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Integrated Optimization of Battery Sizing,Charging, and Power Management in Plug-inHybrid Electric VehiclesXiaosong Hu, Scott J. Moura, Nikolce Murgovski, Bo Egardt, Dongpu CaoAbstract—This brief presents an integrated optimizationframework for battery sizing, charging, and on-road powermanagement in plug-in hybrid electric vehicles (PHEVs). Thisframework utilizes convex programming (CP) to assess interactions between the three optimal design/control tasks. Theobjective is to minimize carbon dioxide (CO2 ) emissions, from theon-board internal combustion engine and grid generation plantsproviding electrical recharge power. The impacts of varying dailygrid CO2 trajectories on both the optimal battery size andcharging/power management algorithms are analyzed. We findthat the level of grid CO2 emissions can significantly impactthe nature of emissions-optimal on-road power management.We also find that the on-road power management is the mostimportant design task for minimizing emissions, through a varietyof comparative studies.Index Terms—Component Sizing, Charging Control, EnergyManagement, Convex Optimization, Plug-in Hybrid ElectricVehicle, Sustainable Transportation.I. I NTRODUCTIONPlug-in hybrid electric vehicles (PHEVs) potentially reducefossil fuel dependence while enabling synergies between vehicles and the electric grid [1], [2]. The performance, economics,and environmental benefits of PHEVs are, however, considerably influenced by their charging patterns, power managementstrategies, and energy storage system sizes. These three aspectsare typically considered in isolation, as discussed next.Researchers have examined PHEV charging schedule designs for objectives such as load following/stabilization [3],enhanced grid network efficiency [4], and battery health [5].These dynamically updated charging schedules may use realtime information on electricity price, green house gas (GHG)emissions, and so forth. Many optimization methodologieshave been employed for this problem, such as convex programming (CP) [6], dynamic programming (DP) [7], linearprogramming (LP) [8], and game theory (GT) [9].In parallel, researchers have examined on-road power management strategies that can be organized into two categories:X. Hu is with the National Active Distribution Network TechnologyResearch Center, Beijing Jiaotong University, Beijing 100044, China, andwith Energy, Controls, and Applications Laboratory, University of California,Berkeley, CA 94720, USA. (e-mail: [email protected]).S. J. Moura is with Energy, Controls, and Applications Laboratory, University of California, Berkeley, CA 94720, USA. (e-mail: [email protected]).N. Murgovski, and B. Egardt are with the Department of Signals andSystems, Chalmers University of Technology, Gothenburg 41296, Sweden.(e-mails: [email protected]; [email protected]).D. Cao is with the Center for Automotive Engineering, Cranfield University,Bedford MK 43 0AL, UK. (e-mail: [email protected]).charge depleting-charge sustaining (CD-CS) and blended approaches [10]. In the CD-CS strategy, PHEVs first operate ina pure electric mode until a minimum battery State-of-Charge(SOC) threshold is reached. Then the controller switchesto a charge sustenance mode. In the blended strategy, anoptimal control problem is typically solved, which results insimultaneous operation of the on-board power sources overtime. Various optimization approaches have been studied togenerate optimal power split algorithms for HEVs/PHEVs,such as DP [10], instantaneous optimization (e.g., Equivalent Consumption Minimization Strategy (ECMS) [11], [12],Pontryagins Minimum Principle (PMP) [13], [14]), modelpredictive control (MPC) [15], [16], and CP [17].All the foregoing studies focus on either charging controlor on-road power management. These two aspects, however,are strongly coupled [18]. To fully investigate interactionsbetween the two optimal control problems, a simultaneousoptimization framework is needed. DP has been used toimplement such a framework, where global optimality isachieved at the cost of tremendous computational complexity[19]. Other studies perform the on-road power managementand charging schedule optimization sequentially [20]. Batterysize also substantially impacts the economic and environmental advantages of PHEVs. Most of previous work evaluatedcombined on-road power management and component sizingoptimization in a bi-loop manner (the outer loop is for sizing,and the inner loop for power management)[21], [22]. Diverseheuristic optimization algorithms were used for the outer-loopsizing optimization with heavy computational burden, such asparticle swarm optimization (PSO) [23] and DIRECT [24]. Toincrease computational efficiency, simultaneous power management and sizing optimization was also reported, e.g., CP[17]. A survey of optimal design strategies for HEVs/PHEVsis given in [25].The tradeoffs between battery sizing, charging control, andthe power management strategy remain insufficiently examined and merit further exploration. For example, what is theoptimal battery size to minimize PHEV GHG emissions?How do dynamic grid emission profiles impact battery size,charging schedule, and power management strategy?This brief extends our previous work on combined PHEVbattery sizing/power management optimization [17] by incorporating charging schedule optimization to minimize the totalamount of daily CO2 emissions. Its overarching goal is toenable a systematic evaluation of the interplay between the

three optimal design/control problems. Two key contributionsare added to the related literature. First, a CP framework isformulated to enable rapid, globally optimal solutions. Theoptimal battery size, charging patterns, and power managementstrategy in a 24-hour horizon can be efficiently extractedin seconds. This facilitates online updates of the controlstrategies, given appropriate forecast information. Second, theimpact of variable daily grid CO2 profiles on the optimal design/control is analyzed. The optimality loss of the optimizedsolution at a medium CO2 level is quantified when applied todifferent CO2 levels.The remainder of the brief is arranged as follows. Section IIdetails the modeling of a PHEV propulsion system and brieflyintroduces a grid emissions model. The CP framework isdescribed in Section III. The optimization results are discussedin Section IV followed by conclusions presented in Section V.II. PHEV AND G RID E MISSION M ODELSxsubject to𝑃𝑃𝑏𝑏 , 𝑆𝑆𝑆𝑆𝑆𝑆BatteryFuel tankf (x)(1)g(x) 0,(2)h(x) 0.(3)A convex program is the special case where f (x) is convex,g(x) is convex, and h(x) is affine with respect to the optimization variable x, over the feasible set [26]. Consequently, themodel equations and constraints are derived to satisfy theseproperties. As will become evident in the model formulation,the optimization variables are Pb,k , Pbt,k , Eb,k , n, Pegu,k , Tk ,which represent the electrochemical battery power, electrical battery power, battery energy, number of cells, enginegenerator unit (EGU) power, and electric motor (EM) torque,respectively. Note that n is a design variable, and the remaining optimization variables are controls. Therefore, weformulate a combined design/control optimization problem asa convex programming problem such that both design andcontrols can be simultaneously solved. This is different fromDP and PMP where only controls are solved sequentiallyin time. To consider design optimization, an outer loop isrequired. A contrast between CP and bi-loop scenario issketched in Fig. 2. Symbol k {0, · · · , N } indexes discretetime, and is herein dropped to reduce notational clutter, exceptto emphasize dynamics. We consider a td 1 sec. time stepwhen the vehicle is driving td Md , and tc 1 min whenit is charging tc Mc . Here, Md and Mc are the sets ofdiscrete time steps for driving and charging, 𝑒𝑒𝑒𝑒𝑒𝑃𝑃𝑓𝑓ICE𝜔𝜔, 𝑇𝑇𝑣𝑣 , 𝑇𝑇𝑏𝑏𝑏𝑏𝑏𝑏GENEGUFig. 1.Plug-in hybrid propulsion system.User Inpute.g., driving cycleUser Inpute.g., driving cycleDesign Optimizatione.g., battery size nCombined Design/ControlOptimizationConvex programming (1)-(3)min f x Objective function f(x)We consider a series plug-in hybrid electric powertrainarchitecture shown in Fig. 1. In the following subsections, wedetail a convex model formulation of this drivetrain and gridCO2 emissions. Then we formulate the optimization program,which takes the form of the canonical nonlinear programmingproblem,minElectric grid𝑃𝑃𝑎𝑎𝑎𝑎nxs.t., g x 0,Optimal Control Probleme.g., DP and PMPh x 0.n, Pbt, Pegu, etc.Pbt, Pegu, etc.Bi-loop ScenarioConvex programmingFig. 2. Contrast between convex programming and bi-loop scenario for thecombined design/control optimization in PHEVs.engine-generator unit (EGU) with a power rating of 35 kW.Convex modeling formulations of these components are provided next.1) Battery Pack: The lithium-ion battery pack comprisesstrings connected in parallel. Each battery cell is modeled as anopen circuit voltage (OCV), U (soc), in series with a resistor,R, and the power at the terminals is given byPbt Pb nRi2 ,(4)where Pb nU (soc)i, and i is the cell current. The dischargepower is positive and charge power is negative. An affineapproximation of the OCV function given byU (soc) Qsoc U0 ,C(5)A. PHEV Modelis employed [27], where Q and C have the interpretationof charge capacity and capacitance (their values are listed inTable II). This affine approximation is determined by fittingexperimental LiFePO4 cell data and is reasonably accuratewithin the 20%-80% SOC window. One may express thebattery pack energy as Z socZ soc QEb nQU (σ)dσ nQσ U0 dσC 0 0 QnC nQsoc2 U0 soc U 2 (soc) U02 . (6)2C2The PHEV consists of a lithium-ion battery pack, 35 kWpermanent-magnet synchronous electric motor (EM), and anBy solving for U (soc) as a function of Eb in (6) and pluggingthis expression into the electrochemical pack power Pb , we

Eb,k 1 Eb,k t · Pb,k ,RCPb2 0,Pbt Pb 2Eb U02 Cnn Ecell,min Eb n Ecell,max ,n 0,(8)761.5651540.54320 503050300450 001210 200 1000100EM Torque,T [Nm]0200 30 2010008100300600Original EM data (speed [rpm])Convex EM model0020Using the properties of convex functions (§3.2 [26]), it is easyto verify the left-hand side of (7) is convex w.r.t. Pbt , Pb , Eb , nfor non-negative Eb , n. When dimensioning the battery pack,the optimal energy capacity is assumed to take non-quantizedvalues, which is fulfilled by relaxing n to a continuouslyvalued variable. We assume battery cells can be fabricatedin accordance with the optimized pack power and energy[27]. The convex battery model constraints representing parkedcharging and propulsion during driving are given by7 1001020EM power, Pem [kW]2000(7)8100300600 0100RCPb2Pbt Pb 0.2Eb U02 CnEM power loss, Ploss,em [kW]attain i as a function of Pb and Eb , which is ultimatelysubstituted into (4), yielding equality constraint0030 045030Fig. 3. Original EM power data and approximate model with power loss asa quadratic function of torque.preserving convexity of functions (§3.2 [26])). Eventually,Lemma 1 holds for (12), and its convexity is thereby satisfied.As in [19], we enforce net zero energy transfer over the 24hour period, so that today’s optimal control does not sacrificeperformance tomorrow by depleting the battery,(9)Eb,0 Eb,N ,(10)where k 0 and k N represent the initial and final timesteps of the 24-hour PHEV operation.2) Electric Motor (EM): The electrical power balanceequation during driving is given by(11)where Ecell,min and Ecell,max represent the min/max allowablecell energy levels. Equation (8) encodes the battery energystorage dynamics and clearly produces an affine equalityconstraint w.r.t. all optimization variables [27]. Inequality (9)is a relaxed form of equality constraint (7), and is necessary topreserve convexity. One can analytically show this constraintis active at the optimal solution [27], thus incurring zeroerror. The battery energy limits (10) also produce convexinequality constraints (as proved in Lemma 1 and Remark1), and the number of cells (11) must be nonnegative. Inaddition, the battery cell current is limited during both drivingand recharging [19]. Consequently, battery power constraintsare formulated for the driving and charging modes as qq 22 i22 min n C Eb,k U0 n Pb,k imax n C Eb,k U0 n ,Pem Ploss,em Pau Pbt Pegu(13)(14)where Pem is the EM power, Ploss,em denotes parasitic lossesin the EM, and Pau represents vehicle auxiliary loads (e.g.HVAC). The required EM power depends on the drive cyclevelocity and vehicle mass (i.e. battery size) according toPem T ω Tv ω (A(ω)n B(ω))ω(15)where Tv is the torque demand on the shaft between the EMand the final drive, which is an affine function of the batterysize n, as detailed in the Appendix. Symbol ω is the EMangular shaft speed and is proportional to wheel speed. EMtorque may be less than the required torque for deceleration,since the difference can be provided by frictional brake torque.for k Md , The EM losses Ploss,em are modeled by a convex quadratic Pfor k Mc , function of EM torque T ,bt,min Pbt,k 0,(12)Ploss,em a2 (ω)T 2 a1 (ω)T a0 (ω),(16)where imin , imax are the cell current limits for power management during driving, and Pbt,min is the charging power where a2 , a1 , a0 are coefficients that depend on ω. Notelimit while parked. Again, (12) produces convex inequality a2 (ω) 0 uniformly in ω to preserve convexity. EM powerconstraint functions with respect to Pb , n, Eb , as proved in loss data and the convex regression are shown in Fig. 3.Additionally, the EM torque is limited by angular speedLemma 1 and Remark 1.Lemma 1 (§3.1 [26]): The inequality constraint H(x) 0 dependent bounds given byis convex if and only if the domain of H(x) is a convex set,and its Hessian 2 H(x) is positive semidefinite.Remark 1: For nEcell,min Eb and Eb nEcell,max ,Lemma 1 clearly holds, and thus the convexity of (10) isguaranteed. Similarly, Lemma 1 clearly holds for the batterypowerconstraints during charging. According to Lemma 1,q2n( C Eb,k U02 n) is concave with respect to n and Eb .qSince imin 0 and imax 0, imin n( C2 Eb,k U02 n) andq imax n( C2 Eb,k U02 n) are convex (as per basic operationsTmin (ω) T Tmax (ω).(17)3) Engine-Generator Unit (EGU): The rate of gasolineenergy consumption, i.e., “gasoline power” Pf , along the EGUoptimal operating line (OOL) is described by2Pf b2 Pegu b1 Pegu b0(18)where b2 0 to preserve convexity. The regression fit isdisplayed in Fig. 4. The EGU power Pegu is also boundedby limits0 Pegu Pmax,egu .(19)

EGU efficiency [%]Gasoline power, Pf [kW]80604020302000102030EGU net power, Pegu [kW]Original EGU dataConvex EGU model100102030EGU net power, Pegu [kW]Grid CO2, cc [kg/kWh]Fig. 4. Original gasoline consumption data and approximate model withpower loss as a quadratic function of EGU net power.0.80.70.604High level81216Time [h]Medium level2024Low levelFig. 5. Grid CO2 emission trajectories, i.e, cc,k in (23) for optimization(data is taken from [28]).Finally, a heuristic engine on/off control signal is determinedby a threshold policy( 1, for Tv ω Pon,e (20)0, otherwise.That is, the engine shuts off when vehicle power demand . The best threshold valueis smaller than a threshold Pon Pon is achieved by iteratively solving the convex optimization values inproblem ((24)-(26) in Section III), over different Ponan outer loop. We observe from simulations that this heuristiccould give a solution close to the global optimum from DP(see Table III).B. Grid CO2 ModelWe adopt an economic grid dispatch model for power plantsin Michigan [28] to generate marginal grid CO2 emissionsassociated with PHEV charging. This provides the appropriatecoefficients for the objective function, to be defined in SectionIII. Given a total load demand, the model performs griddispatch and calculates generation costs and CO2 emissions.The resultant CO2 emissions per unit energy are within arange from 0.55 to 0.85 [kg/kWh], which is independent ofthe grid load (the inclusion of PHEVs). The reason is thatthe underlying power plants are assumed to remain the same,inducing the same range of average CO2 . Three 24-hour CO2traces reflecting days of high, medium, and low CO2 emissionscaused by different generation mixes (see Fig. 5) are chosenfor the subsequent integrated optimization problem (SectionIII). More details on the grid model are given in [19], [28].III. I NTEGRATED O PTIMIZATION F RAMEWORKWe consider daily PHEV operation composed of two identical driving trips (at 8 a.m. and at 5 p.m.) and parking, whichVelocity [m/s] Velocity [m/s]4010020 (a)10005001000Time [s]402000(b)200150004812Time [h]162024Fig. 6. FTP-75 driving cycle and timing of the daily PHEV operation: (a)FTP-75 cycle; (b) Timing of the 24-hour PHEV usage.is representative of workday commutes. The Federal TestProcedure (FTP-75) is chosen to simulate city driving patterns.The FTP-75 driving cycle and commute times over 24-houroperation are indicated in Fig. 6. Although specific routesfrom Fig. 6 are employed here, the integrated optimizationframework is equally applicable to other velocity profiles, triplengths, and commute times. The impact of trip profiles andtraffic conditions on PHEV energy consumption is elaboratedin [29], [30].The objective function F is formulated to minimize thetotal amount of daily CO2 emissions associated with gasolineconsumption and recharging the PHEV from the grid.whereF Fgas Fgrid Fgas Fgrid ,cg td XPf (Pegu,k ),Lg ρgk Md tc Xcc,k Pbt,k ,ηc(21)(22)(23)k Mcwhere cg is the CO2 produced by combusting gasoline in[kg/L], Lg is the lower heating value of gasoline in [J/g], andρg is the gasoline density in [g/L]. The time-varying grid CO2in [kg/kWh] is represented by cc,k , and ηc denotes averagecharger efficiency.The optimization variables are Pb,k , Pbt,k , Eb,k , n, Pegu,k ,Tk , where k Md Mc indexes time. The time indices aregrouped into the set of driving time Md and charging timeMc index sets. Consequently, we summarize the optimizationproblem asminPb ,Pbt ,Eb ,n,Pegu ,Tsubject to:F Fgas Fgrid(24)(8) (13), (15) (20),(25)Pem Ploss,em Pau Pbt Pegu .(26)Since convex equality constraints (14), (16) do not produceconvex feasible sets, we relax this equality to an inequalityin (26). One can analytically show this constraint is active atthe optimal solution, uniformly in time [27]. Additionally, weremark that it is possible to eliminate optimization variablesvia the equality constraints. This, however, destroys the convexstructure that we desire. The tradeoff is added dimensions tothe decision space. This is worthwhile, however, due to theefficiency of numerical CP solvers relative to general purposenonlinear programming algorithms [27]. Finally, we note that

The optimal trajectories under a high grid CO2 scenarioare showcased in Fig. 7. The PHEV effectively operates incharge-sustaining mode during the two trips, and there is nouse of electrical grid energy (i.e. the PHEV operates as aseries HEV). As demonstrated in Fig. 8, the EGU operatesnear the maximum efficiency point such that gasoline usageis less carbon intensive per unit power. The optimal trajectoryunder the low CO2 level is shown in Fig. 9. In this case,the PHEV exerts pure electric mode for driving and rechargeswhen grid CO2 is lowest. Reduced grid CO2 relative to engineCO2 discourages gasoline consumption. The result under themedium CO2 level is similar to the case of the low CO2 level.The optimized battery sizes, CO2 emissions, and computational times under the three grid CO2 levels are summarizedin Table III, where DP results are also provided for benchmarking purposes. Given the same battery size optimized byCP, DP guarantees the globally optimal charging and powermanagement strategy and incorporates binary engine on/offcontrol. The CP and DP results are comparable. In particular,the greatest loss of optimality is 0.6% under the high grid CO2case, which is attributed to the convex modeling assumptionsand sub-optimal engine-off control designed in Section II. Dueto pure electric-mode driving under both the medium and lowgrid CO2 levels, the associated losses are about 0.3%. CPrequires 2-3 orders of magnitude less computational time thanDP. DP requires more than 2 hours to solve, even withoutincluding the battery sizing task.Additionally, the optimal battery size does not necessarilyincrease, as CO2 levels decrease. In order to further explorethe coupling between battery sizing and CO2 emissions, theoptimization results with respect to different battery sizesValue2.000Aerodynamic drag coefficient cdAir density ρ [kg/m3 ]0.300Rolling resistance coefficient crWheel radius r [m]0.0101.1840.308ParameterVehicle mass excludingbattery pack mv [kg]EM inertia Iem [kgm2 ]Inertia of final drive andwheels I [kgm2 ]Vehicularauxiliarypower, Pau [kW]Final gear λcc 410(d) 1040EGU power, Pegu(e)0 2077.17.27.5TABLE IIM AIN S PECIFICATIONS OF THE O NBOARD P OWER S OURCES350000.8000.014.000Cell mass mb [kg]0.071.0867.4under the high grid CO2 case are exhibited in Fig. 10. Thedependency of CO2 emissions on battery size is nonlinear.Furthermore, the globally optimal solution corresponds tohigh EGU efficiency. Enlarging the battery past a criticalpoint will increase gasoline consumption and consequentlyCO2 emissions, since the PHEV mass increase outweighs themarginal improvement of the average EGU efficiency. ThePHEV does not recharge in this case, for any battery size.The optimal battery sizing under the low grid CO2 case isdepicted in Fig. 11. When the battery is less than 3 kWh, thePHEV blends battery and engine power, whereas the resultantCO2 emissions are not minimized with respect to battery size.On the other hand, overly large batteries lead to unnecessaryelectricity consumption as a consequence of increased mass.Using the established optimization framework, differentMaximum EGU powerPmax,egu [W]Nominal cell capacity Q[As]Cell equivalent capacitance C [F]Cell resistance R [ohm]0.1007.3Time [h]Fig. 7. Optimal trajectories under the high grid CO2 level: (a) Grid CO2 ;(b) Cumulative CO2 ; (c) SOC trajectory; (d) Recharging power; (e) powersplit in the first trip.Value2.320Value1155Battery terminal power, Pbt20ParameterGasoline-related CO2cg [kg/l]Gasoline lower heatingvalue Lg [J/g]Gasoline density ρg [g/l]TABLE IK EY V EHICLE PARAMETERSParameterFrontal area Af [m2 ]Power [kW] SOC, soc [%] CO2 [kg]IV. R ESULTS & D ISCUSSION0.850.80.75 (a)0.7032 (b)10075(c)70Power [kW]the objective function (24) is not explicitly dependent on alloptimization variables. However, all variables are coupled viaconstraints.The CVX solver [26], is applied to parse the problem,yielding a general semi-definite program (SDP) that can beefficiently solved by SeDuMi [31]. Theoretical and algorithmicdetails of convex programming are discussed in [26]. Thekey parameters of the small-size PHEV are listed in Table I,while the main specifications of the on-board power sourcesare given in Table II. The additional mass resulting frompackaging and circuitry is assumed to account for 12.3% ofthe total mass of the battery pack [27].42600749828051782ParameterMaximum cell dischargecurrent imax [A]Maximum cell chargecurrent, imin [A]Initial cell SOC soc0[%]Maximum cell SOCsocmax [%]Minimum cell SOCsocmin [%]Maximum pack recharging power Pbt,min [W]Averagechargerefficiency ηc [%]Nominal cell voltage [V]Value70-35709030-1000983.3

TABLE IIIO PTIMIZATION RESULTS UNDER VARIOUS GRID CO2 EMISSIONSCENARIOS .Efficiecny [%]403020Efficiency curveEGU operating pointMaximum efficiency1000102030EGU net power, Pegu [kW]Fig. 8.HighDP2.51 kgCP2.53 kgDP2.48 hTime CP17.60 sBattery sizeCP & DP3.78 kWh 2.9 GHz processor with 4 GBCO240MediumLow2.36 kg2.16 kg2.37 kg2.17 kg2.32 h2.16 h13.91 s12.42 s5.17 kWh 5.07 kWhRAM was used.EGU efficiency.0CO2 0122468Battery size [kWh]101238(b)37.5055 (c)3000 1Fig. 10. Optimization results with respect to different battery sizes under thehigh grid CO2 level: (a) Total amount of daily CO2 emissions; (b) AverageEGU efficiency.(d) 0.5Power [kW]82 (b)102.6 (a)2.54Efficiency [%]0.70.650.6 (a)0.550Power [kW] SOC, soc [%] CO2 [kg]cc [kg/kWh]2.70EGU power, Pegu4020Battery terminal power, Pbt(e)ment when day-ahead predictions of grid CO2 is available,we consider updated control policies with fixed battery size.As illustrated in Fig. 13, the updated control policies recovernearly all of the optimality loss ( 1%), which underscores theimportance of on-road power management.0 207V. C ONCLUSION7.17.27.3Time [h]7.47.5Fig. 9. Optimal trajectories under the low grid CO2 level: (a) Grid CO2 ; (b)Cumulative CO2 ; (c) SOC trajectory; (d) recharging power; (e) power splitin the first trip.charging/power management strategies can be conveniently assessed by changing the optimization constraints. For instance,a comparison is conducted between the optimal solution andthree heuristic charging/power management strategies underthe high grid CO2 level (see Fig. 12). In this case, notethat optimizing on-road power management (i.e., high EGUefficiency) is critical to reducing CO2 emissions, while intuitive full charging before each trip is a non-optimal decision.Analogous analysis can be made for the medium and low CO2levels.In practice, one cannot optimize PHEV design & control forvarying daily grid CO2 traces. Namely, adjusting battery packsize is often difficult, particularly for traditional integratedbattery pack designs. One alternative is to obtain the optimalsolution under the medium grid CO2 trace, and then evaluateits performance on other traces. This approach, nevertheless,results in loss of optimality with respect to CO2 minimization,which is quantified in Fig. 13. The approach incurs about 14%CO2 increase in the case of high grid CO2 level. Since the CPframework enables rapid updates of charging/power manage-This brief develops an integrated optimization frameworkfor battery dimensioning, charging, and on-road power management of PHEVs that minimizes the total amount of dailyCO2 emissions. A convex programming problem is formulatedto unify the three important optimal design/control problemsfor systematically evaluating their interactions.Three cases with high, medium, and low grid CO2 levels arestudied for a small-size PHEV. The results reveal that as gridCO2 decreases, the PHEV increasingly depends on electricityusage, and its recharging occurs in the vicinity of lowestcarbon time. The loss of CO2 reduction caused by simulatingthe optimal solution in the medium CO2 case for the highCO2 case is up to approximately 14%. The computationaladvantages of the framework permit a rapid and efficientday-ahead update of charging/power management control law,which noticeably mitigates such loss.R EFERENCES[1] Z. Ma, D. Callaway, and I. Hiskens, “Decentralized charging controlof large populations of plug-in electric vehicles,” IEEE Transactions onControl Systems Technology, vol. 21, no. 1, pp. 67–78, Jan 2013.[2] S. Moura, H. Fathy, D. Callaway, and J. Stein, “A stochastic optimalcontrol approach for power management in plug-in hybrid electricvehicles,” IEEE Transactions on Control Systems Technology, vol. 19,no. 3, pp. 545–555, May 2011.[3] D. Callaway and I. Hiskens, “Achieving controllability of electric loads,”Proceedings of the IEEE, vol. 99, no. 1, pp. 184–199, Jan 2011.

TABLE IVN OMENCLATURE2.6Description [unit]Coefficients in convex electric motor model in (16)Frontal area [m2 ]Coefficient in the torque-demand function in (15)Coefficients in convex engine-generator unit model in (18)Coefficient in the torque-demand function in (15)Grid-related CO2 [kg/kWh]Aerodynamic drag coefficientGasoline-related CO2 [kg/l]Rolling resistance coefficientEquivalent Cell capacitance [F]Engine on/off control signalBattery energy [J]Minimum cell energy [J]Maximum cell energy [J]Convex objective function in (1)Total amount of daily CO2 emissions to minimize [kg]CO2 emissions related to gasoline consumption [kg]CO2 emissions related to recharge [kg]Gravity [m/s2 ]Convex inequality constraint in (2)Affine equality in (3)Inequality constraint in Lemma 1Cell current [A]Maximum Cell discharge current [A]Maximum Cell charge current [A]Inertia of final drive and wheels [kgm2 ]Electric motor inertia [kgm2 ]Time indexGasoline lower heating value [J/g]Cell mass [kg]Vehicle mass excluding battery pack [kg]Time set for chargingTime set for drivingNumber of battery attery size [kWh]10120.80.60.40.2012Gasoline [kg]Efficiecny [%]CO2 [kg]Symbolai , i 0, 1, 2AfA(ω)bi , i 0, 1, 2B(ω)cccdcgcrCeEbEcell,minEcell,maxf n[8][9][10]Fig. 11. Optimization results with respect to different battery sizes under thelow grid CO2 level: (a) Total amount of daily CO2 emissions, (b) AverageEGU efficiency and gasoline consumption.[11][4] K. Clement-Nyns, E. Haesen, and J. Driesen, “The impact of chargingplug-in hybrid electric vehicles on a residential distribution grid,” IEEETransactions on Power Systems, vol. 25, no. 1, pp. 371–380, Feb 2010.[5] Z. Ma, S. Zou, and X. Liu, “A distributed charging coordination forlarge-scale plug-in electric vehicles considering battery degradationcost,” IEEE Transactions on Control Systems Technology, vol. PP, no. 99,pp. 1–1, 2015.[6] Y. He, B. Venkatesh, and L. Guan, “Optimal scheduling for emPmax,egu PonQrRsoctctdTTmaxTminTvU (soc)U0vxβρρg t tc td 2λωηcDescription [unit]Vehicular auxiliary power [W]Electrochemical battery power [W]Electrical battery power [W]Charging power limit while parked [W]Engine-generator unit power [W]Electric motor power [W]Parasitic

battery sizing/power management optimization [17] by incor-porating charging schedule optimization to minimize the total amount of daily CO 2 emissions. Its overarching goal is to enable a systematic evaluation of the interplay between the. three optimal design/control problems. Two key contributions