6. Define the design and stochastic spaces

In this section, the characterization of the model parameters and design variables is described. The design variables are variables that are controllable by the designer within a certain range. This range should be provided, to shape the search space for the optimizer. A model parameter corresponds to a parameter that usually cannot be controlled by the designer (e.g. the cost of a photovoltaic panel), or the decision on this parameter value is fixed (e.g. a fixed amount of photovoltaic panels on the roof). Such a parameter can be considered deterministic or uncertain. The design variables and model parameters are characterized in two files: design_space.csv and stochastic_space.csv. These files are present in the case folder for the specific case considered (e.g. CASES\H2_FUEL for the H2_FUEL case). In design_space.csv, the deterministic values for the model parameters and the range for the design variables are provided. In stochastic_space.csv, the uncertainty is allocated to the specific model parameters and design variables. When a deterministic design optimization is performed, only the design_space.csv file is required. In the other cases, i.e. uncertainty quantification and robust design optimization, both files are required.

6.1. The design_space.csv file

In the design_space.csv file, the design variables and the model parameters which need a quantification in your model are defined. When performing uncertainty quantification, the design_space.csv file consists only of model parameters. In the case of deterministic design optimization or robust design optimization, the design_space.csv file requires design variables. Additionally, if some model parameters require a quantification outside the model, the design_space.csv file includes both Characterizing the design variables and Characterizing the model parameters.

6.1.1. Characterizing the design variables

To define a design variable, the set-up in the design_space.csv file is as follows:

name,feature_type,lb,ub

where:

  • name: name of the variable;

  • feature_type: a design variable is indicated with var;

  • lb: lower bound value for the design variable;

  • ub: upper bound value for the design variable.

An example for a configured design variable des_var_1 with a range between 10 and 50 is:

des_var_1,var,10,50

6.1.2. Characterizing the model parameters

In this file design_space.csv, the deterministic value (or mean value when the parameter is considered uncertain) is provided. The configuration of a model parameter is similar to the configuration of a design variable:

name,feature_type,value

where:

  • name: name of the variable;

  • feature_type: a parameter is indicated with par;

  • value: deterministic value (or mean value when the parameter is stochastic).

An example of a configured model parameter par_1 with a mean value of 0.03 is:

par_1,par,0.03

6.1.3. Example of design_space.csv

An example of a configured design_space.csv file, which consists of 3 model parameters (par_1, par_2 and par_3) and 2 design variables (design_var_1 and design_var_2), is presented:

design_var_1,var,1,3
design_var_2,var,10,100
par_1,par,4
par_2,par,2.5
par_3,par,175

6.2. The stochastic_space.csv file

The uncertainty on the design variables and model parameters can be allocated through the file stochastic_space.csv. This file is required when performing robust design optimization and uncertainty quantification, where several parameters are subjected to uncertainty. For every design variable and model parameter defined in design_space.csv, an uncertainty can be defined.

6.2.1. Characterizing the uncertainties

Defining the uncertainty can be done as follows:

name,abs_rel,distribution,deviation

where:

  • name: name of the parameter or variable, equal to the name of the parameter or variable in design_space.csv;

  • abs_rel: absolute or relative uncertainty to the mean, defined with absolute or relative, respectively;

  • distribution: The distribution of the uncertainty;

  • deviation: uncertainty on the mean.

The following distributions are available:

  • Uniform

  • Gaussian

The meaning of deviation at the end of the line depends on the distribution. When a Uniform distribution is considered, the deviation refers to the absolute (or relative) difference between the upper bound of the Uniform distribution and the mean: for \(\mathcal{U}(a,b)\), \(deviation = (b-a)/2\)). When a Gaussian distribution is considered, the value corresponds to the standard deviation: \(\mathcal{N}(mean,deviation)\). Keep always in mind that the mean value of the design variables is the deterministic value assigned by the optimizer in each iteration. In the case of model fixed parameter the mean value corresponds to the fixed value as it is assigned in the model definition.

An example of a configured uncertain parameter par_2, characterized by a Uniform distribution with a \(\pm 1\) deviation from the mean value:

par_2,absolute,Uniform,1

Note that it is not required to allocate an uncertainty to every design variable and model parameter defined in design_space.csv. In other words, when a parameter (or variable) is defined in design_space.csv, but not in stochastic_space.csv, the parameter (or variable) is considered deterministic. Moreover, it is not necessary to keep the same order of appearance of parameters and variables in design_space.csv stochastic_space.csv files.

6.2.2. Example of stochastic_space.csv

In summary, a stochastic_space.csv file corresponding to the illustrative design_space.csv example file in Example of design_space.csv looks like this:

par_1,relative Gaussian,0.5
par_2,absolute Uniform,1
design_var_2,relative,Uniform,0.1

Where the model parameter par_3 and design variable design_var_1 are considered deterministic, par_1 is characterized by a Gaussian distribution with a relative standard deviation of 0.5 (i.e. \(\mathcal{N}(4,2)\)), par_2 is characterized by a Uniform distribution with an absolute deviation of 1 (i.e. \(\mathcal{U}(1.5,3.5)\)) and design_var_2 is characterized by a Uniform distribution with a relative deviation of 0.1. For design_var_2, the actual Uniform distribution depends on the mean value selected by the optimizer for each evaluated design.