**5.2 Separation on the basis of the time constant**

Our objective is to establish the static value of the activation voltage as a function of current. The problem is that if the current scanning is too slow the variation of the state of charge will corrupt the measure.In such condition we can never reach the static value. Typical results are given in fig 17, compare to charge driven dynamical models as discussed in section 4.

### **5.3 Correction of battery voltage connected to the state of charge**

A first hypothesis is that for slowly varying current the voltage drift is a function of the stored charge Q, computed by summation of the current. 3D plots are made as a function of the couple I, Q, with current steps to identify the relaxation time and asymptotic value as representative of the storage voltage or activation steps. (Fig.18)

By combining models obtained (diffusion phenomena/Storage, activation and input cell) and experimental measurements, we propose a simple and effective model of the battery

s V

**5. Overvoltage model and experimental validation** 

(R≈2.5 10-3 Ω, γ=400F, C0=2.3 104F, c=3.2 104F, r1=2.5 10-3 Ω, c1=Cx=5 105F, Rx=0.01 Ω)

experimental methodology presented is centered on this non linearity topic.

**5.3 Correction of battery voltage connected to the state of charge** 

representative of the storage voltage or activation steps. (Fig.18)

For a non linear system, it is difficult to find a general approach.

Identification of a linear model may be delicate, but there are a lot of classical well trained

For most cases, it is possible to separate steady state non linear set point positioning, then

For battery, the set point should be defined by the state of charge and the operating current. But the fact that when you apply a non zero operating current, the state of charge is no longer fixed. This is an important practical problem, all the more critical as there is a the strong dependence of the activation impedance with respect to the current. The

Our objective is to establish the static value of the activation voltage as a function of current. The problem is that if the current scanning is too slow the variation of the state of charge will corrupt the measure.In such condition we can never reach the static value. Typical results are given in fig 17, compare to charge driven dynamical models as

A first hypothesis is that for slowly varying current the voltage drift is a function of the stored charge Q, computed by summation of the current. 3D plots are made as a function of the couple I, Q, with current steps to identify the relaxation time and asymptotic value as

Fig. 16. Overvoltage model of lead acid battery

**5.2 Separation on the basis of the time constant** 

**5.1 Experimental analysis** 

local small signal linear investigation.

methods for this.

discussed in section 4.

voltage.

Fig. 17. Experimental analysis and simulation of activation phenomena

Fig. 18. Highlight of the diffusion and activation process

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 145

Characterization of dynamical non linear systems is quite complex. Purely phenomenological description may lead to a huge number of parameters for imperfect



For non specialists, the "Warburg impedance" may look a very esoteric electrochemical topic. We introduce an analogy with a subject quite trivial for electronic engineers, capacitive transmission lines. Approximation of such distributed parameter device by lumped RC network has been the subject of a lot of papers (Kuhn et al., 2006 and Mauracher et al., 1997). We propose an efficient approximation with RC cells in geometric progression.

We consider that our analysis constitutes an important contribution to the understanding of battery operation, in particular for electronic engineers. It opens the way for inclusion in efficient control of complex systems, either in the field of power managing or signal

Bard A. (2000). Electrochemical Methods, Fundamental and Applications, 2nd ed., Harris D.,

Bisquert, J.,Compte, A. (2001). Theory of the electrochemical impedance of anomalous diffusion,Journal of Electroanalytical Chemistry, vol. 499, pp. 112-120. Blanke, H., Bohlen, O., Buller S., De Doncker, R.W. Fricke, B., Hammouche, A., Linzen, D.,

hybrid electric vehicles," Journal of Power Sources,vol. 144, pp. 418-425. Coleman, M.; Chi Kwan Lee; Chunbo Zhu; Hurley, W.G.(2007). State-of-Charge

Coupan F., Sadli I., Marie-Joseph I., Primerose A., Clergeot H.(2010). New Battery dynamic

Thele, M., et Sauer ,D.U. (2005), "Impedance measurements on lead-acid batteries for state-of-charge, state-of-health and cranking capability prognosis in electric and

Determination From EMF Voltage Estimation: Using Impedance, Terminal Voltage, and Current for Lead-Acid and Lithium-Ion Batteries, IEEE Transactions on

Model: Application to lead-acid battery, The 2nd International conference on computer and automation Engineering. Electrical and energy systems of IEEE

From our physical analysis, nonlinearities are summarized by two simple elements:

(charge driven model with given relaxation time)

**6.2 Efficient lumped model for distributed parameters systems** 

(Ed), John Wiley & Sons, ISBN 0-471-04372-9.

Industrial electronics, Vol. 54, pp. 2550 - 2557.

volume 5 pp 140-145, Singapore, 26-28 Feb, 2010.


**6.1 Investigation of nonlinearities** 

results.

impedance

**6.3 Possible extensions** 

processing.

**7. References** 

### **5.4 Improved storage voltage estimation**

Instead of using Q as a reference for the storage voltage, we use the voltage given by the optimized RC storage net. Again the plot is made for steps between constant values of the current. We can see that between curves corresponding to two given values of the current, there remains a variation of the activation voltage.

Fig. 19a. General Validation of diffusion/stockage and activation phenomena

### **5.5 Effect of the non linear voltage transformation**

The same points are plotted using the pseudo-voltage after exponential transform. We see that now the plot between two given values of the current, which constitutes a good validation of the non linear model.

Fig. 19b. Reduction of activation voltage variations by linearisation
