Application of genetic algorithm in pyrolysis model parameter estimation. Anna Matala 60968U

Transcription

1 Application of genetic algorithm in pyrolysis model parameter estimation Anna Matala 60968U 1

2 Contents 1 Introduction 3 2 Small scale experiments 3 3 Pyrolysis Modeling 7 4 Genetic Algorithm Basic idea Fitness function Results Test materials Effect of the GA parameters Effect of fitness function Consistency of resulting Application on the real cable material Finding other parameters than A and E Using the algorithm Conclusions 28 2

3 1 Introduction The use of numerical fire simulation programs has become an important part of fire safety engineering and research because the full scale tests are usually far too expensive or even impossible to put into practice. For the deterministic simulation of fire spread, the behavior of the condensed phase materials must be modeled. This modeling should consider both heat transfer and the condensed phase reactions taking place in the material. The parameters of the equations describing these reactions determine the kinetics of material and are strongly material dependent. These parameters are usually not known and they may be difficult to find. Finding the values of the reaction parameters based on some experimental data is a parameter estimation problem. Typically this problem can be formulated as an optimization problem aiming at smallest possible error between experimental observation and model output. Many of the common optimization methods do not work effectively in cases of many variables and complex time series. Some of the methods (e.g. gradient method) may be unable to find the global minimum in presence of the local minima. Genetic Algorithms (GA) are offering an effective method for big problems. It can find several variables at the same time and thanks to its structure, it always tries to find better solutions outside of the local minimum area. GA have recently been used for the parameter estimation of condensed phase reactions by Lautenberger et al. [1] and Rein et al. [2]. The purpose of this work is to study the practical aspects of GA application in the estimation of pyrolysis model parameters. These aspects include the effects of the various GA parameters and the definition of the error function to be minimized. Finally, the algoritm is used to find the reaction parameters of a real cable material used in the nuclear power plant applications. The results are provided in the form usable by the version 5 of the Fire Dynamics Simulator software [3]. 2 Small scale experiments In general to get good simulation results, it is important to define the boundary conditions as accurate as possible. In fire simulation, this means knowing how 3

4 the burning material is behaving when heated. There are many small scale techniques that can be used to get information about material behavior in heated environment. TGA (Thermogravimetric Analysis) [4] needs typically some mg of sample material. The sample is placed to furnace that is heated slowly (heating rate 2-20 K/min). The slow heating ensures that the material is in equilibrium with its surroundings all the time. The material is connected to a balance that measures the weight of the sample. The data is collected by computer and the result tells how the mass of sample changes during the heating. By analyzing the shape of the TGA graph one can find the amount of reactions and the temperature area they are occurring. The unit of TGA data is mass or mass percent. Figure 1 shows a TGA graph and its gradient for a cable material at heating rate 2 K/min and when the surrounding gas is air. Figure 1: TGA graph and its gradient to a cable material. DSC (Differential Scanning Calorimetry) [4] is similar to TGA test but instead of mass reduction it measures the changes of energy. It also needs only a small piece of sample material and heating happens uniformly. In DSC, the temperatures of the sample and the reference material are kept identical and the amount of heating energy needed for this is measured. In figure 2 is shown a DSC graph of the previous cable material. In DSC graph one can see whether the reaction is endothermic or exothermic. Endothermic reaction absorbs energy while 4

5 exothermic releases it. The enthalpy of reaction can be calculated as an integral over the selected temperature range. The reactions of a material should be seen in TGA and DSC graph in equal temperatures. Figure 2: DSC graph for a cable material. DTA (Differential Thermal Analysis) [4] is practically the same as DSC but the samples are not kept in same temperature and instead the changes of it are measured during the heating. STA (Simultaneous Thermal Analysis) is a device that measures TGA and DSC/DTA data in same experiment. Cone calorimeter (ISO ) can be used to test samples in order of magnitude larger scales than TGA. The sample size is 10 x 10 cm and it is set under a cone heater and ignited. During the combustion several quantities are simultaneously measured and saved to computer. Some of the most important quantities are burning rate, heat release rate and the production of various gases. It is giving a better idea what really is happening during the combustion. In figure 3 is shown a burning rate of birch in cone test. 5

6 Figure 3: Burning rate of birch in cone test. 6

7 3 Pyrolysis Modeling FDS (Fire Dynamics Simulator) is modeling the kinetics of the materials condensedphase using the Arrhenius equation. It can be written d dt ( ρ ρ 0 ) = A( ρ ρ 0 ) n e E RT, (1) where ρ is density of material, ρ 0 is density of original material in the beginning and R is a gas constant. A, E and n are depending of the material. A is called frequency factor and can have values from wide range, [10 10,10 25 ] s 1. E is the activation energy and its values are typically between [100,500] kj/mol. n is the reaction order and has values near 1. This equation models one reaction of a material. If material undergoes many reactions, the parameters have to be found for every reaction separately. 4 Genetic Algorithm 4.1 Basic idea Genetic Algorithms (GA) are based on the idea of evolution [5]. Assume that we have a model with some unknown variables, and we want to use GA to find the best match with the reality. In other words, we are comparing the experimental data to simulated data. In this work, the simulated data is generated by FDS5 using these variables. In the language of genetic algorithms, these variables are called genes. They can be binary or real values. If they are binary, they may have some encoding function which converts the value of gene to real output. This is also called genotype-phenotype mapping. However, this work concentrates on finding real value variables. Different values of a gene are called alleles. A set of genes (or set of variables of the model) is called chromosome or individual. A set of individuals is called population. As in real evolution, many different populations may co-exist. These population groups are called subpopulations and they are isolated. This helps to maintain the variety in the whole population. To get new genes to subpopulations, they migrate every now and then between other subpopulations. The GA has most of its terminology from evolution biology and the explanations are listed here: 7

8 Initial population A matrix of population size (number of individuals x number of genes) created by generating random numbers of selected range. To some variables with large range of possible values, logarithmic scale can be used. Fitness value The goodness of an individual, how well the candidate model matches the experimental data. Calculated using fitness function. Stopping condition Condition of stopping the algorithm. maximum number of generations or achieving certain fitness. Can be a Ranking To give values to individuals according their fitness values. The better individual, the better value in ranking. The ranking value can be understood as a probability to survive to the next generation. Selection To choose the parents for the next generation. The individuals with high ranking value can be parents for many offspring while the individuals with worst ranking values should not be able to survive. The selection is based on stochastic methods. Crossover The production of offspring. The chosen parents are set to pairs and each pair are changing alleles according the conditions of crossover. Mutation A gene in a chromosome is replaced by new random number from the defined area. The mutation occurs relatively seldom but is really important to maintain the variety in population. By mutation the algorithm can find new and better alleles that leads to global minimum. Migration Individuals or genes of different subpopulations are changing place bringing new alleles to subpopulations. Next generation The next generation consists of the best offspring and best parents so that the number of individuals in population stays constant. All the actions (selection, crossover, mutation and migration) occur stochastically. Before running GA, the user should define the rates of actions or the probabilities that the actions are taking place. Normally the crossover rate is quite high, 0.7 or bigger and mutation and migration rates considerably smaller. The procedure of GA is as follows: 1. Create initial population and calculate fitness values. 8

9 2. Determine stopping condition and crossover, mutation and migration rates. 3. While the stopping condition is not true, repeat: Ranking. Selection. Crossover. Mutation. Calculate new fitness values. Create the next generation. Migration. 4. When algorithm stops running, it returns the best solution found. 4.2 Fitness function The most important part of creating an effective genetic algorithm is to find the best way to calculate the fitness of the model, the fitness values. In general, the idea is to minimize difference between the experimental and modeled data. To calculate this difference, Least Mean Squares (LMS) is commonly used. It can be scaled to maximum of 1 by using following equation [1] (Texp R = 1 T exp ) 2 (T exp T mod ) 2 (Texp T (2) exp ) 2 where the summing goes over the time series T. Exp denotes to experimental data, mod to model and T to average of time series T. The latter is used to scale the fitness value. To find the best fitness function, different ways to calculate the fitness value are studied. In addition to the actual time series, its gradient may also be used in the computation of the fitness value. In TGA test the residual mass is the most important result and therefore comparing them may also be useful. The fitness function is now ov = w1(1 +w2(1 (Mexp M exp ) 2 (M exp M mod ) 2 (Mexp M exp ) 2 ) (Gexp Ḡexp) 2 (G exp G mod ) 2 (Gexp Ḡexp) 2 ) (3) +w3((r exp R mod ) 2. where M is TGA data, G its gradient and R residual mass in the end. w=[w1 w2 w3] is weight vector and the sum of cells is 1. The effect of each part can be 9

10 studied by changing the weights. Other parameters studied are the mutation rate and the number of subpopulation. The latter effects on one hand to diversity of population but on the other hand every individual increases calculation time. A good compromise should be found between too small and too large. Heating rate affects to the shape of the graph so if possible, the fitness values should be calculated as mean with all available heating rates. In figure 4 is shown an example of model that succeeds to explain one rate well but the other ones not. Figure 4: The reason to compare with more than one heating rates. In one rate model can be good but in the others totally wrong. The model should explain every rate at the same time. Solid line stands for the original data and the dash line for the solution of the algorithm. 10

11 5 Results 5.1 Test materials In this work, three different materials are used to test the performance of genetic algorithm. Two of them are generated by simulation to make it easier to compare parameters, and one is a real TGA data. The most simple material is a solid of which the water is evaporated. The reaction is SOLID + BOUNDW AT ER k SOLID + W AT ERGAS. This data is generated by simulator using parameters that are A=4.56E15 and E=1.62E5. The other parameters of FDS input file are presented in table 1. n 1 emissivity 1 density 1000 conductivity 1 specific heat 1 residue yield 0 fuel yield 0 water yield 1 Table 1: Water evaporation: Parameters for FDS input file. The second material is cellulose [6] and its mechanism is presented in figure 5. Figure 5: The chemical reaction of cellulose. The cellulose reaction is called Broido-Shafizadeh scheme and can be used to describe the kinetics of wood as the composition of wood is 50 % of cellulose and 25 % of hemicellulose. The mechanism accounts for two competing reaction pathways. The reaction k2 is predominating in low temperatures and leads to 11

12 char and gas, and in air to smoldering combustion. The reaction k3 is predominating in high temperatures. The residual of this reaction is tar and in air the combustion is flaming. The propotion of the residuals depends on the temperatures and of course, on the Arrhenius parameters. This data is also generated by simulator and the parameters used are listed in table 2. The other parameters to build FDS input file are presented in table 3 A1 E1 A2 E2 A3 E3 2.8E19 s E5 J/mol 1.3E10 s E5 J/mol 3.23E14 s E5 J/mol Table 2: Cellulose: The known parameters. n 1 1 n 2 1 n 3 1 emissivity 1 density 400 conductivity 0.17 specific heat 2.3 residue 1 yield 1 residue 2 yield 0.35 residue 3 yield 0 Table 3: Cellulose: FDS input file parameters. The third material is used for cable sheaths. This material is a mixture of combustible cable material and non-combustible additive that is used for flame retardant of the cable. When the material is heated, the additive material degrades releasing mainly water gas. The degradation reaction is endothermic, and thus delays or prevents the heating of the cable to temperatures required for the degradation of the solid. The reactions are BOUNDW AT ER k1 W AT ERGAS SOLID k2 F UEL 12

13 The data of this material is produced in real TGA test and therefore, the correct values of the reaction parameters are not known. The FDS input file parameters are listed in table 4. Parameter BOUND WATER SOLID n 1 1 emissivity 1 1 density conductivity specific heat residue yield fuel yield water yield 1 0 Table 4: Cable sheath: FDS input file parameters. 13

14 5.2 Effect of the GA parameters GA needs many parameters and some of the most interesting of them are the number of subpopulation and the mutation rate. They may effect considerably to performance of algorithm and therefore they are studied carefully. The tests are started using equation 3. It soon becomes clear that at least few subpopulations are needed to maintain enough variety in the population. In figure 6 it is seen how fast the average is converging near the best individual even the solution could be much better. In this test, the population size was 50. The later tests were made with 4 subpopulations of 20 individuals in each. Figure 6: Average and the best fitness value of every generation. The average converges really fast. The effect of the mutation rate is not as clear. Originally it is set to 1/6 (one divided by the number of variables) and it works fine. Too high mutation rate causes so many mutations that even the best chromosomes are lost. On the other hand the low mutation rate can not prevent the fast convergence to wrong value. 14

15 5.3 Effect of fitness function Simulation of the cellulosic material was to used to study the proper weighting of the fitness function. Many combinations of weights were tested and good solutions were found using weights [1/2 1/2 0] and [1/3 1/3 1/3]. However, it may not be necessary to take into account all the parts of the fitness function. Actually, calculating the fitness values only according the first part of the fitness function (w=[1 0 0]), the solutions are very good. The best individual is shown in figure 7 and its gradient in figure 8. Even the gradient was not included directly to the fitness function, the algorithm found quite good fit. Figure 7: Cellulose: The best individual when only TGA data is compared. 15

16 Figure 8: Cellulose: The gradient when only TGA data is compared with heating rate 2 K/min. 16

17 5.4 Consistency of resulting The consistency of the GA results was studied by repeating the GA for cellulose several times, and comparing best individuals to each other. The comparison showed that resulting parameters may be different even though the corresponding TGA graphs look very similar. This demonstrates that the parameters producing the TGA curve are not unambiguous. Figure 9 shows three parameters A of reactions of cellulose in eight tests. The tests 3 and 6 evoked good results and it can be seen how ratios of A s are similar to those of the original data. The A s of less fitting solutions are having a different ratios. The same happens with parameter E. This suggests that the value of parameter does not matter - but the ratio between parameters does. Figure 9: Cellulose: The best A s of different tests. The water evaporation with only one reaction reveals more about the dependency of parameters. A solution found by GA is shown in figure 10 and the parameters are A = E+17 and E = 1.79E+5. There is a strong dependence, also known as compensation effect, between A and E of same reaction. Figure 11 shows the surface of fitness values calculated through the space of A and E. It can be seen that the smallest fitness values form a straight line in A-E phase. This means that all the good solutions for this problem have the same ratio of A and E. Now it is interesting to know how the fitness value behaves if this ratio changes. In figure 12 is presented fitness 17

18 Figure 10: Water evaporation: The solution by GA. value as function of E when A is kept constant. The fitness value changes quite linearly and fast when the ratio changes. Of course this ratio is different to every reaction so there can not be found any general rule. If there can be various parameters that model similar TGA graph, what would be the optimal choice? As the goal is to be able to model the real fires, it would be important that the same parameters model correctly bigger scale tests as well. An easy way to check this is to simulate a cone test using parameters from the best solutions. In figure 13 is plotted cone tests for cellulose with the best parameters of the tests. The solutions are good also for cone test, but a different amount of fluctuations can be observed. Significant fluctuation can make the large scale fire simulation unstable and so the best solution would have it as little as possible. Usually smaller parameters A are causing less fluctuation. Therefore the next step is to make the algorithm prefer smaller A s without loosing the fitness of solutions. To make the algorithm prefer smaller A s, a penalty is added to fitness function. The penalty is linear in logarithmic scale of A so that A = has zero penalty and A = has a penalty of 1. If there are several significant reactions every A should have a penalty and the final penalty of A s is their average. This penalty should be weighted appropriately so that it does not dominate the GA. 18

19 Figure 11: Water evaporation: Surface of fitness values. The effect of A should be of the same order than the differences in fitness values, order of Many weights are studied and good solutions are found with weights of the area [1/1000, 20/1000]. Figure 14) shows a solution with weight 1/1000 (parameters presented in table 5). The cone test (figure 15) shows that the parameters are a good model and the fluctuation is not anymore remarkable. A1 E1 A2 E2 A3 E E+14 s E+5 J/mol 2.95E+10 s E+5 J/mol E+15 s E+5 J/mol Table 5: Cellulose: Optimal parameters when A has weight 1/

20 Figure 12: Water evaporation: The values of fitness function when A is constant and E changes. Figure 13: Cellulose: Cone tests with the parameters of best solutions. 20

21 Figure 14: Cellulose: Best solution when the penalty of A is having a weight of 1/1000. Figure 15: Cellulose: Cone test of solution when A has weight of 1/

22 5.5 Application on the real cable material The algorithm was applied on the estimation reaction parameters of the cable material described in section 5.1. Until now in the examples all the reactions have taken place in the same area of temperature and there were not more than one peaks in the gradient. In cable material, two reaction happens one after another. In addition, the data was not generated by simulator but in real TGA test. Figure 16 shows a model to this data found by genetic algorithm and the parameters are listed in table 6. Even the solution found by algorithm follows the experimental data mostly correctly, the reactions seem to be sharper than they should. The problem remains the same even the test is repeated many times, the model is always more angular than the experimental data. This can mean that the real material behavior is not modeled properly in FDS. There may be some additional unknown reaction or some of the other parameters is not correct. The gradients do not look quite same either (figure 17). In experimental data there is much more noise that causes less smooth gradient. The area of integrals are more or less equal and the peaks are in right place anyway. A1 E1 A2 E E+12 s E+05 J/mol 1.85E+10 s E+05 J/mol Table 6: Cable sheath: Parameters found by GA. 22

23 Figure 16: Cable sheath: Comparison of experimental and predicted TGA curves. Figure 17: Cable sheath: The gradients. 23

24 5.6 Finding other parameters than A and E The tests so far have concentrated on finding parameters A and E of Arrhenius equation. However, there can be many other unknown parameters in FDS input file as well. One of them is the residue mass. If the reactions overlap in temperature, the residuals in the graph can not be found. When the residue of reaction k2 of cellulose was set to be variable (originally 0.35), the algorithm found solution shown in figure 18. The parameters are shown in table 7. The residual of the reaction k2 is now found to be even the real value should be The TGA graph looks correct anyway and so does the cone test graph (figure 19). The residual mass depends on which of the two reactions is dominating. Only the reaction k2 has a residual. Here the algorithm has found a solution where all the residual mass is from the reaction k2 so that reaction is fully dominating. As the algorithm compares only the graphs, it can not find any better solution without knowing more about the material and reactions. A1 1.0E+10 s 1 E1 1.33E+5 J/mol A E+14 s 1 E2 2.04E+5 J/mol r A E+17 s 1 E E+5 J/mol Table 7: Cellulose: Parameters when also residual of second reaction is variable. Another studied parameter is the reaction order n. Usually n is getting values between [0.5, 1.5] but there is no reason why it could not be smaller or bigger as well. n determines how steep is the TGA graph in the temperature space. In figure 20 is plotted cable sheath data simulated with 3 different reaction order n 1 (0.5, 1, 1.5) when n 2 was set to 1. To study how the algorithm manages to find these parameters, n 1 and n 2 was set to be variables between [0,3]. In figure 21 is shown the solution that algorithm found for a problem where n 1, n 2 and the residual mass of second reaction are variables. The parameters are listed in table 8. The model is a bit better than without variating these parameters (figure 16). 24

25 Figure 18: Cellulose: Best individual when residual of second reaction is also variable. A E+14 s 1 E1 1.94E+5 J/mol A E+15 s 1 E2 2.47E+5 J/mol n n r Table 8: Cable sheath: Parameters when n 2 was also variable. 25

26 Figure 19: Cellulose: Cone test when residual mass is found by algorithm. Figure 20: Cable sheath: The effect of first reaction order. 26

27 Figure 21: Cable sheath: Solution of algorithm when also reaction orders n 1, n 2 and residual mass r 2 were variables. 27

28 5.7 Using the algorithm The algorithm seems to work well even for big problems. It is however stochastic so it is not guaranteed that it always finds a good solution, nor that the solution is found during the first 500 generations. If the best model of the current generation is plotted while the algorithm is running, user can observe the process. If the solution seems not to be even close after hundreds of generations, it may be better to stop it and start from the beginning. On the other hand, if the solution is already satisfying, there is no need to continue. The algorithm is comparing graphs and if the graph looks good enough, the solution would not be later any better anyway. The disadvantage of the algorithm is that it is computationally expensive. It calls FDS for every chromosome and to every burning rate of the original data separately while calculating the fitness values. To run this algorithm effectively a fast computer is needed. Another difficulty is that the material behavior must be known to be able to write the FDS input file. At least the amount of reactions and some of the material parameters must be correct. 6 Conclusions The goal was to apply genetic algorithm to find the kinetic parameters of materials for modeling them in FDS5. The GA worked well especially when the material behavior was exactly known. It found many suitable sets of parameters among of which the smallest ones were preferred because they affect less fluctuation. It seems that the kinetic parameters of material should not be understood as any fundamental constant but as parameters that model the material correctly. It is not proved yet that the parameters found by algorithm are really modeling big scale fires correctly. The smaller tests and simulations seem to work fine anyway and that is promising. The next step is to validate parameters by making bigger scale tests and simulations. 28

29 References [1] Lautenberger, C., Rein, G., Fernandez-Pello, C. The application of a genetic algorithm to estimate material properties for fire modeling from bench-scale fire test data. Fire Safety Journal 41 (2006) [2] Rein, G., Lautenberger, C., Fernandez-Pello, C., Torero, J., Urban, D. Application of genetic algorithms and thermogravimetry to determine the kinetics of polyurethane foam in smoldering combustion. Combustion and Flame 146 (2006) [3] McGrattan, K., Hostikka, S., Floyd, J., Baum, H. and Rehm, R. Fire Dynamics Simulator (Version 5) Technical Reference Guide. National Instititute of Standards and Technology, MD. USA. NIST Special Publication p. (Draft: August 26, 2007). [4] Kellner, R., Mermet J.-M., Otto, M., Valcárcel, M., Widmer H. M. Analytical Chemistry: A Modern Approach to Analytical Science, Second editition. (2004) [5] Reeves, Colin R., Rowe, Jonathan E. Genetic algorithms - principles and perspective (2002). [6] Di Blasi, C. Physico-chemical processes occuring inside a degrading two-dimensional anisotropic porous medium. Internationa Journal of Heat and Mass Transfer 41 (1998)