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A comparison of PMP-based Energy Management Strategies for Plug-in-Hybrid Electric Vehicles
9th IFAC International Symposium on Advances in Automotive 9th IFAC International Symposium on Advances in Automotive 9th Control 9th IFAC IFAC International International Symposium Symposium on on Advances Advances in in Automotive Automotive Control online at www.sciencedirect.com Control Orléans, France, June Symposium 23-27, 2019 onAvailable 9th IFAC International Advances in Automotive Control Orléans, France, June 23-27, 2019 Orléans, France, France, June June 23-27, 23-27, 2019 2019 Control Orléans, Orléans, France, June 23-27, 2019
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IFAC PapersOnLine 52-5 (2019) 592–597
A comparison of PMP-based Energy A A comparison comparison of of PMP-based PMP-based Energy Energy Management Strategies for Plug-in-Hybrid A comparison of PMP-based Energy Management Strategies for Plug-in-Hybrid Management Electric Strategies for Plug-in-Hybrid Vehicles Management Electric Strategies for Plug-in-Hybrid Vehicles Electric Vehicles Vehicles ∗ ∗∗∗ Roland Schmid Electric urger ∗∗ ∗ Johannes B¨ ∗∗ Naim Bajcinca ∗∗∗
Roland Schmid ∗∗ Johannes B¨ u rger ∗∗ Naim Bajcinca ∗∗∗ Roland u Roland Schmid Schmid Johannes Johannes B¨ B¨ urger rger ∗∗ Naim Naim Bajcinca Bajcinca ∗∗∗ ∗ ∗∗ ∗∗∗ ∗Roland Schmid Johannes B¨ urger Naim Bajcinca M¨ u nchen, Germany (e-mail: ∗ BMW AG, Petuelring 130, 80788 BMW AG, Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: ∗ ∗ BMW AG, Petuelring 130, u [email protected]). BMW AG, Petuelring 130, 80788 80788 M¨ M¨ unchen, nchen, Germany Germany (e-mail: (e-mail: [email protected]). ∗ ∗∗ [email protected]). AG, Petuelring 130, 80788 M¨ uunchen, Germany (e-mail: BMW AG, Petuelring 130, 80788 M¨ nchen, Germany (e-mail: [email protected]). ∗∗BMW BMW AG, Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: ∗∗ ∗∗ BMW 130, 80788 M¨ u nchen, Germany (e-mail: [email protected]). [email protected]). BMW AG, AG, Petuelring Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: [email protected]). ∗∗∗∗∗ [email protected]). BMW AG, Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: Mechatronics in Mechanical and Automotive ∗∗∗ Department [email protected]). of Mechatronics in Mechanical and Automotive ∗∗∗ ∗∗∗ Department Department of Mechatronics in Mechanical and Automotive [email protected]). Engineering, Technical University of Kaiserslautern, Postfach 3049, Department of Mechatronics in Kaiserslautern, Mechanical andPostfach Automotive Engineering, Technical University of 3049, ∗∗∗ Engineering, Technical University of Kaiserslautern, Postfach 3049, Department of Mechatronics in Kaiserslautern, Mechanical andPostfach Automotive 67663 Kaiserslautern, Germany (e-mail: [email protected]). Engineering, Technical University of 3049, 67663 Kaiserslautern, Germany (e-mail: [email protected]). 67663 Kaiserslautern, Germany (e-mail: [email protected]). Engineering, Technical University of Kaiserslautern, Postfach 3049, 67663 Kaiserslautern, Germany (e-mail: [email protected]). 67663 Kaiserslautern, Germany (e-mail: [email protected]). Abstract This paper presents a of energy management strategies based on PonAbstract This paper presents aa comparison comparison of energy management strategies based on PonAbstract This paper presents comparison of energy management strategies based on Pontryagin’s Minimums Principle (PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles Abstract This paper presents a comparison of energy management strategies based on Pon(PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles tryagin’s Minimums Principle tryagin’s Minimums Principle (PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles Abstract This paper presents a comparison of energy management strategies based on Ponis minimized by considering the power split and electrical driving decision as optimization tryagin’s Minimums Principle (PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles the power split and electrical driving decision as is by considering is minimized minimized bypaper considering the power split and electrical driving decision as optimization optimization tryagin’s Minimums Principle (PMP). Thesplit fuelbetween consumption ofdriving Plug-in-Hybrid Electric Vehicles variables. This discusses the trade-off optimality and computational efficiency is minimized by considering the power and electrical decision as optimization between variables. This paper discusses the trade-off optimality and computational efficiency variables. This paper discusses thepower trade-off between optimality and decision computational efficiency is minimized bypaper considering the splitbetween and electrical driving as optimization of analytical and map-based PMP methods. Thereby aa decision guidance for further research variables. This discusses the trade-off optimality and computational efficiency of analytical and map-based PMP methods. Thereby decision guidance for further research of analytical and map-based PMP methods. Thereby aa contrasting decision guidance for further research variables. This paper discusses the trade-offmethods between optimality andfuel computational efficiency is presented which evaluates the discussed by consumption, state of of analytical and map-based PMP methods. Thereby decision guidance for further research is presented which evaluates the discussed methods by contrasting fuel consumption, state of is presented which evaluates the discussed methods by contrasting fuel consumption, state of analytical and map-based PMP methods. Thereby a decision guidance for further research charge trajectory, split and engine on/off decision and computational efficiency. is presented whichpower evaluates the discussed methods by contrasting fuel consumption, state of of charge trajectory, power split and engine on/off decision and computational efficiency. charge trajectory, power split the anddiscussed engine on/off on/off decision and computational computational efficiency. state of is presented whichpower evaluates methods by contrasting fuel consumption, charge trajectory, split and engine decision and efficiency. © 2019,trajectory, IFAC (International Federation of Automatic by Elsevier Ltd.efficiency. All rights reserved. charge power split and engine on/off Control) decisionHosting and computational Keywords: Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, Keywords: Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, Keywords: Minimums Principle (PMP), Energy Management Strategy, Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based Keywords: Pontryagin’s Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based Keywords: Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based PMP 1. INTRODUCTION in terms of optimality. To find an optimal control strategy strategy 1. INTRODUCTION in terms of optimality. To find an optimal control 1. in terms of optimality. To find an optimal control strategy for Hybrid-Electric-Vehicles (HEV) based on PMP, map1. INTRODUCTION INTRODUCTION in terms of optimality. To find an optimal control strategy for Hybrid-Electric-Vehicles (HEV) based on PMP, mapfor Hybrid-Electric-Vehicles (HEV) based on PMP, map1. INTRODUCTION in terms of analytical optimality.methods To find an optimal control strategy based and can be applied. Map-based for Hybrid-Electric-Vehicles (HEV) based on PMP, mapand analytical methods can be applied. Map-based Finite oil oil resources resources and and the the harmful harmful effects effects of of local local and and based based and analytical methods can be applied. Map-based for Hybrid-Electric-Vehicles based ondirectly PMP, mapPMP methods construct the(HEV) Hamiltonian from Finite based and analytical methods can be applied. Map-based PMP methods construct the Hamiltonian directly from Finite oil and the harmful local global traffic emissions have resulted ineffects a high highof demand of PMP Finite traffic oil resources resources andhave the resulted harmful in effects ofdemand local and and methods construct the Hamiltonian directly from based and analytical methods can be Jager applied. Map-based fuel consumption maps (compare de et al. (2013)). global emissions a of PMP methods construct the Hamiltonian directly from consumption maps (compare de Jager et al. (2013)). global emissions have resulted in aa high demand of Finite oil resources andand theefficient harmful local and alternative, ecofriendly propulsion systems for global traffic traffic emissions have resulted ineffects highofsystems demand of fuel fuel consumption maps (compare de Jager et al. (2013)). PMP methods construct the Hamiltonian directly from In contrast to this, analytical approaches approximate alternative, ecofriendly and efficient propulsion for fuel consumption maps (compare de Jager et al. (2013)). contrast to this, analytical approaches approximate alternative, and propulsion systems global trafficecofriendly emissions haveefficient resulted in a high demandfor of In transportation. To address address this problem, Plug-in-Hybrid alternative, ecofriendly and efficient propulsion systems for In contrast to this, analytical approaches fuel consumption maps (compare de Jager etapproximate al.functions (2013)). engine/electrical efficiency maps with quadratic transportation. To this problem, Plug-in-Hybrid In contrast to this, analytical approaches approximate efficiency maps with quadratic functions transportation. To address this problem, Plug-in-Hybrid alternative, ecofriendly and efficient propulsion systems for engine/electrical Electric Vehicles (PHEV) can be deployed. The hybrid transportation. To(PHEV) address this be problem, Plug-in-Hybrid engine/electrical efficiency maps with functions In contrast analytical approaches approximate of the degreetoof of this, freedom (e.g. Koot et quadratic al. (2005)) (2005)) and are are Electric Vehicles can deployed. The hybrid engine/electrical efficiency maps with quadratic functions of the degree freedom (e.g. Koot et al. and Electric Vehicles can deployed. The hybrid transportation. To(PHEV) address this be problem, Plug-in-Hybrid powertrain consists of a combination of an electric motor Electric Vehicles (PHEV) can be deployed. The hybrid of the degree of freedom Koot et al. (2005)) and are engine/electrical efficiency(e.g. maps with quadratic functions computationally efficient. Although numerous publications powertrain consists of a combination of an electric motor of the degree of freedom (e.g. Koot et al. (2005)) and are efficient. Although numerous publications powertrain consists of of an motor Electric Vehicles (PHEV) can offers be deployed. The hybrid and a combustion combustion engine and degrees of freedom freedom powertrain consists of aa combination combination of an electric electric motor computationally computationally efficient. Although numerous publications of the degree of freedom (e.g. Koot et al. for (2005)) and are discuss map-based or analytical methods optimization and a engine and offers degrees of computationally efficient. Although numerous publications map-based or analytical methods for optimization and combustion and offers degrees of powertrain consists of a combination of an electric motor (load point shift andengine electrical drive decision) which can be discuss and aapoint combustion engine anddrive offers degrees of freedom freedom discuss map-based analytical methods for computationally Although numerous publications of the power power splitefficient. ofor HEV, to date, date, few consider the (load shift and electrical decision) which can be discuss map-based orHEV, analytical methods for optimization optimization of the split of to few consider the setup setup (load point shift and electrical drive decision) which can be and a combustion engine and offers degrees of freedom used to reduce fuel consumption and/or emissions. Incan order (loadto point shift and electrical drive decision) whichIn be of the power split of to date, few consider setup discuss map-based orHEV, analytical methods for optimization of PHEVs and optimize both the power split in the combinaused reduce fuel consumption and/or emissions. order the power split of HEV, to date, few consider the setup of PHEVs and optimize both the power split in combinaused to reduce fuel consumption and/or emissions. order (load point shift and electrical drive decision) whichIn can be of to exploit the full potential of a hybrid powertrain strucused to reduce fuel consumption and/or emissions. In order PHEVs and optimize both the power split in combinathe power split of on/off HEV, to date, few consider the setup tion with an engine decision. Schmid et al. (2018) to exploit the full potential of a hybrid powertrain strucof PHEVs and optimize both the power split in combinawith an engine on/off decision. Schmid et al. (2018) to the of hybrid powertrain used to fuel potential consumption Instrucorder ture, anreduce intelligent energy management management strategy (EMS) is tion to exploit exploit the full full potential of aaand/or hybridemissions. powertrain struction with an engine on/off decision. Schmid et al. (2018) of PHEVs and optimize both the power splitan in analytical combinapresented a PMP-based EMS which uses ture, an intelligent energy strategy (EMS) is tion with an engine on/off decision. Schmid et al. (2018) aa PMP-based EMS which uses an analytical ture, an energy management strategy (EMS) is to exploit the different full potential ofofa EMS hybrid powertrain strucrequired. The classes have been discussed discussed ture, an intelligent intelligent energy management strategy (EMS) is presented presented PMP-based EMS which uses an analytical tion withfor an engine on/off decision. Schmid et al. (2018) method co-state computation based onan trip-preview required. The different classes of EMS have been presented a co-state PMP-based EMS which uses analytical method for computation based on trip-preview required. The different classes of EMS have been discussed ture, an intelligent energy management strategy (EMS) is in numerous surveys and publications (e.g. Enang and required. The surveys different and classes of EMS have been discussed method for co-state computation based on trip-preview presented a and PMP-based EMS which uses an analytical information combined it with the iterative approach in numerous publications (e.g. Enang and method for co-state computation based on trip-preview and combined it with the iterative approach in surveys and publications (e.g. Enang and information required. The different classes of EMS have been discussed Bannister (2017) and Rizzoni Rizzoni and Onori (2015)). in numerous numerous surveys and publications (e.g. Enang and information and combined it with the iterative approach method for co-state computation based onmethod trip-preview of Elbert et al. (2014). It is still open which offers Bannister (2017) and and Onori (2015)). information and combined it with the iterative approach Elbert et al. (2014). It is still open which method offers Bannister (2017) and Rizzoni and Onori (2015)). in numerous surveys and publications Enang and of One can distinguish distinguish rule-based EMS and(e.g. (online/offline) Bannister (2017) andrule-based Rizzoni and Onori (2015)). of Elbert et al. It is still open method offers information and(2014). combined it with thewhich iterative approach the best trade-off between optimality and computational One can EMS and (online/offline) of Elbert et al. (2014). It is still open which method offers best trade-off between optimality and computational One rule-based EMS and (online/offline) Bannister (2017) andEMS. Rizzoni and Onori (2015)). optimization based Rule-based EMS (e.g. Amb¨ Amb¨ uhl hl the One can can distinguish distinguish rule-based EMS and (online/offline) the best trade-off between and computational of Elbert et al. (2014). It is optimality still open which method offers efficiency. optimization based EMS. Rule-based EMS (e.g. u the best trade-off between optimality and computational efficiency. optimization based EMS. Rule-based EMS (e.g. hl One distinguish rule-based EMS et al.can (2010)) rely on on predefined rule and sets (online/offline) which are u oboptimization based EMS. Rule-based EMS (e.g. Amb¨ Amb¨ u hl efficiency. the best trade-off between optimality and computational This paper therefore provides decision guidance to evalevalet al. (2010)) rely predefined rule sets which are obefficiency. paper therefore provides aaa decision guidance to et al. (2010)) rely on predefined rule sets which are oboptimization based EMS. Rule-based EMS (e.g.based Amb¨ u hl This tained from offline cycle calibration and are on et al. (2010)) rely on predefined rule sets which are obThis paper therefore provides decision guidance to evalefficiency. uate map-based PMP versus two variants of analytical tained from offline cycle calibration and are based on This paper therefore provides a decision guidance to evalmap-based PMP versus two variants of analytical tained from offline and are based on et al. (2010)) rely oncycle predefined rule sets are obengineering experience. Tocalibration get guarantees guarantees of optimality optimality tained from experience. offline cycle calibration and which are based on uate uate map-based PMP versus variants of analytical This paper therefore provides atwo decision guidance to evalPMP. For this reason, we utilize the approximation preengineering To get of uate map-based PMP versus two variants of analytical For this reason, we utilize the approximation preengineering experience. To get of tained from offline cycleconsumption) and are based on PMP. (e.g. with respect to fuel fuel optimization based engineering experience. Tocalibration get guarantees guarantees of optimality optimality PMP. For this reason, we utilize the approximation preuate map-based versus twoextend variants analytical sented by Koot Koot etPMP al. (2005) (2005) and the of approach by (e.g. with respect to consumption) optimization based PMP. For this reason, we utilize the approximation presented by et al. and extend the approach by (e.g. to optimization based engineering experience. ToThe getglobal guarantees ofsolution optimality approaches are attractive. optimal can sented (e.g. with with respect respect to fuel fuel consumption) consumption) optimization based by Koot et al. (2005) and extend the approach by PMP. For this we utilize the approximation preintroducing thereason, explicit engine on/off decision presented approaches are attractive. The global optimal solution can sented by Koot et al. (2005) and extend the approach by introducing the explicit engine on/off decision presented approaches are attractive. The optimal solution can (e.g. with by respect to fuel consumption) optimization based be found dynamic programming (DP) as proposed proposed by introducing approaches are attractive. The global global(DP) optimal solution can the explicit engine decision presented sented by Koot et al. (2005) andon/off extend the the approach by in Schmid et al. (2018). Further we compare resulting be found by dynamic programming as by introducing the explicit engine on/off decision presented Schmid et al. (2018). Further we compare the resulting be found by dynamic programming (DP) as proposed by approaches attractive. TheDP global optimal solutioncomcan Yuan et al. al. (2013). However, requires extensive be found byare dynamic programming (DP) as proposed by in in Schmid et al. (2018). Further we compare the resulting introducing the explicit engine on/off decision presented approach with the approach of Schmid et al. (2018) and Yuan et (2013). However, DP requires extensive comin Schmid et al. (2018). Further we compare the resulting with the approach of Schmid et al. (2018) and Yuan et (2013). However, DP extensive combe found byressources dynamic programming (DP) as proposed by approach putational and hence isrequires not feasible for online online Yuan et al. al. (2013). However, DPis requires extensive comapproach the approach of et al. (2018) and in map-based Schmidwith et al. (2018). Further we compare resulting PMP method. We analyse the methods by putational ressources and hence not feasible for with the approach of Schmid Schmid etthe al.the (2018) and aaaapproach map-based PMP method. We analyse methods by putational and hence not feasible for online Yuan et al.ressources (2013). However, DPis requires extensive comapplication. As an alternative approach, Pontryagin’s Minputational ressources and hence is not feasible for online map-based PMP method. We analyse by approach with the approach ofstate Schmid etthe al.methods (2018) and contrasting fuel consumption, of charge trajectory, application. As an alternative approach, Pontryagin’s Mina map-based PMP method. We analyse the methods by fuel consumption, state of charge trajectory, application. As alternative Pontryagin’s Minputational ressources and is not to feasible foroptimal online imum Principle (PMP) canhence beapproach, applied find an an application. As an an alternative approach, Pontryagin’s Min- contrasting contrasting consumption, state of charge trajectory, acontrasting map-basedfuel PMP method. We analyse the on/off methods by computation time, power splits and engine deciimum Principle (PMP) can be applied to find optimal fuel consumption, state of charge trajectory, time, power splits and engine on/off deciimum (PMP) can applied find an application. As an alternative approach, Pontryagin’s control strategy (e.g. Kim et be al. (2014)).to PMP can be Minused computation imum Principle Principle (PMP) can be applied to find can an optimal optimal computation time, power splits and decicontrasting fuel consumption, state ofengine chargeon/off trajectory, sions. control strategy (e.g. Kim et al. (2014)). PMP be used computation time, power splits and engine on/off decicontrol strategy (e.g. et al. (2014)). can be imum toPMP find an optimal to findPrinciple candidates for Kim thecan optimal solution by applying applying the sions. control strategy (PMP) (e.g. Kim et be al. applied (2014)). PMP can be used used sions. computation power andThe engine on/offofdeciThe paper is is time, organized as splits follows: modelling the to find candidates for the optimal solution by the sions.paper organized as follows: The modelling of the to find candidates for the optimal solution by applying the control strategynecessary (e.g. Kim et al. (2014)). PMP can be used corresponding conditions, which are computato find candidates for the optimal solution by applying the The The paper is organized as follows: The modelling of the sions. PHEV is presented in section 2. In section 3 the map-based corresponding necessary conditions, which are computaThe paper is organized as follows: The modelling of the is presented in section 2. In section 33 the map-based corresponding computato find candidates foroften the conditions, optimal byare applying the PHEV tionally efficientnecessary and close to tosolution DPwhich (Yuan et al. (2013)) corresponding necessary conditions, which are computaPHEV is presented in section 2. In section the map-based The paper is organized as follows: The modelling of the tionally efficient and often close DP (Yuan et al. (2013)) PHEV is presented in section 2. In section 3 the map-based tionally and close (Yuan et (2013)) corresponding computationally efficient efficientnecessary and often oftenconditions, close to to DP DPwhich (Yuanare et al. al. (2013)) PHEV is presented in section 2. In section 3 the map-based tionally efficient and often close to DP (Yuan et al. (2013)) 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 592 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 592 Control. Peer review© under responsibility of International Federation of Automatic Copyright 592 Copyright © © 2019 2019 IFAC IFAC 592 10.1016/j.ifacol.2019.09.094 Copyright © 2019 IFAC 592
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and analytical methods are discussed. Section 4 presents an evaluation of the resulting EMS and provides a detailed comparison. Finally, section 5 gives a conclusion. 2. SYSTEM MODELLING 𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕
-
+
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫
𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪
𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻
593
the chemical power arising from the combustion of fuel can be derived as ˙ c (t), n(t)). (2) Pf (Pc (t), B(t), n(t)) = B(t)hm(P Here, h is the lower heating value of fuel. Ps (Pe (t), n(t)) represents the electrical power provided by the electrical storage. The losses of the electrical engine in combination with the electrical losses of the battery are defined by the corresponding efficiency map depending on n(t) and Pe (t). Fig. 3 shows a normalized extract (valid for one given engine speed) of the efficiency maps of the considered system.
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬 𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎
Figure 1. System structure of parallel Plug-in-Hybrid Electric Vehicles. By deploying a combination of an electric motor with a combustion engine, PHEVs provide an advantage over conventional vehicles: They can be propelled by the utilization of chemical energy (obtained from the combustion of fuel) and/or electrical energy (obtained from the battery). We consider a parallel hybrid powertrain as shown in Fig. 1 which enables a mechanical coupling of the combustion engine by a clutch. This allows a combined operation if the clutch is closed (hybrid driving), or a sole operation of the electric motor if the clutch is opened and the combustion engine is switched off (pure electric driving). Furthermore it is possible to charge the vehicle’s battery by increasing the load of the combustion engine or during deceleration phases of the vehicle through the recuperation of energy. To set up an EMS for PHEVs, a mathematical description of the system is required. The vehicle power split (depicted in Fig. 2) is modelled as Preq (t) = Pc (t) + Pe (t), (1) where Preq (t) is the power requested from the vehicle and Pc (t) and Pe (t) are the mechanical powers delivered by the combustion engine and the electric motor. Fuel consumpiton
Energy consumption
𝑷𝑷𝒇𝒇 (𝒕𝒕) = 𝒉𝒉𝒎𝒎(𝒕𝒕)
Combustion engine
𝑷𝑷𝒔𝒔 (𝒕𝒕)
Electric motor
𝑷𝑷𝒄𝒄 (𝒕𝒕)
Clutch with boolean decision variable 𝑩𝑩(𝒕𝒕)
Figure 3. Extract of the normalized efficiency maps of the electric system and the combustion engine depicted for a given engine speed n. Using (1), (2) and Ps (Pe (t), n(t)), the fuel consumption can be expressed as a function of the degrees of freedom Ps (t) and B(t) and the external parameters Preq (t) and n(t) as Pf (Ps (t), Preq (t), B(t), n(t)). In this paper, we consider the battery energy level E(t) as the state and the input as [Ps (Pe (t), n(t)), B(t)]T . In combination with the initial and terminal value of the battery energy level, this results in the following description of the system dynamics and boundary conditions as: (3) E˙ = −Ps (Pe (t), n(t)) (4) E(0) = Estart E(T ) = Eend . (5) 3. PMP-BASED OPTIMIZATION OF PLUG-IN-HYBRID ELECTRIC VEHICLES 3.1 PMP-based Energy Management Strategy
𝑷𝑷𝒓𝒓𝒓𝒓𝒓𝒓 (𝒕𝒕)
𝑷𝑷𝒆𝒆 (𝒕𝒕)
Figure 2. Modelling of the vehicle power flows in the hybrid powertrain.
Given the system modelling of section 2, PMP can be applied to optimize the fuel consumption over a driving cycle by minimizing the cost T J = Pf (Ps (t), Preq (t), B(t), n(t)) dt. (6) 0
In this context it is assumed, that mechanical losses arising from the gearbox, the clutch and the differential are already considered in Preq (t). Moreover a selection of the gears is executed by a separate gear selection strategy and is not explicitly optimized. The vehicle’s fuel consumption is represented by the fuel consumption map m(P ˙ c (t), n(t)). The fuel mass flow depends on the current engine operation point (defined by the current engine speed n(t) and Pc (t)) and a boolean decision variable B(t) indicating the engine’s status (B(t) = 1 - engine is on and running, B(t) = 0 - engine is switched off). Accordingly, 593
PMP states that a candidate for an optimal control input for minimization of (6) is found, if the control input minimizes the Hamiltonian H(Ps (t), Preq (t), B(t), n(t), s(t)) = (7) Pf (Ps (t), Preq (t), B(t), n(t)) + s(t) · Ps (t) including the co-state s(t) = −λ(t), while (3)-(5) and ∂H(Ps (t), Preq (t), B(t), n(t), s(t)) (8) s˙ = − ∂E(t) are satisfied. Due to the system modelling of section 2, the Hamiltonian in (7) is independent of the state E(t). This
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implies that the optimal co-state is constant (compare Kim et al. (2011)). ∂H(Ps (t), Preq (t), B(t), n(t), s(t)) =0 s˙ = − ∂E(t) (9) !
→ s = const. The optimal, constant co-state can be computed using route-preview-information received by a navigation system. For that purpose map-based and analytical methods can be deployed as presented in the following. To increase the readability of this document, in the following, we simplify the notations by dropping the time index. 3.2 Map-Based PMP Map-based approaches construct the Hamiltonian directly by interpolating the exact engine efficiency maps of the considered system. This results in a map-based stage cost (Pf (Ps , Preq , B, n)) which can be applied to derive the Hamiltonian as: H(Ps , Preq , B, n, s) = Pf (Ps , Preq , B, n) + s · Ps . (10) Based on this Hamiltonian a three-step procedure for finding the optimal solution can be used: First, the optimal value of Ps is calculated for each time step by minimizing the corresponding Hamiltonian for some co-state. Thereby the case of a running engine (B = 1) is considered: ∂ ! H(Ps , Preq , B, n, s) = 0 → Ps,opt . (11) ∂Ps Second, the optimal engine on/off decision of every future time step is calculated by applying the control law of (12): 1 if Hon < Hof f B= (12) 0 if Hon ≥ Hof f with
and
Hon (Ps,opt (Preq , n, s), Preq , B, n, s) = Pf (Ps,opt (Preq , n, s), Preq , B, n)+ s · Ps,opt (Preq , n, s)
(13)
Hof f (Ps,of f (Preq , n), Preq , B, n, s) = s · Ps,of f (Preq , n).
(14)
Here, Ps,of f (Preq , n) represents the required electrical power which has to be provided by the battery to enable pure electrical driving. The optimal engine on/off decision in each time step is found by a comparison of the corresponding Hamiltonians resulting for the cases that the engine is either turned on (B = 1, Hon (Ps,opt (Preq , n, s), Preq , B, n, s)) or turned off (B = 0, Hof f (Ps,of f (Preq , n), Preq , B, n, s)) (compare Elbert et al. (2014)). The given co-state implies a specific state of charge trajectory which may not satisfy the boundary conditions. Consequently, in step 3, we update the costate using an iterative procedure (e.g. bisection) until the state of charge trajectory satisfies the boundary conditions (Shooting Method - compare Kim et al. (2014)) . 3.3 Analytical PMP Analytical PMP approaches utilize approximations of the exact efficiency maps to achieve an analytical description 594
of the stage cost. For that purpose, two different approximation methods can be applied, a separate approximation of the fuel consumption map and the efficiency map of the electrical system or a direct approximation of the stage cost which both lead to quadratic approximations of the Hamiltonian and are therefore computationally appealing. Quadratic approximation of the stage cost - Separate approximation of combustion engine and electric motor To derive an analytical description of the stage cost, the separate efficiency maps considered in the system modelling of section 2 can be approximated by simple functions (compare Fig. 3). For that purpose, in de Jager et al. (2013) an affine approximation of the engine efficiency map is presented as: Pf (Pc , B, n) ≈ B · (α0 (n) + α1 (n)Pc ).
(15)
Pe (Ps , n) ≈ β0 (n) + β1 (n)Ps + β2 (n)Ps2 .
(16)
Pf (Ps , Preq , B, n) ≈ B · (α0 (n)+ α1 (n)(Preq − β0 (n) − β1 (n)Ps − β2 (n)Ps2 )).
(17)
Furthermore, a quadratic approximation of the efficiency map of the electrical system is applied as: Here, the coefficients used for the approximation are depending on the current engine speed. Combining the system power split shown in (1) with (15) and (16), this results in a representation of the stage cost as:
Accordingly the system’s approximate Hamiltonian can be derived as H(Ps , Preq , B, n, s) ≈ B · (α0 (n) + α1 (n)(Preq − (18) β0 (n) − β1 (n)Ps − β2 (n)Ps2 )) + s · Ps .
Overall we obtain a convex, quadratic Hamiltonian. For the case of a running engine (B = 1) this provides an unique solution which approximates the solution of (11) as: s − α1 (n)β1 (n) . (19) Ps,opt (n, s) = 2α1 (n)β2 (n) Quadratic approximation of the stage cost - Direct approximation An alternative method is to apply a direct approximation of the exact stage cost as presented in Koot et al. (2005). For this purpose a quadratic approximation of the exact stage cost depending on the system’s efficiency maps is applied as Pf (Ps , Preq , B, n) ≈ B · (γ0 (Preq , n)+ γ1 (Preq , n)Ps + γ2 (Preq , n)Ps2 )
(20)
with coefficients depending Preq and n. For better visualization this approximation is depicted in Fig. 4. Accordingly the Hamiltonian can be derived by applying (20) as: H(Ps , Preq , B, n, s) ≈ B · (γ0 (Preq , n)+ γ1 (Preq , n)Ps + γ2 (Preq , n)Ps2 ) + s · Ps
(21)
which is also convex and quadratic and hence provides an unique solution which approximates the solution of (11) in the case of a running engine (B = 1) as: Ps,opt (Preq , n, s) =
−γ1 (Preq , n) − s . 2γ2 (Preq , n)
(22)
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Sampling Points of P (P ,P f
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Figure 5. Velocity profile considered for the evaluation.
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4.1 Fuel Consumption First we consider the cumulated fuel consumption in Fig. 6 and the total fuel consumption in table 1. We can conclude that map-based PMP results in the lowest fuel consumption as expected (Benchmark).
0 -1.5
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Figure 4. Quadratic approximation of the stage cost Normalized Depiction. Analytical solution By applying an analytical description of the stage cost, an analytical derivation of the optimal co-state based on future trip informations can be achieved. For that purpose, an iterative algorithm can be applied (compare Schmid et al. (2018)): First, the optimal power split in each time step of the future horizon is derived by applying an analytical calculation of the optimal co-state for the case of a continuously running engine (B = 1). Given the assumptions of section 3.1, the optimal control input Ps,opt (which is valid at each time step) can be calculated by minimizing the corresponding Hamiltonian. This results in a linear function depending on the engine speed, the requested power and the co-state which can be calculated easily (see (11),(19) and (22)). Subsequently, an analytical calculation of the optimal co-state can be applied. For that purpose (11) is combined with the given boundary conditions ((3)-(5)) and the preview information gained from the navigation system. Summing up the resulting system behaviour over the complete preview horizon results in a linear equation of the co-state which can be solved easily. Accordingly, the resulting co-state can be used for the calculation of the optimal power split. Second, the optimal engine on/off decision of every future time step can be found by applying the control law presented in (12). Finally, the optimal value of the co-state, the corresponding optimal power splits and optimal engine on/off decisions are recalculated for the remaining time steps of the future horizon where the engine remains switched on. This procedure is repeated until the algorithm converges and the optimal solution is found (see Schmid et al. (2018) for details). 4. EVALUATION This section considers the simulation of the strategies discussed in section 3. We consider a simulation of a given real world driving cycle depicted in Fig. 5 of 45 kilometres including urban, highway and rural street types. The main focus is to contrast fuel consumption, state of charge trajectory, power split, electrical driving decision and computational efficiency of the approaches. 595
Figure 6. Accumulated fuel consumption - Normalized Depiction.
Further one can notice that the application of a direct quadratic approximation of the stage cost results in a slightly increased fuel consumption which is very close to the benchmark solution. The application of a separate approximation of the engine efficiency map results in an increased fuel consumption (by ≈ 1%). Table 1. Fuel Consumption Separate approx. Direct approx. Map-based (Benchmark)
Fuel Consumption 101.193% 100.3452% 100%
4.2 State of Charge Trajectory, Power Split and Electrical Driving To analyse the deviations between the different state of charge trajectories depicted in Fig. 7, we consider the power splits and electrical drive decisions of each method.
State of Charge Trajectory - Normalized Depiction [%]
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which in fact has a non-negligible curvature. Further, according to the fact that the electrical drive decision is defined by a comparison of the Hamiltonians (compare (12)), the electrical drive decision differs for the approaches as well (see Fig. 9).
100 Separate approximation of the stage cost Direct approximation of the stage cost Map-based stage cost
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Figure 7. State of charge trajectories - Normalized Depiction.
(1) (2) (4) (3)
(5)
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(7)
Figure 9. Electrical driving decisions. The power split is defined by the location of the minimum of the corresponding Hamiltonian (compare (11)). Consequently the corresponding Hamiltonians defined in (10), (18) and (21) have to be examined in detail. For that purpose the corresponding Hamiltonians are depicted in Fig. 8 for two cases: 1.) Electric motor assists the combustion engine. 2.) Battery is charged by using an increased engine load point. 1.) Assisting the combustion engine
Location of the Minimum
To explain the varying electrical drive decisions, the electrical driving decision boundaries resulting from the different modelling approaches are depicted in Fig. 10.
2.) Charging the battery
Location of the Minimum
Figure 8. Comparison of the different Hamiltonians for two representative engine operation points - Normalized Depiction. The Hamiltonian resulting from a direct approximation of the stage cost is very similar to the Hamiltonian resulting from map-based stage costs. Further it is possible to notice that the minima of these approaches are very similar. When using separate approximations of the stage costs, this results in a significant perturbation of the Hamiltonian and it’s minimum. Overall the direct approximation offers a method closer to the benchmark. The disadvantages of using a separate approximation of engine efficiency maps can be explained by the inaccuracies of an affine approximation of the fuel consumption (compare Fig. 3) 596
Figure 10. Electrical driving decision boundaries - Normalized Depiction. The electrical driving decision boundaries are determined by a numerical method which compares the resulting Hamiltonians in each engine operation point to determine the engine on/off decision (see (12)). Accordingly the electrical driving decision boundaries can be derived from the received information. Considering the electrical driving decision boundaries shown in Fig. 10, it is possible to notice that the application of directly approximated stage costs results in an electrical driving decision boundary which is very close to the benchmark solution. Furthermore the corresponding electrical driving decision boundaries resulting from map-based and directly approximated stage costs
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show higher gradients compared to the electric driving decision boundary arising from separately approximated engine efficiency maps. This effect can be explained by the inaccuracies arising from the affine approximation of the fuel consumption as follows: In the case of low engine loads, the applied affine approximation overestimates the fuel consumption. Consequently pure electric driving will be favoured (electrical driving decision boundary is above the benchmark solution). Furthermore, in the case of high engine loads, the fuel consumption will be underestimated. This results in moderate electrical driving (electrical driving decision boundary is below the benchmark solution). To visualize the varying tendency of starting the engine, we tagged representative time steps in Fig. 9 and Fig. 10 where hybrid driving is favoured by map-based and directly approximated costs, while a separate approximation tends to favour electrical driving. Depending on the considered driving cycle, this results in a varying proportion between electrical and hybrid driving as exemplary shown for the considered driving cycle in table 2. Table 2. E-Drive Proportion Separate Approx. Direct Approx. Map-based
E-Drive Proportion 50.96% 46.25% 44.54%
4.3 Computation To evaluate the computational efficiency of the methods, the computation time of the algorithms were measured in Matlab R2015b on a standard office laptop including an Intel(R) Core (TM) i7-6600U CPU. The resulting computation time for one iteration of the algorithm (update of the co-state and determination of the optimal power split and engine on/off decision for every time step - compare section 3) and the total computation time are shown in table 3. Table 3. Computation Time Separate Approx. Direct Approx. Map-based
Time per Iteration 0,22 s 0,31 s 18,71 s
Iterations 37 20 7
Total 8,14 s 6,20 s 130,97 s
The increased computation of the map-based approach is due to two dimensional interpolations of the engine efficiency maps at each prediction step and the iterative simulation of the complete driving cycle (shooting method). Further the computational effort for one iteration of the analytical approach based on separately approximated engine efficiency maps is lower compared to the analytical approach based on a direct approximation of the stage cost. This can be explained by the fact that a separate approximation of the engine efficiency maps requires only one dimensional look up tables while the application of a direct approximation requires a two dimensional look-up table (compare dependencies of the applied coefficients). However, even if the separate approximation of the engine efficiency maps results in the lowest computational burden for one iteration, this method requires an increased number of iterations. This results in a higher total execution 597
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time compared to the method based on a direct approximation of the stage cost. 5. CONCLUSION This paper presents a comparison of PMP-based EMS for PHEVs and provides a decision guidance to evaluate mapbased PMP and two variants of analytical PMP. The application of map-based PMP results in the best fuel economy, but is computationally expensive. Analytical approaches on the other hand are computationally efficient, but sacrifice optimality with respect to fuel consumption in general. It is remarkable, however, that the direct approximation of the stage cost results in a fuel consumption extremely close to map-based PMP, but at a fraction of the computation time. Therefore this approach clearly offers the best tradeoff between fuel economy an computational efficiency. REFERENCES Amb¨ uhl, D., Sundstr¨om, O., Sciarretta, A., and Guzzella, L. (2010). Explicit optimal control policy and its practical application for hybrid electric powertrains. Control Engineering Practice, 18, 1429–1439. de Jager, B., van Keulen, T., and Kessels, J. (2013). Optimal Control of Hybrid Vehicles. Elbert, P., N¨ uesch, T., Ritter, A., Murgovski, N., and Guzzella, L. (2014). Engine on/off control for the energy management of a serial hybrid electric bus via convex optimization. IEEE Transactions on Vehicular Technology, 63, 3549–3559. Enang, W. and Bannister, C. (2017). Modelling and control of hybrid electric vehicles (a comprehensive review). Renewable and Sustainable Energy Reviews, 74, 1210–1239. Kim, N., Cha, S., and Peng, H. (2011). Optimal control of hybrid electric vehicles based on pontryagin’s minimum principle. IEEE Transactions on Control Systems Technology, 19, 1279–1287. Kim, N.W., Lee, D.H., Zheng, C., Shin, C., Seo, H., and Cha, S.W. (2014). Realization of pmp-based control for hybrid electric vehicles in a backward-looking simulation. International Journal of Automotive Technology, 15, 625–635. Koot, M., Kessels, J.T.B.A., de Jager, B., Heemels, W.P.M.H., van den Bosch, P.P.J., and Steinbuch, M. (2005). Energy management strategies for vehicular electric power systems. IEEE Transactions on Vehicular Technology, 54, 771–782. Rizzoni, G. and Onori, S. (2015). Energy management of hybrid electric vehicles: 15 years of development at the ohio state university. Oil & Gas Science and Technology Revue IFP Energies nouvelles), 70, 41–54. Schmid, R., B¨ urger, J., and Bajcinca, N. (2018). Efficient optimal control of plug-in-hybrid electric vehicles including explicit engine on/off decisions. European Control Conference. URL https://www.mv.uni-kl. de/fileadmin/mec/Dokumente/pdfs/ECC18_0080_FI. pdf. Yuan, Z., Teng, L., Fengchun, S., and Peng, H. (2013). Comparative study of dynamic programming and pontryagin’s minimum principle on energy management for a parallel hybrid electric vehicle. Energies, 6, 2305–2318.
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A comparison of PMP-based Energy A A comparison comparison of of PMP-based PMP-based Energy Energy Management Strategies for Plug-in-Hybrid A comparison of PMP-based Energy Management Strategies for Plug-in-Hybrid Management Electric Strategies for Plug-in-Hybrid Vehicles Management Electric Strategies for Plug-in-Hybrid Vehicles Electric Vehicles Vehicles ∗ ∗∗∗ Roland Schmid Electric urger ∗∗ ∗ Johannes B¨ ∗∗ Naim Bajcinca ∗∗∗
Roland Schmid ∗∗ Johannes B¨ u rger ∗∗ Naim Bajcinca ∗∗∗ Roland u Roland Schmid Schmid Johannes Johannes B¨ B¨ urger rger ∗∗ Naim Naim Bajcinca Bajcinca ∗∗∗ ∗ ∗∗ ∗∗∗ ∗Roland Schmid Johannes B¨ urger Naim Bajcinca M¨ u nchen, Germany (e-mail: ∗ BMW AG, Petuelring 130, 80788 BMW AG, Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: ∗ ∗ BMW AG, Petuelring 130, u [email protected]). BMW AG, Petuelring 130, 80788 80788 M¨ M¨ unchen, nchen, Germany Germany (e-mail: (e-mail: [email protected]). ∗ ∗∗ [email protected]). AG, Petuelring 130, 80788 M¨ uunchen, Germany (e-mail: BMW AG, Petuelring 130, 80788 M¨ nchen, Germany (e-mail: [email protected]). ∗∗BMW BMW AG, Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: ∗∗ ∗∗ BMW 130, 80788 M¨ u nchen, Germany (e-mail: [email protected]). [email protected]). BMW AG, AG, Petuelring Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: [email protected]). ∗∗∗∗∗ [email protected]). BMW AG, Petuelring 130, 80788 M¨ u nchen, Germany (e-mail: Mechatronics in Mechanical and Automotive ∗∗∗ Department [email protected]). of Mechatronics in Mechanical and Automotive ∗∗∗ ∗∗∗ Department Department of Mechatronics in Mechanical and Automotive [email protected]). Engineering, Technical University of Kaiserslautern, Postfach 3049, Department of Mechatronics in Kaiserslautern, Mechanical andPostfach Automotive Engineering, Technical University of 3049, ∗∗∗ Engineering, Technical University of Kaiserslautern, Postfach 3049, Department of Mechatronics in Kaiserslautern, Mechanical andPostfach Automotive 67663 Kaiserslautern, Germany (e-mail: [email protected]). Engineering, Technical University of 3049, 67663 Kaiserslautern, Germany (e-mail: [email protected]). 67663 Kaiserslautern, Germany (e-mail: [email protected]). Engineering, Technical University of Kaiserslautern, Postfach 3049, 67663 Kaiserslautern, Germany (e-mail: [email protected]). 67663 Kaiserslautern, Germany (e-mail: [email protected]). Abstract This paper presents a of energy management strategies based on PonAbstract This paper presents aa comparison comparison of energy management strategies based on PonAbstract This paper presents comparison of energy management strategies based on Pontryagin’s Minimums Principle (PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles Abstract This paper presents a comparison of energy management strategies based on Pon(PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles tryagin’s Minimums Principle tryagin’s Minimums Principle (PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles Abstract This paper presents a comparison of energy management strategies based on Ponis minimized by considering the power split and electrical driving decision as optimization tryagin’s Minimums Principle (PMP). The fuel consumption of Plug-in-Hybrid Electric Vehicles the power split and electrical driving decision as is by considering is minimized minimized bypaper considering the power split and electrical driving decision as optimization optimization tryagin’s Minimums Principle (PMP). Thesplit fuelbetween consumption ofdriving Plug-in-Hybrid Electric Vehicles variables. This discusses the trade-off optimality and computational efficiency is minimized by considering the power and electrical decision as optimization between variables. This paper discusses the trade-off optimality and computational efficiency variables. This paper discusses thepower trade-off between optimality and decision computational efficiency is minimized bypaper considering the splitbetween and electrical driving as optimization of analytical and map-based PMP methods. Thereby aa decision guidance for further research variables. This discusses the trade-off optimality and computational efficiency of analytical and map-based PMP methods. Thereby decision guidance for further research of analytical and map-based PMP methods. Thereby aa contrasting decision guidance for further research variables. This paper discusses the trade-offmethods between optimality andfuel computational efficiency is presented which evaluates the discussed by consumption, state of of analytical and map-based PMP methods. Thereby decision guidance for further research is presented which evaluates the discussed methods by contrasting fuel consumption, state of is presented which evaluates the discussed methods by contrasting fuel consumption, state of analytical and map-based PMP methods. Thereby a decision guidance for further research charge trajectory, split and engine on/off decision and computational efficiency. is presented whichpower evaluates the discussed methods by contrasting fuel consumption, state of of charge trajectory, power split and engine on/off decision and computational efficiency. charge trajectory, power split the anddiscussed engine on/off on/off decision and computational computational efficiency. state of is presented whichpower evaluates methods by contrasting fuel consumption, charge trajectory, split and engine decision and efficiency. © 2019,trajectory, IFAC (International Federation of Automatic by Elsevier Ltd.efficiency. All rights reserved. charge power split and engine on/off Control) decisionHosting and computational Keywords: Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, Keywords: Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, Keywords: Minimums Principle (PMP), Energy Management Strategy, Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based Keywords: Pontryagin’s Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based Keywords: Pontryagin’s Minimums Principle (PMP), Energy Management Strategy, PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based PMP Plug-in-Hybrid Electric Vehicle, Engine on/off decisions, Analyical PMP, Map-based PMP 1. INTRODUCTION in terms of optimality. To find an optimal control strategy strategy 1. INTRODUCTION in terms of optimality. To find an optimal control 1. in terms of optimality. To find an optimal control strategy for Hybrid-Electric-Vehicles (HEV) based on PMP, map1. INTRODUCTION INTRODUCTION in terms of optimality. To find an optimal control strategy for Hybrid-Electric-Vehicles (HEV) based on PMP, mapfor Hybrid-Electric-Vehicles (HEV) based on PMP, map1. INTRODUCTION in terms of analytical optimality.methods To find an optimal control strategy based and can be applied. Map-based for Hybrid-Electric-Vehicles (HEV) based on PMP, mapand analytical methods can be applied. Map-based Finite oil oil resources resources and and the the harmful harmful effects effects of of local local and and based based and analytical methods can be applied. Map-based for Hybrid-Electric-Vehicles based ondirectly PMP, mapPMP methods construct the(HEV) Hamiltonian from Finite based and analytical methods can be applied. Map-based PMP methods construct the Hamiltonian directly from Finite oil and the harmful local global traffic emissions have resulted ineffects a high highof demand of PMP Finite traffic oil resources resources andhave the resulted harmful in effects ofdemand local and and methods construct the Hamiltonian directly from based and analytical methods can be Jager applied. Map-based fuel consumption maps (compare de et al. (2013)). global emissions a of PMP methods construct the Hamiltonian directly from consumption maps (compare de Jager et al. (2013)). global emissions have resulted in aa high demand of Finite oil resources andand theefficient harmful local and alternative, ecofriendly propulsion systems for global traffic traffic emissions have resulted ineffects highofsystems demand of fuel fuel consumption maps (compare de Jager et al. (2013)). PMP methods construct the Hamiltonian directly from In contrast to this, analytical approaches approximate alternative, ecofriendly and efficient propulsion for fuel consumption maps (compare de Jager et al. (2013)). contrast to this, analytical approaches approximate alternative, and propulsion systems global trafficecofriendly emissions haveefficient resulted in a high demandfor of In transportation. To address address this problem, Plug-in-Hybrid alternative, ecofriendly and efficient propulsion systems for In contrast to this, analytical approaches fuel consumption maps (compare de Jager etapproximate al.functions (2013)). engine/electrical efficiency maps with quadratic transportation. To this problem, Plug-in-Hybrid In contrast to this, analytical approaches approximate efficiency maps with quadratic functions transportation. To address this problem, Plug-in-Hybrid alternative, ecofriendly and efficient propulsion systems for engine/electrical Electric Vehicles (PHEV) can be deployed. The hybrid transportation. To(PHEV) address this be problem, Plug-in-Hybrid engine/electrical efficiency maps with functions In contrast analytical approaches approximate of the degreetoof of this, freedom (e.g. Koot et quadratic al. (2005)) (2005)) and are are Electric Vehicles can deployed. The hybrid engine/electrical efficiency maps with quadratic functions of the degree freedom (e.g. Koot et al. and Electric Vehicles can deployed. The hybrid transportation. To(PHEV) address this be problem, Plug-in-Hybrid powertrain consists of a combination of an electric motor Electric Vehicles (PHEV) can be deployed. The hybrid of the degree of freedom Koot et al. (2005)) and are engine/electrical efficiency(e.g. maps with quadratic functions computationally efficient. Although numerous publications powertrain consists of a combination of an electric motor of the degree of freedom (e.g. Koot et al. (2005)) and are efficient. Although numerous publications powertrain consists of of an motor Electric Vehicles (PHEV) can offers be deployed. The hybrid and a combustion combustion engine and degrees of freedom freedom powertrain consists of aa combination combination of an electric electric motor computationally computationally efficient. Although numerous publications of the degree of freedom (e.g. Koot et al. for (2005)) and are discuss map-based or analytical methods optimization and a engine and offers degrees of computationally efficient. Although numerous publications map-based or analytical methods for optimization and combustion and offers degrees of powertrain consists of a combination of an electric motor (load point shift andengine electrical drive decision) which can be discuss and aapoint combustion engine anddrive offers degrees of freedom freedom discuss map-based analytical methods for computationally Although numerous publications of the power power splitefficient. ofor HEV, to date, date, few consider the (load shift and electrical decision) which can be discuss map-based orHEV, analytical methods for optimization optimization of the split of to few consider the setup setup (load point shift and electrical drive decision) which can be and a combustion engine and offers degrees of freedom used to reduce fuel consumption and/or emissions. Incan order (loadto point shift and electrical drive decision) whichIn be of the power split of to date, few consider setup discuss map-based orHEV, analytical methods for optimization of PHEVs and optimize both the power split in the combinaused reduce fuel consumption and/or emissions. order the power split of HEV, to date, few consider the setup of PHEVs and optimize both the power split in combinaused to reduce fuel consumption and/or emissions. order (load point shift and electrical drive decision) whichIn can be of to exploit the full potential of a hybrid powertrain strucused to reduce fuel consumption and/or emissions. In order PHEVs and optimize both the power split in combinathe power split of on/off HEV, to date, few consider the setup tion with an engine decision. Schmid et al. (2018) to exploit the full potential of a hybrid powertrain strucof PHEVs and optimize both the power split in combinawith an engine on/off decision. Schmid et al. (2018) to the of hybrid powertrain used to fuel potential consumption Instrucorder ture, anreduce intelligent energy management management strategy (EMS) is tion to exploit exploit the full full potential of aaand/or hybridemissions. powertrain struction with an engine on/off decision. Schmid et al. (2018) of PHEVs and optimize both the power splitan in analytical combinapresented a PMP-based EMS which uses ture, an intelligent energy strategy (EMS) is tion with an engine on/off decision. Schmid et al. (2018) aa PMP-based EMS which uses an analytical ture, an energy management strategy (EMS) is to exploit the different full potential ofofa EMS hybrid powertrain strucrequired. The classes have been discussed discussed ture, an intelligent intelligent energy management strategy (EMS) is presented presented PMP-based EMS which uses an analytical tion withfor an engine on/off decision. Schmid et al. (2018) method co-state computation based onan trip-preview required. The different classes of EMS have been presented a co-state PMP-based EMS which uses analytical method for computation based on trip-preview required. The different classes of EMS have been discussed ture, an intelligent energy management strategy (EMS) is in numerous surveys and publications (e.g. Enang and required. The surveys different and classes of EMS have been discussed method for co-state computation based on trip-preview presented a and PMP-based EMS which uses an analytical information combined it with the iterative approach in numerous publications (e.g. Enang and method for co-state computation based on trip-preview and combined it with the iterative approach in surveys and publications (e.g. Enang and information required. The different classes of EMS have been discussed Bannister (2017) and Rizzoni Rizzoni and Onori (2015)). in numerous numerous surveys and publications (e.g. Enang and information and combined it with the iterative approach method for co-state computation based onmethod trip-preview of Elbert et al. (2014). It is still open which offers Bannister (2017) and and Onori (2015)). information and combined it with the iterative approach Elbert et al. (2014). It is still open which method offers Bannister (2017) and Rizzoni and Onori (2015)). in numerous surveys and publications Enang and of One can distinguish distinguish rule-based EMS and(e.g. (online/offline) Bannister (2017) andrule-based Rizzoni and Onori (2015)). of Elbert et al. It is still open method offers information and(2014). combined it with thewhich iterative approach the best trade-off between optimality and computational One can EMS and (online/offline) of Elbert et al. (2014). It is still open which method offers best trade-off between optimality and computational One rule-based EMS and (online/offline) Bannister (2017) andEMS. Rizzoni and Onori (2015)). optimization based Rule-based EMS (e.g. Amb¨ Amb¨ uhl hl the One can can distinguish distinguish rule-based EMS and (online/offline) the best trade-off between and computational of Elbert et al. (2014). It is optimality still open which method offers efficiency. optimization based EMS. Rule-based EMS (e.g. u the best trade-off between optimality and computational efficiency. optimization based EMS. Rule-based EMS (e.g. hl One distinguish rule-based EMS et al.can (2010)) rely on on predefined rule and sets (online/offline) which are u oboptimization based EMS. Rule-based EMS (e.g. Amb¨ Amb¨ u hl efficiency. the best trade-off between optimality and computational This paper therefore provides decision guidance to evalevalet al. (2010)) rely predefined rule sets which are obefficiency. paper therefore provides aaa decision guidance to et al. (2010)) rely on predefined rule sets which are oboptimization based EMS. Rule-based EMS (e.g.based Amb¨ u hl This tained from offline cycle calibration and are on et al. (2010)) rely on predefined rule sets which are obThis paper therefore provides decision guidance to evalefficiency. uate map-based PMP versus two variants of analytical tained from offline cycle calibration and are based on This paper therefore provides a decision guidance to evalmap-based PMP versus two variants of analytical tained from offline and are based on et al. (2010)) rely oncycle predefined rule sets are obengineering experience. Tocalibration get guarantees guarantees of optimality optimality tained from experience. offline cycle calibration and which are based on uate uate map-based PMP versus variants of analytical This paper therefore provides atwo decision guidance to evalPMP. For this reason, we utilize the approximation preengineering To get of uate map-based PMP versus two variants of analytical For this reason, we utilize the approximation preengineering experience. To get of tained from offline cycleconsumption) and are based on PMP. (e.g. with respect to fuel fuel optimization based engineering experience. Tocalibration get guarantees guarantees of optimality optimality PMP. For this reason, we utilize the approximation preuate map-based versus twoextend variants analytical sented by Koot Koot etPMP al. (2005) (2005) and the of approach by (e.g. with respect to consumption) optimization based PMP. For this reason, we utilize the approximation presented by et al. and extend the approach by (e.g. to optimization based engineering experience. ToThe getglobal guarantees ofsolution optimality approaches are attractive. optimal can sented (e.g. with with respect respect to fuel fuel consumption) consumption) optimization based by Koot et al. (2005) and extend the approach by PMP. For this we utilize the approximation preintroducing thereason, explicit engine on/off decision presented approaches are attractive. The global optimal solution can sented by Koot et al. (2005) and extend the approach by introducing the explicit engine on/off decision presented approaches are attractive. The optimal solution can (e.g. with by respect to fuel consumption) optimization based be found dynamic programming (DP) as proposed proposed by introducing approaches are attractive. The global global(DP) optimal solution can the explicit engine decision presented sented by Koot et al. (2005) andon/off extend the the approach by in Schmid et al. (2018). Further we compare resulting be found by dynamic programming as by introducing the explicit engine on/off decision presented Schmid et al. (2018). Further we compare the resulting be found by dynamic programming (DP) as proposed by approaches attractive. TheDP global optimal solutioncomcan Yuan et al. al. (2013). However, requires extensive be found byare dynamic programming (DP) as proposed by in in Schmid et al. (2018). Further we compare the resulting introducing the explicit engine on/off decision presented approach with the approach of Schmid et al. (2018) and Yuan et (2013). However, DP requires extensive comin Schmid et al. (2018). Further we compare the resulting with the approach of Schmid et al. (2018) and Yuan et (2013). However, DP extensive combe found byressources dynamic programming (DP) as proposed by approach putational and hence isrequires not feasible for online online Yuan et al. al. (2013). However, DPis requires extensive comapproach the approach of et al. (2018) and in map-based Schmidwith et al. (2018). Further we compare resulting PMP method. We analyse the methods by putational ressources and hence not feasible for with the approach of Schmid Schmid etthe al.the (2018) and aaaapproach map-based PMP method. We analyse methods by putational and hence not feasible for online Yuan et al.ressources (2013). However, DPis requires extensive comapplication. As an alternative approach, Pontryagin’s Minputational ressources and hence is not feasible for online map-based PMP method. We analyse by approach with the approach ofstate Schmid etthe al.methods (2018) and contrasting fuel consumption, of charge trajectory, application. As an alternative approach, Pontryagin’s Mina map-based PMP method. We analyse the methods by fuel consumption, state of charge trajectory, application. As alternative Pontryagin’s Minputational ressources and is not to feasible foroptimal online imum Principle (PMP) canhence beapproach, applied find an an application. As an an alternative approach, Pontryagin’s Min- contrasting contrasting consumption, state of charge trajectory, acontrasting map-basedfuel PMP method. We analyse the on/off methods by computation time, power splits and engine deciimum Principle (PMP) can be applied to find optimal fuel consumption, state of charge trajectory, time, power splits and engine on/off deciimum (PMP) can applied find an application. As an alternative approach, Pontryagin’s control strategy (e.g. Kim et be al. (2014)).to PMP can be Minused computation imum Principle Principle (PMP) can be applied to find can an optimal optimal computation time, power splits and decicontrasting fuel consumption, state ofengine chargeon/off trajectory, sions. control strategy (e.g. Kim et al. (2014)). PMP be used computation time, power splits and engine on/off decicontrol strategy (e.g. et al. (2014)). can be imum toPMP find an optimal to findPrinciple candidates for Kim thecan optimal solution by applying applying the sions. control strategy (PMP) (e.g. Kim et be al. applied (2014)). PMP can be used used sions. computation power andThe engine on/offofdeciThe paper is is time, organized as splits follows: modelling the to find candidates for the optimal solution by the sions.paper organized as follows: The modelling of the to find candidates for the optimal solution by applying the control strategynecessary (e.g. Kim et al. (2014)). PMP can be used corresponding conditions, which are computato find candidates for the optimal solution by applying the The The paper is organized as follows: The modelling of the sions. PHEV is presented in section 2. In section 3 the map-based corresponding necessary conditions, which are computaThe paper is organized as follows: The modelling of the is presented in section 2. In section 33 the map-based corresponding computato find candidates foroften the conditions, optimal byare applying the PHEV tionally efficientnecessary and close to tosolution DPwhich (Yuan et al. (2013)) corresponding necessary conditions, which are computaPHEV is presented in section 2. In section the map-based The paper is organized as follows: The modelling of the tionally efficient and often close DP (Yuan et al. (2013)) PHEV is presented in section 2. In section 3 the map-based tionally and close (Yuan et (2013)) corresponding computationally efficient efficientnecessary and often oftenconditions, close to to DP DPwhich (Yuanare et al. al. (2013)) PHEV is presented in section 2. In section 3 the map-based tionally efficient and often close to DP (Yuan et al. (2013)) 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 592 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 592 Control. Peer review© under responsibility of International Federation of Automatic Copyright 592 Copyright © © 2019 2019 IFAC IFAC 592 10.1016/j.ifacol.2019.09.094 Copyright © 2019 IFAC 592
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and analytical methods are discussed. Section 4 presents an evaluation of the resulting EMS and provides a detailed comparison. Finally, section 5 gives a conclusion. 2. SYSTEM MODELLING 𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕
-
+
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫
𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮𝑮 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪
𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻
593
the chemical power arising from the combustion of fuel can be derived as ˙ c (t), n(t)). (2) Pf (Pc (t), B(t), n(t)) = B(t)hm(P Here, h is the lower heating value of fuel. Ps (Pe (t), n(t)) represents the electrical power provided by the electrical storage. The losses of the electrical engine in combination with the electrical losses of the battery are defined by the corresponding efficiency map depending on n(t) and Pe (t). Fig. 3 shows a normalized extract (valid for one given engine speed) of the efficiency maps of the considered system.
𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬 𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎
Figure 1. System structure of parallel Plug-in-Hybrid Electric Vehicles. By deploying a combination of an electric motor with a combustion engine, PHEVs provide an advantage over conventional vehicles: They can be propelled by the utilization of chemical energy (obtained from the combustion of fuel) and/or electrical energy (obtained from the battery). We consider a parallel hybrid powertrain as shown in Fig. 1 which enables a mechanical coupling of the combustion engine by a clutch. This allows a combined operation if the clutch is closed (hybrid driving), or a sole operation of the electric motor if the clutch is opened and the combustion engine is switched off (pure electric driving). Furthermore it is possible to charge the vehicle’s battery by increasing the load of the combustion engine or during deceleration phases of the vehicle through the recuperation of energy. To set up an EMS for PHEVs, a mathematical description of the system is required. The vehicle power split (depicted in Fig. 2) is modelled as Preq (t) = Pc (t) + Pe (t), (1) where Preq (t) is the power requested from the vehicle and Pc (t) and Pe (t) are the mechanical powers delivered by the combustion engine and the electric motor. Fuel consumpiton
Energy consumption
𝑷𝑷𝒇𝒇 (𝒕𝒕) = 𝒉𝒉𝒎𝒎(𝒕𝒕)
Combustion engine
𝑷𝑷𝒔𝒔 (𝒕𝒕)
Electric motor
𝑷𝑷𝒄𝒄 (𝒕𝒕)
Clutch with boolean decision variable 𝑩𝑩(𝒕𝒕)
Figure 3. Extract of the normalized efficiency maps of the electric system and the combustion engine depicted for a given engine speed n. Using (1), (2) and Ps (Pe (t), n(t)), the fuel consumption can be expressed as a function of the degrees of freedom Ps (t) and B(t) and the external parameters Preq (t) and n(t) as Pf (Ps (t), Preq (t), B(t), n(t)). In this paper, we consider the battery energy level E(t) as the state and the input as [Ps (Pe (t), n(t)), B(t)]T . In combination with the initial and terminal value of the battery energy level, this results in the following description of the system dynamics and boundary conditions as: (3) E˙ = −Ps (Pe (t), n(t)) (4) E(0) = Estart E(T ) = Eend . (5) 3. PMP-BASED OPTIMIZATION OF PLUG-IN-HYBRID ELECTRIC VEHICLES 3.1 PMP-based Energy Management Strategy
𝑷𝑷𝒓𝒓𝒓𝒓𝒓𝒓 (𝒕𝒕)
𝑷𝑷𝒆𝒆 (𝒕𝒕)
Figure 2. Modelling of the vehicle power flows in the hybrid powertrain.
Given the system modelling of section 2, PMP can be applied to optimize the fuel consumption over a driving cycle by minimizing the cost T J = Pf (Ps (t), Preq (t), B(t), n(t)) dt. (6) 0
In this context it is assumed, that mechanical losses arising from the gearbox, the clutch and the differential are already considered in Preq (t). Moreover a selection of the gears is executed by a separate gear selection strategy and is not explicitly optimized. The vehicle’s fuel consumption is represented by the fuel consumption map m(P ˙ c (t), n(t)). The fuel mass flow depends on the current engine operation point (defined by the current engine speed n(t) and Pc (t)) and a boolean decision variable B(t) indicating the engine’s status (B(t) = 1 - engine is on and running, B(t) = 0 - engine is switched off). Accordingly, 593
PMP states that a candidate for an optimal control input for minimization of (6) is found, if the control input minimizes the Hamiltonian H(Ps (t), Preq (t), B(t), n(t), s(t)) = (7) Pf (Ps (t), Preq (t), B(t), n(t)) + s(t) · Ps (t) including the co-state s(t) = −λ(t), while (3)-(5) and ∂H(Ps (t), Preq (t), B(t), n(t), s(t)) (8) s˙ = − ∂E(t) are satisfied. Due to the system modelling of section 2, the Hamiltonian in (7) is independent of the state E(t). This
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implies that the optimal co-state is constant (compare Kim et al. (2011)). ∂H(Ps (t), Preq (t), B(t), n(t), s(t)) =0 s˙ = − ∂E(t) (9) !
→ s = const. The optimal, constant co-state can be computed using route-preview-information received by a navigation system. For that purpose map-based and analytical methods can be deployed as presented in the following. To increase the readability of this document, in the following, we simplify the notations by dropping the time index. 3.2 Map-Based PMP Map-based approaches construct the Hamiltonian directly by interpolating the exact engine efficiency maps of the considered system. This results in a map-based stage cost (Pf (Ps , Preq , B, n)) which can be applied to derive the Hamiltonian as: H(Ps , Preq , B, n, s) = Pf (Ps , Preq , B, n) + s · Ps . (10) Based on this Hamiltonian a three-step procedure for finding the optimal solution can be used: First, the optimal value of Ps is calculated for each time step by minimizing the corresponding Hamiltonian for some co-state. Thereby the case of a running engine (B = 1) is considered: ∂ ! H(Ps , Preq , B, n, s) = 0 → Ps,opt . (11) ∂Ps Second, the optimal engine on/off decision of every future time step is calculated by applying the control law of (12): 1 if Hon < Hof f B= (12) 0 if Hon ≥ Hof f with
and
Hon (Ps,opt (Preq , n, s), Preq , B, n, s) = Pf (Ps,opt (Preq , n, s), Preq , B, n)+ s · Ps,opt (Preq , n, s)
(13)
Hof f (Ps,of f (Preq , n), Preq , B, n, s) = s · Ps,of f (Preq , n).
(14)
Here, Ps,of f (Preq , n) represents the required electrical power which has to be provided by the battery to enable pure electrical driving. The optimal engine on/off decision in each time step is found by a comparison of the corresponding Hamiltonians resulting for the cases that the engine is either turned on (B = 1, Hon (Ps,opt (Preq , n, s), Preq , B, n, s)) or turned off (B = 0, Hof f (Ps,of f (Preq , n), Preq , B, n, s)) (compare Elbert et al. (2014)). The given co-state implies a specific state of charge trajectory which may not satisfy the boundary conditions. Consequently, in step 3, we update the costate using an iterative procedure (e.g. bisection) until the state of charge trajectory satisfies the boundary conditions (Shooting Method - compare Kim et al. (2014)) . 3.3 Analytical PMP Analytical PMP approaches utilize approximations of the exact efficiency maps to achieve an analytical description 594
of the stage cost. For that purpose, two different approximation methods can be applied, a separate approximation of the fuel consumption map and the efficiency map of the electrical system or a direct approximation of the stage cost which both lead to quadratic approximations of the Hamiltonian and are therefore computationally appealing. Quadratic approximation of the stage cost - Separate approximation of combustion engine and electric motor To derive an analytical description of the stage cost, the separate efficiency maps considered in the system modelling of section 2 can be approximated by simple functions (compare Fig. 3). For that purpose, in de Jager et al. (2013) an affine approximation of the engine efficiency map is presented as: Pf (Pc , B, n) ≈ B · (α0 (n) + α1 (n)Pc ).
(15)
Pe (Ps , n) ≈ β0 (n) + β1 (n)Ps + β2 (n)Ps2 .
(16)
Pf (Ps , Preq , B, n) ≈ B · (α0 (n)+ α1 (n)(Preq − β0 (n) − β1 (n)Ps − β2 (n)Ps2 )).
(17)
Furthermore, a quadratic approximation of the efficiency map of the electrical system is applied as: Here, the coefficients used for the approximation are depending on the current engine speed. Combining the system power split shown in (1) with (15) and (16), this results in a representation of the stage cost as:
Accordingly the system’s approximate Hamiltonian can be derived as H(Ps , Preq , B, n, s) ≈ B · (α0 (n) + α1 (n)(Preq − (18) β0 (n) − β1 (n)Ps − β2 (n)Ps2 )) + s · Ps .
Overall we obtain a convex, quadratic Hamiltonian. For the case of a running engine (B = 1) this provides an unique solution which approximates the solution of (11) as: s − α1 (n)β1 (n) . (19) Ps,opt (n, s) = 2α1 (n)β2 (n) Quadratic approximation of the stage cost - Direct approximation An alternative method is to apply a direct approximation of the exact stage cost as presented in Koot et al. (2005). For this purpose a quadratic approximation of the exact stage cost depending on the system’s efficiency maps is applied as Pf (Ps , Preq , B, n) ≈ B · (γ0 (Preq , n)+ γ1 (Preq , n)Ps + γ2 (Preq , n)Ps2 )
(20)
with coefficients depending Preq and n. For better visualization this approximation is depicted in Fig. 4. Accordingly the Hamiltonian can be derived by applying (20) as: H(Ps , Preq , B, n, s) ≈ B · (γ0 (Preq , n)+ γ1 (Preq , n)Ps + γ2 (Preq , n)Ps2 ) + s · Ps
(21)
which is also convex and quadratic and hence provides an unique solution which approximates the solution of (11) in the case of a running engine (B = 1) as: Ps,opt (Preq , n, s) =
−γ1 (Preq , n) − s . 2γ2 (Preq , n)
(22)
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Sampling Points of P (P ,P f
s
re q
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1 ,B,n))
P f (Ps ,P req ,B,n)
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Figure 5. Velocity profile considered for the evaluation.
0.5
4.1 Fuel Consumption First we consider the cumulated fuel consumption in Fig. 6 and the total fuel consumption in table 1. We can conclude that map-based PMP results in the lowest fuel consumption as expected (Benchmark).
0 -1.5
-1
-0.5
0
0.5
1
Ps
Figure 4. Quadratic approximation of the stage cost Normalized Depiction. Analytical solution By applying an analytical description of the stage cost, an analytical derivation of the optimal co-state based on future trip informations can be achieved. For that purpose, an iterative algorithm can be applied (compare Schmid et al. (2018)): First, the optimal power split in each time step of the future horizon is derived by applying an analytical calculation of the optimal co-state for the case of a continuously running engine (B = 1). Given the assumptions of section 3.1, the optimal control input Ps,opt (which is valid at each time step) can be calculated by minimizing the corresponding Hamiltonian. This results in a linear function depending on the engine speed, the requested power and the co-state which can be calculated easily (see (11),(19) and (22)). Subsequently, an analytical calculation of the optimal co-state can be applied. For that purpose (11) is combined with the given boundary conditions ((3)-(5)) and the preview information gained from the navigation system. Summing up the resulting system behaviour over the complete preview horizon results in a linear equation of the co-state which can be solved easily. Accordingly, the resulting co-state can be used for the calculation of the optimal power split. Second, the optimal engine on/off decision of every future time step can be found by applying the control law presented in (12). Finally, the optimal value of the co-state, the corresponding optimal power splits and optimal engine on/off decisions are recalculated for the remaining time steps of the future horizon where the engine remains switched on. This procedure is repeated until the algorithm converges and the optimal solution is found (see Schmid et al. (2018) for details). 4. EVALUATION This section considers the simulation of the strategies discussed in section 3. We consider a simulation of a given real world driving cycle depicted in Fig. 5 of 45 kilometres including urban, highway and rural street types. The main focus is to contrast fuel consumption, state of charge trajectory, power split, electrical driving decision and computational efficiency of the approaches. 595
Figure 6. Accumulated fuel consumption - Normalized Depiction.
Further one can notice that the application of a direct quadratic approximation of the stage cost results in a slightly increased fuel consumption which is very close to the benchmark solution. The application of a separate approximation of the engine efficiency map results in an increased fuel consumption (by ≈ 1%). Table 1. Fuel Consumption Separate approx. Direct approx. Map-based (Benchmark)
Fuel Consumption 101.193% 100.3452% 100%
4.2 State of Charge Trajectory, Power Split and Electrical Driving To analyse the deviations between the different state of charge trajectories depicted in Fig. 7, we consider the power splits and electrical drive decisions of each method.
State of Charge Trajectory - Normalized Depiction [%]
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which in fact has a non-negligible curvature. Further, according to the fact that the electrical drive decision is defined by a comparison of the Hamiltonians (compare (12)), the electrical drive decision differs for the approaches as well (see Fig. 9).
100 Separate approximation of the stage cost Direct approximation of the stage cost Map-based stage cost
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Figure 7. State of charge trajectories - Normalized Depiction.
(1) (2) (4) (3)
(5)
(6)
(7)
Figure 9. Electrical driving decisions. The power split is defined by the location of the minimum of the corresponding Hamiltonian (compare (11)). Consequently the corresponding Hamiltonians defined in (10), (18) and (21) have to be examined in detail. For that purpose the corresponding Hamiltonians are depicted in Fig. 8 for two cases: 1.) Electric motor assists the combustion engine. 2.) Battery is charged by using an increased engine load point. 1.) Assisting the combustion engine
Location of the Minimum
To explain the varying electrical drive decisions, the electrical driving decision boundaries resulting from the different modelling approaches are depicted in Fig. 10.
2.) Charging the battery
Location of the Minimum
Figure 8. Comparison of the different Hamiltonians for two representative engine operation points - Normalized Depiction. The Hamiltonian resulting from a direct approximation of the stage cost is very similar to the Hamiltonian resulting from map-based stage costs. Further it is possible to notice that the minima of these approaches are very similar. When using separate approximations of the stage costs, this results in a significant perturbation of the Hamiltonian and it’s minimum. Overall the direct approximation offers a method closer to the benchmark. The disadvantages of using a separate approximation of engine efficiency maps can be explained by the inaccuracies of an affine approximation of the fuel consumption (compare Fig. 3) 596
Figure 10. Electrical driving decision boundaries - Normalized Depiction. The electrical driving decision boundaries are determined by a numerical method which compares the resulting Hamiltonians in each engine operation point to determine the engine on/off decision (see (12)). Accordingly the electrical driving decision boundaries can be derived from the received information. Considering the electrical driving decision boundaries shown in Fig. 10, it is possible to notice that the application of directly approximated stage costs results in an electrical driving decision boundary which is very close to the benchmark solution. Furthermore the corresponding electrical driving decision boundaries resulting from map-based and directly approximated stage costs
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show higher gradients compared to the electric driving decision boundary arising from separately approximated engine efficiency maps. This effect can be explained by the inaccuracies arising from the affine approximation of the fuel consumption as follows: In the case of low engine loads, the applied affine approximation overestimates the fuel consumption. Consequently pure electric driving will be favoured (electrical driving decision boundary is above the benchmark solution). Furthermore, in the case of high engine loads, the fuel consumption will be underestimated. This results in moderate electrical driving (electrical driving decision boundary is below the benchmark solution). To visualize the varying tendency of starting the engine, we tagged representative time steps in Fig. 9 and Fig. 10 where hybrid driving is favoured by map-based and directly approximated costs, while a separate approximation tends to favour electrical driving. Depending on the considered driving cycle, this results in a varying proportion between electrical and hybrid driving as exemplary shown for the considered driving cycle in table 2. Table 2. E-Drive Proportion Separate Approx. Direct Approx. Map-based
E-Drive Proportion 50.96% 46.25% 44.54%
4.3 Computation To evaluate the computational efficiency of the methods, the computation time of the algorithms were measured in Matlab R2015b on a standard office laptop including an Intel(R) Core (TM) i7-6600U CPU. The resulting computation time for one iteration of the algorithm (update of the co-state and determination of the optimal power split and engine on/off decision for every time step - compare section 3) and the total computation time are shown in table 3. Table 3. Computation Time Separate Approx. Direct Approx. Map-based
Time per Iteration 0,22 s 0,31 s 18,71 s
Iterations 37 20 7
Total 8,14 s 6,20 s 130,97 s
The increased computation of the map-based approach is due to two dimensional interpolations of the engine efficiency maps at each prediction step and the iterative simulation of the complete driving cycle (shooting method). Further the computational effort for one iteration of the analytical approach based on separately approximated engine efficiency maps is lower compared to the analytical approach based on a direct approximation of the stage cost. This can be explained by the fact that a separate approximation of the engine efficiency maps requires only one dimensional look up tables while the application of a direct approximation requires a two dimensional look-up table (compare dependencies of the applied coefficients). However, even if the separate approximation of the engine efficiency maps results in the lowest computational burden for one iteration, this method requires an increased number of iterations. This results in a higher total execution 597
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time compared to the method based on a direct approximation of the stage cost. 5. CONCLUSION This paper presents a comparison of PMP-based EMS for PHEVs and provides a decision guidance to evaluate mapbased PMP and two variants of analytical PMP. The application of map-based PMP results in the best fuel economy, but is computationally expensive. Analytical approaches on the other hand are computationally efficient, but sacrifice optimality with respect to fuel consumption in general. It is remarkable, however, that the direct approximation of the stage cost results in a fuel consumption extremely close to map-based PMP, but at a fraction of the computation time. Therefore this approach clearly offers the best tradeoff between fuel economy an computational efficiency. REFERENCES Amb¨ uhl, D., Sundstr¨om, O., Sciarretta, A., and Guzzella, L. (2010). Explicit optimal control policy and its practical application for hybrid electric powertrains. Control Engineering Practice, 18, 1429–1439. de Jager, B., van Keulen, T., and Kessels, J. (2013). Optimal Control of Hybrid Vehicles. Elbert, P., N¨ uesch, T., Ritter, A., Murgovski, N., and Guzzella, L. (2014). Engine on/off control for the energy management of a serial hybrid electric bus via convex optimization. IEEE Transactions on Vehicular Technology, 63, 3549–3559. Enang, W. and Bannister, C. (2017). Modelling and control of hybrid electric vehicles (a comprehensive review). Renewable and Sustainable Energy Reviews, 74, 1210–1239. Kim, N., Cha, S., and Peng, H. (2011). Optimal control of hybrid electric vehicles based on pontryagin’s minimum principle. IEEE Transactions on Control Systems Technology, 19, 1279–1287. Kim, N.W., Lee, D.H., Zheng, C., Shin, C., Seo, H., and Cha, S.W. (2014). Realization of pmp-based control for hybrid electric vehicles in a backward-looking simulation. International Journal of Automotive Technology, 15, 625–635. Koot, M., Kessels, J.T.B.A., de Jager, B., Heemels, W.P.M.H., van den Bosch, P.P.J., and Steinbuch, M. (2005). Energy management strategies for vehicular electric power systems. IEEE Transactions on Vehicular Technology, 54, 771–782. Rizzoni, G. and Onori, S. (2015). Energy management of hybrid electric vehicles: 15 years of development at the ohio state university. Oil & Gas Science and Technology Revue IFP Energies nouvelles), 70, 41–54. Schmid, R., B¨ urger, J., and Bajcinca, N. (2018). Efficient optimal control of plug-in-hybrid electric vehicles including explicit engine on/off decisions. European Control Conference. URL https://www.mv.uni-kl. de/fileadmin/mec/Dokumente/pdfs/ECC18_0080_FI. pdf. Yuan, Z., Teng, L., Fengchun, S., and Peng, H. (2013). Comparative study of dynamic programming and pontryagin’s minimum principle on energy management for a parallel hybrid electric vehicle. Energies, 6, 2305–2318.