NAPs optimization

Even though the aforementioned NAPs have proven their effectiveness in reducing the impact of aircraft noise, those procedures are implemented individually as there is no certain knowledge of how they should be combined. Consequently, raises the need for an optimal noise abatement procedure that takes into account all the variables playing a role in the trajectory definition. Due to the fact that the optimization of NAPs is a relatively new field, a big variety of proposals and strategies can be found. On the following section, a brief review of the currently available literature will be exposed, addressing four main characteristics of NAPs optimizers; noise models, measurement grids and cost functions strategies and different numerical approaches.

1.2.2.1. Noise models

When implementing a noise model, a trade-off between the computational cost and the accuracy of the model must be taken into account. As [18] states; “The high mathematical complexity associated with the computation of emission and propagation of aircraft noise limits the adoption of accurate models to offline optimization studies, typically performed

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to optimize the departure and approach paths to aircraft, which are the portions of the flight in close proximity to the ground”. In addition, explains that, due to some opportune assumptions, the integration of simplified models such as the Noise-Power-Distance (NPD) models with a simplified worst-case emission and with a Demographic Distribution (D3) and Digital Terrain Elevation (DTED) databases can enable a quasi-real-time noise model implementation. The most widely used NPD model is the Integrated Noise Model (INM) from the Federal Aviation Agency (FAA), which can be found in several studies such as [19,5,18].

The INM is based on the European Civil Aviation Conference (ECAC) document 29 3rd Edition from the ICAO and uses a grid-based approach for computing noise footprints.

An NPD table is given from which noise is computed for a given distance and power set-ting. The values are calculated based on interpolation from the NPD table, which contains empirical measurements for each aircraft type under reference conditions.

A very similar new implementation of the ECAC rules is given by the Eurocontrol Aircraft Noise and Performance (ANP) databases. The ANP is the free of cost alternative to the INM even though is also based on the same above-mentioned ICAO document. Although it has the same structure as the INM, the ANP database is not as complete as the INM database and has fewer settings options. As an example, the INM database offers 4 flight aerodynamic configurations, whereas the ANP database only offers Arrival and Departure configurations.

For more details on the ANP and INM databases refer to [7,20].

1.2.2.2. Measurement grids and cost functions

Once the noise model is implemented, a cost function must be defined in order to optimize the aircraft trajectory. When dealing with NAPs optimization, one of the crucial aspects to take into account is the allocation of the observer(s), also known as measurement grid. As noise models like the INM are specially conceived for computing noise prints by means of a grid-based approach. The first intents implemented a grid of points, or “microphones”, where noise was measured [2,3]. Although this is one of the most accurate ways of esti-mating the noise print over an area, the computational cost of this kind of implementation is quite considerable.

Trying to minimize it, [21, 22] implemented a set of sensitive areas which borders were defined by a set of points (Figure 1.5(b)). By doing this, while ensuring that the noise constraints were fulfilled inside the sensitive areas, the number of points was considerably reduced as well as the computational cost. Other studies opt for a less accurate but yet more simple solution, such as placing one microphone for each town located near the airport [23] or just imposing a Noise Sensitive Area (NSA) which should not be overflown (Figure1.5(a)).

In cases where more than one point of measurement is used, a proper way of combining them must be achieved. As more than one objective has to be taken into account, the optimization problem falls into the multi-objective optimization problem class. First efforts, just evaluated the noise print as a whole, adding all the measures, and displaying the mean as the cost function [2]. Others applied different weightings to each of the measurements, basing the weighting in the density of the population or the number of awakenings [3,4].

Some studies added more complex algorithms in order to add equity to the cost function.

For instance, Ref. [21], applies what he calls the min-max approach: whereas most of the multi-objective cost functions present in the literature tried to minimize the whole impact of the trajectory, Prats centered his approach in reducing the noise impact there where it was worst, as he stated that this was the fairest way of addressing noise abatement procedures.

(a) NSA-based optimization [18] (b) Noise sensitive areas definition [5]

Figure 1.5: Different NAPs optimizations

1.2.2.3. Numerical Methods

According to [5], “the optimization of an aircraft trajectory, as a 4-dimensional continuum, is a constrained optimal control problem”. These kind of problems are not easy to solve, due to nonlinear functions appearing in the definition of the optimization objective and constraints. For this reason, there exist several techniques currently being implemented aiming to solve this optimization problem in the most efficient way possible.

First intents tried to solve this type of problems analytically, offering feasible solutions for small and simplified optimization problems [24]. Unfortunately, NAPs optimization does not fall into this category. That is why, as computers started to be more powerful, also numerical methods started to become the preferred solution in order to solve this kind of problems. A lot of different numerical methods can be found aiming to solve the problem stated before; Stochastic methods and gradient-based approaches, are within the most commonly found methods.

Stochastic methods incorporate random elements in the algorithm with the purpose of finding an optimal solution. Those kinds of methods can be divided into several different types, such as genetic and ant colony algorithms, which are inspired by biological and natural processes. Nevertheless, when an optimal solution is found those methods are not good proving that the corresponding solution is the global optimal and the computational burden is also considerable.

Therefore, gradient-based approaches will be chosen for this project. There are two main categories concerning gradient-based algorithms; direct shooting and direct collocation

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methods, where direct collocation methods have been elected as the approach imple-mented in this project.

Direct collocation methods consist on a discretization of the whole equations set defining the problem in order to transform the initial continuous optimal control problem into a set of discretized states and controls which with the different constraints imposed, define the set of variables and constraints of a new Non-Linear Programming (NLP) problem. Then, the remaining NLP problem can be solved using non-linear programming algorithms such as the ones implemented in the CONOPT solver, which will be introduced later.

However, it is worth mentioning that the main drawback of direct methods is the lack of cer-tainty when the problem to solve is non-convex since one cannot guarantee that a globally optimal solution is attained. Furthermore, the solution obtained will be very sensitive to the input parameters (guesses) given at the beginning of the optimization [5].

CHAPTER 2. MODEL

When performing the optimization of a noise abatement trajectory, it is important to define accurately the model describing the aircraft noise emission and its propagation, as well as, the aircraft dynamics. That being said, in this chapter, the adopted method for computing aircraft noise will be explained, in addition to a new technique for measuring the noise impact of an aircraft trajectory. Finally, the model describing the aircraft dynamics and performance will be introduced.