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Mathematical homogenization and stochastic modeling of energy storage systems
Journal Pre-proof Mathematical homogenization and stochastic modeling of energy storage systems Chigoziem A. Emereuwa PII:
S2451-9103(20)30016-8
DOI:
https://doi.org/10.1016/j.coelec.2020.01.009
Reference:
COELEC 503
To appear in:
Current Opinion in Electrochemistry
Received Date: 30 December 2019 Revised Date:
17 January 2020
Accepted Date: 20 January 2020
Please cite this article as: Emereuwa CA, Mathematical homogenization and stochastic modeling of energy storage systems, Current Opinion in Electrochemistry, https://doi.org/10.1016/ j.coelec.2020.01.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier B.V. All rights reserved.
Mathematical homogenization and stochastic modeling of energy storage systems Chigoziem A. Emereuwa Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa. Email: [email protected]
Abstract Mathematical homogenization theory as a multiscale modeling strategy for deriving macroscopic models is gaining relevance in modeling electrochemical energy storage systems (ESS) for its ability to capture the detailed microstructural properties of a material. Stochastic modeling on the other hand, captures molecular fluctuations and uncertainties associated with ESS. In this short review, modeling ESS using both tools are presented. Integrating the two provides an effective tool for deriving macroscopic models that accurately predict various macroscopic behaviour and electrochemical properties of ESS, to enable optimization and manufacturing of high performance ESS.
Introduction Mathematical modelling of electrochemical energy storage and conversion systems (i.e., batteries, electrochemical capacitors, and fuel cells) has continued to revolutionize the research, development and innovation in this field. Specifically, mathematical modelling is critical for predicting the different electrochemical processes that occur in energy storage systems (ESS) [1], [2], [3] such as ionic conductivity or diffusivity, thermal conductivity, cycling efficiency, operation lifetime, mechanical properties and safety issues, to mention a few. Various ESS models have been developed, including electrochemical models [4], analytical models [5], electric circuit models [6], [7] and stochastic models [8], [9]. Generally, ESS consist of different length scales, and modeling of processes at the pore scale in lithium-ion batteries (LIBs), for example, could be cumbersome, resulting in the need to derive macro-scale models that also account for microscopic processes, [10], [11]. Figure 1 clearly shows a graphical description of the different length scales within ESS and the benefits of various proposed models. Recently, with the growing interest in developing efficient and accurate macroscropic models to aid experimental scientists in designing ESS with effective electrochemical and storage properties, tools such as mathematical homogenization theory and stochastic modeling are becoming popular. This is because of their unique ability to capture; (1) the effects of microscopic mechanisms which enables predictability of the systems’ chemical and structural properties [10] and (2) the influence of molecular fluctuations on the systems [12]. The roles of these tools in modelling the behaviour of ESS is the focus of this short review.
Figure 1: Schematic representation of the study of mechanical properties of energy storage systems (notably, lithium-ion batteries, LIBs) indicating that it involves multiple scales and disciplines, and the various models that have been proposed to characterize their physico-chemical and mechanical properties at each length scale. Figure adapted from [13] with permission.
Multi-scale modelling of energy storage systems There are various strategies for obtaining macroscopic models of multi-scaled ESS materials, they either follow the top-down approach or the bottom-up approach (Figure 2(A)). Some of these strategies include mathematical homogenization [14] and the Volume Averaging Method (VAM) [15], [16]. Homogenization theory follows a bottom up approach as it systematically captures and scales up detailed descriptions of microstructural processes which is then used as parameters in macroscopic models [17]. The VAM involves describing the macroscopic equations in terms of averaged variables derived by taking the spatial averages of the microscopic equations, over a representative elementary volume. The macroscopic electrochemical battery model developed by Doyle, Fuller and Newman [4], [18] was derived using VAM. It is called the pseudo two-dimensional (P2D) model and is commonly used to predict battery dynamics. Despite the success of the VAM in predicting battery behaviour, there are some limitations. For instance, it’s an oversimplification to assume a homogeneous medium within the electrode [10], the local microstructure may influence the electrochemical reaction at the electrode-electrolyte interface [19] and geometric features such as pore shape and properties that go beyond porosity and tortuosity characterization are neglected even though transport characteristics such as surface area and tortuosity are accounted for [20]. The microstructural properties not captured, makes VAM an insufficient strategy for use in applications involving active material utilization and phase separating electrodes [21]. Mathematical homogenization theory, on the other hand, aims at deriving macroscopic models from microscopic processes. It considers at least one smaller length scale to account for geometric features such as electric permittivity/conductivity or pore shape. The multi-scaled material is assumed to be homogeneous and the homogenized/macroscopic model is obtained by an asymptotic analysis of the microscopic models as the length scale of the microstructure approaches zero as illustrated in Figure
2(B). The homogenized model and parameters (e.g. effective conductivity or ironic diffusivity [22]) obtained account for the microscopic processes but represent the overall behaviour of the material. This strategy provides a tool for improving LIB particle design [17] and is described as the most computationally effective [13] .
Figure 2: (A): Schematic representation of the different scales within electrochemical energy storages (batteries and supercapacitors) resolving the processes that take place at different length and time-scales. Figure adapted from [17] with permission. (B): Left: A multiscale material with microstructure length r. Middle: Passing to the limit (r → 0), Right: The macroscopic model is obtained by mathematical homogenization. Figure adapted from [23] with permission.
Arunachalam and Onori [21] used a homogenized macroscopic model to overcome the limitations of the Newman model in predicting voltage at high temperatures of cell operation. Their results showed that at low state-of-charge (SOC), the predictability of the Newman model deteriorates when trying to predict the voltage response for high temperature. They formulated and resolved sensitivity functions for the partial differential equations (PDEs) of the homogenized model and results indicate that parameter identifiability depends on the battery SOC. Hennessy and Moyles [24] used homogenization theory to derive a thermal model of a battery consisting of several connected lithiumion cells. There are various mathematical homogenization techniques such as the two-scale convergence method [25], Tartar’s energy method and the asymptotic expansions method (AEM) also known as multiple scale expansions to mention a few. However, the AEM is commonly used for macroscopic modeling of various behaviours of electrochemical systems [24] such as batteries. As an illustration of AEM, consider the following diffusion equation: , = ∆, where represents the quantity under investigation, e.g. concentration of a chemical or temperature, is the diffusion coefficient, > 0 represents time and is the macroscopic coordinate. The first step is to write the micromodel according to the geometry of the microstructure with suitable initial and interface conditions,
, = ∆ , #1 is the ratio of the characteristic length of the microstructural and macroscopic features. The idea behind AEM is to assume that the solution assumes an asymptotic expansion of the form , = , , + , , + , , + ⋯
with = representing the coordinate at the microstructure. This expansion is substituted into the
micromodel 1 and equal powers of are equated and solved consequently. After rigorous analysis, the homogenized equation , = ∇∇ut, x is obtained, where is expressed in terms of the microstructure. Lai and Ciucci [26] showed a detailed homogenization of battery models using AEM.
Homogenization of battery models The AEM has been used in various literature for macroscopic modeling of batteries. Periodicity of the microstructure is often assumed in this method. The method provides a systematic way of deriving macroscopic models that account for mechanisms that occur at the quasiperiodic microstructure [27]. In LIBs, the electrolyte can seep through the anode and cathode which are porous matrices made of active materials and binder-a polymer with conductive carbon black- compressed together. Lai and Ciucci [26] used AEM to derive micro/macro battery models similar to the one developed by Doyle, Fuller and Newman [4] for lithium batteries. The strength of this approach is that the geometric description of the microstructure can be used as input into a set of PDE, solved to estimate parameters such as porosity and tortuosity. In principle, knowing the detailed nature of the microstructure can help determine all fitting parameters pertaining to the geometry like porosity, tortuosity and effective volumetric area available for reaction. The authors recovered the Newman’s model by assuming that the microstructure has a spherical shape. Using AEM, Schmuck [20] derived a homogenized composite cathode equation from microscopic composite cathode equations considering the ionic transport in the polymer electrolyte, Li-intercalation undergoing a possible phase transformation in the solid phase, electron transport, and Butler-Volmer reactions across the solid electrolyte interface. A microscopic model accounting for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler-Volmer reaction kinetics was proposed by Richardson et al. [28]. Using AEM, the coefficients in the electrode scale model was derived in terms of the microstructural features (such as particle shape and size) of the electrode. These can be used to investigate the effects of changes in particle design.
The theory of homogenization can be combined with other approaches to enhance the efficacy of the results. Arunachalam et al., [29] presented macroscopic mass and transport equations derived using AEM for LIB [30]. A closure approach was used to calculate the effective diffusion and conductivity coefficients of the homogenized model, i.e. formulating and solving a closure variable in the electrode microstructure. According to the authors, the closure-homogenization approach will enable a more accurate modeling capability of lithium-ion transport in non-spherical active particle electrodes. Sagiyama and Garikipati [31] presented an approach to numerical homogenization. According to the authors, from the standpoint of machine learning methods, direct numerical simulation can represent the correct homogenized response that arises from complex microstructures. A quantitative
micromechanical model for overall electrical conductivity of cathode material in LIBs was developed by Vilchevskayav and Sevostianov [32]. In the model, they accounted for the effect of the pore shape using Maxwell homogenization technique. The results were compared with experimental data and according to the authors, the pore shape produces very strong effects on the overall conductivity which may explain the variation in experimental data. Gully et al. [22] used mathematical homogenization approach based on AEM, image processing, numerical computation and electron microscopy to determine the values of different effective transport coefficients such as diffusivity of a species and electric conductivity. They considered three constituent phases of an electrode material, which are electrolyte, active particles, and carbon-doped binder, Figure 3. Their work demonstrated that the effective transport coefficients of a given material with a complex microstructure can be determined based on the properties of the individual phases.
Figure 3: (A) A schematic of a cross section of a lithium-ion cell. (B): A schematic of the microstructure within the cathode. Three distinct material components are observed, the electrolyte ΩE, the active particle ΩAP and the polymer binder ΩB.
Figure adapted from [33] with permission.
Foster et al. [33] considered a complicated three-dimensional microstructure comprising of three different material phases, the active material, electrolyte and binder. They used homogenization theory, image analysis and image acquisition to calculate the effective electronic conductivity of commercial LIB cathode, before and after cycling. The homogenization theory used is also based on AEM. Three constituent phases of the electrode material and three length scales: the length scale of the electrolyte pore, the length scale of the active particle and the length scale of the entire cathode were considered in their approach. The homogenization performed revealed the effects of the length scales on the conductivity, and their model showed that the geometric changes that occur through cycling, affect the conduction of electronic current through the electrode. Hunt et al. [34] presented thermal electrochemical models for porous electrode batteries using asymptotic homogenization. They considered the thermal and diffusion effects in the electrode particles and anisotropy of the material based on the microstructure.
Stochastic modelling of energy storage systems At a microscopic scale, the properties of the ESS are also affected by the presence of molecular fluctuations [35]. In comparison to other macroscopic models, stochastic models, particularly Markov process can model batteries and battery-powered systems as a whole [36], [37]. Dolatabadi et al. [38] proposed a stochastic programming method for optimal sizing of a hybrid ship power system with energy storage system, photovoltaic power and diesel generator. Electrochemical reactions occurring within batteries during charge and discharge processes are influenced by random use profile and random ambient temperature [39], [40]. Stochastic models describe this discharge and recovery effect using stochastic processes [41]. Yu et al. [42] proposed a stochastic approach to consider uncertainty in energy profiles. In summary, the stochastic nature of batteries and battery powered systems can be
fully characterized using stochastic models that capture the fluctuations taking place at small scales. PDEs used for multiscale modeling [43] that capture these random nature and fluctuations are called stochastic PDEs (SPDEs). For uncertainty quantification and mechanical property analysis, Xu and Bae [44] proposed a stochastic reconstruction algorithm to generate random but statistically equivalent 3D microstructure models. The model gives some insights on the microstructure features and deformation patterns under external loads. The proposed model was applied to a commercial separator and the results validated by experimental data. Guhlke et al. [45] developed a new model for many particle electrodes. This model was then applied to LiFePO4(LFP) particles of nano-meter size. The phase transition from a lithium-poor to a lithium-rich phase led to a system of stochastic differential equations. According to the authors, the model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate-limiting capacity. Feinauer et al. [46] proposed a unified workflow which integrates generating a 3D microstructure of porous electrodes using stochastic models with a microscopic 3D spatial-resolved physical model for the electrochemical behaviour of the lithium-ion cells, taking lithium platting and stripping into account as a step towards computer aided engineering for the development of efficient lithium ion cells. Stochastic modeling and mathematical homogenization are both powerful tools for modeling ESS. Hence an integration of both tools is crucial for developing macroscopic models that effectively capture microstructural properties including molecular fluctuations [47], [48]. The assumption of a periodic microstructure in AEM is often insufficient, hence new approaches for the homogenization of SPDEs are being proposed as stochastic models not only capture molecular fluctuations [49] but also the random nature of the microstructure [50]. However, these theories need to be further explored and incorporated into modeling electrochemical properties of energy storage systems. To further enhance the ability of the models to predict the behaviour of ESS and their components, it is also important to explore and incorporate computational techniques involving machine learning (ML) [31], [51], [43] and artificial intelligence (AI). These techniques encode physics based knowledge into ML and AI using PDEs and the approaches can be extended to SPDEs [43]. Ryan and Murkherjee [52] provided a recent review on mesoscale modeling in electrochemical devices such as batteries with a focus on particle methods and fine-scale computational fluids dynamics based direct numerical simulation techniques.
Concluding remarks In this short review, stochastic modeling and macroscopic modeling using homogenization theory was discussed. Although Homogenization theory as a strategy for deriving macroscopic models is less popular in computational electrochemistry in comparison to VAM, it has recently attracted significant research interest and has been recognized as a powerful tool that provides a better computationally efficient way to capture microstructural electrochemical properties (such as thermal conductivity and ionic diffusivity) in macroscale models than VAM. Hence in future research, it is crucial that this theory is further explored and integrated with stochastic modelling and machine learning theories, as this can provide new insights into designing ESS with optimized microstructural components and properties. Figure 4 elegantly illustrates how the various benefits of machine learning and multiscale modeling can be incorporated to maximize the benefits of both tools. Stochastic modelling (uncertainty quantification) is emerging as a critical tool in machine learning and artificial intelligence. As we get into the 4th Industrial Revolution, it is no doubt that this field will revolutionize the way we predict, design, optimize and manufacture energy storage materials and systems.
Figure 4: A schematics of the mutual benefits of the integration of machine learning and multiscale modeling. Figure is adapted from [43].
References Papers of particular interest, published within the review period have been highlighted as follows: * Papers of special interest. ** Papers of outstanding interest.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S2451-9103(20)30016-8
DOI:
https://doi.org/10.1016/j.coelec.2020.01.009
Reference:
COELEC 503
To appear in:
Current Opinion in Electrochemistry
Received Date: 30 December 2019 Revised Date:
17 January 2020
Accepted Date: 20 January 2020
Please cite this article as: Emereuwa CA, Mathematical homogenization and stochastic modeling of energy storage systems, Current Opinion in Electrochemistry, https://doi.org/10.1016/ j.coelec.2020.01.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier B.V. All rights reserved.
Mathematical homogenization and stochastic modeling of energy storage systems Chigoziem A. Emereuwa Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa. Email: [email protected]
Abstract Mathematical homogenization theory as a multiscale modeling strategy for deriving macroscopic models is gaining relevance in modeling electrochemical energy storage systems (ESS) for its ability to capture the detailed microstructural properties of a material. Stochastic modeling on the other hand, captures molecular fluctuations and uncertainties associated with ESS. In this short review, modeling ESS using both tools are presented. Integrating the two provides an effective tool for deriving macroscopic models that accurately predict various macroscopic behaviour and electrochemical properties of ESS, to enable optimization and manufacturing of high performance ESS.
Introduction Mathematical modelling of electrochemical energy storage and conversion systems (i.e., batteries, electrochemical capacitors, and fuel cells) has continued to revolutionize the research, development and innovation in this field. Specifically, mathematical modelling is critical for predicting the different electrochemical processes that occur in energy storage systems (ESS) [1], [2], [3] such as ionic conductivity or diffusivity, thermal conductivity, cycling efficiency, operation lifetime, mechanical properties and safety issues, to mention a few. Various ESS models have been developed, including electrochemical models [4], analytical models [5], electric circuit models [6], [7] and stochastic models [8], [9]. Generally, ESS consist of different length scales, and modeling of processes at the pore scale in lithium-ion batteries (LIBs), for example, could be cumbersome, resulting in the need to derive macro-scale models that also account for microscopic processes, [10], [11]. Figure 1 clearly shows a graphical description of the different length scales within ESS and the benefits of various proposed models. Recently, with the growing interest in developing efficient and accurate macroscropic models to aid experimental scientists in designing ESS with effective electrochemical and storage properties, tools such as mathematical homogenization theory and stochastic modeling are becoming popular. This is because of their unique ability to capture; (1) the effects of microscopic mechanisms which enables predictability of the systems’ chemical and structural properties [10] and (2) the influence of molecular fluctuations on the systems [12]. The roles of these tools in modelling the behaviour of ESS is the focus of this short review.
Figure 1: Schematic representation of the study of mechanical properties of energy storage systems (notably, lithium-ion batteries, LIBs) indicating that it involves multiple scales and disciplines, and the various models that have been proposed to characterize their physico-chemical and mechanical properties at each length scale. Figure adapted from [13] with permission.
Multi-scale modelling of energy storage systems There are various strategies for obtaining macroscopic models of multi-scaled ESS materials, they either follow the top-down approach or the bottom-up approach (Figure 2(A)). Some of these strategies include mathematical homogenization [14] and the Volume Averaging Method (VAM) [15], [16]. Homogenization theory follows a bottom up approach as it systematically captures and scales up detailed descriptions of microstructural processes which is then used as parameters in macroscopic models [17]. The VAM involves describing the macroscopic equations in terms of averaged variables derived by taking the spatial averages of the microscopic equations, over a representative elementary volume. The macroscopic electrochemical battery model developed by Doyle, Fuller and Newman [4], [18] was derived using VAM. It is called the pseudo two-dimensional (P2D) model and is commonly used to predict battery dynamics. Despite the success of the VAM in predicting battery behaviour, there are some limitations. For instance, it’s an oversimplification to assume a homogeneous medium within the electrode [10], the local microstructure may influence the electrochemical reaction at the electrode-electrolyte interface [19] and geometric features such as pore shape and properties that go beyond porosity and tortuosity characterization are neglected even though transport characteristics such as surface area and tortuosity are accounted for [20]. The microstructural properties not captured, makes VAM an insufficient strategy for use in applications involving active material utilization and phase separating electrodes [21]. Mathematical homogenization theory, on the other hand, aims at deriving macroscopic models from microscopic processes. It considers at least one smaller length scale to account for geometric features such as electric permittivity/conductivity or pore shape. The multi-scaled material is assumed to be homogeneous and the homogenized/macroscopic model is obtained by an asymptotic analysis of the microscopic models as the length scale of the microstructure approaches zero as illustrated in Figure
2(B). The homogenized model and parameters (e.g. effective conductivity or ironic diffusivity [22]) obtained account for the microscopic processes but represent the overall behaviour of the material. This strategy provides a tool for improving LIB particle design [17] and is described as the most computationally effective [13] .
Figure 2: (A): Schematic representation of the different scales within electrochemical energy storages (batteries and supercapacitors) resolving the processes that take place at different length and time-scales. Figure adapted from [17] with permission. (B): Left: A multiscale material with microstructure length r. Middle: Passing to the limit (r → 0), Right: The macroscopic model is obtained by mathematical homogenization. Figure adapted from [23] with permission.
Arunachalam and Onori [21] used a homogenized macroscopic model to overcome the limitations of the Newman model in predicting voltage at high temperatures of cell operation. Their results showed that at low state-of-charge (SOC), the predictability of the Newman model deteriorates when trying to predict the voltage response for high temperature. They formulated and resolved sensitivity functions for the partial differential equations (PDEs) of the homogenized model and results indicate that parameter identifiability depends on the battery SOC. Hennessy and Moyles [24] used homogenization theory to derive a thermal model of a battery consisting of several connected lithiumion cells. There are various mathematical homogenization techniques such as the two-scale convergence method [25], Tartar’s energy method and the asymptotic expansions method (AEM) also known as multiple scale expansions to mention a few. However, the AEM is commonly used for macroscopic modeling of various behaviours of electrochemical systems [24] such as batteries. As an illustration of AEM, consider the following diffusion equation: , = ∆, where represents the quantity under investigation, e.g. concentration of a chemical or temperature, is the diffusion coefficient, > 0 represents time and is the macroscopic coordinate. The first step is to write the micromodel according to the geometry of the microstructure with suitable initial and interface conditions,
, = ∆ , #1 is the ratio of the characteristic length of the microstructural and macroscopic features. The idea behind AEM is to assume that the solution assumes an asymptotic expansion of the form , = , , + , , + , , + ⋯
with = representing the coordinate at the microstructure. This expansion is substituted into the
micromodel 1 and equal powers of are equated and solved consequently. After rigorous analysis, the homogenized equation , = ∇∇ut, x is obtained, where is expressed in terms of the microstructure. Lai and Ciucci [26] showed a detailed homogenization of battery models using AEM.
Homogenization of battery models The AEM has been used in various literature for macroscopic modeling of batteries. Periodicity of the microstructure is often assumed in this method. The method provides a systematic way of deriving macroscopic models that account for mechanisms that occur at the quasiperiodic microstructure [27]. In LIBs, the electrolyte can seep through the anode and cathode which are porous matrices made of active materials and binder-a polymer with conductive carbon black- compressed together. Lai and Ciucci [26] used AEM to derive micro/macro battery models similar to the one developed by Doyle, Fuller and Newman [4] for lithium batteries. The strength of this approach is that the geometric description of the microstructure can be used as input into a set of PDE, solved to estimate parameters such as porosity and tortuosity. In principle, knowing the detailed nature of the microstructure can help determine all fitting parameters pertaining to the geometry like porosity, tortuosity and effective volumetric area available for reaction. The authors recovered the Newman’s model by assuming that the microstructure has a spherical shape. Using AEM, Schmuck [20] derived a homogenized composite cathode equation from microscopic composite cathode equations considering the ionic transport in the polymer electrolyte, Li-intercalation undergoing a possible phase transformation in the solid phase, electron transport, and Butler-Volmer reactions across the solid electrolyte interface. A microscopic model accounting for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler-Volmer reaction kinetics was proposed by Richardson et al. [28]. Using AEM, the coefficients in the electrode scale model was derived in terms of the microstructural features (such as particle shape and size) of the electrode. These can be used to investigate the effects of changes in particle design.
The theory of homogenization can be combined with other approaches to enhance the efficacy of the results. Arunachalam et al., [29] presented macroscopic mass and transport equations derived using AEM for LIB [30]. A closure approach was used to calculate the effective diffusion and conductivity coefficients of the homogenized model, i.e. formulating and solving a closure variable in the electrode microstructure. According to the authors, the closure-homogenization approach will enable a more accurate modeling capability of lithium-ion transport in non-spherical active particle electrodes. Sagiyama and Garikipati [31] presented an approach to numerical homogenization. According to the authors, from the standpoint of machine learning methods, direct numerical simulation can represent the correct homogenized response that arises from complex microstructures. A quantitative
micromechanical model for overall electrical conductivity of cathode material in LIBs was developed by Vilchevskayav and Sevostianov [32]. In the model, they accounted for the effect of the pore shape using Maxwell homogenization technique. The results were compared with experimental data and according to the authors, the pore shape produces very strong effects on the overall conductivity which may explain the variation in experimental data. Gully et al. [22] used mathematical homogenization approach based on AEM, image processing, numerical computation and electron microscopy to determine the values of different effective transport coefficients such as diffusivity of a species and electric conductivity. They considered three constituent phases of an electrode material, which are electrolyte, active particles, and carbon-doped binder, Figure 3. Their work demonstrated that the effective transport coefficients of a given material with a complex microstructure can be determined based on the properties of the individual phases.
Figure 3: (A) A schematic of a cross section of a lithium-ion cell. (B): A schematic of the microstructure within the cathode. Three distinct material components are observed, the electrolyte ΩE, the active particle ΩAP and the polymer binder ΩB.
Figure adapted from [33] with permission.
Foster et al. [33] considered a complicated three-dimensional microstructure comprising of three different material phases, the active material, electrolyte and binder. They used homogenization theory, image analysis and image acquisition to calculate the effective electronic conductivity of commercial LIB cathode, before and after cycling. The homogenization theory used is also based on AEM. Three constituent phases of the electrode material and three length scales: the length scale of the electrolyte pore, the length scale of the active particle and the length scale of the entire cathode were considered in their approach. The homogenization performed revealed the effects of the length scales on the conductivity, and their model showed that the geometric changes that occur through cycling, affect the conduction of electronic current through the electrode. Hunt et al. [34] presented thermal electrochemical models for porous electrode batteries using asymptotic homogenization. They considered the thermal and diffusion effects in the electrode particles and anisotropy of the material based on the microstructure.
Stochastic modelling of energy storage systems At a microscopic scale, the properties of the ESS are also affected by the presence of molecular fluctuations [35]. In comparison to other macroscopic models, stochastic models, particularly Markov process can model batteries and battery-powered systems as a whole [36], [37]. Dolatabadi et al. [38] proposed a stochastic programming method for optimal sizing of a hybrid ship power system with energy storage system, photovoltaic power and diesel generator. Electrochemical reactions occurring within batteries during charge and discharge processes are influenced by random use profile and random ambient temperature [39], [40]. Stochastic models describe this discharge and recovery effect using stochastic processes [41]. Yu et al. [42] proposed a stochastic approach to consider uncertainty in energy profiles. In summary, the stochastic nature of batteries and battery powered systems can be
fully characterized using stochastic models that capture the fluctuations taking place at small scales. PDEs used for multiscale modeling [43] that capture these random nature and fluctuations are called stochastic PDEs (SPDEs). For uncertainty quantification and mechanical property analysis, Xu and Bae [44] proposed a stochastic reconstruction algorithm to generate random but statistically equivalent 3D microstructure models. The model gives some insights on the microstructure features and deformation patterns under external loads. The proposed model was applied to a commercial separator and the results validated by experimental data. Guhlke et al. [45] developed a new model for many particle electrodes. This model was then applied to LiFePO4(LFP) particles of nano-meter size. The phase transition from a lithium-poor to a lithium-rich phase led to a system of stochastic differential equations. According to the authors, the model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate-limiting capacity. Feinauer et al. [46] proposed a unified workflow which integrates generating a 3D microstructure of porous electrodes using stochastic models with a microscopic 3D spatial-resolved physical model for the electrochemical behaviour of the lithium-ion cells, taking lithium platting and stripping into account as a step towards computer aided engineering for the development of efficient lithium ion cells. Stochastic modeling and mathematical homogenization are both powerful tools for modeling ESS. Hence an integration of both tools is crucial for developing macroscopic models that effectively capture microstructural properties including molecular fluctuations [47], [48]. The assumption of a periodic microstructure in AEM is often insufficient, hence new approaches for the homogenization of SPDEs are being proposed as stochastic models not only capture molecular fluctuations [49] but also the random nature of the microstructure [50]. However, these theories need to be further explored and incorporated into modeling electrochemical properties of energy storage systems. To further enhance the ability of the models to predict the behaviour of ESS and their components, it is also important to explore and incorporate computational techniques involving machine learning (ML) [31], [51], [43] and artificial intelligence (AI). These techniques encode physics based knowledge into ML and AI using PDEs and the approaches can be extended to SPDEs [43]. Ryan and Murkherjee [52] provided a recent review on mesoscale modeling in electrochemical devices such as batteries with a focus on particle methods and fine-scale computational fluids dynamics based direct numerical simulation techniques.
Concluding remarks In this short review, stochastic modeling and macroscopic modeling using homogenization theory was discussed. Although Homogenization theory as a strategy for deriving macroscopic models is less popular in computational electrochemistry in comparison to VAM, it has recently attracted significant research interest and has been recognized as a powerful tool that provides a better computationally efficient way to capture microstructural electrochemical properties (such as thermal conductivity and ionic diffusivity) in macroscale models than VAM. Hence in future research, it is crucial that this theory is further explored and integrated with stochastic modelling and machine learning theories, as this can provide new insights into designing ESS with optimized microstructural components and properties. Figure 4 elegantly illustrates how the various benefits of machine learning and multiscale modeling can be incorporated to maximize the benefits of both tools. Stochastic modelling (uncertainty quantification) is emerging as a critical tool in machine learning and artificial intelligence. As we get into the 4th Industrial Revolution, it is no doubt that this field will revolutionize the way we predict, design, optimize and manufacture energy storage materials and systems.
Figure 4: A schematics of the mutual benefits of the integration of machine learning and multiscale modeling. Figure is adapted from [43].
References Papers of particular interest, published within the review period have been highlighted as follows: * Papers of special interest. ** Papers of outstanding interest.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: