System optimization of turbofan engines using genetic algorithms

System optimization of turbofan engines using genetic algorithms A. Homaifar Department USA

of Electrical

Engineering,

North

Carolina

A&T

State

University,

Greensboro,

NC,

H. Y. Lai Department NC, USA

of Mechanical

Engineering,

North

Carolina

A&T

State

University,

Greensboro,

Triangle

Park, NC, USA

E. McCormick Research

Triangle

Institute,

Centerfor

Systems

Engineering,

Research

This paper presents an application of genetic algorithms to the system optimization of turbofan engines. Genetic algorithms are relatively new general-purpose optimization algorithms that apply the rules of natural genetics to explore a given search space. In order to characterize the many measures of aircraft engine performance, two d&erent criteria are chosen for evaluation. These criteria are thrust per unit mass flow rate and overall eficiency. These criteria are optimized using four key parameters including Mach number, compressor pressure ratio, fan pressure ratio, and bypass ratio. After observing how each parameter inJEuences objective functions independently, the two objective functions are combined to examine their interaction in a multiobjective function optimization. Numerical results indicate that genetic algorithms are capable of optimizing a complex system quickly. The resultant parameter values agree well with previous studies.

Keywords: genetic algorithms,

optimization

Introduction Modern aircraft engine design is an exceptionally complex task. Today’s engines contain literally tens of thousands of parts. Hill and Peterson,’ Zucrow,’ Sheperd,3 Bathie,4 and Hale5 have described many different criteria used to judge the performance of an engine, and these criteria are subject to optimization. This paper uses genetic algorithms (GA) to assist in the optimization of this design process for a turbofan engine. The premise behind the turbofan engine comes from the observation that thrust increases with airflow rate for a given fuel consumption rate. An uneconomical method for increasing airflow rate would be simply to increase the engine size. This, however, would result in an unnecessarily heavy engine that might offset any

added benefits. A turbofan engine uses a turbojet with a larger fan area that divides the incoming airflow into two streams. The “hot” stream passes through the engine and is burned with fuel. The “cold” stream bypasses the engine and is compressed in the bypass stage. These two streams result in two different exit velocities, U: and U%.

The turbofan engine consists of seven fundamental stages,’ as shown in Figure I. These stages include: l-2 2-3 3-4 4-5 5-6

Address reprint requests to Dr. Homaifar at the Dept. of Electrical Engineering, North Carolina A&T State University, Greensboro, NC 27411, USA. Received 6 October 1992; revised 5 July 1993; accepted 28 July 1993

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Inlet diffuser and ducting stage-decreases air velocity as air is carried to compressor inlet. Compressor stage-compresses air. Combustion stage-heats air through mixing and burning of fuel. Turbine stage--expands air to obtain power to drive the compressor. Afterburner stage-additional heating of air through more mixing and burning of fuel. Exhaust nozzle stage-accelerates and exhausts air. Fan bypass stage-bypasses some air around the engine’s internal operating parts.

0 1994 Butterworth-Heinemann

System

a

1

Figure 1.

2

Stages

3

of a turbofan

4

5

6

7

engine.

Genetic algorithms are used in this study because they are able to quickly optimize the objective functions involving subfunctions of multivariates. The resultant parameter values are accurate and can be used for design optimization in the early engine design stages. The details of GA are presented in this context.

Genetic algorithms Genetic algorithms are general-purpose optimization algorithms with a probabilistic component. They provide a means to search poorly understood, irregular spaces. Genetic algorithms have successfully been applied to a variety of function optimization, parameter searches, and machine learning applications. John Holland6 originally developed GA and provided its theoretical foundations. GA was developed to simulate some of the processes observed in natural evolution, a process that operates on chromosomes (organic devices for encoding the structure of living beings) rather than on living beings. Natural selection links chromosomes with the performance of their decoded structures. The processes of natural selection cause those chromosomes that encode successful structures to reproduce more often than those that do not. Recombination processes create different chromosomes in children by combining material from the chromosomes of the two parents. Mutation may cause the chromosomes of children to be different from those of their parents. GA appropriately incorporates these features of natural evolution in computer algorithms to solve difficult problems in the way that nature has done-through evolution. GA requires the problem to be maximized (or minimized) in the form of an objective (cost) function. In GA, a set of variables for a given problem is encoded into a binary string or any other coding structure, analogous to a chromosome in nature. These strings are converted to a numerical value and then linearly mapped over the range allowed for the variable. This value is then used to evaluate the objective function, yielding a “fitness.” GA selects parents from a pool of strings (population) according to the basic criteria of survival of the fittest. It reproduces new strings by recombining the parts of the selected parents in a random manner. The repopulation of the next generation is done using three methods: reproduction, crossover, and mutation. Reproduction means simply that strings with high fitness should receive multiple copies in the next generation

optimization

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while the strings with low fitnesses receive a lower number of copies or even none at all. Crossover refers to splitting a string into two parts at a randomly generated crossover point and recombining it with another string that has been split at the same crossover point. This procedure serves to promote changes in the best strings that will give them even higher fitnesses. Mutation is the random alteration of a bit in the string. This assists in keeping diversity in the population. In explaining the mechanisms of GA, a few terms need to be defined. Because we are dealing with binary strings, a notation must be developed to denote similarity subsets (schemata). A schema is a similarity subset with strings that contain similarities at some bit positions. We can expand this format even further with the introduction of a wild card character, *, in addition to the binary set (0, l}. For example, the set {0001,0101, OOll} can be described by the similarity template O**l. Using this notation, we can now specify a schema’s order and deJining length. For a given schema, h, its order o(h) is defined as the number of fixed bit positions within that schema. The defining length of a schema, 6(h), is the distance between the outermost fixed positions of a schema. As an example, the schema Ol****O has order 3, defining length 5, and can represent 16 different individuals. With these definitions we can now present the fundamental theorem of genetic algorithms, the schema theorem.’ The schema theorem enables us to calculate a lower bound on the expected number of a particular schema, h, following reproduction, crossover, and mutation. This theorem is stated as 1

m(h, t + 1) 2 m(h, t) 50

6(h) cE p,o(h) 1

5

(1)

where m is the expected number of schemata, t is the generation index, 1 is the overall string length, c(h) is the average fitness of those strings representing the subset h, 5 is the average fitness of the entire population, pc and P,,, are, respectively, the crossover and mutation probabilities. Examining the schema theorem, we see that it states that a schema will grow when it is short, has low order, and has above-average fitness. Genetic algorithms have many advantages over other methods. Currently most literature defines three main types of search methods: calculus-based, enumerative, and random.8 Calculus-based methods can be divided into two classes: indirect and direct. In indirect methods, local extrema are determined by finding where the gradient of the objective function is equal to zero. Direct methods follow along the objective function in a direction related to the local gradient. Both classes share two main disadvantages that greatly limit their usefulness. First, both are local in scope; that is, if a function has multiple local maxima, these methods may drive toward one of these values without ever approaching the global maxima. The second deficiency of calculus-based methods is their dependence on the existence of derivatives. However, many “real world” functions are discontinuous and noisy, and therefore do not work well with a method

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that prefers smooth, continuous functions. Enumerative methods offer an attractive advantage-simplicity-but this advantage carries a high cost. Enumerative schemes basically take a discretized search space and examine the objective function at every point. Although simple in technique, this brute force method is quite inefficient, and its execution times become too long as the search space becomes larger. The last method is the random search method. In many respects, this method can perform as poorly as enumerative methods because of its inefficiency. Randomly searching through a space and saving the best results can become time-consuming as the space becomes large. The primary advantage of GA is its robustness. GA works through function evaluation, not through differentiation or other such means. Although GA begins with a randomly generated set of points, it exploits the information contained in those points to drive it through the search space. Because GA is based on function evaluation, it can be applied to all types of optimization problems as long as they are properly coded. Because of its robustness, GA has been used in optimization problems as diverse as image analysis by Grefenstette and Fitzpatrick9 and gaming strategy by Axelrod.” Another practical engineering example of GA’s application is Goldberg’s study” of a system of 10 pipes and 10 pumping stations. Also, expert systems can be improved upon by using GA, as shown by Davis and Coombs.iz The software used for this paper was developed by Goldberg.8 The simple genetic algorithm (SGA) program, written in Pascal, is modified to allow the evaluation of multiple parameters and multiple objective functions. Multiple parameters are incorporated by letting each string be divided into substrings that represent the different variables. For example a string of length 18 could be used to represent three substrings of length 6, each of which represents a different parameter. In this evaluation, the string length is set at 36 and the number of substrings is four. The precision for each substring is determined from the equation 7[

=

u m=x -

Umin 2”’ - 1

(2)

where rr is the precision of the parameter, U,,, and Urnin are the maximum and minimum of the allowed value of the parameter, and A1 is the length of the substring. Obviously, using multiple parameters requires increasingly longer string lengths if each variable is to have an acceptable precision. Multiple objective functions are incorporated into the program by initially evaluating the population using the first objective function. These fitnesses are then normalized by dividing by the maximum fitness in the population for that objective function. In the same manner, the second objective function is evaluated, normalized, and added to the normalized result from evaluation of the first objective function. After dividing by two, we have a fitness that varies from 0.0 to 1.0. Also, by choosing this method it

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et al. would be possible to show bias toward one is considered to be more important

one function if than the other.

Algorithm performance The main purpose of this computational experiment is to examine the ability of GA to perform a multiparameter, multiobjective function optimization on a real-world problem. Although there are many different measurements of engine performance, in this study the thrust per unit mass flow rate (TMA) and the overall efficiency, q, are the two functions under investigation. TMA is an important consideration in turbofan engine design because size constraints limit an engine’s mass flow rate. Therefore, we seek to get as much thrust as possible for this mass flow rate. Overall efficiency is also important to engine design. Overall efficiency is the product of two other efficiencies-propulsion efficiency and thermal efficiency. Propulsion efficiency is the ratio of an engine’s thrust power to the rate of production of propellant kinetic energy. Thermal efficiency is the ratio of the rate of addition of kinetic energy to the propellant to the total energy consumption rate. Overall efficiency is an important part of engine design because flight range is directly proportional to overall efficiency. To obtain these two functions, we must examine each stage of the engine and derive expressions for the total (stagnation) temperature and pressure ratios across these stages. Using the conservation equations for mass, momentum, and energy, we derived equations governing the behavior of each engine stage. These equations are given in the appendix. Using these equations, we can evaluate expressions for TMA and overall efficiency. The resultant equations are ‘T = (1 + f)ub + pz.4,”- (1 + #l?)u &a

ro = (1 + PI

(3)

(ii, - u)u fQR

(4)

When evaluating these functions, engine performance is considered to be a four-parameter problem. The parameters used in evaluating these functions are: Mach number, M; compressor pressure ratio, P,,/P,,; fan pressure ratio, P,,/P,,, and bypass ratio, p. It is obvious from the equations derived in the appendix that there are many other possible parameters to add to this list, but these variables are chosen because there are data readily available for comparison. Reasonable values are therefore assumed for such parameters as free-stream pressure and temperature and the stage efficiencies. The initial ranges for the parameters, their individual string lengths, and their respective accuracies as shown in Table 1. The ranges for the two pressure ratios are typical upper and lower bounds for each. The limits on Mach number were made to ensure a reasonable stall speed and to prevent shock formation. Finally, the upper limit on bypass ratio was set partially by the increase in weight penalty for larger fan and cowling sizes. The lower limit on bypass ratio is the lowest reasonable value for an engine to be considered a true turbofan. The string

System optimization Table 1.

Variables to be optimized

Parameter

u max

u I?%”

M

0.9 25.0 3.0 6.0

0.3 5.0 1.0 0.1

PO31

PO2

P&I Pea B

String length

Accuracy

8 10 9 9

0.0024 0.0196 0.0039 0.0115

lengths were then chosen to give the best possible resolution while trying to minimize computation time. The variables that had to be defined for GA are population size, crossover probability, mutation probability, and maximum number of generations. Their respective values are 300, 0.65, 0.001, and 1000. Population sizing is an important requirement in GA. Populations must be large enough to provide adequate diversity. However, the larger the population, the greater the number of calculations. The population used here is based on Goldberg’s study on minimum population size. 13

Analysis With the proper variables set and the range for the parameters determined, four different cases were analyzed. Case 1 was done using TMA as the objective function with the Mach number required to be subsonic. Case 2 examined overall efficiency with Mach number subsonic. Case 3 examined the two functions combined and the Mach number subsonic. Case 4 reexamined overall efficiency and TMA when Mach number was allowed to be supersonic.

Case 1 The initial run was made with TMA as the objective function. The results for each parameter and the objective function are shown in Figures 2(a)-2(e). The figures show the parameter values for the most fit individual of each generation. As the figures show, GA tended to converge to its final answer within approximately 300 generations for each parameter. With a population of 300, this equates to 90,000 function evaluations to convergence. On a 25MHz 80386 PC with a 80387 math coprocessor, each generation took approximately 5 sec. This yields a computer time to convergence of approximately 25 min. The only exception with respect to convergence is the compressor pressure ratio, P,,/P,,. Although this value tended to fluctuate between 16 and 18, the smoothness of the objective function shows that this small variation has little influence on the final result. Work done by Pinkel and Karpi4 examined the influence of Mach number, compressor pressure ratio, and combustor temperature on TMA. Although their experiments examined only one variable at a time with other parameters being fixed, they provided us with confirmation of GA’s strength. Because they examined more than one value of each parameter, we can infer how each

using genetic algorithms:

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particular parameter’s value will affect the other parameters. In one experiment, P,,/P,, was allowed to vary and To4 was held constant at 1089 K. Also, two different Mach numbers were used, 0.66 and 0. For M = 0.66, a maximum TMA of approximately 400 N-s/kg was obtained with P,,fP,, z 6. For M = 0, a maximum TMA of approximately 60 N-s/kg was obtained with Po3/ P,, E 8. In both cases, TMA behaved quadratically in nature on a log scale, increasing to a maximum and then decreasing. This supports the fact that GA drove M as low as it could. Pinkel and Karp’s second experiment kept M constant at 0.66 and used three different To4 values: 811 K, 1089 K, and 1367 K. For 811 K, TMA was a maximum of approximately 245 N-s/kg with P,,jP,, E 3. For 1089 K, TMA was a maximum of 400 N-s/kg with Po3/ P,, E 6. For 1367 K, TMA was a maximum of approximately 610 N-s/kg with P,,/P,, E 9. As before, TMA behaved quadratically on a log scale, but, as x4 increased, the peak became flatter giving an indication as to why GA fluctuated between 16 and 18 for its maximum TMA. Given the combustor temperature used in our work of 1555 K, and the lower M that GA chose, the value of P,,/P,, E 17 becomes a reasonable solution. To see what effect crossover and mutation probability had on convergence, an additional run was made using crossover and mutation probabilities of 0.5 and 0.005, respectively. The results, shown in Figures 3(a)-3(e) closely match those in Figures 2(a)-2(e). However, there appears to be greater fluctuation in the parameter values of the most fit individual from one generation to the next, accompanied by a lower convergence value for TMA. This can be explained by the fact that the reduced crossover probability is not allowing GA to piece together good copies often enough, thereby decreasing the convergence value. While this action is taking place, the increased mutation probability is creating more random bit changes, assisting in the fluctuation from one generation to the next.

Case 2 The second case is made with overall efficiency as the objective function. The results are shown in Figures 4(a)d(e). Again, GA converged rather quickly to its final solution. In examining the equation for thermal efficiency, differentiation with respect to u would imply that thermal efficiency is maximized when u z u,/2. For the flight conditions specified, this would entail driving Mach number as high as possible, which is what happened with GA. This fact also tended to drive the other parameters to their final values. One should also notice that the highest overall efficiency obtained is approximately 0.22. Because this value tended to peak as Mach number peaked, it suggests that even higher Mach numbers could lead to higher overall efficiencies.

Case 3 The third run is made while optimizing the two functions together. The two functions were combined into a single

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function as described earlier in the sectic .3n genetic algorithms. Output of this run is show in Figures 5(a)-5(f). Analysis of this run gives an ,&%cation of which parameters have the greatest influence on each individual objective function. Although Mach number needed to be low for optimum TMA and high for optimum overall efficiency, it can be seen that if both functions are considered equally important, the requirement for overall efficiency wins out and drives the Mach number higher. To a lesser extent, the same is true for the bypass ratio and TMA. Although overall efficiency dictates a moderate value for bypass ratio, TMA becomes greater as bypass ratio becomes greater. This necessity remains after the two functions are combined. Also, compressor ratio appears to be more relevant to TMA than overall efficiency. Notice that the fan pressure ratio tends to level out to the same value for both objective functions. This shows that while the fan pressure ratio has a bearing on both, the effects of the “hot” stream dominate both functions. When examining Figures 5(e) and S(f), one can surmise that while a low Mach number may be essential for a high overall efficiency, it does indeed have a detrimental effect on TMA, which resulted in a nearly 40% drop from the TMA obtained when it was the sole objective function. In the same way, the changes in bypass ratio and compressor ratio led to a 10% drop in overall efficiency when compared with the value obtained when optimizing overall efficiency alone. These losses, especially the dramatic change for TMA, show the delicate balancing GA performed to obtain the best overall solution. The best overall solution, however, may not lead to the most well-balanced solution, and therefore the two functions may need to be weighted so each bears an equal burden of loss.

tion, Figures 7(a)-7(e) bear this fact out. With this in mind, we must conclude that for the results obtained for overall efficiency shown in Figures 6(a)d(e), the location of a shock wave on the engine inlet has a drastic influence on the parameters we used. When one considers the physical consequences of a shock wave, the results tend to make more sense. The occurrence of a normal shock leads to an increase in stagnation pressure downstream of the shock. With this natural increase in pressure, one can see that higher compressor pressure ratios are no longer needed. The shock also leads to the negation of the effect of bypass ratio. Because the effect of bypass ratio becomes negligible, this in turn eliminates the influence of fan pressure ratio, explaining why the erratic results for this parameter have no effect on overall efficiency.

Conclusion In this study, genetic algorithms have been applied to a complex turbofan engine design problem. Although under most circumstances a multiparameter, multiobjective function optimization problem would be considered a difficult task, GA handled the task quite successfully. Although the model used here to represent a turbofan engine is a relatively simple one, the procedure would be exactly the same with a more elaborate model. Results of assorted runs agreed with experimental and singleparameter optimization results. Of course, realistic design would include many more parameters, but for GA this would only mean increasing the string length to incorporate these additional parameters. The ability of GA to handle a problem of this nature could make it a very important tool in the future of engine design.

Acknowledgment Case 4 As noticed earlier, overall efficiency is apparently limited by the constraint on the upper bound of the Mach number. To see if this is true, an additional run was made with an upper bound on Mach number that exceeded unity. The new upper bound on Mach number was set to 1.8. If the Mach number exceeded unity, the program was modified to assume a normal shock wave as per across the engine inlet instead of allowing Anderson” the possibility of the engine swallowing the shock. The results for this run are shown in Figures 6(a)d(e). Overall efficiency did indeed increase slightly, but the effect on the parameters is more pronounced. Optimum compressor ratio decreased by approximately one third, while bypass ratio changed from being as high as possible to being as low as possible. Also, the optimum fan pressure ratio became highly erratic and did not converge after 1000 generations. To validate that the algorithm for supersonic Mach numbers was correct, another run was made with TMA as the objective function. Because TMA requires Mach number to be as low as possible, there should be no difference between this run and Case 1. Given slight changes due to the probabilities of crossover and muta-

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This work is partially supported by grants from the NASA Center of Research Excellence under grant number NAGW-2924 and Honeywell Inc. under grant number 48057. The authors wish to thank them for their support.

Nomenclature CPC

‘ph L

Y Y1 qb

%

constant-pressure specific heat (cold), kJ/kg-K constant-pressure specific heat (hot), kJ/kg-K fuel to air ratio flight Mach number atmospheric pressure, Pa stagnation pressure at stage *, Pa fuel heat of reaction, kJ/kg universal gas constant, kJ/kg-K atmospheric temperature, K stagnation temperature at stage *, K flight velocity, m/s bypass ratio ratio of specific heats (cold) ratio of specific heats (hot) burner efficiency compressor efficiency

System optimization

References

2 3 4 5 6 I 8

9

10

11

12

13

14 15

A. Homaifar

diffuser efficiency fan efficiency nozzle efficiency turbine efficiency

qd

Ylr %I %

1

using genetic algorithms:

Hill, P. G. and Peterson, C. R. Mechanics and Thermodynamics of Propulsion, 3rd ed. Addison-Wesley Publishing Co., Reading, MA, 1970 Zucrow, M. J. Principles of Jet Propulsion and Gas Turbines. John Wiley & Sons, New York, 1948 Sheperd, D. G. Aerospace Propulsion. American Elsevier Publishing Company, New York, 1972 Bathie, W. W. Fundamentals of Gas Turbines. John Wiley & Sons, New York 1984 Hale, F. J. Introduction to Aircraft Performance, Selection, and Design. John Wiley & Sons, New York, 1984 Holland, J. H. Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor, MI, 1975 Goldberg, D. E. Genetic algorithms and Walsh functions: Part I, A gentle introduction. Complex Systems 1989b, 3, 129-152 Goldberg, D. E. Genetic Algorithms in Search, Optimization, & Machine Learning. Addison-Wesley Publishing Co., Reading, MA, 1989 Grefenstette, J. J. and Fitzpatrick, J. M. Genetic search with approximate function evaluations. Proceedings of an International Conference on Genetic Algorithms and Their Applications, ed. J. J. Grefenstette, Pittsburgh, 1985 Axelrod, R. The evolution of strategies in the iterated prisoner’s dilemma. Genetic Algorithms and Simulated Annealing, ed. L. Davis. Pitman, London, and Morgan Kaufmann Publishers, 1987 Goldberg, D. E. Computer-aided gas pipeline operation using genetic algorithms and rule learning. Ph.D. Thesis, Technical Publication BER-84001SP, University of Alabama, 1986 Davis, L. and Coombs, S. Optimizing network link sizes with genetic algorithms. Conference on Computer Simulation and Modeling, Tucson, AZ, 1987b Goldberg, D. E. Sizing populations for serial and parallel genetic algorithms. University of Alabama, Tuscaloosa, The Clearinghouse for Genetic Algorithms, TCGA Report No. 88004, 1988 Pinkel, B. and Karp, 1. M. A thermodynamic study of the turbojet engine. NACA Report No. 891, 1947 Anderson, J. D. Modern Compressible Flow. McGraw-Hill Book Company, New York, 1982

et al.

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Appendix The following is a list of the equations used in the formulation of the thrust per unit mass flow and overall efficiency objective functions. They describe the relationship among pressure ratios, temperature ratios, and other properties for the turbofan engine, as given in Figure I. The derivation of these equations is given in Hill and Peterson.’

(A171

Appl. Math.

Modelling,

1994,

Vol. 18, February

83