TAILIEUCHUNG - Sustainable Growth and Applications in Renewable Energy Sources Part 8

Tham khảo tài liệu 'sustainable growth and applications in renewable energy sources part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 131 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 For currents the anti symmetry property Is L-z - Is z For densities the symmetry property ns L-z ns z Voltage and concentration 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 ns L nH L 4 aPbSO4 L kT I n 0 I kJ .V A1 ln I _ I ln 2e 1 aPbS04 0 J 2e 8a In this relation we may use the fact that According to the neutrality condition section nH 2nS Due to the symmetry of concentrations ns L ns 0 . Concerning PbSO4 activity it is equal to one unless we are very close to full charge this will not be considered here . In such conditions the expression of battery voltage may be set in the form V A2 3T ln ns 0 8b Let n0 be a reference sulfate concentration and E0 the corresponding Nernst voltage then the relation may be written in the form V E0 V ln T. VL e 8c 132 Sustainable Growth and Applications in Renewable Energy Sources This result corresponds to point d in introduction . 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 Eo according to figure 5 9 The pseudo voltage may then be obtained by an exponential transformation of the original voltage according to the expression I IV-E I _ I I V-E V E0 Vl I expl ự - 1-11 E0-Vl VLI expl ự - 10 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 ns 0 ns L ns - L 1

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