4.2.3. Additional Stability Improvements and Model Acceleration

In addition to the pressure oscillations, certain unusual cases exist where masses would reach small negative values causing the model to crash. This was rectified by using the 'NonNegative' option of the ODE15s solver to insure the necessary adjustments of time-steps when masses approach zero with a steep gradient and the adjustment of some empirical model parameters, especially in several hyperbolic tangent functions that reduce chemical reaction rates when the mass of a reactant is small. Furthermore, the model was accelerated by making the integration more efficient. The ODE15s solver numerically calculates a Jacobian matrix that leads to a high number of function evaluations with a single variable changing. By detecting this behavior and using previous results for expensive analytical calculations, such as the determination of chemical activities and radiative heat transfer, if none of the relevant inputs have changed, the simulation speed is increased significantly. Furthermore, the number of function evaluations could be reduced by using the 'JPattern' option. Overall this made the simulation more stable and significantly faster [39]. Crashes or non-physical results such as negative masses have not been observed with the new model, even with unrealistic inputs and extreme conditions.

### **5. Results and Discussion**

Validation of the EAF process model using extensive measured process data from a 140 t direct current (DC) furnace can be found in previous publications where the thermochemistry of the gas phase [34] and liquid phases [39], the behavior of different carbon carriers [33], the heat transfer [32], as well as the performance of the complete model [37], were evaluated thoroughly, which is not reproduced here. The model has also been tested with data from additional furnaces including both alternating current (AC) and DC technology and has given satisfactory results after adjustment of the

empirical parameters during preliminary research. Instead, various simulations where both were run in automatic control and simulator mode were set up to evaluate the model stability and speed as well as its capability to reproduce existing operation strategies and adjust them for an altered set of boundary conditions.
