The development of lithium-ion battery technology has ensured that battery thermal management systems are an essential component of the battery pack for next-generation energy storage systems. Using dielectric immersion cooling, researchers have demonstrated the ability to attain high heat transfer rates due to the direct contact between cells and the coolant. However, feedback control has not been widely applied to immersion cooling schemes. Furthermore, current research has not considered battery pack plant design when optimizing feedback control. Uncertainties are inherent in the cooling equipment, resulting in temperature and flow rate fluctuations. Hence, it is crucial to systematically consider these uncertainties during cooling system design to improve the performance and reliability of the battery pack. To fill this gap, we established a reliability-based control co-design optimization framework using machine learning for immersion cooled battery packs. We first developed an experimental setup for 21700 battery immersion cooling, and the experiment data were used to build a high-fidelity multiphysics finite element model. The model can precisely represent the electrical and thermal profile of the battery. We then developed surrogate models based on the finite element simulations in order to reduce computational cost. The reliability-based control co-design optimization was employed to find the best plant and control design for the cooling system, in which an outer optimization loop minimized the cooling system cost while an inner loop ensured battery pack reliability. Finally, an optimal cooling system design was obtained and validated, which showed a 90% saving in cooling system energy consumption.

As the popularity of consumer electronics, electric vehicles (EVs), and renewable electricity generation grows, the demand for lithium-ion batteries (LIBs) is increasing.1 Recent advances in battery technology enable better LIBs for higher-performance EVs and provide better grid-scale balancing of renewable electricity generation.2 In addition, the reliable protection of vehicles or the power grid has led to the further development of battery thermal management systems (BTMSs).3 An effective BTMS ensures the safety of the LIBs while maximizing their lifetime and power capability and mitigating the potential for thermal runaway by maintaining the LIB temperature within an optimal range.4 The primary motivation for utilizing BTMSs is to remove heat from a battery pack, which leads to more flexibility in controlling the operational temperature and helps to achieve a more uniform temperature within individual cells and among different cells within the pack. Various BTMSs have been utilized for decades, from air cooling systems to indirect liquid-based methods employing cold plates and heat pipes.5–7 

Air cooling systems emerged as the first and most employed BTMSs. Air cooling BTMS is favored for its straightforward design and implementation, lightweight construction, ease of maintenance, and cost-effectiveness. Two general types of air cooling for LIBs were identified: natural convection and forced convection.8,9 Natural convection air cooling takes advantage of natural airflow through the battery pack, achieving an approximately 10 W/(m2 · K) heat transfer coefficient for heat rejection.10 Forced convection air cooling is more effective than natural convection air cooling due to the forced airflow ensuring better heat transfer. However, forced air cooling systems are heavier and occupy a larger space due to the requirement of additional equipment and energy requirements.9 While forced air cooling enhances performance, air-cooled BTMSs still face challenges in handling extreme conditions, including high or low ambient temperatures, abrupt battery failures, and prolonged high C-rate operation.11–13 Liquid-based systems (direct or indirect) typically exhibit a greater capacity to absorb and reject heat when compared to air-cooled systems. Many studies have focused on LIB indirect-cooled BTMS development.14,15 Typically, these systems comprise of cold plates with multiple LIBs sandwiched between them. Indirect liquid-based systems frequently demonstrate better ability to absorb and dissipate heat when compared to air-cooled systems, thanks to their enhanced convective heat transfer performance.16,17 However, the compromise for this increased performance is increased system complexity and mass. Furthermore, liquid-cooled systems have encountered challenges sustaining a uniform temperature distribution of LIBs. A promising substitute to traditional methods is dielectric immersion cooling, wherein cells come into direct contact with an electrically insulating fluid. This method takes significant advantage of the heat capacity of fluid to attain exceptionally high heat transfer rates due to the direct contact between the cell walls and the fluid. This approach provides liquid access to a broader cooling surface area and increases the proximity of the fluid to hot spots.18 

The control from the battery management system can cause abnormal batteries to shut off.19 This control process is the same independent of the cooling method used. Immersion cooling can prevent both thermal runaway of batteries and thermal runaway propagation thanks to the use of an inflammable coolant.20 Currently, battery cells are connected using spot welding in battery packs, which makes replacement of individual cells challenging. Immersion cooling increases the difficulty of replacement due to the direct contact of battery cells with the cooling fluid. The immersion fluid must be carefully managed to prevent leakage or contamination during maintenance. Additionally, removing and replacing a single battery cell can be complex and may require draining the coolant or isolating the affected section to prevent system-wide disruptions. This complexity can increase maintenance time and cost. Therefore, it is crucial to design immersion cooling systems with modular and accessible battery arrangements to facilitate easier maintenance and replacement. Ensuring robust sealing and fluid management strategies can also help mitigate these challenges and enhance the practicality of immersion cooling systems in real-world applications.

However, there is a lack of control-oriented studies that focus on dielectric fluid immersion cooling of LIBs.21 The majority of dielectric immersion cooling studies use a constant coolant flow rate, which cannot provide the most efficient cooling solution for LIBs.22–24 Due to the dynamic heat generation rate during the usage of the LIBs, a constant coolant flow rate can provide neither optimal nor reliable cooling in order to maintain a uniform LIB temperature. Thus, feedback control becomes a feasible solution for LIBs dielectric immersion cooling.25 Although some research has used feedback control for LIB dielectric immersion cooling, the distance between the battery cells has not been considered while designing the cooling system.26–28 Past research has considered the distance between the battery cells to be plant design; however, optimization was processed separately for the plant design and the control design.29 This led to suboptimal results, which cannot comprehensively optimize the BTMS considering cooling and cost. To tackle these challenges, this study aims to optimize the total energy consumption of a representative BTMS while ensuring the safety and reliability of the LIBs by simultaneously optimizing the distance between the battery cells and the target temperature of the feedback control scheme.

Using a novel control co-design method, this study jointly considers the plant design and control design of a BTMS. The uncertainties inherent to the cooling equipment cause fluctuations of coolant temperature and flow rate and must be considered during optimization. Traditional deterministic design optimization models can achieve optimal designs, which can find the designs near constraint boundaries of the design space. However, the lack of consideration of the uncertainties inherent in the cooling equipment leads to reduced BTMS reliability, causing significant cooling performance variability and potential LIB failure. As a solution, reliability-based design optimization (RBDO) methods can incorporate these uncertainties into the design process to ensure the reliability of the BTMS and to generate robust and reliable designs.30 

Traditional trial and error approaches are not feasible for current BTMS designs due to the complexity of the thermal system.31 The reliability-based control co-design problem is expensive to solve even with the help of multiphysics finite element (FE) simulations. To address this issue, this study adopts combined optimization methods with BTMS experiments, multiphysics FE simulations, and a data-driven surrogate approach using machine learning to accelerate the optimization process for BTMSs. Using experimental results from LIB dielectric immersion cooling, the electrical and thermal properties of the LIB and coolant are obtained. The experimental data were further used to refine the multiphysics FE model. Based on the FE simulations, a more complicated real-world battery pack was built. In addition, using symmetry of the face-centered cubic layout of cylindrical cells within the battery pack, the multiphysics FE model was simplified to include only a part of two-column LIBs as shown in Fig. 1. Using the high-fidelity FE model, the battery pack under harsh loading conditions was simulated using a cost-effective and flexible approach when compared to physical experiments.32 However, due to the complexity of the BTMS, the computational cost of the multiphysics FE simulation ensured that the optimization process was slow. Machine learning (ML) provides a promising solution, reducing the computational cost by providing surrogate models which represent complex systems.33–36 Furthermore, ML shows excellent potential in quantifying uncertainties in simulations, which can provide more reliable results.37 Various ML methods have been employed in engineering problems, including support vector machines,38 response surface methods,39,40 artificial neural networks,41,42 and Gaussian process (GP) models.43,44 Among these methods, the GP model is a promising candidate for complex systems due to its superior performance.45 Furthermore, the GP model can estimate model uncertainty, making it able to be integrated into RBDO problem-solving methodologies. With the help of Monte Carlo simulations (MCSs) and sampling methods such as Latin hypercube sampling (LHS),46 the GP model can predict the outputs of a complex system for any desirable design input through the regression model and random process. Using these techniques, our work shows potential of ML approaches for future BTMS design.

FIG. 1.

Battery thermal management system with direct contact immersion cooling.

FIG. 1.

Battery thermal management system with direct contact immersion cooling.

Close modal

This study leverages ML to improve BTMS through reliability-based control co-design optimization. The primary aim is to lower the overall BTMS energy consumption by optimizing the plant design (LIB battery distance) and control design (target temperature) of the BTMS while considering uncertainties related to BTMS operation. We use experimental data to develop high-fidelity FE models to simulate a BTMS under harsh loading conditions. A training data set was generated from the FE models within the design space using the LHS method. The data set was fed to the ML model for RBDO, leading to the identification of an optimal BTMS design. The results of this study highlight the effectiveness of employing surrogate models within the RBDO framework. Moreover, we provide a framework for BTMS designs that can meet reliability criteria while significantly enhancing computational efficiency. The paper is organized as follows: Sec. II explains the immersion cooling experiment; Sec. III develops the FE model; Sec. IV outlines the Kriging-based surrogate models used for the FE model and the RBDO framework, as well as implementation to find the optimum design parameters; and Sec. V shows validation of the surrogate models and the optimal BTMS designs. This study provides a framework for the design of next-generation BTMSs.

Figure 2(a) depicts the schematic layout of the battery immersion cooling setup. One 21700 cylindrical battery cell (INR21700-50E, Samsung SDI) was tested. The test bench includes a circulation bath (AD07R-40-A11B, PolyScience), a battery tester (LBT21084, Arbin Instruments), a Coriolis flow meter (CMF050M, Emerson Electric), a motorized needle valve (ENV-0225, Enfield Technologies), a valve driver (D5-02-U01, Enfield Technologies), and a 3D-printed chamber housing the battery. The specially designed 3D-printed enclosure emulates the environment for battery immersion cooling, ensuring a controlled experimental setup. The chamber was printed using a Formlabs Form 3 SLA Printer with Formlabs clear resin. Transparent printing material enables optical access for the experiments, ensuring solid attachment of the thermocouples to the battery surface while also being compatible with the working fluid without causing any deformation. The chamber also features a 5.08 mm diameter inlet and outlet. The inlet and outlet face opposite sides of the battery. As shown in Fig. 2(b), the inlet is lower than the outlet, which ensures immersion conditions for the battery cell body. A 20 mm diameter central hole at the top of the chamber allows easy insertion of the battery cell. A small section (∼2 mm height) of the battery protrudes outside of the chamber at the top and bottom to solder the bus bars (nickel strips having 0.2 mm thickness) to the battery tabs. The radial gap between the cell and the chamber edge is filled with anti-corrosion sealant (Marine Weld 8272, JB-Weld), preventing any fluid leakage, and stabilizing the cell from rotating or moving in the liquid flow around it. Three calibrated thermocouples (TC-TT-K-40-36, Omega Engineering) were attached to the cell body to monitor local temperatures during the charging/discharging processes. Three calibrated thermocouples (TC-TT-K-40-36, Omega Engineering) were attached using tiny pieces of highly conductive aluminum tape to minimize the additional thermal resistance between the cells and coolant. Thermocouples were placed in equal distances from each other (∼35 mm) at the highest, middle, and lowest points on the cell body to monitor local temperatures during the charging/discharging processes to investigate temperature uniformity. Each thermocouple had a ±0.268 °C uncertainty in the measuring span of 0–260 °C. For data acquisition and logging, LabVIEW software interfaces with a National Instruments DAQ system (cDAQ-9174, CompactDAQ chassis), employing an NI 9213 PLC module for temperature measurements, an NI9203 analog current input module for flow rate measurements, and an NI9265 C-Series DAQ to generate current-output signals operating the needle valve. The circulation bath, filled with 3M Novec 7300 dielectric fluid, maintains temperature stability of ±0.1 °C and provides consistent pumping power for coolant circulation. The motorized needle valve, driven by a current-output signal actuator, enables precise flow rate control from 0.16 to 1.23 kg/min, programmable through the LabVIEW program. The Coriolis flow meter, having a ±0.1% accuracy of the reading flow rate, directly measures the mass flow rate of the coolant, ensuring precise data collection.

FIG. 2.

(a) Schematic layout of the battery immersion cooling experimental setup. Schematic not to scale. (b) Photograph of the one-cell battery immersion cooling setup showing the 3D-printed chamber and the blue LiB cell inserted inside.

FIG. 2.

(a) Schematic layout of the battery immersion cooling experimental setup. Schematic not to scale. (b) Photograph of the one-cell battery immersion cooling setup showing the 3D-printed chamber and the blue LiB cell inserted inside.

Close modal

To emulate harsh loading condition for the battery pack, the test was conducted using a 2C constant current (CC) discharging (9.8 A, the highest continuous discharge limit on the battery datasheet) with a cutoff voltage of 2.5 V. Prior to each discharging process, the battery used constant current constant voltage (CCCV) charging at 1C with a 98 mA cutoff current to ensure the same capacity of the battery cell. The needle valve was calibrated using the reading from the Coriolis flow meter. Different flow rates were tested for the 2C discharging processes. Both the battery tester and NI DAQ system recorded the timestamp, and the temperature data were later processed with the discharging profile based on the timestamp. The temperature readings of the three thermocouples were almost the same, similar to our previous temperature readings from multiple thermocouples and an infrared (IR) camera of the battery cell with natural convection.25 At higher flow rates, cooling of the battery cell was better. Also, open-circuit voltage (OCV) testing was conducted. The battery state-of-charge (SOC) was continually measured. These data was processed to obtain the OCV-SOC curve of the battery for building the FE model. For each discharge test, the batteries must be fully charged and then the user must wait for the batteries to cool down. This process takes much longer than the battery discharge time. In addition, whenever we change the plant design parameters, a new container must be 3D printed, and the batteries must be manually sealed, which is time-consuming and expensive. Thus, simulation is needed to speed up the process and lower the time and monetary cost.

High-fidelity FE models were built in COMSOL Multiphysics 6.0 using the obtained experiment data as tuning parameters. According to the similarity of the temperature data from different thermocouples in the vertical direction, a simplified 2D axisymmetric model was adopted to accelerate the simulation process. Furthermore, experiments using battery cell cooling with natural convection shoed that the temperature of the top and the bottom of the cell is typically lower than the middle part of the cell.47 Because, there is no active material on the top and the bottom of the cell, and majority of heat is generated within the active material. Moreover, we confirmed using an IR camera that the weld on top and bottom of the battery did not cause unwanted temperature rise. Hence, the 2D axisymmetric model is a good representation of the battery pack with cylindrical cells.

We built the battery cell model based on the geometry data obtained from the datasheet and cross-sectional measurements. As shown in Fig. 1, there are four layers within the INR21700-50E battery cell: the mandrel, the active material, the canister, and the wrap. According to the safety datasheet, the cathode is lithium nickel cobalt aluminum oxide (NCA), the anode is graphite and silicon, and the electrolyte salt is lithium hexafluorophosphate. Copper and aluminum serve as the current collectors. A lumped battery module was adopted with input from the experiment OCV-SOC data, which uses a battery equivalent circuit with a small set of lumped parameters. The cell voltage ( E c e l l ) can be obtained by the OCV based on SOC and temperature [ E OCV ( SOC , T ) ], ohmic overpotential ( η a c t ), and activation overpotential ( η a c t ) as
E c e l l = E OCV ( SOC , T ) + η I R + η a c t ,
(1)
where the OCV at a specific temperature can be calculated by the OCV at the reference temperature [ E OCV ( SOC , T r e f ) ]. This can be obtained from OCV-SOC data at a reference temperature ( T r e f ),
E OCV ( SOC , T ) = E OCV ( SOC , T r e f ) + ( T T r e f ) E OCV ( SOC ) T .
(2)
In the simulation, the SOC of the battery was calculated through time evolution based on the battery cell capacity ( Q c e l l , 0 ) and current ( I c e l l ),
S O C t = I c e l l Q c e l l , 0 .
(3)
The ohmic overpotential ( η I R ) was obtained from the ohmic overpotential at 1C ( η I R , 1 C ), with the current at 1C ( I 1 C ),
η I R = η I R , 1 C I c e l l I 1 C .
(4)
The activation overpotential ( η a c t ) on electrode surfaces can be calculated as
η a c t = 2 R T F asinh ( I c e l l 2 J 0 I 1 C ) ,
(5)
where R denotes the molar gas constant, F is Faraday's constant, and J 0 is the dimensionless charge exchange current. Only the active material layer is defined in the lumped battery module for the battery cell. Heat transfer in the solids and fluids model is adopted, and the temperatures of other components are calculated by solving for heat transfer in solid, with consideration of the specific heat capacity ( C p ), density ( ρ ) of each material, absolute temperature ( T ), the velocity vector of translational motion ( u ), heat flux by conduction and radiation ( q , q r ), coefficient of thermal expansion ( α ), second Piola–Kirchhoff stress tensor ( S ), and additional heat sources ( Q ),
ρ C p ( T t + u T ) + ( q + q r ) = α T : d S d t + Q .
(6)
We consider the dielectric immersion cooling system. 3M Novec 7300 dielectric fluid has a thermal conductivity of 0.063 W/(m K)48 and a specific heat capacity of 880 J/(kg K).49 The fluid domain heat transfer is calculated using the energy equation by
ρ C p ( T t + u T ) + ( q + q r ) = α p T ( p t + u p ) + τ : u + Q ,
(7)
where α p is the coefficient of thermal expansion. The boundary of the battery pack is thermally insulated. The inlet coolant temperature and flow rates are set to be the same as those observed in the experimental setup.

Based on the experiment setup, the Reynolds number (Re) can be calculated. With the lowest flow rate, the Reynolds number can be obtained as 69 770, which is higher than 2300. So, turbulence model should be adopted in FE simulation.50 After comparing different turbulence models, the k–ɛ turbulence model was adopted in the turbulent flow module due to its high accuracy for the one-cell scenario. Also, the k–ɛ turbulence model is a popular choice for industrial applications due to its good convergence rate and relatively low memory requirements.51 While building the FE model, some parameters could be directly tested through experiments. Other parameters that could not be directly tested through experiments were obtained through a parameter estimation process. In the parameter estimation process, the experimental thermal and electrical behavior of the battery was set as the target valve that simulation results need to reach. By adjusting the input parameters such as ohmic overpotential at 1C, dimensionless charge exchange current, and diffusion time constant, simulation results were able to get closer to experimental thermal and electrical behavior of the battery. Figure 3 shows the electrical and thermal profiles of the simulation and experiment, demonstrating that the FE model achieves high fidelity when compared to the experiments.

FIG. 3.

(a) Battery voltage and (b) temperature as a function of SOC for the FE simulations and experiments. Experiments conducted using the 2C discharge processes at Novec flow rates of 2.15 × 10 6 and 1.17 × 10 5 m 3 / s with an inlet temperature of 25 °C.

FIG. 3.

(a) Battery voltage and (b) temperature as a function of SOC for the FE simulations and experiments. Experiments conducted using the 2C discharge processes at Novec flow rates of 2.15 × 10 6 and 1.17 × 10 5 m 3 / s with an inlet temperature of 25 °C.

Close modal

After successfully implementing the single-cell immersion cooling model, we scaled up the simulation for a battery pack. As shown in Fig. 1, the battery pack has four rows of batteries with a face-centered cubic layout. The distance between battery cells (d) is equal to the distance between battery cells and the walls of the battery pack. There is one inlet and one outlet facing each column of the cells. Due to the symmetry of the battery pack, part of the two columns of battery cells was chosen for simulation, and the boundaries on the sides were set as symmetry conditions (adiabatic).

A proportional integral differential (PID) controller was added to the model with the reading of the highest temperature of the battery cells ( T m a x ) at a certain time. The feedback control adjusts the coolant flow rate to ensure T m a x stays within the optimal range of 15–35 °C to maintain good performance.52 The coolant inlet flow rate ( Q ˙ i n l e t ) can be adjusted based on the target temperature ( T t a r g e t ), propositional control gain ( K P ), and the bias ( Q ˙ b i a s ), according to
Q ˙ i n l e t = K P ( T t a r g e t T m a x ) + Q ˙ b i a s .
(8)

The boundary of the battery pack is thermally insulated, and Tref was set to 25 °C. The battery pack under harsh loading conditions (2C discharging) was simulated. According to industrial applications, the distance between battery cells within a battery pack ranges from 1 to 5 mm.53 Moreover, the target temperature for cooling control of a battery pack is typically set to 25–35 °C.54 From the simulations, the pressure drop of the battery pack can be obtained, which can be used for the cooling system energy consumption calculation.

For the BTMS design, the battery pack layout and the control parameters need to be optimized simultaneously to optimize the BTMS considering the combined effects of cooling and cost. This study is focused on the distance between battery cells and the target temperature of the feedback control. The control co-design method can connect the control and plant designs efficiently.55 We study the BTMS of hybrid electric vehicles under harsh loading scenarios. The energy consumed by the BTMS ( W t o t a l ) is consisted of two parts: energy consumed by pumping the coolant ( W p ) and energy consumed by carry the weight of coolant ( W c ),
W t o t a l = W p + W c .
(9)
The control co-design problem can be formulated as
min d , T t a r g e t W t o t a l ( d , T t a r g e t ) , s . t . T m a x 35 ° C 0.
(10)
For design parameters d and T t a r g e t, they are allowed to range from 1 to 5 mm and 25–35 °C. The energy consumed by pumping the coolant can be calculated by the coolant flow rate ( Q ˙ ) and the pressure drop ( Δ p ),
W p = Q ˙ Δ p d t .
(11)
When the distance between battery cells changes, the size of the battery pack will change with the same number of cells. As a result, the mass of the coolant will change due to the change in the battery pack volume. For electric vehicles, the mass directly determines the range of the vehicle as more energy needs to be consumed for a heavier battery pack. Curb weight versus energy consumption from commercial EVs was collected and analyzed to evaluate the energy consumed to carry the mass of the coolant.56 Approximately, 36.3 kJ needs to be consumed to carry 1 kg of the weight with an EV for every 100 km. The typical range of an EVs during the harsh discharge is 40 miles (64.27 km), according to data on commercially available vehicles.57 For the two columns of battery cells, the mass of the coolant (M) is dependent on the distance between battery cells. Furthermore, the energy consumed to carry the coolant of the battery pack during the discharge process can be calculated based on the mass of the coolant and coefficient (k = 23.4kJ/kg),
W c = k M .
(12)

Though control co-design methods can provide the solution to comprehensively consider the plant design and control design, the uncertainties of the cooling equipment cause fluctuations of inlet coolant temperature and flow rate and have not been considered in prior optimization studies. Traditional methods often treat optimization problems deterministically, potentially ignoring uncertainties in the system parameters, operating conditions, and modeling inaccuracies.58 Thus, they obtained the optimal design that perform well under nominal conditions but fail to meet reliability requirements under variability.59 RBDO methods explicitly incorporate uncertainties into the optimization process. Thus, RBDO is needed to find robust and reliable design solutions.

In RBDO, an outer design loop utilizes the objective function from the co-design method. In addition, an inner loop achieves the evaluation of probabilistic constraints related to reliability and uncertainty propagation processes. This approach can improve the reliability and robustness of BTMS designs while considering the complex interplay of uncertainties in the design process. The RBDO model for BTMS design can be expressed as
{ min ( d , μ X ) W t o t a l ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) s . t . Pr { T max ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) 35 ° C } [ R i ] = 1 [ p f i ] , i = 1 , 2 , , n g , d L d d U , T t a r g e t L T t a r g e t T t a r g e t U , T i n l e t N ( μ T i n l e t , σ T i n l e t 2 ) , k Q ˙ i n l e t N ( μ k Q ˙ i n l e t , σ k Q ˙ i n l e t 2 ) .
(13)

In the model described above, the following components are present:

  1. d and T t a r g e t are the design variables, while d L, T t a r g e t L and d U, T t a r g e t U represent the lower bounds (1 mm, 25 ° C) and upper bounds (5 mm, 35 ° C).53,54

  2. T i n l e t and k Q ˙ i n l e t are independent random parameters that are beyond the control strategy and are inherent in the BTMS. These parameters cause fluctuation of the inlet coolant temperature and flow rate.

  3. W t o t a l ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) is the objective function (total energy consumption of BTMS) considering the fluctuation of coolant temperature and flow rate.

  4. T m a x ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) 35 ° C is a constraint function. The battery temperature needs to stay within the optimal range of 15 35 ° C to maintain high performance.52 So, the maximum temperature during the battery discharging process ( T m a x ) should not exceed 35 ° C with a desired reliability [ R i ] or 1 [ p f i ], where [ p f i ] represents the prescribed allowable probability of failure. Here, the reliability is set as 0.99 ( [ p f i ] = 1 % ).

  5. T i n l e t is the coolant inlet temperature in a realistic scenario, which is set to be 25 ° C. However, due to uncertainties inherent to the chiller, it is not exactly 25 ° C at all times. Based on a past experiment studying the BTMS of EV,60 the standard deviation is set to 0.9 ° C.

  6. k Q ˙ i n l e t is the coefficient of inlet coolant flow rate. The flow rate is adjusted based on feedback control and k Q ˙ i n l e t should always be 1. However, due to the uncertainty of the pump, there exists an error from the pump. According to the tolerance of ISO 9906, the standard deviation of k Q ˙ i n l e t is 0.017.

The overall process used for solving the RBDO model is shown in Fig. 4. The procedure initiates this with the outer optimization loop, in which the design variables are varied iteratively to find the optimal solution to minimize W t o t a l. While finding the optimal solution of the outer loop, a reliability analysis is conducted to assess the probability of system failure through the inner loop. This inner loop analyzes reliability based on the uncertainties inherent to the BTMS. During each iteration of the outer loop, the calculated likelihood of failure is checked against predetermined reliability standards. If the calculated likelihood of failure does not meet these specified reliability standards, the design variables are modified based on the reliability analysis results. This iterative process of refining the design in the outer loop and evaluating reliability in the inner loop continues until a design meets performance and reliability requirements. The design will be finalized if the optimization process converges to a solution that meets all constraints. As a result, the optimal set of design variables is returned.

FIG. 4.

Reliability-based design optimization flowchart. The blue shaded region represents the inner loop.

FIG. 4.

Reliability-based design optimization flowchart. The blue shaded region represents the inner loop.

Close modal
The MCS method was employed to solve the reliability analysis. It can approximate multidimensional integration by sampling the random input space with a sufficiently large sample size, effectively reducing the complexity of multidimensional integration of the joint probability density function across an unknown failure domain in the system random inputs. The sample size of the MCS is based on the desired coefficient of variation for the reliability estimate. The MCS method evaluates numerous sample points across the random input space for reliability analysis. This allows for estimation of the probability of failure by comparing the number of sample points within the failure region to the total number of sample points used. The indicator function is often used mathematically to check if a sample point falls within the failure region,
R i = Pr { T m a x ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) 35 ° C } , = T m a x ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) 35 ° C I ( x ) f T i n l e t , k Q ˙ i n l e t ( T i n l e t , k Q ˙ i n l e t ) d T i n l e t d k Q ˙ i n l e t , = E ( I ( x ) ) ,
(14)
where f T i n l e t , k Q ˙ i n l e t ( T i n l e t , k Q ˙ i n l e t ) is the joint probability density function of the system random inputs and I ( x ) represents an indicator function, which can be defined as
I ( x ) = { 1 , if T m a x ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) 35 ° C , 0 , if T m a x ( d , T t a r g e t , T i n l e t , k Q ˙ i n l e t ) > 35 ° C .
(15)

The expectation of the indicator function across the random input space can be represented as a system reliability probability. This expectation can be approximated by employing a substantial quantity of sample points derived from d, T t a r g e t, T i n l e t, and k Q ˙ i n l e t, randomly drawn from the input space. However, due to the complex and nonlinear relationships between BTMS design and battery pack performance, solving the problem solely through FE models is challenging and computationally expensive. Employing the MCS method for reliability analysis for the battery pack immersion cooling FE model demands many performance function evaluations, resulting in a prohibitively high computational cost. To solve this problem, ML enhanced surrogate models can be adopted to replace the battery pack immersion cooling FE models, enabling more efficient system reliability estimation by applying the MCS method based on the developed surrogate models.

Due to the complexity of the FE model, the simulation is time-consuming (more than 12 min for each simulation on a server with 48 CPUs). If each iteration directly relies on the FE model for the optimization process, it will be computationally expensive and time-consuming. By adopting surrogate models, the outcomes can be predicted in a much shorter time than finite element simulations.61 Even considering the training process of the machine learning models, the surrogate models still greatly improved computational efficiency. Thus, surrogate models are needed to speed up the optimization process. The GP-based surrogate model is able to generate an estimated surface from a scattered set of points via a covariance-governed GP interpolation method. Using this property, it can use the training samples to build the model and predict new response sample points. Besides, the GP-based surrogate model provides probabilistic predictions, offering point estimates and uncertainty bounds for these predictions.62 GP-based modeling also has the flexibility to be adapted to the complexity of the data without being constrained by a predefined structure, so it can be suitable for a wide range of problems and simulation data types. The GP-based surrogate model is applied as
Y ( x ) = T ( x ) + S ( x ) ,
(16)
where Y ( x ) is the prediction results of the performance function at point x using the GP model, T ( x ) is a polynomial term of x that interpolates the input sample points, and S ( x ) represents a Gaussian stochastic process with zero mean and variance σ 2. The polynomial term T ( x ) is the product of the regression basis function f and the regression coefficients β,
T ( x ) = f T ( x ) β ^ .
(17)
The covariance function between two arbitrary input points x i and x j can be defined as
Cov [ S ( x i ) , S ( x j ) ] = σ 2 R ( x i , x j ) ,
(18)
where R ( x i , x j ) represents the process variance and R ( ) is the correlation function. It can be calculated by x i k and x j k, which are the kth components of x i and x j. Here, dim is the dimension of x i and θ k is a parameter that indicates the correlation between the points in dimension k,
R ( x i , x j ) = exp [ k = 1 d i m θ k ( x i k x j k ) 2 ] .
(19)
GP provides not only the prediction at an untried point but also the variance of the prediction thanks to the stochastic characteristics. The variance indicates the uncertainty of the prediction. At an untried point x, the GP predictor g ^ ( x ) follows a Gaussian distribution denoted by
R g ^ ( x ) N ( μ Y ( x ) , σ Y 2 ( x ) ) ,
(20)
where μ Y ( x ) and σ Y 2 ( x ) are the prediction and its variance,
μ Y ( x ) = f T ( x ) β ^ + r ( x ) R 1 ( y F β ^ ) ,
(21)
σ Y 2 = σ ^ Y 2 { 1 r ( x ) T R 1 r ( x ) + [ F T R 1 r ( x ) f ( x ) ] T × ( F T R 1 F ) 1 ( F T R 1 r ( x ) f ( x ) ) } ,
(22)
where y is a vector of responses at the TPs, F is a m × p matrix with rows f T ( x ), m is the number of TPs, and r(⋅) is the correlation vector containing the correlation between x and each of the TPs,
r ( x ) = [ R ( x , x 1 ) , R ( x , x 2 ) , , R ( x , x m ) ] T .
(23)
Here, R is the correlation matrix, which is composed of correlation functions evaluated at each possible combination of m TPs, defined as
R = [ R ( x i , x j ) ] T , 1 i m , 1 j m .
(24)
In addition, β ^ is the least-square estimate of β, defined as
β ^ = ( F T R 1 F ) 1 F T R 1 y .
(25)
Lastly, σ ^ Y 2 can be determined by
σ ^ Y 2 = 1 m ( y F β ^ ) T R 1 ( y F β ^ ) .
(26)

The parameters θ k are determined through the maximum likelihood estimation. Due to the computational cost of the FE simulation model, we employed GP-based surrogate models to replace the FE simulation models. A key aspect of surrogate modeling is choosing a suitable sampling strategy to generate random sample points for building a high-quality surrogate model. The LHS method is commonly used for this task.63 Using LHS, the random input space is evenly filled with sample points, allowing for a comprehensive understanding of the true model.

In our study, the input design space comprised 300 sampled points obtained through the LHS method. These points were used as design parameters to construct the FE model of the BTMS. After that, the GP-based surrogate models were established, which take the design variables ( d and T t a r g e t) and independent random variables ( T i n l e t and k Q ˙ i n l e t) as input parameters, and predict the output parameters ( W t o t a l and T m a x).

To validate the surrogate model, we conducted FE simulations on specific randomly selected parameter points to obtain W t o t a l and T m a x. During simulation, W t o t a l and T m a x were obtained from the surrogate models using the same parameter points. The predicted W t o t a l and T m a x have relatively high accuracy with absolute percentage errors between the surrogate models and FE simulations lower than 5%, confirming the high fidelity of the developed surrogate models.

After validation of the surrogate model, reliability-based control co-design was conducted. The convergence history during the optimization process is shown in is shown in Fig. 5. After eight iterations, the values of the W t o t a l and T m a x stabilized. Though the constraint for T m a x of 35 ° C, the results show that T m a x converges to 33.1 ° C. This is due to the consideration of the uncertainties inherent in the BTMS and to ensure reliability.

FIG. 5.

Convergence history during the optimization process.

FIG. 5.

Convergence history during the optimization process.

Close modal

From the optimization, we found the optimal design for the BTMS (Table I). The optimal battery cell distance is not near the lower bound (1 mm) because it would dramatically increase the pressure drop and pump energy consumption. Additionally, the optimal cell distance is not large since larger distances result in a bigger and heavier battery pack. Although the energy consumed to carry the weight of the coolant is not as high as that consumed by pumping the coolant, a larger battery pack is not preferred according to our objective function. This trend also aligns with real-world practices, as car manufacturers aim to increase battery pack energy density.

TABLE I.

The optimal design.

d (mm)Ttarget (°C)SurrogateFE
Wtotal (kJ)Tmax (°C)Wtotal (kJ)Tmax (°C)
3.1269 31.8461 1.8103 33.13 1.7319 32.98 
d (mm)Ttarget (°C)SurrogateFE
Wtotal (kJ)Tmax (°C)Wtotal (kJ)Tmax (°C)
3.1269 31.8461 1.8103 33.13 1.7319 32.98 

Following the optimization, the optimal design was validated using the FE simulation. The results of the FE simulation show that both W t o t a l and T m a x from the surrogate models have high fidelity. Also, the temperature during the harsh loading condition of the battery pack is shown in Fig. 6. The temperatures of all battery cells increase quickly during the beginning of the discharging process because the feedback control does not actively increase the flow rate when the temperatures of the battery cells are lower than T t a r g e t. The battery cell closest to the inlet has a relatively high temperature compared to other battery cells. This is mostly because the cell is too close to the inlet, and turbulence causes the flow rate around the majority of components of the cell to be relatively low. The simulations show that four rows of cells will not result in a big temperature increase of the coolant while passing through the cells. Due to the symmetry of the battery pack, the temperature distribution of the whole battery pack can be obtained from this partial simulation result. From the 300 initial samples, the average W t o t a l was determined to be 33 kJ. The optimized design successfully lowers W t o t a l to 1.8 kJ, which lowers the energy consumption of the cooling system by over 90% for the BTMS using feedback control.

FIG. 6.

FE simulation results of the BTMS optimal design.

FIG. 6.

FE simulation results of the BTMS optimal design.

Close modal

This study provides a framework for the control co-design of BTMS under the uncertainties of coolant temperature and flow rate. Experimental testing of cell immersion cooling provided the properties of battery and coolant used for multiphysics FE simulations. Although some properties of the battery could not be directly obtained from the experiment, these parameters were estimated from the results of experimental testing. After improving the multiphysics FE model, we demonstrate that it is a high-fidelity representation of electrical and thermal profiles of a LIBs undergoing dielectric cooling. Following individual cell FE simulations, a full-size BTMS model was built and simplified based on symmetry of the system, and feedback control was added to the BTMS to improve efficiency. Based on the initial points generated through LHS of the design space, a parametric sweep was conducted with the outcome of the training data set. An ML surrogate model was adopted to reduce the computational cost for RBDO. The RBDO approach involved an outer optimization loop to minimize the cooling system energy consumption, while an inner loop ensured battery pack reliability. Finally, the optimal design of the cooling system was obtained and validated, which showed a 90% savings in the cooling system energy consumption. This study shows a feasible approach to designing BTMSs for different form factors of batteries through an RBDO-assisted ML approach. Our methods provide a framework to accelerate the design for various BTMSs with different battery form factors. Following the same sequential of our method: experiment, simulation, surrogate modeling, and RBDO, the optimal designs for various BTMSs can be obtained. With battery research focusing on safety and reliability, this research opens potential pathways for future BTMS design with complex systems to increase battery pack efficiency. Furthermore, our design approach ensures the reliability and safety of the next-generation battery packs using immersion cooling.

This work was supported by the National Science Foundation Engineering Research Center for Power Optimization of Electro-Thermal systems (POETS) under Cooperative Agreement No. EEC-1449548. This work was also funded by the National Science Foundation through the award (CMMI-2037898).

The authors have no conflicts to disclose.

Zheng Liu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (lead); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Pouya Kabirzadeh: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Hao Wu: Formal analysis (supporting); Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Wuchen Fu: Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting). Haoyun Qiu: Data curation (supporting); Validation (supporting). Nenad Miljkovic: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Yumeng Li: Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Pingfeng Wang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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