Numerical simulations are crucial for modeling complex systems, but calibrating them becomes challenging when data are noisy or incomplete and likelihood evaluations are computationally expensive. Bayesian calibration offers an interesting way to handle uncertainty, yet computing the posterior distribution remains a major challenge under such conditions. To address this, we propose a sequential surrogate-based approach that incrementally improves the approximation of the log-likelihood using Gaussian Process Regression. Starting from limited evaluations, the surrogate and its gradient are refined step by step. At each iteration, new evaluations of the expensive likelihood are added only at informative locations, that is to say where the surrogate is most uncertain and where the potential impact on the posterior is greatest. The surrogate is then coupled with the Metropolis-Adjusted Langevin Algorithm, which uses gradient information to efficiently explore the posterior. This approach accelerates convergence, handles relatively high-dimensional settings, and keeps computational costs low. We demonstrate its effectiveness on both a synthetic benchmark and an industrial application involving the calibration of high-speed train parameters from incomplete sensor data.

The railway world is undergoing major changes. The advent of new technologies allows us to rethink the train system and face new challenges, but one must not forget all the ecological constraints that are now accentuated by the increase in energy costs. This paper focuses on the optimization of the driver commands to limit the energy consumption of the trains under punctuality and security constraints. This problem falls within the framework of control optimization problems for nonlinear dynamic mechanical systems in the presence of constraints and uncertainties. A four-step approach is then proposed in this paper to solve this problem: (1) the introduction of simplified and fast-to-evaluate models to model the nonlinear dynamic behavior of the train and its energy consumption; (2) the identification in a Bayesian formalism of the parameters on which these models depend from on-track measurements on commercial trains; (3) the reformulation of the optimization problem so that it integrates the uncertainties related to an imperfect knowledge of these estimated parameters; (4) the resolution of the optimization problem using evolutionary algorithms. The main specificity of this work lies in the fact that not only the objective function to be minimized, here the energy consumed by the train, is impacted by the uncertainties, but also the admissibility constraints of the solution, here punctuality and operating safety. The integration of the uncertainties in the search for the control function is thus not trivial and requires several original adaptations in order to make the final optimization problem well posed.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Driver's control optimization under uncertainties to reduce energy consumption of high-speed trainsJulien Nespoulous, Christian Soize, Christine Funfschilling, Guillaume

The speed profile of a train plays an important role in energy consumption and resulting costs. The industrial objective of this work is thus to develop a method to reduce the energy consumed by a train over a journey by playing on the driver commands (traction and braking forces) while respecting punctuality constraints. First, a rigid body approach (Lagrangian formalism) is introduced to characterise the dynamics of the train. In particular, the aerodynamic (including the wind effect), traction, and braking forces are taken into account, and a special attention is paid to the vertical and lateral characteristics of the track as they play a key role in the train dynamics. Second, a model for energy consumption and recovery (thanks to dynamic braking) is introduced. Experimental measurements of a high-speed line are then used to estimate the parameters on which the two previous models are based and to validate their predictive capacities. The optimisation problem under constraints is finally solved using an evolutionary algorithm where the constraints are implemented using an augmented Lagrangian formalism. The performance of the proposed method in terms of speed optimisation and energy consumption reduction is compared to measurements associated with commercial trains.

The train is a complex nonlinear system, whose dynamic behavior is difficult to predict accurately because of its environmental sensitivity. Indeed, in spite of a relative fine modeling of the vehicle and its rolling environment (track and wind), the slightest uncontrolled disturbance can modify the dynamic comportment of the train. For this reason, uncertainty must be considered in the physical models. The industrial objective of this work is twofold. Firstly, the construction of a longitudinal dynamic model for high-speed trains able to take into account the fluctuations inherent to the system. Secondly, the optimization under uncertainty of the driver’s command with the objective of reducing the energy consumed by the train, under a set of punctuality and physical nonlinear constraints (speed limitation, final speed, and final position constraints).

: Controlling energy consumption has turned to be an important challenge of the 21 st century and par-ticularly in the railway world since the transport sector constitutes one of the largest consumers. For this reason, the railway companies pay close attention to their energy consumption and seek to reduce it. Recently, this objective has become even more crucial because of the growing demand stemming from the increase of the train frequency, as well as their speed. To achieve this consumption reduction, three levers can be activated: modify the rolling environment, the vehicle characteristics, or its speed profile. The present work focuses on the optimization of the speed profile. This latter must fulfill a number of constraints. First it has to respect the speed limitation on the track to assure the passengers’ security; the involved constraints are called the security constraints . Second the train has to arrive in the train station at a certain time and a specific position with an appropriate speed; these are called the punctuality constraints . Third, the passengers should not be subjected to violent accelerations or jolts; these conditions are regrouped in the comfort constraints . Consequently, the mathematical problem consists in an optimization problem under deterministic non-linear constraints . However, the speed profile is driven by the longitudinal behavior of the train on the track. The optimization supposes thus to construct a precise model to describe the train longitudinal dynamics and its energy consumption. But the mechanical system includes quantities that are more

This paper presents a Bayesian calibration method for a simulation-based model with stochastic functional input and output. The originality of the method lies in an adaptation involving the representation of the likelihood function by a Gaussian process surrogate model, to cope with the high computational cost of the simulation, while avoiding the surrogate modeling of the functional output. The adaptation focuses on taking into account the uncertainty introduced by the use of a surrogate model when estimating the parameters posterior probability distribution by MCMC. To this end, trajectories of the random surrogate model of the likelihood function are drawn and injected in the MCMC algorithm. An application on a train suspension monitoring case is presented.

This paper presents a review of various works that highlight the importance of introducing the variability of the road-track/vehicle system into dynamic simulations as soon as this latter is meant to be predictive. The first section of the paper presents the Uncertainty Quantification, Verification and Validation method (UQ-VV). This latter proposes tools to model uncertainties, to associate a confidence to the prediction of quantities of interest and to estimate the probability of occurrence of different scenarios. The method is illustrated by various examples mainly from the rail domain but also from the road sector. The second section summarises application examples of predictive modelling, robust optimisation and calibration.

This paper presents a novel method for the state health monitoring of high-speed train suspensions from in-line acceleration measurements by embedded sensors, for maintenance purposes. We propose a model-based method relying on a multibody simulation code. It performs the simultaneous identification of several suspension mechanical parameters. It is adapted to the introduction of uncertainties in the system and to the exploitation of numerous high-dimensional measurements. The novel method consists of a Bayesian calibration approach using a Gaussian process surrogate model of the likelihood function. The method has been validated on numerical experiments. We demonstrate its ability to detect evolutions of the health state of suspension elements. It has then been tested on actual acceleration measurements to study the time evolution of the suspension parameters.

Train suspension elements ensure its stability and play a key role in the ride safety and passengers comfort. They undergo damage throughout their lifetime, which may influence the train dynamic behavior. Consequently, they require regular maintenance, usually based on visual inspection or mileage criteria. However, a better knowledge of the actual health state of the suspensions would allow for providing maintenance closer to the real needs. This work deals with the development of a remote diagnosis method for high-speed train suspensions, which consists in the inverse identification of the suspension mechanical parameters from in-service measurements of the train dynamic behavior by embedded accelerometers. The excitation source of a rolling train is the track geometric irregularities, which consist of small displacements of the rails relatively to the theoretical track design. Track geometry also undergoes damage because of railway traffic. Consequently, the irregularities evolve through time. Because the train dynamic behavior is very dependent on them, sole acceleration measurements are not sufficient to correctly identify the suspensions mechanical parameters. Measurements of the track irregularities must be taken into account along with the corresponding measurements of the train dynamic behavior. This implies the use of a train dynamics software, in order to simulate the train dynamic behavior on a specific track geometry. For this work, we relied on the commercial multibody code Vampire. The studied vehicle is a French TGV Reseau. Accelerometers are located at the connections between carbodies, above and on the shared bogie. For each connection, carbody and bogie vertical and lateral accelerations are measured. The various acceleration signals are studied in the frequency domain. Seven mechanical parameters of various suspension types are simultaneously identified: dampers, airsprings, elastomer stiffnesses… Measurements are performed without interruption during the ride. Consequently, for a single inverse identification, joint measurements of the track geometric irregularities and of the train dynamic behavior on several hundreds of kilometers of track are generally available. The large quantity of data as well as the uncertain nature of the different physical quantities of interest encourage a statistical approach of the problem. The inverse identification is performed thanks to a Bayesian calibration procedure. The principle of Bayesian calibration is to update the initial knowledge about the system parameters using measurements of the system output. This procedure provides the distribution of probable values of the parameters. Such information allows for estimating of the accuracy of the inverse identification, through the computation of confidence intervals for instance. Because it requires simulation runs on the hundreds of kilometers of track for numerous values of the parameters, the classical Bayesian calibration procedure would be computationally unaffordable. An adaptation of the procedure relying on the approximation of the expensive likelihood function by a Gaussian process surrogate model has been developed to address this numerical cost issue. The impact of the use of a random surrogate model has been studied, in particular the influence of the surrogate model uncertainty, which represents the error inherent in the approximation of the likelihood function. The inverse identification procedure has first been validated on a numerical experiment. The principle of a numerical experiment is to generate artificial acceleration signals thanks to simulation, using a vehicle model with known degraded suspension parameters. They can then be used as if they were measurements to perform a mock identification. Since the parameters values are known, the quality of the identification can be measured. In such a case, the inverse identification displayed very satisfying results, with identification errors below 5% of the admissible interval for every parameter. The inverse identification procedure has then been tested on actual measurements of the train dynamic behavior. A significant evolution can be observed from the parameters nominal value. Since the real value of the suspension parameters remains unknown, no comparison could be performed in this case. The influence of the surrogate model uncertainty is also emphasized by this study. Indeed, when it is taken into account in the identification procedure, the size of the confidence intervals for the identified parameters significantly increases. This means that the accuracy of the identification tends to be overestimated if the surrogate model uncertainty is ignored. Transportation systems are nowadays more and more equipped with various kinds of sensors that allow for monitoring its different component. They make remote diagnosis method possible, which can be a precious tool for maintenance optimization. For train suspensions, embedding sensors remains difficult because of the variety and number of suspension elements. The advantage of an approach relying on accelerometers is to provide monitoring with a limited number of sensors. It however requires investing bigger efforts on data processing and on the identification method. Indeed, the expected information, the suspensions state, is not directly accessible in the measurements. We developed a statistical inverse identification method using measurements of the train dynamic behavior by embedded accelerometers. It involves train dynamics simulations in order to take into account track geometric irregularities measurements. The method shows very promising results on numerical experiments as well as on actual measurements. A subsequent step now consists in developing criteria on the suspension parameters to trigger maintenance operations. Concerning the mathematical aspects of the method, it is based on a Bayesian calibration procedure, which allow for estimating the identification accuracy. It also uses Gaussian process surrogate modeling in order to reduce computational costs. By applying the identification procedure on measurements performed at different time steps (with a time gap of several months), the evolution of the suspensions parameters with time and thus the gradual degradation of the suspension elements could be studied.

The work presented here deals with the development of a state health monitoring method for high-speed train suspensions using in-service measurements by embedded accelerometers. Mathematically, it consists in solving a statistical inverse problem. A rolling train is a dynamic system excited by the track geometric irregularities. They consist of small displacements of the rails relatively to the theoretical track design. The suspension elements play a key role for the ride safety and comfort. The train dynamic response being dependent on the suspensions mechanical characteristics, information about the suspensions state can be inferred from acceleration measurements in the train. This information would allow for providing a more efficient maintenance. Track geometry is subject to damage caused by railway traffic and to maintenance operations. Consequently, it evolves through time. Because of the high sensitivity of the train dynamic response to the track geometric irregularities, their evolution must be taken into account through the use of train dynamics simulation. Because the system input (the track geometric irregularities) and output (the train dynamic response) are stochastic quantities, the inverse problem is solved in the Bayesian framework. The monitoring method thus consists in performing a Bayesian calibration of a simulation-based model using joint measurements of the system input and output. Its objective is to identify the posterior distribution of the model parameters describing the suspensions mechanical characteristics. Classical Bayesian calibration implies the computation of a likelihood function using a stochastic model and experimental data. This likelihood function is then used to estimate the posterior distribution of the model parameters. This step can be performed by Markov Chain Monte Carlo (MCMC) algorithms, which require numerous calls to the likelihood function. If the latter is expensive to compute, it may result in unaffordable computational costs, which is the case here. To address this issue, we propose to rely on surrogate models. They are usually used to provide an algebraic approximation of the system output. However, in the present case, the output is functional, which makes a surrogate model difficult to build. Instead, we propose a calibration method based on a Gaussian surrogate model of the scalar likelihood function. We present how such a random surrogate model can be used to estimate the model parameters distribution, how the new uncertainty it introduces can be taken into account to correctly evaluate the calibration accuracy, and the results of the method applied to our railway monitoring case.

The objective of the work presented here is a bayesian calibration of parameters describing the mechanical characteristics of high-speed train suspensions for maintenance purposes. This calibration is achieved by comparing simulation results to on-track acceleration measurements. It requires the estimation on the multidimensionnal admissible set of the parameters of the likelihood function of the train dynamic response. This estimation is achieved thanks to the identification of a kriging metamodel of this likelihood function to reduce the numerical cost. From this metamodel, the posterior probability density function of the parameters is estimated using an MCMC algorithm.