# Optimal Control of a Spark Ignition Engine Including Cold Start Operations for Consumption/Emissions Compromises

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## Abstract

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_{x}) as a function of engine operating parameters. Such parameters include engine speed, intake manifold pressure, equivalence ratio, and spark advance. The proposed models provide accurate predictions over a large range of engine operating conditions. The adequate accuracy and low computational burden of the models are promising in the context of optimal control theory. Dynamic programming is applied in order to find the best operating parameters that define trade-off between fuel consumption and emissions over driving cycles.

## 1. Introduction

_{x}), carbon monoxide (CO) and particulate matters. More and more efficient and expensive after-treatment systems have been introduced in order to reduce pollutant emissions from internal combustion engines such as the three way catalyst (TWC), the Diesel particulate filter (DPF) and the selective catalytic reduction (SCR). These devices are highly efficient but require specific operating conditions, as well as long and complex calibration procedures which induce a very high development cost of the internal combustion engine [4]. In the past, only new vehicles had to go through homologation procedures in a vehicle test bench over a predefined speed cycle with controlled environments. Now, Real Driving Emissions (RDE) measurements are also performed and this makes calibration procedures even more complex and expensive.

_{x}) emissions. CO and HC emissions mainly depend on the physical development of the flame front during the combustion process. The accurate prediction of pollutant emissions requires the coupling of 3D CFD model with chemical kinetics mechanisms in order to fully describe their production inside the combustion chamber on a crank-angle basis [10]. Such models lead to very high computation costs, unacceptable with a multi disciplinary approach such as vehicle design. Even 0D/1D models reduce the complexity of the problem but still have a computing time that cannot fit with optimization process over an integrated approach [11].

- A simple backward model of the drivetrain (i.e., from drive cycle to engine) as described by [18] to calculate at each time step the engine speed and torque.
- Simple forms to describe the behavior of emissions with a semi empirical model based on experimental data.
- A catalytic converter model describing its thermal and efficiency behaviors with only one spatial discretization as proposed by Pandey et al. [6].
- A dynamic programming algorithm to calculate the optimal control parameters of the engine over an entire drive cycle by minimizing a weighted sum of pollutant emissions and fuel consumption. This implementation was derived from [20].

## 2. Models and Methods

#### 2.1. Experimental Setup and Procedure

#### 2.2. Emissions Model

#### 2.2.1. CO Model

_{2}. The coefficient of determination is 0.99 and the mean error between the data and the predictions is 31%.

#### 2.2.2. HC Model

#### 2.2.3. NO Model

- Early NO are formed at the flame front and in fuel-rich areas. They result from the combination of nitrogen N
_{2}with hydrocarbon radicals (C, C2, CH_{2}or CH_{2}) to give the intermediate products HN, HCN, CN or CNH_{2}. These can recombine with oxygen to give nitrogen oxide NO [21]. - The NO-fuel are derived from the oxidation of the nitrogen atoms present in the fuel [26].

_{2}via reactions that require less energy. The concentration of NO is therefore at a maximum and decreases in both rich and lean mixes. This corresponds well to the behavior observed in Figure 8 that we have modeled by a second degree function ${(\varphi -0.9)}^{2}$.

#### 2.3. Internal Combustion Engine Model

#### 2.3.1. Fuel Flow

#### 2.3.2. Brake Torque

#### 2.3.3. Exhaust Temperature

#### 2.4. Catalyst Model

#### 2.5. Vehicle Dynamics

#### 2.6. Problem Formulation and Discretization Values

## 3. Results

#### 3.1. Pollution Centered Scenario

_{x}emissions before light-off are 11 times lower than in the nominal configuration.

- Catalyst light-off: during this period, we observe a late spark advance (10 to 15° compared to optimal ignition). This is combined with a lean mixture, $\mathsf{\Phi}=0.9$. These two parameters increase the exhaust gas temperature. For constant engine torque, the reduced fuel conversion efficiency associated with late combustion requires greater fuel and air flow rate (see intake pressure in Figure 26). As a consequence, exhaust enthalpy and convective effects are high. This shortens the period where catalyst efficiency is low.
- warm catalyst period: the strategy adopts a stoichiometric combustion which represents a good compromise to oxidize CO and HC while reducing NO
_{x}. In the same time, we observe a relative delay in spark advance to lower exhaust gas temperature and reduce NO (and HC to a lesser extent) formation. The emission of CO depends only on the fuel/air ratio (see Equation (1)) of the mixture, and stoechiometry is a good compromise for this pollutant. This delay increases with the brake power as it can be seen in the extra urban part of the NEDC cycle, again to lower NO emissions.

#### 3.2. Trade-off between Fuel Consumption and Pollutant Emissions

#### 3.3. Worldwide Harmonized Light-Duty Vehicles Test Cycle

## 4. Discussion

#### 4.1. Originality of the Work

- defining the maps to be used by real time control strategies for both hot and cold catalyst
- serving as reference values for comparison with online methods

#### 4.2. Discussion of the Method

#### 4.3. Challenges and Limitations

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Acronyms and chemical components | |

BTDC | Before Top Dead Center |

CAD | Crank Angle Degree |

CFD | computational fluid dynamics |

CO | carbon monoxide |

HC | unburned hydrocarbons |

ICE | internal combustion engine |

NEDC | New European Driving Cycle |

NO | nitrogen oxide |

NO_{x} | nitrogen oxides |

TWC | three way catalyst |

WLTC | Worldwide harmonized Light-duty vehicles Test Cycle |

Symbols used in the equations | |

$\left[CO\right]$ | CO concentration in ppm |

$\left[HC\right]$ | HC concentration in ppm |

$\left[{N}_{2}\right]$ | nitrogen equilibrium concentration |

$\left[NO\right]$ | NO concentration in ppm |

${\left[NO\right]}_{min}$ | saturation value for [NO] model |

$\left[{O}_{2}\right]$ | oxygen equilibrium concentration |

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

$\delta T$ | discretization step for catalyst temperature (state variable) in K |

$\mathsf{\Delta}{T}_{cata}$ | variation of catalyst temperature over a step time in K |

${\mathsf{\Delta}}_{SA,min}$ | adjusting parameter (effect of spark advance on exhaust temperature) |

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

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

${\dot{Q}}_{amb}$ | thermal flux of catalyst with ambient in W |

${\dot{Q}}_{gaz}$ | thermal flux of catalyst with exhaust gas in W |

${\dot{Q}}_{reac}$ | thermal flux of catalyst corresponding to chemical reaction in W |

${\eta}_{\varphi}$ | fuel/air eq ration’s contribution to engine efficiency |

${\eta}_{c0}$ | reference combustion efficiency |

${\eta}_{comb}$ | combustion efficiency |

${\eta}_{DF}$ | differential efficiency |

${\eta}_{fi}$ | indicated fuel efficiency |

${\eta}_{GB}$ | gear box efficiency |

${\eta}_{SA}$ | spark advance’s contribution to engine efficiency |

${\eta}_{vol,\varphi}$ | Equivalence ratio’s contribution to volumetric efficiency |

${\eta}_{vol}$ | volumetric efficiency of the engine |

${\mathsf{\Gamma}}_{f}$ | friction torque in N·m |

${\mathsf{\Gamma}}_{ICE}$ | brake torque of internal combustion engine in N·m |

${\mathsf{\Gamma}}_{wheel}$ | Wheel torque in N·m |

${\omega}_{ICE}$ | engine speed in rad·s^{−1} |

${\omega}_{idle}$ | idle speed in rad·s^{−1} |

$\varphi $ | fuel/air equivalence ratio |

$\rho $ | air density in kg·m^{−3} |

${a}_{\varphi}$ | adjusting parameter (effect of fuel/air eq ratio on exhaust temperature) |

${a}_{N}$ | adjusting parameter (effect of engine speed on exhaust temperature) |

${a}_{Pintake}$ | adjusting parameter (effect of intake pressure on exhaust temperature) |

${a}_{SA}$ | adjusting parameter (effect of spark advance on exhaust temperature) |

${b}_{\varphi}$ | adjusting parameter (effect of fuel/air eq ratio on exhaust temperature) |

${b}_{N}$ | adjusting parameter (effect of engine speed on exhaust temperature)) |

${b}_{Pintake}$ | adjusting parameter (effect of intake pressure on exhaust temperature) |

${b}_{SA}$ | adjusting parameter (effect of spark advance on exhaust temperature) |

${C}_{p,gaz}$ | thermal capacity of exhaust gas in J·K^{−1}·kg^{−1} |

${C}_{p}$ | thermal capacity of the catalyst in J·K^{−1}·kg^{−1} |

${C}_{x}$ | drag coefficient |

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

${F}_{res}$ | rolling resistance force in N |

g | acceleration of gravity in m·s^{−2} |

$h{c}_{X}$ | adjusting parameters for the [HC] model |

${J}_{ICE}$ | Engine inertia in kg·m^{2} |

${J}_{veh}$ | vehicle inertia in kg·m^{2} |

${J}_{wheel}$ | Inertia of a wheel in kg·m^{2} |

${k}_{DF}$ | differential gear ratio |

${k}_{GB}$ | gear box ratio |

${k}_{N}$ | effect of engine speed on exhaust temperature |

${k}_{phi}$ | effect of fuel/air eq. ratio on exhaust temperature |

${k}_{Pintake}$ | effect of intake pressure on exhaust temperaturet |

${k}_{rol}$ | rolling coefficient |

${k}_{SA}$ | effect of spark advance on exhaust temperature |

$LHV$ | low heating value of the fuel in $\mathrm{W}\xb7\mathrm{s}\xb7{\mathrm{g}}^{-1}$ |

${m}_{cata}$ | catalyst mass in kg |

${\dot{m}}_{f,ref}$ | average fuel mass flow of the reference vehicle running the NEDC test cycle in g·s^{-1} |

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

${m}_{veh}$ | vehicle mass in kg |

${\dot{m}}_{X,ref}$ | average mass flow of pollutant species X on the NEDC test cycle in g·s^{−1} |

${\dot{m}}_{X}$ | mass flow of each pollutant X in g·s^{−1} |

N | engine speed in rpm |

$n{o}_{X}$ | adjusting parameters for the [NO] model |

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

${R}_{c}$ | number of revolutions per combustion cycle (2 for four stroke engine) |

${R}_{gaz}$ | perfect gas constant in J·mol^{−1}·K^{−1} |

${R}_{tire}$ | tire radius in m |

${S}_{F}$ | frontal area of the vehicle en m^{2} |

$SA$ | spark Advance (CAD BTDC) |

$S{A}_{opti}$ | optimal spark advance (maximum torque) in CAD BTDC |

T | temperature of the gas in K |

t | time in s |

${T}_{amb}$ | ambiant temperature in K |

${T}_{exh}$ | exhaust temperature in K |

${T}_{intake}$ | intake temperature in K |

v | vehicle speed in m·s^{−1} |

${V}_{d}$ | displacement volume in liter |

${({m}_{air}/{m}_{fuel})}_{st}$ | air to fuel ratio at stoichiometry |

## Appendix A. Model Parameters

#### Appendix A.1. Engine Parameters

${\mathbf{co}}_{1}$ | 164.073 | ${\mathbf{hc}}_{1}$ | −0.0866 | ${\mathbf{no}}_{1}$ | 4.666 |

${\mathbf{co}}_{\mathbf{2}}$ | 0.00217 | ${\mathbf{hc}}_{\mathbf{2}}$ | 4976 | ${\mathbf{no}}_{\mathbf{2}}$ | −73.687 |

${\mathbf{hc}}_{\mathbf{3}}$ | 5.864 | ${\mathbf{no}}_{\mathbf{3}}$ | 105.7 | ||

${\mathbf{hc}}_{\mathbf{4}}$ | −0.1995 | ${\mathbf{no}}_{\mathbf{4}}$ | −2669 | ||

${\mathbf{hc}}_{\mathbf{5}}$ | 480.4 | ${\left[\mathbf{NO}\right]}_{\mathbf{min}}$ | 300 |

N (rpm) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

500 | 750 | 1000 | 1500 | 2000 | 2500 | 3000 | 3500 | 3750 | 4000 | 4500 | 5000 | 5500 | 6000 | ||

${\mathbf{P}}_{\mathbf{intake}}$(mBar) | 200 | 29 | 30 | 33 | 36 | 38 | 40 | 40 | 40 | 44.1 | 40 | 40 | 40 | 40 | 40 |

300 | 30 | 30 | 30 | 33 | 36 | 40.1 | 40.1 | 38,1 | 42.1 | 40.1 | 40 | 40 | 40 | 40 | |

400 | 28 | 28 | 30 | 32 | 33.3 | 35.1 | 35.1 | 34.1 | 34.1 | 34.1 | 34 | 34 | 34 | 34 | |

500 | 22 | 25 | 28 | 30 | 31.1 | 33.1 | 31.1 | 29.1 | 32.1 | 31.1 | 31 | 31 | 31 | 31 | |

600 | 20 | 23 | 25 | 27 | 28.1 | 29.1 | 28.1 | 27.1 | 29.1 | 30.1 | 29 | 29 | 29 | 29 | |

700 | 19 | 20 | 20 | 21 | 23.1 | 27.1 | 28.1 | 26.1 | 26.1 | 26.1 | 26 | 26 | 26 | 26 | |

800 | 21 | 20 | 20 | 24 | 21.1 | 26.1 | 28.1 | 24.1 | 25.1 | 24.1 | 25 | 25 | 25 | 25 | |

900 | 20 | 22 | 20 | 22 | 21.1 | 24.1 | 24.1 | 24.1 | 24.1 | 24.1 | 24 | 24 | 24 | 24 | |

1000 | 20 | 20 | 20 | 17 | 19.1 | 21.1 | 21.1 | 20.1 | 23.1 | 24.1 | 21 | 21 | 21 | 29 |

${\mathit{R}}_{\mathit{c}}$ | 2 | ${\mathit{V}}_{\mathit{c}\mathit{y}\mathit{l}}$ | 1.8 |

${\eta}_{c0}$ | 0.98 | ${\eta}_{fi}$ | 0.43 |

A | 300 | ${R}_{gaz}$ | 8.314 |

B | 2000 | $LHV$ | 44,000 |

${a}_{SA}$ | 5.507 · 10^{−5} | ${({m}_{air}/{m}_{fuel})}_{st}$ | 14.7 |

${b}_{SA}$ | 0.311 | ${\mathsf{\Delta}}_{SA,min}$ | 20 |

${C}_{p,gaz}$ | 1.2 · 10^{3} | ${a}_{N}$ | 1.468 · 10^{−4} |

${a}_{\varphi}$ | −0.771 | ${b}_{N}$ | 0.251 |

${b}_{\varphi}$ | 1.503 | ${a}_{Pintake}$ | 0.00108 |

${b}_{Pintake}$ | 0.805 |

#### Appendix A.2. Vehicle Parameters

${\mathit{\eta}}_{\mathit{D}\mathit{F}}$ | 0.97 | ${\mathit{k}}_{\mathit{r}\mathit{o}\mathit{l}}$ | 0.008 |

${\eta}_{GB}$ | 0.95 | ${S}_{F}$ | 0.728 |

$\rho $ | 1.1841 | ||

${k}_{DF}$ | 4.7647 | ${C}_{x}$ | 1 |

${k}_{GB}$ | [3.4545 1.8667 1.2903 0.9512 0.7447] | ${m}_{veh}$ | 1499.3 |

${J}_{wheel}$ | 0.7 | ||

${R}_{wheel}$ | 0.3069 | ${\omega}_{idle}$ | 78.53 |

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**Figure 2.**CO concentration as a function of control parameters: (

**a**) engine speed, (

**b**) fuel/air equivalence ratio, (

**c**) spark advance and (

**d**) intake pressure. Each curve represents a series of measurements where only the parameter in abscissa changes.

**Figure 4.**Comparison between $\left[CO\right]$ measurement and $\left[CO\right]$ prediction based on $\left[CO\right]$ model.

**Figure 5.**HC concentration as a function of control parameters (each curve represents a series of measurements where only the parameter in abscissa changes).

**Figure 7.**Comparison between $\left[HC\right]$ measurement and $\left[HC\right]$ prediction based on $\left[HC\right]$ model.

**Figure 8.**NO concentration as a function of control parameters (each curve represents a series of measurements where only the parameter in abscissa changes).

**Figure 10.**Comparison between $\left[NO\right]$ measurement and $\left[NO\right]$ prediction based on $\left[NO\right]$ model.

**Figure 11.**Emission model at ($N=2000$ rpm, $\varphi =0.9$ Bar)—relative spark advance ${\mathsf{\Delta}}_{SA}$ varies from −20° to 20° (see Equation (14)).

**Figure 13.**Comparison between ${\dot{m}}_{fuel}$ measurement and ${\dot{m}}_{fuel}$ prediction based on ${\dot{m}}_{fuel}$ model.

**Figure 15.**Comparison between ${\mathsf{\Gamma}}_{ICE}$ measurement and ${\mathsf{\Gamma}}_{ICE}$ prediction based on ${\mathsf{\Gamma}}_{ICE}$ model.

**Figure 16.**Exhaust temperature as a function of control parameters (each curve represents a series of measurements where only the parameter in abscissa changes).

**Figure 18.**Comparison between ${T}_{exh}$ measurement and ${T}_{exh}$ prediction based on ${T}_{exh}$ model.

**Figure 30.**Cumulated emissions (WLTC); yellow star: TWC light-off for the reference strategy-purple star: TWC light-off for the pollution centered strategy.

Intake manifold pressure (bar) | 0.3 0.5 0.7 0.9 1 |

Engine speed (rpm) | 800 1000 1500 2000 3000 4000 |

fuel/air eq. ratio (-) | 0.7 0.8 0.85 0.9 0.95 1 1.1 1.2 |

Spark Advance (°BTDC) | −10: 5: 50 |

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

catalyst temperature | 0.1 K | not bounded | |

Fuel/air eq. ratio | 0.01 | 0.7 | 1.1 |

Relative spark advance | 2° | −30° | 10° |

Step time | 1 s | NA |

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

Internal combustion engine | 1.8 L | Mean Value Engine Model |

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

Chassis (Peugeot 308 SW-2009) | 1470 kg | longitudinal forces |

Ref. Strategy ${\mathit{SA}}_{\mathit{opti}}$ $\mathit{\varphi}=1.0$ | $\mathit{\alpha}=5$ $\mathit{SA}\phantom{\rule{2pt}{0ex}}\mathit{var}$ $\mathit{\varphi}\phantom{\rule{2pt}{0ex}}\mathit{var}$ | Relative Variation | |
---|---|---|---|

Consumption (L/100 km) | 6.57 | 7.59 | +15% |

CO (g/km) | 0.53 | 0.144 | −73% |

HC (g/km) | 0.056 | 0.024 | −57% |

NO (g/km) | 0.212 | 0.035 | −84% |

Ref. Strategy ${\mathit{SA}}_{\mathit{opti}}$ $\mathit{\varphi}=1.0$ | $\mathit{\alpha}=5$ $\mathit{SA}\phantom{\rule{2pt}{0ex}}\mathit{var}$ $\mathit{\varphi}\phantom{\rule{2pt}{0ex}}\mathit{var}$ | Relative Variation | |
---|---|---|---|

Consumption (L/100 km) | 6.86 | 8.02 | +17% |

CO (g/km) | 0.2554 | 0.128 | −50% |

HC (g/km) | 0.029 | 0.013 | −55% |

NO (g/km) | 0.179 | 0.040 | −77% |

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## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/app11030971

**AMA Style**

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. *Applied Sciences*. 2021; 11(3):971.
https://doi.org/10.3390/app11030971

**Chicago/Turabian Style**

Jeanneret, Bruno, Alice Guille Des Buttes, Jérémy Pelluet, Alan Keromnes, Serge Pélissier, and Luis Le Moyne. 2021. "Optimal Control of a Spark Ignition Engine Including Cold Start Operations for Consumption/Emissions Compromises" *Applied Sciences* 11, no. 3: 971.
https://doi.org/10.3390/app11030971