Data Science-Based Battery Operation Management II

Abstract

This chapter focuses on the data science-based management for another three key parts during battery operations including the battery ageing/lifetime prognostics, battery fault diagnosis, and battery charging. For these three key parts, their fundamentals are first given, followed by the case studies of deriving various data science-based solutions to benefit their related operation management.

This chapter focuses on the data science-based management for another three key parts during battery operations including the battery ageing/lifetime prognostics, battery fault diagnosis, and battery charging. For these three key parts, their fundamentals are first given, followed by the case studies of deriving various data science-based solutions to benefit their related operation management.

5.1 Battery Ageing Prognostics

Battery would inevitably degrade during the operation period, further affecting its safety and efficiency. In this context, it is of extreme importance for developing effective data science-based solutions to benefit battery ageing/lifetime prognostics. This section would first introduce battery ageing mechanism and related stress factors, then the framework of performing Li-ion battery ageing prediction with data science is given, followed by two case studies of deriving different data science-based solutions to achieve cyclic ageing and lifetime predictions of Li-ion battery.

5.1.1 Ageing Mechanism and Stress Factors

5.1.1.1 Li-Ion Battery Ageing Mechanism

Li-ion battery ageing is a complicated and long-term procedure. Understanding mechanisms of battery ageing is the prerequisite for designing data science-based tools and methodologies for battery ageing prognostics. Research has been conducted to analyse the essential reasons for battery degradation [1, 2]. The most effective and ideal way is to translate battery ageing knowledge into a mathematical form. We will give a brief introduction of the most common battery ageing mechanisms in this subsection.

The main degradation mode in Li-ion battery can be divided into three categorizations, that is, the loss of lithium inventory (LLI), the loss of active material (LAM) in the electrodes, and the increase of internal resistance. The side reactions in Li-ion battery will consume lithium inventory, and then only fewer Li-ion is available for the charging or discharging process. The related side reactions include: electrolyte decomposition reactions or lithium plating, the formulation of the solid electrolyte interface (SEI) on the surface of the graphite negative electrode. Regarding LAM, the electrodes structure changes with the volume of active materials during cycling. Then, the mechanical stress is induced by the above process, which causes particle cracking and thus reduces the density of lithium storage. In addition, the chemical decomposition and dissolution reactions of transition metals into the electrolyte and SEI modification also have an effect on the LAM. The internal resistance increase will be caused by the formation of the SEI and the loss of electrical contact inside the porous electrode [3].

Li-ion battery deteriorates in both cycling and storage conditions, which implies the cycling ageing and calendar ageing of battery [4, 5]. Generally, mechanical strains of the active materials in the electrode or lithium plating are the main reasons for cycling ageing, while the evolution of passivation layers is the dominant ageing mechanism of calendar ageing [6]. This chapter will not distinguish the cycling and calendar ageing of a battery in detail, and readers may refer to [7] for more information.

5.1.1.2 The Stress Factors for Li-Ion Battery Degradation

A variety of external factors affect battery ageing process. Besides the high SoC and temperature, overcharge/discharge, current rate, and cycling depths all have influences on battery degradation [8]. Notably, those factors are not linearly characterized with battery health status, which significantly complicates the battery ageing.

The related stress factors are outlined in Fig. 5.1 and will be introduced one by one in the following content.

Fig. 5.1

Battery ageing stress factors and the related degradation modes, reprinted from [7], with permission from Elsevier

• (1) High temperatures: Extremely high temperature may easily lead to “thermal runaway” of a cell, which is an ultimate threat to battery operation management. Moreover, high temperature also accelerates the side reactions, for example, the SEI layer grows faster on the anode, and then the LLI and internal resistance are increased. Additionally, the metal dissolution from cathode and electrolyte decomposition also speeds up the LAM and LLI.

• (2) Low temperatures: Low temperature slows down the Li-ion transport in the electrodes and electrolyte. Attempting to fast charging at low temperature may lead to the crowding of Li-ions. Thus, LLI appears with lithium plating of graphite, which eventually causes the growth of lithium dendrites and further penetrates the separator and short circuit inside a battery.

• (3) Overcharge/discharge: In the case of overcharging, none active Li-ion is available on the cathode and also not enough room is left for Li-ion in the anode. Then, the structure of cathode material irreversibly changes with overdelithiated. Afterwards, the active material decomposition and the dissolution of transition metal ions occur inside the battery. Thus, when the battery undergoes an overcharging process, the electrolyte is decomposed and the total resistance is significantly increased. Considering the heat generated by side reactions at electrodes, overcharging a battery usually generates a lot of heat. Additionally, there is an abnormal increase in the anode potential, followed by the anodic dissolution of copper current collector and the formation of copper ions. Therefore, a risk of internal short circuit exists, because the reverse reactions can form copper dendrites.

• (4) High currents: A large current will cause localized overcharge and overdischarge inside the cell, and a high current usually accompanies more heat generated. In this thread, Li-ion battery with organic electrolytes can prone the rapid temperature increase in comparison with water-based electrolytes. In addition, fast charging will accelerate the metallic Li-plating as the graphite presents a very limited capability to accept Li-ion in this condition.

• (5) Mechanical stresses: The mechanical stresses of cell come from different aspects, such as electrode material expansion, gas evolution in mechanically constrained cells, and external loading during service. The highest stress inside cell comes from the electrode particles near the separator, which leads to the risk of cracking and fracture. Once exceeding a certain limitation, a material failure occurs in the electrode. In this condition, the performance of the cell significantly deteriorates.

For a large-scaled battery-based energy storage system, battery power flow needs to be managed according to the requirement of the load. Moreover, battery operation management system has to ensure the safety and manage the lifetime of battery. For example, the charge or discharge of battery has to be suspended, once a battery reaches its fully states; an optimal charging profile can be planned to achieve a better trade-off between the battery capacity fade and charging time. Hence, fully understanding of the impact factors is essential to establish a reliable data science-based battery ageing prognostic method. A large amount of highly quantity dataset is necessary for battery ageing prediction methods with big data. Some stress factors are more important than others in a specific application, which means that the factor has the greatest impact should be considered when designing the experimental testing matrix.

5.1.2 Li-Ion Battery Lifetime Prediction with Data Science

Li-ion battery lifetime prediction with big data will be introduced and discussed in this subsection. The recently proposed data science methods for battery lifetime prognostics are overviewed at the beginning. Afterwards, machine learning (ML) methods for battery lifetime prediction are introduced. Two case studies of our previous work using modified Gaussian process regression (GPR) and hybrid data science model will be detailed, and the conclusions are summarized at the end of this subsection.

5.1.2.1 Overview of Battery Lifetime Prognostics Methods

The battery lifetime prediction is a key part to indicate the remaining service time of battery. The remaining useful lifetime (RUL) is typically reached when the predefined battery degradation index arrives at a specific threshold. RUL can be obtained by calculating the estimated lifespan of a training unit minus the current life position. The relationship between the diagnosis and the prognostics is shown in Fig. 5.2.

Fig. 5.2

Data science-based battery health diagnostics and prognostics framework, reprinted from [7], with permission from Elsevier

In the framework of Fig. 5.2, two typical data science-based models are often used for battery lifetime prediction, which is analytical models and ML models. Analytical model needs the development of an ageing model by fitting a mathematical function to describe a large set of measurement datasets under laboratory conditions. On the contrary, ML model can directly learn from the ageing dataset itself to predict battery lifetime [9].

This subsection will mainly introduce the analytical model with data fitting. Analytical model tries to use a mathematical function representing the connections of a battery and its service time or cycling number. It can be divided into two categories: semi-empirical lifetime prediction model and empirical ageing model with filtering. Semi-empirical model is an open-loop approach, where model performance largely depends on the fitting process of the ageing dataset. Once the model is constructed, its parameters cannot be varied anymore. Meanwhile, the empirical model with filtering is able to be updated according to the new dataset with the benefit of a closed-loop structure.

• (1) Semi-empirical life prognostics models

In order to establish a mathematical expression describing the battery lifetime performance, semi-empirical lifetime prediction models have been used to directly capture the relationship between ageing stress factors and battery health status. Mostly, the semi-empirical model is constructed by interpolating and fitting dataset collected from a specific experimental test [10]. For a better accuracy, those datasets should come from a wide range of operation conditions. In reality, it is rather difficult to consider the effects of all the related factors as previously described. Therefore, only the most important factors are considered for simplification.

Lots of studies build the cycling and calendar ageing model of battery independently, and the combination of these two models can generate the prediction under dynamic load profiles [11]. In order to obtain the models, the cells are cycled or stored under a well-designed test condition so that the influence of different ageing factors, such as temperature, SoC and current rates, can be deeply investigated. Afterwards, the capacity loss is calculated as the function of time, cycle numbers, and Ah-throughput. Ah-throughput is the amount of charge from one electrode to the other. The selection of fitting equations relies on the measured battery capacity trajectory. In this way, the parameters of the lifetime model are determined by fitting a large amount of ageing dataset. Once the model is constructed, the parameters are unable to be changed. Ah-throughput and current cycle number should be registered as the input for the battery future capacity prediction during operation. In addition, the battery usage conditions and loads are fed to the model if the battery lifetime is predicted.

In calendar ageing, the capacity loss is usually proportional to power law relation with time, which can be expressed as follows [12],

$$ Q_{{\text{loss}}}^{{\text{Cal}}} (t) = Q(t) - Q(0) = k_{{\text{Cal}}} \left( {T,{\text{ SoC}}} \right) \cdot t^{Z_{{\text{Cal}}} } $$

(5.1)

where $Q_{{\text{loss}}}^{{\text{Cal}}}$ is battery capacity loss during calendar ageing, $Q\left( t \right)$ and $Q\left( 0 \right)$ stand for battery capacity values at time point $t$ and its start life, $Z_{{\text{Cal}}}$ is a dimensionless constant, $k_{{\text{Cal}}}$ represents a stress factor related to battery temperature $T$ and ${\text{SoC}}$. In general, the dependence of $k_{{\text{Cal}}}$ on temperature $T$ can be empirically captured by using the Arrhenius equation as [13]:

$$ k_{{\text{Cal}}} = A \cdot e^{ - E_a /RT} $$

(5.2)

where $A$ denotes a pre-exponential factor and $E_a$ represents the activation energy. Additionally, the SoC dependence on calendar ageing lifetime is typically fitted by the linear functions [14], exponential functions [15], or the Tafel equation [12].

On the other hand, the battery cycling ageing is sensitive to the operation profile. Thus, the prediction of cycle life is more complicated than calendar life prediction because more variables are involved in this condition. These factors include temperature, cycle number/time, charge/discharge current rate, cycling voltage range, and average SoC during cycling. Cycle number is often used as a measure of time for cycle lifetime modelling. One common cycling ageing model is shown in Eq. (5.3), which uses power law to express the capacity loss as:

$$ Q_{{\text{loss}}}^{{\text{Cyc}}} (L) = Q(L) - Q(0) = k_{{\text{Cyc}}} \left( {T,I,{\text{ DoD}}} \right) \cdot L^{Z_{{\text{Cyc}}} } $$

(5.3)

where $Q_{{\text{loss}}}^{{\text{Cyc}}}$ reflects battery capacity loss during cycling ageing, which means the overall capacity difference over time/cycles, $L$ is either cycle number or Ah-throughput, $k_{{\text{Cyc}}}$ represents the effects of the ageing factors on capacity degradation, $I$ is the cycling current. ${\text{DoD}}$ is the depth of discharge during cycling. The parameters in Eq. (5.3) can be fitted from experimental dataset. Arrhenius equation can be used to empirically account for the effect of temperature. In addition, the current rate and depth-of-discharge (DoD) dependency on cycling ageing can also be expressed using exponential or polynomial functions. For example, the polynomial functions can be used to describe the capacity fade with DoD and cycling number as follows:

$$ Q_{{\text{loss}}}^{{\text{Cyc}}} (L) = \sum\limits_{i = 0,j = 0}^{n,m} {\left( {a_i \cdot L^i + b_j \cdot {\text{DoD}}^j } \right)} $$

(5.4)

where $L$ is the cycle number, $a_i$ and $b_j$ are the fitting coefficients, $n$ means the order of $L$-factor, and $m$ is the order of ${\text{DoD}}$-factor.

• (2) Empirical ageing model with filtering

Empirical ageing models with filtering are able to update their parameters once new data is available. A preliminary prediction model is constructed first by fitting the experimental data to a suitable function describing the capacity degradation. The generally used linear, exponential, and polynomial functions are listed in Table 5.1. Then, the filtering methods should be applied to update the parameters of the model during battery degradation process, when the new measurement arrives. In this way, the models can be adjusted to provide a more accurate RUL prediction.

Table 5.1 Models and filters used for the battery RUL prediction, reprinted from [7], with permission from Elsevier

From Table 5.1, we can find that Kalman filter (KF), particle filter (PF), and their variants can enable the dynamic update of the prediction model. The observation is applied to estimate and update the parameters according to the form of a probability density function (PDF) in Bayesian inference. The filter can be chosen by the dynamic of the system and the noise distributions. For example, KF is often a good choice for linear system with Gaussian noise. A linear capacity fade model with KF is used to predict the RUL of the valve-regulated lead–acid battery in [28]. Since the Li-ion battery fading process is often nonlinear, variant KF like extended KF and unscented KF can be used to address this issue. However, for the KF family, the state space PDF is still Gaussian distributed at each iteration. In reality, the errors of RUL may not come from multiple sources for the data acquisition and transmission. Thus, the noise cannot always follow the Gaussian distribution. In this case, the KF algorithms may cause the divergence of the prediction.

For the purpose of solving non-Gaussian distribution with a nonlinear system, PF is more widely used in this thread. PF belongs to the sequential Monte Carlo method which utilizes the Bayesian inference and the importance sampling method [29]. The Bayesian update is able to deal with the particles that have the probability information of unknown parameters. When new measurement is arrived, the posterior from the previous step acts as the prior information for the current step. Therefore, the parameters are updated by multiplying it with the likelihood [30]. In the area of battery RUL prediction, numerous studies related to PF and its variants have been carried out [31].

The sum of the impedance parameters shows a linear correlation with battery capacity in [32]. An impedance growth function is used to describe the battery ageing behaviour, which is combined with the PF framework to implement the battery RUL prediction in Fig. 5.3. It shows that the prediction accuracy is improved by increasing the considered dataset. It should be noted that the performance of the empirical ageing model with filtering is highly dependent on the fitted prediction model. Only one model may not enough for the complex ageing behaviour. Thus, Ref. [16] proposes two empirical models for the prediction of the Li-ion battery degradation.

Fig. 5.3

Schematic illustration of RUL prediction with PF, reprinted from [7], with permission from Elsevier

5.1.2.2 Comparisons of Battery Lifetime Prediction with Data Science

The data science methods for RUL prediction have some differences in complexity, prediction accuracy, and the ability to produce confidence intervals. The main features of data science-based battery lifetime prediction methods are summarized in Table 5.2. The prediction accuracy of semi-empirical lifetime models relies on the developed mathematical function. However, this type of lifetime model has open-loop nature, which is established according to a large amount of ageing dataset from laboratory test. These models have a lower computational burden and can be easily applied to the hardware for battery operation management. It should be noted that the methods do not have a recalibration mechanism because of an open-loop structure. The prediction accuracy is also related to the amount of dataset. The empirical ageing models with filtering belong to the closed-loop type. Thus, they can adaptively adjust to the desired prediction results online. In addition, the parameters of the model can also be updated during the operation for tuning the prediction. However, the structure of those models may limit the capability of the prediction under more complex battery ageing conditions. In this way, we recommend to employ hybrid approaches. For example, a lifetime model can be combined with an adaptive filter to update the parameters of the model for a more reliable prediction. In this case, the model can be used for real battery degradation prediction in different cycling conditions.

Table 5.2 A comparison of battery lifetime prediction methods, reprinted from [7], with permission from Elsevier

ML methods do not rely on any explicit mathematical model, and mostly the information from the historical test dataset is utilized to describe battery ageing behaviour. The non-probabilistic methods cannot give the uncertainty level for prediction at each estimated point. However, considering the uncertainty from measurements, state estimation algorithm, and future cycling profile, the uncertainty level is critical for battery lifetime prediction. Thus, the probabilistic methods with the ability to produce PDF are able to predict the results and also give the confidence bounds. We recommend probabilistic ML methods in the battery lifetime estimation. Another issue here is that most existing researches validate their methods with the same dataset in the model training phase. There exists a question about the generalization of those models in real applications, in which the cycling profile is completely different. Of course, the generalization of the models can be improved by training under more complex ageing conditions. Moreover, suitable structures and parameters should be investigated to develop a self-adaptive battery lifetime predictor in future [33].

In order to accelerate the development and optimization of battery technologies, some methods are proposed to accurately predict the lifetime of battery in a very early stage. Reference [34] tries to solve this issue using lasso and elastic-net regression based on a comprehensive training dataset that includes 124 commercial LFP/graphite cells. The best regression model can predict the lifetime for 90.9% of the tested cell within the first 100 cycles. Moreover, the classification model could classify cells with the first five cycles with the error of 4.9%. This work proves the great potential of applying ML techniques for battery lifespan prediction. Through coupling migration concept into GPR, a migrated GPR-based data science solution is designed in [35] to predict battery future two-stage ageing trajectory, while the knee point effect can be considered just using a small portion of starting ageing data.

5.1.3 Case 1: Li-Ion Battery Cyclic Ageing Predictions with Modified GPR

In this case study, a data science-based solution by devising the modified GPR is developed to predict the future capacity of Li-ion battery with the consideration of various cyclic cases (temperatures and DoDs) [36]. Specifically, a GPR-based data science model structure is first proposed by involving inputs of cyclic battery temperature and DoD. Then after coupling the typical Arrhenius law and empirical polynomial equation with the compositional kernel, a novel GPR model is derived to integrate both electrochemical and empirical elements of battery ageing. This is the first known data science application by constructing GPR’s kernel function with electrochemical and empirical knowledge of battery ageing for future cyclic capacity predictions.

5.1.3.1 Cyclic Ageing Dataset

Table 5.3 details the cyclic ageing test matrix of tested cells with the same middle-SoC of 50% but under various cycling DoDs (50%, 80%, and 100%) and temperatures (35 and 45 °$\mathrm{C}$). More detailed experimental information can be found in [36], which is not repeated here due to space limitations. According to this test matrix, battery capacity dataset under six cyclic cases can be obtained, as shown in Fig. 5.4. To ensure the derived model could study enough mapping mechanism, four cases including Case 1, Case 2, Case 3, and Case 4 are utilized as the training dataset, while other two cases (Case 5 and Case 6) are adopted for the validation purpose.

Table 5.3 Cyclic ageing test matrix of tested cells, reprinted from [36], with permission from IEEE

Fig. 5.4

Cyclic capacity dataset under various DoD and temperature conditions, reprinted from [36], with permission from IEEE

5.1.3.2 Model Structure for Battery Cyclic Capacity Prediction

According to this dataset, a GPR model structure through involving a series of capacity terms $C_{{\text{bat}}} \left( {t - i} \right)$ (here $i$ represents the previous time period), cyclic temperature $T_{{\text{cyclic}}}$ and DoD ${\text{DOD}}_{{\text{cyclic}}}$ is designed to perform future cyclic capacity prediction, as illustrated in Fig. 5.5.

Fig. 5.5

Model structure for battery cyclic capacity prediction, reprinted from [36], with permission from IEEE

In the training process, after the combination of training dataset (here four cases are included), model’s input vector and output vector are $\left[ {C_{{\text{bat}}} \left( {t - i} \right), \ldots ,C_{{\text{bat}}} \left( t \right),T_{{\text{cyclic}}} ,{\text{ DOD}}_{{\text{cyclic}}} } \right]$ and $C_{{\text{bat}}} \left( {t + 1} \right)$, respectively. Both $T_{{\text{cyclic}}}$ and ${\text{DOD}}_{{\text{cyclic}}}$ are constant for each specific cycling case. After training GPR model, both one-step $C_{{\text{bat}}} \left( {t + 1} \right)$ and multi-step $C_{{\text{bat}}} \left( {t + k} \right)$ predictions are performed for Case 5 and Case 6. To capture battery future multi-step capacity, a recursive process by using the previously predicted capacity value as next input point for further predicting a new capacity point under same $T_{{\text{cyclic}}}$ and ${\text{DOD}}_{{\text{cyclic}}}$ is adopted. As the previous and current capacity points are also involved, this model structure has an ability to consider battery ageing trend of different cycling cases.

5.1.3.3 Modified GPR

As $T_{{\text{cyclic}}}$ and ${\text{DOD}}_{{\text{cyclic}}}$ are two key elements for determining battery cyclic ageing dynamics, their influence is thus needed to be considered carefully. To this end, an attempt has been done here to modify GPR’s kernel for developing a novel data science model (labelled as Model B) that could take the electrochemical or empirical elements of Li-ion battery ageing into account. To be specific, the related components within GPR’s kernel are modified separately to reflect cyclic temperature, DoD, and battery capacity.

Temperature dependency: For battery cyclic degradation, the Arrhenius equation $f_{{\text{Arr}}} \left( T \right)$, which shows that battery side reaction will decrease with reduced temperature exponentially, has been reported in numerous literatures [37] to reflect temperature effect as:

$$ f_{Arr} (T) = a \cdot \exp \left( { - E_A /RT} \right) $$

(5.5)

where $a$ is a pre-exponential parameter. $R$ means the ideal gas constant. ${E}_{A}$ represents the activation energy of electrochemical reactions. $T$ stands for battery operational temperature.

According to the Arrhenius equation, a component $k_{T{\text{cyc}}} \left( {x_T ,x_T^{{\prime}} } \right)$ related to $T_{{\text{cyclic}}}$ within GPR’s kernel could be modified with the similar exponential form as:

$$ k_{\text{Tcyc}} \left( {x_T ,x_T^{\prime} } \right) = l_T \cdot \exp \left( { - \frac{1}{\sigma_T }\left\| {\frac{1}{x_T } - \frac{1}{{x_T^{\prime} }}} \right\|} \right) $$

(5.6)

where ${l}_{T}$ and ${\sigma }_{T}$ are two hyperparameters. It should be known that this temperature dependency is described by an isotropic form to reflect the relevance degree between outputs generated by the difference of temperatures ${x}_{T}$ and ${x}_{T}^{{\prime}}$. In this context, GPR model successfully couples the Arrhenius law to capture temperature dependency.

DoD dependency: Based upon numerous experimental conclusions [38], the effects of DoD on battery cycling degradation usually show a polynomial or linear trend. That is, DoD dependency can be empirically captured by using the polynomial equation. In this context, a specific component $k_{{\text{DOD}}} \left( {x_{{\text{DOD}}} ,x_{{\text{DOD}}}^{{\prime}} } \right)$ is modified with the polynomial form to describe DoD dependency as:

$$ k_{\text{DOD}} \left( {x_{\text{DOD}} ,x_{\text{DOD}}^{\prime} } \right) = \left( {l_D \cdot x_{\text{DOD}}^T x_{\text{DOD}}^{\prime} + c_D } \right)^{d_D } $$

(5.7)

where ${l}_{D}$, ${c}_{D},$ and ${d}_{D}$ are three related hyperparameters. It should be noted that this component is a none stationary kernel, which could benefit computation effort as the none stationary kernel generally needs a small number of data to train.

Capacity dependency: To describe dependency of battery capacity, a squared exponential (SE) kernel with hyperparameters ${l}_{c},{\sigma }_{c}$ is used to describe the difference of capacity terms ${x}_{c}$ and ${x}_{c}^{{{\prime}}}$ as:

$$ k_{C{\text{bat}}} \left( {x_c ,x_c^{\prime} } \right) = l_c^2 \exp \left( { - \mathop \sum \limits_{c = 1}^{i + 1} \frac{{\left\| {x_c - x_c^{\prime} } \right\|^2 }}{2\sigma_c^2 }} \right) $$

(5.8)

At this point, all components of GPR’s kernel have been formulated to consider the electrochemical or empirical knowledge of Li-ion battery ageing. Then a novel modified kernel for “Model B” is formulated as:

$$ \begin{aligned} k_{{\text{modified}}} \left( {{{\varvec{x}}},{{\varvec{x}}}^{\prime} } \right) & = k_{C{\text{bat}}} \left( {x_c ,x_c^{\prime} } \right) \cdot k_{T{\text{cyc}}} \left( {x_T ,x_T^{\prime} } \right) \cdot k_{{\text{DOD}}} \left( {x_{{\text{DOD}}} ,x_{{\text{DOD}}}^{\prime} } \right) \\ & = l_f^2 \cdot \exp \left( { - \frac{1}{\sigma_T }\left\| {\frac{1}{x_T } - \frac{1}{{x_T^{\prime} }}} \right\|} \right) \cdot \left( {x_{{\text{DOD}}}^T x_{{\text{DOD}}}^{\prime} + c_D } \right)^{d_D } \cdot \\ & \exp \left( { - \mathop \sum \limits_{c = 1}^{i + 1} \frac{{\left\| {x_c - x_c^{\prime} } \right\|^2 }}{\sigma_c^2 }} \right) \\ \end{aligned} $$

(5.9)

where ${{\varvec{x}}} = \left( {{ }\begin{array}{*{20}c} {x_c ,} & {x_T ,} & {x_{{\text{DOD}}} } \\ \end{array} } \right)$. Based upon the model structure in Fig. 5.5, $x_c$ is $\left[ {C_{{\text{bat}}} \left( {t - i} \right), \ldots ,C_{{\text{bat}}} (t)} \right]$, $x_T$ is $T_{{\text{cyclic}}}$, $x_{{\text{DOD}}}$ is $DOD_{{\text{cyclic}}}$. $l_f ,\sigma_T ,c_D ,d_D$ and $\sigma_C$ are five related hyperparameters.

5.1.3.4 Results and Discussions

Next, the performance of modified GPR is explored based on real cyclic dataset. Figure 5.6 illustrates its training results for four cases. Through deriving the modified kernel to consider battery electrochemical and empirical degradation knowledge, modified GPR could well capture battery capacity ageing dynamics. Quantitatively, the maximum ME, MAE, and RMSE of all these cases are just 0.1689, 0.0557, and 0.0790 Ah, indicating the satisfactory fitting ability can be achieved by using modified GPR.

Fig. 5.6

Training results by using modified GPR for each cyclic cases from training dataset, reprinted from [36], with permission from IEEE

After the training process, both one-step and multi-step tests are carried out to explore the extrapolative prediction performance of modified GPR. According to the one-step prediction results as shown in Fig. 5.7, the well-trained model could capture battery cyclic capacity ageing trends for both Case 5 and Case 6, as indicated by the satisfactory matches among output points and real capacity data. Table 5.4 illustrates its corresponding performance indicators. Not surprisingly, the ME, MAE, and RMSE for one-step prediction of these two cases are all within 0.15Ah, indicating that modified GPR could provide highly accurate performance for one-step cyclic capacity prognostics.

Fig. 5.7

One-step prediction results by using modified GPR for testing dataset, reprinted from [36], with permission from IEEE

Table 5.4 Performance indicators for one-step and multi-step prediction results by modified GPR, reprinted from [36], with permission from IEEE

Next, the evaluation of multi-step prediction of modified GPR is carried out. As illustrated in Fig. 5.8, the predicted capacities also well match the real data, which implies that satisfactory multi-step prediction accuracy can be obtained by using the designed model. For Case 5 in Fig. 5.8a, although a few mismatches happen at the large local fluctuation conditions, the global capacity trend could be also well captured. Here the MAE and RMSE are just 0.0680 and 0.0873 Ah. Similarly, more efficient multi-step prediction results can be obtained in Case 6. Quantitatively, the ME, MAE, and RMSE values are all within 0.2004Ah for cyclic capacity predictions of Case 6.

Fig. 5.8

Multi-step prediction results by using modified GPR for testing dataset, reprinted from [36], with permission from IEEE

Comparison with other GPR models: To further explore the performance of modified GPR model (here named as Model B), a typical SE-based GPR model (SEGM) with two hyperparameters ($\sigma_f$ and $\sigma_l$) in the form of Eq. (5.10) and an automatic relevance determination (ARD)-SE-based GPR model (Model A) with four hyperparameters ($\sigma_f$, $\sigma_T$, $\sigma_{{\text{DOD}}}$, $\sigma_c$) in the form of Eq. (5.11) are also adopted here. Table 5.5 shows the optimized hyperparameters of these GPR models.

$$ k_{{\text{SE}}} \left( {x,x^{\prime} } \right) = \sigma_f^2 \exp \left( { - \frac{{\sum_{c = 1}^{i + 1} \left\| {x_c - x_c^{{\prime} } } \right\|^2 + \left\| {x_T - x_T^{{\prime} } } \right\|^2 + \left\| {x_{{\text{DOD}}} - x_{{\text{DOD}}}^{{\prime} } } \right\|^2 }}{2\sigma_l^2 }} \right) $$

(5.10)

$$ \begin{aligned} k_{{{\text{ARD}}\;{\text{SE}}}} \left( {\varvec{x},\varvec{x}^{\prime } } \right) & = \sigma _{f}^{2} \exp \left[ { - \frac{1}{2}\left( {\frac{{\left\| {x_{T} - x_{T}^{\prime } } \right\|^{2} }}{{\sigma _{T}^{2} }} + \frac{{\left\| {x_{{{\text{DOD}}}} - x_{{{\text{DOD}}}}^{\prime } } \right\|^{2} }}{{\sigma _{{{\text{DOD}}}}^{2} }}} \right.} \right. \\ & \quad \left. {\left. { + \sum\limits_{{c = 1}}^{{i + 1}} {\frac{{\left\| {x_{c} - x_{c}^{\prime } } \right\|^{2} }}{{\sigma _{c}^{2} }}} } \right)} \right] \\ \end{aligned} $$

(5.11)

Table 5.5 Hyperparameters of GPR-based models, reprinted from [36], with permission from IEEE

• (1) Comparison with the Training Results: Fig. 5.9 shows the performance indicators of training results for SEGM, Model A and Model B. Obviously, through using the GPR technique with ARD kernel and modified kernel, the training results of Model A and Model B are both better than those from SEGM. Quantitatively, after coupling GPR models with improved kernels, the related training performance can be enhanced nearly twice in comparison with the SEGM.

  Fig. 5.9

  

  Indicators of using different model types for all training dataset, reprinted from [36], with permission from IEEE

• (2) Comparison with the Prediction Results: To further evaluate the multi-step prediction performance of each model type, the corresponding performance indicators for total testing dataset are compared and illustrated in Fig. 5.10. It can be seen that the ME and RMSE for Model A and Model B are within 0.71 and 0.31 Ah, which are 32.6 and 13.6% less than those of SEGM. In addition, in comparison with Model A owns the ARD-SE kernel, Model B also provides the significant improvement for multi-step cyclic capacity predictions. The RMSE here becomes 0.0835 Ah (72.3% decrease), indicating the superiority of coupling electrochemical or empirical knowledge into GPR.

  Fig. 5.10

  

  Indicators of using different model types for multi-step prediction, reprinted from [36], with permission from IEEE

Based upon the above results, Model B outperforms Model A and SEGM for both the one-step and multi-step predictions. This suggests that the data science model considering battery electrochemical or empirical knowledge is promising for predicting battery future capacities under different cycling cases.

5.1.4 Case 2: Li-Ion Battery Lifetime Prediction with LSTM and GPR

In this subsection, we will introduce a long short-term memory (LSTM) and Gaussian process regression (GPR)-based hybrid data science model to predict battery future capacities and RUL during its cyclic conditions [39]. To achieve reliable future capacities and RUL prediction, three key points need to be concerned: First, the raw capacity data shows the highly nonlinear trend with regeneration phenomenon, which would significantly affect the accuracy of battery health prognosis. Second, capturing the interactions of battery capacity time-series is crucially important to understand its long-term dependencies. Third, prediction uncertainty would occur frequently and should not be ignored.

To handle these challenges, the utilized hybrid data science-based model mainly contains three parts: an empirical mode decomposition (EMD) part to decompose the raw capacity data, a LSTM submodel part to learn the long-term fading dependence of capacity, and a GPR submodel part to quantify the uncertainties of prediction results.

5.1.4.1 Hybrid Data Science Framework for Future Ageing Prediction

The framework of using this hybrid data science-based model as well as the workflow to predict future capacities and RUL of battery is shown in Figs. 5.11 and 5.12, respectively.

Fig. 5.11

Framework of using hybrid data science-based model to predict future capacities and RUL of battery, reprinted from [39], open access

Fig. 5.12

Workflow of using the hybrid LSTM + GPR model to predict future capacities and RUL of battery, reprinted from [39], open access

The utilized hybrid data science-based model can be divided into two parts. For the battery future capacities prediction, with the current and historical capacity vector $\left\{ {C_{{\text{bat}}} \left( {t - i} \right), \ldots , C_{{\text{bat}}} \left( {t - 1} \right), C_{{\text{bat}}} \left( t \right)} \right\}$ as the inputs of model, the future capacity $C_{{\text{bat}}} \left( {t + k} \right)$ could be predicted through using GPR submodel and LSTM submodel to study the potential mappings of intrinsic mode functions (IMFs) and residual after using the EMD to decouple the raw capacity data. Here $k$ and $i$ represent the future step as well as previous step, respectively. The details of EMD technique can be found in [40] for the readers of interest. For the battery RUL prognostics, a recursive prediction process which adopts the previously predicted capacity as the next model input to further predict new capacity value is carried out iteratively until the end-of-life (EoL) of battery is reached. Then the corresponding RUL (${\text{RUL}}_{{\text{bat}}}$) could be calculated. It should be known that the capacity prediction is carried out just based on the historical capacity information. Detailed prediction workflow is described as follows:

• Step 1: Preprocessing: for the data preprocessing before any training processes, an efficient and simple normalization approach [41] is adopted to convert the raw capacity data $C_{{\text{bat}}}$ into a normalized level $C_{{\text{bat}}}^{{\prime}}$ through using an equation: $v^{\prime} = v/C_{{\text{new}}}$. Here $C_{{\text{new}}}$ stands for the fresh capacity value of a battery. $v^{\prime}$ and $v$ are the data samples in $C_{{\text{bat}}}^{{\prime}}$ and $C_{{\text{bat}}}$, respectively. Then the data will be decomposed into several IMFs and a residual through adopting the EMD technique. For the hybrid model part, select the suitable kernel function (here is rational quadratic kernel [42]) for GPR. Set the structure as well as initialize the parameters for LSTM and GPR submodels.

• Step 2: Models training: for a decomposed residual sequence, train the LSTM submodel to fit the residual sequence. For the obtained IMFs, train the GPR submodels to fit each IMF sequence.

• Step 3: Estimation of battery future capacities: for the long-term signal part, using the well-trained LSTM submodel to predict the future residual value. For the regeneration signal part, applying the well-trained GPR models to capture the mean and covariance values of each IMF. Afterwards, the predicted battery future capacities along with the corresponding uncertainty quantification can be obtained through combining these results.

• Step 4: Predicting battery RUL: calculating the battery capacity when it reaches the EoL as $C_{{\text{EOL}}}$. Repeating the prediction step until the predicted future capacity fades below $C_{{\text{EOL}}}$. Then output the predicted RUL of battery.

Following this workflow, the capacities in future cycles of battery could be estimated. Then the battery RUL can be also predicted to provide valuable information for the maintenance decision of aged battery, while the uncertainties of predicted results could be quantified accordingly.

Based upon the above-mentioned workflow, to evaluate the extrapolation performance of this hybrid data-driven model, multi-step ahead prediction tests by using different horizons of 6, 12, and 24 steps are carried out. For these tests, inputs are obtained using 10 historical capacity data up to current cycle, and the prediction is carried out at the cyclic $k$-step ahead of the current cycle.

5.1.4.2 Results and Discussions

Figure 5.13 shows the $k$-step ahead prediction results for some open-source batteries [43]. It can be seen that some short-period mismatches occur in the multi-step ahead prediction cases, which should be mainly caused by the lack of priori information for the future large local fluctuations of battery capacity. However, the predicted capacity values would gradually rematch the true test ones again due to the efficient information decomposition and the strong long-term capture abilities of the utilized hybrid model. Interestingly, as the prediction step increases, the 95% confidence range would distribute in a wider region, implying that the prediction uncertainties become larger. This is hardly surprising given that the relative long-step predictions contain much more uncertainties. Even so, the max uncertainty value is still less than $\pm 10\mathrm{\%}$ capacity range, indicating that the prognostic results are reliable. Here the uncertainty boundaries are mainly related to the so-called scope compliance uncertainty, which is used to quantify “how confident” the GPR model felt when performing predictions.

Fig. 5.13

Results of $k$-step ahead battery capacity predictions, reprinted from [39], open access

Table 5.6 illustrates the performance metrics for the battery multi-step ahead prediction. According to Table 5.6, the maximum RMSE for B05, B06, and B07 are 0.0041, 0.0059, 0.0052 for the 24-step ahead prediction cases, which are 13.9, 20.4, 4% more than those of the 12-steps case, and 7.9, 15.7, and 40.5% larger than those of the 6-steps case. However, all these values are less than 0.006, indicating that satisfactory overall capacity predictions can be obtained for such cases. In the light of this, the proposed LSTM + GPR hybrid model presents a good extrapolation performance for battery multi-step ahead prediction.

Table 5.6 Performance metrics for the battery multi-step ahead prediction, reprinted from [39], open access

According to the requirements of battery health diagnosis system, predicting the future battery RUL as early as possible with a reliable accuracy level is more meaningful for battery real-world application. In such a case, it is critically important to predicting the RUL of battery at an early cycle stage. To further investigate the recursive prediction performance and the robustness of our proposed LSTM + GPR model, the open-source batteries from NASA are tested. Table 5.7 illustrates the quantitative results for all RUL prediction cases of battery. Here, the left and right bounds of uncertainties are defined by the first and last time instant when the obtained confidence range reaches the predefined EoL value of the battery, respectively.

Table 5.7 Quantitative results of RUL predictions for all battery cases, reprinted from [39], open access

For these NASA batteries, to investigate the effects of various EoL values, the predefined EoLs of B05, B06, and B07 are set as 75, 66, and 77%, respectively. Figure 5.14 shows their predicted RUL results. It can be seen that for various batteries with different defined EoL values, the predicted capacities present similar trends with the real capacity curves. According to Table 5.7, the actual EoL values of B05, B06, and B07 are 126, 127, and 142 cycles, respectively. When implementing the RUL prediction at the 34th cycle (here is the first 1/5 proportion), the predicted RUL for B06 is 94, which is only 1 cycle (1.1%) later than the actual RUL. The predicted RULs for B05 and B07 are both 3 cycles (3.3 and 2.7%) later than their actual RULs. Meanwhile, all the RUL uncertainty bounds of these predictions cover the real RUL values effectively.

Fig. 5.14

RUL prediction results for NASA batteries. a B05, b B06, c B07, reprinted from [39], open access

5.2 Battery Fault Diagnosis

To meet the endurance requirements of electric vehicles (EVs), the battery manufacturers pursue large capacity and high energy density by filling more active materials in battery manufacturing, which will make Li-ion battery more prone to faults and safety accidents [44]. In recent years, fire accidents in EVs caused by battery faults such as thermal runaways have occurred frequently. Therefore, the battery fault diagnosis has attracted considerable attention worldwide. Generally, one obvious fault named thermal runaway is caused by mechanical, electrical, or thermal abuses. These abuses will further lead to internal short circuits (ISCs) to a certain extent [45, 46]. That is, the ISC fault is the common cause of battery thermal runaway. Therefore, the early ISC fault diagnosis is important for improving battery safety.

5.2.1 Overview of Data Science-Based Battery Fault Diagnosis Methods

For battery fault diagnosis, related data science-based approaches would directly analyse and explore the battery operation data to detect faults without the requirements of accurate analytical models and expert experiences [47]. Here the battery fault detection process will be simplified without taking complex fault mechanisms and system structures into account, especially for thermal runaways being affected by different factors. However, data science-based battery fault diagnosis methods generally require a suitable preprocess stage to handle raw battery data. As fault mechanisms are usually ignored, it becomes difficult to perform faults explanations based on this method. Besides, several data science-based battery fault diagnosis methods also present inherent drawbacks such as the requirement of numerous battery historical fault data to further cause large computational effort and complexity. In general, the data science-based approaches utilized in the battery fault diagnosis domain can be divided into the types of signal processing, ML, and information fusion.

For signal processing-based fault diagnosis, various signal processing techniques are adopted to extract feature parameters of faults, such as the deviation, variance, entropy, and correlation coefficient. Then these faults will be detected through parameter comparisons with the values from a normal state. NNs and SVMs are two popular utilized data science tools. NN-based fault diagnosis would study the implicit logics from a given input–output pair during an offline training stage and then generates a nonlinear black-box model for the utilizations in an online operation phase. The logic of SVM-based fault diagnosis is to first transform inputs into a high-dimensional space based on the kernel functions and then to search the optimal hyperplane of this high-dimensional space. Through treating Li-ion battery fault diagnosis as the sample classification issue, an accurate SVM-based data science model could be trained by using related historical data. For the information fusion-based data science method, it would use the uncertain information of battery faults to make decisions. According to the analyses of multiple source information, more reliable battery faults can be detected.

Table 5.8 illustrates a comparison of these three data science-based approaches for battery fault diagnosis. For signal process one, due to the nature of neglecting battery dynamics, it is easy to be implemented but becomes difficult to locate battery faults directly considering the strong-coupled battery faults. For the ML one such as NN, it is able to well match and extract knowledge from training samples through setting suitable parameters. However, lacking enough Li-ion battery fault data would also lead to overfitting issues. In comparison with NN, SVM would present better generalization results under the small sample cases. This could become suitable for Li-ion battery cases with limited fault data. However, the hyperparameters of kernel functions within SVMs must be well-optimized or selected for the specific battery fault diagnosis issue. To make full use of multiple sources from Li-ion batteries for improving their fault diagnosis accuracy, a reliable information fusion solution becomes essential.

Table 5.8 A comparison of data science-based approaches for battery fault diagnosis

5.2.2 Case: ISC Fault Detection Based on SoC Correlation

In this subsection, a data science-based battery ISC detection and diagnosis method through using the battery charging data is introduced [48]. In real applications, as battery cells are strictly screened to ensure consistency before being grouped, the characteristic parameters (such as voltage, internal resistance, SoC) between cells in series should show similar trends during charging and discharging, and these parameters have a high degree of correlation between cells. When the ISC occurs in a cell, its characteristics are quite different from those of other cells due to the extra power consumption of the ISC resistor. Therefore, the correlation between the cell with ISC and the normal cells would decrease, making the ISC can be detected from the correlation of cell parameters. When a cell is charged or discharged, its voltage is an easily detected dominant signal. However, battery voltage fluctuates under dynamic conditions, and its internal resistance is difficult to be calculated online. In this context, these two parameters become unsuitable for online ISC detection.

From several existing research [49, 50], SoC estimated by EKF presents less fluctuation even under dynamic conditions. The reason for this phenomenon is that the voltage and current of the battery are the signals that change rapidly under dynamic conditions, while the SoC changes slowly at a long-term scale. Therefore, the correlation of SoC is a competitive candidate for estimating ISC. Figure 5.15 illustrates the proposed online ISC detection based on SoC correlation. Taking three batteries in series as an example, suppose there is an ISC in Cell 3. The cells in series show almost the same SoC during charging and discharging, but the ISC cell’s SoC becomes slightly different from that of normal cells. Specifically, the ISC cell exhibits a faster SoC drop due to additional power loss when discharging, and the SoC of the ISC cell would increase more slowly due to power loss when charging. In addition, the SoC difference between the normal cells and the ISC cells increases with time. Therefore, the difference in SoC can be used to detect ISC online with high sensitivity.

Fig. 5.15

Schematic diagram of the proposed data science-based ISC detection method, reprinted from [48], with permission from Elsevier

To reduce the influence of SoC estimation cumulative error on the correlation, a method of calculating the SoC correlation coefficient in a moving window is adopted to improve the robustness. At a given moment, the SoC correlation coefficient is calculated in a fixed period, and then it is updated in real time with the moving window. As shown in Fig. 5.15, $L$ is the size of moving window, and $r_{{\text{thr}}}$ is the set threshold. Obviously, the SoC correlation coefficient between normal cells should get close to 1, whereas that between a cell with ISC and normal cells should tend to 0. To improve this misjudgment, only the SoC correlation coefficients of a cell and two adjacent cells are less than the threshold before the ISC warning is performed.

5.2.2.1 ISC Detection Algorithm

Figure 5.16 describes the proposed ISC detection algorithm based on SoC correlation, which consists of four steps: In step 1, the voltage, total current, and temperature of each cell are collected in real time. In step 2, the first-order RC (1RC) model and EKF are used to estimate the SoC. In step 3, the SoC correlation coefficient between each two adjacent cells is calculated. Assume that the first and last cells in the series-connected batteries are adjacent. Therefore, each cell has two correlation coefficients relative to its two neighbouring cells. In step 4, the correlation coefficient is compared with a predefined threshold to determine ISC and identify the ISC cell.

Fig. 5.16

ISC online detection algorithm flow based on SOC correlation, reprinted from [48], with permission from Elsevier

Figure 5.17 illustrates the SoC estimation method based on the EKF algorithm, where − represents the prior value, + represents the posterior value, and the subscript k represents the time step. Specifically, the method can be described as follows:

Fig. 5.17

SoC estimation process based on EKF, reprinted from [48], with permission from Elsevier

• Step 1. A priori estimation. First, a priori SoC value at time k is calculated by the ampere-hour counting method as follows.

  $$ {\text{SOC}}_k^- = {\text{SOC}}_{k - 1}^+ + \frac{{\Delta t_{k - 1} }}{C_Q \cdot 3600}I_{k - 1} $$

  (5.12)

  where $C_Q$ is the battery capacity.

• Step 2. Error calculation. The voltage error at time k $\left( {E_k } \right)$ is obtained by comparing the model terminal voltage with the measured terminal voltage as follows:

  $$ E_k = \tilde{U}_k - U_{t,k} $$

  (5.13)

• Step 3. Calculation of Kalman gain matrix. The Kalman gain matrix ${{\varvec{L}}}_k$ at time step k is calculated as follows:

  $$ \left\{ {\begin{array}{*{20}l} {{\bf{P}}_k^- = {\bf{A}}_k {\bf{P}}_{k - 1}^+ {\bf{A}}_k^T + {\bf{Q}}} \hfill \\ {{\bf{L}}_k = {\bf{P}}_k^- {\bf{H}}_k^T /({\bf{H}}_k {\bf{P}}_k^- {\bf{H}}_k^T + {\bf{R}})} \hfill \\ {{\bf{P}}_k^+ = (1 - {\bf{L}}_k {\bf{H}}_k ){\bf{P}}_k^- } \hfill \\ {{\bf{H}}_k = \left( {\begin{array}{*{20}c} {\left. {\frac{{\partial U_{{\text{ocv}}} }}{{\partial {\text{SOC}}}}} \right|_{{\text{SOC}} = {\text{SOC}}_k^- } } & { - 1} \\ \end{array} } \right)} \hfill \\ {{\bf{A}}_k = \left. {\left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & {\exp \left( { - \Delta t/\tau_{1,k} } \right)} \\ \end{array} } \right)} \right|_{{\text{SOC}} = {\text{SOC}}_k^- } } \hfill \\ \end{array} } \right. $$

  (5.14)

  where ${\bf{L}}_k$ is the Kalman gain matrix at time k, ${\bf{P}}$ is the covariance matrix of system, ${\bf{Q}}$ is the system noise covariance, ${\bf{R}}$ is the measurement noise covariance. The larger ${\bf{Q}}$ is, the smaller the weight of SoC in the final estimated SoC is. ${\bf{R}}$ is closely related to voltage correction. The larger ${\bf{R}}$ is, the smaller the influence of voltage correction on SoC estimation results.

  • Step 4. A posteriori estimation. The posteriori SoC is updated with ${\bf{L}}_k$ and error as follows:

    $$ {\text{SOC}}_k^+ = {\text{SOC}}_k^- + L_k E_k $$

    (5.15)

The correlation coefficient is often used to study the linear consistency between two variables. It can be expressed as follows:

$$ \left\{ {\begin{array}{*{20}l} {r_{X,Y} = \frac{{{\text{cov}} (X,Y)}}{{\sqrt {{\text{Var}}(X) \cdot {\text{Var}}(Y)} }} = \frac{{\sum_{i = 1}^n {(X_i - \mu_X )(Y_i - \mu_Y )} }}{{\sqrt {\sum_{i = 1}^n {(X_i - \mu_X )^2 } } \sqrt {\sum_{i = 1}^n {(Y_i - \mu_Y )^2 } } }}} \hfill \\ {\mu_{\text{X}} = \frac{1}{n}\sum\limits_{i = 1}^n {X_i } } \hfill \\ {\mu_Y = \frac{1}{n}\sum\limits_{i = 1}^n {Y_i } } \hfill \\ \end{array} } \right. $$

(5.16)

where $r_{X,Y}$ is the correlation coefficient of variables $X$ and $Y$, ${\text{cov}} \left( {X,Y} \right)$ is the covariance of $X$ and $Y$, $Var(X)$ and ${\text{Var}}(Y)$ are the variance of variables $X$ and $Y$, respectively. $\mu_X$ and $\mu_Y$ are the mean value of variables $X$ and $Y$, respectively. n represents the number of samples, $r_{X,Y}$ is the correlation coefficient between X and Y, and its value range is −1 to 1. When $r_{X,Y} = 0$, it means that the two variables are completely unrelated. When $r_{X,Y} < 0$, it means that the two variables are negative correlation, and when $r_{X,Y} > 0$, it means that the two variables are positive correlation.

For the convenience of calculation, Eq. (5.16) can be rewritten as follows:

$$ r_{X,Y} = \frac{{n\sum_{i = 1}^n {X_i Y_i } - \left( {\sum_{i = 1}^n {X_i } } \right)\left( {\sum_{i = 1}^n {Y_i } } \right)}}{{\sqrt {n\sum_{i = 1}^n {X_i^2 } - \left( {\sum_{i = 1}^n {X_i } } \right)^2 } \sqrt {n\sum_{i = 1}^n {Y_i^2 } - \left( {\sum_{i = 1}^n {Y_i } } \right)^2 } }} $$

(5.17)

To facilitate the algorithm implementation, Eq. (5.17) can be discretized as follows:

$$ \left\{ {\begin{array}{*{20}l} {a_k = a_{k - 1} + X_k Y_k } \hfill \\ {b_k = b_{k - 1} + X_k } \hfill \\ {d_k = d_{k - 1} + Y_k } \hfill \\ {f_k = f_{k - 1} + X_k^2 } \hfill \\ {g_k = g_{k - 1} + Y_k^2 } \hfill \\ {(r_{X,Y} )_k = \frac{na_k - b_k d_k }{{\sqrt {nf_k - b_k^2 } \sqrt {ng_k - d_k^2 } }}} \hfill \\ \end{array} } \right. $$

(5.18)

where $a_k$ is the cumulative term of the product of two variables, $b_k$ and $d_k$ are the cumulative terms of variables $X$ and $Y$, respectively, and $f_k$ and $g_k$ are the cumulative terms of the power of variables $X$ and $Y$, respectively.

As described above, a moving window is used to calculate the SoC correlation coefficient to improve the robustness. Therefore, Eq. (5.18) can be rewritten as follows:

$$ \left\{ {\begin{array}{*{20}l} {A_k = a_k - a_{k - L} } \hfill \\ {B_k = b_k - b_{k - L} } \hfill \\ {D_k = d_k - d_{k - L} } \hfill \\ {F_k = f_k - f_{k - L} } \hfill \\ {G_k = g_k - g_{k - L} } \hfill \\ {(r_{X,Y} )_k = \frac{LA_k - B_k D_k }{{\sqrt {LF_k - B_k^2 } \sqrt {LG_k - D_k^2 } }}} \hfill \\ \end{array} } \right.\;\;(k \ge L) $$

(5.19)

where $L$ is the size of the moving window.

Noted that if $L$ is significantly large, the detection sensitivity would be reduced. Therefore, an appropriate value of L should be selected to achieve high detection accuracy and short detection time, and it is set to 600 in this study.

5.2.2.2 Experimental Set-Up and Process

The real ISC is concealed inside the battery, and it is generally difficult to trigger it quantitatively. Common ISC experiments are carried out through some simulation experiments. The ideal ISC equivalent experiment needs to meet the following requirements [51]: (a) It can simulate the thermal and electrical behaviours of the battery; (b) The ISC resistance, trigger form, and time are controllable; (c) The battery damage is consistent with the actual situation; (d) High repeatability. At present, there is no ideal method that can meet all the above requirements. The commonly used method is to simulate the ISC with the external parallel resistance of the battery. This method has high repeatability and controllability, but it fails to reflect the thermal characteristics of ISC. Since the proposed ISC detection method does not involve the thermal characteristics of the battery, it is reasonable for us to use an external parallel resistor to simulate ISC. The experimental set-up is shown in Fig. 5.18, in which seven commercial ternary cells with the standard capacity of 50 Ah, cut-off voltage of 4.25 V, and discharge cut-off voltage of 2.8 V are connected in series for charging and discharging experiments. In addition, a resistor is connected in parallel to Cell 2 as the equivalent ISC resistance and controlled by a switch.

Fig. 5.18

Experimental device, reprinted from [48], with permission from Elsevier

According to the Chinese National Standard GB/T 31484-2015, a normal battery should maintain 85% of its full capacity after being rest for 28 days in open circuit. Therefore, the critical ISC resistance can be roughly calculated as follows:

$$ R_{{\text{ISC}}} = \frac{U_0 }{{{{C_0 *15\% } / {(28*24)}}}} = \frac{3.65}{{{{50 \cdot 15\% } / {\left( {28{*}24} \right)}}}}{ = }327\Omega $$

(5.20)

where $U_0$ is the nominal voltage of the test cell; $C_0$ is the nominal capacity.

In the experiment, different ISC equivalent resistors (1, 10, and 100 Ω) are used to simulate different degrees of ISC. Specifically, the resistor of 100 Ω is used to simulate early slight ISC, resistor of 10 Ω is used to simulate developing ISC, and resistor of 1 Ω is used to simulate severe ISC. The charging and discharging schemes of this experiment can be described as follows. First, the experimental battery pack is charged to the cut-off voltage using a constant current, and then discharged under the New European Driving Cycle (NEDC) dynamic condition. The ISC control switch is closed at the same time, and the ISC resistor starts to work until the discharge cut-off voltage is reached. Repeat the above steps four times. Due to the extra power consumption in the external resistor, the voltage difference between the Cell 2 and the normal cell increases. Therefore, when using different ISC resistors to simulate different ISCs, the Cell 2 needs to be taken out from the module and charged it separately to ensure that its voltage is equivalent to the normal cell.

5.2.2.3 ISC Fault Diagnosis Results

Figure 5.19 shows the charge and discharge results of three equivalent ISC tests. It can be observed that the voltage of the Cell 2 is consistent with the initial voltage of other normal cells, indicating that the power loss caused by the ISC in the initial stage cannot be clearly distinguished. After the ISC is activated for a period, the voltage of Cell 2 is significantly lower than that of other normal cells, and the difference increases over time. In addition, increasing the severity of the ISC (i.e. reducing the ISC resistor) will produce an earlier voltage difference. As shown in Fig. 5.20b, when the cell is discharged to the cut-off voltage, the voltage difference between the Cell 2 and the normal cells reaches the maximum. This is because the open-circuit voltage of the cell at a low SoC shows the fastest downward trend, and the cell with ISC enters the low SoC region before the normal cells, thereby accelerating the voltage drop and further increasing the voltage difference.

Fig. 5.19

Voltage curves for different ISC resistors under NEDC dynamic condition. a $R_{{\text{ISC}}} = 100\;{\Omega }$; b $R_{{\text{ISC}}} = 10\;{\Omega }$; c $R_{{\text{ISC}}} = 1\;{\Omega }$, reprinted from [48], with permission from Elsevier

Fig. 5.20

SoC estimation results for different ISC resistors under the NEDC dynamic condition. a $R_{{\text{ISC}}} = 100\;{\Omega }$; b $R_{{\text{ISC}}} = 10\;{\Omega }$; c $R_{{\text{ISC}}} = 1\;{\Omega }$, reprinted from [48], with permission from Elsevier

In conclusion, these experimental results show that ISC can be detected and fault cells can be identified online through the difference of charging and discharging curves of the series cells. However, the voltage curve fluctuates greatly, which may lead to misjudgment. In addition, these voltage curves are obtained without considering sensor error and noise. Note that the voltage sensor error in the actual EVs is quite large, and the fluctuation of the actual voltage curve may be greater, which may affect the ISC detection accuracy. Therefore, it is not feasible to judge ISC directly through voltage curves.

ISC detection using SoC correlation coefficient. Based on the above experimental data and the EKF algorithm, the SoC of each cell is estimated, and then the SoC difference under different ISC resistors is studied. Figure 5.20 shows the SoC estimation results under NEDC dynamic conditions. As expected, the SoC curves are similar to the voltage curves described in Fig. 5.20, but they fluctuate much less than the voltage curves. Therefore, it is reasonable and feasible to detect ISC online by estimating the difference of SoC and setting a reasonable threshold, which can greatly reduce the false positive probability.

The SoC correlation coefficients of each cell and two adjacent cells are calculated, and the results are shown in Fig. 5.21. Here, $r_{i,j}$ is the SoC correlation coefficient between cells $i$ and $j$, and the green dotted line indicates the ISC detection threshold. Only the two SoC correlation coefficients corresponding to a cell are simultaneously below the threshold, the cell can be judged as the ISC cell. To ensure the high accuracy and short detection time of ISC, the appropriate threshold is determined by offline calibration. From the experimental results, when the equivalent ISC resistance is 100 Ω and the SoC difference is higher than 2.5%, the correlation coefficient is 0.7, and thus, 0.7 is selected as the threshold in this study. Moreover, the initial SoC estimation at EV startup may have a large error to cause false alarms. Therefore, the car does not start the ISC detection until a period of time after it has been started. In this study, this time is set to 1000 s.

Fig. 5.21

ISC detection results based on the SoC correlation coefficient. a $R_{{\text{ISC}}} = 100\;{\Omega }$; b $R_{{\text{ISC}}} = 10\;{\Omega }$; c $R_{{\text{ISC}}} = 1\;{\Omega }$, reprinted from [48], with permission from Elsevier

The enlarged graph on the right of Fig. 5.21 shows the SoC correlation coefficients between Cell 1 and Cell 2 (blue curve) and between Cell 2 and Cell 3 (red curve). In addition, when the two SoC correlation coefficients related to Cell 2 are lower than the threshold, an ISC alarm will be triggered, which is indicated by a red triangle. The SoC correlation coefficient of normal cells is above 0.8, indicating that the SoC between these cells is highly correlated. On the other hand, the SoC correlation coefficient of ISC cells with low SoC suddenly drops, indicating that the SoC correlation between the ISC cell and the normal cells is low. As shown in Fig. 5.21b, when the ISC resistance is less than $10\;\Omega$, the SoC correlation coefficient is 0, indicating that the SoC of the ISC cell is completely different from that of the normal cells.

In general, it can be observed that the proposed online ISC detection algorithm is very sensitive and accurate under dynamic conditions. In addition, as shown in Fig. 5.21a, for the early ISC with a large equivalent resistance, although the detection time is longer, the method is still effective under dynamic conditions, and the detection time is significantly shorter than the latency of the early ISC. Therefore, the proposed method is effective for the online detection of early ISC.

Comparison with other ISC detection methods. The proposed data science-based ISC detection method is compared with the other three ISC detection methods to confirm its advantages, namely the static leakage, SoC difference, and voltage difference methods. The static leakage method is the simplest and most direct ISC detection method. According to GB/T31484-2015 [48], the battery should maintain 85% of its full capacity after being idle for 28 days in an open-circuit state. When the equivalent ISC resistance is 100, 10, and 1 Ω, the time required for the 15% capacity loss can be calculated as 202.7, 20.3, and 2.03 h according to Eq. (5.21), which is the ISC detection time.

$$ t_d = \frac{C_0 \cdot 15\% }{{28 \cdot 24 \cdot U_0 /R_{{\text{ISC}}} }} $$

(5.21)

where $t_d$ is the time required to detect the ISC.

The SoC difference method estimates the ISC based on the difference between the SoC of each cell and the average SoC. Since the cell with the smallest SoC is more likely to have an ISC failure, this cell is not considered when calculating the average value:

$$ \Delta {\text{SOC}}_i = {\text{SOC}}_i - \left( {\sum_{i = 1}^7 {{\text{SOC}}_i - \min \left\{ {{\text{SOC}}_1 , \ldots {\text{SOC}}_7 } \right\}} } \right)/6 $$

(5.22)

where $\Delta \text{SOC}_i$ is the difference between SoC of the ith cell and the average SoC.

In this study, a 5% SoC difference is set as the threshold for ISC detection; that is, if the cell's SoC difference is higher than 5%, it is considered that the cell has an ISC failure. Figure 5.22a shows that the SoC difference of the Cell 2 with an ISC resistance of 100 Ω increases very slowly and does not reach the threshold after 24 h, indicating that this method is not suitable for early ISC detection and the detection time is very long. Nevertheless, as the degree of ISC increases, the SoC difference increases significantly over time, thereby reducing the detection time.

Fig. 5.22

ISC detection results using the SOC difference. a $R_{{\text{ISC}}} = 100\;{\Omega }$; b $R_{{\text{ISC}}} = 10\;{\Omega }$; c $R_{{\text{ISC}}} = 1\;{\Omega }$, reprinted from [48], with permission from Elsevier

Figure 5.23 shows the ISC detection results based on the voltage difference method. The threshold of voltage difference is set to 0.5 V in this study. It can be observed that the detection time using the voltage difference method for the early ISC is very long, and the voltage difference fluctuates greatly. Therefore, the voltage difference method is not suitable to detect the early ISC. It can be concluded that the proposed SoC correlation coefficient method has better detection accuracy and shorter detection time than the other three methods.

Fig. 5.23

ISC detection results using the voltage difference. a $R_{{\text{ISC}}} = 100\;{\Omega }$; b $R_{{\text{ISC}}} = 10\;{\Omega }$; c $R_{{\text{ISC}}} = 1\;{\Omega }$, reprinted from [48], with permission from Elsevier

The battery has rich charge and discharge data in the service process, and the ISC will cause the change of battery voltage, which makes it possible to detect the ISC online and in real time based on the charge and discharge data. To improve the reliability of the ISC detection, a data-driven ISC detection method based on SoC correlation coefficient is proposed in this section. The experimental results show that the proposed data science-based solution has the advantages of good real time, high accuracy and excellent robustness. It can detect the early ISC and greatly improve battery safety.

5.3 Battery Charging

Battery charging is also a key part required to be managed during battery operation [52]. Technical issues facing the development of efficient battery charging solutions arise from different charging objectives, hard constraints, and charging termination, as illustrated in Fig. 5.24. In this context, this section first introduces several key objectives that need to be considered during battery charging, then two case studies through designing suitable data science-based solutions for both battery cell charging and pack charging are detailed and analysed.

Fig. 5.24

Structures of charging infrastructure, objective as well as termination

5.3.1 Battery Charging Objective

In general, the objective of designing suitable charging solution for Li-ion batteries is to provide a good capacity utilization, a short time for charging process, a high charging efficiency with less energy loss, while maintaining a long battery cycle life [53]. In addition, battery temperature will rise dramatically during the charging process especially in high power applications. Overheat temperature would result in battery failures so the temperature rise is also a critical objective for battery charging. Therefore, suitable battery operation management that provides the proper charging patterns to balance these charging objectives is indispensable for battery applications.

Short charging time: Charging time is one of the key aspects for the battery applications especially in transportation such as EVs. On the one hand, a long charging time will inevitably affect the convenience of EV usage and limit its acceptance by customers [54]. It is necessary to improve charging speed for EVs especially in some public charging conditions that are similar to gasoline refuelling for conventional vehicles. However, too fast charging will lead to significant energy loss and battery performance degradation, further decreasing battery performance or causing safety problems [55]. It is therefore rational to consider charging time as one of the key factors in designing battery charging solutions.

High energy efficiency: Battery energy efficiency is the ratio of the charged energy to the energy required to be discharged to the initial state prior to charging [56]. The energy efficiency during the battery charging process would be affected by many factors such as current, internal resistance, SoC, and temperature. Large energy loss implies low efficiency of energy conversion in battery charging, which needs to be addressed [57]. It is critical to develope optimal charging strategy that can decrease the energy loss caused by battery internal resistance and control charging or discharging currents appropriately to achieve high energy efficiency during the battery charging process.

Low-temperature rise: Temperature affects battery performance in many ways such as round trip efficiency, energy and power capability, cycle life, reliability, and charge acceptance [58]. Both the battery surface and internal temperatures may exceed permissible levels when it is charged with high current especially in high power applications [59], and the overheating temperatures may intensify battery ageing process and even cause explosion or fire in severe situations [60]. In this context, the temperature rise of battery becomes an important factor that needs to be considered in battery operation management and many strategies are researched to achieve battery charging with low-temperature rise.

Long cycle life: The cycle life of battery is the amount of the complete charging/discharging cycles that a battery works until its capacity falls below 80% of its nominal capacity. Even if the same charging current rate is applied, the cycle life of Li-ion battery would be also extremely influenced by different charging ways. Fast charging leads to the accelerated fading of battery capacity due to the related increase of a surface layer and the loss of Li+ ions. This process is also associated with the lithium plating onto the battery anode as well as the polarization at the electrode–electrolyte surface. According to the analysis of battery electrochemistry, a suitable charging current profile plays the important role in prolonging battery service life and needs to be carefully considered.

Other objectives: Some other objectives are also crucial for achieving efficient battery charging. Battery polarization, which means the variation of the equilibrium potential in a battery electrochemical reaction, has a tremendous impact on the battery charging performance. The battery charging current induces losses due to its polarization. Both the battery charging speed and efficiency would be enhanced by controlling battery polarization. In addition, decreased polarization contributes to the reduction of battery capacity fade, because the temperature rising rate can be restricted. So the charging polarization is also selected as an important objective to achieve battery health-conscious charging [61]. Some researchers also focus on developing charging strategies to increase battery available capacity [62], which could be achieved by the smaller increase of internal resistance or lower temperature rise.

Conflicting objectives: Developing an advanced charging strategy for battery operation management is not a simple task and usually implicates the trade-off among different coupled but conflicting objectives such as charging time and energy efficiency [63], while also pursuing temperature rise minimization, energy loss minimization, long service life, and low normalized discharged capacity. Therefore, it is essential to involve the optimization of multiple conflicting objectives when evaluating the cost-effectiveness of battery charging patterns in real battery charging applications.

The typical combinations of multi-objective for Li-ion battery charging can be divided into some main parts. For multi-stage constant-current charging, the double-objective optimization is often considered which combines the battery charging time and the normalized discharged capacity. Besides, the temperature rise, efficiency, and cycle life are also selected as the charging objectives in some published works [64, 65]. For the constant-current–constant-voltage charging, the main multi-objectives are often composed of battery charging time and energy efficiency. Besides, the temperature rise, cycle life, and capacity fading are also noticeable points for developing optimal charging strategies. For other charging strategies in published works, the same multi-objectives can be adopted as the optimization targets for improving battery charging performance during its operation management.

5.3.2 Case 1: Li-Ion Battery Economic-Conscious Charging

In this study, a data science-based framework through using multi-objective optimization solutions for economy-conscious charging is introduced, as shown in Fig. 5.25 [66]. Given the predefined battery electrothermal-ageing model and the economic price model from [67, 68], three important charging objective functions including battery charging time, average temperature, and particularly charging cost can be created. Then the suitable charging pattern is designed to charge battery with effective energy and time management.

Fig. 5.25

Overall multi-objective optimization framework for economic-conscious charging, reprinted from [66], with permission from Elsevier

5.3.2.1 MCC Profile

In this research, the typical multi-stage constant-current (MCC) charging pattern is explored. This MCC charging pattern generally consists of some CC phases, as illustrated in Fig. 5.26. Due to Li-ion battery being usually less susceptible to lithium plating at low SoC conditions, a relatively large CC rate $I_{{\text{CC}}1}$ could be utilized at the beginning of MCC charging process to transfer enough energy throughput into a battery. Then a series of stepwise reduced CC phases would be adopted until reaching the last CC phase with a CC rate $I_{{\text{CCN}}}$. During this process, a CC phase would turn into another CC phase when the battery terminal voltage rises up to the predefined cut-off voltage $V_{{\text{cut}}}$.

Fig. 5.26

Charging current and terminal voltage for MCC profile, reprinted from [66], with permission from Elsevier

Although this MCC pattern is convenient to be applied in EV applications, a key but challenging issue is to set suitable battery current and voltage values during its entire charging process. In theory, MCC pattern’s charging time is mainly affected by battery cut-off voltage $V_{{\text{cut}}}$ and current rates in each CC phase ($I_{{\text{CC}}1}$, $I_{{\text{CC}}2}$,…, $I_{{\text{CCN}}}$). As recommended by Li-ion battery manufacturers, $V_{{\text{cut}}}$ is generally set as its maximum level to improve battery capacity utilization. For charging current rate, a large value could directly speed up the charging process, but also lead to severe issues such as lithium plating, increased energy loss, and overheating of battery, further significantly affecting battery service life. Then the economic cost caused by the wasted electricity and faded battery capacity will be increased. In this context, it is vital to optimizing current rates of MCC pattern for efficient equilibration among the time, temperature, and particularly economic cost during the battery charging process.

5.3.2.2 Charging Cost Function

Based upon the coupled battery electrothermal model and related economic price model from [66], several crucial but conflicting cost functions could be formulated.

For battery charging time (BCT), less BCT represents that the battery charging process could be finished with a faster speed. In the MCC pattern, the cost function for BCT is expressed as:

$${\text{JMCC}}_{\text{BCT}}=\Delta t*{t \text{CC}}_{N}/60$$

(5.23)

where $\Delta t$ represents the sampling time period, ${t \text{CC}}_{N}$ denotes the total amount of sample points when a battery is charged from its initial SOC to its final target SOC Here, ${\text{JMCC}}_{\text{BCT}}$ has an unit of minute (M).

For battery average temperature (BAT), less BAT could protect battery from overheating under same ambient temperature. The cost function ${\text{JMCC}}_{\text{BAT}}$ for BAT during MCC pattern is described as:

$${\text{JMCC}}_{\text{BAT}}=\Delta t*\left[\sum \limits_{t=0}^{{ t \text{CC}}_{1}}{T}_{a}\left(t\right)+\dots + \sum \limits_{t{\text{CC}}_{N-1}+1}^{t{\text{CC}}_{N}}{T}_{a}\left(t\right)\right]/{t \text{CC}}_{N}$$

(5.24)

where ${t \text{CC}}_{1}$, …,${t \text{CC}}_{N}$ are sample points when each CC phase is ended, respectively, ${T}_{a}(t)$ represents a radial average battery temperature at time point $t$.

For battery charging cost (BCC) that is divided into two main types including battery electricity loss cost (BEC) and battery ageing cost (BAC), its objective function ${\text{JMCC}}_{\text{BCC}}$ during MCC pattern can be described by:

$${\text{JMCC}}_{\text{BCC}}={\text{JMCC}}_{\text{BEC}}+{\text{JMCC}}_{\text{BAC}}$$

(5.25)

where ${\text{JMCC}}_{\text{BEC}}$ denotes the charging cost caused by electricity loss and could be further described by:

$$ \left\{ {\begin{array}{*{20}l} {{\text{JMCC}}_{{\text{BEC}}} = \Delta t*\left[ {\sum\limits_{t = 0}^{t{\text{CC}}_1 } {f_{{\text{EC}}} } (t) + \cdots + \sum\limits_{t{\text{CC}}_{N - 1} + 1}^{t{\text{CC}}_N } {f_{{\text{EC}}} (t)} } \right]} \\ {f_E (t) = a(t)*\left[ {I^2 (t)R_0 (t) + V_1^2 (t)/R_1 (t) + V_2^2 (t)/R_2 (t)} \right]} \\ \end{array} } \right. $$

(5.26)

where ${f}_{\text{EC}}\left(t\right)$ represents electrical energy loss cost occurred at $t$, $a\left(t\right)$ denotes the corresponding TOU price at $t$. It should be known that the value of $a\left(t\right)$ remains constant for a long time period and would be affected by the time instant $T$ of a day with a relation as follows:

$$ a\left( t \right) = f\left( {t \in T} \right) = \left\{ {\begin{array}{*{20}l} {\frac{{1.1946}}{{3.6 \times 10^{6} }}} & {T \in \left( {23:00 - 7:00} \right)} \\ {\frac{{1.4950}}{{3.6 \times 10^{6} }}} & {\begin{array}{*{20}l} {T \in \left( {7:00 - 10:00} \right)\;{\text{or}}} \\ {T \in \left( {15:00 - 18:00} \right)\;{\text{or}}} \\ {T \in \left( {21:00 - 23:00} \right)} \\ \end{array} } \\ {\frac{{1.8044}}{{3.6 \times 10^{6} }}} & {\begin{array}{*{20}l} {T \in \left( {10:00 - 15:00} \right)\;{\text{or}}} \\ {T \in \left( {18:00 - 21:00} \right).} \\ \end{array} } \\ \end{array} } \right. $$

(5.27)

For battery degradation cost during MCC charging, its cost function ${\text{JMCC}}_{\text{BAC}}$ could be obtained as:

$$\left\{\begin{array}{l}{\text{JMCC}}_{\text{BAC}}=\frac{{B}_{\text{new}}-{B}_{\text{used}}}{{N}_{\text{BAC}}}\\ {N}_{\text{BAC}}=\frac{{T}_{\text{Ah}}}{{E}_{\text{Ah}}}\end{array}\right.$$

(5.28)

where ${B}_{\text{used}}$ stands for the cost value of utilized battery, ${N}_{\text{BAC}}$ represents the total amount of charging cycle when a battery’s capacity degrades to its end-of-life (EoL). In addition, ${T}_{\text{Ah}}$ is the total throughput in Ah for all charging cycles while ${E}_{\text{Ah}}$ is that of one cycle. For EV applications, Li-ion battery’s EoL is generally achieved when the capacity of battery cell degrades to 80% of its nominal value. Supposing that a battery is charged by using the same MCC pattern during each charging cycle, ${T}_{\text{Ah}}$ would be mainly affected by the average $\tilde{I}$, $\widetilde{{{\text{soc}}}}$, and $\widetilde{T_a }$ of this specific MCC pattern. Then ${T}_{\text{Ah}}$ could be obtained by:

$$ T_{{\text{Ah}}} = \left[ {\frac{20}{{\sigma_{{\text{funct}}} \left( {\tilde{I},\widetilde{{{\text{soc}},}}\widetilde{T_a }} \right)}}} \right]^{1/z} $$

(5.29)

$$ \left\{ {\begin{array}{*{20}l} {\tilde{I} = \sum\limits_{t = 0}^{t{\text{CC}}_N } I (t)/{t \text{CC}}_N } \\ {\widetilde{{{\text{soc}}}} = \sum\limits_{t = 0}^{t{\text{CC}}_N } {{\text{soc}}} (t)/t{\text{CC}}_N } \\ {\widetilde{T_a } = \sum\limits_{t = 0}^{t{\text{CC}}_N } {T_a } (t)/t{\text{CC}}_N .} \\ \end{array} } \right. $$

(5.30)

For the ${E}_{\text{Ah}}$ of a specific MCC pattern, its charging throughput is also obtained as:

$$ E_{{\text{Ah}}} = \frac{\Delta t}{{3600}}\sum_{t = 0}^{t{\text{CC}}_N } {I(t)} . $$

(5.31)

In this study, once the current rates of each CC phase are optimized, all elements of the cost functions ${\text{JMCC}}_{\text{BCT}}$, ${\text{JMCC}}_{\text{BAT}}$, and ${\text{JMCC}}_{\text{BCC}}$ could be obtained. Namely, current rates ${I}_{\text{CC1}}$, ${I}_{\text{CC2}}$,…, ${I}_{\text{CCN}}$ become the decisive variables that require to be carefully optimized for battery economy-conscious charging.

5.3.2.3 Optimization Procedure

The optimization of MCC charging pattern belongs to a highly nonlinear and strongly coupled process. Numerous time-varied parameters such as capacitors and resistances within coupled battery model are strongly associated with battery temperatures and SoCs. Battery capacity degradation would be also highly affected by its electrical and thermal dynamics. Moreover, different constraints including the charging current, terminal voltage, battery SoC, and temperature need to be considered during optimization as follows:

$$\left\{\begin{array}{c}\begin{array}{c}{I}_{\text{min}}\le I(t)\le {I}_{\text{max}}\\ {V}_{min}\le V(t)\le {V}_{\text{max}}\end{array}\\ {s}_{I}\le \text{SoC}(t)\le {s}_{T}\\ {T}_{\text{min}}\le {T}_{a}\left(t\right)\le {T}_{\text{max}}.\end{array}\right.$$

(5.32)

For this time-varied and complex optimization problem, an effective multi-objective optimization tool is required to optimize MCC pattern and equilibrate these key but conflicting charging objectives. Due to the advantages of being immune from the NP-hard and highly nonlinear issues, meta-heuristic optimization method becomes a powerful tool. Among different meta-heuristic multi-objective optimizers, non-dominated sorting genetic algorithm II (NSGA-II) has been widely used in many real applications for handling complicated optimization issues due to its outstanding ability to keep elitism optimization and preserve diversity. In the light of this, NSGA-II approach is herein employed to search the optimal MCC patterns, with the purpose of achieving economy-conscious charging under various priorities.

Figure 5.27 illustrates the optimization flowchart of searching the proper MCC pattern to equilibrate battery charging time, average temperature, and particularly economic cost. Each process is detailed as follows:

Fig. 5.27

Optimization flowchart by searching the proper MCC profile to equilibrate charging objectives, reprinted from [66], with permission from Elsevier

• Step 1: Set battery charging constraints. These constraints include: (1) the initial and target SoCs:${s}_{I}$ and ${s}_{T}$; (2) the current, voltage, and temperature limitations: ${I}_{\text{min}}$, ${I}_{\text{max}}$,${V}_{\text{min}}$, ${V}_{\text{max}}$, ${T}_{\text{min}}$, and ${T}_{\text{max}}$.

• Step 2: Set MCC pattern’s parameters. These parameters include: (1) the cut-off voltage: ${V}_{\text{cut}}$; (2) the number of CC charging phases: ${N}_{\text{CC}}$.

• Step 3: Select charging objectives with the consideration of different user requirements. According to the priority of battery application, formulating a proper cost function with a combination of BCT, BAT, and BCC.

• Step 4: Initialize NSGA-II’s parameters. The main parameters of NSGA-II include: (1) the number of generation: ${G}_{m}$; (2) the population size: ${N}_{p}$.

• Step 5: For $k=1$ to ${k}_{\text{max}}$, do

• (1) For each CC phase, calculating the cost function combining the selected charging objectives until battery SoC reaches a target ${s}_{T}$. When battery terminal voltage goes up to its ${V}_{cut}$, a CC phase would be terminated and then jump into another one.

• (2) Evaluate ${\text{JMCC}}_{\text{BCT}}$, ${\text{JMCC}}_{\text{BAT}}$, and ${\text{JMCC}}_{\text{BCC}}$ of whole charging process through summing the cost functions of each CC phase. Then analyse the optimized MCC pattern’s performance based on the optimal sets from related Pareto frontier.

• (3) Search a new MCC pattern again if the obtained MCC pattern is unsatisfactory. This optimization process will be terminated when an end criterion ${k}_{\text{max}}$ is reached.

Through following this data science framework, economy-conscious battery charging could be achieved based on the optimized MCC pattern. Moreover, battery charging time and average temperature could be also balanced.

5.3.2.4 Results and Discussion

Then the sensitivity of key parameters including battery cut-off voltage, convection resistance, and ambient temperature is first analysed to explore their effects on the MCC pattern optimization. The optimal set is drawn by the form of Pareto frontier to equilibrate conflicting objectives for various user demands. Constant parameters are set as: $\Delta t=1s$, ${s}_{I}=0.1$, and ${s}_{T}=0.95$, respectively. Temperatures are set as: ${{T}_{s}\left(0\right)=T}_{I}\left(0\right)={T}_{\text{amb}}$. Battery parameter limitations are set as: ${I}_{\text{min}}=0\,\text{A}$, ${I}_{max}=10\,\text{A}$, ${V}_{\text{min}}=3.0\,\text{V}$, ${V}_{\text{max}}=3.6\,\text{V}$, ${T}_{\text{min}}=15\;^{\circ}{\text{C}}$, and ${T}_{\text{max}}=45\;^{\circ}{\text{C}}$. The population size and generation number of NSGA-II are set as ${N}_{p}=360$ and ${G}_{m}=60$.

Sensitivity Analysis for the MCC Profile

For the multi-objective optimizations, the Pareto frontier could provide a series of optimal strategies to graphically demonstrate cases that one cost function cannot be enhanced without making other cost functions worse. If the Pareto frontier from a specific case could distribute evenly and widely, while the points in this Pareto frontier are close to the origin of coordinate, the optimal MCC pattern for such case becomes better. In the light of this, sensitivities of cut-off voltage, convection resistance, and ambient temperature are analysed via the Pareto frontier to explore their influences on battery charging.

Sensitivity of cut-off voltage: in theory, a small cut-off voltage ${V}_{\text{cut}}$ of battery would restrain battery capacity utilization, but too large ${V}_{\text{cut}}$ can also cause damages or safety issues to battery. In this test, ${I}_{\text{CC1}}$ is initially set within 5A (2C) and 10A (4C). The convection resistance ${R}_{u}$ and ambient temperature ${T}_{\text{amb}}$ are set as 3.08 ${\text{KW}}^{-1}$ and 25 °C, respectively. Six ${V}_{\text{cut}}$ cases (3.60, 3.58, 3.56, 3.54, 3.52, and 3.50 V) are utilized to explore battery cut-off voltage’s sensitivity.

Figure 5.28 illustrates the Pareto frontiers for the optimized MCC patterns with various ${V}_{\text{cut}}$. Obviously, a better optimal MCC set could be obtained through using a larger ${V}_{\text{cut}}$. As ${V}_{\text{cut}}$ reduces, the number of obtained optimal points would also decrease, and the Pareto frontier tends to move to the right, leading to a fact that more time is spent for battery charging. In addition, the values of both ${\text{JMCC}}_{\text{BAT}}$ and ${\text{JMCC}}_{\text{BCC}}$ would become less with the reduced ${V}_{\text{cut}}$. It can be concluded that a small ${V}_{\text{cut}}$ leads to the low charging current, and hence, the average temperature rise of battery is also restrained, further causing less cost of charging, but charging speed would be sacrificed accordingly.

Fig. 5.28

Effects of cut-off voltages on the optimized MCC profile via Pareto frontier, reprinted from [66], with permission from Elsevier

Sensitivity of convection resistance: another key parameter is the heat convection resistance ${R}_{u}$ that reflects heat convection between ambient condition and battery shell. This parameter could also reflect battery thermal management operation such as the air fan or liquid cooling. For this test, ${I}_{\text{CC1}}$ is also set within 5A and 10A. ${V}_{\text{cut}}$ is fixed as 3.6 V while ${T}_{\text{amb}}$ is set as $25\;^{\circ}{\text{C}}$. Five cases of ${R}_{u}$ (1, 3.08, 5, 10, and 15 ${\text{KW}}^{-1}$) are utilized to explore ${R}_{u}$’s sensitivity.

After the optimization, the Pareto frontiers for optimized MCC pattern under different ${R}_{u}$ are illustrated in Fig. 5.29. It can be seen that as ${R}_{u}$ increases, the Pareto frontier would move to the upper. On the contrary, battery charging process would speed up when increasing ${R}_{u}$. To summarize, a lower ${R}_{u}$ favours battery average temperature as well as the charging cost, at a sacrifice of charging time.

Fig. 5.29

Effects of convection resistances on the optimized MCC profile via Pareto frontier, reprinted from [66], with permission from Elsevier

Sensitivity of ambient temperature: for the test of ambient temperature ${T}_{\text{amb}}$, ${I}_{\text{CC1}}$ is initially set between 5 and 10A. ${V}_{\text{cut}}$ and ${R}_{u}$ are set as 3.6 V and 3.08 ${KW}^{-1}$, respectively. Six ${T}_{\text{amb}}$ cases ($15\;^{\circ}{\text{C}}$, $20\;^{\circ}{\text{C}}$, $25\;^{\circ}{\text{C}}$, $30\;^{\circ}{\text{C}}$, $35\;^{\circ}{\text{C}},$ and $40\;^{\circ}{\text{C}}$) are utilized to explore ${T}_{amb}$’s sensitivity.

Figure 5.30 illustrates the influences of ${T}_{amb}$ on the optimized MCC pattern. Obviously, larger ${T}_{amb}$ will lead the obtained Pareto frontier move to the left-upper, indicating that charging time is reduced but both the battery average temperature and charging cost would become larger adversely. Therefore, ${T}_{\text{amb}}$ with small value favours achieving low charging cost as well as average temperature rise, but it would also lead battery internal resistance become larger, further inducing a reduced charging speed of battery.

Fig. 5.30

Effects of the ambient temperature on the optimized MCC profile via Pareto frontier, reprinted from [66], with permission from Elsevier

In summary, the optimized MCC charging pattern is critically associated with these three parameters. Large ${V}_{\text{cut}}$ benefits battery charging time. Low ${R}_{u}$ and small ${T}_{\text{amb}}$ favour the charging cost and average temperature of battery. It is vital to carefully select suitable parameters, based on specific requirements from various battery applications.

Tests for Various Charging Cases

Next, the optimization results using different two charging objective combinations are first analysed, followed by several balanced MCC charging patterns to explore their efficacies on the trade-offs of various charging objectives. In this test, ${I}_{\text{CC1}}$ is initially fixed within 5 and 10 A. The current decrement of CC phases is set between 1.5 and 2.5 A. ${V}_{\text{cut}}$ is set as 3.6 V, while ${R}_{u}$ and ${T}_{\text{amb}}$ are set as $3.08 {\text{KW}}^{-1}$ and $25 ^\circ \text{C}$, respectively. After using NSGA-II to optimize MCC pattern, the corresponding Pareto frontiers of various two charging objective combinations are shown in Fig. 5.31.

Fig. 5.31

Optimization results of the MCC profile with two objectives via Pareto frontier, reprinted from [66], with permission from Elsevier

From Fig. 5.31a, b, the optimal particles in the Pareto frontiers could be distributed uniformly in a wide region, indicating that both charging cost and battery average temperature conflict with charging speed. Not surprisingly, a quicker charging speed can be obtained through adopting a larger current rate in the MCC pattern. However, this increased charging current would also result in the larger thermal reactions happening within a battery, hence increasing the battery average temperature. On the other hand, both the average SoC and current during the whole charging process would be higher under a larger current rate case. This would aggravate the electrical energy loss and battery fading during charging. In this context, related economic charging costs will be intensified. From Fig. 5.31c, it can be seen that the optimal particles finally converge into a single point, indicating that there is no conflict between charging cost and battery average temperature.

Balanced charging: Fig. 5.32 illustrates the optimization results of MCC patterns with three balanced objectives. It can be seen that the objective functions ${\text{JMCC}}_{\text{BAT}}$ and ${\text{JMCC}}_{\text{BCC}}$ conflict with ${\text{JMCC}}_{\text{BCT}}$. Through sacrificing charging time, both battery temperature rise and economic loss could be substantially restrained. According to five different cases of balanced charging, Case 1 and Case 2 represent the quick charging solutions that spend less time to reach SoC target. Case 4 and Case 5 prefer lower charging cost and average temperature rise. Case 3 denotes a neutral position. Then Fig. 5.33 illustrates electrical dynamics while Fig. 5.34 shows the thermal dynamics of these balanced charging cases, respectively.

Fig. 5.32

Optimization results of the MCC profile with three objectives via Pareto frontier, reprinted from [66], with permission from Elsevier

Fig. 5.33

Electrical dynamics of the MCC profile for the balanced charging: a current, b voltage, and c SoC, reprinted from [66], with permission from Elsevier

Fig. 5.34

Thermal dynamics of the MCC profile for the balanced charging: a average temperature, (b surface temperature, and c internal temperature, reprinted from [66], with permission from Elsevier

According to Figs. 5.33 and 5.34, as the required charging time becomes larger from Case 1 to Case 5, the initial current in the first CC phase inevitably decreases. The total CC phase number within MCC pattern is also reduced gradually. Quantitatively, compared with Case 1 owns the quick charging solution that takes 15.72 min, Case 5 spends 22.10 min charging time (41% increase). However, battery average temperature and total charging cost in such a case, respectively, decrease to 26.41 °C (4.5% decrease) and 0.0033 $Yuan$ (19.5% decrease). Through shortening charging time, both the average SoC and temperature of battery will increase dramatically to accelerate electricity loss and battery degradation, further leading to the increased economic cost during charging. Moreover, there is a range that increasing charging time linearly would decrease charging cost and average battery temperature significantly. Outside this range, further linearly speeding up the charging process would cause less effect to reduce charging cost and battery temperature. From this test, MCC patterns before Case 3 could be adopted to effectively decrease both economic cost and average temperature rise.

5.3.3 Case 2: Li-Ion Battery Pack Charging with Distributed Average Tracking

In this study, a data science-based charging solution for Li-ion battery pack is explored with low computational burden [69]. To be specific, based upon the typical Rint models for battery cells, an optimized average SoC trajectory could be first generated by constructing and handling a multi-objective optimization considering both user demand and the energy loss of battery pack. Then a distributed charging solution is derived to make the SoCs from cells could track this trajectory, where the model bias observers of each cell are designed for online compensation.

5.3.3.1 User-Involved Data Science Charging Framework for Battery Pack

Figure 5.35 illustrates the explored multi-module charger for Li-ion battery pack with $n$ cells connected in serial. For this multi-module charge, each cell could be charged by an independent module, bringing the benefits to avoid the overcharging issue. Due to the superiorities in terms of easy to be implemented, reasonable costs and size, this type of charger has been successfully adopted in many real applications.

Fig. 5.35

Multi-module charger for battery packs, reprinted from [69], with permission from IEEE

According to the concept of leader–followers in the multi-agent system [70], a distributed average tracking framework can be designed to charge each cell of battery pack, as illustrated in Fig. 5.36. Specifically, an average charging trajectory is first derived as the leader role. Then all cells as the follower role could track this trajectory simultaneously. To achieve this, a typical cell Rint model with an average initial SoC is adopted to produce an optimized average charging trajectory firstly as:

Fig. 5.36

Framework of derived user-involved data science charging solution, reprinted from [69], with permission from IEEE

$$ \left\{ {\begin{array}{*{20}l} {x_{0} \left( {k + 1} \right) = x_{0} \left( k \right) + b_{0} u_{0} \left( k \right)} \\ {y_{0} \left( k \right) = f_{0} \left( {x_{0} \left( k \right)} \right) + h_{0} \left( {x_{0} \left( k \right)} \right)u_{0} \left( k \right)} \\ \end{array} } \right. $$

(5.33)

where ${x}_{0}$, ${y}_{0},$ and ${u}_{0}$ represent Rint model’s state, output, and input, respectively. ${b}_{0}$, $f_0 ( \cdot )$, and $h_0 ( \cdot )$ are related nominal values. Here $b_i$ is a Coulomb efficiency parameter, $f_i ( \cdot )$ and $h_i ( \cdot )$ ($1\le i\le n$) stand for the nonlinear relations between SoC and OCV as well as internal resistance, respectively. The initial value ${x}_{0}\left(0\right)$ is the average value of all cells’ initial SoC. Then the optimal average SoC trajectory could be obtained by handling an optimization issue as:

$$ \underbrace {\min }_{u_0 \left( 0 \right), \ldots ,u_0 \left( {N - 1} \right)}\gamma_1 \left( {x_0 (N) - x_s } \right)^2 + \gamma_2 \sum_{k = 0}^{N - 1} {h_0 } \left( {x_0 (k)} \right)u_0^2 (k) $$

(5.34)

$$ \begin{aligned} & \begin{array}{*{20}l} {{\text{s.t.}}} & {x_0 \left( {k + 1} \right) = } \\ \end{array} x_0 (k) + b_0 u_0 (k) \\ & f_0 \left( {x_0 \left( {k + 1} \right)} \right) + h_0 \left( {x_0 \left( {k + 1} \right)} \right)u_0 (k) \le y_M \\ & \begin{array}{*{20}l} {0 \le u_0 (k) \le u_M ,} & {x_0 \left( {k + 1} \right) \le x_M } \\ \end{array} \\ \end{aligned} $$

(5.35)

Supposing $U={\left[{u}_{0}\left(0\right),\dots ,{u}_{0}\left(N-1\right)\right]}^{T}$, ${H}_{k}=\left[{1}_{k}^{T}, {0}_{N-k}^{T}\right]$, Eq. (5.34) can be expressed as ${x}_{0}\left(k\right)={x}_{0}\left(0\right)+{b}_{0}{H}_{k}U$ and the related optimization issue could be further rewritten as:

$$ \left\{ {\begin{array}{*{20}l} {\underbrace {\min }_U\;J_1 \left( U \right)} \\ {\begin{array}{*{20}l} {{\text{s.t.}}} & {C\left( U \right) \le 0} \\ \end{array} } \\ \end{array} } \right. $$

(5.36)

with

$$ \begin{aligned} J_1 \left( U \right) & = U^T \left( {\gamma_1 b_0^2 H_N^T H_N + \gamma_2 G\left( U \right)} \right)U \\ & + 2\gamma_1 b_0 \left( {x_0 \left( 0 \right) - x_s } \right)H_N U + \gamma_1 \left( {x_0 \left( 0 \right) - x_s } \right)^2 \\ \end{aligned} $$

(5.37)

$$C\left(U\right)=\left[\begin{array}{c}F\left(U\right)+{G}_{1}\left(U\right)U-{Y}_{M}\\\Phi U-{U}_{M}\\ MU-{X}_{C}\end{array}\right]$$

(5.38)

where $U$ represents the optimization variable. Other variables are expressed as:

$$ \left\{ {\begin{array}{*{20}c} {G\left( U \right) = {\text{diag}}\left\{ {h_{0} (x_{0} \left( 0 \right), \ldots ,h_{0} \left( {x_{0} \left( 0 \right) + b_{0} H_{{N - 1}} U} \right)} \right\}} \\ {F\left( U \right) = \left[ {f_{0} \left( {x_{0} \left( 0 \right) + b_{0} H_{1} U} \right), \ldots ,f_{0} \left( {x_{0} \left( 0 \right) + b_{0} H_{1} U} \right)} \right]^{T} } \\ {G_{1} \left( U \right) = {\text{diag}}\left\{ {h_{0} \left( {x_{0} \left( 0 \right) + b_{0} H_{1} U} \right), \ldots ,h_{0} \left( {x_{0} \left( 0 \right) + b_{0} H_{N} U} \right)} \right\}} \\ {Y_{M} = y_{M} 1_{N} } \\ {{{\Phi }} = \left[ {\begin{array}{*{20}c} {I_{{N,~}} } & { - I_{{N~}} } \\ \end{array} } \right]^{T} } \\ {U_{M} = \left[ {\begin{array}{*{20}c} {u_{M} 1_{N}^{T} ,} & {0_{N}^{T} } \\ \end{array} } \right]^{T} } \\ \end{array} } \right. $$

(5.39)

where ${I}_{N}$ is an identity $N\times N$ matrix. Then the optimal average SoC trajectory ${x}_{0}^{r}(k)$ ($1\le k\le N$) could be scheduled as:

$${x}_{0}^{r}\left(k\right)={x}_{0}\left(0\right)+{b}_{0}{H}_{k}{U}^{r}$$

(5.40)

5.3.3.2 Distributed Battery Charging Based on SoC Tracking

For the $i$th ($1\le i\le n$) cell, a SoC-tracking-based solution is derived to charge battery, as shown in Fig. 5.37, where a model bias observer is designed as:

Fig. 5.37

Diagram of SoC-tracking-based charging solution for $i$th battery cell, reprinted from [69], with permission from IEEE

$$ \left\{ {\begin{array}{*{20}l} {\widehat{{w_{i} }}\left( {k + 1} \right) = \widehat{{w_{i} }}\left( k \right) + l_{i} \left( {y_{i} \left( k \right) - \widehat{{y_{i} }}\left( k \right)} \right)} \\ {\widehat{{y_{i} }}\left( k \right) = f_{0} \left( {x_{i} \left( k \right)} \right) + h_{0} \left( {x_{i} \left( k \right)} \right)u_{i} \left( k \right) + \widehat{{w_{i} }}\left( k \right)} \\ \end{array} } \right. $$

(5.41)

where $\widehat{{w}_{i}}\left(k\right)$ and ${l}_{i}$ represent estimated result and the observer gain, respectively.

To ensure the SoC ${x}_{i}\left(k\right)$ of $i$th cell could track the obtained trajectory ${x}_{r}^{0}(k)$ while also meet the constraint requirements, the SoC-tracking-based solution can be formulated as:

$$ \underbrace {\min }_{u_i (k)}\frac{\gamma_3 }{2}\left( {x_i (k) - x_0^r (k)} \right)^2 + \frac{\gamma_4 }{2}\gamma_2 u_i^2 (k) $$

(5.42)

$$ \begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{s.t.}}} & {x_i \left( {k + 1} \right) = } \\ \end{array} x_i (k) + b_i u_i (k)} \\ {f_0 \left( {x_i \left( {k + 1} \right)} \right) + h_0 \left( {x_i \left( {k + 1} \right)} \right)u_i (k) + \widehat{w_i }\left( {k + 1} \right) \le y_M } \\ {\begin{array}{*{20}c} {0 \le u_i (k) \le u_M ,} & {x_i \left( {k + 1} \right) \le x_M } \\ \end{array} } \\ \end{array} $$

(5.43)

where ${\gamma }_{3}$ and ${\gamma }_{4}$ denote weight parameters that should be over zero. Then battery charging current ${u}_{i}\left(k\right)$ can be derived through handling the optimization issue with constraints at each $k$ to make battery cells could track their optimized trajectories.

5.3.3.3 Results and Discussions

In this study, a battery pack with 10 cells connected in serial is utilized to explore the performance of designed data science-based charging solution. Table 5.9 gives the detailed information of cell’s capacity and initial SoC. Nonlinear relations of SoC and OCV $f_0 ( \cdot )$ as well as SoC and internal resistance $h_0 ( \cdot )$ are described by:

Table 5.9 Capacities and initial SoC of different cells within a pack, reprinted from [69], with permission from IEEE

$$ f_0 ( \cdot ) = \left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {a_1 {\text{SoC}} + b_1 ,} & {{\text{if}}\;0 \le {\text{SoC}} < 0.05} \\ \end{array} } \\ {\begin{array}{*{20}l} {a_2 {\text{SoC}} + b_2 ,} & {{\text{if}}\;0.05 \le {\text{SoC}} < 0.20} \\ \end{array} } \\ {\begin{array}{*{20}l} {a_3 {\text{SoC}} + b_3 ,} & {{\text{if}}\;0.20 \le {\text{SoC}} < 1.00} \\ \end{array} } \\ \end{array} } \right. $$

(5.44)

$$ h_0 ( \cdot ) = \left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {a_4 SoC + b_4 ,} & {if\;0 \le SoC < 0.05} \\ \end{array} } \\ {\begin{array}{*{20}l} {b_5 ,} & {if\;0.05 \le SoC < 0.95} \\ \end{array} } \\ {\begin{array}{*{20}l} {a_6 SoC + b_6 ,} & {if\;0.95 \le SoC < 1.00} \\ \end{array} } \\ \end{array} } \right. $$

(5.45)

where ${a}_{1}=3.61$, ${b}_{1}=3.13$, ${a}_{2}=1.21$, ${b}_{2}=3.2$, ${a}_{3}=0.8$, ${b}_{3}=3.282$, ${a}_{4}=-0.46$, ${b}_{4}=0.057$, ${b}_{5}=0.034$, ${a}_{6}=2.06$, ${b}_{6}=-1.923$. The maximum current and terminal voltage of each cell are 3C and 4.2 V. Here the weight coefficients are set as: ${\gamma }_{1}={10}^{4}$, ${\gamma }_{2}=0.1$, ${\gamma }_{3}={10}^{4},$ and ${\gamma }_{4}={10}^{-3}$. Sample periods ${T}_{1}$ and ${T}_{2}$ are set as 300 s and 1 s, respectively.

Charging results: in this study, the target SoC ${x}_{s}$ and charging time ${T}_{s}$ are set as 100% and 120 min. Figure 5.38a–c illustrates the charging results of SoC, current and terminal voltage of each cell through using the designed data science strategy. It can be seen their constraints are all well guaranteed. According to the related battery pack SoC, cell SoC difference, and energy loss as shown in Fig. 5.38d–f, the SoC of battery pack could successfully track the scheduled average trajectory to reach the desired SoC of 98.48%. In addition, the SoC difference of cells would converge from 9.08 to 0.64%, further validating the effectiveness of the designed charging strategy. The cells’ SoC difference would become larger when the cell is nearly fully charged. This is not surprising as a constant-voltage stage is obtained at this region to meet voltage constraint, which would further lead to the inconsistent current to charge each cell.

Fig. 5.38

Charging results for battery pack: a SoCs of battery cells, b terminal voltages of battery cells, c Charging currents of battery cells, d SoC of battery pack, (e) SoC difference of cells, f energy loss with ${x}_{s}=100\%$ and ${T}_{s}=120$ min, reprinted from [69], with permission from IEEE

Tests of various user settings: to further investigate the performance of the proposed charging strategy under various user settings, two tests in terms of various charging durations and desired SoCs are carried out. For test of various charging durations, ${x}_{s}$ is set as 100%, charging duration ${T}_{s}$ is set as 120 min, 90 min, 60 min, and 30 min, respectively. For test of desired SoC, ${T}_{s}$ is set as 120 min, while ${x}_{s}$ is set as 100%, 90%, and 80%, respectively. Based upon these settings, the corresponding SoC responses of battery pack for these two tests are shown in Fig. 5.39. Table 5.10 also illustrates the results of battery pack SoC, energy loss as well as SoC difference at the end of battery charging. Quantitatively, the SoC of battery pack can be rapidly charged to 77.76% with a relatively larger energy loss of 134.8 kJ through using a tight charging duration of 30 min. On the contrary, through using a long charging duration of 120 min, battery pack could be charged to 79.93% with a relatively lower energy loss of 36.5 kJ. This indicates another benefit of using the derived charging strategy, as the charging current could be tuned based on different user requirements to reduce the energy loss of pursuing rapid charging blindly.

Fig. 5.39

SoC responses of battery pack: a various charging durations, b various desired battery SoCs, reprinted from [69], with permission from IEEE

Table 5.10 Results of the end of charging process under various user settings, reprinted from [69], with permission from IEEE

Tests of various weight selections: the coefficients ${\gamma }_{1}$ and ${\gamma }_{2}$ reflect the weights of user demand as well as energy loss. To explore the effects of different weights on battery charging performance, ${\gamma }_{1}$ is fixed in this study while ${\gamma }_{2}$ is set as 0.01, 0.1, 10, 100, and 1000, respectively. Figure 5.40 shows the related results of battery pack SoC and energy loss. From Fig. 5.40, obviously increasing ${\gamma }_{2}$ could result in less energy loss, but would adversely affect the performance of charging battery pack to a predefined value. It can be seen that a proper trade-off could be achieved by setting ${\gamma }_{2}$ as 0.1. In the light of this, ${\gamma }_{2}=0.1$ is selected for designing charging solution in this study.

Fig. 5.40

Battery pack charging results: a battery pack’s SoC, b energy loss under various weight coefficients, reprinted from [69], with permission from IEEE

5.4 Summary

This chapter describes another three key aspects of data science-based battery operation management including battery ageing prognostics, fault diagnosis, and charging. For battery ageing prognostics, Li-ion battery ageing mechanism and stress factors are first introduced, followed by the description of the data science framework and classical methods to achieve battery ageing/lifetime prediction. Then two data science-based case studies through deriving modified GPR and a hybrid data science model to predict future cyclic capacity degradation and battery RUL are given and discussed. For battery fault diagnosis, after overviewing three typical types of data science-based methods, a data science-based case study of deriving a battery ISC fault detection strategy through using SoC correlation is described. For battery charging, several key and conflicting objectives during battery charging are first introduced, then two data science-based case studies through designing a multi-objective optimization-based battery cell economic-conscious charging and a distributed average tracking-based battery pack charging are introduced, respectively. All these case studies indicate that through designing suitable data science-based strategies, satisfactory results of battery ageing prognostics, fault diagnosis, and charging can be achieved for effective battery operation management.