A Suite of Reduced-Order Models of a Single-Layer Lithium-ion Pouch Cell
Scott G. Marquis, Robert Timms, Valentin Sulzer, Colin P. Please, S. Jon Chapman
AA Suite of Reduced-Order Models of a Single-LayerLithium-ion Pouch Cell
Scott G. Marquis ∗ Robert Timms ∗† Valentin Sulzer ‡ Colin P. Please ∗† S. Jon Chapman ∗† Abstract
For many practical applications, fully coupled three-dimensional models describing the be-haviour of lithium-ion pouch cells are too computationally expensive. However, owing to the smallaspect ratio of typical pouch cell designs, such models are well approximated by splitting theproblem into a model for through-cell behaviour and a model for the transverse behaviour. Inthis paper, we combine different simplifications to through-cell and transverse models to developa hierarchy of reduced-order pouch cell models. We give a critical numerical comparison of eachof these models in both isothermal and thermal settings, and also study their performance onrealistic drive cycle data. Finally, we make recommendations regarding model selection, takinginto account the available computational resource and the quantities of interest in a particularstudy.
Lithium-ion batteries are the most popular form of energy storage for many modern devices, withapplications ranging from portable electronics to electric vehicles [1, 2, 3]. Improving both the perfor-mance and lifetime of these batteries by design changes that increase capacity, reduce losses and delaydegradation effects is a key engineering challenge. Mathematical models allow possible improvements ∗ Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK ([email protected],[email protected], [email protected], [email protected]). † The Faraday Institution, Quad One, Becquerel Avenue, Harwell Campus, Didcot, OX11 0RA, UK. ‡ Department of Mechanical Engineering, Univ of Michigan, 2044 WE Lay Auto Lab, 1231 Beal Ave, Ann Arbor MI48109-2133 ([email protected]). a r X i v : . [ phy s i c s . c o m p - ph ] A ug o be explored in an efficient manner before expensive and time consuming physical experiments areperformed. However, a major hurdle is that existing battery models are computationally expensive,especially when seeking to resolve space- and time-dependent coupled electrical, thermal, and degra-dation effects within a battery. As a result, there is considerable interest in developing models thataccount for the key physical behaviours, but at a significantly reduced computational cost. In this pa-per, we derive a set of such simplified models, which are valid in various physically-relevant parameterregimes, by systematically reducing a detailed coupled electrochemical–thermal model of a lithium-ionpouch cell.In [4], we introduced a detailed fully-coupled 3D Doyle–Fuller–Newman (DFN) model of a lithium-ionpouch cell based upon the classical 1D DFN model developed by Newman and collaborators [5,6,7]. Anumber of other similar higher-dimensional DFN-based models exist in the literature [8,9,10,11,12,13]).This 3D DFN model allows for non-uniform behaviour in the current and temperature profiles in thetransverse dimensions of the pouch. Capturing these non-uniform effects is of great importance forunderstanding how the cell may degrade in an uneven manner [14, 15], which in turn adversely affectscell performance and lifetime [8]. However, the full 3D DFN model is very computationally expensive,motivating us to develop simpler models which still retain the essential physics. By exploiting the largegeometric aspect ratio of the cell, and the large ratio of the thermal voltage to the typical Ohmic dropin the current collectors, we derived the simplified “2+1D” DFN model. The 2+1D DFN comprises acollection of 1D DFN models describing the through-cell electrochemistry, coupled via a 2D electricalproblem in the current collectors and a 2D thermal problem. Similar 2+1D models, also referredto as “potential pair” models, have been exploited in an ad-hoc way in many battery simulations.Within the 2+1D structure, various models of have been employed to represent the 1D through-cellelectrochemistry, these include: the 1D DFN [16]; a nonlinear resistor fitted to an electrochemicalmodel [17]; and data [12]. In this paper, we begin from the 2+1D DFN model and, via asymptoticmethods, systematically derive a suite of computationally simpler models.The models that we compare in this paper are displayed pictorially in Figure 1. The 2+1D DFN,the most complex model we consider, is shown in the top-left panel. This model consists of a two-dimensional problem for the potentials in each current collector and a two-dimensional thermal prob-lem, coupled to a one-dimensional DFN model at each point representing the through-cell electrochem-istry. Proceeding downwards in Figure 1, represents making simplifications to the transverse model(i.e. the current collectors). The first simplification (the middle row) gives rise to a set of models we2abel ‘CC’. These models consist of a single ‘average’ through-cell electrochemical model and a seconddecoupled two-dimensional problem for the current collector resistances (hence the name ‘CC’) whichcan be solved ‘offline’ before solving the through-cell model. On the bottom row of Figure 1, theeffects of the current collectors are neglected entirely, with only a single representative through-cellelectrochemical model solved. We refer to these as 0D transverse models. The model equations asso-ciated with each of these three transverse models are presented in §
2. Moving to the right in Figure 1represents making simplifications to the through-cell electrochemical model. The three through cellmodels that we consider are the DFN model, the single particle model with electrolyte (SPMe), andthe single particle model (SPM). Formal derivations of the SPMe and SPM from the DFN can be foundin [18]. The DFN model comprises one-dimensional equations for the electrode potentials, the elec-trolyte potentials, and the electrolyte concentrations that are posed on the macroscale (the thicknessof the electrode). At each point in the model there is a radial problem posed on the microscale (theradius of an active particle) for the lithium concentrations in the particles. The DFN model is oftenreferred to as “pseudo two-dimensional” (P2D) since at each point in the one-dimensional macroscalethere is an additional one-dimensional microscale problem (i.e there is a particle at each point in theelectrode). The SPMe replaces the collection of problems for the particles with a single average particlein each electrode. The equations for the average particles are coupled to a one-dimensional equationfor the electrolyte concentration, and the electrode and electrolyte potentials are then recovered viaalgebraic expressions. Finally, the SPM discards the problem in the electrolyte, and consists of onlya single representative particle in each electrode. The model equations for each of these through-cellelectrochemical models are presented in §
3. By combining any transverse model with any through-cellelectrochemical model, we arrive at one of the nine models in Figure 1. For example, by choosing the‘CC’ transverse model and the SPMe for the through-cell model, we arrive at the SPMeCC model,which consists of an ‘average’ SPMe model and decoupled current collector problem. Alternatively, bychoosing a 0D transverse model and the SPM through-cell model, we arrive at the standard 1D SPMmodel. We shall compare each of the nine models in both the isothermal and thermal cases, and inparticular assess the predictions of key output variables from each model (e.g. the terminal voltage,average temperature, etc.). By doing this, we aim to aid the reader in choosing the appropriate modelfor a particular application with given requirements of computational speed and accuracy. Our resultsare summarized in § Before stating the governing equations we comment on our notation. We denote electric potentials by φ , current densities by i , lithium concentrations by c , molar fluxes by N , and temperatures by T . Todistinguish potential, fluxes and concentrations in the electrolyte from those in the solid phase of theelectrode, we use a subscript e for electrolyte variables and a subscript s for solid phase variables. Toindicate the region within which each variable is defined, we include an additional subscript k, whichtakes one of the following values: n (negative electrode), p (positive electrode), cn (negative currentcollector), cp (positive current collector), or s (separator). For example, the notation φ s,n refers to theelectric potential in the solid phase of the negative electrode. When stating the governing equations,we take the region in which an equation holds to be implicitly defined by the subscript of the variables.These regions are given byΩ cn = [ − L cn , × Ω , Ω n = [0 , L n ] × Ω , Ω s = [ L n , L x − L p ] × Ω , Ω p = [ L x − L p , L x ] × Ω , Ω cp = [ L x , L x + L cp ] × Ω , corresponding to the negative current collector, negative electrode, separator, positive electrode, andpositive current collector, respectively, where Ω = [0 , L y ] × [0 , L z ] is the projection of the cell ontothe ( y, z )-plane. Here L k (k ∈ { cn, n, s, p, cp } ) are the thicknesses of each component of the cell, L x = L n + L s + L p is the distance between the two current collectors, and L y and L z are the cell widthand height, respectively.We also introduce the notation ∂ Ω tab,k to refer to the negative and positive tabs (k ∈ { cn, cp } ), ∂ Ω ext,k to refer to the external boundaries of region k ∈ { cn, n, s, p, cp } , and ∂ Ω k , k to refer to theinterface between regions k and k . For instance, the notation ∂ Ω n,s refers to the interface betweenthe negative electrode and the separator. Finally, for k ∈ { cn, cp } we use ∂ Ω tab,k, ⊥ to denote theprojection of the tabs onto the ( y, z )-plane, and ∂ Ω ext,k, ⊥ = ∂ Ω \ ∂ Ω tab,k, ⊥ to denote the non-tabregion of the boundary of the projection.Derivatives in x and r are written out explicitly, and we define the gradient in the transverse direction In the electrolyte c denotes the lithium-ion concentrations. +1D DFN (P4D) 2+1D SPMe (P3D) 2+1D SPM (P3D)DFNCC (P2D) SPMeCC (P1D) SPMCC (P1D)DFN (P2D) SPMe (P1D) SPM (P1D) Figure 1: Schematic diagram of the models we consider, and their complexity. DFN corresponds toone-dimensional problem through the cell, at each point of which there is a radial problem for theconcentration of Li in the active material. Thus, such a model is pseudo-two-dimensional (P2D). In2+1 DFN, at every point of the current collectors a one-dimensional DFN model is solved, leading toa pseudo-four-dimensional (P4D) model. In the DFNCC, a single DFN model is solved, along with anuncoupled two-dimensional problem in the current collectors. SPM considers a single active particlein each electrode, leading to a one-dimensional model (P1D). In the SPMe, an extra one-dimensionalequation for the electrolyte is added; such a model is still P1D.5s ∇ ⊥ ≡ ∂∂y e + ∂∂z e . (1.1)We use an overbar to denote an average in the x -direction, so that in each electrode¯ f n := 1 L n (cid:90) L n f n d x , ¯ f p := 1 L p (cid:90) L x L x − L p f p d x . (1.2)We use angled brackets to denote an average in the ( y, z )-directions, (cid:104) f (cid:105) = 1 L y L z (cid:90) Ω f d y d z . (1.3)Quantities averaged over both the x -direction of an electrode and the y - z directions are then denotedby (cid:104) ¯ f k (cid:105) . Our starting point, the “2+1D DFN model”, is a set of through-cell one-dimensional models DFNcoupled to a two-dimensional problem for the boundary conditions. It was derived in [4] from a fullythree-dimensional DFN model in the limit of large current collector conductivity and small aspectratio. A further limit, in which the current collector conductivity is even larger, lead to a model inwhich only a single through-cell one-dimensional problem needs to be solved, with an additional two-dimensional problem needed to calculate an in-series resistance. We refer to this simpler model as the“DFNCC”, where the “CC” stands for current collectors.The reduced models in [4] were derived using the DFN model to describe the electrochemistry.However the simplifications to the transverse behaviour are independent of the choice of model for thethrough-cell behaviour. In the following we summarise the simplifications discussed in [4]. We thenshow that, in the limit in which the current collector conductance is extremely large, the effect of thecurrent collectors can be neglected entirely, and we recover the standard one-dimensional descriptionof cell behaviour at the macroscale.
In the limit in which the aspect ratio is small, the effective current collector conductance is large, andthe cell cooling is moderate, the full 3D pouch cell model reduces to a collection of through-cell models6oupled via two-dimensional problems in the transverse direction for the distribution of potential inthe current collectors and the balance of energy. In terms of dimensional parameters, this model issuitable in the regime L x L ⊥ (cid:28) , I app FL cn σ ck RT ∞ ∼ , h face L ⊥ λ eff L x ∼ , h edge L ⊥ λ eff L x ∼ , where L ⊥ is a typical transverse dimension (e.g. L y , L z or ( L y L z ) / ), σ ck is the current collectorconductivity (k ∈ { n,p } ), R is the molar gas constant, T ∞ the ambient temperature, I app is the appliedcurrent, and λ eff the effective thermal conductivity. The heat transfer coefficient h may in general bea function of space (e.g. for tab cooling [20]), and we allow for the situation where the coefficient onthe edges h edge (which acts on an area of typical size L x L ⊥ ) may be larger than that on the faces h face (which acts on an area of typical size L ⊥ ).In this limit the model is the two-dimensional pair-potential problem L cn σ cn ∇ ⊥ φ s,cn = I , L cp σ cp ∇ ⊥ φ s,cp = −I , in Ω (2.1a)with boundary conditions φ s,cn = 0 on ∂ Ω tab,cn, ⊥ , ∇ ⊥ φ s,cn · n = 0 on ∂ Ω ext,cn, ⊥ (2.1b) − σ cp ∇ ⊥ φ s,cp · n = I app A tab,cp on ∂ Ω tab,cp, ⊥ , ∇ ⊥ φ s,cp · n = 0 on ∂ Ω ext,cp, ⊥ , (2.1c)where I ( φ s,cn , φ s,cp , T ) is the through-cell current density, given by any one-dimensional electrochemicalmodel. This is coupled to the two-dimensional thermal problem ρ eff ∂T∂t = λ eff ∇ ⊥ T + ¯ Q − ( h cn + h cp ) L ( T − T ∞ ) , in Ω (2.1d) − λ eff ∇ ⊥ T · n = h eff ( T − T ∞ ) on ∂ Ω (2.1e)with initial condition T = T , where the x-averaged heat source is¯ Q = ¯ Q + L cn L σ cn |∇ ⊥ φ s,cn | + L cp L σ cp |∇ ⊥ φ s,cp | , (2.1f)7ith ¯ Q being the x-averaged heat source from the one-dimensional electrochemical model with thecurrent density given by I = I app / ( L y L z ). In the above I app is the applied current, A tab,cp is thepositive tab area, ρ eff is the effective volumetric heat capacity, L is the total cell thickness ( L x + L cn + L cp ), h eff the effective edge heat transfer coefficient, and h cn and h cp are the heat transfer coefficients onthe faces of the negative and positive current collectors, respectively. The effective edge heat transfercoefficient is h eff = 1( L x + L cn + L cp ) (cid:90) L x + L cp − L cn h edge d x , and the effective volumetric heat capacity and thermal conductivity are ρ eff = (cid:80) k ρ k c p,k L k (cid:80) k L k , λ eff = (cid:80) k λ k L k (cid:80) k L k , (2.1g)respectively. Here ρ k , c p,k and λ k are the the density, specific heat capacity and thermal conductivity,respectively, of each component. In the limit where the current collector conductance is even larger, the potential in the current collectorsis approximately uniform, and it suffices to solve a single volume-averaged through-cell model, ratherthan a collection of through-cell models. Further, if the edge heat transfer coefficient h eff is suitablysmall, and the face heat transfer coefficients h cn and h cp are uniform in space, the cell temperature isapproximately uniform. In terms of dimensional parameters, this simplified model is suitable when L x L ⊥ (cid:28) , I app FL cn σ ck RT ∞ (cid:28) , h face L ⊥ λ eff L x ∼ , h edge L ⊥ λ eff L x (cid:28) , The “CC” model consists of an algebraic expression for the terminal voltage and a single differentialequation for the average cell temperature V = V ( I , (cid:104) T (cid:105) ) − R cp I app − R cn I app , (2.2a) ρ eff ∂ (cid:104) T (cid:105) ∂t = ¯ Q ( I , (cid:104) T (cid:105) ) − ( h cn + h cp ) L ( (cid:104) T (cid:105) − T ∞ ) − ( (cid:104) T (cid:105) − T ∞ ) L y L z (cid:90) ∂ Ω h eff d s (2.2b)+ H cn I + H cp I in Ω , V and ¯ Q are the voltage and x -averaged heat source terms determined from the solution of asingle one-dimensional electrochemical model. The current collector resistances are computed as R cn = (cid:104) f n (cid:105) L y L z L cn σ cn , R cp = 1 L y L z σ cp A tab,cp (cid:90) ∂ Ω tab,cp, ⊥ f p d s (2.2c)and the coefficients related to Ohmic heating in the current collectors are H cn = L cn (cid:104)|∇ ⊥ f n | (cid:105) L ( L y L z L cn ) σ cn , H cp = L cp (cid:104)|∇ ⊥ f p | (cid:105) L ( L y L z L cp ) σ cp , (2.2d)where f n and f p satisfy the auxilliary equations ∇ ⊥ f n = − , ∇ ⊥ f p = 1 in Ω , (2.2e) f n = 0 on ∂ Ω tab,cn, ⊥ , ∇ ⊥ f n · n = 0 on ∂ Ω ext,cn, ⊥ , (2.2f) ∇ ⊥ f p · n = L y L z L cp A tab,cp on ∂ Ω tab,cp, ⊥ , ∇ ⊥ f p · n = 0 on ∂ Ω ext,cp, ⊥ , (cid:104) f p (cid:105) = 0 . (2.2g)The potential distribution in the current collectors can be determined from f n and f p via φ s,cn = − I app f n L y L z L cn σ cn , φ s,cp = V + I app f p L y L z L cp σ cp . (2.2h)An approach sometimes used in the literature is to retain the spatial derivatives in the energybalance equation, but use heat source terms from the averaged one-dimensional electrochemical model(see e.g. [11]). Such an approach corresponds to replacing ¯ Q ( I , T ) with ¯ Q ( I , (cid:104) T (cid:105) ) and replaces (2.2b)with ρ eff ∂T∂t = λ eff ∇ ⊥ T + ¯ Q ( I , (cid:104) T (cid:105) ) − ( h cn + h cp ) L ( T − T ∞ ) (2.3a)+ L cn L σ cn |∇ ⊥ φ s,cn | + L cp L σ cp |∇ ⊥ φ s,cp | , in Ω − λ eff ∇ ⊥ T · n = h eff ( T − T ∞ ) on ∂ Ω (2.3b)This model captures the variation due to Ohmic heating in the current collectors and cooling at theboundaries, but neglects the spatial variation of the heat source within the cell. In the following resultswe use (2.3) in favour of (2.2b). 9 .3 The extremely large conductance limit: 0D Model
In the CC model the conductivity of the current collectors is high enough that the potential is ap-proximately uniform across them, and the resistances due to the current collectors are calculated as aperturbation. If this perturbation is so small as to be negligible, these correction terms can be ignored.In that case the potential in the current collectors is uniform φ s,cn = 0 , φ s,cp = V, (2.4)and we arrive at a model in which the effects of the current collectors are ignored entirely. In thismodel the cell behaviour is uniform in ( y, z ), and the temperature, which is now a function of timeonly, is governed by the ODE ρ eff ∂T∂t = ¯ Q ( I , T ) − ( h cn + h cp ) L ( T − T ∞ ) − ( T − T ∞ ) L y L z (cid:90) ∂ Ω h eff d s in Ω , (2.5)with initial condition T = T . The CC model includes terms of order ( I app F /L cn σ ck RT ∞ ) but neglectsthose of order ( I app F /L cn σ ck RT ∞ ) , while the 0D model neglects terms of order ( I app F /L cn σ ck RT ∞ ). Here we give the governing equations for the DFN, SPMe and SPM. In [18] it was shown that the SPMand SPMe can be derived from the DFN via asymptotic analysis in the limit in which the electricalconductivity in the electrodes and electrolyte is large and the timescale for the migration of lithiumions in the electrolyte is small relative to the typical timescale of a discharge. For clarity, we outlinethe requirements for the simplified models to be valid, but we omit the details of the derivation.10 .1 DFN
The one-dimensional DFN model comprises equations for charge conservation
I − i e,k = − σ k ∂φ k ∂x , k ∈ { n, p } (3.1a) ∂i e,k ∂x = a k j k , k = n, p0 , k = s , (3.1b) i e,k = (cid:15) bk κ e ( c e,k , T ) (cid:18) − ∂φ e,k ∂x + 2(1 − t + ) RTF ∂∂x (log( c e,k )) (cid:19) , k ∈ { n, s, p } ; (3.1c)mass conservation ∂c s,k ∂t = − r ∂∂r (cid:18) r N s,k (cid:19) , N s,k = − D s,k ( c s,k , T k ) ∂c s,k ∂r , k ∈ { n, p } , (3.1d) (cid:15) k ∂c e,k ∂t = − ∂N e,k ∂x + 1 F ∂i e,k ∂x , k ∈ { n, s, p } , (3.1e) N e,k = − (cid:15) bk D e ( c e,k , T k ) ∂c e,k ∂x + t + F i e,k , k ∈ { n, s, p } (3.1f)and electrochemical reactions j k = j , k sinh (cid:18) F η k R g T k (cid:19) , k ∈ { n, p } , (3.1g) j , k = m k ( T k )( c s,k ) / ( c s,k,max − c s,k ) / ( c e,k ) / (cid:12)(cid:12) r = R k k ∈ { n, p } , (3.1h) η k = φ s,k − φ e,k − U k ( c s,k , T ) (cid:12)(cid:12) r k = R k , k ∈ { n, p } ; (3.1i)along with boundary conditions relating to charge conservation φ s,n (cid:12)(cid:12) x =0 = φ s,cn , φ s,p (cid:12)(cid:12) x = L x = φ s,cp , (3.1j) i e,n (cid:12)(cid:12) x =0 = 0 , i e,p (cid:12)(cid:12) x = L x = 0 , (3.1k) φ e,n (cid:12)(cid:12) x = L n = φ e,s (cid:12)(cid:12) x = L n , i e,n (cid:12)(cid:12) x = L n = i e,s (cid:12)(cid:12) x = L n , (3.1l) φ e,s (cid:12)(cid:12) x = L x − L p = φ e,p (cid:12)(cid:12) x = L x − L p , i e,s (cid:12)(cid:12) x = L x − L p = i e,p (cid:12)(cid:12) x = L x − L p ; (3.1m)11ass conservation in the electrolyte N e,n (cid:12)(cid:12) x =0 = 0 , N e,p (cid:12)(cid:12) x = L x = 0 , (3.1n) c e,n (cid:12)(cid:12) x = L n = c e,s | x = L n , N e,n (cid:12)(cid:12) x = L n = N e,s (cid:12)(cid:12) x = L n , (3.1o) c e,s | x = L x − L p = c e,p | x = L x − L p , N e,s (cid:12)(cid:12) x = L x − L p = N e,p (cid:12)(cid:12) x = L x − L p ; (3.1p)and mass conservation in the electrode active material N s,k (cid:12)(cid:12) r k =0 = 0 , N s,k (cid:12)(cid:12) r k = R k = j k F , k ∈ { n, p } . (3.1q)In addition, the following initial conditions are prescribed for the lithium concentrations in the solidand electrolyte c s,k ( x, y, z, r,
0) = c s,k,0 , k ∈ { n, p } , (3.2a) c e,k ( x, y, z,
0) = c e,0 , k ∈ { n, s, p } . (3.2b)The heat source term Q k is computed as Q k = Q Ohm,k + Q rxn,k + Q rev,k , (3.3)and accounts for Ohmic heating Q Ohm,k due to resistance in the solid and electrolyte, irreverisbleheating due to electrochemical reactions Q rxn,k , and reversible heating due to entropic changes in thethe electrode Q rev,k [21]. In the electrodes these terms are computed as Q Ohm,k = − (cid:18) i s,k ∂φ s,k ∂x + i e,k ∂φ e,k ∂x (cid:19) , k ∈ { n, p } , (3.4) Q rxn,k = a k j k η k , k ∈ { n, p } , (3.5) Q rev,k = a k j k T ∂U k ∂T (cid:12)(cid:12)(cid:12)(cid:12) T = T ∞ , k ∈ { n, p } . (3.6)However, in the separator there is no heat generation due to electrochemical effects, and we need onlyconsider the Ohmic heat generation term given by Q Ohm,s = − i e,s · ∂φ e,s ∂x . (3.7)12he x-averaged through-cell heat generation is given by¯ Q = 1 L (cid:88) k L k Q k . (3.8) The one-dimensional single particle model with electrolyte (SPMe) was derived in [18] by employingasymptotic methods. The limit taken is that of a short timescale for lithium-ion diffusion in theelectrolyte relative to the discharge timescale, alongside high electron conductivity in the electrodesand high ion conductivity in the electrolyte. The conditions for application are summarised in Table 1.The model includes terms of order C e , but neglects those of order C , where C e is defined in Table 1.Parameter combination Required size Interpretation C e = I typ L/ ( D e,typ F c n,max ) (cid:28) RT σ k / ( F I typ L ) (cid:29) RT κ e,typ / ( F I typ L ) (cid:29) R k ) I typ / ( D s,k F c n,max L ) (cid:28) / C e Solid diffusion occurs on ashorter or similar timescale to adischarge I typ / ( m k a k ( c e,typ ) / c n,max L ) (cid:28) / C e Reactions occur on a shorter orsimilar timescale to a dischargeTable 1: The key conditions to be satisfied for the application of the SPMe, with I typ = I app /L y /L z . Inaddition, it is required that C e (cid:28) L k /L (cid:28) / C e , C e (cid:28) c p,max /c n,max (cid:28) / C e , and C e (cid:28) c e,typ /c n,max (cid:28) / C e , which are true in practical situations.In this paper we use a slightly modified version of the SPMe from that introduced in [18]. Thesemodifications are discussed in Appendix B. The SPMe comprises equations for mass conservation in a13ingle “average” particle in each electrode ∂c s,k ∂t = − r ∂∂r (cid:18) r N s,k (cid:19) , N s,k = − D s,k ( c s,k , T k ) ∂c s,k ∂r , k ∈ { n, p } , (3.9) N s,k (cid:12)(cid:12) r k =0 = 0 , k ∈ { n, p } , N s,k (cid:12)(cid:12) r k = R k = I F a n L n , k = n , − I F a p L p , k = p , , (3.10) c s,k ( y, z, r,
0) = c s,k,0 , k ∈ { n, p } ; (3.11)and mass conservation in the electrolyte (cid:15) k ∂c e,k ∂t = − ∂N e,k ∂x + I F L n , k = n , , k = s , − I F L p , k = p , , (3.12) N e,k = − (cid:15) bk D e ( c e,k , T ) ∂c e,k ∂x + xt + I F L n , k = n ,t + I F , k = s , ( L − x ) t + I F L p , k = p , (3.13) N e,n (cid:12)(cid:12) x =0 = 0 , N e,p (cid:12)(cid:12) x = L = 0 , (3.14) c e,n | x = L n = c e,s | x = L n , N e,n (cid:12)(cid:12) x = L n = N e,s (cid:12)(cid:12) x = L n , (3.15) c e,s | x = L − L p = c e,p | x = L − L p , N e,s (cid:12)(cid:12) x = L − L p = N e,p (cid:12)(cid:12) x = L − L p (3.16) c e,k ( x, y, z,
0) = c e,0 , k ∈ { n, s, p } . (3.17)The through-cell current density, I , and local voltage V = φ s,cp − φ s,cn are related through V = U eq + η r + η c + ∆Φ Elec + ∆Φ
Solid , (3.18)where U eq is the electrode-averaged open-circuit voltage U eq = U p (cid:16) c s,p (cid:12)(cid:12) r = R p , T (cid:17) − U n (cid:16) c s,n (cid:12)(cid:12) r = R n , T (cid:17) , (3.19)14 r is the electrode-averaged reaction overpotential and is given by the difference of the average positiveand negative electrode reaction overpotentials as η r = η p − η n (3.20) η n = 2 RTF sinh − (cid:18) I a n j L n (cid:19) , (3.21) η p = − RTF sinh − (cid:18) I a p j L p (cid:19) , (3.22)where j and j are the average negative and positive exchange current densities given by j = 1 L n (cid:90) L n m n ( T )( c s,n ) / ( c s,n,max − c s,n ) / ( c e,n ) / d x, (3.23) j = 1 L p (cid:90) LL − L p m p ( T )( c s,p ) / ( c s,p,max − c s,p ) / ( c e,p ) / d x. (3.24)The remaining terms in (3.18) are the electrolyte concentration overpotential η c = 2(1 − t + ) RTF log (cid:18) c e,p c e,n (cid:19) , (3.25)the electrolyte Ohmic losses ∆Φ Elec = − I κ e ( c e,0 , T ) (cid:18) L n (cid:15) b n + L s (cid:15) b s + L p (cid:15) b p (cid:19) , (3.26)and the solid-phase Ohmic losses ∆Φ Solid = − I (cid:18) L p σ p + L n σ n (cid:19) . (3.27)If the through-cell current I is provided then we can a-posteriori read V from (3.18). However, if V is provided then (3.18) provides an algebraic constraint that must be enforced during the solutionprocess.The remaining key variables are given by the following algebraic expressions. The negative and15ositive electrode potentials are given by φ s,n = φ s,cn + I x σ n L n (2 L n − x ) , (3.28) φ s,p = φ s,cp + I ( x − L x )( L x − L p − x )2 σ p L p ; (3.29)the electrolyte potential is given by φ e,n = Φ e + 2(1 − t + ) RTF log (cid:18) c e,n c e,n (cid:19) − I κ e ( c e,0 , T ) (cid:18) x − L (cid:15) b n L n + L n (cid:15) b s (cid:19) , (3.30) φ e,s = Φ e + 2(1 − t + ) RTF log (cid:18) c e,s c e,n (cid:19) − I xκ e ( c e,0 , T ) (cid:15) b s , (3.31) φ e,n = Φ e + 2(1 − t + ) RTF log (cid:18) c e,p c e,n (cid:19) − I κ e ( c e,0 , T ) (cid:32) x (2 L x − x ) + L − L x (cid:15) b p L p + L x − L p (cid:15) b s (cid:33) , (3.32)where Φ e = φ s,n − η n + I L n κ e ( c e,0 , T ) (cid:18) (cid:15) b n − (cid:15) b s (cid:19) ; (3.33)and the average negative and positive interfacial current densities are given by j n = I F a n L n , j p = − I F a p L p . (3.34)The heat source term Q is computed using (3.3)–(3.8). The SPMe retains terms of order C e but neglects terms of order C . If C e is so small that the terms oforder C e may themselves be neglected, the result is the one-dimensional single particle model (SPM)[18]. Thus, the SPM is less accurate than the SPMe, but also slightly simpler. The SPM consists only16f equations for mass conservation in an average particle in each electrode ∂c s,k ∂t = − r ∂∂r (cid:18) r N s,k (cid:19) , N s,k = − D s,k ( c s,k , T k ) ∂c s,k ∂r , k ∈ { n, p } , (3.35) N s,k (cid:12)(cid:12) r k =0 = 0 , k ∈ { n, p } , N s,k (cid:12)(cid:12) r k = R k = I F a n L n , k = n , − I F a p L p , k = p , (3.36) c s,k ( y, z, r,
0) = c s,k,0 , k ∈ { n, p } . (3.37)The through-cell current, I , and local voltage V are related through V = U eq + η r , (3.38)where U eq and η r are given by (3.19) and (3.20), respectively. The negative and positive electrodepotentials are given by φ s,n = φ s,cn , φ s,p = φ s,cp , (3.39)the electrolyte potential is given by φ e = φ s,n − U n (cid:16) c s,n (cid:12)(cid:12) r = R n , T (cid:17) − η n , (3.40)and the electrolyte concentration is given by c e,k = c e,0 . (3.41)Finally, the heat source term Q is computed using (3.3)–(3.8). However, since the potentials in theSPM are all constant in space, the only non-zero contributions to the heating come from the termsdue to electrochemical reactions Q rxn,k and Q rev,k . The Ohmic heating terms appear at higher order,and are included in the SPMe.Although the SPM is simpler than the SPMe, we note that in terms of computational complexitythey are similar—the SPMe essentially comprises three uncoupled one-dimensional partial differentialequations in comparison to the two uncoupled one-dimensional partial differential equations of theSPM. In contrast the DFN model, as already noted, is pseudo-two-dimensional.17 Critical Comparison of Isothermal 1+1D Models
In this section, we provide a numerical comparison of the models depicted in Figure 1. For easeof exposition we consider the case in which all variables are uniform in y , so that each model hasone-dimensional current collectors (in z ) (in the “extremely large conductance” limit the effects ofthe current collectors are ignored). In total we consider 9 models: 1+1D DFN, 1+1D SPM, 1+1DSPMe, DFNCC, SPMeCC, SPMCC, DFN, SPMe and SPM. Each model is implemented within theopen source battery modelling software PyBaMM (Python Battery Mathematical Modelling) [19].PyBaMM is a modular open-source framework that allows users to build and solve physics-basedmodels of lithium-ion cells by selecting different physical effects for each component of the cell, so iswell-suited to comparing a suite of different models like those in Figure 1. The model equations arediscretised in space using the finite-volume method and integrated in time using an adaptive, variable-order backward differentiation formula. For the spatial discretisation we use the following number ofgrid points: N z = 30 in the z -direction of the current collectors; N r n = 20 and N r p = 20 in r -directionof the negative and positive particles, respectively; and N x n = 35, N x s = 20, and N x p = 35 in the x -direction negative electrode, separator and positive electrode, respectively. Unless otherwise stated,results are presented for a 1C constant current discharge. To demonstrate the relative computationalcomplexity of each model we present the number of states in each model alongside the solve timein Table 2. We observe a dramatic decrease in both memory requirements and solve times as thecomplexity of the model is decreased. For example, the SPMeCC requires around 0.2% of the memoryrequired by the 1+1D DFN and is over 400 times faster to solve than the 1+1D DFN. Model 1+1D DFN 1+1D SPMe 1+1D SPM DFN(CC) SPMe(CC) SPM(CC)Number of states 49561 3961 1261 1651 131 41Solve time [ms] 8308 1514 105 259 18 13
Table 2: The number of states required for the time integration component of the isothermal versionof each model and associated time integration solve time (i.e. excluding any initial time independentsolves) for a 1 C constant current disharge with N z = 30, N x n = 35, N x s = 20, N x p = 35, N r n = 20,and N r p = 20. Since we have ignored the time-independent part of the problem, the results for the“CC” and 1D (no current collector effects) models are the same.Despite the clear computational benefits of employing a reduced-order model, modelling errors areintroduced and must be understood. To help quantify spatial and temporal errors, we use two measures. Here we refer to the number of states required by part of the model that is integrated in time. That is, we ignorethe states in the “CC” part of the problem which is solved “offline” and is separate from the time integration. z defined for a variable Φ to be (cid:15) z = max t | Φ − Φ Reduced | , (4.1)where Φ is the value of variable predicted by the 1+1D DFN model and Φ Reduced is the valueof the variable predicted by relevant the reduced-order model. The other measure of the error that weuse is the maximum absolute error at each point in time in the discharge defined for each variable Φas (cid:15) t = max z | Φ − Φ Reduced | . (4.2)Both (cid:15) z and (cid:15) t automatically inherit the units of the variable, Φ, under study.The results and conclusions presented below are specific to the parameter values in Tables 6 and 7,and one should keep in mind that the most appropriate model is a function of the particular parametervalues being used. In Section 5, we will discuss changes to key parameters that are likely to have alarge impact upon the conclusions of this section. The most commonly used output of lithium-ion battery models is typically the predicted terminalvoltage. In Figure 2 (a), and (b), we present the 1 C terminal voltage predicted by each model and theassociated errors between the reduced-order model and the 1+1D DFN. At a glance, with the exceptionof the SPM, the reduced-order models are able to recover the behaviour predicted by the 1+1D DFNat 1 C. However, in Figure 2 (b), we observe between a one and two order of magnitude decrease in theerror when using the DFN as the through-cell model, irrespective of the choice of transverse model.This suggests that simplifications in the the through-cell electrochemistry result in a greater increasein error than simplifications in transverse direction. In Figure 2 (c), the maximum absolute error inthe terminal voltage across the discharge is presented as a function of the C-rate. As expected, theerror increases with increasing C-rate for all of the models considered. Additionally, the error in themodels which use the SPM and SPMe increases more quickly with C-rate when compared with themodels that use the DFN. This suggests that at higher C-rates retaining a more complex through-cellmodel becomes increasingly important.If the terminal voltage is the only quantity of interest then our results suggest that employing areduced-order model that does not couple the current collector and electrochemical effects is the most19ppropriate model choice, in the sense the voltage can be accurately predicted at a greatly reducedcomputational cost. In particular, if one has a larger computational budget and has access to efficientdifferential algebraic equation (DAE) solvers then making use of the 1D DFN or DFNCC gives thebest prediction of the voltage. However, if the computational budget is more limited the 1D SPMeor SPMeCC may be more appropriate. Finally, we note that the errors between experimental dataand more complex models (i.e. DFN) can be on the order of 10 − V [22], so in practice introducinga modelling error on the order of 10 − V, such as in the SPMeCC, may not affect the ability of themodel to replicate experimental results. .
00 0 .
25 0 . Discharge capacity [A h] . . . . . V [ V ] (a) .
00 0 .
25 0 . Discharge capacity [A h] − − − A b s o l u t ee rr o r [ V ] (b) C-rate − − − M a x a b s o l u t ee rr o r [ V ] (c) Figure 2: Comparison of the predicted terminal voltage: (a) 1 C discharge voltage profile predictedby each model; (b) 1 C discharge voltage profile absolute error between each reduced-order model andthe 1+1D DFN at each time in the discharge; and (c) the maximum absolute error (across all timesduring the discharge) between each reduced-order model and the 1+1D DFN at a range of C-rates.The results for the 1+1D SPMe, SPMeCC, and SPMe are almost indistinguishable and the results forthe 1+1D SPM, SPMCC, and SPM are indistinguishable.
In Figure 3(a), we display the variation in z of the surface concentration of the through-cell averaged( x -averaged) negative particle predicted by the 1+1D DFN throughout the full discharge. Initially thesurface concentration is uniform in z . However, as the discharge proceeds, the upper portion of thecell near the tabs becomes more depleted than the lower portion of the cell. This is due to currentpreferentially travelling through the upper portion of the cell as the path of least resistance. When20he upper portion of the cell is almost fully depleted, the resistances associated with discharging theupper portion of the cell increase (i.e. it becomes harder to remove more lithium from the negativeelectrode and harder to insert lithium into the positive electrode) to a sufficient level such that currentpreferentially travels through the less utilized portions of the cell, thus leading to the final uniformparticle surface concentration. The effect of this on the current can be observed in Figure 4(a).In Figure 3(b), we plot the x -averaged negative particle surface concentration at 0 .
17 A h predicted bythe 1+1D DFN, the 1+1D SPM, the 1+1D SPM, and the average concentration which is predicted bythe models that do not include the z -dependence of the concentrations (i.e. the “CC” and 1D models).Given that the variation in surface concentration is on the order of 15 mol m − , employing a z -averagedmodel recovers the surface concentrations to 0 .
1% accuracy. However, as shown in Figures 3(c) and(d), an order of magnitude decrease in the error can be achieved when using a z -resolved model suchas the 1+1D SPM or 1+1D SPMe. For situations in which resolving the spatial inhomogenieties (in z ) in the surface concentration are crucial, a model at least as detailed as the 1+1D SPM should beemployed.A further comparison of the surface concentration variables is provided in Table 3. Instead ofcomparing the x -averaged surface concentration, we now compare the surface concentration at every( x , z )-location in the cell. Here, we observe that the root mean square error (RMSE) in the particlesurface concentration is in fact larger for the 1+1D SPM(e) than the 1D DFN and DFNCC. This isbecause 1+1D SPMe considers a single x -averaged particle in the through-cell direction, and in thisinstance the through-cell variation in the particle surface concentration is more significant than the z -direction variation. This highlights the fact that the best combination of through-cell and transversesimplifications depends on which quantities are of interest: for the results here we find that the 1+1DSPMe best recovers the through-cell averaged particle surface concentration, but the 1D DFN betterapproximates the surface concentration at each ( x , z ) location. In Figure 4(a), we present the through-cell current density predicted by the 1+1D DFN as a functionof z and discharge capacity. As discussed in our description of the particle surface concentration vari-ation, we observe that the current preferentially travels through the top of the cell with the exceptionof particular times where particle concentrations are such that the through-cell resistance makes itpreferential for the current to flow more uniformly through the cell (at 0 .
35 A h) and through the21 . . . . Discharge capacity [A h] z [ mm ] (a) z [mm] ¯ c s , n (cid:12)(cid:12) r = R n [ m o l m − ] (b) . . . . Discharge capacity [A h] − − − (cid:15) t [ m o l m − ] (c) z [mm] − − − (cid:15) z [ m o l m − ] (d) − − − ¯ c s , n (cid:12)(cid:12) r = R n − h ¯ c s , n i (cid:12)(cid:12) r = R n [mol m − ] Figure 3: Comparison of the x -averaged negative particle surface concentration: (a) Variation inthe x -averaged negative particle surface concentration predicted by the 1+1D DFN; (b) Snapshot at0.17[A.h]; (c) Maximum (over z ) absolute error at each time in the discharge; (d) Maximum (over theentire discharge) absolute error at each z location.22ottom of the cell (at the end of discharge). In Figure 4(b), we show the through-cell current densityas a function of z at 0 .
17 A h through the discharge. We observe that the average current (as predictedby the “CC” and 1D models) provides a good first approximation of the through-cell current densitywhich is reasonably uniform (only varying by about 0 .
04 A m − ). However, the 1+1D SPM and 1+1DSPMe both provide an improved estimate of the spatial variation in the through-cell current density,and can give an order of magnitude decrease in the error, as demonstrated in Figures 4(c) and (d).In applications where accurately resolving the current distribution is a key objective and the compu-tational budget is limited, the 1+1D SPMe is the best choice in terms of offering good accuracy atlow computational costs. One such application is in studying the effects of degradation, where deter-mining the distribution of current and therefore the overpotentials is crucial for determining rates ofdegradation. . . . . Discharge capacity [A h] z [ mm ] (a) z [mm] . . . . . I [ A m − ] (b) . . . . Discharge capacity [A h] − − − − − (cid:15) t [ A m − ] (c) z [mm] − − − − − (cid:15) z [ A m − ] (d) .
90 23 .
95 24 .
00 24 .
05 24 . I [A m − ] Figure 4: Comparison of the through-cell current: (a) Through-cell current predicted by the 1+1DDFN; (b) Snapshot at 0.17 [A h]; (c) Maximum (over z ) absolute error at each time in the discharge;(d) Maximum (over the entire discharge) absolute error at each z location.23 .4 Comparison of negative potentials In Figure 5(a), we present the potential in the negative electrode at the point 0 .
17 A h in the discharge(we display this instead of the current collector potential through time because the current collectorpotential is approximately constant throughout a 1 C discharge). In the upper left corner of theelectrode, next to the tab, the potential takes values close to the reference value of zero. However, aswe move through the cell in the x -direction or down the current collector in the z -direction we observea drop in the potential. In particular, a greater potential drop is observed in the current collectordirection and so current collector Ohmic losses are of greater importance than through-cell electrodeOhmic losses. In Figures 5(b), (c), and (d), we compare the negative current collector potential at0 .
17 A h as predicted by the 1+1D and “CC” models. Note that the 1D models predict that thepotential will simply take on the reference value of 0 everywhere in the current collector. We observeexcellent agreement between all models.
In this section, we have provided a critical comparison of some of the key outputs predicted by the1+1D DFN model and the reduced-order models introduced in this paper. In the interest of brevity,we have only presented detailed results for a limited set of variables, but results for other key variablesare provided in Table 3 for reference.The key result is that in the isothermal case, choosing a reduced-order model that simplifies the z -direction behaviour (e.g. the DFNCC) instead of the x -direction behaviour (e.g. 1+1D SPMe)provides a better allocation of computational resources. Only in situations where the z -directionvariation in variables, such as the surface concentrations and through-cell current, are essential shouldsimplifications to the through-cell model be made in favour of simplifications to the transverse model.In such situations, the 1+1D SPMe typically provides a good balance of accuracy and computationalcost. We now compare the thermal versions of the models under the conditions of a 3 C discharge and tabcooling. Tab cooling is simulated by applying λ ∇ T · n = − h tab ( T − T ∞ ) (4.3)24 .
000 0 .
025 0 .
050 0 .
075 0 . x [mm] z [ mm ] (a) z [mm] − . − . − . − . − . − . − . . φ s , c n [ m V ] (b) . . . . Discharge capacity [A h] − − − − − − (cid:15) t [ V ] (c) z [mm] − − − − − − (cid:15) z [ V ] (d) − . − . − .
05 0 . φ s , n [mV] Figure 5: Comparison of negative potentials: (a) Snapshot at 0.17 [A.h] of the negative electrodepotential predicted by the 1+1D DFN; (b) Snapshot of the negative current collector potential at0.17[A.h]; (c) Maximum (over z ) absolute error in the negative current collector potential at eachtime in the discharge; (d) Maximum (over the entire discharge) absolute error in the negative currentcollector potential at each z location. The results of the DFNCC, SPMeCC, and SPMCC are allrepresented by CC. 25 ariable Units Typical values 1+1D SPMe 1+1D SPM DFNCC SPMeCC SPMCC DFN SPMe SPM V V 3.4 – 3.8 3 . × − . × − . × − . × − . × − . × − . × − . × − φ s, cn V − . × − – 0 1 . × − . × − . × − . × − . × − . × − . × − . × − φ s,n V − . × − – 0 2 . × − . × − . × − . × − . × − . × − . × − . × − φ e,k V − .
26 – − .
17 2 . × − . × − . × − . × − . × − . × − . × − . × − φ s,p V 3.4 – 3.8 3 . × − . × − . × − . × − . × − . × − . × − . × − φ s,cp V 3.4 – 3.8 3 . × − . × − . × − . × − . × − . × − . × − . × − I A m − . × − . × − . × − . × − . × − . × − . × − . × − c s,n (cid:12)(cid:12) r = R n mol m − . × . × .
77 9 . × . × .
77 9 . × . × c s,p (cid:12)(cid:12) r = R p mol m − . × . × .
33 4 . × . × .
33 4 . × . × c e,k mol m −
800 – 1200 1 . × . × . × − . × . × . × − . × . × i e,k A m − .
65 1 .
65 2 . × − .
65 1 .
65 2 . × − .
65 1 . j k A m − − . . × − . × − . × − . × − . × − . × − . × − . × − η n V 0.0014 – 0.008 1 . × − . × − . × − . × − . × − . × − . × − . × − η p V − .
12 – − .
06 4 . × − . × − . × − . × − . × − . × − . × − . × − Table 3: RMSE of key model variables in each of the reduced-order models vs. the 1+1D DFN for a1 C constant current discharge.on the tabs and λ ∇ T · n = 0 (4.4)on all other boundaries. For the purposes of this comparison, both tabs are placed at the top of thecell and the value of h tab is set to 1000 W m − K − so that a temperature variation on the order ofa few degrees Kelvin is observed across the cell; this is in accordance with experimental results [20].This approach was taken as a simple way to induce a variation in the temperature in the z -directionthat is similar to the variation seen in experiments. However, a proper treatment of tab cooling wouldinvolve a more complete model of the tab which has been shown to be a major heat transfer bottleneck [20].Figure 6 we show the volume-averaged temperature predicted by each model. Owing to the effectsof tab cooling, it is important to account for the current collectors in order to accurately predict thetemperature. However, a large contribution to the error comes from ignoring electrolyte effects andassuming uniform (in x ) electrode potentials, as evidenced by the poor performance of all of the modelsthat use the SPM. For the parameters in Tables 6 and 7 it is better to include a more complex modelof the electrochemistry with a simpler model of the transverse behaviour in order to accurately predictthe temperature. For instance, it is a better use of computational resources to choose the DFNCCrather than the 1+1D SPMe.The DFN, SPMe and SPM employ a lumped thermal model with cooling proportional to the dif-ference between the average temperature and the ambient temperature. Since the actual temperatureat the tab is lower than the average temperature, these models overpredict the cooling rate, giving alower volume-averaged cell temperature, as observed in Figure 6.Figure 7 shows the temperature as a function of space and time throughout the discharge. Again26 . . . . Discharge capacity [A h] h ¯ T i [ K ] (a) . . . . Discharge capacity [A h] − − − − A b s o l u t ee rr o r [ K ] (b) Figure 6: Comparison of the volume-averaged cell temperature: (a) predicted volume-averaged celltemperature; (b) absolute error between reduced model and 1+1D DFN at each time during discharge.we observe that the DFNCC best predicts the temperature profile of the 1+1D DFN, and the greatesterror is introduced by using the SPM. Note that the error for the 1+1D and “CC” models is similar,backing up the suggestion to use a simpler transverse model (that still retains some z dependence).In Figure 8, we break down the volume-averaged heating into its individual components to helpdiagnose where errors in the various reduced models arise. We observe that across all forms of heating,the DFNCC is almost indistinguishable from the results of the 1+1D DFN. Whilst one might initiallythink that the 1D DFN should produce the same irreversible and reversible reaction heating as theDFNCC, this is not not the case because the temperature predicted by the 1D DFN is lower, asmentioned in the discussion of Figure 6. The temperature dependence of the electrochemical reactionsthen means that the 1D DFN actually overpredicts the reaction heating (an effect which increasesas the temperatures diverges further through the discharge). The SPMeCC and 2+1D SPMe bothcapture the general behaviour of the Ohmic and irreversible reaction heating, however they fail tocapture fluctuations in this general behaviour, particularly in the Ohmic heating. This is a result offailing to capture though-cell variations in the reaction overpotentials, as well as the more detailedvariations in the electrolyte potentials. Despite this, both the SPMeCC and 1+1D SPMe performreasonably well at recovering the total heating. The main failing of the SPM and 1+1D SPM isthat neither accounts for any though-cell Ohmic heating, with the 1+1D SPM only accounting forcurrent collector Ohmic heating. As a result, these models significantly under predict the cell heating.Therefore it is recommended that a more complicated model than the SPM is used for the through-cell27 . . . . Discharge capacity [A h] z [ mm ] (a) z [mm] ¯ T [ K ] (b) . . . . Discharge capacity [A h] − − − − (cid:15) t [ K ] (c) z [mm] − − − − (cid:15) z [ K ] (d) − − ¯ T − h ¯ T i [K] Figure 7: Comparison of temperature profiles: (a) x -averaged cell temperature variation through timepredicted by the 1+1D DFN; (b) x -averaged cell temperature at 0.17 [A h]; (c) Maximum (over z )absolute error at each time in the discharge; (d) Maximum (over the entire discharge) absolute errorat each z location. 28odel for thermal studies, unless the model parameters vary significantly from those in this study. . . . . Discharge capacity [A h] h ¯ Q t o t i [ W m − ] (a) . . . . Discharge capacity [A h] h ¯ Q O h m i [ W m − ] (b) . . . . Discharge capacity [A h] h ¯ Q r x n i [ W m − ] (c) . . . . Discharge capacity [A h] h ¯ Q r e v i [ W m − ] (d) Figure 8: Comparison of volume-averaged cell heating: (a) total heating; (b) Ohmic heating; (c)reaction (irreversible) heating; (d) irreversible heating.In Figure 9 (a), we present the total heating predicted by the 1+1D DFN as a function of z and thedischarge time. Initially the cell heats near the top, but as the discharge proceeds heating mainly occursnear the bottom of the cell. This may seem in contrast to what we would expect given the isothermalcurrent profiles in Figure 4. However, as shown in Figure 10, the current profile can in fact be verydifferent for the tab cooling scenario considered here. We now see that the current now preferentiallytravels through the bottom of the cell for most of the discharge, and the increased current leads toincreased heating. This is an effect of temperature dependence of the parameters in the through-cellmodels: higher temperature leads to lower resistance. In Figures 7 (b), (c), and (d), we compare theheat generation predicted by each reduced order model with that of the 1+1D DFN. We observe thatall reduced models give rise to around a 10% error in the total heating.29 . . . . Discharge capacity [A h] z [ mm ] (a) z [mm] ¯ Q t o t [ W m − ] (b) . . . . Discharge capacity [A h] (cid:15) r [ W m − ] (c) z [mm] (cid:15) z [ W m − ] (d) − −
500 0 500 1000 ¯ Q tot − h ¯ Q tot i [W m − ] Figure 9: Comparison of x -averaged total heating, ¯ Q tot : (a) The variation in the heat generation profilethroughout the discharge predicted by the 1+1D DFN; (b) x -averaged heat generation at 0.17[A.h];(c) Maximum (over z ) absolute error at each time in the discharge; (d) Maximum (over the entiredischarge) absolute error at each z location. . . . . Discharge capacity [A h] z [ mm ] (a) z [mm] I [ A m − ] (b) I [A m − ] Figure 10: Current distribution during discharge of tab-cooled cell.30 .7 Drive cycle comparison
The constant current discharges that we have investigated throughout this paper are useful for com-paring models. However, they are not fully representative of a realistic usage scenario. To give anexample of how the models perform in more realistic conditions, we compare the performance of eachmodel on a portion of the US06 drive cycle. Here, we just consider the measured terminal voltage andthe average temperature of the cell, as shown in Figure 11. We observe that similar to our previousresults, the main errors are introduced by making simplifications to the through-cell electrochemicalmodel. Therefore to most accurately capture the temperature rises, computational effort should beplaced upon using a more detailed through-cell model like the DFN with a CC or 0D transverse model,rather than in using a detailed 1+1D transverse model and a simplified through-cell model like theSPM.
In the previous sections, we have provided a detailed comparison of the nine models in Figure 1. InTable 4, we have aimed to condense this information into a concise and clear format. Whilst ourrecommendations presented in Table 4 are informed by the quantitative performance of the models ineach of the comparisons in the previous sections, they are to some degree qualitative in nature.The first two columns in the table correspond to the solve time and the number of states requiredto solve that model. The solve times should only be considered relative to one another, as one couldachieve speed ups by employing different numerical methods or hardware; they are representative ofthe time complexitity of each model. Similarly, the number of states should only be considered asrepresentative of the spatial complexity of each model as different discretization methods may lead toa different number of states. Solve times in green are on the order of 10ms, orange on the order of100ms, and red on the order of 1000ms, so that a green model is approximately 100 times faster than ared model. We similarly colour the states of each model so that a red model requires around 10 timesmore memory than a orange model and around 100 times more memory than a green model.The second set of columns summarises the ability of each model to accurately predict five key outputvariables during a 1 C discharge: the terminal voltage, V ; the current distribution, I ; the negativecurrent collector potential, φ s,cn ; the x -averaged negative particle surface concentration, ¯ c s,n ; and theaverage cell temperature, (cid:104) ¯ T (cid:105) . We adopt a traffic light system for each of these variables as described31 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . . . . . V [ V ] .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . − − A b s o l u t ee rr o r [ V ] .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . − − I a pp [ A ] .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . h T i [ K ] .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . − − A b s o l u t ee rr o r [ K ] Figure 11: Comparison of model performance on US06 drive cycle. Solve times were: 1+1D DFN154s, 1+1D SPMe 83s, 1+1D SPM 11s, DFN(CC) 25s, SPMe(CC) 1s, SPM(CC) 0.23s32n Table 5. The system is designed to divide the predictions of each variable into three categories,where green is most accurate and red is least accurate. This is done with reference to the resultspresented throughout this paper. Generally, moving from one color to the next represents an order ofmagnitude difference in error, but this is not always the most appropriate division and we refer thereader to Table 5 for precise details. To be clear, red does not indicate that the model should not beused, but rather that this model performs poorly relative to the other models. Since we refer to the1+1D DFN model for calculation of errors in models, the 1+1D DFN is always coloured green.The third set of columns considers the ability of each model to predict the terminal voltage underlow current, medium current, and high current conditions. We again adopt a traffic light system wherea maximum absolute error of < × V is coloured green, < × − V is coloured orange, and < Model Solve time [ms] States 1 C 3 C V I φ s,cn ¯ c s,n (cid:104) ¯ T (cid:105) I app < < I app < I app > C Table 4: Qualitative evaluation of model. In the variables, we employed the traffic light systemdescribed in Table 5. For the final set of columns (current dependence), we make use of the resultsin Figure 2(c) with the following traffic light system for the maximum absolute errors: < × − V(green), < × − V (orange), < V [V] I [A m − ] φ s,cn [V] ¯ c s,n [mol m − ] (cid:104) ¯ T (cid:105) [K] < − < × − < − < < − < − < − < − < < < < < − < < V , I , φ s,cn , and ¯ c s,n these errors refer to theisothermal case. For the variable T , the errors refer to the thermal case.We now interpret the summary presented in Table 4. Firstly, by employing the 1+1D SPMe insteadof the 1+1D DFN, we can achieve similar performance in terms of predicting the current, current33ollector potential, and y-z concentration variation while reducing memory requirements by an orderof magnitude. This is achieved at the expense of a slight reduction in accuracy of the terminal voltageprediction and average cell temperature. The model is most appropriate in the low to medium C-rate range. Employing the 1+1D SPM sees a similar reduction in memory requirements, but with anadditional reduction in solve time by an order of magnitude. However, this is achieved at the expenseof less accurate predictions of all of the variables as well as being limited to low C-rates.The DFNCC offers a reduction in both solve time and memory requirements by an order of magnitudewithout loss of accuracy in both the terminal voltage and average cell temperature. Further, there isonly a small reduction in accuracy of the current collector potential. However, it achieves this reductionin solve time and memory requirements at the expense of accuracy in the estimates of the through-cellcurrent density, and x -averaged negative particle concentration. The DFNCC remains applicable acrossthe range of C-rates we investigated. The SPMeCC drops the solve time and memory requirementsby a further order of magnitude relative to the DFNCC. To achieve this a small amount of accuracyis sacrificed in the terminal voltage and average temperature estimates. Additionally, the SPMeCCis limited to low and medium C-rates. The solve time and memory requirements of the SPMCC aresimilar to that of the SPMeCC, but the predictions of the voltage and average temperature are lessaccurate. Further, the SPMCC is limited to low C-rates.The DFN, SPMe, and SPM all offer orders of magnitude reductions in solve time and memoryrequirements compared to the models that account for current collector effects, but all do so at theexpense of accuracy in the predictions of through-cell current density, current collector potential, x -averaged concentration, and average temperature. However, the DFN can accurately predict theterminal voltage, and performs well across the full range of C-rates we considered. The SPMe performedmoderately at recovering the terminal voltage, and moderately well at low to medium C-rates. TheSPM was the worst performing model across all variables, and is best applied at low C-rates.Table 4 highlights the trade-off that must be made between computational complexity and accuracy.However, we see that through the appropriate choice of a model, one can choose where accuracy issacrificed in favour of reduced complexity. For example, in a study of spatially dependant degradationwithin a lithium-ion pouch cell, it makes more sense to employ the 1+1D SPMe or the 1+1D SPMinstead of the DFN or DFNCC, because they retain greater accuracy in the y-z dependent variables fora similar computational budget. Alternatively, in pack or module level simulations where one is onlyinterested in the average temperature and voltage outputs of a cell, the DFNCC or SPMeCC are more34ppropriate. Further, where computational constraints are really strict such as in model-based control,simple models such as the SPM and SPMe are most appropriate (the SPMCC and the SPMeCC couldalso be used in a limited form). In applying these models it is important to understand their limitationsso that they can be used appropriately. As a general rule, if integrated quantities, such as the terminalvoltage, are of most importance then the most detailed through-cell model that the budget can affordshould be used. On the other hand, if capturing variations in the transverse directions is importantthen a 1+1D model is an appropriate choice. In this paper we have presented a hierarchy of reduced-order models of a lithium-ion pouch cell ofvarying fidelity and computational complexity. Simplifications to the transverse and through-cell be-haviour, derived via asymptotic analysis [4, 18], have been combined in a consistent way, so that theycan be used in any combination to suit the particular application. In particular, we note that thetransverse and through-cell model can be selected independently, and the framework allows for ad-ditional effects to be included in a straightforward manner. The asymptotic approach allows for apriori estimates of the modelling error to be made so that the appropriateness of applying a partic-ular reduced-order model can be assessed prior to implementation. In addition, we have provided anumerical comparison of the reduced-order models and the full 1+1D DFN model.Through a series of comparisons it was demonstrated that the choice of reduced-order model dependson the variables of interest for a particular application. For instance, in many control or systems-levelapplications, one is only concerned with obtaining integrated quantities, such as the terminal voltageand volume-averaged cell temperature. In such cases it is best to select the highest fidelity through-cell models the computational budget allows, combined with a simpler transverse model (e.g. theSPMeCC). However, if distributed (in y - z ) quantities are of interest, such as in trying to model non-uniform degradation, then it is necessary to choose a more complicated transverse model (e.g. the1+1D SPMe). 35 Parameter Values
Parameter Units Description cn n s p cp L k µ m Region thickness 25 100 25 100 25 L tab,ck mm Tab width - 40 - 40 - c e,typ mol m − Typical lithium concentration in electrolyte - 1 × × × - D e,typ m s − Typical electrolyte diffusivity - 5 . × − . × − . × − - (cid:15) k - Electrolyte volume fraction - 0 . . c s,k,max mol m − Maximum lithium concentration in solid - 2 . × - 5 . × - σ k Ω − m − Solid conductivity 5 . ×
100 - 10 3 . × D s,k,typ m s − Typical solid diffusivity - 3 . × − - 1 × − - R k µ m Particle radius - 10 - 10 - a k µ m − Electrode surface area density - 0.18 - 0.15 - m k,typ A m − (m mol − ) . Typical reaction rate - 2 × − - 6 × − - ρ k kg m − Density 8954 1657 397 3262 2707 c p,k J kg − K − Specific heat capacity 385 700 700 700 897 λ k W m − K − Thermal conductivity 401 1.7 0.16 2.1 237 E m k J mol − Activation energy for reaction rate - 3 . × - 3 . × - E D e J mol − Activation energy for electrolyte diffusivity 3 . × E κ e J mol − Activation energy for electrolyte conductivity 3 . × c k, mol m − Initial lithium concentration in solid - 1 . × - 3 . × - T K Initial temperature 298.15 F C mol − Faraday’s constant 96487 R g J mol − K − Universal gas constant 8.314 T ∞ K Reference temperature 298.15 b - Bruggeman coefficient 1.5 t + - Transference number 0.4 L x µ m Cell thickness 225 L y mm Cell width 207 L z mm Cell height 137 I app A Applied current 0.681 h W m − K − Heat transfer coefficient 10 ρ eff J K − m − Lumped effective thermal density 1 . × λ eff W m − K − Effective thermal conductivity 59.396
Table 6: Typical dimensional parameter values taken from [23]. The parameters are for a carbonnegative current collector, graphite negative electrode, LiPF in EC:DMC electrolyte, LCO positiveelectrode, and aluminium positive current collector. B Modifications to the SPMe
The governing equations for the SPMe used here are slightly modified from the version stated in [18].Firstly, we have extended the model to account for thermal effects. Secondly, in [18], a linear diffusionequation for the electrolyte concentration is used by taking the lithium-ion flux in the electrolyte tobe of the form N e,k = − (cid:15) bk D e ( c e,0 , T ) ∂c e,k ∂x + xt + I F L n , k = n ,t + I F , k = s , ( L − x ) t + I F L p , k = p . (B.1)36 arameter Units Description U k = U k,ref + ( T k − T ∞ ) ∂U k ∂T k (cid:12)(cid:12)(cid:12)(cid:12) T k = T ∞ k ∈ { n, p } V Open circuit potential U n,ref = 0 .
194 + 1 . (cid:16) − . c n c s,n,max (cid:17) + 0 . (cid:16) ( c n c s,n,max − . / . (cid:17) − . (cid:16) ( c s,n c s,n,max − . / . (cid:17) − . (cid:16) ( c s,n c s,n,max − . / . (cid:17) − . (cid:16) ( c s,n c s,n,max − . / . (cid:17) − . (cid:16) ( c s,n c s,n,max − . / . (cid:17) − . (cid:16) ( c s,n c s,n,max − . / . (cid:17) − . (cid:16) ( c s,n c s,n,max − . / . (cid:17) +0 . (cid:16) ( c s,n c s,n,max − . / . (cid:17) V Reference open circuit potential U p,ref = 2 . . (cid:16) . − . c s,p c s,p,max (cid:17) + 2 . (cid:16) . − . c s,p c s,p,max (cid:17) − . (cid:16) . − . c s,p c s,p,max (cid:17) +0 . (cid:16) . − . c s,p c s,p,max (cid:17) +0 . (cid:16) ( − c s,p c s,p,max + 0 . / . (cid:17) − . (cid:16) ( c s,p c s,p,max − . / . (cid:17) V Reference open circuit potential ∂U n ∂T n (cid:12)(cid:12)(cid:12)(cid:12) T n = T ∞ = − . . /c s,n,max )exp (cid:16) − . c s,n c s,n,max (cid:17) +(0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) − (0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) − (0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) − (0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) − (0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) − (0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) − (0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − )+(0 . / (0 . c s,n,max ))((cosh (cid:16) ( c s,n c s,n,max − . / . (cid:17) ) − ) VK − Entropic change ∂U p ∂T p (cid:12)(cid:12)(cid:12)(cid:12) T p = T ∞ = 0 . − . /c s,p,max )((1 . / cosh (cid:16) . − . c s,p c s,p,max (cid:17) ) ) +2 . − . /c s,p,max )((cosh (cid:16) . − . c s,p c s,p,max (cid:17) ) − )+0 . . /c s,p,max )((cosh (cid:16) . − . c s,p c s,p,max (cid:17) ) − ) − . . /c s,p,max )((cosh (cid:16) . − . c s,p c s,p,max (cid:17) ) − ) − (0 . / . /c s,p,max )((cosh (cid:16) ( − c s,p c s,p,max + 0 . / . (cid:17) ) − ) − (0 . / . /c s,p,max )((cosh (cid:16) ( c s,p c s,p,max − . / . (cid:17) ) − ) VK − Entropic change D e = (5 . × − ) × exp (cid:0) − . c e × − (cid:1) × exp (cid:16) E D e R g (cid:16) T ∞ − T k (cid:17)(cid:17) m s − Electrolyte diffusivity D s,k = D s,k,typ k ∈ { n, p } m s − Solid diffusivity κ e = (cid:0) . . c e × − − . c e × − ) + 0 . c e × − ) (cid:1) × exp (cid:16) E κ e R g (cid:16) T ∞ − T k (cid:17)(cid:17) Ω − m − Electrolyte conductivity m k = m k,typ exp (cid:16) E m k R g (cid:16) T ∞ − T k (cid:17)(cid:17) k ∈ { n, p } Am − (m mol − ) . Nominal reaction rate
Table 7: Model coefficients taken from Scott Moura’s DFN model (based on DUELFOIL parameters).The coefficients are for a graphite negative electrode, an LiPF in EC:DMC electrolyte, and a LCOpositive electrode.in place of (3.13). The only difference is the dependence of the diffusion coefficient upon c e,0 insteadof c e,k . By asymptotically expanding (3.13) and (B.1) in powers of C e (as defined in Table 1), it can beseen that the two are asymptotically equivalent up to terms of size O ( C ). This means that formally thesame order of error is introduced by using either version. However, through numerical experimentation,we have found that in some situations, particularly at higher C-rates, utilizing (3.13) instead of (B.1)more accurately recovers the electrolyte concentration and therefore other key variables in the model.The other modification is to retain the ‘log’ terms in (3.25), (3.30), (3.31), and (3.32) instead oflinearising as in [18]. Therefore we convert as follows1 c e,0 ( c e, α − c e, β ) → log (cid:18) c e, α c e, β (cid:19) . (B.2)By making asymptotic expansions of these expressions in terms of powers of C e , we can again showthat the two forms are asymptotically equivalent up to terms of size O ( C e ).In Figure 12, we present the electrolyte concentration predicted by the DFN, SPMe (linear) from [18],and the SPMe (nonlinear) employed in this paper using the parameters in [22] and a C-rate of 5. Weobserve that for this parameter set and C-rate the nonlinear version of the SPMe performs significantlybetter at recovering the electrolyte concentration. However, if we examine Table 8, we observe that for37MSE (mV)1C 2.5C 5C 7.5CSPMe (Linear) 0.6898 2.767 5.109 -SPMe (Nonlinear) 0.7080 2.8334 5.6791 12.2611Table 8: RMSE for each version SPMe compared to the DFN at different C-rates. Note that theerror for SPMe (Linear) at 7.5C is omitted as the model breaks down before the discharge is complete.These results are for the parameter set in [22].lower C-rates both versions of the SPMe recover the terminal voltage to a similar degree of accuracy.Therefore, in some circumstances the slightly simpler version of the SPMe as written [18] may be moreappropriate. x [ µ m] c e , k [ m o l. m - ] DFNSPMe (Linear)SPMe (Nonlinear) 0 50 100 150 x [ µ m] − − A b s o l u t ee rr o r [ m o l. m - ]
5C discharge (at 576.0 seconds)
Figure 12: Comparison of electrolyte concentration predicted by the DFN, SPMe (Linear) from [18],and SPMe (Nonlinear) for a 5 C discharge using the parameter set in [22]: (a) electrolyte concentrationprofile; (b) maximum absolute errors in electrolyte concentration at every time in discharge.38 eferences [1] R. Van Noorden. The rechargeable revolution: A better battery.
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