Energy-Optimal Control of Plug-in Hybrid Electric Vehicles for Real-World Driving Cycles

Transcription

1 Energy-Optmal Control of Plug-n Hybrd Electrc Vehcles for Real-World Drvng Cycles Stephane Stockar, Vncenzo Marano, Marcello Canova, Gorgo Rzzon, Fellow, IEEE, and Lno Guzzella, Fellow, IEEE 1 Abstract Plug-In Hybrd Electrc Vehcles (PHEVs) are today recognzed as a promsng soluton for reducng fuel consumpton and emssons, due to the ablty of storng energy through drect connecton to the electrc grd. Such benefts can be acheved only wth a supervsory energy management strategy that optmzes the energy utlzaton of the vehcle. Ths control problem s partcularly challengng for PHEVs, due to the possblty of depletng the battery durng usage and the vehcle-to-grd nteracton durng recharge. Ths paper proposes a model-based control approach for PHEV energy management that s based on mnmzng the overall CO 2 emssons produced - drectly and ndrectly - from the vehcle utlzaton. A supervsory energy manager s formulated as a global optmal control problem and then cast nto a local problem by applyng the Pontryagn s Mnmum Prncple. The proposed controller s mplemented n an energybased smulator of a prototype PHEV, valdated on expermental data. A smulaton study s conducted to calbrate the control parameters and to nvestgate the nfluence of vehcle usage condtons, envronmental factors and geographc scenaros on the PHEV performance, usng a large database of regulatory and real-world drvng profles. C nom E f H I batt J K L ṁ CO2 ṁ equv NOMENCLATURE Battery Nomnal Capacty Energy Torque Splt Factor Hamltonan Battery Current Cost Functonal Fnal State Penalty Term Lagrangan CO 2 Mass Flow Rate Equvalent Mass Flow Rate S. Stockar (Correspondng Author), V. Marano, M. Canova and G. Rzzon are wth the Oho State Unversty Center for Automotve Research, Columbus, OH 43212, USA. Emal: stockar.1@osu.edu. L. Guzzella s wth the Department of Mechancal and Process Engneerng, ETH Zurch, 892 Zurch, Swtzerland. ṁ f Q LHV P batt R s S t T u V batt V oc x Fuel Mass Flow Rate Fuel Lower Heatng Value Battery Power Battery Internal Resstance Equvalency Factor Target Set for the Fnal State Tme Torque Control Law Battery Voltage Open Crcut Voltage State Varable η Effcency η ch Battery Charger Effcency λ Lagrange Multpler λ Intal Condton for the Lagrange Multpler κ Specfc CO 2 content µ l Scalar Lagrange Multpler τ CS Fracton of Drvng Cycle n CS Mode ω Angular Velocty Ω x Feasble Set for the State AER BSA CS CD ECMS EM HEV ICE PHEV SDP SoC SoE All Electrc Range Belted Starter Alternator Charge Sustanng Charge Depleetng Equvalent Consumpton Mnmzaton Strategy Electrc Motor Hybrd Electrc Vehcle Internal Combuston Engne Plug-n Hybrd Electrc Vehcle Stochastc Dynamc Programmng State of Charge State of Energy

2 I. INTRODUCTION Plug-n Hybrd Electrc Vehcles (PHEVs) are today consdered a soluton to reduce fuel consumpton and CO 2 emssons n the transportaton sector. Compared to conventonal hybrd vehcles, the hgh-capacty energy storage system of PHEVs and the ablty to recharge the battery through connecton to the electrc grd provde the opportunty to control the battery depleton durng vehcle utlzaton, ultmately mprovng the fuel economy. Varous studes have shown that the performance of PHEVs depend on several factors, many of whch have lttle or no nfluence on charge-sustanng hybrds and conventonal vehcles [1], [2], [3], [4], [5], [6], [7]. To name a few, the length of the drvng path, the contrbuton of the electrcty on the overall energy consumpton of the vehcle, the cost of the electrc energy and ts specfc CO 2 content have been recognzed as predomnant factors n the assessment of fuel economy and emssons for PHEVs. A subject of strong nterest on the part of the automotve ndustry s to understand the mplcatons of dfferent energy management strateges on fuel consumpton, CO 2 emssons, battery lfe, range and performance. To ths extent, one of the crtcal challenges for control desgn s to properly account for the grd energy n the vehcle energy optmzaton problem. Some methods have been so far proposed to desgn supervsory controllers for PHEVs, ncludng the mnmzaton of an equvalent fuel consumpton, or the vehcle operatng costs or the cumulatve CO 2 emssons [2], [3], [4], [8], [9]. Due to the complexty of the control problem, heurstc methods have been often consdered [8], [1], [11], [12]. Although rule-based methods are often successfully employed n ndustry, t s generally found that the controller desgn process s cumbersome, tme-consumng and ts results are lmted to a specfc vehcle desgn and usage condtons. For ths reason, model-based approaches can mprove on all of these drawbacks and yeld more cost-effectve solutons. Stochastc Dynamc Programmng (SDP) appears more ndcated, especally when a small number of reference drvng profles can be found as statstcally representatve of the vehcle utlzaton [13]. Recent results show mprovements n fuel economy, operaton costs and emssons [8], [14]. The SDP approach however requres sgnfcant amount of data to provde a statstcally relevant valdaton framework. Further, the polcy evaluaton typcally requres large computaton tme, partally overcome by estmatng the control polcy off-lne and tabulate the results n the actual mplementaton. The Equvalent Consumpton Mnmzaton Strategy (ECMS) s a well-known approach for on-lne energy management of HEVs that has recently been adapted to the supervsory control of PHEVs [15], [16], [17], [18], [19], [2]. The proposed approach s based on assumng that the energy expended by the vehcle can be converted nto an equvalent consumpton of fuel. The results presented lead to the concluson that nearoptmal fuel economy can be acheved f the control algorthm depletes the battery proportonally to the drvng dstance. However, ths mples that the vehcle velocty profle must be known a pror. Such condton prevents the ECMS to be generalzed, requrng calbraton of the equvalency factor for each drvng profle. Furthermore, the assumpton of convertng the battery energy nto an equvalent fuel mass flow rate, s not formally applcable to PHEVs, snce the electrc energy stored from the grd depends on the energy generaton mx. Ths paper presents a novel supervsory energy management strategy for charge-depletng hybrd vehcles that accounts for the vehcle prmary energy consumpton, ncludng the fuel energy and the electrc energy from the grd. The structure of the proposed algorthm s general and adaptable to dfferent vehcle archtectures (seres, parallel, seres-parallel) and to any number of power splts. The proposed approach s based on the formulaton of a global optmal control problem that mnmzes the global CO 2 emssons produced (drectly and ndrectly) by the vehcle use. The Pontryagn s Mnmum Prncple s then appled to obtan a local mnmzaton problem. The control strategy s appled to a forward-orented smulator of a seres-parallel PHEV and used to conduct vehcle performance analyss, evaluatng the mpact of the control parameters for a varety of vehcle utlzaton and envronmental scenaros. The paper s organzed as follows: after an overvew of the hybrd vehcle confguraton and the model adopted for the control study, a descrpton of the energy management strategy and ts mplementaton nto a control algorthm are gven. Smulaton results are presented to evaluate the senstvty of the proposed control strategy to the vehcle usage condtons, envronmental and geographc scenaros, also provdng an assessment of vehcle performance, fuel consumpton and CO 2 emssons. 2

3 II. DESCRIPTION OF THE VEHICLE AND OF THE SIMULATOR The vehcle consdered n ths study s a seres-parallel prototype PHEV, bult on a md-sze SUV platform [21], [22]. As shown n Fgure 1, the vehcle drvetran ncludes a downszed Desel engne coupled to a Belted Starter Alternator (BSA) and a 6 speed automatc transmsson on the front axle, and an Electrc Motor (EM) wth on the rear axle. Table I descrbes the man vehcle components. The confguraton chosen for ths vehcle allows for a varety of operatng modes such as pure electrc drve, electrc launch, engne load shftng, motor torque assst, and regeneratve brakng [22]. Fg. 1. Dagram of the prototype PHEV drvetran. TABLE I DESCRIPTION OF THE VEHICLE DRIVETRAIN COMPONENTS. Component Type Specfcatons Chasss Md-sze SUV 25 Chevrolet Equnox Engne Desel 1.9l, Inlne 4, 4rpm, 2rpm Starter/ Alternator Permanent Magnet 1.6kW Nomnal, 8Nm Peak Torque, 415r/mn Max Speed Energy Storage L-Ion 3.2V, 2.3Ah (nomnal) per cell; 9 cells n seres, 15 modules n parallel; 1kWh pack energy Transmsson 6 Speed 45Nm torque capacty Auto Electrc Mototon AC Induc- 32kW Nomnal, 185 Nm Peak Torque block computes the tractve force, whch s the nput to a vehcle longtudnal dynamcs model that predcts the vehcle velocty. Fg. 2. Informaton flows n the forward-orented vehcle smulator The drvetran components ncluded n the vehcle powertran are detaled n Fgure 3. The energy-based (quas-statc) modelng approach s adopted to predct the overall vehcle fuel consumpton over a drvng cycle, neglectng hgh-frequency dynamc effects [25]. The engne model s based on ts steady-state fuel consumpton map, mplemented n the smulator as a functon of the engne speed and nput torque. Smlarly, the electrc machnes are modeled as statc elements, wheren the effcency s mapped as a functon of ther speed and nput torque. The combned engne and BSA torque s transmtted through a torque converter and a 6-speed automatc transaxle, whle the EM s coupled to the rear axle through a fxed gearbox. Losses n the transmsson components are accounted for through the defnton of effcency terms. The gear shftng strategy s determned by a smple schedulng controller based on the engne speed and accelerator command. A forward-orented, energy-based smulator was developed and valdated usng a combnaton of drvng tests and laboratory test data [21], [22], [23], [24]. Fgure 2 descrbes the nformaton flows wthn the vehcle smulator [25], [26], [27]. The accelerator and brake pedal poston commands from the drver are nput to the controller, whch determnes the torque commands to the engne, electrc motor and BSA. The powertran Fg. 3. Block dagram of the drvetran power flows. A. Energy-Based Model of the Battery As shown n Table I, the vehcle ncludes a 1kWh L- Ion battery pack, whch enables for an All Electrc Range (AER) of 25km [1]. A smplfed model of the battery 3

4 was bult accordng to the equvalent crcut analogy [24], [28], [26]. In partcular, a zero-order model s here consdered to descrbe the battery voltage output: V batt (t)= V oc (SoC(t)) R(SoC(t)) I batt (t) (1) The open crcut voltage V oc and nternal resstance R are polynomal functons of the battery SoC [28]. The state of charge of the battery s defned as: SoC(t)=SoC 1 t I batt (t)dt (2) C nom where C nom represents the nomnal battery capacty, ndcated n Table I. Although the above model represents a strong approxmaton of the real battery behavor, t s consstent wth the energy-based formulaton, whch lmts the analyss to the steady-state and low-frequency behavor of the system. More complex and accurate battery models, for example ncludng the hgh-frequency dynamcs and the effects of temperature, can be ncluded n the same vehcle smulator, for example to study the effects of drvng condtons and energy management control on battery agng [29]. The battery model s here utlzed to formulate the energy management control problem. In ths case, the battery dynamcs s descrbed by the state equaton: and d dt SoC(t)= η I batt(t) C nom (3) η = η batt 1 η batt f I batt (t) f I batt (t)> where the battery effcency s defned as: η batt (SoC(t),t)= V batt(t) V oc (SoC(t)) (4) (5) For the zero-order model descrbed by Equaton 1, the battery effcency can be explctly calculated: η batt (SoC(t),t)=1 R(SoC(t))I batt(t) V oc (SoC(t)) III. OVERVIEW OF THE ENERGY MANAGEMENT PROBLEM FOR CHARGE-SUSTAINING HYBRID VEHICLES For a charge sustanng HEV, the supervsory controller formulates a control law u(t) that mnmzes a cost functon over a perod of tme [t a,t b ]. Commonly, the cost functon s the vehcle fuel consumpton: J HEV (u)=m f = tb (6) t a ṁ f (t)dt (7) calculated over the entre msson of the vehcle. A constrant on the battery state of charge at the end of the msson s also ncluded to ensure nomnally charge sustanng operatons: SoC(t b )=SoC(t a ) (8) The control sequence u(t) that satsfes the above state constrant s the soluton of the optmal control problem [3], [31], [32], [25], [33], [34]. By applyng the Pontryagn s Mnmum Prncple [35], [36], the constraned global optmzaton problem presented above s cast nto a local mnmzaton problem gven by a Hamltonan functon defned by the vehcle equvalent fuel consumpton [25], [37]: ṁ equv (t)=ṁ f (t)+s(t) Pbatt(t) Q LHV (9) where s(t) s a fuel energy equvalency factor (nondmensonal), P batt s the net power drawn from the battery and Q LHV s the fuel lower heatng value. The cost functonal defned by Equaton (9) s mnmzed at each tme step. Ths allows one to fnd a soluton to the optmal control problem that can be mplemented on a vehcle. Note that the approxmaton of convertng an electrcal energy utlzaton nto a fuel mass flow rate ntroduces the equvalency factor s(t). Ths calbraton parameter has a consderable mpact on the battery SoC durng a drvng path. For ths reason, the equvalency factor must be optmzed or adapted based on the specfc vehcle drvng profle consdered, n order to acheve optmal fuel economy and charge sustanng operatons [27], [37]. Expermental results however show that the ECMS performs close to the global optmum wth modest calbraton effort, wth the advantage of beng mplementable on-lne [31], [21]. IV. OPTIMAL CONTROL PROBLEM FORMULATION FOR PHEV ENERGY MANAGEMENT In order to defne a supervsory energy management strategy for PHEVs, an optmal control problem for charge-depletng systems s here formulated. Compared to the energy management problem formulaton presented above, the constrant on the fnal SoC defned n Equaton (8) must be elmnated to enable charge-depletng operatons. Further, the equvalence between the battery energy usage and the fuel mass flow rate shown n Equaton (9) s formally ncorrect for PHEVs, where the battery energy s mostly provded by the grd, hence decoupled 4

5 from the fuel energy. Ths mples that the cost functon must be redefned for PHEVs. In ths study, the cost functon s defned to account for the prmary energy consumed by the vehcle durng a drvng path. The most representatve ndcator of the well-to-wheel energy utlzaton of a PHEV s gven by the cumulatve CO 2 emssons produced by the vehcle utlzaton: J PHEV (u)= tb t a ṁ CO2, f (t)+ṁ CO2,e(t)dt (1) where m CO2, f represents the mass CO 2 produced by the consumpton of the fuel (when the engne s utlzed) and m CO2,e results from the consumpton of the electrc energy stored on-board. In order to apply the optmal control theory to the PHEV energy management, the varables m CO2, f and m CO2,e must be related to vehcle system varables as follows: ṁ CO2, f(t)=κ 1 P f uel (t) ṁ CO2,e(t)=κ Pbatt(t) (11) 2 η ch where, accordng to Fgure 3, P f uel s the power assocated to the fuel utlzaton and s determned as follows, assumng a lower heatng value (Q LHV = 43MJ/kg): P f uel (t)=ṁ f (t) Q LHV (12) The term η ch =.86 n Equaton (11) represents the battery charger effcency (when the vehcle s connected to the grd) [38], and κ 1 and κ 2 are defned as the specfc CO 2 content n the fuel and electrcty per kwh. Note that κ 1 corresponds to the engne Brake Specfc CO 2 (BSCO 2 ), whch can be readly calculated from fuel consumpton data. The term κ 2 can be reasonably estmated based on the average CO 2 content of the electrcty generaton mx for a specfc geographc regon [39]. To account for the energy stored n the battery, the State of Energy (SoE) s ntroduced: SoE(t)= E batt(t) E nom (13) where E nom = C nom V oc s the nomnal battery energy (kw h). Consderng the SoE as the new state varable nstead of the SoC, the state equaton of the system becomes: d dt SoE(t)= η(soc(t)) P batt(t) E nom (14) where η s defned accordng to Equaton (5), and P batt s the battery power, defned postve f dschargng. Note that, f V batt (t)= V oc, then SoE = SoC. Based on the state equaton above, the control varable u(t) for the energy management problem can be defned as the vector: [ u(t)= P batt (t); P ] EM,el(t) (15) P batt (t) where the second element represents the power splt between the rear EM and the BSA electrc power outputs. Snce the mechancal power demand to the drvetran s known, the electrc EM and BSA power can be obtaned from the effcency maps of the two components and smple energy balances, accordng to the power flow dagram n Fgure 3. In order to respect the physcal lmtatons mposed by the drvetran components, the control and state varables are subject to constrants. In partcular, the battery SoE and power must be lmted to prevent abuse and agngrelated ssues [4]: SoE mn SoE(t) SoE max P batt,mn P batt (t) P batt,max (16) where, usually SoE mn =.25 and SoE max =.95. Further constrants stem from the power lmts of the drvetran components: P EM,mn P EM (t) P EM,max P BSA,mn P BSA (t) P BSA,max (17) where the power lmts are functons of the EM and BSA speed. V. SOLUTION OF THE PHEV ENERGY MANAGEMENT PROBLEM The optmzaton problem defned above s tackled usng Pontryagn Mnmum Prncple [35], [36], whch, n prncple, allows one to obtan closed-form expressons for locally optmal control sgnals. In the general case, an explct control sgnal can only be found solvng a two-pont boundary value problem. For the specfc problem at hand, an optmal soluton can be found adoptng the reasonable smplfcatons shown below. The startng pont s a descrpton of the system dynamcs: dx(t) = f (x(t),u(t),t) (18) dt wth the cost functonal: tb J(u)= L(x(t),u(t),t)dt+ K(x b,t b ) (19) t a The theorem ntroduces the Hamltonan functon: H(x(t),u(t),λ(t),t)=L(x(t),u(t),t)+λ(t) f(x(t),u(t),t) (2) 5

6 whch has to be mnmzed at each tme t to provde the optmal control polcy. If u o (t) s the optmal control polcy, then the followng necessary condtons must be satsfed: v) µ o l For the PHEV control problem, an extended Hamltonan functon s defned, based on the cost functonal n Equaton (1) and the state constrants on the battery SoE: H(x(t),u(t),λ(t), µ(t),t)=κ 1 P f uel (t)+ [ κ2 +P batt (t) λ(t) η(soc(t)) µ(t) η(soc(t)) ] η ch E nom E nom (21) wth: µ l f SoE(t) SoE max µ(t)= µ l f SoE(t) SoE mn (22) else where P f uel can be calculated from the engne fuel consumpton maps, as n Fgure 3 and µ l s the scalar Lagrange multpler for the nequalty constrants on the SoE. The extended Hamltonan functon allows one to nclude the state constrants wthn the same optmal control problem. Note that Equaton (21) provdes necessary condtons for optmalty accordng to the prevously mentoned condtons. Such formulaton, however, can lead to sub-optmal results f the state constrants are actve. When ths occurs, the optmal value for the parameter µ l s unknown and should be determned by applyng condtons )-v). Snce the tme ntervals durng whch the state s sldng along of the upper or lower boundary are lmted n occurrence, the value for µ l s determned here by a tral and error procedure [35]. The necessary condton for the co-state λ o (t) s: dx o dλ o (t) (t) ) = dt λ H o = f (x o (t),u o = x H o = ( κ1 P f uel (t) ) κ2 P batt (t) (t),t) dt x x( η ch dλ o (t) + ( ) Pbatt (t) η(soc(t)) (λ ) = x H o (t)+ µ(t)) o x E dt nom ) x o (23) (t a )=x a wth the ntal condton λ o (t = t a ) = λ. Snce no v) x o (t b ) S R n explct condton s gven for λ, ths parameter needs v) H(x o (t),u o (t),λ o (t),t) H(x o (t),u(t),λ o (t),t) to be calbrated. The ODE for the co-state λ o (t) can be further smplfed snce P f uel (t) and P batt (t) do not depend on the If the state x(t) s bounded, namely: v) x o (t) Ω x (t) t [t a,t b ], battery SoE (or SoC). However, the above assumpton s Ω x (t)={x R n G(x,t) ;G : R n not vald for the battery effcency. In fact, the battery x[t a,t b ] R} power s P batt (t)=i batt (t) V batt (t), whle the maxmum where G(x, t) defnes the nequalty constrants, an addtonal term s ntroduced n the Hamltonan functon n V oc (SoC(t),t). Ths wll further penalze any operaton battery power durng dschargng s P max (t) = I batt (t) order to account for ths lmtaton. The correspondng at low SoE, when the battery effcency s lower. Lagrange multpler s a scalar denoted by µ l and subject Insertng ths expresson n Equaton (23), the co-state to the Kuhn-Tucker condton: equaton can be rewrtten as follows: 6 P batt (t) (λ dλ o (t) o (t)+ µ(t)) η batt = E nom x dt P batt(t) (λ o (t)+ µ(t)) E nom η batt (SoC(t),t) 2 η batt x (24) Accordng to the mnmum prncple, the control polcy denoted by u o (t) s optmal f H(x o (t),u(t),λ o (t),t) presents a global mnmum wth respect to u o (t). As a fnal remark, a proof of equvalence between the ECMS and the soluton of the optmal control problem through the Pontryagn Mnmum Prncple was obtaned for the charge-sustanng HEV case n [31], [27]. Ths proof s here extended to the charge-depletng PHEV case. In fact, the ECMS formulaton presented n Equaton (9) can be made equvalent to the Hamltonan functon defned by Equaton (21) f the equvalency factor s(t) s defned as: ) + f P batt (t)<; f P batt (t) ; s(t)= κ 2 κ 1 η ch η(soc(t),t) E nom κ 1 (λ(t)+ µ(t)) (25) VI. IMPLEMENTATION OF THE ENERGY MANAGEMENT STRATEGY The soluton of the optmal control problem defned by Equaton (21) can be appled to forward-orented models or to a vehcle control system. Fgure 4 llustrates a procedure for the mplementaton of the soluton nto a control algorthm. Note that, although the vehcle drvetran ncludes three propulson systems (namely an

7 engne and two electrc motors), the proposed mplementaton allows for the optmal torque splt between an arbtrary number of power generaton elements. Fg. 4. Flow chart descrbng the mplementaton of the energy management algorthm. Accordng to Fgure 4, the varables f ICE and f BSA defne the fracton of the torque demand to the drvetran (T req ) that s commanded to the engne and to the BSA, respectvely. By conductng an energy balance on the system n Fgure 3, three matrces contanng all the possble torque combnatons that satsfy the drvetran demand are generated: T ICE (t)= f ICE T req (t) R nxm T BSA (t)= f BSA (1 f ICE ) T req (t) R nxm T EM (t)=(1 f BSA ) (1 f ICE ) T req (t) R nxm (26) where the dmensons m and n are related to the chosen resoluton for the factors f ICE and f BSA. The torque request at the drveshaft, T req s evaluated usng the drver accelerator and brake commands α and β as follows: T req (t)=α(t) T + max+ β(t) T max (27) where T max + s the maxmum postve torque that the powertran can generate combnng ICE, BSA and EM, whle Tmax s the maxmum negatve torque that can be absorbed by the electrc machnes (BSA and EM), accountng for battery power lmtatons. The torque delvered by each component s then lmted accordng to Equaton (17). Note that the torque varables defned are consdered as mechancal, hence calculated at the shaft of each component. The electrcal power provded by the battery and the the power assocated to the engne fuel utlzaton are then computed to evaluate the Hamltonan functon n Equaton (21). Specfcally P f uel s determned from the engne fuel consumpton, accordng to Equaton (12), whle the the power of the electrc machnes s computed from the effcency maps for the BSA and EM: P EM,el (t)=t EM (t) ω EM (t) η EM,el P BSA,el (t)=t BSA (t) ω ICE (t) η BSA,el (28) where, for the rear electrc motor, η EM,el = 1/η EM f the machne s workng as a motor, and η EM,el = η EM f as a generator. For each torque splt combnaton that satsfes the above constrants, the Hamltonan functon s defned based on Equaton (21). In dong so, the expresson η batt SoE n Equaton (24) s explctly calculated, accordng to Equaton (5): dη batt (SoC(t)) I batt (t) = dsoe V oc (SoC(t)) R(SoC(t)) I batt (t) ( R(SoC(t)) R(SoC(t)) ) V oc (SoC(t)) SoC(t) V oc (SoC(t)) SoC(t) (29) Note that, snce the parameters V oc and R are contnuous pecewse polynomal functons [28], they can be dfferentated n the entre SoC range. At any tme step, the combnaton fice o and f BSA o that mnmzes the Hamltonan functon matrx s chosen as the soluton of the optmzaton problem. It s worth observng that the proposed algorthm, although sutable for mplementaton nto forward-orented smulators or hardware-n-the-loop systems for control development and testng, can not be drectly appled to real-tme control due to the requred computaton and numercal optmzaton of the Hamltonan functon at each tme step. However, the computaton effort can be sgnfcantly decreased by pre-computng the Hamltonan functon and mportng the results as maps n the vehcle control system. A smlar approach was adopted for the mplementaton of an ECMS to a charge-sustanng HEV [33], [21], [37]. VII. RESULTS AND ANALYSIS The energy management algorthm was appled to the forward-orented PHEV smulator to conduct an evaluaton of the vehcle performance for a varety of usage condtons. 7

8 The focus of the study conducted s on the effects of the control parameters on the vehcle fuel economy and CO 2 emssons, and the nfluence of drvng condtons and energy generaton scenaros. Swtzerland (κ 2 = 142 [g/kwh]) 4% 1% 51% 44% Unted States (κ 2 = 567 [g/kwh]) 3% 5% 2% A. Vehcle Drvng Scenaros The characterstcs of the drvng profle have a strong mpact on the calbraton of the PHEV control algorthm [1], [11], [15], [41]. In ths study, a rch set of drvng profles was adopted as a valdaton framework for the energy management control algorthm, analyzng scenaros consstent wth the drvng behavor of customers and mprovng the generalty of the results. The smulatons were conducted on a set of regulatory and real-world drvng profles extracted from a database of fleet studes data to statstcally represent typcal usage condtons of a PHEV, ncludng urban, extra-urban and hghway segments wth varable drvng length [42]. Table V n the Appendx lsts the man characterstcs (velocty and energy demand at the wheel) of all the the drvng cycles consdered n ths study. The cycles are all characterzed by a drvng dstance greater than the vehcle all electrc range. Ths allows for the possblty of depletng the battery, dependng on the calbraton of the energy management strategy. Velocty [km/h] Tme [mn] Fg. 5. Example of velocty profle for the controller valdaton (ndcated as Path 3 n Table V). Fgure 5 shows the velocty profle of one of the nonregulatory cycles consdered. Ths pattern s representatve of mxed-mode drvng condtons, alternatng urban drvng and a hghway segment. B. Electrcty Generaton Scenaros The mpact of the electrcty generaton mx on the PHEV utlzaton was evaluated by varyng the specfc CO 2 emsson coeffcent κ 2 to encompass dfferent energy generaton scenaros, ncludng electrcty producton from both fossl fuel and renewable sources. Four of the values consdered for κ 2 are representatve of the energy generaton mx for USA, Swtzerland, France and Germany, as summarzed n Fgure 6. 48% Germany (κ 2 = 646 [g/kwh]) 4% 9% 12% 27% Coal Hydro Nuclear Gas Other Renewable 11% 8% France (κ 2 = 75 [g/kwh]) 5% 1% 4% Fg. 6. Summary of electrcty generaton mx for four countres (Sources: [43] [44] [45] [46] [47]). For smplcty, t wll be assumed that the grd energy consumed by the PHEV has the same specfc CO 2 content as the generaton mx. Note that ths must be consdered an approxmaton for the European countres, where the open energy market may cause dfferences between the CO 2 content of the electrcty produced by each country and that consumed by the vehcle. C. Defnton of Controller Parameters and Performance Metrcs Based on the optmal control problem n Equaton (21), the parameters requrng calbraton are the ntal condton for the Lagrange multpler λ and the scalar Lagrange multpler µ l (whch vares the penalty on the battery SoE constrants). The mpact of the above parameters wll be evaluated through three dfferent metrcs. Frst, the evoluton of the battery SoE durng the drvng pattern and ts fnal value SoE f nal (λ, µ l ), wll be consdered. Then, the overall CO 2 mass calculated wth Equatons (1-11) wll be evaluated: 79% 19% m CO2 = κ 1 m f Q LHV + κ 2 1 η ch E nom SoE (3) where Q LHV s the fuel lower heatng value, SoE s the dfference between ntal and fnal SoE. Another varable s ntroduced to ndcate whether the vehcle s operatng n Charge Depletng (CD) or Charge Sustanng (CS) mode, hence dentfyng how fast the control strategy depletes the battery. The varable τ CS 8

9 defnes the fracton of the drvng cycle where the vehcle operates n CS mode at ts lower SoE bound: τ CS = t CS t cyc (31) Specfcally, t CS s calculated consderng the tme durng whch the vehcle operates wthn a ±5% wndow around SoE = SoE mn. In the followng results, the battery s assumed at SoE = SoE max = 95% at the begnnng of each cycle. Knowledge of the fracton of the drvng cycle n CS mode s not only relevant for energy optmzaton, but also for relablty, safety and agng ssues [48]. D. Analyss of Smulaton Results for One Drvng Cycle and One Energy Scenaro In order to llustrate the results, one case study wll be analyzed n detal, wth reference to the drvng cycle shown n Fgure 5 and the U.S. energy generaton scenaro. Smulatons were conducted to evaluate the vehcle CO 2 emssons, fuel economy and the utlzaton of the battery energy n relaton wth the control parameters. Fgure 7 reports the the values of the fnal battery SoE obtaned by varyng the parameters λ and µ l. Note that an undesred complete depleton of the battery s possble for certan combnatons of the control strategy parameters. Fnal State of Energy [ ] Parameter λ Constrants on SOE volated Parameter µ l Fg. 7. Fnal value of the battery SoE as functon of λ and µ l for the case study (Cycle Path 3, U.S. scenaro). It s evdent that µ l affects the ablty of the controller to respect the state constrants. In partcular, the SoE exceeds ts boundares when µ l s below a threshold (for the consdered scenaro, µ l < 1). The parameter λ determnes whether the lower or the upper SoE bound s volated. In the frst case, the vehcle uses more battery energy than the one allowed. In the second case, the controller requres the engne to produce more power to further charge the battery. If any of the above cases occurs, the correspondng soluton s not part of the feasble doman and cannot be consdered for the control problem. For ths reason, the ponts volatng the state constrants wll be removed from the followng results. Fgures 8(a) and 8(b) show the contrbuton from the fuel energy and the electrc energy to the total vehcle CO 2 emssons. Comparng wth Fgure 7, a fnal SoE close to ts upper bound mples that most of the energy consumed by the vehcle was suppled by the ICE, leadng to hgh fuel consumpton and engne CO 2 contrbuton. Conversely, a low fnal SoE results n lower engne CO 2 emssons. Combnng the CO 2 from fuel energy and battery energy, t s possble to obtan the overall CO 2 emssons for the PHEV, as shown n Fgure 8(c). The response to the control parameters s almost flat, ndcatng rather lmted benefts on the vehcle CO 2 emssons. Ths behavor can be explaned by the hgh specfc CO 2 content of the electrcty n the scenaro consdered [43] [44]. In fact, the producton of energy from coal (a carbon-rch fuel) and the low well-to-tank effcency of the electrcty generaton ultmately offset the hgher tank-to-wheel effcency of the electrc tracton, to the pont where the CO 2 produced from the battery energy use s comparable to the fuel energy utlzaton. Fgure 9 shows the PHEV fuel consumpton over the sample cycle. Snce the factor κ 1 s constant, the fuel consumpton s drectly related to the CO 2 contrbuton from the fuel energy. Ths mples a smplfcaton n the study, snce the brake specfc CO 2 of the engne vares based on the engne operatng condton. However, for a PHEV such dfferences would be mnmal, as the engne operatng range s lmted compared to a conventonal vehcle. Comparng Fgure 9 wth Fgure 8(c), t s evdent that best engne fuel consumpton s acheved whenever the battery s completely depleted at the end of the cycle. However, the optmal value of the parameter λ s determned by mnmzng the cumulatve CO 2 emssons, whch does not necessarly correspond to best fuel economy. On the other hand, f the specfc CO 2 content of the grd, κ 2, tends to zero, the mnmum of the CO 2 and the best fuel economy would be concdent. Ths corresponds to an deal case where the electrc energy s produced entrely from renewable sources. Fgure 1 shows the fracton of cycle duraton where the vehcle operates n CS mode at ts lower SoE bound 9

10 35 3 CO 2 from Fuel Energy [g/km] Parameter λ Parameter µ l (a) Contrbuton from fuel energy. Fg. 9. Fuel economy of the PHEV as functon of λ and µ l for the case study (Cycle Path 3, U.S. scenaro). CO 2 from Grd Energy [g/km] as a functon of the control parameters. For hgh values of λ, the vehcle s operated n CD-CS mode and the SoE reaches the lower bound before the end of the drvng path. For the drvng cycle consdered, τ CS s slghtly below 4%, meanng that approxmately 6% of the energy requested to the drvetran can be satsfed wth the battery Parameter λ Parameter µ l Overall Vehcle CO 2 [g/km] (b) Contrbuton from electrc energy. 1 5 Parameter λ (c) Combned Parameter µ l Fg. 8. Overall vehcle CO 2 emssons as functon of λ and µ l for the case study (Cycle Path 3, U.S. scenaro). Fracton of Drvng n CS Mode [%] Parameter λ Parameter µ l Fg. 1. Percentage of the cycle n charge sustanng mode at the low SoE bound as functon of λ and µ l for the case study (Cycle Path 3, U.S. scenaro). For λ > 1 the control strategy forces the vehcle to deplete the battery and, when the lower SoE bound s reached, swtches to CS mode. Conversely, as λ decreases, τ CS decreases steeply to zero and, when λ, the control strategy s no longer able to deplete the battery. At ths condton, the fnal SoE s near the same value as the ntal one, hence the control strategy tends 1

11 to operate the system n CS mode at the hgher SoE bound. State of Energy [ ] λ = 1 λ = +4 λ = +6 λ = gco 2 /km 296 gco 2 /km 278 gco 2 /km 289 gco 2 /km Tme [mn] Fg. 11. Battery SoE profle durng the drvng cycle for µ l = 18 varyng λ (Cycle Path 3, U.S. scenaro). Ths s confrmed n Fgure 11, where the evoluton of the battery SoE durng the drvng cycle s represented for four dfferent values of λ, whle µ l s set constant. Intermedate solutons are observed for values of λ ncluded wthn the two bounds. In partcular, a value λ = 6 allows the battery to be gradually depleted durng the cycle, reachng the lower SoE bound only at the end of the drvng pattern and avodng any charge sustanng operatons. Ths operaton, known as Blended Mode, allows for achevng the mnmum vehcle fuel consumpton along a prescrbed drvng cycle [17]. E. Effects of Drvng Cycle Characterstcs For all the vehcle drvng profles lsted n Table V and the U.S. energy generaton scenaro, a full factoral desgn of experments was generated, varyng the control parameters λ and µ l of the supervsory controller. Then, the optmal combnaton (λ, µ l ) opt was determned by mnmzng the CO 2 emssons produced by the PHEV. Fgure 12 summarzes the results of the smulatons, representng the optmal value of the Lagrange multpler λ aganst the vehcle energy demand at the wheel calculated for the drvng cycles consdered n the study. The parameter µ l was set to a constant value so as to ensure the constrants on the battery SoE are always respected. As Fgure 12 shows, the results tend to cluster wthn a lmted range of values for the parameter λ and are almost ndependent on the energy demand at the wheel. A senstvty study was conducted to evaluate the nfluence on the cost functonal J(u) of errors n the optmal value of the control parameter λ. The analyss was conducted wth reference to fve specfc drvng patterns, representng the lmt scenaros for Fgure 12. Optmal Value for λ λ,ft =.27*x [MJ] Energy Demand at the Wheel [MJ] Fg. 12. Optmal value of the ntal condton λ as functon of the vehcle energy demand for dfferent drvng cycles (U.S. scenaro). Drvng Cycle λ opt J opt Var. λ Dev. (g CO2 /km) (%) Mn. Energy (93) 6 26 ±1 2./1.9 Max. Energy (6) ±1.4/.3 Mn. λ opt (94) ±1.8/1.9 Max. λ opt (63) ±1.5/.1 Max. λ opt (81) ±1 1.4/.1 TABLE II SENSITIVITY ANALYSIS OF THE COST FUNCTIONAL J(u) TO THE PARAMETER λ (U.S. SCENARIO). Table summarzes the senstvty results to varatons n λ around the optmal value correspondng to each of the fve drvng cycles consdered. In all cases, the senstvty of the vehcle CO 2 emssons s very lmted. The results confrm that, for the energy generaton scenaro consdered, the control strategy s relatvely nsenstve to the characterstcs of the drvng pattern [26]. The behavor can be justfed by consderng that the parameter λ s the ntal condton of the co-state ODE of the optmal control problem. Therefore, ts nfluence on the optmal soluton decreases progressvely wth the duraton of the drvng cycle, as λ(t) converges. In summary, the smulaton results show that the vehcle CO 2 emssons are relatvely nsenstve to the Lagrange multpler λ, for the consdered energy generaton scenaro. Furthermore, the optmal value of the control parameter, whch allows the vehcle to operate n Blended Mode wth mnmum energy consumpton, s nearly ndependent from the drvng cycle duraton and vehcle energy demand. Conversely, the parameter µ l has no mpact on the vehcle performance, but ensures satsfacton of the constrants on the battery SoE bounds. Specfcally, a threshold value can be dentfed for µ l so that the state constrants are always respected, allowng one to reduce the controller calbraton problem to the sole parameter λ. Ths presents advantages for parameters tunng, as 11

12 near-optmal results can be acheved wth mnmal calbraton effort, n partcular wthout the need of nformaton such as the drvng length. F. Effects of Energy Generaton Scenaros In order to extend the valdaton framework, dfferent scenaros were consdered to evaluate the senstvty of the control parameter λ to dfferent values of the energy generaton mx. As an example, ths analyss was ntally lmted to the sample drvng cycle shown n Fgure 5. Fgure 13 represents the vehcle CO 2 emssons and engne fuel consumpton aganst the parameter λ for the four dfferent energy generaton scenaros shown n Fgure 6. Overall CO 2 [g/km] Fuel economy [l/1km] Swtzerland κ 2 Decreasng Germany USA France Swtzerland Germany USA France Parameter λ Fg. 13. Impact of the energy generaton mx on the CO 2 and fuel consumpton aganst the parameter λ for dfferent energy generaton scenaros (Cycle Path 3). The U.S. and German energy producton scenaros are relatvely smlar, wth the hgh specfc CO 2 content of the electrc generaton mx causng a relatvely flat response of the vehcle overall emssons to the control parameter λ. Conversely, the case of Swtzerland and France s rather dfferent, as the energy generaton s predomnantly composed by renewable or low CO 2 prmary sources. These two scenaros offer promsng opportuntes for a large PHEV penetraton. Here, a hgher senstvty n the vehcle CO 2 emssons can be observed wth respect to the control strategy parameter. Fgure 14 llustrates the nfluence of the specfc CO 2 content of the grd energy on the optmal value of the Lagrange multpler λ, wth reference to the sample drvng cycle, ndcatng a lnear correlaton between κ 2 and λ. Ths suggests that the calbraton of the PHEV supervsory controller could be done when the battery s connected to the grd, based on the specfc CO 2 content of the electrcty generaton durng the chargng operaton. Optmal value for the parameter λ France λ opt (κ2 )=.14*κ Swtzerland USA Germany Specfc CO content n the grd energy κ [g CO /kwh] Fg. 14. Influence of the energy generaton mx parameter κ 2 on the optmal value of the parameter λ (Cycle Path 3). Fgure 15 summarzes the optmal value of the Lagrange multpler λ aganst the vehcle energy demand at the wheel for all the drvng cycles consdered. The low specfc CO 2 content of the electrc energy generaton n Swtzerland causes a dfferent behavor, compared to the the one observed n Fgure 12 for the U.S. scenaro. Although the results are stll clustered n a lmted range of λ, a slghtly ncrease dependence of the optmal parameter value wth the drvng cycle energy demand can be observed. Optmal Value for λ λ,ft =.26*x [MJ] Energy Demand at the Wheel [MJ] Fg. 15. Optmal value of the ntal condton λ as functon of the vehcle energy demand for dfferent drvng cycles (Swss scenaro). Drvng Cycle λ opt J opt Var. λ Dev. (g CO2 /km) (%) Mn. Energy (93) ±1 13.5/23.8 Max. Energy (6) ±1 6./1.2 Mn. λ opt (25) ±1 1./3. Max. λ opt (51) 117 ±1 3.6/11.2 Max. λ opt (92) 229 ±1 2.8/.1 TABLE III SENSITIVITY ANALYSIS OF THE COST FUNCTIONAL J(u) TO THE PARAMETER λ (SWISS SCENARIO). Ths behavor ndcates that the optmalty of the control strategy (and, consequently, the PHEV fuel consumpton and CO 2 emssons) s more affected by the drvng cycle characterstcs as the electrc energy s predomnantly generated from renewable sources. 12

13 Smlar to the U.S. scenaro, a senstvty study was conducted on the cost functonal J(u) varyng the parameter λ for dfferent drvng patterns. It s possble to notce here the ncreased senstvty of the vehcle CO 2 emssons to errors n the optmal value of the control parameter. On the other hand, a consderably large error must be gven to λ n order to detect suffcently hgh varatons n the cost functonal J(u). Ths ndcates the presence of a relatvely large regon around the sweet-spot where the CO 2 emssons and the vehcle performance vary only margnally. VIII. CONCLUSION The paper presents a novel approach to the supervsory energy management of Plug-n Hybrd Electrc Vehcles. The paper addresses the fuel consumpton and CO 2 emssons assocated to the PHEV use through a wellto-wheel energy balance that explctly accounts for the fuel energy and grd energy utlzaton. An optmal control problem was formulated by defnng a cost functonal based on the cumulatve CO 2 produced - drectly and ndrectly - by the vehcle. The Pontryagn s Mnmum Prncple was then appled to reduce a global optmzaton problem to a local mnmzaton, allowng for the control problem to be solved and mplemented n an algorthm. The control algorthm was then mplemented on a valdated energy-based PHEV smulator. Smulatons were conducted to evaluate the senstvty of the supervsory controller to dfferent vehcle utlzaton and energy generaton scenaros. A large database of drvng profles, ncludng regulatory cycles and real-world vehcle velocty profles extracted from fleet studes data, were consdered to provde a valdaton framework. Based on the analyss conducted, the proposed energy management strategy presents a relatvely low senstvty to the drvng profle characterstcs (.e., the energy demand at the wheel or the drvng dstance). Ths result was acheved because of the defnton of a cost functonal that formally accounts for the dfferent mx of prmary energy forms utlzed by the PHEV, representng an mprovement over the conventonal control approaches that approxmate the energy utlzaton wth an equvalent fuel consumpton metrc. In partcular, the vehcle CO 2 emssons show a presence of an optmal condton varyng the control strategy parameter λ, but also a relatvely large sweetspot where only margnal varatons from the optmal condton occur. Conversely, a hgher senstvty to the control parameter λ was observed on the battery SoE profle and, ultmately, the vehcle operatng mode. Furthermore, the senstvty to the vehcle usage condtons and the tradeoff between fuel and electrcal power consumpton are dependent on the specfc CO 2 emssons assocated to the electrcty generaton. In partcular, a hgher senstvty was observed for the energy generaton scenaros characterzed by a low CO 2 content. Whle the present paper does not specfcally address real-tme control developments, ts nsghts are valuable n developng energy management strateges that can lead to more readly tunable algorthms that can address dfferent objectves. In partcular, the analyss presented n ths paper can assst n addressng dfferences n electrcty generatons between dfferent regons and countres, allowng for the development of energy management strateges that can acheve, for example, mnmum CO 2 emssons n the face of a dfferent mx of electrc power generaton feedstocks. Gven the ncreasng use of geographcal nformaton systems and navgaton systems, whch can lead to some degree of a-pror knowledge of the vehcle trajectory, the results presented n ths paper represent a step forward n understandng of the potental of formal optmzaton methods n gudng the desgn of real-tme energy management strateges. REFERENCES [1] S.Stockar, P. Tulpule, V. Marano, and G. 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