Under the background of global energy transition and vehicle electrification, the thermal safety and energy efficiency management of the power battery have become the key bottlenecks that limit the further penetration of electric vehicles. The conventional thermal management system is often challenged by the inherent contradiction between the precise regulation of battery temperature and the energy consumed by the cooling/heating actuators. Such a contradiction is especially evident when the electric vehicle operates under rapidly changing driving profiles and ambient temperatures. To address this issue, I have carried out a systematic study on a high-voltage battery thermal management system for pure electric vehicles, with the central goal of achieving a trade-off between temperature control quality and system energy consumption. In this work, theoretical analysis, experimental calibration, one-dimensional physical simulation, design-of-experiments, model predictive control, and meta-heuristic optimization are integrated in a coherent framework. The final results demonstrate that the proposed layered temperature-energy collaborative management strategy can maintain the high-voltage battery within the desired temperature window under low-temperature, normal-temperature, and high-temperature conditions, while considerably reducing the total parasitic energy consumption of the thermal management loop.
As an introduction to the main contributions, the research begins from the electro-thermal mechanism of a lithium-ion high-voltage battery, then establishes a lumped-parameter heat generation and heat transfer model. The model is coupled with a vehicle longitudinal dynamics model and a high-voltage battery equivalent circuit model to produce a complete pure electric vehicle powertrain simulation environment. Based on AMESim, I build a one-dimensional simulation platform of the electric vehicle powertrain and the thermal management system. The crucial thermodynamic parameters are calibrated with dedicated experiments. After validation, the model accuracy is sufficiently high for control-oriented studies: the maximum error of battery temperature rise is less than 3°C, and the heat exchanger capacity error is less than 5% in the calibrated operating range. With this reliable model, I further use optimal Latin hypercube sampling to explore the parameter space, and variance analysis to identify the most significant control variables. The results show that compressor speed and pump speed dominate the coupled temperature-energy response, while fan speed mainly acts as an auxiliary regulator. Based on these findings, I design a family of control strategies adapted to different ambient temperature categories. For low-temperature environments, a PID controller regulates the positive temperature coefficient heater. For normal-temperature environments, a nonlinear model predictive controller continuously adjusts the pump and fan in a coordinated manner. For high-temperature environments, the nonlinear predictive controller is enhanced by a grey wolf optimizer to separately control the compressor and pump in a precise and energy-efficient way. Extensive comparative simulations with threshold control and PID control confirm that the proposed strategy achieves superior temperature tracking and significant energy saving under all investigated environmental scenarios.
1. Modeling of the High-Voltage Battery Thermal Management System
The high-voltage battery pack in this research consists of multiple cells arranged in a parallel-series configuration. The thermal management system is designed to maintain the high-voltage battery temperature inside a favorable range, usually between 25°C and 35°C, with enough uniformity to prevent accelerated aging and thermal runaway. I first analyze the heat sources inside a single lithium-ion cell. During charging and discharging, several heat components are produced: reaction heat generated by lithium intercalation and deintercalation, polarization heat caused by internal polarization effects, joule heat from the current flowing through the cell resistance, and side-reaction heat. Among them, side-reaction heat is negligible in normal operation. Therefore, the total heat generation rate can be approximated as the sum of the first three components:
$$
\dot{Q}_{total} = \dot{Q}_r + \dot{Q}_p + \dot{Q}_j
$$
In this expression, \dot{Q}_r is the reaction heat rate, \dot{Q}_p denotes the polarization heat rate, and \dot{Q}_j is the joule heat rate. To adopt a control-oriented but physically meaningful formulation, I employed the widely used Bernardi equation. According to Bernardi’s model, the heat generation rate of the high-voltage battery can be calculated from the operating current, internal resistance, open-circuit voltage, and entropy coefficient. The formula is expressed as:
$$
\dot{Q}_{gen} = I \left( I R_e – T_b \frac{dU_{ocv}}{dT_b} \right)
$$
where I is the instantaneous current flowing through the battery, R_e is the equivalent internal resistance, T_b is the battery temperature, U_{ocv} is the open-circuit voltage, and the term dU_{ocv}/dT_b is the entropic coefficient. This equation implies that the total heat generation is not only determined by the I–R loss, but also influenced by the reversible heat associated with the entropy change. In the practical thermal management model, the current is determined by the vehicle power demand and the high-voltage battery state, while the equivalent resistance is dependent on temperature, state of charge, and current direction.
The heat exchange between the high-voltage battery and its surroundings occurs via conduction, convection, and radiation. In the liquid-cooled configuration studied here, conduction dominates inside the battery, convection dominates between the coolant and the cooling plate, and radiation is usually neglected. The convective heat transfer between the cooling fluid and the battery cooling plate is described by:
$$
\dot{Q}_{conv} = h_b A_b \left( T_b – T_{cool} \right)
$$
where h_b is the equivalent heat transfer coefficient, A_b denotes the effective contact area, and T_{cool} is the coolant temperature near the battery. The heat transfer coefficient is evaluated using the Dittus-Boelter style correlation:
$$
Nu = 0.023 Re^{0.8} Pr^{0.3}
$$
$$
h_b = \frac{Nu \lambda_{cool}}{D_h}
$$
where Nu is the Nusselt number, Re is the Reynolds number, Pr is the Prandtl number, \lambda_{cool} is the thermal conductivity of the coolant, and D_h is the hydraulic diameter of the flow channel. The Reynolds number is governed by the coolant velocity, which is controlled by the water pump speed. Hence, the heat removal capability is tightly related to the pump speed. With these equations, I build a lumped-parameter electro-thermal coupled model of the high-voltage battery. The model treats the battery pack as a uniform thermal body, which significantly reduces the computational cost without losing the main dynamic characteristics.
1.1 High-voltage battery model
For the electrical part, I adopt the Rint equivalent circuit model. It consists of an ideal voltage source representing the open-circuit voltage and a resistance representing the equivalent internal resistance. The terminal voltage is:
$$
U_b = U_{ocv} – I_b R_e
$$
and the power balance is expressed by:
$$
P_b = U_b I_b
$$
The state of charge is obtained by current integration:
$$
SOC = SOC_0 – \frac{1}{C_b} \int_0^t I_b(\tau) d\tau
$$
where SOC_0 is the initial state of charge and C_b is the nominal capacity of the high-voltage battery pack. The open-circuit voltage is a function of SOC, while the equivalent internal resistance depends on SOC and temperature. I obtained these dependencies experimentally and described them with response surface maps.
1.2 Electric-vehicle powertrain model
To produce realistic battery thermal loads, I integrate the high-voltage battery model with a vehicle longitudinal dynamics model. The vehicle motion equation includes rolling resistance, aerodynamic drag, grading resistance, and acceleration resistance:
$$
F_{traction} = m_{veh} \frac{dv}{dt} + \frac{1}{2} \rho_a C_D A_f v^2 + m_{veh} g f \cos\theta + m_{veh} g \sin\theta
$$
where m_{veh} is the vehicle mass, v is the vehicle speed, \rho_a is the air density, C_D is the drag coefficient, A_f is the frontal area, f is the rolling resistance coefficient, and \theta is the road slope. For the test scenarios, the road slope is assumed to be zero. Table 1 summarizes the main vehicle parameters used in the powertrain simulation.
| Parameter | Symbol | Value |
|---|---|---|
| Vehicle mass | m_veh | 1445 kg |
| Rolling resistance coefficient | f | 0.01 |
| Aerodynamic drag coefficient | C_D | 0.31 |
| Frontal area | A_f | 2.09 m² |
| Rotational inertia factor | δ | 1.4 |
The driver model is a longitudinal speed-tracking controller that outputs acceleration and braking commands to follow a prescribed speed profile. The motor model converts electrical power into mechanical torque, considering its efficiency map. With these elements, the high-voltage battery supplies power to the electric motor and thermal management system; therefore, the battery current is not simply defined by the driving cycle alone but also depends on the thermal management load. This coupling is important because the parasitic power consumption of the cooling system reduces the electric range and increases the thermal load of the high-voltage battery.
1.3 Thermal management architecture
The designed high-voltage battery thermal management system is composed of three interconnected loops. The first loop is the coolant loop that directly cools or heats the battery pack. Inside this coolant loop, a water pump drives a mixture of water and ethylene glycol through the cooling plate. A positive temperature coefficient heater is placed along the coolant path so that the coolant can be warmed during cold weather. The second loop is the radiator loop, which is connected to the main coolant loop through a two-position three-way valve. A fan blows ambient air across the radiator when the ambient temperature allows direct cooling of the coolant. The third loop is the vapor-compression refrigeration loop, which consists of a compressor, a condenser, an expansion valve, and a chiller. The chiller is a plate heat exchanger that transfers heat from the coolant to the refrigerant. When ambient temperature is high and the radiator alone cannot maintain the high-voltage battery below the upper temperature limit, the refrigeration loop is activated. The entire architecture is schematically shown in the image above.
2. Experimental Parameter Identification and Model Validation
I established a dedicated test bench for the high-voltage battery cell experiments. The battery under test is a square lithium iron phosphate cell designed for automotive use. The bench includes a charge-discharge testing system, a temperature-controlled environmental chamber, T-type thermocouples, and a data acquisition module. Through systematic tests, I identified the open-circuit voltage curve, the equivalent internal resistance, the heat capacity, and the thermal conductivity of the cell. These values are necessary to construct the AMESim simulation model with physical accuracy.
2.1 Open-circuit voltage test
Before each test, the cell was allowed to rest at a fixed temperature to reach equilibrium. A very low current is generally used to approximate the open-circuit voltage if the direct measurement is not practical. In this work, I used a low-rate discharge procedure. The cell was fully charged, and then discharged at 2 A while the terminal voltage was recorded until the cutoff voltage. The resulting open-circuit voltage versus SOC curve shows a decreasing trend, which agrees with the expected thermodynamic behavior of the high-voltage battery. The measured data were then imported into the battery model as a lookup table.
2.2 HPPC test and internal resistance identification
The equivalent internal resistance of the high-voltage battery strongly depends on the temperature and SOC. I applied the Hybrid Pulse Power Characteristic test to extract the resistance values at different SOC levels. In each pulse, a constant current discharge is applied for a short period, followed by a rest period. The voltage drop caused by the series resistance is evaluated from the sudden voltage change at the beginning of the pulse. The resistance is calculated as:
$$
R = \frac{U_1 – U_0}{I_{pulse}}
$$
where U_0 is the voltage just before the pulse, U_1 is the voltage immediately after the pulse is applied, and I_{pulse} is the pulse current. In the test, I measured the internal resistance at 40°C, 30°C, and 20°C, with SOC descending from 100% down to 0%. A response surface of resistance with respect to SOC and temperature was then constructed. Table 2 gives an excerpt of the measured resistance map at selected SOC and temperature points.
| SOC | 40°C | 30°C | 20°C |
|---|---|---|---|
| 90% | 0.85 | 1.03 | 1.32 |
| 70% | 0.92 | 1.12 | 1.45 |
| 50% | 1.05 | 1.28 | 1.72 |
| 30% | 1.22 | 1.55 | 2.18 |
| 10% | 1.68 | 2.17 | 3.05 |
It is apparent that the internal resistance increases significantly at low SOC and low temperature. This reveals that the high-voltage battery is more vulnerable to overheating under cold-start conditions, while at high temperature and low SOC the resistance rise causes more joule heat and may accelerate thermal runaway if not properly managed.
2.3 Heat capacity measurement
The heat capacity of the high-voltage battery cell determines the transient thermal response. I placed heating films on all six surfaces of the cell and wrapped the cell in adiabatic foam to minimize heat loss. A constant heating power of 40 W was applied for 600 seconds. The average surface temperature was recorded by multiple thermocouples. If the heat loss is negligible, the equivalent heat capacity is:
$$
C_b = \frac{P_{heat} \Delta t}{T_{end} – T_{initial}}
$$
where P_{heat} is the heating power, \Delta t is the heating duration, T_{initial} and T_{end} are the average cell temperatures before and after heating. In the experiment, the temperature rose from 14.5°C to 35.4°C. Consequently, I obtained the equivalent heat capacity as 1142 J/K.
2.4 Model validation of battery temperature response
After the model parameters were identified, I performed an independent validation test at 26°C ambient temperature. The cell was discharged according to a time-dependent current profile, and the simulated battery temperature and voltage responses were compared with the experimental measurements. The maximum temperature deviation during the entire discharge test remained smaller than 3°C. This result is sufficiently accurate for a lumped-parameter high-voltage battery model, because the lumped model intentionally ignores internal gradients and local aging variations. Therefore, the battery model can be embedded into the thermal management simulation for further investigations.
3. Calibration of Thermal Management Components
The complete thermal management model also needs accurate submodels of the condenser and the chiller. Their heat transfer performance influences the refrigerant-side cooling capacity and thus the temperature response of the high-voltage battery. I calibrated these heat exchangers using dedicated test data provided by component-level experiments.
3.1 Condenser calibration
The condenser is a micro-channel parallel-flow heat exchanger. Its main structural parameters are summarized in Table 3. In the AMESim model, the heat transfer coefficient is adjusted by modifying the Nusselt correlation coefficients. I conducted several experiments with different air inlet velocities and refrigerant mass flow rates. For each test point, the heat rejection capacity was measured by the air-side enthalpy difference method. The data were then imported into the calibration module of AMESim. The calibration tool iteratively adjusts the unknown coefficients until the simulation results match the experimental heat transfer rate.
| Parameter | Value |
|---|---|
| Flow-circuit distribution | 20–20–20–20 |
| Geometric dimension | 260×210×43 mm |
| Fin thickness | 0.07 mm |
| Fin pitch | 1.4 mm |
| Number of microchannels | 18 |
| Flat-tube height | 1.5 mm |
| Flat-tube width | 18 mm |
| Flat-tube pitch | 6.2 mm |
Table 4 lists four representative calibration conditions and the corresponding heat rejection results. At the same operating conditions, the difference between simulation and experimental heat capacity is less than 5%. This confirms that the calibrated condenser model is reliable enough for system-level simulations.
| Case | Air velocity (m/s) | Air temperature (°C) | Refrigerant flow (kg/h) | Inlet pressure (bar) | Experimental heat (W) | Simulation heat (W) | Relative error |
|---|---|---|---|---|---|---|---|
| 1 | 1.5 | 26 | 101.3 | 2 | 3725 | 3862 | 3.7% |
| 2 | 2.5 | 26 | 130.4 | 2 | 4688 | 4821 | 2.8% |
| 3 | 4.5 | 26 | 158.9 | 2 | 6102 | 5945 | 2.6% |
| 4 | 7 | 26 | 191.3 | 2 | 7350 | 7187 | 2.2% |
3.2 Chiller calibration
The chiller, also called a plate heat exchanger, transfers heat between the refrigerant loop and the coolant loop. I also collected data under four different conditions. The inlet pressure and mass flow of the refrigerant were varied while the coolant-side flow rate and temperature were fixed. The AMESim calibration module fitted the key heat transfer coefficients. The maximum relative error between the simulated and measured heat exchange rate was lower than 5%, as shown in Table 5. The calibrated chiller model is therefore suitable for analyzing the dynamic cooling of the high-voltage battery pack.
| Case | Coolant flow (m³/h) | Coolant temp (°C) | Refrigerant flow (kg/h) | Inlet temp (°C) | Inlet pressure (bar) | Experimental heat (W) | Simulation heat (W) | Relative error |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.49 | 34 | 98.3 | 42.2 | 11 | 3208 | 3321 | 3.5% |
| 2 | 0.73 | 34 | 124.3 | 61.4 | 11 | 4312 | 4518 | 4.8% |
| 3 | 0.92 | 34 | 132.6 | 72.2 | 11 | 4830 | 4944 | 2.4% |
| 4 | 1.24 | 34 | 157.7 | 84.3 | 11 | 5685 | 5810 | 2.2% |
With the validated component models, the full-vehicle thermal management model reproduces the actual thermal dynamics of the high-voltage battery pack under realistic driving cycles. Thus, it provides a reliable virtual test environment for the later control design and comparative simulation.
4. Identification of Critical Control Variables Based on Optimal Latin Hypercube Sampling
The high-voltage battery thermal management system is a multi-variable and strongly coupled system. The actuator inputs include compressor speed, water pump speed, and fan speed. In a conventional control design, it is not clear which input should be prioritized to simultaneously satisfy the constraints on battery temperature and system energy consumption. I therefore performed a systematic factor screening study.
4.1 Single-factor analysis
First, I evaluated the sensitivity to each actuator independently. In these simulations, the ambient temperature was set to 30°C, the vehicle speed to 60 km/h, and the battery initial temperature to 30°C. The compressor speed was varied from 1000 to 5000 rpm while the pump was fixed at 1500 rpm and the fan was off. The results showed that a higher compressor speed clearly increases the cooling capacity. However, compressor power consumption also increases with speed in a non-linear way, meaning that blindly increasing the compressor speed would lead to a serious energy penalty. The pump speed was then varied from 500 to 2500 rpm while the compressor was fixed at 2000 rpm. A higher pump speed improves the heat transfer between the high-voltage battery and the coolant, but beyond approximately 2000 rpm the temperature benefit becomes negligible while pump power still grows with the cubic law. The fan speed was investigated using the same method. Raising the fan speed from 750 to 4500 rpm increased fan power from only 3 W to more than 670 W, but the reduction in high-voltage battery peak temperature was merely 0.1–0.2°C. This clearly indicates that the fan has a limited effect on the battery temperature response in the investigated scenarios. Therefore, fan speed can be treated as an auxiliary variable.
4.2 Optimal Latin hypercube sampling design
To quantify the coupled influences of compressor speed, pump speed, and fan speed, I designed a three-dimensional factor space. The range of each variable was chosen from the single-factor analysis and from practical actuator limits. The ranges are shown in Table 6.
| Factor | Symbol | Minimum | Maximum |
|---|---|---|---|
| Compressor speed (rpm) | N_comp | 2000 | 4000 |
| Water pump speed (rpm) | N_pump | 1000 | 2000 |
| Fan speed (rpm) | N_fan | 1500 | 2250 |
The standardized factor space is defined as follows:
$$
X_1 = \frac{N_{comp} – 2000}{2000}, \quad X_2 = \frac{N_{pump} – 1000}{1000}, \quad X_3 = \frac{N_{fan} – 1500}{750}
$$
I generated 30 samples using the optimal Latin hypercube sampling algorithm. Compared with conventional Latin hypercube sampling, the optimal version minimizes the possible clustering of samples in the design space by maximizing the distance between neighboring points. To verify the quality, I examined the pairwise correlation coefficients and the distribution of sample distances. All absolute correlation coefficients among the three factors were lower than 0.2, meaning that the factors can be treated as nearly orthogonal to each other. The sample point distances followed a narrow distribution around an average distance of 0.69. These two indicators confirm that the sampling design is sufficiently uniform and independent for variance analysis.
4.3 ANOVA for battery temperature response
Each of the 30 samples was simulated under identical boundary conditions: ambient temperature of 30°C, initial battery temperature of 30°C, initial SOC of 90%, a constant vehicle speed of 60 km/h, and a simulation duration of 1200 seconds. The battery average temperature at the end of each simulation was treated as the temperature response. The variance analysis results are presented in Table 7.
| Source | Sum of squares | Degrees of freedom | Mean square | F value | P value |
|---|---|---|---|---|---|
| Model | 2.07 | 3 | 0.6915 | 152.10 | <0.0001 |
| N_comp | 0.5501 | 1 | 0.5501 | 121.01 | <0.0001 |
| N_pump | 1.08 | 1 | 1.08 | 237.13 | <0.0001 |
| N_fan | 0.002 | 1 | 0.002 | 0.4384 | 0.5137 |
| Residual | 0.1182 | 26 | 0.0045 | ||
| Total | 2.19 | 29 |
The model F value is 152.10 and the overall P value is less than 0.0001, indicating that the linear regression is highly significant. The coefficient of determination R² is 0.9461, meaning that the model explains 94.61% of the total variation. The F values of compressor speed and pump speed are both highly significant, with P values lower than 0.0001. Among the two, pump speed has the larger F value, which means that under the given constant-speed driving condition, the pump speed has a slightly stronger influence on the final battery temperature than the compressor speed. In contrast, fan speed has an F value of only 0.44 and a P value of 0.5137, which is not statistically significant. Therefore, fan speed is not a controlling factor for high-voltage battery temperature in the studied range.
4.4 ANOVA for system energy consumption response
The total energy consumption of the thermal management system is the sum of the energy consumed by the compressor, the pump, and the fan. Table 8 lists the ANOVA results for the total power consumption at the end of the simulations.
| Source | Sum of squares | Degrees of freedom | Mean square | F value | P value |
|---|---|---|---|---|---|
| Model | 5.074E6 | 3 | 1.691E6 | 1994.38 | <0.0001 |
| N_comp | 4.331E6 | 1 | 4.331E6 | 5105.98 | <0.0001 |
| N_pump | 79683.92 | 1 | 79683.92 | 93.95 | <0.0001 |
| N_fan | 40376.73 | 1 | 40376.73 | 47.61 | <0.0001 |
| Residual | 22051.40 | 26 | 848.13 | ||
| Total | 5.097E6 | 29 |
The regression model for energy consumption is also highly significant, with an R² value of 0.9957. In this case, compressor speed is the dominant factor, with an extremely large F value of 5105.98. Pump speed and fan speed are also significant, as their P values are below 0.0001, but their contributions are much smaller than that of the compressor speed. Combining this result with the temperature analysis, one can conclude that compressor speed and pump speed are the key variables that affect both temperature and energy consumption. Fan speed mostly increases energy consumption without providing a proportional temperature reduction. Hence, the fan should be controlled as an auxiliary device, while the main coordinated control effort should focus on the compressor and pump.
5. Temperature-Energy Coordinated Control Strategies
According to the ambient temperature range, I divided the operation of the high-voltage battery thermal management system into three modes: low-temperature mode, normal-temperature mode, and high-temperature mode. In the low-temperature mode, the system heats the high-voltage battery with a positive temperature coefficient heater. In the normal-temperature mode, the system uses the radiator and coolant loop to dissipate heat, with the fan and pump as the main actuators. In the high-temperature mode, the chiller can actively cool the coolant, requiring the compressor and pump to be coordinated. The objective is always to maintain the high-voltage battery temperature in a suitable window while minimizing the energy consumed by thermal management. In addition, the control strategy must avoid excessive overshoot, frequent on-off switching, and unnecessary high-speed operation.
5.1 Control objective formulation
For the low-temperature mode, precise temperature tracking is not necessary as long as the battery temperature remains between about 0°C and 10°C. However, the system should not overheat the battery. The heating power consumed by the positive temperature coefficient heater and the pump should be minimized. Since the single positive temperature coefficient heater can be easily controlled by a simple feedback law, I used both threshold control and PID control. The control action is determined by the error between the target temperature and the measured battery temperature:
$$
u_{PID}(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}
$$
where e(t)=T_ref-T_b(t). This controller outputs the required heating power.
For the normal-temperature mode, I formulated a constrained finite-horizon optimal control problem. Let the state vector be x=[T_b, T_c, T_rad]^T, the control vector be u=[\dot{m}_{cool}, V_{air}]^T, and the disturbance be the battery heat generation rate Q_gen. The dynamics are written in general nonlinear form:
$$
\dot{x}(t) = f_c(x(t), u(t), d(t))
$$
The objective function over the prediction horizon N_p is:
$$
J = \min_{u} \left\{ \alpha_1 \sum_{i=1}^{N_p} \left(T_b(k+i)-T_{ref}(k+i)\right)^2 + \alpha_2 P_{fan}(k+i) + \alpha_3 P_{pump}(k+i) \right\}
$$
subject to the actuator constraints:
$$
0 \le n_{pump} \le 3000\ \mathrm{rpm}
$$
$$
0 \le n_{fan} \le 4500\ \mathrm{rpm}
$$
where \alpha_1, \alpha_2, \alpha_3 are weights, k denotes the current sampling instant, and P_{fan} and P_{pump} are the fan and pump power consumptions at time step i. The nonlinear model is used to predict the future high-voltage battery temperature, and the optimizer determines the best pump and fan speed sequences. Only the first element of the optimal sequence is implemented; the optimization is repeated at the next sampling instant.
For the high-temperature mode, the control variables become compressor speed and pump speed, while the fan speed is set to a constant value because of its auxiliary role. The state-space model is:
$$
x = \left[ T_b, T_{c,in} \right]^T, \quad u = \left[ n_{comp}, n_{pump} \right]^T
$$
The battery dynamics are described by:
$$
c_{p,b} m_b \frac{dT_b}{dt} = Q_{gen} + c_{p,c} \dot{m}_c \left( T_{c,in} – T_{c,out} \right)
$$
The outlet coolant temperature is expressed using the effectiveness-NTU relation:
$$
T_{c,out} = T_b – (T_b – T_{c,in}) \exp\left( \frac{-h_b A_b}{c_{p,c} \dot{m}_c} \right)
$$
The coolant temperature at the inlet of the cooling plate is governed by the energy balance in the chiller:
$$
c_{p,c} m_{c,all} \frac{dT_{c,in}}{dt} = c_{p,c} \dot{m}_c \left( T_{c,out} – T_{c,in} \right) – Q_c(n_{comp})
$$
where Q_c is the cooling capacity provided by the refrigeration cycle. I fitted this capacity as a cubic polynomial of compressor speed. For the high-voltage battery thermal management model used in this study, the fitted equation is:
$$
Q_c = 43 n_{comp}^3 – 241 n_{comp}^2 + 414 n_{comp} + 3939
$$
with n_{comp} in thousands of revolutions per minute and Q_c in watts. The coefficient of determination is 0.9978, which confirms an excellent fit.
The objective function for the high-temperature mode is:
$$
J = \min \left\{ \beta_1 \sum_{i=1}^{N_p} \left(T_b(k+i)-T_{ref}(k+i)\right)^2 + \beta_2 P_{comp}(k+i) + \beta_3 P_{pump}(k+i) \right\}
$$
subject to:
$$
0 \le n_{pump} \le 3000\ \mathrm{rpm}
$$
$$
0 \le n_{comp} \le 4000\ \mathrm{rpm}
$$
The nonlinear model predictive controller is solved by the grey wolf optimizer. GWO is a population-based metaheuristic inspired by the social hierarchy and hunting behavior of gray wolves. In the algorithm, the alpha wolf represents the best solution, beta the second best, delta the third best, and omega wolves update their positions around these leading solutions according to the following equations. For each prey direction, the distance is computed:
$$
D_{\alpha} = |C_1 X_{\alpha} – X|, \quad D_{\beta} = |C_2 X_{\beta} – X|, \quad D_{\delta} = |C_3 X_{\delta} – X|
$$
Then the updated position is:
$$
X_1 = X_{\alpha} – A_1 D_{\alpha}
$$
$$
X_2 = X_{\beta} – A_2 D_{\beta}
$$
$$
X_3 = X_{\delta} – A_3 D_{\delta}
$$
$$
X(t+1) = \frac{X_1 + X_2 + X_3}{3}
$$
In my implementation, the grey wolf optimizer was configured with 20 search agents and 20 iterations per sampling step. To improve convergence, the last optimal pump speed and compressor speed were injected into the initial population, thus making the search start near the previous solution. The optimization process was executed in Simulink as an S-function-like module, which communicates with AMESim at every sampling time.
5.2 Validation of the prediction models
The nonlinear model predictive controller relies on a reduced-order prediction model. Before applying it to closed-loop control, I verified the prediction accuracy against the high-fidelity AMESim model. In the normal-temperature verification, the ambient temperature was set to 25°C, and the fan speed and pump speed were varied according to arbitrary step-like profiles. The high-voltage battery temperature predicted by the reduced-order model was compared to the AMESim simulation result. The maximum temperature difference was below 0.5°C. In the high-temperature verification, the ambient and initial battery temperatures were set to 35°C, and the compressor speed was varied. The reduced-order model also predicted the battery temperature within 0.6°C of the high-fidelity model. These errors are acceptable because the high-voltage battery temperature response is characterized by large time constants; a small model mismatch can be compensated by the feedback mechanism of the predictive controller.
6. Comparative Simulation Results and Discussion
To evaluate the proposed strategies, I simulated the complete electric vehicle with an AMESim-Simulink co-simulation platform. The driving cycle was the New European Driving Cycle, repeated several times to create sustained thermal loads. The high-voltage battery model was integrated in AMESim, while the control algorithms were implemented in Simulink. This co-simulation arrangement provides both physical fidelity and algorithmic flexibility. The simulation results are organized below according to the ambient temperature mode.
6.1 Low-temperature heating results
The ambient temperature was set to -10°C. The target battery temperature was 5°C, and the simulation lasted 2400 seconds. In the first baseline scenario without any heating, the high-voltage battery remained close to the ambient temperature and slowly warmed only by internal heat generation. This condition is unacceptable for battery performance. In the threshold control scenario, the positive temperature coefficient heater was either fully on or fully off depending on whether the battery temperature exceeded 6°C or fell below 5°C. The positive temperature coefficient heater consumed a constant power of 4000 W when it was switched on. In the PID control scenario, the heater power was continuously modulated according to the temperature error, which reduced the average heating intensity. The simulation results are summarized in Table 9.
| Strategy | Final battery temperature (°C) | Temperature overshoot | Heater energy consumption (kJ) |
|---|---|---|---|
| No heating | -6.2 | – | 0 |
| Threshold control | 5.4 | 0.4°C | 8837 |
| PID control | 5.1 | 0.1°C | 7921 |
The PID-controlled positive temperature coefficient heater reduced the energy consumption by 10.37% compared with the threshold control, while keeping the high-voltage battery temperature closer to the target. This demonstrates that even a simple PID feedback can effectively smooth the heating action and improve the temperature-energy trade-off.
6.2 Normal-temperature cooling results
The ambient temperature was set to 25°C, and the initial battery and coolant temperatures were also 25°C. I repeated five New European Driving Cycle loops to accumulate heat in the high-voltage battery. Three control strategies were compared: threshold control, dual-loop PID control, and nonlinear model predictive control. In the threshold strategy, the pump and fan were switched on when the battery temperature exceeded 28°C and turned off when it dropped below 27°C. In the PID strategy, two independent PID controllers adjusted the pump and the fan based on the battery temperature error. The NMPC strategy coordinated both actuators by minimizing the defined objective function over the prediction horizon.
Without thermal management, the high-voltage battery temperature continuously increased to 34.54°C. With threshold control, the temperature was limited to about 28.67°C, with a visible overshoot of 0.67°C because the control actions were discrete and had a delayed effect. With PID control, the maximum temperature was 28.72°C, and the overshoot was about 1.72°C relative to the 27°C target. The PID controller produced relatively large oscillations because the two individual PID loops did not properly coordinate the pump and fan. With the proposed NMPC, the maximum battery temperature was 28.37°C and the maximum overshoot was 1.37°C. Moreover, NMPC returned the temperature to the target faster after a sudden load change. The temperature fluctuation of the high-voltage battery under NMPC was smaller and more robust than the two conventional strategies. Table 10 presents the cumulative energy consumption and relative comparison.
| Strategy | Total energy consumption (kJ) | Compared to threshold control | Compared to PID control |
|---|---|---|---|
| Threshold control | 10000.87 | – | – |
| Dual PID control | 8585.67 | -14.15% | – |
| NMPC | 7613.71 | -23.87% | -11.32% |
NMPC achieves the lowest energy consumption among the three. Compared with threshold control, NMPC reduces the energy consumption by 23.87%, and compared with PID control, it reduces energy consumption by 11.32%. Similar to the temperature response, the actuator speed profiles show interesting differences. In the threshold strategy, the fan and pump operate mostly at maximum speed in frequent on-off bursts. In the PID strategy, the pump and fan speeds vary continuously, but the two profiles are not synchronized efficiently. In the NMPC strategy, the pump and fan speeds are thoughtfully balanced: when the thermal load is small, the controller reduces the pump speed; when the radiator cooling is sufficient, the controller avoids overspeeding the fan. This coordination is the main source of the energy saving.
6.3 High-temperature cooling results
The most challenging scenario is the high-temperature environment, because the vapor-compression refrigeration system consumes a significant part of the high-voltage battery energy. I set the ambient temperature to 35°C and the initial high-voltage battery temperature to 35°C. The target temperature in the cabin-independent battery thermal management was set to 27°C. Three strategies were investigated: threshold control, PID control, and GWO-optimized nonlinear model predictive control. In the threshold strategy, the compressor and pump were turned on when the battery temperature exceeded 28°C and turned off when the temperature fell below 27°C. Because the compressor draws a large current, this on-off control inevitably causes voltage fluctuations and energy waste. In the PID strategy, both compressor and pump were adjusted by feedback controllers. In the GWO-NMPC strategy, the predictive controller was solved by the grey wolf optimizer to generate smooth and coordinated speed commands.
The simulated high-voltage battery temperature histories show that all three strategies can bring the initial 35°C battery down to around 27°C within about 300 seconds. After that, the vehicle enters the urban phase of the driving cycle. The threshold control exhibits recurrent temperature fluctuations of roughly ±0.5°C because the compressor is discontinuously switched. PID control performs better at rejecting the intermittent load changes, but its compressor speed fluctuates more actively and has a slight oscillatory tendency. The GWO-NMPC strategy keeps the battery closest to the target, with a maximum absolute error smaller than approximately 1°C, and avoids the unnecessary sudden compressor actions.
Figure 1 shows the compressor speed results of the three controllers. The threshold control behaves like a relay: the compressor is either at full speed or zero. The PID control produces frequent corrections, leading to a jagged speed profile. The GWO-NMPC control outputs a smooth speed profile that follows the slowly varying thermal load. Similar behavior is seen for the pump speed. The overall energy consumption in the high-temperature mode is summarized in Table 11.
| Strategy | Total energy consumption (kJ) | Compared to threshold control | Compared to PID control |
|---|---|---|---|
| Threshold control | 23561.3 | – | – |
| PID control | 13135.4 | -44.25% | – |
| GWO-NMPC | 11473.5 | -51.31% | -12.62% |
Because the compressor consumes much more power than the pump and fan, energy saving in high-temperature mode is more pronounced. The threshold strategy wastes energy through repeated startup and shutdown transients. PID control reduces energy compared with threshold control, but the continuously oscillating compressor speed also consumes extra energy. The GWO-NMPC strategy, with its smooth predictive optimization, reduces total energy by 51.31% relative to threshold control and by 12.62% relative to PID control. More importantly, these savings are achieved without compromising the temperature tracking performance. The high-voltage battery is maintained in a narrow band around the target, which is beneficial for cell lifetime and safety.
7. Discussion on Temperature-Energy Synergy for High-Voltage Batteries
The results of this study confirm that a high-voltage battery thermal management system should not be viewed merely as a temperature-protection device; it is an energy system that interacts with the vehicle power split and driving range. For a pure electric vehicle, every watt of compressor power comes from the same high-voltage battery that supplies the traction motor. Thus, reducing thermal management energy consumption directly increases the miles that can be driven for the same battery capacity. The temperature-energy synergistic control philosophy is therefore essential.
Several important design insights can be derived from the simulation results. First, for cooling the high-voltage battery, the pump speed should be adjusted according to the heat load. At low heat generation, a low pump speed is sufficient and saves pump energy. When vehicle acceleration or high ambient temperature increases the heat rejection requirement, the pump speed should be gradually increased, not abruptly to full speed. Second, the compressor is the most energy-intensive component in high-temperature operation. A nonlinear model predictive controller can anticipate the future power demand of the high-voltage battery and pre-cool it before the battery temperature reaches a dangerous threshold. In contrast, threshold control only reacts after the temperature limit is crossed, causing overshoot and frequent cycling. Third, fan speed has a weak influence on the high-voltage battery temperature in the studied configurations. The major role of the fan is to improve the condenser heat rejection. It should be adjusted only when the compressor is already operating at a high speed and the ambient temperature permits effective heat exchange with the radiator.
The relation between temperature and energy is not simply linear. The objective functions used in this study penalize the squared temperature deviation and the actuator power simultaneously. The weighting coefficients \alpha_i and \beta_i provide a way to tune the controller according to the operating strategy of the electric vehicle. In a range-priority mode, more weight can be placed on the energy consumption terms, allowing the battery temperature to drift closer to its upper boundary. In a performance-priority mode, the temperature error can be weighted more heavily, ensuring that the high-voltage battery operates in the most favorable temperature range for high power output. This hierarchical tuning possibility makes the proposed framework suitable for integration into a vehicle-level energy management controller.
Additionally, the study highlights the importance of sampling-based factor screening in the development of thermal management control systems. Without the ANOVA results, it would be tempting to use all three actuators as independent control variables in the predictive controller. However, because the fan speed has almost no influence on the high-voltage battery temperature in the tested ranges, including it in the controller would increase the computational burden and possibly lead to unnecessary actuation. The simplified control structure that I selected after the screening study—pump and fan in normal-temperature, pump and compressor in high-temperature—reduces the optimization search space to two dimensions while preserving all critical physical behavior. This simplification contributes to a stable and real-time implementable controller.
8. Conclusion and Future Work
In this research, I have established a complete development chain for a high-voltage battery thermal management system that integrates temperature control and energy consumption optimization. The work covers system modeling, parameter identification, factor screening, control strategy design, and comparative validation. The main conclusions can be summarized as follows.
First, the lumped-parameter electro-thermal model of the high-voltage battery, calibrated through experimental tests, is able to reproduce the measured temperature rise with errors below 3°C. The calibrated condenser and chiller models have heat transfer errors below 5% in the investigated operating range. This level of accuracy is sufficient for system-level thermal management studies and for the evaluation of control strategies.
Second, the optimal Latin hypercube sampling design combined with variance analysis offers a systematic method to identify critical variables in a multi-parameter thermal management system. The compressor speed and pump speed are significant for both the battery temperature and system energy consumption, while the fan speed has a negligible effect on battery temperature and an adverse effect on energy consumption. Based on this evidence, the fan was treated as an auxiliary input, simplifying the controller without sacrificing performance.
Third, the temperature-energy coordinated control strategies show clear advantages over conventional approaches. In the low-temperature environment, PID control of the positive temperature coefficient heater reduces energy consumption by 10.37% compared with threshold control. In the normal-temperature environment, the nonlinear model predictive controller reduces total energy consumption by 23.87% compared with threshold control and by 11.32% compared with dual PID control, while maintaining the high-voltage battery temperature at 27°C with smaller oscillations. In the high-temperature environment, the grey-wolf-optimized nonlinear model predictive controller reduces total energy consumption by 51.31% compared with threshold control and by 12.62% compared with PID control. The high-voltage battery temperature remains close to 27°C with no unacceptable overshoot, proving that the proposed approach can handle the nonlinear and coupled nature of the thermal system.
There are several opportunities for future work. The current model assumes a uniform temperature distribution inside the high-voltage battery pack. A more detailed 3D thermal model could capture temperature gradients and hot spots, allowing the controller to preserve battery health more accurately. Another direction is to include battery degradation and fast-charging scenarios in the objective function. During fast charging, the heat generation rate of the high-voltage battery is extremely high and the cooling system must operate at its limit. Predictive control can be extended to coordinate the charger power with the cooling power. Finally, the computational efficiency of the nonlinear predictive controller can be further improved by using tailored optimization algorithms or approximate dynamic programming, enabling real-time deployment on production vehicle control units. With these improvements, the temperature-energy synergistic design will become even more practical for next-generation electric vehicles.
Through the systematic methodology and simulation evidence presented in this paper, I have demonstrated that an advanced high-voltage battery thermal management system can deliver both precise thermal regulation and significant energy saving. The proposed temperature-energy synergistic framework is expected to provide a valuable reference for the engineering design and control development of high-voltage battery thermal management systems in modern and future electric vehicles.