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Complete Glossary & Equations

This glossary compiles all the key terms, variables, and mathematical models used throughout the SPARK simulator.

1. Environment & Physics

Ambient Pressure (\(P\))

The pressure of the air at a specific altitude, computed using the barometric formula.

\[ P = P_0 \cdot \left(1 - \frac{L \cdot h}{T_0}\right)^{\frac{g \cdot M}{R \cdot L}} \]
  • \(P_0\): Standard sea-level pressure (101325 Pa)
  • \(T_0\): Standard sea-level temperature (288.15 K)
  • \(L\): Temperature lapse rate (0.0065 K/m)
  • \(h\): Altitude in meters
  • \(g\): Gravitational acceleration (9.80665 m/s²)
  • \(M\): Molar mass of Earth's air (0.0289644 kg/mol)
  • \(R\): Universal gas constant (8.3144598 J/(mol·K))

Air Density (\(\rho\))

The mass per unit volume of the Earth's atmosphere, computed using the ideal gas law.

\[ \rho = \frac{P}{R_{\text{specific}} \cdot T} \]
  • \(R_{\text{specific}}\): Specific gas constant for dry air (287.058 J/(kg·K))
  • \(T\): Temperature at altitude in Kelvin

Ground Speed (\(V_{ground}\))

The horizontal speed of the aircraft relative to the Earth's surface, accounting for wind vectors.

\[ V_{ground} = \sqrt{(V_{TAS} \cos(\psi) + V_{wind,x})^2 + (V_{TAS} \sin(\psi) + V_{wind,y})^2} \]
  • \(V_{TAS}\): True Airspeed
  • \(\psi\): Aircraft heading
  • \(V_{wind,x}\) / \(V_{wind,y}\): Wind vector components

2. Aerodynamics

Lift (\(L\)) & Drag (\(D\))

The primary aerodynamic forces. Lift counteracts the aircraft's weight, while Drag opposes its motion through the air.

\[ L = \frac{1}{2} \rho V_{TAS}^2 S C_L \]
\[ D = \frac{1}{2} \rho V_{TAS}^2 S C_D \]
  • \(S\): Wing reference area
  • \(C_L\): Lift coefficient
  • \(C_D\): Drag coefficient

Required Shaft Power (\(P_{\text{shaft\_kW}}\))

The mechanical power needed at the propeller shaft to overcome drag and propel the aircraft.

\[ P_{\text{shaft\_kW}} = \frac{D \cdot V_{TAS}}{\eta_p \cdot 1000.0} \]
  • \(\eta_p\): Propeller efficiency
  • \(D\): Drag (which equals Thrust \(T_{req}\) in steady flight)

3. Battery Dynamics

Terminal Voltage (\(V\))

The voltage output of the battery pack, accounting for open-circuit voltage and internal resistance drops.

\[ V = V_{oc}(SoC) - I \cdot R_{internal} \]
  • \(V_{oc}(SoC)\): Open-circuit voltage as a function of State of Charge
  • \(I\): Current drawn
  • \(R_{internal}\): Internal resistance of the battery cells

Degradation Rate (\(k_{DoD}\))

The non-linear scaling of battery wear based on the Depth of Discharge. Often modeled as a power law where \(k_{DoD} = 2.0\) signifies that doubling the DoD quadruples the wear.

4. Economics

Capital Recovery Factor (CRF)

A ratio used to calculate the present value of an annuity. It converts the initial capital expenditure into a stream of annualized costs.

\[ CRF = \frac{r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
  • \(r\): Annual discount rate
  • \(n\): Expected operational lifespan (in years)

Total Profit (profit_total_eur)

The net financial gain calculated over a specific operational period (e.g., annually).

\[ \text{profit_total_eur} = (\text{Revenue} - \text{OPEX}_{\text{per_flight}}) \cdot \text{Annual_Flights} - \text{CAPEX}_{\text{annualized}} \]

Revenue Lost

The opportunity cost incurred when passengers are offloaded to respect the Maximum Takeoff Weight (MTOW) constraint due to battery mass.

\[ \text{Revenue Lost} = \text{pax_offloaded} \cdot \text{ticket_price_eur} \]

5. Environment & Optimization

Effective CO2 (\(CO_{2,\text{effective}}\))

The total climate impact, including direct emissions and non-CO2 effects (e.g., contrails, NOx).

\[ CO_{2,\text{effective}} = CO_{2,\text{direct}} \cdot M_{\text{non-CO2}} \]
  • \(M_{\text{non-CO2}}\): Climate forcing multiplier

Objective Function (\(J\))

The multi-objective cost function minimized by the Differential Evolution algorithm.

\[ J = (w_{\text{eco}} \cdot \text{co2_ratio}) + (w_{\text{econ}} \cdot \text{profit_penalty}) \]
  • \(w_{\text{eco}}\): Weight for ecological impact
  • \(w_{\text{econ}}\): Weight for economic impact
  • \(\text{co2_ratio}\): \(CO_{2,\text{hybrid}} / CO_{2,\text{thermal}}\)
  • \(\text{profit_penalty}\): \((\text{Profit}_{\text{thermal}} - \text{Profit}_{\text{hybrid}}) / \text{Profit}_{\text{thermal}}\)

Tech Risk Cost (tech_risk_cost_eur(t))

The financial premium applied to novel technologies, which decays exponentially over time.

\[ \text{tech_risk_cost_eur}(t) = \text{Initial_Premium} \cdot e^{-\lambda \cdot (t - t_0)} \]
  • \(\lambda\): Decay constant (tech_risk_lambda)
  • \(t_0\): Baseline year