Robustness of ECMS-based Optimal Control in Parallel Hybrid Vehicles

7th IFAC Symposium on Advances in Automotive Control The International Federation of Automatic Control September 4-7, 2013. Tokyo, Japan

Robustness of ECMS-based Optimal Control in Parallel Hybrid Vehicles ? Chris Manzie ∗ Olivier Grondin, Antonio Sciarretta and Gianluca Zito ∗∗ ∗

Department of Mechanical Engineering, The University of Melbourne, Victoria, Australia 3010 (e-mail: [email protected]). ∗∗ Control, Signals and Systems Department, IFP Energies Nouvelles, 1 et 4, avenue de Bois-Preau, Rueil-Malmaison, France 92500

Abstract: Control algorithms for hybrid vehicles have undergone extensive development leading to near-optimal techniques being employed and demonstrated over the previous decade. The different optimal controllers are inevitably linked through the assumed knowledge of the powertrain. For alternative fuelled engines, this assumed knowledge becomes circumspect due to composition variability, leading to uncertainty in the models used by the hybrid powertrain controller. The robustness of a map-based optimal controller using an equivalent charge maintenance strategy (ECMS) is consequently investigated in terms of theoretical fuel consumption losses under certain restrictive assumptions. The potential real world impact of variable composition fuels in hybrid powertrains is also assessed through two case studies involving significantly different prototype flex fuel hybrid vehicles. Keywords: Hybrid vehicles, flex-fuel, optimal control, robustness. 1. INTRODUCTION

other proposed schemes (Manzie et al., 2012; Zhang et al., 2011).

The ability of a hybrid vehicle to maximise fuel economy is indelibly linked to the control strategy used to select the utility of the combustion engine and the electric motor. Initial work in this regard utilised rule-based control strategies (Buntin and Howze, 1995; Salman et al., 2000; Schouten et al., 2002), while the integration of optimisation (Borhan et al., 2012) and optimal control methodologies (Delprat et al., 2004; Sciarretta et al., 2004; Wei et al., 2007; Serrao et al., 2009) have led to improved fuel economy, particularly in the knowledge of part (Manzie et al., 2012) or all (Back et al., 2002, 2004) of the future driving conditions. These optimal control techniques have been readily extended to cope with additional issues facing hybrid electric vehicles including battery charging and catalyst temperature (Serrao et al., 2011), while a lack of knowledge about the drive cycle has been partially addressed through the incorporation of speed (Kim et al., 2009) or state-of-charge (Musardo et al., 2005) dependence on the equivalence factor, typically through the integration of a proportional feedback term. Common to the optimal control approaches is the need to inform the controller of certain characteristics about the engine and motor being controlled. The required information is typically quasi-steady, and includes e.g. fuel consumption maps in the case of ECMS (Sciarretta et al., 2004; Stockar et al., 2011) or efficiency contours under ? The authors acknowledge the support of the Australian Research Council through FT100100324, as well as support provided by IFP Energies Nouvelles and the ACART centre at the University of Melbourne.

978-3-902823-48-9/2013 © IFAC

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Aside from powertrain electrification, there is also significant interest in the use of alternatives to gasoline and diesel to aid in fuel security and improve tailpipe CO2 . Fuels such as liquefied petroleum gas (LPG), compressed natural gas (CNG) and gasoline-ethanol blends have attracted different levels of regional interest depending on local availability and distribution networks. However, the composition of all these fuels in an engine is subject to high variability. For instance, LPG can vary from almost pure propane to equal proportions of propane and butane; while ethanol engines may encounter fuel blends ranging from E85 to E0. On a vehicle with only an alternative-fuel internal combustion engine, this compositional variation leads to a detuning of the engine calibration to ensure sufficient robustness for all possible fuels that will be encountered. If the alternative fuelled engine is integrated into a hybrid vehicle, there is another level of consideration as the aforementioned model based controllers all rely on quasisteady maps that are potentially incorrect. Given the possible CO2 advantages afforded by marrying hybrid powertrains with an alternative fuel, the robustness of the existing algorithms, and indeed their ability to deliver close to optimal performance in the presence of map uncertainty, is an open question. This paper sets out to analyse the general problem of using uncertain fuel models in an ECMS controller. The degradation in performance is then quantified for two prototype flex-fuel hybrid vehicles developed to run on ethanol blends ranging from E5 to E85. 10.3182/20130904-4-JP-2042.00120

IFAC AAC 2013 September 4-7, 2013. Tokyo, Japan

2. LOCAL ROBUSTNESS ANALYSIS OF ECMS

1 a0 (Ne (t))u2 + b0 (Ne (t))u + c0 (Ne (t)) (7) 2 1 g(u, t) = a1 (Nm (t))u2 + b1 (Nm (t))u + c1 (Nm (t)) (8) 2 1 ∆f (u, t) = a2 (Ne (t))u2 + b2 (Ne (t))u + c2 (Ne (t)) (9) 2 f (u, t) =

Quasi-steady models for fuel and charge consumption, mf and q respectively, used in a general Hamiltonian-based controller design for a hybrid powertrain with control inputs u are described in the following general form: Z mf = f (t, u)dt (1) Z q = g(t, u)dt (2) Given the quasi-steady assumption, the control variables are the engine and motor operating points. In a Pontryagin’s Minimum Principle-based controller such as described in (Sciarretta et al., 2004), the resulting Hamiltonian and optimal equivalent charge management strategy given an allowable input range contained in the (potentially time varying) set Ut may then be described as H(f, g, u, t) = f (t, u) + s(t)g(t, u) (3) u∗ (t) = arg min H(f, g, u, t) (4) u∈Ut

Note that s is termed the equivalence factor as it represents the fuel-electricity equivalence. If the drive cycle is known a priori, s(t) := s∗ can be determined numerically as the constant that drives battery state of charge at the end of the cycle to a desired level. Assumption 1 The values of u∗ (t) form dense subsets of the allowable input range, u∗ ∈ Ut∗ ⊂ Ut containing the point of maximum engine efficiency, η peak , and zero engine torque. Furthermore, the average engine efficiency within the subset containing η peak is given by ηav = (1 − γ)η peak , where γ is a small positive number. Ideally, the optimal inputs will be clustered around regions of high engine efficiency whenever the engine is on. The density of the inputs in the efficiency space will be influenced by the vehicles torque demand arising from the drive cycle as well as the presence and severity of any operational constraints placed on the controller. The value of γ will be affected both by the density of the inputs and the gradients in both the fuel and electrical usage maps used by the controller, f (t, u) and g(t, u). Now consider that the quasi-steady fuel consumption map used in the controller is perturbed by a static map, ∆f (t, u), i.e. the map used by the controller is f˜(t, u) = f (t, u) + ∆f (t, u). Uncertainties in the electrical path and driveline of hybrid vehicles are much less common, and so perturbations to g(t, u) are not considered here. The resulting (assumed optimal) control, u ˜∗ , obtained using the ECMS strategy is described by ˜ f˜, g, u, t) = f˜(t, u) + s˜(t)g(t, u) H( (5) ∗ ˜ ˜ u ˜ (t) = arg min H(f , g, u, t) (6) u∈Ut

The following assumption is now applied for all three maps relevant to the performance of the ECMS controller. Assumption 2 The quasi-steady maps are sufficiently well approximated at each speed by the following equations: 128

The assumption of quadratic maps in practice restricts discussion to a local analysis, albeit in conjunction with Assumption 1 this will be about the point of maximum engine efficiency. Allowing higher order polynomials to represent ∆f invokes the potential for discontinuities in the arg min H operations, that are not readily analysed. Also, note that the coefficients of f and g will ensure that one function is increasing in u while the other is decreasing. For ease of notation, the speed-dependence of the coefficients in (7) - (9) is omitted in the following discussion. The following additional assumptions are now placed on the perturbation map and drive cycle. Assumption 3 The parameters of the perturbation map a2 , b2 and c2 are small. This is in keeping with the local nature of the analysis. Assumption 4 The drive cycle is of length T and completely known a priori. This allows constant equivalence factors, s∗ (f, g) and s˜∗ (f + ∆f, g) to be used in the analysis. These optimal equivalence factors may be charge sustaining in the case of parallel hybrid operation, but optimal charge depletion for plug-in hybrid vehicles is also possible. Theorem 1. Under Assumptions 1-4, using an ECMS strategy with an incorrect fuel map parameterised by a2 , b2 and c2 will lead to a fuel consumption penalty on a known drive cycle of: ∆m(a2 ,b2 , c2 , γ) = (1 − γ) Z T × (O(a2 ) + O(b2 ) + O(c2 )) κ(t)dt (10) 0

where κ(t) = 1 if the engine is on and κ(t) = 0 otherwise. Proof. Following from Assumption 2 the Hamiltonians for the true and the assumed systems are respectively: H(f, g, t) = f (u, t) + s∗ g(u, t) (11) 1 = (a0 + s∗ a1 )u2 + (b0 + s∗ b1 )u + (c0 + s∗ c1 ) 2 (12) ˜ f˜, g, t) = 1 (a0 + s˜∗ a1 + a2 )u2 H( 2 + (b0 + s˜∗ b1 + b2 )u + (c0 + s˜∗ c1 + c2 ) (13) In the absence of any input or state constraints, the Hamiltonian-minimising controls calculated from (4) and (6) are given by b0 + b1 s∗ umin (t) = − (14) a0 + a1 s∗ b0 + b1 s˜∗ + b2 (15) u ˜min (t) = − a0 + a1 s˜∗ + a2 The resulting optimal constrained controls are then obtained by projecting these into the allowable sets, i.e.:

IFAC AAC 2013 September 4-7, 2013. Tokyo, Japan

u∗ (t) = projUt umin (t) ˜min u ˜∗ (t) = projUt u

(16) (17)

the known ECMS result for constant engine and motor efficiencies that s∗ → ηηfe during propulsion.

As both s∗ and s˜∗ are constant by Assumption 4, in the case of parallel and series hybrids they will be charge sustaining by definition. Hence there is no need for state of charge correction in calculating the increase in fuel used over the drive cycle by adopting the suboptimal policy u ˜∗ (t). The fuel penalty for using incorrect map, ∆mf , is subsequently given by:

Furthermore, in the event that the drive cycle is unknown and the equivalence factor is estimated online, the dependence on the ∆f parameters may change as there are additional feedback dynamics involved. The actual performance degradation will be additionally influenced by both the structure of the s∗ -estimator algorithm and the gains used within it.

Z ∆mf :=

3. OVERVIEW OF ENGINE AND SIMULATOR f (˜ u∗ (t)) − f (u∗ (t))dt

(18)

Defining κ(t) = 1 if the engine is on and κ(t) = 0 otherwise, then from Assumption 1 it follows that the fuel penalty can be expressed as: " 2 # 2  Z a0 κ(t) b0 + b1 s˜ + b2 b0 + b1 s∗ ∆mf = − 2 a0 + a1 s˜ + a2 a0 + a1 s∗   b0 + b1 s∗ b0 + b1 s˜ + b2 + b0 κ(t) − dt (19) a0 + a1 s∗ a0 + a1 s˜ + a2 While (23) shows an explicit dependence on the curvature and gradient of the map perturbation, there is only an implicit dependence on c2 and γ that arises through the different equivalent factors, s˜∗ and s∗ . This is made explicit through consideration of the engine efficiency, η(Ne , u) := kNe u f (u) . If the efficiency gradients are relatively constant with distance from η peak , from Assumption 1 the averaged operating point within Ut∗ can be related to the maximum efficiency operating point, up according to:   ∂η u∗ = (1 − γ) up (20) ∂u up The maximum efficiency point, up , can be found by apdη = 0. The left hand side of plying Assumption 2 with du (20) is obtained (14) and (15) for each case, and hence the following solutions for it s∗ and s˜∗ are obtained: √ √ ∂η b0 a0 + (1 − γ) ∂u a0 c0 s∗ = − √ (21) √ ∂η b1 a0 + (1 − γ) ∂u a1 c0 √ √ ∂η (b0 + b2 ) a0 + a2 + (1 − γ) ∂u (a0 + a2 ) c0 + c2 ∗ s˜ = − √ √ ∂η b1 a0 + a2 + (1 − γ) ∂u a1 c0 + c2 (22) Since ∆mf (0, 0, 0) ≡ 0, by substituting (21) and (22) into (19) and applying Assumption 3 provides the truncated Taylor series expression: ∂∆mf ∆mf (a2 ,b2 , c2 , γ) = ∆mf (0, 0, 0) + a2 ∂a2 (0,0,0) ∂∆mf ∂∆mf + b2 + c2 (23) ∂b2 (0,0,0) ∂c2 (0,0,0)

Two vehicles are used as real world case studies. The first vehicle (Vehicle A) is a parallel hybrid demonstrator developed at IFP Energies Nouvelles for flex fuel operation. On the electrical side it has a 37.7kW, 35Nm electric motor capable of 20,000 rpm. The battery pack consists of LiIon cells with 7.6 kWh capacity. This is relatively large, but reflects potential plug in capability being incorporated, although only charge sustaining operation is considered here. The battery state of charge is allowed to vary within the range 0.35 to 0.75, with a desired setpoint of 0.55. There is also an integrated starter-alternator enabling regenerative braking and engine-off capability. The second vehicle (Vehicle B) was also developed within IFP and has a more conventional electrical configuration for parallel hybrid operation, with a similar size motor but much smaller Li-Ion battery pack (42 kW and 1.5 kWh respectively). On the mechanical side, the engine used in Vehicle A is a naturally aspirated, 4-cylinder, 1.346 litre spark ignition engine and a five speed automated transmission. On the other hand, Vehicle B has a turbocharged, 4-cylinder 2.0 litre spark ignition engine with slightly lower compression ratio. The key parameters of each vehicle are summarised in Table 1. Table 1. Case study vehicle summaries.

Engine Type Total displacement Air charging Compression ratio Max engine power Max engine torque Plug-in capable Max motor power Battery energy Vehicle mass

Vehicle A ET3J4 1.346 l Naturally aspirated 11:1 65 kW @5250 rpm 133 Nm @3250 rpm Yes 37.7 kW 7.6 kWh 1700 kg

Vehicle B F4Rt 2l Twin-scroll TC 10.55:1 150 kW @5000 rpm 300 Nm @2750 rpm No 42 kW 1.5 kWh 1930 kg

The result of the theorem then follows directly.

Static calibrations for the warm engines were performed on a dynamometer for ethanol blends ranging from E5 to E85. The difference in fuel flow rate maps for the two compositions, ∆f := fE5 − fE85 , is shown in Figure 1 for different engine speeds. From these maps it appears that limiting the approximation of ∆f to a polynomial in torque of order two for fixed engine speed is not unreasonable, and appears to be relevant across the entire torque range, with the possible exception of high torque and high speed operating points on the F4R engine.

It is worth noting that for the case where a0 , c0 → 0 (i.e. the engine efficiency is constant for all torque), from (21) that s∗ → − bb01 . It is readily shown that this equates to

For further insight, the efficiency contour maps are shown in Figure 2 and 3. From these it is clear that despite a relatively linear characteristic for the ∆f contours,

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IFAC AAC 2013 September 4-7, 2013. Tokyo, Japan

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Before considering the effect of the incorrect fuel blend information in the ECMS performance explicitly, the result of Theorem 1 was tested by using a quadratic approximation of f (u) derived from the ET3 engine running on E5 fuel at 2500rpm. This curve was used to build a speedindependent map that could be used in the ECMS controller, with a peak in efficiency at approximately 96Nm of torque. This map was subjected to small perturbations in the coefficients of 10% the original value, leading to the efficiency peak shifting by less than 5Nm. Maintaining small perturbations is consistent with Assumption 3, and also ensures that the ECMS controller’s operation is not significantly varied through encountering significantly different state constraints during operation on each fuel blend. To aid in isolating the map effect, some controller constraints, including limitations on gear changes etc, were relaxed during this phase of the testing. The simulated fuel consumption was corrected for state of charge deviations, and the subsequent fuel losses (expressed as a percentage) for different small perturbations over the NEDC are shown in Table 2. Table 2. Effect on NEDC corrected fuel economy of small perturbations on fuel map Perturbation a2 = −10% b2 = −10% c2 = −10% a2 , b2 , c2 = −10%

∆mf 0.6% 1.6% 0.3 % 2.3%

Perturbation a2 = 10% b2 = 10% c2 = 10% a2 , b2 , c2 = 10%

∆mf 0.7% 3.7% 1.0% 5.2%

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Fig. 3. Efficiency contours for the F4R engine running on (left) E5 and (right) E85 fuel the maximal efficiency for each composition is obtained under quite different operating points in each engine. For example, the ET3 engine operating on E5 has a degradation in efficiency past a torque of around 85Nm across all speed ranges, while under E85 the peak efficiency is obtained at the highest torques. Similarly, the efficiency peak for the F4R engine is at a medium torque and high speed running on E5, but this shifts to a medium speed and high torque when E85 fuel is used. The simulator used in this work is a version of the HyHiL environment developed within IFP Energies Nouvelles and used extensively in the development and testing of hybrid powertrain controllers, e.g. Del Mastro et al. (2009); Chasse et al. (2009a,b). It includes detailed electrical and mechanical modelling capability, with efficiencies of both paths realistically prescribed from experimental testing. To partially alleviate drive-cycle dependant results, several drive cycles were employed in the testing. Those chosen included two standard regulatory cycles, the well-known NEDC and the FTP cycles, and one more representative of real world driving, the World Harmonized Light Duty 130

These results suggest that the controller appears reasonably robust to small variations in any particular parameter, as the percentage of fuel losses are smaller than the variation in any coefficient. There is also a slight asymmetry in the results, with positive parameter perturbations leading to slightly larger observed fuel loss. This is a mapdependant characteristic influenced by the small changes of the map gradients in the regions of operation, and also influenced by the electrical map, g. Concurrent variations in the coefficients also lead to approximately linear perturbations in the fuel mass, although given the difficulty in compensating for state of charge variation at high precision, it is problematic to ascertain whether there are potential higher order effects that are being observed, or whether the deviations in multiple parameters have complementary effects. 4.2 Vehicle case studies Attention now turns to assessing the real world significance of fuel map perturbations. From Theorem 1, it is expected that small perturbations in the fuel maps will lead to proportional degradation in the fuel economy, with the level of proportionality dependent on the size of the perturbations in the quadratic surfaces. The total fuel economy degradation is also affected by the average operating efficiency

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Table 3. Simulated fuel consumption using ET3 hybrid running with incorrect fuel maps and equivalence ratio

FTP FTP WLTP WLTP

Ass fuel E5 E85 E85 E5 E5 E85 E85 E5 E5 E85 E85 E5

s 3.05 4.22 4.22 3.05 3.15 4.41 4.41 3.15 3.04 4.17 4.17 3.04

Fuel cons. 4.61 4.66 6.46 6.51 4.26 4.32 5.97 6.00 5.21 5.27 7.31 7.40

∆mf

Drive cycle NEDC

1.1% 0.6% 1.5% 0.5% 1.2% 0.7%

Actual fuel E5 E5 E85 E85 E5 E5 E85 E85 E5 E5 E85 E85

NEDC FTP FTP WLTP WLTP

in the constrained region of operation. To assess the likely implications for ethanol blended gasoline, simulations with the maps for E5 and E85 from Section 3 as both the true and assumed fuels are conducted for each of the two vehicles described in Section 3. The parameter variations in the fuel maps are engine speed dependant, although are larger in the case of Vehicle B as might be expected from Figure 3.

Consistent with Theorem 1, complete knowledge of the drive cycle is assumed to enable the ECMS to use the optimal fuel-electricity equivalence factor, s∗ , (i.e. one that ensures no battery state of deviations on completion of the cycle). The fuel maps and equivalence factor were synchronised, so that use of the incorrect fuel information involves both the matched (but incorrect) map and equivalence factor. This is more realistic and also means that only marginal state of charge correction is required in evaluating the fuel use. The results of the simulations over three drive cycles are given in Table 3 and Table 4 for Vehicles A and B respectively. For Vehicle A, it is clear that the fuel penalty incurred is not significant with no combination of drive cycle and incorrect information yielding greater than 1.5% worse economy. To investigate further it is useful to consider the selected engine operating points over one of the drive cycles. Figure 4 show the engine speed and torque over the WLTP cycle, superimposed on the efficiency contours for E5 operation, when the engine is running on E5 fuel but the ECMS controller has both the correct and incorrect fuel information. 131

s

Fuel cons. 4.61 4.67 6.47 6.61 4.11 4.43 5.69 5.76 5.13 5.73 7.12 7.21

3.18 4.53 4.53 3.18 3.68 4.89 4.89 3.68 3.41 4.52 4.52 3.41 0.3

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Full controller constraints are also included explicitly, and play a role in determining the validity of Assumption 1. The imposed constraints are typical of those in real world implementation, and include limiting gear selection, consideration of alternator limitations and state of charge usage to ensure driveability and battery management requirements are maintained. However these are limited to constraints at the ECMS-controller level - due to the quasi-steady nature of the simulation environment, it is not possible to consider the impact of engine-level control constraints, for example knock limits, and so these are assumed to be already captured within the quasi-steady fuel consumption maps.

Ass fuel E5 E85 E85 E5 E5 E85 E85 E5 E5 E85 E85 E5

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Table 4. Simulated fuel consumption using F4R hybrid running with incorrect fuel maps and equivalence ratio

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Fig. 4. ET3 engine operating points superimposed on efficiency contours for the WLTP running on E5 fuel with the ECMS using (left) E5 and (right) E85 information only From this Figure, it is clear that Assumption 1 regarding the denseness of the operating points about the maximum efficiency point is not well supported, primarily due to the imposition of constraints on the ECMS controller invoked by the available hardware. The large overlap between the regions occupied by the engine operating points in Figure 4, coupled with the relatively flat efficiency contours in the occupied regions leads to similar fuel consumptions. In the context of Theorem 1, this implies the parameter γ is relatively large and therefore the fuel losses are expected to be reduced. To partially isolate the effect of the imposed controller constraints on Vehicle A, it is calculated the ECMS with correct fuel information results in an average engine efficiency for the WLTP cycle of 35.4%. As the peak engine efficiency is 38.2% for this engine running on E5 fuel, there appears to be close to 8% fuel penalty imposed by the constraints of the ECMS controller if the WLTP cycle is assumed to be sufficiently rich. The average operating efficiency degrades to only 35.1% when the wrong fuel information is supplied to the controller. Thus, in this case study the ECMS appears quite robust to the fuel uncertainty, although different constraint imposition made possible through changes to the hardware integration may lead to better fuel economy with the correct fuel information, and subsequently a more appreciable difference between the ECMS utilising different fuel maps. In the case of Vehicle B the fuel economy degradation is much more severe in two of the drive cycles when E5 fuel is used but the controller assumes E85 fuel is

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present, with 8 and 12% losses incurred. The asymmetry in the results (i.e. there is no significant fuel economy degradation using an assumed fuel of E85 on any drive cycle) is predicted from the peak efficiency points. The low degradation in the case of the NEDC cycle is primarily due to the comparatively low number of operating points in the cycle not allowing a comprehensive exploitation of the degrees of freedom available to the controller, and is not entirely unexpected. The operating points for Vehicle B on the WLTP cycle using both correct and incorrect fuel information are shown in Figure 5, and appear to show a higher density of points in both cases. This in turn leads to greater fuel penalties when the wrong fuel composition is assumed. 0.25 0.25

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Fig. 5. F4R engine operating points superimposed on efficiency contours for the WLTP running on E5 fuel with the ECMS using (left) E5 and (right) E85 information only 5. CONCLUSION The optimal control implementations of ECMS in hybrid powertrains appear to demonstrate a robustness to small perturbations in the maps. However, case studies performed on real vehicles exhibited larger perturbations and consequently the fuel consumption differences may become significant. Further work will consider how to characterise the likely losses for real vehicles, and the development of possible compensatory strategies. REFERENCES Back, M., Simons, M., and Kirschbaum, F. (2002). Predictive control of drivetrains. In 15th Triennial IFAC World Congress. Back, M., Terwen, S., and Krebs, V. (2004). Predictive powertrain control for hybrid electric vehicles. In IFAC Symposium on Advances in Automotive Control. Borhan, H., Vahidi, A., Phillips, A.M., Kuang, M.L., Kolmanovsky, I.V., and Di Cairano, S. (2012). Mpc-based energy management of a power-split hybrid electric vehicle. IEEE Transactions on Control Systems Technology, 20(3), 593–603. Buntin, D. and Howze, J. (1995). A switching logic controller for a hybrid electric/ICE vehicle. In American Control Conference, volume 2, 1169 –1175. Chasse, A., Hafidi, G., Pognant-Gros, P., and Sciarretta, A. (2009a). Supervisory control of hybrid powertrains: an experimental benchmark of offline optimization and online energy management. In E-COSM ’09 - IFAC Workshop on Engine and Powertrain Control, Simulation and Modelling. Rueil-Malmaison, France. 132

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