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1 742 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005 Impedance-Based Simulation Models of Supercapacitors and Li-Ion Batteries for Power Electronic Applications Stephan Buller, Member, IEEE, Marc Thele, Rik W. A. A. De Doncker, Fellow, IEEE, and Eckhard Karden, Member, IEEE Abstract To predict performance of modern power electronic systems, simulation-based design methods are used. This work employs the method of electrochemical impedance spectroscopy to find new equivalent-circuit models for supercapacitors and Lithium-ion batteries. Index Terms Lithium-ion (Li-ion) batteries, simulation models, supercapacitors (SCs). I. INTRODUCTION SIMULATION-BASED development methods are increasingly employed to cope with the complexity of modern power electronic systems. For these methods, suitable submodels of all system components are mandatory. However, compared to the submodels of most electric and electronic components, accurate dynamic models of electrochemical energy storage devices are rare. Therefore, this paper employs the method of electrochemical impedance spectroscopy (EIS) to extent the physics-based, nonlinear equivalent circuit models of supercapacitors (SCs) [2] to describe Lithium-ion (Li-ion) batteries. The following section briefly introduces the method of electrochemical impedance spectroscopy and presents measured impedance spectra. From these spectra, appropriate equivalent-circuit models are deduced. After this, the Matlab/Simulink implementation of the new simulation models is discussed and simulation results as well as verification measurements are provided. Finally, conclusions are drawn and future perspectives of the new impedance-based modeling approach are outlined. Paper IPCSD , presented at the 2003 Industry Applications Society Annual Meeting, Salt Lake City, UT, October 12 16, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Power Electronics Devices and Components Committee of the IEEE Industry Applications Society. Manuscript submitted for review July 1, 2003 and released for publication March 3, S. Buller was with the Institute for Power Electronics and Electrical Drives (ISEA), Aachen University of Technology (RWTH-Aachen), D Aachen, Germany ( M. Thele and R. W. A. A. De Doncker are with the Institute for Power Electronics and Electrical Drives (ISEA), Aachen University of Technology (RWTH-Aachen), D Aachen, Germany ( E. Karden is with Energy Management, Ford Research Center Aachen (FFA), D Aachen, Germany ( Digital Object Identifier /TIA II. IMPEDANCE SPECTRA OF SCS AND LI-ION BATTERIES Electrochemical impedance spectroscopy can be performed either in a galvanostatic or in a potentiostatic mode. Following the first approach, a small ac current flows through the storage device under investigation and its ac voltage response is measured. From the ac current and the measured ac voltage response, the storage impedance is determined online using discrete Fourier transforms (DFTs). Superimposed with the ac excitation signal, a dc current (charging or discharging) defines the overall working point of the cell. Due to the pronounced nonlinearity of most electrochemical storage systems, especially of batteries, the differential impedance is usually not equal to the quotient. In these cases, modeling the large-signal behavior of an energy storage device requires impedance measurements at several working points followed by integration of the differential impedance with respect to current, i.e.,. In addition, the impedance of storage devices usually depends on temperature and state of charge. Therefore, sets of impedance spectra have to be analyzed systematically [3] [5]. Due to mass transport phenomena, dynamic battery performance during continuous discharging or charging of batteries differs significantly from that during dynamic microcycling with frequent changes between charging and discharging. As the latter is typical for many practical battery applications (e.g., hybrid-electric vehicles or stop/start vehicles), EIS on Li-ion batteries has been performed using a specific microcycle technique [5]. During the investigation of the SCs, impedance spectra have been recorded at four different voltages and five temperatures. As an example, Fig. 1 shows the complex-plane representation of impedance spectra of a 1400-F SC at Vina frequency range from 70 Hz down to 160 mhz. This frequency range corresponds to typical time constants in most high-power applications, e.g., cranking of a vehicle with a combustion engine. In the high-frequency range Hz, the SCs show inductive behavior. Then, at approximately m, the impedance plots intersect the real axis. For intermediate frequencies, the complex-plane plots form an angle of approximately 45 with the real axis. This angle is explained by the limited current penetration into the porous structure of the electrodes (which has been discussed in [2]). For lower frequencies, the spectra approach a nearly vertical line in the complex plane, which is typical of ideal capacitors /$ IEEE

2 BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES 743 Fig. 3. Equivalent-circuit model of the Li-ion battery. Fig. 1. Complex-plane diagram of impedance spectra of a 1400-F SC manufactured by Montena Components SA, U = 1:25 V. Fig. 4. Complex-plane impedance diagram of measured and modeled impedance data of the Li-ion battery at # =25 C, 80% SOC, I =0Aand I =1Acharge. III. EQUIVALENT-CIRCUIT MODELS Fig. 2. Impedance spectra of an Li-ion battery (I =0A, # =25 C). Fig. 2 shows measured impedance spectra of a Li-ion battery (Saft LM , 3.6 V/5 Ah) at room temperature for different states of charge (in this case with zero dc current). Impedance data have been recorded for eight frequencies per frequency decade starting at 6 khz. For all spectra, some characteristic frequencies are given. At approximately, the real axis intersection of the impedance spectra is observed. For lower frequencies, all spectra show two capacitive semicircles. The first semicircle is comparably small and slightly depressed, whereas the second one is larger, nearly nondepressed, and grows remarkably with decreasing state of charge. Finally, at the low-frequency end of the depicted spectra, diffusion becomes visible. The diffusion impedance shows a 45 slope, which is typical of a so-called Warburg impedance. Due to the boundary condition for diffusion of Li ions in the electrodes, the diffusion branch of the spectrum approaches a capacitor-like impedance spectrum 90 for even lower frequencies. Impedance spectra of valve-regulated lead-acid batteries (VRLA) with different superimposed dc currents have also been measured at several state of changes (SOCs) and temperatures. These results are beyond the scope of this paper but can be found in [1], and [4]. The discussion of the model topology and the general modeling principle in this section concentrates on the Li-ion battery technology. In the case of SCs, excellent agreement with the measured impedance spectra was achieved using a ladder network model, consisting of the resistance of the pore electrolyte and the nonlinear double-layer capacitance of the phase boundary electrode/electrolyte. Detailed results as well as the lumped-element representation of the ladder network are reported in [1] and [2]. To model the recorded impedance spectra, suitable equivalent-circuit topologies have to be defined. Based on the underlying physical processes, the equivalent circuits should allow an optimum representation of the measured spectra with a minimum set of model parameters. In a second step, the model parameters have to be calculated. To minimize the deviations between modeled data and measured spectra, a least-square fitting algorithm is employed. In Fig. 3, the electric equivalent circuit of an Li-ion battery is depicted. This circuit consists of an inductance, an ohmic resistance, a so-called element representing a depressed semicircle in the complex-plane [1], [3], a nonlinear RC circuit ( and ) as well as of a Warburg impedance. Using the depicted model topology, the observed ac behavior of an Li-ion battery can be described accurately. The following adaptation of the model to the measured impedance spectra shows that, despite several simplifications, all relevant processes including porosity, charge transfer and diffusion are modeled with sufficient precision. Fig. 4 compares measured and calculated impedance data of the Li-ion battery at 25 C and 80% SOC. For all frequencies and both depicted dc currents, the corresponding curves show nearly perfect agreement.

3 744 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005 Fig. 5. Nonlinearity of the resistance R (# =25 C, 50% SOC). Fig. 6. Open-circuit voltage as a function of the state of charge (Li-ion). Obviously, the diameter of the low-frequency semicircle, i.e., parameter, strongly depends on the dc current that is superimposed during the impedance measurement. The nonlinearity of with dc current is depicted in Fig. 5. Apart from the data points, which have been determined from the measured impedance spectra, Fig. 5 also shows a calculated curve which models the current dependency of. The relation between the dc current and the corresponding overvoltage at the impedance element can be described by a Boltzmann-type equation. In electrochemistry, this type of equation is known as Butler Volmer equation (1). The constants are the exchange current, the number of transferred elementary charges, the symmetry coefficient, and the thermal voltage ( mv if K). The nonlinear charge transfer resistance as (1) can be calculated Equation (2) can be solved analytically for the special cases, (irreversible reactions), and (symmetric kinetics). In all other cases, numeric calculation is required. Therefore, to determine the Butler Volmer parameters,, and from the data points in Fig. 5, a second fitting algorithm is employed. The best approximation of the data points is obtained for A,, and. For classic redox reactions, represents the change in oxidation number of the reaction ions. Hence, integer values are expected. However, for the Li-ion battery, a noninteger value for was found to be best. This result might be explained by the specific nature of the Li intercalation reaction or might be due to the simplifications that are necessary to allow a parameterization of the simulation model without reference electrode measurements. An electrochemical investigation of this finding is not required for the further development of the dynamic battery model and is therefore considered beyond the scope of this work. Finally, Fig. 6 shows the open-circuit battery voltage as function of the state of charge. The Li-ion battery was partly (2) Fig. 7. Approximation of a ZARC element by RC circuits. discharged and the open-circuit voltage was measured after a minimum rest period of five hours. The curve in Fig. 6 can be stored into a lookup table. IV. MODEL IMPLEMENTATION So far, the dynamic behavior of the modeled energy storage devices is still described in the frequency domain. The time-domain behavior of the equivalent-circuit model can be calculated by solving a set of ordinary differential equations. For this calculation, simulation tools like Matlab/Simulink can be employed. However, not all complex impedance elements (e.g., ZARC elements and Warburg impedances) can directly be implemented in a common circuit simulation tool. For these elements, appropriate approximations by means of RC circuits or RC ladder network topologies have to be found first [1]. As an example, the basic idea for the representation of a ZARC element, employed to model a depressed, capacitive semicircle in the complex-plane diagram, is depicted in Fig. 7. The approximation is based on a series connection of nonlinear RC circuits. All RC circuits are fully determined by the parameters of the ZARC element. Thus, the number of experimental parameters remains constant. With an increasing number of RC circuits, the approximation of the ZARC elements becomes more and more precise. However, the calculating time increases. Thus, an appropriate compromise between simulation accuracy and computation effort has to be found. This question is thoroughly discussed in [1]. The specific model parameters which are needed for the simulation of a certain battery are stored in a separate file. To allow a linear adaptation of a parameterized battery model to differently sized batteries of the same technology, all parameters are defined with respect to the battery s nominal current and the number of battery cells connected in series. Furthermore, due to the nonlinearity of some impedance elements, the original battery current in the simulation model is replaced by the relative current.

4 BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES 745 Fig. 8. Current profile for the verification of the SC model. Fig. 10. Comparison of the measured voltage response and the data obtained from different simulation models. TABLE I COMPARISON OF MEASURED AND SIMULATED EFFICIENCY DATA.WORKING POINT: 1.5 V, ROOM TEMPERATURE, CYCLE DEPTH 615% Q Fig. 9. Measured and simulated voltage response to the current profile. V. VERIFICATION AND APPLICATION OF THE MODELS As a final step, the results of the simulation models are compared with measured data in the time domain. Both, the SC model as well as the model of the Li-ion battery have been verified in detail [1]. In this section, some examples of these verification measurements are given. For the verification of the SC model, the current profile depicted in Fig. 8 has been employed [2]. The imposed charging and discharging pulses model a highly dynamic load at the beginning as well as deeper charging and discharging periods at the end of the evaluation. The corresponding voltage curves are depicted in Fig. 9. The measured and the calculated data show excellent agreement. The influence of the porous structure of the SC electrodes can be illustrated by means of a comparison of the full simulation model with the voltage response of the simplified model which only consists of a series connection of the ohmic resistance and the capacitance of the SC. For this comparison, Fig. 10 provides an enlarged view of the first current pulses of the verification profile. Once more, the excellent agreement of the measured and simulated voltage data becomes obvious. In addition, remarkable deviations due to the neglect of porosity are observed for the simplified model. For low frequencies, i.e., for comparably long relaxation times, these deviations could be overcome by replacing the ohmic resistance by the larger dc resistance with being the resistance of the electrolyte in the pores of the electrodes [2]. In this case however, the fast voltage transients would not be well represented anymore. One advantage of SCs used as energy storage devices is their good energy efficiency. This efficiency is also influenced by the porous structure of the electrodes which means that the increasing real part of the impedance with decreasing frequency has to be taken into account. To compare measured and simulated efficiency data, an SC is partly charged and discharged with constant dc currents of various amplitudes. The cycle depth is chosen to be 540 A s which corresponds approximately to. For each current amplitude, the charge/discharge cycle is repeated ten times but only the last five cycles, which start and finish at the same internal conditions (quasi-stationary), are used for the efficiency calculation. In a second step, the same current profile is simulated by means of the newly developed capacitor model. Measured and simulated efficiency data are compared in Table I. Again, very good agreement is observed. Next, a simulation example of the Li-ion battery model is presented. For the model verification, the dynamic discharge current profile depicted in Fig. 11 has been selected. The comparison of the simulated and the measured voltage response to this current profile at a state of charge of 77.5% and room temperature is shown in Fig. 12. Excellent agreement of the measured and the simulated voltage curves is found. The outstanding accuracy of the simulation model is due to the exact representation of the complex battery impedance including all important nonlinearities. An important precondition for the high quality of the simulation results is the nearly perfect reproducibility of the battery

By this, the simulation of long-lasting constant-current charging or discharging periods will become possible.

Combined with the mechanisms of heat transport and dissipation, the thermal behavior of batteries can be simulated.

ACKNOWLEDGMENT The authors are grateful to the Ford Research Center Aachen (FFA), especially to Dr. D. Kok and Dr. L. Gaedt for supporting this research project. Fig. 12.

5 746 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005 Fig. 11. Current profile for the verification of the Li-ion battery model. storage technologies, e.g., NiMH batteries or even fuel-cell stacks in the future. This versatility will allow the combined simulation of different energy storage devices for the evaluation of new storage-hybridization concepts. Furthermore, by means of additional submodels, e.g., describing mass transport in VRLA batteries, the validity range of the existing models can be further enlarged. By this, the simulation of long-lasting constant-current charging or discharging periods will become possible. Another interesting future application of the impedancebased simulation models is a detailed thermal battery design. All mechanisms of heat generation of SCs or batteries can be precisely represented. Combined with the mechanisms of heat transport and dissipation, the thermal behavior of batteries can be simulated. Consequently, the influence of different future cooling concepts, for example, on life-cycle costs, could be evaluated. ACKNOWLEDGMENT The authors are grateful to the Ford Research Center Aachen (FFA), especially to Dr. D. Kok and Dr. L. Gaedt for supporting this research project. Fig. 12. Measured and simulated voltage response of the Li-ion battery (77.5% SOC, 25 C). behavior during operation as well as the lack of parasitic reactions. From this point of view, Li-ion batteries are especially suited for any kind of model-based description. Compared to Li-ion batteries, the simulation of lead acid batteries turns out much more difficult. Nevertheless, the described simulation approach could also be successfully adapted to this battery technology [1], [4]. VI. CONCLUSION AND FUTURE PERSPECTIVES This paper has shown that nonlinear, lumped-element equivalent-circuit models meet the accuracy requirements for simulation models of energy storage devices. To demonstrate the power of this modeling concept, Li-ion batteries and SCs were selected. For the determination of suitable equivalent-circuit topologies as well as for the parameterization of these models, the method of EIS was employed. After the implementation of the models, the simulation results were compared to test-bench data. Excellent agreement of simulated and measured voltage data was found. Due to the versatility the impedance-based modeling approach, the described concept can also be employed for other REFERENCES [1] S. Buller, Impedance-based simulation models for energy storage devices in advanced automotive power systems, Ph.D. dissertation, ISEA, RWTH Aachen, Aachen, Germany, [2] S. Buller, E. Karden, D. Kok, and R. W. De Doncker, Modeling the dynamic behavior of supercapacitors using impedance-spectroskopy, IEEE Trans. Ind. Appl., vol. 38, no. 6, pp , Nov./Dec [3] E. Karden, Using low-frequency impedance spectroscopy for characterization, monitoring, and modeling of industrial batteries, Ph.D. dissertation, ISEA, RWTH Aachen, Aachen, Germany, [4] S. Buller, M. Thele, E. Karden, and R. W. De Doncker, Impedancebased nonlinear dynamic battery modeling for automotive applications, J. Power Sources, vol. 113, pp , [5] E. Karden, S. Buller, and R. W. De Doncker, A method for measurement and interpretation of impedance spectra for industrial batteries, J. Power Sources, vol. 85, pp , Stephan Buller (M 97) received the Ph.D. degree from the Institute for Power Electronics and Electrical Drives (ISEA), Aachen University of Technology (RWTH-Aachen), Aachen, Germany, in He joined ISEA in 1997, spending five years as a Research Associate. From 2002 until 2004, he was a Chief Engineer at ISEA. His research activities were mainly in the area of batteries and other energy storage systems. In January 2005, he joined an international consulting company. Marc Thele received the Diploma in Electrical Engineering from the Institute for Power Electronics and Electrical Drives (ISEA), Aachen University of Technology (RWTH-Aachen), Aachen, Germany, in In June 2002, he joined ISEA as a Research Associate. His research activities are in the area of battery simulation models of different technologies.

During 1987, he was appointed as a Visiting Associate Professor at the University of Wisconsin, Madison. In December 1988, he joined the General Electric Company Corporate R&D Center, Schenectady, NY.

6 BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES 747 Rik W. A. A. De Doncker (M 87 SM 99 F 01) received the Doctor of Electrical Engineering degree from the Katholieke Universiteit Leuven, Leuven, Belgium, in During 1987, he was appointed as a Visiting Associate Professor at the University of Wisconsin, Madison. In December 1988, he joined the General Electric Company Corporate R&D Center, Schenectady, NY. In 1994, he joined Silicon Power Corporation as Vice President. In October 1996, he became a Professor at Aachen University of Technology (RWTH-Aachen), Aachen, Germany, and Head of the Institute for Power Electronics and Electrical Drives (ISEA). Eckhard Karden (M 00) received the Ph.D. degree from Aachen University of Technology (RWTH-Aachen), Aachen, Germany. He is a Research Engineer for storage systems in the Energy Management Group of the Ford Research Center Aachen (FFA), Aachen, Germany. Before he joined Ford in 2002, he was Chief Engineer at the Institute for Power Electronics and Electrical Drives (ISEA), RWTH-Aachen.

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