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Optimization of complex powertrain systems for fuel economy and emissions Ilya Kolmanovsky ∗ , Michiel van Nieuwstadt, Jing Sun Ford Research Laboratory, Dearborn, MI, USA Received 20 December 1998

Keywords: Automotive applications; Powertrain control; Optimization; Optimal control

1. Introduction Rising customer expectations for fuel economy and performance, tightened government emission regulations and increasing competitive pressures have forced the automotive industry to seriously consider and introduce advanced powertrain technologies. To illustrate the extent of growing environmental pressures, consider the European legislation. In 1992 Stage I regulations imposed on passenger gasoline cars limited the combined emissions of nitric oxides (NOx ) and hydrocarbons (HC) to 0.97 g/km. The emissions are collected and averaged over an 11 km drive cycle that simulates urban and suburban driving conditions and lasts for about 1200 s, see Fig. 1. The Stage IV emission regulations that become eective around the year 2004 aim at limiting emissions of HC to 0.1 g/km and emissions of NOx to 0.08 g/km, a reduction by more than a factor of 5 as compared to Stage I! Analogous trends exist for fuel economy, cost and performance requirements. Examples of advanced technologies that are called to the rescue include hybrid electric vehicles, gasoline direct injected engines, fuel cells, alternative fuel vehicles and continuously variable transmissions. The ability to rapidly assess feasibility (in terms of emissions, fuel economy, performance and cost), perform cost/bene t analysis and de ne subsystem level speci cations for a proposed powertrain system is very important early on in the design stage. The assessment of powertrain feasibility is clearly not possible unless not only its physical ∗ Correspondence address: 20000 Rotunda Drive, SRL, MD 2036, P.O. Box 2053, Dearborn, MI 48121-2053, USA. Tel.: +1-313-845-1040; fax: +1-313-845-0962. E-mail address: [email protected] (I. Kolmanovsky).

1468-1218/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 2 1 2 - 6

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Fig. 1. Vehicle speed pro le (km/h) versus time (s) over the Euro cycle.

parameters but also its operation, including control strategy, are optimized. It is this aspect of the problem, optimization of powertrain operation to improve fuel economy and emissions, that we concentrate on here. An assumption underlying our discussion is that predictions of a new powertrain system response to control strategy changes can be made from either experimental data or a simulation model. There does exist a signi cant body of literature directed towards generation of optimized powertrain strategies or optimized calibrations. Representative approaches rely on the use of the Lagrange Multiplier Rule [4,12,14,16], Dynamic Programming [2], Integer Linear Programming [15], solving Pontryagin maximum principle experimentally [7] and direct optimization [19]. These references make extensive use of the “quasi-static” assumption that substantially simpli es the optimization problem. This “quasi-static” assumption is that the internal combustion engine fuel consumption and feedgas emission values at any given time instant are static functions of engine speed, engine torque and control variables (such as fueling rate, spark timing, EGR valve position, etc.) at the same time instant. Steady-state engine mapping data generated experimentally or from a high delity simulation model are typically used to develop these static functions. Although the “quasi-static” assumption may not be accurate during the cold-start [6], it is accurate for establishing trends and relative eects for the warmed-up operation, see the discussion in [15]. Once the optimized strategies have been generated under the “quasi-static” assumption the actual numbers for emissions and fuel economy are, typically, validated either experimentally or on a more detailed powertrain simulation model that incorporates transient eects. This “quasi-static” assumption is also used as a basis for several simulation and modelling packages, see e.g. [5,9,23].

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Fig. 2. Partitioning the engine speed/engine torque plane into cells.

To illustrate the use of the “quasi-static” assumption consider, for example, the optimization procedure for a conventional port-fuel injected spark ignition engine equipped with the three-way catalyst (TWC). Similar treatment is possible for conventional diesel engines. First, the engine speed and engine torque trajectories are derived from the vehicle speed pro le in Fig. 1 using the estimates of the vehicle mass, tire radius, aerodynamic drag coecient, frontal area, rolling resistance coecient, gear ratios, shift schedule, idle speed value and power losses in the drivetrain. The time trajectory of engine speed, N and engine torque, is then a prescribed 2-vector # " w1 (t) N = : w(t) = w2 (t) Next, we partition the engine speed/engine torque plane into rectangular cells C i ; i = 1; : : : ; M , see Fig. 2. Let T (i) be the total time the engine spends in the cell C i over the drive cycle while wi is the speed/torque point corresponding to the center of the cell C i . For each of the cells C i we have to prescribe the values of the control inputs ui (fueling rate, spark timing, EGR rate, etc.) so that the fuel consumption over the drive-cycle is minimized while the emission constraints are met: M X

Wf (ui ; wi )T (i) → min

(1)

Wsj (ui ; wi )T (i) ≤ gj :

(2)

i=1

subject to M X i=1

Here Wf (u; w) denotes the fueling rate in gram per second, Wsj (u; w) is the mass ow rate of the jth regulated emission species (oxides of nitrogen (NOx ), carbon monoxide

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(CO) and hydrocarbons (HC)) into the tailpipe and gj is the emission limit for the jth species, j = 1; 2; 3. The representation of the objective function and constraints as a sum of independent terms has been made possible by the “quasi-static” assumption on the engine operation and a similar assumption on “quasi-static” behavior of the TWC conversion eciencies (valid for engine operation around the stoichiometry). Ref. [19] optimizes directly over the values of ui ; i = 1; : : : ; M . The approach followed in [4,12,14,16] is slightly dierent, based on the Lagrange duality and the observation that the objective function and constraints are separable functions. Hence, the problem is reduced to a two-stage optimization problem. In the rst stage, for each cell C i the cost function of the form Wf (ui ; wi ) +

j=3 X

j Wsj (ui ; wi ) → min

j=1

is minimized where j are the Lagrange multipliers. The same values of j are used for every cell. Hence, a calibration is generated that prescribes the values of the control inputs ui as functions of engine speed, engine torque and Lagrange multipliers j ; j = 1; 2; 3. The second stage of the optimization is to adjust the Lagrange multipliers to attempt to achieve the objectives (1) and (2). The feasibility is established if the constraints (2) are met. In this paper we describe two new case studies that lead to dynamic optimal control problems. The common feature of these two case studies is that fuel consumption and tailpipe emissions are determined not just by present operating conditions of the powertrain but also by the past operating history. This is because critical to operation of these powertrains are emission or energy storage mechanisms. Hence, the separability property that was crucial for ecient generation of the calibration for the conventional PFI engine is lost and the problem has to be treated as a dynamic optimal control problem. The rst case study (Section 2) concerns a gasoline direct injected engine equipped with a lean NOx trap. There is an emission storage mechanism in the trap due to NOx storage in the trap under some operating conditions and NOx release from the trap under some other operating conditions. The second case study (Section 3) focuses on a parallel hybrid powertrain con guration encompassing a turbocharged diesel engine and a supplemental electric motor/generator with a battery. Here an energy storage mechanism due to the battery operation is present. We also consider in Section 4 an example where both emission and energy storage mechanisms are present when a diesel particulate lter is added to the parallel hybrid con guration of Section 3. The diesel particulate lter is used to reduce the emissions of particulate matter. The solution approach relies on a xed structure optimization whereby reasonable powertrain operating policies are assumed and parametrized with a small number of parameters; then the values of the parameters are determined via a numerical solution of the resulting mathematical programming problem. This approach is adopted to make the problem computationally tractable and also to generate policies that do not depend explicitly on time. The latter property is necessary for the strategies to be implementable, see the discussion in [6] in the context of a cold start optimization problem for a PFI engine. In addition, optimized xed structure policies are often less

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Fig. 3. DISI engine.

sensitive to parameter variations and uncertainties then the actual optimal policies, see e.g. [8] for a discussion of this in the xed structure controller design context.

2. Optimizing DISI engine operation Our rst case study is a gasoline direct injection spark ignition. The DISI engine can operate at extremely lean overall air-to-fuel ratios (up to 40 : 1 as compared to 14:64 : 1 for stoichiometric operation) due to its ability to run strati ed. This reduces pumping losses and increases thermodynamic eciency of the cycle thereby improving fuel economy. The transition between homogeneous combustion mode and strati ed combustion mode is accomplished by changing the fuel injection timing from early injection to late injection. See [1,20] for more information on the operation of strati ed charge engines and refer to Fig. 3 for the schematics of the engine. During lean operation, however, the conventional three-way catalyst (TWC) is ineective in reducing NOx emissions and increased exhaust gas recirculation (EGR) rates and additional exhaust aftertreatment devices such as a lean NOx trap (LNT) are required. The LNT is only capable of trapping NOx and as it becomes lled its trapping eciency decreases. Hence, the LNT has to be periodically purged o the stored NOx in a manner that the stored NOx (pollutant) is converted to nitrogen and carbon dioxide. The purge is accomplished by reverting to an engine air-to-fuel ratio richer than stoichiometry for several seconds. Although in the lean conditions the TWC is not very eective for NOx reduction, it does remain very eective for HC and CO reduction. Our objective is to determine the best fuel economy that a given DISI engine and aftertreatment con guration can provide subject to the constrained NOx emissions and to determine the potential fuel economy advantage of operating the DISI powertrain lean with LNT as opposed to stoichiometric without LNT. We restrict our treatment to NOx emissions only to simplify the exposition and also because it is a more dicult control problem for the lean burn operation where the CO and HC emissions can be eectively handled by the TWC. To simplify the exposition we also assume the fully warmed up engine operation.

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2.1. DISI engine and aftertreatment modelling The model that we use for the optimization is based on the “quasi-static” assumption for the engine fuel consumption and feedgas properties while the dynamics are due to the exhaust aftertreatment (TWC and LNT) storage mechanisms. Speci cally, the model has the form x˙ = f(x; v);

(3)

y = H (x; v; w); where

" x=

xTWC xLNT

# ;

Wtot

T v= WNOx WCO

;

" w=

N e

#

" ;

y=

Wf Wˆ NOx

# ;

WHC and where xTWC is the mass of oxygen stored in the TWC, xLNT is the mass of NOx stored in the LNT, Wtot is the mass ow rate of the exhaust gas out of the engine (feedgas), is the feedgas air-to-fuel ratio, T is the feedgas temperature, WNOx ; WCO ; WHC are the mass ow rates of feedgas NOx , CO and HC out of the engine, respectively, N is the engine speed, e is the engine torque, Wf is the engine fueling rate and Wˆ NOx is the mass ow rate of NOx out of the tailpipe. The vector v represents engine exhaust feedgas properties and, under the “quasi-static” assumption, can be related to the engine operating variables by a static nonlinear model of the form v = r(u; w);

(4)

where the vector v is de ned as E u= Â ; tinj E is the exhaust gas recirculation (EGR) rate, Â is the spark timing and tinj is the injection timing. In the static model it is assumed that the fueling rate Wf is speci ed implicitly by prescribing the torque value e and u. The injection timing tinj can take one of the two discrete values that correspond to early injection for the homogeneous mode or late injection for the strati ed mode. The range of variables u is constrained due to limits on the intake manifold pressure to be less than the atmospheric pressure, knock and mis re limits and constraints on the feasible ow quantities through the electronic throttle and EGR valve. These restrictions can be represented by inequality

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211

constraints of the form h(u; w) ≤ 0:

(5)

The model (3) is based on the standard control-oriented representation for the oxygen storage dynamics of the TWC (see [3,17]) and NOx storage dynamics of the LNT (see [22]). See also [10,11] for more information about LNT operation and modelling, including the chemistry and the eects of temperature and sulfur poisoning on the LNT trapping eciency. Since the LNT trapping eciency depends on its temperature, a standard heat transfer model for the turbulent circular pipe ow connecting the TWC and the LNT has also been incorporated. To facilitate the application of ODE solvers and numerical optimization algorithms some adjustments to the structure of the models [3,17,22] have been made to ensure that the right-hand sides of the dierential equations are smooth. The static submodel (4) is described in a companion paper [20]. The objective of minimizing total fuel consumption over the speci ed drive-cycle subject to constraints on the tailpipe NOx emissions can be represented as Z

T 0

Z 0

T

Wf (t) dt → min; Wˆ NOx (t) dt ≤ gNOx ;

where T is the duration of the drive cycle and gNOx is the NOx emission constraint. 2.2. Operating policies The engine and aftertreatment operating policies are speci ed and parametrized using the following steps that involve generation of three dierent calibrations for normal mode, a calibration for purge mode and a transition policy between the normal mode and the purge mode. First, several calibrations for operating the engine in the normal mode are generated: homogeneous lean u = uhl (w; R) and strati ed lean u = stoichiometric u = ust (w; R), sl u (w; R) that can be used to trade-o NOx emissions out of the TWC estimated from “steady-state” TWC conversion eciencies 1 and fuel consumption at a speci ed engine speed/engine torque operating point w. The dierence in these three calibrations is due to assumed ranges of the air-to-fuel ratio and injection timing. For example, the stoichiometric calibration corresponds to the air-to-fuel ratio near stoichiometry and early injection, the homogeneous lean calibration corresponds to lean air-to-fuel ratio and early injection while the strati ed lean calibration corresponds to lean air-to-fuel ratio and late injection. For a given w not all of these calibrations may exist. For example, the strati ed operation is only possible for low engine speed and low engine torque values. 1

Steady-state conversion eciencies are calculated assuming that xTWC is at equilibrium for a given u.

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To be more speci c, the calibrations for the normal mode are generated by minimizing for each wi ; i = 1; : : : ; N , on the selected grid the fuel consumption subject to a constraint that the ratio W˜ NOx ; R= Wf Here W˜ NOx is the estimated mass ow rate does not exceed a speci ed upper bound R. of NOx out of the TWC assuming “steady-state” conversion eciencies of the TWC. R s ; s = 1; : : : ; L, is also assumed and the optimization A discrete grid on the values of R; problem takes the form: Wf (u; wi ) → minu ; h(u; wi ) ≤ 0; s R(u; wi ) ≤ R

(6)

i

q(u; w ) ≤ 0; u j ≤ u ≤ u j : The function q in the inequality representation, q(u; w) ≤ 0, is selected appropriately to ensure that the engine feedgas properties such as temperature and exhaust gas ow rate can sustain eective maximum trapping capacity of the LNT [10,11] above a prescribed threshold value. The bounds u j ≤ u ≤ u j restrict the ranges of the air-to-fuel ratio and injection timing for each of the three calibration ust ; uhl ; usl corresponding to j = 1; 2; 3, respectively. The values for each of the three calibrations obtained on a grid of points wi ; i = s 1; : : : ; M, and R ; s = 1; : : : ; L, are interpolated to generate the values of ust (w; R); hl sl u (w; R) and u (w; R) for arbitrary w and R. During the normal mode operation if uhl (w; R) and ust (w; R) the one w and R are given then out of the three values usl (w; R); is selected that provides the least fuel consumption. If w falls outside the existence range for one of the calibrations then this calibration is not taken into consideration. Next, the purge calibration, up , is generated by maximizing the estimated mass ow rate of CO out of the TWC, W˜ CO , assuming the “steady-sate” conversion eciencies of the TWC. Since the LNT purge involves reactions of CO with NOx stored in the trap, maximizing W˜ CO ensures that LNT purge is accomplished as rapidly as possible. In addition, purge can only take place when the engine is operated under homogeneous charge conditions and with rich of stoichiometry air-to-fuel ratio. Hence, a constraint on the air-to-fuel ratio of the form u1 = ≤ s where s is the stoichiometric air-to-fuel ratio is imposed. The optimization problem is thus W˜ CO (u; wi ) → maxu ; h(u; wi ) ≤ 0;

(7)

u1 ≤ s : The values for the purge calibration obtained on a grid of points wi ; i = 1; : : : ; M , are interpolated to generate the values of up (w) for arbitrary w. Note that the optimization problems (6), (7) are independent for dierent values of and, hence, can be solved in parallel. The problems (6), (7) are nonlinear and w and R,

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Fig. 4. Normal to purge mode transition diagram.

Fig. 5. Total tailpipe NOx emissions versus total fuel consumption.

nonconvex optimization problems. To facilitate the numerical solution of these global optimization problems a Multi-Level Single Linkage algorithm [13] in combination with Sequential Quadratic Programming (SQP) for local constrained minimization was employed. Finally, a transition policy between normal and purge mode is selected and parametrized by a parameter P so that whenever xLNT exceeds P during the normal mode the transition to the purge mode is activated. The normal operation resumes when xLNT falls below a speci ed threshold that in this study is considered xed, see Fig. 4. The operating policies, parametrized by two parameters R and P are examined over the drive cycle, and the policy that yields the least fuel consumption under the speci ed constraint on NOx emissions is selected. 2.3. Optimization results Fig. 5 shows the total fuel consumption and total tailpipe NOx emission values that Each curve can be attained over the drive cycle by varying the parameters P and R.

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Fig. 6. Time trajectory of xLNT .

corresponds to a constant value of R and varying P. Decreasing the value of P results in more frequent purges thereby resulting in less tailpipe NOx emissions and higher fuel consumption. Less tailpipe NOx emissions are generated because the LNT lls less on average and, therefore, its trapping eciency is higher on average. The increase in fuel consumption under more frequent purging is due to less time spent operating in the lean regime where the fuel economy is improved. The eect of decreasing the value of R is to decrease total tailpipe NOx emissions and increase total fuel consumption. This is because small values of R tend to shift the engine operation to stoichiometry where the TWC is eective in converting NOx while the fuel economy advantage of the lean operation is lost. The appearance of the curves changes only slightly when initial conditions for the state of the TWC and the state of the LNT are varied. The dashed lines in Fig. 5 correspond to Stage III and Stage IV NOx emission levels. From the plots the least fuel consumption constrained by Stage IV NOx emission levels is about 478 g. The trajectory of the state of the LNT, xLNT , corresponding to this fuel consumption and NOx emission levels is shown in Fig. 6, where transitions between normal and purge modes are clearly visible. To quantify the fuel economy bene t of the LNT, the optimization was repeated without the LNT. The fuel consumption without the LNT totalled about 521 g under State IV NOx emission constraint. Hence, the relative fuel economy bene t of the LNT can be estimated at about 8%. If the engine operation is constrained to the stoichiometric air-to-fuel ratio (like in conventional PFI engines), the fuel consumption is 527 g. Hence, relative fuel economy improvement due to lean operation without the LNT as opposed to stoichiometric operation without the LNT is estimated at only about 1%. This improvement is due to the fact that even without the LNT the engine can be run lean at some operating conditions where feedgas levels of NOx are fairly low. It is also of interest to examine the case when the LNT is included in the powertrain con guration but no active purge policy is undertaken. Refs. [10,11] show that even when the LNT is completely full it has some NOx conversion eciency (about 20% on

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215

an average) attributed to the second sorbent phase conversion eect. Also occasional (unsolicited or passive) purge may occur for higher loads of the engine operation. For this case the least fuel consumption constrained by the same NOx emission levels was about 510 g translating to about 3% improvement relative to stoichiometric operation. P can be made As a further attempt to re ne the conclusions, the parameters R; functions of engine speed and engine torque (i.e. w) de ned in terms of several other parameters; these other parameters are then optimized to provide the best emission constrained fuel economy. This approach has been tried but for the parametrization that we considered the changes in the numbers were not signi cant.

3. Optimizing parallel hybrid operation In this section we consider a parallel hybrid vehicle which is equipped with a small turbocharged compression ignition (CI) internal combustion engine (ICE), with peak power of 61 kW and maximum torque of 178 N m. An electric motor/generator of power 10 kW is connected directly to the crankshaft. The battery has a power of 11:3 kW. See Fig. 7. The electric components are fairly small and designed for short power boosts only. The objective is to quantify the fuel economy bene t (without emission constraints) of the parallel hybrid powertrain as opposed to the stand-alone diesel engine con guration. 3.1. Parallel hybrid modelling The “quasi-static” assumption is used for the diesel engine modelling. Speci cally, v = r(u; w);

(8)

where

Wf

W NOx v= WCO WHC WPM

;

prail

egr u= vgt tinj

;

w=

N ; e

and where Wf is the diesel engine fueling rate, WNOx ; WCO ; WHC ; WPM are the mass

ow rates of feedgas NOx ; CO; HC and PM out of the engine, respectively, prail is the common rail pressure, egr is the EGR valve position, vgt is the variable-geometry turbocharger vane position, tinj is the injection timing, N is the engine speed and e is the torque generated by the diesel engine. In the static model it is assumed that the fueling rate Wf is speci ed implicitly by prescribing the torque value e and u.

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Fig. 7. Parallel hybrid powertrain.

The dynamics are due to the state of charge of the battery denoted by Q. If em is the torque delivered by the electric motor, then Q˙ = f(Q; Pbt );

(9)

where Pbt is the power requested from or supplied to the batery. This power is related to the electric motor torque, em , by the relationship (10) Pbt = Nem − Ploss ; 30 where power losses, Ploss , depend on N and em . The battery itself is modelled as an internal resistance with open-circuit voltage, both dependent on state of charge, while its hysteresis is modelled by a recharge eciency. See [9,18,21]. Since the powertrain is hybrid and comprises two torque sources, the vehicle speed trajectory on the Euro cycle is mapped to the engine speed trajectory N and demanded torque trajectory, d , that is to be delivered by these two torque sources. The 5 speed manual transmission and mass of the battery were included in the derivation of N and d trajectories. The operating strategy described in the following section speci es the torque demand for the electric motor em and the torque demand for the diesel engine e in a manner that always ensures that d = em + e : 3.2. Operating policies The parallel hybrid strategy is essentially that of Cuddy and Wipke [5] with the dierence that we use engine speed as opposed to vehicle speed: 1. When braking, always use the electric motor to charge the battery. 2. When the engine speed is below N0 , shut o the internal combustion engine and power only with the electric motor (provided the battery is suciently charged). 3. When the engine speed is above N0 and the requested torque d is below 0 (N ), still generate 0 (N ) with the ICE and use the surplus of energy to charge the battery (provided the battery is not already fully charged).

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217

4. When the battery state of charge is below a fraction Q0 of the maximum state of charge, deliver maximal torque with the ICE and use the surplus of energy to charge the battery. 5. When the requested torque is more than the maximum torque the ICE can deliver, use the electric motor to assist the ICE. 3.3. Optimization results For optimal fuel economy one needs to de ne the optimal strategy when to engage and disengage the electrical power source, which is determined by the function 0 (N ) and parameters N0 and Q0 . The function 0 (N ) is implemented as a 1D lookup table with three break points at engine speeds of 1000; 2000; 3000 RPM, respectively. This givies a total of ve parameters to be optimized. Note that the eect of this hybrid strategy is to only operate the ICE at higher speeds and loads, where it has better eciency. To calculate the fuel consumption we have to account for energy stored in the battery, which is in general not the same at the beginning and end of cycle. We simulate once starting at a battery state of charge Q(0)=0:2, once at Q(0)=1, and interpolate linearly between the respective fuel consumptions: k=

Q02 − Qf 2 ; (Q02 − Qf 2 ) − (Q01 − Qf1 )

(11)

FC = kFC1 + (1 − k)FC2 ; where Q0i is the initial state of charge of the ith run, Q is the nal state of charge of the ith run, and FCi is the fuel consumption of the ith run. We set up a brute force numerical optimization problem where the cost function is the fuel consumption over the Euro cycle, and where at each iteration we simulate the Euro cycle. We do not account for cold start eects. The simplex method was used for parameter optimization. The optimal parameters in the hybrid strategy turn out to be N0 = 1256;

0 (1000) = −10:5;

0 (3000) = 11:0;

Q0 = 0:47:

0 (2000) = 13:6; (12)

Table 1 compares the fuel consumption and emissions of the hybrid strategy with those of a vehicle with internal combustion engine only. It can be seen that the hybrid strategy has a positive eect both on fuel economy and emissions. Fig. 8 shows the battery state of charge for the parallel hybrid strategy over the Euro cycle. Fig. 9 shows engine speed and load over the Euro cycle for the ICE and parallel hybrid strategy. It is observed that the eect of the strategy is to move engine operation away from low speeds and loads.

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Fig. 8. Battery state of charge for the parallel hybrid strategy over the Euro cycle.

Fig. 9. Engine speed and load distribution for the ICE (crosses) and parallel hybrid (circles) strategies over the Euro cycle.

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219

Table 1 Fuel economy (1=100 km) and emissions (g/km) of parallel hybrid compared to ICE

ICE PH

FC

NOx

HC

PM

CO

3.69 3.45

0.40 0.33

0.13 0.13

0.034 0.029

1.25 0.96

4. Optimizing diesel particulate ÿlter regeneration Now we study the addition of a diesel particulate lter to the parallel hybrid of Section 3. The lter is modelled as a perfect storage of particulate matter, with regeneration taking place as a function of the temperature of the exhaust gas. If this ◦ temperature is above 350 C, and we have a cerium additive in the fuel, the trap will regenerate by burning all stored carbon. Once enough energy has been supplied to the trap to start the reaction, the reaction is self-sustaining. The DPF has an eect on the fuel economy by increasing the back pressure, hence increasing the pumping losses. Based on experimental data, we roughly model this eect as p3 = 2:5 · Qdpf ; 0:25 Wf ; = p3 −3 + (53 × 103 )=N

(13)

where Qdpf is the trap loading in grams, p3 is the exhaust back pressure in kPa, Wf is the fuel ow in kg/h and N is the engine speed in RPM. The increased back pressure aects the engine in a more profound way than a mere increase in fuel consumption: the increased back pressure will aect EGR ow and turbine RPM and eciency, and hence mass air ow into the intake manifold. We assume here that the operating engine strategy corrects for these eects at the expense of a fuel consumption penalty only. The eect of trap loading on back pressure and of back pressure on fuel consumption can be modelled more accurately by having the coecients depend on engine speed and load, but the quintessential eect is similar to that of Eq. (13). Since trap regeneration is governed by temperature, we need to augment the ICE model with an exhaust temperature and mass ow, modelled as 2D lookup tables over speed and load. We assume we measure the back pressure p3 to monitor the trap loading. For diagnostic purposes it would be preferable to measure quantities after the trap, from which we would need to infer the trap loading based on a more complicated trap model. In principle, we could conduct a study as for the lean NOx trap, if it were not for the small amounts of particulate matter generated. The total amount of particulate matter generated over a Euro cycle is less than 0.5 g, corresponding to an increase in back pressure of about 1 kPa, which has a negligible eect on fuel consumption. Hence optimization of DPF strategies over drive cycles starting with an empty trap is problematic. To illustrate the procedure we assume that we start with a full trap (20 g) and add the following rule to the parallel hybrid strategy:

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I. Kolmanovsky et al. / Nonlinear Analysis: Real World Applications 1 (2000) 205 – 221 Table 2 Eect of light-o RPM on fuel consumption (1=100 km) over the Euro cycle Ndpf FC

1000 3.87

1500 3.84

2000 4.01

2500 5.22

3000 5.03

1. If p3 ¿ p30 and N ¿ Ndpf generate maximum torque with the ICE, and use the surplus to charge the battery. The reason we restrict this rule to higher engine speeds, is that only at higher engine speeds we can generate enough heat ux to light o the DPF. This value is dependent ◦ on the temperature drop in the exhaust system. We assume here that this drop is 200 C, and look at the eect on fuel consumption of dierent values for Ndpf in Table 2. For a meaningful study we assume that the initial DPF load causes an increase in back pressure greater than p30 . The general trend is that fuel consumption increases as we delay DPF light o. The big jump in fuel consumption between 2000 and 2500 RPM is due to the fact that engine speeds in the urban part of the drive cycle are below 2500 RPM. During the nal high-speed stretch (v = 120 km/h), the engine generates enough heat to light o the DPF without intervention of the strategy. Acknowledgements Thanks to Jessy Grizzle, Je Hepburn, Shankar Raman and Yanying Wang for helpful suggestions that aided our work, and to Diana Brehob, Je Cook, Jim Kerns and Barry Powell for helpful discussions References [1] R.W. Anderson, J. Yang, D. Brehob, Vallance, R.M. Whiteakar, Understanding the thermodynamics of direct injection spark ignition (DISI) combustion systems: an analytical and experimental investigation, SAE Paper 962018. [2] J.E. Auiler, J.D. Zbrozek, P.N. Blumberg, Optimization of automotive engine calibration for better fuel economy — methods and applications, SAE Paper 770076. [3] E.P. Brandt, Y. Wang, J.W. Grizzle, A simpli ed three-way catalyst model for use in on-board SI engine control and diagnostics, Proceedings of the ASME Dynamic Systems and Control Division, Vol. 61, 1997, pp. 653– 659. [4] J.F. Cassidy, A computerized on-line approach to calculating optimum engine calibrations, SAE Paper 770078. [5] M.R. Cuddy, K.B. Wipke, Analysis of the fuel economy bene t of drivetrain hybridization, SAE Paper 970289. [6] A.I. Cohen, K.W. Randall, C.D. Tether, K.L. Van Voorhies, J.A. Tennant, Optimal control of cold automotive engines, SAE Paper 840544. [7] A.R. Dohner, Transient system optimization of an experimental engine control system, over the federal emissions driving schedule, SAE Paper 780286. [8] R.S. Erwin, D.S. Bernstein, Discrete-time H2 =H∞ control of an acoustic duct: delta-domain design and experimental results, Proceedings of the 36th conference on Decision and Control, 1997, pp. 281–282. [9] L. Guzzella, A. Amstutz, F. Grob, Optimal operation strategies for hybrid powertrains, Proceedings of IFAC workshop on Advanced in Automotive Control, Mohican State Park, OH, February, 1998.

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