*2.4. Simulated Annealing*

The basic principle of the methaheuristic optimization method of SA according to [23] describes the quality of a possible solution or an individual *x* by means of a fitness function *f* . The goal of the optimization is to minimize the fitness *f*(*x*). Starting from an initial solution *x*0, a random selection of new individuals *x*k+<sup>1</sup> within the solution space is performed. A new individuum is accepted and not discarded if one of the constraints

$$f(\mathbf{x}\_{k+1}) \le f(\mathbf{x}\_k) \tag{3}$$

$$\text{cor}\quad P(f(\mathbf{x}\_{k+1}) - f(\mathbf{x}\_k)) = \boldsymbol{\varepsilon}^{-\frac{f(\mathbf{x}\_{k+1}) - f(\mathbf{x}\_k)}{\boldsymbol{I}\_k}}\tag{4}$$

is satisfied. Thereby (4) describes, based on Boltzmann statistics, the probability *P* with which an intermediate result of the optimization may deteriorate. This realizes the possibility to leave a local optimum in the search for the global optimum. For this purpose, a temperature *T*<sup>k</sup> is introduced, which is successively reduced over the iterations. The lower the temperature, the lower the probability of accepting a worse solution. Accordingly, the algorithm converges with the "cooling" of the solution space.

#### Application of SA for the IM Optimization

The initial solution *x*<sup>0</sup> of the SA process in the presented IM optimization process is a roughly designed machine geometry. The rough design is based on problem-specific design parameters, such as the rated power, and is computed according to [29]. An individual of the SA optimization is defined by a chromosome. This contains all geometry parameters described in the optimization variables. The optimization variables are determined by the parameter selection approach in [27].

In the SA, as well as in the ES and PS method, the parameters of the chromosome are changed, resulting in new designs of the IM. The new machine designs are calculated, like the initial solution according to the chromosome and [29]. The evaluation of an individual based on its chromosome is done by a fitness function, which will be discussed in Section 2.6.

In this work, new individuals are determined using a normally distributed random vector - *X*, whose normalization results in the random vector*y*

$$
\vec{X} \sim \mathcal{N}(0, 1) \in \mathbb{R}^n \tag{5a}
$$

$$\vec{y} = \frac{\vec{X}}{||\vec{X}||'}\tag{5b}$$

with an expected value of zero and a standard deviation of one. The number of dimensions is equal to the number of optimization parameters *n*. From the chromosome*x*<sup>k</sup> of the current individual and the updated temperature *T*k, the new individual is determined by

$$\vec{\pi}\_{\mathbf{k}+1} = \vec{\pi}\_{\mathbf{k}} + \sqrt{T\_{\mathbf{k}}} \frac{\vec{\pi}\_{\mathbf{k}}}{||\vec{\pi}\_{\mathbf{k}}||} \* \vec{y}\_{\mathbf{}} \tag{6}$$

where operator (∗) describes an element-wise vector multiplication. The temperature adaptation is performed according to the Boltzmann annealing by

$$T\_{\mathbf{k}} = \frac{T\_0}{\ln k'} \tag{7}$$

since this method guarantees global convergence for sufficiently large starting temperatures *T*<sup>0</sup> [30]. However, since only very high starting temperatures ensure global convergence, but these result in low convergence rates, the technique of Very Fast Re-Annealing (VFR) is used. This increases the temperature after a given number of iterations to avoid the method converging to local minima [31].

If one or more dimensions of the resulting chromosome lie outside the solution domain, they are set to the upper or lower boundary conditions, depending on which boundary was violated. This results in the chromosome*x*∗ <sup>k</sup>+<sup>1</sup> which, in a convex combination with the current chromosome, realizes a valid solution

$$\vec{\text{x}}\_{\mathbf{k}+1} = a\vec{\text{x}}\_{\mathbf{k}+1}^{\*} + (1 - a)\vec{\text{x}}\_{\mathbf{k}} \tag{8}$$

via a random variable *α* evenly distributed between zero and one. The from (6) or (8) resulting chromosome can subsequently be accepted or rejected analogous to the criteria described in Section 2.4.

By considering the step-wise cooling temperature in the description of the new chromosome, local convergence is realized. The termination criterion of the SA is a given number of iterations at which the change in fitness of the best individual is less than a defined tolerance.
