A reduced order approach for a battery model is introduced by White and his former Ph.D. student Cai, cf. [CR09], [Cai10]. Cai uses the Newman model which is discretised by finite elements. He generates a reduced order model with the information of a snap-shot to a certain parameter set for a 1C-rate discharging, that means that the battery is completely discharged within one hour. Using this snapshot, he creates a reduced order model via POD and shows that (for the same parameter set) the reduced and the true results for the discharge curve agree for other discharge rates, too, e.g. a 10C-rate, where the battery is discharged within six minutes.
Volkwein and his former Ph.D. student Lass use the same approach but in a more mathematical way: Lass works on the Newman model, too. The equation system as well as the proof for the existence and uniqueness of a solution can be found in [WXZ06]. Lass applies the POD method to obtain a reduced order model which he uses for a (parameter) optimisation. The parameters in this context are generally restricted. So he checks if he has got the “most important” true solutions by considering the residual. In order to understand the results of the parameter estimation he also performs a sensitivity analysis. In contrast to Cai he does not just variate the current and his objective is not the state of charge (SoC) but the terminal voltage. Thus the results are hardly comparable. But Lass shows that the POD method is convenient to generate a reduced order model for the Newman model of the battery.
1.3. Outline
The aim of the present thesis is 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. An important tool for this is an error estimator which estimates the error between the FV and the RB solution.
The thesis is organised as follows.
• After we set the topic of the present thesis into the context of (mathematical) model order reduction and the numerical treatment of battery models in the pre-vious sections, in Chapter 2 we collect some mathematical preliminaries, which are required for the following chapters.
• In Chapter 3 we introduce our considered battery model. The elliptic-parabolic nonlinear partial differential equation system describes the mass and charge trans-port in a lithium-ion battery. We discretise this equation system for block elec-trodes in one dimension via the cell-centred FVM and the backward Euler scheme.
We implement the discretised system in Matlab and validate the numerical solver by comparing the results to the results which are generated by a commercial solver with the same input data.
1.3. Outline
• Chapter 4 is concerned with the model order reduction by employing the RBM to the discretised system. For the basis generation we use the POD-Greedy algorithm and need therefore POD. The nonlinearities are approximated by (D)EIM. For the (POD-)Greedy algorithm we present an error estimator which estimates the error between the RB and FV solution. For linear problems the error estimator is well known. In anticipation of difficulties for an error estimator for the transport equa-tions of the battery we linearise the corresponding equation system. Furthermore we give an outlook what to do with our reduced order model: two measurements of interest are the terminal voltage and the state of charge (SoC). We define both quantities and set the basics for the later parameter estimation and sensitivity analysis.
• In the next two chapters we study the RBM for the elliptic and the parabolic sub-problem separately.
– In Chapter 5 we examine the elliptic equation on the positive electrode with constant concentration and adjusted boundary conditions. This equation sys-tem should represent the charge transport in the positive electrode. We de-velop an error estimator for a certain parameter constellation. Unfortunately this parameter constellation can be easily violated in the battery context.
Therefore we linearise the original nonlinear equation system and develop an error estimator for the linearised problem. Afterwards we compare these two estimators in a numerical test for parameter settings where both are valid.
The RB model is used for parameter estimation and sensitivity analysis with respect to the terminal voltage.
– In Chapter 6 we follow the same steps for the parabolic equation on the posi-tive electrode. We adjust the boundary conditions and this time the potential is fixed. This system should represent the mass transport in the positive elec-trode. We develop an error estimator which is valid for a certain parameter constellation. Since this parameter constellation is generally violated in the battery context, we again linearise the equation system and construct an es-timator which is valid for an arbitrary (battery) parameter set. We compare both estimators numerically. For a parameter set, where solely the estimator for the linearised problem is valid, we run different tests, where the battery parameter set is always the same, but the settings in the Greedy algorithm vary. Since we consider in this chapter a highly nonlinear model we apply (D)EIM.
This time we consider the SoC in our parameter estimation and sensitivity analysis using the RB model.
• In Chapter 7 we use the knowledge of the previous two chapters and directly linearise the coupled system of equations. The error estimator for this problem is valid if a certain condition for the ratio of the step size in time and space holds.
In general this condition can not be ensured in a real battery context. Therefore we linearise the elliptic and the parabolic equation separately and develop an error estimator for both equations separately. We use these estimators generating an RB model for our parameter estimation and sensitivity analysis. There we focus
again on the terminal voltage and vary the diffusion coefficients in the electrodes and in the electrolyte.
• In Chapter 8 we summarise the results and give an outlook to possible future work on this topic.
2. Preliminaries
In this chapter we provide some mathematical tools which we apply in the following chapters. For more details as given below we refer mainly to the publications [HPUU09], [Zei92], [Wer11], [NW06], [Han09] and [Deu04].