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ScienceDirect Procedia Engineering 185 (2017) 212 – 219

6th Russian-German Conference on Electric Propulsion and Their Application Evaluation of the efficiency of propulsion systems usage with energy accumulation at spacecraft

S. A. Ishkov, A. A. Khramov* Samara University, 34, Moskovskoye Shosse, Samara, 443086, Russia

Abstract The problem of cooperative optimization of trajectory and main design parameters of spacecraft, equipped with propulsion systems with energy accumulation, during low Earth orbit formation and correction in non-central gravitational field is considered. The working principle of these propulsion systems is a periodic cycle of energy accumulation from the energy source at the passive stages of orbital transfer, and its discharge to supply power to the engine during the active stages of orbital transfer. The special characteristics of propulsion systems with energy accumulation is limited operation time per one switching, which determines by the energy, stored in an accumulator. For design-ballistic analysis, the mass model of the spacecraft with an uncontrollable power-limited engine with energy accumulator is used. The general problem is separated into dynamic and parametric parts. Using averaging mathematical models of motion and Pontriagin’s maximum principle, the calculation method of the ballistic characteristics of approximately-optimal orbital transfer is obtained. The method of design-ballistic optimization is obtained as an iterative procedure. It was found, that inclusion of an energy accumulator in a propulsion system provides an increasing of orbital transfer efficiency. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2017 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of RGCEP – 2016. Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application Keywords: spacecraft; propulsion system with energy accumulation; design model; averaging method; optimal control program;

1. Introduction One method of increasing orbital transport operation efficiency, for spacecraft transferring from injection to working orbit or orbital correction, is to use engines with high specific impulse. Specific impulse can be increased by using new working fluids or an additional thermal or electrical energy supply. Widely used electric jet engines (EJE) of a low-thrust, can provides a significant increase in payload mass or enable the spacecraft to be equipped with a lighter rocket. At same time, EJE have the disadvantage of a significantly longer duration of orbital transfer. Nowadays, perspective propulsion systems with energy accumulation devices are being designed, characterized by a thrust order of about dozens of Newton and specific impulse which average between EJE and liquid engines (LE). At the Keldysh Research Center, a solar thermal propulsion system (STPS) with heating of the working fluid is being designed. The operational principle of STPS is next. During the sunlit portion of an orbital transfer, solar panels can be used to transfer solar energy to heat energy, which can be stored in a thermal accumulator (TA).

* Corresponding author. Tel.: +7-846-267-4507. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application

doi:10.1016/j.proeng.2017.03.302

213

S.A. Ishkov and A.A. Khramov / Procedia Engineering 185 (2017) 212 – 219

Before fuel (hydrogen) supply to the engine, it is heating in TA that increases specific impulse. STPS can operate at LE mode with cold components (oxygen - hydrogen). The special characteristic of this propulsion system is limited operation time per one switching, determined by the TA working capacity, and the higher thrust order, that allow the orbital transfer duration to be decreased. 2. Problem statement The main optimization problem in spacecraft dynamics is the maximal payload mass mPL transferring to working orbit for determined boundary conditions, initial spacecraft mass m0 and orbital transfer duration T . For problem solving, the design model of a spacecraft, equipped with an uncontrollable power-limited engine with energy accumulator [2] is used. It is assumed that propulsion system operates in two modes. Firstly, the engine is switchedon, thrust P and exhaust velocity c are constants and the energy accumulator is discharged with constant power N E . Secondly, the engine is switched-off, parameters P and c are equal to zero and the energy accumulator is charged from solar panels with constant power N S . For analysis, a spacecraft mass model is introduced, as a sum of spacecraft components mass: m0

mPL mC mE mS men mwf m fs ,

where mPL – payload mass, mC – spacecraft structural mass, mE – mass of TA, mS – solar panel mass, men – engine mass, mwf – working fluid mass, m fs – fuel storage and fuel transfer system mass. The components mass can be written as functions: mC

PC m0 , mE

J E E0 , mS

J S N S , men

J en P , mwf

P Tm , m fs c

E fs mwf ,

where E 0 – maximal energy, stored in the accumulator (accumulator operational capacity), N S – solar panel power,

P – engine thrust, c – exhaust velocity, Tm – duration of active areas of orbital transfer, PC , J E , J S , J en , E fs – respectively specific mass characteristics, assumed as constant values. For payload mass, we can write: mPL

§ § V m0 1 PC J E E0 J S N S J en P 1 E fs m0 ¨1 exp ¨ x © c ©

·· ¸¸ , ¹¹

(1)

where Vx – characteristic velocity of orbital transfer. The vector of optimized design parameters contained engine thrust, solar panel power and TA operational capacity: \ P, NS , E0 . If we fix the design parameters, the maximum of payload mass mPL correspons to minimal characteristic velocity Vx . The main variational problem is divided into dynamic and design parts. The dynamic problem involves determining the optimal control program, which minimizes characteristic velocity, taking into account limitations to control vector u and state vector x:

uopt t arg min ª¬Vx u, x0 , xk , T T =fixe, x0

fixe, xk

uU , xX

fixeº¼ .

The design part is formalized in the following equation:

mPL

arg max mPL \ Vx \
2[ (S D )@ ,

D · §D · § sin ¨ [ ¸ cos ¨ ¸ cos(K ) bk , P S 2¹ © ©2¹ A w0

(2)

A w0

§ D· ¨1 ¸ w0 , © S¹

dVx dt

where A – semi-major axis of the orbit, q and k – the components of vector Laplace, [ – half width of acceleration area with argument of latitude of its center K , D – passive area duration by the argument of latitude, which we assume as constant value, P – Earth’s gravitational constant, b

B B | , Am A3,5 Am3,5

A0 Ak , B 2

H (5cos2 i 1) , 2 P

km5 , i – orbit inclination. s2 For determination of the optimal control program, we use Pontriagin’s maximal principle. Hamiltonian function of system (2) takes the form:

H

2, 6333 1010

H

S D · D D ½ A w0 § ® A\ A ¨ [ ¸ sin([ ) cos \ q cosK \ k sinK ¾ b q\ k k\ q 1 , 2 ¹ 2 2 © ¿

4

P S ¯

where \ A , \ q , \ k – co-state variables. Its equations take form:

d\ A dt d\ q dt

2

S D · 1 § D· D ½ A w0 § ®3\ A ¨ [ ¸ sin ¨ [ ¸ cos \ q cosK \ k sinK ¾ , 2 ¹ A © 2¹ 2 P S ¯ © ¿

b\ k ,

d\ k dt

(3)

b\ q .

Control programs are determined from the necessary condition of the Hamiltonian function maximum:

dH d[

4

dH dK

4

A w0 ª D · §D · º § A\ A cos ¨ [ ¸ cos ¨ ¸ \ q cosK \ k sin K » « 2¹ P S ¬ © ©2¹ ¼ A w0

P S

D · §D · § sin ¨ [ ¸ cos ¨ ¸ \ q sin K \ k cosK 2¹ © ©2¹

0,

0.

On rearrangement, taking into account limitation to active areas duration on orb, the optimal duration of the acceleration area takes the form:

[ opt

S D 2

arcsin

A\ A

\

2 k0

\

2 q0

,

at

A\ A

§D · d cos 2 ¨ ¸ ; ©2¹ \ 1 tg K 2 q0

2

215


1 sign \ A

[opt

2

S D ,

at

A\ A

§D · ! cos 2 ¨ ¸ . ©2¹ \ 1 tg K 2 q0

(4)

2

In a general case, thecontrol program for [ t can contain three stages. The first contains once the acceleration area on orb: [ S D const . The second contains the acceleration and deceleration areas on orb: 0 [ S D , [ var . And the third contains once the deceleration area: [ 0 . From the maximum condition for parameter K , we have:

\k \q

tgKopt

tg (K0 bt ) ,

Kopt

K0 bt .

(5)

Consequently, the optimal location of the acceleration area center relative to node line K is linear depending on time. System (2) and (3) with determined control programs (4) and (5) at given boundary condition forms is a boundary value problem. Unknown parameters are the initial values of co-state variables \ A0 , \ q 0 , \ k 0 and orbital transfer duration T . As residual errors, displacement of final values of state variables and value of Hamiltonian function, which must equal to zero at optimal trajectory are used. The boundary value problem was solved approximately. For first and third stages, equations of motion (2) have an analytical solution:

A0

A

1 I t

2

,

q

e0 cos(Z0 bt ) \ ln 1 I t cos(K0 bt ) ,

k

e0 sin(Z0 bt ) \ ln 1 I t sin(K0 bt ) , k0 q0 , Z0

where e0

\

r2

arctg

k0 – eccentricity and argument of perigee of initial orbit, I q0

(6)

r w0

A0 § D · , 1 P ¨© S ¸¹

sin D , sign “+” corresponds to first stage, sign “–“ to third. S D

For determination of spacecraft motion at second stage, we need the explicit form of parameter [ t . Using the system (2) and its co-state system (3) we obtained integral:

d A\ A dt

H 1 b q\ k k\ q

const .

2

Since H { 0 , we can write:

A0\ A0

A\ A

1 b q0\ k 0 k0\ q 0 2

\ k20 \ q20 cos

D

t .

2

Then, the control program for duration of acceleration area has the form:

[

S D 2

arcsin I1 I2 t ,

(7)

216


A0\ A0

where I1

\ k20 \ q20 cos

, I2

D

1 b q0\ k 0 k\ q 0 2 \ k20 \ q20 cos

D

.

2 2 For determined control program of parameter [ , the analytical solution can be obtained only for semi-major axis of the orbit:

A

A0 § A0 w0 f t · ¨¨1 2 ¸ P S I2 ¸¹ ©

where f t

2

,

(8)

I1 I2t arcsin I1 I2t I1 arcsin I1

1 I1 I2t 1 I12 . 2

We can write approximately time dependence for Laplace vector components, if we assume that duration of active areas on the orb is small and change of semi-major axis not significant: A t | Am const . In addition, taking (7) into account, we can write:

S D · § sin ¨ [ ¸ 2 ¹ ©

sin I1 I2t .

sin y

For small values y d 50 : sin y | y q

e0 cos(Z0 bt )

O 2b

I1 I2t . Then, the desired solution takes the form:

u

° sin K0 sin bt · I22 § § 4b 2 § u ®I1I2 ¨ t sin K0 bt ¸ 2b ¨¨ ¨1 I 2 b © ¹ 2 °¯ ©© k

e0 sin(Z0 bt )

O 2b

½° · cosK0 sin bt · I22 2 ¸¸ t sin K0 bt ¾ , ¸ t cos K0 bt b ¹ °¿ ¹ 2

u

° ½° cosK0 sin bt · I22 § § 4b 2 · sin K0 sin bt · I22 2 § u ®I1I2 ¨ t cos K0 bt ¨ ¨1 2 ¸ t sin K0 bt ¸¸ t cos K0 bt ¾ , ¸ ¨ b b I2 ¹ © ¹ 2b © © °¯ °¿ ¹ 2

where O

Am w0

(9)

D

. 2 Using solutions (6), (8) and (9), an approximately-optimal control program and the orbital transfer trajectory are determined by the algorithm which contains the following iteration steps: 1. Set the initial value of coefficient I1 , initial value of acceleration area center relative to node line K0 and orbital transfer duration T ;

4

P S

cos

2. Taking into account H { 0 , coefficients I2 is calculated: I2

2

A0 w0

P S

1 I ; 2 1

S D I1 , [k I 1 I2 T ; 2 2 4. Taking into account the limitation to active area duration on the orb ( 0 d [ d S D ), we determine duration of 3. Initial and final value of acceleration area duration are calculated: [ 0

every stage: tn

S D

[ 0 n [ kn , where n = 1..3 – stage number, [ kn , [ 0n – final and initial value of nth acceleration area I2

duration; 5. In accordance with (6), (8) and (9), we consequently determine the value of state parameters at the end of every stage Akn , qkn , k kn . Furthermore, the final parameter values for the previous stage is the initial for next; 6. Displacement between founded values of orbit and predetermined parameters were determined. Using

217


displacement values, we determine next approximation for coefficient I 1 , argument of latitude of acceleration area center K0 and orbital transfer duration T .

§ D· ¨1 ¸ w0T . © S¹ In the presence of passive areas on the orb control program (4), (5) cover limited region of existence of orbital transfers. We can determine the region of the existence border from the control programs, contained once maximal duration active area (acceleration or deceleration) on the orb: Characteristic velocity as determined as: Vx

S D tO , sin D where O

(10)

ln( Ak / A0 ) e 2ek e0 cos 'Za e02 2 k

, 'Za

Zk Z0 bT – argument of perigee changing due to engine thrust. For

the small active area duration ( D o S ) condition (10) a manifold of intersecting initial and final orbits are determined. With the increase in duration of active areas, the region of existence of orbital transfers extends and captures part of the manifold of non-intersecting orbits. In the limit, when using the control program without passive areas ( D 0 ) this manifold contained all closed orbits. If condition (10) doesn’t satisfy, the optimal control program contained maximal acceleration or deceleration area on the orb, with switching of argument of latitude of its center to 180 degrees at time moment t1 . Here, control program parameters determined as:

[

1 sign Ak A0 2

S D ,

K1 K 0 bt , K2 K 0 S b t1 t ,

§ 'k · A0 1§ arctg ¨ ¨¨1 ¸ bT – argument of latitude of active area center at initial moment, T ' q A I © ¹ k © transfer duration, K1 , K2 – program of location centers of active area before and after

where K0

ek sin Zk e0 sin Z0 bT , 'q

'k

· ¸¸ – orbital ¹ switching,

ek cos Zk e0 cos Z0 bT .

Moment of switching t1 determined by formula:

t1

ª 1« 1 I« ¬«

§ 'k 2 'q 2 A0 exp ¨ ¨ \ Ak ©

·º ¸» . ¸» ¹ ¼»

Characteristic velocity, in this case, depends only on required change of semi major axis:

Vx

§ P P · ¨¨ ¸ sign Ak A0 . Ak ¸¹ © A0

4. Method of design-ballistic optimization The approximately obtained optimal control programs are able to solve the problem of design parameter optimization. The optimal design parameters of spacecraft in accordance with the criteria of maximal payload mass should be determined taking into account the specified orbital transfer duration and boundary conditions. The optimization problem reduced to the iteration procedure, where the optimized parameter is the propulsion system thrust. Each iteration step is contained the following stages. 1. The value of the propulsion system thrust was set at Pmin d P d Pmax , initial spacecraft mass m0 , boundary conditions x0 , xk and orbital transfer duration T . The minimal value of propulsion system thrust Pmin is limited by

218


orbital transfer duration without passive areas, maximal Pmax

100 N, by technical limitations of STPS.

2. The optimal control program and orbital transfer trajectory are determined. Characteristic velocity Vx , maximal value of semi major axis Amax and duration of passive area on the orb D is calculated. 3. In the condition of effective TA utilization (accumulator is full discharged after maximal duration active area), the accumulator operational capacity was calculated: E0

'ta max N E

S D

3 Amax Pc

P KT

, where 'ta max – maximal

operation time of the propulsion system, N E – power, transferring from TA to the engine, K T – jet propulsive efficiency. 4. In accordance with the fully charged condition of the TA after the operating cycle of propulsion system, the power 'ta max S D Pc , NE , where Tp – orbital period. of solar panel is calculated: N S Tp 'ta max D 2KT 5. The payload mass is calculated in accordance with (1). 5. Numerical simulations The results of the design-ballistic optimization problem, during working orbit formation for initial data are shown in tables 1 and 2, and plot in fig. 1. Working orbit formation is considered in two stages. Firstly, transfer from injection orbit to circular orbit, height 300 km, due to cold component in LE mode (mode 1). Second, many orb transfer to working orbit in mode without hydrogen afterburning, heated in TA (mode 2). Table 1. Spacecraft parameters, equipped with STPS.

J E , kg/MJ

J S , kg /kW

J en , kg/N

E fs

0,1

1,18

50 (STPS-1) 20 (STPS-2)

0,39

0,5

KT , %

m0 , kg

PLE , N

c , m/s

234

4532 (mode 1) 7456 (mode 2)

PC

Parameter Value Parameter Value

80

7700

Table 2. Injection and working orbit parameters during orbit formation. Injection orbit

A0 , km 6578

Working orbit

e0

Z 0 , deg.

Ak , km

ek

Z k , deg.

0,00205

0

7858

0,00248

0

STPS-1

STPS-2

LE

6000 5800

Payload mass, kg

5600 5400 5200 5000 4800 4600 0

5

10

15

20

25

30

35

40

45

Orbital time duration, days

Fig. 1 – The payload mass depending on duration of the orbital transfer during working orbit formation

The results of the design-ballistic optimization problem, during orbit correction in STPS mode without hydrogen afterburning, are plotted in fig. 2, the plots in table 1, 3 show the initial data. The initial spacecraft mass is equal to 6570 kg, orbit inclination during correction is 69,9 degrees.

219

S.A. Ishkov and A.A. Khramov / Procedia Engineering 185 (2017) 212 – 219 Table 3. Initial and final orbit parameters during orbit correction. Initial orbit

Final orbit

A0 , km

e0

Z 0 , deg.

Ak , km

ek

Z k , deg.

6750

0,0149

120

6896

0,0239

0

STPS-1

STPS-2

LE

5800 5700

Payload mass, kg

5600 5500 5400 5300 5200 5100 5000 0

5

10

15

20

25

30

35

Orbital time duration, days

Fig. 2. The payload mass depending on duration of the orbital transfer during orbit correction.

6. Conclusions Numerical simulation results show, that due to an increase in orbital transfer duration, STPS provide injection of a greater payload mass compared with oxygen-hydrogen LE. The efficiency of STPS is largely determined by the energy-mass characteristics of the spacecraft power supply system. The design parameter optimization method, obtained in this work, can be used in initial stages of STPS spacecraft design. References [1] A.S. Koroteev et al., Solar thermal rocket engine, The patent of the Russian Federation [in Russian], Russian patents, 1999, available at: http://rupatent.info/21/25-29/2126493.html. [2] G.L. Grodzovskii, Ju.N. Ivanov, V.V. Tokarev, Mechanics of Space Flight (Problems of Optimization) [in Russian], Moscow: Nauka, 1975, 704 pp. [3] A.A. Khramov, Optimum programs of correction of quasielliptical and circular orbits of spacecraft with the limited-thrust engine [in Russian], Vestnik of Samara University, 2011, Vol. 26, No. 2, pp. 112 – 122.