Optimal 4-D Aircraft Trajectories in a Contrailsensitive

Transcription

1 Optimal 4-D Aircraft Trajectories in a Contrailsensitive Environment Bo Zou 1, Gurkaran Singh Buxi 2, Mark Hansen 2 1 Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago 60607, USA 2 Department of Civil and Environmental Engineering, University of California at Berkeley, Berkeley 94720, USA Abstract: Aircraft induced contrails present an important source and a growing concern for climate change in aviation. This paper develops a methodology to determine optimal flight trajectories that minimize the total flying cost in a dynamic, contrail-sensitive environment. The total flying costs consist of costs due to fuel burn, crew, passenger travel time, CO 2 emission, and contrail formation. By constructing a multi-layer hexagonal grid structure to represent the airspace, we formulate the single aircraft trajectory optimization problem as a binary integer program that allows for flight altitude and heading adjustment, and contrail information update. Various cost factors are quantified, in particular the one corresponding to aviation-generated contrails, using the Global Warming Potential concept. Computational analyses show that optimal trajectories depend critically upon the time horizon choice for calculating the CO 2 climate impact. Shifting flights to periods with low contrail effect is not justified, given the limited benefit but potentially large passenger schedule delay cost increase. The analyses are further extended to determining the optimal trajectories for multiple flights using a successive optimization procedure. Keywords: Contrail; Flight Trajectory; Optimization; Climate Impact 1

2 1 Introduction Air transportation contributes a small but growing share of global anthropogenic climate change impact. Today, emissions of CO 2 from aircraft operations account for approximately 2% of the global total, and are expected to grow 3-4% annually (ICAO, 2012). The projected growth, in contrast to emission reductions in other industrial and agricultural sectors, will lead to a larger share of emissions from the aviation sector in the future. RCEP (2003) estimates that by 2050 the aviation sector will be responsible for 6% of total radiative forcing (RF) from human activities. While CO 2 is the most widely known greenhouse gas agent derived from aviation, emissions from aircraft engines include other constituents that contribute, via the formation or destruction of atmospheric constituents, to climate change. Emissions of NO x tend to increase tropospheric ozone and reduce methane. The increase in radiative forcing associated with ozone is largely offset by the methane reduction, resulting in a relatively small net positive NO x impact compared to the CO 2 impact (Williams et al., 2003). Of increasing concern is another aviation-induced climatic driver contrails, which are line-shaped clouds composed of ice particles and formed in the wake of jet aircraft at high altitude where the ambient temperature is very low. Like natural high clouds, contrails affect the atmosphere near the surface in two opposing ways: by reducing the amount of earth-emitted radiation escaping to space, and by increasing the amount of solar radiation reflected. For contrails, the effect of outgoing terrestrial radiation is larger, resulting in a warming at the surface (Williams et al., 2002). Contrails evaporate quickly if the ambient air is dry, but can persist if the ambient air is humid enough. While a number of studies have examined the climate impact of persistent contrails with varying estimated results, the general conclusion is that the magnitude of contrail climate impact is non-negligible comparable to that of CO 2 (Penner et al., 1999; Mannstein and Schumann, 2003; Waitz et al., 2004; Schumann, 2005). Among a range of operational, technological, economic, and regulatory options that aim to mitigate the climate impact of aviation-induced contrails (Penner et al., 1999; Waitz et al., 2004; Williams et al., 2007), this study will focus on the operational aspect. Specifically, in this paper we will develop a 4-D (three dimensions in space plus the time dimension) aircraft trajectory design model that allows one or multiple flights to fly optimally in the presence of airspace that is favorable to persistent contrail formation. Nevertheless, it is worth noting that exploring the operational possibilities is closely intertwined with considerations on the technological, economic, and regulatory fronts. For instance, decisions on the optimal flight paths depend critically upon aircraft fuel consumption parameters, how climate impacts from CO 2 and contrails are valuated, as well as air traffic management standards and norms. Several previous modeling efforts have attempted to address the issue of mitigating aviationincluded contrails. Some use simulation tools to consider vertical displacement of flight paths to avoid the persistent contrail formation areas (PCFAs) in the airspace (Williams et al., 2002, 2003; Fichter et al., 2005; Sridhar et al., 2012; Chen et al., 2012). They find altitude adjustment to be an effective strategy to significantly reduce contrail production. Sridhar et al. (2011) present the optimal control approach to investigate the horizontal adjustment of flight paths to minimize contrail-inclusive flying cost. Mixed integer programming techniques are also considered to minimize total aircraft fuel cost while avoiding the formation of persistent contrails (Campbell et al., 2008). These studies yield similar conclusion that travel time through PCFAs can be eliminated entirely or reduced by a substantial amount at a relative small price of fuel consumption increase. Our work extends the above research and makes two important contributions. Firstly, we consider horizontal and vertical aircraft maneuvering in the presence of evolving PCFAs over time, and the difference in contrail impact by time of day. By formulating the problem as a binary integer program, our approach permits greater flexibility in flight path design and enables optimal 4-D 2

3 flight routing in a dynamic, contrail-sensitive environment. Secondly, different from the previous studies which focus on contrail formation alone, or in conjunction with fuel burn, we take a comprehensive view by quantifying a more inclusive set of climatic and cost components in aircraft operations. We introduce a novel approach that converts flight climate impacts into dollar values, and consider the optimization of all climatic and cost impacts, including CO 2 emissions, contrail formation, flight operating expenses, and passenger travel time, in an integrated manner. Results from the analyses offer more explicit insights into the critical factors and tradeoffs in the trajectory design. The remainder of the paper is organized as follows. In Section 2, we conduct a review of the existing aviation-contrail related research. Section 3 introduces a grid structure to represent the airspace and PCFA regions. Building upon this grid structure, a binary integer program is formulated in the ensuing section to identify the optimal trajectory for a single flight. Relevant cost factors used in the program are determined in Section 5, followed by computational analyses in Section 6. The model is then extended to determining trajectories for a sequence of flights. The solution procedure and numerical investigations are presented in Section 7. Section 8 concludes and provides future research directions. 2 Literature review and research contributions This section reviews existing aviation-related research. Understanding the physics of contrail formation will be helpful before we proceed. The contrail formation is well documented and described first by the Schmidt-Appleman criterion (Schmidt, 1941; Appleman, 1953). Contrails are clouds that form when a mixture of warm engine exhaust gases and cold ambient air reaches saturation with respect to water (i.e. the relative humidity with respect to water, RH w, is greater than a critical value r contr), forming liquid drops, which quickly freeze to form ice crystals (Sridhar et al., 2011). While contrails are short lived in dry air, when the ambient air is supersaturated with respect to ice (i.e. the Relative Humidity with respect to Ice, RH i, is greater than 100%), the rate of condensation into ice is faster than the rate of evaporation, and the ice crystals will grow to form persistent visible contrails. Like natural high clouds, persistent contrails modify the radiation budget of the earth-atmosphere system by reducing the outgoing terrestrial radiation more than they reflect solar radiation, resulting in warming of the earth s surface (Williams et al., 2002). The regions of the airspace that are favorable to persistent contrail formations, i.e. RH w > r contr and RH i > 100%, are called Persistent Contrail Formation Areas (PCFAs). For brevity, in the remainder of the paper "contrail formation" means the formation of persistent contrails. The design of optimal aircraft trajectories in a persistent contrail-sensitive environment requires knowledge about the temporal and spatial characteristics of persistent contrails, and air traffic constraints when considering multiple aircraft. In addition, effective strategies to mitigate contrail formation should not be conceived alone but jointly with consideration of other climate impacts and cost components germane to aircraft operations. The following four subsections provide a review of existing aviation-related contrail studies with focus on: the time-of-day effect of persistent contrails, the spatial distribution of PCFAs and associated aircraft routing strategies, cost factors related to flight operation, and multiple-trajectory optimization. We highlight the contribution of the present study given the shortcomings in the literature. 2.1 Time-of-day effect Persistent contrails affect the earth's radiation balance through radiative forcing because they trap outgoing long-wave radiation emitted by the earth and atmosphere at a greater rate than they reflect incoming, shortwave solar radiation. This radiative forcing effect has been reported to vary diurnally, mainly due to the variation of shortwave radiation (Meerkotter et al., 1999). RF is at minimum around 6am and 6pm, and reaches the maximum during night and early morning 3

4 (Meerkotter et al., 1999). Forster and Stuber (2007) estimate the ratios of night/day radiative forcing to be 1.5 and 3.2 under all-sky and clear-sky conditions. As a consequence, night traffic, while only accounting for 25% of daily air traffic, contributes to 60-80% of the total contrail RF (Stuber et al., 2006). Neglecting the diurnal variation in air traffic/contrail cover and assuming a mean contrail cover can overestimate the global mean radiative forcing by up to 30% (Forster and Stuber, 2006). Stordal and Myhre (2001) suggest shifting the peak traffic periods towards sunrise or sunset to reduce the contrail impact. Such time adjustment, however, entails cost. Some passengers would be forced to depart at times farther away from their preferred departure time, creating more schedule delay. Airlines may need additional aircraft to execute the schedule because of the added time constraints. In addition, as flight traffic is more concentrated around hours with low contrail RF effect, congestion would be exacerbated both at airports and in the airspace. Consideration of such consequences remain largely absent in the literature. In the present study, we will show that the benefit from shifting flight departure times is limited vis-àvis the potential cost. 2.2 Spatial adjustment of flight path Given the uneven vertical distribution of PCFAs in the airspace, there exist potentials to change aircraft cruise altitude to reduce contrail formation and the overall climate impact. Persistent contrails usually form in a layer below the tropopause, where it is cold and moist enough to fulfill the Schmidt-Appleman criterion. At lower altitudes, it is too warm, such that even short-lived contrails would not appear. At higher levels, particularly in the stratosphere, it is usually too dry for contrails to form (Sausen et al., 1998). On the other hand, adjusting flying altitude would affect fuel consumption. The impact on fuel consumption and contrail coverage of flying at different cruise altitudes has been investigated in Williams et al. (2002, 2003), Fichter et al. (2005), and more recently, Chen et al. (2012). Fichter et al. (2005) conclude that global contrail coverage can be reduced by 45% by an altitude displacement of -6,000 ft. Under this extreme altitude change, no contrails would form at latitudes above 30 N. In Europe, restricting flying altitude of short-haul, intra-europe flights would increase average journey time and fuel burn by less than 1 minute and 3.9% respectively with a near total elimination of contrail production (Williams et al., 2002). In the U.S., altitude adjustment for short-distance (<500 miles) and medium-distance ( miles) flights contributes respectively the least and most to total persistent contrail reduction (Chen et al., 2012). Longer-haul air traffic occurs predominantly in the cold and dry lowermost stratosphere. A small decrease in flight altitude would shift the flights into the upper troposphere where ambient conditions may be more favorable for contrail formation (Fichter et al., 2005); whereas larger altitude decrease would yield reduction in contrail formation. The benefits from such reduction, in most cases, are not likely to be offset by the resulting CO 2 emissions increase (Williams et al., 2003). Aircraft maneuverings other than altitude adjustment are also important to mitigate the contrail effect. An aircraft can choose to deviate from a contrail-free optimal path while staying at the same flight level to partially or fully avoid PCFAs. Sridhar et al. (2011) adopt an optimal control approach to model such horizontal flight path adjustment to minimize contrail-inclusive total flying cost, and identify the optimal trajectories for a range of altitudes. A more complete strategy is to simultaneously allow horizontal and vertical trajectory adjustment, which, however, makes trajectory design computationally more challenging. The optimal control approach, for example, would be extremely intensive computationally when the problem is extended to three-dimension space. To model the adjustment in both altitudes and headings of the flight trajectory, we employ a grid network to structure the airspace and solve for the optimal trajectory by formulating the problem as a binary integer program. Compared to a recent study (Campbell et al., 2008) which adopts a similar approach, we account for both spatial and temporal distribution of PCFAs, and 4

5 relax the hard constraints of complete contrail avoidance by making explicit tradeoff between contrail formation and other climatic and cost effects. Our approach permits greater flexibility in flight path design and enables optimal 4-D flight trajectories in a contrail-sensitive environment. 2.3 Cost factors While there is little doubt about the importance of appropriate estimates for different climatic and cost components in identifying optimal flying strategies, this question has received rather limited attention. Existing studies evade explicit calculation of the cost/benefit from changes in flight trajectory. Campbell et al. (2008) in effect assume infinite cost when a flight flies into PCFAs; justification of flight altitude change in Williams et al. (2003) rests on the argument that the overall RFs from CO 2 and contrails are of comparable magnitude. There are two major problems with these assumptions. First, operational strategies should be approached from a broader perspective rather than simply the CO 2-contrail tradeoff. If flight travel time increases as a result of contrail avoidance, not only will extra fuel consumption impose penalties on aircraft operations, but flight crew will work longer hours, also increasing the operating cost to airlines. Moreover, since climate impact from CO 2 and contrail formation would clearly be borne by the society rather than air carriers alone, passenger travel time cost should also be included in the overall cost consideration. Second, as admitted by Williams et al. (2002) and Forster and Stuber (2007), using RF as the measure for climate impacts is questionable, since RF includes the impact of all historic flights, and does not account for the resident timescales of emissions. Given the distinct physical characteristics of different greenhouse gas agents, global warming potential (GWP) and global temperature change (GTP) represent better metrics to quantify the true climate impact of different gases. While contrail is not categorized as a gas agent, our study makes a first attempt to unify the climate impact of CO 2 and contrail formation using the GWP concept, 1 and incorporate them along with cost related to fuel, crew, and passenger travel time into the flight trajectory design process. 2.4 Optimizing trajectories for multiple flights While vehicle dispatching and routing have been examined extensively for non-aviation modes (e.g. Albercht, 2009; D Ariano and Pranzo, 2009; Escuín et al., 2012; Li et al., 2011), explicit optimization of multi-aircraft trajectories with consideration of air traffic constraints and flow management is relatively limited in the literature. To our knowledge, only simulation tools have been used (Williams et al., 2002; William and Noland, 2005; Sridhar et al., 2012; Chen et al., 2012) to investigate cruise altitude adjustment for many flights to reduce contrail formation. It is recognized that potential altitude restriction could result in an increasing number of separation minima violations between aircraft, adding to air traffic controllers' workload to resolve the trajectory conflict issues (Williams et al., 2002; Williams and Noland, 2005). In the present study, we take a further step and approach the multi-flight case from a strategic flow planning perspective. Anticipating the potential flight conflicts, we develop a successive optimization procedure to determine the trajectories of individual flights based on the departure schedule of flights. We also allow aircraft to wait on the ground, providing additional flexibility for flights to adjust their operations in response to PCFA regions and minimum separation constraints in the airspace. 1 Ng et al. (2011) use GWP to unify the effect of CO 2, H 2O, and NO x emissions, but not contrail formation. Independent of our study, Sridhar et al. (2012) employ average GTP to assess the climate impact of contrails and CO 2 in air traffic operations. 5

6 3 Representation of the airspace 3.1 Network structure We represent the airspace as a network comprised of nodes and links. Each node can be perceived as a waypoint in the airspace and a link is a line segment joining two neighboring waypoints. An aircraft travels on the link between two connected nodes. This representation is a discrete approximation of the continuous airspace. The approximation is reflected in the possible flight trajectories in which an aircraft can turn sharper angles than commonly seen in real situations. The network considered in our study covers the airspace between 90 W to 75 W and 38 N to 43 N, from ft to ft in altitude. Operational instructions (Delta, 2003) and simulation evidence (Williams et al., 2002) suggest that ft may present a fairly good proxy for the maximum cruise altitude. 2 We divide the altitude into seven evenly separated flight levels, with the distance between two neighboring ones being 1500 ft. 3 At a given flight level, the airspace is modeled as a network of interconnected hexagons. Each flight level has 1225 nodes (35 35 nodes between the longitude and latitude boundaries), leading to a total of 8575 (1225 7) nodes for the seven flight levels. Because of the symmetry, an arbitrary node in the network can be both a vertex of one hexagon and the center of another hexagon. The distance between two neighboring nodes on the same longitude and flight level is approximately 19 km, while the distance between two (staggered) nodes on the same flight level and adjoining longitudes is approximately 39 km. Because of the spherical shape of the earth, the above two distances vary slightly by longitude and latitude. A top view of the node distribution for part of the airspace is shown in Figure 1. All the flight levels are identical with respect to the location of the nodes. Any interior node at an intermediate flight level is connected to eighteen closest nodes: six nodes of the same level, six nodes on one level above and below (Figure 2). Both directions of travel are allowed on a link. Fig. 1 Nodes in the hexagonal structure of airspace 2 Nonetheless, the maximum altitude of feet can certainly be relaxed by considering higher flight levels. 3 The vertical separation distance is just a modeling parameter and can be configured with any other value. While 1000 feet is often used as the actual separation distance in civil aviation, here we choose 1500 feet mainly to reduce the network size and save computational time. 6

7 Fig. 2 Representation of nodes and links 3.2 PCFAs and update This sub-section specifies the distribution of PCFAs in the airspace. Due to the lack of real world data, a hypothetical PCFA distribution will be considered in the subsequent optimal trajectory design process. The hypothetical distribution is, nonetheless, based on some empirical evidence. Specifically, we use the PCFA coverage at ft shown in Figure 3 of Sridhar et al. (2011) as the basis to generate PCFAs in the present study. For simplicity, we modify the shape of the PCFA region at each flight level into two rectangles: a larger rectangle (at higher latitudes) bordered with a smaller rectangle (at lower latitudes), as shown in Figure 3. We determine the PCFA region for each flight level between ft and ft in three steps. In the first step, we calculate the fraction of sky suitable for contrail formation by linearly interpolating the isoquants of PCFA regions mentioned in Williams et al. (2003) between different altitudes. In step 2, we calculate the ratio between the fraction of PCFAs for each altitude (from Step 1) and the fraction of PCFAs at ft. In general, the ratio is greater than one. In step 3, for each altitude we multiply the area of PCFA at ft by the ratio from step 2. Following Sridhar et al. (2011), we assume no PCFAs at ft and ft flight levels. It can be observed, from both the isometric and bird views in Figure 3, that the PCFAs which are represented by the rectangle areas are larger at lower altitudes. To capture the temporal dynamics, we further assume that the PCFA regions shift 1 E after 30 minutes, and that the update is known precisely beforehand. Fig. 3 Isometric and bird views of the PCFAs at different altitudes 7

8 4 Model for single aircraft trajectory We assume that the flight trajectory starts and ends with predetermined waypoints o and q, at ft. The optimal trajectory is a set of consecutive links flown by the aircraft that yield the minimal flying cost. We assume that the aircraft arrives at q within 10T seconds, which is divided into T, 10-second (i.e. deca-second) time intervals. T is introduced only for programming convenience, and does not necessarily impose constraints on total flight time. Here we assume T = 6000 or the aircraft should reach q within seconds ( 17 hours) after starting from o, which is much greater than normal flight times. In the remaining part of this section, a moment in time is represented by t {0,1,2.., T}, measured in deca-seconds. Let N = {, i, } denote the set of nodes in the network. Each i N is associated with a latitude, a longitude, and an altitude. The link from node i to node j is represented as (i, j) A, where A is defined as the set of links. Each link (i, j) is associated with a link travel time h i,j (t) and a unit travel cost c i,j (t) (in $/sec) when the aircraft leaves i for j at time point t. In principle, the link travel time h i,j (t) depends on the speed of the aircraft and wind, both of which can vary with time. Due to the lack of real wind information, however, we assume no wind in the airspace in the present study. Real wind information, if it becomes available, can certainly be incorporated in the calculation of h i,j (t). We further assume a constant aircraft speed at 235m/s ( 0.78 Mach at feet). While aircraft typically cruise at a constant Mach number, which implies different cruise speeds at different altitudes, the speed variations within the altitude range considered in the study are relatively small. The difference is always less than 4% of the maximum ground speed (at ft), assuming no wind presence. On the other hand, while the constant 235 m/s tends to underestimate the true aircraft speed at lower altitudes with a constant 0.78 Mach number, making traveling at lower flight levels appear longer, the BADA formulas suggest that the consequent change in fuel burn rate also depends upon opposing effects brought by speed change. Therefore, potential bias in the estimated cost of flying through a link at lower altitudes under the constant speed assumption is likely to be small. This assumption can nevertheless be relaxed by introducing more detailed, altitude-dependent speed to calculate h i,j (t). 4 In the present study, since h i,j (t) is determined by dividing the length of (i, j) by the constant aircraft speed, h i,j (t) = h i,j t. An aircraft leaving node i at time t arrives at node j at time t + h i,j. We measure h i,j in deca-seconds and round it to nearest integer deca-second. The unit travel cost c i,j (t) is the sum of the unit costs of fuel burn, crew, passenger travel time, and CO 2 emission, when flying outside PCFAs. The unit costs of crew and passenger travel time are invariant to (i, j) and t. Because of the constant aircraft speed, the unit costs of fuel burn and CO 2 emission are also constant for a given (i, j). When (i, j) is within PCFAs (i.e. both i and j are within PCFAs), c i,j (t) should further include the unit cost of contrail formation. The values of these different cost components are discussed in the next section. Because PCFA regions shift over time, we assume that if the aircraft is traversing a link between two nodes when the contrail update occurs, the unit travel cost of the link remains unchanged. The effect of contrail updates would not be experienced by the aircraft until the aircraft enters a new link. We formulate the following 0-1 integer program to determine the optimal trajectory of a single aircraft. In the program, x i,j (t), (i, j) A, t {0,1,2.., T} are a set of 0-1 binary decision variables. x i,j (t) = 1 if the aircraft leaves node i at time t for node j; x i,j (t) = 0 otherwise. 4 We thank one of the reviewers for pointing this out. 8

9 T min c i,j (t)h i,j x i,j (t) t=0 (i,j) A (1) subject to x j,k (t) x i,j (θ) = 0 j N, t {0,1,.., T} k:(j,k) A i:(i,j) A θ:θ+h i,j =t (2) x o,j (0) = 1 j:(o,j) A x i,q (t) = 1 i:(i,q) A t:t+h i,q T x i,j (t) {0,1} (i, j) A, t {0,1,.., T} (3) (4) (5) The objective function (1) minimizes the total cost of flying. c i,j (t)h i,j is the total cost of flying on link (i, j) when the aircraft leaves node i for node j at time t (in which case x i,j (t) = 1), measured in dollars. Equation (2) is a flow conservation constraint stating that if the aircraft enters node j at time t, it should leave node j at the same time, i.e. waiting is not allowed at any node. θ denotes a time such that the aircraft leaving node i at θ arrives at node j at t. Equation (3) states that the aircraft leaves the starting node o at t = 0. Equation (4) states that the aircraft should have arrived at q by time T. Finally, (5) states that the decision variables are binary. The optimization problem can be solved by generating a time-space graph where each node and link are replicated T + 1 times, one for each t {0,1,, T}. However, the space-time network approach to determine the optimal trajectory is computationally expensive. Instead, we employ a specialized algorithm proposed by Chabini and Abou-Zeid (2003), which improves the computational efficiency of an earlier algorithm (Cai et al., 2001) in finding the minimum cost path in a dynamic network. The key difference between the two algorithms developed by Chabini and Abou-Zeid (2003) and Cai et al. (2001) is that the former computes lower bounds on the minimum travel cost to reduce the search area in the dynamic network, therefore avoiding visiting every node in the time-space network without affecting the optimality of the solution. Chabini and Abou-Zeid (2003) show that introducing the lower bounds in their algorithm can significantly save computational time compared to the one in Cai et al. (2001). Details about the algorithm are provided in Appendix A. 5 Cost factor determination Solving the above optimization problem requires estimates of the unit cost, which contains costs associated with fuel, crew, passenger travel time, CO 2 emission, and contrail formation if the aircraft flies in a PCFA. The former four categories of cost can be either directly obtained or derived from the literature. On the contrary, quantification of the contrail formation cost factor has not yet been seen in existing studies. We introduce a new approach in this section to estimate the unit climate impact of contrail formation while an airplane flies in PCFAs, and unify the impact with the other four cost categories. A set of assumptions will be made to infer the contrail 9

10 formation cost impacts. We present multiple contrail cost factor estimation results, which reflect the uncertainties associated with the assumptions, and also allow for gauging the sensitivity of optimal aircraft trajectories to those assumptions. In this section, all cost components are measured in dollars per second, based on a B aircraft. 5.1 Fuel Standard formulas as prescribed by the Eurocontrol BADA database are used to determine thrust and thrust specific fuel consumption rates (gallons/sec) at different altitudes for cruise and descent. Because relevant data were unavailable when we were first performing the analysis, we assume that fuel burn rate in climbing is twice that in cruise. Certainly, this can be adjusted should more accurate data become available. 5 These rates are multiplied by the unit jet fuel cost, for which we take the Oct 7, 2011 price at $2.909/gallon (IATA, 2011), to obtain the unit fuel cost in $/sec, as shown in Table 1 below. Table 1: Estimated unit fuel cost ($/sec) Altitude (ft) Climbing Cruise Descent Crew Unit crew cost is calculated by dividing the total pilot and flight attendant costs by the total block time across all B aircraft operated in the U.S. in Using data from the Bureau of Transportation Statistics Form 41 P-5.2 database, the unit crew cost for B in 2010 is estimated at $0.1573/sec Passenger travel time The travel cost per unit time for all passengers onboard is the product of passengers' value of time (VOT), aircraft seat capacity, and load factor. Following the U.S. Department of Transportation guidance on the economic value of passenger travel time (U.S. DOT, 2011), we use wage rate as a proxy for passenger VOT. The mean wage rate of $21.35/hr in the U.S. in 2010 is used (BLS, 2010). Assuming 160 seats on a typical B and an 80% load factor, the unit travel time cost for all passengers onboard amounts to $0.759/sec. 5.4 CO2 emission The estimation of CO 2 cost per second is based on the fuel consumption rate (in gallons/sec) as calculated in 5.1, the density of jet fuel, the fixed ratio of jet fuel consumption and CO 2 emission, and estimates of the social cost of carbon (SCC). We consider 0.82 kg/l as the density of jet fuel, 5 While the BADA fuel consumption model is the commonly used tool for estimating aircraft specific fuel burn, it is also known that the BADA fuel consumption model works well in cruise, but performs less accurately in climbing and descent (Abramson and Ali, 2012). 6 In contrast to a recent study (George O'Neill et al., 2012) which includes a total operating cost factor. we believe that other cost components than fuel and crew, such as materials and capital, are not sensitive to flight travel time, therefore not considered in the cost function. 10

11 based on Air BP (2000). The fixed ratio of jet fuel consumption and CO 2 emission follows U.S. EIA (2012), which states that burning one gallon of jet fuel emits 9.57 kilograms of CO 2. The emission rate of CO 2 therefore also depends upon flying altitude. We adopt $35 from Greenstone et al. (2011) as the mean social cost for one ton of CO 2 emitted. This social cost estimate is very close to that in a similar study (George O'Neill et al., 2012). 5.5 Contrail formation Quantifying the unit cost of contrail formation involves two steps: first estimating the climate impact of formed contrails for one second of flight time, and then converting that into dollars. As mentioned in section 2, radiative forcing is not suitable as a metric when comparing the climate impact of emissions of one substance relative to the emission of some reference gas. Instead, we use GWP to quantify the climate impact of formed contrails. For trace gas i, the original definition of GWP is the time-integrated commitment to radiative forcing from the instantaneous release of one kilogram of gas i relative to that of one kilogram of CO 2 (reference gas), expressed as: GWP i (H) = H 0 RF i H 0 RF CO2 H 0 (t)dt (t)dt = a ic i (t) dt H a CO2 c CO2 (t) dt 0 (6) where H is the time horizon, which is a user choice, and t is the time. RF CO2 (t) and RF i (t) denote the specific radiative forcing of CO 2 and gas i (in Wm -2 kg -1 ), respectively, at time t. a CO2 and a i are the instantaneous radiative forcing from one kilogram emission of CO 2 and gas i; c CO2 (t) and c i (t) the remaining abundance of CO 2 and gas i at time t from their respective impulse of one kilogram of CO 2 and gas i, thus describing the decaying of the gas agents over time. We comply with the International Panel on Climate Change (IPCC) recommendations and consider three time horizons (H): 20, 100, and 500 years (Fuglestvedt et al., 2010). The choice of time horizon can be interpreted as a discounting rate, and reflects different value judgments about the importance of climate impacts in the far future (Fuglestvedt et al., 2003). Choice of time horizon is particularly important as it determines the relative effectiveness of mitigations that focus one short-lived or long-lived species, such as contrail vs. CO 2. While using GWP to quantify the climate impacts between different gas agents is straightforward, it is less clear when comparing the impacts from contrail and CO 2, since contrail is not a type of physical gas. To circumvent this issue, Fuglestvedt et al. (2010) introduce a mass-based contrail GWP metric, which relates the total forcing due to contrails to the total aviation emissions of CO 2 by the aviation fleet and calculates the GWP of contrails for the average of the present-day fleet on a per kilogram of CO 2 basis, assuming a 5-hour lifetime of contrails. The GWP contrail (H) is given by GWP contrail (H) = RF contrail(t) dt 0 H M CO2 RF CO2 (t) dt H 0 T (7) where RF contrail T (t) is the total RF in a year from all aviation-induced contrails at time t; M CO2 the total amount of CO 2 (in kg) emitted for the same year. Values estimated in Fuglestvedt et al. (2010) are presented on the first row in Table 2. It can be seen that, despite the short lifetime of contrails (5 hours) chosen in the calculation), the GWPs remain non-negligible for H = 20 and 100 years, because the initial radiative forcing is significant. 11

12 Since flight-specific GWP contrail (H) values are unavailable in the limited body of literature on contrail formation at the individual flight level, we rely on the above aggregate GWP contrail (H) as representative values in the subsequent trajectory optimization process. However, these values cannot be directly used, in that persistent contrails can only be produced within PCFA regions whereas CO 2 emission occurs throughout the flight. In order to compare the unit climate impact between contrail and CO 2 when the aircraft flies in a PCFA region, the above GWP contrail (H) values need to be scaled up to reflect the actual amount of time flights spent in PCFAs. We introduce the following adjusted GWP contrail (H): GWP contrail (H) = GWP contrail(h) η (8) where η is the aggregate portion of time the global fleet spent in the air. Minnis et al. (1999) estimate that airliners cruise 10 to 20% of the time in air masses that are humid enough for the formation of persistent contrails. Schumann (2005) and Gierens et al. (1999) point out that airliners fly on average 15% of their time in ice-supersaturated air masses. In line with these estimates, we choose η =15%. The second row of Table 2 shows the calculated GWP contrail (H). Table 2: GWP for contrail with different time horizon choices H = 20 yrs H = 100 yrs H = 500 yrs GWP contrail (H) GWP contrail (H) Source: Fuglestvedt et al. (2010) and self-calculation. Following the discussion in sub-section 2.1, we use different GWP values for contrails formed in the day and night to reflect the time-of-day effect. The derivation of day- and night-specific GWP contrail (H)'s is based on the assumptions that night traffic accounts for 25% in the total and that the contrail RF during night is on average about twice that in the day (Meerkotter et al., 1999). Specifically, let GWP contrail,day (H) and GWP contrail,night (H) denote respectively the adjusted global warming potential during day and night. We have: GWP contrail,night (H) = 2GWP contrail,day (H) (9) GWP contrail (H) = 0.25GWP contrail,night (H) GWP contrail,day (H) (10) Through simple algebra, (9) and (10) lead to GWP contrail,day (H) = 0.8GWP contrail (H) and GWP contrail,night (H) = 1.6GWP contrail (H). If more detailed, location-specific data became available (e.g. sunrise/sunset times, local time dependencies with longitude), the GWP contrail,night (H) and GWP contrail,day (H) values derived from the above assumptions could be adjusted. We further assume that the ratio of the social cost 12

13 of formed contrails over that of emitted CO 2 per unit time when the aircraft flies inside PCFAs is equal to the corresponding GWP contrail (H). Therefore, the unit contrail formation cost is the unit CO 2 emission cost multiplied by the day-/night-specific GWP contrail (H) values. Recall that the unit CO 2 emission cost depends upon the fuel burn rate and varies by flying altitude. The same applies to the unit contrail formation cost. As an illustration, the last three columns of Table 3 report the calculated unit contrail costs when the aircraft cruises in the day with different horizon choices. The range of contrail cost factors considered here is wider than that in a previous study (George O'Neill et al., 2012). 7 The second column provides the unit cost from all other components, i.e. the sum of fuel, crew, passenger travel time, and CO 2 emission costs. Altitude (feet) Table 3: Cost factors for cruise in the day with different horizon choices Other cost factor ($/second) than contrail formation Contrail cost factor ($/second) 20 years 100 years 500 years Numerical analysis of a single aircraft trajectory In this section, we incorporate the cost factor values derived above into the program (1)-(5) to solve for the optimal single aircraft trajectory under various scenarios. (41.50 N, W) and (38.96 N, W) are chosen as the starting and ending waypoints, roughly corresponding to the airspace around ORD and IAD. We first present optimal trajectories both with and without PCFAs in the airspace, and then explore the sensitivity of results with PCFAs to the time-of-day effect and the different time horizon choices for calculating contrail GWP. Figures 4-6 show the optimal trajectories for a B aircraft with and without PCFAs. The nodes in the PCFA region are denoted by circles for the first 30 minutes and by crosses afterwards. 8 The base case has no PCFA presence, with H=20 years. The optimal trajectory is described by the dashed line. On the vertical profile, the aircraft will climb continuously to ft and cruise, taking advantage of lower fuel consumption at higher cruising altitude. The aircraft starts to descend only when it approaches the ending waypoint (IAD). On the longitude-latitude plane, the aircraft does not fly along a straight line joining ORD and IAD, because such a straight path does not exist in the grid network. Instead, the aircraft will first fly south while climbing to ft, because this path provides the fastest way to reach ft while flying towards IAD. After achieving the highest altitude, the aircraft will take a sharp turn, which is attributable to the use of the grid network to represent flyable air routes, and head towards the southeast. In reality, the aircraft would take smoother, more gradual turns. 7 In George O'Neill et al. (2012), the social cost of contrails considered ranges between $0.10 to $1.65 per nautical mile of contrail formed. 8 A given node in the grid network is identified as a PCFA node as long as it falls into the PCFA region specified in sub-section 3.2. The overlapped PCFA region is shown by nodes with both circles and crosses. 13

14 With the same H=20 years, the trajectory starting with a solid line and continuing with squares in Figures 4-6 represents the optimal trajectory when PCFAs persist in the airspace and the aircraft flies in the night. The solid line depicts the flight path for the first half an hour; the part with squares the trajectory afterwards. We observe that each heading change from southeast to south is associated with an altitude change, which is not by coincidence. Just as in the non-pcfa case, the aircraft first climbs up to ft. The climb, however, is interrupted by the presence of PCFAs. Since contrail formation is costly, the optimal route always avoids the PCFA region. The aircraft will first fly to the level which is right below the PCFAs (25500 ft), and then cruise to the point where climbing up while still avoiding PCFAs becomes possible. This is at the east most boundary of the PCFA region, when the time is still within the first 30 min. (This can be best seen in Figure 5.) The aircraft will climb one level up (to ft) and then change heading angle towards the southeast, because if keeping heading south the aircraft would fly into the PCFAs. The horizontal flying distance towards the southeast at ft will be minimum only one link long in order to just avoid PCFAs. Then, the aircraft will head south and accomplish the rest of the climb, all the way to ft, in the fastest way without producing contrails. Note that after 30 minutes the aircraft would be outside the shifted PCFA region and therefore never experience the penalty from generating contrails. The descent profiles of the aircraft are identical in PCFA and non-pcfa cases. The minimum cost of flying in the base case (with no PCFAs) amounts to $6950 per trip. When PCFAs exist, the optimal total cost would increase to $7050 per trip, because the aircraft now spends more time at lower altitude to avoid PCFAs and bears higher fuel consumption ( kg vs kg). Since the optimal path never touches PCFA, it can be regarded as flying a suboptimal route when no PCFA exists. However, the total travel time in both cases remain the same (4320 seconds) due to the constant aircraft speed and the almost identical flying distances (difference only due to different flying altitudes) despite different paths. We examine the sensitivity of the above results to two input parameters, time of day and the time horizon H, in the presence of PCFAs. We are interested in the time-of-day effect because of the different RF properties of contrails. Sensitivity analyses of the time horizon, on the other hand, are primarily due to the uncertainty about the relative climate impact between emitted CO 2 and contrails in the far future. Each input is given two choices: day vs. night (time-of-day), and 20 vs. 500 years (H), resulting in four different scenarios. The total cost, travel time, fuel burn, and time spent in PCFAs are reported in Table 4. We observe two types of trajectories for the four scenarios. The first type of trajectory the same as the dashed line in Figures 4-6 passes through the PCFAs at ft. This corresponds to the two scenarios with H = 500 years. As indicated in Table 3, the contribution of contrail formation to the total cost factor is minimal when a 500-year horizon is chosen. Not surprisingly, the trajectory is indifferent to PCFAs and, in effect, identical to the base case trajectory, with total cost only 0.2% higher. Because the contrail effect is so small, differentiating night and day travel does not yield different flight paths. This is obtained assuming no winds. Certainly, travel time may be path dependent if winds which vary in the course of a day are considered. When the GWP calculation is based on H = 20 years, the contrail effect becomes non-negligible. The optimal trajectory is also insensitive to the time-of-day effect, with identical trajectory as shown by the first-solid-line-then-square in Figures 4-6. This is because producing contrails would become so expensive that the best flying strategy is to avoid PCFAs altogether, resulting in the same optimal trajectory and cost for both the daytime and nighttime cases. 14

15 Altitude Latitude ORD IAD Longitude Fig. 4 Longitude-latitude view for the baseline and PCFA-present trajectories (note: circles denote PCFAs in the first 30 min; crosses denote PCFAs afterwards; dashed line denotes the optimal trajectory without PCFAs; solid line denotes the optimal trajectory with PCFAs in the first 30 min; squares denote the optimal trajectory with PCFAs afterwards. The same in Figures 5-6) ORD IAD Longitude Fig. 5 Longitude-altitude view for the baseline and PCFA-present trajectories 15

16 Altitude ORD Latitude IAD Fig. 6 Latitude-altitude view for the baseline and PCFA-present trajectories (note: because the PCFA region only moves along the longitude axis, from the altitude-latitude angle we will not perceive any change. In other words, the PCFA region after 30 min is hidden behind the PCFA region before 30 min.) Table 4: Total costs, travel time and fuel for different scenarios Time of day H Cost Travel time Fuel Time in PCFAs (year) ($) (second) (kilogram) (second) Day Night Day Night The above results suggest that optimal flight trajectories depend critically upon the choice of GWP horizon. On the other hand, distinguishing contrail effects by time of day would not produce different trajectories, nor significant cost difference. The cost difference for each daynight pair everything else being equal range between 0 and $20. In contrast, the potential passenger schedule delay increase resulting from flight schedule shift can be large. Taking $15.7/hour as the value of schedule delay per passenger (Adler et al., 2005), even an hour increase in schedule delay for each passenger would lead to 160seats 0.8 1hr $15.8/hr = $ additional schedule delay cost. It is therefore not justified to shift flights to the daytime solely for the reason of reducing contrail impact. 7 Extension to multiple-aircraft trajectories The above single-aircraft trajectory design problem can be extended to determining multiple flight trajectories. The added complexity lies primarily on minimum separation requirement in the airspace. We will demonstrate a procedure that successively optimizes trajectories for a sequence of flights flying between different origin-destination pairs, assuming that a flight scheduled to depart earlier possess higher priority over all later flights. In the case of a potential conflict between a low priority and a high priority flight, the low priority flight would have to deviate and allow the higher priority flight to proceed unimpeded. We will present a numerical example using 16

17 this successive procedure and historic flight schedules, and discuss the implications from the results. 7.1 Problem Setup We consider a sample of historic flights flying between 5pm and 9 pm (Eastern Daylight Time) on May 24, 2007, among five airports in the U.S.: John F. Kennedy International Airport (JFK), Detroit Metro Airport, Lambert St Louis International Airport (STL), ORD, and IAD. This includes 44 flights, whose detailed schedules (origin, destination, scheduled departure time) are provided in Appendix B. As fuel burn and emission parameters for some aircraft types are not available to us, we assume that all flights were flown by B aircraft. While details of the results may vary should we use the actual aircraft types when relevant fuel performance data become available, the basic insights drawn from the multi-trajectory analysis will remain intact. Additional assumptions need to be made and input and input parameters to be specified before proceeding to computing trajectories. First, to reflect the minimum separation requirement, we assume that no two flights are allowed to be closer than two minutes. Second, facing potential trajectory conflicts, a flight may want to wait on the ground at the origin airport, which entails a smaller unit cost than additional flying time. The ground delay option is offered in the multipleflight trajectory modeling process. In light of the ratio values in the existing literature (e.g. Ball et al., 1999; Mukherjee and Hansen, 2007), we assume that each second of ground delay entails one third of the unit cost while flying at feet ($/sec, excluding any contrail effect). Third, we assume the same initial PCFA distribution as in the single flight case, and that the PCFAs shift 1 E every 60 minutes in the airspace. Finally, given the still poor understanding of the contrail impact on climate change, we consider not only the cost factor with contrails occurring in the night (with a 20-year time horizon) but also its multipliers (5 and 10), to assess the sensitivity of flight trajectories to potentially more substantial climate impact of contrails. Higher contrail climate impact could arise due to uncertainties of the details about particle properties, cloud location, temperature, and time of year (Meerkotter et al., 1999), and with respect to the magnitude of contrail forcing and the forcing efficacy on climate response (Fuglestvedt et al., 2010). 7.2 Successive optimization algorithm We adopt a successive optimization procedure with priority among flights set in accordance with the schedule departure times. For each individual flight, the trajectory determination algorithm employed in the single aircraft case is augmented by accounting for the trajectories of all previous flights, such that the minimum separation requirement is not violated. To avoid trajectory conflicts and contrail formation, each flight can fly a more circuitous route in the airspace, or wait on the ground at the origin airport, or both. Note that in the single aircraft case, the trajectory optimization is confined to traveling between two waypoints in the airspace. By allowing for ground delay, we implicitly extend the analysis scope to the full trip of each flight between the origin and destination airports, with the underlying assumption that the trajectories during takeoffs and landings remain constant in the optimization process. Full details about the algorithm to compute the optimal trajectories are provided in Appendix C. This multiple aircraft trajectory algorithm, essentially a "greedy" approach, reflects in part the current sequential flight dispatching practice. However, the trajectory design procedure also suggests a departure from the state-of-the-practice, in that when optimizing a given flight's trajectory, the flight trajectory planner/dispatcher would have full information about all preceding trajectories, regardless of whether the flights belong to the same airline. Such a procedure is certainly more realistic, and computationally much cheaper, than determining all trajectories simultaneously to achieve the "system optimum". On the other hand, the implied information sharing among dispatchers may represent a future, more collaborative flight dispatching 17