# I. INTRODUCTION

Lithium batteries have many advantages such as high energy density, high power density, long cycle life and so on. They are widely used as the energy storage elements of electric vehicles. In order to improve the safety and reliability of a battery pack, fully play its efficiency, and prolong its life, the battery pack must be managed effectively. Battery state of charge (SOC) estimation is important since it is the basis of battery management systems (BMS). However, battery is a kind of nonlinear uncertain system. As a result, it is very difficult to accurately estimate battery SOC.

The common methods of SOC estimation are: 1. Current integration [1]: SOC is estimated by the time integral of the current with the initial SOC provided. However, the error increases with time. 2. Open circuit voltage (OCV) [2]: SOC is estimated based on the functional relationship between open circuit voltage and SOC. However, the battery must be set aside for a period of time. Unfortunately, this is not suitable for real-time SOC estimation. 3. Neural network [3]-[6]: a large number of samples with comprehensive data are required for model training and the sample data as well as the training methods affect the accuracy. 4. Kalman filter [4], [7]-[10]: SOC is estimated with algebraic iterative method. However, its accuracy is affected by model accuracy. 5. Predictive electromotive force (EMF) or OCV [11], [12]: these methods can estimate SOC accurately. However, the formulas involve many parameters, and the computation is complicated. 6. Mathematical fitting [13]: the function relationship SOC and the current, voltage, temperature and so on can be concluded by experiments. However, the function relationship is limited by the applicable conditions. 7. Impedance analysis [14]: SOC is estimated by battery electrochemical impedance spectroscopy analysis. However, this is only suitable for lab research.

Sliding mode observer is a kind of iterative algorithm which can be used in battery SOC estimation [15]-[20]. The design of previous sliding mode observers for battery SOC estimation includes: 1. Battery dynamic model is established in matrix form. Base on this model, an observer is designed which needs to solve the matrix equation. When the order of the matrix is relatively high, the calculations are more complicated [15], [16]. 2. Battery dynamic model is established in differential equations form. One observer is designed on the basis of each equation, which is needed to design multiple observers for all of the state components [17], [18]. Battery model parameters are changing in terms of some influence factors such as temperature, SOC, current rate and so on. Therefore, sometimes the model parameters are represented by variables which are functions of the above influence factors. The design method of the changing parameters system observer is the same as that for the above two methods [19], [20].

To improve methods for estimating battery SOC, this paper employs a novel sliding mode observer for SOC estimation. Firstly, based on battery Thevenin circuit model, taking the battery polarization voltage and open circuit voltage as state components, with the difference between the terminal voltage of the battery and the ohm resistance voltage drop as the observation, a two-dimensional state space equation is established to simplify the battery state equation. Secondly, based on the battery state equation, a novel observer, which does not possess the limiting condition of existing observers, is designed. The design of the new observer for SOC estimation does not need a solving matrix equation or multiple observers for all of the state components. Therefore, it can reduce the calculations in design and is easy to use in engineering.

The new observer can be used for both time-variant systems and time-invariant systems, and the observer design methods of the two kinds of systems are the same. The observer’s robustness is proved by Lyapunov stability theorem. The characteristics of the new observer, based on the changing parameters model and the unchanging parameters model, are verified by tests. The verification shows that the new observer is robust and has a simple design. It also shows that improving the battery model accuracy can decrease the estimation error.

# II. BATTERY MODELING

The Thevenin circuit model is shown in Fig. 1. It is simple and well simulated to the external features of batteries.

**Fig. 1.**Battery Thevenin model.

In Fig. 1, vocis the OCV, which is equal to the electromotive force of the battery; r0 is the ohm internal resistance of the battery; c1 and r1 are polarization capacitance and resistance; vt is terminal voltage of the battery; v1 is the polarized voltage; and it is the input current. The results are as follows:

Therefore, the differential equation of v1 is:

There exist a function relationship voc = g(Soc) between OCV and SOC, where Soc represents SOC, and the function does not experience a significant change with temperature [17], [18], [21]. The test battery OCV vs. SOC curves at -10℃, 0℃, 10℃, 20℃, 30℃ and 40℃ are shown in Fig. 2. From Fig. 2, it can be seen that the test battery OCV curves have small differences. In addition, the battery test was operated in thermostat. Therefore, this paper take it as an invariant function. As a result, the differential equation with t as the time is:

**Fig. 2.**OCV vs. SOC at different temperatures.

In equation (3), k is the derived function of the OCV with respect to the SOC and it is changed with the SOC accordingly; η is the discharge efficiency as well as a variable; and Qn is the battery capacity. Based on equations (2) and (3), the battery model is:

Equation (5) is the result of k and η in (3) after being substituted with 1, the error is merged into the rear nonlinear uncertain function. The two bounded functions f1 and f2 are the error-sum of linearization, noise and other factors. In line with the current direction in Fig. 1, the system terminal voltage is:

Since the terminal voltage vt, current it and ohm internal resistance r0 can be obtained by measurement, it is possible to choose vy = vt - itr0 as an observation. Then the observation equation is:

Equations (4), (5) and (7) are battery state space equations where the state variable is x = [v1,voc]T, the observation is y = vy, and the input is u = it. Therefore, the coefficient matrices are:

The system state is estimated with a sliding mode observer which needs the system’s linear part to be observable. Based on the necessary and sufficient condition, the result is as follows:

Since rankH = 2 . Then the linear part of the battery state equation is observable.

The relationship between the SOC as an independent variable and the OCV as a function is shown in Fig. 2. If the OCV is an independent variable and the SOC is a function, the result at 20℃ is shown in Fig. 3. The OCV can be estimated by the sliding mode observer. Based on the relationship between the SOC and the OCV Soc = g-1(voc), the SOC can be estimated with the OCV as an independent variable and the SOC as a function.

**Fig. 3.**SOC vs. OCV at 20℃.

In applications, it is possible to obtain the battery state components polarization voltage and the OCV by algebraic iteration of the novel sliding mode observer. The SOC is estimated based on the function relationship between the OCV and the SOC. The SOC can be estimated in real-time by the novel observer, which meets the demands of electric vehicles.

# III. THE NOVEL SLIDING MODE OBSERVER METHOD FOR SOC ESTIMATION

## A. Underlying Theory

The following is a kind of nonlinear uncertain system:

Where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rl×n, n ≥ l ≥ m and u(t) ∈ Rm is known system control variables, with full-rank B , C , observable (A,C) ,as well as bounded, nonlinear, and uncertain function f(x,u,t) = [f1, f1,..., fn]T ∈ Rn. The existing sliding mode observer for the system must satisfy f(x,u,t) = Bξ(x,u,t) , where ξ(x,u,t) ∈ Rm is an arbitrary bounded function. Compared with the existing observers, the novel observer, which does not have the limiting condition above, is proposed for SOC estimation in this paper.

For equations (9) and (10), the design of the new observer is as follows [22], [23]:

is the system state vector estimation, G ∈ Rn×l is the designed matrix, and v(t) ∈ Rn×R+ → Rn is the control variable designed for the observer. If the error of the system state vector estimation is the design for the sliding mode surface is as follows:

Where ŷ(t) ∈ Rl is the system observation estimation, F is the designed matrix, which makes the state variables in the surface slid to the zero equilibrium point, and M = FC . The control variable v is designed as [22], [23]:

Where β ≥ 0, and 0 ≥ η ≥ 1.

Based on equations (9) and (11), the following equation for the estimation error is:

Where A0 = A - GC. If the error is partitioned as e = [e1;e2] with e1 ∈ Rn-m and e2 ∈ Rm, the error state equations in partitioned matrix form are as follows:

Accordingly, the sliding mode surface in partitioned matrix form is as follows:

In the above expressions, M ∈ Rn×n , M = [M1,M2] , M1 ∈ Rn×(n-m) , M2 ∈ Rn×m , A011 , A012 , A021 , and A022 are partitions of A0 , A011 ∈ R(n-m)×(n-m) , A012 ∈ R(n-m)×m , A021 ∈ Rm×(n-m) , A022 ∈ Rm×m , B = [B1;B2] , B1 ∈ R(n-m)×m , B2 ∈ Rm×m , f = [f2;f1], f1 ∈ Rn-m , and f2 ∈ Rm .

For the error components are not linear correlation, the linear transformation in the error system will not change the convergence of the error system. Therefore, the error system (15), (16) and the sliding mode surface in the partitions form (17) can be transformed as:

The designed observer shows robustness against modeling errors, when the following three assumptions for the sliding mode observer (11)-(13) are satisfied.

A1: Let

For the sliding mode observer (11)-(13), the matrixes M and G are designed to make A0 satisfy λmax(A0) ≤ 0 and to make AS satisfy λmax(AS) ≤ 0, where λmax(A0) and λmax(AS) are the matrix maximum eigenvalues of A0 and AS, respectively.

A2: In equation (13), parameter ρ is satisfied by ρ║MB║≥║Mf║.

Satisfy assumptions A1 and A2, and the control variable (13) can make the system state vector x(t) move to the sliding mode surface (12).

A3: Let:

The matrix M' is designed to make AM the same as a Hurwitz matrix, and the sliding mode observer can make the error vector converge to an equilibrium point e(t) = 0.

## B. Verifying the Robustness of the Novel Sliding Mode Observer

The following is to verify the robustness of the novel sliding mode observer (11)-(13) against the nonlinear functions f(x,u,t) in system (9) and (10).

Take the Lyapunov function as:

The derivative is as follows with t as the time:

If the parameter is ρ║MB║≥║Mf║ and S ≠ 0, then:

Now the state vectors will converge to the sliding mode surface. When the vector reaches the surface, it will slide along the surface accompanied by up and down quivering. The expression of the sliding mode surface (20) shows: if it is substituted into equation (18), and equations (12) and (13) can tell that v = 0 on the sliding mode surface. Finally, it can be known that the dynamic feature of the error state vector on the sliding mode surface is:

If AM is a Hurwitz matrix, the error state will be converged to an equilibrium point e(t) = 0 . Therefore, the sliding mode observer is stable. The speed of the convergence will be affected by the eigenvalue of AM.

The proving processes are suitable for both time-varying systems and time-invariant systems. Therefore, the new observer method can be used in the two systems. The design methods for the time-invariant system observer and the time-varying system observer are the same. Battery model parameters are functions of temperature, current rate, SOC and so on, and the functions can be obtained by testing.

## C. Sliding Mode Observer Design

The design procedures of the new sliding mode observer method for SOC estimation are as follows:

D1. According to the error state function (14), the matrix function A0 = A - GC is known. The matrix is designed so that G = 0. Then . As a result, λmax(A0) ≤ 0. The matrix was designed so that F = 1. Then M = FC = [1 1]. According to equation (21) AS = 2A0 . Therefore, λmax(AS) ≤ 0

D2. The following parameter was designed so that ρ = 10, which makes ρ║MB║≥║Mf║ in this test.

D3. Meanwhile, A012 = A021 = A022 = 0, and the coefficient matrices of the error equation (18) is . According to expression (22) , which is a Hurwitz matrix.

The observer estimators are OCV voc and the polarization voltage v1. In addition, the whole algorithm estimator is the SOC. For the test in this paper, the designed parameters values of the new observer are shown in Table I.

**TABLE I**THE DESIGNED PARAMETERS VALUES

The three assumptions mentioned in previous are satisfied according to the above designs. Therefore, the new observer is robust to the uncertain part f(x,u,t) in the model. The discrete form of the new observer is:

Where and

Where the sampling period Δt = 1s. Taking the coefficients of equations (4), (5) and (7) and the designed parameters values into the discrete form of the observer equations (26) and (27), the OCV as the system state vector component can be estimated by iterative computation, and the SOC can be estimated by the function of SOC vs. OCV. The overall SOC estimation algorithm is shown in Fig. 4.

**Fig. 4.**The whole SOC estimation algorithm.

For the design of the exiting slide mode observer it is necessary to solve the matrix equation or to design multiple observers for all of the state components, and the design calculation is complex. Through its design processes, the new observer does not need to solve the matrix equation or design many observers. Therefore, it is simple and convenient for engineering applications.

# IV. SOC ESTIMATION BY THE NEW OBSERVER METHOD

To verify the performance of the novel sliding mode observer method for battery SOC estimation, a test under preset conditions for a lithium battery is operated, in which the SOC estimation and the ideal value are compared. The test battery shown in Fig. 5 is a 25Ah Lithium power battery.

**Fig. 5.**The test battery.

The testing platform consists of the Arbin BT2000 battery tester for discharging and charging batteries shown in Fig. 6, the incubator shown in Fig. 7, a host computer and monitoring software for the working processes. The monitoring software of the host computer can set the battery state while monitoring and collecting information on the current, voltage and temperature. The incubator can guarantee a set temperature for the battery.

**Fig. 6.**Arbin BT2000 battery test equipment.

**Fig. 7.**Incubator.

## A. Battery Circuit Model Parameter Extraction

Extracting the parameters of Thevenin circuit model is necessary when using the new sliding mode observer method for SOC estimation. In this case the pulse current is used for the extraction. At room temperature, the pulse current become 0A suddenly from the C/3 discharge rate, and the battery terminal voltage change curve at SOC=0.537 is shown in Fig. 8.

**Fig. 8.**Battery terminal voltage after pulse current.

From Fig. 1, it can be seen that the ohm internal resistance of the battery Thevenin model is r0 = Δu0/i, and that the polarization internal resistance is r1 = Δu1/i.

According to the dynamic feature of the RC circuit, the time constant is τ = r1c1. After the current pulse discharging for τ, the battery terminal voltage increases to 63.2% Δu1.

The model parameters can be computed based on the above expression, and the terminal voltage curve of the battery after a pulse is shown in Fig. 8. The results of the model parameters at SOC=0.537 are: r0 = 1.24mΩ, r1 = 1.46mΩ and c1 = 1.24×105F .

## B. Simulation FUDS Condition Estimation Result

The performance of the new sliding mode observer will be verified under the simulated FUDS condition with the battery shown in Fig. 5. The test is designed as follows: At room temperature, a battery with an initial SOC of 100% is selected. The simulated FUDS condition includes discharge and charge processes. To prevent the battery from being over charged, 1AH is discharged at the C/3 rate. Then the battery is run under the simulated FUDS condition set by the host computer, and stopped the test when the battery terminal voltage reaches 3v. At this point, the SOC may not be 0. The current and the terminal voltage under the simulated FUDS condition are shown in Fig. 9.

**Fig. 9.**Discharge current and terminal voltage.

The battery SOC is estimated with a new observer under the simulation FUDS condition, during which the initial value of the open circuit voltage can be set randomly within the stable range of the observer. If the initial polarized voltage v1 = 0 and the initial open circuit voltage is 3v, then x0 = [0;3]. After iterative computation, the estimated values and the measured values of the observation are shown in Fig. 10.

**Fig. 10.**The estimated values and the measured values of the observation.

From Fig. 10, it can be seen that the estimated observation converges to the measured observation rapidly in the initial stage, after stabilization, the estimated values shake up and down along with the measured values. If the initial states are set like the other values, similar results can be obtained.

The initial value of the polarized voltage is v1 = 0 , and the initial values of the open circuit voltage voc are 3v, 4v and 5v. After the iterative computation, the open circuit voltage estimation results are shown in Fig. 11. They have almost equal convergent results.

**Fig. 11.**Battery OCV estimation by the new sliding mode observer.

The SOC can be estimated based on the relationship between the OCV and the SOC in Fig. 3. Under the simulated FUDS condition, the battery discharge efficiency is η ≠ 1. After the test and a rest for 2 hours, the OCV changed to 3.2216v. As a result, the SOC is 0.0079 instead of 0, with 0.2122AH left. The sum of the discharge coulomb, counted by the discharging efficiency η = 1, and the remaining capacity is almost the full capacity of the battery. Therefore, the average discharging efficiency is set as 1. The real SOC can be computed with the current integration method since the initial SOC is 100%. The estimated SOC and the real value are shown in Fig. 12 and the estimation error is shown in Fig. 13.

**Fig. 12.**Battery SOC estimation.

**Fig. 13.**Battery SOC estimation error.

Fig. 12 shows that the SOC estimation by the new sliding mode observer method converges to the ideal value under simulation FUDS condition in spite of different initial open circuit voltages.

Fig. 13 shows the corresponding SOC estimation error. After stabilization, when the initial voltage is set to 3v, 4v and 5v, the estimation errors are all within 4.5%. Likewise, similar results can be obtained when the initial state, which is in the stable range of the observer, is set to different values. Test analysis shows that the estimation results are not affected by the initial values, which are within the stable range. The test with the simulated FUDS condition shows that the new sliding mode observer method is robust to nonlinear uncertain systems.

## C. Changing Parameters Model Estimation Result

Battery is a kind of complex time-varying system, and many influence factors, such as SOC, current rate, temperature and so on, can cause changes in the model parameters, which affects the SOC estimation. The function relationships between the parameters and influence factors can be obtained by testing.

Battery is a kind of complicated nonlinear uncertain system. Therefore, the design guidelines of the new observer consider the parameters changes which effect the SOC estimation, and some of the major factors that influence parameters changes. In the new observer, r0 has a significant influence on the SOC estimation. r0 changes with the above influence factors, and the function r0 = r0(Soc,i,T) can be concluded by testing, where T is the battery temperature. In the test, the battery was operated in thermostat, and the current rate was not more than 2C. Therefore, the effects of the current rate and temperature were not considered. The battery SOC was changed from 100% to 0%, and the function r0 = r0(Soc) is shown in Fig. 14. It can be seen from Fig. 14 that r0 is increased greatly when the SOC decreases form 100% to 0%. Therefore, the model parameter r0 = r0(Soc) is chosen in this paper.

**Fig. 14.**The relationship between ohm internal resistance and SOC.

The verifying processes show that the new observer is suitable for both time-varying systems and time-invariant systems. The function r0 = r0(Soc) is substituted in the system equation, and the changing parameter r0(Soc) is used in the new observer for the SOC estimation, which is a circulation iterative approach algorithm [19], [20]. The algorithm schematic diagram is shown in Fig. 15. The observer design methods of the two systems are the same [19], [20].

**Fig. 15.**Circulation iterative algorithm of the new observer.

The analysis shows, that when the design parameters of the time-varying battery system applied to the hardware experiment are same as those in Table I, the new observer is robust. According to the identified model parameters above, the time-varying battery system model parameters are shown in Table II.

**TABLE II**BATTERY TIME-VARYING SYSTEM MODEL PARAMETERS

Similarly, the OCV is the observer estimator, and the SOC is the whole algorithm estimator. By taking the designed parameters and the model parameters into functions (26) and (27), the OCV can be obtained.

Let the initial state x0 = [0;4]. The SOC estimation of the hardware realization is based on both the changing and unchanging parameter models, which are shown in Fig. 16, and the estimation error is shown in Fig. 17.

**Fig. 16.**SOC estimation based on time-invariant model and time-variant model.

**Fig. 17.**SOC estimation error base on time-invariant and time-variant systems.

From Figs. 16 and 17, it can be seen that updating parameters in real time can improve the accuracy of the estimation. Similar results can be obtained if the initial open circuit voltage is 3v and 5v. When the initial voltage is set as 3v, 4v and 5v, the maximum estimation errors are about 3.4%.

A battery is a strongly nonlinear uncertain system. Therefore, it is very difficult to accurately establish battery model. The SOC estimation used the new observer is on the basis of the battery model, and it is worth further study in terms of establishing a more precise battery model to improve estimation accuracy. The battery model, measuring accuracy, battery nonlinear characteristics and other characteristics contributed to estimation errors. In order to further decrease estimation errors, it is necessary to improve the precision of measurement, battery performance and so on.

# V. CONCLUSION

(1) A novel sliding mode observer was proposed in this paper. It was developed from existing observers, for battery SOC estimation. Through the design of the new sliding mode observer, it can be seen that it is simpler than the existing observer design method and convenient for engineering applications.

(2) The robustness of the new observer is verified by Liapunov stability theory and simulation FUDS condition testing.

(3) The new observer can be used for both changing and unchanging parameters model SOC estimations, and the observer design methods are the same. Through tests, it can be seen that the estimation accuracy is improved when the model parameters are updated in real-time.

(4) In order to improve estimation precision, there needs to be further study on battery performance. In addition, more accurate models need to be established in future research.