Duc Ngoc Pham A Framework of a Power Management System for a Hybrid Electric VTOL Aircraft using Optimal Control - DSpace@MIT

A Framework of a Power Management System for a
 Hybrid Electric VTOL Aircraft using Optimal
 Control
 by
 Duc Ngoc Pham
 B.S., United States Air Force Academy (2020)
 Submitted to the Department of Aeronautics and Astronautics
 in partial fulfillment of the requirements for the degree of
 Master of Science in Aeronautics and Astronautics
 at the
 MASSACHUSETTS INSTITUTE OF TECHNOLOGY
 May 2022
 © Massachusetts Institute of Technology 2022. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Department of Aeronautics and Astronautics
 May 17, 2022

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Choon Sooi Tan
 Senior Research Engineer
 Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Jonathan P. How
 R.C. Maclaurin Professor of Aeronautics and Astronautics
 Chair, Graduate Program Committee

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A Framework of a Power Management System for a Hybrid
 Electric VTOL Aircraft using Optimal Control
 by
 Duc Ngoc Pham

 Submitted to the Department of Aeronautics and Astronautics
 on May 17, 2022, in partial fulfillment of the
 requirements for the degree of
 Master of Science in Aeronautics and Astronautics

Abstract
The operation of a hybrid fixed-wing vertical takeoff and landing unmanned aerial
vehicle (VTOL UAV) is assessed using an optimal control framework for the most
energy-efficient and time-efficient trajectories. The UAV is equipped with a modu-
lar hybrid propulsion system (MHPS) where the electrical and carbon-fuel system
components are interchangeable on a mission-to-mission basis, enabling aircraft per-
formance flexibility. The framework is used to assess the effects of MHPS electric
power hybridization, energy hybridization mass ratio, and peak power output on
UAV performance during demanding phases of flight, which include takeoff, landing,
and hover. Results showed that the most time and energy efficient takeoff trajecto-
ries involved minimizing the vertical displacement gained during the transition from
vertical to horizontal flight. The power management strategy for minimum energy
consumption during takeoff, landing, and hover was largely dictated by propulsion
component efficiencies; maximizing the electric motor power and minimizing the car-
bon fuel power would reduce energy consumption. As peak power output and electric
power hybridization increased, takeoff and landing energy consumption decreased.
Minimum time takeoff and landing trajectories and power management strategy were
only dependent on power-to-weight ratio; a higher peak power output reduced takeoff
and landing time. The power management strategy for efficient hover mirrored that
of takeoff and landing; a higher peak power output resulted in less energy consumed.
A preliminary assessment of the tradeoffs of electrification was conducted using a
takeoff/cruise endurance non-dimensional performance group given as the ratio of
the product of peak power output and cruise endurance to the total energy consumed
during takeoff. Increasing electric power hybridization adversely affected overall air-
craft performance, due to the reduction in cruise range and endurance. For each
selected value of electric power hybridization, there is an optimal peak power output
that strikes the best balance between takeoff and cruise performance. For short, de-
manding mission segments, like takeoff or landing, the energy hybridization mass ratio
has a smaller impact than electric power hybridization, since range or mass changes
are not of concern when analyzing independent takeoff or landing performance.

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Thesis Supervisor: Choon Sooi Tan
Title: Senior Research Engineer

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Acknowledgments
First and foremost, I would like to thank my research advisor Dr. Tan. His optimism,
careful guidance, and endless patience were invaluable over the last two years. I feel
lucky to have a strong technical advisor with such empathy and flexibility. Without
his enthusiasm and support, I would not have been able to complete this thesis.
 I would like to give a special thanks to the Northrop Grumman Corporation for
supporting this research. Program manager Rob Talbert, Dr. Robert Gamache, and
Alan Stull were instrumental in making this possible.
 Additional appreciation goes to the students, staff, and faculty of the MIT Gas
Turbine Lab. Special thanks to Laurens Voet for his immense help and constant
support across countless tutorial and debugging sessions; I hope we can go skiing
again soon. Thanks to Alex Kunycky for introducing me to MIT and for laying the
foundations for this research.
 Many thanks to Albert; I couldn’t have asked for a better pandemic roommate.
Thanks to Manwei and friends for the weekday park skiing and gym sessions, Marek
for being a fellow film enthusiast and NTU comrade. Finally, I would like to thank
my family. Thank you Mom, Dad, Cathy, Angela, and Coco for always welcoming
me home for much needed breaks and for encouraging me to be the best person I can
be. I am forever grateful.

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Contents

1 Introduction 15
 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
 1.1.1 Aircraft Propulsion Systems . . . . . . . . . . . . . . . . . . . 15
 1.1.2 Vertical Takeoff and Landing Unmanned Aerial Vehicles . . . 16
 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
 1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 1.3.1 Modular Hybrid Propulsion System . . . . . . . . . . . . . . . 18
 1.3.2 Power Management for Hybrid Electric Aircraft . . . . . . . . 19
 1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 1.5 Thesis Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 1.6 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 22
 1.7 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Technical Approach 25
 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
 2.2 Aircraft model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
 2.3 Modular Hybrid Propulsion System . . . . . . . . . . . . . . . . . . . 27
 2.4 Hybrid Propulsion System Performance . . . . . . . . . . . . . . . . . 29
 2.5 VTOL Specific Phases of Flight . . . . . . . . . . . . . . . . . . . . . 30
 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Mission Assessment Framework 33
 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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3.2 Dymos and OpenMDAO Software . . . . . . . . . . . . . . . . . . . . 34
 3.3 Dymos Phases and Trajectories . . . . . . . . . . . . . . . . . . . . . 34
 3.4 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 36
 3.5 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
 3.5.1 Defining ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 37
 3.5.2 Controls and Parameters . . . . . . . . . . . . . . . . . . . . . 38
 3.5.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
 3.5.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . 39
 3.6 Mission Assessment Framework . . . . . . . . . . . . . . . . . . . . . 39
 3.6.1 Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 40
 3.6.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
 3.6.3 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
 3.6.4 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
 3.6.5 Power Management . . . . . . . . . . . . . . . . . . . . . . . . 43
 3.6.6 Fuel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
 3.6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Results and Discussion 47
 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
 4.2 Takeoff Trajectory Results . . . . . . . . . . . . . . . . . . . . . . . . 48
 4.2.1 Takeoff Minimum Energy . . . . . . . . . . . . . . . . . . . . 48
 4.2.2 Takeoff Minimum Time . . . . . . . . . . . . . . . . . . . . . . 52
 4.2.3 Effect of Hybridization Level on Energy-Optimal Takeoff . . . 55
 4.2.4 Effect of Hybridization Level on Time-Optimal Takeoff . . . . 57
 4.3 Landing Trajectory Results . . . . . . . . . . . . . . . . . . . . . . . 58
 4.3.1 Landing Minimum Energy Consumed . . . . . . . . . . . . . . 59
 4.3.2 Landing Minimum Time . . . . . . . . . . . . . . . . . . . . . 62
 4.3.3 Effect of Hybridization Level on Energy-Optimal Landing . . . 64
 4.3.4 Effect of Hybridization Level on Time-Optimal Landing . . . . 65
 4.4 Hover Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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4.4.1 Minimum Energy Consumed . . . . . . . . . . . . . . . . . . . 67
 4.4.2 Effect of Hybridization Level on Energy-Optimal Hover . . . . 70
 4.5 Preliminary Assessment of Electrification Tradeoffs . . . . . . . . . . 70
 4.5.1 Cruise Range/Endurance Estimation . . . . . . . . . . . . . . 71
 4.5.2 Takeoff Performance Group Results . . . . . . . . . . . . . . . 73
 4.6 An Integrated Mission for Demonstrating Framework Capability and
 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
 4.6.1 Minimum Energy Simple Mission . . . . . . . . . . . . . . . . 75
 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Summary and Conclusion 81
 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
 5.2 Primary Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6 Recommendations for Future Work 83

A Additional Figures 85

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List of Figures

 1-1 A single airframe for a GA aircraft could potentially take on several
 diverse roles with a MHPS. . . . . . . . . . . . . . . . . . . . . . . . . 19
 1-2 Parallel hybrid drivetrain model. . . . . . . . . . . . . . . . . . . . . 20

 2-1 MHPS Architecture and component groups. . . . . . . . . . . . . . . 28
 2-2 Vertical landing of tail-sitter unmanned aerial vehicle. . . . . . . . . . 32

 3-1 Extended design structure matrix of the trajectory model to design the
 energy management system of the hybrid UAV. . . . . . . . . . . . . 41
 3-2 Free Body Diagram (FBD) of the VTOL UAV during the cruise phase. 42
 3-3 Suter TOA 288 Fuel Consumption. . . . . . . . . . . . . . . . . . . . 44

 4-1 Optimal states and control for an energy-efficient tailsitter takeoff. . . 51
 4-2 Optimal states and control for an time-efficient tailsitter takeoff. . . . 54
 4-3 Effect of power hybridization level on energy-optimal takeoff. . . . . . 56
 4-4 Effect of power hybridization level on time-optimal takeoff. . . . . . . 58
 4-5 Optimal states and control for an energy-efficient tailsitter landing. . 61
 4-6 Optimal states and control for an time-efficient tailsitter landing. . . 63
 4-7 Effect of power hybridization level on energy-optimal tailsitter landing. 65
 4-8 Effect of power hybridization level on time-optimal tailsitter landing. 66
 4-9 Optimal states and control for an energy-efficient tailsitter hover. . . 69
 4-10 Effect of power hybridization level on energy-optimal hover . . . . . . 70
 4-11 Energy-Optimal Takeoff/Cruise Performance . . . . . . . . . . . . . . 73
 4-12 Optimal states for an energy-efficient UAV mission. . . . . . . . . . . 77

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4-13 Optimal states and control for an energy-efficient UAV mission. . . . 78

A-1 Additional states for an energy-optimal tailsitter takeoff ( = 0.2). . 85
A-2 Additional states for an time-optimal tailsitter takeoff ( = 0.2). . . 86
A-3 Additional states for an energy-optimal tailsitter landing ( = 0.2). 87
A-4 Additional states for an time-optimal tailsitter landing ( = 0.2). . 88

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List of Tables

 3.1 Mission Assessment Framework and State Variables . . . . . . . . . . 40
 3.2 Equation of Motion State Variables and Rate Sources . . . . . . . . . 40

 4.1 Takeoff Trajectory definition: a combination of a vertical climb and
 transition phase with boundary constraints (I: initial; F: final), path
 constraints and controls. . . . . . . . . . . . . . . . . . . . . . . . . 48
 4.2 Landing Trajectory definition: a combination of a transition and verti-
 cal descent phase with boundary constraints (I: initial; F: final), path
 constraints and controls. . . . . . . . . . . . . . . . . . . . . . . . . 59
 4.3 Hover Trajectory definition: 15-minute hover phase with boundary
 constraints (I: initial; F: final), path constraints and controls. . . . . 67
 4.4 Takeoff Trajectory definition: a combination of a vertical climb and
 transition phase with boundary constraints (I: initial; F: final), path
 constraints and controls. . . . . . . . . . . . . . . . . . . . . . . . . 75

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Chapter 1

Introduction

1.1 Background

1.1.1 Aircraft Propulsion Systems

The majority of modern aircraft rely on carbon-fuel based propulsion systems; ex-
amples include internal combustion engines (ICEs), turboprops, turbojets and tur-
bofans. While carbon-fuel based propulsion systems offer high specific energy, they
suffer from mechanical complexity, relatively low rate of technological improvement,
and scale-dependent specific power [17].
 Electrified propulsion systems have been suggested as a solution to the deficien-
cies of traditional carbon-fuel propulsion systems. Benefits include lower mechanical
complexity, reduced environmental impact, and relatively scale-independent specific
power levels. These design advantages, namely lower mechanical complexity, have en-
abled electrified propulsion systems to be utilized in novel designs such as swiveling
engines and distributed propulsion schemes. However, the major challenge for electric
aircraft is the low energy density of batteries compared to liquid fuel – carbon fuel
specific energies are superior to battery specific energies by a factor of approximately
60 [16]. This limitation restricts the range and endurance of an electric aircraft to a
fraction of what is capable for a carbon fuel powered aircraft.
 To utilize the advantages provided by electrical propulsion systems while main-

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taining the flight time/range of carbon fuel propulsion systems, electric-carbon fuel
hybrid propulsion systems have increasingly come under assessment followed by re-
search and development for use in aircraft. Hybrid propulsion systems of this type
utilize carbon-fuel to provide the majority of energy while utilizing electric propulsion
system components to provide power and design flexibility. While hybrid propulsion
systems show promise in enabling step changes in aircraft performance enhancement,
there are few flown hybrid aircraft and there is no existing set of unified design stan-
dards or practices.

1.1.2 Vertical Takeoff and Landing Unmanned Aerial Vehicles

Electrical complexity, thermal management, and lack of experience limit practical
usage of electrified propulsion on larger aircraft. As a result, small aircraft, such as
unmanned aerial vehicles (UAVs), stand to benefit most from the implementation
of hybrid-electric propulsion systems. Unmanned aerial vehicles are aircraft flown
via self-control or remote-control without a human pilot on board. In recent years,
demand for vertical takeoff and landing (VTOL) UAVs has grown significantly due to
their low cost and flexibility. UAV missions consist of many phases, including takeoff,
climb, cruise, loiter, dash, etc. As a result, UAV airframes and propulsion systems
must be capable of accomplishing a wide range of missions.
 Conventional fixed-wing UAVs have long endurance and range, but require ded-
icated infrastructure for takeoff and landing. Fixed-wing VTOL aircraft are a solu-
tion to this issue, as they are capable of both hover and forward flight using aerody-
namic surfaces. However, transitioning from vertical to horizontal flight for fixed-wing
VTOL aircraft is a demanding process. During transition, the propulsion system must
provide sufficient thrust to support the weight of the tilting aircraft while also acceler-
ating the aircraft to a speed and configuration in which sufficient lift can be provided
by the wings.

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1.2 Motivation
Among UAV manufacturers, there is a growing need for aircraft with diverse perfor-
mance capabilities that can accomplish a wide range of missions. An example would
be a UAV that is required to fly large-payload, high endurance, and short-takeoff mis-
sions. Traditional propulsion systems are often ill-suited to meet these requirements
due to their fixed, designed-for "worst-case" nature [17]. As a result, many tradi-
tional aircraft propulsion systems are oversized for most missions, which translates
to a smaller payload and lower energy efficiency.
 A novel modular hybrid propulsion system (MHPS) is a potential solution to
the growing need for flexible aircraft propulsion systems. In an aircraft outfitted
with a MHPS, components of the powertrain can be removed and changed between
flights to alter propulsion system performance. Modularity provides the capability
to tailor the propulsion system to a variety of aircraft missions, enabling increased
aircraft performance flexibility. The resulting configurations allow a single airframe
to be optimally suited for a variety of missions with different requirements for range,
power, and payload.
 However, the addition of an electrical power source in a hybrid system presents
a new challenge: the need for an energy management control policy that determines
which energy source is utilized at each point in time during flight. Furthermore,
because of the increase in empty weight due to the additional components in a hybrid-
electric UAV (e.g. electric motors, batteries, power electronics), it is critical that
the propulsion system is optimized for fuel consumption and operational efficiency.
A strategy to improve hybrid propulsion system efficiency is through optimization
of the power split between the carbon fuel and battery sources, i.e. through the
selection of the energy consumption from each available source along the trajectory.
This power management can be done in real-time or predefined before a mission,
depending on the selected control methodology. When integrated in the aircraft
design methodology, the optimal energy management is used in each step of the
aircraft sizing loop, minimizing the fuel consumption for the entire mission.

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Despite the perceived operational advantages, onboard energy management in
hybrid aircraft is not yet in widespread use. Most energy management optimization
studies involving hybrid aircraft fix the power split between the fuel power and electric
power during the optimization run [9]. Notable exceptions use dynamic programming
(DP) algorithms to optimize the energy management of an aircraft during flight [4].
However, aircraft models with more states, such as speed or altitude, are difficult
to quantify using DP due to computational limitations [9]. In this thesis, a direct
method, namely collocation, is used. The use of a direct method like collocation
allows for more complex aircraft models, giving new insights on optimal trajectories
in addition to optimal energy management.

1.3 Literature Review

1.3.1 Modular Hybrid Propulsion System

The modular hybrid propulsion system considered in this thesis refers to a propul-
sion system consisting of electrical and carbon fuel subsystems in a parallel hybrid
configuration, where modularity refers to the ability to change propulsion system
components. The mechanical simplicity of electrical components, especially in the
power transfer method (electrical wiring vs. mechanical shafts), enables interchange-
ability in hybrid architectures. With modular components, the propulsion system’s
performance can be tailored to each mission, maximizing mission-specific performance
or payload. This is accomplished by ensuring that the propulsion system is properly
sized, such that there is no "dead" mass and ensuring operational requirements are
met.
 Prior research demonstrated that, compared to a hybrid-electric propulsion system
with fixed components, a MHPS with 10% of aircraft weight reserved for modular
power system components could increase aircraft cruise time by up to 90% and reduce
takeoff distance by up to 50%, depending on the installed configuration. With a large
range of performance potential, a MHPS has the ability to enable a single airframe to

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meet a diverse set of mission requirements and accomplish a wide variety of missions.

Figure 1-1: A single airframe for a GA aircraft could potentially take on several
diverse roles with a MHPS.

1.3.2 Power Management for Hybrid Electric Aircraft

The rather low specific energies of current batteries (when compared to carbon fuels),
has motivated an increased interest in carbon-fuel electric hybrid propulsion systems.
Hybrid propulsion systems allow designers to leverage electric propulsion architecture
benefits while maintaining the endurance and range of traditional propulsion systems;
hybrid propulsion systems can utilize a central carbon-fuel system to provide the ma-
jority of energy required for flight while using the electric system to either supplement
or fully-supply the power required to start and maintain flight. This presents a unique
power management challenge since hybrid aircraft have multiple energy sources that
can compete to meet power and thrust requirements. As a result, there exists a con-
tinuous range of carbon fuel and battery power settings to produce the same required
power. Choosing when and how much battery power to supplement the carbon fuel
engine can influence total mission fuel burn, even when the total battery power used
remains the same.
 Power management of hybrid-electric configurations has been addressed in the au-

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Figure 1-2: Parallel hybrid drivetrain model.

tomobile industry with multiple control schemes, ranging from Stochastic Dynamic
Programming to simple hard coded rules. Power management strategies can be classi-
fied as either rule-based or optimization-based. Rule-based techniques rely on heuris-
tics, intuition and human experience to develop a series of rules for control manage-
ment, while optimization-based strategies rely on maximizing/minimizing the value
of a certain objectives such fuel consumed or efficiency. [14].

1.4 Research Objectives
The overall research objective is to assess the effect of hybridization on the perfor-
mance of a fixed-wing VTOL UAV across multiple phases of flight, and to determine
the optimal energy management strategy. An effective strategy to determine the
optimal trajectory and energy management for a hybrid aircraft is presented and as-
sessed. The present work will also investigate the main factors influencing optimal
energy management for a fixed-wing VTOL aircraft. The optimization objective is
to minimize the energy or time required to complete phases of flight such as takeoff
and landing. In this thesis, the following research questions will be addressed:

 1. What are the attributes of optimal flight trajectories for VTOL UAVs (with
 respect to minimizing energy and time)?

 2. What is the optimal energy management strategy for different phases of flight
 and across an entire mission?

 3. What advantages or disadvantages does a hybrid propulsion system offer during

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demanding phases of flight?

 4. What level of performance can be achieved with an optimized hybrid propulsion
 system relative to traditional propulsion systems?

 5. How should the results be cast in an appropriate set of non-dimensional param-
 eters so that a generic scaling rule can be established? How does the attributes
 of hybrid-propulsion system scale with flight vehicle characteristics and mission
 requirements?

1.5 Thesis Scope
In this thesis, no attempt will be made in integrating optimal energy management in
the design of a hybrid aircraft. Rather, an effective strategy to determine the optimal
energy management for a hybrid aircraft is presented and assessed. A collocation
approach is used, i.e. a direct optimization method based on an implicit integration
technique. The use of a direct method like collocation allows for more complex aircraft
models, giving new insights on optimal trajectories in addition to optimal energy
management.
 This work discusses a representative modular hybrid propulsion system intended
for use on a fixed-wing VTOL UAV. The flight phases of focus are takeoff, landing,
transition, and hover due to their demanding peak power requirements. A simple mis-
sion from takeoff to landing is also assessed. To determine the effect of hybridization
and other propulsion system design choices on UAV performance, the optimal control
framework is used to compute the most time or energy-efficient flight trajectories and
power profiles for various permutations of propulsion system components. The re-
sults are used to develop guidelines for propulsion system configurations for different
missions and to inform the development of onboard energy management strategies.
However, no steps are taken to integrate specific energy management policies or con-
trol laws into the practical design of an aircraft.
 To quantify the dependence of VTOL UAV performance on power hybridization

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and propulsion system design choices, a set of quantitative performance relationships
are developed for VTOL-specific phases of flight (vertical takeoff/landing, transition,
hover).

1.6 Original Contributions
Contributions of this thesis are:

 1. Criteria for selecting the extent of electrified propulsion contribution relative to
 carbon-fuel based propulsion contribution. These criteria include the conditions
 under which to use carbon-fuel or electrified systems, based on the optimal
 trajectories and power profiles obtained from the optimization framework.

 2. A set of quantitative takeoff, landing, and hover performance relationships for
 modular hybrid VTOL aircraft. These performance relationships quantify the
 functional dependence of performance on power hybridization level and choices
 in aircraft design. Furthermore, the performance relationships enable MHPS
 sizing and determination of MHPS constituent components.

 3. Guidelines to configure modular hybrid propulsion systems for a variety of
 VTOL UAV missions. A selection guideline for determining the power hy-
 bridization ratio is proposed. The optimal values of peak power and energy
 hybridization are provided by a parametric assessment of a set of representative
 optimized mission phase completion times and energies. Using these guidelines,
 operators can select propulsion components that enable the modular hybrid air-
 craft to maximize performance during different phases of flight (takeoff, landing,
 hover, transition).

 Additionally, the optimal flight trajectories and MHPS control profiles produced
by the optimization framework represent original contributions to this area of re-
search. Attributes of optimal flight trajectories for an MHPS-powered VTOL UAV
executing takeoff, landing, hover, and a simple complete mission are presented and
analyzed.

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1.7 Thesis Organization
This thesis is organized as follows. Chapter 2 outlines the technical approach taken
to address the stated research questions. Chapter 3 includes the development of the
optimal control framework. Chapter 4 details findings from performance assessments
of takeoff, landing, hover, and a simple complete mission. Chapter 5 provides a
summary of the primary findings and implications for aircraft propulsion system
usage and design. Chapter 6 outlines recommendations for future work.

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Chapter 2

Technical Approach

2.1 Introduction
The Modular Hybrid Propulsion system introduces the use of an electrical subsystem
(hybridization) to provide propulsion functionalities and the ability to swap power
production and energy storage components (electrical or carbon fuel) to tailor the
propulsion system to different kinds of missions (modularity). To address the research
questions posed in Chapter 1, a representative UAV is chosen for analysis using a
mission assessment optimization framework. This section details the optimization
framework used to analyze hybrid UAV performance and the simplified models used
to simulate different phases of flight (takeoff, landing, hover, etc.) for the hybrid UAV
of interest. The term simplified models refers to the computationally inexpensive
first-principles based low-order models that illustrate key trends.

2.2 Aircraft model
A fixed-wing VTOL UAV platform is utilized to evaluate MHPS performance against
traditional propulsion system performance since these aircraft typically have high
performance flexibility requirements. The airframe has a maximum takeoff mass of
50 kilograms. The total mass consists of 8 kg of total energy storage, in the form of
carbon fuels or batteries; up to 9 kg for power production (i.e. internal combustion

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engine and electric motor), 16 kg airframe components and avionics, and an 17 kg
payload. The MHPS is described in further detail in Section 2.3. Airframe and
payload masses represent similar mass fractions as compared to other small- and
medium-sized UAVs [17]. Thrust is provided by a single propeller, mounted in a
pusher configuration at the rear of the aircraft.
 Pusher-prop UAV designs are commonly seen in large-sized UAVs such as the
General Atomics MQ-1, or similar medium sized UAVs like the NextTech Atlas-V
or the Brumby Mark III. The representative UAV is tail-sitter, a type of VTOL
aircraft that takes off and lands on its tail, then tilts horizontally for forward flight.
Conventional fixed-wing UAVs have long endurance and range, but require dedicated
infrastructure for takeoff and landing. Fixed-wing VTOL aircraft, such as tail-sitters,
are a solution to this issue, as they are capable of both hover and forward flight using
aerodynamic surfaces. However, transitioning from vertical to horizontal flight for
fixed-wing VTOL aircraft is a demanding process. During transition, the propulsion
system must provide sufficient thrust to support the weight of the tilting aircraft
while also accelerating the aircraft to a speed and configuration in which sufficient
lift can be provided by the wings.
 The airfoil employed is a NACA 4418, providing high lift and stable stall charac-
teristics [11]. This airfoil has undergone testing at extreme angles of attack, so the
representative aircraft and mission framework would meet the requirements of both
conventional and VTOL aircraft operation. The wings have a relatively-large wetted
area (3 2 ) to keep wing loading low and are comparable to other endurance aircraft,
such as small UAVs and gliders. The aircraft model module employed is entirely
generic - aircraft attributes such as airfoil type, carbon fuel/battery mass fractions
can be altered by substituting values in the aircraft model component of the mission
assessment framework.

 26

2.3 Modular Hybrid Propulsion System
The representative MHPS considered in this thesis is a parallel hybrid system where
the carbon fuel engine is also connected to the propeller, unlike a serial hybrid. With
this additional connection, propulsive power can be provided by both the electric
motor and carbon fuel engine. The representative MHPS includes three groups of
components:

 1. Energy Storage System which includes two sources of energy, a fuel tank for
 carbon fuel energy and a battery stack for electrical energy

 2. Power Production mass which includes mass from an electric motor and an
 internal combustion engine

 3. Thrust Production components include the mass of the propeller and asso-
 ciated machinery

 The MHPS converts energy into thrust through two primary paths. In the carbon
fuel path, gasoline is burned in the internal combustion engine, converting chemical
energy into mechanical energy and power which is transferred through a shaft and
gearbox to the propeller where thrust is produced. In the second path, electricity is
discharged from the batteries to power an electric motor, which then provides shaft
power to drive the propeller and produce thrust. Since the electric motor and ICE
are connected to the propeller in a parallel configuration, both sources can deliver
power to the propeller simultaneously.
 The modularity of the MHPS introduces the ability to swap propulsion system
components in between flights. Thus, the effects of hybridization can be assessed
by changing propulsion system component attributes such as electric motor size or
battery capacity. Optimizations and analyses can then be conducted to assess the
impact of hybridization and to determine the optimal propulsion system configuration
for each type of mission. In this thesis, the energy storage system components (fuel
tank and battery size) and power production components (ICE and electric motor) are
considered modular. Optimizations are carried out for various MHPS configurations

 27

Figure 2-1: MHPS Architecture and component groups.

to determine the advantages or disadvantages that hybridization offers for different
VTOL UAV missions.
 Specifically, the propulsion system design choices utilized are:

 • Electric motor peak power, 

 • Carbon fuel engine peak power, 

 • Electric energy hybridization factor, = + 
 (where is total
 energy stored in batteries and is total energy stored in carbon fuel)

 • Electric power hybridization factor, = 
 + 

 Two powerplants serve as reference power production components. The reference
carbon-fuel engine is based on the Suter Industries TOA 288, a 9.2 kg two-cylinder
ICE. The reference electric motor is based on the MGM-Compro REX30 motor, a
5.2 kg electric propulsion system (including motor and motor controller). The peak
power output from both systems is 18 kW, in line with UAVs of similar size and
requirements.
 Power hybridization factor and carbon fuel/electric motor peak power is varied
to determine the effect of changing power production components. Variation of the
power hybridization factor translates to a change in the size of the power produc-
tion components. As increases, the size and power output of the electric motor

 28

increases. If total power is held constant, the size and power output the ICE must
also decrease.
 The Energy Storage System components are varied through the Energy Hybridiza-
tion Mass Ratio, , a different form of electric energy hybridization factor. The En-
ergy Hybridization Mass Ratio is defined in Equation 2.1, where is the battery
mass, is the energy storage system (ESS) mass, and is carbon fuel mass.

 = = (2.1)
 + 
 As increases from 0 to 1, the portion of total energy stored in the batteries
increases. As an effect of their lower energy density, the total energy stored in the ESS
decreases as increases. The strategy of varying component mass while maintaining
the total ESS mass highlights the specific impact of battery and carbon fuel energy
densities on the aircraft capability and performance. Analysis of is primarily used
in this thesis to analyze the effects of electrification on longer mission segments like
cruise range. For short, demanding mission segments, like takeoff or landing, the 
has a smaller impact than , since range or mass changes are not of concern when
analyzing independent takeoff or landing performance.

2.4 Hybrid Propulsion System Performance
Hybrid propulsion system performance is analyzed and quantified for takeoff, landing,
and hover mission segments. Focus is placed upon establishing the functional depen-
dence of hybrid component technological parameters and design choices on aircraft
performance.
 The primary technological parameters that affect hybrid propulsion system per-
formance are:

 • Electric motor specific power, = 
 
 • Carbon fuel engine specific power, = 
 
 • Electric motor efficiency, 

 29

• Carbon fuel engine efficiency, 

 • Battery specific energy, 

 • Carbon fuel specific energy, 

 In addition to new design challenges, the MHPS introduces operational challenges
due to the presence of the electrical subsystem. The first challenge is efficient opera-
tion of the propulsion system over the course of a mission. To efficiently accomplish
mission objectives, battery and carbon fuel energy sources must be properly managed.
With an appropriate energy management policy, carbon fuel and battery energy can
be utilized in a manner that satisfies optimization objectives, such as minimum en-
ergy consumed or minimum time to complete a mission phase, while meeting mission
operational requirements. The energy management policy referred to here determines
the power output and operation point of each EM, generator and carbon-fuel engine.
The chosen operating points in turn determine the power drawn from the battery
and carbon-fuel stores at each point of time during a mission. This thesis formulates
and develops an effective framework for determining an optimal energy management
policy for energy-efficient and time-efficient trajectories.

2.5 VTOL Specific Phases of Flight
As noted in Chapter 1, UAV airframes must also be capable of accomplishing a wide
range of missions. Typical UAV missions consist of a variety of phases, including take-
off, climb, cruise, loiter, dash, etc. Conventional fixed-wing UAVs have long endurance
and range, but require dedicated infrastructure for takeoff and landing. Fixed-wing
VTOL aircraft are a solution to this issue, as they are capable of both hover and
forward flight using aerodynamic surfaces. However, transitioning from vertical to
horizontal flight for fixed-wing VTOL aircraft is a demanding process. During tran-
sition, the propulsion system must provide sufficient thrust to support the weight of
the tilting aircraft while also accelerating the aircraft to a speed and configuration

 30

in which sufficient lift can be provided by the wings. Thus, this transition is an im-
portant consideration for the design of these types of aircraft. Out of the different
phases of flight for a VTOL UAV mission, the takeoff, landing, transition and hover
phases are singled out for investigation in this thesis due to their demanding power
requirements.
 Takeoff for a fixed-wing VTOL aircraft begins with a vertical climb from the
ground, followed by a transition from vertical to horizontal flight. During this ma-
neuver, the aircraft and propulsion system must be operated efficiently while meeting
takeoff requirements. To minimize energy consumed, the aircraft must minimize time
spent where the propulsion system provides the entirety of the lift. However, if the
aircraft completes the takeoff maneuver too fast, it risks consuming excessive energy.
Therefore, there exists an optimal takeoff trajectory and energy management strat-
egy. During a tail-sitter UAV landing, the UAV executes a pitch up maneuver to
transition from horizontal flight to vertical flight, followed by a controlled vertical
descent. Although the maneuver is less demanding than takeoff, since lift is already
being generated by the wings, efficiency gains can be made by determining the op-
timal parameters (i.e. velocity, flight path angle, angle of attack, etc.) to begin the
pitch up maneuver, or the flight path itself. During transition from vertical to hori-
zontal flight, kinetic energy is traded for potential energy as the aircraft loses velocity
during the climb upwards. During the hover and vertical descent, the only source
of lift is the propulsion system; as a result, an energy-efficient power management
strategy is essential.
 Hover is another demanding VTOL-unique mission phase. During hover, the tail-
sitter is oriented vertically with zero velocity, with the propulsion system providing a
thrust force equal to the weight of the aircraft.
 While the mission assessment framework is used to determine the operation of
the UAV, such as optimal takeoff/landing trajectories and energy management strat-
egy, the results produced can also be used to inform design decisions about MHPS
configurations. Propulsion characteristics, such as hybridization level or component
efficiencies, can drastically impact UAV performance. For example, during missions

 31

like cargo delivery with multiple takeoffs and landings; the MHPS can be configured
with power production and energy storage system components beneficial for vertical
flight.

 Figure 2-2: Vertical landing of tail-sitter unmanned aerial vehicle.

2.6 Summary
In summary, a representative UAV is selected and defined as a research platform
for assessing the impact of hybridity and modularity on aircraft performance. The
assessment consists of first defining the propulsion system architecture followed by se-
lecting the specific components and properties that will vary with system modularity.
A mission framework is formulated and defined using three open-source Python pack-
ages (Dymos, OpenMDAO, and IPOPT) to determine the aircraft dynamics while the
UAV completes the phases of flight. VTOL-unique phases of flight are defined for as-
sessment, and optimization objectives are identified for these mission segments. The
formulation of the optimization framework is outlined in the following chapter.

 32

Chapter 3

Mission Assessment Framework

3.1 Introduction
A mission assessment and optimization framework is developed for the representative
tail-sitter UAV described in Chapter 2. The framework is used to determine the tra-
jectory and power schedule for a specified phase or phases of flight for an optimization
objective. The optimization objective for each assessment is either minimum energy
consumed, or minimum time required for completion of the selected flight mission
and or phase. The formulation of the mission assessment framework is outlined in
this chapter.
 D. Yang recommended dynamic programming to seek the optimal power man-
agement of an MHPS. However, dynamic programming has significant practical lim-
itations, which stem from the exponentially-increasing dimensionality of the opti-
mization problem that results when assessing complex systems. In practice, dynamic
programming can only be used for problems with either a small number of state vari-
ables to optimize, or a small number of timesteps. As a result, aircraft models with
more states, such as speed or altitude, are difficult to quantify using DP [17].
 Therefore the above noted computational limitations have motivated the selec-
tion of the Python package Dymos as the framework to determine the optimal power
management and flight trajectory for the MHPS-equipped tailsitter UAV. Dymos is
a library for optimizing control schedules for dynamic systems – sometimes referred

 33

to as optimal control or trajectory optimization. Dymos is focused on direct opti-
mization methods based on an implicit integration technique, collocation. The use of
a direct method such as collocation allows for more complex aircraft models, giving
new insights on optimal trajectories in addition to optimal energy management [9].

3.2 Dymos and OpenMDAO Software
Dymos, a library for the optimal control of dynamic multidisciplinary systems, en-
ables optimization of tightly connected subsystem designs as well as the operation
of the overall system. In this case, the term multidisciplinary refers to more gen-
eral optimization problems where dynamics are only one part in a larger system-level
model with additional – potentially computationally expensive – calculations that
come before and after the dynamic calculations. While Dymos can optimize standard
optimal control problems, it can also optimize systems in which a trajectory is just
one part of the overall optimization. Many optimization packages rely on the param-
eterization of hardware models, for example, an approximation of an engine’s mass
as a function of its thrust level. Rather, Dymos allows users to impose higher-fidelity
design considerations and apply the resulting subsystem designs to the trajectory
profile [5]. Dymos is built upon the OpenMDAO framework, which enables highly
effective/efficient computation of accurate derivatives. The OpenMDAO framework
has a modular derivative system, which allows users to select from finite differencing,
complex step, hand differentiated, and algorithmic differentiation.

3.3 Dymos Phases and Trajectories
A trajectory or flight path is the path that an object with mass in motion follows
through space as a function of time. Dymos breaks the trajectory into portions of
time called phases. Most real-world use cases of optimal control involve complex tra-
jectories that cannot be modeled with a single phase. For example, a trajectory may
consist of multiple phases that may have different equations of motion, different path

 34

constraints or, different control parameterizations. Dymos uses the concept of phases
to support intermediate boundary constraints and path constraints on variables in
the system. Each phase represents the trajectory of a dynamical system, and may
be subject to different equations of motion, force models, and constraints. Multiple
phases may be assembled to form one or more trajectories by enforcing compatibility
constraints between them.
 The procedure for subdividing a single trajectory into multiple phases provides
several capabilities. First, intermediate constraints along a trajectory can be enforced
by applying a boundary constraint to a phase that begins or ends at the time of
interest. For example, during takeoff, a tail-sitter UAV may be required to vertically
climb to a certain altitude before it can pitch over and transition to horizontal flight.
Additionally, path constraints can be applied to within each phase to bound some
performance parameter within that phase. For instance, the tail-sitter UAV may need
to adjust its trajectory to avoid stall during cruise.
 For implicit and explicit phases, the equations-of-motion or process equations are
defined via an ordinary differential equation where x is the vector of state variables
(the variable being integrated), t is time (or time-like), u is the vector of parameters
(an input to the ODE), and f is the ODE function.

 = ( , , ) (3.1)
 
 Each phase within a trajectory can use its own separate set of ordinary differential
equations (ODE). For example, an aircraft with vertical takeoff and landing capability,
such as the representative tail-sitter UAV, may use different ODEs for vertical flight
and horizontal flight. ODEs are implemented as standard OpenMDAO models which
are passed to phases at instantiation time with some additional annotations to identify
the states, state-rates, and control inputs.
 Each phase in the trajectory uses its own specific time discretization tailored to
the dynamics present in that portion the trajectory. As a result, the time evolution
of the trajectory can be split into multiple phases with separate time discretizations.

 35

If one part of a trajectory has fast dynamics and another has slow dynamics, the time
evolution can be broken into two phases with separate time discretizations. Mem-
bers of the optimal-control community use many different techniques to discretize
the continuous optimal control problem into a form that can be solved by nonlin-
ear optimization algorithims. Each discretization technique is called a transcription.
Two different collocation transcriptions are supported by Dymos: high-order Gauss-
Lobatto and Radau. Both transcription methods represent state and control variable
trajectories as piecewise of at least third order. They are formulated in such a manner
that enables efficient computation of the required quantities to perform integration
in a numerically rigorous manner.

3.4 Optimization Algorithm
Since Dymos is not distributed with an optimizer, it relies on optimizers that are
available in the OpenMDAO installation. OpenMDAO contains an interface to the
optimizers in SciPy, and an additional wrapper for the pyoptsparse library which
has more powerful optimizer options such as SNOPT, and IPOPT. Through OpenM-
DAO, users are allowed to integrate their own optimizer of choice, which Dymos can
then use without any additional modifications. However, higher-quality optimizers
are important for getting good performance, especially on more challenging optimal-
control problems. Dymos is primarily designed to work primarily with gradient-based
algorithms. In general, optimal-control and co-design problems will have both a very
large number of design variables and a very large number of constraints. Both issues
make gradient-based methods the strongly preferred choice [5].
 The Interior-Point OPTimizer (IPOPT) package was chosen as the optimizer for
the mission assessment framework. IPOPT is an interior point method implemen-
tation with a filter line-search algorithm that is designed specifically for large-scale
nonlinear programming. The combination of Dymos, OpenMDAO, and IPOPT is
used to organize the trajectory and ODEs, determine the system derivatives, and
seek the optimal solution.

 36

3.5 Optimal Control
Optimal control deals with finding a control for a dynamical system over a length
of time such that an objective function is optimized; in other words, optimal control
implies the optimization of a dynamical system. Usually, the optimization comes in
the form of a trajectory in which system states evolve over time. A system state
variable is one of the set of variables that are used to describe the mathematical
"state" of a dynamical system. In a mechanical system, such as an aircraft in flight,
the position coordinates and velocities are typical state variables. The evolution of
these states is governed by an ordinary differential equation (ODE). In Dymos, all
dynamics are characterized as an ODE. When using optimal control software, the
dynamics of the system are usually defined as a set of ODEs that utilize explicit
functions to compute the rates of state variables to be time integrated.

3.5.1 Defining ODEs

In optimal control problems, ordinary differential equations (ODE) or differential
algebraic equations (DAE) dictate the evolution of the state variables of the system.
Typically, the evolution of the system state variables occurs in time, and the ODE
represents the system’s equations of motion (EOM). However, the equations of motion
are not limited to mechanical systems; they can define a variety of different systems.
Equations of motion are commonly referred to as process equations in other fields.
Dymos uses a standard OpenMDAO System to represent equations of motion. The
OpenMDAO system takes a set of variables as input, and computes outputs that
include time-derivatives of the state variables.
 Finally, the system dynamics may be subject to some set of controllable parame-
ters. In Dymos these are categorized into the dynamic controls and the static param-
eters, which are covered in the next section. Dymos also needs to know how state,
time, and control variables are to be connected to the System and needs to know
which outputs to use as state time-derivatives.

 37

3.5.2 Controls and Parameters

Control variables are used to impact the behavior of the system. Controls may be
allowed to vary with time, such as the power provided by an electric motor or the
angle-of-attack of an aircraft during flight. These variables are referred to as dy-
namic controls. Other controllable parameters might be fixed with time, such as the
wingspan or airfoil shape of an aircraft. These variables are referred to as parame-
ters, although in the literature they may also be referred to as static controls. The
endpoints in time, state values, control values, and design parameter values define
the independent variables for the optimization problem. In Dymos, these variables
are discretized in time to convert a continuous-time optimal control problem into a
nonlinear programming problem.

3.5.3 Constraints

When optimizing problems, constraints on the system are always present. Some
constraints can be imposed as bounds on the independent variables. For example,
the initial conditions of a trajectory can be fixed by bounding the initial states and
time to the desired value.
 Constraints on a variable placed at the start or end of a phase are called bound-
ary constraints. There are a few ways to impose boundary constraints in a Dymos
optimization problem, each with slightly different behavior. One way to constrain
fixed values is to remove them from the optimization problem altogether. This can
be done for time, state, and control variables. Although removing constrained values
from the optimization problem simplifies computation, the optimizer lacks freedom
to move these values around, which can lead to failure modes that are not obvious.
 The second class of constraints supported by Dymos are path constraints, so called
because they are imposed throughout the entire phase. Like boundary constraints,
path constraints can be placed upon design variables by using simple bounds.

 38

3.5.4 Objective Function

Dymos may be used to both simulate and optimize dynamical systems. The phase
construct is generally used in optimization contexts. Within each phase, the user can
set the objective:
 = (¯ ¯
 , , ¯, ) (3.2)

The objective may be any output within the phase, as with constraints. Phases
can also be incorporated into larger models wherein the objective is defined in some
subsystem outside of the phase [5].

3.6 Mission Assessment Framework
The mission assessment framework models the dynamics of the representative UAV
aircraft in flight at each time step in the simulated mission. These physics-based cal-
culations are broadly grouped into three categories: aerodynamics, propulsion, and
flight dynamics. Aerodynamic forces are modeled based on the aircraft state, includ-
ing velocity, mass, altitude, flight path angle, and angle of attack. Power consumption
and thrust production are calculated based on velocity, ICE power setting, and EM
power setting. These forces acting on the aircraft are then balanced to calculate the
resulting aircraft flight dynamics (acceleration and pitch rate). The UAV operation
is affected by three control variables (angle of attack, ICE power setting, and EM
power setting), which correspond to typical pilot controls so as to realistically model
ability to control the aircraft.
 Each state variable has a rate source. A state variable is one of the set of variables
that are used to describe the mathematical "state" of a dynamical system. Intuitively,
the state of a system describes enough about the system to determine its future
behaviour in the absence of any external forces affecting the system.
 The extended design structure matrix (XDSM) [8] of the trajectory model is shown
in Figure 3-1. The equations of motion governing each module of the XDSM are
outlined in the following sections. Analytical partial derivatives of the trajectory

 39

Table 3.1: Mission Assessment Framework and State Variables

 Type Variable Unit
 Electrical energy consumed J
 Carbon fuel energy consumed J
 Distance travelled m
 State Variables Altitude m
 Velocity m/s
 Flight path angle deg
 Aircraft mass kg
 Angle of attack deg
 Control Variables Carbon fuel engine power W
 Electric motor power W

 Table 3.2: Equation of Motion State Variables and Rate Sources

 State Variable Rate Source
 Electrical energy consumed EM power 
 Carbon fuel energy consumed ICE power 
 Distance travelled Horizontal velocity ˙
 Altitude Vertical velocity ˙
 Velocity Acceleration ˙
 Flight path angle Pitch rate ˙
 Aircraft mass Fuel burn rate ˙ 

equations of motion are implemented explicitly in the trajectory model to support
gradient-based optimization.
 The trajectory model for the phases of flight is developed using the NASA Dymos
package [5] within the NASA OpenMDAO library [6]. Each phase of the trajectory
is discretized using a Gauss-Lobatto pseudo-spectral transcription method using 15
segments per phase and a third order transcription per segment [7]. The following
subsections provide an overview of the mission assessment framework components
and equations of motion.

3.6.1 Flight Dynamics

To determine the trajectory of the aircraft, we solve for the aircraft’s position, velocity,
and flight path angle as functions of time, given the control variables. A free body
diagram of the forces acting on the VTOL UAV during the cruise is shown in Figure

 40

Figure 3-1: Extended design structure matrix of the trajectory model to design the
energy management system of the hybrid UAV.

3-2. The equations of motion are expressed as follows:
 In equations 3.3-3.6, the aircraft mass, flight path angle, and angle of attack are
given by , , and , respectively.

 = cos − − sin (3.3)
 
 cos 
 = sin + − (3.4)
 
 = cos (3.5)
 
 = sin (3.6)
 
3.6.2 Aerodynamics

The lift and drag forces on the aircraft are computed using:

 1
 = 2 (3.7)
 2

 41

Figure 3-2: Free Body Diagram (FBD) of the VTOL UAV during the cruise phase.

 1
 = 2 (3.8)
 2
 For this study, the aerodynamic force coefficients, and , are evaluated as
functions of from public wind tunnel data for the NACA 4418 airfoil. Smooth
fits of lift and drag coefficient data were produced through cubic spline interpolation,
with continuity of derivatives between segments. In the case study presented here, the
wing surface area is equal to 3.0 2 , a representative wing surface area for comparable
medium-sized UAVs.

3.6.3 Atmosphere

The 1976 US Standard Atmospheric (USSA) model [15] is used, given by Equations
3.9-3.12. The model computes the ambient temperature, 0 , pressure, 0 , density,
 0 , and speed of sound, 0 at altitude, , given the sea level conditions (referenced
by subscript sl). In Equations 3.9-3.12, the ratio of specific heats, the gravitational
constant, the air gas constant and the atmospheric temperature lapse rate are given
by , , , and , respectively.

 0 = − (3.9)

 42

(︂ 
 )︂ 
 0
 0 = (3.10)
 
 0
 0 = (3.11)
 0

 (3.12)
 √︀
 0 = 0

3.6.4 Propulsion

The net thrust from the propeller is computed as a function of power using Equation
3.13, based on disk actuator theory [3].

 (︃ √︃ )︃
 ∞ ∞2 
 = = ∞ + + + (3.13)
 2 4 2 0 

 In Equation 3.13, is the power supplied to the propeller disk and is the
thrust, is the disk area, and is a correction factor to account for induced-power
losses related to nonuniform inflow, tip effects, and other simplifications made in
momentum theory [3]. In this paper, = 1 for ideal power, and = 0.125 2 .
Finally, ∞ is the component of freestream velocity normal to the propeller disk, i.e.
 ∞ = cos . Power is a control variable in the optimization problem, and we use the
Newton-Raphson method to solve this non-linear equation for thrust with power
as an input [3].

3.6.5 Power Management

The available power to the propulsor is computed using the following:

 = · + · (3.14)

 Carbon fuel engine efficiency is equal to 0.3, and electric motor efficiency
 is equal to 0.9. These values are estimates from representative electric motors

 43