**3.1 Input impedance cell**

126 Sustainable Growth and Applications in Renewable Energy Sources

proposed in terms of equivalent electrical circuits (Bard, 2000). In particular, associated to diffusion phenomenon, the Warburg impedance Zw has been introduced, involving integration with a non integer order. In Laplace notation (where p denotes the equivalent derivation operator) the Warburg impedance has the form: Zw = A p- ½. In a previous communication, we demonstrated that the effective cell capacity reduction described by Peucker's law may be connected to the step response associated to the Warburg impedance

Anyway, some midway solution must obviously be found between underlying fundamental physics and the need of the engineers for a computationally efficient simplified model. In this chapter, we discuss the major processes resulting in a voltage drop that occurs during a redox reaction sitting in storage electrochemical. The phenomena of diffusion/storage and activation are identified as the main factors for the voltage drop in the batteries (Esperilla et al.,2007). These phenomena occur when the battery is subjected to an electric current, which is to say when there is mass transport in electrochemical interface; they are called faradic phenomena. Focusing particularly on transport mechanism of carriers in the battery, we observed strong similarities between electrochemical interfaces and PN junction diodes (Coupan et al., 2010). Based on the approximation of the physics of semiconductor PN

Along these lines, in this chapter, we introduce a dynamical model of the battery, which explains in terms of a simplified equivalent circuit how the total stored charge is distributed along a cascade of individual elements, with increasing availability time delays. This explains why short cycling makes use only of the closer elements in the chain. It opens the way to a wise design of systems combining short delay storage (for instance super-

At steady state (without current), according to the electrical charges of the reactants in the redox reaction, the chemical potential gradient across the interface may be balanced by an electrical potential gradient. This electric field, integrated across the interface, results in the

When a current is applied to the electrochemical cell, the electronic flow in the metal terminals corresponds to an ionic flow, in proportion defined by the redox reaction stoechiometry at the electrolyte interface. Corresponding carriers which are present in the electrolyte can then move either under the effect of an electrical potential gradient (migration) or the effect a concentration gradient (diffusion). Occasionally, electrolyte transport by convection may also be of influence (Linden et al., 2001). This movement of carriers causes a change in battery voltage compared to the steady sate potential, called over-potential. Note that it is a nonlinear function of the current, depending not only on the present value of the current but on its past variations: it is termed a dynamical non linear relationship. The phenomena responsible for this over-potential involve a number of different and complex processes that overlap each other: that is to say, the kinetics of electron transfers, mass transfers, but also ohmic effect and other non-faradic effects. In this

 The diffusion/storage overvoltage is connected to variation of the ionic concentrations in the electrolyte: average value related to the state of charge, and gradient related variation at the interface in presence of current. However this phenomenon always

junction, we propose a physical analysis coupled to experimental investigation.

capacities) and conventional batteries used for long term full range cycling.

equilibrium potential given by the Nernst relationship (Marie-Joseph, 2003).

study, we focus on the phenomena of diffusion/storage and activation.

(Marie-Joseph et al., 2004).

**2. Analysis methodology** 

With a simplified assumption of symmetrical electrochemical impedance for the electrodes (denoted *Z'/2*), we can infer equivalent circuit of fig 1-a, being an inter-plates capacitance, *R* the electrolyte resistance and *2C0* the double layer capacitance of the interface. The corresponding reduced input circuit is given fig 1-b.

Elements of the input cell are easily identified experimentally at small operating currents and high enough frequencies. Due to the activation threshold, impedance *Z'* is quite high at low current, so that the double layer impedance *C0* dominates for frequencies greater than about 0.1 Hz.

Once the elements of the cell are known, current and voltage may easily be corrected for. In the following, we are interested only in the internal electrochemical impedance *Z'*.

Fig. 1. Input impedance cell (simplified symmetric plates model)

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 129

In the electrolyte, as can be seen in figure 2, there is an inversion of the sulfate ions flow along the z axis. More precisely according to the simultaneous equations (2), we obtain the

IS(0)=I and IH (0) = 0 (3a)

 IS (L) = -I and IH (L) = 2I (3b) As it will be seen in section 3.2 the constant current case corresponds to a stationary solution of the dynamical case with *∂2I/∂z2 = 0,* which implies a linear variation of the current between the given limits. The profile of currents *IS(z*) and *IH(z)* is then obtained according to

a. The total current is equal to the sulfate ion current at the negative electrode (see

c. According to b), we will establish that there is a linear relationship between sulfate concentration and density current (trough linear Partial Derivative Equations) d. The Nernst cell voltage may be expressed as a non linear function of the sulfate

As a consequence, for given boundary conditions, from a) and c) we deduce that there exists a relation of linear filtering between the total current *I(t)* and the sulfate anode concentration

According to d), we find that the cell voltage *V(t)* may be directly expressed as a (non linear) logarithmic function of this concentration (sect 3.1.3). We propose a linearization of the problem, by the use of an exponential mapping on *V(t)*: in this way we introduce a "pseudopotential" proportional to the sulfate concentration (sect 3.2.1.3). This pseudo-potential is then related to the current by linear impedance. This impedance may be simplified in terms

boundary conditions at the electrodes:

Fig. 3. Linear model of current IS (z) et IH (z) **Main steps in diffusion phenomena analysis** 

concentration for z=0 (section 3.2.1.3)

equation 3.a)

of a RC network (3.2.2.4).

*ns(0,t)*.

The mains steps in our analysis will be the following:

b. Sulfate ion motion is dominated by diffusion (see next section)

Figure 3:
