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Generalized Characterization Methodology for Performance Modelling of Lithium-Ion Batteries Daniel-Ioan Stroe *, Maciej Swierczynski, Ana-Irina Stroe and Søren Knudsen Kær Department of Energy Technology, Aalborg University, 9220 Aalborg East, Denmark; [email protected] (M.S.); [email protected] (A.-I.S.); [email protected] (S.K.K.) * Correspondence: [email protected]; Tel.: +45-30-62-25-89 Academic Editor: Sheng S. Zhang Received: 30 September 2016; Accepted: 22 November 2016; Published: 1 December 2016

Abstract: Lithium-ion (Li-ion) batteries are complex energy storage devices with their performance behavior highly dependent on the operating conditions (i.e., temperature, load current, and state-of-charge (SOC)). Thus, in order to evaluate their techno-economic viability for a certain application, detailed information about Li-ion battery performance behavior becomes necessary. This paper proposes a comprehensive seven-step methodology for laboratory characterization of Li-ion batteries, in which the battery’s performance parameters (i.e., capacity, open-circuit voltage (OCV), and impedance) are determined and their dependence on the operating conditions are obtained. Furthermore, this paper proposes a novel hybrid procedure for parameterizing the batteries’ equivalent electrical circuit (EEC), which is used to emulate the batteries’ dynamic behavior. Based on this novel parameterization procedure, the performance model of the studied Li-ion battery is developed and its accuracy is successfully verified (maximum error lower than 5% and a mean error below 8.5 mV) for various load profiles (including a real application profile), thus validating the proposed seven-step characterization methodology. Keywords: lithium-ion (Li-ion) battery; characterization; methodology; performance modelling; electrochemical impedance spectroscopy; DC pulses; validation

1. Introduction Recent developments of lithium-ion (Li-ion) batteries based on new and improved chemistries have resulted in batteries with high performance, long lifetime and increased safety [1,2]. Thus, Li-ion batteries have become the key energy storage technology for e-mobility applications [3,4]. Furthermore, energy storage systems based on Li-ion batteries are evaluated in different projects where they are used in grid ancillary service applications [5–7] or for renewables’ grid integration [6,8,9]. Consequently, Li-ion batteries are expected to become the major player in utility-scale applications as stated in different surveys [10,11]. Nevertheless, Li-ion batteries are characterized by a high investment cost in comparison to other energy storage technologies (e.g., lead acid and nickel metal hydride (NiMH) batteries) and their performance behavior is highly influenced by the operating conditions: temperature, load current and state-of-charge (SOC). Thus, in order to gain from the aforementioned advantages, the Li-ion batteries have to be operated in an efficient and cost-effective manner. This objective can be achieved by relying on accurate performance models, which are able to estimate accurately the dynamic behavior of Li-ion batteries independently of the operating conditions; thus, precise energy management strategies can be defined and accurate sizing of the Li-ion batteries based systems can be realized. Moreover, by relying on accurate performance models, Li-ion batteries can be tested by simulations considering different scenarios, thereby reducing the laboratory testing efforts that are usually cost demanding and time consuming.

Batteries 2016, 2, 37; doi:10.3390/batteries2040037

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Depending on the modelling approach followed, Li-ion battery performance models are divided into three main categories: electrochemical models [12,13], mathematical models [14,15], and electrical models [16–18]; furthermore, combinations between these three modelling approaches were proposed [19,20]. These approaches are characterized by different accuracy levels and degrees of complexity. Electrical models represent the most suitable solutions for Li-ion battery modelling when they need to be integrated in system-level simulations because of their features such as high accuracy and average computation complexity [21]. Thus, an electrical-based performance modelling approach is detailed in this paper. Electrical models use an equivalent electrical circuit (EEC), composed of combinations of resistors, capacitors, inductors, constant phase elements etc., to describe the performance (dynamic) behavior of Li-ion batteries. Two methods are traditionally used to parametrize the elements of the EEC. The first method involves applying a DC pulse to the Li-ion battery and measuring the voltage response of the battery [16,22–24]. The DC pulse-based electrical models, also referred as Thévenin-based electrical models, use a series resistance and several resistance-capacitance (RC) parallel networks to predict the response of the Li-ion battery to transient load events. The accuracy of the Thévenin-based models depends on the number of RC networks used to estimate the voltage of the Li-ion battery (i.e., the higher the number of RC parallel networks, the better the accuracy of the model), as shown in [22,24]. For a thorough parameterization of the EEC, the DC pulses have to be applied at different SOCs, load currents, and temperatures since the parameters of the EEC are highly dependent on the operating conditions as illustrated in [16,25,26]. One of the disadvantages of the DC pulse technique is represented by the fact that high charging and discharging current pulses cannot be applied at high and low SOC levels, respectively, for low temperatures since the maximum and minimum battery voltage values will be reached. Consequently, a full parameterization of the EEC is not possible. Furthermore, the elements of the EEC, which are determined with this method, mostly lack physicochemical meaning. All these disadvantages are overcome when the second method based on the electrochemical impedance spectroscopy (EIS) technique is used for parameterization of the EEC [27]. EIS has become in the last decade an established technique for characterizing Li-ion batteries and for modelling and parameterizing their performance behavior [17,18,28,29]. The models developed with this technique, referred to as impedance-based models, have the advantage that their EEC’s elements can be easily related with physicochemical processes that occur inside the battery, such as charge transfer and diffusion [28,30,31]. In order to parameterize an impedance-based model, EIS measurements have to be performed at various battery SOC levels and temperatures. Furthermore, it is possible to perform EIS measurements with superimposed DC current, in order to obtain the dependence of the EEC elements on the load current [29]; however, these measurements can be performed only for small currents (reduced C-rates) because, since the EIS measurements are time demanding, they will result in changing the Li-ion battery’s SOC. Thus, the impedance-based performance model will not be able to predict accurately the voltage of the Li-ion batteries for high C-rates, which are characteristic for Li-ion batteries used in many e-mobility and grid applications. This paper proposes a new approach to parameterize the EEC, which combines the advantages of the two traditional methods. Therefore, the proposed hybrid method uses the EIS technique to parameterize the EEC and express the dependence of its elements on the SOC and temperature and the DC pulse technique to express their dependence on the load current. Furthermore, a physicochemical interpretation of the EEC elements and their corresponding values can be realized. Nevertheless, building and parameterizing a performance model for a Li-ion battery is a complex process and an extensive laboratory characterization of the targeted battery has to be performed in order to obtain information about various parameters such as: capacity, open-circuit voltage (OCV) etc. and their dependence on the operating conditions such as: SOC, temperature, and load current. Thus, besides proposing a new approach of parameterizing the EEC, this paper presents a comprehensive seven-step methodology for characterizing and modelling the performance behavior of a Li-ion battery. The present paper is structured as follows. Section 2 gives brief information about the Li-ion battery used

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in this work and presents the experimental set-up used for characterizing the Li-ion battery. Section 3 presents the proposed seven-step methodology for characterizing and parameterizing the performance model of the studied Li-ion battery. Based on the obtained results from the characterization test, Batteries 2016, 2, 37 3 of 19 the performance model of the Li-ion battery is developed in Section 4. In Section 5, extensive results Li-ion Section 3 presents the proposed seven-step model methodology for characterizing and obtained frombattery. the validation of the proposed performance are presented, while concluding parameterizing the performance model of the studied Li-ion battery. Based on the obtained results remarks are given in Section 6. from the characterization test, the performance model of the Li-ion battery is developed in Section 4. In Section 5, extensive results obtained from the validation of the proposed performance model are 2. Experimental Set-Up presented, while concluding remarks are given in Section 6.

At present, a wide variety of Li-ion battery chemistries are available on the market that are, 2. Experimental Set-Up depending on their characteristics, suitable for a broad range of applications from electro-mobility present, variety ofgrid Li-ion battery chemistries are available onnone the market thatchemistries are, to back-up At power anda wide renewables’ integration [32]. Nevertheless, of these depending on their characteristics, suitable for a broad range of applications from electro-mobility to cycle, is superior to the others from all perspectives (e.g., energy density, specific energy, cost per back-up power and renewables’ grid integration [32]. Nevertheless, none of these chemistries is etc.) [33]. Therefore, the characterization methodology presented in this work and the proposed superior to the others from all perspectives (e.g., energy density, specific energy, cost per cycle, etc.) [33]. performance model can be applied independently of the type of the selected Li-ion battery chemistry. Therefore, the characterization methodology presented in this work and the proposed performance For exemplifying theapplied proposed characterization methodology, a commercial 2.5 chemistry. Ah batteryFor based on model can be independently of the type of the selected Li-ion battery the lithium iron phosphate/graphite (further methodology, referred to asa commercial LFP/C), is2.5 used in this based work.on the exemplifying the proposed characterization Ah battery lithium iron phosphate/graphite (further referred as LFP/C), used ininside this work. During the whole characterization process, thetobattery wasisplaced a temperature-controlled During the whole characterization process, the battery was placed a temperaturechamber, as illustrated in Figure 1, and its temperature was continuouslyinside monitored using a PT100 controlled chamber, as illustrated in Figure 1, and its temperature was continuously monitored using sensor. Furthermore, the temperature of the chamber was adjusted in order to obtain the desired a PT100 sensor. Furthermore, the temperature of the chamber was adjusted in order to obtain the temperature on the surface of the LFP/C battery; thus, the temperature values mentioned in this work desired temperature on the surface of the LFP/C battery; thus, the temperature values mentioned in are the ones of the ambient temperature values of the chamber. this workthe arebattery the ones and of thenot battery and not the ambient temperature values of the chamber.

+ EIS + Vdc PT100 + Idc - Vdc

- Idc

- EIS

Figure 1. Lithium iron phosphate/graphite (LFP/C) battery during the characterization procedure. EIS:

Figure 1. Lithium iron phosphate/graphite (LFP/C) battery during the characterization procedure. electrochemical impedance spectroscopy. EIS: electrochemical impedance spectroscopy. 3. Lithium-Ion Battery Characterization Methodology

3. Lithium-Ion Characterization Methodology The Battery main objective of the proposed characterization methodology was to measure the parameters of the studied battery and to determine their dependence on the operating Theperformance main objective of the proposed characterization methodology was to measure the performance conditions; the operating conditions cover the sum of the conditions given by the load current, parameters of the studied battery and to determine their dependence on the operating conditions; temperature, and SOC. Based on the obtained results, which are presented throughout this section, the operating conditions cover theLFP/C sum battery of the was conditions given by the load current, temperature, and the performance model of the developed and parameterized (see Section 4). SOC. BasedInonorder the obtained results, which arebehavior presented throughout this section, performance to determine the performance of the tested LFP/C battery, the the seven-step methodology, is was presented in Figure 2, parameterized was proposed. This model of the LFP/Cwhich battery developed and (seemethodology Section 4). is composed of various tests, which were carried out following the sequence illustrated Figure 2; the firstthe six tests In order to determine the performance behavior of the tested in LFP/C battery, seven-step are presented throughout the next subsections, while the last step (i.e., verification test) is extensively methodology, which is presented in Figure 2, was proposed. This methodology is composed of various discussed in Section 5. tests, which were carried out following the sequence illustrated in Figure 2; the first six tests are presented throughout the next subsections, while the last step (i.e., verification test) is extensively discussed in Section 5.

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Figure 2. Test sequence for characterization of the LFP/C battery. OCV: open-circuit voltage; SOC:

Figure 2. Test sequence for characterization of the LFP/C battery. OCV: open-circuit voltage; state-of-charge; and HPPC: hybrid pulse power characterization. SOC: Figure state-of-charge; and HPPC: hybrid pulse of power characterization. 2. Test sequence for characterization the LFP/C battery. OCV: open-circuit voltage; SOC: state-of-charge; Test and HPPC: hybrid pulse power characterization. 3.1. Preconditioning

3.1. Preconditioning Test

The objective of the preconditioning test was to remove any possible passivation to which the 3.1. Preconditioning Test

The objective of the preconditioning was to remove to which battery was subjected to, between the test manufacturing time any and possible the initialpassivation tests. Moreover, an the The objective of between the test was to remove any possible passivation tointerface which the additional goal of to, this testpreconditioning was to the battery capacity since the solid electrolyte of battery was subjected thestabilize manufacturing time and the initial tests. Moreover, an additional battery was subjected to, between the manufacturing manufacturing process time andand theitsinitial tests. Moreover, an the batteries is not shaped after the porosity/shape changes goal of this test was to stabilize the battery capacity since the solid electrolyte interface of the batteries additional goalafter of this wasfew to stabilize the battery capacity since thetest solid electrolyte interface of thetest first cycles. Thus, preconditioning was composed of five is notmeaningfully shaped after isthenot manufacturing process and the its porosity/shape changes meaningfully after the the batteries shaped after the manufacturing process and its porosity/shape changes successive cycles performed 1C-rate of (i.e., A) at 25 °C [21]. The resultscycles first few cycles. charge-discharge Thus, thethe preconditioning testThus, waswith composed five2.5 successive meaningfully first few cycles. the preconditioning test wascharge-discharge composed of five obtained for theafter case of the discharging capacity of the tested battery are presented in Figure 3. ◦ C [21]. The results obtained for the case of performed withcharge-discharge 1C-rate (i.e., 2.5cycles A) at 25 successive performed with 1C-rate (i.e., 2.5 A) at 25 °C [21]. the Thedischarging results capacity of thefor tested battery in Figure 3. tested battery are presented in Figure 3. obtained the case of theare discharging capacity of the 2.6presented

Capacity Capacity [Ah] [Ah]

2.6 2.58 2.58 2.56 2.56 2.54 2.54 2.52 2.52 2.5 0

1

2.5 0

1

2

3

4

5

6

2

3

4

5

6

Measurement Number

Figure 3. Discharging capacity of the LFP/C battery measured during the preconditioning test (1CMeasurement Number rate, T = 25 °C). Figure 3. Discharging capacity of the LFP/C battery measured during the preconditioning test (1CFigure 3. Discharging capacity of the LFP/C battery measured during the preconditioning test (1C-rate, rate, T = 25 °C). The ◦ measured capacity of the LFP/C battery was stable, with a tendency of slight monotonic

T = 25 C). increase of approximately 0.6% during the five performed cycles; nevertheless, the battery was The measured capacitysince of the battery was stable,more withthan a tendency of slight monotonic considered preconditioned its LFP/C capacity did not change 3% during two consecutive The measured capacity of the LFP/C battery was stable, with a tendency of slight monotonic increase of approximately 0.6% during the five performed cycles; nevertheless, the battery was discharges [34]. considered preconditioned since its capacity did not change more thannevertheless, 3% during twothe consecutive increase of approximately 0.6% during the five performed cycles; battery was discharges [34]. considered preconditioned since its capacity did not change more than 3% during two consecutive 3.2. Relaxation Test

discharges [34]. the behavior of the Li-ion batteries is altered by the concentration gradients of the ionic Usually,

3.2. Relaxation Test charge carriers immediately after the load current is switched off [35]. Consequently, a relaxation 3.2. Relaxation Test behavior of the Li-ion batteries is altered by the concentration gradients of the ionic Usually, period, which the will allow the battery to reach thermodynamic stability, has to be applied between the charge carriers immediately afterthe the load current is switched off [35]. Consequently, a relaxation switch off of the load current measurement. In the order to determine the optimal length Usually, the behavior of theand Li-iondesired batteries is altered by concentration gradients of the ionic period, which will allow the battery to reach thermodynamic stability, has to be applied between the of this relaxation period, the behavior of the OCV of the LFP/C battery was investigated; the battery charge carriers immediately after the load current is switched off [35]. Consequently, a relaxation switch off of the load current and the desired measurement. In order to determine the optimal length was fully charged and then discharged at three different SOCs (i.e., 80%, 50% and 20%), where it was period, will allow thethe battery to reach hasinvestigated; to be applied of which this period, behavior of thethermodynamic OCV the voltage LFP/Cstability, battery thebetween battery the kept at relaxation OCV conditions for 24 h, respectively [21].ofThe values was measured, with one-second switchwas offfully of the load current and the desired measurement. In(i.e., order to50% determine thewhere optimal length charged and then discharged at three different SOCs 80%, and 20%), it was resolution, during the relaxation period were related to the OCV value measured after 24 h (when the of this relaxation period, the behavior of the OCV of the LFP/C battery was investigated; the battery kept at OCV conditions for 24 h, respectively [21]. The voltage values measured, with one-second battery was considered quasi-stabilized); thus, the OCV error was computed according to Equation (1). resolution, during relaxation period to the OCV(i.e., value80%, measured after20%), 24 h (when was fully charged and the then discharged at were threerelated different SOCs 50% and wherethe it was battery consideredfor quasi-stabilized); thus, the OCVThe error was computed to Equation (1). kept at OCVwas conditions 24 h, respectively [21]. voltage values according measured, with one-second

resolution, during the relaxation period were related to the OCV value measured after 24 h (when the battery was considered quasi-stabilized); thus, the OCV error was computed according to Equation (1).

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|Vi − OCV | · 100 (1) OCV where εOCV represents the OCV error and Vi represents the voltage measured at every second during the 24 hBatteries relaxation. 2016, 2, 37 5 of 19 Based on the obtained results, which are summarized in Table 1, it was concluded that the OCV value is only slightly influenced byV a− OCV relaxation time, which varies between 15 min i (1) battery = ε 100 OCV hour was ⋅considered and 24 h. Thus, a relaxation time of one enough for the LFP/C OCV to reach quasi-thermodynamic stability and to allow for accurate measurement of the desired where εOCV represents the OCV error and Vi represents the voltage measured at every second during performance parameters. the 24 h relaxation. εOCV =

Based on the obtained results, which are summarized in Table 1, it was concluded that the OCV

Table 1. isVoltage measured at 20%, and 80% after various relaxation time value only slightly influenced by a50%, relaxation time,SOC which varies between 15 min and 24 periods h. Thus, and a corresponding OCVoferrors. relaxation time one hour was considered enough for the LFP/C battery to reach quasithermodynamic stability and to allow for accurate measurement of the desired performance SOC = 20% parameters. Relaxation Time Voltage (V) OCV Error (%)

SOC = 50% Voltage (V)

SOC = 80%

OCV Error (%)

Voltage (V)

OCV Error (%)

Table 1. Voltage measured at 20%, 50%, and 80% SOC after various relaxation time periods and 3.157 2.80 3.230 2.03 3.270 2.00 corresponding OCV errors.

1s 15 min 3.236 0.37 1 h Relaxation 3.241 0.23 SOC = 20% 2h 0.17Error (%) Time 3.243 Voltage (V) OCV 24 h 3.248 3.157 0 2.80 1s

3.3.

3.292 0.15 3.295 SOC = 50% 0.07 3.295 (V) OCV Error 0.05(%) Voltage 3.297 3.230 2.030

3.325 0.37 3.335 0.07 SOC = 80% 0.04 Voltage 3.336 (V) OCV Error (%) 0 3.2703.337 2.00

15 min 3.236 0.37 3.292 0.15 3.325 0.37 1h 3.241 0.23 3.295 0.07 3.335 0.07 Capacity Test 2h 3.243 0.17 3.295 0.05 3.336 0.04 24 h 3.248 0 3.297 0 3.337 0 applied load The capacity of Li-ion batteries is dependent on the operating temperature and on the

current [15,17]. Moreover, as presented by various researchers, the capacity of Li-ion batteries degrades 3.3. Capacity Test over time [36–38]; however, this analysis is out of the scope of the present paper. The objective of the The capacity of Li-ion batteries is dependent on the operating temperature and on the applied capacity test was to determine the dependence of the charged and discharged capacity of the LFP/C load current [15,17]. Moreover, as presented by various researchers, the capacity of Li-ion batteries battery degrades on the two parameters. Thus, of of thethe battery measured for overaforementioned time [36–38]; however, this analysis is the out capacity of the scope presentwas paper. The ◦ six different C-rates C/4, 1C, 2C, 3C the anddependence 4C) and at different (i.e., 15 C, objective of the(i.e., capacity testC/2, was to determine offour the charged and temperatures discharged capacity ◦ C), following of◦the LFP/C on the twothe aforementioned parameters. Thus, theincapacity 25 ◦ C, 35 C and 45battery procedure which is presented [21]. of the battery was for sixof different C-rates (i.e., C/4,discharging C/2, 1C, 2C, 3C and 4C) and at four different Themeasured dependence the LFP/C battery’s capacity on the C-rate and temperatures on the temperature (i.e., 15 °C, 25 °C, 35 °C and 45 °C), following the procedure which is presented in [21]. is illustrated in Figure 4. The discharged capacity of the LFP/C battery measured at 25 ◦ C decreases as The dependence of the LFP/C battery’s discharging capacity on the C-rate and on the the applied current increases; however, the decrease of the capacity for the considered C-rate interval temperature is illustrated in Figure 4. The discharged capacity of the LFP/C battery measured at 25 (C/4–4C) lower than Thiscurrent behavior suggests a Peukert number very closefortothe 1, considered which is in good °Cis decreases as the1%. applied increases; however, the decrease of the capacity agreement with the results for a1%. similar batterysuggests chemistry by Omar et al. [39]. Furthermore, C-rate interval (C/4–4C)reported is lower than This behavior a Peukert number very close to 1, whichin is Figure in good 4, agreement with the results reported for a similara battery chemistryof bythe Omar et al. battery as presented for the considered temperature interval, slight increase LFP/C Furthermore, presented in Figure 4, for the considered temperature interval, a slight capacity[39]. with increasingastemperature was measured, which might have been caused byincrease the increased of the LFP/C battery capacity with increasing temperature was measured, which might have been electronic and ionic conductivity in the electrode and electrolyte at high temperatures; similar results caused by the increased electronic and ionic conductivity in the electrode and electrolyte at high were reported in [17,39,40]. temperatures; similar results were reported in [17,39,40].

Figure 4. Dependence of the discharged capacity on temperature and load current.

Figure 4. Dependence of the discharged capacity on temperature and load current.

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Batteries 2016, 2, 37 3.4. Open-Circuit Voltage versus State-of-Charge Test

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3.4. Open-Circuit versus State-of-Charge Test Similar to most Voltage of the Li-ion battery parameters, the OCV is strongly dependent on the operating conditions (i.e., temperature, SOC, etc.). Therefore, the test was to measure OCV-SOC 3.4. Open-Circuit Voltage versus State-of-Charge Test Similar to most of the Li-ion battery parameters, thegoal OCVofisthis strongly dependent on the the operating characteristic of LFP/C battery at different for and discharging conditions (i.e., temperature, SOC, etc.). Therefore,temperatures the ofisthis test both was tocharging measure OCV-SOC Similar tothe most of the Li-ion battery parameters, thegoal OCV strongly dependent on the the operating characteristic of the LFP/C battery at different temperatures for both charging and discharging conditions; the OCV was measured in steps of 5% SOC following the methodology described in [21]. conditions (i.e., temperature, SOC, etc.). Therefore, the goal of this test was to measure the OCV-SOC ◦ C, is presented conditions; the OCV was measured in steps of 5% SOC following the methodology described in [21]. For instance, the OCV-SOC characteristic of the tested battery, measured at 25 in characteristic of the LFP/C battery at different temperatures for both charging and discharging For instance, the OCV-SOC characteristic of the tested battery, measured at 25 °C, is presented Figure 5. For the characteristic, a hysteresis effect is observed, which described is more pronounced conditions; themeasured OCV was measured in steps of 5% SOC following the methodology in [21]. in Figure 5. For thethe characteristic, a hysteresis effect is observed, which pronounced For instance, OCV-SOC characteristic of battery, measured atis25more °C, is presented for the 10%–40% SOCmeasured interval. This behavior ofthe thetested OCV is intrinsic for Li-ion batteries based on forFigure the 10%–40% interval. This behavior of the OCV isisintrinsic for Li-ionisbatteries based on in 5. For theSOC measured hysteresis effect observed, more pronounced active materials, which performcharacteristic, a two-phaseatransition lithium insertionwhich and extraction process, as is active materials, which perform This a two-phase transition lithium insertion and extraction process, is for the 10%–40% SOC interval. behavior ofeffect the OCV is intrinsic for Li-ion batteries basedason the case of the LFP material [41]. The hysteresis ismainly mainly influenced by the previous history the case of the LFP material [41]. The hysteresis effect is influenced by the previous history active materials, which perform a two-phase transition lithium insertion and extraction process, as is (e.g., (e.g., charging/discharging C-rate and relaxation time)ofofthe the LFP/C battery and can be modelled charging/discharging andhysteresis relaxationeffect time) battery be modelled the case of the LFP materialC-rate [41]. The is mainlyLFP/C influenced byand the can previous history following various approaches reported following various approaches reportedininliteratures literatures [42,43]. [42,43]. (e.g., charging/discharging C-rate and relaxation time) of the LFP/C battery and can be modelled following various approaches reported in literatures [42,43]. 3.6

Voltage [V] [V] Voltage

3.6 3.4 3.4 3.2 3.2 3 3 2.8 2.8 2.6 0

20

2.6 0

20

40

60

40

60

State-of-Charge [%]

Charging Discharging Charging 80 100 Discharging 80

100

Figure 5. The OCV-SOC characteristic of the LFP/C State-of-Charge [%] battery measured at 25 °C.

Figure 5. The OCV-SOC characteristic of the LFP/C battery measured at 25 ◦ C. Figure 5. The OCV-SOC characteristic of the LFP/C battery measured at 25 °C.

3.5. Hybrid Pulse Power Characterization Test

3.5. Hybrid Pulse Power Characterization Test

3.5. Hybrid Pulse Power Characterization Test The impedance represents a very important performance parameter of the Li-ion batteries since

it defines the power capability of the battery. Moreover, it is the parameter which describes thesince The impedance represents a avery important performance parameter of the Li-ion batteries The impedance represents very important performance parameter of the Li-ion batteries since dynamic behavior of Li-ion batteries. As most of the Li-ion battery parameters, the impedance shows it defines the power capability of the battery. Moreover, it is the parameter which describes it defines the power capability of the battery. Moreover, it is the parameter which describes the the a non-linear behavior depending on operating conditions such as SOC,the temperature, C-rate dynamic behavior of Li-ion batteries. As most of the theLi-ion Li-ion battery parameters, the impedance shows dynamic behavior of Li-ion batteries. Asthe most battery parameters, impedance shows [44,45]. Furthermore, the battery impedance is highly dependent on the battery’s state-of-health a non-linear behavior depending on operating the operating conditions such as SOC, temperature, C-rate a non-linear behavior depending on the conditions such as SOC, temperature, C-rate [44,45]. (SOH) [44]; however, thisbattery is out of the scopeisofhighly the present work.on In the order to account for all the [44,45]. Furthermore, dependent battery’s state-of-health Furthermore, the battery the impedanceimpedance is highly dependent on the battery’s state-of-health (SOH) [44]; aforementioned dependences, the impedance of the LFP/C battery was measured following (SOH) [44]; however, this is out of the scope of the present work. In order to account for all thea however, this is out of the scope of the present work. In order to account for all the aforementioned modified version of the hybrid pulse power characterization (HPPC) test. The measurement profile aforementioned dependences, the impedance of the LFP/C battery was measured following a dependences, the of thepresented LFP/C battery was measured a modified version applied toversion the impedance LFP/C Figure 6 and consistsfollowing of consecutive charging and of modified of thebattery hybrid is pulse power in characterization (HPPC) test. The measurement profile the hybrid pulse power characterization (HPPC) test. The measurement profile applied to the LFP/C discharging DCLFP/C current pulsesisofpresented different in C-rates 0.1C,consists 0.5C, 1C, 2C, 3C and 4C); the pulse applied to the battery Figure(i.e., 6 and of consecutive charging and battery is presented in18Figure 6 and consists oftime consecutive charging and3Cdischarging duration wasDC setcurrent to s [34] and relaxation between two consecutive pulses setcurrent to discharging pulses ofthe different C-rates (i.e., 0.1C,each 0.5C, 1C, 2C, and 4C); was theDC pulse 15 min. pulses of different C-rates (i.e., 0.1C, 0.5C, 1C, 2C, 3C and 4C); the pulse duration was set to 18 duration was set to 18 s [34] and the relaxation time between each two consecutive pulses was set tos [34] and the relaxation time between each two consecutive pulses was set to 15 min. 15 min. 4C Discharge Discharge

3C

43CC 2C

32CC 21CC

-0.1C

-1 0C -0.1C

-2CC -1 -3CC -2

0.1C

18 s

1C

18 s

1C

4C

3C

2C

0.5C

0.1C

1 C0

Charge Charge

C-rateC-rate

4C

0.5C -0.5C -1C 15 min -0.5C -1C

-2C

15 min -2C

-4CC -3

Time

-4 C

Time

-3C -3C

-4C

Figure 6. Pulse profile for measuring the internal resistance of the LFP/C battery. -4C Figure 6. Pulse profile for measuring the internal resistance of the LFP/C battery.

Figure 6. Pulse profile for measuring the internal resistance of the LFP/C battery.

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Furthermore,ininorder orderto to consider consider the dependence onon SOC and temperature, Furthermore, dependenceofofthe theimpedance impedance SOC and temperature, test profilepresented presented Figure wasthe applied to SOC a 5% SOC Furthermore, in order to consider dependence of the impedance on(considering SOC and temperature, thethe test profile ininFigure 66 was applied tothe the5%–95% 5%–95% SOCinterval interval (considering a 5% SOC ◦ ◦ ◦ ◦ resolution) and for the same temperatures (i.e., 15 °C, 25 °C, 35 °C and 45 °C) considered during the test profile presented in Figure 6 was applied to the 5%–95% SOC interval (considering a 5% SOC resolution) and for the same temperatures (i.e., 15 C, 25 C, 35 C and 45 C) considered during measurement the other performance parameters. resolution) and forother the same temperatures (i.e., 15 °C, 25 °C, 35 °C and 45 °C) considered during measurement ofofthe performance parameters. The typical voltage response of a Li-ion pulse is is illustrated in in measurement of the other performance parameters. The typical voltage response of a Li-ion battery batteryto toaadischarging dischargingcurrent current pulse illustrated FigureThe 7; atypical similarvoltage voltageresponse behaviorof is aobtained for a charging current pulse. As pulse presented in the ISO Li-ion battery to a discharging current is illustrated in Figure 7; a similar voltage behavior is obtained for a charging current pulse. As presented in the ISO 12405-1:2011 Standard [34], the charging andfor discharging of As thepresented battery could be Figure 7; a similar voltage behavior is obtained a charging impedance current pulse. in the ISO 12405-1:2011 Standard [34], the charging and discharging impedance of the battery could be calculated calculated for Standard various pulse lengths (i.e., and 0.1 s,discharging 2 s and 18impedance s). Furthermore, the impedances 12405-1:2011 [34], the charging of the battery could be for various pulse lengths (i.e., 0.1 s, 2 s and 18 s). Furthermore, the impedances corresponding to corresponding these pulse caused electrochemical processes calculated for tovarious pulselengths lengthsare (i.e., 0.1 s,by/related 2 s and to18different s). Furthermore, the impedances these pulse lengths are caused by/related to different electrochemical processes inside the battery [44]. inside the batteryto[44]. Nevertheless, in this determined for a current length corresponding these pulse lengths are work, causedimpedance by/relatedvalues to different electrochemical processes Nevertheless, in this work, impedance values determined for a current length of 18 s are presented. of 18 s are on the current voltage measurements, the impedance of the Li-ion inside the presented. battery [44].Based Nevertheless, in thisand work, impedance values determined for a current length Based the current based and voltage measurements, the impedance of thethe Li-ion batteryofisthe computed battery computed onon Ohm’s law, as:and voltage of 18on s is are presented. Based the current measurements, impedance Li-ion based on Ohm’s law, as: battery is computed based on Ohm’s law, as: V V −− ΔV VV ∆V = 11 0 0 = Ri R= (2) (2) i = ΔΔIV V1I1−IV ∆I 10 Ri = = (2) I1 ΔI ∆V represents where RiRrepresents the impedance of the battery, the change in the voltage due to the i represents the impedance of the battery, ∆V represents the change in the voltage due to the where applied pulse, ∆I represents the in∆V therepresents current,VV represents the voltage measured applied pulse, ∆Iimpedance representsof the change in the current, 1 1represents voltage measured i represents the thechange battery, the change inthe the voltage due to the wherecurrent Rcurrent after 1818 s, s,Vcurrent measured justbefore before applying current pulse, and I1measured represents 0 0represents after V represents the voltage measured just current pulse, and I1 represents applied pulse,the ∆I voltage represents the change in the applying current, Vthe 1the represents the voltage thethe amplitude current pulse. amplitude ofthe theapplied applied currentmeasured pulse. just before applying the current pulse, and I1 represents after 18 s, V0 of represents the voltage the amplitude of the applied current pulse.

Figure Theoreticalvoltage voltageand and current current of measurement with thethe Figure 7.7.Theoretical of Li-ion Li-ion battery batteryduring duringimpedance impedance measurement with current pulse technique. Figure 7. Theoretical voltage and current of Li-ion battery during impedance measurement with the current pulse technique. current pulse technique.

The voltage response of the LFP battery to the current profile measured at 50% SOC is illustrated The voltage response ofofthe profile measured measuredatat50% 50%SOC SOCis is illustrated in Figure 8. The voltage response theLFP LFPbattery batteryto to the the current current profile illustrated in Figure 8. in Figure 8. 3.35

3.35

Voltage Voltage [V] [V]

3.3 3.3

3.25 3.25 3.2 3.2 3.15 0 3.15 0

0.1 C-rate 0.5 C-rate 0.1 C-rate 1 C-rate 0.5 C-rate 2 C-rate 1 C-rate 3 C-rate 2 C-rate 4 C-rate 3 C-rate 410 C-rate 20 10

30

Time [s]

20

30

40 40

50 50

Time [s] Figure 8. Measured voltage response of the LFP battery at discharging current pulses of different Crates (SOC = 50%, T =voltage 25 °C). response of the LFP battery at discharging current pulses of different CFigure 8. Measured Figure 8. Measured voltage response of the LFP battery at discharging current pulses of different rates (SOC = 50%, T = 25 °C). C-rates (SOC = 50%, T = 25 ◦ C).

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Based on battery’s voltage response (similar to those presented in Figure 8), and considering Based Based on on battery’s battery’s voltage voltage response response (similar (similar to to those those presented presented in in Figure Figure 8), 8), and and considering considering Equation (2), the charging and discharging impedance of the tested LFP/C battery was Equation (2), the charging and discharging impedance of the tested LFP/C battery was determined Equation (2), the charging and discharging impedance of the tested LFP/C battery was determined determined for all the considered conditions; for example, Figure presents the measured impedance of the for for all all the the considered considered conditions; conditions; for for example, example, Figure Figure 999 presents presents the the measured measured impedance impedance of of the the ◦ LFP/C during pulse charging and discharging with 1C-rate (i.e., 2.5 A) at T = 25 °C. Both impedance LFP/C during pulse charging and discharging with 1C-rate (i.e., 2.5 A) at T = 25 C. Both impedance LFP/C during pulse charging and discharging with 1C-rate (i.e., 2.5 A) at T = 25 °C. Both impedance characteristics are showing aa parabolic dependence on the nearly flat region characteristics a parabolic dependence on the following a nearlyaa flat region characteristicsare areshowing showing parabolic dependence onSOC, the SOC, SOC, following following nearly flatbetween region between 20% and 90% SOC. 20% and 90% between 20% SOC. and 90% SOC. 0.025 0.025

Charging Charging Discharging Discharging

[ Ω] ] RR [Ω internal internal

0.023 0.023 0.021 0.021 0.019 0.019 0.017 0.017 0.015 0.0150 0

20 20

40 40

60 60

SOC SOC [%] [%]

80 80

100 100

Figure 9. 9. Measured impedance impedance of the the LFP/C battery charging and and discharging current current Figure battery during during 18 18 sss charging Figure 9. Measured Measured impedance of of the LFP/C LFP/C battery during 18 charging and discharging discharging current ◦ pulse (1C-rate, T = 25 °C). pulse C). pulse (1C-rate, (1C-rate, TT == 25 25 °C).

3.6. Electrochemical Impedance Spectroscopy Spectroscopy Test 3.6. Electrochemical Impedance Spectroscopy Test Test EIS has developed as a reliable method for characterization and modelling the performance EIS has has developed developed as as aa reliable reliable method method for for characterization characterization and and modelling modelling the the performance performance behavior of of the Li-ion batteries is by behavior the Li-ion Li-ion batteries. batteries.The Thesmall-signal small-signalimpedance impedance Li-ion batteries is determined of the the Li-ion batteries. The small-signal impedance ofof thethe Li-ion batteries is determined determined by applying a small sinusoidal current (galavantostatic mode) or voltage (potentiostatic mode) and by applying a small sinusoidal current(galavantostatic (galavantostaticmode) mode)or orvoltage voltage (potentiostatic (potentiostatic mode) and applying a small sinusoidal current measuring the amplitude and phase the or current, This measuring and phase shiftshift of theof voltagevoltage or current, This procedure measuring the theamplitude amplitude and phase shift ofoutput the output output voltage or respectively. current, respectively. respectively. This procedure is repeated for a sweep of frequencies, and thus the battery impedance spectrum is obtained. is repeatedisfor a sweepfor ofafrequencies, and thus the spectrumspectrum is obtained. procedure repeated sweep of frequencies, andbattery thus theimpedance battery impedance is obtained. In this this work, mode and In work, the the EIS were performed performed in in galvanostatic and for EIS measurements measurements were galvanostatic mode for the the frequency frequency range 10 kHz–10 mHz. Furthermore, all the EIS measurements were performed without range mHz. Furthermore, all the EIS were performed superimposed range 1010kHz–10 kHz–10 mHz. Furthermore, all measurements the EIS measurements werewithout performed without superimposed DC current; thus, the influence of the current on the small-signal AC impedance was DC current; thus, influence ofthe the influence current onofthe ACsmall-signal impedance was determined. superimposed DCthe current; thus, thesmall-signal current on the AC not impedance was not determined. As for the other considered performance parameters, the small-signal impedance As the other As considered performance parameters, the parameters, small-signalthe impedance wasimpedance measured notfor determined. for the other considered performance small-signal was measured at temperatures and for entire interval (0%–100% with 5% at different temperatures for the entire interval (0%–100% SOC with 5%SOC SOC resolution). was measured at different differentand temperatures and SOC for the the entire SOC SOC interval (0%–100% SOC with 5% SOC SOC resolution). The dependence of the LFP/C battery’s impedance spectra on the SOC and temperature The dependence of the LFP/Cofbattery’s impedance on the SOC and temperature is presented resolution). The dependence the LFP/C battery’sspectra impedance spectra on the SOC and temperature is Figures 10 in Figures 10in and 11, respectively. is presented presented in Figures 10 and and 11, 11, respectively. respectively.

-ImaginaryZZ[m [m ΩΩ -Imaginary ]]

8 8 6 6 4 4

5 5

6 6

7 7

8 8

2 2 0 0 4 4

6 6

8 8

10 10

Ω Real Real Z Z [m [mΩ]]

SOC=20% SOC=20% SOC=50% SOC=50% SOC=80% SOC=80% 12 14 12 14

Figure 10. Dependence of the impedance spectra on the SOC; T = 25 ◦°C. Figure 10. 10. Dependence Dependence of of the the impedance impedance spectra spectra on on the theSOC; SOC; TT == 25 25 °C Figure C..

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-Imaginary -Imaginary Z [mΩ] Z [mΩ]

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8 10 6 8 4 6

o

T=15 C o

2 4 0 2 4

T=25 C o

T=35 C o T=15oC C T=45 o

6

8

0

10

12

Real Z [mΩ]

14

T=25 C

16

o

T=35 C

18

o

T=45 C

Figure 11. Dependence of the impedance spectra on the temperature; SOC = 50%.

4 of the 6 impedance 8 10 spectra 12 on 14the temperature; 16 18 Figure 11. Dependence SOC = 50%.

4. Lithium-Ion Battery Performance Model

Real Z [mΩ]

Figure 11. Dependence of the impedance spectra on the temperature; SOC = 50%. 4. Lithium-Ion Battery Performance Model

Li-ion battery performance models can be classified into three main categories: electrochemical,

Li-ion battery performance models can be classified into threeand main categories: electrochemical, electrical, and mathematical models, depending on the development parameterization approach. 4. Lithium-Ion Battery Performance Model These modelling approaches are characterized by different degrees of complexity, different accuracy electrical, and mathematical models, depending on the development and parameterization approach. Li-ion battery performance models can be classified into three main categories: electrochemical, levels and could be used for different purposes (e.g., battery design, system simulation). Theseelectrical, modelling approaches are characterized by different degrees of complexity, different accuracy and mathematical models, depending on the development and parameterization approach. this work has focusedpurposes on the development anddesign, parameterization of the LFP/C battery levelsNevertheless, and could be used for different (e.g., battery system simulation). Nevertheless, These modelling approaches are characterized by different degrees of complexity, different accuracy performance model using an electrical modelling approach. this work focused development parameterization of the LFP/C battery performance levelshas and could on be the used for differentand purposes (e.g., battery design, system simulation). Nevertheless, this work has focused on the development and parameterization of the LFP/C battery model using an electrical modelling approach. 4.1. Equivalent Electrical Circuit Model performance model using an electrical modelling approach.

Li-ionElectrical batteries’Circuit electrical models generally use EECs, composed of simple (e.g., resistors, 4.1. Equivalent Model capacitors, inductors) more Model complex elements (e.g., constant phase elements, Warburg elements, 4.1. Equivalent ElectricalorCircuit

Li-ion electrical models use EECs, composed of simple (e.g., of resistors, etc.) andbatteries’ a voltage source to express thegenerally dynamic behavior of batteries. The basic configuration an Li-ion batteries’ electrical modelselements generally(e.g., use constant EECs, composed of simple Warburg (e.g., resistors, capacitors, inductors) or more complex phase elements, elements, EEC-based performance model is presented in Figure 12. The structure of the EEC can have different capacitors, inductors) or more complex elements (e.g., constant phase elements, Warburg elements, etc.) and a voltagedepending source to on express dynamic of batteries. The basic configuration configurations variousthe aspects, such behavior as: model accuracy, model computation time, Li- of etc.) and a voltage source to express the dynamic behavior of batteries. The basic configuration of an ion batteryperformance chemistry, etc. an EEC-based model is presented in Figure 12. The structure of the EEC can have different EEC-based performance model is presented in Figure 12. The structure of the EEC can have different configurations depending on aspects, such as:as:model computationtime, time,Li-Li-ion configurations depending various on various aspects, such modelaccuracy, accuracy, model model computation battery etc. ionchemistry, battery chemistry, etc.

Figure 12. Basic configuration of an EEC-based performance model for Li-ion batteries.

According to the configuration shown in Figure 12, the voltage of the battery is computed Figure 12. Basic configuration of an EEC-based performance model for Li-ion batteries. Figure 12. Basic (3). configuration of an EEC-based performance model for Li-ion batteries. according to Equation

Vbatin= V According to the configuration shown Figure 12, the voltage of the battery is computed (3) OC ± VEEC According the configuration shown in Figure 12, the voltage of the battery is computed according toto Equation (3). where Vbat represents the voltage of the battery, VOC represents the OCV, and VEEC represents the according to Equation (3). = VOC ±the VEECdynamics of the battery. (3) voltage drop across the EEC, which is usedVto bat model Vbat = VOC ± VEEC (3)

The voltage drop across EEC’s impedance is obtained as the sum of various over-voltages, which where Vbat represents the voltage of the battery, VOC represents the OCV, and VEEC represents the are caused by different voltage processesofoccurring inside the represents battery with different time during the wherevoltage V bat represents the battery, V OC OCV, and constants V EEC represents drop acrossthe the EEC, which is used to model the dynamics the of the battery. charging or discharging [46]: voltage drop EEC, which used to model the dynamics battery. The across voltagethe drop across EEC’sisimpedance is obtained as the sumof of the various over-voltages, which The voltage drop across EEC’s impedance is obtained as the sum of various V = V ± V ± V are caused by different processes occurring constants during (4)which EEC inside ohmic the chtrbattery diff with different time over-voltages, charging discharging [46]: occurring inside the battery with different time constants during are caused byordifferent processes where Vohmic represents the ohmic over-voltage (caused by the resistance of the poles, current charging or discharging [46]: VEEC = Vohmic the ± Vchtrcharge ± Vdiff transfer over-voltage (caused (4) collectors, electrolytes, etc.), Vchtr represents by

VEEC = Vohmic ± Vchtr ± Vdiff

(4)

where Vohmic represents the ohmic over-voltage (caused by the resistance of the poles, current wherecollectors, V ohmic represents the ohmic (caused the resistance the poles, current electrolytes, etc.), over-voltage Vchtr represents the by charge transfer of over-voltage (causedcollectors, by

electrolytes, etc.), V chtr represents the charge transfer over-voltage (caused by electrochemical reaction

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Batteries 2016, 2, 37and V diff represents the diffusion over-voltage (caused by a deficit or surplus 10 of 19 at inner surfaces), of Batteries 2016, 2, 37 10 of 19 reactants at the reaction’s location). electrochemical reaction at inner surfaces), and Vdiff represents the diffusion over-voltage (caused by Thus, to model reaction the voltage behavior of the battery, the contributions of the V OC andbyof the electrochemical at inner andLFP/C Vlocation). diff represents the diffusion over-voltage (caused a deficit or surplus of reactants atsurfaces), the reaction’s V EEC acomponents have to be known. The contribution of the OCV to the battery voltage was obtained deficit or to surplus reactants the reaction’s location). Thus, modelofthe voltageat behavior of the LFP/C battery, the contributions of the VOC and of the for various conditions (i.e., SOCs and temperatures) during the characterization, as of presented to model thetovoltage behavior of the LFP/Cof battery, thebattery contributions of the Vwas OC and the VEEC Thus, components have be known. The contribution the OCV to the battery voltage obtained in Section 3.4. To obtain contribution of the V EEC the LFP/C voltage,was the EEC had to V EECvarious components havethe to be known. contribution oftothe OCV to thebattery battery voltage obtained for conditions (i.e., SOCs andThe temperatures) during the battery characterization, as presented for various3.4. conditions (i.e., and temperatures) the battery characterization, as presented be parameterized. in Section To obtain theSOCs contribution of the VEECduring to the LFP/C battery voltage, the EEC had to be in Section 3.4. To obtain the contribution of the V EEC to the LFP/C battery voltage, the EEC had to be parameterized. The parameterization of the EEC was realized by curve fitting the impedance spectra of the LFP/C parameterized. The parameterization of the EEC was realized by curve fitting the impedance spectra of the of battery, which were measured at different conditions as presented in Section 3.6. The configuration Thebattery, parameterization of measured the EEC was realized by curve fitting the impedance spectra of the LFP/C were different conditions as presented in Section the EEC which was which used for curve fitting,atand consequently to express the dynamics of3.6. theThe LFP/C LFP/C battery, which were measured at different conditions as presented in Section 3.6. The configuration of the EEC which was used for curve fitting, and consequently to express the dynamics battery, is illustrated in Figure 13. The curve fitting procedure was based on a complex non-linear configuration of the EEC which wasinused for13. curve consequently to express of the LFP/C battery, is illustrated Figure Thefitting, curve and fitting procedure was based the on adynamics complex least-square (CNLS) algorithm, whereinboth real13. and imaginary components of the measured impedance of the LFP/C battery, is illustrated Figure The curve fitting procedure was based on a complex non-linear least-square (CNLS) algorithm, where both real and imaginary components of the were non-linear fitted simultaneously using a least square minimization; an example of the impedance least-squarewere (CNLS) where botha least real and imaginary components ofspectrum the measured impedance fitted algorithm, simultaneously using square minimization; an example of curvemeasured fitting results is shown in Figure 14. impedance were fitted simultaneously using a least square minimization; an example of the impedance spectrum curve fitting results is shown in Figure 14.

the impedance spectrum curve fitting results is shown in Figure 14.

Figure 13. Equivalent electrical circuit (EEC) configuration based on ZARC elements used for fitting

Figure 13. Equivalent electrical circuit (EEC) configuration based on ZARC elements used for fitting Figure 13. Equivalent electrical circuit on ZARC the measured impedance spectra of the(EEC) LFP/Cconfiguration battery. CPE:based constant phase elements element. used for fitting the measured impedance spectra of the LFP/C battery. CPE: constant phase element. the measured impedance spectra of the LFP/C battery. CPE: constant phase element. 6

-Imaginary Z [mZΩ[m ] Ω] -Imaginary

6 4 4 2 2 0 0 5

7

5

7

9

Real Z9 [mΩ] Real Z [mΩ]

Measured spectrum Fitted spectrum Measured spectrum Fitted 11 spectrum 13 11

13

Figure 14. Measured and fitted impedance spectrum using the EEC presented in Figure 13. Figure 14. Measured and fitted impedance spectrum using the EEC presented in Figure 13.

Figure 14. Measured and fitted impedance spectrum using the EEC presented in Figure 13. As presented in Figure 14, the EEC configuration based on a series inductance Ls, a series As presented in Figure the EEC based on Ls, a phase series resistance Rs, and two ZARC 14, elements (i.e.,configuration parallel connections of aa series resistorinductance and a constant resistance Rs, and two ZARC (i.e., parallel connections of a resistor aLsconstant phase As presented in Figure 14, theelements EEC configuration based on spectrum a series inductance ,LFP/C a series resistance element) was able to accurately fit the measured impedance of the and tested battery. element) was to accurately the spectrumand ofimpedance the tested spectra LFP/C battery. Rs , and two ZARC elements (i.e., fit parallel connections a constant phase Therefore, thisable EEC configuration wasmeasured used to impedance fit all of thea resistor measured (atelement) the Therefore, this EEC configuration was used to fit all the measured impedance spectra (at the conditions discussed in Section 3.6) and thus to model the performance behavior of the LFP/C battery. was able to accurately fit the measured impedance spectrum of the tested LFP/C battery. Therefore, conditions discussed inselected Section 3.6) thusmeasured to model the performance behavior of the LFP/C battery. The configuration impedance of the EEC is by: this EEC was used to fitand allgiven the impedance spectra (at the conditions discussed The impedance of the selected EEC is given by: in Section 3.6) and thus to model the performance Rbehavior of the LFP/C battery. The impedance of R2 1 + Z EEC = jω Ls + Rs + (5) N1 N2 R R the selected EEC is given by: Z = jω L + R + 1 + ( jω )1 Q1 R1 + 1 + ( jω )2 Q2 R2 EEC s s (5) N N 1 + ( jω ) 1 Q1 R1 1 + ( jω ) 2 Q2 R2 where R1, Q1, and N1 represent the resistance, theRgeneralized capacitance R2 and the depression factor 1 ZNEEC = jωL + resistance, Rs + +capacitance (5) sthe where R1, QZARC 1, and element, 1 represent the explanations generalized the depression N1 N2and of the first respectively; the same are valid theR second ZARC factor of the 1 + ( jω) Q1 R1 1 + ( jω)for Q 2 2 of the first ZARC element, respectively; the same explanations are valid for the second ZARC of the EEC. EEC. of the EIS techniquethe over other parameterization is represented where R1 ,AQmain N1 represent the resistance, generalized capacitancetechniques and the depression factor of 1 , andadvantage A feature main advantage of theelements EIS technique over other parameterization techniques is represented by its to relate EEC to the physicochemical process that occur inside the battery the first ZARC element, respectively; the same explanations are valid for the second ZARC of the EEC. by its feature to EEC elements the term physicochemical thattooccur inside battery [28,30]. in relate the present work, thetofirst in Equation process (5) is used describe thethe inductive A main Thus, advantage of the EIS technique over other parameterization techniques is represented by [28,30]. in the present work, the first term Equation (5) is the used to describe caused the inductive behaviorThus, at high frequencies, the second term is in used to calculate over-voltage by the its feature to relate EEC elementsthe to the physicochemical that inside the battery [28,30]. behavior at high frequencies, second toprocess calculate theoccur over-voltage caused by ohmic resistance (Vohmic), the third term isterm usedistoused calculate the over-voltage corresponding to the the Thus,ohmic in theresistance present work, the first term in Equation (5) is used to describe the inductive behavior (Vohmic), the third term is used to calculate the over-voltage corresponding to the at

high frequencies, the second term is used to calculate the over-voltage caused by the ohmic resistance

charge-transfer process (Vchtr), while the last term is used to determine the over-voltage corresponding to the diffusion process (Vdiff). Consequently, based on the aforementioned aspects, the battery voltage Equation (3) was rewritten as given in Equation (6). Furthermore, considering Equation (6), the configuration of the EEC, which was used to model the performance behavior of the Batteries 2016, 2, 37 11 of 20 LFP/C battery, is presented in Figure 15. V

=V

+ I ( jω L + R +

R1

+

R2

)

OC bat s s (6)19 Batteries 2016, 2, 37 11 of N N (V ohmic ), the third term isbatused to calculate the to the charge-transfer + ( jω ) Q1 R1 1corresponding + ( jω ) Q2 R2 1 over-voltage process (V chtr ), whileprocess the last term is used to determine the over-voltage corresponding to the diffusion charge-transfer (Vchtr), while the last term is used to determine the over-voltage process (V diff ). Consequently, based on the aforementioned voltage Equation corresponding to the diffusion process (Vdiff ). Consequently,aspects, based onthe thebattery aforementioned aspects, (3) was rewritten as given in Equation (6). Furthermore, considering Equation (6), the configuration the battery voltage Equation (3) was rewritten as given in Equation (6). Furthermore, considering of the EEC, which used to model theEEC, performance of the battery, is presented Equation (6),was the configuration of the which was behavior used to model theLFP/C performance behavior of the in LFP/C Figure 15. battery, is presented in Figure 15. R R2 R 1 Vbat = VOC + Ibat ( jωLs + Rs + ) (6) + R2 N Vbat = VOC + I bat (jω Ls + Rs +1 + ( jω1)N N1 Q +R )2Q R (6) N 1 + jω ( ) 2 2 1 1 1 + ( j ω ) Q R 1 + ( jω ) Q R 1

1

2

2

1 1

2

2

Figure 15. Basic electrical configuration of the performance model of the LFP/C battery.

4.2. Dependence on Load Current Besides being dependent on SOC and temperature—dependences obtained from the EIS measurements (see Section 3.6)—the parameters of the EEC are also dependent on the load current as illustrated in Figure 15. Nevertheless, because all the EIS measurements were performed without superimposed DC current (see Section 3.6), the effect of the load current on the parameters of the EEC was not determined. Among the parameters of the EEC, the charge transfer resistance R1 shows a Figure 15. Basic electrical configuration of the the performance model ofof thethe LFP/C battery. 15. Basic electrical of performance model LFP/C battery. highlyFigure non-linear dependence onconfiguration the load current. This non-linear dependence of the charge-transfer resistance, and of the corresponding over-voltage, on the LFP/C battery current is presented in Figure 4.2. Dependence on Load Current

16. 4.2. Dependence on Load Current

Theoretically, exponential between the Li-ion battery current andfrom the potential Besides beingthe dependent on dependence SOC and temperature—dependences obtained the EIS

Besides being(see dependent on SOC and temperature—dependences obtained from the EIS can be determined from the 3.6)—the Butler-Volmer equation given in are Equation (7) [47]. on the load current measurements Section parameters of the EEC also dependent measurements (see Section 3.6)—the parameters of the EEC are also dependent on the load current as illustrated in Figure 15. Nevertheless, because all the EIS measurements were performed without (1−α ) nF  αnF  ⋅η − ⋅η as illustrated in Figure 15. Nevertheless, because all theRTEIS measurements were performed without superimposed DC current (see Section 3.6), effect load  e RT  current on the parameters of the EEC i = i0the − eof the (7)  charge superimposed DC current (see Section 3.6), the of the the the parameters was not determined. Among the parameters of effect the EEC, transferon resistance R1 shows of a the  load current highly non-linear dependence on the load current. This non-linear dependence of the charge-transfer EEC was not determined. Among the parameters of the EEC, the charge transfer resistance R1 shows where i is the current density, i0 is the exchange current density, α is is the symmetry factor, resistance, andelectrode of the corresponding the LFP/C battery current presented in Figure a highly non-linear dependence on the over-voltage, load current.onThis non-linear dependence of the charge-transfer −1·mol−1), F is the Faraday n is the number of electrons, R is the universal gas constant (8.3144 J·K 16. resistance, and of the corresponding over-voltage, on the the LFP/C battery currenttheis electrode presented in −1), η is the constant (96,485.339 overpotential between Theoretically, theC·mol exponential dependence between(i.e., the Li-iondifference battery current and the potential Figure 16. potential and equilibrium potential). can be determined from the Butler-Volmer equation given in Equation (7) [47].

 αnF ⋅η − (1−α ) nF ⋅η   i = i0  e RT − e RT    

(7)

where i is the electrode current density, i0 is the exchange current density, α is the symmetry factor, n is the number of electrons, R is the universal gas constant (8.3144 J·K−1·mol−1), F is the Faraday constant (96,485.339 C·mol−1), η is the overpotential (i.e., the difference between the electrode potential and equilibrium potential).

Figure 16. Measured non-linear dependence of the charge transfer resistance and of the corresponding

Figure 16. Measured non-linear dependence of the charge transfer resistance and of the corresponding over-voltage on the current of the LFP/C battery (note: positive current stands for charging). over-voltage on the current of the LFP/C battery (note: positive current stands for charging).

Theoretically, the exponential dependence between the Li-ion battery current and the potential can be determined from the Butler-Volmer equation given in Equation (7) [47].   (1−α)nF αnF i = i0 e RT ·η − e− RT ·η

Figure 16. Measured non-linear dependence of the charge transfer resistance and of the corresponding over-voltage on the current of the LFP/C battery (note: positive current stands for charging).

(7)

where i is the electrode current density, i0 is the exchange current density, α is the symmetry factor, n is the number of electrons, R is the universal gas constant (8.3144 J·K−1 ·mol−1 ), F is the Faraday constant

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(96,485.339 C·mol−1 ), η is the overpotential (i.e., the difference between the electrode potential and equilibrium potential). Batteries 2016, 2, 37 12 of 19 However, information about various parameters given in Equation (7) were not available and Batteries 2016, 2, 37 laboratory equipment in order to determine them. Consequently, in 12 order of 19 require specialized to However, information about various parameters given in Equation (7) were not available and determine the dependence of the LFP/C battery voltage on the load current, the following approach require specialized laboratory equipment in order to determine them. Consequently, in order to However, information about various parameters given in Equation (7) were not available and was followed: a comparison was between battery voltagethe response measured determine the dependence of theperformed LFP/C battery voltage the on the load current, following approach over require specialized laboratory equipment in order to determine them. Consequently, in order to was of followed: athe comparison was performed between the battery voltage3.5) response measured overvoltage a a period 18 s in laboratory (during the HPPC tests—see Section and the simulated determine the dependence of the LFP/C battery voltage on the load current, the following approach period of 18 s in the laboratory (during the HPPC tests—see Section 3.5) and the simulated voltage response. The obtained difference was considered to describe the voltage impact response of the current on the was followed: a comparison was performed between the battery measured overvoltage a response. The obtained difference was considered to describe therepeated impact of for the current on the voltage of the LFP/C battery. The aforementioned procedure was all SOCs, temperatures, period of 18 s in the laboratory (during the HPPC tests—see Section 3.5) and the simulated voltage of the LFP/C battery. The aforementioned procedure was repeated for all SOCs, temperatures, and and C-rates, were difference considered the to battery impedance measurement by voltage HPPC test. response.which The obtained wasduring considered describe the impact of the current on the C-rates, which were considered during the battery impedance measurement by HPPC test. The of the LFP/C battery. The aforementioned procedure waseffect repeated for all SOCs,on temperatures, The obtained voltage correction factors, assigned to the of the current the LFP/Cand battery obtained voltage correction factors, assigned to the effect of the current on the LFP/C battery voltage C-rates, which were considered during the battery impedance measurement by HPPC test. Thetable voltage were stored as a 3D look-up table; for instance, Figure 17 presents the look-up were stored as a 3D look-up table; for instance, Figure 17 presents the look-up table implementation obtained voltage correction factors, assigned to the current on the LFP/C battery voltage implementation the voltage correction factors for effect the 25of◦the C dimension. of the voltageofcorrection factors for the 25 °C dimension. were stored as a 3D look-up table; for instance, Figure 17 presents the look-up table implementation of the voltage correction factors for the 25 °C dimension.

Figure 17. Dependence of the voltage correction factor on the SOC and current for T = 25 °C.

Figure 17. Dependence of the voltage correction factor on the SOC and current for T = 25 ◦ C. Figure 17. Dependence of the voltage correction factor on the SOC and current for T = 25 °C.

4.3. Model Implementation

4.3. Model Implementation

4.3. Model BasedImplementation on the voltage Equation (6) and considering the dependences of the performance

Based on theon voltage Equation conditions, (6) and considering the dependences the performance parameters the operating the performance model ofof the LFP/C batteryparameters was Based on the voltage Equation (6) and considering the dependences of the performance implemented according tothe theperformance block diagrammodel proposed in Figure on the operating conditions, of the LFP/C18. battery was implemented according parameters on the operating conditions, the performance model of the LFP/C battery was to theimplemented block diagram proposed inblock Figure 18. proposed in Figure 18. according to the diagram Q1, N1 Rs

Rs

SOC = SOC i −

1 Capacity

SOC = SOC i −

 I bat ⋅ dt

1 Capacity

 I bat ⋅ dt

Q2, N2

Ls

Ls

Q1, N1 R1

Q2, N2 R2

R1

R2

ZEEC⋅ Ibat

ZEEC⋅ Ibat

Figure 18. Block diagram implementation of the performance model for the LFP/C battery. Figure 18. Block diagram implementationof ofthe the performance performance model forfor thethe LFP/C battery. Figure 18. Block diagram implementation model LFP/C battery. load The inputs of the proposed performance model for the LFP/C battery are the applied current, the initial SOC and the temperature, while the output of the model is the estimated battery The inputs of the proposed performance model for the LFP/C battery are the applied load voltage; additionally, the model can returnmodel information about thebattery batteryare SOC every moment. The inputs the proposed performance LFP/C theatestimated applied load current, current, theof initial SOC and the temperature, whilefor thethe output of the model is the battery Furthermore, since no thermal model was developed for the LFP/C battery, the temperature of the the initial SOC and the temperature, while the output of the model is the estimated battery voltage; voltage; additionally, the model can return information about the battery SOC at every moment. battery is considered constant during the whole simulation. Furthermore, since no was developed forbattery the LFP/C battery, themoment. temperature of the additionally, the model canthermal returnmodel information about the SOC at every Furthermore, The parameterization of the proposed performance model was performed based on the results is considered during thefor whole since battery no thermal modelconstant was developed thesimulation. LFP/C battery, the temperature of the battery is obtained from the characterization test, which was presented in detail in Section 3. The parameters The parameterization of whole the proposed performance model was performed based on the results considered constant during simulation. of the LFP/C battery werethe implemented as 2D look-up tables (i.e., capacity, OCV, EEC’s parameters) obtained from the characterization test, which was presented in detail in Section 3. The parameters of the LFP/C battery were implemented as 2D look-up tables (i.e., capacity, OCV, EEC’s parameters)

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The parameterization of the proposed performance model was performed based on the results obtained from Batteries 2016, 2, 37the characterization test, which was presented in detail in Section 3. The parameters 13 of 19 of the LFP/C battery were implemented as 2D look-up tables (i.e., capacity, OCV, EEC’s parameters) and 3D-look-up 3D-look-up table (i.e., Butler-Volmer Butler-Volmer correction correction voltage) voltage) in in order order to to consider consider their their dependences dependences and on the the operating operating conditions conditions (i.e., (i.e., SOC, SOC, temperature, temperature, and and load load current). current). Because Because for for similar similar operating operating on conditions, the batteries is different between charging andand discharging case case (i.e., conditions, the capacity capacityofofthe theLi-ion Li-ion batteries is different between charging discharging Coulombic efficiency different thanthan 100%), two two separate look-up tables were implemented for the (i.e., Coulombic efficiency different 100%), separate look-up tables were implemented for charging and discharging capacity, respectively. Because of the hysteresis effect (see Figure 5), which the charging and discharging capacity, respectively. Because of the hysteresis effect (see Figure 5), is inherent to LFP/C batterybattery chemistry, a similar approach was followed for implementing the OCV which is inherent to LFP/C chemistry, a similar approach was followed for implementing the characteristic; however, for other Li-ionLi-ion battery chemistries, a single look-up table istable enough to model OCV characteristic; however, for other battery chemistries, a single look-up is enough to the OCV for both and discharging cases. cases. model thecharacteristic OCV characteristic forcharging both charging and discharging During the the fitting fitting process process of of the the measured measured impedance impedance spectra, spectra, ititwas was found foundout outthat thatonly onlyRRss,, R R11, During R22, Q11, and Q22 are had been been R are dependent dependent on on the the SOC and temperature at which the EIS measurements had performed. Consequently, these parameters were implemented into the performance performance model as as 2D 2D performed. look-up tables tablessince sincetheir theirvalues valuesare aredependent dependent SOC and temperature; Figure 19 exemplifies look-up onon thethe SOC and temperature; Figure 19 exemplifies the the look-up implementation ofseries the series resistance Rs. other The other parameters the EEC, N1, look-up tabletable implementation of the resistance Rs . The parameters of theofEEC, Ls , N1L,s,and − 8 −8 and N2 were found be constant, independent of the operating conditions: = 1.8·× H,NN11 == 0.5, N found to betoconstant, independent of the operating conditions: Ls =Ls1.8 × 1010 H, 2 were and N N22 == 0.8. and 0.8.

Figure 19. 19. Two-dimensional Two-dimensional look-up look-up table table implementation implementationof ofRRss of of the the EEC. EEC. Figure

Finally, the estimated voltage of the LFP/C battery was obtained by summing up the Finally, the estimated voltage of the LFP/C battery was obtained by summing up the contributions contributions from the OCV, the voltage drop across the EEC, and the Butler-Volmer correction from the OCV, the voltage drop across the EEC, and the Butler-Volmer correction voltage, as illustrated voltage, as illustrated in Figure 18. in Figure 18. 5. Validation Validation of of the the Performance PerformanceModel Model 5. In order order to to verify verify the battery and to In the accuracy accuracy of of the thedeveloped developedperformance performancemodel modelofofthe theLFP/C LFP/C battery and validate the proposed characterization procedure, different load current profiles were considered. to validate the proposed characterization procedure, different load current profiles were considered. The accuracy accuracy of of the the developed developed performance performance model model was was evaluated evaluated by by computing computing the the coefficient coefficient of of The 2 which was obtained by comparing the measured and estimated battery’s voltage 2 determination R determination R which was obtained by comparing the measured and estimated battery’s voltage profiles, according accordingto toEquation Equation(8). (8). profiles, n

(V ( x ) − V ( x )) 2 ∑(Vmodel ( x ) − Vmeas ( x )) n

2

model

meas

R 2 = 1 − x =1 2 R2 = 1 − x=1n n Vmodel ( x ) − Vmeas
2 V x − V ( ) ∑ meas x =1 model

(

)

(8) (8)

x =1

where Vmodel is the estimated battery voltage, Vmeas represents the measured battery voltage, x where V model is the estimated battery voltage, V meas represents the measured battery voltage, x represents the present observation, and n represents the total number of observations. represents the present observation, and n represents the total number of observations. Moreover, the mean ( ε ) and the maximum (εmax) deviation between the measured and estimated battery voltage profiles were computed according to Equations (9) and (10) for the three considered verification cases.

ε = Vmeas (x) − Vmodel (x)

(9)

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Moreover, the mean (ε) and the maximum (εmax ) deviation between the measured and estimated battery voltage profiles were computed according to Equations (9) and (10) for the three considered Batteries 2016, 2, 37 14 of 19 verification cases. Batteries 2016, 2, 37 14 of 19 (9) |ε| = |Vmeas ( x ) − Vmodel ( x )|   ( ) − ( ) V x V x   meas model ε max = max 100|V ⋅ meas Vmodel (x ) ( x )| Vmeas(Vxx))−− Vmodel (10) εmax =εmax 100 · (10) meas (x )  max = max100 ⋅ (10) Vmeas(x() x )  meas   5.1.5.1. Pulse Discharge Pulse Discharge 5.1. Pulse Discharge The first validation modelwas wascarried carriedout outusing usinga adischarging discharging The first validationofofthe thedeveloped developed performance performance model The first validation ofaathe developed performance model waswas carried using discharging current pulse profile. From fully charged thethe LFP/C battery discharged witha 1C-rate (i.e., current pulse profile. From fully chargedstate, state, LFP/C battery was out discharged with 1C-rate current pulse profile. From a fully charged state, the LFP/C battery was discharged with 1C-rate (i.e., 2.5 A) in steps of 10% SOC; between two consecutive discharging current pulses, a relaxation period (i.e., 2.5 A) in steps of 10% SOC; between two consecutive discharging current pulses, a relaxation 2.5 A) in steps of 10% SOC; between two consecutive discharging current pulses, a relaxation period of 15 min was applied. The measured and estimated voltage profiles of the LFP/C for this verification period of 15 min was applied. The measured and estimated voltage profiles of the LFP/C for this of 15 are minillustrated was applied. measured voltage performance profiles of the LFP/C for verification case in The Figure Theand developed performance model was ablewas tothis estimate very verification case are illustrated in20. Figure 20.estimated The developed model able to estimate case are illustrated in Figure 20. The developed performance model was able to estimate very max = 3.66% and a mean accurately the voltage of the tested LFP/C battery, with a maximum error ε very accurately the voltage of the tested LFP/C battery, with a maximum error εmax = 3.66% and accurately the voltage of the tested LFP/C battery, with a maximum error εmax = 3.66% and a mean ε = 6.8 error a mean error ε =mV. 6.8 mV. error ε = 6.8 mV. 3.6

Measurement Simulation Measurement

3.6 3.4

Voltage Voltage [V][V]

3.4 3.2

Simulation

3.2 3 3 2.8

3.3 3.3 3.25

2.8 2.6 2.6 2.4 2.4 2.2 2.2 2 0 2 0

3.25 3.2 6000 3.2 6000

6500

7000

7500

6500

7000

7500

2000

4000

6000

[s] 4000 Time 6000

2000

Time [s]

8000

10000

12000

8000

10000

12000

Figure Measuredand andestimated estimatedvoltage voltage profiles profiles during during pulse battery Figure 20.20. Measured pulse discharging dischargingofofthe theLFP/C LFP/C battery Figure Measured and estimated voltage profiles during pulse discharging of the LFP/C battery (I bat = 2.520. A, T = 25 °C). ◦ (Ibat = 2.5 A, T = 25 C). (Ibat = 2.5 A, T = 25 °C).

The distribution of the voltage deviations, expressed as relative error values, is presented in The distribution ofofthe deviations, expressed as relative relativeerror errorthe values, presented The thevoltage voltage deviations, expressed as values, isispresented in in Figure 21,distribution showing that the model has a slight tendency of over-estimating voltage behavior of Figure 21, showing that the model has a slight tendency of over-estimating the voltage behavior Figure 21, showing that thethe model a slight tendency over-estimating the voltage behavior of the tested LFP/C battery; mainhas contribution of thisoftendency is represented by the voltage of estimation the tested LFP/C battery; themain main contribution thistendency tendency represented thevoltage voltage tested error LFP/Cafter battery; the contribution ofofthis isisrepresented bybythe the current pulse is cut off. The dynamic behavior of the performance model estimation after the pulse cut off. behavior of theperformance performance model estimation error after thecurrent current pulse cut off. The The dynamic behavior the might be error improved (thus the accuracy ofisis the model willdynamic further increase) byof considering an EECmodel with might be improved (thus the accuracy of the model will further increase) by considering an EEC with might be improved (thus the accuracy of the model will increase) by considering an EEC more ZARC elements (or R-C parallel networks) [24]; however, this will cause a high increase ofwith the more ZARC elements (orR-C R-Cparallel parallelnetworks) networks) [24]; [24]; however, however, this ofof the more ZARC elements this will willcause causeaahigh highincrease increase the computation time of (or the performance model. computation time the performancemodel. model. computation time of of the performance 1000

Number of Occurances Number of Occurances

1000 800

Battery cell voltage over-estimation Battery cell voltage over-estimation

Battery cell voltage under-estimation Battery cell voltage under-estimation

800 600 600 400 400 200

εmax = 3.66% εmax = 3.66%

200 0 -4 0 -4

-2

0 [%] -2 Relative Error 0

Relative Error [%]

2

4

2

4

Figure 21. Distribution of the relative error obtained during pulse discharging verification case. Figure Distributionofofthe therelative relativeerror errorobtained obtained during during pulse Figure 21.21. Distribution pulse discharging dischargingverification verificationcase. case.

5.2. Dynamic Pulse Charging and Discharging 5.2. Dynamic Pulse Charging and Discharging In real applications (e.g., e-mobility, grid support etc.), Li-ion batteries are subjected to complex In real applications (e.g., e-mobility, grid the support etc.), Li-ion batteriesmodel are subjected to complex charging-discharging profiles. Consequently, developed performance was verified for a charging-discharging profiles. Consequently, the developed performance model was verified for a

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current profile, which was composed of charging and discharging pulses of different C-rates, over the whole SOC interval; the considered current profile is presented in Figure 22. To allow for a non5.2. Dynamic Pulse Chargingthe and Discharging biased verification, C-rates, which were used for this dynamic profile, are different than the Crates used to compute the Butler-Volmer correction voltage (see Sections 3.5 and 4.2). In real applications (e.g., e-mobility, grid support etc.), Li-ion batteries are subjected to complex The measured and estimated voltage profiles obtained by applying the dynamic current profile, charging-discharging profiles. developed presented in Figure 22, are Consequently, illustrated in Figurethe 23. For this case, theperformance proposed model model is able to was predictverified for

a current profile, which was composed of acharging pulses oferror different over the the voltage of the LFP/C battery with mean errorand of ε discharging = 6.4 mV and maximum of εmax =C-rates, 4.89%. distribution current of the relative error, presented ininFigure 24,22. shows in this for caseaanon-biased whole SOC Furthermore, interval; thetheconsidered profile is presented Figure To allow tendency of the developed model to under-estimate the voltage of the battery; as shown in the verification, the C-rates, which were used for this dynamic profile, are different than the C-rates used zoomed view of Batteries 2016, 2, 37of Figure 23, this under-estimation of the voltage was mainly caused by the inability 15 of 19 to computethe themodel Butler-Volmer correction Sections 3.5 andto4.2). to react to sudden and largevoltage variation(see of C-rates (i.e., 3.5C-rate 1C-rate). Batteries 2016, 2, 37

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Current [A]

current profile, which was composed of charging and discharging pulses of different C-rates, over 10 the wholeprofile, SOC interval; the considered current profile presented inpulses Figureof22. To allow for a noncurrent which was composed of charging andisdischarging different C-rates, over biased verification, the C-rates, which were usedprofile for this dynamic profile, are22. different than Cthe whole SOC interval; the considered current is presented in Figure To allow forthe a non5 rates used to compute the Butler-Volmer correction voltage (see Sections 3.5 and 4.2). biased verification, the C-rates, which were used for this dynamic profile, are different than the Cmeasured andthe estimated profiles obtained applying the 0 voltage correction ratesThe used to compute Butler-Volmer voltageby (see Sections 3.5dynamic and 4.2).current profile, presented in Figure 22, illustrated in Figure 23. For this case, the proposed model is current able to predict The measured andare estimated voltage profiles obtained by applying the dynamic profile, -5 the voltage in of Figure the LFP/C battery with a in mean error = 6.4 mVthe andproposed maximum errorisofable εmaxto = 4.89%. presented 22, are illustrated Figure 23.ofForε5this case, model predict Furthermore, the distribution of the relative error, presented in Figure 24, shows in this a 0 the voltage of the LFP/C battery withCharging a mean error of ε = 6.4 mV and maximum error of εmax =case 4.89%. to -10 SOC tendency of thethedeveloped model tonextunder-estimate the voltage the battery; as shown thea Furthermore, distribution of the relative error, -5presented in of Figure 24, shows in this in case zoomed view of Figure 23, thismodel under-estimation of the voltage was mainly caused by the inability of tendency of the developed to under-estimate the voltage 2000 2200 of the battery; as shown in the -15 0large variation 1000 2000 3000 (i.e., 40003.5C-rate 5000 to 1C-rate). the model to react to sudden and of C-rates zoomed view of Figure 23, this under-estimation of the voltage was mainly caused by the inability of Charging/Discharging profile applied at each SOC

Time [s]

the model to react to sudden and large variation of C-rates (i.e., 3.5C-rate to 1C-rate). 10

Figure 22. Dynamic current profile used for verification of the LFP/C battery performance model.

Figure 22. Dynamic current profile 10 used for verification of the LFP/C battery performance model. 5 3.6 Measurement

5

Current Current Voltage [V][A] [A]

Simulation 0 The measured and estimated voltage profiles obtained by applying the dynamic current profile, 3.4 presented in Figure 22, are illustrated 0in Figure 23. For this case, the proposed model is able to predict -5 5 3.2 the voltage of the LFP/C battery with a mean error of 0ε = 6.4 mV and maximum error of εmax = 4.89%. -5 5 Charging to 3.4 -5 3 Furthermore, the distribution of the -10 relative error, presented in Figure 24, shows in this case a tendency next SOC 0 Charging to -10 nextthe SOC of the developed model to under-estimate voltage of the battery; as shown in the zoomed view of 3.3 -52000 2200 -15 2.8 0 1000 2000 3000 4000 5000 Figure 23, this under-estimation of the voltage wasTime mainly [s] 2000 caused 2200 by the inability of the model to react -15 3.2 0 1000 2000 3000 3800 4000 40005000 3600 2.6 to sudden and large variation of C-rates (i.e., 3.5C-rate to 1C-rate). Time [s] 0 1000 for verification 2000 3000of the4000 5000 Figure 22. Dynamic current profile used LFP/C battery performance model. Charging/Discharging profile applied at each SOC

Charging/Discharging profile applied at each SOC

Time [s]

Figure 22. Dynamic current3.6profile used for verification of the LFP/C battery performance model. Figure 23. Measured and estimated voltage profiles during dynamic pulse charging and discharging Measurement 3.6 T = 25 Simulation °C). of the LFP/C battery (Ibat = 2.5 A, Measurement Simulation

3.4

Number of Occurances Voltage Voltage [V] [V]

1200 3.4 3.2 1000 3.2 3 800 3 2.8 600

3.4 3.4 3.3

2.8 400 2.6 0

3.3 3.2 3600 3800 4000 4000 5000 3.2 3000 3600 3800 4000 Time [s] 2000 3000 4000 5000

1000

2.6 200 0

2000

1000

Number of Occurances Number of Occurances

Time [s] dynamic pulse charging and discharging Figure 23. Measured and estimated voltage profiles during 0voltage profiles during dynamic pulse charging and discharging Figure 23. Measured and estimated -2 0 2 4 bat = 2.5 A, T = -4 25 °C). ofFigure the LFP/C battery (I 23. Measured and estimated voltage profiles during dynamic pulse charging and discharging of the LFP/C battery (Ibat = (I2.5 A, T = 25 ◦ C). Relative Error [%] bat = 2.5 A, T = 25 °C). of the LFP/C battery 1200 Figure 24. Distribution of the relative error obtained during dynamic pulse charging and discharging 1200 verification case. 1000 1000 800 800 600 600 400 400 200 200 0 0

-4 -4

-2

0

Relative Error [%]

-2

0

2 2

4 4

Relativeduring Error [%] Figure 24. Distribution of the relative error obtained dynamic pulse charging and discharging verification case. Figure 24. Distribution of the relative error obtained during dynamic pulse charging and discharging

Figure 24. Distribution of the relative error obtained during dynamic pulse charging and discharging verification case. verification case.

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5.3. Field-Measured Profile 5.3.5.3. Field-Measured Field-MeasuredProfile Profile 5.3. Field-Measured Profile of the developed performance model was performed using a realistic The third verification The third verification of the developed performance model was was performed usingusing a realistic current The third verification of the developed performance model performed a realistic current profile, which was of measured on field on LFP/C batteries providing primary frequency The third the developed performance modelprimary was performed using a realistic profile, which wasverification measured field on LFP/C batteries providing frequency regulation (PFR) current profile, which wasonmeasured on field on LFP/C batteries providing primary frequency regulation (PFR) in the Danish energy market [5]. The considered current profile with a length of two current profile, which was measured on field on LFP/C batteries providing primary frequency in the Danish(PFR) energy [5]. energy The considered current profile with a length of two hours is illustrated regulation in market the Danish market [5]. The considered current profile with a length of two hours is illustrated Figure 25.energy The comparison the estimated and measured of two the regulation (PFR) inin the Danish market [5].between The considered current profile with avoltage length of hours is25. illustrated in Figurebetween 25. The comparison between the estimated andofmeasured voltage of when the in Figure The comparison the estimated and measured voltage the LFP/C battery, LFP/C battery, when the field-measured current profile was considered, is presented in Figure hours is illustrated in Figure 25. The comparison between the estimated and measured voltage of 26. the battery, when the profile field-measured current profile was considered, is presented in Figuremodel 26. theLFP/C field-measured current was considered, is presented in Figure 26. The performance The performance model the LFP/C battery estimates with good accuracy (inε Figure = 8.4 mV LFP/C battery, when theoffield-measured current profilegenerally was considered, is presented 26. of the batterymodel estimates generally with good accuracy (ε = 8.4with mV good and εmax = 4.54%) TheLFP/C performance of the LFP/C battery estimates generally accuracy ( ε =the 8.4voltage mV and ε max = 4.54%) the voltage of the battery. Nevertheless, the model has the tendency to over-estimate The performance model of the LFP/C battery estimates generally with good accuracy ( ε = 8.4 mV and εmax = 4.54%) the voltagethe of the battery. the model has the tendency to over-estimate of the the voltage battery. Nevertheless, model has Nevertheless, the tendency to over-estimate the response of the response of the LFP/C sudden and large variation of voltage the load current have and εmax = 4.54%) the voltage of thebattery, battery.when Nevertheless, the model has the tendency to over-estimate the voltage response of the LFP/C battery, when of sudden andcurrent large variation of the load current have LFP/C battery, when sudden and large variation the load have occurred (i.e., changes ininthe occurred (i.e., changesofin current fromwhen belowsudden 1C-rateand to 4C-rate); this behavior is illustrated the voltage response thethe LFP/C battery, large variation of the load current have occurred (i.e., changes in the current from below 1C-rate to 4C-rate); this behavior is illustrated in current from below 1C-rate to 4C-rate); thisrelative behavior is illustrated in 27, where the distribution Figure 27,(i.e., where the distribution of the obtained byFigure comparing the is measured and occurred changes in the current from below error 1C-rate to 4C-rate); this behavior illustrated in Figure 27, where the distribution of the relative error obtained by comparing measured and of estimated the relative error obtained by comparing the measured and estimated voltage the profiles is plotted. voltage profiles is plotted. Figure 27, where the distribution of the relative error obtained by comparing the measured and estimated voltage profiles is plotted. estimated voltage profiles is plotted. 10 10 10

Current Current Current [A][A][A]

5 5 5 0 0 0 -5 -5 -5 -10 -100 -100 0

1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 Time4000 [s] 5000 6000 7000 1000 2000 3000 5000 6000 7000 Time 4000 [s]

Voltage Voltage Voltage [V][V][V]

Figure Field-measuredcurrent current profile used usedTime for [s] the the model. Figure 25.25. the verification verificationof theperformance performance model. Figure 25.Field-measured Field-measured currentprofile profile used for for the verification ofofthe performance model. Figure 25. Field-measured 3.6 current profile used for the verification of the performance model. 3.6 3.6 3.4 3.4 3.4 3.2 3.2 3.2 3 3 3 2.8 2.8 2.8 2.6 0 2.6 0 2.6 0

Measurement Measurement Simulation Measurement Simulation Simulation

3.3 3.3 3.2 3.3 3.2 3.1 3.2 3.1 4000 5000 6000 1000 2000 30003.14000 7000 40005000500060006000 Time 4000 [s]400050005000 1000 2000 3000 6000 6000 7000 Time 4000 [s] 1000 2000 3000 5000 6000 7000

Number of Occurances Number Occurances Number of of Occurances

Time [s] Figure 26. Measured and estimated voltage profiles obtained for the field-measured current profile at Figure 26. Measured and estimated voltage profiles obtained for the field-measured current profile at Figure Measured and estimated voltage profiles obtained for the thefield-measured field-measuredcurrent currentprofile profile TFigure = 2526. °C. 26. Measured and estimated voltage profiles obtained for at at T = 25 °C. ◦ C. T =T25 = 25 °C. 3000 3000 3000 2500 2500 2500 2000 2000 2000 1500 1500 1500 1000 1000 1000 500 500 500 0 0 0

-4 -4 -4

-2 0 2 [%] 2 -2Relative Error 0 [%] 2 -2Relative Error 0

4 4 4

Relative Error [%] the field-measured profile verification Figure 27. Distribution of the relative error obtained during Figure 27. Distribution of the relative error obtained during the field-measured profile verification case. Figure 27. Distribution of the relative error obtained during the field-measured profile verification Figure case. 27. Distribution of the relative error obtained during the field-measured profile verification case. case.

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The developed performance model is able to accurately estimate the voltage behavior of the tested LFP/C battery—regardless of the applied current profile—returning a maximum error lower than 5% and a mean error below 8.5 mV as summarized in Table 2. Table 2. Performance model accuracy for different load profiles. PFR: primary frequency regulation. Profile

Mean Error, |ε|

Maximum Error, εmax

R2

Pulse discharging Dynamic profile PFR profile

6.8 mV 6.4 mV 8.4 mV

3.66% 4.89% 4.54%

0.9917 0.9255 0.7724

6. Conclusions A seven-step characterization methodology for measuring the performance characteristics of the Li-ion batteries was proposed in the first part of this work. For exemplification of the procedure, a commercially available 2.5 Ah LFP/C battery was used. The results obtained during the characterization test were used to develop and parameterize the performance model of the LFP/C battery. In order to model the dynamic behavior of the battery, a novel hybrid procedure was proposed. Thus, the developed performance model combines information obtained from the electrochemical impedance spectroscopy (EIS) test (i.e., dependence of the impedance on SOC and temperature) and HPPC test (i.e., dependence on the load current), respectively. The proposed hybrid performance model was verified using different representative dynamic current profiles. The obtained results, maximum error lower than 5% and a mean error below 8.5 mV, have suggested that the developed model is able to estimate with high accuracy the voltage of the LFP/C battery, independent of the considered test conditions. Thus, it can be concluded that the proposed seven-step generalized characterization procedure for Li-ion batteries was validated. Even though the proposed characterization procedure was illustrated for a certain Li-ion battery chemistry (i.e., LFP/C), it can be applied to any type of Li-ion battery chemistry with the amendment that the values of the test conditions (i.e., temperature and load current levels) have to be adjusted in order to match the manufacturer data-sheets. Author Contributions: Daniel-Ioan Stroe and Maciej Swierczynski conceived, designed, and performed the experiments. Daniel-Ioan Stroe and Ana-Irina Stroe analyzed the data; Søren Knudsen Kær reviewed the paper; and Daniel-Ioan Stroe wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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