Tabulación de Resultados Experimentación con Tribunal

The aircraft weight is a very important factor in aircraft performance. Although aircraft weight changes continuously during the flight, because of fuel consumption, the biggest impact of aircraft weight during the arrival and approach phases comes from the large variations in aircraft landing weight from flight to flight within the same aircraft type. Aircraft lift coefficient under any nominal operational condition is directly related to aircraft weight. Thus, variation in aircraft weight will affect the lift/drag ratio under any given nominal operational conditions. In most cases, aircraft flap extension speeds are also determined by the aircraft weight. As a result, the FMS computed VNAV path would be different for different landing weights. The difference would mostly be the in the location of deceleration point, and in the flight path angle of a non-geometric segment. The dynamics of the aircraft will also be different for different weight.

Variations in the landing weight of a given aircraft type are mostly due to uncertainties in local market demand (payload) and operating conditions (fuel consumption). Historical data can be used to model these aircraft weight variations. Table 3-1 shows UPS Boeing 757-200 and Boeing 767-300 package cargo aircraft design weight parameters and statistics of the corresponding aircraft landing weight data collected between April 20 and May 18, 2004 at KSDF. Those statistics were obtained from data of 163 B757-200 and 139 B767-300 arrivals. The probability mass functions based on the aforementioned landing weight data for the two aircraft types are shown as bar charts in Fig. 3-6 and 3-7 respectively.

Table 3-1 Landing weight parameters of UPS cargo aircraft^a, 1000 lb.

Aircraft Type

Parameters B757-200 B767-300

Spec Operating Empty 114.000 188.000

Maximum Design Landing 210.000 326.000

Minimum 146.617 229.271

Maximum 194.534 298.183

Mean 167.539 262.205

Standard Deviation 11.000 18.000

a Statistics based on data of 163 B757-200 and 139 B767-300 arrivals to KSDF between 20-Apr-2004 and 18-May-2004.

0.00 0.05 0.10 0.15 0.20 0.25

116.5 121.6 126.6 131.7 136.7 141.8 146.8 151.9 156.9 162.0 167.1 172.1 177.2 182.2 187.3 192.3 197.4 202.4 207.5

Landing Weight, 1000 lb

Probability Mass

Data Normal Beta

Figure 3-6 UPS B757-200 landing weight distribution.

0.00 0.05 0.10 0.15 0.20 0.25

191.6 198.9 206.2 213.4 220.7 227.9 235.2 242.5 249.7 257.0 264.3 271.5 278.8 286.1 293.3 300.6 307.8 315.1 322.4

Landing Weight, 1000 lb

Probability Mass

Data Normal Beta

Figure 3-7 UPS B767-300 landing weight distribution.

In each of the figures, the left boundary of the horizontal axis gives the spec operating empty weight of the corresponding aircraft. The right boundary of the horizontal axis gives the maximum design landing weight of the corresponding aircraft. By definition, landing weights are always within the range defined by those two limits for the specific aircraft type. As it can be seen, the landing weights resembled the normal distribution. Thus, a simple model of aircraft landing weight would be a normally distributed random variable for each aircraft type. The random variable should be bounded by the corresponding aircraft’s historical minimum landing weight and maximum landing weight to avoid nonrealistic values.

Model parameters such as the mean, standard deviation, minimum, and maximum can be estimated using sample statistics. Normal distribution landing weight models based on the sample mean and standard deviation values listed in Table 3-1 are shown in Fig. 3-6 and 3-7 as solid black curves. It is seen that the normal distribution models captured the historical data well. The dashed curves in Fig. 3-6 and 3-7 are beta distribution models, which will be discussed later.

Landing weights of passenger aircraft were also examined. For the purpose of comparison, Delta Air Lines Boeing 757-200 and Boeing 767-300 passenger aircraft landing weights collected between August 1 and August 30, 2005 at Hartsfield-Jackson Atlanta International Airport (KATL) were used. The design weight parameters and statistics of landing weights for these two passenger aircraft are listed in Table 3-2. The corresponding probability mass functions are shown in Fig. 3-8 and 3-9 respectively.

Again, in each of the figures, the left boundary of the horizontal axis gives the spec operating empty weight of the corresponding aircraft type. The right boundary of the horizontal axis gives the maximum design landing weight of the corresponding aircraft type.

Table 3-2 Landing weight parameters of Delta passenger aircraft^a, 1000 lb.

Aircraft Type

Parameters B757-200 B767-300

Spec Operating Empty 130.860 186.380

Maximum Design Landing 198.000 300.000

Minimum 140.620 202.500

Maximum 197.872 294.977

Mean 180.258 266.422

Standard Deviation 10.041 16.318

a Statistics based on data of 2759 B757-200 and 1184 B767-300 arrivals to KATL between 1-Jul-2005 and 31-Jul-2005.

It is seen that for the same aircraft type, the difference between the spec operating empty weight and the maximum design landing weight is smaller for the passenger version than the cargo version. The historical landing weight data also indicate that the passenger versions had slightly higher mean but lightly lower standard deviation than the cargo versions. However, the most striking difference between the passenger versions and the cargo versions is in the probability mass functions as shown by the figures.

The probability mass functions of the passenger aircraft are heavily skewed to the right-hand side.

Similar to the treatment of the cargo aircraft, a normal distribution model was built from the sample mean and sample standard deviation for each of the two passenger aircraft types. These normal distribution models are shown as solid black curves in Fig. 3-8 and 3-9. It is seen that the normal distribution models did not capture the data well for the passenger aircraft. To be bounded by the corresponding aircraft’s historical minimum landing weight and maximum landing weight to avoid nonrealistic values, a big chuck of the right hand tail of the normal distribution has to be cut off. This will further decrease the modeling accuracy.

0.00 0.05 0.10 0.15 0.20 0.25

132.6 136.2 139.7 143.2 146.8 150.3 153.8 157.4 160.9 164.4 168.0 171.5 175.0 178.6 182.1 185.6 189.2 192.7 196.2

Landing Weight, 1000 lb

Probability Mass

Data Normal Beta

Figure 3-8 Delta B757-200 landing weight distribution.

0.00 0.05 0.10 0.15 0.20 0.25

189.4 195.4 201.3 207.3 213.3 219.3 225.3 231.2 237.2 243.2 249.2 255.2 261.1 267.1 273.1 279.1 285.1 291.0 297.0

Landing Weight, 1000 lb

Probability Mass

Data Normal Beta

Figure 3-9 Delta B767-300 landing weight distribution.

A commonly used distribution to model variables that are lower and upper bounded is the beta distribution⁷. The probability density function (pdf) of beta distribution is defined on the interval [0, 1]

1

The mean and standard deviation of the beta random variable X with parameters D and E are

E

If the estimated mean E^ (X) and the estimated standard deviation ST^ D(X) are known, the two parameters of the beta distribution can be estimated

For an aircraft with spec operating empty weight mOEW and maximum design landing weight mMLW, the normalized aircraft landing weight x for a given landing weight m can be defined as

OEW

The estimated mean E^{^}(X) and the estimated standard deviation ST^{^}D(X) of the normalized aircraft landing weight can then be computed from the estimated mean E^{^}(M) and the estimated standard deviation ST^{^}D(M) of the aircraft landing weight as

OEW

With the normalized estimates from Eq. (3-10) and Eq. (3-11), using Eq. (3-7) and Eq. (3-8), estimates of parameters D ^and E of the beta distribution model can be obtained. The parameters of the beta

distribution models for each of the four aircraft types examined thus far are listed in Table 3-3. The modeled beta distributions are shown as dashed curves in Fig. 3-6, Fig. 3-7, Fig. 3-8, and 3-9 respectively.

Table 3-3 Parameters of beta distribution landing weight models.

Aircraft Type D^{^} E^{^}

UPS B757-200 11.5102 9.1285

Delta B757-200 5.6602 2.0330

UPS B767-300 13.3679 11.4924

Delta B767-300 6.4060 2.6874

It is seen that the beta distribution models were very close to the normal distribution models for the two UPS cargo aircraft types. The beta distribution models were much better than the normal distributions for the two Delta passenger aircraft types. Moreover, a beta distribution is bounded by the lower and upper boundaries (spec operating empty weight and maximum landing weight in this case) – one does not have to worry about nonrealistic values generated during the simulation. Thus, the beta distribution models are a better choice in modeling aircraft landing weight distributions. However, the simulation of a beta distribution model is more difficult than a normal distribution model. For aircraft landing weight distribution similar to that of the UPS B757-200 and B767-300 aircraft, normal distribution models would be comparably good but very easy to implement in the simulation.