*4.2. SOC Estimation*

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Usually, the battery state of charge and the remaining useful life are considered as two important parameters to quantify and monitor the present battery state. In this study, long-term low temperature is the main factor a ffecting the battery capacity and the remaining useful life. References [21,22] have reported joint/dual extended Kalman filter and unscented Kalman filter with an enhanced self-correcting model, which can simultaneously estimate the SOC and capacity. The SOC estimation in this study is realized based on study on low-temperature characteristics of battery. The relationship between battery voltage and battery capacity at low temperatures is shown in Figure 5.

The mathematical model of the relationship between battery voltage and battery capacity at di fferent temperatures can be expressed as follows.

*U*−<sup>50</sup> ◦C = 5.5 × 10−<sup>8</sup> × *SOC*<sup>6</sup> −50 ◦C − 2.51 × 10−<sup>5</sup> × *SOC*<sup>5</sup> −50 ◦C + 4.72 × 10−<sup>3</sup> × *SOC*<sup>4</sup> −50 ◦C − 0.471 × *SOC*<sup>3</sup> −50 ◦C +26.24 × *SOC*<sup>2</sup> −50 ◦C − 711.8 × *SOC*−<sup>50</sup> ◦C + 9362.2 *U*−<sup>40</sup> ◦C = 2.9 × 10−<sup>8</sup> × *SOC*<sup>6</sup> −40 ◦C − 1.36 × 10−<sup>5</sup> × *SOC*<sup>5</sup> −40 ◦C + 2.58 × 10−<sup>3</sup> × *SOC*<sup>4</sup> −40 ◦C − 0.261 × *SOC*<sup>3</sup> −40 ◦C +14.64 × *SOC*<sup>2</sup> −40 ◦C − 434.1 × *SOC*−<sup>40</sup> ◦C + 5306.9 *U*−<sup>30</sup> ◦C = 1.3 × 10−<sup>8</sup> × *SOC*<sup>6</sup> −30 ◦C − 6.04 × 10−<sup>6</sup> × *SOC*<sup>5</sup> −30 ◦C + 1.15 × 10−<sup>3</sup> × *SOC*<sup>4</sup> −30 ◦C − 0.116 × *SOC*<sup>3</sup> −30 ◦C +6.487 × *SOC*<sup>2</sup> −30 ◦C − 192.4 × *SOC*−<sup>30</sup> ◦C + 2353.8 *U*−<sup>20</sup> ◦C = 1.5 × 10−<sup>8</sup> × *SOC*<sup>6</sup> −20 ◦C − 6.82 × 10−<sup>6</sup> × *SOC*<sup>5</sup> −20 ◦C + 1.31 × 10−<sup>3</sup> × *SOC*<sup>4</sup> −20 ◦C − 0.132 × *SOC*<sup>3</sup> −20 ◦C +7.508 × *SOC*<sup>2</sup> −20 ◦C − 244.2 × *SOC*−<sup>20</sup> ◦C + 2760.1 *U*−<sup>10</sup> ◦C = 8.0 × 10−<sup>9</sup> × *SOC*<sup>6</sup> −10 ◦C − 3.72 × 10−<sup>6</sup> × *SOC*<sup>5</sup> −10 ◦C + 7.13 × 10−<sup>4</sup> × *SOC*<sup>4</sup> −10 ◦C − 0.072 × *SOC*<sup>3</sup> −10 ◦C +4.098 × *SOC*<sup>2</sup> −10 ◦C − 122.4 × *SOC*−<sup>10</sup> ◦C + 1508.8 *U*0 ◦C = 4.66 × 10−<sup>9</sup> × *SOC*<sup>6</sup> 0 ◦C − 2.13 × 10−<sup>6</sup> × *SOC*<sup>5</sup> 0 ◦C + 4.01 × 10−<sup>4</sup> × *SOC*<sup>4</sup> 0 ◦C − 0.0401 × *SOC*<sup>3</sup> 0 ◦C<sup>+</sup> 2.229 × *SOC*<sup>2</sup> 0◦C− 25.5895 × *SOC*0 ◦C + 797.2 (11)

where *U*−50◦C, *U*−40◦C, *U*−30◦C, *U*−20◦C, *U*−10◦<sup>C</sup> and *U*0◦C are the battery voltages at −50 ◦C, −40 ◦C, −30 ◦C, −20 ◦C, −10 ◦C, 0 ◦C, respectively; *SOC*−50◦C, *SOC*−40◦C, *SOC*−30◦C, *SOC*−20◦C, *SOC*−10◦<sup>C</sup> and *SOC*0◦C are the values of state of charge at −50 ◦C, −40 ◦C, −30 ◦C, −20 ◦C, −10 ◦C, 0 ◦C, respectively.

We evaluated the relationship between battery voltage and battery capacity from −50 ◦C to 0 ◦C. However, the values at the non-measured battery voltage and battery capacity could be predicted by interpolation. Equation (11) can be described as follows.

$$\begin{aligned} \mathcal{U}(T\_i) &= a(T\_i) \mathcal{S} \mathcal{O} \mathcal{C}^6(T\_i) + b(T\_i) \mathcal{S} \mathcal{O} \mathcal{C}^5(T\_i) + c(T\_i) \mathcal{S} \mathcal{O} \mathcal{C}^4(T\_i) \\ &+ d(T\_i) \mathcal{S} \mathcal{O} \mathcal{C}^3(T\_i) + c(T\_i) \mathcal{S} \mathcal{O} \mathcal{C}^2(T\_i) + f(T\_i) \mathcal{S} \mathcal{O} \mathcal{C}(T\_i) + \mathcal{g}(T\_i) \end{aligned} \tag{12}$$

where *U*(*Ti*) is the battery voltages at di fferent temperatures from −50 ◦C to 0 ◦C; *SOC*(*Ti*) is the battery capacity at di fferent temperatures from −50 ◦C to 0 ◦C; *a*(*Ti*), *b*(*Ti*), *c*(*Ti*), *d*(*Ti*), *e*(*Ti*), *f*(*Ti*), and *g*(*Ti*) are the coe fficients of the Equation (12), which have dependences of low temperatures.

The coe fficients *a*(*Ti*), *b*(*Ti*), *c*(*Ti*), *d*(*Ti*), *e*(*Ti*), *f*(*Ti*) and *g*(*Ti*) from −50 ◦C to 0 ◦C are shown in Figure 6.

**Figure 5.** (**<sup>a</sup>**–**f**) show the relationship between battery voltage and battery capacity at −50 ◦C, −40 ◦C, −30 ◦C, −20 ◦C, −10 ◦C, 0 ◦C, respectively.

**Figure 6.** (**<sup>a</sup>**–**g**) show the coefficients *a*(*Ti*), *b*(*Ti*), *c*(*Ti*), *d*(*Ti*), *e*(*Ti*), *f*(*Ti*) and *g*(*Ti*) *from* −50 ◦C to 0 ◦C, respectively.

As shown in Figure 6, the temperature dependence of the coefficients of *a*(*Ti*), *b*(*Ti*), *c*(*Ti*), *d*(*Ti*), *e*(*Ti*), *f*(*Ti*) and *g*(*Ti*) is generally in the form of a cubic functions. The coefficients *a*(*Ti*), *b*(*Ti*), *c*(*Ti*), *d*(*Ti*), *e*(*Ti*), *f*(*Ti*) and *g*(*Ti*) in Equation (12) at different temperatures could be predicted as follows.

$$\begin{cases} \begin{aligned} a(T\_i) &= -9.96 \times 10^{-13} T\_i^3 - 4.8 \times 10^{-11} T\_i^2 - 9.28 \times 10^{-10} T\_i + 4.27 \times 10^{-9} \\ b(T\_i) &= 4.52 \times 10^{-10} T\_i^3 + 2.19 \times 10^{-8} T\_i^2 + 4.28 \times 10^{-7} T\_i - 1.96 \times 10^{-6} \\ c(T\_i) &= -8.46 \times 10^{-8} T\_i^3 - 4.11 \times 10^{-6} T\_i^2 - 8.17 \times 10^{-5} T\_i + 3.69 \times 10^{-4} \\ d(T\_i) &= 8.41 \times 10^{-6} T\_i^3 + 4.1 \times 10^{-4} T\_i^2 + 8.25 \times 10^{-3} T\_i - 0.037 \\ c(T\_i) &= -4.66 \times 10^{-4} T\_i^3 - 0.023 T\_i^2 - 0.4657 T\_i + 2.066 \\ f(T\_i) &= 0.0137 T\_i^3 + 0.675 T\_i^2 + 13.8716 T\_i - 60.908 \\ g(T\_i) &= -0.1654 T\_i^3 - 8.2046 T\_i^2 - 170.3108 T\_i + 741.803 \end{aligned} \tag{13}$$
