Lithium-ion Batteries in Hybrid Vehicles: Modeling and Technical Characteristics

In a Fuel Cell Hybrid Electric Vehicle (FCHEV) powertrain, we know that Fuel Cells respond relatively slowly to sudden load changes. Therefore, to enable the vehicle to accelerate smoothly and recover energy during braking, we need another “companion” that can react more swiftly: the Battery.

In an FCHEV system, the Battery acts as the Energy Storage System (ESS) or an energy buffer.

This article will not delve too deeply into chemistry but will focus on the physical quantities and mathematical formulas necessary for you to build a Battery Model on a computer (e.g., using MATLAB/Simulink). 🔋

1. Structure and Principle: The “Rocking Chair”

Fundamentally, a Lithium-ion battery operates based on the movement of Lithium ions ($Li^+$) between two electrodes. This back-and-forth movement is why it is often referred to as a “rocking chair” battery.

Structure

• Cathode (Positive Electrode): Typically made from metal oxides (such as LFP, NMC). This is where Li-ions reside when the battery is fully discharged.
• Anode (Negative Electrode): Typically made from Graphite. This is where Li-ions are stored when the battery is fully charged.
• Electrolyte: The medium that allows $Li^+$ ions to travel back and forth between electrodes.
• Separator: A permeable membrane that prevents the two electrodes from touching (which would cause a short circuit) but allows ions to pass through.

Working Principle

Imagine Lithium ions as commuters:

• Charging: An external electric current forces the $Li^+$ ions to leave their home (Cathode) to travel to their workplace (Anode) and “intercalate” (lodge) within the graphite layers.
• Discharging: The $Li^+$ ions want to return home. They travel back from the Anode to the Cathode, creating an electric current flowing through the external circuit to power the motor.

2. Core Simulation Parameters

To simulate a battery in MATLAB/Simulink, we are less concerned with detailed chemical reactions and more interested in the following macroscopic variables:

2.1. Capacity ($Q$)

This represents the ability of the battery to store electric charge.

• Unit: Ampere-hour ($Ah$) or Coulomb ($C$). ($1 Ah = 3600 C$).
• Meaning: A $50Ah$ battery can theoretically provide a current of 1A for 50 hours, or 50A for 1 hour.

2.2. State of Charge (SOC)

This is the “fuel gauge” of the battery, indicating the percentage of energy remaining.

• $SOC = 100\%$: Fully charged.
• $SOC = 0\%$: Empty.

Calculation Formula (Coulomb Counting): To calculate SOC at time $t$, we integrate the charging/discharging current over time:

Where:

• $SOC(t_0)$: Initial SOC.
• $Q_{nom}$: Nominal capacity of the battery ($Ah$).
• $I(t)$: Current ($A$). Convention: Discharging is positive (+), Charging is negative (-).

2.3. Depth of Discharge (DOD)

The inverse of SOC.

2.4. Open Circuit Voltage (OCV)

This is the battery’s voltage when no current is flowing (at rest).

• OCV is not a constant (like the 12V or 3.7V we often say); it depends non-linearly on the SOC.
• Characteristic: When fully charged ($SOC=100\%$), OCV is highest. As the battery drains, OCV drops.
• Equation: $V_{oc} = f(SOC)$. (In simulation, this is usually implemented using a Lookup Table).

2.5. Internal Resistance ($R$)

A battery is not an ideal voltage source; it has internal resistance.

• Larger $R$ $\to$ More heat loss ($P_{loss} = I^2 R$) and greater voltage drop.
• $R$ varies with Temperature and SOC. (Low temperature $\to$ Internal resistance increases $\to$ Battery becomes “weak”).

3. Equivalent Circuit Model (ECM)

For Energy Management System (EMS) algorithms, we often use the R-int Model (the simplest Internal Resistance Model).

The terminal voltage equation ($V_{term}$) under load is:

• During Discharging ($I > 0$): $V_{term} < V_{oc}$ (Voltage sag).
• During Charging ($I < 0$): $V_{term} > V_{oc}$ (Voltage swell).

Battery Power ($P_{batt}$): $P_{batt} = V_{term} \cdot I = V_{oc} \cdot I - I^2 \cdot R_{int}$

4. Case Study: Toyota Mirai Battery Pack 🚗

In the Toyota Mirai (Gen 2), the Lithium-ion battery is not used for long-range driving but rather to support acceleration and capture regenerative braking energy.

Assumed Specifications (for the High Voltage Pack):

• Type: Lithium-ion.
• Nominal Capacity ($Q$): 4.0 Ah.
• Nominal Voltage ($V_{nom}$): 310.8 V (Consisting of 84 cells in series, 3.7V per cell).
• Total Energy ($E$): $\approx 1.24 \text{ kWh}$.
• Internal Resistance ($R_{int}$): 0.5 Ohm.

Simulation Scenario: The car is cruising with an initial SOC of 60%. The driver accelerates hard to overtake, demanding a discharge current of $I = 50 A$ for 10 seconds.

Calculations:

1. SOC Drop:

   • Charge consumed: $\Delta Q = I \times t = 50 \times 10 = 500 \text{ Coulombs}$.
   • Convert to Ah: $\frac{500}{3600} \approx 0.138 \text{ Ah}$.
   • Percentage lost: $\frac{0.138}{4.0} \times 100 \approx 3.47\%$.
   • $\Rightarrow SOC_{new} = 60\% - 3.47\% = 56.53\%$.

2. Terminal Voltage ($V_{term}$):

   • Assume at $SOC=60\%$, the lookup table gives $V_{oc} \approx 320V$.
   • Voltage drop due to resistance: $V_{drop} = I \times R = 50 \times 0.5 = 25V$.
   • $\Rightarrow V_{term} = 320 - 25 = 295V$.

3. Actual Power ($P_{batt}$):

   • $P_{batt} = 295V \times 50A = 14,750W = 14.75 \text{ kW}$.

Conclusion: In just 10 seconds of acceleration, the battery lost nearly 3.5% of its capacity, and the voltage sagged by 25V. This illustrates why the EMS must carefully coordinate power flow to avoid “shocking” or rapidly depleting the battery.