energies Article
Impedance Characterization and Modeling of Lithium-Ion Batteries Considering the Internal Temperature Gradient Haifeng Dai 1,2, *, Bo Jiang 1,2 and Xuezhe Wei 1,2 1 2
*
National Fuel Cell Vehicle & Powertrain System Research & Engineering Center, No. 4800, Caoan Road, Shanghai 201804, China;
[email protected] (B.J.);
[email protected] (X.W.) School of Automotive Studies, Tongji University, No. 4800, Caoan Road, Shanghai 201804, China Correspondence:
[email protected]; Tel.: +86-21-6958-3847
Received: 30 December 2017; Accepted: 13 January 2018; Published: 17 January 2018
Abstract: Battery impedance is essential to the management of lithium-ion batteries for electric vehicles (EVs), and impedance characterization can help to monitor and predict the battery states. Many studies have been undertaken to investigate impedance characterization and the factors that influence impedance. However, few studies regarding the influence of the internal temperature gradient, which is caused by heat generation during operation, have been presented. We have comprehensively studied the influence of the internal temperature gradient on impedance characterization and the modeling of battery impedance, and have proposed a discretization model to capture battery impedance characterization considering the temperature gradient. Several experiments, including experiments with artificial temperature gradients, are designed and implemented to study the influence of the internal temperature gradient on battery impedance. Based on the experimental results, the parameters of the non-linear impedance model are obtained, and the relationship between the parameters and temperature is further established. The experimental results show that the temperature gradient will influence battery impedance and the temperature distribution can be considered to be approximately linear. The verification results indicate that the proposed discretization model has a good performance and can be used to describe the actual characterization of the battery with an internal temperature gradient. Keywords: lithium-ion battery; impedance characterization; temperature gradient; discretization model
1. Introduction Currently, the world faces increasingly severe environmental problems, including the greenhouse effect and industrial pollution [1]. It is also beset by the shortage of resources, especially fossil fuel energy. To resolve these urgent problems, people consider green cars including battery electric vehicles (BEVs), fuel cell electric vehicles (FCEVs), hybrid EVs (HEVs), and so on, to be critically important [2]. As key components of green cars, batteries have attracted great attention in recent years. Among the current commercial batteries, lithium-ion batteries (LIBs) are considered as the best choice for automobile application, due to their high power/energy density, long life cycle, safety, and environmental friendliness [3–5]. However, safe operating conditions are hard to ensure for LIBs, so the battery management system (BMS) is essential when LIBs are used in vehicles [6,7]. To monitor and better manage LIBs in the battery pack, the pack voltage, pack current, cell voltage, cell temperature, cell impedance and so on must be detected. Battery impedance has been regularly employed to investigate battery parameters and states. Buller et al. [8] proposed an impedance-based simulation model for supercapacitors and LIBs, and the
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results showed that the simulated and measured voltage data were in excellent agreement. Xu et al. [9] analyzed the impedance spectra of LIBs and derived a simplified impedance model for the estimation of battery state of charge (SOC). Based on the different value of the simplified equivalent circuit elements at different SOCs, Westerhoff et al. [10] developed a novel method to estimate the battery SOC, and this method could be considered as complementary to the ampere-hour integral method. Moreover, battery impedance has been employed to estimate the battery’s internal temperature and determine the thermal model parameters [11–14]. Considering the importance of battery impedance, many researchers have studied the impedance characterization during practical applications. Schönleber et al. [15] derived the impedance of a LIB electrode from the intercalation reaction of lithium and studied its dependency on SOC, and the results showed that the electrolyte diffusion impedance is independent of the SOC. Huang et al. [16] systematically analyzed how different SOCs of 7.5~93.0% can influence battery impedance, and the results indicated that both charge transfer resistance and diffusion resistance increased dramatically when SOC is below 26.5%. Jiang et al. [17] investigated the impedance of batteries with different life cycles (up to 520 cycles) and found that when the frequency is higher than 100 Hz, the impedance phase of different, aging batteries would be different to each other. Other than the influence of SOC and battery aging status on battery impedance, battery temperature also has a great impact on impedance. Andre et al. [18] investigated the influence of temperature and SOC on LIBs and, through the cell impedance, the non-linear correlation between temperature and impedance was found. Raijmakers et al. [19] studied the impedance characterization of a pristine nickel cobalt aluminum oxide battery (NCA) cell at different temperatures and found that ohmic resistance would decrease as the temperature increases and inductive characteristics barely change. Xiao et al. [20] found that the battery impedance is sensitive to temperature because the ions’ mobility and permittivity are dependent on temperature. During operation, heat generation caused by charge/discharge may contribute to the temperature rise, and undoubtedly, the temperature rise is inhomogeneous. Usually, the cell internal temperature will always be higher, and the temperature will decrease from inside to outside, which can be called the temperature gradient. It is a common phenomenon, but very few studies have investigated the influence of the temperature gradient. In references [21,22], a lumped-parameter electro-thermal model was proposed and applied. The parameters of the electrical model were modified according to the average battery temperature. However, the core temperature was obviously higher than the surface temperature, and the influence of the temperature gradient was ignored. The influence of the temperature gradient on an LIB’s impedance characterization has not been extensively studied either. Troxler et al. [23] designed an artificial temperature gradient for the battery and analyzed the performance of LIBs. For a cell with similar average temperature, the larger the temperature gradient, the greater the behavior deviation from a cell with a uniform temperature will be. However, that study only focused on the charge transfer resistance. In reference [11], the overall resistance of the battery with inhomogeneous temperature distribution was similar to the resistance of the battery with homogeneous temperature distribution, and the homogeneous temperature was the mean value of the inhomogeneous temperature. The objective of this paper is to investigate battery impedance characterization considering the internal temperature gradient, and then prove that the model considering the temperature gradient will perform better. We will propose a novel discretization model and experiments with the design of artificial temperature gradients for the battery with the help of a heating plate. The remaining part of this paper is organized as follows: Section 2 will introduce the battery impedance, and a discretization model, based on the non-linear impedance model, is proposed to study the influence of the battery internal temperature gradient on impedance. In Section 3, several experiments are conducted to investigate the impact on battery impedance, and a heating plate is used to construct the artificial temperature gradient. Section 4 will provide some experimental results and discuss the influencing factors on battery impedance. The verification results of the proposed discretization model are also
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Energies 2018, 11, 220 2. Methodology
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2.1. Battery Non-Linear Impedance Model
shown in order to verify the model performance. Finally, some conclusions are drawn and summations Electrochemical impedance spectroscopy (EIS) is the method most used to obtain the provided in Section 5.
impedance characterization of LIBs, due to its accuracy and non-damaging effects. During EIS measurement, the electrochemical system is loaded with a specific amplitude but different 2. Methodology frequency sine current; after measuring the corresponding electrical feedback voltage signal, of the electrochemical 2.1. characterization Battery Non-Linear Impedance Model system will be obtained. Figure 1 shows the impedance spectroscopy of a lithium-ion battery at 50% SOC and ambient temperature if 0 °C. The Electrochemical impedance spectroscopy (EIS)tois10 thekHz. method most used obtainimpedance the impedance measurement frequency ranges from 0.01 Hz Obviously, the towhole characterization of LIBs,ofdue to main its accuracy non-damaging effects. During EIS measurement, spectroscopy consists three regions: and the low-frequency region, middle-frequency region and the high-frequency region [9]. Thewith low-frequency region ranges from 0.01 Hz to 0.16 the after electrochemical system is loaded a specific amplitude but different frequency sineHz, current; middle-frequency region ranges from 0.16 Hz to 630 Hz, and the high-frequency region ranges from measuring the corresponding electrical feedback voltage signal, characterization of the electrochemical 630 Hz 10obtained. kHz. system willtobe Figure 1 shows the impedance spectroscopy of a lithium-ion battery at 50% In the low-frequency region, impedance spectroscopy looks like a straight line with a constant SOC and ambient temperature if 0 ◦ C. The measurement frequency ranges from 0.01 Hz to 10 kHz. slope, a typical phenomenon of diffusion [24]. In the middle-frequency region, the impedance Obviously, the whole impedance spectroscopy consists of three main regions: the low-frequency region, spectroscopy is a depressed semicircle caused by the charge transfer reaction at the electrode middle-frequency and high-frequency region [9]. The low-frequency from 0.01 surfaces. In theregion high-frequency region, the impedance spectroscopy forms aregion small ranges curve and Hz to 0.16 Hz, thethe middle-frequency ranges from 0.16 Hz tothe 630battery’s Hz, andinductance the high-frequency intersects with abscissa axis. Theregion high-frequency region reflects and region ranges from 630 Hz to 10 kHz. ohmic resistance.
Figure1.1.Impedance Impedance spectroscopy ofof a battery at 50%, 0 °C.0 ◦ C. Figure spectroscopy a battery at 50%,
According to the above battery impedance spectroscopy and the battery equivalent circuit In the low-frequency region, impedance spectroscopy looksinlike a straight line with a constant models widely used, we employ the following model shown Figure 2 to analyze battery slope, a typical phenomenon of diffusion [24]. In the middle-frequency region, the impedance impedance. The model consists of four parts to represent the physical and electrochemical reaction spectroscopy is a depressed semicircle caused Lbyand theresistance charge transfer reaction at the electrode processes inside the battery. The inductance Ro describe the battery behaviors surfaces. at high frequency, where the the resistance Ro equals the valueforms of thea small intersection theintersects impedance In the high-frequency region, impedance spectroscopy curve of and with the spectroscopy and the abscissa axis. To better capture the non-linear behavior of ohmic the battery, a new abscissa axis. The high-frequency region reflects the battery’s inductance and resistance. equivalent circuit element with a constant, frequency-independent phase has been widely applied According to the above battery impedance spectroscopy and the battery equivalent circuitinmodels lithium-ion equivalent models [23–25]. middle-frequency region is modeled a constant The widely used, we employcircuit the following modelThe shown in Figure 2 to analyze batterybyimpedance. phase element (CPE) (CPE1) and a resistance (R1) connected in parallel. In the low-frequency region, model consists of four parts to represent the physical and electrochemical reaction processes inside the another CPE (CPE2) instead of a Warburg impedance is used to describe the diffusion battery. The inductance L and resistance Ro describe the battery behaviors at high frequency, where the characterization, because the Warburg impedance may lead to fitting failure and a CPE can also resistance Rothe equals the value of intersection of the impedance spectroscopy and the abscissa axis. produce characterization of the an infinite-length Warburg element [25].
To better capture the non-linear behavior of the battery, a new equivalent circuit element with a constant, frequency-independent phase has been widely applied in lithium-ion equivalent circuit models [23–25]. The middle-frequency region is modeled by a constant phase element (CPE) (CPE1 ) and a resistance (R1 ) connected in parallel. In the low-frequency region, another CPE (CPE2 ) instead of a Warburg impedance is used to describe the diffusion characterization, because the Warburg impedance may lead to fitting failure and a CPE can also produce the characterization of an infinite-length Warburg element [25].
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Figure 2. Equivalent circuit of the battery impedance model.
Figure 2. Equivalent circuit of the battery impedance model.
The impedance of a CPE can be expressed in fractional calculus as:
The impedance of a CPE can be expressed in fractional calculus as: Z CPE ( s ) =
1 Qsα1
ZCPE (s) =
(1)
(1)
Qsoperator; Q is the fractional coefficient; and where ZCPE is the impedance of the CPE; s is the Laplace α is the fractional order, α ∈ R , 0 ≤ α ≤ 1 . It should be noted that the CPE represents a resistance wherewhen ZCPEα is the impedance of the CPE; s is the Laplace operator; Q is the fractional coefficient; and α = 0, and represents a capacitance when α = 1. is the fractional order, α ∈ expression R, 0 ≤ α ≤of1.the It impedance should be of noted that the CPE represents a resistance when The mathematical the whole equivalent circuit model is: α
α = 0, and represents a capacitance when α = 1. R1 1 Z mod ) = Ls + Ro + of the + The mathematical expression of the el ( simpedance αwholeβ equivalent circuit model is:(2) 1+R1Q1 s
Q2 s
1L and Ro are the inductance R1 of Ls the+equivalent circuit model; where Zmodel denotes the impedance + (2) Zmodel (s) = Ro + α 1 + R1in Q1the s middle-frequency Q2 s β resistance in the high-frequency region; R1 is the resistance region; Q1 and α are the fractional coefficient and order of the CPE1; and Q2 and β are the fractional coefficient and where Zmodel denotes order of the CPE2. the impedance of the equivalent circuit model; L and Ro are the inductance
resistance in the high-frequency region; R1 is the resistance in the middle-frequency region; Q1 and α are2.2. theDiscretization fractional coefficient and order of the CPEGradient Model Considering the Temperature 1 ; and Q2 and β are the fractional coefficient and order of the CPE . 2 As mentioned above, heat will be generated during charging and discharging, and the heat will
lead to a temperature difference between core and surface. Usually, a battery consists of several electrode layers in which the positive layer and the negative layer are separated by the separator. Thementioned positive andabove, negativeheat tabswill are connected in parallel andcharging then constitute a battery. According to heat As be generated during and discharging, and the this view, we can consider one electrode layer or several electrode layers as a cell unit, and several will lead to a temperature difference between core and surface. Usually, a battery consists of several cell units form a battery. The thickness of each cell unit is small so that the temperature of each cell electrode layers in which the positive layer and the negative layer are separated by the separator. The unit can be considered as uniform. Every cell unit’s temperature is different if there exists a positive and negative tabs are connected in parallel and then constitute a battery. According to this temperature gradient internal to the battery, and every unit’s temperature can be determined if the view,internal we can temperature consider one electrode layer or several electrode layers as a cell unit, and several cell units distribution is obtained. form a battery. The thickness of each cell unit is small the temperature of eachdistribution cell unit can be In Schmidt’s [11] study, three assumptions aboutso thethat battery internal temperature considered as uniform. Every cell unit’s temperature is different if there exists a temperature gradient perpendicular to the electrodes have been made, including the constant distribution, linear internal to the battery, and everydistribution. unit’s temperature be determined if the internal temperature distribution, and polynomial Constant can distribution is that the temperature of each electrodeisisobtained. a constant equal to the mean value of the temperature of two surfaces. For polynomial distribution distribution, temperature distribution inside about the battery is a polynomial In Schmidt’sthe [11] study, three assumptions the battery internalfunction. temperature distribution In our study, we assume the temperature distribution inside the is linear. On distribution, the one perpendicular to the electrodes have been made, including the constant battery distribution, linear hand, considering the battery thickness, the curvature of the temperature distribution line is small, and polynomial distribution. Constant distribution is that the temperature of each electrode is a and it can be approximated as linear. On the other hand, the calculation of each cell unit’s constant equal to the mean value of the temperature of two surfaces. For polynomial distribution, the temperature will become more complicated if the temperature distribution line is non-linear. The temperature distribution inside the battery is a polynomial function. assumption is shown below:
2.2. Discretization Model Considering the Temperature Gradient
In our study, we assume the temperature distribution inside the battery is linear. On the one hand,Assumption considering the temperature battery thickness, the curvature of the temperature distribution line is small, 1. The distribution is linear in the direction vertical to the battery tabs if the battery far less than theas battery height. and itthickness can be is approximated linear. On the other hand, the calculation of each cell unit’s temperature will become more complicated if the temperature distribution line is non-linear. The assumption is shown below: Assumption 1. The temperature distribution is linear in the direction vertical to the battery tabs if the battery thickness is far less than the battery height.
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Figure 3 shows discretization.InInFigure Figure battery is considered as Figure 3 shows thediagram diagramof ofthe the battery battery discretization. 3a,3a, thethe battery is considered Energies 2018, 11, 220 the 5 of 19 as a continuous battery, and T i is the internal temperature and T sis is the surface temperature. In most a continuous battery, and T is the internal temperature and T the surface temperature. In most case, s i the temperature ofdiagram the two of surfaces is similar, isInthe maximum temperature. Figure 3 shows battery discretization. Figure 3a, the battery battery is considered the case, temperature of the the two surfaces isthe similar, and Ti and is theTimaximum battery temperature. According According to the above assumption, the temperature distribution is linear and symmetrical. In3b, we as a continuous battery, and T i is the internal temperature and T s is the surface temperature. In most to the above assumption, the temperature distribution is linear and symmetrical. In Figure Figure 3b, we discretize the continuous battery into individual cell units and the battery can be case, the the twointo surfaces is similar, and Tiand is the battery temperature. discretize thetemperature continuousofbattery individual cell units themaximum battery can be considered as these considered as the these several cell units in distribution parallel. Every cell unit has a uniform According to above assumption, theconnected temperature is linear and symmetrical. In several cell units connected in parallel. Every cell unit has a uniform temperature. temperature. Figure 3b, we discretize the continuous battery into individual cell units and the battery can be considered as these several cell units connected in parallel. Every cell unit has a uniform temperature.
Figure Diagram of of battery battery discretization: (a)(a) continuous battery; (b) discrete battery. Figure 3.3.Diagram discretization: continuous battery; (b) discrete battery.
In Figure 4, we a heating source on the left of the battery, anddiscrete Ts1 is the temperature of Figure 3. place Diagram of battery discretization: (a)side continuous battery; (b) battery. Inbattery’s Figure 4, placeand a heating on theofleft of the battery, and Ts1heating is the source temperature the leftwe surface Ts2 is thesource temperature the side battery’s right surface. If the of the battery’s left surface and T is the temperature of the battery’s right surface. If the heating s2 source provides enough heat anda heating the battery does produce itself, Ts1and canTs1be as the In Figure 4, we place on not the left side ofheat the battery, is considered the temperature of source enough heat the does notthe produce heat considered maximum temperature ofand theand and Ts2 is the minimum. From Ts1itself, to Ts2, Tthe temperature inside as the s1Ifcan the provides battery’s left surface Tbattery s2 is thebattery temperature of battery’s right surface. thebe heating source the battery is in a linear Every unit’s temperature can be determined we obtain Ts1 inside maximum temperature the the battery anddoes Tcell the minimum. Ts1s1 to Tbe theif temperature provides enough heatofdistribution. and battery not produce heat From itself, can as the s2 is s2 , considered Ts2. After determination of the and temperature of eachtemperature cell From unit, the unit’s impedance will be obtain maximum of the battery Ts2 is theunit’s minimum. Ts1 cell to Ts2 ,be thedetermined temperature the and battery is temperature inthe a linear distribution. Every cell can ifinside we determined. For example, in Figure 4, the impedance of the leftmost cell unit performs like the the battery is in a linear distribution. Every cell unit’s temperature can be determined if we obtain Ts1 Ts1 and Ts2 . After the determination of the temperature of each cell unit, the cell unit’s impedance continuous battery at high temperature, and the impedance of thethe rightmost cell unit acts will as the and T s2. After the determination of the temperature of each cell unit, cell unit’s impedance be like will be determined. For example, in Figure 4, the impedance of the leftmost cell unit performs continuous at low temperature. Theimpedance continuousofbattery is made of several cell units determined.battery For example, in Figure 4, the the leftmost cellupunit performs like the the continuous battery at high temperature, and the impedance of the rightmost cell unit acts as connected parallel.at high temperature, and the impedance of the rightmost cell unit acts as the continuousinbattery
the continuous battery at low temperature. The continuous battery is made up of several cell units continuous battery at low temperature. The continuous battery is made up of several cell units connected in parallel. connected in parallel.
Figure 4. Diagram of battery discretization with a heating source. Figure4.4.Diagram Diagram of with a heating source. Figure of battery batterydiscretization discretization with a heating source.
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AfterAfter dividing a battery with sourcelike likeFigure Figure 4 into N cell units, dividing a battery withananexternal external heating heating source 4 into N cell units, the the temperature of each cell unit will be as follows: temperature of each cell unit will be as follows: i −1 − i − 1 (Ts1 − T− i = T− s 2 )T ) Ti =TT s2 s1 s1 N − 1 ( T N − 1 s1
(3)
(3)
where Ti is the temperature of the ith cell unit, and the 1st is the leftmost cell unit.
where Ti is the temperature of the ith cell unit, and the 1st is the leftmost cell unit. According to the relationship of each cell unit, the impedance of the discretization model can be According to the relationship of each cell unit, the impedance of the discretization model can be listed as: listed as: 1 1 )==N ZD−Zmodel (4) D-model((ss) N (4) ∑Zi (Zs)i (s) i =1i =1
is the impedance discretizationmodel, model,and andZZi iisisthe theimpedance impedance of the ith cell cell unit. ZD-model wherewhere ZD-model is the impedance of of thethe discretization unit. In the following sections, the influence of temperature on battery impedance will be investigated. In the following sections, influence of temperature impedance will be After the relationship between Zi andthe Ti in ith cell unit is obtained, on the battery impedance of the discretization investigated. After the relationship between Zi and Ti in ith cell unit is obtained, the impedance of model will be known according to Equations (3) and (4). the discretization model will be known according to Equations (3) and (4). According to the above methodology, the flowchart of the discretization modeling of the battery According to the above methodology, the flowchart of the discretization modeling of the impedance be illustrated in Figure in 5. Figure 5. batterycan impedance can be illustrated
Figure 5. Flowchart of the discretization modeling of a battery.
Figure 5. Flowchart of the discretization modeling of a battery.
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Energies 2018, 11, 220 3. Experiment
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3. Experiment EIS experiments are undertaken to investigate the battery’s characteristics. A pouch-type LiFePO4
battery with nominal capacity of 40 Ahtois investigate used in thethe experiment, and there are 51 electrode layers EIS a experiments are undertaken battery’s characteristics. A pouch-type stacked inside the battery. size of the battery 133 × 184 mm. LiFePO 4 battery with a The nominal capacity of 40 is Ah is mm used×in16.1 the mm experiment, and there are 51 electrode layerswith stacked inside the gradients battery. The the battery isand 133 conducted mm × 16.1 mm × 184 mm. In the Experiments temperature orsize not of are designed in this section. Experiments temperature gradients or not are and conducted in this In experiment without awith temperature gradient, usually littledesigned heat is generated in the EIS section. experiment, so the experiment without a temperature gradient, usually little heat is generated in the EIS the environmental chamber temperature is considered as the battery temperature. In the experiment so gradients, the environmental chamber is considered as the battery temperature. In withexperiment, temperature a heating plate temperature is used to construct an artificial temperature gradient in the experiment with temperature gradients, a heating plate is used to construct an artificial order to obtain an accurate and qualitative result. One side of the battery is covered with the heating temperature gradient in order to obtain an accurate and qualitative result. One side of the battery is plate, and the other side is exposed to the environment. Apparently, due to heat convection, the real covered with the heating plate, and the other side is exposed to the environment. Apparently, due to temperature of the side coated with the of heating may bethe lower thanplate the heating platethan temperature, heat convection, the real temperature the sideplate coated with heating may be lower the and due to heat real temperature of the side to the environment may betohigher heating plateconduction, temperature,the and due to heat conduction, the exposed real temperature of the side exposed thanthe the environment environmental Wethe useenvironmental thermocouples to obtain the temperature and maytemperature. be higher than temperature. Weactual use thermocouples to the obtain thegradient actual temperature and the temperature gradient of the addition to this, that in the temperature of the battery. In addition to this, in order tobattery. confirmInthe assumption order to confirm the assumption that the temperature gradient is linearly distributed, two temperature gradient is linearly distributed, two thermocouples are placed inside the battery. thermocouples arethe placed insideof the battery. Figure 6 shows position the thermocouples inside the battery. T-type thermocouples are Figure 6 shows the position of the thermocouples inside the battery. T-type are used in the experiments, and the diameter of the thermocouples is 0.28 mm.thermocouples One thermocouple is used in the experiments, and the diameter of the thermocouples is 0.28 mm. One thermocouple is placed in the halfway point within the battery, and another is situated at the quarter point within the placed in the halfway point within the battery, and another is situated at the quarter point within the battery. The battery cell under experiment is shown in Figure 7a. battery. The battery cell under experiment is shown in Figure 7a.
Figure 6. Position of thermocouples the thermocouples inside battery: (a) quarter point of the battery; Figure 6. Position of the inside the the battery: (a) quarter point of the battery; (b) (b) halfway halfway within the battery; (c) diagram of the thermocouples inside the battery. within the battery; (c) diagram of the thermocouples inside the battery.
Figure 7 shows the EIS experimental setup. An electrochemical workstation, as shown in Figure Figure 7 shows EIS experimental setup. workstation, shown in Figure 7b, 7b, including thethe electrochemical interface and An the electrochemical frequency response analyzer, isas used to generate including the electrochemical interface and the frequency response analyzer, is used to generate the sinusoidal current and then analyze the frequency response of the cell in order to obtain the the sinusoidal currentAand then analyze the7c) frequency response of thecurrent cell ingenerated order to by obtain battery impedance. power booster (Figure can enlarge the sinusoidal the the electrochemical A temperature station 7d), including the wirelessby the battery impedance.workstation. A power booster (Figuremeasurement 7c) can enlarge the(Figure sinusoidal current generated temperature workstation. unit and the wireless logging station, is used to obtain temperature and log the data. electrochemical A temperature measurement stationthe (Figure 7d), including the wireless The battery cell is placed in an environment chamber, as shown in Figure 7e. temperature unit and the wireless logging station, is used to obtain the temperature and log the data.
The battery cell is placed in an environment chamber, as shown in Figure 7e. 3.1. Experiments without the Temperature Gradient To acquire the characterization of battery impedance and the relationship between the parameters of the battery’s non-linear impedance model and temperature, the EIS experiment without the temperature gradient is conducted first. Figure 8 shows the diagram of battery EIS experiment without the temperature gradient.
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Figure 7. Experimental setup: (a) the battery cell; (b) the electrochemical workstation; (c) the power booster; (d) the temperature measurement station; (e) the environment chamber.
3.1. Experiments without the Temperature Gradient To acquire the characterization of battery impedance and the relationship between the parameters of the battery’s impedance model and temperature, the experiment Figure 7. Experimental setup:non-linear (a) the battery cell; (b) the electrochemical workstation; (c) EIS the power Figure 7. Experimental setup: (a) the is battery cell; (b) the Figure electrochemical workstation; power without the(d)temperature gradient conducted first. 8 shows the diagram (c) of the battery EIS booster; the temperature measurement station; (e) the environment chamber. booster; (d) the temperature measurement station; (e) the environment chamber. experiment without the temperature gradient. 3.1. Experiments without the Temperature Gradient To acquire the characterization of battery impedance and the relationship between the parameters of the battery’s non-linear impedance model and temperature, the EIS experiment without the temperature gradient is conducted first. Figure 8 shows the diagram of battery EIS experiment without the temperature gradient.
8. Diagram of battery electrochemical impedancespectroscopy spectroscopy (EIS) without the the FigureFigure 8. Diagram of battery electrochemical impedance (EIS)experiment experiment without temperature gradient. temperature gradient.
Table 1 shows the detailed EIS experiment steps without the temperature gradient. Enough rest
Table the before detailed without Enough time 1isshows necessary theEIS testexperiment to improve steps the accuracy ofthe the temperature experiment. Agradient. sinusoidal currentrest time iswith necessary before the test to improve the accuracy of the experiment. A sinusoidal current with 2 A amplitude and a frequency ranging from 0.01 Hz to 10 kHz is loaded on the battery. Figure and 8. Diagram of battery electrochemical impedance spectroscopy (EIS) experiment without the 2 A amplitude a frequency ranging from 0.01 Hz to 10 kHz is loaded on the battery. temperature gradient.
Table 1 shows the detailed EIS experiment steps without the temperature gradient. Enough rest time is necessary before the test to improve the accuracy of the experiment. A sinusoidal current with 2 A amplitude and a frequency ranging from 0.01 Hz to 10 kHz is loaded on the battery.
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Table 1. EIS experiments without the temperature gradient. CC-CV: constant current-constant voltage. Energies 2018, 11, 220
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StepTable 1. EIS experiments without the temperature Experiment Description gradient. CC-CV: constant current-constant
3.2.
1 voltage. Rest the battery for 4 h at 25 ◦ C 2 Charge the battery to 100% SOC in CC-CV mode at 25 ◦ C Step Experiment Description 3 Rest the battery for 4 h at 30 ◦ C 1 Rest the battery for 4 h at 25 °C 4 EIS Test 2 Charge the battery to 100% SOC in CC-CV◦mode at◦25 °C ◦ Set the chamber temperature to −20 C, −15 C, −10 C, −5 ◦ C, 0 ◦ C, 10 ◦ C and 20 ◦ C, 5 3 Rest the 4 h at for 30 °C andbattery rest thefor battery 4 h, repeat Step 4 46 EIS Test Rest the battery for 4 h at 25 ◦ C Set the chamberthe temperature −20SOC, °C, −15 −10 50% °C, −5 °C, 030% °C, SOC, 10 °C and and 10% 20 °C, andrepeat rest Discharge battery to to 90% 70%°C, SOC, SOC, SOC, 57 the battery 4 h, repeat Step 4 Step 3for to Step 6 Endbattery for 4 h at 25 °C 68 Rest the Discharge the battery to 90% SOC, 70% SOC, 50% SOC, 30% SOC, and 10% SOC, repeat Step 3 7 to Step 6 Experiments with Artificial Temperature Gradients 8 End
As mentioned above, a heating plate is used to construct the temperature gradient. Figure 9 3.2. Experiments with Artificial Temperature Gradients shows the diagram of the battery EIS experiment with temperature gradients. One side of the As mentioned above, a heating plate is used to construct the temperature gradient. Figure 9 battery cell is covered with a heating plate. Besides the thermocouples inside the battery, there shows the diagram of the battery EIS experiment with temperature gradients. One side of the battery are thermocouples stuckwith to the surface of the battery. Although the heating plate temperature and cell is covered a heating plate. Besides the thermocouples inside the battery, there are thermocouples stuck to the of the battery. Although the heating plate temperature the surface the chamber temperature can be surface set to specific values, the actual temperature of theand battery can be will set tobe specific the actual of thegradient. battery surface measured chamber by the temperature thermocouples usedvalues, to analyze thetemperature temperature The battery measured by the thermocouples will be used to analyze the temperature gradient. The battery maintains 50% SOC during the experiment with an artificial temperature gradient in order to focus on maintains 50% SOC during the experiment with an artificial temperature gradient in order to focus the correlation the temperature gradient. on theof correlation of the temperature gradient.
9. Diagram of battery EIS experiment with the temperature gradients. Figure 9.Figure Diagram of battery EIS experiment with the temperature gradients.
Table 2 shows the detailed EIS experiment steps with a temperature gradient. Enough rest time also necessary before the testexperiment to improve the accuracy experiment. gradient. Enough rest time is Table 2is shows the detailed EIS steps withof athe temperature also necessary before the test to improve the accuracy of the experiment.
Table 2. EIS experiment with the temperature gradients. Step 1 2 3 4 5 6 7 8 9 10
Experiment Description Rest the battery for 4 h at 25 ◦ C Charge the battery to 100% SOC in CC-CV mode at 25 ◦ C Rest the battery for 4 h at 25 ◦ C Discharge the battery to 50% SOC in 0.5 C (1 C = 40 A ) at 25 ◦ C Set the heating plate temperature to 30 ◦ C and the chamber temperature to 20 ◦ C Rest the battery for 4 h at 25 ◦ C EIS Test Change the heating plate temperature (every 5 ◦ C), and repeat Step 6 to Step 7 Set the chamber temperature to −20 ◦ C, −15 ◦ C, −10 ◦ C, −5 ◦ C, 0 ◦ C, 10 ◦ C and 15 ◦ C, and rest the battery for 4 h, repeat Step 5 to Step 8 End
4 5 6 7 8
Discharge the battery to 50% SOC in 0.5 C (1 C = 40 A ) at 25 °C Set the heating plate temperature to 30 °C and the chamber temperature to 20 °C Rest the battery for 4 h at 25 °C EIS Test Change the heating plate temperature (every 5 °C), and repeat Step 6 to Step 7 Energies 2018, 11, 220 10 of 18 Set the chamber temperature to −20 °C, −15 °C, −10 °C, −5 °C, 0 °C, 10 °C and 15 °C, and rest 9 the battery for 4 h, repeat Step 5 to Step 8 10 End
4. Results and Discussion
4. Results and Discussion
4.1. Battery EIS Characterization without the Temperature Gradient 4.1. Battery EIS Characterization without the Temperature Figure 10 shows the EIS experiment results of the Gradient battery with a uniform temperature. Figure 10a
shows theFigure EIS of10the battery different temperatures, and from this can see temperature. that, with theFigure increase of shows theat EIS experiment results of the battery withwe a uniform temperature, the impedance becomes smaller when the frequency is low. While in the high-frequency 10a shows the EIS of the battery at different temperatures, and from this we can see that, with the increase of temperature, impedance becomes smaller when theeach frequency is low. in thein the region, the EIS curves with the different temperatures coincide with other and thisWhile is clearer high-frequency region, theThis EIS indicates curves with different temperatures coincide with each other and diffusion this enlarged part of Figure 10a. that the temperature mainly influences the battery is clearer in the enlarged part of Figure 10a. This indicates that the temperature mainly influences and charge transfer reaction processes, while the battery’s inductance and ohmic resistancethe are not battery diffusion and charge transfer reaction processes, while the battery’s inductance and ohmic affected by the temperature. Figure 10b shows the EIS of a battery with the same temperature but resistance are not affected by the temperature. Figure 10b shows the EIS of a battery with the same different SOCs. From this picture, we can see that all the curves are close to each other in the middle temperature but different SOCs. From this picture, we can see that all the curves are close to each and high frequencies, and in the low-frequency region, the most obvious difference is the slope. With other in the middle and high frequencies, and in the low-frequency region, the most obvious the increase ofisthe slope the straight line becomes This indicates that SOC mainly difference theSOC, slope.the With the of increase of the SOC, the slopelarger. of the straight line becomes larger. impacts the battery diffusion process. This indicates that SOC mainly impacts the battery diffusion process.
Figure batterywith with uniform uniform temperature: temperature: (a)(a) 50% state of charge (SOC), different Figure 10. 10. EISEIS of ofbattery 50% state of charge (SOC), different temperatures; (b) 10 °C, different SOCs. ◦ temperatures; (b) 10 C, different SOCs.
The EIS results of the battery with uniform temperature are shown in Figure 11 with the The EIS results of the battery temperature shown inand Figure 11 with frequency frequency as a variable; Figurewith 11a,buniform shows the impedanceare magnitude phase of thethe battery as a variable; Figuretemperatures, 11a,b shows while the impedance magnitude and phase of the battery under under different Figure 11c,d is the impedance magnitude and phase of different the battery at while different SOCs. Figure 11aimpedance shows thatmagnitude the battery and impedance the temperatures, Figure 11c,d is the phase ofwill theincrease battery as at different temperature decreases, and the the battery battery impedance at will −20 °C is almost the impedance at 20 and SOCs. Figure 11a shows that impedance increase as 10 thetimes temperature decreases, ◦ ◦ °C at the frequency of 0.01 Hz to 0.1 Hz. The impedance phase curves shown in Figure 11b have a Hz the battery impedance at −20 C is almost 10 times the impedance at 20 C at the frequency of 0.01 great discrimination between 1 Hz and 10 Hz, and that is why the impedance phase can be used to to 0.1 Hz. The impedance phase curves shown in Figure 11b have a great discrimination between 1 estimate battery temperature [12]. In a similar way to Figure 10b, the impedance magnitude curves
Hz and 10 Hz, and that is why the impedance phase can be used to estimate battery temperature [12]. In a similar way to Figure 10b, the impedance magnitude curves in Figure 11c and the phase curves in Figure 11d are close to each other in all frequency ranges, and it can be concluded that the SOC has little impact on the battery impedance. 4.2. Battery EIS Characterization with Temperature Gradients Temperature distribution inside the battery is analyzed before obtaining the EIS experiment results of the battery with the artificial temperature gradients. Figure 12 shows the temperature distribution inside the battery with the artificial gradients. Due to the battery’s thickness and heat transfer, the actual temperature gradient may be different to the temperature gradient that is set. Six groups of data are shown in this figure, including three groups with a smaller set temperature gradient (10 ◦ C), and three groups with a larger set temperature gradient (35 ◦ C). To get a more intuitive view of these data, we give them a linear fit, and the fitting results are also shown in the figure. From this, we can see that the fitting results are excellent, and the R-Square of all fitting results are greater than 0.98.
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It can beinconcluded thatthe thephase temperature is to linearly distributed, and the assumption Figure 11c and curves ininside Figurethe 11dbattery are close each other in all frequency ranges, and proposed in Section 2 is reasonable. it can be concluded that the SOC has little impact on the battery impedance.
Figure EIS battery of the battery in plots Bode with plots uniform with uniform temperature: impedance magnitude Figure 11. EIS 11. of the in Bode temperature: (a,b)(a,b) impedance magnitude and and phase of battery at 50% SOC, different temperatures; (c,d) impedance magnitude and phase of phase of battery at 50% SOC, different temperatures; (c,d) impedance magnitude and phase of battery battery at 10 °C, different SOCs. at 10 ◦ C, different Energies 2018, 11, 220 SOCs. 12 of 19
4.2. Battery EIS Characterization with Temperature Gradients Temperature distribution inside the battery is analyzed before obtaining the EIS experiment results of the battery with the artificial temperature gradients. Figure 12 shows the temperature distribution inside the battery with the artificial gradients. Due to the battery’s thickness and heat transfer, the actual temperature gradient may be different to the temperature gradient that is set. Six groups of data are shown in this figure, including three groups with a smaller set temperature gradient (10 °C), and three groups with a larger set temperature gradient (35 °C). To get a more intuitive view of these data, we give them a linear fit, and the fitting results are also shown in the figure. From this, we can see that the fitting results are excellent, and the R-Square of all fitting results are greater than 0.98. It can be concluded that the temperature inside the battery is linearly distributed, and the assumption proposed in Section 2 is reasonable.
Figure 12.Temperature Temperature distribution distribution inside battery. Figure 12. inside battery.
As mentioned above, the temperature has a great impact on battery impedance. The next point
As mentioned above, the temperature has a great impact on battery impedance. The next point to study is the influence of the temperature gradient on the impedance. Figure 13 shows four sets of to study the influence of the temperature gradient on the impedance. Figure 13 shows four sets of data,isincluding one experiment at a 10 °C uniform temperature and three experiments with different ◦ C uniform temperature and three experiments with different data, temperature including one experiment at a 10 gradients. In the experiment at a 10 °C uniform temperature, the battery temperature temperature Inasthe experiment 10 ◦ Cthree uniform temperature, the battery temperature gradient gradients. is considered zero. Although at theaother experiments are implemented in different environmental temperature, have one thing in common, the actual heating plate temperature is gradient is considered as zero.they Although the other three experiments are implemented in different about 10 °Ctemperature, (10.59 °C, 10.28 °C,have 9.76 one °C), thing so these areactual considered to have similar environmental they in experiments common, the heating plateatemperature ◦ C, 10.28 ◦ C,figure, maximum temperature. In the black line to the magenta line, thetotemperature is about 10 ◦ C (10.59 9.76 ◦from C), sothe these experiments are considered have a similar gradient becomes larger. Figure 13a shows that battery impedance will increase as the temperature maximum temperature. In the figure, from the black line to the magenta line, the temperature gradient gradient becomes greater from 0.01 Hz to 10 Hz, and Figure 13b shows that the battery impedance becomes larger. Figure 13a shows that battery impedance will increase as the temperature gradient phase will decrease as the temperature gradient becomes greater from 1 Hz to 100 Hz. It may be becomes greater from 0.01 Hz to 10 Hz, and Figure 13b shows that the battery impedance phase will concluded that if the maximum battery temperature is fixed, the smaller the minimum temperature, the greater the battery impedance magnitude.
to study is the influence of the temperature gradient on the impedance. Figure 13 shows four sets of data, including one experiment at a 10 °C uniform temperature and three experiments with different temperature gradients. In the experiment at a 10 °C uniform temperature, the battery temperature gradient is considered as zero. Although the other three experiments are implemented in different environmental temperature, they have one thing in common, the actual heating plate temperature is about 10 °C (10.59 °C, 10.28 °C, 9.76 °C), so these experiments are considered to have a similar Energies 2018, 11, 220 temperature. In the figure, from the black line to the magenta line, the temperature 12 of 18 maximum gradient becomes larger. Figure 13a shows that battery impedance will increase as the temperature gradient becomes greater from 0.01 Hz to 10 Hz, and Figure 13b shows that the battery impedance decrease asphase the temperature becomes greater fromgreater 1 Hz from to 100 Hz. concluded that will decrease asgradient the temperature gradient becomes 1 Hz to It 100may Hz. be It may be concluded that if the maximum battery temperature is fixed, theminimum smaller the minimum temperature, if the maximum battery temperature is fixed, the smaller the temperature, the greater the the greater the battery impedance magnitude.
battery impedance magnitude.
Figure 13. EIS of battery in Bode plots with different artificial temperature gradients: (a) impedance
Figure 13. EIS of battery in Bode plots with different artificial temperature gradients: (a) impedance magnitude; (b) impedance phase. magnitude; (b) impedance phase. There is another point worth noting in Figure 12: the magenta line and the olive line intersect at of the point battery.worth According to the themagenta temperature is linearly Therethe is half another noting inassumption Figure 12:that the line and thedistributed, olive linetheintersect at temperature at the battery’s center can be considered as the average battery temperature. A situation the half ofwhereby the battery. According to the assumption that the temperature is linearly distributed, the several experiments under different temperature gradients but the same battery average temperature at the battery’s center canimpedance be considered as the average temperature. temperature demonstrate similar characterization should be battery verified and discussed. A situation We experiments select some experiments with approximate average temperature, thebattery lower average whereby several under different temperature gradients butincluding the same average temperature (1 °C) and the higher average temperature (21 °C). Besides these, we also temperature demonstrate similar impedance characterization should be verified and discussed. supplement the experiments with those of 1 °C and 21 °C uniform temperature. Figure 14 shows the We select some experiments with approximate average temperature, including the lower average temperature (1 ◦ C) and the higher average temperature (21 ◦ C). Besides these, we also supplement the Energies 2018, 11, 220 13 of 19 experiments with those of 1 ◦ C and 21 ◦ C uniform temperature. Figure 14 shows the experimental EIS experimental EIS of a battery withtemperature, approximate average temperature, and in Figure 14b,d, thephase of a of a battery with approximate average and in Figure 14b,d, the impedance impedance phase of a battery with different temperature gradients, but in both cases the similar battery with different temperature gradients, but in both cases the similar average temperatures almost average temperatures almost coincide with each other. In Figure 14a,c, the impedance magnitude of coincide with each other.gradient In Figure theother impedance ofand each temperature each temperature differs14a,c, from each in the low magnitude frequency range is unrelated to the gradient temperature. It may be concluded that batteries with similar average temperature possess similar It may be differs from each other in the low frequency range and is unrelated to the temperature. characterizations. concludedimpedance that batteries with similar average temperature possess similar impedance characterizations.
EIS of battery approximateaverage average temperature: (a,b)(a,b) impedance magnitude and Figure 14. Figure EIS of14.battery with with approximate temperature: impedance magnitude and phase of battery with approximate 1 °C average temperature; (c,d) impedance magnitude and phase ◦ phase of battery with approximate 1 C average temperature; (c,d) impedance magnitude and phase of of battery with approximate 21 °C average temperature. battery with approximate 21 ◦ C average temperature.
Some summaries and conclusions about the battery impedance characterization can be drawn according to the above experimental results: (1) Temperature mainly influences the diffusion and charge transfer reaction processes, while the battery inductance and the ohmic resistance are not affected. SOC only has an impact on the battery diffusion process. In general, the temperature has a greater impact on battery impedance than SOC.
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Some summaries and conclusions about the battery impedance characterization can be drawn according to the above experimental results: (1)
(2)
(3)
Temperature mainly influences the diffusion and charge transfer reaction processes, while the battery inductance and the ohmic resistance are not affected. SOC only has an impact on the battery diffusion process. In general, the temperature has a greater impact on battery impedance than SOC. For batteries with the same maximum temperature, battery impedance will increase as the temperature gradient becomes greater, especially in the low-frequency region, and the battery impedance phase will decrease as the temperature gradient become greater in the middle-frequency region. In our experiment with artificial temperature gradients, we find that the temperature distribution can be approximately considered to be linear, so the central temperature is equal to the average temperature. The experiments with similar average temperature possess similar impedance characterization, and this is consistent with the idea of lumped parameter model.
4.3. Determination of the Parameters of a Battery Non-Linear Impedance Model The results of experiments without the temperature gradient are used to fit the equivalent circuit Energies 11, 220 14 of 19 model, and one2018, of the fitting results is shown in Figure 15. This indicates that the experimental data Energies 2018, 11, 220 14 of 19 and the fitting data almost coincide at all the experimental frequencies, and Figure 15a shows that in experimental data data and and the the fitting fitting data data almost almost coincide coincide at at all all the the experimental experimental frequencies, frequencies, and and experimental the high-frequency region the is not a straight line toline theperpendicular abscissa axis; so, maybe a Figure 15a 15a shows shows that in EIS the high-frequency high-frequency region theperpendicular EIS is is not not aa straight straight to the the Figure that in the region the EIS line perpendicular to resistance abscissa parallelaxis; with L canparallel improve fitting.LLOverall, thethe fitting results are abscissa axis; so,inductance maybe aa resistance resistance parallel withthe inductance can improve improve the fitting. Overall, theacceptable so, maybe with inductance can fitting. Overall, the fitting results are acceptable and can be used in the discretization model. and can befitting usedresults in theare discretization model. acceptable and can be used in the discretization model.
Figure 15. Fitting result: (a) EIS; (b) impedance magnitude; (c) impedance phase.
Figure 15. Fitting result: EIS; (b)impedance impedance magnitude; (c) impedance phase. phase. Figure 15. Fitting result: (a) (a) EIS; (b) magnitude; (c) impedance
Figure 16 16 shows shows the the fitting fitting parameters parameters of of the the battery battery model, model, and and R Roo and and R R11 decrease decrease Figure
exponentially with temperature increase, while Q1,1, and and Q22 increase increase exponentially when exponentially the Figureexponentially 16 shows the fitting parameters of the battery model, and Roexponentially and R1 decrease with temperature increase, while Q Q when the temperature increases. increases. temperature with temperature increase, while Q1, and Q2 increase exponentially when the temperature increases.
Figure 16. Parameters of battery non-linear impedance model: (a) inductance L; (b) resistance Ro; (c)
16. Parameters of battery non-linear impedance model: (a) inductance L; (b) resistance Ro; (c) Figure 16.Figure Parameters battery non-linear impedance model: (a) inductance L; (b) resistance Ro ; fractional coefficient of CPE 1, Q1; (e) fractional coefficient of CPE2, Q2; (f) fractional resistance R1; (d)of resistance R1; (d) fractional coefficient of CPE1, Q1; (e) fractional coefficient of CPE2, Q2; (f) fractional (c) resistance R of; (d) of CPE1 , Q1 ; (e) fractional coefficient of CPE2 , Q2 ; (f) fractional order CPEfractional 1 and CPE2, coefficient α and β. order 1of CPE1 and CPE2, α and β. order of CPE1 and CPE2 , α and β.
We employ employ aa polynomial polynomial fit fit to to get get the the relationship relationship between between the the battery battery impedance impedance model model We parameters and and temperature, temperature, and and the the formula formula can can be be listed listed as: as: parameters -11 L == 4.66 4.66 ××10 10-11 4.58××10 10-8-8 L tt ++ 4.58 R = -1.95 × 10-9-9 t33 + 1.27 × 10-7-7 t 22 − 4.10 ×10-6-6 t + 1.3 × 10-3-3 Roo = -1.95 × 10 t + 1.27 × 10 t − 4.10 ×10 t + 1.3 × 10 R R1 = -1.01× 10-6-6 t33 + 5.66 × 10-5-5tt 22 −−1.1 1.1××10 10-3-3tt+8.9 +8.9 ××10 10-3-3 1 = -1.01× 10-4 t3 + 5.66 × 10 Q = 3.89 × 10-4 t3 + 7.8 × 10-3-3t 22 + 0.29t + 11.22
(5)
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We employ a polynomial fit to get the relationship between the battery impedance model parameters and temperature, and the formula can be listed as: L = 4.66 × 10−11 t + 4.58 × 10−8 Ro = −1.95 × 10−9 t3 + 1.27 × 10−7 t2 − 4.10 × 10−6 t + 1.3 × 10−3 −6 3 −5 2 −3 −3 R1 = −1.01 × 10 t + 5.66 × 10 t − 1.1 × 10 t + 8.9 × 10 Q1 = 3.89 × 10−4 t3 + 7.8 × 10−3 t2 + 0.29t + 11.22 Q2 = 1.9 × 10−3 t3 + 0.62t2 + 25.19t + 354.13 α = 0.82 − 4.2 × 10−3 t β = 1.6 × 10−3 t + 0.48
(5)
where t is the temperature in centigrade, and L, Ro , R1 , Q1 , Q2 , α and β are the model parameters. Energies 2018, 11, 220 15 of 19 The above work has enabled the relationship between the battery impedance model parameters thework temperature berelationship obtained, between and thethe relationship will bemodel usedparameters in the battery Theand above has enabledtothe battery impedance and the temperature to be obtained, and the relationship will be used in the battery discretization discretization model. model.
4.4. Verification of Battery Impedance Model Considering the Temperature Gradient 4.4. Verification of Battery Impedance Model Considering the Temperature Gradient
In most cases, the battery’s surface temperature or average temperature is used to modify the In most cases,However, the battery’s surface or average temperature is used to modify the battery’s impedance. the abovetemperature analysis makes it obvious that the internal temperature battery’s impedance. However, the above analysis makes it obvious that the internal temperature gradient has a great influence on battery impedance. Figure 17 shows the comparison of the gradient has a great influence on battery impedance. Figure 17 shows the comparison of the experimental results and lumped model impedance modified by the surface temperature. The experimental results and lumped model impedance modified by the surface temperature. The parameters of the lumped model Equation(5)(5) battery surface parameters of the lumped modelare areobtained obtained according according totoEquation andand battery surface temperature. Plainly, there are large deviations between the experimental results and the lumped temperature. Plainly, there are large deviations between the experimental results and the lumped model impedance modified byby thethesurface nomatter matterwhat what maximum surface model impedance modified surfacetemperature, temperature, no thethe maximum surface or or minimum surface temperatures are. minimum surface temperatures are.
Figure 17. Comparison of the experimentalresults results and and lumped impedance modified by the Figure 17. Comparison of the experimental lumpedmodel model impedance modified by the surface temperature: (a) EIS; (b) impedance magnitude; (c) impedance phase. surface temperature: (a) EIS; (b) impedance magnitude; (c) impedance phase.
According to the discretization model shown in Equation (4) and the relationship between the
According to the parameters discretization model shown in Equation (4) impedance and the relationship between impedance model and temperature in Equation (5), the of a battery with artificial temperature gradientsand can temperature be obtained. in Three experiments a higher of temperature the impedance model parameters Equation (5), thewith impedance a battery with gradient (the actual temperature battery surfaces greatertemperature than 20 °C) gradient are artificial temperature gradients can bedifference obtained. between Three experiments with aishigher ◦ chosen to verify the battery discretization model. Figure 18 shows the verification of the battery (the actual temperature difference between battery surfaces is greater than 20 C) are chosen to verify discretization model. The discretization model considering the temperature gradient and the the battery discretization model. Figure 18 shows the verification of the battery discretization model. lumped model considering only the average temperature are also compared. The discretization model considering the temperature gradient and the lumped model considering Overall, the proposed battery discretization model is closer to the actual experimental data than only the average temperature are also compared. the lumped model. From Figure 18b,e,h, we can see that both the proposed model and the lumped Overall, the proposed battery discretization is closer toHz. theThis actual experimental data model deviate from the actual experimental datamodel from 0.01 Hz to 10 may be because that the than the lumped model. Fromtemperature Figure 18b,e,h, we can seeorder that both the proposed model theFrom lumped relationship between and the fractional of CPE 1 and CPE2 may not beand linear. modelFigure deviate frominthe actual fromimpedance 0.01 Hz tophase 10 Hz. This may beexperimental because that the 18c,f,i, most cases,experimental the proposeddata model’s is closer to the data, which proves the accuracyand of the proposed model. relationship between temperature the fractional order of CPE1 and CPE2 may not be linear. From Figure 18c,f,i, in most cases, the proposed model’s impedance phase is closer to the experimental data, which proves the accuracy of the proposed model.
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Figure 18. Verification of battery discretizationmodel: model: (a–c) magnitude andand phase of of Figure 18. Verification of battery discretization (a–c)EIS, EIS,impedance impedance magnitude phase battery with heating plate temperature set at 25 °C and −15 °C environmental temperature; (d–f) EIS, ◦ ◦ battery with heating plate temperature set at 25 C and −15 C environmental temperature; (d–f) EIS, impedance magnitude and phase of battery with heating plate temperature set at 30 °C and −10 °C impedance magnitude and phase of battery with heating plate temperature set at 30 ◦ C and −10 ◦ C environmental temperature; (g–i) EIS, impedance magnitude and phase of battery with heating plate environmental temperature; (g–i) EIS, impedance magnitude and phase of battery with heating plate temperature set at 35 °C and −5 °C environmental temperature. temperature set at 35 ◦ C and −5 ◦ C environmental temperature.
Figure 19 shows the errors of the discretization model and the lumped model. Obviously, the magnitude error ofthe the errors discretization is smallermodel in the whole frequency range, whileObviously, the phase the Figure 19 shows of the model discretization and the lumped model. errors in Figure 19d–f show that the lumped model performs a little better than the discretization magnitude error of the discretization model is smaller in the whole frequency range, while the phase in the low-frequency Apparently, lower athe temperature, the errorsmodel in Figure 19d–f show that region. the lumped model the performs little better thanthe thegreater discretization magnitude error. In Figure 19a–c, the experiment with the lower average temperature has a greater model in the low-frequency region. Apparently, the lower the temperature, the greater the magnitude error in the low frequency range. In the high-frequency region, the relationship between the phase error. In Figure 19a–c, the experiment with the lower average temperature has a greater error in the error and the frequency is approximately linear and independent of temperature. This may be due to low frequency In themodel’s high-frequency region, thea relationship between therepresent phase error and the the batteryrange. impedance inherent defect that single inductor may not the real frequency is approximately linear and independent of temperature. This may be due to the battery characterization of battery impedance in the high-frequency region. In the whole frequency range, impedance model’s inherent magnitude defect thaterror a single may not represent characterization the maximum impedance of theinductor battery discretization model isthe 1.46real mOhm, and the maximum impedance phase error is 6.96°. region. In the whole frequency range, the maximum of battery impedance in the high-frequency Table 3 compares the of statistical data of the discretization model and mOhm, the average impedance magnitude error the battery discretization model is 1.46 andtemperature the maximum ◦ model. Obviously, the accuracy of the discretization model is higher, especially the impedance impedance phase error is 6.96 . magnitude of the proposed model. Among three experiments different temperature Table 3 compares the statistical data of thethe discretization model with and the average temperature gradients, the root mean squared error (RMSE) of the proposed model is at least 15% smaller than model. Obviously, the accuracy of the discretization model is higher, especially the impedance that of the average temperature model. For the model impedance phase, the proposed model is magnitude of the proposed model. Among the three experiments with different temperature gradients, slightly superior. the root mean squared error (RMSE) of the proposed model is at least 15% smaller than that of the average temperature model. For the model impedance phase, the proposed model is slightly superior. Through the above analysis, the performance of the proposed model is proved to be accurate and effective. Compared with the lumped model with the average temperature, the proposed model is identical to the actual battery impedance. With the help of the discretization model, we can obtain a more accurate characterization of the battery.
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Figure 19. Errors of the discretizationmodel modeland and the the lumped lumped model: impedance magnitude Figure 19. Errors of the discretization model:(a–c) (a–c) impedance magnitude errors; (d–f) impedance phase errors. errors; (d–f) impedance phase errors. Table 3. Statistical data of the discretization model and the average temperature model.
Table 3. Statistical data of the discretization model and the average temperature model. Root Mean Squared Error (RMSE)
Root Mean Set:Error 25~−15 °C Squared Set: 30~−10 °C (RMSE) Set: 35~−5 °C
Magnitude Error (mOhm) Discretization Average Temperature Magnitude Error (mOhm) Model Model 0.67 0.77 Average Discretization 0.16 0.32 Temperature Model 0.15 0.23
Model
Phase Error (°) Discretization Average Temperature (◦ ) Model Phase Error Model 2.20 2.54 Average Discretization 1.77 1.84 Temperature Model 2.22 2.34
Model
◦C
Set: 25~ −15 the above analysis, 0.67 the performance 0.77of the proposed 2.20 2.54 Through model is proved to be accurate ◦C Set: 30~ − 10 0.16 0.32 1.77 1.84 model and effective.◦ Compared with the lumped model with the average temperature, the proposed Set: 35~−5 C 0.15 0.23 2.22 2.34 is identical to the actual battery impedance. With the help of the discretization model, we can obtain a more accurate characterization of the battery.
4.5. Limitations and Future Work 4.5. Limitations and Future Work
In the proposed discretization model, the temperature distribution in the direction parallel to the In the proposedIndiscretization model, the distribution temperature distribution in the direction to The battery taps is ignored. fact, the temperature is inhomogeneous in this parallel direction. the battery taps is ignored. In fact, the temperature distribution is inhomogeneous in this direction. temperature of the part close to the battery taps is higher. Moreover, this model may not be suitable for The temperature of the part close to the battery taps is higher. Moreover, this model may not be cylindrical-type batteries because the volume of inner and outer layers is different and the calculation suitable for cylindrical-type batteries because the volume of inner and outer layers is different and of each layer impedance become complex. the calculation of eachwill layer impedance will become complex. The proposed discretization model batterynon-linear non-linear impedance model. The proposed discretization modelisisbased based on on the the battery impedance model. It is It is worthworth notingnoting that this is generic for batteries with similar characterizations like thelike impedance thatmodel this model is generic for batteries with similar characterizations the impedance spectroscopy Figure 1. For other lithium-ion different battery the spectroscopy shown in Figureshown 1. Forinother lithium-ion batteries withbatteries differentwith battery chemistries, chemistries, theimpedance battery non-linear model may different and should be determined battery non-linear model impedance may be different and be should be determined first. In that case, first. In that case, the discretization method can be applied to batteries with various the discretization method can be applied to batteries with various chemical materials.chemical Verification materials. Verification of the adaptability of this discretization method to other battery chemistries of the adaptability of this discretization method to other battery chemistries shall be conducted in shall be conducted in future work. future work. Considering the thickness of the battery used in this paper, the temperature distribution is Considering the thickness of the battery used in this paper, the temperature distribution is assumed to be linear. Unlike the artificial temperature gradient in this paper, the core and surface assumed to be linear. Unlike the artificial temperature gradient in this paper, the core and surface temperatures will construct the temperature gradient in actual applications of batteries. Future work temperatures will construct the temperature gradient in actual applications of batteries. Future will focus on the temperature distribution of the battery with a larger size. Based on anwork will focus on the temperature distribution of the with aonline, largerand size. Based on anparameters electrothermal electrothermal model, the core temperature willbattery be estimated then the model will modified according to be theestimated proposed online, discretization model. The modified modelwill parameters model, thebecore temperature will and then the model parameters be modified will eventually improve the precision of the batteryThe electrothermal model. parameters will eventually according to the proposed discretization model. modified model improve the precision of the battery electrothermal model.
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5. Conclusions This paper proposed a battery discretization model and designed some experiments to investigate the impedance characterization of a battery with internal temperature gradients. Some conclusions and summations are drawn below: (1)
(2) (3)
Temperature has a more significant impact on battery impedance than SOC. In the low-frequency region, the temperature gradient will influence the impedance magnitude, and in the middle-frequency region, the temperature gradient will influence the impedance phase. For batteries with the same maximum temperature, battery impedance will increase as the temperature gradient gets greater, especially in the low-frequency region. The temperature distribution inside the battery can be approximately considered to be linear, and batteries with the similar average temperature have a similar impedance characterization. A discretization model was proposed to investigative the impedance of a battery with temperature gradients, and experimental results show that the model performs well. The maximum impedance magnitude error is 1.46 mOhm. Compared to the lumped model with average temperature, the impedance produced by the proposed model is identical to the actual battery impedance, and the RMSE of the proposed model is at least 15% smaller than that of the average temperature model.
Acknowledgments: This work is financially supported by the National Natural Science Foundation of China (NSFC, Grant No. 51677136) and the National Key Research and Development Program of China (Program No. 2017YFB0103105). Author Contributions: Haifeng Dai designed the experiments and wrote the paper; Bo Jiang performed the experiments and analyzed the results; Xuezhe Wei reviewed and revised the paper. All authors read and approved the manuscript Conflicts of Interest: The authors declare no conflict of interest.
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