# Noise-Aware and Equitable Urban Air Traffic Management:
An Optimization Approach

Zhenyu Gao111Corresponding Author. Postdoctoral Fellow, Department of
Aerospace Engineering and Engineering Mechanics, Yue Yu222Postdoctoral Fellow,
Oden Institute for Computational Engineering and Sciences, Qinshuang
Wei333Postdoctoral Fellow, Oden Institute for Computational Engineering and
Sciences, Ufuk Topcu444Temple Foundation Endowed Professorship No. 1, and
John-Paul Clarke555Professor and Ernest Cockrell, Jr. Memorial Chair in
Engineering The University of Texas at Austin, Austin, TX 78712, United States

###### Abstract

Urban air mobility (UAM), a transformative concept for the transport of
passengers and cargo, faces several integration challenges in complex urban
environments. Community acceptance of aircraft noise is among the most
noticeable of these challenges when launching or scaling up a UAM system.
Properly managing community noise is fundamental to establishing a UAM system
that is environmentally and socially sustainable. In this work, we develop a
holistic and equitable approach to manage UAM air traffic and its community
noise impact in urban environments. The proposed approach is a hybrid approach
that considers a mix of different noise mitigation strategies, including
limiting the number of operations, cruising at higher altitudes, and ambient
noise masking. We tackle the problem through the lens of network system
control and formulate a multi-objective optimization model for managing
traffic flow in a multi-layer UAM network while concurrently pursuing demand
fulfillment, noise control, and energy saving. Further, we use a social
welfare function in the optimization model as the basis for the efficiency-
fairness trade-off in both demand fulfillment and noise control. We apply the
proposed approach to a comprehensive case study in the city of Austin and
perform design trade-offs through both visual and quantitative analyses.

###### keywords:

Urban Air Mobility , Noise , Air Traffic Management , Multi-Objective
Optimization , Fairness

††journal: Transportation Research Part C: Emerging Technologies

##  1 Introduction

Urban air mobility (UAM) is a game-changing concept set to transform urban
transportation by extending the current system into a third domain – the sky.
The advancement of UAM hinges on several key factors: novel aircraft
configurations, advanced battery technologies, strides in autonomy and
robotics, augmented air traffic management systems, and innovative business
models [1]. As envisioned, UAM will primarily operate electric vertical take-
off and landing (eVTOL) aircraft, hybrid-electric ultra-short takeoff and
landing (eSTOL) aircraft, and unmanned aerial vehicles (UAVs) at lower
altitudes within or traversing metropolitan areas. These operations will
support a range of urban services, including passenger mobility, cargo
delivery, infrastructure monitoring, public safety, and emergency response
[2]. Undoubtedly, UAM will revolutionize the transport of passengers and cargo
and promote public welfare, particularly in currently underserved local and
regional environments.

However, incorporating UAM into existing urban environments introduces a range
of intricate challenges. Drawing from the lessons of past urban helicopter
services and recent analyses of UAM’s operational constraints, it becomes
clear that community acceptance, system safety, and infrastructure are
critical obstacles during the implementation or scale-up of UAM systems [3, 4,
5]. To promote community/societal acceptance of UAM, solutions to mitigate its
community noise impact are of utmost importance [6, 5, 1]. Given that UAM
operates in closer proximity to urban residents, the range of noises generated
by UAM aircraft poses significant health risks, including psychological
discomfort, sleep disruption, and an increased risk of cardiovascular diseases
[7]. Therefore, modeling and mitigation of UAM noise are now at the forefront
of research in the aerospace community [8, 9, 10, 11, 12, 13].

Overall, there are many operational strategies for mitigating UAM’s community
noise impact, such as: (1) limiting the number of operations, (2) cruising at
higher altitudes, and (3) ambient noise masking which concentrates flights
over less noise-sensitive areas. Practical approaches to implementing these
noise mitigation strategies include air traffic flow management and flight
trajectory planning. In addition, UAM will operate as a network system,
wherein aerial vehicles navigating through flight corridors that link nodes
comprising vertiports and waypoints. On the control of UAM traffic flow in a
network system, existing studies [14, 15, 16, 17, 18, 19] have explored
various objectives, such as demand fulfillment, congestion relief, risk
management, traffic complexity reduction, and fairness. Of the three
identified strategies for UAM noise mitigation, most of the current research
has centered exclusively on one specific strategy and has not yet rigorously
incorporated the intriguing concept of ambient noise masking into a solution.
A more holistic solution would be to employ a hybrid approach and utilize a
mix of different noise mitigation strategies. This requires a more integrated
approach that manages UAM operations in the context of control of network
systems, which remains an unaddressed research gap in the literature.

In this study, we aim to develop an optimization approach that considers a mix
of all three noise mitigation strategies, with an emphasis on ambient noise
masking, to reduce UAM’s community noise impact in urban environment. We
consider a scenario where an urban area consists of many communities, and the
ambient noise level for each community is accessible through real-world data.
We then manage the UAM traffic flow within a multi-layer network over the city
to control UAM’s noise impact, while striving to accommodate more service
demand and consume less energy. Our goal is to ensure that the impact of UAM
noise is both (1) limited – not exceeding a maximum threshold above each
community’s ambient noise level, and (2) equitable – maintaining fairness in
the increased community noise levels. We summarize the four main contributions
of this work as follows:

  1. 1.

Formulating an optimization problem which leverages a hybrid approach to
manage UAM’s community noise impact. This multi-objective optimization
formulation assigns air traffic flow in a UAM network while concurrently
pursuing demand fulfillment, noise control, and energy saving. It can also
perform efficiency-fairness trade-off in both demand fulfillment and noise
control using a social welfare function.

  2. 2.

Proposing a numerical method that solves the optimization problem which has
nonsmooth and nonconvex functions in both its objectives and constraints. This
method replaces the nonsmooth functions with linear differentiable ones by
introducing auxiliary variables and iteratively approximates the nonconvex
function via the convex-concave procedure.

  3. 3.

Applying the proposed methodology to a real-world application case for the
city of Austin and demonstrating its efficacy. This case study includes real-
world geospatial and ambient noise data for 292 census tracts (communities) in
Austin and a multi-layer UAM network design that contains three cruising
altitudes, 19 vertiports, 270 links, and 62 origin-destination (O-D) pairs.

  4. 4.

Conducting design trade-offs in multiple aspects via visual and quantitative
analytics. These include (i) intuitive data visualizations, (ii) efficiency-
fairness trade-offs for demand fulfillment and noise increase, (iii) three-
dimensional trade-offs between the average demand fulfillment, noise increase,
and energy consumption, and (iv) detailed comparisons between various
candidate design schemes.

We organize the remainder of the paper as follows. In Section 2 we review the
literature in three relevant streams and identify research gaps. In Section 3
we introduce details of noise measures, aircraft noise modeling, and community
noise concerns. In Section 4 we describe the multi-layer Austin UAM network
for the case study. In Section 5 we formulate a network model and an
optimization model, and a procedure for solving the optimization model. In
Section 6 we present results of the case study via design visualizations and
quantitative comparisons. In Section 7 we discusses the limitations and
extensions of the study before Section 8 concludes the paper.

##  2 Background and Literature Review

###  2.1 UAM Operational Challenges

By elevating urban transportation into the sky, UAM will unlock numerous
benefits for the evolution of urban life and development. Yet its integration
into urban environments also presents a variety of complex challenges. In the
past, urban helicopter services struggled with challenges in community
acceptance, safety concerns, and financial viability [3]. Recently,
researchers from MIT [4, 6] deduced that community acceptance of aircraft
noise, availability of takeoff and landing area, and scalability of air
traffic control are the three most stringent constraints during the growth of
a UAM system. Similarly, an analysis by European researchers [5] identified
five areas of implementation and operational challenges for UAM: technology,
societal considerations, infrastructure, services, and policies. Every
thorough examination of UAM’s operational challenges places community/societal
acceptance high on the list of stringent obstacles.

Among the aspects crucial for societal acceptance, community noise stands out
as a particularly prominent factor. Study [6] pinpointed aircraft noise as the
only UAM operational constraint that has the highest severity level across all
development stages; study [5] listed noise reduction as a challenge
necessitating the most extended timeframe for resolution. Given the essential
need for UAM systems to be environmentally and socially sustainable,
addressing community noise has emerged as one of the top priorities.

###  2.2 UAM Noise Modeling and Mitigation

Efforts to model and mitigate UAM aircraft noise are now at the forefront of
research in the aerospace community. Precise modeling of aircraft noise is
essential for identifying effective noise mitigation solutions [20]. UAM
aircraft generate different types of noise, with the primary source being
rotor Blade-Vortex Interaction (BVI) noise generated from unsteady pressure
fluctuations on blades [21]. On the noise modeling approach, the current
practice predominantly depends on acoustic simulation due to the scarcity of
measurement data for novel eVTOL aircraft configurations. Researchers from
NASA [8, 9] leveraged advanced acoustic models to predict the noise patterns
of UAM aircraft and generated noise data for aviation noise analysis tools
[22]. Such results have informed several studies on UAM’s community noise
impact in urban environment [10, 11, 23]. Other studies conducted detailed
modeling of UAM’s noise footprints in complex urban environments, such as
those near vertiports [12] and 3D flight trajectories [13].

There are three main types of operational strategy for UAM noise mitigation:
(1) limiting the number of operations, (2) maintaining a greater distance from
objects, and (3) ambient noise masking which concentrates flights over less
noise-sensitive areas such as areas with large ambient noise levels. While the
first strategy focuses on limiting the number of operations/services, the
latter two strategies focus on adjusting flight trajectory. Noise-aware flight
trajectory planning has been an effective approach for noise reduction
regardless of aircraft technologies. Within commercial aviation, noise
abatement trajectories such as the continuous descent approach (CDA) [24] and
the noise abatement departure procedures (NADP) [25] have been extensively
studied. Similar ideas have also been applied to design flight trajectories
for helicopter [26] and UAM [13] operations. Most recently, a new paradigm [1]
was introduced for noise mitigation through urban airspace management, i.e.,
defining noise-sensitive no-fly zones for urban noise control.

###  2.3 UAM Network and Traffic Management

The future UAM system, when operational in urban and suburban environments, is
expected to function as a network system. Network design and air traffic
management are two crucial problems to ensure effective and sustainable UAM
operations. Both problems encompass multiple objectives in the design process
and have garnered interest from researchers across a range of fields such as
aerospace engineering, transportation, operations research, and urban
planning. Existing studies in the literature have explored UAM network design
and traffic flow management for various objectives, such as serving passenger
demand [14], minimizing ground traffic congestion [15], managing traffic
overflows due to physical and operational disruptions [16], incorporating the
multi-leg operational strategy [17], reducing air traffic complexity [18], and
serving the demands of urban communities in a fair manner [19].

In both air traffic management and the larger scope of operations management,
the importance of equity and fairness is on the rise. Through incorporating
fairness consideration, planners strive to equitably distribute both merits
(e.g., resources) and downsides (e.g., delays) of a system among all
participants. In the field of air transportation, the concept of fairness has
been integrated into optimization problems related to service distribution
[19], delay management [27, 28], airport slot allocation [29, 30], conflict
resolution [31], and separation management [32]. In other important fields
such as healthcare, fairness has also been considered in problems including
kidney transplantation [33] and healthcare facility location [34]. In future
system design problems involving multiple participants/parties, the
efficiency-fairness trade-off [35] is likely to emerge as a new norm.

###  2.4 Research Gaps

After an in-depth review of the literature in the three most pertinent
research streams, we have identified the following research gaps:

  1. 1.

Within the three UAM noise mitigation strategies, the majority of existing
studies have focused solely on the effects of operation volume and the
distance between aircraft and ground structures or buildings. The potential of
ambient noise masking in managing UAM noise remains an unexamined area so far.

  2. 2.

The existing approaches to noise-aware UAM operations design have solely
focused on a single noise mitigation strategy. To the best of our knowledge,
there has been no prior exploration into the use of a mixed approach of
different noise mitigation strategies.

  3. 3.

No previous research has investigated UAM noise management problem in the
context of control and optimization of network systems. Our effort marks the
first step toward combining UAM noise management with control of network
system.

To address these research gaps, in this work we consider a noise-aware UAM
traffic flow management problem in a multi-layer service network.
Specifically, we simultaneously consider a mix of all three noise mitigation
strategies, emphasizing on a novel strategy ambient noise masking, to control
community noise in the urban environment. In addition, we also consider
multiple trade-offs involving demand fulfillment, noise control, energy
consumption, and equity in various aspects.

##  3 UAM and Community Noise

The noise consideration in this work refers to the unwanted sound produced by
UAM aircraft or its components in flight. For the typical configurations of
UAM aircraft, the three primary sources of noise are engine noise (from both
propeller and mechanical components), aerodynamic noise, and aircraft systems
noise [36]. UAM noise can cause community annoyance, disrupt sleep, and could
elevate the risk of cardiovascular disease in urban residents [37]. Unlike
conventional aviation noise—which primarily affects people living in the
vicinity of airports—UAM noise could broadly impact most communities
throughout a city. In this section, we introduce the computation of UAM
aircraft noise, metrics for both single and cumulative noise events, and
aspects of community noise in urban environments.

###  3.1 Single Event and Cumulative Noise Measures

Before computing single event noise and the cumulative effect of multiple
noise events, we first introduce multiple fundamental concepts in noise and
discuss the differences among them. Decibel (dB) is the unit used to quantify
sound intensity. Decibels are measured on a logarithmic scale, reflecting how
our ears perceive sound pressures. A-weighted decibel (dBA) is the unit that
considers the way the human ear perceives different pitches or frequencies. It
emphasizes the portions of the frequency spectrum most sensitive to our
hearing and is used by the Federal Aviation Administration (FAA) to measure
aviation noise. Maximum sound level (Lmax) is the maximum sound level in the
duration of a particular noise event.

Figure 1: Single event aircraft noise: sound exposure level (SEL), maximum
noise level (Lmax), and duration

On the noise evaluation of a single event, sound exposure level (SEL) is the
most commonly used noise measure for an aircraft flyover. It captures both the
intensity and duration of a sound event in one numerical quantity and
compresses the event into a reference duration of one second:

$\text{SEL}=10\log_{10}\left(\frac{1}{T_{0}}\sum_{i=1}^{n}10^{L_{i}/10}\right)$
(1)

where $T_{0}=1s$ is the reference duration, $n$ is the number of seconds
during the measurement period, and $L_{i}$ is the sound level (in dBA) for the
i-th one-second time period. Figure 1 illustrates the computation of SEL for a
single event, where the measurement period covers the portion of the duration
when the sound level is within 10 dB below Lmax.

Equivalent sound level (Leq) measures the average acoustic energy during a
specific time frame to account for the cumulative effect of multiple noise
events, such as the aggregate sound level at a location with aircraft flyovers
throughout the day. Given the individual SEL values of various noise events,
Leq is obtained by:

$\text{Leq}=10\log_{10}\left(\sum_{i=1}^{n}10^{\text{SEL}_{i}/10}\right)-10\log_{10}\left(\frac{T}{T_{0}}\right)$
(2)

where $\text{SEL}_{i}$ is the SEL value for the i-th event, $n$ is the number
of events, $T$ is the time interval, and $T_{0}=1s$ is the SEL reference
duration. In the scope of this study, we are interested in the cumulative
noise levels over either 1 hour or 24 hours. When $T$ equals to 3,600 and
86,400 seconds respectively, we have:

$\text{Leq}_{1h}=10\log_{10}\left(\sum_{i=1}^{n}10^{\text{SEL}_{i}/10}\right)-35.56$
(3)
$\text{Leq}_{24h}=10\log_{10}\left(\sum_{i=1}^{n}10^{\text{SEL}_{i}/10}\right)-49.37$
(4)

Day-night average sound level (DNL) and community noise equivalent level
(CNEL) are both similar to Leq yet differ in how a noise event is treated
during evening and nighttime. To account for higher noise sensitivity levels,
DNL adds a 10 dBA “penalty” to each nighttime event (10 pm to 7 am); CNEL adds
an additional 4.77 dBA penalty to each evening event (7 pm to 10 pm). CNEL is
obtained by:

$\text{CNEL}=10\log_{10}\left(\sum_{i=1}^{n}10^{\text{SEL}_{i}/10}+\sum_{i=n+1}^{m}10^{(\text{SEL}_{i}+4.77)/10}+\sum_{i=m+1}^{r}10^{(\text{SEL}_{i}+10)/10}\right)-49.37$
(5)

where the three terms in the bracket represent day, evening, and night events,
respectively.

###  3.2 UAM Aircraft Noise Modeling

Aircraft noise modeling has been a crucial problem in aerospace engineering
and air transportation. The approach for noise computation depends on the
modeling range and terrain complexity. When modeling the noise exposure of an
aircraft operation in complex urban terrain, such as downtown areas, high-
fidelity methods such as acoustic ray tracing [1] are necessary to capture
complex acoustics phenomena such as atmospheric absorption, reflection,
diffraction, and scattering over the ground surfaces and buildings. On the
other hand, the modeling of aircraft noise on larger scales, such as community
noise exposure in urban and suburban areas, often employs more efficient and
generalized aircraft noise models, such as noise spheres and noise-power-
distance (NPD) data. For our study, NPD data is the favorable choice for
modeling the community noise of UAM in a city.

NPD data describes the relationship between the noise level of an aircraft and
its slant distance to the receiver. Currently, NPD data are available for
close to 300 fixed-wing aircraft and 26 helicopter types [38]. For
helicopters, the engine power variable in NPD data is replaced by operational
mode (i.e., departure, approach, level-fly) due to different type
characteristics. For each aircraft type, the NPD data come from either noise
certification tests or controlled tests executed under stringent international
standard procedures [39]. Separate sets of NPD data are available to account
for different operational modes and in-flight directivity (left, right, and
centre).

Since there is no existing NPD data for emerging UAM aircraft designs like
eVTOL aircraft, recent studies and models on UAM noise have mainly adopted two
approaches to estimate NPD data for potential UAM aircraft configurations. The
first approach utilizes existing NPD data for helicopters as basis and applies
noise reduction or scaled-down to account for the expectation or prediction
that eVTOL aircraft are quieter than existing helicopters. For example, a
recent work [10] started with the NPD data of the Robinson R44 helicopter and
applied two reduction levels on the baseline noise profile: 10 dBA and 15 dBA;
another work [40] selected the Bell 206 helicopter as a reference vehicle for
specifying a dBA offset. The second approach conducts acoustic simulation
which can capture the 3D directivity of noise near a UAM aircraft
configuration [9]. The simulation result is then processed and converted into
noise sphere or NPD data for modeling UAM’s community noise exposure.

(a) The NASA RVLT quadrotor reference vehicle

(b) The example noise hemispheres generated by simulation

(c) Noise simulation scenarios to obtain SEL NPD data for the three modes

Figure 2: The vehicle configuration considered in this study and noise
simulation methods (sources: [8, 22])

Researchers from NASA Langley Research Center [8, 22] have used acoustic
modeling tools and flight simulation to generate NPD data for novel UAM
aircraft configurations. In this study, we adopt the noise models for the NASA
Revolutionary Vertical Lift Technology (RVLT) quadrotor reference vehicle
[41], shown in Figure 2(a). This all-electric vehicle has four three-bladed
rotors, can carry up to six passengers, and can operate at a maximum airspeed
of 109 knots true airspeed (KTAS). The noise data of this vehicle
configuration is generated using ANOPP2’s Aeroacoustic ROtor Noise (AARON)
tool. Figure 2(b) shows examples of the noise hemispheres for the RVLT
quadrotor vehicle, which are then used in flight simulations to generate NPD
data for different operational modes. Figure 2(c) displays the simulation
scenarios for generating SEL NPD data for the three operational modes: level
flyover (L), and (dynamic) departure (D) and approach (A). For each
operational mode, measurements are collected along the centerline below the
track and at 45 degrees azimuth angles to capture lateral directivity. Figure
3 displays the resulting NPD data for the NASA RVLT quadrotor vehicle.

Figure 3: Simulated NPD data for the NASA RVLT quadrotor vehicle, under normal
(left) and log (right) distance scales

Because NPD data is only available at certain distance levels from 200 ft to
20,000 ft, practitioners often use logarithmic interpolation to estimate noise
levels at other distance levels. In this work, we fit regression models to the
NPD curves in Figure 3. For each operational mode and measurement position
combination, our regression model has the following form:

$L_{\text{AE}}(d)=a_{0}+a_{1}\log_{10}{d}+a_{2}(\log_{10}{d})^{2}$ (6)

where $L_{\text{AE}}$ is noise in A-weighted SEL and $d\in[200,20000]$ is
distance in ft. Figure 4 visualizes the goodness-of-fit for the six NPD
curves, where each regression model is plotted with the original NPD data.
Table 1 contains the regression coefficients for the six NPD curves in Figure
4.

Figure 4: Goodness-of-fit for NPD curves

Condition | $a_{0}$ | $a_{1}$ | $a_{2}$
---|---|---|---
Mode L - Centerline | 88.09 | 3.21 | -2.62
Mode L - Side | 78.01 | 7.26 | -3.39
Mode D - Centerline | 84.05 | 8.76 | -4.18
Mode D - Side | 77.34 | 11.34 | -4.72
Mode A - Centerline | 93.35 | 5.17 | -2.86
Mode A - Side | 85.55 | 6.83 | -3.14

Table 1: Regression coefficients for NPD curves

To better model the lateral sound patterns, we add two additional lateral
adjustments to the NPD data: (i) lateral attenuation adjustment, and (ii)
lateral directivity adjustment [40]. The lateral attenuation adjustment takes
into account ground reflection and refraction. First, we approximate the
general ground attenuation component using the following equation:

$E_{g}(l)=\begin{cases}11.83\cdot\left(1-e^{-0.0009\cdot
l}\right),&\text{if}~{}0\leq l\leq 3,000~{}\text{ft}\\\
10.86,&\text{if}~{}l>3,000~{}\text{ft}\end{cases}$ (7)

where $l$ is the lateral sideline distance (distance between the lateral
receiver and the centerline). Then, the refraction component can be computed
using the following equation:

$\Lambda(\beta)=\begin{cases}10.86,&\text{if}~{}\beta\leq 0\\\
1.137-0.0229\cdot\beta+9.72\cdot
e^{-0.142\cdot\beta},&\text{if}~{}0^{\circ}<\beta\leq 50^{\circ}\\\
0,&\text{if}~{}50^{\circ}<\beta\leq 90^{\circ}\end{cases}$ (8)

where $\beta$ is the elevation angle of the aircraft with respect to the
horizontal plane of the receiver. With these two components, the overall
lateral attenuation adjustment can be computed as:

$L_{\text{LA-ADJ}}(l,\beta)=\frac{E_{g}(l)\cdot\Lambda(\beta)}{10.86}$ (9)

The lateral directivity adjustment takes into account the change in
directivity through linear interpolation between the two NPD curves where
$\beta$ equals to 90∘ and 45∘, respectively. It can be computed as:

$L_{\text{LD-
ADJ}}(d,\beta)=\left(L_{\text{AE},90^{\circ}}(d)-L_{\text{AE},45^{\circ}}(d)\right)\cdot\left(\frac{90-|\beta|}{90-45}\right)$
(10)

Finally, with NPD curves and the two additional lateral adjustments, the SEL
noise that a flight operation poses to a receiver on the ground is calculated
using the following equation:

$L(d,l,\beta)=L_{\text{AE},90^{\circ}}(d)-L_{\text{LD-
ADJ}}(d,\beta)-L_{\text{LA-ADJ}}(l,\beta)$ (11)

###  3.3 Community Noise

Noise pollution in urban communities is a complex problem influenced by a
variety of sources and their variabilities. A rigorous treatment of this
problem, including the measurement, prediction, and control of community noise
requires a systematic approach. When operating UAM services in urban
environments, a crucial factor in noise control is recognizing that different
communities within the city exhibit varying levels of noise sensitivity. Urban
communities exhibit different noise acceptance levels due to two primary
factors: (1) land use, and (2) ambient noise. The FAA’s 14 CFR Part 150666FAA,
Airport Noise Compatibility Planning (14 CFR Part 150),
https://www.faa.gov/airports/environmental/airport_noise classifies land use
into five categories: residential, public use, commercial use, manufacturing
and production, and recreational. Within each category, there are 3 to 6
detailed sub-categories that may have various sensitivities to noise
intrusions. The ambient noise in an urban space refers to the background sound
produced by numerous sources in urban areas such as transportation,
construction, human activity, aircraft, and natural phenomena.

In this work, we consider ambient noise as the primary factor for determining
the noise acceptance level of a community in the city for two reasons. First,
even within a small area in a city, land use could be highly heterogeneous.
For example, a small downtown neighborhood could well be a mixture of
residential, public use, and commercial use sites; a suburban neighborhood
could contain industrial, commercial use, and public use lands that are
intertwined. Second, one of the most effective noise mitigation strategies is
“ambient noise masking”, i.e., concentrating flights over less noise-sensitive
areas such as areas with large traffic volume and industrial parks [6].

Table 2: The five categories of urban residential areas, their daytime noise levels in dBA, and the amounts of correction added to measured CNEL (source: [42]). Description | Daytime Median Noise Level in dBA |  Correction Added to Measured CNEL in dB
---|---|---
Typical Range | Average
Quiet Suburban | 41 to 45 | 43 | +10
Normal Suburban | 46 to 50 | 48 | +5
Urban | 51 to 55 | 53 | 0
Noisy Urban | 56 to 60 | 58 | -5
Very Noisy Urban | 61 to 65 | 63 | -10

We next look at the classification of urban communities based on ambient noise
and community reactions to noise levels above ambient noise. On the former,
Table 2 contains how the U.S. Environmental Protection Agency (EPA) [42]
classified urban residential areas into five categories, according to their
ambient noise levels. The five classes of urban communities in Table 2 are
widely adopted; the daytime median noise level in dBA is also representative
of a community’s ambient noise level. Considering the increased urban noise
levels since these standards were first made, we use the upper bounds of the
typical ranges in Table 2, namely, 45 dBA, 50 dBA, 55 dBA, 60 dBA, and 65 dBA
to represent the ambient noise levels of the five community classes. Table 3
displays the result of a related study that models the expected community
reaction as a function of noise exposure above the ambient noise level. We
estimate that when UAM’s cumulative noise is 10 dB above the ambient noise,
widespread complaints would happen; and the reaction becomes threats of legal
action when the difference increases to 20 dB. We will use information from
Table 2 and Table 3 as evidence to build evaluation functions for community
noise annoyance.

Table 3: Summary of expected community reaction and approximate annoyance as a function of CNEL and ambient noise levels (source: [42]). Expected Community Reaction |  | Difference Between
---
CNEL and Average Daytime
Median Noise Level in dB
Approximate % Very Much Annoyed |  Approximate % Little or Not Annoyed
Mean | Range of Data
No reaction | 2 | 0 to 8 | 10 | 45
Sporadic complaints | 6 | 3 to 8 | 5 | 37
Widespread complaints | 11 | 7 to 19 | 0 | 26
Threats of legal action | 21 | 18 to 24 | -5 | 14
Vigorous action | 28 | 23 to 34 | -10 | 7

In this study we choose the city of Austin as the use case. To partition an
urban area into communities, the two most common methods are zip code and
census tract. Zip code areas are determined based on factors including mail
volume, geographical features, and delivery routes. The city of Austin can be
partitioned into 46 zip code areas. Zip code is not a natural organization of
communities in every city and should be evaluated on a case-by-case basis.
Although zip code can provide a fine separation of communities in the city of
Austin, we choose census tract in this study for two reasons. First, census
tract provides partition at a more detailed level. Figure 5(a) displays the
partition of 292 census tracts in the city of Austin, where the position of
downtown Austin is marked by the red dot. Second, it is more convenient to
find ambient noise and population data for a census tract, as shown in Figure
5(b) and Figure 5(c). To obtain the ambient noise levels of the 292 Austin
communities, we refer to the open source data National Transportation Noise
Exposure Map [43], which contains the most recent noise levels modeled by the
Bureau of Transportation Statistics (BTS) for road, rail and aviation traffic.
We then utilize the noise exposure data to classify the communities into five
classes, following the guidelines in Table 2. In the community noise
classification result in Figure 5(c), a darker color indicates a higher
ambient noise level. Population is another critical factor in the evaluation
of aviation noise exposure. Figure 5(a) displays the 5-year (2016-2020) total
population estimates from the Census Bureau’s American Community Survey (ACS).

(a) The Austin communities and their total populations

(b) The Austin communities and their populations densities

(c) The Austin communities classified by ambient noise level

Figure 5: Partition and aspects of the 292 communities in the city of Austin

##  4 The Multi-layer UAM Network

We choose the city of Austin as the case study to demonstrate our overall
approach. Figure 6(a) displays a UAM network in Austin which contains 19
destinations (vertiports), 45 undirectional links (90 directional links) in
each layer, 62 origin-destination (O-D) pairs, and 292 communities. This UAM
network is an outcome of a systematic design that takes into account the
following factors: (1) the current Austin public transportation network, (2)
the future Austin Public Transit Plan777A New Transit Plan for Austin,
https://www.projectconnect.com/, (3) the current city traffic patterns, (4)
the locations of communities and city sub-centers, (5) the distribution of
city residents, and (6) features of urban topography. Each node in the network
represents an actual location within a city center/sub-center that is
geographically and societally reconfigurable into a vertiport. We select the
set of O-D pairs through a preliminary demand analysis that involves the
public transportation plans and the functions of communities surrounding each
node. The set of links represents flight corridors between vertiports and has
a simplified topological structure that has no intersection between the links
and can efficiently serve all O-D pairs, i.e., there are multiple routes for
each O-D pair, with the distance of each route being marginally longer than
the direct straight-line distance between the O-D pair.

(a) The Austin UAM service network

(b) The three-layer flight corridors

Figure 6: Schematic of the three-layer UAM network design for the city of
Austin

The concept of flight corridors, the 3D sections of airspace designated for
aerial vehicle operations [44], is classic and long-established in air
transportation. For both commercial aviation and UAM, these virtual highways
in the sky can provide navigation, reduce the complexity and increase
predictability of airspace, structure air traffic to enhance operational
safety and efficiency, and minimize the impact on other operations. As urban
air traffic management literature [45, 44, 18] and multiple UAM airspace
design concepts [46] have envisioned, flight corridors are integral parts of
the UAM system infrastructure.

Multi-layer network or system [47, 44, 48, 18] is another prevailing concept
in air transport system modeling and planning. In this work, a multi-layer
network has a significant implication – it provides flight corridors at
multiple altitudes such that UAM operations can mitigate community noise
impact through adjusting cruising altitude.

Figure 6(b) displays the three-layer UAM network in Austin, where the three
cruising altitudes are 1,000 ft, 2,000 ft, and 3,000 ft above ground level
(AGL), respectively. We use 500 ft mean sea level (MSL) as the reference
altitude for Austin and the resulting cruising altitudes in MSL are 1,500 ft,
2,500 ft, and 3,500 ft, respectively. The design of such a multi-layer network
and flight corridors in urban airspace must consider several factors. The
three most influential factors are urban topography (e.g., natural terrains,
buildings), controlled airspace around airport (e.g., Class B, C, D airspace),
and prohibited and restricted areas (e.g., restricted military area). In the
Austin scenario, the lowest level of Figure 6(b) can maintain a 250 ft
clearance distance from any objects on the ground. The Austin-Bergstrom
International Airport locates at the southeast corner of Figure 6(a), whose
Class C airspace intersects part of the UAM network. We assume that UAM
aircraft can operate within designated sections of the Austin airport’s Class
C airspace through ‘geofencing’, a method that establishes virtual boundaries
to separate UAM operations from those of commercial and general aviation. In
Section 5, we explore the management of UAM traffic flow within the network in
Figure 6(b).

##  5 Problem Formulation

###  5.1 Network Model

In this subsection we introduce the basic definitions of an air mobility
network, the additional considerations in this model, and the constraints in
the optimization problem.

####  5.1.1 Nodes, Links, Routes, and Communities

We use a directed graph $\mathcal{G}$ to model the air mobility network. The
network $\mathcal{G}$ consists of a set of nodes
$\mathcal{N}=\\{1,2,...,n_{n}\\}$, where each node corresponds to either a
vertiport (take-off/landing site on the ground) or a way point (intersection
of flight corridors in the air), and a set of links
$\mathcal{L}=\\{1,2,...,n_{l}\\}$, where each link corresponds to a flight
corridor. Within the set of nodes, we let $\mathcal{V}=\\{1,2,...,n_{v}\\}$
denote a set of vertiports in the network. Note that in a $k$-level air
mobility network, we have $n_{n}=k\cdot n_{v}$. A link is an ordered pair of
distinct nodes, where the first and second nodes of the link correspond to its
“tail” and “head”, respectively. We let $\mathcal{R}=\\{1,2,...,n_{r}\\}$
denote a set of routes in the network, where each route is a sequence of links
connected in a “head-to-tail” fashion. We let
$\mathcal{O}=\\{1,2,...,n_{o}\\}$ denote a set of O-D pairs which have demand
for air mobility services in the city. Lastly, we let
$\mathcal{C}=\\{1,2,...,n_{c}\\}$ denote a set of communities in the city.

####  5.1.2 Incidence Matrices

We use incidence matrices to encode the topology of network $\mathcal{G}$ and
the relationships between elements in the network. First, we use the node-link
incidence matrix $E\in\mathbb{R}^{n_{n}\times n_{l}}$ to describe the
incidence relationship between nodes and links. The entry $[E]_{ij}$ in matrix
$E$ associates node $i$ and link $j$ as follows:

$[E]_{ij}=\begin{cases}1,&\text{if node $i$ is the head of link $j$,}\\\
-1,&\text{if node $i$ is the tail of link $j$,}\\\
0,&\text{otherwise.}\end{cases}$ (12)

We then use the link-route incidence matrix $F\in\mathbb{R}^{n_{l}\times
n_{r}}$ to describe the incidence relationship between links and routes. The
entry $[F]_{ij}$ in matrix $F$ associates link $i$ and route $j$ as follows:

$[F]_{ij}=\begin{cases}1,&\text{if route $j$ contains link $i$,}\\\
0,&\text{otherwise.}\end{cases}$ (13)

We use the O-D pair-route incidence matrix $H\in\mathbb{R}^{n_{o}\times
n_{r}}$ to describe the incidence relationship between O-D pairs and routes.
The entry $[H]_{ij}$ in matrix $H$ associates O-D pair $i$ and route $j$ as
follows:

$[H]_{ij}=\begin{cases}1,&\text{if route $j$ serves the demand of O-D pair
$i$,}\\\ 0,&\text{otherwise.}\end{cases}$ (14)

Lastly, to formulate the vertiport capacity constraints later, we use the
destination-route incidence matrix $J\in\mathbb{R}^{n_{v}\times n_{r}}$ to
describe the incidence relationship between destinations and routes.
Specifically, the entry $[J]_{ij}$ in matrix $J$ associates destination $i$
and route $j$ as follows:

$[J]_{ij}=\begin{cases}1,&\text{if node $i$ is the destination of route
$j$,}\\\ 0,&\text{otherwise.}\end{cases}$ (15)

####  5.1.3 Network Flow Constraints

We introduce two sets of network flow constraints to describe the relation
between the traffic flow on each link, each route, and traverse each node. The
first set of network flow constraints ensures the flow balance conditions at
the nodes. They state the conservation of flow, i.e., the flow (total number
of flights) out of a node equals to the flow into a node at any time. This
flow conservation condition is strictly true at each way point. Although each
vertiport has a capacity (the physical space that can contain a certain number
of aircraft), we still apply the flow conservation condition because the
magnitude of capacity is relatively small compared to the number of flights in
the long run. We use a link-flow vector $\mathbb{y}\in\mathbb{R}^{n_{l}}$ to
denote the number of flights on different links per unit time, where the entry
$[\mathbb{y}]_{i}$ denotes the number of flights that fly through link $i$. We
then formulate the flow balance constraints as follows:

$E\mathbb{y}=\boldsymbol{0}_{n_{n}},\quad\mathbb{y}\geq\boldsymbol{0}_{n_{l}}$
(16)

where we let $\boldsymbol{0}_{n}$ denote the $n$-dimensional zero vector for
any $n\in\mathbb{N}$.

Next, we introduce the overlapping link constraints. In air mobility network
operations, multiple routes can share a common link. Therefore, the number of
flights assigned to each route is constrained by the overlapping links. We
further define a route-flow vector $\mathbb{z}\in\mathbb{R}^{n_{r}}$ where
each entry $[\mathbb{z}]_{i}$ denotes the number of flights that fly route
$i$. With the link-route incidence matrix $F$, we write the overlapping link
constraints as follows:

$F\mathbb{z}=\mathbb{y},\quad\mathbb{z}\geq\boldsymbol{0}_{n_{r}}$ (17)

####  5.1.4 Capacity Constraints

To ensure safe and feasible operations in the air mobility network, we must
also consider the capacity constraints at the links, vertiports, and way
points. For example, the throughput of a vertiport per unit of time is limited
by its feasible space and air traffic control conditions nearby. Therefore,
there exists an upper bound of the total inflow (number of arrival flights)
that each vertiport can take. We use $\mathbb{c}_{v}\in\mathbb{R}_{\geq
0}^{n_{v}}$ to denote the vertiport capacity vector, where the entry
$[\mathbb{c}_{v}]_{i}$ denotes the maximum inflow entering vertiport $i$ per
unit time. We formulate the vertiport capacity constraints as follows:

$J\mathbb{z}\leq(1-\epsilon)\mathbb{c}_{v}\,.$ (18)

where $\epsilon\in[0,1]$ is a tolerance parameter to control the safety
margin. Similarly, we use $\mathbb{c}_{l}\in\mathbb{R}_{\geq 0}^{n_{l}}$ to
denote the link capacity vector, where the entry $[\mathbb{c}_{l}]_{i}$
denotes the maximum number of flights link $i$ can take per unit time. We then
formulate the link capacity constraints as follows:

$\mathbb{y}\leq(1-\epsilon)\mathbb{c}_{l}\,.$ (19)

Lastly, we consider the way point capacity constraints. Here, we further
define a matrix $K\in\mathbb{R}^{n_{n}\times n_{l}}$ such that each entry in
$K$ is $[K]_{ij}=\max\\{[E]_{ij},0\\}$. We use
$\mathbb{c}_{w}\in\mathbb{R}_{\geq 0}^{n_{n}}$ to denote the way point
capacity vector, where the entry $[\mathbb{c}_{w}]_{i}$ denotes the maximum
inflow entering way point $i$ per unit time. We then formulate the way point
capacity constraints as follows:

$K\mathbb{y}\leq(1-\epsilon)\mathbb{c}_{w}\,.$ (20)

###  5.2 Noise Impact Matrix

The main objective of this work is to control the UAM system’s noise impact
through regulating the traffic flow in the links of the air mobility network
$\mathcal{G}$. Therefore, a core component in this process is a matrix that
describes the relationships between link traffic flow and its noise impacts on
the communities. We use matrix $N\in\mathbb{R}^{n_{l}\times n_{c}}$ to
describe this relationship, where the entry $[N]_{ij}$ denotes the magnitude
of single event noise (in SEL) that a flyover through link $i$ poses to
community $j$. We use the noise modeling approach described in Section 3.2 to
compute all entries in matrix $N$. This process also takes into account the
ambient noise level of a community, such that $[N]_{ij}=0$ if the magnitude of
single event noise from link $i$ to community $j$ is less than or equal to the
ambient noise level of community $j$. Hence, matrix $N$ is expected to be a
sparse matrix whose majority of the entries are zero.

With the noise impact matrix $N$ and link flow vector $\mathbb{y}$, we can now
compute the cumulative noise that a certain air traffic flow condition poses
to each community in the city. We then use $\mathbb{n}\in\mathbb{R}^{n_{c}}$
to denote the cumulative community noise result, where the entry
$[\mathbb{n}]_{j}$ denotes the cumulative noise at community $j$. According to
Equation (3), the $j$-th entry is

$[\mathbb{n}]_{j}=10\log_{10}\left(\sum_{i=1}^{n_{l}}y_{i}\cdot
10^{N_{ij}/10}\right)-10\log_{10}\left(\frac{T}{T_{0}}\right)\,.$ (21)

We then further define a processed matrix $M\in\mathbb{R}^{n_{l}\times n_{c}}$
such that each of its element $[M]_{ij}=10^{[N]_{ij}/10}$. The cumulative
community noise vector $\mathbb{n}$ is then computed as:

$\mathbb{n}=10\log_{10}\left(M^{\top}\mathbb{y}\right)-10\log_{10}(T/T_{0})\cdot\boldsymbol{1}_{n_{c}}\,.$
(22)

###  5.3 Energy Consumption

Cruising at a higher altitude serves as an effective method to reduce the UAM
noise impact on communities, yet this approach comes at a price. A prominent
price is the increased energy consumption. Climbing to and descending from
higher altitudes consumes more energy, potentially reducing the overall
efficiency of the UAM system. Since energy is among the crucial aspects of
sustainability, we also consider energy consumption in the trade-off study.
Section A includes a detailed analysis of the energy consumption of eVTOL
aircraft in a standard flight profile with five segments: vertical takeoff,
climb, cruise, descent, and vertical landing. The main takeaway is that, in
the range of UAM missions, flying at a higher altitude often results in
increased energy consumption. In our case study for the city of Austin,
depending on the route distance, cruising at 2,000 ft rather than 1,000 ft AGL
consumes 7.28% to 18.34% more energy; cruising at 3,000 ft rather than 1,000
ft AGL consumes 14.70% to 36.97% more energy.

To incorporate energy consumption in the optimization model, we use
$\mathbb{p}\in\mathbb{R}^{n_{r}}$ to denote the additional energy consumption
due to cruising at altitudes higher than 1,000 ft AGL, where the entry
$[\mathbb{p}]_{i}$ denotes the additional energy consumption of route $i$.
$[\mathbb{p}]_{i}=0$ if route $i$ is at 1,000 ft AGL. In our study, we use
“average additional energy consumption” $p_{a}$ below as the indicator of the
energy efficiency of the system:

$p_{a}\coloneqq\frac{\mathbb{p}^{\top}\mathbb{z}}{\boldsymbol{1}_{n_{r}}^{\top}\mathbb{z}}\,.$
(23)

where the denominator $\boldsymbol{1}_{n_{r}}^{\top}\mathbb{z}$ is the total
route flow in the network. In the optimization problem, one can incorporate
$p_{a}$ either in the objective function or in the constraints.

###  5.4 Fairness and Social Welfare Function

Incorporating fairness criteria into optimization models has become a topic of
rising interest. In the design and operation of UAM systems, fairness embodies
social equity, a key factor in fostering societal acceptance of UAM. Compared
to traditional optimization objectives such as cost and efficiency, fairness
presents a more diverse array of understanding and formulation possibilities.

Overall, formulating and solving an optimization model with fairness
consideration involves three steps: (1) select an appropriate
fairness/inequality criterion according to the context; (2) incorporate the
fairness criterion in a social welfare function (SWF), which serves as either
the objective function or a constraint in the optimization formulation; (3)
convert the optimization formulation into a computationally tractable
mathematical programming model.

Throughout this section, we let $\mathbb{x}$ denote the system design outcome
and $\mathbb{u}$ denote the associated ulitity outcome. We assume that
$\mathbb{x}$ yields a distribution of utilities
$\mathbb{u}=\\{u_{1},...,u_{n}\\}$ to parties $1,...,n$, which are determined
by a vector-valued utility function
$\mathbb{u}=U(\mathbb{x})=\left(U_{1}(\mathbb{x}),...,U_{n}(\mathbb{x})\right)$
[49]. A SWF $\mathcal{F}(\mathbb{u})$ aggregates the utility vector
$\mathbb{u}$ into a scalar value representing the desirability of utilities
$\mathbb{u}$. In this study, we favor SWFs that facilitate a trade-off between
efficiency and fairness. Such SWFs are designed to achieve a balance between
utilitarian (maximizing the total or average welfare across all parties –
system efficiency) and maximin (maximizing the welfare of the least advantaged
parties – system fairness) criteria. Here we discuss two SWFs that can achieve
this trade-off: fairness-threshold, and alpha-fairness.

This fairness-threshold SWF is initially utilitarian, but switches to maximin
criterion for certain groups when unfairness surpasses a predefined level. In
the 2-party setting, the fairness-threshold SWF is

$\mathcal{F}_{\Delta}(u_{1},u_{2})=\begin{cases}\min\\{u_{1},u_{2}\\}+\Delta,&\text{if}~{}|u_{1}-u_{2}|\geq\Delta\\\
\frac{1}{2}(u_{1}+u_{2}),&\text{otherwise}\end{cases}$ (24)

where parameter $\Delta$ is a pre-defined threshold that influences the
fairness features of the solution. A recent work [50] generalized the
fairness-threshold SWF to $n$ parties:

$\mathcal{F}_{\Delta}(\mathbb{u})=\Delta+\frac{1}{n}\sum_{i=1}^{n}\min\\{u_{i}-\Delta,u_{\min}\\}$
(25)

where $u_{\min}\coloneqq\min_{j}u_{j}$. In (25), the utilitarian criterion is
applied to parties whose utilities are within $\Delta$ of the lowest utility,
and that other parties whose utilities are above this level are not viewed as
deserving greater utility under the current distribution. To this end,
function $\mathcal{F}_{\Delta}(\mathbb{u})$ evaluates the sum of $n_{c}$
terms, one for each party. For the $i$-th party, if
$u_{i}-u_{\min}\leq\Delta$, this term takes the value $u_{i}$, otherwise it
takes the value $u_{\min}$.

Existing literature has extensively investigated the theoretical properties of
another popular option, the alpha-fairness SWF [51, 35, 33], which has the
form

$\mathcal{F}_{\alpha}(\mathbb{u})=\begin{cases}\displaystyle\sum_{i=1}^{n}\frac{1}{1-\alpha}u_{i}^{1-\alpha},&\text{if
$\alpha>0$, $\alpha\neq 1$}\\\
\displaystyle\sum_{i=1}^{n}\log(u_{i}),&\text{if $\alpha=1$}\end{cases}$ (26)

where $\alpha$ is a control parameter that can change the nature of the
function from utilitarian criterion ($\alpha=0$) to maximin criterion
($\alpha\rightarrow\infty$). While both fairness-threshold and alpha-fairness
SWFs introduce a parameter $p$ for efficiency-fairness trade-off, we mainly
apply the fairness-threshold SWF in Section 5.5 because it offers improved
interpretability of the objective function, since the parameter $\Delta$
carries significant implications for policy-making.

Inequality measure is another crucial element in formulating and assessing
fairness in a decision-making process. In scenarios where the objective is to
establish an egalitarian distribution for utilities $\mathbb{u}$, system
designers often achieve greater fairness by reducing inequality. Many
statistics are available for quantifying inequality. In this study, we employ
the Gini coefficient
$\text{Gini}(\mathbb{u})=(1/2n^{2}\bar{u})\sum_{i=1}^{n}\sum_{j=1}^{n}|u_{i}-u_{j}|$
as the primary inequality measure due to its broad acceptance in economics,
sociology, and engineering.

###  5.5 The Optimization Model

With the network model, operational constraints, noise impact computation,
energy consumption computation, and SWF, we formally formulate an optimization
model to manage UAM traffic flow within the three-layer network in Figure
6(b). We further assume that the overall demand for UAM services exceeds
supply due to the limited UAM resources and/or network throughput.

Table 4: The complete list of parameters and decision variables in the optimization model Network Parameters | $M\in\mathbb{R}^{n_{l}\times n_{c}}$ | The processed noise impact matrix
---|---|---
$E\in\mathbb{R}^{n_{n}\times n_{l}}$ | The node-link incidence matrix
$F\in\mathbb{R}^{n_{l}\times n_{r}}$ | The link-route incidence matrix
$H\in\mathbb{R}^{n_{o}\times n_{r}}$ | The O-D pair-route incidence matrix
$J\in\mathbb{R}^{n_{v}\times n_{r}}$ | The destination-route incidence matrix
$K\in\mathbb{R}^{n_{n}\times n_{l}}$ | The inflow indication matrix
$\mathbb{a}\in\mathbb{R}_{\geq 0}^{n_{c}}$ | A vector with community ambient noise levels
$\mathbb{e}\in\mathbb{R}_{\geq 0}^{n_{o}}$ | A vector with the estimated demands for all O-D pairs
$\mathbb{c}_{v}\in\mathbb{R}_{\geq 0}^{n_{v}}$ | The vertiport capacity vector
$\mathbb{c}_{l}\in\mathbb{R}_{\geq 0}^{n_{v}}$ | The link capacity vector
$\mathbb{c}_{w}\in\mathbb{R}_{\geq 0}^{n_{n}}$ | The way point capacity vector
$\mathbb{p}\in\mathbb{R}_{\geq 0}^{n_{r}}$ | The additional energy consumption vector
Design Parameters | $\Delta_{n,max}\in\mathbb{R}_{\geq 0}$ | The maximum allowable noise above the ambient level
$T\in\mathbb{R}_{\geq 0}$ | The time interval (in seconds)
$m_{u}\in\mathbb{R}_{\geq 0}$ | The upper bound of average increased noise
$p_{u}\in\mathbb{R}_{\geq 0}$ | The upper bound of additional energy consumption
Decision Variables | $\mathbb{y}\in\mathbb{R}^{n_{l}}$ | The link flow vector
$\mathbb{z}\in\mathbb{R}^{n_{r}}$ | The route flow vector
$\mathbb{d}\in\mathbb{R}^{n_{o}}$ | The demand fulfillment vector
$\mathbb{n}\in\mathbb{R}^{n_{c}}$ | The absolute community noise impact vector
$\mathbb{n}^{\prime}\in\mathbb{R}^{n_{c}}$ | The increased community noise impact vector

We introduce our proposed optimization model, which includes three core
components: decision variables, objective, and constraints. The overall
objective is to serve more demand while limiting community noise impact and
energy consumption. In addition, we seek fairness in two aspects: demand
fulfillment and community noise impact. On the former, we aim to achieve a
fair distribution of the percentage of demand satisfied across all O-D pairs.
On the latter, we aim to achieve a fair distribution of community noise
annoyance/reaction, which is mainly indicated by the noise increase beyond a
community’s ambient noise level (see Table 3), across all communities. Table 4
is a complete list of all parameters and decision variables in the
optimization model.

In the multi-objective optimization formulation (27), the objective function
is a convex combination of two SWFs for demand fulfillment $\mathbb{d}$ and
community noise increase $\mathbb{n}^{\prime}$ respectively.

$\displaystyle\underset{\mathbb{y},\mathbb{z},\mathbb{d},\mathbb{n},\mathbb{n}^{\prime}}{\mbox{maximize}}\quad$
$\displaystyle\omega\mathcal{F}_{\Delta_{1}}(\mathbb{d})+(1-\omega)\mathcal{F}_{\Delta_{2}}\left(\boldsymbol{1}_{n_{c}}-\frac{1}{\Delta_{n,max}}\mathbb{n}^{\prime}\right)$
(27a) subject to $\displaystyle
E\mathbb{y}=\boldsymbol{0}_{n_{n}},~{}F\mathbb{z}=\mathbb{y},$ (27b)
$\displaystyle
J\mathbb{z}\leq(1-\epsilon)\mathbb{c}_{v},~{}\mathbb{y}\leq(1-\epsilon)\mathbb{c}_{l},~{}K\mathbb{y}\leq(1-\epsilon)\mathbb{c}_{w},$
(27c) $\displaystyle\mathbb{d}=\text{diag}(\mathbb{e})^{-1}H\mathbb{z},$ (27d)
$\displaystyle\mathbb{n}=10\log_{10}\left(M^{\top}\mathbb{y}\right)-10\log_{10}(T/T_{0})\cdot\boldsymbol{1}_{n_{c}},$
(27e)
$\displaystyle\mathbb{n}^{\prime}=\max\\{\mathbb{n}-\mathbb{a},\boldsymbol{0}_{n_{c}}\\},$
(27f)
$\displaystyle\mathbb{p}^{\top}\mathbb{z}/\boldsymbol{1}_{n_{r}}^{\top}\mathbb{z}\leq
p_{u},$ (27g)
$\displaystyle\mathbb{d}\leq\boldsymbol{1}_{n_{o}},~{}\mathbb{n}^{\prime}\leq\Delta_{n,max}\cdot\boldsymbol{1}_{n_{c}},$
(27h)
$\displaystyle\mathbb{y}\geq\boldsymbol{0}_{n_{l}},~{}\mathbb{z}\geq\boldsymbol{0}_{n_{r}}.$
(27i)

where in the objective function (27a), a parameter $\omega\in[0,1]$ controls
the relative weights between the two objectives for design trade-offs. Both
objectives use the fairness-threshold SWF from (25).

Optimization model (27) encompasses the following sets of constraints. The
constraints in (27b), first introduced in (16) and (17), ensure the
conservation of traffic flow at nodes and links. The constraints in (27c),
first introduced in (18), (19), and (20), ensure the satisfaction of capacity
constraints at vertiports, links, and waypoints. The constraints in (27d)
define the vector of demand fulfillment ratio for all O-D pairs, denoted by
$\mathbb{d}$, where $\mathbb{e}$ is a vector with the estimated demands for
all O-D pairs. Each entry $[\mathbb{d}]_{i}$ in the range of $[0,1]$. The
constraints in (27e), first introduced in (22), capture the relation between
the cumulative community noise and link traffic flow. The constraints in (27f)
describe the relation between the cumulative community noise $\mathbb{n}$, the
community noise increase, denoted by vector $\mathbb{n}^{\prime}$, and the
community ambient noise levels denoted by vector $\mathbb{a}$ where the
entries $[\mathbb{a}]_{i}$ and $[\mathbb{n}^{\prime}]_{i}$ denote the ambient
noise level (in dB) and noise increase for community $i$, respectively. In
particular, the entry $[\mathbb{n}^{\prime}]_{i}$ denotes the magnitude to
which the cumulative UAM noise posed to the community $i$, $[\mathbb{n}]_{i}$,
is higher than the community’s ambient noise $[\mathbb{a}]_{i}$. Should the
cumulative UAM noise be less than the community’s ambient noise level, it
becomes imperceptible to the residents, resulting in
$[\mathbb{n}^{\prime}]_{i}=0$. The constraints in (27g), first introduced in
(23), upper-bound the additional energy consumption due to cruising at higher
altitudes. The constraints in (27h) upper bound constraints the demand
fulfillment ratio and the noise increase. Entries in $\mathbb{d}$ should not
exceed 1 as achieved supply should not exceed demand. In addition, for each
community, we set an upper limit on the increased noise level as the
limitation of community noise impact. In this study, we choose
$\Delta_{n,max}=25$ dB as the maximum allowable UAM noise above the ambient
level. Finally, The constraints in (27i) are nonnegativity constraints imposed
on decision variables $\mathbb{y}$ and $\mathbb{z}$, also from (16) and (17).

###  5.6 Solving the Optimization Problem

Optimization problem (27) is challenging to solve due to the nonsmooth and
nonconvex functions that appear in its objective and constraints. In
particular, the pointwise maximum function appeared in (27a) and the noise
increase function in (27f) are both piecewise linear and nonsmooth. On the
other hand, the logarithm function appeared in (27e) is nonlinear and
nonconvex, making optimization (27) a nonconvex nonlinear program not solvable
by any commercial linear program solvers.

We propose a numerical method that solves optimization (27) by iteratively
solving linear programs. This method replaces the nonsmooth functions in (27)
with linear differentiable ones by introducing auxiliary variables.
Furthermore, it iteratively approximates the logarithm function in (27) via
the _convex-concave procedure_ , an algorithm designed for a special class of
nonlinear programs with structured nonconvexity. This algorithm guarantees
convergence to local optimality and has empirically demonstrated rapid
convergence in a wide range of applications. For further details on the
convex-concave procedure, we refer the interested readers to [52] and
references therein.

We start with transforming the nonsmooth functions in (27) into linear ones
using additional auxiliary variables. For the noise increase function, we
introduce an auxiliary variable $\mathbb{w}\in\mathbb{R}^{n_{c}}$, and make
the following transformation

$\mathbb{n}^{\prime}=\mathbb{w}.$ (28)

In addition, we add the following constraints to (27)

$\mathbb{w}\geq\mathbb{n}-\mathbb{a},~{}\mathbb{w}\geq\mathbf{0}_{n_{c}}.$
(29)

For function $\mathcal{F}_{\Delta_{1}}(\mathbb{d})$, we introduce two
auxiliary variable $\mathbb{u}\in\mathbb{R}^{n_{o}}$ and
$d_{\min}\in\mathbb{R}$, and make the following transformation

$\mathcal{F}_{\Delta_{1}}(\mathbb{d})=\Delta_{1}+\frac{1}{n_{o}}\mathbf{1}_{n_{o}}^{\top}\mathbb{u}.$
(30)

In addition, we add the following constraints to (27)

$\mathbb{u}\leq\mathbb{d}-\Delta_{1}\mathbf{1}_{n_{o}},~{}\mathbb{u}\leq
d_{\min}\mathbf{1}_{n_{o}},~{}d_{\min}\mathbf{1}_{n_{o}}\leq\mathbb{d}.$ (31)

Similarly, for function
$\mathcal{F}_{\Delta_{2}}(\mathbf{1}_{n_{c}}-\frac{1}{\Delta_{n,\max}}\mathbb{n^{\prime}})$,
we introduce two auxiliary variable $\mathbb{v}\in\mathbb{R}^{n_{c}}$ and
$s_{\min}\in\mathbb{R}$, and make the following transformation

$\mathcal{F}_{\Delta_{2}}\left(\mathbf{1}_{n_{c}}-\frac{1}{\Delta_{n,\max}}\mathbb{n}^{\prime}\right)=\Delta_{2}+\frac{1}{n_{c}}\mathbf{1}_{n_{c}}^{\top}\mathbb{v}.$
(32)

In addition, we add the following constraints to (27)

$\mathbb{v}\leq\mathbf{1}_{n_{c}}-\frac{1}{\Delta_{n,\max}}\mathbb{n}^{\prime}-\Delta_{2}\mathbf{1}_{n_{c}},~{}\mathbb{v}\leq
s_{\min}\mathbf{1}_{n_{c}},~{}s_{\min}\mathbf{1}_{n_{c}}\leq\mathbf{1}_{n_{c}}-\frac{1}{\Delta_{n,\max}}\mathbb{n}^{\prime}.$
(33)

The idea of the above transformations is to replace pointwise maximum and
minimum functions with upper and lower bounds with additional auxiliary
variables, respectively. We note that these transformations are lossless since
the objective in (27) aim to maximize $\mathcal{F}_{\Delta}(\mathbb{d})$ and
$\mathcal{F}_{\Delta_{2}}(\mathbf{1}_{n_{c}}-\frac{1}{n,\max}\mathbb{n}^{\prime})$.

Next, we approximate the nonlinear nonconvex logarithmic function in (27) via
linearization. Given a linearization point $\hat{\mathbb{y}}$, we can make the
following approximation

$\displaystyle\mathbb{n}$
$\displaystyle=10\log_{10}\left(M^{\top}\mathbb{y}\right)-10\log_{10}(T/T_{0})\cdot\boldsymbol{1}_{n_{c}}$
(34) $\displaystyle\approx
A(\hat{\mathbb{y}})(\mathbb{y}-\hat{\mathbb{y}})+b(\hat{\mathbb{y}})$

where

$\displaystyle A(\hat{\mathbb{y}})$
$\displaystyle\coloneqq\frac{10}{\ln(10)}\mathop{\rm
diag}(M\hat{\mathbb{y}})^{-1}M,$ (35a) $\displaystyle b(\hat{\mathbb{y}})$
$\displaystyle\coloneqq
10\log_{10}\left(M^{\top}\hat{\mathbb{y}}\right)-10\log_{10}(T/T_{0})\cdot\boldsymbol{1}_{n_{c}}.$
(35b)

With the above steps, we approximate optimization (27) using the following
linear program:

$\displaystyle\underset{\begin{subarray}{c}\mathbb{y},\mathbb{z},\mathbb{d},\mathbb{w}\\\
\mathbb{u},d_{\min},\mathbb{v},s_{\min}\end{subarray}}{\mbox{maximize}}\quad$
$\displaystyle\frac{\omega}{n_{o}}\mathbf{1}_{n_{o}}^{\top}\mathbb{u}+\frac{1-\omega}{n_{c}}\mathbf{1}_{n_{c}}^{\top}\mathbb{v}$
(36a) subject to $\displaystyle
E\mathbb{y}=\boldsymbol{0}_{n_{n}},~{}F\mathbb{z}=\mathbb{y},~{}\mathbb{d}=\text{diag}(\mathbb{e})^{-1}H\mathbb{z},$
(36b) $\displaystyle
J\mathbb{z}\leq(1-\epsilon)\mathbb{c}_{v},~{}\mathbb{y}\leq(1-\epsilon)\mathbb{c}_{l},~{}K\mathbb{y}\leq(1-\epsilon)\mathbb{c}_{w},$
(36c) $\displaystyle\mathbb{w}\geq
A(\hat{\mathbb{y}})(\mathbb{y}-\hat{\mathbb{y}})+b(\hat{\mathbb{y}}),~{}\mathbb{w}\geq\mathbf{0}_{n_{c}},$
(36d) $\displaystyle\frac{1}{n_{c}}\mathbf{1}_{n_{c}}^{\top}\mathbb{w}\leq
m_{u},~{}\mathbb{w}\leq\Delta_{n,max}\cdot\boldsymbol{1}_{n_{c}},~{}\mathbb{p}^{\top}\mathbb{z}/\boldsymbol{1}_{n_{r}}^{\top}\mathbb{z}\leq
p_{u},~{}\mathbb{d}\leq\boldsymbol{1}_{n_{o}},$ (36e)
$\displaystyle\mathbb{u}\leq\mathbb{d}-\Delta_{1}\mathbf{1}_{n_{o}},~{}\mathbb{u}\leq
d_{\min}\cdot\mathbf{1}_{n_{o}},~{}d_{\min}\cdot\mathbf{1}_{n_{o}}\leq\mathbb{d},$
(36f)
$\displaystyle\mathbb{v}\leq\mathbf{1}_{n_{c}}-\frac{1}{\Delta_{n,\max}}\mathbb{w}-\Delta_{2}\cdot\mathbf{1}_{n_{c}},~{}\mathbb{v}\leq
s_{\min}\cdot\mathbf{1}_{n_{c}},~{}s_{\min}\cdot\mathbf{1}_{n_{c}}\leq\mathbb{w},$
(36g)
$\displaystyle\mathbb{y}\geq\boldsymbol{0}_{n_{l}},~{}\mathbb{z}\geq\boldsymbol{0}_{n_{r}}.$
(36h)

Notice that the quality of approximating (27) with (36) depends heavily on the
choice of $\hat{\mathbb{y}}$ in (35).

To solve optimization (27) using the approximating linear program in (36), we
propose the convex-concave procedure, an iterative method to solve difference-
of-convex programs via linearization [52]. We summarize this procedure in
Algorithm (1). The idea is to solve the linear program in (36), then use the
solution to construct a new linearization point and the corresponding linear
program, and finally repeat this procedure until convergence.

Algorithm 1 Convex-concave procedure

The parameters listed in Table 4, an initial linearization point
$\hat{\mathbb{y}}$, stopping tolerance $\epsilon$.

$f_{0}=-\infty$, $f_{1}=\infty$, $k=1$.

Compute $A(\hat{\mathbb{y}})$ and $b(\hat{\mathbb{y}})$ using (35).

while $|f^{k+1}-f^{k}|>\epsilon$ do

Solve the linear program in (36), obtain optimal solution
$(\mathbb{y}^{k},\mathbb{z}^{k},\mathbb{d}^{k},\mathbb{w}^{k},\mathbb{u}^{k},,d_{\min}^{k},\mathbb{v}^{k},,s_{\min}^{k})$
with optimal value $f^{k}$.

Compute $A(\hat{\mathbb{y}})$ and $b(\hat{\mathbb{y}})$ using (35) with
$\hat{\mathbb{y}}=\mathbb{y}^{k}$.

$k=k+1$

end while

$(\mathbb{y}^{k},\mathbb{z}^{k},\mathbb{d}^{k})$

##  6 Results

In this section we present and discuss detailed results of the case study of
the city of Austin. We consider an urban air traffic management problem for a
$T=3,600$ seconds (1 hour) time interval. We conduct trade-off studies between
the three aspects – demand fulfillment, noise control, and energy consumption,
with efficiency and fairness trade-off considerations in the first two
aspects. We first display some representative results via data visualizations
on flow and noise distributions, followed by quantitative comparisons using
multiple metrics.

Figure 7: The community noise reaction score function

###  6.1 Result Visualizations

To intuitively demonstrate the interactions between the three decision-making
aspects and the implications of fairness, we first use three groups of
visualizations, Figure 8(a), 8(b), and 8(c). The visualizations contain 8
design results in total, while each result consists of four plots. As in
Figure 8(a), the first plot depicts the air traffic flows within the three-
layer UAM network in Figure 6(b). We utilize color coding to indicate the flow
magnitude within each link so that one can observe the traffic flow intensity
and distribution in each case. The second plot depicts the overall community
noise exposure in 292 Austin communities caused by UAM operations, where a
progressive intensification in color shade signifies an escalation in noise
impact. The third plot depicts the community noise reaction, which is a
function of noise increase above a community’s ambient noise level.
Specifically, to quantify a community’s annoyance to UAM noise, we employ a
nonlinear and concave function in Figure 7 which reflects the study results in
Table 3. In the third plot, a darker hue indicates a higher level of community
annoyance. The fourth plot is a histogram that shows the distribution of the
percentage of demand fulfillment across the 62 O-D pairs.

(a) Maximizing demand fulfillment – utilitarian

(b) Maximizing demand fulfillment – egalitarian

(c) Low level of demand fulfillment – example

Figure 8: Visualizations of UAM traffic flow distribution and noise impact
under different demand fulfillment scenarios

Figure 8 displays the first group of visualization results which focuses on
different demand fulfillment scenarios. Figure 8(a) shows the result when
demand fulfillment is the only objective, i.e., $\omega=1$ in (27) and (36).
This is the result with the greatest overall demand fulfillment (utilitarian).
Through the first three plots in Figure 8(a), we can observe considerable
traffic flows in all three network layers, extensive noise impacts throughout
the city, and intense noise reactions in numerous communities. The last plot
in Figure 8(a) shows that under the utilitarian criterion, the distribution of
% demand fulfillment is unfair. While over 35 O-D pairs receive 100% demand
fulfillment, 10 O-D pairs receive no demand fulfillment at all. We then apply
an egalitarian criterion in the optimization of demand fulfillment and
generate the result in Figure 8(b). In Figure 8(b), we observe reduced traffic
flows in the network, as well as slightly reduced noise impacts and community
noise reactions. More importantly, the last plot of Figure 8(b) shows that
every O-D pair can now receive 45% to 60% demand fulfillment, which is
apparently fairer compared to the utilitarian case. Figure 8(c) displays a
scenario when the UAM system’s demand fulfillment is at a relatively low
level. We can see that traffic flows are light in the network, and that all
O-D pairs receive only 5% to 20% demand fulfillment. Consequently, we notice
very low noise impacts and minor noise reactions from communities.

(a) Minimizing noise impact under moderate demand fulfillment

(b) Minimizing energy consumption under moderate demand fulfillment

(c) Considering both noise impact and energy consumption under moderate demand
fulfillment

Figure 9: Visualizations of UAM traffic flow distribution and noise impact
under different trade-offs between noise impact and energy consumption

Figure 9 displays a group of visualization results that highlights the trade-
off between noise impact and energy consumption. Compared to results in Figure
8, we start to implement noise control such that the demand fulfillment
outcomes in Figure 9(a) to Figure 9(c) are all at a moderate and constant
level. Figure 9(a) shows the result when noise control takes a significant
role without constraint on energy consumption. Under this scenario, most air
traffic flows go through the highest layer (altitude) of the network,
resulting in excellent noise control effects yet high energy consumption due
to climbing to the top layer. Figure 9(b) shows a scenario contrary to Figure
9(a). In this scenario, we aim to provide a similar level of supply with
minimal energy consumption. As a result, all air traffic flows remain in the
bottom layer that is the closest to the ground surface. Unsurprisingly, this
worsens both the overall noise impact and the community noise reactions.
Figure 9(c) presents a trade-off between Figure 9(a) (noise control) and
Figure 9(b) (energy saving). In this scenario, we observe traffic flows in all
three network layers and a noise impact outcome that is in between the two
extreme cases in Figure 9(a) and Figure 9(b).

(a) Controlling noise impact – utilitarian

(b) Controlling noise impact – egalitarian

Figure 10: Visualizations of UAM traffic flow distribution and noise impact
under different noise impact fairness levels

Figure 10 displays the final group of visualization results centered around
noise fairness. Under approximately the same demand fulfillment level, Figure
10(a) shows the result when a utilitarian criterion is applied to noise
impact. Through the community noise reaction map, we observe that for some
communities at the center of the Austin area, the noise annoyance levels are
extremely high. This indicates that the current noise management scheme is
unfair to those communities. On the contrary, when we apply an egalitarian
criterion on community noise impact, we obtain a much more homogeneous
community noise reaction result in Figure 10(b). This noise fairness concept
is one of the core concepts in this study. Eventually, our objective is to
achieve fairness in both demand fulfillment (across 62 O-D pairs) and
community noise reaction (across 292 communities).

###  6.2 Design Trade-offs

Most sustainable system design tasks have multiple, and often conflicting,
objectives. Therefore, the design trade-off is the kernel of such topics. In
this study, we explore trade-offs of two distinct types. First, we conduct
efficiency-fairness trade-offs for design aspects that have multiple
parties/entities, including demand fulfillment and community noise reaction.
Second, we conduct a trade-off between the three optimization objectives:
demand fulfillment, noise control, and energy saving.

We first present the efficiency-fairness trade-off results for demand
fulfillment and noise impact. The fundamental principle of the efficiency-
fairness trade-off is the inability to concurrently achieve both system
efficiency (represented by total or average benefits) and fairness
(represented by equality or low variability). We employ Gini coefficient to
measure inequality, where a lower Gini coefficient is indicative of enhanced
equality or fairness. The left plot of Figure 11 shows the efficiency-fairness
trade-off result for demand fulfillment. In the ideal case, the goal is to
achieve a high average demand fulfillment along with a low Gini coefficient.
This ‘Ideal Corner’ is indicated with a red star on the plot. The left plot of
Figure 11 displays the efficiency-fairness trade-off curves for demand
fulfillment under four average noise increase levels. We first observe that
overall, the design points get closer to the ideal point at the cost of higher
average noise impacts. For every level of average noise increase, one cannot
achieve a lower Gini coefficient without compromising on average demand
fulfillment. Similarly, the right plot of Figure 11 shows the efficiency-
fairness trade-off result for community noise increase. Aside from the
unattainability of the ideal corner without concessions, an interesting
observation is that the four trade-off curves, corresponding to different
demand fulfillment levels, fall on a single line which captures the pronounced
correlation between noise impact and demand fulfillment. Overall, the
efficiency-fairness trade-off results in Figure 11 are logical and provide
insights on the price of fairness in this specific problem.

Figure 11: The efficiency-fairness trade-off for demand fulfillment (left) and
noise increase (right)

We then investigate the trade-off between the three factors in the
optimization problem: demand fulfillment, noise control, and energy saving.
Here, we fully explore the design space by running 2,500 cases through
optimization problem (35) under different parameter and constraint settings.
Each design point has three attributes: average demand fulfillment (%),
average noise increase (dB), and extra energy consumption (%). The left plot
Figure 12(a) shows the distribution of generated design points in a three-
dimensional space. An ideal design is expected to have high demand
fulfillment, low noise increase, and low extra energy consumption. While the
ideal corner (near $(0,0,80)$, the upper corner pointing outwards) is
unattainable, the distribution of points in the left plot Figure 12(a) forms a
three-dimensional Pareto surface for multi-objective decision making. The
right plot of Figure 12(a) is an overhead view of the three-dimensional Pareto
surface, where the magnitude of average demand fulfillment is indicated by
color coding and contours. Through the contours, we observe that average noise
increase is a dominating factor in determining average demand fulfillment,
especially when the latter is below 50%. When both average demand fulfillment
and average noise increase are high, energy consumption begins to play a role
in the three-dimensional trade-off.

(a) The three-dimensional trade-off between demand fulfillment, noise control,
and energy saving

(b) The eight chosen designs on the Pareto surface

Figure 12: The three-dimensional trade-off and design options

With the three-dimensional Pareto surface in the left plot of Figure 12(a), we
proceed to choose 8 candidate designs near the ideal corner for further
analyses and comparisons. Figure 12(b) shows the locations of the 8 chosen
designs on the Pareto surface and the contours. For each design, we further
generate a utilitarian result and an egalitarian result, simultaneously for
both demand fulfillment and noise increase. Figure 13 shows the boxplot
comparison between the 8 chosen designs. Each compact boxplot displays a
distribution’s median (black and white dot), first quartile, third quartile,
minimum, and maximum. In addition, metrics from all three objectives are
normalized to be within $[0,1]$. For each design, we display its properties on
five aspects: extra energy, demand fulfillment under utilitarian criterion
(U), demand fulfillment under egalitarian criterion (E), noise increase under
utilitarian criterion (U), noise increase under egalitarian criterion (E).
While extra energy is represented by a single value, the other four properties
are represented by distributions. Within each design, one can clearly observe
the efficiency-fairness trade-off through comparing between the utilitarian
and egalitarian results. When we compare across the 8 utilitarian designs,
Design 1 and Design 3 both have high demand fulfillment and moderate noise
increase, while Design 3 consumes more energy (15% to 10%). When we compare
across the 8 egalitarian designs, Design 1, Design 3, and Design 8 are good
candidates on demand fulfillment and noise increase, while Design 1 consumes
the least energy.

Figure 13: Boxplot comparison between the 8 chosen designs

In the final set of analyses, we employ radar chart to compare the 8 designs
from a slightly different angle. Figure 14 displays the two radar chart
results, one for utilitarian designs, and another for egalitarian designs.
Compared to Figure 13, we now convert the five metrics such that a larger
value is preferable for all five dimensions of Figure 14. The five new
dimensions are: demand fulfillment mean, demand fulfillment fairness, noise
mitigation mean, noise mitigation fairness, and energy saving. From the left
plot of Figure 14, we can see that Design 1 (red) is the most balanced design
among the utilitarian designs. Design 2 (yellow) and Design 3 (light green)
are two other potential options. Among the egalitarian designs in the right
plot of Figure 14, Design 1 (red), Design 2 (yellow) and Design 8 (magenta)
are potentially good options. When examining the differences between
utilitarian and egalitarian designs in their entirety, the most striking
observation is that in egalitarian designs, noise mitigation fairness is
improved significantly at the cost of considerable reduction in demand
fulfillment mean. Because of the multi-objective nature of most real-world
design and planning problems, no design scheme is high performing in all
dimensions. Among the 8 candidate designs, Design 1 (which is also at a
‘central’ location on the Pareto surface) is balanced in all dimensions. Other
designs can be very strong in a subset of the dimensions. For example, Design
8 has top performances in noise mitigation mean and fairness; Design 6 has top
performances in demand fulfillment mean and fairness. In a scenario where each
dimension is equally valued, from results in Figure 14 one could find a single
quantitative metric that represents the ‘total’ benefit of a design scheme.

Figure 14: Radar chart comparison between the 8 chosen designs

##  7 Remarks

While the framework proposed is holistic in addressing various elements of
urban air traffic management, it retains certain limitations that necessitate
continued exploration and research. In this section, we briefly discuss three
future avenues that will continue to advance urban air traffic control and
noise management.

First, while the existing framework incorporates three key aspects–demand
fulfillment, noise control, and energy consumption–there are additional
perspectives in traffic flow management that could also be taken into account.
For example, traffic complexity is one such perspective. The concern of
traffic complexity arises as the demands in air traffic networks increase and
challenge the ability of the system to maintain safe operations under various
conditions [53]. Existing studies on air traffic complexity take airspace
structures, traffic flow, and operational constraints into account, and
compute the complexity metrics based on linear [45, 18] or nonlinear dynamical
systems [54]. Incorporating traffic complexity into the problem formulation
can further improve the overall performance of air traffic management.

The second future avenue is the optimization of flight corridors. The current
network structure in Figure 6 considers aspects such as system efficiency,
safety, and navigation. Nevertheless, to maximize the potential of ambient
noise making, there is room for additional optimization in the layout of
flight corridors. The fundamental approach is to reconfigure the flight
corridors to align more directly with the city’s ambient noise patterns.
Furthermore, introducing variabilities in network structures across different
layers could be beneficial. The current three-layer network has the same
network structure in all three layers. Adapting the network structure to be
more flexible and variable could continuously enhance the system in various
areas, including noise mitigation, safety, and efficiency.

The third avenue for future exploration involves incorporating additional
considerations of societal impact. Aside from community noise impact, other
societal impacts of UAM include third-party risk, perceived risk, privacy,
etc. Through a combination of physics-based and data-driven modeling, one can
also develop those societal impact maps for a city. By doing so, a more
thorough spectrum of societal impact insights in complex urban environments
can be achieved, enhancing UAM’s societal acceptance through projects like
network design, trajectory planning, and traffic flow management.

##  8 Conclusion

To promote the societal acceptance of UAM and mitigate UAM’s community noise
impact, this paper presents a holistic optimization approach for noise-aware
and equitable urban air traffic management. The overall approach concurrently
considers a mix of different noise mitigation strategies (reducing operation
volume, cruising at higher altitudes, ambient noise masking), conducts trade-
off between three main design objectives (demand fulfillment, noise control,
energy saving), and performs efficiency-fairness trade-off in aspects that
involve multiple parties (demand fulfillment and noise control). Furthermore,
it utilizes a city’s ambient noise data to fully explore ambient noise
masking, a less examined strategy in the literature. The optimization problem
leverages social welfare function in objectives and seeks to optimally assign
air traffic flow in a multi-layer network over a city. A numerical method
involving convex-concave procedure is proposed to solve the optimization
problem. A comprehensive case study using the city of Austin demonstrates the
proposed approach’s benefits in managing UAM’s community noise impact in
desired ways and performing various design trade-offs. Enhancements to the
current approach can be achieved by integrating additional air traffic
management aspects, optimizing flight corridors, and considering more societal
impacts.

## Acknowledgements

This work was partially supported by the National Aeronautics and Space
Administration (NASA) University Leadership Initiative (ULI) program under
project “Autonomous Aerial Cargo Operations at Scale”, via grant number
80NSSC21M071 to the University of Texas at Austin. Any opinions, findings,
conclusions, or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the project sponsor.

## References

  * Gao et al. [2023] Z. Gao, A. Porcayo, J.-P. Clarke,  Developing virtual acoustic terrain for urban air mobility trajectory planning,  Transportation Research Part D: Transport and Environment 120 (2023) 103794\.
  * Cohen et al. [2021] A. P. Cohen, S. A. Shaheen, E. M. Farrar,  Urban air mobility: History, ecosystem, market potential, and challenges,  IEEE Transactions on Intelligent Transportation Systems 22 (2021) 6074–6087.
  * Vascik et al. [2018] P. D. Vascik, R. J. Hansman, N. S. Dunn,  Analysis of urban air mobility operational constraints,  Journal of Air Transportation 26 (2018) 133–146.
  * Vascik and Hansman [2017] P. D. Vascik, R. J. Hansman,  Constraint identification in on-demand mobility for aviation through an exploratory case study of los angeles,  in: 17th AIAA Aviation Technology, Integration, and Operations Conference, p. 3083.
  * Pons-Prats et al. [2022] J. Pons-Prats, T. Živojinović, J. Kuljanin,  On the understanding of the current status of urban air mobility development and its future prospects: Commuting in a flying vehicle as a new paradigm,  Transportation Research Part E: Logistics and Transportation Review 166 (2022) 102868\.
  * Vascik and Hansman [2017] P. D. Vascik, R. J. Hansman,  Evaluation of key operational constraints affecting on-demand mobility for aviation in the los angeles basin: ground infrastructure, air traffic control and noise,  in: 17th AIAA Aviation Technology, Integration, and Operations Conference, p. 3084.
  * Gao et al. [2022] Z. Gao, S. I. Kampezidou, A. Behere, T. G. Puranik, D. Rajaram, D. N. Mavris,  Multi-level aircraft feature representation and selection for aviation environmental impact analysis,  Transportation Research Part C: Emerging Technologies 143 (2022) 103824\.
  * Rizzi et al. [2022] S. A. Rizzi, S. J. Letica, D. D. Boyd, L. V. Lopes,  Prediction-based approaches for generation of noise-power-distance data with application to urban air mobility vehicles,  in: 28th AIAA/CEAS Aeroacoustics 2022 Conference, p. 2839.
  * Rizzi et al. [2021] S. A. Rizzi, J. A. Page, R. Cheng,  Comparison of two community noise models applied to a nasa urban air mobility concept vehicle,  in: 50th International Congress and Exposition on Noise Control Engineering (Inter-Noise 2021).
  * Rimjha et al. [2021] M. Rimjha, A. Trani, S. Hotle,  Urban air mobility: Preliminary noise analysis of commuter operations,  in: AIAA Aviation 2021 Forum, p. 3204\.
  * Ng [2022] H. K. Ng,  Noise impact analysis for urban air mobility in dallas-fort worth metroplex,  in: AIAA AVIATION 2022 Forum, p. 3404\.
  * Yunus et al. [2023] F. Yunus, D. Casalino, F. Avallone, D. Ragni,  Efficient prediction of urban air mobility noise in a vertiport environment,  Aerospace Science and Technology 139 (2023) 108410.
  * Bian et al. [2021] H. Bian, Q. Tan, S. Zhong, X. Zhang,  Assessment of uam and drone noise impact on the environment based on virtual flights,  Aerospace Science and Technology 118 (2021) 106996.
  * Kai et al. [2022] W. Kai, A. Jacquillat, V. Vaze,  Vertiport planning for urban aerial mobility: An adaptive discretization approach,  Manufacturing & Service Operations Management 24 (2022) 3215–3235.
  * Yu et al. [2023] Y. Yu, M. Wang, M. Mesbahi, U. Topcu,  Vertiport selection in hybrid air-ground transportation networks via mathematical programs with equilibrium constraints,  IEEE Transactions on Control of Network Systems (2023) 1–12.
  * Wei et al. [2023] Q. Wei, Z. Gao, J.-P. Clarke, U. Topcu,  Risk-aware urban air mobility network design with overflow redundancy,  arXiv preprint arXiv:2306.05581 (2023).
  * Willey and Salmon [2021] L. C. Willey, J. L. Salmon,  A method for urban air mobility network design using hub location and subgraph isomorphism,  Transportation Research Part C: Emerging Technologies 125 (2021) 102997\.
  * Wang et al. [2022] Z. Wang, D. Delahaye, J.-L. Farges, S. Alam,  Complexity optimal air traffic assignment in multi-layer transport network for urban air mobility operations,  Transportation Research Part C: Emerging Technologies 142 (2022) 103776\.
  * Yu et al. [2023] Y. Yu, Z. Gao, S. H. Li, Q. Wei, J.-P. Clarke, U. Topcu,  Alpha-fair routing in urban air mobility with risk-aware constraints,  arXiv preprint arXiv:2310.00135 (2023).
  * Gao et al. [2022] Z. Gao, T. G. Puranik, D. N. Mavris,  Probabilistic representatives mining (prem): A clustering method for distributional data reduction,  AIAA Journal 60 (2022) 2580–2596.
  * Yu [2000] Y. H. Yu,  Rotor blade–vortex interaction noise,  Progress in Aerospace Sciences 36 (2000) 97–115.
  * Rizzi and Rafaelof [2023] S. A. Rizzi, M. Rafaelof,  On the modeling of uam aircraft community noise in aedt helicopter mode,  in: AIAA AVIATION 2023 Forum, p. 3363\.
  * Rizzi and Rafaelof [2022] S. A. Rizzi, M. Rafaelof,  Second generation uam community noise assessment using the faa aviation environmental design tool,  in: AIAA SCITECH 2022 Forum, p. 2167\.
  * Clarke et al. [2004] J.-P. B. Clarke, N. T. Ho, L. Ren, J. A. Brown, K. R. Elmer, K.-O. Tong, J. K. Wat,  Continuous descent approach: Design and flight test for louisville international airport,  Journal of Aircraft 41 (2004) 1054–1066.
  * Lim et al. [2020] D. Lim, A. Behere, Y.-C. D. Jin, Y. Li, M. Kirby, Z. Gao, D. N. Mavris,  Improved noise abatement departure procedure modeling for aviation environmental impact assessment,  in: AIAA Scitech 2020 Forum.
  * Greenwood [2017] E. Greenwood,  Helicopter flight procedures for community noise reduction,  in: American Helicopter Society Annual Forum, NF1676L-25361.
  * Montlaur and Delgado [2020] A. Montlaur, L. Delgado,  Flight and passenger efficiency-fairness trade-off for atfm delay assignment,  Journal of Air Transport Management 83 (2020) 101758.
  * Chin et al. [2021] C. Chin, K. Gopalakrishnan, H. Balakrishnan, M. Egorov, A. Evans,  Efficient and fair traffic flow management for on-demand air mobility,  CEAS Aeronautical Journal (2021) 1–11.
  * Ribeiro et al. [2018] N. A. Ribeiro, A. Jacquillat, A. P. Antunes, A. R. Odoni, J. P. Pita,  An optimization approach for airport slot allocation under iata guidelines,  Transportation Research Part B: Methodological 112 (2018) 132–156.
  * Zografos and Jiang [2019] K. G. Zografos, Y. Jiang,  A bi-objective efficiency-fairness model for scheduling slots at congested airports,  Transportation Research Part C: Emerging Technologies 102 (2019) 336–350.
  * Barnhart et al. [2012] C. Barnhart, D. Bertsimas, C. Caramanis, D. Fearing,  Equitable and efficient coordination in traffic flow management,  Transportation science 46 (2012) 262–280.
  * Guo et al. [2022] Y. Guo, M. Hu, B. Zou, M. Hansen, Y. Zhang, H. Xie,  Air traffic flow management integrating separation management and ground holding: An efficiency-equity bi-objective perspective,  Transportation Research Part B: Methodological 155 (2022) 394–423.
  * Bertsimas et al. [2013] D. Bertsimas, V. F. Farias, N. Trichakis,  Fairness, efficiency, and flexibility in organ allocation for kidney transplantation,  Operations Research 61 (2013) 73–87.
  * Shehadeh and Snyder [2023] K. S. Shehadeh, L. V. Snyder,  Equity in stochastic healthcare facility location,  in: Uncertainty in Facility Location Problems, Springer, 2023, pp. 303–334.
  * Bertsimas et al. [2012] D. Bertsimas, V. F. Farias, N. Trichakis,  On the efficiency-fairness trade-off,  Management Science 58 (2012) 2234–2250.
  * Gao and Mavris [2022] Z. Gao, D. N. Mavris,  Statistics and machine learning in aviation environmental impact analysis: A survey of recent progress,  Aerospace 9 (2022) 750.
  * Basner et al. [2017] M. Basner, C. Clark, A. Hansell, J. I. Hileman, S. Janssen, K. Shepherd, V. Sparrow,  Aviation noise impacts: state of the science,  Noise & health 19 (2017) 41.
  * John A. Volpe National Transportation Systems Center [2022] John A. Volpe National Transportation Systems Center, Noise Modeling Methods for Urban Air Mobility Vehicles in the Federal Aviation Administration’s Aviation Environmental Design Tool, Technical Report DOT-VNTSC-NASA-22-02, U.S. Department of Transportation, 2022.
  * Gao et al. [2022] Z. Gao, Y. Li, T. G. Puranik, D. N. Mavris,  Minimax and multi-criteria selection of representative model portfolios for complex systems analysis,  AIAA Journal 60 (2022) 1505–1521.
  * Woodcock et al. [2022] J. Woodcock, S. Hasan, J. Garcia Paje, R. Biziorek, P. Henning, A. Guthrie, V. Jurdic, A. L. Maldonado, D. Hiller,  Development of a noise-based route optioneering tool for advanced air mobility (aam) vehicles,  in: 28th AIAA/CEAS Aeroacoustics 2022 Conference, p. 2836.
  * Silva et al. [2018] C. Silva, W. R. Johnson, E. Solis, M. D. Patterson, K. R. Antcliff,  Vtol urban air mobility concept vehicles for technology development,  in: 2018 Aviation Technology, Integration, and Operations Conference, p. 3847.
  * USEPA [1971] USEPA, Community Noise, Technical Report NTID300.3, U.S. Environmental Protection Agency (EPA), 1971.
  * Seto and Huang [2023] E. Seto, C.-H. Huang,  The national transportation noise exposure map,  medRxiv (2023) 2023–02.
  * Muna et al. [2021] S. I. Muna, S. Mukherjee, K. Namuduri, M. Compere, M. I. Akbas, P. Molnár, R. Subramanian,  Air corridors: Concept, design, simulation, and rules of engagement,  Sensors 21 (2021) 7536.
  * Wang et al. [2021] Z. Wang, D. Delahaye, J.-L. Farges, S. Alam,  Air traffic assignment for intensive urban air mobility operations,  Journal of Aerospace Information Systems 18 (2021) 860–875.
  * Bauranov and Rakas [2021] A. Bauranov, J. Rakas,  Designing airspace for urban air mobility: A review of concepts and approaches,  Progress in Aerospace Sciences 125 (2021) 100726.
  * Zhao et al. [2019] M. Zhao, X. Wang, W. Pan, X. Zhang,  The vulnerability analysis of the multi-layer air transport system,  in: 2019 5th International Conference on Transportation Information and Safety (ICTIS), IEEE, pp. 956–962.
  * Rodriguez-Deniz et al. [2022] H. Rodriguez-Deniz, M. Villani, A. Voltes-Dorta,  A multilayered block network model to forecast large dynamic transportation graphs: An application to us air transport,  Transportation Research Part C: Emerging Technologies 137 (2022) 103556\.
  * Xinying Chen and Hooker [2023] V. Xinying Chen, J. Hooker,  A guide to formulating fairness in an optimization model,  Annals of Operations Research (2023) 1–39.
  * Elçi et al. [2022] Ö. Elçi, J. Hooker, P. Zhang, Structural properties of equitable and efficient distributions, Technical Report, Technical report, Carnegie Mellon University, submitted, 2022\.
  * Lan et al. [2010] T. Lan, D. Kao, M. Chiang, A. Sabharwal,  An axiomatic theory of fairness in network resource allocation,  in: 2010 Proceedings IEEE INFOCOM, pp. 1–9.
  * Lipp and Boyd [2016] T. Lipp, S. Boyd,  Variations and extension of the convex–concave procedure,  Optimization and Engineering 17 (2016) 263–287.
  * Histon et al. [2002] J. M. Histon, R. J. Hansman, G. Aigoin, D. Delahaye, S. Puechmorel,  Introducing structural considerations into complexity metrics,  Air Traffic Control Quarterly 10 (2002) 115–130.
  * Delahaye et al. [2003] D. Delahaye, S. Puechmorel, J. Hansman, J. Histon,  Air traffic complexity based on non linear dynamical systems,  in: ATM 2003, 5th USA/Europe Air Traffic Management Research and Development Seminar, pp. pp–xxxx.
  * Kasliwal et al. [2019] A. Kasliwal, N. J. Furbush, J. H. Gawron, J. R. McBride, T. J. Wallington, R. D. De Kleine, H. C. Kim, G. A. Keoleian,  Role of flying cars in sustainable mobility,  Nature communications 10 (2019) 1555.
  * Johnson [2012] W. Johnson, Helicopter theory, Courier Corporation, 2012.
  * Stoll et al. [2014] A. M. Stoll, E. V. Stilson, J. Bevirt, P. P. Pei,  Conceptual design of the joby s2 electric vtol pav,  in: 14th AIAA Aviation Technology, Integration, and Operations Conference, p. 2407.
  * Stoll and Bevirt [2022] A. Stoll, J. Bevirt,  Development of evtol aircraft for urban air mobility at joby aviation,  in: The Vertical Flight Society’s 78th Annual Forum FortWorth, TX, USA.

##  Appendix A Energy Consumption of eVTOL Aircraft

The analysis of energy/power consumption during a UAM mission starts with a
standard flight profile, depicted in Figure 15(a). Typically, there are five
segments in a UAM flight profile: vertical takeoff, climb, cruise, descent,
and vertical landing. The total energy consumption of the mission is the
summation of energy consumptions in all five segments. Below we analyze the
energy consumption model for each segment.

Both the vertical takeoff and landing segments can be approximated as hover,
which is the most energetically intensive segment of the flight profile [55].
Using the momentum theory for helicopter [56], we compute the hover power of
an eVTOL aircraft with:

$P_{\text{hover}}=\frac{mg}{\eta_{h}}\sqrt{\frac{\delta}{2\rho}}\,.$ (37)

where $m$ (kg) is the aircraft takeoff mass, $g=9.81~{}\text{m}/\text{s}^{2}$
is the acceleration due to gravity, $\eta_{h}$ is the hover system efficiency,
$\delta$ is the disk loading, and $\rho$ is the air density. Based on
estimated data for a tilt-propeller configuration eVTOL aircraft [57, 58], we
assume $\delta=580~{}\text{N}/\text{m}^{2}$, $m=1800~{}\text{kg}$,
$\eta_{h}=0.7$, and have:

$P_{\text{hover}}=\frac{1800~{}\text{kg}\cdot
9.81~{}\text{m}/\text{s}^{2}}{0.75}\sqrt{\frac{580~{}\text{N}/\text{m}^{2}}{2\cdot
1.225~{}\text{kg}/\text{m}^{3}}}=362.3~{}\text{kW}\,.$ (38)

The AGL altitude where vertical takeoff ends (and vertical landing starts)
depends on the environment and topography surrounding the vertiport. Here we
assume that for both hover segments, the aircraft takes 30 seconds to
climb/descent 250 ft (76 m). The energy consumption for vertical takeoff and
landing is

$E_{\text{hover}}=P_{\text{hover}}\cdot
t_{\text{hover}}=362.3~{}\text{kW}\cdot 60~{}\text{s}=21.7~{}\text{MJ}$ (39)

which is constant for missions at all three altitudes in this study.

Next, we consider the climb segment. We assume a small flight path angle
($\gamma=10~{}\text{deg}$) and that weight is approximately equal to lift
during this process. We have:

$P_{\text{climb}}=\frac{TV_{\text{climb}}}{\eta_{c}}=\frac{V_{\text{climb}}}{\eta_{c}}(mg\sin{\gamma}+D)$
(40)

where $\eta_{c}$ is the climb system efficiency, and the drag $D$ is

$\displaystyle D=\frac{1}{2}\rho
V_{\text{climb}}^{2}S\left(C_{D_{0}}+KC_{L}^{2}\right)$
$\displaystyle=\frac{1}{2}\rho
V_{\text{climb}}^{2}S\left(C_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{(1/2\rho
V_{\text{climb}}^{2}S)^{2}}\right)$ (41) $\displaystyle=\frac{1}{2}\rho
V_{\text{climb}}^{2}SC_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{1/2\rho
V_{\text{climb}}^{2}S}$

where $C_{D_{0}}$ is the coefficient for skin friction and form drags of the
aircraft (zero-lift drag coefficient), $S$ is the reference area, and
$(L/D)_{\text{max}}$ is the maximum lift-to-drag ratio. Then, we have the
climb power of an eVTOL aircraft:

$P_{\text{climb}}=\frac{V_{\text{climb}}}{\eta_{c}}\left(mg\sin{\gamma}+\frac{1}{2}\rho
V_{\text{climb}}^{2}SC_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{1/2\rho
V_{\text{climb}}^{2}S}\right)\,.$ (42)

Note that when climbing from a low altitude near the ground surface to the
cruising altitude, the air density $\rho$ is not a constant and decreases with
altitude. However, when the altitude is below 10,000 ft MSL, a linear model
can accurately represent the change in air density. Consequently, we use the
mid-point air density $\rho_{\text{mid-h}}$ to represent the air density
during climb. For example, when the cruising altitude is $h=$ 2,000 ft MSL,
$\rho_{\text{mid-h}}$ is the air density at $(2000-250)/2+250=1125$ ft. In
this work, we assume that the climb rate is ROC = 1,000 ft/min.

For the cruise segment, the power is modeled in a similar way as:

$P_{\text{cruise}}=\frac{TV_{\text{cruise}}}{\eta_{r}}=\frac{V_{\text{cruise}}}{\eta_{r}}D=\frac{V_{\text{cruise}}}{\eta_{r}}\left(\frac{1}{2}\rho
V_{\text{cruise}}^{2}SC_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{1/2\rho
V_{\text{cruise}}^{2}S}\right)$ (43)

where $\eta_{r}$ is the cruise system efficiency. Lastly, for the descent
segment, we assume that the engine power is reduced to 40% of the power
required for the cruise segment. In this work, we assume that $\eta_{c}=0.75$,
$\eta_{r}=0.8$, $(L/D)_{\text{max}}=20$, $C_{D_{0}}=0.03$,
$S=30~{}\text{m}^{2}$. With $V_{\text{climb}}=1000/(\sin(10~{}\text{deg})\cdot
60)=96~{}\text{ft/s}$, $V_{\text{cruise}}=135~{}\text{ft/s}$, we have the
following power estimations:

  * •

At cruising altitude 1,000 ft:

$P_{\text{climb}}=154.1~{}\text{kW},\quad
P_{\text{cruise}}=40.8~{}\text{kW},\quad P_{\text{descent}}=16.3~{}\text{kW}$
(44)

  * •

At cruising altitude 2,000 ft:

$P_{\text{climb}}=154.0~{}\text{kW},\quad
P_{\text{cruise}}=39.9~{}\text{kW},\quad P_{\text{descent}}=16.0~{}\text{kW}$
(45)

  * •

At cruising altitude 3,000 ft:

$P_{\text{climb}}=154.0~{}\text{kW},\quad
P_{\text{cruise}}=39.0~{}\text{kW},\quad P_{\text{descent}}=15.6~{}\text{kW}$
(46)

(a) The eVTOL aircraft flight profile

(b) Distribution of Extra Energy Consumption

Figure 15: Energy consumption information

We then calculate the energy consumption for the 4 segments. The energy
consumption for climb is

$\displaystyle E_{\text{climb}}$ $\displaystyle=P_{\text{climb}}\cdot
t_{\text{climb}}$ (47)
$\displaystyle=\frac{V_{\text{climb}}}{\eta_{c}}\left(mg\sin{\gamma}+\frac{1}{2}\rho_{\text{mid-h}}V_{\text{climb}}^{2}SC_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{1/2\rho_{\text{mid-h}}V_{\text{climb}}^{2}S}\right)\frac{(h-250)}{\text{ROC}}$

For the cruise segment, we need to first calculate the actual cruise distance.
During the climb and descent segments, since the rate of climb/descent is
1,000 ft/min, the horizontal speed is $1000/\tan(10~{}\deg)=5671$ ft/min =
94.5 ft/s. Therefore, the distance traveled for climb and descent combined is

$d_{\text{climb+descent}}=94.5~{}\text{ft/s}\cdot\frac{(h-250)\cdot
60}{1000}s\cdot 2=11.34(h-250)~{}\text{ft}$ (48)

The remaining distance for cruise is

$d_{\text{cruise}}=d_{\text{total}}-d_{\text{climb+descent}}$ (49)

Therefore, the energy consumption for cruise is

$E_{\text{cruise}}=P_{\text{cruise}}\cdot
t_{\text{cruise}}=\frac{V_{\text{cruise}}}{\eta_{r}}\left(\frac{1}{2}\rho
V_{\text{cruise}}^{2}SC_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{1/2\rho
V_{\text{cruise}}^{2}S}\right)\frac{d_{\text{cruise}}}{V_{\text{cruise}}}$
(50)

Lastly, the energy consumption for descent is

$E_{\text{descent}}=0.4P_{\text{cruise}}\cdot
t_{\text{descent}}=\frac{0.4V_{\text{cruise}}}{\eta_{r}}\left(\frac{1}{2}\rho
V_{\text{cruise}}^{2}SC_{D_{0}}+\frac{1}{4C_{D_{0}}(L/D)_{\text{max}}^{2}}\frac{(mg)^{2}}{1/2\rho
V_{\text{cruise}}^{2}S}\right)\frac{(h-250)}{\text{ROC}}$ (51)

Finally, the total energy consumption for the entire mission is

$E_{\text{total}}=E_{\text{hover}}+E_{\text{climb}}+E_{\text{cruise}}+E_{\text{descent}}$
(52)

Through the analysis, we observe that when operating at a higher altitude, the
total energy consumption $E_{\text{total}}$ could either increase or decrease,
depending on the total travel distance. On one hand, cruising at a higher
altitude requires greater energy consumptions during climb
($E_{\text{climb}}$) and descent ($E_{\text{descent}}$). On the other hand,
the thinner air density at a higher altitude results in reduced drag and
therefore less energy consumption during cruise $E_{\text{cruise}}$ (this
applies to aircraft configurations that rely mainly on fixed wings to generate
lift during cruise). Hence, when the cruise distance is long enough, flying at
a higher altitude may become beneficial in energy consumption. Overall, within
the range of UAM, flying at a higher altitude still in general requires more
energy. Figure 15(b) displays the distributions of extra energy consumption
(in %) when cruising at 2,000 ft and 3,000 ft AGL. Depending on the route
distance, cruising at 2,000 ft rather than 1,000 ft AGL results in between
7.28% to 18.34% extra energy; cruising at 3,000 ft rather than 1,000 ft AGL
results in between 14.70% to 36.97% extra energy. In the longer Regional Air
Mobility (RAM) operations, this extra energy consumption could become energy-
saving due to the larger portion of the cruise segment.