect parent assets, such as the long-term degradation of the stiffness of the materials that form the track support system and earthworks (Drainage Asset Policy, unpublished report, 2017). An impaired drainage system can thus result in damage to parent assets, and hence further disruption to train operation as well as higher maintenance costs and risk to human safety. It is NR’s major concern to eliminate the safety consequences of drainage failure, such as derailments and injuries of passengers and NR workers.Flooding would occur when there is a lack of local hydraulic capacity in the system, which could be caused by inadequately designed capacity, asset degradation, change in land use/land cover, or increased load due to climate change. All drainage assets are expected to be designed and built in accordance with NR’s design standard with the hydraulic capacity to operate for a rainfall event of a specified return period and duration. A 1 in 10-year return period is the lowest standard; therefore, all the drainage systems are expected to withstand a 1 in 10-year rainfall event. However, results from flooding events analysis of the 2,250 cases of flooding incidents recorded during the last 5 years show that around 95% of flooding happened with precipitation less than the expected rainfall volume of a 1-day-duration, 10-year return period rainfall event. Since design standards can be expected to be followed in a regulated industry, this preliminary analysis provides evidence that this flooding could be either due to poor design or due to the degradation of installed drainage assets from their original condition.Degradation is reflected by changes in asset condition. At NR, the asset condition is split into two parts: the structural condition and the service condition: •Structural condition: the fabric of the asset and the severity of structural defects that affect its integrity. Structural defects can be addressed by repairing or replacing the asset.•Service condition: defects that affect the performance of the asset and the severity of the defects that reduce its hydraulic capacity below the original design level. These defects may be independent of the structural condition or may be linked. Service defects can be addressed by maintenance of the asset such as cleansing or vegetation clearance.As stated in CIRIA C714 (CIRIA 2014), the effective management and maintenance of the drainage network requires knowledge of the asset inventory, its previous and current condition, hydraulic capacity, historical performance, and current status. NR is in the process of improving its drainage asset knowledge by scheduling surveys and inspections to verify the existing data record and identify unrecorded assets, achieving a 25,000 asset inventory increase in the last control period (April 1, 2014–March 31, 2019). The degradation process controls how hydraulic performance has changed since the last inspection, or will change in the future. It is therefore important to study the degradation process of the assets and develop appropriate modeling tools to enable a better understanding and estimation of the status of drainage assets.Literature ReviewThere are many factors thought to influence the rate of infrastructure asset deterioration such as asset type, age, size, material, and local soil characteristics. Railway drainage systems are composed of buried drainage pipes linked at catchpits (chambers that provide inspection access and allow sediment to settle), which drain via outlets to adjacent surface water bodies; they operate by gravity and so are analogous to stormwater sewers. There is little information on the deterioration of railway drainage systems. In Ana et al. (2009), an investigation into the important factors affecting pipe deterioration in the sewer network of Leuven (Belgium) was carried out using logistic regression. It revealed that out of the 10 variables considered, age, material, and length are the only three that significantly affected the pipe service condition. However, by comparing results with similar studies in UK (Ariaratnam et al. 2001) and Canadian networks (Davies et al. 2001), they found that each of the studied networks has a slightly different set of significant variables, and thus concluded that there is no single set of variables that can explain sewer deterioration; it seemed to vary from one network to another.Due to the many uncertainties in the deterioration process such as unobservable explanatory variables and measurement errors, deterioration is often predicted using a probabilistic model to capture its stochastic nature. Although deterioration models for railway drainage systems have not been developed, studies of other piped systems such as sewer systems and stormwater pipes have been published.Some of the models only describe two states, deficient or nondeficient, such as the cohort survival model proposed by Herz (1998), which is based on the Herz distribution, determining the lifetime probability distribution derived from the current stock of pipes. It has been applied to drinking water distribution networks to predict the future rehabilitation need. However, these methods only provide information on when assets are expected to fail; they lack the condition information in between functioning and failure, which is critical for asset managers in formulating a planned maintenance regime.The Markov approach is a probabilistic model widely used for simulating infrastructure deterioration that can describe systems with multiple condition states. Micevski et al. (2002) developed a Markov model for the structural deterioration of stormwater pipe infrastructure, where the Markov transition probabilities were estimated using the Metropolis-Hastings algorithm. Both Baik et al. (2006) and Wirahadikusumah et al. (2001) presented the use of a Markov chain–based deterioration model in sewer pipes. While Baik et al. (2006) used the ordered probit model to estimate the probability of deterioration, Wirahadikusumah et al. (2001) used nonlinear optimization focused only on structural deterioration. Kleiner et al. (2010) introduced the nonhomogeneous Poisson process (NHPP) for future prediction of the structural failure for an individual water main, considering both static factors (i.e., pipe intrinsic) and dynamic factors (e.g., climate, cathodic protection, breakage history). Markov models have also been applied to many infrastructures other than piped systems such as bridges and pavements. Mizutani et al. (2017) used a Markov model to predict reinforced concrete bridge elements deterioration due to chloride-induced corrosion of the reinforcement, using Bayesian statistics as an estimate for transition probabilities when there is little to no available time series inspection information. Wellalage et al. (2015) presented a Metropolis-Hasting algorithm–based Markov chain Monte Carlo simulation approach to calibrating Markovian bridge deterioration models using inspection data for 15 years on Australian railway bridges. Surendrakumar et al. (2013) provided a Markovian probability process to predict the future condition of the pavement, which can be used to design a decision support system for pavement maintenance management.Neural networks are thought to be of relevance because they are particularly effective in dealing with data that have high volatility and nonconstant variance. Tran et al. (2006) used neural networks to predict the condition of stormwater pipes and Najafi and Kulandaivel (2005) used them on sewer networks. The probabilistic neural network (PNN) model developed by Tran et al. (2006) was tested with snapshot-based sample data and compared with a traditional parametric model using discriminant analysis. The data set is consistent of 650 data points taken by closed-circuit television (CCTV) inspections, obtained from 27 km of the total 800 km of stormwater pipes in the City of Greater Dandenong, Australia. The structural and hydraulic conditions are graded into three levels: (1) good, (2) fair, and (3) poor. Results show it slightly outperforms others in terms of prediction performance; however, the accuracy of the model is still not high because the percentage of correct prediction of PNN is only 66.9%, and also the key factors for prediction are difficult to interpret.Markov models and neural networks are both widely used for modeling infrastructure degradation. However, neural network models essentially classify the assets into different condition groups based on some input factors mostly consisting of asset characteristics and surrounding geographical conditions; they do not actually simulate the degradation process, and hence would not be able to capture the stochasticity in the degradation process. Also, because they rely heavily on the quality and quantity of the input factors, they are not suitable for systems that have a limited amount of such information.There currently does not exist any serviceability deterioration model for railway drainage pipe systems. Hence, this paper will be focused on predicting the service condition of drainage assets using the Markov model, so as to eventually be able to predict the impact of condition deterioration on the long-term loss of service performance. Although Network Rail records both structural condition and service condition, only service condition is used in this paper since it relates more directly to hydraulic performance and thus flood risk. However, the same approach used here could be applied to the structural condition scores to model structural degradation.MethodologyInput DataAs stated in the Drainage Asset Policy (unpublished report, 2017), the service condition is measured on a 1–5 grading system as illustrated in Table 1. The system adopted is compatible with guidance from CIRIA (2014). NR has service conditions recorded for 88% of their drainage assets. These condition scores will be used to build a model for predicting future states of drainage assets. In the “Case Study” section, this will be applied to an exemplar group of railway drainage assets.Table 1. Description of service condition scoreTable 1. Description of service condition scoreService conditionDescription1Clear2Superficial deposits with no loss of capacity3Capacity slightly reduced4Capacity severely reduced5Blocked or unsafe conditionMarkov Model FrameworkIn this study, a Markov chain approach is used to model the degradation rate, which gives an estimation of transition probability from one state to a lower state. This decision is made under the assumption that the probability of degradation depends only on the current condition of the asset. Such an assumption is made based on expert opinion and will be verified subsequently. Since drainage assets could degrade to a worse state any time during the year, in order to correctly estimate the adverse effect of degradation on the drainage capacity throughout the year, a Markov model with continuous time steps is chosen because it is believed to better reflect the degradation process of railway drainage assets.Because the change of condition is expected to happen at any time during the useful working life, a continuous time Markov chain (CTMC) is used, which is described by a stochastic process X={X(t)|0≤t} with discrete state space S={s1,s2,…,sn} that satisfies the following for any time s,t≥0, and i,j∈S: (1) P(X(s+t)=j|X(s)=i,{X(u):0≤ui+1 is also included in the model. With the transition rate matrix Q, the probability matrix P for any arbitrary time interval s to t can be obtained by P(s,t)=exp((s−t)Q).Verification of the Markov PropertyFor the Markov property to stand, it is necessary to prove that the probability of an asset degrading into score j with a given current score i is not related to its previous conditions.This can be done by analyzing the three state transition sequence (Xt|Xt−1,Xt−2) of the historical data set, where (Xt=i|Xt−1=j,Xt−2=k)=(i|j,k) represents an asset condition jump from j to i, given that the previous condition before j is k, i.e., the condition transfer from state k to state j and then state i. If the Markov property holds, for any given i and j, there would be no difference in the probability of the sequence (i|j,Xt−2) to exist, for all Xt−2

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