##
470248 Multi-Rate Moving Horizon Estimation for an Electric Arc Furnace Steelmaking Process

Due to the harsh operating conditions, EAFs lack measurements and most of the states are not directly measured. Another issue arises due to the measurements being taken at different sampling rates. State knowledge is essential in real-time control applications. Very limited applications of state estimation for EAF operation have been reported. The extended Kalman filter (EKF) was applied in [1] to the refining stage of EAF operation but the model had 4 states, which are insufficient to capture the detailed process dynamics. An EKF was used in [2] to identify the arc current parameter for obtaining the electrical properties of the EAF load. However, the EAF model only involved the power system. A constrained multi-rate EKF was implemented in [3] to estimate the states of EAF system using plant measurements. The EKF showed an acceptable performance in tracking the true states of the process, even in the presence of parametric plant-model mismatch. Although different versions of the Kalman filter such as the EKF, unscented Kalman filter etc. are employed by some researchers, moving horizon estimation (MHE) is gaining popularity due to the ability to handle constraints and to use computationally efficient numerical optimization algorithms [4]. In a previous work, we presented a parameter estimation based multi-rate MHE framework for EAF operation [5]. The MHE problem consists of solving a nonlinear dynamic optimization problem subject to the nonlinear model under consideration and some other constraints specified by the user. It uses a finite set of past available measurements to reconstruct the full state of the process, thus keeping the optimization problem numerically tractable. The use of a finite size window of measurements by MHE provides a straightforward way to include measurements with different sampling rates [6].

In this work, we present a rigorous multi-rate MHE solution strategy for the EAF process. A novel initialization method for MHE problems based on a background solve to improve solution time is also introduced and implemented. The optimization-based strategy is particularly suitable for applications incorporating large-scale differential-algebraic equation (DAE) systems. The study is conducted using a first-principles dynamic EAF model developed originally in [7], in which the EAF is partitioned into four zones (gas, solid scrap, slag and molten metal). Chemical equilibrium is assumed within the slag and gas zones, with mass and energy transfer across the zone interfaces. The MHE problem built around the highly nonlinear model is solved using direct dynamic optimization approaches to estimate states for an EAF heat (batch). We provide an overview and a description of the MHE application that includes the multi-rate measurements handling, initialization of the online MHE problem, and investigation of different DAE optimization paradigms in the solution of the MHE optimization problem. The performance of the MHE under different conditions and using different optimization strategies is illustrated through application to several case studies. Additionally, avenues for future research will be identified and some perspective provided on the real-time application of the proposed strategy as a decision support tool for the operators of EAFs.

References

S. Billings, F. Boland and H. Nicholson, "Electric arc furnace modelling and control,"

*Automatica,*vol. 15, no. 2, pp. 137-148, 1979.F. Wang, Z. Jin, Z. Zhu and X. Wang, "Application of extended Kalman filter to the modeling of electric,"

*Neural Networks and Brain, 2005. ICNN\&B'05. IEEE International Conference,*vol. 2, pp. 991-996, 2005.Y. Ghobara, "Modeling, Optimization and Estimation in Electric Arc Furnace,"

*Master's thesis,*no. McMaster University, 2013.F. Allgöwer, T. A. Badgwell, J. S. Qin, J. B. Rawlings and S. J. Wright, "Nonlinear predictive control and moving horizon estimation. An introductory overview,"

*Advances in Control,*no. Springer, pp. 391--449, 1999.S. Shyamal and C. L. E. Swartz, "A Multi-rate Moving Horizon Estimation Framework for Electric Arc Furnace Operation," in

*Proc. 11th DYCOPS-CAB, (accepted)*, Trondheim, Norway, 2016.R. López-Negrete and L. T. Biegler, "A moving horizon estimator for processes with multi-rate measurements: A nonlinear programming sensitivity approach,"

*Journal of Process Control,*vol. 22, no. 22, pp. 677-688, 2012.R. D. MacRosty and C. L. E. Swartz, "Dynamic modeling of an industrial electric arc furnace,"

*Ind.Eng.Chem.Res.,*vol. 44, pp. 8067-8083, 2005.

**Extended Abstract:**File Not Uploaded

See more of this Group/Topical: Computing and Systems Technology Division