Nonlinear Double-Capacitor Model for Rechargeable Batteries: Modeling, Identification and Validation

1 Nonlinear Double-Capacitor Model for Rechar geable Batteries: Modeling, Identification and V alidation Ning T ian, Student Member , IEEE , Huazhen Fang, Member , IEEE , Jian Chen, Senior Member , IEEE , and Y ebin W ang, Senior Member , IEEE Abstract —This paper proposes a new equivalent circuit model for r echargeable batteries by modifying a double-capacitor model proposed in the literature. It is known that the original model can address the rate capacity effect and energy recovery effect inherent to batteries better than other models. Howev er , it is a purely linear model and includes no repr esentation of a battery’ s nonlinear phenomena. Hence, this work transforms the original model by introducing a nonlinear-mapping-based voltage source and a serial RC circuit. The modification is justified by an analogy with the single-particle model. T wo offline parameter estimation approaches, termed 1.0 and 2.0, are designed f or the new model to deal with the scenarios of constant-current and variable-current charging/discharging, r espectively . In particular , the 2.0 appr oach proposes the notion of Wiener system identification based on the maximum a posteriori estimation, which allows all the parameters to be estimated in one shot while ov ercoming the noncon vexity or local minima issue to obtain physically reasonable estimates. An extensive experimental ev aluation shows that the proposed model offers excellent accuracy and predicti ve capability . A comparison against the Rint and Thevenin models further points to its superiority . With high fidelity and low mathematical complexity , this model is beneficial for various real-time battery management applications. Index T erms —Batteries, equivalent cir cuit model, nonlinear double-capacitor model, parameter identification, experimental validation. I . I N T RO D U C T I O N R ECHARGEABLE batteries have seen an ev er-increasing use in today’ s industry and society as power sources for systems of different scales, ranging from consumer electronic devices to electric vehicles and smart grid. This trend has motiv ated a growing body of research on advanced battery management algorithms, which are aimed to ensure the per- formance, safety and life of battery systems. Such algorithms generally require mathematical models that can well charac- terize a battery’ s dynamics. This has stimulated significant This work was supported in part by the National Science Foundation under A wards CMMI-1763093 and CMMI-1847651. Ning T ian and Huazhen Fang are with the Department of Mechanical Engineering, University of Kansas, Lawrence, KS 66045, USA (e-mail: ning.tian@ku.edu, fang@ku.edu). Jian Chen is with the State Ke y Laboratory of Industrial Control T ech- nology , College of Control Science and Engineering, Zhejiang University , Hangzhou 310027, China, and also with the Ningbo Research Institute, Zhejiang University , Ningbo 315100, China (e-mail: jchen@zju.edu.cn). Y ebin W ang is with the Mitsubishi Electric Research Laboratories, Cam- bridge, MA 02139, USA (e-mail:l yebinwang@ieee.org). attention in battery modeling during the past years, with the current literature offering a plethora of results. There are two main types of battery models: 1) electrochem- ical models that b uild on electrochemical principles to describe the electrochemical reactions and physical phenomena inside a battery during charging/discharging, and 2) equiv alent circuit models (ECMs) that replicate a battery’ s current-voltage char- acteristics using electrical circuits made of resistors, capacitors and voltage sources. With structural simplicity , the latter ones provide great computational ef ficiency , thus more suitable for real-time battery management. Howe ver , as the other side of the coin, the simple circuit-based structures also imply a difficulty to capture a battery’ s dynamic behavior at a high accuracy . Therefore, this article aims to dev elop a new ECM that offers not only structural parsimony but also high fidelity , through transforming an e xisting model in [1]. The work will systematically inv estigate the model construction, parameter identification, and experimental v alidation. A. Literatur e Re view 1) Review of Battery Modeling: As mentioned abov e, the electrochemical models and ECMs constitute the majority of the battery models av ailable today . The electrochemical mod- eling approach seeks to characterize the physical and chemical mechanisms underlying the charging/discharging processes. One of the best-known electrochemical models is the Doyle- Fuller-Ne wman model, which describes the concentrations and transport of lithium ions together with the distribution of sepa- rate potential in porous electrodes and electrolyte [2–4]. While delineating and reproducing a battery’ s behavior accurately , this model, like many others of similar kind, in volv es many partial differential equations and causes high computational costs. This has dri ven the development of some simplified versions, e.g., the single-particle model (SPM) [4, 5], and various model reduction methods, e.g., [6–8], tow ard more efficient computation. By contrast, the ECMs are generally considered as more competitiv e for real-time battery monitoring and control, having found their way into v arious battery management systems. The first ECM to our knowledge is the Randles model proposed in the 1940s [9]. It rev eals a lead-acid bat- tery’ s ohmic and reactive (capacitiv e and inductiv e) resistance, demonstrated in the electrochemical reactions and contributing 2 to various phenomena of voltage dynamics, e.g., voltage drop, recov ery and associated transients. This model has become a de facto standard for interpreting battery data obtained from electrochemical impedance spectroscopy (EIS) [10]. It also provides a basis for building di verse ECMs to grasp a bat- tery’ s voltage dynamics during charging/discharging. Adding a voltage source representing the open-circuit voltage (OCV) to the Randles model, one can obtain the popular Thevenin model [11–13]. The Thevenin model without the resistance- capacitance (RC) circuit is called as the Rint model, which includes an ideal voltage source with a series resistor [12]. If more than one RC circuit is added to the Thevenin model, it becomes the dual polarization (DP) model that is capable of capturing multi-time-scale v oltage transients during charg- ing/discharging [12]. The literature has also reported a few modifications of the Thev enin model to better characterize a battery’ s dynamics. Generally , they are based on two approaches. The first one aims to describe a battery’ s voltage more accurately by incor- porating certain phenomena, e.g., hysteresis, into the voltage dynamics, or through different parameterizations of OCV with respect to the state of charge (SoC) [14–20]. Some literature also models the resistors and capacitors as dependent on SoC, as well as some other factors like the temperature or rate and direction of the current loads in order to improv e the accuracy of battery voltage prediction [21, 22]. The second approach sets the focus on improving the runtime prediction for batteries. In [23], a battery’ s capacity change due to cycle and temperature is considered and parameterized, and the dependence of resistors and capacitors on SoC also charac- terized. A similar in vestigation is made in [24] to improve the Thevenin model, which proposes to capture the nonlinear change of a battery’ s capacity with respect to the current loads. An ECM that shows emerging importance is a double- capacitor model [1, 25]. It consists of two capacitors con- figured in parallel, which correspond to an electrode’ s bulk inner part and surface region, respectiv ely , and can describe the process of charge dif fusion and storage in a battery’ s electrode [26]. Compared to the Thev enin model, this circuit structure allows the rate capacity effect and charge recovery effect to be captured, making the model an attracti ve choice for charging control [26, 27]. Howe ver , based on a purely linear circuit, this model is unable to grasp nonlinear phenomena innate to a battery—for instance, the nonlinear SoC-OCV relation is beyond its descriptive capability—and thus has its applicability limited. The presented work is motiv ated to remov e this limitation by rev amping the model’ s structure. The effort will e ventually lead to a ne w ECM that, for the first time, can capture the charge diffusion within a battery’ s electrode and its nonlinear voltage behavior simultaneously . 2) Review of Battery Model Identification: A key problem associated with battery modeling is parameter identification, which pertains to extracting the unknown model parameters from the measurement data. Due to its importance, recent years hav e seen a growth of research. The existing methods can be divided into two main categories, experiment-based and data-based. The first category conducts experiments of charging, dischar ging or EIS and utilizes the experimental data to read a model’ s parameters. It is pointed out in [28, 29] that the transient voltage responses under constant- or pulse- current char ging/discharging can be lev eraged to estimate the resistance, capacitance and time constant parameters of the Thev enin model. In addition, the relation between SoC and OCV is a defining characteristic of a battery’ s dynamics. It can be experimentally identified by charging or discharging a battery using a very small current [30], or alternatively , using a current of normal magnitude but intermittently (with a sufficiently long rest period applied between two discharging operations) [31, 32]. The EIS experiments have also been widely used to identify a battery’ s impedance properties [33– 35]. While inv olving basic data analysis, the methods of this category generally put emphasis on the design of experiments. In a departure, the second category goes deeper into under- standing the model-data relationship and pursues data-driven parameter estimation. It can enable provably correct identi- fication ev en for complex models, thus often acknowledged as better at extracting the potential of data. It is proposed in [36] to identify the Thev enin model by solving a set of linear and polynomial equations. Another popular means is to formulate model-data fitting problems and solve them using least squares or other optimization methods to estimate the parameters [37–42]. When considering more complicated elec- trochemical models, the identification usually in volves large- size nonlinear noncon vex optimization problems. In this case, particle swarm optimization and genetic algorithms are often exploited to search for the best parameter estimates [3, 43–45]. A recent study presents an adaptive-observ er-based parameter estimation scheme for an electrochemical model [46]. While the abo ve works focus on identification of physics-based models, data-dri ven black-box identification is also examined in [47–49], which construct linear state-space models via subspace identification or nonparametric frequency domain analysis. A topic related with identification is experiment design, which is to find out the best input sequences to excite a battery to maximize the parameter identifiability . In [50, 51], optimal input design is performed by maximizing the Fisher information matrices—an identifiability metric— in volv ed in the identification of the Thev enin model and the SPM, respecti vely . The presented work is also related with the literature on W iener system identification, because the model to be de- veloped has a W iener-type structure featuring a linear dy- namic subsystem in cascade with a static nonlinear subsystem. W iener systems are an important subject in the field of parameter identification, and a reader is referred to [52] for a collection of recent studies. W iener system identification based on maximum likelihood (ML) estimation is inv estigated in [53, 54], which shows significant promises. Howe ver , the optimization procedure resulting from the ML formulation can easily conv erge to local minima due to the presence of the nonlinear subsystem. This hence yields a motiv ation to enhance the notion of ML-based identification in this work to achiev e more effecti ve battery parameter estimation. B. Statement of Contrib utions This work presents the follo wing contributions. 3 • A new ECM, named the nonlinear double-capacitor (NDC) model , is dev eloped. By design, it transforms the linear double-capacitor model in [1] by coupling it with a nonlinear circuit mimicking a battery’ s voltage behavior . W ith this piv otal change, the NDC model introduces two advantages ov er existing ECMs. First, it can simulate not only the charge diffusion characteristic of a battery’ s electrochemical dynamics, b ut also the critical nonlinear electrical phenomena. This unique feature guarantees the model’ s better accurac y , which comes at only a v ery slight increase in model complexity . Second, the NDC model can be interpreted as a circuit-based approximation of the SPM. This further justifies its soundness while inspiring a refreshed look at the connections between the SPM and ECMs. • Parameter identification is in vestigated for the proposed model. This begins with a study of the constant-current charging/dischar ging scenario, with an identification ap- proach, termed 1.0, de veloped by fitting parameters with the measurement data. Then, shifting the focus to the sce- nario of variable-current charging/dischar ging, the study introduces a Wiener perspectiv e into the identification of the NDC model due to its W iener-type structure. A W iener identification approach is proposed for the NDC model based on maximum a posteriori (MAP) estimation, which is termed 2.0. Compared to the ML-based coun- terparts in the literature, this new approach incorporates into the estimation a prior distribution of the unknown parameters, which represents additional information or prior knowledge and can help driv e the parameter search tow ard physically reasonable values. • Experimental validation is performed to assess the pro- posed results. This in volv es multiple experiments about battery discharging under different kinds of current pro- files and a comparison of the NDC model with the Rint and Thev enin models. The validation shows the considerable accuracy and predictiv e capability of the NDC model, as well as the effecti veness of the 1.0 and 2.0 identification approaches. C. Or ganization The remainder of the paper is organized as follows. Sec- tion II presents the construction of the NDC model. Section III studies parameter identification for the NDC model in the constant-current charging/discharging scenario. Inspired by W iener system identification, Section IV proceeds to dev elop an MAP-based parameter estimation approach to identify the NDC model. Section V offers the experimental validation. Fi- nally , Section VI gathers concluding remarks and suggestions for future research. I I . N D C M O D E L D E V E L O P M E N T This section develops the NDC model and presents the mathematical equations governing its dynamic behavior . T o begin with, let us revie w the original linear double- capacitor model proposed in [1]. As shown in Figure 1(a), this model includes two capacitors in parallel, C b and C s , 𝑅 0 𝑅 𝑠 𝑅 𝑏 𝐶 𝑠 𝐶 𝑏 𝐼 𝑉 (a)  0           = ℎ     1  1      (b) Figure 1: (a) The original double-capacitor model; (b) the proposed NDC model. each connected with a serial resistor , R b and R s , respectiv ely . The double-capacitor structure simulates a battery’ s electrode, providing storage for electric charge, and the parallel con- nection between them allows the transport of charge within the electrode to be described. Specifically , one can consider the R s - C s circuit as corresponding to the electrode surface region exposed to the electrolyte; the R b - C b circuit represents an analogy of the bulk inner part of the electrode. As such, this model has the following features: • C b  C s and R b  R s ; • C b is where the majority of the charge is stored, and R b - C b accounts for low-frequenc y responses during charg- ing/discharging; • C s is much smaller, and its v oltage changes at much f aster rates than that of C b during charging/discharging, making R s - C s responsible for high-frequency responses. In addition, R 0 is included to embody the electrolyte re- sistance. This model was designed in [1] for high-po wer lithium-ion batteries, and its application can naturally extend to double-layer capacitors that are widely used in hybrid energy storage systems, e.g., [55]. As pointed out in [26], the linear double-capacitor model can grasp the rate capacity effect, i.e., the total char ge absorbed (or released) by a battery goes down with the increase in charging (or dischar ging) current. T o see this, just notice that the terminal voltage V mainly depends on V s (the voltage across C s ), which changes faster than V b (the voltage across C b ). Thus, when the current I is large, the fast rise (or decline) of V s will make V hit the cut-of f threshold earlier than when C b has yet to be fully charged (or discharged). Another phenomenon that can be seized is the capacity and voltage recov ery effect. That is, the usable capacity and terminal voltage would increase upon the termination of discharging due to the migration of charge from C b to C s . Howe ver , this model by nature is a linear system, unable to describe a 4 defining characteristic of batteries—the nonlinear dependence of OCV on the SoC. It hence is effecti ve only when a battery is restricted to operate conservati vely within some truncated SoC range that permits a linear approximation of the SoC- OCV curve. T o overcome the above issue, the NDC model is proposed, which is shown in Figure 1(b). It includes two changes. The primary one is to introduce a voltage source U , which is a nonlinear mapping of V s , i.e., U = h ( V s ) . Second, an RC circuit, R 1 - C 1 , is added in series to U . Next, let us justify the above modifications from a perspectiv e of the SPM, a simplified electrochemical model that has recently attracted wide interest. Figure 2 gi ves a schematic diagram of the SPM. The SPM represents an electrode as a single spherical particle. It describes the mass balance and diffusion of lithium ions in a particle during charging/dischar ging by Fick’ s second law of dif fusion in a spherical coordinate system [5]. If subdividing a spherical particle into two finite volumes, the bulk inner domain (core) and the near -surface domain (shell), one can simplify the diffusion of lithium ions between them as the charge transport between the capacitors of the double- capacitor model, as proven in [26]. For SPM, the terminal voltage consists of three elements: the difference in the open- circuit potential of the positiv e and negativ e electrodes, the difference in the reaction overpotential, and the voltage across the film resistance [4]. The open-circuit potential depends on the lithium-ion concentration in the surface region of the sphere, which is akin to the role of V s here. Therefore, it is appropriate as well as necessary to introduce a nonlinear function of V s , i.e., h ( V s ) , as an analogy to the open-circuit potential. W ith U = h ( V s ) , the NDC model can correctly show the influence of the charge state on the terminal voltage, while inheriting all the capabilities of the original model. Furthermore, the NDC model also contains an RC circuit, R 1 - C 1 , which, together with R 0 , simulates the impedance- based part of the voltage dynamics. Here, R 0 characterizes the linear kinetic aspect of the impedance, which relates to the ohmic resistance and solid electrolyte interface (SEI) resistance [56]; R 1 - C 1 accounts for the voltage transients related with the charge transfer on the electrode/electrolyte interface and the ion mass diffusion in the battery [57]. This work finds that one RC circuit can offer sufficient fidelity , though it is possible to connect more RC circuits serially with R 1 - C 1 to gain better accuracy . The dynamics of the NDC model can be expressed in the state-space form as follows:            ˙ V b ( t ) ˙ V s ( t ) ˙ V 1 ( t )   = A   V b ( t ) V s ( t ) V 1 ( t )   + B I ( t ) , V ( t ) = h ( V s ( t )) − V 1 ( t ) + R 0 I ( t ) , (1a) (1b) where A =    − 1 C b ( R b + R s ) 1 C b ( R b + R s ) 0 1 C s ( R b + R s ) − 1 C s ( R b + R s ) 0 0 0 − 1 R 1 C 1    , B =    R s C b ( R b + R s ) R b C s ( R b + R s ) − 1 C 1    . In above, I > 0 for charging, I < 0 for discharging, and V 1 refers to the voltage across the R 1 - C 1 circuit. One can parameterize h ( V s ) as a polynomial. A fifth-order polynomial is empirically selected here: h ( V s ) = α 0 + α 1 V s + α 2 V 2 s + α 3 V 3 s + α 4 V 4 s + α 5 V 5 s , where α i for i = 0 , 1 , . . . , 5 are coefficients. Note that h ( V s ) should be lower and upper bounded, depending on a battery’ s operating voltage range. This implies that V b and V s must also be bounded. For any bounds selected for them, it is always possible to find out a set of coefficients α i ’ s to satisfy h ( · ) . Hence, one can straightforwardly normalize V b and V s to let them lie between 0 V and 1 V , without loss of generality . In other words, V b = V s = 1 V at full charge ( SoC = 1 ) and that V b = V s = 0 V for full depletion ( SoC = 0 ). Follo wing this setting, SoC is giv en by SoC = Q a Q t = C b V b + C s V s C b + C s , (2) where Q t = C b + C s denotes the total capacity , and Q a = C b V b + C s V s the av ailable capacity , respectiv ely . It is easy to verify that the SoC’ s dynamics is gov erned by ˙ SoC = h C b C b + C s C s C b + C s 0 i   ˙ V b ˙ V s ˙ V 1   = 1 Q t I . (3) Meanwhile, it is worth noting that the SoC-OCV function would share the same form with h ( · ) . T o see this point, recall that OCV refers to the terminal voltage when the battery is at equilibrium without current load. For the NDC model, the equilibrium happens when V b = V s , V 1 = 0 V and I = 0 A, and in this case, V s = SoC according to (2), and OCV = h ( V s ) . This suggests that OCV = h (SoC) . In addition, the internal resistance R 0 is also assumed to be SoC- dependent following the recommendation in [58], taking the form of R 0 = γ 1 + γ 2 e − γ 3 SoC + γ 4 e − γ 5 (1 − SoC) . (4) The rest of this paper will center on developing parameter identification approaches to determine the model parameters using measurement data and apply identified models to exper- imental datasets to ev aluate their predictive accuracy . I I I . P A R A M E T E R I D E N T I FI C A T I O N 1 . 0 : C O N S T A N T - C U R R E N T C H A R G I N G / D I S C H A R G I N G This section studies parameter identification for the NDC model when a constant current is applied to a battery . The discharging case is considered here without loss of generality . In a two-step procedure, the h ( · ) function is identified first, and the impedance and capacitance parameters estimated next. A. Identification of h ( · ) The SoC-OCV relation of the NDC model is given by OCV = h (SoC) , as aforementioned in Section II. Hence, one can identify h ( · ) by fitting it with a battery’ s SoC-OCV data. T o obtain the SoC-OCV curve, one can discharge a battery using a small current (e.g., 1/25 C-rate as suggested 5 Shell Core Li + Li + Li + Li + Negative Electrode Positive Electrode Separator e − e − Li + Discharge Current Collector Current Collector  0  1 Solid Particles Figure 2: The single-particle model (top), and a particle (bottom) subdi vided into two volumes, core and shell, which correspond to R b - C b and R s - C s , respecti vely . in [30]) from full to empty . In this process, the terminal voltage V can be taken as OCV . Immediately one can see that α 0 = V and P 5 i =0 α i = V , where V and V are the minimum and maximum value of V in the process. Therefore, OCV = h (SoC) can be written as a function of α i for i = 1 , 2 , . . . , 4 as follo ws: OCV = V + 4 X i =1 α i SoC i + V − V − 4 X i =1 α i ! SoC 5 , where OCV can be read directly from the terminal voltage measurements. By (3), SoC can be calculated using the coulomb counting method as follows: SoC = 1 + 1 Q t I t. From abov e, one can observe that α i for i = 1 , 2 , . . . , 4 can be identified by solving a data fitting problem, which can be addressed as a linear least squares problem. The identification results are unique and can be easily obtained. Then with α 0 = V and α 5 = V − V − P 4 i =1 α i , the function h ( · ) becomes explicit and ready for use. B. Identification of Impedance and Capacitance Now , consider discharging the battery by a constant cur- rent of normal magnitude to determine the impedance and capacitance parameters. The identification can be attained by expressing the terminal voltage in terms of the parameters and then fitting it to the measurement data. 1) T erminal V oltage Response Analysis: Consider a battery left idling for a long period of time, and then discharge it using a constant current. According to (1a), V s can be derived as V s ( t ) = V s (0) + I t C b + C s + C b ( R b C b − R s C s ) I ( C b + C s ) 2 ·  1 − exp  − C b + C s C b C s ( R b + R s ) t  , (5) where V s (0) is known to us as it can be accessed from SoC(0) when the battery is initially relaxed. Howe ver , it is impossible to identify C b , R b , C s and R s altogether . This issue can be seen from (5), where V s depends on three param- eters, i.e., 1 / ( C b + C s ) , C b ( R b C b − R s C s ) / ( C b + C s ) 2 and ( C b + C s ) / [ C b C s ( R b + R s )] . Even if the three parameters are known, it is still not possible to extract all the four individual impedance and capacitance parameters from them due to the parameter redundancy . Therefore, one can sensibly assume R s = 0 , as recommended in [42]. This is a tenable assumption for the NDC model since R s  R b as aforementioned. As a result, (5) reduces to V s ( t ) = V s (0) + β 1 I t + β 2 I  1 − e − β 3 t  , (6) where β 1 = 1 C b + C s , β 2 = R b C 2 b ( C b + C s ) 2 , β 3 = C b + C s C b C s R b . Here, β 1 is known because Q t has been calibrated by coulomb counting in Section III-A. When β 2 and β 3 are also av ailable, C b , C s and R b can be reconstructed as follows: C b = β 2 β 3 β 1 ( β 1 + β 2 β 3 ) , C s = 1 β 1 + β 2 β 3 , R b = 1 β 1 β 3 C b C s . Further , in the abov e constant-current discharging scenario, the ev olution of V 1 follows V 1 ( t ) = e − β 5 t V 1 (0) − I β 4  1 − e − β 5 t  , (7) where β 4 = R 1 , β 5 = 1 R 1 C 1 . Since the battery has idled for a long period prior to discharg- ing, V 1 (0) relaxes at zero and can be remov ed from (7). Then, combining (1b), (4), (6) and (7), the terminal voltage response is given by V ( θ ; t ) = 5 X i =0 α i V i s ( θ ; t ) + I θ 3  1 − e − θ 4 t  + I θ 5 + I θ 6 e − θ 7 SoC( t ) + I θ 8 e − θ 9 (1 − SoC( t )) . (8) with θ =  β 2 β 3 β 4 β 5 γ 1 γ 2 γ 3 γ 4 γ 5  > , and V s ( θ ; t ) = V s (0) + I t/Q t + θ 1 I  1 − e − θ 2 t  , SoC( t ) = SoC(0) + I t/Q t . 2) Data-F itting-Based Identification of θ : In above, the terminal voltage V is expressed in terms of θ , allowing one to identify θ by minimizing the difference between the measured voltage and the voltage predicted by (8). Hence, a data fitting problem similar to the one in Section III-A can be formulated. It should be noted that the resultant optimization will be nonlinear and noncon vex due to the presence of h ( · ) . As a consequence, a numerical algorithm may get stuck in local minima and ev entually giv e unreasonable estimates. A promising way of mitigating this challenge is to constrain the numerical optimization search within a parameter space that 6 is believ ably correct. Specifically , one can roughly determine the lower and upper bounds of part or all of the parameters, set up a limited search space, and run numerical optimization within this space. W ith this notion, the identification problem can be formulated as a constrained optimization problem: ˆ θ = arg min θ 1 2 [ y − V ( θ )] > Q − 1 [ y − V ( θ )] , s.t. θ ≤ θ ≤ θ , (9a) (9b) where ˆ θ is the estimate of θ , θ and θ are the pre-set lower and upper bounds of θ , respecti vely , y the terminal voltage measurement vector , Q an M × M symmetric positi ve definite matrix representing the covariance of the measurement noise, with M being the number of the data points. Besides, y =  y ( t 1 ) y ( t 2 ) · · · y ( t M )  > , V ( θ ) =  V ( θ ; t 1 ) V ( θ ; t 2 ) · · · V ( θ ; t M )  > . Multiple numerical algorithms are av ailable in the literature to solve (9), a choice among which is the interior-point-based trust-region method [39]. I V . P A R A M E T E R I D E N T I FI C A T I O N 2 . 0 : V A R I A B L E - C U R R E N T C H A R G I N G / D I S C H A R G I N G While it is not unusual to charge or discharge a battery at a constant current, real-world battery systems such as those in electric vehicles generally operate at v ariable currents. Motiv ated by practical utility , an interesting and challenging question is: W ill it be possible to estimate all the parameters of the NDC model in one shot when an almost arbitrary current profile is applied to a battery? Having this question addressed will greatly improve the av ailability of the model, even to an on-demand level, for battery management tasks. This section offers a study in this regard from a Wiener identification perspectiv e. It first un veils the NDC model’ s inherent W iener- type structure and then de velops an MAP-based identification approach. Here, the study assumes R 0 to be constant for con venience. A. W iener-T ype Strucutre of the NDC Model The NDC model is structurally similar to a W iener system— the double RC circuits constitute a linear dynamic subsystem, and cascaded with it is a nonlinear mapping. The follo wing outlines the discrete-time W iener-type formulation of (1). Suppose that (1a) is sampled with a time period ∆ T and then discretized by the zero-order-hold (ZOH) method. The discrete-time model is expressed as x ( t k +1 ) = A d x ( t k ) + B d I ( t k ) , (10) where k is the discrete-time inde x with t k = k ∆ T , and A d = e A ∆ T , B d = Z ∆ T 0 e Aτ d τ ! B . Let us use t instead of t k to represent the discrete time instant in sequel for notational simplicity . Then, (10) can be written as x ( t ) = ( q I 3 × 3 − A d ) − 1 B d I ( t ) + ( q I 3 × 3 − A d ) − 1 q x (0) , where q is the forward shift operator, and I 3 × 3 ∈ R 3 × 3 is an identity matrix, respectiv ely . Since V s ( t ) =  0 1 0  x ( t ) and V 1 ( t ) =  0 0 1  x ( t ) , one can obtain the follo wing after some lengthy deriv ation: V s ( t ) = G 1 ( q ) I ( t ) + G 2 ( q ) V s (0) , V 1 ( t ) = G 3 ( q ) I ( t ) + G 4 ( q ) V 1 (0) , (11) (12) where G 1 ( q ) = ( β 1 + β 2 ) q − 1 − ( β 1 β 3 + β 2 ) q − 2 1 − (1 + β 3 ) q − 1 + β 3 q − 2 , G 2 ( q ) = 1 1 − q − 1 , G 3 ( q ) = β 4 q − 1 1 + β 5 q − 1 , G 4 ( q ) = 1 1 + β 5 q − 1 , with β 1 = A 21 B 11 + A 12 B 21 A 12 + A 21 ∆ T , β 2 = A 21 ( B 21 − B 11 ) ( A 12 + A 21 ) 2 (1 − β 3 ) , β 3 = e − ( A 12 + A 21 )∆ T , β 4 = − ( β 5 + 1) B 31 / A 33 , β 5 = − e A 33 ∆ T . Note that the notation β is slightly abused above without causing confusion. Assume that the battery has been at rest for a sufficiently long time to achiev e an equilibrium state before a test. In this setting, V s (0) = SoC(0) , V 1 (0) = 0 V , and G 4 ( q ) V 1 (0) = 0 . Besides, one can also see that the same parameter redundancy issue as in Section III-B occurs again— only three parameters, β 1 through β 3 , appear in (11), but four physical parameters, C b , C s , R b and R s , need to be identified. T o fix this, let R s = 0 as was done before. Then β 1 through β 3 reduce to be β 1 = ∆ T C b + C s , β 2 = R b C 2 b (1 − β 3 ) ( C b + C s ) 2 , β 3 = e − C b + C s C b C s R b ∆ T . If β 1 through β 5 become av ailable, the physical parameters can be reconstructed as follows: C b = ∆ T β 1 − C s , C s = (1 − β 3 ) ∆ T β 1 − β 1 β 3 − β 2 log β 3 , R b = − (∆ T ) 2 C b C s β 1 log β 3 , R 1 = − β 4 β 5 + 1 , C 1 = − ∆ T log( − β 5 ) R 1 . Finally , it is obvious that V ( t ) = h [ G 1 ( q ) I ( t ) + G 2 ( q ) V s (0)] − G 3 ( q ) I ( t ) + R 0 I ( t ) . (13) The above equation reveals the block-oriented W iener-type structure of the NDC model, as depicted in Figure 3, in which the linear dynamic model G 1 ( q ) and the nonlinear function h ( V s ) are interconnected sequentially . Giv en (13), the next pursuit is to estimate all of the parameters simultaneously , which include α i for i = 1 , 2 , . . . , 4 , β i for i = 1 , 2 , . . . , 5 , and R 0 . Here, α 0 and α 5 are free of identification as they can be expressed by α i for i = 1 , 2 , . . . , 4 (see Section III-A). 7  1 (  )  (  )  (  ) Linear Dynamic Model  3 (  )  (  )  2 (  )   (0) ℎ     (  ) Nonlinear Function  0  (  ) Figure 3: The W iener-type structure of the nonlinear double-capacitor (NDC) model. B. MAP-Based W iener Identification Consider the following model based on (13) for notational con venience: z ( t ) = V ( θ ; u ( t )) + v ( t ) , (14) where u is the input current I , z the measured voltage, v the measurement noise added to V and assumed to follow a Gaussian distrib ution N (0 , σ 2 ) , and V ( θ ; u ( t )) = h [ G 1 ( q , θ ) u ( t ) + G 2 ( q ) V s (0) , θ ] − G 3 ( q , θ ) u ( t ) + θ 10 u ( t ) , with θ =  α 1 α 2 α 3 α 4 β 1 β 2 β 3 β 4 β 5 R 0  > . The input and output datasets are denoted as u =  u ( t 1 ) u ( t 2 ) · · · u ( t N )  > ∈ R N × 1 , z =  z ( t 1 ) z ( t 2 ) · · · z ( t N )  > ∈ R N × 1 , where N is the total number of data samples. A combination of them is expressed as Z =  u z  . An ML-based approach is developed in [53] to deal with W iener system identification. If applied to (14), it leads to consideration of the following problem: ˆ θ = arg max θ p ( Z | θ ) . Follo wing this line, one can deri ve a likelihood cost function and perform minimization to find out ˆ θ . Howe ver , this method can be vulnerable to the risk of local minima because of the nonconv exity issue resulting from the static nonlinear function h ( · ) . This can cause unphysical estimates. While carefully selecting an initial guess is suggested to alleviate this problem [59], it is often found inadequate for many practical systems. In particular, our study showed that it could hardly deliv er reliable parameter estimation when used to handle the NDC model identification. MAP-based W iener identification thus is proposed here to ov ercome this problem. The MAP estimation can incorpo- rate some prior knowledge about parameters to help drive the parameter search tow ard a reasonable minimum point. Specifically , consider maximizing the a posteriori probability distribution of θ conditioned on Z : ˆ θ = arg max θ p ( θ | Z ) . (15) By the Bayes’ theorem, it follo ws that p ( θ | Z ) = p ( Z | θ ) · p ( θ ) p ( Z ) ∝ p ( Z | θ ) · p ( θ ) . In above, p ( θ ) quantifies the prior information av ailable about θ . A general way is to characterize it as a Gaussian random vector following the distribution p ( θ ) ∼ N ( m , P ) . Based on (14), p ( z | θ ) ∼ N ( V ( θ ; u ) , R ) , where R = σ 2 I and V ( θ ; u ) =  V ( θ ; u ( t 1 )) · · · V ( θ ; u ( t N ))  > . Then, p ( Z | θ ) · p ( θ ) ∝ exp  − 1 2 [ z − V ( θ ; u )] > R − 1 [ z − V ( θ ; u )]  · exp  − 1 2 ( θ − m ) > P − 1 ( θ − m )  . If using the log-likelihood, the problem in (15) is equiv alent to ˆ θ = arg min θ J ( θ ) , (16) where J ( θ ) = 1 2 [ z − V ( θ ; u )] > R − 1 [ z − V ( θ ; u )] + 1 2 ( θ − m ) > P − 1 ( θ − m ) . For the nonlinear optimization problem in (16), one can exploit the quasi-Ne wton method to numerically solve it [53]. This method iterati vely updates the parameter estimate through θ k +1 = θ k + λ k s k . (17) 8 Here, λ k denotes the step size at iteration step k , and s k is the gradient-based search direction giv en by s k = − B k g k , (18) where B k ∈ R 10 × 10 is a positiv e definite matrix that approx- imates the Hessian matrix ∇ 2 J ( θ k ) , and g k = ∇ J ( θ k ) ∈ R 10 × 1 . Based on the well-kno wn BFGS update strategy [60], B k can be updated by B k =  I − δ k γ > k δ > k γ k  B k − 1  I − γ k δ > k δ > k γ k  + δ k δ > k δ > k γ k , (19) with δ k = θ k − θ k − 1 and γ k = g k − g k − 1 . In addition, g k = −  ∂ V ( θ k ; u ) ∂ θ k  > R − 1 [ z − V ( θ k ; u )] + P − 1 ( θ k − m ) , (20) where each column of ∂ V ( θ ; u ) ∂ θ ∈ R N × 10 is gi ven by ∂ V ( θ ; u ) ∂ θ i = x ◦ i − x ◦ 5 for i = 1 , 2 , . . . , 4 , ∂ V ( θ ; u ) ∂ θ 5 = Σ ◦ q − 1 − θ 7 q − 2 1 − (1 + θ 7 ) q − 1 + θ 7 q − 2 u , ∂ V ( θ ; u ) ∂ θ 6 = Σ ◦ q − 1 − q − 2 1 − (1 + θ 7 ) q − 1 + θ 7 q − 2 u , ∂ V ( θ ; u ) ∂ θ 7 = Σ ◦ θ 6 q − 2 − 2 θ 6 q − 3 + θ 6 q − 4 (1 − (1 + θ 7 ) q − 1 + θ 7 q − 2 ) 2 u , ∂ V ( θ ; u ) ∂ θ 8 = − q − 1 1 + θ 9 q − 1 u , ∂ V ( θ ; u ) ∂ θ 9 = θ 8 q − 2 1 + 2 θ 9 q − 1 + θ 2 9 q − 2 u , ∂ V ( θ ; u ) ∂ θ 10 = u , with x = G 1 ( q , θ ) u + G 2 ( q ) V s (0) 1 , Σ = 4 X i =1 iθ i x ◦ ( i − 1) + 5 V − V − 4 X i =1 θ i ! x ◦ 4 . Here, x ◦ u denotes the Hadamard product of x and u , x ◦ 2 denotes the Hadamard power with x ◦ 2 = x ◦ x , and 1 ∈ R N × 1 denotes a column vector with all elements equal to one. Finally , note that λ k needs to be chosen carefully to make J ( θ ) decrease monotonically . One can use the W olfe conditions and let λ k be selected such that J ( θ k + λ k s k ) ≤ J ( θ k ) + c 1 λ k g > k s k , ∇ J ( θ k + λ k s k ) > s k ≥ c 2 ∇ J ( θ k ) > s k , (21a) (21b) with 0 < c 1 < c 2 < 1 . For the quasi-Newton method, c 1 is usually set to be quite small, e.g., c 1 = 10 − 6 , and c 2 is typically set to be 0.9. The selection of λ k can be based on trial and error in implementation. One can start with picking a number and check the W olfe conditions. If the conditions are not satisfied, reduce the number and check again. An interested reader is referred to [60] for detailed discussion about the λ k selection. Summarizing the above, T able I: Quasi-Ne wton-based implementation for MAP-based W iener identification. Initialize θ 0 and set the con vergence tolerance repeat Compute g k via (20) if k = 0 then Initialize B 0 = 0 . 001 1 k g 0 k I else Compute B k via (19) end if Compute s k via (18) Find λ k that satisfies the W olfe conditions (21) Perform the update via (17) until J ( θ k ) con ver ges retur n ˆ θ = θ k T able I outlines the implementation procedure for the MAP- based W iener identification. Remark 1: While the MAP estimation has enjoyed a long history of addressing a v ariety of estimation problems, no study has been reported about its application to Wiener system identification to our knowledge. Here, it is found to be a very useful approach for providing physically reasonable parameter estimation for practical systems, as it takes into account some prior knowledge about the unknown parameters. In a Gaussian setting as adopted here, the prior p ( θ ) translates into a regularization term in J ( θ ) , which prev ents incorrect fitting and enhances the robustness of the numerical optimization against nonconv exity . Remark 2: The proposed 2.0 identification approach requires some prior knowledge of the parameters to be av ailable, which can be developed in sev eral ways in practice. First, R 0 can be roughly estimated using the voltage drop at the beginning of the discharge, to which it is a main contributor . Second, the polynomial coefficients of h ( · ) can be approximately obtained from an experimentally calibrated SoC-OCV curve if there is any . Third, one can derive a rough range for C b + C s if a battery’ s capacity is approximately known. Finally , as the parameters of batteries of the same kind and brand are usually close, one can take the parameter estimates acquired from one battery as prior knowledge for another . Remark 3: In general, a prerequisite for successful iden- tification is that the parameters must be identifiable in a certain sense. Follo wing along similar lines as in [18, 61], one can rigorously define the parameters’ local identifiability for the considered W iener identification problem and find out that a sufficient condition for it to hold is the full rankness of the sensitivity matrix ∂ V ( θ ; u ) /∂ θ , which can be used for identifiability testing. Using this idea, our simulations consistently showed the full rankness of the sensitivity matrix under v ariable current profiles such as those in Figure 8, indicating that the NDC model can be locally identifiable. Related with identification is optimal input design, which concerns designing the best current profile to maximize the parameter identifiability [50, 51]. It will be part of our future research to explore this interesting problem for the NDC 9 model. Remark 4: It is worth mentioning that the 2.0 identification approach can be readily e xtended to identify some other ECMs that hav e a Wiener -like structure like the Rint and Thev enin models. One can follow similar lines to de velop the computational procedures for each, and hence the details are skipped here. Remark 5: The 1.0 and 2.0 identification approaches are designed to perform offline identification for the NDC model, each with its own advantages. The 1.0 approach is designed for in-lab battery modeling and analysis, using simple two- step (trickle- and constant-current discharging) battery testing protocols. While requiring a long time for experiments, it can offer high accuracy in parameter estimation. More sophisti- cated by design, the 2.0 approach can extract the parameters all at once from data based on variable current profiles. It can be conv eniently exploited to determine the NDC model for batteries operating in real-world applications. V . E X P E R I M E N T A L V A L I DA T I O N This section presents experimental validation of the pro- posed NDC model and parameter identification 1.0 and 2.0 approaches. All the e xperiments in this section were conducted on a PEC R  SBT4050 battery tester (see Figure 4). It can support charging/discharging with arbitrary current-, voltage- and po wer-based loads (up to 40 V and 50 A). A specialized server is used to prepare and configure a test offline and collect experimental data online via the associated software LifeT est TM . Using this facility , charging/discharging tests were performed to generate data on a Panasonic NCR18650B lithium-ion battery cell, which was set to operate between 3.2 V (fully discharged) and 4.2 V (fully charged). A. V alidation Based on P arameter Identification 1.0 This validation first extracts the NDC model from training dataset using the 1.0 identification approach in Section III and then applies the identified model to validation datasets to assess its predictiv e capability . As a first step, the cell was fully charged and relaxed for a long time period. Then, a full discharge test was applied to the cell using a trickle constant current of 0.1 A (about 1/30 C-rate). W ith this test, the total capacity is determined to be Q t = 3 . 06 Ah by coulomb counting, implying C b + C s = 11 , 011 F . Further , from the SoC-OCV curve fitting, we obtain OCV = 3 . 2 + 2 . 59 · SoC − 9 . 003 · SoC 2 + 18 . 87 · SoC 3 − 17 . 82 · SoC 4 + 6 . 325 · SoC 5 , which establishes h ( · ) immediately . The measured and iden- tified SoC-OCV curves are compared in Figure 5. Next, the cell was fully charged again and left idling for a long time. This was then followed by a full discharge using a constant current of 3 A to produce data for estimation of the impedance and capacitance parameters. The identification was achiev ed by solving the constrained optimization problem in (9). The computation took around 1 sec, performed on a Dell Precision T ower 3620 equipped with 3 GHz Inter Xeon CPU, 16 Gb RAM and MA TLAB R2018b. T able II summarizes the initial Figure 4: PEC R  SBT4050 battery tester . 0 0.2 0.4 0.6 0.8 1 SoC 3.2 3.4 3.6 3.8 4 4.2 OCV Measured OCV Predicted OCV Figure 5: Identification 1.0: parameter identification of h ( · ) that defines SoC-OCV relation. guess, lower and upper bounds, and obtained estimates of the parameters. The physical parameter estimates are extracted as: C b = 10 , 037 F , C s = 973 F , R b = 0 . 019 Ω , R s = 0 , R 1 = 0 . 02 Ω , C 1 = 3 , 250 F , and R 0 = 0 . 0531 + 0 . 1077 e − 3 . 807 · SoC + 0 . 0533 e − 7 . 613 · (1 − SoC) . The model is now fully av ailable from the two steps. Figure 6 shows that it accurately fits with the measurement data. While an identified model generally can well fit a training dataset, it is more meaningful and rev ealing to examine its predictiv e performance on some different datasets. Hence, fiv e more tests were conducted by discharging the cell us- ing constant currents of 1.5 A, 2.5 A and 3.5 A and two variable current profiles, respectiv ely . Figure 7 shows what the identified model predicts for discharging at constant cur- rents. An ov erall high accuracy is observed, ev en though the prediction is slightly less accurate when the current is 1.5 A, probably because the parameters are current-dependent to a certain extent. The variable current profiles are portrayed in Figures 8(a) and 9(a), which were created by scaling the Urban Dynamometer Driving Schedule (UDDS) profile in [62] to span the ranges of 0 ∼ 3 A and 0 ∼ 6 A, respectiv ely . Fig- ures 8(b) and 9(b) present the predictive fitting results. Both of them illustrate that the model-based voltage prediction is quite close to the actual measurements. These results demonstrate the excellent predictiv e capability of the NDC model. B. V alidation Based on P arameter Identification 2.0 Let us now consider the 2.0 identification approach devel- oped in Section III, which treats the NDC model as a Wiener - 10 T able II: Identification 1.0: initial guess, bound limits and identification results. Name β 2 β 3 β 4 β 5 γ 1 γ 2 γ 3 γ 4 γ 5 Initial guess 0.02 0.05 0.005 1/100 0.05 0.2 8 0.07 12 θ 0.005 0.005 0.001 1/800 0.01 0.05 1 0.01 1 θ 0.2 0.2 0.03 1/10 0.09 0.35 15 0.12 15 ˆ θ 0.0163 0.0575 0.02 1/65 0.0531 0.1077 3.807 0.0533 7.613 Note: quantities are giv en in SI standard units in T ables II and III. T able III: Identification 2.0: initial guess, prior knowledge and identification results. Name α 1 α 2 α 3 α 4 ˘ β 1 ˘ β 2 ˘ β 3 β 4 β 5 R 0 Initial guess 2.59 -9.003 18.87 -17.82 9 . 078 × 10 − 5 8 . 914 × 10 − 4 0.964 − 4 . 938 × 10 − 4 − 0 . 9753 0.08 m - - - - 9 . 078 × 10 − 5 8 . 914 × 10 − 4 0.964 − 4 . 938 × 10 − 4 − 0 . 9753 0.08 p diag( P ) - - - - 0 . 001 × m 5 0 . 15 × m 6 0 . 15 × m 7 0 . 15 × m 8 0 . 15 × m 9 0 . 15 × m 10 ˆ θ 2.32 -8.15 19.345 -20.78 9 . 082 × 10 − 5 9 . 227 × 10 − 4 0.982 − 4 . 859 × 10 − 4 − 0 . 8153 0.069 0 500 1000 1500 2000 2500 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) Measured terminal voltage Predicted terminal voltage Figure 6: Identification 1.0: model fitting with the training data obtained under 3 A constant-current discharging. 0 2000 4000 6000 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) 1.5 A 2.5 A 3.5 A Measured terminal voltage Predicted terminal voltage Figure 7: Identification 1.0: predictiv e fitting o ver v alidation data obtained by discharging at different constant currents. type system and performs MAP-based parameter estimation. This approach advantageously allows all the parameters to be estimated in a con venient one-shot procedure. Follo wing the manner in Section V -A, one can apply the 2.0 approach to a training dataset to extract an NDC model and then use it to predict the responses over sev eral other different datasets. The validation here is also set to ev aluate the NDC model against the Rint model [12] and the Thevenin model with one serial RC circuit [12], which are commonly used in the literature. The comparison also extends to a basic version of the NDC model (referred to as “basic NDC” in sequel), one with a constant R 0 and without R 1 - C 1 circuit, with the purpose of examining the utility of the NDC model when it is reduced to a simpler form. Note that, ev en though the NDC model is the most sophisticated among them, all of the four models offer high computational efficienc y by requiring only a small number of arithmetic operations. These four models are all Wiener -type, so the 2.0 identi- fication approach can be used to identify them on the same training dataset, i.e., the one shown in Figure 8, thus ensuring a fair comparison. The parameter setting for the NDC model identification and the estimation result are summarized in T able III. The computation took around 4 sec. The resultant physical parameter estimates are given by: C b = 10 , 031 F , C s = 979 F , R b = 0 . 063 Ω , R s = 0 , R 1 = 0 . 003 Ω , C 1 = 2 , 449 F and R 0 = 0 . 069 Ω . The identification results for the Rint, Thevenin model and basic NDC models are omitted here for the sake of space. Figure 10(a) depicts how the identified models fit with the training dataset. One can observe that the NDC model and its basic version show excellent fitting accuracy , overall better than the Rint and Thev enin models. A more detailed comparison is giv en in Figure 10(b), which displays the fitting error in percentage. It is seen that the Rint model shows the least accuracy , followed by the The venin model. The NDC model and its basic version well outperform them, with the NDC model performing slightly better . Proceeding forward, let us in vestigate the predictiv e per- formance of the four models over several validation datasets. First, consider the datasets obtained by constant-current dis- charging at 1.5 A, 2.5 A and 3.5 A, as illustrated in Figure 7. Figure 11 demonstrates that the NDC model and its basic ver - sion can predict the voltage responses under different currents much more accurately than the Rint and Thev enin models. Next, consider the dataset in Figure 9 based on v ariable-current discharging. Figure 12 shows that the prediction accuracy of all the models is lo wer than the fitting accuracy , which is understandable. Ho wev er , the NDC model and its basic version are still again the most capable of predicting, with the error 11 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s) -3 -2 -1 0 Current (A) (a) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) Measured terminal voltage Predicted terminal voltage 3100 3200 3300 3.7 3.75 3.8 (b) Figure 8: Identification 1.0: predicti ve fitting over validation dataset obtained by discharging at varying currents (0 ∼ 3 A). (a) Current profile. (b) V oltage fitting. 0 500 1000 1500 2000 2500 3000 3500 4000 Time (s) -6 -4 -2 0 Current (A) (a) 0 500 1000 1500 2000 2500 3000 3500 4000 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) Measured terminal voltage Predicted terminal voltage 1200 1250 1300 3.7 3.75 3.8 3.85 (b) Figure 9: Identification 1.0: predictiv e fitting o ver v alidation dataset obtained by discharging at varying currents ( 0 ∼ 6 A). (a) Current profile. (b) V oltage fitting. mostly lying below 1% . As a contrast, while the Thev enin model can offer a decent fit with the training dataset as shown in Figure 10, its prediction accuracy ov er the validation dataset is not as satisfactory . This implies that it is less predictiv e than the NDC model. Another ev aluation of interest is about the SoC-OCV rela- tion. As mentioned earlier, the 2.0 approach can estimate all the parameters, including the function h ( · ) . This allows one to write the SoC-OCV function directly based on the identified h ( · ) as it also characterizes the SoC-OCV relation. That is, OCV = 3 . 2 + 2 . 32 · SoC − 8 . 15 · SoC 2 + 19 . 345 · SoC 3 − 20 . 78 · SoC 4 + 8 . 222 · SoC 5 . Identification of the other three models can also lead to estimation of this function. Figure 13 compares them with the benchmark shown in Figure 5, which is obtained exper - 12 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) Real Rint Thevenin Basic NDC NDC 3100 3200 3300 3.7 3.75 3.8 (a) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s) 0 1 2 3 4 Error (%) Rint Thevenin Basic NDC NDC (b) Figure 10: Identification 2.0. (a) Model fitting with training dataset. (b) Fitting error in percentage. 0 2000 4000 6000 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) 1.5 A 2.5 A 3.5 A Real Rint Thevenin Basic NDC NDC Figure 11: Identification 2.0: predicti ve fitting ov er v alidation datasets obtained by discharging at different varying currents. imentally by discharging the cell using a small current of 0.1 A. It is obvious that the SoC-OCV curves obtained in the identification of the NDC model and its basic version are closer to the benchmark overall. This further shows the benefit of the NDC model as well as the efficacy of the 2.0 approach. Summing up the above validation results, one can draw the following observations: • The NDC model is the most competent among the four considered models for grasping and predicting a battery’ s dynamic beha vior , justifying its v alidity and soundness. • The basic NDC model can of fer fitting and prediction accuracy almost comparable to that of the full model. It thus can be well qualified if a practitioner wants to use a simpler NDC model yet without much loss of accuracy . • The 2.0 identification approach is effecti ve in estimating all the parameters of the NDC model as well as the Rint and Thevenin models in one shot from variable- current-based data profiles. It can not only ease the cost of identification considerably but also provide on-demand model a vailability potentially in practice. V I . C O N C L U S I O N The growing importance of real-time battery management has imposed a pressing demand for battery models with high fidelity and low complexity , making ECMs a popular choice in this field. The double-capacitor model is emer ging as a fa vorable ECM for div erse applications, promising sev eral advantages in capturing a battery’ s dynamics. Ho wev er, its linear structure intrinsically hinders a characterization of a battery’ s nonlinear phenomena. T o thoroughly improve this model, this paper proposed to modify its original structure by adding a nonlinear-mapping-based voltage source and a serial RC circuit. This dev elopment was justified through an analogous comparison with the SPM. Furthermore, two offline parameter estimation approaches, which were named 1.0 and 2.0, respectiv ely , were designed to identify the model from current/voltage data. The 1.0 approach considers the constant-current charging/dischar ging scenarios, determining the SoC-OCV relation first and then estimating the impedance and capacitance parameters. W ith the observation that the NDC model has a Wiener -type structure, the 2.0 approach was derived from the W iener perspective. As the first of its kind, it lev erages the notion of MAP to address the issue of local minima that may reduce or damage the performance of the nonlinear W iener system identification. It well lends itself to the variable-current charging/dischar ging scenarios and can desirably estimate all the parameters in one shot. The experimental ev aluation demonstrated that the NDC model outperformed the popularly used Rint and Thev enin models 13 0 500 1000 1500 2000 2500 3000 3500 4000 Time (s) 3.2 3.4 3.6 3.8 4 4.2 Voltage (V) Real Rint Thevenin Basic NDC NDC 1200 1250 1300 3.6 3.7 3.8 (a) 0 500 1000 1500 2000 2500 3000 3500 4000 Time (s) 0 1 2 3 4 Error (%) Rint Thevenin Basic NDC NDC (b) Figure 12: Identification 2.0. (a) Predicti ve fitting ov er v alidation dataset obtained by discharging at varying currents between 0 A and 6 A. (b) Predictive fitting error in percentage. 0 0.2 0.4 0.6 0.8 1 SoC 3.2 3.4 3.6 3.8 4 4.2 OCV Real Rint Thevenin Basic NDC NDC Figure 13: Identification 2.0: identification of the SoC-OCV relation based on different models, compared to the truth. in predicting a battery’ s behavior , in addition to showing the effecti veness of the identification approaches for extracting parameters. Our future work will include: 1) enhancing the NDC model further to account for the effects of temperature and include the voltage hysteresis, 2) in vestigating optimal input design for the model, and 3) b uilding ne w battery estimation and control designs based on the model. R E F E R E N C E S [1] V . H. Johnson and A. A. Pesaran, “T emperature-dependent battery mod- els for high-po wer lithium-lon batteries, ” National Rene wable Energy Laboratory , T ech. Rep. NREL/CP-540-28716, 2000. [2] M. Doyle, T . F . Fuller , and J. Newman, “Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell, ” Journal of The Electr ochemical Society , vol. 140, no. 6, pp. 1526–1533, 1993. [3] J. C. Forman, S. J. Moura, J. L. Stein, and H. K. Fathy , “Genetic identi- fication and fisher identifiability analysis of the Doyle–Fuller–Newman model from experimental cycling of a LiFePO 4 cell, ” Journal of P ower Sour ces , vol. 210, pp. 263 – 275, 2012. [4] N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. K ojic, “ Algorithms for advanced battery-management systems, ” IEEE Control Systems Magazine , vol. 30, no. 3, pp. 49–68, 2010. [5] M. Guo, G. Sikha, and R. E. White, “Single-particle model for a lithium- ion cell: Thermal behavior , ” J ournal of The Electr ochemical Society , vol. 158, no. 2, pp. A122–A132, 2011. [6] P . W . C. Northrop, B. Suthar , V . Ramadesigan, S. Santhanagopalan, R. D. Braatz, and V . R. Subramanian, “Efficient simulation and reformulation of lithium-ion battery models for enabling electric transportation, ” Journal of The Electr ochemical Society , vol. 161, no. 8, pp. E3149– E3157, 2014. [7] C. Zou, C. Manzie, and D. Ne ˇ si ´ c, “ A framew ork for simplification of PDE-based lithium-ion battery models, ” IEEE T ransactions on Contr ol Systems T echnology , vol. 24, no. 5, pp. 1594–1609, 2016. [8] X. Hu, D. Cao, and B. Egardt, “Condition monitoring in advanced bat- tery management systems: Moving horizon estimation using a reduced electrochemical model, ” IEEE/ASME Tr ansactions on Mechatr onics , vol. 23, no. 1, pp. 167–178, 2018. [9] J. E. B. Randles, “Kinetics of rapid electrode reactions, ” Discussions of the F araday Society , vol. 1, pp. 11–19, Mar. 1947. [10] A. I. Zia and S. C. Mukhopadhyay , Impedance Spectr oscopy and Exper- imental Setup . Cham, Switzerland: Springer International Publishing, 2016, pp. 21–37. [11] S. M. G. and M. Nikdel, “V arious battery models for various simulation studies and applications, ” Renewable and Sustainable Energy Reviews , vol. 32, pp. 477–485, 2014. [12] H. He, R. Xiong, and J. Fan, “Evaluation of lithium-ion battery equiv- alent circuit models for state of charge estimation by an experimental approach, ” Ener gies , vol. 4, pp. 582–598, 2011. [13] G. L. Plett, Battery Management Systems, V olume 1: Battery Modeling . Artech House, 2015. [14] C. W eng, J. Sun, and H. Peng, “ A unified open-circuit-voltage model of lithium-ion batteries for state-of-charge estimation and state-of-health monitoring, ” Journal of P ower Sour ces , vol. 258, pp. 228–237, 2014. [15] X. Lin, H. E. Perez, S. Mohan, J. B. Siegel, A. G. Stefanopoulou, Y . Ding, and M. P . Castanier, “ A lumped-parameter electro-thermal model for cylindrical batteries, ” Journal of P ower Sources , vol. 257, pp. 1–11, 2014. [16] M. Gholizadeh and F . R. Salmasi, “Estimation of state of charge, unknown nonlinearities, and state of health of a lithium-ion battery based on a comprehensiv e unobservable model, ” IEEE T ransactions on Industrial Electr onics , vol. 61, no. 3, pp. 1335–1344, 2014. [17] H. E. Perez, X. Hu, S. Dey , and S. J. Moura, “Optimal charging of Li-ion batteries with coupled electro-thermal-aging dynamics, ” IEEE 14 T ransactions on V ehicular T echnolo gy , vol. 66, no. 9, pp. 7761–7770, 2017. [18] H. Fang, Y . W ang, Z. Sahinoglu, T . W ada, and S. Hara, “State of charge estimation for lithium-ion batteries: An adaptive approach, ” Control Engineering Practice , vol. 25, pp. 45–54, 2014. [19] G. L. Plett, “Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification, ” Journal of P ower Sources , v ol. 134, no. 2, pp. 262–276, 2004. [20] X. Hu, S. Li, and H. Peng, “ A comparative study of equiv alent circuit models for Li-ion batteries, ” Journal of P ower Sources , vol. 198, pp. 359–367, 2012. [21] K. Li, F . W ei, K. J. Tseng, and B.-H. Soong, “ A practical lithium- ion battery model for state of energy and voltage responses prediction incorporating temperature and ageing effects, ” IEEE T ransactions on Industrial Electr onics , vol. 65, no. 8, pp. 6696–6708, 2018. [22] K.-T . Lee, M.-J. Dai, and C.-C. Chuang, “T emperature-compensated model for lithium-ion polymer batteries with extended Kalman filter state-of-charge estimation for an implantable charger , ” IEEE Tr ansac- tions on Industrial Electr onics , vol. 65, no. 1, pp. 589–596, 2018. [23] M. Chen and G. A. Rincon-Mora, “ Accurate electrical battery model capable of predicting runtime and I-V performance, ” IEEE T ransactions on Energy Con version , vol. 21, no. 2, pp. 504–511, 2006. [24] T . Kim and W . Qiao, “ A hybrid battery model capable of capturing dynamic circuit characteristics and nonlinear capacity effects, ” IEEE T ransactions on Energy Conver sion , vol. 26, no. 4, pp. 1172–1180, 2011. [25] V . Johnson, “Battery performance models in ADVISOR, ” Journal of P ower Sources , v ol. 110, no. 2, pp. 321–329, 2002. [26] H. Fang, Y . W ang, and J. Chen, “Health-aware and user-in volved battery charging management for electric vehicles: Linear quadratic strategies, ” IEEE T ransactions on Contr ol Systems T echnology , vol. 25, no. 3, pp. 911–923, 2017. [27] H. Fang, C. Depcik, and V . Lvovich, “Optimal pulse-modulated lithium- ion battery charging: Algorithms and simulation, ” Journal of Energy Storag e , vol. 15, pp. 359–367, 2018. [28] B. Schweighofer , K. M. Raab, and G. Brasseur , “Modeling of high power automotiv e batteries by the use of an automated test system, ” IEEE T ransactions on Instrumentation and Measur ement , vol. 52, no. 4, pp. 1087–1091, 2003. [29] S. Abu-Sharkh and D. Doerffel, “Rapid test and non-linear model characterisation of solid-state lithium-ion batteries, ” Journal of P ower Sour ces , vol. 130, no. 1, pp. 266–274, 2004. [30] M. Dubarry and B. Y . Liaw , “Dev elopment of a universal modeling tool for rechargeable lithium batteries, ” Journal of P ower Sour ces , vol. 174, no. 2, pp. 856–860, 2007. [31] H. He, R. Xiong, X. Zhang, F . Sun, and J. Fan, “State-of-charge estimation of the lithium-ion battery using an adaptive extended Kalman filter based on an improved Thevenin model, ” IEEE T ransactions on V ehicular T echnology , vol. 60, no. 4, pp. 1461–1469, 2011. [32] Y . T ian, B. Xia, W . Sun, Z. Xu, and W . Zheng, “ A modified model based state of charge estimation of power lithium-ion batteries using unscented Kalman filter , ” Journal of P ower Sour ces , vol. 270, pp. 619–626, 2014. [33] P . Mauracher and E. Karden, “Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification, ” Journal of P ower Sources , v ol. 67, no. 1-2, pp. 69–84, 1997. [34] K. Goebel, B. Saha, A. Saxena, J. R. Celaya, and J. P . Christophersen, “Prognostics in battery health management, ” IEEE Instrumentation & Measur ement Magazine , vol. 11, no. 4, 2008. [35] C. Birkl and D. Howey , “Model identification and parameter estimation for LiFePO 4 batteries, ” in Proceedings of IET Hybrid and Electric V ehicles Conference , 2013, pp. 1–6. [36] T . Hu and H. Jung, “Simple algorithms for determining parameters of circuit models for charging/dischar ging batteries, ” Journal of P ower Sour ces , vol. 233, pp. 14–22, 2013. [37] Z. He, G. Y ang, and L. Lu, “ A parameter identification method for dynamics of lithium iron phosphate batteries based on step-change current curves and constant current curves, ” Energies , vol. 9, no. 6, p. 444, 2016. [38] J. Y ang, B. Xia, Y . Shang, W . Huang, and C. Mi, “Improved battery pa- rameter estimation method considering operating scenarios for HEV/EV applications, ” Ener gies , vol. 10, no. 1, p. 5, 2016. [39] N. Tian, Y . W ang, J. Chen, and H. Fang, “On parameter identification of an equivalent circuit model for lithium-ion batteries, ” in Contr ol T echnology and Applications (CCT A), 2017 IEEE Confer ence on . IEEE, 2017, pp. 187–192. [40] T . Feng, L. Y ang, X. Zhao, H. Zhang, and J. Qiang, “Online identifica- tion of lithium-ion battery parameters based on an improved equiv alent- circuit model and its implementation on battery state-of-power predic- tion, ” Journal of P ower Sour ces , vol. 281, pp. 192–203, 2015. [41] G. K. Prasad and C. D. Rahn, “Model based identification of aging parameters in lithium ion batteries, ” Journal of P ower Sources , vol. 232, pp. 79–85, 2013. [42] M. Sitterly , L. Y . W ang, G. G. Y in, and C. W ang, “Enhanced iden- tification of battery models for real-time battery management, ” IEEE T ransactions on Sustainable Energy , vol. 2, no. 3, pp. 300–308, 2011. [43] L. Zhang, L. W ang, G. Hinds, C. L yu, J. Zheng, and J. Li, “Multi- objectiv e optimization of lithium-ion battery model using genetic al- gorithm approach, ” Journal of P ower Sources , vol. 270, pp. 367–378, 2014. [44] M. A. Rahman, S. Anwar, and A. Izadian, “Electrochemical model parameter identification of a lithium-ion battery using particle swarm optimization method, ” Journal of P ower Sources , vol. 307, pp. 86–97, 2016. [45] Z. Y u, L. Xiao, H. Li, X. Zhu, and R. Huai, “Model parameter identification for lithium batteries using the coev olutionary particle swarm optimization method, ” IEEE T rans. Ind. Electr on , vol. 64, no. 7, pp. 5690–5700, 2017. [46] D. W . Limoge and A. M. Annaswamy , “ An adapti ve observer design for real-time parameter estimation in lithium-ion batteries, ” IEEE Tr ansac- tions on Control Systems T echnology , 2018, in press. [47] Y . Hu and S. Y urkovich, “Linear parameter varying battery model identification using subspace methods, ” Journal of P ower Sources , vol. 196, no. 5, pp. 2913–2923, 2011. [48] Y . Li, C. Liao, L. W ang, L. W ang, and D. Xu, “Subspace-based model- ing and parameter identification of lithium-ion batteries, ” International Journal of Ener gy Research , vol. 38, no. 8, pp. 1024–1038, 2014. [49] R. Relan, Y . Firouz, J. Timmermans, and J. Schoukens, “Data-driven nonlinear identification of li-ion battery based on a frequency domain nonparametric analysis, ” IEEE T ransactions on Control Systems T ech- nology , vol. 25, no. 5, pp. 1825–1832, 2017. [50] M. J. Rothenberger , D. J. Docimo, M. Ghanaatpishe, and H. K. Fathy , “Genetic optimization and experimental validation of a test cycle that maximizes parameter identifiability for a Li-ion equiv alent-circuit bat- tery model, ” Journal of Energy Storage , vol. 4, pp. 156–166, 2015. [51] S. Park, D. Kato, Z. Gima, R. Klein, and S. Moura, “Optimal input design for parameter identification in an electrochemical Li-ion battery model, ” in Proceedings of American Control Conference , 2018, pp. 2300–2305. [52] F . Giri and E.-W . Bai, Block-Oriented Nonlinear System Identification . London, UK: Springer, 2010, vol. 1. [53] A. Hagenblad, L. Ljung, and A. Wills, “Maximum likelihood identifica- tion of Wiener models, ” Automatica , vol. 44, no. 11, pp. 2697 – 2705, 2008. [54] L. V anbeylen, R. Pintelon, and J. Schoukens, “Blind maximum- likelihood identification of Wiener systems, ” IEEE T ransactions on Signal Pr ocessing , vol. 57, no. 8, pp. 3017–3029, 2009. [55] S. Dey, S. Mohon, B. A yale w , H. Arunachalam, and S. Onori, “ A novel model-based estimation scheme for battery-double-layer capacitor hybrid energy storage systems, ” IEEE T ransactions on Contr ol Systems T echnology , vol. 27, no. 2, pp. 689–702, 2019. [56] A. Mamun, A. Siv asubramaniam, and H. Fathy , “Collectiv e learning of lithium-ion aging model parameters for battery health-conscious demand response in datacenters, ” Energy , vol. 154, pp. 80–95, 2018. [57] D. Andre, M. Meiler , K. Steiner , C. Wimmer , T . Soczka-Guth, and D. Sauer, “Characterization of high-power lithium-ion batteries by electrochemical impedance spectroscopy . I. Experimental inv estigation, ” Journal of P ower Sources , v ol. 196, no. 12, pp. 5334–5341, 2011. [58] M. Dubarry , N. V uillaume, and B. Y . Liaw , “From single cell model to battery pack simulation for Li-ion batteries, ” Journal of P ower Sour ces , vol. 186, no. 2, pp. 500–507, 2009. [59] A. Hagenblad, “ Aspects of the identification of W iener models, ” Ph.D. dissertation, Department of Electrical Engineering, Link ¨ oping Uni ver- sity , Sweden, 1999. [60] S. Wright and J. Nocedal, Numerical Optimization . New Y ork, US: Springer-V erlag New Y ork, 1999, vol. 35, no. 67–68. [61] J. F . V an Doren, S. G. Douma, P . M. V an den Hof, J. D. Jansen, and O. H. Bosgra, “Identifiability: from qualitati ve analysis to model structure approximation, ” IF AC Pr oceedings V olumes of the 15th IF AC Symposium on System Identification , vol. 42, no. 10, pp. 664 – 669, 2009. [62] “The EP A Urban Dynamometer Driving Schedule (UDDS) [Online], ” A vailable: https://www .epa.gov/sites/production/files/2015- 10/uddscol.txt. 15 Ning Tian recei ved the B.Eng. and M.Sc. degrees in thermal engineering from Northwestern Polytechnic Univ ersity , Xi’an, China, in 2012 and 2015. Cur- rently , he is a Ph.D. candidate at the Univ ersity of Kansas, Lawrence, KS, USA. His research in- terests include control theory and its application to advanced battery management. Huazhen Fang (M’14) receiv ed the B.Eng. degree in computer science and technology from North- western Polytechnic University , Xi’an, China, in 2006, the M.Sc. degree in mechanical engineering from the University of Saskatchewan, Saskatoon, SK, Canada, in 2009, and the Ph.D. degree in mechanical engineering from the Univ ersity of Cal- ifornia, San Diego, CA, USA, in 2014. He is currently an Assistant Professor of Me- chanical Engineering with the Univ ersity of Kansas, Lawrence, KS, USA. His research interests include control and estimation theory with application to energy management, coop- erativ e robotics, and system prognostics. Dr . Fang received the 2019 National Science Foundation CAREER A ward and the awards of Outstanding Reviewer or Revie wers of the Y ear from Au- tomatica , IEEE T ransactions on Cybernetics , and ASME J ournal of Dynamic Systems, Measurement and Contr ol . Jian Chen (M’06-SM’10) received the B.E. degree in measurement and control technology and instru- ments and the M.E. degree in control science and engineering from Zhejiang University , Hangzhou, China, in 1998 and 2001, respectively , and the Ph.D. degree in electrical engineering from Clemson Univ ersity , Clemson, SC, USA, in 2005. He was a Research Fellow with the University of Michigan, Ann Arbor, MI, USA, from 2006 to 2008, where he was in volved in fuel cell modeling and control. In 2013, he joined the Department of Control Science and Engineering, Zhejiang Uni versity , where he is currently a Professor with the College of Control Science and Engineering. His research interests include modeling and control of fuel cell systems, vehicle control and intelligence, machine vision, and nonlinear control. Y ebin W ang (M’10-SM’16) receiv ed the B.Eng. degree in Mechatronics Engineering from Zhejiang Univ ersity , Hangzhou, China, in 1997, M.Eng. de- gree in Control Theory & Control Engineering from Tsinghua University , Beijing, China, in 2001, and Ph.D. in Electrical Engineering from the University of Alberta, Edmonton, Canada, in 2008. Dr . W ang has been with Mitsubishi Electric Re- search Laboratories in Cambridge, MA, USA, since 2009, and now is a Senior Principal Research Scien- tist. From 2001 to 2003 he was a Software Engineer , Project Manager , and Manager of R&D Dept. in industries, Beijing, China. His research interests include nonlinear control and estimation, optimal control, adaptiv e systems and their applications including mechatronic systems.