London calling (probably) Parameters and stochastic behaviour of braking force generation and transmission Prof. Dr. Raphael Pfaff Aachen University of Applied Sciences pfaff@fh-aachen.de www.raphaelpfaff.net @RailProfAC December 14, 2016 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 1 / 25
User: Infrastructure Manager Executive Henry Job: ETCS Expert Employer: Infrastructure manager Challenges: Ensure safety, maintain or increase capacity Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 2 / 25
User: Infrastructure Manager Executive I need to ensure that signals are practically never overrun while at the same time, the load on my network increases every year. Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 2 / 25
ETCS provides the answer Wikimedia: Sansculotte/Lonaowna Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 3 / 25
User: Infrastructure Manager Executive With the moving block system, I can improve infrastructure utilisation - I only need to find the braking curves! Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 4 / 25
What is a braking curve? CCS systems rely on braking curves to describe the train s braking capability. To supervise train velocity, CCS systems predict the future braking capability of the train 30 25 20 v/m/s 15 10 5 0 0 100 200 300 400 500 600 700 s/m Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 5 / 25
What is a braking curve? CCS systems rely on braking curves to describe the train s braking capability. To supervise train velocity, CCS systems predict the future braking capability of the train However, there is not the braking capability 30 25 20 v/m/s 15 10 5 0 0 100 200 300 400 500 600 700 s/m Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 5 / 25
What is a braking curve? CCS systems rely on braking curves to describe the train s braking capability. To supervise train velocity, CCS systems predict the future braking capability of the train However, there is not the braking capability Braking curves exhibit a randomised behaviour v/m/s 30 25 20 15 10 5 0 0 100 200 300 400 500 600 700 s/m Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 5 / 25
How to obtain a braking curve? To obtain a braking curve, the stochastic behaviour of the system needs to be analysed, typically by help of a Monte Carlo Simulation. x 1 x 2 y = f (x) y x 3 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 6 / 25
White Box Modelling of the braking system Which parameters can be identified and which effect do they have on the braking distance? Brake pipe: propagation velocity, flow resistances, train length Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 7 / 25
White Box Modelling of the braking system Which parameters can be identified and which effect do they have on the braking distance? Brake pipe: propagation velocity, flow resistances, train length Distributor valve: Filling time, brake cylinder pressure Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 7 / 25
White Box Modelling of the braking system Which parameters can be identified and which effect do they have on the braking distance? Brake pipe: propagation velocity, flow resistances, train length Distributor valve: Filling time, brake cylinder pressure Braking force generation: efficiency, brake radius (for disc brakes), pad/block friction coefficient Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 7 / 25
White Box Modelling of the braking system Which parameters can be identified and which effect do they have on the braking distance? Brake pipe: propagation velocity, flow resistances, train length Distributor valve: Filling time, brake cylinder pressure Braking force generation: efficiency, brake radius (for disc brakes), pad/block friction coefficient Wheel/rail contact: rail surface, contaminants, slip,... Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 7 / 25
White Box Modelling of the braking system Which parameters can be identified and which effect do they have on the braking distance? Brake pipe: propagation velocity, flow resistances, train length Distributor valve: Filling time, brake cylinder pressure Braking force generation: efficiency, brake radius (for disc brakes), pad/block friction coefficient Wheel/rail contact: rail surface, contaminants, slip,... Also discrete failure events need to be considered Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 7 / 25
Random variables Uniform distribution U I All events on an interval I have the same probability 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 4 3 2 1 0 1 2 3 4 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 8 / 25
Random variables Uniform distribution U I All events on an interval I have the same probability Triangular distribution T I As combined probability of two Uniform distributions 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 4 3 2 1 0 1 2 3 4 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 8 / 25
Random variables Uniform distribution U I All events on an interval I have the same probability Triangular distribution T I As combined probability of two Uniform distributions Normal (gaussian) distribution N (µ,σ) Approximation of a high number of U I -distributed 0.00 4 3 2 1 0 1 2 3 4 variables 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 8 / 25
Random variables Uniform distribution U I All events on an interval I have the same probability Triangular distribution T I As combined probability of two Uniform distributions Normal (gaussian) distribution N (µ,σ) Approximation of a high number of U I -distributed 0.00 4 3 2 1 0 1 2 3 4 variables 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 8 / 25
Random variables Uniform distribution U I All events on an interval I have the same probability Triangular distribution T I As combined probability of two Uniform distributions Normal (gaussian) distribution N (µ,σ) Approximation of a high number of U I -distributed variables Markov Chains Discrete state changes, e.g. defects p 10 S 1 S 2 p 33 S 3 S 0 p 00 p20 p 22 p 01 p 12 p 11 p 23 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 8 / 25
Brake pipe parameters Brake pipe parameters determine the distribution of the brake command along the train. Propagation velocity: Required: c 250 m s May be considered lower limit Flow resistances: Flow resistance in the individual wagons determine filtering behaviour of BP Train length: Non-random input parameter Wagon position: Distribution of braked mass in train and effective filling time influence overall braking distance Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 9 / 25
Distributor valve parameters Distributor valve parameters determine the effectivity of the brake command. Filling time t f : Brake modes P/R: (4 ± 1) s Brake modes P/R: (24 ± 6) s Uniform distribution (conservative) Brake cylinder pressure p C : Required: p C = ( 3.8 +0.2 0.1) bar Uniform distribution (conservative) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 3.6 3.7 3.8 3.9 4.0 4.1 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 10 / 25
Braking force generation parameters Parameters of the braking force generation subsystem determine the propagation of braking effort between p C and wheel/disc. Efficiency Typical dynamic efficiency: η [0.75, 95] Depending on maintenance state Assumed uniform distribution Brake radius Systematic variation with pad wear, not relevant for block brakes Pad/block friction coefficient µ B Mean friction coefficient depending on v 0 Stochastic variation of instantaneous coefficient Normal distribution appropriate frequency 600 500 400 300 200 100 0 0.110 0.115 0.120 0.125 0.130 0.135 0.140 0.145 0.150 mu/1 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 11 / 25
Wheel-rail surface parameters Rail surface: According to Hertzian theory Non-Hertzian contacts due to hunting Contaminants: Empirical estimation due to network Mostly dry braking curves simulated Slip: Curving motion, hunting impose 3D-slip on contact patch Adhesion budget gets used y/m 10 3 Adhesion area 5 0 5 1 0.5 0 0.5 1 x/m 10 2 F t/b = 10 kn F t/b = 15 kn F t/b = 20 kn F t/b = 25 kn F t/b = 30 kn Elliptical contact Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 12 / 25
User: Infrastructure Manager Executive Looks like the simulation model is quite complex? Can we do this online? Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 13 / 25
Approaches to obtain braking distance distributions Error-propagation: Conservative: assumes normal distribution for all parameters Complex: requires explicit function formulation and partial differentiation (Standard) Monte-Carlo-Simulation: Efficient (in terms of confidence): returns shortest (also asymmetric) confidence interval Inefficient (in terms of computational effort): For rare event ε 1, N 100 trials required ε Typical according to CSM: ε [ 10 7... 10 9] N 10 11 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 14 / 25
Approaches to obtain braking distance distributions Error-propagation: Conservative: assumes normal distribution for all parameters Complex: requires explicit function formulation and partial differentiation (Standard) Monte-Carlo-Simulation: Efficient (in terms of confidence): returns shortest (also asymmetric) confidence interval Inefficient (in terms of computational effort): For rare event ε 1, N 100 trials required ε Typical according to CSM: ε [ 10 7... 10 9] N 10 11 ERA proposes to precalculate braking curves for limited number of train formations Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 14 / 25
Approaches to obtain braking distance distributions Error-propagation: Conservative: assumes normal distribution for all parameters Complex: requires explicit function formulation and partial differentiation (Standard) Monte-Carlo-Simulation: Efficient (in terms of confidence): returns shortest (also asymmetric) confidence interval Inefficient (in terms of computational effort): For rare event ε 1, N 100 trials required ε Typical according to CSM: ε [ 10 7... 10 9] N 10 11 ERA proposes to precalculate braking curves for limited number of train formations Freight trains to be handled using braked weight and correction factor Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 14 / 25
User: Infrastructure Manager Executive OK, basic Monte-Carlo is too complex to be calculated for each freight train. I fear a correction factor may be too conservative for well maintained wagon fleets. Are there any means to overcome this? Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 15 / 25
ETCS: γ vs. λ braked trains Typical γ-braked trains: Multiple units, other fixed formations Braking curve specification via deceleration values Typical λ-braked trains: Any in-service configurable trains, especially freight trains Braking curve using correction factors (K dry,rst, K wet,rst ) to calculate based on brake weight Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 16 / 25
ETCS: γ vs. λ braked trains Typical γ-braked trains: Multiple units, other fixed formations Braking curve specification via deceleration values Typical λ-braked trains: Any in-service configurable trains, especially freight trains Braking curve using correction factors (K dry,rst, K wet,rst ) to calculate based on brake weight The distribution of braking distances for freight trains of the same braked weight may be large: Empty/loaded selection vs. automatic load detection Maintenance state Tread vs. disc brake Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 16 / 25
ETCS: γ vs. λ braked trains Typical γ-braked trains: Multiple units, other fixed formations Braking curve specification via deceleration values Typical λ-braked trains: Any in-service configurable trains, especially freight trains Braking curve using correction factors (K dry,rst, K wet,rst ) to calculate based on brake weight The distribution of braking distances for freight trains of the same braked weight may be large: Empty/loaded selection vs. automatic load detection Maintenance state Tread vs. disc brake It may be of advantage to run certain λ trains as γ trains Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 16 / 25
Proposed solution (part 1): use importance sampling Importance sampling (IS) increases the probability of desired outcomes in Monte-Carlo-Simulations. Typical IS approaches: Stratification: select only relevant strata of the sampling range Scaling: Scale random variable Translation: Move random variable to more relevant part of sampling space Change of random variable: Replace random variable by one more likely to produce outcomes in the relevant range Adaptive approaches Effect: higher number of samples in region of interest Correction factor: Likelihood ratio L(y) = f(y) f(y) Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 17 / 25
Application of IS to braking curves Step 1: Select relevant variables for IS. mu/1 eta/1 0.15 0.14 0.13 0.12 0.11 0.94 0.92 0.90 0.88 0.86 0.84 900 950 1000 1050 s/m 900 950 1000 1050 s/m C/bar tf/s 27 26 25 24 23 3.90 3.88 3.86 3.84 3.82 3.80 900 950 1000 1050 s/m 900 950 1000 1050 s/m Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 18 / 25
Application of IS to braking curves Step 2: Change identified random variables, in the case at hand µ B 1400000 1200000 1000000 Basic MC IS: mu + l*sigma IS: k * sigma frequency 800000 600000 400000 200000 0 800 850 900 950 1000 1050 1100 1150 1200 1250 s/m Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 19 / 25
Application of IS to braking curves Step 3: Analyse for rare events, here braking distances in excess of 1100 m. N = 5 10 7 Relative frequency 0.012 0.010 0.008 0.006 0.004 0.002 Basic MC IS: mu + l*sigma IS: k * sigma 2500000 2000000 1500000 1000000 500000 Occurrences (dashed) 0.000 1000 1050 1100 1150 0 1200 s/m Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 20 / 25
Application of IS to braking curves Step 3: Analyse for rare events, here braking distances in excess of 1100 m. N = 5 10 7 s n U p U n IS,1 p IS,1 n IS,2 p IS,2 1000 24400 4.89 10 3 2.27 10 6 1.14 10 2 3.11 10 5 1.77 10 3 1050 2 4 10 7 6.66 10 4 2.02 10 4 1.48 10 4 1.59 10 4 1100 0 0 115 2.04 10 7 419 7.50 10 6 1150 0 0 0 0 15 3.88 10 7 1160 0 0 0 0 7 2.05 10 7 1170 0 0 0 0 5 1.46 10 7 1180 0 0 0 0 4 1.16 10 7 1190 0 0 0 0 1 2.90 10 8 Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 20 / 25
User: Infrastructure Manager Executive Well, this reduces the required Monte Carlo iterations by far, however handling braking curves for each wagon during brake assessment doesn t appear feasible. Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 21 / 25
Solution (part 2): Connect the wagon subsystem The Wagon 4.0 offers sensing and connectivity as well as cloud representation. Sensing: Accelerometers to record deceleration, brake cylinder pressure sensor to measure braking force Connectivity: send braking data to cloud Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 22 / 25
Solution (part 3): Big data analytics Big data analytics can be applied to separate train and wagon braking performance Record brake deceleration for wagons (in trains) in cloud Use big data analytics to derive individual wagon braking performance distribution Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 23 / 25
User: Infrastructure Manager Executive Great, the approach to use Importance Sampling, IoT-technologies and Big Data analytics to gain the braking curves of each individually composed train improves our performance compared to running λ-trains. Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 24 / 25
Thank you! Prof. Dr. Raphael Pfaff Rail vehicle engineering pfaff@fh-aachen.de www.raphaelpfaff.net Raphael Pfaff (Aachen UAS) London calling (probably) December 14, 2016 25 / 25