# State of Charge Estimation of a Composite Lithium-Based Battery Model Based on an Improved Extended Kalman Filter Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Battery Model

_{Q}is the remaining capacity, and the C

_{I}is the rated capacity when the battery is discharged with a constant current I.

- Selecting the battery model (A novel composite electrochemical model).
- Determining the different parameters to be identified based on the selected model.
- Performing a series of characteristics tests on the battery.
- Linear fitting of the experimental data (offline identification).
- Building the model in MATLAB.
- Obtaining the equation of state.
- Developing the online estimation algorithm (iEKF).
- Inputting the current, temperature and SoC (simulations in static and dynamic conditions).

_{tI}) definition is amended as [26,27].

_{ti}is the total quantity of electric charge/discharge during the period t, I is the current over a time interval 0 to t and η is the efficiency coefficient which includes both the charge/discharge rate (η

_{i}) and the temperature influence coefficient (η

_{T}).

#### The Composite Battery Model

_{k}is the voltage of the battery model. The composite model is built based on three different electrochemical models as follows [28,29]:

_{0}is the OCV when the battery is fully charged and R is the internal resistance which will change with different charge/discharge status and SoC, i

_{k}is the instantaneous current at time k (negative when the battery is charging and positive when discharging). K

_{1}to K

_{4}are the matching parameters to be identified through battery experiments. The state equation based on the composite battery model is described as follows:

_{0}is the OCV of the fully charged battery and has the same physical meaning as E

_{0}. However, E

_{0}in Equations (3)–(5) is the actual measured value while K

_{0}is obtained by the identification based on OCV-SoC experimental data (see in Section 3.4). Equations (3)–(5) are combined in Equation (7). The electrochemical models of Equations (3)–(5) reflect the relationship between the terminal voltage and the SoC (${x}_{k}$). R is the internal resistance (Ohmic resistance) and changes with the charging/discharging state of the battery ($R{i}_{k}$). ${K}_{1}/{x}_{k}$ from Equation (3) and ${K}_{2}\xb7{x}_{k}$ from Equation (4) reflect the polarization resistances of the battery. ${K}_{3}\xb7\text{}\mathrm{ln}{x}_{k}$ and ${K}_{4}\xb7\mathrm{ln}\left(1-{x}_{k}\right)$ from Equation (5) represent the influence of the internal temperature and material activity during the electrochemical reaction of the battery, respectively.

_{1}to K

_{4}.

## 3. The Experiments and Offline Parameter Identification

_{0}, K

_{1}, K

_{2}, K

_{3}, K

_{4}) of the composite battery model shown in Equations (6) and (7) require a validation. The initial identification of the parameters is an offline and static estimation, obtained by polynomial curve fitting of the experimental data (see Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8). The charge/discharge rate factor ${\eta}_{i}$ was obtained by the charge and discharge test. Similarly, the temperature influence coefficient ${\eta}_{T}$ was obtained from temperature characteristic test. The other parameters (R, K

_{0}, K

_{1}, K

_{2}, K

_{3}, K

_{4}) were obtained through a Hybrid Pulse Power Characterization (HPPC) test and OCV-SoC test [36]. All the tests were conducted on a LiFePO

_{4}battery (single cell) with 206 Ah rated capacity and 3.2 V rated voltage.

#### 3.1. Charge and Discharge Rate Test

_{4}battery are shown in Figure 2 and Figure 3, respectively. From Figure 2 and Figure 3, it is clear that the actual capacity of the battery will decrease as the charge/discharge rate increases.

#### 3.2. Temperature Characteristics Test

_{4}battery at different temperature are shown in Figure 4.

_{4}battery decreases with the decreasing temperature [37].

#### 3.3. Hybrid Pulse Power Characterization (HPPC) Test

_{1}/1 is 1C charge/discharge rate.

_{before}and V

_{after}are the measured voltage values before and after charging/discharge test, respectively. DCR values are rounded to two decimal places.

#### 3.4. OCV-SoC Test

_{0}. The operating characteristics of the battery show that a proportional relationship exists between OCV and SoC. The OCV is roughly regarded as a linearized function of SoC in a simplified system [38]. For example, the OCV rises with the increase in SoC. The current SoC of the battery can be calculated through a model relationship between OCV and SoC. The experimental steps used to obtain an approximate OCV value are shown in Figure 6.

#### 3.5. Offline Parameter Identification

_{4}battery (206 Ah, 3.2 V). A convincing offline parameters identification was required to achieve a reliable mathematical battery model for the simulation of KF. Eight parameters (${\eta}_{i}$, ${\eta}_{T}$, R, K

_{0}, K

_{1}, K

_{2}, K

_{3}, K

_{4}) were identified by the approach of linear fitting and recursive least squares (RLS) in MATLAB. The experimental data was input into MATLAB in the form of different sets of data points. In terms of ${\eta}_{i}$, relationship between the actual capacity and the charge/discharge rate were quantified by linear fitting functions in MATLAB. A second-order polynomial linear fitting equation for ${\eta}_{i}$ was obtained by the use of the polyfit function:

_{0}, R, K

_{1}, K

_{2}, K

_{3}, K

_{4}) were obtained by the method of RLS. Figure 7 shows that the vales of parameters begin to converge after 1500 iterations. The values of the parameters were as follows: R = 0.0048, K

_{1}= −0.000268, K

_{2}= 0.1495, K

_{3}= 0.111 and K

_{4}= −0.01955. The value of K

_{0}was 3.191 and is not shown in the figure. The accuracy in the estimation of parameters was high, of the order of ±0.3%. The mathematical model based on linear fitting from real experimental data is therefore reliable for SoC estimation using iEKF.

## 4. The SoC Estimation Based on an Improved EKF Algorithm

#### 4.1. Analysis of the KF and EKF Algorithm

_{k}, the real output y

_{k}and the unmeasurable state x

_{k}. The system model includes the same input u

_{k}, the known state x

_{k}and the output y

_{x}based on the specific battery model. The optimal estimation is obtained through a comparison between the y

_{k}and y

_{x}to amend the prediction estimation. The state variable x

_{k}of the system model is closer to the real value of y

_{k}. The state-space system model of the discrete-time standard KF is expressed as follows:

_{k}, u

_{k}and y

_{k}are the state variables of the input and the output of the system, respectively. w

_{k}is the process noise variable which is used to describe the superimposed noise and error during state transition. v

_{k}is the measurement noise variable which is used to describe the generated noise and error when the input is measured. A

_{k}, B

_{k}, C

_{k}and D

_{k}are the equation matching coefficients reflecting the dynamic characteristics of the system. Figure 8 shows the state-space model of the discrete simple KF.

_{k}and the mean square error of estimation P

_{k}are made at each sampling interval. For example, the first-time predictive estimate of ${x}_{k}^{-}$ is obtained by the iterative recursion using the state equations based on ${x}_{k-1}^{+}$. The predictive estimates of ${x}_{k}^{-}$ and ${P}_{k}^{-}$ are completed before the y

_{k}measurement. The calculation of the optimal estimates of ${x}_{k}^{+}$ and ${P}_{k}^{+}$ start after the measurement of y

_{k}is processed. To obtain the optimal estimation of ${x}_{k}^{+}$ and ${P}_{k}^{+}$, the predictive estimates of ${x}_{k}^{-}$ and ${P}_{k}^{-}$ will be amended after the calculation of y

_{k}.

- Initial value of ${x}_{0}^{+}$ and ${P}_{0}^{+}$:$${x}_{0}^{+}=\mathrm{E}\left[{x}_{0}\right]$$$${P}_{0}^{+}=E\left[\left({x}_{0}-{x}_{0}^{+}\right){\left({x}_{0}-{x}_{0}^{+}\right)}^{T}\right]$$
- Predictive estimate of the ${x}_{k}^{-}$ and ${P}_{k}^{-}$:$${x}_{k}^{-}={A}_{k-1}{x}_{k-1}^{+}+{B}_{k-1}{u}_{k-1}$$$${P}_{k}^{-}={A}_{k-1}{P}_{k-1}^{+}{A}_{k-1}^{T}+{D}_{w}$$
- KF gain L
_{k}(weighting coefficient matrix)$${L}_{k}={P}_{k}^{-}{C}_{k}^{T}{\left({C}_{k}{P}_{k}^{-}{C}_{k}^{T}+{D}_{v}\right)}^{-1}$$ - Optimal estimate of the ${x}_{k}^{-}$ and ${P}_{k}^{-}$:$${x}_{k}^{+}={x}_{k}^{-}+{L}_{k}\left({Y}_{k}-{y}_{k}\right)$$$${P}_{k}^{-}=\left(1-{L}_{k}{C}_{k}\right){P}_{k}^{-}$$
_{w}and D_{v}in Equations (14) and (15) are the covariance of the process noise w_{k}and the measurement noise v_{k}, respectively.

#### 4.2. The SoC Estimation Model Corrected by EKF Based on Composite Model

_{i}) and temperature (η

_{T}) are obtained by Equations (8a), (8b) and (9), respectively.

- Model establishment: Use Equations (6) and (7).
- Determination of system parameters:$${A}_{k-1}={\frac{\partial f\left({x}_{k-1},{u}_{k-1}\right)}{\partial {x}_{k-1}}|}_{{x}_{k-1}={x}_{k-1}^{+}}=1$$$${C}_{k}={\frac{\partial {y}_{k}}{\partial {x}_{k}}|}_{{x}_{k}={x}_{k}^{-}}={K}_{1}/{\left({x}_{k}^{-}\right)}^{2}-{K}_{2}+{K}_{3}/{x}_{k}^{-}-{K}_{4}/\left(1-{x}_{k}^{-}\right)$$
- Initialization of the state variable and the covariance.$${x}_{0}^{+}=So{C}_{0},{P}_{0}^{+}=var\left({x}_{0}\right)$$
- Iterative calculation of the EKF.$$\{\begin{array}{c}{x}_{k}^{-}={x}_{k-1}^{+}-\left(\frac{{\eta}_{i}\Delta t}{{\eta}_{T}{Q}_{n}}\right){i}_{k-1}\\ {y}_{k}={K}_{0}-R{i}_{k}-{K}_{1}/{x}_{k}^{-}-{K}_{2}{x}_{k}^{-}+\\ {K}_{3}\mathrm{ln}\left({x}_{k}^{-}\right)+{K}_{4}\mathrm{ln}\left(1-{x}_{k}^{-}\right)\\ {P}_{k}^{-}={A}_{k-1}{P}_{k-1}^{+}{A}_{k-1}^{T}+{D}_{w}\\ {L}_{k}=\frac{{P}_{k}^{-}{C}_{k}^{T}}{{C}_{k}{P}_{k}^{-}{C}_{k}^{T}+{D}_{v}}\\ {x}_{k}^{+}={x}_{k}^{-}+{L}_{k}\left({Y}_{k}-{y}_{k}\right)\\ {P}_{k}^{+}=\left(1-{L}_{k}{C}_{k}\right){P}_{k}^{-}\\ k=1,2,3\dots \end{array}$$

_{k}

_{−1}and C

_{k}are defined in Step 2 by using Equations (6), (7) and (10). The SoC

_{0}is calculated based on the remaining charge (after charging/discharging) in the previous state and the OCV in the current state. ${P}_{0}^{+}$, D

_{w}and D

_{v}relate to the performance of the battery and the data collection system. In order to update the status of the system, the sampling frequency is set in the Simulink equal to 2.5 times the bandwidth of the sampled signal as per the “Nyquist-Shannon sampling” criterion [43].

## 5. The Simulation Validation of the Improved EKF Algorithm

#### 5.1. The Validation of the Improved EKF Algorithm

#### 5.2. The Simulation Results

^{−7}.

^{−6}), thereby providing credibility to the iEKF method. The error covariance with the iEKF under static operation condition was of the order of 10

^{−7}. Table 5 gives the calculated estimation error based on the error curves shown in Figure 13 and Figure 15a,b.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Xiong, R.; Zhang, Y.; Wang, J.; He, H.; Peng, S.; Pecht, M. Lithium-ion battery health prognosis based on a real battery management system used in electric vehicles. IEEE Trans. Veh. Technol.
**2018**, 68, 4110–4121. [Google Scholar] [CrossRef] - Lu, L.; Han, X.; Li, J.; Hua, J.; Ouyang, M. A review on the key issues for lithium-ion battery management in electric vehicles. J. Power Sources
**2013**, 226, 272–288. [Google Scholar] [CrossRef] - Un-Noor, F.; Padmanaban, S.; Mihet-Popa, L.; Mollah, M.; Hossain, E. A comprehensive study of key electric vehicle (EV) components, technologies, challenges, impacts, and future direction of development. Energies
**2017**, 10, 1217. [Google Scholar] [CrossRef] - Rezvanizaniani, S.M.; Liu, Z.; Chen, Y.; Lee, J. Review and recent advances in battery health monitoring and prognostics technologies for electric vehicle (EV) safety and mobility. J. Power Sources
**2014**, 256, 110–124. [Google Scholar] [CrossRef] - Cuma, M.U.; Koroglu, T. A comprehensive review on estimation strategies used in hybrid and battery electric vehicles. Renew. Sustain. Energy Rev.
**2015**, 42, 517–531. [Google Scholar] [CrossRef] - Berecibar, M.; Gandiaga, I.; Villarreal, I.; Omar, N.; van Mierlo, J.; van den Bossche, P. Critical review of state of health estimation methods of Li-ion batteries for real applications. Renew. Sustain. Energy Rev.
**2016**, 56, 572–587. [Google Scholar] [CrossRef] - Hu, C.; Jain, G.; Zhang, P.; Schmidt, C.; Gomadam, P.; Gorka, T. Data-driven method based on particle swarm optimization and k-nearest neighbor regression for estimating capacity of lithium-ion battery. Appl. Energy
**2014**, 129, 49–55. [Google Scholar] [CrossRef] - Malkhandi, S. Fuzzy logic-based learning system and estimation of state-of-charge of lead-acid battery. Eng. Appl. Artif. Intell.
**2006**, 19, 479–485. [Google Scholar] [CrossRef] - Antón, J.Á.; Nieto, P.G.; de Cos Juez, F.J.; Lasheras, F.S.; Vega, M.G.; Gutiérrez, M.R. Battery state-of-charge estimator using the SVM technique. Appl. Math. Model.
**2013**, 37, 6244–6253. [Google Scholar] [CrossRef] - Zheng, Y.; Lu, L.; Han, X.; Li, J.; Ouyang, M. LiFePO
_{4}battery pack capacity estimation for electric vehicles based on charging cell voltage curve transformation. J. Power Sources**2013**, 226, 33–41. [Google Scholar] [CrossRef] - Xu, J.; Cao, B.; Chen, Z.; Zou, Z. An online state of charge estimation method with reduced prior battery testing information. Int. J. Electr. Power Energy Syst.
**2014**, 63, 178–184. [Google Scholar] [CrossRef] - Dai, H.; Wei, X.; Sun, Z.; Wang, J.; Gu, W. Online cell SOC estimation of Li-ion battery packs using a dual time-scale Kalman filtering for EV applications. Appl. Energy
**2012**, 95, 227–237. [Google Scholar] [CrossRef] - Simon, D. Kalman filtering with state constraints: A survey of linear and nonlinear algorithms. IET Control Theory Appl.
**2010**, 4, 1303–1318. [Google Scholar] [CrossRef] - Xiong, R.; He, H.; Sun, F.; Zhao, K. Evaluation on state of charge estimation of batteries with adaptive extended Kalman filter by experiment approach. IEEE Trans. Veh. Technol.
**2013**, 62, 108–117. [Google Scholar] [CrossRef] - Pavković, D.; Krznar, M.; Komljenović, A.; Hrgetić, M.; Zorc, D. Dual EKF-Based State and Parameter Estimator for a LiFePO
_{4}Battery Cell. J. Power Electron.**2017**, 17, 398–410. [Google Scholar] [CrossRef] - Yousefizadeh, S.; Bendtsen, J.D.; Vafamand, N.; Khooban, M.H.; Dragičević, T.; Blaabjerg, F. EKF-based Predictive Stabilization of Shipboard DC Microgrids with Uncertain Time-varying Load. IEEE J. Emerg. Sel. Top. Power Electron.
**2018**, 7, 901–909. [Google Scholar] [CrossRef] - Fang, Y.; Xiong, R.; Wang, J. Estimation of Lithium-Ion Battery State of Charge for Electric Vehicles Based on Dual Extended Kalman Filter. Energy Procedia
**2018**, 152, 574–579. [Google Scholar] [CrossRef] - Guo, F.; Hu, G.; Xiang, S.; Zhou, P.; Hong, R.; Xiong, N. A multi-scale parameter adaptive method for state of charge and parameter estimation of lithium-ion batteries using dual Kalman filters. Energy
**2019**, 178, 79–88. [Google Scholar] [CrossRef] - Sun, F.; Hu, X.; Zou, Y.; Li, S. Adaptive unscented Kalman filtering for state of charge estimation of a lithium-ion battery for electric vehicles. Energy
**2011**, 36, 3531–3540. [Google Scholar] [CrossRef] - Zhang, J.; Xia, C. State-of-charge estimation of valve regulated lead acid battery based on multi-state Unscented Kalman Filter. Int. J. Electr. Power Energy Syst.
**2011**, 33, 472–476. [Google Scholar] [CrossRef] - He, Z.; Gao, M.; Wang, C.; Wang, L.; Liu, Y. Adaptive state of charge estimation for Li-ion batteries based on an unscented Kalman filter with an enhanced battery model. Energies
**2013**, 6, 4134–4151. [Google Scholar] [CrossRef] - Gao, S.; Kang, M.; Li, L.; Liu, X. Estimation of state-of-charge based on unscented Kalman particle filter for storage lithium-ion battery. J. Eng.
**2019**, 2019, 1858–1863. [Google Scholar] [CrossRef] - Xu, J.; Mi, C.C.; Cao, B.; Deng, J.; Chen, Z.; Li, S. The state of charge estimation of lithium-ion batteries based on a proportional-integral observer. IEEE Trans. Veh. Technol.
**2013**, 63, 1614–1621. [Google Scholar] - Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. J. Power Sources
**2004**, 134, 262–276. [Google Scholar] [CrossRef] - Barré, A.; Deguilhem, B.; Grolleau, S.; Gérard, M.; Suard, F.; Riu, D. A review on lithium-ion battery ageing mechanisms and estimations for automotive applications. J. Power Sources
**2013**, 241, 680–689. [Google Scholar] [CrossRef][Green Version] - He, H.; Xiong, R.; Zhang, X.; Sun, F.; Fan, J. State-of-charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model. IEEE Trans. Veh. Technol.
**2011**, 60, 1461–1469. [Google Scholar] - Barillas, J.K.; Li, J.; Günther, C.; Danzer, M.A. A comparative study and validation of state estimation algorithms for Li-ion batteries in battery management systems. Appl. Energy
**2015**, 155, 455–462. [Google Scholar] [CrossRef] - He, H.; Xiong, R.; Guo, H.; Li, S. Comparison study on the battery models used for the energy management of batteries in electric vehicles. Energy Convers. Manag.
**2012**, 64, 113–121. [Google Scholar] [CrossRef] - Xiong, R.; Gong, X.; Mi, C.C.; Sun, F. A robust state-of-charge estimator for multiple types of lithium-ion batteries using adaptive extended Kalman filter. J. Power Sources
**2013**, 243, 805–816. [Google Scholar] [CrossRef] - Piller, S.; Perrin, M.; Jossen, A. Methods for state-of-charge determination and their applications. J. Power Sources
**2001**, 96, 113–120. [Google Scholar] [CrossRef] - Plett, G. LiPB dynamic cell models for Kalman-filter SOC estimation. In The 19th International Battery, Hybrid and Fuel Electric Vehicle Symposium and Exhibition; Pusan, Korea, 2002; pp. 1–12. Available online: https://scholar.google.com.hk/scholar?hl=zh-CN&as_sdt=0%2C5&q=.+LiPB+dynamic+cell+models+for+Kalman-filter+SOC+estimation.+In+The+19th+International+Battery%2C+Hybrid+and+Fuel+Electric+Vehicle+Symposium+and+Exhibition&btnG= (accessed on 31 October 2019).
- Han, H.; Xu, H.; Yuan, Z.; Zhao, Y. Modeling for Lithium-ion Battery Used in Electric Vehicles. In Proceedings of the 2014 IEEE Conference and Expo Transportation Electrification Asia-Pacific (ITEC Asia-Pacific), Beijing, China, 31 August 2014; pp. 1–5. [Google Scholar]
- He, H.; Xiong, R.; Fan, J. Evaluation of lithium-ion battery equivalent circuit models for state of charge estimation by an experimental approach. Energies
**2011**, 4, 582–598. [Google Scholar] [CrossRef] - Hu, X.; Li, S.; Peng, H. A comparative study of equivalent circuit models for Li-ion batteries. J. Power Sources
**2012**, 198, 359–367. [Google Scholar] [CrossRef] - Chang, W.-Y. The state of charge estimating methods for battery: A review. ISRN Appl. Math.
**2013**, 2013, 953792. [Google Scholar] [CrossRef] - Hunt, G.; Motloch, C. Freedom Car Battery Test Manual for Power-Assist Hybrid Electric Vehicles; INEEL, Idaho Falls: Idaho Falls, ID, USA, 2003. [Google Scholar]
- Doughty, D.H.; Crafts, C.C. FreedomCAR: Electrical Energy Storage System Abuse Test Manual for Electric and Hybrid Electric Vehicle Applications; Sandia National Laboratories: Albuquerque, NM, USA, 2006. [Google Scholar]
- Hu, Y.; Yurkovich, S. Battery cell state-of-charge estimation using linear parameter varying system techniques. J. Power Sources
**2012**, 198, 338–350. [Google Scholar] [CrossRef] - Julier, S.J.; Uhlmann, J.K. New extension of the Kalman filter to nonlinear systems. In Signal Processing, Sensor Fusion, and Target Recognition VI; SPIE Digital Library: Bellingham, WA, USA, 1997; pp. 182–194. [Google Scholar]
- Ristic, B.; Arulampalam, S.; Gordon, N. Beyond the Kalman Filter: Particle Filters for Tracking Applications; Artech House: Norwood, MA, USA, 2003. [Google Scholar]
- Best, M.C.; Gordon, T.; Dixon, P. An extended adaptive Kalman filter for real-time state estimation of vehicle handling dynamics. Veh. Syst. Dyn.
**2000**, 34, 57–75. [Google Scholar] - Pérez, G.; Garmendia, M.; Reynaud, J.F.; Crego, J.; Viscarret, U. Enhanced closed loop State of Charge estimator for lithium-ion batteries based on Extended Kalman Filter. Appl. Energy
**2015**, 155, 834–845. [Google Scholar] [CrossRef] - Marvasti, F. Nonuniform Sampling: Theory and Practice; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Pang, S.; Farrell, J.; Du, J.; Barth, M. Battery state-of-charge estimation. In Proceedings of the 2001 American Control Conference (Cat. No. 01CH37148), Arlington, VA, USA, 25–27 June 2001; pp. 1644–1649. [Google Scholar]
- Plett, G.L. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation. J. Power Sources
**2004**, 134, 277–292. [Google Scholar] [CrossRef] - Bizeray, A.M.; Zhao, S.; Duncan, S.R.; Howey, D.A. Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter. J. Power Sources
**2015**, 296, 400–412. [Google Scholar] [CrossRef][Green Version] - Wang, T.; Chen, S.; Ren, H.; Zhao, Y. Model-based unscented Kalman filter observer design for lithium-ion battery state of charge estimation. Int. J. Energy Res.
**2018**, 42, 1603–1614. [Google Scholar] [CrossRef] - Yuan, S.; Wang, Y.; Li, C. SOC Performance Evaluation Analysis in Electric Vehicle Power Battery Operation. Inn. Mong. Electr. Power
**2016**, 34, 81–84. [Google Scholar] - He, H.; Qin, H.; Sun, X.; Shui, Y. Comparison study on the battery SoC estimation with EKF and UKF algorithms. Energies
**2013**, 6, 5088–5100. [Google Scholar] [CrossRef] - Liu, C.; Liu, W.; Wang, L.; Hu, G.; Ma, L.; Ren, B. A new method of modeling and state of charge estimation of the battery. J. Power Sources
**2016**, 320, 1–12. [Google Scholar] [CrossRef]

**Figure 5.**The HPPC test [36]. (

**a**) the HPPC test profile, (

**b**) the start of HPPC test sequence, (

**c**) the complete HPPC sequence.

**Figure 13.**Comparison of error covariance between iEKF model and Ah counting model under static operation conditions.

**Figure 14.**MATLAB inputs of (

**a**) The input current efficiency and (

**b**) The input temperature for the SOC estimation under dynamic operating conditions.

**Figure 15.**Comparison of error covariance between iEKF and Ah models under dynamic operating conditions: (

**a**) over a time of 1000 s (

**b**) from 0 to 100 s. The inset in both the figures shows the magnified plot of the error covariance of the iEKF model only. Note the extremely small value of error covariance of the iEKF model.

Charge Rate Test | 0.2 C | 0.5 C | 1.0 C | 2.0 C | 3.0 C |
---|---|---|---|---|---|

Capacity/Ah | 211.45 | 208.95 | 206.88 | 204.07 | 194.76 |

Capacity retention rate/% | 102.21 | 101.00 | 100.00 | 98.64 | 94.14 |

Energy/Wh | 707.76 | 706.36 | 708.04 | 712.64 | 692.15 |

Energy retention rate/% | 99.96 | 99.76 | 100.00 | 100.65 | 97.76 |

Charge Rate Test | 0.2 C | 0.5 C | 1.0 C | 2.0 C | 3.0 C |
---|---|---|---|---|---|

Capacity/Ah | 214.31 | 212.94 | 211.84 | 211.28 | 211.08 |

Capacity retention rate/% | 101.16 | 100.52 | 100.00 | 99.74 | 99.64 |

Energy/Wh | 687.83 | 672.22 | 657.33 | 641.41 | 628.45 |

Energy retention rate/% | 104.64 | 102.26 | 100.00 | 97.58 | 95.61 |

Temperature Characteristic Test | −20 °C | −10 °C | 0 °C | 25 °C | 45 °C |
---|---|---|---|---|---|

Capacity/Ah | 194.18 | 201.14 | 205.49 | 211.27 | 211.35 |

Capacity retention rate/% | 91.91 | 95.20 | 97.26 | 100.00 | 100.00 |

Energy/Wh | 523.18 | 566.18 | 600.52 | 657.80 | 668.46 |

Energy retention rate/% | 79.54 | 86.07 | 91.29 | 100.00 | 101.62 |

SoC | 206Ah 4C Discharge/3C Charge | |||||
---|---|---|---|---|---|---|

Discharge | Charge | |||||

V_{before} (mV) | V_{after} (mV) | DCR (mΩ) | V_{before} (mV) | V_{after} (mV) | DCR (mΩ) | |

90% | 3328.30 | 2910.70 | 0.45 | 3277.20 | 3613.20 | 0.49 |

80% | 3327.40 | 2893.70 | 0.47 | 3267.20 | 3612.90 | 0.50 |

70% | 3327.40 | 2881.60 | 0.48 | 3258.90 | 3610.10 | 0.51 |

60% | 3296.70 | 2857.40 | 0.48 | 3244.90 | 3586.90 | 0.50 |

50% | 3289.00 | 2829.80 | 0.50 | 3230.40 | 3580.40 | 0.51 |

40% | 3288.00 | 2805.90 | 0.52 | 3217.30 | 3577.30 | 0.52 |

30% | 3284.30 | 2775.60 | 0.55 | 3201.50 | 3570.10 | 0.53 |

20% | 3254.50 | 2707.40 | 0.59 | 3169.90 | 3539.40 | 0.53 |

10% | 3209.60 | 2422.10 | 0.86 | 3110.40 | 3497.00 | 0.56 |

Operation Condition | Maximum Error (%) | Average Error (%) | Relative Error (%) |
---|---|---|---|

Static | 2.39 | 1.43 | 1.20 |

Dynamic | 6.76 | 3.94 | 2.15 |

**Table 6.**The comparison of relative error (%) between SoC estimation based on the iEKF and Ah counting methods under static/dynamic condition.

Estimation Error | Initial SoC of 20% | Initial SoC of 50% | Initial SoC of 70% | ||||
---|---|---|---|---|---|---|---|

Operation | Ah Counting | iEKF | Ah Counting | iEKF | Ah Counting | iEKF | |

Static | 14.9 | 0.8 | 15.3 | 1.0 | 15.0 | 1.2 | |

Dynamic | 17.7 | 1.7 | 18.5 | 2.0 | 19.7 | 2.1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ding, N.; Prasad, K.; Lie, T.T.; Cui, J. State of Charge Estimation of a Composite Lithium-Based Battery Model Based on an Improved Extended Kalman Filter Algorithm. *Inventions* **2019**, *4*, 66.
https://doi.org/10.3390/inventions4040066

**AMA Style**

Ding N, Prasad K, Lie TT, Cui J. State of Charge Estimation of a Composite Lithium-Based Battery Model Based on an Improved Extended Kalman Filter Algorithm. *Inventions*. 2019; 4(4):66.
https://doi.org/10.3390/inventions4040066

**Chicago/Turabian Style**

Ding, Ning, Krishnamachar Prasad, Tek Tjing Lie, and Jinhui Cui. 2019. "State of Charge Estimation of a Composite Lithium-Based Battery Model Based on an Improved Extended Kalman Filter Algorithm" *Inventions* 4, no. 4: 66.
https://doi.org/10.3390/inventions4040066