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Economy-focused PHEV battery lifetime management through optimal fuel cell load sharing Franc¸ois Martel

Yves Dub´e and Sousso Kelouwani

Kodjo Agbossou

Hydrogen Research Institute and Hydrogen Research Institute and Hydrogen Research Institute and Department of Electrical Engineering Department of Mechanical Engineering Department of Electrical Engineering Universit´e du Qu´ebec a` Trois-Rivi`eres Universit´e du Qu´ebec a` Trois-Rivi`eres Universit´e du Qu´ebec a` Trois-Rivi`eres Trois-Rivi`eres, Qu´ebec, Canada Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected]

Abstract—This paper demonstrates an optimal plug-in hybrid electric vehicle (PHEV) energy management strategy aimed at actively controlling the degradation of a given vehicle’s main energy carriers: in the proposed scenario, these consist of a lithium-ion battery pack and a polymer electrolyte membrane fuel cell (PEMFC). This is achieved by optimizing said vehicle’s operating cost during an active driving cycle while including each energy carrier’s usage-sensitive lifetime in addition to fuel and grid recharge expenses. The proposed scenario will lead to detailed solutions for the optimal process’s response and performance, which will be exploited to evaluate its performance versus alternative strategies. Our results demonstrate the economic gains of our solution versus common rule-based charge depleting PHEV energy management strategies as well as insights into the mechanisms at play during the process. Index Terms—Batteries, degradation, electric vehicles, optimal control, dynamic programming, energy management, vehicle dynamics, power system economics.

I. I NTRODUCTION Recent years have seen increasingly pressing signs that our society’s dependence on fossil fuels is unsustainable: their very use is firmly believed to be at the root of the now widely-acknowledged global climate changes and, even within skeptical parties, it is recognized that the resource’s supplies are dwindling, making its extraction and use ever more economically, socially and environmentally challenging. One of the main pathways to alleviate this creeping issue is through electrification of the transportation industry as a cleaner alternative to ubiquitous internal combustion engine-powered (ICE) vehicles. However, many technological breakthroughs are still necessary for this alternative to become economically competitive with ICEs and reach mass acceptance by the general public. A central issue holding back this critical achievement is the durability of the currently available energy carriers used by electric vehicles: namely, battery packs and hydrogen-fueled polymer electrolyte membrane fuel cells (PEMFCs). These electrochemical components are, by their very nature, prone to irreversible degradation phenomena, leading to accelerated performance losses and shortened lifetimes, especially when submitted to harsh operating conditions aboard hybrid electric vehicles (HEVs). However, it is possible to exercise an active

control over many such operating conditions like battery current load and temperature and in turn mitigate some of the deleterious effects of degradation. Since each of these control measures carries a price, it follows that methods able to optimize this process and ensure its performance must be devised. Our work will present such a process, which will produce an economically-optimal PHEV energy management solution over a given driving cycle while simultaneously balancing fuel and energy expenses with battery and fuel cell degradation. This solution and its performance will then be evaluated in detailed fashion and compared with commonly-used alternatives. This paper is structured as follows: the current section lays the groundwork on which this research is founded, Section II presents the various plug-in HEV (PHEV) models necessary for this study, Section III details the optimization strategy and the process’ framework, Section IV provides a discussion on comparative results obtained from its execution and Section V concludes on our findings. A. Context Battery degradation is a major hurdle to the successful commercialization of electric vehicles, more than the reported gradual losses in performance [1], reduced autonomy [2] and impaired vehicle driving characteristics [3] would suggest. These phenomena, when poorly managed, lead to battery lifetimes well below those expected by vehicle users [4]. The expense incurred by frequent battery pack replacement is a significant deterrent to prospective consumers [5]. Many mechanisms and stress factors are attributed to battery degradation [6]–[8], often tied directly to their chemistries and highly dependent on their usage profile and environmental factors. Of these many, a few can be directly controlled by means available on-board typical HEVs, the most potent being their charge/discharge pattern [9] and their temperature [10]. Fuel cells, such as hydrogen-fueled PEMFCs, exhibit degradation phenomena of their own [11], governed by different, but often controllable, mechanisms. Outside of stoichiometry-related issues [12], membrane hydration [13] and fuel purity issues [14], simpler-to-control macro-mechanisms include limiting

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start-stop cycles [15] and drastic load changes [16]. Of the few degradation-themed PHEV-centric papers that have been published, we note studies of stochastic battery health management [17], different approaches to battery degradation modeling [18], [19] and their inclusion within the context of HEVs. Part of the work presented here has seen prior publication [20], [21], where we demonstrated the sensitivity of our optimal process to economic variations; furthermore, these reports present a completely different vehicle and battery configuration based on an experimental PHEV platform. The direct management of battery degradation is rare, though it is often included as a desirable side-effect of some other objective, such as autonomy extension [1], general HEV energy management [22]–[24] or vehicle-to-grid operation [25]. Where optimization is relevant, discrete dynamic programming (DDP) has proven to be an effective and widely used technique [3], [26]–[28]. Of the currently published papers, little information is available on the combined management of multiple HEV energy carrier degradation. B. Problem statement We consider a mid-size PHEV which includes a lithiumion battery pack as a main energy carrier as well as plugin recharge capabilities. This battery pack experiences degradation, which is accelerated by controllable factors such as discharge current and depth-of-discharge (DOD). A driving load will be represented by running the vehicle through a naturalistic driving cycle. Through timely on-board recharge intervals via a secondary energy carrier, in this case a PEMFC, we propose an optimal solution capable of actively supporting the battery pack in a manner that will simultaneously: • Obtain a minimal PHEV operating cost taking into account the degradation, loss of performance and eventual replacement cost of its batteries and PEMFC; • Extend the useful lifetime of its battery pack and fuel cell system; • Optimize the PHEV’s fuel and grid recharge energy consumption. This will be achieved through numerical simulation modeling of the various PHEV components and their degradation mechanisms, all of which to be utilized by a specially-designed DDP algorithm aimed at achieving a minimal operating cost during an active driving cycle.

TABLE I G ENERAL PHEV SPECIFICATIONS Dimensions Wheels Mass Aerodynamic drag

L4.49m W1.78m H1.43m R17 1721 kg 0.287

Motor PEMFC Electric range

111 kW (peak) 62 kW (peak) 61 km

Battery type Number of cells Nominal cell voltage Nominal system voltage Nominal cell capacity Nominal pack capacity

LiF eP O4 lithium-ion 288 3.2V 307.2 V 15 Ah 45 Ah

to define the relationship between each component (1) within the vehicle’s architecture (Fig. 1). Eb (k + 1) = Eb (k) + [PF C (k) + Pg (k) − Pmotor (k)]Δt (1) where Eb is battery energy, PF C , Pg and Pmotor are power values imposed on the battery pack by the fuel cell, public grid and electric traction motor, respectively, for a simulation step k of sampling time Δt. B. Driving cycle The driving cycle represents the load requested from the electrical drivetrain during vehicle operation and as such is one of its main degradation factors. The cycle used for this research (Fig. 2) was designed with specific attributes in mind: • Stimulate battery degradation by demanding deep discharge driving conditions; • Remain within the PHEV’s all-electric range to focus recharge response on the degradation management issue; • Follow a naturalistic pattern representative of typical urban driving habits.

II. PHEV MODELS Below are summarized the various models used in the realization of this work. Part of these were previously published in [21], where the interested reader will find each presented in full detail. The PHEV’s general specifications mirror those of the Chevrolet Volt, a commercial mid-sized sedan PHEV, [29] and are listed in Table I. A. Energy balance The proposed solution relies on the energy management of the various energy carriers aboard the PHEV: its battery pack, PEM fuel cell and plug-in grid recharge; as such, it is essential

Fig. 1. PHEV series architecture

transmission, traction motor and controller and their respective instantaneous values at a given timestep k. E. Plug-in recharge circuit This component reproduces the efficiency losses, grid power consumption and recharge pattern of a typical PHEV “smart charging” circuit during grid-powered plug-in battery recharge which occurs, in this scenario, outside the active 10-hour driving cycle, i.e. when the vehicle is presumably parked at its user’s residence. The energy costs and degradation incurred during this recharge period are fully accounted for during optimization (4). Pg (k) = Vb,cell (k)Ib,cell (k)ηcharger (k)

Fig. 2. Naturalistic driving cycle

Allow for maximum resolution using the DDP optimization algorithm. The resulting cycle was built from a succession of standard driving cycles: the US06, UDDS and NYCC [30] in an effort to replicate a typical urban commute to-and-from work. To allow increased resolution during the optimal process, inactive driving periods during this 10-hour long cycle were compressed, though time-temperature dependant battery degradation during these intervals is fully accounted for. Preliminary optimization runs have demonstrated that recharge intervention during these pauses was non-existent in all scenarios. The cycle’s total length is 56 km out of the PHEV’s 61 km (approximately 92%) nominal autonomous battery capacity. •

C. Mechanical drivetrain The PHEV mechanical drivetrain is represented using the linear kinematic approach [31], meaning that the speed and acceleration profile from a given driving cycle are directly converted to mechanical power requirements through elementary force, torque and speed equations (2). 1 ˙ + ρair Cd Av v(k)2 Pmec (k) = [mv v(k) (2) 2 +mv gμcos(θ(k))]rw ωw (k) where Pmec is vehicle propulsion mechanical power, mv is vehicle mass, v is vehicle speed, v˙ its acceleration, ρair is air density, Cd is the vehicle’s aerodynamic coefficient, Av is its frontal area, g is gravitational acceleration, μ is wheel friction, rw is wheel radius and ωw is wheel angular speed, each for a given timestep k. D. Electrical drivetrain This next step translates the mechanical power demand profile into an electrical current load to be shared between the battery pack and the PEMFC. This is done through linear conversion using (3), which includes efficiency maps for the electrical motor and power electronics involved. Pmotor (k) = Pmec (k)[ηt (k)ηmotor (k)ηc (k)]−1


where Pmotor is the propulsion motor’s electrical power load, ηt , ηmotor and ηc are efficiency maps for the vehicle’s


where Vb,cell (k) and Ib,cell (k) are battery cell tension and charger-controlled current, respectively, and ηcharge is the charger’s efficiency curve, all for a given timestep k. F. PEMFC model 1) Power and fuel consumption: The functional fuel cell model uses a system-level efficiency curve represents the energy losses measured relative to the power demand from the optimal decision process (5). A second part tallies the hydrogen fuel spent to provide said power output, using hydrogen’s higher heating value (HHV) to provide a conservative estimate of this energy-to-fuel mass conversion (6). PF C (k) = u∗ (k)φF C ηconv


mH2 (k + 1) =mH2 (k)+   u∗ (k)φF C Δt ηF C (k)(1 − δF C )HHVH2


where u∗ is the optimal power demand, φF C is the PEMFC’s maximum power output, ηconv is its power converter’s efficiency curve, mH2 is the consumed hydrogen mass, ηF C is the fuel cell system efficiency, δF C is its degradation percentage and HHV H2 is hydrogen’s higner heating value (HHV), over given step k of length Δt. 2) Voltage loss and degradation: PEMFC degradation is accounted for according to continuous load variations and damaging events such as start-stop cycles and drastic power variations using the model provided by [31] whose parameters are reported in Table II. In addition to measuring degradation costs and remaining fuel cell lifetime, this value represents a gradual loss of cell voltage; this loss is translated into decreasing cell efficiency in (7). k+1 

δF C (k + 1) = δF C (k) +


 kF C [n1 V1 + n2 V2 + Uload Δt]

Vnom,cell (1 − EoLF C )

(7) where n1 , n2 , V1 and V2 are event counters and instantaneous voltage degradation rates for start-stop and drastic load  is a continuous load-dependent changes, respectively, Uload degradation rate, Vnom,cell is nominal cell voltage and EoLF C is fuel cell end-of-life voltage loss level, set at 90% as defined in [31], over a step k of length Δt.


Voltage degradation rate

Start-stop Load change Low power idling High power load kF C factor

V1 = 13.79μV /cycle V2 = 0.4185μV /cycle  Uload = 8.662μV /h  Uload = 10.00μV /h 1.72

Fig. 4. Lithium-ion battery lifecycles vs DOD [37]

SOC(k + 1) = SOC(k) −

Qe (k + 1) Cδ (k + 1)

DOD(k) = 1 − SOC(k)

Fig. 3. Lithium-ion battery equivalent circuit model diagram [32]

G. Lithium-ion battery model 1) Electrical behaviour: The battery model used for this work is based on the research of [32] and is represented in Fig. 3. A similar model was employed and characterised in a previous publication [33] using lead-acid batteries to better represent an experimental platform used for validation [34]. The same model’s general structure was updated to represent lithium-ion batteries using the experimental parameters provided by the author of [32]. This model is adapted to our structure by defining an equation for battery energy evolution Eb (8) and rearranging the general energy balance found in (1) to include battery discharge current Ib (9), itself a direct function of battery voltage Vb (10) as represented in Fig. 3. Eb (k + 1) = Eb (k) − Vb (k)Ib (k)Δt   Pmotor (k) − PF C (k) − Pg (k) Ib (k) = Vb (k) Vb (k) = Em (SOC, θb ) − Ib (k)R0 (SOC, θb ) − R1 (SOC, θb )Im (k)e(−t/τ1 )

(8) (9) (10)

where Vb and Ib are battery voltage and current, Em is battery electromotive force, SOC is state-of-charge, θb is electrolyte temperature, R0 and R1 are internal resistance values, Im is main branch circuit current and τ1 is a time delay factor over elapsed time t. This battery model is further tied to our DOD-based degradation model through battery electrical charge Qe (11) and state-of-charge (SOC) (12) / depth-of-discharge (DOD) (13). Qe (k + 1) = Qe (k) + Ib (k)Δt


(12) (13)

where Qe , Cδ and DOD are values for battery charge, capacity and depth-of-discharge, respectively, for a given step k of length Δt. 2) Battery degradation: Battery degradation is modeled using our own previously-published model [21], built on the proven Ah-throughput and Arrhenius time/temperature reaction rates to account for the effects of discharge current, depthof-discharge, temperature, calendar life and gradual performance losses. The model’s flexibility allowed us to adapt said model from lead-acid to lithium-ion chemistries by updating its various reference tables and parameters to their lithium-ion homologues [35]–[37]. We exploit battery lifecycle data (Fig. 4) to define a stress factor, φDOD , meant to reflect the negative influence of discharge levels on battery degradation rates. φDOD (DOD, k) = 1 +

(κmax − κ(DOD, k)) κmax

κ(DOD, k) = Cδ (k) × DOD(k) × ncycles (DOD) κmax = max(κ(DOD, k))

(14) (15) (16)

where φDOD is the discharge level stress factor, κ and κmax are battery lifetime recoverable energy values for instantaneous and optimal conditions, respectively, while ncycles is the number of cycles achievable at a given DOD level. The partial degradation model δb,I is designed to monitor the fraction of the maximum lifetime recoverable battery energy κmax expended through discharge currents Ib . We expand the original modeling principle to include stress factor φDOD instead of average values for κmax .  tf   Ib (k)φDOD (DOD, k) Δt k=t0 δb,I (Ib , DOD, k) = (17) κmax The second half of the battery degradation model accounts for Arrhenius-based degradation rates, which are dependant on elapsed time t and electrolyte temperature θb . The stress

factor φDOD is included as well to accelerate this phenomenon accordingly. (18). δb,θ (DOD, θb , k) =    tf  −Ea  Ae R(θb (k)−θb,ref ) φDOD (DOD, k) Δt

instantaneous battery SOC and the grid-based recharge cost required to compensate.   M  (m ˙ H2 )Δti αH2 (25) CH2 (xH2 , u, k) = xH2 (k) +




λb where A is an exponential prefactor, Ea is activation energy for degradation reactions, R is the univeral gas constant, θb and θb,ref are instantaneous and reference values for electrolyte temperature and λb is battery lifetime estimate. Both degradation models of (17) and (18) are summed up to define total degradation, as defined by the relationship presented in (19) below. δb (Ib , DOD, θb , k) =





III. O PTIMAL PROCESS The optimization process used for this work is based on discrete dynamic programming theory [38] and was designed to use precalculated data tables and omit calculation points out of timestep-dependent admissible state boundaries to improve its computational efficiency.

Cδb (xδb , k) = xδb (k) +

 (δb )Δti αb

CQe (xQe ) = xQe (N )αg   M  (δF C )Δti αF C CδF C (xδF C , u, k) = xδF C (k) +

xH2 (k) ∈ XH2 (k)


xδb (k) ∈ Xδb (k)


xQe (k) ∈ XQe (k)


xδF C (k) ∈ XδF C (k)


This process is based on the active control of battery degradation through optimal recharge using a supporting energy source; this is represented by a single control variable, the power demand to the PEMFC (24) used for battery support. (24)

B. Performance criteria We define criteria to evaluate the relative performance of each admissible decision made by the DDP algorithm. These are directly related with each state variable detailed above and quantify the impact of said decisions on their evolution. Being based on economic performance, these criteria can be summed up as the monetary cost associated with each decision in regards to hydrogen fuel expense (25), battery degradation (26), battery grid recharge (27) and PEMFC degradation (28). All state variables and criteria are positive, steadily increasing functions save for CQe (27) which is a linear relation between

(27) (28)


C. Objective function The various performance criteria enumerated above are combined into a single objective function, providing a unique performance score for each decision made by the algorithm. Once again, this function can be described as the sum of each individual monetary expense incurred during a given decision interval (29). J(xs , u, k) =

N   CH2 (xH2 , u, k) + Cδb (xδb , k)

+ CδF C (xδ F C, u, k) + CQe (xQe , N )


The DDP process requires state variables to account for the evolving parameters involved. These represent the various consumables to be minimized in concert during PHEV operation, namely hydrogen fuel, battery degradation, remaining battery charge and PEMFC degradation, as shown by (20)-(23) below.



A. State and control variables

u(k) ∈ UF C



D. Constraints It is essential to properly define the range of admissible values for each variable involved in such a process to ensure coherent results. First, we define those limits for each state and control variable in play (30)-(33).  max xH2 (k) ∈ XH2 (k) = xmin H2 (k), xH2 (k)  max xδb (k) ∈ Xδb (k) = xmin δb (k), xδb (k)  max xQe (k) ∈ XQe (k) = xmin Qe (k), xQe (k)  max xδF C (k) ∈ XδF C (k) = xmin δF C (k), xδF C (k)

(30) (31) (32) (33)

The limits of each state variable are predetermined by running the PHEV models at boundary conditions for a given driving cycle. However, the PHEV models make use of many more variables not directly involved in the optimal algorithm, but nonetheless constrained by physical limitations; these require proper constraints as well, as shown by (34)-(38). SOCmin ≤ SOC(k) ≤ SOCmax


Vb,min ≤ Vb (k) ≤ Vb,max


Ib,min ≤ Ib (k) ≤ Ib,max


Pmotor,min ≤ Pmotor (k) ≤ Pmotor,max


Pgmotor,min ≤ Pmotor (k) ≤ Pmotor,max



δb ($/kW h)

Qe ($/kW h)

δF C ($/kW )





IV. R ESULTS AND DISCUSSION Here is defined a precise scenario based on our original problem definition, where we run a high-resolution optimal process over a single driving cycle (Fig. 2); it is followed by a comparative study of our optimal solution versus alternative rule-based PHEV management solutions. A. Proposed scenario We run the proposed PHEV models through the DDP optimization process over a single naturalistic driving cycle representative of a 10-hour urban driving pattern to-and-from work, as illustrated in (Fig. 2). From the outset, the PHEV’s battery pack is fully charged and its hydrogen fuel tank is full. This driving cycle is designed so that the battery pack has just enough capacity to complete it without assistance; therefore, PEMFC fuel expense is expected to occur solely for degradation management purposes. Our models (19) predict that battery degradation management is achievable through PEMFC-assisted load sharing by limiting battery discharge current and DOD during active use. The PEMFC itself degrades through use as well (7) and the related cost is accounted for (28). Temperature is kept constant (20◦ C) for Arrhenius-based time/temperature battery degradation purposes (18) as active temperature management is not yet implemented in our process. Plug-in grid recharge is not available during the active driving cycle, but is tallied according to battery DOD at its final step to represent postcycle, “night-time” home recharge; this energy cost (27), as well as the battery degradation occurring during post-cycle recharge, is accounted for in the decision process (29). We consider a complete cycle to be a full 24-hour ”day”: 10 hours are spent in active driving conditions (Fig. 2) and the remaining 14 hours is allowed for grid-based battery recharge and storage. As such, the optimal process is expected to produce the minimum PHEV operating cost necessary to complete this driving cycle while considering hydrogen fuel (25), battery degradation (26), grid recharge (27) and PEMFC degradation (28) costs. Operating cost itself is measured in monetary terms (29): hydrogen [14] and grid recharge [39] is a straightforward dollar cost per expended kg and kWh, respectively, while battery pack [40] and fuel cell [41] degradation expenses are represented by an equivalent loss percentage of the cost of the component itself, both of which will require replacement upon reaching 100% degradation; these costs are presented in Table III.

B. Performance comparison vs. rule-based strategies We propose a detailed look at the performance of our optimal process in comparison to the two rule-based PHEV energy management systems detailed below: the charge depleting/charge sustaining energy management (CDCS) and the simple recharge (SR) strategies; as a comparative basis, an U nmanaged option will be presented as well. • CDCS rule-based strategy: The CDCS alternative strategy prioritizes battery power until a given SOC level (set at 25% as per industry standards) is reached, at which point a secondary energy carrier kicks in at a variable rate to maintain said SOC at a constant value for the remainder of the driving cycle. • SR rule-based strategy: The SR strategy allows the depletion of the battery pack until a set SOC level is reached (25% as per industry standards), after which it initiates a constant, full-power recharge of the battery pack instead of sustaining its current SOC level. • Unmanaged strategy: This option is really no strategy at all: in this scenario, no recharge effort is supplied by the fuel cell during active driving, thus relying solely on the battery pack’s charge for propulsion. 1) Technical analysis: This section presents our results to highlight the optimal solution’s performance in terms of energy consumption and component degradation, beginning with Fig. 5 which displays the optimal power split between battery pack and PEMFC. A general feature of note is that the majority of the optimal response is located late in the driving cycle: this indicates a clear prioritization of battery discharge over fuel consumption, even with the pack’s degradation factored in. The response to a particularly intense power spike early in the cycle does highlight that an early recharge response is sometimes necessary to obtain an optimal outcome. Given our battery model’s tendency to accelerate degradation according to DOD (14), such a response appears to be economically advantageous only at particularly intense discharge events. This demonstrates that, at least in this scenario and particular economic parameter equilibrium, it remains economically optimal to address the bulk of battery degradation late in the

Fig. 5. Optimal power split between battery pack and PEMFC over the course of a driving cycle, showing total driving load (black) and its sharing proportions between batteries (blue) and fuel cell (red)


Optimal CDCS SR Unmanaged

H2 (kg)

δb (10−3 %)

Qe (kW h)

δF C (10−3 %)

0.08 0.10 0.32 0

0.93 0.95 0.92 0.96

11.45 11.45 8.18 13.18

0.063 0.046 0.159 0

cycle, in similar fashion to simpler rule-based alternatives. Figure 6 focuses on the final, but most active, instants of the driving cycle and offers a comparison of the battery SOC levels obtained by each strategy. At a glance, the behavior of the CDCS and SR strategies upon reaching the 25% SOC limit is precisely as described above, as is the non-existing response from the U nmanaged option. The optimal response from Fig. 6 does differ from the simpler rule-based alternatives in that it occurs much earlier in the cycle at around 38% SOC (excluding the early peak response mentioned above), and promotes a slower, more controlled discharge of the battery pack from this point onward. This behavior maintains DOD levels higher for an extended period, resulting in the desired optimal balance of component degradation and energy expense. The overall performance of the proposed solutions is broken down in Table IV. The fuel consumption of the optimal solution, H2 , is shown to be the lowest among the active alternatives (the U nmanaged option does not use fuel at all, obviously) while displaying an advantageous balance of battery degradation δb , close behind the recharge-heavy SR strategy, and fuel cell degradation δF C second only to the CDCS alternative; as for grid-based recharge, it is on par with the CDCS option, which is not surprising given that both finish their cycles at similar SOC levels. The optimal solution’s performance regarding battery degradation δb is only surpassed by 1% by the SR strategy owing to the latter’s highintensity battery recharge; however, this rule-based strategy is shown to be highly wasteful, as its fuel consumption H2 and its PEMFC degradation δF C are shown to be 300% and 152% higher that the optimal outcome, respectively, for comparable degradation results. Meanwhile, the balanced CDCS strategy consumes 25% more hydrogen than the optimal but appears to do very little in terms of battery degradation management. The optimal solution’s second-rank performance in fuel cell degradation δF C is directly related to its usage profile, as it is used for longer periods than the leading CDCS strategy and causes more damaging load-changing events, as modeled by (7). The U nmanaged option appears to be the most resourceefficient alternative, as it does not exhibit any fuel expense or PEMFC degradation but displays the highest δb values and the totality of its battery recharge Qe is relegated to post-cycle plug-in grid recharge. 2) Economic performance: This section examines the performance of the optimal solution in economic terms: the operating costs of the optimal solution are broken down in

Fig. 6. Comparative SOC trajectories highlighting the major divergence paths between each strategy, occuring in the final portion of the driving cycle

Fig. 7. This graph displays the progression of each cost as the driving cycle unfolds; it can also be interpreted as the expense trajectory of each consumable (H2 , δb , Qe and δF C ) weighted by the economical cost of each component, which are found in Table III. The main active periods are located at both ends of the cycle, where the cycle’s acceleration profile is most demanding (Fig. 2); the slower, inner-city driving portions occurring in the middle are shown to cause comparatively less battery degradation and discharge. Fuel consumption and PEMFC degradation, as previously described, occurs mainly at the end of the cycle; this graph displays the same progression in relative monetary values. The costs of post-cycle battery degradation, each a function of the conditions of the PHEV at the active cycle’s final step, are not shown in Fig. 7; however, they are added to obtain the total values for each solution in Table V. The individual values for H2 , δb , Qe and δF C are proportionally similar to those from Table IV; however, the addition of economical weights allows the evaluation of the total operating cost of the PHEV, the minimization of which being the objective function J of our optimal process (29). Table V clearly shows the optimal solution to be the cheapest out of all proposed strategies. It is interesting to note, however, that its closest competitor is the U nmanaged solution, which displays an almost identical total operating cost. The optimal solution’s behavior previously hinted that battery energy, even while including their degradation, remained the cheapest option available aboard this PHEV; this result reaffirms this statement. However, our optimal solution displays an improvement of 3% in battery degradation performance over the U nmanaged solution for an almost identical operating cost. Some limitations apply to the proposed results. First, they are highly dependent on the cost of each component, as we demonstrated before in [21]; different component costs will dramatically alter this outcome. Second, our choice of driving cycle and battery capacity is designed specifically to be completable using only battery power; driving conditions requiring more energy than is available within the battery

degradation management; these efforts are well underway towards publication in the near future. The results presented here also raise specific questions, such as the response of our system when battery recharge is a necessity rather than an option; simulations are underway as well to provide these answers. ACKNOWLEDGMENT

Fig. 7. Optimal PHEV operating costs breakdown between hydrogen fuel H2, battery degradation δb (labelled Deg), grid recharge Qe and fuel cell degradation δF C (labelled P EM F C) over the course of a single driving cycle, excluding post-cycle degradation costs

This work is made possible by contributions from the Fondation de l’Universit´e du Qu´ebec a` Trois-Rivi`eres as well as the Fonds de recherche Qu´ebec – Sciences et technologies (FRQNT) and Hydro-Qu´ebec. R EFERENCES


Optimal CDCS SR Unmanaged




δF C


0.24 0.28 0.93 0

6.40 6.54 6.35 6.67

1.26 1.26 0.90 1.43

0.19 0.14 0.48 0

8.09 8.22 8.66 8.10

pack would invalidate the presented U nmanaged alternative. This choice, however, ensures that the response observed here is solely guided by economically-focused component degradation management and not to satisfy the demands of a particularly intense driving cycle. This allows our results to address a fundamental question: PHEV battery degradation management through fuel-based recharge can indeed be an economically advantageous solution, even with no recharge imperative stemming from external operating conditions. V. C ONCLUSION This paper aims to evaluate a proposed optimal PHEV energy management strategy including, at its core, the degradation of its main energy carriers. By expanding on a previouslypublished effort, the results presented in this paper have achieved the following objectives aimed at from the outset: 1) To sucessfully adapt our optimal strategy to a different PHEV configuration and battery chemistry as a demonstration of its versatility. 2) To present a detailed breakdown of the optimal solution’s impact on core PHEV operating parameters, such as battery SOC and driving power split. 3) To demonstrate the technique’s potential by comparing its economical performance to rule-based strategies commonly used as benchmarks in HEV research. Following this work, we intend to fully exploit the versatility of the proposed technique to expand the results presented in this article. Our choice of a structure built around full 24hour cycles is primarily motivated by our objective to optimize vehicle operation over a long-term horizon and provide meaningful insight into the dynamics of long-term battery

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