Journal of Power Sources 293 (2015) 416e428
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
On-line optimization of battery open circuit voltage for improved state-of-charge and state-of-health estimation Shijie Tong, Matthew P. Klein, Jae Wan Park* Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA
h i g h l i g h t s SoC and SoH estimation achieved by extended Kalman filter and recursive least squares. Novel SoHC-SoCV function and OCV-SoCV function for battery degradation. Parameter varying approach for optimization of battery OCV curve.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 October 2014 Received in revised form 1 March 2015 Accepted 25 March 2015 Available online 27 March 2015
A battery management system (BMS) ensures performance, safety and longevity of a battery energy storage system in an embedded environment. One important task for a BMS is to estimate the state of charge (SoC) and state of health (SoH) of a battery. The correlation between battery open circuit voltage (OCV) and SoC is an important reference for state estimation. The OCV-SoC correlation changes with respect to battery degradation. To improve the accuracy of state estimation, it is important to have the OCV-SoC correlation updated periodically. This work presents a solution by proposing a novel SoH(SoC) correlation as part of the battery equivalent circuit model (ECM). On-line optimization of SoH(SoC) correlation implicitly optimizes the OCV(SoC) correlation, as well as the capacity of a battery. An associated state and parameter dual estimator is proposed incorporating an Extended Kalman Filter (EKF) as a state observer, Recursive Least Square (RLS) algorithm as an internal resistance identifier, and Parameter Varying Approach as the SoH(SoC) correlation identifier. Battery experiment and simulation results validate that updating the SoH eSoC correlation effectively tracks battery SoH on-line. Furthermore, it implicitly updates OCV(SoC) function, further improving SoC estimation accuracy by 0.5%~3%. © 2015 Published by Elsevier B.V.
Keywords: Lithium ion battery Equivalent circuit model SoC SoH OCV Kalman filter
1. Introduction Lithium-ion battery systems are a promising energy storage solution for plug-in hybrid electric vehicles (PHEVs) and plug-in electric vehicles (PEVs) due to their advantages in energy and power density, cycle durability, and low environmental impact. Continued increases in performance and reducing cost are improving the appeal for large-scale stationary energy storage applications, such as smart grid and off-grid renewable energy storage. Varieties of applications require close monitoring of SoC and SoH, two states that indicate the level of charge and degradation of the battery. A SoC and SoH estimator takes battery current
* Corresponding author. E-mail address:
[email protected] (J.W. Park). http://dx.doi.org/10.1016/j.jpowsour.2015.03.157 0378-7753/© 2015 Published by Elsevier B.V.
and temperature measurement as input, voltage measurements as output, and refers to a model based estimator to evaluate internal states. Also, the estimator optimizes the parameters, e.g. internal resistance and capacity. Choices of battery models for state estimator design include: zero order single resistance model, ordinary differential equation (ODE) based equivalent circuit model (ECM), or partial differential equation (PDE) based electrochemical model. Many commercial automotive applications reported by licensed patents use a zero order model or simple ECM based model for SoC and SoH estimator design [1e4]. Higher order ECM based model features a set of ordinary differential equations with extra states describing battery charge transfer dynamics [5e20]. They are gaining popularity in estimator design as it achieves good simulation accuracy and to some extent are physically justifiable providing limited insights into the electrochemical reactions [20e28]. A comprehensive review of their applications for different
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battery chemistries is provided by Hu et al. [29]. Many ECM based estimator designs with comprehensive model parameters were reported [6,11,30e32]. Battery electrochemical model based estimator designs take a reference model based on partial differential equations (PDEs). It provides the most physical insight of a battery system [33,34]. PDE based estimator designs were reported to handle extreme battery management tasks with intensive computational cost, and great effort for system identification [34,35]. Overall, an estimator's computational cost increases as model complexity increases, with the benefit being increased physical insight. Among the choices of battery estimator designs, the ECM based battery estimators have the best trade-off between estimation accuracy, and computational efficiency. They are conveniently applicable in an embedded environment, and appeal to a wide range of battery management applications. Battery model parameters such as internal resistance, capacity, and the OCV-SoC correlation slowly vary over the course of degradation. As a result, estimator design often combines a state observer [9,11,13,20,36] and a parameter identifier [14,20,37,38]. Plett et al. [20] proposed a state and parameter dual estimator using an Extended Kalman filter (EKF). Parameters included in the optimization were OCV, internal resistance, coulombic efficiency, and capacity. He et al. [31] proposed an EKF combined with online recursive least square (RLS) for dual estimation. The purpose of the RLS algorithm is to estimate the OCV value and establish an OCV-SoC correlation function. Using a second order ECM as a reference, the approach achieved 0 such that the nonlinear function 4ð; ; ; Þ and cð; ; ; Þ are bounded via
þ 2 x k ; uk ; m2;k k4 xk b x k 4 xk ; b
(54)
2 x k ; uk ; m2;k kc xk b x k c xk ; b
(55)
þ
þ2
2
for x; b xk ; b x k εℝp ; uεℝq with xk b x k < ε4 and xk b x k < ε4 , respectively. Appendix II. Battery parameters optimizer convergence analysis Adaptive estimator design optimizes the model parameters and states trying to match the actual system input and output. In the case of battery, there is no guarantee that the SoC and SoH estimation converges to values with physical sense, and it is very important that the estimator be enforced to do so [20]. The function of OCV-SoCV are established for a battery to be yield for life time, so in a long-term dynamic environment, as long as battery are operated under certain voltage range, estimation of SoCV converges as battery OCV being filtered out as dc gain:
d ¼ OCV 1 U c c U U I R SoC V L S BATT BATT o
(56)
The convergence of states [Us, UL] and parameters [RS, RL, RO] are inherently insured using filter with zero DC gain (battery transient dynamics). After filtered out the zero input DC gain as OCV, the DC gain under current load is resolved as IBATTRO, the fluctuation due to the dynamic current load is resolved as US and UL. Note that the time constants tS and tL are fixed to be 1s and 30s in the estimator design section of this paper. It might be an unrealistic assumption as battery dynamics tend to have increasing relaxing time at high and low SoC, but they are helpful constrains to ensure convergence of the RS, RL optimization value at two fixed frequency response.
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Symbols UBATT: battery voltage, V IBATT: battery current, A RS/L: equivalent resistor, U tS/L: equivalent time constant, sec RO: Ohmic resistance, U SoCV: OCV based state-of-charge SoCC: Ah based state-of-charge SoHC: Ah based state-of-health CCAP: battery capacity f: ECM state update function g: ECM output update function x: ECM state variables y: ECM output variables u: ECM input variables q: ECM parameters A,C: ECM A/C matrix w,v: EKF process/measurement errors L: EKF gain 4: RLS measurement update ε: RLS cost function error z: RLS cost function a,b: RLS forgetting factors P: RLS gain g: PVA updating gain Abbreviations BMS: Battery Management System DoD: Depth of Discharge EES: Electricity Energy Storage EKF: Extended Kalman Filter EV: Electric Vehicle ICE: Internal Combustion Engine OCV: Open Circuit Voltage PHEV: Plug-in Hybrid EV PVA: Parameter Varying Approach RLS: Recursive Least Squares SoC: State of Charge SoH: State of Health ECM: Equivalent circuit model ODE: Ordinary Differential Equation PED: Partial Differential Equation Subscripts BATT: battery C: Coulomb counting based V: OCV observation based j,k: iterration index n: battery SoHC: function index
