Overview

The SI combustion model follows a structure similar to that of [46], with extensive modifications for additional functionalities and better agreement with data. The basic premise of the model is to carry out a modified Otto cycle, with polytropic compression and expansion processes and an instantaneous combustion. The model’s most challenging tasks are to obtain the necessary quantities for the thermodynamic cycle calculation to be carried out, and to introduce realistic effects into the idealized Otto cycle so that the model can accurately predict outputs seen in experiment.

Cycle k-1 Cycle k

EVO(k-1)

IVO(k-1) EVC(k-1) IVC(k-1) Crank

Angle

Cylinder Pressure

EVO(k) IVO(k) EVC(k) IVC(k)

EVO(k-2) Cycle k+1 Control Action k Control Action k-1

Exhaust Valve Profile Intake Valve Profile [mf(k-1) θsp(k-1) θevc(k-1) θivo(k-1)] [mf(k) θsp(k) θevc(k) θivo(k)]

Figure 2.7: Cycle definition of the SI combustion model.

following the mean value modeling approach. Note that this approach assumes that the engine speed is known and changes slowly relative the cycle duration. The combustion model cycle is divided at the exhaust valve opening (EVO) event as depicted in Fig. 2.7. It is clear from the diagram 2.7 that with this cycle definition, the EVC timing for the current cycle is dictated by the EVO timing for the past cycle,

θevc(k) = θevo(k − 1) + ∆EV (2.34)

where ∆EV is the exhaust cam duration and k is the cycle index. This EVC convention is not a consequence of the valve event’s underlying physical process, but rather a necessity for implementation of the mean value model in simulation. Also indicated in Fig. 2.7 are the instants where control inputs can be calculated and applied to the combustion model, so the interface between a controller and the model can be seen. Each control action is assumed to occur at the end of the combustion model cycle, so that the cycle completes and any combustion outputs can be fed back to the controller to calculate the control inputs to be applied to the next cycle. Note that the valve timings θevc and θivo are technically states and not input commands due to the actuator

dynamics (see Sec. 2.2.3), but are taken to be externally specified by the air path model and so are input once per cycle as with the other input variables. The injection timing θsoi is not counted

amongst the inputs to the SI model as it is not used in control of the SI combustion mode, as it has been found to have minimal effect on combustion for a wide range of timings in the intake stroke. Lastly, note that both the intake and exhaust valves in Fig. 2.7 follow the high lift profile with PVO; in a later Section, SI/HCCI mode transitions will be considered where the SI combustion operates with a low lift intake cam profile and high lift exhaust cam profile, which is chosen based on the

mode transition strategy. For SI/HCCI transitions with that strategy, the SI combustion model will be parameterized assuming a low lift intake cam profile, and any adjustments to the model with the high lift intake profile presented here will be described.

Before going onto describe the combustion model equations, we first note the following assump- tions employed by the combustion model.

Model Assumptions

1. The mixture can be treated as an ideal gas with constant specific heats. The specific heat during the intake and compression strokes is equivalent to the specific heat of atmospheric air; the specific heat of the gases during combustion is indeterminate, and can be chosen to give a reasonable temperature rise due to combustion.

2. The compression and expansion strokes can be treated as polytropic processes with a constant polytropic coefficient, and the combustion can be treated as constant volume, adiabatic heat addition.

3. The point of instantaneous combustion is taken at |θ50− θ50,M BT| as in [49], where θ50 is

the crank angle where 50% of the fuel mass has burned and θ50,M BT is the θ50 timing for

maximum brake torque, taken = 7◦ aTDC. This selection logic causes the cylinder volume during combustion to be larger as θ50 moves further from its max brake torque position, which

causes the extracted work and hence torque output to reduce in a trend similar to that seen in experiments as spark is advanced/retarded from its optimal position.

4. Cylinder to cylinder variations can be neglected - the combustion model is parameterized to a single cylinder only.

5. Cycle to cycle couplings are negligible in the SI combustion mode due to a low quantity of recycled exhaust gas. The combustion can be modeled as a static nonlinear mapping that is independent of previous cycles, and so contains no states. To justify this assumption, it is noted that the maximum residual gas fraction as processed from steady-state actuator sweep data in SI/HCCI transtion relevant conditions is ≈ 25%, and will in general tend to be lower than this value because of the EVC placement necessary to attain the residual gas fraction this large. Also note that this is only about half of the minimum residual gas fraction observed in HCCI steady-state sweep data of ≈ 47%.