18. - 20. 5. 2011, Brno, Czech Republic, EU
STOCHASTIC APPROACH AND OPTIMAL CONTROL OF CONTINUOUS STEEL CASTING PROCESS BY USING PROGRESSIVE HEDGING ALGORITHM Lubomír KLIMEŠ a, Tomáš MAUDER a, Josef ŠTĚTINA a a
Faculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, 616 69 Brno, Czech Republic,
[email protected]
Abstract The purpose of the article is to introduce a method for solving stochastic optimization problem in optimal control of the continuous steel casting process. This technique is used to control a production of steel slabs in order to reach maximum productivity with a given quality of casted products. Moreover, all real engineering processes are also influenced by various random effects that cannot be precisely predicted to occur. In our case, the main attention is aimed at the scenario-based model in which a sudden breakdown of water nozzles in the secondary cooling zone can occur with given probability. For this purpose, stochastic optimization approaches are favourably used with the progressive hedging algorithm suggested by American mathematicians in 1980s. This algorithm allows to solve scenario-based stochastic optimization problems by decomposition technique and each scenario describing a particular situation in the production is solved separately, whereas to solve the entire model may exceed computational capability. Another great benefit of the progressive hedging algorithm is the possibility of parallelization which leads to the significant speed up of computations. In this paper, the original parallel implementation of the progressive hedging algorithm using GAMS optimization software is introduced and tested for the continuous steel casting model based on the Fourier-Kirchhoff equation discretized by using the finite difference method and on the enthalpy approach due to the phase and structural changes during the cooling process. The model is modified to the two-stage structure and it comprises the situation in which the total breakdown occurs in the secondary cooling zone. The results of that stochastic model solved by using the progressive hedging algorithm are presented, discussed and compared with deterministic model results. Finally, the future development and research is adumbrated. Key words: stochastic optimization, continuous steel casting, progressive hedging algorithm, parallelization 1. INTRODUCTION Numerical modelling and optimal control of the continuous steel casting process belongs to the area of largescale and complex engineering problems, see, e.g., [1, 2, 3]. Due to the requirements on the performance, quality and competitiveness, the production in steelworks and its control are being required to be efficient and optimal. To fulfil these demands, the mathematical model of the continuous casting process has to be described sufficiently in details. Moreover, in order to reach the best results, the model should also embrace random influences and uncertainty that are naturally involved in all real applications and that model has to be solved by using appropriate methods and techniques. The benefit of solving the model taking into account the random influences is the fact that its solution gives better and more valuable results than ordinary approaches. Due to the reasons discussed above, we utilize stochastic programming approach and especially, the scenario-based optimization technique, see [4, 5]. By using that approach, random faults and breakdowns in the secondary cooling zone in the continuous steel casting machine may be effectively described and modelled. The breakdown in the secondary cooling zone, which can suddenly happen during the production, can be caused, for instance, by a blockage of cooling zone or by an unpredictable defect of pump that pumps water through the cooling zone. Furthermore, the breakdown may be either partial or total. The partial breakdown represents the situation in which the cooling effect is temporarily reduced, but the cooling still partially works. On the other hand, in the case of total breakdown the cooling is completely out of order. Nevertheless, the
18. - 20. 5. 2011, Brno, Czech Republic, EU control of the casting machine should be able to immediately react to these non-standard situations in order to keep the production in optimum. For solving the assembled model mentioned above, the progressive hedging algorithm, see [6, 7], which is a decomposition method for solving stochastic optimization problems, is used to find the optimal setting of the continuous steel casting machine and its secondary cooling zone. Furthermore, the decomposition technique of the algorithm allows to implement the parallel computing schema and this utilization may significantly speed up the computational procedures, and thereby the computing time and other expenses may also be substantially spared. For this purpose, an original parallel implementation of the progressive hedging algorithm is adumbrated and the results gained by using that approach are presented and annotated. 2. STOCHASTIC OPTIMIZATION APPROACH The model of given optimization problem involving only fully-known parameters and data is said to be the deterministic model. The term “fully-know” in this context means that all the parameters, coefficients and other data are given by particular values in the moment when the problem solving starts. Nevertheless, real engineering problems are not entirely deterministic and they also naturally involve a part of random influences and uncertainty. There are several methods how the decision maker can model randomness of given problem. One approach is to model and describe these influences by using the random variables with a prescribed probability distribution. This method is called stochastic programming approach, see [4]. The general form of the stochastic optimization programme is as follows,
min f (x, ) : g (x, ) 0, h(x, ) 0, x X
(1)
x
where f is an objective function, g and h are inequality and equality vector function, respectively, and ξ is a random vector generated by random variables. Although the programme (1) is syntactically correct, its meaning is not obvious since the interpretation of random part is not clear. Because of these reasons, the socalled deterministic equivalents with correct both the syntactics and semantics have to be utilized, see [4]. In this case, the expected objective equivalent
min E ( f (x, )) : g (x, ) 0 a. s., h(x, ) 0 a. s., x X x
(2)
is chosen where the expected value of objective function is minimized and constraint functions are prescribed to be almost surely satisfied. In the programme (2) the probability distribution of random vector may be arbitrary. However, for engineering applications it is very convenient and simultaneously sufficient to assume the discrete probability distribution of random vector having only a finite number of particular realizations. It means that the probability distribution is not continuous, but the number L of random vector outcomes can be arbitrarily (finitely) large. This approach is called the scenario-based technique, see [4, 5], and each particular realization (particular values of random variables) of random vector ξ is called a scenario. Each scenario si is also linked with the particular probability 0 pi 1 , p i 1 , representing how the situation described by a given scenario is likely to happen. Hence, by the probability the problem framework can be easily modified. In the case of scenariobased stochastic optimization, the following problem (cf. Eq. (2)), which hedges against all possible outcomes of random elements, see [7], is considered,
min p k f ( x, k ) : x C i : Ci x : g ( x, k ) 0, h( x, k ) 0, x X x
L
L
k 1
i 1
(3)
However, the main advantage and simultaneously the feature of stochastic programming approach is the facility to model and solve problems where several decisions are taken successively in different moments in time whereas the information about outcomes of random elements is also gradually spread in time and becomes known. This technique is called multi-stage stochastic programming, see [4, 5].
18. - 20. 5. 2011, Brno, Czech Republic, EU In this paper, the model of the continuous steel casting is assembled in the two-stage structure. The first stage decision represents the initial setting of cooling parameters in the moment when the production starts. Then the realization of random event – whether the breakdown in the secondary cooling zone occurs or not – is observed and after that the second stage decision, which is a change of cooling setup, is taken. It is obvious that there are two different types of decisions regarding randomness: the first stage decision is taken before the random outcome is observed, whereas the second stage decision is taken after the observation. 3. PROGRESSIVE HEDGING ALGORITHM AND ITS PARALLEL IMPLEMENTATION To solve the scenario-based stochastic optimization problem directly may be a very difficult and complicated task because with an increasing number L of scenarios the problem (3) rapidly grows up. It should be noticed that even the problem with only one scenario can be substantially a large-scale problem requiring powerful hardware and long computing time. Due to these reasons, an effective decomposition method was suggested for solving these problems. This method, which was proposed by Rockafellar and Wets in 1980s, is called the progressive hedging algorithm (PHA), see [6, 7], and is designated for solving scenario-based stochastic optimization problems with multistage structure. The mentioned algorithm is based on the decomposition technique which means that the original problem with L scenarios is decomposed into smaller subproblems and each subproblem corresponds to a particular scenario. These subproblems are then solved separately and the algorithm blends iteratively the scenario solutions according to the aggregation principle to average solutions that converge to the solution of problem (3), see [7]. Hence, regarding the decomposition, it is not necessary to be able to solve the entire problem, but it is sufficient only the ability to solve significantly simpler subproblems. This property is very important since it allows to solve problems that are unsolvable as an unit. Below we present the progressive hedging algorithm in the version for problems with the two-stage structure, see [7, 8]. Two-Stage PHA. Let S be a set of all scenarios, S L ,and for all i 1, , L let pi be the probability for a particular scenario si. Choose the penalty parameter > 0 and the termination parameter > 0 . Set
ˆ ( s ) xˆ 0 ( s ), xˆ 0 ( s) 0, 0 for all s S and j 1 . w 0 ( s ) 0 for all s S , X 0 1 2
1. For each scenario s S solve the problem
T
min f (x1 , x 2 , ) w j 1 ( s ) x1 x
1 2
j 1 2
x1 xˆ 1 j
: g ( x1 , x 2 , s ) 0, h(x1 , x 2 , s ) 0, ( x1 , x 2 ) X
j
and denote its solution as X j ( s ) (x1 ( s ), x 2 ( s )) . ˆ ( s ) (xˆ j ( s ), xˆ j ( s )) for each scenario s S according to the formulas 2. Calculate “average” solutions X j 1 2 j j j j j xˆ 1 ( s ) xˆ 1 sS p s x1 ( s ) and xˆ 2 ( s ) x 2 ( s ) . If the termination condition
j 1 j L xˆ 1 xˆ 1
2
sS
j 1 xˆ 2 ( s )
2 j xˆ 2 ( s )
sS
j p s x1 ( s )
j 2 xˆ 1
1
2
ˆ is the desired solution to the original problem with the given tolerance. Otherholds, then stop, X j
wise, update the weights w j ( s ) w j 1 ( s ) (x1 ( s) xˆ 1 ) for all scenarios, set j j 1 and go to 1. j
j
As it was already mentioned, the progressive hedging algorithm decomposes the original problem into smaller subproblems. The crucial feature of decomposition is that all these scenario subproblems are independent to each other. This means that there is not a rule prescribing in what order the subproblems should be solved. The standard approach is to solve them in a serial order, one-by-one. However, by using the multi-processors hardware the parallel computing can be utilized and several subproblems can be solved simul-
18. - 20. 5. 2011, Brno, Czech Republic, EU taneously on different processors or computers. This arrangement may considerably speed up the computations and therefore, the computing time and other loads can be greatly spared. For this purpose, an original parallel implementation of the progressive hedging has been utilized, see [8, 9]. This implementation is based on using the Message Passing Interface (MPI), which is an API library providing the parallel programming facility in ordinary programming languages. For solving all scenario subproblems, the optimization software GAMS, which is a general algebraic modelling system for solving various optimization problems comprising miscellaneous optimization solvers, has been used. To drive the algorithm and operate the MPI and GAMS software, the main program in C++ has been written. 4. OPTIMAL CONTROL OF CONTINUOUS STEEL CASTING The continuous steel casting is a modern production method of steel in steelworks. By this method, molten steel transferred from a furnace or from a converter is transformed to solid semi-finished products that are designated for next processing. To fulfil the ambitious demands on quality, productivity and competitiveness, accurate models and sophisticated numerical techniques and algorithms have to be used. Stochastic optimization and corresponding algorithms represent a great tool for the continuous steel casting modelling influenced by random events, breakdowns or faults. By using these methods, the production can be efficiently controlled according to particular situations. The simplified schema (in the one half view due to the horizontal symmetry) of the continuous steel casting process is shown in Fig. 1. This model describes a horizontal construction and contains only four cooling zones in the secondary cooling instead of larger number of zones that are installed in real machines. The yellow color represents molten steel that enters to the continuous steel casting machine through the mould (position 1 in Fig. 1). The purpose of the mould is to cool down the surface of casted product in order to form a solid crust. From the mould, steel continues through the secondary cooling zones (positions 2 - 5 in Fig. 1). Their purpose is the further cooling down of casted billet or slab so that the output of machine reaches a product that is completely solidified in its cross-section.
Fig. 1. The simplified schema of the continuous steel casting process The mathematical model describing the continuous steel casting method is based on the Fourier-Kirchhoff equation comprising the enthalpy function due to the phase and structural changes during the production. Thus, the process behaves according to the equation
T T 2 2 t z y 2
H
2
H v z z
in (0,
(4)
where T ( y, z , t ) and H ( y, z , t ) are the temperature and the enthalpy, respectively, is the heat conductivity and v z is the casting velocity. The appropriate initial and boundary conditions, see e.g. [1, 2], are linked to the equation (4), and they together form the mathematical model of continuous steel casting. The most important boundary condition is the following one,
T n
htc j htc red Tsurface T
(5)
describing the heat transfer beneath each cooling nozzle j in the secondary cooling zone where htcred represents the reduced heat transfer coefficient due to the radiation. The model is completed by the tabulated
18. - 20. 5. 2011, Brno, Czech Republic, EU temperature-enthalpy relationship, see [2], which is unique for given steel, and by other technological requirements. The mathematical model above has been discretized by using the finite difference method and properly modified to the two-stage structure. It is assumed that during the production the total breakdown in the second cooling zone (position 3) of secondary cooling can occur with the probability p2. The total breakdown means that the heat transfer coefficient htc2 in the second cooling zone suddenly becomes to zero. Thus, to describe this problem the model including two scenarios has been utilized. The first scenario s1 with the probability p1 = 0.95 describes the normal situation in the production without any fault. On the other hand, the second scenario s2 represents the situation when the total breakdown in the second cooling zone occurs in a certain time with the probability p2 = 0.05. The aim of solving the stochastic optimization problem with two scenarios mentioned above is to maximize the casting velocity vz and to find the corresponding setting of cooling parameters htcj. The control of the machine is interested in these questions: To reach the maximum casting velocity, how the cooling parameters should be initially (in a moment when the machine starts) set up regarding the possible breakdown in the future? And in the case the total breakdown in the second cooling zone occurs, how the setting of cooling parameters should be immediately changed to keep the best conditions for production? 5. RESULTS AND DISCUSSION The above adumbrated two-stage scenario-based stochastic optimization model was solved by using the suggested parallel implementation of the progressive hedging algorithm on a computer with two CPU cores. Therefore, each CPU core was used for solving one particular scenario. The optimal initial setting of cooling parameters gained by using the PHA is x1 (htc1 , htc2 , htc3 , htc4 )
T
T
2
(476.32, 249.11, 309.11, 0) W/m K
with the optimal casting velocity vz = 2.2156 m/min. Obviously, if no breakdown in the second cooling zone occurs, no change in the setting of cooling parameters is done. However, in the case of breakdown in the secondary cooling zone the initial setting x 1 of cooling parameters should be immediately changed to T
T
2
x 2 (htc1 , htc2 , htc3 , htc4 ) (500, 0, 68.14, 0) W/m K
and the optimal casting velocity drops to vz = 1.8366 m/min. The temperature courses in the core (blue color) and on the surface (red color) of casted product for both scenario situations are shown in Fig. 2 and Fig. 3, respectively, where the time ˆ represents the time in which 95 seconds elapsed after the breakdown in the scenario s2. To evaluate these results, the value of stochastic solution, VSS, see [4], which is a frequently utilized indicator of using the stochastic programming approach, can be used. The value of stochastic solution represents the possible profit that can be reached by using the stochastic optimization programme (2) and especially (3) (which may be solved by the progressive hedging algorithm) instead of simpler, so-called EEV programme in which the random parameters are replaced by their expected values. Note that to solve EEV programme is significantly simpler than to solve programmes (2) or (3), but on the other hand, they al-
Fig. 2. Temperatures for scenario s1 and time ˆ
Fig. 3. Temperatures for scenario s2 and time ˆ
18. - 20. 5. 2011, Brno, Czech Republic, EU ways give better results. For the above continuous steel casting problem, the value of stochastic solution takes VSS = 0.0258 m/min, which makes over one kilometer per month in 24-hours production regime and regarding the price of casted steel, over one kilometer extra in production every month is not negligible. 6. CONCLUSION The presented paper deals with the optimal control of the continuous steel casting via stochastic programming approach. The scenario-based model derived by the finite difference method and involving the possibility of total breakdown in the secondary cooling zone is assembled. To solve it, the progressive hedging algorithm belonging to the class of decomposition methods is introduced and its original parallel implementation for effective computing is utilized. The results gained by using suggested methods and stochastic optimization are presented and compared with results that may be obtained by another simpler approach based on expected values of random parameters. In conclusion, it has been shown that stochastic optimization and related methods can be effectively utilized for the optimal control of the continuous steel casting. Further research will be aimed at the successive model tuning, its more realistic geometry and scenario setup. Acknowledgement. The research leading to the presented results was supported by grants of the Grant Agency of the Czech Republic GAČR 106/09/0940, GAČR 106/08/0606 and GAČR P107/11/1566. The main author, the holder of Brno PhD Talent Financial Aid sponsored by Brno City Municipality, also gratefully acknowledges for that financial support. REFERENCES [1] MAUDER, T.; KAVIČKA, F.; ŠTĚTINA, J.; FRANĚK, Z.; MASARIK, M. A mathematical & stochastic modelling of the concasting of steel slabs. In Sborník konference. Tanger. Hradec nad Moravicí: Tanger, s.r.o., 2009. s. 41-48. ISBN: 978-80-87294-10-9. [2] MAUDER, T.; ŠTĚTINA, J.; KAVIČKA, F. Optimal control of the continuous slab casting process based on mathematical programming methods. In Sborník konference METAL 2010. Tanger s.r.o., 2010. s. 130-136. ISBN: 978-80-87294-03-1. [3] KAVIČKA, F.; ŠTĚTINA, J.; SEKANINA, B.; STRÁNSKÝ, K.; DOBROVSKÁ, J.; HEGER, J. The optimization of a concasting technology by two numerical models. Journal of Materials Processing Technology, 2007, roč. 185, č. 1- 3, s. 152- 159. ISSN: 0924- 0136. [4] BIRGE, J. R.; LOUVEAUX, F. Introduction to Stochastic Programming. New York : Springer Verlag, 1997. 421 s. ISBN 0-387-98217-5. [5] KALL, P.; WALLACE, S. W. Stochastic Programming. 2nd ed. Chichester : Wiley, 1994. 317 s. ISBN 0471-95158-7. [6] ROCKAFELLAR, R. T.; WETS, R. J.-B. Scenarios and policy aggregation in optimization under uncertainty. In Mathematics of Operation Research, sv. 16, s. 119-147. INFORMS, Linthicium, 1991. ISSN 0364-765X. [7] WETS, R. J.-B. An aggregation principle in scenario analysis and stochastic optimization. In S. W. Wallace, editor, Algorithms and Model Formulations in Mathematical Programming. Springer Verlag, New York, 1989. ISBN 0-387-50842-2. [8] KLIMEŠ, L. Stochastic Programming Algorithms. Brno, 2010. 95 s. Diplomová práce. Vysoké učení technické v Brně, Fakulta strojního inženýrství. [9] KLIMEŠ, L.; POPELA, P. An implementation of progressive hedging algorithm for engineering problems. In MENDEL 2010 - 16th International Conference on Soft Computing. Brno: BUT, 2010. s.459-464. ISBN: 978-80-214-4120-0.
