Robust Identification of Time-Invariant Electrical Equivalent Circuit Models of Graphene Batteries

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1 Robust Identification of Time-Invariant Electrical Equivalent Circuit Models of Graphene Batteries Ashraf Mostafa, Sami Oweis, KaC Cheok Abstract This paper presents results of a study of recursive techniques for robust identification of time-invariant electrical equivalent circuit models on graphene batteries. An indirect discrete time technique is studied here. The results of this study indicate indirect discrete time technique is robust for identification of time-invariant circuit models from clean voltage-current data. Similar conclusions can also be drawn for identification of equivalent circuit models in the presence of noise. Index Terms Robust Identification, Recursive Parameter Estimation, Electrical Equivalent Circuit Model, Graphene Battery I. INTRODUCTION Many researchers put an enormous amount of research and development efforts have been spent to date on new lithium-ion (Li-ion) battery technology because they are extensively used in most portable consumer electronic devices, electric and hybrid electric vehicles, and industrial instruments as well as machineries with few focus on graphene batteries, graphene battery technology is considered one of the promising battery technology due to its high cycling performance, high reversibility and lower cost [1 4] over traditional Li-ion batteries. The real graphene battery breakthroughs is the graphene-lithium-ion hybrid chemistries incorporated into the cathodes of lithium-sulfur cells. This type of technology is still under intensive research. The more innovative graphene battery technologies will require significant R&D expenditures. Graphene battery innovation has a comparable structure to conventional batteries in that they have two anodes and an electrolyte answer to facilitate ion transfer. The primary distinction between solid-state batteries and graphene-based batteries is in the composition of one or both electrodes. The change fundamentally lies in the cathode; however, carbon allotropes can be used in the anode too. It's ultra-light, only Manuscript received July 30, Ashraf Mostafa, Oakland University ( mostafa@oakland.edu). Sami Oweis, Oakland University ( sami.oweis@gmail.com). KaC Cheok, Oakland University ( cheok@oakland.edu). an atom thin, but then it's 200 times more grounded than steel. It's flexible, transparent, and more conductive than copper. Researchers have been promising more grounded, lighter, flexible items, speedier transistors, bendable telephones, and numerous other achievement graphene devices for over decade. As we know batteries modeling is found to be very useful for a wide variety of applications and can be used also for modeling graphene batteries, such as estimation of state-of-charge and state-of-health, optimum battery charger design, and management of battery systems. The available models can be divided into three broad categories, namely, electrochemical, electrical and empirical. A good summary of these models can be found in [5], [6]. Electrochemical models and their approximations [7]-[10] are very accurate when used to identify the parameters and constraints that limit a battery cell s performance, but they are very complex and involve time-varying partial differential equations. They are mostly useful for actual physical design of such batteries. Empirical models [5], [6] use mathematical methods to predict system level behavior of such batteries, such as battery capacity and efficiency, but they only work for specific applications and often provide inaccurate results. Also, they are not very useful for modeling the voltage-current characteristics of such batteries. Electrical equivalent circuit (EEC) models that use a combination of capacitors, resistors and voltage and current sources have proved to be a reliable way of modeling the voltage-current characteristics of Li-ion batteries. Chen and Mora s model [11] is a good example of such models. In view of their modeling accuracy and ease of model identification, EEC models have found widespread applications in estimating a battery s state of charge (), state of health (SOH), and predicting its end-of-discharge time [12]-[16]. In view of this, we focus on EEC models in this paper. Numerous uses of EEC models require an accurate estimation of its model parameters. In addition, the parameter estimation method is required to be robust in the presence of both unmodeled dynamics and measurement noise. The issue of robustness in presence of unmodeled dynamics is important because almost all EEC models represent a reduced order model of a very high order, relatively complex battery system. Similarly, the issue of 40

2 robustness in presence of noise is also important because most applications call for estimation of model parameters from noisy voltage-current measurements. A review of EEC modeling literature reveals that the estimation of model parameters is almost always performed using an indirect discrete time (IDT) system identification method. This paper focus on recursive parameter estimation methods using time-invariant systems because industries prefer using low cost processing technology rather than high cost processing technology. We also investigate the robustness of the parameter estimation methods in presence of both noise and unmodeled dynamics. Using voltage-current data from a simulated battery model, an IDT identification method. This paper presents a novel approach by utilizing discrete-time system identification methods on graphene batteries and verified this approach by simulation and experiment. This discrete-time approach has not been used on graphene battery applications. This paper is organized as follows. Section 2 summarizes discrete time system identification methods investigated on graphene battery in this paper. Section 3 summarizes a simulated battery model. The main results of graphene battery parameter estimation are presented in Section 4. Finally, some concluding remarks are given in Section 5. II. SYSTEM IDENTIFICATION METHODS (2), where T denotes the sampling interval. Thus, we obtain an equivalent DT system of the following form: where u(k) and y(k) denote the samples of y(t) and u(t), respectively, at the sampling instant, t k = kt. It should be noted here that the relationship between DT model parameters, {c k, 1 k n, d k, 1 k m}, and the original CT model parameters, {a k, 1 k n, b k, 1 k m}, is usually governed by a set of complex nonlinear equations, which must be inverted later to obtain the estimates of the original CT system from those of the equivalent DT system. To use RLS, first rewrite equation (1a) in the following predictive form: (4a) where denotes the regression vector and, (3) (4b) (4c) denotes the parameter vector. Then use the following RLS parameter estimation algorithm: As mentioned earlier, the main focus of this paper concerns a comparative study of robustness of indirect DT (IDT) time system identification methods on graphene battery. Only a summary of their basic principles is presented here. Details of these methods can be found in [18]-[23]. A. An Indirect Discrete Time Method An indirect discrete time (IDT) system identification method based on recursive least squares (RLS) parameter estimation method is illustrated here. Consider a continuous time (CT) system characterized by the following constant coefficient ordinary differential equation: (1a) where u(t) and y(t) denote the input and output of the system, respectively. Also, A(d) and B(d) denote polynomials in the differential operator, d, of the following form: (1b) ( ). (1c) Suppose the above system is discretized using a CT to DT transformation method, such as bilinear transform, (4d) (4e) (4f) (4g) where denotes an initial estimate of, and is chosen to be a large positive number [19]. Finally, as mentioned earlier, the estimates of the original CT system (1) are obtained by solving a set of equations that are usually nonlinear and define the relationship between the original CT model parameters and the equivalent DT system parameters. III. BATTERY MODEL USED FOR SIMULATION STUDIES As mentioned earlier, we used a simulated battery data for a comparative evaluation of the two system identification methods discussed above. The simulated data was generated using Chen and 41

3 Mora s battery model [11]. A brief description of this model is provided below. (10) Chen and Mora s battery model, shown in Figure (11) 1, consists of two circuits, an energy balance circuit and a voltage response circuit. (12) (14) (13) Figure 1. Chen and Mora s Second-Order EEC Battery Model [7] The above circuit models the cell capacity, the self-discharge, the and the run-time of the battery. The capacitor, C Capacity, represents the total charge stored in the battery. V represents the voltage across C Capacity which has a value between 0 and 1 V. V equals 1 V when the battery is fully charged and equals 0 V when it is fully discharged. Therefore, V OC is a function of. The value of V depends on the magnitude and the direction of I Batt. In the stand-alone state, the self-discharge resistor, R self-discharge, represents the loss of charge when the battery is idle. A voltage response circuit describes how the cell voltage, V Batt, responds to a given current, I Batt. The open-circuit voltage, V OC, is a voltage-controlled voltage source dependent on. A series resistor, R Series, represents the battery s internal resistance and two parallel RC networks are used to model the transient response. Here R Transient_S and C Transient_S are used to model the transient short-time constant characteristics of the battery, whereas R Transient_L and C Transient_L are used to model the long-time transient constant transient characteristics. Henceforth we refer to the above parameters as R TS, C TS, R TL and C TL, respectively. The open-circuit voltage, V OC, is the electrical potential difference between battery terminals when there is no external load connected, and it is calculated using the following equation: The values of R Series, R TS, C TS, R TL and C TL as functions of can be calculated using the following equations: (9) Although Chen and Mora s original model is a second-order model consisting of five circuit elements, a number of follow-up researchers [13]-[16] have found that a simplified first-order model consisting of three circuit elements perform almost as well as the original second-order model. In view of this, they proposed a simplified first-order model shown in Fig. 2 below. Figure 2. A Simplified First-Order Chen and Mora s Model [13] The above models are henceforth referred to as second-order Chen-Mora model (CMM) and first-order CMM, respectively. IV. EVALUATION OF IDT SYSTEM IDENTIFICATION METHOD The main purpose of this study is to show the robustness of IDT system identification method for time-invariant systems, in the absence/presence of noise and/or unmodeled dynamics. With this in mind, a number of experiments were performed on Turnigy Graphene 800mAh 2S 20C Lipo Pack batteries. A. Comparison of accuracy of estimates for first-order CMM In this set of experiments, we simulated a time-invariant first-order CMM shown in Figure 2 above, estimated the model parameters using IDT method in the absence/presence of noise, and compared their relative accuracy. The results obtained are as follows. B. Time-invariant first-order CMM, clean data A sample of the results obtained for first-order time-invariant circuit model are illustrated in Figures 3, 4 42

4 RTL (Ohm) RTL (Ohm) Rseries (Ohm) Rseries (Ohm) CTL (Farad) and 5 below. These show estimates of R series, R TL and C TL from clean data as varies, and seem that IDT perform well in the absence of noise Est. CTL(x) 4000 Act CTL(x) Finally, in this set of experients, we test the trusworthiness IDT by examining the parameter estimates obtainined using IDT from an actual battery discharge data collected in the lab from a 800 mah two cell, discharged at a constant current The estimated values of R series, R TL and C TL, obtained using IDT, for both first and second order battery models are displayed in Figures 3, 4 and 5 below. An examination of these figures indicate that IDT work well in estimating these parameters Figure 5. Comparison of Time invariant C TL estimates C. Time-invariant first-order CMM, noisy data Est. Ro(x) Act Ro(x) Although this set of experiments was performed for various levels of signal-to-noise ratio (SNR), for the sake of brevity, we illustrate the results for SNR = 35 db only, in figures 6, 7 and 8 below. These seem to indicate that the IDT work well in the presence of noise Est. Ro(x) Act Ro(x) Figure 3. Comparison of Time invariant Rseries estimates Est. RTL(x) Act RTL(x) Figure 6. Comparison of Time invariant Rseries estimates Est. RTL(x) Act RTL(x) Figure 4. Comparison of Time invariant R TL estimates

5 CTL (Farad) Figure 7. Comparison of Time invariant R TL estimates Figure 8. Comparison of Time invariant C TL estimates V. DISCUSSION AND SUMMARY OF RESULTS The conclusions drawn from the above results can be summarized as follows. Considering the choice of the model order our results indicate that the accuracy of the parameter estimates using the IDT method, and its robustness in the presence of noise. VI. CONCLUSION This paper presents the results of a study of a recursive parameter estimation method IDT on graphene battery, for estimating parameters of time-invariant electrical equivalent circuit battery model. IDT represents an indirect discrete time approach. Based on both simulated and actual battery data, this investigation shows that while IDT is robust for estimating time-invariant battery models from clean data and noisy data. 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Little, Electrochemical impedance study of initial lithium ion intercalation into graphite powders, Electrochim. Acta 46 (2001) , [4] C.E.L. Foss, A.M. Svensson, S. Sunde, F. Vullum-Bruer, Electrochemical impedance spectroscopy of a porous graphite electrode used for Li-ion batteries with EC/PC based electrolytes Carl Erik Lie Foss, Electrochem. Soc. 41 (2012) 1 6. [5] Ramadesigan, V., Northrop, P., & De, S., Santhanagopalan, S., Braatz, R., & Subramanian, V. (2012). Modeling and Simulation of Lithium-ion Batteries from a Systems Engineering Prespective, Journal of the Electrochemical Society, pp. R31-R45. Ashraf Mostafa Graduate student at Oakland University. Sami Oweis PhD, MSc and BSc in Electrical and Computer Engineering from Oakland University. KaC Cheok Professor at Oakland University. 44