_{x}

^{1}

^{*}

^{2}

^{2}

^{2}

^{1}

^{1}

_{x} emissions of a diesel HEV. In addition, they performed the simulations, they conducted the experimental validation, and they wrote the article. Giorgio Mancini played a key role in the setup and calibration of the internal combustion engine and the test bench. He was also partially involved in the drafting of the article. Christopher Onder was the technical supervisor, while Nicolò Cavina and Lino Guzzella were the project supervisors. All supervisors were involved in exchanging ideas on the development of the theoretical framework, its practical application, and/or the experimental validation. Moreover, all supervisors participated in the internal review of the article draft.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Motivated by the fact that the real driving NO_{x}_{x}_{x}

Light-duty diesel vehicles are known for their low fuel consumption, as compared to gasoline vehicles. However, due to legislative restrictions, vehicle manufacturers continuously have to make considerable efforts to reduce the pollutant emissions of diesel vehicles. Although the legislative limits have been continuously reduced over the last decade, the real driving emissions, which are the emissions emitted during every-day driving, can far exceed the legislative limits, even for Euro 6 certified light-duty vehicles, as shown in several studies [

One option to cope with such a radical change would be to continuously monitor and control the pollutant emissions by an appropriate exhaust aftertreatment system. Another option is provided by electric hybridization of the vehicles, which not only offers a reduction of pollutant emissions, but also a simultaneous reduction of the CO_{2} emissions. Since hybrid electric vehicles (HEVs) have an additional degree of freedom for the control of the energy flows in the powertrain, the trade-off between fuel consumption and pollutant emissions can be further influenced.

Some studies can be found in the literature about the control of pollutant emissions for HEVs. For example, the authors of [_{x}_{2} emissions, while guaranteeing charge-sustaining conditions for various driving cycles. The weighting factors between the various components of the target cost function are constant and considered to be tuning parameters. For a diesel HEV equipped with a selective catalytic reaction (SCR) system, a noncausal extended ECMS is proposed in [_{x}

Dynamic programming (DP) has also been applied to address the problem of building a supervisory control system for the fuel and emission reduction. Examples are given in [

A general approach based on optimal control theory is proposed in [_{x}

Other studies focus on the control of transient emissions, especially regarding hybrid powertrains relying on a diesel combustion engine. An optimal energy management strategy is provided by the authors of [_{x}

However, a control strategy that includes online adaptation of the weighting factors for the objective pollutant emission, to take into account real-world driving conditions and a possible modification of the emissions target level, has not been demonstrated yet. Therefore, in this paper, an energy management strategy that allows for the tracking of a specific NO_{x}

The paper is structured as follows: after explaining the vehicle model in detail in Section 2; the energy management strategy, which takes into account the real driving emissions, is derived in Section 3; then, Section 4 presents an experimental validation of the method proposed, as well as a simulation case study that quantifies the fuel savings potential compared to a standard method not accounting for real driving emissions.

The vehicle under investigation is a fictitious executive class sedan. The powertrain architecture is of the pre-transmission parallel type, as illustrated in

More details on the modeling of the vehicle are given in the following. All the values needed to parameterize the model are listed in

The equation for the longitudinal dynamics of the vehicle is described by [_{w} denotes the wheel torque; _{w} denotes the wheel radius; ρ_{air} denotes the air density; _{d} denotes the aerodynamic drag coefficient; _{r} denotes the rolling friction coefficient; _{v} denotes the nominal vehicle weight; Θ_{w} and Θ_{EM} denote the inertia of the four wheels and the electric motor, respectively; a_{g} denotes the gravitational acceleration; and γ denotes the road slope.

The rotational speed of the gearbox input shaft, ω_{g}, is given by:
_{g}_{g}

The torque delivered to the wheels, _{w}, is calculated by:
_{g}_{g}_{,0}, η_{g}_{,1}, ω_{g}_{,1} being the parameters to model the speed-dependent gearbox losses, which account for the increased friction at higher gearbox input speeds [

The input torque of the gearbox is:
_{m} is the motor torque; _{e} is the engine torque; and _{c} is the clutch state command determined by the energy management. The value _{c} = 1 means that the clutch is closed, and the value _{c} = 0 means that the clutch is open. The clutch command is assumed to be realized instantaneously.

The desired torque for the electric motor, determined by the energy management, is assumed to be realized instantaneously without any delays. The power of the electric motor, including the power electronics, is given by a steady-state efficiency map, depicted in

The minimum and maximum torque values for the electric motor, as well as the minimum and maximum speeds are defined by:

The desired torque of the engine is determined by the energy management. However, the rate of change of the engine torque is limited by 100 N m/s in order to prevent the formation of excessive soot and NO_{x}

The rotational speed of the engine, ω_{e}, is defined by (for brevity, the notation of time is omitted):
_{e} is the engine on/off command, with 0 standing for “off” and 1 standing for “on”; and _{c} is the clutch state command, with 0 standing for “open” and 1 standing for “closed”.

The mass flow rate of the fuel consumed,
_{x}

The map for the fuel efficiency and the map for the NO_{x}

At all times, the engine is only allowed to operate within its limits, defined by maximum torque and speed range conditions,

The battery is modeled as an equivalent circuit with a constant open-circuit voltage, _{oc}, in series with a constant internal resistance, _{i}_{4} type [

The equation of the dynamics of the battery SOC, ξ, is given by:
_{b} is the battery current, and _{aux} is a constant power consumed by electric auxiliary units. Although in this paper, _{oc} and

The battery current and the battery SOC are constrained to:

The driver model consists of a proportional-integral (PI) controller. The output of the model is a preliminary throttle position, θ̃, calculated by:
_{p,D} and _{I,D} being the proportional and integral controller parameters with the manually tuned values of 0.9 and 11 s, respectively. The final throttle position, θ, is then obtained by saturating the preliminary throttle signal as follows:

Due to the saturation and the integral control part of the driver model, an anti-windup scheme is implemented [

The mapping of the throttle position, θ, to a desired torque command, _{req}, is designed, such that:

θ = 100% corresponds to the maximum possible traction torque provided by both the engine and the motor at the current vehicle speed;

θ = 0% corresponds to a zero torque request; and

θ = −100% corresponds to the maximum possible brake torque.

The intermediate torque commands are obtained by linear interpolation of the throttle position.

The energy management defines the set points for the gear number to be engaged, _{g}_{c}, the engine on/off state, _{e}, and the torque split, _{ts}, between the engine and the motor.

The gear command, _{g}

The clutch command, _{c}, is defined as follows: if the engine is off, the clutch is assumed to be open. If the engine is on and the desired engine torque is larger than zero, the clutch is closed. Otherwise, if the engine is on and the desired engine torque is equal to zero, the clutch is open, and the engine is assumed to be running idle.

The torque split, _{ts}, is defined as the ratio of the motor torque and the requested torque from the driver:

Then, the torques of the motor and the engine are obtained by:

Recharging during standstill is not considered here; however, this feature could be implemented easily.

The commands for the engine on/off, _{e}, and the torque split, _{ts}, are determined by an online optimization method commonly known as the ECMS [

This section presents a detailed description of the controller developed to optimize the fuel consumption of a diesel hybrid vehicle under a constraint for the NO_{x}

The hybrid vehicle system can be described by a system of first-order ordinary differential equations:
_{NOx}_{x}_{e} the engine on/off command and _{ts} the torque split command, as previously introduced in Section 2.7. The control input for the gear number, _{g}, is decided independently of _{c}, is defined by a known function of _{e}. Therefore, _{g} and _{c} are not subject to optimization and, hence, not included in

Notice that the vehicle speed is not considered to be a state variable, because the speed is assumed to be perfectly tracked. Based on the actual vehicle speed and the driver's torque request, the required torques and the shaft speeds can be determined using a backwards calculation [

Using these definitions, the model for the system dynamics is derived based on the description presented in the previous section. The state dynamics equations become:

The choice for the integral of the emissions as a state variable has an advantage compared to the specific emission level, as defined in legislation as:
_{NOx}_{x}

These properties play a role in the following derivation.

According to the general methodology introduced in [

The closed final-time set, _{NOx}

The optimal solution is found using Pontryagin's minimum principle (PMP) [^{o}

The co-state vector must stay within the normal cone, ^{o}

The Hamiltonian of the system must be minimized for all times with respect to all admissible inputs

The dynamics of the co-states can be rewritten, separating the battery SOC and the cumulated NO_{x}

Since the fuel mass flow and the NO_{x}

Moreover, since neither the system dynamics nor the fuel mass flow depend explicitly on the cumulated NO_{x}

Combining

The importance of the choice of cumulated emissions is here demonstrated, since the term,
_{x}

The Hamiltonian in _{NOx}

The latter new expression for the Hamiltonian in

For the reformulated optimization problem, λ_{NOx}_{x}

The objective of this section is to present a feedback controller derived for online control of the cumulative NO_{x}_{x}_{x}

To achieve the goal of generating a charge and emissions sustaining strategy, two terms can be added to _{ref}, and deviations from the reference values for the normalized cumulated emission, _{x}_{ref}, thus obtaining a new formulation for the cost functional:

The extended cost

Since the additional terms of the extended Hamiltonian do not explicitly depend on the control inputs, they will be minimized by the optimal policy, ^{o}

To calculate the co-state dynamics, λ, the Hamilton–Jacobi–Bellman equations provide the following expression [^{o}^{o}

Since the optimal cost-to-go is not known

The five terms are explained in the following:

_{f}_{1,ξ}(ξ,_{x}

_{f}_{1},_{NOx}_{x}_{x}

_{f}_{2}: a fuel consumption that is supposed to be independent of both the current SOC and the current emission level, needed to drive the rest of the driving mission with correct reference values;

_{ξ}(ξ): denotes the penalty for SOC deviations from the reference value;

_{NOx}_{x}_{NOx}

Following the approach described in [_{ref}

Since in the future, this energy must be compensated for using the thermal path, a certain amount of fuel will be saved/consumed to discharge/charge the battery. Such a quantity will clearly depend on the future efficiencies of the engine and the electric path, which, in turn, depend on the future engine operating points. Moreover, the used engine operating points will also depend on the cumulated NO_{x}_{c}, is a function of _{NOx}_{l}

An expression for the additional fuel cost of saving NO_{x}_{x}

The second term in _{x}_{x}_{x}_{h}

The cost for the penalty is obtained by integrating this trajectory as follows:

Similarly to the treatise in Section 3.3.3, the costs of the third term in _{k}

The total sub-optimal cost-to-go

Accordingly, the sub-optimal cost-to-go function is time-invariant, and so will be the sub-optimal co-states, which are given by:

The partial derivatives of

By analyzing _{FCN}, that instead depends on the operating points occurring in the period considered and another term that we suppose to be negligible, under the hypothesis that the dynamics of the average charging/discharging efficiency does not depend directly on the cumulated emissions. The simplified approach followed in this section is to replace the constant equivalence terms by a factor, λ_{NOx}_{,0}, and to add an integrator, which is used to online adapt it during operation, to respect the average emissions target:

_{NOx}

_{l}_{b} representing the inner electrochemical battery power and the electrical energy equivalence factor [

The combination of

Since the average conversion efficiency, η_{c}, will vary depending on the operating points of the components involved (engine, electric motor, battery) and the operating points will vary as a function of the driving cycle and of the NO_{x}_{x}_{NOx}_{i,ξ} as follows:

Such a PI controller for the electrical energy equivalence factor was also proposed by the authors of [

The final structure of the controller is presented in Section 3.7. Since the controller is able to control the real driving NO_{x}

By normalizing the Hamiltonian function in _{NOx}_{NOx}_{NOx}

The introduction of the reformulated weighting factor, α_{NOx}_{x}

The proportional gains of the PI controllers then become:

To prevent excessively frequent engine starts and stops that can arise due to the application of optimal control-based methods [

The function, ℐ_{e}, denotes an indicator function detecting a change request for the engine on/off state. If a change is requested, the indicator function is one and zero otherwise. By manual tuning, a value for the penalty, δ, of 0.1 × 10^{−3} kg × 43 MJ/kg proved to yield a reasonable performance. However, a penalty alone cannot ensure a minimum engine on/off dwell time, which is desired for comfort and emissions considerations.

Therefore, an additional heuristic engine on/off comfort function is implemented similarly to the one presented in [

Due to these measures, the average number of engine starts and stops, on the four here considered driving cycles, could on average be reduced to a reasonable amount of 2.1 starts per minute compared to 4.6 starts per minute without any measure to prevent frequent starts and stops. The loss in fuel economy due to this comfort function amounts on average to 3.8% compared to a theoretical value for the fuel consumption obtained without any comfort function. The minimum engine on/off dwell time amounts to 5 s in almost any case for the driving scenarios considered.

The goal of this section is to describe the methodology applied to identify the dependency of _{0} on α_{NOx}_{NOx}_{0}. In this case, PMP is adopted, since it is more suitable for the present application of a forward-facing vehicle model, including many input and state variables. This procedure is applied for several different values of α_{NOx}_{0}, that ensures a charge sustaining condition.

The methodology is applied to various driving scenarios, in this case to the four well-known driving cycles, New European Driving Cycle (NEDC), Federal Test Procedure 75 (FTP-75), Worldwide Harmonized Light Vehicles Test Procedure (WLTP) and California Unified Cycle (LA92). The simulation results of the identification procedure are depicted in _{NOx}_{0} is almost linear. However, a linear fit can lead to _{0}-values, which yield considerable deviations of the final SOC when simulating the vehicle on certain driving cycles without feedback. A quadratic fit, as indicated by the black dashed line, turned out to be more adequate for generating a unique relationship between the equivalence factors for the driving cycles considered.

_{x}_{0}-values for each α_{NOx}

Based on the mathematical derivation of the controller presented in the previous sections, the desired controller, to be tested in simulation and experimental tests in the following sections, is illustrated in

The two PI controllers of _{ref}(_{0}, to enforce a charge-sustaining constraint. Alternatively, the reference value can take into account that the current kinetic and potential energy of the vehicle can be recuperated in the future and stored as electrical energy with certain efficiencies, η_{c,K}, η_{c,P}, resulting in the following expression [_{0} is the height difference between the current altitude and some reference altitude. The numerical values for η_{c,K} and η_{c,P} are both 0.65, which were obtained by manual tuning.

The reference cumulative emissions can be computed in a simple way by defining a specific emission level, _{NOx}

The value for _{NOx}_{x}_{NOx} should be chosen that is below the actual limit value issued, for example, by legislation.

Further, _{0}, can be replaced by a driving distance-dependent reference trajectory, as proposed by the authors of [_{x}

First, an experimental validation of the models and of the methodology, as presented in Sections 2 and 3, is shown. Then, a case study is presented in which the benefit of using an RDE-ECMS compared to an ECMS with a constant emission-related equivalence factor, α_{NOx}

The goal of this subsection is first to show that the RDE-ECMS presented in Section 3 works also in practice, and second, that the quasi-static modeling for the fuel consumption and the NO_{x}

For the experimental validation of the RDE-ECMS, the method presented in Section 3 was applied both in simulation and in HIL experiments. In the HIL experiments, only the engine was used in real hardware. The longitudinal dynamics, as well as the vehicle components were simulated on a computer. This setup allowed for the measurement of the real fuel consumption and the real NO_{x}

In the HIL experiments here, the desired torque command, which is calculated by the energy management controller, is sent to the electronic control unit (ECU) of the engine, while the desired engine speed command is sent to the dynamometer of the engine test bench. The NO_{x}

We used the following setup to compare the simulation results to the results obtained with the HIL experiments:

driving cycle: WLTP (Class 3 Cycle);

NO_{x}_{NOx}

SOC reference signal: 60% plus a correction as shown in

initial condition for the emission-specific equivalence factor α_{NOx}(0) = 0;

controller parameters _{p,ξ} = 2, T_{i}_{,ξ} = 480, p = 1, _{p},_{NOx}_{i}_{NOx}

_{x}_{NOx}_{x}

As can be seen from the figure, the vehicle speed trajectories of the simulation results and the HIL results are identical. The SOC trajectories of the simulation and the HIL results are similar; both are charge-sustaining at around the same SOC reference level. Furthermore, the NO_{x} trajectories of both are very similar; after an initial transient phase, the trajectories become more stable, and they approach the desired reference level. A similar behavior is observed for the specific fuel consumption that, in addition, exhibits a visible offset that is explained below. Further, the dynamics of the equivalence factor, α_{NOx}_{x}_{NOx}_{x}_{NOx}_{NOx}

However, as can be seen from the figure, the RDE-ECMS cannot guarantee maintaining the NO_{x}

The offset of the trajectories of the simulation and the HIL results is a consequence of some of the neglected dynamics in the engine model of the simulation. For example, the thermal dynamics are not considered in the simulation, although in practice, they can have an influence on the formation of pollutant emissions. In fact, such effects influence the behavior of the SOC and the NO_{x}_{x}_{x} emissions. For a fair comparison, the equivalent NO_{x} emissions and the equivalent fuel consumption have to be calculated.

To make a fair comparison between the simulation and the experimental results, the NO_{x} emissions and the fuel consumption have to be corrected to take into account the different levels of the SOC. Here, the correction is made based on the following equations:

_{fuel,eq} denotes the distance specific, battery charge equivalent fuel consumption:
_{fuel,norm} stands for a normalization value;

_{NOx}_{,eq} denotes the distance specific, battery charge equivalent NO_{x}_{NOx}_{,norm} stands for a normalization value;

Δ_{fuel,eq} and Δ_{NOx}_{,eq} are the equivalent fuel mass and the equivalent NO_{x}_{b}, η̄_{m}, η̄_{e} denoting the averaged efficiencies of the battery, the motor and the engine, while the superscript, ^{(c)} denotes “charging phase” and ^{(}^{d}^{)} denotes “discharging phase”.

Δ_{b} is the amount of net energy stored in the battery at the end of the driving cycle:

_{x}_{NOx}_{NOx}_{NOx}_{x}_{NOx}_{NOx}

According to the figure, the results obtained with the simulation underestimate the results obtained with the HIL experiments. The error in all the simulations compared to the experimental data is well below 5%. Therefore, quasi-static models for the fuel consumption and the NO_{x}

By now, it was shown that the RDE-ECMS can simultaneously control the SOC and the NO_{x}

Assume that the RDE have to be lower than a specific value, say 1.02 or 102% for the specific NO_{x}_{x}_{NOx}_{x}

The non-adaptive ECMS was tuned to respect the NO_{x}_{p}_{,ξ} = 2, _{i}_{,ξ} = 480, _{NOx}_{p}_{,ξ} = 2, _{i}_{,ξ} = 480, _{p}_{NOx}_{i,NOx}_{NOx}_{p}_{,ξ} and _{i}_{ξ}, of the RDE-ECMS were taken from the non-adaptive ECMS. The two other parameters, _{p,NOx}_{i,NOx}_{NOx}

These two strategies were applied on each of five repetitions of the four different driving cycles, NEDC, FTP-75, WLTP and LA92. _{x}_{NOx}_{NOx}

As seen from _{x}_{NO}_{x}_{x}

_{x}_{NOx}_{x}_{x}

Overall, the RDE-ECMS proved to minimize the fuel consumption, while tracking a reference NO_{x}

This paper presents an energy management strategy to account for NO_{x}_{x}_{x}_{x}

So far, the presented RDE-ECMS has been applied to warm engine conditions. A future evolution of this strategy will be to investigate the control of tailpipe NO_{x}_{x}

The authors declare no conflicts of interest.

_{x}

_{x}

_{2}and NO

_{x}

_{x}emission potential investigation and trade-off of a hybrid electric vehicle based on dynamic programming

_{x}

_{x}

_{x}

A hybrid electric vehicle (HEV) architecture considered in this paper: pre-transmission parallel HEV.

Efficiency map (in %) of the electric motor, including the inverter losses. The underlying measurement data were obtained for a 25-kW electric motor. The torque axis was then scaled linearly with the nominal power [

Experimental data for the fuel efficiency and the NO_{x}_{x}

Heuristic gearshift table.

Relationship between the equivalence factors and the trade-off between the fuel consumption and NO_{x}_{NOx}_{0}; and (_{x}_{NOx}

Controller structure.

Comparison of results obtained with the simulation and with the hardware-in-the-loop (HIL) experiment on three repetitions of the Worldwide Harmonized Light Vehicles Test Procedure (WLTP).

Comparison of the simulation results to the hardware-in-the-loop (HIL) experiments. The equivalent NO_{x}_{NOx}_{,eq}) and the equivalent fuel consumption (_{fuel,eq}) take into account the correction based on the SOC deviation between the initial and the final SOC. (_{NOx}_{NOx}_{NO}_{x}_{NOx}

Comparison of the results obtained with the non-adaptive equivalent consumption minimization strategy (ECMS) (“α_{NOx}_{NO}_{x}_{fuel,eq}; (_{m̄}_{fuel,eq} of the RDE-ECMS over the non-adaptive ECMS; and (_{x}_{NOx}_{,eq}.

Performance of the RDE-ECMS and the non-adaptive ECMS compared to the non-causal optimal trade-off.

Nominal data of the vehicle and of the powertrain components. SOC: state of charge.

Wheel radius | _{w} |
0.32 m |

Air density | ρ_{air} |
1.24 kg/m^{3} |

Effective frontal area | _{d} · |
0.60 m^{2} |

Rolling friction coefficient | _{r} |
0.012 |

Gravitational constant | a_{g} |
9.81 m/s^{2} |

Total vehicle mass | _{v} |
1827 kg |

Total inertia of the wheels | Θ_{w} |
4.78 kg m^{2} |

Inertia of the motor | Θ_{m} |
0.0435 kg m^{2} |

Gear ratios | _{g} |
[10.8, 7.1, 4.7, 3.4, 2.5, 2.0, 1.8] |

Gearbox efficiency parameter | η_{g}_{,0} |
0.95 |

η_{g}_{,1} |
0.02 1/(rad/s) | |

ω_{g,1} |
400 rad/s | |

Nominal motor power | - | 40 kW |

Maximum motor speed | ω_{m,max} |
628 rad/s |

Nominal engine power | - | 170 kW |

Minimum engine speed | ω_{e}_{,min} |
105 rad/s |

Maximum engine speed | ω_{e,max} |
471 rad/s |

Maximum battery capacity | _{0} |
7.64 A h |

Open circuit voltage | _{oc} |
263 V |

Battery internal resistance | _{i} |
0.24 Ω |

Minimum battery current | _{b,min} |
−166 A |

Maximum battery current | _{b,max} |
229 A |

Minimum SOC | _{min} |
0.20 |

Maximum SOC | _{max} |
0.80 |

Auxiliary power demand | _{aux} |
400 W |