Design and Control Co-Optimization for Hybrid Powertrains: Development of Dedicated Optimal Energy Management Strategy

Design and Control Co-Optimization for Hybrid Powertrains: Development of Dedicated Optimal Energy Management Strategy

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Preprints, 8th 8th IFAC IFAC International International Symposium Symposium on on Preprints, Advances in Control Preprints, 8th IFAC International Advances in Automotive Automotive Control Symposium on Preprints, 8th IFAC International Symposium on June 19-23, Sweden Preprints, 8th IFACNorrköping, International Symposium ononline at www.sciencedirect.com Available Advances in2016. Automotive Control June 19-23, 2016. Norrköping, Sweden Advances Automotive Control Advances in2016. Automotive Control June 19-23,in Norrköping, Sweden June June 19-23, 19-23, 2016. 2016. Norrköping, Norrköping, Sweden Sweden

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IFAC-PapersOnLine 49-11 (2016) 277–284 Design Design and and Control Control Co-Optimization Co-Optimization for for Hybrid Hybrid Powertrains: Powertrains: Development Development of of Design and Control Co-Optimization for Hybrid Powertrains: Development Dedicated Optimal Energy Management Strategy Design and Control Co-Optimization for Hybrid Powertrains: Development of of Dedicated Optimal Energy Management Strategy Dedicated Optimal Energy Management Strategy Dedicated Optimal Energy Management Strategy Jianning ZHAO11, Antonio SCIARRETTA11

Jianning ZHAO , Antonio SCIARRETTA Jianning ZHAO111, Antonio SCIARRETTA111 Jianning Antonio SCIARRETTA SCIARRETTA Jianning ZHAO ZHAO ,, Antonio 1 1 IFP Energies nouvelles, 92852 Rueil Malmaison, France (e-mail: {jianning.zhao, antonio.sciarretta}@ifpen.fr) IFP Energies nouvelles, 92852 Rueil Malmaison, France (e-mail: {jianning.zhao, antonio.sciarretta}@ifpen.fr) 1 11 IFP Energies nouvelles, 92852 Rueil Malmaison, France (e-mail: {jianning.zhao, antonio.sciarretta}@ifpen.fr) IFP IFP Energies Energies nouvelles, nouvelles, 92852 92852 Rueil Rueil Malmaison, Malmaison, France France (e-mail: (e-mail: {jianning.zhao, {jianning.zhao, antonio.sciarretta}@ifpen.fr) antonio.sciarretta}@ifpen.fr) Abstract: Computationally-efficient Computationally-efficient optimal optimal energy-management energy-management strategies strategies are are required required for for the the optimal optimal Abstract: design of hybrid electric vehicles with respect to fuel economy and other criteria. In this paper, novel Abstract: Computationally-efficient optimal energy-management strategies are required for thethe optimal design of hybrid electric vehicles with respect to fuel economy and other criteria. In this paper, the novel Abstract: Computationally-efficient optimal energy-management strategies are required for optimal Abstract: Computationally-efficient optimal energy-management strategies are required for the the optimal Selective Hamiltonian Minimization (SHM) technique is introduced. Based on Pontryagin’s Minimum design of hybrid electric vehicles with respect to fuel economy and other criteria. In this paper, the novel Selective Hamiltonian Minimization (SHM) technique is introduced. Based on Pontryagin’s Minimum design of electric vehicles respect to economy and criteria. In paper, novel design of hybrid hybrid electricMinimization vehicles with with(SHM) respect to fuel fuel economy and other other criteria. In this thismodes, paper, the the novel Principle, the developed explicitly calculating possible and Selective technique is introduced. Based oncontrol Pontryagin’s Principle, Hamiltonian the SHM SHM is is Minimization developed by by(SHM) explicitly calculating possible optimal optimal modes, Minimum and then then Selective Hamiltonian Minimization (SHM) technique is introduced. introduced. Based on oncontrol Pontryagin’s Minimum Selective Hamiltonian technique is Based Pontryagin’s Minimum selecting the that function. Parametric analytical models of powertrain powertrain Principle, theone SHM isminimizes developedthe byHamiltonian explicitly calculating possible optimal control modes, and then selecting the one that minimizes the Hamiltonian function. Parametric analytical models of Principle, the SHM is developed by explicitly calculating possible optimal control modes, and Principle, theare SHM developed byHamiltonian explicitly calculating possible optimal control modes, and then then components developed to Hamiltonians in form. Engine battery are selecting the one thatisminimizes the function. Parametric models of models powertrain components are developed to compute compute Hamiltonians in closed closed form. analytical Engine and and battery models are selecting the one that minimizes the Hamiltonian function. Parametric analytical models of powertrain selecting the one that minimizes the Hamiltonian function. Parametric analytical models of powertrain further expressed in terms of their main design parameters and technologies thanks to a statistical components are developed to compute Hamiltonians in closed form. Engine and battery models are further expressed in terms of their main design parameters and technologies thanks to a statistical components are developed to compute Hamiltonians in closed form. Engine and battery models are components are developed compute Hamiltonians in closed form. Engine and battery are analysis over components. An example of optimization is to illustrate the further in sample terms to of their main parameters technologies thanks amodels statistical analysisexpressed over several several sample components. Andesign example of design designand optimization is presented presented to illustrate the further expressed in terms of their main design parameters and technologies thanks to aa statistical further expressed in terms of their main design parameters and technologies thanks to statistical effectiveness the approach. analysis over of several sample components. example of design optimization is presented to illustrate the effectiveness of the proposed proposed approach. An analysis several sample An analysis over over of several sample components. components. An example example of of design design optimization optimization is is presented presented to to illustrate illustrate the the effectiveness the proposed approach. Keywords: HEV, EMS, SHM, co-optimization, parametrization. effectiveness of the proposed approach. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: HEV, EMS, SHM, co-optimization, parametrization. effectiveness of the proposed approach. Keywords: HEV, EMS, SHM, co-optimization, parametrization. Keywords: HEV, HEV, EMS, EMS, SHM, SHM, co-optimization, co-optimization, parametrization. parametrization. Keywords: Nüesch Nüesch et et al. al. (2014). (2014). This This technique technique allows allows to to optimize optimize the the 1. 1. INTRODUCTION INTRODUCTION EMS and sizing of HEV components simultaneously. Nüesch et al. (2014). This technique allows to optimize the EMS and sizing of HEV components simultaneously. Nüesch et al. (2014). This technique allows to optimize the 1. INTRODUCTION Nüesch et al. (2014). This technique allows to optimize the However, it sizing has toofbe be combined with either a heuristic heuristic Hybrid (HEV) EMS and HEV components simultaneously. 1. 1. INTRODUCTION INTRODUCTION it has to combined with either a Hybrid electric electric vehicles vehicles (HEV) have have proved proved to to be be an an However, EMS and sizing of HEV components simultaneously. EMS and sizing of HEV components simultaneously. controller or with another optimization method, such as DP effective means to tovehicles reduce fuel fuel consumption of road roadtotransport. transport. it with has to be combined withmethod, either such a heuristic Hybrid electric (HEV) have proved be an However, controller or another optimization as DP effective means reduce consumption of it has be combined with either aa heuristic Hybrid vehicles (HEV) have to be an However, it with has to to be non-convex combined with either such heuristic Hybrid electric electric vehicles (HEV) have proved proved bewith an However, or PMP, to deal with powertrain models or However, their higher inherent complexity controller or another optimization method, as DP effective means to reduce fuel consumption ofcompared roadtotransport. or PMP, to deal with non-convex powertrain models or However, their higher inherent complexity compared with controller or with another optimization method, such as DP effective to reduce fuel of road transport. controller or with another optimization method, such as and DP effective means means to reduce fuel consumption consumption ofcompared road transport. discrete control variables, such as gear shifting law conventional powertrains requires careful design and control. or PMP, to deal with non-convex powertrain models or However, their higher inherent complexity with discrete control variables, such as gear shifting law and conventional powertrains requires careful design and control. PMP, to deal with powertrain models or However, their higher inherent complexity compared with or PMP, to control. dealvariables, with non-convex non-convex powertrain models or However, their higher inherent complexity compared with or engine on/off Although heuristic design and control rules are widely discrete control such as gear shifting law and conventional powertrains requires careful design and control. on/off control. Although heuristic design and careful control design rules and are control. widely engine discrete control variables, such as gear shifting law and conventional powertrains requires discrete control variables, such as gear shifting law and conventional powertrains requires careful design and control. employed in the automatic Although heuristic design industry, and control rules optimization are widely engine on/off control. employed in the automotive automotive industry, automatic Hofman and Steinbuch engine Although heuristic design and rules are engine on/off on/off control. Although heuristic design and control control rules optimization are ofwidely widely Hofman and control. Steinbuch (2014) (2014) proposed proposed an an integrated integrated techniques could further improve the effectiveness HEV employed in the automotive industry, automatic optimization techniques could further improve the effectiveness of HEV optimization method composed of sequential quadratic Hofman and Steinbuch (2014) proposed an integrated employed in the automotive industry, automatic optimization employed in the automotive industry, automatic optimization optimization method composed of sequential quadratic design and reduce their development time. At least two Hofman and Steinbuch (2014) proposed an techniques could further improve the effectiveness of HEV Hofman and method Steinbuch (2014) proposed an integrated integrated design and reduce their development time. At least two programming nested with DP to optimize topology, EMS, optimization composed of sequential quadratic techniques could further improve the effectiveness of HEV techniques could further improve the effectiveness of HEV programming nested with DP to optimize topology, EMS, coupled optimization layers are relevant see optimization method method composed composed of of sequential sequential quadratic quadratic design and reduce their development time.for At HEV, least two optimization coupled optimization layers are relevant for HEV, see and sizing parameters of a parallel HEV. Millo et al. (2015) programming nested with DP to optimize topology, EMS, design and reduce their development time. At least two design and reduce their development time. At least two and sizing parameters of a parallel HEV. Millo et al. (2015) Guzzella optimization and Sciarretta Sciarrettalayers (2013, are p. 42). 42). Control optimization nested with DP to optimize topology, EMS, coupled relevant foroptimization HEV, see programming programming nested with DP tosizing optimize topology, EMS, Guzzella and (2013, p. Control optimized EMS, architecture and parameters of various and sizing parameters of a parallel HEV. Millo et al. (2015) coupled optimization layers are relevant for HEV, see coupled optimization layers are relevant foroptimization HEV, see optimized EMS, architecture and sizing parameters of various refers to the best possible Energy Management Strategy and sizing parameters of aa parallel HEV. Millo et al. (2015) Guzzella and Sciarretta (2013, p. 42). Control and sizing parameters of parallel HEV. Millo et al. (2015) refers to the best possible Energy Management Strategy HEV components using Algorithm (GA)-ECMS optimized EMS, architecture and sizing parameters of various Guzzella and Sciarretta (2013, p. 42). Control optimization Guzzella and Sciarretta (2013, p. includes 42).Management Control optimization HEV components using aa Genetic Genetic Algorithm (GA)-ECMS (EMS), optimization the EMS, and sizing parameters of various refers towhile the design best possible Energy Strategy optimized optimized EMS, architecture architecture andPorandla sizing parameters of various (EMS), while design optimization includes the best best definition definition coupled approach. Gao and (2005) compared HEV components using a Genetic Algorithm (GA)-ECMS refers to the best possible Energy Management Strategy refers to the best possible Energy Management Strategy coupled approach. Gao and Porandla (2005) compared of HEV architecture and of the sizing parameters of its HEV components using a Genetic Algorithm (GA)-ECMS (EMS), while design optimization includes the best definition HEV components using a Genetic Algorithm (GA)-ECMS of HEV architecture and of the sizing parameters of its different design optimization optimization methods, including DIRECT coupled approach. Gao and methods, Porandla including (2005) compared (EMS), while design includes the (EMS), design optimization optimization the best best definition definition design DIRECT components. coupled approach. Gao and Porandla (2005) compared of HEVwhile architecture and of theincludes sizing parameters of its different coupled Gaosimulated and methods, Porandla (2005) compared components. (DIvided RECTagnles), annealing and GA for different approach. design optimization including DIRECT of HEV architecture and of the sizing parameters of its of HEV architecture and of the sizing parameters of its (DIvided RECTagnles), simulated annealing and GA for aa different design optimization methods, including DIRECT components. differentHEV. design optimization methods, including DIRECT parallel In these applications, computation burden In the past decades, the control optimization for a fixed (DIvided RECTagnles), simulated annealing and GA foron a components. components. HEV. In these applications, computation In the past decades, the control optimization for a fixed parallel (DIvided RECTagnles), simulated annealing and GA for aa (DIvidedPC RECTagnles), simulated annealing and burden GA foron typical machines range from tens of minutes to many hybrid powertrain on a given drive cycle has been extensively parallel HEV. In these applications, computation burden on In the past decades, the control optimization for a fixed machines from tens of minutesburden to many hybrid on a the given drive cycle has been for extensively parallel PC HEV. In these theserange applications, computation burden on In the the powertrain past decades, decades, the control optimization for a fixed fixed typical parallel HEV. In applications, computation on In past control optimization a depending on size the design and studied, see and Guzzella Dynamic typical PC machines range from of parameters minutes toset many hybrid powertrain on a given cycle has(2007). been extensively hours, depending on the the size of of the tens design and studied, see Sciarretta Sciarretta anddrive Guzzella (2007). Dynamic hours, typical PC range from tens of minutes to many hybrid powertrain on aa given drive cycle has been typical PCofmachines machines range from tens of parameters minutes toset many hybrid powertrain on(e.g., given drive cycle has been extensively extensively the length the drive cycle. The key factor to reduce such programming (DP) Sundström and Guzzella, 2009) has hours, depending on the size of the design parameters set and studied, see Sciarretta and Guzzella (2007). Dynamic the length of the drive cycle. The key factor to reduce such programming (DP) (e.g., Sundström and Guzzella, 2009) has hours, depending on the size of the design parameters set and studied, see Sciarretta and (2007). Dynamic hours, depending on the size“function ofThe the key design parameters setsuch and studied, seeused Sciarretta and Guzzella Guzzella (2007).the Dynamic time appears to be a faster evaluation” time, e.g., been widely since it approximates precisely optimal the length of the drive cycle. factor to reduce programming (DP) (e.g., Sundström and Guzzella, 2009) has time appears to bedrive a faster “function evaluation” time, such e.g., been widely used since it Sundström approximates precisely the optimal the length of the cycle. The key factor to reduce programming (DP) (e.g., and Guzzella, 2009) has the length of the drive cycle. The key factor to reduce such programming (DP) (e.g., Sundström and Guzzella, 2009) has time to EMS optimization. trajectories. computation can time appears to be a faster “function evaluation” time, e.g., been widely However, used sinceits it approximates precisely therelevant. optimal the the time to solve solve EMS optimization. trajectories. its computation load load can be be appears to aa faster “function been widely used since it precisely the optimal time appears to be be faster “function evaluation” evaluation” time, time, e.g., e.g., been widely However, used since it approximates approximates precisely therelevant. optimal time To overcome such drawback, the Equivalent Consumption the time to solve EMS optimization. trajectories. However, its computation load can be relevant. To overcome such drawback, the Equivalent Consumption This paper presents SHM to optimize the EMS of parallel and the time to solve EMS optimization. trajectories. However, its computation load can be relevant. the time to solve EMS optimization. trajectories. However, its computation load can be relevant. This paper presents SHM to optimize the EMS of parallel and Minimization Strategy (ECMS),thederived derived from Consumption Pontryagin’s To overcome Strategy such drawback, Equivalent Minimization (ECMS), from Pontryagin’s series HEV while accounting for sizing parameters of various This paper presents SHM to optimize the EMS of parallel and To overcome such drawback, the Equivalent Consumption To overcome such drawback, the Equivalent Consumption series HEV while accounting for sizing parameters of various Minimum Principle (PMP), was initially introduced by This paper presents SHM to the EMS parallel and Minimization Strategy(PMP), (ECMS), derived fromintroduced Pontryagin’s This paper presents SHM Parametric to optimize optimize theparameters EMS of of parallel and Minimum Principle was initially by powertrain components. modeling of powertrain series HEV while accounting for sizing of various Minimization Strategy (ECMS), derived from Pontryagin’s Minimization Strategy (ECMS), fromintroduced Pontryagin’s powertrain components. Parametric modeling of powertrain Paganelli et al. (2000) and Brahma et al. (2000). In spite of its series HEV while accounting for sizing parameters of various Minimum Principle (PMP), wasderived initially by series HEV while accounting for sizing parameters of various Paganelli et al. (2000) and Brahma et al. (2000). In spite of its is in Section presents the powertrain components. powertrain Minimum Principle (PMP), was initially introduced by components Minimum (PMP), was DP, initially introduced components is described described Parametric in section section 2. 2.modeling Section 33of the higher computation efficiency than ECMS still requires components. Parametric modeling ofpresents powertrain Paganelli etPrinciple al. (2000) and Brahma al. (2000). spite of by its powertrain powertrain components. Parametric powertrain higher computation efficiency thanet DP, ECMS In still requires SHM technique, while SHM SHM performance is assessed assessed against components is described in section 2.modeling Section 3ofpresents the Paganelli et al. (2000) and Brahma et al. (2000). In spite of its Paganelli et al. (2000) and Brahma et al. (2000). In spite of its SHM technique, while performance is against significant effort to find the optimal co-state. components is described in section 2. Section 3 presents the higher computation efficiency than DP, ECMS still requires components is described in section 2. Section 3 its presents the significant effort to find the optimal co-state. the PMP-based commercial tool HOT matricial SHM technique, while SHM performance isand assessed against higher computation efficiency than DP, ECMS still requires higher computation efficiency than DP, ECMS still requires the PMP-based commercial tool HOT and its matricial SHM technique, while SHM performance is assessed against significant effort to find the optimal co-state. SHM technique, while SHM performance is assessed against version vHOT (Sciarretta et al., 2013, 2015) in Section 4. In order to efficiently perform design optimization of HEV in the PMP-based commercial tool HOT and its matricial significant effort to find find the optimal optimal co-state. significant to the vHOT (Sciarretta et al., 2015) 4. In order to effort efficiently perform designco-state. optimization of HEV in version the PMP-based commercial tool HOT and its matricial the PMP-based commercial tool2013, HOT and in itsSection matricial Next, an exemplary engine parametric optimization is shown reasonable computation time, an approximated yet fast EMS version vHOT (Sciarretta et al., 2013, 2015) in Section 4. In order to efficiently perform design optimization of HEV in Next, an exemplary engine parametric optimization is shown reasonable computation time, an approximated yet fast EMS vHOT (Sciarretta et al., 2015) in Section 4. In order perform design optimization of HEV in version vHOT (Sciarretta etparametric al., 2013, 2013, 2015) in in Section Section 4. In order to to efficiently efficiently perform design optimization offast HEV in version in Section 5. Eventually, conclusions are drawn 6. optimization method is required. Ambühl et al. (2010) Next, an exemplary engine optimization is shown reasonable computation time, an approximated yet EMS in Section 5. Eventually, conclusions areoptimization drawn in Section 6. optimization method istime, required. Ambühl etyet al.fast (2010) Next, an exemplary engine parametric is shown reasonable computation an approximated EMS Next, an exemplary engine parametric optimization is shown reasonable computation time, an approximated yet fast EMS proposed an explicit optimal control policy and the Section 5. Eventually, conclusions are drawn in Section 6. optimization required. Ambühl al. set (2010) proposed an method explicit is optimal control policyet and the in 2. POWERTRAIN MODELING in 5. are in optimization method is required. Ambühl et al. (2010) in Section Section 5. Eventually, Eventually, conclusions conclusions are drawn drawn in Section Section 6. 6. optimization method is required. Ambühl etMinimization al. set (2010) 2. PARAMETRIC PARAMETRIC POWERTRAIN MODELING foundations of the Selective Hamiltonian proposed an explicit optimal control policy and set the foundations of the Selective Hamiltonian Minimization 2. PARAMETRIC POWERTRAIN MODELING proposed an explicit optimal control policy and set the proposed an explicit optimal control policy and set the (SHM) technique. did it powertrain models and 2. POWERTRAIN MODELING foundations of theHowever, Selectivethey Hamiltonian Minimization 2. PARAMETRIC PARAMETRIC POWERTRAIN MODELING (SHM) technique. they did not not employ employ it for for The The parametric parametric powertrain models for for control control and design design foundations of Selective Hamiltonian Minimization foundations of the theHowever, Selective Hamiltonian Minimization parametric optimization. Convex optimization has been optimization purpose can be classified into two levels: 1) (SHM) technique. However, they did not employ it for The parametric powertrain models for control and design parametric optimization. Convex optimization has been optimization purpose can be classified into two levels: 1) (SHM) technique. However, they did not employ it for The parametric powertrain models for control and design (SHM) technique. they etoptimization didal. not employ for The parametric powertrain models fordescribe control and design applied to HEV HEV byHowever, Pourabdollah (2013, 2015) and descriptive analytical models that the energy parametric optimization. Convex has itbeen optimization purpose can be classified into two levels: 1) applied to by Pourabdollah et al. (2013, 2015) and descriptive analytical models that describe the energy parametric optimization. Convex optimization has been optimization purpose can be classified into two levels: 1) parametric optimization. Convex etoptimization has been purpose can be classified into twothelevels: 1) applied to HEV by Pourabdollah al. (2013, 2015) and optimization descriptive analytical models that describe energy applied to HEV by Pourabdollah et al. (2013, 2015) and descriptive analytical models that describe the energy applied to HEV by Pourabdollah et al. (2013, 2015) and descriptive analytical models that describe the energy Copyright © 2016, 2016 284 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2016 IFAC IFAC 284 Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 284Control. Peer review©under of International Federation of Automatic Copyright © 284 Copyright © 2016 2016 IFAC IFAC 284 10.1016/j.ifacol.2016.08.042

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where ̅   is the NA full-load mean effective pressure (bmep) defined by Nam and Giannelli (2005).

characteristics of a specific component (efficiency), and 2) predictive sizing models that estimate the characteristics of the component from its design parameters and technology. The overall sizing parameters set Π is given by

Π = Π  ∪ Π ∪ Π .

Parameters …  are identified as functions of main sizing parameters. The engine technology is denoted by a discrete design variable  . Other typical sizing parameters are engine displacement  , peak torque  at speed   , and peak power  at speed  . Therefore, the set of engine design parameters is defined as

(1)

where subscripts , , and  stand for engine, motor and battery, respectively. 2.1 Internal combustion engine (ICE)

Π =  ,  ,  ,  .

Engine technologies considered in this study are classified as naturally-aspirated (NA) and turbocharged (TC) regarding the charging method, spark-ignition (SI) and compression ignition (CI) in terms of the ignition type, stoichiometric (ST) and lean-burn (LB) with regard to the combustion technology. Parametric modeling of the ICE includes the descriptions of fuel consumption and full-load performance.

Parameters ,…, are evaluated by taking the average value of various engines of the same technology.

Then, parameters ,, are found by imposing that fullload torque: 1) at launch speed (1000 rpm) matches ̅ , 2) at   matches the value  , and 3) at  matches  .

Parameters  and  are determined by imposing that full-load torque: 1) at launch speed matches a given bmep, and 2) at   matches the value  . Parameters  and  are determined by imposing that full-load torque: 1) at a technology-dependent speed  matches the value  , and 2) at  matches  .

2.1.1 Descriptive analytical model Based on the Willans line method (Rizzoni et al. 1999), fuel power  is approximated in function of engine brake power  by a bilinear curve,   

 =  

,

+

  

,

− ≤  ≤   ≤  ≤ 

,

(2)

2.1.3 Validation

The proposed model parameters are identified from a set of fourteen sample engines (Table 1). Both the descriptive and the predictive models are validated in terms of their relative modeling errors as depicted in Fig. 1. The mean errors (  for the descriptive analytical model that is calibrated for each engine,  ∗ for the predictive sizing model whose coefficients are evaluated for the whole identification set) are computed from the discrepancy between computed fuel power at a grid of engine operating points and the original mapped data. The straight lines are the average errors of the two levels of models over the whole identification set.

where   ,   , and    are the friction power, the corner power and the full-load power that are functions of engine speed  ;    and  denote the indicated efficiency when the brake power is less and greater than the corner power, respectively. At this descriptive modeling level, these parameters can be identified for each engine from its original mapped data. 2.1.2 Predictive sizing model Parameters in (2) are expressed as follows:  = 

  ,

(NA)

 min  ,   ,

(TC)

,

 =  +   +   ,

 =  ,

 =





 +     .

Results show that the predictive sizing model introduce larger errors than the descriptive one, as expected. However, these errors in average are less than 6% and can be considered as acceptable for the design optimization purpose.

(3) (4)

Table 1. Engine identification set.

(5) ID

(6)

1 2 3 4 5 6 7

In these equations,  is the full-load power defined by  =   ,

(7)

The full-load torque  is determined by  = 

and 

 =

 +   +   , min +   ,  ,  +   , ̅   

,

(10)

(NA) , (TC) (8)



CI/TC CI/TC CI/TC CI/TC CI/TC CI/TC CI/TC

 (L) /  (Nm) /  (kW)

2.2 / 297 / 90 1.6 / 242 /80 2.0 / 324 / 98 2.2 / 327 / 88 1.5 / 208 / 78 2.0 / 368 / 121 1.2 / 145 /43

ID

8 9 10 11 12 13 14



SI/TC SI/NA/LB SI/NA/LB SI/TC SI/TC SI/NA/ST SI/NA/LB

 (L) /  (Nm) /  (kW)

0.9 / 145 / 58 1.5 / 120 / 60 1.9 / 166 / 82 2.0 / 302 / 150 1.8 / 312 / 148 1.0 / 95 / 54 1.4 / 128 / 70

2.2 Electric motor/generator (EMG) EMG technologies considered in this study comprise interior permanent-magnet synchronous motors (IPMSM) and induction motors (IM). Modeling of EMG includes the descriptions of the electric consumption and the full-load performance for both motoring and generating regimes, and

(9)

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the predictions of such consumption by knowing the typical sizing parameters.

279

and charging regimes, and the predictions of such losses by knowing the typical sizing parameters. Table 2. Motor identification set.  (Nm)

ID

1 2 3 4 5 6 7 8 9 10

Fig. 1. Mean modeling errors of descriptive analytic model   and predictive sizing model  ∗ for engine identifications.

52 63 73 83 94 104 115 125 160 190

 (kW)

10 13 16 19 21 24 27 30 25 30

ID

11 12 13 14 15 16 17 18 19 -

 (Nm)

210 250 133 200 267 300 250 250 100 -

 (kW)

35 40 36 54 73 88 96 125 40 -

2.2.1 Descriptive analytical model

At this level, electric power of EMG is expressed as a quadratic function of mechanical power  ,   =  +   +   ,  ≤  ≤  , (11)

where    and    are the minimal and maximal power, while parameters ,,   are identified from original mapped data.

Fig. 2. Mean modeling errors of descriptive analytical model   for motor identifications.

2.2.2 Predictive sizing model

A discrete variable  describes EMG technology. Other typical sizing parameters include rated torque  and rated power  . Therefore, the set of EMG design parameters is Π =  ,  ,  .

2.3.1 Descriptive analytical model

The electrochemical power of a battery pack is modeled as  =  +   +   ,

(12)

 = max− , −  ,

(15)

where  is the terminal power;   and   are minimal and maximal admissible terminal powers that are in function of state of charge (SOC) , while ,,  are coefficients that can be identified from original mapped data.

In this terms, full-load power is expressed as  = min ,   ,

 ≤  ≤  ,

(13) (14)

2.3.2 Predictive sizing model

while the predictive sizing model of the EMG electric consumption is still under calibration.

Predictive sizing model of the battery is established at the cell level under the assumption that a battery pack is composed of identical cells connected in series and in parallel. Apart from the battery technology  , the sizing parameters also include nominal capacity  , and total cell number of the battery pack  , which is summarized by

2.2.3 Validation The investigated EMG samples are exclusively IPMSM, whose sizing parameters  and  are listed in Table 2. The original data of each motor are created by the motor design tool developed by Le Berr et al. (2012) and Abdelli et al. (2014).

Π =  ,  ,  .

(16)

The descriptive analytical model is validated by showing its mean error   in Fig. 2. The figure shows that an average error below 5% is obtained when calibrating model (11) for each single motor. That fixes a lower bound for the modeling error to be expected from the predictive sizing models that are still under development.

Parameters in (15) are expressed as follows:

2.3 Battery

with the dependency on the battery state of charge  having been neglected, and

 =  +   +    ,  =  +   +

 =

The battery technology considered in this study is composed of high energy (HE) and high power (HP) lithium-ion pouch type cells. Modeling of battery include the descriptions of inner losses and full-load performance for both discharging





  ,

 +   +   ,

 =   +  ,

 =   +   +    .

286

(17) (18) (19)

(20)

(21)

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2.3.3 Validation The battery samples of HE and HP lithium-ion cells are listed in Table 3. Original data are extracted from manufacturer sheets (discharge curves at various C-rates and same cut-off voltage). The mean errors ϵ and ϵ∗ (same definition as in 2.1.3) are shown in Fig. 3. With an average error below 2%, the predictive sizing model seems promising for further EMS and design optimization. Table 3. Battery identification set. ID 

1 HE 25

 (Ah)

2 HE 31

3 HE 40

4 HE 53

5 HE 75

6 HP 31

7 HP 40

8 HP 53



The procedure of SHM method is illustrated in the flowchart shown in Fig. 4. In the ‘Control modes’ block each relevant mode is calculated as a function of sizing parameters Π, test cycle information , and co-state . Hamiltonian function calculated by (25) is in the form of , , . Subsequently, minimizing Hamiltonian w.r.t. the control mode dimension converts relevant variables into the form , . Finally, the optimal co-state  is calculated by a root-finding algorithm that fulfills the terminal constraint of the state variable, see Sciarretta et al. (2015).

(22)

The control variable is , which represents motor power for parallel HEV and battery power for series HEV. Without considering thermal dynamics, the system state equation is   = , ,  =





 =  ,

(26)

Additionally, SHM is based on a matricial implementation, thus constraints on the state variable  are not enforced. However, a special treatment, like the Picard method (Sciarretta et al., 2015), could be implemented.

The optimal energy management for a hybrid electric powertrain is formulated by considering the fuel consumption as the performance index 

= 0 →  =  .

However, the control variable is constrained by its physical limits. In addition to that, the physical limits of other components introduce more bounds on the control variable through the HEV power balance. Besides constraints, the analytical model of the engine is a continuous bilinear function with two discontinuities, i.e., at zero power ( = 0) and at the corner power ( =  ). In short, a finite set of possible optimal solutions is identified. In this paper, all the possible optimal solutions for the control variable are referred to as control modes. The eventual optimal solution is found by selecting the mode for which the respective Hamiltonian is minimized. As in Ambühl et al. (2010), this method is referred to as the Selective Hamiltonian Minimization.

3. OPTIMAL ENERGY MANAGEMENT STRATEGIES



Since parametric models in Section 2 are either linear, or quadratic functions, and so is the Hamiltonian function, the unconstrained optimal control is found in closed form by setting



9 HP 75

Fig. 3. Mean modeling errors of descriptive analytic model   and predictive sizing model  ∗ for battery identifications .

  =    , .

The co-state , also referred to as the equivalence factor, can be regarded as a weighting factor that transforms the battery power into fuel power. Due to the fact that  =  does not depend on the state variable  in the predictive parametric model of the battery (Sect. 2.3), the equivalence factor is constant over time ∀ ∈ 0,  .

(23)

where  =  is the electrochemical energy of the battery.

The control problem over time 0,   is subject to several constraints including the system dynamics (23), control constraints, and the global constraint on initial and final states 0 =   =  .

Fig. 4. Flowchart of the SHM approach.

(24)

3.1 Parallel hybrid powertrain

The control constraints of the parallel HEV differ e.g. from those of the series HEV due to the intrinsic characteristics of such architectures and they will be given separately in 3.1 and 3.2. The global optimal energy management problem, which consists of minimizing (22) subject to (23), (24) and the aforementioned control constraints, is reduced to the instantaneous minimization of the Hamiltonian function

Figure 5 illustrates the power balance of a parallel HEV. The block Electric Drive Unit (EDU) is composed of the battery and the EMG connected in series. According to (11) and (15), the mathematical EDU model    would be a fourthorder polynomial function. Since this functionality would be too complex for SHM, a quadratic function (27) is used instead,

, ,  =  ,  +  , , .

 ,  =  +   +  

(25)

287

(27)

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where ,, are simply re-tuned from the predicted data by (11), (13) to (15), and (17) to (21).

281

In a series HEV (Fig. 6), series-connected engine and EMG (functioning as a generator) are regarded as an Auxiliary Power Unit (APU). Considering the linearity of engine model (2) and the quadratic functionality of EMG model (11), the analytical model of APU is represented by the quadratic function  ,  =  +   +  

where parameters ,, are in function of  . However, the method of optimal operation line (OOL) is employed to express  as a function of  and thus eliminate it from (35).

Fig. 5. Power balance of parallel HEV. For the parallel HEV, the control variable is defined as the mechanical power of EMG, i.e.,  =  . The power balance at the mechanical node is given by   +   =  .

(35)

(28)

Hence, the Hamiltonian function is =

  

 +  +   +   .

(29)

The state constraint is not considered throughout this work by assuming that the battery can supply sufficient energy during the operation. The physical constraints of the engine and motor yield   ∈ 0,   ∪  , , ,   ∈ , , , .

Fig. 6. Power balance of series HEV The control variable in this case is defined as the terminal battery power, i.e.,  =  . The power balance in this case is given by

(30)

  +   =  ,

(31)

where    is the electric power of the traction motor and  is the electric power provided by APU.

By taking the operating power balance (28) into account, the admissible range of the control variable   is either

Thereupon, the Hamiltonian function is

  ∈ max , ,   −    , or

 =  +   −   +  −   +  +   +   .

min  , ,  (32a)

min  , , . (32b)

The physical constraints of the APU and the battery yield

The unconstrained solution to the Hamiltonian minimization is determined as  = , =

 

.

  ∈ 0, , ,

(38)

  ∈ , , , .

(33)

(39)

By accounting for the operating limit of  , the admissible operating range for the control variable  is derived as

In addition, the control constraints given by (32) must be fulfilled. Eventually, seven possible optimal control modes are identified and summarized as     ,  ,   ,  =   =   = ,  −   ,     −   0 

(37)

Possible optimal control modes for the series HEV (Guzzella and Sciarretta, 2013, p. 319-329 and Formentin et al., 2015) are determined according to the same procedure as in Sect. 3.1.

  ∈ max , ,   − ,   ,

 

(36)

  = max , ,   − ,  ,

min  , , . (40)

The unconstrained solution to the Hamiltonian minimization is determined as

(34)

 = , =

      

.

(41)

By taking the admissible operating bounds on the control variable into account, the possible optimal control modes are summarized as

where  = 1,, 7.

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    ,      =   =   = , , ,   − ,  0 

where  = 1,, 6.

On the one hand, SHM finds the smallest co-state  yet a similar FE compared with other approaches for both P-GSG and O-GSG. On the other hand, SHM presents the best computational efficiency for both gear shifting strategies as expected, taking benefit of an effective reduction of the search space and matrix manipulation instead of iterative procedures.

(42)

Contributions of each possible control mode in terms of time and energy proportions over the cycle are depicted in Fig. 7 (left and right) for P-GSG and O-GSG, respectively. The modes adopted in both cases, see (34), are 0 (vehicle standstill), 1 (purely EV mode), 2 (hybrid mode with battery charged since , < 0), 4 (maximal recharge mode), 5 (maximal engine power), and 6 (best engine efficiency). The time and energy proportions in the case of O-GSG are similar to the ones of P-GSG. However, mode 2 in the O-GSG case is used less than in the P-GSG case. As a result, the fuel consumption in the O-GSG case is less than in the P-GSG case as expected.

4. RESULTS OF SHM

In this section, two baseline parallel and series HEV are defined, whose specifications are listed in Table 4. In the parallel HEV the motor is placed between the engine and the gearbox with a reduction gear (the purely electric mode is performed in the second gear). Moreover, the overall gear ratios take into the gearbox and the differential. As for the series HEV, single gears are used between ICE and generator, and between motor and driven shaft. In addition, the battery is assumed to be the same for both types of powertrains. Table 4. Specifications of baseline parallel and series HEV. Vehicle architecture test weight Transmission motor gear ratio overall ratios Engine displacement peak power peak torque Motor peak power peak torque Generator peak power peak torque Battery capacity state of charge

single-shaft parallel 1814 kg

series 1400 kg

27.2 {15.5, 8.2, 5.8, 4.4, 3.5}

9.8 0.67 (ICE/generator)

1.4 L 60 kW,5250 rpm 131 Nm, 3250 rpm

0.65 L 44 kW, 7500 rpm 66 Nm, 6000 rpm

37 kW 28 Nm

96 kW 250 Nm

-

37 kW, 4000 rpm 90 Nm, 4000 rpm

7 kWh charge-sustaining

7 kWh* charge-sustaining

Table 5. Results of EMS optimizations over the NEDC with SHM for the baseline parallel HEV.

HOT vHOT SHM

predefined gears FE time s [L/100km] [s] 3.04 4.87 523 3.06 4.88 1.52 2.92 4.89 0.11

optimized gears FE time s [L/100km] [s] 3.06 4.85 1428 3.12 4.86 4.63 2.95 4.85 0.44

Fig. 7. Contribution of each control mode in terms of time and energy proportion over the NEDC with SHM; predefined (left) and optimized (right) gear shifting strategy.

*Practically, battery of a larger capacity is applied in series HEV. In this application, however, it is assumed to be the same as in the parallel HEV.

4.2 Series hybrid powertrain

The SHM is run on a i7-4810QM CPU @ 2.80 GHz machine with 16 GB RAM. The test cycle is chosen as NEDC. SHM results are compared with the implementation of standard PMP (HOT, see Sciarretta, 2013) and its faster matrix counterpart (vHOT, see Sciarretta et al., 2015) in terms of fuel economy and computation time. While HOT and vHOT use the original map data for optimization, the SHM optimizes fuel economy (FE) by using parametric models.

For the baseline series HEV, results of EMS optimization via the three mentioned methods are summarized in Table 6. Again, SHM takes the least computation time and presents similar fuel consumption compared with HOT and vHOT. The time and energy contributions of each mode are depicted in Fig. 8. Here, the adopted modes are 1 (battery EV mode), and 2 (hybrid mode with battery charged or discharged depending on the sign of , ).

4.1 Parallel hybrid powertrain

5. IMPLEMENTATION OF SHM FOR PARAMETRIC OPTIMIZATION OF A HYBRID POWERTRAIN

For the baseline parallel HEV, results of two separate EMS optimizations are presented in Table 5 for predefined gear shifting (P-GSG) and optimized gear shifting strategies (OGSG), respectively.

In order to assess the effectiveness of SHM in allowing sizing optimization in a reasonable computing time, a parallel HEV design is exemplified in this section. The objective is to 289

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improve the fuel economy by exclusively optimizing the parameters set of ICE. The design parameters set is defined as Π = Π while the considered engine technology is limited to NA/ST and NA/LB. The other HEV components remains the same as the parallel baseline configuration (Table 4). Typical design constraints of the vehicle, such as the top speed, gradeability, and acceleration time, are not taken into account.

The trajectory of the objective function value, i.e. the fuel economy, as a function of iteration and function evaluation is given in Fig. 9. As iteration increases, the evaluated functions also significantly augment. The exit criterion is defined by heuristic rules which could impact the eventual optimality. The trajectories of the engine design parameters are depicted in Fig. 10.The optimal engine sizing parameters are shown as  = /  = 0.965 ∙ 10 Π =  = 99    = 42 ∙ 10  

Table 6. Results of EMS optimization over the NEDC with SHM for the baseline series HEV. HOT vHOT SHM

s 3.77 3.81 3.33

FE [L/100km] 4.88 4.89 4.86

283

Computation time [s] 205 0.77 0.14

(47)

Table 7. Main features of the design and EMS cooptimization.

iterations 60

J [L/100km] 4.57

function evaluations 12203

time [min] 21

Fig. 8. Contribution of each control mode in terms of time and energy over the NEDC with SHM for the series HEV. To optimize the engine parameters, the DIRECT (Dividing RECTangles) algorithm (Gao, 2005) is used. The objective function is given by 

min  =    , Π ,  ,

 ,

Fig. 9. Trajectory of the cost function in the example design– EMS co-optimization.

(43)

subject to

 ∈ NA/ST, NA/LB   ∈ 0.6,1.4 Π = ,   ∈ 0.7,1.3  ∈ 0.7,1.3   

(44)

  where  ,  and  are the baseline parallel HEV values in Table 4. To limit the design variables of the ICE to meaningful combinations, additional constraints (45, 46) are applied.

 ∈ 0.9, 1.1

 ∈ 0.9,1.1

 



 







Fig. 10. Trajectory of the engine sizing parameters in the example design–EMS co-optimization. 6. CONCLUSIONS

(45)

Descriptive and predictive parametric models presented in this paper are oriented to HEV control and sizing cooptimization. In particular, this approach allows for the development of SHM as a technique to optimize the EMS. SHM has proved to be more efficient in computation (about 10 times less than vHOT) and to have similar precision as the standard PMP-based techniques based on mapped component data. As shown by the design example, this feature allows typical constrained optimizations (four parameters) in a reasonable time (tens of minutes).

(46)

The gear shifting strategy is pre-defined by the drive cycle of NEDC in this case. The features of the co-optimization example are tabulated in Table 7. The fuel economy is about 6.5% less than the baseline parallel HEV by optimizing the engine design variables. While the computation time of each function evaluation is about 0.10 s that is in line with results shown in Sect. 4.1.

Additionally, the main HEV sizing parameters explicitly appear in the developed parametric models. This feature paves the way for explicit calculation of gradients and other 290

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semi-analytical techniques to further reduce the computation time of the design optimization for HEV.

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