Design and Optimization of Neural Networks To ... - ACS Publications

Energy & Fuels 2007, 21, 2627-2636

2627

Design and Optimization of Neural Networks To Estimate the Chamber Pressure in Internal Combustion Engines by an Indirect Method Fernando Cruz-Peragoń*,† and Francisco J. Jimeńez-Espadafor‡ Department of Mechanics and Mining Engineering, Escuela Polite´ cnica Superior, UniVersidad de Jae´ n, Paraje Las Lagunillas s/n, Jae´ n 23071, 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, SeVilla 41092, Spain ReceiVed March 9, 2007. ReVised Manuscript ReceiVed July 9, 2007

A particular type of artificial neural network (ANN), with the aim of estimating the indicated pressure into cylinders from instantaneous angular speed measurement, has been developed: radial basis function (RBF). This is the main component of a methodology where input and output curves are parametrized. A modified RBF network from its general structure is designed to reduce the size of the network, approaching the hidden layer weights by means of polynomial functions. It makes it possible to use it in medium requirement PCs, such as in automobile applications. Results corresponding to different operating conditions of a single-cylinder diesel engine (DE), a three-cylinder spark ignition engine (SIE), and a 16-cylinder vee power plant DE (V-16 DE) are presented. Results show the accuracy and fast response of the network, making it feasible to use in online diagnostics and control systems in engines, with very low cost (removing the use of piezoelectric sensors).

Introduction To determine the pressure into the chambers of internal combustion engines, apart from direct measurements using piezoelectric transducers, some non-intrusive methods have been developed, by means of vibrations analysis or measuring angular speed in one working cycle.1 These methods require mathematical models to describe the behavior of the gases into the cylinder (combustion models) and the moving pieces of the engine (dynamic models). The direct problem consists of determining the instantaneous angular speed from the pressure into cylinders. These pressures are obtained from the analysis of combustion parameters. On the other hand, when the purpose is to determine the pressure from the angular speed, the procedure is just the opposite and it is called an inverse problem. In this case, it must be solved via optimization techniques, to evaluate parameters related to the combustion process, providing a pressure curve similar to the real one.2,3 This inverse problem is also applied to other important features of engines, such as fuel injection.4-6 An interesting way to solve this problem is using artificial neural networks (ANNs).7 This technique was applied in the combustion engine field some time ago to obtain or control * To whom correspondence should be addressed. Telephone: +34-953212367. Fax: +34-953-212870. E-mail: [email protected]. † Universidad de Jae ´ n. ‡ Universidad de Sevilla. (1) Williams, J. SAE Paper 960039, 1996. (2) Connolly, F. T.; Yagle, A. E. Mech. Syst. Sig. Proc. 1993, 8 (1), 1-19. (3) Cruz-Peragoń, F.; Palomar, J. M.; Munõz, A.; Jimeńez-Espadafor, F. J. An. Ing. Mec. 2000, 13 (3), 1861-1866. (4) Kegl, B. Proc. Inst. Mech. Eng. 1995, 209, 135-141. (5) Kegl, B. J. Mech. Des. 2004, 126, 703-710. (6) Palomar, J. M.; Cruz-Peragoń, F.; Jimeńez-Espadafor, F. J.; Dorado, M. P. Energy Fuels 2007, 21, 110-120. (7) Fausett, L. Fundamentals of Neural Networks; Prentice Hall: New York, 1994.

processes where global parameters are used, such as torque, temperature, power, or fuel efficiency consumption.8,9 Then, the main target of this work is to establish a procedure via ANN to determine the pressure curve, by the instantaneous angular speed measurements, in one operating cycle of an internal combustion engine. As Figure 1 shows, this procedure is composed of three stages: first of all, a pattern generation procedure describes the direct problem for a large amount of different operating conditions. Second, a training procedure establishes a particular combination of internal values of the designed network. Finally, the net provides the instantaneous pressure into each cylinder when both the instantaneous angular speed signal ω and the torque TL are entered. Experimental Section The most common learning methods used are the backpropagation method10 and radial basis function network (RBF).11 The backpropagation method uses a linear regression approach, which is more familiar, being an extension of the “best fit” straight line through a set of points.12 Different first- and second-order methods are suitable for learning, and modifications of them are currently used by many researchers in different fields. Their target is to obtain faster convergence times to avoid the mete-optimization phase, which, in general, optimizes the performance of the learning method.13,14 In this case, the analysis is more complex, because certain pressure curves must be obtained from another velocity curve along the working cycle of the engine, appearing as high nonlin(8) Osman, K. A.; Higginson, A. M. Proc. Intell. Eng. Syst. Artif. Neural Networks 1995, 5, 1007-1012. (9) Kalogirou S. T. Prog. Energy Combust. Sci. 2003, 29, 515-566. (10) Scaife, M. N.; Charlton, S. J.; Mobley, C. SAE Paper 930861, 1993. (11) Lin, W.; Wu, M. H.; Duan, S. Proc.-Inst. Mech. Eng. 2003, 217 (D), 489-497. (12) Mayes, I. W. Proc.-Inst. Mech. Eng. 1994, 208 (A), 272. (13) Battiti, R. Neural Computation 1992, 4, 162. (14) Fodslette, M. Neural Networks 1993, 6, 525-533.

10.1021/ef070122d CCC: $37.00 © 2007 American Chemical Society Published on Web 08/22/2007

2628 Energy & Fuels, Vol. 21, No. 5, 2007

Cruz-Perago´ n and Jime´ nez-Espadafor

Figure 1. Basic scheme of inverse problem solving by ANN training.

Figure 2. GRNN network basic structure (adapted from refs 22 and 23).

earity. Therefore, the parameter estimated may become trapped at a local minimum of the chosen optimization criterion during the learning procedure.15 The results for this kind of network show that they are not useful to the desired approach.16 On the other hand, the RBF approach produces a curved surface in n-dimensional space. This network uses nonlinear functions centered throughout the data and alters the contribution that each function makes to the surface. They do not themselves have a direct physical significance as to what parameters the system is sensitive to but determine the shape of the surface in n-dimensional space, which is the mathematical model.12 The major advantage is that neither combustion nor the dynamic model is necessary, removing adjusting problems caused by mechanical losses or coupling engine dynamometer. The objective is to establish a relationship between some input (velocities) and output (pressures) curves (see Figure 1). Nevertheless, as long as a high number of samples are needed, the design becomes unsuitable. For this reason, the used models produce the input and output vectors of the network, making feasible the optimum structure of the net.16 Some applications of this kind of approach predict the air/fuel ratio in internal engines17 or online vibration monitoring of gas turbines.12 Jacob and Gu18,19 designed a RBF network to determine the cylinder pressure in a four-cylinder engine by the measurement of the angular speed and torque. Six parameters have been taken into consideration for the curve reconstruction, related to the curve shape, next to the mean value of pressure. It shows a promising method to determine faults in engines. If mathematical modeling is employed for pressure reconstruction instead of shape evaluation, some important parameters, such as fuel consumption or injection (or ignition) timing could be estimated. Additionally, a procedure for minimizing the number of nodes of the input layer (angular speed) is necessary. For the training of these networks, two submodels are used for speed-pressure estimation.20 The direct problem consists of determining the instantaneous angular speed from the pressure in cylinders and a dynamic submodel. Previously, each pressure profile in one cycle is obtained from combustion and a gas exchange

submodel.21 The main component of this model is the heat release rate, which depends upon five parameters per cylinder if a spark ignition engine (SIE) is analyzed or seven parameters for a diesel engine (DE). The total number of parameters to evaluate is critical for the effectiveness of the method, and it will change depending upon the number of cylinders. This net is composed of a hidden layer with activation functions with radial basis (Gaussians) and an output layer with continuous activation functions. To optimize the structure for the function approximation, a variant of RBF, called generalized regression neural network (GRNN), is introduced,22 as Figure 2 shows. Signals from input (R-element column vector ip) to the hidden layer (S1element column vector in1) are obtained from distances (absolute value of the difference) between input data and weights of that layer (matrix IW), which are multiplied element by element. Each neuron of the hidden layer obtains an output value (that belongs to column vector on1) depending upon the fitting of the input vector to the weighting vector associated to that node. Those hidden neurons with weighting values very different from input data will give output signals near zero. Otherwise, if these values are similar, output signals will give a value close to unity. Each element (15) Chen, S.; Cowan, C. F. N.; Grant, P. M. IEEE Trans. Neural Networks 1991, 2 (2), 302-309. (16) Cruz-Peragoń, F.; Jimenez-Espadafor, F. An. Ing. Mec. 2004, 15, 1907-1915. (17) O’Reilly, P.; Thompson, S. Eng. Syst. Anal. 1994, 64 (6), 191198. (18) Jacob, P. J.; Gu, F.; Ball, A. D. Proc.-Inst. Mech. Eng. 1999, 213 (D1), 73-81. (19) Gu, F.; Jacob, P. J.; Ball, A. D. Proc.-Inst. Mech. Eng. 1999, 213 (D2), 135-143. (20) Cruz-Peragoń, F.; Carvajal, E.; Cantador, J.; Castillo, A.; JimeńezEspadafor, F.; Munõz, A.; Sańchez, T. Proceedings of the 7th SETC Small Engine Technology Conference and Exhibition, SAE International, Pisa, Italy, 2001; pp 665-673, SAE International Paper 2001-01-1790/4212. (21) Heywood, J. B. Internal Combustion Engine Fundamentals; McGrawHill: New York, 1988. (22) Wasserman, P. D. AdVanced Methods in Neural Computing; Van Nostrand Reinhold: New York, 1993.

Neural Networks Estimation of the Chamber Pressure

Energy & Fuels, Vol. 21, No. 5, 2007 2629

Table 1. Characteristics of Analyzed Engines characteristics number of cylinders bore (mm) stroke (mm) maximum torque (N m) maximum power (kW)

Deutz Dieter Lkrs-a (DE) 1 85.0 100.0 34 (to 2400 rpm) 9.5 (to 3000 rpm)

Maruti 800 (SIE)

Caterpillar 3516 (V-16 DE)

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

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

Table 2. Combustion Parameters Per Cylinder on Engines term

diesel engines

term

Mf θi

mass fuel consumption (kg/cycle) angle of beginning of injection (degrees before top dead center, degrees BTDC); injection timing ignition timing (degrees BTDC) duration of the premixed combustion phase (degrees) duration of the diffusive combustion phase (degrees) form factor for premixed phase (dimensionless) form factor for diffusive phase (dimensionless) specific energy released in the premixed phase (dimensionless)

Mf

mass fuel consumption (kg/cycle)

θi

angle of beginning of combustion (degrees BTDC); ignition timing

θc

duration of combustion (degrees)

M

form factor for combustion (dimensionless)

A

form factor for combustion (dimensionless)

θic θp θd mp md qp

belonging to the column vector in2 (with size S2 × 1) is the result of the dot product (single row) of the LW matrix and the input vector on1. Assigned values to output layer weights will be identical to the output signal (column vector on2 with length S2) of training patterns. The implementation of the method is possible in a mediumrequirement computer, not demanding a workstation, because a basic online device is intended in the future, even for automotive applications. The network programming is made with the help of the Neural Network Toolbox of MatLab.23 The whole method will be applied to a single-cylinder DE and a three-cylinder SIE, using real data from different operating conditions, which are logged by a test bed and a validated model.24 Also, a 16-cylinder vee power plant diesel engine (V-16 DE) is analyzed theoretically. Table 1 shows the main characteristics of these engines.

Network Characteristics and Structure Selection 1. Pressure Curve Characterization (Output Layer Vector). To prevent a large amount of pressure and speed velocity data through the cycle of the engine, the net is trained from parametrized curves. The high number of necessary learning samples requires considering a different number of parameters that characterize the pressure curve. The pressure curves are obtained from a combustion model depending upon seven parameters for a DE and only five for a SIE.20,21 This model is of the single-zone type and no swirl, mainly because of the application of it to diagnostics. The first law of thermodynamics for an open system applied to the limits of the combustion chamber is

dQ

dV -p + dt dt

dU

∑i m˘ ihi ) dt

(1)

where dQ/dt is the heat-transfer rate across the chamber boundaries, sum of the fuel heat-release rate dQf/dt and the heat losses dQL/dt through the wall, p(dV/dt) is the work transfer rate, m˘ i is the mass flow rate in and out of the system at location i, and U is the internal energy of the gases inside the system boundary. In the model, the following conditions have been taken into consideration: (a) The engine heat losses model has (23) Demuth, H.; Beale, M. Neural Network Toolbox User’s Guide; The MathWorks, Inc.: Natick, MA, 1998; Chapter 6. (24) Cruz-Peragoń, F. Analysis of Intelligent Optimization Techniques for Determining Cylinder Pressure in Internal Combustion Engines by Nonintrusive Methods (in Spanish). Ph.D. Thesis, University of Sevilla, Sevilla, Spain, 2005.

spark ignition engines

been included through the Woschni correlation.25 (b) The flow of gases through the rings has been modeled as an isoentropic and quasiestationary flow. (c) The thermodynamic properties of the combustion gases have been included with an analytical ideal gas model, controlled by the composition and the temperature of the mixture (air, vapor fuel, and combustion products). (d) To describe the combustion processes, a Wiebe’s function26 and the fuel consumption Mf (from the equivalence ratio)21 have been used for the heat-release rate, as Table 2 shows, next to eq 2a for SIE and eq 2b for DE, depending upon the crank angle θ.

( ) [ ( ) ]

dQf (m + 1) θ - θi ) Mf A dθ θc θc dQf ) dθ

m

exp -A

θ - θi θc

m+1

(2a)

( ) [ ( ) ] ( ) [ ( ) ]

qp θ - θic mp θ - θic mp+1 exp -6.9 + Mf 6.9 (mp + 1) θp θp θp qd θ - θic md θ - θic md+1 exp -6.9 (2b) Mf 6.9 (md + 1) θd θd θd

(e) For DE, the ignition timing θic can be calculated from the injection timing θi if the ignition delay time ∆θi is known. This delay has been considered through Sitkei’s correlation,27 which accounts for cold and blue flames with a dependence of pressure and temperature. (f) For the gas exchange process, a filling and emptying model was developed. This type of model was accurate enough because of the relatively low influence of the pressure curve during this process on the dynamics of the engine. The submodel described above serves to obtain the pressure curve in one cycle when their parameters are well-known. Figure 3 shows the comparison between a pressure curve obtained from the whole model (direct problem) and the corresponding one for a certain real condition (target), where a close-fitting result can be observed. These two lines are presented for all of the validation tests conducted with similar results.24 Then, the single-cylinder DE uses the parameters described in Table 2 to characterize the output layer of the network. (25) Woschni, G.; Fieger, J. SAE Paper 670931, 1967. (26) Miyamoto, N.; Chikahisa, T.; Murayama, T.; Sawyer, R. SAE Paper 850107, 1985. (27) Sitkei, G. Kraftstoffaufbereitung und Verbrennung bei Dieselmotoren; Springer-Verlag: Berlin, Germany, 1964.


Cruz-Perago´ n and Jime´ nez-Espadafor Table 4. RBF Network for Three-Cylinder SIE learning patterns hidden layer neurons weight value on the hidden layer biases weights output layer neurons weight value on the output layer input layer vector length associate weights to the hidden layer activation function of the hidden layer associate weights to the output layer activation function of the output layer

Figure 3. Examples of validation tests for (a) single-cylinder DE (2600 rpm and 27 N m) and (b) one cylinder of the three-cylinder SIE (3400 rpm and 34 N m). Table 3. RBF Network for Single-Cylinder DE learning patterns hidden layer neurons weight value on the hidden layer biases weights output layer neurons weight value on the output layer input layer vector length associate weights to the hidden layer activation function of the hidden layer associate weights to the output layer activation function of the output layer

20 000 (20 × 1000) 1000 polynomial approximation 1000 7 the same for any average speed 44 1000 × 44 Gauss (µ ) 0; σ ) 0.05) 7000 (7 × 1000) linear

Nevertheless, the three-cylinder SIE requires 15 terms (5 per cylinder), and the V-16 DE needs 112 terms. This high number of parameters increases the network complexity. Then, the characterization is as follows: First, for the three-cylinder SIE, the five parameters that characterize the combustion process (see Table 2) are calculated from validation tests data, by leastsquares fitting.24 As Figure 4 shows, there exists a dependence of these terms with the mean angular speed and the maximum value of the pressure curve. Therefore, the characterization of

10 000 (10 × 1000) 1000 polynomial approximation 1000 × 1 3 the same for any average speed 61 1000 × 61 Gauss (µ ) 0; σ ) 0.05) 3 × 1000 linear

each chamber pressure is made of one term, and the network output layer of the engine needs only three nodes. Finally, for the V-16 DE, there are not real data to compare. Therefore, the basic used model describes the pressure from two superimposed profiles: a constant pressure curve for the motored engine, added to a variable curve component depending upon the maximum reached pressure, such as previously developed by other researchers.2 Then, the output layer only needs 16 nodes, instead of 112. 2. Speed Characterization (Input Vectors). Minimization of the Number of Nodes of the Input Layer. The second submodel, called the dynamic model, results in the instantaneous angular speed for one engine cycle, from the indicated pressure and the knowledge of its geometric and dynamic characteristics.24 To characterize the velocity curves, several points corresponding to significant crank angle degree positions along the cycle are chosen. They correspond to special points, such as maximums, minimums, inflection points, etc., as other researchers use for engine behavior characterization.28,29 Additional points around TDC with combustion are necessary to consider, because the geometry of the engine tends to make indeterminate the model results in that zone.24 All of these sample points must be capable of reproducing the instantaneous angular speed curve by cubic splines. According to this, a large amount of instantaneous velocity curves have been evaluated, as shown in Figure 5. As a result of this analysis, 43 points are necessary for a single-cylinder DE approach (see Figure 6), 60 points are necessary for a three-cylinder SIE, and 90 points are necessary for a V-16 DE. 3. Input-Output Data Scaling. Neural networks, similar to other approximation techniques, make better results if the values for associated vectors are in the same domain.30 All of the input and output values are scaled individually such that the overall variance in the data set is maximized. The scaling used is in the range from -1 to 1. 4. Number of Learning Samples: Network Structure. Radial functions have a Gaussian shape; therefore, they are characterized by a medium value (center, µ) and standard deviation (σ). Dispersion determines the space area of the input vector where neurons respond correctly. Their values must be wide enough for strong responses in neurons overlapping regions (28) Yang, J.; Pu, L.; Wang, Z.; Zhou, Y.; Yan, X. Mech. Syst. Sig. Proc. 2001, 5 (3), 549-564. (29) Moro, D.; Cavina, N.; Ponti, F. J. Eng. Gas Turbines Power 2002, 124 (1), 220-225. (30) Gill, P. E.; Murray, W.; Wrigth, M. H. Practical Optimization; Academic Press: London, U.K., 1997.



Figure 4. Approximation mesh of different parameter values of the combustion model, depending upon the mean speed and maximum pressure. Data evaluated from the validation tests of this model for the three-cylinder SIE: (a) ignition timing θi (degrees BTDC), (b) duration of combustion θc (degrees), (c) form factor A (dimensionless), (d) form factor m (dimensionless), and (e) equivalence ratio (dimensionless).

of the input space. This number of centers is directly related to the number of input and output nodes. Moreover, the RBF network can be sensitive to a change in the assignment of input nodes.31 Therefore, one RBF characteristic is that there are no hard and fast rules about choosing the number of centers. The only way is to try a different number of centers, increasing them until convergence of the model fit is achieved.12 (31) Billings, S. A.; Zheng, G. L. Mech. Syst. Sig. Proc. 1999, 13 (2), 348.

For this purpose, the basic structure (see Figure 2) was applied to a single-cylinder DE and trained with a different number of learning samples (from 500 to 14 000), with only one average speed. The higher complexity of the combustion model of a DE made it possible to extrapolate the conclusions to a SIE without question. The initial results saw that the maximum capacity of this network was for 10 000 samples. These outcomes demonstrate that each viable net (up to 10 000 samples) was capable of perfectly adjusting the pressure curves



vectors of the net), being a number of 1000 for only one average speed (as Figure 2 shows). Later, the same output combinations are used N times, each one for other average speeds (the same indicated torque produces two different instantaneous speed curves when two different average angular speeds are considered) The N new designed networks have the same architecture and the same weight values at the output layer, varying only the weight values of the hidden layer. Therefore, the definitely net maintains the same structure shown in Figure 2 for one average speed but varies the weights values of the hidden layer, which are dependent upon the mean angular speed n. It is demonstrated that they can be approached by polynomial functions of eq 3, as shown in Figure 8.

IWj,k ) aj,k + bj,kn + cj,kn2 + dj,kn3 + ej,kn4 + fj,kn5 + gj,kn6 j ) 1, ..., S1 k ) 1, ..., R (3)

Figure 5. Superposition of different derivatives of instantaneous angular speed curves (first derivative, dω/dθ; second derivative, d2ω/ dθ2; and third derivative, d3ω/dθ3).

Similar structures are carried out in the three-cylinder SIE and V-16 DE approaches, varying the number of output parameters, such as described before. Tables 3-5 show the main characteristics for the three evaluated final networks. The last case (V-16 DE) uses 1500 neurons, because there is only one mean speed (power plant engine) and the number of output parameters is higher. This approach assumes to store the polynomial coefficients for each neuron separately, making the computation time a bit higher than for only one mean angular speed. Nevertheless, the drastic complexity reduction of the network that presents an extremely high number of neurons makes up for it. For example, the proposed network applied to a single-cylinder DE needs a hidden layer consisting of 44 000 different weights (see Table 3). Whereas, a conventional GRNN with the same number of training patterns would need a hidden layer with 880 000 weights. Results and Discussion

Figure 6. Example of speed profile parametrization (mean angular speed of 1625 rpm).

of the training patterns. Later, other nontrained pressures (1000) were evaluated for each structure, appearing around 20% of unsatisfactory pressure curves at any case with more than 900 learning samples. Then, a hidden layer with 1000 neurons was chosen to be the main component of the network.16 The second step considered 1000 learning samples with five different average speeds. Here, all of the nontrained net results (pressures) were misadjusted when average speeds were not close to the trained speeds. Furthermore, the experiment was repeated for 3000, 5000, and 10 000 learning samples (with the same number of neurons in the hidden layer), obtaining then very similar conclusions. It entailed that more learning samples were necessary, including a higher number of average speeds. On the other hand, a network with 1000 neurons at the hidden layer constituted the simplest structure with the same accuracy as with a higher number of neurons. The agreement to this contradiction is settled as follows (see Figure 7): First, one net is designed and trained with a certain group of vectors that contain the pressure parameters (output

Once the network is designed (see Tables 3-5), it is trained with a very high number of theoretical patterns, for a direct validation, because there is a perfect fitting between net results and training patterns. Later, a large amount of random theoretical data is generated to describe the initial effectiveness of the net. Their instantaneous angular speed curves are applied to the network, and then, the pressure curves are generated. With the purpose of improving the results in those cases where real data are evaluated, the net is trained additionally with some real curves (engine speed and pressures). These new training patterns substitute some weight values into the hidden and output layers. Once these processes are executed, new experimental data are evaluated, applying the speed curve to the input of the net, thus obtaining the modeled pressure curve into each cylinder. Different deviation errors () are determined, comparing both approximated and target pressure curves, with the aim of evaluating the net effectiveness. For the single-cylinder DE and the three-cylinder SIE, two main parameters will be evaluated, as well as the pressure curve profile estimation: the mass fuel consumption Mf, and the injection (for DE) or ignition (SIE) timing θi. Other parameters with less importance are presented too, such as maximum pressure or combustion slope. 1. Single-Cylinder Diesel Engine. With the net described in Table 3, 10 000 random conditions are analyzed and the deviation errors are statistically evaluated, applying a Kolmog-



Figure 7. General procedure to minimize the architecture of the RBF network applied to an engine.

Figure 8. Values for three weights associated to the first neuron of the hidden layer (single DE net). Table 5. RBF Network for the 16-Cylinder vee DE learning patterns hidden layer neurons output layer neurons input layer vector length associate weights to the hidden layer activation function of the hidden layer associate weights to the output layer activation function of the output layer

1500 1500 16 91 1500 × 91 Gauss (µ ) 0; σ ) 0.05) 16 × 1500 linear

orov-Smirnov test.32 Their values present a probability density function with a similar shape to a χ2 type, such as the example shown in Figure 9. Therefore, Table 6 presents, for different deviation errors (rows), the next statistical results (columns): the first and second columns show the mean µ and standard σ, respectively. Next, the maximum errors for both 95 and 99% (32) Kleinbaum, D. G.; Kupper, L. L.; Muller, K. E. Applied Regression Analysis and Other MultiVariable Methods, 2nd ed.; Duxbury Press: Belmont, CA; 1988.

Figure 9. Probability density function for the deviation error of the indicated mean effective pressure (Imep) in theoretical analysis for the single-cylinder DE.

levels of confidence (R) are presented. Finally, three levels of confidence related with a maximum error of 1, 5, and 10% are also shown. The statistical indicators show a high confidence level in the main parameters that describes the curves and the combustion processes. The analysis of these theoretical curves assures a perfect fitting between real and predicted curves for compression and expansion strokes far from the combustion TDC of the engine. The relation between indicated pressure and engine dynamics fails when the piston passes through that point, where uncertainties exist. The learning samples for training the network have been generated from physical models, and above, the nonunique character of the combustion parameters has been indicated for pressure and speed production. For these reasons, the values obtained from weights produce the nondesired results of the network. In any case, 80% of pressure curves can be estimated.



Table 6. Statistical Indicators for the Estimation in the Single-Cylinder DE with Theoretical Data indicator

µ

σ

maximum with R ) 0.95


R for e 1%

R for e 5%

R for e 10%

percent error Imep percent maximum pressure error absolute error in maximum pressure (MPa) absolute error in maximum pressure location (degrees) percent combustion slope error (premixed phase) percent pressure when input valve closes error percent fuel consumption error injection timing error (degrees BTDC)

0.093 0.56 0.1 0.13 0.5 0.39 2.72 1.48

13.73 8.62 0.63 2.23 13.42 2.8 18.34 0.01

14 8 2 3 19 3 8 2

16 17 2 3 21 3 10 2

23.27 24.2

62.83 68.9

82.63 96.33

24 62.77 33.27

47.6 95.9 73.33

65.77 99.36 88.27

Table 7. Deviation Errors for Different Real Operating Conditions in the Analyzed Single-Cylinder DE torque (N m)

n (rpm)

(%) Imep

(%) maximum pressure

(%) pressure when input valve closes

(%) fuel expense

(%) combustion slope (premixed phase)

(degrees BTDC) injection timing

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

2030 2130 2150 2400 2400 2400 1570 2600 2200 1800 2200

3.6 2.6 1 -0.9 0.4 -3.7 0.8 -5.9 -7.8 -5.6 1.3

-0.1 -1.25 -5.3 -6.4 0.64 -2.4 -8.8 11 -9.4 -8.8 -2.3

12 0.1 -0.1 0.1 -0.3 0.1 1.12 6.9 13.6 14.5 9.8

0.1 0.3 7.3 -1.8 -6 6.2 -5 -8 -7 -4

6.5 -0.3 -9.2 14 5.98 -9.1 7.3 -11.3 -12.4 -8

1.5 -0.3 1 -0.13 0.12 1.1 0.9 1.4 1.9 1.6

Later, the analysis is completed with real data in this application. The deviation errors are included in Table 7, and Figure 10 shows an example of the net results. In most of the cases, the accuracies of the pressure curves are correct, presenting nearly the same feasibility as theoretical results, although the estimation for individual parameters is worse. The main reason is a certain inaccuracy between both real and modeled angular speed curves, which increases the discrepancies. Nevertheless, 80% of pressure curves have been well-estimated also for theoretical data. 2. Three-Cylinder SIE. Once the described net is trained (see Table 4), 10 500 nontrained random theoretical new pressures (related to 3500 different speed combinations) will be tested. In a similar way, as for the case of theoretical results for the single cylinder, approximations have a high degree of feasibility. However, similar accuracies are more difficult, because of a higher number of cylinders. Table 8 presents different statistical indicators from this initial analysis, in the same way as Table 6. One important feature is that the ignition timing deviation error shows higher values than the previous

engine, demonstrating a strong correlation between this one and the other combustion parameters (see Table 2). Finally, 70% of pressure curves can be considered as correct. Once the net is trained with some real data, results considering several real operating conditions were evaluated, as shown in Table 9 and Figure 11. Here, errors are slightly higher than theoretical data, mainly caused by two reasons: first, the five combustion parameters per cylinder were approximated such as Figure 4 shows. This means that the net only has to determine one new parameter (the maximum pressure), instead of those other five. Because of the low number of disposal data, approximation (Figure 4) is for guidance only and there is not enough accuracy, which results in those higher errors. For a good agreement of them, a different experimental setup must be performed, taking into account many measurements (various hundreds), something that is long away from the targets of this work. Second, some overlapping effects of indicated torques between cylinders appear, as well as a little misadjusting in the dynamic engine response. It is important to remember that the training has been

Figure 10. Pressure results for single-cylinder DE, 2400 rpm and 13 N m.

Figure 11. Pressure results for three-cylinder SIE, 3300 rpm and 14 N m.



Table 8. Statistical Indicators of the Deviation Errors in the Three-Cylinder SIE with Theoretical Data indicator

µ

σ



R for e 1%

R for e 5%

R for e 10%

percent error Imep percent maximum pressure error absolute error in maximum pressure (MPa) absolute error in maximum pressure location (degrees) percent combustion slope error percent fuel consumption error percent pressure when input valve closes error injection timing error (degrees)

2.89 4.42 0.138 -0.326 10.05 4.1 0.87 -0.11

8.6 12.4 0.592 2.31 52.5 14.55 16.7 2.75

7.1 11.3 0.7 1 42 11 19 6.72

17 27 0.8 1 56 22 25 7.92

21.1 23.84

67.05 56.53

98.2 86.7

16 30.9 22.7

31.95 57.7 50

47.74 86.5 71.44

Table 9. Deviation Errors for Different Real Operating Conditions in the Analyzed Three-Cylinder SIE torque (N m)

n (rpm)

17

2500

6

2550

14

3300

44

4500

25

4500

20

5580

49

5410

Nr Cil.

(%) Imep

(%) maximum pressure

(%) pressure when input valve closes

(%) combustion slope

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

-3.4 1.4 2.3 -2.4 2.8 -2.8 3.2 3.8 3.4 4 3 -6.2 -4.8 2.1 -5.8 -4.6 3.6 -4.1 -3.5 -2 -4.5

-6.1 -3.3 -4.2 -3.2 -4.8 -3.5 3.1 2.8 2.4 -2.7 -1.7 -5.2 -3.9 3.6 -4.5 -2.8 3.5 -2.8 -4 -3.4 -4.7

-3.6 2.1 -4.7 -9.5 -12.4 -8.8 3.9 3.4 2.3 4.2 11 11 2.3 -2.9 7.8 -9 -0.6 1.4 8 -7 -8

-3 1.5 -1.9 3.8 4.34 1.1 1.9 5.3 8.14 -16 -11.2 7.3 -9.4 9.7 12.2 -7.3 9 -6.1 -2.91 -1.5 -10.7

(%) fuel expense (overall) 9 6 8.7 -3.2

4 -2.2 -7.5

(degrees BTDC) ignition timing 6.6 2.9 6.9 -3.6 9 3 -1.97 -1.7 -0.5 4.4 8.4 -1.1 7.6 1.1 1.7 -5.8 8.1 -5.5 8.9 2.5 0.01

Table 10. Deviation Errors in the V-16 DE error type

µ

σ

maximum error associated with R ) 0.95

acceptable R

percent maximum pressure (dimensionless) maximum absolute pressure deviation (MPa)

0.493 -0.000 25

0.6996 0.5

1 0.7

99.99 (for e 10%) 97.1 (for e 1 MPa)

done with a mathematical model, translating some disadvantages of it into the net responses. The net parameter combinations make a good agreement in most of the cases: nearly 70% of the analyzed conditions present pressure curves very well-fitted to the real profiles. Nevertheless,

the method shows a good ability to sort the size of the three curves considering the maximum pressure on most operating conditions. 3. 16-Cylinder vee Power Plant Diesel Engine. Here, 60 new random and theoretical operating conditions of the engine are evaluated, once the corresponding net is designed and trained, resulting in 960 different pressure curves. The errors are restricted to the maximum pressure, using a simplified pressure model. The stochastic analysis is shown in Table 10. Figure 12 shows the net response for one of the random analyzed conditions. Here, the confidence levels are very high. This is because of the simplicity of the used pressure model, next to only one mean value for the angular speed in all of the evaluated operating conditions. Conclusions

Figure 12. Pressure results for V-16 DE and theoretical objective pressure function, 7640 N m at 1500 rpm (approximately 1.2 MW).

To solve the inverse problem using ANN, only nets based on a nonlinear training algorithm are suitable. On the other hand, the computer memory limit represents another problem to be solved. According to these statements, a modified RBF network (GRNN) has been developed. The proposed network needs a small number of neurons, and it is able to present results with important time saving. Thus, the methodology is not dependent upon engine dynamics. Moreover, it shows a good ability to estimate the pressure curve in one cycle for different engines,


independent of the number of cylinders or behavior principle. Nevertheless, the high number of required samples makes it necessary to train the net via simulations, which are modeldependent. The main advantages of this method are as follows: first, the net provides results very quickly, and second, the confidence level is too high, estimating the pressure curves very well. Therefore, it can be incorporated to diagnosis or control systems on engines with very low costs, removing a very sensitive and expensive collection of transducers. Here, only an angular (position or speed) sensor is necessary. Theoretical results show some effects derived from model limits, such as uncertainties near TDC, where ignition occurs, or indicated torque overlapping between cylinders. Later, results with a real operating measurement show some higher errors, caused by a little misalignment of the dynamic model from the real behavior. This initial misadjusting between theoretical and real data has been overcome by incorporating some real data


as training patterns. They complete the network, modifying some internal weight values (in both hidden and output layers). Work in different ways is necessary to improve these results. First, the model implementation: although combustion parameters provide the pressure curve from a model, the number of them must be reduced. In this sense, a partial correlation indicates that some of them can be derived from others. Second, the network design: measurements to consider can be combinations of network structures, variations in activation functions, and the number of hidden layers, focusing the general hidden layer of the RBF network as the main component of a neuronal complex network. This work indicates a starting point to solve one extended problem in the engine world where conventional methods use expensive and delicate equipment. EF070122D

Design and Optimization of Neural Networks To ... - ACS Publications

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