Two Time-Scaled Battery Model Identification With Application to ...

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Two Time-Scaled Battery Model Identification With Application to Battery State Estimation Yiran Hu and Yue-Yun Wang

Abstract— Electrified propulsion systems have now become an increasingly popular option for automotive companies to meet the more stringent emissions standards. A well-designed battery state estimation (BSE) system, which includes state-of-charge and state-of-health estimation, is one of the most important aspects of a successful electrified propulsion system design. Among different methods, model-based state estimation has proven to be very successful in their accuracy and implementability. A relatively newer approach to model-based BSE is to identify the battery model parameters (typically a low-order control-oriented model) in real time. This allows the battery model parameter to adjust to changing characteristics of the battery, and thus further improving the robustness of the design. However, standard identification algorithms used have very limited capability in performing this identification successfully due to the frequency response characteristics of the battery. In this brief, we describe a two time-scaled battery model parameter identification method, where the slower and faster battery dynamics are identified separately. Compared with standard approach to real-time battery model identification, where no such separation is made, this method can generate a model whose frequency response is much closer to that of the actual battery. Furthermore, this method uses the standard least squares regression method, which can be easily implemented in real time in the form of recursive least squares. Using this identification method, we show how battery SoC can be estimated. Laboratory battery cell data is used to illustrate the difference between this method and the more standard approach. Then, battery pack collected from a test vehicle is used to demonstrate the SoC estimation capability. Index Terms— Lithium-ion battery, model identification, state-of-charge (SoC) estimation.

I. I NTRODUCTION

W

ITH the increasingly stringent emissions standards, more and more automotive manufacturers are turning to electrified propulsion systems as a potential method for meeting these standards. The battery system inside an electrified vehicle is not only very costly but can also be damaged relatively easily if improperly managed [1]. As such, a well-designed battery management system (BMS) is crucial to the success of the entire electrified propulsion system. Battery state estimation (BSE) is one of the most important functions of a BMS. It refers to the real-time estimation of battery states, such as the state-of-charge (SoC) and state-ofhealth. Accurate estimation of these battery states is essential Manuscript received June 30, 2014; accepted August 26, 2014. Date of publication October 14, 2014; date of current version April 14, 2015. Manuscript received in final form September 8, 2014. Recommended by Associate Editor A. Chiuso. The authors are with the Propulsion Systems Research Laboratory, General Motors Research and Development, Warren, MI 48090 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2014.2358846

in determining the optimal real-time operating strategy of the entire propulsion system. Various methods have been proposed to perform BSE (see [2] for a quick summary). In particular, model-based approaches have been shown to provide good performance as well as easy calibration [3]–[7]. In these methods, typically a battery model is identified a priori and then used in real time. This type of approach has the drawback of being unable to compensate for battery variations and aging. In more recent works, such as [8], [9], and [16]–[18], it is shown that realtime model parameter identification can be used to provide compensation to variations and aging. In such approaches, typically the parameters in a lower order linear ordinary differential equation (ODE) based battery model (typically derived from equivalent circuits) are estimated online based on sensor measurements from the battery pack (e.g., current, voltage, and temperature). Identification of lower order ODE models is the focus of this brief, but we should also mention that some more recent work (e.g., [20]) have also explored real-time parameter adaptation of the more complicated partial differential equations (PDEs) based models that offer better accuracy at the cost of higher computational complexity. Once model is identified, the battery states can be extracted from the identified parameters. The adaptability of this type of algorithm makes them very desirable among the other model-based approaches. The identification technique used is typically recursive least squares type of method, which is easy to implement and relatively fast to compute in real time. Other methods, such as subspace parameter estimation, can also be used [7], [16] though more difficult to implement online. Real-time identification of battery parameters can be quite challenging. First, we note that true lithium-ion battery dynamics are very complex. Based on electrochemistry [10], [11], true battery dynamics should be described by a set of nonlinear PDEs. A low order linear ODE model can only be used as a lumped approximation of the nonlinear PDEs. As such, the identified model parameters can be very data dependent. Second, lithium-ion battery contains dynamics that occur on very different time scales. Considering charge transfer and diffusion (two of the most important effects in battery dynamics), for example, charge transfer occurs on a time scale of 0.1–100 Hz, while diffusion occurs from around 1 Hz down to 0.001 Hz. The traditional prediction error methods (PEMs), which includes the often used least squares approach, tends to over-emphasize modes that affect the overall model response more heavily, especially when the model form is inexact [12]. This results in models that do not work well over all excitation frequencies typically seen in vehicle operation. Last but not least, measurement noise can compound the previous

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HU AND WANG: TWO TIME-SCALED BATTERY MODEL IDENTIFICATION

issues to result in further model inaccuracies. Some of the previous work in this area has attempted to addressed some of these aspects. For example, Sitterly et al. [17] found that despite having a properly setup regression structure, noise can generate bias in the identified coefficients. However, if noise characteristics are known, then this bias can be compensated. In [18], a different strategy was used to combat the same noise issue. This time, voltage and current signals, which are used for the model identification, are sampled faster and then window-averaged before use to remove bias resulting from measurement noise. In this brief, we propose a new BSE method based on a modified method for real-time model identification [15]. This identification method is still based on the least squares method, making it easily implementable in real time. To address the problems described previously, this method propose to identify battery parameters using two separate least squares procedures: one for parameters associated with the slower dynamics and one for parameters associated with the faster dynamics. Separating the slower and faster dynamics allows the identified model to be less data-dependent and more accurate in lower frequency regions. To do this, battery frequency response (also called impedance) is used as a guide to design filters to separate dynamics into these two groups. No impedance information, however, is required in the realtime parameter estimation process. Then, battery parameters can be regressed based on the filtered data using recursive least squares approach, which allows the algorithm to have relative low computational complexity. Once the model is identified, we show how it can be used to calculate the open circuit voltage (OCV) of the battery. Because the OCV has an almost one-to-one relationship with the SoC of the battery [2], the SoC can be estimated based on the estimated OCV. In the following, we first describe the typical approach to control-oriented battery modeling and introduce the model that we will use to model the battery dynamics. Then, the model identification method is described and illustrated using lithium-ion battery data collected on a cycler. Last but not least, an SoC estimation scheme using the identified model is proposed, and then validated using battery pack data from an experimental vehicle. The batteries used for the experimental validation are LG Chem battery cells used in the production GM Chevy Volt, which have 4.2 V nominal voltage and 20-Ah capacity. The battery pack used in the test vehicle (production Chevy Volt) is consisted of 96 cells by three strings of the same type of battery cells. II. BATTERY M ODEL In this section, a model to be used for real-time identification is described. Since the identification procedure is intended to be used in real time, we limit the type of model to be control oriented model whose coefficients can be identified using linear methods. The most used approach to arrive at such a model form is to start with an equivalent circuit model (ECM). In the following, we will briefly review the most common ECM and then provide a generalization of it that is suitable for the work in the remainder of this brief.

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Fig. 1.

Commonly used equivalent circuit.

A. Equivalent Circuit Model Fig. 1 shows the most commonly used ECM. This ECM consists of an OCV, an internal resistance, and n RC circuit pairs. The RC circuit pairs are used to approximate the overvoltage dynamics of the battery. Typically, two to three RC pairs are used, which is a good compromise between complexity and accuracy. For a generic RC circuit pair whose resistor value is R and capacitor value is C, if we denote the voltage across this element as V RC and current through it as I , then the current to voltage relationship can be written as V RC 1 d V RC =− + I. dt RC C This equation can be discretized as  Ts Ts  V RC [k + 1] = e− RC V RC [k] + R 1 − e− RC I [k]

(1)

(2)

where Ts is the sampling time. The voltage across the internal resistance R0 , denoted by V R0 , is given as V R0 = R0 I.

(3)

The OCV, denoted by Voc is described as a static function of the SoC (z) and temperature (T ) Voc = f (z, T ).

(4)

This map can be measured in the laboratory, and is thus assumed to be available for real-time usage. The SoC, defined as the ratio of the amount of ampere-hours available in the battery to the total amount of ampere-hours that the battery can hold (aka capacity of the battery), can be written as 1 dz = I (5) dt 36Cn where Cn is the nominal capacity of the battery. Consequently, via the chain rule and assuming d T /dt is negligible, we can write a dynamic equation for Voc as δf 1 d Voc = I. dt δz 36Cn

(6)

This equation can be discretized as Voc [k + 1] = Voc [k] +

δ f Ts I. δz 36Cn

(7)

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Finally, the battery voltage Vb is given as Vb = Voc + V R0 + V RC .

(8)

Equations (2), (3), (7), and (8) can be collected to form the discrete state space description of the battery dynamics as ⎤⎡ ⎡ ⎤ ⎡1 ⎤ 0 ... 0 Voc [k + 1] Voc [k] Ts − ⎢ ⎥ ⎢ V RC1 [k + 1]⎥ ⎢0 e R1 C1 . . . V RC 1 [k]⎥ 0 ⎥⎢ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎢ ⎥ = ⎢. ⎥ .. .. . . .. .. ⎥ .. ⎣ ⎦ ⎣ .. ⎦ . . . ⎦⎣ − RnTsCn [k] V RC n [k + 1] V RC n 0 0 ... e ⎡ ⎤ δf Ts

n ⎢  δz 36C ⎥ T ⎢ R 1 − e− R1 sC1  ⎥ ⎢ 1 ⎥ +⎢ ⎥ I [k] .. ⎢ ⎥ . ⎣ ⎦ Ts   Rn 1 − e− Rn Cn ⎤ ⎡ Voc [k] ⎥

⎢ ⎢V RC 1 [k]⎥ Vb [k] = 1 1 . . . 1 ⎢ ⎥ + R0 I [k]. .. ⎦ ⎣ . V RC n [k]

Fig. 2.

Typical lithium-ion battery impedance Nyquist plot [11].

model can be transformed back into the ECM shown in Fig. 1 (9) by simply obtaining a diagonalized state space representation using the identified coefficients, which is identical to (9). Note that given a battery model of the form (12), it is relatively straightforward to separate the OCV dynamics from the rest. Due to space constraint, we will omit this detail. B. Generalized Model If we assume that δ f /δz in (9) is a quasi-constant para- However, it suffices to say that this process can be derived meter (since SoC changes relatively slowly, this is a valid easily in the transfer function domain as it is equivalent assumption), then (9) can be viewed as a linear single-input to dividing the denominator polynomial by z − 1, which single-output system. As such, there exists an equivalent corresponds to the integrator dynamics of the OCV. input–output representation from I to Vb . This representation can be generically written as III. I MPROVED M ODEL I DENTIFICATION M ETHOD Vb [k] = a1 Vb [k −1] + a2 Vb [k − 2] + · · · + an+1 Vb [k − n − 1] + b0 I [k] + b1 I [k −1] + · · · + bn+1 I [k − n −1] (10) where the order of the system is n + 1. We note that the dynamics of OCV always has a pole at 1 in discrete domain, which is evident in the state matrices in (9). Therefore, to be consistent, we place a constraint that the characteristic polynomial of (10) must always have a pole at 1. This constraint is mathematically equivalent to 1 = a1 + a2 + · · · + an+1 . (11) n Therefore, if we substitute 1 − i=1 ai for an+1 in (10), then we obtain Vb [k]−Vb [k −n −1] = a1 (Vb [k −1]−Vb[k −n −1]) + · · · +an (Vb [k − n] − Vb [k − n − 1]) +b0 I [k] + b1 I [k − 1] + · · · +bn+1 I [k − n − 1].

(12)

This equation can be considered as a generic linear input– output dynamics equation that is suitable for describing the dynamics of a battery. It is again in an easily regressable form for the parameters a1 through an and b0 thru bn+1 . Note that because the form (10) is simpler to use than (12), we can always solve for an+1 using (11) and then use (10) directly. Equation (12) is more general than the ECM shown in Fig. 1 since it can contain complex eigenvalues. However, if we restrict the eigenvalues of (12) to be positive and real, then this

Standard model identification algorithm applies least squares regression to a model formulation such as (12) directly to obtain the model parameters. Note that for the sake of briefness, we will not provide the equations for recursive least squares since it is available a variety of literatures. In this section, we provide a two time-scaled identification scheme that seeks to improve the standard approach in terms of identification accuracy. Part of this was briefly discussed in [15] but more details are added here to be complete. To do this, we first look at battery frequency response and its relationship to the battery model described in the previous section. A. Frequency Perspective on Battery Model Frequency response of a battery is often used to help understand the behavior and impact of different parts of the battery have on the overall current-to-voltage response of the battery [11]. Typically, battery voltage is considered to be composed of two portions: the OCV and the overvoltage. OCV represents the battery terminal voltage under zero-current equilibrium and is a function of the SoC and temperature of the battery. Overvoltage is the voltage required to overcome electrochemical impedances (which is equivalent to the frequency response) so that a current can flow. The dominant impedances come from the internal resistance, charge transfer, and diffusion. Fig. 2 shows graphically the contribution of each effect to the total impedance of the battery.

HU AND WANG: TWO TIME-SCALED BATTERY MODEL IDENTIFICATION

Based on (7), the dynamic relationship between OCV and current is a time varying integration. The frequency response of an integrator is a mean pass filter, where the filter gain is infinite at zero frequency and then decrease toward zero as frequency goes to infinity. For practical input signals (which are all finite in duration), this mean pass filter becomes a low-pass filter with very narrow passband. The voltage drop over the internal resistance can be viewed as a direct feed-through from the current. Therefore, it is an all-pass filter. The voltage due to charge transfer reaction and diffusion effects both have low-pass filter effect on the current [3]. However, their bandwidths are drastically different. From Fig. 2, charge transfer typically affects frequencies between few hundred Hz to 0.1 Hz, while diffusion dominates between 0.1 Hz and even below 0.001 Hz. When RC circuits are used to approximate such dynamics, the required time constants of each RC circuit can be quite separated. Combining with the fact that this model representation is only an approximation, a direct identification of the entire model will provide coefficients that bias toward the higher frequency [12]. A different model identification strategy is needed to ensure that the identified model can capture the different time constants, and thus better approximating the overall dynamic response. Note that one natural way of obtaining a model that approximates the system behavior in a wide-frequency range is to directly fit the model in frequency domain. This way, all frequencies components are treated with the same weight. However, despite some good attempts [13], [14], there is currently no practical solutions for real-time measurement of battery impedance. This makes real-time fitting in the frequency domain impractical. B. Two Time-Scaled Identification Since a good identification algorithm for this problem must be able to identify time constants that are quite far apart, it is logical then to separate the identification problem into different portions, where the influence of each time constant is isolated from the others. The main theoretical basis of separated identification is as follows [12]. 1) Suppose the measured input and output signals are highpass filtered. Then, a model identified using these filtered signals only matches the frequency response contained in the data above the cutoff frequency of the filter. 2) Suppose the measured input and output signals are low-pass filtered. Then, a model identified using these filtered signals only matches the frequency response contained in the data up to the cutoff frequency of the filter. 3) When there is a sufficient separation between the time constants of a model, high-pass, and low-pass filters can be designed so that models identified using the input/output signals filtered by each filter only match the model in the respective frequency ranges. 4) The feed-through gain of the model identified using the low-pass filtered signals should equal to the steadystate gain of the model identified using the high-pass

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filtered signals. This way the two models can be combined together in a consistent fashion. To apply these facts to battery model identification, we first separate the battery dynamics into a slower group and a faster group. Note that slower and faster can be thought of as narrow and wide bandwidth, respectively. Based on the discussion about the bandwidth of different battery effects, we can make the following selection: in the faster group, we include the overvoltage due to the internal resistance and the charge transfer dynamics; in the slower group, we include the diffusion dynamics and the OCV dynamics. Based on this selection, the models for each group can now be formulated based on the generalized models described in Section II-B. First, denote the overvoltage due to the faster dynamics as Vf and the voltage due to the slower dynamics as Vs . Since the faster dynamics Vf does not include the OCV, model form (10) can be used without constraints. If the order of Vf dynamics is selected as n, then its dynamics can be written as V f [k] = α1 V f [k − 1] + · · · + αn V f [k − n] +β0 I [k] + · · · + βn I [k − n].

(13)

Since Vs contains the OCV, we use the model form (12). Let the order of Vs be m + 1 (i.e., mth order overvoltage). Then, Vs has the difference equation given by Vs [k]−Vs [k −m −1] = a1 (Vs [k −1]−Vs [k −m −1]) +· · ·+ am (Vs [k −m]−Vs [k −m −1]) + b0 I [k]+· · ·+bm+1 I [k −m −1]. (14) Equations (13) and (14) represent the fast and slow dynamics separately. In order for them to work together, we must make sure that they represent the system dynamics in a consistent manor. Mathematically, this means that the steady-state gain for Vf must be equal to the infinite frequency gain of Vs . This constraint comes from the fact that the separation methodology applies singular perturbation theory [12] to derive the slower dynamics model. When singular perturbation approximation is applied to a system to generate a model that approximates the slower dynamics, the faster dynamics of the system is viewed as instantaneous. As such the effect of the faster dynamics on the resulting approximation basically reduces to that of an infinite frequency gain. Hence, the steady-state gain of the faster dynamics must be equal to the infinite frequency gain (feed-through gain) of the slower dynamics. This can be expressed mathematically as b0 =

β0 + · · · + βn . 1 + α1 + · · · + αn

(15)

Note, however, that when the underlying system has well separated dynamics and the filters used to separate this dynamics are designed properly, then the equality described in (15) will happen naturally. The equations for Vf and Vs can be combined to form a single dynamic equation. The exact coefficients for the combined model are a bit cumbersome to write out. However, one simple way to visualize the combination is through the use of transfer functions. The transfer function

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for Vf can be written as Vf β0 z n + · · · + βn = n . I z − α1 z n−1 − · · · − αn

(16)

Similarly, the transfer function for Vs can be written as b0 z m+1 + · · · + bm+1 Vs . = m+1 I z − a1 z m − · · · − am+1

(17)

Assuming all the constraints are met previously, then the battery voltage Vb is given by the transfer function Vs Vb V f = × I I I (β0 z n +· · ·+βn )(b0 z m+1 +· · ·+bm+1 ) . = n (z −α1 z n−1 −· · ·−αn )(z m+1 −a1 z m −· · ·−am+1 ) (18) The equivalent difference equation for this transfer function can be obtained by expanding the numerator and denominator polynomials in (18). As we will show later, for real-time battery SoC estimation, this combined form is not needed. This combined model is only needed if one wants to simulate the battery dynamic response. Given this analysis, we outline the final identification methodology in a cookbook style as follows. 1) Design a low-pass filter to isolate the slower mode dynamics. 2) Design a high-pass filter to isolate the faster mode dynamics. 3) Filter the voltage and current measurements using the high-pass filter. 4) Use recursive least squares to fit model (13). 5) Filter the voltage and current measurements using the low-pass filter. 6) Downsample the low-pass filtered voltage and current appropriately (this depends on the cutoff frequency used in the low-pass filter). 7) Use recursive least squares to fit model (14). 8) If desired, combine the low- and high-pass filtered models by applying the constraint (15) to obtain a full model (18). C. Filter Design The filters we need to design for this algorithm is based on the separation of the time constants inherent in the battery response. Fig. 3 provides a graphical illustration of how to choose the cutoff frequencies for the two filters. In simple terms, the cutoff frequency of the low-pass filter should be high enough to encompass the passband of the slower mode, but be low enough to contain as little of the passband of the faster mode as possible. Note that this construction is intuitive since when the passband of the filter overlaps with only the low frequencies portion of the passband of the faster modes, the faster modes effectively become a constant gain. As such the dynamics of the slower mode can effectively be identified alone. On the other hand, the passband of the high-pass filter needs to overlap with passband of the faster dynamics but has as little overlap with the passband of

Fig. 3.

Filter design illustration.

Fig. 4.

Current profile used to excite the battery.

the slower dynamics as possible. This is so that the signals filtered this way will not be affected by the slower mode dynamics. Note that the specific type of filter to use depends on the user. As always, there will be a tradeoff between the order of the filter and sharpness of cutoff. Therefore, the user can experiment with different filters to see the effect on the identification. In our experience, we find that simple Butterworth filter of first- or second-order generally works well enough. D. Validation of the Model Identification Method In this section, we briefly illustrate this model identification process using a dataset collected on an LG Chem manganese-based prismatic lithium-ion cell (4.2 V nominal voltage with 20-Ah capacity). This data is collected in the laboratory using a battery cycler to enforce a desired current profile. An environmental chamber is used to maintain the temperature of the battery during this test, which for this test is 0 °C. The current profile selected is consists of square waves of frequencies 0.1, 0.01, and 0.001 Hz (Fig. 4), which covers majority of the frequency range often seen in vehicles. The nominal SoC for this test is 50%. It should be noted

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Fig. 5. Comparison of overvoltage response between real battery (blue solid line), third-order model from standard identification (red dashed line) and fourth-order model from two time-scaled identification (red solid line). Top left: overall model response. Top right: zoomed-in view of response to 0.1 Hz excitation. Bottom left: zoomed-in view of response to 0.01 Hz excitation. Bottom right: zoomed-in view of response to 0.001 Hz excitation.

that battery dynamics slows down as temperature is lowered. Temperature of 0 °C is a realistic operating condition, where the standard identification method have especially difficult time identifying the correct coefficients. Therefore, it is a good temperature to use to demonstrate the performance of the two time-scaled identification. To test this identification methodology, we first fitted third-order models directly (i.e., no separation of dynamics). In other words, the overvoltage dynamics is represented by a second-order model. After fitting the model, the identified eigenvalues are examined. The two eigenvalues corresponding to the overvoltage dynamics are 0.9977 and −0.2650, respectively. While the 0.9977 eigenvalue, which corresponds to an approximately 43-s time constant, is reasonable in this context, the −0.2650 eigenvalue makes no sense. If the dynamics corresponding to the −0.2650 eigenvalue is isolated and simulated, it adds almost nothing to the overall dynamics. Thus, this mode is extraneous. As such the model identified is effectively a second order. We tried higher orders, such as fourth and once again only 1 useful mode (aside from the OCV mode) is identified. Next, we tried the proposed method described in the previous section. Once again we start with a third-order model overall. Because dynamics are identified separately in this method, we use a first-order model in the form of (13) for fast dynamics and a second-order model in the form of (14), where (n = 1) for the slower dynamics.

This time, the identified eigenvalue for the faster dynamics is 0.8705 (time constant of 0.79 s), while the eigenvalue for slower dynamics (overvoltage portion) is 0.9993 (time constant of 159 s). Upon simulation, both modes contributes to improve the overall model response as compared with the measured response. Next, we tried a fourth-order model, where the slower dynamics is selected to be third-order and faster dynamics first order. This time the eigenvalues identified are 0.8705, 0.9973, and 0.9998, corresponding to time constants of 0.79, 40.6, and 622.6 s, respectively. Fig. 5 shows the dynamic response comparison between the measured, directly identified third-order model, and the fourth-order model identified using the two time-scaled approach. Note, here, that the dynamic response shown is that of the overvoltage only. The identified OCV dynamics is not simulated because it is an integrator which is subject to drift in a long dataset when there is even a small amount of inaccuracy in the identified integrator coefficient. For the measured dataset, OCV is removed manually based on the OCV to SoC map, which is generated a priori. As we can see from this figure, the directly identified model provides relatively poor performance especially at 0.001 Hz excitation. On the other hand, the model identified using the two time-scaled approach is able to approximate the battery response over all three frequencies very well, suggesting that it captured the frequency response of the battery from 0.1 Hz down to 0.001 Hz.

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Frequency response comparison.

We also compared the frequency responses of the two time-scaled identified models and the directly identified model to the frequency response calculated from the voltage and current data (which represents the true frequency response of the battery). As Fig. 6 shows, the model identified here almost perfectly captures the frequency response contained in the data, while the model identified previously provided a poor match especially in the lower frequency regions. This example demonstrated very clearly the advantage of this purposed identification algorithm as opposed to the direct identification approach. IV. BATTERY S TATE E STIMATION The most important use for real-time battery model identification is in BSE. Accurate estimation of the battery state is crucial for better battery control and diagnostics. In the following, we discuss a scheme in which SoC estimation can be accomplished when battery parameters are regressed using this two time-scaled approach in real time. A. SoC Estimation The SoC estimation scheme, we use here follows the relatively standard assumption that OCV has a one-to-one relationship with SoC. As such, we can reduce the problem of SoC estimation to OCV estimation instead. We should note here that in reality, OCV to SoC relationship often exhibits a hysteresis effect (certain chemistries such as LiFePO4 this effect can be especially large [19]). We will not model this hysteresis effect here as it is a separate and significant topic in itself. However, we do want to note that neglecting this effect can result in a small amount of SoC estimation error based on the charging and discharging history.

Because we employed this frequency-separated model identification procedure, two models identified in real time. Only the slower dynamics model contains information regarding the OCV. Thus, only the slower dynamics model is needed for the purpose of SoC estimation. The main advantage of this approach is that because the slower dynamics model is identified using heavily low-pass filtered voltage and current signals, measurement noises as well as unmodeled higher frequency dynamics are effectively eliminated from the estimated OCV. In addition, we can process this identification at a slower sample rate to reduce computational complexity since low-pass filtered signal can be downsampled, while still preserving its information content. This can be useful in a realtime environment where computational resource is limited. To use the lower frequency model, we need to separate it into two separate pieces, with one piece representing the OCV dynamics and another the overvoltage. Both are causal models that can be run in an open loop fashion in real time to provide an estimate of its output (which in this case are OCV and overvoltage). This naturally provides two different ways of estimating the OCV. One possibility is to run the OCV model to directly obtain an estimate for OCV. The problem with this method is that the identified OCV model is a time varying integrator, which is not asymptotically stable. As such if there are inaccuracies in the model parameters or noise in the measurements, then the estimate can diverge from the true value without any ability to correct itself. Doing it this way carries all the drawbacks of the standard approach of the current integration. Therefore, this is not a preferred method. The second possibility is to use the overvoltage model to obtain an estimate for the overvoltage, and then subtract this overvoltage from the low-pass filtered voltage. Logically, the only thing that remains must be an estimate of the OCV.

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Fig. 7.

Battery SoC estimation structure.

Fig. 8.

Data collected from a battery pack during a driving profile. Left: current data. Right: voltage data.

This method is more robust than the direct estimation method because the overvoltage dynamics are asymptotically stable. Therefore, if there are inaccuracies in the model initialization, this inaccuracy will die out over time. After that, the model output would provide a reasonable estimate of the overvoltage. Because of this, we select this second approach. This method is graphically shown in Fig. 7. In Fig. 7, Vfilt and Ifilt are the low-pass filtered voltage and current. We use (14) to model the input–output relationship between Vfilt and Ifilt . The coefficients of (14) can be regressed using recursive least squares approach. Using the regressed coefficients, the dynamics associated with the overvoltage is extracted (as discussed previously, we will not derive this here). This overvoltage, denoted by Vov can be expressed as Vov [k] = a1 Vov [k −1]+· · ·+an Vov [k −n] + b0 Ifilt [k] + · · · + bn Ifilt [k − n].

(19)

where ai and b j are the identified coefficients. Then, Voc can be estimated by Voc [k] = Vfilt [k] − Vov [k].

(20)

Note that the scheme presented here is a relatively simple scheme. More complicated scheme, for instance, an extended Kalman filter similar to the one shown in [3] can also be used, where the Kalman filter model coefficients are based on the identified model parameters. Such approach can be even more robust to noise, initialization errors, and coefficient inaccuracies. Further studies need to be done to see if enough benefits can be gleaned from the added complexity. To illustrate this SoC estimation scheme, we use sample vehicle data from a production Chevy Volt, where the battery pack (consisted of 96 × 3 LG Chem manganese-based cells) is used as the primary propulsion energy source. Fig. 8 plots the

Fig. 9.

Battery SoC estimation results.

current and voltage through the battery pack during a driving period, respectively. Then, in postprocessing, we generate the SoC of the battery during this drive cycle by using a calibrated current integration, which adjusts for initial condition and capacity deviation from nominal. Using this SoC, we also compute the OCV based on the SoC to OCV map for the batteries used in the pack. The computed SoC and OCV trajectories are used as the reference values to test accuracy of the estimated SoC. Fig. 9 shows the estimated VOC and SoC as compared with the reference values. Note that the low-pass filter used to separate the slower dynamics is the same filter used in Section III-D on the cell data. The major difference is that rather than using a batch least squares approach, a recursive least squares method is used with a forgetting factor of 0.9. We selected this value as it appears to show good balance between robustness and sensitivity. As we can see from this, the error between the estimated and the truth value is

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always less than 5%, which is generally recognized as a good accuracy for this type of SoC estimation. We note that there are some small jumps in the estimate. Those are mainly the result of inaccuracies in the OCV–SoC map caused by the hysteresis effect when the batteries goes to heavy charging after being discharged for significant amount of time (or vice versa). However, as this chemistry is not heavily affected by hysteresis, the overall error still does not exceed 5%. V. C ONCLUSION In this brief, we described a two time-scaled approach to real-time battery model identification and its application to BSE. The major contribution of this method over the standard approach is that the slower and faster dynamics of the battery are identified separately. This separation can achieve better frequency response matching without increasing computational complexity. We also demonstrated methods that use the model identified to estimate battery state estimates, including SoC and power. Simulations based on the real cell and pack data showed that good accuracy in state estimation can be achieved using this framework. R EFERENCES [1] E. Meisner and G. Richter, “The challenge to the automotive battery industry: The battery has to become an increasingly integrated component within the vehicle electric power systems,” J. Power Sour., vol. 144, no. 2, pp. 438–460, 2005. [2] S. Piller, M. Perrin, and A. Jossen, “Methods for state-of-charge determination and their applications,” J. Power Sour., vol. 96, no. 1, pp. 113–120, 2001. [3] G. L. Plett, “Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification,” J. Power Sour., vol. 134, no. 2, pp. 262–276, Aug. 2004. [4] B. S. Bhangu, P. Bentley, D. A. Stone, and C. M. Bingham, “Nonlinear observers for predicting state-of-charge and state-of-health of lead-acid batteries for hybrid-electric vehicles,” IEEE Trans. Veh. Technol., vol. 54, no. 3, pp. 783–794, May 2005. [5] I.-S. Kim, “The novel state of charge estimation method for lithium battery using sliding mode observer,” J. Power Sour., vol. 163, no. 1, pp. 584–590, Dec. 2006.

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