Dependence of Supercapacitor Peukert Constant on Voltage, Aging, and Temperature

Abstract

This paper investigates the dependence of the supercapacitor Peukert constant on its terminal voltage, aging condition, and operating temperature. Recent studies show that the charge delivered by a supercapacitor during a constant current discharge process increases when the discharge current decreases if the discharge current is above a certain threshold, i.e., Peukert’s law applies. By conducting extensive experiments using three supercapacitor samples with different rated capacitances from different manufacturers, this paper reveals that the Peukert constant increases when the initial voltage of the constant current discharge process is lower, the supercapacitor is more heavily aged, or the operating temperature is lower. The physical mechanisms accounting for the Peukert constant dependence are illustrated by analyzing an RC ladder circuit model. When the supercapacitor terminal voltage is higher, the aging condition is lighter, or the operating temperature is higher, more charge is stored in the supercapacitor. Consequently, when the same discharge current is applied, the discharge time is longer and the branch capacitors are more deeply discharged. Therefore, the relaxation effects of the slow branches are reduced and the supercapacitor behaves more like a single capacitor rather than a distributed capacitor network, which ultimately leads to a lower Peukert constant.

I. Introduction

Energy storage is becoming an increasingly critical asset in many systems especially in smart grid and electric vehicles. Among various energy storage technologies, supercapacitors are advantageous in several aspects such as high power density and long cycle life. In fact, supercapacitor-based energy storage systems have been employed by multiple types of microgrids to implement a wide range of control and management functionalities to enhance the efficiency, reliability, and resiliency of microgrids [1].

To exploit the supercapacitor technology, a comprehensive and in-depth understanding of its characteristics at the device level is crucial. Therefore, modeling and characterization of supercapacitors have been of great interest. Supercapacitors are usually constructed using porous carbon electrodes [2]. The porous electrode theory [3] suggests that an electrode can be modeled as an RC transmission line [4]. A ladder circuit composed of multiple RC branches with different time constants can be used to capture the distributed nature of the supercapacitor capacitance and resistance [5]. Various equivalent circuit models [6]-[13] have been proposed to reduce the complexity of the generic ladder circuit model. On the other hand, accurate estimation of the supercapacitor state of charge (SOC) is important at both the device and system levels. Although the supercapacitor terminal voltage is a natural indicator of its state, accurate SOC estimation is still challenging because the supercapacitor capacitance and equivalent series resistance (ESR) are affected by multiple factors [14] such as the terminal voltage, operating temperature, and aging condition in a complex manner. Numerous frameworks [15]-[18] have been developed to identify supercapacitor parameters and estimate supercapacitor states.

Among various aspects of supercapacitor physics, charge redistribution is a process of particular interest. Charge redistribution is a relaxation process originated from the porous structure of the electrodes [3], [4]. Because of this process, the supercapacitor terminal voltage may decay or recover after a charge or discharge process [19]. The physical mechanisms leading to this process have been revealed: the electrode pore sizes are nonuniform. Therefore, the ions in the electrolyte need extra time to penetrate the middle-size mesopores and small-size micropores compared to the large-size macropores [20]-[24]. The effects of charge redistribution on power management strategies in wireless sensor nodes employing supercapacitor-based energy storage systems [25], [26], supercapacitor terminal voltage behavior [27], [28], supercapacitor energy delivery capability [29], [30], and supercapacitor capacitance characterization methods [31] have been extensively studied.

In particular, the impact of charge redistribution on the amount of charge delivered by a supercapacitor during one or multiple constant current discharge processes has been examined [32], [33]. One of the key findings is that the delivered charge increases when the discharge current decreases if the discharge current is above a certain threshold and the energy loss due to self-discharge is negligible, which is consistent with Peukert’s law originally developed for lead-acid batteries [34]. For batteries, the charge capacity loss associated with a larger discharge current is due to the reduced number of active centers in the positive active material and the increased resistance of the interface between the active material and the grid [34]. For supercapacitors, Peukert’s law applies because of the porous electrode structure and the charge redistribution process [32], [33]. Furthermore, a preliminary study [35] explores the feasibility of predicting the supercapacitor discharge time during a constant current discharge process using Peukert’s law and shows that the prediction error can be significantly reduced if the Peukert constant is properly estimated.

This paper develops [32], [33], [35] and studies the dependence of the supercapacitor Peukert constant on its terminal voltage, aging condition, and operating temperature. Experimental results show that the Peukert constant increases when the initial voltage of the constant current discharge process is lower, the supercapacitor is more heavily aged, or the operating temperature is lower. The underlying physical mechanisms are investigated by analyzing an RC ladder circuit model. This analysis reveals that a lower Peukert constant corresponds to a case in which the slow branch capacitors are more deeply discharged, which means that the middle-size mesopores and small-size micropores in the porous electrode structure are more responsive to the discharge current. Consequently, the supercapacitor behaves more like a single capacitor rather than a distributed capacitor network.

The remainder of this paper is organized as follows. Section II demonstrates the applicability of Peukert’s law to supercapacitors. Section III illustrates the methodology for estimating the supercapacitor Peukert constant. Section IV shows the experimental results for the dependence of Peukert constant on voltage, aging, and temperature. Section V provides a physical explanation for the dependence of Peukert constant on voltage. Section VI explains the dependence of Peukert constant on aging and temperature. Section VII concludes this paper.

II. Applicability of Peukert’s Law to Supercapacitors

A. Methodology

Originally developed for lead-acid batteries, Peukert’s law [34] relates the delivered charge to the discharge current as follows:

$$I^{k}t=Q_0,$$ (1)

where Q₀ is the nominal charge capacity rated at a particular discharge current, I is the actual discharge current, t is the actual discharge time, and k is the Peukert constant. This empirical law states that the delivered charge of a battery depends on the discharge current: the larger the discharge current, the less the delivered charge because k > 1. It should be noted that the term “delivered charge” refers to the amount of charge released during one discharge process. Due to the presence of relaxation processes, the total available charge stored in the battery cannot be completely released during one discharge process. Consequently, multiple discharge processes are needed to estimate the amount of the total available charge stored in the battery. Therefore, the applicability of Peukert’s law is limited to one discharge process and this relationship characterizes the dependence of the delivered charge on the discharge current. On the other hand, the total charge estimated using multiple discharge processes is approximately equal when the discharge current varies [34]. This paper adopts a similar methodology to investigate Peukert’s law and the Peukert constant dependence for supercapacitors.

As elaborated in [32], [33], Peukert’s law applies to supercapacitors when the discharge current is above a certain threshold. Specifically, two types of supercapacitor charge capacity are investigated in [32]: the charge released during one discharge process, which is referred to as the “utilized” charge capacity and the total available charge stored in the supercapacitor that can be completely released during multiple discharge processes, which is referred to as the “total” charge capacity. The utilized charge capacity is further examined in [33]: the applicability of Peukert’s law to supercapacitors and the physical mechanisms accounting for this relationship are studied. Note that Peukert’s law for supercapacitors is also associated with the utilized charge capacity, as in [34] for batteries. This paper develops [32], [33] and investigates the dependence of the supercapacitor Peukert constant on various parameters. Therefore, the “delivered charge” mentioned in this paper refers to the “utilized” charge capacity, not the “total” charge capacity.

B. Experiments

The applicability study of Peukert’s law is conducted in [33] using the three supercapacitor samples with different rated capacitances from different manufacturers listed in Table I. They are tested using an automated Maccor Model 4304 tester at room temperature: 20-25 °C, which is represented by 23 °C in this paper. For each sample, a set of constant current discharge experiments is performed when the initial voltage of the discharge process is fixed at a particular value and the cutoff voltage is fixed at 0.01 V. The rated voltage is the same for the three samples and the initial voltage is approximately linearly swept: 2.7, 2, 1.35, and 0.7 V. The experiments and results are summarized as follows.

TABLE I.

Supercapacitor Samples.

Sample	1	2	3
Manufacturer	Eaton	AVX	Maxwell
Model	HV1030-2R7106-R	SCCV60B107MRB	BCAP0350
Capacitance (F)	10	100	350
Voltage (V)	2.7	2.7	2.7

To illustrate the experiment design, Fig. 1(a) shows the measured supercapacitor terminal voltage during a 10 A constant current discharge experiment for sample 2 when the initial voltage of the discharge process is 2.7 V. During this experiment, the supercapacitor is first conditioned by ten charging-redistribution-discharging cycles to minimize the effect of residual charge [36]. Note that this conditioning phase is not the process used to establish the supercapacitor voltage although a certain amount of charge is injected into the supercapacitor during this phase. Immediately following this conditioning phase, the supercapacitor is charged by a constant voltage source of 2.7 V for 3 hours, which is designed to fully charge the supercapacitor. After that, a 10 A constant discharge current is applied and the supercapacitor is discharged to 0.01 V. The discharge termination voltage is set as 0.01 instead of 0 V for safety considerations. Taking 2.7 V as the initial voltage and 0.01 V as the cutoff voltage, the charge delivered during this constant current discharge process is calculated as

$$Q=It,$$ (2)

where I is the discharge current and t is the discharge time. For this experiment, the delivered charge is 252.4 C.

Fig. 1.

Illustration of experiment design using supercapacitor sample 2 when initial voltage of discharge process is 2.7 V. (a) A 10 A constant current discharge experiment. (b) Relationship between delivered charge and discharge current.

The experiment continues to estimate the total available charge stored in the supercapacitor, which is the amount of the charge injected into the supercapacitor during the conditioning phase and the constant voltage charge process. After the supercapacitor voltage reaches the discharge termination condition of 0.01 V, the discharge current is disconnected and the supercapacitor experiences charge redistribution, which results in an increase in the terminal voltage. Once the terminal voltage increase rate is less than 0.01 V per 5 minutes, the charge redistribution process is considered complete and the discharge current is applied again. This discharging-redistribution cycle is repeated ten times to extract the charge stored in the supercapacitor to the maximum extent possible. For this experiment, the sum of the charge released during the ten discharging-redistribution cycles is 293.7 C, which is referred to as the total charge capacity. It is clear that the two types of charge capacity are different: 252.4 C for utilized and 293.7 C for total. Also note that the discharge process (i.e., the ten discharging-redistribution cycles) instead of the charge process is used to estimate the total charge capacity because the charge injected into the supercapacitor during the conditioning phase and the constant voltage charge process cannot be readily determined.

Depending on the supercapacitor sample specifications [37]-[39] and the supercapacitor tester capabilities, a set of constant discharge currents is swept for each sample. For sample 2, nine currents are selected: 10, 5, 1, 0.5, 0.1, 0.05, 0.01, 0.005, and 0.0025 A. While the maximum continuous current is not specified in the datasheet for this sample [38], the value of 10 A is determined according to the datasheets for similar 100 F samples tested in [32]. The discharge current is then scaled down based on the maximum value of 10 A. As elaborated in [32], this set of experiments is repeated once. For the 10 A experiment shown in Fig. 1(a), the second run results in a delivered charge of 248.7 C. The average of these two runs of experiment is then 250.55 C, which is used as the delivered charge corresponding to the 10 A discharge current. For the total charge capacity, the second run results in 290.7 C and the average is therefore 292.2 C. Again, note that the difference between the average utilized and total charge capacities is significant: 250.55 versus 292.2 C The delivered charge results associated with the other eight discharge currents are measured by performing experiments similar to that for the 10 A current: first a conditioning stage, then a constant voltage charge phase, and finally a constant current discharge process.

C. Results

When the initial voltage of the discharge process is 2.7 V, the relationship between the delivered charge (i.e., the utilized charge capacity) and the discharge current for sample 2 is plotted in Fig. 1(b), which is partitioned into two pieces: Peukert’s law applies when the discharge current is above a certain threshold and does not apply anymore when the discharge current is below the threshold. Specifically, when the discharge current decreases from 10 to 0.01 A, the delivered charge increases from 250.55 to 299.41 C and Peukert’s law applies. On the other hand, the delivered charge decreases from 299.41 to 292.62 C when the discharge current decreases from 0.01 to 0.0025 A and Peukert’s law does not apply anymore. This pattern is due to the combined effects of the three aspects of supercapacitor physics: porous electrode structure, charge redistribution, and self-discharge, as elaborated in [33]. Specifically, because of the porous electrode structure, or equivalently, the distributed nature of the supercapacitor capacitance and resistance, slow branch capacitors with large time constants are accessed during the extended discharge process when a lower discharge current is applied, which results in an increase in the delivered charge. In the meantime, the unidirectional charge redistribution from slow branches to fast branches decelerates the voltage drop in the main branch with the smallest time constant and prolongs the discharge time, which also contributes to the increase in the delivered charge. The impact of self-discharge on the delivered charge is negligible when the discharge current is relatively large. If the discharge current is sufficiently low, the energy loss due to self-discharge is significant, which results in a drop in the delivered charge.

While the utilized charge capacity shows strong dependence on the discharge current (i.e., Fig. 1(b)), the total charge capacity estimates corresponding to different discharge currents are approximately equal. For sample 2, Table II lists the total charge capacity estimates denoted as Q_I, where the subscript _I means the discharge current used in the experiment. This quantity estimates the amount of the charge injected into the supercapacitor during the conditioning and constant voltage charge phases. For instance, when the initial voltage of the discharge process is 2.7 V and the discharge current is 10 A, the total charge capacity estimate Q₁₀ is 292.20 C, which is the average of two experiment runs, as elaborated in Section II-B. For brevity, the estimates associated with the other eight discharge currents drop the units of C. Based on these estimates, three statistics are calculated: mean, standard deviation, and coefficient of variation, which are denoted as $\overline{\mathit{Q}}$, s, and cv, respectively. Compared to the strong dependence of the utilized charge capacity on the discharge current, the total charge capacity can be assumed to be independent of the discharge current.

TABLE II.

Total Charge Capacity Estimates for Supercapacitor Sample 2.

Voltage (V)	Q10 (C)	Q5	Q1	Q0.5	Q0.1	Q0.05	Q0.01	Q0.005	Q0.0025	$\overline{\mathit{Q}}$ (C)	s (C)	cv
2.7	292.20	297.00	301.41	301.52	300.51	300.70	302.51	294.99	293.62	298.27	3.87	0.0130
2	205.30	209.40	213.74	213.40	213.08	214.12	216.05	214.80	212.66	212.51	3.25	0.0153
1.35	131.80	136.20	140.52	140.71	140.31	140.17	142.01	143.01	142.37	139.68	3.54	0.0254
0.7	62.00	66.35	70.28	70.18	70.24	70.36	71.26	72.59	71.64	69.43	3.27	0.0471

In fact, a more accurate estimate of the total charge capacity with a much smaller data dispersion in terms of cv can be readily obtained by excluding the measurements associated with the relatively large and sufficiently low discharge currents. Specifically, when the discharge current is sufficiently low, the energy loss due to self-discharge is significant and the total charge capacity is underestimated. Therefore, Q_0.005 and Q_0.0025 should be excluded, as analyzed in [33]. On the other hand, when the discharge current is relatively large, the voltage change due to ESR is significant and the estimation error is greater because of the limited time resolution of the supercapacitor tester, as elaborated in [32]. Therefore, the estimation accuracy can be further improved by excluding Q₁₀ and Q₅. If these four measurements are excluded, the three statistics calculated using the remaining five estimates are 301.33 C, 0.79 C, and 0.0026, respectively. Clearly, the data dispersion is improved in terms of cv (i.e., 0.0026 versus 0.0130) and the mean serves as a better estimate of the total charge capacity. In summary, Fig. 1(b) and Table II reveal that the behaviors of the utilized and total charge capacities are different when it comes to the discharge current dependence: the former shows strong dependence and the latter is independent, which justifies the study conducted in this paper: Peukert’s law and the Peukert constant dependence associated with the utilized charge capacity.

III. Methodology for Estimating Peukert Constant

Since Peukert’s law applies to supercapacitors, it follows that the supercapacitor discharge time can be predicted using this relationship, as demonstrated in [35]. It is also shown in [35] that the prediction error depends on the Peukert constant. Therefore, this paper examines the dependence of the supercapacitor Peukert constant on its terminal voltage, aging condition, and operating temperature.

The methodology for estimating the Peukert constant is illustrated using the data shown in Fig. 1(b). First, the discharge current range between which Peukert’s law applies is identified, which is 10-0.01 A in this case. The numerical results for the delivered charge are listed in Table III, which have also been reported in [32]. Note that the delivered charge uses the average of two experiment runs (e.g., 250.55 C for 10 A).

TABLE III.

Fitted and Optimal Peukert Constant for Supercapacitor Sample 2 When Initial Voltage of Discharge Process is 2.7 V.

Current	Charge	Time	Fit: k = 1.021		Optimal: k = 1.023
I (A)	Q (C)	tm (s)	tp (s)	δ (%)	tp (s)	δ (%)
10	250.55	25.055	25.917	3.44	25.798	2.97
5	258.93	51.786	52.594	1.56	52.425	1.23
1	272.01	272.01	272.01	0.00	272.01	0.00
0.5	276.37	552.74	552.00	0.13	552.76	0.00
0.1	287.09	2870.9	2854.9	0.56	2868.0	0.10
0.05	291.80	5836.0	5793.4	0.73	5828.3	0.13
0.01	299.41	29941	29963	0.07	30240	1.00
			Fit: $\overline{\delta }$ = 0.93%		Optimal: $\overline{\delta }$ = 0.78%

Next, a preliminary estimate of the Peukert constant is determined using the following curve fitting function:

$$t=\frac{Q_0}{I^k}.$$ (3)

The discharge current and the corresponding discharge time are used to fit the Peukert constant. The charge delivered during the 1 A experiment is used as the nominal charge: Q₀ = 272.01 C. Fig. 2(a) shows the measured discharge time and the fit determined using MATLAB. The fitted Peukert constant is k = 1.021.

Fig. 2.

Fitted and optimal Peukert constant for supercapacitor sample 2 when initial voltage of discharge process is 2.7 V. (a) Curve fitting to determine fitted value. (b) Sweeping of Peukert constant to determine optimal value.

Finally, the Peukert constant is fine-tuned and the value leading to the minimum discharge time prediction error is determined, which is referred to as the optimal value and used as the estimate of the real Peukert constant. The prediction error for each individual experiment is evaluated as follows:

$$\delta =\frac{\mid t_p-t_m\mid }{t_m}\times 100\%,$$ (4)

where t_p is the prediction and t_m is the measurement. The average of the errors for the seven experiments is denoted as $\overline{\delta }$, which is used as the metric to identify the optimal Peukert constant value. Fig. 2(b) sweeps the Peukert constant and shows that the minimum average error is obtained when the optimal value takes k = 1.023. Table III lists the numerical results associated with the fitted and optimal values of the Peukert constant. By default, the Peukert constant mentioned in the remainder of this paper refers to the optimal value.

IV. Dependence of Peukert Constant on Voltage, Aging, and Temperature

A. Dependence of Peukert Constant on Voltage

This section studies the dependence of the supercapacitor Peukert constant on the initial voltage from which the supercapacitor is discharged. The rated voltage is the same for the three supercapacitor samples and the initial voltage of the constant current discharge process is approximately linearly swept at room temperature: 2.7, 2, 1.35, and 0.7 V. In fact, this study reuses the delivered charge results reported in [33]. The experiments and results for sample 2 are summarized as follows. Fig. 1(b) shows the delivered charge when the initial voltage of the discharge process is 2.7 V. For the other three voltages (i.e., 2, 1.35, and 0.7 V), the experiment procedures are similar to those for 2.7 V. Specifically, the initial conditioning phase of ten charging-redistribution-discharging cycles and the final constant current discharge process use the same settings (i.e., nine discharge currents are swept and the discharge termination voltage is set to be 0.01 V). However, the constant voltage charge phase uses the value corresponding to the specific voltage examined. For example, Fig. 3(a) shows a 0.1 A experiment when the initial voltage of the discharge process is 2 V. Different from Fig. 1(a), the constant voltage charge phase uses 2 V. This set of nine experiments is performed once and Fig. 3(b) shows the relationship between the delivered charge and the discharge current. similarly, constant current discharge experiments are performed for 1.35 and 0.7 V. Fig. 4 shows the delivered charge results. As in Fig. 1(b), the delivered charge refers to the utilized charge capacity in Figs. 3(b) and 4.

Fig. 3.

Experimental results: delivered charge for supercapacitor sample 2 when initial voltage of discharge process is 2 V (a) A 0.1 A constant current discharge experiment. (b) Relationship between delivered charge and discharge current.

Fig. 4.

Experimental results: delivered charge for supercapacitor sample 2 when initial voltage of discharge process is 1.35 and 0.7 V. (a) Initial voltage is 1.35 V. (b) Initial voltage is 0.7 V.

For 2, 1.35, and 0.7 V, the total charge capacity is also estimated and the results are listed in Table II. As in the 2.7 V case, the statistics for the 2, 1.35, and 0.7 V cases are calculated using all the nine measurements and it can be assumed that the total charge capacity is constant when the discharge current varies. More accurate estimates can also be obtained by excluding Q₁₀, Q₅, Q_0.005, and Q_0.0025. Moreover, it is clear that the total charge capacity decreases when the initial voltage of the discharge process decreases.

For the utilized charge capacity, Figs. 1(b), 3(b), and 4 show that the delivered charge decreases for a given discharge current when the initial voltage of the discharge process decreases. For both the total and utilized charge capacities, the drop in the charge is mainly due to the charge-voltage relationship of capacitors: the charge stored in a capacitor is proportional to the voltage (i.e., Q = CV). For supercapacitors, the drop is accelerated by the voltage dependence of the capacitance and the relatively large ESR, as elaborated in Section V. These figures also show that for sample 2, Peukert’s law applies for the current range of 10-0.01 A when the initial voltage is 2.7 and 2 V. The current range is 10-0.005 A for 1.35 and 0.7 V. For consistency, the narrower range of 10-0.01 A is considered for all the four voltages and the Peukert constant associated with each voltage is estimated using the methodology established in Section III. Fig. 5 plots the Peukert constant results for all the three samples. Clearly, the Peukert constant increases when the voltage decreases for all samples. For sample 2, it increases from 1.023 to 1.036 when the voltage drops from 2.7 to 0.7 V.

Fig. 5.

Experimental results: dependence of Peukert constant on voltage for all supercapacitor samples.

B. Dependence of Peukert Constant on Aging

This section investigates the dependence of Peukert constant on the supercapacitor aging condition at room temperature. The initial voltage of the discharge process is fixed at the rated voltage of 2.7 V. The effects of aging on supercapacitor parameter degradation such as increase in ESR and decrease in capacitance have been examined [40]-[45]. In these studies, supercapacitors usually undergo accelerated current, power, or energy cycling during a relatively short period of time so that the impact of aging on the state of health and calendar life can be investigated. Unlike these dedicated experiments designed for expediting the aging of supercapacitors, the three supercapacitor samples examined in this paper are aged through the normal use during experiments conducted for investigating other characteristics of them (e.g., constant current discharge experiments reported in this paper for studying the Peukert constant dependence).

For all samples, three aging conditions are examined and the three corresponding datasets are denoted as S1, S2, and S3. The aging condition is characterized by the “hours of use” since the device is used for the first time. For example, sample 2 was purchased in November 2016 and first used in May 2017 The S1 dataset was obtained in July 2017. Prior to that, it was used (i.e., charge and discharge processes in current, voltage, and power modes and minutes-long rests between charge/discharge processes) for approximately 700 hours. In fact, this S1 dataset is the one shown in Fig. 1(b) and the 10 A experiment shown in Fig. 1(a) belongs to this dataset.

The aging time in terms of the hours of use is calculated as follows. For the 10 A experiment shown in Fig. 1(a), it started on July 7, 2017 at 15:38 and completed at 23:04 on the same day. The experiment duration was 7 hours and 26 minutes, which is counted in the hours of use. The supercapacitor was left idle until 8:34 on July 8, 2017 and then a 5 A experiment started. The rest time of 9 hours and 30 minutes is not included in the hours of use. Therefore, the hours of use include the experiment durations and exclude the intervals between experiments. As for the experiments contributing to the hours of use, in addition to the constant current discharge experiments reported in this paper and in [32], [33], [35], other types of experiments have also been performed, which include the constant power discharge experiments used to study the supercapacitor energy delivery capability [29], [30] and the experiments designed to examine the supercapacitor capacitance characterization methods [31]. The hours of use contributed by these experiments are calculated using the same way as that for the 10 A experiment shown in Fig. 1(a). The S2 dataset was obtained in April 2018 and sample 2 was used for approximately 3000 more hours since S1. Finally, after another 400 hours of use, the S3 dataset was obtained in May 2018 Therefore, sample 2 was more heavily aged from S1 to S2 and finally to S3. Fig. 6(a) plots the delivered charge results corresponding to the three aging conditions for sample 2. It can be observed that the delivered charge decreases when the supercapacitor is more heavily aged, which is consistent with other studies [40]-[45] that reveal the effects of aging on supercapacitor capacitance degradation. The Peukert constant results for sample 2 are shown in Fig. 6(b): it increases when the supercapacitor is more heavily aged. Specifically, the Peukert constant increases from 1.023 to 1.031 because of the 3000 hours of use between S1 and S2. From S2 and S3, the Peukert constant remains unchanged because of the relatively short use time of 400 hours.

Fig. 6.

Experimental results: dependence of Peukert constant on aging when initial voltage of discharge process is 2.7 V. (a) Delivered charge for supercapacitor sample 2. (b) Peukert constant for all supercapacitor samples.

The Peukert constant results for samples 1 and 3 are also plotted in Fig. 6(b). Their aging conditions are as follows. Sample 1 was purchased in April 2017 and first used in August 2017 to generate the S1 dataset. After 1800 hours of use, the S2 dataset was obtained in April 2018. The S3 dataset was obtained in May 2018 after another 300 hours of use. Sample 3 was purchased in November 2016 and first used in September 2017 to generate the S1 dataset. The S2 dataset was obtained in April 2018 after 3000 hours of use. Finally, the S3 dataset was obtained in May 2018 after another 500 hours of use. Note that while the hours of use for the S1 datasets are denoted as zero for samples 1 and 3 in Fig. 6(b), they were still lightly aged due to the natural and spontaneous aging mechanisms although they were not used to perform any experiments prior to S1. As for the Peukert constant results for samples 1 and 3, a significant increase is observed from S1 to S2 while the increase from S2 to S3 is mild, which is similar to the results for sample 2.

C. Dependence of Peukert Constant on Temperature

To examine the dependence of the supercapacitor Peukert constant on operating temperature, the supercapacitor tester is used together with a Maccor Model MTC-020 temperature chamber that generates temperatures between −20 and 100 °C. The typical operating temperature range of the three supercapacitor samples is between −40 and 65 °C [37]-[39]. Considering the temperature chamber capabilities, four temperatures were swept during January through April 2018 in the descending order: 60, 40, 0, and −18 °C when the initial voltage of the discharge process was fixed at 2.7 V. After that, the room temperature dataset S2 was obtained in April 2018. Therefore, this section reuses the 23 °C: S2 dataset reported in Section IV-B. It should be noted that although the intention of this study was to investigate the dependence of Peukert constant on temperature, the effects of aging on Peukert constant also need to be taken into account considering the relatively long experiment durations. For instance, sample 2 was used for approximately 135 hours during a complete set of nine constant current discharge experiments, which means that it was more heavily aged during a succeeding set of experiments compared to a preceding set (e.g., more heavily aged at 40 °C compared to 60 °C).

The delivered charge results for sample 2 are plotted in Fig. 7(a). First, consider the four temperatures swept in the descending order: 60, 40, 0, and −18 °C. The delivered charge drops when the temperature decreases. This is due to the combined effects of temperature and aging. In general, a lower temperature results in a drop in the supercapacitor capacitance, as specified in the supercapacitor datasheets [37]-[39] and reported in various studies [40]-[45]. From 60 to −18 °C, the temperature decreased and the capacitance was smaller, which led to a drop in the delivered charge. In the meantime, the supercapacitor was more heavily aged when the temperature decreased, which also contributed to the capacitance degradation and ultimately the drop in the delivered charge.

Fig. 7.

Experimental results: dependence of Peukert constant on temperature when initial voltage of discharge process is 2.7 V. (a) Delivered charge for supercapacitor sample 2. (b) Peukert constant for all supercapacitor samples.

Next, consider the −18 and 23 °C: S2 datasets, which can be used to more clearly demonstrate the effects of temperature on the delivered charge. The −18 °C set of experiments was performed first and then the 23 °C: S2 set. Therefore, the supercapacitor was more heavily aged in the 23 °C: S2 set, which could result in a drop in the delivered charge. On the other hand, the temperature was higher and the delivered charge could increase. The actual change in the delivered charge was determined by the overall effect of these two competing mechanisms. If the effects of temperature dominated, the delivered charge would increase, which was the case for sample 2. If the effects of aging were more significant, the delivered charge would decrease. Again, note that the effects of aging cannot be completely excluded from the experiments conducted for studying the effects of voltage and temperature because the aging process is irreversible in general although a phenomenon called “capacitance recovery” [40]-[42] may be observed when the supercapacitor cycling process is interrupted.

The Peukert constant results for all samples are shown in Fig. 7(b). In general, the Peukert constant increases when the temperature is lower although the change is moderate. For sample 2, it strictly increases from 1.029 to 1.039 when the temperature decreases from 60 to −18 °C. For sample 1, the Peukert constant increases when the temperature decreases from 60 to −18 °C although it flattens between 40 and 0 °C. The temperature ranges between which the Peukert constant remains unchanged are 60-40 °C and 23-0 °C for sample 3.

In addition to the dependence of the supercapacitor Peukert constant on voltage, aging, and temperature elaborated in Sections IV-A through IV-C, another observation can be made based on the results shown in Figs. 5, 6(b), and 7(b). Specifically, the Peukert constant seems to be dependent on the supercapacitor capacity: the lower the rated capacitance, the larger the Peukert constant, i.e., the Peukert constant for sample 1 (10 F) is the largest and sample 3 (350 F) is associated with the smallest Peukert constant. To examine if this observation is generally applicable and furthermore to reveal the physical mechanisms leading to this observation, extensive experiments with a much wider range of supercapacitor samples and a systematic investigation of the supercapacitor electrochemical processes are needed. In fact, a relevant observation on the discharge current threshold above which Peukert’s law applies has been reported in [33]: the lower the rated capacitance, the smaller the current threshold. An ongoing work is being conducted to further investigate these two observations.

V. A Physical Explanation for Dependence of Peukert Constant on Voltage

A. Effects of Supercapacitor Physics on Peukert Constant

To investigate the physical mechanisms leading to the dependence of Peukert constant on voltage, a generic RC ladder circuit model for 100 F supercapacitors is analyzed, as shown in Fig. 8, which is originally conceived in [32], [33] to examine the impact of supercapacitor physics on its charge capacity. This model includes five RC branches (R₁ through C₅) to capture the distributed nature of the supercapacitor capacitance and resistance, which is a result of the porous electrode structure and also the origin of the charge redistribution process. The parallel leakage resistor R₆ is used to represent the self-discharge process. The supercapacitor terminal voltage is denoted as V_T, which is a measurable parameter. In fact, V_T equals the voltage across the first RC branch composed of R₁ and C₁. When a source or load is applied to the supercapacitor terminals, the capacitor of each RC branch is accessed through a series connection of all the resistors from the supercapacitor terminals to the branch in question. The time constant of each RC branch can be written as

$${\tau }_i=C_i\sum _{k=1}^i{R}_k,$$ (5)

and the porous electrode theory gives that

$${\tau }_1<{\tau }_2<\cdots <{\tau }_n.$$ (6)

Fig. 8.

A five-branch RC ladder circuit model for 100 F supercapacitors.

The component values of the five RC branches are tuned to generate time constants that can be used to characterize the supercapacitor behavior on various time scales: τ₁ = 1.05, τ₂ = 10, τ₃ = 100, τ₄ = 1000, and τ₅ = 10000 s. The total capacitance of the five branch capacitors is 100 F. The C₁ capacitance is 70% of the total capacitance because the first branch is the main branch. The capacitances are 16, 8, 4, and 2 F for the remaining four branches with a scale factor of 0.5 based on the fact that a slower branch makes a smaller contribution to the total capacitance. As for the resistors, the first branch resistance R₁ uses the typical ESR value specified in the sample 2 datasheet [38]. The other four branch resistances are calculated using (5) based on the conrresponding time constants and capacitances. The value of the parallel leakage resistor R₆ is estimated based on the rated voltage and the leakage current specified in the sample 2 datasheet [38].

The dependence of the supercapacitor Peukert constant on its terminal voltage can be illustrated by analyzing this model, which is implemented and simulated in LTspice. As in Section IV-A, four voltages are swept: 2.7, 2, 1.35, and 0.7 V. For each voltage, a set of nine constant current discharge simulations is run. Take the 2.7 V case for instance. The initial voltages of the five branch capacitors are set to be 2.7 V and the discharge current is swept: 10, 5, 1, 0.5, 0.1, 0.05, 0.01, 0.005, and 0.001 A. For each discharge current, the delivered charge associated with the cutoff voltage of 0.01 V is determined, as shown in Fig. 9(a). For all the four voltages, the delivered charge patterns are similar to the experimental results shown in Figs. 1(b), 3(b), and 4: Peukert’s law applies when the discharge current is above a certain threshold and does not apply anymore if the discharge current is below the threshold. Moreover, the delivered charge associated with a particular discharge current decreases when the voltage decreases. The current range between which Peukert’s law applies is 10-0.01 A for 2.7 and 2 V. For 1.35 and 0.7 V, it is wider: 10-0.005 A. For all the four voltages, the Peukert constants are estimated for the narrower current range of 10-0.01 A, as plotted in Fig. 9(b). Consistent with the experimental results shown in Fig. 5, the Peukert constant increases from 1.027 to 1.045 when the voltage decreases from 2.7 to 0.7 V.

Fig. 9.

Simulation results to illustrate effects of supercapacitor physics on Peukert constant using Fig. 8. (a) Relationship between delivered charge and discharge current. (b) Dependence of Peukert constant on voltage.

An intuitive explanation for the dependence of Peukert constant on voltage is as follows. Physically, Peukert’s law describes the fact that the charge stored in a battery cannot be fully released during a relatively fast discharge process because chemical reactions take time to complete [34]. Ideally, the delivered charge is constant and does not vary with the discharge current, which means that the Peukert constant is k = 1 in this case. For practical batteries, k > 1 because nonideal processes exist. A larger Peukert constant means that the battery is more relaxed, or equivalently, the chemicals are less responsive during the discharge process. Therefore, Peukert constant is a measure of the responsiveness of the battery during a discharge process: a smaller Peukert constant means that the battery is more responsive, or equivalently, less relaxed. For supercapacitors, the physical meaning of Peukert constant is the same. Referring to the model shown in Fig. 8, the four slow branches (R₂ through C₅) represent the middle-size mesopores and small-size micropores in the porous electrode structure, which are harder to be accessed during a discharge process compared to the main branch (R₁ and C₁) representing the large-size macropores [20]-[24]. This is the origin of the relaxation process that makes the supercapacitor behavior deviate from the ideal case in which the supercapacitor can be completely discharged regardless of the discharge current magnitude. For a practical supercapacitor, when the initial voltage of the discharge process is higher, the discharge time is longer and the slow branch capacitors are more deeply discharged. Therefore, they are more responsive to the discharge current, or equivalently, less relaxed during the discharge process, which ultimately results in a smaller Peukert constant and makes the supercapacitor behave more like a single capacitor with k = 1 rather than a distributed capacitor network with k > 1.

To verify if the Peukert constant is affected by the depth of discharge (DOD) of the slow branch capacitors, Fig. 10(a) plots the simulated supercapacitor terminal and branch capacitor voltages when the initial voltage of the discharge process is 2.7 V and the discharge current is 10 A. The branch capacitor DOD is calculated as

$$\mathit{DOD}=\frac{V_{\mathit{initial}}-V_{\mathit{final}}}{V_{\mathit{initial}}}\times 100\%.$$ (7)

Fig. 10.

Simulation results to illustrate effects of initial voltage of discharge process on branch capacitor DODs using Fig. 8. (a) Supercapacitor terminal and branch capacitor voltages when initial voltage is 2.7 V and discharge current is 10 A. (b) Effects of initial voltage on branch capacitor DODs.

In this simulation setup, the initial voltage is V_initial = 2.7 V for C₁ – C₅. At the end of the discharge process (t = 20.2109 s), the terminal voltage V_T reaches 0.0101 V (note that the cutoff voltage is set to be 0.01 V) and the final voltage V_final is 0.1601, 1.2378, 2.5849, 2.6993, and 2.6973 V for C₁ – C₅, respectively. Therefore, the DOD for C₁ – C₅ is 94.1, 54.2, 4.26, 0.03, and 0.10%, respectively. Fig. 10(b) plots the DOD results for this voltage as well as the results for the other three voltages: 2, 1.35, and 0.7 V. The discharge current is fixed at 10 A in all the four setups.

As shown in Fig. 10(b), for a specific branch capacitor, its DOD decreases when the initial voltage of the discharge process decreases. For instance, the DOD for C₂ decreases from 54.2 to 13.7% when the initial voltage decreases from 2.7 to 0.7 V. This is because the discharge time is shortened when the initial voltage is lower. The discharge time is mainly determined by the first branch composed of R₁ and C₁. For an ideal capacitor with a zero ESR, the discharge time is proportional to the initial voltage of the discharge process due to the relationship of Q = CV. For supercapacitors, the relatively large ESR also contributes to the decrease in the discharge time when the initial voltage of the discharge process is lower, which is more significant when the discharge current is relatively large. For the 10 A discharge current, the discharge time is 3.9347 s when the initial voltage is 0.7 V, which is 19.5% of 20.2109 s corresponding to 2.7 V. On the other hand, the ratio of the supercapacitor terminal voltage drop is 25.7% (0.69 versus 2.69 V, note that the cutoff voltage is 0.01 V instead of 0 V). The discharge time ratio (19.5%) is lower than the supercapacitor terminal voltage drop ratio (25.7%) because of the relatively large ESR. Specifically, the voltage drop due to R₁ is 0.15 V when the discharge current is 10 A, which is a significant portion of the supercapacitor terminal voltage drop: 21.7 and 5.6% when the initial voltage is 0.7 and 2.7 V, respectively. Consequently, the voltage drop contributed by C₁ is 0.54 and 2.54 V when the initial voltage is 0.7 and 2.7 V, respectively, which give a ratio of 21.3%. This ratio is lower than the supercapacitor terminal voltage drop ratio of 25.7% because the relatively large ESR accelerates the decrease in the discharge time (and consequently the drop in the delivered charge) when the initial voltage is lower, as shown in the experimental results (Figs. 1(b), 3(b), and 4) and simulation results (Fig. 9(a)). In summary, when the initial voltage of the discharge process is lower, the discharge time is shorter and the branch capacitor DODs are lower, which makes them less responsive or more relaxed during the discharge process and finally results in a larger Peukert constant.

For a given initial voltage of the discharge process, in general, the branch capacitor DOD decreases when the branch time constant increases. As shown in Fig. 10(b), for all the four initial voltages, the DOD pattern is C₁ > C₂ > C₃ > C₅ > C₄. While the DOD for C₁ – C₃ decreases when the branch time constant increases, the DOD for C₅ is greater than that for C₄. This is because the discharge current (10 A) is large and the discharge time (20.2109 s) is short. Therefore, the voltage drops in C₄ and C₅ due to the discharge current are small. On the other hand, C₅ is directly connected to R₆, which also contributes to its voltage drop and results in a larger DOD. For the nine discharge currents swept, the DOD pattern for the 5 A case is the same as that for the 10 A case. For the other seven discharge currents (1-0.001 A), the DOD pattern is C₁ > C₂ > C₃ > C₄ > C₅ because the discharge time is relatively long and C₄ is more deeply discharged by the discharge current than C₅ even though the voltage drop in C₅ also includes contributions from R₆.

B. Effects of Voltage-Dependent Capacitance on Peukert Constant

As analyzed in Section V-A, the dependence of Peukert constant on voltage is originated from the distributed nature of the supercapacitor capacitance and resistance, which makes the supercapacitor behavior deviate from an ideal capacitor. To further explore the impact of the supercapacitor capacitance and ESR characteristics on Peukert constant, this section examines a smaller and voltage-dependent capacitance while Section V-C studies a larger ESR. The supercapacitor model shown in Fig. 8 is modified and the new model is shown in Fig. 11. Specifically, the first branch capacitance C₁ is changed from a constant value of 70 F to the following relationship:

$$C_1=43+10\times V_1.$$ (8)

Fig. 11.

A modified five-branch RC ladder circuit model to illustrate effects of voltage-dependent capacitance on Peukert constant.

Like the other components in the model, the parameters in (8) are assumed with certain arbitrariness for a qualitative study of the impact of voltage-dependent capacitance on Peukert constant. When the voltage is 2.7 V, (8) gives C₁ = 70 F. For other voltages, (8) results in a capacitance lower than 70 F. Note that C₁ is expressed as Q = 43 * x + 5 * x * x in LTspice, where Q is the charge and x is the voltage across C₁. Therefore, C₁ defined in (8) is usually referred to as the differential capacitance: C_diff = dQ/dV [6]. On the other hand, the equivalent capacitance of the linear capacitor [6] holding the same charge as C₁ is C_eq = 43 + 5 × V₁, which gives C_eq = 56.5 F at 2.7 V. To generate an equivalent capacitance of C_eq = 43 + 10 × V₁ (i.e., C_eq = 70 F at 2.7 V), (8) can be modified as C₁ = 43 + 20 × V₁, which is Q = 43 * x + 10 * x * x in LTspcie. Both the differential capacitance and equivalent capacitance cases are analyzed and the results are similar. This section focuses on the differential capacitance case with (8) and also includes the results for the equivalent capacitance case with C_eq = 43 + 10 × V₁. With a voltage-dependent capacitance, the model shown in Fig. 11 can be seen as an extended version of the models proposed in [6], [8], [10]: more branches are included and the supercapacitor behavior on more time scales is considered.

The model shown in Fig. 11 is simulated using the same setups as in Section V-A and the results are plotted in Fig. 12. As shown in Fig. 12(a), the delivered charge pattern is similar to that in Fig. 9(a): Peukert’s law applies when the discharge current is above a certain threshold and the threshold varies with the initial voltage of the discharge process. For the same initial voltage and the same discharge current, the delivered charge is less in Fig. 12(a). For example, the delivered charge peaks at 226.85 C when the initial voltage is 2.7 V and the discharge current is 0.01 A in Fig. 12(a) while it is 263.23 C in Fig. 9(a). The Peukert constants associated with Fig. 12(a) are calculated and the results are plotted in Fig. 12(b): the Peukert constant increases from 1.034 to 1.064 when the voltage decreases from 2.7 to 0.7 V. The Peukert constant in Fig. 12(b) is larger at each of the four voltages compared to its counterpart in Fig. 9(b), e.g., 1.034 versus 1.027 at 2.7 V. On the other hand, for the equivalent capacitance case with C_eq = 43 + 10 × V₁, the Peukert constants are 1.028, 1.035, 1.044, and 1.060 for 2.7, 2, 1.35, and 0.7 V, respectively. This pattern is similar to the one for the differential capacitance case with (8) (i.e., Fig. 12(b)).

Fig. 12.

Simulation results to illustrate effects of voltage-dependent capacitance on Peukert constant using Fig. 11. (a) Relationship between delivered charge and discharge current. (b) Dependence of Peukert constant on voltage.

To illustrate the impact of the voltage-dependent capacitance on Peukert constant, Fig. 13 shows the supercapacitor voltage and branch capacitor DOD results when the discharge current is 10 A. Specifically, Fig. 13(a) compares the supercapacitor terminal voltage V_T and the second branch capacitor C₂ voltage V₂ simulated using two setups when the initial voltage is 2.7 V. The “Constant C₁” setup uses the model with C₁ = 70 F (Fig. 8) and the “Variable C₁” setup uses the model with C₁ =43 + 10 × V₁ (Fig. 11). Since C₁ is lower than 70 F in the “Variable C₁” setup, the discharge time is shorter: 16.6262 versus 20.2109 s. Consequently, C₂ is less deeply discharged and the final value of its voltage V₂ is higher: 1.4440 versus 1.2378 V, which results in a smaller DOD: 46.5 versus 54.2%. Fig. 13(b) compares the DOD results for C₂ for all the four voltages. It can be observed that for each of the four voltages, the DOD is smaller in the “Variable C₁” setup because the discharge time is shorter. Similar observations hold for the other branch capacitors. Therefore, the Peukert constant is larger in the “Variable C₁” setup, as shown in Fig. 12(b).

Fig. 13.

Comparisons of simulation results using Figs. 8 and 11. (a) Supercapacitor terminal voltage V_T and second branch capacitor C₂ voltage V₂. (b) DOD for second branch capacitor C₂.

It should be noted that the decrease in the discharge time (and consequently the drop in the delivered charge) is accelerated by the smaller and voltage-dependent capacitance C₁ when the initial voltage of the discharge process is lower. For the “Variable C₁” setup, the ratio of the discharge time is 16.0% (2.6605 versus 16.6262 s) for the initial voltage of 0.7 and 2.7 V. On the other hand, the discharge time ratio is 19.5% (3.9347 versus 20.2109 s) for the “Constant C₁” setup, as analyzed in Section V-A.

C. Effects of ESR on Peukert Constant

This section investigates the impact of ESR on Peukert constant. The model shown in Fig. 8 is modified as follows: the R₁ value is changed from 0.015 to 0.045 Ω. Again, the new value is assumed with certain arbitrariness. Physically, the ESR increases if the supercapacitor is more heavily aged. Fig. 14 shows the modified model. Note that the values of the other components including C₁ are the same as those in Fig. 8.

Fig. 14.

A modified five-branch RC ladder circuit model to illustrate effects of ESR on Peukert constant.

The modified model is simulated and the results are plotted in Fig. 15. As shown in Fig. 15(a), the delivered charge peaks at 263.20 C when the initial voltage is 2.7 V and the discharge current is 0.01 A while it is 263.23 C in Fig. 9(a). While the drop in the delivered charge is negligible when the discharge current is relatively low (0.5-0.001 A), it is significant for large discharge currents (10-1 A). When the initial voltage is 2.7 V and the discharge current is 10 A, the delivered charge is 176.70 and 202.11 C in Figs. 15(a) and 9(a), respectively. This is because the voltage drop due to the larger R₁ (0.045 Ω in Fig. 14 versus 0.015 Ω in Fig. 8) is an even more significant portion of the supercapacitor terminal voltage drop when the discharge current is relatively large. Consequently, the voltage drop contributed by C₁ is lower and less charge is released. Referring to the model shown in Fig. 14, the voltage drop due to R₁ is 0.45 V when the discharge current is 10 A. The voltage drop contributed by C₁ is therefore 2.24 V when the initial voltage of the discharge process is 2.7 V. On the other hand, when R₁ is 0.015 Ω (Fig. 8), the voltage drop due to R₁ and C₁ is 0.15 and 2.54 V, respectively, as analyzed in Section V-A. Therefore, the discharge time is shorter and the delivered charge is less in Fig. 15(a). The Peukert constants associated with Fig. 15(a) are plotted in Fig. 15(b): it increases from 1.031 to 1.069 when the voltage decreases from 2.7 to 0.7 V. At each of the four voltages, the Peukert constant is greater than its counterpart in Fig. 9(b), e.g., 1.031 versus 1.027 at 2.7 V.

Fig. 15.

Simulation results to illustrate effects of ESR on Peukert constant using Fig. 14. (a) Relationship between delivered charge and discharge current. (b) Dependence of Peukert constant on voltage.

As in Section V-B, the supercapacitor terminal voltage V_T and the second branch capacitor C₂ voltage V₂ simulated using two setups are compared in Fig. 16(a). The initial voltage is 2.7 V and the discharge current is 10 A. The “Small R₁” setup uses R₁ = 0.015 Ω (Fig. 8) and the “Large R₁” setup uses R₁ = 0.045 Ω (Fig. 14). When R₁ is larger, a larger voltage drop occurs at the beginning of the discharge process and the discharge time is shorter: 17.6703 versus 20.2109 s. As a result, the final value of the C₂ voltage V₂ is higher: 1.4967 versus 1.2378 V and its DOD is smaller: 44.6 versus 54.2%, as shown in Fig. 16(b). This observation also holds for the other three voltages and the other branch capacitors. In summary, a larger ESR results in a shorter discharge time, which reduces the branch capacitor DODs and ultimately leads to a larger Peukert constant.

Fig. 16.

Comparisons of simulation results using Figs. 8 and 14. (a) Supercapacitor terminal voltage V_T and second branch capacitor C₂ voltage V₂. (b) DOD for second branch capacitor C₂.

For a relatively large discharge current, the decrease in the discharge time (and consequently the drop in the delivered charge) is also accelerated by the large ESR (R₁) when the initial voltage of the discharge process is lower. For the“Large R₁” setup, the ratio of the discharge time is 9.7% (1.7117 versus 17.6703 s) for the initial voltage of 0.7 and 2.7 V. On the other hand, the discharge time ratio is 19.5% (3.9347 versus 20.2109 s) for the “Small R₁” setup, as analyzed in Section V-A.

VI. A Physical Explanation for Dependence of Peukert Constant on Aging and Temperature

The effects of aging condition and operating temperature on Peukert constant are considered jointly in this section because their impacts on the supercapacitor capacitance and resistance are similar. Specifically, a more heavily aged supercapacitor has a lower capacitance and a higher resistance [40]-[45], which can also be due to a lower operating temperature [37]-[39]. The model with the component values shown in Fig. 11 is used as a baseline setup, which is denoted as C_B.

When the operating temperature is higher, the aging condition is lighter, or the impact of temperature outweighs that of aging if they are competing, the supercapacitor capacitance is higher and the resistance is lower. To model these effects, two setups are constructed: the capacitances are scaled up and the resistances are scaled down. In one setup, the capacitances are 1.1 times of those in the baseline setup and the resistances are 0.9 times, which is denoted as 1.1C_B. For example, C₁ is modified to (43 + 10 × V₁) × 1.1 = 47.3 + 11 × V₁. In the meantime, R₁ is changed to 0.015 × 0.9 = 0.0135 Ω. Similarly, in the 1.2C_B setup, the capacitances are scaled up by a factor of 1.2 and the resistances are scaled down by a factor of 0.8.

On the other hand, the supercapacitor capacitance decreases and the resistance increases if it is more heavily aged, the operating temperature is lower, or because of the combined effects of these two factors. Therefore, in another two setups, the capacitances are scaled down and the resistances are scaled up. In the 0.9C_B setup, the capacitances are 0.9 times of those in the baseline setup and the resistances are 1.1 times. Finally, the 0.8C_B setup scales down the capacitances by a factor of 0.8 and scales up the resistances by a factor of 1.2.

These four setups are simulated using the same settings as those for the baseline setup: nine discharge currents are swept, the initial voltage of the discharge process is fixed at 2.7 V, and the cutoff voltage is set to be 0.01 V. Fig. 17(a) shows the delivered charge results. It is clear that the delivered charge increases from the down-scaled 0.8C_B setup to the up-scaled 1.2C_B setup because of the increasing capacitance and decreasing resistance. The Peukert constant results are plotted in Fig. 17(b). A setup is characterized by its normalized capacitance:

$$C_N=\frac{C}{C_B},$$ (9)

where C is the total capacitance of a setup and C_B is the total capacitance of the baseline setup. When the normalized capacitance increases from 0.8 to 1.2 (the normalized resistance decreases from 1.2 to 0.8 in the meantime), the Peukert constant decreases from 1.037 to 1.031, which is consistent with the experimental results shown in Figs. 6(b) and 7(b): the Peukert constant is lower when the supercapacitor capacitance is larger because of a lighter aging condition or a higher operating temperature. Note that the results shown in Fig. 17 are obtained using the differential capacitance of C₁ = 43 + 10 × V₁ (Fig. 11). The results for the equivalent capacitance case with C_eq = 43 + 10 × V₁ are similar: the Peukert constant decreases from 1.030 to 1.026 when the normalized capacitance increases from 0.8 to 1.2.

Fig. 17.

Simulation results to illustrate effects of aging and temperature (characterized by normalized capacitance) on Peukert constant using Fig. 11. (a) Relationship between delivered charge and discharge current. (b) Dependence of Peukert constant on normalized capacitance.

The supercapacitor terminal voltage profiles are plotted in Fig. 18(a). Note that the discharge current is 1 A. As the normalized capacitance decreases from 1.2 to 0.8, the discharge time decreases from 245.5116 to 160.0077 s. Therefore, the branch capacitors are less deeply discharged and Fig. 18(b) shows that the DODs decrease, which means that they are less responsive to the discharge current and ultimately a larger Peukert constant is observed. Also note that for each of the five setups, the capacitor DOD decreases when the branch time constant increases (i.e., the DOD pattern is C₁ > C₂ > C₃ > C₄ > C₅), as mentioned in Section V-A. In particular, the DOD for C₄ is greater than that for C₅ (e.g., 5.3 versus 1.3% for the 1.2C_B setup).

Fig. 18.

Simulation results to illustrate effects of aging and temperature (characterized by normalized capacitance) on branch capacitor DODs using Fig. 11. (a) Supercapacitor terminal voltage. (b) Branch capacitor DODs.

In summary, Sections V and VI demonstrate that when the initial voltage of the discharge process is higher, the aging condition is lighter, or the operating temperature is higher, more charge is stored in the supercapacitor. Therefore, when a discharge current is applied, the discharge time is extended, which results in higher branch capacitor DODs and ultimately a lower Peukert constant. In this case, the supercapacitor behavior is closer to that of an ideal capacitor with k = 1.

VII. Conclusion

This paper studies the dependence of the supercapacitor Peukert constant on its terminal voltage, aging condition, and operating temperature. Experimental results show that the Peukert constant is lower when the initial voltage of the constant current discharge process is higher, the supercapacitor is relatively lightly aged, or the operating temperature is higher. The physical mechanisms leading to these observations are analyzed. At a higher terminal voltage, more charge is stored in the supercapacitor. A lighter aging condition or a higher operating temperature also results in a larger supercapacitor capacitance and consequently an increase in the charge stored in the supercapacitor. When a discharge current is applied to bring the supercapacitor terminal voltage down to the cutoff voltage, the discharge time is primarily determined by the main branch with the smallest time constant. Since more charge is stored in the supercapacitor, the discharge time is extended and the branch capacitors are more deeply discharged. Therefore, the relaxation effects of the slow branches are reduced and they are more responsive to the discharge current, which makes the supercapacitor behave more like a single capacitor rather than a distributed capacitor network and ultimately results in a lower Peukert constant.

Biography

Hengzhao Yang (M’ 13) received the B.S. degree in optoelectronics from Chongqing University, Chongqing, China, in 2005, the M.S. degree in microelectronics and solid-state electronics from the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China, in 2008, and the Ph.D. degree in electrical and computer engineering from the Georgia Institute of Technology, Atlanta, GA, USA, in 2013. Since 2016, he has been an Assistant Professor with California State University, Long Beach, CA, USA. He was a Postdoctoral Fellow at the Georgia Institute of Technology from 2013 to 2015 and a Visiting Assistant Professor with Miami University from 2015 to 2016. His current research interests include supercapacitor modeling and characterization, design and control of energy storage systems, and power electronics for energy storage applications.