Performance identification marks our third layer of pattern recognition (referencing our layered representation).
In our methodology, Performance specifically pertains to fuel consumption. Here, we elaborate
on how this consumption is calculated, leveraging lower-level patterns alongside an additional
performance pattern (the slope diagram, detailed below).
At this stage, selecting an existing performance model is imperative. Presently, we have retained two performance models: BADA and OPENAP.
The BADA model has been elaborated by Eurocontrol (Refer to this link
for an introductory paper on the BADA Model). It serves as a self-contained tool, implementing a complete trajectory 4D profile.
In our case, we possess aircraft trajectories gathered from surveillance (radar, ADS-B), and our
objective is to determine the performance (essentially, fuel consumption). Therefore our use ot the BADA model is limited to some functionalities (namely the drag drag and the maximum thrust).
The OPENAP model has been developed by Dr Junzi Sun as a continuation of his PhD work (at Delft University) on aircraft performance modeling using open data (2015-2019).
Using BADA, it is possible to estimate the fuel cost from the value of the thrust. Using OPENAP, not only the fuel but also the emissions can be estimated. The use of BADA is more restrictive, since the license requires aceptance frome Eurocontrol. On the other hand, OPENAP is open source.
Both model rely on a fundamental equation, denoted as the Total Energy Model (TEM), that we introduce below.
The derivation of the TEM involves considering the conservation of energy for an aircraft in
flight.
As depicted in the figure, the power of the aircraft is split into horizontal and
vertical components. Let's introduce our notations: the Air Speed of the aircraft is
denoted by $V$, representing the horizontal speed, while the vertical speed is denoted by
$V_Z$. Apart from its own weight, the aircraft experiences two external forces: Thrust,
denoted by $T$, and Drag, denoted by $D$. We assume that these forces are horizontal,
neglecting the small angle between the aircraft's direction and the horizontal plane (with drag
opposing the speed and thrust aligned with the speed).
Now, we can assess the total energy transferred to the aircraft, which is $(T-D) V$. This energy is
then converted into a horizontal component related to acceleration or deceleration, $m V
\frac{\mathrm{d}V}{\mathrm{d}t}$, and a vertical component related to vertical speed, $m g V_{Z}$.
The comprehensive equation illustrating these quantities is provided . The three terms in the equation correspond
to the .
We reproduce below the fundamental equation of the TEM, to which we will often refer to in the sequel:
\begin{equation}
(T-D)V = m V \frac{\mathrm{d}V}{\mathrm{d}t} + m g V_{Z}
\label{badaf}
\end{equation}
When the aircraft is at a steady altitude, its vertical speed $V_{Z}$ is null, and the fundamental equation of the TEM \ref{badaf} reduces to \begin{equation} (T-D)V =m V \frac{\mathrm{d}V}{\mathrm{d}t} \label{badasteady} \end{equation} We use the temporal speed schedule (the left illustration, with axes Time and IAS), from which we derive the air speed. We have identified maneuvers in the speed diagram, and when the aircraft is steady these maneuvers are either constant speed or change of IAS speed. Both cases lead to a simple derivation of the acceleration (the acceleration is null for constant speed and equal to $\frac{V_{end}-V_{init}}{t_{end}-t_{init}}$ for change of IAS speed). The performance model gives us the Drag $D$, so we determine the thrust by using Equation \ref{badasteady}, which gives us the Thrust \begin{equation} T = D + m \frac{\mathrm{d}V}{\mathrm{d}t} \label{tsteady} \end{equation} From this Thrust the performance model gives us the fuel consumption, which is what we wanted. We have assumed that the mass $m$ was known, we will show later how to determine it.
When the aircraft is in vertical evolution, we can rephrase Equation \ref{badaf} with the following
trick:
\begin{align}
(T-D)V & = m V \frac{\mathrm{d}V}{\mathrm{d}t} + m g V_{Z} \nonumber\\
& = m V \left( \frac{\mathrm{d}V}{\mathrm{d}H} \frac{\mathrm{d}H}{\mathrm{d}t} \right) + m g V_{Z}
\text{
(where } H \text{
is the altitude)} \label{vh} \\
& = m V \left( \frac{\mathrm{d}V}{\mathrm{d}H} V_Z \right) + m g V_{Z} \nonumber\\
& = m V_Z \left( V \frac{\mathrm{d}V}{\mathrm{d}H} + g \right) \label{si1}
\end{align}
We notice that our trick has enabled us to get rid of the acceleration
$\frac{\mathrm{d}V}{\mathrm{d}t}$, which, in the case of an evolutionary aircraft, can be replaced by
the
product $V_Z \frac{\mathrm{d}V}{\mathrm{d}H}$. Actually, the term $\frac{\mathrm{d}V}{\mathrm{d}H}$
makes
sense since the aircraft is in vertical evolution, so its altitude $H$ is varying together with
its
air speed $V$.
Equation \ref{si1} can be rewritten as
\begin{equation}
T = D + m \left( \frac{V_Z}{V} \right)\left( V \frac{\mathrm{d}V}{\mathrm{d}H} + g \right)
\label{tmd}
\end{equation}
And we now detail how to estimate all the terms to the right of this Equation, in order to estimate the
left
part $T$. As in the steady aircraft case, once we have estimated the thrust $T$, we derive the fuel
consumption with the help of the performance model.
In our discussion of lower-level patterns, we introduced the speed schedule, providing the Indicated Air Speed as a function of Altitude. When we closely examine this speed schedule, as depicted on the left, the of the speed schedule concerning altitude yields the derivative $\frac{\mathrm{d}IAS}{\mathrm{d}H}$. Utilizing this derivative and an analytical expression of Air Speed as a function of IAS and altitude $V(IAS,H)$, we can easily derive the term $\frac{\mathrm{d}V}{\mathrm{d}H}$. Further details on this point are not provided.
Here, we elaborate on how the term $\left(\frac{V_Z}{V}\right)$ in Equation \ref{tmd} can be approximated. To achieve this, we introduce a new diagram illustrating distance (horizontally) and altitude (vertically). Let's clarify our definition of distance.
For a landing aircraft, distance refers to the distance to the runway, computed by summing up the lengths of all route segments until reaching the final runway. This diagram is presented on the left, where identified maneuvers correspond to segments with a constant slope, as depicted in the figure. Upon landing, we can recognize the .
Referred to as the slope diagram, this diagram captures slope changes. At any point on this diagram, it is evident that . The slope diagram can also be constructed for an airplane during takeoff, where the distance is measured from the runway. While we won't delve into further details regarding the slope diagram's usage here, we will utilize it in subsequent discussions, particularly when determining optimal scenarios.
The determination of the mass is presented below and the Drag $D$ is provided by the performance model. So we have all the terms to the right of Equation \ref{tmd}, which allow us to determine the thrust $T$, and finally the fuel consumption (thanks, again, to the performance model).
In our methodology the determination of the mass is made for aircraft which are either landing or taking off. The algorithm is different for these two cases.
As we shall see, the vertical profile is very sensitive to the value of the mass, and we use this property. When the aircraft takes off, the thrust corresponds to the maximal thrust, which is provided by the performance model. Using Equation \ref{tmd} we can express the vertical speed as : \begin{equation} V_Z = V \frac{\left(\frac{T-D}{m}\right)}{\left(V \frac{\mathrm{d}V}{\mathrm{d}H} + g\right)} \label{vvz} \end{equation} All the terms in this Equation are known except the mass (the term $\frac{\mathrm{d}V}{\mathrm{d}H}$ is given by the speed schedule as explained above, together with the air speed $V$). So, for an aircraft taking off, from the vertical speed we extrapolate the vertical profile from the take off till a certain duration, for several candidate values of the mass, and we retain the value of the mass which "sticks" most to the actual profile. The Figure to the left illustrates our algorithm for several choices of the mass:
In the example of the Figure our algorithm returned a value of 72000 kg, for a duration of 500 seconds after the take off (the blue dashed line on the Figure).
For a landing aircraft, we rewrite Equation \ref{si1} as
\begin{equation}
m = \frac{T-D}{\left(\frac{V_Z}{V} \right) \left(V \frac{\mathrm{d}V}{\mathrm{d}H} + g \right)}
\label{eqm}
\end{equation}
and we estimate the mass from the values of the terms at the landing of the aircraft. For
the
Thrust and the Drag, we retain the values provided by the performance model.
The ratio $\frac{V_Z}{V}$ corresponds (as we have seen above) to the slope of the slope diagram,
with
the
particularity that we are at the landing, and this ratio is given by the ILS slope of 3 degrees, as
.
For
the term $\frac{\mathrm{d}V}{\mathrm{d}H}$ we proceed as explained above, with
the
particularity that we are at the landing, and we retain the constant IAS maneuver at the
landing (illustrated ).