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Air Traffic Management based on 4D Trajectories: A Reliability Analysis using Multi-State Systems Theory
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Transportation Research Procedia 33 (2018) 355–362 www.elsevier.com/locate/procedia
XIII Conference on Transport Engineering, CIT2018 XIII Conference on Transport Engineering, CIT2018
Air Traffic Management based on 4D Trajectories: A Reliability Air Traffic Management based on 4D Trajectories: A Reliability Analysis using Multi-State Systems Theory Analysis using Multi-State Systems Theory
Álvaro Rodríguez-Sanz a,a, **, David Álvarez Álvarez aa, Fernando Gómez Comendador aa, Álvaro Rodríguez-Sanz , David Álvarez Álvarez ,aFernando Gómez Comendador , a b Rosa Arnaldo Valdés , Javier Pérez-Castán and Mar Najar Godoy Rosa Arnaldo Valdés a, Javier Pérez-Castán a and Mar Najar Godoy b a Universidad Politécnica de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040, Spain. a Universidad de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040, Spain. ISDEFE (Ingeniería dePolitécnica Sistemas para la Defensa de España), Calle Beatriz de Bobadilla, N3, Madrid 28040, Spain. b ISDEFE (Ingeniería de Sistemas para la Defensa de España), Calle Beatriz de Bobadilla, N3, Madrid 28040, Spain. b
Abstract Abstract The current Air Traffic Management (ATM) functional approach is changing. Both SESAR (Single European Sky ATM Research) The current Air Traffic Management (ATM) functionalSystem) approach is changing. Both SESAR (Single European Sky ATM Research) and NEXTGEN (Next Generation Air Transportation support the four-dimension (4D) trajectory implementation within and Generation Air Transportation System) support the four-dimension (4D)integrated trajectoryas implementation theirNEXTGEN operational(Next concepts. Apart from the three classical spatial dimensions, ‘time’ is now an additional within fourth their operational Apart fromflights the three dimensions, is now as can an additional fourth dimension, whichconcepts. will restrict aircraft over classical indicatedspatial waypoints along the‘time’ trajectory. 4Dintegrated trajectories be understood as dimension, which willSystems restrict aircraft over waypoints alongand the usage trajectory. 4D trajectories can beanalysis understood as complex Multi-State (MSS) flights that rely onindicated environmental, internal conditions. A reliability of the complex Systemsthat (MSS) thatthe rely environmental, internal usageand conditions. reliability analysis of the waypointsMulti-State and time windows describe 4Don trajectory may allow airline and operators air traffic A service providers to establish waypoints andindicators time windows describe metrics. the 4D trajectory maydevelops allow airline operators and air traffic service providers to establish performance and that compliance This paper a model to evaluate potential ‘malfunctions’ of a 4D performance indicators and compliance metrics. This paper develops a model to evaluate potential ‘malfunctions’ a 4D trajectory, based on the MSS reliability theory. This is a natural extension of classical binary-state evaluation: trajectoriesofpresent trajectory, based on thelevels MSS reliability theory. is a natural extensionrange). of classical binary-statereliability evaluation: trajectorieswhich present different performance and several failureThis modes (a degradation The operational assessment, is different performance levels and several (a degradation range). The reliability assessment, which is achieved with Monte Carlo simulation andfailure randommodes processes (Markov) methods, offersoperational a framework to predict how probable is for achieved with Monte simulation andWe random processes (Markov) methods, offers a framework to predict howand probable is for the trajectory to enterCarlo a degraded state. use this analysis to quantify the 4D trajectory level of variability to propose the trajectory to enter degraded state.trajectory We use degradations this analysis or to unplanned quantify thesituations. 4D trajectory level of variability and tothrough proposea corrective measures to asolve potential The methodology is validated corrective measures to solve trajectory or unplanned situations. The methodology is validated through a practical case study. The mainpotential contribution of thisdegradations paper is to provide a methodology to evaluate the robustness of 4D trajectories practical case study. main contribution this paper isintotraffic provide a methodologyand to evaluate the robustness of 4D trajectories and to deal with theirThe perturbation, which is of a cornerstone synchronization conflict resolution. and to deal with their perturbation, which is a cornerstone in traffic synchronization and conflict resolution. © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. © © 2018 The Authors. by Elsevier Ltd. This is an an open accessPublished article under under the CC CC BY-NC-ND BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is open access article the license This is an and openpeer-review access article underresponsibility the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) responsibility scientific committee of of the the XIII XIII Conference Conference on Transport Transport Engineering, Engineering, Selection under of the scientific committee Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. CIT2018. Keywords: 4D trayectories; Reliability analysis; Multi-state systems; Markov Chain processes Keywords: 4D trayectories; Reliability analysis; Multi-state systems; Markov Chain processes
* Corresponding author. Tel.: +34 626574829. * E-mail Corresponding Tel.: +34 626574829. address:author. [email protected]. E-mail address: [email protected].
2352-1465 © 2018 The Authors. Published by Elsevier Ltd. 2352-1465 © 2018 Thearticle Authors. Published by Elsevier Ltd. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND licensecommittee (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the scientific of the XIII Conference on Transport Engineering, Selection CIT2018. and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 2352-1465 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 10.1016/j.trpro.2018.11.001
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1. Introduction and problem statement The recent increase in air traffic demand provides a challenging operational situation for the current European Air Traffic Management (ATM) system (SESAR, 2015). Achieving accurate prediction of trajectories is a fundamental condition for reliable detection and resolution of conflicts. The future ATM system relies on the Trajectory Based Operations (TBO) concept (FAA, 2016; SESAR, 2014). TBO involves separating aircraft through a definition of a strategic trajectory (long-term), rather than the currently practicing tactical (short-term) conflict resolution. This approach is based on a four dimensional framework, composed by the three spatial dimensions and a time constraint (Mutuel et al., 2013). The trajectory predictions need to consider external disturbances to the aircraft and internal uncertainty sources (Casado et al., 2012; Garcia-Chico et al., 2008; Pabst et al., 2013). These disturbances and uncertainties may cause a degradation in the trajectory. Therefore, the management and study of degradation is a key element when ensuring the accuracy of the trajectory. However, real measures (over the trajectory) will be required to improve reliability, react to unplanned conditions and thus maintain the expect capacity. Trajectories are degraded due to environmental and operational uncertainties (Casado et al., 2012; Garcia-Chico et al., 2008). Sharing, updating and coordinating changes in trajectory becomes necessary to ensure reliability. Hence, these measurements will help to appraise the required adjustments to correct trajectory degradation. This degradation increases over time if appropriate actions are not taken. Reliability analysis provides theoretical and practical tools to test the behavior and performance of trajectories in a scenario of uncertainties. The properties of Markov-chains have been used to model the interdependent effects between sequential procedures of aircraft turnarounds (Wu and Caves, 2004). In Air Traffic Control (ATM), a MultiState System (MSS) theory and a Markov model have been applied to measure the efficiency and reliability of an Air Traffic Automatic System (ATCAS) (Wang and Liu, 2012). The estimation of potential benefits of new ATM tools has also been appraised using MSS reliability through Markov models (Liu and Hwang, 2011). Monte Carlo simulations were implemented to estimate the range of uncertainties in model parameters and technology performance accuracy. Monte Carlo simulation allows, with a simple computation procedure, modelling system operating scenarios to assess the reliability systems (Aven and Jensen, 1999; Trivedi, 2016). The purpose of this study is to define a model to assess flight operational efficiency. This is achieved by defining a series of indicators based on trajectory influence parameters, which allow the evaluation of the 4D trajectory operational state at that given moment. This can be applied in a predictive way, thus being able to anticipate the trajectory degradation in order to apply corrective actions. The paper is structured as follows: first, we introduce the methodology (MSS theory to characterize 4D trajectories and a Markov-Chain method to model their performance evolution); and then, we study the reliability of 4D trajectories. 2. Methods and materials 2.1. Multi-state systems A system is designed to perform its functions in a given environment. Traditionally, the way to evaluate performance and reliability was through a binary system, with two states: it works, or it does not work (Natvig, 2011; Trivedi, 2016). This perspective limits the study. In many cases, systems can perform their functions with many efficiency values that are generally known as performance rates. A system that has a finite number performance rates is called a multi-state system (MSS) (Lisnianski et al., 2010). Usually, a multi-state system is composed of elements that can also be considered as multi-state systems. An element is a system entity that has no more subdivisions. This does not imply that the element cannot be made up of parts, but it means that, in a reliability study, it will be studied as an entity (Lisnianski, 2007). The performance rates of the elements range from perfect operation to complete failure. Faults that can lead to a reduction in the performance of an element are called partial faults. After a partial failure, the elements continue to operate at reduced performance, and after a total failure, the elements are disabled and do not perform their function (Lisnianski et al., 2010). The performance rate of an element at any time is a random variable that takes its values from the probabilities associated with the different states (rate of return) of the system’s element.
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2.2. Markov-Chains Markov-chains are a special type of discrete stochastic process in which the probability of an event only depends of the precedent state of the system. This type of systems is memoryless and satisfy the Markov property (Aven and Jensen, 1999). The probability of transition between state 𝑋𝑋𝑛𝑛−1 = 𝑖𝑖 and 𝑋𝑋𝑛𝑛 = 𝑗𝑗 is given by 𝛾𝛾𝑖𝑖,𝑗𝑗 , where 𝑛𝑛 is the number of transitions. The matrix 𝑷𝑷 represents collectively these transition probabilities: 𝛾𝛾1,1 𝛾𝛾2,1 𝑷𝑷 = ( ⋮ 𝛾𝛾𝑘𝑘,1
𝛾𝛾1,2 𝛾𝛾2,2 ⋮ 𝛾𝛾𝑘𝑘,2
⋯ ⋯ ⋯ ⋯
𝛾𝛾1,𝑘𝑘 𝛾𝛾2,𝑘𝑘 ⋮ ) 𝛾𝛾𝑘𝑘,𝑘𝑘
The vector 𝜋𝜋𝑛𝑛𝑇𝑇 defines the probability that the system is in a specific state on the 𝑛𝑛-th transition: 𝜋𝜋𝑛𝑛𝑇𝑇 = [𝜋𝜋1,𝑛𝑛 𝜋𝜋2,𝑛𝑛 ⋯
𝜋𝜋𝑘𝑘,𝑛𝑛 ]
(1)
(2)
where 𝜋𝜋𝑘𝑘,𝑛𝑛 is the probability that the system is in state 𝑘𝑘 on the 𝑛𝑛-th transition. The probabilities of the state for each transition are determined iteratively as follows: 𝑇𝑇 𝜋𝜋𝑛𝑛𝑇𝑇 = 𝜋𝜋𝑛𝑛−1 𝑷𝑷
(3)
The random evolution of a Markov-chain is completely determined by its transition matrix P and its initial density distribution 𝜋𝜋𝑇𝑇0 . Therefore, with the transition matrix obtained, the study of Markov-chains is reducible to the linear algebra study of its transition matrix and state vector (Eq 3). 3. Reliability analysis 3.1. Trajectory modelling In the first part of the study, the aircraft performance and the 4D trajectory were modelled in order to obtain data regarding its evolution. The aircraft chosen was a Boeing 737-900ER, one of the most frequent aircraft in shortmedium range flights (OAG, 2017). Next, 4D trajectories were modelled using the EUROCONTROL’s BADA (Base of Aircraft Data) methodology (EUROCONTROL, 2017a). In order to develop the model, it was necessary to characterize 4D trajectories and identify their influence parameters, which are shown in Table 1 (aircraft performances and variables related to the scenario). These functional relationships between the parameters are obtained directly from the BADA manual. The BADA aircraft model is based on a mass-varying, kinetic approach to aircraft performance modelling. It is structured in three parts: the Aircraft Performance Model (APM), the Airline Procedure Model (ARPM) and the Aircraft Characteristics Model (ACM) (EUROCONTROL, 2014). These three elements represent the Aircraft Dynamic Model (ADM), which is combined with the Atmosphere Model (AM) to obtain the state variables given the control ones. The appraised scenario belongs to the cruise phase, as this phase allows considering a basis aircraft configuration, and therefore building the global study from the simplest phase of modelling to stages that are more complex. The initial hypothesis for the study is that the aircraft is following a 4D trajectory, within the new SESAR operational concept (SESAR, 2014). The trajectory was resembled by a flight in two dimensions, with the following main characteristics: rectilinear trajectory, level changes and no turning. The 4D trajectory model was implemented and generated using the MATLAB© software (MathWorks, 2017). Model validation was performed (in order to check its accuracy) by comparing the MATLAB© simulated trajectory with real data flights extracted from the tool NEST of EUROCONTROL (EUROCONTROL, 2017b), using different intra-European routes (with similar characteristics to the scenario of study). The flights chosen for comparison were those that present similar characteristics. The test error regarding time and position presents an average value of 7%, reaching less than 5% during the stabilized flight level sections.
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Table 1. Modelling parameters and dependencies. Block
Parameter
Atmosphere Model
Dependencies
Description
Pressure
p = f [T(h), ρ(h)]
T is the temperature, ρ is the density and h is the altitude.
Speed of sound
a0 = f [k, R, T, M]
M is the flight Mach, T is the temperature, R is the universal gas constant and k is the adiabatic air coefficient.
w = f [φ, λ, h]
φ is the latitude, λ is the longitude and h is the altitude.
CL = f [δ, p0, k, S, M, m, φ, g0]
δ is the pressure ratio, p0 is the pressure at mean sea level, S is the he wing surface, m is the aircraft mass and g0 is the acceleration of gravity at mean sea level.
L= f [δ, p0, k, S, M, CL]
-
CD = f [CL, δ, d1… d15, Mmax, p0, k, S, M, m, φ, g0]
d1… d15 are characteristic parameters of each aircraft.
D= f [δ, p0, k, S, M, CD]
-
CT = f [ti1…ti12, a1…a36, M, δ, δT]
ti1…ti12 and a1…a36 are characteristic parameters of each aircraft and δT is the throttle ratio.
Th= f [δ, mref, Wmref, CT]
mref, Wmref are the aircraft reference mass and weight for each aircraft model.
CF= f [δ, θ, M, fi1 …fi9, CT]
fi1 …fi9 are characteristic parameters of each aircraft and θ is the temperature ratio.
F= f [δ, θ, mref, Wmref, a0, Lhv, CF]
fi1 …fi9 are characteristic parameters of each aircraft and θ is the temperature ratio.
Wind Lift coefficient Aerodynamic Forces Model
Lift Drag coefficient Drag Thrust coefficient
Propulsive Forces Model
Thrust Fuel consumption coefficient Fuel consumption
Once the model was developed and validated, the next step was to perform a variation of the influence parameters in order to appraise different potential situations. This is achieved through a Monte Carlo simulation approach (Rubinstein and Kroese, 2016), which approximate the model to reality, obtaining data that represent real situations. In the Monte Carlo simulation, the following input variables are sampled: aircraft mass, temperature, pressure, density and wind. The model estimates aerodynamic and propulsive variables from mass with functional relationships show in Table 1. We performed 10,000 simulations.
(a)
(b)
Fig. 1. (a) Ground speed evolution. (b) Causal relationships and sensitivity analysis between influence parameters.
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The results obtained with the simulations show a degradation in trajectory parameters as the distance flown by the aircraft grows. Figure 1 (a) shows how ground speed is degraded due to uncertainties (Casado et al., 2012; Margellos and Lygeros, 2013). The trajectory is strongly affected by different uncertainties: the actual performances of the aircraft (modelled with BADA), the operation throughout the trajectory and the weather/atmospheric conditions affecting the flight (input variables sampled in Monte Carlo simulation) 3.2. Causal analysis A sensitivity analysis was formulated to measure the influence of modelling parameters on flight time. A Bayesian Network (BN) technique was used to perform a causality analysis. A BN is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG) (Darwiche, 2010). The BN for the trajectory prediction model consist of ten interrelated nodes. Each node corresponds to a parameter or influence factor in the evolution of the trajectory. The sample values for each parameter are distributed in different discrete states. The number of states at each node is adjusted to reflect the value of the specific parameter and its variability. Each state is assigned a probability whose value is defined by the number of times that the variable can be found in this state over the simulation experiments (Monte Carlo simulation). This BN structure was built using the modelling tool GeNIe (BayesFusion, 2017). In this process, a combination of data (simulations) with knowledge of functional relationships (Table 1) was used. Figure 1 (b) represents the results, the causal structure and the intensity of the relationships between parameters. In Figure 1 (b), (s1, s2, …, sn) are the default names given by the modelling tool for the nodes’ states. As can be seen, the most influential parameters are speed, aircraft mass, temperature and thrust. These parameters will be used in the reliability model. 3.3. Reliability model Regarding the reliability analysis, we define the modelled 4D trajectory as the system to be studied. The global performance of this system will be given by the performance of the system´s partial elements, which in this case will be the different parameters that define the trajectory. Therefore, it is necessary to define states or rates of system performance and states or rates of performance for the elements, considering the interrelationships and crossinfluences. In this phase, the system (4D trajectory) is considered as a system with a state vector, composed of three states: optimal, acceptable and degraded, with their associated performance rates, 100%, 50% and 60%, respectively. As already mentioned in the methodology, the study of Markov-chains can be reduced to the algebraic study of the properties of transition matrices. These transition matrices or probabilities of being and/or going from one state to another of the system, are calculated from the data obtained from the Monte Carlo simulations. Following a statistical study, we obtain that parameters’ distributions can adequately be represented by Normal distributions. The choice of states is therefore based on the confidence intervals defined by a Normal distribution (Table 2) Table 2. Intervals defining system states. State Optimal Acceptable Degraded
Interval 68.3 % 27.3 % 4.4 %
We developed the Markov-chain model using the MATLAB© software. Simulated 4D trajectories have been used to estimate 𝜇𝜇 and 𝜎𝜎 (mean and standard deviation) for each influence variable. 𝜇𝜇 arises from the deterministic model of the trajectory, obtaining a 𝜇𝜇 value for each instant of time and for each study parameter. As for the 𝜎𝜎 value, the standard deviation is considered at the initial instant. Therefore, the initial instant is taken as a reference point to study the degradation of the trajectory over time. Once we have calculated 𝜇𝜇 and 𝜎𝜎, we define the intervals corresponding to each of the states into which the trajectory has been divided in Table 2.
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3.4. Transition matrix Once the states are established at defined intervals, the next step is to obtain the transition matrix of the model. Firstly, we obtain the transition matrix of the partial parameters and then the transition matrix of the global model. Transition matrices are calculated for all parameters included in the Monte Carlo simulation (mass, speed, temperature, thrust and range). Table 3. Trajectory global transition matrix. Optimal Acceptable Degraded
Optimal 0.9822 0.0182 2.9920e-04
Acceptable 0.0174 0.9687 0.0086
Degraded 4.0418e-04 0.0131 0.9911
The global transition matrix (Table 3) is generated with the five parameters: mass, speed, temperature, thrust and range. In addition, two partial blocks are defined. These two blocks were called fundamental (speed and range) and non-fundamental (temperature, thrust and mass), and allowed us to add a priority in the calculation of the global transition matrix. We chose those fundamental parameters due to their impact (Figure 1 (b)) in the trajectory state (sensitivity and intensity of relationships was appraised through the causal model). If one of these two parameters are degraded, the trajectory is degraded. If not, the state is calculated considering the non-fundamental parameters’ state. 5. Results and functionalities We use three indicators to describe the system performance: mean instantaneous performance ( Et ), mean instantaneous deficiency (Dt ) and mean instantaneous reliability (R). In order to obtain indicators that characterize the average MSS output performance, we can use the performance expectation. The mean value of MSS instantaneous output performance (Eq. 4) at time 𝑡𝑡 is determined as (Lisnianski and Levitin, 2003): 𝐸𝐸𝑡𝑡 =
∑𝑡𝑡𝑘𝑘=1 𝑔𝑔𝑘𝑘 𝑝𝑝𝑘𝑘 (𝑡𝑡)
(4)
𝐷𝐷𝑡𝑡 =
∑𝑁𝑁𝑖𝑖=1 𝑝𝑝𝑖𝑖 (𝑡𝑡) 𝑚𝑚𝑚𝑚𝑚𝑚(𝑤𝑤 − 𝑔𝑔𝑖𝑖 ; 0)
(5)
Being 𝑁𝑁 the total number of states, 𝑔𝑔𝑘𝑘 is the performance rate associated with state 𝑘𝑘 and 𝑝𝑝𝑘𝑘 (𝑡𝑡) is the probability that the system is at state 𝑘𝑘 at time 𝑡𝑡. Figure 2 (a) shows the mean instantaneous performance evolution. The system is initially functioning perfectly (the initial state is selected as a reference with a 100% performance rate), and then evolves towards the mean instantaneous performance value for stationary distribution (43.35 %). The mean instantaneous deficiency (Eq. 5) or deviation is defined as a weighted average between the system probability to be found in each state and the service levels associated to these states. A weighted average of the value of a random variable where the probability function provides weights can be understood as the expected value (Lisnianski and Levitin, 2003). In case the difference is negative the average is weighted with a zero. That is because in those cases the system is meeting the expected demand and the aim of the index is to assess the cases when the system is not fulfilling the demand.
Where 𝑝𝑝𝑖𝑖 (𝑡𝑡) is the probability that the system is in state i at t-th time, w is the expected demand and 𝑔𝑔𝑖𝑖 is the level of performance associated to state i. The value of mean deficiency for stationary distribution (steady-state or long-term) is 21.33 (Figure 2 (b)). This indicator shows the evolution towards degradation of the trajectory. Starting from a state defined as optimal or ideal that satisfies the expected demand, it evolves to a degraded and unacceptable state. This confirms the outputs obtained from the mean instantaneous performance indicator. 𝑅𝑅 = 1 − ∑𝑖𝑖𝑗𝑗=1 𝑝𝑝𝑗𝑗 (𝑡𝑡)
(6)
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(a)
(b)
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(c)
Fig. 2. (a) Mean instantaneous performance. (b) Mean instantaneous deficiency. (c) Mean instantaneous reliability.
The reliability of a system (R) is defined (Eq. 6) as the system’s ability to remain in acceptable states during the operation period (Figure 2 (c)). Therefore, the reliability function can be defined as the probability that the system is not in its unacceptable states (Lisnianski and Levitin, 2003). In equation 6, 𝑖𝑖 is the number of unacceptable states. Figure 2 (c) shows the evolution of the probability that the 4D trajectory is in the correct states (optimal and acceptable). Reliability first decreases rapidly with time and later reaches a stable value (stationary distribution) of 57.35%. 6. Conclusion 4D trajectories and their associated Trajectory Based Operations (TBO) concept require high accuracy and reliability in trajectory monitoring and forecasting. In the first stage of the study, we defined a 4D trajectory prediction model and studied the influences of different parameters in the estimation of checkpoints. This model and a Monte Carlo technique allow us to perform 10,000 simulations and evaluate, through the information obtained from the simulations, the evolution of trajectory’s parameters through a stochastic approach. We developed a reliability analysis using multi-state systems theory and a Markov-Chain model. First, we defined the trajectory as a multi-state system, composed of the most influential parameters that were identified in the 4D trajectory model. These parameters (mass, speed, thrust, range and temperature) were identified with the help of the causal analysis performed in the first part of the study. In addition, the causal analysis allowed us to define different blocks or elements of study. The performance rates were defined as: optimal, acceptable and degraded. Subsequently, a Markov Chain approach was used to define the transition between the different states (instant times) of the system. For the global model, to define the global transition matrix, we introduced two blocks of parameters: fundamental and non-fundamental. The fundamental block is the most restrictive: if it is degraded, the trajectory state is considered degraded. In case the fundamental block is not degraded, the other parameters are considered for the calculation of the global transition matrix. We defined as the correct state the one that is imposed by the optimal and acceptable state, and the degraded state being the incorrect one. The reliability analysis showed the system evolution in time. With this analysis, we found a huge degradation towards an incorrect state. The probability of being in a correct state is 57.35%. Another output of the reliability model is the mean performance ratio in the stationary distribution, which is 43.35%, below acceptable status. Finally, time to achieve this stationary state is 698 seconds (note that we do not consider correction or deviation control measures along the trajectory). In conclusion, the aircraft suffers disturbances during the flight that cause a great degradation of the trajectory. Hence, it is necessary to take appropriate actions to meet the agreed targets within the TBO concept. The main contribution of this paper is the development of a tool for assessing 4D trajectories reliability, which is associated with trajectory degradation. This model can be useful for the flight operator or the air traffic management service provider to predict and manage 4D trajectories, as the model makes it possible to assess, through different
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flight parameters, the evolution of trajectory degradation. With this information, it is possible to make trajectory updates at the appropriate moment, to help deviation management and predictability of 4D trajectories. Our initial results indicate that MSS theory provides a promising approach for the reliability assessment of 4D trajectories. Future work will be focused on refining and deepening the analysis of the different system elements. Moreover, potential corrective actions should be identified to improve reliability of the global system by acting on the different influence parameters. In addition, further research is required to extend the model to more complex trajectory stages (like evolving flight) and thus obtain a reliability study for the complete 4D trajectory. References Aven, T., Jensen, U., 1999. Stochastic models in reliability, Applications of mathematics. https://doi.org/10.1007/978-1-4614-7894-2 BayesFusion, 2017. GeNIe Modeler. Casado, E., Goodchild, C., Vilaplana, M., 2012. Identification and Initial Characterization of Sources of Uncertainty Affecting the Performance of Future Trajectory Management Automation Systems. Int. Conf. Appl. Theory Autom. Command Control Syst. Darwiche, A., 2010. Bayesian networks. Commun. ACM 53, 80–90. https://doi.org/10.1145/1859204.1859227 EUROCONTROL, 2017a. BADA Technical Documentation and Datasets. EUROCONTROL, 2017b. NEST modelling tool. EUROCONTROL, 2014. User manual for the Base of Aircraft Data (BADA) Family 4. EEC Technical Scientific Report No. 12/11/22-58. Brussels: European Organisation for the Safety of Air Navigation. FAA, 2016. The Future of the NAS. Garcia-Chico, J., Vivona, R. a, Cate, K.T., 2008. Characterizing Intent Maneuvers from Operational Data–A Step towards Trajectory Uncertainty Estimation, in: AIAA Guidance, Navigation and Control Conference and Exhibit. Lisnianski, A., 2007. Extended block diagram method for a multi-state system reliability assessment. Reliab. Eng. Syst. Saf. 92, 1601–1607. https://doi.org/10.1016/j.ress.2006.09.013 Lisnianski, A., Frenkel, I., Ding, Y., 2010. Multi-state system reliability analysis and optimization for engineers and industrial managers, 1st ed, Multi-State System Reliability Analysis and Optimization for Engineers and Industrial Managers. https://doi.org/10.1007/978-1-84996320-6 Lisnianski, A., Levitin, G., 2003. Multi-State System Reliability: Assessment, Optimization and Applications, 1st ed. World Scientific, Singapore. Liu, W., Hwang, I., 2011. Probabilistic Trajectory Prediction and Conflict Detection for Air Traffic Control. J. Guid. Control. Dyn. 34, 1779– 1789. https://doi.org/10.2514/1.53645 Margellos, K., Lygeros, J., 2013. Toward 4-D Trajectory Management in Air Traffic Control: A Study based on Monte Carlo Simulation and Reachability Analysis. IEEE Trans. Control Syst. Technol. 21, 1820–1833. https://doi.org/10.1109/TCST.2012.2220773 MathWorks, 2017. MATLAB. www.mathworks.com/products/matlab. Mutuel, L.H., Neri, P., Paricaud, E., 2013. Initial 4D Trajectory Management Concept Evaluation, in: Tenth USA/Europe Air Traffic Management Research and Development Seminar. Natvig, B., 2011. Multistate Systems Reliability with Applications, Wiley Seri. ed. John Wiley & Sons. OAG, 2017. OAG Traffic Analyser [WWW Document]. OAG public website. URL http://www.oag.com/Insight/OAG-Traffic-Analyser Pabst, T., Kunze, T., Schultz, M., Fricke, H., 2013. Modeling external disturbances for aircraft in flight to build reliable 4D trajectories, in: 3rd International Conference on Application and Theory of Automation in Command and Control Systems (ATACCS’2013). Naples, Italy. Rubinstein, R.Y., Kroese, D.P., 2016. Simulation and the Monte Carlo Method, Wiley. New York: Wiley. https://doi.org/10.1111/j.17515823.2009.00074_8.x SESAR, 2015. European ATM Master Plan - Edition 2015, The Roadmap for Sustainable Air Traffic Management. Luxembourg: Publications Office of the European Union. https://doi.org/10.2829/512525 SESAR, 2014. SESAR Concept of Operations Step 2 Edition 2014, 01.01.00. ed. Brussels. Trivedi, K.S., 2016. Probability and Statistics with Reliability, Queuing and Computer Science Applications, Probability and Statistics with Reliability, Queuing and Computer Science Applications. https://doi.org/10.1002/9781119285441 Wang, X., Liu, W., 2012. Research on Air Traffic Control Automatic System Software Reliability Based on Markov Chain. Phys. Procedia 24, 1601–1606. https://doi.org/10.1016/j.phpro.2012.02.236 Wu, C.L., Caves, R.E., 2004. Modelling and simulation of aircraft turnaround operations at airports. Transp. Plan. Technol. 27, 25–46. https://doi.org/10.1080/0308106042000184445
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XIII Conference on Transport Engineering, CIT2018 XIII Conference on Transport Engineering, CIT2018
Air Traffic Management based on 4D Trajectories: A Reliability Air Traffic Management based on 4D Trajectories: A Reliability Analysis using Multi-State Systems Theory Analysis using Multi-State Systems Theory
Álvaro Rodríguez-Sanz a,a, **, David Álvarez Álvarez aa, Fernando Gómez Comendador aa, Álvaro Rodríguez-Sanz , David Álvarez Álvarez ,aFernando Gómez Comendador , a b Rosa Arnaldo Valdés , Javier Pérez-Castán and Mar Najar Godoy Rosa Arnaldo Valdés a, Javier Pérez-Castán a and Mar Najar Godoy b a Universidad Politécnica de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040, Spain. a Universidad de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040, Spain. ISDEFE (Ingeniería dePolitécnica Sistemas para la Defensa de España), Calle Beatriz de Bobadilla, N3, Madrid 28040, Spain. b ISDEFE (Ingeniería de Sistemas para la Defensa de España), Calle Beatriz de Bobadilla, N3, Madrid 28040, Spain. b
Abstract Abstract The current Air Traffic Management (ATM) functional approach is changing. Both SESAR (Single European Sky ATM Research) The current Air Traffic Management (ATM) functionalSystem) approach is changing. Both SESAR (Single European Sky ATM Research) and NEXTGEN (Next Generation Air Transportation support the four-dimension (4D) trajectory implementation within and Generation Air Transportation System) support the four-dimension (4D)integrated trajectoryas implementation theirNEXTGEN operational(Next concepts. Apart from the three classical spatial dimensions, ‘time’ is now an additional within fourth their operational Apart fromflights the three dimensions, is now as can an additional fourth dimension, whichconcepts. will restrict aircraft over classical indicatedspatial waypoints along the‘time’ trajectory. 4Dintegrated trajectories be understood as dimension, which willSystems restrict aircraft over waypoints alongand the usage trajectory. 4D trajectories can beanalysis understood as complex Multi-State (MSS) flights that rely onindicated environmental, internal conditions. A reliability of the complex Systemsthat (MSS) thatthe rely environmental, internal usageand conditions. reliability analysis of the waypointsMulti-State and time windows describe 4Don trajectory may allow airline and operators air traffic A service providers to establish waypoints andindicators time windows describe metrics. the 4D trajectory maydevelops allow airline operators and air traffic service providers to establish performance and that compliance This paper a model to evaluate potential ‘malfunctions’ of a 4D performance indicators and compliance metrics. This paper develops a model to evaluate potential ‘malfunctions’ a 4D trajectory, based on the MSS reliability theory. This is a natural extension of classical binary-state evaluation: trajectoriesofpresent trajectory, based on thelevels MSS reliability theory. is a natural extensionrange). of classical binary-statereliability evaluation: trajectorieswhich present different performance and several failureThis modes (a degradation The operational assessment, is different performance levels and several (a degradation range). The reliability assessment, which is achieved with Monte Carlo simulation andfailure randommodes processes (Markov) methods, offersoperational a framework to predict how probable is for achieved with Monte simulation andWe random processes (Markov) methods, offers a framework to predict howand probable is for the trajectory to enterCarlo a degraded state. use this analysis to quantify the 4D trajectory level of variability to propose the trajectory to enter degraded state.trajectory We use degradations this analysis or to unplanned quantify thesituations. 4D trajectory level of variability and tothrough proposea corrective measures to asolve potential The methodology is validated corrective measures to solve trajectory or unplanned situations. The methodology is validated through a practical case study. The mainpotential contribution of thisdegradations paper is to provide a methodology to evaluate the robustness of 4D trajectories practical case study. main contribution this paper isintotraffic provide a methodologyand to evaluate the robustness of 4D trajectories and to deal with theirThe perturbation, which is of a cornerstone synchronization conflict resolution. and to deal with their perturbation, which is a cornerstone in traffic synchronization and conflict resolution. © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. © © 2018 The Authors. by Elsevier Ltd. This is an an open accessPublished article under under the CC CC BY-NC-ND BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is open access article the license This is an and openpeer-review access article underresponsibility the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) responsibility scientific committee of of the the XIII XIII Conference Conference on Transport Transport Engineering, Engineering, Selection under of the scientific committee Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. CIT2018. Keywords: 4D trayectories; Reliability analysis; Multi-state systems; Markov Chain processes Keywords: 4D trayectories; Reliability analysis; Multi-state systems; Markov Chain processes
* Corresponding author. Tel.: +34 626574829. * E-mail Corresponding Tel.: +34 626574829. address:author. [email protected]. E-mail address: [email protected].
2352-1465 © 2018 The Authors. Published by Elsevier Ltd. 2352-1465 © 2018 Thearticle Authors. Published by Elsevier Ltd. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND licensecommittee (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the scientific of the XIII Conference on Transport Engineering, Selection CIT2018. and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 2352-1465 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 10.1016/j.trpro.2018.11.001
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1. Introduction and problem statement The recent increase in air traffic demand provides a challenging operational situation for the current European Air Traffic Management (ATM) system (SESAR, 2015). Achieving accurate prediction of trajectories is a fundamental condition for reliable detection and resolution of conflicts. The future ATM system relies on the Trajectory Based Operations (TBO) concept (FAA, 2016; SESAR, 2014). TBO involves separating aircraft through a definition of a strategic trajectory (long-term), rather than the currently practicing tactical (short-term) conflict resolution. This approach is based on a four dimensional framework, composed by the three spatial dimensions and a time constraint (Mutuel et al., 2013). The trajectory predictions need to consider external disturbances to the aircraft and internal uncertainty sources (Casado et al., 2012; Garcia-Chico et al., 2008; Pabst et al., 2013). These disturbances and uncertainties may cause a degradation in the trajectory. Therefore, the management and study of degradation is a key element when ensuring the accuracy of the trajectory. However, real measures (over the trajectory) will be required to improve reliability, react to unplanned conditions and thus maintain the expect capacity. Trajectories are degraded due to environmental and operational uncertainties (Casado et al., 2012; Garcia-Chico et al., 2008). Sharing, updating and coordinating changes in trajectory becomes necessary to ensure reliability. Hence, these measurements will help to appraise the required adjustments to correct trajectory degradation. This degradation increases over time if appropriate actions are not taken. Reliability analysis provides theoretical and practical tools to test the behavior and performance of trajectories in a scenario of uncertainties. The properties of Markov-chains have been used to model the interdependent effects between sequential procedures of aircraft turnarounds (Wu and Caves, 2004). In Air Traffic Control (ATM), a MultiState System (MSS) theory and a Markov model have been applied to measure the efficiency and reliability of an Air Traffic Automatic System (ATCAS) (Wang and Liu, 2012). The estimation of potential benefits of new ATM tools has also been appraised using MSS reliability through Markov models (Liu and Hwang, 2011). Monte Carlo simulations were implemented to estimate the range of uncertainties in model parameters and technology performance accuracy. Monte Carlo simulation allows, with a simple computation procedure, modelling system operating scenarios to assess the reliability systems (Aven and Jensen, 1999; Trivedi, 2016). The purpose of this study is to define a model to assess flight operational efficiency. This is achieved by defining a series of indicators based on trajectory influence parameters, which allow the evaluation of the 4D trajectory operational state at that given moment. This can be applied in a predictive way, thus being able to anticipate the trajectory degradation in order to apply corrective actions. The paper is structured as follows: first, we introduce the methodology (MSS theory to characterize 4D trajectories and a Markov-Chain method to model their performance evolution); and then, we study the reliability of 4D trajectories. 2. Methods and materials 2.1. Multi-state systems A system is designed to perform its functions in a given environment. Traditionally, the way to evaluate performance and reliability was through a binary system, with two states: it works, or it does not work (Natvig, 2011; Trivedi, 2016). This perspective limits the study. In many cases, systems can perform their functions with many efficiency values that are generally known as performance rates. A system that has a finite number performance rates is called a multi-state system (MSS) (Lisnianski et al., 2010). Usually, a multi-state system is composed of elements that can also be considered as multi-state systems. An element is a system entity that has no more subdivisions. This does not imply that the element cannot be made up of parts, but it means that, in a reliability study, it will be studied as an entity (Lisnianski, 2007). The performance rates of the elements range from perfect operation to complete failure. Faults that can lead to a reduction in the performance of an element are called partial faults. After a partial failure, the elements continue to operate at reduced performance, and after a total failure, the elements are disabled and do not perform their function (Lisnianski et al., 2010). The performance rate of an element at any time is a random variable that takes its values from the probabilities associated with the different states (rate of return) of the system’s element.
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2.2. Markov-Chains Markov-chains are a special type of discrete stochastic process in which the probability of an event only depends of the precedent state of the system. This type of systems is memoryless and satisfy the Markov property (Aven and Jensen, 1999). The probability of transition between state 𝑋𝑋𝑛𝑛−1 = 𝑖𝑖 and 𝑋𝑋𝑛𝑛 = 𝑗𝑗 is given by 𝛾𝛾𝑖𝑖,𝑗𝑗 , where 𝑛𝑛 is the number of transitions. The matrix 𝑷𝑷 represents collectively these transition probabilities: 𝛾𝛾1,1 𝛾𝛾2,1 𝑷𝑷 = ( ⋮ 𝛾𝛾𝑘𝑘,1
𝛾𝛾1,2 𝛾𝛾2,2 ⋮ 𝛾𝛾𝑘𝑘,2
⋯ ⋯ ⋯ ⋯
𝛾𝛾1,𝑘𝑘 𝛾𝛾2,𝑘𝑘 ⋮ ) 𝛾𝛾𝑘𝑘,𝑘𝑘
The vector 𝜋𝜋𝑛𝑛𝑇𝑇 defines the probability that the system is in a specific state on the 𝑛𝑛-th transition: 𝜋𝜋𝑛𝑛𝑇𝑇 = [𝜋𝜋1,𝑛𝑛 𝜋𝜋2,𝑛𝑛 ⋯
𝜋𝜋𝑘𝑘,𝑛𝑛 ]
(1)
(2)
where 𝜋𝜋𝑘𝑘,𝑛𝑛 is the probability that the system is in state 𝑘𝑘 on the 𝑛𝑛-th transition. The probabilities of the state for each transition are determined iteratively as follows: 𝑇𝑇 𝜋𝜋𝑛𝑛𝑇𝑇 = 𝜋𝜋𝑛𝑛−1 𝑷𝑷
(3)
The random evolution of a Markov-chain is completely determined by its transition matrix P and its initial density distribution 𝜋𝜋𝑇𝑇0 . Therefore, with the transition matrix obtained, the study of Markov-chains is reducible to the linear algebra study of its transition matrix and state vector (Eq 3). 3. Reliability analysis 3.1. Trajectory modelling In the first part of the study, the aircraft performance and the 4D trajectory were modelled in order to obtain data regarding its evolution. The aircraft chosen was a Boeing 737-900ER, one of the most frequent aircraft in shortmedium range flights (OAG, 2017). Next, 4D trajectories were modelled using the EUROCONTROL’s BADA (Base of Aircraft Data) methodology (EUROCONTROL, 2017a). In order to develop the model, it was necessary to characterize 4D trajectories and identify their influence parameters, which are shown in Table 1 (aircraft performances and variables related to the scenario). These functional relationships between the parameters are obtained directly from the BADA manual. The BADA aircraft model is based on a mass-varying, kinetic approach to aircraft performance modelling. It is structured in three parts: the Aircraft Performance Model (APM), the Airline Procedure Model (ARPM) and the Aircraft Characteristics Model (ACM) (EUROCONTROL, 2014). These three elements represent the Aircraft Dynamic Model (ADM), which is combined with the Atmosphere Model (AM) to obtain the state variables given the control ones. The appraised scenario belongs to the cruise phase, as this phase allows considering a basis aircraft configuration, and therefore building the global study from the simplest phase of modelling to stages that are more complex. The initial hypothesis for the study is that the aircraft is following a 4D trajectory, within the new SESAR operational concept (SESAR, 2014). The trajectory was resembled by a flight in two dimensions, with the following main characteristics: rectilinear trajectory, level changes and no turning. The 4D trajectory model was implemented and generated using the MATLAB© software (MathWorks, 2017). Model validation was performed (in order to check its accuracy) by comparing the MATLAB© simulated trajectory with real data flights extracted from the tool NEST of EUROCONTROL (EUROCONTROL, 2017b), using different intra-European routes (with similar characteristics to the scenario of study). The flights chosen for comparison were those that present similar characteristics. The test error regarding time and position presents an average value of 7%, reaching less than 5% during the stabilized flight level sections.
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Table 1. Modelling parameters and dependencies. Block
Parameter
Atmosphere Model
Dependencies
Description
Pressure
p = f [T(h), ρ(h)]
T is the temperature, ρ is the density and h is the altitude.
Speed of sound
a0 = f [k, R, T, M]
M is the flight Mach, T is the temperature, R is the universal gas constant and k is the adiabatic air coefficient.
w = f [φ, λ, h]
φ is the latitude, λ is the longitude and h is the altitude.
CL = f [δ, p0, k, S, M, m, φ, g0]
δ is the pressure ratio, p0 is the pressure at mean sea level, S is the he wing surface, m is the aircraft mass and g0 is the acceleration of gravity at mean sea level.
L= f [δ, p0, k, S, M, CL]
-
CD = f [CL, δ, d1… d15, Mmax, p0, k, S, M, m, φ, g0]
d1… d15 are characteristic parameters of each aircraft.
D= f [δ, p0, k, S, M, CD]
-
CT = f [ti1…ti12, a1…a36, M, δ, δT]
ti1…ti12 and a1…a36 are characteristic parameters of each aircraft and δT is the throttle ratio.
Th= f [δ, mref, Wmref, CT]
mref, Wmref are the aircraft reference mass and weight for each aircraft model.
CF= f [δ, θ, M, fi1 …fi9, CT]
fi1 …fi9 are characteristic parameters of each aircraft and θ is the temperature ratio.
F= f [δ, θ, mref, Wmref, a0, Lhv, CF]
fi1 …fi9 are characteristic parameters of each aircraft and θ is the temperature ratio.
Wind Lift coefficient Aerodynamic Forces Model
Lift Drag coefficient Drag Thrust coefficient
Propulsive Forces Model
Thrust Fuel consumption coefficient Fuel consumption
Once the model was developed and validated, the next step was to perform a variation of the influence parameters in order to appraise different potential situations. This is achieved through a Monte Carlo simulation approach (Rubinstein and Kroese, 2016), which approximate the model to reality, obtaining data that represent real situations. In the Monte Carlo simulation, the following input variables are sampled: aircraft mass, temperature, pressure, density and wind. The model estimates aerodynamic and propulsive variables from mass with functional relationships show in Table 1. We performed 10,000 simulations.
(a)
(b)
Fig. 1. (a) Ground speed evolution. (b) Causal relationships and sensitivity analysis between influence parameters.
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The results obtained with the simulations show a degradation in trajectory parameters as the distance flown by the aircraft grows. Figure 1 (a) shows how ground speed is degraded due to uncertainties (Casado et al., 2012; Margellos and Lygeros, 2013). The trajectory is strongly affected by different uncertainties: the actual performances of the aircraft (modelled with BADA), the operation throughout the trajectory and the weather/atmospheric conditions affecting the flight (input variables sampled in Monte Carlo simulation) 3.2. Causal analysis A sensitivity analysis was formulated to measure the influence of modelling parameters on flight time. A Bayesian Network (BN) technique was used to perform a causality analysis. A BN is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG) (Darwiche, 2010). The BN for the trajectory prediction model consist of ten interrelated nodes. Each node corresponds to a parameter or influence factor in the evolution of the trajectory. The sample values for each parameter are distributed in different discrete states. The number of states at each node is adjusted to reflect the value of the specific parameter and its variability. Each state is assigned a probability whose value is defined by the number of times that the variable can be found in this state over the simulation experiments (Monte Carlo simulation). This BN structure was built using the modelling tool GeNIe (BayesFusion, 2017). In this process, a combination of data (simulations) with knowledge of functional relationships (Table 1) was used. Figure 1 (b) represents the results, the causal structure and the intensity of the relationships between parameters. In Figure 1 (b), (s1, s2, …, sn) are the default names given by the modelling tool for the nodes’ states. As can be seen, the most influential parameters are speed, aircraft mass, temperature and thrust. These parameters will be used in the reliability model. 3.3. Reliability model Regarding the reliability analysis, we define the modelled 4D trajectory as the system to be studied. The global performance of this system will be given by the performance of the system´s partial elements, which in this case will be the different parameters that define the trajectory. Therefore, it is necessary to define states or rates of system performance and states or rates of performance for the elements, considering the interrelationships and crossinfluences. In this phase, the system (4D trajectory) is considered as a system with a state vector, composed of three states: optimal, acceptable and degraded, with their associated performance rates, 100%, 50% and 60%, respectively. As already mentioned in the methodology, the study of Markov-chains can be reduced to the algebraic study of the properties of transition matrices. These transition matrices or probabilities of being and/or going from one state to another of the system, are calculated from the data obtained from the Monte Carlo simulations. Following a statistical study, we obtain that parameters’ distributions can adequately be represented by Normal distributions. The choice of states is therefore based on the confidence intervals defined by a Normal distribution (Table 2) Table 2. Intervals defining system states. State Optimal Acceptable Degraded
Interval 68.3 % 27.3 % 4.4 %
We developed the Markov-chain model using the MATLAB© software. Simulated 4D trajectories have been used to estimate 𝜇𝜇 and 𝜎𝜎 (mean and standard deviation) for each influence variable. 𝜇𝜇 arises from the deterministic model of the trajectory, obtaining a 𝜇𝜇 value for each instant of time and for each study parameter. As for the 𝜎𝜎 value, the standard deviation is considered at the initial instant. Therefore, the initial instant is taken as a reference point to study the degradation of the trajectory over time. Once we have calculated 𝜇𝜇 and 𝜎𝜎, we define the intervals corresponding to each of the states into which the trajectory has been divided in Table 2.
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3.4. Transition matrix Once the states are established at defined intervals, the next step is to obtain the transition matrix of the model. Firstly, we obtain the transition matrix of the partial parameters and then the transition matrix of the global model. Transition matrices are calculated for all parameters included in the Monte Carlo simulation (mass, speed, temperature, thrust and range). Table 3. Trajectory global transition matrix. Optimal Acceptable Degraded
Optimal 0.9822 0.0182 2.9920e-04
Acceptable 0.0174 0.9687 0.0086
Degraded 4.0418e-04 0.0131 0.9911
The global transition matrix (Table 3) is generated with the five parameters: mass, speed, temperature, thrust and range. In addition, two partial blocks are defined. These two blocks were called fundamental (speed and range) and non-fundamental (temperature, thrust and mass), and allowed us to add a priority in the calculation of the global transition matrix. We chose those fundamental parameters due to their impact (Figure 1 (b)) in the trajectory state (sensitivity and intensity of relationships was appraised through the causal model). If one of these two parameters are degraded, the trajectory is degraded. If not, the state is calculated considering the non-fundamental parameters’ state. 5. Results and functionalities We use three indicators to describe the system performance: mean instantaneous performance ( Et ), mean instantaneous deficiency (Dt ) and mean instantaneous reliability (R). In order to obtain indicators that characterize the average MSS output performance, we can use the performance expectation. The mean value of MSS instantaneous output performance (Eq. 4) at time 𝑡𝑡 is determined as (Lisnianski and Levitin, 2003): 𝐸𝐸𝑡𝑡 =
∑𝑡𝑡𝑘𝑘=1 𝑔𝑔𝑘𝑘 𝑝𝑝𝑘𝑘 (𝑡𝑡)
(4)
𝐷𝐷𝑡𝑡 =
∑𝑁𝑁𝑖𝑖=1 𝑝𝑝𝑖𝑖 (𝑡𝑡) 𝑚𝑚𝑚𝑚𝑚𝑚(𝑤𝑤 − 𝑔𝑔𝑖𝑖 ; 0)
(5)
Being 𝑁𝑁 the total number of states, 𝑔𝑔𝑘𝑘 is the performance rate associated with state 𝑘𝑘 and 𝑝𝑝𝑘𝑘 (𝑡𝑡) is the probability that the system is at state 𝑘𝑘 at time 𝑡𝑡. Figure 2 (a) shows the mean instantaneous performance evolution. The system is initially functioning perfectly (the initial state is selected as a reference with a 100% performance rate), and then evolves towards the mean instantaneous performance value for stationary distribution (43.35 %). The mean instantaneous deficiency (Eq. 5) or deviation is defined as a weighted average between the system probability to be found in each state and the service levels associated to these states. A weighted average of the value of a random variable where the probability function provides weights can be understood as the expected value (Lisnianski and Levitin, 2003). In case the difference is negative the average is weighted with a zero. That is because in those cases the system is meeting the expected demand and the aim of the index is to assess the cases when the system is not fulfilling the demand.
Where 𝑝𝑝𝑖𝑖 (𝑡𝑡) is the probability that the system is in state i at t-th time, w is the expected demand and 𝑔𝑔𝑖𝑖 is the level of performance associated to state i. The value of mean deficiency for stationary distribution (steady-state or long-term) is 21.33 (Figure 2 (b)). This indicator shows the evolution towards degradation of the trajectory. Starting from a state defined as optimal or ideal that satisfies the expected demand, it evolves to a degraded and unacceptable state. This confirms the outputs obtained from the mean instantaneous performance indicator. 𝑅𝑅 = 1 − ∑𝑖𝑖𝑗𝑗=1 𝑝𝑝𝑗𝑗 (𝑡𝑡)
(6)
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(b)
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(c)
Fig. 2. (a) Mean instantaneous performance. (b) Mean instantaneous deficiency. (c) Mean instantaneous reliability.
The reliability of a system (R) is defined (Eq. 6) as the system’s ability to remain in acceptable states during the operation period (Figure 2 (c)). Therefore, the reliability function can be defined as the probability that the system is not in its unacceptable states (Lisnianski and Levitin, 2003). In equation 6, 𝑖𝑖 is the number of unacceptable states. Figure 2 (c) shows the evolution of the probability that the 4D trajectory is in the correct states (optimal and acceptable). Reliability first decreases rapidly with time and later reaches a stable value (stationary distribution) of 57.35%. 6. Conclusion 4D trajectories and their associated Trajectory Based Operations (TBO) concept require high accuracy and reliability in trajectory monitoring and forecasting. In the first stage of the study, we defined a 4D trajectory prediction model and studied the influences of different parameters in the estimation of checkpoints. This model and a Monte Carlo technique allow us to perform 10,000 simulations and evaluate, through the information obtained from the simulations, the evolution of trajectory’s parameters through a stochastic approach. We developed a reliability analysis using multi-state systems theory and a Markov-Chain model. First, we defined the trajectory as a multi-state system, composed of the most influential parameters that were identified in the 4D trajectory model. These parameters (mass, speed, thrust, range and temperature) were identified with the help of the causal analysis performed in the first part of the study. In addition, the causal analysis allowed us to define different blocks or elements of study. The performance rates were defined as: optimal, acceptable and degraded. Subsequently, a Markov Chain approach was used to define the transition between the different states (instant times) of the system. For the global model, to define the global transition matrix, we introduced two blocks of parameters: fundamental and non-fundamental. The fundamental block is the most restrictive: if it is degraded, the trajectory state is considered degraded. In case the fundamental block is not degraded, the other parameters are considered for the calculation of the global transition matrix. We defined as the correct state the one that is imposed by the optimal and acceptable state, and the degraded state being the incorrect one. The reliability analysis showed the system evolution in time. With this analysis, we found a huge degradation towards an incorrect state. The probability of being in a correct state is 57.35%. Another output of the reliability model is the mean performance ratio in the stationary distribution, which is 43.35%, below acceptable status. Finally, time to achieve this stationary state is 698 seconds (note that we do not consider correction or deviation control measures along the trajectory). In conclusion, the aircraft suffers disturbances during the flight that cause a great degradation of the trajectory. Hence, it is necessary to take appropriate actions to meet the agreed targets within the TBO concept. The main contribution of this paper is the development of a tool for assessing 4D trajectories reliability, which is associated with trajectory degradation. This model can be useful for the flight operator or the air traffic management service provider to predict and manage 4D trajectories, as the model makes it possible to assess, through different
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flight parameters, the evolution of trajectory degradation. With this information, it is possible to make trajectory updates at the appropriate moment, to help deviation management and predictability of 4D trajectories. Our initial results indicate that MSS theory provides a promising approach for the reliability assessment of 4D trajectories. Future work will be focused on refining and deepening the analysis of the different system elements. Moreover, potential corrective actions should be identified to improve reliability of the global system by acting on the different influence parameters. In addition, further research is required to extend the model to more complex trajectory stages (like evolving flight) and thus obtain a reliability study for the complete 4D trajectory. References Aven, T., Jensen, U., 1999. Stochastic models in reliability, Applications of mathematics. https://doi.org/10.1007/978-1-4614-7894-2 BayesFusion, 2017. GeNIe Modeler. Casado, E., Goodchild, C., Vilaplana, M., 2012. Identification and Initial Characterization of Sources of Uncertainty Affecting the Performance of Future Trajectory Management Automation Systems. Int. Conf. Appl. Theory Autom. Command Control Syst. Darwiche, A., 2010. Bayesian networks. Commun. ACM 53, 80–90. https://doi.org/10.1145/1859204.1859227 EUROCONTROL, 2017a. BADA Technical Documentation and Datasets. EUROCONTROL, 2017b. NEST modelling tool. EUROCONTROL, 2014. User manual for the Base of Aircraft Data (BADA) Family 4. EEC Technical Scientific Report No. 12/11/22-58. Brussels: European Organisation for the Safety of Air Navigation. FAA, 2016. The Future of the NAS. Garcia-Chico, J., Vivona, R. a, Cate, K.T., 2008. Characterizing Intent Maneuvers from Operational Data–A Step towards Trajectory Uncertainty Estimation, in: AIAA Guidance, Navigation and Control Conference and Exhibit. Lisnianski, A., 2007. Extended block diagram method for a multi-state system reliability assessment. Reliab. Eng. Syst. Saf. 92, 1601–1607. https://doi.org/10.1016/j.ress.2006.09.013 Lisnianski, A., Frenkel, I., Ding, Y., 2010. Multi-state system reliability analysis and optimization for engineers and industrial managers, 1st ed, Multi-State System Reliability Analysis and Optimization for Engineers and Industrial Managers. https://doi.org/10.1007/978-1-84996320-6 Lisnianski, A., Levitin, G., 2003. Multi-State System Reliability: Assessment, Optimization and Applications, 1st ed. World Scientific, Singapore. Liu, W., Hwang, I., 2011. Probabilistic Trajectory Prediction and Conflict Detection for Air Traffic Control. J. Guid. Control. Dyn. 34, 1779– 1789. https://doi.org/10.2514/1.53645 Margellos, K., Lygeros, J., 2013. Toward 4-D Trajectory Management in Air Traffic Control: A Study based on Monte Carlo Simulation and Reachability Analysis. IEEE Trans. Control Syst. Technol. 21, 1820–1833. https://doi.org/10.1109/TCST.2012.2220773 MathWorks, 2017. MATLAB. www.mathworks.com/products/matlab. Mutuel, L.H., Neri, P., Paricaud, E., 2013. Initial 4D Trajectory Management Concept Evaluation, in: Tenth USA/Europe Air Traffic Management Research and Development Seminar. Natvig, B., 2011. Multistate Systems Reliability with Applications, Wiley Seri. ed. John Wiley & Sons. OAG, 2017. OAG Traffic Analyser [WWW Document]. OAG public website. URL http://www.oag.com/Insight/OAG-Traffic-Analyser Pabst, T., Kunze, T., Schultz, M., Fricke, H., 2013. Modeling external disturbances for aircraft in flight to build reliable 4D trajectories, in: 3rd International Conference on Application and Theory of Automation in Command and Control Systems (ATACCS’2013). Naples, Italy. Rubinstein, R.Y., Kroese, D.P., 2016. Simulation and the Monte Carlo Method, Wiley. New York: Wiley. https://doi.org/10.1111/j.17515823.2009.00074_8.x SESAR, 2015. European ATM Master Plan - Edition 2015, The Roadmap for Sustainable Air Traffic Management. Luxembourg: Publications Office of the European Union. https://doi.org/10.2829/512525 SESAR, 2014. SESAR Concept of Operations Step 2 Edition 2014, 01.01.00. ed. Brussels. Trivedi, K.S., 2016. Probability and Statistics with Reliability, Queuing and Computer Science Applications, Probability and Statistics with Reliability, Queuing and Computer Science Applications. https://doi.org/10.1002/9781119285441 Wang, X., Liu, W., 2012. Research on Air Traffic Control Automatic System Software Reliability Based on Markov Chain. Phys. Procedia 24, 1601–1606. https://doi.org/10.1016/j.phpro.2012.02.236 Wu, C.L., Caves, R.E., 2004. Modelling and simulation of aircraft turnaround operations at airports. Transp. Plan. Technol. 27, 25–46. https://doi.org/10.1080/0308106042000184445