Optimization problem definition (Electrical Machine)

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In general, optimization means to find the best solution for a problem under the consideration of given constraints and it does not mean to select the best out of a number of given solutions. In other words the definition of an optimum is:

Define a pointtmpF-65_thumbwith the independent variablestmpF-66_thumb tmpF-67_thumbin such a way that by their variation inside the admissible space the value of a quality functiontmpF-68_thumbreaches a maximum or a minimum. The pointtmpF-69_thumbis described as the optimum.

This definition in mathematical terms: Minimise a quality function

tmpF-75_thumb

considering

tmpF-76_thumb

The gj are called inequality and thetmpF-77_thumbequality constraints. Any constraint can be determined in one of these forms. Constraints represent limitations on the behaviour or performance of the design and are called behaviour or functional constraints, whereas physical limitations on the design variables (e.g. availability, manufacturability) are known as geometric or side constraints. If an optimisation problem with only inequality constraintstmpF-78_thumb(Fig. 7.2) is considered, all sets of values x that satisfy the equationtmpF-79_thumbform a (JV-l)-dimensional hyper-surface

of the design surface, the constraint surface. The constraint surface splits the design surface into two basic regions: the feasible or acceptable region withtmpF-80_thumb, and the infeasible or unacceptable region with tmpF-81_thumb, If, during the progress of the optimisation, a design vector lies on a particular constraints surface, this constraint is called an active constraint.

Constraint surfaces in a hypothetical two-dimensional design space, with side constraintsand behaviour constraints

Fig. 7.2. Constraint surfaces in a hypothetical two-dimensional design space, with side constraintstmpF-88_thumband behaviour constraintstmpF-89_thumb

The independent variables are the design parameter or object variables. Fig. 7.3 shows the shape of a two-dimensional quality function with the global optimum and difficulties such as saddle points and’ local extremum.

Quality function with two object variables.

Fig. 7.3. Quality function with two object variables.

To obtain commensurable criteria for the generation of the design variations and to support a simplified formulation of the stopping criteria of the algorithm, the design variables should be transferred into a normalised form:

tmpF-95_thumb

wheretmpF-96_thumbis the original parameter with its given physical dimension,tmpF-97_thumb the lower bound of the parameter variation range, whiletmpF-98_thumbdenotes the actual parameter variation range. If no lower or upper bound of the parameter is given, the design variable can be normalised to its initial valuetmpF-99_thumb

The appropriate formulation of the quality function represents a particular problem. All design aims must be formulated in this single function and all object variables must be implemented. Multiobjective optimisation extends the optimisation theory by permitting multiple objectives to be optimised simultaneously. It is also known under different names, such as Pareto optimisation, vector optimisation, efficient optimisation, multicriteria optimisation, etc. One way of formulating a single objective function is a weighted linear combination of the q different objective functions:

tmpF-104_thumb

wheretmpF-105_thumbdenotes a weighting factor best formulated with the properties

tmpF-107_thumb

andtmpF-109_thumbare the individual objective functions. In practice, the choice of the weighting factors may already influence the result of the optimisation. It is often not straightforward to select a single fixed weighting factor for each objective, especially if the objective function is erroneous or if no particular preference is given to one of the objectives.