Analysis of modelling and simulation methodologies for vehicular propulsion systems. Theo Hofman, Dennis van Leeuwen and Maarten Steinbuch

Transcription

1 Int. J. Powertrains, Vol. 1, No. 2, Analysis of modelling and simulation methodologies for vehicular propulsion systems Theo Hofman, Dennis van Leeuwen and Maarten Steinbuch Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Abstract: In this paper, three different modelling and simulation methods (forward dynamic, quasi-static backwards, and inverse dynamic) will be compared and the simulation results using these methods will be analysed. From a Forward Dynamic (FDM) model, a Forward Quasi-static (FQM) and a Backward Quasi-static Model (BQM) for the engine will be derived. The difference in simulation results for the vehicle used on representative driving cycles will be discussed. The forward dynamic model contains a scalable engine model based on physical laws. The accuracy of this model will be investigated by comparing simulation results with measured quasi-static efficiency data of actual engines. Keywords: modelling; simulation methods; engine scaling; computation time; powertrains; propulsion systems. Reference to this paper should be made as follows: Hofman, T., van Leeuwen, D. and Steinbuch, M. (2011) Analysis of modelling and simulation methodologies for vehicular propulsion systems, Int. J. Powertrains, Vol. 1, No. 2, pp Biographical notes: Theo Hofman received his MSc (with honours) and PhD Degree in Mechanical Engineering from Eindhoven University of Technology, Eindhoven. The Netherlands, in 1999 and 2007 respectively. From 1999 to 2003 he was Research and Development Engineer with Thales Cryogenics BV. Since 2010, he is an Assistant Professor with the Control Systems Technology group. He is Associate Editor of Int. Journal of Electric and Hybrid Vehicles. His research interests are modelling, design, and control of electric and hybrid technologies for propulsion systems. Dennis van Leeuwen received his BEng Degree in Automotive Engineering from Hogeschool van Arnhem en Nijmegen, Arnhem, The Netherlands, in He is currently pursuing the MSc Degree in Mechanical Engineering in Eindhoven University of Technology, Eindhoven, The Netherlands. Copyright 2011 Inderscience Enterprises Ltd.

2 118 T. Hofman et al. Maarten Steinbuch received his MSc (cum laude) and PhD degree from Delft University of Technology, Delft, The Netherlands, in 1984 and 1989, respectively. From 1987 to 1999, he was Researcher and Groupleader with Philips Electronics. Since 1999, he is Full Professor in Systems and Control, and Head of the Control Systems Technology group of the Mechanical Engineering Department of Eindhoven University of Technology. Currently, he is Editor-in-Chief of IFAC Mechatronics and Associate Editor of Int. Journal of Powertrains. His research interests are modelling, design and control of motion systems, robotics, automotive powertrains and control of fusion plasmas. 1 Introduction Modelling and simulation of longitudinal vehicle dynamics is widely used for the design and optimisation of drive trains and drive train control strategies. Depending on the way the components are modeled, a vehicle model can either be a quasi-static or dynamic model (Guzzella and Sciarretta, 2005). In addition, vehicle models can be classified as forward looking or backward facing models based on the direction of calculation (from engine to wheel and vica versa) (Wipke et al., 1999; Gao et al., 2007). There are several vehicle simulation tools available, based on these two methods of modelling and simulation, i.e., forward dynamic and quasi-static backwards simulation method. Both methods have their own characteristics, which makes them more or less suited for a certain application. These characteristics are described in detail in literature (see Guzzella and Sciarretta, 2005; Wipke et al., 1999; Verdonck et al., 2008). Recently, in literature (Fröberg and Nielsen, 2008), a new method based on the combination of the favourable characteristics of the existing simulation methods is proposed, i.e., Inverse Dynamic simulation Method (IDM). Essentially, there are two different methods for obtaining models of dynamic systems (Bosgra, ): Experimental modelling: I.e., finding systematic relationships between variables of the real system measured during experiments Theoretical or physical modelling: Describing the behaviour of the system with a mathematical model, based on theory of underlying sciences. Modelling yields a trade-off in the amount of assumptions (or, sometimes uncertainties) and the amount of time, or effort required to set-up a sufficient accurate model. In addition, there is also a trade-off between the required level of model detail and the computation time required to run a simulation with the model (see Gao et al., 2007, for more details). Vehicle models typically consist of equations based on physical laws as well as experimental data and engineering assumptions (Gao et al., 2007). Design exploration and optimisation of a power train typically takes place without the real components being available. For optimisation purposes, components have to be sized and tested in different configurations. This requires models which can describe the system behavior and can be dimensioned with sufficient accuracy. For these

3 Analysis of modelling and simulation methodologies 119 reasons, models based on physical laws, which can be scaled by device parameters are favourable (Guzzella and Sciarretta, 2005). Vehicle simulation platforms are usually based on a modular modelling approach, meaning that they consist of a set of drive train component models, each simulated as a stand-alone subsystem (Verdonck et al., 2008). This in contrary to the Inverse Dynamic Method (IDM). During a simulation, the components (subsystems) transmit power by pairs of signal variables via mutual connections. These variables are potential p(t) and flow f(t) variables, e.g., representing torque and rotational speed, or voltage and current, etc. The power transfer is a result of an interaction between the subsystems, one of the variables f(t) and p(t) acts as a cause (input) leading to an effect (output) (Bosgra, ). In forward modelling one variable at a connection acts as an input, the other as output, the connections are causal. In backward modelling, the variables at a connection are both inputs or outputs, here the power transfer is imposed on the subsystems ( effect-cause ) and not a result of an interaction (Verdonck et al., 2008; Bosgra, ). For Hardware-In-the-Loop (HIL) simulation, i.e., realtime simulation including one or more hardware parts, causality is very important (Bouscayrol, 2009). The basic character of a conservation law is an open integrator; a response (reaction) on a certain input (action) cannot start before the input starts. Cause and effect are coupled by an integrator, and not by a differentiator. 1.1 Contribution of the paper In this paper, inspired by Fröberg and Nielsen (2008), and, Verdonck et al. (2008), three different modelling and simulation methods (forward dynamic, quasi-static backwards, and inverse dynamic) will be compared and the simulation results using these methods will be analysed. In literature, inverse dynamic simulation is already compared to forward dynamic simulation (see, Fröberg and Nielsen, 2008). However, in this paper a Backwards Quasi-static Model (BQM), derived from a FDM applying the method described in Verdonck et al. (2008), is added, enabling a comparison of all three methods. The base line vehicle used in this paper consists of a conventional drive train with a naturally aspirated engine. From the Forward Dynamic Model (FDM), a Forward Quasi-static Model (FQM) and a Backward Quasi-static Model (BQM) for the engine will be derived. The difference in simulation results for the base line vehicle used on a representative drive cycles (New European Drive Cycle (NEDC), Federal Test Procedure 75 (FTP75)) will be discussed. The work presented in this paper will form the basis to develop a modelling, simulation and design method which can be used for quick (alternative, or hybrid) drive train specification, or (supervisory) control calibration with sufficient accuracy (see, also Hofman et al. (2008, 2009)). Accurate scalable energy conversion models or storage models can be used, e.g., in HIL systems reducing the calibration time, or eliminating the requirement of building a full-scale experimental drive train test set-up, and therefore reducing costs. 1.2 Outline of the paper In Section 2, the FDM, with an engine model based on physical laws will be described. This model will be used to derive a BQM in Section 3, and an inverse

4 120 T. Hofman et al. dynamic model in Section 4 respectively. The simulation results with these three models are discussed in Section 5 as well as the comparison of the scalable engine model with measured data of two engines with different displacement volumes. Finally, in Section 6 the conclusions are described. 2 Forward Dynamic Model For reasons of comparison, the FDM presented here and its parameter values, listed in Appendix 1, are adopted from Fröberg (2005). This is a basic model and it consists of a naturally aspirated engine and a simple (stiff) drive train. The dynamic engine model only includes the dynamics of the intake manifold pressure, which cannot be described using quasi-static modelling. The dynamics of the electronic throttle actuator are neglected. Input of the model is the effective throttle area A eff. Intake manifold pressure p i and vehicle speed v are the states, where the latter one is selected as output. This is depicted in Figure 1 (see, Nomenclature). Figure 1 FDM consisting of a dynamic engine model and stiff drive line. DR = driver model; EN = engine; TR = transmission; WH = wheel; CH=chassis 2.1 Naturally aspirated engine model The rotating parts of the engine are modelled as a rotating inertia and the force balance at the flywheel holds, J e ω e = T cr T e, (1) here J e is the engine/flywheel inertia, T cr is the produced combustion torque from the pistons on the crankshaft and T e is the net torque output at the crankshaft. The torque T cr is modelled according a standard mean value engine model, i.e., a model with a time scale typically larger than one engine cycle. The Brake Mean Effective Pressure (BMEP) can be described by BMEP = IMEP g FMEP PMEP, (2) where the Indicated, Friction (Soltic, 2000), and Pump Mean Effective Pressure are, BMEP = 4πT cr V d, (3) IMEP g = m f q hv η th V d, (4)

5 Analysis of modelling and simulation methodologies 121 [( ( ) ) 1.8 Sωe FMEP = π bl 10 5 π πT ] cr ξ aux V d B, (5) PMEP = p e p i. (6) With the engine displacement V d, engine fuel charge m f, fuel heating value q hv, thermal efficiency η th, engine stroke S, engine speed ω e, boost layout π bl, friction coefficient ξ aux, engine bore B, exhaust manifold p e and inlet manifold pressure p i respectively. Assuming that the engine runs stoichiometric, i.e., normalised air fuel ratio λ =1, the fuel charge m f is given by the air mass per engine cycle m ac and the stoichiometric air fuel ratio (A/F ) S, m ac m f = ( A F ) S λ. (7) The inducted air mass charge is described by (Hendricks et al., 1996), m ac = η vol p i V d RT i =(s i p i y i ) V d RT i. (8) with the volumetric efficiency η vol, the gas constant R, the intake manifold temperature T i, the manifold pressure slope s i and the manifold pressure intercept y i. The difference between the mass flow passing the throttle ṁ at and the mass flow into the cylinders ṁ ac gives the pressure derivative in the intake manifold according the following intake manifold pressure state equation: ṗ i = RT i V i ṁat ω e m ac 2πn }{{ r, (9) } ṁ ac where V i is the intake manifold volume and n r is the number of revolutions per engine cycle. The mass flow passing the throttle is modelled according (Heywood, 1998), ṁ at = p a A eff Ψ(p r ). (10) RTa With the pressure ratio p r = pi p a, the ambient pressure and temperature upstream of the throttle p a and T a and flow function Ψ(p r ). Assuming the fluid behaves as an ideal gas and for fixed p a and T a at a certain downstream pressure p i the mass flow ṁ at will reach a maximum and will not further increase with decreasing p i (Heywood, 1998). The pressure ratio p r where the mass flow reaches its maximum is called critical pressure ratio, and is given by, ( ) γ 2 γ 1 p r =, (11) γ +1

6 122 T. Hofman et al. where γ is the specific heat ratio. For p r less than or equal to the critical pressure ratio, i.e., for choked flow or critical flow, Ψ(p r ) is given by, ( ) γ+1 2 γ 1 Ψ(p r )= γ, (12) γ +1 else, i.e., for sub-critical flow, ( 2γ Ψ(p r )= p 2 γ r γ 1 p γ+1 γ r ). (13) 2.2 Drive train model The transmission is modelled as a gear with ratio i and constant efficiency η tm, ω e = iω w, (14) T e iη tm = T w. (15) The wheel is modelled without slip. Wheel torque T w and speed ω w are related to tractive force F and vehicle speed v according to, T w = Fr, (16) ω w r = v. (17) With the wheel radius denoted as r. The motion of the vehicle is described by the following force balance: m v = F 1 2 c dρav 2 mgc r, (18) with the vehicle mass m, vehicle acceleration v, air drag coefficient c d, air density ρ, vehicle cross sectional area A, gravitational acceleration g and rolling friction coefficient c r. 3 Deriving a quasi-static model The FDM presented in the previous section is used to generate a BQM of the naturally aspirated engine. The derivation of a BQM consists of two steps (Verdonck et al., 2008): 1 In the first step, the system of differential equations of the FDM is turned into static equations describing its steady state. This results in a quasi-static model, still with forward causality, i.e., a FQM. The purpose of this model is determination of the admissible inputs for the BQM. 2 During the second step the BQM is derived by changing the causality by selecting different input(s) and output(s). These two steps result in a BQM, which is based on physical laws and depends on the same (sub)set of parameters as the original FDM (Verdonck et al., 2008).

7 Analysis of modelling and simulation methodologies Forward quasi-static model The static equations describing the steady state behaviour of the FDM consist of equations (2) and (7) till equation (10), where ṗ i in equation (9) is set to zero. Besides constant parameters, these five equations contain seven unknowns: ṁ at, A eff, m ac, p i, ω e, T cr and m f respectively. Basically, if two of the unknowns are given, the other five variables can be determined. For the FQM, A eff and ω e are imposed. The effective throttle area A eff is varied from up to [m 2 ] and the engine speed from 50 up to 650 [rad/s]. Given A eff and assuming the flow through the throttle is choked, ṁ at can be computed according equation (10). Since, in case of choked flow, Ψ(p r ) is given by equation (12) and is thus only a function of the (constant) specific heat ratio γ. Subsequently p i can be found using equations (8) and (9). The found p i can be used to check the assumption of choked flow with equation (11). If this assumption holds, m f is computed with equation (7) and T cr with equation (2). Else, the unknown variables can be found by applying an iteration routine. Since, the flow is sub-critical, Ψ(p r ) has to be computed using equation (13). Substitution of Ψ(p r ) in equation (10) and subsequently combining equations (8) (10) results in a non-linear function θ(p i )=ṗ i =0: ṁ at {}}{ θ(p i ):= RT ( i p a ( A γ ) 2 ( ) γ+1 ) pi γ pi γ V i eff RTa γ 1 p a p a ω e V d (s i p i y i ) 2πn r RT }{{ i =0. (19) } ṁ ac The zero of this equation can be found by applying Newton s method (see, Heath, 2002; Vuik et al., 2004). As the engine is naturally aspirated, the intake pressure p i will be smaller than ambient pressure p a. It was found, that the iteration routine converges if the initial guess for p i is set just below ambient pressure. The iteration is stopped by setting the desired relative tolerance to and by limiting the maximum number of iteration steps to a value of 10. After the intake pressure has been found, m f and T cr are computed. The iteration routine is implemented in a Matlab script and is used to generate a FQM of the forward dynamic engine model presented earlier. The computed torque lines as function of engine speed ω e for different throttle areas A eff are plotted in Figure 2. The generated map is validated using the FDM. The same combinations of constant throttle area and engine speed are imposed on the dynamic engine model, i.e., the engine block in Figure 1. After 10 s simulation time the (steady state) torque values are collected. The relative error between the torque values resulting from the FDM and FQM is always below 1%, so the iteration routine works properly.

8 124 T. Hofman et al. Figure 2 FQM torque lines plotted as function of engine speed for different effective throttle areas (see online version for colours) The maximum effective throttle area of the modeled engine is not specified. For the BQM generation it will be assumed that the maximum throttle area is [m 2 ]. The torque values from the FQM corresponding with this throttle position will be used as maximum torque line to limit the BQM. This concludes the first step in the derivation of the BQM. 3.2 Backward quasi-static model In the second step, T cr and ω e are imposed. Using these variables none of the five equations can be solved. A certain value for the intake pressure p i is used as initial guess in an iteration routine to find the solution of δ(p i )=0, where the function δ is given by combining the equations (2), (7) and (8): δ(p i ):= (s ip i y i ) q hv η ( th A ) F λrt S i [( ( ) ) 1.8 Sωe π bl 10 5 π πT ] cr ξ aux V d B p e + p i 4πT cr V d =0. (20) Afterwards the fuel mass flow can be computed. This results in the fuel consumption map in Figure 3.

9 Analysis of modelling and simulation methodologies 125 Figure 3 BQM fuel consumption map [kg/s] (see online version for colours) The generated fuel consumption map from Figure 3 is inserted in a 2D lookup table in Simulink to model the engine. Next the equations of the stiff drive train are also added. 4 Deriving an inverse dynamic model Unlike the forward dynamic and backwards quasi-static simulation method, inverse dynamic simulation is not based on a modular modelling approach. A FDM in the affine form (see Slotine and Li (1991) for the definition) forms the basis for an inverse dynamic model. As the name already states, inverse dynamic simulation is based on system inversion (stable inversion of non-linear systems, using a standard change of coordinates), i.e., writing the input(s) as function of the output(s) (Fröberg and Nielsen, 2008; Fröberg, 2005). For the sake of clarity, the derivation of the IDM as in Fröberg (2005) is presented here again, yet a shorter explanation is used without loss of comprehension. First the FDM is brought to the affine form, i.e., the equations (1) (18) are combined to bring the system to the following state space form: ẋ 1 (t) =c 1 x 2 1(t)+c 2 x (t)+c 3 x 2 (t)+c 4, ( ) x2 (t) ẋ 2 (t) =c 5 x 1 (t)+c 6 x 1 (t)x 2 (t)+c 7 Ψ u(t), (21) y(t) =x 1 (t). With the input u(t) =A eff (t), the states x 1 (t) =v(t), x 2 (t) =p i (t), output y(t) =v(t) and the lumped parameter coefficients c i, as listed in Appendix 2. The inversion procedure for this Single-Input Single-Output (SISO) system is similar to p a

10 126 T. Hofman et al. feedback linearisation, a method to transform non-linear system dynamics into a (partly) linear one (Slotine and Li, 1991). The inversion procedure for the system of equation (21) consists of three steps: 1 the first step is finding the relative degree of the system, i.e., determination of the number of times the output y has to be differentiated before the input u appears explicitly 2 the second step is transforming the system into an equivalent linear system (Khalil, 2002) by a coordinate change 3 the final step is computing the required input (and corresponding state trajectories) for a certain desired output. The relative degree is found using the following definition; a system has relative degree r at a point x 0 if, for all x in the neighbourhood of x 0 (Slotine and Li, 1991), L g L k f h(x) =0 0 k<r 1, (22) L g L r 1 f h(x) 0, (23) implying, if the rth time derivative of the output y depends on the input u explicitly and none of the lower derivatives of y does. For this definition, the so-called Lie derivatives are used, a compact notation for the expressions obtained by repeatedly differentiating y(t) (Slotine and Li, 1991; Khalil, 2002). The Lie derivatives of h(x) with respect to f(x) and g(x) are defined as, L f h(x) = h(x) f(x), (24) x L g h(x) = h(x) g(x). (25) x Computing the relative degree of the system described by equation (21) gives, L g h(x) =0, x; (26) ( ) x2 (t) L g L f h(x) =c 3 c 7 Ψ 0, x {x (x 2 p a )}, (27) p a so the relative degree is r =2. In this case the relative degree equals the system order. A problem may occur when the intake manifold pressure x 2 (t) equals the ambient pressure p a because in this case the relative degree is undefined. But as the engine is naturally aspirated it will never reach this operating point (Fröberg, 2005). The next step is (partially) linearising the system. This done by a coordinate transformation z = T (x) using the output and its first r 1 derivatives as new state coordinates. This gives, z = T (x) = [ = [ z1 z 2 ] [ ẏ = = y] [ ] x1 x 1 x 1 c 1 x c 2 x c 3 x 2 + c 4 ], (28)

11 Analysis of modelling and simulation methodologies 127 which state equations are described by, [ ] [ẏ ] [ ] z1 z 2 = = z 2 ÿ L r f h(x)+l gl r 1. (29) f h(x)u(t) The transformation has to be a diffeomorphism, i.e., continuously differentiable with a continuously differentiable inverse (Khalil, 2002). The third step is to calculate the required input u d (t) necessary to generate a desired output trajectory y d (t), i.e., the throttle signal to follow the prescribed driving cycle (Fröberg, 2005), where, u d (t) = ( L g L r 1 f h(x)) 1 ( L r f h(x)+ÿ d (t) ) (30) L r f h(x) =L 2 f h(x) =(2c 1 x c 2 x ) (c 1 x c 2 x c 3 x 2 + c 4 )+c 3 (c 5 x 1 + c 6 x 1 x 2 ), (31) L g L r 1 f h(x) =L g L f h(x) =c 3 c 7 Ψ ( x2 (t) p a ). (32) The state equations expressed in the new z-coordinates required to compute the input u d (t) with equation (30) are found by inverting the coordinate transformation. They can be used to find the state trajectories x d (t) generating the desired output y d (t), x = T 1 (z) = [ x1 x 2 ] [ = z 1 1 c 3 ( z2 c 1 z 2 1 c 2 z 1.8 ] ) 1 c 4. (33) In this case in order to find the states x d (t) and input u d (t) a set of algebraic equations has to be solved. 4.1 Driver model To compute the required input, the output has to be differentiated as many times as the relative degree of the system. Computing the derivative of the velocity profile of the NEDC leads to discontinuities. This will lead to infinite accelerations and thus infinite high torques which will never be present in a real power train. Therefore the velocity profile has to be smoothed to make it continuously differentiable and to get smooth torque signals. Smoothing of the velocity profile to make it continuously differentiable can be done in several ways. In literature, the author proposes the following convolution integral to obtain a smooth trajectory v d (t) from the original drive cycle v c (t) (Fröberg and Nielsen, 2008; Fröberg, 2005), v d (t) = 1 C g(t τ)v c (τ)dτ. (34)

12 128 T. Hofman et al. With the following choice for the convolution kernel g(t), which makes v d (t) infinitely many times continuous differentiable, g(t) = {e a2 a 2 t 2, t a, 0, otherwise. (35) The derivatives of v d (t) are computed with the same convolution using the analytical derivatives of g(t). This mathematical smoothing can be performed off-line and it describes the behaviour of the driver. Therefore, tracking of the drive cycle is independent of the power train modelled and it can be adjusted by setting the time constant a. A smaller value of this time constant results in tighter tracking of the drive cycle (Fröberg and Nielsen, 2008; Fröberg, 2005). The speed points of a drive cycle typically are prescribed at 1 s intervals. In order to apply the convolution integral (see, equation (34)) for smoothing, the speed points of the driving cycle are connected using a piecewise linear function. This integral is numerically approximated and solved using an adaptive Lobatto quadrature algorithm (Gander and Gautschi, 2000). 5 Simulation results 5.1 Simulation setup The three different models presented in the preceding sections are simulated using the New European Drive Cycle (NEDC) and Federal Test Procedure (FTP75). The models do not contain brakes, or a clutch, and the gear shifting points are prescribed, which means that the model does not change gear at a given engine speed, but at a fixed moment in time. For the computation of the fuel consumption only the part of the drive cycle where the engine is providing force to the vehicle is relevant. The simulation models do not describe braking and idling, during these periods the model switches to an idle consumption. The total fuel consumption is computed as the integral of the maximum of the actual fuel mass flow and this idle consumption, which is set to [kg/s]. All simulations are carried out in Matlab/Simulink using the ode45 solver with a relative tolerance of The maximum step length for the IDM is set to 1 s in order to get trajectories more similar to the forward dynamic simulation. For the dynamic models, two different driver models are used (see Table 1). These settings are for comparison reasons taken from Fröberg (2005) and are designed such that they have comparable response time. Table 1 Driver model parameters used for simulations dynamic models Driver model Forward Inverse 1 K p = K i = a =5 2 K p = K i = a =1

13 Analysis of modelling and simulation methodologies Comparison of the simulation methods Tracking of the driving cycle is different for the three simulation methods, as is shown in Figure 4 on a part of the NEDC cycle. The BQM velocity exactly matches the driving cycle, whereas the IDM shows exact tracking on most parts of the driving cycle, except at discontinuities where the driver model smoothes the driving cycle. In both the BQM and IDM, tracking of the drive cycle is independent of the power train modeled, thus changing configuration or parameters does not influence tracking behaviour, unlike in the forward dynamic method. Figure 4 Velocity on a part of NEDC cycle Figure 5 shows the effective throttle area A eff (t) on a part of NEDC, which forms the input of the power train model. The inverse method shows a nice smooth throttle signal in contrast to the forward method, which shows oscillations. This is due to the fact the forward dynamic method uses a PI controller to control the non-linear power train model. The oscillations occur during vehicle standstill and have no physical meaning, since idling behavior is not included in this basic model. In a more extensive model this can be resolved by opening the clutch during these periods and including an idling mode for the engine model. The fuel mass flow on a part of the NEDC cycle is shown in Figure 6. It can be observed that the trajectories ṁ f (t) for all three methods are quite similar. All three methods produce nearly the same result for the fuel consumption estimation on both the NEDC and FTP75 (maximum difference less than 4%), as can be seen in Table 2. These results show that tighter tracking of the driving cycle, i.e., using driver model 2 instead of driver model 1, leads to a higher fuel consumption. Besides, FDM leads to slightly higher fuel consumption compared to IDM. On the NEDC cycle, driver model 2 gives the most similar results for the three methods, whereas on the FTP cycle driver model 1 does. Although, the BQM cannot describe the dynamics of the intake manifold pressure, it can predict the fuel consumption quite accurate, which

14 130 T. Hofman et al. Figure 5 Effective throttle area during a part of NEDC Figure 6 Fuel mass flow during a part of NEDC confirms that the inlet manifold dynamics of an SI engine are not relevant for the estimation of fuel consumption, as stated in Guzzella and Sciarretta (2005). Table 3 shows the simulation time and number of computation steps required to obtain the fuel consumption results. The recorded simulation times of the different Simulink models include all actions starting from the driving cycle resulting in the calculation of the fuel consumption. Except for the inverse dynamic model, where the filtering by the driver model is done off-line. Compared to the forward dynamic

15 Analysis of modelling and simulation methodologies 131 Table 2 The influence of the simulation method on the fuel consumption Method Driver model NEDC [L/100 km] FTP [L/100 km] Quasi-static (BQM) n.a % % Forward (FDM) % % Inverse (IDM) % % Forward (FDM) % % Inverse (IDM) % Table 3 Simulation time and number of computation steps for the different methods Method Driver model Cycle No. of steps Time [s] Quasi-static (BQM) n.a. NEDC % FTP % Forward (FDM) 1 NEDC % FTP % Forward (FDM) 2 NEDC % FTP % Inverse (IDM) 1 NEDC % FTP % Inverse (IDM) 2 NEDC % FTP method, the backwards quasi-static and IDM offer a significantly higher simulation speed. This is favourable in case of power train design and optimisation processes requiring a large number of simulations. However, the differences in simulation time between the FDM and IDM on both the NEDC and FTP75 are significantly smaller (factor of 100) compared to the results as shown in Fröberg and Nielsen (2008), and Fröberg (2005). By limiting the maximum step size to 1 s for the IDM, the difference in the number of simulation steps between the FDM and IDM is also significantly smaller. In order to explain this more detailed information regarding the modelling and simulation set up is required. 5.3 Engine scaling To investigate the accuracy of the scalable engine model presented in Section 2.1, this model will be compared with measured data of actual engines with different displacement volumes. These engines are the Geo Metro 1.0L (41kW) SI engine and the Saturn 1.9L (63kW) SOHC SI Engine. This measured data is originating from the Advisor files FCSI41emis and FCSI63emis. The routine described in Section 3.2 is used to generate a fuel consumption map, using the torque and speed values from the data files. Engine scaling is done by changing the engine bore B, stroke S and inertia J e. The measured and generated maps for both engines are shown in Figure 7. This figure shows that the fuel consumption values and distribution of these maps are quite similar for both engines. Next, to compare the engine maps in a more quantitative manner, the measured and generated fuel consumption maps are inserted in the BQM to compare the fuel consumption on both the NEDC and FTP75 cycle. For these simulations the vehicle mass (and frontal area for 1.0L case) and transmission gear ratios are adjusted.

16 132 T. Hofman et al. Figure 7 Engine maps 1.0L engine (top) and 1.9L engine (bottom) (see online version for colours) The results are shown in Table 4. These results show that the generated map is sufficiently accurate (error up to 4.6%) for the engine operating points on both the NEDC and FTP75 cycle. Table 4 Fuel consumption [l/100km] on NEDC and FTP75 of quasi-static model with both measured and generated fuel consumption map NEDC FTP Engine Measured Generated Error Measured Generated Error 1.0L % % 1.9L % % 6 Conclusion In this paper, three different modelling and simulation methods (forward dynamic, quasi-static backwards, and inverse dynamic) were analysed and compared.

17 Analysis of modelling and simulation methodologies 133 Simulation of a power train model consisting of a dynamic engine model and a stiff drive train showed a maximum relative difference in fuel consumption of 3.9% for the three simulation methods. Comparing the recorded simulation times shows that the time required to run a simulation with the FDM is between 3 and 15 as long as for the BQM. In addition, the time required to run a simulation with the IDM is between 0.4 and 1.4 as long as for the BQM. Simulation results with a scaled dynamic engine model and an engine efficiency model based on measured data showed a maximum relative error of 4.6% for the engine operating points on both the NEDC and FTP75 cycle. Future work will focus on modelling, simulation and design methods, which can be used for quick hybrid drive train specification (including scalable electrical components, batteries, electric machines), or (supervisory) control calibration with sufficient accuracy. References Bosgra, O. ( ) Lecture Notes Physical Modelling for Systems and Control (4k560), Lecture Notes, Eindhoven University of Technology, Bouscayrol, A. (2009) Tutorial Hardware-In-the-Loop (HIL) Simulation, EVS 24 Stavanger. Fröberg, A. (2005) Extending the Inverse Vehicle Propulsion Simulation Concept to Improve Simulation Performance, Licentiate Thesis, Linköping Institute of Technology, Linköping, Sweden. Fröberg, A. and Nielsen, L. (2008) Efficient drive cycle simulation, IEEE Transactions on Vehicular Technology, Vol. 57, No. 3, pp Gander, W. and Gautschi, W. (2000) Adaptive quadrature revisited, BIT, Vol. 40, No. 1, pp Gao, D., Mi, C. and Emadi, A. (2007) Modeling and simulation of electric and hybrid vehicles, Proc. of the IEEE, Vol. 95, No. 4, pp Guzzella, L. and Sciarretta, A. (2005) Vehicle Propulsion Systems Introduction to Modeling and Optimization, Springer-Verlag, Berlin Heidelberg. Heath, M. (2002) Scientific Computing: An Introductory Survey, McGraw-Hill, New York. Hendricks, E., Chevalier, A., Jensen, M., Sorensen, S., Trumpy, D. and Asik, J. (1996) Modelling of the Intake Manifold Filling Dynamics, SAE Technical Paper Series SP-1149, pp Heywood, J. (1988) Internal Combustion Engine Fundamentals, McGraw-Hill, New York. Hofman, T., Steinbuch, M., van Druten, R. and Serrarens, A. (2008) Hybrid component specification optimization for a medium-duty hybrid electric truck, International Journal of Heavy Vehicle Systems, Vol. 15, Nos. 2 4, pp Hofman, T., Steinbuch, M., van Druten, R. and Serrarens, A. (2009) Design of CVT-based hybrid passenger cars, IEEE Transaction on Vehicular Technology, Vol. 58, No. 2, pp Khalil, H. (2002) Nonlinear Systems, 3rd ed., Prentice-Hall, Upper Saddle River, New Jersey. Slotine, J-J. and Li, W. (1991) Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, New Jersey. Soltic, P. (2000) Part-Load Optimized SI Engine Systems, PhD Dissertation, Swiss Federal Institute of Technology.

18 134 T. Hofman et al. Verdonck, N., Sciarretta, A. and Chasse, A. (2008) Automated model generation for hybrid vehicles within a hardware-in-the-loop environment, Proc. of Les rencontres scientifiques de l IFP Advances in Hybrid Powertrains, Paris, France, pp Wipke, K., Cuddy, M. and Burch, S. (1999) ADVISOR 2.1: a user-friendly advanced powertrain simulation using a combined backward/forward approach, IEEE Transactions on Vehicular Technology, Vol. 48, No. 6, pp Vuik, C., van Beek, P., Vermolen, F. and van Kan, J. (2004) Numerieke Methoden voor Differentiaalvergelijkingen, Delft Institute of Applied Mathematics, Delft, The Nederlands. Nomenclature A A ( eff A ) F S B BMEP c d c r F FMEP g IMEP g i J e m m ac ṁ at m f n r p a p e p i p r PMEP q hv r R s i S T a T i T cr T e T w Vehicle cross sectional area Effective throttle area Stoichiometric air fuel ratio Engine bore Brake mean effective pressure Air drag coefficient Rolling resistance coefficient Wheel traction force Friction mean effective pressure Gravitational acceleration Gross indicated mean effective pressure Gear ratio Engine inertia Vehicle mass Engine air charge Air mass flow past throttle Engine fuel charge Revolutions per engine cycle Ambient pressure Exhaust manifold pressure Intake manifold pressure Pressure ratio Pump mean effective pressure Fuel heating value Wheel radius Gas constant Slope of normalised air charge Engine stroke Ambient temperature Intake manifold temperature Crank shaft torque Engine flywheel torque Wheel torque

19 Analysis of modelling and simulation methodologies 135 v V d V i y i γ η th η tm η vol λ ξ aux π bl ρ ω e ω w Vehicle speed Engine displacement volume Intake manifold volume y-intercept of the normalised air charge Specific heat ratio Thermal efficiency Transmission efficiency Volumetric efficiency Normalised air fuel ratio Friction coefficient Boost lay-out Air density Engine rotational speed Wheel rotational speed Appendix 1 Parameter values Quantity Value Dimension A 2.5 [m 2 ] ( A ) 14.8 [ ] F S B 0.09 [m] c d 0.33 [ ] c r 0.01 [ ] g 9.81 [m/s 2 ] i [ , , , , ] [ ] J e 0.2 [kg m 2 ] m 1700 [kg] n r 2 [ ] p a [N/m 2 ] p e [N/m 2 ] q hv [J/kg] r 0.3 [m] R 286 [J/(kgK)] s i 0.95 [ ] S 0.09 [ ] T a 293 [K] T i 293 [K] V d [m 3 ] V i [m 3 ] y i [ ] γ 1.26 [ ] η th [ ]

20 136 T. Hofman et al. Appendix 1 (continued) Parameter values Quantity Value Dimension η tm 0.9 [ ] λ 1 [ ] ξ aux 1.34 [ ] π bl 1 [ ] ρ 1.29 [kg/m 3 ] Appendix II Parameter coefficients c 1 = c 2 = c 3 = 1 2 c dρa m + Jei2 ηtm r 2 iη tm ( Si πr ( ) ( r m + Jei2 η tm 4π r 2 V d ( iη tm ( ) ( r m + Jei2 η tm 4π r 2 V d ) 1.8 πbl 10 5 ξ aux B ) ξ aux B ) s iq hv η th +1 ( A F ) λrt i s ) ξ aux ( ) y iη iq hv η th tm π ( c 4 = A F ) λrt bl ξ aux i B + p e s ( ) ( ) r m + Jei2 η tm 4π r 2 V d ξ aux B + mgc r m + Jei2 η tm r 2 ) B c 5 = iy iv d 2πn r rv i c 6 = is iv d 2πn r rv i c 7 = RT ip a V i RTa.