# A Comprehensive Survey on Climate Optimal Aircraft Trajectory Planning

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## Abstract

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_{2}) and other non-CO

_{2}effects, and the associated climate impact is expected to soar further. The mitigation of CO

_{2}contributions to the net climate impact can be achieved using novel propulsion, jet fuels, and continuous improvements of aircraft efficiency, whose solutions lack in immediacy. On the other hand, the climate impact associated with non-CO

_{2}emissions, being responsible for two-thirds of aviation radiative forcing, varies highly with geographic location, altitude, and time of the emission. Consequently, these effects can be reduced by planning proper climate-aware trajectories. To investigate these possibilities, this paper presents a survey on operational strategies proposed in the literature to mitigate aviation’s climate impact. These approaches are classified based on their methodology, climate metrics, reliability, and applicability. Drawing upon this analysis, future lines of research on this topic are delineated.

## 1. Introduction

_{2}) emissions and of non-CO

_{2}effects. The non-CO

_{2}effects comprise nitrogen oxide (NO

_{x}) emissions that are causing a concentration change in ozone and methane, water vapor emissions, aerosols, and persistent contrail and contrail-cirrus formation. CO

_{2}and non-CO

_{2}effects induce a change in the Earth’s radiation balance between incoming solar radiation and thermal outgoing radiation. This radiative imbalance is referred to as radiative forcing (RF). The overall global aviation RF of non-CO

_{2}effects is positive and thus warming [3]. Moreover, it is important to note that non-CO

_{2}emissions are responsible for two-thirds of the global aviation RF [1,4].

_{2}effects (it can also be beneficial to reduce emissions such as NO

_{x}and soot)), and the avoidance of climate-sensitive regions at an operation level (i.e., reducing non-CO

_{2}effects).

_{2}emissions, is linked to implementation time scales of comparable magnitude to the contemplated horizon, associated both with the availability of new products within the aeronautical industry and the life cycle of current aircraft. The same applies to alternative propulsion. The feasibility of electric aircraft (all-electric or hybrid) in particular relies on technological enablers such as an increase in battery-specific energy. The specific energy density of the battery is estimated to increase by a factor of four for all-electric flights up to 500 nautical miles with 180 passengers on board [6], a factor which can be however reduced by resorting to a hybrid solution. Considering initially a short-haul application of electric aircraft, another key consideration directly affecting the profitability of this technology is charging time. Additionally, one has to consider the carbon intensity of electric energy generation. In [7], a hypothetical trend for the specific energy of batteries suggests that the viability of short-haul narrow-body hybrid aircraft eventually resides on a shift towards renewable energy production, nowadays requiring investment. Another alternative in the spotlight is hydrogen-powered aircraft, which do not emit CO

_{2}but, in turn, emit more water vapor, which may lead to a higher formation of persistent contrails compared to kerosene-engine exhausts. To explore the mitigation potential of hydrogen aircraft, a better understanding of their climate impact due to contrail effects is required.

_{2}emissions, the consideration of which results in significantly lesser mitigation [9]. Nonetheless, while the use of alternative propulsion, jet fuels, and the continuous improvement of aircraft efficiency present themselves as necessary developments towards the achievement of sustainable aviation, these solutions lack immediacy. Aside from large implementation time scales, they require substantial investment in research, production, testing, and certification. Consequently, employing alternative solutions that can bridge this time horizon is crucial. Here is where climate-friendly aircraft operations can make a difference.

_{2}emissions, non-CO

_{2}effects highly depend on the geographical location, altitude, time of day, and current spatial and synoptic conditions. By taking into account the dependencies of non-CO

_{2}effects in the aircraft trajectory planning, operational mitigation towards climate optimized aircraft trajectories is possible. Thus, to consider the climate impact of aviation in the aircraft path planning, information on the climate-sensitive regions, i.e., regions where those non-CO

_{2}effects are significantly enhanced, needs to be available. Moreover, aircraft dynamical models and the optimization approach are crucial factors affecting the performance and mitigation potential of the optimized trajectories. Numerous studies on climate-optimized trajectories exist. However, these studies differ, e.g., in the optimization algorithm, incorporation of climate impacts as the objective function, number of flights, and maneuvres. That is why our study aims to provide a comprehensive review of the state-of-the-art studies for the past two decades, i.e., 2000–2021, considering these factors, e.g., aircraft dynamical model, climate metrics, and optimization approaches. Of course, there exist some valuable surveys. For example, there are two surveys on the methods adopted for aircraft trajectory optimization (ATO) [10,11], and there is another review on contrail avoidance techniques [12]. Additionally, climate mitigation strategies were partially reviewed in the study of Zou et al. in 2013 (online version) [13]. The most recent survey, considering the time period 2000–2018, was conducted on the mathematical optimization in enhancing the sustainability of aircraft trajectory [14]. Here, in addition to climate impacts, the authors also considered the effects of noise. However, the authors in [14] did not review the climate impacts of aviation in sufficient detail. For instance, only two studies regarding contrail mitigation strategies were reported, and also the focus was on optimal control and meta-heuristic as solution approaches. Nevertheless, at the time of publication of the above-mentioned survey, some additional papers had investigated the reduction of contrail induced climate impacts, and used different techniques, including path planning and mathematical programming, to mitigate climate impacts of non-$\mathrm{C}{\mathrm{O}}_{2}$ emissions. In our proposed survey, this review is conducted so that all the mentioned aspects regarding climate impacts of non-$\mathrm{C}{\mathrm{O}}_{2}$ emissions and solution approaches are addressed in detail. After reviewing recent studies, existing scientific gaps are identified to propose a future road map of eco-efficient flight planning.

## 2. Operational Mitigation Strategies for non-CO_{2} Climate Impact

_{2}RF effect is mainly caused by contrail cirrus and $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions lead both to the chemical destruction of methane and to the chemical formation of ozone; however, the resulting net $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ effect is positive, as the ozone formation dominates. Overall the RF of contrail cirrus, in spite of its large uncertainty, has the largest contribution to positive net RF. Water vapor and aerosol–radiation interactions from soot particles are estimated to have a slight impact.

#### Operational Mitigation Strategies

- How to integrate climate effects in aircraft trajectory planning?
- Which methods to generate optimized trajectories considering an objective function expanded by climate effects?

## 3. Trajectory Optimization

#### 3.1. Optimal Control Approach

#### 3.1.1. Interpretation of Aircraft Trajectory Optimization as Optimal Control Problem

#### Dynamical Model of Aircraft

#### Path and Boundary Constraints

#### Objective Function

- Direct emission: Some studies in the literature considered the emissions such as $\mathrm{C}{\mathrm{O}}_{2}$ and $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ in aircraft path planning [36]. Direct emission does not provide suitable insight into the climate impacts since, for instance, the effects of 1 kg emission of $\mathrm{C}{\mathrm{O}}_{2}$ on climate are highly different from that of $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ [33]. Instead, these emissions are usually inputted to other climate metrics to estimate the climate impacts caused by them. To calculate different emissions, the corresponding emission indices are required. There exist various approaches to calculate emission indices. Boeing fuel flow method 2 (BFFM2) is an extensively employed approach in the literature to calculate emission indices for $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$, CO, and HC [37,38].
- Generation of persistent contrails: A great majority of studies focused on avoiding areas that are sensitive to the formation of persistent contrails (e.g., [39,40]). In these so-called ice-supersaturated regions (ISSR), the formation of persistent contrail cirrus is possible [41]. In some cases, in addition to the ISSR, Schmidt–Appleman criterion (SAC), stating the formation of contrails at sufficiently low temperatures and sufficiently moist (relative to liquid water) environments is adopted [42,43,44]. In the literature, the consideration of both ISSR and SAC is called persistent contrail formation areas (PCFA) [13]. Within the numeric global climate model, such areas can also be identified by means of potential contrail cirrus coverage, a fraction of the grid box which contrails can maximally cover under the simulated atmospheric condition [29,45,46]. Such fractional representation is mainly due to the relatively large grid box size within climate models. Usually, time or distance flown in contrail-sensitive areas is defined as the objective to be minimized. In addition, the areas that are favorable for contrails are inputted to those metrics that quantify their corresponding climate impact, as the condition, determining the existence of the contrails.
- Radiative forcing: In some studies, RF has been used to quantify the climate impacts of aircraft emissions [47,48]. However, RF is not a direct measure of climate change. Instead, it measures the energy imbalance caused by changes in the Earth’s radiation balance between incoming solar radiation and thermal outgoing radiation. Such radiative impact has the potentiality to evolve the atmosphere temperature, estimated by RF [33].
- Global warming potential: One climate metric that allows comparing the climate impacts of all agents (i.e., greenhouse gases) is the global warming potential (GWP) [34,49]. GWP estimates how much energy (calculated using time-integrated RF) is absorbed for the emission of a trace gas compared to that of 1 kg carbon dioxide ($\mathrm{C}{\mathrm{O}}_{2}$) over a given period. Thus, the larger the GWP, the more a given gas warms the earth in relation to $\mathrm{C}{\mathrm{O}}_{2}$ over that period. Depending on the objective and application, some factors need to be considered to define suitable metrics. For instance, the emission scenario (e.g., pulse, sustained, or future emission scenario) and time horizon (e.g., 20 years or 100 years) are to be specified. The sustained emission scenario assumes constant emission of gas for the considered period, while pulse emission regards the emission of gas for one year and zero thereafter. The time period is specified with the selection of the time horizon. The studies [33,34,49] have discussed the selection of these factors based on different objectives. As an example, the pulse emission scenario for the time horizon of 20 years can be a suitable option for representing the short-term climate impacts. In contrast, 50 and 100 years time horizons can be used to capture medium-range and long-term climate impacts, respectively.
- Global temperature change potential: Unlike the GWP, estimating heat absorbed over a given period caused by a greenhouse gas emission, global temperature change potential (GTP) provides the temperature change at the end of the period [34]. This metric adapts a linear system for modeling the global temperature response to aviation emissions and contrails. In this metric, similar to the GWP, the changes are estimated compared to $\mathrm{C}{\mathrm{O}}_{2}$. For the GTP, the emission scenario and the time horizon need to be specified.
- Average temperature response: Another metric that measures the climate impact in terms of temperature change is average temperature response (ATR) [33,49]. ATR is a derivative metric of GTP which combines the integrated temperature change for different emission scenarios and time horizons. Some functions (called climate change functions) have been developed within the EU-projects REACT4C, ATM4E, and FlyATM4E to quantify the climate impacts of each agent in terms of ATR. In Section 4, the employment of such functions for climate optimal trajectory planning will be reviewed.

#### 3.1.2. Solution Approaches

#### 3.2. Non-Optimal Control Approach

## 4. Review of Climate Optimal Trajectory Planning Studies

#### 4.1. Simulation-Based Strategies

#### 4.2. Non-Optimal Control Methods

#### 4.2.1. Mathematical Programming

#### 4.2.2. Meta-Heuristics

^{2}less than the cost-optimal one. The authors in [46], using the EMAC model coupled with AirTraf 1.0 and a contrail submodel, optimized flight time and the flight distance through contrail forming regions modeled using potential contrail coverage (potcov) (see [45] for the calculation of potcov).

#### 4.2.3. Path Planning

#### 4.3. Optimal Control Methods

#### 4.3.1. Indirect Optimal Control

_{x}, and ${\mathrm{H}}_{2}\mathrm{O}$ and the avoidance of PCFA for contrails. The indirect optimal control approach was employed to solve the optimization problem. The fuel burn and climate impact of cross-polar air traffic flying different routing options such as great circle, wind-optimal, and contrail-avoidance were calculated for 15 origin–destination pairs. The trade-off between the formation of persistent contrails and additional GWP of aircraft emissions was also explored. It is worth mentioning that by altitude optimization in this section, we mean that the optimization is performed only considering lateral path and repeated for different altitudes, not simultaneously optimizing both lateral and vertical profiles. Lührs et al. in [85] employed a similar approach considering flight time and climate impacts as optimization objectives. The climate impacts were quantified using climate change functions (CCF) developed within EU-project REACT4C [68], estimating near-surface temperature changes in terms of ATR20. The CCFs were developed for $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emission, $\mathrm{C}{\mathrm{O}}_{2}$, contrails, and water vapor. In all the mentioned studies, the authors target the cruise phase of the aircraft at a constant altitude and velocity such that they adopt a 2D formulation, having latitude and longitude or the x-y coordinates of a horizontal plane respectively as state variables, and the heading angle as the only control. Additionally, Ref. [20] considers as well the evolution of the mass. However, the amount of simplifications in the dynamical model (constant velocity and altitude), as well as the use of the heading as control, results in the dynamics of the aircraft being represented by kinematic variables, such that the inclusion of a dynamical equation for the evolution of the mass of the aircraft is believed to be of limited contribution. In a sense, it seems not to introduce additional information into the system. Another effect of these simplifications is that it is generally not needed to introduce constraints, as altitude and velocity are kept fixed, and the control, heading, is an angle. However, solutions may be produced that require rates of change of heading which are not flyable.

#### 4.3.2. Direct Optimal Control

#### 4.4. Remarks

- The mitigation potential of non-$\mathrm{C}{\mathrm{O}}_{2}$ climate impacts using operational strategies is promising.
- The use of trajectory optimization techniques is increasing due to the capability to produce more efficient trajectories (in the sense of mitigation potential).
- Although the optimal control is known as one of the most reliable approaches to solve trajectory optimization problems (at least considering the free-route concept), due to some drawbacks with the implementation, they share almost the same portion with non-optimal control methods to the total trajectory optimization-based techniques.
- The focus has always been on conventional jet-powered aircraft.
- The mitigation strategies using ATO techniques reviewed in this paper were mostly performed on single flights.
- The aircraft performance, meteorological variables, and climate and cost indices were all considered deterministic in the reviewed studies.

## 5. Discussion and Challenges

#### 5.1. Objective Function: Physical Understanding and Predictability of Aviation Climate Impacts

#### 5.2. Aircraft Dynamics and Constraints: New Models for H_{2} and Hybrid Vehicles

_{2}driven propulsion, hybrid artifacts that may combine electric/H

_{2}driven engines with kerosene jet, and the use of sustainable aviation fuels (SAFs). In all cases, additional studies of emissions and their impact would be needed, together with new dynamical models to capture adequately the dynamical behavior of such systems.

_{2}-powered aircraft. According to the EU’s Horizon Europe and the EU’s Clean Aviation funding programs, hydrogen propelled aircraft are thought to play a leading role in what concerns environmentally friendly aeronautics [101]. Hydrogen can potentially overcome issues related to the low capacity (mainly specific energy) of current and forthcoming batteries technology. There are two main ways to use hydrogen as an energy provider; the first one leverages on fuel cells, devices that use the chemical energy of hydrogen (or other fuels) to cleanly and efficiently produce electricity; the second strategy would use hydrogen as propellant directly in the combustion chamber of the modified engines. In both cases, hydrogen needs to be stored in tanks. Even though hydrogen-propelled aircraft are referred to as zero-carbon aircraft, this does not mean their environmental impact is negligible: Their contribution to non-$\mathrm{C}{\mathrm{O}}_{2}$ emissions, especially when it comes to water vapor and the potential formation of linear and persistent contrails, can play an equally or even larger noxious role to the environment when compared to kerosene-engine exhausts.

_{2}-powered aircraft trajectories is missing in the literature. More efforts in the understanding of their emissions, the associated impact, and the modeling of the equations of motion and constraints are needed (e.g., see [102] for the study on the contrail coverage of hybrid-electric aircraft). When it comes to hybrid-powered vehicles, hybrid dynamical systems may be needed [103].

#### 5.3. Solution Approach: Development of Efficient Deterministic/Stochastic Dynamical Optimization Solvers

#### 5.4. Network-Scale Climate Optimal Trajectories

## 6. Conclusions

- Physical understanding and predictability of aviation climate impacts, particularly understanding and quantifying the uncertainties associated with climate science and meteorological forecast.
- Better understanding of H
_{2}- and hybrid-powered aircraft emissions, the associated climate impacts, and the modeling of the equations of motion. - Developing high-performance dynamical optimization solvers to generate robust eco-efficient trajectories with acceptable accuracy in a computationally fast manner.
- Identifying climatic hotspots and incorporating them into network-wide models and solution approaches for problems related to, e.g., demand and capacity balancing, network complexity, and resiliency.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ATO | Aircraft trajectory optimization |

ATM | Air traffic management |

OC | Optimal control |

TO | Trajectory optimization |

NTO | Non-trajectory optimization |

CSR | Climate sensitive region |

PCFA | Persistent contrail formation areas |

ISSR | Ice-supersaturated |

SAC | Schmidt–Appleman criteria |

CCF | Climate change function |

aCCF | Algorithmic climate change function |

eCCF | Emission-based climate change function |

GWP | Global warming potential |

GTP | Global temperature change potential |

AGTP | Absolute global temperature potential |

ATR | Average temperature response |

RF | Radiative forcing |

PMP | Pontryagin’s minimum principle |

HJB | Hamilton–Jacobi–Bellman |

SOC | Simple operating cost |

COC | Cash operating cost |

DOC | Direct operating cost |

NLP | Nonlinear programming |

2PBVP | Two-point boundary value problems |

DAE | Differential algebraic equations |

FACET | Future air traffic management concepts evaluation tool |

BFFM2 | Boeing fuel flow method 2 |

SQP | Successive quadratic programming |

AEDT | Aviation environmental design tool |

FAA | Federal aviation administration |

MIP | Mixed-integer programming |

MILP | Mixed-integer linear programming |

MIQP | Mixed-integer quadratic programming |

BIP | Binary integer programming |

TOMATO | Toolchain for multicriteria aircraft trajectory optimization |

MOTO | Multi-objective trajectory optimization |

GA | Genetic algorithm |

EMAC | ECHAM/MESSy atmospheric chemistry |

TOM | Trajectory optimization module |

IPT | Interior-point |

RAMS | Air traffic control mathematical simulator |

## Appendix A. Optimal Control Methodologies

#### Appendix A.1. Pontryagin’s Minimum Principle (PMP)

- Euler–Lagrange equations:$$\frac{\mathrm{d}{\mathbf{x}}^{*}}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial \lambda},\phantom{\rule{1.em}{0ex}}\frac{\mathrm{d}{\lambda}^{*}}{\mathrm{d}t}=-\frac{\partial \mathcal{H}}{\partial \mathbf{x}}$$
- Transversality conditions:$$\lambda \left({t}_{f}\right)={\left[\frac{\partial \mathcal{M}}{\partial \mathbf{x}}+{l}^{T}\frac{\partial \Psi}{\partial \mathbf{x}}\right]}_{t={t}_{f}}$$

#### Numerical Approach (Indirect Optimal Control)

#### Appendix A.2. Hamilton-Jacobi-Bellman (HJB)

#### Numerical Approach (Dynamic Programming)

#### Appendix A.3. Direct Optimal Control

`fmincon`function in MATLAB, KNITRO [117], DONLP2 [118], GAMS [119], and IPOPT [120] exist. After solving the NLP problem, an optimal decision vector will be obtained (${\Lambda}^{o}$), which is to be interpolated to be reconverted back to the continuous domain.

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**Figure 1.**Classification of methods employed in the literature to mitigate the climate impact of non-$\mathrm{C}{\mathrm{O}}_{2}$ emissions.

**Figure 2.**Different objectives usually considered within cost/climate optimal trajectory planning. Each block can be considered as an objective in the objective function.

**Figure 3.**A classification of the optimal control methods used to solve aircraft trajectory optimization problems.

**Figure 4.**Overview of investigated operational strategies for mitigating the climate impact of aviation.

**Figure 5.**Overview of investigated optimization-based strategies for mitigating climate impacts of aviation.

**Table 1.**Cost and climate metrics used in the literature for cost/climate optimal trajectory planning.

Cost Metrics | |
---|---|

Metric | Description |

SOC | Estimates cost with linear relation to the flight time and fuel consumption [31]. |

COC/DOC | Estimates cost considering other aspects in addition to the flight time and fuel consumption, including flight crew, cabin crew, landing fee, fuel, insurance, etc. [29,32]. |

Climate Metrics | |

Metric | Description |

RF | Measures energy imbalance caused by changes in the Earth’s radiation balance between incoming solar radiation and thermal outgoing radiation [33]. |

GWP | Measures how much energy is absorbed for the emission of a trace gas compared to that of equivalent $\mathrm{C}{\mathrm{O}}_{2}$ over a time horizon [34]. |

GTP | Measures global mean surface temperature change compared to an equivalent amount of $\mathrm{C}{\mathrm{O}}_{2}$ at the end of a time horizon [35]. |

ATR | Measures average temperature response over a time horizon (a derivative metric of GTP) [33]. |

**Table 2.**Non-optimal control strategies proposed in the literature for climate optimal trajectories.

Non-Optimal Control Approach | ||||||
---|---|---|---|---|---|---|

Work | Module/Model | Method | Climate Variables (Metric) | Objective Function | Maneuvers (DoF) | No. Flights |

Campbell et al. (2008) [71] | - | MILP/MIQP | Contrails (ISSR) | Contrail avoidance, fuel | 3D | Single |

Campbell et al. (2009) [19] | - | MILP | Contrails (ISSR) | Contrail avoidance, fuel | 3D | Single |

Wei et al. (2012) [72] | LP | Contrails (ISSR) | Contrail avoidance, fuel | 1D (Altitude) | Multiple | |

Campbell et al. (2013) [73] | - | MILP | Contrails (ISSR) | Contrail avoidance, fuel | 3D | Single |

Zou et al. (2014) [13] | - | IP | Contrails (SAC + ISSR → GWP${}^{*}$) | Costs due to fuel burn, crew, passenger travel time, CO _{2} emission, contrail formation | 4D with constant speed | Multiple (44) |

Celis et al. (2014) [36] | - | MADS and GA | NO_{x} emission | NO_{x} emission, time, fuel | 3D | Single |

Lim et al. (2015) [74] | MOTO | NLP (fmincon in MATLAB) | Contrails (ISSR → RF) | RF of contrails | 3D (2D + T) | Single |

Lim et al. (2016) [47] | MOTO | NLP (fmincon in MATLAB) | Contrails (ISSR → RF) | RF of contrails | 3D (2D + T) | Single |

Foster et al. (2016) [30] | TOMATO | A${}^{*}$ for lateral path | (Contrails (ISSR), NO_{x}, H_{2}O, CO_{2},CO, Black carbon, H _{2}SO_{4}) (GWP) | DOC, GWP${}_{total}$ | 3D | Single |

Rosenow et al. (2017) [75] | TOMATO | A${}^{*}$ for lateral path | (Contrails (ISSR), NO_{x}, H_{2}O, CO_{2},CO, Black carbon, H _{2}SO_{4}) (GWP) | DOC, GWP${}_{total}$ | 3D | Multiple (13,584) |

Lim et al. (2017) [48] | MOTO | NLP (fmincon in MATLAB) | Contrails (ISSR → RF) | RF of contrails, distance, time, RF of CO _{2} emission | 3D (2D + T) | Single |

Yin et al. (2018) [76] | EMAC/AirTraf 1.0 | GA | Short-term NO_{x} effects (aCCF → ATR20) | Cost (time and fuel), ATR20${}_{{O}_{3}}$ | 3D (with constant speed) | Multiple (85) |

Yin et al. (2018) [46] | EMAC/AirTraf 1.0 | GA | Contrails (potcov) | Time + contrail distance | 3D (with constant speed) | Multiple (103) |

Yamashita et al. (2020) [29] | EMAC/AirTraf 2.0 | GA | (Contrails (potcov), O_{3}, CH_{4}, H_{2}O, CO_{2})(aCCF → ATR _{20}) | Time, fuel, SOC, COC, NO_{x} emission,H _{2}O emission, contrail formation,ATR20${}_{total}$ | 3D (with constant speed) | Multiple (103) |

Rosenow et al. (2019) [32] | TOMATO | A${}^{*}$ for lateral path | (Contrails (ISSR), NO_{x}, H_{2}O, CO_{2},CO, Black carbon, H _{2}SO_{4}) (GWP) | DOC, GWP${}_{total}$ | 3D | Multiple (13,584) |

Yamashita et al. (2021) [31] | EMAC/AirTraf 2.0 | GA | (Contrails (potcov), O_{3}, CH_{4}, H_{2}O, CO_{2})aCCF → ATR20) | Time, fuel, COC, Contrail formation, ATR20${}_{total}$ | 3D (with constant speed) | Multiple (103) |

Optimal Control Approach | ||||||
---|---|---|---|---|---|---|

Work | Module/Model/Software | Method | Climate Variables (Metric) | Objective Function | Maneuvers (DoF) | No. Flights |

Sridhar et al. (2010) [20] | - | Indirect (shooting) | Contrails (ISSR) | Contrail avoidance, fuel, time | 2D | Multiple (24, 12 city pairs) |

Sridhar et al. (2010) [84] | - | Indirect (shooting) | Contrails (ISSR) | Contrail avoidance, fuel, time | 2D | Multiple (24, 12 city pairs) |

Ng et al. (2011) [51] | - | Indirect (shooting) | Contrails (SAC + ISSR), (CO_{2}, NO_{x}H _{2}O) (GWP) | Contrail avoidance, GWP of CO_{2},NO _{x}, H_{2}O, fuel, time | 2D | Multiple (15) |

Sridhar et al. (2011) [40] | - | Indirect (shooting) | Contrails (SAC + ISSR) | Contrail avoidance, fuel, time | 2D | Multiple (24, 12 city pairs) |

Soler et. al (2014) [39] | BONMIN (MINLP), IPOPT (NLP) | Mixed-integer optimal control | Contrails (SAC + ISSR → GWP${}^{*}$) | Costs due to fuel burn, crew, passenger travel time, CO _{2} emission, and contrail avoidance | 4D | Single |

Lührs et al. (2014) [85] | - | Indirect (shooting) | (Contrails, O_{3}, CH_{4}, H_{2}O,CO _{2}) (CCF → ATR20) | Time, ATR20_{total} | 2D | Single |

Hartjes et al. (2016) [86] | GPOPS (transcription) / SNOPT (NLP) | Direct (pseudospectral collocation, SQP for NLP) | Contrails (SAC + ISSR) | DOC, contrail avoidance | 3D | Single |

Lührs et al. (2016) [87] | TOM (GPOPS (transcription) / IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails, O_{3}, CH_{4}, H_{2}O,CO _{2}) (CCF → ATR20) | COC, ATR20_{total} | 2D/3D | Multiple (9) |

Niklaß et al. (2016) [88] | TOM (GPOPS (transcription) / IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails, O_{3}, CH_{4}, H_{2}O,CO _{2}) (eCCF → ATR100) | COC, ATR100_{total}(Climate-restricted airspace) | 4D | Single |

Matthes et al. (2017) [89] | TOM (GPOPS (transcription) / IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails, O_{3}, CH_{4}, H_{2}O,CO _{2}) (ECF → ATR(20,100)) | COC, ATR(20,100)_{total} | 4D | Multiple |

Niklaß et al. (2017) [16] | TOM (GPOPS (transcription)/ IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails, O_{3}, CH_{4}, H_{2}O,CO _{2}) (eCCF → ATR100) | COC, ATR100_{total}(Climate-restricted airspace) | 4D | Multiple (9) |

Lührs et al. (2021) [90] | TOM (GPOPS (transcription) / IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails (ISSR), O_{3}, CH_{4}, H_{2}O,CO _{2}) (aCCF → ATR20) | Fuel, ATR20_{total} | 4D | Multiple (13,000) |

Matthes et al. (2020) [91] | TOM (GPOPS (transcription) / IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails (ISSR), O_{3}, CH_{4}, H_{2}O,CO _{2}) (aCCF → ATR20) | Fuel, ATR20_{total} | 4D | Multiple (2000) |

Niklaß et al. (2021) [92] | TOM (GPOPS (transcription) / IPOPT (NLP)) | Direct (pseudospectral collocation, IPT for NLP) | (Contrails, O_{3}, CH_{4}, H_{2}O,CO _{2}) (eCCF → ATR100) | COC + ATR100_{total}(Climate-charged airspace) | 4D | Single |

Vitali et al. (2021) [93] | - | Direct (Chebyshev pseudospectral collocation) | (Contrails (ISSR), CO_{2}, NO_{x},H _{2}O, soot, SO_{2}) (GWP) | DOC, GWP${}_{\mathrm{total}}$(20, 50 and 100 years) | 4D | Single |

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Simorgh, A.; Soler, M.; González-Arribas, D.; Matthes, S.; Grewe, V.; Dietmüller, S.; Baumann, S.; Yamashita, H.; Yin, F.; Castino, F.; Linke, F.; Lührs, B.; Meuser, M.M. A Comprehensive Survey on Climate Optimal Aircraft Trajectory Planning. *Aerospace* **2022**, *9*, 146.
https://doi.org/10.3390/aerospace9030146

**AMA Style**

Simorgh A, Soler M, González-Arribas D, Matthes S, Grewe V, Dietmüller S, Baumann S, Yamashita H, Yin F, Castino F, Linke F, Lührs B, Meuser MM. A Comprehensive Survey on Climate Optimal Aircraft Trajectory Planning. *Aerospace*. 2022; 9(3):146.
https://doi.org/10.3390/aerospace9030146

**Chicago/Turabian Style**

Simorgh, Abolfazl, Manuel Soler, Daniel González-Arribas, Sigrun Matthes, Volker Grewe, Simone Dietmüller, Sabine Baumann, Hiroshi Yamashita, Feijia Yin, Federica Castino, Florian Linke, Benjamin Lührs, and Maximilian Mendiguchia Meuser. 2022. "A Comprehensive Survey on Climate Optimal Aircraft Trajectory Planning" *Aerospace* 9, no. 3: 146.
https://doi.org/10.3390/aerospace9030146