Active Adaptive Battery Aging Management for Electric Vehicles

The battery pack accounts for a large share of an electric vehicle cost. In this context, making sure that the battery pack life matches the lifetime of the vehicle is critical. The present work proposes a battery aging management framework which is capable of controlling the battery capacity degradation while guaranteeing acceptable vehicle performance in terms of driving range, recharge time, and drivability. The strategy acts on the maximum battery current, and on the depth of discharge. The formalization of the battery management issue leads to a multi-objective, multi-input optimization problem for which we propose an online solution. The algorithm, given the current battery residual capacity and a prediction of the driver's behavior, iteratively selects the best control variables over a suitable control discretization step. We show that the best aging strategy depends on the driving style. The strategy is thus made adaptive by including a self-learnt, Markov-chain-based driving style model in the optimization routine. Extensive simulations demonstrate the advantages of the proposed strategy against a trivial strategy and an offline benchmark policy over a life of 200 000 (km).

the other side, it is amenable to be sided by a learning algorithm. 132 We exploit these features developing an active adaptive battery 133 aging management systems that adapts to the way of driving of 134 the driver. 135 This work extends a previous contribution [24] in two direc-  and the recharging strategy. Section III describes the driver's be-145 havior model and its learning mechanism. Section IV introduces 146 the offline and online battery aging management strategies. 147 Sections V and VI analyze the results and draw the final conclu-148 sions.

150
The model aims to quantify the battery aging for different 151 driving conditions. The reference model is a compact car as the 152 one modeled in [25]. Table I summarizes the main parameters 153 used in the simulation. The model, as depicted in Fig. 1, has 154 three main components: the vehicle longitudinal dynamics, the 155 powertrain model and the battery pack with its thermal and aging 156 dynamics.

158
Mainly two approaches exist for powertrain modeling [26]. 159 In the backward facing approach, the powertrain states are 160 computed starting from the vehicle velocity profile, which is 161 an input. Conversely, the forward facing approach follows the 162 more natural powertrain causality using the throttle position 163 as an input. In this work, the model is based on a mixed 164 forward-backward facing approach, as highlighted by Fig. 1. 165 The forward portion models the driver's response to a desired 166 reference speed and the longitudinal dynamics of the vehicle. 167 The backward facing part, starting from the power requested for 168 the vehicle motion, computes the power drawn from or provided 169 to the battery. This mixed modeling allows for the modeling of 170 the driving performance losses that the battery aging algorithm 171 will necessarily introduce. Starting from the forward facing 172 portion, the driver's action on the accelerator pedal, in order 173 to follow a desired speed, is modeled with a simple proportional 174 regulator [24]. Thus, traction torque requests (T req m ) are com-175 puted based on the difference between the reference, which is 176 the driver's desired longitudinal velocity, and the actual vehicle 177 speed: with k p the proportional gain. The motor current is computed as 179 follows: where k m is the electric motor torque constant. The requested 181 motor current is then saturated to I max , 1 which is computed from 182 the output of the aging management strategy I cell max according to: 183 whereη m is the electric motor efficiency, V b the battery pack with M and v the vehicle mass and speed, r t the gear ratio,

193
R w the wheel radius, F r the rolling resistance, C x the drag 194 coefficient, ρ a the air density, and A the vehicle cross sectional 195 area. Therefore, the power provided by the traction motor for 196 the motion is modeled as follows: with T sat m the motor torque. In the backward facing portion, 198 the electric machine is modeled as an efficiency map which 199 computes the battery power: with η m the motor efficiency. Thus, the cell power request is 201 given by P cell = P b /n cell with n cell = n s × n p the total num-   Therefore, as soon as the battery reaches the limit State of 215 Charge, the vehicle is decelerated from the current desired speed 216 v ref to 0 (km/h) and then recharged at a power limited by the 217 maximum cell current I cell max . Once the charge is completed, the 218 vehicle is accelerated again tov ref . Rather than reasoning in 219 terms of SoC, the aging strategy employs the DoD as the control 220 variable. We define the DoD to be symmetric with respect to a 221 SoC of 50%. For instance, a DoD of 70% (Fig. 2) denotes a 222 battery SoC varying between 15% and 85%. Note that, on the 223 optimization scales of thousands of kilometers, neglecting the 224 exact position of the charging points is reasonable.

226
As pointed out in the introduction, modeling aging phenom-227 ena is a complex task, most contributions rely on semi-empirical 228 models. In our framework, we follow the same path in order 229 to design a control-oriented model. The model leverages some 230 simplifying assumptions: 3) Even though temperature is accounted for in the model, 236 we assume the BMS to be equipped with an energy man-237 agement system.

238
Under these assumptions, the battery pack is modeled as a 239 single large cell with its electrical equivalent circuit: a voltage 240 source v oc and a resistance R cell accounting for Joule losses. 241 The battery open circuit voltage is function of the SoC, while 242 its resistance is generally depending on aging and temperature.

243
Thus, the cell current is given by: The cell SoC dynamics [26] takes the following expression: with Q the cell capacity, decreasing with aging. As already 246 shown in [24], the battery aging model is derived from [9] and 247 extended from the HEV's scenario to the EV's one. Therefore, 248 the rate of capacity loss with respect to the processed Ah is 249 described as follows: with the second equation modeling the Ah throughput as the total 251 current processed by the cell. The parameters E a and R g are the 252 activation energy, equal to 31.5 (kJ/mol), and the universal gas 253 constant. η and z are identified from experimental data. α SoC 254 is a penalizing factor that accelerate the aging for low and high 255 SoC [10], [27]: with SoC min , SoC max , b, c, and d empirically determined shap-257 ing parameters (Fig. 3) with k res derived from the experimental data of [28]. Eventually, 265 recalling that the temperature dependency of the cell internal 266 resistance is expressed by [29]: with R cell,0 the nominal cell resistance and T 1 , T 2 identified 268 parameters, the following is obtained: It should be noted that the proposed model is empirical in 270 nature and thus subject to variation depending on the actual 271 characteristic of the cell in use. The framework, while needing 272 an aging model, does not exploit any specific features of the 273 proposed model.

274
Thermal management. Temperature is one of the stress fac-275 tors increasing battery aging. For this reason, a common practice 276 in automotive companies is to introduce a battery cooling system 277 in order to control the temperature to a desired value T . Here, 278 we consider an air-cooled battery pack. Fig. 4 depicts the high 279 level architecture. Givenṁ fl the mass flow rate of the air forced 280 by the fan into the cooling system evaporator, the heat exchange 281 is given by: where C p,f l is the specific heat capacity of air, T i and T o,1 the 283 air temperature at the evaporator input and output respectively. 284 Thus, the cooled air is forced into the battery pack, leading to 285 the following energy balance: whereQ b is the heat exchange between battery and air, and T o,2 287 the air temperature after the battery pack. Under the assumption 288 of a uniform temperature distribution T and modeling the heat 289 generated by the battery pack as R b i 2 b ,Q b is rewritten as follows: 290 with T room the room temperature, R b the total battery pack 291 resistance, i b the battery pack current, and R conv the thermal 292 resistance between the battery and the surroundings. Assuming 293 no other heat exchange takes place and that T i = T o,2 , the 294 following equality holds true: Therefore, the per-cell electric power absorbed by the cooling 296 circuit, i.e., by the compressor, to control the battery temperature 297 at T is given by: where COP is the coefficient of performance of the cooling 299 system [30]. Eventually, the total power requested to each cell 300 is increased by P cool , leading to the following equation:   Fig. 6. Effect of the DoD on the cell capacity. The maximum current is limited to 2.5 (C-rate) and the DoD is varying between 60% and 80%. Fig. 7. Effect of the current limitation I cell max on the cell capacity. The DoD is fixed at 70% and the I cell max is varying between 1 (C-rate) and 5 (C-rate).

III. DRIVER'S BEHAVIOR LEARNING
336 The battery aging model shows that the cell current is one 337 of the stress factors to be reckoned with. The cell current 338 heavily depends on the instantaneous driver's torque request. 339 The driving style has an impact on the aging dynamics. This 340 section proposed a model to describe the driver's behavior and 341 a learning mechanism that allows the model to adapt to changes 342 in the driving style.  probability matrix T ∈ R s×s : (20) for all n, m ∈ {1, . . . , s}.

353
The transition matrix can be learnt offline or online as new  the driver's desired speed to be modeled as an Artemis Rural 378 driving cycle.

379
Once a transition probability is known, the desired speed 380 profile can be randomly generated as a realization of the Markov 381 chain. Fig. 9 compares the speed and acceleration distributions 382 of the Artemis Rural driving cycle and of a driving cycle gen-383 erated from the transition probability matrix of Fig. 8, over a 384 traveled distance of 500 (km). From figure, one concludes that 385 the Artemis Rural driving cycle and the generated profile are 386 equivalent in terms of speed and acceleration distributions but 387 will be different in terms of time domain behavior.

389
This section develops two battery aging management ap-390 proaches: an offline approach that relies on the perfect knowl-391 edge of future driving cycle, and an implementable approach. 392 Both optimization algorithms modulate the DoD and the max-393 imum current I cell max in the attempt of minimizing the capacity 394 degradation while guaranteeing acceptable driving performance 395 in terms of range, charging time, and fulfillment of a desired 396 speed profile.

397
The performance index J quantifies the above considerations 398 and translates a multi-objective optimization problem into a 399 single objective one:

404
The minus before the last cost component denotes that only

405
J range must be maximized.

406
J accounts for several objectives. The first term penalizes the 407 capacity degradation over the traveled kilometers N : with Q(0) = Q nom the nominal capacity.
with t(N ) the time horizon, i.e., the time to travel N kilometers.

412
The terms J charge and J range respectively take into account 413 the charging time and the driving range: The weights (28) are chosen to make the magnitude of the asso-460 ciated cost components comparable. Moreover, for the problem 461 at hand, varying only α l is reasonable because, while managing 462 the battery life, the major concern is monitoring its capacity 463 degradation over time. Thus, Fig. 10 represents the optimization 464 results in terms of components of the cost function. Increasing 465 α l allows one to focus more on the battery aging minimization 466 instead of on guaranteeing satisfactory driving performances. 467 Therefore, in terms of control actions, this leads to lower DoD 468 values, which reduce the driving range, and to a lower I cell max , 469 which increases both the charging time and the error between the 470 driver's desired speed and the actual vehicle velocity. Eventually, 471 for a reasonable trade-off, a value of α l = 2.7 × 10 8 (km/Ah), 472 in correspondence of the Pareto fronts elbows, is chosen.

B. Online Optimization 474
The offline optimization is carried out under the assumption of 475 complete knowledge of the driver's behavior along the traveled 476 distance N . This makes the result not applicable in practice, here 477 we introduce the online optimization technique. As summarized 478 by Fig. 11, the idea is to introduce a Model Predictive Control 479 (MPC) like procedure based on PSO. First, a prediction horizon 480 N p and a control discretization step N u are selected, on a 481 distance base, satisfying the following inequality: N u ≤ N p . 482 Therefore, at each optimization step k, with k the MPC index, 483   11. Online optimization. At k, the objective function is minimized over a prediction horizon N p assuming a control discretization step N u . Only the control input computed from k to k + 1 is applied, before rerunning the optimization at k + 1.
where t(k + k p ) − t(k) is the time to travel N p kilometers and 487 E(k + k p ) − E(k) the number of charging events between k 488 and k + k p . The minimization of (29) is still obtained rely-489 ing on PSO. Therefore, only the first pair of control inputs 490 [DoD(k), I cell max (k)] T is applied from k to k + 1 (i.e., over a 491 traveled distance N u ). The system state at k, provided as input 492 for the prediction, takes the following form: where Q(k), SoC(k), and Ah(k) are respectively the residual 494 capacity, the battery SoC, and the Ah processed till k. Note that 495 this yields a closed-loop term. Eventually, the next optimization 496 step is performed at k + 1.

497
Since for each optimization step a prediction over N p is 498 needed, an estimate of the future driver's behavior, in terms 499 of desired speed, is necessary. We consider two cases. The 500 Online approach assumes the Artemis Rural driving cycle to 501 be a description of the average driver's behavior. In this case, 502 the prediction over N p is the repetition of the aforementioned 503 driving cycle; the driving cycle is applied regardless of the actual 504 driving style. The Online MC approach on the other hand tries 505 to consider the current drive style. It relies on the Markov chain 506 model. It uses the online learnt transition matrix to generate the 507 desired speed profile over N p . This second approach thus adapts 508 to changes in the driving style. Fig. 12 summarizes the online 509 optimization architecture.

510
As already mentioned, at each step k, the strategies solve an 511 optimization over N p . A careful selection of prediction horizon 512 and control discretization step is fundamental to solve the op-513 timization problem in a reasonable time, without affecting the 514 found strategy. In the following, we use N p = 6000 (km) and 515 N u = 2000 (km) together with a particle swarm size p #,2 = 60. 516 The next section better illustrates the validity of the choice.

518
To guarantee repeatability and fairness of comparison, the 519 validation uses two deterministic driving styles: (a) the Artemis 520 Rural driving cycle and (b) the Highway Fuel Economy Test, 521 both illustrated in Fig. 13. Note that the Highway Fuel Economy 522 Test is scaled in order to reach a maximum speed of 120 (km/h), 523 a realistic speed limit in the European Union. The two driving 524 cycles have complementary characteristics, the Artemis Rural 525 has a lower average speed but a higher maximum acceleration 526 than the Highway Fuel Economy Test; the Artemis Rural cap-527 tures a more dynamic driving style, comprising both urban, and 528 highway driving.  Table II summarizes the value 532 of the cost functions. The table also computes the cost function 533 for the open-loop case with no active battery degradation man-534 agement strategy, see Appendix A. The comparison considers a 535 horizon N of 200 thousand kilometers, over which the driver's 536 behavior is assumed to be fixed (i.e., either scenario (a) or (b), 537 exclusively). From these results, one can draw the following 538 conclusions for the Artemis Rural driving cycle: Fig. 12. Online optimization architecture. The first approach assumes the Artemis Rural driving cycle to be a good description of the driver's model. Thus, the Markov chain branch (dashed line) is deactivated. Conversely, the second approach relies on Markov chains to first learn the driver's behavior and then to generate the desired speed profile over the prediction horizon N p . The online optimization is triggered each control discretization step N u . the Online MC strategies lead to results close to the offline 549 benchmark. As a matter of fact, the Online solution is com-550 puted assuming the future driver's behavior to be modeled 551 as an Artemis Rural driving cycle, which is the truth for 552 this first study. The learning mechanism in this context 553 does not bring any advantage. If anything, it actually leads 554 to slightly worse performances. The loss of performance is 555 due to the non perfect description of the driving cycle that 556 the Markov chains achieves;  The Online MC approach leads to results close to the offline 582 benchmark with an average optimization step time of 25 (min) 583 for scenario (a), a reasonable computational time for the low 584 dynamics under investigation and over a control discretization 585 step N u = 2000 (km). Indeed, 25 (min) corresponds to an av-586 erage traveled distance of 24 (km), which is negligible for aging 587 monitoring purposes. 2 Furthermore, the average computational 588 time decreases with the increment of the driving cycle average 589 speed. This is reasonable because, at each PSO optimization 590 step, the model is simulated over N p for p #,2 different configu-591 rations of the control variables DoD and I cell max . Thus, the higher 592 the average velocity the quicker the simulation. The complexity 593 introduced by the Markov-chain-based learning mechanism, 594 in terms of computational time increment with respect to the 595 Online strategy, is acceptable. Eventually, it must be noted that, 596 while the proposed online strategies are demanding in terms of 597 computational power, there is not hard real time constraints that 598 would force a local computation. As a matter of fact, given the 599 slow dynamics of battery aging, and the rising trend in intercon-600 nected vehicles, the optimal battery management strategy can be 601 computed relying on cloud services without any computational 602 power limitation. result of an optimization. If a DoD of 75% is used instead 612 of the optimal one, the gain in terms of kilometers grows 613 to 48,000 (km); 614 r The proposed strategy is also adaptive. This means that the 615 Markov chain is capable of adapting to modifications of 616 the driver's behavior, thus increasing robustness.

618
The paper proposes a strategy for battery aging management 619 for EV's. Battery aging management in electric vehicles is 620 particularly complex because it entails a modification of the 621 vehicle performance.

622
The first part of the paper introduces the main features and 623 specifics of the problem. A mixed forward/backward electric 624 vehicle model defines the main stress factors affecting battery 625 aging and with them the control variables. Subsequently, these 626 results inform the definition of a cost function that quantifies 627 battery aging along with the loss of performance. The opti-628 mization problem is first solved offline relying on PSO. Then, 629