A Genetic Algorithm for Determining Cylinder Pressure in Internal

2600

Energy & Fuels 2007, 21, 2600-2607

A Genetic Algorithm for Determining Cylinder Pressure in Internal Combustion Engines Fernando Cruz-Peragoń*,† and Francisco J. Jimeńez-Espadafor‡ Department of Mechanics and Mining Engineering, Escuela Polite´ cnica Superior de Jae´ n, UniVersidad de Jae´ n, Paraje Las Lagunillas s/n, 23071, Jae´ n, Spain, and Department of Energetic Engineering, Escuela Te´ cnica Superior de Ingenieros, UniVersidad de SeVilla, Camino de los Descubrimientos s/n, Isla de la Cartuja, 41092, SeVilla, Spain ReceiVed NoVember 3, 2006. ReVised Manuscript ReceiVed May 22, 2007

In this paper, a genetic-algorithm- (GA-) based method has been developed to determine the instantaneous pressure in cylinders of internal combustion engines. Some parameters associated with a pressure curve in each cylinder are optimized from two validated submodels which reproduce the engine behavior using a genetic algorithm. In this process, the profile of both measured and modeled instantaneous angular speeds of the engine are compared. According to a typical design of a genetic algorithm, different operators, such as the selection process, crossover, and mutation, have been evaluated and applied over theoretical data. The best results define the adopted GA structure, and it is applied to three different engines: single-cylinder diesel engine(DE), three-cylinder spark ignition engine, and a 16-cylinder vee power plant diesel engine. This algorithm makes it possible to estimate a good quality pressure profile in cylinders for any kind of engine (operation basis and the number and disposition of cylinders), using indirect measurements and low-cost equipment.

Introduction Defects in a reciprocating engine, which are related to the chamber pressure, can produce some problems, such as a lack of power, nonallowed emission levels, nonuniform power distribution at each cylinder, vibrations and noise, and so forth. Thus, knowledge of the internal pressure in the cylinder could identify these malfunctions. The internal pressure is obtained by direct measurement in the combustion chamber using a piezoelectric sensor. For this purpose, some nonintrusive methods have been developed. One of them measures the instantaneous engine speed,1,2 to develop an inverse problem (see Figure 1). The formulation of mathematical models together with an optimization process based on a genetic algorithm (GA) has been extensively used to optimize several engineering systems.3,4 In particular, very interesting applications related to combustion and internal combustion engines have been recently conducted,5 in the fields of chemical reaction mechanisms and emissions. In the current paper, an application of the GA method based on the study of both angular speed and the pressure in the chambers during an engine cycle is presented. The combustion process is characterized by the mass fraction burned.6 Some * Corresponding author. Phone: +34 953 212367. Fax: +34 953 212870. E-mail: [email protected]. † Universidad de Jae ´ n. ‡ Universidad de Sevilla. (1) Williams, J. SAE Paper 960039; SAE: Warrendale, PA, 1996. (2) Lim, B.; Lim, I.; Park, J.; Son Y.; Kim, E. SAE Paper 940145; SAE: Warrendale, PA, 1994. (3) Obayashi, S.; Takanashi, S. Genetic Algorithms in Engineering Systems: InnoVations and Applications; Conference Publication, IEEE: Sheffield, U.K., 1995; pp 7-12. (4) Fisher, K. A. Genetic Algorithms in Engineering Systems: InnoVations and Applications; Conference Publication, IEEE: Sheffield, U.K., 1995; pp 18-22. (5) Kalogirou, S. T. Prog. Energy Combust. Sci. 2003, 29, 515-566.

difficulties appear when directly determining the pressure from angular speed. The most important issue is a discontinuity on the top dead center (TDC) of the engine that prevents the knowledge of some important points in the cycle. So, the inverse problem is developed by an optimization procedure over the direct problem (angular speed from pressures).6,7 Figure 2 shows the structure of the method for the estimation of the combustion pressure curve in each cylinder. Starting from some parameters, a combustion model is applied to obtain the pressure curve in cylinders. This curve determines the instantaneous angular velocity by means of an engine dynamic model. This velocity is compared with the target one by an objective function (Ψ) that is as follows in eq 1:

Ψ ) (1/θ720°)

∫θθ +720° [$(θ) - ω(θ,P(θ))]2 dθ o

(1)

o

where θ720° is the engine cycle (four strokes, two crank revolutions), θo is the initial crank angle of the cycle, θo+720° is the final crank angle of the cycle, ω j corresponds to the instantaneous measured angular velocity, ω is the instantaneous modeled angular velocity, and P is the modeled pressure curve in each cylinder. This function reaches a minimum value when both instantaneous velocity curves converge. The convergence criteria considers any of the three following items: first, the difference between two consecutive objective functions is less than a certain value; second, the difference between two consecutive values of the parameters to optimize (vector (6) Cruz-Peragoń, F.; Palomar, J. M.; Munõz, A.; Jimeńez-Espadafor, F. Anales de Ingenierıá Meca´ nica 2000, 13 (3), 1861-1866. (7) Cruz-Peragoń, F.; Carvajal, E.; Cantador, J.; Castillo, A.; JimeńezEspadafor, F.; Munõz, A.; Sańchez, T. Proc. of 7th SETC Small Engine Technology Conference and Exhibition, Pisa, Italy, 2001; SAE International: Warrendale, PA, 2001; Vol. 2, pp 665-673; SAE International Paper 2001-01-1790/4212.

10.1021/ef0605495 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/17/2007

Cylinder Pressure in Internal Combustion Engines

Energy & Fuels, Vol. 21, No. 5, 2007 2601

Figure 1. Inverse problem schematic process.

Figure 2. Optimization method.

modulus) is less than a certain value; and finally, the number of iterations reaches a maximum value.8 During this procedure, a high nonlinearity appears, and an optimization method must be carefully defined. One of the methods employed is a second derivative, the Newton method.8 In this method, a quadratic model of the objective function is obtained by taking the first three terms of the Taylor-series expansion at the current point. Searching strategies related to robustness and rapid convergence have been successfully improved, reducing computing time and minimizing computing errors (Gauss-Newton, Levenberg-Marquardt, quasi-Newton approximations, etc.). Nevertheless, these methods take into account the continuity of the objective function derivatives. The initial values of the parameters to optimize are also very important. The implementation of an optimization strategy is highly dependent on the initial values, since the numerical method can reach a local minimum. So, the method cannot accurately determine the pressure in the cylinders. For this purpose, it is necessary to adopt a numerical method that does not take into account the continuity of the objective function derivatives. The GA method uses mathematical models to reproduce the system behavior, using the objective function (eq 1) to optimize all the parameters of the pressure curves. The process evolves until the objective function reaches a minimum, through (8) Gill, P. E.; Murray, W.; Wrigth, M. H. Practical Optimization; 11th Impression, Academic Press: London, 1997.

successive iterations (generations). Increasing the number of cylinders requires an increase in the number of generations. Continuity of the objective function derivatives is not required. Moreover, the method is quite robust with complex systems with many variables. The method searches for the final value of the parameters for the whole function domain.9,10 Methodology 1. Engines and Measurements. The main characteristics of the evaluated engines are shown in Table 1. In order to analyze their working behavior, different measurements at different operating conditions have been done using an engine test bed. 2. Engine Model. The direct problem consists of the determination of the instantaneous angular speed from the pressure in the cylinders. First, the pressure in one cylinder is obtained from a combustion and gas exchange submodel, based on the first law of thermodynamics applied to the limits of the combustion chamber. It was also considered to be a no-swirl and single-zone type.7,11 The main component of this model is the mass fraction burned, by means of Wiebe’s function, obtained from the parameters of Table 2, which originates the heat release rate from the fuel. The kind of engine and the mass fuel consumption per cylinder and cycle (MC) are also two important parameters. So, if a spark ignition engine (9) Gen, M; Cheng, R. Genetic Algorithms & Engineering Design; John Wiley & Sons: New York, 1997. (10) Pham, D. T.; Karaboga, D. Intelligent optimisation techniques, Springer: London, 2000. (11) Heywood, J. B. Internal Combustion Engine Fundamentals; McGrawHill: New York, 1988.

2602 Energy & Fuels, Vol. 21, No. 5, 2007

Cruz-Perago´ n and Jime´ nez-Espadafor Table 1. Characteristics of Analyzed Engines

characteristics number of cylinders bore (mm) stroke (mm) maximum torque (Nm) maximum power (kW)

Deutz Dieter LKRS-A (DE)

Maruti 800 (SIE)

1 85.0 100.0 34 (to 2400 rpm) 9.5 (to 3000 rpm)

3 68.5 72.0 102 (to 3500 rpm) 47 (to 6000 rpm)

Caterpillar 3516 (16V-DE) 16-V 170.0 190.0 10 000 (to 1500 rpm) 1500 kW (to 1500 rpm)

Table 2. Wiebe’s Function Parameters per Cylinder on Engines term

diesel engines

term

spark ignition engines

θi

angle of beginning of injection (degrees) duration angle of the premixed combustion phase (degrees) duration angle of the diffusive combustion phase (degrees) form factor for premixed phase (dimensionless) form factor for diffusive phase (dimensionless) specific energy released in premixed phase (dimensionless)

θi

angle of beginning of combustion (degrees) duration angle of whole combustion (degrees) form factor for combustion (dimensionless) form factor for combustion (dimensionless)

θp θd mp md qp

(SIE) is analyzed, five parameters per cylinder will be considered, and seven parameters if a diesel engine (DE) is also taken into account. When these parameters are known and put into the combustion model, they yield to the pressure in the combustion chamber, which is one of the inputs to the dynamic submodel. Additional assumptions of this last submodel are a flexible crankshaft and viscous and elastic coupling, completed by an instantaneous friction loss model.12,13 With these particularities, a torque balance equation system defines the engine dynamics.14 Then, the determination of the angular speed is carried out by first-order differential-equationbased methods.15 Later, the whole model is validated for each of them.16 As a result of a sensitivity analysis done over the whole model, only two input parameters can be considered as main components of the combustion submodel: the fuel ratio (proportional to the mass fuel consumption, MC) and injection (DE) or ignition (SIE) timing. These two factors can be well approximated by an optimization method; however, the other parameters are affected by high partial correlation indexes, which causes, although the pressure curve could be well adjusted, their values to not be so well-estimated.17 3. Particularities of This GA. The algorithm starts from a population of initial parameters, whose values are randomly searched. This population is composed by various parameter combinations, called chromosomes, consisting of vectors. Each chromosome contains the values of the parameters that generate the pressure curve (five for SIE and seven for DE), called gens.9 All of these values are scaled, so its domain is [0, 1], preventing a scaling problem.8 The direct problem solution provides the instantaneous angular speed of the engine for each parameter combination included in each chromosome. Next, the population is subjected to certain control operators, such as mutation and crossover. (12) Rezeka, S. F.; Henein, N. A. SAE Paper 840179; SAE: Warrendale, PA, 1984. (13) Zweiri, Y. H.; Whidborne, J. F.; Sereviratne, L. D. Proc. Instn. Mech. Engrs. 2001, 215 D, 1197-1216. (14) Connolly, F.; Yagle, A. Mech. Syst. Signal Proc. 1994, 8 (1), 1-19. (15) Rizzoni, G.; Zhang, Y. Mech. Syst. Sign. Proc. 1994, 8 (3), 275287. (16) Cruz-Peragoń, F. Ana´lisis de metodologıás de optimizacioń inteligentes para la determinacioń de la presioń en ca´mara de combustioń de motores alternativos de combustioń interna por me´todos no intrusivos. Ph.D. Thesis (in Spanish), University of Seville, Spain, 2005. (17) Cruz-Peragoń, F.; Jimeńez-Espadafor, F. Anales de Ingenierıá Meca´ nica. 2004, 15, 1891-1896.

θc mp a

In this work, the control mutation is developed by a random process applied to gen k in one chromosome i of the population, Z(i,k), for the step j, where the value for the new generation j + 1 is individually generated by the next expression (eq 2): Zij+1(k) ) Zij(k) + λN(0,1)

(2)

where Zij(k) is the previous value of gen k in chromosome i, λ is the mutation rate in that chromosome, and N(0,1) is a random normal distribution value, whose mean value is 0 and whose standardized dispersion value is 1. When the process evolves, the range of these mutations decreases, so high changes are produced in the initial stages, and local adjustments are made in the last generations. CrossoVer is applied by two methods, from two chromosomes randomly chosen from the population, Zlj and Zmj (parents): first, one chromosome, Zij+1, is generated by the next arithmetic expression (eq 3): Zij+1 ) βZlj + (1 - β)Zmj

(3)

where β is a random value between 0 and 1 that changes in each generation. The position i of this chromosome in the population depends on the selection process. This arithmetic operator produces a high selective pressure, so inheritors assume the parents’ characteristics, making the approximation of a global minimum in the solution space of this work that is very wide difficult. To prevent it, different values in the rate of probability of crossover must be evaluated. The second application is called BLX-R,18,19 which produces h new chromosomes from Zlj and Zmj, where each component k, Zq(k), is randomly selected from the interval described in eq 4: [min - (IR), max + (IR)] with I ) max - min

(4)

where min and max are the minimum and maximum values of Zlj(k) and Zmj(k), respectively; that is, if Zlj(k) is the maximum of both terms, Zmj(k) is the minimum of them. R is also a value that begins near 1, and it decreases when the process evolves. From these h new chromosomes, only one or two of them, the best ones, are selected for the new generation. The BLX-R operator is based on (18) Eshelman, L. J.; Shaffer, J. D. Real coded Genetic Algorithms and Interval Schemata. Foundations of Genetic Algorithms, 2; Whitley, L. D., Ed.; Morgan Kaufman Publishers: San Mateo, CA, 1993; pp 187-202. (19) Nomura, T.; Shimohara, K. EVol. Comput. 2001, 9 (3), 283-308.



Figure 3. Results of the control parameter study of GA applied to a single-cylinder DE theoretical model (2400 rpm and full load): (a) fitness evolution for arithmetic crossover, (b) fitness evolution for BLX-R crossover, (c) curve results for arithmetic crossover, and (d) curve results for BLX-R crossover. Table 3. Minimum Objective Function in the Optimization Process, 2400 rpm, Full Load crossover

case evaluated minimum value objective function (fitness)

second-order method

7.53 × 10-2

genetic algorithm population size Sp; mutation rate λ; probability mut. or cross., φ

arithmetic (×102)

BLX-R (×102)

60 - 0.2 - 0.6 40 - 0.2 - 0.6 80 - 0.2 - 0.6 60 - 0.02 - 0.6 60 - 0.4 - 0.6 60 - 0.2 - 0.5 60 - 0.2 - 0.9

2.14 2.31 1.47 2.54 2.35 3.3 1.82

1.20 1.20 1.17 1.31 1.10 1.22 1.63

a random generation of gens from an associated setting of the parents. This method produces chromosomes that can differ from their parents. So, this operator produces a low selective pressure over the daughters. In the next step, by means of these two genetic operators, gens of chromosomes are modified by a random process, varying some values of them. At each step, only mutation or crossover is produced in one only chromosome. A random value controls this selection, affecting only one chromosome. This task is necessary because the calculation process requires considerable time that affects the computation time of each generation. Then, the instantaneous angular velocity of each chromosome is obtained from the whole engine model described previously and compared with the target angular velocity, resulting in the objective function (eq 1). Later, the next genetic operator, called the selection process, searches for the chromosome whose angular speed fits better to

the real one, producing the minimum objective function. Similar to the mutation and crossover processes, this operator is used in two different ways: First, if the fitness value is less than the minimum reached until then, the process takes that chromosome, erasing the worst one. At this point, a random component is included, to prevent a local minimum, resulting in another new population of chromosomes. Second, if the BLX-R operator is used, its two best new chromosomes substitute for the two worst ones of the population, even if these new chromosomes have a higher fitness than the others. As the process evolves, chromosomes vary, reducing its corresponding objective function, and the medium error of the population follows the tendency of the minimum error of the chromosomes. After many steps, both errors are practically the same. 4. Improvement of the GA Optimization Method. Theoretical Initial Results. To determine the effectiveness of the GA, different operators of the GA structure are modified, developing the two


Cruz-Perago´ n and Jime´ nez-Espadafor

Figure 4. Detailed optimization process applied to inverse problem solving by GA in combustion engines. Table 4. Deviation Errors for Different Real Operative Conditions in the Analyzed Single Cylinder DE torque (Nm)

n (rpm)

(%) Imep

(%) maximum pressure

(%) pressure when input valve closes

(%) fuel expense

(%) combustion slope (premixed phase)

(deg BTDC) injection timing

-12 4 9 4 10 13 0 27 27.5 27.4 12 17 16.7

2030 2130 2150 2400 2400 2400 2600 2600 2200 1800 2600 2200 1800

1.9 1.66 3.4 0.9 0.85 0.54 3.4 1 -1.8 0.6 -0.2 5.4 2.8

-3.8 -7.5 -3.7 -9.4 -3.7 -4.8 -0.6 -0.2 -3.4 0.8 -4.3 0.7 -3.6

-11.1 12.9 -12.13 -2.7 -0.25 13 12 0.1 9.8 -13.5 14.8 9

-5.9 1.5 3.3 5.4 6.3 -4.8 -4.4 6.1 2.7 -8.9 5.1 2.86

7.5 -0.7 0 -0.7 -0.8 13.9 5.9 -2.1 11.7 0.8 8.5 0.5

0.4 -4.5 1.9 -8.8 -0.2 -5.5 2.2 -0.5 0.5 2.1 -1.6 -1.5

crossover processes presented and changing different constants in the algorithm. The calculus is made using theoretical data over the single-cylinder DE model, whose results are compared with a second-order optimization method. As said before, the main feature

is that both modeled and target curves are compared, resulting in an objective function. Later, a termination criterion determines if the algorithm finishes, or if the structural parameters must be changed as well. For the current engine, 5000 is the maximum


Figure 5. Pressure approximation results for single DE, 2600 rpm and 27 Nm.

Figure 6. Pressure approximation results for single DE, 1800 rpm and 27.4 Nm.

number of iterations. These variations are evaluated in the process several times, and for different conditions: full load (100%), half load (50%), and part load (25%), each one for 1800, 2400, and 3000 rpm. Results for one particular condition are shown in Table 3, where the final fitness value of the process is obtained, presenting the best value of the fitness, once the combustion parameters are optimized. Table 3 demonstrates that the fittings via the GA method are better than those based on the second-order method. For example, results from the analysis referred to in this table, where Sp ) 40, λ ) 0.2, and φ ) 0.6, are presented in Figure 3. Figure 3a and b correspond to the objective function evolution along the optimization process for arithmetic and BLX-R crossover, respectively. Figure 3c and d show the target and optimized pressure curves for the cases of Figure 3a and b, respectively. The selective pressure of the arithmetic crossover is minimized, varying its rate, preventing a local minimum, as Figure 3a shows, where fitness reduces when the process evolves. With BLX-R crossover (see Figure 3b), sometimes, the objective function associated with the new incorporated chromosome of the population makes a mean value higher than in the last step; it seems that the value goes away from the correct solution. Nevertheless, it approximates more quickly to the minimum value than it does with the arithmetic crossover.


Figure 7. Pressure approximation results for single DE, 2600 rpm and 12 Nm.

These preliminary results show that BLX-R crossover is more useful to implement into this structure. It approximates more quickly to the minimum value, extending the searching field and reducing the selective pressure in relation with the arithmetic crossover. So, it causes the GA to reach a very low value of the fitness in fewer generations than the other operator does, that is, fewer stages are necessary to obtain the optimized pressure in the cylinders. The other operating conditions have a similar tendency, as Table 3 shows, giving the best result with a population of 60 chromosomes, a mutation rate λ of 0.4, and giving a slightly higher probability of producing mutation than crossover (φ ) 0.6). So, the GA is applied taking into account a BLX-R crossover, with those rates. The number of chromosomes of the population can be reduced to 40, allowing a faster algorithm, and with better results than a conventional method. The optimizing algorithm works with an objective function, associated with the angular speed curve profile in one engine cycle. With theoretical data, both curves (target and optimized) are nearly the same, so the evolutions of the pressure curves make them very close when the process finishes. In some cases, little errors appear around the TDC, because the dynamic submodel presents uncertainties, and the indicated torque is very low. This particularity demonstrates that, even with the same angular speed curves (both target and optimized), the approximation in this zone may be incorrect. The fuel consumption is the main model parameter, being very well fitted, with a maximum relative error of 2.5%. Another parameter with good theoretical agreement is the injection timing, with 2° of misadjusting. The rest of the parameters allow a burned mass fraction distribution close to the target one. The final algorithm for this application is shown in Figure 4.

Results The procedure has been applied to experimental data to obtain the modeled pressure curve in each cylinder. The effectiveness of the method has been analyzed by comparing both approximated and target pressure curves, by each indicated mean effective pressure (Imep). For the engines with one and three cylinders, apart from the pressure curve profile estimation, two main parameters will be also evaluated: the mass fuel consumption per cylinder and cycle (called here the fuel expense) and the injection (for DE) or ignition (SIE) timing. Other less important parameters are also presented, such as the maximum pressure or the combustion slope. The comparisons are made via deviation errors.


Figure 8. Pressure approximation results for three-cylinder SIE, 3500 rpm and 60 Nm.

1. Single-Cylinder Diesel Engine. The computation time for solving the direct problem in this application is near 8 s (Pentium IV, 2.3 MHz). This time corresponds to each generation, as Figure 4 shows, as well. Although the maximum number of generation is 5000, the convergence is fast because of the advantages of the BLX-R operator. So, although the maximum required time to carry out the optimization process is near 28 h, the effective mean time is equal to 14 h. The number of combustion parameters to be optimized is 7 (single-cylinder DE). Next, results for experimental data are presented. First, Table 4 contains the estimation errors when real and optimized pressure curves are compared. Figures 5-7 show the results of three curve profiles, for some operating conditions defined in Table 4. The estimation errors are higher than those obtained when using theoretical data. The modeled chamber pressures are close to the real one, with a slightly worse fitness, because of the particular dynamic complexities, such as frictional losses, or the speed acquisition system. On the other hand, it is more difficult to approximate the combustion parameters correctly, although in many cases, the approximation of the fuel consumption and the injection timing are also very close. 2. Three-Cylinder Spark Ignition Engine. In this case, the number of parameters to be optimized is 15 (five per cylinder in a SIE). The structure is the same as before, with a maximum number of generations of 15 000. The computation time of each generation is over 13 s, which carries out a mean process time of 60 h, with an average number of generations of 8000. Previous to real conditions, theoretical data are evaluated. As a result, pressure curves are well-approximated, as well as the fuel consumption per cycle. Nevertheless, two new aspects appear: first, there is a higher disagreement related with the ignition angle, and second, there is a higher misadjusting in the compression and expansion strokes in the pressure curve profiles. The addition of all the indicated torque curves over the engine dynamic submodel causes certain new indeterminacies associated with the overlapping of those curves over the response of the direct problem (engine speed). Results over different real operating conditions appear in Figures 8-10 and Table 5. Here, there are worse estimations than those obtained for one cylinder. This is because of the overlapping of the indicated torque explained previously; thus, the same effects appear for theoretical data. In spite of it, there is very good agreement between pressure curves, providing relatively feasible information about fuel consumption.

Cruz-Perago´ n and Jime´ nez-Espadafor



3. 16-Cylinder Vee Power Plant Diesel Engine. Theoretical Analysis. As in previous cases, the same methodology is applied, but in this case, only one constant angular speed is considered: 1500 rpm. The engine analyzed belongs to a power plant, connected to a four-pole generator, providing a 50 Hz electrical signal. With the same considerations as the other engines, the number of parameters to optimize is 112, making a high number of generations necessary (a maximum of over 80 000), making a feasible process with a conventional computer impossible. So, the number of generations per day is about 2000 with the same computer characteristics as previously described. The direct problem needs 40 s per generation. The numerical calculation is very slow, since it takes a 40 days’ process time. Initially, one condition was analyzed with these assumptions, obtaining poor results. So, although the angular speed curves are very similar (both the target and optimized ones), the final pressure curves acquire a great misadjustment. To demonstrate the effectiveness of the GA method with this kind of engine, the pressure model is replaced by another simple one, where only one parameter controls the curve profile per cylinder (maximum pressure).14 So, the number of parameters to optimize drops drastically from 112 to 16. The results are very good, as Figure 11 shows, with a 3% maximum error. The



Table 5. Deviation Errors for Different Real Operative Conditions in the Analyzed Three-Cylinder SIE

torque (Nm)

n (rpm)

number cylinders

(%) Imep

(%) max. pressure

(%) pressure when input valve closes

17

2540

6

2550

14

3300

60

3500

44

4550

25

4540

20

5580

49

5410

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

-6 1.6 5.4 -0.52 2.8 -0.3 4.5 4.13 3.13 2.1 4.5 -1.1 3.65 5.45 1.99 -1.1 5.6 -0.6 -2.84 3.4 0.65 4.17 -0.016 -1.426

-6.9 0.43 5.2 -2.43 6.7 -5.5 2.1 5.2 0.06 -1.64 4.21 -4.08 2.25 4 -0.4 -3.03 5.04 -2.4 0.74 4.1 0.62 3.26 -1.88 -4.36

-12.27 0.75 12.35 -2 8.6 -5.2 2.25 7.95 -1.9 -1.7 5.7 -5.2 3.4 7.15 0.06 -2.62 7.66 -2.22 1.1 5.96 0.95 3.27 -1.75 -4.25

computation time is reduced down to 10 s per iteration, with an average whole time required of 72 h. The results show a good level of agreement. Conclusions This paper presents a very low cost methodology, where only one encoder is needed as a sensor, compared to conventional systems, where a piezoelectric sensor, with a high cost and short useful lifespan, is necessary for each cylinder. Also, results for both theoretical and real data show that the GA optimization methodology fits the chamber pressure in each cylinder better than other conventional methods, which depend on the derivatives of the objective functions. It allows the real pressure in the cylinders to be obtained with a very low degree of error (which increases if a multicylinder engine is considered, but with acceptable results). This optimization is performed by improving control parameters of the GA, using finally random mutation and BLX-R crossover processes.

Figure 11. Pressure approximation results for 16-V DE and theoretical objective pressure function, 750 kW at 1500 rpm.

(%) combustion slope

(%) fuel expense (overall)

-12.27 0.75 12.35 -2 8.6 -5.2 2.25 7.95 -1.9 -1.7 5.7 -5.2 3.4 7.15 0.06 -2.62 7.66 -2.22 1.1 5.96 0.95 3.27 -1.75 -4.25

6 -7 -5.9

3.3 -0.1 -4.8 -8 -5

(deg BTDC) ignition timing -9.2 7.9 8.6 5.35 8.2 -2 7 8 6 4.5 9 4.4 7 16 8 9.9 8.6 4.4 -3.2 5.7 9.22 -1.14 -2.9 -6.14

Engine dynamics play an important role in the accuracy of the final results; that is, both real (target) and modeled curves of angular speed must be nearly or exactly the same when the optimization process has finished. Most of the investigations over multicylinder engines show results where pressures in the different chambers are the same, except in one of the cylinders. An important contribution of this work is that all cylinders can show different pressure curves independently, such as in Figures 8-11, showing a more realistic scenario. As the different individual indicated torques in multicylinder engines are covered up, differences between modeled (and optimized) and target (and real) curves grow proportionally with the number of cylinders. This effect does not depend of the kind of engine (DE or SIE). The major misadjustments that are shown in the pressure curve profiles after the optimization process are caused by the uncertainties around the TDC when the mixture burns. Related to the individual combustion model parameters, the fuel consumption and injection timing are wellestimated. The combination of the rest of the parameters gives in most of the cases a correct mass fraction burned. The main disadvantage of this method is that it requires a high amount of computation time, so the diagnosis is ineffective in some of the different environments where engines are used. In this sense, a good way to improve this time is to increase the convergencecriteria-associated error. Moreover, the continuous hardware improvement and the actual development of multiprocessor modules will reduce the computation time. Additionally, the GA method can be applied to establish an initial combination of parameters, and later, a conventional Newton-based method finishes the searching of the target values. Some interesting applications can be done with this method, such as at the end of the production line as a diagnosis tool for engines. Knowledge of the real pressure with a cheap methodology can be used to control the combustion process, increasing the final power, reducing the pollutant emissions, and so forth, in power plants, or in automotive engines. EF0605495

A Genetic Algorithm for Determining Cylinder Pressure in Internal

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