**3.2 Diffusion overvoltage**

For the sake of clarity, a good part of the analysis will be carried in the stationary case, corresponding to constant current. We use a one dimensional battery model, the variable being the abscissa *z* between the negative (*z=0*) and the positive plate (*z=L*). Results are then extended with a constant cross section S. to the general dynamical case, including time dependency.

## **3.2.1 Constant current analysis**

### **3.2.1.1 General presentation**

During the discharge of the lead acid battery, sulfate ions are "swallowed" by both electrodes according to chemical reactions:

Positive electrode:

$$PbO\_2 + SO\_4^{2-} + 4H^+ + 2e^- \xrightarrow{Discharge} PbSO\_4 + 2H\_2O \tag{1}$$

Negative electrode:

$$Pb + SO\_4^{2-} \longrightarrow \overset{D^{is}}{\longrightarrow} \overset{\text{arc}}{\longrightarrow} Pb \text{ SO }\_4 + \text{ } \text{2 } e^-$$

Figure 2 illustrates the transport of ions along axis Oz associated with the two half-reactions at the electrodes (inter-electrode distance *L*):

Fig. 2. Battery operation: case of discharge

So, two types of ions are responsible for current transport through the electrolyte. Those are sulfate ions (subscript S) and hydrogen ions (subscript H). In terms of currents:

$$I(z) = I\_{H}(z) + I\_{S}(z) \tag{2a}$$

Let S be the section area between the plates (constant for à one dimensional model). The same relation holds in terms of current densities:

$$f(z) = \frac{I}{S} = f\_H(z) + f\_S(z) \tag{2b}$$

For the sake of clarity, a good part of the analysis will be carried in the stationary case, corresponding to constant current. We use a one dimensional battery model, the variable being the abscissa *z* between the negative (*z=0*) and the positive plate (*z=L*). Results are then extended with a constant cross section S. to the general dynamical case, including time dependency.

During the discharge of the lead acid battery, sulfate ions are "swallowed" by both

2 4 2*H SO*

So, two types of ions are responsible for current transport through the electrolyte. Those are

 () () () *H S Iz I z I z* (2a) Let S be the section area between the plates (constant for à one dimensional model). The

> () () () *H S <sup>I</sup> Jz J z J z <sup>S</sup>*

sulfate ions (subscript S) and hydrogen ions (subscript H). In terms of currents:

2 4 4 2 4 2 2 *Disch e PbO SO H e PbSO H O* (1)

2 <sup>4</sup> *S O*

(2b)

4*H*

2 arg

2 arg 4 4 2 *D isch e Pb SO PbSO e* Figure 2 illustrates the transport of ions along axis Oz associated with the two half-reactions

> 2 <sup>4</sup> *S O*

**3.2 Diffusion overvoltage** 

**3.2.1 Constant current analysis 3.2.1.1 General presentation** 

Positive electrode:

Negative electrode:

electrodes according to chemical reactions:

at the electrodes (inter-electrode distance *L*):

Fig. 2. Battery operation: case of discharge

same relation holds in terms of current densities:

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 boundary conditions at the electrodes:

$$\mathbf{I\_{S}(0)=I \quad \text{and} \quad \mathbf{I\_{H}(0)=0}}\tag{3a}$$

$$\mathbf{I}\_{\mathbb{S}}\begin{pmatrix}\mathbf{L}\end{pmatrix} = \mathbf{-I} \quad \text{and} \quad \mathbf{I}\_{\mathbb{H}}\begin{pmatrix}\mathbf{L}\end{pmatrix} = \mathbf{2}\mathbf{I} \tag{3b}$$

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 Figure 3:

Fig. 3. Linear model of current IS (z) et IH (z)

### **Main steps in diffusion phenomena analysis**

The mains steps in our analysis will be the following:


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 *ns(0,t)*.

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 of a RC network (3.2.2.4).

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 131

Let us point out this symmetry property which will generalize for the dynamical case.

*n nn V A*

According to the electrochemical model defined above, while applying Nernst's equation (Marie-Joseph, 2003); we obtain the expression of voltage as a function of the limit

Concerning PbSO4 activity, it is equal to one, unless we are very close to full charge

Let n0 be a reference sulfate concentration and E0 the corresponding Nernst voltage, then the

0

*<sup>n</sup> VEV*

3

*kT <sup>V</sup> e*

*L*

  <sup>2</sup> <sup>3</sup> ln (0) *<sup>s</sup> kT e*

*<sup>s</sup> <sup>L</sup>*

0 (0) ln

*n*

4 4

2 2 (0) ( ) ln ln *s s <sup>H</sup> PbSO PbSO kT kT L L e e a a L*

<sup>4</sup> (0) () ()

. (8a)

*VA n* (8b)

(8c)

Fig. 4. Concentration profile (Stationary 1D Model)

**3.2.1.3 Voltage and concentration** 

 *.*

In this relation, we may use the fact that:

(this will not be considered here).

relation may be written in the form:

concentrations in the form:

Following the boundary condition (3a and 3b) we find:

For densities, the **symmetry** property: ns(L-z) = ns(z)

1

Due to the symmetry of concentrations, ns(L)=ns(0).

According to the neutrality condition (section 3.2.1.2), nH = 2nS

In such conditions, the expression of battery voltage may be set in the form:

For currents, the **anti symmetry** property: *IS(L-z) = - IS(z)*

#### **3.2.1.2 Diffusion fields and currents**

In the electrolyte, the carriers are transported under the influence of an electric field E and the diffusion field ξ, connected to the concentration gradient. For the two types of carrier (k: Boltzmann constant; e: charge of one electron):

$$\begin{aligned} \xi\_S &= -\frac{kT}{q\_S} \frac{1}{n\_S} \frac{dn\_S}{dz} \\ \xi\_H &= -\frac{kT}{q\_H} \frac{1}{n\_H} \frac{dn\_H}{dz} \end{aligned} \tag{4}$$

Note that, from the relation: qS = -2qH = -2e, and the neutrality condition, we get the relation between concentrations: nH = 2 nS . By substitution in (4), we derive the corresponding relation between the diffusion fields:

$$
\xi\_\mathsf{H}/\xi\_\mathsf{S} = \mathsf{q}\_\mathsf{S}/\mathsf{q}\_\mathsf{H} = \mathsf{:}\mathsf{2}\tag{5}
$$

The corresponding expression of the currents, for each type of carrier, is then given by the relation:

$$\begin{aligned} \mathbf{J}\_{\rm s} &= \mu\_{\rm s} \eta\_{\rm s} n\_{\rm s} \left( \mathbf{E} + \boldsymbol{\tilde{\tau}\_{\rm s}} \right) \\ \mathbf{J}\_{\rm H} &= \mu\_{\rm H} \eta\_{\rm H} n\_{\rm H} \left( \mathbf{E} + \boldsymbol{\tilde{\tau}\_{\rm H}} \right) \end{aligned} \tag{6}$$

In this relation, *JH* and *JS* have a similar magnitude (see fig 3). The mobility of hydrogen ions being much higher than the sulfate ions, this implies that *E + ξH* is very small, so that: *E ≈ - ξH*. From this result and (5), we find that the current densities may be expressed in terms of the diffusion field ξS alone:

$$\mathbf{E} \approx \mathbf{-\xi\_H} = \mathbf{2\xi\_S} \mathbf{E} + \xi\_S \approx \mathbf{3\xi\_S}$$

Whence

$$\mathbf{J}\_{\mathbb{B}} = \mathfrak{ps} \,\mathrm{gs} \,\mathrm{res}\,(\mathbf{3} \,\mathrm{\mathcal{G}}) \tag{7a}$$

Or, according to (4):

$$J\_s \text{ (z)} = 3 \,\mu\_s kT \left( \frac{dn\_s}{dz} \right) \tag{7b}$$

And from (2):

$$J = J\_s \text{ (0)} = 3 \,\mu\_s kT \left(\frac{d\mathbf{n}\_s}{dz}\right)\_0 \tag{7c}$$

This establishes the step c) of our diffusion analysis exposed in section 3.2.1.1 We may introduce in (7b) the linear profile of the current, valid in the stationary case. We then derive a parabolic symmetric profile of the concentration of sulfate ions (Fig. 4), with *nS(0) = nS(L).*

Fig. 4. Concentration profile (Stationary 1D Model)

Let us point out this symmetry property which will generalize for the dynamical case. Following the boundary condition (3a and 3b) we find:


#### **3.2.1.3 Voltage and concentration**

130 Sustainable Growth and Applications in Renewable Energy Sources

In the electrolyte, the carriers are transported under the influence of an electric field E and the diffusion field ξ, connected to the concentration gradient. For the two types of

<sup>1</sup> -

*S*

*H*

*S*

*H*

  (4)

ξH/ξS = qS/qH = -2 (5)

(6)

(7b)

(7c)

<sup>1</sup> -

*S S*

*kT dn q n dz kT dn q n dz*

*H H*

Note that, from the relation: qS = -2qH = -2e, and the neutrality condition, we get the relation between concentrations: nH = 2 nS . By substitution in (4), we derive the corresponding

The corresponding expression of the currents, for each type of carrier, is then given by the

*S SSS S H HHH H*

In this relation, *JH* and *JS* have a similar magnitude (see fig 3). The mobility of hydrogen ions being much higher than the sulfate ions, this implies that *E + ξH* is very small, so that: *E ≈ - ξH*. From this result and (5), we find that the current densities may be expressed in

E ≈ - ξH = 2 ξS E+ ξ<sup>S</sup> ≈ 3 ξ<sup>S</sup>

JS = μS qS nS (3 ξS) (7a)

() 3 *<sup>S</sup> S S dn J z kT dz* 

(0) 3 *<sup>S</sup> S S dn J J kT dz* 

We may introduce in (7b) the linear profile of the current, valid in the stationary case. We then derive a parabolic symmetric profile of the concentration of sulfate ions (Fig. 4), with

This establishes the step c) of our diffusion analysis exposed in section 3.2.1.1

0

*J qn E J qn E* 

**3.2.1.2 Diffusion fields and currents** 

relation between the diffusion fields:

terms of the diffusion field ξS alone:

relation:

Whence

Or, according to (4):

And from (2):

*nS(0) = nS(L).*

carrier (k: Boltzmann constant; e: charge of one electron):

According to the electrochemical model defined above, while applying Nernst's equation (Marie-Joseph, 2003); we obtain the expression of voltage as a function of the limit concentrations in the form:

$$\dots V = A\_1 + \frac{kT}{2e} \text{Im} \left( \frac{n\_s(0)}{a\_{PbSO\_4}(0)} \right) + \frac{kT}{2e} \text{Im} \left( \frac{n\_s(L)n\_s(L)^4}{a\_{PbSO\_4}(L)} \right). \tag{8a}$$

In this relation, we may use the fact that:


In such conditions, the expression of battery voltage may be set in the form:

$$V = A\_2 + \frac{\Im kT}{e} \ln \left( n\_s(0) \right) \tag{8b}$$

Let n0 be a reference sulfate concentration and E0 the corresponding Nernst voltage, then the relation may be written in the form:

$$\begin{cases} V = E\_0 + V\_L \ln\left(\frac{n\_s(0)}{n\_0}\right) \\ V\_L = \frac{3kT}{e} \end{cases} \tag{8c}$$

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 133

3kT Q -θ<sup>I</sup> V = E + ln q Q

Relation (12) may be written in terms on an RC model valid only for constant current charge

 0 *S S L D D*

*Q- I <sup>Q</sup> V V -R I Q C* 

In the general case, current densities and concentrations densities depend both on z and t.

3 *n J s s z kT s*

<sup>s</sup> Js <sup>n</sup> = - = 2e zt t

These two coupled Partial Derivative Equations define the diffusion process (Lowney et al.,

 

0

<sup>θ</sup><sup>=</sup> <sup>18</sup> *<sup>s</sup> L kT*

 

or discharge in the form:

With CD = Q0/VL and RD = θ/CD

V

**3.2.2 Dynamical model for time varying current 3.2.2.1 General diffusion equations (one dimension)** 

We may add the charge conservation equation:

The driving condition is given by relation:

1980).

Equation (7) may be written in term of partial derivative:

Fig. 6. RC equivalent circuit for constant current (after linearization)

And the bounding condition resulting of the current anti symmetry:

2

S

0

(13)

(14)

(15)

( ) (0, ) ( ) *I t J t Jt <sup>s</sup> <sup>S</sup>* (16)

*J Lt J t* ( , ) (0, ) *s s* (17)

(12)

This result corresponds to point *d)* in introduction (3.2.1.1).

### **3.2.1.4 "Linearised" pseudo-voltage using an exponential transformation**

We suggest to introduce a "pseudo-voltage" which is à linear function of the concentration, and which aims to the voltage *V* when it is close to the reference voltage *E0*, according to figure 5:

$$\tilde{\mathbf{V}} = \mathbf{E}\_0 + \mathbf{V}\_\mathbf{l} \left( \frac{n\_\text{S} - n\_\text{o}}{n\_\text{o}} \right) = \mathbf{E}\_\mathbf{o} - \mathbf{V}\_\mathbf{l} + \mathbf{V}\_\mathbf{l} \left( \frac{n\_\text{S}}{n\_\text{o}} \right) \tag{9}$$

Fig. 5. Linearised Pseudo-voltage

The pseudo voltage may then be obtained by an exponential transformation of the original voltage according to the expression:

$$\tilde{\mathbf{V}} = \mathbf{E}\_0 + \mathbf{V}\_\mathrm{l} \left( \exp\left(\frac{V - \mathbf{E}\_0}{\mathbf{V}\_\mathrm{l}}\right) - \mathbf{1} \right) = \mathbf{E}\_0 \cdot \mathbf{V}\_\mathrm{l} + \mathbf{V}\_\mathrm{l} \left( \exp\left(\frac{V - \mathbf{E}\_0}{\mathbf{V}\_\mathrm{l}}\right) \right) \tag{10}$$

#### **3.2.1.5 Constant current equivalent circuit**

According to figure 4, the limit concentrations (for z=0 and z=L) are easily expressed, and may be related to the total stored charge QS and the internal current I:

$$\begin{cases} \mathbf{n}\_s(\mathbf{0}) = \mathbf{n}\_s(L) = \langle \mathbf{n}\_s \rangle - \frac{L}{6} \Big( \frac{d\mathbf{n}\_s}{dz} \Big)\_0 \\\\ \mathbf{Q}\_s = SL\langle \mathbf{n}\_s \rangle \\\\ I = \mathbf{I}\_s(\mathbf{0}) = \mathbf{S} \mathbf{J}\_s(\mathbf{0}) = \mathbf{3} \mathbf{S} \mu\_s kT \Big( \frac{d\mathbf{n}\_s}{dz} \Big)\_0 \end{cases} \tag{11}$$

According to equations (8) and (11), the voltage may be expressed in terms of the stored charge Qs and the internal current I according to:

 

Relation (12) may be written in terms on an RC model valid only for constant current charge or discharge in the form:

$$\tilde{V} = V\_L \frac{\left(Q\_s \text{ - } \theta I\right)}{Q\_0} = \frac{Q\_s}{C\_D} \text{ - } R\_D I \tag{13}$$

With CD = Q0/VL and RD = θ/CD

132 Sustainable Growth and Applications in Renewable Energy Sources

We suggest to introduce a "pseudo-voltage" which is à linear function of the concentration, and which aims to the voltage *V* when it is close to the reference voltage *E0*, according to

> 0 0 L 0L L

The pseudo voltage may then be obtained by an exponential transformation of the original

0 0 - 0 L 0L L

According to figure 4, the limit concentrations (for z=0 and z=L) are easily expressed, and

<sup>s</sup> ss s

dn n0 n n

 

*I S S kT*

<sup>s</sup> s s

*<sup>L</sup> <sup>L</sup>*

dn I (0) J (0) 3 *<sup>s</sup>*

According to equations (8) and (11), the voltage may be expressed in terms of the stored

may be related to the total stored charge QS and the internal current I:

s s

*SL*

Q n

charge Qs and the internal current I according to:

E E V = E + V exp 1 E V V exp V V

L L

6

0

*dz*

*dz*

0

(11)

 *V V* (10)

V=E V E V V + -

0 0

(9)

*n n s s n n n* 

This result corresponds to point *d)* in introduction (3.2.1.1).

figure 5:

*V*

Fig. 5. Linearised Pseudo-voltage

voltage according to the expression:

**3.2.1.5 Constant current equivalent circuit** 

*V*

**3.2.1.4 "Linearised" pseudo-voltage using an exponential transformation** 

Fig. 6. RC equivalent circuit for constant current (after linearization)

### **3.2.2 Dynamical model for time varying current**

#### **3.2.2.1 General diffusion equations (one dimension)**

In the general case, current densities and concentrations densities depend both on z and t. Equation (7) may be written in term of partial derivative:

$$\frac{\partial n\_{\rm S}}{\partial z} = \frac{J\_{\rm S}}{\Re \mu\_{\rm S} kT} \tag{14}$$

We may add the charge conservation equation:

$$\frac{\partial \mathbf{J}\_S}{\partial \mathbf{z}} = \mathbf{\hat{z}} \frac{\partial \rho}{\partial \mathbf{t}} = \mathbf{2e} \frac{\partial \mathbf{n}\_s}{\partial \mathbf{t}} \tag{15}$$

These two coupled Partial Derivative Equations define the diffusion process (Lowney et al., 1980).

The driving condition is given by relation:

$$J\_S(0, t) = J(t) = \frac{I(t)}{S} \tag{16}$$

And the bounding condition resulting of the current anti symmetry:

$$J\_S(L, t) = -J\_S(0, t) \tag{17}$$

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 135

For linear systems, we look for solutions in the form A exp(pt) for inputs, or A(z) exp(pt)

( , ) ( )exp ( , ) ( )exp *I z t I z pt V z t V z pt*

> *dV s I s dz*

 

2

*dI*

*s pV s dz*

<sup>2</sup> *p p <sup>L</sup>* 

2

2

 

() - <sup>2</sup> *<sup>L</sup> I z Ish z <sup>s</sup>* 

<sup>1</sup> ( ) -

*Vs Z p <sup>I</sup> <sup>L</sup> <sup>s</sup> th*

<sup>1</sup> ( )

*Z p Cp*

*<sup>L</sup> <sup>C</sup>* 

 

2

 

*<sup>L</sup> V z Ich z <sup>s</sup>* 

We then get the Laplace impedance at the input of the equivalent circuit:

 

 

(19)

(21)

(22)

(23)

*z* . If we impose Is(L/2)=0, then

(24)

(25)

(20)

**3.2.2.3 Equivalent impedance solution** 

Let:

Whence

Whence:

if 1

2 2 d I

s = I s dz *p*

along the line, p being any complex constant.

Then we obtain the simplified set of equations

Let = /L2. Then - and be the solutions of :

Then solution for Is(z) is a linear combination of exp

(0) ( ) (0)

2 2

*L L L th* 

#### **3.2.2.2 General electrical capacitive line analogy**

In the diffusion equations (14) et (15), making use of relation (9), the sulfate ion density *nS* may be expressed in terms of the pseudo potential *V* , and the current densities *Js* may be replaced by currents *Is = S Js*. We then obtain a couple of joint partial derivative equations between the pseudo voltage *V (z,t)* and the sulfate current *Is(z,t)* :

$$\begin{cases} \frac{\partial \tilde{\mathbf{V}}\_{\rm S}}{\partial \mathbf{z}} = \frac{\mathbf{V}\_{\rm L}}{\mathbf{n}\_{0}} \frac{\partial \mathbf{n}\_{\rm S}}{\partial \mathbf{z}} = \frac{\mathbf{V}\_{\rm L}}{\mathbf{n}\_{0} \mu\_{\rm S} kT} \left(\frac{\mathbf{I}\_{\rm S}}{S}\right) \\\\ \frac{\partial \mathbf{I}\_{\rm S}}{\partial \mathbf{z}} = \mathbf{S} \frac{\partial \mathbf{I}\_{\rm S}}{\partial \mathbf{z}} = 2eS \left(\frac{\mathbf{n}\_{0}}{\mathbf{V}\_{\rm L}} \frac{\partial \tilde{\mathbf{V}}\_{\rm S}}{\partial t}\right) \end{cases} \tag{18a}$$

These are the equations of a capacitive transmission line with linear resistance and linear capacity as defined below. (Bisquert et al., (2001).

$$\begin{aligned} \frac{\partial V\_S}{\partial z} &= \rho I\_S & \rho &= \frac{1}{\mu\_S eSn\_0} \\ \frac{\partial I\_S}{\partial z} &= \gamma \frac{\partial V\_S}{\partial t} & \gamma &= \frac{2en\_0 S}{V\_L} \end{aligned} \tag{18b}$$

Taking in account the symmetry of the concentrations, we obtain an equivalent circuit consisting in a length L section of transmission line, driven on its ends with symmetric voltages. The current is then 0 in the symmetry plane at L/2. The input current is the same as for a L/2 section with open circuit at the end.

(Open circuit capacitive transmission line of length L/2) Fig. 7. Equivalent electrical circuit for the pseudo-voltage

### **3.2.2.3 Equivalent impedance solution**

134 Sustainable Growth and Applications in Renewable Energy Sources

In the diffusion equations (14) et (15), making use of relation (9), the sulfate ion density *nS* may be expressed in terms of the pseudo potential *V* , and the current densities *Js* may be replaced by currents *Is = S Js*. We then obtain a couple of joint partial derivative equations

> S S L L 0 0

znzn <sup>μ</sup><sup>s</sup>

*S eS*

V n V V I

 

<sup>s</sup> J nV <sup>2</sup> zz V

These are the equations of a capacitive transmission line with linear resistance and linear

*s z eSn*

*s s en S zt V*

Taking in account the symmetry of the concentrations, we obtain an equivalent circuit consisting in a length L section of transmission line, driven on its ends with symmetric voltages. The current is then 0 in the symmetry plane at L/2. The input current is the same

S 0S L

1

 

2

  s

0 0

*L*

*s*

(18a)

(18b)

*kT S*

*t*

**3.2.2.2 General electrical capacitive line analogy**

capacity as defined below. (Bisquert et al., (2001).

as for a L/2 section with open circuit at the end.

(Open circuit capacitive transmission line of length L/2)

Fig. 7. Equivalent electrical circuit for the pseudo-voltage

between the pseudo voltage *V (z,t)* and the sulfate current *Is(z,t)* :

I

*V s I*

*I V*

For linear systems, we look for solutions in the form A exp(pt) for inputs, or A(z) exp(pt) along the line, p being any complex constant. Let:

$$\begin{cases} I(z,t) = I(z) \exp(pt) \\ V(z,t) = V(z) \exp(pt) \end{cases} \tag{19}$$

Then we obtain the simplified set of equations

$$\begin{cases} \frac{dV\_S}{dz} = \rho I\_S\\ \frac{dI\_S}{dz} = \gamma p \, V\_S \end{cases} \tag{20}$$

Whence 2 2 d I s = I s dz *p*

Let = /L2. Then - and be the solutions of :

$$
\rho \alpha^2 = p \rho \gamma = p \frac{\tau}{L^2} \tag{21}
$$

Then solution for Is(z) is a linear combination of exp*z* . If we impose Is(L/2)=0, then

$$I\_S(z) = Ish\left(\alpha\left(z \cdot \frac{L}{2}\right)\right) \tag{22}$$

Whence:

$$V\_S(z) = \frac{1}{a} \, \rho \, \text{lch}\left(a \left(z \cdot \frac{L}{2}\right)\right) \tag{23}$$

We then get the Laplace impedance at the input of the equivalent circuit:

$$Z(p) = \frac{V\_S(0)}{I\_S(0)} = \frac{\rho}{\alpha t h \left(\frac{aL}{2}\right)}\tag{24}$$

if 1 2 2 *L L L th* 

$$\begin{cases} Z(p) \approx \frac{1}{\mathbb{C}p} \\ \mathbb{C} = \frac{\mathbb{Y}L}{2} \end{cases} \tag{25}$$

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 137

This approximation introduces a high frequency limit equal to the cutoff frequency of the

The total capacity will be equal to the total L/2 line capacity. The frequency limit for the

For instance for k =3 this may result in a drastic reduction of the number of cells for a given

In practice, the open circuit line model will be valid only if the entire electrolyte is between the cell plates. In practice this is usually not true. For our batteries, about one half of the electrolyte volume was beside the plates. In such case there is an additional transversal transport of ions, with still longer time constants. This could be accommodated by an

Satisfactory preliminary results for model validation were obtained with a much simplified

network, with an experimental fitting of the component values (Fig 11).

propose to use (*M*+1) cascaded sections but with impedance in geometric progression.

*rc.* Drawing from the previous example concerning Warburg impedance, we

Fig. 9. Elementary approximation by a cascade of identical RC cells

Fig. 10. Approximation by a cascade of RC cells in geometric progression

approximation remains given by the first cell cutoff frequency.

**3.2.4 Practical RC model used for experimentations** 

additional RC cell connected at the output of the line.

Fig. 11. Transmission line with additional RC cell

cells *fN = 1/2*

quality of approximation.

In case of harmonic excitation (p = j) this corresponds to small frequencies (<<1). The impedance is the global capacity of the line section. In practice for batteries (Karden et al., 2001), this corresponds to very small frequencies (10-5Hz)

$$\begin{array}{ll}\text{if } \left| \alpha L \right| >> 1 & \quad \hbar \left| \left( \frac{\alpha L}{2} \right) \approx 1 \right.\\\\ &Z(p) \approx \sqrt{\frac{\rho}{\gamma}} p^{-1/2} & \quad \tag{26} \end{array} \tag{26}$$

In case of harmonic excitation (p = j) this corresponds to high enough frequencies (>>1) the impedance is the same as for an infinite line (Linden, D et al., 2001), corresponding to the Warburg impedance.

### **3.2.2.4 Approximation of the Warburg impedance in terms of RC net**

An efficient approximation of a p-½ transfer function is obtained with alternate poles and zeros in geometric progression. In the same way, concerning Warburg impedances an efficient implementation (Bisquert et al., 2001) is achieved by a set of RC elements in geometric progression with ratio k, as represented in figure 8.

Let 0 = 1/RC. Note that the progression of the characteristic frequencies is in ratio k2.

Fig. 8. RC cells in geometric progression

Let Y() be the admittance of the infinite net. It is readily verified that


It can be verified by simulation that the fitting is quite accurate, even for values up to k=3.

### **3.2.3 Approximation of a finite line in terms of RC net**

The simplest approximation would to use a cascade of N identical RC cells simulating successive elementary sections of line of length *L =L/2N*. with: *r* = *L* and *c = L*

In case of harmonic excitation (p = j) this corresponds to small frequencies (<<1). The impedance is the global capacity of the line section. In practice for batteries (Karden et al.,

> -1/2 *Zp p* ( )

In case of harmonic excitation (p = j) this corresponds to high enough frequencies (>>1) the impedance is the same as for an infinite line (Linden, D et al., 2001), corresponding to the

An efficient approximation of a p-½ transfer function is obtained with alternate poles and zeros in geometric progression. In the same way, concerning Warburg impedances an efficient implementation (Bisquert et al., 2001) is achieved by a set of RC elements in

Let 0 = 1/RC. Note that the progression of the characteristic frequencies is in ratio k2.

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2001), this corresponds to very small frequencies (10-5Hz)

**3.2.2.4 Approximation of the Warburg impedance in terms of RC net** 

Let Y() be the admittance of the infinite net. It is readily verified that


**3.2.3 Approximation of a finite line in terms of RC net** 

successive elementary sections of line of length


It can be verified by simulation that the fitting is quite accurate, even for values up to

The simplest approximation would to use a cascade of N identical RC cells simulating

*L =L/2N*. with: *r* =

*L* and *c =* 

*L*

geometric progression with ratio k, as represented in figure 8.

if 1 1

*<sup>L</sup> L th*

Warburg impedance.

2

Fig. 8. RC cells in geometric progression

/4)

k=3.

Fig. 9. Elementary approximation by a cascade of identical RC cells

This approximation introduces a high frequency limit equal to the cutoff frequency of the cells *fN = 1/2rc.* Drawing from the previous example concerning Warburg impedance, we propose to use (*M*+1) cascaded sections but with impedance in geometric progression.

Fig. 10. Approximation by a cascade of RC cells in geometric progression

The total capacity will be equal to the total L/2 line capacity. The frequency limit for the approximation remains given by the first cell cutoff frequency.

For instance for k =3 this may result in a drastic reduction of the number of cells for a given quality of approximation.
