All electric aircraft: A reality on its way

Abstract

In this paper, the history and future opportunities along with various challenges and progress in electric aviation and for more electric aircraft (MEA) has been discussed. In introduction, the significant needs and ideas of MEA is covered. Impacts of aviation on environment and the application of electricity in airplane along with the idea of More Electric Aircraft (MEA) is discussed with an overview of hybrid and all electric aircraft. Here, the effort is made to show the difficulties faced by the engineers to successfully run the electric aircraft for commercial purpose. The progress in the field of converting conventional aircraft into electric is reviewed and analyzed in this paper. Different aspects and challenges faced in making the aviation sector greener are also shown in this paper. Finally, striving ideas to overcome the challenges, to develop the field of electric aircraft and to make the sky cleaner and greener along with the future scope and research possibilities are put forward in this paper.

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1
3All electric aircraft: A reality on its way
4
5
6Nischal Thapa,Sidharth Ram,Sonu Kumar,Jimmy Mehta
7Department of Mechanical Engineering, Manav Rachna Institute of Research and Studies, Faridabad, India
8
9
11
article info
12 Article history:
13 Received 17 May 2020
14 Received in revised form 24 October 2020
15 Accepted 19 November 2020
16 Available online xxxx
17 Keywords:
18 Aircraft power systems
19 Electric System
20 Emission
21 Environment Control
22 More Electric Aircraft
23 History of Electric Aircraft
24 All Electric Aircraft Future Scopes
25 Progress and Research Areas
26
27
abstract
28
In this paper, the history and future opportunities along with various challenges and progress in electric
29
aviation and for more electric aircraft (MEA) has been discussed. In introduction, the significant needs
30
and ideas of MEA is covered. Impacts of aviation on environment and the application of electricity in air-
31
plane along with the idea of More Electric Aircraft (MEA) is discussed with an overview of hybrid and all
32
electric aircraft. Here, the effort is made to show the difficulties faced by the engineers to successfully run
33
the electric aircraft for commercial purpose. The progress in the field of converting conventional aircraft
34
into electric is reviewed and analyzed in this paper. Different aspects and challenges faced in making the
35
aviation sector greener are also shown in this paper. Finally, striving ideas to overcome the challenges, to
36
develop the field of electric aircraft and to make the sky cleaner and greener along with the future scope
37
and research possibilities are put forward in this paper.
38
Ó2020 Elsevier Ltd. All rights reserved.
39
Selection and peer-review under responsibility of the scientific committee of the 1st International Con-
40
ference on Energy, Material Sciences and Mechanical Engineering.
41
42
43
44
1. Introduction
45
The climate of the earth is changing rapidly and our behaviors
46
must also change in order to fix this which means we need to move
47
towards cleaner, more efficient, forms of transport. Higher energy
48
efficiencies, lower greenhouse emission and lower audible noise
49
for aircraft have become critical issues that need to be addressed
50
in this modern world due to the threats on quality of life and even
51
survivability of our future generations. To achieve these, people
52
have focused on the idea of MEA or aircraft with its complete
53
energy and powertrain dependent on electrical energy whose
54
enhancement is the common understanding of the society of
55
human beings [1]. There are research works going on to replace
56
all mechanical, electrical, hydraulic or pneumatic energy systems
57
used in the aircraft with an electric energy system gained through
58
the generator of engines or through auxiliary power units (APU)
59
through MEA. The energy of the battery, the efficiency and weight
60
of the aircraft, sound produced, challenges or key technological dif-
61
ficulties faced in the growing electric aircraft field including bat-
62
tery energy density upgrade and performance improvement, high
63
lift-to-drag ratio aerodynamic aircraft design, electric aircraft
64
material and structure design and manufacture, high efficiency
65
electric propulsion system design are need to be addressed and
66
worked on.Fig. 1.Fig. 2.Fig. 3.Fig. 4.Fig. 5.Fig. 6.Fig. 7.Fig. 8.Fig. 9.
67
Fig. 10.
68
2. History
69
Electric flight is not a new technology. The only thing, was the
70
mass of the aircraft due to which, it was not really possible for
71
many years. In 1940, Fred Militky was the first who used electric
72
motors to operate model aircraft, but he was unsuccessful due to
73
heavy motors and lead batteries to obtain good results. He contin-
74
uously believed in his idea until he was finally able to introduce a
75
small model of airplane in 1960. There were highly efficient elec-
76
tric motor and small lead batteries in this model. With the intro-
77
duction of Ni-Cd batteries with higher power densities in 1972,
78
he was successful in making the first radio-controlled model which
79
was commercially produced [2]. After that he started his work on
80
producing the first electric flight of manned aircraft. On 21 October
81
1973, for the first time a 23-year-old aircraft manufacturer named
82
Heino Brditschka flew a manned aircraft nearly 300 m above the
83
ground of the airfield in Wels, near Linz which was a modified ver-
84
sion of the HB-3 power glider. Brditschka and the ideas provided
85
by Fred Militky were the two minds who made the dream of flying
86
with an electric engine possible [3]. Further development in this
87
was restricted due to the lack of better battery systems. However,
88
there is constant progress in this field which is ultimately leading
89
us to greener aviation.
https://doi.org/10.1016/j.matpr.2020.11.611
2214-7853/Ó2020 Elsevier Ltd. All rights reserved.
Selection and peer-review under responsibility of the scientific committee of the 1st International Conference on Energy, Material Sciences and Mechanical Engineering.
Q1Q3
Q4
Q5
Q6
Materials Today: Proceedings xxx (xxxx) xxx
Contents lists available at ScienceDirect
Materials Today: Proceedings
journal homepage: www.elsevier.com/locate/matpr
MATPR 20308 No. of Pages 9, Model 5G
28 December 2020
Please cite this article as: N. Thapa, S. Ram, S. Kumar et al., All electric aircraft: A reality on its way, Materials Today: Proceedings, https://doi.org/10.1016/j.
matpr.2020.11.611
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90
Following timeline will provide you some idea regarding the
91
history and progress in this field of electric aircraft [1 4]:
92
93
Some Previous works in different fields of Electric Aviation
94
When we need to talk about some previous research works of
95
other authors in this field of electric aviation, we can look after ref-
96
erence [5–6] where the power management and distribution for
97
the electric aircraft have been studied. Moreover, the study of var-
98
ious electric machines and drives used in the aircraft generation
99
along with their different challenging applications have also been
100
studied previously which can be found in references [7–8].Some
101
more previous works are as below:
102
I. Battery
103
104
Research works are being carried out in all possible ways to
105
power electric aircraft. Among them, we are here focused mainly
106
on batteries. We are discussing about researchers who are working
107
to improve the energy density of the batteries and to make it light
108
weight. A group of researchers (Mohd Tariq, Ali Iftekhar Maswood ,
109
Chandana Jayampathi Gajanayake and Amit Kumar Gupta) has said
110
in their research paper named ’Aircraft Batteries: Current Trend
111
towards More Electric Aircraft’ that a battery should be able to
112
deliver power reliably, be certifiable safe, be light weight, have a
113
consistent power output over their operating environment and
114
have a reasonably long life in order to qualify to be used in any
115
type of electric aircraft.
Fig. 1. Step wise progression [2].
Fig. 2. Fours forces in aircraft.
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II. Propulsion
117
118
Researchers Ion Boldea, Lucian N. Tutelea, Leila Parsa, and David
119
Dorrell, had suggested in their paper named ’Automotive Electric
120
Propulsion Systems With Reduced or No Permanent Magnets: An
121
Overview’ that combining the motor and the generator with the
122
gearing to remove the mechanical planetary gearbox and combine
123
the motor and the generator in the propulsion system will be effec-
124
tive in increasing the efficiency of the system. But this method
125
comes with need in more maintenance which will be a bit
126
disadvantage.
127
III. Fuselage Materials
128
129
Fuselage materials used to construct any type of electric aircraft
130
plays a significant role in determining the weight of the aircraft. It
131
has been suggested that a fuselage structure for an aircraft fuselage
132
is to be made in composite material and an aircraft including such
133
a fuselage structure. This provides savings in mass while allowing
134
grounding of electric systems, along with mechanical stiffness. Air-
135
bus researchers Guillaume Gallant, Laurent Giuseppin, and Damien
136
Aguera told this in their patented document for fuselage structure
137
of electric aircraft.
138
IV. Aerodynamic Design
139
140
Appropriate aerodynamic design of electric aircraft is must in
141
order to make them efficient. It is mainly concerned with the max-
142
imum utilization of power generated via aerodynamically sound
143
body structure. Researchers Michael D. Patterson, Matthew J. Dask-
144
ilewiczy, and Brian J. Germanz of Georgia Institute of Technology
145
had told in their named, ’’Conceptual Design of Electric Aircraft
146
with Distributed Propellers: Multidisciplinary Analysis Needs and
147
Aerodynamic Modeling Development’’ that aerodynamic analysis
148
methods including lifting line methods, lifting surface methods
149
(e.g., the vortex lattice method), and computational fluid dynamics
150
(CFD) are must to be while designing. Also wing analysis and anal-
151
ysis of the interaction between propeller and wing should be done
152
appropriately during designing process.
153
V. Thermal Management
154
155
Heat generated in these types of electric aircraft should be man-
156
aged properly. In a paper named, ’’Dynamic Thermal Management
157
System Modeling of a more Electric Aircraft’’ by K. McCarthy, E.
158
Walters, and A. Heltzel, various thermal management components
159
are discussed like FTMS (Fuel Thermal Management System), heat
160
exchanger and others are must on board of any electric aircraft
161
which help in significant heat sink and helps in proper manage-
162
ment of generated heat.
163
3. Need of electrification of aircraft
164
It is certain that aircraft emissions cause pollution. According to
165
recent reports of European Union, nearly three percent of total
166
greenhouse gases produced globally are from aviation sector. Air-
167
craft related emissions has a modest share (for instance; CO2 emis-
168
sion from aircraft are more than other CO
2
emission from others)
169
[9]. As of current records the world air traffic has growth rate of
170
4–5% so situation will only get worse if we can’t anything for it
Fig. 3. Pipistrel -Alpha electro [16].
Fig. 5. Power sources on a conventional civilian aircraft [32].
Fig. 4. Cessna Skyhawk 172[15].
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[10]. Increased aircraft operations activity in or around where an
172
airport is located also produces emissions that degrade the air
173
quality [1]. Besides this, there is another reason which is cost of
174
operation [2]. Aircrafts have huge operating cost due to which they
175
are an expensive mode for travelling permitting access to limited
176
individuals. For reference, an Airbus A380 costs around 305,735
177
USD for a 14hrs flight from Sydney to los angles out of which
178
244,534 is fuel cost alone. However, electric plane has a low cost
179
of operation due to low maintenance cost and secondly, they use
180
renewable energy resources which are comparatively cheaper than
181
jet fuel [11]. ‘Noise’ is another environmental hindrance which our
182
conventional aircraft are unable to overcome but electrification
183
promises to do so [12].
184
4. Challenges in the field of electric aviation
185
Now the question is when we know the benefits of electric air-
186
crafts over gasoline ones why aren’t we having them, what are the
187
challenges that are in the way of electric aviation?
188
Challenges in the field of electric aviation:
189
I. Low energy density of batteries
190
II. Limitations to the distance travelled
191
192
The above two are the major challenges or we can say only chal-
193
lenges in the field of electric aviation. Although there are some
194
other challenges also but those are only the consequences which
Fig. 6. Concept for the power sources on a MEA [32].
Fig. 8. Full Hybrid Architecture [1].
Fig. 7. More Electric Hybrid Architecture [1].
Fig. 9. All Electric Architecture [1].
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we have to overcome due to these two hurdles. By technology here
196
I mean to say that we need highly efficient motors and energy
197
convertors.
198
I. Low energy density
199
200
Energy density is one of the major hurdles in the field of electric
201
aviation. Our best ‘lithium-ion’ batteries are having the energy
202
density of 265 Wh/kg while energy density of jet fuel is 11.95
203
kwh/kg. We can clearly see the huge difference between the energy
204
densities of both the sources [13].Table 1: below shows the differ-
205
ence in power generated at the shaft due to different energy
206
sources.Table 2.
207
Although the above table clearly shows the effect on power at
208
the shaft due to difference in energy density of jet fuel and battery
209
but still power is not the factor that limits electric aviation it is the
210
energy density.
211
How does this energy density effect our aircraft?
212
Firstly, we should know that all the fuel or energy required for
213
flying need to be stored on board. Now if we want to run an aircraft
214
by electricity (batteries) alone which was previously powered by
215
jet fuel then we should know that we will be requiring more bat-
216
teries then jet fuel(approximately 50 times more) for supplying
217
power equivalent to that of jet fuel which means the mass of our
218
fuel (i.e. batteries) will increase by 50 times which in turn will
219
increase the mass of our aircraft also (due to extra mass of batter-
220
ies) so now we will be requiring more power to lift that extra mass
221
and to fulfil the power demand we need to add more batteries but
222
this again will increase our aircraft mass but in case of jet fuel the
223
situation is exactly opposite aircraft mass decreases as it travel
224
some distance due to the consumption of fuel. By this we can
225
understand our current obstacles [1].
226
II. Distance travelled
227
228
The low energy density also limits our distance which we can
229
travel with an electric aircraft in one charge. IFB-Stuttgart E-
230
Genius is a 2-seater electric aircraft which in 2013 set a record of
231
travelling about 400 km distance in one charge. When compared
232
with our conventional aircrafts electric aircraft are far behind them
233
in the distance factor. And distance is one of the major factors that
234
influence the use of any transport.
235
III. Other challenges in the field
236
We need to improve Efficiencies of our motors and energy con-
237
vertors so that we can minimise the losses and increase our
238
overall efficiency.
239
To overcome the obstacle of distance either we need to decrease
240
mass of our aircraft or increase the energy density of our batter-
241
ies. In our case we need to do both. So, mass of aircraft is
242
another challenge.
243
244
5. Progress in the field of electric aviation
245
As of now it is clear that having a full electric commercial flight
246
with the current resources and advancements we have, will take
247
quite some time.
248
For making electric aviation possible we need to focus on some
249
parameters such as:
250
Increasing energy density of batteries.
251
Decreasing mass of plane.
252
Decreasing drag force on the plane.
253
Increasing lift force of the plane.
254
255
In case of energy density, we are having our best lithium ion
256
batteries and improving them further may take quite some time.
257
While for other parameter, many aviation giants including Airbus,
258
Boeing, NASA and many start-ups are working on this concept of
259
reducing mass of plane in various ways such as by using lighter
260
and durable material for the body of aircraft, reducing the weight
261
of other instruments. And reducing the weight or dry mass of elec-
262
tric motor. Siemens and MagniX are among top electric motor
263
manufacturer in this field. Siemens have successfully developed
264
an electric motor which weights around 50 kg and can produce
265
260 kW power. By increasing wingspan of plane (although it has
266
its own limitation) and improving design of plane to make it more
267
aerodynamic we can work on our other two parameters also. In
268
fact, the smaller electric planes which we are having right now
269
are based on these concepts. But there is still a limitation of dis-
270
tance which we can travel by any electric aircraft which roughly
271
has a range of 400 km [14]. May be the distance is a barrier but still
Fig. 10. Electric Taxi Demonstration [47].
Table 2
Comparison between CESSENA SKYHAWK and PIPISTREL-ALPHA ELECTRO.
Specifications Cessena Skyhack
(Conventional Aircraft)
Pipistrel-Alpha
Electro
Length 8.3 m6.5 m
Height 2.7 m2.05 m
Wingspan 11.0 m10.5 m
Takeoff Distance 497 m225 m
Landing Distance 407 m460 m
Empty Weight 762 kg 251 kg(incl. PRS)
Range 1185 km 160 km
Max. Cruise Speed 230 km/hr 200 km/hr
Table 1
Power given to Shaft per kg.
Energy density (Wh/kg) Global efficiency Shaft power (W/kg)
Jet fuel 11,950 0.55 6572
Battery 265 0.95*0.90 227
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272
with certain advancements these electric planes are capable of
273
replacing the regional commercial flights which will not only
274
decrease the carbon emission but also make the regional trans-
275
portation cheaper.
276
Pipistrel-alpha electro is a 2 seater all electric aircraft with a
277
range of about 160 km in one charge and has an operating cost
278
of roughly 25 USD per hour although it’s not suited for travelling
279
commercially or over long distances but in coming future it is
280
expected to replace gasoline training aircrafts such as Cessna Sky-
281
hawk 172 which is one of the leading aircraft for pilot training.
282
These aircrafts not only cause pollution but also has an operating
283
cost of around 150–200 USD per hour. The below table shows
284
the basic difference between leading gasoline trainer aircraft and
285
all electric aircraft [15 –16].
286
6. More electric aircraft (MEA)
287
Aircrafts are being powered by mechanical power, hydraulic
288
power or pneumatic power sources because of which the idea of
289
more electric aircraft is introduced where electrical power systems
290
are given more emphasis. This concept is now being used for envi-
291
ronmental control systems, aircraft actuation systems, fuel pump-
292
ing, and wing ice protection. Great effort to make more fuel
293
efficient and quieter aircraft which contribute to greener environ-
294
ment is done because of this concept [17]. In aircraft, different sys-
295
tems requires the use of different types of energy which can be
296
electrical, hydraulic, mechanical or pneumatic energy which have
297
their own drawbacks and also our engine total efficiency is also
298
effected in our energy conversion process which is solved due to
299
the introduction of this idea[18]. In the MEA, the use of electric sys-
300
tem instead of conventional one will dramatically improve our air-
301
crafts maintenance ,efficiency, reliability, supportability, fuel
302
conservation and also it will reduce our aircraft weight and can
303
also increase it accommodation capability[19–21].The challenge
304
for MEA includes storing maximum amount of energy in a given
305
amount of weight [22] so as to maintain its light weight and max-
306
imum accommodation capability and another challenge is conver-
307
sion and control of power with highly efficient power electronics
308
[23–24]. MEA’s first (commercial) flight was made by the Boeing
309
787 Dreamliner. A350 by Airbus Corporation is another example
310
of recent development in the field of more electric [25].
311
7. Replacement of various aircraft systems by electric system
312
For various types of systems such as the avionics systems,
313
entertainment and lighting while on board, the electrical system
314
in the aircraft has been used. The pneumatic system, a system of
315
compressed air, [26] provides air for other systems such as the sys-
316
tems controlling the environment (mainly pressurization and air
317
conditioning) and wing ice protection [27]. Power electronic con-
318
verters, which are important for electro-mechanical /electrohy-
319
draulic actuators, constitute center of motor drives which plays a
320
crucial role in electrifying the aircraft [28–29]. All the aircraft actu-
321
ation system covered by the hydraulic system in aircraft, for pump-
322
ing the fuel and oil the mechanical system is driven from engine
323
gearbox. Thus, to maintain aircraft weight, power electronic device
324
should be designed in a manner that its size and weight should be
325
less so it won’t affect the aircraft overall weight [30–31]. So, this
326
MEA concept is introduced where it clearly means that using single
327
type of power source from engine will be more effective. Here elec-
328
trical power is chosen seeing its more advantages in its application
329
and flexibility [17]. The overall purpose of this change in technol-
330
ogy is to decrease the environmental impact of air travel, operating
331
costs, fuel consumption. Efficiency of aircraft can be improved by
332
removing the pneumatic system which removes the bleed air
333
system on the gas turbine. Also, some weight from the overall
334
weight of aircraft is reduced when we remove the hydraulic and
335
mechanical system and leading us to realize the full potential of
336
this concept [32].
337
8. Future trends and opportunities
338
8.1. Electric system architecture
339
An objective to design the electric system is to obtain an oper-
340
ation performance, safety and maintenance [33]. There are three
341
potential system architectures i.e. ‘More Electric – Hybrid’ to ‘Full
342
Hybrid’ to ‘All Electric’. There is the requirement of varying devel-
343
opment for each system component and great challenge for the
344
complete systems. Though there are many proposals for hybrid
345
systems, those presented here are considered the most feasible at
346
present. Selection of architecture may also be affected by the need
347
of different thrust to air borne, climb and cruise, as well as different
348
aerodynamic of the aircraft, varying altitude, and varying aircraft
349
design are taken under consideration [1]. Besides, ways to monitor
350
the bidirectional power flow for starting engine, to regulate the bus
351
voltage distribution in response to varying loads and to ensure the
352
efficient power system operation throughout the aircraft are other
353
important areas for research [34–35].
354
8.2. Hybrid gas-electric propulsion aircraft
355
Audible noise, NOx emissions, and fuel and energy utilization
356
are decreased in the objective for future generation subsonic fixed
357
wing aircraft developed by NASA [36]. However, the challenge with
358
noncryogenic motors is to get high efficiency with volume and
359
weight constraints [36]. As a result, opportunities and obstacles
360
exist for developing and applying new technologies, including
361
the new machine topologies, conductors with greater resistivity
362
than copper, and fault redundancy. Additionally, new opportuni-
363
ties for research are the development of lightweight and high
364
strength composite materials, structural advances, new cooling
365
techniques and materials, and better insulation materials with
366
high thermal conductivity [37].
367
8.3. Electric taxi
368
Different systems have been proposed to reduce the use of main
369
engines on ground for making aircraft efficient. Though taxiing
370
with only one engine has also been proposed, electric taxi is chosen
371
to be best way to reduce more use of engine in ground [38–39].
372
Electric taxi capability is one of the more futuristic technologies
373
for electrifying aircraft. Aircraft use thrust from the engine to steer
374
through the airport runways, once they are out of the gate which is
375
undesirable because main engines burn significant amounts of fuel
376
causing high emission [40]. In order to solve this problem, it has
377
been suggested to cut off the engine emission by providing the
378
electric motors on the main gears and nose wheel so that the air-
379
craft can taxi [41] without turning on the engine before take-off.
380
When the aircraft lands, it will be able to access electric taxi to
381
the gate through electric motors and shutting off the engines. A
382
motor, power electronic converter, controls, cockpit contacts, and
383
APU power are needed to achieve this. Power from the APU would
384
need to be conditioned for the traction motor by using power elec-
385
tronic converters [42] and this technology is known as e-taxi or
386
green taxi [43]. This growing subsystem has the capacity to suc-
387
ceed in reducing fuel use, lowering emissions and increasing oper-
388
ational capability. When fuel cells and batteries are used instead of
389
regular engine source to power the APU for electric taxi, it will sig-
390
nificantly decrease the emission during taxiing. Pilot’s technical
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391
and procedural work to be in command during taxi is also solved
392
with eco-friendly system. [44]. Many safety critical and system
393
requirements are need to be looked after to make this concept a
394
reality which includes reliability, efficiency, cost, fault tolerance
395
and weight. Recent research practices in this field include the
396
direct wheel actuators [45] and heat considerations for potential
397
direct wheel with electric motors [46].
398
8.4. Fuel cell technology
399
Fuel cell have its own supremacy, its harmful gas emission is
400
very low and it is more efficient than diesel or gas engine and even
401
more silent. In contrast of with battery, it’s a light weight that’s
402
why fuel cell has come as the first preference to power the aircraft
403
instead of batteries [48]. In neoteric years, fuel cell had gained the
404
interest of many popular agencies for clean energy [49]. Less
405
weight of fuel cell helps in reducing the overall weight and size
406
of APU’s, and this will make aircraft even more efficient with suf-
407
ficient energy production. Fuel cell or battery and fuel cell hybrid
408
system can replace the APU’s with more efficiency. To achieve
409
the dream of commercial electric aircraft, even more improvement
410
has to be needed in other equipment too. Aircraft with higher effi-
411
ciency and extremely low emission is still needed. To study more,
412
refer reference [50] which gives the thorough study of different
413
tactics for a fuel cell hybrid energy system under emergency
414
situation.
415
9. Conclusion
416
Although there has been a lot of progress in the field of aviation
417
but the fact is that there is still miles to go before we can proudly
418
say that the aviation industry has been fully electrified and has
419
been made greener and cleaner, but until then we are not stopping
420
to grow, research and develop in this field. Other conclusions of
421
this paper are given below in bulleted form:
422
Electrification of aircraft has become the most important thing
423
to do in this modern world where the emission of greenhouse
424
gases and pollution are high.
425
There are currently many challenges in the field of electric avi-
426
ation where lots of previous research had already been done but
427
still requires further research work for the improvement of per-
428
formance of the aircrafts.
429
This paper also concludes that there has been a significant pro-
430
gress in areas like increasing the energy density of battery,
431
decreasing aircraft mass, using advance electrical and mechan-
432
ical technologies and so on.
433
This paper also gives insight of future trends and research
434
opportunities in the field of electric aviation
435
436
which will definitely help engineers, designers and researchers
437
to explore today’s technological barriers and get the goal to
438
decrease the size and weight of current machines, and improve
439
the efficiency and reliability of the aircraft even further and ulti-
440
mately contributing for green aviation.
441
Declaration of Competing Interest
442
The authors declare that they have no known competing finan-
443
cial interests or personal relationships that could have appeared
444
to influence the work reported in this paper.
445
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