Battery Technology Project: Mathematical Model of a Lead Acid Battery

Aim:

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

Theory:

In photovoltaic systems, lead acid batteries are the most commonly used type of battery. Although lead acid batteries have a low energy density, low efficiency, and high maintenance requirements, they have a long lifespan and low costs when compared to other battery types. Lead acid batteries have the distinct advantage of being the most commonly used type of battery for most rechargeable battery applications (for example, starting car engines), and thus have a well-established established, mature technology base. Lead-acid batteries can be classified as secondary batteries. Reversible chemical reactions occur in secondary cells. By passing current through the battery, the reactants that generate an electric current in these batteries (via chemical reactions) can be regenerated (recharging). Discharging is the chemical process of extracting current from a secondary battery (forward reaction). Charging is the process of regenerating active material.

Working of Lead Acid Battery:

Figure: Working of Lead Acid Battery [electrical4u, link]

A storage battery, also known as a secondary battery, is a type of battery in which electrical energy is stored as chemical energy and then converted to electrical energy as needed. Charging a battery is the process of converting electrical energy into chemical energy by using an external electrical source. Discharging of secondary battery refers to the conversion of chemical energy into electrical energy for the purpose of supplying an external load. When a battery is charged, current flows through it, causing chemical changes inside the battery. During their formation, these chemical changes absorb energy. When the battery is connected to an external load, the chemical changes reverse, releasing the absorbed energy as electrical energy and supplying it to the load.

Dipping lead peroxide plates and sponge lead plates in dilute sulfuric acid creates the lead acid storage battery. An external load is connected between these plates. The molecules of diluted sulfuric acid split into positive hydrogen ions (`H^+`) and negative sulphate ions (`SO_4^(2−)`). When hydrogen ions come into contact with a `PbO_2` plate, they receive electrons from it and transform into hydrogen atoms, which then attack `PbO_2` and form `PbO_2` and `H_2O` (water). This `PbO` combines with `H_2SO_4` to form `PbSO_4` and `H_2O`. Because `SO_4^(2−)` ions are freely moving in the solution, some of them will reach a pure `Pb` plate, where they will give up their extra electrons and become radical `SO_4`. Because the radical `SO_4` cannot exist on its own, it will attack `Pb` and form `PbSO_4`. Because `H^+` ions take electrons from the `PbO_2` plate and `SO_4^(2−)` ions give electrons to the `Pb` plate, there is an electron imbalance between these two plates. As a result, current would flow through the external load between these plates to balance the electron inequality. This is known as lead acid battery discharging.
The colour of lead sulphate (`PbSO_4`) is whitish.

When discharging:

1. PbSO4 has been applied to both plates.
2. The specific gravity of sulfuric acid solution decreases due to water formation during the reaction at the PbO2 plate.
3. As a result, the rate of reaction slows, implying that the potential difference between the plates narrows during the discharging process.

If the load is disconnected, connect the positive terminal of an external DC source to `PbSO_4` covered with `PbO_2` plate and the negative terminal of that DC source to `PbO_2` covered with `Pb` plate. The density of sulfuric acid decreases during discharging, but sulfuric acid remains in the solution. This sulfuric acid is also present in the solution as `H^+` and `SO_4^(2−)` ions. Positively charged hydrogen ions (cation) move to the electrode (cathode) connected to the negative terminal of the DC source. Each `H^+` ion takes one electron from this to form a hydrogen atom. These hydrogen atoms then react with `PbSO_4` to produce lead and sulfuric acid.`SO_4^(2−)` ions (anions) move towards the electrode (anode) connected to the positive terminal of the DC source, where they give up their extra electrons and transform into radical `SO_4`. Because this radical `SO_4` cannot exist on its own, it reacts with the anode's `PbSO_4` to form lead peroxide (`PbO_2`) and sulfuric acid.

Anode reaction: `Pb+SO_4^(2−) harr PbSO_4+2e^−`
Cathode reaction: `PbO_2+4H^++2e^−+SO_4^(2−)harr2H_2O+PbSO_4`
Overall reaction: `PbO_2+Pb+2H_2SO_4harr2PbSO_4+2H_2O`

Understanding Battery Parameters and Specifications:

Before modelling a battery, the various parameters used to characterise the cells or batteries, as well as the cell or battery's behaviour under various operating conditions, must be explored.

• C rates: Discharge current is frequently expressed as a C-rate in battery descriptions to normalise against battery capacity, which varies greatly between batteries. A C-rate is a measure of how quickly a battery is discharged in comparison to its maximum capacity. A 1C rate indicates that the discharge current will completely discharge the battery in one hour. This equates to a discharge current of 100 amps for a battery with a capacity of 100 amp-hours. This battery's 5C rate is 500 amps, and its C/2 rate is 50 amps. An E-rate, on the other hand, describes the discharge power. A 1E rate is the discharge power required to completely discharge a battery in one hour.
• State of Charge (SOC): The current battery capacity expressed as a percentage of maximum capacity. To determine the change in battery capacity over time, SOC is typically calculated using current integration.
• Depth of Charge (DoD): The percentage of discharged battery capacity expressed as a percentage of maximum capacity. A deep discharge is defined as a discharge of at least 80% DOD.
• Terminal Voltage: The voltage between the battery terminals with a load applied. The terminal voltage varies with SOC and discharge/charge current.
• Open Circuit Voltage: With no load applied, the voltage between the battery terminals. The open-circuit voltage is affected by the battery's state of charge and rises with it.
• Internal Resistance: The resistance within the battery, which varies with charging and discharging, is also affected by the battery's state of charge. As internal resistance rises, so does battery efficiency, and thermal stability suffers as more charging energy is converted to heat.
• Nominal Voltage: The reported or reference voltage of the battery, also sometimes thought of as the “normal” voltage of the battery.
• Cutt-off Voltage: The minimum allowable voltage. It is this voltage that generally defines the “empty” state of the battery.
• Capacity or Nominal Capacity (Ah for a specific C rate): The total Amp-hours available when the battery is discharged at a specific discharge current (specified as a C-rate) from 100% state-of-charge to the cut-off voltage. The capacity is calculated by multiplying the discharge current (in Amps) by the discharge time (in hours), and it decreases as the C-rate increases.
• Energy or Nominal Energy (Wh for a specific Crate): The battery's "energy capacity," defined as the total Watt-hours available when discharged at a specific discharge current (specified as a C-rate) from 100% state-of-charge to the cut-off voltage. The discharge power (in Watts) is multiplied by the discharge time to calculate energy. Energy, like capacity, decreases as C-rate increases.
• Cycle Life (number for a specific DOD): The number of discharge-charge cycles a battery can go through before failing to meet certain performance criteria. For specific charge and discharge conditions, cycle life is estimated. The rate and depth of cycles, as well as other factors such as temperature and humidity, all have an impact on the battery's actual operating life. The lower the cycle life, the higher the DOD.
• Specific Energy (Wh/kg): The nominal battery energy per unit mass, also known as the gravimetric energy density. The specific energy of a battery is a property of its chemistry and packaging. It determines the battery weight required to achieve a given electric range, along with the vehicle's energy consumption.
• Specific Power (W/kg): The maximum power available per unit mass. Specific power is a feature of battery chemistry and packaging. It calculates the battery weight required to meet a given performance goal.
• Energy Density (Wh/L): The nominal battery energy per unit volume, also known as the volumetric energy density. The specific energy of a battery is a property of its chemistry and packaging. It determines the battery size required to achieve a given electric range, along with the vehicle's energy consumption.
• Power Density (W/L): The maximum power available per unit volume. Specific power is a feature of battery chemistry and packaging. It calculates the battery size required to meet a given performance goal.
• Maximum Continuous Discharge Current: The maximum current at which the battery can be continuously discharged. This limit is typically set by the battery manufacturer to prevent excessive discharge rates from damaging the battery or reducing its capacity. This, along with the maximum continuous power of the motor, defines the vehicle's top sustainable speed and acceleration.
• Maximum 30-sec Discharge Pulse Current: The maximum current at which the battery can be discharged for up to 30 seconds in pulses. This limit is typically set by the battery manufacturer to prevent excessive discharge rates from damaging the battery or reducing its capacity. This, along with the peak power of the electric motor, defines the vehicle's acceleration performance (0-60 mph time).
• Charge Voltage: When fully charged, the voltage at which the battery is charged. Charging schemes typically involve constant current charging until the battery voltage reaches the charge voltage, followed by constant voltage charging with the charge current tapering until it is very small.
• Float Voltage: The voltage at which the battery is maintained after being charge to 100 per cent SOC to maintain that capacity by compensating for self-discharge of the battery.
• Charge Current (Recommended): The ideal current at which the battery is initially charged (to roughly 70 per cent SOC) under constant charging scheme before transitioning into constant voltage charging.
• Internal Resistance (Maximum): The resistance within the battery, generally different for charging and discharging.

Discharge Curves:

Energy cells have been developed for a wide range of applications using a variety of different technologies, resulting in a wide range of performance characteristics being available. The graphs below depict some of the key factors that an applications engineer should consider when specifying a battery to meet the end product's performance requirements.

• Cell Chemistry:
  The electrochemical properties of the active chemicals used in the cell, known as cell chemistry, determine the nominal voltage of a galvanic cell. The actual voltage appearing at the terminals at any given time, as with any cell, is determined by the load current and the internal impedance of the cell, which varies with temperature, charge state, and cell age.
  
  Figure: Cell voltage vs discharged capacity for various cell chemistry [link]
  
  The X-axis displays cell characteristics normalised as a percentage of cell capacity, allowing the graph's shape to be displayed independently of the actual cell capacity. The length of each discharge curve would be proportional to the nominal capacity of the cell if the X-axis was based on discharge time.
  
  
• Temperature Characteristics:
  Temperature has a significant impact on cell performance. At the low end, in aqueous electrolyte batteries, the electrolyte itself may freeze, lowering the operating temperature. Lithium batteries suffer from anode lithium plating at low temperatures, resulting in a permanent reduction in capacity. At the extreme, the active chemicals may degrade and destroy the battery. In the range between these limits, cell performance generally improves with increasing temperature.
  
  Figure: Voltage vs Discharge Time at different temperatures [link]
  
  The graph above depicts how the performance of Lithium-Ion batteries degrades as the operating temperature drops. More importantly, for both high and low temperatures, the further the operating temperature is from room temperature, the shorter the cycle life.
  
  
• Discharge Rates:
  The discharge curves for a Lithium-Ion cell shown below show that if the cell is discharged at very high rates, the effective capacity of the cell is reduced. This is known as capacity offset, and it occurs in almost all cell chemistries.
  
  If the discharge occurs over several hours, as in some high rate applications such as electric vehicles, the effective capacity of the battery can be up to double the specified capacity at the C rate. The Society of Automotive Engineers (SAE) defines the standard condition for measuring the Amp-hour capacity of automotive batteries as a 20-hour discharge (0.05C). The graph below shows that as the discharge rate is reduced from 1.0C to 0.05C, the effective capacity of a deep discharge lead-acid battery nearly doubles. The effective capacity drops dramatically for discharge times less than one hour (High C rates).
  
  Two conclusions to be drawn from two graphs depicting discharge rate. When comparing battery capacity specifications, exercise caution to ensure that comparable discharge rates are used. In an automotive application, using high current rates for hard acceleration or hill climbing on a regular basis reduces the vehicle's range.
  
  
• Self Discharge Characteristics:
  
  The self-discharge rate is a measurement of how quickly a cell loses energy while sitting on the shelf as a result of unwanted chemical reactions within the cell. The rate is affected by cell chemistry and temperature. The rate of unwanted chemical reactions that cause internal current leakage between the positive and negative electrodes of the cell increases with temperature, as does the rate of battery self-discharge.
  
• Internal Impedence
  The internal impedance of a cell determines its current-carrying capability. Low internal resistance allows high currents.
  
  Effects of Internal Impedence:
  • When current flows through the cell, an IR voltage drop occurs across the cell's internal resistance, lowering the terminal voltage during discharge and increasing the voltage required to charge the cell, reducing its effective capacity and charge/discharge efficiency. Higher discharge rates cause higher internal voltage drops, which explains why the voltage discharge curves at high C rates are lower.
  • The physical properties of the electrolyte influence the internal impedance; the lower the impedance, the smaller the granular size of the electrolyte material. In a milling process, the grain size is controlled by the cell manufacturer. The spiral design of the electrodes is frequently used to increase surface area and thus reduce internal impedance. This lowers heat generation and allows for faster charge and discharge rates. As a result, we can say that cell capacity is directly proportional to electrode area and internal resistance is inversely proportional to electrode area. As a result, we can say that internal resistance is inversely proportional to battery capacity and state of charge.
  • Most cell chemistries' internal resistance increases significantly near the end of the discharge cycle as the active chemicals are converted to their discharged state and thus effectively used up. This is the primary cause of the rapid drop in cell voltage at the end of the discharge cycle.
  • A galvanic cell's internal resistance is temperature-dependent, decreasing as temperature rises due to increased electron mobility. The graph below is an example of this.
    
    Thus, the cell may be inefficient at low temperatures, but efficiency improves at higher temperatures due to lower internal impedance as well as increased rate of chemical reactions. However, because of the lower internal resistance, the self-discharge rate increases. Furthermore, at high temperatures, cycle life degrades. To achieve the best performance in high power applications, some form of heating and cooling may be required to keep the cell within a specific temperature range.
    Furthermore, the Joule heating effect of the `I^2R` losses in the cell's internal resistance will cause the cell's temperature to rise.
  • The resistance of the electrolyte tends to increase as the cell ages. The surface of the electrodes deteriorates with age, and the contact resistance increases, while the effective area of the plates decreases, reducing capacitance. All of these effects raise the cell's internal impedance, reducing its ability to perform. Comparing a cell's actual impedance to its impedance when it was new can provide a measure or representation of a cell's age or effective capacity. Such measurements are far more convenient than discharging the cell and can be performed without destroying the cell under test.
  • Internal resistance influences a cell's effective capacity as well. The greater the internal resistance, the greater the losses during charging and discharging, particularly at higher currents. This means that for high discharge rates, the available capacity of the cell is reduced. If it is discharged over a long period of time, the AmpHour capacity increases. This is significant because some manufacturers specify the capacity of their batteries at very low discharge rates, making them appear much larger than they are.

Mathematical Model of a Lead-Acid Battery:

The first step in creating an accurate mathematical model of a battery is to construct and parameterize an equivalent circuit that reflects the battery's nonlinear behaviour and dependencies on temperature, SOC, SOH, and current. These dependencies are specific to the chemistry of each battery. A battery's mathematical model is determined by the experimental values obtained from various test cases. The laboratory experiments on various cell chemistries contribute to the formation of the mathematical model. A battery cell's nominal voltage is an experimental value that gives us an approximate value of the voltage when the cell is delivering electrical power. In practise, the voltage fluctuates because when the cell is discharged, the voltage drops and when it is charged, the voltage rises. This cell behaviour can be expressed using the battery's internal resistance, which is known as a simple equivalent circuit model.

Figure: Simple Equivalent Circuit of a Battery

Applying Kirchoff's Law:
`V=E−IR`

where, `E` is open circuit voltage, `I` is the current flowing through the circuit, `V` is the voltage across the load and `R` is the internal resistance of the battery. If the current is zero, this indicates that the electromotive force (EMF) equals the open-circuit voltage. The above circuit predicts the in-use battery voltage by assuming that the open-circuit voltage `E` and battery internal resistance `R` are constant, but they are not because their values are affected by other factors such as charge state, operating temperature, and so on. For a 1Ah cell, the internal resistance of a lead-acid cell is 0.022ohm.

Internal Resistance of a Battery pack can be given as,
`R=n*(0.022//C_10)` ohm

where, `n` is the no of cells.

• Self-discharging of the battery:
  The batteries are known to convert chemical energy to chemical energy and vice versa. The rate of charge and discharge is determined by the rate of these reactions. Impurities in the cell material, as well as external environmental factors such as temperature, humidity, and so on, all contribute to the reactions that occur inside the cell. As a result, these unwanted reactions contribute to the cell's unwanted discharge. Thus, the open-circuit voltage can be defined as the battery EMF observed when the battery is discharged at a very low C-rate (C/20 or lower). This experimental study can be used to determine the open-circuit voltage values at various battery charge left or SOC values.
  
  Figure: The open-circuit voltage (OCV) of a sealed lead acid battery changes with the state of charge (SOC=1-DOD)
  
  Based on the above experimental results, a linear polynomial function can be derived to estimate the open-circuit voltage based on the depth of discharge value (DOD).
  ie, `E=n[2.15−DoD*(2.15−2)]`, where `n` is the no of cells.
  
  The dynamic behaviour of the battery example cannot be considered in the above equation because when a load is connected, the battery voltage `E` immediately changes to a lower value, but in reality, the battery takes some time to settle down. As a result, a more accurate dynamic behaviour will include time and temperature in the equation.
  When the current is drawn at a higher C-rate, the capacity of the battery is reduced; this phenomenon is important in estimating the battery capacity. The Peukert battery model is used to estimate battery capacity when the discharge rate is typically higher, such as in the case of an electric vehicle.
  
  
• Peukert Equation:
  

  It is a common misconception that the energy that the battery does not deliver due to Peukert's Law is "lost" (as heat for example). In fact, once the load is removed, the battery voltage will recover, allowing more energy to be drawn from the battery. This is due to the fact that the law only applies to batteries discharged at constant current down to the cutoff voltage. Because the battery can no longer deliver that current without falling below the cutoff voltage, it is considered discharged at that point, despite the fact that significant energy remains in the battery.

  The Peukert equation is a useful mathematical tool for characterising cell behaviour and quantifying capacity offset. This is an empirical formula which approximates how the available capacity of a battery changes according to the rate of discharge.
  `C_p=I^kT`
  where, `C_p` is the theoretical capacity of the battery expressed in Amp-hours, `I` is current, `T` is time, and `k` is the Peukert Number, a constant for the given battery. The equation shows that at higher currents, there is less available energy in the battery. The Peukert Number is directly related to the internal resistance of the battery. Higher currents mean more losses and less available capacity.
  
  The Peukert number indicates how well a battery performs when subjected to continuous high currents. When the battery is discharged at high currents, a value close to 1 indicates that it performs well; the higher the number, the more capacity is lost. A battery's Peukert number is calculated empirically. For Lead-acid batteries, the Peukert number is typically between 1.3 and 1.4.
  
  
  

  The graph above shows that at very high continuous discharge rates, the effective battery capacity is reduced. However, with intermittent use, the battery has time to recharge during idle periods when the temperature returns to normal. Because of this recovery potential, the capacity loss is less and the operating efficiency is higher if the battery is used intermittently, as shown by the dotted line. This is the opposite of how an internal combustion engine behaves, which operates most efficiently with continuous steady loads. In this regard, electric power is a better solution for delivery vehicles that face constant interruptions.

  The Peukerts effect shows that if the current `I` flow from the battery, then from battery point of view the amount for current flow our is `I^k`. As long as the values of `I` and `k` are greater than 1, the value of `I^k` will be larger than `I`. Thus the loss charge in time step `deltat`can be estimated as,
  `"Loss of Charge"=δt⋅(I^k−I)`
  
  The apparent or effective charged removed from the battery in time `deltat`can be found using the below equation,
  `CR_(n+1)=CR_n+(δt⋅I^k)/3600`in Amp-hours
  
  where, `CR_n` is the total charge removed till `n^(th)`time step, `deltat` is the time step in seconds `(deltat)//3600` will be time in hours).
  
  Similarly, the charge supplied by the battery in time `deltat`can be found using the below equation,
  `CS_(n+1)=CS_n+(δt⋅I^k)/3600`in Amp-hours
  
  where, `CS_n` is the total charge removed till `n^(th)`time step, `deltat` is the time step in seconds `(deltat)//3600` will be time in hours).
  
  The depth of discharge of the battery can be expressed in terms of Peukert capacity and charged removed from the battery,
  `DoD_n=(CR_n)/C_p`
  
  
  

  • Calculating Peukart Coefficient:
    

    The method to find the Peukaert coefficient is fairly easy and only requires battery capacity at two different discharge times.

    The two different ratings give two different rated currents:
    `I_1=C_1/T_1` and `I_2=C_2/T_2`
    
    The respective Peukert capacity for the above can be written as:
    `C_p=I_1^k⋅T_1` and `C_p=I_2^k⋅T_2`
    
    Since the Peukert capacity is the same and equating both the equation,
    `I_1^k⋅T_1=I_2^k⋅T_2`
    `(I_1/I_2)^k=T_2/T_1`
    
    Taking log on both sides,
    `k=(logT_2-logT_1)/(logI_1-logI_2)`

MATLAB CODE for Mathematical model of lead acid battery:

The programming is done in MATLAB.

%% Mathematical Model of lead-Acid battery
% By: Jibin Jafar

clear all 
close all 
clc 

%% Defining Inputs 
% Variable T is used for defining time steps
T = (0:50:500000);       %Total time 10001 steps with each time space of 50 sec. 
I = [10,50,100,150,200]; %Current in Amphers 
n = 1;                   %No.of cells 
C = 50;                  %Capacity in Ah 
k = 1.045;               %.045 represents loss of capacity due to discharge 

%% Initializing Parameters
CR = zeros(1,length(T)); %Effective Charge removed (CR)
DOD = zeros(1,length(T));%Depth of discharge
V = zeros(1,length(T));  %Voltage 
CS = zeros(1,length(T)); %Charge supplied
E = zeros(1,length(T));  %Operating circuit voltage 

dT = (T(2)-T(1))* ones(1,length(T)); % Finding Time step
R_in = (0.022/C)*n;      %Internal Resistance 
E(1)= 2.15;              %open circuit voltae

%% Calculations
for i = 1:length(I)
    V(i) = n*(2.15-I(i)*R_in); %finding voltage//Kirchoffs law
    
    %Define the voltage equation for OCV 
    for n = 2:length(T) 
        dT(n) = T(n)-T(1);
        Cp = (I(i)^k)*(dT(n)); %Peukert's capacity 
        CR(n) = (CR(n-1)+(dT(n)*(I(i)^k))/3600);%effective charge removed at deltat (at t=n) 
        DOD(n) = (CR(n)/Cp); %depth of recharge at time n
        E(n) = 2.15-DOD(n)*(2.15-1.75); %open circuit voltage at t=n and I= i
        V(n) = E(n)-I(i)*R_in; %voltage

    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')
set(gca,'FontSize',12)
end
% Plot: DoD vs voltage for different 
plot(DOD, E)
lgd = legend('I=10A ~ C/5','I=50A ~ 1C','I=100A ~ 2C','I=150A ~ 3C','I=200A ~ 4C','Open Circuit Voltage (E)','FontSize',18)
lgd.FontSize = 9;
grid on;

% Plot: DoD vs E
figure(2)
plot(DOD,E)
hold on
plot(DOD, V) 
legend('E-OCV','V-Voltage')
xlabel('DoD') 
ylabel('Voltage')
grid on

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

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

The cooments are provided with the code for the better understanding.

• Explanation of the Code:
  • Firstly, Total time and time steps have been defined. (10001 steps are defined, each with a time step of 50 seconds)
  • Variables defined for CR,CS,V,DOD,Cp,E,n,I,C,k
  • I is the discharge current and is given as 10,50,100,150,200.
  • There are various discharge rates such as C/5, 1C, 2C, 3C, and 4C.
  • Used in calculation loops:
    The first for loop is used to compute the inner commands for each I value.
    The second for loop, which is used to calculate CS,CR,E,V, and DOD, repeats the simulation until the time limit is reached.
  • If condition is used ensure that voltage and OCV are within their limit. Voltage and OCV will drop as soon as DOD exceeds 0.99, and the value is then considered to be 1, indicating that the battery is completely discharged.
  • Plot command was used to plot the results.

Outputs and Observations:

• Voltage variation with respect to DoD for various discharge rates:
  
  The above plot clearly shows that as the discharge rate increases, the battery's discharge capacity decreases. The top line represents the open circuit voltage, which is the maximum possible voltage that a battery can achieve ideally. In practise, however, this is impossible, and a close value to OCV can be obtained by decreasing the discharge rate.
  
  
• Voltage variation with respect to DoD:
  
  The variations of the battery's OCV and Nominal Voltage with respect to the depth of discharge are given here. The difference seen here is due to internal resistance and other losses (due to temperature, chemistry, etc). The voltage vs depth of discharge characteristics show that OCV is directly proportional to the battery's state of charge. This also demonstrates the effect of internal resistance increasing with depth of discharge, causing the voltage drop to increase with depth of discharge.
  
  
• Voltage variations with respect to Charge Supplied:
  
  The discharge curves show that the effective capacity of the cell is reduced if the cell is discharged at very high rates (or conversely increased with low discharge rates). This phenomenon is called the capacity offset.
  
  
• Voltage variations with respect to Charge Released:
  
  Voltage vs Charge removed (CR) and Voltage vs Charge supplied (CS) plots show that the charge removed is greater than the charge supplied to the system.
  
  

Conclusion:

The mathematical model of a lead-acid battery is completed successfully using the Matlab script coding platform.

References:

1. http://mocha-java.uccs.edu/BMS1/CH2/ESCtoolbox.zip
2. https://www.researchgate.net/figure/Lead-acid-battery-chemistry-a-during-discharging-b-during-charging-and-c-LA_fig1_343746458
3. Advantages of Lead Acid Battery,disadvantages of Lead Acid Battery (rfwireless-world.com)
4. Lithium-Hybrid (zwerfcat.nl)
5. https://en.wikipedia.org/wiki/Peukert%27s_law#:~:text=Peukert's%20law%2C%20presented%20by%20the,approximately%20according%20to%20Peukert's%20law.