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I
EXECUTIVE SUMMARY
INTRODUCTION/BACKGROUND
Steel industries face two main problems: steel quality and energy consumption.
Both problems are linked to the reheating furnace. Many scholars have proposed
several mathematical models for the furnace and heating process. To study these
problems, construct a mathematical model for furnace and steel slab temperature is
necessary. This paper applies the control algorithm on proposed model and evaluates
the performance in MATLAB/SIMULINK which is a cheap way to study before put
into the actual process. The control strategies include PID feedback control and expert
experience feed forward control.
AIMS AND OBJECTIVES
Project Aim:
1. To find an appropriate heating curve for slab.
2. To look at what performance can be achieved with PID approaches and feed
forward control based on simulating model
Project Objectives:
1. To obtain an appropriate model of walking-beam furnace and the slab heating
model in MATLAB.
2. Research PID controller and feed forward control.
3. Implement PID and feed forward control on the model.
4. Evaluate potential approaches (strengths and weaknesses).
I
ACHIEVEMENTS
This paper proposes furnace and slab temperature model and applies PID
feedback control and expert experience feed forward control for that. Finally, simulate
that in the MATLAB/SIMULINK.
CONCLUSIONS / RECOMMENDATIONS
This paper first constructs furnace and slab temperature model and then simuliate
in MATLAB/SIMULINK and then proposes dynamic optimization control strategy of
the reheating furnace set value based on PID feedback control and expert experience
feed forward compensation. Finally, compare these two control strategies result and
prove the latter one is better.
Although this paper proposed the mathematical model of the furnace and the
furnace temperature dynamic optimization settings for a certain research, the
following areas for further work will be done due to the complexity of the heating
process.
I
ABSTRACT
To improve slab quality and decrease the energy consumption, it is necessary to
optimize the heating curve of slab. This paper presents a study of slab and furnace
temperature modeling and corresponding control strategies. Considering the cost and
the difficulties in applying control strategies in real production line, modeling in
MATLAB/SIMULINK first is the best choice. When the model is constructed, PID
with the feed forward compensation is applied for the model which performance is
satisfactory.
Keyword:
Reheating furnace model, Slab temperature model, PID, Feed forward
compensation.
TABLE OF CONTENTS
ABSTRACT .............................................................................................................................. 3
Chapter 1- Introduction ...................................................................................................... 1
1.1. Background and Motivation ...................................................................................... 1
1.1.1 The development of furnace .............................................................................. 1
1.1.2 Furnace temperature control .............................................................................. 2
1.2 Literature review .............................................................................................................. 3
1.3 Problem ............................................................................................................................ 6
1.4 Aims and Objectives ........................................................................................................ 6
1.5 Project Management ........................................................................................................ 7
Chapter 2- Model of the walking-beam reheating furnace .............................................. 9
2.1 Introduction of walking-beam reheating furnace ............................................................. 9
2.2 The model of slab temperature....................................................................................... 10
2.2.1 The optimization of slab temperature curve............................................................ 10
2.2.2 Slab temperature tracking ....................................................................................... 12
2.3 The model of furnace temperature ................................................................................. 15
2.4 Optimal setting for furnace temperature ........................................................................ 18
2.5 Simulation ...................................................................................................................... 20
2.5.1 The entire model ..................................................................................................... 20
2.5.2 Set point of furnace temperature ............................................................................. 21
2.5.3 Slab temperature model .............................................................................................. 22
2.5.4 Furnace temperature ................................................................................................ 25
2.5.5 Disturbance ............................................................................................................. 29
2.5.6 Feedback control ..................................................................................................... 30
2.6 Conclusion and new problem ......................................................................................... 32
Chapter 3- The optimization of furnace temperature set point based on feed forward
compensation
3.1 Introduction of feed forward compensation ................................................................... 33
3.2 Theory of feed forward compensation ........................................................................... 33
3.3 Simulation ...................................................................................................................... 36
3.3.2 Result ...................................................................................................................... 41
3.4 Validation ....................................................................................................................... 43
3.5 Conclusion ..................................................................................................................... 44
Chapter 4- Summary ......................................................................................................... 45
Reference ................................................................................................................................ 47
1
Chapter 1- Introduction
1.1. Background and Motivation
In steel industries, reheating furnace brings a huge amount of energy
consumption. Many scholars worked on energy saving control problem of reheating
furnace and come out with variety of optimal control strategies [1-11]. However, due
to the characteristics of steel industry, there always be unpredictable parameters
during the heating furnace design, or advanced control algorithms cannot be tested
directly in the actual process. Hence, it is necessary to develop a mathematical model
of reheating furnace which can be used not only to determine some undetectable
parameters according to the real process but also to apply some advanced control
algorithm in offline simulation providing foundation for online control.
1.1.1 The development of furnace
Reheating furnace is the main equipment used in steel industry rolling slab by
heating to a certain temperature distribution. According to the different ways of
heating it can be divided into two types: cycle and continuous. Continuous heating
furnace is most widely used in the current production. Cycle furnace is heating slab in
a fixed position; it does not apply to the case of mass production. A continuous
heating furnace is heating slab during the slab moves from the furnace entrance has
been moved to its outlet, and in this process.
With the increase in international demand for steel, mill toward high efficiency
and large capacity development, which corresponds to increase furnace load in the
limited space increased heating capacity. Therefore, continuous furnace is developed
towards the multi-stage, and thus appeared the walking-beam reheating furnace.
2
1.1.2 Furnace temperature control
With the rapid development of computer technology, scholars from various
countries made use of computer technology to optimize the metallurgical furnace
control, such as computer-controlled mathematical furnace model and the optimal
combustion control, and achieved some economic benefits. However, due to the
complexity of the actual production system and complex slab heating process affected
by various factors, the model is non-linear distributed parameter system. In general,
advanced computer-controlled furnace strategy is not mature and there is rarely
success application in furnace control.
Furnace mathematical models generally can be classified into empirical ones and
theoretical ones: empirical model of the furnace is to obtain main factors may reflect
the furnace based on analyzing a large number of field experiments and statistical data,
which is relatively simple with narrow applications and inability to meet modern
multi steel production; while theoretical one is to formulate mechanism model of slab
heating process through the finite element analysis or discrete the slab by finite
difference, then the unknown parameters in the equation can be determined according
to the experiment results. However, there are great difficulties to establish an accurate
mathematical model due to the complexity of the heating process and modeling
methods shortcomings.
The model of furnace is actually a mathematical description of thermal processes,
which can reveal the basic law of thermal processes occurring in the furnace to
determine the quantitative relationship between the parameters of heating process. It
can be used to study the thermal theory, furnace design and thermal process
optimization by computer. In steel rolling production, in order to meet the slab heating
quality and yield requirements, it is necessary to establish the reliable automatic
control system to monitor the temperature of the slab accurate and directly. Since the
3
temperature distribution inside the slab still cannot be achieved on real-time online
testing, the steel temperature is estimated by slab heating process mathematical
models so that optimal control of slab heating temperature can be achieved.
1.2 Literature review
Continuous furnace mathematical model have been utilized for online control in
a few developed countries in the late 1960s. In the mid-1980s, research on
mathematical models began to be more advanced; the study focused more on the
automatic control strategy. However, continuous furnace mathematical model is still
the basis to achieve optimal control of the furnace. [1, 2]
Slab heating process involves the gas flow, fuel combustion, heat transfer,
thermal conductivity, and billet internal oxidation, decarbonization and other complex
physical and chemical phenomena, which depends on the furnace structure,
production operations and many other relevant factors. Such a complex thermal
process is difficult to be described by simple mathematical equations and also more
difficult to solve with internal and external random factors interfering and large inertia,
pure hysteresis nonlinearity distribution. Therefore, no matter for online controlling or
offline design or calculation, the model should be simplified according to the purpose.
For different subjects and research purposes, many scholars have proposed
various forms of furnace mathematical models. Misaka. J.& Takahashi.R.[3] made use
of the total heat absorption rate method to establish a mathematical model for the
prediction of the slab temperature and achieved some energy savings. Pike, H. E.&
Citron, S. J.[4] utilized distributed parameter theory for modeling and applied
approximate lumped parameter model to study the static and dynamic optimization of
furnace model. Wick, H. J.[5] took use of Kalman filter technology to estimate slab
temperature distribution, but the surface temperature of the slab can be obtained by
4
this method which limits its further use.
A.Kusters [6] proposed a parameter estimation using multivariate methods and
established a multisession walking-beam furnace ARX model. This method claims
that the furnace is divided into six zones based on the structural characteristics of the
furnace, the furnace temperature model of each zone is established taking into account
the mutual coupling of each zone. Finally, least square method is applied to obtain
various parameters.
Yoshitani, N.& Ueyama,T.& Usui, M. [7] developed an optimal furnace control
system consist of the best heating state model, slab temperature model when unloaded
and optimal set point to solve the problem of slab variety specification large changes
or the precise control of temperature and temperature uniformity. They proposed two
method to reduce the energy consumption and improve the quality of the slab heating:
first one is to modify the temperature of the slab heating curve in real time by the
online simulator based on a nonlinear mathematical and distributed parameter model;
Second one is adopting some means of accelerating optimal process to make the
control effect more obvious. Such model is widely used.
Yongyao Yang, YongZai Lv, etc. [8-9] proposes furnace discrete state space
model and optimal control theory based computer control strategy. First, list PDE and
its associated two-dimensional boundary for slab thermal heat conduction, then
transform a series of subsystems associated with large discrete state equations using
the system decomposition and discretization methods, and the slab temperature
distribution at any position at each time can be calculated in real time according to the
actual parameters furnace and each section slab initial parameters. Furthermore taking
this model as a basis, design optimization strategy based on heuristic continuous
furnace computer control system. Control strategy is divided into two parts: the steady
state optimization calculation and dynamic compensation setting value for furnace
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temperature. Steady-state optimization used heuristic search methods, while the
dynamic compensation is based on feed forward - feedback principle in order to
achieve lower power consumption.
With modern rolling to a continuous, large-scale, high-speed, high precision and
multi-species direction, originally simply applied some of the control algorithm DDC
controls have not meet production needs. Because these algorithms are combustion
control value from the perspective of the control design, and the advanced computer
control of the furnace, since the set value is in accordance with certain indicators
calculated optimal performance, and its combustion control with strong servo system
features. For optimal control settings, because the furnace complex conditions of
production, the control system under different conditions may be different quality
requirements, the use of a single setting optimization mode often can not make it in a
variety of conditions have reached the optimal control performance. To this end,
Yangyong Yao, Liang Jun, etc. [10] proposed a multi-mode control scheme furnace
settings. Based on the fundamental theorem of heat transfer and energy balance
principle, the use of time, space discretization technique established for estimating the
temperature distribution in the furnace billet heating furnace nonlinear discrete state
space model, based on the optimization of certain propositions, using heuristic search
strategy path extension to solve it. Meanwhile, in the furnace temperature setpoint
based on optimal control settings increase the fuel flow control, in order to achieve
multi-mode oven setting control.
Dirk. S.& Arend. K. [11] proposed the system which has two components: the
slab temperature calculation model and the controller to achieve the desired
temperature. The error of output of the mathematical model is less than ± 20 degrees
out of approximate 95% of the slab temperature calculation, adjust the actual
temperature of the slab according to the slab condition in the furnace in each zone and
6
the slab ideal heating curve. System has two control loops: the main loop is the
furnace temperature set point of each zone, which is obtained by calculated by the
actual temperature and the target temperature; auxiliary loop is set value of fuel
calculation. The system achieved better performance in Tata Iron & Steel Company
(India).
Other bachelors [12-15] also studied the furnace temperature set point dynamic
compensation, which is based on feed forward - feedback principle, to compensate
furnace temperature setting point according to the selected quasi-steady-state
conditions and the current differences between the actual working conditions and set
point. The main factors considered are rolling rhythm, the mean temperature
difference of slab in preheating zone and heating zone, the estimated and the optimum
temperature distribution of the slab and the detect signals of the slab surface
temperature. Utilizing feed-forward compensation to correct for furnace billet speed
fluctuations, and using state feedback to amend for other factors caused the slab
temperature distribution and optimum temperature distribution deviation. In order to
decrease the impact of model error and reinforce the adaptability and robustness of
the system, the real-time signal of slab surface temperature when unloading is
collected as a secondary feedback signal further to compensate.
1.3 Problem
As discussed in section 1.2, there are quite a lot of strategies proposed by many
scholars. Here comes the problem: which strategy should be applied leading to better
performance? As the difficulties in obtaining data in real process and applied the
strategy for actual system, simulating in computer seems to be important.
1.4 Aims and Objectives
Project Aim:
7
1. To find an appropriate heating curve for slab.
2. To look at what performance can be achieved with PID approaches and feed
forward control based on simulating model
Project Objectives:
1. To obtain an appropriate model of walking-beam furnace and the slab heating
model in MATLAB.
2. Research PID controller and feed forward control.
3. Implement PID and feed forward control on the model.
4. Evaluate potential approaches (strengths and weaknesses).
1.5 Project Management
Chapter one introduces the control strategies of furnace, analyzed and
summarized the furnace optimization control research status and significance,
pointing out the problems, and list the structure of whole project.
Chapter two introduces the furnace structure and analyzes the necessity of
optimizing the slab heating curve which is because of the difficulty to measure the
actual temperature distribution directly, so that the slab temperature
prediction/tracking model is introduced. The furnace dynamic model and set point of
the furnace temperature is also proposed. Finally, simulate the basic model in
MATLAB/SIMULINK.
Chapter three claims that there is large time delay causing normal control cannot
work on time, and proposes a feed forward dynamic compensation for furnace
temperature set point. Then simulation results shows better performance than model
proposed in Chapter 2.
8
Chapter 4 summarizes the above chapter and gives a conclusion and future target
of the project.
9
Chapter 2- Model of the walking-beam reheating furnace
This chapter will discuss the theory of furnace and slab temperature calculation
for modeling and propose the basic walking-beam reheating furnace temperature
model. The optimal slab heating curve and corresponding furnace temperature
distribution will be discussed as well.
2.1 Introduction of walking-beam reheating furnace
Walking-beam furnace reheating furnace, one of the main equipment in the steel
rolling industry, is used to heat slabs with less energy consumption and more accuracy
in temperature based on controlling the temperature of furnace, air-fuel ratio, air and
fuel pressure and furnace pressure, etc. The slabs are loaded from the furnace head
and driven towards furnace end by the walking beam in a specific speed. In other
words the zones slabs pass through are preheating zone, heating zone I, heating zone
II and soaking zone respectively.
The structure of the walking-beam reheating furnace is shown in Fig 1. There are
several burners in each zone except preheating zone to maintain the furnace
temperature. Hence there is no control in preheating zone which effect by other zone.
In need of analysis, other three zones can be divided into 6 parts, which are upper
heating zone I, lower heating zone I, upper heating zone II, lower heating zone II,
upper soaking zone, lower soaking zone.
10
Figure 1 The structure of the walking beam reheat furnace [16]
The slab once pushed into preheating zone will be heated by waste gas from
heating zone I. Then heating zone I and II will do the heating job making the slab
reaching the required temperature rapidly. Sent into soaking zone, the surface and
center temperature of slab will reach a balance which leads to improvement of
strength, hardness and toughness of the slab.
2.2 The model of slab temperature
This chapter will describe the model of slab temperature including ideal heating
curve and heat transfer function of slab.
2.2.1 The optimization of slab temperature curve
During the heating process, the reheating furnace provides sufficient heat for the
slab to ensure that the slab can be heated to the specified temperature range. To reach
the target and minimize furnace fuel consumption and oxidation loss of the slab, an
optimal temperature curve which depends on the kinds of the slab will be present. The
best performance will be ensured only if the slab following the optimal curve.
The optimal slab temperature curve corresponds to the choice of heating method.
There are three kinds of heating method which are shown in Fig 2.
11
Figure 2 Illustration of different methods for slab heating [17]
( -the surface temperature of slab, -the mean temperature of slab)
For method (a), the temperature of slab raises slow in the preheating zone
followed by the large temperature gradient in heating zone which will lead to the large
temperature difference of the surface and inner of the slab. This situation is not
acceptable because the uneven heating of the slab will make the rolling process
become more difficult.
For method (c), the temperature of slab soars in the preheating zone causing the
large temperature difference of the surface and inner of the slab. Moreover, most slabs
are still in elastic state under 500-600°C which means large thermal stress caused by
soaring temperature will lead to defect of slab as well as reduction of yield.
For method (b), the heating temperature is moderate in preheating zone resulting
in low thermal stress. Then the slab is in plasticity state after reaching 600°C while
the furnace temperature increases rapid enough making the surface of the slab reach
the required temperature in heating zone. Pushed to the soaking zone, the center
temperature of the slab approaches to the surface gradually. Because of the low
difference of surface and center, such slab is good to be rolled. Oxidation loss of the
slab decreases as well in terms of less time in heating zone.
12
In a word, the method (b) is the best one for heating slab.
2.2.2 Slab temperature tracking
Thermocouples are installed to detect the furnace temperature surrounding slabs
rather than measuring the slab temperature directly because of the difficulty in direct
measurement. To solve this problem, a mathematical model of the furnace and slab
temperature tracking and distribution are proposed using measurable data to estimate
the slab temperature and its linkages with the preset furnace temperature value.
Based on mechanism knowledge and experiments, the slab temperature
prediction model can be established. For further application in on-line calculation, the
model should not be too complicated. Hence, there are several assumptions to
simplify the heating process:
(1) The furnace temperature is one-dimensional linear distribution along the
direction of furnace of furnace while slab temperature being along the slab
thickness direction.
(2) Assume the furnace temperature corresponding to the position of slab to be
the basis temperature calculation of heat transfer.
(3) Ignore the heat transfer between slab and walking beam, slab and fixed beam.
(4) Since heat radiation in furnace can be absorbed within short distance and
walls are installed between each zone, heat radiation between neighboring
zones can be ignored.
(5) The specific heat capacity of each layer is considered to be equal.
Based on assumptions above, asymmetric one-dimensional heating conduction
equation of slab can be described as below:
13
{
𝛿𝑇
𝛿𝑡=𝑎
𝛿2𝑇
𝛿𝑡2
𝑇(𝑥,𝑡)| 𝑡=0=𝑇0(𝑥)
𝜆𝛿𝑇
𝛿𝑥| 𝑥=
𝐻2
=𝑞𝑢
𝜆𝛿𝑇
𝛿𝑥| 𝑥=−
𝐻2
=−𝑞𝑑
(1)
Where H is the thickness of slabs, (𝑥, 𝑡) is slab temperature distribution along
the thickness direction, 𝜆 is heat diffusion coefficient, 𝑞𝑢 and 𝑞𝑑 is heat flux of slab
upper and lower surface respectively. One slab is divided into 5 layers along thickness
direction, as figure 3 shows:
Figure 3 Slab layers
𝑞𝑢 and 𝑞𝑑 can be calculated by equation (2):
𝑞𝑢 = εσ [(T𝑓𝑢 + 273)4− (T1 − 273)4]
𝑞𝑑 = εσ[(T𝑓𝑑 + 273)4− (T5 − 273)4]
Where σ is Boltzman constant which equals to 4.88*10-8
, ε represents radiation
coefficient, T1 and T5 is top and bottom surface temperature of slabs.
Using central difference method indicating that difference is expressed by the
mean value of forward difference and backward difference [5], equation (1) and (2)
can be rewrite as follows to calculate the distribution of slab temperature.
(2)
14
qdTTTTTTTTTTTqdTT
TTTTT
dx
ux
N
N
N
N
N
m
m
m
m
m
m
m
m
m
m
2)(
)(2
)(2
)(2
2)(
000
)(200
0)(20
00)(2
000
0
5
0
4
0
5
0
4
0
3
0
4
0
3
0
2
0
3
0
2
0
1
0
2
0
1
5
4
3
2
1
(3)
where m = 𝐶𝑝 ∗ 𝛾 ∗ 𝑑𝑥2 Δ𝑡⁄ , 𝜆 is the heating transfer coefficient, 𝑑𝑥 is the
thickness of each layer which equals to H/4, 𝐶𝑝 is the specific heat capacity, 𝛾 is the
specific gravity, 𝑖𝑁 represents the slab temperature at current time and 𝑖
0
represents slab temperature at last time, 𝑞𝑢 represents heat flux of upper slab surface,
𝑞𝑑 represents heat flux of lower slab surface, Δ𝑡 is the differential model calculation
step which yields that Δ𝑡 𝐶𝑝 ∗ 𝛾 ∗ 𝑑𝑥2⁄ ≤ 0.25 to ensure no shock on the boundary
value[6]. 𝐶𝑝 𝑎𝑛𝑑 𝜆 are functions of slab temperature, which is shown in equation (4):
𝐶𝑝 = 408.7 + 0.199 + 810.9exp (−𝛼| − 768|)
𝜆 = 55.85 − 31.23/𝑐ℎ[0.003( − 1208 + 273)]
Where, when T < 768°C, ∝= 0.0099; when T ≥ 768°C, ∝= 0.0261.
The mean temperature of slab can be calculated as equation (5) [9] shows:
=𝐶1𝑇1
𝑁+2(𝐶2𝑇2𝑁+𝐶3𝑇3
𝑁+𝐶4𝑇4𝑁)+𝐶5𝑇5
𝑁
𝐶1+2(𝐶2+𝐶3+𝐶4)+𝐶5 (5)
From the relationship of slab (A3) temperature and specific heat capacity as
figure 4 shows, when the temperature difference is small, the specific heat capacity
gradient is small, hence the assumption of 𝐶1 ≈ 𝐶2 ≈ 𝐶3 ≈ 𝐶4 ≈ 𝐶5 at the same time
(4)
15
of different parts of slab.
Figure 4 Relationship between slab temperature and specific heat capacity
Hence, equation (5) can rewrite as follows,
=𝑇1𝑁+2(𝑇2
𝑁+𝑇3𝑁+𝑇4
𝑁)+𝑇5𝑁
8 (6)
Based on model proposed above and furnace temperature distribution, the slab
heating curve and tapping temperature can be obtained.
2.3 The model of furnace temperature
The furnace temperature is commonly referred to the temperature detected
directly by the thermocouple. The set point of temperature for temperature control
loop regulate valve opening to control gas flow causing the change of furnace
temperature and corresponding slab temperature. Therefore, it is necessary to
establish the dynamic model of the furnace temperature to find the optimal furnace
temperature set point based on slab heating indicator and economic indicator.
Furnace temperature modeling is essentially a time-varying and nonlinear heat
transfer problem, including radiation, conduction and convection. This heat transfer
problem holds lag time and time constant varying from the furnace load (the numbers
of slab). However, the numbers of slab is always the maximum one during the actual
production due to maximizing the profit and efficiency. Hence, the assumption that
the lag time an time constant do not change during the whole process is reasonable.
16
Basically, this process can be converted into a multi-volume with pure time-delay
process, that is to say bigger time constant becomes dominant time constant of the
piecewise process while smaller ones are combined into one being equivalent to a
pure time delay. As Equation (6) shows:
𝐺𝑝(𝑠) =𝐾𝑝
∏ 𝑇𝑗 +1𝑛𝑗=1
⇒𝐾𝑝𝑒
−𝜏𝑠
𝑇 +1 (7)
Equation (3) is based on piecewise controllability of each zone. Actually, the
model of furnace temperature is complicated and hard to demonstrate by math model.
To simplify the model, what to be controlled is the average temperature of a region
near sensors rather than the entire furnace temperature field.
According to furnace heat transfer characteristics, the furnace can be divided into
4 parts which are mentioned in section 2.1, and there is no control in preheating zone.
Since each zone of the furnace linked by the open loop, there is a strong coupling
between adjacent zones. For example, fuel flux changes in heating zone I will not
only affect the temperature of heating zone I, but also heating zone II and preheating
zone, and so on.
Based on the analysis of heat transmission characteristic in furnace, heat flux is
mainly transmits from unloading side towards loading side. Therefore, assume that
there is only unidirectional coupling between adjacent zones, that is to say a furnace
zone temperature changes that only affect the loading side one. Thus, the thermal
transfer characteristics can be approximated in figure 5.
17
Figure 5 Illustration of the thermal transmission in furnace
Where zone 1.3.5 represent lower heating zone I, lower heating zone II, lower
soaking zone respectively, while zone 2.4.6 represent upper heating zone I, upper
heating zone II, upper soaking zone respectively. Fi (i=1,...6) is fuel flow for each
zone; Tfi (i=1,...6) is furnace temperature for each zone.
The model shown in Figure 4 divides the furnace into 3 subsystems which can be
modeled separately. The advantage of this modeling approach is to fully consider the
furnace coupling effects between the various zones, so that the model can describe the
characteristics of the furnace more accurately. According to Figure 4, the
multivariable furnace temperature dynamic model is described by the following heat
balance equation:
𝑑𝑇𝑓𝑖(𝑡)
𝑑𝑡= 𝑏𝑖1 ( 𝑓(𝑖+2)(𝑡 − 𝜏𝑖1) − 𝑓𝑖(𝑡)) + 𝑏𝑖2 ( 𝑓𝑗(𝑡 − 𝜏𝑖2) − 𝑓𝑖(𝑡)) + 𝑏𝑖3Δ𝐹𝑖(𝑡 −
𝜏𝑖3)
𝑑 𝑓𝑗(𝑡)
𝑑𝑡= 𝑏𝑗1 ( 𝑓(𝑗+2)(𝑡 − 𝜏𝑗1) − 𝑓𝑗(𝑡)) + 𝑏𝑗2 ( 𝑓𝑖(𝑡 − 𝜏𝑖2) − 𝑓𝑗(𝑡)) + 𝑏𝑗3Δ𝐹𝑗(𝑡
− 𝜏𝑗3)
Where 𝑓𝑖(i=1.3.5), 𝑓𝑗(j=2.4.6) represent the furnace temperature of ith zone,
Δ𝐹𝑖 (i=1.3.5), Δ𝐹𝑗 (j=2.4.6) represent corresponding changes of fuel flux,
𝑏𝑖𝑘, 𝑏𝑗𝑘(i=1.3.5;j=2.4.6;k=1.2.3) is constant, 𝜏𝑖𝑘, 𝜏𝑗𝑘 (i=1.3.5;j=2.4.6;k=1.2.3) is pure
(8)
18
delay time. Finally, the furnace temperature dynamic model can be identified as long
as getting enough data from actual process.
2.4 Optimal setting for furnace temperature
Section 2.2.1 has discussed the optimal curve of heating slab. To achieve this,
optimal furnace temperature distribution need to be proposed. That is to say, the
optimal temperature curve of the slab corresponds to the optimal distribution curve of
furnace. Therefore, it is necessary to find the optimal furnace temperature distribution
curve. Once the optimal furnace temperature distribution curve is obtained, the
optimum furnace temperature distribution can be achieved through the furnace
combustion control system. The purpose of optimizing the furnace control, in fact, is
to find the best value of the furnace in each zone within the allowable range which is
the furnace temperature set point, in order to minimize heat energy consumption and
to meet the slab requirements. Furthermore, the furnace temperature applied is
changing along the furnace length direction rather than detecting by thermocouples or
preset one. Hence, the furnace temperature distribution needs to be done first.
Zhang.D.H. [8] has proposed a quadratic function along the furnace length
direction with constrains for the furnace temperature, as equation (9)
𝑓(𝑡) = 𝑑 + 𝑒𝑡 + 𝑓𝑡2 (9)
Where 𝑓(𝑡)the furnace temperature, t is the heating time. Equation (9) should
follow constrains below simultaneously:
1. The temperature of lower heating zone should reach a certain value.
2. At the time entering soaking zone (t1), the furnace should meet a required
temperature.
3. At the time unloading (tf), the slab should reach a certain temperature.
That is to say,
19
1. When t=0, 𝑓1 𝑖𝑛 ≤ 𝑑 ≤ 𝑓1 𝑎𝑥
2. When t= t1, 𝑓2 𝑖𝑛 ≤ 𝑑 + 𝑒𝑡 + 𝑓𝑡2 ≤ 𝑓2 𝑎𝑥
3. When t= tf, 𝑓3 𝑖𝑛 ≤ 𝑑 + 𝑒𝑡 + 𝑓𝑡2 ≤ 𝑓3 𝑎𝑥
𝑓1 𝑖𝑛, 𝑓1 𝑎𝑥 , 𝑓2 𝑖𝑛, 𝑓2 𝑎𝑥, 𝑓3 𝑖𝑛, 𝑓3 𝑎𝑥 are determined by actual process.
Considering the quality of the slab and minimization the energy cost, the
objective function is proposed to meet the requirements of slab quality as less energy
consumption as possible, and this equation can be solved by fmincon function in
MATLAB:
min J = min {1
2𝑃[ (𝑡𝑓) −
∗ (𝑡𝑓)]2+
1
2𝑄[ (𝑡𝑓) − 𝑐(𝑡𝑓)]
2+
1
2𝑅 ∫ 𝑢(𝑡)2
𝑡𝑓𝑡=0
}
Constrains are shown below:
1. T(t + Δt) = F(T(t), 𝑓(t + Δt))
2. (t + Δt) − (t) ≤ Δ 𝑎𝑥
3. (t) − 𝑐(t) ≤ Δ 𝑐 𝑎𝑥
4. | (𝑡𝑓) − ∗ (𝑡𝑓)| ≤ Δ 𝑜𝑢𝑡
5. 𝑖𝑛(𝑡𝑖) ≤ (𝑡𝑖) ≤ 𝑎𝑥(𝑡𝑖)
6. 𝑓 𝑖𝑛(𝑡𝑖) ≤ 𝑓(𝑡𝑖) ≤ 𝑓 𝑎𝑥(𝑡𝑖)
7. 𝑓 𝑡 𝑖𝑛 ≤ 𝑓 𝑡 ≤ 𝑓 𝑡 𝑎𝑥
Where,
t : Heating time, or the corresponding position of slab in the furnace.
t𝑖 : Some key position of furnace, such as head and tail of each zone.
(10)
20
t𝑓 : The whole heating time, or the length of the furnace.
(t), (t), 𝑐(t): Slab mean temperature, surface temperature ( 1) and center
temperature ( 3) at time t respectively.
Δ 𝑎𝑥, 𝑐 𝑎𝑥 , Δ 𝑜𝑢𝑡 : Maximum permission of slab heating rate, section
temperature difference and slab temperature difference when unloading respectively.
∗ (𝑡𝑓) : Expectation of slab mean temperature.
𝑎𝑥(𝑡𝑖), 𝑖𝑛(𝑡𝑖) : Maximum and minimum of slab mean temperature at t𝑖.
𝑓 𝑎𝑥(𝑡𝑖), 𝑓 𝑖𝑛(𝑡𝑖) : Maximum and minimum of furnace temperature at t𝑖.
𝑓 𝑎𝑥(𝑡𝑖), 𝑓 𝑖𝑛(𝑡𝑖) : Maximum and minimum of furnace temperature set point.
P, Q, R are weighted coefficient.
For equation (10), 1
2[ (𝑡𝑓) −
∗ (𝑡𝑓)]2
represents the requirement of slab
temperature when unloading, 1
2[ (𝑡𝑓) − 𝑐(𝑡𝑓)]
2 indicates slab temperature
difference between surface and center, 1
2∫ 𝑢(𝑡)2𝑡𝑓𝑡=0
shows the energy cost of furnace.
Hence, the weighted coefficient P, Q, R can be set according to the actual requirement.
The bigger P/Q/R is, the higher corresponding requirement is needed. What needs to
be concerned is P, Q ≫ R.
2.5 Simulation
Last 4 sections have proposed an almost complete static model for furnace and
slab temperature which can be regarded as steady state of the whole system. To verify
the reliability of this model, simulating in MATLAB/SIMULINK is a direct and
convenient approach.
2.5.1 The entire model
The structure of original system is shown in Fig 6. It is consist of three
21
subsystems: furnace temperature model, furnace temperature distribution model and
slab temperature model. The input of the system is set point of furnace temperature
and output is slab temperature (mean slab temperature, slab temperature of each layer,
cross section difference).
Figure 6 Original model
2.5.2 Set point of furnace temperature
Taking A3 steel slab (200mm *200mm *3000mm) as an example. Assuming slab
heating interval is 90 seconds and walking step is 500mm, which corresponds to this
heated slab heating furnace is about 1.5 hours. Hence, dt and dx are 90s and 50mm
respectively. The density of slab is 7800kg/m3. Radiation coefficient ε equals 0.35.
Specific heat capacity can be calculated by equation (4). The parameters m and lam_s
are calculated by the block shown on the bottom of the model. The expected mean
temperature of the slab when unloading is 1085°C. The whole simulation is cover 1.5
hours (5400 second) which is the time one slab stays in the furnace.
According to requirement of actual process, the set point of furnace temperature
along the furnace length corresponding to the slab heating time is shown in Figure 7.
The furnace specific parameters are listed: effective length of the furnace is 29348mm;
the length of preheating zone, heating zone I, heating zone II, and soaking zone is
22
13598mm, 5000mm, 6300mm, and 4450mm respectively. Code in Matlab file, the set
point curve along furnace length is plotted according to the cost function (7). The
range of each zone temperature is 1100 ± 5°C, 1150 ± 5°C, 1130 ± 5°C .
Figure 7 Set point of furnace temperature
2.5.3 Slab temperature model
Slab temperature model can be divided into two parts: radiative heat flux ,
temperature of each layer, slab mean temperature calculation and online parameter
calculation as figure 8 to 11 show. Radiative heat flux model is based on equation (2),
while temperature of each layer model and other two is based on equation (3) (5) and
(4) respectively.
0 500 1000 1500 2000 2500 3000 3500600
700
800
900
1000
1100
1200
Furnace temperature
23
Figure 8 Top and bottom surface radiative heat flux
Figure 9 Temperature of each layer
Figure 10 Slab mean temperature
24
Figure 11 Parameter calculation
The whole slab temperature tracking model is shown in figure 12.
Figure 12 Slab temperature tracking model
Hence, the optimal slab heating curve is simulated as figure 13 shows. Figure 13
demonstrates that the slab temperature rises as method (b) discussed in section 2.2.1
under such furnace temperature distribution. The controller designed on following
section will be based on the optimal curve.
25
Figure 13 Optimal heating curve
2.5.4 Furnace temperature
Considering a 6-zone walking-beam furnace, as Figure 5 shows, it is clear that the
furnace heat is transferred from tail to head. The couple between neighboring zones is
not taken into account although they are coupled actually
Utilizing the decoupled furnace model proposed in reference [2]:
upper heating zone I: 𝐺𝑇𝑓𝑝1 =0.0051
𝑠 + 0.0105
lower heating zone I: 𝐺𝑇𝑓𝑝2 =0.0011
𝑠 + 0.0039
upper heating zone II: 𝐺𝑇𝑓𝑝3 =0.0005
𝑠 + 0.0026
lower heating zone II: 𝐺𝑇𝑓𝑝4 =0.0004
𝑠 + 0.0019
upper soaking zone: 𝐺𝑇𝑓𝑝5 =0.0047
𝑠 + 0.0015
0 1000 2000 3000 4000 5000 60000
200
400
600
800
1000
1200
time（ s）
tem
pera
ture
°C()
Slab temperature
Section temperature difference
Set point
26
lower soaking zone ∶ 𝐺𝑇𝑓𝑝3 =0.0027
𝑠 + 0.0056
The control flow of each furnace zone is shown in Figure 14,
Figure 14 Control flow of each zone
The inner loop is fuel flow control loop while the outer loop is zone temperature
control loop, where fsi is the optimal set point of ith zone, GTfci and GFci are furnace
and fuel flow controller, GFVi is a fuel control valve, GTfMi and GFMi are measure and
transfer devices, GFpi and GTfpi are furnace pipes and furnace models. Transfer
function of each zone is shown below:
𝐺𝑇𝑓𝑐𝑖 = 𝑘𝑝1 +1
𝑘𝑖1𝑠+ 𝑘𝑑𝑠
𝐺𝐹𝑐𝑖 = 𝑘𝑝2 +1
𝑘𝑖2𝑠
𝐺𝐹𝑉𝑖 =1
2𝑠 + 1
𝐺𝐹𝑝𝑖 =5
8𝑠 + 1
𝐺𝐹𝑀𝑖 =1
𝑠 + 1
𝐺𝑇𝑓𝑀𝑖 =1
10𝑠 + 1
Considering the maximum flow of the fuel pipe, it is necessary to add a saturation
27
to limit the fuel flow. Hence, the system will be more reliable. According to actual
process, the fuel flow limitation of each zone is shown in table 2:
Zone Maximum fuel flow
Upper heating zone I 2500
Lower heating zone I 5000
Upper heating zone II 7000
Lower heating zone II 7000
Upper soaking zone 3000
Lower heating zone 3000
Table 2 Maximum fuel flow
Take lower heating zone I as an example, as figure 15 shows:
Figure 15 The Simulink model of lower heating zone I
PI controller parameters are listed in table 3,
Kp Ki
Fuel flow control loop 0.9 0.01
28
Upper heating zone I 70 0.2
Lower heating zone I 28 0.1
Upper heating zone II 60 0.15
Lower heating zone II 78 0.25
Upper soaking zone 50 0.13
Lower soaking zone 48 0.1
Table 3 Parameters of PI controllers
Large numbers of experiments show that the smaller the gain of integrator and
the larger proportion between the gain of proportion and integrator are, the better
performance is. The furnace temperature and fuel flow of each zone are shown in
figure 16.
Figure 16 The furnace temperature and fuel flow of each zone
0 1000 2000 3000 4000 5000 60000
1000
2000
3000
time（ s）
T/F
low
Upper Heating Zone I
0 1000 2000 3000 4000 5000 60000
2000
4000
6000
T/F
low
Lower Heating Zone I
0 1000 2000 3000 4000 5000 60000
2000
4000
6000
8000
time（ s）
T/F
low
Upper Heating Zone II
0 1000 2000 3000 4000 5000 60000
2000
4000
6000
8000
time（ s）
T/F
low
Lower Heating Zone II
0 1000 2000 3000 4000 5000 60000
1000
2000
3000
time（ s）
T/F
low
Upper Soaking Zone
0 1000 2000 3000 4000 5000 60000
1000
2000
3000
time（ s）
T/F
low
Lower Soaking Zone
Fuel Flow(Nm3/h)
Furnace temperature(°C)Fuel Flow(Nm3/h)
Furnace temperature(°C)
Fuel Flow(Nm3/h)
Furnace temperature(°C)
Fuel Flow(Nm3/h)
Furnace temperature(°C)
Fuel Flow(Nm3/h)
Furnace temperature(°C)
Fuel Flow(Nm3/h)
Furnace temperature(°C)
29
As spoken at previous section, there is no control in the preheating zone, hence
the response at the first 13598mm which corresponds to interval from 0 to 2448
second is not taken into consideration. As shown, every time furnace temperature
changes, which can be regarded as step response, the overshoot is less than 15%
which is acceptable.
Use zone choosing model to combine all 6 zones temperature into final furnace
temperature. Zone choosing model is consist of 3 square waves and 1 step signal to
make the set point of different zone control specific zone temperature according to the
length of each zone mentioned in section 2.5.2, which is shown in figure 17.
Figure 17 Zone choosing model
2.5.5 Disturbance
As known, there are lots of factors will influence the quality of the slab. To
30
simplify the disturbance, the assumption that all effect can be gathered to a total
changes of slab temperature brought by all disturbance is made. For the furnace model
above, assuming at time t, the heating process is effect by some disturbance causing a
large slab temperature error, model and result are shown in Fig 18.
2.5.6 Feedback control
The main control loops include the heating flux and furnace temperature control
loops of six zones which is proposed in section 2.5.4 and slab temperature control
loop being discussed in this section. The furnace temperature, heating flux and slab
temperature control loops form the cascade control loops. All controllers of these
three loops are PID controller. The structure of slab temperature control loop is shown
in figure 18. The parameter of PI controller is 0.12 and 0.0001 respectively without
overshoot based on energy saving principle.
31
Figure 18 Feedback control
Figure 19 Slab temperature with disturbance
It is seen that the disturbance appears in the exit of heating zone I while heating
zone II and soaking zone have not been affected yet. The first slab effected moves
slowly in the furnace, when it reaches the exit of the furnace this error will always
0 1000 2000 3000 4000 5000 60000
200
400
600
800
1000
1200
time（ s）
Tem
pera
ture
(°C
)
slab with disturbance
furnace temperature
Ideal T Slab
cross section
T Slab with disturbance
32
exist, by the end this slab cannot meet the requirements.
2.6 Conclusion and new problem
In last section, the basic model of slab and furnace temperature is contrasted
followed by a problem that the system ability of resisting disturbance is weak. The
reason why PID controller is not applied to control the slab temperature directly is
that this model will be used for online controlling leading to the simpler model the
better, while using PID controller for slab temperature will definitely increase the
calculating time which will make the time delay become larger and the accuracy
become lower. One of the most important reasons is that if the furnace temperature is
controlled in good performance, there will be no need to control slab temperature any
more. That is because the furnace temperature and slab temperature are related, once
one of each is done, the other will be done as well.
Due to the long time heating process, feedback control of the whole process
brings large time delay and cannot reflect the change of the temperature on time. That
is to say, when the effect of disturbance is detected, there has been a lot of slab
unqualified. Hence, applying feedback control cannot reach the requirement. What
can be done is introducing the feed forward compensation based on the first slab
effect by disturbance,
In a word, the feed forward compensation is needed for the system to eliminate
the error of following slab temperature. The feed forward compensation will be
discussed at chapter 3.
33
Chapter 3- The optimization of furnace temperature set point
based on feed forward compensation
This chapter applies feed forward compensation for the furnace temperature set
point and simulate in MATLAB/SIMULINK. Compare the result of PID feedback
control and feed forward compensation at the end of this chapter.
3.1 Introduction of feed forward compensation
Walking-beam reheating furnace is important equipment in the steel industry,
which main function is to heat the slab loaded and make its temperature and
uniformity meet certain requirements. If the temperature is below standard one, there
will be difficulties of rolling and damage to rolling equipment; in the contrary, if the
slab temperature is too high, there will be excessive oxidation of the surface and large
energy waste. Even if the slab temperature meets the requirements, the energy
consumption might not be the smallest. Hence，it is necessary to set the value of the
steady state dynamic calibration furnace.
Based on the theory proposed by Zhongjie Wang[29], a PID controller-based,
supplemented by expert experience compensating controls is proposed. Considering
all aspects of the impact between the furnace zones, feed forward control is applied to
control each furnace zone through dynamic compensation, simulation results in the
end of this chapter will prove it to be a better approach.
3.2 Theory of feed forward compensation
First, set steady furnace temperature value according to the static model built in
chapter 2. Then according to the actual measured slab temperature at exit and section
temperature difference, utilizing expert experience and fuzzy method to make
dynamic compensation for set point of each zone. [29] The structure is shown in Fig
34
19.
Figure 20 The control structure based on feedback
Assuming EC is section temperature difference, surface temperature difference is
ES. The maximum allowable values of EC and ES are ECT and EST respectively,
which are about 25°C. If EC and ES values are not satisfactory, expert experience is
applied to compensate furnace temperature correction settings.
Based on practical experience and simulation results, the compensation strategy
is formed as follows. Δ 𝑓 1, Δ 𝑓 2, Δ 𝑓 3 are the temperature compensation for
heating zone I, II and soaking zone respectively.[25]
1. If EC − ECT ≤ 0, ES < −EST, then
Δ 𝑓 1 =|𝐸𝐶 − 𝐸𝐶 |
0.3, Δ 𝑓 2 =
|𝐸𝐶 − 𝐸𝐶 |
1.2+ |𝐸𝑆|,
Δ 𝑓 3 = |𝐸𝐶 − 𝐸𝐶 | +|𝐸𝑆|
2,
2. If EC − ECT ≤ 0, ES > EST, then
Δ 𝑓 1 = 0, Δ 𝑓 2 = 1.25(−ES + 3(EC − ECT)),
Δ 𝑓 3 = −𝐸𝑆 − (𝐸𝐶 − 𝐸𝐶 )
3. If EC − ECT ≤ 0, −EST < ES < EST, then
Δ 𝑓 1 = 0, Δ 𝑓 2 = 0, Δ 𝑓 3 = 0,
35
4. If 0 < EC − ECT ≤ 10, ES < −EST, then
Δ 𝑓 1 = 0, Δ 𝑓 2 = 𝐸𝐶 − 𝐸𝐶 , Δ 𝑓 3 = |𝐸𝑆|,
5. If 0 < EC − ECT ≤ 10, ES > EST, then
Δ 𝑓 1 = 2|𝐸𝐶 − 𝐸𝐶 |, Δ 𝑓 2 = −(𝐸𝐶 − 𝐸𝐶 ), Δ 𝑓 3 = −𝐸𝑆,
6. If 0 < EC − ECT ≤ 10,−EST < ES < EST, then
Δ 𝑓 1 = 𝐸𝐶 − 𝐸𝐶 , Δ 𝑓 2 = 0, Δ 𝑓 3 = 0,
7. If EC − ECT > 10, ES < −EST, then
Δ 𝑓 1 = 𝐸𝐶 − 𝐸𝐶 , Δ 𝑓 2 = 2(𝐸𝐶 − 𝐸𝐶 ), Δ 𝑓 3 = 0,
8. If EC − ECT > 10, ES > EST, then
Δ 𝑓 1 = 2(𝐸𝐶 − 𝐸𝐶 ), Δ 𝑓 2 = 1.25(−2𝐸𝑆 + 2(𝐸𝐶 − 𝐸𝐶 )),
Δ 𝑓 3 = −𝐸𝑆 − (𝐸𝐶 − 𝐸𝐶 ),
9. If EC − ECT > 10,−EST < ES < EST, then
Δ 𝑓 1 = 2(𝐸C − ECT), Δ 𝑓 2 = 0, Δ 𝑓 3 = 0.
As slab heating process is irreversible, the previous compensation based on the
expert experience for slab heating is mainly according to the information to guide the
subsequent slab heating. However, because of a variety of disturbances, for a single
slab heating process, if the slab temperature distribution is far away from the ideal
heating curve, it will result in the irreparable impact, especially large deviations
making the slab into the waste directly.
Based on the rules above, the feedback compensation can be transferred into feed
forward compensation. Because of the slab temperature tracking model, the slab
temperature and corresponding error at the exit of each zone can be calculated.
According to this error, compensate the set value of each zone for the furnace
temperature. The structure is shown in figure 20.
36
Figure 21 The structure of feed forward control
When the error is greater than a given ΔTmax, the compensate value should
assigned to the several following zones, it is necessary to add a coordinating upper
layer for feed forward compensate to correct the heating curve of the ideal condition
based on the actual furnace so that target value of each zone can be determined. Feed
forward compensation is mainly to calculate ΔTfsi corresponding to ES based on a
series of rules from experience which is listed before, or determine a local
optimization using objective function is as follows:
min 𝐽𝑖 = 𝑚𝑖𝑛 {1
2𝑃 ∗ [ (𝑡𝑓𝑖) −
∗ (𝑡𝑓𝑖)]2 +
1
2𝑅 ∫ 𝑓𝑖(𝑡)
2𝑡𝑓𝑖+1
𝑡=𝑡𝑓𝑖𝑑𝑡} (11)
To simplify the procedure, the experience rule is taken for the feed forward
compensation.
3.3 Simulation
The model and structure are the same as what is in chapter 2. Add the
compensator, as figure 21 and 22 shows.
37
Figure 22 Structure of whole system
38
Figure 23 Feed forward compensator
3.3.1 Feed forward compensation model
According to compensation strategy opposed in section 3.2, feed forward control
model is obtained. Each case is shown below.
Figure 24 Case 1.2.3
Using if and if action model can be achieved easily. Figure 24 shows the
structure of case 1, 2, and 3 which judge the range of ES first and subsystem case 1,
case 2 and case 3 decide by the range of EC-ECT. When case 3 take place, there is no
change in temperature of three zones. Hence the action constant is zero. Case 1 and
case 2 is shown blow.
39
Fig 24-1 Case 1
Fig 24-2 Case 2
Subsystem case 4, 5, 6 and case 7, 8, 9 are shown in figure 25 and 26. The theory
is the same as case 1, 2, 3. For space reason, there is no need to discuss.
Figure 25 Case 4.5.6
40
Fig 25-1 Case 4
Fig 25-2 Case 5
Fig 25-3 Case 6
Figure 26 Case 7.8.9
41
Fig 26-1 Case 7
Fig 26-2 Case 8
Fig 26-3 Case 9
The subsystem in case 1 to 9 is the same as the zone choosing model proposed in
figure 17. EC and ES equal 25 and 30 respectively. Adjust the gain of EC-ECT and
ES to make the slab temperature difference as less as possible.
3.3.2 Result
The result of slab temperature with feed forward compensation is shown in
figure 27.
42
Figure 27 Feed forward curve
Compared to figure 19, the error of slab temperature of both PID feedback
control and expert experience feed forward control is shown in figure 28. As
mentioned in chapter 2, when the temperature of slab is in plasticity state after
reaching 600°C which corresponds to approximately 3000 second, the requirement for
slab temperature error is lower than when unloading. It is clear that the performance
with feed forward compensation is much better than that with PID feedback control.
The error of slab temperature with feed forward control is almost zero when
unloading. It is easily to be proved that expert experience feed forward compensation
is better than PID feedback control for system with large time delay and various
disturbances.
0 1000 2000 3000 4000 5000 60000
200
400
600
800
1000
1200
1400
time（ s）
Tem
pera
ture
(°C
)
slab with disturbance
furnace temperature
Ideal T Slab
cross section
T Slab with disturbance
43
Figure 28 Comparison of FB and FF
3.4 Validation
The actual slab temperature when unloading is collected by Shengli Oilfield
Highland Company. Using the set point of the actual process and put into the
simulation model, the result of simulated slab temperature when unloading is shown
in figure 29 comparing to the actual slab temperature. Where the solid line is the
actual slab temperature, the dash line represents the simulated slab temperature. The
maximum error is 7.3 °C while the mean error is 2.1°C. From the error between the
actual and simulation, it is easily seen that the model is approximately close to the
actual process.
0 1000 2000 3000 4000 5000 60000
20
40
60
80
100
120
140
time（ s）
Tem
pera
ture
(°C
)
slab temperature error
PID feedback
feedforward
44
Figure 29 Simulation result and Actual temperature
3.5 Conclusion
During the actual heating slab process, disturbances such as rolling rhythm
mutations, the furnace pressure and others will eventually result in the slab heating
temperature distribution curve deviates from the ideal, and there is a big time delay
using feedback compensation based on large system leading to weakness in reflect
changes in working conditions on time. In view of this, a method based on feed
forward control dynamic compensation for furnace temperature set value is propose.
Each furnace zone of the furnace is regarded as a subsystem. Considering the
interaction between the various subsystems, dynamic compensation is applied for
furnace temperature according to the slab temperature deviation at the exit of former
zone. The simulation results show that when the disturbance happens, the method of
the feed forward compensation can compensate on time to reduce the number of
failed slab, while the accuracy is improved to some extent compared to feedback
control.
0 10 20 30 40 50 60 70 80 90 1001065
1070
1075
1080
1085
1090
1095
1100Slab temperature
Time
Tem
pera
ture
Predict temperature
Actual temperature
45
Chapter 4- Summary
Scholars from various countries have done many researches on furnace to reduce
energy cost since energy crisis in the 1970s. Along with the modern mill toward large,
continuous, high-speed, high precision and multi-species direction, traditional furnace
based combustion control cannot meet the above requirements, which requires
furnace model and optimization of control being in higher quality. Rolling process
requires to be provided suitable temperature slab, while traditional combustion control
or temperature control is unable to complete this process task. Therefore, the
advanced modeling and control techniques should be used for the actual furnace
temperature and the cross section temperature controlling.
However, the furnace is a typical complex industrial control object, including
thermodynamics, chemical and physical processes of all kinds, with a multi-variable,
time-varying, nonlinear, strongly coupled, distributed parameter, large inertia and time
delay and other characteristics, hence production targets cannot be achieved with the
conventional control methods. In particular, the slab temperature distribution in the
furnace cannot be detected directly and continuously measured; it is to achieve this
purpose to control the furnace. This paper is mainly around reheating furnace
temperature control optimization settings and control strategies studies, a feed
forward compensation based on the reheating furnace setting is proposed.
For optimal control of furnace feedback compensation loop with a time delay in
reflecting the real working conditions, this paper proposes dynamic optimization
control strategy of the reheating furnace set value based on feed forward
compensation. Each furnace section of the furnace is regarded as a series of
subsystems considering the interaction between the various subsystems, and then
decoupled these subsystems.
46
Although this paper proposed the mathematical model of the furnace and the
furnace temperature dynamic optimization settings for a certain research, the
following areas for further work will be done due to the complexity of the heating
process:
1. Establish a precise mathematical model of furnace is a very difficult task, the
current models are constructed based on certain assumptions and need to be
further improved.
2. Study fewer computation and online real-time optimization for furnace
temperature setting method for the actual production to meet the
requirements.
3. Make the furnace optimization setting control technology into actual
products, and further extended to more industrial processes.
47
Reference
[1] F.Hollander and S.PA.Zuurbier (1982), Design, development and performance of
on-line computer control in a 3-zone reheating furnace, J. Iron and Steel Engineer,
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50
Appendix
1. Set point of furnace temperature.
function f=objfun(n)
x=0;
T_heat1=1100 ;
T_heat2=1150 ;
T_soak=1130
f(1)=640
for i=1:3000
f(i+1)=f(i)+0.46-2.1*10^-4*x;
if x<1359.8
f(i+1)=f(i)+0.46-2.1*10^-4*x;
else if x>=1359.8&x<=1859.8 %%heating 1
if f(i+1)<=T_heat1-5
f(i+1)=T_heat1-5;
else if f(i+1)>=T_heat1+5
f(i+1)=T_heat1+5;
end
end
else if x>=1859.8&x<2489.8 %%heating 2
if f(i+1)<=T_heat2-5
f(i+1)=T_heat2-5;
else if f(i+1)>T_heat2+5
f(i+1)=T_heat2+5;
end
end
else %% Soaking
if f(i+1)>T_soak+5
f(i+1)=T_soak+5 ;
else if f(i+1)<T_soak-5
f(i+1)=T_soak-5 ;
end
end
end
end
end
x=x+1;
end
2. Validation
%% Clear up the environment variable
clc
clear
51
%% Set network parameters
load slab_temperature input output input_test output_test
M=size(input,2); %the number of input nodes
N=size(output,4); %the number of output nodes
nhid=8; % Number of invisible nodes
lp1=0.01; %Learning Probability
lp2=0.001; %Learning Probability
nit=100; %Iteration
%Initialize the weights
Wjk=randn(nhid,M);Wjk_1=Wjk;Wjk_2=Wjk_1;
Wij=randn(N,nhid);Wij_1=Wij;Wij_2=Wij_1;
a=randn(1,nhid);a_1=a;a_2=a_1;
b=randn(1,nhid);b_1=b;b_2=b_1;
%Initialize the node
y=zeros(1,N);
net=zeros(1,nhid);
net_ab=zeros(1,nhid);
%Initialize incremental weight learning
d_Wjk=zeros(nhid,M);
d_Wij=zeros(N,nhid);
d_a=zeros(1,nhid);
d_b=zeros(1,nhid);
%% Normalize input and output data
[inputn,inputps]=mapminmax(input');
[outputn,outputps]=mapminmax(output');
inputn=inputn';
outputn=outputn';
%% Network prediction
for i=1:nit
%Error accumulation
error(i)=0;
% Training
for kk=1:size(input,1)
x=inputn(kk,:);
yqw=outputn(kk,:);
52
for j=1:nhid
for k=1:M
net(j)=net(j)+Wjk(j,k)*x(k);
net_ab(j)=(net(j)-b(j))/a(j);
end
temp=mymorlet(net_ab(j));
for k=1:N
y=y+Wij(k,j)*temp;
end
end
%Calculate the sum of errors
error(i)=error(i)+sum(abs(yqw-y));
%Weight adjustment
for j=1:nhid
%Calculate d_Wij
temp=mymorlet(net_ab(j));
for k=1:N
d_Wij(k,j)=d_Wij(k,j)-(yqw(k)-y(k))*temp;
end
%Calculate d_Wjk
temp=d_mymorlet(net_ab(j));
for k=1:M
for l=1:N
d_Wjk(j,k)=d_Wjk(j,k)+(yqw(l)-y(l))*Wij(l,j) ;
end
d_Wjk(j,k)=-d_Wjk(j,k)*temp*x(k)/a(j);
end
%Calculate d_b
for k=1:N
d_b(j)=d_b(j)+(yqw(k)-y(k))*Wij(k,j);
end
d_b(j)=d_b(j)*temp/a(j);
%Calculate d_a
for k=1:N
d_a(j)=d_a(j)+(yqw(k)-y(k))*Wij(k,j);
end
d_a(j)=d_a(j)*temp*((net(j)-b(j))/b(j))/a(j);
end
%Update weight parameter
Wij=Wij-lp1*d_Wij;
53
Wjk=Wjk-lp1*d_Wjk;
b=b-lp2*d_b;
a=a-lp2*d_a;
d_Wjk=zeros(nhid,M);
d_Wij=zeros(N,nhid);
d_a=zeros(1,nhid);
d_b=zeros(1,nhid);
y=zeros(1,N);
net=zeros(1,nhid);
net_ab=zeros(1,nhid);
Wjk_1=Wjk;Wjk_2=Wjk_1;
Wij_1=Wij;Wij_2=Wij_1;
a_1=a;a_2=a_1;
b_1=b;b_2=b_1;
end
end
%% Network prediction
%Normalize predicted input
x=mapminmax('apply',input_test',inputps);
x=x';
%Network prediction
for i=1:100
x_test=x(i,:);
for j=1:1:nhid
for k=1:1:M
net(j)=net(j)+Wjk(j,k)*x_test(k);
net_ab(j)=(net(j)-b(j))/a(j);
end
temp=mymorlet(net_ab(j));
for k=1:N
y(k)=y(k)+Wij(k,j)*temp ;
end
end
yuce(i)=y(k);
y=zeros(1,N);
net=zeros(1,nhid);
net_ab=zeros(1,nhid);
54
end
%Anti-normalization of predicted output
ynn=mapminmax('reverse',yuce,outputps);
%%
figure(1)
plot(ynn,'-.')
hold on
plot(output_test,'r')
title('Slab temperature')
legend('Predict temperature','Actual temperature')
xlabel('Time')
ylabel('Temperature')
