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Modelling technique

Computer Aided Design (CAD) and Computer Aided Engineering (CAE) are used in many industrial domains. They allow simulating the behaviour of physical systems. Therefore, computer simulations are widely used by engineers when designing products. Development cost can be significantly decrease since the application of the trail-and-error process has a lower cost, when using these computer simulations as virtual prototypes.
In the last decades, scientists and engineers have put a lot of eﬀort to model engineering systems based on the knowledge of basic physical phenomena e.g. magnetic, electrical and mechanical laws. Experiments are necessary in order to identify the model parameters, verify modelling hypotheses, and assess the model accuracy. The modelling can be very simple or very complex depending on details needed and the complexity of the system itself i.e. the level of interaction between the physical domains and the number of components to be simulated simultaneously. The obtained models are used afterwards to simulate or study the behaviour and performances of the studied systems.
In some cases, mathematics modelling techniques replace physical-based modelling, for example when:
• Physical phenomena are very diﬃcult to understand or not well established.
• The system being studied is very complex.
• Experiments are available.
Mathematics modelling techniques are closely related to the statistics field. A well-known technique is the Design Of Experiment (DOE) methodology. It allows modelling a real system based on a limited number of experiments. The DOE is popular and has proved its eﬀectiveness in a wide range of applications such as in chemical and industrial processes [9]. In statistics literature, another technique can also be stated such as the Bayesian network or the Kriging method [13]. Outside the field of statistics, famous mathematics modelling techniques are for example the Neural network [65] and the Splines [42].
Although mathematics modelling techniques are initially developed for “Real-world” ex-periments, they can also be used to model “computer simulation”. When doing so, the term of Surrogate model or Metamodel1 are employed. In this research, we emphasize our interest in the modelling of experiments done on a computer (i.e. simulation). The modelling technique and mathematical formulation are shared between the Mathematics modelling and the Surrogate modelling. The main diﬀerence is that the Mathematics model is applied to real experiments and the Surrogate model is applied to computer simulations obtained by using a physical-based model, which can be reproduced without an experimental error. The idea be-hind the Surrogate model approach is discussed in Section 2.2. This reflects our main concern on CAD/CAE and optimal design processes. As computer technology has been improved on a daily basis, engineers tend to develop more and more complex physical models, which take into account many phenomena. However it is not very practical to include these models in the optimization process because the computations are usually time-consuming. Therefore, an approach using surrogate models is necessary. This kind of model is constructed using information from a physical-based high fidelity model. It replaces the high fidelity model in the optimization process and its suﬃciently low computation cost allows to speed-up the process.
Figure 2.1 depicts a summary of the idea introduced earlier. It also gives a summary of various physical-based modelling techniques usually employed in electrical engineering domains as well as several surrogate modelling techniques presented in the following sections.

Physical-Based Modelling

The physical-based modelling technique is certainly the most common approach used by engineers. A system or process is modelled based on the knowledge of physical phenomena2 and equations describing them. The experiment is always carried out to identify some model parameters, to validate the hypothesis posed and to verify the accuracy of the model.
In electrical engineering and related fields, three main techniques are used frequently; analytical, numerical, and semi-numerical modelling. Each technique has got its own advan-tages and drawbacks. There is no technique that is better than another. It depends on how it is used and what application is made of it. For example, an analytical model of motor is very suitable for pre-sizing and a numerical model such as the Finite Element Analysis (FEA) is used for detailed design phase.

Analytical model

Analytical models are expressed explicitly in terms of analytical equations. These analytical equations can be solved without any diﬃculty. The analytical model can be represent a system at diﬀerent detail levels. This depends on the applications and needs. Again, there is not any model better than another. A model is more or less suitable in a context. For example, a motor can be very low detailed with a circuit model as used in control application and more detailed with an electromagnetic model with linked to geometries in a sizing application. Because of their nature, explicit models are very cheap regarding their computation time. Therefore, they can be integrated successfully in an optimal design process [60].
An example in electrical engineering applications, especially in motor designs, is presented in Figure 2.2. This example shows an analytical model with two interconnected sub-models. A simple circuit model shown in Figure 2.2a represents the electrical domain and an equiva-lent magnetic circuit model with linked to the geometry (Figure 2.2b) describes the magnetic domain. Both models are used together to simulate performances of a motor. Many hypothe-ses are considered e.g. the magnetic flux is non-saturated, the phase current and flux density in the air gap are sinusoidal and the leakage flux is introduced via an empirical coeﬃcient.

Numerical model

To solve the nonlinear characteristics of a numerical model, an iterative method is required.
They are for example, numerical, semi-numerical and time-domain simulation models.
Many numerical techniques are used nowadays in our activity field: the Finite Element Method (FEM) [115], the Boundary Element Method (BEM) [91] and the Finite Volume Method (FVM) [25]. The FVM is mainly applied in the field of Computation Fluid Dynamic (CFD). The FEM is commonly used in the application of magnetic field computation (mag-netic domain) and stress analysis (mechanical domain). These methods consist in dividing an object into small elements via a mesh generation process. Governing partial diﬀerential equations are then discretized over this mesh and solved numerically. Such governing equa-tions are for example Maxwell equations in magnetic field computation [11], the elasticity theory for stress analysis [16] and the Navier-Stoke in CFD [110]. Numerical models allow taking into account nonlinear characteristics of materials and many phenomena, which are neglected in explicit models. This increases the precision and accuracy of the analysis results. They can also give local information on the system e.g. the magnetic saturation in a small zone of tooth. Furthermore, the combination of several physical domains in the same solver is possible i.e. solving the coupled magnetic, thermal and mechanical domains simultaneously [24, 68, 90].
The development of numerical software packages seems to be a demanding task. Fortu-nately, many commercial software packages are available on the market. Some of them can be listed: Opera, Flux, ANSYS, FLUENT, FloTHERM, CosmosWorks and COMSOL. In the point of view of the user/designer, using such software to model a system can reduce sig-nificantly their modelling eﬀort, while no deep knowledge of the theory is needed. However, the computation time can be very high. Figure 2.3 shows modelling and analysis results of a motor using FEM package Vector Field Opera 3D [22].
Unlike numerical models, a semi-numerical model is expressed by using analytical equa-tions with some nonlinear characteristics, which are needed to be solved iteratively [11]. Such a model oﬀers a good compromise between computation time and precision because it does not require a huge number of assumptions as in an explicit model (due to its nonlinear property) nor provide a lot of local information. Well-known semi-numerical models are the permeance network (magnetic field computation) and the lumped parameter thermal model (thermal, heat transfer). Figure 2.4 shows the permeance network of a synchronous generator taken from [88].

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Summary

The model development is always a very long and delicate task. It requires a lot of time, financial resources, knowledge and experiences in the application domain. This is always an experts job. For analytical models and semi-numerical models, this development phase is realised by experts. The models are usually proprietary or in-house coded. For numer-ical models, designers commonly use software packages already developed by the software company.
Four criteria are compared in Figure 2.5:
(i) Modelling details: The more the model is described in detail, the easier it is to access the local information. It is supposed that if the local information is available, the global one is also obtained.
(ii) Accuracy: The model can be build with diﬀerent levels of accuracy.
(iii) Computation time: The model should be selected on the basis of the needs of the designer. The model with too much information may not be suitable for certain appli-cations.
(iv) System description: The ability of each component to react and to interacts with the system and the other components around it.
The numerical technique is more accurate but it does not describe the interactions with the system very well. The analytical technique is rapid and able to describe the system. However, it hardly provides detailed local information. The developing trend is to combine these techniques in order to obtain both high accuracy and description of the system. Several examples of hybrid model can be listed:
• The transient finite element analysis, which uses an electric circuit model and mechani-cal dynamic model (both are time-domain simulation models) and the FEM to simulate the performance of a motor in transient state.
• A system of two components. One is described by an analytical model and the other by a numerical model.
The integration of modelling techniques can be done directly and transparently in one software package or indirectly via an external process such as multidisciplinary design analysis (see Section 3.2.1).
A prototype is also compared in Figure 2.5. It allows validating the modelling due to the fact that all physical phenomena and all interactions can be observed. However, development time and cost are the main drawbacks.

Table of contents :

Introduction
Part I Tools
1 Optimal design of complex systems
1.1 Complex system design methodology
1.1.1 Conventional approach
1.1.2 Design problem as inverse problem
1.1.3 Complex system decomposition
1.1.4 Global system design
1.1.5 Multi-criteria design problem
1.1.5.1 Pareto optimality definition
1.1.6 Optimal design
1.1.6.1 Specification definition
1.1.6.2 Design problem formulation
1.1.6.3 Optimization problem formulation
1.1.6.4 Problem solving
1.1.6.5 Result analysis
1.1.6.6 Decision making process
1.2 Optimal design in the railway industry
1.2.1 Overview of railway traction systems
1.2.1.1 Power supply
1.2.1.2 DC supply
1.2.1.3 Electric braking
1.2.1.4 Variable speed converter
1.2.1.5 Traction motor
1.2.2 Decomposition of train systems
1.2.3 Corporate design process
1.2.4 Standardisation
1.2.5 Design criteria
1.2.6 Model and analysis tools
1.2.7 Optimal design consideration
1.3 Conclusion
2 Modelling technique
2.1 Physical-Based Modelling
2.1.1 Analytical model
2.1.2 Numerical model
2.1.3 Summary
2.2 Surrogate modelling
2.2.1 Overview
2.2.2 Sampling technique
2.2.2.1 Classical experimental design
2.2.2.2 Monte Carlo Sampling
2.2.2.3 Latin Hypercube Sampling
2.2.2.4 Number of sample point
2.2.2.5 Summary
2.2.3 Surrogate modelling technique
2.2.3.1 Polynomial
2.2.3.2 Radial basis function
2.2.3.3 Kriging
2.2.4 Surrogate model accuracy assessment
2.2.4.1 Assessment methodology
2.2.4.2 Results
2.2.5 Non-conventional sampling and modelling technique
2.2.5.1 Global response surface methodology
2.2.5.2 Kriging-assisted sampling technique
2.3 Multimodel approach
2.4 Conclusion
3 Optimization technique
3.1 Global system approach
3.1.1 Single-objective
3.1.1.1 Gradient method
3.1.1.2 Genetic algorithm
3.1.1.3 Hybrid algorithm
3.1.2 Multi-objective
3.1.2.1 Transformation
3.1.2.1.1 Weighted-sum
3.1.2.1.2 Epsilon-constraint
3.1.2.1.3 Illustration example
3.1.2.2 Multi-objective algorithm
3.1.3 Surrogate-assisted optimization algorithm
3.1.3.1 Overview
3.1.3.2 Main algorithm
3.1.3.3 Initial point selection
3.1.3.4 Infill criteria for single-objective optimization
3.1.3.5 Constraint handling
3.1.3.5.1 Probability method
3.1.3.5.2 Direct integration into sub-optimization problems
3.1.3.5.3 Constraint handling illustration example
3.1.3.6 Multi-objective optimization
3.1.3.6.1 Infill criteria for multi-objective optimization
3.1.3.6.2 Acceptance condition for multi-objective optimization
3.1.3.7 Mathematical test example
3.1.3.7.1 Single-objective noisy function optimization
3.1.3.7.2 Multi-objective problem
3.2 Decomposition approach
3.2.1 Multidisciplinary Design Optimization
3.2.1.1 Overview
3.2.1.2 Multidisciplinary feasibility
3.2.1.3 Individual disciplinary feasibility
3.2.1.4 All at once
3.2.1.5 Mathematical example
3.2.1.6 Conclusion
3.2.2 Target cascading optimal design
3.2.2.1 Overview
3.2.2.2 TC formulation
3.2.2.3 Mathematical example
3.2.2.4 Conclusion
3.3 Conclusion
Part II Applications
4 Optimal design of traction motors
4.1 Traction motor design methodology
4.1.1 Traction motor modelling
4.1.1.1 Thermal module
4.1.1.2 Temperature feedback
4.1.1.3 Control strategy
4.1.2 Design problem formulation
4.2 Definition of required torque and speed
4.2.1 Torque and speed characteristics
4.2.1.1 Approach using rated torque and speed
4.2.1.2 Approach using torque versus speed curve
4.2.1.3 Approach using load cycle
4.2.2 Comparative study methodology
4.2.3 Results
4.2.3.1 Case I and II
4.2.3.2 Case III
4.2.3.3 Case IV and V
4.2.3.4 Case VI
4.2.3.5 Computation time
4.2.4 Conclusion on the comparative study
4.3 Multidisciplinary Design Optimization
4.3.1 MDO techniques comparison
4.3.1.1 MDF case
4.3.1.2 IDF case
4.3.1.3 AAO case
4.3.2 SMPM design results
4.3.3 Conclusion on the MDO
4.4 Multi-criteria optimal design problem
4.4.1 Design problem definition
4.4.2 Results
4.4.3 Conclusion on the multi-criteria optimal design
4.5 Use of high fidelity tools
4.5.1 Multi-physical model
4.5.2 Single objective problem (SOP)
4.5.3 Multi-objective problem (MOP)
4.5.4 Conclusion on the optimal design using high fidelity tools
4.6 Conclusion
5 Optimal design of railway traction systems
5.1 Tram traction system re-design problem
5.1.1 Problem description
5.1.2 Modelling of traction systems
5.1.3 Optimal design problem
5.1.4 Comparative results
5.1.4.1 SQP optimization results
5.1.4.2 GA optimization results
5.1.4.3 Hybrid method (GA & SQP) results
5.1.4.4 Summary
5.1.5 Traction system design results
5.1.6 Conclusion on the re-design problem
5.2 Multilevel tram traction system design
5.2.1 Multilevel optimal design problem
5.2.2 Tram traction system modelling
5.2.2.1 Modelling of traction systems
5.2.2.2 Modelling of heat sinks
5.2.2.3 Modelling of traction motors
5.2.3 Results
5.2.4 Conclusion on the multilevel problem
5.3 Conclusion
Conclusion
Appendices