1. Introduction

2. Thermal management system modeling

2.2. Battery model

Compared with the battery first-order RC and electrochemical models, the Rint model requires fewer parameters and is faster to compute [23,24]. As shown in Figure 2, to simplify the model, the battery Rint model is considered in this paper (see Table 3 for the main parameters). The battery pack is assumed to be a mass block with uniform temperature and consists only of an open-circuit voltage source and internal resistance. Therefore, the battery terminal voltage during charging is calculated as:

$$U_{\mathrm{T}}=U_{\mathrm{ocv}}+I_{\mathrm{bat}}\cdot R_{\mathrm{bat}}$$

(1)

where $U_{\mathrm{T}}$ is the battery pack terminal voltage, $U_{\mathrm{ocv}}$ is the battery pack open circuit voltage, $I_{\mathrm{bat}}$ is the battery pack charging current (battery charging current is positive), $R_{\mathrm{bat}}$ is the battery pack charging internal resistance.

Li-ion batteries’ internal resistance and open-circuit voltage are nonlinear functions of temperature and SOC, and the temperature and SOC of the battery are correlated with battery charging-preheating costs. Therefore, modeling of battery heat production and charging is required. The thermodynamic processes of the battery pack in the low-temperature charging-preheating scenario mainly include:

• heat production by the internal resistance of the battery;
• effective heating by the PTC heater;
• heat transfer between the battery pack and the environment.

The effective heat transferred to the battery can be divided into two parts: one part is used for heating the battery mass block, and the other is dissipated into the environment. Thus, the battery heat balance equation is as follows:

$${\dot{Q}}_{\mathrm{bat}}+{\dot{Q}}_{\mathrm{Ph}}-{\dot{Q}}_{\mathrm{env}}=\mathrm{c}\cdot \mathrm{m}\cdot {\dot{T}}_{bat}$$

(2)

where ${\dot{Q}}_{\mathrm{bat}}$ is the heat-generating power of the battery pack, ${\dot{Q}}_{\mathrm{Ph}}$ is the effective power of the PTC heater, ${\dot{Q}}_{\mathrm{env}}$ is the heat exchange power between the battery pack and the environment, $\mathrm{c}$ is the specific heat capacity of the battery pack, $\mathrm{m}$ is the total mass of the battery pack, ${\dot{T}}_{\mathrm{bat}}$ is the average temperature of the battery pack.

Taking into account ${\dot{Q}}_{\mathrm{bat}}={I_{\mathrm{bat}}}^2\cdot R_{\mathrm{bat}}$, ${\dot{Q}}_{\mathrm{Ph}}=P_h\cdot {\eta }_{\mathrm{ptc}}$, and ${\dot{Q}}_{\mathrm{env}}=\mathrm{h}\cdot \mathrm{A}\cdot \left(T_{\mathrm{bat}}-{\mathrm{T}}_{\mathrm{env}}\right)$, the specific expression for Equation 2 is as follows :

$${I_{\mathrm{bat}}}^2\cdot R_{\mathrm{bat}}+P_{\mathrm{h}}\cdot {\mathsf{\eta }}_{\mathrm{ptc}}-\mathrm{h}\cdot \mathrm{A}\cdot \left(T_{\mathrm{bat}}-{\mathrm{T}}_{\mathrm{env}}\right)=\mathrm{c}\cdot \mathrm{m}\cdot {\dot{T}}_{bat}$$

(3)

where $P_{\mathrm{h}}$ is the electric power of the PTC heater, ${\mathsf{\eta }}_{\mathrm{ptc}}$ is the cumulative heating efficiency of the PTC heater and the water-water heat exchanger, $\mathrm{h}$ is the heat transfer coefficient of the battery pack, $\mathrm{A}$ is the heat exchange area of the battery pack, ${\mathrm{T}}_{\mathrm{env}}$ is the ambient temperature.

The SOC of the battery during the charging process is related to the charging current, and its expression is as follows:

$$S\dot{O}C=\frac{I_{\mathrm{bat}}}{Q_{\mathrm{bat}}\cdot 3600}$$

(4)

where $Q_{\mathrm{bat}}$ is the battery pack capacity.

Table 3. Main parameters of battery pack.

Table 3. Main parameters of battery pack.

Parameters	Value	Unit
Total mass	240	kg
Specific heat capacity	1140	j/(kg·k)
Capacity	163	A·h
Heat exchange area	0.7474	m2
Heat transfer coefficient	10	W/(m2 ·K)
Cumulative heating efficiency	0.9	—
Series-parallel connection	96S1P	—

Figure 2. Schematic diagram of the battery model.

Figure 2. Schematic diagram of the battery model.

3. Battery AC Charge-Preheat Strategy

3.2. Dynamic programming solution

Dynamic programming algorithms are widely used in hybrid vehicle energy management systems to obtain global optimal control strategies for given driving cycles[27,28]. The algorithm can convert a multi-stage decision-making problem into a series of single-stage decision-making problems, which are gradually solved inversely from back to front according to the state transfer equation. As shown in Figure 4, the battery’s initial and termination states are fixed, and the user’s target departure time is set to $t_{\mathrm{desire}}$. The state variables and time domain are discretized, and the discrete state mesh shown in Figure 4 can be obtained. Among them, the simulation step $\Delta t$ is set to 600s, and the number of discrete grids in the time domain is $\frac{t_{\mathrm{desire}}}{\Delta t}+1$. The number of discrete grids for battery temperature and SOC is 201, and the discretization accuracies of the battery temperature and SOC are 0.2736℃ and 0.0037, respectively. The specific solution flow of dynamic programming is as follows:

• Calculate the cost at the termination time :

  $$J\left(x_N\right)=L\left(x_N,u_N\right)$$

  (15)
• Calculate theremaining cost $J\left({x_m}^{i,j},{u_m}^q\right)$. According to equation (10), the control variables $I_{bat}\left(k\right)$ and $P_h\left(k\right)$ required to transfer from state $x_m$ to $x_{m+1}$ can be solved directly, and thus the instantaneous transfer cost $L\left({x_m}^{i,j},{u_m}^q\right)$ between these two states can be solved. Therefore, the expression for the remaining cost is as follows:

  $$J\left({x_m}^{i,j},{u_m}^q\right)=L\left({x_m}^{i,j},{u_m}^q\right)+J^{*}\left(x_{m+1}\right)$$

  (16)

  where ${x_m}^{i,j}$ represents the state at the position in the grid (i , j) at the k=m step, ${u_m}^q$ represents the No.q control variable of ${x_m}^{i,j}$ at the k=m step, $J^{*}\left(x_{m+1}\right)$ is the minimum remaining cost of ${x_m}^{i,j}$ at the k=m+1 step, $J\left({x_m}^{i,j},{u_m}^q\right)$ is the remaining cost of state ${x_m}^{i,j}$ to terminate state with input control ${u_m}^q$ at the k=m step.
• Calculate the minimum remaining cost at the k=m step.

  $$J^{*}\left(x_m\right)=\underset{u\in U,x\in X}{\min }(L(x_m,u_m)+J^{*}\left(x_{m+1}\right))$$

  (17)
• Calculate the optimal control sequence at the k=m step.

  $$u^{*}\left(x_m\right)=\arg \underset{u\in U,x\in X}{\min }(L(x_m,u_m)+J^{*}\left(x_{m+1}\right))$$

  (18)
• Store the optimal control sequence and the minimum remaining cost value for each step and repeat steps b to d until the computation ends at k=1 step, terminating the backward iterative process.
• Reproduce the control variables of the inverse solution positively.

Figure 4. Principle diagram of the dynamic programming algorithm.

Figure 4. Principle diagram of the dynamic programming algorithm.

4. Results and discussion

4.1. Conventional preheating strategy

As shown in Figure 5, most electric vehicle owners choose to charge at night. The battery is left in the environment for a long time after charging, and the battery temperature becomes the same as the ambient temperature. Before traveling, the high-power PTC heater is turned on to heat the battery to the target temperature. The conventional preheating strategy does not utilize time-of-use electricity price and charging waste heat. In addition, the grid cost consumed by the PTC heater is closely related to the user’s travel time. For example, when the user travels in the peak price zone, the grid cost consumed by the PTC heater is much higher than that in the valley price zone.

When the battery remains in the environment after charging for a long time, the heat produced by the charging internal resistance in the conventional preheating strategy is negligible (Assuming $I_{\mathrm{bat}}$ is set to 0), and the temperature rise function of the battery is derived from Equation (3):

$$T_{\mathrm{bat}}=\frac{P_{\mathrm{h}}\cdot {\mathsf{\eta }}_{\mathrm{ptc}}\cdot \left(1-e^{-\frac{\mathrm{A}\cdot \mathrm{h}\cdot t}{\mathrm{c}\cdot \mathrm{m}}}\right)+\mathrm{h}\cdot \mathrm{A}\cdot {\mathrm{T}}_{\mathrm{env}}}{\mathrm{h}\cdot \mathrm{A}}$$

(19)

The battery preheating target temperature is set to 25°C. The maximum power of the PTC heater is 7000 W. Further, according to the above equation, the preheating time and energy consumption required for the conventional preheating strategy at different ambient temperatures can be found. The results are shown in Table 4.

The cost of the conventional preheating strategy is solved by integrating the grid energy consumed to heat the battery during the preheating time and the time-of-use electricity price for that period:

$$c_{\mathrm{heat},\mathrm{now}}={{\displaystyle \int }}_{t_{\mathrm{desire}}-t_{\mathrm{heat},\mathrm{now}}}^{t_{\mathrm{desire}}}{c}_{\mathrm{ele}}\left(t\right)\cdot P_{\mathrm{h},\max }\mathrm{d}t$$

(20)

where $c_{\mathrm{heat},\mathrm{now}}$ is the cost of the conventional preheating strategy, $t_{\mathrm{desire}}$ is the user’s target departure time, $t_{\mathrm{heat},\mathrm{now}}$ is the preheating time for the conventional preheating strategy, $c_{\mathrm{ele}}(\cdot )$ is the time-of-use electricity price function, $P_{\mathrm{h},\max }$ is the maximum power of the PTC heater.

The cost-saving rate can be calculated based on the cost of the optimized strategy and the cost of the conventional preheating strategy with the following expression.

$$c_{\mathrm{rate}}=\frac{c_{\mathrm{heat},\mathrm{now}}-J^{*}\left(x_1\right)}{c_{\mathrm{heat},\mathrm{now}}}*100\%$$

(21)

where $c_{\mathrm{rate}}$ is the cost-saving rate, $J^{*}\left(x_1\right)$ is the cost of the optimized preheating strategy.

Table 4. Preheating time and energy consumption under different ambient temperatures.

Table 4. Preheating time and energy consumption under different ambient temperatures.

Ambient temperature/°C	Preheating time/s	Energy consumption/(kw·h)
-20	2008.4	3.91
-15	1779.7	3.46
-10	1552.5	3.02
-5	1326.6	2.58
0	1102.1	2.14

Figure 5. Flowchart of conventional preheating strategy and optimized preheating strategy.

Figure 5. Flowchart of conventional preheating strategy and optimized preheating strategy.

5. Conclusions

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