4.4. (Discrete) Empirical Interpolation Method
7. RBM Applied to a Coupled System
7.3. Error Estimator for the Single Equations
7.4.3. Weak Greedy with EIM
For the weak Greedy using EIM with 10 magic points the Greedy algorithm stops after the 5th Greedy step because of numerical instabilities. We get Nc =5 basis vectors for the concentration and Nφ=5 for the potential, cf. Figure 7.9. For the construction of the basis we need 4 different FV solutions for the concentration and 5 for the potential.
Completely 7 different FV solutions are needed.
10 20 30 40 50
Figure 7.9.: Numerical test: weak Greedy with EIM: basis vectors for the concentration (left) and for the potential (right).
The decay of the error and its estimators is plotted in Figure 7.10 in the maximum norm and in Figure 7.11 in the Euclidean norm for the concentration as well for the potential.
The error for the potential in the maximum as well as in the Euclidean norm does not decay. The estimated error for the potential increases, decreases, increases and again decreases in both norms. The error for the concentration stagnates in the beginning and then decreases in both norms. For the concentration the error decreases after the third step and the estimated (error est. lin.) after the forth for both norms. These facts let us assume that the achievable accuracy for the Greedy algorithm with EIM is about 10−3 for this setting.
The parameter choices, position and value of the estimated error, in the Greedy algorithm can be found in the Appendix D.2, cf. Table D.30 for the maximum norm and Table D.31 for the Euclidean norm. The parameter choices for the estimator and the error
1 2 3 4 5 10−3
10−2 10−1 100 101
Greedy Step Number
Accuracy in the Maximum Norm - Concentration error
residual error est. lin.
error est. lin. eps.
1 2 3 4 5
10−6 10−4 10−2 100 102 104
Greedy Step Number
Accuracy in the Maximum Norm - Potential error
residual error est. lin.
error est. lin. eps.
Figure 7.10.: Numerical test: weak Greedy with EIM: decay of the error and its estima-tors in the maximum norm for the concentration (left) and for the potential (right).
differ. The estimator and the residual mainly choose the same parameter. The value of the estimator is closer to the error itself than the residual. For the concentration in the maximum norm it is appropriate except for the first Greedy step. The parameters which are used to generate the basis vectors are listed in Appendix D.2, cf. Table D.32.
We tested the weak Greedy algorithm with the usage of EIM with different settings:
changing the maximum number of magic points, as well as the initial parameter set.
For this training set all tested settings yield the same result: the error in the maximum norm does not get smaller than 10−4 for the concentration and not smaller than 10−3 for the potential, respectively.
We did also tests for different training set where we variate the conductivity in the three different parts of the battery. The qualitatively results are the same.
7.5. Parameter Estimation
In this section we use our constructed RB model of the previous section to estimate parameters for a required terminal voltage. In order to avoid an excursion into the field of the multi-objective optimisation, cf. e.g. [Hil01], we do the parameter estimation solely for the terminal voltage. Further the state of charge and the terminal voltage do correlate which follows directly from the formulation of the transport equations (3.1).
Forµ∈Dand tm∈[0, tf],m∈{1, . . . ,Nt}the terminal voltage (4.25) for the discretised problem is given by
Uter(tm;µ)=φN
,m(µ)−φ
1,m(µ)=∶Uterm (µ).
For a required (discrete) terminal voltageUreq∈RNt the (discrete) cost function is given
7.5. Parameter Estimation
Accuracy in the Euclidean Norm - Concentration error
Accuracy in the Euclidean Norm - Potential error
residual error est. lin.
error est. lin. eps.
Figure 7.11.: Numerical test: weak Greedy with EIM: decay of the error and its estima-tors in the Euclidean norm for the concentration (left) and for the potential (right).
The corresponding optimisation problem is given by: for a required terminal voltage Ureq∈RNt solve the following problem
µmin∈Dad
JˆN,ter(µ) subject to µ∈Dad. (7.12) The admissible parameter domain Dad agrees in the following tests with the area of validity of the RB model.
The optimisation problem using the RB model reads the same, but instead of the FV solutions the RB solutions are used, cf. Section 4.8:
µmin∈Dad
JˆN,ter(µ) subject to µ∈Dad. (7.13)
7.5.1. Numerical Tests
We use the RB models which we generated in Section 7.4 via the weak Greedy algorithm without and with EIM and estimate the diffusion coefficients to approximate the required terminal voltage. The results we get with the RB models is compared to the results we get with the FV model. In the following we list different numerical tests: we consider
FVM ROM ROM EIM Table 7.4.: Parameter estimation: test 1
reason for the stop offmincon, required parameter set, initial parameter set, computed parameter set, computational time, number of iterations, number of function evaluations, residual for the three models: FV, RB and the RB using EIM.
the parameter domain of the previous section. In the first test we start with three small diffusion coefficients and the required ones are bigger, cf. Table 7.4. In the second test it is the other way around, cf. Table 7.5. The required voltage is generated by evaluating µreq = (De,areq, De,ereq, Dreqe,c) in the FV model. The lower bound for the parameters is chosen to be µlb = (1.00 ⋅10−10,7.50⋅10−8,3.90 ⋅10−11) and the upper bound µub=(1.00⋅10−8,7.50⋅10−6,3.90⋅10−9).
For the Matlab routine fminconwe use the following settings. We apply the sequential quadratic programming algorithm. If the change of the evaluated function between the last two steps is smaller than 10−10(TolFun) the algorithm stops. The maximum number of optimisation steps should be less than 100 (MaxIter). As tolerance for the smallest step size, we set 10−15 (TolX) and we set for the constraints violation 10−6 (TolCon).
Further, we set a user defined gradient.
In the first numerical test the identified parameters are the same no matter if we use the FV, the RB model or the not so accurate RB model with EIM. All three optimisation processes need the same amount of iterations and function evaluations and stop because the step size for the optimisation process gets smaller than the step size toleranceTolX= 10−15, cf. Table 7.4. The estimated parameters are appropriate since the residuals, i.e.
the cost function evaluated at the estimated parameters are smaller than 10−5 and thus smaller than the error between the RB and the FV solution of the potential.
In the second numerical test we have the same qualitative result for the FV model compared with the RB model where EIM is used. The optimisation using the RB model without EIM computes a different optimal parameter set and needs more function evaluations for that: 10 times more computational time is needed than using the FV model. The residual in all three optimisation procedures, i.e. the cost functions evaluated at the estimated parameters is smaller than 10−4, so smaller than the error between the
7.5. Parameter Estimation Table 7.5.: Parameter estimation: test 2
reason for the stop offmincon, required parameter set, initial parameter set, computed parameter set, computational time, number of iterations, number of function evaluations, residual for the three models: FV, RB and the RB using EIM.
Figure 7.12.: Terminal voltage for the estimated parameters for the different models:
test 1 (left) and test 2 (right), the black (exact) line is covered by the red line (FVM).
FV and the RB solution for the potential. Summarising the results, we can proclaim that even our not so accurate RB model which uses EIM is accurate enough to estimate the optimal terminal voltage in comparison to the FV model. Further, the diffusion coefficients are difficult to identify. It seems that the change of two parameters (De,e, De,a) can influence the terminal voltage like a certain change of one parameter (De,c) can do, cf. Table 7.5 for the results of the FV method compared with the results of the RB model without EIM. For this we refer to the sensitivity analysis on the subsequent section.
Additionally, we should mention some issues of the present optimisation problems: in general we can not guarantee that there exists a unique solution. The parameters we want to estimate, i.e. the diffusion coefficients are in the scale 10−11 to 10−6. Hence they are very small in comparison to the Newton tolerance ǫNewton=10−10 and to the Greedy tolerance ǫGreedy=10−6.
Another issue is that the same estimated parameters with different models create differ-ent terminal voltage curves, cf. Figure 7.12 - even if they differ in the scale of 10−3V. But the required voltage is constructed with the FV model. Further, the absolute difference of the estimated terminal voltage is very small and definitely below the discretisation errorǫdis=10−2.
7.6. Sensitivity Analysis
In this section we do some sensitivity studies to get to know what influence the diffusion coefficients have on the terminal voltage. Regarding the interpretation we face one difficulty: for battery specific parameters the three coefficients are on different scales especially for our choices in Section 7.4. This means we can not change the parameters by a fixed value and have a look if they have quantitatively the same change to the terminal voltage. In [Las14, Chapter 3.2] the parameters are changed in the same way.
Here also the terminal voltage is considered for the nonlinear heat equation.
For the sensitivity equation (2.16) we need the derivative of F1N and F2N with respect to cN⋅,m+1,φN
⋅,m+1,De,c,De,e and De,a, cf. (3.10) and the derivatives of the corresponding RB functions (4.5) respectively. We assume that the diffusion coefficients are constant in each subdomain of the battery.
7.6.1. Numerical Tests
We proceed as in Section 5.6 but now have time dependent sensitivities.
We consider the parameter set range of the previous sections. So the diffusion coefficients areDe,c∈[1.0⋅10−10,1.0⋅10−8],De,e∈[7.5⋅10−8,7.5⋅10−6]andDe,a∈[3.9⋅10−11,3.9⋅10−9], cf. Table 7.2. Our reference parameter set is µref =(De,cref, De,eref, Drefe,a) =(1.0⋅10−9,7.5⋅ 10−7,3.9⋅10−10). Further we have some additional fixed parameter sets where we change only one parameter in comparison to the reference parameter set: µ(1),+=(1.0⋅10−8,7.5⋅ 10−7,3.9⋅10−10),µ(1),−=(1.0⋅10−10,7.5⋅10−7,3.9⋅10−10),µ(2),+=(1.0⋅10−9,7.5⋅10−6,3.9⋅ 10−10), µ(2),−=(1.0⋅10−9,7.5⋅10−8,3.9⋅10−10), µ(3),+ =(1.0⋅10−9,7.5⋅10−7,3.9⋅10−9) and µ(3),− = (1.0⋅10−9,7.5⋅10−7,3.9⋅10−11). We use the same RB model as for the parameter estimation of the previous section, the one which is generated by the weak
7.6. Sensitivity Analysis
Greedy algorithm without using EIM. In Figure 7.13 we plot the terminal voltage for those different (fixed) parameter sets and the sensitivities for the terminal voltage at the reference parameter set in µ1,µ2 and µ3 direction.
−2x 10−3Terminal Voltage for Different Parameters
U[V]
(a) Terminal voltage for six different parameters:
µref= (1.0⋅10−9,7.5⋅10−7,3.9⋅10−10),µ(1),+= (1.0⋅
0.5x 104 Sensitivities for the Terminal Voltage
t[s]
∂∆U/∂µ1
∂∆U/∂µ2
∂∆U/∂µ3
(b) Sensitivities of the terminal voltage in the three parameters direction at the parameter point µref = (1.0⋅10−9,7.5⋅10−7,3.9⋅10−10).
Figure 7.13.: Terminal voltage to different parameters as well as the sensitivity of the three parameters at a reference parameter.
For a detailed examination we consider the sensitivities in different directions. Therefore we divide starting from the reference parameter setµref=(1.0⋅10−9,7.5⋅10−7,3.9⋅10−10) the intervals [1.0⋅10−10,1.0⋅10−8], [7.5⋅10−8,7.5⋅10−6], [3.9 ⋅10−11,3.9⋅10−9] into nine subintervals. With this we compute the sensitivities of µ1, µ2 and µ3 in all three directions. The results are plotted in Figure 7.14 in dependency on µ1, in Figure 7.15 in dependency on µ2 and in Figure 7.16 in dependency on µ3 in all three directions.
7.6.2. Conclusion
If we increase one of the parameters in comparison to the reference parameter set the terminal voltage increases, but it decreases absolutely. If we decrease one of the param-eters it is the other way around, cf. Figure 7.13a. The biggest effect has the change of the first and the second parameter, i.e. changing the diffusion coefficients in the positive electrode and in the electrolyte. But we should keep in mind that all three coefficients work on different scales in our numerical example. Further the increasing/decreasing of the diffusion coefficients influence the terminal voltage in the same way. In Figure 7.13b we see that the change of the diffusion coefficients in the electrodes have the biggest influence on the terminal voltage for our reference parameter set and the diffusion coef-ficient in the electrolyte has the fewest. The diffusion coefcoef-ficient in the electrolyte is so
10−10
Sensitivities of the Terminal Voltage inµ1-direction
t[s] 10
Sensitivities of the Terminal Voltage inµ2-direction
t[s] Sensitivities of the Terminal inµ3-direction
t[s]
Figure 7.14.: Sensitivities of the terminal voltage in µ1-,µ2- and µ3-direction in depen-dency on µ1.
large that it can hardly influence the lithium-ion transport as well as the charge trans-port. The diffusion coefficients in the electrodes are smaller and have an influence on the transports in the battery and thus on the terminal voltage. The normalised eigenvalues of the Fisher information matrix are
λFisher=⎛
So two parameters have much more influence on the terminal voltage than the third one.
In Figures 7.14, 7.15 and 7.16 we see that the change of one parameter does only influence the sensitivity in its own direction. So if we decrease the diffusion coefficient in one of the electrodes or in the electrolyte the sensitivity also decreases and increases absolutely.
7.6. Sensitivity Analysis
10−8
10−6
10−4 0
50 100−3
−2
−1 0 1
x 104
µ2
Sensitivities of the Terminal Voltage inµ1-direction
t[s] 10
−8 10−6
10−4 0
50
−4000100
−3000
−2000
−1000 0 1000
µ2
Sensitivities of the Terminal Voltage inµ2-direction
t[s]
10−8
10−6
10−4 0
50
−20000100
−15000
−10000
−5000 0 5000
µ2
Sensitivities of the Terminal Voltage inµ3-direction
t[s]
Figure 7.15.: Sensitivities of the terminal voltage in µ1-, µ2- and µ3-direction in depen-dency on µ2.
10−12
10−10
10−8 0
50 100−3
−2
−1 0
x 104
µ3
Sensitivities of the Terminal Voltage inµ1-direction
t[s] 10
−12 10−10
10−8 0
50
−100100
−50 0 50
µ3
Sensitivities of the Terminal Voltage inµ2-direction
t[s]
10−12
10−10
10−8 0
50 100−6
−4
−2 0 2
x 104
µ3
Sensitivities of the Terminal Voltage inµ3-direction
t[s]
Figure 7.16.: Sensitivities of the terminal voltage in µ1-,µ2- and µ3-direction in depen-dency on µ3.
8. Conclusion
There exist in general two kinds of solvers, which describe the transport processes of a lithium-ion battery: computationally expensive ones, which represent the physics very well and cheap ones where the representation of the physical and chemical processes is simplified due to the fast computation. For parameter estimations and sensitivity studies both kinds of solvers are not convenient: one is too slow and the other one too inexact.
The aim of the present thesis was to generate a suitable reduced order model via the RBM for the transport equations of the lithium-ion battery, which is applicable for parameter estimations and sensitivity studies. The RB model bases on slow but concerning physical and chemical processes a good approximated solver. To apply this method efficiently we need an error estimator which estimates the error between the RB and the FV solution to a certain parameter.
We executed the following steps in the present thesis.
• For practical reasons we considered the transport equations of the lithium-ion battery in one dimension with block electrodes. We discretised these equations with this certain geometrical structure via the FV method and the backward Euler scheme and implemented it in Matlab. We showed that the results to a certain parameter set of the one-dimensional Matlab solver are the same compared to a three dimensional solver in which the same equations with the same discretisation scheme was implemented.
• After we had introduced the RBM we considered two smaller sub-problems which are related to the full battery model: the elliptic equation and the parabolic equa-tion were studied separately with adjusted boundary condiequa-tions on the positive electrode, representing the charge and mass transport on the positive electrode.
For both problems we developed an a posteriori error estimator which estimates the error between the RB and FV solution to a certain parameter. The bottleneck of these error estimators is that they are solely valid for certain parameter constel-lations which can be infringed by battery specific parameters. In order to obtain a more general valid error estimator we linearised these two equations and developed error estimators to these problems. Numerical tests, where we compared these two estimators (with respect to the maximum norm), showed that the results do not differ too much and in some cases the one to the linearised problem was even more appropriate because it has a better effectivity; this means it is closer to the real error but is still an upper bound. If we consider an estimator to the nonlinear elliptic problem with respect to the Euclidean norm the error is much more over-estimated than the estimator to the linearised problem due to the fact that we have a pre-factor which depends on the discretisation. If we set this pre-factor 1
we would get the same qualitative results as with respect to the maximum norm.
For the parabolic problem we can avoid this pre-factor which depends on the num-ber of discretisation points. We work with another one due to the fact that the concentration is bounded. In our numerical test this pre-factor was close to one.
– For the elliptic problem we considered for the parameter estimation the ter-minal voltage. The results using the RB model were the same as using the FV model. The hypothesis, that two (of four) parameters are not identifiable, was confirmed in the sensitivity analysis.
– For the parabolic problem considering a parameter set, where solely the esti-mator to the linearised model is valid, we did some numerical tests. In those tests we varied the Greedy settings, the remaining data was in every run the same. As a result we can proclaim that in every Greedy step only one POD-mode should be added, further the nonlinearities should be approximated by EIM and not by DEIM. Additionally, the error should be estimated in the maximum norm.
The parameter estimation was done for a required SoC. The results for the RB and FV models agreed in most of our tests, but some differences occurred.
These differences arose, because some parameters seem to influence the SoC in the same way. The sensitivity studies again confirmed this assumption.
• In awareness of these results of the (decoupled) equations on the positive electrode we applied the RBM on the coupled elliptic-parabolic equation system on the complete battery. We linearised both equations and developed an error estimator to this linearised problem. This error estimator has no constraints on the parameter domain, but for the ratio of the stepsize in time and space. This condition is not valid for our standard parameter set. Consequently we needed to linearise the two equations separately. The obtained error estimators have the same structure as in the elliptic (for the potential φ) and in the parabolic (for the concentration c) case which was considered on the positive electrode only. Our numerical tests showed that the error estimators are suitable. A side result was, that, as expected, applying EIM the accuracy of the RB model can not get as accurate as without applying it.
For our parameter estimation we wanted to best approximate a required terminal voltage varying the diffusion coefficients in the electrode and in the electrolyte.
Comparing the results using the FV model our RB model seems to be applicable.
Differences in the estimated terminal voltage occurred because the three diffusion coefficients seem to influence it in the same way.
All error estimators can use an arbitraryp-vector norm which induces a sub-multiplicative matrix norm (lub-norm). Further they are valid in one, two and three dimensions. The conclusion we did is based on numerical results which are only valid for a certain pa-rameter set (range).
Outlook
There are a lot of possible future projects based on the present thesis. E.g. we can apply many RB tools, we mentioned in Chapter 1, but skipped in the present work for practicability reasons: If we want to generate an RB model which is applicable for a wide range of parameters it can make sense to work with dictionaries. To reduce the offline time we can use a Greedy algorithm which bases on an optimisation problem.
There are a lot of possible future projects based on the present thesis. E.g. we can apply many RB tools, we mentioned in Chapter 1, but skipped in the present work for practicability reasons: If we want to generate an RB model which is applicable for a wide range of parameters it can make sense to work with dictionaries. To reduce the offline time we can use a Greedy algorithm which bases on an optimisation problem.