# Measurement and Model-Based Control of Solidification in Continuous Casting of Steel Billets
## Appendix A  
The implemented procedure for dynamic control of the crater end position can be summarised by the following steps to be carried out within each cyclic calculation of the strand state (the names of the involved variables and parameters are explained in Section 3.3.2):  
* After achievement of defined minimum casting length: Initialise target casting parameters based on regression Formula (7) for given target crater end position with default reference value.
* Check stationary casting conditions, i.e., variations of casting speed, spray water flows and simulated crater end position are within defined tolerances.
* Instationary state \(\rightarrow\) keep target casting parameters from previous cyclic calculation.  
* Stationary state and \(\mid\)Apos_CE \(\mid\) within defined tolerance: \(\rightarrow\) keep actual casting parameters: SPW_i_tar = SPW_i_v_c_tar = v_c_* Stationary state and \(|\)Apos_CE\(|\) outside defined tolerance: \(\rightarrow\) calculate target casting parameters from regression Formula (7):  
* d_SPW + a
* pos_CE_ref: i. d_pos_CE_tar \(\geq\) d_pos_CE_min and d_pos_CE_tar \(\leq\) d_pos_CE_max: d_SPW_tar = b/(2c)
- (b\({}^{2}\)/(4c) + d_pos_CE_tar/c)\({}^{1/2}\) d_v_c_tar = 0 ii. d_pos_CE_tar < d_pos_CE_min: d_SPW_tar = d_SPW_max d_v_c_tar = (d_pos_CE_tar
- d_pos_CE_min)/a iii. d_pos_CE_tar > d_pos_CE_max: d_SPW_tar = d_SPW_min d_v_c_tar = (d_pos_CE_tar
* Check correction of target casting parameters based on comparison with actual ones: \(\Delta\)SPW = 100
* SPW_tar)/SPW_ref, \(\Delta\)v_c = v_c
- v_c_tar with thresholds \(\varepsilon_{\text{SPW}}\), \(\varepsilon_{\text{v\_c}}>0\) to be exceeded in case of significant difference \(|\)Apos_CE\(|>\varepsilon_{\text{CE}}\): i. \(\Delta\)pos_CE\(|>\varepsilon_{\text{CE}}\) and \(\Delta\)SPW >
- \(\varepsilon_{\text{SPW}}\) and \(\Delta\)v_c < \(\varepsilon_{\text{v\_c}}\): d_SPW_tar\({}^{\prime}\) = d_SPW_tar + \(\varepsilon_{\text{SPW}}\), d_v_c_tar\({}^{\prime}\) = d_v_c_tar if d_SPW_tar\({}^{\prime}\) > d_SPW_max: d_SPW_tar\({}^{\prime}\) = d_SPW_max, d_v_c_tar = d_v_c_tar
- \(\varepsilon_{\text{v\_c}}\) ii. \(\Delta\)pos_CE <
- \(\varepsilon\)CE and \(\Delta\)SPW < \(\varepsilon\)SPW and \(\Delta\)v_c >
- \(\varepsilon\)v_c: d_SPW_tar\({}^{\prime}\) = d_SPW_tar
- \(\varepsilon_{\text{SPW}}\), d_v_c_tar\({}^{\prime}\) = d_v_c_tar if d_SPW_tar\({}^{\prime}\) < d_SPW_min: d_SPW_tar\({}^{\prime}\) = d_SPW_min, d_v_c_tar\({}^{\prime}\) = d_v_c_tar + \(\varepsilon_{\text{v\_c}}\) iii. In the remaining cases, no correction is necessary: d_SPW_tar\({}^{\prime}\) = d_SPW_tar, d_v_c_tar\({}^{\prime}\) = d_v_c_tar
* Calculate new target spray water flows and casting speed: SPW_i_tar = SPW_i_ref
* (1 + d_SPW_tar\({}^{\prime}\)/100) v_c_tar = v_c_ref + d_v_c_tar\({}^{\prime}\)  
* SPW_i_tar v_c_tar\({}^{\text{av}}\) = \(\alpha^{\ast}\)v_c_tar\({}^{\text{av}}\)prev + (1
- v_c_tar where SPW_i_tar\({}^{\text{av}}\)prev and v_c_tar\({}^{\text{av}}\)prev are the respective average values of the previous cyclic step.  
The developed dynamic process control for the ESF caster to adjust the crater end position to its given target value uses the moving average values for the target casting speed and spray water flow rates as set-points to be displayed to the operator.