Multi-objective Genetic Algorithms for Thermodynamic Optimization of Turbojet Engines

aK. Atashkari, aN. Nariman-zadeh , bX. Yao, aA. Pilechi aDepartment of Mechanical Engineering,Engineering Faculty, The University of Guilan P.O. Box 3756, Rasht, IRAN bSchool of Computer Science, The University of Birmingham Edgbaston, Birmingham B15 2TT, U.K.

Abstract:- A multi-objective genetic algorithm (GAs) is used for Pareto based optimization of thermodynamic cycle of ideal turbojet engines considering different pairs of objective functions, namely, specific thrust (ST), specific fuel consumption (SFC), propulsive efficiency (p), and thermal efficiency (t) for two-objective optimization processes. This provides the best Pareto front of different two-objective optimization from the space of design variables, which are Mach number and pressure ratio, to the space of the above-mentioned objective functions. In this way, a new diversity preserving algorithm is proposed to enhance the performance of multi-objective evolutionary algorithms (MOEAs) in optimization problems with multi-objective functions.

Key-Words:- Thermodynamics, Pareto Optimization, Jet Engines, Multi-objective Optimization, Genetic Algorithms. solutions that are non-dominated to each other but are superior to the rest of solutions in search space. 1 Introduction This means that it is not possible to find a single In the optimization of complex real-world problems, solution to be superior to all other solutions with there are several objective functions or cost respect to all objectives. In other words, changing functions (a vector of objectives) to be optimized the vector of design variables in such Pareto front (minimized or maximized) simultaneously. consisting of these non-dominated solutions could Optimization in engineering design has always been not lead to the improvement of all objectives of great importance and interest particularly in simultaneously. Consequently, such change leads to solving complex real-world design problems. deteriorating of at least one objective to an inferior Recently, some heuristic optimization methods such one. Thus, each solution of the Pareto set includes at as Genetic Algorithms (GAs) have been used least one objective inferior to that of another extensively for such real-world optimisation solution in that Pareto set, although both are problems. Such nature-inspired evolutionary superior to others in the rest of search space. It must algorithms [1-2] differ from other traditional be noted that this non-dominancy does exist in calculus based techniques. The population-based different levels, forming different ranked Pareto approach of evolutionary algorithms is very fruitful fronts, although the first Pareto front is the most to solve many real-world optimal design or decision important and will be the ultimate solution. The making problems which are indeed multi-objective inherent parallelism in evolutionary algorithms optimization issues. In these problems, there are makes them suitably eligible for solving MOPs. several objective or cost functions (a vector of There has been a growing interest in devising objectives) to be optimized (minimized or different evolutionary algorithms for MOPs. Among maximized) simultaneously. These objectives often these methods, the Vector Evaluated Genetic conflict each other so that improving one of them Algorithm (VEGA) proposed by Schaffer [2], will deteriorate another objective function. Fonseca and Fleming’s Genetic Algorithm (FFGA) Therefore, there is no single optimal solution as the [2], Non-dominated Sorting Genetic Algorithm best with the respect to all the objective functions. (NSGA) by Srinivas and Deb [1], and Strength Instead, there is a set of optimal solutions, well Pareto Evolutionary Algorithm (SPEA) by Zitzler known as Pareto optimal solutions or Pareto front, and Thiele [2], and the Pareto archived evolution which distinguishes significantly the inherent strategy (PEAS) by Knowles and Corne [2] are the natures between single-objective and multi- most important ones. A very good and objective optimization problems. The concept of comprehensive survey of these methods has been Pareto front or set of optimal solutions in the space presented in [2]. Basically, both NSGA and FFGA of objective functions in multi-objective as Pareto-based approaches use the revolutionary optimization problems (MOPs) stands for a set of non-dominated sorting procedure originally proposed by Goldberg [3]. Besides, the diversity values to all objective functions [2]. In general, it issue and the lack of elitism was also a motivation can be mathematically defined as: for modification of that algorithm to NSGA-II [4] in which a direct elitist mechanism, instead of sharing * * * * T find the vector X  x1 , x2 ,..., xn  to optimize mechanism, has been introduced to enhance the T population diversity. This modified algorithm has F(X )   f1 (X ), f2 (X ),..., fk (X ) , (1) been known as the state-of-the-art in evolutionary subject to m inequality constraints MOPs. A comparison study among SPEA and other evolutionary algorithms on several problems and gi (X )  0 , i  1 to m test functions showed that SPEA clearly outperforms the other multi-objective EAs [2]. , (2) Some further investigations developed in reference [5] demonstrated, however, that the elitist variant of and p equality constraints NSGA (NSGA-II) equals the performance of SPEA. h j (X )  0 , j  1 to p , (3) The thermal systems, like many other real-world where X * n is the vector of decision or design engineering design problems, are highly complex, variables, and F(X ) k is the vector of non-convex, and multi-objective in nature [6]. The objectives in thermal systems are usually conflicting objective functions which each of them be either and non-commensurable, and thus Pareto solutions minimized or maximized. However, without loss of provide more insights into the competing objectives. generality, it is assumed that all objective functions Recently, there has been a growing interest in are to be minimized. Such multi-objective evolutionary Pareto optimization in the thermal minimization based on Pareto approach can be systems. A thermoeconomic analysis has been conducted using some definitions: performed by Toffolo and Lazzareto [5] in which two exergetic and economic issues in a cogeneration DEFINITION OF PARETO DOMINANCE power plant have been considered as conflicting A vector U  u ,u ,...,u k is dominance to objectives. A similar point of view has also been 1 2 k k considered by Wright, et al [7] in a multi-criterion vector V  v1,v2 ,...,vk  (denoted by U  V ) optimization of building thermal design problem. A if and only if i 1,2,...,k, ui  vi  monetary multi-objective optimization of a combined cycle power system has been studied by j 1,2,..., k : u j  v j . In other words, there is at Roosen, et al [8]. least one u j which is smaller than v j whilst the rest u ’s are either smaller or equal to corresponding v In this paper, an optimal set of design variables in ’s. turbojet engines, namely, the input Mach number

Mo and the pressure ratio of compressor c are found using Pareto approach to multi-objective DEFINITION OF PARETO OPTIMALITY optimization. In this study, different pairs of A point X *   (  is a feasible region in n conflicting objectives in ideal subsonic turbojet satisfying equations (2) and (3)) is said to be Pareto engines are selected for optimization. These include optimal (minimal) with respect to the all X   if a combination of thermal efficiency (t) and and only if F(X * )  F(X ) . Alternatively, it can propulsive efficiency ( ) together with specific fuel p be readily restated as consumption (SFC) and specific thrust (ST). In this i 1,2,..., k,X   {X *} f (X * )  f (X ) way, a new preserving diversity algorithm called є- i i * elimination diversity algorithm is proposed to  j 1,2,..., k : f j (X )  f j (X ) . In other enhance the performance of NSGA-II in terms of words, the solution X * is said to be Pareto optimal diversity of population and Pareto fronts. (minimal) if no other solution can be found to dominate X * using the definition of Pareto 2 Multi-Objective Optimization dominance. Multi-objective optimization which is also called DEFINITION OF PARETO SET multicriteria optimization or vector optimization has is a set in the ٭ For a given MOP, a Pareto set been defined as finding a vector of decision Ƥ variables satisfying constraints to give acceptable decision variable space consisting of all the Pareto ∄ | {X   ٭optimal vectors Ƥ X   :F(X )  F(X )}. In other words, there is no other X  as a vector of number of objective functions (particularly for more than two objectives) in MOPs. Such modified .٭decision variables in  that dominates any X ∈Ƥ MOEA is then used for a four-objective thermodynamic optimization of subsonic turbojet DEFINITION OF PARETO FRONT engines and the results are compared with those of .is a set of original NSGA-II ٭For a given MOP, the Pareto front ƤŦ vector of objective functions which are obtained using the vectors of decision variables in the Pareto 3 є-elimination diversity algorithm In the є-elimination diversity approach that is used ٭that is ƤŦ ,٭setƤ In to replace the crowding distance assignment .{٭ {F(X )  ( f1(X ), f2 (X ),...., fk (X )): X Ƥ approach in NSGA-II [4], all the clones and/or - is a set of the є ٭other words, the Pareto front ƤŦ similar individuals are recognized and simply .٭ vectors of objective functions mapped from Ƥ eliminated from the current population. Therefore, based on a pre-defined value of as the elimination Evolutionary algorithms have been widely used for є multi-objective optimization because of their natural threshold (є=0.001 has been used in this paper) all properties suited for these types of problems. This is the individuals in a front within this limit of a mostly because of their parallel or population-based particular individual are eliminated. It should be search approach. Therefore, most of difficulties and noted that such є-similarity must exist both in the deficiencies within the classical methods in solving space of objectives and in the space of the multi-objective optimization problems are associated design variables. This will ensure that eliminated. For example, there is no need for either very different individuals in the space of design several run to find the Pareto front or quantification variables having -similarity in the space of of the importance of each objective using numerical є weights. In this way, the original non-dominated objectives will not be eliminated from the sorting procedure given by Goldberg [3] was the population. The pseudo-code of the є-elimination basic motivation for emerging different versions of approach is depicted in figure (1). Evidently, the multi-objective optimization algorithms. However, clones or є-similar individuals are replaced from the it is very important that the genetic diversity within population with the same number of new randomly the population be preserved sufficiently [9]. This generated individuals. Meanwhile, this will main issue in MOPs has been addressed by many additionally help to explore the search space of the related research works. Consequently, the premature given MOP more efficiently. convergence of MOEAs is prevented and the solutions are directed and distributed along the true Pareto front if such genetic diversity is well 4 Multi-Objective Thermodynamic provided. The Pareto-based approach of NSGA-II Optimization of Turbojet Engines [4] has been recently used in a wide area of Turbojet engines use air as the working fluid and engineering MOPs because of its simple yet produce thrust based on the variation of kinetic efficient non-dominance ranking procedure in energy of burnt gases after combustion. The study yielding different level of Pareto frontiers. However, of thermodynamic cycle of a turbojet engine the crowding approach in such state-of-the-art involves different thermo-mechanical aspects such MOEA is not efficient as a diversity-preserving as developed specific thrust, thermal and propulsive operator, particularly in problems with more than efficiencies, and specific fuel consumption. A two objective functions. In fact, the crowding detailed description of the thermodynamic analysis distance computed by routine in NSGA-II [4] may and equations of ideal turbojet engines are given in return an ambiguous value in such problems. The [10]. Such introductory thermodynamic model is reason for such drawback is that sorting procedure basically enough to capture the principles of of individuals based on each objective in this behaviour and interactions among different input algorithm will cause different enclosing hyper-box. and output parameters in a multi-objective optimal Thus, the overall crowding distance of an individual sense. Furthermore, this provides a suitable real- computed in this way may not exactly reflect the true measure of diversity or crowding property. In this work, a new method is presented to modify NSGA-II so that it can be safely used for any Pseudo-code of є-elimination є-elim=є-elimination (pop) //pop includes Evidently, it can be observed that p, t, and ST are design variables and objective functions// maximized whilst SFC is minimized in those sets of define є //Define elimination threshold objective functions. A population size of 100 has k=1 //Front No. been chosen in all runs with crossover probability Pc i=1 until i+1

IF {║F(X(i), F(X(j))║< є ⋀ ║X(i), X(j)║< є} Figure (2) shows the Pareto front of two objectives, propulsive efficiency and specific fuel consumption ٭X(i), X(j) ∈ Ƥk ٭F(X(i), F(X(j))∈ ƤŦk (p, SFC), obtained using both the approach of this THEN pop= pop\ pop( j ) // Remove the є-similar individual work. It is clear from this figure that choosing appropriate values for the decision variables, r_new_ind = make_new_random_individual namely Mach number (M0) and pressure ratio (c),

//Generate new random to obtain higher value of p would normally cause individual higher value of SFC. However, if the set of decision pop=pop ∪ r_new_ind //Add new randomly generated variables are selected based on this Pareto front, it will consequently lead to the best possible Figure 1: Pseudo-code of є-elimination for preserving genetic diversity Propulsion Efficiency vs Specific Fuel Thermal Efficiency vs Specific Fuel Consumption Consumption

0.6 0.72 world engineering benchmark for comparison 0.7 0.5 y 0.68 c y n purpose between MOEA using the new diversity c e 0.66 i 0.4 n c e i i f c

f 0.64 i e f preserving mechanism of this work with NSGA-II. f

n 0.3 e 0.62

l o i a

s 0.6 l m u 0.2 r e p 0.58 o h t r

p 0.56 Input parameters involved in such thermodynamic 0.1 0.54 analysis in an ideal turbojet engine given in 0 0.52 Appendix A are Mach number (M ), input air 2.E-05 4.E-05 6.E-05 8.E-05 2.1E-05 2.2E-05 2.3E-05 2.4E-05 2.5E-05 0 specific fuel consumption (kg/sec/N) specific fuel consumption (kg/sec/N) temperature (T0, ºK), specific heat ratio ( ), heating Figure 2: Pareto front of Figure 3: Pareto front of value of fuel (hpr, kj/kg), exit burner total propulsive efficiency and thermal efficiency and specific temperature (T , K), and pressure ratio,  . Output specific fuel consumption in 2- fuel consumption in 2-objective t4 c optimization parameters involved in the thermodynamic analysis objective optimization in the ideal turbojet engine given in [10]are, specific combination of propulsive efficiency and specific thrust, (ST, N/kg/sec), fuel-to-air ratio (f), specific fuel consumption ( , SFC). In other words, if any fuel consumption (SFC, kg/sec/N), thermal p other pair of decision variables M0 and c is chosen, efficiency (t), and propulsive efficiency (p). However, in multi-objective optimization study, the corresponding values of p and SFC will locate some input parameters are already known or a point inferior to this Pareto front. Such inferior  assumed as, T0 = 216.6 K, =1.4, hpr =48000 kj/kg, area in the space (plane in this case) of p and SFC and Tt4 = 1666 K. The input Mach number 0 < M0 ≤ is conspicuously the bottom/right side of this Pareto 1 and the compressor pressure ratio 1 ≤ c ≤ 40 are front. In figure (2) two sections can be clearly considered as design variables to be optimally found distinguished. The corresponding values of decision based on two-objective optimization of 4 output variables and objective functions in the first section -5 parameters, namely, ST, SFC, ηt, and ηp. of figure (2-a) are as c = 40, 0 < Mo ≤1, 2.096x10 -5 ≤ SFC ≤2.43x10 , and 0 < p ≤0.39. These values in

the second section of that figure are as 1.07 ≤c ≤ -5 -5 In order to investigate the optimal thermodynamic 8.25, Mo ≈1., 3.16x10 ≤ SFC ≤6.8x10 , and 0.4 < behaviour of subsonic turbojet engines, 5 different p ≤0.55. Therefore, this Pareto front provides set each including two objectives of output optimal solutions for decision variables M0 and c if parameters, namely, ST, SFC,  , and p, are t the two-objective optimization of p and SFC is of considered individually. Such pairs of objectives to importance to the designer. Figure (3) shows the be optimized separately have been chosen as (p, Pareto front of two objectives, thermal efficiency and specific fuel consumption ( , SFC), obtained SFC), (p, ST), (t, SFC), (t, ST), and (p, t). t using the approach of this work. It is clear from this Propulsion Efficiency vs Specific Thrust Thermal Efficiency vs Specific Thrust figure that choosing appropriate values for the 0.6 0.7 0.69 0.5 decision variables, namely Mach number (M0) and y 0.68 c y n c e

0.4 n i e c 0.67 i i f pressure ratio (c), to obtain higher value of  c f t i f e f

0.3 e 0.66 n

l o i a s l would normally cause higher value of SFC. m 0.65 r u

0.2 e p h o t r 0.64 However, if the set of decision variables are selected p 0.1 based on this Pareto front, it will consequently lead 0.63 0 0.62 to the best possible combination of thermal 0 500 1000 1500 850 950 1050 1150 specific thrust (N/kg/sec) specific thrust( N/kg/sec) efficiency and specific fuel consumption (t, SFC). In other words, if any other pair of decision Figure 4: Pareto front of Figure 5: Pareto front of propulsive efficiency and thermal efficiency and specific variables M0 and c is chosen, the corresponding specific thrust in 2-objective thrust in 2-objective optimization optimization values of t and SFC would locate a point inferior to this Pareto front. Such inferior area in the space efficiency and specific thrust (t, ST), obtained

(plane in this case) of t and SFC is conspicuously using the approach of this work. It is clear from this the bottom/right side of this Pareto front. The figure that choosing appropriate values for the corresponding values of decision variables and decision variables, namely Mach number (M0) and objective functions of figure (3) are as c = 40, .01 < pressure ratio (c), to obtain higher value of t -5 -5 Mo ≤1, 2.1x10 ≤ SFC ≤2.43x10 , and 0.65 ≤ t ≤ would normally cause lower value of SFC. 0.71. Therefore, this Pareto front provides optimal However, if the set of decision variables are selected solutions for decision variables M0 and c if the two- based on this Pareto front, it will consequently lead to the best possible combination of thermal objective optimization of t and SFC is of importance to the designer. Figure (4) shows the efficiency and specific thrust (t, ST). In other

Pareto front of two objectives, propulsive efficiency words, if any other pair of decision variables M0 and and specific thrust (p, ST), obtained using the c is chosen, the corresponding values of t and ST approach of this work. It is clear from this figure will locate a point inferior to this Pareto front. Such that choosing appropriate values for the decision inferior area in the space (plane in this case) of t variables, namely Mach number (M0) and pressure and ST is conspicuously the bottom/left side of this ratio (c), to obtain higher value of p would Pareto front. The corresponding values of decision normally cause lower value of ST. However, if the variables and objective functions of figure (5) are as set of decision variables are selected based on this 37.3 ≤ c ≤ 40, 0 < Mo ≤1, 890 ≤ ST ≤1169, and 0.64

Pareto front, it would consequently lead to the best ≤ t ≤ 0.7. Therefore, this Pareto front provides possible combination of propulsive efficiency and optimal solutions for decision variables M0 and c if specific thrust ( , ST). In other words, if any other p the two-objective optimization of t and ST is of pair of decision variables M0 and c is chosen, the importance to the designer. corresponding values of p and ST will locate a point inferior to this Pareto front. Such inferior area Figure (6) depicts the Pareto front of two objectives, thermal efficiency and propulsive efficiency ( , p), in the space (plane in this case) of p and ST is t conspicuously the bottom/left side of this Pareto obtained using the approach of this work. It is front. In figure (4), two sections can be clearly obvious from this figure that choosing appropriate distinguished. The corresponding values of decision values for the decision variables, namely Mach variables and objective functions in the first section number (M0) and pressure ratio (c), to obtain higher of figure (4) are as 13.5 ≤ c ≤ 39.3, 0 < Mo ≤1, value of t would normally cause lower value of p. 817.8 ≤ ST ≤1169.4, and 0 < p ≤0.41. These values However, if the set of decision variables are selected in the second section, where p is more than 0.4, are based on this Pareto front, it will consequently lead to the best possible combination of thermal as 1.22 ≤c ≤ 4.28, 0 ≤ Mo ≤ 1, 515.1 ≤ ST ≤ 906., and 0.41 < p ≤0.51. Therefore, this Pareto front efficiency and propulsive efficiency (t, p). In provides optimal solutions for decision variables M0 other words, if any other pair of decision variables and c if the two-objective optimization of p and M0 and c is chosen, the corresponding values of t

ST is of importance to the designer. Figure (5) and p will locate a point inferior to this Pareto shows the Pareto front of two objectives thermal, front. Such inferior area in the space (plane in this Propulsion Efficiency vs Thermal efficiency It should be noted that these relationships are valid when the corresponding two-objective optimization 0.6 of such functions is of importance to the designer

y 0.55 and, in fact, demonstrates the optimal compromise c n e

i of such pair of objectives.

c 0.5 i f f e

n 0.45 o i 5 Conclusion s l u

p 0.4 A new diversity preserving mechanism called є- o r

p elimination algorithm has been proposed and 0.35 successfully used with the Pareto approach of 0.3 MOEAs for thermodynamic cycle optimization of 0.1 0.3 0.5 0.7 ideal turbojet engines. Such multi-objective thermal efficiency optimization led to important relationships and Figure 6: Pareto front of propulsive efficiency and useful optimal design principles in thermodynamic thermal efficiency in 2-objective optimization optimization of ideal turbojet engines in the space of objective functions. These optimal principles could case) of t and p is conspicuously the bottom/left not have been discovered without the use of multi- side of this Pareto front. The corresponding values objective optimization process. of decision variables and objective functions of figure (6) are as 1 ≤  ≤ 8.78, M =1, 0.4 ≤  ≤ c o p References: 0.56, and 0.16 ≤ t ≤ 0.55. Such values for the single point in this figure are as,  = 40, M =1,  = [1]Srinivas, N. and Deb, K., “ Multiobjective c o p optimization Using Nondominated Sorting in 0.4, and  =0.71, respectively. t Genetic Algorithms”, Evolutionary Computation, Vol. 2, No. 3, pp 221-248, 1994. Evidently, figures (2-6) reveal some important and [2]Coello Coello, C.A., “A comprehensive survey of interesting optimal relationships of such evolutionary based multiobjective optimization thermodynamic parameters in ideal thermodynamic techniques”, Knowledge and Information Systems: cycle of turbojet engines that may have not been An Int. Journal, (3), pp 269-308, 1999. known without a multi-objective optimization [3]Goldberg, D. E., “Genetic Algorithms in Search, approach. For example, figures (2-4) demonstrate Optimization, and Machine Learning”, Addison- that the optimal behaviours of  with respect to p Wesley, 1989. both SFC and ST are approximately linear. [4]Deb, K., Agrawal, S., Pratap, A., Meyarivan, T., However, the corresponding relationships of t with “A fast and elitist multi-objective genetic algorithm: respect to both SFC and ST from figures (3-5) can NSGA-II”, IEEE Trans. On Evolutionary be readily represented by Computation 6(2):182-197, 2002. [5]Toffolo, A., and Lazzaretto, A., “Evolutionary 9 2 t=(7x10 )(SFC) –323350 (SFC)+4.1524, (4) algorithms for multi-objective energetic and economic optimization in thermal system design”, with R2=0.9931, and (4) Energy 27:549-567, 2002. [6]Bejan, A., Tsatsaronis, G., and Moran, M., -7 2 t =(6x10 )(ST) +0.0014 (ST)+1.5233, (5) “Thermal & Design Optimization”, John Wiley & (5) Sons, NY, 1996. [7]Wright, J.A., Loosemore , H.A., and Farmani, R., with R2=0.9986. “optimization of building thermal design and control by multi-criterion genetic algorithm”, Energy and Moreover, figure (6) also represents a non-linear Buildings 34:959-972, 2002. optimal relationship of t and p in the form of [8]Roosen, P., Uhlenbruck, S., and Lucas, K., “Pareto optimization of a combined cycle power 2 p =0.8047 (t) +9.874(t)+0.6998, (6) system as a decision support tool for trading off (6) investment vs. operating costs”, Int. J. of Thermal Sciences 42: 553-560, 2003. with R2=0.9982. [9]Yao, X., “Evolving Artificial Neural Networks”, Proceedings of IEEE, 87(9):1423-1447, Sept., (1999). [10] Mattingly, J.P., “Elements of Gas Turbine Propulsion”, Mc-Graw Hill, 1996.