Optimal cold start calibration of spark ignition engines

Transcription

1 Milano (Italy) August 28 - September 2, 211 Optimal cold start calibration of spark ignition engines Farzad Keynejad Chris Manzie Department of Mechanical Engineering, The University of Melbourne, Australia ( keynejad@unimelb.edu.au, manziec@unimelb.edu.au). Abstract: This paper investigates optimal strategies for minimising the conflicting objectives of fuel consumption and warm up duration following an engine cold start. Using a reduced order model, parametric solutions for the optimal air-fuel ratio, valve timing, cam timing and idle speed setpoint are sought in the presence of input constraints. The results are then validated using a dynamic programming approach on a high fidelity engine model, and compared experimentally with a production calibration for an inline 6-cylinder, SI gasoline engine. The methodology used can be readily extended to other mixed objective problems, and can be a useful tool in decreasing engine calibration time. 1. INTRODUCTION Extensive calibration of spark ignition engines is required to meet both emissions legislation and to ensure smooth engine operation in a wide range of operating conditions, and represents one of the most time consuming phases of vehicle development. Consequently, there has been considerable interest in the past to use both model based (for example see Prabhakar et al. (1977); Rishavy et al. (1977); Auiler et al. (1977); Rao et al. (1979); Dohner (1981); Tennant et al. (1983); Sun and Sivashankar (1997); Kang et al. (21); Maloney (29)) and model free (Popovic et al. (26)) approaches to automate the calibration process. However, this extensive body of work is largely characterised by engine-specific results and typically assumes warm engine operation only. Cold start operation is of significant interest as tailpipe emissions and fuel consumption are typically highest during this period of operation. With this in mind, recently there have been significant developments in mean value engine models that encompass a thermal aspect, as discussed in Roeth and Guzzella (21); Manzie et al. (29) and Keynejad and Manzie (21*). The existence of these models allows optimal control techniques to be employed for engine calibration over cold start conditions, which is useful for several reasons - firstly, experimental calibration over the cold start region is extremely slow due to temperature soaks limiting testing to often three cold starts per day; secondly, the increased dimensionality of the problem when temperature is included in the steady state maps used in calibration (a problem that is exacerbated by the increased functionality of modern engines in terms of numbers of actuators); and thirdly, there are conflicting objectives associated with engine cold start which arise from the desire to minimise fuel consumption, but also the need for sufficient thermal performance to ensure catalyst light-off and provide heat to the cabin. The advantage of using optimal control approaches on analytic engine models is that they provide a generic solution methodology, that can then be very quickly tuned for an individual engine using either simplified numeric techniques or reduced experimentation (or a combination of both). Here, to simplify the presentation of the methodology, the proposed calibration approach does not explicitly consider tailpipe emissions during cold start, instead assuming that these will map to temperature setpoints and constraints imposed on the engine control variables (such as spark timing and lambda during the warm up duration). Explicit consideration of emissions during cold start can be achieved by augmenting the engine model with emissions capability and an appropriate aftertreatment system model. Static approaches to emissions modelling (e.g. Hafner and Isermann (23); Fiengo et al. (23)) may be appropriate following light off, however models capturing catalyst dynamics (e.g. Andrianov et al. (21)) are likely to be required to enable accurate cold start emissions optimisation. The key contribution of this work is demonstration that analytic (parametric) results for a mixed objective optimisation problem (in this case fuel and warm up time) on a general engine enable the numeric (exact) solution for a specific engine to be found quickly. The implication is that engine calibration can be made a less time consuming process through the incorporation of the methodology, leading to reduced development costs for new vehicle models. 2. ANALYTIC OPTIMISATION To obtain parametric results, a three-state model from Keynejad and Manzie (21*) is used. This model was previously shown to be capable of capturing cumulative fuel consumption and thermal transients to within 5% of experimentally obtained data over the NEDC cycle. As they are critical to the analysis, the underlying model assumptions and equations are reproduced in this section. Assumption 1. The controllers responsible for maintaining air-fuel ratio, engine idle speed and cam timing provide perfect setpoint tracking at all times. Copyright by the International Federation of Automatic Control (IFAC) 1322

2 Milano (Italy) August 28 - September 2, 211 Assumption 1 decouples controller and engine dynamics to enable the best possible performance to be predicted irrespective of control structure employed. This assumption will be relaxed in the numerical analysis later, when local controllers are included. The engine inputs are constrained to limit imposed by emissions and driveability considerations. Consequently, the inputs to be analysed are air fuel ratio, β [β min, β max ; spark timing, θ sa [θ sa,min, θ sa,max ; valve overlap, θ vo [θ vo,min, θ vo,max ; and idle speed setpoint, ω idle [ω idle,min, ω idle,max. The following two assumptions will be made about the engine indicated and volumetric efficiency for engine inputs in these allowable ranges, which characterise low- and part-load operation: Assumption 2. Indicated efficiency, η i, increases monotonically with both spark advance, θ sa, and air-fuel ratio, β. Assumption 3. Volumetric efficiency, η vol, increases monotonically with engine speed, ω crank, and decreases monotonically with valve overlap angle, θ vo. Assumptions 2 and 3 imply a unique mapping between the engine input vector, [u, ω idle = [β, θ vo, θ sa, ω idle and a virtual input vector of schedulable setpoints, v = [β, η i, η vol, ω idle. 2.1 Engine model The outputs of interest when considering cold start fuel consumption are the fuel flow rate, Ṁ f, and a representative engine temperature, T rep. This latter output does not model the temperature of a physical state, but instead represents a single lumped temperature that characterises of all the solid and fluid masses in the engine system. Implicitly, this represents an assumption that the temperature of any physical state is monotonically related to T rep. In Keynejad and Manzie (21*) for example, the actual oil temperature is related to the representative engine temperature by ˆT oil = T amb + K (T rep T amb ), where K was an identified constant. The internal dynamics of the reduced order model can be written in terms of three states: the intake manifold pressure P im, a representative engine temperature, T rep, and the engine speed, ω crank, with dynamics given by the following equations: P im = RT amb [Ṁin (T amb, P amb, P im, α th ) V im Ṁout (T amb, P im, ω crank, u) T rep = 1 [ Q cyl,rep (T amb, P im, ω crank, u) C rep (1) + Q fr,rep (P amb, P im, ω crank, T rep ) Q rep,amb (T amb, T rep ) (2) ω crank = 1 [ τ ind (T amb, P amb, P im, ω crank, u) J total τ fr (P amb, P im, ω crank, T rep ) τ dc (3) In (1)-(3), there are boundary conditions set by the ambient pressure and temperature, P amb and T amb and the drive cycle, ω dc or τ dc. The time constants of the equations are given by the manifold volume and constant gas V properties, im RT amb, the engine inertia, J total, and the representative temperature equivalent heat capacitance, C rep. A consequence of Assumption 1 is the throttle is adjusted so that the net torque delivered precisely tracks the defined drive cycle torque, τ dc, (or equivalently, the engine speed ω crank always approximately matches the drive cycle speed, ω dc ) irrespective of the chosen engine controls u. In effect, this means that the effective area of the throttle is an increasing function of ω dc, represented by Ā (ω dc, ω crank ). These simpications means (3) effectively forms a constraint equation away from idle, rather than a dynamic equation requiring treatment in the analytical optimisation to follow. Note however, that under idle conditions the crank speed can be scheduled without violating Assumption 1, meaning that ω idle remains a schedulable input to the system. It is now useful to define K 1 := N cylv s 4π and K 2 := 1 RT amb, and represent the exhaust gas heat transfer as a function of engine speed, α(ω crank ) (it can be shown using Nusselt number arguments that α is monotonic in ω crank ). The right hand side terms of (1)-(3) can now be expressed as functions of known constants and static maps of the states and iinputs according to: Ṁ in (T amb, P amb, P im, α th ) = K 2 P amb Ā (ω dc, ω crank ) ψ Ṁ out (T amb, P im, ω crank, u) ( Pamb P im ) (4) βη vol (ω crank, θ vo ) = K 1 K 2 ω crank P im (5) Q cyl,rep = α (ω crank ) [1 η i (θ sa, β) Ṁ f Q lhv (6) Q fr,rep (P amb, P im, ω crank, T rep ) = K 1 P fme (P amb, P im ω crank, T rep ) ω crank (7) Q rep,amb (T amb, T rep ) = G rep,amb (T rep T amb ) (8) τ ind (T amb, P im, ω crank, u) η i (θ sa, β) η vol (ω crank, θ vo ) = K 1 K 2 Q lhv P im K 1 P pme (P amb, P im ) (9) τ fr (P amb, P im, ω crank, T rep ) = K 1 P fme (P amb, P im, ω crank, T rep ) (1) While T rep is one of the outputs required, the other output of interest in the reduced order model is the fuel used. Applying Assumption 1 to the air fuel ratio controller means the instantaneous fuel flow rate follows from (5): Ṁ f (P im,ω crank, u) = 1 β Ṁout 2.2 Cost function η vol (ω crank, θ vo ) = K 1 K 2 ω crank P im (11) As discussed earlier, cold starting an engine may be considered a tradeoff between minimising fuel use and minimising the time to reach a desired temperature. Furthermore, the relative importance of fuel use and warm up duration may 1323

3 Milano (Italy) August 28 - September 2, 211 change depending on the circumstances since fuel penalties are associated with faster warmup which is unavoidable in the case of emissions, but a small fuel penalty may be tolerable if cabin heat is requested on a cold day. To capture this tradeoff, the following cost function is proposed: tf J = w F Ṁ f dt + (T des (t f ) T rep (t f )) 2 (12) The constant w F represents a weight applied to the fuel use relative to the warm-up time, while T des (t f ) represents the desired temperature at the end of period of optimisation. In the next section, the set of virtual control variables v (and by implication the actual inputs u and ω idle ) that optimise the cost function (12), subject to the plant equations of (1) - (11) and input constraints are sought. 2.3 Parametric solution for optimal cold start engine control The Hamiltonian for the reduced order system can be expressed in terms of the states, x = [P im, T rep, ω crank, Lagrange multipliers, p = [p 1, p 2, p 3 T, and virtual inputs, v, as: H(x, v, p) = w F Ṁ f + p 1 P im + p 2 T rep + p 3 ω crank (13) Substitution of (1), (2) and (11) leads to the expression of the Hamiltonian as: η vol H(x, v, p) = w F K 1 K 2 () ω crankp im + p [ K2 ( ) 1 P ambā K 2 V (ω Pamb dc, ω crank) ψ im P ( ) im βηvol K 1 K 2 ω crank P im + p [ 2 α (ω crank ) (1 η i ) η vol K 1 K 2 Q lhv ω crank P im C rep +K 1 P fme (P amb, P im, ω crank, T rep ) ω crank G eng,amb (T rep T amb ) + p [ 3 η i η vol K 1 K 2 Q lhv J total () P im K 1 P pme (P amb, P im ) K 1 P fme (P amb, P im, ω crank, T rep ) τ dc (14) For the derived policy to be optimal (denoted by optimal inputs, v, states, x and costate vector, p ), the following necessary conditions apply from Kirk (197): ẋ (t) = p H (x (t), v (t), p (t), t) (15) ṗ (t) = x H (x (t), v (t), p (t), t) (16) H (x (t), v (t), p (t), t) H (x (t), v(t), p (t), t) (17) To establish the optimal policy for the virtual inputs, v, only constant idle operation will be considered in entirety due to space constraints, however a discussion of the more general case will also be included. To begin, the following quantities are defined for brevity of the proceeding arguments: γ := w F K 1 K 2 ω crankp im (18) γ 1 := K 1ω crank P im V im (19) γ 2 := K 1K 2 Q lhv α (ω crank ) ω crank P im C rep (2) γ 3 := K 1K 2 Q lhv Pim (21) J total ( ) P amb ψ Pamb Pim γ 4 := (22) V im K2 Applying the necessary conditions of (17) with the Hamiltonian of (14) leads to the following inequality which must hold for optimality: [γ γ 1 p 1β + γ 2 p 2 (1 ηi ) + γ 3 p 3ηi ηvol β γ 4 p 1Ā (ω idle, ωcrank) [γ γ 1 p 1β + γ 2 p 2 (1 η i ) + γ 3 p η vol 3η i + γ 4 p 1Ā (ω idle, ωcrank) (23) This sets up four different switching conditions for each of the four virtual control inputs. These switching conditions form surfaces in the p-space, which can be derived as shown in Table 1. To aid in visualising these results, the conditions in the p 1 -p 2 plane for some arbitrary p 3 > are illustrated in Figure 1. The trajectory through the p-space is given by the solution of (16), which is nontrivial explicitly as this represents three coupled equations with time-varying coefficients. However, some important general aspects of the solution can be implied from the structure of the regions outlined in Table 1: From Assumptions 2 and 3 the virtual control inputs can now be mapped back to the physical controls. Maximising indicated efficiency corresponds to maximum spark advance, i.e. running at MBT spark, and minimising indicated efficiency corresponds to maximum retard from MBT. Similarly maximising volumetric efficiency corresponds to minimising valve overlap in order to minimise internal exhaust gas recirculation. Each of the control inputs will switch from low to high efficiency operation at different times for optimality of the solution. For realistic values of the parameters γ,..., γ 5, there will be at most one switch of each control inputs. The resulting control trajectories consist of: 1) a maximum heat production policy where the engine runs at it s most inefficient characterised by u = [β min, θ vo,max, θ sa,min, ω idle,max ; 2) a maximum efficiency policy where the engine runs to conserve fuel with u = [β max, θ vo,min, θ sa,max, ω idle,min ; and 3) a transition policy, which covers the switching of the control variables between the other two policies. The shape of the overall control trajectories is illustrated in Figure 2. The weighting on fuel use, w F, is directly related to the size of the region corresponding to the maximum efficiency policy in the co-state space, since it appears in γ. 1324

4 Milano (Italy) August 28 - September 2, 211 Table 1. Switching conditions for virtual engine controls as a function of co-state variables Condition If p 1 > Result ωidle = ω idle,min a consequence, the engine controls will switch earlier in time. 3. IMPLEMENTATION AND RESULTS If p 1 < ω idle = ω idle,max 3.1 Simulation results p 2 If p 2 > γ 3 γ 2 p 3 If p 2 < γ 3 γ 2 p 3 If p 2 > γ +γ 1 p 1 β max γ 3 p 3 η i γ 2 (1 η i ) η i = η i,max η i = η i,min [η vol, β = [η vol,min, β max If p 2 < γ +γ 1 p 1 β max γ 3 p 3 η i γ 2 (1 η i ) [η vol, β = [η vol,max, β min p 2 = $ 3 p 3 / $ 2 # i,max # i,min " max # vol,min! idle,max! idle,min " min Maximum heat production region p 1 Maximum efficiency region # vol,max Fig. 1. One two-dimensional segment (for arbitrary p 3 > ) of the decision planes for the virtual engine controls in the co-state space 7-1*,-26'!"#$%&%'()"*'' +,-.&/-1'+-2$/3' β min θ vo,max θ sa,min ω idle,max 5,"16$-1' +-2$/3' β max θ vo,min θ sa,max ω idle,min!"#$%&%')4/$)1/3' '+-2$/3' 5$%)' Fig. 2. General optimal cold start strategy for a SI engine If non-idle operating conditions are considered, the same general results eventuate, although the progression of the co-state variables in time is clearly different. This is completely intuitive, as running the engine away from idle will naturally generate more heat, leading to less need to generate heat by running the engine a low efficiency. As To validate the optimality of the results via simulation, a high-order engine model (also from Keynejad and Manzie (21*)) is used along with local PI controllers on each of the actuators. A modified dynamic programming algorithm was used to determine the optimal control trajectories given the cost function (12) with the modification T rep = T coolant, as the coolant temperature is a state in the higher order model. Two cases were considered, the first being an optimal fuel policy with no consideration of temperature (i.e. w F >> 1) while the second case uses a balanced policy that attempts to match the engine coolant temperature of the modelled engine at idle after 3 seconds running using the production engine calibration, (i.e. T coolant (3) = 325K), with the weight on total fuel use set to w F = 1. The resulting state and output trajectories are shown in Figure 3, where it is clear that there is a significant fuel penalty relative to the best possible to meet the temperature requirement implied by the production temperature setpoint after 3 seconds. The inputs to achieve these state trajectories are in Figure 4 and it is seen the analysis in Section II holds. For the fuel optimal case, the engine operates at maximum efficiency at all points in time. Meanwhile, when a balanced objective is desired, the engine starts off running inefficiently and the control variables switch at different points in time. Although not shown explicitly here, small perturbations about the derived strategy result in higher cost calculated using (12). This provides some further validation that the developed policies are at least locally optimal for the given problem formulation. The importance of this validation is that once the control problem is formulated, a reduced computation numerical optimisation can be performed to either: (1) Search for the p() that satisfies the boundary conditions t [, t f and develop the control policy using (15)-(16) and Table 1, or alternatively; (2) Search only for the switching times of each of the control variable using a dynamic programming approach. 3.2 Experimental results To obtain experimental validation of the results, a 4.l, inline 6-cylinder production engine was connected a 46kW transient engine dynamometer and the production calibration was adjusted via an ATI Vision interface. The production engine has no actuation authority of the cam at low temperature so only three engine control inputs were used. An optimal fuel policy was derived by minimising (12), with the temperature under the production calibration (which meets legislated emissions levels) used to set the reference level after 3s. For comparison purposes, an 1325

5 Fuel use [g Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 211 %(! %%! The engine was run for 3 seconds at constant idle conditions, using each of the three (two optimal and one production) policies. The results for measured temperatures and fuel consumption are show in Figure 5. %$! ),+1,23452,"-67 %#! -67"" %!! Coolant temperatures (solid) Fast warm up policy $'! Oil temperatures (dashed) $&!! #!!" $!!" %!!" )*+,"-.,/"! Temperature [ K [ K 3 Minimum fuel policy Production calibration Time [sec Time [sec Fig. 3. Simulation results at idle for minimum fuel policy (grey lines) and faster warm up policy achieving T coolant (3s) = 325K (black lines). Fuel use [g [g Minimum fuel policy Fast warm up policy Engine idle speed [rpm Cam angle [CA AFR/AFRs Spark angle [CA Time [ sec Fig. 4. Optimal idle engine inputs obtained numerically from simulation for minimum fuel policy (grey lines) and faster warm up policy achieving T coolant (3s) = 325K (black lines). additional balanced policy that achieved 1K higher temperature at the final time was also sought. 4 2 Production calibration Time [sec Fig. 5. Experimental results at constant idle operation under the three policies. (Top) Coolant (solid) and oil (dashed) temperatures, (Bottom) Cumulative fuel consumption. Not surprisingly, when the production calibration and the optimal fuel consumption policy are compared in Figure 5 the overall results differ only marginally. The oil and coolant temperatures at the end of the interval match well, validating the use of a single representative engine temperature state in the analysis. Similarly, the cumulative fuel use under both policies is very similar, although the derived optimal policy has only marginally better fuel consumption. This closeness of solutions is not unexpected since given the development effort typically associated with engine calibration it is reasonable to assume that the outcome will be at least very close to locally optimal. This result provides some real world validation of the derived policy. The faster warm up policy represented by the black line in Figures 5 (and which can be considered as representative of tighter emissions constraints) also clearly demonstrates 1326

6 Milano (Italy) August 28 - September 2, 211 that there is an associated fuel penalty. Although not shown here due to space constraints, testing was also conducted over the NEDC drive cycle. Under this scenario, the fuel penalty associated with the higher temperature setpoint is less pronounced, because additional heat available when running the engine at higher speeds and loads facilitates an earlier transition from the maximum heat policy to the maximum efficiency policy. Finally, it is worth noting is that the time to develop the faster warm up policy using the methodology outlined in this paper is significantly shorter. Thus once the development time associated with calibrating the engine models is invested, it is relatively straightforward to generate new policies to suit any desired objective. This implies that different optimal policies can be generated for different conditions (e.g. when the cabin heater is requested, it may be desirable to sacrifice some initial fuel to achieve faster warm up) and the look up tables in existing engine controllers automatically populated. In essence, this is similar in spirit to explicit model predictive control approaches. 4. CONCLUSIONS AND FURTHER WORK The proposed approach to engine cold start calibration is to use a low order model to develop parametric strategies, which can then be used to reduce the computational burden associated with numerical approaches on higher fidelity engine models. Of course, it is possible that the calibration time associated with the development of the models may be significant, and thus exact identification of the optimal strategy may be significant. In this instance, there may be benefits in incorporating both a model based approach to optimal calibration that uses less accurate engine models that are faster to define, and coupling this with model free optimisation approaches (such as Popovic et al. (26); Nesic et al. (21a) or Nesic et al. (21b)) that can fine tune the solution to find a local optima. Further extensions include with augmenting the reduced order model with static maps representing engine-out emissions and modelling the three-way catalyst thermal and chemical dynamics in a control oriented fashion. The augmented model can then be used to set additional explicit constraints when optimising the cost function (12). ACKNOWLEDGEMENTS The authors acknowledge the ARC funding of Linkage Project LP and the in-kind support provided by the Ford Motor Company of Australia. Furthermore, the resources within the ACART research centre ( were instrumental to the experimental work. REFERENCES Andrianov, D., Keynejad, F., Dingli, R., Voice, G., Brear, M., Manzie, C., 21. A cold-start emissions model of an engine and aftertreatment system for optimisation studies (SAE Paper ). Auiler, J. E., Zbrozek, J. D., Blumberg, P. N., Optimization of automotive engine calibration for better fuel economy methods and applications. SAE Paper Dohner, A. R., Optimal control solution of the automotive emission-constrained minimum fuel problem. Automatica 17 (3), Fiengo, G., Glielmo, L., Santini, S., Serra, G., 23. Control oriented models for TWC-equipped spark ignition engines during the warm-up phase. In: American Control Conf. Hafner, M., Isermann, R., 23. Multiobjective optimization of feedforward control maps in engine management systems towards low consumption and low emissions. Trans. Inst. of Measurement & Control 25, Kang, J. M., Kolmanovsky, I., Grizzle, J. W., 21. Dynamic optimization of lean burn engine aftertreatment. 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