Improvements to Temperature, Warburg Impedance, and Voltage Computations for a Design-Based Predictive Model for Lithium-Ion Capacitors

An earlier study manipulated the Butler-Volmer equation to effectively model a lithium-ion capacitor’s (LIC) energy storage as a function of its constituent components and charge current. However, this model had several shortcomings: computed temperature values were too low, voltage was inaccurate, and the model required Warburg impedance values that were two orders of magnitude higher than experimental results. This study began by analyzing the model’s temperature and voltage computations in order to justify output values. Ultimately, these justifications failed. Therefore, in situ temperature rise was measured during charge cycles. Experimental results indicated that temperature increases minimally during a charge cycle (·kg-1) temperature increase is negligible. After it was found that LIC temperature change is minimal during a charge cycle, the model accurately computed LIC voltage during the charge cycle and computed Warburg impedance that agreed with values derived from earlier experimental studies, even falling within the measurements’ precision error.


Introduction
Renewable energy and electrification of the energy sector have surged in popularity as the environmental, economic, and sustainability concerns associated with fossil fuels have called for alternative energy sources. Because renewable energy sources are intermittent and relatively unpredictable, their wholesale integration requires substantial energy storage infrastructure. Additionally, an electric or hybrid electric vehicle requires energy storage. These factors have caused energy storage technology to also see significant attention.
Lithium-ion batteries (LIB) are the most common type of electrochemical energy storage device under development and in use. LIBs are characterized by high specific energy (150 -200 W•h•kg −1 ) but low specific power (often below 1 kW•kg −1 ) and relatively low cycle lives (2000 cycles for lithium-sulfur, 5000 cycles for lithium titanium dioxide, often lower for other chemistries) [1] [2]. Lithium-ion capacitors (LIC) are a variation of LIB technology that incorporates many aspects of electrochemical double-layer capacitor (EDLC, aka supercapacitor or ultracapacitor) technology. LICs can simultaneously exploit the Faradaic reaction characteristic of a LIB and the non-Faradaic reaction characteristic of an EDLC [3]. Thus LICs are characterized by high specific power (10 kW•kg −1 ), long cycle life (e.g. 300,000 cycles in laboratory experiments at General Capacitor) [4] and low specific energy (5 -10 Wh•kg −1 ) [5]. The advantage of LICs is their ability to exploit the high specific energy of LIBs, the high specific power of EDLCs, and the long cycle life of EDLCs.
Often customers seeking energy storage solutions do not base their products on available energy storage technology but have a product under development for which they seek a niche solution. Usually, when a customer considers LIC technology for an application a cell must be prototyped for evaluation, including designing a new form factor and estimating its energy storage, power, and other performance characteristics. There is often trial and error in this process. Building and testing prototypes can be time-consuming, expensive, and therefore a substantial barrier to market entry. Therefore, the LIC industry needs a predictive model that estimates a LIC's performance based upon its design parameters and operating conditions. Recent research has developed models that can describe both LIB and EDLC voltage and energy storage dynamics as a function of charge or discharge (dis/charge) current and time [6] [7]. But these models describe the performance of an existing energy storage device inside a system as a function of system performance.
Few inroads have been made to dynamically predict LIC energy storage or voltage a function of the electrochemical interactions between the materials that comprise the constituent components of the device. Voltage has been demonstrated as a tool for estimating state of charge in LIBs, while studying their heat generation during dis/charge [8]. And models have been developed to dynamically describe energy storage and dynamic voltage in EDLCs [9]. This EDLC model relies upon the framework of an equivalent circuit model (ECM), like the present study and its antecedent. However, it assumes known impedances and capacitances in the device and does not compute these values from the properties of the constituent components, as the present model does. Because LICs exhibit  [12]. The Randles ECM attempts to describe an electrochemical device from data on the flow of electricity between its electrolyte and electrodes. This data is usually collected via electrochemical impedance spectroscopy (EIS) and can be interpreted to describe the energy storage device in terms of a series resistance (R S ), a double layer capacitance (C dl ), a charge transfer resistance (R ct ), and a capacitive element with its own resistance, referred to as the Warburg element (R W ) [13] [14] [15]. A basic Randles ECM topology is shown in Figure 1. R W can be represented in the time domain as shown in Figure 2 [4] [10].
The earliest LIC modeling studies considered LIC performance as part of a larger system. In 2013 Omar et al. [16]   There are three Ohmic resistors, denoted R electrode from the electrodes and R connection from the hardware connecting the energy storage device to the external circuit. These may be combined to a single variable R s . The loop between G and the energy storage device induces some inductance (L connection ), although L connection often has a small order of magnitude and is ignored. PE electrode is an assumed ideal voltage source. The movement of electrical charge in the electrolyte is described by R W , R ct , and C dl .  [27]. Because LICs employ reactions characteristic of both LIBs and EDLCs, modeling research in both of these fields may apply to a LIC model. Of particular concern in LICs is how their internal resistance decreases at high temperatures [21] and that LICs exhibit an inverse relationship between charge and discharge power and energy [23].

The Previous Version of the Model
This study is preceded by another that developed a model to predict LIC performance as a function of constituent components [15].
The Butler-Volmer temperature increase and the LIC's exchange current, in turn, affected the capacitance of the LIC as follows where C W is the Warburg capacitance, c i is the ionic concentration of the electrolyte in moles per kilogram of electrolyte, A s is the surface area, and l is the thickness of the electrode's active layer [3]. Once these relationships were understood, the model was able to predict energy storage as a function of charge power. Additionally, the model was able to estimate LIC voltage and therefore state of charge.
This model computed energy storage within 4% accuracy over a wide range of charge powers, as shown in Figure 3. Figure 3. Chart comparing experimental and modeled LIC energy storage over a wide range of charge powers, ranging from 0.3 -120 A. Notice that for a LIC energy storage varies as much as 300%. Also notice that energy storage increases slightly at very higher power [10].
where i is current and A is the surface area, Equation (3) implies Many efforts have been applied to model known energy storage devices as a function of temperature and other variables (e.g. [16] [29]), but this is the study and its predecessor are the first attempts to predict a LIC's performance as a function of its constituent components. Another potential concern is hysteresis, as temperature effects on a device may not be observed immediately [30]. A cornerstone of this study is Uno and Tanaka's previous work demonstrating that elevated temperature in LICs degrades them in a similar manner as elevated current does [5]. This relationship and its implications are a cornerstone of this study.

Deficiencies in the Previous Version of the Model
Voltage in the model should be computed as a sum of all voltage drops across major RC elements in the Randles ECM depicted in Figure 2. Voltage drops across all series (V S ), double layer (V dl//CT ), and Warburg (V W ) elements are given by Equations (7)-(9), respectively.
And Equation (9) is rewritten for every branch, n, of the equivalent R W elements as follows So that The total voltage increase, V T , can be expressed by Reinvestigation identified some discrepancies in temperature computations.
The temperature was computed from the model's computed capacitance (T mod ) and from experimental energy storage (T exp ), using Equation (4) and a modification of the relationship between capacitance and energy where E is energy stored, C is capacitance, and ΔV is the change in voltage. In this situation C will be replaced by C T , a total capacitance variable, which includes both C W and a non-temperature dependent double-layer capacitance (C dl ) as follows where ρ is the density of the electrode material, M is the mass of the active material, ε is the porosity of the electrode material, l is the thickness of the electrode, and l S is the thickness of the separator material between the electrodes. These results showed moderate agreement between T mod and T exp , except at high power, as shown in Figure 4. However, all tests were performed in a room with an ambient temperature of 300 K. Thus the low power experimental temperature values and all model temperature computations, which hovered near 150 K, were suspect and called for a reassessment of the model's computations.
In order to understand how T a impacts V, a fit equation was computed from T exp in order to determine T as a function of P as follows 3 2 0.0006 0.0617 3.1543 131.0 Use of this equation improved V calculations as shown in Figure 5. Of note, improvements in the accuracy of T a improve the model's computation of V. However, improving T a does not eliminate all error, especially at low power.

Abbreviations and Acronyms
This data indicated that error in T computations only partially addressed the error in V. Equation (2), used to compute T a , computes T a as a function of η, i d , and i o , but is unaffected by R W . By contrast, T exp relies upon Equation (3), which considers R W . Subsequent experiments reverted to computing T a using Equation (2) and studied R W 's effects upon the circuit because R W had previously received little attention. Upon examination it was hypothesized that because R W is divided into four smaller RC circuits, each of which has its own impact on temperature (Figure 2), T exp may have been misinterpreted as a temperature change In order for this to be accurate, R W had to be adjusted to change with charge current. If changes in the model's R W input led to matches in experimental and modeled E and V, and if Equation (17) is correct, then the R W value may be computed as the only missing variable. Under this assumption R W showed a predictable relationship with respect to charge power, but this relationship broke down around 30 W, spiking at 60 W, as shown in Figure 6.  values. These values are two orders of magnitude higher than those reflected in the previous study's EIS experiments and were eventually determined to be a major source of the error shown in Figure 4 and Figure 5.
This relationship can be approximated as The reason for this spike in R W at 60 W is probably the same as the culprit for C T 's increases at very high power. This phenomenon of increasing C T is probably due to electrolyte breaking down at high temperature and increasing the number of ions that could carry charge inside the LIC [10] [28]. An increase in C T can be mathematically accommodated by an increase in R W in accordance with Equations (3) and (12). Earlier studies indicate R W increases with temperature, but these studies employed low charge currents (0.2 mA/cm 2 ) and studied relatively low ambient temperatures (−20˚C -+70˚C) [31], so they may have not incorporated many of the capacitive aspects of a LIC. By contrast, at 30 W the charge current is approximately 360 mA/cm 2 and the T variable has increased by 30% over its values below 10 W. Experimentalists do not yet understand these high power dynamics. Below 30 W charge power, R W can be approximated as These R W values identified in Figure 6 and explained in Equation (18) and Equation (19) all rendered less than 1% error in voltage. They also yielded good agreement between experimentally-computed temperature (T exp ) and temperature computed from the model's C W (T check ). However, as shown in Figure 7, T exp and T check did not agree with T in accordance with Equation (17). In order to verify T a 's accuracy, temperature increase was computed across each of the four R W C W elements in series (R 1 C 1 , R 2 C 2 , R 3 C 3 , R 4 C 4 ) using Equation (17) and R ct C dl using a modified Equation (2), as follows   Figure 6 to be accurate T check and T exp must agree with T. Because they do not, serious doubt is cast upon the R W values used.
This temperature increase was designated T check2 . T check2 and T check must obey Equation (17) in order to be valid. R ct C dl 's contribution to T a is negligible at all charge powers. For example, at 60 W, where the temperature was computed to be highest, 8 5 0.09 1 96485 7.6 10 7.9 10 8.314 Consequently, the following approximation is valid for T check2 T mod , T check , T exp , and T check2 are compared in Figure 8. T check2 is four times T check at all charge powers, satisfying Equation (17). Also, T check2 agrees well with T at all powers until P is 30 W, after which error increases, maximizing at 60 W, where R W spikes. At 60 W and 120 W T check2 continues to be four times as large as T exp and T check . But T decreases to within 50% of T exp and T check . Consequently, Equation (17) is only reliably satisfied below 30 W, where T is reliable. Further research into electrolyte breakdown at high temperatures and/or current is needed to provide more insight into this phenomenon and to enable effective modeling.
Using the methods described above, computing T from Equation (2), and computing R W from Equations (18) or (19), error in V's final value was less than 1%, and error in E remains negligible. Although the model gave accurate results regarding energy storage and voltage, it still had two serious deficiencies: • There was no in-situ data to validate the computation of T. All T values had been theoretically computed from the Butler-Volmer equation and various derivations, Equations (1)-(3), (13), and (20). Some direct, experimental temperature data was needed in order to assess this. • In order for the model to work properly, R W had to be on the order of 1.5 Ω, as shown in Figure 6. But experimental EIS data from the previous study indicated R W should be on the order of 0.015 Ω.

Temperature Change Study
As in the previous study [10], a 200F LIC made by General Capacitor (product number LCA200G1, shown in Figure 9, was used to provide experimental data as a baseline to compare against the model. The reasons for this choice are twofold.
• At the time General Capacitor's 200F product was widely marketed and was commercially available. • The previous version of the model computed a theoretical temperature increase for LCA200G1, which needed to be validated or disproven.
LCA200G1 design specifications are shown in Table 1. All experimental data in this study came from a LCA200G1 made using the method described by Cao et al. [27]. Because LCA200G1 was a commercial, flagship product, much data had already been collected about its performance, including cycle life data, which can take months, if not years to collect. Additionally, many components and operational parameters, including separator material, electrolyte, operating voltage range, and current collector tab position and welding methods were well-known and likely to be repeated by General Capacitor. This eliminated much of the variability often encountered in laboratory-made LICs.  (T check2 ), computed by the model in order to verify C w (T check ), and experimentally (T exp ). The agreement between T check2 , T check (using Equation (17)). T exp (using Equation (17)   LICs were handled shortly after testing, it was doubtful that the actual ΔT was any higher than a few Kelvin. The initial hypothesis was that the actual ΔT was only slightly lower than the experimental value because of thermal insulating effects inside the LIC. Results indicated the temperature question is more complex than this, as will be explained. Results are shown in Figure 10.
At high power charges no temperature increase was observed, as shown by the 10 W charge in Figure 9 and as anticipated by Equation (6). The ΔT values observed at all of these charge powers are compared in Figure 11. Notice ΔT peaks then lowers. This is probably a factor of current adjusting for voltage. Notice ΔT equals 0 for the 10W charge. Figure 11. Comparison of total ΔT increase as a function of power charge. Notice a strong correlation between peak ΔT and charge power until 3 W, nominally 1 A. This correlation breaks down at higher power. In Figure 9 ΔT peaks then decrease again. This is probably a byproduct of current changes due to Ohm's law. Because of voltage changes during a charge, a constant power charge will begin with a high current when the voltage is low and end with a low current when the voltage is high. Therefore, near the end of a charge, there is less current to affect the temperature of the LIC. In order to more directly assess the effects of current upon temperature, constant current measurements were made. At currents below 1.0 A, these results showed a linear temperature increase with time. At a higher current, 5.0 A, no temperature increase was measured. 5.0 A was the limit of test equipment. These results are shown in Figure 12. They indicate the Butler-Volmer equation's temperature increase appears to accurately link current and temperature at relatively low current, but this relationship breaks down at higher currents, where more capacitive behavior dominates. There appears to be a hysteresis as charge current does not immediately affect temperature. Above a certain threshold the charge cycle finishes before current's effects on cell temperature can be felt. The reason for this is not understood.
If the 5.0 A charging data is ignored, the slope of the charge temperature increases linearly with charge current and can be predicted by an equation

0.0043
which is plotted in Figure 13. In terms of current density, this is as demonstrated in Figure 14.
The results shown in Figures 9-14 indicated what the value of ΔT used in the model should be. These experimental ΔT values were used to calibrate the model. First, the initial ambient temperature was set at 298 K, the nominal ambient temperature of the environmental test chamber where the study was conducted. As the maximum ΔT was 2.24 K, the change in the T variable is less than 1% when operating at room temperature. Thus T may be treated as a constant in many circumstances.

RW Improvements
As discussed earlier, another problem with the previous model is that R W was considered static. In fact, it is not. The values found in Figure 6 and Equations (16) and (17) were two orders of magnitude higher than the experimental value of 0.011 Ω computed from potentiostatic EIS data, using a maximum current of 0.1 A. In order to improve the model, another approach was adopted, found by Greenleaf et al. in an earlier study [32]. This approach computes R W as follows where D s is the species' diffusion coefficient, δ is the Nernst diffusion coefficient, found from the effective particle radius (r) of the negative electrode material as     r δ = (26) r for the hard carbon material in this anode is approximately 50 μm, making δ equal to 10 μm.
R W values calculated using Equation (20) and experimental ΔT data agreed with experimental values found from the previous study's EIS data that were used to build the previous study's model. This data anticipates R W values on an order of magnitude similar to the EIS data. Results are compared in Figure 15.
This R W computation method was combined with the revised method of calculating ΔT. Once R W was computed by Equation (20) and used accurate ΔT values, error in V diminished but was not eliminated. When the lowest computed R W value was used, 0.013 Ω, error in the final V value was less than 5%. R W and energy computations matched experimental values. Energy computations never appreciably changed. The voltage drop across R S is very small, 5.4 mV, so it does not appreciably affect ΔV. These results also indicate that R W should be treated as a constant and does not change with current. Voltage error is compared in Figure 16.
One limitation of this method, however, is that the model initially increases V very quickly with high error, although V eventually levels off, as shown in Figure 17. Upon examination of V S , V dl//CT , V W , it is found V dl//CT is the primary source of this error, as it increases by 1 V during the first several seconds of a charge but is then constant. More research is needed to determine why V dl//CT behaves in this manner and to correct it. Because the problem has been isolated to V dl//CT , it is believed the model may struggle with the non-Faradaic, capacitor behavior in an LIC more so than the Faradaic, battery behavior.

Model Adjustments from Temperature Change
The only other unknown left in Equation (2) was η. η values that agreed with experimental results shown in Figures 9-11 were found and are shown in Figure   18. The method of computing R W had minimal impact on η. Because T is relatively constant these results caused η to decrease by two orders of magnitude when compared to the previous model.

Conclusions
The previous study's model was improved with experimental data and additional theoretical relationships. The first step was to compute temperature values from the model's predicted energy storage and from experimentally measured energy storage. This method used energy storage's implications on Warburg capacitance and Warburg capacitance's implications on cell temperature. These temperature values were analyzed, but they gave values that were lower than expected. Experimental measurements were taken and used to validate the temperature variables. The experimental results indicated that although the temperature does increase as an LIC is charged, this increase is small, <1%. However,