Conclusions

Comparison of Physics-Based, Semi-Empirical and Neural Network-Based Models for Model-Based

6. Conclusions

A comparison of the performance of physics-based, semi-empirical and artificial neural network (ANN)-based models has been carried out in this paper to estimate the main combustion metrics for an FPT Euro VI 3.0 L F1C diesel engine. The models were developed for combustion control-oriented

applications. The combustion metrics that are predicted by the models are MFB50, PFP, and BMEP.

These metrics are interesting for control-oriented applications, since they are closely related to the thermal efficiency and to the pollutant formation process of the engine. Four different types of modeling approaches have been investigated. The first approach is based on a previously developed low-throughput mean-value physics-based model, which is capable of simulating the heat release rate, the in-cylinder pressure, and the related metrics on the basis of the accumulated fuel mass principle.

In this approach, the tuning parameters of the model are estimated on the basis of physically-consistent correlations, which are modeled using power-law functions. The second investigated approach is based on the joint use of artificial neural networks (ANNs) and physical modeling: instead of using the power-law functions, ANNs have been used to identify the parameters of the physics-based model.

The third investigated approach is based on a direct estimation of MFB50, PFP, and BMEP on the basis of semi-empirical correlations. The fourth approach is based on the use of ANNs to directly estimate MFB50, PFP, and BMEP. The performance of the four models was evaluated under steady-state conditions and transient operation over WHTC.

Moreover, a new methodology, which is based on the sequential use of the Pearson correlation and partial correlation analysis, has been developed to identify the optimal set of input model parameters in order to a priori identify the correlation variables that should be excluded and the most robust correlation variables.

The main results can be summarized as follows:

(1) The new methodology set up for the identification of the model input parameters has allowed an improvement to be obtained in the calibration of the baseline physics-based model. Although the accuracy in the estimation of BMEP is quite similar, an improvement in the estimation of PFP, and MFB50 can be obtained, since the related RMSE values decrease from 2.47 bar to 1.82 bar and from 0.86 deg to 0.63 deg, respectively, at steady-state conditions. Moreover, it allows a saving in the model calibration computational time to be achieved.

(2) The direct ANN model has shown the best accuracy in the estimation of MFB50, PFP, and BMEP under both steady-state conditions (RMSE=0.25 deg, 0.85 bar and 0.071 bar, respectively) and transient operation over the WHTC (RMSE=1.1 deg, 9.6 bar and 0.7 bar, respectively);

the accuracy of the baseline physics-based model is slightly worse than that of the direct ANN model (RMSE=0.63 deg, 1.8 bar, 0.15 bar under steady-state conditions, RMSE=1.2 deg, 10.5 bar, 0.8 bar over WHTC).

(3) The accuracy of the ANN physics-based model is higher than that of the baseline physics-based model under steady-state operations (RMSE=0.48 deg, 1.6 bar, 0.17 bar), but it shows a marked deterioration for transient operation (RMSE=1.2 deg, 13.8 bar, 0.8 bar), and this approach, therefore, seems to lack robustness.

(4) The accuracy of the semi-empirical model is worse than that of the physics-based model and of the direct ANN model under steady-state conditions (RMSE=0.63 deg, 3 bar, 0.26 bar) and over the WHTC (RMSE=1.4 deg, 11.7 bar, 0.8 bar). However, the accuracy can still be considered acceptable, in view of the simple mathematical structure of this kind of model, and the low number of tuning parameters that are necessary (and therefore of experimental data needed for calibration).

(5) The computational time required for the direct ANN model and semi-empirical model is less than 50μs on an ETAS ES910 rapid prototyping device, while that of the baseline physics-based model is of the order of 350μs.

Overall, the direct ANN model can be considered the best candidate for control-oriented applications, since:

• it is much simpler in structure and theory and easier to build than the physics-based model;

• it does not require any detailed knowledge or modeling of the physical and chemical processes;

• it is robust not only under steady-state conditions, but also in transient operation;

• its accuracy is high, even when a not so large dataset of experimental tests (410) is used for training;

• it features the best accuracy under steady-state and transient operating conditions

• the required computational time is lower than that of the physics-based model.

The main drawback of this kind of model is that it is of the black-box type. Thus, it does not provide detailed information about the physical and chemical processes that occur inside the combustion chamber, such as the heat release trend, in-cylinder pressure, or temperatures.

Author Contributions:The authors contributed equally to the preparation of the paper. Conceptualization, S.H., R.F., S.d.; Methodology, S.H., R.F., S.d.; Software, S.H.; Formal Analysis, S.H., R.F., S.d.; Data Curation, A.M.

(Andrea Manelli), L.V., S.d.; Writing-Original Draft, S.H., R.F.; Writing-Review and Editing, R.F., M.M., A.M.

(Antonio Mittica); Supervision, R.F., S.d., A.M. (Antonio Mittica), H.W., Y.W.

Funding:FPT is kindly acknowledged for its support in the activities. This work was also supported by the China Scholarship Council (Grant No. 201706680011) and Marine Low-Speed Engine Project-Phase I (CDG01-KT0302).

The authors would like to thank Omar Marello for his efforts in discussing and carrying out this research.

Conflicts of Interest:The authors declare no conflict of interest.

Abbreviations

ABDC After Bottom Dead Center AFst stoichiometric air-to-fuel ratio ANN artificial neural network

BMEP Brake Mean Effective Pressure (bar)

CFD Computer Fluid-Dynamics

DoE Design of Experiment

ECU Engine Control Unit EGR Exhaust Gas Recirculation

EOC End of Combustion

EOI End of Injection

EVO Exhaust Valve Opening

FMEP Friction Mean Effective Pressure (bar) H_L lower heating value of the fuel

HCCI Homogeneous Charge Compression Ignition ICE Internal Combustion Engines

IMEP360 gross Indicated Mean Effective Pressure (bar) IMEP720 net Indicated Mean Effective Pressure (bar) IVC Intake Valve Closing

K combustion rate coefficient

m mass

m_air trapped air mass

m_EGR trapped EGR mass

m_f,inj injected fuel mass (mg per cycle/cylinder) m._f,inj fuel injection rate

MFB50 crank angle at which 50% of the fuel mass fraction has burned (deg) n polytropic coefficient for the compression phase

n’ polytropic coefficient for the expansion phase N engine rotational speed (1/min)

O2 intake charge oxygen concentration (%)

p pressure (bar)

PCCI Premixed Charge Compression Ignition p_EMF exhaust manifold pressure (bar abs) p_f injection pressure (bar)

PFP peak firing pressure

p_IMF intake manifold pressure (bar abs) PMEP Pumping Mean Effective Pressure (bar) q injected fuel volume quantity (mm³) Q_ch chemical heat release

Q_f,evap fuel evaporation heat

q_main total injected fuel volume quantity of the main pulse per cycle/cylinder q_tot total injected fuel volume quantity per cycle/cylinder

q_tot,pil total injected fuel volume quantity of the pilot pulses per cycle/cylinder Q_fuel chemical energy associated with the injected fuel

Q_ht,glob global heat transfer of the charge with the walls Qnet net heat release

R² squared correlation coefficient RAF relative air-to-fuel ratio

RMS root mean square

RMSE root mean square error SOC start of combustion SOI electric start of injection SVP support vector machine

t time

T temperature (K)

T_EMF exhaust manifold temperature T_IMF intake manifold temperature

T_SOC charge temperature at start of combustion T_SOI charge temperature at start of injection

V volume

V2X Vehicle-to-Everything

Valve_EGR opening position of the high pressure EGR valve VGT Variable Geometry Turbine

WHTC Worldwide Harmonized Heavy-duty Transient Cycle

X_r,EGR EGR rate

Greek Symbols

Δpexh-int difference between the exhaust and intake manifold pressure

γ isentropic coefficient

ρ density

ρIMF density in the inlet manifold

ρSOC charge density at the start of combustion ρ_SOI charge density at the start of injection τ ignition delay coefficient

Appendix A Correlation Analysis for the Selection of the Model Input Variables

In order to make the correlation analysis more synthetic, the considered variables have been associated with indexing numbers (IN), as shown in TableA1.

For the sake of clarity, FigureA1reports the temporal sequence of the calculation of the different variables used in the physics-based model.

The Pearson correlation coefficient was calculated, using Matlab R2016b, in order to identify the correlations between the variables identified in TableA1. The results are reported in FigureA2.

Table A1.Collected variables and corresponding index number (IN).

Variable Name IN Variable Name IN Variable Name IN Variable Name IN

N 1 Valve_EGR 13 T_SOC,main 25 Qnetmax 37

p_f 2 XrEGR 14 SOC_pil 26 ΔpIMF 38

SOI_pil 3 RAF 15 SOCmain 27 n 39

SOI_main 4 O₂ 16 Δpexh−int 28 n 40

q_tot,pil 5 ρ_IMF 17 τ_pil 29 PFP 41

q_main 6 ρSOI,pil 18 τ_main 30 IMEP360 42

q_tot 7 ρSOI,main 19 K_pil 31 PMEP 43

p_IMF 8 ρSOC,pil 20 K_1,main 32 FMEP 44

T_IMF 9 ρ_SOC,main 21 K_2,main 33 IMEP720 45

p_EMF 10 T_SOI,pil 22 MFB50 34 BMEP 46

T_EMF 11 T_SOI,main 23 Q_ht,glob 35

m_air 12 T_SOC,pil 24 Q_f,evap 36

The numbers in the rows and columns in FigureA2are the indexing numbers of the variables shown in TableA1. The background color changes from green to white and from white to red when the Pearson coefficient value varies in the (−1, 0) range or in the (0, 1) range, respectively. The linear relationship between two variables is considered strong when their Pearson coefficient value is in the (−1.0,−0.5) or (0.5, 1) ranges, moderate when it is in the (−0.5,−0.3) or (0.3, 0.5) ranges, weak when it is in the (−0.3,−0.1) or (0.1, 0.3) ranges, and non-existent or very weak when from it is in the (−0.1, 0.1) range [43].

Figure A1.Temporal calculation sequence of the physics-based model variables.

The Pearson coefficient can provide useful information for the selection of:

• the input parameters of the correlations which are used to estimate the calibration parameters of the physics-based models (i.e., those reported in Table3), or the input parameters of the ANNs that are used to estimate the same calibration parameters;

• the input parameters of the semi-empirical correlations that are used to directly estimate MFB50, PFP, BMEP;

• the input parameters of the ANNs that are used to directly estimate MFB50, PFP, BMEP.

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Figure A2.Pearson correlation coefficients for all the possible combinations of the variables reported in TableA1.

As can be seen in FigureA2, the groups of variables [IN 3, IN 4], [IN 6, IN 7], [IN 13, IN 14], [IN 17, IN 19], [IN 20, IN 21], [IN 24, IN 25] and [IN 26, IN 27], namely [SOIpil,SOImain] [qmain,qtot], [ValveEGR, Xr,EGR], [ρIMF,ρSOImain], [ρSOC,pil,ρSOC,main], [TSOC,pil,TSOC,main] and [SOCpil,SOCmain], are very closely correlated to each other, since their crossed Pearson coefficient is almost equal to 1. Therefore, during the identification of the input variables of the models, it is sufficient to select only one of them as representative of each variable group. Moreover, it should be noted that the variables which are calculated at an earlier stage are expected to be closely correlated to the variables that are calculated in a subsequent stage (see FigureA1).

The following rules were identified to select the set of input variables to use in the correlations of the physics-based model or in the semi-empirical and ANN-based models, on the basis of the information derived from the Pearson correlation analysis:

• the temporal sequences of the variables in the model have to be taken into account, i.e., variables cannot be used as the independent variables if they are calculated or occur after the dependent variable, according to the scheme in FigureA1;

• the directly measured engine operation variables (namelyN,pf,SOI, fuel injection quantity,pIMF, T_IMF,Valve_EGR) should be considered as independent variable candidates with top priority, because they are the primary cause of the changes in engine operation. Moreover, these variables are usually very precise because they are measured directly in real-time. The generated parameters (ρSOI,TSOI), based on SOI, should be considered with higher priority than those based on SOC (ρSOC,TSOC), because of the uncertainty chain for the estimation of SOC on the basis of SOI;

• the dependent variables (such asτpil,τmain, SOC,Kpil,K1,main,K2,main,Qf,evap,Qht,glob,n,n’,ΔpIMF, PMEP, FMEP) should be considered with a lower priority as correlation parameters, because they are usually less precise than the directly measured parameters. TheX_r,EGR,RAF,O₂,ρSOC

andTSOCgenerated parameters should also be considered with a lower priority because of their relatively low precision;

• some variable groups (e.g., [SOIpil,SOImain] [qmain,qtot], [ValveEGR,Xr,EGR], [pIMF,mair,ρIMF], [ρSOC,pil,ρSOC,main], [TSOC,pil,T_SOC,main] and [SOCpil,SOC_main]) are constituted by two variables which are closely correlated to each other. Therefore, only one variable should be selected as representative of each group;

• the total number of independent variable candidates should not be too high (in this study a maximum of 15 variables was chosen for the estimation of each dependent one), in order to ensure that the partial coefficient order is not too large and to make the model more robust to input uncertainty.

Some considerations should also been made concerning them_airandp_EMFvariables. Although mairis measured directly, it is usually not a precise or stable variable, since the accuracy of the engine air mass flow sensor is not very high (~5%), and its use, therefore, needs to be evaluated carefully.

Moreover, many medium and heavy-duty engines are not generally equipped with air mass flow sensors and the exhaust manifold pressure is generally not measured in production engines, so its use as an input variable for the model should also be evaluated carefully (in this study it was avoided).

On the basis of the rules explained above, an analysis of the Pearson correlation coefficients was carried out to select independent variable candidates for each dependent variable. A summary is reported in TableA2.

It should be recalled that some of the dependent variables are the calibration parameters of the physics-based model (see Table3), and they need to be modeled through correlations or ANNs, while other dependent variables (i.e., MFB50, PFP, BMEP) are the direct outcome of the semi-empirical model or of the direct ANN model, i.e., the metrics to be used for combustion control applications.

After the previous step, the partial correlation coefficient between each input variable of the initial candidate set reported in TableA2and the corresponding dependent variable was calculated.

The results are reported in TableA3, where the previously estimated Pearson correlation coefficients are also reported. The simultaneous analysis of the Pearson correlation and of the partial correlation coefficients is very useful, since it allows the least and most robust correlation variables to be identified.

Table A2.Initial candidate set of the input variables for each output parameter, identified on the basis of the Pearson correlation analysis.

Dependent

Variable Initial Candidate Set of the Independent Variables Selected Using Pearson Correlation Analysis RAF N,p_f,SOI_pil,q_tot,pil,qtot,pIMF,TIMF,X_rEGR,ρSOI,pil,ρSOI,main

τpil N,p_f,SOI_pil,q_tot,pil,pIMF,TIMF,Valve_EGR,RAF,ρSOI,pil,T_SOI,pil

τmain N,p_f,SOI_pil,q_tot,pil,qtot,TIMF,m_air,X_rEGR,RAF,ρSOI,pil,ρSOI,main,ρSOC,pil,T_SOI,pil,Δpexh−int,τpil

K_pil N,p_f,SOI_pil,q_tot,pil,pIMF,TIMF,Valve_EGR,ρSOI,pil,T_SOI,pil,τpil,Δpexh−int

K_1,main N,p_f,SOI_main,q_tot,pil,qtot,pIMF,TIMF,Valve_EGR,RAF,ρSOI,main,ρSOC,pil

K_2,main N,p_f,SOI_pil,q_tot,pil,qtot,TIMF,m_air,X_rEGR,RAF,ρSOI,pil,ρSOI,main,T_SOI,main,K_1,main Q_ht,glob N,p_f,SOI_pil,q_tot,pil,qtot,TIMF,m_air,X_rEGR,RAF,ρSOI,pil,ρSOC,pil,τmain,K_2,main Q_f,evap N,p_f,SOI_pil,q_tot,pil,qtot,T_IMF,m_air,ρSOI,pil,ρSOI,main,RAF,Valve_EGR,T_SOI,pil,Δpexh−int

ΔpIMF N,p_f,SOI_pil,q_tot,pil,q_main,p_IMF,X_rEGR,ρSOI,pil,ρSOI,main,T_SOI,pil,ρSOC,main,Δpexh−int

n N,p_f,SOI_pil,q_tot,pil,qtot,p_IMF,T_IMF,Valve_EGR,ρSOI,pil,ρSOI,main,T_SOI,pil,T_SOI,main,T_SOC,main,Δpexh−int

n N,p_f,q_tot,pil,q_main,SOI_pil,T_IMF,Valve_EGR,RAF,ρIMF,ρSOI,main,T_SOI,pil,Δpexh−int,K_1,main,K_2,main MFB50 N,p_f,SOI_main,q_tot,pil,q_main,T_IMF,X_rEGR,RAF,ρIMF,ρSOI,pil,ρSOI,main,T_SOI,pil

PFP N,p_f,q_tot,pil,q_main,SOI_main,T_IMF,X_rEGR,RAF,ρIMF,ρSOI,pil,ρSOI,main,T_SOI,main PMEP N,p_f,q_tot,pil,SOI_pil,p_IMF,T_IMF,Valve_EGR,τpil,T_SOI,pil,Δpexh−int

FMEP N,p_f,SOI_pil,q_tot,pil,q_main,p_IMF,T_IMF,Valve_EGR,ΔpIMF,PFP,ρSOI,mian,T_SOI,pil,Δpexh−int

BMEP N,p_f,q_tot,pil,qtot,SOI_main,T_IMF,m_air,X_rEGR,RAF,ρSOI,pil,ρSOI,main

In fact, input variables that are characterized by low Pearson and partial correlation coefficients can be removed from the candidate list, while input variables which are characterized by a high value of the Pearson correlation and the partial correlation coefficient at the same time can be selected as very robust correlation variables. Particular attention should also be paid to the input variables which, although showing a strong correlation with the dependent variables, are characterized by a limited precision (since they are derived parameters or measured parameters with low accuracy sensors).

On the basis of the previous considerations, the least robust correlation variables are highlighted in TableA3with a gray background, while the highly correlated variables are marked with an orange background, and finally the highly correlated variables characterized by a low precision are marked with a blue background.

It was found that some input variables (i.e., those marked with a white background in TableA3) are characterized by an intermediate value of the correlation coefficients. In order to understand whether they should have been included or excluded from the input candidate list, a sensitivity analysis was carried out, considering all the possible combinations of these variables and estimating the accuracy of the related models, as reported in Section4.

As reported in the same section, it should be noted that the proposed approach, i.e., the preliminary selection of the input variables through Pearson and partial correlation coefficients, and a subsequent sensitivity analysis carried out on a reduced set of variables, is more computationally efficient than a pure sensitivity analysis approach which includes the entire set of input variables.

TableA3.Pearsonandpartialcorrelationcoefficientsfortheinputvariablesoftheinitialcandidateset. DependentVariableInitialCandidateoftheIndependentVariables RAFIndependentvariableNpfSOIpilqtot,pilqtotpIMFTIMFXrEGRρSOI,pilρSOI,main Pearsoncoefficient0.10−0.43−0.28−0.41−0.77−0.51−0.080.31−0.62−0.57 Partialcoefficient−0.46−0.56−0.47−0.48−0.29−0.240.19−0.500.420.21 τpilIndependentvariableNpfSOIpilqtot,pilpIMFTIMFValveEGRRAFρSOI,pilTSOI,pil Pearsoncoefficient0.800.29−0.89−0.730.060.11−0.010.43−0.71−0.81 Partialcoefficient0.44−0.37−0.13−0.06−0.150.13−0.160.12−0.050.44 τmainIndependentvariableNpfSOIpilqtot,pilqtotTIMFmairXrEGRRAFρSOI,pilρSOI,mainρSOC,pilTSOI,pilΔpexh−intτpil Pearsoncoefficient0.31−0.32−0.50−0.40−0.700.18−0.580.450.58−0.78−0.63−0.58−0.400.330.68 Partialcoefficient0.26−0.47−0.270.03−0.18−0.01−0.05−0.01−0.130.34−0.290.060.000.080.26 KpilIndependentvariableNpfSOIpilqtot,pilpIMFTIMFValveEGRρSOI,pilTSOI,pilτpilΔpexh−int Pearsoncoefficient0.350.10−0.47−0.30−0.04−0.140.25−0.40−0.510.650.35 Partialcoefficient−0.450.360.02−0.010.13−0.15−0.07−0.130.100.76−0.16 K1,mainIndependentvariableNpfqtot,pilqtotSOImainpIMFTIMFValveEGRRAFρSOI,mainρSOC,main Pearsoncoefficient−0.13−0.44−0.22−0.510.18−0.39−0.150.140.75−0.33−0.36 Partialcoefficient−0.340.13−0.200.04−0.44−0.190.15−0.26−0.12−0.490.69 K2,mainIndependentvariableNpfSOIpilqtot,pilqtotTIMFmairXrEGRRAFρSOI,pilρSOI,mainTSOI,mainK1,main Pearsoncoefficient−0.020.320.110.360.52−0.160.36−0.41−0.750.340.30−0.16−0.68 Partialcoefficient0.01−0.06−0.09−0.17−0.200.13−0.06−0.500.21−0.27−0.10−0.270.01 Qht,globIndependentvariableNpfSOIpilqtot,pilqtotTIMFmairRAFXrEGRρSOI,pilρSOC,pilτmainK2,main Pearsoncoefficient−0.230.250.400.600.94−0.350.70−0.72−0.610.840.60−0.700.52 Partialcoefficient0.33−0.480.40−0.040.93−0.44−0.080.340.14−0.43−0.24−0.420.09 Qf,evapIndependentvariableNpfSOIpilqtot,pilqtotTIMFmairRAFValveEGRρSOI,pilρSOI,mainTSOI,pilΔpexh−int Pearsoncoefficient0.37−0.04−0.54−0.42−0.430.31−0.360.340.36−0.57−0.33−0.380.32 Partialcoefficient0.07−0.14−0.310.04−0.01−0.07−0.270.03−0.030.260.09−0.060.07 ΔpIMFIndependentvariableNpfSOIpilqtot,pilqmainpIMFXrEGRρSOI,pilρSOI,mainTSOI,pilρSOC,mainΔpexh−int Pearsoncoefficient0.570.86−0.36−0.330.750.98−0.520.450.90−0.350.950.47 Partialcoefficient−0.09−0.130.06−0.03−0.020.42−0.02−0.090.16−0.12−0.150.18 nIndependentvariableNpfSOIpilqtot,pilqtotpIMFTIMFValveEGRρSOI,pilρSOI,mainTSOI,pilTSOI,mainTSOC,mainΔpexh−int Pearsoncoefficient−0.49−0.210.390.350.29−0.03−0.45−0.270.390.130.21−0.25−0.47−0.54 Partialcoefficient−0.450.09−0.15−0.280.24−0.190.19−0.01−0.010.13−0.050.03−0.42−0.16 nIndependentvariableNpfSOIpilqtot,pilqmainTIMFXrEGRRAFρIMFρSOI,mainTSOI,pilΔpexh−intK1,mainK2,main Pearsoncoefficient−0.48−0.710.27−0.06−0.770.190.680.65−0.76−0.720.36−0.270.69−0.53 Partialcoefficient−0.69−0.13−0.04−0.24−0.360.050.04−0.040.070.05−0.030.250.520.20 MFB50IndependentvariableNpfSOImainqtot,pilqmainTIMFXrEGRRAFρIMFρSOI,pilρSOI,mainTSOI,pil Pearsoncoefficient0.110.380.540.170.690.57−0.42−0.540.580.730.730.26 Partialcoefficient0.76−0.640.840.070.350.13−0.10−0.100.07−0.100.160.13 PFPIndependentvariableNpfSOImainqtot,pilqmainTIMFXrEGRRAFρIMFρSOI,pilρSOI,mainTSOI,main Pearsoncoefficient0.300.76−0.210.020.870.89−0.65−0.640.900.550.85−0.24 Partialcoefficient−0.680.59−0.690.210.36−0.32−0.10−0.130.35−0.20−0.250.30