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The model assumes that there is one primary reactant species in an excess of supporting electrolyte, and that a simultaneous side reaction may occur. Results are compared with the experimental data observed by various authors for the deposition of copper from sulfate solutions with the simultaneous generation of dissolved hydrogen. For an upstream counterelectrode, distributions of reaction rate for both single and multiple reactions , concentration, and potential describe the detailed system behavior. User Name Password Sign In. This Article doi:

The base case third harmonic spectrum, shown in Figure Model newmanexhibits signal in the Model newman mass-transport and thermodynamic regime as well as the mid-frequency regime dominated by the reaction kinetics, much like the standard linear electrochemical impedance spectrum shown in Figure 1a. Each of these derivatives are evaluated at the lithium concentration in the solid present at the particular state of charge being evaluated, c s 0. Figure 1. Want to try it yourself? Introducing higher harmonic spectra into the analysis of an electrochemical system provides additional informational content over the linear response alone. However, the homogeneous model equations can be somewhat expensive to solve for a 3D spiral cell or prismatic cell geometries. Reddy, Matthias K. A Nyquist plot for the two model geometries nwwman Figure 3 and Figure 4. The overlapping diffusional time constants remain a challenge for extracting individual coefficients from impedance spectra. Singh P.

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The pseudo-two-dimensional P2D model of lithium-ion batteries couples a volume-averaged treatment of transport, reaction, and thermodynamics to solid-state lithium diffusion in electrode particles.

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The pseudo-two-dimensional P2D model of lithium-ion batteries couples a volume-averaged treatment of transport, reaction, and thermodynamics to solid-state lithium diffusion in electrode particles. Here we harness the linear and nonlinear physics of the P2D model to evaluate the fundamental linear and higher harmonic nonlinear response of a LiCoO 2 LiC 6 cell subject to moderate-amplitude sinusoidal current modulations.

An analytic-numeric approach allows the evaluation of the linearized frequency dispersion function that represents electrochemical impedance spectroscopy EIS and the higher harmonic dispersion functions we call nonlinear electrochemical impedance spectroscopy NLEIS.

Base case simulations show, for the first time, the full spectrum second and third harmonic NLEIS response. The effect of kinetic, mass-transport, and thermodynamic parameters are explored. The nonlinear interactions that drive the harmonic response break some of the degeneracy found in linearized models. We show that the second harmonic is sensitive to the symmetry of the charge transfer reactions in the electrodes, whereas EIS is not.

In short, NLEIS has the potential to increase the number of physicochemical parameters that can be assessed in experiments similar in complexity to standard EIS measurements. Electrochemical impedance spectroscopy EIS is a commonly utilized tool for the noninvasive analysis of a wide range of electrochemical systems. The condition of linearity and ease of fitting equivalent circuit analogs has led to the widespread adoption of EIS for both characterization 3 — 5 and prognostic 6 — 8 applications within lithium-ion battery research.

In particular, EIS has been used to garner insight into important physicochemical processes including the growth of the solid electrolyte interphase SEI layer, 9 mass transfer and kinetics in electrode materials, 10 as well as other degradative and capacity loss mechanisms. Fundamentally, linearization reduces the information content of a given EIS measurement. The consequence of linearization is model degeneracy, in that, one EIS dataset can be represented equally well by different physical models 12 and different circuit analogs, 13 because of the loss of information inherent with linearization.

This degeneracy and loss of information may be an especially important limitation for the study of lithium-ion batteries if we hope to carefully dissect degradative interfacial processes. The issue of linear EIS model degeneracy and information loss is particularly challenging for sealed batteries.

Analytical solutions for the pseudo-two-dimensional P2D model impedance response under stationary conditions show that linearized EIS is unable to uniquely determine many of the model parameters needed for simulating charge-discharge curves. Changes to the battery potential state of charge do not alter the inaccessibility of unique transfer coefficients.

Alternative ways to measure unique transfer coefficients using EIS, such as passing a mean current that drives the electrode kinetics into the Tafel regime, introduce the challenge of maintaining a stationary state during EIS measurements in a sealed cell. An approach that breaks some of the degeneracy of linearized EIS without violating the requirement for stationarity is the use of moderate-amplitude current modulations around a zero-mean current.

Moderate-amplitude modulations are used to drive a weakly nonlinear response, as evidenced by higher harmonics in the frequency response of the system. Typically, higher harmonics in impedance experiments have been treated as noise, to be eliminated as a way of ensuring the linearity of EIS measurements. Extending EIS to probe the full-spectrum nonlinear response requires a more sophisticated approach than fitting lumped-parameter equivalent circuit analogs.

Physics-based models of the full electrochemical system offer the flexibility and interpretability required for a meaningful analysis. Here we extend the P2D impedance model developed by Newman et al. Later papers show that the extra information available in an NLEIS experiment comes at a modest increase in experimental difficulty.

To the best of our knowledge, full-spectrum nonlinear EIS has not been applied to the study of lithium-ion batteries. Yet, the ability to distinguish between different physical models, noninvasively probe nonlinear internal states, and investigate the symmetry and reversibility of degradative reactions make NLEIS a candidate for providing information-rich insights that can move the field forward.

The widely adopted P2D battery model has been a powerful tool for understanding, 35 optimizing, 36 , 37 and controlling 38 lithium-ion batteries for many decades. The P2D model is a physics-based representation of the coupled nonlinear mass transport, thermodynamic, and reaction processes inside a lithium-ion battery.

Porous electrode theory treatment of the electrodes requires volume averaging over the particles and electrolyte pores leading to two macrohomogeneous solid- and solution-phase continua superimposed in the x direction across the cell sandwich thickness. Here we take the solid-phase diffusivity, D s , to be independent of concentration in the solid, c s.

In all cases, effective transport coefficients diffusivities, conductivities are modifications of the intrinsic transport coefficients using a Bruggeman-type tortuosity factor to account for the porous character of the medium. The governing equations described above are both coupled and nonlinear. If we assume the system of equations is stable to finite amplitude current perturbations, then nonlinear theory suggests a pure single frequency sinusoidal input perturbation will generate an output response with spectral components at the excitation frequency fundamental or 1 st harmonic and integer higher harmonics 2 nd , 3 rd , and 4 th harmonic of the fundamental frequency.

We have shown previously that a fast-computing analytic-numeric method can be used for analyzing the linear and nonlinear response of many different electrochemical and transport-reaction systems when they are stable to finite amplitude perturbations. The resulting governing equations are shown in the Appendix.

The higher harmonic equations contain terms which depend on the lower harmonic solutions in a hierarchical manner such that the harmonics are solved sequentially not simultaneously. We chose to mesh the solid-phase equations A1. Mesh refinement in the interfacial regions with high gradients was used so that the solution converged for all parameter and frequency combinations resulting in approximately 9, total elements.

Further mesh refinement produced negligible changes in the computed results. Nyquist representations of the a linear impedance, b second harmonic, and c third harmonic spectra for the base parameters given in Table I.

The high frequency kinetics arc and low frequency mass transport tail are characteristic of a lithium-ion battery. If the governing equations and boundary conditions representing the physics of a battery were linear, then the standard electrochemical impedance shown in Figure 1a would be the only response of the system to a sinusoidal current perturbation of any magnitude.

However, because the P2D model is a coupled nonlinear set, it also produces a frequency-dependent higher harmonic response that we can compute. The five characteristic transport and reaction timescales, and the high frequency ohmic limit, described above, can be used to help interpret the higher order harmonic response.

As we will show in the next section, this lack of kinetically-driven second harmonic response is an artifact of the particular literature values used in the base case scenario symmetric transfer coefficients on both electrodes ; under more general conditions, the second harmonic is indeed sensitive to charge transfer.

The base case third harmonic spectrum, shown in Figure 1c , exhibits signal in the low-frequency mass-transport and thermodynamic regime as well as the mid-frequency regime dominated by the reaction kinetics, much like the standard linear electrochemical impedance spectrum shown in Figure 1a.

For both the second and third harmonic, the nonlinear impedance asymptotes to zero in the limit of high frequencies. At high frequencies, the response of the battery is dominated by ohmic processes that are governed by intrinsically linear physics.

As a result, the nonlinear harmonics vanish at high frequencies whereas the linear impedance remains finite. The base case values of 0. There is no intrinsic reason to assume charge transfer symmetry on either electrode. The linear response is insensitive to the transfer coefficients with all five curves superimposed upon one another.

While only the reaction symmetry of the positive electrode was varied here, a similar dependence on the negative electrode charge transfer symmetry also exists. Figure 2b shows that symmetry in charge transfer is clearly important for the second harmonic response. Moreover, the direction of the kinetic arc depends on the direction of the charge transfer asymmetry. As a result, the second term on the right-hand side RHS of both equations dominates the harmonic response in the second order flux density.

This sensitivity to reaction asymmetry has been demonstrated experimentally by Xu and Riley 41 who studied the ferri—ferrocyanide redox couple as well as Heubner et al. The third harmonic shown in Figure 2c is also sensitive to charge transfer symmetry, but in an inverse sense from the second harmonic. The largest third harmonic kinetic loop is observed when there is symmetry, and then the kinetic loop shrinks as the symmetry is broken.

It is also noteworthy that the shift in third harmonic is insensitive to whether the charge transfer asymmetry favors oxidation or reduction. The results of Figure 2 show that there is new information to be gleaned by measuring the second and third harmonics.

In the case of the transfer coefficients, the standard linear EIS spectra Figure 2a are insensitive to the specific values, whereas the second harmonic provides information about the magnitude of the asymmetry and direction charge or discharge that is kinetically more facile.

The third harmonic just gives insights into the magnitude of the asymmetry. Figure 3 shows the effect of the exchange current densities, i 0, i , and double-layer capacitances, C dl , i , for the positive and negative electrodes. The remaining discussion of individual parameter effects on the linear and nonlinear harmonic response will focus on only the first and second harmonic, for clarity. Nyquist representations of the first a, c, e, g and second b, d, f, h harmonic impedance spectra for different values of exchange current density, i 0 , and double-layer capacitance, C dl.

The characteristic timescales of the system result in changes to these kinetic parameters only affecting the mid-frequency arc in both the first and second harmonic response.

The linear impedance spectra in Figures 3a and 3c show the sensitivity of the kinetic arc width to the changes in exchange current density on each electrode.

The increasing size of the kinetic arc is normally interpreted through the charge transfer resistances, R ct , with smaller exchange current densities resulting in larger R ct. The width of the second harmonic kinetic arc is dependent on both the asymmetry of the charge transfer as well as on the value of the exchange current density. Varying the double-layer capacitance, C dl , changes the characteristic frequency for each electrode without changing the overall charge transfer resistance of the electrode.

Figures 3e — 3h show that independently varying the capacitances by four orders of magnitude separates the timescales for each electrode kinetic process and introduces distinct arcs for the two electrodes. In many ways, the linear and second harmonic show similar behaviors, namely, they go from being one distinctive arc to two clear arcs.

Because the timescales between the kinetic, transport, and thermodynamic regimes are well separated for the base case parameters, changing either the exchange-current densities or the double-layer capacitances has little effect on the low frequency response of any of the harmonics for the values shown here.

In equivalent circuit approaches to impedance analysis, the low frequency tail of the linear EIS response is often used to extract information about the diffusional processes. In their original description of the P2D impedance model on which this work builds, Doyle et al.

The overlapping diffusional time constants remain a challenge for extracting individual coefficients from impedance spectra. Here we explore if the nonlinear second harmonic spectra may offer more sensitivity. Figure 4 shows the effects of varying the solution- and solid-phase diffusion coefficients over several orders of magnitude. Figures 4b , 4d , 4f shows the effect of the mass-transport parameters on the second harmonic response.

Similarly to the linear impedance response, the high frequency arc remains constant due to the well separated timescales between the kinetic and diffusion regimes.

Figures 4b and 4d show that varying either the negative or positive solid-phase diffusion coefficients has the effect of changing the magnitude and phase of the low frequency response like the linear impedance. For particularly low values of the solution-phase diffusivity, however, Figure 4f shows that the structure of the low frequency second harmonic rearranges as this diffusional impedance dominates the response. In nearly all cases, the low frequency changes in second harmonic impedance are more dramatic than in the linear response, suggesting a full physics analysis of experimental linear and second harmonic data may result in a much greater ability to fit low frequency processes.

We are exploring this currently. Nyquist representations of the first a, c, e and second b, d, f harmonic impedance spectra for different values of solution- and solid-phase diffusion coefficients, D and D s , i. The second harmonic shows a more exaggerated response to changes in the mass-transport parameters. The thermodynamics of the cell determine the impedance response as the frequency goes to zero.

The important thermodynamic parameters in the low frequency range are the derivatives of the open circuit potential, U c s , with respect to the concentration of lithium in the electrodes. Each of these derivatives are evaluated at the lithium concentration in the solid present at the particular state of charge being evaluated, c s 0.

The sensitivity to second and higher order derivatives could be useful for applications such as estimation of state of charge or probing fundamental thermodynamic models predictions for lithium intercalation. The sign of the second derivative the curvature of U c s changes in between different phase-change plateaus in the open circuit potential. Figure 5 shows the dependence of these open circuit potential derivatives around the base case parameters on the linear and second harmonic spectra.

As with the mass transfer case, one sees similar effects between the linear and second harmonic impedance spectra, though with exaggerated effects in the second harmonic, when the first derivative of the open circuit potential is varied Figures 4a — 4d. In contrast, the first harmonic is completely insensitive to the second derivative of open circuit potential, whereas the second harmonic is sensitive at low frequencies. As the first derivative is varied a - d , a similar effect is seen in the linear and second harmonic response, with a slight exaggeration in the second harmonic response.

The linear impedance is insensitive to the changing second derivative. Physics-based impedance models are an important component of understanding and interpreting impedance spectra due to their flexibility and physically meaningful parameters.

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This paper aims to solve the parameter identification problem to estimate the parameters in electrochemical models of the lithium-ion battery. The parameter estimation framework is applied to the Doyle-Fuller-Newman DFN model containing a total of 44 parameters. The DFN model is fit to experimental data obtained through the cycling of Li-ion cells.

The parameter estimation is performed by minimizing the least-squares difference between the experimentally measured and numerically computed voltage curves. The minimization is performed using a state-of-the-art hybrid minimization algorithm. This approach saves much time in parameterization of models with a high number of parameters while achieving a high-quality fit. Reddy, S. Reddy, Matthias K. Published by Emerald Publishing Limited. Anyone may reproduce, distribute, translate and create derivative works of this article for both commercial and non-commercial purposes , subject to full attribution to the original publication and authors.

The applications of lithium-ion batteries have drastically increased over the past decade. Accurate analysis of the battery can sometimes require the internal state of the cell to be known. This internal state can include abstract quantities e. Some of these quantities can be measured through experimentation. In several cases, the material properties of the cell can also be of interest. These material properties sometimes cannot be measured directly and must be estimated, often non-intrusively.

This gives rise to the traditional parameter estimation problem. Parameter estimation techniques attempt to identify certain parameters in a model using only the model response. Parameter estimation techniques can be non-intrusive and non-destructive depending on whether the model response can be obtained non-intrusively and non-destructively. The parameter estimation problem in this work can be stated as follows: Given only the voltage, how can the material properties and model parameters of the lithium-ion cell model be estimated?

This literature review predominantly focuses on studies that perform the parameter estimation offline. Schmidt et al. They also used the Fisher information to determine the identifiable parameters. Speltino et al. Santhanagopalan et al. Scharrer et al.

Their work made use of a Morris-One-At-A-Time sensitivity analysis to identify the three most sensitive parameters in the model. Forman et al. To date, this is the latest attempt in estimating a significant number of parameters in the DFN model. Recently, Jin et al. They then used Levenberg—Marquardt algorithm to estimate the values of these five parameters.

A parallel genetic algorithm was used by Zhang et al. They reported a computing time of Uddin et al. This work drastically accelerates the parameter estimation of several parameters in the single particle and Newman models. In the work of Forman et al. This great speed-up is due to the sophisticated minimization algorithm used to perform the parameter estimation.

This work makes use of several minimization algorithms and continuously switches between them to accelerate convergence and avoid local minima. The dynamics of lithium-ion batteries is of a highly multi-physics nature. For this reason, an efficient implementation of the mathematical model is needed. Owing to the nature of materials inside a cell several simplifying assumptions have been made, often applied in the field of battery modelling, to enable computational simulation of the electrical and chemical processes inside a cell.

This results in a simplified one-dimensional diffusion equation, which implies uniform superficial current and constant isotropic diffusion inside. Thus, the entire equation system may be quickly solved in contrast to full 3D-simulations, while accurately describing the insertion process.

Figure 1 shows the model domain schematically, including the layered structure of a cell, as well as the sub-domains annotations. The DFN model : Each electrode is represented by homogenously distributed spherical particles as the limiting factor, connected via electrolyte. Owing to the homogenous distribution of the particles and the assumption of a small dimension orthogonal to the layered structure, a single one-dimensional cut through the electrolyte domain models the electrolyte geometry, i.

We assume constant behavior of the electronic quantities in the solid domain. The measured voltage data obtained through cycling a Panasonic NCRB commercial cell is used in this work. The cell is then charged to 3. At each level a set of current pulses are applied such that the short term dynamic behavior of the cell is reflected as much as possible in the voltage. Figure 2 shows the voltage and current measured throughout this time of roughly 2.

The traditional approach to solve the parameter identification problem involves minimizing the difference between the measured response and predicted response. An algorithm that can efficiently minimize the error with a few model evaluations is very appealing. This minimization algorithm must be robust and should be able to avoid local minima.

For this reason, a newly developed hybrid optimizer is used to solve the above optimization problem. Owing to the large computational time required to solve the mathematical model and because of the non-linearity of the cost-function space, an efficient and robust minimization technique is needed.

The minimization technique in this work is a single objective hybrid optimizer SOHO. The SOHO algorithm features three individual algorithms. It is well known from the no free lunch theorem that no single algorithm is superior over another for an entire problem set.

This means that the superiority of one algorithm over another for a problem set is paid for by the loss of its superiority over another problem set. This drives the need to couple several optimization algorithms to increase their robustness over a larger set of problems. The SOHO is initialized with one of the three previously stated algorithms. Each algorithm operates till convergence. If stagnation is detected, an alternative algorithm is selected randomly from the remaining two.

This random selection of algorithm adds a stochastic nature to search process and avoids user bias. Figure 3 shows the three algorithms and the switching scheme. This hybridization allows SOHO to avoid local minima and increases the convergence rate to the global minimum. The parents to be mated are selected randomly from the entire population set.

More details on these algorithms may be found in Deb The DFN-model used in this work is defined using 44 parameters. Table AI in the Appendix shows the parameters to be identified for the model. The parameters to be estimated are: the separator resistance, along with the particle radii, diffusion coefficients, reaction rates and active mass of both the cathode and the anode, electrode area, separator porosity and the tortuosity of the cathode, anode and separator.

Thus, the total number of RK coefficients is 29 for both the anode and the cathode. As previously mentioned, the parameter estimation problem is solved by minimizing the L 2 -norm of the difference between the calculated and measured voltage curves. The calculated curve is obtained by solving the mathematical model while the measured voltage curve is obtained experimentally.

The minimization is performed using the SOHO algorithm. The SOHO algorithm will search for the parameters, within a user-specific bound, that best minimizes this L 2 -norm.

The lower and upper bounds for each of the parameters to be estimated in the model is given in Table AI. It should be mentioned that the bounds on each variable are conservative and larger than usual. This is to mimic the lack of prior knowledge about the parameters. Owing to the large allowable range for most parameters, the optimization algorithm should first efficiently search a large parameter space but then must focus its search on a smaller region where there is a greater chance of finding the global minimum.

In this respect, Forman et al. Solving two separate optimization problems greatly increases the computational cost and time. This work makes efficient use of the recombination operators to solve both optimization problems in a single run. A higher value of each index leads to a solution that is closer to its parents. The trade-off between global and local search is controlled by adapting the distribution indices as a function of generations.

Each distribution index linearly increased from a value of 1 to 50 as function of generations. This leads to a more global search at the beginning which then gradually becomes a local search. The SOHO algorithm was run for a total of 1, generations, although in all cases, the minimum was found in less than generations. The SOHO algorithm is parallelized in a master-slave arrangement.

The master node performs all optimization computation recombination, selection, etc. A total of parallel runs i. The maximum allowable time was set as twice the average computing time. This greatly reduces computing time as infeasible parameter combinations runs are not evaluated. These termination criteria add additional degree of non-linearity and discontinuity to the cost function space.

For this reason, a gradient based method will find it very difficult to converge to the correct values of the model parameters. The SOHO algorithm is not affected by any of these function space modifications.

Previous results show that a single particle model is able to accurately model the battery response. For certain cases, e.

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