Model predictive control for maintenance operations planning of railway infrastructures,

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1 Delft University of Technology Delft Center for Systems and Control Technical report 5- Model predictive control for maintenance operations planning of railway infrastrctres Z. S, A. Núñez, A. Jamshidi, S. Baldi, Z. Li, R. Dollevoet, and B. De Schtter If yo want to cite this report, please se the following reference instead: Z. S, A. Núñez, A. Jamshidi, S. Baldi, Z. Li, R. Dollevoet, and B. De Schtter, Model predictive control for maintenance operations planning of railway infrastrctres, in Comptational Logistics (Proceedings of the 6th International Conference on Comptational Logistics (ICCL 5), Delft, The Netherlands, Sept. 5) (F. Corman, S. Voß, and R.R. Negenborn, eds.), Lectre Notes in Compter Science, Cham, Switzerland: Springer, ISBN , pp , 5. Delft Center for Systems and Control Delft University of Technology Mekelweg, 68 CD Delft The Netherlands phone: (secretary) fax: URL: This report can also be downloaded viahttp://pb.deschtter.info/abs/5_.html

2 Model Predictive Control for Maintenance Operations Planning of Railway Infrastrctres Zho S, Alfredo Núñez, Ali Jamshidi, Simone Baldi, Zili Li, Rolf Dollevoet, and Bart De Schtter Delft Center for Systems and Control & Section of Railway Engineering, Delft University of Technology, 68 CD, Delft, the Netherlands tdelft.nl Abstract. This paper develops a new decision making method for optimal planning of railway maintenance operations sing hybrid Model Predictive Control (MPC). A linear dynamic model is sed to describe the evoltion of the health condition of a segment of the railway track. The hybrid characteristics arise from the three possible control actions: performing no maintenance, performing corrective maintenance, or doing a replacement. A detailed procedre for transforming the linear system with switched inpt, and recasting the transformed problem into a standard mixed integer qadratic programming problem is presented. The merits of the proposed MPC approach for designing railway track maintenance plans are demonstrated sing a case stdy with nmerical simlations. The reslts highlight the potential of MPC to improve condition-based maintenance procedres for railway infrastrctre. Keywords: Health Condition Monitoring and Maintenance, Model Predictive Control(MPC), Track Maintenance, Railway Engineering Introdction A railway track infrastrctre system is composed of a set of different assets. All of those assets are distribted and interconnected over the railway network, continosly working together to keep the railway service reliable, safe, and fast. Each asset has a different need for maintenance, at different times and according to its degradation process, which is inflenced by geographic position, tonnage of the track, health condition of the rolling stock, among many other factors. Ths, to sstainably manage railway assets, a step forward from the crrent policy of find and repair towards a more integrated methodology containing condition-based monitoring and predictive maintenance is reqired to improve the entire whole system performance. In the Netherlands, over forty percent of the maintenance costs is related to track maintenance []. De to this fact, a condition-based maintenance decision spport system can facilitate the infrastrctre manager to decide where and which type of maintenance shold be performed. Moreover, if a prediction capability is incorporated in the decision making, we can expect maintenance actions that will anticipate problems and will take corrective measres before a failre become costly or nsafe for the sers. This stdy proposes a model based predictive maintenance strategy sing conditionbased monitoring. We in particlar show how this strategy can be applied to maintenance planning for ballast degradation and for treating sqats. Theroleofballastistoprovidespporttothetrackswithhardstonesaimingtodistribteloads over the sleepers while the train is passing. It also allows rain and snow to drain, thermal expansion, and weight variance; and it inhibits the growth of weed and vegetation. In the deterioration process of ballast, some stones may be displaced de to the vibrations and some others will get a white ronded shape losing their properties (see Figre a). When the ballast is in a bad condition, it will be reflected in the rail geometry. In order to keep the performance level at a satisfactory condition, tamping (packing the track ballast nder the railway track) and ballast replacement are the principal maintenance actions to consider. In the literatre, different stdies have been proposed on how to predict changes in the track geometry condition. In some stdies the ballast deterioration is

3 Zho S et al modeled deterministically based on the average growth of track irreglarities [,,4]. Other stdies have linked stochastic degradation process with the effects of possible maintenance and renewal options [5,6]. Sqats are a type of Rolling Contact Fatiges (RCFs) that initiate on the rail srface and evolve into a network of cracks beneath the srface of the track that when not treated on time can evolve into rail breakage. Treating RCFs to avoid a redced life cycle of the track is very expensive. Early stage sqats can be efficiently treated with grinding, while the only soltion for late stage sqats is track replacement(see Figre b). There are different methods to detect sqats, like non-atomatic inspection sing hman inspectors, photo/video records, and non-destrctive testing sch as ltrasonic and eddy crrent test [7,8]. For atomatic detection of sqats in an early stage, axle box acceleration (ABA) systems can be efficiently employed [9]. Predictive and robst models for sqats evoltion have been proposed in []. In [], it is sggested to consider clsters of sqats to facilitate grinding maintenance operations; however, the maintenance actions were obtained nder static scenarios. In this paper, the main contribtion is to propose a sitable model that incorporates the dynamics in the decision process of railway track maintenance operations, together with a rolling horizon methodology that can deal with discrete integer control actions. The major benefit of applying Model Predictive Control (MPC) to maintenance operations planning is that the reslting strategy is flexible. By pdating the degradation model sing health condition monitoring methodologies reglarly, the maintenance plans can be adapted dynamically. Especially when severe problems are predicted within the prediction horizon, MPC will sggest more freqent or more effective maintenance operations, reslting in a more efficient plan. This is a big step from crrent practice in railway maintenance, where cyclic preventive maintenance prevails, which, as a myopic strategy, is nable to predict the evoltion of the degradation process, and treats severe problems only when they occr. Another merit of the proposed predictive methodology is that the objective fnction explicitly captres the trade-off between maintenance costs and the health condition of the track. This is crcial for railway infrastrctre managers, who reqire transparent tools that facilitate the decision making process. Moreover, other factors concerning the management of railway infrastrctres, like closre time de to maintenance, can also be conveniently added to the MPC optimization problem. In addition, limits on the admissible degradation level can be inclded effortlessly as constraints in MPC, which is especially sefl for the maintenance of safety-critical components. This paper is organized as follows: first a brief introdction to MPC is presented in Section ; then Key Performance Indicators (KPIs) and maintenance options for railway infrastrctres are explained in Section ; the proposed MPC approach is explained in detail in Section 4 and illstrated by the case stdy in Section 5; finally a short smmary and remarks on ftre work is provided in Section 6. Model Predictive Control (MPC) Model Predictive Control (MPC) is an advanced design methodology for control systems, which has gained wide poplarity in the process indstry since the last decade. MPC was pioneered simltaneosly by Richalet et al. [,] and Ctler and Ramaker [4] in the late 97s. The main reasons for the sccess of MPC in the process indstry are: Easy handling of mlti-inpt-mlti-otpt (MIMO) processes, non-minimm phases processes, processes with large time delay, and nstable processes; Easy tning of parameters (in principle only three parameters need to be tned); Natral embedding of constraints in a systematic way; Easy handling of strctral changes by reglarly pdating the process model. Five elements are essential for MPC:. A process model. A cost criterion

4 MPC for Railway Maintenance Operations (a) Loose ballast indicated by the presence of white dst (b) Severe sqats (type C) Fig. : Two different defects in railway infrastrctres. Constraints 4. Optimization algorithms 5. Receding horizon principle. For maintenance operations planning in railway network, the process model can be the degradation model or the performance indicator of an asset, the cost criterion can be a trade-off between track condition and maintenance cost, and the constraints can be an pper bond on the maximm degradation level and bdget. Depending on the process model, which might contain continos and discrete variables, the cost criterion, and the imposed constraints, the comptation of a seqence of ftre control actions at each sample step will reslt in different optimization problems, which mst be solved by some mathematical programming algorithm. For railway maintenance, since the degradation of the track condition is a continos process and maintenance operations are intrinsically discrete, the planning of maintenance operations shold in general reslt in a mixed integer programming problem. This optimization problem mst be solved at each sample step, providing a seqence of optimal discrete control actions. The length of the seqence is called prediction horizon when sing a receding horizon approach. Instead of sing the whole seqence of predicted control actions, only the first entry is applied, and a new seqence is compted at the next sample step with pdated information, e.g. new measrements of the track condition. Despite its sccess in the process indstry, the applications of MPC in railway maintenance operations are scarce, althogh the application of mathematical models and optimization is not ncommon in maintenance [5,6]. The process model associated with most operations planning problems sally contains both continos and discrete dynamics. In [7] an MPC scheme is applied to plan risk mitigation actions together with other control variables, sing a mixed integer qadratic formlation. For complex decision making problems, a hierarchical or distribted approach is often applied to render the problem tractable. See [8] for an application of hierarchical distribted MPC to risk management of a network of irrigation canals, and [9] for applications of hybrid MPC to interventions in behavioral health and inventory management in spply chains. As a model-based decision making approach, MPC can be viewed as an extension of conditionbased maintenance, as it does not only take into consideration the crrent track condition, bt also predicts the track condition, sing pdated track measrements, as well as knowledge on the process, e.g. degradation models. Railway Maintenance A brief introdction on railway maintenance is presented, explaining how the track condition is measred in practice, as well as typical operations on track maintenance.

5 4 Zho S et al. Key Performance Indicators (KPIs) for maintenance planning To ensre the proper fnctioning of railway tracks, both temporal and spatial characteristics need to be considered in the maintenance decisions. For this prpose, Key Performance Indicators (KPIs) are developed to captre the dynamics of track deterioration and the evoltion of defects. These KPIs sally consider a broad set of measrements from different sorces and define the health condition of the track as a single nmber. When normalized, wold mean a healthy track, while a track with very bad condition. This health nmber changes over time, and when reaching some threshold corrective maintenance is performed. In the case of ballast, the track bed positioning and alignment changes in terms of speed and nmber of passing trains have to be considered in the design of KPI. The following measrements are sally considered [,]: () cyclic top, which is a measre of resonant freqencies, () rail spacing compared to the standard gage, () a measre of cant variation over m and 5 m, (4) lateral geometry (alignment) - rail alignment, averaged over the left and right rail, expressed as short and long wave standard deviations. In [] some KPIs are proposed for maintenance operations related to sqats sing axle box acceleration measrements. The nmber of sqats of type A (light sqats in early stage), nmber of sqats of type B or C (severe sqats), nmber of potential risk points, and density of sqats are the KPIs proposed. Field observations also revealed that different sqats have different rates of growth, ths three different evoltion scenarios were proposed: slow growth, average growth, and fast growth. De to the big nmber of KPIs, temporal dependence and scenarios, one global KPI sing a fzzy inference system was proposed to facilitate maintenance decisions. This global KPI for sqats represents the health condition of a clster of defects and it is represented by a score between zero (representing a healthy state of the track) and two (indicating an nhealthy condition of the track).. Maintenance options This paper evalates three possible maintenance options: () to do nothing, () corrective maintenance: tamping for ballast or grinding for sqats, () replacement: fll ballast replacement or track replacement. Next, a short description of the maintenance actions is given. In the case of tamping, the track geometry is adjsted when the track alignment is otside the accepted tolerances. Tamping machines are able to pack the ballast nder the sleepers in order to correct the alignment of the rails sing measrements of the track geometry, estimating the needed adjstments, lifting/inserting the track and at the same time vibrating tamping arms []. Tamping can improve the track condition. However, in the worst case, vibrating arms into the ballast cold lead some break-p of the stones which can accelerate the ballast degradation. When the ballast has reached the end of its sefl life, then ballast renewal mst be carried ot. In the case of grinding, the Dtch railways se a cyclic grinding strategy that is not based on the condition of the track. This mean that sections that are healthy are grond, and also sections that need replacement are grond in spite of its inefficiency. Sqats in early stage can be removed by grinding, becase they have not developed the network of crack beneath yet. Therefore, early detection of RCF rail defects is cost-effective when a grinding campaign is schedled appropriately in time. Frthermore, grinding is not efficient for cracks deeper than 5-7 mm. Grinding severe sqats can delay rail replacement bt may accelerate the sqat evoltion, becase the cracks are not yet removed. When the conditions of either ballast or track has worsened, and tamping or grinding is ineffective, then replacement will be the only option. With predictive models that incorporate the degradation process of the ballast or the track, it is possible to estimate the life cycle of the track and to inform the infrastrctre manager abot the tentative time for replacing. This is crcial information becase replacement campaigns are extremely costly and the track needs to be navailable for a long period. In the case of ballast replacement, keeping a similar total volme of stones before and after the replacement is essential. Long segments with deteriorated ballast will reqire a higher packing level of the damaged ballast []. In the case of rails, they are typically

6 MPC for Railway Maintenance Operations 5 replaced according to predefined estimated life cycle, tonnage, wear limit, fatige, and weather conditions. Determining the optimal rail replacement interval is a critical isse for rail indstry, de to its high cost and conseqences over the entire performance of the infrastrctre. Figre shows a generic KPI and the typical effect of corrective maintenance or replacement. While a corrective maintenance (grinding or tamping) will improve the performance, a general drop of the performance is sally observed. In the case of replacement, while more expensive than corrective maintenance, it sally leads into a healthy stats for a longer period of time. In the next section, a hybrid MPC methodology is proposed to captre the principal dynamics of railway maintenance operation KPI clster i Grinding or tamping α Ballast or track replacement Time (Months) Fig. : Generic KPI evoltion over time. 4 MPC for Maintenance Operation Planning A maintenance model describing the degradation dynamic of the track condition is presented, with three maintenance operations as inpts. This linear model with discrete inpts is then transformed into a standard Mixed Logic Dynamic (MLD) system, and an MPC scheme is designed for this MLD system. 4. Maintenance Model Description We consider the following discrete-time state space model for the degradation dynamic (nhealthy condition) of a certain asset (e.g. track with sqats, or ballast) in a segment of track: x (k +) = a x (k)+f (x(k), (k)) x (k +) = x (k)+f (x(k), (k)) () with constraint m x i (k) M i {, } () The state vector is x(k) = [ x (k) x (k) ] T. In particlar, x represents the level of degradation of a part of the track regarding a certain type of falt, e.g. the average length of the sqats in a clster. The parameter a is the degradation rate of the track, which is assmed to be larger than so that the track condition deteriorates exponentially as observed in late stage of degradation. The state x records the level of degradation pon the last corrective maintenance operation. This is necessary becase corrective operation will be nlikely to bring the track back to a condition healthier than the previos corrective operation. The states are bonded in the interval [m, M] by the constraint (), with m and M representing the lowest and highest admissible degradation level, respectively.

7 6 Zho S et al The inpt (k) {,, } is a discrete inpt representing the three major maintenance operations: do nothing, corrective maintenance, and replacement, respectively. The interpretation of the inpt is given in Table. Table : Interpretation of maintenance operations and the corresponding vale of inpt Inpt vale Maintenance operation Effect on the track condition do nothing no effect corrective maintenance bring the track condition to a healthier level, bt sally not as healthy as the condition pon the last corrective maintenance replacement bring the track condition to a state with no degradation The effects of the maintenance operations are captred by the discontinos fnctions f and f, which take the following representation: if (k) = f (x(k), (k)) = a x (k)+x (k) if (k) = a x (k)+m if (k) = () and if (k) = f (x(k), (k)) = α if (k) = x (k)+m if (k) = (4) with α, which sally takes a small positive vale, indicating the offset of the effect of one corrective maintenance from the last corrective maintenance, i.e. a corrective maintenance can only bring the condition to a level which is α worse than that jst after the previos corrective maintenance operation. 4. Transformation into an Mixed Logic Dynamic (MLD) Systems The system () can be viewed as a hybrid system with linear plant model (degradation process) and switching control. The difficlty of designing a controller for sch systems lies in the different conditions triggered by different vales of the control inpt (eqations () (4)). By associating the three conditions triggered by the three control actions with two binary variables δ (k) and δ (k), the system () can be transformed into a Mixed Logic Dynamic (MLD) system. The translation from the control inpt (k) to the binary variables δ (k) and δ (k) is shown in Table. Table : Translation from control inpt to binary variables (k) δ (k) δ (k) We eliminate the forth option (δ = δ = ) by adding the constraint: δ (k)+δ (k) (5)

8 MPC for Railway Maintenance Operations 7 After translating the control inpt (k) into binary variables δ (k) and δ (k), system () can be rewritten as x (k +) = a x (k)+δ (k)( a x (k)+x (k))+δ (k)( a x (k)+m) = a x (k) a δ (k)x (k) a δ (k)x (k)+δ (k)x (k)+δ (k)m x (k +) = x (k)+αδ (k)+δ (k)( x (k)+m) = x (k) δ (k)x (k)+mδ (k)+αδ (k) (6) The system (6) is non-linear in the state x(k) and in the two binary variables δ (k) and δ (k). The non-linear system can be transformed into the following linear system: [ ] x (k +) = x (k +) [ a ][ ] x (k) + x (k) by introdcing for axiliary variables [ m m α ][ ] δ (k) δ (k) [ a a + ] z (k) z (k) z (k) (7) z 4 (k) z (k) = δ (k)x (k) z (k) = δ (k)x (k) z (k) = δ (k)x (k) z 4 (k) = δ (k)x (k) The eqation for each axiliary variable z p (k) = δ i (k)x j (k) p {,,, 4}, i, j {, } is eqivalent to the following for ineqality constraints [4,5]: z p (k) Mδ i (k) z p (k) mδ i (k) z p (k) x j (k) m( δ i (k)) z p (k) x j (k) M( δ i (k)) (8) which reslts in sixteen ineqality constraints in total. Finally, the mixed logical dynamic (MLD) system [4] described by the linear dynamics (7) and linear constraints (), (5), (8) can be formlated in the following compact form : x(k +) = Ax(k)+B δ(k)+b z(k) (9) E x(k)+e δ(k)+e z(k) g () with δ(k) = [ δ (k) δ (k) ] T and z(k) = [ z (k) z (k) z (k) z 4 (k) ] T 4. The MLD-MPC Problem and its soltion via MIQP Consider the MLD system (9) (). Denote by ˆx(k + j k) the estimated state at sample step k with information available at sample step k+j. Likewise, define ẑ(k+j k) as the estimate of the axiliary variable at sample step k +j. Let N p be the prediction horizon, and define x(k) = [ˆx T (k + k)... ˆx T (k +N p k) ] T Since the introdced binary variable δ is a fll replacement of the original discrete control inpt, which no longer appears in the reslting MLD system, we treat δ as control inpt for the MLD-MPC problem Here we set the control horizon eqal to the prediction horizon

9 8 Zho S et al as the estimates of the ftre state at sample step k. The estimates of inpt and axiliary variables ( δ(k)and z(k))canbedefinedinasimilarway.gropingtheinptandaxiliaryvariablestogether we frther define Ṽ(k) = [ δt (k) z T (k) ] T After sccessive sbstittion of (9), the state prediction eqation can then be written in the following compact form x(k) = M Ṽ(k)+M x(k) () The goal of maintenance operations planning is to minimize degradation with the lowest possible maintenance cost, which can be formlated by the following objective fnction J(k) = J Perf (k)+λj Cost (k) () with J Perf (k) and J Cost (k) representing the performance and the cost of maintenance operations, respectively, while the weighting parameter λ > captres the trade-off of the two conflicting objectives. More specifically, if a weighted -norm is considered for performance, i.e. J Perf (k) = x(k) P, with P positive definite weighting matrix, a mixed integer qadratic programming (MIQP) problem will be obtained. Alternatively, a -norm or an norm can also be applied, with J Perf (k) = P x(k) or J Perf (k) = P x(k), and P a matrix with non-negative entries. If a -norm or an norm is applied, the optimization problem can be recast as a mixed integer linear programming (MILP) problem, which is generally easier to solve than an MIQP problem. The cost of maintenance operations J Cost (k) has the following linear representation: J Cost (k) = R δ(k) = QṼ(k) with R a matrix with non-negative entries assigning weights to different maintenance operations, i.e. the cost of corrective maintenance and replacement. Since no weights are assigned to the axiliary variables, we have Q = [ R ]. For illstration prposes, we consider now a weighted -norm for J Perf. Given the weighting matrices P and Q, the objective fnction can be rewritten as: T J(k) = x(k) P x(k)+λqṽ(k) () = (M Ṽ(k)+M x(k)) T P(M Ṽ(k)+M x(k))+λqṽ(k) (4) = Ṽ T (k)s Ṽ(k)+ ( S +x T (k)s )Ṽ(k)+S4 (5) Note that S 4 is a constant and can be removed from the objective fnction withot changing the soltion of the optimization problem. Finally we can formlate the MLD-MPC problem as a standard MIQP problem with decision variable Ṽ(k): minṽ T (k)s Ṽ(k)+(S +x T (k)s )Ṽ(k) (6) Ṽ(k) s.t. F Ṽ(k) F +F x(k) (7) MIQP problems are identified as NP-hard [6], indicating that in practice the soltion time might grow exponentially with the problem size. Among the major algorithms for solving MIQP and MILP problems, branch-and-bond methods [7] are generally regarded as the most efficient ones. 5 Case Stdy Now we consider a simple case stdy to illstrate the proposed approach. An MPC controller for the MLD system (9) () with qadratic objective fnction (6) is implemented in Matlab,

10 MPC for Railway Maintenance Operations 9 and the MIQP optimization problem at each sample step is solved sing the Grobi Optimizer 5.6., which can solve MIQP and MILP problems. The parameters for the degradation dynamics, as well as the performance criteria (in particlar, the parameters for the objective fnction) are given in the appendix. We present the simlation reslts with three different prediction horizons (N p = 5,, 5), and three different weights (λ =.,, ). A simple cyclic preventive maintenance approach with the same objective fnction is also implemented and optimized in Matlab for a comparison. The two decision variables for the cyclic approach are the period for corrective maintenance (T c ) and the period for replacement (T r ). The latter is sally fixed as a mltiple of the former, i.e. T r = nt c, ths we determine the mltiple n instead of T r. The planning horizon is the whole simlation time, and an iteration of all possible combinations of T c and n is applied to generate the best possible cyclic maintenance plan for a given cost criterion. x x (a) N p = 5, λ =. x x (b) N p =, λ =. x x (c) N p = 5, λ =. Fig.: Predicted state x and control inpt for different prediction horizons and a small λ x x (a) N p = 5, λ = x x (b) N p =, λ = x x (c) N p = 5, λ = Fig.4: Predicted state x and control inpt for different prediction horizons and a medim λ x x (a) N p = 5, λ = x x (b) N p =, λ = x x (c) N p = 5, λ = Fig.5: Predicted state x and control inpt for different prediction horizons and a big λ The simlation reslts from the MPC approach are given in Figre 5 and the reslts from the cyclic approach are given in Figre 6. The flexibility of MPC is clearly demonstrated by the reslting maintenance plans, which all sggest more freqent corrective maintenance as the track condition deteriorates, despite the differences in weighting parameters and the prediction horizon.

11 Zho S et al x x (a) λ =. x x (b) λ = x x (c) λ = Fig. 6: Maintenance plans and the corresponding effect on the track degradation level with different λ Figre 5 also indicate that a longer prediction horizon reslts in a more catios maintenance plan, i.e. earlier replacement and lower overall degradation level. The weight λ in the objective fnction represents the trade-off between the track condition and the maintenance cost. This is also shown in Figre 5, where it can be noticed that for a larger λ, the first replacement is sggested at a later time step, for a given N p. On the contrary, the cyclic approach sggests the same optimal maintenance plan for the three different λ (corrective maintenance every 65 days and replacement every days). The vales of the closed-loop objective fnction for both approaches are given in Table. It can be seen that with a long enogh prediction horizon, the maintenance plan generated by MPC always ot performs the plan generated by the cyclic approach. Table : Evalation of closed-loop objective fnction for MPC (J MPC ) and cyclic approach (J CYC ). For a given λ, the lowest objective fnction vale is marked in bold. J MPC λ J N CYC p = 5 N p = N p = Conclsions In this contribtion a hybrid Model Predictive Control (MPC) approach with discrete inpt has been developed to spport decision making in railway maintenance. We have provided a detailed procedre to illstrate how to transform the model-based optimization problem into a standard mixed integer qadratic programming problem. Nmerical simlations have been performed for a case stdy with different parameter settings, and the reslts indeed demonstrate the potential of MPC as an optimization-based approach to aid decision making in maintenance of railway infrastrctres. Ftre work incldes developing a more extensive process model with parameters obtained from track measrements, and extending the crrent approach from maintenance operations planning for a single segment of track to joint decision making for mltiple segments. Acknowledgement Research sponsored by the NWO/ProRail project Mlti-party risk management and key performance indicator design at the whole system level (PYRAMIDS), project 48--, which is partly financed by the Netherlands Organisation for Scientific Research (NWO).

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14 MPC for Railway Maintenance Operations Appendix The vales and interpretation of the parameters for the degradation dynamics () are given in Table 4. The initial condition is x() = [mm] T, and the sampling time is 5 days. Table 4: Parmaters for degradation dynamics Parameter Vale Interpretation a.9 degradation rate α. offset for corrective maintenance m. minimm degradation level M maximm allowed degradation level The matrix P is a sqare matrix consisting of diagonal replications of the diagonal matrices P Sb = diag(, ): P = diag P Sb,...,P }{{ Sb } N p times The matrix Q is a block-row matrix containing N p horizontal replications of the diagonal matrix Q Sb = diag(r,,,,, ): Q = Q Sb,...,Q }{{ Sb } N p times where the parameter r is the ratio of the cost of one replacement over the cost of one corrective maintenance. For the case stdy we assme replacement is times more costly than corrective maintenance, ths r =.