# Measurement and Model-Based Control of Solidification in Continuous Casting of Steel Billets
## 3 Results
### Vibrometer Measurements of Crater End Position at ESF
#### 3.2.2 Vibrometer Measurement Campaign with Varying Casting Speeds  
By increasing the casting speed, the crater end position is expected to move through a fixed measurement position of the laser vibrometer mounted at a distance of about 16-19 m below the meniscus. For that purpose, the measurement conditions were chosen as follows:  
* Continuous measurement at 17.5 m below the meniscus with seven casting speeds increasing from \(\mathrm{v_{c}}^{1}\) to \(\mathrm{v_{c}}^{7}\). * Two 30 min measurements after changes in casting speed to measure during transitional and stationary conditions. The pre-processing steps were similar to the first measurement campaign, but additionally, FFTs with 10, 30 and 60 s measurement intervals were analysed. In order to detect the passing of the crater end at the measurement position, unsupervised learning methods in terms of clustering approaches were used for this campaign, which are supposed to reveal significant differences in the frequency spectrum of fully solidified strands and those with a liquid core. The number of optimal clusters was derived using the elbow method with the classical K-means algorithm, which led to a total of four clusters. Three clustering algorithms were applied overall:  
* K-means;
* Gaussian mixture model;
* Hierarchical clustering with Ward linkage. The results are depicted in Figures 8-10 for FFTs with a 10 s measurement interval, as this configuration revealed the best results. The upper figures show the number of measurement points (i.e., 10 s intervals) that each identified cluster contains, and the bottom figures show the temporal development of the cluster prediction during the measurement campaign. The casting velocity was increased incrementally towards the end of the campaign. Thus, it was expected that the crater end moves during the campaign from a position before the vibrometer, where the strand at the vibrometer is completely solidified, to a position behind the vibrometer, where the strand at measurement position has a liquid core. Figure 7: Confusion matrix for first measurement campaign based on test data set. The K-means algorithm clusters data by trying to separate samples in different groups of equal variance, minimising a criterion known as the inertia. This algorithm requires the number of clusters to be specified. It scales well to a large number of samples and has been used across a wide range of application areas in many different fields. A Gaussian mixture model is a probabilistic model that assumes that all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters. One can think of mixture models as generalising K-means clustering to incorporate information about the covariance structure of the data as well as the centres of the latent Gaussians. Hierarchical clustering is a general family of clustering algorithms that build nested clusters by merging or splitting them successively. This hierarchy of clusters is usually represented as a tree (or dendrogram). The root of the tree is the unique cluster that gathers all the samples, the leaves being the clusters with only one sample. Ward linkage minimises the sum of squared differences within all clusters. It is a variance-minimising approach and in this sense similar to the K-means objective function but tackled with an agglomerative hierarchical approach. Figures 8-10 show that all three approaches detect  
* Transient effects during the beginning of the measurement campaign with significant population of more than two clusters, which reflects the associated unsteady strand conditions of the start of casting phase;
* A transition in population of the two most frequently predicted clusters after a measurement time between 16,200 and 18,000 s (4.5-5.0 h, indicated by red vertical lines), where the casting speed increased from \(\mathrm{v_{c}}^{5}\) to \(\mathrm{v_{c}}^{6}\). Figure 10: Size of clusters and temporal evolution of cluster predictions using hierarchical clustering with Ward linkage.