Development of a Conceptual Design Model for Aircraft Electric Propulsion with Efficient Gradients

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This is a preprint of the following article:
B. J. Brelje and J. R. R. A. Martins. Development of a Conceptual Design Model for Aircraft Electric Propulsion with Efficient
Gradients. Electric Aircraft Technologies Symposium, 2018.
The published article may differ from this preprint. Copyright 2018 Benjamin J. Brelje and Joaquim R.R.A Martins, All Rights Reserved
Development of a Conceptual Design Model for Aircraft
Electric Propulsion with Efficient Gradients
Benjamin J. Brelje∗† and Joaquim R.R.A. Martins‡
University of Michigan, Ann Arbor, Michigan
Research on electric aircraft propulsion has greatly expanded in the last decade, revealing
new insights on the unique features of the electric aircraft design problem, and identifying short-
comings in existing analysis techniques and tools. In this paper, we survey currently-available
analysis codes for aircraft with electric propulsion. We introduce a new conceptual design
and optimization toolkit—OpenConcept—built for aircraft incorporating electric propulsion.
OpenConcept consists of three parts: a library of simple, conceptual-level models of common
electric propulsion components; a set of analysis routines necessary for aircraft sizing and
optimization; and several example aircraft models. All of OpenConcept’s codes have been
analytically differentiated, enabling the use of OpenMDAO 2’s efficient Newton solver, as well
as gradient-based optimization methods. OpenConcept supports parametric cost modeling
and waste heat management at the component level, enabling realistic thermal and economic
constraints in optimization studies. We present a case study involving the electrification of
existing turboprop airplanes. We model the Daher TBM 850 and Beechcraft King Air C90GT
in OpenConcept, and validate the sizing, weights, fuel burn, and takeoff field length analyses.
We then define a series hybrid electric propulsion architecture for the King Air, and perform
a retrofit study. Finally, we perform multidisciplinary design optimization to minimize both
fuel burn and trip cost for varying design ranges and assumed battery specific energy levels.
We ran more than 750 multidisciplinary optimization cases with full mission analysis. Each
optimization runs in approximately 2 minutes on a typical notebook PC. We demonstrate that
OpenConcept is a flexible and efficient way of performing conceptual-level analysis of aircraft
with unconventional propulsion architectures.
Nomenclature
η= efficiency
γ= flight path angle
µ= tire rolling or braking friction coefficient
a= acceleration
BFL = balanced field length
c= cost
CD0= zero-lift drag coefficient
CL= lift coefficient
CP= propeller power coefficient
D= drag force
dprop = propeller diameter
∗PhD Candidate, Department of Aerospace Engineering, AIAA Student Member
†The author is also an employee of The Boeing Company; this article is written in a personal capacity.
‡Professor, Department of Aerospace Engineering, AIAA Associate Fellow
1

e(b)= (battery) specific energy (energy / mass)
E= energy
EASA = European Aviation Safety Agency
EP = electric propulsion
FAA = Federal Aviation Administration
g= gravitational acceleration
HE= degree of hybridization with respect to energy (percentage of mission energy from electricity)
ho= takeoff obstacle clearance height
ISA = International Standard Atmosphere
J= propeller advance ratio
KIAS = knots indicated airspeed
L= lift force
m= mass
MDO = multidisciplinary design optimization
MLW = maximum landing weight
MTOW = maximum takeoff weight
OEI = one engine inoperative
OEM = (aircraft) original equipment manufacturer
p= specific power (power / mass)
P= power
PSFC = power specific fuel consumption (fuel flow rate per unit power)
r= distance or range
R= residual (equation to be driven to zero)
Sre f = wing reference area
SLSQP = sequential least squares programming
T= thrust force
t= time
TMS = thermal management system
TOW = takeoff weight
V= velocity
V0= airspeed at the start of the takeoff roll
V1= takeoff decision speed
V2= takeoff safety speed (minimum OEI climb speed)
VR= takeoff rotation speed
W= weight
I. Introduction
In
the last decade, aircraft electric propulsion (EP) has become an important research topic. Established industry firms,
startups, government agencies, and academia have all conducted conceptual design studies and flown low-power
prototypes. For designers and analysts, aircraft EP technology can cause difficulties with existing analysis tools, because
many were developed on the assumption that energy would be stored as only fuel. This paper surveys existing tools for
analyzing the mission and economic performance of aircraft with electric propulsion, and describes the development of
a new open-source conceptual design tool with efficient gradients, component-level parametric cost models, and basic
thermal analysis capabilities.
II. The Electric Aircraft Design Problem
Though the field of aircraft EP is still relatively new, insights have emerged about what makes the EP design problem
unique.
Design freedom
Electric propulsion removes constraints that limited where conventional propulsors could be
placed in an aircraft, due in part to the scaling properties of electric motors [
1
]. This leads to more degrees of
freedom in the tradespace and a higher likelihood of finding unconventional configurations with merit.
Close coupling
Designers are using electric propulsion for boundary-layer ingestion [
2
] or lift augmentation [
3
].
2

Recent findings indicate that these design features introduce close coupling between aerodynamics and propulsion [
4
,
5], and between the propulsion, electrical, and thermal management systems [6].
Spatial integration
While spatial or geometric integration is important for aircraft design generally, electric
propulsion introduces more challenging spatial constraints (such as the volumetric energy density of batteries,
which is 18 times lower than fuel [
7
]). The greater design freedom may enable new ways of alleviating spatial
constraints.
Constraints outside the “hard” aerosciences
Past modeling and optimization studies of conventional transport
aircraft [
8
] generally began with a conventional architecture (such as “tube-and-wing with podded engines”) and
made incremental changes based on analysis of the traditional aerosciences disciplines, such as aerodynamics and
propulsion. This approach ensures that regulatory, operational, and safety requirements can be satisfied during the
detailed design phase without having to model them during conceptual design. With a greatly expanded tradespace
and little historical data, each new EP aircraft architecture must be analyzed for safety, operational feasibility, and
cost.
Thermal constraints
Conventional propulsors generally benefit from exhausting their waste heat into the free
stream. Although EP is generally highly efficient, MW-scale electrical power systems generate significant waste heat,
which must be managed locally and at the airplane level [
6
]. Thermal constraints are time- and path-dependent [
9
].
The weight, cost, and power consumption of the thermal management systems (TMS) necessary to handle this
waste heat cannot be adequately addressed using empirical data on conventional environmental control systems.
The weight of TMS alone can be significant—as much as 5% of the weight of the entire EP system even at low
power levels [10].
These features of the electric propulsion design problem introduce competing requirements for models. Low-cost,
moderate-fidelity analysis is necessary in order to explore the broad tradespace; high-fidelity analysis and optimization
is necessary in order to explore close coupling and spatial integration.
III. State-of-the-art in Electric Aircraft Analysis
While “textbook” methods and computer codes for conceptual aircraft design have been refined over decades,
fundamental assumptions of their formulas and empirical datasets break down when electric propulsion is introduced.
For example, the Breguet range equation no longer applies when battery power is used [
7
]. The problem is particularly
pronounced in the area of mission analysis, performance, and sizing codes (admittedly, an imprecise/broad category).
Some researchers have attempted to adapt existing computational tools (based on combustable fuels) to the EP
problem. Perullo et al. [
11
] and Gladin et al. [
12
] adapted NPSS (an engine cycle modeling environment) by adding
medium-fidelity electrical component models. Welstead et al. [
13
] managed to work around limitations in the time-tested
aircraft performance code FLOPS during the trade study for the X-57.
Nevertheless, the trend seems to be to develop built-for-purpose analysis tools to handle the EP problem. At the
“textbook” end of the spectrum, Bauhaus Luftfahrt has developed closed-form expressions for electric aircraft sizing [
14
].
In the realm of analysis codes, NASA has abandoned FLOPS for future EP studies, opting to develop a new code,
LEAPS, which natively supports mission analysis using energy from non-fuel sources [
13
]. At the same time, individual
NASA projects have developed their own EP models [9, 15, 16].
Table 1 summarizes the major, published electric fixed-wing aircraft analysis frameworks, color-coded by level of
fidelity (there are most probably other frameworks in industry that are not public). The table notably excludes electric
VTOL modeling frameworks, which have generally been either low-fidelity or proprietary [
17
,
18
]. Table 1 shows
that certain disciplines have received a great deal of attention (electrical, propulsion), while nearly no cost modeling
has been incorporated (though one-off studies have tackled the issue [
18
]). While one-off safety analyses have been
conducted, safety has not been incorporated into the analysis process in an automated fashion.
There is significant duplication of effort in the research community, particularly within the area of electrical system
modeling and mission analysis. Several codes with similar levels of fidelity for integrating energy used over a mission
have been announced [
9
,
13
,
16
,
19
–
25
], but none have been open-sourced or made publicly available (with the exception
of Stanford’s SUAVE, which includes some support for EP modeling). NASA’s LEAPS is being developed with the
intention of open-sourcing the code, but the release timeline is not clear [13].
A current capability gap is that no publicly-available EP mission analysis and sizing code supports thermal analysis.
Multiple industry and government studies have already demonstrated the need to include thermal constraints in analysis
and optimization at the conceptual level [
6
,
9
,
19
,
26
]. Although the LEAPS energy integration method supports
electrical and fuel energy storage options, thermal analysis is not included.
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Table 1 Electric aircraft modeling and simulation [27]
GT-HEAT [11, 12, 19] NASA X-57 NASA N-3X ESAero [28] Bauhaus Luftfahrt
Aerodynamics FLOPS/drag polar; BLI benefit based on
flat-plate momentum thickness
Design using vortex lattice/boundary layer codes; some
CFD for analysis [3, 16, 29]
CFD results from similar configuration, with increment
for BLI [30]
Drag polar L/D correction methods from Toren-
beek [31]
Structures NA 6 DOF beam FEM [16] NA NA (for MDAO); detailed analysis of split-wing
published in NASA report
NA
Weights FLOPS tops-down methods Parametric wing weight (from Raymer) [3], sized beam
model [16]
WATE for propulsion flowpaths; tops-down kg/kW esti-
mates for electrics/TMS [15, 32]
WATE for fan weight [33]; low-fidelity radiator
model; tops-down empirical for all others
Semi-empirical structural methods; tops-
down kg/kW methods forelectrics [14]
GNC Engine, motor, TMS control variables for
on- and off-design analysis
Full-mission optimal control [9] NA; some discussion of off-design conditions in [34] NA NA
Electrical Moderate fidelity motor/inverter loss mod-
eling; equivalent-circuit battery
Transient battery model based on Theveninequiv circuits
(cell-level). Assumed efficiencies for wire/motors [9]
Conceptual: efficiency stackup method with estimates for
future tech. Transient: RLC circuit model in SimPower-
Systems [15, 35]
Efficiency stackup; battery model unclear Low-fidelity efficiency stackup with empir-
ical battery discharge curve [14]
Turbo/Propulsion NPSS Propeller map from manuf.; prop efficiency from the-
ory [9], blade element momentum theory [16]
NPSS [36] 2D fan analysis using velocity triangles [33]; effi-
ciency maps for turbomachinery
Single prop efficiency parameter [14]
Thermal TMS sizing considering various heat
sources and types of heat sinks
Analytical model for optimization; thermal FEM of mo-
tor [9, 37]
Coolant system load based on efficiency stackup (assume
100% to heat) [15]
Cooling based on flight cond. [38]; TMS model
discussed in [6]
NA
Operating Cost NA exceptfor fuel/energy NA exceptfor fuel/energy NA NA Considers relative cost of fuel/elec; cash
operating cost [39]
Noise
NA NA
ANOPP noise simulation prompted redesign [32]
NA NA
Safety
NA
Comprehensive FMEA [40, 41] FMEA and FTA for loss of thrust; more work needed for
other hazards [42, 43]
Qualitative
NA
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The open-source SUAVE conceptual design tool also does not incorporate thermal management analyses.
Another need exists for an EP model with efficient gradients. When high-fidelity disciplinary analysis codes and
hundreds of design variables are used, the computational cost can be minimized by using gradient-based optimization [
44
].
However, using finite-difference gradients significantly increases computational cost. SUAVE does not support analytic
or automatic gradients, and it does not appear that LEAPS will either. Other NASA electric aircraft studies demonstrate
the benefits of efficient gradients in mission analysis codes [9, 16].
Falck et al. [
9
], and Hwang and Ning [
16
] developed electric aircraft mission analysis codes with moderate fidelity
and efficient gradients. The two codes are similar and rely on optimal control theory and collocation methods to
calculate trajectories, energy usage, and thermal states. Using OpenMDAO, the codes provide efficient gradients for use
in large scale optimization [
16
]. However, optimal control-based methods sometimes introduce robustness problems.
For example, the problem may not converge if initial guesses of the states and trajectories are not close enough. There
may also be cases where non-optimal trajectories form constraints on the design problem (e.g., whenever a human pilot
is in the loop). Neither model supports parametric cost estimates, and neither model is currently publicly available.
Therefore, a need exists for an electric aircraft mission performance and sizing tool with:
•Thermal analysis
•Component-based parametric cost
•Public availability
•Efficient gradients for use with high-fidelity multidisciplinary design optimization (MDO)
To meet these needs, we introduce our conceptual design toolkit, under the working title “OpenConcept”.
OpenConcept is a Python-based library built on top of the NASA-led OpenMDAO 2 framework [
45
,
46
]. At the highest
level, it consists of three parts: a library of propulsion modeling components; a set of reusable, analytically-differentiated
mission analysis codes; and a set of example aircraft models capable of analysis only, simple resizing, or full MDO.
IV. Propulsion Models
We have developed a set of simple conceptual-level electrical and turbomachinery models based on OpenMDAO’s
ExplicitComponent
class. These can be connected together to form all-electric, conventional, series hybrid, parallel
hybrid, or turboelectric architectures. We currently have developed:
•SimpleBattery: electrical power source with constant specific power and specific energy
•SimpleMotor:constant assumed efficiency
•SimpleGenerator: constant assumed efficiency
•SimpleTurboshaft: constant assumed power-specific fuel consumption (PSFC)
•SimplePropeller: shaft power to thrust based on empirical efficiency map
•Splitter: combines or divides power sources or loads
Each component has one or more required scalar sizing design variables (such as motor horsepower or battery weight),
which remain constant throughout the mission. The components also use vectorized inputs and outputs, such as throttle
setting or shaft power in. The user may optionally override default technology parameters (representative of the
state-of-the-art today) with different assumptions (e.g., cost per motor kW, or battery specific energy).
Vectorization is an important concept in OpenMDAO. Instead of invoking a model function once for each flight
condition with scalar inputs, all of the flight conditions for all of the analysis points are fed in at once as
numpy
vectors,
and model computations are handled as vectorized operations. This is a good design practice in Python, since repetitive
computation is performed by the compiled
numpy
code. While we made the design decision not to use an optimal
control formalism for our mission analysis code, our vectorized propulsion model components are fully compatible with
NASA Glenn’s soon-to-be open source optimal control tool, Dymos, which also runs on top of OpenMDAO.
Our design philosophy is that, in general, mechanical components “push” and electrical components “pull”. A
turboshaft engine pushes shaft power proportional to its rated power and throttle setting. An electrical motor pushes
shaft power in the same way, but also pulls an electrical power demand on any connected upstream source, which might
include a generator or a battery.
Where mechanical power (typically, a turboshaft) drives a generator, there is an implicit gap where power in must
equal power out. OpenConcept uses a Newton solver to find the throttle setting for the engine by driving the following
residual equation to zero at every flight condition:
®
Rgen =®
Pgen −®
Preq (1)
Figure 1 shows a twin-motor series hybrid propulsion system built in OpenConcept and used for the case study in
5

Section VI. The “examples” folder of the first software release will contain the code for this architecture as well as
single and twin engine conventional turboprop architectures.
Turboshaft
Generator
Motor
Propeller
Battery
Splitter
Mechanical
Electrical
Motor
Propeller
Hybrid %
Implicit Gap
Throttle
Throttle
Control Param
Set by optimizer subject
to SOC minim.
Set by Newton
solver for steady flt
Set by Newton
solver so generator
meets elec load
Fig. 1 Example of a twin-motor series hybrid electric propulsion model in OpenConcept.
Each component also provides outputs useful for economic optimization or thermal analysis:
•heat_out
(vector) tracks the waste heat produced by the component at each flight condition based on the current
operating power and the component efficiency.
•component_sizing_margin (vector) is P/Prated at each flight condition.
•component_cost (scalar) represents the non-recurring cost of the component in USD.
•component_weight (scalar) is the component’s contribution to airplane empty weight
The
Simple
components generate heat at a rate proportional to their instantaneous power level and their (constant)
inefficiency. They also have linear cost and weight estimates based on the rated power and technological assumptions.
Structuring weight, cost, and thermal data in a standard way makes it simple to connect analysis tools to entire propulsion
layouts.
While OpenConcept is not multifidelity, it is simple for a user to increase model fidelity in a modular way as an
aircraft’s design matures. Using OpenMDAO’s native surrogate modeling classes, it is easy to upgrade the simple
constant performance models with empirical data and maintain accurate partial derivatives. We validated this with the
SimplePropeller component, which uses empirical propulsive efficiency data at a grid of advance ratios and power
coefficients (Figure 4). For example, future users may easily upgrade the motor component with a proprietary efficiency
map based on RPM and torque. However, care must be taken to only use empirical data when the domain of data is
large. Newton solvers and optimizers have a tendency to walk outside the bounds of a surrogate model which can cause
convergence failures.
The
Simple
components can also be replaced with more complex physics-based OpenMDAO models, as long as
they have accurate and efficient partial derivatives. For example, pyCycle is an OpenMDAO-based cycle analysis tool
similar to NPSS. pyCycle can, in principle, accept flight conditions and control inputs from OpenConcept and return
medium-fidelity fuel flows and thrusts [
47
,
48
]. Blade element momentum theory propeller modeling was implemented
in OpenMDAO in [16]. Compiled non-OpenMDAO codes can be “wrapped” in OpenMDAO.
Upcoming work will greatly extend the thermal analysis and design capabilities of OpenConcept, including tracking
component temperatures across mission time points and calculating power required to maintain heat equilibrium using a
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cooling system.
V. Analysis Routines
OpenConcept currently has two primary analysis modules:
takeoff
and
mission
. Together, they provide sufficient
objective and constraint data to perform conceptual-level sizing and MDO of a wide variety of conventional and
unconventional aircraft.
A. Takeoff Analysis
The
takeoff
module calculates balanced field length (BFL) and propulsion system states during the takeoff run,
using methods and assumptions presented in [
49
]. For this flight phase, control inputs are specified by the user (e.g.
100% throttle) and accelerations are determined using a force balance equation. In order to compute balanced field
length, takeoff divides the takeoff into five segments:
1) Takeoff roll at full power from V0to V1
2) Takeoff roll at one-engine-inoperative (OEI) power from V1to VR
3) Rejected takeoff with zero power and max braking from V1to V0
4) Transition in a steady circular arc to the OEI climb-out flight path angle and speed
5) Steady climb at V2speed and OEI power until an obstacle height hois reached
During the takeoff roll (segments 1, 2, and 3), the force balance equation is:
®
dV
dt
=®
T−®
D−µ(mg−®
L).(2)
The accelerate-go distance combines segments 1, 2, 4, and 5, while the accelerate-stop distance includes 1 and 3.
By default,
VR=
1
.
1
Vstall
and
V2=
1
.
2
Vstall
, where stall speed is calculated by a separate model component as a
function of MTOW; these default multipliers (from [
49
]) can be overridden.
V0
is assumed to be 1 m/s in order to
avoid singularities in analysis codes at zero forward speed. Default
µ
is 0.03 during the takeoff roll, and 0.4 during
emergency braking in a rejected takeoff, though this can be also be overridden to simulate wet or snowy runways or
improved aircraft brake systems. The obstacle clearance height is set at 35 feet by default (14 CFR 23), but can be
trivially changed to 50 feet to model a Part 25 transport aircraft.
Equation
(2)
is an ordinary differential equation and must be integrated to obtain distances for segments 1, 2 and 3.
For example, the distance travelled during run up to decision speed (segment 1) is:
RV1=∫V1
V0
dr
dt
dt
dV dV =∫V1
V0
V
adV .(3)
OpenConcept uses an implementation of Simpson’s Rule for numerical integration with analytic derivatives. An integral
can be approximated using Simpson’s rule as follows:
∫xU
xL
f(x)dx ≈1
3∆x(f0+4f1+2f2+4f3+2f5+... +2f2N−2+4f2N−1+f2N)(4)
∆x=xU−xL
2N,(5)
where
N
is the number of Simpson subintervals and
∆x
is the constant spacing between the points
®
f
. This method always
requires evaluating a function at 2N+1points. Simpson’s rule integrates polynomials up to third order exactly [50].
Integration requires known endpoints of the differential variable. During the takeoff phase, OpenConcept ‘knows’
the indicated airspeeds (
V0
,
V1
,
VR
) and integrates numerically with respect to
dV
over each segment. Since acceleration
depends on drag, which depends on velocity but is not time dependent, the acceleration can be computed over a
equally-spaced range of velocities and then integrated using Simpson’s rule.
The
takeoff
module uses a Newton solver to vary the chosen
V1
speed until the accelerate-go and accelerate-stop
distances are equal, or until the accelerate-go distance is longer than the accelerate-stop distance and
V1=VR
. The
accelerate-go distance is then equal to the balanced field length, which can be used as an optimization or sizing
constraint.
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The takeoff module does not consider the slight change in aircraft weight as fuel is burned during takeoff, but it
does track the total fuel (and electricity, if applicable) consumed during the takeoff roll. A limitation of integration
with respect to velocity is that spuriously low takeoff field length results may occur when negative accelerations are
fed to the integrator. This only occurs when the aircraft cannot physically accelerate through the limits of integration.
OpenConcept provides a helper function,
takeoff_check
, to verify that all accelerations are positive during the takeoff
roll. If
takeoff_check
returns an error, the user should specify a higher thrust setting or reduce aircraft weight and
drag until takeoff is physically possible.
B. Mission Analysis
The
mission
module accomplishes two main tasks: setting condition-dependent control inputs necessary for steady
flight, and integrating quantities such as fuel burn and energy over the mission profile. An OpenConcept mission
currently consists of three segments, but can easily be extended to six to include a reserve mission:
•Climb at constant vertical speed and indicated airspeed to the cruise altitude.
•Cruise at constant indicated airspeed and altitude.
•Descent at constant indicated airspeed and vertical speed to the landing altitude.
Figure 2 illustrates flight conditions and aircraft states for a representative hybrid-electric aircraft mission.
0
10000
20000
30000
Altitude (ft)
250
500
750
1000
1250
Motor shp
150
200
250
true airspeed (kias)
200
400
600
Turbine shp
130
140
150
160
170
ind airspeed (ktas)
100
200
300
400
500
600
Gen elec hp
0 20 40 60 80 100 120 140
time (min)
12100
12200
12300
12400
12500
weight (lb)
0 20 40 60 80 100 120 140
time (min)
0.0
0.2
0.4
0.6
0.8
1.0
Battery SOC
Fig. 2 Representative mission profile (from Section VI case study; eb=500 Wh/kg, range=500 nmi)
During mission analysis, the aircraft is treated as a point mass, which changes as fuel burns. At each flight condition,
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the mission.ThrustResidual component calculates the value of the residual equation
®
Rthrust =®
T−®
D−®
mgsin(®
γ).(6)
OpenMDAO’s Newton solver drives these residuals to zero at every flight condition in the mission by varying the
primary thrust control parameter (usually either motor or engine throttle). If more than one independent thrust control
parameter is available (for example, high-lift and cruise propellers as in the X-57), the user can specify some of the
parameters and let the Newton solver find the remaining one. Alternatively, an optimizer can find the optimal value for
every control parameter by treating the thrust residual as an equality constraint.
The steps in an OpenConcept mission analysis during each Newton iteration are:
1) Generate vectors representing the flight condition at each point in time during the mission.
2) Calculate atmospheric properties [51].
3) Compute climb and descent phase distances and times to obtain cruise distance and time.
4)
Run an OpenConcept propulsion model at each flight condition to obtain fuel flows, battery loads, thrusts, and
constrained quantities like heat output.
5)
Integrate fuel flow and battery load with respect to time using Simpson’s rule to obtain aircraft weight and battery
state of charge (SOC) vectors.
6) Calculate flight CLand drag.
7) Calculate the thrust-drag residual.
Many conventional mission analysis codes include mission range as an output, assuming that the aircraft will fly
until the fuel weight is zero. OpenConcept uses mission range as an input, since the limiting all-electric case has no
change in weight to integrate. Therefore, the cruise phase range and time depend on the climb and descent phases
as well as the specified total range. Because of the current mission specification scheme, OpenConcept “knows” the
beginning and end time points of each phase; fuel and battery states are then integrated with respect to time using
Simpson’s rule. An alternative, more flexible integration strategy could include integrating with respect to altitude for
the climb and descent segments, and with respect to distance for the cruise segment.
Once the Newton solver has converged the mission, thrust balances drag (and weight, if climbing or descending), lift
matches weight, and any hybrid turbomachinery components are producing the correct shaft power to meet electrical
loads.
The
mission.ComputeMissionResiduals
component then assesses whether the aircraft’s design weights are
consistent with the mission being flown. The residuals are:
RTOW =WTO −Wfuel −Wempty −Wpayload
Rvol =Wfuel,max −Wfuel.
(7)
For aircraft with batteries, the
mission.ComputeMissionResidualsBattery
class should be used instead. In this
case, the residuals are:
RTOW =WTO −Wfuel −Wempty −Wpayload −Wbatt
Rbatt =Ebatt,max −Ebatt,used (8)
Rvol =Wfuel,max −Wfuel
The Newton solver does not automatically drive these high-level equations to zero, which enables analysis of aircraft
where not all of the fuel or battery is consumed during a mission. However, it can be convenient to use OpenConcept in
“sizing” mode, where these residuals are posed as inequality constraints to the optimizer. TOW and battery weight are
then set to the minimum required to fly the mission. When using the
mission
module without an optimizer in the loop,
the user must manually ensure that the mission weights are feasible.
Partial derivatives for all of the
mission
analysis methods have been verified using OpenMDAO’s
check_partials
method. Users of OpenConcept should always verify partial derivatives when implementing new components or
modifying existing ones. Simple errors in partial derivatives inevitably cause major Newton and optimizer convergence
issues.
The number of Simpson subintevals in each mission segment is a major driver of the size of the linear algebra
problem solved by OpenMDAO. We performed a convergence study of the integration method using a representative
hybrid aircraft model which uses significant battery and fuel energy (described in Section VI). Table 2 illustrates that
very accurate fuel burn and BFL results can be obtained with a relatively minimal number of points per mission segment.
Five intervals per segment is the default and integrates fuel burn and BFL nearly exactly; using three intervals runs
significantly faster and may be a good lower bound for running numerous optimizations.
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Table 2 Simpson integration convergence
Simpson Intervals Points Fuel Burn FB Error BFL BFL Error
15 31 314.18975 0.000% 1357.0047 0.00%
8 17 314.18975 0.000% 1357.0047 0.00%
5 11 314.19058 0.000% 1357.0053 0.00%
4 9 314.19259 0.001% 1357.1920 0.01%
3 7 314.19254 0.001% 1357.2149 0.02%
2 5 314.19330 0.001% 1356.2602 −0.05%
1 3 314.31917 0.041% 1357.5849 0.04%
VI. Case Study: Design of an Electric Aircraft for Minimum Operating Cost
To validate the code, we conducted a case study where the notional goal is to convert a Beechcraft King Air C90GT
to series hybrid electric propulsion.
A. Conventional Baseline
To test OpenConcept’s propulsion modeling and analysis routines on a simple case, we first modeled a single
engine turboprop, the SOCATA/Daher TBM 850. Structural and system weights were estimated using textbook
formulas [
49
,
52
], with a constant factor of 1.6 applied to structural weight in order to match published empty
weights [
53
]. Cruise drag was estimated using a drag polar formulation, with induced drag and zero-lift drag estimated
using tops-down methods [
49
].
CLmax
was set to match the nominal takeoff rotation speed to handbook values [
54
].
Initial model runs using the a priori estimate of
CD0
resulted in a fuel burn total close to the published value; we
adjusted CD0to match fuel burn and maximum range.
Propeller maps from manufacturers are closely held. For our case study, propeller efficiency was estimated from a
published map [
55
]. We “compressed” this map in the
CP
axis so that the peak efficiency point better matched our
anticipated operating point, and visually extrapolated the map into the higher
CP
region. We also adjusted the very low
speed propulsive efficiency downward to reduce spuriously high thrust levels during the takeoff roll. The balanced field
length for the single-engine TBM is simply the takeoff distance with full takeoff power since the one-engine-inoperative
takeoff distance is not defined.
We modeled the Beechcraft King Air C90GT in a nearly identical way, except that the King Air has two engine
and propeller components and a structural factor of 2.0 [
56
]. On the King Air, the PT6A-135A engine is derated by
about 25% (from 750 hp to 550 hp) for structural reasons. Balanced field length was calculated using 25% derated
takeoff power, and zero power in Engine 2 following
V1
. Table 4 illustrates input and model output data for the TBM
850 and King Air. Balanced field length for the King Air matched published figures quite closely, but the simulated fuel
burn was about 25% lower. It is possible that the propeller map is not representative of the King Air at the cruise flight
condition, or that in reality the derated PT6A-135A engine is operating significantly below peak efficiency at our cruise
throttle setting and altitude. We did not attempt to vary
CD0
upward enough to match the published (higher) fuel burn
for the King Air.
10

(a) Daher TBM 850 (photo by Gyrostat, CC-BY-SA) (b) Beechcraft King Air C90GTi (photo by Joao Carlos
Medau, CC-BY)
Fig. 3 OpenConcept benchmark aircraft
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Advance ratio (V/nD)
0.0
0.5
1.0
1.5
2.0
2.5
Power coefficient
cp
(
P
/
n
3
D
5)
0.00
0.12
0.24
0.36
0.48
0.60
0.72
0.84
0.96
Fig. 4 Propeller efficiency map for the case study (notional; not to be used for industry studies).
B. Sizing the Propulsion System of a Series-Hybrid Conversion
Next, we examine the feasibility of a drop-in replacement of the twin turboprop architecture of the King Air with a
series hybrid system. The pupose of this study is not to prove or disprove the viability of electric propulsion for light
aircraft retrofit; we are simply illustrating the use of OpenConcept for an aircraft study. A modern clean-sheet design
would have weight and drag advantages which may make electric propulsion feasible at longer ranges and lower eb.
We assume that any single component of the propulsion system may fail and that the aircraft must be able to continue
safe flight and landing during the takeoff phase. To achieve this, the series hybrid architecture includes the following
features:
•Two motors and propellers (providing redundancy in the event of motor or propeller failure)
•
Batteries split into at least two independent packs, with battery power alone used for takeoff (providing redundancy
against single engine failure on takeoff)
These conditions ensure a level of safety on takeoff equivalent to a twin turboprop. Specific power, efficiency, and cost
11

assumptions for individual powertrain components are stated in Table 3.
Table 3 Powertrain technology assumptions
Component Specific Power (kW/kg) Efficiency Cost PSFC (lb/hp/hr)
Battery 5.0 – $50/kg –
Motor 5.0 97% $100/hp –
Generator 5.0 97% $100/hp –
Turboshaft/Prop 7.15∗– $775/shp 0.6
The
scipy.optimize
SLSQP algorithm was used to size the propulsion system components (motor, engine, and
generator sizing) for minimum fuel burn on the design mission. MTOW, wing area, and all other parameters remain
equal to the King Air baseline. The optimization problem is formulated as:
minimize: fuel burn
by varying:
Wbattery
Pmotor (rated)
Pturboshaft (rated)
Pgenerator (rated)
HE(degree of hybridization w.r.t energy)
subject to scalar constraints:
RTOW =WTO −Wfuel −Wempty −Wpayload −Wbatt ≥0
Rbatt =Ebatt,max −Ebatt,used ≥0
BFL ≤4452 ft (no worse than baseline)
and vector constraints:
®
Pmotor ≤1.05Pmotor (rated)
®
Pturboshaft ≤Pturboshaft (rated)
®
Pgenerator ≤Pgenerator (rated)
®
Pbattery ≤Wbattery ·pb
The optimizer successfully sized the motor, generator, battery, and turboshaft and found the mix of electric and fuel
energy (degree of hybridization,
HE
) which minimized fuel burn. Design variables and simulation outputs are listed in
Table 4. Using battery specific energy of 750 Wh/kg, the series conversion could not meet the 1000nmi design range
of the original King Air. At a maximum range of 762nmi, the optimizer converged on a design with essentially the
minimum allowable amount of battery (sized by power, not energy, at takeoff). Since the 762 nmi mission was at the
very limit of the airplane’s capability, we changed the design range to 500 nmi and resized the propulsion system again.
This time,
eb
significantly affected the sizing (at 250, 500, and 750 Wh/kg). All three designs had identical motor sizing
(to meet the takeoff constraint at MTOW), but the generators and engine power increased with decreasing
eb
due to the
larger fraction of power from fuel. The 750 Wh/kg case burned 38% less fuel than the 250 Wh/kg case.
To further validate the speed and flexibility of OpenConcept, the first author conducted a rough feasibility study of
an all-electric conversion of the Cessna C208B Grand Caravan. The objective was to assess the feasibility of using
all-electric propulsion to handle cargo flights of one hour or less, as proposed by the start-up firm MagniX. From
start to finish (including gathering input data on the Grand Caravan online and assessing control surface areas using
photogrammetric methods), the study took less than 90 minutes. The results are not tabulated here, but indicated that
the idea could be feasible with current technology.
∗does not include 104kg base wt
12

C. Multidisciplinary Design Optimization for Minimum Fuel Burn
Following successful demonstration of the simple sizing capability, we increased the design freedom of the optimizer
to include MTOW, fuel volume, wing area, and prop diameter. The optimization problem is:
minimize: fuel burn +0.01MT OW
by varying:
MTOW
Sre f
dpr op
Wbattery
Pmotor (rated)
Pturboshaft (rated)
Pgenerator (rated)
HE(degree of hybridization w.r.t energy)
subject to scalar constraints:
RTOW =WTO −Wfuel −Wempty −Wpayload −Wbatt ≥0
Rbatt =Ebatt,max −Ebatt,used ≥0
Rvol =Wfuel,max −Wfuel ≥0
BFL ≤4452 ft (no worse than baseline)
Vstall ≤81.6kt (no worse than baseline)
and vector constraints:
®
Pmotor ≤1.05Pmotor (rated)
®
Pturboshaft ≤Pturboshaft (rated)
®
Pgenerator ≤Pgenerator (rated)
®
Pbattery ≤Wbattery ·pb
Results for 250, 500, 750, and 1000 Wh/kg on a 500 nmi design mission are listed in Table 4. In these four
optimizations, we see some hints of discontinuities in the hybrid electric design space. At 1000 and 750Wh/kg, the
airplane prefers to use no fuel and fly the design mission completely on batteries. At 500 Wh/kg, the optimizer hits
the MTOW upper bound (5700kg, above which EASA and the FAA require pilots to obtain a type rating). It uses as
much battery as possible, supplementing with just enough fuel to meet the required range. At 250 Wh/kg, the optimizer
prefers to reduce battery weight to the takeoff power-constrained minimum and reduce MTOW below the baseline.
This fuel burn optimization was repeated over a grid of 252 individual combinations of design range and
eb
(Figure 5). Only one out of 252 optimization runs failed to converge, and that run was easily rectified by changing the
starting “guess” to a more realistic set of design weights and component sizes.
The primary finding from this set of optimizations is that, for an airplane with King Air-like structural efficiency and
aerodynamics, hybrid propulsion is generally only preferable to all-fuel or all-electric operation when an upper limit
exists on MTOW. Practical MTOW limits might include a retrofit application with an existing airframe, or regulatory
limits (as mentioned above). This finding may not apply generally to more aerodynamically and structurally efficient
clean-sheet designs.
In the upper left corner (short range, high
eb
), the optimizer can eliminate fuel altogether. Since fuel burn is zero
everywhere in this triangular region, using pure fuel burn as the objective function will fail to converge on a reasonable
airplane. A small additional term proportional to MTOW was added to the objective function in order to encourage the
optimizer to reduce MTOW (and therefore, battery weight) as much as possible, once fuel burn is reduced to zero.
D. Multidisciplinary Design Optimization for Minimum Cost
We ran an additional grid of 252 optimizations with respect to operating cost. The optimization problem is
formulated as follows:
13

300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Fuel mileage (lb/nmi)
0.0
0.3
0.6
0.9
1.2
1.5
1.8
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Trip DOC (USD) per nmi
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Degree of hybridization (electric percent)
0
20
40
60
80
100
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Maximum Takeoff Weight (lb)
8000
9000
10000
11000
12000
Fig. 5 Minimum fuel burn MDO results.
14

minimize: trip cost
by varying:
MTOW
Sre f
dpr op
Wbattery
Pmotor (rated)
Pturboshaft (rated)
Pgenerator (rated)
HE(degree of hybridization w.r.t energy)
subject to scalar constraints:
RTOW =WTO −Wfuel −Wempty −Wpayload −Wbatt ≥0
Rbatt =Ebatt,max −Ebatt,used ≥0
Rvol =Wfuel,max −Wfuel ≥0
BFL ≤4452 ft (no worse than baseline)
Vstall ≤81.6kt (no worse than baseline)
and vector constraints:
®
Pmotor ≤1.05Pmotor (rated)
®
Pturboshaft ≤Pturboshaft (rated)
®
Pgenerator ≤Pgenerator (rated)
®
Pbattery ≤Wbattery ·pb
A notional trip cost model was constructed as follows:
Trip cost =cfuel +celectricity +cbattery +cdepreciation
where:
cfuel =($2.50/gal)Wfuel/ρfuel
celectricity =($36.00/MWh)Ebatt, used
cbattery =($50.00/kg)Wbatt /nbatt cycles
cdepreciation =caircraft/nflights,daily/365 days/nyears
and:
caircraft =(OEM premium)cairframe +cengine +cmotors +cgenerator
cairframe =($277/kg)OEW −Wengines
cengine =($775/shp)Pengine (rated)
cmotors =($100/shp)Pmotors (rated)
cgenerator =($100/shp)Pgenerator (rated)
nbatt cycles =1500
nflights,daily =5
nyears =15
OEM premium =1.1
15

300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Fuel mileage (lb/nmi)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Trip DOC (USD) per nmi
0.200
0.275
0.350
0.425
0.500
0.575
0.650
0.725
0.800
0.875
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Degree of hybridization (electric percent)
0
20
40
60
80
100
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Maximum Takeoff Weight (lb)
8000
9000
10000
11000
12000
Fig. 6 Minimum Cost MDO Results
16

Fuel and electricity prices were picked as representative wholesale values for 2018. Battery cost was estimated
assuming $200 per kWh, and specific energy of 250 Wh/kg. Airframe cost factor includes everything except propulsion
components and was estimated based on general light turboprop pricing trends (in current USD currency) Engine
price per shaft horsepower was estimated based on new PT6A prices listed on various online marketplaces. The OEM
premium assumes a 10% operating profit above Tier 1 supplier costs. Note that the cost model does not contain any
contribution from maintenance cost (for either conventional or electric propulsion), nor does it include crew costs or
landing fees. Crew cost is neglected, as the missions being tested are equal in block time. Maintenance and crew cost
could be easily added to the cost calculation for future studies.
The price of aerospace-certified propulsion motors and generators, and the cycle life of aerospace-grade batteries,
are all currently unknown since they are not commercially available. The maintenance cost of electric propulsion is also
unknown, although it can be expected to be lower than for turboprop engines on an hourly basis. The effect of these
economic assumptions on optimal design can be easily tested using OpenConcept.
Figure 6 shows the cost, fuel burn, degree of hybridization, and MTOW for the cost-optimized aircraft. Compared to
Figure 5, optimizing for cost has moved the electric propulsion feasibility line up and to the left. Under this (debatable)
set of economic assumptions, electric propulsion is more favorable for fuel burn reduction than for economics, at least at
moderate specific energy levels.
We ran an additional 21 optimizations of a King Air-like conventional twin turboprop, but allowed the optimizer to
perform full MDO (same rules as for the hybrid MDO study, but without the electric propulsion design varibles or
constraints). The purpose of these cases was to provide a fair comparison between a conventional architecture and the
series hybrid architecture. Figure 7 shows the relative difference between the cost-optimzed hybrid electric aircraft
compared to cost-optimized coventional aircraft at the same design range. In the lower right half, the series hybrid
(effectively turboelectric) design actually costs more to operate than a conventional twin turboprop. A breakeven point
runs nearly linearly from lower left to upper right; conventional and electric are economically equivalent along this
line. Above the breakeven line (in the hybrid regime), costs fall rapidly as the optimizer can trade fuel for batteries and
reduce the size of the turbogenerator system. Once turbogenerator power is reduced to zero, the (low) costs remain
relatively stable even as battery specific energy improves. Figure 7 illustrates that the potential for cost savings is high if
a significant proportion of battery power can be used. Turboelectric propulsion by itself is not an efficient replacement
for turboprop engines, at least when no ancillary aerodynamic or propulsive efficiency benefit can be realized. If
maintenance costs were modeled, we could expect the breakeven line to move down and to the right.
VII. Using OpenConcept for Technology Assessment
A fundamental challenge faced by aircraft designers in industry is finding the right time to incorporate new technology
onto an aircraft family. We learned in Section VI that a tipping point exists in the design space: an optimal conventional
airplane will be as light as possible, but once batteries are economically favored, the best aircraft is as heavy as possible
with batteries. This tendency makes retrofitting a hybrid or all-electric powertrain to an existing airframe infeasible,
assuming the user wishes to fly similar mission ranges. Therefore, clean sheet designers must be able to predict at what
point in time electrical component and battery technology will become economically favored for the chosen mission.
OpenConcept can conduct the sort of low-cost, moderate-fidelity tradespace exploration required to answer this question.
The Section VI hybrid King Air airframe was heavy and had relatively high parasitic drag. To simulate improved
airframe technology, the
CD0
was reduced from 0.022 to 0.018, and the structural weight factor reduced from 2.0
to 1.5 (a 33% structural weight reduction; for example, using carbon composites and reclaiming margin). We then
ran an additional 252 MDO scenarios on the same range–versus–
eb
grid, using the minimum cost optimization rules
presented in Section VI. We also ran the full span of design ranges on the conventional twin turboprop model, using the
same structural weight and drag reductions, to provide a fair comparison. The total time (including setup) to run the
optimizations and analyze results was approximately 9 hours on a standard Windows laptop. Figure 8 illustrates the fuel
burn, cost, weight, and energy.
We found that, as predicted, the lower-weight and lower-drag airplane favored electric propulsion at lower
eb
across
all ranges. The economic breakeven line moved down about 50 Wh/kg, nearly into the present-day feasibility zone for a
300 nmi mission (Figure 9). The line is also shallower, meaning that the effect of the weight and drag reduction is felt
more strongly at longer ranges.
The published literature has contributed to industry understanding of the primary technical and performance drivers
of electric flight. However, results from published studies are not general enough to tell a design team when to implement
electric propulsion for their particular set of requirements. Using OpenConcept, designers can rapidly test their internal
17

300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Fuel Burn - Percent Change
100
75
50
25
0
25
50
75
100
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
DOC (ex-maintenance) - Percent Change
60
48
36
24
12
0
12
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Trip Energy Cost - Percent Change
100
75
50
25
0
25
50
75
100
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
MTOW - Percent Change
100
75
50
25
0
25
50
75
100
Fig. 7 Minimum Cost Hybrid vs Minimum Cost Conventional (MDO results)
18

300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Fuel Burn - Percent Change
100
75
50
25
0
25
50
75
100
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
DOC (ex-maintenance) - Percent Change
60
48
36
24
12
0
12
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Trip Energy Cost - Percent Change
100
75
50
25
0
25
50
75
100
300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
MTOW - Percent Change
100
75
50
25
0
25
50
75
100
Fig. 8 Minimum Cost Hybrid vs Minimum Cost Conventional with reduced structural weight and drag (MDO
results)
19

300 400 500 600 700 800
Design range (nmi)
300
400
500
600
700
800
Specific energy (Whr/kg)
Reduced OEW, Drag
Section VI Case Study
DOC (ex-maintenance) - Percent Change
60
48
36
24
12
0
12
Fig. 9 Improved airframe technology makes electric propulsion economically feasible at lower specific energies
assumptions and generate large amounts of trade study data with modest setup time and computational resources.
Similar studies could be conducted with respect to specific power or efficiency of individual electrical components,
accounting for time-dependent heating and cooling constraints.
20

Table 4 Baseline analysis, component resizing, and MDO results (minimum fuel burn objective)
TBM850
(Model)
TBM850
(Published)
King Air
C90GT
(Model)
King Air
C90GT
(Published)
Hybrid
Conversion -
Max Range
Hybrid
Conversion
750
Hybrid
Conversion
500
Hybrid
Conversion
250
Hybrid MDO
1000
Hybrid MDO
750
Hybrid MDO
500
Hybrid MDO
250
King Air
MDO
1000nmi
King Air
MDO 500nmi
Optimization Rules Analysis Analysis Analysis Analysis Comp Sizing Comp Sizing Comp Sizing Comp Sizing MDO MDO MDO MDO MDO MDO
Specific Energy (Wh/kg) – – – – 750 750 500 250 1000 750 500 250 – –
Design Range (nmi) 1250 1150+100 1000 894+100 761.6 (max) 500 500 1000 500
MTOW (lb) 7392 7392 10100 10100 10100 10156 12505 12566 8912.8 9367 7941
OEW (lb) 4756 4762 7177 7150 7433.9 7197 7328 7410 6755 7594 7967 6828.5 6827 6279
Max Fuel Wt (vol imit, lb) 2000 2000 2570 2570 2570 1102 1102 1102 1102 1540 1102
MLW (lb) 7000 7000 9600 9600 9600 9600 9600
Rated TO SHP (each) 850 850 550 550 527.2 527.2 572.2 527.2 519.4 640.0 649.9 456.6 511.5 434.6
Turboshaft SHP (each) 850 850 550 550 1061.0 629.0 867.6 1018.2 – – 641.1 940.0 511.5 434.6
Generator SHP – – – – 1029.2 610.6 841.5 987.6 – – 630.4 907.9 – –
Prop Diameter (ft) 7.58 7.58 7.50 7.50 7.50 7.22 7.22 7.22 7.22 7.22 7.22
PSFC (lb/hp/hr) 0.60 0.60 0.60 0.6 0.6
Wing Ref Area ( ft2)193.8 193.8 294.0 294.0 294.0 296.0 364.6 366.3 259.8 273.1 231.5
Wingspan (ft) 41.5 41.5 50.2 50.2 50.2 50.4 55.9 56.1 47.2 48.4 44.6
Aspect Ratio 8.95 8.95 8.58 8.58 8.58 8.58 8.58
Flaps-Down CLmax 1.70 1.52 1.52 1.52 1.52
Oswald Efficiency 0.78 0.80 0.80 0.80 0.80
CD0at Cruise 0.0205 0.0220 0.0220 0.0220 0.0220
CD0at Takeoff 0.0300 0.0290 0.0290 0.0290 0.0290
Takeoff Rotation Speed (kias) 89.6 90 89.8 90 89.8 89.8 89.8 89.8 89.8 89.8 89.8 89.8 89.8 89.8
Battery Wt (lb) – – – – 381.4 1397.9 1079.2 754.6 2400.8 3911.4 3077.6 330.3 – –
Takeoff Battery % – – – – 100% 100% – –
Cruise Battery % – – – – 5.0% 43.7% 22.3% 8.6% 100.0% 100.0% 52.5% 3.6% – –
Design Payload (lb) 1000 1000 1000 1000 1000 1000 1000
Cruise Speed (KIAS) 201 201 170 170 170 170 170
Cruise Altitude (ft) 28000 28000 29000 29000 29000 29000 29000
Climb Rate (ft/min) 1500 1500 1500 1500 1500
Climb Speed (KIAS) 124 124 124 124 124
Descent Speed (KIAS) 140 130 130 130 130
BFL, SL, ISA+0 (ft) 2844 2838 4452 4519 4452 4452 4452 4452 4452 4452 4452 4452 4452 4452
BFL % Error 0.2% -1.5%
Cruise Fuel Flow (lb/hr) 414 415 468 612
Cruise Fuel % Error -0.3% -23.5%
Design Mission Fuel (lb) 1605.8 1663.8 1284.2 504.5 692.7 811.0 0.0 0.0 520.0 754.0 1540.3 663.4
Design Mission Fuel (lb/nmi) 1.285 1.664 1.686 1.009 1.385 1.622 0.000 0.000 1.040 1.508 1.540 1.327
Bold numbers indicate an active design variable bound or constraint
21

VIII. Conclusions
This paper examined unique features of the electric aircraft design problem and briefly surveyed the current state of
electric aircraft modeling tools. We identified gaps in current capabilities. In particular, a need existed for an open-source
mission analysis tool compatible with electric propulsion, including cost and thermal analysis. We introduced a new,
open-source mission analysis and conceptual sizing tool—OpenConcept—written in Python and running atop of the
OpenMDAO framework. OpenConcept’s analytic gradients enable the use of Newton solvers and efficient gradient-based
optimization. This enabled us to run hundreds of individual optimizations without high-performance computing
hardware, and rapidly explore the electric propulsion tradespace for a twin series hybrid concept. More than 750
individual MDO cases across a wide range of battery specific energy levels and design ranges revealed discontinuities
and tipping points in the design space. We demonstrated the ability to rigorously quantify the influence of operating
economics on optimal fixed-wing hybrid electric aircraft design. While many published studies use fuel burn as an
objective function, we find that this tendency may be overstating the economic benefit realizable through electric
propulsion, especially at lower specific energies. Using operating cost as an objective function balances the weight gain
due to electric propulsion with the fuel burn reduction in a more realistic way. Finally, we explore the effect of airframe
technology on the conventional versus electric economic tipping point. A more efficient aerostructure substantially
reduces the technological requirements necessary for electric propulsion to be economically favorable. The authors note
that tipping points and trends in the design space for this airframe and architecture may not apply to other conceptual
designs. Using OpenConcept, other researchers and designers may rigorously and rapidly examine the tradespace for
each architecture, mission, and set of economic inputs.
Acknowledgements
The first author was supported by the National Science Foundation Graduate Research Fellowship Program under
Grant DGE 1256260. Any opinions, findings, and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work was also
supported in part by the U.S. Air Force Research Laboratory (AFRL) under the Michigan-AFRL Collaborative Center
in Aerospace Vehicle Design (CCAVD), with Dr. Richard Snyder as the task Technical Monitor. Justin Gray at NASA
Glenn provided extensive advice and support for OpenMDAO.
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