State-Of-Charge Monitoring and Battery Diagnosis of Different Lithium Ion Chemistries Using Impedance Spectroscopy

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Article State-of-Charge Monitoring and Battery Diagnosis of Different Lithium Ion Chemistries Using Impedance Spectroscopy

Peter Kurzweil 1,* and Wolfgang Scheuerpﬂug 2

1 Electrochemistry Laboratory, University of Applied Sciences (OTH), Kaiser-Wilhelm-Ring 23, 92224 Amberg, Germany 2 Diehl Aerospace GmbH, Donaustraße 120, 90451 Nürnberg, Germany; wolfgang.scheuerpﬂ[email protected] * Correspondence: [email protected]; Tel.: +49-9621-482-3317

Abstract: For lithium iron phosphate batteries (LFP) in aerospace applications, impedance spectroscopy is applicable in the ﬂat region of the voltage-charge curve. The frequency-dependent pseudocapacitance at 0.15 Hz is presented as useful state-of-charge (SOC) and state-of-health (SOH) indicator. For the same battery type, the prediction error of pseudocapacitance is better than 1% for a quadratic calibration curve, and less than 36% for a linear model. An approximately linear correlation between pseudocapacitance and Ah battery capacity is observed as long as overcharge and deep discharge are avoided. We verify the impedance method in comparison to the classical constant-current discharge measurements. In the case of ﬁve examined lithium-ion chemistries, the linear trend of impedance and SOC is lost if the slope of the discharge voltage curve versus SOC changes. With nickel manganese cobalt (NMC), high impedance modulus correlates with high SOC above 70%.

Keywords: battery life testing; capacitance; state-of-charge determination; state-of-health; aging; impedance spectroscopy; pseudocharge; lithium-ion battery

Citation: Kurzweil, P.; Scheuerpﬂug, W. State-of-Charge Monitoring and Battery Diagnosis of Different 1. Introduction Lithium Ion Chemistries Using Because of the high demands on the reliability of emergency power supplies in Impedance Spectroscopy. Batteries aircrafts, planned take-offs can be delayed. After parking for a long time without a power 2021, 7, 17. https://doi.org/10.3390/ batteries7010017 supply, the state-of-charge (SOC) of aircraft batteries drops because of self-discharge. Indeed, capacity determination and recharging of a 2-Ah battery using the constant current

Received: 28 December 2020 discharge method and other diagnosis measures take roughly 1.5 to 2 h according to the Accepted: 15 February 2021 state-of-the-art in air traffic. In the aviation sector, fast charging is only permitted in an Published: 4 March 2021 emergency. As an expensive precaution, freshly charged batteries must be kept ready. Short maintenance intervals require a reliable method for fast battery diagnosis that reflects at Publisher’s Note: MDPI stays neutral least the upper SOC range. with regard to jurisdictional claims in Based on our earlier work [1,2] on SOC determination using electrochemical impedance published maps and institutional affil- spectroscopy [3–6], we have investigated various chemistries of lithium ion batteries under iations. normal ambient conditions and overcharging. We tried to find early indicators for thermal overload and critical overcharge in the impedance spectrum of “healthy” batteries. This work focuses on lithium iron phosphate (LFP) [7,8], which is less sensitive to thermal runaway and fire than cobalt-based systems (LCO, NMC, NCA) and manganese spinel (LMO).

Copyright: © 2021 by the authors. For emergency power supplies in aircrafts, LFP combines a couple of positive properties Licensee MDPI, Basel, Switzerland. such as inherent safety on thermal runaway in the event of overcharging and overheating. This article is an open access article Lithium ions migrate through the linear channels of the olivine lattice Li1-xFePO4. The distributed under the terms and material is readily available, strategically uncritical and almost not harmful. Unfortunately, conditions of the Creative Commons the cell voltage of 3.6 V is lower than the 4.2 V of most other lithium technologies. The Attribution (CC BY) license (https:// capacity is prone to degradation if long periods of time are operated either in the upper or creativecommons.org/licenses/by/ lower potential range and at low temperatures [9–11]. 4.0/).

Batteries 2021, 7, 17. https://doi.org/10.3390/batteries7010017 https://www.mdpi.com/journal/batteries Batteries 2021, 7, 17 2 of 16

Batteries 2021, 7, x FOR PEER REVIEW 2 of 16 1.1. Battery State Indicators

Rated capacity [12] is the electric charge QN that is stored by a new battery under defined1.1. Battery conditions. State Indicators The state-of-charge (SOC) [13] describes the relationship between the currently available capacity, Q(t) = α·β·QN, and the total capacity Q0 at the previous full charge.Ratedα = 1capacity (100% SOC) [12] representsis the electric the fullcharge charge, QN andthatα is= stored 0 (0% SOC)by a isnew the battery emptybattery. under However,defined conditions. lithium-ion The batteries state-of-charge are not allowed (SOC) [13] to be describes discharged the below relationship the cut-off between voltage. the currently available capacity, Q(t) = α·β·QN, and the total capacity Q0 at the previous full

charge. α = 1 (100% SOC) representsSOC the= fullQ/ Qcharge,0 = α and α = 0 (0% SOC) is the empty(1) battery. However, lithium-ion batteries are not allowed to be discharged below the cut- off voltage.The state-of-health (SOH) [13] considers the capacity loss using the remaining capacity in aged batteries Q0 in relation to the nominal capacity QN of the new battery. SOC = 𝑄/𝑄 =𝛼 (1) The state-of-health (SOH) [13] considersSOH = Q the0/ QcapacityN = β loss using the remaining capacity(2) in aged batteries Q0 in relation to the nominal capacity QN of the new battery. The state-of-function (SOF) reflects the deterioration in performance (power fading). For SOC determination [14,15], voltageSOH = 𝑄 measurements/𝑄 =𝛽 have been common practice(2) since the 1930s. Accuracy is poor for relatively flat voltage curves, as shown in Figure1. The state-of-function (SOF) reflects the deterioration in performance (power fading). Hung et al. [16] determined the SOC by the help of a line, which is defined by the dynamic For SOC determination [14,15], voltage measurements have been common practice resistance ∆U/∆I during incremental charging and discharging. since the 1930s. Accuracy is poor for relatively flat voltage curves, as shown in Figure 1. Hung et al. [16] determined the SOC∆(∆U by/ ∆theI) help of a line, which is defined by the dynamic = a · SOC + b (3) resistance ΔU/ΔI during incremental∆SOC charging and discharging.

(a) (b) FigureFigure 1.1. Lithium iron phosphatephosphate batterybattery (LithiumWerks(LithiumWerks 2.62.6 Ah):Ah): ((aa)) ChargeCharge atat constantconstant currentcurrent andand constantconstant voltage.voltage. ( b) Flat(b) Flat discharge discharge proﬁle. profile. EIS EIS measurements measurements were were performed performe fromd from 100% 100% to 30% to 30% state-of-charge state-of-charge (SOC) (SOC) in 2% in steps.2% steps.

Impedance spectroscopy [17], coulomb counting [18], bookkeeping methods [19,20], Δ(Δ𝑈⁄ Δ𝐼) and look-up tables [21] have been established =𝑎⋅SOC+𝑏 since the mid-1970s, and have been supple-(3) ΔSOC mented in the past decade by fuzzy logic, Kalman ﬁlters, learning algorithms, predictive methodsImpedance [15], and spectroscopy the analysis [17], of relaxation coulomb timescounting [22]. [18], At present, bookkeeping there ismethods still no univer-[19,20], saland method look-up of tables precisely [21] have determining been establishe the usabled since capacity the mid-1970s, of a battery and without have been completely supple- dischargingmented in the the past battery. decade by fuzzy logic, Kalman filters, learning algorithms, predictive methods [15], and the analysis of relaxation times [22]. At present, there is still no univer- 1.2.sal method State-of-Health of precisely Indicators determining the usable capacity of a battery without completely dischargingImpedance the battery. spectroscopy displays changes in the resistance and capacitance of a battery without destroying the cell. Reliable methods of health assessment and fault diagnosis1.2. State-of-Health such as lithiumIndicators plating, short circuit, overcharge, and deep discharge require a Impedance spectroscopy displays changes in the resistance and capacitance of a battery without destroying the cell. Reliable methods of health assessment and fault diagnosis such as lithium plating, short circuit, overcharge, and deep discharge require a deep Batteries 2021, 7, 17 3 of 16

deep understanding of broadband impedance and modeling, as the impedance spectrum is strongly inﬂuenced by battery states and operating conditions [23], See Section 3.7. The impedance at a certain frequency is useful in diagnosing the thermal runaway. The electrolyte resistance, RS = Re Z(ω → ∞), the intersection on the real axis, does not decrease strictly linearly with rising temperature. Eddahech et al. [24] suggest the real part at 0.1 Hz for the estimation of remaining life. As well, Galeotti et al. [25] correlate the available capacity and the SOH with the ohmic resistance. Howey et al. [26] evaluate the range between 1 Hz and 2 kHz with both multisine and noise excitation signals. M. Spielbauer et al. [27] studied the mechanical deformations using computer tomography; internal short circuits cause a drop in ohmic resistance at frequencies above 100 Hz. Srinivasan et al. [28] use the abrupt rise of the phase shift part of cell impedance as a quick indicator, seconds before the upcoming cell venting and the thermal runaway some minutes later; the phase shift ϕ depends only slightly on the battery size and Ah-capacity. The cell voltage remains constant until after the cell has been vented. The current state-of-the-art does not provide a universal and rapid method of determining the state-of-health of a battery without carefully examining the degradation of hundreds of full charge/discharge cycles. This work tries a novel approach in order to link the actual battery charge with a quickly measurable impedance quantity.

1.3. Pseudocapacitance and Pseudocharge Each impedance value consists of a real part (ohmic resistance) and an imaginary part (reactance). The real part of impedance (resistance R = Re Z) and the modulus |Z(ω)| reflect the electrolyte and the kinetic inhibitions of the electrode processes. We add the pseudocapacitance C(ω)[1,29] according to Equation (4), as a unique measure for the activity of the electrode/electrolyte interface. As a qualitative indicator of the battery’s state-of-charge, this quantity depends mainly on the imaginary part of impedance (reactance, X = Im Z).

dQ Im Y(jω) −Im Z(jω) C(ω) = = Re C(jω) = = (4) dU jω ω · |Z(jω)|2

At high frequencies (ω → ∞), pseudocapacitance tends to the geometric double-layer capacitance CS of the interface. The ohmic resistance of the electrolyte solution, RS = Re Z(ω → ∞), is found as the intersection of the complex plane plot with the real axis. The approximation in Equation (5) holds for high frequencies, when the polarization resistance of the battery is negligible [30]. This is true when the DC resistance of the battery is not much greater than the electrolyte resistance RS.

−Im Z(ω) −1 CS = lim C(ω) = lim ≈ (5) ω→∞ ω→∞ h 2 2i ω · Im Z(ω) ω · [Re Z(ω) − RS] + [Im Z(ω)]

The pseudocapacitance C(ω) can be calculated for individual data points in all areas of the impedance spectrum. The differently fast processes at the electrode-electrolyte interface appear in certain frequency ranges. The electrolytic double layer, absorbed ions on the electrode surface, and ions intercalating into the porous electrodes cause the capacitive properties of cell impedance. Depending on the frequency range, pseudocapacitance according to Equation (4) can be interpreted as double-layer capacitance (at high and medium frequencies) or as capacitance that is involved in ion adsorption and mass transport phenomena (at low frequencies). Common equivalent circuits for batteries describe this mass transport capacitance by transmission line networks that contain resistances and capacitances or the so-called Warburg impedance. This work dispenses with the curve fitting to equivalent circuit diagrams. The diagram of frequency-dependent capacitance C(ω) versus resistance R is useful for the direct comparison of storage batteries (see Section3). Batteries 2021, 7, 17 4 of 16

Since the impedance method does not provide Ah-capacities, we introduce the pseudocharge Q(ω) as a measure for the available electric charge of the battery. U(t) is the cell voltage at time t of the impedance measurement.

Q(ω, t) = C(ω) · U(t) (6)

Note that there is a scaling factor between the true electrical charge (battery capacity) and the frequency-dependent pseudocharge Q(ω). See Section 3.3.

2. Experimental Setup This lithium-ion batteries of the LFP type of different manufacturers were investigated in the course of long-time tests under real conditions as in the airplane. A. VoltSolar: 18,650 type, 3.2 V, 1.4–1.5 Ah, charge max. 3.65 V, cut-off voltage 2.0 V. B. Sony: US18650FT cylindrical, 3.2 V, 1.05–1.1 Ah, C. LithiumWerks (formerly A123) 26,650 cell, ANR26650M1B: 3.3 V, 2.6 Ah. Different cell chemistries and manufacturers were compared (see Table1).

Table 1. Lithium-ion batteries in this study. Nominal data according to manufacturers’ data sheets. Approximate slope of the voltage-SOC curve (see Section 3.6). Cubic and linear approximation of the capacitance-capacity curve (see Section 3.2).

∆U Chemistry Cell Voltage Capacity C Rate ∆SOC Correlation of Pseudocapacitance U (V) Q Charge and V per (in F) and Capacity (in Ah) 3 2 (Ah) Discharge 100 % CS=aQ +bQ +cQ+d

1 LTO BE Power 18,650 (Li4Ti5O12) 2.8 . . . 1.5 1.3 6 1.5 – – – – – 2 LFP SONY US18650 FTC 3.6 . . . 2 1.1 1.1 30 0.21 – – – – 3 LithiumWerks ANR26650M1B 3.6 . . . 2 2.6 4 70 0.18 – – – – 4 VoltSolar 18,650 IFR 3.6 . . . 2 1.4 – 3 – – – – – Samsung INR18650-20R 2.73 −12.5 19.3 −9.34 5 LMO 4.2 . . . 2.5 2 4 20 1.15 (LiNiCoMnO2) – – 302 6.2 6 NMC Samsung ICR18650-22P (MnNi) 4.2 . . . 2.5 2.15 2.15 10 – – – – – 7 Samsung INR18650-25R (NiMn) 4.2 . . . 2.5 2.5 4 20 – – – – – 1018 −5798 11,032 −6540 8 LG ICR18650HE2 (Ni Mn Co) 4.2 . . . 2.0 2.5 4 20 1.0 – – 182 132 9 LG 18650-HG2 (Co Ni Mn) 4.2 . . . 2.5 3 4 20 – – – – – 516 −2644 4599 −2349 10 LCO Panasonic UR18650 FK 4.2 . . . 2.5 2.5 1.75 5 0.81 – – 231 −59 11 NCA SONY US18650VTC5 4.2 . . . 2.5 2.6 2.5 20 1.0 – – – – 12 LG INR18650MH1 4.2 . . . 2.5 3.2 3.1 10 – – – – – 13 Panasonic NCR18650GA 4.2 . . . 2.5 3.3 10 10 0.93 – – – – 14 Samsung INR18650-35E LiNiCoAlO2 4.2 . . . 2.65 3.35 2 8 – – – – – 15 Panasonic NCR18650 B 4.2 . . . 2.5 3.4 1.62 3.4 – – – – –

2.1. Test Procedure In order to ﬁnd the desired correlation between pseudocapacitance and real remaining battery capacity (SOC), the fully charged battery was discharged in 2% steps using constant current. After a rest period to set a stationary cell voltage, the impedance spectrum was measured. 1. Capacity determination by coulomb-counting: Each cell was ﬁrst charged at 1 C rate (CC) to the upper cutoff voltage, then at constant voltage (CV) until the current dropped below 0.1 A. The discharge took place at 25 ◦C and 1 C rate under a constant current load until the cut-off voltage was reached. Q is the withdrawn electrical charge. Then the battery was recharged as above. 2. Impedance measurements were taken 20 min after each constant-current discharge step, from SOC 100%, 98%, 96%, down to 30%. The VoltSolar cell was tested down to 50% SOC. The EIS measurement took 80 s for six values per frequency decade in the Batteries 2021, 7, x FOR PEER REVIEW 5 of 16

Batteries 2021, 7, 17 5 of 16 2. Impedance measurements were taken 20 min after each constant-current discharge step, from SOC 100%, 98%, 96%, down to 30%. The VoltSolar cell was tested down to 50% SOC. The EIS measurement took 80 s for six values per frequency decade in the fre- frequency range from 1 kHz to 0.1 Hz. The measuring arrangement is schematically quencyshown inrange Figure from2. 1 kHz to 0.1 Hz. The measuring arrangement is schematically shown in Figure 2.

FigureFigure 2.2. ((aa)) SchematicSchematic measurementmeasurement setup:setup: ImpedanceImpedance spectrometerspectrometer BIM2BIM2 (BRS(BRS GmbH,GmbH, Stuttgart,Stuttgart, Germany),Germany), datadata loggerlogger (Agilent(Agilent 34972A), 34972A), electronic electronic load load (ET (ET Systems Systems ELP/DCM ELP/DCM 9712C), DC powerpower sourcesource (Elektro-Automatik(Elektro-Automatik EA-PSEA-PS 2342-10B),2342-10B), andand relayrelay boxbox residereside inin a a climatic climatic chamber chamber (Vötsch (Vötsch VT7021). VT7021). DataData acquisition:acquisition: Labview.Labview. ((bb)) BasicBasic modelmodel assumptionassumption andand datadata evaluation.evaluation.

2.2. Measurement Parameters 2.2. Measurement Parameters The battery is periodically charged and discharged by a small AC excitation signal that isThe superimposed battery is periodically on the almost charged constant and cell discharged voltage, so by that a small no net AC charge excitation or discharge signal thatof the is superimposedcell occurs. The on AC the amplitude almost constant must be cell small voltage, in order so that to meet no net the charge criterion or discharge of small- ofsignal the cellexcitation, occurs. but The large AC enough amplitude to find must a compromise be small in orderwith the to meetduration the of criterion the meas- of small-signalurement periods. excitation, An adequate but large integration enough to time find of a compromise8 to 10 cycles with was the chosen duration in order of the to measurementobtain smooth periods. impedance An adequatespectra. The integration current flowing time of through 8 to 10 cycles the cell was was chosen negligible in order (20 tomA obtain AC, smooth<10 mA impedanceDC). It would spectra. take The hundre currentds of flowing impedance through measurements the cell was negligibleor several (20hours mA to AC, significantly <10 mA DC). discharge It would the take battery. hundreds The cell of impedancevoltage did measurementsnot change by ormore several than hours0.05 V to during significantly the impedance discharge measurement the battery. Theaccording cell voltage to the did data not logger. change by more than 0.05 VThe during equilibration the impedance time after measurement setting the SOC according (2% step) tothe is important data logger. to avoid outliers. In the eventThe equilibration of a non-equilibrium, time after setting the battery the SOC discharges (2% step) against is important the measurement to avoid outliers. device, In theand event the individual of a non-equilibrium, cells in a battery the battery pack discharges interfere againstwith each the other. measurement Measurement device, time and the individual cells in a battery pack interfere with each other. Measurement time must be must be invested if the data points scatter. The frequency point at 0.15 Hz takes roughly invested if the data points scatter. The frequency point at 0.15 Hz takes roughly a minute a minute (6.7 s times ten cycles) for the measurement, which is still fast compared to a 1 C (6.7 s times ten cycles) for the measurement, which is still fast compared to a 1 C constant constant current discharge profile (1 h) and the subsequent recharging of a 2 Ah battery current discharge profile (1 h) and the subsequent recharging of a 2 Ah battery (fast CC (fast CC and slow CV charging takes 1.5 h according to the above Figure 1a). This proce- and slow CV charging takes 1.5 h according to the above Figure1a). This procedure seems dure seems appropriate because the aviation batteries are measured after a period of stor- appropriate because the aviation batteries are measured after a period of storage. That is, age. That is, the impedance measurement can be started without delay. the impedance measurement can be started without delay.

3.3. ResultsResults andand DiscussionDiscussion 3.1.3.1. FrequencyFrequency ResponseResponse ofof PseudocapacitancePseudocapacitance TheThe NyquistNyquist diagramdiagram inin FigureFigure3 3 can can be be regarded regarded as as composed composed of of two two views views [ [31],31], whichwhich showshow thethe frequencyfrequency responseresponse ofof resistanceresistanceR R= = ReRe ZZand and reactancereactanceX X= = ImIm ZZ.. BatteriesBatteries 20212021, 7, ,x7 FOR, 17 PEER REVIEW 6 of6 16 of 16

FigureFigure 3. Frequency 3. Frequency response response of impedance of impedance of the cell ofC LithiumWerks the cell C LithiumWerks (2.6 Ah). The electrolyte (2.6 Ah). The S 𝐶 = −1⁄ (𝜔𝑋), resistanceelectrolyte R = R resistance(333 Hz) wasRS subtracted.= R(333 Capacitance Hz) was subtracted. versus frequency: Capacitance versusincluding frequency: leakage: 𝐶 = −𝑋⁄ 𝜔(𝑅 +𝑋 ), and corrected by the electrolyte2 2 resistance: 𝐶 = CS = −1/(ωX), including leakage: CP = −X/ ω R + X , and corrected by the electrolyte resis- ⁄ −𝑋 𝜔((𝑅 −𝑅 ) +𝑋h) according2 to Equation2i (4). Imaginary part of impedance: X = Im Z(ω). tance: CPR = −X/ ω (R − RS) + X according to Equation (4). Imaginary part of impedance: X = Im Z(ω). Some parts of the spectra show more or less linear regions. The data of the other tested cellsSome are parts comparable of the spectra and do shownot provide more orany less additional linear regions. information The dataat this of stage the other of evaluation.tested cells The are different comparable courses and of do the not Nyquis providet plot any for additional the same information battery chemistry at this stageraise of theevaluation. urgent question The different of which courses physical of thevariable Nyquist can plot be formeaningfully the same battery evaluated chemistry for SOC raise monitoring.the urgent The question Nyquist of diagrams which physical in Figure variable 4 show can that be the meaningfully state-of-charge evaluated affects for both SOC themonitoring. real part and The the Nyquist imaginary diagrams part of in imp Figureedance.4 show Resistance that the state-of-charge drops and capacitance affects both in- the creasesreal partwith and SOC. the Unlike imaginary the 18,650 part of cells, impedance. the 26,650 Resistance design exhibits drops andsignificant capacitance inductance, increases i.e.,with positive SOC. imaginary Unlike the parts 18,650 occur cells, at high the 26,650 frequencies design above exhibits 333 signiﬁcant Hz. The stored inductance, residual i.e., electricpositive charge imaginary Q (capacity) parts of occur a battery at high is frequenciesexpected to abovecorrelate 333 with Hz. the The pseudocapaci- stored residual tanceelectric according charge toQ the(capacity) definition of a C battery = dQ/d isU expected. The pseudocapacitance to correlate with theaccording pseudocapacitance to Equa- tionaccording (4) is coined to the by deﬁnitionthe reactance,C = therefor dQ/dUe. Thethe imaginary pseudocapacitance part of impedance according is toexpected Equation to show(4) is coinedthe state-of-charge by the reactance, as well. therefore What is theneeded, imaginary however, part is of a impedancequantity that is maps expected the to state-of-chargeshow the state-of-charge as linearly as as possible well. What over isa needed,large measuring however, range. is a quantity Pseudocapacitance that maps the C(ωstate-of-charge) combines all asinformation linearly as about possible resistance, over a largereactance, measuring and frequency. range. Pseudocapacitance The electrolyte resistance,C(ω) combines which is all of information little use for about SOC resistance,determination, reactance, is subtracted and frequency. from the The measured electrolyte realresistance, parts. The which pseudocapacitance is of little use corrected for SOC determination, in this way is then is subtracted directly related from theto the measured elec- tricreal charge. parts. The pseudocapacitance corrected in this way is then directly related to the electric charge. BatteriesBatteries2021 2021, 7,, 17 7, x FOR PEER REVIEW 7 of7 16of 16

Im Z /mΩ C / F -20 50 A 100 90 80 70 45 0.1 Hz 40 -15 35

0.1 Hz 30 -10 25 3 20 15 -5 1 10 1 kHz 5 1 kHz 2 0 0 50 55 60 65 70 75 55 60 65 70 75 Re Z / mΩ Re Z / mΩ

Im Z / mΩ C / F -20 30 B 98 90 80 70 25 -15 20

-10 15

0.1 Hz 10 -5 5

1 kHz 0 0 40 45 50 55 60 65 40 45 50 55 60 65 Re Z / mΩ Re Z / mΩ

Im Z /mΩ C / F -4 30 C 100 90 80 70 -3 25 0.1 Hz

-2 20 capacitive -1 15

0 10

1 5 inductive 1 kHz 2 0 65 67 69 71 73 40 45 50 55 60 65 Re Z / mΩ Re Z / mΩ

Figure 4. SOC monitoring by impedance spectroscopy of lithium iron phosphate (LFP) batteries at rest: (A) VoltSolar (1.5 Ah),Figure (B) Sony 4. SOC (1.1 monitoring Ah), (C) LithiumWerks by impedance (2.6 spectroscopy Ah). Pseudocapacitance of lithium ironC(ω phosphate) according (LFP) to Equation batteries (4), at where rest: (A the) VoltSolar electrolyte (1.5 Ah), (B) Sony (1.1 Ah), (C) LithiumWerks (2.6 Ah). Pseudocapacitance C(ω) according to Equation (4), where the electro- resistance RS = R(333 Hz) was subtracted from the real parts of impedance. Frequency range 0.1 Hz to 1 kHz. Regions: 1 lyte resistance RS = R(333 Hz) was subtracted from the real parts of impedance. Frequency range 0.1 Hz to 1 kHz. Regions: Electrolyte resistance, 2 Charge transfer, 3 mass transport. 1 Electrolyte resistance, 2 Charge transfer, 3 mass transport.

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3.2.3.2. CorrelationCorrelation ofof PseudocapacitancePseudocapacitance andand BatteryBattery CapacityCapacity InIn the the following following the the signiﬁcance significance of of the the pseudocapacitance pseudocapacitanceC C(ω(ω)) for for the the actual actual electric electric chargechargeQ Qand and energy energy content content of of the the battery battery is is shown. shown. The The linear linear relationship relationship between between the the pseudocapacitancepseudocapacitance andand thethe SOCSOC inin FigureFigure5 5is is best best at at a a frequency frequency around around 0.15 0.15 Hz. Hz. This This meansmeans thatthat thethe capacitancecapacitance doesdoes notnot necessarilynecessarily havehave toto bebe measuredmeasured downdown toto extremely extremely lowlow frequencies.frequencies. The The chosen chosen frequency frequency of of 0.15 0.15 Hz Hz covers covers a part a part of the of the mass mass transport transport lim- limitedited charge charge transfer transfer reaction reaction that that is proportional is proportional to tothe the genuine genuine charge charge storage storage process process by bylithium lithium ion ion intercalation intercalation at atvery very low low frequencies. frequencies.

(a) (b)

FigureFigure 5. 5.( a(a)) State-of-charge State-of-charge monitoring monitoring of of a VoltSolara VoltSolar LFP LFP cell cell (1.5 (1.5 Ah) Ah) using using capacitance capacitance at selected at selected frequencies. frequencies. Example: Example:C/F C/F = 0.1925·SOC + 11.64 at 0.15 Hz. SOC = Q/Q0 is based on Ah counting. EIS measurements were performed from 100% = 0.1925·SOC + 11.64 at 0.15 Hz. SOC = Q/Q0 is based on Ah counting. EIS measurements were performed from 100% to 50% SOCto 50% in 2% SOC steps. in 2% (b )steps. Linear (b relationship) Linear relationship between capacitance between ca andpacitance battery and capacity: batteryC capacity:= 14.65·Q C+ = 11.64 14.65 (fit⋅Q quality:+ 11.64 (fit 95.3%). quality: 95.3%). We conclude that pseudocapacitance reflects the state-of-charge better than any other quantityWe obtainedconclude from that pseudocapacitance impedance spectroscopy. reflects The the linearity state-of-charge between better pseudocapacitance than any other andquantity electric obtained charge isfrom improved, impedance when spectros the electrolytecopy. The resistance linearityRS isbetween corrected pseudocapaci- in Equation (6).tance Overcharging and electric disrupts charge theis improv linear trended, when between the electrolyte capacity and resistance SOC. The R abruptS is corrected increase in inEquation capacity (6). at SOCOvercharging = 1, however, disrupts could the be lin usedear to trend indicate between that thecapacity battery and is fullySOC. charged The ab- beforerupt increase there is in a riskcapacity of overcharging. at SOC = 1, however, could be used to indicate that the battery is fully charged before there is a risk of overcharging. 3.3. Verification and Validation of the Pseudocapacitance Model 3.3. VerificationThe correlation and Validation of pseudocapacitance of the PseudocapacitanceC at 0.15 Model Hz and state-of-charge (SOC) was verifiedThe by correlation impedance of measurementspseudocapacitance on three C at 0.15 different Hz and LithiumWerks state-of-charge 2.6 (SOC) Ah batteries. was ver- Theified “true” by impedance SOC values measurements were determined on three by di Ahfferent counting. LithiumWerks Each battery 2.6 wasAh batteries. tested three The times“true” in SOC a row values in the were SOC determined range from by 100% Ah co tounting. 30% in Each 2% steps. battery Analogous was tested to three Figure times5a, congruentin a row inC the(SOC) SOC curves range were from obtained 100% to 30% with in a repeatability2% steps. Analogous of about to 2%. Figure 5a, congruent CThree(SOC) batteries curves were of the obtained same type with were a repeatability tested on different of about days 2%. (see Table2). The pseu- docapacitanceThree batteries values of had the a same precision type ofwere better tested than on 12% different for each days SOC (see value. Table According 2). The pseu- to thedocapacitance David test, the values data had obey a aprecision normal distribution. of better than A 12% linear for model each isSOC a good value. approximation, According to havingthe David a regression test, the data coefficient obey a of normal 98.6%. dist Accordingribution. to A the linear Mandel model test, is a however, good approxima- a square 2 modeltion, having is better a regression (R = 99.3%). coefficient The linear of 98.6 trend%. getsAccording better ifto the the SOC Mandel range test, is consideredhowever, a upsquare to 98% model (instead is better of 100%),(R2 = 99.3%). because The overcharge linear trend phenomena gets better if occur the SOC at full range charge. is consid- The predictionered up to error 98% is(instead less than of 1%100%), for thebecause quadratic overcharge model phenomena (average ± 0.4%),occur andat full less charge. than 36%The forprediction the linear error model. is less than 1% for the quadratic model (average ± 0.4%), and less than 36% for the linear model. Batteries 2021, 7, 17 9 of 16

Batteries 2021, 7, x FOR PEER REVIEW 9 of 16 Table 2. Validation of the linear correlation between state-of-charge and pseudocapacitance for three lithium-iron phosphate batteries of the same manufacturer.

TableBattery 2. Validation of the linear correlation Battery 1between state-of-charge and Battery pseudocapacitance 2 for three lithium-iron Battery 3 phos- phateMeasurement batteries of the same manufacturer. 1 2 3 1 2 3 1 2 3 RepeatabilityBattery of 1.42Battery 1.37 1 1.36 1.48Ba 1.15ttery 2 1.36 1.56 Battery 1.35 3 1.34 slopeMeasurement∆SOC/∆C 1 1.38 ±2 0.023 1 1.33 ±2 0.133 1 1.41 ±2 0.09 3 Repeatability of 1.42 1.37 1.36 1.48SOC = 1.4121.15·C − 220.41.36 1.56 1.35 1.34 Linear model slope ΔSOC/ΔC 1.38 ± 0.02 or C = 0.6995·SOC − 1.33152.8 ± for0.13 SOC in % and C in F. 1.41 ± 0.09 SOC = −SOC0.01120 = 1.412·C2 +⋅C 5.877 − 220.4·C − 657.6 QuadraticLinear model model or C =or 0.6995C = 0.003748⋅SOC −·SOC 152.82 + for 0.2198 SOC·SOC in % + 166.6and C in F. SOC = −0.01120⋅C2 + 5.877⋅C − 657.6 Quadratic model If the SOC determinedor C = by 0.003748 Ah counting⋅SOC2 + is 0.2198 assumed⋅SOC to + be166.6 the “true” value, the model predicts SOC values that are good enough to estimate whether a battery of the same type and manufacturerIf the SOC determined is full, three by quarterAh counting full, half-full, is assumed quarter to be full, the or“tru empty.e” value, the model predicts SOC values that are good enough to estimate whether a battery of the same type 3.4.and Propertiesmanufacturer of Pseudocharge is full, three quarter full, half-full, quarter full, or empty. We introduce the pseudocharge Q(ω) = C(ω)·U(t), so that the capacitance C(ω) ob- tained3.4. Properties by impedance of Pseudocharge spectroscopy can be better compared with the actual electrical charge Q (batteryWe introduce capacity). the This pseudocharge quantity includes Q(ω) the= C( instantaneousω)⋅U(t), so that cell the voltage capacitanceU(t), which C(ω) ob- for itselftained is by not impedance a precise SOC spectroscopy indicator (see can Figure be be6ttera). Thecompared voltage with exhibits the severalactual plateauselectrical whilecharge the Q pseudocharge(battery capacity). is fairly This linear quantity with in respectcludes tothe the instantaneous SOC. Furthermore, cell voltage the pseu-U(t), dochargewhich for canitself be is normalized not a precise to SOC the fully indicator charged (see battery, Figure 6a). so that The a voltage SOC scale exhibits between several 0% andplateaus 100% while is obtained. the pseudocharge is fairly linear with respect to the SOC. Furthermore, the Q(ω, t) C(ω, t) · U(t) pseudocharge can be normalizedSOC(t )to∼ the fully charged∼ battery, so that a SOC scale between(7) Q (ω) C (ω) · U 0% and 100% is obtained. 0 0 0 Q(ω) /As Q / As 120 3.5 5000

100 4000 0.15 Hz U / V 80 3.4 3000 60

2000 40 3.3 0.22 Hz 0.15 Hz 0.1 Hz

1000 20

0 3.2 0 50 60 70 80 90 100 0 50 100 150 200 SOC in % Q(ω) / As

(a) (b)

FigureFigure 6.6. VoltSolar LFP battery. ( a)) Pseudocharge Pseudocharge QQ(0.15(0.15 Hz) Hz) == C C(0.15(0.15 Hz) Hz)⋅U·U andand cell cell voltage voltage U Uversusversus the the “true” “true” state-of- state- of-chargecharge Q Qmeasuredmeasured by by Ah Ah counting. counting. (b (b) )Linear Linear correlation correlation between between pseudocharge pseudocharge QQ((ωω)) and and “true” battery capacitycapacity atat differentdifferent dischargedischarge states.states. Example:Example: QQ/As/As = 66.92⋅·Q(0.15 Hz) − 2226.2226. LPF LPF batteries from different manufacturersmanufacturers showshow qualitatively the same results. qualitatively the same results.

There is a scaling factor between the𝑄 real(𝜔, electric 𝑡) 𝐶(𝜔,𝑡) charge and ⋅ 𝑈(𝑡) the pseudocharge which in practice must be determined bySOC calibration(𝑡) ∼ (see Section∼ 3.3). Nevertheless, the pseudocharge(7) 𝑄(𝜔) 𝐶(𝜔) ⋅ 𝑈 (in As) at a selected frequency is a linear function of “true” battery capacity (see Figure6b). There is a scaling factor between the real electric charge and the pseudocharge which in practice must be determined by calibration (see Section 3.3). Nevertheless, the pseudocharge (in As) at a selected frequency is a linear function of “true” battery capacity (see Figure 6b). At high state-of-charge, overcharge effects cause a slight deviation from the Batteries 2021, 7, 17 10 of 16

Batteries 2021, 7, x FOR PEER REVIEW 10 of 16

At high state-of-charge, overcharge effects cause a slight deviation from the generally linear relationship. The linearity is coined by the steps in the voltage-capacity curve. generally linear relationship. The linearity is coined by the steps in the voltage-capacity Example: The fully charged LithiumWerks battery (2.6 Ah, 3.35 V) exhibits a pseu- curve. docharge of: Q(0.15 Hz) = CU = 201.6 F·3.35 F = 675.4 As = 0.188 Ah. Therefore, there is a correctionExample: factor The offullyQ =charged 13.9·Q (0.15LithiumWerks Hz), which ba isttery required (2.6 Ah, to compare3.35 V) exhibits Ah counting a pseu- withdocharge the impedance of: Q(0.15 method. Hz) = C The U = “true” 201.6 energyF·3.35 F of = W675.4= 3.35 As V= ·0.1882.6 Ah Ah. = 8.7Therefore, Wh or 9360 there Ws is a requirescorrection the correction:factor of Q W= 13.9= 4.1⋅Q·Q(0.15(0.15 Hz), Hz). which is required to compare Ah counting with the impedance method. The “true” energy of W = 3.35 V⋅2.6 Ah = 8.7 Wh or 9360 Ws re- 3.5.quires Conversion the correction: of Impedance W =Data 4.1⋅Q to(0.15 Energy Hz). and Power Pseudocharge Q(0.15 Hz) allows to calculate an equivalent for the stored energy W and3.5. the Conversion speciﬁc energy of Impedance of the Data battery to Energy (energy and divided Power by mass), as shown in Figure7a. Pseudocharge Q(0.15 Hz) allows to calculate an equivalent for the stored energy W and the specific energy of the batteryW(ω, t) (ener= Qgy(ω ,dividedt) · U(t) by mass), as shown in Figure 7a.(8)

𝑊(𝜔)⁄ Wh 𝑊 ⁄Wh 𝑊 ⁄Wh 10 Wh/kg 10 10 9 SOC 9 9 8 8 8 200 C 7 C 7 C 7 6 6 6 5 5 5 4 4 4 A A 100 A 3 3 3 2 2 2 50 B B 1 B 1 1 0 0 0 60 65 70 75 05100 1 10 100 ⁄ C rate P / W 𝑅(𝜔) Ω (a) (c) (b)

FigureFigure 7. (7.a )(a Energy) EnergyW Wof of three three types types of of LFP LFP cells cells determined determined by by the the pseudocharge pseudochargeQ =QCU = C atU at 0.15 0.15 Hz Hz against against the the real real part part ofof impedance impedance (ohmic (ohmic resistance) resistance) at 0.15at 0.15 Hz. Hz. A VoltSolarA VoltSolar 3.2 3.2 V/1.4 V/1.4 Ah, Ah, B SonyB Sony 3.2 3.2 V/1.1 V/1.1 Ah, Ah, C LithiumWerksC LithiumWerks 3.3 3.3 V/2.6 V/2.6 Ah. Ah. (b)(b For) For comparison: comparison: Peukert Peukert diagram diagram of true of true energy energy versus versus C rate, C rate, (c) Ragone (c) Ragone plot: plot: true true energy ener versusgy versus power power (measured (measured by by current and voltage). current and voltage).

To prove the quality of the impedance𝑊(𝜔, method, 𝑡) = 𝑄(𝜔, we compared 𝑡) ⋅𝑈(𝑡) the pseudoenergy with(8) data from conventional electrical stress tests, in which LFP cells were discharged at different currents.To Theprove available the quality energy of the is compiledimpedance in me thethod, Peukert we compared plot against the the pseudoenergy applied C rate with (Figuredata 7fromb), and conventional in the Ragone electrical plot (Figure stress 7testc) againsts, in which the withdrawn LFP cells were electric discharged power. at dif- ferentWith currents. a high current The available (C rate) energy and high is compiled discharge in power, the Peukert the battery plot willagainst be exhausted the applied inC less rate time. (Figure As can 7b), be and seen, in the the energy Ragone agrees plot very (Figure well 7c) with against the results the withdrawn using impedance electric data,power. at least on the order of magnitude. With a high current (C rate) and high discharge power, the battery will be exhausted 3.6.in Impactless time. of Cell As Chemistrycan be seen, the energy agrees very well with the results using impedance data,The at determinationleast on the order of aof generalized magnitude. absolute state-of-charge is complicated by the various cell chemistries of lithium-ion batteries. Therefore, we compared different lithium- ion3.6. batteries Impact asof Cell compiled Chemistry in Table 1 to demonstrate the quality of capacitance C( ω) for SOC monitoring.The determination The impedance of spectraa generalized (Figure 8absolute) show three state-of-charge regions: is complicated by the 1. variousElectrolyte cell chemistries and solid-electrolyte of lithium-ion interface batteries. (SEI) Therefore, at high frequencies; we compared different lith- 2. ium-ionCharge-transfer batteries as at compiled medium in frequencies; Table 1 to demonstrate the quality of capacitance C(ω) 3. for PoreSOC diffusionmonitoring. and The intercalation impedance at spectr frequenciesa (Figure below 8) show 0.01 Hz.three regions: 1. Electrolyte and solid-electrolyte interface (SEI) at high frequencies; 2. Charge-transfer at medium frequencies; 3. Pore diffusion and intercalation at frequencies below 0.01 Hz. Batteries 2021Batteries, 7, x FOR2021 ,PEER7, 17 REVIEW 11 of 16 11 of 16

Im 𝑍⁄mΩ LCO Im 𝑍⁄mΩ NCA -20 -6

100 100 90 -15 90 80 80 -4 70 50 -10

-2 -5 0.1 Hz

1 kHz 0 0 0 5 10 15 20 0246 (Re Z - R ) / mΩ S (Re Z - RS) / mΩ

Im 𝑍⁄mΩ NMC Im 𝑍⁄mΩ NCA -8 -10

100 100 90 -8 90 -6 80 80 70 70 50 -6 50 30 -4 -4

-2 -2

0 0 46 48 50 52 54 56 54 56 58 60 62 64 66 Re Z / mΩ (Re Z - R ) / mΩ S

Figure 8. FigureImpedance 8. Impedance spectra of spectra different of different lithium-i lithium-ionon chemistries chemistries at rest atin rest defined in deﬁned charge charge states. states. NMC NMC as measured as measured to to show show the theinfluence inﬂuence of SOC of SOC on onelectrolyte electrolyte resistance. resistance. LCO LCO and and NC NCAA are are shifted shifted on on the the real real axis axis to tothe the electrolyte electrolyte resistance resistance of SOC of SOC = =1. 1.Frequency Frequency range range 0.1 0.1 Hz Hz to to 1 1kHz. kHz.

It is usefulIt is to useful subtract to subtractthe electrolyte the electrolyte resistance resistance (intersection (intersection with the withreal theaxis) real to axis) to create a setcreate of congruent a set of congruent curves, which curves, are which independent are independent of contact of and contact cable and errors. cable errors.

𝑅(𝜔) =ReR(ω) 𝑍=(𝜔Re) −ReZ(ω 𝑍) (−𝜔→∞Re Z()ω → ∞) (9) (9)

The frequencyThe frequency is chosen is below chosen 10 below Hz. With 10 Hz. the With corrected the corrected real part, real the part, modulus the modulus 2 |Z(ω)| and|Z( ωthe)| pseudocapacitance, and the pseudocapacitance, C(ω) = –ImC(ω Z)/( =ω –Im |ZZ|/(), ωare| Zcalculated.|2), are calculated. The shape The of shape of diffusiondiffusion impedance impedance at low frequencies at low frequencies depends depends on whether on whether the lithium-ion the lithium-ion are mobile are mobile in in linear channels (Li1-xFePO4), in areas of the layer lattice (Li1-xCoO2, NMC) or in the void linear channels (Li1-xFePO4), in areas of the layer lattice (Li1-xCoO2, NMC) or in the void spaces of a spinel (Li1-xMn2O4, LMO). spaces of a spinel (Li1-xMn2O4, LMO). The qualitativeThe qualitative impact of impact the state-of-charge of the state-of-charge on the impe on thedance impedance spectrum spectrum is compiled is compiled in Table in3. TableThe Nyquist3. The Nyquist diagram diagram of the LCO of the chemistryLCO chemistry makes SOCmakes monito SOCring monitoring difficult difﬁcult because becausethe curves the are curves more are or more less orcongruent; less congruent; the real the part real of part impe ofdance impedance (resistance) (resistance) at at any any frequencyfrequency does does not notallow allow a reliable a reliable SOC SOC determination. determination. Better Better than than the the ohmic ohmic re- resistance, sistance, thethe imaginaryimaginary partpart ImIm ZZ((ωω) and pseudocapacitance CC((ωω)) at at low low frequencies frequencies re- reﬂect the flect the state-of-charge,state-of-charge, althoughalthough there is no clear trend between SOC and reactance. At most in most in thethe SOC SOC range range from from 80% 80% to to 100%, 100%, a ahigh high capacitance capacitance shows shows a ahigh high state-of-charge. state-of-charge. Batteries 2021, 7, 17 12 of 16

Batteries 2021, 7, x FOR PEER REVIEW 12 of 16

Table 3. Qualitative changes in the impedance spectrum in different state-of-charge areas: R resistance of the high-frequency arc, X reactance at 0.1 Hz, C capacitance at 0.1 Hz. Table 3. Qualitative changes in the impedance spectrum in different state-of-charge areas: R resistance of the high-frequency arc, X reactance atFull 0.1 ChargeHz, C capacitance at 0.1 Hz. Medium State-of-Charge Low State-of-Charge Cell # SOC > 0.9 SOC ≈ 0.5 SOC < 0.5 Cell # Full Charge Medium State-of-Charge Low State-of-Charge No signiﬁcant impact on R < 40 mΩ drops slightly. R ≈ 40 mΩ drops slightly. X(0.1 5SOC LMO > 0.9 SOC ≈ 0.5 SOC < 0.5 resistance. C ∼ SOC Reactance increases: X ∼ SOC. Hz) = constant. No significant impact on resistance. R < 40 mΩ drops slightly. Reactance R ≈ 40 mΩ drops slightly. X(0.1 Hz) 5 LMO No signiﬁcant impact on R and X are slightly higher when 10C LCO ∼ SOC increases: X ∼ SOC. = constant.R and X are high when SOC is resistance R and X are SOCslightly is lower. higher when SOC R and X are high when SOC is low. 10 LCO No significant impact on resistance low. Steeper U(Q) curve. is lower. Steeper U(Q) curve. Resistance and impedance R and X are slightly higher when 11 1113Resistance NCA and impedance increase: R and X are slightly higher when SOC R and XR areand highX are when high SOC when is low. SOC is NCA increase: R ∼ SOC SOC is lower. 13 R ∼ SOC is lower. Steeper low.U(Q) Steeper curve. U(Q) curve.

In the case of medium and low state-of-charge, an illogical order occurs that pretends In the case of medium and low state-of-charge, an illogical order occurs that pretends a higher SOC. It makes the evaluation easier if the frequency response of the imaginary a higher SOC. It makes the evaluation easier if the frequency response of the imaginary part is related to the fully charged battery. However, this relative quantity, Im Z(SOC)/Im part is related to the fully charged battery. However, this relative quantity, Im Z(SOC)/Im Z(SOC = 1), misrepresents overcharging and aging phenomena. Z(SOC = 1), misrepresents overcharging and aging phenomena. LMO chemistry shows a slight increase of impedance |Z(ω)|with SOC in the linear LMO chemistry shows a slight increase of impedance |Z(ω)|with SOC in the linear range of the flat voltage-charge curve. Below SOC = 0.5, impedance increases strongly. range of the flat voltage-charge curve. Below SOC = 0.5, impedance increases strongly. With NMC and NCA chemistry, high impedance |Z| (in F) and high residual battery With NMC and NCA chemistry, high impedance |Z| (in F) and high residual battery capacity (in Ah) correlate for high state-of-charge above 70%. At low SOC, the resistance capacity (in Ah) correlate for high state-of-charge above 70%. At low SOC, the resistance and capacitance increase. Pseudocapacitance is not clearly a linear function of SOC. and capacitance increase. Pseudocapacitance is not clearly a linear function of SOC. FigureFigure 9 demonstrates9 demonstrates that that pseudocapacitanc pseudocapacitancee reflects reflects the the state-of-charge state-of-charge in the correctcor- rectorder order of of magnitude. magnitude. Unfortunately, Unfortunately, the the connex connex is is not not linear. linear. Rather Rather two two linear linear regions regions can canbe be distinguished distinguished from from SOC SOC = = 60% 60% to to 100% 100% for for all all battery battery chemistries, chemistries, which which can can be be fitted fittedby by a 3rd a 3rd order order polynomial polynomial (coefficients (coefficients see Tablesee Table1). The 1). linearityThe linearity between between impedance imped- and anceSOC and holds SOC holds as long as aslong the as slope the slope of the of discharge the discharge curve curve∆U/ ∆ΔSOCU/ΔSOC does does not change.not change. This is Thisnot is not surprising, surprising, because because the the impedance impedance represents represents the the slope slope of the of the current–voltage current–voltage curve, curve,|Z ||Z =| d =U d/dU/dI. TheI. The steeper steeper the the voltage-charge voltage-charge curve, curve, the the more more resistance resistance and and reactance reac- tanceincrease increase when when the the battery battery is depleted.is depleted.

U / V 1001 4.2

4 0.990 LCO 𝐶 (SOC) 3.8 𝐶(100%) NCA 0.880 3.6 LMO NMC

3.4 0.770 L 𝑄(SOC) 3.2 LFP 𝑄(100%) 0.660 3 60 70 80 90 100 30 40 50 60 70 80 90 100 SOC in % SOC in %

(a) (b)

FigureFigure 9. (a 9.) NMC(a) NMC battery battery #8: #8:Normalized Normalized pseudocapacitance pseudocapacitance and and pseudocharge pseudocharge at at0.1 0.1 Hz Hz (reference: (reference: 589 589 F Fand and 2470 2470 As As at at SOCSOC = = 100%) 100%) versus versus “true” “true” SOC SOC from from Ah counting. (b) CellCell voltagevoltage ofof differentdifferent cell cell chemistries chemistries versus versus state-of-charge state-of-charge from fromAh Ah counting. counting. Batteries Batteries under under test test see see Table Table1. 1. Batteries 2021, 7, x FOR PEER REVIEW 13 of 16 Batteries 2021, 7, 17 13 of 16

The linear relationship between the state-of-charge (SOC) and the pseudocapacitance The linear relationship between the state-of-charge (SOC) and the pseudocapacitance C from impedance spectroscopy can be qualitatively understood by the following depend- C from impedance spectroscopy can be qualitatively understood by the following de- encies. Q0 is the capacity of the fully charged battery, which is determined by Ah counting. pendencies. Q0 is the capacity of the fully charged battery, which is determined by Ah counting. 𝑄 𝑈 SOC = =𝐶 (10) Q 𝑄 U𝑄 SOC = = C (10) Q0 Q0 𝜕𝑈 d𝑄 𝑄 𝑄 d𝑈 ∂U =dQ Q= =Q dU ∼𝑍 𝜔 𝑄 (11) 𝜕SOC d(𝑄⁄ 𝑄 ) 0 𝐶 0𝐼 d𝑡 = = = ∼ |Z| ω Q0 (11) ∂SOC d(Q/Q0) C I dt Equation (10) describes that the pseudocapacitance for all battery chemistries de- pendsEquation directly (10) on describesthe SOC, thatalthough the pseudocapacitance the linearity may forbe disturbed all battery by chemistries individual depends steps in directlythe voltage-charge on the SOC, curve. although Equation the linearity (11) states may that be the disturbed slope of by the individual voltage-SOC steps curve in the de- voltage-chargepends on the kinetic curve. Equationinhibitions (11) (impedance states that | theZ|) slope of the of cell the reaction. voltage-SOC curve depends on the kinetic inhibitions (impedance |Z|) of the cell reaction. 3.7. Impact of Aging 3.7. Impact of Aging This section is intended to show briefly that the proposed pseudocapacitance can This section is intended to show briefly that the proposed pseudocapacitance can correctly reflect the state-of-health of a battery even if the shape of the impedance spectra correctly reflect the state-of-health of a battery even if the shape of the impedance spectra changes significantly during cycle aging. With aging and progressive SEI formation, the changes significantly during cycle aging. With aging and progressive SEI formation, the electrolyte resistance at high frequencies becomes larger and larger until two arcs become electrolyte resistance at high frequencies becomes larger and larger until two arcs become visible in the impedance spectrum (Figure 10). The internal resistance increases until the visible in the impedance spectrum (Figure 10). The internal resistance increases until the endend of of service service life life is is reached. reached. The The electrolyte electrolyte resistance resistance is not is anot good a good SOH SOH indicator, indicator, because be- thecause passive the passive layer formation layer formation in some in new some batteries new batteries leads to leads an apparent to an apparent improvement improve- of conductivityment of conductivity during the during first charge-discharge the first charge-discharge cycles until cycles the gradual until the deterioration gradual deteriora- follows. Fortion combined follows. SOCFor combined and SOH measurements,SOC and SOH themeasurements, pseudocapacitance the pseudocapacitance is still appropriate. is Asstill shownappropriate. above, As it is shown useful above, to normalize it is useful C(ω )to to normalize the fully chargedC(ω) to batterythe fully (SOC charged = 1) battery or the new(SOC battery = 1) or (SOH the new = 1). battery (SOH = 1).

Im Z / mΩ -25

-20

-15 SOH = 100% SOH = 80%

-10 100 300 1 500 1000 cycles

-5

0 60 65 70 75 80 85 90 95 100 105 110 115 120 Re Z / mΩ

(a)

Figure 10. Cont. BatteriesBatteries 20212021,, 77,, 17x FOR PEER REVIEW 1414 ofof 1616

Cs(0.1 Hz)/ F R(1 kHz) / mΩ 600 100

500 80

400 60 300 40 200

100 20

0 0 2300 2100 1900 1700 1500 2300 2100 1900 1700 1500 Q / mAh Q / mAh (b) (c)

FigureFigure 10.10. AgingAging ofof anan LCOLCO batterybattery atat thethe samesame state-of-chargestate-of-charge (Panasonic(Panasonic UR18650UR18650 FK).FK). ((aa)) ImpedanceImpedance spectraspectra ofof fullyfully chargedcharged batteriesbatteries duringduring agingaging byby aa thousandthousand charge-dischargecharge-discharge cycles.cycles. ((bb)) PseudocapacitancePseudocapacitance correlatescorrelates linearlylinearly withwith thethe actualactual batterybattery capacitycapacity determined determined by by Ah Ah counting: counting:C C= = 141 141·Q⋅Q+ + 191. 191. Fit Fit quality quality 0.92. 0.92. (c )(c Increase) Increase of of electrolyte electrolyte resistance resistance in in time. time. 4. Conclusions 4. ConclusionsPrior art methods for SOC monitoring suffer from some specific drawbacks: In order to reliablyPrior artdetermine methods the for actual SOC monitoring remaining sufferbattery from capacity some at specific a certain drawbacks: point in Intime, order an toAmpere reliably hour determine counting the must actual be carried remaining out du batteryring a complete capacity atdischarge. a certain The point cell involtage time, anis an Ampere uncertain hour measure counting of the must “true” be carried electricout charge. during So far, a complete there is no discharge. general and The com- cell voltageprehensive is an model uncertain or equivalent measure ofcircuit the “true” that is electric able to charge. display So and far, predict there is the no state-of- general andcharge comprehensive and state-of-health model of or a equivalentbattery in all circuit system that states is able over to the display entire and service predict life of the a state-of-chargebattery. and state-of-health of a battery in all system states over the entire service life ofThe a battery. empirical concept of pseudocapacitance might overcome some of the shortcom- ings,The albeit empirical new challenges concept arise. of pseudocapacitance There is a significant might linear overcome correlation some between of the shortcom- pseudo- ings,capacitance albeit newand challengesremaining capacity arise. There for the is a ma significantin types linearof lithium-ion correlation batteries. between Our pseu- ap- docapacitanceproach does not and require remaining specific capacity model assumptions for the main in types advance, of lithium-ion such as predefined batteries. equiv- Our approachalent circuits does or notnumerical require simulations, specific model whic assumptionsh are often unclear in advance, and complicate such as predefined the system equivalentanalysis during circuits the or operation. numerical simulations, which are often unclear and complicate the systemThe analysis pseudocapacitance during the operation. reacts to electrodes that change over time and signs of aging. However,The pseudocapacitance the proposed method reacts is not to electrodes as sensitive that as changethe Ah overcounting. time andFor batteries signs of aging.of the However,same type theand proposed manufacturer method the isSOC not forecast as sensitive quality as theis about Ah counting. 1 percentage For point batteries based of theon a same quadratic type calibration and manufacturer curve (C the versus SOC SOC). forecast The quality method is is about suitable 1 percentage for high and point me- baseddium charge on a quadratic states. The calibration SOC detection curve using (C versus pseudocapacitance SOC). The method works is best suitable when for the high cur- andrent–voltage medium chargecurve is states. flat, which The SOC is true detection for LFP using batteries. pseudocapacitance With some lithium-ion works best chemis- when the current–voltage curve is flat, which is true for LFP batteries. With some lithium-ion tries below 50% SOC, the pseudocapacitance can correlate unclearly with the “true” state- chemistries below 50% SOC, the pseudocapacitance can correlate unclearly with the “true” of-charge. state-of-charge. Further research is necessary to improve the method so that steps in the voltage-ca- Further research is necessary to improve the method so that steps in the voltage- pacity curve, and overcharging effects can be properly managed. Long-term SOH moni- capacity curve, and overcharging effects can be properly managed. Long-term SOH toring, aging scenarios, and simulations have to be postponed to future work. monitoring, aging scenarios, and simulations have to be postponed to future work. Author Contributions: Writing—original draft preparation, review and editing, both authors. All Author Contributions: Writing—original draft preparation, review and editing, both authors. All authors have read and agreed to the published version of the manuscript. authors have read and agreed to the published version of the manuscript. Funding: This work was supported by Diehl Aerospace GmbH. Funding: This work was supported by Diehl Aerospace GmbH. Institutional Review Board Statement: Not applicable. The study did not involve humans or ani- Institutional Review Board Statement: Not applicable. The study did not involve humans or mals. animals. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. Batteries 2021, 7, 17 15 of 16

Abbreviations

C pseudocapacitance (F) f frequency (Hz) Q electric charge, battery capacity (Ah) Q0 capacity of a fully charged battery (Ah) R ohmic resistance, real part of impedance (Ω) RS electrolyte resistance (Ω) U cell voltage (V) Y complex admittance: Y = Z–1 (Ω–1) Z complex impedance (Ω) ω angular frequency: ω = 2πf (s–1) Ah Ampere hour: 1 Ah =√ 3600 C j imaginary operator: −1 LCO lithium cobalt oxide LMO lithium manganese spinel LFP lithium iron phosphate N subscript: nominal value NCA nickel cobalt aluminum NMC nickel manganese cobalt SOC, α state-of-charge SOH, β state-of-health

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