The heat generation inside battery usually has great impact on the battery temperature, which could consequently seriously influence the whole battery operation performance.^[41] To accurately model the heat generation process, Bernardi et al.^[42] proposed a general equation, which can calculate the heat generation rate from different heat sources, including electrochemical reactions, phase changes, mixing effects, and Joule heating. Although the general equation has high accuracy, it is usually difficult and time-consuming to accurately obtain all heat generation sources. Hence, a simplified version of Bernardi’s equation has been developed. By neglecting the heat generations from phase changes and mixing effects, and regarding the heat capacity as a constant, the heat sources in the simplified Bernardi’s equation could be summarized into two parts, i.e., reversible and irreversible heat. The irreversible heat is mainly caused by over potential during operation including ohmic losses, charge-transfer over potentials, and mass transfer limitations. The reversible heat, also called as entropic heat, is related to the entropy change of the active material at both positive and negative electrodes.

Battery degradation occurs not only during the operation^[43,44] but also during the storage,^[45] which are referred as cycle aging and calendar aging, respectively.^[46,47] A battery consists of several components, i.e., current collectors, positive electrode, negative electrode, separator, and electrolyte. Due to the internal and external factors, each component has the risk of failure during the battery service life. However, no matter what the specific failure mechanism, they all induce two phenomena, i.e., capacity fading and resistance increasing.^[17,48–50] At the negative electrode, the formation, growth, and updating of the SEI could consume the cycleable lithium and electrolyte, which would cause the reduction of capacity and liquid phase volume fraction as well as the increase of the transportation resistance in electrolyte. Moreover, lithium plating occurring at negative electrode would contribute to the capacity fading, especially at the end of battery life.^[51] In addition, the crack evolution caused by the mechanical stress during lithium insertion/extraction could result in capacity loss.^[52–55] At the positive electrode, the main degradation mechanisms include the solvent oxidation and the active material loss. The solvent oxidation occurs as a result of high voltage operation, which may cause film formation at the interphase of positive electrode/electrolyte. The low electrical conductivity of the formatted interface film may lead to the loss of electronic contact between primary particles and result in resistance increase.^[56] The active material loss is caused by the material dissolution, structural degradation, and fracture propagation throughout the lithium insertion/extraction.^[57,58]

To simulate the multi-physics response of batteries, different types of multi-physics coupled models have been developed. Figure 2 shows the coupling strategies among different models. The electrical model usually simulates the current–voltage response behavior, and calculates the heat generation rate and electrical dependent side reaction rate for providing the inputs of thermal model and aging model, respectively. The thermal model obtains the heat generation rate from both electrical model and aging model, and provides the temperature dependent electrical and aging parameters to the electrical model and aging model, respectively. The aging model simulates the aging behavior and provides the aging dependent electrical parameters and heat generation rate for the electrical model and the thermal model, respectively.

Fig. 2.

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	Fig. 2. Coupling strategies among battery multi-physics models.

Parameters identification is a vital step for constructing the battery model. In a battery model, only a few parameters can be measured directly, while others need to be estimated based on experimental data, e.g., measured current–voltage response profiles or current–voltage–temperature response profiles. Hence, it is essential to properly carry out experiments to obtain data for parameter estimation. A generalized electrical and thermal parameter estimation problem can be formed as the following weighted least-square optimization problem:

In practical application, the parameter identification procedure is usually different according to the different battery model type. For the ECM model, the OCV–SOC relationship is usually modeled by a predefined analytical function and the OCV model parameters could be independently estimated by OCV–SOC experimental data.^[107] Other state dependent circuit parameters in the ECM model, e.g., ohmic resistance, charge transfer resistance, and capacitance, are usually estimated by the charge/discharge current–voltage response profiles. Both one- and two-step approaches have been utilized to the circuit parameters estimation. In the two-step approach, the circuit parameters are first estimated one-by-one from the pulse charge/discharge current–voltage response profiles at different SOC values and then the circuit parameters look-up table is established with respect to SOC. The circuit parameters at unseen SOC are obtained by the interpolation method. Another way is to construct analytical functions, e.g., polynomial or exponential function, by means of the circuit parameters and SOC data in look-up tables.^[108–110] In the one-step approach, the coefficients of the predefined functions for state dependent circuit parameters are directly estimated by fitting the current–voltage response profiles over a whole SOC range. Although the parameter estimation optimization problem in the one-step approach is much more complex, the established model is usually more accurate and stable.

The number of parameters in the electrochemical model is much larger than that of parameters in the ECM, making the parameter estimation process more difficult. Different optimization algorithms have been adopted for the optimal parameter identification of the electrochemical model, such as genetic algorithm,^[28,73] particle swarm algorithm,^[111] and heuristic algorithm.^[112] Experiments based on half-cells are usually conducted for the essential model information such as the OCV–SOC relationship at each electrode. Even though the parameter identification based on the half-cell testing can reduce the number of model parameters, how to obtain a global optimal solution is still difficult as there exhibits multiple local optima in the parameter optimization problem. In addition, during the solution process of the parameter optimization problem, the time-consuming electrochemical model needs to be simulated repeatedly. Therefore, the efficient global optimization algorithm should be paid more attention in future.

When identify the thermal parameters in a multi-physics electrical–thermal coupled model, there are usually two ways. One way is to use the experimental data from various thermal property tests such as accelerating rate calorimeter,^[113] thermal impedance spectroscopy,^[114] and heat flow calorimeter.^[115] Another way is to estimate according to the temperature evolution profiles based on a thermal model.^[73] Although the thermal parameters might vary with the battery states such as SOC,^[116,117] temperature,^[118] and aging,^[119] the effects of battery state on thermal parameters are usually ignored in the electrical–thermal coupled model. It should be noted that the effective estimation of the aging dependent thermal parameters is still a challenging task and needs to be further explored.

In general, the parameters on electrode scale mainly contain the thickness and porosity of positive and negative electrodes, and the particle size distribution. The optimization of the electrode scale parameters is usually conducted based on the P2D based model. In the ECM, the electrode scale parameters are not considered, while in the SP model, the non-uniformity in the thickness direction is neglected, which make them seldom used in the electrode scale parameter optimization. According to the various requirements on the battery performance improvement, the optimization parameters of electrode scale mainly include electrode thickness,^[120–122] electrode porosity,^[123] positive and negative electrode capacity ratio,^[124] and particle size.^[125]

To maximize the specific energy or discharge capacity, some works proposed to optimize the electrode thickness and porosity based on the P2D model. Srinivasan et al.^[126] carried out the optimization of the electrode thickness and porosity of graphite/LiFePO₄ battery, which achieved maximum specific energy at given discharge time (from 10 h to 2 min). In this study, the porosity of the negative electrode remained constant while the thickness of the negative electrode was changed according to the ratio of the positive and negative active materials. The similar strategy was adopted by Appiah et al.^[127] to optimize the electrode thickness and porosity of graphite/LiNi_0.6Co_0.2Mn_0.2O₂ battery, in which the P2D model was modified to include the effect of contact internal resistance difference. De et al.^[128] proposed a multi-step optimization approach to optimize the electrode thickness and porosity of both electrodes, in which the optimization variables were added one by one during the optimization procedure. Dai et al.^[129] maximized the energy density of the battery by optimizing the porosity of the positive and negative active materials and the thickness of the positive electrode, meanwhile a direct search method was adopted for optimization. The results of graded electrode porosity design were then compared with the results of uniform porosity design.

When conducting the particle size optimization, the P2D model must include the stress and the side reaction effects, otherwise the particle size of the active materials would favor to be as small as possible for decreasing the solid diffusion polarization. Golmon et al.^[130] combined the P2D model with a mechanical model to simulate the stress introduced by lithium diffusion, in which the stress level was bounded by a maximum value to control the capacity fading rate. In this study, the local porosity and local particle radii at each finite element node of both electrodes greatly increased the optimization variables, hence, the globally convergent method of moving asymptotes of Svanberg was adopted to handle the complex optimization problems. Liu et al.^[131] used an P2D electrochemical–degradation coupled model to simultaneously maximize specific energy and specific power and minimize the capacity loss with respect to electrode thickness, electrode porosity, and particle size. The elitist non-dominated sorting genetic algorithm (NSGA-II) was adopted to solve the multi-objective optimization problem.

For large format batteries, the effects of electrode scale parameters on battery thermal performance are usually taken into account. Zhao et al.^[75] studied the effects of electrode thickness on battery performance using P2D elelctrochemcial–3D thermal model. Mei et al.^[76] studied the effects of electrode thickness, electrode porosity, and particle size on battery electrochemical and thermal performance. In addition, some studies have focused on designing structured electrode using electrochemical model. Cobb et al.^[132,133] proposed co-extruded electrode structures to improve the battery energy density, and simulation study was taken using a dimension-extended P2D model. Nemani et al.^[134] proposed a bi-tortuous, anisotropic graphite anodes structure to improve the ion transportation, and the equations of the P2D model were extended to 2D to carry out the simulation. However, the industrial production of such structured electrode remains quite difficult. In general, given predefined current profile, most optimal design problems on electrode scale can be formulated as single objective optimization problems with respect to maximize the specific energy or discharge capacity. If the charge/discharge current is considered to be an additional optimization variable, multiple objectives need to be simultaneously optimized for balancing the tradeoff between specific energy and specific power.^[135]

The design parameters on cell scale mainly refer to electrode size, tab size and location, number of stacks, etc. Usually, with the scale up of battery size, the non-uniformity of both electrical and thermal distributions will become more severe, and then the battery charge/discharge performance and service life will deteriorate significantly.^[136,137] Hence, the design strategy on the cell scale is quite important, especially for large format batteries.

In general, both 2D and 3D models can be employed for the optimization of cell scale parameters. Based on the ECM–thermal coupled model, Kim et al.^[138] studied the effects of the electrode size and tab location on the potential, current density, and temperature distributions with a 2D electrical–thermal coupled model. Kosch et al.^[139] studied the effects of tab location and size on the temperature distribution of a pouch type cell using a 2D ECM–thermal coupled model. A 2D ECM–3D thermal coupled model was proposed by Wu et al.^[67] and used for the thermal design of pouch cell,^[140] in which the temperature distributions of cells with different electrode sizes and different tab locations were systematically compared.

Also, there have been studies about the optimization of cell scale parameters based on the electrochemical model. Mei et al.^[141] used a P2D electrochemical–3D thermal model to study the effects of tab dimension on the thermal performance of pouch type cell. Zhang et al.^[142] studied the impacts of tab locations and electrode size on the temperature distribution of pouch type cell, in which the uniformity of temperature distribution under different design configurations was studied by a P2D electrochemical–2D thermal coupled model. Samba et al.^[30] studied the influence of the position of the tab on the current density, voltage, and temperature distribution by establishing a 3D electrochemical–3D thermal coupled model for a pouch cell with single electrode plate pair. Rieger et al.^[29] used a 3D electrochemical–2D thermal–2D stress coupled model to study the effects of tab position on the battery performance, except for the uniformity of temperature and current density distribution, the mechanical criteria were also introduced for rating different designs.

It is found that the optimization of design parameters on cell scale is usually realized by parametric studies, while the optimization algorithms are seldom explicitly used to perform design optimization. The main reason is that the simulation of 2D or 3D coupled model used on cell scale is time-consuming, which hinders the direct application of the optimization algorithm to some extent. In addition, the simultaneous consideration of multiple performance criteria such as energy density, service life, uniformity, and cost would further increase the optimization difficulty greatly. Hence, more efficient global optimization algorithms are expected for such complex multi-objective battery design optimization problems.

A battery pack usually consists of dozens even hundreds of cells, which makes the thermal issues increasingly prominent. To guarantee the safe operation and extend battery life, the battery pack should be carefully designed not only to avoid local heat accumulation but also to improve the temperature consistency among cells. On the pack scale design optimization, both cell arrangement and cooling strategy usually need to be simultaneously optimized. In general, a simple electrical–thermal coupled model,^[143] or even empirical thermal model,^[144] is used to simulate the cell heat generation characteristics. By the combination of such simple thermal models with computational fluid dynamics simulation, the gap spacing between cell, the flow speed of air or liquid can be optimized to achieve uniform temperature distribution and lower maximum temperature rise. Although these simplified battery models are helpful to reduce the computational complexity, their generalization ability and the effectiveness of optimal pack design schemes need to be further verified.

Besides, more precise battery models have been introduced to assist in the optimal design on the pack scale. Tong et al.^[145] studied the effects of operational and design parameters on the pack performance using liquid cooling strategy, in which the internal physical property distribution of each battery was simulated with an electrochemical–thermal coupled model. Based on that, an electrochemical–thermal coupled model combined with 2D thermal–fluid conjugate model was proposed to study the effects of different cell arrangements on the temperature distribution in a battery module.^[146] A 2D electrochemical–2D thermal coupled model was used by Bandhauer et al.^[147] to study different cell designs (8 A⋅h and 20 A⋅h) and cooling strategies on the temperature and current distributions during dynamic loads operation. Darcovich et al.^[148] compared different cooling plate configurations of battery pack, in which an SP electrochemical–3D thermal model was used to simulate the battery performance. An air cooling strategy for battery pack was developed by Sun et al.^[149] based on a 3D battery module thermal model, in which each cell was simulated using an ECM–3D thermal coupled model. The similar model was also used to develop a liquid cooling system for a battery module,^[150] moreover, an analytical optimization approach was developed for the identification of optimal design concept. Zhao et al.^[151] studied the effects of different cooling strategies on the temperature rise and gradient of a pouch cell using a 2D ECM–2D thermal coupled model, in which both surface and tab cooling strategies were compared.

The application of more accurate battery model is expected to increase the effectiveness and reliability of the battery pack design. However, the introduction of accurate battery would significantly increase the model complexity on the pack scale and consequently might result in very high simulation computational cost, especially combined with optimization algorithms. To tackle this issue, the advanced modeling strategies, which possess high accuracy and high efficiency, are expected to be developed. Moreover, with the intend of reducing the computational cost, the coupling strategies between the cell scale and pack scale requires more attention. In addition, like the design on the cell scale, the multi-objective optimization method on the pack scale needs to be further studied. The references in battery design on different scales are summarized in Table 1, in which the methods are classified based on the adopted battery model.

Table 1.

Table 1.

Table 1. Summary of model based battery design methods on electrode, cell, and pack scales. . Scale Model type Design variable Design objective Methods and reference Electrode P2D Electrode thickness and porosity Specific energy; Specific power; Parametric study[126,127] Optimization[128,129] P2D-stress/P2D-degradation Electrode thickness and porosity Particle size Specific energy; Specific power; Stress Optimization[130,131] P2D-thermal Electrode thickness; Electrode porosity Specific energy; Specific power; Temperature distribution; Maximum temperature; Parametric study[75] Optimization[76] Cell ECM-thermal Electrode size; Tab location; Tab size Potential, current, Temperature, SOC distributions. Parametric study.[138–140] Electrochemical-thermal Electrode size; Tab location; Tab size Potential, current, Temperature, SOC distributions. Parametric study[30,141,142] Electrochemical-thermal combined with stress Electrode size; Tab location; Tab size Potential, current, Temperature, SOC distributions. Parametric study[29] Pack ECM-thermal cell arrangement related parameters; cooling strategy related parameters; Maximum temperature; Temperature distribution; Parametric study[143,145–148,151] Electrochemical-thermal cell arrangement related parameters; cooling strategy related parameters; Maximum temperature; Temperature distribution; Parametric study[149] Optimization[150]

Table 1. Summary of model based battery design methods on electrode, cell, and pack scales. .

As illustrated by Li et al.,^[161] the SOC of battery could be defined from different perspectives. From the thermodynamics perspective, the SOC describes the lithium ion concentration in an electrode or in two electrodes of a single cell. From the engineering perspective, the SOC is used to describe the remaining useful capacity of battery, which can be defined as

The A⋅h-counting method is the simplest way for SOC determination. With the measured current and known initial SOC, the SOC at any moment can be calculated by

Although the A⋅h-counting method is easily implemented, it has several defects. First, the current measurement noise integrates during the iteration process. Second, the battery capacity fades with aging cannot be corrected. Moreover, the accurate initial SOC is hard to obtain in real applications. Although some works proposed the SOC–OCV relationship to improve the initial SOC accuracy, this method is only available under long-term standing conditions. Furthermore, for the batteries with very flat OCV curve like LiFePO₄, the SOC estimation from SOC–OCV relationship may also have great errors. Some other specific operation conditions can also be used for SOC calibration, for example, when the battery is charged or discharged to preset cut-off certification conditions, the SOC can be determined as 100% and 0%, respectively. It should be pointed out that, the calibration methods mentioned above face the risk of failure, because the application scenarios of the battery and the usage habits of specific users vary widely.

To solve this problem, the model-based SOC estimation methods have been developed, which are the combinations of battery model and state estimation algorithms. Figure 3 gives a general framework of model based battery SOC estimation methods. During battery state estimation, a battery model is used to predict the battery terminal voltage. The model predicted terminal voltage is then compared with the measured value, the errors are then taken as the input of the estimation algorithms. The obtained battery SOC is used to update the input of the battery model and sate equation.

Fig. 3.

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	Fig. 3. General flow chart of model based battery SOC estimation method.

The ECM model and SP model are two main model types for SOC estimation. Based on ECM, different estimators have been constructed by adopting different estimation algorithms. The mostly used one is the Kalman filter (KF) family, including extended Kalman filter (EKF),^[162–164] unscented Kalman filter (UKF),^[165,166] and sigma-point Kalman filter (SPKF).^[167,168] Other methods such as particle filter (PF),^[169] sliding mode observer,^[170] H-infinity observer,^[171] proportional-integral observer,^[172] Luenberger observer,^[173] moving horizon estimation (MHE)^[174,175] combined with ECM for SOC estimation have also been developed. Based on the SP model, the EKF,^[176,177] UKF,^[178] PF,^[179] Luenberger observer,^[180,181] adaptive PDE observer^[182] based estimators were studied. In addition, some researchers have tried to use the combination of P2D model and KF based algorithms to realize the SOC estimation.^[183–185] Although the KF based estimator has proven the good performance, it still has several challenges in real applications. First, the noise covariances are difficult to determine. The unsuitable choice of noise covariances would not only reduce the estimation accuracy, but also result in the risk divergence. Moreover, the hard constrains cannot be added to the algorithms directly. Hence, adaptive EKF (AEKF),^[186] adaptive UKF (AUKF),^[187] and adaptive PF (APF)^[188] have been proposed to update the noise covariances matrix.

To improve the SOC estimation accuracy under wide operation temperature region, different methods have been proposed to integrate the effect of temperature on parameters. One method is to adopt a temperature-compensated model, in which the temperature is measured and treated as an input to correct the model parameters. From this view, the combinations of temperature-compensated ECMs and EKF,^[189] UKF,^[190] PF,^[191] the combination of temperature-compensated SP models and Luenberger observer,^[86] are proposed successively. Another method is to adopt an electrical–thermal coupled model, in which the temperature and SOC are estimated simultaneously during the SOC estimation. Following this way, Bizeray et al.^[70] proposed an P2D electrochemical–thermal coupled model and EKF based battery state estimation method. Notice that the time-step of EKF was set as 5 s to reduce the computation time.

Table 2 compares different SOC estimation methods, in which the advantages and disadvantages are compared between A⋅h-counting method, OCV based method, and model based methods. Numerous model based SOC estimation methods have been proposed over the past decades. However, it is hard to determine which one is most suitable for real applications. Since the evaluation of SOC estimation methods is multi-dimensional, containing the computational complexity, estimation accuracy, and robustness, more comparative studies need to carry out.^[192–196] From the perspective of model, it is widely recognized that although the P2D model is most accurate, it is too complex to implement in the real-time applications. By contrary, the ECM is much simpler and can meet the usual accuracy requirement. This makes it widely accepted by researchers from different fields. The SP model provides a compromise between the accuracy and the complexity, and shows a good estimation stability in the long-term.

Table 2.

Table 2.

Table 2. Comparison between different SOC estimation methods. . Method Advantage Disadvantages A⋅h-counting Easy to implement. Accuracy depends on the initial SOC value. Fault with long term current measurement accumulative error. Self-discharge rate, coulomb efficiency needed. OCV based Easy to implement. Long rest time required. Fault when OCV-SOC curve is flat. Model based Initial SOC value insensitive. No rest time needed. More efforts on the modeling and estimator design. Software demand increasing.

Table 2. Comparison between different SOC estimation methods. .

SOH is an indicator for battery aging state, which can be defined by capacity fading and internal resistance increasing. The following equations give respectively the definition of SOH by battery capacity and internal resistance:

In the past decades, different model-free methods have been proposed for the SOH estimation, herein, model refers specifically to the model described in Section 2. In some works, the capacity loss or the capacity serves as the SOH indicator, which is presented as a function of time and operation conditions such as current, temperature, DOD, and cycle number.^[197–199] Besides that, some researchers used the feature extraction based on battery charge/discharge curve to obtain the battery capacity.^[200–203] Since the operation conditions of battery are quite unpredictable, this method is limited for the on-line usage.

Model based SOH estimation method is regarded as an effective way to enhance the estimation accuracy. When the internal resistance serves as the SOH indicator, the EIS based SOH estimation methods are widely studied. By establishing the frequency-domain ECM, the internal resistance associated with internal reaction and transfer phenomena can be identified, and then the SOH can be evaluated.^[204] Because of the high cost of test equipment,^[205] the difficulty of excluding the effects of SOC and temperature on internal resistance, the implementation of EIS for on-line usage is still challenging.

Either use the capacity or the internal resistance as the SOH indicator, the alternative way for the SOH estimation is the combination of battery model and observer/filtering algorithms. Usually, joint or dual estimation techniques are adopted, which can obtain the estimation of SOC and SOH simultaneously.^[206] Figure 4 compares the general frameworks of joint and dual estimators for SOC and SOH. In a joint estimation approach, a single sate vector is used for both parameters and state estimation. The internal resistance or capacity, as an indicator of SOH, is added in the state equation. Then both SOC and SOH are estimated simultaneously in each iteration. While in a dual estimation approach, two estimators are used in parallel, one is used for the SOC estimation and the other one is used for the SOH estimation.

Fig. 4.

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	Fig. 4. Comparison of joint and dual estimation methods for SOC and SOH: (a) joint estimation method; (b) dual estimation method.

Based on ECM, the EKF,^[207,208] least squares estimation,^[209] PF,^[210] dual sliding mode observer^[211] for SOH estimation have been proposed. Notice that, unlike SOC, SOH usually varies slowly during the operation, therefor, the battery SOH can be estimated on a longer time scale. Combined estimation methods such as two EKF,^[212] EKF combined with recursive least squares^[213,214] have also been developed. Based on the SP model, the SOH estimation is realized using PF,^[215] least squares method,^[216] sliding mode observers,^[217] adaptive output-injection observer,^[218] adaptive PDE observer,^[182] and trinal proportional-integral observers.^[219]

SOP is defined as the maximum power that a battery can deliver or receive on upcoming time scale, generally several seconds, meanwhile, ensuring the safe operation of battery.^[220] It is an essential battery state especially for vehicle usage during operations such as acceleration, climbing, or regenerative braking. SOP depends on many other battery states such as SOC, temperature, and SOH. Therefore, the SOP estimation performance is strongly affected by the accuracy of the SOC, temperature, and SOH estimation.^[221]

A direct way to obtain SOP is to establish the relationship of SOP with SOC, temperature, and SOH. Generally, the empirical function or look-up table is first constructed off-line based on the pre-designed laboratory tests. Then, the SOP estimation is conducted by bringing the online measurement data into the preset equation or look-up table. This method is simple and easy to implement. However, since the effect of the load history is neglected, there might be great estimation errors in real applications.^[222]

Model-based method is an effective way to improve the estimation accuracy of SOP. Based on ECM,^{[220,223,224]} ECM–thermal coupled model,^[225,226] and SP model,^[227] various SOP estimation methods have been proposed and compared in details.^[228] In these methods, the battery model is used to obtain the maximum power capability such that the SOC and the simulated voltage are kept within the preset region, meanwhile, temperature constrains are satisfied based on a thermal model.^[226] In order to reduce the calculation complexity and meet the real-time requirement, battery models have to be simplified, which might introduce the errors into the estimation process. Therefore, how to choose a proper model complexity is critical in the real applications. When conducting the SOP estimation, the boundary method is most widely used, which is directly based on the design boundaries and the operation boundaries. Besides that, considering the nonlinear properties of the battery model, the SOP estimation can also be treated as an optimization problem, where the objective is the maximum battery power within the preset time region, and the constraints are the allowed SOC, voltage, and temperature. By solving this problem, the SOP estimates could be obtained effectively. Although the adoption of optimization algorithms such as Karush–Kuhn–Tucker (KKT) conditions^[229] and genetic algorithm^[230] allows more accurate estimation, its computational cost is higher when compared to the traditional methods.

In most existing BMS, the battery temperature is directly determined by the sensor measurement at battery surface. However, in practice, the battery internal temperature could greatly differ from the surface temperature, especially under extreme operation conditions such as large current rate.^[231,232] It means that even the surface temperature is within the allowed region, the unexpected reactions still possibly happen inside due to the high internal temperature. Hence, to extend battery life and reduce thermal safety risks, it needs to monitor the internal temperature in real-time.

To realize the real-time monitoring of battery internal temperature, some researchers tried to use embedded thermocouples or fiber sensors to obtain the internal temperature directly.^[233,234] Although it is intuitive, the installation of thermocouples and fiber sensors is difficult during the assembly process and would affect the battery performance. Alternatively, some non-intrusive methods like EIS^[235,236] have been used, however, they are restricted in the practical application because of the complicated test procedure and the expensive equipment. The model-based method provides another way to get an accurate estimation of the internal temperature. Lin et al.^[237] proposed an adaptive observer for the core temperature estimation based on a two-state lumped parameter thermal model, in which the reversible heat generation and SOC dependence of internal resistance were neglected. Zhang et al.^[238] proposed a KF based internal temperature estimation method, in which a simplified ECM–thermal coupled model was used while the SOC effects on the ECM parameters were neglected. The similar thermal model was also adopted and combined with KF by Sun et al.^[239] Kim et al.^[240] proposed a 1D thermal model and dual KF based method for internal temperature monitoring, in which polynomial approximation was used for the thermal model reduction. Dai et al.^[241] proposed an equivalent electrical network thermal model combined with adaptive KF approach for the internal temperature estimation. In these works, the SOC is an essential parameter for a model based internal temperature estimator, because both reversible and irreversible heat generation rates are calculated based on it. Table 3 summaries the applications of battery models in battery state estimation.

Table 3.

Table 3.

Table 3. Summary of battery state estimation methods used in literature. . State Models adopted Methods and references SOC ECM based model EKF,[162–164] UKF,[165,166] SPKF,[167,168] PF,[169] Sliding mode observer,[170] H-infinity observer,[171] Proportional-integral observer,[172] Luenberger observer,[173] MHE,[174,175] AEKF,[186] AUKF,[187] APF[188] SP based model EKF,[176,177] UKF,[178] PF,[179] Luenberger observer,[86,180,181] Adaptive PDE observer[182] P2D based model Linear KF,[183] EKF,[70] PF,[184] output error injection observer[185] SOH ECM based model EKF,[207,208] least squares estimation,[209] PF,[210] dual sliding mode observer,[211] two EKF,[212] EKF with recursive least squares[213,214] SP based model PF,[215] LS,[216] sliding mode observer,[217] adaptive output-injection observer,[218] adaptive PDE observer,[182] trinal proportional-integral observers[219] SOP ECM based model Constraints based calculation,[220,223–226] Optimization based estimation[229,230] SP based model SP model[227] Internal temperature ECM-thermal Adaptive observer,[237] KF,[238,239] Dual KF,[240] Adaptive KF[241]

Table 3. Summary of battery state estimation methods used in literature. .