# To charge in-flight or not: an inquiry into parallel-hybrid electric
aircraft configurations
via optimal control

Mengyuan Wang, and Mehran Mesbahi Mengyuan Wang and Mehran Mesbahi are with
the William E. Boeing Department of Aeronautics and Astronautics, University
of Washington, Seattle, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We examine two configurations for parallel hybrid electric aircraft, one with,
and one without, a mechanical connection between the engines and the electric
motors. For this two designs, we then review the power allocation problem in
the context of aircraft energy management for a 19-seat conceptual Hybrid
Electric Aircraft. We then represent the original optimal control problem as a
finite-dimensional optimization and validate the second-order sufficient
conditions for global optimality of the obtained solution. This is then
followed by a sensitivity analysis of the fuel consumption on the initial
aircraft weight and flight endurance. Our simulation and theoretical results
clarify the limited benefit of charging the battery in-flight for this class
of hybrid electric aircraft to reduce $\mbox{CO}_{2}$ emissions.

###### Index Terms:

Hybrid-Electric Aircraft; hybrid architectures; optimal control; sensitivity
analysis

##  I Introduction

Aircraft electrification highly depends on the development of more efficient
battery technologies, as the current energy density of most advanced batteries
falls short of the desired levels for commercial aircraft. Consequently,
developing Hybrid-Electric Aircraft (HEA) has become a viable strategy for
next-generation aviation to satisfy reduced $\mbox{CO}_{2}$ emissions
requirements in the coming decades.

The existing literature on hybrid architectures for aircraft propulsion [1, 2,
3, 4] mostly borrows principles from hybrid electric ground vehicles, often
assuming that the gas turbine and electric motor should be connected
mechanically, allowing the turbine to charge the battery during flight. This
connective mechanism has several potential benefits: (a) The engine can run in
its most efficient region to conserve fuel by using the electric motor as an
extra load, and (b) The electric motor can work in“one-engine-inoperative”
(OEI) mode by charging the battery to its fully charged state after the
climbing phase. These benefits are compatible with series, parallel, and
series-parallel configurations. On the other hand, MagniX–a manufacturer of
electric propulsion systems for electric aircraft–has proposed a novel
parallel hybrid structure in 2021, in which the engine and electrical paths
are independent (without a mechanical connection in place). In this structure,
the electric motor and the engine work together during the climbing phase, and
the engine handles the cruise phase and the descent phase. In this
configuration, one can bypass the use of the engine and transition the HEA to
an All-Electric Aircraft (AEA)–when the energy density of the battery reaches
the desired level. In this paper, these two structures are compared in the
context of the aforementioned benefits, and specifically, through the lens of
optimal power allocation.

With two available energy sources for the propulsion system, the importance of
power allocation algorithms cannot be overstated for HEA. In this directon,
Leite and Voskuijl [5] utilized dynamic programming to obtain an optimal
energy management scheme for HEA for a given flight profile; Doff-Sotta et al.
[6] on the other hand has developed a convex formulation for optimal HEA
energy management. The latter work, in particular, showed that optimal control
strategy significantly reduces fuel consumption as compared with heuristic
methods. Donateo et al. [7] in the meantime, applied dynamic programming for
energy management for a lightweight rotorcraft, where the corresponding
optimal control was formalized in terms of a shortest path problem on a graph,
followed by the application of Dijkstra algorithm for its solution.

Although there are a few manufactured HEA on the market [8, 9, 10, 11],
publicly accessible technical data and characteristics for these aircraft are
rather limited. In the meantime, there are a number of studies that have
explored the design space of HEA. For example, De Vries et al. [12] provided a
comprehensive approach for the conceptual design process of HEA. Multiple
hybrid-electric structures, including the all-electric propulsion system, have
also been considered. Isikveren et al. examined the design of vertical take-
off and landing aircraft [13]. Finger et al. compared two design approaches
utilizing an existing 19-seat conventional aircraft as the baseline model
[14].

In this work, we first review the power allocation problem for a given flight
profile, formalized in terms of minimizing the corresponding fuel consumption
by solving an optimal control problem (OCP). Subsequently, using an
approximate linear fuel flow rate model, this OCP is represented as a finite-
dimensional optimization problem. This formulation of the power allocation
problem is then applied to a 19-seat conceptual aircraft model, where the
second-order sufficient conditions are also examined for the corresponding
optimization problem. In this case, the optimal power allocation strategy is
tailored specifically for this mid-size aircraft with turboprop engines.
Lastly, two hybrid structures (with and without mechanical connection in the
propulsion system) are compared in the context of fuel optimized power
allocation, allowing us to evaluate the utility of the mechanical connection
to charge the battery in-flight.

The remainder of the paper is organized as follows. In Section II, two
parallel hybrid-electric architectures and the technical data of the
conceptual aircraft model are introduced. In Section III, the longitudinal
aircraft dynamics and simplifications made for the formulation of optimal
control problems are provided; the power relation in the propulsion system is
also discussed. In Section IV, the previously proposed optimal control problem
is reviewed and re-written based on a linear approximation of the engine’s
flow rate. Subsequently, this optimal control setup is formulated as a finite-
dimensional optimization problem. Using the power allocation optimization
formulation in Section V, and the corresponding sensitivity analysis, the fuel
consumption between charging and not charging the battery during the cruise
phase is then compared. This is then followed by evaluating the aircraft fuel
consumption with different battery configurations (leading to different take-
off weights). These two facets shed light on our evaluation for the question
posed in the title of the manuscript. Conclusions and future work are then
provided in Section VI.

##  II Hybrid-electric architectures and a conceptual aircraft model

In this section, two parallel hybrid-electric architectures are discussed in
Section II-A; technical data for the conceptual aircraft model and components
in the propulsion system are provided in Section II-B.

###  II-A Hybrid-electric architectures

For a conventional aircraft, relying on engines for propulsion, it is
challenging to devise means of operating the engine in its peak efficiency
regime; this is primary due to considerable differences in power requirements
for the climb and the cruise phases of a typical flight profile. One of the
primary advantages of hybrid electric aircraft is the ability to downsize the
engine and use the electric motor as a supplemental propulsion, allowing the
engine to operate at its most efficient regime during flight. This point of
view when adopted for developing aircraft hybrid-electric architectures, is
often combined with design principles from ground vehicles; as such, HEA
designers often assume a mechanical connection in the propulsion system. The
most commonly used architectures include series, parallel, and series-parallel
configurations. The parallel architecture is the most popular configuration
due to its relatively higher efficiency and lower weight (in comparison with
the series configuration), as well as its simpler energy management (as
compared with the series-parallel configuration). In general, a mechanical
connection in the propulsion system enables conventional engines to work more
efficiently, either by decoupling the engine from the propeller (in the series
configuration) or by taking the electric motor as an extra load for the engine
(in the parallel configuration). Nevertheless, to the best of our knowledge,
this potential advantage of having a mechanical connection in HEA has not been
validated, neither theoretically nor experimentally.

The mechanical connection between the conventional engine and the electric
motor often necessitates a complicated clutch/gearing mechanism and advanced
control algorithm for the hybrid propulsion system. In addition, charging the
battery during flight introduces new challenges, such as the need for more
sophisticated cooling systems and the possibility of battery pack degradation.

MagniX has recently proposed a novel hybrid structure for aircraft electric
propulsion. In particular, the Magnix design has the conventional engine and
electric paths mechanically independent, with completely separate control
systems for two propulsion paths. With this architecture, the hybrid electric
propulsion system can still be advantageous as it allows efficient engine
operation. In addition, the absence of mechanical connections reduces the
aircraft’s total weight and may improve safety due to less complexity in the
mechanical system.

In this paper, we examine the standard parallel architecture (referred to as
the connected configuration), and the MagniX architecture, which is called the
independent configuration. In the connected configuration, the internal
combustion engine/gas turbine and the electric motor are both connected to the
propeller, as illustrated in Fig. 1. As a result, this configuration supports
the following operating modes:  a) engine-alone mode,  b) motor-alone mode,
c) combined mode (in which the engine and the motor drive the propeller
simultaneously) and  d) power split mode (in which the engine drive the
propeller and charge the battery).

In an independent configuration in Fig. 2, the propulsion system retains the
first three operating modes; the only difference is that the engine cannot
charge the battery during flight. This paper provides a complete comparison
between the fuel consumption of these two architectures using the power
allocation problem formulation and the corresponding algorithm and sensitivity
analysis [15].

Figure 1: Connected parallel hybrid architecture Figure 2: Independent
parallel hybrid architecture

Before delving into the power allocation problem formulation and its
consequences for the main theme of this manuscript, we provide an overview of
the aircraft model and its key components and subsystems of relevance to the
power allocation problem and subsequent analysis.

###  II-B HEA propulsion system model

We use the conceptual HEA propulsion system model developed by Finger, et al.
[14] for subsequent analysis. Finger, et al. considered an existing 19-seat
commuter aircraft as the baseline and examine two approaches for the HEA
preliminary design. The conventional aircraft was re-sized to accommodate the
hybrid electric propulsion technology. Some of the key characteristics of one
of the configurations used in [14] are shown in Table I. Note that parameters
with ∗ are only estimated values.

TABLE I: Conceptual HEA technical data, including estimated parameters Parameters | Values
---|---
Wing area | 32 $\mbox{m}^{2}$
Maximum take-off weight | 6385 kg
Cruise altitude | $3000\,\mbox{m}$
Battery capacity∗ | 400 Ah
Battery pack nominal voltage∗ | 540 V
Propeller efficiency∗ | 0.7
Zero-lift drag coefficient∗ | 0.024
Induced drag coefficient∗ | 0.056

In [14], the authors did not specify the engine model, battery model, or
electric motor model; instead, the reference weight and some characteristics
for these components were provided. We have chosen the following models for
the propulsion system–with some parameters slightly different from those in
[14]:

  1. 1.

Turboprop engine: the PT6A-135A engine by Pratt & Whitney is selected for our
analysis; the weight of this engine is $153\mbox{kg}$; its maximum output
power is $560\mbox{kW}$; and its average specific fuel consumption (SFC) is
$356\mbox{g/kW/h}$. In this paper, it is assumed that the engine runs in a
fixed rotational speed; the SFC and the fuel flow rate can be represented as
functions of the output power of the engine, shown in Fig. 3.

(a)  (b)

Figure 3: Fuel consumption characteristics of the engine: (a) SFC vs output
power; (b) fuel flow rate vs output power

  2. 2.

Electric motor: the Magni250 by MagniX is selected for our analysis; it weighs
$72\mbox{kg}$; its maximum output power is $280\mbox{kW}$; it is assumed that
the total efficiency of this motor is $95\%$;

  3. 3.

Battery cells: in [14], the authors assumed a futuristic value for the energy
density of the battery cell, significantly greater than the current state of
battery technology. In this paper, an existing battery model from a Korean
battery company Kokam is used for the analysis. The technical data of this
battery is given in Table. II.

TABLE II: Technical data of an existing NMC cell Model | Capacity | Weight |  Energy Density
---|---|---|---
SLPB140460330 | 200Ah | 3.96 kg | 189Wh/kg

For the battery pack configuration, it is assumed that all battery cells
operate identically, and that the parallel and series configurations of the
battery pack do not affect the output voltage and the state of charge of each
cell. In addition, it is assumed that battery cell does not have an internal
state dynamics, resulting in a constant output voltage of $3.6\mbox{V}$. For
the power allocation algorithm in Section IV, it is assumed that the battery
pack has 2 parallel paths and 150 cells connected in series in each path. This
configuration has an output voltage of $540\mbox{V}$ and a total capacity of
$400\mbox{Ah}$. The total weight of the battery pack is $600\mbox{kg}$, which
is greater than the value in [14]. Other battery pack configurations are also
examined in Section V.

##  III Flight dynamics and power relation in the propulsion system

Developing effective power allocation algorithms for HEA hinges upon
interdependence between the aircraft flight dynamics and its propulsion
system. In this section, we provide a brief overview of these two subsystems;
the flight dynamics is introduced in Section III-A and the power relation in
the aircraft propulsion system in Section III-B.

###  III-A Longitudinal dynamics during the cruise phase

The complete longitudinal dynamics can be found in [15]; in this section, we
focus on the optimal velocity profile during the cruise phase using the
following assumptions:

  1. 1.

The aircraft is flying from one point to another at a constant altitude;

  2. 2.

The angle of attack is small and as such $\sin{\alpha}\approx 0$ and
$\cos{\alpha}\approx 1$.

The simplified dynamics during the cruise phase is,

$\displaystyle\dot{y}$ $\displaystyle=v$ (1a) $\displaystyle\dot{v}$
$\displaystyle=\frac{1}{m}(T-D)$ (1b) $\displaystyle L$ $\displaystyle=W,$
(1c)

where $y$ is the horizontal position, $v$ is the velocity, $T$ is the thrust
force, $D$ is the drag force, and $m$ is the total mass of the aircraft.
Moreover, $W$ is the total weight of the aircraft and $L$ is the lift force.
The drag and lift forces can be expressed as,

$\displaystyle D$ $\displaystyle=\frac{1}{2}\rho SC_{\small D}v^{2}$ (2a)
$\displaystyle C_{\small D}$ $\displaystyle=C_{\small D,0}+KC_{\small L}^{2}$
(2b) $\displaystyle L$ $\displaystyle=\frac{1}{2}\rho SC_{\small L}v^{2},$
(2c)

where $\rho$ is the air density; $S$ is the wing area; $C_{\small D}$ is the
drag coefficient; $C_{\mbox{\small D,0}}$ is the zero lift drag coefficient;
$K$ is the induced drag coefficient; all of these parameters are constant
during the cruise phase.

###  III-B HEA Propulsion system

Then we discuss the power demand and power available in the aircraft
propulsion system, which are required to subsequently formulate the power
allocation problem.

####  III-B1 Power demand during the cruise phase

The power demand $P_{\mbox{req}}$ during the cruise phase is

$P_{\mbox{\small req}}=Tv,$ (3)

####  III-B2 Power provided by the gas turbine

We follow the assumption in [12] that the output power of the engine is given
by

$P_{\mbox{\small e,out}}=\tau P_{\mbox{\small e,max}},$ (4)

where $\tau$ is the throttle of the engine, and $P_{\mbox{\small e,max}}$ is
the maximum output power of the engine.

####  III-B3 Power provide by the electric motor

It is assumed that the transfer efficiency of the electric motor is a constant
value; hence, the output power from the electric motor is given as,

$P_{\mbox{\small m,out}}=\eta_{m}P_{\mbox{\small bat}}=\eta_{m}UI,$ (5)

where $\eta_{m}$ is the total efficiency of the electric motor;
$P_{\mbox{\small bat}}$ is the output power of the battery pack; $U$ is the
voltage of the battery pack; $I$ is the current passing through the battery
pack.

####  III-B4 HEA Power relation

The power relation in the propulsion system can be presented as

$P_{\mbox{\small req}}=\eta_{p}(P_{\mbox{\small e,out}}+P_{\mbox{\small
m,out}}),$ (6)

where $\eta_{p}$ is the transfer efficiency of the propeller, assumed to be a
constant value for our analysis. Using Eq.(3)- Eq.(6), we thus have the
following useful relation,

$Tv=\eta_{p}(\tau\cdot P_{\mbox{\small e,max}}+\eta_{m}UI).$ (7)

##  IV Power allocation algorithm for HEA

In this section, the HEA power allocation problem (originally represented as
an optimal control problem) is formulated as a finite-dimensional
optimization; then the second-order sufficient conditions for the solution of
this problem are verified in Section IV-A. Numerical simulations supporting
the theoretical analysis are then provided in Section IV-B.

###  IV-A Theoretical analysis

In [15], a power allocation algorithm for HEA, providing optimal power
distribution between the engine and the electric motor for a given flight
profile, has been proposed. In this context, it was assumed that the aircraft
has a connected hybrid structure and the engine can charge the battery during
flight. More specifically, the optimal control problem was formulated as,

$\displaystyle J=$ $\displaystyle\min_{\tau}\quad-m(t_{f})$ (8a) s.t.
$\displaystyle\dot{m}=-c$ (8b) $\displaystyle\dot{\theta}=-\frac{I}{Q}$ (8c)
$\displaystyle m(t_{0})=m_{0};\quad\theta(t_{0})=\theta_{0}$ (8d)
$\displaystyle 0.1\leq\tau\leq 1$ (8e) $\displaystyle m_{\min}\leq m\leq
m_{\max},\quad\theta_{\min}\leq\theta\leq\theta_{\max},$ (8f)

where $\theta$ is the state of charge of the battery pack (SOC), and the
current $I$ is a function of $m$ and $\tau$, deduced from Eq. (7). In this
paper, the flow rate $c$ is nearly a linear function of the output power of
the engine. Consequently, instead of using SFC and power to compute the flow
rate, a representation of the flow rate as a function of the output power is
derived and the dynamics of the aircraft mass is re-parameterized as a linear
function of the control input as,

$\dot{m}=-(c_{1}P_{\mbox{e,out}}+c_{2}P_{\mbox{e,out}})=\tilde{c}_{1}\tau+\tilde{c}_{2}.$
(9)

In order to study the impact of the battery pack capacity on the fuel
consumption, instead of the SOC of the battery pack, the remaining charge $q$
in the battery pack is taken as a state, whose dynamic is given by,

$\dot{q}=-I.$ (10)

The performance and solution of an optimal control problem are, in theory,
invariant with respect to the magnitude of the states and the control input.
In practice, however, numerical issues can arise in the solver when these
magnitudes are highly different; in Problem (8), the aircraft mass is
approximately $6000\mbox{kg}$, and the range of the state of charge of the
battery pack is 0 to 1. Hence, we use the standard remedy of applying the
affine variable transformations to the aircraft mass and the remaining charge
in the battery pack as,

$\displaystyle\hat{m}$ $\displaystyle=a_{1}m+b_{1}$ (11a)
$\displaystyle\hat{q}$ $\displaystyle=a_{2}q+b_{2};$ (11b)

we now define the scaled time $\hat{t}\in[0,1]$; the dynamics of “scaled
states” with respect to the “scaled time” is now written as,

$\displaystyle\dot{\hat{m}}$ $\displaystyle=k_{11}\tau+k_{10}$ (12a)
$\displaystyle\dot{\hat{q}}$
$\displaystyle=k_{21}\tau+k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m},$ (12b)

where both states are linear in the control input $\tau$. The scaled optimal
control problem is now written as,

$\displaystyle J=$ $\displaystyle\min_{\tau}\quad-\hat{m}(\hat{t}=1)$ (13a)
s.t. $\displaystyle\dot{\hat{m}}=k_{11}\tau+k_{10}$ (13b)
$\displaystyle\dot{\hat{q}}=k_{21}\tau+k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m}$
(13c) $\displaystyle\hat{m}(0)=1;\quad\hat{q}(0)=\hat{q}_{0}$ (13d)
$\displaystyle\hat{q}(1)=\hat{q}_{f}$ (13e) $\displaystyle 0\leq\tau\leq 1$
(13f) $\displaystyle 0\leq\hat{m}\leq 1,\quad 0\leq\hat{q}\leq 1.$ (13g)

This formulation of the power allocation problem is in the form of bang-bang
control problem with state constraints. When solving this problem using Tomlab
for a reasonable flight profile, the only active state constraint is
$\hat{q}=1$; hence, we subsequently only consider the constraint $\hat{q}\leq
1$. This constraint is denoted as,

$S(\hat{q}(\hat{t}))=\hat{q}-1\leq 0.$ (14)

$S(\hat{q}(t))\leq 0$ is denoted as $S$ for simplicity in the sequel. An
interval $[\hat{t}_{1},\hat{t}_{2}]\in[0,1]$ is called a boundary arc if
$S(\hat{q}(t))\equiv 0$ holds for all $t\in[\hat{t}_{1},\hat{t}_{2}]$. Hence
on the boundary arc the time derivative of $S$ with respect to the scaled time
$\hat{t}$ should be zero for all $t\in[\hat{t}_{1},\hat{t}_{2}]$, where,

$\frac{dS}{d\hat{t}}=\frac{\partial
S}{\partial\boldsymbol{x}}\frac{\partial\boldsymbol{x}}{\partial\hat{t}}=[0~{}1]\begin{bmatrix}\dot{\hat{m}}\\\
\dot{\hat{q}}\end{bmatrix}=\dot{\hat{q}},$ (15)

with $\boldsymbol{\mbox{x}}=[\hat{m}~{}\hat{q}]^{\top}$. Hence, the boundary
control $\tau_{b}$ can be computed from $\frac{dS}{dt}=\dot{\hat{q}}=0$ as the
following feedback expression

$\tau_{b}=-\frac{1}{k_{21}}(k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m}).$ (16)

The augmented Hamiltonian of Problem (13) is given as,

$\displaystyle H$
$\displaystyle=\lambda_{1}\cdot\dot{\hat{m}}+\lambda_{2}\cdot\dot{\hat{q}}+\eta\cdot
S$
$\displaystyle=\lambda_{1}k_{10}+\lambda_{2}(k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m})+\eta(\hat{q}-1)$
$\displaystyle+(\lambda_{1}k_{11}+\lambda_{2}k_{21})\tau$
$\displaystyle=H_{1}+\sigma(\hat{t})\tau.$ (17)

Notice that the Hamiltonian is linear in the control $\tau$; from the
Pontryagin’s minimum principle, the optimal control law on interior arcs with
$S<0$ is obtained as,

$\tau^{*}=\Big{\\{}\begin{array}[]{cc}\tau_{\max}&\mbox{if}~{}\sigma(t)<0\\\
\tau_{\min}&\mbox{if}~{}\sigma(t)>0.\end{array}$ (18)

For a boundary arc, the optimal control has been computed as
$\tau_{\min}<\tau_{b}<\tau_{\max}$, since the optimal control should minimize
the Hamiltonian, this minimum principle yields,

$\sigma(\hat{t})=0,\quad\forall\hat{t}\in[\hat{t}_{1},\hat{t}_{2}].$ (19)

This relation is interpreted as the property that a boundary control behaves
formally like a singular control by Maurer in [18]. When solving Problem (13)
using a numerical solver, the optimal control follows a “max-boundary-min”
structure. Combining the numerical results with the preceding theoretical
analysis, the following optimal control structure is thus obtained,

$\tau^{*}=\Bigg{\\{}\begin{array}[]{cc}\tau_{\max}&0\leq\hat{t}\leq\hat{t}_{1}\\\
\tau_{b}&\hat{t}_{1}\leq\hat{t}\leq\hat{t}_{2}\\\
\tau_{\min}&\hat{t}_{2}\leq\hat{t}\leq 1.\end{array}$ (20)

In order to study the second order sufficient conditions and the sensitivity
to initial conditions of states, this optimal control problem is then
formulated as a (finite-dimensional) optimization problem. This approach is
particular useful when the structure of the optimal control, i.e., the
sequence of bang-bang and boundary arcs, has been derived. The arc-
parametrization method in [19] is applied to formulate the optimization
problem with variables
$\textbf{x}=[\xi_{1},~{}\xi_{2},~{}\xi_{3},~{}z]^{\top}$(different
$\boldsymbol{x}$ than in Eq. (15)), as,

$\displaystyle\xi_{1}$ $\displaystyle=\hat{t}_{1}-0,$
$\displaystyle\xi_{2}=\hat{t}_{2}-\hat{t}_{1}$ $\displaystyle\xi_{3}$
$\displaystyle=\hat{t}_{f}-\hat{t}_{2},$ $\displaystyle
z=\hat{m}(\hat{t}_{2});$

the last variable is introduced for the simplicity formulation of constraints.
The power allocation problem can now be represented as an optimization
problem,

$\displaystyle\min_{\tau}~{}G\coloneqq$ $\displaystyle~{}-z-k_{10}\xi_{3}$
(22a) $\displaystyle\mbox{s.t.}~{}\Phi_{1}\coloneqq$ $\displaystyle
h_{13}\xi_{1}^{3}+h_{12}\xi_{1}^{2}+h_{11}\xi_{1}+h_{10}-1=0$ (22b)
$\displaystyle\Phi_{2}\coloneqq$ $\displaystyle
g_{13}\xi_{3}^{3}+g_{12}z\xi_{3}^{2}+g_{11}\xi_{3}^{2}+k_{20}\xi_{3}$
$\displaystyle~{}+k_{22}z^{2}\xi_{3}+k_{23}z\xi_{3}+1-\hat{q}_{f}=0$ (22c)
$\displaystyle\Phi_{3}\coloneqq$
$\displaystyle~{}z-\frac{-\tan(\frac{r_{0}\xi_{2}}{2})+f_{1}m_{0}+f_{2}\xi_{1}+f_{3}}{f_{1}(1+\tan(\frac{r_{0}\xi_{2}}{2})(f_{1}m_{0}+f_{2}\xi_{1}+f_{3}))}$
$\displaystyle~{}+\frac{k_{31}}{2k_{32}}=0$ (22d)
$\displaystyle\Phi_{4}\coloneqq$ $\displaystyle~{}\xi_{1}+\xi_{2}+\xi_{3}=1;$
(22e)

the detailed process of deducing this above optimization problem from the
original optimal control problem is provided in Appendix VI-A. Note that the
Lagrangian for Problem Eq. (22) is

$L\coloneqq
G+\rho_{1}\Phi_{1}+\rho_{2}\Phi_{2}+\rho_{3}\Phi_{3}+\rho_{4}\Phi_{4},$ (23)

that is useful to derive second-order sufficient conditions in Appendix VI-A.

###  IV-B Numerical case study

We now consider the following flight conditions to examine the power
allocation problem for the HEA: $y_{d}=400\mbox{km}$, $v=100\mbox{m/s}$,
$\theta_{0}=0.7$, $m_{0}=6350\mbox{kg}$, $\theta_{f}=0.4$; we use Tomlab to
solve Problem (13) and re-scale the results back to original states. The
results are given in Fig. 4 and Fig.5. The fuel consumed for this example is
310.8kg. Fig. 5 shows that the optimal control structure follows the optimal
control structure in Eq. (20).

(a)  (b)

Figure 4: (a) Remaining charge in the battery pack; (b)Total mass of the
aircraft

(a)

(b)

Figure 5: (a) Control input; (b) Approximate Specific Fuel Consumption

Then the corresponding optimization problem is solved using ipopt; the optimal
variables are determined as,
$[\xi_{1},~{}\xi_{2},~{}\xi_{3},~{}z]^{\top}=[0.1807,0.6801,0.1392,0.3706]$,
and the optimal objective value is $-\hat{m}(1)=-0.3525$. Hence the aircraft
mass at the final time is m=6039.2kg, and the fuel consumed is 310.8kg. In
this numerical case, the dual variables are also provided by ipopt, namely,
$[\rho_{1},\rho_{2},\rho_{3},\rho_{4}]=[-0.1253,-0.1272,0.9971,-0.6783]$; the
first order necessary condition for optimality can now be checked, i.e.,
$L_{\boldsymbol{x}}=0$; moreover, the Hessian $L_{\boldsymbol{xx}}$ of the
Lagrangian for $\boldsymbol{x}=[0.1807,0.6801,0.1392,0.3706]$ is

$L_{\boldsymbol{xx}}=\begin{bmatrix}-0.0186&-0.0188&0&0\\\
-0.0188&-0.0144&0&0\\\ 0&0&-0.0028&0.0211\\\ 0&0&0.0211&0.0002,\end{bmatrix}$
(24)

and

$\Phi_{\boldsymbol{x}}=\begin{bmatrix}1.6741&0&0&0\\\ 0&0&-4.3085&-0.0231\\\
0.8907&0.6803&0&1\\\ 1&1&1&0\end{bmatrix},$ (25)

with $\mbox{\bf rank}\,\Phi_{\boldsymbol{x}}=4$ and $L_{\boldsymbol{xx}}$ is
positive definite. Hence, the second-order sufficient conditions are
satisfied, and the optimal solution is unique and, in fact, globally optimal.
With this transition to the finite-dimensional optimization problem, one can
conduct sensitivity analysis on the optimal solution, a technique that proves
instrumental to address the question posed in the title of this manuscript.

##  V To Charge in-flight or not: comparison of two parallel hybrid
architectures

The mechanical connection between the engine and the electric motor in the HEA
propulsion system allows the engine to charge the battery pack in-flight. For
the climb phase, the power demand is usually higher than the maximum output
power of the engine–hence the propulsion system can only run in the combined
mode. The power-split mode is only applicable during the cruise phase and the
descent phase. In this section, it is demonstrated that the fuel saving by
charging the battery during the cruise phase to steer the engine to operate
more efficiently is negligible. The second potential advantage of charging the
battery in-flight is that the electric motor can work in OEI mode by charging
the battery to its fully charged state after the climb phase. It has been
shown that under certain flight conditions, this advantage can be achieved
instead by carrying more battery packs before take-off.

###  V-A To Charge or not in-flight during the cruise phase

First, fuel consumption between charging and not charging the battery pack in
the cruise phase is compared. By solving Problem (13) with an extra constraint
$\dot{\hat{q}}\leq 0$, one can determine the optimal fuel consumption when
charging the battery pack during the flight is not allowed (the current
passing through the battery pack can only be positive). In this direction, it
is assumed that all other conditions and constraints remain the same.

(a)

(b)

Figure 6: (a) Remaining charge in the battery pack; (b) Approximate Specific
Fuel Consumption

Observing Fig. 6, when not allowing charging of the battery during the flight,
the optimal power distribution strategy yields an “engine-alone” then “motor-
alone” structure. The total fuel consumed is $311.08\mbox{kg}$, which is only
$0.33\mbox{kg}$ more than the case when the battery was allowed to charge in-
flight. Comparing the first segment of Fig. 5b and Fig. 6b, we note that when
charging the battery on $[0,\hat{t}_{1}]$ in Fig. 5b, the SFC is about
$0.35\mbox{kg/kw/h}$, which is lower than the case when battery was not
charging ($0.367-0.369\mbox{kg/kw/h}$) in Fig.6b. However after charging the
battery to its full state, the aircraft will run in the engine alone mode
again, where the SFC is about $0.367-0.369\mbox{kg/kw/h}$ in Fig. 5b, same as
the case when charging was not not permissible.

Charging the battery during flight can make the engine work more efficiently,
but the duration of this efficient operation depends on the initial and
terminal constraints of the battery pack. After charging the battery to its
full capacity, the engine will go back to its inefficient operational region.
In fact, overall, the fuel saving by charging the battery to steer the engine
operation more efficiently is rather negligible.

###  V-B Carrying more battery in-flight

The second benefit of charging the battery is that the electric motor can work
in the OEI mode. In order to verify this, we compare the total fuel
consumption for the combination of the climb and the cruise phase when the
take-off weight is different due to different battery pack configurations. In
the previous section, it is assumed that the battery pack has 2 parallel paths
with a capacity of $400\mbox{Ah}$. Each parallel path in the battery pack has
a capacity of $400\mbox{Ah}$ and weighs $600\mbox{kg}$. Different battery pack
configurations are achieved by changing the number of parallel paths.
Adding/removing parallel paths only increases/decreases the total capacity,
not affects the output voltage of the battery pack. First, the fuel
consumption during the climb phase with a fixed flight path angle is computed,
and the terminal condition for the climb phase is when the altitude reaches
$3000\mbox{m}$.

TABLE III: Fuel consumption of the climb phase with different battery pack configuration Number of parallel paths | 1 | 2 | 3
---|---|---|---
Total capacity of the battery pack (Ah) | 200 | 400 | 600
MTOW (kg) | 5785 | 6385 | 6985
SOC after climb | 0.38 | 0.69 | 0.79
Remaining charge (Ah) | 76 | 276 | 474
Fuel consumed during climbing (kg) | 32.84 | 37.55 | 42.02

From Table. III, it is observed that the consumed fuel difference is only
$10\mbox{kg}$ when the take-off weight difference is $1200\mbox{kg}$ (between
1 path and 3 paths). The optimal power allocation solutions show that during
the climb phase, the electric motor operates at its maximum output power, and
the engine runs based on the power demand.

For the cruise phase, we compare the following cases:

  1. 1.

case 1: $Q=200\mbox{Ah}$, $m_{0}=5750\mbox{kg}$, $\theta_{0}=0.4$,
$\theta_{f}=1$;

  2. 2.

case 2: $Q=400\mbox{Ah}$, $m_{0}=6350\mbox{kg}$, $\theta_{0}=0.7$,
$\theta_{f}=1$;

  3. 3.

case 3: $Q=400\mbox{Ah}$, $m_{0}=6350\mbox{kg}$, $\theta_{0}=0.7$,
$\theta_{f}=0.5$, and the engine cannot charge the battery;

  4. 4.

case 4: $Q=600\mbox{Ah}$, $m_{0}=6950\mbox{kg}$, $\theta_{0}=0.8$,
$\theta_{f}=0.67$, and the engine cannot charge the battery.

It is assumed that case 1 has the connected architecture; hence the engine can
charge the battery in-flight. The terminal condition for the battery is to
charge it to $200\mbox{Ah}$. The same architecture and terminal conditions are
assumed for case 2. Cases 3 and 4 have independent architectures, the terminal
condition is that the battery pack has $200\mbox{Ah}$ left. Then the cruise
range from $150\mbox{km}$ to $400\mbox{km}$ is swept to compute the fuel
consumption under the four cases listed above. The cruise speed is fixed as
$80\mbox{m/s}$ for all cases. The simulation result is given in Fig. 7.

Figure 7: Fuel consumption with different battery configurations; case 1 and
case 2 are connected configurations; case 3 and case 4 are independent
configurations

It is observed that the fuel consumption is nearly a linear function of the
flight distance for each case (with different slope for each case). The
sensitivity derivative shows that the derivative of the objective function
with respect to the terminal time $\hat{t}_{f}$ is a constant value, but this
value changes slightly as the terminal time changes. For example, in case 2,
$\frac{dG}{dt_{f}}=0.6498$ when $t_{f}=0.375$; $\frac{dG}{dt_{f}}=0.6363$ when
$t_{f}=1$; this value decreases as the final time $t_{f}$ increases. In the
simulation result, the slope of the red line in Fig. 7 is 0.6433. Reference
[17] provides explicit formula to compute the sensitivity derivatives,

$\begin{bmatrix}\frac{d\boldsymbol{x}^{p}}{dp}\\\
\frac{d\rho^{p}}{dp}\end{bmatrix}=-\begin{bmatrix}L_{\boldsymbol{xx}}&\Phi_{\boldsymbol{x}}^{*}\\\
\Phi_{\boldsymbol{x}}&\boldsymbol{0}\end{bmatrix}\begin{bmatrix}L_{\boldsymbol{x}p}\\\
\Phi_{p}\end{bmatrix},$ (26)

where $\boldsymbol{x}^{p}$ and $\boldsymbol{\rho}^{p}$ are the optimal and
dual variables corresponding to parameter $p$. Then the derivative of the
objective function $G$ with respect to the parameter $t_{f}$ is computed as

$\frac{d\mbox{G}}{dt_{f}}=\frac{\partial
F}{\partial\boldsymbol{x}}\frac{\partial\boldsymbol{x}}{\partial t_{f}}.$ (27)

The following observations are obtained when comparing cases with the same
terminal battery charge condition (after charging for the connected case;
beginning of the cruise phase for the independent case):

  1. 1.

Comparing case 2 and case 3, it is observed that these two cases have the same
initial aircraft mass and battery pack capacity; the only difference is the
terminal condition for the SOC of the battery pack. Case 2 requires the
battery to remain in a fully-charged state ($400\mbox{Ah}$) after a short
period of the cruise flight; case 3 requires the remaining charge in the
battery pack is more than $200\mbox{Ah}$. Simulation results show case 2 needs
about $30\mbox{kg}$ more fuel consumed than case 3 for all flight ranges.

  2. 2.

Comparing case 1 and case 3 (the blue line and the green line), the remaining
charge in the battery pack is $200\mbox{Ah}$ for both cases. Even though case
1 has lighter initial aircraft mass, charging the battery to its fully-charged
state requires more fuel; carrying more battery pack can save fuel under this
remaining charge condition.

  3. 3.

Comparing case 2 and case 4, same as the comparison between case 1 and case 3;
the remaining charge in the battery pack is $400\mbox{Ah}$ for both cases.
Carrying more battery can still save fuel.

Hence, with the same remaining battery charge requirement, carrying more
battery before take-off is more favorable than charging the battery in-flight
in terms of fuel saving and $\mbox{CO}_{2}$ emissions.

##  VI Conclusions

Two parallel hybrid electric architectures are compared in terms of fuel
consumption. The connected architecture has the mechanical connection between
the engine and the electric motor, allowing the engine to charge the battery
in-flight in order to improve the engine’s fuel efficiency. The independent
architecture does not have a mechanical connection, which allows an easy
transition from hybrid aircraft to all-electric aircraft. An optimal power
allocation algorithm is applied to compare the fuel usage of these two
architectures. Both theoretical and simulation results show that the fuel
saving by charging the battery is negligible for the 19-seat conceptual
aircraft. The other potential advantage of charging the battery during flight
is also examined. The simulation results show that with the same requirement
on the remaining battery charge, carrying more battery before taking-off is
more fuel efficient than charging the battery during the flight. As such, the
connected architecture cannot bring significant fuel savings compared with the
independent configuration. In addition, the mechanical connection increases
the system complexity and generally requires sophisticated control strategies.
This architecture also makes it difficult to transition to an all-electric
aircraft configuration. Lastly, charging the battery in the harsh flight
environment requires accurate thermal management system as well as causing
battery degradation. Our assessment–in the context of the question posed in
the title of the manuscript– is therefore implicit from these observations.

## Appendix

In this section, we provide details on some of the analytic steps used for our
analysis.

###  VI-A Transcription to an optimization problem

The explicit functions of the optimal control on each arc are as follows:

  1. 1.

On $\hat{t}\in[0,\hat{t}_{1}]$, the optimal control is
$\tau^{*}=\tau_{\max}=1$. Then the dynamics of states can be written as,

$\displaystyle\dot{\hat{m}}$ $\displaystyle=k_{11}+k_{10}$ (28a)
$\displaystyle\dot{\hat{q}}$
$\displaystyle=k_{21}+k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m}.$ (28b)

We therefore obtain the explicit state function of time as,

$\displaystyle\hat{m}(\hat{t})$ $\displaystyle=m_{0}+(k_{11}+k_{10})\hat{t}$
(29a) $\displaystyle\hat{q}(\hat{t})$
$\displaystyle=h_{13}\hat{t}^{3}+h_{12}\hat{t}^{2}+h_{11}\hat{t}+h_{10},$
(29b)

where,

$\displaystyle h_{13}$ $\displaystyle=\frac{1}{3}k_{22}(k_{11}+k_{10})^{2}$
$\displaystyle h_{12}$
$\displaystyle=(k_{22}m_{0}+\frac{1}{2}k_{23})*(k_{11}+k_{10})$ $\displaystyle
h_{11}$ $\displaystyle=k_{21}+k_{20}+k_{22}m_{0}^{2}+k_{23}m_{0},$

and $h_{10}=\hat{q}_{0}$ is computed from $\hat{q}(\hat{t}=0)=\hat{q}_{0}$.

  2. 2.

On $\hat{t}\in[\hat{t}_{1},\hat{t}_{2}]$, $\hat{q}(\hat{t})=0$ holds for all
$\hat{t}$, and the optimal control is
$\tau_{b}=-\frac{1}{k_{21}}(k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m})$. Hence,
the dynamics of the aircraft mass can be written as,

$\dot{\hat{m}}=-\frac{k_{11}}{k_{21}}(k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m})+k_{10};$
(31)

the initial condition for this interval is
$\hat{m}(\hat{t}_{1})=m_{0}+(k_{11}+k_{10})\hat{t}_{1}$. Thereby, by
separation of variables, we obtain the explicit function of the aircraft mass
on the boundary arc as,

$\hat{m}(\hat{t})=\frac{1}{2k_{32}}[r_{0}\tan(-\frac{r_{0}(\hat{t}-\hat{t}_{1})}{2}+C_{0})-k_{31}],$
(32)

where $k_{30}=\frac{k_{11}k_{20}}{k_{21}}-k_{10}$
$k_{31}=\frac{k_{11}k_{23}}{k_{21}}$, $k_{32}=\frac{k_{11}k_{22}}{k_{21}}$,
and

$\displaystyle C_{0}$
$\displaystyle=\frac{-2\arctan(\frac{2k_{32}m(\hat{t}_{1})+k_{31}}{r_{0}})}{r_{0}}$
$\displaystyle r_{0}$ $\displaystyle=\sqrt{4k_{32}k_{30}-k_{31}^{2}}.$

We denote $\hat{m}(t_{2})$ as $z$, which is the third constraint in Problem
Eq. (22).

  3. 3.

On $\hat{t}\in[\hat{t}_{2},1]$, the optimal control is
$\tau^{*}=\tau_{min}=0$, then the dynamics of states can be written

$\displaystyle\dot{\hat{m}}$ $\displaystyle=k_{10}$ (33a)
$\displaystyle\dot{\hat{q}}$
$\displaystyle=k_{20}+k_{22}\hat{m}^{2}+k_{23}\hat{m}$ (33b)

where the initial states on this arc are $[z,1]$; the explicit functions of
the states are,

$\displaystyle\hat{m}(\hat{t})$
$\displaystyle=\hat{m}(\hat{t}_{2})+k_{10}(\hat{t}-\hat{t}_{2})$ (34a)
$\displaystyle\hat{q}(\hat{t})$
$\displaystyle=g_{13}(\hat{t}-\hat{t}_{2})^{3}+g_{12}z(\hat{t}-\hat{t}_{2})^{2}+g_{11}(\hat{t}-\hat{t}_{2})^{2}$
$\displaystyle+k_{20}(\hat{t}-\hat{t}_{2})+k_{22}z^{2}(\hat{t}-\hat{t}_{2})+k_{23}z(\hat{t}-\hat{t}_{2})+1.$
(34b)

The first constraint for the optimization problem is when $q$ hits the
boundary at the junction time $\hat{t}=\hat{t}_{1}$, given as

$\Phi_{1}\coloneqq\hat{q}(\hat{t}_{1})=1.$ (35)

When we substitute this constraint into Eq. (29b), we obtain,

$h_{13}\xi_{1}^{3}+h_{12}\xi_{1}^{2}+h_{11}\xi_{1}+h_{10}=1$ (36)

The second constraint for the optimization problem is the terminal condition
for $\hat{q}$, $\hat{q}(1)=\hat{q}_{f}$, after substituting into Eq. (34b), we
obtain

$g_{13}\xi_{3}^{3}+g_{12}z\xi_{3}^{2}+g_{11}\xi_{3}^{2}+k_{20}\xi_{3}+k_{22}z^{2}\xi_{3}+k_{23}z\xi_{3}+1-\hat{q}_{f}=0.$
(37)

###  VI-B Verification of SSC and sensitivity analysis

We refer Theorem 3.1 in [16] pertaining to the second-order sufficient
condition for the optimization problem Eq. (22). Let $\boldsymbol{x}$ be a
feasible solution for Problem Eq. (22). Suppose the following conditions hold:

  1. 1.

$L_{\boldsymbol{x}}=0$ (first order necessary condition).

  2. 2.

$\mbox{rank}[\Phi_{\boldsymbol{x}}]=r$, where r is the number of constraints

  3. 3.

$v^{\top}L_{\boldsymbol{xx}}v>0,\forall v\in\mathbb{R}^{n},v\neq
0,\Phi_{\boldsymbol{x}}v=0$, where $n$ is the number of variables in Problem
Eq. (22).

Let $\Delta=1+\tan(\frac{r_{0}}{2}\xi_{2})(f_{1}x_{10}+f_{2}\xi_{1}+f_{3})$
and $f_{4}=f_{1}x_{10}+f_{2}\xi_{1}+f_{3}$.

The $L_{\boldsymbol{x}}$ is now computed as,

$\displaystyle L_{\xi_{1}}$
$\displaystyle=\rho_{1}(3h_{13}\xi_{1}^{2}+2h_{12}\xi_{1}+h_{11})+\rho_{4}$
$\displaystyle+\frac{-\rho_{3}f_{2}}{f_{1}}\frac{(1+\tan^{2}(\frac{r_{0}}{2}\xi_{2}))}{\Delta^{2}}$
(38a) $\displaystyle L_{\xi_{2}}$
$\displaystyle=\frac{\rho_{3}r_{0}}{2f_{1}}\frac{(1+f_{4}^{2})\sec^{2}(\frac{r_{0}}{2}\xi_{2})}{\Delta^{2}}+\rho_{4}$
(38b) $\displaystyle L_{\xi_{3}}$
$\displaystyle=-k_{10}+\rho_{4}+\rho_{2}(3g_{13}\xi_{3}^{2}+2g_{12}z\xi_{3}$
$\displaystyle+2g_{11}\xi_{3}+k_{20}+k_{22}z^{2}+k_{23}z)$ (38c)
$\displaystyle L_{z}$
$\displaystyle=-1+\rho_{3}+\rho_{2}(g_{12}\xi_{3}^{2}+2k_{22}\xi_{3}z+k_{23}\xi_{3}).$
(38d)

The $L_{\boldsymbol{xx}}$ is represented as,

$\displaystyle L_{\boldsymbol{xx}}$
$\displaystyle=\begin{bmatrix}\frac{\partial
L_{\xi_{1}}}{\partial\xi_{1}}&\frac{\partial
L_{\xi_{1}}}{\partial\xi_{2}}&\frac{\partial
L_{\xi_{1}}}{\partial\xi_{3}}&\frac{\partial L_{\xi_{1}}}{\partial z}\\\
\frac{\partial L_{\xi_{2}}}{\partial\xi_{1}}&\frac{\partial
L_{\xi_{2}}}{\partial\xi_{2}}&\frac{\partial
L_{\xi_{2}}}{\partial\xi_{3}}&\frac{\partial L_{\xi_{2}}}{\partial z}\\\
\frac{\partial L_{\xi_{3}}}{\partial\xi_{1}}&\frac{\partial
L_{\xi_{3}}}{\partial\xi_{2}}&\frac{\partial
L_{\xi_{3}}}{\partial\xi_{3}}&\frac{\partial L_{\xi_{3}}}{\partial z}\\\
\frac{\partial L_{z}}{\partial\xi_{1}}&\frac{\partial
L_{z}}{\partial\xi_{2}}&\frac{\partial L_{z}}{\partial\xi_{3}}&\frac{\partial
L_{z}}{\partial z}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}L_{11}&L_{12}&0&0\\\ L_{21}&L_{22}&0&0\\\
0&0&L_{33}&L_{34}\\\ 0&0&L_{43}&L_{44}\end{bmatrix},$ (39)

where

$\displaystyle L_{11}$ $\displaystyle=\rho_{1}(6h_{13}\xi_{1}+2h_{12})$
$\displaystyle+\frac{2\rho_{3}f_{2}^{2}}{f_{1}}\frac{(\tan(\frac{r_{0}}{2}\xi_{2})+\tan^{3}(\frac{r_{0}}{2}\xi_{2}))}{\Delta^{3}}$
(40a) $\displaystyle L_{12}$
$\displaystyle=\frac{-\rho_{3}r_{0}f_{2}}{f_{1}}\frac{\sec^{2}(\frac{r_{0}}{2}\xi_{2})(\tan(\frac{r_{0}}{2}\xi_{2})-f_{4})}{\Delta^{3}}$
(40b) $\displaystyle L_{21}$ $\displaystyle=L_{12}$ (40c) $\displaystyle
L_{22}$
$\displaystyle=\frac{\rho_{3}r_{0}^{2}(1+f_{4}^{2})\sec^{2}(\frac{r_{0}}{2}\xi_{2})}{2f_{1}}$
$\displaystyle\frac{(\tan(\frac{r_{0}}{2}\xi_{2})(f_{4}\tan(\frac{r_{0}}{2}\xi_{2})+1)-f_{4}\sec^{2}(\frac{r_{0}}{2}\xi_{2}))}{\Delta^{3}}.$
(40d)

The function $\Phi_{\boldsymbol{x}}$ is now computed as,

$\displaystyle\frac{\partial\Phi_{1}}{\partial\xi_{1}}$
$\displaystyle=3h_{13}\xi_{1}^{2}+2h_{12}\xi_{1}+h_{11},\quad\frac{\partial\Phi_{1}}{\partial\xi_{2}}=0$
$\displaystyle\frac{\partial\Phi_{1}}{\partial\xi_{3}}$
$\displaystyle=0,\quad\frac{\partial\Phi_{1}}{\partial z}=0;$ (41a)
$\displaystyle\frac{\partial\Phi_{2}}{\partial\xi_{1}}$
$\displaystyle=0,\quad\frac{\partial\Phi_{2}}{\partial\xi_{2}}=0$
$\displaystyle\frac{\partial\Phi_{2}}{\partial\xi_{3}}$
$\displaystyle=3g_{13}\xi_{3}^{2}+2g_{12}z\xi_{3}+2g_{11}\xi_{3}+k_{20}+k_{22}z^{2}+k_{23}z$
$\displaystyle\frac{\partial\Phi_{2}}{\partial z}$
$\displaystyle=g_{12}\xi_{3}^{2}+2k_{22}\xi_{3}z+k_{23}\xi_{3};$ (41b)
$\displaystyle\frac{\partial\Phi_{3}}{\partial\xi_{1}}$
$\displaystyle=\frac{-f_{2}(1+\tan^{2}(\frac{r_{0}}{2}\xi_{2}))}{f_{1}\Delta^{2}},~{}\frac{\partial\Phi_{3}}{\partial\xi_{2}}=\frac{r_{0}\sec^{2}(\frac{r_{0}}{2}\xi_{2})(1+f_{4}^{2})}{2f_{1}\Delta^{2}}$
$\displaystyle\frac{\partial\Phi_{3}}{\partial\xi_{3}}$
$\displaystyle=0,\quad\frac{\partial\Phi_{3}}{\partial z}=1;$ (41c)
$\displaystyle\frac{\partial\Phi_{4}}{\partial\xi_{1}}$
$\displaystyle=1,\quad\frac{\partial\Phi_{4}}{\partial\xi_{2}}=1,\quad\frac{\partial\Phi_{4}}{\partial\xi_{3}}=1,\quad\frac{\partial\Phi_{4}}{\partial
z}=0,$ (41d)

## ACKNOWLEDGMENT

The authors would like to thank Kuang-Ying Ting, Yue Yu, Professor Mohammad
Reza Soltani, and Professor Krishna A. Shah for helpful discussions.

## References

  * [1] Y. Xie, A. Savvarisal, A. Tsourdos, D. Zhang, J. GU, “Review of hybrid electric powered aircraft, its conceptual design and energy management methodologies,” Chinese Journal of Aeronautics, vol. 34, no. 4, pp.432-450, 2021, doi: 10.1016/j.cja.2020.07.017.
  * [2] T. Wall, R. Meyer, “A survey of hybrid electric propulsion for aircraft,” 53rd AIAA/SAE/ASEE Joint Propulsion Conference, pp.4700, 2017, doi: 10.2514/6.2017-4700.
  * [3] C. Friedrich, P.A. Robertson, “Hybrid-electric propulsion for automotive and aviation applications,” CEAS Aeronautical Journal, vol. 6, no. 2, pp. 279-290, 2015, doi: 10.1007/S13272-014-0144-X.
  * [4] B.J. Brelje, J.R. Martins, “Electric, hybrid, and turbo-electric fixed-wing aircraft: A review of concepts, models, and design approaches,” Progress in Aerospace Sciences, vol. 104, pp. 1-19, 2019, doi: 10.1016/j.paerosci.2018.06.004.
  * [5] J.P.S. Pinto Leite, M. Voskuijl, “Optimal energy management for hybrid-electric aircraft”, Aircraft Engineering and Aerospace Technology, Vol. 92 No. 6, pp. 851-861, doi: 10.1108/AEAT-03-2019-0046.
  * [6] M. Doff-Sotta, M. Cannon, M. Bacic, “Optimal energy management for hybrid electric aircraft,” IFAC-PapersOnLine, vol. 53, no. 2, pp.6043-6049, 2020, doi: 10.1016/j.ifacol.2020.12.1672
  * [7] T. Donateo, A. Ficarella, L. Spedicato, “Applying dynamic programming algorithms to the energy management of hybrid electric aircraft,” Turbo Expo: Power for Land, Sea, and Air, vol. 51043, pp. V003T06A015, 2018, doi: 10.1115/GT2018-76500.
  * [8] F. Martini, “World’s first serial hybrid electric aircraft to fly at Le Bourget,” AXX20110666e, Siemens, Munich, 2011.
  * [9] W. Wittmann, “Flying with Siemens integrated drive system,” IDT2013064084e, Siemens, Nuremberg, Germany, 2013.
  * [10] J.W. Langelaan, A. Chakrabarty, A. Deng, K. Miles, V. Plevnik, J. Tomazic, T. Tomazic, G. Veble, “Green flight challenge: aircraft design and flight planning for extreme fuel efficiency,” Journal of Aircraft, vol. 50, no.3, pp. 832-846, 2013, doi: 10.2514/1.C032022.
  * [11] M. Caujolle, R. Gage, “Airbus, Rolls-Royce, and Siemens team up for electric future Partnership launches E-Fan X hybrid-electric flight demonstrator,” 2017. Available: http://www. airbus. com/newsroom/press-releases/en/2017/11/airbus–rolls-royce–and-siemens-team-up-for-electric-future-par. html
  * [12] R. De Vries, M. Brown, R. Vos, “Preliminary sizing method for hybrid-electric distributed-propulsion aircraft,” Journal of Aircraft, vol. 56, no. 6, pp. 2172-2188, 2019, doi: 10.2514/1.C035388.
  * [13] A.R. Isikveren, S. Kaiser, C. Pornet, P.C. Vratny, “Pre-design strategies and sizing techniques for dual-energy aircraft,” Aircraft Engineering and Aerospace Technology: An International Journal, 2014.
  * [14] D.F. Finger, R. de Vries, R. Vos, C. Braun, C. Bil, “A comparison of hybrid-electric aircraft sizing methods,” AIAA Scitech 2020 Forum, pp. 1006, 2020, doi: 10.2514/6.2020-1006.
  * [15] M. Wang, M. Mesbahi, “Power Allocation For Hybrid Electric Aircraft via Optimal Control during Climb, Cruise, and Descent,” AIAA Scitech 2021 Forum, pp. 0640, 2021, doi: 10.2514/6.2021-0640.
  * [16] H. Maurer, I. Altrogge, N. Goris, “Optimization methods for solving bang-bang control problems with state constraints and the verification of sufficient conditions,” Proceedings of the 44th IEEE Conference on Decision and Control, pp. 923-928, 2005.
  * [17] J.H. Kim, H. Maurer, “Sensitivity analysis of optimal control problems with bang-bang controls,” 42nd IEEE International Conference on Decision and Control, vol. 4, pp. 3281-3286, 2003.
  * [18] H. Maurer, “On optimal control problems with bounded state variables and control appearing linearly,” SIAM Journal on Control and Optimization, vol. 15, no. 3, pp. 345-362, 1977.
  * [19] C.Y. Kaya, J.L. Noakes, “Computational method for time-optimal switching control,” Journal of optimization theory and applications, vol. 117, no. 1, pp. 69-92, 2003.


* [99] J. Chen, X. Song, Liqiang Nie, X. Wang, H. Zhang, and Tat-Seng Chua. 2016. Micro tells macro: predicting the popularity of micro-videos via a transductive model. In MM. 898–907.
  * [100] Cimiano, P., Hotho, A., & Staab, S. (2005). Learning Concept Hierarchies from Text Corpora using Formal Concept Analysis. Journal of Artificial Intelligence Research 2005 DOI:10.1613/jair.1648
  * [101] M Collery(2022), Learning binary classification rules for sequential data, https://strl2022.github.io/files/short1.pdf.
  * [102] M. Collery et al.(2023), Neural-based classification rule learning for sequential data, arXiv:2302.11286.
  * [103] Comim, F. A Poset-Generalizability Method for Human Development Indicators. Soc. Indic. Res. 2021, 158, 1179–1198.
  * [104] M. Crampes, J. Oliveira-Kumar, S. Ranwez, J. Villerd. Visualizing Social Photos on a Hasse Diagram for Eliciting Relations and Indexing New Photos. IEEE Computer Graphics and Applications, 2009, 15 (6), pp.985-992.
  * [105] V. Dankers, E. Bruni, and D. Hupkes. 2022. The Paradox of the Compositionality of Natural Language: A Neural Machine Translation Case Study. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 4154–4175, Dublin, Ireland. Association for Computational Linguistics.
  * [106] De Loof K, De Baets B, De Meyer H, Brüggemann R. A hitchhiker’s guide to poset ranking. Comb Chem High Throughput Screen. 2008 ;11(9):734-44. doi: 10.2174/138620708786306032.PMID:18991576.
  * [107] della Queva, S. (2017). Analysis of Social Participation: A Multidimensional Approach Based on the Theory of Partial Ordering. In M. Fattore, R. Bruggemann Partial Order Concepts in Applied Sciences, Springer.
  * [108] Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statistics. 38 (2): 325–339. doi:10.1214/aoms/1177698950.
  * [109] T. Denoeux, M-H Masson (2010), Dempster-Shafer Reasoning in large Partially Ordered Sets: Applications in Machine Learning, Advances in Intelligent and Soft Computing, book series (AINSC) volume 68. Springer, Berlin, Heidelberg.
  * [110] Di Brisco, A., & Farina, P. (2018). Measuring gender gap from a poset perspective. Social Indicators Research, 136, 1109–1124.
  * [111] Diaz et al., Predicting Proteome-Early Drug Induced Cardiac Toxicity Relationships (Pro-EDICToRs) with Node Overlapping Parameters (NOPs) of a new class of Blood Mass-Spectra graphs ,The 11th International Electronic Conference on Synthetic Organic Chemistry session Computational Chemistry https://sciforum.net/paper/view/1371
  * [112] Diego Colombo, Marloes H Maathuis, Markus Kalisch, and Thomas S Richardson. Learning high-dimensional directed acyclic graphs with latent and selection variables. The Annals of Statistics, pages 294–321, 2012.
  * [113] di Bella, E., Corsi, M., Leporatti, L. (2017). POSET Analysis of Panel Data with POSAC. In: Fattore, M., Bruggemann, R. (eds) Partial Order Concepts in Applied Sciences. Springer, Cham.https://doi.org/10.1007/978-3-319-45421-4_11.
  * [114] S. Dong, P. Wang, and K. Abbas. 2021. A survey on deep learning and its applications. Comput. Sci. Rev. 40, C (May 2021).
  * [115] B.J. Dutka, 1, K. Jones, 1, K.K. Kwan, 1, H. Bailey , 2, R. McInnis 1. Use of microbial and toxicant screening tests for priority site selection of degraded areas in water bodies, Water Research, Vol. 22; 4, 1988, 503-510. https://doi.org/10.1016/0043-1354(88)90047-4
  * [116] W. Dyer, Weighted Posets: Learning surface order from dependency trees.Proceedings of the 18th International Workshop on Treebanks and Linguistic Theories (TLT, SyntaxFest 2019) https://api.semanticscholar.org/CorpusID:204885369
  * [117] U. Ergashev, E. C. Dragut, W. Meng, Learning To Rank Resources with GNN, WWW ’23: Proceedings of the ACM Web Conference 2023, pp. 3247–3256, https://doi.org/10.1145/3543507.3583360
  * [118] Fattore, M. (2008). Hasse diagrams, poset theory and fuzzy poverty measures.  Rivista Internazionale Di Scienze Sociali, Anno, 116(1), 63–75.
  * [119] Fattore, M., Brueggemann, R., OWSIŃSKI, J., (2011). Using poset theory to compare fuzzy multidimensional material deprivation across regions. In: Ingrassia S., Rocci R. and Vichi, M. New Perspectives in Statistical Modeling and Data Analysis. Berlin: Springer-Verlag.
  * [120] Fattore, M., Maggino, F. and Colombo, E. (2012) “From Composite Indicators to Partial Orders: evalu-ating socio-economic phenomena through ordinal data”. In: Maggino F, Nuvolati G (eds.) Quality of Life in Italy: research and reflections. Social Indicators Research Series, 48, 41–68.
  * [121] Fattore, M., Arcagni, A., Barberis, S., (2014). Visualizing Partially Ordered Sets for Socioeconomic analysis. Revista Colombiana de Estadistica, 34(2), pp. 437–450.
  * [122] Fattore, M., Arcagni, A., (2014). PARSEC: an R package for poset-based evaluation of multidimensional poverty. In: Bruggemann R., Carlsen L. and Wittmann J. Multi-Indicator Systems and Modelling in Partial Order. Berlin: Springer.
  * [123] Fattore M.; Maggino, F.; Arcagni, A.(2015)Exploiting Ordinal Data For Subjective Well-Being Evaluation The Measurement of Subjective Well-Being in Survey Research Vol. 16, No. 3, pp. 409–428.
  * [124] Fattore M., Maggino F., (2015). A new method for measuring and analyzing suffering - Comparing suffering patterns in Italian society. In Anderson R. E. (Eds.). World Suffering and the Quality of Life. New York: Springer.
  * [125] Fattore, M. (2016). Partially ordered sets and the measurement of multidimensional ordinal deprivation. Social Indicators Research, 128(2), 835–858.
  * [126] Fattore M., Maggino F., Arcagni A. (2016) “Non-aggregative assessment ofsubjective well-being”, In G. Alleva, A. Giommi (eds.), Topics in Theoretical and Applied Statistics, Springer International Publishing Switzerland,
  * [127] M. Fattore, R. Bruggemann. Partial Order Concepts in Applied Sciences, Springer, Cham, 2017, pp. 71–86. DOI10.1007/978-3-319-27274-0_20.
  * [128] Fattore M. (2018) “Non-aggregated indicators of environmental sustainability”, Silesian Statistical Review, 16(22), 7-22.
  * [129] Fattore, M., & Arcagni, A. (2018). F-FOD: Fuzzy first order dominance analysis and populations ranking over ordinal multi-indicators system. Social Indicators Research, 144(1), 1–29.
  * [130] Fattore M., Zenga Ma. (2019) “New posetic tools for the evaluation of financialliteracy”, In Bianco A., Gnaldi M., Conigliaro. P. (eds.), Italian Studies on Quality of Life, Springer, 978-3-030-06021-3.
  * [131] Fattore, M., Arcagni, A., & Maggino, F. (2019). Optimal scoring of partially ordered data,with an application to the ranking of smart cities. https://research.uniroma1.it/pubblicazioni/49725.
  * [132] F. Feng, X. He, Y. Liu, L. Nie, and T-S Chua. 2018. Learning on Partial-Order Hypergraphs. In  WWW 2018: The 2018 Web Conference, April 23–27, 2018, Lyon, France. ACM, New York, NY, USA 10 Pages.
  * [133] Fisher, D.H., & Langley, P. (1985). Approaches to Conceptual Clustering. International Joint Conference on Artificial Intelligence.
  * [134] H. Fisher. Knowledge Acquisition Via Incremental Conceptual Clustering. Machine Learning 2:139–172, 1987.
  * [135] Finn, V.K., and others (1982). Many valued 1ogics as fragments formalize semantics. Acta phi- losophica Fennica, vol.35.
  * [136] V. K. Finn, M. I. Zabezhailo and O. M. Anshakov On a Computer-Oriented Formalization of Plausible Reasoning In F. Bacon-J. S. Mill’S Style (Main Principles and Computer Experiments) IFAC Artificial Intelligence. Leningrad. USSR 1983. .
  * [137] Finn, V.K.: Plausible Reasoning in Systems of JSM Type. Itogi Nauki i Tekhniki, Seriya Informatika 15, 54–101 (1991).
  * [138] A. Formica, Ontology-based concept similarity in Formal Concept Analysis, Information Sciences, Volume 176, Issue 18, 2006, Pages 2624-2641, https://doi.org/10.1016/j.ins.2005.11.014.
  * [139] Francés, O.; Abreu-Salas, J.; Fernández, J.; Gutiérrez, Y.; Palomar, M. Multidimensional Data Analysis for Enhancing In-Depth Knowledge on the Characteristics of Science and Technology Parks. Appl. Sci. 2023, 13, 12595. https://doi.org/10.3390/app132312595
  * [140] Lerche, D. and P. Sorensen (2003) Evaluation of the ranking probabilities for partial orders based on random linear extensions. Chemosphere 53, 981-992.
  * [141] Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer-Verlag (1999).
  * [142] H. Gao et al. “Graph-augmented Learning to Rank for Querying Large-scale Knowledge Graph.” ArXiv abs/2111.10541 (2021):
  * [143] Garriga, G.C. (2011). Formal Concept Analysis. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning. Springer, Boston, MA.
  * [144] Gennari, J., Langley, P., & Fisher, D. (1989). Models of incremental concept formation. Artificial Intelligence, 40:12–61.
  * [145] P. Gepner, Machine Learning and High-Performance Computing Hybrid Systems, a New Way of Performance Acceleration in Engineering and Scientific Applications, Proceedings of the 16th Conference on Computer Science and Intelligence Systems,ACSIS, Vol. 25, pages 27–36 (2021).
  * [146] R. Godin, R. Missaoui, H. Alaoui. Incremental concept formation algorithms based on Galois(concept) Lattice, Computational IntelligenceVolume 11: 2 ,pp. 246–267.
  * [147] Goncalves, B. and Porto, F. Research lattices: towards a scientific hypothesis data model. In Conference on Scientific and Statistical Database Management, SSDBM ’13, Baltimore, MD, USA, July 29 - 31, 2013.
  * [148] E. Goodwin et al., Compositional Generalization in Dependency Parsing. arXiv:2110.06843 (cs).
  * [149] G. Grätzer, General Lattice Theory: Academic Press, Inc., New York. 1978.
  * [150] Y. Guo, Z. Lin, J-G Lou, D. Zhang, Hierarchical Poset Decoding for Compositional Generalization in Language, $34^{th}$ conference on Neural Processing Systems(NeurlPS 2020) Vancouver Canada.
  * [151] Guo et al. RankDNN: Learning to Rank for Few-Shot Learning,(2023) The Thirty-Seventh AAAI Conference on Artificial Intelligence (AAAI-23)
  * [152] Hadzikadic, M., & Yun, D. (1989). Concept formation by incremental conceptual clustering. Proceedings of the Eleventh International Joint Conference on Artificial Intelligence (pp. 831–836). Detroit, MI: Morgan Kaufmann.
  * [153] M. R. Hacene, A. Napoli, P. Valtchev, Y. Toussaint and R. Bendaoud, ”Ontology Learning from Text Using Relational Concept Analysis,” 2008 International MCETECH Conference on e-Technologies (mcetech 2008), Montreal, QC, Canada, 2008, pp. 154-163, doi: 10.1109/MCETECH.2008.29.
  * [154] M. Hajij et al., Higher-Order Attention Networks (2022), https://arxiv.org/abs/2206.00606v1.
  * [155] M. Hajij et al., Topological Deep Learning: Going Beyond Graph Data (2023), arXiv: arXiv:2206.00606v3.
  * [156] E. Halfon, M.G. Reggiani, On ranking chemicals for environmental hazard  Environ. Sci. Technol. 1986, 20, 11, 1173–1179.
  * [157] Halfon, E. (2006). Hasse Diagrams and Software Development. In: Brüggemann, R., Carlsen, L. (eds) Partial Order in Environmental Sciences and Chemistry. Springer, Berlin, Heidelberg.
  * [158] Hanson, S., & Bauer, M. (1989). Conceptual clustering, categorization, and polymorphy. Machine Learning, 3:343–372.
  * [159] B. A. Hassan, Ontology Learning Using Formal Concept Analysis and WordNet, arXiv:2311.14699, 2023.
  * [160] Hilckmann, A., Bach, V., Bruggemann, R., Ackermann, R., & Finkbeiner, M. (2017). Partial order analysis of the government dependence of the sustainable development performance in Germany’s federal states. In M. Fattore & R. Bruggemann (Eds.), (2017) Partial Order Concepts in Applied Sciences. Springer.
  * [161] S. Hira, P.S. Deshpande, Data Analysis using Multidimensional Modeling, Statistical Analysis and Data Mining on Agriculture Parameters, Procedia Computer Science, Vol. 54, 2015, pp. 431-439, https://doi.org/10.1016/j.procs.2015.06.050.
  * [162] T. Hirai a, F. Comim, Measuring the sustainable development goals: A poset analysis, Ecological Indicators, Vol. 145, 2022.
  * [163] Huchard, M., Rouane-Hacène, M., Roume, C., Valtchev, P.: Relational concept discovery in structured datasets. Ann. Math. Artif. Intell. 49(1-4) (2007) 39-76.
  * [164] D. I. Ignatov, Introduction to Formal Concept Analysis and Its Applications in Information Retrieval and Related Fields(2017), arXiv:1703.02819
  * [165] Ikeda, M., & Yamamoto, A. (2013). Classification by Selecting Plausible Formal Concepts in a Concept Lattice.
  * [166] Ip, E., Chen, S.-H., & Quandt, S. (2016). Analysis of multiple partially ordered responses to belief items with don’t know option. Psychometrika, 81(2), 483–505.
  * [167] T. Ivanciuc, O. Ivanciuc, D. J. Klein, Posetic Quantitative Superstructure/Activity Relationships (QSSARs) for Chlorobenzenes, J. Chem. Inf. Model. 2005, 45, 4, 870–879. https://doi.org/10.1021/ci0501342
  * [168] Ivaldi, E., Ciacci, A., & Soliani, R. (2020). Urban deprivation in Argentina: A poset analysis. Papers in Regional Science, 99(6), 1723-1747.
  * [169] S. D. Langhans, P. Reichert, N. Schuwirth, The method matters: A guide for indicator aggregation in ecological assessments, Ecological Indicators, Volume 45, 2014, pp. 494-507, https://doi.org/10.1016/j.ecolind.2014.05.014.
  * [170] S. Jabin, Machine Learning methods and applications using Formal Concept Analysis, International Journal of New Technologies in Science and Engineering Vol. 2: 3, 2015,
  * [171] Jaeger J, Tatsuoka C, Berns S, Varadi F, Czobor P, Uzelac S: Associating functional recovery with neurocognitive profiles identified using partially ordered classification models. Schizophr Res 2006, 85:40-48.
  * [172] Jaeger J, Tatsuoka C, Berns SM, Varadi F: Distinguishing neurocognitive functions in schizophrenia using partially ordered classification models. Schizophr Bull 2006, 32:679-691
  * [173] M.F. Janowitz., Semi-Flat Cluster Methods, Discrete Mathematics 21 (1978) , 47-60.
  * [174] M.F. Janowitz., An order theoretic model for cluster analysis. SIAM Journal on Applied Mathematics,34:55–72, 1978.
  * [175] Janowitz, M.F. (2007). Cluster Analysis Based on Posets. In: Brito, P., Cucumel, G., Bertrand, P., de Carvalho, F. (eds) Selected Contributions in Data Analysis and Classification. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg.
  * [176] M. F. Janowitz: Ordinal and Relational Clustering. Interdisciplinary Mathematical Sciences 10, World Scientific 2010, pp. 1-200.
  * [177] R. Jardine, Posets, Metric Spaces, Topological Data Analysis(2020),http://www.sci.brooklyn.cuny.edu/~noson/CTslides/Jardine20.pdf
  * [178] W. Kim, F. Memoli, Persistence Over Posets, Notices of American Mathemtical Society, Vol 20, No. 8, (2023).
  * [179] Jia, H., Newman, J., Tianfield, H. (2009). A new Formal Concept Analysis based learning approach to Ontology building. In: Sicilia, MA., Lytras, M.D. (eds) Metadata and Semantics. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77745-0_42.
  * [180] I. Jonyer, L. B. Holder, and D. J. Cook. Graph-Based Hierarchical Conceptual Clustering. Proceedings of the Florida Artificial Intelligence Research Symposium, pp. 91-95, 2000.
  * [181] I. Jonyer, D. J. Cook , L. B. Holder, Graph-Based Hierarchical Conceptual Clustering, Journal of Machine Learning Research 2 (2001) 19-43.
  * [182] Kainz, W., Egenhofer, M. J., and Greasley, I. (1993). Modelling spatial relations and operations with partially ordered sets. International Journal of Geographical Information Systems, 7(3):215–229.
  * [183] H-M. Kaltenbach, Teaching Design of Experiments using Hasse diagrams arXiv:1912.08567v1
  * [184] Kang, X., Li, D., & Wang, S. (2011). A multi-instance ensemble learning model based on concept lattice. Knowl. Based Syst., 24, 1203-1213.
  * [185] S. Kardaetz, T. Strube, R. Bruggemann, G. Nützmann, Ecological scenarios analyzed and evaluated by a shallow lake model, J. Environ. Manag. 88 (2008) 120–135.
  * [186] J. Karhunen, T. Raiko, K. Cho, Unsupervised deep learning: A short review, Advances in Independent Component Analysis and Learning Machines, Academic Press, 2015, Pages 125-142,https://doi.org/10.1016/B978-0-12-802806-3.00007-5.
  * [187] Y. Kashnitsky and D. I. Ignatov, “Can FCA-based Recommender System Suggest a Proper Classi er ? 2 Multiple Classi er Systems,” FCA do Artif. Intell. 2014), p. 17, 2014.
  * [188] M. Kendall, A new measure of rank correlation, Biometrika, Volume 30, Issue 1-2, June 1938, Pages 81–93 (doi:10.1093/biomet/30.1-2.81)
  * [189] Kickmeier-Rust, M.D., Albert, D. (2013). Using Hasse Diagrams for Competence-Oriented Learning Analytics. In: Holzinger, A., Pasi, G. (eds) Human-Computer Interaction and Knowledge Discovery in Complex, Unstructured, Big Data. HCI-KDD 2013. Lecture Notes in Computer Science, vol 7947. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39146-0_6
  * [190] Kim, W., Memoli, F.: Persistence over posets. Notices of theAmerican Mathematical Society 2761, 1214–1224 (September 2023).
  * [191] T. N Kipf and M. Welling. 2017. Semi-supervised classification with graph convolutional networks. ICLR (2017).
  * [192] Klein, D. J. Similarity and Dissimilarity in Posets. J. Math. Chem. 1995, 18, 321-348.
  * [193] Klein, D. J.; Babic, D. Partial Orderings in Chemistry. J. Chem. Inf. Comput. Sc. 1997, 37, 656–671.
  * [194] D. J. Klein, Prolegomenon on partial orderings in chemistry, MATCH Commun. Math.Comput. Chem. 42 (2000) 1–290.
  * [195] Klein, D. J., & Ivanciuc, T. (2006). Directed reaction graphs as posets. In R. Bruggemann & L. Carlsen (Eds.), Partial order in environmental sciences and chemistry (pp. 35–57). Berlin: Springer.
  * [196] Khatri, Minal, ”Formal Concept Analysis for Image Classification and Machine Learning Models for Anti-CRISPR Protein Discovery in Bioinformatics” (2023). Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2024–. 47.
  * [197] Kotsianti, S.B., Kanellopoulos, D. (2007). Combining Bagging, Boosting and Dagging for Classification Problems. In: Apolloni, B., Howlett, R.J., Jain, L. (eds) Knowledge-Based Intelligent Information and Engineering Systems. KES 2007. Lecture Notes in Computer Science(), vol 4693. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74827-4_62
  * [198] V. Kristina Et Al. , “Features of PyHasse software used for the evaluation of chemicals in human breast milk samples,” Simulation in Umwelt- und Geowissenschaften , vol.141, Hamburg, Germany, pp.169-180, 2012.
  * [199] R. Kurtz(2020), Contributions to Semantic Dependency Parsing. https://www.diva-portal.org/smash/get/diva2:1457198/FULLTEXT02.
  * [200] S. O. Kuznetsov, Mathematical aspects of concept analysis, Journal of Mathematical Sciences 80(2),(1996).
  * [201] Kuznetsov, S.O. Machine Learning on the Basis of Formal Concept Analysis. Automation and Remote Control 62, 1543–1564 (2001). https://doi.org/10.1023/A:1012435612567
  * [202] Kuznetsov, S.O, Machine Learning and Formal Concept Analysis Conference: Concept Lattices, Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, February 23-26, 2004.
  * [203] S. Kuznetsov, Ordered Sets for Data Analysis(2019), arXiv:1908.11341 .
  * [204] G. Lebanon, J. Lafferty, Conditional models on the ranking poset, NIPS’02: Proceedings of the 15th International Conference on Neural Information Processing Systems, January 2002, pp. 431–438.
  * [205] Lebowitz, M. (1986). Concept learning in a rich input domain: generalization-based memory. In R.S. Michalski, J.G. Carbonell, & T.M. Mitchell (Eds.), Machine Learning: An Artificial Intelligence Approach (Vol. 2). Morgan Kaufmann, San Mateo, CA.
  * [206] C. Lei et al., The Application of Multidimensional Data Analysis in the EIA Database of Electric Industry, 2011 3rd International Conference on Environmental Science and Information Application Technology (ESIAT 2011), In: Procedia Environmental Sciences 10 ( 2011 ) 1210 – 1215.
  * [207] Levy, S. (1985). Partial order analysis of crime indicators. Social Indicators Research, 16(2), 195–199.
  * [208] Linear Extensions https://mathworld.wolfram.com/LinearExtension.html.
  * [209] J. Liu, Q. Zhang, W. Wang, L. McMillan, J. Prins, Clustering pair-wise dissimilarity data into partially ordered sets. In Proceedings of the Twelfth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. (2006) 637–642.
  * [210] Liu, Q., An, S., Lou, J.-G., Chen, B., Lin, Z., Gao, Y., Zhou, B., Zheng, N., and Zhang, D. Compositional generalization by learning analytical expressions. In Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS’20, Red Hook, NY, USA, 2020.
  * [211] Maarek, Y.S. (1990). An Incremental Conceptual Clustering Algorithm that Reduces Input-Ordering Bias. In: Golumbic, M.C. (eds) Advances in Artificial Intelligence. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9052-7_7
  * [212] H. Mannila, C. Meek, Global partial orders from sequential data, KDD’00: Proceedings of the sixth ACM SIGKDD international conference on knowledge discovery and data mining, August 2000, pages 161–168.
  * [213] L. Markowsky and G. Markowsky, “Lattice Data Analytics and an Exploratory Analysis of the Carver2 Dataset,” to appear in IDAACS 2019.
  * [214] Markov, A lattice-based approach to hierarchical clustering. Proceedings of the Florida Artificial Intelligence Research Symposium, pp. 389–393, 2001.
  * [215] Martin, J., & Billman, D. (1994). Acquiring and combining overlapping concepts. Machine Learning, 16(1– 2):121–155.
  * [216] McKusick, K., & Langley, P. (1991). Constraints on tree structure in concept formation. Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence (pp. 810–816). Sydney, Australia: Morgan Kaufmann.
  * [217] Meddouri, N., Maddouri, M. (2009). Boosting Formal Concepts to Discover Classification Rules. In: Chien, BC., Hong, TP., Chen, SM., Ali, M. (eds) Next-Generation Applied Intelligence. IEA/AIE 2009. Lecture Notes in Computer Science(), vol 5579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02568-6_51.
  * [218] N. Meddouri , H. Khoufi , M. Maddouri, Parallel Learning and Classification for Rules based on Formal Concepts Procedia Computer Science Volume 35, 2014, pp. 358-367.
  * [219] J-P. M$\acute{e}$tivier et al., Discovering Structural Alerts for Mutagenicity Using Stable Emerging Molecular Patterns, J. Chem. Inf. Model. 2015, 55, 5, 925–940, 2015 https://doi.org/10.1021/ci500611v.
  * [220] R.S. Michalski, Knowledge Acquisition Through Conceptual Clustering: A Theoretical Framework and an Algorithm for Partitioning Data into Conjunctive Concepts. International Journal for Policy Analysis and Information Systems. Vol. 4, No. 3, 1980.
  * [221] Michalski and R. E. Stepp. Learning From Observation: Conceptual Clustering. In R.S.Michalski, J.G. Carbonell, and T.M. Mitchell (Eds.), Machine Learning: An Artificial Intelligence Approach, Volume 1, Tioga Publishing Company, pp. 331-363, 1983.
  * [222] V. Mnih, N. Heess, A. Graves, and k. kavukcuoglu, “Recurrent models of visual attention,” in 27th Annual Conference on Neural Information Processing Systems (NIPS 2014). Curran Associates, Inc., 2014, pp. 2204–2212.
  * [223] A. M. Mwafise, G-S Cheon, H. J. Choi, S. Giraudo, Operad Structure of Poset Matrices (2024), arXiv:2401.06814.
  * [224] Mollica, C., & Tardella, L. (2020). PLMIX: an R package for modelling and clustering partially ranked data. Journal of Statistical Computation and Simulation, 90(5), 925–959. https://doi.org/10.1080/00949655.2020.1711909.
  * [225] M. Nabeel et al., A survey of ontology learning techniques and applications, Database, Volume 2018, 2018, bay101, https://doi.org/10.1093/database/bay101
  * [226] Nardo, M., M. Saisana, A. Saltelli and S. Tarantola: 2005b, Tools for Composite IndicatorsBuilding. European Commission, report EUR 21682 EN (Joint Research Centre, Ispra,Italy).
  * [227] Newlin, J.T., Patil, G.P. Application of partial order to stream channel assessment at bridge infrastructure for mitigation management. Environ Ecol Stat 17, 437–454 (2010). https://doi.org/10.1007/s10651-010-0162-8.
  * [228] Nicholls GK, Lee JE, Karn N, Johnson D, Huang R, Muir-Watt A, Bayesian inference for partial orders from random linear extensions: power relations from 12th Century Royal Acta, , 2022. arXiv: 2212.05524.
  * [229] Onishchenko, A.A., Gurov, S.I. Classification based on formal concept analysis and biclustering: possibilities of the approach. Comput Math Model 23, 329–336 (2012). https://doi.org/10.1007/s10598-012-9141-2.
  * [230] Ouyang, C.; Liu, Y. , Formal Concept Analysis Supporting Ontology Learning From Database, Advanced Science Letters, Volume 7 2012, pp. 473-477(5).
  * [231] V. Padhye, K. Lakshmanan, A deep actor critic reinforcement learning framework for learning to rank, Neurocomputing, Volume 547, (2023.) https://doi.org/10.1016/j.neucom.
  * [232] Pakkar, M.S. A Hierarchical Aggregation Approach for Indicators Based on Data Envelopment Analysis and Analytic Hierarchy Process. Systems 2016, 4, 6. https://doi.org/10.3390/systems4010006
  * [233] G. P. Patil, C. Taillie, Multiple indicators, partially ordered sets, and linear extensions: Multi-criterion ranking and prioritization, Environ. Ecol. Stat. 11 (2004) 199–228.
  * [234] G. Patil, W. L. Myers, R. Bruggemann, Multivariate datasets for inference of order: some considerations and explorations, in: R. Bruggemann, L. Carlsen, J. Wittmann (Eds.), Multi-Indicator Systems and Modelling in Partial Order, Springer, New York, 2014,pp.13–45.
  * [235] L. Pellegrina, C. Cousins, F. Vandin, M. Riondato. MCRapper: Monte-Carlo Rademacher Averages for Poset Families and Approximate Pattern Mining. Extended version. https://arxiv.org/abs/2006.09085.
  * [236] Popescu-Spineni, S. (1998). Hierarchy Techniques of Multidimensional Data Analysis (MDA) in Social Medicine Research. In: Rizzi, A., Vichi, M., Bock, HH. (eds) Advances in Data Science and Classification. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-72253-0_87
  * [237] O. Prokasheva, A. Onishchenko, S. Gurov, Classification Methods Based on Formal Concept Analysis, Conference: FCAIR 2013 – Formal Concept Analysis Meets Information Retrieval. Workshop co-located with the 35th European Conference on Information Retrieval (ECIR 2013)At: Moscow.
  * [238] S. Pudenz, ProRank – software for partial order ranking, MATCH Commun. Math. Comput. Chem. 54 (2005) 611–622.
  * [239] Qiu, J., Wu, Q., Ding, G. et al. A survey of machine learning for big data processing. EURASIP J. Adv. Signal Process. 2016, 67 (2016). https://doi.org/10.1186/s13634-016-0355-x.
  * [240] N.R Rashidah, A Comparison Between Single Linkage and Complete Linkage in Agglomerative Hierarchical Cluster Analysis for Identifying Tourists Segments, IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics.
  * [241] A. Raveh and S. F. Landau, Partial Order Scalogram Analysis with Base Coordinates (POSAC): Its Application to Crime Patterns in All the States in the United States, Journal of Quantitative Criminology Vol. 9, No. 1., pp. 83-99 (1993).
  * [242] G. Restrepo & R. Brüggemann, Ranking regions using cluster analysis, Hasse diagram technique and topology , $3^{rd}$ International Congress on Environmental Modelling and Software, Burlington, Vermont, USA 2006.
  * [243] G. Restrepo, R. Bruggemann, M. Weckert, S. Gerstmann, H. Frank, Ranking patterns, an application to refrigerants, MATCH Commun. Math. Comput. Chem. 59 (2008) 555–584.
  * [244] Rimoldi S.M.L., Arcagni A., Fattore M., Barbiano di Belgiojoso E.. (2020) “Targeting policies for multidimensional poverty and social fragility relief among migrants in Italy, using F-FOD analysis”, Social Indicators Research \- doi10.1007/s11205-020-02485-7
  * [245] Rimoldi S.M.L., Arcagni A., Fattore M., Terzera T. (2020) “Social and material vulnerability of the Italian municipalities: comparing alternative ap- proaches”, Social Indicators Research- doi:10.1007/s11205-020-02330-x.
  * [246] Sagi O, Rokach L. 2018. Ensemble learning: a survey. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 8(4):e1249
  * [247] G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrsch. Verw. Gebiete 2 (2964) 340-368.
  * [248] Rouane-Hacene, M., Valtchev, P., Nkambou, R.: Supporting Ontology Design through Large-Scale FCA-Based Ontology Restructuring. In: ICCS 2011. (2011) 257–269.
  * [249] M. D. Ruiz, E. Hüllermeier, A formal and empirical analysis of the fuzzy gamma rank correlation coefficient, Information Sciences,Vol. 206, 2012, pp. 1-17, https://doi.org/10.1016/j.ins.2012.04.006 .
  * [250] F. Ruskey, Generating linear extensions of posets by transpositions, Journal of Combinatorial Theory, Series B, Volume 54, Issue 1, 1992, Pages 77-101,https://doi.org/10.1016/0095-8956(92)90067-8.
  * [251] I.M. Sabara, F. Rozi, M. N. Jauhari, Agglomerative Hierarchical Clustering Analysis Based on Partially-Ordered Hasse Graph of Poverty Indicators in East Java, Proceedings of the 12th International Conference on Green Technology (ICGT 2022).
  * [252] K. Sachs, O. Perez, D. Lauffenburger, G. Nolan, Causal protein-signaling networks derived from multiparameter single-cell data. Science. https://www.science.org/doi/10.1126/science.1105809#supplementary-materials.
  * [253] Salman, H.E. Leveraging a combination of machine learning and formal concept analysis to locate the implementation of features in software variants 2023, Information and Software Technology, vol. 164.
  * [254] Sahami, M. (1995). Learning classification rules using lattices (Extended abstract). In: Lavrac, N., Wrobel, S. (eds) Machine Learning: ECML-95. ECML 1995. Lecture Notes in Computer Science, vol 912. Springer, Berlin, Heidelberg.https://doi.org/10.1007/3-540-59286-5_83.
  * [255] Sangroya, A., Anantaram, C., Rawat, M., & Rastogi, M. (2019). Using Formal Concept Analysis to Explain Black Box Deep Learning Classification Models. FCA4AI@IJCAI.
  * [256] A. Santana, E. Colombini, Neural Attention Models in Deep Learning: Survey and Taxonomy(2021), arXiv:2112.05909v1.
  * [257] Semeraldi et al., Partial Order Rank Features in Colour Space,Appl. Sci. 2020, 10(2), 499; https://doi.org/10.3390/app10020499
  * [258] Sen, A. (1970b). Interpersonal aggregation and partial comparability. Econometrica, 38(3), 393–409.
  * [259] Seymour, R. B. (1960). Missing Data in Non-Linear Trend Analysis of Repeated Measurements on the Same Individuals. The Journal of Educational Research, 54(4), 141–144. http://www.jstor.org/stable/27530403.
  * [260] Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976).
  * [261] Silan M, Boccuzzo G, Arpino B. Matching on poset‐based average rank for multiple treatments to compare many unbalanced groups. Statistics in Medicine. 2021;40(28):6443–6458. doi: 10.1002/sim.9192.
  * [262] U. Simon, R. Bruggemann, S. Mey, S. Pudenz, METEOR- application of a decision support tool based on discrete mathematics, MATCH Commun. Math. Comput. Chem. 54 (2005) 623–642.
  * [263] Simon, U.; Brüggemann, R.; Behrendt, H.; Shulenberger, E.; Pudenz, S. METEOR: a step-by-step procedure to explore effects of indicator aggregation in multi criteria decision aiding – application to water management in Berlin, Germany. Acta hydrochim. Hydrobiol. 2006, 34, 126-136.
  * [264] Song et al., Graph-based Semi-supervised Learning: A Comprehensive Review 2021.
  * [265] Sorensen, P. B.; Mogensen, B. B.; Carlsen, L.; Thomsen, M. The Influence on Partial Order Ranking from Input Parameter Uncertainty Definition of a Robustness Parameter. Chemosphere 2000, 595–600.
  * [266] P. B. Sorensen, R. Bruggemann, L. Carlsen, B. B. Mogensen, J. Kreuger, S. Pudenz, Analysis of monitoring data of pesticide residues in surface waters using partial order ranking theory, Envir. Tox. Chem. 22 (2003) 661–670.
  * [267] Sorensen, P. B. ; Brüggemann; M. Thomsen; D. B. Lerche, Application of multidimensional rank-correlation. MATCH Communications in Mathematical and in Computer Chemistry, 54(3), 643-670.
  * [268] M. Sugiyama, Machine Learning and Information Geometry II, https://mahito.info/files/Sugiyama_NII_IISS_2018_02.pdf.
  * [269] J.L. Sullivan & S. Feldman Multiple Indicators - An Introduction (1979), https://www.ojp.gov/ncjrs/virtual-library/abstracts/multiple-indicators-introduction.
  * [270] Taeb, A. ; Buhlmann, P. ; Chandrasekaran, V. Model Selection over Partially Ordered Sets(2024), ArXiV: 2308.10375 .
  * [271] Tatsuoka C: Sequential classification on partially ordered sets. PhD thesis Cornell University, Statistics Department; 1996.
  * [272] Tatsuoka C. Data analytic methods for latent partially ordered classification models.  Applied Statistics (Journal of the Royal Statistical Society Series C). 2002;51:337–350.
  * [273] Tatsuoka C, Ferguson T: Sequential classification on partially ordered sets. Journal of the Royal Statistical Society, Series B 2003, 65:143-157.
  * [274] Tatsuoka C, Corrigendum:Data analytic methods for latent partiallyordered classification models, Appl.Statist.(2005) 54,Part2,pp.465–467
  * [275] Tatsuoka C. Diagnostic models as partially ordered sets. Measurement. (2009) 7:49–53.
  * [276] Tatsuoka C, Varadi F, Jaeger J. Latent partially ordered classification models and normal mixtures. J Edu Behav Stat. (2013) 38:267–94.
  * [277] Tatsuoka C, Tseng H, Jaeger J, Varadi F, Smith MA, Yamada T, Smyth KA,Lerner AJ. com AsDNInn: Modeling the heterogeneity in risk of progres-sion to Alzheimer’s disease acrosscognitive profiles in mild cognitiveimpairment. Alzheimer’s Res Ther. 2013;5:1–19.
  * [278] Tatsuoka C. Sequential classification on lattices with experiment-specific response distributions. Sequent Anal Design Methods Appl. (2014) 33:400–20.
  * [279] Tatsuoka C, McGowan B, Yamada T, Espy KA, Minich N, Taylor HG. Effects of extreme prematurity on numerical skills and executive function in kindergarten children: an application of partially ordered classification modeling. Learn Individ Differ. (2016) 49:332–40.
  * [280] K. M. Ting, I. Witten, Stacking Bagged and Dagged Models (1997),https://www.researchgate.net/publication/2516354_Stacking_Bagged_and_Dagged_Models
  * [281] F Tombari (2023), Tame representations in Topological Data Analysis, https://www.diva-portal.org/smash/get/diva2:1759244/FULLTEXT01.pdf.
  * [282] F. J. Torres-Rojas & Castro-Mora, Partially Ordered Sets and Logical Clocks for Distributed Systems
  * [283] Tovar, M., Pinto, D., Rendón, A.M., Serna, J.G., & Ayala, D.V. (2014). Identification of Ontological Relations Using Formal Concept Analysis. LANMR.
  * [284] Tsakovski, V. Simeonov, Hasse diagrams as explanatory tool in environmental data mining: A case study, in: J. Owsinski, R. Bruggemann (Eds.) Multicriteria Ordering and Ranking: Partial Orders, Ambiguities and Applied Issues, Sys. Res. Inst. Polish Acad. Sci., Warsaw, 2007, pp. 50–68.
  * [285] Pirintsos, S., Bariotakis, M., Kalogrias, V., Katsogianni, S., Brüggemann, R. (2014). Hasse Diagram Technique Can Further Improve the Interpretation of Results in Multielemental Large-Scale Biomonitoring Studies of Atmospheric Metal Pollution. In: Brüggemann, R., Carlsen, L., Wittmann, J. (eds) Multi-indicator Systems and Modelling in Partial Order. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8223-9_11
  * [286] van Engelen, J.E., Hoos, H.H. A survey on semi-supervised learning. Mach Learn 109, 373–440 (2020). https://doi.org/10.1007/s10994-019-05855-6
  * [287] K. Voigt, G. Welzl, R. Bruggemann, Data analysis of environmental air pollutant monitoring systems in Europe, Environmetrics 15(2004) 577–596.
  * [288] K. Voigt, R. Bruggemann, Ranking of pharmaceuticals detected in the environment: Aggregation and weighting procedures, Combin. Chem. High Through. Screen. 11 (2008) 770–782.
  * [289] K. Voigt, R. Bruggemann, M. Kirchner, K.-W. Schramm, Influence of altitude concerning the contamination of humus soils in the German Alps: a data evaluation approach using PyHasse, Environ. Sci. Pollut. Res. 17 (2010) 429–440.
  * [290] K. Voigt, R. Bruggemann, H. Scherb, H. Shen, K. H. Schramm, Evaluating therelationship between chemical exposure and cryptorchidism by discrete mathematical method using PyHasse software, J. Environ. Model. Soft. 25 (2010) 1801–1812.
  * [291] Voigt, K.; Brüggemann, R. Water contamination with pharmaceuticals: data availability and evaluation approach with Hasse diagram technique and METEOR. MATCH Commun. Math. Comput. Chem. 2005, 54, 671-689.
  * [292] Voigt, Kristina et al. “A multi-criteria evaluation of environmental databases using the Hasse Diagram Technique (ProRank) software.”  Environ. Model. Softw. 21 (2006): 1587-1597.
  * [293] K. Voigt et al., Application of the PYHASSE Program Features: Sensitivity, Similarity, and Separability for Environmental Health Data, Statistica & Applicazioni - Special Issue, 2011, pp. 155-168.
  * [294] X. Wang and A. Gupta. Unsupervised learning of visual representations using videos. In ICCV, 2015.
  * [295] X. Wang, P. Li, M. Zhang, Improving Graph Neural Networks on Multi-node Tasks with Labeling Tricks, (2023) arXiv:2304.10074. Available in: https://arxiv.org/abs/2304.10074
  * [296] C. Wendler, Machine Learning on non-Euclidean domains: powersets, lattices, posets, (2023) ETH Zurich.
  * [297] R. Wille, Restructuring lattice theory: An approach based on hierarchies of concepts, in: I. Rival (Ed.), Ordered Sets, D. Reidel Publishing, Dordrecht, 1982, pp. 445–470.
  * [298] J. Wittmann, J. Markert, S. Plura, R. Brüggemann, A Software Platform Towards a Comparison of Cars-A Case Study for Handling Ratio-Based Decisions, EnviroInfo 2011: Innovations in Sharing Environmental Observations and Information, Shaker Verlag Aachen. Available in: http://enviroinfo.eu/sites/default/files/pdfs/vol6919/0467.pdf.
  * [299] Wu et al., A Comprehensive Survey on Graph Neural Networks,(2019) arXiv:1901.00596.
  * [300] Wu, C., Li, H., & Ren, J. (2021). Research on hierarchical clustering method based on partially-ordered Hasse graph. Future Generation Computer Systems. Doi: https://doi.org/10.1016/j.future.2021.07.025
  * [301] Z. Xie, W. Hsu, Z. Liu, M. L. Lee: Concept Lattice based Composite Classifiers for high Predictability. Artificial Intelligence, vol. 139, pp. 253–267, Wollongong, Australia (2002).
  * [302] Xiao, S., Yan, J., Farajtabar, M., Song, L., Yang, X., & Zha, H. (2017). Joint Modeling of Event Sequence and Time Series with Attentional Twin Recurrent Neural Networks. ArXiv, abs/1703.08524.
  * [303] Y. Yadav, A. Samal, E. Saucan, A Poset-Based Approach to Curvature of Hypergraphs, Symmetry, 14(2):420, 2022.
  * [304] M. Yang, R. Wang, Y. Shen, H. Qi and B. Yin, “ Breaking the Expression Bottleneck of Graph Neural Networks,” in IEEE Transactions on Knowledge and Data Engineering, vol. 35, no. 6, pp. 5652-5664, 1 June 2023, doi: 10.1109/TKDE.2022.3168070.
  * [305] Yoneda, Y., Sugiyama, M., & Washio, T. (2018). Learning Graph Representation via Formal Concept Analysis.  ArXiv, abs/1812.03395.
  * [306] H. Yu, Y. Chen, P. Lingras, G. Wang, A three-way cluster ensemble approach for large-scale data, International Journal of Approximate Reasoning, Volume 115, 2019, Pages 32-49, https://doi.org/10.1016/j.ijar.2019.09.001.
  * [307] L. A. Zadeh, Fuzzy Sets, Information and Control, 8, 338–353 (1965).
  * [308] J. Zhang , Learning to Rank Cases with Classification Rules(2008), http://www.ecmlpkdd2008.org/files/pdf/workshops/pl/8.pdf.
  * [309] L. Zhang, D. Wang , X. Liu, Computing Concept Lattices with Clustering Approaches.
  * [310] Zhao, M., Zhang, S., Li, W. et al. Matching biomedical ontologies based on formal concept analysis. J Biomed Semant 9, 11 (2018). https://doi.org/10.1186/s13326-018-0178-9.
  * [311] Zia, A., Khamis, A., Nichols, J. et al. Topological deep learning: a review of an emerging paradigm. Artif Intell Rev 57, 77 (2024).https://doi.org/10.1007/s10462-024-10710-9.
  * [312] Y. Zuo and R. Serfling, General Notions of Statistical Depth Function, The Annals of Statistics, Vol. 28, No. 2 (Apr., 2000), pp. 461-482.
  * [313] UCI machine learning repository, http://www.ics.uci.edu/~mlearn/MLRepository.html.
  * [314] Kohonen, Teuvo (January 2013). ”Essentials of the self-organizing map”. Neural Networks. 37: 52–65. doi:10.1016/j.neunet.2012.09.018.
  * [315] MPII Human Pose Dataset, http://human-pose.mpi-inf.mpg.de/.
  * [316] Leeds Sports Poset Dataset https://datasets.activeloop.ai/docs/ml/datasets/lsp-dataset/.
  * [317] Olympic Sports Dataset http://vision.stanford.edu/Datasets/OlympicSports/.
  * [318] Compositional Freebase Questions (CFQ) Dataset,https://github.com/google-research/google-research/tree/master/cfq
  * [319] Universal Dependencies Corpora https://www.tensorflow.org/datasets/catalog/universal_dependencies
  * [320] Istat. (2018a). Indicators for the measurement of equitable and sustainable well-being. https://www.istat.it/en/well-being-and-sustainability/the-measurement-of-well-being/indicators.
  * [321] Eurobarometer dataset for service performance https://www.gesis.org/en/eurobarometer-data-service/
  * [322] Python implementaion of deep supervised visual similarity learning(2017) In :https://github.com/asanakoy/deep_unsupervised_posets .
  * [323] C. Squires, causaldag (2018), https://github.com/uhlerlab/causaldag.
  * [324] Poset-Hypergraph-Curvature Datasets, https://github.com/asamallab/Poset-Hypergraph-Curvature.