Jia-xue LIU, Bo LI, Run-xia GUO

(College of Electronic Information and Automation, Civil Aviation University of China, Tianjin 300300, China)

Abstract: In terms of the problem “surplus fuel will increase fuel consumption” in Civil aviation transportation, together with the practical significance of flight time and fuel consumption per hour, this paper proposed a calculation method for the fuel capacity upon flight time. We analyzed the cases which include the collected flight time and fuel consumption of passenger airplane B737-800 on 2 000 kilometers routes, explored the numerical features of flight time by applying the chaotic time series theory, selected the neighborhood points by hamming distance and similarity, then proposed an improved adding-weight zero-rank local region prediction model especially for small datasets. Simulation results show that our proposed method could provide a better fit in multi-steps prediction than state-of-art adding-weight zero-rank prediction algorithms. The airline flight case of B737-800 and the predicted flight time were collected, and the proposed method has been validated. Numerical experiments show that this new method could save fuel capacity from 790 kg to 2 795 kg and reduce the fuel consumption from 0.6% to 1.47% during the same voyage.

Key words: Fueling charge, Flight hours, Chaotic time series, Similarity

1 Introduction

“Surplus fuel increases fuel consumption” phenomenon refers to the increase of aircraft fuel consumption caused by the increase of aircraft gross weight, which is owing to excessive fueling charge. The flight remaining oil after landing is applied as one of the flight efficiency evaluation indicators in international civil aviation industry. Nearly 25% flights of one famous domestic airline remain more than 2-hour fuel consumption after landing, while the internationally recognized standard is only 1.25-hour. Take the B737 flight for an example, the corresponding internationally recognized landing remaining oil is about 3 tons, while the same flight in domestic airline remains about 4 tons. Therefore, the domestic refueling programs urgently need to be optimized. Under the premise of ensuring flight safety, the optimization of fueling charge can effectively alleviate the problem of fuel wasting.

Fuel consumption and flight hours are affected by a lot of factors [1]. At the macro level, routine distance, air traffic flow influenced by the airport and airspace capacity are the main factors to determine the fuel consumption and flight time. Microscopically, flight model, airport schedules, weather, airline production plan and passengers will also have an impact on the fuel consumption and flight hours. Because of many uncertainties, the establishment of a suitable model for fuel consumption and flight hours to calculate the fueling charge is significant for flight planning and overall planning of airlines[2].

In the chaotic system, sample data are sensitive to the initial value, which makes the output reflect the input change immediately. Chaotic theory provides a nonlinear analysis method which is much more realistic, and are commonly applied in transportation research [3-6]. Those work starts from the nonlinear characteristics of the traffic flow, predict the time series via the chaotic time series prediction methods based on small dataset, studies the inner order of the stochastic system, which verifies the feasibility of chaotic system in predicting for a short term. The flight hours and fuel consumption data sampled also have some nonlinear characteristics, so the short-term prediction can be carried out on the basis of chaotic characteristics verification.

This paper set a single chaotic time sequence of unknown system (the flight hours of B737-800 on the 2000 Kilometers flight segment) to be the study object. The state space of flight hours is reconstructed based on the chaotic characteristics of flight hour series. A novel weighted zero-order local prediction model for small data is proposed, and the neighborhood points are selected by Hamming distance and similarity. According to the national airworthiness regulations and operation rules of airlines, this paper established a method to calculate the fuel capacity upon the flight hours and fuel consumption per hour. The model has been verified based on the QAR (Quick Access Recorder) data with aircraft B737-800. The remaining of this paper is organized as follows. In Section 2, the chaotic characteristics of flight hour series will be introduced. Section 3 will give the details of classical weighted zero-order local prediction method and analyze how to improve this model. Section 4 presents the improved fueling charge calculation method, and the performance evaluation is shown in Section 5. The perspective of future work for this paper is concluded in Section 6.

2 Chaos of flight hours

2.1 The judgement of chaos

Only when the time series of flight hours have chaotic characteristics, the prediction model of chaotic theory can be used. The Lyapunov exponent reflects the sensitive dependence of chaotic dynamical system on initial conditions. If the maximum Lyapunov exponent is greater than zero, then the dynamical system is set to be chaotic.

The Wolf method is used to extract the maximum Lyapunov exponent of one-dimension data in this paper. The kernel of Wolf method is to estimate the Lyapunov exponent via the evolution of phase trajectory, phase plane and phase volume. The method is summarized as follows:

Step1 The time series are denoted asx1,x2,…,xn(xix(ti)). Given embedding dimensionmand time delayτ, the phased points can be generated asN=n-(m-1)τ. The dimension of reconstructed time series ism×N.Then, the phase point at timeticould be expressed as:

Y(ti) = (x(ti),x(ti+τ),…,x(ti+(m-1)τ)),

i=1,2,…,N

(1)

Step3 Select the time stepK, in the corresponding time period, the system linear exponential growth rate could be calculated as:

(2)

Step4 Set the average of exponential growth rate to be the estimated value of Lyapunov exponent.

(3)

Increase the embedding dimensionmone by one, then repeat the step 2～4 until the Lyapunov exponent becomes stable, and finally the maximum Lyapunov exponent of the time series could be obtained.

2.2 Phase space reconstruction of chaotic time series

Phase space reconstruction is the base of chaotic time series analysis. The C_C algorithm [7] is used to calculate the optimal time delayτand embedding dimensionmin this paper.

Fig.1 The change of lyapunov exponent in time series

The reconstruction procedure is summarized as below:

Step2 Divide the time series intopsubsequences of lengthL=[n/p], where,p=1,…,n. Then theithsubsequence is represented as:

Xp,I={xI,xI+p,…,xI+(m-1)p},I=1～p

(4)

Definition1 The correlation integrationpsubsequences with embedding dimensionmis defined as:

(5)

Where,M=n-(m-1)p,r>0 is the correlation radius, anddp,ij=||Xp,i-Xp, j|| is the Euclidean distance between theithand thejthcolumn duringpsubsequences.θ(v) is a jump function, which can be expressed as:

(6)

Definition2 The statisticS(m,n,rj,p) is defined as:

(7)

For given embedding dimensionmand the total number of subsequencesp, the value ofS(m,n,rj,p) could vary with the different radius of r. The difference between the maximum and minimumS(m,n,rj,p) is defined as:

ΔS(m,p)=max{S(m,n,r,p)}-min{S(m,n,r,p)}

(8)

Where, ΔS(m,p) measures the maximum deviation of radiusr. Take the first zero ofS(m,n,r,p) or the minimum value of ΔS(m,p) as the first local maximum of the time series, the corresponding subsequence index as the optimal time delay,τ*=p, which indicates that the reconstructed attractor orbits are fully expanded in the phase space.

Step3 The average ofS(m,n,rj,p) is obtained from the following formula:

(9)

(10)

So, the optimal embedding time windowtwand embedding dimension m can be calculated by:

(11)

(12)

3 Weighted zero-order local predictionmodel

3.1 Weighted zero-order local region method

Chaotic time series prediction methods can be divided into three categories: global region method, local region method and the largest Lyapunov exponent prediction method. All the points in the trajectory are treated as fitting objects, so the global region method is of low accuracy in precision and only possible in theory. The local method sets the previous point of the predicted one as the central value, then obtains the development directions of the value, finally selects the closest one from the date in development directions as the predicted value. Compared with global method using all the data, the local method is more suitable to fit the points in most cases.

In the local region method, the fitting of prediction point is based on the neighborhood of the center point. During the fitting procedure, the Euclidean distance between the neighborhood points and the center points only takes the average process, and the influence of Euclidean distance on the center points is neglected. Therefore, the Euclidean distance between center point and neighborhood point is regarded as the final modifier, which can improve the prediction accuracy to a certain extent. The improved weighted zero-order local prediction method is defined as follows:

(13)

3.2 Improved neighborhood selection

The neighborhood pointYkis an important basis to forecast in local prediction method. The prediction trend could be inferred from the trend of neighborhood points. The appropriate neighborhood points can improve the prediction accuracy. It is easily found that the macro factors affecting the flight hours are relatively stable in a certain period of time under the analysis of numerical characteristics of flight hours, for example, the validity of airline flight plans expires at least three months once generated. Additionally, some key elements, such as flight height, routine and waypoint are fixed within a time period. Microscopically, most of influence factors are uncontrollable, such as weather, whose impact on flight hours is within a certain range.

According to the characteristics of both macroscopic and microscopic factors, the change of the microscopic factors is more obvious in the change trend of sample points. If the similar micro factors can be selected, the prediction results will be better [8]. In allusion to the time series in small dataset, Hamming distance is used to judge the similarity between each phase point, then the similarity could be used as evaluation criterion to measure the correlation of each phase point in this paper. The implementation process is shown as follows:

(14)

(15)

2) Calculate the average Hamming distancediNand semblancesiNbetween all the phase points and the central points.

(16)

(17)

3) Set the distance and similarity as the criteria to select the nearest neighborhood points:

DiN=diN+siN

(18)

It is obvious that the lowerDiNis, the better neighbor point is selected. The smallest q points inDiNare the so-called selected neighborhood points.

3.3 Algorithm

We summarize the algorithm as follows, and give the implements step by step:

4 Fueling charge calculation methodbased on flight hours

Fueling charge, also called task fuel weight (WF), can be divided into two items:

WF=WFused+WFres

(19)

Where,WFusedis the actual fuel consumption,WFresis the reserve fuel weight.

Usually, there are two cases when calculating the task fuel weightWF:

(1) The reserve fuel is proportional to the used fuel, and it could be written as:

WFres=MresWFused

(20)

Where,Mresis the reserve fuel factor, usually given in the flight plan requirements.

(2) Task fuel is determined by flight plan. According to the mission requirements, the transit range and waiting time are given [9-10]. Take the Chinese trunk aircraft for an example, according to the CCAR25 requirements (transport aircraft airworthiness standards-the 25 part of Chinese civil aviation regulations), transit range is 370.4 Kilometers, and the flight flies standby for 45 minutes with minimum fuel flow in air. The phases of flight mission can be divided into eleven stages as shown in Fig.2. The fuel used in 1th～7thand 11thstage is actual fuel consumptionWFused, in 8th-11thstage is the so-called reserve fuelWFres. Therefore, the total flight mission time can be expressed as:

TF=TFnorm+TFres1+TFres2

(21)

(22)

(23)

Where,TFnormis the normal arrival flight hour,TFres1is the transit flight hour which means that the total flight hours from destination airport to the farthest alternate[11].TFres2is the air holding time, totally 45 minutes.Tiindicates the flight hour of each mission stage.

Fig.2 Flight Mission Analysis

The fuel charge calculation method based on the flight hours could be defined as follows:

WFused=TFnorm×MF

(24)

WFres=(TFres1+TFres2)×MF

(25)

Where,MFis the average fuel consumption per hour of the route.

5 Performance evaluation

The experiment par took the flight statistical data of flight B737-800 on the 2 000 Kilometers flight segment per quarter from 2006 to 2014, which includes 41 pieces of data totally, as the sample, then predicted the data of next four quarters, and compared the results with the actual observed data. The adopted performance index for prediction method are shown in Table 1.

Table 1 Predictive performance indicators

IndexCalculation formulaRMSEErmse=∑Ni=1(xi-^xi)N-1SMAPEEsmape=1N∑Ni=1xi-^xixi+^xiNMAEEnmae=∑Ni=1xi-^xiNMAEEmae=max(xi-^xi)

The comparisons of predicted value and ground truth for two different point selection methods are shown in Fig.3. As can be seen from this figure, the two methods can obtain better prediction values. Combined with the actual operation of airline, the errors are within allowable range. However, the prediction method based on Hamming distance is closer to the real data. The average relative percentage error of two investigated methods are shown in Fig.4. With the increase of the prediction points, the error of the improved method could be obviously reduced, especially, the error fluctuation is getting smaller. All of these results suggest that the approach presented here is superior to the neighborhood selection method based on the Euclidean distance.

The error values of two methods are shown in Table 2. It can be observed that the use of Hamming distance for similarity calculation could decrease the error of weighted zero-order local method. The model based on the neighborhood selected by similarity outperforms the original model.

Fig.3 Comparison of predictive value and real value of two selection methods

Fig.4 Comparison of average relative percent error of two selection methods

Table 2 Error comparison

IndexEuclidean distanceHaiming distanceErmse2.606 7190.996 656Esmape0.004 9380.001 797Enmae1.738 2600.635 020Emae4.654 0001.845 100

Take the flights between A airport and B airport as the example, the actual distance is between 1 770.1 km and 1 859 km according to the QAR data, which belongs to the 2 000 km segment. The farthest alternate airport of A is C airport, and the corresponding transit range is 450km. The farthest alternate airport of B is D airport, and the corresponding transit range is 380 km. After checking the QAR data between A airport and C airport, one could find that the total time for 8th, 9th, 10thstage is about 50～55 min,TFres1=0.83 h. Take the predicted flight hours of 2 000 km segment as the normal arrival time between A airport and B airport,TFnorm=2.93 h, and calculate the average fuel consumption per hour viaMF=2.41 t/h. One could obtain the pre-refuelling for the route airline from A airport to B airport is 10.87 tons via Equations (19)～(25).

Table 3 shows the comparison experimental results of the refueling based on the flight hour. The present method could reduce the refueling [12-13] by 790 ～2 795 kg for each flight. And the remaining oil after landing could provide at most 1.86h endurance. Thus, the oil consumption per hour could be reduced by 0.6%～1.47%, and it means that 98kg oil will be saved at least. Totally, the flight could save oil 27 tons per year.

Table 3 Comparison of actual refuelling and pre- refuelling of a route airline

Flight No.actual charge/tactual use/tpre charge/treduced fuel/tNo.111.6596.46810.870.79No.213.6656.73310.872.795No.311.8186.39310.870.948No.412.8076.96710.871.937No.512.1456.43710.871.275No.612.2856.85310.871.415

6 Conclusion

This paper collected the actual flight hours of one famous domestic airline as the sample data, and proposed a flight hour prediction method for the civil aircraft by applying chaotic time series theory. After that, a calculation model was established for the fuel capacity upon flight time, together with the practical significance of flight time and fuel consumption per hour. The experimental results show that this model could predict the pre-refueling of flight B737-800 in 2 000 km segment accurately, which could provide a reference for the flight schedules [14] and energy-saving emission reduction.

It is worth pointing out that the air traffic is becoming complex along with the larger-scale air lines, so there are some limitations in studying the flight hour from the aspects of airplane performance indicators and environment variables. Therefore, how to analyze the variation characteristics of flight time macroscopically could compensate for the aforementioned limitations, and fit the elaborate management of civil aviation transportation well.

Acknowledgements

Supported by Sponsored by Natural Science Foundation of China(No.61603395 and U1433102).