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Figure 3. Frequency response of the Padé approximants and transcendental transfer functions of the four variables within the negative electrode

Figure 3 Frequency response of the Padé approximants and transcendental transfer functions of the four variables within the negative electrode

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Figure 1. Schematic diagram of a lithium ion cell embodying the P2D model based on Ref. 9 and transfer function of each variable in each domain based on Ref.
Figure 1. Schematic diagram of a lithium ion cell embodying the P2D model based on Ref. 9 and transfer function of each variable in each domain based on Ref.
and s is the frequency parameter in the Laplace domain. For the molar flux in the negative electrode: For the phase potential difference $, in the negative electrode: linear form of the model. The details of this are given by Lee, Chemistruck and Plett.° The set-point, p*, used by these authors is given by the initial conditions of the cell, defined as p* = {9s,e a= = Uocr(es 0), Cs,e Cs,05 Ce Ce01 J O}, where ds.e = Os — &.. These authors also define a set of transfer functions for the cell, shown in their respective domains in Figure 1, where the dimension- less distance z is related to the spatial variable x via the transfor- mations, z = x/L™®(0 < x < L™S) and z = (L'™ — x)/LPS(L™? + L*? < x < L'"). Subsequently, the “~” notation denotes the variables Ose = = 5,6 — Uocp(Cs,0), and €;,¢ = Cs,e — Cs,9. The transcendental transfer functions are formed by first linearizing the governing equa- tions of the P2D model around the set point p* and then taking Laplace transforms of these linearized equations. The details of these transfer functions are as follows.
and s is the frequency parameter in the Laplace domain. For the molar flux in the negative electrode: For the phase potential difference $, in the negative electrode: linear form of the model. The details of this are given by Lee, Chemistruck and Plett.° The set-point, p*, used by these authors is given by the initial conditions of the cell, defined as p* = {9s,e a= = Uocr(es 0), Cs,e Cs,05 Ce Ce01 J O}, where ds.e = Os — &.. These authors also define a set of transfer functions for the cell, shown in their respective domains in Figure 1, where the dimension- less distance z is related to the spatial variable x via the transfor- mations, z = x/L™®(0 < x < L™S) and z = (L'™ — x)/LPS(L™? + L*? < x < L'"). Subsequently, the “~” notation denotes the variables Ose = = 5,6 — Uocp(Cs,0), and €;,¢ = Cs,e — Cs,9. The transcendental transfer functions are formed by first linearizing the governing equa- tions of the P2D model around the set point p* and then taking Laplace transforms of these linearized equations. The details of these transfer functions are as follows.
ES i Ws es The nonlinear corrections for the internal variables are computed as follows: The advantage of these transfer functions is that they model the P2D variables at any spatial location in the cell domain based only on the applied current/,,,(s). The transfer functions given in Equa- tions 9-17 and Equations Al—A5 (in the Appendix) are complicated and computationally expensive to calculate which, like the P2D model from which they are derived, makes them not amenable to real time ap- plication. The Discrete Time Realization algorithm (DRA) proposed by Lee, Chemistruck and Plett”* can reduce the transcendental transfer functions into an optimized, reduced order, discrete-time, state-space model. In the DRA, Lee et al. first compute the discrete-time pulse response of the transcendental, continuous-time, transfer functions (Equations 9-17 and Equations Al—A5). They then apply the Ho- Kalman algorithm to produce the state-space realization from this discrete-time impulse response. These authors show that the resulting simple, state-space model produced from the DRA can reduce the computation time requirements by more than 5000 times, compared to the P2D model. However, the DRA requires about 11 min on a desktop computer to produce the reduced state-space model from the transcendental transfer functions for a given set of parameters (e.g. Reims 1", Ds, Dz, &e and &,). There have been two major improvements to the classical DRA in terms of reducing the computation overhead, the first by Gopalakrishnan et al.7* who implemented an improved singular value decomposition scheme on a virtual Hankel matrix to Here i, is the k eigenvalue, and (x; ,) the corresponding eigen- function, for C.(x, 5)/Iapp(s) and K is the order of the expansion.° The transfer functions, J,’ es" (5) [Tapp(syand Jp °s" (5) /Tapp(s) as well as those for the electrolyte potential in each domain are given in the Appendix.
ES i Ws es The nonlinear corrections for the internal variables are computed as follows: The advantage of these transfer functions is that they model the P2D variables at any spatial location in the cell domain based only on the applied current/,,,(s). The transfer functions given in Equa- tions 9-17 and Equations Al—A5 (in the Appendix) are complicated and computationally expensive to calculate which, like the P2D model from which they are derived, makes them not amenable to real time ap- plication. The Discrete Time Realization algorithm (DRA) proposed by Lee, Chemistruck and Plett”* can reduce the transcendental transfer functions into an optimized, reduced order, discrete-time, state-space model. In the DRA, Lee et al. first compute the discrete-time pulse response of the transcendental, continuous-time, transfer functions (Equations 9-17 and Equations Al—A5). They then apply the Ho- Kalman algorithm to produce the state-space realization from this discrete-time impulse response. These authors show that the resulting simple, state-space model produced from the DRA can reduce the computation time requirements by more than 5000 times, compared to the P2D model. However, the DRA requires about 11 min on a desktop computer to produce the reduced state-space model from the transcendental transfer functions for a given set of parameters (e.g. Reims 1", Ds, Dz, &e and &,). There have been two major improvements to the classical DRA in terms of reducing the computation overhead, the first by Gopalakrishnan et al.7* who implemented an improved singular value decomposition scheme on a virtual Hankel matrix to Here i, is the k eigenvalue, and (x; ,) the corresponding eigen- function, for C.(x, 5)/Iapp(s) and K is the order of the expansion.° The transfer functions, J,’ es" (5) [Tapp(syand Jp °s" (5) /Tapp(s) as well as those for the electrolyte potential in each domain are given in the Appendix.
Figure 2. A low order Padé approximation model of the linear P2D model.
Figure 2. A low order Padé approximation model of the linear P2D model.
Figure 5. Comparison between the Padé approximation model and the P2D model for a pulse current profile.
Figure 5. Comparison between the Padé approximation model and the P2D model for a pulse current profile.
Figure 6. Comparison between the Padé approximation model and the P2D model for the UDDS applied current profile.
Figure 6. Comparison between the Padé approximation model and the P2D model for the UDDS applied current profile.
Figure 7. Comparison between the Padé approximation model with the full (nonlinear) P2D model for the applied current profile associated with the wind farm application, as depicted in Figure 4a.
Figure 7. Comparison between the Padé approximation model with the full (nonlinear) P2D model for the applied current profile associated with the wind farm application, as depicted in Figure 4a.
Table I. Cell parameters used in the simulation. Table II. Comparison of simulation time of the proposed approximation model to compute all cell variables at all discrete. spatial locations.
Table I. Cell parameters used in the simulation. Table II. Comparison of simulation time of the proposed approximation model to compute all cell variables at all discrete. spatial locations.
to a state of charge (SOC) of 93.2%. This SOC will continue to increase under the constant voltage regime, but will do so very slowly. From Figure 8b, however, we can see that at the point of switching the potential difference o, — , atx = Lis greater than OV. Therefore, under the method proposed by Chaturvedi et al. we would continue to charge at a constant current until d, — >. = 0, yielding a 6.8% extra SOC. The additional time required for gaining 6.8% extra SOC in this constant current regime is 243s, which is much faster than the time required to gain the same amount of charge if we had switched to a constant voltage regime (which would take on the order of hours). The pivotal point here is that the values of 6, — 6, at x = L"®predicted by our Padé approximation model compare very favorably to those predicted by the P2D model but have the advantage of being easily computed. An accurate and fast prediction of the 6; — , values at x = L™®, which cannot be easily measured empirically, means that our Padé approximation method could be useful for control purposes in fast charging. and the higher order approximations given in Figure 9 for our three previous applied current profiles (i.e. the pulse current profile, UDDS, grid-scale application). Table IV shows the root mean square value of the voltage error over the length of each current profile. From these figures we observe that the accuracy of the higher order approxima- tion model increases by some 5.4% and 7.5% for the pulse current profile and the UDDS profile, respectively. In the pulse current pro- files and the grid-scale profile, there are negligibly small differences between the low and higher order approximations models. However, as can also be seen in Table III, the computational time requirement of the higher order approximation model increases by approximately 20% for the pulse and UDDS current profiles and 7.7% for the grid current profile. We see that the higher order approximation model does not result in significant improvements for any of these applied current profiles and therefore, increasing the approximation order is unnecessary.
to a state of charge (SOC) of 93.2%. This SOC will continue to increase under the constant voltage regime, but will do so very slowly. From Figure 8b, however, we can see that at the point of switching the potential difference o, — , atx = Lis greater than OV. Therefore, under the method proposed by Chaturvedi et al. we would continue to charge at a constant current until d, — >. = 0, yielding a 6.8% extra SOC. The additional time required for gaining 6.8% extra SOC in this constant current regime is 243s, which is much faster than the time required to gain the same amount of charge if we had switched to a constant voltage regime (which would take on the order of hours). The pivotal point here is that the values of 6, — 6, at x = L"®predicted by our Padé approximation model compare very favorably to those predicted by the P2D model but have the advantage of being easily computed. An accurate and fast prediction of the 6; — , values at x = L™®, which cannot be easily measured empirically, means that our Padé approximation method could be useful for control purposes in fast charging. and the higher order approximations given in Figure 9 for our three previous applied current profiles (i.e. the pulse current profile, UDDS, grid-scale application). Table IV shows the root mean square value of the voltage error over the length of each current profile. From these figures we observe that the accuracy of the higher order approxima- tion model increases by some 5.4% and 7.5% for the pulse current profile and the UDDS profile, respectively. In the pulse current pro- files and the grid-scale profile, there are negligibly small differences between the low and higher order approximations models. However, as can also be seen in Table III, the computational time requirement of the higher order approximation model increases by approximately 20% for the pulse and UDDS current profiles and 7.7% for the grid current profile. We see that the higher order approximation model does not result in significant improvements for any of these applied current profiles and therefore, increasing the approximation order is unnecessary.
Table III. Comparison of simulation time of the proposed approximation model to compute only cell voltage.
Table III. Comparison of simulation time of the proposed approximation model to compute only cell voltage.
Figure 8. Comparison between two charging methods (a). Cell voltage during charging. (b). The over potential g; — , at x = L™®
Figure 8. Comparison between two charging methods (a). Cell voltage during charging. (b). The over potential g; — , at x = L™®
model predictions match very closely with those obtained from the full (nonlinear) P2D model but in a fraction of the computational time. Importantly, variables can be evaluated at specific discrete lo- cations within the domain, without the need to compute all values of the variable at all discrete locations, as is the case with the FVM im- plementation of the P2D model. This can result in further significant time savings and enhance the flexibility of the model for a variety of applications. within a one-dimensional representation of the battery domain. Cor- rections for nonlinear behavior are also incorporated into the reduced model. The proposed model substantially reduces the complexity of the P2D model whilst maintaining the functionality of predicting all of the variables of that model. The form of the Padé approximations, once resolved, only require relatively simple rational functions to be evaluated, which leads to significant computational savings when solving the reduced model. Our results demonstrate that the reduced Figure 9. Higher order Padé approximation model of the linear P2D model.
model predictions match very closely with those obtained from the full (nonlinear) P2D model but in a fraction of the computational time. Importantly, variables can be evaluated at specific discrete lo- cations within the domain, without the need to compute all values of the variable at all discrete locations, as is the case with the FVM im- plementation of the P2D model. This can result in further significant time savings and enhance the flexibility of the model for a variety of applications. within a one-dimensional representation of the battery domain. Cor- rections for nonlinear behavior are also incorporated into the reduced model. The proposed model substantially reduces the complexity of the P2D model whilst maintaining the functionality of predicting all of the variables of that model. The form of the Padé approximations, once resolved, only require relatively simple rational functions to be evaluated, which leads to significant computational savings when solving the reduced model. Our results demonstrate that the reduced Figure 9. Higher order Padé approximation model of the linear P2D model.
Table IV. Comparison of RMS error of cell voltage and the computational workload between low order and higher order Padé approximation models.
Table IV. Comparison of RMS error of cell voltage and the computational workload between low order and higher order Padé approximation models.
Figure 10. Cell voltage error obtained from low order and higher order Padé approximation models when compared to the predictions of the P2D model.
Figure 10. Cell voltage error obtained from low order and higher order Padé approximation models when compared to the predictions of the P2D model.
The transfer functions used to compute the electrolyte concentration is given as,
The transfer functions used to compute the electrolyte concentration is given as,
Ngoc Tham Tran © https://orcid.org/0000-0001-9994-3620 Troy Farrell © https://orcid.org/0000-0002-6629-4174 For the electrolyte potential in the positive electrode (L™ + LSP < x < L'):
Ngoc Tham Tran © https://orcid.org/0000-0001-9994-3620 Troy Farrell © https://orcid.org/0000-0002-6629-4174 For the electrolyte potential in the positive electrode (L™ + LSP < x < L'):
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