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Concepts and terminology

Mathematical formulation🔗

When working on calibration problems following concepts are relevant:

Measurement data (\(MD\)) – includes values of system parameters, inputs and outputs that are set and observed in the physical laboratory settings. Measurement data may contain several measurements for the same settings that give different results due to measurement errors and general uncontrolled variability of the system. Measurement data may contain variables that are not directly present in the model and require transformation before it can be applied.

Operating conditions (\(OC\)) – contains values of simulation parameters and inputs of a model mapped or calculated based on measurement data.

Reference output (\(RO\)) – contains expected values for simulation output variables of a model mapped or calculated based on measurement data without considering model equations.

Simulation result (\(SR\)) - contains values for simulation output variables calculated based on given inputs and outputs. We assume that simulation results are reduced to scalar KPIs in the model. In case of dynamic processes where transitive effects are relevant that KPIs may be calculated by, e.g., by some additional reduction such as running average or min/max value in certain time intervals.

Calibration parameters (\(CP\)) or correction factors – model parameters that can be adjusted to reach agreement between simulation results and reference output.

Given the above definitions calibration oven \(N\) operating conditions can be formulated as an optimization problem:

\[ \text{argmin}_{CP} \sum_{i=1}^{N} w_{i}^{OC} \, \text{norm}_{W^R} \left[ SR(CP, OC_i) - RO(OC_i) \right] \]

Where \(\text{norm}_{W^R}\) is either absolute or squared weighted norm over different results and reference diffs. \(w_{i}^{OC}>0\) are user specified weights assigned to the different operating conditions.

Calibration library numerical specifics🔗

Weights🔗

In Calibration library, the weights are assigned as follows:

  • \(\text{norm}_{W^R}\) - the squared weighted norm is used.
  • \(w_{i}^{OC}>0\) - all operating conditions are weighted equally, i.e. \(w_{i}^{OC} = 1\).

Calculation of \(SR\) and \(RO\)🔗

In Calibration library, the simulation result (\(SR\)) is always fetched as the last result point for a given analysis. This means that if a dynamic simulation is used, only the last result point will be used for the comparison with the reference output.

Furthermore, the results are normalized using the nominal attributes from the model to normalize the contribution of each variable.

Calibration library terminology🔗

To perform the calibration in Modelon Impact a number of concepts have been introduced in the Calibration library. It is important to understand what they are and how they relate to the above mathematical formulation to work efficiently with the library.

The below concepts are used as input to the Calibrate-function.

Reference variables🔗

These are the variables from the model for which the difference against the values found in the reference result are to be minimized. In the above formulation, they are the outputs that are compared between the simulation result and the reference output (\(SR - RO\)).

Calibration parameters🔗

These corresponds directly to the Calibration parameters (\(CP\)) in the above formulation.They are model parameters that are subject to change from the calibration algorithm.

Baseline experiment🔗

The experiment to be calibrated. The output of the model is not only dependent on the model itself, but also the analysis and settings used to simulate as well. The analysis and settings used to produce the baseline experiment will be re-used during calibration. Corresponds to simulation used to produce \(SR\).

Reference result🔗

A result which contains the reference variables to be calibrated. Operating conditions (\(OC\)) are applied as parametrization to each case of the reference result. The reference output (\(RO\)) is taken from this result.

This is a multi-sim result with one case per operating condition (\(OC\)). There are three ways to generate a reference result: