2012 IEEE 7th International Power Electronics and Motion Control Conference - ECCE Asia June 2-5, 2012, Harbin, China
High Capacity LiFePO4 Battery Model with Consideration of Nonlinear Capacity Effects A new approach to model the nonlinear capacity effects of high capacity battery
Low Wen Yao, Aziz, J. A. Department of Energy Conversion, Faculty of Electrical Engineering, Universiti Teknologi Malaysia 81300 Skudai, Johor, Malaysia.
[email protected],
[email protected] A strict management on battery is vital for the purpose of applying battery in the applications of high power system. An accurate battery model is the first issue that should be realized by the circuit designers in order to enable the optimize control on battery’s charge/discharge processes. An accurate battery model is helpful to improve the safety of usage and prolong the life of battery [2].
Abstract—Lithium Ferro Phosphates (LiFePO4) batteries have high potential to be widely employed in the future due to its stability on chemical and thermal characteristics. Therefore, an accurate battery model of high capacity LiFePO4 battery is important for a circuit designer to handle and optimize battery usage. However, nonlinear capacity effects of the battery are usually not been considered. The ignorance of nonlinear capacity effects cause the existing battery models unable to perform accurately. In this paper, nonlinear capacity effects of high capacity battery are studied. A new battery model is developed with consideration of nonlinear capacity effects. Developed battery model is validated from the experimental results. Comparison between experimental and simulation results shows that the developed model is capable to capture nonlinear capacity effects of the high capacity battery.
Remarkable efforts are committed to the development of battery model over the past decades. The battery models are divided to three categories; they are electrochemical model, mathematical model and electrical model. Electrochemical model uses interrelated partial differential equations to describe chemical process of battery [3-4]. It is accurate but complex and time consuming. Mathematical model uses empirical formula to simulate dynamic behaviors of battery. It is simple but still unable to provide accurate I-V information [2, 5]. Thus, both of them are less suitable to be applied in system design. Electrical model uses an equivalent circuit to simulate the characteristics of battery. It is suitable for system design since I-V information of a battery is readily available [5]. Furthermore, mathematical equations that describe the dynamic behaviors of battery can be simply derived from its electrical circuit [6]. Consequently, an effective controller for a battery powered system can be designed by using electrical model.
Keywords-battery model; energy storage; nonlinear capacity effects;
I.
INTRODUCTION
Rechargeable batteries are used as an energy storage element for portable electronic devices, electric vehicle, distributed generation and avionics system. Majority of portable electronic devices are using lithium-ion batteries due to their high energy density and the compact size [1]. Lithium Ferro Phosphate (LiFePO4) battery which using phosphate as its cathode, is expected to be widely utilized as energy storage element in future since it is excellent in thermal and chemical stabilities.
Fig. 1 shows the circuit diagram of electrical battery model which proposed in [5]. The accuracy of this battery model has been proven with the experimental results as discussed in [5] and [7]. However, the model is considered less suitable for high capacity battery since it did not consider all the dynamic behaviors of high capacity battery [8]. For instance, the nonlinear capacity effects are not considered in this electrical battery. The ignorance of these effects would cause the battery model unable to accurately capture the dynamic characteristics of battery [4, 9-10]. Nonlinear capacity effects are discussed in [3] as illustrated in Fig. 2. At the full charged condition, electrode is surrounded by maximum concentration of active species as shown in Fig. 2(a). When a load is applied, active species at the electrode surface are consumed due to the chemical reaction. The consumed active species are then refilled by
Figure 1. VRC battery model as proposed in [5].
This work was supported by Fundamental Research Grant Scheme (FRGS) of Malaysia and Universiti Teknologi Malaysia under project “Fundamental Study of Electric Vehicle Charging System”. 182
978-1-4577-2088-8/11/$26.00 ©2012 IEEE
electrolyte diffusion. However, the rate of diffusion is unable to keep pace with the chemical reaction, especially when high current is loaded. Thus, concentration gradient is formed as illustrated in Fig. 2(b). Cut-off voltage is reached when the concentration of active material of the electrode surface drop to a certain level as shown in Fig. 2(c). At this point, the chemical process is no more available and the remaining capacity is unusable. However, when the battery is rest until its equilibrium state, the electrode would once again be surrounded with active species as shown in Fig. 2(d). The remaining capacity is now usable. The recovery of the usable capacity is called capacity recovery effect [3]. References [9] and [10] proposed an enhanced battery model with nonlinear capacity effects consideration by combining both electrical battery model and Rakhmatov’s diffusion analytical model. However, the model is less suitable to be applied in real time condition due to the complexity in diffusion analytical model [4]. A hybrid battery model with nonlinear capacity effects consideration is also proposed in [4]. In this model, the nonlinear capacity effects are represented with kinetic battery model (KiBaM) whereas the dynamic I-V characteristic is captured by VRC battery model. The findings in [4] show that the consideration on nonlinear capacity effects can effectively reduce the modeling error, especially at low SOC region.
Figure 2. Capacity Recovery Effect [3].
rest for 2 hours and 18 minutes. In this context, battery is loaded with 0.5C (9A) for 720 seconds (i.e. 12 minutes) to reduce 10% of battery’s SOC per each cycle. The discharge and rest cycle is repeated until the battery voltage reach its cutoff voltage. A long period of rest time (i.e. 10 times larger than RC transient) is implemented for accurately determining the parameters of the battery model as suggested in [2]. On the other hand, the CDT test is done by continuously discharge the battery with constant current. In this paper, CDT tests for 0.5C (9A) is made to investigate the nonlinear capacity effects of battery. The typical voltage profile for PDT and CDT battery tests are shown in Fig. 3.
In this paper, the nonlinear capacity effects of battery are focused comprehensively. The aim of this paper is to develop an accurate battery model with consideration of nonlinear capacity effects. A new approach of capturing nonlinear capacity effects using additional SOC increment during unloaded condition is proposed. The SOC increment is assumed as an exponential reaction. The performance of this battery model is discussed by comparing the simulated results of the battery model with the experimental data carried out from the battery tests. The rest of the paper is organized as follow. Section II describes the experimental set up and procedures. The model extraction is presented in Section III. Section IV discusses on the model validation and Section V concludes the paper. II.
EXPERIMENTAL SET UP AND PROCEDURES
A.
Experimental Set Up The test system consists of LiFePO4 battery. The nominal capacity, nominal voltage and discharge cut-off voltage are 18Ah, 3.2V and 2V respectively. A DAQ device, NI9219 by National Instruments, is interfaced with a computer with LabVIEW 2010 to collect experimental data. Battery tests is carried out by using electronic load IT8514C which produced by ITECH with the rating of 120V, 240A and 1200W.
(a)
B.
Procedures of Battery Test Pulse discharge (PDT) and continuous discharge (CDT) tests are applied on LiFePO4 battery for identification of parameter and model validation. PDT test is done by intermittently discharge a battery with constant current. In this paper, the period of PDT cycle is set to 2.5 hours, where the battery is loaded with 9A (i.e. 0.5C) for 12 minutes and then
(b) Figure 3. Typical voltage profile for (a) PDT and (b) CDT tests.
183
III.
MODEL EXTRACTION
γ0 −
γk =
In this paper, the electrical battery model as illustrated in Fig. 4 is used for battery modeling. As can be seen from Fig. 4, voltage-controlled voltage source corresponds to open circuit voltage (OCV) of the battery, two RC parallel networks is used to characterize transient responses of the electrochemical process, a series resistance (RS) represents the charge/discharge energy losses of the battery, and another series resistance (∆RS) is used to characterize the voltage drop that caused by the nonlinear capacity effects.
1 CN
∫
tk
0
I L dt
γ α + Δγ ( γ α )[1 − exp( −τt )]
, IL≠0
, IL=0 (1)
∆γ is dependent on the γα. In this paper, the capacity recovery is assumed as a first order exponential response. The recovered SOC, ∆γ is reached exponentially within the rest time. Since the rest time for this paper is set to 8280 seconds (equivalent to 2 hours and 18 minutes), the value of time constant, τ is set to 2000 seconds by using the following equation, where TS is the settling time of first order exponential response.
A.
Nonlinear Capacity Effects Fig. 5 illustrates two voltage curves; blue line represents the voltage curve for 0.5C CDT test whereas red line represents the voltage curve for 0.5C PDT test. By referring to Fig. 5, there is a voltage deviation between these voltage curves. The runtime of battery under PDT test is longer than the runtime under CDT test. In this aspect, the capacity of the battery is found to be nonlinear. The capacity recovery took place during relaxation in PDT test.
τ=
Ts , 4
(2)
The value ∆γ and ∆Rs can be identified by analyzing the
In order to capture nonlinear capacity effects, the SOC of the battery is expressed as (1), where CN is the nominal capacity, γk is the SOC at the time tk, γ0 is the initial value of SOC, γα is the initial SOC of unloaded condition, and ∆γ(γα) is the recovered SOC that caused by capacity recovery. The value of CN is set to 64800 Ampere-second (equivalent to 18Ah).
Figure 6. Deviation between curves caused by capacity recovery.
∆γ Figure 4. Proposed battery model.
Figure 7. Shifted PDT curve.
∆V=IL×∆RS=0.063V Figure 5. Comparison between CDT and PDT curves for 0.5C without consideration of nonlinear capacity effects.
Figure 8. Calculate the value of ∆RS from vertical divergence.
184
The function of ∆γ(γα) can be identified via SOC recoverySOC relationship curve as illustrated in Fig. 9. The curve of ∆RS versus γα is shown in Fig. 10.
curves deviation of Fig. 5. In this context, ∆γ represents the horizontal curve deviation while ∆RS characterizes the vertical curve deviation. The value of ∆γ is calculated by shifting the PDT curve (red) and then the value of ∆RS is calculated based on the divergence between CDT curve (blue) and the adjusted PDT curve. An example of PDT curve shifting for 6th PDT cycle is described as follows:
B.
Open Circuit Voltage, OCV In this paper, the OCV is determined after the battery is at rest for 2 hours and 18 minutes. The OCV-SOC curve is shown in Fig. 11 and the relationship is expressed by (3), where γ represents SOC.
(i) Due to the capacity recovery, PDT curve at the 6th cycle is deviated with CDT curve as shown in Fig. 6. (ii) The PDT curve is shifted until the shifted PDT curve has a similar voltage drop’s rate with CDT curve as shown in Fig. 7. The shifted SOC is represented as ∆γ. At this point, the shapes of CDT curve and shifted PDT curve are identical. However, a vertical gap still existed between them. The vertical deviation of these curves is characterized by using ∆RS as shown in the Fig. 8.
OCV( γ ) = (0.6367) γ 0.2441 + (3.433 × 10 −8 ) γ 4 − (8.689 × 10 −6 ) γ 3 (3) + (7.976 × 10 − 4 ) γ 2 − 0.036 γ + 2.256
C.
RC Parallel Networks By referring to proposed model in Fig. 4, during unloaded condition (rest), the terminal voltage, Vt can be denoted as (4). Vk1 and Vk2 are the voltages across RC parallel networks during unloaded condition, whereas Vk1(α) and Vk2(α) are the voltage level across RC parallel networks during the beginning of unloaded condition, α. τk1 and τk2 represent the time constants of the transient response of Vk1 and Vk2 respectively.
(iii) For this case, the vertical variation is 0.063V. The value of ∆RS can be calculated based on this vertical divergence.
Vt = OCV − Vk1 − Vk 2 ⎛−t⎞ ⎟⎟ Vk1 = Vk1 (α) exp⎜⎜ ⎝ τ k1 ⎠ ⎛ −t ⎞ ⎟⎟ Vk 2 = Vk 2 (α) exp⎜⎜ ⎝ τk2 ⎠
Figure 9. SOC recovery-SOC relationship curve.
,t>α
(4)
∆RS(ohm)
The parameters of RC parallel networks are calculated using (5). Fig. 12 illustrates the parameters of RC parallel networks. The parameter identification of RC parallel networks is done by using MATLAB curve-fitting tools. In order to simplify the process of parameter identification, the value of SOC is assumed as constant in the unloaded condition. The extracted parameters for unloaded condition are denoted as (6). γα(%) Figure 10. ∆RS versus γ0.
R1 =
Vk 1 ( α ) IL
R2 =
Vk 2 ( α ) IL
C1 =
τ1 R1
C2 =
τ2 R2
(5)
R 1 ( γ ) = 4048 exp(− 0.3166γ ) + 0.02186 exp(− 0.02246γ )
R 2 ( γ ) = 191.9 exp(− 0.237 γ ) + 0.01518 exp(− 0.02378γ ) 8
C1 ( γ ) = (−1.887 ×10 ) γ
Figure 11. OCV-SOC relationship curve.
4
− 2.375
+ (3.787 ×10 )
C 2 ( γ) = (2.936 ×10 ) γ − (9.396 ×10 5 )
185
4
(6)
Since the voltage responses for loaded and unloaded condition are different, the parameters for unloaded condition are not suitable to be implemented for loaded condition [11]. The time constant for relaxation curve is larger than the discharging curve. If the larger time constant is applied on loaded condition, the voltage across capacitances would not reach the value of Vk1(α) and Vk2(α) by the end of discharging pulse, thus lead to modeling error. In this paper, in order to simplify the calculation, the function of R2(γ) is maintained while the larger time constant, τk2 is set to 144 seconds to make sure the value of Vk2(α) can be achieved within 720 seconds of loaded condition. Then, the function of C2(γ) for loaded condition is calculated as (7).
C 2 (γ) =
144 R 2 (γ)
, for loaded condition
(7)
D.
Series Resistance, Rs RS is usually determined based on the instantaneous voltage drop when the discharge started or the instantaneous voltage rise when the relaxation started in the PDT test as discussed in [7]. It is usually the first parameter that can be directly determined. However, the value of RS that extracted from PDT test would not satisfy the voltage curve of CDT due to the divergence between CDT and PDT curve. In this work, the identification of RS is put at the last. The nonlinear RS can be identified by comparing experimental result of CDT with the simulation result with RS=0 as shown in Fig. 13. The values of Rs on several SOC can be determined from the voltage deviation between these two curves. Rs can be expressed by (8). R S ( γ ) = (1.1289 × 10 3 ) exp(− 0.2754 γ ) + 0.023256 exp(− 0.01251γ ) (8)
IV.
MODEL VALIDATION
In order to validate the developed model, the experimental data of PDT and CDT tests are used to compare with the simulation result. Fig. 14 shows the comparison between experimental and simulation results for the CDT test. As shown in the figures, there is a close agreement between experimental and simulation results. Fig. 15 shows the comparison between
∆V=IL×RS
Figure 13. Identification of series resistance, RS.
Figure 14. Comparison between experimental and simulation results for 0.5C CDT test.
Figure 12. Extracted parameters for unloaded condition.
186
the results shows that the proposed approach can effectively capture the nonlinear capacity effects. V.
CONCLUSION
In this paper, an accurate and simple electrical battery model has been successfully developed. The considerations of nonlinear capacity effects for electrical battery model have been successfully applied to reduce the modeling error. The close agreement between simulation and experimental results shows that the characteristics of battery can be accurately simulated with the developed model. Moreover, the simple methodology in capturing the nonlinear capacity effects enables the researcher to build up the high accuracy battery model easily. With the proposed modeling approach, the battery modeling method can be simplified and then will eventually help the development of battery management system and energy storage system for green technology applications.
Figure 15. Comparison between experimental and simulation results for 0.5C PDT test.
REFERENCES [1]
K. M. Tsang, L. Sun, and W. L. Chan, “Identification and modelling of Lithium ion battery,” Energy Conversion and Management, vol. 51, pp. 2857-2862, 2010. [2] H. Zhang and M. Chow, “Comprehensive Dynamic Battery Modeling for PHEV Applications”, IEEE Power and Energy Society General Meeting, 2010. [3] R. Rao, S. Vrudhula, and D. Rakhmatov, “Battery modeling for energy aware system design”, IEEE Computer Society, vol. 36, no. 12, pp. 7787, Dec. 2003. [4] T. Kim and W. Qiao, “A Hybrid Battery Model Capable to Capturing Dynamic Circuit Characteristics and Nonlinear Capacity Effects”, IEEE Transactions on Energy Conversion, vol. 26, No. 4, 2011. [5] M. Chen and G. A. Rincon-Mora, “Accurate Electrical Battery Model Capable of Predicting Runtime and I-V Performance”, IEEE Transactions on Energy Conversion, vol.21, No.2, 2006. [6] I. S. Kim, “The novel state of charge estimation method for lithium battery using sliding mode observer”, Journal of Power Sources, vol. 163, pp. 584-590, 2006. [7] L. Chenglin, L. Huiju and W. Lifang, “A Dynamic Equivalent Circuit Model of LiFePO4 Cathode Material for Lithium Ion Batteries on Hybrid Electric Vehicle”, IEEE Vehicle Power and Propulsion Conference, VPPC, 2009. [8] A. Hentunen, T. Lehmuspelto and J. Suomela, “Electrical Battery Model for Dynamic Simulations of Hybrid Electric Vehicles”, IEEE Vehicle Power and Propulsion Conference , VPPC, 2011. [9] J. Zhang, S. Ci, H. Sharif and M. Alahmad, “An Enhanced CircuitBased Model for Single-Cell Battery”, IEEE Applied Power Electronics Conference and Exposition (APEC), 2010. [10] J. Zhang, S. Ci, H. Sharif and M. Alahmad, “Modeling Discharge Behavior of Multicell Battery”, IEEE Transactions on Energy Conversion, vol.25, No.4, 2010. [11] K. H. Norian, “Transient-boundary voltage method for measurement of equivalent circuit components of rechargeable batteries”, Journal of Power Sources, vol.16, Issue 4, pp. 2360-2363, 2011.
Figure 16. Comparison between experimental and simulation results for 0.5C PDT test at low SOC region.
Figure 17. Comparison between PDT and CDT curves for 0.5C with consideration of capacity recovery.
experimental and simulation results for the PDT test whereas Fig. 16 shows the comparison at low SOC region. By referring to the figures, there is a good matching between experimental and simulation results, the model can perform accurately at the low SOC region. Fig. 17 shows the capacity recovery in PDT test. By comparing the voltages curves in Fig. 5 and Fig. 17,
187
