This annex is attached in order to complement the given information in Section 3 associated with the fuel consumption algorithm. The entire procedure including all strategies and assumptions are detailed at following [3].
The used engine for this analysis is a twin-jet one and its fuel consumption is described for nominal and idle thrust conditions. In Eq. (8.1) is represented the general expression for the nominal fuel-flow rate.
"#$% = '()* · ' = ("-+ "/0) · ' = 1
As it can be observed in Equation (8.1), the nominal fuel flow can be obtained by the product between thrust (T) and its specific fuel consumption (TSFC). This last one is computed as a linear function of airspeed (V). The BADA coefficients
!"#$%& !"( are the nominal fuel-flow rate model coefficients and depend on the aircraft type.
Thrust idle is the minimum thrust obtained when the engine power control lever is set to its minimum position. So, the minimum fuel-flow rate for idle thrust is given Equation 8.2. rate model coefficients and depend on the aircraft type.
Equation (8.1) and (8.2) show that the jet engine fuel-flow rate computations need parameters such as altitude, airspeed, thrust:
• Altitude is available from the radar traces data.
• Airspeed is estimated using position and the wind states.
• Thrust is estimated using the equations of motion.
Integrating the fuel-flow rate, the amount of fuel consumed can be computed. It can be seen in Eq. (8.3).
"# = %&' (). +)
-.
/
where, ! is the fuel-flow rate given in kg/s !" is the flight final time
• Equations of motion
Through the equations of motion, the aircraft can be modelled considering it as a mass point.
" = 1
(& + ℎ)· +,- (.. 0)
" = 1
% + ℎ · )*+,· -./ (1. 3)
ℎ = $% ('. ))
where, ! is the latitude ! is the longitude
ℎ is the geometric altitude
! is the mean radius of the Earth
!"# is the north component of the ground speed !"# is the east component of the ground speed
!" is the climb or descent rate depending if it is positive or negative
In Figure 8.1. the impact of the wind on the aircraft performance is depicted.
Figure 8.1. Speed and Wind components
The variables observed in Figure 8.1. are listed below:
• !" : ground speed
• !" : wind velocity
• !": magnitude of the horizontal component of the airmass-relative aircraft
velocity
• !" : heading angle of the ground-relative aircraft speed with respect to the local north direction
• ! : heading angle of the airmass-relative aircraft velocity with respect to the local north direction
• !": heading angle of the wind velocity with respect to the local north
direction
• Also the different wind velocity components are: !" as the north, !" as the east and !" as the vertical component.
Actually, the magnitude of the airmass-relative speed (! ) is the true airspeed (TAS). At following, the acceleration is obtained using its derivation and the flight principles modelling the aircraft as a point of mass.
" = $ cos ( -*
+ -, · sin ( -01cos 2 cos 3 -04sin 2 cos 3- 05sin 3 (7. 9)
where, ! is thrust " is drag
! is angle of attack ! is mass
! is gravitational acceleration ! is the flight path angle
Projecting the true airspeed and using the flight path angle, the horizontal component !" is computed as:
"# = " cos ( (*. *) The kinetic equations of airmass-relative heading angle and flight path angle are: gravitational acceleration and slowly mass variation. The altitude rate obtained in Equation (8.6) can be rewritten in terms of airspeed, flight path angle and the wind vertical component as in Eq. (8.11).
ℎ = $% = $ sin ) + +% (-. //)
The wind states vary with time and also with position, so the time derivative of each wind component cab be computed as:
"# =%"&
Equation (8.13) shows its dependence on parameters like drag, mass, airspeed or the wind states among others. If these last states are null, considering a calm atmosphere model, the thrust expression of Equation (8.14) is obtained.
" = $ + &' + &(ℎ
' (+. -.)
In order to maintain the balance in Equation (8.14) it is observed that the thrust during descent can be less than the minimum one generated by the engines due to the errors of the drag model or the aircraft weight. So, the minimum thrust is established in Eq. (8.15) for a twin-jet engine aircraft.
"#$% = '(-'*· ℎ
The aerodynamic drag force is obtained as a function of the drag coefficient and the dynamic pressure as:
In order to maintain the course in presence of wind, the across-track of the wind has to be cancelled. In this way, the following equations are used:
"#cos ' + )* = ",* (.. 01) and
"#sin ' +)* = ",* (.. 0.)
In this way, the heading angle is obtained:
" = tan-( )*+-,+
)*--,- (/. 12)
The resultant magnitude of the horizontal component of the airmass-relative aircraft velocity is:
"# = "%&-(& )+ "%+-(+ ) (-. /0)
Using Eq. (8.20) together with Eq. (8.8) and Eq. (8.11), the true airspeed can be calculated in terms of wind velocity components as:
" = "$%-'% ( + "$*-'* ( + (ℎ-'-)( (/. 12)
Prescribing " is obtained basing it on the aircraft trajectory and differentiating Eq. (8.19).
" = $%&-(& $%)-() - $%)-() $%&-(&
$%&-(& * + $%)-() * (-. //)
Substituting " from Eq. (8.22) and using the above relations of Eq. (8.8), (8.17), (8.18) and (8.20) the following relation is obtained:
" sin & = "( = )*+,-+ )/+,0 (2. 45)
Separating !" $%& !' as:
"# = -& '()-*)
'()-*) ++ '(--*- +
(/. 12)
and
"# = % &'(-*(
&'(-*( #+ &',-*, #
(.. 01)
In order to compute the other component of ! Eq. (8.11) is differentiated:
" = ℎ-&'-( sin "
( cos " (/. 12) Substituting Eq. (8.26) in Eq. (8.10) and using Equations (8.11), (8.17) and (8.18):
" cos & = "( = )*+,-+ )/+,0+ )1ℎ + )3+,-ℎ + )4+,0ℎ + )15 (7. 9:) Separating !", !$, !%, !& ()* !+ as:
!" = $ %&'-)' )*
%&'-)' + + %&--)- + %&'-)' + + %&--)- + + (ℎ-)*)+
(2. 42)
!" = $ %&'-)' )*
%&+-)+ , + %&'-)' , %&+-)+ ,+ %&'-)' , + (ℎ-)*),
(2. 45)
"# = % &'(-*( + + &'--*- +
&'(-*( + + &'--*- + + (ℎ-*0)+
(2. 45)
!" = -% &'(-)(
&'(-)( *+ &',-), * &'(-)( * + &',-), * + (ℎ-)/)*
(2. 45)
and
!" = -% &'(-)(
&'*-)* + + &'(-)( + &'*-)* + + &'(-)( + + (ℎ-)/)+
(2. 45)
So, the lift force and the bank angle can be computed using Eq. (8.23) and Eq.
(8.27).
" = "$%+ "%' (). ++)
" = tan-( )*
)+ (-. /0)
!" can be computed using Eq. (33) and knowing that:
"# = 2&
'()* (,. ./)
Drag force can be computed using Equations (8.16) and thrust can be computed using Eq. (8.13).
Finally, the fuel-flow rate, which is the main objective of this study, is determined using Eq. (8.1) and (8.2) and also the above computed variables depending on its nominal or idle state.
• State Estimation
In order to estimate the above aircraft states a Kalman Filter is employed. It is applied to Equations (8.4), (8.5) and (8.6). The needed observations are based on the aircraft trajectory (latitude, longitude and altitude). In Figure 8.2 a Proportional-Integral-Derivative (PID) controller-based estimator is shared for the altitude parameter.
Figure 8.2. Altitude estimator
The proportional gain (KP), the integral gain (KI) and the derivative gain (KD) are set and combined in order to reach an optimal state. The poles of the closed-loop are located in the left side of the s-plane.
The main objective is to estimate the altitude rate and the vertical acceleration.
In this way, the controller uses a double integrator of aircraft with altitude rate and altitude as measurements for feedback.
At following, the first and second derivatives of the altitude using the Taylor Series approximation for the time-step k, which is separated by Δt from the next time step k+1, are obtained.
ℎ # + 1 = ℎ # + 1 -ℎ #
∆) (+. -.) and
ℎ # + 1 = ℎ # + 1 -2ℎ # + ℎ #-1
∆*+ (-. /0)
Thus, the estimated value is:
ℎ # + 1 = '(∆*++ ',∆*- ℎ # + 1 + 2 + '/∆* ℎ # -ℎ #-1 + ∆*+1(#)
(1 + '/∆* + '(∆*++ ',∆*-) (4. 64)
and
" # + 1 = " # + '(∆* ℎ # + 1 -ℎ # + 1 (.. 01)
The Eq. (8.39), implements the integral feedback term. Meanwhile, the first terms of Eq. (8.38) are initialized with:
ℎ" = ℎ 1 &'( ℎ) = ℎ(2)
The altitude rate and vertical acceleration can be estimated using Equations (8.36) and (8.37) based on the altitude estimation of Eq. (8.38). The altitude rate and the acceleration can be computed in another way, which is using the altitude observations instead of the estimated ones. But, there is a big problem based on the introduced noise. Noise would be amplified by the differencing process so, using the PID filter, the noise of the altitude measurement is reduced obtaining in this way a soother response of the altitude rate and the acceleration.
The implemented estimator can be applied to the others parameters. For example, the acceleration terms using derivatives of Eq. (8.4) and (8.5) are given below:
"#$ = & + ℎ ) + ℎ) (+. -.)
"#$ = & + ℎ cos ,- - & + ℎ sin ,,- + 123 ,ℎ- (5. 78)