Peukert’s Law for Supercapacitors with Constant Power Loads: Applicability and Application

Abstract

This paper examines the applicability of Peukert’s law to supercapacitors with constant power loads and the application of this relationship in predicting the supercapacitor discharge time during a constant power discharge process. Originally developed for lead-acid batteries, Peukert’s law states that the delivered charge increases when the discharge current decreases. This paper shows that Peukert’s law applies to supercapacitors when the discharge power is above a certain threshold and does not apply anymore when the discharge power is sufficiently low. This pattern is due to the combined effects of three aspects of the supercapacitor physics: porous electrode structure, charge redistribution, and self-discharge. Based on the applicability study, this paper demonstrates the effectiveness of Peukert’s law in predicting the supercapacitor discharge time during a constant power discharge process by conducting extensive experiments using three supercapacitor samples with different rated capacitances from different manufacturers at various voltages.

I. Introduction

Energy storage is becoming an increasingly critical asset in many systems especially in smart grid and electric vehicles. For instance, 1749 operational or announced projects totaling a rated power of 195.75 GW have been reported to the DOE Global Energy Storage Database [1] as of August 2018. The significant growth of global energy storage installation is due to the huge technical and economic benefits introduced by a variety of applications and use cases of these systems. In fact, 17 energy storage applications grouped into five categories [2] have been identified and analyzed.

Energy storage technologies are different in terms of characteristics such as energy density, power density, cycle life, leakage rate, and ramp rate [3]. In the electrochemical technology category, batteries and supercapacitors are complementary in many aspects: high power density, low energy density, and long cycle life for supercapacitors versus the opposite for batteries. Therefore, in addition to using supercapacitors or batteries separately, some systems combine them and adopt hybrid configurations. As a matter of fact, supercapacitor-based energy storage systems have been employed in microgrids for various applications [4]. Typical use cases include balancing the power mismatch between source and load [5]–[7], sharing fast and slow power components between supercapacitors and batteries [8]–[10], and improving the power quality [11]–[13].

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. Various equivalent circuit models [14]–[20] have been proposed to reduce the complexity of the generic ladder circuit model. Moreover, numerous frameworks [21]–[24] have been developed to identify supercapacitor parameters and estimate supercapacitor states.

Recently, the impact of the supercapacitor physics especially the charge redistribution process on various aspects of the supercapacitor behavior has been investigated. Charge redistribution is a relaxation process originated from the porous structure of the electrodes. In addition to the physical mechanisms [25] leading to this process, its effects on power management strategies in wireless sensor networks [26], [27], supercapacitor terminal voltage behavior [28], [29], supercapacitor charge capacity [30]–[32], and supercapacitor capacitance characterization methods [33] have been extensively studied. In particular, the supercapacitor behavior during a constant power discharge process has been investigated [34], [35]. A constant power load is present when a power converter tightly regulates its output, which introduces a destabilizing effect usually referred to as the negative impedance instability [36], [37]. Therefore, a better understanding of the behavior of supercapacitors with constant power loads can facilitate designing and implementing more efficient, reliable, and resilient energy storage systems.

This paper develops [34], [35] in two aspects. First, the impact of charge redistribution on the amount of energy delivered during a constant power discharge process is investigated in [34] and it is found that the delivered energy increases when the discharge power decreases, i.e., Peukert’s law [38] applies. The swept discharge power range is extended in [35] and it is revealed that Peukert’s law only applies when the discharge power is above a certain threshold because self-discharge results in a significant energy loss if a sufficiently low discharge power is applied. While the initial voltage of the constant power discharge process is fixed at the rated voltage in [34], [35], this paper examines other initial voltages from which the supercapacitor is discharged. Second, this paper further demonstrates that the supercapacitor discharge time during a constant power discharge process can be predicted using Peukert’s law at various initial voltages while the initial voltage is fixed at the rated voltage in [34]. These extensions lead to new results and observations, as elaborated in the rest of this paper.

The remainder of this paper is organized as follows. Section II studies the applicability of Peukert’s law to supercapacitors when a constant power load is applied. Section III demonstrates the application of Peukert’s law in predicting the supercapacitor discharge time during a constant power discharge process. Section IV concludes this paper.

II. Applicability of Peukert’s Law to Supercapacitors with Constant Power Loads

This section examines the applicability of Peukert’s law to supercapacitors when a constant power load is applied. The three supercapacitor samples listed in Table I are tested using an automated Maccor Model 4304 tester at room temperature. Section II-A investigates the relationship between the delivered energy and the discharge power for sample 2 when it is discharged from its rated voltage of 2.7 V. This relationship is explained by analyzing a five-branch RC ladder circuit model and the effects of the supercapacitor physics on the delivered energy pattern are elaborated in Section II-B. Section II-C examines the delivered energy patterns for another two initial voltages (2.35 and 2 V) from which sample 2 is discharged. Finally, Section II-D presents the results for samples 1 and 3 at the three initial voltages: 2.7, 2.35, and 2 V.

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

A. Relationship Between Delivered Energy and Discharge Power for Supercapacitor Sample 2 at 2.7 V

Originally developed for lead-acid batteries, Peukert’s law [38] 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.

As revealed in [34], [35], Peukert’s law applies to supercapacitors with constant power loads if the discharge power is above a certain threshold. The main results are summarized as follows. To study the applicability of Peukert’s law, a set of constant power discharge experiments is performed for each sample. The constant discharge power is swept depending on the supercapacitor sample specifications and the supercapacitor tester capabilities. Take sample 2 for instance. Fig. 1(a) shows the measured supercapacitor voltage when the constant discharge power is 1 W. During this experiment, the supercapacitor is first conditioned by ten charging-redistribution-discharging cycles to minimize the effect of residual charge. It is then charged by a constant voltage source of 2.7 V for 3 hours, which is designed to fully charge the supercapacitor. After that, a 1 W constant discharge power is applied and the supercapacitor is discharged to half of the rated voltage (i.e., 1.35 V), which is a typical minimum operating voltage for supercapacitors in practice. Taking 2.7 V as the initial voltage and 1.35 V as the cutoff voltage, the energy delivered during this constant power discharge process is calculated as

$$E=Pt,$$ (2)

where P is the discharge power and t is the discharge time. For this experiment, the delivered energy E is 271.08 J.

Fig. 1.

Experimental results for supercapacitor sample 2 when initial voltage of discharge process is 2.7 V. (a) A 1 W constant power discharge experiment. (b) Relationship between delivered energy and discharge power.

The discharge power is swept and the delivered energy during each experiment is calculated. The highest discharge power is determined based on the maximum continuous current of sample 2 and the minimum terminal voltage during the experiment. While the maximum continuous current is not specified in the datasheet for this device, the value of 10 A is determined according to the datasheets for similar 100 F supercapacitors tested in [30]. The highest discharge power is therefore 13.5 W: P_max = I_maxV_min = 10 × 1.35 = 13.5 W. This value is then scaled down as follows: 6.75, 1.35, 0.675, 0.135, 0.0675, 0.0135, and 0.00675 W. Together with 13.5 and 1 W, nine power levels are swept and the delivered energy results are shown in Fig. 1(b). The delivered energy pattern is partitioned into two pieces: Peukert’s law applies when the discharge power is above a certain threshold and does not apply anymore when the discharge power is below the threshold. Specifically, the delivered energy increases from 233.01 to 293.15 J when the discharge power decreases from 13.5 to 0.0135 W. On the other hand, the delivered energy decreases from 293.15 to 292.78 J when the discharge power decreases from 0.0135 to 0.00675 W. Therefore, Peukert’s law is applicable only when the discharge power is within a certain range, which is 13.5-0.0135 W in this case. When a constant power load is applied, Peukert’s law can be written as

$$P^{k}t=E_0,$$ (3)

where E₀ is the nominal energy delivered at a certain discharge power P₀.

B. Effects of Supercapacitor Physics on Delivered Energy

To illustrate the effects of the supercapacitor physics on the delivered energy, the five-branch RC ladder circuit model for 100 F supercapacitors shown in Fig. 2 is analyzed, which is conceived in [30], [31] to investigate the impact of the supercapacitor physics on the charge delivered during a constant current discharge process. The supercapacitor terminal voltage is denoted as V_T. The five RC branches (R₁ through C₅) 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 models the self-discharge behavior. 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

$${\tau }_i=C_i\sum _{k=1}^i{R}_k,$$ (4)

and the porous electrode theory gives that

$${\tau }_1<{\tau }_2<\cdots <{\tau }_n.$$ (5)

Fig. 2.

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 equivalent series resistance (ESR) value specified in the sample 2 datasheet. The other four branch resistances are calculated using (4) 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 of sample 2.

This model is implemented and simulated in LTspice. The initial voltages of the five branch capacitors are set to be 2.7 V and the discharge power is swept. The delivered energy associated with the cutoff voltage of 1.35 V is calculated. Fig. 3 plots the simulated relationship between the delivered energy and the discharge power, which is consistent with the experimental results shown in Fig. 1(b). Specifically, Peukert’s law applies when the discharge power decreases from 27 to 0.02 W and does not apply anymore when the discharge power decreases from 0.02 to 0.00338 W.

Fig. 3.

Simulated relationship between delivered energy and discharge power.

The effects of the supercapacitor physics on the delivered energy can be illustrated as follows. Consistent with [30], [31], three aspects of the supercapacitor physics contribute to the delivered energy pattern: porous electrode structure, charge redistribution, and self-discharge. Due to the distributed nature of the supercapacitor capacitance and resistance, a slower RC branch with a larger time constant will be accessed during the extended discharge phase when a lower discharge power is applied, which means that slow branch capacitors are more deeply discharged and therefore more energy is released during this process. In the meantime, since the supercapacitor is fully charged at the beginning of the discharge process, charge is redistributed from slow branches to fast branches during the discharge phase, which decelerates the voltage drop in fast branches and extends the discharge phase. Therefore, charge redistribution also results in an increase in the delivered energy when the discharge power is lower. On the other hand, when a sufficiently low discharge power is applied, the discharge phase is significantly extended and self-discharge leads to a noticeable energy loss, which ultimately results in a drop in the delivered energy.

To further illustrate the impact of the supercapacitor physics, Fig. 4 shows the simulated terminal and branch capacitor voltages when the discharge power is 1.35 and 0.00675 W, respectively. When a discharge power is applied to bring the supercapacitor terminal voltage down to the cutoff voltage, the charge stored in the branch capacitors is extracted progressively from the first branch with the smallest time constant to the last branch with the largest time constant. A lower discharge power requires a longer time to reduce the supercapacitor terminal voltage (mainly the voltage across the first branch capacitor) to the cutoff voltage. As the discharge time extends, more charge stored in slow branches with large time constants is extracted. In the meantime, the charge stored in slow branches redistributes to the first branch, which decelerates the drop in the terminal voltage and prolongs the discharge time. As shown in Fig. 4(a), the terminal voltage V_T almost overlaps with V₁. Charge is mainly extracted from C₁ and its voltage drops rapidly. Because of the increasingly larger time constant of the corresponding branch, the drop in V₂ – V₅ becomes smaller at a particular point of time compared to the drop in V₁. Given the voltage differences between V₂ – V₅ and V₁, charge is unidirectionally redistributed from C₂ – C₅ to C₁ during the entire discharge process, which decelerates the drop in V₁. When the discharge power is lower, these effects are more significant and therefore more charge is extracted from the supercapacitor. During the entire discharge phase, energy is dissipated by R₆ representing the self-discharge process. Since the discharge time is relatively short (i.e., 175.0213 s) when the discharge power is 1.35 W, the energy loss due to self-discharge is insignificant. In summary, when the discharge power is relatively high, the charge redistribution process dominates and the energy loss due to self-discharge is negligible, which lead to a relationship between the delivered energy and the discharge power that can be described by Peukert’s law.

Fig. 4.

Simulated terminal and branch capacitor voltages. (a) Discharge power is 1.35 W. (b) Discharge power is 0.00675 W.

On the other hand, Fig. 4(b) shows the simulation results when the discharge power is 0.00675 W. All the branch capacitor voltages are closer to the cutoff voltage of 1.35 V at the end of the extended discharge phase (i.e., 38558.0630 s) compared to Fig. 4(a). Therefore, a lower discharge power discharges the branch capacitors more deeply, which tends to result in an increase in the delivered energy. In the meantime, the impact of self-discharge becomes significant during the extended discharge phase, which may lead to a noticeable drop in the delivered energy. The overall effect of the three aspects of the supercapacitor physics determines the discharge power threshold below which the delivered energy decreases when the discharge power decreases, i.e., the relationship between the delivered energy and the discharge power is not consistent with Peukert’s law anymore.

C. More Experimental Results for Supercapacitor Sample 2

Based on the experiment procedure illustrated in Section II-A, this section examines the relationship between the delivered energy and the discharge power for sample 2 when the initial voltage of the discharge process is 2.35 and 2 V, as plotted in Fig. 5. Together with 2.7 V, the initial voltage is linearly swept. In general, the delivered energy pattern is similar when the initial voltage varies: Peukert’s law applies when the discharge power is above a certain threshold and does not apply anymore if the discharge power is below the threshold. Although the discharge power is relatively sparsely swept, it can still be observed that the discharge power threshold is different when the initial voltage varies. Specifically, the threshold above which Peukert’s law applies is 0.0135 W for 2.7 V, as shown in Fig. 1(b). For 2.35 and 2 V, Fig. 5 shows that Peukert’s law still applies when the discharge power decreases from 0.0135 to 0.00675 W, i.e., the applicable discharge power range is wider when the initial voltage is lower. To verify if this observation is generally applicable and furthermore to reveal the physical mechanisms accounting for it, more work needs to be conducted.

Fig. 5.

Relationship between delivered energy and discharge power for supercapacitor sample 2. (a) Initial voltage of discharge process is 2.35 V. (b) Initial voltage of discharge process is 2 V.

D. Experimental Results for Supercapacitor Samples 1 and 3

While Sections II-A and II-C focus on sample 2, this section presents the results for the other two samples listed in Table I. For brevity, Fig. 6 only shows the delivered energy results for samples 1 and 3 when the initial voltage of the discharge process is 2.7 V. For both samples, the delivered energy pattern is partitioned into two pieces for all the three initial voltages examined. Moreover, the discharge power threshold differs as the initial voltage varies. For sample 1, the threshold is 0.00675 W when the initial voltage is 2.7/2.35 V and 0.00135 W for 2 V. For sample 3, the threshold is 0.09 W for 2.7/2.35 V and 0.018 W for 2 V. Since the three samples are made by different manufacturers and their specifications vary in terms of parameters such as rated capacitance and leakage current, the similarities in their delivered energy patterns suggest that the relationship between the delivered energy and the discharge power can be utilized by a variety of supercapacitors with different ratings. As for the discharge power threshold above which Peukert’s law applies, a rigorous and systematic study needs to be conducted to examine the effects of various factors such as supercapacitor manufacturer, device physics, rated capacitance, rated voltage, leakage current, operating temperature, and aging condition on this parameter.

Fig. 6.

Relationship between delivered energy and discharge power for supercapacitor samples 1 and 3 when initial voltage of discharge process is 2.7 V. (a) Sample 1. (b) Sample 3.

III. Application of Peukert’s Law in Supercapacitor Discharge Time Prediction

A. Problem Statement

Since Peukert’s law applies to supercapacitors when the constant power load is above a certain threshold, it follows that the supercapacitor discharge time during a constant power discharge process can be predicted using this relationship, as demonstrated in [34] using four 100 F supercapacitor samples at the rated voltage of 2.7 V. This paper develops this study and considers other rated capacitances (i.e., from 100 F only to three values: 10, 100, and 350 F) and other initial voltages of the constant power discharge experiment (i.e., from 2.7 V only to three values: 2.7, 2.35, and 2 V). The supercapacitor samples listed in Table I are examined and the results presented in Section II are utilized.

The problem considered in this section states as follows. For all the three supercapacitor samples, first determine the discharge power range between which Peukert’s law applies when the initial voltage of the discharge process varies. Then use a subset of the experiments within this power range to estimate the Peukert constant. Finally, predict the discharge time for the remaining experiments within this power range and evaluate the prediction accuracy. Take sample 2 for instance. Peukert’s law applies for the power range of 13.5-0.0135 W for the initial voltage of 2.7 V. The power range is 13.5-0.00675 W for 2.35 and 2 V. For consistency, the narrower range of 13.5-0.0135 W is considered for all the three initial voltages. For each voltage, five among the eight experiments are used to estimate the Peukert constant (i.e., the 13.5, 1.35, 1, 0.135, and 0.0135 W experiments form the training set), which is then used to predict the discharge time for the remaining three experiments (i.e., the 6.75, 0.675, and 0.0675 W experiments form the testing set). Note that the training set is selected in a way that its power range covers that of the testing set to ensure that the Peukert constant estimated using the training set is applicable to the testing set. In particular, the 1 W experiment is included in the training set because the energy delivered during this experiment is used as the nominal energy. This setup emulates a typical application scenario in which the Peukert constant determined through a set of constant power discharge experiments at a particular voltage is used to predict the discharge time when a different discharge power is applied to the supercapacitor at the same voltage.

B. Methodology

This section illustrates the supercapacitor discharge time prediction methodology using the 2.7 V dataset of sample 2 shown in Fig. 1(b). As indicated by (3), the Peukert constant and the nominal energy are needed to predict the discharge time. To estimate the Peukert constant, two curve fitting functions are adopted. The first is a direct application of (3):

$$t=\frac{E_0}{P^k},$$ (6)

which uses the discharge power and the corresponding discharge time to fit the Peukert constant. The energy delivered during the 1 W experiment (i.e., P₀ = 1 W) is taken as the nominal energy E₀.

The second function utilizes the normalized discharge power (i.e., P/P₀) and the normalized energy (i.e., E/E₀). Specifically, Peukert’s law in (3) can be rewritten as

$$(Pt)(P^{k-1})=E_0.$$ (7)

Note that for the actual discharge power P, (2) holds: Pt = E. Therefore, (7) can be rearranged as

$$\frac{E}{E_0}=P^{1-k}.$$ (8)

Given P₀ = 1 W, the right-hand side of (8) can be rewritten as follows for consistency, i.e., both sides are normalized:

$$\frac{E}{E_0}={\left(\frac{P}{P_0}\right)}^{1-k}.$$ (9)

For both functions, Fig. 7 shows the curve fitting results in MATLAB. The fit determined using (6) and (9) is denoted as “Fit: Direct” and “Fit: Normalized”, respectively. The fitted Peukert constant is 1.018 and 1.025, respectively, as listed in Table II. For brevity, “Fit: Direct” and “Fit: Normalized” are written as Fit_D and Fit_N in this table, respectively.

Fig. 7.

Fitted Peukert constant for supercapacitor sample 2 when initial voltage of discharge process is 2.7 V. (a) Fit: Direct. (b) Fit: Normalized.

TABLE II.

Discharge Time Prediction Setups for Supercapacitor Sample 2 at 2.7 V.

	Rated	Constant	FitD	FitN	Optimal
k	1	1	1.018	1.025	1.021
E0 (J)	273.375	271.08	271.08	271.08	271.08
tp1 (s)	40.5	40.16	38.80	38.29	38.58
tp2 (s)	405	401.6	404.5	405.6	404.9
tp3 (s)	4050	4016	4216	4296	4250
δ1 (%)	9.70	8.78	5.10	3.70	4.50
δ2 (%)	0.23	0.61	0.09	0.37	0.21
δ3 (%)	4.55	5.35	0.65	1.25	0.16
$\overline{\delta }$ (%)	4.83	4.91	1.95	1.77	1.62

The discharge time for the 6.75, 0.675, and 0.0675 W experiments in the testing set is then predicted using (6). The prediction accuracy is evaluated as follows:

$$\delta =\frac{\mid t_p-t_m\mid }{t_m}\times 100\%,$$ (10)

where t_p is the predicted discharge time and t_m is the measurement. For the 6.75, 0.675, and 0.0675 W experiments, the measured discharge time is 36.92, 404.08, and 4243.14 s, respectively. As listed in Table II, the predicted discharge time for the three experiments is denoted as t_p1, t_p2, and t_p3, respectively. Correspondingly, the prediction error is denoted as δ₁, δ₂, and δ₃, respectively. In addition, the average of the three errors is calculated and denoted as $\overline{\delta }$, which is used as the metric to evaluate the overall prediction accuracy. The predicted discharge time and the prediction errors associated with the two fitted Peukert constants are listed in the Fit_D and Fit_N columns in Table II, respectively.

While the two Peukert constants fitted using the training set serve as good estimates of the actual value, it remains unclear if they result in the minimum error. Therefore, to determine the optimal Peukert constant that leads to the minimum error, Fig. 8 sweeps the Peukert constant and plots the corresponding average error. The minimum average error is observed at k = 1.021, which is referred to as the optimal value. The prediction results associated with this value are listed in the “Optimal” column in Table II.

Fig. 8.

Optimal Peukert constant for supercapacitor sample 2 when initial voltage of discharge process is 2.7 V.

For comparisons, Table II also shows the results for another two setups in which the Peukert constant is set as k = 1. In this case, the delivered energy does not vary with the discharge power, i.e., Peukert’s law is not utilized. This assumption does not take into account the impact of the supercapacitor physics on the delivered energy. Therefore, it can be anticipated that this assumption leads to significant errors when it comes to predicting the supercapacitor discharge time, which is indeed the case, as shown in Table II. As for the nominal energy, the “Constant” setup uses the energy delivered during the 1 W experiment (i.e., E₀ = 271.08 J). The “Rated” setup uses the energy calculated based on the supercapacitor rated capacitance (100 F) and the beginning (V₁ = 2.7 V) and termination (V₂ = 1.35 V) voltages of the constant power discharge process: E_Rated = 0.5C($V_1^2-V_2^2$) = 273.375 J. Table II shows that the prediction error is significantly reduced when the fitted or optimal Peukert constant is used compared to the cases in which Peukert’s law is not utilized. As for the overall prediction accuracy, the optimal Peukert constant results in an average error of 1.62%, which is much smaller than that in the “Rated” (4.83%) or “Constant” (4.91%) setup.

C. Results

Using the methodology illustrated in Section III-B, the prediction errors for the other two datasets of sample 2 are evaluated and the results are summarized in Table III. For brevity, only the average errors are included. Tables IV and V list the results for samples 1 and 3, respectively. The main observation on Table III-V is that for all the three samples at all the three initial voltages, the average error associated with the Fit_D, Fit_N, or “Optimal” setup is smaller than that resulted from the “Rated” or “Constant” setup, which demonstrates the effectiveness of predicting the supercapacitor discharge time using Peukert’s law within the applicable discharge power range. These results also show that the fitted Peukert constants usually deviate from the optimal value that leads to the minimum error, which suggests that more effective methods are needed to estimate the optimal Peukert constant.

TABLE III.

Discharge Time Prediction Results for Supercapacitor Sample 2 at Three Voltages.

Voltage		Rated	Constant	FitD	FitN	Optimal
2.7 V	k	1	1	1.018	1.025	1.021
	E0 (J)	273.375	271.08	271.08	271.08	271.08
	$\overline{\delta }$ (%)	4.83	4.91	1.95	1.77	1.62
2.35 V	k	1	1	1.026	1.047	1.032
	E0 (J)	185	162.46	162.46	162.46	162.46
	$\overline{\delta }$ (%)	19.61	11.69	7.11	6.23	6.16
2 V	k	1	1	1.034	1.061	1.053
	E0 (J)	108.875	92.58	92.58	92.58	92.58
	$\overline{\delta }$ (%)	28.04	17.28	11.05	9.94	9.93

TABLE IV.

Discharge Time Prediction Results for Supercapacitor Sample 1 at Three Voltages.

Voltage		Rated	constant	FitD	FitN	optimal
2.7 V	k	1	1	1.035	1.037	1.040
	E0 (J)	27.3375	26.07	26.07	26.07	26.07
	$\overline{\delta }$ (%)	6.84	8.07	1.29	0.87	0.34
2.35 V	k	1	1	1.055	1.059	1.062
	E0 (J)	18.5	14.59	14.59	14.59	14.59
	$\overline{\delta }$ (%)	12.69	12.52	2.36	1.54	1.09
2 V	k	1	1	1.067	1.071	1.074
	E0 (J)	10.8875	7.85	7.85	7.85	7.85
	$\overline{\delta }$ (%)	17.72	15.12	3.10	2.28	1.67

Another observation on Table III-V is that the supercapacitor Peukert constant increases when the initial voltage of the constant power discharge process decreases, which applies to all the three samples. For instance, the optimal Peukert constant for sample 2 increases from 1.021 to 1.032 and finally to 1.053 when the initial voltage decreases from 2.7 to 2.35 and finally to 2 V. This dependence of the supercapacitor Peukert constant on the initial voltage of the constant power discharge process is similar to the one for supercapacitors during constant current discharge processes. In fact, the dependence of the supercapacitor Peukert constant associated with a constant current discharge process on its terminal voltage, aging condition, and operating temperature has been investigated [39]. Experimental and simulation 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. Moreover, 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. For the supercapacitor Peukert constant associated with a constant power load, a study of its dependence on voltage, aging, and temperature is being conducted.

The results in Table III-V also suggest that the Peukert constant is smaller when the supercapacitor rated capacitance is larger. At each of the three initial voltages, the Peukert constant decreases when the supercapacitor rated capacitance increases from 10 to 350 F. For example, the optimal Peukert constant at 2.7 V is 1.040, 1.021, and 1.009 for samples 1-3, respectively. A similar observation holds for the Peukert constant associated with a constant current discharge process [39]. To examine if these two observations are generally applicable and furthermore to reveal the physical mechanisms accounting for them, extensive experiments with a much wider range of supercapacitor samples and a systematic investigation of the supercapacitor electrochemical processes are needed.

TABLE V.

Discharge Time Prediction Results for Supercapacitor Sample 3 at Three Voltages.

Voltage		Rated	Constant	FitD	FitN	Optimal
2.7 V	k	1	1	1.008	1.011	1.009
	E0 (J)	956.8125	1049.06	1049.06	1049.06	1049.06
	$\overline{\delta }$ (%)	8.69	1.28	0.35	0.40	0.33
2.35 V	k	1	1	1.017	1.019	1.017
	E0 (J)	647.5	641.83	641.83	641.83	641.83
	$\overline{\delta }$ (%)	2.17	1.99	0.47	0.48	0.47
2 V	k	1	1	1.026	1.028	1.026
	E0 (J)	381.0625	363.29	363.29	363.29	363.29
	$\overline{\delta }$ (%)	3.98	3.08	0.69	0.69	0.69

Finally, it should be noted that while this paper demonstrates the usefulness of Peukert’s law in predicting the discharge time for supercapacitors with constant power loads, this law has also been used to predict the supercapacitor discharge time during a constant current discharge process [40]. Based on the work with constant power and current loads reported in this paper and [40], the supercapacitor discharge time prediction methodologies are being extended to application scenarios in which arbitrary loads are applied to supercapacitors (i.e., a load composed of varying current and/or power). For batteries, various techniques have been developed to convert the varying discharge currents to “effective” constant discharge currents so that Peukert’s law can be utilized [38]. For supercapacitors, similar extensions are being made.

IV. Conclusion

This paper first investigates the applicability of Peukert’s law to supercapacitors when a constant power load is applied. Experimental results show that Peukert’s law applies when the discharge power is above a certain threshold and does not apply anymore when the discharge power is sufficiently low. This is due to the combined effects of three aspects of the supercapacitor physics: porous electrode structure, charge redistribution, and self-discharge. Specifically, the porous electrode structure and the charge redistribution process result in an increase in the delivered energy when a lower discharge power is applied. The impact of self-discharge is negligible when the discharge power is relatively high. If the discharge power is sufficiently low, self-discharge results in a significant energy loss and consequently a drop in the delivered energy.

Based on the applicability study, this paper further demonstrates that Peukert’s law can be utilized to predict the supercapacitor discharge time during a constant power discharge process. Evaluation results show that the prediction error can be significantly reduced when the Peukert constant is properly determined. Moreover, this paper shows that the Peukert constant increases when the initial voltage of the constant power discharge process decreases, which stimulates further research on the dependence of the Peukert constant on various factors such as the supercapacitor terminal voltage, aging condition, and operating temperature.

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 with 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.