Improving Railway Track Maintenance Using Power Spectral Density (PSD)

IMPROVING RAILWAY TRACK MAINTENANCE USING POWER SPECTRAL DENSITY (PSD) ABDUR ROHIM BOY BERAWI A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN TRANSPORT SYSTEMS Supervisor: Professor Doctor Raimundo Delgado Co-Supervisor: Professor Doctor Rui Calçada AUGUST 2013 i GENERAL TABLE OF CONTENTS GENERAL TABLE OF CONTENTS .............................................................................. i ABSTRACT .................................................................................................................... iii RESUMO ......................................................................................................................... v ACKNOWLEDGEMENT .............................................................................................. vii TABLE OF CONTENTS ................................................................................................. ix LIST OF FIGURES ....................................................................................................... xiii LIST OF TABLES ......................................................................................................... xix ABBREVIATIONS ....................................................................................................... xxi LIST OF NOTATIONS ............................................................................................... xxiii 1. INTRODUCTION .................................................................................................... 1 2. LITERATURE REVIEW .......................................................................................... 9 3. RESEARCH METHODOLOGY ............................................................................ 79 4. THE APPLICATION OF POWER SPECTRAL DENSITY (PSD) IN TRACK QUALITY ASSESSMENTS ................................................................................... 89 5. CORRELATION ANALYSIS OF RAILWAY TRACK GEOMETRY ............... 111 6. THE APPLICATION OF AN OPTIMIZATION MODEL FOR TRACK MAINTENANCE .................................................................................................. 147 7. RESULTS AND DISCUSSIONS .......................................................................... 189 REFERENCES ............................................................................................................. 199 APPENDICES .............................................................................................................. 209 ii iii ABSTRACT The implementation of Power Spectral Density (PSD) for assessing the track quality condition is relatively new in the area of railway tracks. Most of the Infrastructure Managers (IM‟s) tend to use the Track Quality Index (TQI) method, which is typically a statistical function of the standard deviation of each geometrical defect. In comparison with the PSD technique, TQI has some obvious disadvantages since, for example, it cannot indicate a specific problem that exists on a track whereas PSD can. This research was conducted in response to a need for a rigorous approach to the development of a track degradation model for the purpose of track maintenance. The Power Spectral Density (PSD) forms the core focus for this research since it provides a systematic technique for evaluating track quality condition. To achieve the underlying objective, the research was divided into two major phases. The first phase attempted to examine the application of power spectral density in the track quality assessments. The investigation was then further continued by evaluating the existing relationship between various track geometry parameters. This phase is of particular importance towards establishing a reasonably accurate model of track degradation, which takes into account the interactions among various geometry variables. The second phase was conducted by developing a predictive degradation model which may capture the evolution of track quality in terms of statistical index and frequency spectrum. The results obtained from this model together with a track recovery model were then applied to analyze different maintenance scenarios. The research findings indicate that some variables of track geometry are closely related. For instance, the strongest positive relationship can be found between the left and the right rails, in both longitudinal profile and alignment. Based on the coherence analysis, the variations of longitudinal profile in both rails are similar for wavelengths longer than 6 m while alignment exhibits a strong relationship for wavelengths longer than iv 66 m. Typically, the most detrimental wave among various track geometries can be found at a wavelength band between 6-30 m. In correspondence with the application of the maintenance model, the results showed that the influence of the proposed track quality criteria (with TQI or PSD limits), the adopted variables of geometry defect (longitudinal profile or alignment) and selected maintenance strategies (preventive, delay, or the combination of regular and corrective maintenance) are key factors in determining the maintenance decision. From the analysis results, the preference for the track quality criteria of TQI may reduce the number of maintenance actions by 7% in relation to the use of PSD. A reduction in the number of tamping actions might also be achieved if the track maintenance solely considered defects in the longitudinal profile instead of in the alignment variable. The declination ranges up to 52% with respect to PSD criteria and 22% with respect to TQI criteria. Finally, the selected strategy of delayed maintenance has proven to be more efficient for tamping decision than preventive and the combination of regular and corrective maintenance. Keywords: power spectral density; track quality index; ballasted track; track maintenance v RESUMO A implementação de espectros de densidade de potência (PSD) para avaliação da condição da via é relativamente recente na área de vias ferroviárias. A grande maioria dos Gestores de Infraestruturas (IM‟s) utiliza o método do Índice de Qualidade da Via (TQI), o qual é tipicamente uma função estatística do desvio-padrão de cada defeito geométrico. Relativamente ao método PSD, a aplicação do TQI possui algumas desvantagens como, por exemplo, não ser possível indicar um problema específico existente na via. Por seu lado, o PSD consegue dar resposta a este tipo de problemas. A presente investigação foi conduzida no sentido de satisfazer a necessidade de uma abordagem rigorosa ao desenvolvimento de um modelo de degradação da via para manutenção da via. O método PSD representa o principal foco de interesse da presente investigação uma vez que envolve uma metodologia sistemática para determinação da condição da via. Para atingir estes objectivos, o trabalho encontra-se dividido em 2 fases. A primeira fase contempla a análise da aplicação do método PSD na avaliação da qualidade da via. O trabalho continua com a avaliação da relação existente entre os vários parâmetros relativos à geometria da via. Esta fase assume especial importância para establecer um modelo de degradação da via de forma apropriada e precisa, uma vez que considera a interação entre as várias variàveis da geometria da via. A segunda fase do trabalho foca-se no desenvolvimento de um modelo preditivo de degradação com capacidade de captar a qualidade da via no que diz respeito aos parâmetros estatisticos e espectro de frequência. Os resultados obtidos deste modelo juntamente com o modelo de processo de recuperação são então aplicados para análise de diferentes cenários de manutenção. As recentes descobertas indicam que algumas das variáveis da geometria da via estão relacionadas. Por exemplo, existe uma forte relação entre o alinhamento da direita e da esquerda, e também entre o perfil e o alinhamento. Baseado na análise, para comprimentos de onda maiores que 6 m a variação do perfil nos dois alinhamentos é vi semelhante, enquanto o alinhamento apenas exibe uma forte relação para maiores que 66 m. Tipicamente, os comprimentos de onda entre 6 e 30 m, para várias geometrias de via, são os mais danosos. Em correspondência com a aplicação do modelo de manutenção, os resultados mostram que a influência do critério proposto de qualidade da via (com limites obtidos através de TQI ou PSD), das variáveis de defeitos geométricos de preferência (perfil ou alinhamento) e determinadas estratégias de manutenção (preventiva, tardia, ou a combinação entre a manutenção regular e corretiva) são os factores-chave na determinação da decisão de manutenção. Através da análise dos resultados, o critério do índice de qualidade da via TQI poderá reduzir o número de ações de manutenção até 7% menos do que se usar o PSD. A redução do número de manutenções também poderá ser alcançado se for apenas considerado o perfil de irregularidade em vez do alinhamento. Esta redução pode atingir os 52% com o critério PSD e os 22% com o critério TQI. Finalmente, a estratégia adotada de manutenção tardia provou ser mais eficiente para decisões de manutenção do que a manutenção preventiva e a combinação entre a manutenção regular e corretiva. Palavras-chave: espectros de potência; indice de qualidade da via; via balastrada; manutenção da via vii ACKNOWLEDGEMENTS Many people have contributed to the work described here, and the author would like to take the opportunity to thank them. First and foremost, a big thank you to my supervisor Prof. Raimundo Delgado for the patient guidance, encouragement and valuable advice he has provided throughout my time as his student. His positive outlook and confidence in my research inspired me and gave me confidence. Grateful appreciation is also expressed to my co-supervisor, Prof. Rui Calçada, for his assistance and constructive comment with many insightful suggestions. He spent countless hours proofreading my thesis and discussing new research ideas. His encouragement and kindness will never be forgotten. I also wish to thank to Dr. Cecilia Vale for her advice and technical input during various stages of my study. Without her involvement, this work would not have been possible. I gratefully acknowledge the funding received towards my PhD from the Fundação para a Ciência e a Tecnologia (FCT). The support and direction offered by the MIT Portugal Program are greatly appreciated. The author would also like to thank the support of the Portuguese Railway Administration (REFER) for the access to railway information data. Utmost gratitude is also forwarded to Prof. Alvaro Costa and Prof. Jorge Pinho, Director of MIT Portugal – University of Porto Program for their supports. Thanks must also go to Prof. Isabel Ribeiro for helping me when needed. I would like to thank Prof. Cristiana who spent endless hours proofreading my thesis. xiv 3 RESEARCH METHODOLOGY 79 Figure 3.1 – Schematic Diagram of the Research Methodology ............................... 84 4 THE APPLICATION OF POWER SPECTRAL DENSITY (PSD) IN TRACK QUALITY ASSESSMENTS 89 Figure 4.1 – Comparison of Various PSD Standards - PSD Longitudinal Profile* ... 91 Figure 4.2 – Comparison of Various PSD Standards - PSD Alignment* .................. 91 Figure 4.3 – Comparison of Various PSD Standards - PSD Cross Level or Superelevation irregularity* .................................................................. 92 Figure 4.4 – Comparison of Various PSD Standards - PSD Gauge* ......................... 92 Figure 4.5 – Simulated track irregularities ................................................................ 97 Figure 4.6 – Comparison of PSDs ............................................................................. 97 Figure 4.7 – Transformation PSD for China 120 km/h ............................................. 98 Figure 4.8 – Transformation PSD for China 160 km/h ............................................. 98 Figure 4.9 – Transformation PSD for China 200 km/h ............................................. 98 Figure 4.10 – The Influence of Wavelength in the Artificial Longitudinal Profile Irregularity - FRA Class 6 ................................................................ 100 Figure 4.11 – The Influence of Wavelength in the Artificial Longitudinal Profile Irregularity - German Low disturbance ............................................ 100 Figure 4.12 – Comparison between Actual and Artificial Longitudinal Profile Irregularities – German low PSD ..................................................... 102 Figure 4.13 – Comparison between Actual and Artificial Longitudinal Profile Irregularities– German PSD ............................................................. 102 Figure 4.14 – Comparison between Actual and Artificial Longitudinal Profile Irregularities – FRA PSD ................................................................. 103 Figure 4.15 – Sample of fit. curve Longitudinal Profile D1 .................................... 105 Figure 4.16 – Sample of fit. curve Alignment. D1 .................................................. 105 xv Figure 4.17 – Sample of fit. curve Super-elevation ................................................. 105 Figure 4.18 – Sample of fit. curve Gauge ................................................................ 105 Figure 4.19 – Comparison of profile D1 .................................................................. 106 Figure 4.20 – Comparison of alignment D1 ........................................................... 106 Figure 4.21 – Comparison of super-elevation ......................................................... 106 Figure 4.22 – Comparison of gauge ......................................................................... 106 5 CORRELATION ANALYSIS OF RAILWAY TRACK GEOMETRY 111 Figure 5.1 – Track Characteristics of Sample Track Segment ................................ 117 Figure 5.2 – Samples of Track Geometry (KM 200.00-201.50) ............................. 118 Figure 5.3 – Curvature Geometrical Characteristics (KM 200.00-233.40) ............. 119 Figure 5.4 – Cross-Correlation Analysis for Longitudinal Profile and Alignment . 120 Figure 5.5 – Cross-Correlation Analysis for the Left and the Right Rails .............. 121 Figure 5.6 – Cross-Correlation Analysis between Super-elevation and other Track Geometry Variables ............................................................................ 123 Figure 5.7 – Cross-Correlation Analysis between Twist and other track geometry variables .............................................................................................. 125 Figure 5.8 – Cross Correlation Analysis between Gauge and other track geometry variables .............................................................................................. 126 Figure 5.9 – Cross-Correlation Analysis between Curvature and other track geometry variables .............................................................................................. 127 Figure 5.10 – Left and Right Longitudinal Profile .................................................. 131 Figure 5.11 – Left and Right Alignment.................................................................. 131 Figure 5.12 – Left Profile and Left Alignment ........................................................ 131 Figure 5.13 – Right Profile and Right Alignment ................................................... 131 Figure 5.14 – Left Profile and Gauge ...................................................................... 132 xvi Figure 5.15 – Right Profile and Gauge .................................................................... 132 Figure 5.16 – Left Profile and Super-elevation ....................................................... 133 Figure 5.17 – Right Profile LD and Super-elevation ............................................... 133 Figure 5.18 – Left Profile and Twist ........................................................................ 133 Figure 5.19 – Right Profile and Twist ..................................................................... 133 Figure 5.20 – Left Alignment and Super-elevation ................................................. 134 Figure 5.21 – Right Alignment and Super-elevation ............................................... 134 Figure 5.22 – Left Alignment and Twist ................................................................. 135 Figure 5.23 – Right Alignment and Twist ............................................................... 135 Figure 5.24 – Left Alignment and Gauge ................................................................ 135 Figure 5.25 – Right Alignment and Gauge .............................................................. 135 Figure 5.26 – Super-elevation and Gauge ............................................................... 136 Figure 5.27 – Twist and Gauge ................................................................................ 136 Figure 5.28 – Super-elevation and Twist ................................................................. 137 Figure 5.29 – Longitudinal Profile autocorrelation - January, 2009 ....................... 140 Figure 5.30 – Alignment autocorrelation - January, 2009 ....................................... 140 Figure 5.31 – Super-elevation Autocorrelation - January, 2009 .............................. 141 Figure 5.32 – Twist Autocorrelation - January, 2009 .............................................. 141 Figure 5.33 – Gauge autocorrelation - January, 2009 .............................................. 142 Figure 5.34 – Longitudinal Profile autocorrelation - March, 2008 ......................... 142 Figure 5.35 – Alignment autocorrelation - March, 2008 ......................................... 142 Figure 5.36 – Super-elevation Autocorrelation - March, 2008 ................................ 143 Figure 5.37 – Twist Autocorrelation - March, 2008 ................................................ 143 Figure 5.38 – Gauge autocorrelation - March, 2008 ................................................ 143 xvii 6 THE APPLICATION OF AN OPTIMIZATION MODEL FOR TRACK MAINTENANCE 147 Figure 6.1 – Relationship Analysis Between PSD and TQI .................................... 153 Figure 6.2 – Location of the Sample Track Segment in Portugal ............................ 156 Figure 6.3 – Sample Before and After Track Adjustment at KM 200.00 – 200.200159 Figure 6.4 – Dead spots on track segment at KM 217.200 – 217.600 ..................... 159 Figure 6.5 – SD of Longitudinal Profile at the initial time instant and European Standard limit – January 2009 ............................................................ 160 Figure 6.6 – SD of Alignment at the initial time instant and European Standard limit – January 2009 .................................................................................... 160 Figure 6.7 – PSD China of Longitudinal Level and PSD Chinese limit – January 2009 .................................................................................................... 160 Figure 6.8 – PSD China of Alignment and PSD Chinese limit – January 2009 ...... 160 Figure 6.9 – Line speed of Track Segments ............................................................ 161 Figure 6.10 – The evolution of SD of track longitudinal profile for Segment 1 (KM 200.00-200.199) ................................................................................. 163 Figure 6.11 – Degradation Rate of longitudinal Profile .......................................... 166 Figure 6.12 – Degradation Rate of Alignment ........................................................ 166 Figure 6.13 – Track Recovery - Longitudinal Profile ............................................. 167 Figure 6.14 – Track Recovery - Alignment ............................................................. 167 Figure 6.15 – Distribution of Total Tamping over time based on TQI limit ........... 178 Figure 6.16 – Distribution of Total Tamping over time based on PSD limit .......... 178 Figure 6.17 – Distribution of Total Tamping over time based on various maintenance strategies ........................................................................................... 179 Figure 6.18 – Evolution of Track Quality for Segment 2 with Preventive Maintenance strategy, based on TQI limit ....................................... 179 xviii Figure 6.19 – Evolution of Track Quality for Segment 2 with Preventive Maintenance strategy, based on PSD limit ...................................... 180 Figure 6.20 – Evolution of Track Quality for Segment 12 with Preventive Maintenance strategy, based on TQI limit ....................................... 181 Figure 6.21 – Evolution of Track Quality for Segment 12 with Preventive Maintenance strategy, based on PSD limit ...................................... 181 Figure 6.22 – Evolution of Track Quality for Segment 145 with Preventive Maintenance strategy, based on TQI limit ....................................... 182 Figure 6.23 – Evolution of Track Quality for Segment 67 with Preventive Maintenance strategy, based on TQI limit ....................................... 182 Figure 6.24 – Evolution of Track Quality for Segment 121 with Preventive Maintenance strategy, based on TQI limit ....................................... 183 Figure 6.25 – Analysis of maintenance strategies to tamping decision for segment, based on TQI limit ........................................................................... 184 xix LIST OF TABLES 2 LITERATURE REVIEW 9 Table 2.1 – Ranges of the coefficients [Lichtberger, 2005] ...................................... 23 Table 2.2 – Allowable deviations for J coefficient [Madejski&Grabczyk, 2002] .... 29 Table 2.3 – Standard Deviation (SD) values [Sadeghi & Asgarinejad, 2008] .......... 31 Table 2.4 – TGI Classification for Maintenance [Talukdar et al., 2006] .................. 31 Table 2.5 – Track Quality Levels .............................................................................. 32 Table 2.6 – The Quality Level defined in the European Standard [Puzavac et al., 2011] ..................................................................................................... 33 Table 2.7 – SD Threshold values for Longitudinal Profile and Alignment – Alert Limit ..................................................................................................... 34 Table 2.8 – Isolated Defects SD for Longitudinal Profile - Mean to peak value ...... 34 Table 2.10 – Allowable limits of parameter defectiveness [Madejski & Grabczyk, 2002] ..................................................................................................... 39 Table 2.11 – Quality Qualifications of Track Lines [Madejski & Grabczky, 2002] . 40 Table 2.12 – Coefficients for Power Spectral Density (PSD) function [Xia, 2002]. . 48 Table 2.13 – Track PSD parameters [Lin et al., 2004] .............................................. 50 Table 2.14 – Spectral Parameters for line speed of 200 km/h [Xianmai et al., 2008] 51 Table 2.15 – Spectral Parameters for line speed of 160 km/h [Xianmai et al., 2008] 52 Table 2.16 – Spectral Parameters for line speed of 120 km/h [Xianmai et al., 2008] 52 Table 2.17 – Summary of Track Degradation Model & PSD– Literature Review ... 55 xx 4 THE APPLICATION OF POWER SPECTRAL DENSITY (PSD) IN TRACK QUALITY ASSESSMENTS 89 Table 4.1 – Comparison between Standard Deviations of Simulated Track Irregularities................................................................................ 101 5 CORRELATION ANALYSIS OF RAILWAY TRACK GEOMETRY 111 Table 5.1 – Summary of Cross-Correlation Analyses ............................................. 128 Table 5.2 – Summary of the coherence analysis ..................................................... 138 6 THE APPLICATION OF AN OPTIMIZATION MODEL FOR TRACK MAINTENANCE 147 Table 6.1 – Areas under spectrum for PSD FRA .................................................... 154 Table 6.2 – Areas under spectrum for PSD Germany ............................................. 154 Table 6.3 – Areas under spectrum for PSD China ................................................... 155 Table 6.4 – Characteristics data of Sample Track Segment .................................... 157 Table 6.5 – Degradation Rates of Track longitudinal Profile ................................. 164 Table 6.6 – Degradation Rates of Track Alignment ............................................... 165 Table 6.7 – Influence of the Track Quality Assessment Criteria and the Consideration of Track Geometry Parameters ........................................................... 175 Table 6.8 – Influence of Various Track Maintenance Strategies ............................ 176 Table 6.9 – Prediction Performance ........................................................................ 185 xxi ABBREVIATIONS AI Alignment Index AL Alert Limit AMPL A Mathematical Programming Language CEN European Committee for Standardization DFT Discrete Fourier Transform FFT Fast Fourier Transform FRA Federal Railroad Administration GI Gauge Index GPS Global Positioning System IAL Immediate Action Limit IL Immediate Limit ILP Integer Linear Programming IM Infrastructure Manager IP Integer Programming LCC Life Cycle Cost LP Linear Programming M&R Maintenance and Renewal MAE Mean Absolute Error xxii MGT Million Gross Ton MILP Mixed Integer Linear Program ORE Office for Research and Experiments PSD Power Spectrum Density RMSE Root Mean Squared Error SD Standard Deviation SNCF Société Nationale des Chemins de Fer Français TGI Track Geometry Index TI Twist Index TQI Track Quality Index TRC Track Recording Car UI Unevenness Index xxiii LIST OF NOTATIONS I. Latin Letters Chapter 2 - Amplitude of auto spectrum density; Indication of the state of rail surface - Roughness coefficient; Scale factor for alignment - Reverse check method - Scale factor for longitudinal profile - Roughness coefficient - Coefficient Chinese PSD; One half of track gauge - Coefficient factor (ORE Model) - Coefficient factor (ORE Model) - Coefficient Chinese PSD; Constant factor (ORE Model); Width of sleeper in longitudinal direction; Rate of deterioration - Constant factor (ITDM Model) - Coefficient Chinese PSD; Total number of section samples - Difference of cross level or superelevation irregularity between two points - Coefficient Chinese PSD - Coefficient Chinese PSD xxx - Center wavenumber - Upper wavelength - Wave band - Stationary stochastic process Chapter 5 - Length of track irregularity signal M - Half width of Hanning window N - Number of data points - Cross power spectrum density between signal and - Power spectrum of signal - Cross correlation coefficient - Auto-correlation of sampled signal s(x) - Auto-correlation of signal s(x) - Cross correlation function - Average power of the input waveform - Space domain of track irregularity - Continuous random of track irregularity - Number of lag sample - Distance xxxi xxxii II. Greek Letters Chapter 2 - Spatial wave-number , - Critical wavenumber - Coefficient (Sato‟s model) - Coefficient (Sato‟s model) - Coefficient (Sato‟s model) - Wear depth - Wavelength - Angle of attack between wheel set to the track - Standard deviation of height / longitudinal profiles - Standard deviation for interaction - Standard deviation limit of longitudinal profile in a given track class - Standard deviation limit for interaction in a given track class xxxiii Chapter 4 - Phase angle - Uppermost limits - Discrete angular wavenumber - Uppermost limit Chapter 5 - Desired level of confidence - Lag distance γ2 - Coherence function - Confidence level 1 INTRODUCTION 1.1 INTRODUCTION In recent years, studies on track degradation in railways have attracted a great deal of attention. Intensive research activities have been carried out by many organizations with the aim of securing a high level of safety and reliability of the infrastructure system. New technologies and stringent safety standards are constantly being introduced for several reasons, not only to prevent the assets from failure or damage but also to minimize the main sources of problems associated with the performance degradation in terms of quality, comfort and safety of each journey. Since failure on the railway system will result in significant economic losses, many Infrastructure Managers (IMs) spend a substantial proportion of budget on Maintenance and Renewal (M&R) of the tracks, which makes up a considerable part of the total railway operating cost, accounting for up to 70% of the total life cycle cost (LCC) of track infrastructure [Jianmin, 2007]. For instance, a single track of 1 km long in typical European countries requires an average of 30,000 euro for a 1-year maintenance period [Gines, 2008]. With this massive amount of financial expenditure, a small reduction in the cost of maintenance will undoubtedly bring a significant impact, particularly to the overall LCC. Chapter 1 2 As degradation is one of the prime issues in this matter, it becomes important to understand the complexity of degradation mechanisms, its likelihood of occurrence in the railway track, as well as the variables affecting the degradation. The recognition of any changes in the track condition over time and the consequences of maintenance actions on the track performance will enable to predict the residual life time of the asset. The accurate life cycle including any necessary maintenance activities throughout the service life can thus be drawn and by doing so, the railway company is capable of systematically reducing the operation and maintenance expenditures without affecting traffic safety. 1.2 PROBLEM DESCRIPTION Nowadays, efforts have been made to develop effective maintenance and renewal policies. The goal of maintenance management is to reduce the adverse effects of failure and to maximize the availability of the railway network at minimum cost [Lofsten, 1999]. In this case, the railway infrastructure managers (IMs) play an important role. They are challenged to optimize each stage of the maintenance procedure and to analyze the best alternative maintenance strategies (inspection frequency, interval of tamping, etc.) with respect to cost effectiveness and safety issues. In order to assure the operational services, there are two main maintenance strategies that may be applied [Holmgren, 2005]. The first one is preventive maintenance, where the intervention is performed at a predetermined interval and/or on a continuous basis. The primary goal of this typical maintenance is to prevent the consequences of failure or the degradation of the functioning items. The second strategy is known as corrective maintenance, where the intervention is carried out after particular equipment has suffered failure. This type of maintenance is unplanned and repairs are intended to bring the system back to work in order; however, it is neither practical nor economically feasible to perform both methods. Regular maintenance of a large infrastructure network is costly and fairly time-consuming because it might be done even when it is unnecessary. On the other hand, the corrective maintenance can be extremely costly since the disruption of traffic operation due to the system failure will have an adverse Introduction 3 effect at an additional cost, namely involving delay costs, train cancellations and penalties imposed by traffic operators to the IMs. To be able to manage these issues, a prediction model of track degradation is needed to make the best decision in maintenance and replacement strategies, to account for costs and risks over the life cycle of a railway track. Analyses and detailed studies on track degradation have been done by many researchers and various predictive models have been proposed, from simple deterministic to the most elaborated stochastic models. Based on the literature review, the main railway degradation models are those developed by Bing and Gross [1983], Shenton [1984], Sato [1995] and TU Muenchen [Demharter, 1982]. The models were built based on empirical analysis, in which the settlements were mainly considered as the controlling degradation factors. More recent developments in track degradation models refer the use of rail geometry data to provide the characterization of track geometry irregularities. By computing the rail variance, for individual or a set of geometrical parameters (such as longitudinal profile, alignment, gauge, twist and cross level or superelevation irregularity), the quantitative value, named Track Quality Index (TQI), can be derived. The progress of changes in TQI may help the track manager to predict the future quality of a unit section [Bing and Gross, 1983]. In spite of its principal role, which is to portray the track condition, TQI may not reflect the wavelength contents of geometry defect, which is inherently related to the particular issue of train-track interaction. In order to overcome this drawback, as an alternative for assessing rail irregularity, power spectrum graph may be used [Zhiping et al., 2009]. Such graph provides information concerning track irregularity in the frequency domain, with horizontal and vertical axes representing the spatial frequency and Power Spectrum Density (PSD), respectively [Zhiping et al. (2009), Zhiping and Shouhua (2009), Zhiqiang et al. (2009)]. Both of these axes can be used as indicators for track quality. The higher the PSD values, the poorer the track quality will be, while the lower the PSD values mean Chapter 1 4 the opposite situation. An analysis on the horizontal axis can also help to detect which wavelengths have contributed to the geometry defect [El-Sibaie and Zhang, 2004]. The PSD application, however, is not as widely used as the previous aforementioned technique. The expertise and knowledge required to process and to interpret information regarding the Power Spectral Density is the main drawback in the development of this method. Following these shortcomings, this dissertation aims to develop a logical model for the deterioration of track geometry and to incorporate the proposed model as basis for optimizing maintenance in practice. The Power Spectral Density (PSD) forms a core focus of the studies since it involves a systematic technique for evaluating track quality condition. For the purpose of analysis, a particular segment of the Portuguese Northern Railway Lines with 33.4 km in length will be considered and multivariate statistical analyses will be employed in some geometric parameters. The results will be used to derive the optimization model for scheduling track maintenance by means of tamping in a given period of time. The application of maintenance model will consider the deterioration of two geometrical parameters that most influence the vehicles and the track dynamics in the vertical and horizontal directions, which are the longitudinal profile and the alignment. Several track maintenance strategies will also be analysed in order to obtain the most efficient approach to reduce the number of maintenance actions while keeping the safety level. In this case, a new proposed maintenance strategy called Delay Maintenance (DM) will be introduced and will be compared with the conventional approaches in the track maintenance management. The core of this strategy is taken from the benefit of delaying the time to perform the maintenance operations, so thus the recovery effectiveness can be much higher than the conventional one. The track maintenance optimization model will be detailed in Chapter 6. Introduction 5 1.3 RESEARCH GOALS AND OBJECTIVES The primary goals of this research are: 1. To investigate the application of Power Spectral Density in the assessment of railway track quality. For this purpose, PSD standards developed in various countries will be analyzed and those methods will then be implemented in real field assessments. 2. To quantify the degree of interdependency and to establish the similarity of one track geometry variable to another. For evaluating the existing relationship between each of them, correlation analyses will be employed in this research. 3. To develop an optimization model for scheduling track maintenance in ballasted tracks. The proposed model will consist of two parts: the predictive degradation model that may able to capture the evolution of track quality in terms of statistical index and frequency spectrum, and the track recovery model due to tamping operations. The results obtained from the optimization model will then be applied to analyze different maintenance scenarios. 1.4 SCOPE AND LIMITATIONS The scope and limitations of this research are as follows. Firstly, the research is focused on developing a degradation model in the context of operation and maintenance stages of a track life cycle system, with a predefined design and situation. Any changes in the design structure and characteristics of the railway components made towards a better performance should be conducted separately. The reason for this limitation is the size and complexity of the research area. Secondly, not all factors that may influence the track degradation process will be considered in the prediction model development. This is due to the insufficiency and unavailability of information related to some of these factors in the database (e.g. environment, type of materials, etc). Chapter 2 12 to the extent and large scale of the railway network. The accuracy of this method is questionable as well, since not all rail defects can be directly detected by the human eye. With the advancement of technology, the application of sophisticated cars has been incorporated as supplement to the previous technique. The use of the track inspection car has proven to be efficient, yielding more outputs in the track information data. In Portugal, track condition monitoring for high speed and conventional lines is performed by the Track Recording Car (TRC) EM 120 (Figure 2.3). This machine is able to acquire traditional geometry parameters, such as gauge, cross level or superelevation irregularity, twist, alignment and longitudinal profiles with a sampling rate of 0.25 m, and provides the calculation of the track quality index from the derived parameters. In order to perform those functions, the machine is embedded with two main modules: a car with real time measurement module and a stationary data storage module [Cacho, 2009]. The first module is designed to read any gap in the geometry parameters from the predefined design and then sends the results to the second module for storage purposes. The computer program, which links to the Global Positioning System (GPS) receiver, will analyze the data and give the information according to customer specifications. Figure 2.3 – Track Recording Car EM 120 [Comboios.org, 2007] Literature Review 13 The irregularity data provided by EM 120 is based on the measurement of a 10 m chordversine. The principle of this technique is to assess the gap corresponding to the observed roughness of a straight-line in the center point of the track in the 10 m distance measurement. This gap is then defined as track geometry irregularity. The following figure illustrates how the track geometry parameters are measured by EM 120. Figure 2.4 – Track Surface Irregularity [Oyama, 2006] 2.4 TRACK DEGRADATION It has been observed that the condition of a railway track degrades rapidly over a period of time. Having the knowledge of the degradation process will aid in the estimate of the future state of a track condition and in the mitigation of the problems associated with operational safety. The following section presents the theoretical framework and the current practices related with railway track degradation. 2.4.1 GENERAL CONCEPT Track degradation is a complex process. The mechanism involves many influencing parameters such as axle load, traffic speed, climate, track characteristics and topography (Figure 2.5). Today, research efforts have been carried out not only to address the degradation problems, but also to determine the contribution of each parameter to the entire process. Surface irregularity 10 mchord 5 m 5 m Sleeper Rail Chapter 2 14 Figure 2.5 – Influencing Parameters to the Track Degradation Ferreira & Murray [1997] have investigated the physical factors that may have impact on track deterioration. According to the results, the authors argue that the declination in the track quality is mainly driven by three parameters, i.e. dynamic forces, axle load and train speed. Speed contributes to the deterioration process by increasing the dynamic forces at high speeds and decreasing those at low speeds. Load contributes to increased rail wear and fatigue, wheel wear, and strains in rails and sleepers. As a consequence, cracks in the rail and sleepers will occur, the railhead will be worn out, the rail fastening will be loosen, and the ballast load will thus be redistributed. These situations will lead to reduce travel safety and comfort, and increase track components deterioration and delays. Rail - Type/Quality - Wear - Age - Rail welding Ballast - Type/Quality - Thickness - Age - Dust Sleeper - Type/Quality - Thickness - Age Train - Type of train - Traffic density - Axle load /tonnage - Speed Maintenance - Lubrication - Ballast cleaning - Tamping - Grinding - Inspection interval Condition Superstructure Track Degradation Substructure - Type /quality - Thickness Environment - Temperature - Humidity - Corrosive Environement Dynamic Load Static Load Track Geometry - Curvature - Elevation Literature Review 15 Later work reported by Larson [2004] has found that wear and fatigue damage are considerably affected by the existence of curvature. The shape and radius of curvature can determine the rail defect with the following relationship: - Narrow curves implies wear (ahead of fatigue) - Tangent track implies fatigue (ahead of wear) Using the Swedish railway data, Larson then attempted to correlate various ranges of curvature with the state of track condition: Figure 2.6 – Wear and Fatigue mechanisms as a function of Curve Radii [Larson, 2004] As indicated in the figure, the narrow curve has caused a higher degradation index compared to the plain track. Higher index means shorter service life of the rails. Wear, in this case, governs the rail degradation with short life span, while fatigue drives the rail degradation with long life span [Zarembski, 1991]. Pita et al. [2004] and Berggren [2005] later investigated the influence of track stiffness to the track degradation. Based on their experimental study, the authors found that an increase in vertical stiffness produces a negative effect, especially in vertical stresses exerted by the vehicles on the rail. However, having a very flexible track may enhance Chapter 2 16 the energy dissipated from vehicles running at high speed. Considering this duality, an attempt was made to define the optimum value for the vertical stiffness, which minimizes the maintenance cost on the one hand, and the cost due to the dissipated energy on the other. The equilibrium of these two costs is achieved when the vertical stiffness of the track stands at about 70-80 kN/mm for lines on which trains run at high speed. Similarly, the application of maintenance (consisting of tamping, grinding, lubrication, etc.) could improve the quality of railway track. When the tamping action is performed, the ballast under the ties is re-compacted and the area of contact increases. The larger the areas of contact, the better the ties distribute the weight of the rail and rolling stock, which in turn may impede the acceleration of track to face deterioration. Likewise, the application of preventive grinding also leads to a significant increase in the rail service life, since it slows down the rail corrugation growth and decreases traffic noise. From the experience gained in practice, the combination of maintenance methods, such as rail grinding and lubrication, may extend the life span and the limit of rail components from 50% to 300% [Judge, 2002]. Some other factors contributing to track deterioration have also been examined by many researchers. Lichberger [2001] discussed the effect of initial track quality to preserve the track from rapid degradation. Johansson et al. [2008], Witt [2008] and Lundvist [2005] argued that the selection of under sleeper pad could help reduce the ground vibration and minimize track misalignment. The preferences in choosing the material quality are also essential to attenuate the distortion on the track performance. Poor materials can cause more track degradation, while good materials will enhance the resistance of the track to failure [Zwanenburg, 2006]. 2.4.2 TRACK DEGRADATION CURVE In order to define where the quality limit is and to decide when the intervention is required, it is therefore necessary to understand the degradation behavior. Normally, the Literature Review 17 degradation line will exhibit a “saw tooth” shape, in which the quality deteriorates between two subsequent maintenance activities [Jovanovich, 2004]. This process is schematically shown in the following figure: Figure 2.7 – General Trend Degradation Model [Lichtberger, 2001] The railway track commences with an initial quality from a newly constructed line or previous maintenance action. During the train operation, the track quality starts to degrade as a result of interaction of several effective parameters, such as the cumulative of track loads (MGT), time, speed, etc. When the defect of the track reaches the threshold limit, tamping should be carried out to reduce the amount of standard deviation, leading the track geometry to deterioration in two major phases. The first phase occurs directly after tamping, around the first 0.5-2 MGT of traffic borne. This period is followed by a rapid exponential track failure, characterized by the breaking off of the points of ballast stone that settle into a more compact position [Lichtberger, 2001]. Once the track has been sufficiently established, the second phase of degradation takes place. The track faults will deteriorate slowly and increase linearly in proportion to the number of load cycles [Lyngby, 2009]. Several mechanisms of ballast and sub-grade behavior are governed during this process. Continuous volume Gaining quality after tamping Maintenance Degradation curve Time or MGT Track Fault Threshold for Action Initial phase Second phase Chapter 2 18 reduction due to particle rearrangement and sub-ballast or sub-grade penetration into ballast voids are examples of such characteristics. Subsequently, the efficiency of maintenance will decrease in time and the period between two tampings becomes shorter. When tamping is considered ineffective to repair the geometry faults, line reconstruction should be carried out. 2.5 TRACK DEGRADATION MODELS: CURRENT PRACTICES AND APPLICATIONS In the past few years, several attempts have been made to build a track degradation model, from the simple one that relies on a single parameter to a comprehensive one which embraces several influencing variables. From the reviews on the available literature, these models can be classified into two different aspects [Sadeghi and Asgarinejad, 2007]: - Track degradation considered from a structural viewpoint - Track degradation considered from a geometrical viewpoint From a structural viewpoint, the model development is based on the progression of defects in the physical structure, such as ballast settlement, wear and corrugation. Shenton [1984], Sato [1997], Chrismer and Selig [1993] and TU Munich [Demharter, 1982] have developed models in this area. From a geometrical viewpoint, the reflection of the actual state of track condition uses geometrical parameters, such as longitudinal profile, alignment, etc. Some of the developers of this model are Bing and Gross [1983] and recently, the practical use of the model has been adopted by many countries. In fact, both viewpoints are correlated. Any deviation in the geometry parameters is known as a result from the track structural problems [Berggren, 2005]. Literature Review 19 2.5.1 MODELS OF STRUCTURAL DEGRADATION OF THE TRACK This section contains a brief and general description of 8 models dealing with the structural degradation of the track. A. Sato track degradation model The study on track deterioration due to ballast settlement has produced an equation proposed by Sato [1995]. The following equation is used to estimate the settlement of the track under repeated loading on both heavy haul narrow gauge and high standard gauge: (2.1) where: = track settlement (mm) = repeated number of loading or tonnage carried by track (cycles or tons) = coefficients Sato divides the model into two major parts. The first part refers to the initial track settlement imposed by the compaction of the ballast, which occurs directly after maintenance. The settlement in this phase is relatively fast and it can be best modeled using an exponential function ( ). As soon as the ballast is consolidated, the second stage will occur. The settlement increases linearly in proportion with the cyclic loading ( , which is called the long term settlement. The severity of settlement depends on the quality and behavior of the ballast, the sub-ballast and the sub-grade. The coefficient expresses the level of settlement and indicates the steepness of the initial function. defines how quickly the track settlement grows in the second phase. Chapter 2 20 B. Shenton settlement model Various parameters influencing track degradation have been investigated by Shenton. The research was conducted on different tracks spread in many countries, such as Britain, USA and several European countries. By combining the theories of functionality followed by hypotheses testing in practice, he derived a general equation that may quantify the ballast settlement [Shenton, 1984]. The proposed model is defined by: [ ] (2.2) where: = ballast settlement (mm) = a factor to describe the track structure ) = equivalent wheel set load (kN) = amount of track improvement given by the tamping machine (mm) = total number of axles (-) Furthermore, Shenton suggested an equivalent wheel-set load ( ) according to the following equation: (∑ ∑ ) (2.3) where: = equivalent wheel-set load (kN) = static wheel-set load type i (kN) = number of wheel-set load type i (-) The implementation of Shenton model confirmed the absence of some influencing parameters, such as train speed and dynamic load. Literature Review 21 C. Sugiyama Model Sugiyama examined the growth of vertical defect within 100 days of train operation in Japan. By applying the regression theory, he proposed the degradation model as a function of passage tonnage, rail factors and vehicle speed [Iwnicki et al., 1999]. (2.4) where: = average growth of irregularities in section (mm /100 days) = passed tonnage (MGT/year) = average running speed (km/h) = structure factor = rail influence factor (1 for CWR and 10 for jointed rail ) = influence factor for sub-grade (1 for good and 10 for bad) The structure factor is derived from the following equation: √ √ (2.5) where: = maximum sleeper pressure due to a wheel load (Pa) = rail pad (N/m) = intermediate mass consisting of sleeper, ballast and subgrade (ton) = flexural rigidity of the rail (Nm2) = track stiffness (N/mm) As indicated in the formula, is defined as a relative value. The lower the value of , the lower the track deterioration will be. Chapter 2 28 = standard deviation of horizontal irregularities (mm) = standard deviation of track twist (mm) = standard deviation of track gauge (mm) The standard deviation for each measured parameter is calculated using the following formula: √ ∑  (2.17) where: = number of signals registered in the analyzed track section (-) = value of parameter at point i (mm)  = average value of track irregularity (mm) This synthetic coefficient also specifies the allowable deviation for different line speeds (Table 2.2). If any values are exceeded, a remedial action is required to bring the track back to the appropriate level. Literature Review 29 Table 2.2 – Allowable deviations for J coefficient [Madejski&Grabczyk, 2002] Speed (km/h) J Coeff. (mm) Speed (km/h) J Coeff. (mm) 80 7 150 2.3 90 6.2 160 2 100 5.5 170 1.7 110 4.9 180 1.6 120 4 190 1.5 130 3.5 200 1.4 140 2.8 220 *) 1.1 *) Calculated through extrapolation B. Track Geometry Index (TGI) The Track Geometry Index (TGI) was developed by the Indian Railways, which aimed to quantify the level of track condition. This model relies on the standard deviation of various geometry parameters over segments of 200 m in length. The average value of such segments per km gives the general TGI value [Talukdar et al., 2006]. TGI can be calculated with the following formula: (2.18) where , and are the indices for unevenness, twist, gauge, and alignment, respectively. The calculations for the different parameters are obtained by: Chapter 2 30 where: = measured standard deviation value of unevenness, twist, gauge and alignment, respectively (mm) = standard deviation prescribed for newly laid track for unevenness, twist, gauge and alignment, respectively (mm) = standard deviation prescribed for maintenance of unevenness, twist, gauge and alignment, respectively (mm) and are obtained from the average of the measured standard deviations of the left and the right rails. (2.19) where : = standard deviation of the left longitudinal profile (mm) = standard deviation of the right longitudinal profile (mm) Literature Review 31 Table 2.3 specifies the SD values used for newly laid track and for urgent maintenance tracks. The classification of track condition with corresponding maintenance is given in Table 2.4. Table 2.3 – Standard Deviation (SD) values [Sadeghi & Asgarinejad, 2008] Table 2.4 – TGI Classification for Maintenance [Talukdar et al., 2006] No TGI Value Maintenance requirement 1 TGI > 80 No maintenance required 2 50 < TGI < 80 Need basic maintenance 3 36 < TGI < 50 Planned maintenance 4 TGI < 36 Urgent maintenance Parameters Chord Length SD for newly laid track SD for maintenance with max. speed ≥ 105 km/h SD for maintenance with max. speed < 105 km/h (m) (mm) (mm) (mm) Unevenness 9.60 2.5 6.2 7.2 Twist 3.60 1.75 3.8 4.2 Gauge 1.00 1.00 3.6 3.6 Alignment 7.20 1.50 3.0 3.0 Chapter 2 32 C. European Regulation Standard [prEN 13848-5:2005] The European Committee for Standardization (CEN) has created a group of standards, prEN 13848, which consist of five parts of technical specifications. This series of standards aims to define a unique approach for evaluating track geometrical quality in various member countries. First part of the integrated standard provides the terminology and a framework for specification of track geometry parameters, including track gauge, longitudinal profile, alignment, superelevation irregularity (or cross level) and twist. Parts 2 to 4 of the standard cover the measuring system, track recording vehicle [Part 2], track construction and maintenance machine [Part 3], and manual and light weight devices [Part 4]. The remaining part of the European Standard, Part 5, specifies the minimum requirements for the quality levels of track geometry, and gives the safety-related limits for each parameter as defined in Part 1. For addressing operational safety and ensuring the interoperability of train services, the Standard has set up three different quality levels. The maintenance strategies are relatively dependent on these levels, as explained in the Table 2.5. The specific parameters assigned to the quality levels are provided in Table 2.6. Table 2.5 – Track Quality Levels Track Quality Levels Alert Limit (AL) If a limit value is exceeded, an action to correct the error has to be considered in the regularly planned maintenance. Intervention Limit (IL) If a limit value is exceeded, an action to correct the error has to be done immediately before the next inspection Safety Limit (IAL) If a limit value is exceeded, an action should be done to reduce the risk of derailment (closing the line, reducing speed, immediate tamping, etc.) Literature Review 33 Table 2.6 – The Quality Level defined in the European Standard [Puzavac et al., 2011] Parameters Nominal to peak value Nominal to mean value Mean to peak value Standard Deviation SL IL AL SL IL AL SL IL AL SL IL AL Longitudinal Profile ✔ ✔ ✔ ✔ Alignment ✔ ✔ ✔ ✔ Gauge ✔ ✔ ✔ ✔ ✔ ✔ Superelevation Irregularity ✔ ✔ ✔ Twist ✔ ✔ ✔ As specified in Table 2.6, the European Standard used the standard deviation of the track geometry irregularity, in either longitudinal profile or alignment with the corresponding wavelengths between 3-25 m (D1). This indicator represents the dispersion of geometry defects (position of the measured points along the track), in relation to the mean signal (mean position of the track) over a 200 m-segment section (see equation 2.17). The higher the value of SD, the poorer the track quality will be; the lower values of SD correspond to the opposite situation. A track quality can also be assessed according to the number of isolated track geometry defects per unit of track length, typically over 1 km or more. It may also be counted over 100 m or 200 m of track [Tzanakakis, 2013]. Three main levels have to be considered; Alert Limit (AL), Intervention Limit (IL) and Safety Limit (SL). For each limit, the standard defines the track geometry quality based on wavelength spans of 3 < λ ≤ 25 m (D1) and 25 < λ ≤ 70 m (D2). Tables 2.7 to 2.9 provide the permissible levels for those aforementioned parameters (longitudinal profile and alignment). Chapter 2 34 Table 2.7 – SD Threshold values for Longitudinal Profile and Alignment – Alert Limit Wavelength domain Speed (km/h) Longitudinal Profile D1 (mm) Alignment D1 (mm) V ≤ 80 2.3 - 3 1.5 – 1.8 80 < V ≤ 120 1.8 – 2.7 1.2 – 1.5 120 < V ≤160 1.4 - 2.4 1.0 – 1.3 160 <V≤ 220 1.2 – 1.9 0.8 – 1.1 220 < V≤ 300 1.0 – 1.5 0.7 – 1.0 *The standard deviations are only given for Alert Limit. Table 2.8 – Isolated Defects SD for Longitudinal Profile - Mean to peak value Speed (km/h) Alert Limit (AL) Intervention Limit (IL) Safety Limit (SL) Wavelength range (in mm) Wavelength range (in mm) Wavelength range (in mm) D1 D2 D1 D2 D1 D2 V ≤ 80 12-18 N/A 16-20 N/A 29 N/A 80 < V ≤ 120 10-16 N/A 12-18 N/A 26 N/A 120 < V ≤ 160 8-15 N/A 10-17 N/A 24 N/A 160 < V ≤ 220 7-12 14-20 9-14 18-23 20 33 220 < V ≤ 300 6-10 12-18 8-12 16-20 17 28 Literature Review 35 Table 2.9 – Isolated Defects SD for AlignmentMean to peak value Speed (km/h) Alert Limit (AL) Intervention Limit (IL) Safety Limit (SL) Wavelength range (in mm) Wavelength range (in mm) Wavelength range (in mm) D1 D2 D1 D2 D1 D2 V ≤ 80 12-15 N/A 14-16 N/A 22 N/A 80 < V ≤ 120 8-11 N/A 10-12 N/A 17 N/A 120 < V ≤ 160 6-9 N/A 8-10 N/A 14 N/A 160 < V ≤ 220 5-8 10-15 7-9 14-17 12 24 220 < V ≤ 300 4-7 8-13 6-8 12-14 10 20 N/A = Not Applicable The analyses of track quality with respect to the isolated defect limits will be mainly discussed in Chapter IV while the quality assessments based on the standard deviation will be conducted in Chapter VI. D. Track Quality Index (TQI) The other quality measurement, TQI, was initiated by the Federal Railroad Administration, in the United States [El-Sibaie and Zhang, 2004]. The basic concept of this TQI is the use of the space curve length to represent track quality. As shown in Figure 2.8, for a specific track segment length, the rougher the track surface, the longer the space curve will be when stretched to a straight line: Chapter 2 36 Figure 2.8 – Theoretical definition for fixed length (Lo) and traced length (Ls) The TQI formula, for each longitudinal profile, alignment, cross level or superelevation irregularity and gauge, is expressed by the following equation: [ ] (2.20) where: = Track Quality Index for each individual track geometry parameter = traced length of space curve (feet) = fixed length of track segment (feet) The traced space curve length is calculated by summing up the distance between any two points within the track segment: ∑√ (2.21) where: = difference between two measurements (feet) = sampling interval along the track (feet) = sequential number (-) = number of data points in the segment (-) Literature Review 37 E. Q value Banverket, the Swedish Railway company, has a number of indices that are used to express the state condition of their infrastructure facilities. Some of the main condition indices are known as K-value and Q-value [Anderson, 2002]. The Q-value is a weighted index of the standard deviation of the geometric parameters from its comfort limits set for a specific track class. The Q value, therefore, is calculated per kilometer track using the following expression: * + (2.22) where: = standard deviation of height / longitudinal profiles (mm) = standard deviation for interaction (calculated as a combined effect from superelevation irregularity and side position of the rail (mm) = standard deviation limit of longitudinal profile in a given track class (mm) = standard deviation limit for interaction in a given track class (mm) The Q-value is represented as a percentage. The lower value of the state condition indicates that the train may shake and be perceived uncomfortable by the passengers and vice versa. F. K Value The other main condition index that has been used by Banverket is the K-value [Anderson, 2002]. However, the application of K-value is not suitable for shorter track sections. The mathematical formulation for K-value is expressed by: Chapter 2 44 For scaling the auto-spectrum, the equation below is used: √ (2.30) where: = amplitude of auto-spectrum density = number of spectral lines Stationary Check The technique of Fast Fourier Transform (FFT) is based on the stochastic (or random) stationary hypothesis. According to this concept, the joint probability distribution of any subset of the sequence of random variables should be invariant with respect to a shift in time or distance. To have a clear view on this concept, a stochastic process is assumed as a finite sequence of random variables, e.g., , and the joint probability distribution is represented by: { } (2.31) The stochastic process is said to be stationary if: { } { } (2.32) for every and shift and for Therefore, the first step to analyze the track irregularity using FFT is to check whether the measured data can be classified stationary or not. The common technique, that is easy to apply and useful for verification, consists of using the turn check method and reversed order check method [Zhiping and Shouhua, 2009]. If the random sign is stationary, the measured data should be stochastic and there will be no trend component. Time shift by 𝑙 Literature Review 45 The procedure for reverse check method is as follows. Consider a sequence of observations of random variable , where the observations are denoted by , in which . Then count the number of times that for . The sum of such inequality is called reverse check method ( ), as defined by Equation (2.33): { then, ∑ (2.33) The total number of from a set of observation is then denoted , as follows: ∑ (2.34) Considering the hypothesis that the observation is independent and there is no trend component, a confidence interval is given to by: (2.35) If the number of turn check ( ) drops into the confidence interval, the measured data is stationary. Waveform Filtering Filters play a vital role in removing selected wavelengths from an incoming waveform and minimizing random contributions called "noise". An ideal filter will have an amplitude response that is unity for the wavelengths of interest (called pass band) and zero everywhere else (called stop band). The wavelength at which the response changes from pass band to stop band is referred to as the “cutoff wavelength”. Figure 2.10 shows the sample of Chebyshev filters used to limit the wave irregularities within the interval of 3 to 25 m. Chapter 2 46 a) Chebyshev Low-Pass Filter b) Chebyshev High-Pass Filter Figure 2.10 – Chebyshev Filters A low-pass Chebyshev filter is designed to pass the low wavelengths of track irregularity, from zero to a certain cut-off wavelength, 3 m (k = 0.33), and to block high wavelengths. A high-pass Chebyshev filter is designed to pass the high wavelengths of track irregularity, from a certain cut-off wavelength, 25 m (k = 0.04), to λ (analysis wavelengths), and to block low wavelengths. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.2 0.4 0.6 0.8 1 Wavenumber (1/m) Amplitude Chebyshev Filter 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.2 0.4 0.6 0.8 1 Wavenumber (1/m) Amplitude Chebyshev Filter Literature Review 47 TRACK POWER SPECTRAL DENSITY (PSD) STANDARDS Comprehensive evaluations of the track irregularity spectrum have been made in several countries such as Britain, Germany, USA and China [Zhipping and Shouhua, 2009]. Various analytical expressions of the PSD function have been proposed, depending on the characteristics of the track measured in each specific country. Those studies are described in the following sections. A. The FRA PSD Standards (United States) The US Federal Railroad Administration (FRA) has classified the railway track into 9 categories of track classes, in which Classes 1 to 6 are designed for ordinary tracks and Classes 7 to 9 are dedicated for high speed railways. For each track classes, the random track irregularity is described using a one-sided power spectral density (PSD) function. It also has to be noted that due to the limitation in the measurement equipment, the function is only applied to the wavelength range of 1.524 m to 304.8 m [Liu et al., 2011]. The empirical formula of PSD is as follows: For vertical alignment: (2.36) For lateral alignment: (2.37) For gauge and superelevation irregularity (cross level): (2.38) Chapter 2 48 where: = PSD of track vertical alignment irregularity [cm2 / (rad/m)] = PDS of track lateral alignment irregularity [cm2 / (rad/m)] = PSD of track gauge or superelevation irregularity (cross level) [cm2 / (rad/m)] = spatial wave-number [rad/m] = critical wavenumber [rad/m] = roughness coefficient related to the line grade [cm2·rad / m] = a determined variable ( 0.25) The spatial wavenumber ( ) is related to the frequency per time unit (Hertz) by the following relation . Table 2.12 presents the parameters used in Equations (2.36) to (2.38). Table 2.12 – Coefficients for Power Spectral Density (PSD) function [Xia, 2002]. Line Grade Max. line speed [cm2.rad / m] [cm2.rad / m] [rad/m] [rad/m] Freight (km/h) Passenger (km/h) 1 16 24 1.2107 3.3634 0.8245 0.6046 2 40 48 1.0181 1.2107 0.8245 0.9308 3 64 97 0.6816 0.4128 0.8245 0.852 4 97 129 0.5376 0.3027 0.8245 1.1312 5 129 145 0.2095 0.0762 0.8245 0.8209 6 177 0.0339 0.0339 0.8245 0.438 Note: the coefficients of track classes 7 to 9 are not defined yet by the FRA Literature Review 49 B. German PSD Standard The German track PSD spectrum is widely used for dynamic simulations of railway vehicles, especially in the European countries [Zhiqiang et al., 2009]. The model is characterized by a single-sided spectrum and is best represented by track irregularities in the range of to (rad/m) [Zhang et al., 2010]. The PSD function is expressed by: For longitudinal profile: (2.39) For lateral alignment: (2.40) For cross level or superelevation irregularity: (2.41) where: = PSD of track longitudinal profile irregularity [m2 / (rad/m)] = PSD of track lateral alignment irregularity [m2 / (rad/m)] = PSD of cross level or superelevation irregularity [m2 / (rad/m)] = ⁄ denotes spatial wavenumber [rad/m] = critical wavenumber [rad/m] = 0.75 m (one half of track gauge) = scale factor for longitudinal profile [m2·rad / m] = scale factor for alignment [m2·rad / m] Chapter 2 50 The parameters for the above equations are given in Table 2.13, which represent the track irregularity with low and high levels of perturbation. Table 2.13 – Track PSD parameters [Lin et al., 2004] Parameters [10-7·m2·rad / m] [10-7·m2·rad / m] [rad/m] Low Disturbance 2.119 4.032 0.820 High Disturbance 6.125 10.80 0.820 Parameters [rad/m] [rad/m] Low Disturbance 0.0206 0.438 High Disturbance 0.0206 0.438 C. Chinese PSD Standard Various spectra of track irregularities (also referred as PSD Standards) were published by the Chinese Academy of Railway Science (CARS). The standards are created for evaluation and diagnosis of track quality, which are suitable for three different operational speed classes: 200 km/h, 160 km/h, and 120 km/h [Xianmai et al., 2008]. For each of these, the spectrum range is given to accommodate the disparity of spectral amplitude that may vary from one track section to another. The track condition with a certain spectrum is usually fitted with one of these ranges. The closer the spectral curve of the track section to the lower limit of the reference value, the higher the track quality; conversely, the closer the spectral curve to the upper limit of the reference value, the lower the track quality. Literature Review 51 The Chinese PSD function is based on single-sided spectrum, which relies on 6 different coefficients as shown below: (2.42) where denotes track irregularity PSD in unit [mm2 / (1/m)] and is wavenumber, often called spatial frequency of a wave, measured in 1/m. The spectral coefficients for Equation (2.42) are given in Table 2.14 to 2.16. Table 2.14 – Spectral Parameters for line speed of 200 km/h [Xianmai et al., 2008] Track Irregularity Gauge Upper 362.2681 0.2393 15370.860 681.2174 10.2670 -0.0007 General 54.0439 0.0357 8254.682 365.8602 5.5139 -0.0004 Lower 119.2536 0.0783 36295.990 1619.269 24.2936 -0.0018 Cross level (Superelevation defect) Upper 951.449 2.1747 47442.79 2121.780 25.473 0.0112 General 35.4842 0.0811 6369.446 284.8838 3.4199 0.0015 Lower 238.6205 0.5418 85347.970 3842.441 45.8306 0.02 Alignment Upper 0.0 0.00699 0.0 1.0 0.01893 0.00003 General 0.0 0.00194 0.0 1.0 0.01894 0.00003 Lower 0.0 0.00097 0.0 1.0 0.01893 0.00003 Longitudinal Profile Upper 0.0 0.00353 0.0 1.0 0.00752 0.0 General 0.0 0.00098 0.0 1.0 0.00788 0.0 Lower 0.0 0.00047 0.0 1.0 0.00783 0.0 Chapter 2 52 Table 2.15 – Spectral Parameters for line speed of 160 km/h [Xianmai et al., 2008] Track Irregularity Gauge Upper 612.3768 0.4046 8660.944 383.8376 5.7852 -0.0004 General 213.1331 0.1408 10851.220 480.9504 7.2484 -0.0005 Lower 187.2267 0.1238 31792.310 1407.659 21.23 -0.0015 Cross level (Superelevation defect) Upper 1890.022 4.2158 19981.090 984.2226 18.5928 0.0011 General 94.9519 0.2118 3613.811 178.0026 3.3627 0.0002 Lower 511.6737 1.1433 65036.82 3191.977 60.4676 0.0036 Alignment Upper 0.0 0.01751 0.0 1.0 0.01893 0.00003 General 0.0 0.00486 0.0 1.0 0.01893 0.00003 Lower 0.0 0.00146 0.0 1.0 0.01893 0.00003 Longitudinal Profile Upper 0.0 0.01016 0.0 1.0 0.00704 0.0 General 0.0 0.0029 0.0 1.0 0.00758 0.0 Lower 0.0 0.00084 0.0 1.0 0.0075 0.0 Table 2.16 – Spectral Parameters for line speed of 120 km/h [Xianmai et al., 2008] Track Irregularity Gauge Upper 640.740 0.4233 6524.507 289.1726 4.3582 -0.0003 General 255.976 0.1691 7819.645 346.5745 5.2233 -0.0004 Lower 325.929 0.2151 28425.14 1261.602 18.9956 -0.0015 Cross level (Superelevation Upper 1830.68 7.3882 20908.35 1028.226 30.9382 0.008 General 110.624 0.44649 2527.1 124.2566 3.73927 0.00097 Literature Review 53 Track Irregularity defect) Lower 1077.07 4.3434 70234.04 3460.220 103.9562 0.027 Alignment Upper 0.0 0.02622 0.0 1.0 0.01893 0.00003 General 0.0 0.00874 0.0 1.0 0.01893 0.00003 Lower 0.0 0.00306 0.0 1.0 0.01893 0.00003 Longitudinal Profile Upper 0.0 0.01351 0.0 1.0 0.00687 0.0 General 0.0 0.00478 0.0 1.0 0.00739 0.0 Lower 0.0 0.00166 0.0 1.0 0.00721 0.0 Note: Although PSD provides a limit range of the spectral amplitude, this does not mean that the track spectrum cannot be lower or higher than the threshold limit value. The range is proposed based on the expected amplitude span of the Chinese track irregularity spectra. D. SNCF PSD Standards (France) Through an investigation on the railway track in France, SNCF proposed a singlesided PSD, which is valid for vertical alignment. The equation, defined as a function of cyclic wavenumber [cycles / m], presents the track irregularities within the range of 2 m ≤ L ≤ 40 m [Broeck (2001), Fryba (1996)]. The SNCF model is as follows: For vertical irregularity: ( ⁄) (2.43) where: = Indication of the state of rail surface [m3] or [m2 / (cycle/m)] 308 0.509·10-6 for a good state 308 1.790·10-6 for a good state Chapter 2 60 The principle of operation of such tamping machine comprises several procedures: A. The tamping machine takes the position over the sleeper to be tamped B. The lifting rollers elevate the sleeper and rails to the adjusted level, leaving a void beneath the sleeper C. The machine arms bars are pushed down vertically into the ballast in either side of the sleepers D. By squeezing and vibrating the arms, the ballast fills the void beneath the sleeper and its packing is improved. E. The arms are withdrawn from the ballast and the machine is moved forward to the next sleeper to repeat the cycle operation Figure 2.12 – Tamping Process [Selig, 1994] Some undesired effects may occur during the tamping procedure. The vibration generated by tamping may, for instance, result in completely disturbed and loosened ballast bed. The disturbed ballast thus leads to lateral track instability, putting the track at risk of safety. In order to reduce this drawback, the infrastructure manager usually performs the subsequent activities of re-compaction of the ballast after tamping, using mechanical stabilization. Literature Review 61 2) Dynamic Track Stabilization The lateral track instability commonly occurs due to the loss of compaction of the ballast as a side effect of the vibration induced during the tamping operation. To mitigate this problem, the dynamic track stabilizer is used to consolidate ballast more densely and to provide an optimum homogenous settlement of the track. By imposing the DTS technique, the track will gain a settlement corresponding to 70,000 ton up to 100,000 ton of train loads [Lichtberger, 2005]. Figure 2.13 – Dynamic Track Stabilizer [Unitedindustrial, 2013] The dynamic track stabilizer consists of 4 axle wagon fitted with a diesel engine and pressurized cylinders on the stabilizing unit. When the stabilizing action is carried out, the machine generates a vertical force beneath the track with an approximate load of 356 kN. The vibration that is transmitted to the ballast, lies in the natural frequency range and caused the stones to settle closer together within the cavities. This method allows the track to settle more uniformly and systematically, resulting in a 30% extension in the maintenance cycle [Grabe & Maree, 1997]. Chapter 2 62 Figure 2.14 – Dynamic track stabilization equipment‟s [Total Track, 2013] 3) Ballast Cleaning Ballast becomes degraded due to the repeated passage of trains and to very intense compression during wheel-track interactions. Such ballast crushes into small particles of stones known as fines. When the fines combine with water, the ballast loses its primary function of support to the track bed, as well as its drainage capability. In order to remove the fines, ballast cleaning can be performed using an automated machine with adjustable excavating chain. The ballast is transferred upwards to the machine frame to be vibrated in order to eliminate the dirt and any other particles smaller than 35 mm. Afterwards, the conveyor arrangement distributes the clean coarse materials back to the ballast bed. Literature Review 63 Figure 2.15 – Ballast Cleaning Machine [Remtech, 2010] 4) Rail Grinding Irregularities in the geometry of the rails can cause a very high dynamic load. These irregularities partly occur due to faults in the manufacturing process or as a result of the train operation activities [Magel & Kalousek, 2002]. This special type of track imperfection is the so called rail corrugation, which is a periodic vertical irregularity on the railhead. Although rail corrugations do not pose a risk of immediate derailment, some undesirable problems can occur, such as increase in noise and in the vibrations experienced by passengers, ballast deterioration and higher maintenance cycles [Kumar, 2006]. At present, grinding can be considered the most effective maintenance practice to remove the irregularities and to restore the original rail profile. There are two types of rail grinding strategies. The first one is preventive grinding, which serves to prevent the development of defects growing from the surface or into the subsurface of the rail. In this method, the maintenance operation relies on the application of one pass of a large production grinder or multiple passes of a lighter grinder. The second strategy is the implementation of corrective grinding with the purpose of removing the defects on the Chapter 2 64 surface after they have shown significant presence in the rail [Sroba, 2004]. The operation usually involves multiple passes of a large production grinder. Typically, the grinding machine consists of a series of vehicles equipped with grinding wheels. As it moves along the track, the equipment performs a grinding operation on the rail surface while it re-profiles the rail [Cope, 1993]. When it is used for grinding operation, several grinding units are blocked in one angle plane while performing the reprofiling operation; the grinding wheels are set at different angles so that a polygonal profile is achieved. Figure 2.16 – Rail Grinding machine [Plasser and Theurer, 2013] An accurate application of the rail grinding will produce several impacts: - Overall improvement in rail life - Reduction in rolling contact fatigue - Reduction in rail wear - Reduction in corrugation - Reduction in energy dissipation - Reduction in noise Literature Review 65 5) Rail Lubrication Rail lubrication is a technique to reduce the friction and wear that occurs between the flange part of the wheel and the gauge side of curved tracks [Alp et al., 1996]. Using lubrication, the wear rate can be reduced about 10 to 15 times in the 300-400 meter curve radius and 2 to 5 times in 600 meter curve radius [Jendel, 1999]. Figure 2.17 shows a machine dedicated to perform lubrication. Lubrication may also be made by automatic applicators which are installed in the track or mounted on the motive stock. The selection of the application method will depend on the combination of economic factors, the nature of the railway network and the traffic levels. Figure 2.17 – Rail lubrication equipment [Memolub, 2013] Chapter 2 66 6) Replacement Traditionally, replacement simply consists of replacing the worn-out track components by new ones. As technology steps forwards, the estimate of the service life of a track structure or of a particular component can be easily determined. The replacement strategy is then conducted based on the prediction of the economic life span of the track materials. In this section, some replacement methods commonly used in the track maintenance will be briefly described.  Rail Replacement Prior to rails replacement, new welded rails are transported to the site using a train vehicle. When these arrive, the rails are brought down and placed beside the defected track for installation. A rail exchanger is then used to take out the old rails from ties and insert the new rails to the sequence.  Ballast Replacement A ballast replacement machine and its technique are quite similar to ballast cleaning. However, when the ballast replacement is carried out, a number of hopper wagons are normally attached in the sequence of the machine as storage and supplier of the new ballast. When the wagon arrives at the excavated site, the bottom of a bucket is then opened and the ballast falls down to the track. 2.7 REVIEW OF OPTIMIZATION MODELS The optimization model for scheduling a preventive maintenance by mathematical programming is a relatively new concept. However, there are already some contributions on this theme by some scholars. For example, Martland et al. [1994] proposed a technique to assist a rail manager in determining the best allocation for maintenance activities by minimizing the maintenance cost. The authors considered route geometry, track conditions, traffic Literature Review 67 volume along with the life cycle costing strategy as decision variables in the target functions. They examined the effect of these costs in an optimal maintenance schedule in a numerical example. A mathematical formulation for optimizing the maintenance works was also developed by Budai et al. [2004]. The objective of their model is to minimize the time required for maintenance, which is expressed by a cost function. For obtaining the nearly–optimal solution, they used a heuristic approach. Redy et al. [2006] presented a simulation model and developed a statistical analysis considering different types of lubrication and grinding strategies. Throughout the simulation, the impacts on various costs, such as grinding cost, operational risk, replacement and lubrication were analyzed to find the proper time interval for interventions. In another study conducted by Grimes [1995], an optimum schedule for track maintenance was obtained using the technique called Genetic Programming (GP). Financial aspects, such as the cost of maintenance and profit for maintaining quality, were the main considerations in generating the intervention action. Based on the comparison with other maintenance tools, GP provided a satisfactory performance. Lyngby et al. [2008] later used the procedure of Markov Chain to determine the optimum number of maintenance interventions required in the track segments. Three aspects were considered in the model development, i.e. punctuality cost, accident cost and extra maintenance cost, due to reduced track quality. In the works presented by Oyama and Miwa [2006], the maintenance schedule was developed through the use of integer programming algorithms. By taking into consideration the cost and the level of degradation, they developed a multi-criteria Chapter 2 68 optimization model to find the optimal preventive tamping intervals for broad railway networks. Another approach was also introduced by Hokstad et al. [2005] with the assistance of a computer software application namely Maple. Utilizing the combination of preventive maintenance and condition monitoring, the generated maintenance was scheduled by minimizing the conflict with train operation hours. Finally, Vale et al. [2010] developed an approach which made use of a mixed integer programming model specifically for scheduling tamping on ballasted tracks. The optimal solution was obtained by considering some technical aspects, such as the track gradual degradation, the track layout, the level of recovery and the allowable limits for intervention. Determining optimal maintenance intervals during the projected horizons, while assuring the safety and satisfying certain constraints are the objectives of this thesis. To achieve these objectives, a mathematical model designed to optimize maintenance schedule is formulated as mixed integer programming (MILP). The fundamental concept and the general nature of this model are described in the next sections. 2.7.1 MIXED INTEGER LINEAR PROGRAMMING (MILP) Linear programming (LP) is a branch of applied mathematics that deals with finding an optimal solution to a given linear function over a set defined by linear inequalities and equations. This technique was developed in 1947 by George Dantzig, an American mathematical scientist who invented an efficient method called simplex algorithms for solving linear problems. Shortly after, many scholars contributed to the field of linear programming in different ways, including theoretical development, computational aspects and exploration of new applications of the subject [Bazaraa et al., 1990]. Literature Review 69 Basically, linear programming contains several essential elements, which are: 1. Decision Variables 2. Linear Objective function 3. Linear Constraints The first element represents the level of quantity undertaken by the respective unknown variables (number of items to produce, amount of money to invest, etc.). These variables are usually represented using symbols, such as The second element deals with the goal or objective of a particular problem, such as minimizing the expenses or maximizing the profits. It consists of a certain number of variables which form a total objective value (Z) equal to The parameter of expresses the contribution of each unit to the objective function. The last element denotes limitations that restrict alternatives available to the decision makers. There are three types of constraints: less than or equal to (≤), greater than or equal to (≥) and simply equal to (=). The constraint “≤” ensures the solution used less than or equal the number of resources available. A “≥” constraint specifies minimum resources that must be utilized in the final solution. And the “=” constraint is more restrictive in the sense that it specifies the amount of some resource variables. Given these definitions, the standard formulation of LP can be written as follows: (2.45) (2.46) (2.47) Chapter 2 76 2.9 SUMMARY Maintaining and controlling the quality of the railway infrastructure are essential to ensure the availability of the system. These aims should be followed by the implementation of a track maintenance strategy with respect to the balance between safety level and economic aspects. In such case, a comprehensive understanding of the track degradation process and knowledge of all causes to rail degradation can help Infrastructure Managers (IM‟s) predict the track change behavior and prevent failures in the system. This chapter provided a general review of the track degradation mechanisms, the analytical models used for assessing the track geometry condition, followed by the common techniques for carrying out track maintenance. In the end of chapter, the fundamental concept and general nature of mathematical integer programming, to be used as a basis for finding the optimal tamping schedule, is described. From a review on the available literature, the assessment of railway track quality can be classified into two different approaches: assessment by considering the structural aspect (consists with settlement, wear and fatigue) and assessment by considering the geometrical aspect. The last approach measures a railway track quality from the progression of statistical and power spectral density of geometry defects (see Table 2.17). Satoh, Shenton, ORE and TU Munich have developed track degradation models from a structural perspective. Settlement is the main consideration in the models, governing the behavior and performance of railway tracks. The expression given for the deterioration is distinguished by two major phases: the first phase is related to the rapid settlement after maintenance and the second phase is associated with the long term settlement. However, this approach lacks the implementation of some influencing parameters such as train speed and dynamic load. The more comprehensive degradation model is then given by Sugiyama, who took into account the factors of train speed and track structure. Literature Review 77 The model is of particular use for predicting track degradation for 100 days of train operation and it considers a cumulative of one year passage tonnage rather than individual axle load. In the TU Graz model, the initial track quality is of the highest importance as it may determine the behavior of the railway track over its entire period of service. To calculate the track quality, this model should be combined with measurements of geometry parameters obtained from the track recording car. Archard wear equation is a simple model and the base for a number of refined wear models. However, the wear coefficient, as used in this model, is difficult to estimate due to the need of detailed information from both laboratory and field tests. An alternative approach for analyzing wear is the ITDM model, which is more sophisticated and more complex. It endeavors to embrace all the major factors which may influence service life of track components such as material hardness, wheel forces and rail lubrication. Further reviews were also made to other models, namely the Track Quality Index (TQI), that utilized geometrical parameters for track quality assessment purposes. Most of the models depend on the statistical evaluation of track geometry defects over a particular distance. The quantitative value of track quality can be varied from one model to the others, due to the diverse type of measurements conducted by railway companies, such as the mid chord of measurements (used to measure deflection of geometry defect) and the weighted value for each single geometry parameter. Similarly to the TQI model, the track quality assessments based on PSD are considerably varied among the countries. Usually the model is developed to represent the railway track spectrum in a particular country. The characteristic features of each PSD model as well as its comparison will be discussed in Chapter 4. In the area of track maintenance, several methods and technologies used to repair and to correct the track defects have been identified. Such maintenance is diverse according to the mechanisms and the consequences of improvement. Tamping, for instance, is widely Chapter 2 78 used to correct the geometry defect caused from deformation in the track structural bed, while rail grinding is used particularly to remove the defect in the rails caused by manufacturing or the nature of operations. The combination of maintenance methods can result in a higher performance of the track infrastructure. Finally, the fundamental concept and general nature of mixed integer programming model are explained. This kind of problem has proved to be useful to address diverse types of problems in planning, routing and scheduling of railway track maintenance. The branch and bound method and cutting plane method are the main techniques in order to derive the optimum solution from the constructed mathematical problems. The application of this problem in an actual railway network will be presented in Chapter 6 3 RESEARCH METHODOLOGY 3.1 INTRODUCTION This chapter outlines the research methodology designed to achieve the aim and objectives of the thesis. It begins with an explanation of the research strategy and objectives, followed by the research structure. This chapter also explains the processes employed and provides a justification for the selection of methodologies and for preparation of the conclusions. As stated in the introductory chapter, this research aims to develop an optimization model for scheduling track maintenance with respect to safety and reliability issues. The Power Spectral Density (PSD) forms a core focus of the studies since it involves a systematic technique for evaluating track quality condition. To achieve the underlying objective, the research begins by examining the application of power spectral density in track quality assessments. Investigation is then continued further by seeking the relationship and degree of interdependency of one geometry variable to the others. These steps are of particular importance towards establishing a foundation for planning an effective maintenance decision as well as for providing a reasonably accurate model of track degradation, which takes into account interactions among various geometrical parameters. The development of a proposed model is then validated in practice. Chapter 3 80 3.2 RESEARCH STRATEGY Research strategy can be defined as the way in which the research objectives are achieved. There are two general strategies within the context of research, namely „quantitative research‟ and „qualitative research‟ [Greene and Caracelli, 1997]. Deciding on which type of research will be conducted depends on the purpose of the study, the available resources, and the type and availability of information [Bouma and Atkinson, 1995].  Quantitative research Quantitative research refers to the systematic empirical investigation of a given problem, based on testing a hypothesis or a theory composed of variables, measured with numbers, and analysed with statistical procedures or computational techniques, in order to determine whether the hypothesis or theory holds true [Creswell, 1994]. Quantitative data is, therefore, not abstract, it is solid and reliable, and presented in numerical format such as statistics, percentages, quantities, etc.  Qualitative Research Qualitative research refers to a method of inquiry employed in many different academic disciplines, which emphasizes meanings, experiences and understanding the complexity of the problems [Strauss and Corbin, 1998]. The information gathered in qualitative research may facilitate the interpretation of the relationships between variables. In order to take advantage of the strengths of both aforementioned methods, this research employed a combination of quantitative and qualitative approaches. The quantitative approach is used since the data analysis comprises many numbers, counts and statistical procedures to derive a base model for maintenance optimization. Research Methodology 81 The qualitative approach is conducted using a case study to help attain a deeper understanding of the interaction of different geometry variables associated with track deterioration. This method also enables to draw conclusions emerged from the data, to form a theory that explains a pattern in the base model and at the same time validates the theories in practice. 3.3 RESEARCH OBJECTIVE Research can be classified into several categories depending upon the knowledge on a certain area and the intended solution [Kumar, 2006]:  Exploratory Research Exploratory research is practically used for a problem that has not been clearly defined or when the researcher does not have sufficient knowledge on the area of study. The focus in this research is to gain a deep insight and familiarity on the issues for further investigation.  Descriptive research This research category attempts to describe the situation or phenomena of an issue in a systematic manner. There are many methods involved in this study, such as conducting surveys to describe the status-quo and developmental studies seeking to observe changes in the behaviour of a phenomenon.  Explanatory research When an issue is already known and there is some description of it, this research category can help identify its “why” and “how”. This type of research looks for causes and reasons for such situation or phenomenon. For example, a descriptive research may discover that the wheel-rail interaction is one of the factors which play an important role in track geometry degradation, whereas the explanatory research is more interested in learning why or how the interaction between wheel and rail can influence the degradation process. Chapter 3 82  Correlational research Correlational research is used to discover the relationship or interdependency between two or more aspects of a situation. It attempts to identify the causal of a phenomenon on one hand and the impact on the other hand, for instance the relationship between the track stiffness and the level of degradation. In the current analysis, the selected methodologies are descriptive and correlational researches. Descriptive research is used in the early studies to describe the mechanism of changing of track performance behaviour as well as to explain the parameters influencing track degradation. The correlational research seeks to identify the relationship among various track geometry parameters as a foundation for developing the optimization model. 3.4 RESEARCH PROTOCOLS A conceptual framework was developed to specifically guide and monitor the activities of the research in a systematic way. The framework contains the descriptions of several main elements on how projects and activities are expected to work to accomplish the objectives. Figure 3.1 shows the sequence of the research methods. The research begins by conducting a comprehensive literature survey to address the research aim and objectives as described in chapter one. The research aim is to develop a logical model for the deterioration of track geometry and to incorporate the proposed model as basis for optimizing maintenance in practice. The research is divided into two phases. The first phase is to investigate the application of power spectral density in the track quality assessments. Various PSD standards are compared in order to define the characteristic features contained in each particular standard. The implementation and procedure used to quantify the state of railway Research Methodology 83 irregularity will be also described with an application of case study. The investigation is then continued further in the second phase. A comprehensive correlation study is conducted to determine the degree of interdependency and to establish the similarity of one geometry variable to another. Using constructive knowledge, a predictive degradation model containing several geometry parameters is then developed. The model is able to forecast the future progress behavior of the track irregularity in the statistical and frequency domain. Each of the research phases contains several main activities. The following section is dedicated to explain how the research aim and objectives are achieved. Chapter 3 84 Model Validation Figure 3.1 – Schematic Diagram of the Research Methodology Yes No Regression analysis Selecting the Track Geometry Statistics Condition Rating Transition degradation Models Causal Parameters Improvement/Recovery Model Analyzing PSD functions and their applications Analysis of variance Maintenance Optimization Case study Data Set for Modelling Correlation analysis Review Existing Models Finish Phase I Phase II Research Aim 1 Research Aim 2 Research Aim 3 Research Methodology 85 Phase I This section discusses the activities shown in Figure 3.1 that relate to phase I of the research methodology. Such activities include literature review, data collection, research approaches and procedures used to develop the optimization model. The literature survey was carried out within (1) the concept of track degradation and (2) the application of existing methodologies for assessing track quality, including track quality index and power spectral density approaches. The aims are to develop a comprehensive understanding of the track degradation mechanism, to systemize the normative aspects that should be taken into account in the framework modelling, and to identify the feasibility of current approaches in adapting the changes in track performance behaviour. In order to obtain this information, different databases from many sources were explored. These data can be classified as primary and secondary data. The data collected by the researcher for the purpose of study through various experiments or from the onsite data recording are called primary data. The data taken by the researcher from secondary sources, internal and external, are called secondary data. In the context of this study, the primary data were collected from REFER databases. It consisted of measurement reports conducted by Track Recording Car EM 120, which comprised all information about the geometrical quality of the track, such as longitudinal profile, horizontal alignment, gauge, super-elevation, network topography, etc. Over 8 years of inspection records were acquired from October 2003 to January 2009. For the secondary data, the full text of many journal articles and books were found in electronic databases, such as Elsevier, Emerald, ASCE, Transportation Research Record, etc. Some technical standards from European Committee for Standardization (CEN) were also examined. The researcher also studied relevant reports, master thesis and PhD dissertations from various universities. Chapter 4 92 Figure 4.3 – Comparison of Various PSD Standards - PSD Cross Level or Superelevation irregularity* Figure 4.4 – Comparison of Various PSD Standards - PSD Gauge* *) the number of PSD standards in each particular geometry variable depends on the availability of PSD functions 10-1 100 101 102 103 10-4 10-2 100 102 104 106 Wavelength (m) PSD [mm2 /(1/mm)] Comparison PSD standards - Track Cross Level FRA Germany China 1 FRA 1 2 FRA 6 3 Germany high 4 Germany low 5 China 120 6 China 200 Germany Wave range FRA Wave range China Wave range 4 1 5 6 2 3 10-1 100 101 102 103 10-4 10-2 100 102 104 106 Wavelength (m) PSD [mm2 /(1/mm)] Comparison PSD standards - Track Gauge FRA China 1 FRA 1 2 FRA 2 3 FRA 3 4 FRA 4 5 FRA 5 6 FRA 6 7 China 120 8 China160 9 China 200 China Wave range FRA Wave range 1 2 3 6 5 7 8 9 4 The Application of Power Spectral Density (PSD) in Track Quality Assessments 93 Figure 4.1 shows the comparison of longitudinal profile among PSD standards. Braun‟s track irregularity has less value than (or is superior to) the other standards, particularly for the short wave defects, of less than 16 m. For wavelength below 4 m, the curve of “Chinese 200” PSD expresses the same magnitude as “German low disturbance” spectrum and is superior to SNCF and FRA power spectral density. An opposite result is given by France spectrum (“SNCF good”), which is stricter for long wave irregularities, especially for those above 16 m. The PSD comparison for track alignment irregularities is further detailed in Figure 4.2, which is supported by the function from FRA, German, and Chinese PSD standards. For wavelengths below 38 m, the magnitude of “German low disturbance” is lower than the other two standards, which indicates more restriction to the allowable tolerance imposed by German high speed lines. Similar PSD value is seen in “Chinese 200”, “German high disturbance” and “FRA PSD 6” (177 km/h), particularly for wavelengths below 10 m. A similar view is also given in Figure 4.3 for PSD Cross level (Superelevation irregularity), where the PSD of “German low disturbance” has lower spectral magnitude (superior) than the PSD of China and USA, for both short and long wave irregularities. For defects below the wavelength of 2 m, the magnitude of the PSD of “Chinese 200” is overlapping the PSD of “German high disturbance” and is lower than the PSD of “FRA 6”, which indicates the superiority of the track quality construction in German and China. Furthermore, the figure also shows that for a line speed of 177 km/h (Class 6), FRA has better control than “China 120”. Figure 4.4 presents the PSD comparison of track irregularities for gauge variable. It is argued that the tolerance value of the Chinese spectrum is likely to be equal to the FRA standard, especially when it is analyzed based on the travelling speed. The spectrum of “FRA 6” (177 km/h), for example, lies between the “Chinese 200” and “Chinese 160” (wavelength < 7 m). Chapter 4 94 The comparisons of all these different standards have revealed that the German PSD is generally stricter to the geometry errors of longitudinal profile, alignment, and cross level or superelevation irregularity, which indicates a better quality control applied by German railway standards. The Chinese and FRA PSD are also showing the same characteristics in terms of curve trends as well as spectra amplitude, particularly for the wave irregularities below 10 m. 4.3 PROCEDURES FOR APPLYING PSD STANDARDS This section deals with the procedure used to generate an artificial irregularity resulting from PSD standards as well as the techniques for finding the best fitting curve of the track irregularity spectra to a particular PSD standard. Both approaches are beneficial to determine the state of the track condition in the later stages. 4.3.1 TRANSFORMATION OF PSD INTO RAIL GEOMETRY IRREGULARITY Broeck [2001], Xia [2002], Lei & Noda [2002], Zhang et al. [2001], Song et al. [2003], Ju et al. [2010] and Gupta [2008] have all employed a similar method for creating an excitation force in the vehicle dynamic analysis. The steps of the procedure are explained below. Supposing a stationary stochastic process with zero mean and variance, , the sample function of the stochastic process can be simulated by: ∑ (4.1) where: = Gaussian random variable with zero mean and variance, , as defined by: The Application of Power Spectral Density (PSD) in Track Quality Assessments 95 = Power Spectral Density Standard = Phase angle distributed between 0 and 2л randomly = Center wavenumber. In order to attain this value, a wave band, , can be defined as: therefore, ( ) where is the total number increments in the range of ( , ). Another method for creating rail track irregularity was inspired by Claus and Schiehlen [1997], Yang et al. [2004] and Dias et al. [2008]. Assume that PSD standard is defined as a function of the spatial wavenumber (rad/m). The random track irregularity can be produced by implementing the trigonometric series, as expressed by: √ ∑ (4.2) where: = total number of discrete angular wavenumber considered = discrete angular wavenumber (rad/m), which defined as: where: and are the upper and the lower limits considered. = phase angle distributed between 0 and 2л randomly = the amplitude coefficient of random series, defined as: Chapter 4 96 , √( ) √( ) √( ) for Checking Procedure In order to use the spectrum standard for assessing track quality, the PSD function is transformed from the wavenumber-based domain to the solution in the spatial-based domain. The result, an artificial irregularity dependent of the travelled distance, is shown in Figure 4.5. Subsequently, to validate the proposed simulation approach, the reverse process is carried out by determining the power spectral density of the artificial track irregularity according to Equation 2.29 and comparing the result with the analytical curve of PSD standard. Figure 4.6 gives a good agreement between these two spectra, which indicates the appropriateness of the method in the rail irregularity simulation. The Application of Power Spectral Density (PSD) in Track Quality Assessments 97 Figure 4.5 – Simulated track irregularities Figure 4.6 – Comparison of PSDs 4.4 FACTORS AFFECTING THE GENERATION OF TRACK IRREGULARITY In order to provide an appropriate method in rail track generation, several factors that may influence to the magnitude of an artificial track irregularity should be clearly defined. For this purpose, some consideration factors such as the lines speed, the preference of wavelength interval (wave band) and the selection of PSD standard are investigated, and the results are given as follows. A process for generating random track irregularity has been presented in the preceding section. Figures 4.7 to 4.9 present the examples of the track geometry simulation obtained from the Chinese power spectral density, corresponding to the line speeds of 120 km/h, 160 km/h and 200 km/h, respectively (see sec. 2.5.3). The simulation generated a longitudinal profile irregularity for general class spectrum with wavelength range between 3-25 m. The figures also show the threshold limits for isolated defect as defined in the European Standard (see sec. 2.5.2). 0100 200 300 400 500 -20 -15 -10 -5 0 5 10 15 20 Position along the track (m) Artificial Track Irregularities (mm) Rail Irregularity Generation Artificial signal 10-3 10-2 10-1 100101 10-6 10-4 10-2 100 102 104 Comparison between PSD of artificial rail irregularity and PSD standard n (cycles/m) PSD, (m2 /(cycles/m)) PSD artificial rail PSD standard Chapter 4 98 Figure 4.7 – Transformation PSD for China 120 km/h Figure 4.8 – Transformation PSD for China 160 km/h Figure 4.9 – Transformation PSD for China 200 km/h Based on Figures 4.7 to 4.9, it can be observed that line speed has greatly contributed to the amplitude of the geometrical defect. The higher the line speed results, the lower the track irregularity obtained from PSD. As a matter of fact, all the rail generation produced from Chinese PSD are below those of European Standard limits. The preference for the use of frequency bandwidths is another issue of interest. This fact is due to the limitation in the measurement equipment/track recording car that may not be able to detect the geometrical defects in some specific intervals. The spatial track irregularities have therefore been generated from various PSD functions, taking into 050 100 150 200 250 300 -25 -20 -15 -10 -5 0 5 10 15 20 25 Transformation PSD China 120 km/h Distance (m) Track Irregularity (mm) Artificial Signal Alert Limit Intervention Limit Safety Limit 050 100 150 200 250 300 -25 -20 -15 -10 -5 0 5 10 15 20 25 Transformation PSD China 160 km/h Distance (m) Track Irregularity (mm) Artificial Signal Alert Limit Intervention Limit Safety Limit 050 100 150 200 250 300 -25 -20 -15 -10 -5 0 5 10 15 20 25 Transformation PSD China 200 km/h Distance (m) Track Irregularity (mm) Artificial Signal Alert Limit Intervention Limit Safety Limit The Application of Power Spectral Density (PSD) in Track Quality Assessments 99 account different wavelength intervals as defined in the European Standard [prEN 13848-5]. - D1: wavelength of irregularities within the interval of 3 < λ ≤ 25 m - D2: wavelength of irregularities within the interval of 25 < λ ≤ 70 m - D3: wavelength of irregularities within the interval of 70 < λ ≤ 200 m Figures 4.10 and 4.11 present the generations of rail track irregularity obtained from FRA 6 and German low disturbance spectra (best classes) using different wavelength ranges, respectively. The rail simulation follows the procedures as described in the Section 4.3.1, with an adjustment on the parameter of waveband ( ) that is corresponding to the particular wave range in interest. a) FRA 6 D1 (3 < λ ≤ 25 m) a) German Low D1 (3 < λ ≤ 25 m) 0200 400 600 800 1000 -30 -20 -10 5 0 5 10 20 30 Position along the track (m) Artificial Track Irregularities (mm) Artificial Track Irregularities Generated from FRA 6 Artificial signal Alert Limit Intervention Limit Safety Limit 0200 400 600 800 1000 -30 -20 -10 -5 0 5 10 20 30 Position along the track (m) Artificial Track Irregularities (mm) Artificial Track Irregularities Generated from Germany Low Artificial signal Alert Limit Intervention Limit Safety Limit Chapter 4 100 b) FRA 6 D2 (25 < λ ≤ 70 m) b) German Low D2 (25 < λ ≤ 70 m) *No threshold limits defined for D3 *No threshold limits defined for D3 c) FRA 6 D3 (70 < λ ≤ 200 m) c) GermanyLow D3 (70 < λ ≤ 200 m ) Figure 4.10 – The Influence of Wavelength in the Artificial Longitudinal Profile Irregularity - FRA Class 6 Figure 4.11 – The Influence of Wavelength in the Artificial Longitudinal Profile Irregularity - German Low disturbance As expected, the magnitude of track irregularities generated by “FRA 6” PSD appears to be greater than the one produced by “German low” track spectrum. This fact confirmed the finding in Figures 4.10a and 4.11a , which indicated the superior quality of the “German low disturbance” in relation to the “FRA 6” PSD. 0200 400 600 800 1000 -30 -20 -10 -5 0 5 10 20 30 Position along the track (m) Artificial Track Irregularities (mm) Artificial Track Irregularities Generated from FRA 6 Artificial signal Alert Limit Intervention Limit Safety Limit 0200 400 600 800 1000 -30 -20 -10 -5 0 5 10 20 30 Position along the track (m) Artificial Track Irregularities (mm) Artificial Track Irregularities Generated from Germany Low Artificial signal Alert Limit Intervention Limit Safety Limit 0200 400 600 800 1000 -30 -20 -10 0 10 20 30 Position along the track (m) Artificial Track Irregularities (mm) Artificial Track Irregularities Generated from FRA 6 Artificial signal 0200 400 600 800 1000 -30 -20 -10 -5 0 5 10 20 30 Position along the track (m) Artificial Track Irregularities (mm) Artificial Track Irregularities Generated from Germany Low Artificial signal The Application of Power Spectral Density (PSD) in Track Quality Assessments 101 Note that the magnitude of track defect grows as the wavelength of interest is becoming longer. For example, one can remark that the maximum irregularity of “German low D1” (Figure 4.11a) is around 2 mm, while the maximum defect of “German low D3” (Figure 4.11c) is about 5 mm. 4.5 THE APPLICATION OF PSD STANDARDS IN TRACK QUALITY ASSESSMENTS The specimen used in this analysis was taken from a particular segment of the Portuguese Northern Railway Lines. The geometrical data was provided by the Track Recording Car (TRC), from successive inspections conducted on March 5, 2007 (before maintenance) and June 25, 2007 (after maintenance). Using an optical measurement system, this car is able to record various geometry parameters such as longitudinal profile, alignment, gauge, superelevation irregularity (cross level), and twist in points spaced by 0.25 m. Although the rail track in this study is only approximately 1 km long for a design speed of 120 km/h, this section mainly intends to show how the PSD method is put into practice in track quality assessments. Rail Track Generation Based on the proposed simulation method (see Chapter 4.3), the track irregularities are generated from German and FRA PSD standards. Each standard is comprised by two different track classes; one is the highest (best) track class of the standard and one is the same class with the real track data. In this case, FRA 6 and German low disturbance are classified in the first category while FRA 4 and German high disturbance are analyzed in the second category. Every rail simulation is generated with respect to a waveband between 3 m to 25 m. The track quality is then assessed by comparing the artificial track irregularity with the real track data. Figure 4.12 gives a comparative sample of track longitudinal profile irregularities for a length of 250 m. Figures 4.13 and 4.14 show detailed comparisons of the artificial irregularities of the German and the FRA PSD, respectively. Chapter 4 108 4.6 CONCLUSIONS The PSD standards obtained from various countries have been briefly described. The comparisons among each of them as well as their application in practice were also reviewed. Several essential facts can be drawn as given in the following points.  Power Spectral Density (PSD) has great advantages in the railway track quality assessment. It can describe a wide range of spectral characteristics of random wave irregularity, indicating peak and cyclic peak in each wave. The wavelength is strongly linked with problems; the short wavelength associates with train safety while the long wavelength corresponds to riding comfort.  There are several functions and methods for transforming a particular PSD standard to the stochastic random series. The procedures are presented in detail in section 4.3. By comparing the inverse of the time series generated from the PSD function and the theoretical PSD, it was found that the applied methodology is concise and acceptable.  In the simulation of rail geometry defect, the magnitude of track irregularity is considerably influenced by the preference of wavelength interval (waveband) and line speed. Longer wavelength intervals result in higher magnitude, whereas shorter wavelength intervals create the opposite. On the contrary, a higher line speed will give a smaller variability of the track defect while a lower line speed produces a larger magnitude.  Based on the comparison among different PSD spectrums, it can be seen that the German PSD is generally stricter for the geometry errors of longitudinal profile, alignment and cross level or superelevation irregularity, which indicates a better quality control applied by the German railway standards. For the curves produced by Chinese and FRA spectrums, the analysis shows a comparable result in terms of The Application of Power Spectral Density (PSD) in Track Quality Assessments 109 their magnitude and tendency especially for wave irregularities below 10 m. The detailed analysis on the comparison of various PSD standards is as follows. A similar characteristic of the longitudinal profile is shown between Chinese 200 and German low disturbance at wavelengths shorter than 4 m, which is superior to the SNCF and FRA power spectral densities. As the wavelength increases, France spectrum (SNCF good) takes it into more consideration especially for the irregularities above 16 m. Note that in this particular wave, the SNCF PSD results correspond to a highest curve. For the PSD comparison of track alignment, the magnitude of German low disturbance is lower than FRA and Chinese PSD at wavelengths lower than 38 m, indicating more restrictions to the allowable tolerance imposed by German high speed lines. At wavelengths below 10 m, the curves presented by Chinese 200, German high disturbance and FRA 6 are close and seem to be almost equal. A superiority of German PSD can also be found in the PSD comparison of cross level or superelevation irregularity at all the investigated wavelengths.  The analysis reveals that the spectral quality of the track before and after tamping is considerably different. The spectra of the track after intervention shows a decrease in power compared with the track spectrum before intervention. This fact thus justifies the applicability of PSD to evaluate the performance indicators obtained from the maintenance work. Chapter 4 110 5 CORRELATION ANALYSIS OF RAILWAY TRACK GEOMETRY 5.1 INTRODUCTION Track geometry is an important factor influencing journey quality and track performance. It consists of several geometry variables such as longitudinal profile, horizontal alignment, cross-level, twist and gauge, which are closely related. Combined track geometry irregularities may cause a severe vehicle-track interaction that affects train safety and derailment resistance. Apart from that, the deterioration of certain track geometry parameters does not stand alone. Current research indicates that a degraded track geometry parameter can induce further degradation of the other parameters [Karttunen, 2012]. It is therefore necessary to understand the role and impact of each variable on the others and the type of relationship, if any, among track geometry parameters, to provide a base for developing a reasonably accurate model of track degradation. Such knowledge could also assist in modeling the input of rail excitation in vehicle-track dynamic simulation [Broeck, 2001]. The analysis has been conducted, using a typical track geometry data from the Northern line of the Portuguese Railways, to establish the actual relationship and statistical Chapter 5 112 correlation that may exist among track geometry variables. The relationship analyses were conducted in the wavelength domain using three different approaches: cross correlation, autocorrelation and coherence analysis. The analyses results will determine the appropriate method to construct the prediction model of deterioration used in the track maintenance optimization problem (Chapter 6). The following sections describe the methodology used and the results of these analyses. 5.2 THE CORRELATION OF TRACK GEOMETRY The correlation among track geometry parameters is applied to quantify the degree of interdependency of one geometry variable to the other, or to establish the similarity between two different datasets. This concept enables to distinguish three correlation categories. The first category is autocorrelation, which describes the general dependency of values of some observations at a certain distance to the values of the same observations at another distance . A symbol of is known as the lag distance between these observations. Equation 5.1 gives an autocorrelation formula for a random continuous track irregularity [ ]: ∫ (5.1) where is the length of the track irregularity signal and represents the amount of lag that should be shifted in distance relative to the original signal For the sampled track irregularity signal, the autocorrelation function is given by the following equation: ∑ (5.2) Correlation Analysis of Railway Track Geometry 113 where space shift, , is quantified by the number of lag samples and is a space domain of track irregularity. The autocorrelation of a random continuous track irregularity may exhibit a greater value at a smaller lag and probably a lower value at a larger lag. The highest peak of the function is identified at zero lag, , which equals to the average power of the input waveform. ∑ (5.3) | | (5.4) where is the average power of the input waveform and constitutes the maximum value of the autocorrelation function. The autocorrelation function is particularly useful in identifying the presence of repetitive patterns or periodicities in a given dataset, which in turn can be beneficial to determine the condition of track geometry [Zhiping and Shouhua, 2009]. The second category is called cross-correlation. The concept is basically similar with autocorrelation. However, instead of correlating a waveform against itself, the cross correlation is performed by taking two different waveforms as a function of a space-lag applied to one of them. Considering two different waveforms and , the cross correlation is given by: ∫ (5.5) Chapter 5 114 or ∫ (5.6) where is the length of the track irregularity signal and represents the amount of lag that should be shifted in distance relative to the original signal For a sampled signal of track irregularity, the cross correlation function is defined as: ∑ (5.7) where is the number of shifted distances or lags. In practice, to determine the degree of similarity between two signals is not sufficient to simply compare the amplitude of the cross-correlation. Normalized cross-correlation is often used to quantitatively assess the quality of the correlation. This value is obtained by normalizing the magnitude by an amount depending on the energy content of the data, as given by: [∑ ∑ ] ⁄ (5.8) The normalized quantity ) will vary between -1 and 1. Zuo and Xiang [2006] have proposed a guideline for the interpretation of correlation coefficients: a) When , the signals s(n) and g(n) are perfectly correlated, while a value of shows that the signals are completely uncorrelated. b) When , a linear relationship exists between the two signals. A higher p value indicates a stronger correlation, while a lower p value means a weaker correlation. Further classifications of this category are: Correlation Analysis of Railway Track Geometry 115  , refers to a weak correlation  , refers to a low correlation  , refers to a significant correlation  , refers to a high correlation c) When , the bond is identified as positive relationship while for , the bond is identified as negative correlation. The last category is called coherence, which measures the linear dependence between two signals as a function of wavelength. The analysis of coherence is of particular importance, since it is able to identify at which wavelengths two stochastic waveforms are coherent and at which wavelengths they are not. Given two sampled signals of track irregularity, and , the coherence is based on the square of the absolute value of the cross-power spectrum divided by the power spectrum of the input signals, as defined by: | | ( ) (5.9) where: = coherence = cross-power spectrum density between signal and which is obtained by: ∑ = power spectrum of signal , which is obtained by Chapter 5 116 ∑ = power spectrum of signal , which is obtained by ∑ The magnitude of the coherence function at any frequency has a range of values between (zero) and (one). The value of one indicates perfect coherence between two different datasets in a particular wavelength, while the value of zero indicates the opposite. 5.3 TEST DATA ANALYSIS A comprehensive track geometry database was prepared to investigate the statistical correlation among rail geometry variables. The data was collected from the Track Recording Car (TRC) EM 120, which provided around 17 measurement surveys from October, 2003 to January, 2009 (1926 days). During the measurement process, the TRC EM 120 took samples once every 0.25 m and counted several geometry variables such as longitudinal profile, alignment, gauge, cross level or superelevation irregularity, curvature, altitude, etc. Figure 5.1 gives the details of the track characteristics of the Portuguese Northern Line railway used in this analysis. The sampled line is located at the midpoint between two cities, Pampilhosa and Aveiro, with a total length of 34 km. In order to have a sufficient sample size for spectrum analysis, the track is partitioned into 34 equal sized sections that are averaged to obtain the value of correlation. Correlation Analysis of Railway Track Geometry 117 Analyses of track geometry irregularities in the vertical and horizontal planes were conducted for each rail separately. There are three types of the mentioned data used in these analyses: track irregularities based on the measurement of 10 m chord length (with no specified wavelength interval), track irregularities D1 (with wavelength range between 3-25 m) and track irregularities D2 (with wavelength range between 25-70 m). Position Start End 200+000.00 233+399.7 5 Length 33.4 km Period of Investigation 2003-2009 (17 Measurement Files) Design Speed 50 - 220 km /h Track Geometrical Characteristics Mixed line [Straight and Curved] Figure 5.1 – Track Characteristics of Sample Track Segment 5.3.1 CROSS CORRELATION Using the methods described in the preceding section, the computation of crosscorrelation was conducted to seek the relationship among the track geometry variables, including:  Left alignment  Right alignment  Left longitudinal profile  Right longitudinal profile  Gauge  Super-elevation  Twist  Curvature Appendices 220 63 NaN NaN NaN NaN 0.0006 0.19 0.50 11 0.00064 64 0.0005 0.57 0.84 4 0.0006 0.11 0.66 9 0.00051 65 0.0002 1.39 0.57 4 0.0007 -0.22 0.61 11 0.00044 66 0.0004 1.30 0.57 6 0.0007 -0.10 0.47 7 0.00055 67 0.0007 -0.09 0.89 8 0.0007 -0.43 0.71 9 0.00071 68 0.0003 0.50 0.53 9 0.0003 0.28 0.84 9 0.00029 69 0.0003 0.59 0.63 10 0.0004 -0.08 0.76 9 0.00035 70 0.0007 -0.14 0.49 11 0.0004 0.01 0.68 9 0.00055 71 0.0003 0.64 0.74 7 0.0002 0.27 0.49 7 0.00023 72 0.0005 0.22 0.68 9 0.0004 0.00 0.76 9 0.00048 73 0.0003 0.38 0.81 8 0.0002 0.48 0.42 7 0.00028 74 0.0003 0.33 0.59 8 0.0003 0.51 0.60 11 0.00029 75 0.0002 0.21 0.47 10 0.0002 0.66 0.68 4 0.00022 76 0.0003 0.19 0.71 11 0.0005 0.46 0.91 4 0.00042 77 0.0004 0.10 0.63 9 0.0004 0.63 0.70 7 0.00041 78 0.0002 0.52 0.54 9 0.0003 0.44 0.89 6 0.00023 79 0.0002 0.33 0.59 6 0.0002 0.72 0.59 8 0.00020 80 0.0001 0.76 0.58 5 0.0002 0.22 0.62 11 0.00016 81 0.0002 0.75 0.65 7 0.0003 0.07 0.77 11 0.00025 82 0.0001 0.77 0.75 7 0.0002 0.18 0.76 11 0.00015 83 0.0002 0.40 0.80 5 0.0003 0.24 0.73 5 0.00024 84 0.0003 0.84 0.89 6 0.0005 0.44 0.70 10 0.00041 Appendices 221 85 0.0003 2.04 0.88 5 0.0003 1.66 0.51 8 0.00033 86 0.0003 2.18 0.31 6 0.0010 1.42 0.39 4 0.00064 87 0.0001 1.57 0.13 5 0.0007 0.36 0.92 4 0.00037 88 0.0004 1.42 0.83 6 0.0011 0.06 0.91 4 0.00074 89 0.0007 1.89 0.66 6 0.0008 2.04 0.75 4 0.00077 90 0.0018 1.31 0.37 7 0.0030 -0.86 0.71 4 0.00240 91 0.0003 0.37 0.69 10 0.0004 0.83 0.86 10 0.00035 92 0.0004 0.85 0.67 10 0.0001 0.63 0.59 10 0.00029 93 0.0004 0.98 0.57 8 0.0003 0.74 0.72 10 0.00037 94 0.0007 0.04 0.94 7 0.0004 0.31 0.51 11 0.00057 95 0.0002 0.43 0.42 7 0.0003 0.22 0.67 7 0.00028 96 0.0006 -0.22 0.51 10 0.0006 0.03 0.72 9 0.00061 97 0.0005 0.15 0.57 6 0.0002 0.73 0.49 7 0.00035 98 0.0004 0.25 0.64 10 0.0002 0.48 0.69 9 0.00030 99 0.0004 0.70 0.75 8 0.0004 0.40 0.72 9 0.00042 100 0.0004 0.28 0.55 8 0.0005 0.19 0.52 8 0.00045 101 0.0007 -0.11 0.67 9 0.0004 0.85 0.45 11 0.00056 102 0.0003 0.44 0.66 10 0.0004 0.23 0.75 10 0.00036 103 0.0003 0.96 0.65 11 0.0003 0.74 0.41 9 0.00027 104 0.0004 0.28 0.68 11 0.0004 0.36 0.77 10 0.00042 105 0.0003 0.14 0.78 7 0.0002 0.76 0.67 5 0.00025 106 0.0003 0.18 0.83 11 0.0002 0.63 0.69 11 0.00029 Appendices 222 107 0.0003 0.36 0.85 5 0.0002 0.45 0.69 7 0.00027 108 0.0003 0.72 0.69 9 0.0003 0.26 0.44 9 0.00031 109 0.0003 0.72 0.63 8 0.0003 0.29 0.68 11 0.00028 110 0.0002 0.87 0.47 8 0.0004 0.26 0.65 8 0.00031 111 0.0003 0.72 0.55 7 0.0005 0.18 0.77 10 0.00040 112 0.0002 0.89 0.66 6 0.0005 0.10 0.80 11 0.00034 113 0.0004 0.93 0.97 4 0.0005 0.47 0.67 8 0.00042 114 0.0005 0.12 0.78 9 0.0002 1.16 0.53 9 0.00034 115 0.0003 0.18 0.66 10 0.0002 0.65 0.68 10 0.00023 116 0.0008 -0.58 0.92 5 0.0010 -0.55 0.99 5 0.00089 117 0.0003 0.12 0.57 8 0.0002 0.67 0.51 7 0.00026 118 0.0002 0.41 0.56 6 0.0003 0.61 0.76 8 0.00027 119 0.0001 0.43 0.65 6 0.0004 0.53 0.66 9 0.00025 120 0.0003 0.32 0.65 6 0.0003 0.74 0.35 8 0.00029 121 0.0007 1.20 0.54 7 0.0006 1.16 0.46 11 0.00063 122 0.0004 0.17 0.79 9 0.0007 -0.02 0.50 9 0.00055 123 0.0001 0.90 0.52 6 0.0005 0.18 0.57 11 0.00030 124 0.0007 0.51 0.79 6 0.0006 0.14 0.60 10 0.00067 125 0.0006 1.50 0.75 6 0.0005 0.86 0.68 10 0.00057 126 0.0005 0.55 0.71 10 0.0005 0.42 0.77 11 0.00051 127 0.0002 1.33 0.63 5 0.0004 1.24 0.48 10 0.00028 128 0.0003 0.74 0.50 6 0.0005 0.37 0.82 7 0.00037 Appendices 223 129 0.0007 0.25 0.94 7 0.0010 -0.23 0.86 9 0.00088 130 0.0004 0.62 0.61 11 0.0006 -0.11 0.77 10 0.00048 131 0.0006 0.25 0.84 9 0.0008 -0.14 0.52 10 0.00072 132 0.0006 1.08 0.61 9 0.0010 -0.46 0.88 6 0.00081 133 0.0002 0.98 0.64 10 0.0006 0.22 0.81 6 0.00036 134 0.0003 0.10 0.52 7 0.0004 0.44 0.60 5 0.00038 135 0.0006 -0.09 0.69 10 0.0002 1.00 0.65 9 0.00040 136 0.0005 0.06 0.59 6 0.0004 0.05 0.55 11 0.00048 137 0.0003 0.57 0.71 8 0.0005 -0.03 0.66 10 0.00041 138 0.0004 0.53 0.76 10 0.0003 0.48 0.60 11 0.00032 139 0.0005 0.15 0.57 11 0.0005 0.38 0.70 10 0.00049 140 0.0016 -1.26 0.72 11 0.0009 -0.03 0.48 11 0.00121 141 0.0003 0.82 0.51 10 0.0007 0.21 0.75 10 0.00046 142 0.0005 0.48 0.87 7 0.0004 0.26 0.49 11 0.00046 143 0.0010 0.34 0.84 11 0.0005 0.30 0.50 11 0.00075 144 0.0004 0.85 0.61 6 0.0003 0.45 0.33 11 0.00035 145 0.0003 0.55 0.74 7 0.0003 0.72 0.79 11 0.00027 146 0.0006 0.00 0.65 11 0.0004 0.46 0.55 8 0.00051 147 0.0006 0.22 0.73 7 0.0009 0.11 0.66 10 0.00072 148 0.0008 -0.36 0.58 10 0.0008 -0.12 0.40 11 0.00080 149 0.0011 -0.27 0.86 8 0.0004 0.27 0.67 5 0.00076 150 0.0007 0.47 0.55 4 0.0004 0.39 0.91 5 0.00055 Appendices 224 151 0.0006 0.15 0.84 9 0.0007 -0.23 0.94 6 0.00067 152 0.0006 0.28 0.67 8 0.0006 0.09 0.84 9 0.00056 153 0.0005 0.28 0.64 8 0.0004 0.50 0.48 9 0.00042 154 0.0017 0.38 0.78 5 NaN NaN NaN NaN 0.00167 155 0.0011 -0.15 0.83 4 0.0010 0.25 0.69 5 0.00106 156 0.0003 1.02 0.95 5 0.0005 1.02 0.62 8 0.00039 157 0.0004 0.31 0.65 4 0.0002 0.34 0.68 6 0.00035 158 0.0006 0.06 0.77 4 0.0007 0.21 0.81 5 0.00065 159 0.0004 1.41 0.54 4 0.0003 2.00 0.17 5 0.00034 160 0.0005 0.87 0.94 4 0.0004 1.00 0.53 7 0.00045 161 0.0001 0.48 0.84 9 0.0002 0.78 0.74 6 0.00016 162 0.0003 0.47 0.83 8 0.0004 1.02 0.52 5 0.00035 163 0.0001 0.89 0.77 6 0.0002 1.21 0.35 6 0.00017 164 0.0002 0.38 0.64 7 0.0004 0.02 0.86 5 0.00027 165 0.0002 0.72 0.70 7 0.0001 0.56 0.93 5 0.00014 166 0.0001 0.26 0.57 6 0.0001 0.22 0.34 8 0.00014 167 0.0002 0.27 0.84 5 0.0002 0.19 0.90 5 0.00023 Average 0.67 Average 0.67 0.00047