Optimality of Standard Flight Procedures of Commercial Aircraft

Optimality of Standard Flight Procedures of Commercial Aircraft Antonio Franco, Alfonso Valenzuela and Damián Rivas Department of Aerospace Engineering Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Seville, 41092, Spain {antfranco,avalenzuela,drivas}@us.es Abstract Standard flight procedures for climb, cruise and descent, commonly flown by commercial aircraft, are analyzed, and the optimality of their performance is assessed by comparison with the performance of optimal trajectories, which are computed using the theory of singular optimal control. Three flight procedures are studied: CAS/Mach climb, constant-Mach cruise with fixed arrival time, and constant-CAS unpowered descent. The optimality criteria selected are minimum fuel consumption for climb and cruise, and maximum horizontal distance travelled for descent. Results are presented for a model of a Boeing 767-300ER. 1 Introduction Trajectory optimization is, from the operational point of view, a subject of great importance in air traffic management (ATM), that aims at defining optimal flight procedures that lead to energy-efficient flights. However, in practice airlines fly standard procedures which in principle are not optimal, such as CAS/Mach climb, constant-Mach cruise or Mach/CAS descent, although they are optimized, that is, the constant values of CAS and/or Mach number are chosen so as to obtain the best possible performance. The use of standard flight procedures is common in the generation of global trajectories. Thus, they are used, among others, by Gill and Maddock [1] in the Experimental Flight Management System of the PHARE programme, by Wu and Zhao [2] who optimize trajectories from liftoff to touchdown and quantify the deviation from actual trajectories due to modeling errors, and by Rivas et al. [3] who present a trajectory computation tool designed for a general aircraft performance model (general drag polar and general engine model), taking into account wind effects and temperature corrections for a non-standard atmosphere. The main objective of this work is to assess the optimality of the following standard flight procedures: 1) CAS/Mach climb between two given points (given speed and altitude), 2) constant-Mach cruise at constant altitude with fixed arrival time between two given points (given speed and distance), and 3) constant-CAS unpowered descent between two given points (given speed and altitude). These procedures are optimized using parametric optimization theory (see Fletcher [4]). The optimality assessment is twofold: first, the procedures themselves are compared with the optimal procedures (which in general are not at constant values of CAS and/or Mach number), and, second, their best performance is compared with the optimal performance. The optimal trajectories are computed using the theory of singular optimal control (see Bell and Jacobson [5]); examples of application of this theory to cruise and descent problems can be found in Refs. [6,7]. The optimality criteria selected for these problems are: minimum fuel consumption for climb and cruise, and maximum horizontal distance travelled for descent. Results are presented for a model of a Boeing 767-300ER (a typical twin-engine, wide-body, long-range transport aircraft). Wind effects are not considered in this work. A. Franco, A. Valenzuela, D. Rivas 2 Problem formulation The equations of motion for a flight in a vertical plane are ˙ V=T−D m−gγ ˙m=−cT ˙ h=V γ ˙x=V (1) where the assumptions γ¿1and V˙γ gnegligible have been taken into account (see Jackson et al. [8]). In these equations, the drag is a general known function D(V, m, h), which takes into account the remaining equation of motion L=mg. The thrust T(V, h)is given by T(V, h) = πTM(V, h)where πis a thrust control parameter, 0< πmin ≤π≤πmax = 1, and TM(V, h)is a general known function. The specific fuel consumption is also a general known function c(V, h). Wind effects are not considered. In this paper three problems are analyzed: 1) Minimum-fuel climb in a vertical plane between two given points (given speed and altitude), in the case of fixed engine rating. The initial values of speed, mass, altitude and distance (Vi, mi, hi, xi), and the final values of speed and altitude (Vf, hf) are given. The final value of mass (mf), distance (xf), and flight time (tf) are unspecified. 2) Minimum-fuel cruise at constant altitude with fixed arrival time between two given points (given speed and distance). The initial values of speed, aircraft mass and distance (Vi, mi, xi), and the final values of speed and distance (Vf, xf)are given. The final value of aircraft mass (mf) is unspecified, and flight time (tf) is fixed. 3) Maximum-range unpowered descent in a vertical plane between two given points (given speed and altitude). The initial values of speed, altitude and distance (Vi, hi, xi), and the final values of speed and altitude (Vf, hf)are given. The final value of distance (xf) and the flight time (tf) are unspecified. The aircraft mass is constant. 3 Optimal problem In the climb problem, the objective is to minimize the following performance index Jcl =−Ztf 0 cTdt, (2) subject to the state equations (1), with given π(π=πcl). The flight-path angle γis the control variable. In the cruise problem, the objective is to minimize the following performance index Jcr =−Ztf 0 cT dt, (3) subject to the state equations (1), with γ= 0 (constant altitude). The thrust control parameter πis the control variable. In the descent problem, the objective is to minimize the following performance index Jd=−Ztf 0 Vdt. (4) subject to the state equations (1), with π= 0. The flight-path angle γis the control variable. In all the problems to be solved, there is one control variable which appears linearly in the equations of motion, as well as on the performance indices to be optimized (γat climb and descent, and πat cruise). As a consequence, the Hamiltonian of the problem is also linear on the control variable, which leads to a singular optimal control problem. Let Hbe the Hamiltonian of the problem and uthe control variable. The derivative Huis called the switching function. The optimal control problem reduces to finding the optimal control u∗that minimizes H. In general, u∗is determined by the necessary condition for optimality Hu= 0 (provided that umin < 2 Optimality of Standard Flight Procedures u∗< umax), but in the problems to be solved the function Hudoes not depend on u(hence, Huu = 0, reason why the problem is called singular). The singular control using now follows from the condition ¨ Hu= 0 (the function ¨ Huhappens to be linear in u). In the three problems considered, the corresponding optimal path lies on a singular manifold in the state space called singular arc, which is of order q= 1 (as defined in Ref. [5]) and can be obtained from the necessary conditions H=const (H= 0 if tfis unspecified), Hu= 0 and ˙ Hu= 0 (see Bryson and Ho [9]). The singular arc is in fact the locus of possible points in the state space on which optimal paths can lie, as well as a switching boundary for the optimal control (see Ben-Asher [10]). Integration along the optimal path leads to optimum performance index, provided that the generalized Legendre-Clebsch condition (see Kelley et al. [11]) given by −∂¨ Hu/∂u ≥0(as q= 1) is satisfied along the singular arc. In this work it is assumed that the initial and final points of the path are given. In that case, the optimal path is formed, in general, by three arcs: one to go from the initial point to the singular arc, the singular arc, and a final arc to go from the singular arc to the final point. The initial and final arcs are defined by the control being at its maximum or minimum value; this type of optimal control is called bang-singular-bang. Bang-singular-bang solutions are subject to additional necessary conditions for the junctions between singular and nonsingular arcs to be optimal: McDanell-Powers condition and Weierstrass-Erdman corner conditions (see Refs. [9,12]). In order to obtain the solution for each problem, a numerical procedure must be defined, which includes the integration of the state equations, first, with either u=umin or u=umax from the initial point (with known initial values) until the singular arc is reached (which defines the first junction point), second, with u=using from the first junction point until the second one is reached, and, third, with either u=umin or u=umax from the second junction point until the final point (with known final values) is reached. Part of the resolution procedure is also to check whether both the assumed structure for the control is correct and the generalized Legendre-Clebsch condition is satisfied. A description of this type of procedure can be found in Refs. [6,7]. Although called optimal trajectories, the solutions are in fact extremals, that is, trajectories that satisfy the necessary conditions for optimality. 4 Standard procedures The standard flight procedures analyzed in this work are described in this section. They are: 1) CAS-Mach climb, 2) constant-Mach cruise, and 3) constant-CAS descent. 4.1 Optimized CAS-Mach climb First, the optimized CAS-Mach climb is analyzed. The CAS-Mach procedure considered in this work is formed by four segments, all of them with constant engine rating: 1) acceleration at constant altitude hi from the initial speed Vito the climb CAS (CASc), 2) climb with constant CAS (CASc) from hito the transition altitude h2at which climb Mach Mcis reached, 3) climb with constant Mach (Mc) from h2to the final altitude hf, and 4) acceleration/deceleration at constant altitude hffrom Mcto the final speed Vf. To solve the equations of motion (1) for each flight segment, a flight constraint must be given so that the control parameter γcan be determined. For the initial and final flight segments the flight constraint is h=const, and, therefore, γ= 0. For the constant-CAS segment, it is CAS =const =CASc, which is in fact a speed law V=VC(h)given by (see Asselin [13]) VC=v u u u t 2 kRaΘ(h) Ã1 + pSL p(h)"µ1 + k 2 ρSL pSL CAS2 c¶1/k −1#!k −1 ,(5) where k= (κ−1)/κ,κ=1.4 is the ratio of specific heats, Ra=287.053 J/(kgK) the gas constant of the air, and p, Θ, ρ the pressure, temperature and density, with pSL, ρSL the reference sea-level values. For the constant-Mach segment, the flight constraint is M=const =Mc, which is in fact a speed law V=VM(h) given by VM=McpκRaΘ(h).(6) At the end of the integration one has the final aircraft mass mf. The fuel consumption is therefore mF= mi−mf. This procedure to obtain the fuel consumption for given values of CAS and Mach can be written in symbolic form as mF=mF(CASc, Mc)(7) 3 A. Franco, A. Valenzuela, D. Rivas The CAS-Mach procedure is now optimized to give minimum fuel consumption, taking CAScand Mcas the optimization parameters. The optimum values of CAScand Mc, say CAS∗ cand M∗ c, are obtained solving the following parametric optimization problem minimize mF(CASc, Mc) subject to CASf≤CASc≤CASi Mf≤Mc≤Mi (8) where CASi,Mi,CASfand Mfare the values of CAS and Mach that correspond to Vi, hiand Vf, hfrespectively. In this work, the optimization solver used is MATLAB’s fmincon, a sequential quadratic programming (SQP) method (see Fletcher [4], for example). 4.2 Constant-Mach cruise Now the constant-Mach cruise is analyzed. The constant-Mach procedure considered in this work is formed by three segments, all of them with constant altitude: 1) acceleration/deceleration from the given initial speed Vito the cruise speed Vcr, with maximum cruise/idle engine rating, 2) cruise with constant speed Vcr, and 3) acceleration/deceleration from Vcr to the given final speed Vf, with maximum cruise/idle engine rating. To solve the equations of motion (1) for each flight segment, a flight constraint must be given so that the control parameter πcan be determined. For the initial and final flight segments the flight constraint is π=πmax or π=πmin. For the constant-Mach segment, the flight constraint is M=const =Mcr, hence the cruise speed is Vcr =Mcra(h)where a(h)is the speed of sound at the given altitude h. To perform the integration, the variables Vcr and x2(distance flown during the second segment) are guessed. At the end of the integration one has values of the flight distance and the flight time which in general are different from xfand tf. One must then iterate on the two free variables Vcr and x2until distance and time coincide with xfand tfto within some prescribed tolerance. The iteration is started with the initial guess Vcr =xf tf and x2=xf. Once the iteration ends, one has the final aircraft mass mf; the fuel consumption is mF=mi−mf. 4.3 Optimized constant-CAS descent Finally, the optimized constant-CAS descent is analyzed. The constant-CAS procedure considered in this work is formed by three segments, all of them with zero thrust: 1) deceleration at constant altitude hifrom the initial speed Vito the descent CAS (CASd), 2) descent with constant CAS (CASd) from hito the final altitude hf, and 3) acceleration/deceleration at constant altitude hffrom CASdto the final speed Vf. As in the CAS-Mach climb, to solve the equations of motion (1) for each flight segment, a flight constraint must be given so that the control parameter γcan be determined. For the initial and final flight segments the flight constraint is h=const, and, therefore, γ= 0. For the constant-CAS segment, it is CAS =const = CASd, which leads to the same speed law V=VC(h)given by Eq. (5), except for considering CASdinstead of CASc. At the end of the integration one has the distance travelled xf. This procedure to obtain the range for a given value of CAS can be written in symbolic form as xf=xf(CASd)(9) The constant-CAS procedure is now optimized to give maximum range, taking CASdas the optimization parameter. The optimum value of CASd, say CAS∗ d, is obtained solving the following parametric optimization problem minimize −xf(CASd) subject to CASf≤CASd≤CASi (10) where CASiand CASfare the values of CAS that correspond to Vi, hiand Vf, hfrespectively. The optimization solver used is again MATLAB’s fmincon. 4 Optimality of Standard Flight Procedures 5 Results The aerodynamic and propulsion models considered in this paper for the numerical applications (corresponding to a Boeing 767-300ER) are described in the Appendix, and the atmosphere model is the International Standard Atmosphere (ISA). Climb results are presented for the case of initial and final γmin-arcs, which require that the initial and final speeds be sufficiently low and high respectively. In particular, γmin = 0 has been considered so that the initial and final arcs are horizontal segments, as in the optimized CAS-Mach procedure, with which the optimum results are to be compared. The initial conditions (corresponding to a hypothetical SID final fix) are CASi= 250 kt, hi= 10000 ft, and the final conditions are Mf= 0.8,hf= 33000 ft. The initial aircraft weight ranges from Wi= 1650 kN to Wi= 1750 kN. Cruise results are presented for a flight defined by a range xf= 8000 km, and by initial and final speeds Vi= 240 m/s and Vf= 180 m/s. The fixed arrival time ranges from tf= 9.17 h to tf= 10 h. The initial aircraft weight ranges from Wi= 1650 kN to Wi= 1750 kN. The cruise altitude is taken to be h= 10000 m. Descent results are presented for the case of two initial and final γmax-arcs, which require that the initial and final speeds be sufficiently high and low respectively. In particular, γmax = 0 has been considered so that the initial and final arcs are horizontal segments, as in the optimized constant-CAS procedure, with which the optimum results are to be compared. The initial conditions are Mi= 0.8,hi= 33000 ft, and the final conditions (corresponding to a hypothetical approach fix within the TMA) are CASf= 210 kt, hf= 9000 ft. The aircraft weight during the descent ranges from Wi= 1100 kN to Wi= 1300 kN. In the following, optimal trajectories and control laws as well as global results (fuel consumption, distance travelled, final time) are presented along with the optimized, standard ones. A comparison between both sets of results is made. The influence of the aircraft weight on the results is analyzed. 5.1 Climb problem In this section the optimized standard CAS-Mach climb is compared with the optimal climb problem. The optimal and the optimized CAS-Mach climb trajectories V(h)are represented in Fig. 1, for different values of the initial aircraft weight. The climb trajectories start and finish with horizontal accelerations. In the optimal trajectories, these horizontal segments correspond to the γmin-arcs, from the given initial point to the singular arc, and from the singular arc to the given final point. In the optimized CAS-Mach climbs, these horizontal segments correspond to the initial and final horizontal accelerations from the given initial CAS to CAS∗ c, and from M∗ cto the final Mach number. The speed continuously increases, reaches a maximum and then slowly decreases. As Wiincreases, the speed along the singular arc and the constant-CAS/Mach segments slightly increases. Note that the first part of the optimal profile is not at constant CAS; in that sense, the CAS/Mach procedure itself is not close to optimal. 2000 4000 6000 8000 10000 12000 140 160 180 200 220 240 260 h[m] V[m/s] Wi Figure 1: V(h)comparison for Wi= 1650,1675,1700,1725 and 1750 kN. Solid lines: optimal climb. Dashed lines: optimized CAS-Mach climb. 5 A. Franco, A. Valenzuela, D. Rivas The optimal control γ(h)and the optimal altitude law h(x)are represented in Fig. 2, along with the flight path angle law and the altitude law corresponding to the optimized CAS-Mach climb, for the same values of the initial aircraft weight. The control is discontinuous: for the optimal trajectories, one has the two arcs with γmin = 0 (hardly seen in the figure) and the singular arc, whereas, for the optimized CAS-Mach trajectories, one has the four constitutive segments. Note that the control γdecreases as the climb progresses. As Wi increases, the control slightly decreases, and the horizontal distance travelled increases. 2000 4000 6000 8000 10000 12000 0 0.5 1 1.5 2 2.5 3 3.5 4 h[m] γ[deg] Wi (a) 0 50 100 150 200 250 300 350 2000 4000 6000 8000 10000 12000 x[km] h[m] Wi (b) Figure 2: γ(h)and h(x)comparison for Wi= 1650,1675,1700,1725 and 1750 kN. Solid lines: optimal climb. Dashed lines: optimized CAS-Mach climb. In Figs. 3 and 4, the minimum fuel consumption, the flight time and the range for both problems are represented as functions of the initial aircraft weight. As expected, heavier aircraft require larger values of fuel consumption, time to climb and distance travelled, as compared to lighter aircraft. One can see that the differences between both sets of results are almost negligible in all cases (less than 3.4 kg in minimum fuel consumption, less than 3.3 s in flight time and less than 188 m in range). Hence, it can be concluded that the performance of the CAS-Mach procedure is very close to optimal, provided that the optimum values of CAScand Mcare used in the climb. 1.65 1.7 1.75 x 106 2500 2600 2700 2800 2900 3000 3100 3200 Wi[N] mF[kg] Figure 3: Minimum fuel consumption vs. initial aircraft weight. Solid lines: optimal climb. Dashed lines: optimized CAS-Mach climb. 6 Optimality of Standard Flight Procedures 1.65 1.7 1.75 x 106 17 18 19 20 21 22 23 Wi[N] tf[min] (a) 1.65 1.7 1.75 x 106 240 250 260 270 280 290 300 310 320 Wi[N] xf[km] (b) Figure 4: Flight time (a) and Range (b) vs. initial aircraft weight. Solid lines: optimal climb. Dashed lines: optimized CAS-Mach climb. 5.2 Cruise problem Now the standard constant-Mach cruise is compared with the optimal cruise problem. The optimal trajectories and the constant-Mach trajectories for different initial aircraft weights and an arrival time tf= 9.5h are shown in Fig. 5a. The corresponding controls are shown in Fig. 5b. For the optimal trajectories, the structure is minimum-thrust arc, singular arc, minimum-thrust arc, in all cases shown. For the constant-Mach trajectories, one has the three constitutive segments. In this problem where the final distance and 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 and the constant-Mach cruise control (which are almost identical) increase as Wiincreases, and they decrease along the singular arc and the constant-Mach segment. 0 2000 4000 6000 8000 0.6 0.65 0.7 0.75 0.8 0.85 x[km] M[-] Wi Wi (a) 0 2000 4000 6000 8000 0 0.2 0.4 0.6 0.8 1 x[km] π[-] Wi (b) Figure 5: Trajectories (a) and control (b) for Wi= 1500,1550,1600,1650,1700 kN (tf= 9.5h, h= 10000 m). Solid lines: optimal cruise. Dashed lines: constant-Mach cruise. 7 A. Franco, A. Valenzuela, D. Rivas To see the influence of the arrival time, the optimal trajectories and the constant-Mach trajectories for different arrival times (tf= 9.17,9.33,9.5,9.67,9.83,10 h) and for Wi= 1600 kN are shown in Fig. 6a. The corresponding controls are shown in Fig. 6b. As expected, the Mach number decreases as the arrival time increases. For small values of tfthe Mach number of the optimal trajectory is roughly constant, however for large values of tfit decreases along the cruise. The results also show that the variation of both the singular control and the constant-Mach cruise control with the arrival time is quite small. 0 2000 4000 6000 8000 0.6 0.65 0.7 0.75 0.8 0.85 x[km] M[-] tf (a) 0 2000 4000 6000 8000 0 0.2 0.4 0.6 0.8 1 x[km] π[-] (b) Figure 6: Trajectories (a) and control (b) for different arrival times (tf= 9.17,9.33,9.5,9.67,9.83,10 h, h= 10000 m, Wi= 1600 kN). Solid lines: optimal cruise. Dashed lines: constant-Mach cruise. The minimum fuel consumption for both problems as a function of the flight time is shown in Fig. 7 for different initial aircraft weights. As expected, heavier aircraft require larger values of fuel consumption, as compared to lighter aircraft. One can see that the differences between the optimal and the constant-Mach results are almost negligible in all cases (less than 62 kg). Hence, it can be concluded that the performance of the constant-Mach procedure is very close to optimal. 9 9.5 10 10.5 3.6 3.8 4 4.2 4.4 4.6x 104 tf[h] mF[kg] Wi Figure 7: Minimum fuel consumption vs flight time, for Wi= 1500,1550,1600,1650,1700 kN (h= 10000 m). Solid lines: optimal cruise. Dashed lines: constant-Mach cruise. 8 Optimality of Standard Flight Procedures 5.3 Descent problem Finally, the optimized standard constant-CAS unpowered descent is compared with the optimal descent problem. The speed laws V(x)corresponding to the optimal problem and to the optimized constant-CAS descent are presented in Fig. 8 for different values of the aircraft weight. The descent trajectories start and finish with horizontal decelerations. In the optimal trajectories, these horizontal segments correspond to the γmax-arcs, from the given initial point to the singular arc, and from the singular arc to the given final point. In the optimized constant-CAS descent, these horizontal segments correspond to the initial and final horizontal decelerations from the given initial Mach number to CAS∗ d, and from CAS∗ dto the final CAS. The speed continuously decreases along the entire trajectory. As the aircraft weight increases, the speed during the singular arc and the constant-CAS segment increases. The differences between both sets of profiles are very small, so that one can conclude that the constant-CAS procedure is very close to optimal. 0 50 100 150 200 120 140 160 180 200 220 240 x[km] V[m/s] W Figure 8: V(x)comparison for W= 1100,1150,1200,1250 and 1300 kN. Solid lines: optimal descent. Dashed lines: optimized constant-CAS descent. The optimal control γ(h)and the flight path angle corresponding to the optimized constant-CAS descent are represented in Fig. 9a for the same values of the aircraft weight. They are discontinuous (one has the two arcs with γmax = 0 and the singular arc). The results show that both the optimal control and the ground path angle are roughly constant, which can be also seen in Fig. 9b, where one has a roughly linear altitude profile during the singular arc. The aircraft weight Whas very little influence both on the singular optimal control and on the flight path angle during the constant-CAS segment. The optimal altitude law and the altitude law corresponding to the optimized constant-CAS descent h(x) are represented in Fig. 9b for the same values of the aircraft weight. The altitude continuously decreases, except for the initial and final decelerations at constant altitude. As the aircraft weight increases, the altitude during both the singular arc and the constant-CAS segment slightly decreases (the influence of Wis very small). The maximum range and the flight time for the optimized constant-CAS descent and for the optimal problem are represented as functions of the aircraft weight in Fig. 10. The influence of Win the maximum range is negligible, however, as the aircraft weight increases, the flight time decreases. One can see that differences between the optimized constant-CAS results and the optimal results are negligible (less than 28 m in maximum range and 0.32 s in flight time). Hence, it can be concluded that the performance of the constant-CAS procedure is very close to optimal, provided that the optimum value of CASdis used in the descent. 9