Projected Gradient and Model Predictive Control : Optimal Energy and Pollutants Management for Hybrid Electric Vehicle

9th IFAC International Symposium on Advances in Automotive 9th IFAC International Symposium on Advances in Automotive Control 9th IFAC International Symposium on Advances in Automotive Control 9th IFAC International Advances in at www.sciencedirect.com Orléans, June Symposium 23-27, 2019 on 9th IFAC France, International Symposium onAvailable Advancesonline in Automotive Automotive 9th IFAC International Symposium on Advances in Automotive Control Orléans, France, June 23-27, 2019 Control Control Control Orléans, France, Orléans, France, June June 23-27, 23-27, 2019 2019 Orléans, Orléans, France, France, June June 23-27, 23-27, 2019 2019

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IFAC PapersOnLine 52-5 (2019) 121–127

Projected Projected Gradient Gradient and and Model Model Predictive Predictive Control Control :: Optimal Optimal Energy Energy and and Pollutants Management for Hybrid Electric Vehicle Projected Gradient and Model Predictive Control : Optimal Energy and Projected Gradient and Model Predictive Control : Optimal Energy Pollutantsand Management for Hybrid Electric Vehicle Projected Gradient Model Predictive Control : Optimal Energy and and Pollutants Management for Hybrid Electric Vehicle Pollutants Management for Hybrid Electric Vehicle , Pollutants Management for Hybrid Electric Vehicle Jean KUCHLY* **, Dominique NELSON-GRUEL*, Jean KUCHLY*,**, Dominique NELSON-GRUEL*,

,, Alain CHARLET*, Yann CHAMAILLARD*, Jean KUCHLY* **, Dominique Dominique NELSON-GRUEL*, Jean KUCHLY* NELSON-GRUEL*, ,**, Alain CHARLET*, Yann CHAMAILLARD*, Jean KUCHLY* NELSON-GRUEL*, ,**, Dominique Cédric NOUILLANT** Jean KUCHLY* **, Dominique NELSON-GRUEL*, Alain CHARLET*, CHARLET*, Yann CHAMAILLARD*, Alain Yann CHAMAILLARD*, Cédric NOUILLANT** Alain CHARLET*, Yann CHAMAILLARD*,  CHAMAILLARD*, Alain CHARLET*, Yann Cédric NOUILLANT** NOUILLANT** Cédric  Cédric NOUILLANT** Cédric NOUILLANT**  Université d’Orléans *Laboratoire PRISME, *Laboratoire PRISME, Université d’Orléans **PSA Peugeot Citroën d’Orléans *Laboratoire PRISME, Université *Laboratoire PRISME, Université **PSA Peugeot Citroën d’Orléans *Laboratoire PRISME, Université d’Orléans *Laboratoire PRISME, Université **PSA Peugeot Peugeot Citroën Citroën d’Orléans **PSA **PSA Peugeot Citroën Peugeot Citroënstrategy for the energy management of a Abstract: This article aims to propose a **PSA real-time compatible Abstract: This article aims to propose a real-time compatible strategy for the energy management of a compression-ignition (HEV), under pollutants emissions constraints. First,ofanaa Abstract: This article article Hybrid aims to to Electric propose Vehicle real-time compatible strategy for for the energy energy management Abstract: This aims propose aaa real-time compatible strategy the management compression-ignition Hybrid Electric Vehicle (HEV), under pollutants emissions constraints. First,of Abstract: This article aims to propose real-time compatible strategy for the energy management ofanaa offline method is applied in order to obtain a performance reference. Then, an online-oriented approach Abstract: This article aims to propose a real-time compatible strategy for the energy management compression-ignition Hybrid Electric Vehicle (HEV), under under pollutants emissions constraints. approach First,ofan an compression-ignition Hybrid Electric Vehicle (HEV), pollutants emissions constraints. First, offline method is applied in order to obtain a performance reference. Then, an online-oriented compression-ignition Hybrid Electric Vehicle (HEV), under pollutants emissions constraints. First, an based on Modelis Predictive Control (MPC) isaa described, andreference. a pollutants projected gradient algorithm usedapproach to solve compression-ignition Hybrid Electric Vehicle (HEV), under emissions constraints. First, an offline method applied in order to obtain performance Then, an online-oriented offline method is applied in order to obtain performance reference. Then, an online-oriented approach based on Model Predictive Control (MPC) is described, and a projected gradient algorithm used to solve offline method is applied in order to obtain a performance reference. Then, an online-oriented approach the successive optimal control problems is explicitly presented. Both approaches are compared through offline method is applied in order to obtain a performance reference. Then, an online-oriented approach based on Model Predictive Control (MPC) is described, and a projected gradient algorithm used to solve based on Modeloptimal Predictive Control (MPC) described, and a projected gradient algorithm used to solve the successive control problems is is explicitly presented. Both approaches are compared through based on Predictive Control (MPC) is described, and gradient used solve simulation and optimal the results are validated through experimentation. based on Model Model Predictive Control (MPC) isexplicitly described, and aa projected projected gradient algorithm algorithm used to to solve the successive control problems is presented. Both approaches are compared through the successive optimal control problems is explicitly presented. Both approaches are compared through simulation and the results are validated through experimentation. the successive optimal control problems is explicitly presented. Both approaches are compared through the successive optimal control problems is explicitly presented. Both approaches are compared through simulation and the results are validated through experimentation. Keywords: Nonlinear System Control, Numerical Optimization, Model Predictive Control, Dynamic simulation and the are validated through experimentation. © 2019, IFAC (International Automatic Control) HostingModel by Elsevier Ltd. AllControl, rights reserved. simulation and the results results areFederation validated through experimentation. Keywords: Nonlinear System Control,ofthrough Numerical Optimization, Predictive Dynamic simulation and the results are validated experimentation. Programming, Hybrid Electric Vehicle, Pollutants management, Energy management Keywords: Nonlinear System Control, Numerical Optimization, Model Predictive Control, Keywords: Nonlinear Nonlinear System Vehicle, Control,Pollutants Numericalmanagement, Optimization, Model Predictive Control, Control, Dynamic Dynamic Programming, Hybrid Electric Energy management Keywords: System Control, Numerical Optimization, Model Predictive Keywords: Nonlinear System Vehicle, Control,Pollutants Numericalmanagement, Optimization, Model Predictive Control, Dynamic Dynamic Programming, Hybrid Electric Electric Vehicle, Pollutants management, Energy management Programming, Hybrid Energy management Programming, Hybrid Electric Vehicle, Pollutants management, Energy management Programming, Hybrid Electric Vehicle, Pollutants management, Energy management 1. INTRODUCTION 1. INTRODUCTION The model used is described exhaustively in Simon (2018). To INTRODUCTION 1.their INTRODUCTION Being able to reduce1. environmental impact has become a The model used is described exhaustively in Simon (2018). To INTRODUCTION it up briefly: 1.their INTRODUCTION Being able to reduce1. environmental impact has become a sum The model used is described exhaustively in Simon (2018). To model used is described exhaustively in Simon (2018). To highly critical challenge toenvironmental face for automotive manufacturers. sum it up briefly: The model used is described exhaustively in Simon (2018). To Being able to reduce their impact has become aa The Being able to reduce their impact has become The model used is described exhaustively in Simon (2018). To highly critical challenge toenvironmental face for automotive manufacturers. sum it up briefly: Being able to reduce their environmental impact has become a sum it up briefly: The past few years in particular have seen strong changes in 1) The mechanical submodel of the vehicle is based on the Being able to reduce their environmental impact has become a sum it up briefly: highly critical challenge to face for automotive manufacturers. highly critical challenge to face for automotive manufacturers. sum it up briefly: The past few years in particular have seen strong changes in 1) The mechanical submodel of the vehicle is based on the highly critical challenge to face for automotive manufacturers. public opinion, but in also in face legalfor regulation. Mentioning only Newton’ssubmodel law, taking intovehicle account highly critical challenge to automotive manufacturers. 1) second The mechanical mechanical submodel of the the vehicle is aerodynamic based on on the the The past few years particular seen strong changes in The past few years particular have seen strong changes in 1) The of is based public opinion, but in also in legal have regulation. Mentioning only second Newton’ssubmodel law, taking intovehicle account aerodynamic The past few years in particular have seen strong changes in 1) The mechanical of the is based Europe, the Worldwide harmonized Light Vehicles (WLTC) and mechanical resistive forces. The past few years in particular have seen strong changes in 1) The mechanical submodel of the vehicle is based on on the the second Newton’s law, taking into account aerodynamic public opinion, but also in legal regulation. Mentioning only public opinion, but also in legal regulation. Mentioning only second Newton’s law, taking into account aerodynamic Europe, the Worldwide harmonized Light Vehicles (WLTC) and mechanical resistive forces. public opinion, but also in regulation. Mentioning only second Newton’s law, into and Real Driving testsVehicles have arose and public opinion, but Emissions also harmonized in legal legal(RDE) regulation. Mentioning only second Newton’sresistive law, taking taking into account account aerodynamic aerodynamic and mechanical mechanical resistive forces. Europe, Worldwide (WLTC) Europe, the Worldwide harmonized Light (WLTC) 2) and The battery submodel is forces. based on Guzella (2007). It is and Realthe Driving Emissions (RDE)Light testsVehicles have arose and Europe, the Worldwide harmonized Light Vehicles (WLTC) and mechanical resistive forces. replaced the old New European Driving Cycle (NEDC), and Europe, the Worldwide harmonized Light Vehicles (WLTC) and mechanical resistive forces. 2) The battery submodel is based on Guzella (2007). It is and Real Driving Emissions (RDE) tests have arose and and Real Driving Emissions have arose and considered assubmodel an ideal voltage source in series(2007). connection replaced old New European(RDE) Drivingtests Cycle (NEDC), and European Realthe Driving Emissions (RDE) tests have arose and 2) The battery is based on Guzella It is 2) The battery submodel is based on Guzella (2007). It is the norm on emissions has become more and more and Real Driving Emissions (RDE) tests have arose and considered as an ideal voltage source in series connection replaced the old New European Driving Cycle (NEDC), and 2) The battery submodel is based on Guzella (2007). It is replaced the old New Cycle (NEDC), and with an internal resistance. Both of these parameters 2) considered The battery submodel is based on Guzella (2007). Itare is the European norm on European emissionsDriving has become more and more replaced the old New European Driving Cycle (NEDC), and as an ideal voltage source in series connection considered as an ideal voltage source in series connection restrictive. Among the solutions found to obtain a satisfying replaced the old New European Driving Cycle (NEDC), and with an internal resistance. Both of these parameters are the European norm on emissions has become more and more considered as an ideal voltage source in series connection the European norm on emissions has become more and more obtained through maps of the State of Charge (SOC) considered as an ideal voltage source in series connection restrictive. Among the solutions found to obtain a satisfying the European norm on emissions has become more and more with an internal resistance. Both of these parameters are with an internal Both of these parameters compromise between consumption, pollutants emission the European norm the on fuel emissions has become more and more obtained throughresistance. maps of the State of Charge (SOC) are restrictive. Among solutions found to obtain aa satisfying with an resistance. Both of these parameters are restrictive. Among the solutions found to obtain satisfying with an internal internal resistance. Both of of these parameters are compromise between fuel consumption, pollutants emission restrictive. Among the solutions found to obtain aa promising satisfying obtained through maps of the State Charge (SOC) obtained through maps of the State of Charge (SOC) and vehicle cost, using a hybrid powertrain is a restrictive. Among the solutions found to obtain satisfying 3) The ICE consists in a static submodel, where maps give compromise between fuel consumption, pollutants emission obtained through maps of the State of Charge (SOC) compromise between fuel consumption, pollutants emission obtained through maps of thesubmodel, State of Charge (SOC) and vehicle cost, using a hybrid powertrain is a promising 3) The ICE consists in a static where maps give compromise between fuel consumption, pollutants emission development. compromise between fuel consumption, pollutants emission the fuel consumption, the submodel, NOx production, and give the and vehicle cost, using aa hybrid powertrain is aa promising 3) The ICE consists in aa static where maps and vehicle cost, using powertrain is 3) The ICE consists in where maps development. the fuel consumption, the submodel, NOx production, and give the and vehicle cost, using aa hybrid hybrid powertrain is aa promising promising 3) The ICE consists in aa static static submodel, where maps give and vehicle cost, using hybrid powertrain is promising exhaust air temperature and flow. 3) The ICE consists in static submodel, where maps give development. the fuel the NOx production, and the development. the fuel consumption, the NOx production, and the Adding an electric engine to a conventional Internal exhaust airconsumption, temperature and flow. development. the fuel consumption, the NOx production, and development. Adding an electric engine to a conventional Internal the fuelair consumption, the flow. NOx production, and the the exhaust temperature and exhaust air temperature and flow. Combustion Engine (ICE) grants benefits through several 4) The depollution system of a Diesel engine is composed Adding an electric engine to aabenefits conventional exhaust air and flow. Adding an electric engine to conventional Internal Combustion Engine (ICE) grants through Internal several exhaust air temperature temperature and flow. 4) The depollution system of a Diesel engine is composed Adding an electric engine to a conventional Internal aspects: energy recovery from deceleration, Stop and Start, Adding an Engine electric engine to abenefits conventional Internal of a depollution Diesel Oxidation Catalyst (DOC) andis acomposed Selective Combustion (ICE) grants through 4) The system of aa Diesel engine Combustion Engine (ICE) grants benefits through several 4) The system of Diesel engine aspects: energy recovery from deceleration, Stop andseveral Start, of a depollution Diesel Oxidation Catalyst (DOC) andis acomposed Selective Combustion Engine (ICE) grants benefits through several 4) The depollution system of aaTheir Diesel engine is composed and charge level of the ICE. The hybrid powertrain takes Combustion Engine (ICE) grants benefits through several Catalytic Reduction (SCR). pollutant conversion 4) The depollution system of Diesel engine is composed aspects: energy from deceleration, Stop and Start, of aa Diesel Oxidation Catalyst (DOC) and aa Selective aspects: energy recovery from deceleration, Stop and Start, of Diesel Oxidation Catalyst (DOC) and Selective and charge levelrecovery of the ICE. The hybrid powertrain takes Catalytic Reduction (SCR). Their pollutant conversion aspects: energy recovery from deceleration, Stop and Start, of a Diesel Oxidation Catalyst (DOC) and a advantage oflevel therecovery differently distributed efficiency theStart, two aspects: energy from deceleration, Stop of and efficiency depends merely on their temperature. Thus, of a Diesel Oxidation Catalyst (DOC) and conversion a Selective Selective and charge of the ICE. The hybrid powertrain takes Catalytic Reduction (SCR). Their pollutant and charge level of the ICE. The hybrid powertrain takes Catalytic Reduction (SCR). Their pollutant conversion advantage of the differently distributed efficiency of the two efficiency depends merely on their temperature. Thus, and charge level of the ICE. The hybrid powertrain takes Catalytic Reduction (SCR). Their pollutant conversion engines, and the reversible aspect of the electric engine. and charge level of the ICE. The hybrid powertrain takes the used submodel solves thermal 0D differential Catalytic Reduction (SCR). Their pollutant conversion advantage of the differently distributed efficiency of the two efficiency depends merely on their temperature. Thus, advantage of the differently distributed efficiency of the two efficiency depends merely on their temperature. Thus, engines, and the reversible aspect of the electric engine. the used submodel solves thermal 0D differential advantage of the differently distributed efficiency of efficiency merely on temperature. advantage ofthe thereversible differentlyaspect distributed efficiency of the the two two equations todepends obtain the temperature evolution of the Thus, DOC efficiency depends merely on their their temperature. Thus, engines, and of the electric engine. the used submodel solves thermal 0D differential engines, and the reversible aspect of the electric engine. the used submodel solves thermal 0D differential This way, a new layer of control is added to the conventional equations to obtain the temperature evolution of the DOC engines, and the reversible reversible aspect isof ofadded the electric electric engine. the used submodel solves thermal 0D differential engines, the aspect the and the SCR, before using maps to obtain their the used submodel solves thermal 0D differential This way,and a new layer of control to the engine. conventional equations to obtain the temperature evolution of the DOC equations to obtain the temperature evolution of the ICE control: the Energy Management System (EMS). Its role and the SCR, before using maps to obtain their This way, aa new layer of control is added to the conventional equations to toefficiency obtain the theand temperature evolution of the the DOC DOC This way, new layer of control is added to the conventional conversion the final NOx emissions. equations obtain temperature evolution of DOC ICE control: the Energy Management System (EMS). Its role This way, aa new layer of control is added to the conventional and before using maps to obtain and the the SCR, SCR, before using maps to obtain their their is to determine the torque split between the two engines in a This way, new layer of control is added to the conventional conversion efficiency and the final NOx emissions. ICE control: the Energy Management System (EMS). Its role and the SCR, before using maps to obtain ICE the Energy Management System (EMS). Its role and the SCR, before using maps to obtain their their is to control: determine the torque split between the two engines in a ICE control: the Energy Management System (EMS). Its role conversion efficiency and the final NOx emissions. conversion efficiency and the final NOx emissions. way that minimizes fuel consumption under environmental ICE control: the Energy Management System (EMS). Its role is to determine the torque split between the two engines in a conversion efficiency and the final NOx emissions. is to determine the torque split between the two engines in a conversion efficiency and the final NOx emissions. way that minimizes fuel consumption under environmental is to torque split between the two engines in constraints, for athe given vehicle approaches is to determine determine the torque splitarchitecture. between under theMany two engines in aa way that minimizes consumption environmental way that fuel consumption under environmental constraints, for a givenfuel vehicle architecture. Many approaches 2.1 Conclusion way been that minimizes minimizes fuel consumption under environmental have proposed: in Rousseau (2008), Nüesch (2014), Beck way that minimizes fuel consumption under environmental constraints, for vehicle Many approaches constraints, for aaa given given vehicle architecture. architecture. Many(2014), approaches have been proposed: in Rousseau (2008), Nüesch Beck 2.1 Conclusion constraints, for vehicle architecture. Many approaches (2007), or proposed: Michel (2014) for example. The point of this article constraints, for a given given vehicle architecture. Many approaches 2.1 Conclusion have been in Rousseau (2008), Nüesch (2014), Beck 2.1 Conclusion have been proposed: in Rousseau (2008), Nüesch (2014), Beck (2007), or proposed: Michel (2014) for example. The point of this article 2.1 Conclusion have been in Rousseau (2008), Nüesch (2014), Beck model includes 3 dynamic variables, excepting vehicle 2.1 Conclusion is to propose a real-time strategy concept for theof EMS, using have been proposed: in Rousseau (2008), Nüesch (2014), Beck The (2007), or Michel (2014) for example. The point this article The includes 3 dynamic variables, excepting vehicle (2007), or Michel (2014) for example. The point of this article is to propose a real-time strategy concept for theofEMS, using speedmodel (2007), or Michel (2014) for example. The point this article whichincludes is imposed by the road cycle: the batteryvehicle SOC, model-based control. The model 33 dynamic variables, excepting (2007), or Michel (2014) strategy for example. The for point ofEMS, this article The model includes dynamic variables, excepting is to propose a real-time concept the using speed which is imposed by the road cycle: the batteryvehicle SOC, is to propose a real-time strategy concept for the EMS, using The model includes 3 dynamic variables, excepting vehicle model-based control. is to propose a real-time strategy concept for the EMS, using and the DOC and SCR temperatures. Other variables of The model includes 3 dynamic variables, excepting vehicle which is imposed by the road cycle: the battery SOC, is to propose acontrol. real-time strategy concept for the EMS, using speed speed which is imposed by the road cycle: the battery SOC, model-based and the DOC and SCR temperatures. Other variables of model-based control. speed which is imposed by the road cycle: the battery SOC, 2. MODEL model-based control. interest such as fuel consumption, NOx production and speed which is imposed by the road cycle: the battery SOC, and the DOC and SCR temperatures. Other variables of model-based control. 2. MODEL and the DOC and SCR temperatures. Other variables of interest such as fuel consumption, NOx production and and the DOC and SCR temperatures. Other variables of MODEL depollution conversion efficiency are static and are obtained and the DOC andfuel SCR temperatures. Other variables of 2. MODEL such as consumption, NOx production and The system object of this2. paper is a hybrid electric vehicle, interest interest such as fuel consumption, NOx production and 2. MODEL depollution conversion efficiency are static and are obtained interest such as fuel consumption, NOx production and MODEL The system object of this2.paper is a hybrid electric vehicle, depollution through look-up tables experimentally obtained. In the interest such as fuel consumption, NOx production and conversion are and are using a compression ignition engine DV6 manufactured by depollution conversion efficiency are static static and are obtained obtained The system object of this paper is aa hybrid electric vehicle, through look-up tablesefficiency experimentally obtained. In the The system object of this paper is electric vehicle, depollution conversion static and using a compression ignition engine DV6 manufactured by through The system object of this paper is aa hybrid hybrid electric vehicle, following figure, 𝝉𝝉𝑰𝑰𝑰𝑰𝑰𝑰 isefficiency the torqueare delivered by are the obtained ICE, 𝝉𝝉𝒆𝒆𝒆𝒆 depollution conversion efficiency are staticobtained. and are obtained look-up tables experimentally In the PSA group. The vehicle uses a parallel architecture for its The system object of this paper is hybrid electric vehicle, through look-up tables experimentally obtained. In the using a compression ignition engine DV6 manufactured by following figure, 𝝉𝝉 is the torque delivered by the ICE, 𝝉𝝉𝒆𝒆𝒆𝒆 using aa compression ignition DV6 manufactured look-up tables experimentally obtained. In the 𝑰𝑰𝑰𝑰𝑰𝑰 PSA The vehicle uses aengine parallel architecture for by its through usinggroup. compression ignition engine DV6 manufactured by through look-up tables experimentally obtained. In is the torque delivered by the electric engine, and 𝝎𝝎 is the 𝑪𝑪𝑪𝑪 is the torque delivered by the ICE, 𝝉𝝉 following figure, 𝝉𝝉 powertrain, and is mild hybrid: the electric engine power using a compression ignition engine DV6 manufactured by 𝑰𝑰𝑰𝑰𝑰𝑰 𝒆𝒆𝒆𝒆 following figure, 𝝉𝝉 is the torque delivered by the ICE, 𝝉𝝉 PSA group. The vehicle uses a parallel architecture for its is the torque delivered by the electric engine, and 𝝎𝝎 is the 𝑰𝑰𝑰𝑰𝑰𝑰 𝒆𝒆𝒆𝒆 PSA group. The vehicle a parallel architecture for its following figure, 𝝉𝝉 is the torque delivered by the ICE, 𝝉𝝉 𝑪𝑪𝑪𝑪 powertrain, and is mild uses hybrid: the electric engine power PSA group. The vehicle uses parallel architecture for crankshaft rotational following figure, 𝝉𝝉𝑰𝑰𝑰𝑰𝑰𝑰 isby the torque delivered by the ICE, 𝝉𝝉𝒆𝒆𝒆𝒆 𝑰𝑰𝑰𝑰𝑰𝑰speed. 𝒆𝒆𝒆𝒆 (10kW) is low compared to theaaICE power. the torque delivered the electric engine, and 𝝎𝝎 the PSA group. The vehicle uses parallel architecture for its its is 𝑪𝑪𝑪𝑪 is is the torque delivered by the electric engine, and 𝝎𝝎 is the powertrain, and is mild hybrid: the electric engine power crankshaft rotational speed. 𝑪𝑪𝑪𝑪 powertrain, and is mild hybrid: the electric engine power is the torque delivered by the electric engine, and 𝝎𝝎 the (10kW) is low compared to the ICEthe power. 𝑪𝑪𝑪𝑪 is powertrain, and is mild hybrid: electric engine power is the torque delivered by the electric engine, and 𝝎𝝎 is the 𝑪𝑪𝑪𝑪 crankshaft rotational speed. powertrain, and is mild hybrid: the electric engine power crankshaft (10kW) is (10kW) is low low compared compared to to the the ICE ICE power. power. crankshaft rotational rotational speed. speed. (10kW) crankshaft rotational speed. (10kW) is is low low compared compared to to the the ICE ICE power. power.

2405-8963 © 2019, IFAC (International Federation of Automatic Control) Copyright 2019 IFAC 121Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 responsibility IFAC 121Control. Peer review©under of International Federation of Automatic Copyright © 2019 2019 IFAC IFAC 121 Copyright © 121 10.1016/j.ifacol.2019.09.020 Copyright © 2019 IFAC 121 Copyright © 2019 IFAC 121

2019 IFAC AAC 122 Orléans, France, June 23-27, 2019

𝑆𝑆𝑆𝑆𝑆𝑆(𝑡𝑡)

𝑇𝑇𝐷𝐷𝐷𝐷𝐷𝐷 (𝑡𝑡)

Jean Kuchly et al. / IFAC PapersOnLine 52-5 (2019) 121–127

Model

𝑇𝑇𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡)

𝑆𝑆𝑆𝑆𝑆𝑆(𝑡𝑡 + 1)

𝑱𝑱 = ∫𝒕𝒕𝒕𝒕 ((𝟏𝟏 − 𝜸𝜸). 𝒎𝒎̇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 (𝒕𝒕) + 𝜸𝜸. 𝒎𝒎̇𝑵𝑵𝑵𝑵𝑵𝑵 𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 (𝒕𝒕)) 𝒅𝒅𝒅𝒅 ( 4 ) 𝒕𝒕𝒕𝒕

𝑇𝑇𝐷𝐷𝐷𝐷𝐷𝐷 (𝑡𝑡 + 1)

The minimization of this criterion on a given cycle, with given initial and final SOC guarantees an optimal compromise between NOx emissions and fuel consumption. Here 𝛾𝛾 is a factor chosen accordingly to the desired weighting between the two values 𝑚𝑚̇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (𝑡𝑡) and 𝑚𝑚̇𝑁𝑁𝑁𝑁𝑁𝑁 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (𝑡𝑡).

𝑇𝑇𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡 + 1)

𝜏𝜏𝐼𝐼𝐼𝐼𝐼𝐼 (𝑡𝑡)

𝑚𝑚̇ 𝑁𝑁𝑁𝑁𝑁𝑁 (𝑡𝑡)

𝜏𝜏𝑒𝑒𝑒𝑒 (𝑡𝑡)

𝑚𝑚̇ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (𝑡𝑡)

𝜔𝜔𝑐𝑐𝑐𝑐 (𝑡𝑡)

This criterion can be used in an offline strategy, since the initial point, the desired final point and the whole cycle are known. The point of this paper, however, is to present an online method that will have to face uncertainty of the future. This above presented criterion will then to be adapted, as described later in the paper.

Fig. 1. Discrete model used in the paper 3. OPTIMAL CONTROL PROBLEM

3.4 Offline strategy 3.1 Control signal

To be able to estimate the performance of an online strategy, an ideal reference is necessary. An offline algorithm allows to obtain this ideal reference since it has access to the knowledge of the whole cycle, and has no computing time constraint. Therefore, for each online strategy result on a given cycle, a Dynamic Programming algorithm (Simon, 2018) will be used to solve the optimal control problem on the cycle and obtain an ideal reference, considering these state constraints:

The role of the EMS is to determine at each time the best torque split between electric and conventional engines: 𝒖𝒖(𝒕𝒕) =

𝝉𝝉𝒆𝒆𝒆𝒆

𝝉𝝉𝑰𝑰𝑰𝑰𝑰𝑰 +𝝉𝝉𝒆𝒆𝒆𝒆

(1)

Knowing the total torque request at the crankshaft 𝜏𝜏𝑐𝑐𝑐𝑐 made by the driver, computing 𝑢𝑢 allows to deduct the torque setpoints of the ICE and the electric engine. 𝑢𝑢(𝑡𝑡) = 1 corresponds to a torque request entirely fulfilled with the electric engine, 0 < 𝑢𝑢(𝑡𝑡) < 1 corresponds to a torque request partially fulfilled with both engines, 𝑢𝑢(𝑡𝑡) = 0 corresponds to a torque request entirely fulfilled with the ICE, and 𝑢𝑢(𝑡𝑡) < 0 corresponds to a battery reload with an ICE torque superior to the torque request.

𝑺𝑺𝑺𝑺𝑺𝑺𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝑫𝑫𝑷𝑷 𝑺𝑺𝑺𝑺𝑺𝑺𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝑫𝑫𝑫𝑫 { 𝑻𝑻𝑫𝑫𝑫𝑫𝑫𝑫 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝑫𝑫𝑫𝑫 𝑻𝑻𝑺𝑺𝑺𝑺𝑺𝑺 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝑫𝑫𝑫𝑫

Bound constraints are defined by the extreme torques admissible by the two engines: (2)

The 𝜏𝜏𝑒𝑒𝑒𝑒 𝑚𝑚𝑚𝑚𝑚𝑚 constraint is related to the generator functioning of the electric engine. A set of constraints on 𝑢𝑢 can thus be deducted: 𝒖𝒖 ≥ 𝟏𝟏 −

{

𝒖𝒖 ≤

𝒖𝒖 ≥

(5)

The final temperatures are free, but the final SOC must be the same as the final SOC of the compared online strategy, for energetic balance purpose. For example, let consider a given online algorithm for the EMS, that is applied to a WLTC cycle. Say it ends with a 58% SOC (it will be the case of the further described algorithm; an exactly null battery balance does not make sense in the online context). In order to evaluate the performances of this online algorithm, a dynamic programming algorithm is also applied to the WLTC cycle, considering a SOC terminal constraint of 58%. The algorithm will generate these torques:

3.2 Constraints 𝝉𝝉𝑰𝑰𝑰𝑰𝑰𝑰 ≤ 𝝉𝝉𝑰𝑰𝑰𝑰𝑰𝑰 𝒎𝒎𝒎𝒎𝒎𝒎 { 𝝉𝝉𝒆𝒆𝒆𝒆 ≤ 𝝉𝝉𝒆𝒆𝒆𝒆 𝒎𝒎𝒎𝒎𝒎𝒎 𝝉𝝉𝒆𝒆𝒆𝒆 ≥ 𝝉𝝉𝒆𝒆𝒆𝒆 𝒎𝒎𝒎𝒎𝒎𝒎

= 𝑺𝑺𝑺𝑺𝑺𝑺𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 = 𝑺𝑺𝑺𝑺𝑺𝑺𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 = 𝑻𝑻𝑫𝑫𝑫𝑫𝑫𝑫 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 = 𝑻𝑻𝑺𝑺𝑺𝑺𝑺𝑺 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐

𝝉𝝉𝑰𝑰𝑰𝑰𝑰𝑰 𝒎𝒎𝒎𝒎𝒎𝒎

𝝉𝝉𝒄𝒄𝒄𝒄 𝝉𝝉𝒆𝒆𝒆𝒆 𝒎𝒎𝒎𝒎𝒎𝒎 𝝉𝝉𝒄𝒄𝒓𝒓 𝝉𝝉𝒆𝒆𝒆𝒆 𝒎𝒎𝒎𝒎𝒎𝒎 𝝉𝝉𝒄𝒄𝒄𝒄

(3)

𝒖𝒖 ≤ 𝟏𝟏

The last constraint is related to the non-reversible aspect of the ICE. 3.3 Global objective A control signal is optimal if it minimizes a given criterion, designed to fit the goals of the control problem. In the case of the EMS, the global criterion is defined as follows: Fig. 2. Electric engine and ICE torques generated by DP 122

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123

The cost function can be a quadratic function of the control input, calling for a Quadratic Programming (QP) algorithm (Chen, 2014), or it can be nonlinear, leading to the resolution of a nonlinear optimization problem (Zhang, 2001).

Here is the SOC obtained, finishing as asked at 58%:

Optimization algorithm control vector guess u

Cost function

Optimal control vector u*

Model + cost computing Fig. 4. Direct method for optimal control In the next section, the formalism of the numerical optimization domain will be adopted: the cost function will be denoted 𝒇𝒇, and its input will be denoted 𝒙𝒙, rather than 𝑢𝑢 in the control formalism. Moreover, the iteration index of the algorithm will be denoted 𝒌𝒌.

Fig. 3 State of Charge signal obtained after using DP In this paper, the quality of an online algorithm will consist in its ability to approach the performances of offline Dynamic Programming in terms of fuel consumption and NOx emission. In the previous example of DP, the values at the end of the WLTC cycle are 996g for the fuel consumption and 2.48g for the NOx emissions.

To achieve accurate convergence in the shortest computing time possible, the effort has to be put in a rigorous implementation of the model, and in an efficient numerical optimization algorithm. Information concerning the different parts of the further algorithm and more generally concerning a wide range of numerical optimization algorithms can be found in Wright (1999).

4. ONLINE STRATEGY A large amount of real-time methods for the EMS have been explored in the literature; a wide description of existent families of methods can be found in Sciaretta (2008). The realtime aspect of these methods are based either on an instantaneous, static optimization like the well-known ECMS (Paganelli, 2002) and its pollutant-including development the ECPMS (Simon, 2015), or on a receding horizon optimization involving a MPC-structure (Koot, 2005), eventually allowing to take into account some of the system’s dynamics.

4.1 Numerical optimization algorithms generality Numerical optimization algorithms can be classified following three categories. First, algorithms that can be classified as first order algorithms, including the basic gradient algorithm. Here the search direction comes from the first order derivatives.

Concerning the optimal control aspects, most of these methods are based on the Pontryagin Minimum Principle (PMP, Pontryagin, 1962), or on the time-consuming Dynamic Programming (DP, Bellman, 1954). Although, direct methods could represent a relevant alternative, considering their computing time performance compared to DP and their flexibility compared to PMP, both in terms of constraints handling and easy modification of the cost function. The point of the following section is to describe general knowledge about direct methods and to propose an example of implementation of a direct EMS algorithm.

Fig. 5. Example of first order behaviour The implementation of such algorithms is usually simple, but their computing time performances are generally not the best. Second order algorithms go further and are based also on information, traditionally estimated, on the second order derivatives:

Direct methods consider an optimal control problem as follows: a function is defined as a discrete model of the system on a given horizon, taking a time-discretized control vector as input. A function call will thus simulate the model on the whole horizon, and compute a user-defined cost value, that will be the output of the function. The direct method consists then in finding numerically the minimum of the defined “cost” function, using classical numerical optimization algorithms based on successive calls to this function. Since the input of the cost function is the control vector, the algorithm should converge to an optimal control vector u*.

Fig. 6. Example of second order behaviour 123

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These algorithms are a reference in the matter of computing time. In particular, for optimization problems of low or medium input or constraints dimensions that typically result from optimal control problems, the Sequential Quadratic Programming algorithm (SQP) is usually considered as a reference. Finally, a third category can be considered, which regroups all the algorithms that do not rely on derivative information. Examples are numerous and diverse: Nelder-Mead algorithm, particle swarm optimization… In particular, the genetic algorithm has already been proposed for the EMS (Piccolo, 2001).

The principle of the conjugate gradient is to give a certain inertia to the direction computing, by adding a term depending on the previous step: 𝒅𝒅(𝒌𝒌) = −𝛁𝛁𝒇𝒇(𝒙𝒙(𝒌𝒌)) + 𝜷𝜷(𝒌𝒌). 𝒅𝒅(𝒌𝒌 − 𝟏𝟏)

(8)

Many methods exist to chose 𝛽𝛽(𝑘𝑘) and one of the most used is the Polak-Ribière (1969) method: 𝛃𝛃(𝐤𝐤) =

𝛁𝛁𝐟𝐟 𝐓𝐓 (𝐤𝐤).(𝛁𝛁𝛁𝛁(𝐤𝐤)−𝛁𝛁𝛁𝛁(𝐤𝐤−𝟏𝟏)) ‖𝛁𝛁𝛁𝛁(𝐤𝐤−𝟏𝟏)‖𝟐𝟐

(9)

A step direction computed using the conjugate gradient method with Polak-Ribière factor offers proper convergence speed. 4.3 Constraints handling As expressed before, the bounding box nature of the constraints makes the handling of the constraints really convenient.

For example, let a constraint be 𝑥𝑥(𝑘𝑘) ≤ 𝑢𝑢𝑢𝑢(𝑘𝑘). In the case of an unfeasible step direction such as 𝑥𝑥(𝑘𝑘) + 𝑑𝑑(𝑘𝑘) > 𝑢𝑢𝑢𝑢(𝑘𝑘), a new, feasible direction will be obtain by a simple projection on the bounding box: 𝑑𝑑̃ (𝑘𝑘) = 𝑢𝑢𝑢𝑢(𝑘𝑘) − 𝑥𝑥(𝑘𝑘).

Fig. 7. Example of other behaviour These algorithms often present poor computing time performance, but allow to solve particular cases problems: non-continuous cost functions, mixed integer problems, etc… As stated before, the SQP is the reference algorithm for direct optimal control, due to its high convergence properties, and its ability to deal with nonlinear constraints. In the case of the EMS optimal control however, the only constraints are bounds on the control value as described in section 3.2. The admissible space is then a bounding box, and the constraints can be easily dealt with by projecting an eventual unfeasible point on this bounding box. This particularity allows to use efficiently a first-order algorithm: the projected gradient. The following paragraphs of this section describe the projected gradient implemented to solve the EMS optimal control problem.

4.4 Step length

In classical numerical optimization algorithms such as projected gradient, the algorithm will start from an initial point of the cost function, and make successive steps 𝒔𝒔, progressing iteratively toward the minimum of the function. Typically, a step is computed this way:

Two major algorithm strategies exist. In trust regions methods, a maximum step length is computed and then a direction vector is obtained within the determined range. In line search methods, a direction is computed first, and then an appropriate displacement in the proposed direction is chosen.

(6)

where 𝛼𝛼(𝑘𝑘) is the scalar length step and 𝑑𝑑(𝑘𝑘) is the step direction vector. The algorithm must have 3 features: compute the step direction, handle the constraints and compute the step length.

In the implemented projected gradient algorithm, a polynomial interpolation line search is performed. Its functioning is described below.

4.2 Step direction

Where 𝐷𝐷(𝑘𝑘) = 𝛻𝛻𝛻𝛻(𝑥𝑥(𝑘𝑘)). 𝑑𝑑(𝑘𝑘) is the derivation in the sense of 𝑑𝑑(𝑘𝑘), and 𝑐𝑐1 is a user-defined positive constant, typically 10−3 . A step satisfying Armijo criterion produce a decrease in the cost function at least equal to a portion set by 𝒄𝒄𝟏𝟏 of the local directional derivative. In practice, the satisfaction of this

𝒙𝒙(𝒌𝒌 + 𝟏𝟏) − 𝒙𝒙(𝒌𝒌) = 𝒔𝒔(𝒌𝒌) = 𝜶𝜶(𝒌𝒌). 𝒅𝒅(𝒌𝒌),

Fig. 8. Projection on a bounding box. (Wright, 1999)

In order to ensure convergence, it is necessary for a step to satisfy Armijo criterion:

𝒇𝒇(𝒙𝒙(𝒌𝒌) + 𝜶𝜶(𝒌𝒌). 𝒅𝒅(𝒌𝒌)) ≤ 𝒇𝒇(𝒙𝒙(𝒌𝒌)) + 𝒄𝒄𝟏𝟏 . 𝜶𝜶(𝒌𝒌). 𝑫𝑫(𝒌𝒌)( 10 )

In the basic gradient algorithm, the step direction is given by the opposite of the gradient: 𝒅𝒅(𝒌𝒌) = −𝛁𝛁𝒇𝒇(𝒙𝒙(𝒌𝒌))

(7)

However, the poor quality of this direction has led to diverse improvements, including the conjugate gradient method. 124

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Set 𝑘𝑘 ← 𝑘𝑘 + 1 If 𝑘𝑘 = 1 Set 𝑑𝑑𝑘𝑘 = −∇𝑓𝑓(𝑥𝑥𝑘𝑘 ) Set 𝑑𝑑𝑘𝑘 using the projection described in section 4.3 Else Set 𝛽𝛽𝐾𝐾 using (36) Set 𝑑𝑑𝑘𝑘 using (35) Project 𝑑𝑑𝑘𝑘 If ∇𝑓𝑓(𝑥𝑥𝑘𝑘 )𝑇𝑇 ∙ 𝑑𝑑𝑘𝑘 > 0 , corresponding to an ascending direction Reset 𝑝𝑝𝑘𝑘 = −∇𝑓𝑓(𝑥𝑥𝑘𝑘 )𝑇𝑇 Project 𝑑𝑑𝑘𝑘 End End Set 𝛼𝛼𝑘𝑘 using the line search described in section 4.4 Set 𝑠𝑠𝑘𝑘 = 𝛼𝛼𝑘𝑘 𝑑𝑑𝑘𝑘 Set 𝑥𝑥𝑘𝑘 ← 𝑥𝑥𝑘𝑘 + 𝑠𝑠𝑘𝑘 End This algorithm is the main part of the proposed implementation of the Model Predictive Control (MPC). Designed in the late seventies (Richalet, 1978), MPC is a convenient method for real-time optimal control. A possible functioning of this method can be summed up as follows: first, an optimal control problem is solved on a given, finite horizon. Then, the first value of the obtained control signal is applied to the system. At the next time, sensors and adequate estimators allow to reinitialize the differential equations of the model, insuring the closed-loop characteristic of the method. The same process can then be repeated, with an initial guess for the optimal control problem based on the previous solution.

criterion thus ensures that each step will produce a decrease in the cost, leading the algorithm to convergence. This need for decrease insurance is due to the local approximation made by the algorithm: the further the step leads, the less reliable the local information become. It may then be often necessary to reduce the step length. A good line search implementation must achieve both proper computing time and quality step length. An efficient way to proceed is the polynomial interpolation method: A merit function is defined:

𝝋𝝋(𝜶𝜶(𝒌𝒌)) = 𝒇𝒇(𝒙𝒙(𝒌𝒌) + 𝜶𝜶(𝒌𝒌). 𝒅𝒅(𝒌𝒌))

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The merit function is equivalent to the cost function, taken at the current iterate and depending only on the step length 𝜶𝜶. A first value 𝛼𝛼0 is tried; in our case 𝛼𝛼0 = 1. If 𝜑𝜑(𝛼𝛼0 ) satisfies Armijo criterion, the line search stops and 𝛼𝛼0 is taken as the next step length. If not, a quadratic interpolation of the merit function is made based on the known information: 𝜑𝜑(0) the current value of the cost function, 𝐷𝐷 = 𝜑𝜑′(0) its directional derivative, and 𝜑𝜑(𝛼𝛼0 ) its newly computed value. We chose the next line search candidate 𝛼𝛼1 as the value of 𝛼𝛼 that minimizes the interpolated polynomial.

In the following simulation process, the optimal control problem is solved using the described direct method, the first value of the solution is applied to the model, and the model gives directly the next initial states of the system. Fig. 9. A new step length candidate is computed

4.6 Cost function

As previously, if 𝛼𝛼1 satisfies Armijo, the line search stops. If not, a new cubic interpolation is performed, using the newly computed value 𝜑𝜑(𝛼𝛼1 ). The minimizing value 𝛼𝛼2 of the cubic polynomial is tried in the line search. If it is still unable to satisfy Armijo criterion, the furthest point is replaced by the newest and new cubic interpolations are performed until the sufficient decrease criterion is satisfied. The polynomial interpolation line search method offers interesting performances both in terms of step quality and computing time.

An offline method like DP process a whole cycle and has access to a total knowledge of it. Therefore, it is possible for such an algorithm to satisfy strictly a SOC terminal constraint. Consequently, there is no need to specify in its minimization criterion a term related to the electric power consumed or to an eventual SOC regulation. Since in online methods the future is uncertain, a more complicated cost function is adopted:

With:

𝑱𝑱 = ∑𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒆𝒆 (𝒊𝒊) + 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑(𝒊𝒊) + 𝒓𝒓𝒓𝒓𝒓𝒓(𝒊𝒊) 𝒊𝒊=𝟏𝟏

𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒆𝒆 (𝒊𝒊) = 𝒎𝒎̇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 (𝒊𝒊) + 𝑨𝑨. (𝑷𝑷𝒆𝒆𝒆𝒆 (𝒊𝒊) + 𝟐𝟐. 𝑷𝑷𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍 (𝒊𝒊))

4.5 Algorithm

( 12 )

( 13 )

This term considers the energy consumption of both the ICE and the electric engine (𝑃𝑃𝑒𝑒𝑒𝑒 ), considering the approximated losses (𝑷𝑷𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍 ) due to the charge and discharge of the battery. It allows to take account of the efficiency of both engines.

The proper implementation of the above described features leads to the following numerical optimization algorithm: Input: 𝑥𝑥0 the initial guess, 𝑙𝑙𝑏𝑏 and 𝑢𝑢𝑏𝑏 the constraints vector Set 𝑘𝑘 = 0 Repeat until a stopping criterion is met

𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑(𝒊𝒊) = 𝑩𝑩. 𝒎𝒎̇𝑵𝑵𝑵𝑵𝑵𝑵 𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 (𝒊𝒊)

This term considers the pollutant emissions. 125

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𝒓𝒓𝒓𝒓𝒓𝒓(𝒊𝒊) = 𝑪𝑪. (𝑺𝑺𝑺𝑺𝑺𝑺(𝒊𝒊) − (𝟓𝟓𝟓𝟓 − 𝒌𝒌𝒓𝒓𝒓𝒓𝒓𝒓 . 𝒗𝒗(𝒊𝒊))𝟒𝟒

The behaviours of both algorithms are similar at low speed, and differ more in the highway part of the WLTC cycle. It is mainly related to the cautious functionning of the MPC algorithm, that does not allow too important SOC variations because of the SOC-regulating part of the cost function (12). Due to its total knowledge of the end of the cycle, the DP algorithm can afford to be more aggressive.

( 15 )

This term allows to obtain a final SOC around 50%. 𝒌𝒌𝒓𝒓𝒓𝒓𝒓𝒓 is a ̇ during constant equal to the mean ratio between 𝒗𝒗̇ and 𝑺𝑺𝑺𝑺𝑺𝑺 the regenerative braking parts of a WLTC cycle. The control algorithm will thus smoothly regulate the SOC below 50% according to the current speed of the vehicle, considering that the energy recovery of a potential braking will regenerate the SOC up to 50%. This feature prevents the SOC to meet extreme values. Besides, A, B and C are constants allowing to take into account both the scale of the different values and the weight put to each term.

Considering the DP results of section 3, the fuel consumption of the online strategy is 3.9% worse than the offline strategy, but the NOx emission is 4.7% better. With a 10-instant horizon, the computing time is 4.6ms per optimization problem. In comparison, the DP algorithm applied on a 10instants part of the WLTC cycle presents a computing time of roughly 200ms. Even if both the performances and the computing time call for improvement, the simulation tends to show that a MPC algorithm using a direct method is suitable in the case of the real-time EMS.

The cost function (12) is directly used in the numerical optimization algorithm, whereas the minimization criterion (4) is a global objective to be achieved over a whole run and under a certain terminal SOC aim, usable in a DP algorithm. 5. RESULTS 5.1 Simulation

5.2 Experimentation

The aim of the simulation is to determine the performance of the previously described method on a WLTC cycle, compared to offline obtained performances.

In order to assess the simulation results, an experimentation is made using a high-dynamic test bench, using a compression ignition engine DV6 from PSA Group. The ICE is constrained to follow a WLTC cycle and controlled to produce the torque signals computed during the two simulations (online and offline strategies).

The MPC algorithm, with a 10-instants horizon and the cost function described above, will be applied to a WLTC cycle. The engine speed and torque are considered known on the upcoming horizon, that lasts thus 10 seconds. The algorithm generates these torque signals:

The test bench allows to measure fuel consumption and the NOx quantity produced by the engine, but does not include a battery nor a depollution system, so the SOC evolution and conversion efficiencies have to be simulated afterwards. The experimentation presents results coherent with the tendencies observed through simulation: the online strategy leads to a fuel consumption 5.6% higher than the offline strategy, and NOx emissions 4.9% lower. The slack between simulation and experimentation is mostly due to the mainly static nature of the model. In particular, the fuel consumption look-up table is unable to take into account the dynamics of the system, leading to unneglectable errors.

Fig. 10. Torque signals generated by MPC The MPC simulation also leads to a final SOC at 58%. Its behaviour and performances can then be conveniently compared to the results of section 3.

Fig. 12. Experimentations results

Fig. 11. SOC simulation with MPC and DP methods 126

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Although the MPC NOx cumulative emissions are very close between simulation and experimentation, it is due to errors compensating each other all along the cycle. The models used for both MPC and DP are indeed equivalent.

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5.3 Perspectives Numerous enhancements have to be brought to the method. First, Kalman filters have to be implemented in order to have access to reliable information on the state variables (SOC, depollution system temperatures) and on the eventual perturbations (slope, vehicle mass…). Also, the performances of the algorithm are directly related to the model accuracy and the computing time of the whole process, that have both to be improved. Beyond being critical for the possibility to implement an algorithm on an embedded system with restricted computing resources, good computing time performances allows also to use a more complex model, and to attain better accuracy in the numerical optimization results by iterating more. Finally, the most important part of the necessary progress will concern prediction of the future. MPC needs indeed a knowledge as accurate as possible on the upcoming horizon, particularly concerning the vehicle speed, from which will be deducted engine speed and torque request. The vehicle speed could be estimated through data fusion between the last values of speed and the predictions of a model based on external information such as the distance and relative speed of the vehicle in front (Lefèvre, 2014). Moreover, knowledge of the further future is also critical in order to achieve optimality of the control on a whole cycle. In particular, the cost function minimized each time by the numerical optimization algorithm could be modified on a regular basis to take into account information like the need to stock some energy or the upcoming possibility to easily refill the battery. Literature contains already examples of work on these subjects: for example, on adaptive-ECMS (Onori, 2011), or on sensibility of MPC to prediction noise (Fu, 2011). REFERENCES Beck, R., Bollig, A., & Abel, D. (2007). Comparison of two real-time predictive strategies for the optimal energy management of a hybrid electric vehicle. Oil & Gas Science and Technology-Revue de l'IFP, 62(4), 635-643. Bellman, R. (1954). The theory of dynamic programming (No. RAND-P-550). RAND Corp Santa Monica CA. Chen, Z., Mi, C. C., Xiong, R., Xu, J., & You, C. (2014). Energy management of a power-split plug-in hybrid electric vehicle based on genetic algorithm and quadratic programming. Journal of Power Sources, 248, 416-426. Fu, L., Ümit, Ö., Tulpule, P., & Marano, V. (2011, June). Realtime energy management and sensitivity study for hybrid electric vehicles. In American Control Conference (ACC), 2011 (pp. 2113-2118). IEEE. Guzella, L., Sciaretta, A., et al. (2007). Vehicle propulsion systems, volume 1. Springer. Koot, M., Kessels, J. T., De Jager, B., Heemels, W. P. M. H., Van den Bosch, P. P. J., & Steinbuch, M. (2005). Energy management strategies for vehicular electric power

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