An Adaptive Strategy for Li-ion Battery SOC Estimation

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1 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 An Adaptive Strategy for Li-ion Battery Estimation D. Di Domenico, E. Prada, Y. Creff IFP Energies nouvelles, Rond-point de l échangeur de Solaize BP 3, 6936 Solaize, France; {domenico.didomenico, eric.prada, yann.creff}@ifpen.fr Abstract: This paper presents an extended Kalman filter (EKF) based on an electro-thermal for the of the state of charge of a lithium-ion battery. In order to compensate for uncertainties in the parameters and the measurements, it is first shown that the filter robustness strongly depends on the range. Then the filter weights are adapted according to the value. This technique is tested using experimental data collected at the batteries testing facilities of IFP Energies nouvelles, from a commercial A23 lithium iron phosphate/carbon (LFP/C) cell. Despite its simplicity, the filter shows good performance, with an average error within 3% range.. INTRODUCTION Thanks to its high specific energy and power, weak memory effect, and a slow loss of charge when not in use, lithium-ion battery technologies are becoming more and more important as power source for new plug-in hybridelectrical vehicles (PHEV) as well as in many new generation electric vehicles (EV) and hybrid electric vehicles (HEV). As a consequence, increasing demand for nontraditional vehicles has resulted in increasing research effort on battery management system (BMS) design. The optimal battery managementprogressivelybecomes one of the most relevant issues in automotive control [Chaturvedi et al., 2]. BMS has to ensure the appropriate use of the battery in providing the electrical power demand, while guaranteeing feasible and safe operations. Indeed, apart from avoiding the overcharge and the thermal abuse, that can cause battery lifetime degradation, permanent damages or evenexplosion, an efficient BMS must include the cell balancing and the battery thermal management. In order to improve the BMS, dynamic battery s are required [Chaturvediet al., 2]. Typically, equivalent circuit s are used for this purpose. However, electrochemical control-orienteds have also been proposed [Santhanagopalan and White, 26, Smith et al., 27]. Based on electrochemical laws, these s are able to describe the physical cell limitations, which have relevant effects during high transient loads. In both electrochemical and equivalent circuit s, the battery dynamics strongly depends on the state of charge (). The main battery operations are then related to the which is usually a key task for BMS [Wang et al., 29, Chaturvedi et al., 2]. Accurate is required in order to achieve high efficiency, slow aging, no damaging and (for PHEV or HEV) pollutant emission reduction. Allowing a correct use of the battery, knowledge of is useful to improve both battery health and vehicle economy through optimal battery sizing [Hu et al., 2]. The literature on battery is quite extensive and several techniques have been proposed, exhibiting advantages or disadvantages [Piller et al., 2]. Among them, the Ampere-hour (Ah) counting is used by many commercial battery BMSs. Consisting in integrating the battery current, this open-loop and non- based method is easy to implement on-line but it is affected by the uncertainty on the initial condition, by the measurement error accumulated during the battery life and by the battery capacity degradation due to usage [Lin et al., 2, Terry and Wang, 25, Wang et al., 29]. Other drawbacks, such as the losses during charge or discharge and the drift caused by the cell self-discharge have also been highlighted [Ng et al., 2]. The Ah counting can be combined with the open circuit voltage (OCV) measurement, useful to re-initialize the counting after long rest periods [Wang et al., 29] or to estimate the battery capacity [Ng et al., 2]. Unfortunately, due to the large characteristic time associated to the battery relaxation, the OCV measurement can be unavailable in automotive applications [Pang et al., 2, Verbrugge and Tate, 24]. Black-box methods, such as fuzzy-logic [Salkind et al., 999] or artificial neural network [Shen, 27, Cheng et al., 28] have alsobeen proposed. Evenif they can reachhigh accuracy, the learning process is computationally heavy and, furthermore, it has to be performedoff-line. Recently the effort to use -based techniques for the battery state, such as extended Kalman filter [Plett, 24, 26, Barbarisi et al., 26, Santhanagopalan and White, 26, Smith et al., 28, Di Domenico et al., 2] or sliding mode [Kim, 26], has been intensified. The accuracy reached by these techniques is about 2%. Despite good results obtained by the EKF, this approach presents some critical points, related to the linearization, to the nonlinear propagation of the white noise and the high sensitivity of the on the ing uncertainty [Julier and Uhlmann, 24]. These limitations result in a classical compromise between the filter robustness and the convergence speed [Wang et al., 29, Santhanagopalan and White, 2]. In this paper, we present an extended Kalman filter for, based on a that integrates the Copyright by the International Federation of Automatic Control (IFAC) 972

2 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 mostrelevantelectrochemicaleffects inalow-orderlumped [Bernard et al., 2, Prada et al., 2]. Even if such a cannot take into accountthe low-frequencies phenomena, this choice was made with the intent to test the level of precision that can be reached by simplifying as much as possible the structure of the estimator. The filter exhibits excellent performance in simulation but, as expected, it suffers, over a large spectrum of, from a lack of robustness which is mostly a consequence of the flatness of the OCV with respect to the and of the neglected dynamics. The analysis of the filter sensitivity allowed us to favor the filter robustness when the sensitivity is higher and, conversely, to privilege the speed of convergence when the filter is more robust. Further, the OCV map is used to initialize the filter, pre-positioning the close to the actual value, which is particularly important when a more robust is required and the filter convergence speed is reduced. The global strategy is then tested using both generated and experimentally collected data for a Hybrid Pulse Power Characterization (HPPC) profile [USCAR, 23]. Despite the simplicity, the error in the, when compared to the prediction, is within 3% and is comparable with the results of the techniques presented so far in literature [Plett, 24, 26, Barbarisi et al., 26, Santhanagopalan and White, 26, Smith et al., 28, Di Domenico et al., 2]. 2. LFP/C CELL MODEL The D impedance-based takes into account the main Li-ion electrochemical phenomena: electromigration inside the electrolyte at high frequencies; Butler-Volmer electrochemical charge transfer kinetics in the medium frequency range associatedwith electrochemical double layer capacitance; diffusion of ionic species in the electrodes in the low frequency range. All the phenomena follow Arrhenius laws. From simplified Butler-Volmer formalism for a redox reaction, charge transfer current density can be expressed as ( ) ( αox nf i f = i (exp RT η exp α )) rednf RT η () with i the exchange current density, n the number of electrons transferred in the redox reaction. A symmetry assumption (α red = α ox ) is made concerning the charge and discharge, giving charge transfer resistance R ct as a function of faradaic current ( ) if R ct = = η i nf RT + ( if 2i ) 2 (2) Up to fast stable dynamics, i f is equal to I, the current demanded to the battery. The diffusive impedance is often represented by a Warburg impedance, the expression of which differs as the mass transport boundary conditions vary [Diard et al., 996]. In the frequency domain, Warburg impedance with specific boundary conditions can be written as sτd ) tanh( (q, T) Z w (s) = R d (q, T) (3) sτd (q, T) where q denotes the, R d is the diffusion resistance, τ D is the characteristic diffusion time of the phenomenon, and s is the Laplace variable. In the time-domain, Z w is usually implemented as a transmission line by a finitelength Cauer or Foster-type electric network [Kuhn et al., 26]. In the present work, a first order transmission line, function of the, has been chosen in order to reduce complexityandcomputationaltime: this introduces a variable V d into the. The energy balance inside the cell can be expressed as mc p dt dt = Q elec,gen(q, T, I) q n (T) (4) where m is the cell mass (kg), C p is the calorific capacity (J/kg/K), T the temperature (K) inside the cell, Q elec,gen the total heat generated (W) by the cell as the current flows through it, and q n the heat exchanged with the ambient (W) [Kim et al., 27]. The total heat generation rate is the sum of reversible and irreversible contributions. The reversible heat generation rate Q rev (W) released by an electrode is given by Q rev = du th (q)it (5) dt where du th /dt is the entropic term, function of. The irreversible heat generation rate, generally known as Joule losses, is always exothermic and is given by Q irrev = R tot (T, I)I 2 + V d 2 (6) R d (T) where R tot (T, I) = R Ω (T) + R ct (T, I) (7) and R Ω is the high frequencies electromigration resistance. In state-space formulation, the can be expressed as q = I C nom ( V d = I V ) d (8) C d (q, T) R d (T) T = (Q irrev (T, I) + Q rev (q, T, I) q n (T)) mc p where C nom is the nominal cell capacity and C d is the diffusive capacitance. Finally the cell voltage is computed as V = U (q, T) + V d + R tot (T, I)I (9) where U is the OCV. The was calibrated and validated based on experimental data performed at the batteries testing facilities of IFP Energies nouvelles on a commercial 2.3Ah LiFePO 4 /C cell. The results are shown in Figures and KALMAN FILTER STATE OF CHARGE ESTIMATION In most cases, battery has to be estimated based on measured the voltage V, the temperature T and on the known demanded current I. As the thermal dynamics are slower than the electric dynamics, the temperature is considered as a slowly varying parameter known from the measurements. The LFP/C cell then reduces to 9722

3 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 current [A] voltage [V] experimental x 4 Fig.. Cell voltage compared with the LFP/C cell prediction. temperature [K] current [A] experimental 36 Fig. 2. Cell skin temperature is compared with the LFP/C cell prediction. q = I C nom V d = C d (q) ( I V ) () d R d and V = U (q) + V d + R tot (I)I () In order to verify the system observability, the observability matrix of the linearized system can be checked. Linearization leads to {ẋ = Ax + Bu (2) y = Cx + Du with x = (q, V d ) T, u = I, y = V and A = ( Cd 2 I V ) d dcd (3) R d dq R d C d B = C = 36C nom C d ( ) du dq D = R tot + R tot I (4) The observability matrix for the system (2)-(6) is (5) (6) O = du dq ( Cd 2 I V d R d ) dcd dq R d C d (7) As O is full rank, the system (2) is observable. The observability of the linearized system implies the non linear system observability [Hermann and Krener, 977]. Based on the 2 nd -order () a Kalman filter can be designed, according to { ˆx = f (ˆx, u) + Ke (y ŷ) (8) ŷ = V (ˆx, u) where ˆx and ŷ are respectively the estimated state and output, f( ) is the nonlinear function deduced from () and V ( ) is the nonlinear output function (). K e is the Kalman gain, obtained as K e = PCR (9) where P is the solution of the Riccati equation P = AP + PA T PCR C T P + Q (2) P() = P and Q and R are weight matrices appropriately tuned offline to minimize the mean square error between the predicted and the. error [V] voltage [V] Fig. 3. Kalman filter simulation test: voltage when noise is added to the measurements % error Fig. 4. Kalman filter simulation test: error on the state of charge when noise is added to the measurements. Figures 3-5 highlightthe filter performance in simulation. A HPPC profile is imposed to the cell and the 9723

4 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Table. Filter robustness. For each parameter, a +% variation is imposed. Parameter MSE Mean of the absolute Standard deviation of the ( 5 ) error ( 3 ) absolute error ( 3 ) Nominal filter R ct R ct R ct R ct R ct C nom R d R d R Ω C d C d C d U U U U V d [V] uncertainty of % with respect to the nominal value was simulated for each filter parameter. Several robustness indicators were considered, namely the mean square error, the mean and the standard deviation of the absolute error. The mean square error was computed as T MSE = q(t) ˆq(t) 2 dt, (2) T where q is the reference value of the and ˆq(t) is the filter..2 Fig. 5. Kalman filter simulation test: V d when noise is added to the measurements. outputs (voltage and temperature) simulate the cell measurements as filter inputs. Band-limited white noise, with an average amplitude of 5 mv and 2 ma, is also added respectively to the current and voltage, in order to simulate the sensors noise. Introducing the sensors noise into the simulation decreases the convergence time and reduces the accuracy. However the performance is satisfying. In Figure 4 the 2 nd -order Kalman filter results are compared with the full order. The error in the initial condition, close to 3%, is fully and quickly recovered, showing that the filter is able to estimate the correct value of the battery state of charge even if its open loop prediction is wrong. Comparison also shows that the error is less than ±5.%. 4. ROBUSTNESS ANALYSIS Due to ing errors, good filter performance in simulation does not imply good performance in experimental test. Parameter uncertainties, mismatch between measured voltage and prediction, neglected dynamics can heavily deteriorate the -based. A robustness analysis is thus required for a sounded simulation test. The analysis results are summarized in tables and 2. Parameters R cti (resp. R di, R Ω, C di and U i ) are used in the definition of R ct (resp. R d, R Ω, C d and U ). An EKF Fig. 6. Kalman filter sensitivity analysis: filter performance if an indetermination of % is imposed on one of the equilibrium potential parameter. The results in table highlight the critical points for the filter sensitivity, showing that an error of % on the parameters may even results in 2% error on the state. As an example, Figure 6 shows the effect on the of the % uncertainty on the parameter U 2. In order to reduce the error, the nominal Kalman filter weights can be reset, reducing the filter speed of convergence in order to increase the robustness. On the other hand, Figure 6 points out that the filter sensitivity is nonhomogeneous and depends on the value. This is confirmed by table 2 that presents the robustness analysis results for five different and disjoint intervals. When tested on the experimental data, 9724

5 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Table 2. Filter robustness. For each parameter, a +% variation is imposed. The MSE is normalized with respect to the nominal value. The global robustness index (computed as the average of the indexes for each parameter divided by ) summarizes, for each interval, the filter robustness with respect to the main parameters indetermination. Parameter MSE (in %) MSE (in %) MSE (in %) MSE (in %) MSE (in %) ǫ[.8, ] ǫ[.6,.8[ ǫ[.4,.6[ ǫ[.2,.4[ ǫ[,.2[ R ct R ct R Ω C d R d U U U U global robustness index % error Fig. 7. Experimental open circuit voltage function of temperature and. the Kalman filter was then reset optimizing the weight matrices R and Q function of the interval. The results are shown in Figure 8. Furthermore, in order to pre-position the close to the actual value, an initialization module is added to the filter based on the experimental OCV map, shown in Figure 7. The open circuit voltage map is the result of a characterization test performed the batteries testing facilities of IFP Energies nouvelles on a commercial 2.3Ah LFP/C cell. The test consisted in measuring the cell voltage after relaxation every 2% of at T = 33K, 293K and 273K. Figure 9 shows the results of the HPPC experimental test for the strategydescribedabove. Evenifthe initial error on the is only partially removed due to the long rest period, the filter pre-positioning is able to increase the filter convergence time, as confirmed by the comparison between Figures 8 and 9. It has to be noted that a good compromise between robustness and speed of convergence is achieved. As zoom of Figure 9 highlights, the convergence delay is compensated for by the filter pre-positioning that predicts the initial condition within a 2% error range. 5. CONCLUSION A Li-ion battery describing thermal, physical and electrical battery properties was presented and validated on a HPPC profile at T=33 K. Based on the, Fig. 8. Experimental test: strategy performance. The is compared with the prediction % error Fig. 9. Experimental test: performance of the strategy with the pre-positioning. The is compared with the prediction. a 2 nd -order extended Kalman filter was designed for the of the state of charge. The filter showed high performance when tested in simulation, but exhibits a strong sensitivity to the parameters indetermination. Furthermore the filter sensitivity analysis highlighted that the filter robustness depends on the range. In order to compensate for the filter sensitivity to the and the measurement uncertainties, the filter was readapted by tuning its weight matrices depending on the robust

6 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 ness degree function of the interval. The strategy was then tested experimentally during a HPPC profile using the data collected on a commercial A23 lithium iron phosphate/carbon cell. Despite its simplicity, the filter showed good performance during the test, with an error within 2% 3%. In order to further improve this work, future developments are in progress, concerning both ing and. The low frequencies accuracy of the is being improved by using a higher order transmission line to approximate the diffusive impedance. Furthermore, the opportunity of the on-line filter update will be explored in order to account for the a priori unknown process and measurements noise covariance values. REFERENCES O. Barbarisi, F. Vasca, and L. Glielmo. State of charge Kalman filter estimator for automotive batteries. Control Engineering Practice, 4: , 26. J. Bernard, A. Sciarretta, Y. Touzani, and V. Sauvant- Moynot. 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