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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 754653, 12 pages

http://dx.doi.org/10.1155/2013/754653

## Parallelized Genetic Identification of the Thermal-Electrochemical Model for Lithium-Ion Battery

School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China

Received 9 September 2013; Revised 21 October 2013; Accepted 21 October 2013

Academic Editor: Xiaosong Hu

Copyright © 2013 Liqiang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The parameters of a well predicted model can be used as health characteristics for Lithium-ion battery. This article reports a parallelized parameter identification of the thermal-electrochemical model, which significantly reduces the time consumption of parameter identification. Since the P2D model has the most predictability, it is chosen for further research and expanded to the thermal-electrochemical model by coupling thermal effect and temperature-dependent parameters. Then Genetic Algorithm is used for parameter identification, but it takes too much time because of the long time simulation of model. For this reason, a computer cluster is built by surplus computing resource in our laboratory based on Parallel Computing Toolbox and Distributed Computing Server in MATLAB. The performance of two parallelized methods, namely Single Program Multiple Data (SPMD) and parallel FOR loop (PARFOR), is investigated and then the parallelized GA identification is proposed. With this method, model simulations running parallelly and the parameter identification could be speeded up more than a dozen times, and the identification result is batter than that from serial GA. This conclusion is validated by model parameter identification of a real LiFePO_{4} battery.

#### 1. Introduction

In recent years, lithium-ion battery has been widely used in electric vechiles due to its relatively high energy and power density. The accurate estimate of battery life or State-Of-Health (SOH) becomes key issues and attracts more and more interest from researchers. The ultimate purpose of our research is to build a Prognostic and Health Management (PHM) characteristics library of Lithium-ion batteries by investigating their internal parameters at different ageing stages [1]. It would be used in the SOH estimation of electric vehicles in the further. The core work is to develop a numerical model which well predicts the battery behavior and identify the model parameters fast and accurately.

There are several existing models for Lithium-ion battery, such as equivalent circuit model (ECM), single-particle model (SPM), and pseudo-two-dimensional (P2D) model. Their complexity, reliability of predictions, and computational requirements are compared in Figure 1.

Many types of ECM were proposed [2] and successfully used for State-of-Charge (SOC) estimation of electric vehicles [3]. Simple structure, fewer parameters, and easy implementation make it popular in electric vehicle applications. But the ECM cannot predict the internal properties due to its lack of physical and chemical process description, and the physical meaning of the parameters is not very clear. So it has some limitations for building our PHM characteristics library. The SPM is a simplified form of P2D model; it can predict the diffusion and intercalation process but the concentration and potential effects in the solution phase are neglected [4]. Although we extended the electrolyte concentration and electrochemical reaction distribution effect into SPM and proposed the extend-SPM, its predictability is still inferior to the P2D model [5, 6], and the thermal behavior has not been included yet making it nonideal for PHM characteristics library studies. P2D model is a type of first principle model, proposed by Doyle et al. firstly [7, 8] and improved by many researchers. It addresses the complex mechanism of physical and chemical processes such as diffusion, transport of ion, ohmic phenomenon, and electrochemical reaction using a group of partial differential equations and some algebraic equations.

The parameters of P2D model have corresponding physical meaning; they can be treated as the indicator of the battery health status and used as PHM characteristics. Zhang and White [9] used the stoichiometric number of electrode material to indicate the stages of capacity fade. Schmidt et al. [10] investigated the relationship between the parameter , and the cycle number during ageing and used them as characteristics to evaluate the SOH of batteries. Ramadesigan et al. [11] found that the parameters and degrade with the cycle number following the power-law. These literatures present some isolated relationships between few parameters and battery health. Our research, by contrast, would investigate a large set of model parameters; therefore a rapid, accurate, and noninvasive identification method is necessary.

Although many different methods have already been reported on parameters identification, it is very difficult to identify the parameters of P2D model, due to its highly nonlinear, large numbers of parameters, complex form, and highly computational complexity, because it can only be implemented by numerical method. Parameter identification of P2D model with genetic algorithms consumes huge computing resource; for example, in [12], although the author optimized the GA and the P2D model, it still took three weeks to complete the optimization of 88 parameters with five quad-core computers (Intel Q8200). This identification speed severely restricts our research process; therefore, we need a high-speed parameter identification scheme with our existing computing resource.

The remainder of this paper is organized as follows. In Section 2, the P2D model is improved to thermal-electrochemical model by coupling the temperature-dependent parameters with thermal effect. Section 3 discusses the time consumption problem of genetic identification with thermal-electrochemical model and then proposes a parallelized GA with MATLAB; speedup performance using different parallelized methods is also investigated. Section 4 presents validation studies for the parallelized genetic identification using a real LiFePO_{4} battery.

#### 2. Thermal-Electrochemical Model

Figure 2 gives the schematic of a lithium-ion battery. It consists of two current collectors, a negative electrode (anode), separator, and a positive electrode (cathode). Both the electrodes and the separator are porous structure. Two inner boundaries (anode/separator interface 2 and separator/cathode interface 3) and two external boundaries (Cu/anode interface 1 and cathode/Al interface 4) are also shown. The electrochemical reactions occurring in the electrodes during discharge and charge processes can be expressed as follows
(1)
where MO_{2} stands for metallic oxide such as CoO_{2} or FePO_{4} and and are the stoichiometric number of anode and cathode which is the ratio of Li^{+} concentration and Li^{+} maximum concentration of active material.

During the discharge process, lithium ions (Li^{+}) diffuse from the inside of anode active material to the surface and then are released into the electrolyte by electrochemical reaction, while an equal number of electrons (e^{−}) are also released. Lithium ions transport in the electrolyte and go towards the cathode through the separator, then insert into cathode active materials. By design, the electrons cannot pass through the separator and so they migrate to the current collectors and the external circuit, providing the electrical work. This is known as “rocking chair” because in charge process these phenomena occur inversely. These physical and chemical processes are described by several partial differential equations in Section 2.1 based on Doyle at al.’s work [8].

On the other hand, some thermal phenomena such as energy conservation, heat generations, and exchange also occur during charge/discharge process, and they can be calculated in P2D model. In addition, some model parameters (e.g., diffusion coefficient, conductivity, and reaction rate) depend on the battery temperature; see Section 2.2 for details. A thermal-electrochemical coupled model is presented by feedback the relationship between temperature dependent parameters and thermal effects to P2D model. This model has great advantages as follows.(1)The battery temperature which can influence some parameters and then influence the terminal voltage is considered. It makes the model more predictable compared to others.(2)Some parameters which depended on battery temperature are updated automatically; the relationships between the parameters and temperature are changeless and can be obtained by parameter identification.(3)The parameter set of thermal-electrochemical model not only comes from the P2D model but also contains some thermal parameters. They have corresponding physical chemistry or thermal meaning and can be used for PHM study. It makes the developed model very suitable for our further research.

##### 2.1. Electrochemical Part

###### 2.1.1. Faraday’s Laws

The electrochemical reaction current density is used to describe the wall flux of Li^{+} on the intercalation particle of electrode as follows:
where is the charge number of lithium ion (all the variables are described in the nomenclature section).

So the ionic current density in solution phase modulation is determined by where is specific surface area of active material particles defined as . And the electron current density in solid phase is

The applied current of external circuit is equal to the product of electron current density at boundary 1 or 4 and the area of electrode as follows:

The total current density inside the battery is composed by electron current density and ionic current density.

###### 2.1.2. Charge Balance and Ohm’s Law

In active materials, the solid phase potential is calculated with Ohm’s law: where effective conductivity of solid phase . is 0 at boundary 1 and the charge flux is isolation at boundaries 2 and 3, since there is no charge flux.

In electrolyte, the solution phase potential is determined by Ohm’s law and ionic transport of charges:

Liquid-junction potential is introduced with expression and is effective conductivity of solution phase. is taken to be continuous at boundaries 2 and 3, while it is set as isolation at boundaries 1 and 4.

###### 2.1.3. Mass Balance and Transport Process

The mass balance of Li^{+} in a spherical particle of active material is described by Fick’s law:
where is the distance from the center of the particle. The lithium concentration at the surface is coupled to and the reaction current density, and the Li^{+} flux is set to zero at the center of the particle, because there is no species source.

The Li^{+} concentration in solution, which varies due to diffusion and reaction at the electrode/electrolyte interface, is calculated from the following equation:

Diffusion takes place in the three regions (anode, separator, and cathode). Current collectors present an impermeable wall to the electrolyte, so the Li^{+} flux is null at external boundaries 1 and 4 and shows a continuity condition at inner boundaries 2 and 3. At initial time, the electrolyte concentration is equal to a constant value .

###### 2.1.4. Electrochemical Kinetics

The electrochemical reaction which occurred at the surface of active material is described by Butler-Volmer equation: where and are transfer coefficients of anode and cathode, is overpotential, which is defined as

is the resistance of SEI film, and the is equilibrium potential, which is a function of the solid phase Li^{+} concentration at particle surface. The of MCMB/LiFePO_{4} system can be obtained from [13].

Exchange current density acts as a bridge connecting concentrations in both solid and liquid phases and indicates the ability of electrode polarization. Its calculation is as follows: where is the electrochemical reaction rate.

All the above equations should be solved with a numerical method, and then the terminal voltage of the battery is calculated by where stands for the contact resistance between electrode and current collection.

##### 2.2. Thermal Part

###### 2.2.1. Energy Balance of the Battery

The energy balance equation contains three terms, namely, the heat capacity, the heat conduction, and the heat generation, respectively, where is the volume averaged density of the battery, is the averaged heat capacity, and is effective thermal conductivity.

The boundary condition of heat conduction term in (14) describes the heat exchange between the battery and ambient: where is the heat exchange rate.

###### 2.2.2. Heat Generation

The heat generation rate is defined by and contains electrochemical reaction heat, entropic heat, and ohm heat. They are calculated using the following equation.

For electrochemical reaction heat, one uses

For entropic heat, one uses
where is the entropy coefficient of electrode, related to the stoichiometric number. The entropy coefficient of MCMB/LiFePO_{4} system can be obtained from [17].

Ohm heat could be divided into three parts: The first part is solid phase ohm heat, generated by the electron current in active material, the second part is the sum of solution phase ohm heat and ionic migration heat, and the third part is the ohm heat caused by contact resistance: where is the total thickness of the battery, and .

###### 2.2.3. Heat Exchange

Heat exchange between the battery and ambient is composed by convection and radiation. Heat convection is expressed by Newton’s cooling law: while the heat radiation is expressed by the Stefan-Boltzmann equation: where is heat transfer coefficient, is the emissivity, is the Stefan-Boltzmann constant, and is ambient temperature.

###### 2.2.4. Temperature-Dependent Parameters

Some parameters in P2D model, such as solid phase diffusion coefficients and , solution phase conductivity , solution phase diffusion coefficient , and electrochemical reaction rates and , are coupled with the battery temperature. These parameters can be updated by Arrhenius’ law as follows: where represents the parameters mentioned above, is the parameter value at reference temperature , and denotes the corresponding activation energy of parameter , respectively.

In addition, the equilibrium potential depends on battery temperature according to the Nernst equation: where is the equilibrium potential at reference temperature.

##### 2.3. Model Implementation

Many methods have been reported on model implementation so far, using several kinds of software or language, such as FORTRAN, C++, MAPLE, and MATLAB. The most common numerical method is finite-difference method (FDM); finite-element method (FEM) and finite-volume method (FVM) are also used by some researchers. Some of them are shown in Table 1.

The FORTRAN code with BANDJ solver is optimum for us, because it has the highest computational efficiency and can be embedded into MATLAB for further usage.

We improved the DUALFOIL, which is a freely available FORTRAN code maintained by Newman’s research group, by adding subroutines of distributed heat generation, heat exchange, battery temperature computation, and parameter update. This program can simulate the value of some internal properties (e.g., concentration distribution of electrolyte, reaction current distribution, etc.) and external performances (e.g., terminal voltage and battery temperature) when inputting a set of parameters and an operating condition.

It takes about 8 seconds to simulate the 1C discharge process on a PC with Intel Core i3-530 CPU (2.93 GHz) using this improved program. The simulation time is relevant to the operating condition time, and it might become slower when the program becomes hard to convergence, because in this case the step size will become smaller automatically according to the variable step algorithm.

#### 3. Parallelization for Parameter Identification

##### 3.1. Genetic Identification

The model parameter set includes design and geometry parameters, kinetic and transport parameters, electrolyte parameters, and some temperature related parameters. Simulation data will fit the experimental data only with the accurate model parameters which can be identified by an optimization algorithm.

Firstly, a real battery is tested on DST condition, which contains various discharge and charge rates and some rest periods; then, the terminal voltage is recorded as experimental data. Secondly, model simulation with a set of parameters is running on the same DST condition, and the model output is also recorded. The sum of squared errors (SSE) of two data sets is treated as the optimization objective: where is the experimental data for DST test, is the simulation data with parameter set , and is the actual time of DST test. The final objective is to find the optimal parameter set which makes the SSE minimum.

Genetic algorithm (GA) is a kind of stochastic searching algorithm, has shown exceptional problem-solving ability for complex problems, and is used for parameter identification here. Figure 3 provides a high-level snapshot of genetic identification scheme.

Genetic identification starts with the initialization of a randomized population, where each member is a model parameter set. Then, the offspring members are produced by the genetic operation such as recombination and mutation from parent members and their objective functions are calculated. After a fitness-weighted roulette game, a new population which contains the fittest members who have the minimum objective function value will be created. This process is repeated until converge to a minimal SSE level or evolve to the maximum generation. The optimal parameter set is obtained from the fittest member of the final population. For details of GA, the reader is referred to [18].

The Genetic Algorithm Toolbox [19] is used for programming the parameter identification in MATLAB. The main time consumption of this program is determined by the call times of objective functions, while other codes consume very little time compared to the former. If we set the population size *NIND* = 200, generation gap *GGAP* = 0.8, and the maximum generation *MAXGEN* = 200, the objective function will be called *NIND* + *NIND* × *GGAP* × *MAXGEN* = 32200 times during identification. It takes 12 s on average to simulate the DST test, so 32200 times call will take about 107 hours. Larger population consumes more identification time; it is unacceptable for our further work.

##### 3.2. Parallelization Based on MATLAB

###### 3.2.1. Build a Computer Cluster

MATLAB provides Parallel Computing Toolbox (PCT) and Distributed Computing Server (DCS), with which a computer cluster could be built with several PCs or servers on a local area network, as shown in Figure 4. Many workers are started on hosts and administrated by a MATLAB job scheduler (MJS). Each worker receives a task from MJS, executes the task, and returns the result to the MJS. When all tasks have been done, the MJS returns the final result to the client session. Existing programs of parameter identification could run in the computer cluster with only slight modification using PCT.

Table 2 shows available computing resources; there are five PCs and one server. The computing powers of these 20 cores are almost the same. We built a computer cluster by the following steps.

*Step 1. *Install MATLAB software and start *mdce* service on all hosts.

*Step 2. *Start one worker on each core and a MJS on host HPC with Admin Center GUI.

*Step 3. *Define a cluster profile in MATLAB and validate it.

Host HPC and LiuC are used for computing dedicatedly, while other four hosts are used daily by students concurrently. Two workers run in the background and utilize 50% of CPU time on each host; there is very little influence on foreground programs. In this way, surplus computing resources in the laboratory can be used efficiently.

###### 3.2.2. Parallelization for GA

In genetic identification, most of the time is consumed by the objective function subroutines, because the traditional program serially calculates each objective function of all members, the total computation time is equal to the sum of each individual computation time of the objective functions. The pseudocode is shown in Pseudocode 1 where ObjFun is the subroutine of the objective function calculation, in which the thermal-electrochemical model is called once, *Chrom* is the population, *Chrom*(*i*) stands for the th member in the population, which is a parameter set, and *ObjV* is the objective function values.

We found that the objective function calculation of each member is independent of others; in other words, the calculation order of them is indifferent. If the subroutines of objective function run parallelly, the time consumption will be reduced observably. There are two methods to parallelize the GA in MATLAB; they are Single Program Multiple Data (SPMD) and parallel FOR loop (PARFOR). Figure 5 shows the flowchart of 200 objective functions parallelly running on 20 workers with two methods.

SPMD method allocates tasks to all workers at the very beginning, and each worker executes the task respectively and submits the result when task finished. The pseudo code is shown in Pseudocode 2 where is the actual number of jobs running on each worker, means 10 serial ObjFun subroutines, *labindex* is an automatically assigned variable in SPMD segment, indicating the number of the worker.

PARFOR is an evolution form of ordinary FOR loop, it can allocate tasks automatically and execute the FOR loop body parallelly. Two nested loops were used to divide 200 members into 10 groups in this paper. In each group, 20 objective functions are running on 20 workers at the same time using PARFOR, but different groups are running serially, a group will never start unless all workers in previous group finish their tasks. The pseudo code is shown in Pseudocode 3 where *Chromtemp* and *ObjVtemp* is temporary data used for grouping.

Theoretically, the running time of above two methods is consumed by 10 times of objective function calculation and some communication time of workers and client.

##### 3.3. Comparison of Different Parallelization Methods

A population initialization subroutine was used for verifying the performance of parallel GA. This subroutine only contains objective function and no genetic operation. The average running time is 2437.1 s, equal to the total computing time of 200 DST simulations, approximately. Table 3 and Figure 6 show the time consumption and speedup ratio of different methods on the cluster with different numbers of workers. The running time is an average value of three independent experiments, and the speedup ration is defined as the ratio of serial time consumption to parallel time consumption.

It is seen that the decreasing trend of time consumption presents a reciprocal law to the number of workers, and the speedup ratio increases with a good linear relationship. SPMD method on 20 workers could reduce the running time more than 14 times. If we have more PCs and workers, the time consumption will be much more reduced.

In addition, SPMD method consumes less time compared to PARFOR method, because of the different mechanism of task allocation of the two methods. Figure 7 shows the difference between the two methods using three workers and nine members.

Time consumption of the DST simulation is affected by the model parameter set, and some improper parameters lead to smaller iteration step size and more simulation time. This phenomenon is especially obvious for initial population because there are many improper parameters generated randomly by GA. In Figure 7, the total time consumption of SPMD method is caused by the slowest worker and once communication, that is, T1 + T2 + T3 + Tc, while the total time consumption of PARFOR method is the sum of time consumed by all groups and three communication periods, that is T2 + T4 + T8 + 3Tc; some CPU time is wasted in the latter method. The computing time of these two methods will be the same only if the slowest members of each group were allocated to the same worker; however, this probability is very little, so parallelized genetic identification with SPMD method is always faster than that with PARFOR method.

#### 4. Parallelized Genetic Identification of a **LiFePO**_{4} Battery

_{4}

Parameters of a real 2.3Ah LiFePO_{4} battery (type ANR26650m1A, manufactured by A123 systems) were identified on computer cluster using parallelized GA. Our previous work [20] has clarified which parameters have to be identified and their searching region. Some of these 27 parameters (such as , and ) may change during ageing; the identification result can be used for the research of PHM characteristics library, and others (such as and ) must be identified to get its values because they are hard to be measured in another way.

The DST test at 25°C is used for identification and the experimental data was acquired by a Battery Testing System (type BTS-5V-20A, manufactured by Neware, China).

SPMD parallelization method is used for identification. We set *NIND* = 600 instead of 200 to improve the search effect, because much parameters and large searching region needs larger population diversity, maximum generation was set to *MAXGEN* = 200. So the estimated time for serial identification is 322 hours, about 2 weeks. Table 4 shows the searching region and identification results with parallelized GA; finally, it takes 22.3 hours to finish this on the computer cluster.

As a comparison, the parameters were also identified with serial GA in three different strategies. In order to control the identification time within 24 hours, the population size and maximum generation were set as in Table 5, the product of *NIND* and *MAXGEN* is about 1/15 of those of parallelized GA, and the final time consumption is also shown.

Figure 8 shows the comparison of identification results with different strategies and the experimental data. Simulation curves with parallelized GA result agree very well with the experimental data; while simulation curves from serial GA will fit the experimental data only at low rate discharge and rest periods, the voltage error is very large at high rate charge and discharge periods because, under this condition, the simulated terminal voltage is very sensitive to some model parameters; inaccurate identified parameters create large error.

Figure 9 shows the voltage error distribution and Figure 10 shows the percentile of absolute voltage error of four identification strategies. For parallelized GA, the 50% percentile of voltage absolute error is 3.16 mV, the 80% percentile of voltage absolute error is 9.7 mV, and the 95% percentile of voltage absolute error is 14.3 mV, while the maximum absolute error is 25.4 mV. For serial GA, voltage errors distribute more dispersed, and the maximum error is much larger, as shown in Table 6. Identified results from serial GA show much worse than that from parallelized GA.

Parallelized GA shows much more advantages than serial GA, because it can finish the calculation of larger population and more evolution generations in the same period of time. Larger population contains more gene patterns, making it much easier to find the global optimum and to avoid falling into local optimum or premature convergence; more evolution generations make the resulting more convergent and closer to the global optimum. There is no theoretical method to determine the detail of *NIND* and *MAXGEN* for GA; they are usually determined by experience. Generally, greater *NIND* and *MAXGEN* will lead to a better identified result but more computing time.

This case shows that the parallelized GA has better performance than serial GA within the same computing time, which is evident from the smaller errors in the voltage. That makes it possible to build the PHM characteristics library based on the internal parameters which identified from the batteries on different ageing pattern or ageing stage.

#### 5. Conclusions

In this paper, a thermal-electrochemical model is improved and a parallelized GA is presented to identify the parameters of lithium-ion battery. First of all, the P2D model is expanded by coupling the temperature-dependent parameters with thermal effect. Then, a GA based parameter identification is proposed and we find that its time consumption is unacceptable for our further research. Secondly, a computer cluster built by several hosts with MATLAB Distributed Computing Toolbox and Parallel Computing Toolbox is presented to speedup the identification. The performance of SPMD and PARFOR methods is also discussed, and the simulation experiments show that parallelized GA with SPMD method can speedup the identification more than 14 times on a cluster of 20 workers. Finally, parameters of a real LiFePO_{4} battery are identified parallelly, the result fits the experimental data very well, and the voltage error never exceeds 25.4 mV, which is much smaller compared to those from serial GA which consumes the same time. It takes 22.3 hours to finish parallelized genetic identification, which speeds up our further research significantly.

#### Nomenclature

: | Specific surface area (m^{−1}) |

: | Area of electrode (m^{2}) |

: | Electrolyte concentration (mol m^{−3}) |

: | Concentration of Li^{+} in the intercalation particle (mol m^{−3}) |

: | Concentration of Li^{+} at the surface of intercalation particle (mol m^{−3}) |

: | Specific heat capacity (J kg^{−1} K^{−1}) |

: | Electrolyte diffusion coefficient (m^{2} s^{−1}) |

: | Li^{+} diffusion coefficient in active material (m^{2} s^{−1}) |

: | Activation energy (KJ mol^{−1}) |

: | Open circuit potential of electrode (V) |

: | Faraday’s constant (=96487 C mol^{−1}) |

: | Heat transfer coefficient (W m^{−2} K^{−1}) |

: | Total current density in electrodes or separator (A m^{−2}) |

: | Applied current (A) |

: | Exchange current density (A m^{−2}) |

: | Solid phase current density (A m^{−2}) |

: | Solution phase current density (A m^{−2}) |

: | Electrochemical reaction current density (A m^{−2}) |

: | Wall flux of Li^{+} on the intercalation particle of electrode (mol m^{−2} s^{−1}) |

: | Electrochemical reaction rate (m^{2.5} mol^{−0.5} s^{−1}) |

: | Thickness of electrodes or separator (m) |

: | Heat exchange rate (W m^{−2}) |

: | Heat generation rate (W m^{−3}) |

: | Ideal gas constant (=8.3143 J mol^{−1} K^{−1}) |

: | Contact resistance (Ω m^{2}) |

: | SEI film resistance (Ω m^{2}) |

: | Intercalation particle radius of electrode (m) |

: | Li^{+} transference number in the electrolyte |

: | Battery temperature (K) |

: | Ambient temperature (K) |

: | Applied potential or terminal voltage of the battery (V) |

: | Stoichiometric number of anode |

: | Stoichiometric number of cathode. |

*Greek Symbols*

: | Transfer coefficient of the electrochemical reactions |

: | Thickness of current collector (m) |

: | Emissivity |

: | Volume fraction of electrolyte in electrode or separator |

: | Volume fraction of active material in electrode |

: | Overpotential (V) |

: | Ionic conductivity of the electrolyte (S m^{−1}) |

: | Thermal conductivity (W m^{−1} K^{−1}) |

: | Density (kg m^{−3}) |

: | Stefan-Boltzmann constant (=5.6704 × 10^{−8} W/m^{2}/K^{4}) |

: | Solid phase conductivity (S m^{−1}) |

: | Solution phase potential (V) |

: | Solid phase potential (V). |

*Subscript*

: | Anode |

: | Cathode |

: | Separator |

0: | Initial value |

MAX: | Maximum value |

ref: | Reference value. |

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was financially supported by the National Natural Science Foundation of China (no. 51107021) and the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2014021). The authors sincerely appreciate the significant help with building the computer cluster by Mr. Chong Niu, currently at Xi’an Jiaotong University.

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