*2.5. Hybrid Optimization: Evolution Strategy and Pattern Search*

The stochastic method used in the hybrid optimization is the ES according to [24]. The method is based on populations of individuals in the solution space, in which a generation with *μ* parents creates *λ* children via crossover and mutation. A distinction is made between the plus strategies (*μ* + *λ*), in which a new generation can be composed of parents and children, and the comma strategies (*μ*, *λ*), which only consider the best descendants for the next generation of parents [24]. The method used in this work is based on the comma strategy, as this introduces a maximum "lifetime" of the individuals and thus counteracts premature local convergence. The method is structured in an initialization, a selection, a mutation, a crossover and an inheritance process. The method, especially in the case of mutation and crossover, is problem-specific. Gaussian distributions can be used as a statistical basis. Their variance is flexibly adapted as a function of various parameters in order to achieve good local convergence while at the same time enabling a global search. This achieves the already mentioned stable convergence in the local group. Due to the generation principle, there is the inherent possibility of parallelization, which results in a reduced computing time compared to other stochastic optimization methods [32].

The deterministic method of the hybrid optimization in this work is the PS method. In contrast to other deterministic optimization methods, PS as a direct optimization method does not require a gradient of the fitness function. In addition to the associated numerical stability avoiding the use of derivatives, a local search for problems that are neither continuous nor differentiable is enabled [1]. In this way, a fast local convergence can be realized especially for complex optimization tasks with problem-specific boundary conditions. In this work, the method according to [1] is used.

## 2.5.1. Application of the Hybrid Optimization Method for the IM Optimization

The implementation of the two optimization methods is as previously described. In the following, the implemented crossover and mutation of individuals based on random distributions in the ES optimization will be discussed. For this purpose, normal distributions are used in the presented optimization environment, whose standard deviations and expected values are adjusted via various parameters depending on the situation.

#### Crossover

Crossover of individuals is performed using the (2,1)-strategy. From two parents separated by their chromosomes*x*p1 and*x*p2, by means of the convex function

$$\vec{\mathbf{x}}\_{\mathbf{c}} = \vec{\mathbf{x}}\_{\mathbf{p}1} + \left(\vec{\mathbf{x}}\_{\mathbf{p}2} - \vec{\mathbf{x}}\_{\mathbf{p}1}\right) \* \vec{X} \tag{9}$$

with the normally distributed random vector

$$\vec{X} \sim \mathcal{N}(\mu, \sigma) \in \mathbb{R}^n,\tag{10}$$

a descendant with the chromosome*x*<sup>c</sup> is generated. The expected value *μ* and the standard deviation *σ* are thereby influenced by different, parent- and population-specific factors. The aim of the crossover is to project the properties of the parent with the lower fitness value *f*(*x*p), more onto the descendant. For this purpose, a fitness factor:

$$FF = \begin{cases} \frac{f(\vec{\mathbf{x}}\_{\mathbb{P}^1})}{f(\vec{\mathbf{x}}\_{\mathbb{P}^2})} & f(\vec{\mathbf{x}}\_{\mathbb{P}^1}) < f(\vec{\mathbf{x}}\_{\mathbb{P}^2})\\ \frac{f(\vec{\mathbf{x}}\_{\mathbb{P}^2})}{f(\vec{\mathbf{x}}\_{\mathbb{P}^1})} & f(\vec{\mathbf{x}}\_{\mathbb{P}^2}) < f(\vec{\mathbf{x}}\_{\mathbb{P}^1}) \end{cases} \tag{11}$$

is introduced, which is derived from the fitness values of the parents and shifts the expected value to the parent with the lower fitness value. Since a larger distance between the two parents increases the uncertainty of how the fitness behaves in the solution space between the parent chromosomes, a distance factor

$$DF = 1 - \frac{d\_{\rm P12}}{d\_{\rm max}} \tag{12}$$

is introduced. This describes the normalized relative deviation *d*p12 of the two parents relative to the maximum occurring value *d*max of all parent pairs in the current generation. This factor shifts the expected value toward the parent with the lower fitness when the parents are far apart and reduces the standard deviation, counteracting the uncertainty in the space between. The closer an individual's fitness is to the minimum fitness *f*(*x*min) of a parent in the current generation, the lower the variation of the descendant should be to allow local convergence. This is achieved by an overall fitness factor:

$$OFF = \begin{cases} 1 - \frac{1}{2} \frac{f(\overline{\mathfrak{X}}\_{\text{min}})}{f(\overline{\mathfrak{X}}\_{\text{p1}})}, & f(\overline{\mathfrak{X}}\_{\text{p1}}) < f(\overline{\mathfrak{X}}\_{\text{p2}}) \\\ 1 - \frac{1}{2} \frac{f(\overline{\mathfrak{X}}\_{\text{min}})}{f(\overline{\mathfrak{X}}\_{\text{p2}})}, & f(\overline{\mathfrak{X}}\_{\text{p2}}) < f(\overline{\mathfrak{X}}\_{\text{p1}}) \end{cases} \tag{13}$$

which affects both the expected value and the standard deviation. The expected value of the random vector - *X* is given by

$$\mu = \begin{cases} \frac{1}{2} \cdot FF \cdot DF \cdot OFF, & f(\vec{\mathfrak{x}}\_{\mathbb{P}1}) < f(\vec{\mathfrak{x}}\_{\mathbb{P}2}) \\\ 1 - \frac{1}{2} \cdot FF \cdot DF \cdot OFF, & f(\vec{\mathfrak{x}}\_{\mathbb{P}2}) < f(\vec{\mathfrak{x}}\_{\mathbb{P}1}) \end{cases} \tag{14}$$

which shifts the expected value of the descendant's chromosome toward the parent with the lower fitness value as a function of the factors introduced. The resulting standard deviation

$$
\sigma = \sigma\_{\text{crossover}} \cdot DF \cdot OFF \tag{15}
$$

considers an adjustable maximum standard deviation *σ*crossover in addition to the factors described. This should be set as a function of the elasticities of the optimization parameters in order to respond to large elasticities with a lower variance, thereby counteracting larger jumps through the solution space and thus improving the convergence behavior in the local group.

## Mutation

Mutation of a selected parent with the chromosome*x*<sup>p</sup> is performed using a normally distributed random vector analogous to (10). The descendant's chromosome*x*<sup>c</sup> is calculated by

$$\vec{\mathfrak{X}}\_{\mathbb{C}} = \vec{\mathfrak{X}}\_{\mathbb{P}} \* \left(1 + \vec{X}\right),\tag{16}$$

where operator (∗) describes an element-wise vector multiplication. Here, the random vector has an expected value of *μ* = 0, whereas the standard deviation, analogous to the crossover of individuals, is adapted depending on various parameters. Through these factors, local convergence is to be realized in particular. For this purpose, the overall fitness factor

$$OFF = 1 - \frac{1}{2} \frac{f(\vec{\mathbf{x}}\_{\text{min}})}{f(\vec{\mathbf{x}}\_{\text{P}})} \tag{17}$$

analogous to (13) is introduced. This reduces the variance as a function of the fitness value. In addition to this, with

$$GF = 1 - \frac{\mathcal{g}\_{\text{k}}}{\mathcal{g}\_{\text{max}}} \tag{18}$$

a generation factor is defined for the current generation *g*k, relative to the maximum number of generations *g*max. This factor reduces the standard deviation across generations, which also supports local convergence of mutant descendants.

The resulting standard deviation of the random vector describing the mutation follows accordingly with

$$
\sigma = \sigma\_{\text{Mutation}} \cdot GF \cdot OFF. \tag{19}
$$

The adjustable maximum standard deviation of the mutation *σ*mutation should be adapted as in the case of the crossover depending on the elasticity of the optimization parameters.

#### *2.6. Fitness Function*

The fitness function is identical for all stages of the optimization environment. It assigns an individual *x* a fitness value *f*(*x*) based on its chromosome, considering geometry and thermal constraints as well as a given driving cycle. This fitness value thereby also categorizes unacceptable machine geometries, depending on the number of fulfilled boundary conditions and the achievable points of the driving cycle. By this categorization, inadmissible solutions can be of different fitness, which improves the convergence behavior of the optimization methods and, in particular, realizes a faster search for admissible solutions. The maximum fitness of an individual is given by

$$f\_{\text{max}} = 2 \cdot \eta\_{\text{BC}} \cdot \eta\_{\text{DC}\_{\text{V}}} \tag{20}$$

where *n*BC represents the number of constraints to be considered and *n*DC represents the number of points in the driving cycle. This allows a clear differentiation and thus categorization of the fitness values. The procedure to evaluate the fitness of an individual is based on a sequential process:


$$f(x) = \frac{f\_{\text{max}}}{n\_{\text{BC,Geo}}(x)}.\tag{21}$$


$$f(\mathbf{x}) = \frac{f\_{\text{max}}}{n\_{\text{BC,Geo}}},\tag{22}$$

with the number of all geometry boundary conditions *n*BC,Geo.

5. Determination of the individual fitness of an admissible individual by means of

$$f(\mathbf{x}) = \frac{1}{\sum\_{i} w\_{i}} \sum\_{j} w\_{\mathbf{j}} \frac{p\_{\mathbf{j,indiv}}}{p\_{\mathbf{j,ref}}}.\tag{2.3}$$

where *p*j,indiv are weighted problem-specific decision parameters, *p*j,ref are the decision parameters of a reference machine, and *w*<sup>i</sup> is the sum of the weighting factors. In case of an invalid solution due to the non-achievement of several operating points in the driving cycle, an additional penalty term

$$f\_{\text{penalty}}(\mathbf{x}) = \frac{f\_{\text{max}}}{2n\_{\text{BC}}} \upsilon\_{\text{DC}}(\mathbf{x}) + \Delta\_{\text{offset}} \tag{24}$$

is considered. Here, *v*DC(*x*) stands for the share of the not reachable operating points in the total number of operating points of the driving cycle and Δoffset for an offset to separate the invalid solutions from the allowed ones.
