A Combined State of Charge Estimation Method for Lithium-ion Batteries Used in a Wide Ambient Temperature Range

Ambient temperature is a significant factor that influences the characteristics of lithium-ion batteries, which can produce adverse effects on state of charge (SOC) estimation. In this paper, an integrated SOC algorithm that combines an advanced ampere-hour counting (Adv Ah) method and multistate open-circuit voltage (multi OCV) method, denoted as " Adv Ah + multi OCV " , is proposed. Ah counting is a simple and general method for estimating SOC. However, the available capacity and coulombic efficiency in this method are influenced by the operating states of batteries, such as temperature and current, thereby causing SOC estimation errors. To address this problem, an enhanced Ah counting method that can alter the available capacity and coulombic efficiency according to temperature is proposed during the SOC calculation. Moreover, the battery SOCs between different temperatures can be mutually converted in accordance with the capacity loss. To compensate for the accumulating errors in Ah counting caused by the low precision of current sensors and lack of accurate initial SOC, the OCV method is used for calibration and as a complement. Given the variation of available capacities at different temperatures, rated/non-rated OCV–SOCs are established to estimate the initial SOCs in accordance with the Ah counting SOCs. Two dynamic tests, namely, constant-and alternated-temperature tests, are employed to verify the combined method at different temperatures. The results indicate that our method can provide effective and accurate SOC estimation at different ambient temperatures.


Introduction
Temperatures in many cities around the world, such as Salt Lake City in the US, Harbin in China, Moscow in Russia, and Vancouver in Canada, can reach below 0 °C in winter.In high-latitude and cold regions, temperatures can even reach −30 °C to −40 °C.To enable electric vehicles (pure electric vehicle (EV), plug-in hybrid electric vehicle (PHEV), and hybrid electric vehicle (HEV), which are collectively called xEVs) to work normally in these areas, the temperature-dependent parameters of energy storage systems should be suitable.Lithium-ion batteries are characterized by high specific energy, high efficiency, and long life.These unique properties have made lithium-ion batteries feasible power sources for xEVs.However, lithium-ion battery technology for xEV applications still has many disadvantages, such as its narrow operational temperature range [1].
Given the high complexity of advanced algorithms, the reliability and robustness of these methods are challenged.In practice, the basic algorithms are more feasible.For batteries in pure EVs, the working conditions (driving, rest, and charging) are simple.With vehicle movement and little braking regeneration, the battery SOC may fluctuate but mostly decreases; this phenomenon is defined as the charge-depleting mode.By contrast, the battery SOC increases with the accumulation of charge when the vehicles are charged.The cells are monotonously discharged or charged, so OCV moves along with the OCV boundary curves.The hysteresis of the OCV is easy to eliminate [30].Therefore, the Ah counting method with the initial SOC correction according to the OCV method could meet the precision requirements of SOC estimation of batteries for pure EVs.For batteries in HEVs, the working conditions are relatively complex.When the vehicles are moving, the current is both charged and discharged to keep the battery SOC within a narrow range, which is defined as charge-sustaining mode.However, the OCV transits between the OCV boundary curves as a consequence of non-monotone loading, so estimating the hysteresis of the OCV is difficult.Therefore, an accurate hysteresis model should be established to determine the OCV of batteries [30].Like EVs, the Ah counting method calibrated with the use of the precisely estimated OCV satisfies the requirements to a certain degree.For batteries in PHEVs, the batteries typically operate in either of the two modes: charge-depleting mode of a pure EV and the shallow, charge-sustaining mode of an HEV [31].Therefore, the algorithm for either pure EVs or HEVs can be employed on the basis of the two modes.
However, several existing problems of Ah counting and OCV methods, such as the available capacity decrease at low temperature, which directly influences the accuracy of SOC estimation, are seldom addressed [32].In addition, the coulombic efficiency is not only a function of current, but is also effected by temperature, thus, the variation of coulombic efficiency at different currents and different temperatures (different conditions) should be considered [3].The temperature dependence of the OCV-SOC is also seldom considered in the initial SOC estimation.The OCV-SOC constructed at a certain temperature (e.g., room temperature) is employed to determine the initial SOC.As a result, a large error is obtained when the battery is rested at other temperatures (i.e., not at room temperature) [20,21].
The current study proposes a correction-integrated algorithm that mainly includes an enhanced Ah counting method calibrated with the use of a multistate OCV method.In the Ah counting method, the available capacity and coulombic efficiency at different conditions are the primary factors.In Section 2, a testing method of available capacity and capacity loss at different temperatures are presented.We then discuss the current factor that influences the available capacity.In Section 3, given the variable energy losses at different conditions, the coulombic efficiency and equivalent coulombic efficiency are considered.In addition, the testing method of coulombic efficiency is introduced, and the calculation process of equivalent coulombic efficiency based on the coulombic efficiency is illustrated.The influence of current and temperature on the coulombic efficiency is also discussed.In Section 4, we provide the definition of the rated/non-rated SOC, which is applied to the batteries with/without a thermal management system (TMS).The Ah counting calculation of the rated/non-rated SOC is developed to satisfy the requirement for the applications at different temperatures.For the OCV method (Section 5) corresponding to the rated/non-rated SOC, the rated/non-rated OCV-SOCs are established to estimate the rated/non-rated initial SOC.To estimate the non-rated initial SOC under different temperature paths, we establish the R-L (from room temperature to low temperature) and the L-L (from low temperature to low temperature) non-rated OCV-SOCs.Finally, two dynamic tests, namely, constant-temperature test and alternated-temperature test, are employed to verify the method at different temperatures.

Available Capacity and Capacity Loss
Battery capacity is sensitive to current and temperature.Therefore, current and temperature values must be specified in the capacity definition.For example, the capacity, which is discharged at rated current R I and rated temperature R T , is denoted as T = 10, 0, −10, −20 °C) are defined as the non-rated condition.The definition and calculation of SOC are closely related to the capacity, which we will explain in a subsequent section.

Experiment Setup
The test bench setup (Figure 1) consists of the following: (1) rectangular lithium-ion cells (LiFePO 4 , nominal voltage 3.3 V, nominal capacity 5 Ah, and upper/lower cut-off voltage 3.6/2.5V.); (2) a thermal test chamber for environment, which has a temperature operation range between −55 and 150 °C; (3) a battery test system (Arbin BT2000 tester, Arbin, college town, TX, USA, which has a maximum voltage of 5 V and maximum charging/discharging current of ±200 A; the measurement inaccuracy of the current and voltage transducer inside the Arbin BT2000 system is within 0.1%); and (4) a PC with Arbins' Mits Pro Software for battery charging/discharging control.The Arbin BT2000 is connected to the battery cell placed inside the thermal chamber to maintain the temperature.The measured data are transmitted to the host computer through TCP/IP ports.

Available Capacity Test at Different Temperatures
Given the temperature dependence of the capacity, the test is conducted from −20 to 20 °C at 10 °C intervals.The test procedures designed by many battery test manuals [33,34] at each temperature are as follows: (1) the cell is fully charged using a constant current of 1/3C rate until the voltage reaches to the upper cut-off voltage of 3.65 V at 20 °C; (2) the cell ambient temperature is decreased to the target temperature T; (3) a suitable soak period is employed for thermal equalization; and (4) the cell is fully discharged at a constant current of 1/3C rate until the voltage reaches the bottom cut-off voltage of 2.5 V. Figure 2 shows the discharge voltage curves at different temperatures.From −20 to 20 °C at an interval of 10 °C, the discharged capacities are 78, 84, 94, 98, and 100 Ah.In addition, the ohmic resistance R is most significantly increased as the temperature decreases.Thus, we may conclude that the poor performance of the lithium-ion batteries at low temperature originates from the substantially higher R, which can be further ascribed to the slow kinetics of the electrode reactions [32].The purpose of the above test procedures is to verify the sustainable performance of capacities at different temperatures.Among these procedures, step ( 1) is implemented at room temperature, which is different from the ambient temperature in real vehicle applications.The operation temperature of the batteries without TMS in real vehicles varies along the ambient temperature.Figure 3 shows the temperature of Harbin in China during a regular day in winter [35].The highest, lowest, and average temperatures that day are −6, −13, and −8.8 °C, respectively.Both charging and discharging processes in real vehicles occur at low temperature.Therefore, the test procedures of available capacity at each temperature are redesigned as follows: (1) the cell ambient temperature is decreased to the target temperature T; (2) a suitable soak period is employed for thermal equalization; (3) the cell is fully charged using a constant current of 1/3C rate until the voltage reaches the upper cut-off voltage of 3.65 V at T, and then a 1 h rest is implemented; (4) finally, the cell is fully discharged at a constant current of 1/3C until the voltage reaches the bottom cut-off voltage of 2.5 V, wherein the available capacity of the cell is the number of ampere-hours that can be drawn from the battery.Steps 1 to 4 are repeated three times.If the error of the experimental results between the maximum and the average is within 2%, then the available capacity test is effective and the average value is considered as the available capacity; however, if the error is >2%, the available capacity test should be repeated.The results of available capacity at each temperature are shown in Table 1.Although the battery is charged/discharged until the voltage reaches the same upper/bottom cut-off voltage at different ambient temperatures, the releasable capacity at each temperature varies.The capacity loss test is used to measure the difference of releasable capacities between two different temperatures.The difference between releasable capacities with fully charged battery at 1 T and 2 T is defined as the loss of full charge ( T T LFC − ).Likewise, the difference between releasable capacities with fully discharged battery at 1 T and 2 T is defined as the loss of full discharge ( The test procedures of capacity loss at each temperature are as follows: (1) the cell is fully charged using a constant current of 1/3C until the voltage reaches the upper cut-off voltage of 3.65 V at 1 T ; (2) the cell ambient temperature is decreased to 2 T , the cell is fully discharged at a constant current of 1/3C until the voltage reaches the bottom cut-off voltage of 2.5 V, and then the discharged capacity ChaT DisT Q − is recorded; (5) the cell ambient temperature is then increased to 1 T , and the cell is fully charged using a constant current of 1/3C until the voltage reaches the upper cut-off voltage of 3.65 V; and (6) the cell is fully discharged at a constant rate of 1/3C until the voltage reaches the bottom cut-off voltage of 2.5 V and the discharged capacity  T T LFC − is calculated as follows: T T LFD − is calculated as follows: The results of LFD, LFC, and available capacity at each temperature with respect to the 20 °C 1/3C rate are presented in Table 2.The percentages of LFD, LFC, and available capacity at different temperatures are shown in Figure 5. Table 2 and Figure 5 show that LFD and LFC are different at the same temperature, in which LFD > LFC.Therefore, both the SOC definition and calculation at different temperatures should be redefined and recalibrated, corresponding to LFD, LFC, and available capacity.

Capacity with Different Currents
Current also influences capacity.A capacity test with different currents is conducted at C/3, C/2, 1C, 1.5C, and 2C rate.The capacities at C/3, C/2, 1C, 1.5C, and 2C (denoted as C 3 , C 2 , C, C 2/3 , and C 1/2 , respectively) are measured at the same temperature (20 °C).The test procedures at each rate are as follows: (1) the cell is fully charged using a constant current rate of C/3 until the voltage reaches the upper cut-off voltage of 3.65 V; (2) 1 h rest is implemented; (3) the cell is fully discharged at a constant current rate of C/3, C/2, 1C, 1.5C, or 2C until the voltage reaches the bottom cut-off voltage of 2.5 V; and (4) then another rest for 1 h. Figure 6 shows the discharge curves at different current rates.Less than 3% of the C 1/2 capacity is not accessible with respect to the C 3 capacity.Compared with the temperature effect, the influence of current on capacity is relatively small.Moreover, in real vehicle applications, the battery pack is charged with a suitable current, as recommended by the manufacturer.Although a sophisticated dynamic current profile is run on the battery pack when the vehicle is working, the large current is limited by BMS within a reasonable range.To satisfy the requirement of power and energy for xEVs, a suitable battery pack size is selected during the design phase.Thus, the battery pack is mainly operated in the high-efficiency area and the large current does not last for a long time.In addition, the influence of current on capacity is temporarily neglected in SOC estimation.

Coulombic Efficiency
Coulombic efficiency is another important parameter of SOC estimation.Battery coulombic efficiency, like capacity, is sensitive to current and temperature.Therefore, current and temperature must also be specified in the definition of coulombic efficiency.In the last section, capacity losses at different currents and temperatures mainly involve changes in the thermodynamic and in the kinetic aspects of a battery [4].Unlike the capacity loss, the coulombic efficiency is mainly caused by the energy loss that occurs during charging and discharging.

Definition of Coulombic Efficiency
Coulombic efficiency is the ratio of the Ah removed from a battery during discharging to the Ah required to restore the battery to the SOC before discharging [33].The coulombic efficiency is defined as: where charge Q is the number of charges that enter the battery during charging and discharge Q is the number of charges extracted from the battery during discharging.
, which is primarily caused by the energy loss along with the charging/discharging process.The energy loss leads to the variation in charging/discharging time, which results in the variation in charging/discharging capacity.The energy loss during charging/discharging is mostly thermal loss.The thermal generation factors are decomposed into three elements, namely, reaction heat value, polarization heat value, and Joule heat value, which vary according to the variations in current and temperature [36].
Energy loss also depends on current and temperature.Therefore, the same number of charge that enters/exits the battery during charging/discharging at different currents and temperatures need different amounts of energy.The equivalent charge/discharge coulombic efficiency is defined as the ratio of the charged/discharged capacity at the non-rated condition to the charged/discharged capacity at the rated condition.The equivalent charge coulombic efficiency is given as: Q is the number of charges extracted from the battery at non-rated condition.

Calculation of Coulombic Efficiency
According to the different currents and temperatures running  3).In Table 3, the charge capacities at different conditions are in the first row and the discharge capacities at different conditions are in the first line.The coulombic efficiencies at different conditions are in the cross location.( , )( , ) , , Q .The calculation method of other coulombic efficiencies in Table 3 is similar to that of the rated coulombic efficiency.
According to the definition of equivalent charge/discharge coulombic efficiency, we can extend it to a more comprehensive equivalent charge/discharge coulombic efficiency, which is not only between non-rated condition and rated condition but also between different conditions.According to the different currents and temperatures running Q charge and Q discharge for various possible configurations, 10 kinds of equivalent charge/discharge coulombic efficiencies can be obtained (Tables 5 and 6).In these two tables, the standard charge/discharge capacity (denominator in equivalent charge/discharge coulombic efficiencies) is on the first row and the non-standard charge/discharge capacity (numerator in equivalent charge/discharge coulombic efficiencies) is on the first line.Correspondingly, the equivalent charge/discharge coulombic efficiencies between different conditions are in the cross location.
For example, the charge capacity at non-rated condition ,  ( , )( , ) , , The equivalent charge/discharge coulombic efficiency cannot be directly measured through experiment.Therefore, we transform , ( , )( , ) , , are coulombic efficiencies that can be calculated through experiment.

Dis Cha Dis Cha Cha I T I T I T I T I T Cha Equ I T I T Cha Dis Cha Cha Dis I T I T I T I T I
The equivalent charge/discharge coulombic efficiency is then calculated using the indirect method.
The calculation method of other coulombic efficiencies in Tables 4 and 5 is similar to that for Table 4. Equivalent charge coulombic efficiency at different conditions.

Influence of Current and Temperature on Coulombic Efficiency
Figure 7 shows the influence of current (C/3, C/2, 1C, 1.5C, and 2C) and temperature (20, 10, 0, −10, and −20 °C) on coulombic efficiency.Figure 7 shows that the current has subtle effects on coulombic efficiency.At 20 °C, the coulombic efficiency at C/3 and 2C is 0.999 and 0.998, respectively.In real vehicle applications, although the loading current is a sophisticated dynamic profile, the duration of the large current is short.Therefore, the influence of the current on the coulombic efficiency is ignored in later sections.However, temperature has more significant effects on coulombic efficiency than current (Figure 6).Under C/3 rate, the coulombic efficiency at 20 and −20 °C are 0.999 and 0.96, respectively.Therefore, the influence of temperature on coulombic efficiency should be considered for the SOC estimation.

SOC Estimation
On the basis of the analysis in Section 2, the available capacity is influenced by current and temperature.Therefore, with regard to the variation in available capacity, SOC should be redefined.In this section, we provide the definition of the rated/non-rated SOC according to the available capacity at the rated and non-rated condition.The calculation of the rated/non-rated SOC is then developed to meet the applications at different conditions. is the releasable capacity when the operating battery is completely discharged at the rated condition.Accordingly, the rated SOC is defined as the percentage of the rated releasable capacity relative to the rated capacity: The rated SOC is based on the rated capacity that is unaffected by current and temperature.On the basis of the analysis in Section 2, the influence of current on available capacity is neglected.Therefore, the rated SOC is applied to the batteries with TMS, which can maintain the temperature of batteries in the rated temperature range.

Definition of the Non-Rated SOC
The non-rated capacity , I T C is the capacity when the fully charged battery is completely discharged at non-rated condition.By contrast, the non-rated releasable capacity , releasable I T

C
is the releasable capacity when the operating battery is completely discharged at the non-rated condition.The non-rated SOC is a relative quantity that describes the ratio of the non-rated releasable capacity to the non-rated capacity of the battery.The non-rated SOC is given as: , , , Compared with the rated SOC, the non-rated SOC is based on the non-rated capacity influenced by current and temperature.Therefore, the non-rated SOC is applied to the batteries without TMS.In this case, the temperature of batteries changes with the ambient temperature.

Calculation of the Rated SOC
The rated SOC is based on the rated capacity

C
. However, real vehicle tests show a sophisticated dynamic current profile with different charge and discharge currents.Therefore, the charged and discharged capacity at different conditions should be converted into the discharged capacity at rated condition.In the charging process, we use the coulombic efficiency , ( , )( , ) , , to convert the non-rated charged capacity into the rated discharged capacity.
In the discharging process, the equivalent coulombic efficiency , ( , )( , ) , , is used to convert the different conditions of discharge capacity into the rated discharged capacity.With a measured charging/discharging current I and the corresponding coulombic efficiency, the rated SOC in an operating period t Δ can be calculated using Equation ( 9): , , When the rated SOC is calculated using Equation ( 9), the coulombic efficiency

Cha Dis I T I T I T Dis Equ I T I T i t i t
where i(t) > 0 is charging and i(t) < 0 is discharging.On the basis of the analysis in Section 3.4, the influence of current on coulombic efficiency is neglected.Therefore, the calculation of coulombic efficiency

Cha Dis I T I T I T Dis Equ I T I T i t i t
Figure 8 shows the charge and discharge coulombic efficiency of , η R R I T , which is measured using the method introduced in Section 3.2.to convert the charged capacity under different conditions into the discharged capacity under the corresponding condition.In the discharging process, given that the discharged capacity is under the current operating condition, the equivalent coulombic efficiency is , ( , )( , ) η 1 Dis Equ I T I T = .With a measured charging/discharging current I and the corresponding coulombic efficiency, the non-rated SOC in an operating period t Δ can be calculated using: ( 1) η When the non-rated SOC is calculated using Equation ( 12), the coulombic efficiency , η I T is as follows: On the basis of the analysis in Section 2.2, the influence of current on capacity is negligible.Therefore, the non-rated SOC is based on in real vehicle application.Based on Section 3.4, the influence of current on coulombic efficiency is also negligible.In addition, the calculation of coulombic efficiency , η I T is based on the rated current R I .The coulombic efficiency , Figure 9 shows the charge and discharge coulombic efficiencies of , η I T , which is measured using the method introduced in Section 3.2.In addition to the influence of temperature on coulombic efficiency, the variation in available capacity at different temperatures should be considered in calculating the non-rated SOC.Although the battery has the same releasable capacities, SOCs are different at different temperatures.Therefore, SOCs between different temperatures should be converted depending on the available capacity and capacity loss.According to Equation ( 12), , ( ) I T SOC t − is the SOC converted from the last temperature to the current temperature.The conversion procedure of SOC between different temperatures is as follows: The battery soaked in the temperature of 1 T and 2 T (supposing 1 T > 2 T ) has the same releasable capacities (Figure 10).At the two temperatures, the corresponding SOCs are T, the releasable capacity is: At temperature 2 T , the releasable capacity is: Given that the releasable capacities are the same at 1 T and 2 T , i.e., , then: When the battery is cooled from 1

Initial SOC Estimation by Multistate OCV
SOC is related to the embedding quantity of lithium-ion in the active material and with static thermodynamics.Therefore, the OCV after adequate resting can be considered to reach balanced potential because one-to-one correspondence exists between OCV and SOC and bear little relation to the service life of batteries; it is also an effective method to estimate SOC of lithium-ion batteries [21,37].The initial SOC in Ah counting method can be revised using the OCV method.
Corresponding to the rated/non-rated SOC in the last section, the rated/non-rated OCV-SOCs are established to estimate the initial SOC.To estimate the non-rated initial SOC under different temperature paths, we further establish the R-L non-rated OCV-SOCs and the L-L non-rated OCV-SOCs.SOC estimation by the above OCV-SOCs at different conditions is defined as multistate OCV method.

Rated OCV-SOCs
The rated OCV-SOCs are established to estimate the rated initial SOC.The relationship between SOC and OCV should be based on the rated capacity The rated SOC is generally applied to batteries with TMS.Although TMS exist in battery packs, vehicles also experience cold cranking in winter.In the cold cranking process, the TMS cannot warm the battery on time, which causes a low internal temperature of the battery.Given the OCV thermosensitivity, gaps exist in the OCV-SOCs at different temperatures.In addition, OCV exhibits pronounced hysteresis phenomena, resulting in the non-overlapped charged and discharged OCV-SOC [30].In the current paper, we only introduce the charged OCV test procedures, and the discharged test is the inverse process of the charged test.
The rated OCV-SOC test procedures are as follows: (1) the cell is fully discharged using a constant current of 1/3C rate until the voltage reaches the cut-off voltage of 2.5 V at 20 °C; (2) After a suitable period of rest (generally more than 3 h), the measured OCV is at SOC = 0%, which is denoted as  3) are repeatedly performed until SOC = 0% to 100% at an interval of 5% has been achieved.Five charged and discharged OCV-SOCs are obtained from −20 °C to 20 °C at an interval of 10 °C. Figure 11 emphasizes the charged and discharged OCV-SOCs between 10% and 90% SOC at different temperatures.The OCVs at the same SOC have lower temperature.The rated OCV-SOC at 1 T is described using: The rated OCV-SOCs can be obtained using the rated OCV-SOC at different temperatures with the fitting method: R R I T rated rated rated rated

SOC OCV T AT A T A OCV A T A T A OCV A T A T A OCV A T A T A
As shown in Figure 10, the charged OCV-SOCs are higher than the discharged OCV-SOCs, indicating that hysteresis phenomenon of the OCV occurs during charging/discharging.The SOC based on the charged OCV-SOC is smaller than that based on the discharged OCV-SOC at the same temperature.Therefore, the effect of the hysteresis should not be ignored.If the battery is charged before rest, then the charged OCV-SOCs are used to estimate the initial SOC; otherwise, the discharged OCV-SOCs are used.In the later sections, we deal with the OCV hysteresis in the same way.

Non-Rated OCV-SOCs
Compared with the rated OCV-SOCs, the non-rated OCV-SOCs are established to estimate the non-rated initial SOC.The non-rated SOC is based on , I T C , which is significantly influenced by temperature.To ensure SOC consistency between Ah SOC and OCV SOC , the effects of temperature on the available capacity and capacity loss should be considered in establishing the non-rated OCV-SOCs.
The non-rated SOC is generally applied to the batteries without TMS.In real vehicle applications, the battery pack without TMS experiences the following two cases of temperature variation: (1) the vehicle works at room temperature during daytime and rests in the evening.Given the large temperature difference during the day, the temperature sharply decreases in the evening.The next morning, the vehicle is restarted at low temperature; (2) The vehicle is operated at low temperature all the time.We define the above two cases as the different temperature paths: (1) From room temperature to low temperature (R-L) and ( 2) from low temperature to low temperature (L-L).
OCV is affected by the different temperature paths that cause errors in SOC estimation.Figure 11 shows the OCV-SOCs at 0 °C in different temperature paths.The R-L charged and discharged OCV-SOCs are measured as 4.2.1 test procedures.However, the L-L charged and discharged OCV-SOCs are measured as follows: (1) the cell is fully discharged using a constant current of 1/3C rate until the voltage reaches the cut-off voltage of 2.5 V at 20 °C; (2) the cell ambient temperature is decreased to the target temperature 0 °C and a suitable soak period is implemented for thermal equalization; (3) the cell is then charged at a constant current of C/3 rate until the Ah reaches , / 20 A suitable period is implemented until it returns to the equilibrium state.
Step (3) is repeatedly performed until the battery reaches the upper cut-off voltage of 3.65 V.The voltages during each rest period are recorded to establish the L-L OCV-SOCs.As shown in Figure 12, the OCV-SOCs in different temperature paths are different even at the same temperature.The charge/discharge OCVs decrease/increase with the increase in ambient temperature, and this phenomenon results from the entropy change [38].To estimate the initial SOC in different temperature paths, we establish the R-L non-rated OCV-SOCs and the L-L non-rated OCV-SOCs, respectively.

R-L Non-Rated OCV-SOCs
The R-L non-rated OCV-SOCs are converted using the rated OCV-SOCs according to the available capacity and capacity loss at different temperatures.Considering the rated OCV-SOC at 1 T as an example to demonstrate the process, the detailed procedures are in Figure 13.We assume that the rated OCV-SOC at R   Figure 14 shows the charged and discharged R-L non-rated OCV-SOCs.The R-L non-rated OCV-SOC at 1 T is described by: The R-L non-rated OCV-SOCs can be obtained using the R-L non-rated OCV-SOC at different temperatures with the use of the fitting method:  T as an example, the L-L non-rated OCV-SOC test procedures at each temperature are as follows: (1) the cell is soaked in the target temperature 1

SOC OCV A T A T A OCV A T A T A OCV A T A T A OCV A T A T A
T for a suitable period for thermal equalization, and then fully discharged using a constant current of 1/3C rate until the voltage reaches the cut-off voltage of 2.5 V; (2) after a suitable period of rest, the measured OCV is at SOC = 0%, which is denoted as Step ( 3) is repeatedly performed until SOC = 0% to 100% at an interval of 5% is achieved.
Figure 15 shows the charged and discharged L-L non-rated OCV-SOCs with respect to the capacity.The start and end points on X axis of OCV-SOC at different temperatures correspond to T is described by: The L-L non-rated OCV-SOCs can be obtained using the L-L non-rated OCV-SOC at different temperatures with the fitting method:

Experimental Results for SOC Estimation
According to whether the battery pack is with/without TMS, we calculate the rated SOC/non-rated SOC, respectively.The calculation procedures of the rated SOC are as follows.When the vehicle is started, the BMS measures the temperature of the battery pack.According to the measured temperature, we propose the application of the rated OCV-SOCs instead of the conventional OCV-SOC, which is often established at room temperature, to estimate the rated initial SOC.The other parameters in the rated SOC algorithm, such as C , can be determined according to the rated condition.For the non-rated SOC, the battery pack temperature is also measured and saved in the BMS.By comparing the current temperature and the previous temperature stored in the memory of BMS, we could obtain the temperature path.According to the temperature path R-L or L-L, the corresponding non-rated OCV-SOCs are selected to estimate the non-rated initial SOC.In the process of calculation, , ( ) A validation test with a sophisticated dynamic current profile, the federal urban driving schedule (FUDS) is conducted to verify the SOC estimation algorithm.In the laboratory test, a dynamic current sequence is transferred from the FUDS time-velocity profile.The current sequence is then scaled to fit the specification of the test battery.A completed FUDS current profile over 1372 s is emphasized in Figure 18.FUDS tests are conducted at different ambient temperatures to emulate the operation conditions.The validation tests are separately implemented under constant and alternated temperatures.Figure 19a shows the voltage curves of constant temperature test at −10, 0, and 20 °C.Although the same Ah is charged before nine FUDS cycles (point 1), the releasable capacity at −10 °C or 0 °C is <20 °C in step (3).This phenomenon occurs because the available capacity decreases at low temperature.Figure 19b shows the SOC curves of constant temperature test at −10, 0, and 20 °C.The cell reaches point 1 through the R-L temperature path.The R-L non-rated initial SOCs at point 1 are estimated using the R-L non-rated OCV-SOCs.7.After nine FUDS cycles and before discharging (point 2), the cell undergoes the L-L temperature path.According to the L-L non-rated OCV-SOCs, we can estimate the L-L non-rated initial SOCs.Considering the truth values of SOC at point 1 as the initial SOCs, the non-rated SOCs can be calculated using Equation (12) during the nine FUDS cycles.Finally, the truth values of SOC at point 2 are measured using the discharge test method, by which the errors of L-L non-rated initial SOCs and non-rated SOCs can be calculated.The results at point 2 are reported in Table 8.
As shown in Tables 7 and 8, the errors of initial SOCs in different temperature paths are <3%.The calculation of non-rated SOCs during nine FUDS cycles produces subtle errors caused by the neglect of the influence of current on coulombic efficiency.
Table 7.The R-L non-rated initial SOC and SOC error at 20, 0, and −10 °C (point 1).(2) nine FUDS cycles are loaded to the cell, and the cell is discharged using the constant current of C/3 rate until bottom cut-off voltage is reached; (3) the cell is soaked at 20 °C for 12 h; and (4) additional two FUDS cycles are loaded to the cell, and then the cell is discharged using the constant current of C/3 rate until bottom cut-off voltage is reached.
In the alternated temperature test, voltage, temperature, and SOC are depicted in Figure 20. Figure 20a shows the voltage curves of alternated temperature test.The cell has been discharged completely at −10 °C.However, the cell can be still discharged after 12 h at 20 °C due to the increase in available capacity. Figure 20b shows the temperature curves of alternated temperature test.The ambient temperature increases from −10 °C to 20 °C during the test.Figure 20c shows the SOC curves of alternated temperature test.After nine FUDS cycles, the cell is discharged and emptied at −10 C. The SOC at this moment is

Conclusions
Ambient temperature significantly influences the characteristics of lithium-ion batteries, such as capacity, coulombic efficiency, and OCV.Such temperature effects cause direct errors in SOC estimation.In this paper, we propose a combined SOC algorithm to address the temperature dependence of battery characteristics.First, our method simply and effectively improves the accuracy of the estimated SOC for lithium-ion batteries at different ambient temperatures.With minimal calculations this method can be used in BMS for on-board estimation.Second, the data of battery characteristics are obtained using the uncomplicated battery test from −20 to 20 °C, with a temperature interval of 10 °C.If the method is flexible in a wider temperature range or higher temperature resolution, the tests are only implemented under the corresponding temperature.Finally, two dynamic loading tests are conducted on the battery under constant and alternated temperatures to assess the SOC estimation performance using the proposed approach.The results indicate that the rated/non-rated initial SOC estimation based on the rated/non-rated OCV-SOCs in different temperature paths provides accurate values to calibrate the SOC estimated by Ah counting.In addition, the conversion of SOC between different temperatures exhibits high accuracy.The two SOC algorithms at different ambient temperatures have good consistency.Thus, this approach could be used in actual vehicle applications.Further studies on the following two aspects are recommended.First, advanced algorithms with the battery model under different temperatures can be applied to estimate on-line, real-time, and closed-loop SOC.Second, if this method is to be developed for SOC estimation of battery packs, other problems, such as the variation of cells, should be considered.
and R T (in this paper, R I = 1/3C, and R T = 20 °C) are defined as the rated conditions.The current I and temperature T ( I = C/2, 1C, 1.5C, 2C;

Figure 1 .
Figure 1.Schematic of the battery test bench.

Figure 2 .
Figure 2. Discharge voltage curves at different temperatures.

Figure 3 .
Figure 3. Temperature of Harbin (China) one day in winter.
recorded; (3) the cell is fully charged using a constant current of 1/3C until the voltage reaches the upper cut-off voltage of 3.65 V; (4) the cell is fully discharged at a constant current of 1/3C until the voltage reaches the bottom cut-off voltage of 2.5 V and the charged capacity 2 2 recorded.The test process of capacity loss is shown in Figure4.

Figure 4 .
Figure 4. Test process of capacity loss at different temperatures.

Figure 5 .
Figure 5.The percentage of LFD, LFC, and available capacity at different temperatures.

Figure 6 .
Figure 6.Discharged curves at different current rates.
of charges that enter the battery at rated condition and , Cha I T Q is the number of charges that enter the battery at non-rated condition.The equivalent discharge coulombic efficiency is given as: of charges extracted from the battery at rated condition and , Dis I T configurations, 16 kinds of coulombic efficiency can be obtained (Table For example, the charge capacity at rated condition , first line.Consequently, the cross location of the first row and the first line is the rated coulombic efficiency ,

Q
is on the first row and the charge capacity at rated condition , first line.Consequently, the cross location of first row and first line is the rated equivalent charge coulombic efficiency ,

Figure 7 .
Figure 7. Influence of current and temperature on coulombic efficiency.

4. 1 .
Definition of SOC 4.1.1.Definition of the Rated SOC The rated capacity , when the fully charged battery is completely discharged at the rated condition.The rated releasable capacity ,

Figure 9 .
Figure 9.The coulombic efficiency , I T η at different temperatures.

Figure 10 .
Figure 10.The conversion of SOC between different temperatures.
guarantees that SOCs estimated by Ah counting method ( Ah SOC ) and OCV method ( OCV SOC ) have good consistency.
is then soaked at 10 °C for a suitable period, and the measured measurement is conducted from −20 °C to 20 °C at an interval of 10 °C.After the OCV measurement, the cell is again soaked at 20 °C for a suitable period; (3) the cell is then charged at a constant current of C/3 rate until the charged Ah reaches

Figure 11 .
Figure 11.Rated charged and discharged OCV-SOCs at different temperatures.

Figure 12 .
Figure 12.Charged and discharged OCV-SOCs in different temperature paths.

Figure 13 .
Figure 13.Process of conversion from the rated OCV-SOCs to the R-L non-rated OCV-SOCs.

Figure 14 .
Figure 14.Charged and discharge R-L non-rated OCV-SOCs at different temperatures.
the cell is then charged at a constant current of C/ that the OCV SOC estimated by L-L non-rated OCV-SOCs and Ah SOC are consistent.Figure16shows the charged and discharged L-L non-rated OCV-SOCs with respect to SOC.The L-L non-rated OCV-SOC at 1

Figure 15 .
Figure 15.Charged and discharged L-L non-rated OCV-SOCs with respect to capacity.

Figure 16 .
Figure 16.Charged and discharged L-L non-rated OCV-SOCs with respect to SOC.
is the SOC converted from the last temperature to the current temperature.Meanwhile, the other parameters in the non-rated SOC algorithm, such as , employed according to the measured ambient temperature.A flow chart based on the developed method is shown in Figure17.

Figure 17 .
Figure 17.Chart of battery SOC estimation at different ambient temperatures.
test is conducted before the constant temperature test.The cell is charged using the constant current of C/3 rate at R T until Ah reaches , cell is then soaked at target temperature T for a suitable period.The test procedures of constant temperature test at each target temperature (−10, 0, and 20 °C) are as follows: (1) pre-test; (2) nine FUDS cycles are loaded to the cell, followed by 12 h rest; (3) the cell is discharged using the constant current of C/3 rate until bottom cut-off voltage is reached.

..
18).Before the two FUDS cycles, the measured OCV at 20 °C is According to the rated OCV-SOCs, we can estimate the rated initial SOC at Finally, the true value of SOC is 21.7%, which is measured by Ah counting until the cell is discharged completely.The errors of rated initial SOC and rated SOC are both <2%.

Table 1 .
Available capacity at different temperatures.

Table 2 .
LFD, LFC, and available capacity at different temperatures with respect to the 20 °C 1/3C rate.

Table 3 .
Coulombic efficiency at different conditions.

Table 5 .
Equivalent discharge coulombic efficiency at different conditions.
The alternated temperature test is used to validate the performance of the Ah counting ( under alternated temperature condition.The procedures of alternated temperature test are as follows: (1) pre-test (the same as the −10 °C pre-test at constant temperature); OCV SOC ) estimation