International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 24-26, 2003
Numerical Techniques for Optimization of 3D Induction Heating Systems S. Galunin, M. Zlobina, Yu. Blinov, B. Nacke, A. Nikanorov, H. Schülbe Abstract Transverse Flux Heating (TFH) is one of the most effective induction heating technologies in the field of flat metal products. However, to take all the advantages of TFH concept is only possible by optimal design of the heating installations. The paper is devoted to application of automatic optimization techniques including Genetic Algorithms (GA) for optimal shape design of TFH systems. Special attention is paid to couple the different optimization algorithms with extremely time consuming 3D numerical models. A multilevel shape design optimisation of TFH system using the GA is described as an example. Introduction Heating of flat metal products like strip and sheets in transverse magnetic flux is one of the most effective induction technologies. Even for thin strip of high conductive material, the TFH concept offers very high electrical efficiency in combination with unique technological flexibility and extremely low floor space required. Numerous advantages make this method beyond competition to be applied in continuous strip production and processing lines. However, in practice all the potential advantages of TFH can be only taken by optimal design and optimal control of the heaters. That is why the TFH systems have been chosen for automatic optimization first of all. Typical TFH system (see Fig. 1) consists of the heated strip and, usually, two induction coils located one above and one below the strip. Induction coils are manufactured from water-cooled copper pipes in one or several turns. To increase the electrical efficiency and reduce the stray field around the installation the inductors are often supplemented by magnetic induction cores or electromagnetic screens induction coils direction coils of different shape. In spite of the of motion direction of motion fact that eddy currents are induced in the plane of the strip, the electromagnetic processes in TFH systems are essentially threedimensional. Temperature profile along the strip width at the outlet strip of the heater is formed by strip extremely non-homogeneous distribution of heat sources inside the strip. Because of temperature field is a final result of Fig. 1. View of the TFH induction system Fig. 1. View of the TFH induction system complicated three-dimensional
163
multi-physical process, correlation between outlet temperature profile in the strip and design of the inductors is absolutely not evident. Taking into account a huge amount of possible configurations of TFH systems, it is impossible to make a choice of good solution by conventional way of design. Even to design a well operating system becomes very difficult without automatic optimization search techniques. At the same time, experience of the last years shows that only mathematical simulation allows to avoid extremely cost intensive way to design the TFH installations by experimental try and error. Existing nowadays 3D numerical models in combination with modern computer facilities clear the way to automatic optimal design of TFH systems using advanced optimization algorithms. 1. Modelling and Optimization Tools At present time, different mathematical models to simulate the processes in TFH systems are within the reach [1]. Group of simplest tools are suitable for preliminary investigations of TFH systems. They are usually based on analytical techniques for calculation of electromagnetic and thermal processes in one or two dimensions and can be mainly used to optimize the integral characteristics of the heaters. The second group of models includes several specially developed numerical codes for calculation of two- and, especially, three-dimensional electromagnetic and thermal fields. These programs have comparably user-friendly uncomplicated interface and acceptable runtime at modern computers. Besides, they can be easily adapted to optimization purposes. Numerous commercial software packages for calculating electromagnetic, thermal and mechanical processes in induction heating systems can be combined into the third group of models. They are suitable for two- as well as for three-dimensional analyses of all kinds of induction systems. However, high-power computers and specially trained staff are necessary for professional use of these tools. Long runtime of the universal packages limits their use in the optimisation procedures. All the groups of the programs mentioned above have been analysed and tested on their ability to be applied for automatic shape design of TFH inductors. The programs belonging to the fist group are used for optimization of integral energy characteristics and finding the optimum value of operating frequency. It was found out that the programs of the second group are the most suitable tools for shape optimization of TFH inductors. Particularly, the results presented in the paper have been received using the problem oriented program package INDHEAT-SEM. Different optimization methods belonging to both deterministic and stochastic groups were investigated to search the optimum design of TFH systems. Finally, the most robust and effective algorithms were chosen for the use. The algorithm based on polynomial interpolation of the goal function was applied for one-dimensional search [2]. In case of more than one variable, the complex method of Box shows good results in parametrical optimization of TFH inductors. In spite of comparably small amount of necessary goal function calculations, this technique has some significant limitations, which are typical for all deterministic methods of search. In case of multi-modal goal function, it often can fail to find the global minimum. To increase the probability that found solution is the global minimum, it is necessary to repeat again and again the optimization procedure starting from different initial points. Additionally, all deterministic methods are sensitive to smoothness of goal function surface, which can’t be guaranteed all the time in case of using the numerical models of the process. More sensible, therefore, seems to perform a sort of global search of a design space by stochastic optimisation using, for example, Genetic Algorithms [3]. The main principle of GA
164
Generate initial population
Selection of fitness
Crossover
Mutation
Accept the best solution
Simple single-point crossover 0000000000 1111111111
0000~000000 ⇒ 1111~000000 1111~111111 ⇒ 0000~000000
1111000000 0000111111
Simple double-point crossover 0000000000 1111111111
0000~000~000 ⇒ 0000~111~000 1111~111~111 ⇒ 1111~000~1 11
0000111000 1111000111
Mutation 0101101101 ⇒ 0100101101 0101101101 ⇒ 0111101001
Fig. 2. Basic structure of the Genetic Algorithm and its binary interpretation
is that of natural evolution. The GA search the design space through the use of simulated evolution, i.e. survival of the fittest strategy. They maintain and manipulate a population of candidate solutions. In general, the fittest individuals of any population tend to reproduce and pass their genes on to the next generation, thus improving successive generations. Basic structure of the GA is shown in Fig. 2. Each solution of the problem is represented as one chromosome. Initial population of solutions is generated randomly or heuristically. Evaluation function represents the competitive environment. Genetic operators (crossover, mutation and selection) are used to generate new sets of optimization variables. GA is tuned by several parameters like population size, probability for applying genetic operators, etc. As a result of many tests, the most effective combination of the parameters has been created to provide the robust
operation of GA in all range of the investigated problems. GA use only the values of the objective function but not its derivatives. It makes them suitable to be applied together with numerical calculations of the goal function. In comparison with deterministic techniques GA need usually a bigger amount of the goal function calculations. Nevertheless, in case of GA the probability to find the global minimum is close to 100%. In practice it means no need to repeat the optimisation search several times. This advantage of GA makes them very powerful tools for optimal shape design of different induction systems. 2. Numerical Models Coupled with Optimization Algorithms Conventional way to organise the optimization procedure is to include the goal function calculation within the optimization search program as a subroutine. In case of induction heating, particularly TFH, this method of coupling is extremely inconvenient and not robust enough. For example, a calculation of temperature distribution in the strip requires to make the three-dimensional electromagnetic and thermal calculations, which could be coupled. These calculations are usually realized as a package of several computer programs of big size with comparably long runtime. To include such package into the optimization search program as a subroutine is very complicate or even impossible. It pushed the authors to create an alternative way to couple the optimization search programs with the codes for calculation
165
Initial data for TFH numerical modell
Control data for optimization algorithm
Preprocessor
TFH numerical modell
Optimization algorithm
Postprocessor
Optimum solution
Fig. 3. Coupling of the optimization algorithm with numerical package for TFH modelling
of the goal function (see Fig. 3). A number of different optimization tools has been prepared in standard way and put into a special library [4]. Now different programs for the goal function calculation or packages of these programs can be used for optimisation purposes without any changes. Each set of variables, generated by the optimization algorithm, is transformed to the input data for the numerical model by a specially created pre-processor, while a special post-processor gives out the corresponding value of the goal function. Exchange of the data is organised using the intermediate files of text format.
3. Examples of Optimization
aie
bi
bm
b
hm
Possibilities of the coupling approach described and the developed tools can be illustrated by examples of optimal shape design of one z TFH induction system (see Fig. 4). The optimized system d consists of the heated thin y stainless steel strip and two flat strip rectangular induction coils each induction coil of one turn. One coil is located above and one below the strip. The coils could be completed by magnetic core strip two magnetic cores of brick shape. The strip is moved with constant speed to provide a steady-state mode of heating. y ai Operating frequency of 1000 Hz was used as an optimum value x for the chosen pole pitch t. The direction t of motion GA as an optimization search tool and the INDHEAT-SEM package as a TFH numerical model have been applied. induction coil Multilevel optimization am of this TFH system has been carried out starting from one and up to six independent design Fig. 4. Geometry of the optimized TFH system
166
variables. These variables were the length of the induction coils bi, the width of the induction coils ai in regular zone and the width of the induction coils aie in edge zones, which is taken into account by the factor ki. Next three parameters were the length of the magnetic core bm, the width of the magnetic core am and the distance hm between the strip and the magnetic core. Typical aim of the technology is to heat the strip as uniform as possible. Therefore, the goal function was calculated in percents as a ratio of maximum deviation in the distribution of heat sources, integrated along the strip length, to its level in the middle of the strip. All the variables and corresponding optimum values of the goal function in different optimisation approaches are shown in Tab. 1. The optimization has been started from onedimensional (1D) formulation, where the length of the induction coils bi was only varied. The best goal function value reached in this search is 15.9%. For 2D optimisation approach the width of the induction coils ai was added as a second varied parameter. As a result, the optimum value of the goal function was reduced down to 13.4%. The next 3D optimisation, where the width of the induction coils aie in edge zones was varied independently, allowed to reach the goal function value of 8.8%. Following three approaches of optimization include also the parameters of magnetic core as additional independent variables. The 4D search with additional varying the length of the core bm reduces the optimum value of the goal function down to 7.5%. Including the width of the magnetic core am, like it was done in the 5D optimisation, allowed to reduce the best value of the goal function down to 6.5%. The last test has been carried out with six independent variables including the distance hm between the strip and the magnetic core. This 6D optimisation search results to the best goal function value of 6.0%. Tab. 1. Results of the optimisation searches with increasing number of independent variables Range of Independent independent 1D 2D 3D 4D 5D 6D variables variables ∆bi, cm ∆ai, cm ∆ki (aie=ki*ai) ∆bm, cm ∆am, cm ∆hm, cm Goal function value
-120 … 100 -11 … 12 0.1 … 2.0 -30 … 40 -150 … 100 -0.5 … 10.0
-8.09
-5.84 1.02
-10.03 5.76 0.77
-9.92 1.81 0.72 -14.18
-9.95 1.87 0.72 -15.47 -13.70
-8.9 2.75 0.65 -17.10 -0.99 1.70
15.9%
13.4%
8.8%
7.5%
6.5%
6.0%
The results of optimization become better with an increasing number of independent variables. The optimum value of each variable is within the corresponding range but not on its boarders. So every time the GA has found the global minimum of the goal function. All the time the optimum solution was received by only one start of the search. Initial and final optimum distributions of the normalized integrated heat sources along the strip width are shown in Fig. 5. The optimum distribution has a double side deviation of the same magnitude up and down from its middle level. From our experience, it is an additional confirmation that the point found is the global minimum.
167
ωp ω po
Conclusions
1.6
Automatic optimal design is only the way to develop new high efficient generation of TFH 1.2 installations. A specially 1 developed numerical technology 0.8 provides robust coupling between Optimum distribution the optimization algorithm and 0.6 numerical package to calculate the 0.4 goal function. Created 0.2 optimization procedure, where the 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 b, ?? shape design of inductor is separated from the positioning of Fig. 5. Initial and optimum distributions of normalized the coils, is much stable and robust integrated heat sources along the strip width in operating than conventional search. Genetic Algorithms together with problem oriented three-dimensional software can be recommended as an effective tool for optimal shape design of TFH inductors. Initial distribution
1.4
References [1] Mühlbauer, A.; Ruhnke, A.; Demidovitch, V.; Nikanorov, A.; Lupi, S.; Dughiero, F.: Methods and Tools for All-Round Optimization of Transverse-Flux Induction Heaters. Proceed. of the 17th ASM Heat Treating Society Conference and the 1st Intern. Induction Heat Treating Symposium, Indianapolis (1997), 865-870. [2] Bunday, B.D.; Garside, G.R.: Optimization Methods in PASCAL. Edward Arnold, London 1987. [3] Goldberg, David E.: Genetic algorithms in search, optimization and machine learning. Library of Congress Cataloging-in-Publication Data, 1989. [4] Galunin, S.; Blinov, Yu.; Nikanorov, A.; Schülbe, H.; Nauvertat, G.; Nacke, B.: Application of genetic algorithms for optimization of transverse flux induction heating systems. Proceed. of the International Seminar on Heating by Internal Sources. Padua, September 12-14, 2001, ISBN 88-86281-64-1, pp. 625-630.
Authors Dipl.-Ing. Galunin, Sergey Dr.-Ing. Zlobina, Marina Prof. Dr.-Ing. Blinov, Yury Department of Electrotechnology and Converter Engineering St.Petersburg Electrotechnical University “LETI” Prof.Popov Str. 5 197373 St.Petersburg, Russia E-mail:
[email protected]
Prof. Dr.-Ing. Nacke, Bernard Dr.-Ing. Nikanorov, Alexander Dipl.-Ing. Schülbe, Holger Institute for Electrothermal Processes University of Hannover Wilhelm-Busch-Str. 4 D-30167 Hannover, Germany E-mail:
[email protected]
168
