OPTIMAL CONTROL OF THE CONTINUOUS

18. ‐ 20. 5. 2010, Rožnov pod Radhoštěm, Česká Republika

OPTIMAL CONTROL OF THE CONTINUOUS SLAB CASTING PROCESS BASED ON MATHEMATICAL PROGRAMMING METHODS Tomáš MAUDER a, Josef ŠTĚTINA a, František KAVIČKA a a

Brno University of Technology, Faculty Mechanical Engineering, Technická 2, Brno, CZECH REPUBLIC, [email protected], [email protected], [email protected]

Abstract This paper deals with suitable stochastic optimization in continuous casting process of steel via mathematical programming. This tool is used for optimization and control of production of steel slabs, with the aim of achieving the maximum possible savings and product quality. The overall quality of the continuous casting process is influenced by various random phenomena. The main focus is on the random phenomena which influence water spray control in secondary cooling zone and how the process reacts to the specific situation. Stochastic-optimization 2D model where continuous casting process is roughly described is presented where the random phenomena which influence water spray control is described by stochastic term. The scenario based approach for this problem is the most promising candidate technique. The random variable should be described by probability distribution, which is close to the real process. It can be reached from historical data. Multi-criteria objective function is used where maximal possible casting speed and surface temperature smoothness are requested. Mathematical model contains 2D Fourier-Kirchhoff equation including boundary conditions. Presence of phase and structural changes is covered by enthalpy approach, where relationship between enthalpy and temperature is fitted by a curve, which describes the relation in the best way. Output is created as a set of optimal control parameters. The software implementation was executed as a link between MATLAB and modelling language GAMS. Future development of this research, which considers more factors, creates a 3D model, and which will make closer to the real process, is presented at the end of this paper. Keywords: Nonlinear stochastic optimization, temperature field, enthalpy approach 1.

PROBLEM DESCRIPTION AND NUMERICAL MODEL OF THE TEMPERATURE FIELD

Continuous casting is the predominant way by which steel is produced in the world. An ambition to improve final slab quality and increase productivity led to study of new effective mathematical methods. The modelling of the real continuous casting process is a complex multi-discipline problem. Primary focus is on the heat and mass transfer phenomenon which have main influence on mechanical and structural properties. The mathematical models that were employed are based on mathematical model of temperature field [1]. Our main interest is in the random phenomena which influence water spray control in secondary cooling zone and how the process reacts to the specific situation. For example, when there is a mechanical breakdown on one circuit and the spray circuit stops cooling the slab partially or completely. The main task is finding appropriate mathematical tools for optimal casting control, thus to examine a simplified 2D model which assumes symmetry on cross section (fig. 1) is satisfactory. Another simplification is considering four cooling circuits instead of thirteen, which can occur in real machine.


Fig. 1. Continuous casting scheme The optimization process requires the best mathematical description possible. From three basic heat transfers we will consider only conduction, which plays a dominant role in the process. Convection and radiation will be considered only as boundary conditions in the secondary cooling zone. The temperature field of the slab can be described by Fourier-Kirchhoff equation [2], where the velocity ingredient vz [m/s] is considered only in the process direction. There are phase and structural changes, thus the use of enthalpy approach with thermo-dynamical function of volume enthalpy H [J/m3] is appropriate [3] ⎛ ∂ 2 T ∂ 2T ∂H = λ ⎜⎜ 2 + 2 ∂τ ∂z ⎝ ∂x

⎞ ∂H ⎟⎟ + v z , ∂z ⎠

(1)

where λ is thermal conductivity [W/mK], T is temperature [K], τ is real time [s], x and z are coordinates. In order to be exhaustive, we have to add initial condition and boundary conditions. The initial condition describes initial temperature distribution in the whole process. As a suitable value can be considered casting temperature or precalculated temperature field for average control values. There are four different boundary conditions (fig. 2). T = Tcasting

(

on

Γ0 × 0,τ f

−λ

∂T =0 ∂n

on

Γ0 , Γs × 0,τ f

−λ

∂T = q& ∂n

on

ΓΙ × 0,τ f

−λ

∂T 4 = htc ,1,K, 4 Tsurface − Tw + σε Tsurface − Tw4 ∂n

on

ΓΙΙ × 0,τ f

(

)

(

)

(

(

(

(2)

Fig. 2. Scheme for boundary conditions Tw is water cooling temperature, ε is average emissivity, σ is Stefan-Boltzmann constant and htc is heat transfer coefficient for certain cooling circuit [1-3]. The primary cooling zone (mould) is considered only as heat flux [W/m2] depends on casting speed. q& = K mτ m−0.5 ⋅ 10 6

(3)


Km is mould constant between values 7-10 and τm is time for slab remaining in the mould. Relationship (1) can be approximated by finite-difference terms [2, 3], on an explicit formula ⎛ T n − 2Ti ,nk + Ti −n1,k Ti ,nk +1 − 2Ti ,nk + Ti ,nk −1 ⎞ H n − H in,k ⎟ + v z Δt i ,k −1 H in,k+1 = H in,k + Δtλ ⎜ i +1,k + ⎜ ⎟ Δz Δx 2 Δz 2 ⎝ ⎠ ,

(4)

where n+1 is the index associated to the future time, n is the index corresponding the actual time, Δt is the increment of the time, x and y are the directions, Δx and Δz are increments in the directions, i=1..M and j=1..N are the positions. In equations (4) is enthalpy present together with temperature, so we need to recalculate enthalpy to temperature for each node and each time. The function of enthalpy is not known as a function, but as a tabular form. Various search techniques are often used in practice. This approach is not appropriate for optimization. The relation between enthalpy

Fig. 3. Relationship enthalpy and temperature with

and temperature can be fitted by a curve, which describes the relation in the best way. For

polynomial interpolation

instance, we can fit the data by interpolating polynomial of the twentieth degree (fig. 3), Fourier series, etc. 2.

OPTIMIZATION AND MATHEMATICAL PROGRAMMING

Optimization is the act of obtaining the best result under given circumstances. In design, construction, and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is either to minimize the effort required, or to maximize the desired benefit. Since the effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, optimization can be defined as a process of finding the conditions that give the maximum or minimum value of a function. A number of optimization methods have been developed for solving different types of optimization problems. The optimum seeking methods are also known as mathematical programming techniques and are generally studied as a part of operations research [4]. Mathematical programming techniques are useful for finding the minimum of a function of several variables under a prescribed set of constraints. Stochastic process techniques can be used to analyze problems described by a set of random variables having known probability distributions. Statistical methods enable one to analyze the experimental data and build empirical models to obtain the most accurate representation of the physical situation. Mathematical model is created from the equation of objective function and the set of constraints. Decision variables are casting speed vz and heat transfer coefficients htc1,..,4 in boundary condition in secondary cooling zone. Dependant variables are objective value, temperature, and enthalpy for each node and each time. In the objective function, the information maximal possible casting speed and surface temperature smoothness is required, which describes compromise between final slab quality and productivity time. Thus ⎧ ⎫ z = min ⎨max {TM ,k − TM , k +1 }− v z ⎬ . ⎩ k =1.. N ⎭

(5)


Surface temperature smoothness is created by minimizing of maximal temperature difference between two neighbouring nodes. As constraints we have kept the metallurgical length, temperature T1 and T2, and heat transfer coefficients (fig. 1) between minimal and maximal values. M min ≤ M ≤ M max

0 ≤ htc1,.., 4 ≤ htc1max,..,4 max

T1min, 2 min ≤ T1, 2 ≤ T1max,2 max

(6)

Next constraints are relationship between enthalpy and temperature, where a20-a0 are polynomial constants, and initial condition for temperature field.

(

Ti ,nk+1 = a 20 H in, k+1

)

20

(

+ a19 H in, k+1

)

19

+ L + a1 H in, k+1 + a0

Ti ,0k = Tcasting

(7)

In this way, we have an objective function (5) and ten constraints (2, 4, 6 and 7) for all nodes including all time. For instance when we consider rough mesh 10x50x1000, we have approximately 1,000,000 constraints and variables, which is a very complex optimization problem in non-linear programming. Computing time in this case can reach more than thirty minutes. Moreover, the problem is extended by considering random influence on water spray circuit. It can be reached by using stochastic variable ξ incorporated in relationship (2). The random

variable

should

be

described

by

probability

distribution, which is close to the real process. It can be attained from real historical data. In the first model iteration the Bernoulli distribution is used. Bernoulli distribution is discrete distribution

having

two

possible

outcomes

labeled

by

n=1("success") and n=0 ("failure") in which occurs with probability p and p-1. It therefore has probability density function P (n) = p n (1 − p )

1− n

.

It this case, stochastic variable ξ is multiplied by heat transfer

Fig. 4. Bernoulli probability density function

coefficient htc,2 in secondary cooling zone. When n=1, the spray circuit works normally and n=0 represents breakdown and the spray circuit stops cooling. Presence of nondeterministic term move model to stochastic optimization problems [4, 5]. As a suitable way how to deal with uncertainty is scenario approach [6, 7] which allowed compute number possible scenarios. The scenario approach presumes a probabilistic description of uncertainty. Uncertainty here is characterized through an ensemble describing the set of admissible situations, and the probability distribution. Final optimal values are taken through all scenarios. We should notice that, if we decide calculate ten scenarios, optimization model will grow ten times, which is computing problem for these large models. Different view can be calculate scenarios separate and results put together. Progressive hedging algorithm [7] looks as a suitable technique for future research. 3.

MODELLING LANGUAGE GAMS, RESULTS AND DISCUSSION

The optimization is solved by modelling language GAMS (General Algebraic Modeling System). GAMS is specifically designed for modelling linear, non-linear and mixed integer optimization problems. The system is especially useful with large, complex problems. GAMS works as an interface between users and solvers (we use a non-linear solver CONOPT). More information can be found at http://www.gams.com/. Furthermore, we use mathematical software MATLAB for calculating the starting temperature field and for drawing (animating) the final temperature field (fig. 5-6).


Now let us consider two possible optimal scenarios. The first scenario (A) shows situation, where all spray circuits work (fig. 5). The second scenario (B) shows situation, where htc,2 = 0, thus this water circuit does not work

(fig.

6).

Table

1

shows

a

set

of

optimal

control

parameters

for

both

scenarios.

Fig. 5. Temperature field, surface and core temperature for scenario A

Fig. 6. Temperature field, surface and core temperature for scenario B Table 1. Optimal control parameters for scenario A and B scenario A B

casting speed [m/min] 0.60978488 0.55638691

htc,1 [W/m2K] 153.9 256.3

htc,2 [W/m2K] 187.9 0

htc,3 [W/m2K] 218.8 170.4

htc,4 [W/m2K] 351.8 270.2

metallurgical length [m] 17.5 17.5

T2 [°C] 800 800

If circuit spray stops cooling, optimal control reaction is an increase in water flow in circuit neighbours and slow down of casting speed for preservation metallurgical length and surface temperature T2 (tab. 1). However change of condition from scenario A to B is a critical point for controlling system. If we assume that the mechanical breakdown is possible, for instance with probability P = 0.75 for scenario A and P = 0.25 for scenario B, we can modify our model (fig. 7). In this case, optimal control reflect the possibility when the circuit spray stops cooling slab, and better react on change of condition from scenario A to B. Optimal control parameters are weighted averages from parameters associated with scenario A and B.


Fig. 7. Temperature field, surface and core temperature for probability approach CONCLUSION The paper deals with suitable mathematical tools for optimizing control of continuous slab casting process. We created the original mathematical model working with uncertainty. Apart from that, stochastic mathematical programming approach seems to be effective method for the continuous casting simulations. The future development of our work is improvement of this model to 3D model and considering more factors, which affect the continuous casting process and come closer to the real process. ACKNOWLEDGEMENT The authors gratefully acknowledge a financial support from the project GA106/08/0606 and GA106/09/0940 founded by the Czech Science Foundation and Junior research project on BTU BD13002. REFERENCES [1]

Štětina, J. The dynamic model of the temperature field of concast slab, Ph.D. Thesis, Technical University of Ostrava, Czech Republic, 2007, 105 p.

[2]

Inkropera, F. P., Dewitt D. P. Fundamentals of Heat Mass Transfer, Fourth Edition, Toronto, Willey & Sons, 1996, ISBN 0-471300460-3, 886 p.

[3]

Stefanescu, D. M. Science and Engineering of Casting Solidification, Second Edition, New York, Springer Science, 2009, ISBN 978-0-387-74609-8, 402 p.

[4]

Rao, S. S. Engineering Optimization Theory and Practice, Fourth Edition, New Jersey, Willey and Sons, 2009, ISBN 978-0-47018352-6, 813 p.

[5] [6]

Kall, P., Wallace, S. W. Stochastic Programming, Second Edition, Chichester, Wiley & Sons, 1994, 317 p. Campi, M. C., Garatti, S., Prandini, M. The Scenario Approach for Systems and Control Design, 17th IFAC World Congress, Seoul, Korea, July 6-11, 2008, pages 381-389

[7]

Wets, R. J. The Aggregation Principle in Scenario Analysis and Stochastic Optimization, Springer-Verlag, Berlin, 1989, pages 91-113.