Analysis of optimal aircraft cruise with fixed arrival time including wind effects

Aerospace Science and Technology 32 (2014) 212–222

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Analysis of optimal aircraft cruise with fixed arrival time including wind effects Antonio Franco, Damián Rivas ∗ Department of Aerospace Engineering, Escuela Técnica Superior de Ingeniería, 41092 Sevilla, Spain

a r t i c l e

i n f o

Article history: Received 12 April 2013 Received in revised form 10 October 2013 Accepted 15 October 2013 Available online 21 October 2013 Keywords: Minimum-fuel cruise Fixed arrival time Singular optimal control Wind effects

a b s t r a c t Minimum-fuel cruise at constant altitude with the constraint of a fixed arrival time is analyzed, including the effects of average horizontal winds. The analysis is made using the theory of singular optimal control. The optimal control is of the bang-singular-bang type, and the optimal trajectories are formed by a singular arc and two minimum/maximum-thrust arcs joining the singular arc with the given initial and final points. The effects of average horizontal winds on the optimal results are analyzed, both qualitatively and quantitatively. The influence of the initial aircraft weight and the given cruise altitude is analyzed as well. Two applications are studied: first, the cost of meeting the given arrival time under mismodeled winds, and, second, the cost of flight delays imposed on a nominal optimal path. The optimal results are used to assess the optimality of cruising at constant speed; the results show that the standard constantMach cruise is very close to optimal. Results are presented for a model of a Boeing 767-300ER. © 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction An important problem in air traffic management (ATM) is the design of aircraft trajectories that meet certain arrival-time constraints at given waypoints, for instance at the top of descent, at the initial approach fix, or at the runway threshold (estimated time of arrival). The final-time constraint may be defined, for example, by a flight delay imposed on the nominal (preferred) trajectory. These are four-dimensional (4D) trajectories, which are a key element in the trajectory-based-operations (TBO) concept proposed by SESAR and NextGen for the future ATM system (for example, Bilimoria and Lee [5] analyze aircraft conflict resolution with an arrival time constraint at a downstream waypoint). Also important in ATM is the design of optimal flight procedures that lead to energyefficient flights. In practice, the airlines consider a cost index (CI) and define the direct operating cost (DOC) as the combined cost of fuel consumed and flight time, weighted by the CI; their goal is to minimize the DOC. When the flight time is fixed, the objective is to minimize fuel consumption. In the analysis of aircraft trajectories with fixed flight time, wind effects are of primary importance, because changes in wind speed modify the flight time (over a given range), and therefore lead to changes in the speed profiles required to keep the finaltime constraint. Minimum-DOC trajectories have been studied by different authors. For example, Barman and Erzberger [1] and Erzberger and

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Lee [11] analyze the minimum-DOC problem for global trajectories (climb-cruise-descent), considering steady cruise and taking the aircraft mass as constant; wind effects are considered in Ref. [1] in the case of short-haul missions (range below 500 km). Burrows [7] also analyzes the minimum-DOC problem for global trajectories, without the assumption of constant mass, but with the assumption that the cruise segment takes place in the stratosphere. The particular case of minimum-fuel cruise (CI equal to zero) has been considered by others. For example, Schultz and Zagalsky [25], Speyer [27], Schultz [24], Speyer [28] and Menon [20] analyze the optimality of the steady-state cruise, taking the aircraft mass as constant; wind effects are not considered. Fuel-optimal trajectories with fixed arrival time are studied by Sorensen and Waters [26], Burrows [8], Chakravarty [10] and Williams [30], who analyze the 4D fuel-optimization problem as a minimum-DOC problem with free final time, that is, the problem is to find the time cost for which the corresponding free-final-time, DOC-optimal trajectory arrives at the assigned time. The effects of horizontal winds are considered in Refs. [10,30]: in Ref. [10] the cost of absorbing delays is analyzed, in a scenario formed by the final cruise (400 nmi) and the descent, for altitude-dependent winds, and in Ref. [30] the effects of mismodeled winds in a scenario formed by the final cruise (200 nmi) and the descent segments are studied, for the case of constant winds; wind effects on the whole cruise segment, however, are not considered. In this paper, an analysis of minimum-fuel cruise with fixed arrival time, at constant altitude, in the presence of horizontal winds, is presented. The analysis is made using the theory of singular optimal control (see Bell and Jacobson [2]). The problem is unsteady, with variable aircraft mass. The initial and final speeds are

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

213

Nomenclature a c D g h H K L m mF M S t

speed of sound specific fuel consumption drag gravity acceleration altitude Hamiltonian constant value of the Hamiltonian lift aircraft mass fuel mass Mach number switching function time

given, so that the structure chosen for the optimal control is bangsingular-bang, with the optimal paths formed by a singular arc and two minimum/maximum-thrust arcs joining the singular arc with the given initial and final points. The singular arc in the case of no winds is studied in Franco et al. [13]. Singular optimal control theory has been used, among other works, to analyze maximum-range cruise at constant altitude (Pargett and Ardema [21], Rivas and Valenzuela [22]), minimum-cost cruise including both the DOC and the arrival-error cost associated to not meeting the scheduled time of arrival (Franco and Rivas [12]), and maximum-range unpowered descents in the presence of altitude-dependent winds (Franco et al. [14]). The main objective of this work is to present a quantitative analysis of the effects of average horizontal winds on the optimal trajectories and control laws that lead to minimum fuel consumption while meeting the final-time constraint. The influence of the initial aircraft weight and the given cruise altitude on the optimal results is also analyzed. From the operational point of view, two applications are studied: first, the fuel penalties associated to mismodeled winds are estimated, that is, the cost of meeting the given time of arrival under mismodeled winds is quantified; and, second, the cost of flight delays imposed on a nominal optimal path is quantified as well. The optimal trajectories define speed laws in which the Mach number varies along the singular arc. These optimal solutions, which are a reference for optimal performance, are used to assess the optimality of the standard constant-Mach cruise procedure commonly used in practice (according to air traffic regulations). The comparison with optimal results shows that the performance of the constant-Mach cruise is very close to optimal. Results are presented for a model of a Boeing 767-300ER (a typical twin-engine, wide-body, long-range transport aircraft), with realistic aerodynamic and propulsive models, and for constant winds, which represent average winds along the cruise. The outline of the paper is as follows: the problem is formulated in Section 2, the numerical procedure used to solve the optimal problem is described in Section 3, the constant-Mach cruise procedure is described in Section 4, the results are presented in Section 5, and some conclusions are drawn in Section 6; the aircraft model is described in Appendix A. 2. Problem formulation As already indicated, because the flight time is fixed, the objective is to minimize the fuel consumption for a given range, that is, to minimize the following performance index

tf T TM V w W x xf

flight time thrust maximum thrust aerodynamic speed wind speed aircraft weight horizontal distance range adjoint variable thrust control parameter flight delay mismodeled wind singular-arc parameter

λ

π t f w Ω

t f J=

cT dt

(1)

0

with t f fixed, subject to the following constraints

V˙ =

1

(T − D ) m ˙ = −cT m x˙ = V + w

(2)

which are the equations of motion for cruise at constant altitude and constant heading, in the presence of a horizontal wind (see Jackson et al. [15]). In these equations, the drag is a general known function D ( V , m), which takes into account the remaining equation of motion L = mg; the thrust T ( V ) is given by T = π T M ( V ), where π models the throttle setting, 0 < πmin  π  πmax = 1, and T M ( V ) is a known function; the specific fuel consumption, c ( V ), is also a known function; and the wind speed w (h) is a known function which depends on the given altitude h. Thus, in this problem there are three states, speed (V ), aircraft mass (m) and distance (x), and one control, π . The initial values of speed, aircraft mass and distance (V i , mi , xi ), and the final values of speed and distance ( V f , x f ) are given. The final value of aircraft mass (m f ) is unspecified. (Note that D, T M and c also depend on the given altitude h.) The Hamiltonian of this problem is given by

H = cπ T M +

λV m

(π T M − D ) − λm c π T M + λx ( V + w )

(3)

where λ V , λm and λx are the adjoint variables. Note that H is linear in the control variable, so that it can be written as H = H + S π , where H and the switching function S are given by

H = −λ V

D

+ λx ( V + w )  λV S= − (λm − 1)c T M 

m

(4)

m

The necessary conditions for optimality are summarized next (see Ross [23]): 1) The equations defining the adjoints:

λ˙ V = −

  ∂H λV ∂ D dT M λV = −λx + − − (λm − 1)c π ∂V m ∂V m dV

+ (λm − 1) λ˙ m = −

dc dV



π TM

∂H λV π T M − D ∂ D = + ∂m m m ∂m



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A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

λ˙ x = −

∂H =0 ∂x

(5)

Note that the last equation leads to λx = constant. 2) The transversality condition (associated to m f being unspecified):

λm (t f ) = 0

(6)

3) The Hamiltonian minimization condition, which states that for the control to be optimal it is necessary that it globally minimize the Hamiltonian. Since H is linear in π and π is bounded, minimization of H with respect to π defines the optimal control as follows

⎧ ⎨ πmax if S < 0 π = πmin if S > 0 ⎩π if S = 0 over a finite time interval sing

(7)

where πsing (πmin < πsing < πmax ) is the singular control (yet to be determined). Trajectory segments defined by πsing are called singular arcs. A necessary condition for the singular control to be optimal is analyzed below in Section 2.2. Moreover, since the Hamiltonian is not an explicit function of time, one has the first integral that states that the Hamiltonian is constant along the optimal trajectory, that is,

H (t ) = K

(8)

where the constant K is unknown. In general the optimal trajectory will be composed of singular arcs (with πsing ) and arcs with πmin or πmax ; whether one has πmin or πmax is defined by the sign of the switching function S. In this problem the solution is expected to be of the bang-singularbang type, as suggested by the results obtained by Bilimoria and Cliff [4], where, using a reduced-order model with different time scales, the trajectory is decomposed into 3 parts: an initial transient, the cruise-dash arc and a terminal transient. Although the underlying aerodynamic and propulsive models might affect the structure of the solution, for the smooth models considered in this work, the bang-singular-bang structure is plausible, and hence it is the one analyzed in this paper. Since the initial and final speeds are fixed, there is a physical criterion to decide whether one has πmin or πmax , just by comparing those speeds with the speeds that correspond to the singular arc. A necessary condition for the junctions between singular and nonsingular arcs to be optimal is analyzed below in Section 2.3. Although called optimal trajectories, they are in fact extremals, that is, trajectories that satisfy the necessary conditions for optimality. 2.1. Singular arc The singular arc is defined by the following three equations (see Bryson and Ho [6])

H = K,

S = 0,

S˙ = 0

(9)

where the function S˙ is given by



S˙ = −

λV m

 − (λm − 1)c

− λx + (λm − 1) D

D dT M m dV

dc dV



TM m

 +

λV m



∂D ∂D + c D − mc ∂V ∂m



(10)

(note that the terms in the control variable π have canceled out of this equation). Hence, the three equations that define the singular arc (9) lead to

−λ V

D m

+ λx ( V + w ) = K

λV

− (λm − 1)c = 0  dc λV ∂ D ∂D − λx + (λm − 1) D + c D − mc =0 m ∂V ∂m dV m

(11)

The singular arc is obtained after eliminating the adjoints, λ V and λm , from these equations. One obtains the following expression



D

1

−c−

Ω+V

1 dc c dV



−

∂D ∂D + cm =0 ∂V ∂m

(12)

which is a family of singular arcs defined by the family parameter

Ω=w−

K

(13)

λx

This family can be written as f (m, V , Ω) = 0. This is the same family obtained in Ref. [13] in the case of no wind, but for a different family parameter. The value of Ω is determined by imposing the final time to be t f (the numerical procedure is described in Section 3). Once Ω is determined, Eq. (12) defines a singular line in the ( V , m)-space, which is in fact the locus of possible points in the state space where singular arcs must lie. This singular line is also a switching boundary for the optimal control (see BenAsher [3]). 2.2. Optimal singular control The function S¨ depends linearly on the control variable π . Since this second total derivative of S is the lowest-order derivative in which π appears explicitly, the order of the singular arc is q = 1 (in general, the order is q when such derivative is of order 2q, as defined in Ref. [2]). Let S¨ = A ( V , m) + B ( V , m)π , therefore, because one also has S¨ = 0 (where S = S˙ = 0), the singular control is obtained from A ( V , m) + B ( V , m)π = 0; one gets the following

πsing =

D TM



1+ Vc

A 1 ( V , m)



(14)

B 1 ( V , m)

where A 1 ( V , m) and B 1 ( V , m) are given by

 ∂D ∂2 D ∂2 D m ∂ D ∂D − m2 c 2 − cD + − mc ∂ m∂ V D ∂m ∂V ∂m ∂m   2 dc d dc 1 c ∂ D V + B 1 ( V , m) = D V c 2 + 3 + 2 V c + dV c dV 2 ∂V c dV  2 2 dc D ∂ D ∂ 2 2 ∂ D − mV c 2 + 3 +V + m c V dV ∂ m ∂V 2 ∂ m2 A 1 ( V , m) = m

− 2V cm

∂2 D ∂ m∂ V

(15)

This expression for the optimal singular control depends implicitly on the parameter of the family of singular arcs, since V and m are related by the singular-arc equation (12) which includes the dependence on Ω . The generalized Legendre–Clebsch condition (see Kelley et al. [16]) establishes that for the singular control to be optimal one must have

(−1)q

  ∂ d2q 0 S ∂ π dt 2q

(16) ¨

which in our case (q = 1) reduces to − ∂∂πS  0. It can be shown numerically that the strengthened generalized Legendre–Clebsch ¨

condition (− ∂∂πS > 0) is satisfied.

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

Step 2. Integrate Eqs. (2) with

215

π = πsing from point 1 (with [n]

known initial values V 1 , m1 , x1 ) until the distance x12 is traveled. At the end of the integration along the singular arc one has V 2 , m2 , and x2 (point 2). Step 3. Integrate Eqs. (2) with either π = πmin or π = πmax from point 2 (with known initial values V 2 , m2 , x2 ) until V = V f . The value πmin or πmax is chosen depending on whether one has V f < [n]

V 2 or V f > V 2 . At the final point one obtains the final values x f [n]

and t f , which in general are different from x f and t f ; in such a [n]

case, one must iterate on the guessed values x12 and Ω [n] , which is done as described next. The procedure defined by steps 1 to 3 defines a function g : n] R2 → R2 , (x[12 , Ω [n] ) → (x[fn] , t [fn] ), so that one searches for the valFig. 1. Sketch of the optimal path.

2.3. Optimality of junctions McDanell and Powers [19] prove that, for the optimality of junctions between singular and nonsingular arcs, the following necessary condition must be satisfied: the sum of the order of the singular arc and the lowest-order time derivative of the control which is discontinuous at the junction must be an odd integer if the strengthened generalized Legendre–Clebsch condition is satisfied at the junction and if the control is piecewise analytic in a neighborhood of the junction (which is the case in this problem). This necessary condition is shown to be satisfied, since the order of the singular arc is q = 1 and the lowest-order time derivative of the control which is discontinuous at the junction is r = 0 (that is, the control itself is discontinuous at the junction). Moreover, one has that at the junctions, where the control variable is discontinuous, the adjoint variables, the Hamiltonian and the switching function are all continuous; hence, the Weierstrass– Erdman corner conditions are satisfied (see Ref. [6]). 3. Numerical procedure The definition of an efficient numerical procedure to obtain the optimal path is facilitated by the knowledge of the structure of the solution (see Maurer [18]). In this case the expected optimal path is of the bang-singular-bang type, as sketched in Fig. 1. Based on this type of path, a procedure is defined to obtain the optimal trajectory. The resolution of the problem must be iterative, because of the final-time constraint. The procedure developed, which is described next, iterates in two variables: Ω and the distance x12 traveled along the singular arc (between points 1 and 2 in Fig. 1), and uses the two final conditions x = x f and t = t f to close the problem. The iterative procedure is started with the following initial guess: Ω [0] = w, because for t f unspecified one has K = 0, [0]

and x12 = x f , because the two bang arcs have very small length.

ues x12 and Ω that satisfy g(x12 , Ω) = [x f , t f ] T . If one defines the function G = g(x12 , Ω) − [x f , t f ] T , one searches for the zero of G. The resolution of G(x12 , Ω) = 0 is performed using Matlab’s fsolve, [0] starting the iteration with the values (x12 , Ω [0] ) defined above, [n]

[n]

and stopping when x f = x f and t f = t f to within some prescribed tolerance. Once the problem is integrated, one has the final optimal value of aircraft mass, m f , which defines the minimum fuel consumption m F = mi − m f , for the given values of range and flight time. In the integration of the state equations, Matlab’s ode45 is used. 3.2. Optimal control It still remains to check whether the assumed structure for the control (bang-singular-bang) is correct. That is, one must check that S > 0 for π = πmin and that S < 0 for π = πmax . This requires the computation of S along the extremal path, which in turn requires the computation of λ V and λm (see Eq. (4)). First, to obtain λ V and λm along the final bang it is necessary to solve a two-point boundary value problem defined by the corresponding adjoint equations (5), in which λx is a parametric unknown, with boundary conditions H (t 2 ) = K , S (t 2 ) = 0 and λm (t f ) = 0. These boundary conditions can be rewritten in terms of λ V and λm , as follows

λ V (t 2 ) =

m2 λx

( V 2 + Ω)

D2

λm (t 2 ) = 1 +

λx c2 D 2

( V 2 + Ω)

λm (t f ) = 0

(17)

where Ω has been already computed. The resolution of this twopoint boundary value problem is performed using Matlab’s bvp4c, starting the iteration with the parameter λx and the constant distributions of λ V and λm that satisfy Eqs. (17), namely

(λm )0 = 0 (λx )0 = −

c2 D 2 V2 + Ω

3.1. Iterative procedure

(λ V )0 = −c 2m2

The following iterative procedure is used in the numerical resolution. [n] Step 0. Guess values Ω [n] and x12 . Step 1. Integrate the state equations (Eqs. (2)) with either π = πmin or π = πmax from the initial point (with known initial values V i , mi , xi ) until the singular arc is reached (point 1 in Fig. 1), that is, until V 1 and m1 satisfy f (m1 , V 1 , Ω [n] ) = 0; at that point one also has x1 . The value πmin or πmax is chosen depending on whether one has V i > V 0 or V i < V 0 , where V 0 is defined by f (mi , V 0 , Ω [n] ) = 0, that is, the speed that corresponds in the singular arc to the initial mass mi (point 0).

Once the final bang is integrated, λ V and λm at any point of the singular arc follow from H = K and S = 0, that is,

λ V (t ) =

mλx D

λm (t ) = 1 +

(18)

( V + Ω) λx cD

( V + Ω)

(19)

Finally, λ V and λm along the initial bang are obtained integrating backwards the first two Eqs. (5) from point 1 to point i, with initial conditions λ V (t 1 ), λm (t 1 ) defined by Eqs. (19). The numerical results show that the control structure is correct.

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A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

Fig. 2. Optimal trajectories and optimal control for w = −15, −10, −5, 0, 5, 10, 15 m/s (t f = 9.5 h, h = 10 000 m, W i = 1600 kN). (a) Optimal trajectories. (b) Optimal control.

4. Constant-Mach cruise In Section 5 it is shown that the optimal solutions define variable-Mach cruise procedures. Even though these procedures may not be flyable (according to common air traffic control practice), they are a reference for optimum performance, and, therefore, can be used to analyze the optimality of standard flight procedures. In this paper, the optimality of the constant-Mach cruise is analyzed, procedure which is described next. Let M c be the cruise Mach, hence the cruise speed is V c = M c a(h) where a(h) is the speed of sound at the given altitude h. The procedure considered is formed by three segments: an initial acceleration/deceleration segment from the given initial speed V i to the cruise speed V c , with maximum cruise/idle engine rating; a main cruise segment with constant speed V c ; and a final acceleration/deceleration segment from V c to the given final speed V f , with maximum cruise/idle engine rating. For the initial segment, the equations of motion (2) are integrated with initial conditions V i , mi and xi until V = V c ; at the end of the segment one has m1 and x1 . For the cruise segment, because the speed is constant, the equations of motion (2) reduce to

˙ = −c ( V c , h) D ( V c , m, h) m x˙ = V c + w (h)

(20)

which are integrated with initial conditions m1 and x1 until the distance x12 is flown; at the end of the segment one has m2 and x2 . For the final segment, Eqs. (2) are integrated with initial conditions V c , m2 and x2 until V = V f ; at the end of the segment one has the final values of aircraft mass, horizontal distance and time, m3 , x3 and t 3 . The flight distance and the flight time are in general different from x f and t f . Hence, one must iterate on the two free variables V c and x12 until x3 = x f and t 3 = t f to within some prescribed tolerance. The iteration is started with the initial guess V c = w and x12 = x f . Finally, the fuel consumption is m F = mi − m3 .

xf tf

−

5. Results The model of the Boeing 767-300ER considered in this paper for the numerical applications is described in Appendix A. With respect to the atmosphere, the ISA (International Standard Atmosphere) model is considered. Results are presented for a cruise

flight defined by a range x f = 8000 km, and by initial and final speeds V i = 240 m/s and V f = 180 m/s, corresponding to hypothetical conditions at the end of the climb and the start of the descent (the same values in all cases studied below). Different values of headwind (HW) and tailwind (TW) are considered, corresponding to negative and positive values of w respectively, ranging from −15 to + 15 m/s; the case of no wind (NW) is included. The flight times range from 8.67 to 10.50 h. The nominal initial aircraft weight is taken to be W i = 1600 kN. In the analysis of the effects of W i , results are presented for a reference case defined by w = 0, t f = 9.5 h and h = 10 000 m. The nominal cruise altitude is taken to be h = 10 000 m. In the analysis of the effects of h, results are presented now for a reference case defined by w = 0, t f = 9.5 h and W i = 1600 kN. In the analysis of the effects of cruise altitude on the optimal results, one can take into account the altitude dependence of the wind. For example, in Ref. [10] a linear wind profile is considered. The theoretical analysis made in this paper is general and valid for any wind profile, so that results could be presented for any choice of profile. For simplicity, a constant profile is considered (as in Ref. [30]). The outline of this section is as follows: the optimal trajectories are analyzed in Section 5.1 and the minimum fuel consumption in Section 5.2; then, two applications are considered: the cost of mismodeled winds is studied in Section 5.3, and the cost of flight delays in Section 5.4; and, finally, the optimality of the constantMach cruise procedure is assessed in Section 5.5. Besides the analysis of the wind effects on the optimal results, which is the main objective of this paper, as already indicated the effects of the initial aircraft weight and of the cruise altitude are analyzed as well. 5.1. Optimal trajectories and optimal control The optimal trajectories (Mach number as a function of flown distance) are shown in Fig. 2(a) for t f = 9.5 h, h = 10 000 m, W i = 1600 kN and different values of wind speed (ranging from −15 to 15 m/s). The corresponding optimal controls are shown in Fig. 2(b). The structure is minimum-thrust arc, singular arc, minimum-thrust arc, in all cases shown except for w = −10, −15 m/s, in which cases the optimal trajectories start with a maximum-thrust arc, required to accelerate the aircraft to the high initial singular-arc speed. The results show that, to meet the given arrival time, the optimal Mach number decreases as the wind speed increases (the optimal cruise speed is larger for HWs than for TWs, as expected); for example, for an HW w = −10 m/s the optimal Mach number is M ≈ 0.815, whereas for a TW w = 10 m/s it ranges from 0.756 to 0.732. In general,

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

217

Fig. 3. Optimal trajectories for t f = 9.17, 9.33, 9.5, 9.67, 9.83, 10 h (h = 10 000 m, W i = 1600 kN). (a) HW w = −10 m/s. (b) TW w = 10 m/s.

Fig. 4. Optimal control for t f = 9.17, 9.33, 9.5, 9.67, 9.83, 10 h (h = 10 000 m, W i = 1600 kN). (a) HW w = −10 m/s. (b) TW w = 10 m/s.

the optimal trajectory calls for a variation of the Mach number along the cruise (for a given t f , one has the largest variations of M along the singular arc for the strongest TWs). However, for a given flight time, there is always a range of wind speeds for which the optimal trajectory along the singular arc is M ≈ const; for example, as shown in Fig. 2(a), for t f = 9.5 h and w = −5 m/s one has M ≈ 0.798. The singular control decreases along the singular arc, and its variation with the wind speed is weak. To analyze the influence of the arrival time, the optimal trajectories for HW w = −10 m/s and TW w = 10 m/s, and different arrival times (ranging from 9.17 to 10 h) are shown in Fig. 3 for h = 10 000 m and W i = 1600 kN. The corresponding optimal controls are shown in Fig. 4. As expected, the optimal Mach number decreases as the arrival time increases; and, for the same arrival time, the optimal Mach number is larger in the case of HW. The results also show that the variation of the singular control with the flight time is very small in the case of TW, and somewhat larger in the case of HW. In fact, the results show that for large speeds (say, Mach numbers larger than 0.8) the dependence of the singular control on speed is large, increasing as M increases, whereas for smaller speeds the dependence is very week (this behavior can be seen also in Fig. 2(b)). The reason for this behavior is that the variation of π with M is very shallow at low M (say 0.7 < M < 0.8) and increases strongly for M > 0.8, following the same trend as the aerodynamic drag; note that, as a first approximation, π ≈ D / T M (as given by Eq. (14)), and that the variation of T M with M at high M is not as strong as the variation of D. Now, to study the influence of the initial aircraft weight, the optimal trajectories for different values of W i (ranging from

1500 to 1700 kN) are shown in Fig. 5(a), for t f = 9.5 h, w = 0 and h = 10 000 m. The corresponding optimal controls are shown in Fig. 5(b). In this problem in which the final distance and final time are fixed, the speed is so constrained that the influence of the initial aircraft weight on the speed profiles is very small (almost negligible). However, the singular control increases as W i increases. Finally, to analyze the influence of the cruise altitude, the optimal trajectories for different values of h (h = 9000, 10 000, 11 000 m) are shown in Fig. 6(a), for t f = 9.5 h, w = 0 and W i = 1600 kN. The corresponding optimal controls are shown in Fig. 6(b). One can see that, as the cruise altitude increases, the optimal Mach number increases (result that is related to the corresponding decrease of the speed of sound). The results also show that the singular control increases significantly as the cruise altitude increases. 5.2. Minimum fuel consumption The minimum fuel consumption as a function of the flight time is shown in Fig. 7 for h = 10 000 m, W i = 1600 kN and different wind speeds (ranging from −15 to 15 m/s). For concreteness, some numerical values are given in Table 1. As expected, HWs require larger values of fuel consumption, as compared to TWs. This effect can be quantified now, for example, for a flight time of 9.5 h, in the nominal case of no wind the minimum fuel consumption is 39 838 kg (see Table 1), whereas for an HW w = −10 m/s it is 43 029 kg and for a TW w = 10 m/s it is 38 265 kg; hence, one

218

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

Fig. 5. Optimal trajectories and optimal control for W i = 1500, 1550, 1600, 1650, 1700 kN (t f = 9.5 h, w = 0, h = 10 000 m). (a) Optimal trajectories. (b) Optimal control.

Fig. 6. Optimal trajectories and optimal control for h = 9000, 10 000, 11 000 m (t f = 9.5 h, w = 0, W i = 1600 kN). (a) Optimal trajectories. (b) Optimal control.

Fig. 7. Minimum fuel consumption vs flight time for w = −15, −10, −5, 0, 5, 10, 15 m/s (h = 10 000 m, W i = 1600 kN).

has a difference of 4764 kg between HW and TW, that is an increase of about 12%. All curves in Fig. 7 present a minimum. These minima are the solutions of the minimum-fuel problem with free final time, that corresponds to K = 0, i.e., Ω = w (in this case one has H (t ) = 0, because there is an additional necessary condition for optimality that states that H (t f ) = 0, see Ref. [6]). The numerical values are given in Table 2 (where m F ,0 is the minimum fuel and t f ,0 the corresponding optimal flight time). As before, HWs give larger

values of minimum fuel consumption, and larger values of flight time, as compared to TWs. For example, for this case of free final time, the difference in minimum fuel consumption between an HW w = −10 m/s and a TW w = 10 m/s is 3034 kg, and the corresponding difference in flight time is 48 min. The effect of the initial aircraft weight on the minimum fuel consumption is shown in Fig. 8, for different pairs of flight time and wind speeds. In particular three cases are considered: TW (t f = 9.17 h and w = 10 m/s), NW (t f = 9.5 h and w = 0), and HW (t f = 10 h and w = −10 m/s). Even though the influence of the initial aircraft weight on the speed profiles is almost negligible (as shown in Fig. 5(a)), for the fuel consumption the behavior is different: one has larger fuel consumption for larger values of W i , as expected. The minimum fuel consumption increases almost linearly when W i increases: going from 38 781 to 44 040 kg for HW, from 37 399 to 42 489 kg for NW, and from 35 942 to 40 894 kg for TW, when W i increases from 1500 to 1700 kN, that is, increases of 5259, 5090, and 4952 kg, respectively (13.56%, 13.61%, and 13.77%); the results give an approximately constant rate of increase of about 2500 kg for each 100 kN. Now, the effect of the cruise altitude on the minimum fuel consumption is shown in Fig. 9, for the same pairs of flight time and wind speeds as before: TW (t f = 9.17 h and w = 10 m/s), NW (t f = 9.5 h and w = 0), and HW (t f = 10 h and w = −10 m/s). In each case there is a best altitude that provides lowest minimum fuel consumption. Hence, appropriate selection of cruise altitude implies a reduction in minimum fuel consumption during cruise. For example, in the three cases represented in Fig. 9 (HW, NW, TW), cruising at h = 11 000 m instead of at the best altitudes

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

219

Table 1 Minimum fuel consumption for different flight times and wind speeds (h = 10 000 m, W i = 1600 kN). m F [kg] t f [h]

8.67 8.83 9.00 9.17 9.33 9.50 9.67 9.83 10.00 10.17 10.33 10.50

w [m/s]

−15

−10

−5

0

5

10

15

– – – – – – 44 172 43 080 42 486 42 189 42 084 42 110

– – – – – 43 029 42 056 41 543 41 305 41 246 41 309 –

– – – – 42 001 41 115 40 664 40 473 40 452 40 546 – –

– – – 41 068 40 243 39 838 39 684 39 693 39 813 – – –

– – 40 217 39 430 39 058 38 933 38 966 39 108 – – – –

– 39 438 38 669 38 318 38 212 38 265 38 424 – – – – –

38 727 37 955 37 613 37 520 37 587 37 761 – – – – – –

Table 2 Minimum fuel consumption and optimal flight time for the free-final-time problem, for different wind speeds (h = 10 000 m, W i = 1600 kN). w [m/s]

−15

−10

−5

0

5

10

15

m F ,0 [kg] t f ,0 [h]

42 080 10.38

41 246 10.15

40 444 9.94

39 672 9.74

38 928 9.54

38 212 9.35

37 520 9.17

Fig. 8. Minimum fuel consumption vs initial aircraft weight, for TW (t f = 9.17 h and w = 10 m/s), NW (t f = 9.5 h and w = 0) and HW (t f = 10 h and w = −10 m/s). (h = 10 000 m.)

Fig. 9. Minimum fuel consumption vs altitude, for TW (t f = 9.17 h and w = 10 m/s), NW (t f = 9.5 h and w = 0), and HW (t f = 10 h and w = −10 m/s). (W i = 1600 kN.)

(9784, 9721 and 9705 m) gives increases in minimum fuel consumption of 996, 1141 and 1064 kg, respectively (2.4%, 2.8%, and 2.7%).

w = −15 m/s and  w = −10 m/s); this same result is obtained in Ref. [30], which is explained by the compressible drag increase at the high Mach numbers required to meet the arrival-time constraint in the case of strong HWs.

5.3. Cost of mismodeled winds 5.4. Cost of flight delays In the presence of mismodeled winds, the optimal results are useful in giving an estimation of the fuel penalty that one might have, that is, an estimation of the cost of meeting the given time of arrival under mismodeled winds. The fuel penalty is defined as the difference in fuel consumption between the cases corresponding to the real wind w +  w and the mismodeled wind w, that is, m F , w = m F ( w +  w ) − m F ( w ). The case of negative values of  w is considered, which means HWs stronger (larger in modulus) than expected, and TWs smaller than expected. In the following, the nominal path is that of minimum fuel consumption in the case of free final time: namely, m F ,0 , with flight time t f ,0 (see Table 2); this flight time is to be maintained under the mismodeled wind. The fuel penalty is represented in Fig. 10 as a function of  w for different values of wind speed. One has that mismodeled HWs have fuel penalties larger than mismodeled TWs for the same wind speed error (it can be as large as m F , w ≈ 2400 kg for

The optimal results are also useful in quantifying the cost of a flight delay t f > 0 imposed on a nominal optimal path with a nominal average wind. Again, the nominal path is that of minimum fuel consumption in the case of free final time: namely, m F ,0 , with flight time t f ,0 . Note that m F ,0 and t f ,0 depend on the wind speed (see Table 2). The cost of the flight delay is defined as the difference in minimum fuel consumption between the cases corresponding to the path for t f = t f ,0 + t f and the nominal path, that is, m F ,t = m F (t f ,0 + t f ) − m F ,0 . The delay cost is represented in Fig. 11 as a function of t f for different values of wind speed. Obviously, the larger the delay, the larger the cost; for instance, the cost of absorbing a flight delay of 30 minutes in the presence of a TW w = 15 m/s is around 500 kg. Moreover, the cost of absorbing a given flight delay is larger in the presence of TWs than in the presence of HWs (the cost increases

220

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

Fig. 10. Increase in minimum fuel consumption vs mismodeled wind for w = −15,

Fig. 12. Optimal path (solid line) and constant-Mach path (dashed line) for w = 15 m/s, t f = 9.5 h, h = 10 000 m, and W i = 1600 kN.

Fig. 11. Increase in minimum fuel consumption vs flight delay for w = −15, −10,

Fig. 13. M c vs wind speed for t f = 9.17, 9.33, 9.5, 9.67, 9.83, 10 h (h = 10 000 m, W i = 1600 kN).

−10, −5, 0, 5, 10, 15 m/s (h = 10 000 m, W i = 1600 kN).

−5, 0, 5, 10, 15 m/s (h = 10 000 m, W i = 1600 kN).

as w increases); this same result is obtained in Ref. [10], where it is explained by the larger percentage of the nominal flight time that the flight delay t f represents in the case of tailwinds (because in this case the flight times are smaller). 5.5. Optimality of constant-Mach cruise In this section, the optimality of the constant-Mach cruise procedure is assessed. The comparison between the optimal and the constant-Mach procedure in the case TW w = 15 m/s, t f = 9.5 h, h = 10 000 m and W i = 1600 kN, is represented in Fig. 12, where the initial and final decelerations are not completely represented in order to better compare both trajectories. The constant Mach number obtained in this case (using the iterative procedure described in Section 4) is M c = 0.7311, and the corresponding fuel consumption is (m F )c = 37 784 kg, value that one can see is very close to the optimal value m F = 37 761 kg. The constant Mach number M c is represented in Fig. 13 as a function of the wind speed for different values of flight time (ranging from 9.17 to 10 h). One has that M c decreases as w increases and as t f increases. The influence of the initial aircraft weight on M c is found to be negligible, as distance flown and flight time are given. And the influence of the cruise altitude is shown in Fig. 14, where M c is represented as a function of cruise altitude for the same pairs of flight time and wind speeds as above: TW (t f = 9.17 h and w = 10 m/s), NW (t f = 9.5 h and w = 0), and HW (t f = 10 h and w = −10 m/s); one has that M c increases as h increases (result that is related to the corresponding decrease of

Fig. 14. M c vs altitude for TW (t f = 9.17 h and w = 10 m/s, dash-dotted line), NW (t f = 9.5 h and w = 0, solid line) and HW (t f = 10 h and w = −10 m/s dashed line). (W i = 1600 kN.)

the speed of sound). Note that if the initial and final decelerations are not considered, M c is given by the following relation

Mc =

1 a(h)



xf tf



−w

(21)

which gives a very good approximation, because the effect of those decelerations in the global problem is small. The difference in fuel consumption m F ,c = (m F )c − m F between the optimal and the constant-Mach procedures is represented in Fig. 15 as a function of wind speed, for different values

A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

Fig. 15. Increase in minimum fuel consumption vs wind speed for t f = 9.17, 9.33, 9.5, 9.67, 9.83, 10 h (h = 10 000 m, W i = 1600 kN). Table 3 Increase in minimum fuel consumption for different flight times and wind speeds (h = 10 000 m, W i = 1600 kN).

m F ,c [kg] t f [h]

8.67 8.83 9.00 9.17 9.33 9.50 9.67 9.83 10.00 10.17 10.33 10.50

w [m/s]

−15

−10

−5

0

5

10

15

– – – – – – 5.3 1.5 0.5 2.7 7.8 14.8

– – – – – 4.1 0 .9 0 .7 3.9 9.5 17.1 –

– – – – 3 .2 0.5 1 .1 4 .9 11.1 19.1 – –

– – – 2 .6 0.3 1 .5 5 .8 12.4 20.7 – – –

– –

– 1 .9 0.3 2 .2 7 .1 14.1 22.9 – – – – –

1 .7 0.2 2 .3 7 .3 14.5 23.4 – – – – – –

2.1 0 .3 1.9 6.5 13.4 22.0 – – – –

of flight time (ranging from 9.17 to 10 h). For concreteness, some numerical values are given in Table 3. One can see that the differences are always very small (almost negligible in some cases, clearly in those in which the optimal M is almost constant). Hence, one can conclude that the performance of the constant-Mach cruise is always very close to optimal, giving fuel consumptions larger than the optimum by less than 25 kg in all cases considered. As already indicated, for a given t f (or for a given w) there is always a range of values of w (or t f ) in which the optimal trajectory is M ≈ const; in these cases the difference between both procedures is negligible (m F ,c < 1 kg). The same trends are obtained for different values of W i and h.

second, the corresponding minimum fuel consumption. The influence of the initial aircraft weight and the given cruise altitude on the optimal results has been analyzed as well. From the operational point of view, if one considers a reference scenario with a given flight time and a nominal average wind, the analysis has allowed to quantify, first, the change in cruise speed required in the case of having a different wind, and, second, the fuel penalty associated, that is, the cost of meeting the given flight time under the mismodeled wind. The results have shown that mismodeled headwinds have fuel penalties larger than mismodeled tailwinds for the same wind speed error. As a second application, the cost of absorbing a flight delay imposed on a nominal optimal path with a nominal average wind has been also quantified: it has been shown that, for a given delay, the cost in the presence of tailwinds is larger than in the presence of headwinds. Although results have been presented for uniform wind profiles, the analysis has been general, and any other altitude-dependent wind profile could be considered as well. Despite their theoretical interest, the optimal variable-Mach solutions may not be flyable (according to common air traffic control practice), however, they are a reference for optimal performance and, hence, have been used to assess the optimality of the standard procedure of cruising at constant speed. The results have shown that the performance of this standard constant-Mach procedure is very close to optimal for all values of flight time, wind, aircraft weight and altitude considered in the analysis (in fact, it has been shown that in some cases optimality is obtained by flying at Mach approximately constant). Acknowledgements This work was not supported by any particular grant or contract. Appendix A. Aircraft model The model of the Boeing 767-300ER used in the computations is described next. It has wing surface area 283.3 m2 and maximum take-off weight 1 833 kN. The aerodynamic model defines the drag polar C D = C D (C L , M ), where C L and C D are the lift and drag coefficients, defined by L = 1 ρ V 2 S w C L and D = 12 ρ V 2 S w C D , respectively, where ρ is the air 2 density and S w the reference wing surface area. The drag polar defined by Cavcar and Cavcar [9] is considered, which is given by

CD =

C D 0,i +



5 



5  j j ¯ ¯ k0 j K ( M ) + C D 1,i + k1 j K ( M ) C L

j =1

+ C D 2,i +

6. Conclusions The problem of minimum-fuel cruise with fixed arrival time has been analyzed, for constant-altitude flight. The analysis of this four-dimensional problem has been made using the theory of singular optimal control. The structure of the optimal control considered has been bang-singular-bang, which is what one expects in this case in which the initial and final values of the speed are given; the optimal trajectories then are formed by a singular arc and two minimum/maximum-thrust arcs that join the singular arc with the given initial and final points. The main objective of this paper has been the analysis of the effects of average horizontal winds on the optimal problem, both qualitative and quantitatively. The analysis has given, first, the optimal cruise speed and the optimal control required to meet the given flight time in the presence of a given average wind, and,

221

5 

k2 j K¯ j ( M ) C L2

j =1

(22)

j =1

where

( M − 0.4)2 K¯ ( M ) = √ 1 − M2

(23)

The incompressible drag polar coefficients are C D 0,i = 0.01322, C D 1,i = −0.00610, C D 2,i = 0.06000, and the compressible coefficients are given in Table 4. This polar is valid for M  0.4; for M  0.4, the incompressible drag polar applies (obtained by setting K¯ = 0 in Eq. (22)). The propulsion model defines the thrust available and the specific fuel consumption. The maximum thrust is defined by (see Torenbeek [29])

T M = W TO δ C T

(24)

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A. Franco, D. Rivas / Aerospace Science and Technology 32 (2014) 212–222

Table 4 Compressible drag-polar coefficients for the model aircraft. j

1

2

3

4

5

k0 j k1 j k2 j

0.0067 0.0962 −0.1317

−0.1861 −0.7602 1.3427

2.2420 −1.2870 −1.2839

−6.4350 3.7925 5.0164

6.3428 −2.7672 0.0000

where W TO is the reference take-off weight, δ = p / p SL is the pressure ratio (p SL being the ISA pressure at sea level), and the thrust coefficient is given by (see Mattingly et al. [17] and Barman and Erzberger [1])

CT =

T SL

 1+

W TO

γ −1 2

M

2

γ γ−1

√ 1 (1 − 0.49 M ) θ

(25)

where γ = 1.4, the maximum thrust at sea level and for M = 0 is T SL = 5.0 × 105 N, and θ = Θ/ΘSL is the temperature ratio (ΘSL being the ISA temperature at sea level). The specific fuel consumption is defined by (see Torenbeek [29])

c=

aSL

√

θ

LH

C C (M )

(26)

with the specific fuel consumption coefficient given by (see Mattingly et al. [17])

C C = c SL

LH aSL

(1.0 + 1.2M )

(27)

where aSL is the ISA speed of sound at sea level, the specific fuel consumption at sea level and for M = 0 is c SL = 9.0 × 10−6 kg/(s N), and the fuel latent heat is L H = 43 × 106 J/kg. In general, C C is a function of M and the thrust coefficient C T , however, the dependence of C C with C T is in practice very weak and can be neglected (see Torenbeek [29]). References [1] J.F. Barman, H. Erzberger, Fixed-range optimum trajectories for short-haul aircraft, Journal of Aircraft 13 (1976) 748–754. [2] D.J. Bell, D.H. Jacobson, Singular Optimal Control Problems, Academic Press, New York, 1975. [3] J.Z. Ben-Asher, Optimal Control Theory with Aerospace Applications, AIAA Education Series, AIAA, Reston, VA, 2010. [4] K.D. Bilimoria, E.M. Cliff, Singular trajectories in airplane cruise-dash optimization, NASA-CR-180636, 1987. [5] K.D. Bilimoria, H.Q. Lee, Aircraft conflict resolution with an arrival time constraint, AIAA paper 2002-4444, 2002.

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