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Online Estimation and Error Analysis of both SOC and SOH of Lithium-ion Battery based on DEKF Method
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ScienceDirect ScienceDirect Energy Procedia 158 Energy Procedia 00(2019) (2017)3008–3013 000–000
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Online Estimation and Error Analysis of both SOC and SOH of Lithium-ion Battery based on DEKF Method The 15th International Symposium onaDistrict Heating and Cooling a a Linlin Fang , Junqiu Li *, Bo Peng
Linlin Fang , Junqiu Li *, Bo Peng
Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Institute of Technology, Beijing, 100081,China. Assessing the feasibility of using the heat demand-outdoor temperature function for a long-term district heat demand forecast Abstract aa
a,b,c
a
a
b
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The state-of-charge (SOC) and state-of-health(SOH) are two indexes ., in B. battery management, O. system I. Andrić *, A. Pina , P. Ferrão , J.critical Fournier Lacarrière Le(BMS) Correfor electric vehicles(EVs). To achieve accurate estimation of SOC and SOH, this paper establishes a battery equivalent circuit model and a IN+ Center Factor for Innovation, Technology and Policy Researchto- Instituto Superior Técnico, Av. Rovisco Paisparameters. 1, 1049-001And Lisbon, Portugal uses Forgetting Recursive Least Squares (FFRLS) realize online identification of model based on the b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel,using 78520Double Limay, France relationship between the ohmic internal resistance and the SOH, a joint estimator extended Kalman filter(DEKF) c Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France algorithm is proposed for the estimation of both SOC and SOH. Then, an error model is established to analyze the influence of the battery OCV-SOC curve, battery capacity and battery parameters on the estimation of the SOC and SOH. The experiment results show that the maximum estimation error of SOC and SOH is 1.08% and 1.52% respectively, which have verified that accurate and robust SOC and SOH estimation results can be obtained by the proposed method. Besides, the OCV-SOC curve has Abstract the greatest influence on the estimation error of SOC and SOH among the three kinds of factors mentioned above. Copyright © 2018networks Elsevier Ltd. All rights reserved. heating are commonly the literature as one of the most effective solutions for decreasing the ©District 2019 The Authors. Published by responsibility Elsevier addressed Ltd. of thein scientific th International Conference on Applied Selection and peer-review under committee of investments the 10th greenhouse gas emissions from thethe building sector. These systems require high which are returned through the heat This is an open access article under CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Energy (ICAE2018). sales. Due to the changed climate conditions and building renovation policies, heat demand in the future could decrease, Peer-review under responsibility of the scientific committee of ICAE2018 – The 10th International Conference on Applied Energy. prolonging the investment return period. Keywords: Lithium-ion battery; Battery equivalent circuit model; Double extended Kalman filter; Error model of SOC and SOH estimation The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand forecast. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district 1.renovation Introduction scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were compared with results from a dynamic heat demand model, previously developed and validated by the authors. Lithium-ion batteries areonly featured highis considered, energy density, lowof self-discharge rate, long cycleapplications life and The results showed that when weather by change the margin error could be acceptable for some environmental friendliness thus have foundforwide applications in the area of electric power supply and (the error in annual demand and was lower than 20% all weather scenarios considered). However,vehicle after introducing renovation scenarios, the error value Battery increasedmanagement up to 59.5% (depending on the have weather anddesigned renovation energy storage systems. systems (BMS) been toscenarios provide combination monitoring, considered). diagnosis Thecontrol value of slope coefficient increased on average within the and functions to enhance the operation of batteries [1].range of 3.8% up to 8% per decade, that corresponds to the decrease the number of heating hours ofone 22-139h heating season on the combination of weather and In the in battery management systems, of theduring mostthe critical issues is (depending to accurately estimate the state-of-charge renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the (SOC) in real-time [2–8]. To obtain an accurate SOC and SOH estimation, many methods have been proposed in coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and literatures such as the current integral method [2], open-circuit voltage method [3], neural network method [4], improve the accuracy of heat demand estimations.
extended Kalman filter (EKF) method [5] and particle filter (PF) method [6–8]. Among the above methods, extended © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. * Corresponding author. Tel.: 086-010-68940589; fax: 086-010-68914842. E-mail address: [email protected]. Keywords: Heat demand; Forecast; Climate change
1876-6102 Copyright © 2018 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the 10thth International Conference on Applied Energy (ICAE2018). 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. 1876-6102 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of ICAE2018 – The 10th International Conference on Applied Energy. 10.1016/j.egypro.2019.01.974
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Linlin Fang et al. / Energy Procedia 158 (2019) 3008–3013 Author name / Energy Procedia 00 (2018) 000–000
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Kalman filtering is widely used for its simple theory and easy to implement. However, with the aging of the battery, the attenuation of capacity will lead to inaccurate estimation of SOC. In Reference [9], an equivalent circuit model is established to track the internal resistance of the battery and the internal resistance change of the aging battery is quantitatively studied, but the specific relationship for the battery internal resistance and SOH has not been determined. Reference [10] uses the Kalman filter algorithm to estimate the SOC and SOH, and the accuracy of SOC estimation is controlled within 1%, but the maximum error of the estimation of SOH is 20%. At present, the evaluation of battery SOH is mostly qualitative analysis. Besides, the effects of the change of OCV-SOC relationship curve on estimation of SOC and SOH during battery aging hasn’t been considered either. Based on the above problems, we establish the equivalent circuit model, use the Forgetting Factor Recursive Least Squares (FFRLS) method to realize the online identification of model parameters and use the DEKF algorithm to achieve online joint estimation of SOC and SOH. Then, the influence of battery OCV-SOC curve, battery capacity and battery model parameters on the estimation of SOC and SOH is analyzed by establishing the error model of SOC and SOH estimation. 2. Battery equivalent circuit model and Online Parameter Identification Dynamic voltage characteristics of lithium-ion battery show abrupt changes and gradual changes, which can be described using the equivalent circuit model. The model circuit and the meaning of each symbol are shown in Fig.1. And the state equation of this model is formula (1). Id
UOC
Rd
Cd ＋ Ud －
Ic
Rc
Cc ＋ UC －
R0 IL
＋ UL －
Fig. 1. battery equivalent circuit model Ud I • − + L U d = Cd Rd Cd Uc • I − + L U c = Cc Rc Cc U L = U oc − U d − U c − R0 I L
(1)
The battery model is transformed into a mathematical form that can be identified by a least-squares method and the Forgetting Factor Recursive Least Squares (FFRLS) algorithm is used to realize online parameter identification. Adopting Laplace transform for equation (1), the equation (2) can be obtained by making U= s U t ( s ) − U oc ( s ) rc ( )
and using bilinear transformation.
z 2U rc ( z ) = a1 zU rc ( z ) + a2U rc ( z ) + a3 z 2 I L ( z ) + a4 zI L ( z ) + a5 I L ( z )
(2)
U t ,k = (1 − a1 − a2 )U oc,k + a1U t ,k −1 + a2U t ,k −2 + a3 I L,k + a4 I L,k −1 + a5 I L,k −2
(3)
Then, we can finally obtain the equation (3) by inverse Z-transformation for equation (2).
And we can get the output matrix yk , Parameter matrix and data matrix of the mathematical model, which are shown in the equation (4). Then, the parameters of the battery model can be solved using FFRLS algorithm, as shown in equation (5). (4) y = U = k
t ,k
T k
k
T = (1 − a1 − a2 ) U oc ,k a1 a2 a3 a4 a5 T =1 U L ,k −1 U L ,k −1 iL ,k iL ,k −1 iL ,k −2
− a3 + a4 − a5 R0 = 1 + a1 − a2 − a3 − a4 − a5 R + R 0 c + Rd = 1 − a1 − a2 T 2 (1 + a1 − a2 ) Rc Cc Rd Cd = 2 (1 − a1 − a2 )
T (1 + a2 ) Rc Cc + Rd Cd = 1 − a1 − a2 T ( a5 − a3 ) R R C + R R C + R R C + R R C = c d d d c c 0 d d 0 c c 1 − a1 − a2
(5)
Linlin Fang et al. / Energy Procedia 158 (2019) 3008–3013 Author name / Energy Procedia 00 (2018) 000–000
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3. Online estimation of both SOC and SOH based on DEKF method 3.1. Relationship between SOH and ohmic resistance of the battery The equation (6) shows the definition of SOH of the battery cell based on the ohmic resistance. SOH =
(6)
Rold − Rnow Rold − Rnew
Where Rnew represents the ohmic resistance of the new battery at the factory, Rnow represents the current ohmic resistance of the battery and Rold represents the ohmic resistance when the battery capacity decays to 80%.
However, for the battery pack, the total ohmic internal resistance includes the ohmic internal resistance of the battery itself, as well as the additional internal resistance of the circuit such as the wire and the contact internal resistance. When calculating SOH, we ignore the change in internal resistance as battery capacity decreases. Then, the SOH of the battery pack can be described by the equation (7). (7) Rold − Rnow ( Rold + rold ) − ( R0 + re ) Rold − R0 = SOH
= = Rold − Rnew ( Rold + rold ) − ( Rnew + rnew ) Rold − Rnew
3.2. Online estimation of both SOC and SOH According to the equivalent circuit model of the battery, a discrete state space equation that reflects changes in the state variables such as battery SOC and voltage is established: SOC(k) = U R C (k ) U R C (k ) c
c
d
d
- T 0 0 1 SOC(k-1) C −T −T 0 0 exp( ) U R C ( k − 1) + Rc (1 − exp( )) i( k − 1) + w( k − 1) c c U (k − 1) −T R C −T 0 0 exp( ) (1 exp( )) − R d d d c
c
d
d
(8)
Where C is the current capacity of the battery; T is the sampling period; c ( c = Rc Cc ) and d ( d = Rd Cd ) are the time constants of the two RC loops respectively. i (k − 1) is the current of the sampling point at the time of k-1, which is positive when the battery is discharged and negative when charging the battery. w(k − 1) is process noise of the system. According to the change of the ohmic resistance of the battery, the discrete state space equation of the internal resistance can be described as: (9) R0 (k= ) R0 (k − 1) + r (k − 1) Where r (k − 1) represents the noise of ohmic resistance. And the output observation equation of the battery model is:
U (k )=U oc SOC (k ) − U RcCc (k ) − U Rd Cd (k ) − i (k ) R0 (k ) + v(k )
Where r (k − 1) represents the noise of output observation. The process of using the DEKF algorithm to estimate the SOC and SOH is as follows: Initialization parameters: E[ x(0)] = 0 T P0 E [ x(0) − 0 ][ x(0) − 0 ] = = E [ R (0)] 0 E [ R(0) − ][ R(0) − ]T = Q0 0 0
Update the state of system state x and the ohmic resistance R0 :
(
= Lk Qk Ck Ck Qk Ck + Vk ) −1 K = Pk− CkxT Ckx Qk− CkxT + Vk + k + − − [ yk − g ( xk , Rk , u k )] R = R + L + k k xk− K k [ yk − g ( xk− , Rk− , uk )] k xk =+ R + − − + Q= Qk − Lk Ck Qk P= Pk− − K k Ckx Pk− k k
(
)
Update the time of the ohmic resistance R0 and x :
−
RT
R
−
RT
−1
(10)
(11)
(12)
4
Linlin Fang et al. / Energy Procedia 158 (2019) 3008–3013 Author name / Energy Procedia 00 (2018) 000–000
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(13)
= Ak xk −1 + Bk uk xk Rk− = Rk+−1 − − + = P Ak Pk+−1 AkT + Wk = + Q Q R k k −1 k k −
+
Finally, based on the relationship between the SOH and ohmic resistance, we can calculate the SOH in real time: R −R R − R (k ) (14) SOH ( k ) = = old
now
old
Rold − Rnew
0
Rold − Rnew
3.3. Error model of SOC and SOH estimation In the process of DEKF algorithm, the priori estimation of the SOC is completed by Ampere-hour integral method. However, the posterior estimation of SOC, that is, the correction of SOC by Kalman filter is mainly through the feedback based on the relationship between SOC and OCV to correct the priori estimation. Thus, the relationship between SOC and OCV directly determines the estimation accuracy of the SOC through the feedback. Therefore, we establish the OCV-SOC mapping to correct the SOC estimation error. The polynomial shown in equation (15) is used to express the relationship between the OCV and the SOC of the battery. This model can enhance the correlation between the SOC and the OCV and improve the convergence speed of the Kalman filter algorithm. 2 3 4 5 (15) U oc ( z ) = 0 + 1 z + 2 z + 3 z + 4 z + 5 z Where z represents SOC and i ( i = 0,1, , 5 ) is the fitting coefficient used to fit the mapping relationship between the OCV and the SOC. When the capacity of the battery changes, the relationship between the OCV and the SOC also changes and the estimation of the battery SOC and SOH will be affected. According to the absolute error transfer formula: f = f (x , x , x ) (16) 1
2
f f = x 1
3
x1 +
f
x2 +
x2
f
x3
x3
Then, combining equations (1), (8) and (10), the absolute error of the SOC estimation can be expressed as: SOC = SOC1 + SOC 2 i 2 SOC1 = C SOC 2 = U = p1 U ( z ) + p2 ( U + U ) ( + z + z + z + z + U ( z ) = oc
oc
c
2
0
oc
1
2
(17)
d
3
4
3
4
5
z
5
)
Where SOC , U , U and U represent the absolute errors of SOC, U oc , U c and U d , respectively. c oc d
SOC 1
represents the
absolute error when the SOC is calculated by the Ampere-hour integral method. And SOC 2 represents the absolute error of feedback correction in SOC estimation. U ( z ) is the absolute error of U oc in SOC and OCV model; oc
p1 and p2 are the weight coefficients of the absolute error.
From equations (7), (10), and (16), the absolute error of SOH estimation can be expressed as: = SOH
Rold
= R0
Rold − Rnew
Rold Rold − Rnew
[ p1 U
oc
(z)
+ p2 ( U + U )] c
(18)
d
Where SOH and R represent the absolute errors of SOH and R0 , respectively. 0
Next, this paper considers the influence of the variation in the OCV-SOC curve, the battery capacity and the battery model parameters during the use of the battery on errors of SOC and SOH estimation. When only considering the influence of the OCV-SOC curve, the value of both SOC1 and p2 are 0, and the value of p1 is 1. Then, the absolute errors of the SOC and SOH estimation can be described as: SOC = SOC 2 = U oc ( z ) = ( 0 + 1 z + 2 z 2 + 3 z 3 + 4 z 4 + 5 z 5 ) Rold Rold U oc (= ( 0 + 1 z + 2 z 2 + 3 z 3 + 4 z 4 + 5 z 5 ) z) SOH = Rold − Rnew Rold − Rnew
(19)
When only considering the capacity change, the absolute errors of SOC and SOH estimation can be expressed as: i (20) = = SOC
SOC 1
2
C R SOH = old [ p1 U Rold − Rnew
oc
(z)
+ p2 ( U + U )] c
d
Linlin Fang et al. / Energy Procedia 158 (2019) 3008–3013 Author name / Energy Procedia 00 (2018) 000–000
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When only considering the influence of the model parameters, the weight coefficient p1 is 0 and p2 is 1. The absolute errors of SOC and SOH estimation can be expressed as: SOC 2 = ( U + U ) Rold ( U + U ) SOH = R − Rnew old c
(21)
d
c
d
To analyse the influence on the error of SOC and SOH estimation by the above three factors, this paper establishes four single-variable battery states, that is, updating online estimation, without updating curve, without updating capacity and offline estimation, as shown in Table 1. Table 1. Four single-variable battery states Variable name OCV-SOC curve Battery capacity online/offline
Updating online estimation updated updated online
Without updating curve non-updated updated online
Without updating capacity updated non-updated online
Offline updated updated online
4. Experimental results and error analysis
100
1
80
0.9
60
0.8
40
0.7 0.6
0
0.04 0.03 0.02 0.01
0.4
-40
0
0.3
-60 -80
0.05
0.5
-20
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time(sec)
Fig. 2 UDDS conditions
0.2 0
reference value updating online estimation without updating curve without updating capacity offline estimation
0.06
R0/Ω
20
0.08 0.07
reference value updating online estimation without updating curve without updating capacity offline estimation
SOC
Current(A)
A battery experimental platform is constructed in the environment of 20℃ to acquire the experimental data. As shown in Fig. 2, the UDDS (urban dynamometer driving schedule) current profile is used to verify the algorithm for online parameters identification, SOC and SOH estimation. And 150 accelerated life cycle experiments were performed on the battery to obtain parameters for different aging conditions. The initial capacity of the battery was 32.37 Ah and the capacity after the accelerated life cycle experiment was 31.56 Ah. OCV-SOC curves are measured before and after the battery accelerated life test, respectively. Besides, the estimation of the SOC and SOH can be performed by the parameters identified in the offline state and the online identified parameters, respectively.
-0.01 1000 2000
3000 4000 5000
6000 7000 8000
9000
-0.02 0
Time/sec
Fig. 3 Results of SOC estimation
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 4 Ohmic resistance estimation
According to the above method, the estimation results of SOC, ohmic resistance and SOH under four states can be obtained, as shown in Fig. 3, Fig. 4 and Fig. 5. And the maximum absolute errors of online estimation are shown in Table 2 and Table 3. The results show that the estimation of the SOC based on the DEKF algorithm is in good agreement with the reference curve and the ohmic internal resistance estimated by the DEKF algorithm is stable and can be used for SOH estimation. The SOC estimation has a maximum absolute error of 1.08% and an average of 0.29%. The maximum absolute error estimated by SOH is 1.54% and the average absolute error is 0.57%. The overall error of the estimation result is small and the estimation effect is good.
0.9
0.05 0.04
0.06
0.8
0.04
0.02
0.02
0.01
0.01
0
0 1000
2000
3000
4000 5000 Time/sec
6000
7000
8000
9000
Fig. 5 Results of SOH estimation
-0.01 0
0.05 0.04
0.03
0.05 0.03
0.85
0.06
updating online estimation without updating curve without updating capacity offline estimation
Er
0.07
updating online estimation without updating curve without updating capacity offline estimation
Ea
1
0.08
Esoc
SOH
0.09
reference value updating online estimation without updating curve without updating capacity offline estimation
0.95
0.75 0
0.06
0.1
1.1 1.05
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 6 Absolute error of SOC
-0.01 0
udating online estimation without updating curve without updating capacity offline estimation
0.03 0.02 0.01 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 7 Absolute error of SOH
-0.01 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 8 Relative error of SOH
Table 2. Absolute error of SOC estimation under four states Type of error Maximum error Average error
Updating online estimation 1.08% 0.29%
Without updating curve 4.72% 4.01%
Without updating capacity 4.11% 1.75%
Offline 3.13% 1.27%
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Linlin Fang et al. / Energy Procedia 158 (2019) 3008–3013 Author name / Energy Procedia 00 (2018) 000–000
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Table 3. Absolute error of SOH estimation under four states Type of error Maximum error Average error
Updating online estimation 1.54% 0.57%
Without updating curve 3.80% 3.13%
Without updating capacity 2.75% 1.72%
Offline 4.99% 3.01%
As shown in Fig. 6, Fig.7 and Fig. 8, when battery capacity decays, the estimation of both SOC and SOH produce large errors caused by the change of OCV-SOC curve and the capacity. And the change of OCV-SOC curve has the greatest impact on the estimation. When the battery parameters are estimated offline, the initial estimation error is small but the later error becomes larger and there is a gradual increase over time. Compared with the first two influencing factors, whether the battery capacity is updated or not has little effect on the error of estimation. 5. Conclusion This paper establishes the equivalent circuit model of the battery, realizes the online parameter identification of the battery model and finally realizes the joint estimation of the SOC and SOH by adopting the DEKF algorithm. The SOC estimation has a maximum absolute error of 1.08% and an average of 0.29%. The maximum absolute error estimated by SOH is 1.54% and the average absolute error is 0.57%. The errors are relatively small, verifying the reliability of this estimation method. In addition, by establishing error model of the SOC and SOH estimation, the impact of OCV-SOC curve, battery capacity and battery model parameters on estimation of SOC and SOH is analysed. Four single-variable battery states are selected for simulation experiments and the results show that the OCV-SOC curve has the greatest impact on the errors of SOC and SOH estimation among the three influencing factors. The error caused by offline parameter identification is small in the initial period but increases with time. The method used in this paper and the established error model of SOC and SOH estimation provide a reference for online joint estimation of SOC and SOH and its error analysis. In addition, when estimating the SOC and SOH of the battery, it is necessary to use online parameter identification methods to ensure the accuracy of the estimation. At the same time, when the battery capacity declines, the OCV-SOC curve of the battery and the battery capacity must be updated in time to ensure the reliability of the SOC and SOH estimation. Acknowledgements The authors would like to thank the Collaborative Innovation Centre of Electric Vehicles in Beijing Institute of Technology for the support of this research project. References [1] Lu L, Han X, Li J. A review on the key issues for lithium-ion battery management in electric vehicles. J Power Sources 2013; 226: 272-88. [2] Ng KS, Moo CS, Chen YP, et al. Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithium-ion batteries. Appl Energy 2009;86(9):1506-11. [3] Xing Y, He W, Pecht M, et al. State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures. Appl Energy 2014; 113: 106-15. [4] Ji Wu, Chenbin Zhang, Zonghai Chen. An online method for lithium-ion battery remaining useful life estimation using importance sampling and neural networks. Appl Energy 2016 173: 134-140. [5] J. Sturm, H. Ennifar, S. V. Erhard, A. Rheinfeld, et al. State estimation of lithium-ion cells using a physicochemical model based extended Kalman filter. Appl Energy 2018; 223: 103-123. [6] Min Ye, Hui Guo, Binggang Cao. A model-based adaptive state of charge estimator for a lithium-ion battery using an improved adaptive particle filter. Appl Energy 2017;190: 740-748. [7] Min Ye, Hui Guo, Rui Xiong, Quanqing Yu. A double-scale and adaptive particle filter-based online parameter and state of charge estimation method for lithium-ion batteries. Energy 2018; 144: 789-799. [8] Liu XT, Chen ZH, Zhang CB, et al. A novel temperature-compensated model for power Li-ion batteries with dual-particle-filter state of charge estimation. Appl Energy 2014; 123: 263-72. [9] REMMLINGER J, BUCHHOLZ M, MEILER M. State-of-health monitoring of lithium-ion batteries in electric vehicles by on-board internal resistance estimation [J]. Journal of Power Sources, 2011, 196: 5357-5363. [10] ANDRE D, APPEL C, GUTH T. Advanced mathematical methods of SOC and SOH estimation for lithium-ion batteries[J]. Journal of Power Sources, 2013, 224: 20-27.
ScienceDirect Available onlineatat www.sciencedirect.com Available www.sciencedirect.com Energyonline Procedia 00 (2018) 000–000 www.elsevier.com/locate/procedia
ScienceDirect ScienceDirect Energy Procedia 158 Energy Procedia 00(2019) (2017)3008–3013 000–000
www.elsevier.com/locate/procedia
Online Estimation and Error Analysis of both SOC and SOH of Lithium-ion Battery based on DEKF Method The 15th International Symposium onaDistrict Heating and Cooling a a Linlin Fang , Junqiu Li *, Bo Peng
Linlin Fang , Junqiu Li *, Bo Peng
Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Institute of Technology, Beijing, 100081,China. Assessing the feasibility of using the heat demand-outdoor temperature function for a long-term district heat demand forecast Abstract aa
a,b,c
a
a
b
c
c
The state-of-charge (SOC) and state-of-health(SOH) are two indexes ., in B. battery management, O. system I. Andrić *, A. Pina , P. Ferrão , J.critical Fournier Lacarrière Le(BMS) Correfor electric vehicles(EVs). To achieve accurate estimation of SOC and SOH, this paper establishes a battery equivalent circuit model and a IN+ Center Factor for Innovation, Technology and Policy Researchto- Instituto Superior Técnico, Av. Rovisco Paisparameters. 1, 1049-001And Lisbon, Portugal uses Forgetting Recursive Least Squares (FFRLS) realize online identification of model based on the b Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel,using 78520Double Limay, France relationship between the ohmic internal resistance and the SOH, a joint estimator extended Kalman filter(DEKF) c Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France algorithm is proposed for the estimation of both SOC and SOH. Then, an error model is established to analyze the influence of the battery OCV-SOC curve, battery capacity and battery parameters on the estimation of the SOC and SOH. The experiment results show that the maximum estimation error of SOC and SOH is 1.08% and 1.52% respectively, which have verified that accurate and robust SOC and SOH estimation results can be obtained by the proposed method. Besides, the OCV-SOC curve has Abstract the greatest influence on the estimation error of SOC and SOH among the three kinds of factors mentioned above. Copyright © 2018networks Elsevier Ltd. All rights reserved. heating are commonly the literature as one of the most effective solutions for decreasing the ©District 2019 The Authors. Published by responsibility Elsevier addressed Ltd. of thein scientific th International Conference on Applied Selection and peer-review under committee of investments the 10th greenhouse gas emissions from thethe building sector. These systems require high which are returned through the heat This is an open access article under CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Energy (ICAE2018). sales. Due to the changed climate conditions and building renovation policies, heat demand in the future could decrease, Peer-review under responsibility of the scientific committee of ICAE2018 – The 10th International Conference on Applied Energy. prolonging the investment return period. Keywords: Lithium-ion battery; Battery equivalent circuit model; Double extended Kalman filter; Error model of SOC and SOH estimation The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand forecast. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district 1.renovation Introduction scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were compared with results from a dynamic heat demand model, previously developed and validated by the authors. Lithium-ion batteries areonly featured highis considered, energy density, lowof self-discharge rate, long cycleapplications life and The results showed that when weather by change the margin error could be acceptable for some environmental friendliness thus have foundforwide applications in the area of electric power supply and (the error in annual demand and was lower than 20% all weather scenarios considered). However,vehicle after introducing renovation scenarios, the error value Battery increasedmanagement up to 59.5% (depending on the have weather anddesigned renovation energy storage systems. systems (BMS) been toscenarios provide combination monitoring, considered). diagnosis Thecontrol value of slope coefficient increased on average within the and functions to enhance the operation of batteries [1].range of 3.8% up to 8% per decade, that corresponds to the decrease the number of heating hours ofone 22-139h heating season on the combination of weather and In the in battery management systems, of theduring mostthe critical issues is (depending to accurately estimate the state-of-charge renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the (SOC) in real-time [2–8]. To obtain an accurate SOC and SOH estimation, many methods have been proposed in coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and literatures such as the current integral method [2], open-circuit voltage method [3], neural network method [4], improve the accuracy of heat demand estimations.
extended Kalman filter (EKF) method [5] and particle filter (PF) method [6–8]. Among the above methods, extended © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. * Corresponding author. Tel.: 086-010-68940589; fax: 086-010-68914842. E-mail address: [email protected]. Keywords: Heat demand; Forecast; Climate change
1876-6102 Copyright © 2018 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the 10thth International Conference on Applied Energy (ICAE2018). 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. 1876-6102 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of ICAE2018 – The 10th International Conference on Applied Energy. 10.1016/j.egypro.2019.01.974
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Kalman filtering is widely used for its simple theory and easy to implement. However, with the aging of the battery, the attenuation of capacity will lead to inaccurate estimation of SOC. In Reference [9], an equivalent circuit model is established to track the internal resistance of the battery and the internal resistance change of the aging battery is quantitatively studied, but the specific relationship for the battery internal resistance and SOH has not been determined. Reference [10] uses the Kalman filter algorithm to estimate the SOC and SOH, and the accuracy of SOC estimation is controlled within 1%, but the maximum error of the estimation of SOH is 20%. At present, the evaluation of battery SOH is mostly qualitative analysis. Besides, the effects of the change of OCV-SOC relationship curve on estimation of SOC and SOH during battery aging hasn’t been considered either. Based on the above problems, we establish the equivalent circuit model, use the Forgetting Factor Recursive Least Squares (FFRLS) method to realize the online identification of model parameters and use the DEKF algorithm to achieve online joint estimation of SOC and SOH. Then, the influence of battery OCV-SOC curve, battery capacity and battery model parameters on the estimation of SOC and SOH is analyzed by establishing the error model of SOC and SOH estimation. 2. Battery equivalent circuit model and Online Parameter Identification Dynamic voltage characteristics of lithium-ion battery show abrupt changes and gradual changes, which can be described using the equivalent circuit model. The model circuit and the meaning of each symbol are shown in Fig.1. And the state equation of this model is formula (1). Id
UOC
Rd
Cd ＋ Ud －
Ic
Rc
Cc ＋ UC －
R0 IL
＋ UL －
Fig. 1. battery equivalent circuit model Ud I • − + L U d = Cd Rd Cd Uc • I − + L U c = Cc Rc Cc U L = U oc − U d − U c − R0 I L
(1)
The battery model is transformed into a mathematical form that can be identified by a least-squares method and the Forgetting Factor Recursive Least Squares (FFRLS) algorithm is used to realize online parameter identification. Adopting Laplace transform for equation (1), the equation (2) can be obtained by making U= s U t ( s ) − U oc ( s ) rc ( )
and using bilinear transformation.
z 2U rc ( z ) = a1 zU rc ( z ) + a2U rc ( z ) + a3 z 2 I L ( z ) + a4 zI L ( z ) + a5 I L ( z )
(2)
U t ,k = (1 − a1 − a2 )U oc,k + a1U t ,k −1 + a2U t ,k −2 + a3 I L,k + a4 I L,k −1 + a5 I L,k −2
(3)
Then, we can finally obtain the equation (3) by inverse Z-transformation for equation (2).
And we can get the output matrix yk , Parameter matrix and data matrix of the mathematical model, which are shown in the equation (4). Then, the parameters of the battery model can be solved using FFRLS algorithm, as shown in equation (5). (4) y = U = k
t ,k
T k
k
T = (1 − a1 − a2 ) U oc ,k a1 a2 a3 a4 a5 T =1 U L ,k −1 U L ,k −1 iL ,k iL ,k −1 iL ,k −2
− a3 + a4 − a5 R0 = 1 + a1 − a2 − a3 − a4 − a5 R + R 0 c + Rd = 1 − a1 − a2 T 2 (1 + a1 − a2 ) Rc Cc Rd Cd = 2 (1 − a1 − a2 )
T (1 + a2 ) Rc Cc + Rd Cd = 1 − a1 − a2 T ( a5 − a3 ) R R C + R R C + R R C + R R C = c d d d c c 0 d d 0 c c 1 − a1 − a2
(5)
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3. Online estimation of both SOC and SOH based on DEKF method 3.1. Relationship between SOH and ohmic resistance of the battery The equation (6) shows the definition of SOH of the battery cell based on the ohmic resistance. SOH =
(6)
Rold − Rnow Rold − Rnew
Where Rnew represents the ohmic resistance of the new battery at the factory, Rnow represents the current ohmic resistance of the battery and Rold represents the ohmic resistance when the battery capacity decays to 80%.
However, for the battery pack, the total ohmic internal resistance includes the ohmic internal resistance of the battery itself, as well as the additional internal resistance of the circuit such as the wire and the contact internal resistance. When calculating SOH, we ignore the change in internal resistance as battery capacity decreases. Then, the SOH of the battery pack can be described by the equation (7). (7) Rold − Rnow ( Rold + rold ) − ( R0 + re ) Rold − R0 = SOH
= = Rold − Rnew ( Rold + rold ) − ( Rnew + rnew ) Rold − Rnew
3.2. Online estimation of both SOC and SOH According to the equivalent circuit model of the battery, a discrete state space equation that reflects changes in the state variables such as battery SOC and voltage is established: SOC(k) = U R C (k ) U R C (k ) c
c
d
d
- T 0 0 1 SOC(k-1) C −T −T 0 0 exp( ) U R C ( k − 1) + Rc (1 − exp( )) i( k − 1) + w( k − 1) c c U (k − 1) −T R C −T 0 0 exp( ) (1 exp( )) − R d d d c
c
d
d
(8)
Where C is the current capacity of the battery; T is the sampling period; c ( c = Rc Cc ) and d ( d = Rd Cd ) are the time constants of the two RC loops respectively. i (k − 1) is the current of the sampling point at the time of k-1, which is positive when the battery is discharged and negative when charging the battery. w(k − 1) is process noise of the system. According to the change of the ohmic resistance of the battery, the discrete state space equation of the internal resistance can be described as: (9) R0 (k= ) R0 (k − 1) + r (k − 1) Where r (k − 1) represents the noise of ohmic resistance. And the output observation equation of the battery model is:
U (k )=U oc SOC (k ) − U RcCc (k ) − U Rd Cd (k ) − i (k ) R0 (k ) + v(k )
Where r (k − 1) represents the noise of output observation. The process of using the DEKF algorithm to estimate the SOC and SOH is as follows: Initialization parameters: E[ x(0)] = 0 T P0 E [ x(0) − 0 ][ x(0) − 0 ] = = E [ R (0)] 0 E [ R(0) − ][ R(0) − ]T = Q0 0 0
Update the state of system state x and the ohmic resistance R0 :
(
= Lk Qk Ck Ck Qk Ck + Vk ) −1 K = Pk− CkxT Ckx Qk− CkxT + Vk + k + − − [ yk − g ( xk , Rk , u k )] R = R + L + k k xk− K k [ yk − g ( xk− , Rk− , uk )] k xk =+ R + − − + Q= Qk − Lk Ck Qk P= Pk− − K k Ckx Pk− k k
(
)
Update the time of the ohmic resistance R0 and x :
−
RT
R
−
RT
−1
(10)
(11)
(12)
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(13)
= Ak xk −1 + Bk uk xk Rk− = Rk+−1 − − + = P Ak Pk+−1 AkT + Wk = + Q Q R k k −1 k k −
+
Finally, based on the relationship between the SOH and ohmic resistance, we can calculate the SOH in real time: R −R R − R (k ) (14) SOH ( k ) = = old
now
old
Rold − Rnew
0
Rold − Rnew
3.3. Error model of SOC and SOH estimation In the process of DEKF algorithm, the priori estimation of the SOC is completed by Ampere-hour integral method. However, the posterior estimation of SOC, that is, the correction of SOC by Kalman filter is mainly through the feedback based on the relationship between SOC and OCV to correct the priori estimation. Thus, the relationship between SOC and OCV directly determines the estimation accuracy of the SOC through the feedback. Therefore, we establish the OCV-SOC mapping to correct the SOC estimation error. The polynomial shown in equation (15) is used to express the relationship between the OCV and the SOC of the battery. This model can enhance the correlation between the SOC and the OCV and improve the convergence speed of the Kalman filter algorithm. 2 3 4 5 (15) U oc ( z ) = 0 + 1 z + 2 z + 3 z + 4 z + 5 z Where z represents SOC and i ( i = 0,1, , 5 ) is the fitting coefficient used to fit the mapping relationship between the OCV and the SOC. When the capacity of the battery changes, the relationship between the OCV and the SOC also changes and the estimation of the battery SOC and SOH will be affected. According to the absolute error transfer formula: f = f (x , x , x ) (16) 1
2
f f = x 1
3
x1 +
f
x2 +
x2
f
x3
x3
Then, combining equations (1), (8) and (10), the absolute error of the SOC estimation can be expressed as: SOC = SOC1 + SOC 2 i 2 SOC1 = C SOC 2 = U = p1 U ( z ) + p2 ( U + U ) ( + z + z + z + z + U ( z ) = oc
oc
c
2
0
oc
1
2
(17)
d
3
4
3
4
5
z
5
)
Where SOC , U , U and U represent the absolute errors of SOC, U oc , U c and U d , respectively. c oc d
SOC 1
represents the
absolute error when the SOC is calculated by the Ampere-hour integral method. And SOC 2 represents the absolute error of feedback correction in SOC estimation. U ( z ) is the absolute error of U oc in SOC and OCV model; oc
p1 and p2 are the weight coefficients of the absolute error.
From equations (7), (10), and (16), the absolute error of SOH estimation can be expressed as: = SOH
Rold
= R0
Rold − Rnew
Rold Rold − Rnew
[ p1 U
oc
(z)
+ p2 ( U + U )] c
(18)
d
Where SOH and R represent the absolute errors of SOH and R0 , respectively. 0
Next, this paper considers the influence of the variation in the OCV-SOC curve, the battery capacity and the battery model parameters during the use of the battery on errors of SOC and SOH estimation. When only considering the influence of the OCV-SOC curve, the value of both SOC1 and p2 are 0, and the value of p1 is 1. Then, the absolute errors of the SOC and SOH estimation can be described as: SOC = SOC 2 = U oc ( z ) = ( 0 + 1 z + 2 z 2 + 3 z 3 + 4 z 4 + 5 z 5 ) Rold Rold U oc (= ( 0 + 1 z + 2 z 2 + 3 z 3 + 4 z 4 + 5 z 5 ) z) SOH = Rold − Rnew Rold − Rnew
(19)
When only considering the capacity change, the absolute errors of SOC and SOH estimation can be expressed as: i (20) = = SOC
SOC 1
2
C R SOH = old [ p1 U Rold − Rnew
oc
(z)
+ p2 ( U + U )] c
d
Linlin Fang et al. / Energy Procedia 158 (2019) 3008–3013 Author name / Energy Procedia 00 (2018) 000–000
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When only considering the influence of the model parameters, the weight coefficient p1 is 0 and p2 is 1. The absolute errors of SOC and SOH estimation can be expressed as: SOC 2 = ( U + U ) Rold ( U + U ) SOH = R − Rnew old c
(21)
d
c
d
To analyse the influence on the error of SOC and SOH estimation by the above three factors, this paper establishes four single-variable battery states, that is, updating online estimation, without updating curve, without updating capacity and offline estimation, as shown in Table 1. Table 1. Four single-variable battery states Variable name OCV-SOC curve Battery capacity online/offline
Updating online estimation updated updated online
Without updating curve non-updated updated online
Without updating capacity updated non-updated online
Offline updated updated online
4. Experimental results and error analysis
100
1
80
0.9
60
0.8
40
0.7 0.6
0
0.04 0.03 0.02 0.01
0.4
-40
0
0.3
-60 -80
0.05
0.5
-20
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time(sec)
Fig. 2 UDDS conditions
0.2 0
reference value updating online estimation without updating curve without updating capacity offline estimation
0.06
R0/Ω
20
0.08 0.07
reference value updating online estimation without updating curve without updating capacity offline estimation
SOC
Current(A)
A battery experimental platform is constructed in the environment of 20℃ to acquire the experimental data. As shown in Fig. 2, the UDDS (urban dynamometer driving schedule) current profile is used to verify the algorithm for online parameters identification, SOC and SOH estimation. And 150 accelerated life cycle experiments were performed on the battery to obtain parameters for different aging conditions. The initial capacity of the battery was 32.37 Ah and the capacity after the accelerated life cycle experiment was 31.56 Ah. OCV-SOC curves are measured before and after the battery accelerated life test, respectively. Besides, the estimation of the SOC and SOH can be performed by the parameters identified in the offline state and the online identified parameters, respectively.
-0.01 1000 2000
3000 4000 5000
6000 7000 8000
9000
-0.02 0
Time/sec
Fig. 3 Results of SOC estimation
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 4 Ohmic resistance estimation
According to the above method, the estimation results of SOC, ohmic resistance and SOH under four states can be obtained, as shown in Fig. 3, Fig. 4 and Fig. 5. And the maximum absolute errors of online estimation are shown in Table 2 and Table 3. The results show that the estimation of the SOC based on the DEKF algorithm is in good agreement with the reference curve and the ohmic internal resistance estimated by the DEKF algorithm is stable and can be used for SOH estimation. The SOC estimation has a maximum absolute error of 1.08% and an average of 0.29%. The maximum absolute error estimated by SOH is 1.54% and the average absolute error is 0.57%. The overall error of the estimation result is small and the estimation effect is good.
0.9
0.05 0.04
0.06
0.8
0.04
0.02
0.02
0.01
0.01
0
0 1000
2000
3000
4000 5000 Time/sec
6000
7000
8000
9000
Fig. 5 Results of SOH estimation
-0.01 0
0.05 0.04
0.03
0.05 0.03
0.85
0.06
updating online estimation without updating curve without updating capacity offline estimation
Er
0.07
updating online estimation without updating curve without updating capacity offline estimation
Ea
1
0.08
Esoc
SOH
0.09
reference value updating online estimation without updating curve without updating capacity offline estimation
0.95
0.75 0
0.06
0.1
1.1 1.05
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 6 Absolute error of SOC
-0.01 0
udating online estimation without updating curve without updating capacity offline estimation
0.03 0.02 0.01 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 7 Absolute error of SOH
-0.01 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 Time/sec
Fig. 8 Relative error of SOH
Table 2. Absolute error of SOC estimation under four states Type of error Maximum error Average error
Updating online estimation 1.08% 0.29%
Without updating curve 4.72% 4.01%
Without updating capacity 4.11% 1.75%
Offline 3.13% 1.27%
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Table 3. Absolute error of SOH estimation under four states Type of error Maximum error Average error
Updating online estimation 1.54% 0.57%
Without updating curve 3.80% 3.13%
Without updating capacity 2.75% 1.72%
Offline 4.99% 3.01%
As shown in Fig. 6, Fig.7 and Fig. 8, when battery capacity decays, the estimation of both SOC and SOH produce large errors caused by the change of OCV-SOC curve and the capacity. And the change of OCV-SOC curve has the greatest impact on the estimation. When the battery parameters are estimated offline, the initial estimation error is small but the later error becomes larger and there is a gradual increase over time. Compared with the first two influencing factors, whether the battery capacity is updated or not has little effect on the error of estimation. 5. Conclusion This paper establishes the equivalent circuit model of the battery, realizes the online parameter identification of the battery model and finally realizes the joint estimation of the SOC and SOH by adopting the DEKF algorithm. The SOC estimation has a maximum absolute error of 1.08% and an average of 0.29%. The maximum absolute error estimated by SOH is 1.54% and the average absolute error is 0.57%. The errors are relatively small, verifying the reliability of this estimation method. In addition, by establishing error model of the SOC and SOH estimation, the impact of OCV-SOC curve, battery capacity and battery model parameters on estimation of SOC and SOH is analysed. Four single-variable battery states are selected for simulation experiments and the results show that the OCV-SOC curve has the greatest impact on the errors of SOC and SOH estimation among the three influencing factors. The error caused by offline parameter identification is small in the initial period but increases with time. The method used in this paper and the established error model of SOC and SOH estimation provide a reference for online joint estimation of SOC and SOH and its error analysis. In addition, when estimating the SOC and SOH of the battery, it is necessary to use online parameter identification methods to ensure the accuracy of the estimation. At the same time, when the battery capacity declines, the OCV-SOC curve of the battery and the battery capacity must be updated in time to ensure the reliability of the SOC and SOH estimation. Acknowledgements The authors would like to thank the Collaborative Innovation Centre of Electric Vehicles in Beijing Institute of Technology for the support of this research project. References [1] Lu L, Han X, Li J. A review on the key issues for lithium-ion battery management in electric vehicles. J Power Sources 2013; 226: 272-88. [2] Ng KS, Moo CS, Chen YP, et al. Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithium-ion batteries. Appl Energy 2009;86(9):1506-11. [3] Xing Y, He W, Pecht M, et al. State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures. Appl Energy 2014; 113: 106-15. [4] Ji Wu, Chenbin Zhang, Zonghai Chen. An online method for lithium-ion battery remaining useful life estimation using importance sampling and neural networks. Appl Energy 2016 173: 134-140. [5] J. Sturm, H. Ennifar, S. V. Erhard, A. Rheinfeld, et al. State estimation of lithium-ion cells using a physicochemical model based extended Kalman filter. Appl Energy 2018; 223: 103-123. [6] Min Ye, Hui Guo, Binggang Cao. A model-based adaptive state of charge estimator for a lithium-ion battery using an improved adaptive particle filter. Appl Energy 2017;190: 740-748. [7] Min Ye, Hui Guo, Rui Xiong, Quanqing Yu. A double-scale and adaptive particle filter-based online parameter and state of charge estimation method for lithium-ion batteries. Energy 2018; 144: 789-799. [8] Liu XT, Chen ZH, Zhang CB, et al. A novel temperature-compensated model for power Li-ion batteries with dual-particle-filter state of charge estimation. Appl Energy 2014; 123: 263-72. [9] REMMLINGER J, BUCHHOLZ M, MEILER M. State-of-health monitoring of lithium-ion batteries in electric vehicles by on-board internal resistance estimation [J]. Journal of Power Sources, 2011, 196: 5357-5363. [10] ANDRE D, APPEL C, GUTH T. Advanced mathematical methods of SOC and SOH estimation for lithium-ion batteries[J]. Journal of Power Sources, 2013, 224: 20-27.