**1. Introduction**

For a standard interconnected electrical power network, the problem of optimal management of production arises from randomness of users demand. When using renewable energies, an additional critical problem is that the resource itself is random. The difficulty is still more pregnant when dealing with small isolated production networks, in locations where photovoltaic systems or wind generators should be a promising solution. To resolve the difficulties induced by intermittent production or consumption, these systems must make a consistent use of the energy storage. For example, in the case of an individual photovoltaic system, storage is essential to the scale of at least 24h, in order to overcome the daily fluctuations.

Among the various methods used to store electrical energy, electrochemical batteries constitute the most readily available, with good performance and a reasonable cost (Riffonneau et al.,2008). Renewable Energies are concerned by stationary storage, for which lead acid batteries are a good choice. Despite decades of use and its apparent simplicity, the battery maintains a complex and poorly understood dynamical behavior. Moreover, possible degradation of the battery is largely related to poor control of periods of deep discharge or full load with gassing. For efficient use of this device, a detailed knowledge of operation, and thus a good electrochemical modeling, is essential. Otherwise, it could constitute the most fragile element in a photovoltaic or wind systems because of premature aging resulting in a loss of capacity or a failure risk (Garche et al.,1997).

A lot has been done in the domain of batteries modeling from two opposite ways.

On the one hand, a purely phenomenological approach has been developed by engineers. In particular, very valuable tests are commonly performed using battery cycling with constant charge and discharge currents. In particular, there appears a reduction of the effective capacity when the cycling current increases (Peucker's law (Manwell Jams, 2003)). These results may have direct application for charge monitoring in systems with alternate charging and discharging sequences (for instance traction vehicles); unfortunately, they do not apply to wind turbines or photovoltaic applications subject to random electrical current variations.

On the other hand, extensive physical studies have been made by electrochemists concerning the physics of electrochemical cells. Descriptions of the cell behavior have been

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 127

 The activation overvoltage may be related to injection of sulfate ions in the oxide film at the surface of electrodes. These constitute solid electrolytes no longer governed by Boltzmann but by Fermi-Dirac statistics. There are strong similarities which the injection of minority carriers in PN junctions. In the literature, this phenomenon is usually described by the semi empirical Butler-Volmer relation. We propose a dynamical model drawn from the charge driven model of PN diodes, with given

 Full description of the battery includes conventional circuit modeling of non faradic effects. This is taken into account by an RC "input cell" including plates electrostatic capacitance, Ohmic resistance and the plates double layer capacitances, with typical time constants between 1s and 100s. High frequency models may include inductive

With a simplified assumption of symmetrical electrochemical impedance for the electrodes

*R* the electrolyte resistance and *2C0* the double layer capacitance of the interface. The

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

Once the elements of the cell are known, current and voltage may easily be corrected for. In

1.a 1.b

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

the following, we are interested only in the internal electrochemical impedance *Z'*.

being an inter-plates capacitance,

response times ranging from 103 to 105s.

effects (Blanke et al, 2005).

**3.1 Input impedance cell** 

about 0.1 Hz.

relaxation time (typically in the order of some 102 s).

**3. Input cell and diffusion voltage for lead acid batteries** 

(denoted *Z'/2*), we can infer equivalent circuit of fig 1-a,

corresponding reduced input circuit is given fig 1-b.

appears in agreement with the Nernst equation. We propose a linearisation by inversion of this relation and a dynamical model drawing from the analogy of diffusion equation with a capacitive transmission line. Diffusion phenomena predominate for

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 (Marie-Joseph et al., 2004).

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

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

### **2. Analysis methodology**

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

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

 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 appears in agreement with the Nernst equation. We propose a linearisation by inversion of this relation and a dynamical model drawing from the analogy of diffusion equation with a capacitive transmission line. Diffusion phenomena predominate for response times ranging from 103 to 105s.

