Optimal design for electromagnetic devices: A synthesis approach using intervals and constraint-based methods

Abstract

In electrical engineering, current methods show limits when it comes to complex design problems with heterogeneous and multidisciplinary requirements in the preliminary steps. Also, the setting and the solving of those problems is a simultaneous challenge that needs to be more explored. It may be necessary to recall the need to re-think the conventional design process when it comes to complex electrical systems. Particularly, by considering first the problem admissibility before its solving or optimization. Being able to detect an ill-posed or an over-constrained problem (causal analysis) can help in finding admissible solutions but also forces the designers to understand the meaning of the problem modeling. A synthesis point of view is proposed to ensure admissible and relevant solutions during the preliminary design (pre-design) process. Intervals and constraint-based methods are shown suitable for proposing efficient modeling, solving tools and addressing optimal design. This paper is illustrated by the case study of a benchmark transformer, taking into account multi-physical requirements. The optimal design includes mixed-type variables and tabulated constraints such as manufacturer datasheets. The computed results are then compared with solutions found in the literature and show the advantages of the implemented method in terms of solutions and computation time.

Keywords

Optimal design interval methods constraints satisfaction problems pre-design of electromagnetic devices

1. Introduction

Industrial requirements are increasingly demanding regarding the efficiency of electric devices. Good integration properties have to match very low costs, energy consumption and volume size. Herewith, effective optimal design tools appear to be necessary for new applications such as renewable energy devices, embedded systems, etc. Thus, the development of efficient and relevant computational tools for optimal design is currently an active area of research in electrical engineering.

Currently, most of the works related to engineering design and optimization use direct models to find possible solutions. These models are then used as simulation tools to evaluate the suitability and admissibility of these solutions and to improve them afterwards in an iterative optimization process.

Very few papers have solved complete design problems with the relevant synthesis point of view, meaning using inverse models instead of direct ones. Furthermore, most of research teams restrained themselves to too simple analytical or semi-analytical sizing models for pre-design, if not numerical models. This way of working is explained by the lack of relevant algorithms to address mixed-type variables, e.g. categorical, integer or continuous variables, and ill-posed inverse problems. Besides, many techniques tansform the initial models to solve the problem, resulting in a loss of the problem structure. During pre-design, the problem setting is often neglected for the benefit of the problem solving. Though properly realized, the problem setting can greatly ease the later, considering an adequate solver. There are two main limitations. Firstly is the high number of constraints, which can also be contradictory or over-restrictive. Secondly is the number of variables (i.e. the size of the problem), that is often over- or under-estimated by the designer, because the problem complexity is not totally understood. These are a part of the problem setting that therefore are the major difficulties encountered in this field. Thus, pre-design should address simultaneously the problem setting and the problem solving.

Regarding optimization, it usually does not occur during the pre-design phases, when the major design choices are made, including the device architecture. As a result, the overall costs of the product are impacted drastically, as in aeronautics (see Fig. 1). Consequently, optimization should be integrated from the beginning of any pre-design process.

Fig. 1.

Leverage of developpement expenditures [4].

An alternative way to consider the design problem with the synthesis angle could be by adopting Artificial Intelligence (AI) concepts, in addition to the classical engineering viewpoint. For this purpose, Interval Methods and Constraint Satisfaction Problems (CSP) currently are one of the most promising ways to consider and formalize design problems during pre-design. Although studied for a long time, AI methods mainly belong to the computer science community and are not so common in other engineering communities. Whereas [1,2] are noticeable exceptions in electrical engineering, the used algorithms were limited in addressing complex and multi-requirements design problems. Although the algorithms were improved by [3], some limits still remain. More advanced algorithms and strategies are required and some of them will be introduced here, as a continuity of authors’ previous works done in mechanics and aeronautics [4,5].

A design methodology based on a synthesis approach, constraint programming and interval methods is proposed to meet the current lack of tools for pre-design in electrical engineering.

The transformer case study may be very common, nonetheless, despite its apparent simplicity, it may not be addressed properly (inadequate method or solver).

2. Interval computation and constraint propagation-based approach

Concerning design, AI concepts solve the conflict between direct and inverse models by formulating the problems as CSP, thanks to the a-causal models they are based on. With a-causality, no inputs or outputs have to be defined. The CSP approach uses a declarative and modular modeling language and provides a natural way for problem descriptions.

The mathematical framework naturally is algebraic synthesis instead of analysis. Thus, this approch is mainly dedicated to pre-design phases, where coarse algebraic models may be relevant, e.g. mean or “worst case” steady state model, unlike partial or ordinary differential equations for fine models.

2.1. Constraint Satisfaction Problem

A CSP is defined by a triplet (X, D, C) such that [6,7]:

$X=\{x_{1},x_{2},x_{3},\ldots ,x_{n}\}$ is a finite set of n constraint variables, with $n\in \mathbb{N}$ ;

$D=\{d_{1},d_{2},d_{3},\ldots ,d_{n}\}$ is a finite set of the domains of the constraint variables such as the domain of x _i is d _i, $\forall ∼i\in \{1,\ldots ,n\}$ . d _i forms the set of possibilities for x _i.

The domains can either be discrete, with a closed and bounded set of possible values $\{x^{1},\ldots ,x^{l}\}$ . $l\in \mathbb{N}$ ; or continuous, as an interval of reals [x ^min, x ^max], with x ^min and x ^max two finite reals.

$C=\{c_{1},∼c_{2},∼c_{3},\ldots ,∼c_{p}\}$ is a finite set of constraints to be satisfied by the constraint variables, with $p\in \mathbb{N}$ , i.e. $\forall ∼j\in \{1,\ldots ,p\},∼\exists !∼X_{i}\subseteq X/c_{j}(X_{i})$ .

A constraint is any type of mathematical relations (linear, quadratic, non-linear, boolean,…) involving at least one variable. It can be logical, explicit, etc.

Every knowledge related to the design, e.g. models, requirements and specifications, is expressed as a constraint to be satisfied, but also design rules based on tables, abacuses, or any type of informal relations.

Solving a CSP consists in finding a consistent value for every x _i in X (e.g. instantiate) that satisfies every constraint c _j. Constraints propagation through x _i can reduce the variable domain d _i. Then, the resulting domain is used to downsize other variables domains.

The CSP solving process depends on the type of constraint variables: methods for solving discrete CSPs, i.e. with discrete variables, are different from those solving continuous (or numerical) CSPs, i.e. with continuous variables. Indeed, it is not possible to ensure the consistency of constraints over real numbers, since the representation of reals in numerical computations is not exact as floating point numbers may lead to rounding errors.

Operational research and AI give the methods for discrete CSP. These discrete CSP methods, of exponential complexity, are based on enumeration and constraint propagation (filtering).

Continuous CSP methods are based on intervals computations. Those techniques are a synthesis between interval-based analysis [8] and CSPs [9]. Several techniques have been developed and good examples are given in [10].

2.2. Interval computation and Branch & Prunes algorithms

Through interval evaluation, interval arithmetic [8] is a simple method to determine an interval ensuring the inclusion of the exact lower and upper bounds of a function. This method particularly allows to factor computer round-off errors in. Interval arithmetic has been extended to many non-linear operators, like quadratic, trigonometric ones, etc. The resultant arithmetic is known as Interval Computation.

For continuous CSP, the contractors, a specific implementation of constraints, allow to use some consistency techniques.

Definition 2.1 (Contractor).

The application $C_{\mathbb{I}}:\mathbb{R}^{n}\mapsto \mathbb{R}^{n}$ is a contractor for the closed subset $\mathbb{I}\subset \mathbb{R}^{n}$ if and only if:

[x] is a set of intervals of $\mathbb{R}^{n}$ , usually called a box,

$\forall ∼[x]\in \mathbb{R}^{n}\left\{\begin{array}{@{}c@{}}C_{\mathbb{I}}([x])\subset [x]\quad \text{(contractor property)}\\ (C_{\mathbb{I}}([x])\cap \mathbb{I})=([x]\cap \mathbb{I})\\ \text{(completeness property)}\end{array}\right.$

Several consistency techniques exist in the literature, all based on the contractor and completeness properties. The two main categories, Hull and Box consistencies have been detailed in [11].

Hull consistency is based on a contractor property that once verified, allows the use of the fixed-point theorem. The corresponding algorithm performs computation on domains satisfying all the constraints of the CSP. A CSP (X, D, C) is said to be hull consistent if for every variables x _i in X, there are two bounding points, x _i (lower) and x _i (upper), in the domain [x _i] that satisfy every constraint c _j in C. Hull consistency uses forward evaluation and backward propagation mechanisms based on a binary tree. This tree is the internal representation of each constraint of the problem.

Box consistency is a relaxation of Hull consistency [12]. The constraint satisfaction test on the real domain is replaced by a refutation procedure on the interval domain. Every constraint is rewritten as c (v ₁, … , v _n) = 0 with v ₁ ∈ D ₁, … , v _k ∈ D _k, … , v _n ∈ D _n. If 0 ∈ c (D ₁, … , D _k−1, D _k, D _k+1, … , D _n), ∀ v _k, then the domain of x _k is box consistent. If a domain D _k is not box-consistent, the goal of the solving procedure is to find the two extreme values in D _k that are consistent. The standard implementation of this procedure relies on a dichotomy splitting of the initial domain. This procedure is allowed because of the monotony property for interval evaluation.

The main advantage of Box consistency over Hull consistency is the ability to manage the decoupling between set computations (elimination problem). Nevertheless, its convergence rate is rather low. Algorithms exist to improve this rate (interval Newton methods [12]) but Hull consistency is equivalent and remains cheaper for decoupled constraint. Usually, Box consistency is more expensive than Hull consistency but the resulting boxes are better reduced. Moreover, Box consistency gives significant reductions on polynomial constraints.

For a continuous CSP being Hull or Box consistent, if an interval solution exists, this one is included in the interval solution of the corresponding Hull or Box algorithm results. This condition is necessary but not sufficient. In practice, additional algorithms are needed to find the effective interval solutions.

2.3. Branch and Prune algorithms

Branch and Prune (BP) algorithms are used to find the interval solutions of discrete CSP. The goals are to obtain, for each constraint variable, an upper and a lower value as close as possible to each solution of the discrete CSP(X, D, C). To achieve this goal, BP algorithms recursively applies a pruning operator (i.e. a hull or box consistency operator). The process starts with a given CSP(X, D, C) and an empty list of interval solutions L. At each step, a variable is chosen and its interval D is bisected in two subintervals D ₁ and D ₂. The pruning operator is applied to each of the two subintervals for all constraints in C. This procedure reduces the intervals for the other variables defined in the CSP. If a good precision is obtained for D, i.e. one solution is reached, the resulting set of intervals is added to L. If not, the splitting process continues until one interval becomes empty. Then the algorithm stops and returns L for the given discrete CSP(X, D, C).

For real-life problems, effective solving processes go through a variable ordering, i.e. the selection of a constraint variable among several possibilities. This first selection process is followed by a second one concerning the ordering of the subintervals of the variable domain called value ordering. A correct ordering drastically impacts the computation time. Several heuristics exist to help deciding such as selecting first the most influential variables. Sometimes, the choice can be efficiently done by design rules and designer knowledges.

2.4. Optimization and CSP

CSPs can be adapted to solve optimization [13]. Usually, a constrained optimization problem on integer, real or mixed variables is modeled as follows:

x = (x ₁, x ₂, …, x _n) a set of n variables. Each value should be in $\mathbb{R}$ or $\mathbb{Z}$ ,

$f:\mathbb{R}^{n}\times \mathbb{Z}^{n-r}\mapsto \mathbb{R}$ a scalar function to be minimized, such that:

$\forall i,\in \{1,\ldots ,p\},∼\exists ∼x\subseteq X/g_{i}(x)\leq 0$ , a set of p inequalities to satisfy,

$\forall j,\in \{1,\ldots ,q\},∼\exists ∼x^{\prime }\subseteq X/h_{j}(x^{\prime })\leq 0$ , a set of q equations to satisfy.

and is usually set as: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\min _{x\in X}∼f(x)\\ g_{i}(x)\leq 0∼\forall i\in \{1,\ldots ,p\}\\ h_{j}(x)=0∼\forall j\in \{1,\ldots ,q\}\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$ Two of the main categories of techniques described in technical literature to minimize f subjected to constraints g _i and h _j present some drawbacks:

Deterministic methods: the issue with methods such as Mix Integer Linear Programming and Sequential Quadratic Programming (SQP) is that they would transform any nonlinear problem into a linear or a quadratic one respectively. In the engineering field however, many functions are nonlinear. For design problems, the structure of the initial model (i.e. the nonlinear equations and inequations) is lost therefore.

Stochastic methods like Genetic Algorithms (GA), Simulated Annealing, etc., give an approximation of the global optimum, but g _i and h _j functions are mainly integrated as penalty functions (i.e. within the objective function) and the structure of the design problem is lost again.

The adopted CSP optimization principle to minimize the value of a criteria f is: f is a real variable, equals to a constraint expression, representing the criteria to be minimized. The key point is to solve by dichotomy a sequence of CSPs in which the set of constraints increases from one CSP to the next. Each CSP is solved by a BP procedure. At each step, a constraint expressing that the next CSP has to be better than the current one according to the minimization of f is added. The process stops when a CSP minimizes f according to a required precision ϵ (see Fig. 2). The optimal solution given by this algorithm is a global solution.

The methodology could be extended to multi-objective optimization through the 𝜖-constraint or the weighting factors methods [14,15].

3. The transformer benchmark design problem

This paper is illustrated with a transformer benchmark optimal design problem, which has already been treated with several optimization techniques by other research teams [16,17].

Fig. 2.

Minimizing a function value using CSP.

Although transformers are not recent devices in industry applications, a lot of researches concern their design nowadays [18].

This benchmark is particularly interesting since it is a constrained mixed-type variables problem, like it is usually the case in industrial applications. The proposed optimal design methodology can equally be applied to the pre-design of any other mechatronic devices, as soon as an algebraic model is available.

3.1. Description of the benchmark

The full description of the transformer can be found in [17] and at http://l2ep.univ-lille1.fr/come/ benchmark-transformer.html.

The problem is the design of a 230 V/24 V, 200 VA, 50 Hz safety isolating transformer with a power factor equals to 0.8. This mono-objective optimization problem consists in minimizing the total mass M _tot of the transformer according to 7 tabulated design variables and 7 inequality constraints as shown on Fig. 3.

Fig. 3.

Transformer design variables and position of the optimization design problem [17].

Several transformer models are available on the website, but analytical algebraic models should be selected during the pre-design steps to consider the complex requirements. The analytical model used is the one with the Kapp hypothesis, strongly coupling two lumped models: a thermal model and an electromagnetic model (see Fig. 4, [16]). This multiphysic coupling is a major difficulty in designing a transformer as it leads to implicit equations, those cannot be solved sequentially. The mathematical model presents algebraic loops (the input of one model is the output of the other) that require the use of implicit solver to simulate the corresponding non-linear behaviour [16].

Fig. 4.

Thermal and electric models of the transformer [17].

This design includes mixed-type variables with values that must be chosen in manufacturer catalogues. Enumerated tables, as presented in catalogues, are a type of constraint that is not new in many domains, but is hardly managed in electrical engineering because they mix continuous and integer variables.

3.2. Optimization using Interval and constraint-based methods

The design problem was solved here with a mixed solver called Constraint Explorer (CE) from Dassault Aviation. CE source code cannot be freely shared but further details can be found in [19].

3.2.1. Constraint Modeling

Modeling equations come either from physical laws (compatibility, equilibrium or constitutive laws) or from the definition of intermediate variables. Design variables are the degrees of freedom of the design problem, e.g. the independent variables, whereas intermediate variables, e.g. dependent variables, are defined through alias (intermediate expressions or definitions). Most of the time, modeling equations is a mix of physical laws and aliases.

Constraint modeling consists in rewriting every mathematical relashionship of the problem ( =, <, ≤, >, ≥, ≠), involving at least one variable, into constraints such as: y = f (x, z) → f (x, y, z) = 0, for an equality constraint, with x, y, z, 3 design variables or aliases.

A causal analysis of the analytical model through CE showed that every variable of the transformer but the design variables set $\{a,b,c,d,S_{1},S_{2},n_{1}\}$ are dependent variables and their values are indirectly set as soon as the design set is instantiated.

3.2.2. Specification constraints

Specification constraints can represent several distinct types of information:

Variables fixed by the specifications of a working condition can be considered as fixed parameter for the optimization procedure. For example: the primary RMS voltage: V ₁ = 230 V.

Working requirements define normalized specification constraints (equality or inequality constraints): y ≥ y _min becomes $\frac{y}{y_{\text{min}}}-1\geq 0$ .

3.2.3. Constraints tables

Usually, optimization problems involve non mathematical specifications such as choices in manufacturer catalogues. In common deterministic and stochastic methods, catalogues are not modeled but processed with programming loops: at each iteration step, an optimization problem is generated and solved.

CSP modeling and solving provide a range of global constraints that can be substituted to common sets of constraints. This substitution replaces a set of constraints by a unique constraint on many variables: a constraint table. Then, propagation algorithms can be used to enforce constraints consistency.

Constaints tables usually are presented as a table and constitute a complex data structure gathering all the admissible combinations between several design variables.

Each line of a constraint table establishes a tuple of consistent values. If one or several values of a constraint variable become inadmissible during the CSP solving process, the whole tuple related to this value is removed from the table.

Here, the core sizes a, b, c and d of the transformer form a constraint table. The set of these 4 design variables is named by the new design variable type.

Table 1 reports the first possible values of the sizing parameters a, b, c and d. If, for example, during the propagation process, a is imposed to 0.0225 then, lines 4,5,6, … are removed from the table, giving a contraction of the domain for the variable type. Then, the domains for variables b, c and d are consequently reduced. This type of constraint table modeling significantly improves the performance of the solving process. Moreover, with such a table, as soon as the value of type is set, the value of a, b, c and d are immediately known. The design variables are then reduced to type, S ₁, S ₂ and n ₁.

Table 1
Catalogue of admissible geometries of iron sheet sizes and corresponding type number

Type a b c d

1 0.0225 0.095 0.04 0.077

2 0.0225 0.095 0.04 0.0668

3 0.0225 0.095 0.04 0.057

4 0.02 0.08 0.035 0.0617

5 0.02 0.08 0.035 0.0923

6 0.02 0.08 0.035 0.0517

⋮ ⋮ ⋮ ⋮ ⋮

Type	a	b	c	d
1	0.0225	0.095	0.04	0.077
2	0.0225	0.095	0.04	0.0668
3	0.0225	0.095	0.04	0.057
4	0.02	0.08	0.035	0.0617
5	0.02	0.08	0.035	0.0923
6	0.02	0.08	0.035	0.0517
⋮	⋮	⋮	⋮	⋮

3.3. Numerical results of the Interval based optimization using CE

Every computation in CE is realized at the computer maximal precision, i.e. 10⁻¹⁴. The machine used is a classical Windows 7 HP Z800 Workstation with 32 GB of RAM. The corresponding CPU data are the following: Intel(R) Xeon(R) X560 @ 3.33 GHz, 3326 MHz, with 6 cores and 12 logic processors. Secondly, it is possible to adjust manually the global accuracy of the intervals bisection, i.e. the size of the interval under which a variable is considered instantiated. This global cutting accuracy was settled at 10⁻⁴. Furthermore, if it is also desired to refine the precision of some specific variables, this can also be done manually on the solver interface. However, regarding this transformer design problem, this option considerably slowed down the computation and thus, was not used. To finish, the optimization problem results have been truncated either to match the other teams results from the benchmark, or to match physical meaning (as for the section of the secondary coil S ₂).

First, the solving process started with minimizing the total mass such as M _tot ≤ 20 kg. This value was presumed wide enough to find solutions and be technically realisable.

During the solving process, causal analysis allowed us to pinpoint an issue in the problem setting. The algorithm detected that there was no solution to this problem, because it was over-constrained.

Herewith, some constraints had to be released to be able to solve the resulting relaxed true mixed-type design problem.

CE causality tree (or constraint graph) became a very useful tool to select the most suitable constraint(s) to release [15].

Thus, the output RMS current value I ₂ was relaxed from the initially equality constraint I ₂ = 8 A to I ₂ ≥ 8 A. This inequality is in fact more realistic to common requirements for electrical industrial devices. This change in the specification of I ₂ explains its value in Table 3.

A first constraints propagation led to a first domains reduction (second column of Table 2).

Table 2
Numerical results

Parameters Initial 1st propagation Optimal solution

Design variables

Type (index of the set $\{a,b,c,d\}$ ) [1,62] [1,62] 27

S ₁ (m²) $\{0.05515e-6,\ldots ,19.635e-6\}$ $\{0.05515e-6,\ldots ,19.635e-6\}$ 2.463e −7

S ₂ (m²) $\{0.05515e-6,\ldots ,19.635e-6\}$ $\{0.05515e-6,\ldots ,19.635e-6\}$ 2.54e −6

n ₁ (turns) [200,1200] [200,1200] 611

Performance parameters

M _tot(kg) [0.00671, 2371.413] [0.0070, 499.46] 2.599

Parameters	Initial	1st propagation	Optimal solution
Design variables
Type (index of the set $\{a,b,c,d\}$ )	[1,62]	[1,62]	27
S ₁ (m²)	$\{0.05515e-6,\ldots ,19.635e-6\}$	$\{0.05515e-6,\ldots ,19.635e-6\}$	2.463e −7
S ₂ (m²)	$\{0.05515e-6,\ldots ,19.635e-6\}$	$\{0.05515e-6,\ldots ,19.635e-6\}$	2.54e −6
n ₁ (turns)	[200,1200]	[200,1200]	611
Performance parameters
M _tot(kg)	[0.00671, 2371.413]	[0.0070, 499.46]	2.599

Table 3

Details of the optimal solution found with the CE solver (M _tot = 2. 599 kg)

Design variables
Type (index of the set $\{a,b,c,d\}$ )	a (mm)	b (mm)	c (mm)	d (mm)	S ₁ (mm²)	S ₂ (mm²)	n ₁ (turns)
27	18	54	18	33.5	0.2463	2.54	611
Intermediate variables
	R ₁ (Ω)	I ₁ (A)	R ₂ (Ω)	I ₂ (A)	n ₂ (turns)
	10.105	0.97778	0.14778	8.0288	69
Specifications
Variables	$\frac{{\Delta}V_{2}}{V_{2}}$	$\frac{I_{10}}{I_{1}}$	𝜂	T _iron (°C)	T _copper (°C)	f ₁	f ₂
Requirements	≤0.1	≤0.1	≥0.8	≤100	≤120	≤0.5	≤0.5
Values	0.08224	0.09835	0.8758	99.783	109.457	0.309648	0.361327

Afterwards, we tried to decrease the value of M _tot with the optimization algorithm proposed in the previous section (see Fig. 2). The best result was found requiring M _tot ≤ 2. 6 kg. Once all the design variables were instantiated, every alias could be deduced. Thus, the only optimal safety transformer is obtained for M _tot = 2. 599 kg (last column of Table 2).

The optimal results computed by CE are reported in Table 3. Computation times depend on the performance of the computer but, in any case, are less than few minutes on standard laptops. Computed on our fixed machine gave a result in less than 8 s.

3.4. Performances and comparison with benchmark results

The performance of our optimal results can be carefully compared with others found in the literature using different optimization techniques (see Table 4, [17]).

Table 4
Results of the other optimisation techniques [17] compared to our results (CE software, Dassault Aviation)

Design variables Continuous optim. SQP Pro@Design EE GA BB CE

a (mm) 12.92 18 18 18 18 18

b (mm) 50.12 54 54 54 54 54

c (mm) 16.61 18 18 18 18 18

d (mm) 43.26 33.5 33.5 33.5 33.5 33.5

S ₁ (mm²) 0.325 0.3318 0.2827 0.2376 0.2827 0.2463

S ₂ (mm²) 2.912 2.835 2.27 2.835 2.27 2.54

n ₁ (turns) 640.77 722 610 614 611 611

Optimisation criteria

M _total(kg) 2.311 2.840 2.594 2.633 2.594 2.599

Time, t (s) 4 38 14 ×10⁶ 3 ×10³ 816 ≤8

Constraints (requirements and specifications)

T _copper (°C) 108.80 103.60 109.20 106.25 109.20 109.45

T _iron (°C) 100.00 94.19 99.58 97.13 99.58 99.78

𝜂 (%) 89.55 88.54 87.60 88.04 87.60 87.58

$\frac{I_{10}}{I_{1}}$ (%) 10.00 4.90 9.98 9.58 9.98 9.83

$\frac{{\Delta}V_{2}}{V_{2}}$ (%) 6.91 8.20 8.20 7.86 8.20 8.22

f ₁ (%) 50.00 49.29 35.50 30.20 35.50 30.96

f ₂ (%) 50.00 47.56 32.20 37.98 32.20 36.13

Design variables	Continuous optim. SQP	Pro@Design	EE	GA	BB	CE
a (mm)	12.92	18	18	18	18	18
b (mm)	50.12	54	54	54	54	54
c (mm)	16.61	18	18	18	18	18
d (mm)	43.26	33.5	33.5	33.5	33.5	33.5
S ₁ (mm²)	0.325	0.3318	0.2827	0.2376	0.2827	0.2463
S ₂ (mm²)	2.912	2.835	2.27	2.835	2.27	2.54
n ₁ (turns)	640.77	722	610	614	611	611
Optimisation criteria
M _total(kg)	2.311	2.840	2.594	2.633	2.594	2.599
Time, t (s)	4	38	14 ×10⁶	3 ×10³	816	≤8
Constraints (requirements and specifications)
T _copper (°C)	108.80	103.60	109.20	106.25	109.20	109.45
T _iron (°C)	100.00	94.19	99.58	97.13	99.58	99.78
𝜂 (%)	89.55	88.54	87.60	88.04	87.60	87.58
$\frac{I_{10}}{I_{1}}$ (%)	10.00	4.90	9.98	9.58	9.98	9.83
$\frac{{\Delta}V_{2}}{V_{2}}$ (%)	6.91	8.20	8.20	7.86	8.20	8.22
f ₁ (%)	50.00	49.29	35.50	30.20	35.50	30.96
f ₂ (%)	50.00	47.56	32.20	37.98	32.20	36.13

Comparing the different methods applied allows to pinpoint their relevance to this type of optimization pre-design problem, as well as highlight the problem setting and the problem solving aspects of a complex design problem.

Several optimal design methodologies have been tested for the mono-objective problem:

Combinatorial optimization: –

Pro@Design software (http://designprocessing.free.fr/?lang=en);

–

Exhaustive Enumeration (EE) method solved by the Matlab distributed computing toolbox;

–

Branch & Bound method (BB) with a modified relaxation method;

–

Genetic Algorithms (GA) with a specific encoding.

Continuous optimization: –

Sequential Quadratic Programming (SQP) (modified fmincon (Matlab Optimization Toolbox)).

Table 4 shows that all inequalities constraints were respected by all the methods, but those results are to be considered with care: they are not really comparable, as the continuous relaxation of some of the other methods led to a different design problem.

Many publications consider that all design parameters can be approximated by continuous variables. It may be helpful sometimes as a first approximation [20] but most times, it is not acceptable for real industry applications or for complex specifications as architecture design or components to be chosen in pre-defined catalogues.

SQP results show that this method is not relevant in terms of optimality for mixed-type problems, as the obtained solution is far from the optimum. These gradient-based techniques indeed have high convergence rates and are extremely fast, but do not manage discrete-type variables or non-convex problems. Moreover, such continuous approximations may violate specifications constraints and then make the pseudo-optimal approximated solution, a non-admissible real solution. Thus, four inequality constraints became active : on T _iron, $\frac{I_{10}}{I_{1}}$ , f ₁ and f ₂, unlike one or two with the other solving methods.

As a commercial software, very few can be found about Pro@Design code. The software appears to be a combination between combinatorial and continuous optimization techniques. Although computation time is relatively short, the optimum obtained is ”poor”, especially regarding n ₁, that was probably continuously approached by the solver.

EE ensures the global solution for purely discrete problems, while our solution is very close to the EE one, the applied EE method is not a true combinatorial optimizer and presents a huge computational time (here 23 days with 8 computers). Besides, n ₁ is different while it should be the same as there is only one possible solution to this optimization problem. Here, two inequalities constraints were almost active: T _iron, $\frac{I_{10}}{I_{1}}$ .

GA manage mixed-type variables but the generated random candidate solutions usually are initially far from the optimum, as a stochastic exploration do not explore the full search space or garanty the full constraint satisfaction and have extremely low convergence rates.

The BB algorithm computes nearly the same global solution as the EE method but with a smallest CPU time. However, the BB algorithm stops at the first found node but could pursue the tree exploration to find the optimal solution [21].

As explained above, this problem could not have a solution as it appeared to be over-constrained. Contrary to our method, this over-constraint state was not detected by the other solving procedures, as they found some outcomes. This fact was probably hidden by the continuous approximation of n ₁, completed by the other solvers. Indeed, difficulties may have been met to solve implicit non-linear mixed-type equations. Besides, regarding the problem solving, the methodology used in this paper did not need either to convert discrete variables into continuous ones or the analytical model to be invertible, as the equations are not solved sequentially. However, one inequality constraint is also almost active: T _iron. Actually, the thermal parameter T _iron appears to be the most constraining constraint, no matter what the used method is, except for Pro@Design, that does not give the optimal solution.

Compared to the other results on this benchmark, not only is the proposed method relevant regarding computation time, even for strongly constrained and mixed-type variables problems, and also concerning the optimality of the solution.

Adopting the CSP formalism makes it possible to ensure, in a deterministic way, that every design constraint will be satisfied by all solutions returned by the algorithm.

4. Conclusion

This paper presents the main challenges related to preliminary design in electrical engineering.

In this paper, a number of new challenging aspects are faced with, including the application field of the method (electrical engineering) and the type of problems considered, i.e. mixed-type problems and the presence of many constraints and variables. Furthermore, the difficulty related to the problem setting and its impact on the problem solving take part in these challenges to cope with, i.e the problem structure, the acausal point of view, dealing with ill-posed and over-constraints problems as well as the management of catalogues and non-linear constraints.

The high number of parameters may also constitute another limitation in the understanding of the design problem and thus, tools such as a causal analysis can be helpful by separating the degrees of freedom from the other dependant variables (further details in [15]).

A design methodology based on CSP and interval methods is proposed to meet the current lack of tools for pre-design.

The use of a-causal models, with modeling physical equations and requirements as constraints, makes easier the description and setting of the design problem. Indeed, corresponding implicit equations may be handled using symbolic computation and AI tools. These tools and data structures greatly facilitate the description of the problem design. It makes possible the reusability of models and libraries for similar design problems. Constraint-based models present inherent a-causal structure that expedites the setting of the design problem. Furthermore, a synthesis approach is highly suitable for pre-design whereas an analysis approach can be adopted thereafter for detailled design.

The importance of setting the problem properly may suggest further the need to introduce a specific design language to describe the problem, as proposed in [22].

The presented methodology has been successfully applied to the design of a transformer benchmark. Those appeares relevant for optimal pre-design. It duly notes multiple requirements and specifications, as well as mixed-type variables problems and non-linear design mathematical relashionships. The methodology is also efficient regarding computation time compared to the other tested methods.

Footnotes

Acknowledgements

The authors would like to thank the Hauts-de-France Region and the European Regional Development Fund (ERDF) 2014/2020 for the funding of this doctoral grant relative to the project HETSPEC. This work was partly funded by the French Research National Agency (Project NEMRO, ANR-14-CE17-0013-01).

References

Messine

Nogarede

and Lagouanelle

J.-L.

, Optimal design of electromechanical actuators: a new method based on global optimization, IEEE Trans. on Magnetics 34(1) (1998), 299–308.

Fitan

Messine

and Nogarede

, The Electromagnetic Actuator Design Problem: A General and Rational Approach, IEEE Trans. on Magnetics 40(3) (2004) 5.1579–1590.

Mazhoud

Hadj-Hamou

Bigeon

and Remy

, The electromagnetic actuator design problem: An adapted interval global optimization algorithm, IEEE Transactions on magnetics 48(2) (2012), 387–390.

Zimmer

and Zablit

, Global aircraft predesign based on constraint propagation and interval analysis, in: CEAS Conference on Multidisciplinary Aircraft Design and Optimization, Köln, Germany, 2001.

Yvars

P. A.

, A CSP approach for the network of product lifecycle constraints consistency in a collaborative design context, Engineering Application of Artificial Intelligence (EAoAI) 22(6) (2009).

Tsang

, Foundations of Constraint Satisfaction, Academic Press, London and San Diego, 1993.

Russell

S. J.

and Norvig

, Artificial Intelligence: A Modern Approach, Prentice Hall, 2010.

Moore

, Interval Analysis, Prentice-Hall, 1966.

Davis

, Constraint propagation with interval labels, Artificial Intelligence 24(3) (1987).

10.

Benhamou

and Granvilliers

, Handbook of Constraint Programming, ch. 16, Continuous and Interval Constraints. Elsevier, 2006, pp. 571–604.

11.

Meyer

and Yvars

P. A.

, Optimization of a passive structure for active vibration isolation: an interval-computation- and constraint-propagation-based approach, Engineering Optimization 44(12) (2012), 1463–1489.

12.

Benhamou

McAllester

and Pascal Van Hentenryck

, Clp(intervals) revisited, in: International Symposium on Logic Programming Bruynooghe

(ed.), Ithaca, New York, USA, 1994, pp. 124–138.

13.

Hansen

and Walster

, Global Optimization Using Interval Analysis, 2nd edn, Marcel Dekker, New York, USA, 2004.

14.

Yvars

P. A.

, Pareto bi-criterion optimization for system sizing: A deterministic and constraint based approach, in: International Conference On Engineering Design, Iced11 15–18 August 2011, Technical University Of Denmark, 2011.

15.

Diampovesa

Hubert

and Yvars

P. A.

, Nouveaux outils de synthèse pour la conception préliminaire et le choix d’architecture: Application à un moteur brushless, in: Symposium de Génie Électrique (SGE 2018), 3–5 Juillet 2018, Nancy, France, 2018.

16.

Ben Ayed

and Brisset

, Multidisciplinary optimization formulations benefits on space mapping techniques, The International Journal for Computation and Mathematics in Electrical and Electronic Engineering (COMPEL) 31(3) (2012), 945–957.

17.

Tran

T. V.

Brisset

and Brochet

, Combinatorial and multi-level optimization of a safety isolating transformer, International Journal of Applied Electromagnetics and Mechanics (IJAEM) 26(3, 4) (2007), 201–208.

18.

Eleftherios

I. A.

Tsili

M. A.

Paparigas

D. G.

and Kladas

A. G.

, Global transformer design optimization using deterministic and nondeterministic algorithms, IEEE Trans. on Industry applications 50(1) (2014), 383–394.

19.

Zimmer

Anglada

Christie

and Grandvillers

, Constraint explorer: a modelling and sizing tool for engineering design, in: Metamodelling and Constraint Based Problem Solving for Embodiment Design. Support Systems in SCI, Orlando, USA, 2004.

20.

Wurtz

Kuo-Peng

and De Carvalho

E. S.

, The concept of imaginary machine for design and setting of optimization problems: Application to synchronous generator, in: XXth Int. Conf. on Electric Machine (ICEM’2012), Marseille, France, September 2–5, 2012, pp. 1461–1466.

21.

Brisset

, Démarches et outils pour la conception optimale des machines électriques. Habilitation à diriger des recherches, Université des sciences et techniques de Lille, Lille, France, 18 décembre 2007.

22.

Yvars

P. A.

and Zimmer

, Deps: Un langage pour la spécification de problèmes de conception de systèmes, in: 10ème Conférence Francophone de Modélisation, Optimisation et Simulation- MOSIM14 5 au 7 Novembre 2014, 2014.

Optimal design for electromagnetic devices: A synthesis approach using intervals and constraint-based methods

Abstract

Keywords

1. Introduction

2.1. Constraint Satisfaction Problem

2.2. Interval computation and Branch & Prunes algorithms

Definition 2.1 (Contractor).

2.3. Branch and Prune algorithms

2.4. Optimization and CSP

3. The transformer benchmark design problem

3.2.1. Constraint Modeling

3.2.2. Specification constraints

3.2.3. Constraints tables

Table 1 Catalogue of admissible geometries of iron sheet sizes and corresponding type number Type a b c d 1 0.0225 0.095 0.04 0.077 2 0.0225 0.095 0.04 0.0668 3 0.0225 0.095 0.04 0.057 4 0.02 0.08 0.035 0.0617 5 0.02 0.08 0.035 0.0923 6 0.02 0.08 0.035 0.0517 ⋮ ⋮ ⋮ ⋮ ⋮

Footnotes

Acknowledgements

References

Table 1
Catalogue of admissible geometries of iron sheet sizes and corresponding type number

Type a b c d

1 0.0225 0.095 0.04 0.077

2 0.0225 0.095 0.04 0.0668

3 0.0225 0.095 0.04 0.057

4 0.02 0.08 0.035 0.0617

5 0.02 0.08 0.035 0.0923

6 0.02 0.08 0.035 0.0517

⋮ ⋮ ⋮ ⋮ ⋮