Skip to content

Battery Model

This page documents the high‑fidelity electro‑thermal battery model in PhlyGreen.

Overview

PhlyGreen’s Battery module provides a dynamic, temperature‑aware, physics‑based representation of a lithium‑ion battery pack suitable for hybrid‑electric aircraft preliminary design.

It simulates:

  • A representative cell, based on a modified Shepherd equation
  • Its thermal behaviour using a lumped‑parameter model
  • A complete battery pack, built via series (S) and parallel (P) configuration
  • Full operational constraint handling (voltage, SOC, current, temperature)
  • A sizing loop integrated into the aircraft’s WTO iteration

The model is based on empirical cell parameters (selected in Cell_Models) and is configurable via user‑provided battery settings.

Battery State Variables

At every timestep, the battery tracks three continuous state variables:

  • Cell current: i
  • Spent charge (Ah):

$$ i_t(t+\Delta t) = i_t(t) + \frac{i\,\Delta t}{3600} $$

  • Temperature: \( T \)

The State of Charge (SOC) is defined as:

\[ \text{SOC} = 1 - \frac{i_t}{Q} \]

where \( Q \) is the cell capacity.

A BatteryError is raised if SOC leaves the allowable range:

\[ \text{SOC}_{\min} \le \text{SOC} \le 1 \]

Pack Architecture

A battery pack is constructed as:

  • S cells in series → increases voltage
  • P cells in parallel → increases current capability

Thus:

\[ V_{\text{pack}} = S \, V_{\text{cell}} \]
\[ I_{\text{pack}} = P \, i_{\text{cell}} \]

Pack-level capacity and energy:

\[ Q_{\text{pack}} = P \, Q \]
\[ E_{\text{pack}} = S \, P \, E_{\text{cell}} \]

Cell Electrical Model

Modified Shepherd Equation (Temperature‑Aware)

The cell voltage is computed using a temperature‑corrected modified Shepherd model:

\[ V = E_0(T) - K(T)\left(\frac{Q}{Q - i_t}\right)i_t + A(T)e^{-B(T)i_t} - i\,R(T) \]

Where:

  • \( E_0(T) \) — open‑circuit voltage (with thermal correction)
  • \( K(T) \) — polarization constant
  • \( A(T), B(T) \) — exponential zone parameters
  • \( R(T) \) — internal resistance (Arrhenius‑corrected)

Solving for Current from a Required Power

Given a required pack power \( P_{\text{pack}} \), the battery solves:

\[ P_{\text{cell}} = \frac{P_{\text{pack}}}{S\,P} \]
\[ P_{\text{cell}} = V(i)\, i \]

This becomes a quadratic equation:

\[ a i^2 + b i + c = 0 \]

Where:

  • \( a = -(R + K_r) \) with \( K_r = K Q/(Q-i_t) \)
  • \( b = E_0 + A e^{-Bi_t} - i_t K_r \)
  • \( c = -P_{\text{cell}} \)

The physically meaningful root is selected:

\[ i = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \]

If the discriminant is negative:

\[ b^2 - 4ac < 0 \quad\Rightarrow\quad \text{BatteryError} \]

Thermal Model

Lumped‑Parameter Single‑Node Cell Temperature

The cell temperature evolves according to:

\[ \frac{dT}{dt} = \frac{Q_{\text{loss}}}{C_{\text{th}}} + \frac{T_{\infty} - T}{R_{\text{th}}\, C_{\text{th}}} \]

Where:

  • \( C_{\text{th}} \) — thermal capacitance
  • \( R_{\text{th}} \) — total thermal resistance

Heat Generation

\[ Q_{\text{loss}} = (V_{\text{oc}} - V)i + \frac{dE_0}{dT} i T \]

Convection Cooling Model

\[ h = \max\left[ 30\left(\frac{\dot{m}/(A_{\text{surf}}\rho)}{5}\right)^{0.8},\; 2 \right] \]

with airflow proportional to losses:

\[ \dot{m} = a \, Q_{\text{loss}} \]

Battery Sizing Loop

Inside the aircraft’s WTO root‑finding solver:

  1. Guess WTO
  2. Guess P_number
  3. Run full mission simulation
  4. During each timestep, battery state is updated
  5. If any violation occurs:
    • SOC too low
    • Voltage too low
    • Current > max
    • Temperature out of range
      → A BatteryError is thrown
  6. Increase P_number and retry
  7. Once feasible, compute pack weight
  8. Continue WTO iteration

This ensures size is based on actual in‑mission behaviour, not just averaged metrics.

Inputs

Battery configuration comes from CellInput:

  • "Class" — "I" (simple) or "II" (electro‑thermal)
  • "Model" — reference into Cell_Models
  • "SpecificEnergy" override
  • "SpecificPower" override
  • "Pack Voltage"
  • "Minimum SOC"
  • "Initial temperature"
  • "Max operative temperature"

Cell models include:

  • Mass, size, geometry
  • OCV constants
  • Polarization coefficients
  • Exponential coefficients
  • Arrhenius constants
  • Thermal properties

Outputs

Available throughout the simulation:

  • Vout — pack voltage
  • cell_Vout — cell voltage
  • I_pack and i_cell — total and per‑cell current
  • SOC
  • T
  • Q_loss — thermal losses
  • Pack mass, volume, energy
  • Maximum deliverable power

Error Handling

The battery throws structured exceptions:

  • "SOC_OUTSIDE_LIMITS"
  • "VOLTAGE_OUTSIDE_LIMITS"
  • "CURR_OUTSIDE_LIMITS"
  • "NEG_BATT_TEMP"
  • "BATT_UNDERPOWERED"
  • "TEMP_OUTSIDE_LIMITS"

These allow the sizing algorithm to automatically increase P.

Usage Example

battery = Battery(aircraft)
battery.SetInput()
battery.Configure(parallel_cells=48)

I = battery.Power_2_current(P=150e3)
dTdt, Qloss = battery.heatLoss(Ta=288, rho=1.225)

battery.it += I * dt / 3600
battery.T  += dTdt * dt

Limitations

  • Single‑node temperature model (no spatial gradients).
  • No ageing/SEI model.
  • Cooling system mass is not sized.
  • Discharge‑only model (no charging).

References

  • Shepherd, C. M. Design of Primary and Secondary Cells: II. An Equation Describing Battery Discharge.
    Journal of The Electrochemical Society, 112(7):657, 1965.

  • Tremblay, O., & Dessaint, L.-A. A Generic Battery Model for the Dynamic Simulation of Hybrid Electric Vehicles.
    Proceedings of the 2007 IEEE Vehicle Power and Propulsion Conference, pp. 284–289.

  • Saw, L., Somasundaram, K., Ye, Y., & Tay, A. Electro-thermal analysis of Lithium Iron Phosphate battery for electric vehicles.
    Journal of Power Sources, 249:231–238, 2014.