energies Article
On the Compatibility of Electric Equivalent Circuit Models for Enhanced Flooded Lead Acid Batteries Based on Electrochemical Impedance Spectroscopy Can Aksakal 1,2 and Altug Sisman 1, * 1 2
*
Energy Institute, Istanbul Technical University, 34469 Istanbul, Turkey;
[email protected] INCI GS YUASA, 45030 Manisa, Turkey Correspondence:
[email protected]; Tel.: +90-212-285-3884
Received: 8 December 2017; Accepted: 31 December 2017; Published: 3 January 2018
Abstract: Electric equivalent circuit (EEC) models have been widely used to interpret the inner dynamics of all type of batteries. Added to this, they also have been used to estimate state of charge (SOC) and state of health (SOH) values in combination with different methods. Four EEC models are considered for enhanced flooded lead acid batteries (EFB) which are widely used in micro hybrid vehicles. In this study, impedance and phase prediction capabilities of models throughout a frequency spectrum from 1 mHz to 10 kHz are compared with those of experimental results to investigate their consistency with the data. The battery is charged, discharged, and aged according to appropriate standards which imitates a lifetime of a micro hybrid vehicle battery under high current partial cycling. Impedance tests are repeated between different charge and health states until the end of the battery’s lifetime. It is seen that adding transmission line elements to mimic the high porous electrode electrolyte interface to a double parallel constant phase element resistance model (ZARC) can increase the model data representing capability by 100%. The mean average percentage error (MAPE) of the conventional model with respect to data is 3.2% while the same value of the transmission line added model found as 1.6%. The results can be helpful to represent an EFB in complex simulation environments, which are used in automobile industry. Keywords: micro-hybrid vehicles; lead acid battery; enhanced flooded battery; impedance spectroscopy; state of charge; state of health
1. Introduction Vehicle emission regulations have been forcing the automotive industry worldwide to reduce its carbon footprint [1]. Environmentally friendly, full electric and plug-in hybrid vehicles are gaining popularity and their sales numbers increase each year but due to limited sales volumes, their effect on fleet emission reduction is limited. The trend in the mass market is also progressing in the same direction and as a result, fuel efficient micro-hybrid vehicle sales are also gaining popularity. According to market research, total micro-hybrid vehicle sales will increase from 8.8 million in 2013 to 55.3 million in 2022 [2]. Lead-acid batteries are still unrivaled for micro-hybrid vehicles as well as conventional vehicles due to their robust and safe design, low cost raw materials, mature and cost optimized manufacturing process, and already established efficient recycling processes [3,4]. A lead acid car battery designed for a micro-hybrid vehicle should have additional duties over a standard battery designed only for starting, lighting, and ignition (SLI) applications. A micro-hybrid vehicle battery should be able to supply energy during idle engine stop periods. It can be charged at high current rates during regenerative braking or even it may help the engine to boost acceleration via alternator/generator. To achieve these new duties, some breakthrough improvements have been adapted to lead acid batteries. Newly developed carbon additives have been applied to negative Energies 2018, 11, 118; doi:10.3390/en11010118
www.mdpi.com/journal/energies
Energies 2018, 11, 118
2 of 14
electrodes to increase the cyclic ability and dynamic charge acceptance [5,6]. Electrode grids have been designed with computer aided optimization techniques to minimize voltage loss at high current rates without increasing the grid weight [7]. High porous and low tortuous separators with low electric resistances are now commercially available and they have been utilized in lead acid batteries to improve their electrical performance such as increased cyclic endurance and higher dynamic charge acceptance. These new type lead acid batteries are called enhanced flooded batteries (EFB) [8–10]. EFB can last much longer than that of regular flooded lead acid batteries under high electrical stress of micro-hybrid vehicles. Unlike conventional vehicles, micro-hybrid vehicles utilize lead-acid batteries in a partial state of charge (SOC) to capture energy during the regenerative braking. The electronic control unit of the micro hybrid vehicle should know the battery SOC to decide to stop the engine or not and to decide the charging current during deceleration. Therefore, in recent studies rival lead acid battery technologies have been focusing to understand the SOC of the battery inside the micro hybrid vehicles [11]. In the lifetime of a lead acid battery, various electrochemical degradation processes like sulphation, acid stratification, grid corrosion, etc. could occur. These degradation processes can result in changes in the crystallographic structure of battery materials, influencing electrode potentials. Thus, it usually misleads the end user or car electronic control unit to get an idea about exact SOC and SOH from open circuit voltages. In literature, there are numerous studies that explain electrochemical degradation processes of a lead acid battery [12] and for other chemistries [13–15] to understand their SOC and SOH levels. Most of these works are based on highly non-linear and complex equations based on the underlying physics of an electrochemical cell. Usually these models need more time to simulate. If there is a need of SOC or SOH information in a real time simulation faster methods become a necessity. Electrochemical impedance spectroscopy (EIS) is a faster technique that has a potential to identify the SOC and SOH of a lead acid battery if data sets have been interpreted with a convenient electric equivalent circuit (EEC) model. Besides SOC and SOH predictions, another reason to use an EEC for energy storage unit of an automobile has emerged in recent years. Batteries are increasingly incorporated into the complex electronic system of a car as hybrid powertrain systems are getting more complicated. The resulting electronic and electrochemical combined systems need to be analyzed as a whole. EEC is an easier and efficient way of representing a battery which is in interaction with other electronic components of a hybrid powertrain in a simulation environment [16]. In this paper, the precisions of previously proposed and commonly used EEC models of lead acid batteries are experimentally investigated for modern enhanced flooded batteries, which are widely using in today’s conventional vehicles. In their early work, Gopikanth and Sathyanamara [17] examined the dependency of the impedance data with respect to SOC for a lead acid battery in a relatively narrow frequency spectrum range. They observed that above 200 Hz, the inductive behavior becomes prominent and decided to concentrate on the range between 100 Hz to 15 Hz. In this range, change in the quantities of electrical elements have been observed with respect to SOC. Due to the parabolic behavior of elements, using these parameters for SOC predictions are found as a limited indicator. Mauracher and Karden [18] have completed a series of EIS tests on lead acid batteries to identify parameters of their EEC battery model based on Randles theory. Their proposed model can give battery terminal voltage as a function of current without taking into account of temperature, SOC and SOH with 0.2% voltage tolerance on their simulations. Blanke and co-workers [19] showed that higher frequency region could also give valuable information on SOC. For a certain frequency, the mixed inductive and capacitive behaviors of a lead acid battery cancel each other and the imaginary part of impedance becomes zero for a given SOC value. This frequency is named as transition frequency (FTr ). They observed that this parameter is highly dependent on SOC and the temperature of the battery. Impedance data are shown in Bode or in Nyquist diagrams. In former one, modulus of real and imaginary parts and phase angle are given with respect to frequency in log scale. In the latter case, the negative values of imaginary impedance
Energies 2018, 11, 118
3 of 14
are shown with respect to real impedances to have capacitive loops in the upper part of the graph. 2018, shape 11, 118 of a Nyquist diagram for a lead acid battery is given in Figure 1 below. 3 of 14 AnEnergies idealized
Figure Shape idealized Nyquist diagram lead acid battery. Figure 1. 1. Shape ofof anan idealized Nyquist diagram forfor lead acid battery.
InIn this work, impedance spectrum analyses onon anan EFB are performed forfor a wide SOC and SOH this work, impedance spectrum analyses EFB are performed a wide SOC and SOH ranges. To To interpret impedance data,data, widely used double ZARC ZARC elementelement model ismodel modified by addingby ranges. interpret impedance widely used double is modified transmission line elements each ZARC for an EFB for first for time. model alsomodel compared adding transmission linetoelements to each ZARC for the an EFB theThis firstnew time. This is new is also with three existing models from literature. compared with three existing models from literature. Experiments areare conducted under zero DC discharging or charging to avoid SOC changes Experiments conducted under zerowithout DC without discharging or charging to avoid SOC during the EIS measurement. Charge, discharge, anddischarge, cycling procedures of European Committee for changes during the EIS measurement. Charge, and cycling procedures of European Electrochemical [20,21](CENELEC) are applied to the battery to change its SOC andto Committee forStandardization Electrochemical(CENELEC) Standardization [20,21] are applied to the battery SOH. Rather than charging discharging with a big of discharge aging the of battery, theseto change its SOC and SOH.and Rather than charging anddepth discharging with atobig depth discharge standards for micro hybrid vehicle for batteries can imitate thebatteries real-lifecan working aging thedesigned battery, these standards designed micro hybrid vehicle imitateconditions. the real-life Anworking equivalent circuit with only simple electrical [22,23], a typical Randles circuit with conditions. An equivalent circuit with elements only simple electrical elements [22,23], a typical anRandles addition of inductance [24], a model with two constant elements (CPE) in elements parallel with circuit with an addition of inductance [24], a modelphase with two constant phase (CPE) resistances [19,25], and finally a circuit with addition of two transmission line elements representing in parallel with resistances [19,25], and finally a circuit with addition of two transmission line forelements both anode and cathode electrode considered to investigate consistencyto representing forporous both anode andsurfaces cathodeareporous electrode surfaces their are considered with experimental data. investigate their consistency with experimental data. Before investigating a relationship between data and introduced models, a data quality analysis Before investigating a relationship between data and introduced models, a data quality analysis method called be sure surethat thatdata dataare arelinear, linear,causal causal and stable thus method calledKramers-Kronig Kramers-Kronig(KK) (KK)isis used used to be and stable thus selfself-consistent [26]. KK relation dictates that real and imaginary parts of any immittance function are consistent [26]. KK relation dictates that real parts of any immittance function are interdependent based onon following equations: interdependent based following equations:
( ) 2 Z ∞ Z ( ()x − Re ) − ZRe ( ω ) = 2ω Z Im ((ω )) = dx π 0 x2−− ω 2 ZRe ((ω )) = = R∞ + +
( ()x− ( ()ω ) 22 Z ∞ xZ Im ) − ωZ Im dx 2 2 − π 0 x −ω
(1)(1) (2)(2)
Following thethe KKKK datadata validation, a software basedbased on simplex algorithm is used to Following validation, a software on simplex algorithm is change used tomodel change 2 ) value of the model with data. Model accuracies with parameters to minimize the chi-square (χ model parameters to minimize the chi-square ( ) value of the model with data. Model accuracies respect to data to aredata analyzed based on MAPE methodmethod for different SOC and SOH It is with respect are analyzed based on MAPE for different SOC andconditions. SOH conditions. shown that a circuit additional transmission line elements the minimum error in comparison It is shown that awith circuit with additional transmission linehas elements has the minimum error in with measurements for the full range of SOC and SOH. Also, it is seen that the precisions of the models comparison with measurements for the full range of SOC and SOH. Also, it is seen that the precisions can and imaginary of impedance as of well as the phase shift. ofbe thedifferent models for canreal be different for realparts and imaginary parts impedance as well as the phase shift.
2. Experimental Setup, Procedures, and Results EIS tests, SOC adjustments, and cycling procedure to aging the battery are realized on the same experimental setup shown in Figure 2 without dismantling the battery from the setup. A current booster which is controlled by an impedance analyzer is added to the system for the high current
Energies 2018, 11, 118
4 of 14
2. Experimental Setup, Procedures, and Results EIS tests, SOC adjustments, and cycling procedure to aging the battery are realized on the same experimental setup shown in Figure 2 without dismantling the battery from the setup. A current booster Energies 2018, 11, 118 4 of 14 which is controlled by an impedance analyzer is added to the system for the high current cycling. A script A is script written sequence these experiments without without interruption between between steps. During tests cycling. is to written to sequence these experiments interruption steps.EIS During and aging procedure, battery is submerged in a temperature controlled water bath inside an incubator EIS tests and aging procedure, battery is submerged in a temperature controlled water bath inside an to keep thetotemperature constant atconstant 25 ◦ C. at 25 °C. incubator keep the temperature
Figure 2. Experimental setup. Figure 2. Experimental setup.
2.1. Impedance Analysis Procedure 2.1. Impedance Analysis Procedure A 12 V EFB sample is used for a series of impedance analysis. Before starting a test, the battery A 12 V EFB sample is used for a series of impedance analysis. Before starting a test, the battery is is charged according to EN 50342-1 CENELEC standard [20]. Following the charging step, 24-h pause charged according to EN 50342-1 CENELEC standard [20]. Following the charging step, 24-h pause period is given for voltage relaxation. Impedance analysis is realized under zero DC to keep the SOC period is given for voltage relaxation. Impedance analysis is realized under zero DC to keep the SOC level of the battery constant during the measurements especially at low frequencies. The root mean level of the battery constant during the measurements especially at low frequencies. The root mean square of the excitation current is set to In/5 where In equals to a nominal discharge current which can square of the excitation current is set to In /5 where In equals to a nominal discharge current which deplete the battery in 20 h. AC signal starts from 104 Hz and goes down to 10−3 Hz where the can deplete the battery in 20 h. AC signal starts from 104 Hz and goes down to 10−3 Hz where the frequency step is set as 10 points per decade (PPD) of frequency. For each frequency point; an AC frequency step is set as 10 points per decade (PPD) of frequency. For each frequency point; an AC signal (PPF) is applied to the battery for three times of the period to avoid coincidental fluctuations signal (PPF) is applied to the battery for three times of the period to avoid coincidental fluctuations on on the voltage response. After each impedance analysis set, the SOC of the battery is decreased 10% the voltage response. After each impedance analysis set, the SOC of the battery is decreased 10% by by discharging with In for 2 h. Following the discharge step, another rest period is given for the discharging with I for 2 h. Following the discharge step, another rest period is given for the voltage voltage relaxation nof the battery and the next impedance analysis starts when open circuit voltage relaxation of the battery and the next impedance analysis starts when open circuit voltage change is change is less than 1 mV/min. This procedure continues down to 20% SOC value and after that, the less than 1 mV/min. This procedure continues down to 20% SOC value and after that, the battery is battery is charged with the same standard. By applying this procedure, the impedance data of the charged with the same standard. By applying this procedure, the impedance data of the battery is battery is collected for all SOC values at a wide frequency spectrum for a certain SOH. collected for all SOC values at a wide frequency spectrum for a certain SOH. 2.2. 2.2. Battery Battery Ageing Ageing Procedure Procedure After series of anEIS EIStest test a certain SOH value, the battery is aged with respect to microAfter series of an forfor a certain SOH value, the battery is aged with respect to micro-hybrid hybrid partial cycling standard EN 50342-6 In this procedure, battery’s is decreased 50% partial cycling standard EN 50342-6 [21]. In[21]. this procedure, battery’s SOC is SOC decreased to 50% to before before beginning the cycling. Following the initial discharge, the battery is cycled between 50% and beginning the cycling. Following the initial discharge, the battery is cycled between 50% and 67.5% 67.5% SOC values with 7 × In charge and discharge currents according to 17.5% endurance cycling SOC values with 7 × In charge and discharge currents according to 17.5% endurance cycling procedure. procedure. This microcycle is repeated 85 times and it is considered as one unit of aging. During the This microcycle is repeated 85 times and it is considered as one unit of aging. During the battery battery aging, test termination criterion is the battery voltage, which should be higher than 10 V during the discharging step. After each aging unit, EIS tests are conducted again to understand the change of ZRe, ZIm and phase shift of the battery at different SOH levels. The flowchart of the applied impedance analysis and the aging procedure is given in Figure 3. Experiments are carried on till the voltage decreases to 10 V at cycling unit 10.
Energies 2018, 11, 118
5 of 14
aging, test termination criterion is the battery voltage, which should be higher than 10 V during the discharging step. After each aging unit, EIS tests are conducted again to understand the change of ZRe , ZIm and phase shift of the battery at different SOH levels. The flowchart of the applied impedance analysis and the aging procedure is given in Figure 3. Experiments are carried on till the voltage decreases V at cycling unit 10. Energies 2018,to11,10 118 5 of 14
Figure Figure 3. 3. Battery Battery ageing ageing and and electrochemical electrochemical impedance impedance spectroscopy spectroscopy (EIS) (EIS) procedure. procedure.
2.3. 2.3. Results Results and and Discussion Discussion Impedance Impedance analysis analysis for for aa certain certain SOH SOH is is consecutively consecutively repeated repeated with with decreasing decreasing SOC SOC values values as as explained in the previous section. Nyquist plots for different SOC values of a new battery are given explained in the previous section. Nyquist plots for different SOC values of a new battery are given in kHz to to 11 mHz. mHz. Higher in Figure Figure 4a, 4a, at at aa range range of of 10 10 kHz Higher frequency frequency region region of of the the same same graph graph is is given given in in Figure 4b. Battery impedance starts with an inductive behavior at the higher frequency region and as Figure 4b. Battery impedance starts with an inductive behavior at the higher frequency region and as frequency frequency decreases decreases the the inductive inductive component component is is first first compensated compensated than than surpassed surpassed by by the the capacitive capacitive components of the battery. components of the battery. EIS resultsofofthe the highest SOC thewith restitswith itsincreasing vastly increasing real and EIS results highest SOC test test differdiffer from from the rest vastly real and imaginary imaginary parts seen in Figure 4a. The situation becomes comprehensible since the AC signal parts seen in Figure 4a. The situation becomes comprehensible since the AC signal acts almost likeacts DC almost at such in low in During the mHz During the positive semi-cycle signal, of the at such like low DC frequencies thefrequencies mHz range. therange. positive semi-cycle of the impedance impedance the current signal causes overcharge of an already the current signal, signal causes to overcharge of an to already fully charged battery.fully This charged leads to battery. a much This leads to a much strong voltage increment than those of others. This situation continues for all SOH values at their full SOC levels.
Energies 2018, 11, 118
6 of 14
strong voltage increment than those of others. This situation continues for all SOH values at their full 66ofof1414 SOC levels.
Energies Energies2018, 2018,11, 11,118 118
(a) (a)
(b) (b)
Figure (a) ofofaanew battery at various states ofofcharge (SOC); (b) AAdetailed view ofofthe Figure4. (a)Nyquist Nyquistof new battery various states charge (SOC); detailed the Figure 4.4.(a) Nyquist a new battery at at various states of charge (SOC); (b) A(b) detailed view view of the high high frequency region. high frequency region. frequency region.
In Inthe theearly earlystage stageofoflifetime, lifetime,increments incrementsofofreal realresistance resistance(Z (ZReRe) )atattransition transitionfrequency frequency(F(FTrTr) )with with In the SOC early levels stage of lifetime, increments ofless realthan resistance (Z frequency (FTr ) with Re ) at transition decreasing are limited. Z Re shows 1 mΩ increment from 5.38 decreasing SOC levels are limited. ZRe shows less than 1 mΩ increment from 5.38mΩ mΩtoto6.20 6.20mΩ mΩ decreasing SOC nominal levels are limited. discharge. ZRe shows This less than 1 mΩ increment from 5.38 mΩ to 6.20 mΩ respect respect toto 80% 80% nominal capacity capacity discharge. This isis aa result result ofof decreasing decreasing conductivity conductivity ofof the the respect to 80% nominal capacity discharge. This is a result of decreasing conductivity of the electrolyte electrolyte due to sulphuric acid consuming double sulphation reaction. In order to show the aging electrolyte due to sulphuric acid consuming double sulphation reaction. In order to show the aging due toon sulphuric acid consumingEIS double sulphation reaction. In ordersame to show the aging effect on the effect effect onthe thebattery batteryimpedance, impedance, EISdata dataatatvarious variousSOC SOClevels levelsofofthe the samebattery batteryare areobtained obtainedafter after battery impedance, cycles EIS data at various SOC levels of the sameofbattery are obtained after fiveofunits of five fiveunits unitsofofmicro micro cycleswhich whichdecreases decreasesthe thebattery batterystate state ofhealth healthtoto50%. 50%.The Theresults results ofthese these micro cycles which decreases the battery state of health to 50%. The results of these experiments are experiments experimentsare areshown shownininFigure Figure5a,b. 5a,b. shown in Figure 5a,b.
(a) (a)
(b) (b)
Figure Figure5. 5.(a) (a)Nyquist Nyquistof ofaaabattery batteryat at50% 50%state stateof ofhealth health(SOH) (SOH)at atvarious variousSOC; SOC;(b) (b)A Adetailed detailedview viewof of Figure 5. (a) Nyquist of battery at 50% state of health (SOH) at various SOC; (b) A detailed view of the high frequency region. the high highfrequency frequencyregion. region. the
Between Betweenthese thesetwo twodifferent differenthealth healthstates, states,ZZReReofofthe thebattery batteryshows showsaaminimal minimalincrement incrementofof Between these two different health states, ZRe of the battery shows a minimal increment of 0.36 mΩ in their full SOC values. The resistance gap between new and old batteries increases 0.36 mΩ in their full SOC values. The resistance gap between new and old batteries increaseswhile while 0.36 mΩ in their full SOC values. The resistance gap between new and old batteries increases while SOC SOCdecreases. decreases.At At30% 30%SOC, SOC,ZZReReofofthe thenew newbattery batteryisis5.81 5.81mΩ mΩwhere whereititincreases increasestoto11.99 11.99mΩ mΩafter after SOC decreases. At 30% SOC, ZRe of the new battery is 5.81 mΩ where it increases to 11.99 mΩ after 55units unitsofofmicro microcycling cyclingfor forthe thesame sameSOC SOCvalue. value.The Thechanges changesofofZZReRefor foran anEFB EFBbattery batteryunder underaa 5 units of micro cycling for the same SOC value. The changes of ZRe for an EFB battery under a controlled controlledaging agingprocedure procedureare aregiven givenininFigure Figure6.6.ItItisisseen seenthat thatthe theincrease increaseininthe theDC DCresistance resistanceisis controlled aging procedure are given in Figure 6. It is seen that the increase in the DC resistance is minimal minimalatatall allSOH SOHlevels levelsatathigher higherSOC SOCregion regiontill tillthere thereisisaamalfunction malfunctionininthe thebattery batteryatatthe theend endofof minimal at all SOH levels at higher SOC region till there is a malfunction in the battery at the end of its its itslifetime. lifetime.On Onthe theother otherhand, hand,the theincrease increaseininthe theresistance resistanceatatthe thelower lowerSOC SOCregion regionisishighly highly lifetime. On the other hand, the increase in the resistance at the lower SOC region is highly remarkable. remarkable. remarkable. During the change of SOC from 100% to 30%, it is observed that ZRe increases only 7.9% for a new battery (100% SOH) while it increases 108% and 162% for 50% SOH and 10 % SOH respectively. At the end of 10th ageing unit, battery voltage decreases below 10 V and the test is over. At that point,
Energies 2018, 11, 118
7 of 14
SOH is assumed 0% and the battery is fully charged again to do a final test of EIS thorough out the capacity remains. Energies 2018, 11, 118 7 of 14
Figure withSOC SOCatatvarious variousSOH SOHlevels. levels. Figure6.6.Change Changeof of internal internal resistance resistance with
During the change of SOC from to 30%, itwith is observed ZRe increases only 7.9% for a 3. Model Accordance Comparisons of100% EEC Models Data forthat Different SOC and SOH newConditions battery (100% SOH) while it increases 108% and 162% for 50% SOH and 10 % SOH respectively. At the end of 10th ageing unit, battery voltage decreases below 10 V and the test is over. At that point, EIS graphs gives a lot of information and clues about the electrochemical processes in a battery if SOH is assumed 0% and the battery is fully charged again to do a final test of EIS thorough out the interpreted with an EEC where the elements have physical meaning and have a good match with data. capacity remains. To extract the useful information and make predictions for SOC and SOH from these data, an electric circuit model, which mimics the essential dynamics a battery, is Different needed. Four models are 3. Model Accordance Comparisons of EEC Models of with Data for SOCdifferent and applied for gathered EIS data to analyze their consistency through a broad spectrum range for an EFB. SOH Conditions EEC models consist only from simple electric elements have been considered in the literature both EIS graphs gives a lot of information and clues about the electrochemical processes in a battery for lead acid batteries [22,23] and for other chemistries [13,27–29] as well. In addition to SOC and SOH if interpreted with an EEC where the elements have physical meaning and have a good match with estimations there are applications of these models as a thermal model for batteries [30]. On the bright data. To extract the useful information and make predictions for SOC and SOH from these data, side, these models require less computational power, thus less time to give a prediction and have a an electric circuit model, which mimics the essential dynamics of a battery, is needed. Four different good consistency with data on the higher frequency range if they have a serial inductance. They can be models are applied for gathered EIS data to analyze their consistency through a broad spectrum range used for to anmimic EFB. the voltage response of batteries for a given load using parallel with a coulomb counting based SOC On the other their impedance response consistency with respect to data EEC calculator. models consist only fromhand, simple electric elements have been considered in the literature areboth lower than those of other models because of difficulties in describing the electric characteristics for lead acid batteries [22,23] and for other chemistries [13,27–29] as well. In addition to SOC of diffusion mechanisms conventional elements. Thus, as the signal decreases and SOH estimations by there are applications of these models as aexcitation thermal model forfrequency batteries [30]. On model consistency withmodels the data decreases. Further studies suchthus as neural network [23] or the bright side, these require less computational power, less time to givetraining a prediction variances filtering with [11] for that typehigher of EEC’s have been the model and haveofa Kalman good consistency data on the frequency rangedeveloped if they havetoa decrease serial inductance. error margin. simple element seen in Figure 6a. for a given load using parallel with a They can beAused to mimic themodel voltageis response of batteries coulomb counting SOC thefinds otherapproval hand, their response A Randles cell,based shown in calculator. Figure 7B,On also to impedance interpret lead acid consistency batteries [24]. with respect tocircuit data are lower those of other models because difficulties in describing thefor Elements of the cover thethan essential dynamics of a lead acidofbattery. A serial resistance electric characteristics of diffusion mechanisms by conventional elements. Thus, as the excitation the frequency independent parts like separator and electrolyte resistances, an inductance for all the signal frequency decreases model consistency with the datafor decreases. Further studies such layer as neural high frequency metallic components, a parallel capacitor the electrochemical double effect, network training [23] or variances of Kalman filtering [11] for that type of EEC’s have been developed a parallel resistance for the faradaic reactions that occur on the electrode’s surface and a Warburg to decrease thediffusion model error margin. simple element model is seen inofFigure 6a. element for the related massAtransport limitation processes the battery. A Randles cell, shown in Figure 7B, also finds approval to interpret lead acid batteries [24]. The non-ideal capacitive behavior of an electrode appears with suppressed semi-circles on the Elements of the circuit cover the essential dynamics of a lead acid battery. A serial resistance for the Nyquist plot of EFB. It is hard to imitate this behavior with parallel RC circuit. Model A has been frequency independent parts like separator and electrolyte resistances, an inductance for all the high modified by using a constant phase element (CPE), which is a capacitor with a leakage parameter of α. frequency metallic components, a parallel capacitor for the electrochemical double layer effect, a Impedance expression of CPE is given as: parallel resistance for the faradaic reactions that occur on the electrode’s surface and a Warburg element for the diffusion related mass transport limitation processes of the battery. 1 = CPE α The non-ideal capacitive behavior of Z an electrode appears with suppressed semi-circles on the(3) C (iω ) Nyquist plot of EFB. It is hard to imitate this behavior with parallel RC circuit. Model A has been
Energies Energies2018, 2018,11, 11,118 118
88of of14 14
modified modified by by using using aa constant constant phase phase element element (CPE), (CPE),which which isis aa capacitor capacitor with with aa leakage leakage parameter parameter of of α. Impedance expression of CPE is given as: α. Impedance expression of CPE is given as: Energies 2018, 11, 118
11 = = ( ) ( )
8 of 14
(3) (3)
if == 111 then then CPE CPE element element becomes becomes aa pure capacitor. When α goes ifif α αα= then CPE element becomes pure capacitor. capacitor. When αα goes to to 0, 0, itit behaves behaves like like aa resistor resistor [31]. [31]. This EEC has been used in several works [19,25] to get a more accurate electrode behavior of ThisEEC EEChas hasbeen beenused usedinin several works [19,25] to get a more accurate electrode behavior of lead lead This several works [19,25] to get a more accurate electrode behavior of lead acid acid batteries compared to first two models. acid batteries compared to first two models. batteries compared to first two models.
7. (A) Electricequivalent equivalentcircuit circuit(EEC) (EEC)consisted consistedfrom from simple electric circuit elements L and Figure simple electric circuit elements R, LLand C; Figure7. 7.(A) (A)Electric Electric equivalent circuit (EEC) consisted from simple electric circuit elements R,R, and C; C; (B) EEC of a Randles Circuit with a serial inductance; (C) EEC with constant phase elements (CPEs); (B) EEC of a Randles Circuit with a serial inductance; (C) EEC with constant phase elements (CPEs); (B) EEC of a Randles Circuit with a serial inductance; (C) EEC with constant phase elements (CPEs); (D) (D)EEC EECwith withCPE CPEand andtransmission transmissionline lineelements. elements.
High electrode surface area is a key parameter to get a high power and dynamic charge acceptance High High electrode electrode surface surface area area isis aa key key parameter parameter to to get get aa high high power power and and dynamic dynamic charge charge as well as low internal resistance. New and highly porous active materials have been developed in acceptance acceptance as as well well as as low low internal internal resistance. resistance. New New and and highly highly porous porous active active materials materials have have been been lead acid market and applied on EFB as well as other electrochemical power sources. EEC models, developed developed in in lead lead acid acid market market and and applied applied on on EFB EFB as as well well as as other other electrochemical electrochemical power power sources. sources. which consider the electrode-electrolyte interface as a planar surface, are based on an are assumption that EEC models, which consider the electrode-electrolyte interface as a planar surface, based EEC models, which consider the electrode-electrolyte interface as a planar surface, are based on on an an the reaction that occurs directly on the surface of the the electrode. For a porous electrode surface, electrode however, assumption assumption that the the reaction reaction occurs occurs directly directly on on the surface surface of of the the electrode. electrode. For For aa porous porous electrode ion diffusion parameters depend on electrolyte concentration and it changes as ion penetrates into the surface, surface,however, however,ion iondiffusion diffusionparameters parametersdepend dependon onelectrolyte electrolyteconcentration concentrationand andititchanges changesas asion ion deeper part of pores. Due to continuous parameter changes through the pore, ion diffusion and charge penetrates penetratesinto intothe thedeeper deeperpart partof ofpores. pores.Due Dueto tocontinuous continuousparameter parameterchanges changesthrough throughthe thepore, pore,ion ion transfer parameters are modelled parameters with infinite number of elements. An illustration of a of transmission diffusion diffusion and and charge charge transfer transfer parameters are are modelled modelled with with infinite infinite number number of elements. elements. line element for ion flux inside a pore is given in Figure 8. An illustration of a transmission line element for ion flux inside a pore is given in Figure 8. An illustration of a transmission line element for ion flux inside a pore is given in Figure 8.
Figure inside pore. Figure8. 8.Transmission Transmissionline lineelements elementsinside insideaaapore. pore. Figure 8. Transmission line elements
The impedance of such a transmission line element for porous electrodes on different media has been calculated in the literature before [32]. Theory explained in that work has been applied
Energies 2018, 11, 118 Energies 2018, 11, 118
9 of 14 9 of 14
The impedance of such a transmission line element for porous electrodes on different media has been calculated in the to literature before solar [32]. Theory explained in that work haswhere been diffusion applied from from super capacitors dye-sensitized cells and many other applications and super capacitors to dye-sensitized solar cells and many other applications where diffusion and recombination processes occur. recombination occur. Due to theprocesses nature of lead acid battery chemistry the substrate surface is not isolated and interacts to the nature of lead acid batteryelement, chemistry the substrate surface is not isolated with Due the electrolyte. Thus, a transmission which also includes interactions on theand poreinteracts bottom, with the electrolyte. Thus, a transmission element, which also includes interactions on the pore is considered here. The widely used EEC in Figure 7C is modified by adding a transmission line bottom, is considered here. The widely used EEC in Figure 7C is modified by adding a transmission element and the model given in Figure 7D is proposed for an EFB for the first time. The results and line element andproposed the modelmodel given with in Figure proposed for an EFB the first by time. The results accuracy of the those7D ofisother EEC models are for compared following the and accuracy of the proposed model with those of other EEC models are compared by following the Kramer-Kronig validation of each data set. Kramer-Kronig validation of each data set. EEC models are applied to experimental data by a commercial software based on simplex EEC models are to applied to experimental data bydata a commercial based simplex algorithm which tries minimize the χ2 value between and model software prediction. χ2 is on defined as: algorithm which tries to minimize the value between data and model prediction. is defined as: h i 71 χ2 = ∑ wi ( Zre(data) − Zre(model ) )2 + ( Zim(data) − Zim(model ) )2 (4) (4) =i=1 ( ( ( )) + ( ( )) )− ( )− where number of data is 71 and wi is: where number of data is 71 and wi is: r wi = =
11 Zre(data) (
2 )
(5) (5)
2 ++ Zim(data) ( )
Nyquist and bode diagrams shown in Figure 9a,b are given to demonstrate the accuracy of data fitting of various EEC models. The graph belongs to the battery almost at the end of its its life life (20% (20% SOH) SOH) and 80% SOC level. It It is is seen seen that that without without applying applying aa further further improvement improvement technique, a simple circuit model (Model A) is not able to express the electrochemical phenomena on an EFB especially for the lower frequencies. A A Randles Randles circuit circuit with with aa CPE CPE (Model (Model B) B) has a good agreement with data at the higher frequency region though its accuracy decreases sharply below 1 Hz. The The third third EEC EEC model, model, which has a serial resistance and an inductance connected to two parallel R—CPE elements elements (model (model C) C) to represent both electrodes, has much better compliance with the experimental results in comparison with model A and model B for the whole frequency range from 10 kHz to 1 mHz. In the fourth and final EEC model, transmission line inclusion to both electrodes (model D) decreases the discrepancy between the data and the model further. further.
(a)
(b)
Figure and (b) (b) Bode Bode diagrams diagrams of ofthe theelectrochemical electrochemicalimpedance impedancespectroscopy spectroscopy(EIS) (EIS)data Figure 9. 9. (a) (a) Nyquist Nyquist and data and model compatibility for 80% SOC and 20% SOH. and model compatibility for 80% SOC and 20% SOH.
To makea quantitative a quantitative comparison onaccuracies the accuracies of the all models, all three parameters measured To make comparison on the of the models, three measured parameters Re, phase ZIm, and theΦphase angle Φwith are compared with those of each model basedpercentage on mean ZRe , ZIm , andZthe angle are compared those of each model based on mean absolute absolute percentage (MAPE) method. calculations Re, ZIm, and Φ are given in error (MAPE) method.error MAPE calculations of ZReMAPE , ZIm , and Φ are givenofinZEquations (6)–(8), respectively. Equations (6)–(8), respectively.
Energies 2018, Energies 2018, 11, 118
10 of 14
MAPE(
)=
1
|
(
)
−
(
)|
| (model ) | 1 N | Zre(|data)( − Z)re MAPE( ZRe ) = ∑ i = 1 N | Zre(data) | | ( 1 )− ( )| )= MAPE( | Z − Z 1 | (model ) | im(|data)( N )im MAPE( Z Im ) = ∑ i = 1 N | Zim(data) | 11 MAPE((Φ)) = MAPE = N
|
N
∑ i =1
− −Φ | | |Φdata model | |Φ | | data
(6) (6)
(7) (7)
(8) (8)
MAPE values of Z ZIm andΦΦare aregiven givenin inFigure Figure10, 10,for forall allfour four models. models. ZRe Re,, Z Im, ,and
Figure 10. 10. The The mean mean average average percentage percentage error error (MAPE) (MAPE) values values of of ZZRe Re,, Z , and for different EEC Figure ZIm Im , and Φ for different EEC models at at 80% 80% SOC SOC and and 20% 20% SOH. SOH. models
Model A has a compatibility with the data only for a limited range. Although it can start with a Model A has a compatibility with the data only for a limited range. Although it can start with a matching real resistance, lacking a serial inductance prevents it to handle the positive imaginary part matching real resistance, lacking a serial inductance prevents it to handle the positive imaginary part of the data for the high frequency part. Parallel R-C and R-L parts are trying to form two semi-circles of the data for the high frequency part. Parallel R-C and R-L parts are trying to form two semi-circles and a sharp diffusion behavior which could not cover the data well enough in most circumstances. and a sharp diffusion behavior which could not cover the data well enough in most circumstances. Similar model-data mismatch continues for the phase shift at the high frequency range and resulting Similar model-data mismatch continues for the phase shift at the high frequency range and resulting a bigger average MAPE value than other models which is 24.6% as can be seen in Figure 8. Model B a bigger average MAPE value than other models which is 24.6% as can be seen in Figure 8. Model can cover the data with a good match for the high frequency region with a resistance inductance B can cover the data with a good match for the high frequency region with a resistance inductance couple in series but highly linear response of the second part decreases the model compatibility as couple in series but highly linear response of the second part decreases the model compatibility as the frequency decreases. Model C, which is extensively used in literature, can cover the data with the frequency decreases. Model C, which is extensively used in literature, can cover the data with 3.2% average value of MAPE and has a good adaptation to the data through the whole frequency 3.2% average value of MAPE and has a good adaptation to the data through the whole frequency spectrum. Similar to the previous model, its serial resistance and inductance elements can handle the spectrum. Similar to the previous model, its serial resistance and inductance elements can handle the high frequency spectrum while two ZARCs in series are matching well enough to model the data on high frequency spectrum while two ZARCs in series are matching well enough to model the data on lower frequency range. Adding transmission line elements prior to each ZARC element on Model C lower frequency range. Adding transmission line elements prior to each ZARC element on Model C decreases the error margin even further down to 1.3%, which is shown in Model D. decreases the error margin even further down to 1.3%, which is shown in Model D. Data fitting procedure is applied to 11 different SOH level for nine different SOC conditions as Data fitting procedure is applied to 11 different SOH level for nine different SOC conditions as explained in Figure 1. Prior to each fitting trial, Kramers-Kronig (KK) test is applied to the gathered explained in Figure 1. Prior to each fitting trial, Kramers-Kronig (KK) test is applied to the gathered data. Except the fully charged states, KK can track data with a goodness of fit value of less than 8 × data. Except the fully charged states, KK can track data with a goodness of fit value of less than 10−6 which indicates the reliability of data sets. At 100% SOC levels data solidity decreases because of 8 × 10−6 which indicates the reliability of data sets. At 100% SOC levels data solidity decreases the overcharging behavior of the lower frequency signal deteriorates the Lissajous curves. Beside that because of the overcharging behavior of the lower frequency signal deteriorates the Lissajous curves. state of charge, KK values are always lower than 1% error range for all three measured parameters. Beside that state of charge, KK values are always lower than 1% error range for all three measured Variations of MAPE values for all four models throughout the whole SOC range as the SOH decreases parameters. Variations of MAPE values for all four models throughout the whole SOC range as are given in Figure 11. It is seen that Model D, with transmission line elements, preserves its lower the SOH decreases are given in Figure 11. It is seen that Model D, with transmission line elements, error margin and the accuracy is always higher than those of other models. preserves its lower error margin and the accuracy is always higher than those of other models.
Energies 2018, 11, 118 Energies2018, 2018,11, 11,118 118 Energies
11 of 14 11of of14 14 11
Figure 11. Variation of MAPE values with SOC % for different models at: (a) SOH 100%; (b) SOH 70%; Figure11. 11.Variation Variationof ofMAPE MAPEvalues valueswith withSOC SOC% %for fordifferent differentmodels modelsat: at:(a) (a)SOH SOH100%; 100%;(b) (b)SOH SOH70%; 70%; Figure (c)SOH SOH40%; 40%;and and(d) (d)SOH SOH10%. 10%. (c)
Basedon onthe theaveraged averagedMAPE MAPEvalue valueof ofall allSOC SOCvalues valuesfor foraaacertain certainSOH, SOH,four fourmodels modelsand andKK KK Based Based on the averaged MAPE value of all SOC values for certain SOH, four models and KK values are compared with respect to each other in Figure 12. Depending on data quality based on KK values inin Figure 12.12. Depending on on data quality based on KK values are are compared comparedwith withrespect respecttotoeach eachother other Figure Depending data quality based on results, the averaged MAPE values of Model D over SOC vary from 1.0% to 3.4% for different SOH results, the averaged MAPE values of Model D over SOC vary from 1.0% to 3.4% for different SOH KK results, the averaged MAPE values of Model D over SOC vary from 1.0% to 3.4% for different values andgives gives 1.6%1.6% double averaged result overover SOH while thethe double averaged value ofKK KK values and 1.6% double averaged result over SOH while the double averaged value of isis SOH values and gives double averaged result SOH while double averaged value of KK about 0.8%. On the other hand, the Model C can represent the impedance data with 3.2% double about 0.8%. OnOn thethe other hand, is about 0.8%. other hand,the theModel ModelCCcan canrepresent representthe theimpedance impedancedata data with with 3.2% 3.2% double double averaged MAPE over whole SOC and SOH ranges. The same quantity is 24.7% and 8.9% for Model averaged MAPE over whole SOC and SOH ranges. The same quantity is 24.7% and 8.9% for Model averaged MAPE over whole SOC and SOH ranges. The same quantity is 24.7% and 8.9% for Model A A and Model B, respectively. A and Model B, respectively. and Model B, respectively.
Impedance Impedance MAPE MAPE % %
100.0% 100.0%
10.0% 10.0%
1.0% 1.0%
ModelAA Model
ModelBB Model
ModelCC Model
ModelDD Model
KKTest Test KK
50% 50%
30% 30%
10% 10%
0.1% 0.1% 100% 100%
90% 90%
80% 80%
70% 70%
60% 60%
40% 40%
20% 20%
0% 0%
SOH%% SOH
Figure12. 12.Variation Variationof ofaverage averageMAPE MAPEvalues valueswith withSOH SOHfor fordifferent differentEEC EECmodels. models. Figure EEC models. Figure 12. Variation of average MAPE values with SOH for different
4.Conclusions Conclusions 4. 4. Conclusions An enhanced enhanced lead-acid lead-acid battery battery with with increased increased cold cold cranking cranking performance performance and and cycle cycle life life is is An An enhanced lead-acid battery with increased cold cranking performance and cycle life is examined by EIS for a wide range of SOH and SOC conditions. Charge, discharge, and aging examined by EIS for a wide range of SOH and SOC conditions. Charge, discharge, and aging examined by EIS for a wide range of SOH and SOC conditions. Charge, discharge, and aging proceduresdefined definedfor formicro microhybrid hybridvehicle vehiclebatteries batteriesare areapplied appliedto tothe thebattery. battery.Changes Changesof ofresistance resistance procedures defined for procedures micro hybrid vehicle batteries are applied to the battery. Changes of resistance arealso alsorecorded recorded for these various SOC and SOH SOH steps through the lifetime lifetime of the the It battery. are also recorded for these various SOC through the of battery. ItIt isis are for these various SOC andand SOH stepssteps through the lifetime of the battery. is observed observed that increment of resistance between fully charged and empty states is limited by 7.9% for observed that increment of resistance between fully charged and empty states is limited by 7.9% for that increment of resistance between fully charged and empty states is limited by 7.9% for a new a new battery. This increment boosts to 162% when the battery reaches to the end of its life. Although a new battery. This increment boosts to 162% when the battery reaches to the end of its life. Although there isis aa correlation correlation between between resistance resistance and and SOC, SOC, this this correlation correlation can can only only be be used used for for SOC SOC there estimationififthere thereisisaacertain certainknowledge knowledgeabout aboutthe theSOH SOHof ofthe thebattery. battery. estimation
Energies 2018, 11, 118
12 of 14
battery. This increment boosts to 162% when the battery reaches to the end of its life. Although there is a correlation between resistance and SOC, this correlation can only be used for SOC estimation if there is a certain knowledge about the SOH of the battery. Four EEC models are applied to the impedance data gathered. Consistencies of models with data are examined for a broad SOH and SOC range. It is shown that the proposed model with transition line elements has better accuracy in comparison with those of other models. Although degree of freedom and computing time increase due to additional elements in the model, this problem is trivial for today’s modern computers. The marginal benefit of the model with transmission line element becomes evident as the battery’s SOH decreases. The double averaged MAPE value is only 1.6% for Model D while the Model C has 3.2%. As a result of this work, it is seen that enhanced flooded lead acid batteries with higher porosity electrodes can be simulated better through the frequency spectrum from 10 kHz to 1 mHz with transmission line element included EEC models. These models have a potential for a better SOC and SOH predictions and they may represent more realistic battery behaviors in complex simulation environments. For future works, an EFB battery at different health and charge states will be tested on a test bench which can simulate high charge and discharge currents of an alternator under real driving conditions of a micro hybrid vehicle. The time domain results of the proposed model with transmission line element will be compared with those results. Acknowledgments: This project is supported by INCI GS YUASA Battery Company and the Republic of Turkey, Ministry of Science, Industry, and Technology under contract number of 00552.STZ.2010-1 and authors are thankful to Burak Ulgut from Bilkent University Chemistry Department for a series of seminal discussions. Author Contributions: Can Aksakal conceived, designed, and performed the experiments; Can Aksakal and Altug Sisman have analyzed the data and wrote the paper together. Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature C Capacitance (F) Cn nominal capacity (Ah) FTR Transition Frequency (Hz) i Imaginary unit In nominal discharge current (A) (Cn /20 h) Ri internal resistance (Ω) Uc charging voltage (V) χ2 Chi-square ω Radial Frequency ZRe real impedance (Ω) ZIm imaginary impedance (Ω) Φ Phase Shift Abbreviations AGM Absorbed Glass Mat CHA Charge DCH Discharge DOD Depth of Discharge EEC Electric Equivalent Circuit EFB Enhanced Flooded Battery EIS Electrochemical Impedance Spectroscopy KK Kramers-Kronig MAPE Mean Absolute Percentage Error SLI Starting Lighting Ignition SOC State of Charge (%) SOH State of Health (%)
Energies 2018, 11, 118
13 of 14
References 1.
2.
3.
4.
5. 6. 7.
8. 9. 10.
11.
12. 13.
14. 15. 16. 17. 18. 19.
20. 21.
EU CO2 Emission Standards for Passenger Cars and Light Commercial Vehicles. Available online: http: //www.theicct.org/sites/default/files/publications/ICCTupdate_EU-95gram_jan2014.pdf (accessed on 17 August 2017). Navigant Research Global Market Analysis and Forecast Report on Stop-Start Vehicles. Available online: https://www.navigantresearch.com/newsroom/sales-of-stop-start-vehicles-will-surpass-55-millionunits-annually-by-2022 (accessed on 28 September 2017). Karden, E.; Warm, A.; Rymond, R.; Nagata, Y.; Shinn, P. Lead/Acid batteries for automotive applications—New requirements and ongoing innovation. In Proceedings of the 14th ELBC European Lead Battery Conference, Edinburg, UK, 9–14 September 2014. Budde-Meiwes, H.; Drillkens, J.; Lunz, B.; Muennix, J.; Rothgang, S.; Kowal, J.; Sauer, D.U. A review of current automotive battery technology and future prospects. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 2013, 227, 761–776. [CrossRef] Pavlov, D.; Nikolov, P. Capacitive carbon and electrochemical lead electrode systems at the negative plates of lead–acid batteries and elementary processes on cycling. J. Power Sources 2013, 242, 380–399. [CrossRef] Moseley, P.T.; Rand, D.; Peters, K. Influence of some nanostructured materials on the performance of lead acid battery negative electrodes. Electrochim. Acta 2014, 144, 147–153. Nakhaie, D.; Benhangi, P.H.; Alfantazi, A.; Davoodi, A. The effect of grid configurations on potential and current density distributions in positive plate of lead–acid battery via numerical modeling. Electrochim. Acta 2014, 115, 189–196. [CrossRef] YUASA AGM & EFB Automotive Batteries Explanation Webpage. Available online: https://www.yuasa.co. uk/info/technical/agm-efb-explained/#efb (accessed on 28 September 2017). VARTA EFB Enhanced Technology Explanation Webpage. Available online: https://www.varta-automotive. com/en-gb/technology/efb-battery-technology (accessed on 28 September 2017). BOSCH Enhanced Flooded Battery Technology Explanation Webpage. Available online: http://za.bosch-automotive.com/en_GB/parts_and_accessories_2/service_parts_1/batteries_4/ batterietechnologien_1/efb/efb_1 (accessed on 28 September 2017). Tran, N.; Khan, A.B.; Choi, W. State of charge and state of health estimation of AGM VRLA batteries by employing a dual extended kalman filter and an ARX model for online parameter estimation. Energies 2017, 10, 137. [CrossRef] Tenno, A.; Tenno, R.; Suntio, T. A method for battery impedance analysis. J. Electrochem. Soc. 2004, 151, A806–A824. [CrossRef] Gao, Z.; Chin, C.S.; Woo, W.L. Lithium-ion battery modeling and validation for the smart power system. In Proceedings of the IEEE International Conference on Computer, Communications, and Control Technology (I4CT), Kuching, Malaysia, 21–23 April 2015; pp. 269–274. Zou, C.; Hu, X.; Wei, Z.; Tang, X. Electrothermal dynamics-conscious lithium-ion battery cell-level charging management via state-monitored predictive control. Energy 2017, 141, 250–259. [CrossRef] Zou, C.; Manzie, C.; Neši´c, D. A framework for simplification of PDE-based LITHIUM-Ion battery models. IEEE Trans. Control Syst. Technol. 2016, 24, 1594–1609. [CrossRef] Tsividis, Y.; Milios, J. A detailed look at electrical equivalents of uniform electrochemical diffusion using nonuniform resistance-capacitance ladders. J. Electroanal. Chem. 2013, 707, 156–165. [CrossRef] Gopikanth, M.L.; Sathyanarayana, S. Impedance parameters and the state-of-charge. II. Lead-acid battery. J. Appl. Electrochem. 1979, 9, 369–379. [CrossRef] Mauracher, P.; Karden, E. Dynamic modeling of lead/acid batteries using impedance spectroscopy for parameter identification. J. Power Sources 1997, 67, 69–84. [CrossRef] Blanke, H.; Bohlen, O.; Buller, S.; de Doncker, R.W.; Fricke, B.; Hammouche, A.; Linzen, D.; Thele, M.; Sauer, D.U. Impedance measurements on lead–acid batteries for state-of-charge, state-of-health and cranking capability prognosis in electric and hybrid electric vehicles. J. Power Sources 2005, 144, 418–425. [CrossRef] European Committee for Electrochemical Standardization. Lead-Acid Starter Batteries—Part 1: General Requirements and Methods of Test; Standard BS EN 50342-1; CEN: Brussels, Belgium, 2006. European Committee for Electrochemical Standardization. Lead-Acid Starter Batteries—Part 6: Batteries for Micro-Cycle Applications; Standard BS EN 50342-6; CEN: Brussels, Belgium, 2015.
Energies 2018, 11, 118
22. 23.
24. 25. 26. 27. 28. 29.
30. 31. 32.
14 of 14
Salkind, A.J.; Singh, P.; Cannone, A.; Atwater, T.; Wang, X.; Reisner, D. Impedance modeling of intermediate size lead–acid batteries. J. Power Sources 2003, 116, 174–184. [CrossRef] Monfared, N.A.; Gharib, N.; Moqtaderi, H.; Hejabi, M.; Amiri, M.; Torabi, F.; Mosahebi, A. Prediction of state-of-charge effects on lead-acid battery characteristics using neural network parameter modifier. J. Power Sources 2006, 158, 932–935. [CrossRef] White, C.; Deveau, J.; Swan, L.G. Evolution of internal resistance during formation of flooded lead-acid batteries. J. Power Sources 2016, 327, 160–170. [CrossRef] Zhu, W.H.; Zhu, Y.; Bruce, J.; Tatarchuk, B.J. A simplified equivalent circuit model for simulation of Pb–acid batteries at load for energy storage application. Energy Convers. Manag. 2011, 52, 2794–2799. [CrossRef] Boukamp, B.A. A linear Kronig-Kramers transform test for immittance data validation. J. Electrochem. Soc. 1995, 142, 1885–1894. [CrossRef] Safwat, I.M.; Li, W.; Wu, X. A novel methodology for estimating state-of-charge of Li-Ion batteries using advanced parameters estimation. Energies 2017, 10, 1751. [CrossRef] Jung, S.; Jeong, H. Extended kalman filter-based state of charge and state of power estimation algorithm for unmanned aerial vehicle Li-Po battery packs. Energies 2017, 10, 1237. [CrossRef] Jia, J.; Lin, P.; Chin, C.S.; Toh, W.D.; Gao, Z.; Lyu, H.; Cham, Y.T.; Mesbahi, E. Multirate strong tracking extended kalman filter and its implementation on lithium Iron Phosphate (LiFePO4) battery system. In Proceedings of the 11th IEEE International Conference on Power Electronics and Drive Systems (PEDS), Sydney, Australia, 9–12 June 2015; pp. 640–645. Gao, Z.; Chin, C.S.; Woo, W.L.; Jia, J. Integrated equivalent circuit and thermal model for simulation of temperature-dependent LiFePO4 battery in actual embedded application. Energies 2017, 10, 85. [CrossRef] Zou, C.; Hu, X.; Dey, S.; Zhang, L.; Tang, X. Nonlinear fractional-order estimator with guaranteed robustness and stability for Lithium-Ion batteries. IEEE Trans. Ind. Electron. 2017, PP, 1. [CrossRef] Bisquert, J. Influence of the boundaries in the impedance of porous film electrodes. Phys. Chem. Chem. Phys. 2000, 2, 4185–4192. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
