put modelling optimization and control of an electric arc

Modelling, Optimization and Control of an Electric Arc Furnace

Modelling, Optimization and Control

of an

Electric Arc Furnace

by

Richard MacRosty, M.Sc (Eng)

A Thesis

Submitted to the School of Graduate Studies

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

McMaster University

c© Copyright by Richard MacRosty, September 20, 2005

DOCTOR OF PHILOSOPHY (2005) McMaster University

(Chemical Engineering) Hamilton, Ontario, Canada

TITLE: Modelling, Optimization and Control

of an Electric Arc Furnace

AUTHOR: Richard D.M. MacRosty, M.Sc(Eng)

(University of Cape Town, South Africa)

SUPERVISOR: Dr. C.L.E Swartz

NUMBER OF PAGES: xiv, 154

ii

ABSTRACT

The main objective of this research was to develop methods to optimally operate an

industrial electric arc furnace (EAF). EAFs are widely used in the steel industry to

recycle scrap steel. Simply stated, steel recycling involves melting down the scrap

metal using both chemical and electrical energy sources and making adjustments to

the chemistry. EAFs are complex processes involving limited automation and are

typically operated based on what has worked well in most situations; however, this

is not necessarily optimal.

The contributions from this work can be divided into three main sections, with each

successive section building on the developments from the previous sections. The first

component of the project involved the development of a comprehensive model, which

makes it possible to explore the complex multivariate interactions and subtle relation-

ships of the process. The next step incorporated the model within an optimization

framework, where the optimal operating input profiles for a range of different con-

ditions and objective functions were explored. The final step was to implement a

nonlinear feedback controller that updated the input profiles online.

The first section of work describes the development of a dynamic model for an indus-

trial steelmaking EAF. The model is sufficiently detailed so as to describe the melting

process and chemical reactions, and account for reagent and energy additions. The

lack of knowledge of reaction mechanisms due to the complex nature of the reacting

system is overcome by modelling the process as a system of equilibrium zones with

mass transport limitations. An important objective in developing this model is to

ensure that it can be used within an optimization framework, thus particular consid-

eration is given to the continuity of model equations and the robustness of the model

over a wide range of conditions. Parameter estimation is carried out using industrial

data and the model performance illustrated through simulation studies.

iii

Development of a computational procedure for determining optimal operating strate-

gies for the EAF involves the incorporation of the dynamic model within a math-

ematical optimization framework to determine the optimal input trajectories. The

optimization is based on an economic criterion and process limitations can be ac-

counted for by including them into the optimization problem as constraints. This

optimization procedure enables trade-offs between the process inputs to be made so

as to maximize profit. The use of mathematical optimization to enhance process

performance is illustrated through a number of case studies.

Process disturbances, model-mismatch and other sources of uncertainty may cause

the nominally optimal profile to be sub-optimal or even cause process infeasibilities

when applied to the actual process. Feedback control in the form of a nonlinear

model predictive controller (NMPC) was implemented to address this issue. The

NMPC algorithm uses an economic objective function and re-optimizes the input

profiles during the batch based on the most recent measurements from the process.

Several case studies were carried out to illustrate the effectiveness of the algorithm

at reducing the effects of model uncertainty.

iv

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to Dr Christopher Swartz for his

enthusiasm, wealth of ideas and continual support throughout the course of this

project. I have learnt much under his guidance and have benefited from the many

opportunities he has given me to extend myself.

I am also grateful to Drs John MacGregor and Gordon Irons for their valuable ideas

and support for this work.

Thanks are extended to the members of the Process Systems Research Group who co-

habited the penthouse at various stages during my stay at McMaster. In particular, I

would like to thank Anthony Balthazaar, Mark-John Bruwer, Benoit Cardin, Kevin

Dunn, Adam Warren, Androniki Zavitsanou and Danielle Zyngier for making my

PhD experience more than what is contained in the pages of this thesis.

I would also like to thank Michael Kempe of Dofasco, John Tomson of the McMaster

Steel Center and Pierre Bruchet of Air Liquide for their involvement with this project.

This thesis is dedicated to my wife, Sarah, to whom I am indebted for her constant

support, patience and so much more.

v

Table of Contents

1 Introduction 1

1.1 Operation of the Electric Arc Furnace . . . . . . . . . . . . . . . . . . 2

1.2 Motivation and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Modelling the Electric Arc Furnace 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Detailed Mathematical Model . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Material Balances . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vi

2.4.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.2 Available Measurements . . . . . . . . . . . . . . . . . . . . . 45

2.4.3 Handling the Raw Data . . . . . . . . . . . . . . . . . . . . . 46

2.4.4 Rigorous Parameter Estimation . . . . . . . . . . . . . . . . . 47

2.5 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5.1 Scenario 1. Base Case . . . . . . . . . . . . . . . . . . . . . . 53

2.5.2 Scenario 2. Effect of preheat duration on final melting time. . 57

2.5.3 Scenario 3. Effect of carbon lancing on the slag composition. . 59

2.5.4 Scenario 4. Effect of lancing strategies on slag foaming perfor-

mance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6 Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3 Process Optimization 63

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Optimization of the Electric Arc Furnace . . . . . . . . . . . . . . . . 64

3.3 Optimization of Differential-Algebraic Equation Systems . . . . . . . 67

3.4 Formulation and Implementation of the EAF Optimization Problem . 70

3.4.1 Numerical Robustness . . . . . . . . . . . . . . . . . . . . . . 73

3.4.2 Model Discontinuities . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.3 Path Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 76

vii

3.5 Process Optimization Case Studies . . . . . . . . . . . . . . . . . . . 77

3.5.1 Case O-1: Optimal Solution . . . . . . . . . . . . . . . . . . . 78

3.5.2 Case O-2: Cost of Power . . . . . . . . . . . . . . . . . . . . . 81

3.5.3 Case O-3: Increased Upper Bound on Power Input . . . . . . 86

3.5.4 Case O-4: Comparison of Objective Criteria . . . . . . . . . . 88

3.5.5 Case O-5: Fixed Preheat Duration . . . . . . . . . . . . . . . 94

3.5.6 Case O-6: Event-activated Constraint Formulation . . . . . . 97

3.5.7 Comparison of Scenarios . . . . . . . . . . . . . . . . . . . . . 102

3.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Nonlinear Model Predictive Control 106

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2 Control of Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.1 EAF Control Applications . . . . . . . . . . . . . . . . . . . . 111

4.2.2 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . 112

4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.3.1 Algorithm and Software Implementation . . . . . . . . . . . . 119

4.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4.1 Case MPC-1: Model Uncertainty . . . . . . . . . . . . . . . . 122

4.4.2 Case MPC-2: Process Disturbance . . . . . . . . . . . . . . . 127

viii

4.4.3 Case MPC-3: Unknown Initial State . . . . . . . . . . . . . . 130

4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Conclusions and Recommendations 138

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.2 Recommendations for Further Work . . . . . . . . . . . . . . . . . . . 139

5.2.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.2.3 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

References 144

A Model Details 155

A.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.2 Non-ideal Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 157

A.3 View Factor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 158

A.4 Melting model derivation . . . . . . . . . . . . . . . . . . . . . . . . . 159

ix

List of Figures

2.1 Schematic of EAF model. . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Hyperbolic tangent function used to incorporate the effect of slag depth. 28

2.3 Stages of the melting used in the radiation model. . . . . . . . . . . . 33

2.4 Coefficients for regression model from the sensitivity analysis. . . . . 44

2.5 Off-gas predictions (mole fractions). . . . . . . . . . . . . . . . . . . . 51

2.6 Endpoint slag composition prediction. . . . . . . . . . . . . . . . . . . 52

2.7 Scenario 1: Furnace input profiles. . . . . . . . . . . . . . . . . . . . . 54

2.8 Scenario 1: Solid scrap and liquid steel profiles. . . . . . . . . . . . . 55

2.9 Scenario 1: Offgas composition profiles. . . . . . . . . . . . . . . . . . 56

2.10 Scenario 1: Slag composition profiles. . . . . . . . . . . . . . . . . . . 57

2.11 Scenario 1: Radiative heat transfer in the furnace. . . . . . . . . . . . 58

2.12 Scenario 2: Scrap melting. . . . . . . . . . . . . . . . . . . . . . . . . 59

2.13 Scenario 3: Carbon injection strategy. . . . . . . . . . . . . . . . . . . 60

2.14 Scenario 4: Slag foaming. . . . . . . . . . . . . . . . . . . . . . . . . . 61

x

3.1 Case O-1: Offgas data. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.2 Case O-1: Input profiles. . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3 Hourly energy cost and demand for Ontario, Canada: March 10, 2005. 83

3.4 Case O-2: Input profiles for comparing scenarios A and B. . . . . . . 84

3.5 Case O-3: Input profiles. . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.6 Case O-3: Wall temperature. . . . . . . . . . . . . . . . . . . . . . . . 87

3.7 Case O-4: Time intervals included as optimization variables. . . . . . 90

3.8 Case O-4: Comparison of inputs for scenario A. . . . . . . . . . . . . 92

3.9 Case O-4: Comparison of inputs for scenario B. . . . . . . . . . . . . 93

3.10 Case O-5: Comparison of inputs. . . . . . . . . . . . . . . . . . . . . 95

3.11 Case O-5: Burner input for test case. . . . . . . . . . . . . . . . . . . 96

3.12 Switching function as a function of cumulative power. . . . . . . . . . 98

3.13 Event-activated constraint. . . . . . . . . . . . . . . . . . . . . . . . . 99

3.14 Case O-6(A): Event-activated constraint. . . . . . . . . . . . . . . . . 101

4.1 Schematic representation of controller formulation. . . . . . . . . . . 121

4.2 Case MPC-1: Input Profiles compared to nominal inputs . . . . . . . 125

4.3 Case MPC-1: Input profiles compared to optimal inputs. . . . . . . . 126

4.4 Case MPC-1: Controller performance. . . . . . . . . . . . . . . . . . . 128

4.5 Case MPC-2: Input profiles compared to nominal inputs. . . . . . . . 129

xi


4.7 Case MPC-3: Input profiles compared to nominal inputs. . . . . . . . 131


4.9 Case MPC-3: Noise model. . . . . . . . . . . . . . . . . . . . . . . . . 134

4.10 Case MPC-3: Comparison of controller performance. . . . . . . . . . 135

xii

List of Tables

1.1 Energy Balance of the EAF . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Adjustable Model Parameters . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Results of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3 Model Prediction Data . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4 Comparison of Preheat Strategies . . . . . . . . . . . . . . . . . . . . 58

3.1 Comparison of Profit Based on Power Cost Relative to Base Case . . 85

3.2 Comparison of Profit for Cases O-1 and O-3 Relative to Base Case . . 88

3.3 Comparison of Different Objective Criteria Relative to the Base Case 91

3.4 Summary of Case-Studies Relative to Base Case . . . . . . . . . . . . 102

4.1 Comparison of Results for Case Study MPC-1 . . . . . . . . . . . . . 124

4.2 Comparison of Different Control Intervals for Case Study MPC-1: . . 127



xiii

4.5 Comparison of Controller Performance . . . . . . . . . . . . . . . . . 135

A.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

xiv

Chapter 1

Introduction

Electric arc furnaces (EAFs) are widely used in the steel industry for recycling scrap

steel. Simply stated, steel recycling involves melting down the scrap metal and adjust-

ing the chemistry to obtain the desired product grade. The steel is melted using both

chemical and electrical energy sources. In modern furnaces, the electrical power is

added to the furnace via three electrodes which transfer energy to the steel in the form

of an electric arc. The chemical contribution is derived from combustion reactions

taking place in the furnace, fueled predominantly by coke, natural gas and oxygen.

Table 1.1 details a typical energy balance for an ultra high power furnace with data

taken from Fruehan (1998). In the steel industry EAFs are run in batches, termed

heats. While processing conditions vary greatly, a typical heat takes between one

and three hours and consumes approximately 400 kWh/ton of steel Fruehan (1998).

A more recent review of industrial practice by Irons (2005) indicates that modern

furnaces are now consuming less than 300 kWh/ton of electrical energy. The lower

dependence on electrical power is a result of operational improvements and also due

to a greater reliance on energy derived from chemical sources. The energy-intensive

nature of EAFs makes these operations attractive candidates for improvement.

1

PhD Thesis - R.D.M MacRosty, Chemical Engineering Section 1.1

Table 1.1: Energy Balance of the EAF

Source:

Electrical Energy 50-60%

Burners 5-10%

Chemical Reactions 30-40%

TOTAL 100 %

Destination:

Steel 55-60%

Slag 8-10%

Cooling water 8-10%

Miscellaneous 1-3%

Offgas 17-28%

TOTAL 100 %

1.1 Operation of the Electric Arc Furnace

The scrap charge for each heat is comprised of a range of different scrap sources.

The particular mix for each heat is selected based on a number of factors such as

the availability of each scrap source and the desired product grade being produced.

The selection and packing of the furnace will also influence the composition of the

liquid steel and slag over the course of the batch as well as the melting behaviour.

The charge could include lime and carbon and/or these could be injected into the

furnace during the heat. Typically, two or three buckets of scrap are charged per

heat depending on the bulk density of the scrap and volume of the furnace. The

capacity of a bucket is typically of the order of 100 tons. The EAF batch processing

recipe involves a series of distinct stages, specifically charging, preheating, melting

and tapping (emptying of molten steel from the furnace).

The preheat stage involves the combustion of natural gas to raise the temperature of

2


the steel. Following the preheat, the electrodes are lowered into the furnace and the

power is turned on. An intermediate voltage tap is selected to maintain a stable arc

while the electrodes bore into the scrap. The voltage can be increased once a pool of

liquid has formed at the base of the arc. During the initial stages of the meltdown,

a long arc (high voltage) is typically selected. A long arc allows more energy to be

transferred via radiation to the scrap surrounding the arc. This has a more global

heating effect in the furnace than a shorter arc which focuses the majority of its

energy to the base of the arc. As flat bath conditions are approached, a shorter arc

is preferred because the furnace walls are now exposed and energy radiated laterally

is essentially lost.

Towards the end of the heat the slag layer is foamed by injecting C and O2 which react

together and with FeO to form CO gas; the gas bubbles through the slag causing it

to foam. The foaming slag covers the arc thereby protecting the furnace walls from

arc radiation and also improving the power transfer to the steel and hence the energy

efficiency. The oxidation reactions occurring in the furnace also serve as a source

of energy. During the heat phosphorus, sulphur, aluminum, silicon, manganese and

carbon are removed from the steel as they react with oxygen and float into the slag.

These reactions can be controlled by lancing oxygen into the bath. O2 lancing is

typically carried out until the carbon content is at the required level for tapping.

After a certain amount of power has been added to the furnace, the bath temperature

and carbon content are measured. This information indicates what further additions

need to be made to reach the desired endpoint specifications. The measurements are

then repeated to ensure these specifications have been met. Once the desired com-

position and temperature have been obtained in the furnace, the tap-hole is opened

and the molten steel is poured into the ladle for transport to the next operation.

Downstream from the furnace is the ladle furnace, where final temperature and com-

position adjustments are made to the molten steel before casting. The molten metal

3


is transported from the ladle furnace to the continuous casting unit where the steel

can be cast into billets, blooms or slabs depending on the final use of the steel. The

final step in the steelmaking process involves rolling the steel into coils in the finishing

mill.

1.2 Motivation and Goals

EAFs typically involve a relatively low level of automation and rely heavily on opera-

tor involvement. As with many industrial processes, operator experience is invaluable

for the operation of the process. However, this experience can be limited due to the

multivariable interactions and subtle relationships that may be easily overlooked. The

understanding of these complexities is confounded by the small number of useful pro-

cess measurements, which make it difficult to infer the current state of the process.

Therefore, in most situations process operating procedures are based upon what has

worked well in the past. The manner in which reagents, scrap and electric power are

added to the furnace may be carried out in many possible ways.

The overall aim of this work is to provide a method to optimally operate an industrial

EAF. This goal can be divided into three main sections, with each successive section

building on the development from the previous section. Detailed process knowledge,

in the form of a model, makes it possible to take advantage of more complex rela-

tionships to provide information such as finding the optimal balance and timing of

the energy contributions from chemical and electrical sources. Therefore the first

component of the project involves the development of a comprehensive model to en-

able exploration of the complex multivariate interactions and subtle relationships of

the process. The next step requires incorporating the model within an optimization

framework, where the optimal operating input profiles for a range of different con-

ditions and objective functions can be explored. The final step is to implement a

4


nonlinear feedback controller that is capable of updating the input profiles online in

order to account for process disturbances and model uncertainty.

A particular challenge for the optimization and control of the EAF is that it is op-

erated as a batch process. Batch processes are typically characterized by a number

of unique features, which render the methods used for a continuous process unsuit-

able. Batch processes typically operate over a wide range of conditions and therefore

methods suitable for batch processes must be able to account for nonlinear behaviour.

Linear models, which simplify the solution for many of the optimization and control

methods applied to continuous processes, may not be appropriate here. Furthermore,

in batch processes the path that is followed during processing may affect the final

product quality and therefore an appropriate model should be able to predict this

behaviour.

For reasons stated above, the modelling work focused on the development of a non-

linear dynamic model. Furthermore, to be used in an optimization framework the

model should be able to accurately predict the process behaviour over a wide range

of conditions; therefore the model was developed from a fundamental basis. This

model is comprised of both differential and algebraic equations; thus the application

of dynamic optimization methods are required to determine the operating conditions

that would maximize a specified performance criterion. The feedback controller must

also be able to account for the strong nonlinearities and therefore a nonlinear control

method is appropriate.

1.3 Main Contributions

The complexity of the electric arc furnace process has hindered the advancement of

the operation of this process, in terms of both its optimization and automation. The

5


complexity discourages experimentation, leading to operating strategies that follow

a strict recipe-type policy despite changes in process conditions and objectives. The

work presented in this thesis attempts to address these issues by providing tools that

can be used to enhance the process operation.

The first contribution of this thesis is the development of a comprehensive dynamic

model of the process. Full details of two other first-principles based EAF dynamic

models have been found in the open literature (Bekker et al., 1999; Matson and

Ramirez, 1999). The model developed in this work extends previous contributions in

the level of detail captured while remaining computationally tractable. Some of the

key features of the model are the following:

• The model is formulated using detailed material and energy balances.

• The model comprises a series of multi-zone chemical equilibrium reactors that

are limited by mass transfer.

• Chemical equilibrium is handled by including the necessary optimality con-

ditions of the Gibbs free energy minimization directly within the differential-

algebraic model.

• Composition and temperature of the solid scrap, molten-metal, slag and gas

phases are predicted.

• A radiation model accounts for radiative heat transfer within the furnace and

accounts for the dynamics of the surface geometry.

• The model is able to predict the slag foam height and its effect on energy transfer

from the arc.

The second contribution is the development of a flexible mathematical optimization

framework that makes use of the information provided by the model to determine the

6


optimal strategy for operating the process. An advantage of formulating the problem

in this manner is that the objective criterion can be easily changed to compare optimal

operating strategies based on different process objectives. A series of case studies is

presented to illustrate the formulation and solution of the problem under various

scenarios and indicate the potential for process improvement. Analysis of the results

reveals that by optimizing a detailed process model, tradeoffs inherent in the EAF

process operation can be quantitatively accounted for.

The final contribution of this thesis is the development of a nonlinear model predic-

tive control strategy. The algorithm is applied under assumptions of perfect state

information and no restrictions on the computation time. However, the results of this

work are used to provide a theoretical bound on performance that can be achieved

using multivariable feedback control. The case studies that are carried out allow the

effect of uncertainty on the nominal solution to be investigated.

1.4 Thesis overview

Chapter 2 – Modelling the Electric Arc Furnace

Previous and current work pertinent to the modelling of the furnace is reviewed

in this section. This is followed by the development of a comprehensive model of

the electric arc furnace based on fundamental principles. Parameter estimation

is then carried out to match the process to industrial process data.

The model is dynamic and is capable of predicting the dynamic chemistry and

temperature profiles in the solid, molten-metal, slag and gas phases.

Chapter 3 – Process Optimization

Previous optimization studies on electric arc furnaces as well as methods suitable

7


for the optimization of differential-algebraic systems are reviewed. Following

the formulation of a general optimization problem for EAF operations, a series

of case study results is presented. The purpose of these studies is twofold;

firstly to indicate improved operating strategies in the furnace and secondly to

demonstrate the flexibility of optimization as a tool for process improvement.

Chapter 4 – Nonlinear Model Predictive Control

Model predictive control and methods suitable for controlling batch processes

are reviewed. The online control problem is formulated using the nonlinear

model predictive control algorithm over a shrinking horizon. The work illus-

trates the benefits of feedback control in dealing with uncertainty and distur-

bances to improve the optimization performance in an online setting.

Chapter 5 – Conclusions and Recommendations

A summary of this thesis is presented in the final chapter, which highlights the

major results. Avenues for future work are also discussed here.

8

Chapter 2

Modelling the Electric Arc Furnace

This chapter details the development of a comprehensive model of the electric arc

furnace. Prior EAF models are first reviewed, followed by an overview of the model

developed in this study after which a more detailed description with relevant equations

is given. Following this, the parameter estimation of the model using plant data is

presented and some illustrative simulations form the final section of this chapter.

2.1 Introduction

The fundamental mechanisms involved in the electric arc furnace process are relatively

poorly understood due to the complexity and extreme conditions of the process. These

factors make the development and validation of a highly detailed model very difficult

and necessitate the application of simplifying assumptions about the process.

Bekker et al. (1999) developed a model of the furnace from fundamental thermody-

namic and kinetic relationships for the purpose of closed-loop control simulation. The

model comprises of 14 states and six inputs, which the authors categorize as 2 control

9


inputs and 4 disturbances. Empirical relationships and simplifying assumptions are

used to describe mechanisms which are not well understood or measurable. The model

assumes that energy from the arc and heat of reaction is added to the liquid metal

which in turn heats the solid component. The gas phase of the furnace is assumed to

be at the same temperature as the liquid steel. The temperature of the liquid steel

is increased through both the chemical and electrical energy additions, with energy

losses occurring through the furnace wall. The melt rate and temperature of the

solid steel are determined by the rate of heat transfer to the solid component. The

ratio of the scrap temperature to the liquid steel temperature is assumed to be the

fraction of energy that is available to melt the steel, with the remainder used to raise

the temperature of the solid scrap. The important metallurgical reactions that are

considered are those involving oxidation of Fe, C, and Si and reduction of FeO. The

reaction rates of dissolved C and Si in the steel are assumed to be proportional to the

difference between their concentrations and approximate equilibrium concentrations.

The production of FeO is determined by the abundance of C and Si, and also oxygen

injection. The model assumes that all O2 fed into the furnace is consumed in the

oxidation of Fe, C and Si.

The next three models described consider the process as being comprised of equilib-

rium zones with mass transport limitations. This is motivated by arguments that

reactions tend to be transport limited at steelmaking temperatures.

Cameron et al. (1998) developed an EAF model for the purpose of dynamic simulation

that could be used to identify improvements in EAF operating practices. The authors

model the process as four phases (metal, slag, organic solid and gas) and consider six

interfaces between the metal, slag, gas and carbon material. Chemical equilibrium is

then assumed at each interface, from which the reaction products are re-distributed

to the bulk phases. Material flow between the bulk phases and interfaces is driven

by a concentration gradient, with the chemical equilibrium state at the interface

10


determined by minimizing the Gibbs Free Energy. The authors validated the model

using off-gas chemistry data. Limited detail of the model is provided, presumably

due to proprietary reasons.

Matson and Ramirez (1999) developed a model that approximates the furnace as two

separate control volumes. The first control volume includes the bath, slag and a small

amount of gas; the second includes the freeboard gas. The authors assume chemical

equilibrium in the individual control volumes and transport limitations between each

control volume to determine the rate of reaction. In each control volume a dynamic

elemental balance is used to track the flow of material. The equilibrium algorithm

minimizes the Gibbs free energy subject to atom balance constraints and considers

the presence of the elements, C, H, O, N and Fe. The quantity of Ca is assumed to

remain constant. The chemical equilibrium problem is solved via a subroutine at each

integration step. Mass transfer between the control volumes is modelled as diffusion

across a concentration gradient. The scrap is modelled as a collection of spheres with

sensible heating of the spheres determining the temperature profile of each sphere

as a function of its radius. At each time step the surface temperature of the scrap

is monitored to determine whether the subsequent step will be a sensible heating

iteration or a melting iteration. During the melting iteration, the energy is used

to overcome the latent heat of melting and the radius of the spheres consequently

changed. Small discretization steps are required to attain an acceptable level of

accuracy with this method. The authors used iterative dynamic programming to

determine the optimal carbon and oxygen additions and also the optimal batch time.

Modigell and coworkers (Modigell et al., 2001a,b; Traebert et al., 1999) also developed

an EAF model for use as a simulation tool. In this case the process is modelled as

four distinct reaction zones that are assumed to be in a state of chemical equilibrium,

with flow of material between reaction zones governed by concentration gradients

and adjustable mass transfer coefficients. However, few specifics of the model are

11


provided. The model was validated using endpoint data as this was the only data

available to the authors.

In contrast to the above-described models that consider the EAF operation as a

whole, a number of modelling efforts have focused on individual components of the

EAF process. Examples of the latter include electrode models (Meng and Irons, 2000;

Billings et al., 1979; Collantes-Bellido and Gomez, 1997), detailed three-dimensional

models to predict the radiative heat transfer in the furnace (Guo and Irons, 2003),

models to predict slag foaming (Jiang and Fruehan, 1991; Gou et al., 1996) and models

for post combustion in the furnace freeboard (Kleimt and Kohle, 1997; Tang et al.,

2003).

2.2 Model Overview

There were five main requirements in developing the model for the furnace:

1. The model should be based primarily on fundamental principles. A key reason

for this is that the intended use of the model is for optimization studies. Empir-

ical models are reliable only within the range of the data from which they were

identified; this makes them less desirable for optimization applications where

the model may be required to be evaluated over a relatively large decision space.

2. The model should be able to predict the nonlinear dynamic behaviour of the

process over the course of the batch operation.

3. The model should be robust and not susceptible to numerical difficulties. An

important objective of developing this model is to ensure that it can be used

within a dynamic optimization framework to determine operating conditions

that would optimize a specified performance criterion such as minimizing the

12


cost per ton of liquid steel or maximizing production rate. Therefore of par-

ticular importance are the continuity of model equations and the robustness of

the model over a wide range of conditions.

4. The model should be developed in such a manner that the number of parame-

ters to be estimated from industrial data is kept to a minimum. A particular

challenge encountered was the lack of useful data for parameter estimation. The

development of the model therefore involved a balance between minimizing the

number of model parameters requiring estimation and ensuring sufficient model

accuracy. The number of zones chosen to model the EAF (four) was felt to be

an appropriate compromise.

5. A further objective was to build the model in such a way that the structure was

flexible, allowing the substitution/addition of more detailed relationships into

the model.

2.2.1 Model Structure

The EAF is modelled as a system of equilibrium zones with each zone approximating

the behaviour of a section of the furnace:

1. Gas Zone: includes all gas in the freeboard volume, i.e. the free space in the

furnace above the scrap material.

2. Slag-Metal Interaction Zone: includes all the slag material and the portion

of iron interacting with the slag.

3. Molten Steel Zone: consists of all metallic elements in their liquid state

excluding that portion included in the slag-metal zone.

4. Solid Scrap Zone: includes the charged scrap that is still in solid form.

13


Each zone is distinguished by its unique composition and conditions. Chemical equi-

librium is assumed to exist in the slag-metal interaction and the gas zones; this equi-

librium assumption is reasonable if one considers the high temperatures within the

system (Fruehan, 1998; Cameron et al., 1998; Modigell et al., 2001a). The reaction

of material is limited by mass transfer between the zones, where the mass transfer

coefficients are treated as adjustable parameters estimated from industrial process

data. Figure 2.1 is a schematic diagram of the model depicting the mass flows be-

tween the above described zones. The chemical species included in each zone and the

material additions are also illustrated. The energy model considers the radiation and

convective heat transfer taking place between different zones, the furnace components

and the arc. A description of each of the four zones follows.

Gas Zone

The gas zone includes all material in the freeboard volume. The species considered in

this zone are: CO, O2, CO2, CH4, H2, H2O, N2 and C9H20. C9H20 is taken to be an average

composition of all the volatile components that may be present in the scrap and is

assumed to vaporize from the scrap in the initial minutes of charging. CH4 and O2 are

added to this zone via the burners. O2, N2 and H2O are introduced from ingressed air

and water-cooling of the electrodes. CO enters from the partial combustion of carbon

in the slag-metal interaction zone. The components within this phase are assumed

to exist in a state of chemical equilibrium. The relationships for determining the

equilibrium state in the model are presented in Section 2.3.1.

Slag-Metal Interaction Zone

This zone consists of the slag material and a portion of the molten-metal phase with

which it is in contact, including metal droplets in the slag. The species considered

in this zone are: Fe, Mn, Mg, Al, Si, FeO, Fe2O3, MnO, MgO, Al2O3, SiO2, CaO, C, CO,

O2, N2. The presence of CO2 in this zone is neglected by the following argument. The

reaction of carbon particles in the slag is controlled by the formation of a gaseous

14


CO

CO2H2O

H2

CH4

O2

N2

20

HC 9

FeSiMn

Al

[C]

SiO2

Fe

FeO

Si

Mn

MnO

CAlO 23

O2

3Fe

O2

Al

[CaO

MgO

]

Bur

ners

Ent

rain

men

t

O/G

SLA

G−

ME

TA

L

MO

LTE

N M

ET

AL

GA

S Z

ON

E

.

Lanc

e C

oke

Lim

e /

Dol

ime

Lanc

e O

2

SO

LID

SC

RA

PS

crap

[]

[]

CO

��

��

��

��

Fig

ure

2.1:

Sch

emat

icof

EA

Fm

odel

.

15


layer around the carbon particles as they oxidize. This gaseous layer limits transport

of the O2 to the carbon particle and thus any CO2 that is produced is quickly reduced

to CO via the Boudouard equilibrium reaction,

CO2 + C � 2CO. (2.1)

The mechanism of this reaction in the slag is discussed further by Morales et al.

(1997). All components in the slag-metal zone are also assumed to exist in a state of

chemical equilibrium.

This zone is in direct contact with the gas zone and the molten metal zone. O2

enters this zone via lancing and diffusion from the gas phase (according to its partial

pressure). The presence of iron oxides in the zone also increases the availability of

oxygen for components with a sufficient reducing potential. Metallic elements (Fe,

Mn, Mg and Al) and non-metallic elements (C, Si, P and S) enter this phase from the

molten-metal phase. Carbon is also added from injection and roof additions. Lime

and dolime (CaO.MgO) added to the furnace are also included in this zone.

All oxides accumulate in this zone, except for CO which leaves this zone and enters

the gas phase as it is produced. Decarburization and oxidation reactions are the most

important and these are limited by the transport of C and O2 into this zone. Lancing

has a dual effect on this zone. First, it provides for the addition of C and O2 to the

zone and second, it results in an increased mixing effect of this zone with the molten-

metal zone. The increased mixing effect is captured by relating the mass-transfer

coefficient to the volumetric flow of O2 added through the lance. More details of this

model are given in Section 2.3.1. It is assumed that the majority of carbon added to

the furnace via the lance will enter this zone with the remaining carbon added to the

molten-metal zone. The division of lance carbon between these zones is an estimated

parameter that is determined from industrial data.

16


Slag foaming in the furnace is an important phenomenon; it protects the furnace

elements from radiative damage and also improves the efficiency of the transfer of

energy to the steel. The foaming depends on the rate of CO evolving from the bath,

which is controlled by the amount of carbon and oxygen available. Oxygen is available

either as FeO or it is lanced directly into the bath. The slag composition is also an

important factor in slag foaming with the correct basicity and viscosity essential for

obtaining a foaming slag.

Molten-Metal Zone

Material enters this zone from the solid-scrap zone as it melts and leaves to enter

the slag-metal zone according to the transport rate to the slag-metal interface. It is

assumed that no reactions occur in this zone because of the absence of O2. Energy from

the arc is added to the molten-metal zone and energy transfer takes place between

it and the solid-scrap and slag-metal zones, driven by the prevailing temperature

gradient.

The presence of the following components are modelled in this zone: Fe, Mn, Al, Si, C.

Mass transfer of material to the slag-metal zone is driven by natural diffusion and also

forced diffusion as a result of lancing. Knowledge of the initial mass of carbon in the

steel allows the model to predict its mass at any time. The initial mass of carbon is

estimated from the composition analysis of the scrap sources constituting the furnace

charge. The mass balance equations keep track of the additions (coke additions,

melting scrap etc.) and consumption reactions (decarburization). Additions from

carbon present in the scrap can be modelled simply as a fraction of the mass of steel

that melts. Decarburization is modelled as the C leaving the molten metal zone and

entering the slag-metal zone where it reacts with O2 or FeO to form CO. The driving

force for the mass transport of C is the concentration gradient between these phases.

The equilibrium reaction in the slag-metal interaction zone thus ultimately controls

the rate of decarburization.

17


Solid-Scrap Zone

The solid-scrap zone is modelled as a mass of steel that melts according to the quantity

of energy transferred from the liquid steel, gas and arc, and the proximity of the steel

temperature to its melting point. As steel liquefies it is removed from the solid steel

zone and added to the molten steel zone.

The model predicts a homogenous temperature in the solid-scrap zone which corre-

sponds to the average temperature of the scrap. However, in reality the temperature

is not homogenous and scrap material melts continuously throughout the heat. A

modified version of the melting model proposed by Bekker et al. (1999) was imple-

mented. The temperature ratio that divides energy between sensible heating and

melting is taken as Ts/Tmelt, where Ts and Tmelt are the scrap temperature and steel

melting point temperatures respectively. This prevents the temperature of the solid

material from exceeding its melting point temperature since the portion of energy

contributing to the temperature increase diminishes to zero as the steel temperature

reaches its melting point. Further detail is given in Section 2.3.2.

2.3 Detailed Mathematical Model

2.3.1 Material Balances

The material in each zone can be tracked with an atom balance,

d

dt(bk,z) = F in

k,z − F outk,z (2.2)

where bk is the molar amount of element k in zone z, and F ink and F out

k are the flow

rates of element k into and out of the zone. The chemical equilibrium for the multi-

reaction systems can be computed by minimizing the Gibbs free energy. The method

used in the model does not require reaction stoichiometry to compute equilibrium

18


and is instead determined by solution of the system of equations corresponding to

the first order necessary conditions for constrained minimization of the Gibbs free

energy: ∑i

niaik = bk (2.3)

∆Gof,i + RT ln ai +

∑k

λkaik = 0 (2.4)

where ni is the number of moles of species i at equilibrium in the specified zone, aik

is the number of atoms of element k in species i, bk is the number of moles of element

k, ∆Gof,i is the Gibbs free-energy of formation, ai is the activity of species i and

λk are Lagrange multipliers. The activity is a function of the system temperature,

pressure and composition. The thermodynamic data required were obtained from the

National Institite of Standards and Technology (NIST) Chemistry Webbook (2004).

The activity of the non-ideal slag was determined using the regular solution formalism

using data for the interaction energies obtained from the literature (Ban-ya, 1993).

The activities for the metal constituents in the slag-metal phase were determined using

the unified interaction parameter model using interaction parameter data also sourced

from the literature (Sigworth and Elliot, 1974). Subscripts indicating the zone have

been omitted to simplify the notation. The benefit of using the system of equations

described in (2.3) and (2.4) is that a nested minimization routine is not required

at each integration step and the minimum is determined at each point by solving

the necessary conditions together with the other model equations. The chemical

equilibrium problem is convex when not combined with the phase equilibrium problem

(Smith and Missen, 1982), providing added justification for this approach.

A particular challenge in the development of the model was to ensure its robustness

over a wide range of conditions. To achieve this, a logarithmic transformation was

applied to the molar quantity at equilibrium, ni by introducing a new variable nLi

19


defined as

nLi = ln(ni). (2.5)

The term, enLi now replaces ni in the formulation. The transformation was found to

improve the scaling of the problem particularly when the equilibrium predicts very

small concentrations of species.

The remainder of this section describes the flows into and out of the various zones.

The flows are given in terms of the compounds and relate to elemental flows described

in (2.2) as follows:

Fk,z =∑

i

aikFi,z (2.6)

where k refers to the element and i to the compound.

Gas zone:

The net molar flows for the species into the gas zone are comprised as follows:

F ini,gas − F out

i,gas =Fburner,i + Fsm−gas,i + Fvolatile,i

− Fo/g,i + FPconst,i. (2.7)

The molar flows FA−B,i indicate the flow of component i from zone A to zone B, with

sm representing the slag-metal zone. Fburner,i accounts for the addition of O2 and CH4

via the burners and Fsm−gas,i accounts for the evolution of CO from the slag as well as

the flow of O2 between the slag-metal zone and the gas zone controlled by the partial

pressure of O2 according to,

Fsm−gas,O2 = kPO2(ysm,O2 − ygas,O2) . (2.8)

The inclusion of volatiles in the scrap is modelled by adding the C9H20 at a constant

feedrate, Fvolatile,i, for a short period of time after the furnace has been charged. This

is done to better approximate the dynamics of the vaporization of the volatiles since

20


the equilibrium model would predict instantaneous vaporization. Fo/g is the amount

of material leaving in the off-gas and has the same composition as the gas in the

furnace.

FPconst,i tracks the air leaking into the furnace and also the gas expelled from the

freeboard when the quantity of gas that is produced and added into the furnace

exceeds the ability of the offgas system to remove it. The expelled air is assumed to

have the same composition of the gas in the furnace and the ingressed air has the

same composition as ambient air. The different compositions of the flows results in

the following discontinuous relationship:

FPconst,i = xfreeboard,i min(0, Fnet) + xair,i max(0, Fnet) (2.9)

where Fnet is the total molar flow between the furnace and the environment through

the furnace openings; xfreeboard,i and xair,i are, respectively, the freeboard and ambient

compositions. FPconst,i could be positive or negative, depending on the direction of

flow into or out of the furnace.

The computation of Fnet is based on a constant pressure in the furnace freeboard.

Constant pressure is assumed since there are a number of large openings in the furnace

and the offgas flowrate is assumed to be constant. The assumption of a constant

offgas flowrate is adopted since the only measured variable is the electric current to

the offgas fan and the prediction of the offgas flow is confounded by the operation

of the offgas system. The offgas system is used for a dual furnace system and also

there are a number of dampers that allow air to be ingressed into the offgas system

to enable complete combustion and to cool the offgas. Furthermore, the length of the

offgas system and the compressible nature of gas make any prediction of the offgas

flowrate very inaccurate.

Fnet is related to the pressure using the ideal gas law, which allows the change in the

molar abundance to also account for changes in the volume and the temperature of

21


the furnace freeboard. This is important since the freeboard volume will change as

the scrap melts and as the temperature in the freeboard varies. The change in volume

of the freeboard is predicted based on the mass of scrap melted and is discussed in

more detail with the radiation model in Section 2.3.2.

The discontinuity introduced by the max and min functions can easily be handled

when integrating the model but causes difficulties in the optimization problem. This

issue will be addressed later when the optimization problem is formulated, in Section

3.4.

Slag-metal (sm) zone:

The net molar flows into the slag-metal zone are comprised as

F ini,sm − F out

i,sm = θL,iFlance,i + Froof,i + Fmm−sm,i + Fgas−sm,i. (2.10)

Flance,i includes the injected carbon and lanced oxygen. The injected carbon enters

both the slag-metal and the molten-metal zones. The distribution of the carbon is

handled through the parameter θL,i, with θL,C estimated from operating data. θL,i

is set to one for O2 and zero for all other components. Froof,i tracks the addition of

fluxes through the roof; Fmm−sm,i accounts for the movement of metallic species from

the metal zone to the slag-metal zone and the flow of reduced materials back to the

molten metal as determined by the concentration gradient.

Molten-metal (mm) zone:

The net molar flows for the molten-metal zone are comprised as

F ini,mm − F out

i,mm = (1− θL,i) Flance,i + Fsm−mm,i + Fss−mm,i, (2.11)

where Fss−mm,i represents the addition of molten steel as a result of the solid scrap

melting.

Solid-steel (ss) zone:

For the solid-steel zone it is more convenient to use units of mass instead of moles

22


and therefore the following balance is used:

d

dt(mss) = Mscrap − Mmelt (2.12)

where Mscrap is the rate of addition of the scrap to the furnace and Mmelt is the rate

of melting. Further detail on the computation of the melt rate is given later in the

discussion of the energy balance.

Mass Transport

The driving force for mass transport between the molten-metal and slag-metal zones

is the concentration gradient across these zones. The mass transfer coefficient is

expressed as the product of two parameters. The value of the first parameter is fixed

and represents the mass transfer properties of the component relative to the other

components, while the second parameter is the same for all components and can be

considered as a base mass transfer coefficient. This latter parameter is adjustable and

its value is estimated using process data. The mass transfer is given by

Fmm−sm,i = βikm (ymm,i − ysm,i) (2.13)

where Fmm−sm,i is the molar flow of species i from the molten-metal zone to the slag-

metal zone, βi is the relative mass transfer coefficient for component i, km is the base

mass transfer coefficient between the molten-metal and the slag-metal zones and yz,i

is the molar concentration of species i in zone z.

The oxidation reactions of Si, Al, Mn, Fe and C are controlled by the presence of

oxygen in the slag. The equilibrium calculation determines the preferential reaction

of the components with the available oxygen according to their reduction potentials.

The oxidation of these components depletes their concentration in the reaction zone

and thus the driving force for mass transport is increased. Similarly, if an oxygen

deficit occurs then a build-up of species i will inhibit the transport of that component

23


into the reaction zone. Therefore depending on the conditions in the furnace, the

oxidation reactions may be limited either by the availability of oxygen or by transport

to the oxygen-rich reaction zone.

The rate of decarburization is dependent on the availability of oxygen, either as O2 or

as FeO, and the rate of mass transport to the reaction interface. This is captured in

the model as the transport of carbon from the molten-metal zone to the slag-metal

zone. The availability of oxygen is controlled by its transport to the slag-metal zone

from lancing directly into this zone and also in the form of FeO. Throughout the heat,

the transport of carbon to the slag-metal zone is controlled by natural diffusion across

a concentration gradient. During lancing there is an additional mixing component

that is dependent on the flow rate of the O2 in the lance. Therefore an additional

term is added to the diffusion relationship in (2.13) to account for the increased effect

of O2 lancing,

Fmm−sm,C = βCkm (ymm,C − ysm,C) + kmL (ymm,C − y∗C) (2.14)

where y∗C is the equilibrium concentration of carbon in steel. Fruehan (1998) suggests

a value of 0.03% as a practical limit in steel-making. kmL is related to the volumetric

flow of O2 through the simple relationship kml = γFO2 , where γ is an adjustable

parameter determined from the process data.

Carbon Addition

Carbon is added to the furnace in two ways. It may be charged through the roof

during the heat or injected into the slag/molten-metal. The roof charging method

involves adding a quantity of carbon at once, compared to the injection which is a

continuous addition made over a period of time.

When the carbon is charged into the furnace through the roof, a large portion initially

floats on the slag surface and enters the bath as it dissolves. Ji et al. (2002) developed

24


a model for the accumulation of carbon in the slag. The carbon is depleted from the

slag due to reaction and an amount floats on the slag, the mass of which is proportional

to the mass of carbon in the slag. A similar approach was used in this work except

from a different perspective. Here, the rate of carbon entering the slag-metal zone

was modelled as being proportional to the mass floating on the slag, giving rise to

the equation,

d

dt(mc,float) = F in

C (1−XC)− kdcmc,float (2.15)

where mc,float is the mass of carbon floating on the slag. F inC is the rate of carbon

charged into the furnace during the heat. The proportionality constant, kdc can be

estimated using industrial data. This parameter will depend on the type, quality

and method of addition of the carbon source. A portion of the carbon will be lost

as dust, some will react in the freeboard and a portion will remain floating on the

slag. Furthermore, the coke source is typically impure due to the presence of ash and

volatiles. The installation on which the parameter estimation in this work is based

uses metallurgical coke as the carbon source, which typically consists of 86-88% fixed

carbon (Fruehan, 1998). If the fraction of actual carbon entering the steel can be

estimated then it can be accounted for in the variable XC . In this work a value of

0.15 was assigned to XC to account for the presence of impurities.

Injected carbon is added at a much lower rate compared to roof charging, thus it is

assumed that all carbon injected into the steel will go into solution. Due to the high

speed at which the carbon is injected, a portion will enter the molten-metal while the

remainder will go into the slag. While in reality the division depends on a number of

factors such as the angle of injection and the particle size distribution of the carbon,

in this work this division is assumed constant and estimated as a parameter (θL) from

the process data.

25


Slag Foaming

Slag foaming is desirable since it improves the efficiency of the electrical energy and

protects the furnace from radiative damage. The viscosity, density and surface tension

of the slag have an important effect on the ability of the slag to form a stable foam,

thus the composition of the slag is important. Jiang and Fruehan (1991) derived

a relationship for the foaming factor, Σ, using dimensional analysis techniques to

relate the ratio of the foam height and the superficial gas velocity to the physical

properties of the slag. This relationship has been widely accepted in the literature

for the prediction of slag foaming in steelmaking:

Σ =115µ√

ρσ=

Hf

V sg

. (2.16)

The slag viscosity µ, was estimated using a model given in Urbain (1987). The density,

ρ, was estimated using the partial molar volumes from the data reported by Mills and

Keene (1987). A simple empirical model, obtained from Morales et al. (1997), was

used to estimate the slag surface-tension, σ. The superficial gas velocity, V sg , was

calculated from knowledge of the evolution of CO from the bath and the furnace

geometry. This model, as with the majority of models in the literature, assumes

that the slag depth is sufficient to have no effect on the foaming height of the slag.

Using a model that is independent of the quantity of slag results in the prediction of

unrealistically large slag foaming heights during the initial stages of the heat when

CO is being produced but the slag volume is not sufficient.

To address the issue of incorporating the effect of the volume of slag (or static slag

height) into the model in a simple yet robust manner, the model was adapted such

that the foam height is essentially as predicted by (2.16) if the static slag depth is

greater than a critical height. For smaller static slag depths, the foam height predicted

from (2.16) is scaled by a fraction that is approximately proportional to the static

26


slag height. The modified relationship takes the form,

Hf = Φ(ΣV s

g

). (2.17)

where

Φ =1

2tanh(α(hs) + β) +

1

2(2.18)

in which α and β are adjustable parameters. The function Φ is illustrated in Figure 2.2

together with a piecewise linear function that it can be thought of as approximating.

The advantage of (2.18) over the more direct piecewise linear function is that the

former is differentiable, making it more suitable for dynamic optimization. It should

be noted that the relationship, given by Φ, between slag depth and foaming is not

developed from fundamental principles since to the best of the author’s knowledge

there has been no research on the relationship between the slag volume and the

foaming index. In general, the volume of slag is not considered a limiting factor of

the foam height, since slag foaming is of interest towards the end of the heat when

there is ample slag material available. However, in this work, the entire heat is of

interest and therefore the model must be able to predict the foam height for the

duration of the heat. Thus it is important to compensate for the fact that there may

be insufficient volume of slag during the initial stages. In the absence of literature or

process data, the parameters in (2.18) were fitted using a value of 20cm as the critical

height for the static depth.

2.3.2 Energy Balance

The following energy balance was implemented for the gas, slag-metal and molten-

metal zones:

d

dt(Ez) = Qz +

n∑i=1

Fi,zHi,z

∣∣∣∣∣in

−n∑

i=1

Fi,zHi,z

∣∣∣∣∣out

. (2.19)

27


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

hs

Φ

Figure 2.2: Hyperbolic tangent function used to incorporate the effect of slag depth.

Here, Qz is the heat flow added to zone z; Fi,z is the molar flow of component i

to/from zone z and Hi,z is the corresponding enthalpy. The energy holdup at any

time is computed as

Ez =n∑

i=1

ni,zHi,z (2.20)

where ni,z corresponds to the number of moles of species i in zone z. The heat flow

term, Qz is now developed for each zone in the following text.

Gas zone:

The gas exchanges energy with the solid scrap and the furnace roof and walls. The

energy transferred to the gas is given by the following equation,

Qgas = − Qgas−ss −Qwall −Qroof (2.21)

where Qgas−ss is the convective heat exchange between the gas and the solid scrap

28


given by the following equation,

Qgas−ss = (hA)(Tgas − Tss). (2.22)

Here, h is a convective heat transfer coefficient and A is the interfacial area between

the gas and solid. The energy exchange between the gas and the solid scrap is

most significant during the initial stage of the heat. To capture this behaviour, the

convective heat transfer coefficient is made proportional to the flowrate of material

entering the furnace through the burners, F burner. The prediction of the effective

surface area of the scrap material is confounded by its random nature and thus a

simplification is made whereby the effective surface area is assumed proportional to

the mass of steel divided by its bulk density. The combined area and convective heat

transfer coefficient are thus computed as

hA = kT3Fburner

(mss

ρbulk

). (2.23)

where the constant, kT3 , is estimated from the data. The wall temperature, Twall, is

given by

d

dtTwall =

−qrad2 + Qwall −Qwater,wall

mwallCp,wall

(2.24)

where qrad2 is the radiative energy from the wall; Qwall = hgsAwall(Tgas − Twall), here

hgs is the convective heat transfer coefficient between the wall and freeboard gas;

(mwallCp,wall) is the heat capacity of the furnace wall. Troof and Qroof are calculated

in a similar manner. Qwater,wall is the heat extracted by the cooling water from the

wall panels,

Qwater,wall = mH2OCp,H2O(Tcw,out − Tcw,in) (2.25)

where mH2O is the mass flowrate of water, Cp,H2O is its heat capacity, and Tcw,in and

Tcw,out are respectively the inlet and outlet cooling water temperatures of the cooling

circuit for the walls.

29


Slag-metal (sm) zone:

The slag-metal zone has limited contact with the solid-scrap and thus heat exchange

between these zones is assumed negligible. The amount of energy added to the slag

via the arc is very complex since it will depend on the volume of slag and also the

force with which the arc strikes the bath, since the arc action will tend to displace

the slag and expose the molten-metal below. The model assumes that arc energy is

not directly added to the slag but instead it receives this energy indirectly through

contact with the molten metal. The exchange between the slag and gas is assumed

to be negligible compared to the other sources of energy transfer and therefore not

explicitly considered. Thus the only source of heat exchange of the slag material,

considered in the model, is with the molten-metal:

Qsm = kT2(Tmm − Tsm) (2.26)

where kT2 is a transfer coefficient that is estimated from process data.

Molten-metal (mm) zone:

As with the slag and gas, the heat exchange between the molten-metal and gas is

assumed negligible compared to the other sources of energy transfer. The energy

flows into the molten-metal zone are given by the following equation:

Qmm = Qpower−mm −Qmm−ss −Qmm−sm − qrad4 (2.27)

where qrad4 is the net loss of energy via radiation from the bath and Qpower−mm is the

energy entering the bath from the arc energy. The molten-metal has contact with

both the solid-scrap and slag-metal zones. The heat transfer from the molten-metal

to the solid-scrap, Qmm−ss is made proportional to the mass of liquid to capture the

increasing heat transfer area as more molten-metal is formed,

Qmm−ss = kT1mmm(Tmm − Tss) (2.28)

where mmm is the mass of molten-metal and kT1 is the heat transfer coefficient between

the molten-metal and solid-steel. The effective heat transfer area between the slag-

30


metal and molten-metal is assumed constant and the heat transfer proportional to

the temperature difference between the molten-metal and slag-metal zones,

Qmm−sm = kT2(Tmm − Tsm) (2.29)

where kT2 is the same heat transfer coefficient as in (2.26).

Solid-scrap (ss) zone:

A methodology similar to that in Bekker et al. (1999) was used for the energy balance

and the melt rate of the solid-scrap; the derivation of which is shown in Appendix

A.4. For the solid-scrap there is no reaction and an energy balance yields

d

dt(Ts) =

Qss (1− Tss/Tmelt)− M inscrap

∫ Ts

ToCp,FedT

[mssCp,Fe] kdT

(2.30)

where Tss and Tmelt are the solid-scrap temperature and the melting point tempera-

ture respectively. Qss(1 − Tss/Tmelt) is the fraction of energy entering the steel that

contributes to sensible heating, with the remaining fraction of energy contributing to

the melting of the scrap. M inscrap

∫ Ts

ToCp,FedT accounts for the energy required to heat

scrap as it is charged into the furnace. mss is the mass of solid steel computed from

the mass balance in (2.12). kdT is an adjustable parameter to compensate for varia-

tions in the bulk density and composition, both of which influence the rate of melting.

The bulk density will effect the heat transfer to the steel and the composition will

impact the sensible heating.

The rate at which the solid scrap melts, Mmelt, can be determined by dividing the

rate of energy available to the scrap for melting by the energy per unit mass required

to melt the scrap at its current temperature,

Mmelt =Qss (Tss/Tmelt)[

∆Hf,Fe +∫ Tmelt

TsCp,Fe(s)dT

]kdm

(2.31)

where (Tss/Tmelt) is the fraction of the energy that contributes to the melting of the

scrap. ∆Hf,Fe is the heat of fusion of Fe and kdm is an adjustable parameter to

compensate for variations in the bulk density and composition in the scrap.

31


The energy transferred to the solid-scrap, Qss, is given by the following equation:

Qss = Qpower−ss + Qmm−ss + Qgas−ss −Qvolatile − qrad3 . (2.32)

Qpower−ss is the portion of the electrical energy from the arc that is transferred

to the solid material, Qmm−ss is as given by (2.28), Qgas−ss is as given by (2.22),

and qrad3 is the net loss of energy from the scrap material via radiation. Qvolatile =

Fvolatile (∆Hvap), which accounts for the energy required to vaporize the volatile com-

ponents present in the scrap.

Radiative Heat Transfer

Radiation is an important mechanism of heat transfer due to the high temperatures in

the furnace. It is thus necessary to be able to predict the contribution of radiation as

a mode of heat exchange in the furnace. This component of the model determines the

radiative heat transfer between the different surfaces based on their surface tempera-

ture, emissivity and surface area. An important characteristic that must be captured

is that as the scrap melts or if more scrap is charged, the surface areas also change.

The dynamics of the changing conditions in the furnace are modelled by relating the

void volume in the furnace to the exposed surface area of the various elements (roof,

walls, scrap and bath) in the furnace.

The dynamic behaviour of melting scrap in the furnace is extremely complex and

varies from heat to heat. A simple geometrical model is therefore proposed in order

to approximate dynamic behaviour as the steel melts. The melting model assumes

an initial cone-frustum shaped void is melted into the scrap by the electrodes, which

increases in volume as more material is melted. Figure 2.3 illustrates the furnace

geometry as the heat progresses from the initial stage to the intermediate and final

stages. Figure 2.3(a) shows the initial exposure of the roof, followed by the progressive

exposure of the walls in Figure 2.3(b), until flatbath conditions are approached in

32


Figure 2.3(c). Initially, the furnace roof and walls will be shielded from the radiation

by the scrap material. As the scrap melts the roof will be progressively exposed,

followed by the walls. The cone-frustum shaped void increases in height and radius,

while maintaining a constant critical angle of repose, until the cone base radius is

equal to that of the furnace. In reality the angle will change continually with the

competing effect of the electrodes boring down and the scrap collapsing. A constant

angle of repose provides a simple mechanism for averaging this apparent random

behaviour. More details of the changing geometry are given later in this section.

(a) (b) (c)

� hw

hs

rb

�

Figure 2.3: Stages of the melting used in the radiation model.

The surfaces within the furnace are treated as grey bodies and thus the net radiative

heat transfer from each surface is,

qradi =

Ebi − Ji

(1− εi)/εiAi

(2.33)

where Ebi is the black body emissive power of surface i determined from the Stefan-

Boltzman law, i.e. Ebi = σT 4i ; σ = 6.676×10−8W/(m2.K4) and Ti is the temperature

of surface i. Ji is the radiosity, which is the rate of radiation leaving a unit area of

surface i; Ai is the surface area and εi is the emissivity of surface i. The destination

of the radiative heat transfer to surfaces in an enclosure is described by the following

33


equation,

qradi =

N∑j=1

AiFij(Ji − Jj) (2.34)

where Fij is the view factor, i.e. the fraction of radiation leaving surface i that is

intercepted by surface j.

For an enclosure the view factors are related as follows,

N∑i=1

Fij = 1. (2.35)

The following reciprocity relationship is always true,

AiFij = AjFji. (2.36)

Three-dimensional models using differential elements to compute the view factors

(Guo and Irons, 2003; Reynolds, 2002) are too computationally intensive to be used

in this work. Therefore simplifying assumptions are made with respect to the internal

geometry of the furnace to avoid the large computational expense in computing the

view factors by integration. Thus the following simplifying assumptions are made

with regard to the four surfaces considered in the model: the roof is modelled as a

dome; the bath as a circular disk; the scrap surface as a cone-frustum and the walls

of the furnace as a cylinder. Reynolds (2002) showed that CO rich atmospheres,

typical of smelting furnace freeboards, contribute less than 5% towards the radiative

energy exchange within the furnace. Absorption and emission of radiative energy

from the gaseous freeboard is therefore not explicitly included in the radiation model.

Any discrepancies as a result of this exclusion from the model would be compensated

for in the estimation of the heat transfer coefficients for the gas phase; this would

effectively convert these parameters to overall heat transfer coefficients from purely

convective heat transfer parameters.

34


The system of equations given by (2.33), (2.34), (2.35) and (2.36) requires a further

N(N−1)2

equations to completely specify the radiation model for given surface temper-

atures, emissivities and areas, where N is the number of surfaces in the enclosure.

Here we consider five surfaces in the furnace: the furnace roof, furnace walls, scrap

material, the bath and the arc, which for convenience are numbered as follows: 1-

roof, 2-wall, 3-scrap, 4-bath, 5-arc. The presence of the arc in the furnace will be

addressed shortly. Analytical solutions for the following view factors, based on the

assumed furnace geometry, were obtained from literature (Siegel and Howell, 2001):

F1,1, F1,2, F2,2, F2,4, F4,1, F4,4; more detail is shown in Appendix A.3.

Next, the presence of the electric arc in the furnace is considered. The arc is assumed

to emit radiative energy as a black body and to be perfectly transparent in receiving

radiative energy, as was assumed by Guo and Irons (2003). Due to the transparency

of the arc we do not need to consider the radiation received by the arc from the other

surfaces in the furnace. Furthermore, the model makes the same assumptions about

energy usage from the arc as were made by Guo and Irons (2003). Specifically, 18%

of the energy from the arc is delivered directly to the steel, 2% is absorbed by the

electrode and 80% is delivered in the form of radiation. Thus the energy radiated by

the arc is modelled as,

qrad5 = (0.80) Qarc (2.37)

where qrad5 is the net radiative heat transfer from the arc. Qarc is the total energy

released from the arc and is related to the active power (Pr) in the primary circuit

through a proportionality constant that is estimated from data, Qarc = kP Pr. A

percentage of the radiated energy will be transmitted to the scrap and the steel

bath while the remainder is “lost” to the furnace walls and roof. The fraction of

radiation from the arc that reaches surface i is determined by multiplying qrad5 by the

appropriate view factor, F5,i, and including its contribution in the energy balance in

35


(2.34),

qradi =

4∑j=1

AiFij(Ji − Jj)− qrad5−i (2.38)

where qrad5−i = F5,iq

rad5 and is the radiation received by surface i from the arc. The

negative sign preceding the qrad5−i term indicates that surface i receives energy from the

arc, since qradi is defined as the amount of radiation leaving surface i.

The view factors from the arc, F5,i are determined based on process knowledge. As the

scrap melts the most important change is that the initial shield of scrap protecting

the walls from the arc melts away exposing the wall to arc radiation. During the

meltdown the arc is buried in the scrap and therefore the view factor, F5,3 between

the arc and scrap is close to unity. The fraction of radiation leaving the arc that

is intercepted by the scrap decreases slowly until the end of the heat is approached

when the layer of scrap protecting the wall disappears very quickly. This behaviour

is approximated as an exponential decay with respect to the radius of the base (rb)

of the cone-frustum, used to approximate the void left by molten material. Due to

the constant angle of repose assumed in the model, the radius of the base will change

with the changing mass of solid-scrap in the furnace. The dynamics of the arc to wall

view factor was approximated as,

F5,3 = 0.9− erb/42. (2.39)

Note that the initial and endpoint conditions ensure that:

0 < rb ≤ rR (2.40)

where rR is the furnace radius and is approximately 3.5m. F5,1, F5,4 are determined by

treating the arc as a cylinder and using analytical expressions from literature (Siegel

and Howell, 2001). F5,5 = 0, therefore F5,2 can be computed using the enclosure

equation for view factors, given in (2.35).

36


Effect of Foaming on Power Transfer:

Slag foaming has a significant effect on the amount of power that is transferred to the

steel. As the foam covers the arc, less energy is lost to the roof and walls and is instead

transported directly to the steel. Reports in literature (Fruehan, 1998) indicate energy

transfer efficiency improvements from 40% efficiency without a foaming slag to 60-

90% efficiency when the slag is foamed. The amount of radiative energy transferred

to the scrap, as computed in the radiation component of the model, is for an ideal

case where the effects of a foaming slag are not considered. This section discusses the

use of an efficiency factor, E∗f , to account for the effect of the foaming slag.

The steel receives 18% of the arc energy (Qarc), radiative energy from the arc and

also a portion of energy recovered as a result of slag foaming. The radiative energy

to the solid-scrap and molten-metal is determined explicitly in the radiation model.

The quantity of energy transferred directly from the arc and the portion recovered

due to the foaming slag is divided between the solid-scrap and the molten-metal in

proportion to their relative mass. The energy transfer to the molten-metal due to the

arc is given by,

Qpower−mm =

(mmm

mmm + mss

)(0.18Qarc + E∗

f

(qrad5−1 + qrad

5−2

))(2.41)

where E∗f is the fraction of radiative energy recovered due to foaming; qrad

5−1 and qrad5−2 are

the net radiative transmission from the arc to the furnace roof and walls respectively.

Similarly, the energy to the solid-scrap is given by,

Qpower−ss =

(1− mmm

mmm + mss

)(0.18Qarc + E∗

f

(qrad5−1 + qrad

5−2

)). (2.42)

The net radiative energy from the roof and walls, given in (2.38), is also altered to

compensate for the effect of foaming,

qradi =

4∑j=1

AiFij(Ji − Jj)−(1− E∗

f

)qrad5−i. (2.43)

37


The efficiency E∗f is comprised of two efficiency factors expressed as fractions between

0 and 1, that together determine the impact of the foaming slag on the energy transfer

in the furnace,

E∗f = E1E2. (2.44)

The height of the foaming slag was considered previously in (2.17). The first efficiency

factor, E1, relates the fraction of the arc that is covered by the foam to the fraction

of radiation that is prevented from reaching the wall and instead transmitted to the

steel. Using the data in the literature (Guo and Irons, 2003) as a guideline, it was

assumed that if the arc is fully covered by the foaming slag, at most 70% of the arc

energy will be blocked from reaching the wall and instead transported to the steel.

Furthermore, it is assumed that the arc length will be constant at approximately 0.5m

and therefore at a slag foam height of 0.5m or greater, E1 will be at its maximum

value of 0.7 or 70%. Below 0.5m, E1 is approximately proportional to the slag foam

height, decreasing toward zero as the foam height approaches zero. This relationship

is modelled with the following hyperbolic tangent function,

E1 = 0.7

(1

2tanh (α1Hf + β1) +

1

2

), (2.45)

with α1 = 5.0 and β1 = −1.25. The shape of the function is similar to that in

Figure 2.2.

The second factor, E2, considers that flat bath conditions are required before foaming

can occur unhindered. The model used to determine the slag height does not take

into account that a significant amount of solid material in the furnace will impact

the foaming. The presence of solid scrap will limit the amount of foaming and also

foaming may occur in the void spaces between the scrap and thus be unable to cover

the arc; subsequently the benefits in terms of efficiency are reduced. It is assumed

that when the scrap material is reduced to less than 20% of the initial charge mass of

scrap material (i.e. 0.01%mscrap < 0.2), E2 will be close to 1 and hence the presence

38


of solid material has a negligible effect on foaming. However, when the mass of scrap

is greater than 20% of its initial charge mass, then E2 decreases with the increasing

fraction of solid material. This is implemented using the relationship,

E2 =1

2tanh (α2(1− 0.01%mscrap) + β2) +

1

2, (2.46)

with α2 = 3.2 and β2 = −1.29.

2.4 Parameter Estimation

As has been discussed previously, an objective of the modelling work was to limit

the number of parameters in the model. This goal has to some extent been achieved

in that the model requires that only a relatively modest number of parameters be

estimated. The parameters considered for estimation and the equations in which

they appear are listed in Table 2.1. The values of the parameters used for the model

are reported in Appendix A.1.

In their review of the Dow parameter estimation problem, Biegler et al. (1986) discuss

the importance of using good starting values and eliminating as many unnecessary

model parameters as possible, both of which lead to better conditioned optimization

problems. To this end, initial estimates of the model parameters were obtained using

information from published literature sources and visually comparing the predictions

against the industrial data. This was followed by a sensitivity analysis, the purpose

of which was to eliminate insensitive parameters from the estimation problem.

Dynamic models require specification of the initial condition of the system. The

initial conditions used for the estimation problem are discussed here. The mass of

each scrap charge is recorded for the individual heats. The charge is composed from

different scrap sources such as pig iron, re-bar and directly reduced iron. The mass

and average chemical compositions of each source are known, which enables the overall

39


Table 2.1: Adjustable Model Parameters

Description

γ (2.14) parameter for mass transfer coefficient due to mixing effect

of lancing

km (2.13),(2.14) mass transfer coefficient between the slag-metal and

molten metal zones

kT1 (2.28) heat transfer coefficient between the molten-metal and

solid scrap

kT2 (2.26) heat transfer coefficient between the molten-metal and

slag-metal

kT3 (2.23) heat transfer coefficient between the gas and the

solid scrap

kP – proportionality constant between active power and arc

power

kdm (2.31) solid melt rate tuning parameter

kdT (2.30) solid temperature tuning parameter

kPO2(2.8) mass transfer coefficient for O2 transfer to the slag

kdc (2.15) dissolution parameter for carbon addition

θL (2.10),(2.11) fraction of lance coke entering the molten-metal phase

40


composition of the charge to be determined. The quantity of volatile material in the

scrap is computed in the same manner. The bulk density of the scrap may vary

significantly depending on the scrap mix and the degree of compactness of the scrap.

The bulk density will affect the ability of the burners to heat the scrap and also the

rate at which the scrap melts. At the end of each heat a portion of the molten-metal,

known as the heel, is left in the furnace to aid the next heat. The initial heel (volume

of liquid steel) is not directly measured but computed through a mass balance and

recorded for each heat. The composition of the heel was assumed to be that of the

endpoint target tap chemistry from the previous heat. A small mass of slag was also

assumed to remain in the furnace with the heel, the composition of which was taken

to be the average composition of the slag for the furnace. The initial composition of

the freeboard was assumed to be that of the ambient air.

2.4.1 Sensitivity Analysis

The purpose of the sensitivity analysis was to isolate the least sensitive parameters

in the model; these could then be set as constant values and removed from the rig-

orous estimation problem. This work is motivated by the lack of useful process data

available.

One-factor-at-a-time experiments fail to consider the effect of any interaction between

the factors. Designed experiments, on the other hand, ensure that the appropriate

data is collected, that it can be analyzed using simple statistical methods and ensures

that valid and objective conclusions can be made (Montgomery, 2000).

A factorial design involves testing each factor at a range of different levels to ensure

the system is sufficiently excited and that the main effects and interaction effects of

the factors can be identified. In this case there are 13 parameters to be investigated;

if the parameters are tested at 2 levels, a total of 213(= 8192) experiments will need

41


to be carried out. However, it was not possible to automate the experiments due

to limitations of the gPROMS software. Furthermore, the higher order interaction

terms are typically small and can often be ignored. Thus a fractional factorial design

was carried out; this method ensures the data is used in the most efficient manner for

identifying the factors and higher order interactions. The fraction of the full factorial

design that is completed will determine the degree of aliasing between the main effects

and interaction effects identified in the regression model. A resolution III fractional

factorial design ensures that all main effects are not aliased with any other main

effects, but 2 factor interactions are aliased with the main effects. A resolution IV

design ensures the main effects and 2 factor interactions are not aliased with any other

main effects. A resolution V design ensures the main effects and 2 factor interactions

are not aliased with any other main effects or 2 factor interactions.

A resolution IV, fractional factorial design was carried out to determine the main ef-

fects of the parameters on the model predictions. This design was sufficient to provide

an indication of the effect of the parameters and also specify the variables that are

most strongly influenced by each parameter. The nominal case was determined using

parameter values obtained from published literature and from manual adjustments

based on visual comparisons with the data. Each parameter was then perturbed

above and below its nominal value by 20% according to the designed experiment. To

quantify the effect of the parameter changes, a combined measure of model perfor-

mance was constructed from a selection of the predicted states. The states whose

values are measured on the actual process were selected for the analysis since these

are the only measurements that would be available for the rigorous estimation work.

In addition, variables important to the performance and accuracy of the model were

included, such as the temperatures of each zone and the mass of solid and liquid steel.

For the analysis, it was necessary to construct a single metric that summarizes the tra-

jectory of each state. The integral square deviation of the predictions from the mean

42


state predictions was calculated to summarize the time-dependent predictions for the

entire heat into a single value. Centering and scaling the data ensures that all variables

have the same weighting. It was therefore possible to construct a combined perfor-

mance measure for the model using all the selected states. This combined measure

of performance summarized the overall model sensitivity to each of the parameters.

The integral square deviation of the individual model states provided information on

the parameters by which they were most strongly influenced.

To analyze the information, a linear regression model was built and standard regres-

sion analysis techniques were implemented to compute the statistics as discussed in

Montgomery and Runger (1999); the analysis was carried out using the commercial

design of experiment software MODDE (Umetrics AB, 2001). The coefficients of the

regression model were examined to determine the significance of the EAF model pa-

rameters. Figures 2.4(a) and 2.4(b) show the coefficient values on the ordinate and

the parameters on the abscissae for the regression model. There are thirteen param-

eters shown in these figures; eleven correspond to the parameters described in Table

2.1, the final two are, respectively, the initial concentration of carbon and the mass

of the heel in the furnace. The coefficients with the greatest magnitude and whose

confidence intervals do not include zero are deemed to be significant to the model.

The coefficients for the combined measurement, as shown in Figure 2.4(a), indicate

that only six of the thirteen parameters are shown to be significant. The CO2 mea-

surement is strongly affected by only four of the thirteen parameters. By studying

the model coefficients of both the combined and individual measurements a good un-

derstanding of the model sensitivity was obtained. This study enabled a number of

parameters to be fixed and hence eliminated from the model estimation problem.

This study is not necessarily conclusive since the results depend on the states that

were chosen to evaluate the model and is based on a linear analysis. It is however

a useful tool that can be implemented to provide a good starting point for difficult

43


estimation problems. The study suggested the following parameters to be the most

important for the model: km, kT1 , kT3 , kP , kdm and kdc.

1 2 3 4 5 6 7 8 9 10 11 12 13-120

-100

-80

-60

-40

-20

0

20

40

60

80

Sca

led

& C

ente

red

Coe

ffici

ent V

alue

Parameters

(a) Combined Measurement

1 2 3 4 5 6 7 8 9 10 11 12 13-25

-20

-15

-10

-5

0

5

10

15

20

25

30

Sca

led

& C

ente

red

Coe

ffici

ent V

alue

Parameters

(b) CO2 Measurement

Figure 2.4: Coefficients for regression model from the sensitivity analysis.

44


2.4.2 Available Measurements

The following measurements were available for the estimation of the model:

1. Electrical and material inputs

• Power

• Scrap addition (composition and mass)

• Carbon, lime and dolime additions

• Carbon, oxygen and lime injection

2. Off-gas chemistry: CO, CO2, O2, H2

3. Endpoint carbon concentration and temperature measurements of the steel

4. Average endpoint slag chemistry

There are a number of factors which complicate the estimation problem, the most

severe of which is the limited amount of data available. The only direct measurements

of the process which are available for the duration of the heat are the off-gas compo-

sition. Four components, namely CO, CO2, H2 and O2 are measured. However, there

is a degree of correlation between these measurements due to the O2 dependence of

the other components in the system. This means that there is less actual information

in these measurements than if they were uncorrelated. Furthermore, as is typical

of industrial data, there is noise in the data. The data were processed through a

low-pass filter with a data-window of 3 measurements; this was deemed sufficient to

reduce the most significant effects of noise without adversely effecting the dynamics

of the signal.

The absence of data for the following variables makes the estimation problem partic-

ularly challenging:

45


• Off-gas flowrate: The industrial system being studied has a single off-gas system

serving a pair of furnaces. The off-gas system also has a number of vents which

entrain the ambient air to aid cooling of the gas. The combination of these

factors makes the prediction of the off-gas flowrate using the suction fan very

difficult. This information would enable better predictions of the amount of

ingressed air to be made and also allow a considerably more accurate mass and

energy balance to be carried out.

• Slag composition: The available data for the slag chemistry is the endpoint

composition. Availability of the composition profiles over the heat duration

would significantly improve the observability of the process.

• Bath temperature: The single temperature measurement at the end of the heat

makes it difficult to accurately calibrate the melting model.

The above measurements would provide a much clearer understanding of the be-

haviour within the furnace and allow the model to be better tuned to the process.

Furthermore, it would be possible to add more detail to the model since there would

be more information available for estimation and validation of modelling work. While

technology exists for all the above measurements, there is an associated capital cost,

reliability issues of equipment and maintenance costs. Clearly, there is a tradeoff

and it is necessary to evaluate whether the benefits of such equipment would justify

the cost. Evaluating the benefit of improved instrumentation is one of the planned

extensions to this work.

2.4.3 Handling the Raw Data

There were two types of anomalies present in the raw off-gas data. The first is

associated with the opening of the furnace roof during the heat to add the second

46


charge. When the roof is swung open the off-gas analyzer measures the surrounding

air composition instead of the furnace conditions. This can be quite easily identified

because of the timing and the composition of O2 spikes to approximately 20% while

all other components drop to zero (N2 is not measured). The second aberration

was a result of the off-gas analyzer probes becoming blocked. The analyzer is able

to sense the blockage and purges the probes with air. However, there is a short

period of unreliable measurement preceding and following the purge that must also

be eliminated from the data. The status of the analyzer is a recorded measurement

and thus this data can be easily removed. These anomalies need to be accounted for

so that they do not adversely affect the estimation of the model parameters.

2.4.4 Rigorous Parameter Estimation

The differential-algebraic furnace model and dynamic data require the parameter es-

timation problem to be solved as a dynamic optimization problem. In the estimation

problem, the model parameters are the decision variables and the deviation of the

model predictions from the data is minimized. Cervantes and Biegler (2001) discuss

several methods for the solution of dynamic optimization problems. The two most

successful techniques address the infinite-dimensional nature of the problem by imple-

menting some level of discretization. The first method involves partial discretization,

where the control vector is parameterized but the states remain continuous. The

alternative method requires both the control inputs and the states to be discretized

and reduce the problem to a standard nonlinear programming problem. The partial

discretization approach is favoured for this work because of the large dimensional-

ity of the model. The estimation of the model parameters was implemented using

the gPROMS/gEST (Process Systems Enterprise Ltd., 2004) software package which

utilizes this solution technique. The particular implementation of the maximum-

47


likelihood function used is given by,

Φ =N

2ln(2π) +

1

2min

θ

NE∑i=1

NVi∑j=1

NMij∑k=1

(ln(σ2

ijk) +(zijk − zijk)

2

σ2ijk

) (2.47)

where N is the total number of measurements for all the experiments; θ is the set

of model parameters to be estimated subject to upper and lower bounds; NE is the

number of experiments performed; NVi is the number of variables measured in the

ith experiment and NMij is the number of measurements of the jth variable in the ith

experiment; σijk is the variance of the kth measurement of variable j in experiment

i; zijk is the kth measured value of variable j in experiment i; and zijk is the model

prediction corresponding to the kth measurement of variable j in experiment i.

The estimation problem as given in (2.47) allows the flexibility of including several

types of variance models. The variance may be assumed constant or related to the

magnitude of the measured or predicted values. A drawback of increasing the flex-

ibility in the variance model in the case of limited data is that it is equivalent to

adding more parameters to the model and therefore may affect the conditioning of

the estimation problem. If the variance model is completely specified, (2.47) reduces

to a weighted least-squares problem. This formulation allows multiple heats to be

considered simultaneously to estimate model parameters.

Parameter Scaling

The condition of the parameter estimation problem may vary significantly during

optimization and correct scaling can help offset this effect. Parameters must be scaled

so that they do not vary over many orders of magnitude. The method of scaling used

in this work is of the general form,

θj =θj − 1

2

(θmax

j − θminj

)12

(θmax

j − θminj

) (2.48)

48


where θj is the scaled parameter value and θmaxj and θmin

j are the bounds of the

unscaled parameter θj.

Parametrization of Input Data

The control-vector parametrization approach used in the solution of the differential-

algebraic equation system requires parameterization of the process inputs. A satis-

factory level of discretization must be selected such that the number of intervals is

reasonable and also so that the inputs are accurately represented. Fortunately, in

the case of the furnace the nature of the input variables makes them suitable for the

piecewise-constant approximation. The process inputs are typically held constant for

quite long intervals during normal operation of the plant. In this study the inputs

were parameterized using 3 minute intervals. Parametrization of the inputs at 3 and

5 minute intervals did not show any appreciable difference.

Analysis of Results

Table 2.2 presents the values of the parameter estimates generated by gPROMS. These

parameters were obtained by carrying out the estimation on 8 batches simultaneously.

The only data available for this study that provides information on the progression

of the heat is the off-gas composition data. Figure 2.5 shows the model prediction

(solid line) of the off-gas composition data for one of the heats used for the parameter

estimation. From these predictions, the model shows reasonable agreement with the

process data. The sharp changes approximately midway through the batch coincide

with the introduction of the second charge. Furthermore, the burner flows are ad-

justed in anticipation of the second charge which also affects the off-gas composition.

49


Table 2.2: Results of Estimation

Parameter Estimate

kP 1.11E+01

kT1 1.35E-02

kT3 1.07E-03

kdc 6.94E-02

km 5.53E+01

kdm 4.43E-01

Table 2.3: Model Prediction Data

Selected Heat Average Validation

MSPE MSPE MSPE

CO 7.62E-03 6.59E-03 6.30E-03

CO2 2.67E-03 2.54E-03 3.70E-03

O2 1.78E-04 4.50E-04 1.66E-04

H2 4.14E-03 4.29E-03 6.26E-03

Table 2.3 summarizes the quality of model predictions in terms of the mean square

prediction error for (i) the heat shown in Figure 2.5, (ii) the average for all the heats

used in the estimation process and (iii) validation data sets not used in the parameter

estimation. These values are computed from composition data, with a typical value of

0.2 for CO, CO2 and H2; O2 typically has a value less than 0.05. Comparing the data for

the selected heat shown in Figure 2.5 against the average of all the heats used in the

parameter estimation indicates that model performance for this heat is fairly typical.

The model was validated in a straightforward manner. The model predictions, with

the parameters fixed at the values in Table 2.2, were compared to data from two

heats not used in the estimation problem. The mean square predicted error results

for these new data sets compare favourably with those from the data used to carry

out the estimation, providing some confidence in the model’s predictive capability.

50


Figure 2.5: Off-gas predictions (mole fractions).

Discrepancies between the model predictions and the plant data are expected due

to a number of factors. This includes the inevitably inaccurate predictions of the

ingressed air and off-gas flow rates which are not measured. During the first and

second scrap charges the model predictions tend to be poorest. Possible reasons are

that an accurate estimate of the initial conditions related to the scrap are difficult

to obtain based on the scrap charge. When material is charged, the furnace is also

cooled significantly and takes several minutes to reach the point where the assumption

of equilibrium is appropriate. Another possible reason for the observed discrepancy

is that the model assumption of homogenous scrap results in carbon being released

in proportion to the melt rate. During the meltdown period the system is very

erratic, due for example to collapsing of a section of the scrap pile. Furthermore, the

51


conditions in the off-gas are affected by the properties of the particular piece of scrap

beneath the arcs at that time. However, as the amount of liquid steel increases these

higher frequency disturbances are removed. It is also very difficult to evaluate the

contribution of the various sources of energy in the absence of data from a designed

experiment and temperature measurements for the duration of the heat.

Figure 2.6 compares the model predictions of the endpoint slag chemistry with the

compositions obtained from the process. The model predicted data presented in this

figure are from the same heat as the data presented in Figure 2.5. The model data are

compared with the average composition of the slag chemistry; the chemistry for each

heat is not measured and thus the average was computed as a means of comparison.

The slag composition in the model is determined by calculating the composition in

the slag-metal zone while ignoring the presence of Fe. It is difficult to assess the

model performance based upon a single data point; nevertheless, the model shows a

good fit with the industrial data.

Figure 2.6: Endpoint slag composition prediction.

52


2.5 Simulation Studies

The model comprises 85 differential variables and 1050 algebraic variables; a simu-

lation takes 10-15 CPU seconds on an Intel Pentium IV 3.0 GHz processor. Several

case-studies are presented in this section to illustrate the potential uses of the model

and its functionality.

2.5.1 Scenario 1. Base Case

Figure 2.7 shows the normalized input profiles for a typical heat. Initially the burners

are fired-up (FO2 and FCH4) to preheat the scrap. Following the preheat, the power

(Parc) is turned on and the scrap will begin to melt. When sufficient space has been

created in the furnace, a second charge is added; this occurs at approximately t = 28

mins in the case shown here. Towards the latter stages of the heat, carbon is injected

(Cinj) into the bath to reduce the FeO and produce a foaming slag. At the same time

O2 is lanced (O2,lnc) into the bath to prevent a buildup of carbon in the steel. During

the course of the heat carbon, lime and dolime are charged through the furnace roof

at specific times, typically just before the second charge and then again once there is

sufficient liquid steel.

Figure 2.8 presents the normalized profiles of the mass of solid and liquid steel, in-

dicating the melting progression. Initially, there is very little melting as one would

expect during the preheat. However, once the power is turned on melting proceeds

rapidly. The addition of the second charge is evident at approximately 30 minutes,

illustrated by the sharp increase in the mass of solid steel. The initial mass of liquid

steel is due to the heel left in the furnace from the previous heat in order to aid

melting.

Figure 2.9 shows the offgas composition profile for this heat. It is evident from the

53


0 10 20 30 40 50 60 700

50

100

Par

c

0 10 20 30 40 50 60 700

50

100F O

2

0 10 20 30 40 50 60 700

50

100

F CH

4

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Figure 2.7: Scenario 1: Furnace input profiles.

graph that all O2 and CH4 are completely consumed. The initial increases in the H2 and

CO concentration are due to the combustion of volatiles that vaporize from the scrap

within the first few minutes of it being charged. The increase seen at the time of the

second charge is sharper due to hotter furnace conditions at this time and therefore

the volatiles will vaporize faster. When a large amount of volatiles are present or CO is

coming from the bath, an O2 deficit results and the equilibrium favours the production

of CO over CO2 as the products of combustion; this trend is also favoured by higher

temperatures in the furnace. Towards the end of the heat a large amount of CO is

given off from the bath due to increased C injection and O2 lancing; evidence of this

is seen in the rising CO and CO2 concentrations in the offgas after t = 55 mins.

The slag compositions shown in Figure 2.10 were determined by excluding the mass

of Fe in the slag-metal zone. Initially the volume of slag is very small and hence

54


0 10 20 30 40 50 60 70

0

10

20

30

40

50

60

70

80

90

100

Time [min]

Nor

mal

ized

Mas

s

Mass SolidMass Liquid

Figure 2.8: Scenario 1: Solid scrap and liquid steel profiles.

subject to large variations with the addition of the initial carbon charge. Once the

lime and dolime are charged into the furnace (at t = 22 mins) the volume of the

slag is sufficient such that the composition is less prone to extreme fluctuations. The

addition of C in the slag prevents the concentration of FeO and Fe2O3 from increasing.

However, towards the end of the heat (from t=65mins) the carbon in the bath is

exhausted and O2 lanced into the bath rapidly increases the concentrations of FeO

and Fe2O3. The presence of SiO2 and Al2O3 can also be seen.

The functionality of the radiation model is illustrated in Figure 2.11, where the net

radiative heat transfer from each surface is illustrated. For simplicity the net radiative

heat to the scrap and the bath are combined into a single variable, steel. In the model,

a negative value indicates a net gain of energy onto the surface. During the heat,

the walls and roof do not heat up at the same rate as the steel due to cooling water

55


0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time [min]

Com

posi

tion

[mol

frac

tion]

COCO

2O

2CH

4H

2

Figure 2.9: Scenario 1: Offgas composition profiles.

pumped through the panels. For the first t = 15 mins, the steel is heated via the

burners and increasingly radiates heat to the walls and roof. After the power is turned

on, the net radiative transfer is dominated by the radiation from the arc to the steel

and other furnace elements. When the arc is boring into the scrap, the arc is shielded

by the scrap and therefore the majority of the energy from the arc is radiated to the

scrap.

From time t = 53 mins, the ability of the scrap to shield the walls decreases quite

rapidly and the incident radiation to the walls increases accordingly. The increase

is a result of the walls being exposed as the steel melts, the dead-time in the wall

exposure is a result of the cone shaped void which results when the electrodes bore

into the steel; thus the walls are largely protected until most of the scrap is melted.

The presence of a foaming slag, which shields the walls from the arc, prevents the

56


0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [min]

Com

posi

tion

[mol

frac

tion]

FeOFe

2O

3MgOSiO2Al2O3CaO

Figure 2.10: Scenario 1: Slag composition profiles.

radiation to the walls from increasing even further. Finally, towards the end of the

heat when the power is turned off, there is a net loss of radiative energy from the

bath to the roof and walls.

2.5.2 Scenario 2. Effect of preheat duration on final melting

time.

In this scenario the pre-heat is reduced from 15 minutes in the base case to just 3

minutes and the subsequent additional electrical power and time necessary to com-

pensate for the reduced preheat are illustrated. For the case where the pre-heat is

reduced to 3 minutes, the first charge will require more electrical energy before there

is sufficient space in the furnace to add the second charge. In this scenario we apply

57


0 10 20 30 40 50 60 70-80

-60

-40

-20

0

20

40

60

80

100

Time [min]

Nor

mal

ized

Rad

iativ

e E

nerg

y

qroof

qwall

qsteel

qarc

Figure 2.11: Scenario 1: Radiative heat transfer in the furnace.

the same initial power trajectory as in the base case, but shifted in time to coincide

with the end of the preheat. The power is also maintained at its maximum value until

there is sufficient space in the furnace to add the second charge. This is illustrated

in Figure 2.12 over the time interval, t = 12− 23mins.

Table 2.4: Comparison of Preheat Strategies

Variable Total

Change

Electrical power 17.5%

Burner O2 -8.9%

Burner CH4 -9.3%

Melt time -5.8%

58


Figure 2.12: Scenario 2: Scrap melting.

Figure 2.12 and Table 2.4 show the results of this case study. In Figure 2.12 it is

evident that the overall batch time is longer in the base case. However, studying the

results in Table 2.4, it is evident that the savings in time and burner fuel consumption

are achieved at the expense of increased usage of electrical power.

2.5.3 Scenario 3. Effect of carbon lancing on the slag com-

position.

Figure 2.13 shows two different carbon injection strategies. In the test case, a much

higher rate of injection is maintained until the end of the batch. The consequence

of keeping the injection rate higher toward the end of the heat is shown in the slag

composition where slag FeO composition is reduced by 9%. Thus there is less iron

lost to its oxidized state, which is desirable.

59


Figure 2.13: Scenario 3: Carbon injection strategy.

2.5.4 Scenario 4. Effect of lancing strategies on slag foaming

performance.

For the appropriate slag composition, the C injection and O2 lancing rates can be

used to manipulate the slag foam height. In Figure 2.14, the solid line represents

the predicted foaming height, with the base case and test case shown as (a) and

(b) respectively. The dashed line corresponds to the foaming efficiency E∗f , given

by (2.44), which represents the percent of radiative energy recovered as a result of

foaming. E∗f has a maximum value of 70% since not all energy can be recovered, this

is discussed earlier with the development of the foaming efficiency relationships.

Comparing the foam height in (a) and (b) shows that the slag is foamed to a greater

degree, in case (b) through the injection of more carbon and lancing of more oxygen.

However, comparing the foaming efficiency in (a) and (b) shows that there is little

60


difference between these scenarios. This illustrates that the greater degree of foaming

in case (b) has no benefit in terms of recovered energy since E∗f was already at its

maximum in (a) during that time period. In fact, excessive foaming could result in

the slag foaming through the electrode ports in the roof, an occurrence which is both

possible and undesirable. Furthermore, there is a financial cost for the additional C

and O2 consumed in increasing the foam height in (b). Determining the correct C

and O2 addition policies would be an interesting optimization problem, whereby the

maximum efficiency is desired for the smallest addition of C and O2.

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

(a)

Ef*

Hf

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

Time [min]

(b)

Ef*

Hf

Figure 2.14: Scenario 4: Slag foaming.

2.6 Model Summary

A detailed model of the electric arc furnace has been developed. This model is based

on fundamental principles, although a degree of empiricism has been introduced to

61


model relationships where the real mechanisms are either too complex to be modelled

or insufficient information is available.

Key model parameters have been estimated using available industrial data. However,

further measurements during the progression of the heat would be useful. It is hoped

that this work together with the optimization studies in the following chapter will

provide incentive for further instrumentation of industrial EAF operations so that

advantage may be taken of these tools.

The model framework presented allows for the inclusion of further detail. Potential

enhancements include detailed models for predicting the melting of scrap in the fur-

nace and improved prediction of decarburization and slag foaming, all of which are

areas of ongoing research. However, the generally limited amount of data available for

parameter estimation in an industrial setting should be carefully considered during

model refinements.

A number of case studies have been presented where operational trade-offs to improve

the profitability or production rate of the process have been illustrated. The purpose

of these studies is to show the workings of the model and also motivate the next section

of work, which investigates the rigorous optimization of the furnace to determine

optimal operation policies by explicitly considering the cost of these trade-offs.

62

Chapter 3

Process Optimization

The focus of this chapter is on the nominal open-loop optimization of the EAF pro-

cess. Here, nominal denotes that model uncertainty is not considered during the

optimization. The model developed in Chapter 2 is incorporated within a mathemat-

ical optimization framework that is used to determine the optimal input profiles for

the process according to a specified performance criterion.

The work of this chapter is motivated in the introduction. Thereafter, prior re-

search on optimization of EAFs and available methods for optimization of differential-

algebraic models are reviewed. The proposed EAF optimization problem formulation

is then presented. Finally, a series of case studies is presented to demonstrate both

the benefits and flexibility of optimization.

3.1 Introduction

The manner in which reagents, scrap and electric power are added to the furnace may

be carried out in multiple ways. Detailed process knowledge, in the form of a model,

63


makes it possible to understand the more complex relationships and evaluate differ-

ent operating strategies. Furthermore, the model enables process experimentation to

be carried out without the possibility of economic risk or the occurrence of danger-

ous situations. Such studies are often carried out by performing a large number of

simulations, testing possible input profiles over a wide range of conditions. Whilst

this approach to process optimization is simple and intuitive, the extent to which

the search space is investigated is severely limited and even finding solutions where

variables remain feasible can be challenging (Bansal et al., 2003). A more system-

atic approach is to solve the problem as a dynamic optimization problem whereby

the optimal input profiles are determined based on a specified objective criterion.

The optimizer uses the model to determine the optimal operating conditions of the

process, such as finding the optimal balance and timing of the energy contributions

from chemical and electrical sources. In this chapter the dynamic model of the EAF,

developed in Chapter 2, is used within a mathematical optimization framework to

evaluate such tradeoffs and determine the nominally optimal input profiles for the

furnace based on an economic objective.

3.2 Optimization of the Electric Arc Furnace

Woodside et al. (1970) used optimal control theory to determine the optimal power

trajectory during the carbon-injection stage of the heat, the objectives being to mini-

mize power and duration subject to the endpoint carbon and temperature constraints.

The model used by the authors contains two states: carbon concentration and tem-

perature. The rate of change in the bath temperature is explicitly related to the

electric power and the rate of reaction of carbon is governed by the bath tempera-

ture. The small scale of the model enabled the authors to optimize the system using

variational methods, i.e. methods implementing the first order necessary conditions

64


for optimality obtained from Pontryagin’s maximum principle (Cervantes and Biegler,

2001), to solve the resulting dynamic optimization problem.

Gosiewski and Wierzbicki (1970) used a simple, single state model that related the

power input to the bath temperature. The manipulated variables were the transformer

tap and the arc current; the authors assumed the transformer tap would be kept at

its maximum and thus only considered the manipulation of the current. An economic

objective comprising of the cost of power and the time of operation was maximized

using Pontryagin’s maximum principle.

Gitgarts and Vershina (1984) constructed an economic objective function comprising

of the labour cost, energy cost and cost of refractory wear. A dynamic statistical

model related the electrical variables to the process states. The first state, in the two

state model, represents the progression of the process through a series of stages and

the second is the molten metal temperature. Minimization of the operating cost is

achieved using Pontryagin’s maximum principle.

Boemer and Roedl (2000) investigated a series of operating parameters involving ma-

nipulating the lancing strategy. Improvements to the current strategy were obtained

through numerical simulations on a model of the lancing process and physical tests

carried out on an experimental rig. Danilov (2003) discusses the operational im-

provements obtained through the addition of specialized equipment to the furnace to

improve delivery of materials to the furnace.

Gortler and Jorgl (2004) implemented a method on an industrial furnace that uses

a fuzzy logic model to relate the arc radiation to the water cooled panel tempera-

tures of the furnace walls and roof. The transformer tap and impedance setpoint

were the manipulated variables and the controlled variables were the temperatures

in the water-cooled panels. The principle of optimization is that the electrical input

parameters for maximum meltdown power will be selected such that the water cooled

65


panels remain at the desired levels to prevent damage to the furnace structure.

Pozzi et al. (2005) discuss a system that has been developed which uses offgas chem-

istry data to update a model and control combustion in the freeboard by manipulating

the burner flow; however, no details of the model or mechanisms to determine inputs

are provided. Several other studies (Jones et al., 1999; Maiolo and Evenson, 2001)

analyse process data and infer better operational policies; here the use of the word

optimization appears to indicate an improvement to the current operation.

Much of the work carried out on EAF optimization tends to be on either overly

simplistic models or on models that do not consider all aspects of the furnace. A

drawback of these studies is that by optimizing one of the sub-process of furnace

operation the other sub-processes may be negatively impacted. To ensure that the

optimal operation of the furnace is achieved it is necessary to consider all the sub-

processes in the furnace simultaneously. The best example of this approach available

in the published literature is the work by Matson and Ramirez (1999).

Matson and Ramirez (1999) developed a comprehensive model and used iterative

dynamic programming to solve the resulting dynamic optimization problem. The

model approximates the furnace as two separate control volumes in which chemical

equilibrium is assumed and is described in Section 2.1. The input variables included

in the optimization problem were the carbon injection, oxygen lancing, burner O2 and

the batch duration. The authors used a weighted objective function to minimize the

amount of CO in the offgas, the final amount of FeO and the batch duration. A penalty

function was also included in the objective function to penalize the bath temperature

if it fell below 1920K (coinciding with the end of the melt).

66


3.3 Optimization of Differential-Algebraic Equa-

tion Systems

To capture the dynamics of the EAF process it has been modelled as a differential-

algebraic equation (DAE) system. Differential variables, also known as states, are

time dependent and may be a function of the external forcing functions, other state

variables, algebraic variables and time. DAE systems are more challenging to solve

than purely algebraic systems due to the presence of the differential states that must

be integrated and the infinite-dimensional search space of the decision variables. Opti-

mization of differential-algebraic equation systems can be cast in the following general

form (Cervantes and Biegler, 2001),

minu(t),tf ,

ϕ (x (tf ) , z (tf ) ,u (tf ) , tf ) (3.1)

subject to:

dx(t)

dt= f (x (t) , z (t) ,u (t) , t) (3.2)

0 = h (x (t) , z (t) ,u (t) , t) (3.3)

0 ≥ g (x (t) , z (t) ,u (t) , t) (3.4)

xL ≤ x (t) ≤ xU (3.5)

zL ≤ z (t) ≤ zU (3.6)

uL ≤ u (t) ≤ uU (3.7)

x0 = x (0) (3.8)

67


where

z(t) algebraic state profile vector

x(t) differential state profile vector

u(t) control state profile vector

f differential equation constraints

h equality constraints

g inequality constraints

The objective function, (3.1), may be composed of the state variables, algebraic vari-

ables, input variables, parameters and time. Typically it is formulated to represent

a performance measure of the system such as the economic cost or the square error

from a desired value. The model equations are included as constraints; (3.2) and

(3.3) represent the differential and algebraic model equations respectively. Bounds

are placed on the optimization problem either in the form of an equation (i.e. (3.4))

or directly on the variables themselves ((3.5) - (3.7)). The differential states require

an initial condition to be provided, viz. (3.8).

A variety of sophisticated techniques have been developed for the optimization of

problems involving DAEs. Variational or indirect methods, which make use of the first

order necessary conditions obtained from Pontryagin’s maximum principle to locate

the optimum, have had limited success with large-scale realistic problems. Methods

which discretize the continuous-time problem to obtain a finite-dimensional problem

have been more successful; Biegler and Grossman (2004) categorize these methods as

partial or complete discretization based on the level of discretization implemented in

the formulation of the problem.

Complete discretization methods discretize the state and control profiles using a tech-

nique such as orthogonal collocation on finite elements to approximate them as piece-

wise polynomials. Typically, Lagrange interpolation polynomials are used since the

coefficients of these polynomials correspond to the value of the states at the collocation

68


points. This property allows meaningful bounds to be implemented directly in the

formulation. The DAE system is converted to a purely algebraic system, which may

then be solved using conventional nonlinear programming (NLP) techniques. This

method is often termed the simultaneous method because the integration and opti-

mization are carried out simultaneously. A drawback of this method is the resulting

NLP problem can be very large and it may be necessary to sacrifice accuracy through

use of a coarse discretization of input and state profiles in order to obtain a compu-

tationally tractable problem. Ongoing research in this area has focused on improving

the computational efficiency of these problems; Cervantes and Biegler (1998, 2000)

used reduced sequential quadratic programming (SQP) techniques to take advantage

of the sparsity of DAE systems where the number of states far exceeds the number

of controls. Biegler et al. (2002) report significant efficiency improvements using an

interior-point (IP) optimization algorithm as an alternative to active-set strategies.

Partial discretization methods involve the discretization of only the control variables;

dynamic programming methods and sequential methods fall into this category. Dy-

namic programming is a method which can in principle achieve the global solution

(Luus, 2000), but the application of this method to large-scale problems has had lim-

ited success due to the computational expense associated with the solution of these

problems. Sequential methods, which solve the optimization problem by following a

sequence of integration and optimization steps have been successfully implemented

on large-scale problems. These methods involve parameterization of the inputs using

piecewise polynomials; in many cases zero or first order polynomials are sufficient

and yield piecewise-constant and piecewise-linear control inputs respectively. Gradi-

ent information, for the optimization step, is obtained through finite difference per-

turbations, integration of the adjoint equations or the integration of the sensitivity

equations (Vassiliadis et al., 1994a) and is then passed to a nonlinear programming

(NLP) solver which determines how to manipulate the control parameters as it iter-

ates to find the optimal control parameters. This is considered a feasible-path method

69


since the integration is carried out at each iteration, as opposed to the complete dis-

cretization methods where the integration is only effectively carried out at the final

solution. The disadvantages of the sequential method are that the integration at each

iteration is expensive which accounts for a substantial part of the computational time

and this method is not suited for use with problems with unstable modes due to

the method’s reliance on obtaining the solution of an initial value problem at each

iteration.

Both complete and partial discretization methods have been employed successfully

on a large range of different problems. In general, sequential methods are better

suited to larger, well-behaved problems whereas simultaneous methods are preferred

otherwise. Simultaneous methods do have the further drawbacks of having to find an

initial feasible solution and if they terminate before optimality is reached the solution

may not satisfy the DAE system. The sequential approach was used in this work, as

implemented in gPROMS/gOPT (Process Systems Enterprise Ltd., 2004). However,

recent developments by Biegler and coworkers (Cervantes and Biegler, 1998, 2000;

Biegler et al., 2002) suggest that either method could be viable for this process.

3.4 Formulation and Implementation of the EAF

Optimization Problem

The profit per batch, in dollars, is

ZP = c0Msteel(tf )−(

c1

∫ tf

0

P dt + c2

∫ tf

0

(FO2,brnr + FO2,lnc) dt

+ c3

∫ tf

0

FCH4,brnr dt + c4

∫ tf

0

FC,inj dt + c5

∫ tf

0

FC,chg dt

+c6

∫ tf

0

Fflux dt + c7

∫ tf

0

Fscrap dt

)(3.9)

70


where P is the electrical power; FO2,brnr and FO2,lnc are the flows of oxygen from the

burner and lance respectively; FCH4,brnr is the flow of natural gas from the burner;

FC,inj and FC,chg are the carbon additions from injection and charging respectively;

Fflux is the addition of lime and dolime; Msteel(tf ) is the mass of liquid steel at the

end of the heat and ci is the associated unit cost of each component.

The optimization problem may be formulated according to several different objective

criteria, depending on the operational priorities of the plant personnel. Two such

criteria involve optimization of the process according to the profit per time:

maxu(t),tf

ZP/tf = ZP

(1

tf

)(3.10)

and the profit per ton of liquid steel:

maxu(t),tf

ZP/ton = ZP

(1

Msteel(tf )

). (3.11)

Using these different criteria to optimize operation results in different operating poli-

cies since the latter is more concerned with profit on a per yield basis, while the

former will optimize the process on a per time basis. The control variables, u(t), in

the optimization problem are P , FO2,brnr, FCH4,brn, FO2,lnc, FC,inj and the mass of the

second charge.

The following constraints are imposed in the optimization:

Model equations:

dx(t)

dt= f (x (t) , z (t) ,u (t) , t) (3.12)

0 = h (x (t) ,u (t) , z (t) , t) (3.13)

Input constraints:

uL ≤ u (t) ≤ uU (3.14)

71


Path inequality constraints:

0 ≥ g (x (t) ,u (t) , z (t) , t) (3.15)

xL ≤ x (t) ≤ xU (3.16)

Endpoint constraints:

x(tf ) ≤ xf (3.17)

The model equations are included as equality constraints in the optimization problem.

The differential equations arise from the mass and energy balances and the algebraic

equations are introduced from constitutive mass and heat transfer relationships and

the equilibrium conditions.

The commercial software gOPT/gPROMS (Process Systems Enterprise Ltd., 2004)

was used to solve the above problem; the acronym gPROMS is derived from general

PROcess Modelling System. The software implements the sequential optimization

approach, similar to the strategy outlined in Vassiliadis et al. (1994a), for the solu-

tion of the DAE optimization problem. A discussion of the sequential approach was

presented earlier in Section 3.3.

The DAE solver used for integrating the model and sensitivity equations implements a

variable time step, backward differentiation technique; this implicit method is suitable

for stiff systems. During the optimization step, the duration of each interval and

value of the control variables in that interval are determined in accordance with the

objective function and constraints. The optimization solver is an implementation of a

sequential quadratic programming (SQP) algorithm, which is an effective strategy for

the solution of nonlinearly constrained problems. SQP algorithms obtain the search

direction based on a second-order Taylor approximation of the Lagrange function and

a linear approximation of active constraints around the best current point.

An important consideration in solving these problems is that the nonlinear problem

is nonconvex and as a result the search space may have multiple local optima, which

72


can result in the optimization terminating at solutions which are not globally op-

timal. Unfortunately, rigorous global optimization techniques are limited to small

scale problems for DAE systems. A method employed to avoid obtaining a poor local

solution is to restart the optimization from a range of different initial solutions to test

if a better objective value can be obtained.

3.4.1 Numerical Robustness

A number of measures were employed to improve the numerical conditioning and to

speed up the solution time of the optimization problem.

• Variable scaling: Ensuring that all variables are of a similar order of magnitude

is a well known strategy for improving the numerical conditioning of a problem.

• Equation scaling: The relative scale of equations in the model can impact the

solution of the optimization problem. Balancing the model equations with re-

spect to each other improves the condition number of the Jacobian matrix of

the model equations. This is important for the optimization algorithm since

the active set of constraints is determined based on the value of the associated

Lagrange multipliers, which depend on the conditioning of this Jacobian matrix.

• Logarithmic transformations: In the solution of the DAE system, the concen-

tration variables may be driven to negative values resulting in a meaningless

solution. Logarithmic transformations (as illustrated in Section 2.3.1) in (2.5))

were thus applied to ensure a positive value was maintained. Furthermore, vari-

ables that tend to zero may become very small but not actually reach zero and

these very small numbers can introduce scaling problems. Logarithmic trans-

formations can greatly reduce this effect and thereby improve the conditioning.

73


• Discontinuous approximations: While the DAE optimization algorithm imple-

mented by gPROMS is capable of handling discontinuities (Process Systems

Enterprise Ltd., 2004), experience with the solver indicated that the model

discontinuities caused severe conditioning problems resulting in optimization

failures. Therefore model discontinuities were removed by approximating them

with continuous functions, allowing the gradient information to be successfully

computed during optimization. More detail of the discontinuous approximation

methods is given in Section 3.4.2.

• Solver integration tolerance: Lowering the integration tolerance greatly reduces

the computational time for integrating the model equations. However, as the

integration tolerance is decreased the number of optimization iterations required

to reach the optimum increases, because the accuracy of the sensitivity informa-

tion is effected by the tolerance. Therefore there is a limit to the amount that

the integration tolerance can be lowered. Furthermore, if the integration toler-

ance is lowered beyond a certain point the quality of the solution may become

unacceptable.

• Relaxing variables bounds: Relaxing the bounds of variables such as the flow

rates, from being strictly greater or equal to zero to being greater than a small

negative number (-1E-5), greatly improved the robustness of the DAE solver.

The logarithmic transformations that are carried out on the concentrations

prevent the values from being less than zero. However, during the solution

procedure other variables may also become negative due to the limits of machine

precision and this relaxation was necessary in order to solve the DAE system.

74


3.4.2 Model Discontinuities

The model includes a discontinuous element to deal with the fact that air may be

sucked in or forced out of the furnace depending on the operation of the air handling

system, the burner operation and the amount of gas being produced from the bath.

Air being sucked in will have the composition of the ambient air, while air being

pushed out has the composition of the furnace freeboard. The simulation of this

model is easily captured using logic functions, such as the max and min mathematical

functions. However, this type of logic poses a problem for optimization because the

discontinuity prevents the computation of the gradient; information that is needed

by the optimizer.

The gPROMS/gOPT software is capable of handling discontinuities for simulation of

the models; the software also claims (Process Systems Enterprise Ltd., 2004) to be

able to deal with discontinuities for optimization problems. However, it was observed

that as the optimal solution was approached the condition number of the Hessian,

used for optimization, deteriorated and as a result the optimization problem failed

to converge. The poor conditioning could likely be attributed to a lack of derivative

information of the discontinuous functions when constructing sensitivity information

for the optimizer.

Therefore to handle this problem the discontinuous function g(x) = max(0, f(x)) is

approximated by a continuously differentiable function (Biegler, 2004),

g(x) =1

2f(x) +

1

2

(f(x)2 + ε2

) 12 . (3.18)

Thus the max function in the last term in (2.9) can be formulated as,

max(0, Fnet) =1

2Fnet +

1

2

(F 2

net + ε2) 1

2 (3.19)

where a value of ε = 1 × 10−3 is a good tradeoff between the sharpness of the ap-

proximation and degree of accuracy and was found to perform well. The inaccuracies

75


that result from using this continuous approximation of the discontinuous function

are negligible, particularly if one considers the amount of uncertainty associated with

the prediction of the offgas flows and the air ingressed into the furnace. Minimum

functions are also easily handled since,

min(f(x), 0) = −max(0,−f(x)). (3.20)

Thus the min function in the first term in (2.9) is formulated as,

min(0, Fnet) = −1

2Fnet +

1

2

(F 2

net + ε2) 1

2 . (3.21)

3.4.3 Path Constraints

In the control vector parameterization approach, path constraints can be imposed

on the states by adding point constraints at the interval boundaries. This method

ensures that constraints are respected at interval boundaries but cannot guarantee

they are respected between the intervals and small violations are possible. However,

strict obedience of the constraints, g(x) ≤ 0, can be ensured through the introduction

of an endpoint constraint where the magnitude of the violation is integrated over the

duration of the process and forced to be equal to zero. The integrated constraint

violation is determined using the max operator,

C1 =

∫[max(g(x), 0)]γdt = 0. (3.22)

The max operator can cause excessive oscillation between feasible and infeasible moves

of the optimizer because when the constraint is inactive the violation measure and

its gradient is zero with respect to the control variables. Vassiliadis et al. (1994b)

advocate the use of both methods together as a hybrid approach to combine the

exactness of the integral approach with the increased information regarding constraint

location provided to the optimizer from the point constraints.

76


Vassiliadis et al. (1994b) recommend using a value of γ = 2 since the function then

has first order continuity. However, in implementing these constraints it was found

that using a value of γ = 2 as a general rule could cause the optimization routine

to fail in cases where the initial solution had a large path constraint violation. The

reason for the failure is attributed to the squared term which significantly increases

the value of the constraint violation making it difficult for the optimizer to find a

feasible solution. In this work, it was found that there was very little computational

benefit gained by using values of γ > 1.

3.5 Process Optimization Case Studies

A series of case studies is presented here; the purpose of these studies is to demon-

strate how the fundamental model can be used within an optimization framework to

improve profitability of the EAF process. An actual heat from the industrial process

on which the model parameters are based serves as the base case scenario for this

study to illustrate typical furnace operation. The case studies are then presented to

illustrate how optimization determines the economically optimal operating strategy of

the furnace. The optimization problem is formulated such that the control variables

may move between their upper and lower bounds in order to maximize the profitabil-

ity of the heat. The manipulated variables include the arc power, oxygen and natural

gas flows to the burner, carbon injection, oxygen lancing, the carbon charge and the

mass of the second scrap charge. The factors that are investigated include the cost

of electricity, increasing the upper bound on the arc energy and consideration of the

effect of batch duration on the profitability of operation.

77


3.5.1 Case O-1: Optimal Solution

This case study investigates how the operation of the furnace may be improved by

allowing the optimizer to determine the optimal manner of operation based on the

economic objective function where the profit is maximized. The formulation of (3.12)

to (3.16) is used, where the elements of the control vector u are P , FO2,brnr, FCH4,brn,

FO2,lnc, FC,inj. The final time, tf is assumed fixed in this case study but is included in

the optimization formulation in Cases O-3 and O-4. The specific constraints imposed,

in addition to the model equations, are

Input constraints:

Pmini (t) ≤ Pi ≤ Pmax

i (t)

Fmini (t) ≤ Fi ≤ Fmax

i (t)

Endpoint constraints:

msolid(tf ) ≤ ε

ycarbon(tf ) ≤ Y maxc

Path constraints:

Twall ≤ Tmax

Vsteel ≤ Vfurnace.

The input constraints ensure that the flows and power addition are maintained within

realistic bounds. The burner input flows are allowed to move between their upper and

lower bounds except when the furnace is charged, at which time the maximum flow

of the O2 is decreased due to the open roof; the flow of natural gas will be adjusted

accordingly in the case of an economic objective and the upper bound is thus left

unchanged. The initiation of carbon injection and the oxygen lancing is constrained

by the fact that there needs to be a base of liquid steel before injection begins. The

time of initiation of these flows, used in this case study, are the same as that used

78


for the base case; however, the flows were allowed to fluctuate between maximum

and minimum bounds. Providing an upper bound for the voltage tap as a function

of the heat’s progression is very complex since it can be affected by a number of

factors that are not accounted for in the model. Examples of such factors include arc

stability, collapsing of the scrap pile and the electrodes arcing to the wall. Therefore

it was assumed that the power trajectory in the base case provides an upper bound

on the power usage for the optimized case, with a lower bound of zero. The endpoint

constraints ensure that all steel is melted and that the carbon concentration is at the

desired level. The path constraints maintain the wall temperature below its maximum

bound and ensure the maximum capacity of the furnace is respected when scrap is

added.

The optimal solution was determined using the objective function given by (3.10).

The effect of using the alternative objective criterion, given in (3.11) is compared

later in Case O-4. The optimal solution, obtained using the objective function in

(3.10), improved profitability of the heat by 21% on a $/time basis compared to the

base case. The major improvement is realized through the optimizer determining the

most efficient quantity and timing that material/energy are to be added during the

heat. However, it should be noted that some of the improvement in this scenario will

be due to fact that the optimizer can predict the exact conditions given by the model

whereas in reality, operators need to slightly over-compensate due to a lack of process

data to ensure that endpoint conditions are met.

Figure 3.1 presents the offgas composition profiles for the base case and the optimal

scenario; where the circles (◦) and the crosses (×) represent the base case and optimal

scenarios respectively. The mole fractions have been scaled to their maximum value.

From this figure it is apparent that much less CO is produced in the optimal scenario.

The optimizer recognizes the energy potential of the CO and therefore conserves CH4

and combusts the CO instead. The net result is an economic saving due to the lower

79


usage of CH4 and also a cleaner and smaller volume of offgas. The model makes some

assumptions with regard to the freeboard being a perfectly mixed reactor, therefore

this combustion could be more difficult to achieve in reality.

0 10 20 30 40 50 60 700

0.5

1

CO

0 10 20 30 40 50 60 700

0.5

1

CO

2

0 10 20 30 40 50 60 700

0.5

1

O2

time [mins]

Base CaseOptimal Case

Figure 3.1: Case O-1: Offgas data.

Figure 3.2 compares the input profiles for the base case and the optimized case. The

data shown in this case study and the rest of the case studies in this section have been

scaled for propriety reasons. The optimal solution indicates that slightly less power is

used in the second charge, this energy is instead obtained from chemical sources. The

FO2 (burner oxygen) profile indicates an initial higher usage than the base case and

then steps down to a lower level as the batch progresses. This behaviour is consistent

with what is expected since the effect of the burners is related to the volume of solid

scrap in the furnace. As melting occurs the effectiveness of the burners decreases

80


and therefore so should their usage. The sudden increase in the burner usage at

approximately t = 25 mins is due to a carbon charge at approximately t = 20 mins.

This charge of carbon results in a large increase in the amount of CO produced from

the bath. The complete combustion of CO to CO2 known as post-combustion, is a

valuable source of energy and thus the optimizer puts the O2 flow at its upper bound

to maximize the use of this energy source. Studying the input profile for FO2 from

time t =45-60 mins, it is evident that a higher amount of O2 is needed in the furnace

than was used in the base case scenario. The optimizer takes advantage of the fact

that CO is again being produced from the bath due to lancing and harnesses this

energy by increasing FO2 . However, during the last stages of lancing (t > 70 min) the

optimizer keeps the burner at its minimum level even though there is a high presence

of CO (see Figure 3.1) since there is no benefit gained from combusting this CO as

all the steel is already at the required temperature. While it may be argued that it

would be better to lance earlier and thus harness this energy, this would result in

production of more FeO thus impacting the yield. One could alternatively argue that

more injection carbon could be added; however, this would be at an additional cost.

This is a major benefit of using optimization with a fundamental model since it is

able to make these economic tradeoffs in determining the most profitable mode of

operation.

3.5.2 Case O-2: Cost of Power

In Ontario, Canada, the cost of power typically fluctuates between $0.03-0.15 through-

out the day according to the grid demand and has been known to reach $0.50 kWh in

extreme cases. The cost of other utilities such as natural gas, oxygen and carbon are

more stable and fluctuate over much longer time horizons. Figure 3.3 below is repro-

duced from data collected from the Independent Electricity System Operator (IESO)

(2005) and indicates the typical daily price fluctuations together with the demand in

81


0 10 20 30 40 50 60 700

50

100

P arc

0 10 20 30 40 50 60 700

50

100

F O2

0 10 20 30 40 50 60 700

50

100

F CH

4

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Base Case Optimal Case Constraints

Figure 3.2: Case O-1: Input profiles.

Ontario, Canada. As expected the price is correlated with demand and can therefore

be predicted with some degree of certainty based on recent market trends and the

time of day.

This case study investigates the impact that the cost of power has on the operation of

the furnace. The time based objective function, given in (3.10), was again used and

the duration of the heat was fixed in these scenarios. Scenario A is obtained using

the input profiles that were optimally determined at an electricity cost of $0.03/kWh.

A second scenario (scenario B) considers the case when the power costs $0.15/kWh.

Figure 3.4 compares the input profiles of scenarios A and B, which illustrates how the

82


0 5 10 15 200.04

0.06

0.08

0.1

0.12

Hou

rly E

nerg

y P

rice

[$/k

Wh]

0 5 10 15 201.7

1.8

1.9

2

2.1

2.2

2.3x 1019

Hou

rly D

eman

d [M

W]

Time of Day

Figure 3.3: Hourly energy cost and demand for Ontario, Canada: March 10, 2005.

optimizer manipulates the operating strategy to compensate for the increased power

cost. As expected, the optimizer attempts to reduce the amount of electrical energy

used when the cost of power is higher; this is evident towards the end of the heat,

after approximately 63 minutes. The optimizer compensates for the more expensive

electric power cost by substituting electrical power with chemical power by using a

higher burner flow in the initial 18 minutes and again between t = 39− 57 minutes.

A third scenario (scenario C) considers the extreme situation where the price of power

increases to the point where it is no longer profitable to operate the process using the

current practice. As the cost of electricity rises, the profitability of the heat decreases

until the break-even point is reached at $0.35/kWh. Re-optimizing the process at

this higher price allows a profitable strategy to again be realized. It is thus possible

to conclude that the optimal mode of operation is dependent on the costs at the time

the heat is carried out.

83


0 10 20 30 40 50 60 700

20

40

60

80

100

120

Par

c

Low cost powerHigh cost powerConstraints

0 10 20 30 40 50 60 700

20

40

60

80

100

120

F O2

time [mins]

Figure 3.4: Case O-2: Input profiles for comparing scenarios A and B.

The value of the objective functions for each case is reported in Table 3.1. The

values are reported relative to the base case for proprietary reasons; they have been

normalized by dividing by the profit for base case study and multiplying by 100.

Therefore numbers larger than 100 indicate batches which are more profitable than

the base case and numbers less than 100 are less profitable; note the power cost in

the base case was $0.05/kWh. Negative values indicate batches that would operate

at a loss. The optimal solution was determined at each of the given power prices;

these numbers are reported in bold in the table. The objective function was then

evaluated for each of the other power costs using the same input trajectories, which

corresponds to the other values reported in the same row in the table. The number

in bold has the highest value in the column, which is expected since the input profiles

have been optimized at that particular value of the cost of power.

Studying the data in the table it appears that while there are benefits to be gained

84


Table 3.1: Comparison of Profit Based on Power Cost Relative to Base Case

$0.05/kWh $0.03/kWh $0.15/kWh $0.35/kWh

Case Study ($/min) ($/min) ($/min) ($/min)

O-1 122.5 129.8 81.9 -17.9

O-2 A 122.1 130.2 80.6 -5.1

O-2 B 122.0 129.3 83.0 2.3

O-2 C 120.4 128.0 82.4 6.5

from optimizing the operating practice when the power cost is low, the major benefit

in changing the operating practice would be realized as the power cost increases to

$0.15/kWh and above. This case study motivates modifying the operating practice

based on the current power costs. The results from these studies indicate that shifting

the current practice to favour increased burner usage during peak demand times and

scaling back again as the power cost drops would result in increased profitability.

The majority of EAF operations are mostly controlled by the operators who aim to

achieve consistency batch after batch through repeating the same operating practice.

Taking this into consideration, it may be desirable to encourage the night-shift oper-

ators to follow an operating practice determined from optimizing at a low power cost

and to give the day-shift operators an operating practice determined at a moderate

power cost. This averaging strategy would theoretically be less effective than consid-

ering the current power levels and changing the practice every heat, but considering

that EAFs involve minimal automation and are still predominantly run by operators,

it would likely yield better overall performance.

85


3.5.3 Case O-3: Increased Upper Bound on Power Input

The purpose of this case study is to increase the bound on the maximum power by

20% after the second charge and include the end time as a variable in the optimization

problem. This study illustrates the potential benefits that may be gained from using

a higher tap setting on the transformer or equivalently the potential benefits that may

be obtained from purchasing a transformer with a higher rating if current practice

is limited by the maximum tap setting available. It is possible that this mode of

operation may not be realizable in practice as other factors not accounted for in the

model might place more stringent constraints on the power input.

The optimization was carried out using the form of the objective function given in

(3.10) and in this case study the duration of the heat was an optimization variable.

The input trajectories for this case study are shown in Figure 3.5. The most interest-

ing variable is the arc power, which appears to favour operating at its upper bound

until about 65 minutes into the heat when it retreats for the last several minutes

of the heat. The temperature profile for the wall is shown in Figure 3.6. The fur-

nace wall temperature imposes a constraint on the operation of the furnace. Good

foaming between 45 and 55 minutes prevents the wall temperature from escalating

even though the power input remains at the upper bound. However, after 55 min-

utes as the oxygen lancing increases and the carbon in the bath is exhausted the

foaming decreases and the wall temperature begins to increase. Subsequently, the

final interval requires the power to be reduced to prevent the wall temperature from

violating the constraint at the end point. If this temperature bound were not present

the solution would require much less foaming to protect the walls and the heat would

be marginally shorter since it would not have to reduce the power level during the

final stages. The amount of power and burner fuel used in this case study was more

than was required in case O-1. However, the profit obtained for this study was much

greater than case O-1; the normalized profit was $138.2/min in this case compared to

86


$122.5/min in case O-1. The main difference in the number was due to the reduction

in the time of the heat, which was approximately 10 minutes.

0 10 20 30 40 50 60 700

50

100

P arc

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Base Case Optimal Case Constraints

Figure 3.5: Case O-3: Input profiles.

Figure 3.6: Case O-3: Wall temperature.

This case indicates that there may be significant benefit to increasing the upper

bound on the power input when the end time is a variable, particularly if a higher

87


production rate is desirable. During times when the steel market is good and high

throughput rates are desirable, this approach does provide some promising opportu-

nities. However, during market slumps when the steel price is low and inventories

are near capacity, throughput rates are not a concern. During such periods it would

not be desirable to follow the strategies advocated in this particular case study since

the cost per ton of liquid steel is greater, due to the larger quantities of power and

burner fuel utilized in reducing the heat time. Evidence of the higher cost is provided

in Table 3.2, which compares the normalized profits from case studies O-1 and O-3

on a per time and a per ton of liquid steel basis.

Table 3.2: Comparison of Profit for Cases O-1 and O-3 Relative to Base Case

Case Study $/min $/ton

O-1 122.5 107.8

O-3 138.2 105.1

3.5.4 Case O-4: Comparison of Objective Criteria

This case study investigates the optimal solution determined using the objective cri-

terion of the profit per minute, given by (3.10). These results are then contrasted

against the results obtained using an objective criterion of profit per ton of liquid

steel, given by (3.11). The results of these two objective criteria were found to yield

similar results when the duration of the batch is not included as an optimization

variable in the problem. Thus the batch duration was included as an optimization

variable in this case study.

In both the partial and full discretization formulations for DAE systems, time may be

included as a variable in the optimization problem by allowing each time interval to

vary between given upper and lower bounds. This enables the optimizer to manipulate

the timing of tasks and also the final time of the batch. As was described earlier, a

88


model for the electrical system has not been included and thus the base case power

trajectory was used as the upper bound for the power input. Therefore only certain

time intervals were allowed to vary; specifically when the power input was non-zero

the time intervals were only allowed to increase or decrease once the power level

had reached its maximum (approximately t=22 minutes in the base case) in the first

charge and again towards the end of the heat. This prevented the optimizer from

reducing the duration of intervals where the power is being stepped up in favour

of the intervals where the power is at the maximum tap setting, thereby ensuring

that the integrity of the upper bound is maintained. The duration of the following

intervals were included as variables:

• The first interval restricts the initiation of the burner flow to a moderate value

and is considered a startup condition, therefore the duration of this interval is

not included as a variable. The next four intervals were allowed to vary between

0 and 3 minutes.

• The interval where the power level is at its maximum during the first charge was

allowed to vary between 0− 10 minutes. This allows more/less arc energy to be

added to the initial scrap as well as timing of the scrap charge to be brought

forward or delayed relative to the start of arcing.

• The last 4 intervals of the heat were allowed to vary between 0 − 3 minutes.

This allows the batch to finish early if endpoint conditions have been met.

This is further illustrated in Figure 3.7, where the 3-minute intervals that were in-

cluded as optimization variables are bounded with circles (◦).

The results of the two scenarios are presented in Table 3.3, where the numbers have

again been normalized using the base case as the reference. The first two columns

compare the objective functions; the numbers in bold correspond to the criterion that

89


0 10 20 30 40 50 60 70 800

20

40

60

80

100

time [mins]

P arc

0 10 20 30 40 50 60 70 800

20

40

60

80

100

time [mins]

Par

cVariable Time Intervals Base Case

Base CaseVariable Time IntervalsFigure 3.7: Case O-4: Time intervals included as optimization variables.

was used in the objective function for that particular scenario. The lower yield in

scenario A compared to scenario B indicates the trade-off made between productivity

and yield. The last two columns summarize the total amounts of power and burner

oxygen used per batch in each scenario. Comparing the profits for the two scenarios it

is evident that there is a large difference obtained in each case. The results highlight

the importance of matching the operating objectives with the appropriate objective

function formulation in the optimization problem. The operating objectives are typ-

ically determined at two levels; at a lower level the objectives may be determined

within the process itself. This may be the case if the EAF is the bottleneck or alter-

natively if one of the downstream units are delaying production. At a higher level,

external factors may be considered when determining operating objectives such as the

market conditions. For example, during periods of strong demand, high throughput

is a priority since all product will be sold. In contrast, when the market is slow it is

more important to maximize the profit of each ton produced. During such periods

it is difficult to sell the product and quality not quantity is more important; during

such time the strategy suggested in scenario B would be appropriate.

Figure 3.8 presents the power and oxygen burner profiles for scenario A. The time

90


Table 3.3: Comparison of Different Objective Criteria Relative to the Base Case

Case Study $/min $/ton Yield∫

Parc

∫FO2

O-4A 155.5 105.0 100.8 101.4 82.4

O-4B 129.0 109.6 101.5 95.9 37.7

intervals corresponding to Figure 3.7 are also displayed; where the interval has been

reduced to zero, only a single circle is shown. In this scenario, time was a priority and

the 15 minutes assigned to preheating in the base case is eliminated except for the

3-minute interval which was not allowed to vary. The time interval just prior to the

second charge was extended from 3 minutes to 4.8 minutes and the four intervals at

the end of the heat were reduced from 3 minutes to 2.7, 0, 0 and 0 minutes respectively

to give an overall batch time of 58.0 minutes. This scenario uses more power than

the base case due to the expanded power-on time during the first bucket but there

is an overall saving in burner usage over the duration of the heat as a result of the

reduction of the preheat time.

Figure 3.9 shows the input profiles for scenario B, where the objective criterion was to

maximize the profit per ton of liquid steel. The 15 minute preheat from the base case

study is reduced to just 6.3 minutes in this case; furthermore, the burner flow is quite

low. The variable time interval just prior to the second charge was extended from 3

minutes to 8.0 minutes and all the time intervals at the end remained unchanged at

3 minutes. The figure shows that the power input terminates after approximately 62

minutes, however, refining continues for another 10 minutes until the end of the heat

is reached at approximately 72 minutes.

This study illustrates some interesting characteristics; firstly when the mass of liquid

steel is incorporated into the objective function, the proportion of electrical energy

contributing to melting the steel is increased in favour of using the burners. An

increased burner usage ultimately increases the amount of FeO produced due to the

91


0 10 20 30 40 50 60 700

20

40

60

80

100

120P ar

c

0 10 20 30 40 50 60 700

20

40

60

80

100

120

F O2

time [mins]

Variable Time Intervals Optimal Case Constraints

Figure 3.8: Case O-4: Comparison of inputs for scenario A.

resulting higher temperature of the scrap, which increases the degree of oxidation

and also due to the increased presence of O2 in the furnace. Thus favouring lower

burner usage is an expected result since, when optimizing on a per ton of product

basis, the yield is important and the duration of the heat has no implication on

profitability. Another interesting result is that by increasing the power-on time in the

first charge, the power-on time for the second charge was reduced to give an overall

reduction, when compared to the base case, in the total amount of power required.

The following two reasons are given as explanations for why extending the power-on

time during the first charge is beneficial:

1. During the initial stages of arcing, the tap setting is kept low because of poor arc

stability, which is a result of the arc striking solid scrap material. Towards the

92


0 10 20 30 40 50 60 700

20

40

60

80

100

120P ar

c

Variable Time Intervals Optimal Case Constraints

0 10 20 30 40 50 60 700

20

40

60

80

100

120

F O2

time [mins]

Figure 3.9: Case O-4: Comparison of inputs for scenario B.

end of the heat the bath is at its hottest and reradiates a significant amount of

energy back to the furnace. Therefore the most efficient net gain of energy into

the steel from the electrical power is during the intermediate stages when the arc

is stable and striking a liquid bath but there is still unmelted scrap present. The

unmelted scrap moderates the bath temperature, which decreases the amount

of energy radiated to the furnace; scrap may also intercept radiative energy

from the bath. Thus the optimizer extends the time when the energy absorbed

per unit of energy input is at its highest. The optimizer took advantage of this

in both scenario A and B.

2. Extending the power-on time for the first charge allows a larger volume of molten

metal to form. The molten metal acts as an efficient energy source since it will

transfer heat to the solid scrap by both conduction and radiation. At the time

93


of the second charge the larger mass of molten metal that has accumulated has

a greater heat capacity, which makes it a more effective energy source since it

will better maintain its temperature. It may be argued that it would be more

efficient to transfer the energy directly from the arc to the solid scrap, instead

of from the arc to the molten metal and then to the solid scrap as is advocated

here. However, the power input is limited when arcing to solid scrap due to

stability constraints. The argument given in point 1 above and the fact that

the molten metal has very good contact with the scrap further substantiates

the explanation why it is beneficial to extend the power-on time during the first

charge.

3.5.5 Case O-5: Fixed Preheat Duration

In this case study a fixed amount of time is made available for the preheat; however,

the furnace is not required to use the burners during this period. The case study

simulates the situation of a twin shell operation that shares a single transformer

and set of electrodes. This is a fairly common industrial installation, consisting of

two furnaces which operate sequentially; while the electrodes are being used on one

furnace, the second furnace will be charged and then preheated. In this case study the

time based objective function, given in (3.10) was again used and the batch duration

was fixed.

In the base case scenario, the preheat time was approximately 15 minutes, in this

case the time available for preheating is extended to 25 minutes. At time zero the

furnace will have a 20 ton hot heel and 122 tons of cold scrap. Figure 3.10 shows the

optimal power and burner input profiles as determined for this case.

As expected the solution does not keep the burners on for the full time period available

for the preheat. The reasons for this are twofold: firstly the burners will be most

94


0 10 20 30 40 50 60 700

50

100P

arc

0 10 20 30 40 50 60 700

50

100

F O2

0 10 20 30 40 50 60 700

50

100

F CH

4

time [mins]

Optimal CaseConstraints

Figure 3.10: Case O-5: Comparison of inputs.

efficient when the scrap is cold and secondly overuse of the burners decreases the

overall yield due to the oxidation of the steel.

A surprising result is that the burner operation is initiated from t = 0 and heats

up the scrap for a period of 5 minutes, then allows it to cool for approximately 15

minutes before being turned on 3 minutes prior to the power being started. This

counter-intuitive result is explained by the fact that the hot heel is also present from

the initial point and loses heat to the cold scrap and furnace over the duration of

the 25-minute period. The heel, however, transfers heat much more effectively to the

scrap than the burner since the transfer takes place via radiation and conduction as

opposed to convective transfer in the case of the burner. Furthermore, the energy

in the heel is a more valuable energy source than the burners since it can overcome

95


the latent heat of fusion to melt the solid scrap. At time t = 0, there exists a large

difference between the temperature of the heel and the temperature of the scrap and

furnace walls resulting in large amounts of energy being transferred from the heel to

the solid scrap. As energy leaves the heel it cools and subsequently its ability to melt

the scrap diminishes. Starting the burner early takes advantage of the fact that heat

transfer from the burners is most efficient when the scrap temperature is low and

also preserves more of the valuable energy in the heel by reducing the temperature

difference between the scrap and the heel; more energy is thus available in the heel

for sensible heating above the range of the burners and also for overcoming the latent

heat of fusion. Thus the net effect of starting the burner at time t = 0 more than

compensates for the cooling that occurs while the burner is switched off.

To verify this result the optimal solution was simulated again except the burner action

specified for the first interval was shifted later to time t = 15 minutes, as shown in

Figure 3.11. The results from the simulation results revealed that the heat was not0 10 20 30 40 50 60 700

20

40

60

80

100

120

F O2

0 10 20 30 40 50 60 700

20

40

60

80

100

120

F O2

time [mins]

Test CaseConstraints

Figure 3.11: Case O-5: Burner input for test case.

able to meet the endpoint conditions in this case and melt all the scrap material

by the final time. This is a significant result because it shows that the timing of

the burner operation and not just the duration are important when long periods are

available for preheating.

96


3.5.6 Case O-6: Event-activated Constraint Formulation

In the previous case studies the manner in which the oxygen lancing and carbon

injection bounds were formulated in the optimization problem was determined from

the base case operation. At approximately 40 minutes the upper bound was relaxed

and the lance/injection rate was allowed to increase to the actuator limit. However

in industry, the plant operators typically begin active lancing i.e. increasing the flow

from its lower bound, once the cumulative power input has exceeded a predetermined

threshold, which for the industrial operation under consideration is 50MWh. The

cumulative power input into the furnace is used as an indicator variable to determine

when lancing may begin. This case investigates a better method of constructing the

upper bound for oxygen lancing and carbon injection by relating it to the cumulative

power input instead of time. The time based objective function, given in (3.10) was

used for the scenarios in this case study and the batch duration was fixed.

Two methods for formulating this constraint are presented here, the first of which is a

more direct representation and can be used with any type of objective function. The

second formulation yields a more efficient solution; however, it is only valid when an

economic criterion is used as the objective function. The formulation development is

shown only for oxygen lancing since the carbon injection formulation is analogous.

These event-activated constraints cannot be related directly to the discrete intervals,

since they must be able to shift along the time horizon across the intervals as the

optimizer tests various solutions. This complication was overcome by implementing a

hyperbolic tangent switching function to limit the flow rate to the minimum flow until

the cumulative total power,∫

PT dt, exceeds 50MWh, after which the upper bound is

97


increased to the maximum actuator value. This was formulated as,

W =1

2tanh

[γ

(∫PT dt− 50

)]+

1

2(3.23)

Fmaxlance,O2 =

(Fmax

A − FminA

)W + Fmin

A (3.24)

where W is the switching variable between 0 and 1 and γ is a factor that controls the

rate of switching. The term,(∫

PT dt− 50)

forces W to switch when the cumulative

total power reaches 50 MWh. Fmaxlance,O2 is the upper bound for the lance oxygen and

FmaxA and Fmin

A are the maximum and minimum actuator values respectively. It is

important to formulate the problem so that the upper bound reduces to a minimum

bound and not zero, since during the heat a minimum flow level must be maintained

to prevent blockages from slag or molten metal solidifying in the nozzle. Figure 3.12

illustrates the switching function described by (3.23) for γ = 50, which forces the

switch to be very sharp.

0 10 20 30 40 50 60 70 80 90

0

0.2

0.4

0.6

0.8

1

W

Cumulative Power [MWh]

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100


Nor

mal

ized

Fla

nce,

O2

Flance,O2max

Constraints

Figure 3.12: Switching function as a function of cumulative power.

In addition to the actuator constraints the flow must be less than or equal to the

upper bound,

Flance,O2 ≤ Fmaxlance,O2. (3.25)

Figure 3.13 illustrates the upper bound as determined in (3.24), together with the

actuator bounds. The inclusion of (3.25) forces the minimum actuator bound to be

98


0 10 20 30 40 50 60 70 80 900

20

40

60

80

100


Nor

mal

ized

Fla

nce,

O2

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100


Nor

mal

ized

Fla

nce,

O2

Flance,O2max

Actuator Bounds

Actuator BoundsF

lance,O2max

Figure 3.13: Event-activated constraint.

followed while the cumulative power is less than 50MWh. Once the energy level has

been exceeded the lance flow is free to move between the original limits.

The input constraint must be converted to an endpoint constraint in the formulation

to allow it to be enforced over the time horizon. The introduction of the following

set of equations force (3.25) to be obeyed:

d

dt(FM

lance,O2) = max(0, Flance,O2 − Fmax

lance,O2) (3.26)

FMlance,O2

(tf ) ≤ ε. (3.27)

FMlance,O2

(tf ) is a variable introduced to track the accumulation of the violation. This

formulation was found to be particularly unreliable, unless a good starting point was

provided to the optimizer. It is postulated that the need to pose the constraint as an

end point constraint led to its inefficiency.

Thus an alternative method for handling the event-activated constraint was formu-

lated. In this formulation a new variable, F ∗lance,O2

, is introduced that will be manip-

ulated by the optimizer between the actuator limits,

FminA ≤ F ∗

lance,O2≤ Fmax

A . (3.28)

The lance oxygen flow rate that actually enters the furnace, Flance,O2 , is related to

99


this new variable as follows,

Flance,O2 =(F ∗

lance,O2− Fmin

A

)W + Fmin

A . (3.29)

W is the switching variable and when equal to 1, then Flance,O2 = F ∗lance,O2

and thus

the value of the flowrate chosen by the optimizer corresponds to the value fed into the

furnace. However, when W = 0 then Flance,O2 = FminA and the value of F ∗

lance,O2is free

to take on any value in the range given in (3.28). To ensure that the integrity of this

formulation is maintained the objective must be to either minimize cost or maximize

profit. The new variable F ∗lance,O2

is substituted into the cost function in place of

the original variable, Flance,O2 . This again forces the equality, Flance,O2 = F ∗lance,O2

,

since otherwise this is equivalent to an economic penalty because a cost is incurred

for material that cannot be used.

In Case O-6A, the constraint was triggered at 50MWh. The objective function ob-

tained from using this formulation is comparable to that obtained in Case O-1; the

normalized objective value was $122.1/min in this case compared to $122.5/min in

Case O-1. The slight difference is expected because the optimal time initiation of

injection and lancing is not expected to coincide exactly with 50MWh; in fact the

base case allowed injection to begin at an accumulative power input of just 46.8MWh.

This constraint formulation encourages more energy to be put into the furnace earlier

in order to reach the 50MWh constraint and initiate lancing sooner; resulting in a

slight variation in the strategy compared to that of Case O-1. Figure 3.14 illustrates

the lancing and injection profiles together with their upper and lower bounds; the

switching variable W is superimposed over the figure to illustrate the activation of

the constraint. The lance injection begins as soon as the cumulative power reaches

50MWh.

This formulation was less efficient, from a computational viewpoint, than formulating

the problem with the constraints directly on the time intervals, as was done in Case

O-1. The optimization problem took 44 iterations and 4613 CPU seconds to solve

100


on a Intel Pentium IV 3.0 GHz processor, compared to 28 iterations and 2830 CPU

seconds for Case O-1.

0 10 20 30 40 50 60 700

20

40

60

80

100

120

Cin

j

0 10 20 30 40 50 60 700

20

40

60

80

100

120

O2,

lnc

time [mins]

Fi

WConstraints

Figure 3.14: Case O-6(A): Event-activated constraint.

In case O-6B, the cumulative power at which the switching function was triggered

was included as an optimization variable. (3.23) is modified as follows

W =1

2tanh

[γ

(∫PT dt− Pswitch

)]+

1

2(3.30)

where Pswitch is now a variable included in the optimization problem. The results

of this scenario indicated that initiating lancing after 44.9MWh of energy have been

added to the system was optimal; giving an objective function value of 122.6 compared

to 122.5 in Case O-1. This formulation took 6001 CPU seconds and 64 NLP iterations

to converge.

101


3.5.7 Comparison of Scenarios

Table 3.4 summarizes the profit per minute, the profit per ton of liquid steel and

indicates the objective function used in the case study. A column indicating whether

final time was included as an optimization variable in the formulation is also shown

together with other key indicators for each scenario. The values have been normalized

relative to the base case, but the units of the original variables are reported as a

source of reference. The value of the objective function is emphasized in bold for

each case.∫

P ,∫O2 are the cumulative power input and cumulative oxygen burner

flow respectively.

Table 3.4: Summary of Case-Studies Relative to Base Case

Objective Profit Yield∫

P∫O2 tf

Function [ $min

] [ $ton

] [%] [MWh] [m3N ] (Y/N?)

Base case (3.10) 100.0 100.0 100.0 100.0 100.0 N

Case O-1 (3.10) 122.5 107.8 101.5 93.3 90.9 N

Case O-2(A) (3.10) 130.2 114.7 101.5 97.5 77.0 N

Case O-2(B) (3.10) 83.0 73.1 100.8 91.9 102.1 N

Case O-2(C) (3.10) 6.5 5.8 100.8 90.7 107.3 N

Case O-3 (3.10) 138.2 105.1 100.9 100.2 96.3 Y

Case O-4(A) (3.10) 155.5 105.0 100.8 101.4 82.4 Y

Case O-4(B) (3.11) 129.0 109.6 101.5 95.9 37.7 Y

Case O-5 (3.10) 107.2 106.6 102.2 105.9 70.3 N

Case O-6(A) (3.10) 122.1 107.6 101.5 94.1 87.0 N

Case O-6(B) (3.10) 122.6 107.9 101.5 93.4 91.1 N

Comparing results between the base case and Case O-1, the main difference is the more

efficient use of the material and power additions. The second case study illustrates

the intuitive result that a higher electricity price favours a decrease in power usage

102


and a greater reliance on burner energy. A greater loss of Fe to FeO is observed

when the burner is more heavily utilized; evidence of this is seen by comparing the

steel yields for scenarios A and B presented in the second case study. The third

case study takes advantage of the less strict upper bound to improve profitability

by reducing the processing time. Case O-4 illustrates the importance of matching

the process objectives with the optimization objective criterion. These case studies

give an indication of the type of trade-offs that can be made in meeting different

objectives. In Case O-4B it is evident that the optimizer reduces the use of the

burner to maximize the profit on a yield basis when compared with Case O-4A. Case

O-5 differed from the other scenarios in that an extended period of time of preheating

was forced to occur. This study showed some interesting results as it forced the burner

to come on early to maximize its benefit by heating the scrap before it was heated

by the heel and also to preserve the more valuable energy in the heel. An interesting

observation is that the yield is greater than for Case O-4B, which considers the mass

of liquid steel in the objective function; however, inspection of the data in the table

reveals that this case utilizes a greater quantity of both electrical power and burner

oxygen, making it less profitable. The strategies obtained in Case O-6 and Case O-1

were very similar; however, Case O-6A shows a slight increase in power usage and

also a slightly smaller profit. The differences are expected due to the difference in

the formulation of the problem. The increase in power consumption in Case O-6A is

likely a result of the optimizer trying to force lancing to happen as soon as possible

by increasing the power input so that the cumulative power exceeds 50MWh sooner.

In Case O-6B, the cumulative power value at which the switch occurs was included

as an optimization variable; the optimal operating strategy closely mimics that of

Case O-1 although a small improvement is realized from initiating lancing earlier, at

44.9MWh instead of at 46.8MWh as was the case for Case O-1.

Each of the optimization case studies take approximately an hour to run, where

approximately 85% of the CPU time is spent on the integration of the sensitivity

103


equations. The actual time in each case depends on the proximity of the initial

solution to the optimum as well as the particular case being studied. In cases where

time is included as a variable in the optimization problem the solution time was

observed to be much longer, approaching 2 hours. The reason for this observation is

the variable time introduces an element of singularity into the problem, requiring the

solver to take smaller step sizes at each iteration. The particular objective function

used was also observed to influence the solution time. Studies using the objective

function based on the tons of liquid steel produced took somewhat longer to solve;

although the time difference was not significant. This is also an expected observation

since the objective function influences the nature of the search-space and the gradient

at each iteration.

3.6 Summary and Discussion

An EAF optimization formulation was presented, and its flexibility and potential for

process improvement illustrated through several case studies. Analysis of the results

reveals that by optimization of a detailed process model, tradeoffs inherent in the

EAF process operation can be quantitatively accounted for.

At the current state of the model, the optimization serves as a very useful tool for

determining the directionality of the process and steering it towards more profitable

operation. Already, case studies presented here have been used by plant personnel

to both motivate and design plant trials. Furthermore, the case studies illustrated

here show the potential of optimization and motivate both further development of

the model and also investment to enable collection of more data.

The data that were available for estimation for the model resulted in a fair amount

of model uncertainty and thus it is not possible to guarantee that the improvements

104


represented in Table 3.4 would be realized on the actual process through the imple-

mentation of the respective strategy. However, the strategies advocated indicate the

direction in which the process should be moved in order to improve profitability and

the corresponding trade-offs that are made.

In this work no consideration of plant-model mismatch or disturbances have been

considered in determining these optimal profiles, both of which may cause the nomi-

nally optimal solution to be suboptimal or infeasible when implemented. The sources

of uncertainty must be considered in the optimization problem if the inputs are to be

implemented directly. An uncertainty description of each source of uncertainty may

be propagated through the problem to ensure that constraints will be met for the

worst case of model parameter values or for the expected-value of parameter values

and so forth (Terwiesch et al., 1994). These formulations provide a degree of robust-

ness to the optimal input profiles ensuring the problem remains feasible subject to a

degree of plant-model mismatch and process disturbances; however, they also tend to

be overly conservative. Another alternative is to investigate implementing a feedback

controller that is capable of updating the inputs based on the current state of the

process; this work is investigated further in the following chapter.

105

Chapter 4

Nonlinear Model Predictive

Control

Following an introduction, a literature survey on methods appropriate for the control

of batch processes is given. A more detailed review of model predictive control is

then presented. This is followed by a description of the formulation and software im-

plementation of an economically driven nonlinear model predictive control algorithm

used for EAF operation. Several case studies are then presented to illustrate the

effectiveness of a feedback mechanism for reducing process uncertainty in the form of

process disturbances and model mismatch. Finally, some conclusions from this work

are drawn.

4.1 Introduction

Uncertainty enters the process as model-mismatch, process disturbances, unknown

initial conditions and measurement noise. In many cases the complete mechanisms of

the processes under study are poorly understood and are thus characterized as lumped

106


parameters in the model, which may result in unmodelled dynamics. Uncertainty

with respect to models may be structural or parametric; structural uncertainty is due

to either an incomplete or incorrect model and parametric uncertainty results from

incorrectly identified model parameters. Disturbances may arise from many sources,

for example feedstock variations, fluctuation in ambient conditions, measurement

biases and so forth. The process is also subject to slow-varying changes over time,

due to wear and other changing factors, which impact its operation. To counter

the effect of these uncertainties and ensure the operating strategy remains feasible a

feedback scheme that periodically updates the process inputs may be implemented.

Optimal operation can only be achieved in practice if the feedback controller updates

the control variable profiles using online optimization or methods that are invariant

to uncertainty.

4.2 Control of Batch Processes

Control problems for batch processes are typically posed as trajectory tracking prob-

lems (Morari and Lee, 1999). The objective in these applications is to control the

process along a pre-specified output trajectory that is based either on results from

offline optimization or gained from process experience. Another area of research ad-

dresses updating the trajectories online to re-optimize the process as it deviates from

the conditions corresponding to the nominal solution; this research area is referred to

as online optimization.

Two key classes of control strategies suited for control of batch processes are differen-

tial geometric methods and nonlinear model predictive control. Differential geometric

methods attempt to find an inverse of the nonlinear process by applying transforma-

tions to the states or control variables to convert the control problem to a linear

one; thereby enabling the application of linear control theory. Kozub and MacGregor

107


(1992b) studied the control of endpoint quality properties for a semi-batch emulsion

polymerization process using a feedback linearization scheme. McAuley and MacGre-

gor (1993) designed a nonlinear feedback controller using an input-output feedback

linearization transformation to control a polyethylene reactor. A drawback of these

methods is that, in general, a suitable transformation for a given nonlinear model

may not exist.

One of the earliest applications of nonlinear model predictive control (NMPC) was car-

ried out by Garcia (1984) who applied dynamic matrix control (DMC) and quadratic

dynamic matrix control (QDMC), for the case where hard constraints are present, to

a batch reactor. This scheme is appropriate for a process with time varying param-

eters and/or strong nonlinearities where the setpoint trajectory is pre-specified. The

nonlinear DAE model is linearized and updated as the states of the process change,

which are then used to modify the step response coefficients in the controller. These

local linearizations performed at each measurement enable the controller to better

handle the nonlinear and time-varying behaviour of the process.

Patwardhan et al. (1990) present one of the first applications of NMPC that uses

the full nonlinear model in the computation of the control variables. The authors

applied their algorithm to the start-up of a non-isothermal, non-adiabatic continuous

stirred-tank reactor (CSTR). Valappil and Georgakis (2002) use a NMPC algorithm

to handle model uncertainty by controlling the endpoint properties to within a control

region instead of a setpoint. The authors applied their algorithm to a simulated emul-

sion polymerization reactor in the presence of unmeasured disturbances and model

uncertainty. Nagy and Braatz (2003) developed a shrinking horizon NMPC algorithm

based on an economic objective function that minimizes a weighted sum of the nom-

inal performance objective, an estimate of the variance of the performance objective

and an integral of the deviation of the control trajectory from the nominal solution.

The method was applied to a simulated batch crystallization process and showed su-

108


perior performance compared to both open-loop optimal control and nominal NMPC

trajectory tracking implementations.

The optimization-based control schemes that have been described involve updating

the input trajectory by optimizing the process after obtaining new process measure-

ments. A particular challenge of these methods is obtaining computationally tractable

solutions that can be implemented in real-time. Eaton and Rawlings (1990) proposed

a method that uses a second-order Taylor approximation of the Karush-Kuhn-Tucker

(KKT) conditions for constrained optimization around the nominal solution. This is

used to estimate the sensitivity of the optimal solution with respect to the control

variables and can be solved as a system of equations to determine corrections to the

nominal control trajectories. Confidence intervals are obtained using the sensitivity

information and then used to determine when the model needs to be updated and

when the Taylor approximation is sufficiently accurate.

Gattu and Zafiriou (1999) developed a method to update the setpoint profile by

carrying out a single iteration of a gradient based search. The motivation of using a

single iteration is based on moving the system in the best direction without delaying

the action by waiting for convergence of the solution. A similar idea was used in

the work by Diehl et al. (2002), in which the online updates are also implemented

after a single iteration in order to obtain real-time updates. The authors used a

multiple-shooting method where Hessian and gradient information are obtained from

the nominal solution and are used to improve the quality of the iterations.

Abel and Marquardt (2003) use scenario-integrated methods together with a short

horizon MPC algorithm to address the online control problem. The algorithm allows

certain failure conditions to be addressed by defining a set of failure scenarios that

can be included into the model to simulate a failure. The computational requirements

for an online application are met by using a short prediction horizon. However, the

authors note that this assumption cannot ensure endpoint constraints will be met

109


and thus the method is limited to processes without endpoint constraints.

Other research has focused on obtaining conditions that are invariant under uncer-

tainty and optimality is achieved by using these conditions as references for the feed-

back control scheme. In these methods the control calculation is typically obtained

from the solution to an algebraic equation and therefore is available in real-time.

Visser et al. (2000) and Bonvin et al. (2001) characterize the nominally optimal so-

lution using Pontryagin’s maximum principle into a series of arcs, which are used to

determine the active constraints during each stage of the process. The inputs can

then be manipulated such that these constraints are kept active at the appropriate

stage in the process.

Sun and Hahn (2004) proposed model reduction techniques for DAE systems by

applying model balancing methods to eliminate the unobservable and uncontrollable

parts of the model, extracting only the input-output component from the model.

Traditional optimization and control techniques can then be performed on the reduced

model. Flores-Cerrillo and MacGregor (2002) use partial least squares (PLS) methods

to reduce the dimensionality of the problem to several variables, known as principal

components or latent variables, and developed a method that computes a single-shot

correction for batch processes in order to meet the endpoint criterion. Flores-Cerrillo

and MacGregor (2004) extended this approach to be able to adjust entire manipulated

variable trajectories. At pre-selected decision points, the final properties are predicted

using the PLS model together with the available online measurements to determine

if the desired specifications will be met. The trajectories are recomputed in the

reduced-space and then converted to the full space for implementation.

Batch-to-batch control takes advantage of the repetitive nature in which batch pro-

cesses operate in order to improve performance. Zafiriou et al. (1995) treat each batch

in the batch-to-batch problem as an iteration towards the optimum. After each batch

a gradient is approximated and a line search is performed to determine the step size

110


in the search direction. Clarke-Pringle and MacGregor (1998) developed a batch-

to-batch scheme for the control of molecular weight distribution (MWD) in a batch

operated polymer process. During each batch, the trajectory of a variable is tracked.

This trajectory is updated between batches using the derivative of the change in the

endpoint MWD with respect to the variable trajectory. A tuning factor is also intro-

duced to moderate corrections. Lee et al. (1999) combines the ideas of batch-to-batch

control with model predictive control. The authors use information from past batch

data together with information from the current batch to apply real-time control.

Srinivasan et al. (2001) proposed a method that updates conditions which are invari-

ant to uncertainty, based on the methods of Visser et al. (2000); Bonvin et al. (2001),

in a batch-to-batch manner.

4.2.1 EAF Control Applications

Craig and co-workers (Oosthuizen et al., 1999; Bekker et al., 2000) approached the

furnace control problem by selecting setpoints for three variables and using a model-

predictive controller to determine the optimal inputs to achieve these values. The

authors used a linearized state-space model; the controlled variables selected were

the steel temperature, CO composition in the offgas and the relative pressure in the

furnace. The setpoint for the steel temperature was its target value at the end of

the heat. The setpoints for the other two variables were selected to achieve a low CO

emission and a relative partial pressure in the furnace of -5kPa. The manipulated

variables used in these study were the offgas fan speed, the slip-gap width. Oost-

huizen et al. (1999) also used the rate of directly reduced iron (DRI) addition as a

variable, whereas Bekker et al. (2000) treated it as a measured disturbance. Oost-

huizen et al. (2004) extended this work to design a linear MPC algorithm for the

EAF based on an economic objective. This was achieved by translating the process

economics into weights based on the cost and the expected range of the controlled and

111


manipulated variables. These weights were then used in a quadratic MPC objective

function. Formulation of the controller in this manner enables economic tradeoffs to

be made during the computation of control actions in meeting the setpoints.

In contrast to the work by Oosthuizen et al. (1999, 2004), there are many applications

where regulatory control of only the electrode system is the key focus. Billings et al.

(1979) present a proportional-derivative controller to manipulate electrode position

with the aim of regulating the power. Morris and Sterling (1981) compared propor-

tional, proportional-integral-derivative and optimal control algorithms for the control

of the electrode position. Nadira and Usoro (1988) also controlled electrode position

in order to regulate power using a model algorithmic control, which is a variation of

model predictive control. King and Nyman (1996) developed a feedforward controller

to reduce power fluctuations by manipulating electrode voltage and current using

neural network models. Boulet et al. (2003) proposed a proportional controller for

the control of current and power levels.

4.2.2 Model Predictive Control

Qin and Badgwell (2003) give the following definition for a model predictive con-

troller: “ ... a class of computer control algorithms that utilize an explicit process

model to predict the future response of the plant.” The model used can be either a

detailed fundamental model or an empirical model. For continuous processes where

the objective is regulation, a linear model is generally used; nonlinear models find

application in batch processes and continuous processes where the servo problem is

of interest.

Model predictive control has found widespread use in industry for the following rea-

sons:

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• It is an intuitive algorithm.

• Multivariate interactions and time delays are explicitly accounted for by using

a process model to predict the future state of the process.

• Constraints on the input and output variables are easily handled in the formu-

lation.

• The control objectives are directly specified, which provides considerable flexi-

bility to the algorithm.

• It is appropriate for both linear and nonlinear control.

The dynamic matrix control (DMC) algorithm presented by Cutler and Ramaker

(1980) is often credited as one of the first MPC algorithms. This algorithm uses a

linear step-response model to relate the predicted outputs to past and future input

changes. The basis of the DMC algorithm is the determination of M future control

variable moves such that the difference between the setpoint and the model predicted

response is minimized over a certain time horizon (prediction horizon) in a least-

squares sense. Garcia and Morshedi (1986) developed the quadratic dynamic matrix

control (QDMC) algorithm; this work extended the DMC algorithm to explicitly

incorporate process constraints on the inputs and the outputs through formulating

the optimization problem as a quadratic program (QP).

The optimization of the process is carried out over the control horizon, M , with

only the first control action being implemented. After the implementation of this

control action, the plant outputs are measured and a disturbance estimate computed.

The optimization is then repeated using the new information. Predictive control

involves solving a sequence of open-loop dynamic optimization problems; the feedback

is introduced through the disturbance estimate after an update of the current process

state is obtained (Bequette, 1991; Qin and Badgwell, 2003). For stable processes

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the disturbance estimate is commonly computed as the difference between the most

recent measured output and the corresponding predicted output, and applied over the

prediction horizon. This method of model updating can provide offset-free setpoint

tracking in the presence of plant-model mismatch and also for any constant additive

disturbances that enter the process.

For unstable processes the model can rapidly diverge from the process and therefore

the bias update technique described cannot be used. Muske and Rawlings (1993)

analyzed the bias update in the context of the Kalman filter and illustrate that

this model provides no feedback for the process states; hence the poor performance

observed in unstable systems. Qin and Badgwell (2003) suggest including state or

input disturbance models in the estimation to address this issue.

Nonlinear Model Predictive Control

Linear control techniques assume constant process gains and dynamics and conse-

quently have limited application on highly nonlinear processes or processes that op-

erate over a wide range of conditions, such as batch processes. Nonlinear control

methods account for the process nonlinearities and are thus able to provide a single

strategy that can be applied over a wide operating range.

Nonlinear model predictive control (NMPC), an extension of the general MPC algo-

rithms, uses nonlinear dynamic models; a major challenge for these problems is ob-

taining the solution of the open-loop optimal control problem at each control point.

At each control calculation the NMPC problem corresponds to a dynamic optimiza-

tion problem and any of the methods discussed in Section 3.3 are suitable for its

solution. Path constraint enforcement may be a problem in the nonlinear formu-

lation; this is typically addressed through constraint softening where the constraint

violation is penalized in the objective function (Qin and Badgwell, 2003). The general

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formulation for the regulatory NMPC problem is given below,

minu

J =N∑

j=1

||y(t + j)− yref ||2Q

+M−1∑j=1

||u(t + j)− uref ||2S +M−1∑j=1

||∆u(t + j)||2R (4.1)

subject to:

x(t + 1) = f(x(t), u(t), t) (4.2)

y(t) = g(x(t), t) (4.3)

y ≤ y(t + j) ≤ y ∀j = 1, . . . , N

u ≤ u(t + j) ≤ u ∀j = 1, . . . ,M − 1

∆u ≤ ∆u(t + j) ≤ ∆u ∀j = 1, . . . ,M − 1

where x is an n-vector of states, y is a p-vector of measured outputs and u is a m-

vector of manipulatable variables. yref and uref are the desired operating points for

y and u; upper and lower bounds are denoted as x and x respectively. The first term

of the objective function penalizes deviation of the future outputs; the second term

penalizes future inputs from the desired trajectories and the final term penalizes rapid

changes is the inputs. The weighting matrices Q, R and S can be manipulated to

obtain the desired control objectives, where the norm terms in the objective function

are defined as:

||x||2P = xT Px. (4.4)

For continuous processes with no fixed end point, the algorithm is most often posed

using a receding horizon strategy. This means that the prediction and control horizon

lengths remain the same as the controller moves from one sampling interval to the

next along the time horizon. In batch processes it is often necessary to use a shrinking

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horizon approach, in which case the prediction and control horizons will decrease as

the controller advances to the end of the time horizon.

In many cases the states are not available and it is thus necessary to infer the cur-

rent states from the available measurements. In order to reconstruct the states the

system must be observable, that is an appropriate relationship between the available

measurements and the state must exist. The most commonly used method for con-

structing the states of nonlinear systems is the extended Kalman filter (EKF) (Muske

and Edgar, 1997). The EKF is based on the theory of the Kalman filter which is used

to reconstruct the states of linear systems. The EKF uses the ordinary Kalman filter

to obtain the updated state when the new observation becomes available and the full

nonlinear differential equation model to obtain the predicted state x(t + 1|t).

Consider the nonlinear system given by (4.2) and (4.3). At each interval the states

are updated using the Kalman filter as each new observation, y(t), becomes available,

x(t|t) = x(t|t− 1) + K(t) [y(t)− g (x(t|t− 1), t)] . (4.5)

K(t) is the Kalman gain computed from:

K(t) = P (t|t− 1)GT (t)[G(t)P (t|t− 1)GT (t) + R

]−1

(4.6)

where R is the process noise covariance matrix, G(t) is the linearized measurement

function obtained by taking the partial derivative of (4.3) with respect to the current

state,

G(t) =∂g(x, t)

∂x

∣∣∣∣x=x(t|t−1)

. (4.7)

P (t|t) is the state covariance matrix,

P (t|t) = [I −K(t)G(t)] P (t|t− 1). (4.8)

Between sampling intervals (t and (t+1)) the state is propagated using the nonlinear

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dynamic system:

x(t + 1|t) = f(x(t|t), u(t), t). (4.9)

The state covariance matrix is updated according to

P (t + 1|t) = F (t)P (t|t)F T (t) + Q (4.10)

where Q is the measurement noise covariance matrix and F(t) is the linearized system

function obtained by taking the partial derivative of (4.2) with respect to the current

state,

F (t) =∂f(x, u, t)

∂x

∣∣∣∣x=x(t|t),u=u(t)

. (4.11)

When the system is not fully observable, the unobservable states are predicted from

the model states and are not updated based on measurements. The impact of this

will vary on a case by case basis because the importance and quantity of states that

are unobservable affects the ability of the model to track the process and hence will

impact the ability of the controller to meet its objectives.

Waldraff et al. (1998) use the observability matrix to determine the best positions

for locating additional sensors. However, observability only indicates whether states

are observable or not with providing information on the quality of the estimates. In

their analysis, Gagnon and MacGregor (1991); Muske and Georgakis (2002) use the

state covariance matrix, P (t), to quantify the state prediction error and investigate

the addition of sensors through minimizing the determinant of this matrix.

Kozub and MacGregor (1992b) show how it is possible to augment the system with

a stochastic state to account for unmeasured disturbances and model mismatch that

are non-stationary in nature. Kozub and MacGregor (1992a) advocate the addition of

the stochastic state to ensure that information from the measurements is fed back to

the model states to remove bias from the predictions. Kozub and MacGregor (1992b)

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show an example where a unobservable state converges to the true state with zero

bias, when the EKF is augmented with a stochastic state. This result is not true in

general, however, it does indicate the improved robustness properties gained by the

estimator through augmenting the system with a stochastic state.

4.3 Problem Formulation

The purpose of this study is to evaluate the effectiveness of a feedback controller for

handling process uncertainty. The predictive control algorithm was selected for this

analysis because of its centralized structure, which considers all inputs and outputs

simultaneously to determine all of the manipulated variables, and also its ability to

handle constraints.

The analysis is carried out using an economically driven nonlinear model predictive

control (NMPC) algorithm. This formulation differs from that shown in (4.1) in that

the objective function is comprised of the economic cost of the process at the end

of the horizon, as opposed to regulation around a predetermined trajectory. In this

work, the controller satisfies an economic objective directly in updating the input

trajectories. In contrast, the formulation given in (4.1) determines the input moves

based on regulation around a setpoint. The drawback of the regulation approach is

that following a trajectory that would have achieved optimal operation in the absence

of disturbances may be sub-optimal or lead to infeasible operation as disturbances

enter the process.

This work shares a number of similarities with that of Oosthuizen et al. (2004),

in that an economic objective function is used to trade off the manipulated inputs

to determine the best control strategy for the furnace. The key differences in this

approach are the explicit computation of the economics in the objective function and

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the use of a nonlinear control strategy.

Two key assumptions have been made in the this work:

1. Full noise-free state measurements of the system are available.

2. The computational time required to carry out the control calculation does not

impact its ability to implement the control actions.

The resulting performance can therefore be considered as an upper bound on the

achievable performance that may be obtained through model-based feedback control.

The controller uses the time based economic objective function (see (3.10)) with a

fixed batch time and optimization formulation that was used in Chapter 3 for the

open-loop optimal control problem solved at each sampling period. The controller

considers the following five inputs:

1. arc power;

2. burner flow - O2;

3. burner flow - CH4;

4. carbon injection flow;

5. oxygen lance flow.

4.3.1 Algorithm and Software Implementation

The NMPC algorithm was implemented by interfacing the following software : Mat-

lab, gproms and Microsoft Excel. The modelling and optimization was carried

out using the gproms software and this framework was maintained for simulating the

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feedback controller. In this discussion, the terms controller and optimizer are used

interchangeably since in this application the controller is performing optimization at

each control iteration.

The gproms models require the special gorun facility (Process Systems Enterprise

Ltd., 2004) to enable them to be compiled and run from a batch script. Two models

were compiled, one representing the true process and the other representing the model

of the process that is used by the controller. In each of the case studies that follow,

it is the true process model that is perturbed and the model for the controller is left

unchanged.

The algorithm is represented pictorially in Figure 4.1. At each control iteration, k, the

model of the true process is integrated, from time t = 0, using the past control inputs

and also the most recent optimization results for the current control variables. The

states corresponding to time of the next control interval, (k + 1), are then obtained

from the results of this simulation. Using Excel as an interface, the state variables

are read into the model that is used by the optimizer/controller. The optimization is

then carried out over a reduced horizon, i.e. from the current control interval to the

end of the heat. A control interval of 3 minutes was used. While a smaller control

interval is possible, it is unlikely that actions on the variables that are considered

would ever be implemented at greater frequency on this process.

At each interval the current state measurement from the process is used as the initial

state for the model controller. The results of the optimization are then exported to

Excel which acts as an interface for the model of the true process to read in the

updated control actions. Matlab is used to combine the individual elements and

iterate through the control points to run the controller/process simulations.

The nominally optimal solution is used as the starting point for the input profiles.

To reduce computational time at subsequent control intervals, the optimization at

120


Controller

Plant

x0

u

t(k) t f

u

t ft(0)

ExcelInterface

ExcelInterface

gPROMS Optimization

gPROMS Simulation

(k)

y

t ft(0) t(k+1)

Incrementk = k +1

t(k)

Figure 4.1: Schematic representation of controller formulation.

control interval k is initiated using the input profiles that were obtained at k − 1.

As the horizon shrinks the degrees of freedom in the optimization problem decrease;

subsequently the computational time at each progressive control iteration decreases

as well.

4.4 Case Studies

A series of case studies is presented here to illustrate the benefits of feedback con-

trol. In each case the true process is changed such that there will be some form of

discrepancy between the model used for control and the model of the true process.

121


The performance of the controller is evaluated by comparing the true cost function

obtained from the plant inputs against the following:

1. The optimal solution obtained from the nominal model as implemented on the

true process. This provides a lower bound on performance, where it is likely

that constraints may be violated and the true plant profit may be suboptimal.

2. The optimal solution based on the true process. This provides an upper bound

on the controller performance.

To ensure ease of comparison between the studies in this thesis, the value for the

objective functions reported in the case studies below are given relative to the optimal

profit obtained from the nominal model in the case study (Case 0-1), presented in

Section 3.5.

4.4.1 Case MPC-1: Model Uncertainty

In this case study the presence of plant-model mismatch is evaluated. A parameter

whose value has a major impact on the quality of the implemented nominal solution is

the parameter controlling the rate of melting, namely kdm (see (2.31)). An important

constraint of the optimization problem is to ensure that all material is melted by the

end of the heat; this constraint is always active at the endpoint. The value of the

parameter kdm for the true process was increased by 10%. The effect of this change is

a slower rate of melting of the solid scrap material. If this parameter were incorrectly

estimated it could result in an unnecessary waste of energy in the case where its

estimated value is less than the true value. If its value were estimated to be greater

than the true value, this may cause process delays, since the power would have to be

turned on again until all material has melted.

122


Implementing the nominal solution gave an objective function value of 96.42 relative

to Case 0-1, which has been normalized to 100. As a result of a smaller quantity of

scrap from the first charge melting, less void space was available in the furnace at

the time of the second charge. Subsequently, a smaller quantity of scrap is able to

be processed, which is the main reason for the lower objective function value. The

nominal inputs also resulted in an infeasible solution due to the presence of solid scrap

material at the end of the heat. Reaching the predicted end of the heat without all

the scrap having melted would be very inefficient since it would require the electrodes

to be powered on again. Furthermore, the extra delay would negatively effect the

throughput rate, which is a concern when the success of the operation is evaluated

on a per time basis.

The theoretical best solution is obtained by performing optimization on the true

process. The objective function value in this case is 98.29, which as expected is

less than the nominal solution obtained in Case O-1 due to the greater resistance to

melting. Comparing this solution to that obtained from implementing the nominal

solution on the perturbed process indicates that the process uncertainty will affect

the optimal operating policy.

Implementing the NMPC algorithm enabled an objective function value of 96.46 to be

obtained. This value is only marginally better than that obtained using the nominal

inputs. However, it is much closer to feasibility; studying the final column of Table

4.1 reveals that the mass of unmelted material at the end of the heat has nearly been

eliminated. It is interesting to note that the feedback controller is unable to achieve

feasibility. The reason for this is that the endpoint mass of steel is a binding constraint

and that model mismatch directly effects this constraint. Therefore the controller will

only achieve complete feasibility as the number of control actions toward the end of

the batch increases. In reality, all solid material must be melted and thus the power

would have to be turned back on, for both the nominal and feedback solutions, to

123


Table 4.1: Comparison of Results for Case Study MPC-1

Profit∫

P∫O2 Charge Violation

[ $min

] [MWh] [m3N ] [tons] [kg]

Nominal Strategy 96.42 99.7 107.5 97.67 30.06

Theoretical Optimum 98.29 106.8 113.6 99.97 0.000

Feedback Controller A 96.46 108.1 94.4 98.14 1.901

complete the heat. In the case of the nominal strategy, the larger mass of unmelted

material would take longer to become molten than in the feedback case and therefore

the difference between profit in each case would be greater. Figure 4.2 compares the

input profiles of the nominal solution with those obtained from the feedback controller.

The major difference between the two strategies is that the controller detects that

not all material has melted as it approaches the end of the heat and it therefore

maintains the arc power at a higher level near the end of the heat. The controller

also implements a small increase in the burner usage during the first charge, which

improves melting and allows the addition of a slightly larger second charge.

The controller was not able to perform as well as the theoretical optimum for the

reason that it continually over-predicts the ability of the furnace to melt the scrap.

The mismatch in this case study represents a very important parameter, since it has an

accumulative effect on the process and determines whether the endpoint conditions

can be met. During the initial stages of the heat very little melting occurs and

therefore the measurements during the initial stages of the heat provide little evidence

of the mismatch and thus the controller deviates little from the nominal solution.

Subsequently, it is only after approximately 15 minutes when the arc is turned on that

the controller is able to discern the difference in the true process. At this stage of the

heat, the burners have been below their maximum capacity for several minutes and the

controller has lost this extra capacity, which could have been used to compensate for

the slower melting rate. Furthermore, since the arc power is at its upper bound until

124


near the end of the heat, the controller is unable increase arc power. The combination

of these two factors prevent the controller from melting the same quantity of scrap

in the first charge as was achieved in the theoretical solution and thus the total mass

of scrap processed is less, resulting in a smaller profit. Evidence of this can be seen

by comparing the input profiles of Parc and FO2 for the feedback controller and the

theoretical optimal solution, shown in Figure 4.3. From these figures, it is evident

that the optimal solution was able to harness the extra capacity of the burners to

compensate for the slower rate of melting. This is again confirmed by the data in

Table 4.1, which shows a greater use of burner O2 in the theoretical optimal solution

compared to that obtained using the feedback controller.

0 10 20 30 40 50 60 700

50

100

P arc

0 10 20 30 40 50 60 700

50

100

F O2

0 10 20 30 40 50 60 700

50

100

F CH

4

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Nominal Solution Online Controller Constraints

Figure 4.2: Case MPC-1: Input Profiles compared to nominal inputs

Figure 4.4 shows the controller objective function value versus time for two different

control interval frequencies. Controller A has a control interval of 3 minutes and

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0 10 20 30 40 50 60 700

50

100

P arc

0 10 20 30 40 50 60 700

50

100

F O2

0 10 20 30 40 50 60 700

50

100

F CH

4

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Theoretical Optimum Online Controller Constraints

Figure 4.3: Case MPC-1: Input profiles compared to optimal inputs.

Controller B an interval of 5 minutes. The objective function value calculated at

each control point is computed from the actual implemented inputs and states up

until the current interval, tk and for the predicted inputs and states for t > tk. The

theoretical and nominal solutions are shown as lines of constant value, since these

results were determined offline they are independent of the current control interval.

The curves for the online controllers decrease monotonically, since at each successive

control interval the mismatch is fed back to the controller which is then able updates

the objective function based on the true process state. The result illustrated here,

where the controller with the greater control interval frequency (Controller A) is

superior, is expected. With more frequent control actions the controller is able to

detect disturbances sooner and thus take corrective action sooner as well. Data

comparing these two scenarios is shown in Table 4.2

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Table 4.2: Comparison of Different Control Intervals for Case Study MPC-1:

Profit∫

P∫O2 Charge Violation

[ $min

] [MWh] [m3N ] [tons] [kg]

Feedback Controller A 96.46 108.1 94.4 98.14 1.901

Feedback Controller B 96.33 108.8 93.8 98.09 2.250

Figure 4.4 also provides further evidence for the controller not being able to detect

the plant-model mismatch until approximately t = 19 minutes when the arc power

is turned on. Initially the control objective function value is close to 100, the value

of the base case (Case O-1) solution with no mismatch. As the heat progresses

the controller detects a very small difference, however it is only when the arcs are

turned on and melting proceeds rapidly that the controller detects the mismatch.

This coincides with the sharp drop in the objective function value between 18 − 21

minutes. After this the objective function value plateaus at the introduction of the

second scrap charge, but begins dropping again after approximately 40 minutes as

the rate of melting increases and the controller detects the mismatch.

4.4.2 Case MPC-2: Process Disturbance

Case MPC-2 investigates the effect of a disturbance entering the process. A likely

source for the disturbance is a bias on the O2 lance flow rate; here a bias of 5% relative

to the magnitude of the flow is introduced into the true process.

Implementing the nominal solution gave an objective function value of 99.32. The

higher than expected lancing flow rate resulted in inefficiency due to more O2 being

utilized than was necessary. However, the major loss is due to a decreased yield

because the increased O2 over oxidizes the steel.

Optimizing the process with the increased lance flow resulted in a different strategy

127


0 10 20 30 40 50 60 70

95

95.5

96

96.5

97

97.5

98

98.5

99

99.5

100

Nor

mal

ized

Con

trolle

r Obj

ectiv

e Fu

nctio

n V

alue

time [mins]

Online Controller AOnline Controller BTheoretical OptimumNominal Solution

Figure 4.4: Case MPC-1: Controller performance.

to that obtained in the nominal optimization study, Case O-1, because the elevated

lance flow allows the bounds on the flow to be violated. Within the level of accuracy

reported here, the objective function value was found to be the same as that of Case

O-1. However, less power and burner O2 were needed due to the increased O2 levels

from the lance.

The feedback controller performed well and was able to get near the theoretical so-

lution. As can be seen from the data in Table 4.3, the controller detects the extra

oxygen in the furnace and adjusts the oxygen burner flow accordingly. The carbon

injection and oxygen lancing profiles are compared in Figures 4.5 and 4.6. In these

figures, the actual implemented value is reported and thus the O2 lancing flow rate is

shown to violate the constraints.

128



Profit∫

P∫

FO2 Charge

[ $min

] [MWh] [m3N ] [tons]

Nominal Strategy 99.32 100.0 100.0 100.0

Theoretical Optimum 100.0 99.72 98.01 100.0

Feedback Controller 99.95 100.3 97.73 99.99

0 10 20 30 40 50 60 700

20

40

60

80

100

120

Cin

j

0 10 20 30 40 50 60 700

20

40

60

80

100

120

O2,

lnc

time [mins]

Nominal SolutionOnline ControllerConstraints

Figure 4.5: Case MPC-2: Input profiles compared to nominal inputs.

129


0 10 20 30 40 50 60 700

20

40

60

80

100

120

Cin

j

0 10 20 30 40 50 60 700

20

40

60

80

100

120

O2,

lnc

time [mins]

Theoretical OptimumOnline ControllerConstraints


4.4.3 Case MPC-3: Unknown Initial State

This case study investigates the effect of an uncertain initial state on the nominal

solution and the ability of the nonlinear predictive controller to adjust the nominal

conditions to re-optimize the process. The initial mass of scrap in the true process is

increased such that it is 5% greater than in the controller model. The change in the

initial mass is associated with a change in the bulk density of the scrap charge.

The nominal solution obtained an objective function value of 100.7, the extra mass

of scrap in the furnace allows the nominal solution to improve on the profitability

relative to Case O-1. The increased initial mass resulted in a greater total amount of

scrap being processed compared to Case O-1. This data is reported in Table 4.4.

The theoretical optimal solution had an objective function value of 102.2. Knowledge

of the increased initial charge allowed the optimizer to alter the operating strategy

130


0 10 20 30 40 50 60 700

50

100P ar

c

0 10 20 30 40 50 60 700

50

100

F O2

0 10 20 30 40 50 60 700

50

100

F CH

4

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Nominal Solution Online Controller Constraints

Figure 4.7: Case MPC-3: Input profiles compared to nominal inputs.

and increase the amount of energy to the scrap during the first charge. Relative to the

nominal strategy the optimal solution was able to increase the amount of available

space in the furnace at the time of the second charge and ensure a greater amount

of total scrap was melted; more burner O2 and also more arc power was required to

achieve this. However, the extra cost of these utilities and scrap was more than offset

by the profit obtained by producing more liquid steel.

The controller receives the error at the first measurement and is quickly able to

make corrections to the control variables to account for an incorrect state value.

Due to the nature of the difference, i.e. it enters the process at the beginning and

does not persist throughout the batch, it is easy for the controller to adjust for the

difference between the model and process and move the process very close to the

theoretical optimum, assuming the first measurements and subsequent action are

131


0 10 20 30 40 50 60 700

50

100P ar

c

0 10 20 30 40 50 60 700

50

100

F O2

0 10 20 30 40 50 60 700

50

100

F CH

4

0 10 20 30 40 50 60 700

50

100

Cin

j

0 10 20 30 40 50 60 700

50

100

O2,

lnc

time [mins]

Theoretical Optimum Online Controller Constraints


completed early in the heat. In the absence of noise and with full state knowledge the

true state of the system will be known after the first measurement is taken. Since this

is a once off disturbance, the controller is able to update the control policy after a

single measurement. Therefore to make this scenario more interesting the first control

action only takes place after 6 minutes, instead of the usual 3 minutes. The results in

Table 4.4 indicate that the controller is nearly able to achieve the theoretical optimal

objective function value. The same amount of scrap material is processed in both of

these cases, however, the controller requires an additional quantity of arc power and

burner fuel to achieve this.

In reality, if the state was unavailable at the initial time it is unlikely to become

available at a later time. Typically an observer such as a Kalman filter or a soft

sensor is used to predict the current state of the system. Error introduced into the

132



Profit∫

P∫O2 Charge

[ $min


Nominal Strategy 100.7 100.0 100.0 100.6

Theoretical Optimum 102.2 101.8 103.2 102.2

Feedback Controller 101.9 103.8 106.5 102.2

prediction of the states makes the problem posed here much more challenging since

the true state is not available. Therefore a further scenario is carried out where noise

is added to the state to simulate the estimation of the state using an observer. The

noise model used was a first order autoregressive model, AR(1), of the form:

Gn(z−1) =1

1− 0.95z−1(4.12)

giving an output,

wk = G(z−1)ak (4.13)

where ak is the forcing function. A random number generator is used as the forcing

function and the output added to the state as follows:

m′ss = mss + wk (0.005mss) (4.14)

where mss is the noise free value of the mass of solid scrap. (4.14) shows the magnitude

of the disturbance is scaled such that it is 0.5% of the magnitude of mss. The random

number is obtained from a normal distribution, with a mean of 0 and variance of

1. The noise model together with the random number forcing function is illustrated

in Figure 4.9, where the autoregressive nature is apparent. Studying this figure, it

can be observed that the measured state will be positively biased initially and then

negatively biased in the latter stages of the heat. The objective function obtained

from the feedback controller in the presence of noise was 100.9. As expected, the

133


0 10 20 30 40 50 60 70-4

-2

0

2

4

a k

time [mins]

0 10 20 30 40 50 60 70-4

-2

0

2

4

Wk

time [mins]

Figure 4.9: Case MPC-3: Noise model.

controller was not able to perform as well as the case where the noise-free state is

available.

Figure 4.10 compares the inputs profiles for the arc power and burner of the controller

with and without noise. The over-prediction of the mass of scrap in the initial stages

of the heat causes the controller to hold the burner at its upper limit. However,

the extra burner energy is less efficiently utilized and is thus effectively wasted. The

measurement at approximately t = 65 mins is again an over-prediction of the mass of

solid steel. The controller reacts to this by increasing the power level relative to the

base case. In subsequent measurements the noise causes the mass to be either under

predicted or close to its true value and thus the power is no longer required, which is

a result of the increased power and burner usage that was applied earlier in the heat.

The data in Table 4.5 compares the results of the controller in the absence of noise to

134


the controller when noise is present. From studying the data and the input profiles

in Figure 4.10 it is possible to see how in the case with noise, performance of the

controller degrades as it over compensates and then backs off again as the error on

the state measurement changes from positive to negative.

Table 4.5: Comparison of Controller Performance

Profit∫

P∫O2 Charge

[ $min


Noise-free Controller 101.9 103.8 106.5 102.2

Controller with noise 100.9 103.2 115.2 102.4

0 10 20 30 40 50 60 700

20

40

60

80

100

120

Par

c

Controller without noiseController with noiseConstraints

0 10 20 30 40 50 60 700

20

40

60

80

100

120

F O2

time [mins]

Figure 4.10: Case MPC-3: Comparison of controller performance.

135


4.5 Chapter Summary

In this chapter a nonlinear model predictive control algorithm was presented to act

as a profit based feedback controller for the electric furnace process. Process distur-

bances and model error can cause the nominally optimal trajectories to be suboptimal

or even cause process infeasibilities. The predictive control algorithm that was im-

plemented used an economic objective function and re-optimizes the input profiles

based on the most recent measurements from the process. Several case studies were

carried out to illustrate the effectiveness of the algorithm to re-optimize the process

and maintain feasible operation.

The use of an economic objective function requires that the controller meet the end-

point conditions and respect path constraints in such a way so as to optimize the

profitability of the furnace. An alternative strategy is to penalize the violation of the

end point quality conditions; however, this cannot guarantee that the strategy will

be optimal in an economic sense.

This work investigated the ability of an online controller to reject disturbances and

maintain the furnace operating in an optimal manner. The case studies show that the

controller was particularly effective in the case where the uncertainty was the result of

an error in initial state, so long as the state in question could be accurately observed

at a later time. In the case of model-mismatch the controller was able to improve

upon the solution obtained from implementing the nominal inputs. The particular

mismatch in this study made it difficult to correct, since the degree of the mismatch

could only be detected as melting progressed and one of the main inputs that could

compensate for this discrepancy, namely the arc power, was already saturated at its

upper bound. The controller also proved its ability to account for the disturbance

that was investigated, namely the bias on the oxygen lancing rate.

This work is based on the assumption of full state knowledge at each measurement

136


point. In reality, the states would have to be estimated from the available measure-

ments. The state prediction covariance matrix is often used as a measure of the

estimation error in the states (Gagnon and MacGregor, 1991; Muske and Georgakis,

2002). Kozub and MacGregor (1992b) show how tighter control can be achieved when

more accurate knowledge of the true states is available. In the event that states are

not observable the effect will likely be more pronounced since no correction based on

process measurements can be used to update these estimates.

137

Chapter 5

Conclusions and Recommendations

5.1 Conclusions

The discussion below summarizes key findings from this work.

A detailed model of the electric arc furnace has been developed. This model is based

on fundamental principles, although a degree of empiricism has been introduced to

model relationships where the real mechanisms are either too complex to be modelled

or where insufficient information is available.

Key model parameters have been estimated using available industrial data. However,

further measurements during the progression of the heat would be useful. It is hoped

that this work will provide incentive for further instrumentation of industrial EAF

operations so that advantage may be taken of these tools.

The model framework presented allows for the inclusion of further detail. Poten-

tial enhancements include detailed models for predicting the melting of scrap in the

furnace and improved prediction of decarburization and slag foaming. However, the

138


generally limited amount of data available for parameter estimation in an industrial

setting should be carefully considered during model refinements.

The model was incorporated into an EAF optimization formulation, and its flexibility

and potential for process improvement illustrated through a number of case studies.

Analysis of the results reveals that by optimization of a detailed process model, trade-

offs inherent in the EAF process operation can be quantitatively accounted for.

Process disturbances, model mismatch and other sources of uncertainty may result in

the nominally optimal profile being sub-optimal or even cause the process to violate

constraints. Feedback control in the form of a nonlinear model predictive controller

was implemented to address this issue. The NMPC algorithm uses an economic

objective function and re-optimizes the input profiles during the batch based on the

most recent measurements from the process. Several case studies were carried out to

illustrate the effectiveness of the algorithm at reducing the uncertainty in the form of

disturbances and plant-model mismatch.

5.2 Recommendations for Further Work

This project opens many opportunities for future work in all of the major sections;

the most interesting avenues for further work are summarized below.

5.2.1 Modelling

With regard to modelling, the framework allows for considerable detail to be in-

cluded. However, the limited number of measurements may restrict the benefits that

are gained by introducing more complexity in the model. Furthermore, the inclusion

of detail in the form of spatial variations results in a distributed system which in

139


general is a computationally intensive problem to solve. An area where the model

performance could be greatly improved is through increasing the quantity of avail-

able measurements. A particular challenge in this work would be to determine which

measurements would improve the observability of the system at the least cost/effort.

The design of soft-sensors from measurements which are only recorded on an infre-

quent basis could also be used in this work. Another interesting problem would be to

preferentially select favourable measurements based on the reduction in uncertainty

that could be achieved through their availability. The reduction in uncertainty could

then be traded off against the cost of obtaining these measurements.

Assuming no further process measurements were available, there are two areas that

could be investigated to address the lack of process information. Firstly, it is quite

possible that many of the parameters in the model are correlated with one another.

Therefore the parameter estimation could be carried out in a reduced dimension. In

this case it may be possible to include all parameters in the model estimation but

only need to estimate fewer total parameters, which would enhance the conditioning

of the estimation problem. A technique such as canonical correlation analysis (CCA)

(Krzanowski, 1988) could be used to determine the reduced-space parameters. Here,

CCA could be used to find the linear combinations of the model parameters most

correlated with linear combinations of the output data of the model. The significant

combinations then define the reduced space. A second option would be to use very

detailed mechanistic models from the literature to estimate particular parameters in

the model. Future modelling work should focus on incorporating the electrical system

into the model. This was not included into this work due to the complexities and

large uncertainties associated with the electrical system. Ideally, this work would

focus on developing a model relating the power inputs (arc length and transformer

tap voltage) not only to the energy usage, but also to the bounds on performance.

More specifically, arc stability is one such factor that would determine the voltage tap

and arc length. However, the arc stability can vary greatly depending on the scrap

140


temperature, the conductivity of the scrap, the arc length and so forth, and thus

is subject to both causal and random effects. Including models of the local control

system of the electrodes is also an important task in order to be able to capture the

operation of the electrical system accurately. Other seemingly random events, such as

scrap cave-ins, make the development of realistic models more challenging. However,

ongoing research focusing on the subprocess in the furnace, such as scrap melting

(Guo and Irons, 2005) and carbon injection (Ji et al., 2002), provide a fundamental

understanding and thus aid in model development.

5.2.2 Optimization

An interesting optimization study would be to investigate the scheduling of the fur-

nace. In particular the usage of scrap mix throughout the day would be an interesting

problem because it effects the overall daily economics since influences the grades pro-

duced and also the resource usage for each heat. For example a particular kind of

scrap that is high quality could be divided between many heats to boost the overall

grade quality by an amount depending on its availability or it could rather be kept to

produce a higher grade steel on its own. This problem would likely be formulated as

a mixed-integer programming problem. In this scheduling problem, the full nonlin-

ear dynamic model would not be necessary and simpler empirical relationships that

capture the timing and efficiency of each stage of operation could be used to obtain

a computationally tractable mixed-integer programming problem.

Another challenging scheduling problem would be to investigate the operation of a

twin shell furnace system. Twin shell furnaces share resources such as electrodes

and cranes for charging scrap, therefore to maximize the throughput requires careful

management of these limited resources. There are two interesting problems that can

be investigated, the first involves the sharing of limited resources such as the electrodes

141


and crane. The second problem is more challenging and investigates scheduling the

furnaces such that the load of the offgas system is not exceeded. For example, there are

limitations on the level of preheating that can take place in one furnace when lancing

and arcing is taking place in the other furnace. This limitation is a result of the high

temperatures that are reached when all these tasks are carried out simultaneously. In

this case it may be optimal to operate the individual furnaces in a suboptimal way so

as to optimize overall performance. Due to the complexity of this problem it would

be necessary to use the full nonlinear dynamic model in order to accurately predict

the effects of lower preheating, lancing and power levels necessary to meet the offgas

constraint limitations.

Another valuable study would be a rigorous sensitivity analysis of the optimized

strategies. In particular it would be valuable to know which variables have a large

impact on the objective function and over which intervals these variables are most

important. Knowledge of when variables have little effect on the objective function

is also valuable information.

5.2.3 Control

An obvious extension to the work section on control would be to remove the assump-

tion of perfect state knowledge and use a state estimator to update the controller

model.

A more interesting area of research would be to investigate model reduction techniques

that would allow the control calculations to be performed in real time. Reduced-space

methods offer a promising approach for providing computationally tractable solutions

to the optimization problems of large, complex process models and hence for making

online control a possibility. The PLS methods developed by Flores-Cerrillo and Mac-

Gregor (2002, 2004) offer an attractive solution for providing real-time online control.

142


However, a drawback of the method is its empirical basis preventing extrapolation be-

yond the data upon which the model has been built and thus optimization is limited.

Therefore an interesting area of research would be to investigate extending the con-

fidence region of the model by augmenting the process data with data obtained from

a validated fundamental model. Augmenting the process data with carefully chosen

simulations and/or optimization scenarios from the fundamental model could signifi-

cantly improve the “knowledge” of process constraints and also increase the range of

the model. An additional benefit of using the PLS method is that it eliminates the

need to explicitly build a state observer; the observer is developed implicity in the

construction of the PLS model. In PLS the model’s confidence region is well defined

and therefore it is possible to impose this as a constraint and use the PLS model

within an optimization framework, as was shown by Yacoub and MacGregor (2003).

The contribution would be an online nonlinear model predictive control algorithm for

a large-scale system.

143

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154

Appendix A

Model Details

A.1 Model Parameters

Table A.1 gives the values of the parameters used in the model.

155

PhD Thesis - R.D.M MacRosty, Chemical Engineering Section A.1

Table A.1: Model Parameters

Parameter Value

γ 8.50E-03

km 5.53E+01

kT1 1.35E-02

kT2 1.00E-04

kT3 1.07E-03

kP 1.11E-03

kdm 4.43E-01

kdT 1.21E-01

kPO25.99E-01

kdc 6.94E-02

θL 7.50E-01

156


A.2 Non-ideal Thermodynamics

The activity is determined using activity coefficients,

ai = xiγi (A.1)

where ai is the activity, xi is the mole fraction and γi is the activity coefficient. The

activity of the non-ideal slag was determined using the regular solution formalism

using data for the interaction energies obtained from the literature (Ban-ya, 1993).

RT ln γi =∑

j

αijX2j +

∑j

∑k

(αij + αik + αjk) XjXk (A.2)

where Xi is the cation fraction and αij is the interaction energy between cations i.e.

(i-cation)-O-(j-cation). The subscripts j and k refer to the solutes other than i in

the solution.

The activity for the metal phase was determined using the unified interaction pa-

rameter model (Bale and Pelton, 1990) using interaction parameter data also sourced

from the literature (Sigworth and Elliot, 1974).

ln γi = ln γoi + ln γsolvent +

N∑j=1

εinXj (A.3)

ln γsolvent = −1

2

∑j=1

∑k=1

εjkXjXk (A.4)

where subscripts j and k refer to the solutes other than i in the solution.

157


A.3 View Factor Calculation

The view factors shown in this section were obtained from the text by Siegel and

Howell (2001). The view factor of the dome-shaped roof remained constant and was

computed from the geometry of a spherical cap as F1,1 = 0.20. The view factor from

the roof to the wall was determined as the base of a right circular cylinder to the

inside surface of the cylinder,

F1,2 =hw

r

√1 +

(hw

2r

)2

− hw

2r

(A.5)

where hw is the exposed wall height and r is the radius of the furnace. The view

factor of the inside walls to themselves was approximated as a right circular cylinder:

F2,2 =

(1 +

hw

2r

)−

√1 +

(hw

2r

)2

(A.6)

The view factor of the inside walls to the bath was determined from the inside walls

of a right circular cylinder to a disk of a smaller radius whose axis is perpendicular

to that of the cylinder,

F2,4 =1

4R(H4 −H2)

((X2 −X4)−

√(X2

2 − 4R2) +√

X24 − 4R2

)(A.7)

where R = r2/r4, Hi = hi/r4 and Xi = Hi + R2 + 1. r2 and r4 are the radii of the

cylinder (walls) and disc (bath) respectively. h2 and h4 are the distances from the

top of the cylinder to the disc and from the bottom of the cylinder to the disc. The

view factor of the bath to the roof was determined as two parallel co-axial discs of

different radii,

F4,1 =1

2

X −

√X2 − 4

(R2

R1

)2 (A.8)

where Ri = ri/hT and X = 1 + (1 + R22)/R

21 and ri is the radius of surface i and hT

is the distance between the bath and the roof. The view factor of the bath to itself

is zero since its a planar surface i.e. F4,4 = 0.

158


A.4 Melting model derivation

The following energy balance was implemented for each zone,

d

dt(Ez) = Qz +

n∑i=1

Fi,zHi,z

∣∣∣∣∣in

−n∑

i=1

Fi,zHi,z

∣∣∣∣∣out

(A.9)

where Qz is the heat flow added to zone z; Fi,z is the molar flow of component i

to/from zone z and Hi,z is the corresponding enthalpy. The energy holdup at any

time is computed as:

Ez =n∑

i=1

ni,zHi,z. (A.10)

A methodology similar to the work of Bekker et al. (1999) was used for the energy

balance and the melt rate of the solid-scrap. For the solid-scrap there is no reaction

and using a mass basis instead of a mole basis (2.19) reduces to,

d

dt(Ts) =

Qss (1− Tss/Tmelt)− M inscrap

∫ Ts

ToCp,FedT

[mssCp,Fe] kdT

(A.11)

where Tss and Tmelt are the solid-scrap and melting point temperatures. Qss (1− Tss/Tmelt)

is the fraction of energy entering the steel that contributes to sensible heating, with

the remaining fraction of energy contributing to the melting of the scrap.

The derivation of (A.11) from an energy balance is given here.

dE

dt= Q−Ws +

∑Fi,0Hi,0 −

∑FiHi (A.12)

Also,

dE

dt=

d∑

NiHi

dt=∑

HidNi

dt+∑

NidHi

dt(A.13)

but

dHi

dt= Cpi

dT

dt(A.14)

159


where Cpi has a molar basis. In the absence of reaction,

dNi

dt= Fi,0 − Fi = Fcharge − Fmelt (A.15)

Thus,

dE

dt=∑

Hi(Fi,0 − Fi) +∑

NiCp,idT

dt(A.16)

Therefore,

Q−Ws +∑

Fi,0Hi,0 −∑

FiHi =∑

Hi(Fi,0 − Fi) +∑

NiCp,idT

dt(A.17)

Simplifies to,

Q−∑

Fi,0(Hi −Hi,0) =∑

NiCp,idT

dt(A.18)

Rearranging,

dT

dt=

Q−∑

Fi,0(Hi −Hi,0)∑NiCp,i

=

Q−∑

Fi,0

T∫T0

Cp,idT∑NiCp,i

(A.19)

(A.11) follows.

160

put modelling optimization and control of an electric arc

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