Set-based State of Charge Estimation for Lithium-ion Batteries

Transcription

1 214 American Control Conference (ACC June 4-6, 214. Portland, Oregon, USA Set-based State of Charge Estimation for thium-ion Batteries Matthias Rausch 1,2, Reinhardt Klein 1,3, Stefan Streif 1, Christian Pankiewitz 2, Rolf Findeisen 1 Abstract Batteries, in particular lithium-ion batteries, are one of the dominant energy storage devices, especially in the field of electric mobility. However, the long-term, safe and robust operation of batteries requires accurate estimates of battery parameters and states, and most importantly of the state of charge (SOC. The estimation of these quantities is difficult due to measurement uncertainties, model mismatch and parameter variations, as well as nonlinearities and large operating regimes. We present a method for set-based estimation of the SOC of lithium-ion batteries. The set-based estimator takes measurement and parameter uncertainties explicitly into account and provides a set of consistent states. The performance is compared to an electrochemistry-based observer employing a detailed distributed model for a realistic drive cycle. I. INTRODUCTION thium-ion batteries are widely used in many applications, especially in automotive applications and power tools. The main reasons for this are their high energy density, low self-discharge, and long lifetime. Long-term and safe operation requires a reliable and robust estimation of the battery parameters and states, especially the state of charge (SOC. Despite considerable progress and the application of modelbased estimators and observers, nonlinearities, measurement uncertainties, wide operation ranges and large variations of the operating conditions, such as temperature, render the estimation task challenging. By now, various estimation approaches for SOC exist. In practice, often simple estimation approaches are used. For instance, the SOC can be determined via integration of the applied current and suitable filtering, see e.g. [1]. Other estimation approaches are Luenberger observers, Kalman filters, extended as well as unscented Kalman filters [1] [5], which are typically based on so-called equivalent circuit models (see [6] and Section II-B. In [7], an observer exploiting a detailed distributed model using output error feedback was proposed. Particle filters using a distributed model are for example described in [8] and [9]. We propose a set-based estimator which allows for the measurement and parameter uncertainty to be explicitly taken into account and which provides outer approximations of the SOC and parameters. Such set-based estimates can be used in a battery management system to predict corridors of safe 1 Laboratory for Systems Theory and Automatic Control, Ottovon-Guericke University, 3916 Magdeburg, Germany. {Rolf. {Findeisen,Stefan.Streif}@ovgu.de; 2 Robert Bosch Battery Systems GmbH, P.O. Box 3 22, 7442 Stuttgart, Germany. 3 Robert Bosch LLC, Research and Technology Center, Palo Alto, CA 9434, USA. Reinhardt.Klein@us.bosch.com operation and to optimize performance, taking the present uncertainty directly into account. The estimator is based on a recently developed set-based feasibility formulation [1], [11] for estimation and model invalidation. The proposed estimation approach uses a sliding window of measurement data, similar to moving horizon estimation [12], and provides guaranteed bounds on the SOC. The remainder of this paper is structured as follows: In Section II, we describe the operation of lithium-ion batteries, present a detailed distributed electrochemical model as well as an equivalent circuit model for lithium-ion batteries, and outline some of the challenges of SOC estimation. In Section III the set-based SOC estimation approach is described. Simulation results and a comparison with an electrochemistrybased observer are presented in Section IV. Section V summarizes the approach and concludes the article. II. LITHIUM-ION BATTERIES: PRINCIPLES, MODELING AND CHALLENGES IN STATE OF CHARGE ESTIMATION thium-ion cells generally comprise four main components: A negative and a positive electrode, a separator and an electrolyte (see Fig. 1 and [13]. The electrodes are electrically separated by the separator, which is permeable for lithium-ions dissolved in the electrolyte. The electrolyte typically consists of a salt and a mix of organic solvents and enables the diffusion of lithium-ions from one electrode to the other. During charging, the lithium-ions deintercalate from the active material in the positive electrode and intercalate in the negative electrode. At the same time, electrons are released in the positive and absorbed in the negative electrode. During discharge, this process reverses. In the following we briefly review a distributed model which is used for data generation and for the distributed observer. Furthermore, an equivalent circuit model is derived that is employed in the set-based estimator. A. Detailed Distributed Simulation Model The dynamics of the charge and discharge can be described by appropriate balance equations [13]. This leads to the following set of equations: ε e c e(x,t t = c s(x,r,t t = 1 r 2 Φ e(x,t Φ s(x,t ( c ε e D e(x,t e + 1 t c F i e(x, t, (1a ( r D s r 2 cs(x,r,t r, (1b ( 1 t c = ie(x,t κ + 2RT F ( 1 + d ln f c/a d ln c e(x,t ln ce(x,t, (1c = ie(x,t I(t σ, (1d /$ AACC 1566

2 I e - sep Lsep - Neg. electrode L- Separator + x 1 Fig. 1. i e(x,t + Electrolyte i e = -I Charging L + + j n (x 1 c s (x 1,r,t r Active material of electrodes R p x-axis x 2 Pos. electrode j n (x 2 c s (x 2,r,t Schematic representation of a lithium-ion cell. = 3εs j n (x, t = i(x,t F ρ avg c p dt (t dt R p F j n (x, t, (1e αaf η(x,t αcf η(x,t (e RT e RT, (1f = h cell (T amb (t T (t + I(tV (t + 3ɛ s R p F j n (x, t U(x, tdx r R p + I e - (1g where U(x, t U( c s (x, t T (t U( cs(x,t T. Here c e (x, t is the lithium concentration state in the electrolyte, c s (x, r, t is the lithium concentration in the active material of the anode and cathode, Φ e (x, t is the potential in the electrolyte, the potential in the anode and cathode are Φ s (x, t, the ionic current density in the electrolyte is i e (x, t, j n (x, t is the molar ionic flux between active material and electrolyte and the average internal temperature of the cell is given by T (t. c s (x, t represents the volume averaged lithium concentration of an active material particle. The input of the model is the current density I(t. The ambient temperature T amb (t is an external parameter. The outputs are the internal temperature T (t and the voltage V (t as potential difference of the electrodes, which are evaluated at the negative and positive current collectors V (t = Φ s ( +, t Φ s (, t. The exchange current density i (x, t and the overpotential η(x, t of the main reaction are incorporated by: i (x, t = r eff c e (x, t αa (c max s c ss (x, t αa c ss (x, t αc, η(x, t = Φ s (x, t Φ e (x, t U(c ss (x, t F R film j n (x, t, where c ss (x, t is the lithium concentration on the particle surface r = R p of the active material. U(c ss (x, t models the open circuit potential and c max s the maximum lithium concentration in the active material. The cell internal temperature is considered averaged [14]. The corresponding boundary conditions can be found in [13]. Suitable parameters may be taken for example from [15], [16]. Considering this detailed distributed model, the SOC, i.e. the energy stored in the battery can be defined as follows: SOC (t = 1 L c (x, t L c max dx, (2 s SOC(t = SOC (t SOC min SOCmax SOC. (3 min Here SOC (t represents the SOC of the negative electrode. SOCmax and SOC min are the maximum and minimum SOCs of the negative electrode. A definition based on the positive electrode leads to the same result. Note that if several materials are present within an electrode, the overall SOC of this electrode does not coincide with the individual SOCs of the individual materials. The described model is able to represent the dynamics of lithium-ion batteries quite well and provides physical insight. However, it is too complex to be used for setbased estimation as the size of the resulting optimization problem (see Section III would become prohibitively large. In the following, we present a reduced model, which we will subsequently use for set-based estimation. B. Equivalent Circuit Model Besides first principle modeling, electrochemical effects are often modeled using different electrical elements which results in the loss of the distributed information. We use a simplified model consisting of a resistor R in series with an RC element consisting of a resistor R 1 and a capacitor C 1 (cf. Fig. 2. The open circuit voltage is modeled using a voltage source Γ and directly depends on the SOC. Fig. 3 shows the characteristic shape of this dependence Γ : R R, which is one-to-one and onto. For a wide range of the SOC, the sensitivity of Γ on SOC is rather small. This complicates output based SOC estimation. Furthermore, the parameters R, R 1 and C 1 are typically not constant but also depend on SOC and temperature. We do not directly model this. Instead, we consider these variations in the form Fig. 2. Equivalent circuit model of a cell. 1567

3 of additional parametric uncertainties. The resulting reduced model takes the form ] [ ] [ ] [ [ẋ1 (t 1 = τ x1 (t 1 ] C u(t, ẋ 2 (t x 2 (t C nom y(t = V (t = Γ ( x 2 (t (4 + x 1 (t + R u(t, with x i ( = x i,. Here x 1 (t represents the voltage of the RC element, x 2 (t is the SOC of the cell, τ = R 1 C 1 is the time constant of the RC element and C nom is the nominal capacity of the cell. The applied current I(t is the input u(t. The described model captures the overall dynamics well, if suitably parametrized [17]. C. Challenges Several challenges arise with respect to SOC estimation both from a theoretical and a practical point of view. Observability of the states is of major importance. However, the SOC may only be poorly observable from cell voltage measurements [7], [18], because the open circuit voltage (cf. Fig. 3 depends on the SOC in form of a nonlinear characteristic curve which is only weakly sensitive to changes over a wide range. The weak sensitivity becomes a limiting Γ(SOC [V] SOC [%] III. SET-BASED STATE OF CHARGE ESTIMATION As outlined, there exists by now a multitude of approaches concerning SOC estimation. However, most of these approaches cannot deliver guarantees concerning the quality of the estimate which might be necessary for safety and secure operation. Others, like Kalman filters, require knowledge about an underlying probability distribution. In the following, we outline a set-based approach overcoming some of the challenges and drawbacks. The estimator is based on a relaxation approach presented in [11], [22]. For the purpose of estimation we consider nonlinear, discrete-time systems in implicit form F ( x(k+1, x(k, u(k, p =, H ( y(k, x(k, u(k, p =, where k denotes the time. States are referred to as x R nx, inputs as u R nu, outputs with y R ny and parameters with p R np. F : R nx R nx R nu R np R nx and H : R ny R nx R nu R np R ny are herein assumed to be polynomial or rational. For continuous time models an appropriate discretization and reformulation or approximation of nonpolynomial or nonrational functions is required. We assume that for all input and output measurements tolerances are given. Thus, measurements are not only given by vector or time-series, but are rather described by sets such as intervals or polytopes. In case where sensor or actuator uncertainties are given in the form of unbounded noise, suitable assumptions have to be made such as bounding the measurements by a multiple of the standard deviation. Here we assume, for the sake of simplicity, that the uncertainties are given by the following inequalities (see also Fig. 4: (5 y i y i y i, i = 1,..., n y, (6a u i u i u i, i = 1,..., n u, (6b p i p i p i, i = 1,..., n p, (6c x i x i x i, i = 1,..., n x. (6d Fig. 3. Open circuit voltage Γ as function of state of charge. factor, especially when trying to estimate individual cell SOCs from measurements spanning multiple cells [19], [2]. Furthermore, this may pose problems when reconstructing the SOC from highly noisy signals. If one is interested, for example, in the physical states (concentrations in the anode or cathode, difficulties may arise due to differing electrochemical properties which lead to a deterioration in observability [18]. Besides observability, the consideration of uncertainty both in the measurement as well as in the parameters complicate the estimation task. Frequently the measurements are noisy and the parameters of the systems can vary over a wide range, especially with respect to changes in the ambient temperature [21]. A suitable estimator should account for these uncertainties and should provide sufficiently robust estimates. Output Time Fig. 4. Set-based uncertainty description and state estimation with guaranteed limits. Intervals represent measurements with uncertainties. The gray tube visualizes states which are consistent with the model and measurements. 1568

4 In the following, we denote by X the set of admissible state values that satisfy all of the inequalities of (6d. Generally speaking, the aim of set-based state estimation is to find the set of states X X for which the model (5 is consistent with the uncertain output measurements (6a, the uncertain input measurements (6b and the uncertain parameters (6c. Note that the same framework can be used for set-based parameter estimation or reachability analysis [22]. To obtain tight bounds on all state variables ξ = [x 1 (1, x 2 (1,..., x nx (n t ] over the time horizon T = {1, 2,..., n t }, one can formulate the estimation problem in form of an optimization problem, taking all measurements into account: Min./Max. ξ i subject to F (x(k+1, x(k, u(k, p = k T +, H(y(k, x(k, u(k, p = k T, Inequalities (6 (7 where T + = T \ n t. By Min./Max. ξ i for each variable ξ i ξ, the lower bound is minimized and the upper bound maximized. This leads to lower and upper bounds ξ i and ξ i that are improved, i.e. tighter, than the initial bounds in (6d. In general, problem (7 is nonconvex, difficult to solve, and increases over time as more measurements become available. To handle the non-convexity, one can use different relaxations to convexify the problem. Several relaxation techniques are used, which are not described here. For details see [1], [23], [24]. It is important to note that the feasible set can only increase due to the relaxtions. This means that by solving (7, no solutions are lost, but the obtained set-estimates might become conservative. To overcome this conservatism, several exploration techniques exist, see [1], [23], [24], which can provide tight bounds on the real sets. As pointed out, all measurements are used in (7, i.e. a batch estimator strategy is used. This easily leads to an excessive number of data points, rendering the problem numerically intractable. To overcome this issue we propose to use a moving window of measurement points, similar to moving horizon estimation [12]. The basic idea is to limit the measurement sequences y(k and u(k to a short window and move this window forward in time (compare Fig. 5. The length of the window and a possible overlapping may be adjusted depending on the problem. Furthermore, a transfer of information from one window to the next, i.e. by limiting the state at the beginning of the window to the estimates obtained at the previous time, is performed. If no solution can be found, the measurement data and model are inconsistent, implying that either the assumptions concerning the uncertainties are too conservative and have to be increased or the measurement data cannot be reproduced by the model. Such a result can be used, among other things, for fault detection purposes (see examples in [22]. Directly using a PDE model for the set-based approach is currently out of reach as the size of the optimization problem would become too large due to the necessary discretization. To overcome this, we use the ECM model Output / Input Length of estimation window Window II Window I Window III Time Fig. 5. The estimation window is shifted over the measurement data resembling a moving horizon approach. At each step, optimization problem (7 taking the measurements of the window into account is solved. described in Section II-B. Model parameters and parameter uncertainties were obtained from pulse sequences and a drive cycle considering the PDE model as reference. The resulting model is used within the set-based estimation framework. IV. SIMULATION STUDY The presented approach is illustrated by simulations considering a real drive cycle and compared to the results of a distributed observer [7]. Fig. 6 shows the applied cell current. In the set-based estimator the derived equivalent circuit model is used, while the real battery is simulated using the distributed model. A. Set-based Observer For the set-based observer we assume a tolerance for the current sensor of [-.3 A,.1 A] and for the voltage sensor of [-2 mv, 2 mv]. The continuous time model is discretized with a sampling time of.1 s. The moving horizon estimation window comprises 2 measurement points. A recalculation solving 7 is performed every 15 new measurement points, i.e. all 1.5 s, leading to an overlap in the estimation window of 5 measurement points. The estimator is initialized with an uncertainty in state of charge of [%, 1%]. The implementation of the proposed methods can subsequently be performed using the toolbox ADMIT [22]. B. Distributed Observer We compare the results to a distributed observer [7] which uses linear error correction by means of a feedback of the output error. The observer is based on a reduced distributed model which preserves certain important properties of the full model, such as mass conservation and equilibrium structure. The design of the observer is physically inspired and as such, the introduced linear correction terms can be interpreted as a virtual flux of lithium-ions from one electrode to the other, where the flux direction is dictated by the sign of the output error. Further details can be found in [7]. 1569

5 I(t [A] 5 5 are assumed or estimated, no statements about the moments of the estimates can be given. Based on the sensitivity of the open circuit voltage w.r.t SOC (see Fig. 3, similar results are expected in the mid-soc range. For the clarity of presentation, results are omitted here. 1 9 High SOC 1 time Zeit [min] Fig. 6. C. Simulation Results Current of the drive cycle. In Fig. 7 and 8 we present the results of both observers, considering the full distributed model as real battery. The gray area between the dashed lines represents the estimated set of consistent SOC which is guaranteed to include the true SOC provided by the set-based estimator. Despite the fact that the estimation was initialized with a rather high uncertainty in SOC, the estimated SOC lies within a guaranteed interval of about 5% (cf. Fig. 8. For the lower SOC estimation, the estimated interval comes quite close to the true SOC at about 2.5 minutes. This may result from the fact that the current load is reduced by a factor of about 1 at this moment. As the impedance is generally higher at lower SOC, this leads to a higher change in measured voltage which entails a correction of the estimated SOC (cf. Fig. 8. The SOC estimates of the distributed observer are shown in Fig. 7 and 8 by the dark gray line. As can be seen, the distributed observer can successfully reconstruct the SOC over the whole operating range. To demonstrate this the observer is initialized with more than ±2% error in SOC and ±5 o C error in temperature. The current and voltage measurements are corrupted with white noise with standard deviations of.1 A and 1 mv respectively. Furthermore, we assume a current offset of.1 A. As can be seen from Fig. 7 and 8, the estimated SOC converges in approximately one minute to a residual error of less than 2%. However, due to the differences between the reference model and the reduced model, especially for high currents, and the constant offset of the current measurement, some residual error is expected to occur. It is important to note that the distributed observer can also provide insight into other internal states, such as potential distributions in the electrolyte and solid phases. The results underline the strength of set-based estimation: Despite large uncertainties regarding the measurements and parameters, a rather small interval for the SOC can be obtained. Furthermore, we can guarantee that all values of SOC that are consistent with the model and the measurements lie within the calculated bounds. As no probability distributions SOC(t [%] Low SOC 1 t [min] Fig. 7. Estimation results of the set-based estimator (gray shaded area and the distributed observer (dark line in comparison with the true SOC values ( which have been determined using the simulation model from Section II. SOC error [%] SOC error [%] 5 5 t [min] 5 5 t [min] Fig. 8. SOC errors of the set-based estimator (gray shaded, black dashed line and the distributed observer (dark grey continuous V. CONCLUSIONS In this article, we present a new approach towards SOC estimation. The estimation problem is formulated in terms of an optimization problem, which allows to take uncertainties explicitly into account. To avoid excessive calculations due to 157

6 an increase in measurements over time, we propose, similar to the moving horizon estimation scheme, the use of a measurement window of data points. The estimator provides a guaranteed corridor for the SOC, which can be directly used in the battery management system for prediction of future performance corridors to ensure safe and reliable operation. In comparison with the distributed observer, we are able to recover the SOC to a similar degree of accuracy with a much simpler model. However, in addition to SOC estimates, the distributed observer also provides estimates of internal states of the battery, which can subsequently be used for advanced functions like time optimal fast charging [25]. Future research will be focused on further development of set-based estimation methods which are applicable to more challenging chemistries, such as lithium iron phosphate. This chemistry shows a particular small sensitivity of the open circuit voltage w.r.t. SOC for a wide range of operation. Another challenge encountered in battery estimation is the detection of changes in the open circuit voltage shape with age. Furthermore, a systematic investigation of the effect of window and overlapping sizing on estimation performance is another topic for future research. Summarizing, the set-based approach allows for a guaranteed estimation of the SOC based on a reduced order model. In a battery management system such an envelope can be used to predict a future SOC horizon in order to adequately react on driving situations. Set-based estimation approaches are particularly well suited for offline applications due to the possibility of directly integrating uncertainties. This can be used to verify safety requirements or for the proper choice of sensors and measurements for fault diagnosis [26]. REFERENCES [1] S. Piller, M. Perrin, and A. Jossen, Methods for state-of-charge determination and their applications, Journal of Power Sources, vol. 96, pp , 21. [2] B. Bhangu, P. Bentley, D. 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