Cycle Optimization of a Turbine Engine: an Approach Based on ...

Cycle Optimization of a Turbine Engine: an Approach Based on Genetic Algotithms S. Borguet1 , V. Kelner2 , O. L´eonard 1 University of Li`ege, Aerospace and Mechanical Engineering Department Chemin des chevreuils 1, 4000 Li`ege, Belgium 1 Turbomachinery Group 2 Vehicle Engineering Group email: {S.Borguet,V.Kelner,O.Leonard}@ulg.ac.be

Abstract— Multi-objective optimization is performed of a turbofan. The objectives are to minimize the Specific Fuel Consumption at cruise conditions, as well as to maximize the Specific Thrust during the take-off phase. Three design variables have been selected: the by-pass ratio, the fan pressure ratio, and the high pressure compressor pressure ratio. The backbones of the optimization approach consist of a genetic algorithm and a modular gas turbine simulation tool. Keywords— Turbine Engine Model, Genetic Algorithm, Multi-Objective Optimization

functions evaluation is not too expensive in terms of calculating time. Therefore GAs are efficient when coupled to approximation methods [13], to parametric reconstructions [8], or like in this study, to 0-D models. The paper is organized as follows. The turbine engine simulation tool and the Genetic Algorithm that has been used in the optimization loop are successively presented in section 2 and section 3. Section 4 yields an illustrative case of a turbine engine optimization. Finally, some conclusions are drawn in section 5.

I. Introduction Turbine engines can be modelled at various levels of detail, ranging from simple algebraic relations to full 3-D description of the gas path. In this contribution, aerothermodynamic models (aka. 0-D models) are considered. A number of reference textbooks exist on this topic: [10], [11], [14]. Due to their low computational burden, such models are massively used by the manufacturers throughout an engine program: for preliminary design and performance prediction, for the synthesis of the control laws, for condition monitoring, as well as for the engine-airframe integration. In the present contribution, we report the development of a modular gas turbine simulation tool in the Matlab environment. One of the main features of MaTES (Matlab Turbine Engine Simulator) is its modularity. In order to perform a multi-point optimization of a turbine engine, this simulation tool has been coupled with a Genetic Algorithm. Genetic Algorithms (GAs), initially developed by Holland [4], remain the most recognized and practiced form of Evolutionary Algorithms which are stochastic optimization techniques that mimic the Darwin’s principles of natural selection and survival of the fittests. As a zero-order optimization method, GAs can be used in the case of discontinuous objective functions, within disjoined and/or non-convex design spaces, and together with discrete, continuous or integer design variables. With respect to local search methods (e.g. gradient-based) GAs minimize the risk to converge to a local optimum thanks to the simultaneous processing of the whole candidate solutions. Moreover, they are particularly suitable for multiobjective optimization problems which are often encountered in real design problems. Because of these advantages, GAs are more and more widely used in various disciplines. However, GAs generally require a large number of iterations and they converge slowly. Optimization using Genetic Algorithms is thus advantageous when the objective

II. Turbine Engine Modelling In the following, we focus on a particular type of jet engine: the separated flows turbofan. With the current level of technology, this one has revealed to be the optimum configuration for high subsonic commercial aircraft [2]. A schematic of the engine is sketched in figure 1. 17 18 19

13

12

V bypass Fan

0

1

25

2

hpt

combustor

hpc

lpc

3

4

V core

lpt

45

7 8

5

9

Fig. 1. Layout of the Separated Flows Turbofan

A. Component Library A library of engine components such as fan, compressor, combustion chamber, turbine or nozzle has been developed. Each component can be seen as an operator the purpose of which is to compute the thermodynamic state of the fluid (typically mass flow W , total temperature T 0 and pressure p0 ) at the outlet of the module based on the inlet conditions and some additional parameters (see figure 2). The modelling relies on the equations for mass, momentum and energy balances and/or on empirical information derived from rig tests or advanced CFD calculations (e.g. compressor and turbine maps). Win T0in p0in

Wout Component

T0out p0out

Fig. 2. Schematic of a Component

A.1 Thermodynamic Properties The working fluids are considered as perfect gases of constant specific heats. Upstream the combustion chamber, for pure air, we assume Cp,c = 1005 J/kg.K and γc = 1.4. Downstream the combustor, for combustion products, we assume Cp,h = 1148 J/kg.K and γh = 1.333. The ratio of the universal gas constant to the molar mass of the fluid is set to a constant value of 287.05 J/kg.K. A.2 Environment and Inlet Ambient pressure and temperature are obtained from the International Standard Atmosphere (ISA) table as a function of altitude h and variation from ISA day temperature ∆TISA : p0 T0

= fp (h) = fT (h) + ∆TISA

(1) (2)

The stagnation conditions at the engine inlet depend on the flight Mach number Mf l and are computed with SaintVenant-Wantzel relations:   γc − 1 2 0 Mf l (3) T1 = T0 · 1 + 2 c  0  γ γ−1 c T1 0 p1 = p 0 · (4) T0 The engine inlet is supposed to be adiabatic, but introduces a pressure loss πd so that the thermodynamic state of the fluid at the inlet exit (station 2) is: T20 p02

= T10 = πd · p01

(5) (6)

A.3 Fan For the present application, a simplified version of the fan module is used. Indeed, the same pressure ratio and efficiency are applied to the root and tip regions of the fan. Applying the conservation equations, we have for the fan tip: W13 p013 0 T13

BP R BP R + 1 = πf an · p02    γc −1 1 γc = T20 · 1 + −1 · πf an ηis,f an = W2 ·

(7) (8) (9)

where BP R stands for By-Pass Ratio, i.e. the ratio of cold and hot mass flows. And for the fan root, under our hypotheses: W25 p025 0 T25

1 = W2 · BP R + 1 = p013 0 = T13

and is positive since it is supplied to the air. At the design stage, the mass flow, pressure ratio and polytropic efficiency are specified. This allows a direct computation of the output variables and the scaling of the fan map. Basically, this map models the behaviour of a reference fan at various operating conditions. Our model uses standard maps with β-lines, extracted from GasTurb. In the prediction mode (off-design calculation), the inlet corrected mass flow, pressure ratio and isentropic efficiency are read from the map with the current values of relative spool speed and β. A.4 High Pressure Compressor The modelling of the high pressure compressor is very similar to that of the fan, except there is no splitting of the mass flow. For sake of brevity, the constitutive equations are not written. The off-design behaviour of the HPC is also described with characteristic curves. A.5 Combustion Chamber The combustion chamber increases the enthalpy of the working fluid through the combustion of fuel. The governing equations for this component are: W4 p04 T40

= W3 + WF = πb · p03 ηb · WF · LHV + W3 · Cp,c · T30 = W4 · Cp,h

(13) (14) (15)

where LHV stands for Low Heating Value of the fuel. At the design stage, the combustor exit temperature is specified, which allows determining the required fuel flow WF . In the prediction mode, the fuel flow is imposed and T40 is obtained from the energy balance. The combustion efficiency ηb expresses the fact that the combustion process is incomplete. The pressure loss πb accounts for the friction loss (hydraulic head loss). The hot loss due to the addition of heat to a moving fluid is here disregarded. A.6 Turbines The same turbine module is used for both the high and low pressure turbines. Currently, no cooling scheme is available for the turbine. The thermodynamic quantities at the turbine outlet (station 45 for the hpt, station 5 for the lpt) are computed according to: Wout p0out 0 Tout

= Win = π h pt · p0in l   γh −1  γh 0 = Tin · 1 + ηis, h pt · πh pt − 1 l

(16) (17) (18)

l

(10) (11) (12)

The power required to drive the fan is given by: 0 0 P W = W13 · Cp,c · (T13 − T20 ) + W25 · Cp,c · (T25 − T20 )

The available power generated by the turbine is given by: 0 0 P W = ηm · Win · Cp,h · (Tout − Tin ) and is negative since it is extracted from the working fluid. The mechanical efficiency ηm accounts for the loss generated in the shaft linking the turbine and its load.

At the design stage, the power extracted from the turbine and the polytropic efficiency are specified. The quantities at the turbine exhaust are thus easily computed and the turbine map is scaled. As for the compressors, these characteristics give the corrected inlet mass flow, pressure ratio and isentropic efficiency as a function of the corrected speed and the β parameter. In the prediction mode, the map is first read and the outlet quantities are then computed. A.7 Nozzles For the engine type at hand, a convergent nozzle is considered. Assuming an adiabatic process, the constitutive equations are: W(1)9 p0(1)9 0 T(1)9

= W(1)7 = πn · =

p0(1)7

0 T(1)7

(19) (20) (21)

where πn represents the pressure loss across the nozzle. Depending on the pressure ratio rp = p0 /p0in , there exist two flow regimes: • if rp > rp,cr , the static pressure in the exit plane pout is equal to the ambient pressure p0 and the exhaust Mach number is less than one. • if rp ≤ rp,cr , the nozzle is chocked. The flow at the exit is sonic and the static pressure in the exit plane pout is equal to the critical pressure. The critical pressure ratio rp,cr defining the boundary between the two regimes is given by:  rp,cr =

2 γ+1

γ  γ−1

The gross thrust generated by the nozzle is defined by: F G = Wout · Vout + Aout · (pout − p0 )

On the other hand, in the prediction mode, the user may not assign an arbitrary value to the design parameters anymore. The engine cycle and performances depend on the operating point defined by the atmospheric conditions (altitude, flight Mach number, ∆TISA ) and the value of the control variables (the fuel flow WF ). Here, the conservation laws translate into a set of nonlinear algebraic equations, also termed compatibility equations. As a result of the expression of the characteristic curves and of the modelling of the nozzle, eight compatibility relations have to be verified in the present case: Power balance for the LP spool Power balance for the HP spool Flow compatibility for the fan Flow compatibility for the hpc Flow compatibility for the hpt Flow compatibility for the lpt Core nozzle flow capacity Bypass nozzle flow capacity

P Wf an = −P Wlpt P Whpc = −P Whpt W2 = W2map map W25 = W25 map W4 = W4 map W45 = W45 des A9 = A9 A19 = Ades 19

where Wxmap refers to the value obtained from the component map and Ades refers to the design value of the nozzle x exit area. The following eight iteration variables are associated to the aforementioned equations: XN L XN H W2 BP R βf an βhpc βhpt βlpt

LP spool rotational speed HP spool rotational speed Engine inlet mass flow Bypass ratio positioning in the fan map positioning in the hpc map positioning in the hpt map positioning in the lpt map

(22)

The nozzle module relies on the same computation sequence for both design and prediction modes. Given the inlet conditions, the pressure loss and the ambient backpressure, the outlet conditions and the flow regime are determined. Then the static pressure, flow speed and area are computed at the nozzle exhaust to obtain the gross thrust. B. Assembling the Components To build the turbofan model, the basic modules presented in the previous section are simply assembled following the general schematic of figure 1. Ensuring the conservation of mass, momentum and energy throughout the engine generates constraints which actually restrict the operating range of each component. In the design mode, the user is allowed to specify the key design parameters of the engine such as corrected inlet mass flow, bypass ratio, fan and hpc pressure ratio, combustor exit temperature. The conservation laws are naturally enforced; for instance, the power required by the fan determines the enthalpy drop in the lpt. This leads to a direct, non-iterative calculation of the engine cycle and performances.

Hence, the performance prediction of the turbine engine comes to finding the roots of a vector-valued function. To this end, a quasi-Newton algorithm with a Broyden update of the Jacobian is used. Now, we define three important performance parameters of a turbine engine that will be used in the following. The Net Thrust F N represents the available force to propel the aircraft. It is the sum of the gross thrust of the core (F G9 ) and bypass (F G19 ) nozzles minus the momentum drag. F N = F G19 + F G9 − W0 · V0 The Specific Thrust SF N is the amount of thrust per unit of mass flow. It is an interesting property to compare engines of various sizes as well as different engine types of similar size. FN SF N = W0 Finally, the Specific Fuel Consumption SF C is the ratio of the fuel flow and the net thrust. It is inversely proportional to the global efficiency of the engine. SF C =

WF FN

20

To assess the quality of our modelling tool, a validation test-case has been designed. For the generation of reference results, we use GasTurb [9], a commercial software dedicated to gas turbine modelling. Table I reports the value of the main parameters of a hypothetical engine imagined for validation purpose.

15

h=0m ∆TISA = 0 K WRf an = 300 kg/s πf an = 1.70 πhpc = 8.00 πb = 0.97 TIT = 1400 K ηp,hpt = 0.91

Mf l = 0.0 πd = 0.99 BP R = 5.50 ηp,f an = 0.91 ηp,hpc = 0.90 ηb = 0.99 ηp,hpt = 0.91 πn = 0.99

These specifications are input in MaTES and GasTurb, and a design point calculation is carried out. Table II provides a comparison between both codes for some cycle parameters, such as fuel flow, net thrust, core and bypass nozzle exit Mach number, pressure ratio of the turbines. The gap between MaTES and GasTurb is less than 5 percents on all selected quantities but M9 . The main explanation of the gap lies in the fact that unlike MaTES, GasTurb assumes that the thermodynamic properties of the fluids depend on the temperature and the fuel to air ratio. TABLE II Comparison at Design-Point

WF [kg/s] F N [kN] M9 [ ] M19 [ ] πhpt [ ] πlpt [ ]

MaTES 1.007 96.19 0.692 0.913 2.590 3.708

GasTurb 0.995 97.37 0.743 0.895 2.557 3.586

Gap % 1.282 -1.220 -6.824 2.030 1.287 3.408

Then, to validate the off-design capability of MaTES, an equilibrium running line is computed for ambient conditions defined as h = 10668 m, Mf l = 0.85, ∆TISA = 0 K. Figure 3 shows the evolution of net thrust and Specific Fuel Consumption versus relative corrected LP spool speed. It can be seen that the results generated by MaTES favourably compare with those from GasTurb. The gap for the thrust is less than 2.5 percents and the gap for the SFC is less than 5 percents. Once again, the hypotheses on the gas properties made in each software are responsible for the gap. The smaller gap obtained on the net thrust can be explained by the fact that for high bypass ratio engines, the bypass stream

10 5 0 MaTES

GasTurb

24 SFC [g/kN*s]

TABLE I Design-Point Main Specifications

FN [kN]

C. Validation of the Engine Model

22 20 18 16 70

75

80

85 PCN1 [%]

90

95

100

Fig. 3. Comparison of Net Thrust and SFC predictions contributes to up to 80 percents of the thrust. This flow is made of pure air and has a moderate temperature. Hence the effect of the temperature and the fuel to air ratio on the specific heats is less important than for the core stream, which is used to calculate the fuel flow. This analysis could have been done on the basis of the design point results, considering the gaps on M9 and M19 . As a conclusion, the developed engine modelling tool MaTES provides satisfactory results for preliminary studies. One of its main advantages comes from its modular structure, allowing a number of different engine configurations to be easily simulated. Also, new capabilities can be brought to specific modules (such as turbine cooling or blow-off valves) without the need of rewriting the entire engine model. III. Optimization Loop A. Mechanics of GAs GAs work with artificial populations of individuals that represent candidate solutions and, in spite of their diversity, most of them are based on the same iterative procedure (figure 4).

Fig. 4. Classical GA flowchart The individuals are characterized by genes, which result from the coding of the parameters of the optimization problem. Each individual is evaluated according to the objectives and to the constraints of the optimization problem. This evaluation is used in the process of selection, which determines the probability that an individual is part of the following generation. Successive new individuals (children) are generated by using the best features of the previous generation (parents) and sometimes, innovating ones. The evolution of those individuals, through the genetic operators, tend to improve the quality of the population and to

converge to a global optimum. B. GAs and Multi-Objective Optimization Problems A Multi-objective Optimization Problem (MOP) can be stated as: min f (x) (23) T

where f (x) = [f1 (x), f2 (x), · · · , fnf o (x)]. Because of the conflicting nature of the objectives, a MOP has no unique solution, but rather a set of compromised solutions that can be classified, with the Pareto dominance concept, into dominated and non dominated ones. The non dominated solutions representing the best compromise are distributed on the so-called Pareto front. Traditional a priori methods [1] are based on a decision phase that transforms the MOP into a single objective one through an aggregating approach (weighted sum, goal programming, weighted min-max, etc). Unfortunately these techniques lead to a unique optimized solution on the Pareto front (depending on the a priori weights for example). Conversely, as a powerful a posteriori method which works with a population of candidate solutions, GAs solve the true MOP and provide as many Pareto-optimal solutions as possible in a single run. Furthermore, GAs are less susceptible to the shape or continuity of the Pareto front: they can approximate concave or non continuous Pareto fronts, which an aggregating approach does not allow. These advantages have made them very popular to solve unconstrained MOPs and numerous Pareto based approaches (MOGA, NSGA, NPGA, etc) have been proposed and compared in the litterature [1], [15]. C. Efficient GA for Complex Optmization Problems The GA that has been used in this study is based on a previous version of a home-made computer code that has already been validated and successfully applied to design problem [3], [6], [7], [8]. MOHyGO (Multi-Objective Hybrid Genetic Optimizer) includes the classical genetic operators, and its main features are the following: a real-valued coding for the decision variables, a BLX-alpha crossover, a mutation operator, and a Pareto based approach coupled with an original constraint-handling technique.

each infeasible solution receives a penalty factor (Rconst ) computed on the basis of the violation of the constraints. At last, a selection, based on a “penalized tournament”, is applied. This consists of randomly choosing and comparing the (generally two) individuals: • if they are all feasible, the best ranked element (according to MOGA) wins, • if they are all infeasible, the one having the lower Rconst value wins, • if one is feasible and the others are infeasible, the feasible individual wins. This technique allows a Pareto dominance approach, even in the case of a constrained optimization problem. Moreover, MOHyGO includes an archiving procedure (figure 5). This operator externally stores the non dominated solutions found at each generation in the following way: 1. Copy all the individuals of the current Pareto front to the archive. 2. Remove any dominated solutions from the archive. 3. If the number of non dominated individuals in the archive is greater than a given maximum Narch : apply a clustering strategy [12]. 4. Continue the genetic process. This archiving procedure has been inspired from the SPEA (Strength Pareto Approach) proposed by Zitzler [15]. However, in our implementation, the individuals stored in the archive do not participate to the selection phase, which results in a less disturbed evolution process. The clustering step (e.g. reducing the size of the archive while maintaining its characteristics) is mandatory: the Pareto front (and the archive) could sometimes contain a huge number of non dominated individuals. However, the designer is not interested in being offered with a too large number of solutions from which he has to choose. IV. Numerical Results In this section, the coupling procedure of the Genetic Algorithm and the engine simulation tool is assessed through one test-case. This application describes a simple optimization problem of a high by-pass ratio, unmixed flow turbofan. A. Description Three design variables are considered in this study: the fan pressure ratio πf an , the HPC pressure ratio πhpc and the by-pass ratio BP R. Their ranges are defined in table III. TABLE III Design variables

Fig. 5. MOHyGO flowchart As we can see in figure 5, the constraints are firstly evaluated for each individual. On the one hand, the feasible solutions are ranked according to the MOGA algorithm proposed by Fonseca and Fleming [1]. On the other hand,

Variable πf an πhpc BP R

min 1.5 3 2

max 1.8 12 8

For this test-case, we consider a Pareto-based optimization of two objective functions. The first one is the Spe-

TABLE IV Design-Point Parameters

−5

2.5

x 10

2.3

2.2

2.1

2

1.9

1.8

1.7 300

h = 10668 m ∆TISA = 0 K ηp,f an = 0.90 πb = 0.99 TIT = 1300 K ηp,hpt = 0.91

Mf l = 0.85 πd = 1.00 ηp,hpc = 0.89 ηb = 0.99 ηp,hpt = 0.91 πn = 0.99

When the genetic algorithm calls the engine model, a design-point calculation is first carried out to get the SFC at cruise. Then, the SFN at take-off is obtained from an off-design calculation. For the off-design part, only the variables defining the operating point are needed. Basically, they consist of the flight conditions and a power setting being here the Turbine Inlet Temperature. The take-off point is characterised by SLS-ISA conditions (i.e. h = 0 m, Mf l = 0, ∆TISA = 0 K) and a TIT which is 250 degrees higher than the TIT at cruise. This increase in turbine temperature of 250 degrees leads to a ratio T40 /T20 which is about 5 percents higher for take-off than for cruise. As reported in [2], this is a typical value for contemporary jet engines.

Last Pareto front Archive

2.4

SFC Cruise [kg/N*s]

cific Fuel Consumption (SFC) at cruise conditions which has to be minimized. The second objective corresponds to the Specific Thrust at take-off rating. Note that those two quantities are independent of the engine size. Moreover, we are dealing with two operating points: cruise and take-off. The cruise regime is chosen as the design point; consequently the take-off rating is seen as an off-design point. The design point is specified with the 3-tuple of design variables and the fixed parameters reported in table IV which reflect a certain level of technology available to the manufacturer.

320

340

360

380 400 420 SFN Take−Off [m/s]

440

460

480

500

Fig. 6. Maximization of SFN with minimization of SFC

Pareto-optimal solutions shown in figure 6 help the designer to select the design parameters that verify the necessary SF N and that minimize at the same time the SF C at cruise. Analyzing the 10 optimal solutions of the archive, listed in table V, we can see that the 3-tuples of optimization variables are characterised by a value of πf an and πhpc which are close to their maximum allowable value (resp. 1.8 and 12). These results are in line with the theory (see [10]). It is stated that with the current level of technology, the SF N is nearly constant with respect to the overall pressure ratio (OPR) while the SF C is an inverse function of the OPR. Increasing πf an leads to an increase in the SF N and a decrease in the SF C. TABLE V Optimal Solutions from the Archive

B. Results The GA has been run with a population of Npop = 200 individuals during Ngen = 50 generations (corresponding to 10 000 calls to the engine simulation tool). Figure 6 shows the 76 Pareto-optimal solutions that have been obtained at the end of the optimization process. The CPU time required by the optimization loop was about 25 800 seconds. The Narch = 10 solutions stored in the archive are also depicted in figure 6. All the non-dominated solutions clearly underline the conflicting nature of the two objectives. As usual in the framework of multi-objective evolutionary optimization, the final choice of a preferred configuration will be a posteriori determined by a smart compromise between the different objectives, based on engineering criteria. For the present application, the goal is actually to minimize the SF C at cruise while ensuring a given level of thrust at take-off. Generally, the engine size is upperbounded for practical reasons e.g. underwing installation. This dimension constraint, coupled with the required takeoff thrust, translates to a lower bound on the SF N . The

SF N (×10−2 ) 3.231 4.319 3.769 4.177 3.595 3.951 3.439 4.520 3.078 4.792

SF C(×105 ) 1.729 2.083 1.872 2.020 1.804 1.931 1.762 2.143 1.724 2.232

BP R 6.620 2.933 4.414 3.267 5.011 3.855 5.623 2.564 7.514 2.131

πf an 1.732 1.775 1.784 1.791 1.793 1.793 1.780 1.795 1.603 1.795

πhpc 11.799 11.857 11.572 11.743 11.838 11.751 11.959 11.846 11.582 11.996

Consequently, the optimal solutions are characterised by different BP R. Moderate BP R give high SF N and SF C because it emphasizes the contribution of the core stream and its high exhaust speed, while greater BP R lower the SF N and the SF C due to the dominant contribution of the slower by-pass stream.

V. Conclusions A simple optimization problem of a jet engine has been investigated. To this end, two home-made codes, namely the Genetic Algorithm (MOHyGO) and the turbine simulation tool (MaTES), have been coupled. The ability of a Genetic Algorithm to provide a family of optimal solution to this particular problem has been demonstrated. However, it has to be realized that the proposed application is rather academic. First, a constrained problem could be considered. Then, additional effects, such as the noise generated by the engine or its pollutant emissions should be introduced in the model to define new figures of merit. As a final improvement, the multidisciplinary nature of engine design optimization should be taken into account: to this end, aerodynamic, mechanic as well as thermal simulations of the engine components should replace the 0-D model. As highlighted in [5], manufacturers are currently striving to cope with this complex problem. References [1] C. Coello. A Comprehensive Survey of Evolutionary-Base Multiobjective Optimization, Knowledge and Information Systems, 1(3), pp. 269–308, 1999. [2] N. Cumpsty. Jet Propulsion, Cambridge University Press, 2000. [3] C. Goffaux, S. Pierret, S. Rossomme, V. Kelner, S. Van Oost and L. Barremaecker. Geometric Optimisation of Grooved Heat Pipes by a Genetic Algorithm Technique, Proc. of the 6th International Conference on Heat Pipes, Heat Pumps and Refrigerators, Minsk, Belarus, 2005. [4] J. Holland. Adaptation in Natural and Artificial Systems, The University of Michigan Press, Ann Arbor, 1975. [5] P.Jeschke, J. Kurzke, R. Schaber and C. Riegler. Preliminary Gas Turbine Design Using the Multidisciplinary Design System MOPEDS, ASME Journal of Engineering for Gas Turbines and Power, 126, pp. 258–264, 2004. [6] V. Kelner, O. L´ eonard, Optimal Pump Scheduling for Water Supply Using Genetic Algorithms. In: G. Bugeda, J.-A. D´ esid´ eri, J. Periaux, M. Schoenauer, G. Winter (eds.), Proc. of the 5th International Conference on Evolutionary Computing for Industrial Applications, Barcelona, Spain, 2003. [7] V. Kelner, O. L´ eonard. Application of Genetic Algorithms to Lubrication Pump Stacking Design, Journal of Computational and Applied Mathematics, 168, pp. 255–265, 2004. [8] G. Grondin, V. Kelner, P. Ferrand and S. Moreau. Robust Design and Parametric Performance Study of an Automotive Fan Blade by Coupling Multi-Objective Genetic Optimization and Flow Parameterization, Proc. of the International Congress on Fluid Dynamics Applications in Ground Transportation, Lyon, France, 2005. [9] J. Kurzke. GasTurb v8.0 User’s Manual, 1998. [10] O. L´ eonard. Propulsion A´ erospatiale – Notes de Cours, Universit´ e de Li` ege, 2005. [11] J.D. Mattingly. Elements of Gas Turbine Propulsion, AIAA Education Series, AIAA, 1996. [12] J.N. Morse. Reducing the Size of the Nondominated Set: Pruning by Clustering, Computers and Operations Research, 7(1-2), pp 55–66, 1980. [13] S. Pierret. Multi-Objectives and Multi-Disciplinary Optimization of Three-Dimensional Turbomachinery Blades, Proc. of the 6th World Congress of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, 2005. [14] P.P. Walsh and P. Fletcher. Gas Turbine Performance, Blackwell Science, 1998. [15] E. Zitzler and L. Thiele. Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach,IEEE Transactions on Evolutionary Computation, 3(4), pp. 257–271, 1999.