Vehicle physical parameters.
During the lifetime of an energy storage system, its health deteriorates from use due to irreversible internal changes to the system. This degradation results in decreased capacity and efficiency of the battery or capacitor. This chapter reviews empirical aging models for lithium-ion battery and ultracapacitor energy storage systems. It will explore how operating conditions like large currents, high temperature, or deep discharge cycles impact the health of the energy storage system. After reviewing aging models, this chapter will then show how these models can be used in vehicle energy management control systems to reduce energy storage system aging. This includes both aging-aware control and control of hybrid energy storage systems (systems that include both a battery and an ultracapacitor).
- electric vehicle
- hybrid vehicle
- energy management
- lithium ion
- battery aging
The internal combustion engine is a major contributor to greenhouse gas emissions and hydrocarbon pollution across the globe. Motor vehicles account for a major portion of pollutants such as carbon monoxide, nitrogen oxide, and volatile organic compounds . Alternative powertrain vehicles (APVs), such as electric vehicles (EVs) and hybrid-electric vehicles (HEVs), are potential technological solutions to reduce transportation-sector emissions and fuel consumption. However, APVs require large amounts of battery-stored energy, which can be cost and weight prohibitive . Degradation of the battery further adds to the lifetime cost of an APV, and battery degradation rate has been shown to be inversely correlated with fuel economy [3, 4]. Technologies that improve battery lifespan and fuel economy will reduce this lifetime cost and hasten the adoption of sustainable transportation.
Lithium-based batteries serve as the current main battery of choice for vehicle transportation because of their high energy density and ability for high cycle life. Improving cycle life of lithium batteries means limiting large currents in and out of the battery as much as possible to lower degradation and heat affects. Ultracapacitors (UC) can be added to vehicles to improve battery life by taking excess power away from the battery and storing it in temporary energy storage . Capacitors can quickly unload power back into the system for high load situations such as a hard acceleration, taking away the need for a high-power drain from the battery.
This chapter will begin with a brief review of existing literature on empirical modeling of lithium-ion battery and ultracapacitor degradation. Then, a few select aging models will be reoriented for use in an APV energy management system (EMS). Finally, an example showing how to utilize these control-oriented models will be shown.
2. Energy storage aging review
2.1 Lithium ion battery aging
Aging of batteries is primarily caused by the formation of substrates in the chemical reaction pathways and the formation of cracks in the electrode materials from repeated stress cycles . These aging mechanisms are accelerated by high charge and discharge rates, extreme battery temperatures, and deep depths of discharge . Aging of the battery causes capacity fade (a decrease in the charge storage capacity) and power fade (a decrease in the battery efficiency). However, models of the cell chemistry that include the thermal and stress/strain relationships used to describe aging are computationally intensive and are ill-suited for use in APV EMSs [6, 8].
Research of battery aging in APVs instead tends to utilize empirical models [4, 9, 10, 11, 12, 13, 14]. Using empirical aging models for vehicle battery degradation analysis provides a good trade-off between precision and complexity. These empirical models do not consider the physical or chemical processes of the battery degradation but instead approximate the battery’s health by fitting experimental data to aging factors like charge throughput, calendar life, and number of charge/discharge cycles.
For instance, Refs. [9, 10, 15, 16] develop aging models that relate charge throughput to degradation, with temperature and current magnitude as additional stress factors. Refs. [17, 18] include depth of discharge as an additional stress factor, while  also distinguishes the impact of charging and discharging currents on battery degradation. The aging models for hybrid vehicle applications in [13, 14] consider a number of charge/discharge cycles and calendar life and use temperature, depth of discharge, and average state of charge as aging stress factors. Other models in the literature such as [8, 19, 20] use simple cycle counting to measure the state of health.
Current research works to integrate battery aging dynamics into these EMSs to form controllers that actively regulate battery degradation. In Ref. , the authors developed an SDP-based EMS for a parallel-HEV passenger vehicle that accounted for battery wear by mapping operating conditions to substrate growth, and associating substrate growth with battery state of health. The authors also analyzed how reducing battery aging increased the fuel consumption. In Refs. [4, 21], the authors developed a deterministic EMS for a parallel-HEV passenger vehicle that regulates battery degradation using a “severity factor” map: the control policy penalizes battery usage by an amount related to the severity of the operating conditions (in terms of temperature and current magnitude). The authors of  also showed an inverse correlation between the battery aging and fuel consumption.
2.2 Ultracapacitor aging
Lithium batteries have a high energy density but low power density, meaning that although they store large amounts of energy, that energy cannot be accessed quickly. Additionally, high currents to and from the battery are a stress factor for battery degradation. A potential solution to these problems is to integrate UCs into the energy storage system. UCs store energy in the electric field of an electrochemical double layer and have a high power density but low energy, allowing them to serve as complements to battery energy storage . By integrating UCs into the powertrain, it becomes possible to meet the vehicle power requirements with a smaller battery and reduce battery degradation by restricting the magnitude of the current going to or from the battery [5, 22]. Aging of UCs is primarily dependent on time, temperature, and cell voltage [23, 24, 25].
Current research is interested in optimal control and sizing of the UC to reduce battery aging , and in particular how battery aging and fuel economy are jointly impacted. Some related work includes Ref. , in which the authors develop an optimal control policy to govern UC behavior and demonstrate clear aging improvements over a passive (uncontrolled) system. Refs. [28, 29] carried out a parametric study on battery degradation versus UC size in EVs, using a rule-based control system to govern power allocation. Ref.  developed a control strategy integrating UCs with lead-acid batteries in a HEV for battery life extension, and found that a 50% increase in battery cycle life would be needed for the UC to be cost-effective. Ref.  experimentally demonstrated a decrease in battery power fade and temperature rise in lithium-ion batteries due to UCs on an EV load profile.
3. Control-oriented aging
Meeting this goal of mitigating energy storage system degradation in APVs through control requires forming simplified models of the battery aging dynamics that can be included in or be used to generate an EMS. This chapter will summarize several approaches in the literature for using energy storage aging models for control applications.
3.1 Power-law model
Ref.  developed a lithium-ion battery empirical aging model for normalized battery capacity loss Qloss, based on an Arrhenius equation. The model uses experimental data to relate battery degradation to on charge throughput Ah (in ampere-hours), current Ib (in C), and temperature T (in K).
Ea, B, and z are fitted parameters, while A(Ib) is a fitted function of current. R is the ideal gas constant. Here, Qloss = 0 indicates a new battery, while Qloss = 0.2, for example, indicates a 20% decrease in the storage capacity. This model treats current and temperature as static values. So, Eq. (1) can be differentiated to form a dynamic aging model as follows:
3.2 Cycle life model
The Palmgren-Miner (PM) rule is a common method for analyzing fatigue life in mechanical systems and has been shown to effectively approximate the battery health over nonuniform charge and discharge cycles [8, 32, 33]. As per the PM rule, each charge and discharge cycle is considered to damage the battery by an amount related to the cycle life at that cycle’s depth of discharge, charge and discharge current, and temperature. Ref. , for instance, models the cycle life of a battery as a function of depth of discharge , charging current , discharging current , and temperature .
Assume there is a charge-discharge cycle with operating conditions , , , and , and so the cycle life for these operating conditions is . Then, under the PM rule, the damage from cycle is assumed to be
For multiple charge and discharge cycles, the damage from each cycle can be added to find the total damage . For the total damage up to cycle ,
Cumulative damage of zero denotes that the battery is unaged while cumulative damage of one means the battery has reached the end of its life. Typically, 20% capacity fade indicates a battery’s end of life. So,
However, the above method does not readily lend itself to use in control; full charge and discharge cycles can take a long time to develop, and the EMS must act at a faster rate. One possibility is that the energy management system could consider how its decision would cause the damage of the current cycle to grow or lessen. For instance, consider a battery to be at operating conditions , , , and . The EMS then makes some decision such that the operating conditions become , , , and . Using Eqs. (3) and (4), the change in damage due to the EMS’s decision can be calculated as
Then, Eq. (7) could be used in formulating an energy management strategy, such that the EMS would seek to minimize the additional damage caused by each decision it makes.
3.3 Ultracapacitor aging
Ref.  provides the following model for ultracapacitor aging, where is the state of aging where 0 indicates start of life and 1 indicates end of life.
where is the UC temperature, is the voltage, and the remaining variables (, , , , , and ) are experimentally fitted parameters. This model is ready to be used for control as is. Ref.  defines the UC end-of-life condition as similar to batteries: when the capacitance of the UC has faded by 20%.
4. Case study: electric vehicle with hybrid energy storage
This section develops a model for a hybrid energy storage system electric vehicle (HESS-EV)—specifically, an electric bus that uses a lithium-ion battery pack for energy storage and an ultracapacitor pack for handling large power requests. This example study will be used to show how active control of aging factors can improve the lifespan of the energy storage system without compromising energy consumption. This system is depicted in Figures 1 and 2.
4.1 Vehicle dynamics
For this study, a backward-facing quasi-static vehicle model  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 load placed on the powertrain to be calculated in advance.
This vehicle model, illustrated in Figure 3, considers inertial forces, aerodynamic drag, and rolling resistance (note that road incline is neglected for this chapter). The drag force is given by
where is the air density, is the frontal area, is the drag coefficient, and is the vehicle velocity. Rolling resistance is given by
where is the vehicle’s total mass, is the acceleration due to gravity, and is the rolling resistance coefficient. In a backward-facing model, the inertial force is determined from the vehicle acceleration and the vehicle mass as
is the mass of the bus plus the equivalent mass due to the rotational inertia of the motor and wheels.
where is the rotational inertia of one wheel, is the rotational inertia of the motor, is the wheel radius, is the final drive ratio, and is the gearbox ratio. The acceleration term in Eq. (11) is approximated from a given velocity profile according to
The inertial, drag, and rolling resistance forces sum together to give the traction force on the bus.
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. Vehicle parameters are estimated from existing literature on bus simulation [35, 36, 37].
|Vehicle mass||Mv||18,181 kg|
|Frontal area||Af||8.02 m2|
|Roll resistance coefficient||CR||0.008|
|Wheel inertia||Jw||20.52 kg-m2|
|Motor inertia||Jm||0.277 kg-m2|
|Wheel radius||Rw||0.48 m|
|Final drive ratio||Nfd||5.1:1|
4.2 Powertrain model
This subsection describes the modeling of the HESS-EV powertrain, including the transmission, motor, battery, and ultracapacitor subsystems, as indicated in Figure 1. The goal of the vehicle model is to capture the primary forces on the vehicle while maintaining model simplicity. Both these make simulation of the system easier and make optimal control methods, such as dynamic programming or model-predictive control, less computationally complex. Otherwise, the energy management system might suffer from the “curse of dimensionality”.
Next, the vehicle speed and traction force are transformed into motor torque and motor speed. Assuming transmission efficiency of , represented as torque losses, the motor torque is given by
and the motor speed is given by
Then, the mechanical power needed to drive the vehicle can be expressed in terms of the motor torque and angular velocity.
Here, positive indicates acceleration. Parameter values for the transmission can be found in Table 1.
4.2.2 Motor and power electronics
The electrical power demand of the motor, , is calculated from Pmech and an efficiency parameter ηmotor. ηmotor is a function of and and is determined from a static efficiency map.
The efficiency map is obtained from the National Renewable Energy Laboratory’s Advanced Vehicle Simulator (ADVISOR) data library  and scaled to the appropriate size using the scaling method in . It includes both the motor efficiency and the efficiency of the power electronics. The modeled vehicle utilizes a 250 kW AC induction motor.
The power demand for the electric motor is provided by battery power and ultracapacitor power . As part of the quasi-static simulation, it is assumed that the power demand is always met.
Because is set by the drive cycle and is a controlled variable, is fixed and dependent on both and . Therefore, Eq. (19) can be rewritten as
4.3 Energy storage systems
The previous subsections detailed how the driver’s electrical power request would be determined. As depicted in Figure 4, the EMS decides how that power is split between the lithium-ion battery and the ultracapacitor. This subsection will first detail the modeling of the battery, followed by modeling of the ultracapacitor.
This HESS-EV uses lithium-ion batteries represented by the simple battery model shown in Figure 5, where is the open-circuit voltage (OCV) of a single battery cell, while represents the combined effects of ohmic resistances, diffusion resistances, and charge-transfer resistances . This quasistatic model requires only a single state variable, state of charge (SOC). The OCV as well as the resistance are considered to vary with SOC per experimental data for a lithium-iron-phosphate battery .
The equivalent resistance of the complete battery pack is given by
where is the number of cells in series, is the number in parallel, and is the resistance of a single cell. The open circuit voltage of the vehicle battery pack is likewise given by
Using the equivalent circuit in Figure 4, the battery terminal voltage can be found from the OCV and battery power as follows:
can then be substituted back into Eq. (23) to obtain the battery current, which can be integrated to obtain the state of charge.
where is the capacity of the battery pack in coulombs and is the timestep.
The parameters for the battery model can be found in Table 2. The number of cells in series was chosen so that the OCV would be in line with the recommendations in . The number of cells in parallel was chosen so that the bus can be driven for 4 hours continuously to meet the power requirements of [40, 41].
|Battery cells in parallel||Npar||400 cells|
|Parallel sets in series||Nser||100 sets|
|Total charge capacity||Qbatt||340 Ah|
For this example, the cycle counting method described in Eqs. (3)–(7) is used, with the aging model in Ref.  used to determine cycle life. This is illustrated in Figure 6. The battery is assumed to operate at a constant C.
The ultracapacitor model is similar in nature to the battery model, so the dynamics here will be presented more briefly. For this study, a second-order equivalent circuit based on the 100F ultracapacitor model in  is used to model the individual ultracapacitors. Parameters for this model are given in Table 3. Like with the battery, the ultracapacitor pack consists of ultracapacitors arranged in series and parallel as shown in Figure 7.
|UC parallel cells||Npc||100|
|UC series sets||Nsc||100|
|Resistor 1||R1||29.6 mΩ|
|Capacitor 1||C1||31.7 F|
|Resistor 2||R2||14.7 mΩ|
|Capacitor 2||C2||74.1781 F|
As shown in Figure 1, the UC is connected to the DC bus through a converter, so that the voltage of the UC pack is independent of the voltage at the DC bus. The ultracapacitor pack takes on total power and has cells in parallel per set and sets of cells in series. Then, the power going to each individual cell is
Let be the charge that corresponds to the capacitor, and let be the charge that corresponds to the capacitor. For a given power , the current and terminal voltage can be found in a similar manner to Eqs. (23)–(26).
can then be substituted back into Eq. (23) to obtain the battery current. Then, the state equations for the two capacitors are
The total charge in the capacitor is given by
4.4 Energy management system
A deterministic dynamic programming (DDP) controller is used for the EMS. DDP is a form of optimal control—in this example, the DDP controller solves the following optimization problem over a known velocity profile that is steps long. For this system, the minimum, target, and maximum charge are , , and , and the minimum and maximum UC power are and so as not to exceed manufacturer specified operating conditions . Note that these are constraints on each cell, not the entire pack. The battery power constraints will be discussed shortly.
In plain terms, the DDP controller finds how to split power between the battery and ultracapacitor in such a way as to
Keep the UC charge near a target value
Minimize the aging of the battery
Ensure that the UC charge, UC power, and battery power stays within given bounds
The method to solve DDP problems can be found in Ref. .
In order to demonstrate the benefit of actively controlling aging, two versions of the controller will be tested:
Load-leveling DDP: is set to zero. A battery power constraint of and per cell prevents large power (and therefore large current) going to the battery, and the cost function will bring the UC charge back to the target afterwards.
Active Aging Control: Battery power is unconstrained, but battery damage is directly penalized. A range of values are used for .
The HESS-EV is simulated on the Manhattan Bus Cycle —an urban bus velocity profile—for 4 hours at a time. After each simulation, the aging for the battery and ultracapacitor is measured. The battery capacity and ultracapacitor capacitances are then updated, and the next simulation begins. This process is repeated until the battery reaches its end of life.
The lifespan of the battery in years is estimated by measuring the total kilometers driven before the battery reached the end of its life, and then using the Federal Highway Administration’s average annual kilometers driven by transit busses  to convert the kilometers driven into an approximate number of years. Additionally, the ultracapacitor degradation and average kWh/km for the HESS-EV over the lifespan of the battery are measured.
Figure 9 shows a comparison between battery lifespan and fuel economy for both controller types. Clearly, the aging-aware control outperforms the load-leveling type controller: In all cases, the battery with actively-controlled aging has a longer lifespan.
Additionally, this figure shows a trade-off between efficiency and battery lifespan: As the battery lifespan increases with greater penalties on battery damage, the energy efficiency of the vehicle drops. This is because greater penalties on aging cause more current to pass through UC in order to reduce the load on the battery. In turn, there are more power losses due to the internal resistances of the capacitor pack. This is illustrated in Figure 10, which shows the charge in the UC for two different values of . One can see how a small penalty on means the controller will focus mostly on keeping the UC charge near the target value; this in turn means less current through the UC, so less power losses from the UCs.
Figure 11 shows the relationship between battery degradation and ultracapacitor degradation. Two things are apparent: one, there is an inverse relationship between the two—increasing battery lifespan comes at the cost of reducing ultracapacitor lifespan. A cause of this can be observed in Figure 10. Second, ultracapacitor degradation happens much more slowly than battery degradation. Despite the battery reaching the end of its lifespan, the ultracapacitor ages no more than 6–11%.
Deterioration of energy storage systems is inevitable, but by understanding the process it becomes possible to control and slow the capacity and efficiency fade. This chapter covered empirical aging models for lithium-ion and ultracapacitor systems and their use in vehicle energy management. First, existing work on different lithium ion and ultracapacitor aging models was reviewed, as well as those models’ application in energy management control strategies. After reviewing aging models and discussing how to adapt empirical aging models for control, a case study was carried out on an ultracapacitor-augmented electric vehicle to show how actively controlling aging can improve an EMS. This case study included the steps necessary to model the vehicle and powertrain dynamics as well as simple or quasistatic models of the battery and ultracapacitor. DDP was generally used in two types of controllers: a load-leveling type controller that was unaware of aging dynamics, and a “smart” controller that incorporated battery aging dynamics into its design. When simulated, the aging-aware controller outperformed the simple controller, offering longer battery lifespan without any cost in fuel economy or vehicle performance. This demonstrates how advanced control—making EMSs aware of energy storage aging dynamics—can improve the efficiency and viability of alternative powertrain vehicles.
This work was supported by the University of California, Davis and by the Vertically Integrated Projects (VIP) program at UC Davis.
Conflict of interest
The authors declare no conflict of interest.