MODELING OF ULTRACAPACITOR SHORT-TERM AND LONG-TERM DYNAMIC BEHAVIOR. A Thesis. Presented to. The Graduate Faculty of The University of Akron

Transcription

1 MODELING OF ULTRACAPACITOR SHORT-TERM AND LONG-TERM DYNAMIC BEHAVIOR A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Yang Wang August, 8

2 MODELING OF ULTRACAPACITOR SHORT-TERM AND LONG-TERM DYNAMIC BEHAVIOR Yang Wang Thesis Approved: Accepted: Co-Advisor Dr. Joan Carletta Dean of the College Dr. George K. Haritos Co-Advisor Dr. Robert Veillette Dean of the Graduate School Dr. George R. Newkome Co-Advisor Dr. Tom T. Hartley Date Department Chair Dr. Alex De Abreu Garcia ii

3 ABSTRACT In this thesis several short-term models and a long-term model have been developed for a NESSCAP35P ultracapacitor. For the short-term ultracapacitor models, first-, second-, third- and fourth-order transfer functions consistent with an RC ladder model are assumed. The transfer function coefficients are identified by a least squares algorithm based on experimental data consisting of time-varying current excitations and the resulting terminal voltage responses. A long-term model with six RC branches is developed by fitting the terminal voltage transient response to an impulse charging current. Hundreds of thousands of terminal voltage data points are recorded and least squares identification is employed to determine the optimal values of the unknown parameters in the long-term model. From the ultracapacitor models derived, terminal voltages under different current profiles can be determined accurately over the time frame of one hour with an error less than. V, the impulse charging and discharging response over a time frame of two months can be simulated with an error less than.8 V, and the instantaneous power available can be calculated. iii

4 ACKNOWLEDGEMENTS First I would like to take this opportunity to express my sincere appreciation to my advisor Dr. Joan Carletta for her valuable guidance and encouragement throughout this thesis. I am always illuminated from talking with Dr. Joan Carletta about solving problems encountered in my research. The helpful and patient advice from my co-advisors Dr. Robert Veillette and Dr. Tom T. Hartley are gratefully acknowledged. Throughout my Masters project, I encountered a lot of problems about how to get a good short term model to represent the dynamic behavior of the ultracapacitor and how to do a good fit for an extremely long time data. Each time Dr. Veillette and Dr. Hartley spend lots of time to give me necessary background and give me wise idea to solve them. Through their help, not only I enlarge my knowledge but also learn the techniques for how to analyze a certain problem and how to solve it. Special thanks are given to Dr. James Grover for his patient help to debug hardware problems with the Microprocessor Interface Board. My deepest gratitude goes to my family who provides love and support more than I could ever expect. iv

5 TABLE OF CONTENTS Page LIST OF TABLES...viii LIST OF FIGURES... ix CHAPTER I. INTRODUCTION.... Advantages of Ultracapacitors.... Disadvantage of Ultracapacitors The Need for Ultracapacitor Models Contributions of Research Thesis Outline... 8 II. BACKGROUND AND RELATED WORK.... Structure of Ultracapacitors..... Composition..... Electric double layer structure The Physical Model of Ultracapacitor Theoretical lumped parameter model One-branch model Three-branch linear model Three-branch non-linear model... v

6 ..5 Transmission line model Models based on frequency response data Summary... 8 III. EXPERIMENTAL SET-UP Test Circuit Data Acquisition Flow Chart IV. SHORT-TERM MODEL Procedure for Fitting a Model of Given Order Example Model Fitting: the Third-order Model The First-order Model The Second-order Model The Third-order Model The Fourth-order Model Comparison of Models Conclusions V. LONG-TERM MODEL Experiments for the Long-term Behavior Long-term Model Parameter Identification Validation Instantaneous Power Discussion Conclusions VI. CONCLUSIONS vi

7 BIBLIOGRAPHY vii

8 LIST OF TABLES Table Page. NESSCAP35P specification Poles for the four short-term models of different orders Zeros for the four short-term models of different orders Integral squared voltage errors in V s for three different datasets viii

9 LIST OF FIGURES Figure Page. Temperature dependence of ultracapacitor parameters, from [9] Ultracapacitor structure.... The structure of activated carbon, from [4]....3 Porous structure of activated carbon, from [5]....4 Stern s electrical double layer (EDL) model, from [] Ultracapacitor theoretical model One-branch model without parallel resistor One-branch model with parallel resistor Three-branch linear model Three-branch non-linear model.... Two-branch non-linear model.... Transmission line model Modified transmission line model Equivalent circuit of ultracapacitor, from [36] Comparison of measured data and modeled data in frequency domain, from [36] Approximation of Z p through N RC circuits, from [36] Frequency, temperature and terminal voltage model, from [35]... 8 ix

10 3. Test circuit set-up Analog to digital converter quality test Flow chart to capture current and voltage data, package them and send them out from UART Ultracapacitor model as RC transmission line Voltage and current data used to identify the coefficients in the transfer function Measured voltage and current for the first validation test Measured voltage and current for the second validation test Measured voltage and current for the third validation test Third-order circuit model Third-order modified circuit model Terminal voltage as a function of time for the experiment to determine the effective parallel resistor Finding coefficient k in Eq. (4.5) First-order circuit model Transfer function coefficient identification for first-order model Comparison of simulated and measured voltages of the first validation for the first-order modified model Comparison of simulated and measured voltages of the second validation for the first-order modified model Comparison of simulated and measured voltages of the third validation for the first-order modified model Second-order circuit model Transfer function coefficient identification for second-order model Comparison of simulated and measured voltages of the first x

11 validation for the second-order modified model Comparison of simulated and measured voltages of the second validation for the second-order modified model Comparison of simulated and measured voltages of the third validation for the second-order modified model Transfer function coefficient identification for the third-order model Comparison of simulated and measured voltages of the first validation for the third-order modified model Comparison of simulated and measured voltages of the second validation for the third-order modified model Comparison of simulated and measured voltages of the third validation for the third-order modified model Fourth-order circuit model Transfer function coefficient identification for the fourth-order model Comparison of simulated and measured voltages of the first validation for the fourth-order modified model Comparison of simulated and measured voltages of the second validation for the fourth-order modified model Comparison of simulated and measured voltages of the third validation for the fourth-order modified model Comparison of the errors generated during the first validation test by the first-, second-, third- and fourth-order modified models Comparison of the errors generated during the second validation test by the first-, second-, third- and fourth-order modified models Comparison of the errors generated during the third validation test by the first-, second-, third- and fourth-order modified models Charging current profile used at beginning of the charging-then-relaxing test Terminal voltage of the ultracapacitor during the charging-then-relaxing test... 7 xi

12 5.3 Discharging current profile used at beginning of the discharging-thenrelaxing test Terminal voltage of the ultracapacitor during the discharging-then-relaxing test Using the summation of exponential functions to curve fit the terminal voltage from charging-then-relaxing test, and comparing the error between them Validating the long term model by exciting it with the charging current profile Using Equation (5.6) multiplied by the discharging current area to approximate the voltage transients from discharging-then-relaxing test, and comparing the error between them Validating the long term model by exciting it with discharging current profile First Foster Form derived from Equation (5.) Corrected Foster Form with ESR First Cauer Form derived from the corrected Foster Form Instantaneous energy calculation with load based on First Cauer Form xii

13 CHAPTER I INTRODUCTION Ultracapacitors were first used in military projects to start the engines of battle tanks and submarines and to replace batteries in missiles. With the maturity of the manufacturing and nano-material technology, the cost of ultracapacitors has fallen, and the nominal capacitances have increased significantly. As a result, ultracapacitors have begun to appear in more applications, such as diesel engine starting, railroad locomotives, actuators and memory backup. More recently, ultracapacitors have become a topic of some interest in the green energy world, where their ability to soak up energy quickly makes them particularly suitable for regenerative braking applications [,,3]; in contrast, batteries have difficulty in this application due to their lower rated charging current and shorter cycle life.. Advantages of Ultracapacitors Ultracapacitors, also known as electrochemical double layer capacitors (EDLC) or supercapacitors, are new energy storage devices that have advantages over other energy storage devices. In terms of energy density, existing commercial ultracapacitors range from to Wh/kg [4]. Power density for ultracapacitors may typically range from to 5 W/kg [4], and some newer ultracapacitors have higher power density. In contrast,

14 the energy density for the bipolar lead-acid battery is typically from 4 to 7 Wh/kg and the power density is around 45 W/kg [5], the energy density for modern lithium-ion batteries is from 5 to Wh/kg and the power density is from 3 to 5 W/kg [6], and for automobile applications gasoline has an energy density around, Wh/kg. Although existing ultracapacitors have energy densities that are only / those of some batteries, their power densities are generally ten to one hundred times greater than those of batteries. This special feature makes ultracapacitors a unique fit for applications that require pulse power, such as burst-mode communication for wireless systems, writing to disk and LCD operation for digital cameras, and starting vehicles. As a result, ultracapacitors are becoming more widely used as energy storage devices. In addition to their high power densities, ultracapacitors have several other advantages over other energy storage devices. They have high efficiency, can operate with high currents and over wide temperature ranges, have long cycle life and are environmentally friendly. Each of the advantages is described in more detail next. Coulombic efficiency is defined as the ratio of the number of electrons discharged to the number of electrons that need to be recharged in order to bring an energy storage device back to its original state of charge (SOC). Coulombic efficiency of ultracapacitors is as high as 99% [7]. In addition, ultracapacitors have high round trip efficiency. The round trip efficiency is defined as the ratio of the electrical energy produced after charging and discharging the storage system to the electrical energy required from the charging source. At a five-second rate (discharging to half rated voltage in five seconds, and recharging at the same rate until the ultracapacitor is fully charged), the round trip efficiency is greater than 7% and at a ten-second rate, it is greater than 8% [7]; this

15 round-trip efficiency is just as high as that of batteries [8]. In contrast, the round trip efficiency of a regenerative fuel cell is about 5% [8]. The ultracapacitor s high round trip efficiency implies that an ultracapacitor-based energy storage system needs less cooling capacity than most other alternative technologies, since ultracapacitors dissipate much less energy in heat. Since the equivalent series resistance (ESR) in ultracapacitors is extremely low, an ultracapacitor can be charged with a very high current; this is not possible in energy storage devices like batteries that have higher ESR, because in those devices current must be limited to avoid overheating. In addition, no chemical reactions are involved in the storage and release of energy from ultracapacitors. This means that charging and discharging can be done with the same high rated current. This feature makes the ultracapacitor a good fit for regenerative braking applications; to successfully absorb energy from braking requires a very high charging current profile. In contrast, batterybased energy systems are not able to successfully absorb as much of the braking energy because their charging current must be limited to avoid damage to the batteries. Generally, ultracapacitors can operate over a wide range of temperatures. The range of operating temperatures for ultracapacitors is determined by the electrolyte. If the temperature is low, the mobility of the ions in the electrolyte will be low; near the freezing point of the electrolyte the mobility of the ions will be affected dramatically. In modern ultracapacitors, an organic solution that has a very low freezing point is employed as the electrolyte. As a result, a typical ultracapacitor can be operated at 3

16 temperatures as low as -45. They can be operated at temperatures as high as 6 [9]. Figure. Temperature dependence of ultracapacitor parameters, from [9] Throughout the range of operating temperatures, ESR and capacitance do not vary much, as shown in Figure. [9]. In contrast, lead-acid and lithium-ion batteries, which of all the battery types are the most tolerant of temperature changes, can be operated only from - to 45 []. Further, for some kinds of batteries such as lithium-ion cells, performance drastically decreases at temperatures below. Industry standards specify that an ultracapacitor s useful life ends when its capacitance decreases by % or its ESR increases by %. As an ultracapacitor is used, its performance continually degrades, and its end of life is when its performance will no longer satisfy the application requirements. The ultracapacitor will have unlimited shelf life if it is stored in a discharged state [9]. The ultracapacitor is good for several hundred thousand charge/discharge cycles; this is many more than can be achieved with batteries, 4

17 some of which are good for only several hundred cycles. In addition, because ultracapacitor operation involves no chemical reaction, its operation produces no environmental pollution. Thus using ultracapacitors in hybrid electric vehicles can improve the fuel economy and decrease vehicle emissions throughout the vehicle life.. Disadvantage of Ultracapacitors The main disadvantage of ultracapacitors is that they can withstand only a low rated voltage. That means that if a high terminal voltage is required, such as the 4-V modules [] used in some new automotive electrical systems, individual ultracapacitors must be connected in series to form an ultracapacitor bank. Even if a bank uses all the same kind of ultracapacitors, there will be differences in the individual capacitances; the manufacturing tolerances on the nominal capacitance can be as high as ± % []. This variation in capacitance places significant limitations on how the ultracapacitor bank is controlled and used. For a series string of ultracapacitors, the current into each ultracapacitor is the same. Assuming a simple capacitive model, the voltage v across the ultracapacitor is governed by the equation dv i = C, (.) dt where C is the nominal capacitance of a particular ultracapacitor in the bank and i is the current flow through the ultracapacitor bank. Mismatch in the nominal capacitances means that ultracapacitors with smaller capacitances will have larger terminal voltage changes dv ; this can cause terminal voltages for some cells to go beyond the rated voltage more quickly than others when charging the ultracapacitor bank. Similarly, when discharging a bank, different ultracapacitors will have different terminal voltages, and 5

18 some may even have potentially negative terminal voltages. It is dangerous to operate cells at negative terminal voltages or at terminal voltages higher than their rated maximum; accordingly, care must be taken in control of series-connected ultracapacitor banks. In order to solve the problem, different kinds of balancing circuits made up of passive resistors, switched resistors, DC/DC converters or other components [3,4,5] may be connected in parallel with each ultracapacitor s terminals. Their function is to bypass current around a cell whenever that cell s terminal voltage exceeds a preset voltage; this prevents the ultracapacitor from overcharging. More complicated balancing circuits produce better control results and dissipate less energy. In [6,7], a bypassing circuit is used so that all the ultracapacitors in a bank can be charged to the same upper voltage; this means that when the bank is fully charged, the cells are balanced. Although differences in cell capacitances will result in voltage imbalance at lower bank voltages, there imbalances will not affect safety and ultracapacitor life as long as the ultracapacitor bank is never discharged too far. For this scheme, the bank must be brought up to its upper voltage periodically to reestablish balance for each cell; this is to avoid the superposition of the imbalance voltages from each cell that can occur after many charging and discharging cycles, which could eventually cause individual cells to have negative terminal voltages..3 The Need for Ultracapacitor Models For controllable bypass circuits, a control scheme must be designed to turn the bypass circuits on or off to balance each cell in the ultracapacitor bank while dissipating the least 6

19 energy. Knowing the terminal voltage of each cell is a minimum requirement for the design of effective control strategies. To meet more stringent design requirements, the terminal voltage should be predicted under some known current profiles. Thus it is important to develop accurate ultracapacitor models. Ultracapacitor models can provide detailed information useful for calculating the required volume of ultracapacitors in an energy storage system, for designing sophisticated control strategies, and even for extending the voltage operating range of ultracapacitors. A simple first-order RC model (a large capacitor in series with a small resistor) can be used to model the behavior of an ultracapacitor and simulate fast charging and discharging in order to determine the instantaneous power available that could be stored into or released from an ultracapacitor bank. Although this model may be sufficient for many applications, an ultracapacitor cell behaves more as a distributed capacitance; accordingly, a first-order model cannot account for long-term behavior nor give any indication of how much of the stored charge should be considered available to do work in a given interval. In addition, to better control ultracapacitors, it is not sufficient to have models that are accurate only in short time frames. A long-term model is also needed to account for the charge redistribution phenomenon, which can happen over time frames of a couple of months. This phenomenon affects the instantaneous power available, since the terminal voltage of the ultracapacitor will change gradually over long time frames. The development of a suitably accurate high-order model of the ultracapacitor cell can: () Accurately predict the terminal voltage under different current profiles; () More closely model the slow transient due to charge redistribution to better account for true stored charge and to calculate the instantaneous power available; 7

20 (3) Allow for a more sophisticated energy storage system control strategy; (4) Improve pack balancing strategies; (5) Extend the voltage operating range in some particular cases..4 Contributions of Research In this thesis we develop two kinds of ultracapacitor models for the NESSCAP35P ultracapacitor, one for short-term behavior and the other for long-term behavior. The specifications for a NESSCAP35P ultracapacitor are listed in Table.. From the short-term model the terminal voltage can be predicted under different current profiles over time frames of one hour with the error less than. V. From the long-term model the slow transient due to charge redistribution is simulated over time frames of two months with the error less than.8 V. Table. NESSCAP35P specification Rated Capacitance 35 F Capacitance Tolerance -% to % Rated Voltage.7 V Rated Current 7 A ESR.5 mω Rated Energy Density 5.9 Wh/kg Rated Power Density 5.44 kw/kg Temperature Range -4 to 6 Weight 67 g Cycle Life (5 ) 5, cycles Life Time years.5 Thesis Outline The research work is presented as a thesis in six chapters. Chapter I gives introductory material motivating the problem addressed and presenting the goals of the research. 8

21 Chapter II provides background, related work and an overview of ultracapacitor models and parameter identification methods. Chapter III describes the test circuit and the data acquisition method used to obtain the experimental data on which the developed models are based. Short-term models, good for time frames of about one hour, are developed in Chapter IV. First-, second-, third- and fourth-order models are presented, as well as the least squares identification method used to find the model coefficients. Simulations of the models are compared to experimental data to obtain the error of each model. In Chapter V, a long-term model, good for time frames of about two months, is developed. The charge redistribution phenomenon through two months is observed experimentally, and a long-term model is derived to fit the observed slow transient. The coefficients for the long-term model are found by least squares identification. Further, the long-term model is tested by comparing simulation to experimental observations using impulse-like currents over long time frames. Chapter VI draws conclusions and makes recommendations for future work in this area. 9

22 CHAPTER II BACKGROUND AND RELATED WORK Theoretical lumped parameter models may be developed based on ultracapacitors physical structure. Although this kind of model represents the ultracapacitor structure, it may include a large number of parameters as well as non-linear characteristics that make it difficult to implement in practice. As a result, a lot of published work to model ultracapacitors tries to derive simple models to represent the dynamic behavior and voltage dependence of ultracapacitors.. Structure of Ultracapacitors Ultracapacitors employ an electric double layer structure and use activated carbon as the electrodes; these give ultracapacitors their extremely high capacitance... Composition Ultracapacitors are composed of three parts: positive and negative electrodes, electrolyte and separator, as shown in Figure.. Ultracapacitors employ activated carbon whose surface area approaches square meters per gram [9] as the electrodes. The reason that activated carbon has a high surface area per unit weight is that it is a powder made up of extremely small and very rough particles, which in bulk form a low density volume of particles with pores between them that resembles a sponge. The huge

23 Figure. Ultracapacitor structure number of pores increases the surface area, which allows many more electrons to be stored in the electrodes. The number of electrons stored in the electrodes is proportional to the ultracapacitor s capacitance. Figure., taken from [8], is a cartoon illustrating the structure of activated carbon material. Figure. The structure of activated carbon, from [8]

24 Since the size of pores is not uniform in the activated carbon, the capacitance is not independent of frequency. The largest pores are macropores with a diameter bigger than 5 nm. Inside the macropores there are still smaller mesopores with diameter between and 5 nm. Inside the mesopores there are micropores with diameter between 4 and angstroms and inside them there are submicropores with diameters less than 4 angstroms. Figure.3, taken from [9], illustrates the different size pores. Thus, ions in the electrolyte cannot charge the entire surface area of the device at all frequencies. Qualitatively, the ions can charge big pores such as the macropores at high frequency but can charge only small pores such as mesopores and micropores at low frequency; this is because the ions encounter more resistance on the way to the smaller pores, and so need a long time to reach them. Thus, the ultracapacitor will present different capacitances at different frequencies. Figure.3 Porous structure of activated carbon, from [9]

25 The electrolyte can be either an aqueous or an organic solution. Each one has its own advantages and shortcomings. For the aqueous electrolytes, the solute is the salt and the solvent is the water. Two example aqueous electrolytes are sulfuric acid (H SO 4 ) and sodium hydroxide (NaOH). They result in a lower ESR and higher power densities than those of non-aqueous double layer capacitors (DLC). However, aqueous DLCs can sustain only relatively low operating voltages and have low energy densities. On the other hand, non-aqueous electrolytes have higher energy densities [], and higher operation voltages, but also higher ESR and lower power densities. The separator, placed in the electrolyte between the positive and negative electrodes, is an electrically insulating membrane through which only ions can pass. When the ions attempt to pass the ion-permeable membrane, they will encounter resistance from the separator. This is one of the sources of ESR... Electric double layer structure Ultracapacitors employ a special structure called the electric double layer (EDL) that arises at the interfaces between the porous activated carbon and the electrolyte. Capacitance is created at this interface. It is the capacitance of this double layer that accounts for the capacitance of the device. The double layer includes a compact layer and a diffused layer [,,3], as shown in Figure.4. The compact layer is formed at the interface where the solid electrode and electrolyte contacts. There are two kinds of surface charge distribution at this interface; the first is of an electronic nature on the electrode side, and the other is of an ionic nature with opposite sign on the electrolyte side. The capacitance formed by the compact layer is proportional to the dielectric 3

26 permittivity of the electrolyte and inversely proportional to the charge separation distance which is determined by the diameter, less than angstroms [9], of the ions in the electrolyte. The diffused layer represents the ionic charge distribution in the electrolyte which results from the random thermal motion. From Figure.4, in electrolyte, the potential of the diffused layer closer to the compact layer is higher than the potential further from the compact layer. It is the capacitance formed by the diffused layer that reflects the voltage dependence of the ultracapacitor s capacitance. The total capacitance of the electric double layer structure is equal to the capacitance formed by the compact layer in series with capacitance formed by the diffused layer. The capacitance resulting from the diffused layer is voltage dependent: at low potential levels, the capacitance formed by the diffused layer contributes significantly to the total capacitance of the electric double-layer, while its capacitance becomes negligible at high potential levels. Figure.4 Stern s electrical double layer (EDL) model, from [] 4

27 . The Physical Model of Ultracapacitor Traditional capacitors such as electrostatic capacitors are often modeled by a single ideal capacitor and a single ideal resistor connected in series, where the resistor represents the ESR that prevents the modeled capacitor from acting like an ideal capacitor. Such a model is not sufficient to model an ultracapacitor except for some particular cases. This section describes various other models... Theoretical lumped parameter model Ideally, a model for an ultracapacitor should be based on its physical structure. In [4], the authors presented the lumped parameter equivalent circuit for an ultracapacitor shown in Figure.5. The model is constituted of an infinite number of RC branches with voltage-dependent capacitances to mimic the activated carbon fibers in the positive and negative electrodes, and resistances corresponding to the electrode material, the R anode C p C p C p3 C pn Rp Rp R p 3 R pn R membrane Cn Cn Cn3 Cnn R cathode Rn Rn Rn3 Rnn Figure.5 Ultracapacitor theoretical model 5

28 electrolyte material, the membrane material and the various sizes of pores. Although this model reflects the true physical structure of ultracapacitors and accounts for physical phenomena such as charge diffusion and voltage-dependent behavior, it is difficult to implement the model in practice because the model has so many parameters to be identified that there is no practical method to obtain all of them. Other related work presents several simplified models, as well as methods for parameter identification for those models... One-branch model Figure.6 shows the classical equivalent one-branch circuit model for an ultracapacitor used in [5,3], comprised of an ideal resistor and an ideal capacitor. Although it cannot be used to reflect the long-term behavior of the ultracapacitor, it can be used to simulate fast charging and discharging behavior. The charge redistribution phenomenon cannot be reflected with this kind of RC model, since the model has only i(t) + R v(t) C Figure.6 One-branch model without parallel resistor 6

29 one time constant; however, the charge redistribution does not dominate in relatively short time frames. In [3], the Challenge X team at the University of Akron employed this model to estimate the instant available energy from the ultracapacitor bank after fast charging and discharging. They also developed a technique for least squares identification for the parameters in the model. The model parameters vary in time; in practice, in time the capacitance will decrease and the resistance will increase. By taking advantage of the simple model and their efficient identification algorithm, the work in [3] periodically updates the parameters in the one-branch model based on recorded current and voltage data in order to get an accurate estimation of instant available energy information from the ultracapacitor bank. Through in-vehicle system-level tests, the model was proven sufficient for simulating fast charging and discharging. Another classic model of an ultracapacitor is shown in Figure.7. It is composed of three ideal circuit elements: a capacitor C, a series resistor R ESR which simulates energy loss during capacitor charging and discharging, and a parallel resistor R EPR which simulates energy loss due to capacitor self-discharge. In [6] the author introduced an i(t) + v(t) R ESR C R EPR Figure.7 One-branch model with parallel resistor 7

30 experimental way to identify the parameters in this model by using standard laboratory instruments such as oscilloscopes and voltmeters. First the ultracapacitor is slowly charged to a certain voltage; then charging is stopped and the ultracapacitor under test is relaxed for three hours. The equivalent parallel resistor R EPR is computed as 8s R EPR =, (.) v( t) ln( ) C v() where v() is the terminal voltage at the point that charging is stopped, and v(t) is the terminal voltage after three hours. In order to get R ESR, voltage and current changes after 5 milliseconds when charging or discharging the ultracapacitor under test are recorded, then the value of computed as R ESR is R ESR ΔV =. (.) ΔI The charge change in the ultracapacitor under test and the associated change in its terminal voltage are measured to calculate the capacitance as ΔQ C =. (.3) Δ V Although this method is easier than others that require specialized equipment such as a spectrum analyzer or a controllable constant current supply, there are still some shortcomings of the method. Firstly, the R EPR value derived in this way is not accurate. The ultracapacitor has very complex physical structure that is composed of a large number of RC branches. When the charging current through the ultracapacitor is stopped, the charge in the fast branch will redistribute to lower branches. It takes several weeks or 8

31 even longer for the ultracapacitor to reach a steady state. As a result, the decrease in voltage seen after three hours is not due solely to self-discharge; energy redistribution also causes a decrease. Secondly, the method to derive R ESR needs to measure instant terminal voltage change in a short time such as 5 milliseconds, and during this time there will be a tiny change in the terminal voltage, and even worse the tiny voltage change may be buried in the environmental noise. So the RESR derived in this way may be not accurate. We did our own experiments based on this method. We found the value of R ESR derived from a charging test may be four to five times bigger than the one derived from a discharging test. And in fact, the charging and discharging process should involve the same resistance...3 Three-branch linear model The simplified three-branch linear model shown in Figure.8 was developed in [7]. Based on this model the ultracapacitor was assumed to have three different time constants, that is, a fast branch, a medium branch and a slow branch. These three branches dominate the behavior of an ultracapacitor in short-term time frames. R f Rm Rs C f C m C s Figure.8 Three-branch linear model 9

32 In [7] a constant current excitation method was used to extract the unknown parameters in the three-branch linear model. Time constants of the fast, medium and slow branches are assumed to be quite different. First, the ultracapacitor under test was excited with a constant current over a time interval shorter than the time constant of the fast branch. During this period all charges were assumed to enter only into the fast branch; the other branches are assumed unaffected during this time. In other words, the voltages of the capacitors in the other branches were assumed to keep their initial values. The terminal voltage was observed to increase. Both terminal voltage and current values were recorded during this time, and used to identify the parameters R f and C f for the fast branch. In addition the total charge delivered was calculated for the next parameter identification for medium and slow branches. After that, the constant current supply was shut down so there was no additional charge going to the ultracapacitor and the terminal voltage was observed to decrease due to charge redistribution. The time constant of the slow branch was assumed much longer than the time constant of the medium branch, so that during the charge redistribution to C m over a time interval close to the time constant of medium branch, any charge redistribution to Cs was neglected. In this stage, terminal voltage values of the ultracapacitor were recorded. The voltage, the total charge and the assumed time constant of medium branch were used to identify the parameters R and for the medium branch. After an interval longer than the time constant of the medium branch, the capacitor in the slow branch began to be charged from its initial condition by the fast and medium branches. The terminal voltage was observed to decrease again for the charge redistribution from the fast and medium branches to the slow branch, and m C m

33 terminal voltage values during this time were recorded. The voltage, the total charge and the assumed time constant of the slow branch were used to identify the parameters C s for the slow branch. Rs and..4 Three-branch non-linear model Figure.9 shows a model proposed in [8] to reflect the behavior of an ultracapacitor within a 3-minute time frame. It is composed of three RC branches and a parallel resistor. The first RC branch with the elements R f, C f and voltage-dependent capacitance C f Vcf models the behavior of the ultracapa citor in the time frame of seconds; non-linear behavior of the ultracapacitor is simplified and has only been assigned to the fast-branch. The second branch with parameters Rm and Cm is the medium branch modeling the behavior in the time frame of minutes; and the third branch with parameters R s and tens of minutes. The parallel resistor C s is the slow branch modeling for the behavior in the time frames of R epr models the self-discharge phenomenon. The general approach in [8] to identify the parameters in the medium and slow branches is the same as in the last section. As far as the parameters in the fast branch are R f Rm Rs + R epr V cf C f C f V cf C m C s Figure.9 Three-branch non-linear model

34 concerned, the definition of differential capacitance C diff, that is, the change in charge at a given voltage, is introduced as dq C diff ( V ) = V. (.4) dv In their case, C diff = f f ( V ) C + C V. (.5) In order to identify C f and C f, a small amount of charge was injected to the ultracapacitor and the resulting change in voltage was measured and recorded at different voltage levels. A simple two-branch non-linear model was presented in [9]. The new model is shown in Figure.. In this model the first branch is composed of a constant resistance cf R, a constant capacitance C and a variable capacitor with capacitance proportional to the voltage at the device terminals. The second branch represents the voltage redistribution phenomena inside the device. Since this model has only two branches, it is easy to do parameter identification based on a constant current excitation lasting for a short period + R R V R epr C K v V C Figure. Two-branch non-linear model

35 of time. For the fast branch with the variable capacitance, the authors set up a secondorder equation to curve fit the voltage data observed from experiments in a period of time less than the time constant of the fast branch. In this stage, they assume no change in the capacitor charge in the slow branch. Then, they use the same technique described in Section..3 to identify the parameters in the slow branch. The parameter identification for the models in Figure.8, Figure.9 and Figure. is straightforward. However, the accuracy of the models depends on the assumed time constants for the medium and slow branches, which are chosen without prior knowledge of the dynamic behavior of the ultracapacitor...5 Transmission line model Another ultracapacitor model based on the physical structure of ultracapacitors has been proposed in [3]. The model is shown in Figure.. It is composed of two parts. The first is a non-linear transmission line connected to the terminals through an access R R Rn C Cn Figure. Transmission line model 3

36 resistor R. This part replaces the fast branch in the three-branch model. The second part is an n-branch RC ladder that mimics the long-term behavior of the ultracapacitor. The sections are organized such that the shortest time constant branch is close to the transmission line block and the time constants get longer and longer for sections farther from the transmission line. Not only can the model reflect the non-linear behavior of the ultracapacitor, but it is also flexible: if a short-term model is needed, the RC branches can be truncated, but if a long-term model is needed, additional RC branches can be added to satisfy the requirements. On the other hand, the transmission line is difficult to simulate because it has a complex expression. This also makes the parameter identification difficult, so implementation of the model in practice is not efficient. Figure. shows a proposed simplification of the short-term part of the transmission line model [3]. Four identical RC branches are used to represent the transmission line. An access capacitor C a was added to improve the short-term behavior of the model and a series inductor was introduced to describe the high-frequency behavior. The behavior of the inductor can be ignored if only the slower terminal voltage terms are of importance. Ls Ra R / 4 / 4 R R / 4 R / 4 C a C(V ) / 4 C(V ) / 4 C(V ) / 4 C(V ) / 4 Figure. Modified transmission line model 4

37 Although there are five branches in this model, the last four branches have the same time constant, so the long-term charge redistribution cannot be reflected very well; thus, this model focuses on the short-term behavior. In [3], both constant-current tests and frequency analyses were used to identify the parameters, taking advantage of the fact that frequency analysis is good for determining the dynamic behavior and constant-current analysis is good for determining the voltage dependence of the ultracapacitor...6 Models based on frequency response data Much of the published work on ultracapacitor modeling is based on frequency response data obtained, for example, using electrochemical impedance spectroscopy (EIS) [3,33,34]. The advantage of using the frequency response to identify parameters is that the dynamic behavior of the ultracapacitor can be modeled well. Further, the parameter identification software is embedded in some EIS systems. The disadvantage is that since the excitation signal is usually a small AC signal with very low or fixed DC bias, the derived model cannot accurately predict the voltage-dependent behavior. Thus, for EIS methods different DC biases should be considered. [33] presented a new approach to model the dynamic behavior of ultracapacitors using EIS. The device under test was modeled by an inductor L, a resistor R i and an impedance Z p, as shown in Figure.3. L Ri Z p Figure.3 Equivalent circuit of ultracapacitor, from [33] 5

38 In tests, four different DC voltages and six different temperature profiles were considered. Take one of the tests for example. The device under test was excited with a small AC current with a known, fixed DC bias. The terminal impedance is plotted as a function of frequency in Figure.4 [33]. The plot in Figure.4 can be divided into two parts. One is the near vertical impedance plot at low frequencies, the behavior of which is like an ideal capacitor. The o o other is the - 45 part that forms a - 45 line at intermediate frequencies. In this model, Z p is responsible for the - 45 o degree slope. The impedance Z p models the porosity of the ultracapacitor s elec trodes. Due to this porosity, the real part of the impedance increases with decreasing frequency and the full capacitance of the ultracapacitor will be seen at DC conditions. The parame ter L can be identified by the impedance from very high frequency. The parameter R i can be identified from the intersection between the Figure.4 Comparison of measured data and modeled data in frequency domain, from [33] 6

39 impedance plot and the real axis. The mathematical expression used for Z p is: τ coth( jωτ ) Z p ( jω) = (.6) C jωτ There are two independent parameters in (.6). A large number of impedance measurements are used to find a best fit for the unknown parameters τ and C. When all the frequency domain model parameters have been identified, the model is transferred to time domain model by expanding the impedance into the RC circuit shown in Figure.5. The experiments in [33] show that models with ten RC branches are in good agreement with the measured impedance. Another model with parameter identification done using EIS was recommended in [3]. This model, shown in Figure.6, has 4 RLC components. The model is used to simulate the behavior of an ultracapacitor as a function of frequency, voltage and temperature together. R v and Z p C v elements are utilized to represent the voltage-dependent behavior of the ultracapacitor at low frequency. Circuit in Figure.6 is introduced to consider the electrolyte ionic resistance in the low frequency range. Circuit is Figure.5 Approximation of Z p through N RC circuits, from [33] 7

40 C i Rp R p L R e R i R v R l R i Ca C v = k v C C p p C R Figure.6 Frequency, temperature and terminal voltage model, from [3] used to modify the medium-frequency behavior and Circuit 3 describes the charge redistribution and leakage current. In addition R denotes the electronic resistance and the inductance L is employed to describe the high-frequency behavior. e.3 Summary Generally speaking, there are two kinds of ultracapacitor models in related work: linear models, and non-linear models. For the parameter identification methods, there are still two major methods for different ultracapacitor models. One is based on the constantcurrent response; the other is based on the frequency response. Although the constantcurrent method can represent the voltage-dependent behavior of an ultracapacitor, quite different time constants of different branches have to be assumed. In contrast, the frequency response method can represent the dynamic behavior of an ultracapacitor, but has difficulty reflecting the voltage dependence. Thus, there are some related work that identify the parameters in their models using both methods. 8

41 The present work considers a transfer function that corresponds to an RC ladder model of an ultracapacitor cell. A least squares identification is used to find the best transfer function coefficients from experimental data gathered using only ordinary laboratory instruments such as multimeters, oscilloscopes and constant voltage supplies; this is in contrast to the controllable constant voltage supplies or EIS used in other techniques. The identification process is done only with time-domain data, and is based on timedependent current profiles generated by manually controlling charging and discharging switches at fifteen-second intervals. Unlike some other techniques, the process requires no initial assumptions about the time constants. 9

42 CHAPTER III EXPERIMENTAL SET-UP Determining the parameters of an ultracapacitor model requires first that current and voltage information be captured from the ultracapacitor under test, and then that least squares identification be done using that current and voltage data. In this chapter, the experimental set-up used to capture the data is described. This includes circuitry to control the charging and discharging of the ultracapacitor, as well as the program to measure and record the voltage and current information. 3. Test Circuit Figure 3. Test circuit set-up 3

43 The test circuit, shown in Figure 3., consists of three NESSCAP35P ultracapacitors in series excited by a 5-V, -W power supply. A 5-A fuse is connected in the test circuit to protect the power supply. The rated voltage for each ultracapacitor cell is.7 V. To avoid exceeding the rated terminal voltages, three identical ultracapacitor cells are connected in series and their initial voltages are made equal. Two kinds of MOSFET switches are employed so that the charging and discharging processes can be controlled separately. Five p-channel MOSFETs, each of which can pass a maximum of 3 A of current, are employed in parallel as the charging switch; when charging, their gates are connected together to the preset -5.5-V constant voltage supply through a three-position switch. Two n-channel MOSFETs, each of which can pass a maximum of 4 A of current, are employed in parallel as the discharging switch; when discharging, their gates are connected to a preset.5-v constant voltage supply through the same three-position switch. Two power resistors (. Ω, %, 5W) are used as the charging and discharging loads, respectively. In order to make the test circuit safe and reliable, a -k Ω resistor is connected between the gate terminal of the p-channel MOSFETs and the positive terminal of the power supply and another -kω resistor is connected between the gate terminal of the n-channel MOSFETs and the ground. These two -k Ω resistors serve to prevent the gate signals of the n-channel and p-channel MOSFETs from floating when the threeposition switch is between positions. When the switch is in the charging position, the gates of both the n-channel and p-channel MOSFETs are at -5.5 V, so that the n-channel MOSFETs are OFF and the p-channel MOSFETs are ON; similarly, when the switch is 3

44 in the discharging position, the gates of both the n-channel and p-channel MOSFETs are at.5 V, so that the n-channel MOSFETs are ON and the p-channel MOSFETs are OFF. In order to observe the transient characteristics of the ultracapacitor, the MOSFET switches are controlled to open and close manually by means of the three-position switch approximately every fifteen seconds during the test; this way a dynamic current profile is generated in the test circuit. 3. Data Acquisition A microcontroller board (dspicdem) was employed for the terminal voltage and current measurements. The charging current is measured indirectly by measuring the voltage across the power resistor in series with the charging source; similarly, the discharging current is measured indirectly by measuring the voltage across the power resistor that serves as the discharging load. This is done by measuring the potentials of both terminals of the power resistor with respect to the ground, subtracting the potentials to find the voltage difference across the resistor, and converting to current through the resistor by dividing the voltage by the resistance. The microcontroller board has twelve-bit analog-to-digital converters (ADC). The reference of the ADC is set to 5 V. The MPI board and the test circuit share a common ground. Imperfections in the linearity of the ADC are a possible source of error, and so it is important to investigate the quality of the conversion. In order to do so, known voltages from V to 5 V produced using a power supply were given to the ADC channel of the MPI board, and then the converted values were translated back to voltages in code 3