# Online Semiparametric Identification of Lithium-Ion Batteries Using the Wavelet-Based Partially Linear Battery Model

^{*}

## Abstract

**:**

## 1. Introduction

_{1}-norm penalized wavelet estimator. However, most existing penalized wavelet estimators of the partially linear model are processed in batch form using iterative algorithms [14,23,27], which are not suitable for online implementation purposes. Recent advances in sparse linear model identification have shown that the cyclic coordinate descent (CCD) algorithm provides an efficient means of solving the penalized least squares (LS) problem [28,29] and can be implemented online in a recursive fashion [30,31]. This feature motivates us to extend this algorithm to the wavelet-based partially linear model and develop a recursive penalized wavelet estimator based on a modified online CCD algorithm. The performance of the proposed semiparametric identification approach for lithium-ion batteries is finally investigated by performing simulations and experiments.

## 2. Modeling of the Lithium-Ion Battery

#### 2.1. Battery Equivalent Circuit Equations

_{s}; (2) the resistor-capacitor (RC) parallel network C

_{p}// R

_{p}(where R

_{p}is the equivalent polarization resistance and C

_{p}is the equivalent polarization capacitance), which is used to simulate transient responses of the battery during charging–discharging transients; and (3) the OCV v

_{oc}(h(t)), which is a nonlinear function of SoC h(t). The equivalent circuit model considers the current as the model control input and the terminal voltage as the measured output.

_{s}, R

_{p}, and C

_{p}in this model are functions of the temperature and SoC. In fact, because the variations in the SoC and temperature with respect to time are both very small, these parameters can be assumed as quasi-stationary, i.e., time invariant over a short observation time window. Further, for real-time application, parameter identification can only be performed over a finite time sliding window or exponentially decaying time window of the most recent measurement. Thus, within some error tolerance, these parameters can be assumed as constant during the identification. In other words, online identification can capture these parameters faster than the temperature or SoC variations.

_{b}(t) and i

_{b}(t) are the terminal voltage and current, respectively; and v

_{c}(t) is the voltage across the RC network, which cannot be measured directly.

#### 2.2. PLBM

_{oc}(h(t)) as an unknown function that belongs to some functional space, which must be identified online, as well as the other model parameters. In this case, v

_{oc}(h(t)) can be simply represented as v

_{oc}(t).

_{oc}(t) and the polarization voltage v

_{c}(t), both of which are unknown. However, the unknown term v

_{c}(t) can be removed from these equations based on the following three assumptions [17]: (1) The variation in SoC with respect to time is very small because the consumed (or regained) energy is very small compared with the total useful capacity; (2) The battery temperature is monitored in real-time and controlled at a predetermined level by a well-designed BMS module. Thus, the temperature variation can be ignored for normal operating conditions; (3) The usage history of the battery has a long-term effect on battery behavior, which can be ignored during on-line identification procedure. These assumptions also have been applied in [21]. With the unknown term v

_{c}(t) removed, the dynamics of the battery model can be approximated as follows:

_{b}(k) and i

_{b}(k) are the samples of v

_{b}(t) and i

_{b}(t), respectively, at time point t

_{k}= kT

_{c}and T

_{c}is the sampling period. θ

_{1}, θ

_{2}, and θ

_{3}are defined as:

_{oc}(k) denote the virtual sample of v

_{oc}(t) at time point t

_{k}= kT

_{c}, which is assumed to be an unknown function because v

_{oc}(k) is unknown.

_{b}(k) and i

_{b}(k) are acquired only via observation, and during the observation, these parameters often suffer additional random disturbance. Let y(k) and x(k) denote the observed values of v

_{b}(k) and i

_{b}(k), respectively. From (5), the discrete observation equation can be obtained as follows:

_{b}(k) and i

_{b}(k) but also the interior noise of the battery system.

**θ**and a nonparametric component u(k) involving the unknown function v

_{oc}(k). Thus, we can naturally describe the dynamics of the battery using the semiparametric partially linear model; this description is useful not only because our understanding of the OCV does not provide us with a specific function or a parametric model of OCV but also because it provides a better understanding of the model structure and the parameters.

## 3. Online Identification of the Wavelet-Based PLBM

**θ**and the nonparametric component u(k) from the input–output data. One attractive approach in addressing this problem is to expand the nonparametric component as a linear combination of the parameterized basis functions. The problem is then reduced to a linear estimation problem because the output is linear not only with the linear components but also with the basis functions. The identification method developed in this section is based on the wavelet MRA expansion of the nonparametric component u(k) of the PLBM.

#### 3.1. Wavelet MRA

_{j0}

_{,m}and d

_{j}

_{,m}are respectively termed the scaling and wavelet coefficients at scale levels j

_{0}; and j; and j

_{0}; is an arbitrary integer that represents the coarsest resolution level (defined by the user). The wavelet MRA representation has localization property in both the time domain (via translations) and the frequency domain (via scale). With this property, wavelets can capture global (low frequency) and local (high frequency) characteristics of any finite energy signal easily.

_{k}= kT

_{c}, k = 1, 2,···, N where N = T / T

_{c}is the length of the sampled signal and assumed to be a power of two for simplicity. Then, the MRA expansion of f(k) can be expressed as:

_{0}≥ 0 is the coarsest resolution level (or the largest timescale) and j

_{max}= log

_{2}(N) is the finest resolution level (or the smallest timescale) where the sampled signal is originally represented.

_{0}, and the wavelet coefficients at scale j carry the information that is different between two approximations to the original signal at scales j and j − 1. When these smaller timescale components at scale j ≥ J make little contribution to the original signal, the original signal f(k) can be approximated by a truncated MRA expansion from scale j

_{0}up to J as:

_{max}are assumed to be zero. The higher the upper resolution level J, the more accurate the approximation. The truncated MRA representation can describe not only low-frequency fluctuations but also some rapid localized changes. The advantage of the truncated MRA expansion is that it provides a parsimonious approximation representation of the original signal and is particularly suitable for original signals that do not vary significantly during a short time interval.

#### 3.2. Truncated Wavelet MRA Expansion of the Nonparametric Component

_{1}), another part involved in the nonparametric component, is time invariant during the identification process. Thus, an appropriate model for the nonparametric component u(k) in (8) can be provided using a truncated MRA expansion as:

^{haar}(t) and mother wavelet ψ

^{haar}(t) can be described as:

^{haar}(t) and ψ

^{haar}(t) as g

_{1}(t) and g

_{2}(t); that is, g

_{1}(t) and g

_{2}(t) are the scaling and wavelet basis functions at the coarsest scale of zero, respectively. All other Haar wavelet basis functions derived from g

_{2}(t) are denoted as:

^{j}− 1 for each scale j = 1,2,···,J. We define

**g**(t) = [g

_{1}(t), g

_{2}(t),···,g

_{L}(t)]

^{T}, where L = 2

^{J+1}. For example, Figure 2 shows the Haar basis functions of

**g**(t) when J = 3. The discrete representation of

**g**(t) is defined as:

_{0}= 0 up to J can be compactly expressed as:

**η**= [η

_{1},η

_{2},···,η

_{L}]

^{T}with η

_{l}as the expansion coefficient that corresponds to the associated Haar wavelet basis function g

_{l}(t).

#### 3.3. Recursive Penalized Wavelet Estimator for Online PLBM Identification

^{J}(k), the observation equation can be rewritten as:

_{i}is the ith entry of

**β**. In (21), the first summation term measures how well the candidate solution fits the observed data in the LS sense, whereas the second summation term is a regularizer that considers an l

_{1}-norm penalty on the wavelet coefficients. λ controls the relative weight of the two terms, which means that the larger λ is, the more the wavelet coefficients will shrink to zero. The penalty term in (21) penalizes only the wavelet coefficients of the nonparametric part of the model and not the scaling coefficients. The penalized wavelet estimator (20) can be regarded as an extension of the wavelet shrinkage estimators, which are typically processed in batch form by iterative soft-thresholding algorithms [14,22,23]. However, the batch estimators suffer from high computational complexity and increased memory requirements as time progresses and are thus not appropriate for online implementation. In contrast to batch estimators, their recursive counterparts offer computational and memory savings and enable tracking of the slowly time-varying system. This feature motivated us to develop a recursive penalized wavelet estimator for PLBM identification.

**R**

^{(k)}and vector

**r**

^{(k)}can be defined recursively as follows:

**β**and can admit a closed-form solution.

**β**is currently optimized. We recall that the penalty in (22) is applied only on the wavelet coefficients (β

_{5}:β

_{P}) and not on the scaling coefficient (β

_{4}) or linear parameters (β

_{1}:β

_{3}). Thus, (25) can be rewritten as:

^{(k)}(p,p) is the (p,p)th entry of the matrix

**R**

^{(k)}, and ${w}_{p}^{\left(k\right)}$ is defined as:

^{(k)}(p) is the pth entry of

**r**

^{(k)}. Being a scalar minimization problem, the solution of (26) can be easily obtained in closed form as:

_{+}= max (a,0). Because the Haar wavelet basis functions are compactly supported, local data processing can be employed to reduce the computational burden [35], i.e., the wavelet coefficients ${\widehat{\beta}}_{p}^{\left(k\right)},p=5,\text{\hspace{0.17em}}6,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}P,$ are updated only when at least one of the incoming inputs falls into the support of the associated wavelet functions. The modified online CCD algorithm for the recursive penalized wavelet estimator is summarized as follows:

**Initialize**${\widehat{\beta}}^{\left(0\right)}={\mathbf{0}}_{P}$, ${\mathbf{R}}^{\left(0\right)}={\mathbf{0}}_{P\times P}$ and ${\mathbf{r}}^{\left(0\right)}={\mathbf{0}}_{P}$.

**For**k = 1,2,...,N

**do**

**R**

^{(k)}and vector

**r**

^{(k)}using (23) and (24), respectively.

**For**p = 1,···,4

**do**

**end for**

**For**p = 5,···,P

**do**

**If**g

_{p − 3}(k/N) = 0

**else**

**end if**

**end for**

**end for**

**θ**and

**η**), the nonparametric component u(k) can be approximately calculated via (17) as follows:

## 4. Simulation and Experimental Results

_{0}= 0 and finest resolution level J = 5. The original input-output data in the observation window [0,T] are normalized into the unit interval [0,1] for convenient implementation [33].

#### 4.1. Simulations

_{1}, θ

_{2}and θ

_{3}are generated according to (6) with R

_{s}= 50 mΩ, R

_{p}= 30 mΩ, and C

_{p}= 2500 F. To illustrate the robustness of the proposed identification method, y(k) is corrupted by an additive white Gaussian noise with zero-mean and variance σ

^{2}. The signal-to-noise ratio (SNR) is defined as 1/σ

^{2}. The OCV is assumed to be monotonically varying throughout the entire simulation process and is modeled as a smooth polynomial for simplicity:

_{oc}(k) = −0.5t

_{k}(1.5 − t

_{k}) + 41.5, t

_{k}= k/N, k = 1,2,···,N

_{s}, R

_{p}, and C

_{p}are shown individually in Figure 3 in comparison with their real values. The time axis is rescaled according to the inverse time mapping [0,1] → [0,T]. The penalty threshold adopted here is the traditional universal threshold value $\lambda =\sigma \sqrt{2\mathrm{log}\left(N\right)}$ derived from [26] and used in [14] and [23] for the wavelet semiparametric estimation. The SNR is set at 23 dB. The estimated OCV follows the true OCV variations quite well, and the other estimated parameters all converge to their real values with different convergence speeds. This result demonstrates that the proposed estimator can accurately identify the PLBM and time-invariant linear parameters.

_{b}(k) − v

_{oc}(k)] = θ

_{1}[v

_{b}(k − 1) − v

_{oc}(k)] + θ

_{2}i

_{b}(k) + θ

_{3}i

_{b}(k − 1)

**Figure 4.**Estimated results of equivalent circuit parameters with the semiparametric and parametric approaches: (

**a**) R

_{s}; (

**b**) R

_{p}; (

**c**) C

_{p}.

_{s}accurately. This is because the fact that the ohmic internal resistance represents the terminal voltage variation caused by a current variation in the battery and the slowly varying OCV bias has very little effect on the terminal voltage variation. However, as can be seen from Figures 4b,c the parametric approach cannot estimate the equivalent polarization resistance and capacitance accurately, and the larger the model bias (i.e., the larger d), the larger estimation error will be. This result indicates that the polarization parameters are much more sensitive to OCV model bias in comparison with the ohmic internal resistance.

_{0}= 100 to ensure a fair comparison. To evaluate the MSE, we conducted 500 replicate Monte Carlo simulations for each SNR. Moreover, to clearly illustrate the sensitivity of each approach to observed noise, the input-output data are not pre-filtered. For the adaptive control approach, the OCV was identified based on the discrete version of Equations (11) and (12) in [17]. The results are shown in Figure 5 and illustrate the sensitivity of the three identification approaches to the observed noise. As expected, the performance of all three approaches increases with the SNR. In the low-SNR regime, the wavelet estimators clearly outperform the adaptive control approach due to the fact that the wavelet estimators identify OCV only using the large scale expansion coefficients as shown in (17), thus the high frequency noise ( spanned by small scale expansion coefficients) can be effectively reduced from the identified OCV. However, the MSE performance of the adaptive control approach increases significantly with increasing SNR. In the high-SNR regime, the MSEs of the adaptive control approach and the proposed recursive penalized wavelet estimator are almost indistinguishable. Additionally, it can be seen that the proposed penalized wavelet estimator outperforms the RLS wavelet estimator because of the power of the l

_{1}-norm penalty.

#### 4.2. Experiments

_{2}O

_{4}lithium-ion cells connected in series Each healthy cell has a nominal voltage of 3.6 V and a nominal capacity of 15 A·h. The experimental setup comprised a Digatron EV battery testing system EVT300-500, a battery management module, a controller area network communication unit, and a host computer running the built-in BMS software BTS-600, as shown in Figure 6. The EVT300-500 system includes a programmable charger/electronic load (which can charge or discharge the battery module with a maximum voltage of 500 V and maximum current of 300 A) and a real-time data acquisition unit that can collect test data, including terminal voltage, outflow current, temperature, and accumulated ampere-hours and watt-hours. The battery management module can also measure the terminal voltage and temperature of each cell in the battery module. The BTS-600 software can deal with the collected data and generate control signals for the EVT300-500 system based on the designed program. To reduce the influence of temperature fluctuation, the experiments were performed under a predefined temperature of 28 °C using a temperature-controlled chamber.

_{s}was much faster than those of the other parameters under a steady value of 25.6 mΩ, which implied that the ohmic internal resistance R

_{s}was less sensitive to the charge–discharge behavior of the lithium-ion battery than the other parameters. The polarized resistance R

_{p}and capacitance C

_{p}converged to 14.3 mΩ and 3900 F, respectively, with some fluctuations.

**Figure 9.**Validation results of the voltage responses in a FUDS cycle: (

**a**) Measured and predicted battery voltage responses; (

**b**) Relative error rate.

## 5. Conclusions

**R**

^{(k)}and

**r**

^{(k)}, similar to the RLS algorithm.

_{1}-norm penalty on the wavelet coefficients to identify the wavelet-based PLBM.

## Nomenclature:

C_{p} | polarization capacitance |

c_{j,m} | scaling coefficient of wavelet MRA expansion |

d_{j,m} | wavelet coefficient of wavelet MRA expansion |

f | any finite energy signal |

g_{i} | Haar wavelet basis function |

h | battery state of charge |

i_{b} | battery outflow current |

j | scale level of wavelet MRA expansion |

j_{0} | the coarsest scale of wavelet MRA expansion |

j_{max} | the finest scale of wavelet MRA expansion |

J | the truncation scale of truncated wavelet MRA expansion |

k | discrete time |

N | length of input/output |

R_{p} | polarization resistance |

R_{s} | ohmic internal resistance |

t | continuous time |

T_{c} | sampling period |

u | nonparametric component of PLBM |

v_{b} | battery terminal voltage |

v_{c} | polarization voltage |

v_{oc} | open circuit voltage |

x | observed value of battery terminal voltage |

y | observed value of battery outflow current |

## Greek Symbols

ε | zero-mean white noise |

η_{i} | the ith wavelet expansion coefficient of the nonparametric component of PLBM |

θ_{i} | the ith parameter of the parametric component of PLBM |

λ | penalty factor |

ϕ | wavelet scaling function |

ψ | wavelet mother function |

## Acronyms

CCD | cyclic coordinate descent |

MRA | multiresolution analysis |

OCV | open circuit voltage |

PLBM | partially linear battery model |

RLS | recursive least square |

SoC | state of charge |

SoH | state of health |

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Mu, D.; Jiang, J.; Zhang, C.
Online Semiparametric Identification of Lithium-Ion Batteries Using the Wavelet-Based Partially Linear Battery Model. *Energies* **2013**, *6*, 2583-2604.
https://doi.org/10.3390/en6052583

**AMA Style**

Mu D, Jiang J, Zhang C.
Online Semiparametric Identification of Lithium-Ion Batteries Using the Wavelet-Based Partially Linear Battery Model. *Energies*. 2013; 6(5):2583-2604.
https://doi.org/10.3390/en6052583

**Chicago/Turabian Style**

Mu, Dazhong, Jiuchun Jiang, and Caiping Zhang.
2013. "Online Semiparametric Identification of Lithium-Ion Batteries Using the Wavelet-Based Partially Linear Battery Model" *Energies* 6, no. 5: 2583-2604.
https://doi.org/10.3390/en6052583