# Optimal Control for Cleaner Hybrid Vehicles: A Backward Approach

^{1}

^{2}

^{*}

^{†}

^{‡}

^{§}

## Abstract

**:**

_{x}) of sustaining hybrid vehicles. Both cold start and normal operations are considered. The problem formulation includes two state variables: battery state of energy and catalyst temperature; and three control variables: torque repartition between engine and motor, spark advance, and equivalence ratio. Optimal results were obtained by delaying the first engine crank after the urban part of the mission. The results show that a quick catalyst light off is performed. Once the catalyst is primed, special control parameters values are adopted to operate the engine.

## 1. Introduction

_{2}emissions increased by 1.9% in 2018 and 0.9% in 2019. The transportation sector is responsible for 21% of these emissions, with a slightly increasing trend. Several of the main CO

_{2}emitting countries reduced their emissions in 2019 compared to 2018, including the European Union (by 3.8%), United States (by 2.6%), Japan (by 2.1%), and Russia (by 0.8%). Conversely, China (3.4%) and India (1.6%) increased their emissions in 2019, representing 30.3%, and 6.8% of the global total, respectively.

_{2}) was linked to 54,000 premature deaths across the 27 EU member states and the United Kingdom. Particulate matter and black carbon also affect human health, from impairing the respiratory system to causing premature death [3,4]. In the case of France, the road transportation sector is responsible for 57% of total NO

_{x}emission [5]. Globally, fleet renewal leads to a decrease in emissions despite traffic growth. Emissions of particulate matter PM10, PM2.5, PM1.0, and black carbon (BC) from the road transportation sector include particulate from vehicle exhausts. It also encompasses PM from wear and tear on roads and certain vehicle components such as tires and brakes. The road sector is responsible for 15 to 20% of total PM emissions depending on their size, and 50% of black carbon is emitted by the combustion of fossil fuels in the vehicles.

## 2. Materials and Methods

#### 2.1. Base Line Application Object

#### 2.2. Optimal Control Theory

#### 2.3. Model Structure

- As a function of catalyst temperature, ${T}_{cat}$;
- Regarding the air to fuel equivalence ratio, $\varphi $.

_{x}. Figure 4 presents the shape of the curves.

- CO, HC, and NO tailpipe concentration;
- Fuel flow behavior;
- Brake torque model;
- Exhaust temperature depending on operating parameters.

#### 2.4. Problem Formulation

_{2}target value (expressed in l/100 km) of the Corporate Average Fuel Economy for cars (CAFE regulation, see [25]). These values are presented on Table 2.

- The battery state-of-charge (SOC) lies between 20% and 80% in order to avoid any damage;
- SOC variation is limited by the maximum and minimum battery currents;
- The torque of the ICE (respectively EM) is limited by torque vs speed curves in both traction and friction (respectively regenerative) modes, see Figure 6;
- Catalyst temperature can vary freely during the driving cycle.

- Initial and final values of the SOC are equal to 50% to ensure a fair comparison of scenarios without energy transfers from the battery to the tank, as the reference vehicle is a self-charging hybrid;
- The initial catalyst temperature is set to ambient;
- The final catalyst temperature is unconstrained; of all possible temperatures corresponding to a null electrical energy balance, the one that minimizes the objective function is selected.

#### 2.5. Algorithm Overview

^{®}and is presented in detail in Appendix A. It consists of calculating the cost-to-go matrix of the three-dimensional graph. This graph is defined by the time of drive cycle in the X-axis, the battery state of charge in the Y-axis, and the TWC temperature in the Z-axis. The optimal policy can only be calculated when the entire drive cycle has been performed, so the graph has to be stored in RAM or HDD in this implementation. As we consider three control variables, the lowest value of the objective function can be obtained by any of the combinations of the control variables, which considerably increases the size of the problem. Hopefully, this minimization can be done at each time step. Finally, the optimal policy is calculated with a backward approach, by breaking the decision problem into smaller subproblems.

#### 2.6. Problem Discretization

#### 2.6.1. Battery Energy Step

#### 2.6.2. Catalyst Temperature STEP

#### 2.6.3. Time Step

#### 2.6.4. Selected Optimization Parameters

- Group A:
- −
- The time step is 1 s;
- −
- The SOE step is 1500 J;
- −
- The temperature step varies linearly to limit grid errors during the natural cooling of the catalyst.

- Group B:
- −
- The time step is 2 s;
- −
- For the SOE grid, the step is 3000 J;
- −
- The temperature step is fixed. It increases from 0.2 K to 0.4 K, but the cooling rate stays at 0.2 K/s due to the increased time step.

## 3. Results

#### 3.1. Consumption Centered Scenario

_{x}(the NO

_{x}conversion efficiency falls from 100% to almost 0 in this narrow window; see Figure 4). Therefore, this strategy will not be studied.

_{2}emissions targets for cars, established at 95 g CO

_{2}/km, and the EU current regulations (EURO6) as given in Table 2.

_{2}emissions. In this example, the WLTC driving cycle consumes 20% more than its NEDC predecessor. This new cycle must be welcomed, as it is closer to real-world conditions, but it makes the 95 g CO

_{2}/km target all the more difficult to achieve.

_{x}level is almost three times higher than EURO6 limit. Figure 10 shows the cumulative emissions, while Figure 11 shows the temporal evolution of the catalyst temperature and efficiency.

_{x}, 15% has already been dispersed in the environment.

#### 3.2. Emission-Centered Scenario

_{x}emission (divided by a factor 20), thus passing the EURO6 rules (0.060 g/km). One can also notice that CO emissions increase, but still stay three times lower than EU limits.

- For the fuel, the major parameters will be the relative spark advance, which has to be close to its optimal value (${\Delta}_{SA}=0$). It is also clear that an intake pressure close to atmospheric pressure eliminates pressure drop at the intake valve, thus maximizing efficiency;
- CO is not influenced by the spark advance, but only by the equivalent air to fuel ratio. A low value of $\varphi $ is better for CO engine emission. That is not what we observed. Several factors explain this result:
- −
- In this narrow window, CO emissions variation are low;
- −
- CO efficiency is very high when catalyst is primed;
- −
- The strategy greatly reduces CO emission before catalyst light-off (see Figure 10).

- The HC level is very low, and its performance index, $\frac{{\dot{m}}_{HC}}{{\dot{m}}_{HC,ref}}$ is less than 0.1 (see Figure 12), so it has no impact on the control parameters;
- NO
_{x}clearly deeply influences the strategy; we observe a slight enrichment of the mixture ($\varphi $ = 1.008) that drives the catalyst efficiency to 0.995 (see Figure 4). Conversely, we notice an alternation of the relative spark advance between 0 and −10°. This last value means firing later than is optimal, thus reducing the temperature in the combustion chamber and the feedgas NO level. Finally, the temporal distribution of the spark advance values is questionable. It seems to be correlated with the driving power; the strategy adopts a null relative spark advance when the driving power is high, and delays ignition when the wheel power decreases. This interpretation is highlighted in Figure 18.

#### 3.3. Parametric Study on Engine Displacement

## 4. Discussion

_{x}emissions by a factor 20, and thus pass the EU6 regulation limits.

_{x}one, a very small change in the equivalence ratio or efficiency S-curve shape can have a significant influence on the results. In practice, it is not possible to control the gas chemistry with such precision in a real engine, especially in the case of transient operations that are common with hybridization. Actual equivalence ratio control strategies usually consist of oscillating around the unit value of equivalence ratio to oxidize CO and HC on the one hand and reduce NO on the other. In addition, the catalyst has oxygen storage capabilities that allow small deviations during transient operations. The model of this equivalence ratio control loop is very complex and involves many state variables, so it is not realistic to take it into account when using dynamic programming.

## 5. Conclusions

- Over a wide range of variation, the equivalence ratio acts on the trade-off between the efficiency of the pollution control system and the fuel consumption because the optimal efficiency of the engine is with a lean mixture. However, when emissions are taken into account, the optimal range of variation of the equivalence ratio is reduced by a stoichiometric proportion. In this small range of variation, the equivalence ratio plays a role in the trade-off between the reduction of NO on one hand and of HC and CO on the other. As previously mentioned, a more precise model of the influence of this variable would be necessary to deepen this analysis of equivalence ratio control;
- The optimal strategy degrades the ignition timing to optimize the catalyst priming phase, as in a conventional vehicle. Indeed, despite the reduction in the efficiency of the combustion engine, the ignition delay has a double positive effect on pollution. A significant delay in relation to optimal ignition increases the temperature of the gases at the engine outlet while reducing the concentrations of HC and NO. This ensures that the catalyst is primed as soon as possible while reducing emissions during this critical phase when the catalyst is not yet active. Once the catalyst is primed, the hybrid strategy taking into account pollutant emissions and therefore often tends to delay the advance. This impacts the temperature in the chamber and thus reduces NO emissions;
- The intake pressure is directly linked to the power developed by the internal combustion engine and acts on the one hand on the concentration of NO and HC at the engine outlet and on the other hand on the efficiency of the engine. This latter effect is predominant in the optimal strategy, as the weighted factor for fuel is high in the objective function.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Acronyms and chemical components | |

BC | Black Carbon |

BSFC | Brake Specific Fuel Consumption in g/kWh |

BTDC | Before Top Dead Center |

CAD | Crank Angle Degree |

CO | Carbon Monoxide |

CPU | Central Processing Unit |

DP | Dynamic Programming |

ECU | Engine Control Unit |

EM | Electric Machine |

HC | Unburned Hydrocarbons |

ICE | Internal Combustion Engine |

NEDC | New European Driving Cycle |

NO | Nitrogen Oxide |

NO_{x} | Nitrogen Oxides |

PM | Particulate Matter |

PMP | Pontryaguin Maximum Principle |

RAM | Random Access Memory |

SA | Spark Advance |

TWC | Three Way Catalyst |

WLTC | Worldwide harmonized Light-duty vehicles Test Cycle |

Symbols used in the equations | |

[CO] | CO concentration in ppm |

[HC] | HC concentration in ppm |

[NO] | NO concentration in ppm |

[NO${}_{min}$] | Saturation value for [NO model] |

$\alpha $ | Weighting factor expressing the relative influence of consumption versus CO emissions |

$\beta $ | Weighting factor expressing the relative influence of consumption versus HC emissions |

$\gamma $ | Weighting factor expressing the relative influence of consumption versus NO emissions |

${\Delta}_{SA}$ | Relative spark advance in CAD BTDC |

$\delta t$ | Discretization step for the time in s |

$\varphi $ | Air to fuel equivalence ratio (/) |

$c{o}_{x}$ | Adjusting parameters for the [CO model] |

$h{c}_{x}$ | Adjusting parameters for the [HC model] |

${\dot{m}}_{f,ref}$ | Reference fuel mass flow in g·s^{−1} |

${\dot{m}}_{fuel}$ | Fuel mass flow in g·s^{−1} |

${\dot{m}}_{x,ref}$ | Reference mass flow of pollutant species X in g·s^{−1} |

${\dot{m}}_{x}$ | Mass flow of pollutant species X in g·s^{−1} |

$n{o}_{x}$ | Adjusting parameters for the [NO model] |

${P}_{intake}$ | Intake pressure in bar |

$SA$ | Spark advance (CAD BTDC) |

$SOC$ | Battery state of charge in % |

$SO{E}_{bat}$ | Battery state of energy in J |

## Appendix A. Algorithm Overview

^{®}and comprises three main parts:

- Constructing the graph: this consists of calculating at each time step the cost-to-go function represented by all the feasible commands that satisfy the constraints of the system. This is the costliest part (in term of CPU and RAM), as all paths between the two time steps have to be calculated in term of SOC and temperature, fuel, emissions, and finally cost-to-go function. The size of certain variables depends on the states and the size others depend on the commands. Finally, they are classified using a unique identifier denoted hereafter:$n=size\left(costToGo\right)=size\left(SOC\right)\ast size\left(Temperature\right)\ast size\left(commands\right)$A minimization is done at this stage; several commands lead to a particular node in the graph, specified by a value of $(SOC,temperature)$. For each node, the command that minimizes the cost-to-go function must be found (see below). This is the place in the algorithm dealing with the largest number of variables. It initially uses MATLAB’s sortrows function. As explained in the help function:[B,I] = sortrows(A,…) also returns an index vector I, which describes the order of the sorted rows, namely, B = A(I,:).By making a vector A of size (n, 2) with the identifier in the first column and the cost-to-go function in the second column, one can ordinate the identifier in B and the corresponding indices in I. At this step, it is easy to get the indices of the minimal cost in the original vectors. The job is done, but the drawback with this function is that the element of Matrix A must be of the same type, i.e., double precision floating numbers (for input arguments and output arguments also). This uses a large amount of memory. By developing our own C-sfunction in MATLAB with a fast top-down implementation of a merge-sort algorithm, we created our own prototype, thus enabling an identifier of type uint32, such as this one (freeing a lot of memory):[id_in_order(uint32), order_cost(uint32)] = csortrows(id(uint32), cost(double))
- Storing the variables: to solve the graph (third part of the algorithm), the vector of the optimal commands($size\left(OptCommands\right)=size\left(soc\right)\ast size\left(Temperature\right)$)has to be stored at each time step along with the previous nodes to know where they come from.
- Finding the optimal path: the optimal path is characterized by the minimal cost-to-go function at the end of the driving cycle that satisfies the SOC constraints. Obtaining this index allows one, starting from the end and going backwards step by step up to the beginning of the cycle, to know the optimal commands and the preceding node. Once this is done, all intermediate variables can be recalculated.

## References

- Crippa, M.; Oreggioni, G.; Guizzardi, D.; Muntean, M.; Schaaf, E.; Lo Vullo, E.; Solazzo, E.; Monforti-Ferrario, F.; Olivier, J.; Vignati, E. Fossil CO
_{2}and GHG Emissions of All World Countries: 2020 report; Publications Office of the European Union: Luxembourg, 2020. [Google Scholar] [CrossRef] - Vincent, C.; Peyaud, V.; Laarman, O.; Six, D.; Gilbert, A.; Gillet-Chaulet, F.; Berthier, É.; Morin, S.; Verfaillie, D.; Rabatel, A.; et al. Déclin des deux plus grands glaciers des Alpes françaises au cours du XXIe siècle: Argentière et Mer de Glace. La Météorol.
**2019**, 1, 49. [Google Scholar] [CrossRef] - European Environment Agency. Emissions of Air Pollutants from Transport; Technical Report; European Environment Agency: Kobenhavn, Denmark, 2019. [Google Scholar]
- European Environment Agency. Air Pollution; Technical Report; European Environment Agency: Kobenhavn, Denmark, 2020. [Google Scholar]
- CITEPA. Gaz à Effet de Serre et Polluants Atmosphériques—Bilan des Emissions en France de 1990 à 2017; Rapport National d’Inventaire: Paris, France, 2019. [Google Scholar]
- Kalghatgi, G. Is it really the end of internal combustion engines and petroleum in transport? Appl. Energy
**2018**, 225, 965–974. [Google Scholar] [CrossRef] - Helmers, E.; Dietz, J.; Weiss, M. Sensitivity Analysis in the Life-Cycle Assessment of Electric vs. Combustion Engine Cars under Approximate Real-World Conditions. Sustainability
**2020**, 12, 1241. [Google Scholar] [CrossRef][Green Version] - IEA. Global Energy Review 2020; Technical Report; IEA: Paris, France, 2020. [Google Scholar]
- Masson-Delmotte, V.; Zhai, P.; Pirani, A.; Connors, S.; Péan, C.; Berger, S.; Caud, N.; Chen, Y.; Goldfarb, L.; Gomis, M.; et al. Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change; Cambridge University Press: Cambridge, UK, 2021. [Google Scholar]
- Leach, F.; Kalghatgi, G.; Stone, R.; Miles, P. The scope for improving the efficiency and environmental impact of internal combustion engines. Transp. Eng.
**2020**, 1, 100005. [Google Scholar] [CrossRef] - International Council on Clean Transportation. Too Low to Be True? How to Measure Fuel Consumption and CO2 Emissions of Plug-in Hybrid Vehicles, Today and in the Future; The National Academies of Sciences, Engineering, and Medicine: Washington, DC, USA, 2017. [Google Scholar]
- Mensing, F.; Bideaux, E.; Trigui, R.; Ribet, J.; Jeanneret, B. Eco-driving: An economic or ecologic driving style? Transp. Res. Part C
**2014**, 38, 110–121. [Google Scholar] [CrossRef] - Pandey, V.; Jeanneret, B.; Gillet, S.; Keromnes, A.; Le Moyne, L. A simplified thermal model for the three way catalytic converter. In Proceedings of the TAP 2016, 21st International Transport and Air Pollution Conference, Lyon, France, 24–26 May 2016; p. 6. [Google Scholar]
- Brandt, E.; Wang, Y.; Grizzle, J.W. Dynamic modeling of a three-Way catalyst for SI engine Exhaust emission control. IEEE Trans. Control. Syst. Technol.
**2000**, 8, 767–776. [Google Scholar] [CrossRef] - Brandt, E.P.; Grizzle, J.W. Three-way catalyst diagnostics for advanced emissions control systems. In Proceedings of the 2001 American Control Conference (Cat. No.01CH37148), Arlington, VA, USA, 25–27 June 2001; Volume 5, pp. 3305–3311. [Google Scholar] [CrossRef]
- Shaw, B.T.; Fisher, G.D.; Hedrick, J.K. A simplified coldstart catalyst thermal model to reduce hydrocarbon emissions. IFAC
**2002**, 35, 312. [Google Scholar] [CrossRef][Green Version] - Hedinger, R.; Elbert, P.; Onder, C. Optimal Cold-Start Control of a Gasoline Engine. Energies
**2017**, 10, 1548. [Google Scholar] [CrossRef][Green Version] - Yusuf, A.A.; Inambao, F.L. Effect of cold start emissions from gasoline-fueled engines of light-duty vehicles at low and high ambient temperatures: Recent trends. Case Stud. Therm. Eng.
**2019**, 14, 100417. [Google Scholar] [CrossRef] - Fontaras, G.; Pistikopoulos, P.; Samaras, Z. Experimental evaluation of hybrid vehicle fuel economy and pollutant emissions over real-world simulation driving cycles. Atmos. Environ.
**2008**, 42, 4023–4035. [Google Scholar] [CrossRef] - Guille des Buttes, A. Optimisation Conjointe de la Consommation d’Essence et des Emissions de Polluants Réglementés pour un Véhicule Hybride Essence-Electrique d’Architecture Parallèle. PhD Thesis, École Doctorale ED160 EEA, Lyon, France, 2021. [Google Scholar]
- Zhang, P.; Yan, F.; Du, C. A comprehensive analysis of energy management strategies for hybrid electric vehicles based on bibliometrics. Renew. Sustain. Energy Rev.
**2015**, 48, 88–104. [Google Scholar] [CrossRef] - Saerens, B.; Diehl, M.; Bulck, E. Optimal Control Using Pontryagin’s Maximum Principle and Dynamic Programming; Springer: London, UK, 2010. [Google Scholar]
- Guille des Buttes, A.; Jeanneret, B.; Kéromnès, A.; Le Moyne, L.; Pélissier, S. Energy management strategy to reduce pollutant emissions during the catalyst light-off of parallel hybrid vehicles. Appl. Energy
**2020**, 266, 114866. [Google Scholar] [CrossRef] - Jeanneret, B.; Guille Des Buttes, A.; Pelluet, J.; Keromnes, A.; Pélissier, S.; Le Moyne, L. Optimal Control of a Spark Ignition Engine Including Cold Start Operations for Consumption/Emissions Compromises. Appl. Sci.
**2021**, 11, 971. [Google Scholar] [CrossRef] - European Commission. Regulation (EU) 2019/631; Technical Report; European Commission: Luxembourg, 2020. [Google Scholar]
- Bellman, R. Dynamic Programming; Dover Books on Computer Science Series; Dover Publications: New York, NY, USA, 2003. [Google Scholar]
- Sundstrom, O.; Guzzella, L. A generic dynamic programming Matlab function. In Proceedings of the 2009 IEEE Control Applications, (CCA) Intelligent Control, (ISIC), St. Petersburg, Russia, 8–10 July 2009; pp. 1625–1630. [Google Scholar] [CrossRef]
- Elbert, P.; Ebbesen, S.; Guzzella, L. Implementation of Dynamic Programming for n-Dimensional Optimal Control Problems with Final State Constraints. IEEE Trans. Control. Syst. Technol.
**2013**, 21, 924–931. [Google Scholar] [CrossRef] - Guzzella, L.; Onder, C.H. Introduction to Modeling and Control of Internal Combustion Engine Systems; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar] [CrossRef]
- Vinot, E.; Scordia, J.; Trigui, R.; Jeanneret, B.; Badin, F. Model simulation, validation and case study of the 2004 THS of Toyota Prius. Int. J. Veh. Syst. Model. Test.
**2008**, 3, 139–167. [Google Scholar] [CrossRef] - Ngo, V.; Hofman, T.; Steinbuch, M.; Serrarens, A. Effect of Gear Shift and Engine Start Losses on Control Strategies for Hybrid Electric Vehicles. World Electr. Veh. J.
**2012**, 5, 125–136. [Google Scholar] [CrossRef][Green Version] - Ngo, V.D.; Navarrete, J.A.C.; Hofman, T.; Steinbuch, M.; Serrarens, A. Optimal gear shift strategies for fuel economy and driveability. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2013**, 227, 1398–1413. [Google Scholar] [CrossRef] - Cedrone, K.; Cheng, W. SI engine control in the cold-fast-idle period for low HC emissions and fast catalyst light off. SAE Int. J. Engines
**2014**, 7, 968–976. [Google Scholar] [CrossRef][Green Version]

**Figure 4.**Catalyst conversion efficiency model (

**a**): regarding air to fuel equivalence ratio; (

**b**): function of catalyst temperature.

**Figure 9.**Operating points for consumption-centered strategy in the WLTC driving cycle. (red squares mean that the catalyst temperature is higher than the light-off temperature; blue squares mean that it is lower). (

**a**): engine; (

**b**): electric motor.

**Figure 10.**Cumulative emissions for consumption-centered strategy in the WLTC driving cycle. (

**a**): Carbon monoxyde; (

**b**): unburned Hydrocarbons; (

**c**): Nytrogen oxides.

**Figure 11.**Temperature evolution and catalyst efficiency for the consumption-centered strategy in the WLTC driving cycle.

**Figure 15.**Operating points for the emission-centered strategy in the WLTC driving cycle. (red squares mean that the catalyst temperature is higher than the light-off temperature; blue squares mean that it is lower). (

**a**): engine; (

**b**): electric motor.

**Figure 16.**Control parameters with the emission-centered strategy in the WLTC driving cycle. (

**a**): intake pressure; (

**b**): relative spark advance; (

**c**): air to fuel equivalence ratio.

**Figure 17.**Pollutant emissions with the emission-centered strategy in the WLTC driving cycle. (

**a**): Carbon monoxyde; (

**b**): unburned Hydrocarbons; (

**c**): Nytrogen oxides.

**Figure 18.**Correlation between relative spark advance and wheel power (zoom during the highway part of the WLTC driving cycle). (

**a**): intake pressure; (

**b**): relative spark advance; (

**c**): wheel power.

**Figure 20.**Comparison of the performance index between Group A and Group B discretization in the WLTC driving cycle.

**Figure 21.**SOC profile evolution between Group A and Group B discretization in the WLTC driving cycle.

**Table 1.**Main vehicle characteristics (Chassis Peugeot 308 SW—Model/Year 2009) and models used in this study.

Component | Size | Type of Model |
---|---|---|

IDI gasoline engine | 1.6 L | Mean value engine model |

Three-way catalytic converter | EURO 6 compliant | 0D model |

PRIUS II type electric motor | resized to 25 kW | Quasistatic map |

(mechanical power) | ||

Kokam Li-ion Battery | 31 kW–1.7 kWh | Voltage source and resistance |

Gear box 20DP42 | 5 gears | Constant efficiency |

Vehicle weight | 1504 kg | Longitudinal forces |

Fuel Cons. | CO_{2} | CO | HC | NO_{x} |
---|---|---|---|---|

l/100 km | g/km | g/km | g/km | g/km |

4.0 | 95 | 1.0 | 0.1 | 0.060 |

$\mathit{\delta}\left({\mathit{SOE}}_{\mathit{bat}}\right)$ | Fuel Cons. | CO | HC | NO_{x} | Form Factor |
---|---|---|---|---|---|

J | l/100 km (%) | g/km | g/km | g/km | |

100 | 4.392 (0.0) | 0,131 | 0.009 | 0.165 | 1.0 |

500 | 4.393 (0.03) | 0.133 | 0.009 | 0.167 | 0.997 |

750 | 4.395 (0.06) | 0.133 | 0.009 | 0.167 | 0.996 |

1500 | 4.398 (0.13) | 0.133 | 0.009 | 0.167 | 0.993 |

3000 | 4.403 (0.26) | 0.132 | 0.009 | 0.166 | 0.987 |

5000 | 4.416 (0.54) | 0.137 | 0.010 | 0.165 | 0.978 |

**Table 4.**Variation of the traction energy over the WLTC cycle as a function of the time step (10 Hz reference).

Step Time (s) | $\mathit{\delta}\mathit{t}=0.1$ | $\mathit{\delta}\mathit{t}=1.0$ | $\mathit{\delta}\mathit{t}=2.0$ | $\mathit{\delta}\mathit{t}=5.0$ |
---|---|---|---|---|

Traction Energy (Wh/km) | 135.9 | 135.2 | 133.0 | 124.7 |

Variation (%) | Reference | −0.5 | −2.2 | −8.2 |

Optim. | Driving | Fuel Cons. | CO_{2} | CO | HC | NO_{x} |
---|---|---|---|---|---|---|

Parameter | Cycle | l/100 km | g/km | g/km | g/km | g/km |

Group A | NEDC | 3.58 | 84 | 0.213 | 0.014 | 0.171 |

Group A | WLTC | 4.40 | 103 | 0.133 | 0.009 | 0.167 |

Variable | Steps | Min. Value | Max. Value |
---|---|---|---|

Optimization parameter | Group A | not applicable | |

Equivalence ratio | 0.005 | 0.993 | 1.008 |

Relative spark advance | 10° | −30° | 0° |

Step time | 1 s | not applicable |

Drive | Fuel Cons. | CO_{2} | CO | HC | NO_{x} | RAM | Calc. |
---|---|---|---|---|---|---|---|

Cycle | l/100 km | g/km | g/km | g/km | g/km | Gb | Time (h) |

NEDC | 3.77 | 89 | 0.302 | 0.007 | 0.009 | 133 | 57 |

WLTC | 4.52 | 106 | 0.306 | 0.007 | 0.008 | 155 | 95 |

Variable | Steps | Min. Value | Max. Value |
---|---|---|---|

Optimization parameters | Group B | not applicable | |

Fuel/air eq. ratio | 0.15 | 0.993 | 1.008 |

Relative spark advance | 10° | −30° | 0° |

Step time | 2 s | not applicable |

Engine | Fuel Cons. | CO_{2} | CO | HC | NO_{x} | RAM | Calc. |
---|---|---|---|---|---|---|---|

Displ. | l/100 km | g/km | g/km | g/km | g/km | Gb | Time (h) |

0.8 | 4.26 | 100 | 0.291 | 0.007 | 0.011 | 19 | 5 |

1.0 | 4.26 | 100 | 0.294 | 0.006 | 0.008 | 22 | 7 |

1.2 | 4.29 | 101 | 0.299 | 0.007 | 0.009 | (Results not available) | 9 |

1.4 | 4.38 | 103 | 0.298 | 0.006 | 0.008 | (Results not available) | 10 |

1.6 | 4.48 | 105 | 0.304 | 0.006 | 0.008 | 34 | 12 |

1.8 | 4.57 | 108 | 0.312 | 0.007 | 0.008 | 40 | 12 |

2.0 | 4.69 | 110 | 0.319 | 0.007 | 0.008 | 40 | 13 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jeanneret, B.; Guille Des Buttes, A.; Keromnes, A.; Pélissier, S.; Le Moyne, L. Optimal Control for Cleaner Hybrid Vehicles: A Backward Approach. *Appl. Sci.* **2022**, *12*, 578.
https://doi.org/10.3390/app12020578

**AMA Style**

Jeanneret B, Guille Des Buttes A, Keromnes A, Pélissier S, Le Moyne L. Optimal Control for Cleaner Hybrid Vehicles: A Backward Approach. *Applied Sciences*. 2022; 12(2):578.
https://doi.org/10.3390/app12020578

**Chicago/Turabian Style**

Jeanneret, Bruno, Alice Guille Des Buttes, Alan Keromnes, Serge Pélissier, and Luis Le Moyne. 2022. "Optimal Control for Cleaner Hybrid Vehicles: A Backward Approach" *Applied Sciences* 12, no. 2: 578.
https://doi.org/10.3390/app12020578