**1. Introduction**

Due to their low operating speeds and frequent stopping and starting, buses are a prime candidate for hybridization or electrification in the goal of reducing transportation sector emissions. The stop-and-go behavior, in particular, means that regenerative braking can recover a large portion of expended power. However, for handling bus loads and ranges, the lithium ion batteries needed for electric vehicles (EVs) and hybrid-electric vehicles (HEVs) can be prohibitively expensive and heavy [1]. Additionally, the large current spikes from acceleration and deceleration can degrade the battery, reducing range, increasing energy consumption [2,3], and, in general, adding new operational costs to such vehicles.

One possible solution to this is to use a Hybrid Energy Storage System (HESS)—a combination of lithium ion energy storage with an ultracapacitor (UC) sized to handle large charge and discharge currents—in place of standard battery energy storage. In general, lithium ion batteries have a high energy density but low power density: they can store a large amount of charge, but cannot access it quickly without degrading. Specifically, large currents to and from the battery cause its capacity to fade and internal resistance to grow. High temperatures and deep discharges also contribute to battery aging. On the contrary, ultracapacitors have a low energy density and high power density [4]. A HESS, then, allows one to obtain the efficient storage of lithium batteries while allowing an ultracapacitor to handle the large currents [4,5]. The aging of UCs does not depend on current magnitude or discharge depth, rather on time, temperature, and cell voltage [6–8]; therefore, there is not necessarily a tradeoff between battery aging and UC aging in HESSs.

**Citation:** Mallon, K.; Assadian, F. A Study of Control Methodologies for the Trade-Off between Battery Aging and Energy Consumption on Electric Vehicles with Hybrid Energy Storage Systems. *Energies* **2022**, *15*, 600. https://doi.org/10.3390/en15020600

Academic Editor: Daniel-Ioan Stroe

Received: 30 November 2021 Accepted: 11 January 2022 Published: 14 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Current research on HESSs considers HEV, EV, and fuel cell vehicle applications. The bulk of the literature, for instance [9–14], is concerned with the optimal sizing of the HESS so as to maximize the cost-effectiveness of such a system. However, battery aging is often not considered directly in this optimization; instead, battery aging factors such as high temperatures and currents are minimized, rather than battery aging directly, and the benefits to overall aging are only assumed [14]. Some related works that does directly address aging are described here. In reference [15], for instance, an optimal control policy is developed to control UC behavior. This policy demonstrated clear aging improvements over an uncontrolled system using passive energy management. Reference [16] used multi-objective optimization while directly incorporating an aging model and using a rule-based control system to govern energy management for a study on HESS sizing in EVs. Reference [17], likewise, carried out a parametric study on battery degradation versus UC size in EVs, using a control system based on fuzzy logic. Reference [18] considered a HESS that used lead-acid batteries rather than lithium ion, and developed an HEV energy management strategy that tuned for battery life extension. Notably, they found that, for the HESS to be cost-effective, a 50% increase in battery cycle life was required. Reference [19] compared the aging benefits of an optimally-sized HESS to the theoretical maximum benefits—battery aging reductions with an infinitely large HESS. These benefits were experimentally verified, with the developed approach decreasing battery power fade and temperature rise in lithium-ion batteries on a vehicle load profile. Most notably, references [20–22] demonstrate an optimal control strategy to directly minimize battery aging in a HESS for a plug-in HEV.

However, the literature is lacking in direct aging control for EVs, in the impact of ultracapacitor aging in the HESS, and in methods to assess the economic benefit of the HESS given UC aging. Although studies on direct aging control for HEVs do exist, for instance [2,3,23,24], EVs pose a unique control problem due to the fewer controlled variables and different component sizes. This research fills these gaps in knowledge: new energy management strategies to control battery aging and to jointly control battery aging, ultracapacitor aging, and energy losses are developed and compared to existing methods. Then, the cost/benefit analysis of a HESS that considers ultracapacitor aging is performed, and the drawbacks of overusing the ultracapacitor are discussed.

This paper begins by developing the aging models for an electric vehicle hybrid energy storage system, with an aside showing why there must necessarily be a trade-off between battery aging and energy consumption for this vehicle configuration. Next, energy management systems (EMS) for aging control are developed, including Deterministic Dynamic Programming (DDP), Stochastic Dynamic Programming (SDP), and Load-Leveling (LL). The vehicle model and aging-aware controllers are then applied to a case study of an electric bus with a HESS, simulated for the lifespan of the battery. Finally, these simulation results are analyzed, and conclusions are drawn regarding the benefits of optimal control and aging-aware control for vehicle energy management.

#### **2. Modeling**

This section develops a model for an electric vehicle with a hybrid energy storage system. Specifically, the model is of a HESS-equipped electric bus using lithium-ion batteries for energy storage and ultracapacitor modules for handling large currents, as depicted in Figures 1 and 2. The first subsection presents the overall vehicle model, including vehicle dynamics, motor efficiencies, battery dynamics, and so on. The next two subsections deal with the battery aging and ultracapacitor aging models. The final subsection discusses why, based on the provided models, aging control can have a negative impact on fuel economy.

**Figure 1.** Powertrain of an EV with a UC.

**Figure 2.** Block diagram for energy management of an EV with a UC.

## *2.1. Vehicle Modeling*

#### 2.1.1. Vehicle Dynamics

For this study, a backwards-facing quasi-static vehicle model [25] is used to represent the vehicle dynamics. In this model, it is assumed that the driver accurately follows the velocity of a given drive cycle, eliminating the need for a driver model and allowing the time-history of the electrical power demand to be calculated in advance.

This research uses a backwards-facing quasi-static vehicle model [25] to simulate the vehicle dynamics. This method assumes that the model accurately follows a specified velocity profile, which allows for calculating the acceleration and, therefore, the electrical power request can be computed in advance, eliminating the need for a driver model.

The vehicle body is affected by inertial forces, aerodynamic drag, and rolling resistance, while gravitational forces (such as those due to driving on inclines) are neglected. The drag force is given by

$$F\_{drag} = \frac{1}{2} \rho A\_f \mathbb{C}\_D(v\_v)^2 \tag{1}$$

where *ρ*, *Af* , *CD*, and *vv* are the air density, frontal area, drag coefficient, and vehicle velocity, respectively. Rolling resistance is given by

$$F\_{roll} = M\_v \mathbb{g} \mathbb{C}\_R \tag{2}$$

where *Mv*, *g*, and *CR* are the vehicle's total mass (including components such as the engine and generator), acceleration due to gravity, and rolling resistance coefficient. In a backwards-facing model, the acceleration and the vehicle mass determine the inertial force on the vehicle as

$$F\_{inertial} = M\_{c\eta} \frac{dv\_{\upsilon}}{dt}.\tag{3}$$

*Meq* is the combined bus mass and equivalent mass due to the rotational inertia of the motor and wheels

$$M\_{c\eta} = M\_{\upsilon} + 4f\_{\text{av}} \left(\frac{1}{R\_{\text{av}}}\right)^2 + f\_m \left(\frac{N\_{fd}N\_{\text{gb}}}{R\_{\text{av}}}\right)^2,\tag{4}$$

where *Jw*, *Rw*, *Jm*, *Nf d*, and *Ngb* are the rotational inertia of a single wheel, the wheel radius, the rotational inertia of the motor, the final drive ratio, and the gearbox ratio for a single-speed gearbox, respectively. The acceleration term in (3) is approximated from a given velocity profile according to

$$\frac{dv\_v}{dt}(t) \approx \frac{v\_v(t + \Delta t) - v\_v(t - \Delta t)}{2\Delta t}.\tag{5}$$

These three forces sum together to give the tractive force on the bus.

$$F\_{transition} = F\_{inertial} + F\_{drag} + F\_{roll} \tag{6}$$

Parameter values for the vehicle model can be found in Table 1. The bus is assumed to be fully loaded and at its maximum allowable weight. The bus's physical parameters are based on the existing literature on bus simulation [26–28].

**Table 1.** Vehicle model physical parameters.


#### 2.1.2. Transmission

Next, the vehicle velocity and tractive force are equated to motor torque and angular velocity. The motor torque is given by

$$\tau\_{\rm H} = \begin{cases} \begin{pmatrix} \frac{R\_{\rm H}}{N\_{fd}N\_{gb}} F\_{\rm transition} \\ \frac{R\_{\rm H}}{N\_{fd}N\_{gb}} F\_{\rm fraction} \end{pmatrix} / \eta\_{\rm trans}, & F\_{\rm fraction} \ge 0 \\\end{pmatrix} \tag{7}$$

*ηtrans* is the transmission efficiency, represented as torque losses. The motor speed is then given by

$$
\omega\_{\rm un} = \frac{N\_{fd} N\_{\rm gb}}{R\_{\rm uv}} \upsilon\_{\rm v} \tag{8}
$$

The mechanical power *Pmech* needed to drive the vehicle is expressed in terms of the above torque and angular velocity.

$$P\_{mech} = \pi\_{\mathfrak{m}} \cdot \omega\_{\mathfrak{m}} \tag{9}$$

In this formulation, *Pmech* is positive during acceleration. Parameter values for the transmission can be found in Table 1.

## 2.1.3. Motor and Power Electronics

The motor torque and angular velocity are used to find the motor efficiency *ηmotor*, which is constrained to 0 < *ηmotor* < 1. The motor efficiency is determined from a static efficiency map from the National Renewable Energy Laboratory's Advanced Vehicle Simulator (ADVISOR) data library [29]. This efficiency includes both the motor itself as well as the associated power electronics. The bus model in this research uses a 250 kW AC induction motor.

Once the motor efficiency is found, it can be used to evaluate the driver's electrical power request, *Preq*.

$$P\_{rcq} = \begin{cases} \begin{array}{cc} P\_{mclch} / \eta\_{motor} & \tau\_{\text{m}} \ge 0 \\ P\_{mclch} \cdot \eta\_{motor} & \tau\_{\text{m}} < 0 \end{array} \tag{10}$$

The electrical power request is met with power from the battery *Pbatt* and ultracapacitor *Puc*. Given that this is a backwards-facing simulation, the power request must always be met.

$$P\_{req} = P\_{batt} + P\_{nc} \tag{11}$$

In later sections, the ultracapacitor power is developed as the energy management system's controlled variable. Then, *Pbatt* is dependent on *Preq* and *Puc*, and Equation (11) is rewritten as

$$P\_{\text{batt}} = P\_{\text{req}} - P\_{\text{nc}} \tag{12}$$

2.1.4. Battery

The EV's lithium-ion battery cells are modeled with the simple equivalent circuit shown in Figure 3, where *Vcell* is the single-cell open-circuit voltage (OCV) and *Rcell* is the single-cell equivalent series resistance [4]. The individual cells are then combined into a larger battery pack. Only one state variable is required for this model, the state of charge (SOC). The OCV and internal resistance are variable parameters dependent on SOC. The formulas for these parameters are given in [30], which develops a lithium-iron-phosphate battery from experimental data.

**Figure 3.** Battery pack equivalent circuit.

The battery pack equivalent resistance *Req* is given by

$$R\_{eq} = R\_{cell} \cdot \frac{N\_{ser}}{N\_{par}} \tag{13}$$

where *Nser* and *Npar* are the number of cells in series and in parallel, respectively. The battery pack OCV is likewise given by

$$V\_{\rm occ} = N\_{\rm ser} \cdot V\_{\rm cell} \tag{14}$$

The battery pack's terminal voltage is found from the OCV and battery power *Pbatt* using the equivalent circuit in Figure 3.

$$I\_{\rm batt} = P\_{\rm batt} / V\_T \tag{15}$$

$$V\_T = V\_{ocv} - I\_{batt} \cdot R\_{eq} \tag{16}$$

Then, substituting the current equation into the voltage equation and solving yields

$$V\_T^2 = V\_{\rm accv} \cdot V\_T - P\_{\rm batt} \cdot R\_{\rm eq} \tag{17}$$

$$V\_T = 1/2\left(V\_{\rm acc} + \sqrt{V\_{\rm acc}^2 - 4 \cdot P\_{\rm batt} \cdot R\_{\rm cq}}\right) \tag{18}$$

Substituting *VT* into Equation (15) allows the battery current to be found explicitly. Then, integrating the battery current yields the state of charge.

$$\text{SOC}(k+1) = \text{SOC}(k) + \Delta t \cdot \frac{I\_{batt}}{Q\_{batt}},\tag{19}$$

where *Qbatt* is the battery pack's charge capacity in coulombs and Δ*t* is the integration time step.

The battery model parameters are given in Table 2. The number of cells in series ensures that the battery pack, the OCV, is in line with the requirements of [31]. The number of parallel cells was chosen so that the bus can meet the power requirements in [31,32] to drive continuously on an urban bus velocity profile for four hours. Note that Table 2 only gives the nominal values for *Rcell* and *Vcell*—in reality, these parameters vary with SOC and other operating conditions [30].

**Table 2.** Battery model parameters.


### 2.1.5. Ultracapacitor

The ultracapacitor modules are modeled as the first-order equivalent circuit shown in Figure 4. The model itself is based on the 100F ultracapacitor model derived in [33]. Ultracapacitor model parameters are given in Table 3. The ultracapacitor pack, similar to the battery pack, consists of ultracapacitors arranged in *Npc* modules in a parallel set and *Nsc* sets in series. The number of modules is variable so that the effectiveness and cost-benefit of the HESS can be considered across a range of designs.

**Figure 4.** Ultracapacitor pack equivalent circuit.

**Table 3.** Ultracapacitor model parameters.


The UC pack is connected to the DC bus through a converter, as shown in Figure 1. The converter allows the UC pack to operate independently of the DC bus voltage. The

ultracapacitor pack power is indicated by *Puc*, where *Puc* is positive while discharging and negative while charging. Then, each individual module has power *Puc*,*module* given by

$$P\_{\text{uc,modulle}} = \frac{P\_{\text{uc}}}{N\_{\text{pc}}N\_{\text{sc}}} \tag{20}$$

For capacitor charge *quc* at some given power *Puc*,*module*, the current and terminal voltage *Iuc* and *VT*,*uc* are found similarly to Equations (15)–(18).

$$I\_{\rm nc} = P\_{\rm nc,module} / V\_{T,\rm nc} \tag{21}$$

$$V\_{T,uc} = q\_{uc} / \mathcal{C}\_{uc} - I\_{uc}R\_{uc} \tag{22}$$

Substituting Equation (21) into Equation (22) and solving yields

$$
\Delta V\_{T,uc}^2 = q\_{uc} / \mathbb{C}\_{uc} \cdot V\_{T,uc} - P\_{uc,module} R\_{uc} \tag{2.3}
$$

$$V\_{T,uc} = \frac{1}{2} \left( \frac{q\_{uc}}{\mathcal{C}\_{uc}} + \sqrt{\frac{q\_{uc}}{\mathcal{C}\_{uc}} - 4P\_{uc,modulle}R\_{uc}} \right) \tag{24}$$

*VT*,*uc* can then be substituted back into Equation (21) to obtain the ultracapacitor current. Then, the state equation for the capacitor is

$$
\dot{q}\_{\text{uc}} = I\_{\text{uc}} - \frac{q\_{\text{uc}}}{R\_{\text{uc}} \mathcal{C}\_{\text{uc}}} \tag{25}
$$

Then, for the complete ultracapacitor pack,

$$I\_{\rm nc,pack} = I\_{\rm nc} \cdot N\_{\rm pc} \tag{26}$$

$$V\_{T,uc,puck} = V\_{T,uc} \cdot N\_{\&} \tag{27}$$

$$R\_{\rm nc,pack} = R\_{\rm nc} \cdot \frac{N\_{\rm sc}}{N\_{\rm pc}} \tag{28}$$

where *Iuc*,*pack* is the total current going to or from the UC pack, *VT*,*uc* is the terminal voltage of the overall UC pack, and *Ruc*,*pack* is the equivalent series resistance of the entire pack.
