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Fig. 1. (a) Battery with inserted and surface mounted thermocouples and (b) exper- imental device conditions around that SOC that the battery exhibits the maximum charge/discharge power). Of course, one could reproduce the same tests at other average SOCs in order to get the SOC-dependence of the model parameters (Cy and R;,). A 2 Hz frequency was selected from an analysis of the impedance spectra of the battery, which shows that this frequency is intermediate between the low fre- quency of the interfacial part and the high frequency of the diffusion part. Output voltages from both thermocouples were recorded using a digital scope (Yokogawa DL716, Fig. 1b) and further con- verted to temperatures. Fig. 2 displays the experimental surface and internal temperature data measurements when current pulses of different magnitudes (+10, +15, and +20A) are applied during one hour (0-3600s in the plots, Fig. 2), after which the current is turned off to allow for temperature relaxation. From the Fig. 2, it is seen that a thermal steady state has been reached after around an hour under current pulses. Furthermore, whatever the magnitude of the current pulses, the temperature dynamics seem to be simi- lar, with a non-zero slope at initial time indicative of a first-order behavior. — Figure 1 (a) Battery with inserted and surface mounted thermocouples and (b) exper- imental device conditions around that SOC that the battery exhibits the maximum charge/discharge power). Of course, one could reproduce the same tests at other average SOCs in order to get the SOC-dependence of the model parameters (Cy and R;,). A 2 Hz frequency was selected from an analysis of the impedance spectra of the battery, which shows that this frequency is intermediate between the low fre- quency of the interfacial part and the high frequency of the diffusion part. Output voltages from both thermocouples were recorded using a digital scope (Yokogawa DL716, Fig. 1b) and further con- verted to temperatures. Fig. 2 displays the experimental surface and internal temperature data measurements when current pulses of different magnitudes (+10, +15, and +20A) are applied during one hour (0-3600s in the plots, Fig. 2), after which the current is turned off to allow for temperature relaxation. From the Fig. 2, it is seen that a thermal steady state has been reached after around an hour under current pulses. Furthermore, whatever the magnitude of the current pulses, the temperature dynamics seem to be simi- lar, with a non-zero slope at initial time indicative of a first-order behavior.

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Fig. 2. Temperature measurements inside and on the surface for current pulses of magnitude +10, +15, and +20A.

Fig. 3. Example of the voltage variation during the thermal cycle for SOC 20%

Fig.4. (a) Temperature coefficient 0U/dT and (b) equilibrium potential at 25 °C U5 0c and enthalpy potential Uy as a function of SOC.

‘ig. 5. Evolution of (V — U)I (blue) and corresponding smoothed curve (red). (For interpretation of the references to color in this figure legend, the reader is referred to th veb version of the article. The main limitation of using such a detailed lumped model that there are a lot of parameters to be determined, which usua requires additional experimental measurements. Since the goal this work is to build a model that is able to get an estimate of t internal temperature of the battery from a simple measurement is ly of he of the lateral surface temperature, simplifications to the equiva- lent circuit of Fig. 6a can be made. Under the assumption that t surface temperature of the battery is uniform (i.e. the surface tem- perature of the terminals is close to the lateral temperature), t ne ne Fig. 4 presents the variation of JU/0T with the SOC, together with that of the equilibrium potential U and the enthalpy potential Uy. The temperature coefficient is negative up to SOC 35%, and becomes positive for higher SOC values. It is interesting to compare U and Uy: while U is monotonic and increases with the SOC as expected, Uy is not. Furthermore, the dependence of dU/dT and Uy with the SOC presents some similitudes with that of graphite/lithium cells [10,12], which is not surprising since in most of the composition ranges explored here, the positive electrode LiFePO, undergoes a two-phase process and therefore no variation of 0U/0T (and Uy) are expected. The fairly abrupt variation between SOC 70% and SOC 80%

Values of Rout, Rin, Rin/Rout, and C, determined from current-pulses experiments with different current magnitudes. source Q flows through R;,, and Rout. Eq. (4) then reads Table 1

When a thermal steady state is reached, the term dT, /dt goes to zero, which means that all the heat flux generated by the heat Fig. 6. Lumped models ((a) complete model and (b) simplified model). Note that this relationship is valid at all times. The values of Rin /Rout determined at thermal steady state are reported in Table 1 for the three experimental values of current magnitude together with

Fig. 7. From top to bottom: Infrared images of the negative terminal, lateral surface, and positive terminal of the battery after a one-hour test (+15 A).

Fig. 8. Estimations of Tj, from T;,,¢ for the current-pulse experiment used for vali- dation. Rin/Rout is set to 0.379 (value for +20 A). A second validation consisted in a high-rate charge/discharge of the battery at a current of 13.8 A (corresponding to 6C). Upon reaching the upper cutoff potential on charge (3.6 V), the voltage was held at this value until the current decayed to 0.5 A. The voltage and current profiles are given in Fig. 9. During the test, the battery was simply in contact with ambient air at a temperature of 24°C.

Finally, the internal temperature is directly estimated from cur- rent and voltage profiles, using the overall energy balance (Eq. 1), where qn is expressed as Fig. 10. Estimations of T;, from T,,,¢ for the full charge/discharge experiment at 13.8 A. Rin/Rout is set to 0.373 (value for +15 A).

Fig. 9. Current and voltage profile during a charge/discharge of the battery at a current of 13.8 A. The constant current-constant voltage (CCCV) protocol was used on charge.

Fig. 11. Estimations of T;, for the full charge/discharge experiment at 13.8 A using the voltage and current. Values of model parameters obtained from current-pulses experiments at +15 A were used.