Fuel and time savings by flying continuous cruise climbs
Estimating the benefit pools for maximum range operations
Ramon Dalmaua, Xavier Pratsa*
a
Telecommunications and Aerospace Engineering School of Castelldefels
Technical University of Catalonia - Barcelona Tech. Catalonia, Spain
Abstract
This paper estimates the benefits, in terms of fuel and time, that continuous climb operations can save during the cruise phase of
the flights, assuming maximum range operations. Based on previous works, a multiphase optimal control problem is solved by
means of numerical optimisation and using accurate aicraft performance data from the manufacturer. Optimal conventional
trajectories (subject to current air traffic management practices and constraints) are computed and compared with ideal
continuous operations only subject to aircraft performance constraints. Trip fuel and time for both concepts of operations are
quantified for two aircraft types (a narrow-body and a wide-body airplane) and a representative set of different trip distances and
landing masses. Results show that the continuous cruise phase can lead to fuel savings ranging from 0.5% to 2% for the Airbus
A320, while for an Airbus A340 the dispersion is lower and savings lie in between 1% and 2%. Interestingly, trip time is also
redudced between 1% and 5%.
Keywords: continuous cruise climbs; unrestricted flight; fuel savings; trajectory optimization
1. Introduction
According to IATA (2010), fuel was the largest single cost item for the global airline industry in 2008, representing
more than the 30% of the total operating cost. An optimal flight vertical profile, in terms of minimizing fuel
consumption, is not composed by level segments at constant (cruise) altitudes. In fact, the optimal profile consists of
a continuous climb, with a climb rate that reduces progressively as long as the aircraft approaches the optimal
altitude, followed by a continuous descent with the engines at idle (Erzberger et al., 1975; Sorensen & Waters, 1981;
Miller, 1993; Lovegren & Hansman, 2011; Soler et al., 2012).
As it is well known, however, in the current concept of operations aircraft are asked to fly at constant cruise
altitudes in order to separate them safely. In this way, it is much easier to apply strategic vertical separation among
them and the workload of air traffic control (ATC) operators is significantly reduced. Regarding crossing aircraft
flows in terminal maneuvering areas (TMA), climbs and descents are usually interrupted by imposing level flight
*
Corresponding author. Tel.: +34934134125.
E-mail addresses: ramon.dalmau@estudiant.upc.edu (R. Dalmau), xavier.prats@upc.edu (X. Prats)
1
segments at constant altitudes. Furthermore, in busy TMAs path stretching and radar vectoring are also common
ATC separation practices that deviate flight trajectories from the reference (optimal) lateral routes.
Several studies in the last decades have assessed optimal flight profiles for commercial aircraft, mainly
focused in TMA operations - i.e. with continuous descent operations (CDO) and continuous climb operations
(CCO). See for instance Clarke et al. (2004), Thompson et al. (2013), Jin et al. (2013) or Zhao et al. (2013) and the
references therein. Optimal cruise operations are assessed, for instance by Betts & Cramer (1995) and Soler et al.
(2012) and the conclusions arising from these works show the advantages of such operations. The actual quantitative
benefits, however, in terms of fuel savings and accurate determination of the optimal vertical trajectories, are hard to
assess mainly due to approximations in aircraft and engine performance models. Miller (1993) and Kaiser et al.
(2011) emphasize the importance to take into account air compressibility effects into aerodynamic drag equations in
order to obtain accurate trajectories. Aircraft performance models typically used in air traffic management (ATM)
studies, such as the widely used Eurocontrol’s Base of aircraft data (BADA, version 3.6 or lower), are not accurate
enough to derive correct fuel consumption figures and present some known limitations (Senzing et al., 2009).
Furthermore, realistic engine models are also very important to take into account, since actual engine performance
and limitations have a great impact on the maximum and optimum flight altitudes and therefore, on the optimal
speed profiles, trip time and derived fuel quantities.
This paper attempts to accurately quantify the benefits of continuous cruise operations by considering the
flight as a whole and assuming it is executed in maximum range conditions (i.e. fuel is minimized). Accurate aircraft
performance data, derived from Airbus Performance Engineering Programs (PEP), have been used to model
aerodynamic drag, engine thrust and fuel flow. It is out of the scope to assess the obvious safety and operational
issues of an eventual wide-scale implementation of such operations. Thus, the aim of this paper is to motivate future
research efforts and technologies to make them possible.
Next section presents the methodology used to simulate current conventional operations, where realistic
ATC operational restrictions are enforced; and continuous operations, where an uninterrupted and continuous climb
is followed by an also uninterrupted continuous descent. Then, in section 3, some numerical results in terms of fuel
and time savings are given for two aircraft types (an Airbus A320 and an Airbus A340), while Section 4 concludes
this paper.
2. Trajectory
computation methodology
2
Flight trajectories are typically computed in such a way that a compound cost function is minimized. This cost
function is the sum of the trip fuel and the trip time multiplied by a scalar: the Cost Index. This factor reflects the
relative importance of the cost of time with respect to fuel costs. Therefore, the higher the Cost Index is, the more
importance will be given to the trip time in the cost function and faster optimal aircraft speeds will result from the
optimization process. In this paper, only fuel consumption has been minimized when computing conventional and
continuous operations, i.e. the Cost Index has been assumed to be zero. This parameter has a significant commercial
importance for aircraft operators and determining its value may be a very complex task, since the cost of time is not
easy to estimate and may also change tactically during the flight (Cook et al., 2009). Furthermore, it should be noted
that nowadays airlines are adopting very low Cost Index values to account for the rising fuel prices (Hansman et al.,
2014) and consequently, flying at speeds close to the maximum range ones.
The optimization of an aircraft trajectory, as a 4 dimensional continuum, can be formulated as a multiphase constrained optimal control problem, where a cost function is minimized at the same time several constraints
on the decision variables are imposed. A particular set of constraints in this kind of problems is expressed by
differential equations that describe the dynamics of the aircraft (see section 2.1). Moreover, in order to obtain
operationally sound trajectories, other constraints must also be specified, as detailed in section 2.2.
2.1. Aircraft model
This paper assumes a nonlinear point-mass representation of the aircraft (where forces are applied at its center of
gravity) reduced to what is commonly called a gamma-command model (where continuous vertical equilibrium is
assumed). Gamma-command point mass models have been reported to provide sufficient fidelity for ATM purposes
and performance studies such the one presented in this paper. Aircraft dynamics are described in the air reference
frame assuming flat non-rotating earth and neglecting wind components, yielding to the following set of differential
equations:
𝑑𝑣
𝑇−𝐷
= 𝑣̇ =
− 𝑔 sin 𝛾
𝑑𝑡
𝑚
𝑑𝑠
= 𝑠̇ = 𝑣 cos 𝛾
𝑑𝑡
𝑑ℎ
= ℎ̇ = 𝑣 sin 𝛾
𝑑𝑡
𝑑𝑚
= 𝑚̇ = −𝐹𝐹.
𝑑𝑡
(1)
The state vector 𝑥⃗ = [𝑣, 𝑠, ℎ, 𝑚] is formed respectively, by the true airspeed, the along path distance, altitude and the
3
mass of the aircraft; 𝑇 is the total thrust; 𝐷 is the aerodynamic drag; 𝑔 is the gravity acceleration (assumed to be
constant); 𝛾 is the aerodynamic flight path angle and 𝐹𝐹 is the fuel flow. The control vector considered is 𝑢
⃗⃗ =
[𝛾, 𝜋], where 𝜋 is the throttle setting.
All aerodynamic and engine parameters are represented by continuous polynomials, that ensure continuity
for the first and second derivatives as it is required for the numerical solvers used here. Aerodynamic Lift and Drag
forces are, respectively, modeled as:
𝐿=
1
𝜌𝑆𝑣 2 𝐶𝐿 ;
2
𝐷=
1
𝜌𝑆𝑣 2 𝐶𝐷 ;
2
(2)
where 𝐶𝐿 and 𝐶𝐷 are, respectively, the Lift and drag coefficients, 𝑆 the wing reference area and 𝜌 the density of the
air. Since vertical equilibrium is assumed (and the effects of the bank angle in turns are also neglected), the Lift
force during the flight is simply 𝐿 = 𝑚𝑔 cos 𝛾, and by using equation (2) the lift coefficient can be computed as:
𝐶𝐿 =
2𝑚𝑔 cos 𝛾
.
𝜌𝑆𝑣 2
(3)
The drag coefficient, in turn, is expressed as a function of the lift coefficient (𝐶𝐿 ), the Mach number (𝑀)
and also the flaps/slats configuration of the aircraft (𝜉). This relationship considers air compressibility effects,
which cannot be neglected for nominal cruising speeds of typical commercial aircraft (between M.78 and M.82
approximately). In this paper, a polynomial fitting similar to the model proposed by Kaiser et al. (2011) is used,
giving us a very accurate approximation of the drag coefficient:
𝐶𝐷 (𝜉, 𝑀, 𝐶𝐿 ) = 𝐶𝐷0 (𝜉, 𝑀) + 𝐾𝑖 (𝜉, M)(𝐶𝐿 − 𝐶𝐿0 (𝜉, M))2 ,
(4)
where different coefficients 𝐶𝐷0 (𝜉, 𝑀), 𝐾𝑖 (𝜉, 𝑀) and 𝐶𝐿0 (𝜉, 𝑀) for each aircraft configuration (𝜉) have been
obtained after a polynomial least square fitting function process using tabulated aircraft aerodynamic data. This data
has been obtained from Airbus PEP software suite, which provided us with accurate (and certified) values for
aerodynamic Drag (and also engine performance as explained below) for different flight conditions. .I
The International Standard Atmosphere specified by ICAO (1993) has been considered, where the density
(𝜌), pressure (𝑝) and temperature (𝜏) magnitudes are expressed as functions of the altitude. The following
normalized variables are also used in this paper:
4
𝛿=
𝑝
;
𝑝0
𝜃=
𝜏
;
𝜏0
𝜎=
𝜌
;
𝜌0
(5)
where 𝑝0 , 𝜏0 and 𝜌0 are respectively, the standard pressure, temperature and density values at sea level.
Models for engine parameters (i.e. thrust, fuel flow, and revolutions of the engine fan) are developed using
the Buckingham Pi technique of dimensional analysis (Air Force Test Pilot School, 1993). The resulting models for
𝑇
𝐹𝐹
𝛿
𝛿√𝜃
reduced thrust ( ), reduced fuel flow (
𝑁
) and reduced revolutions of the engine fan ( 1 ) allow us to combine a
√𝜃
six variable equations for thrust and fuel flow into simpler, three variable equations. These equations easily account
for variations in temperature and pressure and are of the form that engine thrust models are typically provided.
For most turbofan engine equipping typical commercial aircraft, throttle setting (𝜋 ∈ [0,1]) proportionally
commands the revolutions of the engine fan (𝑁1 ):
𝜋=
𝑁1 − 𝑁1 𝑖𝑑𝑙𝑒
.
𝑁1 𝑚𝑎𝑥 − 𝑁1 𝑖𝑑𝑙𝑒
(6)
The maximum revolutions of the engine fan (𝑁1 𝑚𝑎𝑥 ) and the residual revolutions when the throttle is set to
zero (𝑁1 𝑖𝑑𝑙𝑒 ) are modeled with a third degree polynomial approximation as:
3
3
𝑁1 𝑘 = ∑ ∑ 𝑐𝑖𝑗𝑘 𝜃 𝑖 𝑀 𝑗
𝑘 ∈ {𝑚𝑎𝑥, 𝑖𝑑𝑙𝑒}.
(7)
𝑖=0 𝑗=0
where the engine performance tables of Airbus PEP databases have been used to obtain the polynomial coefficients
𝑐𝑖𝑗𝑘 after a least squares function fitting process.
Following the same methodology, Thrust and Fuel Flow are also modeled by a third order polynomial as a
function of the reduced revolutions of the engine fan and 𝑀:
3
3
𝑇 = 𝑛𝑒 𝛿 ∑ ∑ 𝑐𝑖𝑗𝑇 (
𝑖=0 𝑗=0
3
3
𝑖
𝑁1
√𝜃
) 𝑀𝑗
𝑁1
𝑖
(8)
𝐹𝐹 = 𝑛𝑒 𝛿√𝜃 ∑ ∑ 𝑐𝑖𝑗𝐹𝐹 ( ) 𝑀 𝑗
√𝜃
𝑖=0 𝑗=0
5
being 𝑛𝑒 the number of engines of the aircraft.
2.2. Trajectory constraints modeling
Besides the dynamic equations (1), other constraints must be specified in order to model certain operational aspects
or limitations. In this paper, the initial and final conditions of the problem are taken, respectively, at the moment the
slats are retracted (after the take-off) or extended (before the landing). The remaining parts of the take-off and
approach are not optimized because almost no degrees of freedom are left for optimization, since the trajectory is
heavily constrained with operational procedures. For the initial point of the trajectory, the mass of the aircraft is not
fixed (it will be determined by the optimization process itself, since the trip fuel is being minimized), being all the
remaining state variables fixed to typical operational values.
Generic bounding constraints on certain variables are specified as follows:
𝛾𝑚𝑖𝑛 ≤ 𝛾 ≤ 𝛾𝑚𝑎𝑥 ;
0 ≤ 𝜋 ≤ 1;
𝐶𝐴𝑆𝑚𝑖𝑛 ≤ 𝑣𝐶𝐴𝑆 ≤ 𝑉𝑀𝑂;
𝑀𝑚𝑖𝑛 ≤ 𝑀 ≤ 𝑀𝑀𝑂
(9)
where 𝑣𝐶𝐴𝑆 is the calibrated airspeed (CAS) and 𝛾𝑚𝑖𝑛 , 𝛾𝑚𝑎𝑥 , 𝐶𝐴𝑆𝑚𝑖𝑛 , 𝑀𝑚𝑖𝑛 , 𝑉𝑀𝑂 (Velocity Maximum in
Operations) and 𝑀𝑀𝑂 (Mach Maximum in Operations) are aircraft dependent scalars.
For continuous operations, the trajectory has been modeled by enforcing only the constraints depicted in
equations (1) and (9), along with the initial and final conditions explained above. Therefore, no ATC constraints are
imposed (such as minimum rate of climbs, speed limits or constant cruise altitudes). On the other hand, conventional
procedures require to model more complex constraints and the trajectory is split in several phases as shown in Table
1.
A typical climb in current operational context consists of accelerating to an optimal CAS as soon as the
aircraft has completely retracted flaps and slats. Then, this speed is maintained constant up to the crossover altitude,
where the climb is resumed at a constant Mach number. In most controlled airspaces, ATC procedures typically
impose a maximum CAS of 250 kt under FL100 (ICAO,2001). Typical descents are done in a similar way: starting
to descend at a constant Mach number, then at constant CAS, reducing to 250 kt below FL100 (if applicable) and
finally decelerating to the suitable speed to start extending high-lift devices.
The most important constraint of current concept of operations is perhaps the requirement to fly at constant
cruise altitudes. In general, the lower the aircraft mass the higher the most fuel-efficient cruise altitude will be. Thus,
6
since aircraft is continuously burning fuel (and reducing weight) operators can plan in advance one or more stepclimbs for flights that are long enough. These changes of altitude are always subject to an ATC clearance and are
typically performed with 2000 ft intervals. According to the F.A.A. (2014) and most European regulations (see for
instance G.B.C.A authority, 2013), a minimum rate of climb of 500 ft/min is enforced to all aircraft in order to help
controllers to predict flight profiles when maintaining vertical separations. This constraint is applied for the whole
trajectory, but is especially relevant for the climbs between two cruise altitudes, since it can significantly limit the
capability to climb to a higher cruise altitude. In this paper, step-climbs of 2000 ft in between constant Mach and
altitude cruise phases are modeled and, without loss of generality, odd flight levels are considered.
Flight management systems typically impose a minimum cruise distance and/or cruise time when
computing a trajectory, preventing in this way too short cruises. These distances/times are in general aircraft type
dependent and may be also changed according to airline policy. In this paper, a minimum cruise distance of 50NM
and a minimum cruise time of 5 minutes have been taken into account when computing conventional trajectories.
Finally, at the end of the cruise a deceleration phase (still at constant altitude) is introduced allowing the aircraft to
transition from the best cruise speed to the best descent one.
It should be noted that the optimization algorithm will choose the (optimal) values of the different CAS,
Mach and altitudes described above, as well as the number of step climbs (if any) to perform. Table 1 summarizes
all the phases considered and the mathematical details of how the different constraints have been modeled can be
found by a previous publication of Dalmau & Prats (2014).
7
Table 1. Phases used to model operational constraints of conventional operations
Phase
1
2
Description
Take-off acceleration
Constant CAS climb
3
Climb acceleration
4
5
6
Constant CAS climb
Constant Mach climb
Cruise
7
Step climb
8
…
k+1
k+2
…
𝑁−5
𝑁−4
𝑁−3
𝑁−2
Cruise
…
Step climb
Cruise
…
Cruise deceleration
Constant Mach descent
Constant CAS descent
Descent deceleration
𝑁−1
Constant CAS descent
𝑁
Approach deceleration
Remarks
Constant acceleration
CAS < 250 kt
Under FL100
Constant acceleration
Above FL100
Constant Mach
Constant altitude at a odd Flight Level
Cruise time greater than 5 minutes
Cruise distance greater than 50 NM
Constant Mach
Minimum rate of climb of 500 ft/min
Same as phase #6
Same as phase #7
Same as phase #6
Constant altitude at a odd Flight Level
Constant deceleration
Above FL100
CAS < 250 kt
Under FL100
Constant deceleration
2.3. Problem resolution
The goal of the optimization problem formulated here is to find the best control vector function 𝑢
⃗⃗(𝑡) for the whole
trajectory, defined over the time period [𝑡0 , 𝑡𝑓 ], that minimizes the following cost function:
𝑡𝑓
𝐽 = ∫ (𝐹𝐹 (𝑡) + 𝐶𝐼)d𝑡
(9)
𝑡0
under the constraints expressed in equations (1) and (9) and, for conventional operations, also those summarized in
Table1. 𝐶𝐼 in previous equation is the Cost Index, which is assumed zero in this paper (maximum range operations),
unless otherwise stated.
Due to the non-linearity of the functions appearing in the optimization objective and the constraints, an
analytic solution of the problem cannot be found (Betts, 1995). Thus, numerical methods are required, such as direct
8
collocation methods that transform the original continuous (and thus infinite) problem into a discrete (and finite)
nonlinear programming (NLP) optimization problem. The advantage of these methods is the possibility of solving
very complex problems with a minimum effort of mathematical analysis. In fact, only the physical equations need to
be coded and the necessary conditions for optimality do not have to be derived. Details on this methodology can be
found in (Betts, 1998; Betts, 2010), while Dalmau & Prats (2014) present the mathematical details to compute the
optimal continuous and conventional trajectories used in this paper.
3. Numerical results
This section compares the results (in terms of fuel consumption and flight time) between conventional and
continuous operations for several case studies using an Airbus A320, a typical twin-engine, narrow-body aircraft
and an Airbus A340, a typical four-engine, wide-body aircraft.
3.1. Experimental setup
For the A320 analysis a set of representative trip distances between 400 NM and 2400 NM have been chosen, while
for the A340 study the distances considered range from 1000 NM to 6250 NM. For each trip distance, the best
trajectories have been computed for several landing masses between the operative empty mass and maximum
landing mass (MLM). Results were obtained using solvers CONOPT (as NLP) and SBB as MINLP (mixed integer
nonlinear programming), both bundled into the GAMS software suite. Fig. 1 shows two examples of optimal
trajectories computed with the algorithms described above.
(a) A320 with 80% of MLM
(b) A340 with 100% of MLM
Fig. 1. Examples of optimal trajectories
9
These results are consistent with those found in the literature (Lovegren & Hansman, 2011; Soler, 2012),
where unconstrained trajectories follow the so called cruise climb, i.e. a continuous and uninterrupted climb up to
the moment the continuous descent is initiated. For aircraft equipped with jet engines, the higher the altitude the
more fuel-efficient the engine becomes and therefore, the aircraft seeks to achieve the maximum altitude in the
minimum amount of time. An optimal altitude is found where fuel consumption is minimized by flying at the most
efficient speed and engine setting. As fuel is burned and aircraft weight decreases the amount of Lift needed and
consequently the Drag, are reduced meaning that the required thrust is also lower. If throttle is reduced, then the
engine is not longer operating at the most efficient setting. Therefore, the optimal procedure is to maintain the most
efficient speed and power setting and using the excess thrust to slowly climb the aircraft (recall that for these
procedures a minimum rate of climb has not been imposed). Cruise climb ends when the optimum descent path is
intercepted. This path is the result of descending continuously at minimum gradient (or minimum drag) speed,
which allows the aircraft to maximize the flown distance at idle thrust.
Conventional procedures shown in Fig. 1 are also consistent with optimal constrained flight profiles (Betts,
1995; Soler, 2012), where step climbs are performed at the moment this excess thrust allows the aircraft to climb at
the minimum rate of climb required by operational constraints. Due to this restriction, conventional vertical profiles
are always below continuous climb profiles: for a given mass, increasing the altitude decreases the specific excess
power and therefore, the rate of climb performance. Thus a maximum altitude is reached with the minimum allowed
rate of climb. Then, as fuel is burned while cruising at this constant altitude the rate of climb available increases up
to the point a step climb can be performed up to the next available cruise altitude. As it will be seen in next section,
for a given distance, more fuel is needed in conventional operations than in continuous ones. Since the landing mass
is fixed when comparing the two trajectories, aircraft flying conventional operations will be heavier at the end of the
climb phase than those flying continuous operations, leading to lower optimal cruise altitudes. Nevertheless, a
detailed study of the results has shown that the rate of climb limitation is the prevailing factor that explains why
conventional procedures have always lower cruise altitudes.
3.2. Fuel savings
Figs. 2 and 3 show, for each aircraft, the optimal trip fuel for both conventional and continuous operations, as a
function of the considered distances and landing masses. As expected, the amount of fuel needed increases with the
total trip distance and landing mass. Continuous operations, however, require less fuel if compared with the
conventional scenario. Fig. 4 and Fig. 5 show these fuel savings in absolute and relative terms as a function of the
trip distance and landing mass. Fuel savings are bigger for longer trip distances (relative to the typical ranges of the
10
two aircraft models considered) and for large landing masses. The influence of the landing mass in these fuel
savings becomes more important as long as the trip distances increases. This is a consequence of the non-linear
increase of the transport coefficient as a function of the trip distance. The transport coefficient is defined as the
quotient between the increment of take-off mass required to transport a given increment in landing mass; and this
increment of landing mass.
(a) Conventional operations
(b) Continuous operations
Fig. 2. Total trip fuel for the A320 analysis
(a) Conventional operations
(b) Continuous operations
Fig. 3. Total trip fuel for the A340 analysis
11
(a) Total trip fuel saved
(b) Percentage of trip fuel saved
Fig. 4. Trip fuel savings for the A320 analysis
(a) Total trip fuel saved
(b) Percentage of trip fuel saved
Fig. 5. Trip fuel savings for the A340 analysis
Regarding relative savings, shown in Figs. 4(b) and 5(b), larger values are observed for longer ranges in
each aircraft model. In this case, however, it is not obvious to establish a correlation between the relative savings
and the landing mass of the aircraft.
The sharp bends observed in the previous curves are mainly due to the discrete behavior of the
conventional operations. For example, as long as the landing mass decreases progressively, the optimal altitude
12
increases progressively too. Yet, since only discrete cruise altitudes are allowed, this optimal flight altitude will
suddenly change at some landing mass, producing a discontinuity in the fuel consumption.
It should be noted that in our analysis the conventional flight is somehow idealized, since climb and
descent paths have relaxed constraints allowing continuous climbs and descents. Nowadays, these continuous
operations are not usually performed as level-offs and/or path stretching are mandated by ATC in order to maintain
separation, especially in busy TMAs. Moreover, in some controlled airspaces the ATC may also bound upper and/or
lower airspeeds in order to facilitate the traffic flow synchronization. These speed restrictions may also induce some
extra fuel consumption, since climbs and/or descents would not be longer flown at the optimal speeds.
Thompson et al. (2013) studied the effect of level segments in descent procedures, showing average fuel
savings of approximately 200 kg per flight; while Knorr et al. (2011) estimated the total benefit pools that ATM can
potentially influence in the descent phase of flight to an average of additional 1.1 minutes and 30 kg per arrival1
only as consequence to vertical inefficiencies. Another quantitative example is given in (NATS, 2012), where some
flights trials from Edinburg to London were performed flying an uninterrupted climb to cruise altitude followed by a
direct route (at constant altitude) and an uninterrupted descent. For these trials, the total fuel reduction was found to
be around the 10% of the total trip fuel (around 300kg). The observed fuel savings in our study are mainly achieved
by the possibility to fly a continuous cruise climb and should be added to the previous potential savings due to more
efficient vertical profiles trajectories in the TMA.
Finally, it is worth noting that in previous results conventional operations have been computed considering
a maximum range scenario (i.e. the operator aims at minimizing the total trip fuel). Current operations, however, are
performed at higher cruise speeds (thus, incurring extra fuel consumption), since the cost of the time is also
considered for flight planning, having a very significant commercial relevance for some flights. Therefore, fuel
savings of continuous cruise climb operations would be even larger if compared with these cost-based operations.
Table 2 shows a couple of illustrative examples of conventional operations computed at the Cost Indexes such that
the flight time is the same than in the same flight executed in continuous mode. These trajectories are displayed in
Fig. 6 where, as expected, trajectories with a Cost Index greater than zero fly below maximum range trajectories for
both concepts of operations. Work is underway to analyze the sensitivity of the results to different Cost Index
policies.
1 Fuel computations in (Knorr et al. 2011) are performed with BADA aircraft performance data, acknowledging that fuel figures may
be underestimated and suggesting to double the estimated fuel savings.
13
(a) A320 with a 80% of the MLM
(b) A340 with a 84% of the MLM
Fig. 6. Examples of optimal trajectories at different Cost Indexes
Table 2. Trip fuel and time for continuous and conventional operations at different Cost Index examples
A320 with 80% of
MLM and 1500 NM
of trip distance
A340 with 84% of
MLM and 4000 NM
of trip distance
3.3.
CI (kg/min)
Trip fuel (kg)
Trip time
CI (kg/min)
Trip fuel (kg)
Trip time
Continuous operations
0
20
6976
7086
3h 30'
3h 17'
0
40
46679
47141
9h 03'
8h 38'
Conventional operations
0
20
7096
7149
3h 39'
3h 30'
0
40
47471
48026
9h 34'
9h 05'
Time savings
In this study, trip time differences between the two concepts of operation have also been analyzed and Figs. 7 and 8
depict them in absolute and relative terms. It is very interesting to note that continuous operations not only represent
lower fuel consumptions but also shorter trip times.
14
(a) Total trip time saved
(b) Percentage of trip time saved
Fig. 7. Trip time savings for the A320 analysis
(a) Total trip time saved
(b) Percentage of trip time saved
Fig. 8. Trip time savings for the A340 analysis
This paradox (one would expect fuel savings at the expense of longer trip times) can be explained by the
fact that the speed minimizing fuel consumption increases with altitude, as demonstrated for instance by Miller
(1993). For a given mass, an optimal altitude corresponding to an optimal maximum range speed can be found,
which minimizes fuel consumption. Yet, this altitude cannot always be reached due to engine performance
limitations. In continuous operations, the aircraft is following this optimal altitude (that increases as long as the mass
of the aircraft decreases). For conventional operations, however, the aircraft must cruise at a lower altitude in order
to have the required excess thrust needed to fulfill the minimum rate of climb constraint. Thus, the aircraft flies at a
less fuel-efficient altitude, which leads to a lower cruise speed (if compared with the higher speed that it would have
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at the optimal altitude). Consequently, this difference in cruise altitude produces more fuel consumption and more
trip time.
It is interesting to note that total time savings shown in Figs. 7(a) and 8(a) present an almost linear
relationship with the ground distance and have a negligible correlation with the landing mass. A linear regression
with a least squares approach has given a time savings increase of 7.27 min/1000NM for the A320 (𝜎 =
0.079 min/1000NM) and 8.70 min/1000 NM for the A340 (𝜎 = 0.078 min/1000NM). Moreover, absolute trip
time also presents a linear correlation with trip distance with fitting slopes of 136.91 min/1000NM for the A320
(𝜎 = 0.619 min/1000NM) and 136.40 min/1000 NM for the A340 (𝜎 = 0.736 min/1000NM). This explains the
shape of the curves 7(b) and 8(b) where the relative time savings tend to a constant value for large ground distances.
4. Conclusion
The reduction of fuel consumption (and gaseous emissions) is one of the major drivers of current research efforts in
air transportation. Even small amounts of fuel savings become significant at aggregate level, especially when we
consider the high volume of traffic that is operating every day. This paper has focused on the potential savings of the
introduction of eventual continuous cruise climb operations showing already some noteworthy figures in terms of
fuel consumption, especially for longer routes.
Another important remark that arises from this study is that continuous operations not only reduce fuel
consumption, but also the trip time. This is particularly interesting since aircraft operators typically seek to optimize
a trade-off between fuel and time consumption for a given flight. Thus, the economic benefit of such continuous
operations is twofold. If we disregard the shortest flight distances for each aircraft model, time savings are around
4%-5% of the total trip time. Time benefits, however, are not modeled in detail in this paper, since the actual cost of
time for aircraft operators follows a strong non-linear relationship with flight time (or delay) duration (Cook et al.,
2012). In relative terms, it appears that the mass of the aircraft does not influence significantly on time or fuel
savings.
It is clear that with the current air traffic management (ATM) concept of operations the introduction of
such procedures at large scale is not possible, since flying at constant altitudes is one of the key elements to
strategically separate flows of aircrafts. Nevertheless, according to SESAR and NextGen paradigms, new avionic
systems will be able to support trajectory-based operations in the forthcoming years, while more automation in
ATM systems is also foreseen. In a future scenario, we could envisage that aircraft themselves are responsible for
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keeping separation amongst each other, thus delegating air traffic control responsibilities to the pilot by means of
aircraft surveillance applications systems (SESAR Joint Undertaking, 2007). The results shown in this paper
highlight the need to motivate further research efforts in separation assurance to make such continuous cruise climb
operations safe and operationally sound.
Acknowledgements
The authors would like to thank Airbus Industries for the use of PEP (Performance Engineers Program) suite, which
allowed us to undertake realistic aircraft performances simulations.
References
Air Force Test Pilot School, 1993 (Sep). "Cruise performance theory", in Performance phase, vol. 1, ch. 11. Edwards AFB, CA.
J. T. Betts and E. J. Cramer, 1995 (Jan). "Application of direct transcription to commercial aircraft trajectory optimization", Journal of Guidance,
Control, and Dynamics, 18(1), pp. 151–159.
J. T. Betts, 1998 (Mar). Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 21(2), 193–207.
J. T. Betts, 2010 "Practical methods for optimal control and estimation using nonlinear programming", ser. Advances in Design and Control.
Philadelphia, U.E: Society for Industrial and Applied Mathematics (SIAM)
J. B. Clarke, N. T. Ho, L. Ren, J. A. Brown, K. R. Elmer, K. Zou, C. Hunting, D. L. McGregor, B. N. Shivashankara, K. Tong, A. W. Warren,
and J. K. Wat, 2004 (Sep). "Continuous descent approach: Design and flight test for Louisville international airport", Journal of Aircraft, 41(5),
1054–1066.
Cook, A., Tanner, G., Williams, V., & Meise, G. (2009). Dynamic cost indexing – Managing airline delay costs. Journal of Air Transport
Management, 15(1), 26–35.
A. Cook, G. Tanner, A. Lawes, 2012 (May) "The Hidden Cost of Airline Unpunctuality", Journal of Transport Economics and Policy, 46(2) pp.
157-173
Dalmau, R., & Prats, X., 2014 (Jun) "How much fuel and time can be saved in a perfect flight trajectory ? Continuous cruise climbs vs .
conventional operations", in 6th International Conference on Research in Air Transportation (ICRAT2014), Istanbul (Turkey).
H. Erzberger, J. D. Mclean, and J. F. Barman, 1975 (Dec), “Fixed-range optimum trajectories for short-haul aircraft,” NASA, Washington, D.C.,
Technical Report D-8115.
Hansman, John McConnachie, Dominic Wollersheim, Christoph Elke, Matthew Hansen, Mark Chan, Nathan Crépin, Maxime Li, Tao Peterson,
Everett Trani, Antonio, 2014 (Mar), “The impact of Oil Prices on the Air Transportation Industry”, Technical report. National center of
Excellence for Aviation Operations Research (NEXTOR).
G. B. C. A. Authority, 2013. “Aeronautical information publication (AIP) United Kingdom: En-route (ENR) 1.1 - general rules”.
F. A. A., 20142 (Jan), “Aeronautical Iinformation publicationManual (AIMP): Chapter 4 – Subchapter FSection 4 - Part 1091,” pp. 579–844,
U.S. Dept. of Transportation.
ICAO, 2001 (Jul). "International Standards and Recommended practices. Annex 11 to the Convention on Civil Aviation: Air Traffic Services".
Thirtteen Ed. International Civil Aviation Organization, Montreal, Canada.
17
ICAO, 1993. "Manual of the ICAO Standard Atmosphere: Extended to 80 Kilometres (262500 Feet)". International Civil Aviation Organization,
Montreal, Canada.
L. Jin, Y. Cao, and D. Sun, 2013 (Feb). "Investigation of potential fuel savings due to continuous-descent approach", Journal of Aircraft, 50(3),
807–816.
M. Kaiser, M. Schultz, and H. Fricke, 2011. "Enhanced jet performance model for high precision 4D flight path prediction", in Proceedings of the
1st International Conference on Application and Theory of Automation in Command and Control Systems (ATACCS),. 33–40.
Knorr, D., Chen, X., Rose, M., & Gulding, J., 2011 (Jun). "Estimating ATM Efficiency Pools in the Descent Phase of Flight. Potential Savings in
both Time and Fuel," in Proceedings of the 9th USA/Europe Air Traffic Management R&D Seminar (ATM2011). Berling (Germany)
J. A. Lovegren and R. J. Hansman, 2011 (Feb) “Estimation of potential aircraft fuel burn reduction in cruise via speed and altitude optimization
strategies,” MIT International Center for Air Transport (ICAT), Cambridge, USA, Technical Report.
E. L. Miller, 1993 (May), "Optimal cruise performance", Journal of Aircraft, 30(3), 403–405
C. Nutt, 2012 (Jan) “NATS Fuel Efficiency Metric,” National Air Traffic Services. Tech. Rep OA-1161 Version 1.3
D. A. Senzig, , G. G. Fleming, and R. J. Lovinelli, 2009. “Modeling of terminal-area airplane fuel consumption”. Journal of Aircraft 46(4), pp.
1089-1093
SESAR Joint Undertaking, 2007 “SESAR Concept of Operations", Technical report W2.2.2/D.3
M. Soler, A. Olivares, E. Staffetti, and D. Zapata, 2012 (Jan). "Framework for aircraft trajectory planning toward an efficient air traffic
management", Journal of Aircraft, 49(1), 341–348.
J. A. Sorensen and M. H. Waters, 1981 (Mar). "Generation of optimum vertical profiles for and advanced flight management system", NASA,
Mountain View, California, Technical Report 165674.
T. Thompson, B. Miller, C. Murphy, S. Augustine, T. White, and S. Souihi, 2013 (Jun) “Environmental impacts of continuous-descent operations
in Paris and New York regions. Isolation of ATM/airspace effects and comparison of models,” in Proceedings of the Tenth USA/Europe Air
Traffic Management Research and Development Seminar (ATM2013), Chicago, Illinois (USA).
W. Zhao, S. Alam, H.A. Abbass, 2013. “Evaluating ground-air network vulnerabilities in an integrated terminal maneuvering area using coevolutionary computational red teaming”, Transportation Research Part C: Emerging Technologies, 29, pp. 32-54
18