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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008
2027
Nonlinear State of Charge Estimator for Hybrid Electric Vehicle Battery
Il-Song Kim, Member, IEEE
Abstract—A new method for battery state of charge estimation using a sliding mode observer has been developed. A nonlinear battery dynamic modeling technique is established and design methodology with the sliding mode observer is presented. Contrary to the conventional methods using complicated battery modeling, a simple resistor-capacitor battery model was used in this work. The modeling errors caused by the simple model are compensated by the sliding mode observer. The structure of the sliding mode observer is simple, but it shows robust control property against modeling errors and uncertainties. The convergence of the proposed observer has been proved by the equivalent control method. The performance of the system has been veri?ed by the sequence of urban dynamometer driving schedule test. The test results of the proposed observer system shows robust tracking performance under real driving environments. Index Terms—Battery modeling, hybrid electric vehicle, slidingmode observer, state of charge (SOC), state of charge estimation.
Fig. 1. Con?guration of the HEV operation.
NOMENCLATURE Open-circuit voltage as a function of SOC Z. Polarization capacitance Propagation resistor. Diffusion resistor. Ohmic resistor. Cell terminal voltage. Z Polarization voltage. State of charge. Nominal capacitance of the cell. Estimate of the cell terminal voltage. Estimate of the state of charge. Estimate of the polarization voltage. Feedback gain of sliding-mode observer.
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I. INTRODUCTION UE to the high fuel ef?ciency and low-emission requirements, a hybrid electric vehicle (HEV) has great potential as a new alternative means of transportation. A HEV is mainly composed of an internal combustion engine, electric motor and rechargeable battery. The electric motor provides boost energy to assist the combustion engine and acts as a generator when regenerating brake energy, or when the engine has excess power
Manuscript received April 12, 2007; revised November 15, 2007. Published June 20, 2008. Recommended for publication by Associate Editor S. Choi. The author is with Chung-Ju National University, Chung-Ju, Republic of Korea (e-mail: iskim@cjnu.ac.kr). Digital Object Identi?er 10.1109/TPEL.2008.924629
to charge the battery. As can be seen in Fig. 1, the HCU (Hybrid control unit) controls motor power and engine power according to the vehicle conditions for the best fuel performance. The amount of motor power is limited by the maximum available battery charge/discharge power. For a required motor power, the battery should provide available charge/discharge power to meet the power requirement. During the cold cranking or regenerative braking, the discharged or charged power is rated up to 30 [kw] for 10 s, for example. The battery available power is directly obtained from the state of charge (SOC) information and therefore it is very important to accurately obtain its value for the best performance of the HEV. The SOC is calculated from the BMS by the cell voltage and temperature and other information such as polarization effect caused by the electrolyte concentration gradient during high rate charge and discharge period. It is sent to the HCU by CAN communication line. Using this SOC information, the HCU controls motor power for the best performance and safe operation of the battery. Since the SOC is an internal chemical state of battery and thus cannot be directly measured with electric signals, it should be estimated with the aids of physical measurements such as voltage and current of the battery terminal. There have been numerous attempts to estimate the SOC of battery. The most common methods are the coulomb counting and Kalman ?lter approach. The coulomb counting or current integration method measures the amount of charge taken out or put into a battery in terms of ampere-hours [1]. If a suf?ciently accurate current sensor is used, this method is reasonably accurate and inexpensive to implement. However, the coulomb counting is an open loop SOC estimator and thus the errors in the current detector are accumulated by the estimator. The Kalman ?lter method is a well known technology for dynamic system state estimation such as target tracking, navigation, and battery [2], [3]. It provides a recursive solution to optimal linear ?ltering for state observation and prediction problems as well. The unique advantage of the Kalman ?lter is that it optimally estimates states
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Fig. 3. Battery dynamic model structure. Fig. 2. Open-circuit voltage versus SOC over temperature variation of the lithium-polymer battery.
affected by broadband noise contained within the system bandwidth. The main disadvantage of the Kalman ?lter is that it requires high complex mathematical calculation. The gain is obtained by the ?ve steps of the Kalman ?lter algorithm. There can be some possibility of divergence due to imperfect modeling and complex calculation loads. If the processors are not mighty enough, the calculation time could excess the sampling time and thus it can not track the correct state values. Also the Kalman ?lter has some limitations for a real implementation such as perfect modeling of the plant and Gaussian distribution of the external noises. If these constraints are not satis?ed, the performance of the Kalman ?lter would be degraded and thus cannot be used in the real applications [4]. Other reported methods for estimating the SOC have been based on arti?cial neural networks and fuzzy logic principles [5], [6]. The neural network method incurs large computation overload on the BMS, it can be a problem for online implementation. Since the fuzzy logic method relies on the training data battery is operated in unusual way. The empirical method based on the battery chemical characteristics is also reported in recently [7]. In this paper, a new design method with a sliding-mode observer has been presented for the battery SOC estimation. The sliding-mode observer can overcome aforementioned drawbacks by using sliding-mode techniques. The main characteristics of the sliding-mode observer are simple control structure and robust tracking performance under uncertain environments [8], [9]. II. BATTERY MODELING Dynamic state model of the battery is necessary to develop a simulation model for the emulation of battery behavior [10]–[15]. The model is developed from experimental cell data, where open circuit voltage (OCV) tests are performed on successive discharge of the battery, by the application of periodic current discharge. As for the temperature variation to , the OCV of a lithium-polymer battery from (LI-PB) varies nonlinearly over the battery SOC as can be seen in Fig. 2. Therefore, the nonlinear RC models are developed to model nonlinear OCV characteristics of the Li-PB [16], [17].
The proposed model consists of: 1) nonlinear voltage source as a function of SOC Z to represent nonlinear characto model polarization teristics of the OCV; 2) a capacitance to model propagation resiseffect; 3) a propagation resistor as a function of current I; and tance; 4) a diffusion resistor and terminal voltage as shown in 5) an ohmic resistance Fig. 3. The self-discharge resistor does not considered in the model because the self-discharge characteristic of the lithium battery is extremely low compared to other batteries such as nickel cadium, lead-acid, nickel metal hydride type. is denoted as . The The voltage across the capacitance terminal voltage equation is given as (1) (2) where is the instantaneous current (positive for charging, negative for discharging). The SOC is de?ned as a ratio of the remaining capacity to the nominal capacity of the cell, where the remaining capacity is the number of ampere-hours that can be drawn from the cell at room temperature with the C/30 rate before it is fully discharged [3]. Based on this de?nition, the mathematical relation on the SOC is developed as (3) is SOC and is the nominal capacitance of the where cell which is de?ned as the number of ampere-hours that can be drawn from the cell at room temperature at the C/30 rate, starting with the cell fully charged [3]. The time derivative for SOC Z can be expressed as follows: (4) Equating the two voltage equations (1) and (2) with some algebraic manipulation yields (5) From Kirchhoff’s law, it results in (6)

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Then, from (5) and (6), obtain (7) Equating (7) to (4) results in (8) Also, by similar way, substituting (7) into (6), we have (9) The output voltage equation is given from (2) and (7) by
where
This model is not accurate compared with the real cell data. Therefore, the nonlinear unknown disturbances terms are added to the model in order to compensate for the modeling errors (10)
The output voltage is expressed as a nonlinear equation of Z and therefore can be considered as the third state variable. The rate of change of input current can be negligible due to the fast sampling interval when implemented into digital system. Taking the time derivative of the output voltage and assuming gives
(14) not only represent nonlinearities caused where by linearization error and modeling error, but also time-varying terms and internal/external disturbances. III. THEORETICAL BACKGROUND OF THE SLIDING-MODE OBSERVER Consider the observer problem for a continuous-time single and measurement input system , where , is the scalar model represents bounded modeling errors and feedback control, disturbances, and [18]. When an observer for the system is de?ned as (15) where with , is an estimate of , is the signum function, and represent vectors has dynamics of switching gains. The observer error , where , . and the switching function is de?ned as A local sliding regime exists on the surface whenever (16) where is the ?rst column of switching gain . , a local sliding regime exists on Therefore if for . The set is sometimes referred to as the sliding patch. The ideal sliding
(11) The above equation is based on the assumption that the OCV can be considered as a piecewise linear function of the Z and therefore the following relationship is obtained for the piecewise region: (12) From (2) and (7), solving for and then substituting into (10), then the complete state equation including the derivative of the output voltage is obtained as
(13)

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dynamics are determined by Filippov’s solution concept: for ,
Z,
As OCV is monotonically increasing with respect to the SOC can be considered as piecewise linear to the . Therefore (23) The error system for is given as (24)
(17) represents a projection onto the null space of C along where span {L}. The sliding-mode observer shows robust tracking property by the sliding regime. The robustness of sliding-mode observer is to maintain sliding regime in the presence of dynamical disturbances, and to retain robustness properties of the sliding mode. IV. SLIDING-MODE OBSERVER DESIGN Since the observability matrix of (13) has always full rank, the internal state of the battery can be estimated by the observer. The sliding-mode observer design starts from the output equation [19]–[24]. The corresponding sliding-mode observer for the is given by output (18) are the estimates for , and is a constant where positive feedback gain. When de?ning the error with , the following error equation is obtained:
where is a piecewise linear gain and the maximum value is determined from the experimental result. . Select Choose Lyapunov candidate function , the sign of and is opposite. Therefore , as in the previous case, and for all subsequent time. Then the following relationship is obtained as
(25) Finally, the observer for is built as (26) The error system is given as (27)
(19) where . The convergence of error equation can be proved by the Lyapunov candidate function by choosing . , then the sign of is negative if is Select positive regardless of is. The sign of is positive if is negative. Therefore , and after and for all subsequent time. some ?nite time, According to the equivalent control method, the error system is replaced by its equivin sliding mode behaves as if , which can be calculated from (19) alent value and . Once the sliding surface is assuming and reached, then from the equivalent control concept, reduced to zero and the uncertainties vanish, and then the resulting equation from (19) can be written as follows: (20) The next observer equation for is obtained as (21) are the estimates for , and is a constant where positive feedback gain. De?ne error as and , then the following error equation is obtained as
As in the previous case, select , then the error goes to zero if is larger than the uncertainties. The resultant observer equations are given as
(28) The range of feedback gain should be
(29) The boundaries of the uncertainties can be determined by comparing the cell test data with modeling parameters. The sliding-mode observer switching gain can be arbitrarily assigned to attain robustness against disturbances. However, the restriction on the assignment of the switching gain comes from the condition that the observer is stable. The practical system is implemented with a digital system which has a ?nite sampling time and gives rise to the chattering phenomena. The magnitude of chattering is highly dependent on the observer
(22)

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Fig. 5. Block diagram of the proposed observer operation.
Fig. 4. Picture of the test Li-PB.
as voltage recovery time and resultant value is 1.2 s. Then by circuit analysis, (32) The modeling parameters are calculated using circuit values
gain. If the gain is remarkably big, a large amount of ripples may result, causing estimation errors. Therefore, a tradeoff should be made between the robustness and stability of the observer [18]. The design methodology for sliding-mode observer can be summarized as follows. First, build the battery model by state equation including output state. Second, decompose state variable equations into corresponding observer equations. Third, design the feedback switching gain so as to guarantee the sliding-mode regime by equivalent control method. V. EXPERIMENTAL RESULT The cell comprises of a cathode, an arti?cial graphite anode and is designed for high power application. The nominal capacity and voltage is 5.0 Ah and 3.8 V, respectively. The picture of the test LI-PB is shown in Fig. 4. The dimension in mm and the weight of the cell of the cell is is 120 g. The RC model parameters are obtained from the circuit analysis methods. The nominal capacitance is determined by analyzing the amount of stored energy using the following expression:
(30) by the The internal resistance of the cell is measured to be 4 DC-IR test which performs 1-C discharge for 10 s and calculates . It is generally the ratio of voltage and current change corresponds to the 25% of the total internal assumed that resistance [2] (31) and . The is set to be Therefore the same as . The polarization capacitance is based on high frequency excitation test which involves 10-C pulsed discharge for 500-ms intervals to determine the time constant given by the and its associated resistance. The time constant is de?ned
The test was performed using the cell model parameters. The con?guration of experimental setup is shown in Fig. 5. The thermal chamber and the Nittetsu cycler were used as chargedischarge equipments for temperature regulations. Nittetsu cycler has 0 5 V and 0 120 A of voltage and current measurement range, respectively. The cycler’s voltage measurement accuracy is 5 mV and its current measurement accuracy is 200 mA. It also has precision ampere-hour counter for direct SOC calculation. True SOC was directly obtained from this amperehour counter. The test was performed with fully charged condition to set the SOC to one. As the test proceeds, the true SOC was calculated by the ampere-hour counter. The controller has been built with In?neon 16-bit microprocessor XC167-40 [Mhz]. It has been used for automotive parts such as engine control unit, transmission control unit, air bag, and so on. It contains internal A/D converter for 10-bit and other peripherals such as PWM and communications. The total control loop takes less than 10 ms including measurement and calculation time of the sliding-mode observer. Two types of cell tests were performed for the proposed observer. The ?rst type of test comprised a sequence of constant current discharges for 180 s and rests for 3600 s. The cell was fully charged up to 4.2 V before the test begins. The discharge current is 5 A and it corresponds to the 1-C rate of the nominal capacity. This amounts to 5% decrease of SOC for each period. The sampled data is collected every second. The purpose of this test is to set the OCV over the entire SOC range and the test result was shown in Fig. 2. The thermal chamber was set to 25 . This data was used to identify parameters of the cell models. The cell model parameters are obtained by the cell test results and the sliding-mode observer equations are established by current cycle of the LI-PB as shown in Fig. 5. The charge-discharge current is simultaneously applied to the LI-PB and sliding-mode observer. The terminal voltage of the LI-PB is measured as the

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Fig. 6. Current, voltages for true and model cell, and error.
Fig. 8. Estimated voltage and SOC of the sliding-mode observer.
Fig. 7. Polarization resistance R .
output and fed into the sliding-mode observer to compensate for the errors, and the output of the observer is the estimated SOC. The true SOC is obtained directly from the cycler by precision ampere-hour counter. The result comparing the electrical modeling with the cell’s test data is shown in Fig. 6. It shows the discharge current, true cell voltage, modeling cell voltage and the modeling voltage error. The shapes of the true cell response and the model output are similar in general, although many details of the true cell response are different. This is mainly due to the nonlinear charof the true cell and acteristics of the nominal capacitance also to the fact that the values of resistances are changed by the is a nonlinear resistance which varies on the current. SOC. over current is shown in Fig. 7. The plot of
The observer gains are selected to satisfy the condition in (29). The selected observer gains are , , . The results of SOC estimation using sliding-mode observer are shown in Fig. 8. The estimated model output is controlled with respect to cell terminal voltage with switching ripple, and the estimated SOC follows the true SOC although it has deviation at the start/end of the rest period. This is caused by the discontinuous current and is affected by the abrupt change of . In the discontinuous period, the sliding trajectory is away from the sliding surface by the discontinuous function, but the trajectory tracks into the sliding surface in a short time. The one cycle of Fig. 8 is rendered in Fig. 9. The estimated output voltage tracks cell voltage with chattering ripples. The estimated SOC also tracks true SOC with chattering ripples. The average value of the estimated SOC is close to the true SOC. This result shows that the proposed sliding-mode observer can track SOC accurately even in the presence of errors in the cell modeling. To verify the performance of the proposed observer at the real driving situation, the second test was performed as a sequence of 20 urban dynamometer driving schedule (UDDS) cycles. It is operated by series of charge-discharge pulses and 5-min rests, and spread over the 100%-0% SOC range. It can be seen that the SOC decreases by about 5% during each UDDS cycle. Fig. 10 shows the result of overall UDDS cycle current, true cell and

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Fig. 9. One-cycle plot of Fig. 7.
Fig. 11. One UDDS test cycle result for model voltage.
Fig. 12. Estimated SOC and error for UDDS cycles.
Fig. 10. UDDS test result of the model voltage and error.
model cell voltage and their voltage error. The modeling error is less than 20 mV for 20 80% of SOC range. For a clear view, one cycle of UDDS for model voltage is shown in Fig. 11. The proposed sliding-mode observer was applied to the overall UDDS cycle. The resultant SOC for whole UDDS cycle is shown in Fig. 12. The estimated SOC and error for the whole UDDS cycles are shown in the ?gure. The SOC error is bounded to 3% in all cases. The trajectory of the estimated SOC and error for the one UDDS cycle are shown in Fig. 13 in order to show clear view of the sliding-mode observer behavior. The trajectories are always con?ned to the true SOC with the chattering value. The magnitude of chattering is dependent on the sampling time. If the high performance microprocessor is used for the controller, the chattering would be
Fig. 13. One-cycle result of the estimated SOC and error.
smaller. This chattering can be smoothed by a saturation function instead of a sign function. In another way, the average value of the estimated SOC can be close to the true SOC. In this way, the suggested sliding-mode observer can be directly applied to the HEV environment with superior performance.

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Fig. 14. Tracking performance for an incorrect initial condition.
In order to show the tracking performance of the proposed system for an incorrect initial condition, the test was performed to verify the robustness of proposed system. The true SOC is 0.4 and the true cell voltage is 3.75 V at the starting time. The initial value of sliding-mode observer is set to far away from the true value. The initial value of estimated SOC is 0.8 and the estimated cell voltage is 4.17 V. As can be seen in the Fig. 14, the proposed system converges to the true value about 1 min. The convergence time would be shortened for the higher value of charge-discharge current. However, the current integration method fails to converge to the true value. VI. CONCLUSION A new method for battery modeling has been presented to compensate for the nonlinear Li-PB characteristics. The modeling parameters are extracted by a series of tests. The sliding-mode observer equations are obtained from the battery model and the cell output voltage. The design method of sliding-mode observer has been shown step by step and the convergence of observer has been proved by an equivalent control method. The proposed method shows robust tracking performance under modeling uncertainties and noisy environments compared to the conventional methods. The performances of the proposed system are con?rmed by the UDDS cycle test. SOC error is con?ned to the acceptable level, less than 3% in most cases which is applicable to the real environments. REFERENCES [1] S. Piller, M. Perrin, and A. Jossen, “Methods for state-of-charge determination and their applications,” J. Power Source, vol. 116, pp. 113–129, 2001. [2] B. S. Bhangu, P. Bentley, and C. M. Bingham, “Nonlinear observers for predicting state-of-charge and state-of-health of lead-acid batteries for hybrid-electric vehicle,” IEEE Trans. Veh. Technol., vol. 54, no. 3, pp. 783–794, May 2005. [3] G. L. Plette, “Extended kalman ?ltering for battery management system of LiPB-based HEV battery packs—Part 2. Modeling and identi?cation,” J. Power Source, vol. 134, pp. 262–276, Aug. 2004. [4] S. I. Jang, M. C. Shin, C. D. Yoon, and R. C. Campbell, “A study on adaptive autoreclosure scheme with real-time transient stability,” J. Elect. Eng. & Technol., vol. 1, no. 1, pp. 8–15, 2006. [5] C. C. Chan, E. W. Lo, and S. Weixiang, “The available capacity computation model based on arti?cial neural network for lead acid batteries in electric vehicles,” J. Power Source, vol. 87, pp. 201–204, 2000.
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