*2.2. Battery Aging Model*

This research uses the cycle-life aging model presented in Reference [34], and develops it here into an aging model that can be used in dynamic control of the energy storage systems. Reference [34] models the cycle life of a battery as a function of depth of discharge *DoD*, charging current *Ic*, discharging current *Id*, and temperature *T*.

$$CL = \lg(DoD, I\_{c\prime}, I\_{d\prime}, T) \tag{29}$$

This model starts as a simple curve fit of cycle life to depth-of-discharge at a reference point of *Ic* = 1 *C*, *Id* = 1 *C*, *T* = 25 ◦C . This baseline cycle life is denoted as *CLDoD*. Then,

$$CL\_{DoD} = a\_1 e^{a\_2 \cdot DoD} + a\_3 e^{a\_4 \cdot DoD} \tag{30}$$

where the *ai* terms are curve fit parameters. The cycle life is then obtained by modifying *CLDoD* based on the actual operating *Ic*, *Id*, and *T*.

$$CL = CL\_{D \bullet D} \cdot A\_{I\_d} \cdot A\_{I\_c} \cdot A\_T \tag{31}$$

where

$$A\_{I\_d} = \frac{a\_5 e^{a\_6 \cdot I\_d} + a\_7 e^{a\_8 \cdot I\_d}}{a\_5 e^{a\_6} + a\_7 e^{a\_8}}\tag{32}$$

$$A\_{I\_{\varepsilon}} = \frac{a\_{\mathcal{O}}e^{a\_{10} \cdot I\_{\varepsilon}} + a\_{11}e^{a\_{12} \cdot I\_{\varepsilon}}}{a\_{\mathcal{O}}e^{a\_{10}} + a\_{11}e^{a\_{12}}} \tag{33}$$

$$A\_T = \frac{a\_{13}T^3 + a\_{14}T^2 + a\_{15}T + a\_{16}}{25^3a\_{13} + 25^2a\_{14} + 25a\_{15} + a\_{16}}\tag{34}$$

where the *ai* terms are, again, curve fit parameters. The *a*5–*a*<sup>8</sup> parameters are found from a curve fit of cycle life to varying *Id* for *Ic* = 1 *C*, *T* = 25◦, and *DoD* = 100%. The *a*9–*a*<sup>12</sup> parameters are found from a curve fit of cycle life to varying *Ic* for *Id* = 1 *C*, *T* = 25◦, and *DoD* = 100%. The *a*13–*a*<sup>16</sup> parameters are found from a curve fit of cycle life to varying *T* for *Id* = 1*C*, *Ic* = 1 *C*, and *DoD* = 100%.

The cycle life model in [34] assumes uniform charge and discharge cycles over the life of the battery. The Palmgren–Miner (PM) rule can be used, then, to handle the non-uniform cycles of vehicle operation. This method, originally developed for analyzing material fatigue life, has been shown to effectively approximate the battery health over non-uniform charge and discharge cycles [35–37]. Under the assumptions of this method, each charge and discharge cycle damages the battery an amount equal to the inverse of the cycle life at that cycle's operating conditions. In other words, if we assume a cycle *k* with depth of discharge *DoDk*, charge current magnitude *Ic*,*k*, discharge current magnitude *Id*,*k*, and temperature *Tk*, then the cycle life for these operating conditions is *CLk*. Under the PM rule, this cycle damages the battery an amount *Dk* given by

$$D\_k = 1/\mathbb{C}L\_k \tag{35}$$

Damage accumulates linearly for each charge and discharge cycle. Therefore, the damage from each individual cycle can be summed to find the total damage. The total damage *Dtot* through the *k*-th cycle is therefore

$$D\_{tot}(k) = \sum\_{i=1}^{k} D\_i \tag{36}$$

where each *Di* represents the damage from a single cycle with operating conditions *DoDi*, *Ic*,*i*, *Id*,*i*, and *Ti*. In this way, the damage of individual cycles with unique operating conditions is summed to obtain a total measure of battery health. Zero total damage indicates that the battery is at its beginning of life, while total damage of one indicates the battery's end of life. Battery end-of-life corresponds to a 20% capacity fade, therefore the capacity fade *CF* can be put in terms of the damage as

$$CF(k) = 0.2 \cdot D\_{tot}(k)\tag{37}$$

The above method requires full knowledge of the charge and discharge time histories, which is not practical for use in energy management; the EMS must act at a much faster rate than the pace at which these cycles develop. It is possible, however, that the EMS could determine how a control decision might cause the damage from the current cycle to lessen or grow. For instance, imagine a battery operating at conditions of *DoDj*, *Ic*,*j*, *Id*,*j*, *Tj*. Then, let the energy management system make some decision that produces new operating conditions of *DoDk*, *Ic*,*k*, *Id*,*k*, *Tk*. Using Equations (29) and (35), the change in damage Δ*D* due to the EMS's decision can be computed as

$$
\Delta D = D\_k - D\_{\dot{j}} = \frac{1}{\mathcal{g}(DoD\_{k\prime}I\_{c,k\prime}I\_{d,k\prime}T\_k)} - \frac{1}{\mathcal{g}(DoD\_{\dot{j}\prime}I\_{c,\dot{j}\prime}I\_{d,\dot{j}\prime}T\_{\dot{j}})} \tag{38}
$$

Then, an energy management strategy could incorporate Equation (38) for a measure of potential battery damage. In this way, the strategy would try to minimize the damage from the control decision made at each time step. Note that, when controlling aging in this manner, the EMS can only be aware of the *DoD* up until the current point in time and can only assess damage relative to the current *DoD*, while the "true" aging depends on the size of the completed cycle. Despite this discrepancy, this method still proves an effective way to control battery aging, as will be shown in later sections.

The resistance growth model in Reference [34] can be treated in an identical manner to Equations (29)–(38). The capacity fade and resistance growth models both use rainflow counting, as in [38], to determine the aging from the irregular cycling operations experienced while a vehicle is in operation. For simplicity, it is assumed that the battery operates at a constant internal temperature of 35 ◦C.

#### *2.3. Ultracapacitor Aging*

A novel aspect of this research is that ultracapacitor aging is considered in addition to battery aging. Reference [6] provides the following model for ultracapacitor aging. This model is based on Eyring's Law, a chemical rate equation which gives an ultracapacitor lifespan based on the operating voltage and internal temperature where aging increases exponentially as the voltage and temperature increase. Then, the aging rate at an instance in time is based on the inverse of the lifespan at the given operating conditions. In this model, *SoA* is the state of aging that characterizes both capacitance fade and resistance growth, where a *SoA* of 0 indicates start-of-life and of 1 indicates end-of-life.

$$\frac{dSoA}{dt} = \frac{1}{T\_{life}^{ref}} \cdot \exp\left(\ln(2)\frac{\theta\_{\varepsilon} - \theta\_{\varepsilon}^{ref}}{\theta\_0}\right) \cdot \left(\exp\left(\ln(2)\frac{V - V^{ref}}{V\_0}\right) + K\right) \tag{39}$$

where *θ<sup>c</sup>* and *V* are the UC temperature and voltage, respectively, and the remaining variables (*Tref lif e*, *θ ref <sup>c</sup>* , *θ*0, *Vref* , *V*0, and *K*) are parameters fitted to experimental data. Then, from [6], the instantaneous capacitance *Cuc* and internal resistance *Ruc* are given by

$$\mathcal{C}\_{\rm nc} = \mathcal{C}\_{\rm nc;0} \times (0.95 - 0.15 \cdot SoC) \tag{40}$$

$$R\_{\rm uc} = R\_{\rm uc,0} \times \left(1 - 0.3 \cdot SaA\right)^{-1} \tag{41}$$

where *Cuc*,0 and *Ruc*,0 are the initial values of *Cuc* and *Ruc*.

This model can be used in control without modification. The standard end-of-life conditions for ultracapacitors are defined similarly to batteries: when the UC capacitance has faded by 20% [6]. It is assumed that the ultracapacitor operates at a constant internal temperature of 55 ◦C, estimated from the operating conditions found in [39].
