Mathematical Model of a Lead-Acid Battery

Aim: To run a MATLAB script for the mathematical model of lead acid battery.

Introduction:

The lead–acid battery was invented in 1859 by French physicist Gaston Plante and is the earliest type of rechargable battery. Despite having a very low energy-to-weight ratio and a low energy-to-volume ratio, its ability to supply high surge currents means that the cells have a relatively large pwer-to-weight ratio. These features, along with their low cost, make them attractive for use in motor vehicles to provide the high current required by starter motors.

As they are inexpensive compared to newer technologies, lead–acid batteries are widely used even when surge current is not important and other designs could provide higher energy densities. The lead acid battery is most commonly used in the power stations and substations because it has higher cell voltage and lower cost.

Negative Plate reaction:

Pb(s) + HSO⁻
₄(aq) → PbSO
₄(s) + H⁺
(aq) + 2e⁻

The release of two conducting electrons gives the lead electrode a negative charge.

As electrons accumulate they create an electric field which attracts hydrogen ions and repels sulfate ions, leading to a double-layer near the surface. The hydrogen ions screen the charged electrode from the solution which limits further reaction unless charge is allowed to flow out of electrode.

Positive Plate reaction:

PbO
₂(s) + HSO⁻
₄(aq) + 3H⁺
(aq) + 2e⁻ → PbSO
₄(s) + 2H
₂O(l)

The total reaction can be written as

Pb(s) + PbO
₂(s) + 2H
₂SO
₄(aq) → 2PbSO
₄(s) + 2H
₂O(l)

Terms Used:

DOD: A battery's depth of discharge (DoD) indicates the percentage of the battery that has been discharged relative to the overall capacity of the battery.

SOC: State of charge (SoC) is the level of charge of an electric battery relative to its capacity.

OCV: Open-circuit voltage is the difference of electrical potential between two terminals of a device when disconnected from any circuit.

Peukert's Capcity:

`C_p = I^k*T`

here C_p is Peukert's capacity in Ah

k is Peukert's constant

I is s the actual discharge current in A

T is the actual time to discharge the battery

MATLAB Code:

%MATLAB Script to run Mathematical Model of a Lead acid Battery
clear all
close all
clc

%inputs
T = (0:50:500000); %Time=10001 steps with spacing of 50sec
CR = zeros(1,length(T)); %Charge Removed
DOD = zeros(1,length(T)); %Depth of Discharge(DOD)
V = zeros(1,length(T)); %Voltage
CS = zeros(1,length(T)); %Charge Supplied
E = zeros(1,length(T)); %Operating circuit voltage
I = [10,50,100,150,200]; %current(A)
n = 1; %No. of cell
C = 50; %Capacity in Ah
k = 1.045; %Peukert's constant
dT = (T(2)- T(1))* ones(1,length(T)); %Change in time
R_int = (0.06/C)*n; %Internal resistance
E(1) = 2.15; %circuit voltage

for i=1:length(I)
    V(i)=n*(2.15-I(i)*R_int);
    
%Voltage eqn for OCV
for n=2:length(T)
    dT(n) = T(n)-T(1);
    %Peukert's Capacity 
    Cp = (I(i)^k)*(dT(n));
    
    CR(n)= (CR(n-1)+(dT(n)*(I(i)^k))/3600);
    DOD(n)=(CR(n)/Cp);
    E(n)=2.15-DOD(n)*(2.15-1.75);
    V(n)=E(n)-I(i)*R_int; 
    
    if DOD(n)>0.99
        E(n)=0;
        V(n)=0;
    end
    if DOD(n)>1 
        DOD(n)=1;
    end
    if E(n)>0 
        CS(n)=CS(n-1)+((I(i)*dT(n))/3600);
    else
        CS(n)= CS(n-1); 
    end
end
figure(1)
hold on 
plot(DOD,V)
xlabel('DOD')
ylabel('Voltage') 
end
plot(DOD, E)
legend('E-OCV','V-Voltage','I=10A','I=50A','I=150A','I=200A')
grid on

figure(2)
plot(DOD,E) 
hold on 
plot(DOD, V) 
legend('E-OCV','V-Voltage')
xlabel('DOD') 
ylabel('Voltage')
grid on 

figure(3)
plot(CR,E)
hold on
plot(CR, V)
legend('E-OCV','V-Voltage') 
xlabel('CR') 
ylabel('Voltage')
grid on 

figure(4) 
plot(CS,E) 
hold on 
plot(CS, V) 
legend('E-OCV','V-Voltage')
xlabel('CS') 
ylabel('Voltage')
grid on 


%end