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Powertrain Module

The Powertrain module computes powertrain performance throughout the mission and distributes power across thermal and electrical subsystems depending on the aircraft configuration (Traditional, Serial Hybrid, or Parallel Hybrid).

Overview

The Powertrain model:

  • Converts a given propulsive power requirement into fuel power and electric power (the latter if hybrid)
  • To do so, it builds architecture-specific powertrain efficiency-chains using components efficiencies (see the Theoretical Reference)
  • Components efficiencies are computed based on the current flight conditions, if a model is provided. Otherwise, constant, user-specified values are used.
  • Presently, the available component models are:
    • Hamilton model for propeller efficiency, computed as a function of true airspeed, altitude, and power request
    • A surrogate response function of a twin-spool gas-turbine engine thermal efficiency, computed as a function of true airspeed, altitude, and power request
  • Computes powertrain mass from subsystem specific powers
  • Provides a GT engine power-lapse with altitude, to be used for constraint evaluation and engine rating

The powertrain system performance is evaluated at every timestep during the Mission integration.

Power Ratios

The main objective of the powertrain module is to compute the fuel/battery power required to deliver a certain propulsive power. At any instant in time, the required propulsive power is computed by the Mission module (see Mission):

\[ P_{prop}(t)= W_{TO} \left[ \frac{q V}{W_{TO}/S}\, C_D(C_L, M) + \beta P_s \right] \]

where \(q\) is the dynamic pressure, \(W_{TO}\) is the takeoff weight, \(S\) is the wing area, \(C_D(C_L, M)\) is the drag coefficient given by the polar model, \(\beta\) is the weight fraction, and \(P_s\) is the power excess required for climb.

Depending on the powerplant architecture, the required fuel/battery powers are computed as follows:

1. Traditional

The system solves for the following power ratios:

  • \( \displaystyle \frac{P_f}{P_p} \): fuel power to propulsive power
  • \( \displaystyle \frac{P_{gt}}{P_p} \): gas-turbine shaft power to propulsive power
  • \( \displaystyle \frac{P_{gb}}{P_p} \): gearbox output power to propulsive power
  • \( \displaystyle \frac{P_p}{P_p} = 1 \): reference value

These represent the power chain:

Fuel → Gas Turbine → Gearbox → Propeller → Propulsive Power

The power flow is represented as:

\[ A\,x = b , \]

where the upper-triangular matrix \(A\) is:

\[ A = \begin{bmatrix} -\,\eta_{GT} & 1 & 0 & 0 \\ 0 & -\,\eta_{GB} & 1 & 0 \\ 0 & 0 & -\,\eta_{PP} & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad b = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}. \]

where \(\eta_{GT}\), \(\eta_{GB}\), and \(\eta_{PP}\) are the gas-turbine, gearbox, and propeller efficiencies, respectively. The solution vector \(x\) is:

\[ x = \begin{bmatrix} P_f/P_p \\ P_{gt}/P_p \\ P_{gb}/P_p \\ 1 \end{bmatrix}. \]

Because \(A\) is upper-triangular, the solution corresponds to the sequential efficiency chain.

  1. Gearbox <- Propeller
\[ \frac{P_{gb}}{P_p} = \frac{1}{\eta_{PP}} \]
  1. Gas Turbine <- Gearbox
\[ \frac{P_{gt}}{P_p} = \frac{1}{\eta_{GB}\,\eta_{PP}} \]
  1. Fuel <- Gas Turbine
\[ \frac{P_f}{P_p} = \frac{1}{\eta_{GT}\,\eta_{GB}\,\eta_{PP}} \]

Thus, the system computes the inverse cumulative efficiencies needed to deliver \(P_p\). The calculation is performed using the following code, that solves the linear system \(A\,x = b\):

A = np.array([
    [-self.EtaGTmodel(alt, vel, pwr), 1, 0, 0],
    [0, -self.EtaGB, 1, 0],
    [0, 0, -self.EtaPPmodel(alt, vel, pwr), 1],
    [0, 0, 0, 1]
])

b = np.array([0, 0, 0, 1])

PowerRatio = np.linalg.solve(A, b)

yielding the power ratios vector: \(P_f/P_p, \ P_{gt}/P_p, \ P_{gb}/P_p, \ P_p/P_p\). Both \(\eta_{PP}\) and \(\eta_{GT}\) can be computed as functions of the flight conditions or left as constants (default). To use the models, the user must select the following options in the EnergyInput:

EnergyInput = {
    'Eta Propulsive Model': 'Hamilton',
    'Eta GT Model': 'PW127',
    # ... other options
}

2. Serial Hybrid

The system solves for the following power ratios:

  • \( \displaystyle \frac{P_f}{P_p} \): fuel power to propulsive power
  • \( \displaystyle \frac{P_{gt}}{P_p} \): gas-turbine shaft power to propulsive power
  • \( \displaystyle \frac{P_{gb}}{P_p} \): gearbox output power to propulsive power
  • \( \displaystyle \frac{P_{e1}}{P_p} \): electric generator power to propulsive power
  • \( \displaystyle \frac{P_{e2}}{P_p} \): electric motor power to propulsive power
  • \( \displaystyle \frac{P_{bat}}{P_p} \): battery power to propulsive power
  • \( \displaystyle \frac{P_{s2}}{P_p} \): shaft power to propulsive power
  • \( \displaystyle \frac{P_p}{P_p} = 1 \): reference value

with reference to the power chain in the image below:

Power Chain

The power flow is represented as:

\[ A\,x = b , \]

where the matrix \(A\) is:

\[ A = \begin{bmatrix} -\eta_{\mathrm{GT}} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\[6pt] 0 & -\eta_{\mathrm{EM1}} & 0 & 1 & 0 & 0 & 0 & 0 \\[6pt] 0 & 0 & 0 & -\eta_{\mathrm{PM}} & 1 & -\eta_{\mathrm{PM}} & 0 & 0 \\[6pt] 0 & 0 & 1 & 0 & -\eta_{\mathrm{EM2}} & 0 & 0 & 0 \\[6pt] 0 & 0 & -\eta_{\mathrm{GB}} & 0 & 0 & 0 & 1 & 0 \\[6pt] 0 & 0 & 0 & 0 & 0 & 0 & -\eta_{\mathrm{PP}}(h,V,P) & 1 \\[6pt] \phi & 0 & 0 & 0 & 0 & \phi - 1 & 0 & 0 \\[6pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \]
\[ b = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \]

The solution vector \(x\) is:

\[ x = \begin{bmatrix} P_f/P_p \\ P_{gt}/P_p \\ P_{gb}/P_p \\ P_{e1}/P_p \\ P_{e2}/P_p \\ P_{bat}/P_p \\ P_{s2}/P_p \\ P_p/P_p \end{bmatrix}. \]

The calculation is performed using the following code, that solves the linear system \(A\,x = b\):

A = np.array([[- self.EtaGT, 1, 0, 0, 0, 0, 0, 0],
              [0, - self.EtaEM1, 0, 1, 0, 0, 0, 0],
              [0, 0, 0, -self.EtaPM, 1, -self.EtaPM, 0, 0],
              [0, 0, 1, 0,  - self.EtaEM2, 0, 0, 0],
              [0, 0, - self.EtaGB, 0, 0, 0, 1, 0],
              [0, 0, 0, 0, 0, 0, - self.EtaPPmodel(alt,vel,pwr), 1],
              [phi, 0, 0, 0, 0, phi - 1, 0, 0],
                      [0, 0, 0, 0, 0, 0, 0, 1]])

b = np.array([0, 0, 0, 0, 0, 0, 0, 1])

PowerRatio = np.linalg.solve(A,b)

3. Parallel Hybrid

The system solves for the following power ratios:

  • \( \displaystyle \frac{P_f}{P_p} \): fuel power to propulsive power
  • \( \displaystyle \frac{P_{gt}}{P_p} \): gas-turbine shaft power to propulsive power
  • \( \displaystyle \frac{P_{gb}}{P_p} \): gearbox output power to propulsive power
  • \( \displaystyle \frac{P_{e1}}{P_p} \): electric generator power to propulsive power
  • \( \displaystyle \frac{P_{bat}}{P_p} \): battery power to propulsive power
  • \( \displaystyle \frac{P_{s1}}{P_p} \): shaft power to propulsive power
  • \( \displaystyle \frac{P_p}{P_p} = 1 \): reference value

with reference to the power chain in the image below:

Power Chain

The power flow is represented as:

\[ A\,x = b , \]

where the matrix \(A\) is:

\[ A = \begin{bmatrix} -\,\eta_{GT}(h,V,P) & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\,\eta_{EM1} & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\,\eta_{PM} & 1 & -\,\eta_{PM} & 0 & 0 \\ 0 & 0 & 1 & 0 & -\,\eta_{EM2} & 0 & 0 & 0 \\ 0 & 0 & -\,\eta_{GB} & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -\,\eta_{PP}(h,V,P) & 1 \\ \phi & 0 & 0 & 0 & 0 & \phi - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \]
\[ b = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}. \]

The solution vector \(x\) is:

\[ x = \begin{bmatrix} P_f/P_p \\ P_{gt}/P_p \\ P_{gb}/P_p \\ P_{e1}/P_p \\ P_{bat}/P_p \\ P_{s1}/P_p \\ P_p/P_p \end{bmatrix}. \]

The calculation is performed using the following code, that solves the linear system \(A\,x = b\):

A = np.array([[- self.EtaGTmodel(alt,vel,pwr), 1, 0, 0, 0, 0, 0],
              [0, -self.EtaGB, -self.EtaGB, 1, 0, 0, 0],
              [0, 0, 0, 0, 1, -self.EtaPM, 0],
              [0, 0, 1, 0, - self.EtaEM, 0, 0],
              [0, 0, 0, - self.EtaPPmodel(alt,vel,pwr), 0, 0, 1],
              [phi, 0, 0, 0, 0, phi - 1, 0],
              [0, 0, 0, 0, 0, 0, 1]])

b = np.array([0, 0, 0, 0, 0, 0, 1])

PowerRatio = np.linalg.solve(A,b)

Engine Power Lapse With Altitude

The constraint analysis requires a preliminary estimation of the engine power at different altitudes. Thermal engine maximum power decreases with altitude due to reduced air density.

The power lapse ratio is:

\[ \alpha(h) = \left(\frac{\rho(h)}{\rho(0)}\right)^n \]

where \(n=0.75\) is the power lapse exponent. In the code:

def PowerLapse(self,altitude,DISA):
        """ Full throttle power lapse, to be used in constraint analysis. Source: Ruijgrok, Elements of airplane performance, Eq.(6.7-11)"""
        n = 0.75
        lapse = (ISA.atmosphere.RHOstd(altitude,DISA)/ISA.atmosphere.RHOstd(0.0,DISA))**n
        return lapse

References

de Vries, R., Brown, M., & Vos, R. (2019). Preliminary Sizing Method for Hybrid-Electric Distributed-Propulsion Aircraft. Journal of Aircraft: devoted to aeronautical science and technology, 56(6), 2172-2188.