# Metamodel‐based Design Optimization (MBDO)

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## Decision support tools for complex electromagnetic systems design

In this chapter, the optimal design process is introduced and the general aspects of the complex system optimal design are addressed in general lines. The optimal design process is regarded as a three‐step process, consisting of a preliminary phase of problem definition, the main phase of optimization problem solving and a result analysis and decision making phase. The basic mathematical formulations of both the single‐ and the multi‐objective optimization are introduced and the notion of optimality in both the single‐ and the multi‐objective context is discussed. Some classical both deterministic and stochastic optimization algorithms are briefly presented for solving single‐objective optimization problems. Several techniques for transforming a multi‐objective optimization problem into a single‐objective problem are reviewed, which will be used later on within the advanced optimization techniques developed in the manuscript. A number of multi‐ dimensional data representation techniques, meant to assist the designer with the decision making, are presented towards the end of the chapter.

### General aspects of an optimal design process

The optimal design approach of a device, product or system implies three main steps:
‐ Preliminary phase (problem formulation);
‐ Optimization process (algorithm run);
‐ Results visualization and analysis (decision making).
These sequential steps are strongly related one to another and the result of the optimal design process depends on the appropriate addressing of all three steps. The brief description of these steps is presented next.

Preliminary phase (optimization problem formulation)

The preliminary phase of the optimal design approach consists in acquiring and resembling all the necessary information about the device, system or process to be optimally designed. In this phase, the designer tries to gain as much information as possible about the object of his design, for this will help him to appropriately define the optimization problem to be solved. The result of the optimization process depends entirely on the formulation of the optimization purpose, therefore
30 Decision support tools for complex electromagnetic systems design the proper formulation of the optimization problem is crucial for the success of the optimal design process. A number of issues are addressed within this step, such as: the model (or models) of the device which are to be used within the optimization process, the parameters and design variables of the model, the type of design variables (continuous, discrete, unclassifiable), the domain of variation for the design variables, the output variables of the model, the type of optimization problem (linear, non‐linear), etc.
An important action at this stage consists in determining the kind of relationships existing between the input, output and input/output variables of the model, as well as the relative influence of the input variables over the outputs of the model (sensitivity analysis). The influence of the different design variables on the outputs of the considered model are analyzed with the goal of identifying those variables which do not (or have very little) influence upon the outputs of the model, so that they will be ignored in the formulation of the optimization problem. This way, the optimization algorithm is charged with determining the optimal values of only the influent parameters, thus alleging more or less considerably the task of optimization and saving important computational time. A few tools for the analysis of the functional relationships governing a model are presented later on, in paragraph 1.2.

Optimization process (algorithm run)

The second step of the optimal design process is represented by the optimization process itself, which represents the core of the optimal design process. Complex mathematical techniques are employed at this stage for solving the optimization problem previously formulated. During the last two decades, once with the strong development of the computational power and the appearance of the personal computers, a lot of research has been dedicated to the development of optimization algorithms and mathematical optimization techniques. These optimization algorithms make successive calls to the model of the device considered, selecting then new samples to be evaluated based on the values already calculated.
Depending on the type of optimization problem previously formulated, an appropriate optimization algorithm is selected to solve this problem. The optimization problems can be classified based on several different criteria. Thus, if the objective function can be expressed as a linear combination of the design variables, we deal with a linear optimization problem or linear programming. Otherwise, the problem is said to be non‐linear, solved using non‐linear programming techniques (NLP). If the design variables of the problem take discrete values, the optimization problem is discrete or combinatorial. A number of combinatorial techniques with application in electrical engineering are presented in [TRA 09]. If the domain of variation of variables is continuous, then the optimization problem is continuous. It often arrives that an optimization problem presents both continuous and discrete variables. In this case, it is employed the term of mixed‐integer optimization.
The optimization techniques developed and presented further on in Chapter 2 and Chapter 3 of this manuscript belong to the category of NLP techniques. Although several optimization applications addressed in this work have both discrete and continuous variables, in this work all variables of the applications are considered continuous and handled accordingly.

Results visualization and analysis (decision making)

The third and final step of the optimal design process is represented by the analysis and interpretation of the results supplied by the optimization process mentioned in the previous paragraph. When a multi‐objective formulation has been chosen for the optimization problem, the result is a set of optimal trade‐off designs between the considered objective functions. In this case, the final choice for the design to be considered remains at the latitude of the designer, engineers and/or managers, which will select one design to be conceived. Thus, the multi‐objective optimization represents a tool for decision‐making. A number of tools for the representation and visualization of the results, such as bar charts, spider diagrams, bubble plots, parallel coordinates representations, scatter plot matrix, etc. [EST 12], [NOE 12] have as goal to assist the designer with the decision‐making process. Some of these multi‐dimensional data representation techniques will be introduced later on in this chapter, in paragraph 1.4.2.
Now that the steps of the optimal design process were briefly described, the attention is turned to each of them at a time, starting with some means of analyzing the correlation between the variables of a model, in the preliminary phase of the optimal design process.

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#### Variables influence and correlation

An important number of statistical analysis tools exist, assisting the designer in obtaining insights into the functional relationships between the variables of a model. Model analysis techniques and tools such as the screening technique [VIV 02], analysis of variance (ANOVA) [GOU 06], and response surface methodology are based on the realization of different experimental designs and represent notorious statistical tools. Therefore, these techniques will not be addressed here. Nevertheless, some techniques for the qualitative analysis of models, such as the Pearson correlation coefficient and the Spearman correlation coefficient represent powerful tools for gaining insights into the functional relationships of a model, being less employed currently. These coefficients will be next presented in the following paragraphs.

Pearson correlation coefficient

The Pearson correlation coefficient is a measure for the linear relationship between two input/output variables of a model [MAS 06]. The value of the coefficient is calculated based on a set of n samples , , for which the model has been run. Such a set of samples can be obtained using a design of experiments (DOE) technique [GOU 06], also known as experimental design. The value taken by this coefficient is always between ‐1 and 1. If the graphical representation of all pairs of samples is a straight line, it means that there is a strong linear correlation between the two variables. The slope of the straight line gives the sense of the correlation: positive slope gives positive value for the Pearson coefficient, respectively negative slope corresponds to a negative value of the Pearson coefficient. A value of zero for the Pearson correlation coefficient signifies that no linear correlation exists between the corresponding pair of variables of the model.

Introduction
Chapter 1 Decision support tools for complex electromagnetic systems design
1.1 General aspects of an optimal design process
1.2 Variables influence and correlation
1.3 Single and multi‐criteria optimization
1.3.1 Single‐objective optimization
1.3.2 Local vs. global optimality
1.3.3 Deterministic optimization algorithms
1.3.4 Stochastic optimization algorithms
1.3.5 Multiple criteria optimization
1.3.5.1 Pareto optimality
1.3.5.2 Decision making in the optimal design process
1.3.6 Optimization problem transformation techniques
1.3.6.1 Weighted objectives method
1.3.6.2 􀣕‐constraint method
1.3.6.3 Goal‐attainment method
1.3.6.4 Other transformation techniques
1.3.6.5 Example test‐problem
1.3.7 Complex systems specific optimization strategies
1.3.7.1 Metamodel‐based optimization strategies
1.3.7.2 Decomposition‐based optimization strategies
1.4 Pareto front quality assessment tools
1.4.1 Metrics definition
1.4.2 Multi‐dimensional data representation
1.5 Commercial optimization software
1.6 Conclusion
Chapter 2 Metamodel‐based Design Optimization (MBDO)
2.1 Why optimizing using a metamodel?
2.2 Metamodel‐Based Design Optimization strategies
2.3 Sequential metamodel‐based optimization
2.4 Adaptive metamodel‐based optimization
2.4.1 General process
2.4.2 Multiple criteria
2.4.3 Well‐spread sub‐set selection from a metamodel‐optimal n‐dimension Pareto front
2.4.4 Application: LIM device design optimization problem
2.4.4.1 Modeling of the LIM device
2.4.4.2 Optimization problem formulation
2.4.4.3 Optimization results
2.5 Criterion‐based metamodel optimization
2.5.1 Single‐objective infill point selection criteria
2.5.2 Adaptive infill strategies
2.5.3 Constraint handling
2.5.4 Application: single‐objective SMES device optimization
2.5.5 EGO algorithm with multiple objectives (MEGO algorithm)
2.5.5.1 Existing multi‐objective extensions of the EGO algorithm
2.5.5.2 Pseudo‐distance infill point selection criterion
2.5.6 Multi‐objective EGO algorithm (MEGO)
2.5.7 Distributed computation‐suited MEGO
2.5.7.1 Hybrid model approach
2.5.7.2 Weighted pseudo‐distance approach
2.5.8 Development of the MEGO tool – graphical user interface
2.5.9 MEGO coupling with ModeFrontier® optimization software
2.6 Application: SMES device optimization problem
2.6.1 TEAM22 benchmark description
2.6.2 Modeling of the SMES device
2.6.3 Single‐objective optimization of the 3‐parameter TEAM22 benchmark problem
2.6.3.1 Optimal SMES design using classical global optimization strategies
2.6.3.2 Sequential metamodel‐based optimization of the SMES
2.6.3.3 Adaptive metamodel‐based optimization of the SMES
2.6.3.4 Criterion‐based metamodel optimization of the SMES (EGO algorithm)
2.6.4 Bi‐objective optimization of the SMES device
2.6.4.1 Bi‐objective SMES device optimization using MBDO
2.6.4.2 SMES optimization using the multi‐objective MEGO algorithm
2.6.4.3 SMES optimization using MEGO and the distributed computation
2.7 Conclusion
Chapter 3 Decomposition‐based complex system design optimization
3.1 Why need to decompose a complex system?
3.2 Complex system partitioning
3.2.1 Single‐level optimization strategies
3.2.2 Multi‐level optimization strategies
3.2.2.1 Model‐based complex system decomposition
3.2.2.2 Discipline‐based complex system decomposition
3.2.2.3 Object‐based (physical) complex system decomposition
3.3 Output Space Mapping (OSM) multi‐model optimization strategy
3.4 Collaborative Optimization (CO) multi‐discipline optimization strategy
3.4.1 Basic mathematical formulation of CO
3.4.2 Coordination of the CO process
3.4.3 Existing efficiency enhancements of the CO formulation
3.4.4 Efficiency enhancement of the CO process by integration of the Output Space Mapping (OSM) technique
3.5 Analytical Target Cascading (ATC) multi‐component optimization strategy
3.5.1 Basic mathematical formulation
3.5.2 All‐in‐One (AIO) optimization problem formulation
3.5.3 Coordination strategies for ATC decomposed systems
3.5.4 Consistency constraints handling
3.5.5 Relaxation of the consistency constraints of the AIO problem
3.5.6 Penalty functions for the consistency constraint relaxation
3.5.6.1 Quadratic penalty relaxation
3.5.6.2 Lagrangian relaxation
3.5.6.3 Augmented Lagrangian relaxation
3.5.6.4 Augmented Lagrangian relaxation with the alternating direction method of multipliers
3.5.7 Mathematical example
3.5.8 Existing efficiency enhancements of the ATC process
3.5.9 Metamodel approximation integrated to the ATC formulation
3.6 Multi‐level optimization framework
3.7 Application: Optimal design of a low‐voltage single‐phase safety‐isolation transformer171
3.7.1 Transformer optimization benchmark
3.7.2 Transformer representation
3.7.3 OSM – MEGO isolation transformer optimization comparison
3.7.3.1 Single‐objective transformer optimization
3.7.3.2 Bi‐objective transformer optimization
3.7.4 Multi‐disciplinary optimization of the transformer by CO
3.8 Application: Optimal design of ultra‐capacitor energy storage system (UC‐ESS) on‐board a tramway
3.8.1 The ultra‐capacitor energy storage system (UC‐ESS)
3.8.2 Tramway traction system description
3.8.3 UC‐ESS multi‐level optimal design problem formulation
3.8.4 ATC formulation with adaptive‐metamodel approximation
3.9 Conclusion
Conclusion
Appendix A
Bibliography

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