Future spacecraft power systems must be capable of supplying power to various loads. This delivery of power may necessitate the use of high-voltage, high-power dc distribution systems to transmit power from the source to the loads. Using state-of-the-art power conditioning electronics such as dc-dc converters, complex series and parallel configurations may be required at the interface between the source and the distribution system and between the loads and the distribution system. This research will use state-variables to model and simulate a dc spacecraft power system. Each component of the dc power system will be treated as a multiport network, and a state model will be written with the port voltages as the inputs. The state model of a component will be solved independently from the other components using its state transition matrix. A state-space averaging method is developed first in general for any dc-dc switching converter, and then demonstrated in detail for the particular case of the boost power stage. General equations for both steady-state (dc) and dynamic effects (ac) are obtained, from which important transfer functions are derived and applied to a special case of the boost power stage

Berry, F. C.

English

NASA Technical Reports Server

N95- 32423
MODELING OF DC SPACECRAFT POWER SYSTEMS
Final Report
NASA/ASEE Summer Faculty Fellowship Program--1994
Johnson Space Center
Prepared By:
Academic Rank:
University & Department:
NASA/JSC
Directorate:
Division:
Branch:
JSC Colleagues:
Data Submitted:
Contract Number:
F.C. Berry
Associate Professor
Department of Electrical Engineering
Louisiana Tech University
Ruston, Louisiana 71272
Engineering
Propulsion And Power
Power
Tom Jeffcoat
August 5, 1994
NGT-44-005-803
6-1
https://ntrs.nasa.gov/search.jsp?R=19950026002 2020-06-16T06:10:06+00:00Z
displacement than we can measure from the pointing output. The motor
command networks directing eye movements may receive more information
from the vestibularly-derived spatial map than the motor command
networks for pointing movements. Alternatively, vestibular input to both
systems may be identical but the motor command networks for pointing may
not produce as accurate an output as those for eye movements.
Our data provide a measure of an individual's ability to point to a
remembered target location using only vestibular cues. These data can be
contrasted with similar data derived from subjects whose eye-head
coordination and internal spatial map have been perturbed, for example, by
wearing minimizing lenses. In addition, these data can be compared to data
collected from astronauts exposed to the microgravity environment of space.
REFERENCES
1.) Bloomberg J, Jones GM, Segal B (1991) Adaptive modification of
vestibularly perceived rotation. Exp. Brain Res. 84: 47-56.
2.) Watt DGD. Money KE, Bondar RL, Thirsk RB, Garneau M, ScuUy-Power P
(1985). Canadian medical experiments on shuttle flight 41-G. Can Aeronaut.
Space J. 31:215-226.
3.) Young LR, Oman CM, Watt DGD, Money KE, Lichtenberg BK (1984).
Spatial orientation in weightlessness and readaptation to earth's gravity.
Science. 225: 205-208.
5-14
ABSTRACT
Future spacecraft power systems must be capable of supplying power
to various loads. This delivery of power may necessitate the use of high-
voltage, high-power de distribution systems to transmit power from the
source to the loads. Using state-of-the-art power conditioning electronics
such as dc-dc converters, complex series and parallel configurations may be
required at the interface between the source and the distribution system and
between the loads and the distribution system.
This research will use state-variables to model and simulate a dc
spacecraft power system. Each component of the de power system will be
treated as a multiport network, and a state model will be written with the
port voltages as the inputs. The state model of a component will be solved
independently from the other components using its state transition matrix.
A state-space averaging method is developed first in general for any
dc-dc switching converter, and then demonstrated in detail for the particular
case of the boost power stage. General equations for both steady-state (dc)
and dynamic effects (ac) are obtained, from which important transfer
functions are derived and applied to a special case of the boost power stage.
6-2
INTRODUCTION
The basic de-de level conversion function of switching converters is
achieved by repetitive switching between two linear networks consisting of
ideally lossless storage elements, inductances and capacitances. In practice,
this function may be obtained by use of transistors and diodes that operate as
synchronous switches. On the assumption that the circuit operates in the
continuous conduction mode in which the instantaneous inductor current
does not fall to zero at any point in the cycle, there are only two different
states of the circuit. Each state can be represented by a linear circuit model
or by a corresponding set of state-space equations.
Interval Ta Interval Ta,
x = A ix + Blvs x = A2x + B2vs
Yl = ClX Y2 = c2x
R 1 L1 [ -Rza 0
I !+Vt A_=L_I
Rc Rx 0 --1
RclC1
V.m OUtLx I 1
Cx B1 _1 0
= 1
0 RclC1
Fig. 1. Network for formulating A1 and B1.
R 1 L1
)
, [ -RtAR c R 0
Vout I 1
t
c_ B1 = L1
0
* !
Fig. 2. Network for formulating and B1.
0
-1
Rcl C1
-1
L1
1
Rcl C1
The interval Td denotes the interval when the switch is in the on state and Td,
denotes the interval when the switch is in the off state. The state model is
6-3
formulated for a boost circuit assuming the circuit operates in the continuos
conduction mode. The A and B matrices for each of the two linear
networks are given along with the network model that was used to obtain the
A and B matrices.
SIMULATION
In the analysis of a system by the state-space approach, the system is
characterized by a set of ftrst-order differential or difference equations [1].
The dc-dc converter can be described by a set of state-space equations.
These state-space equations have the following form:
x =Ax +By
where x is a vector of state-variables, v is a vector of inputs, and A and B
are matrices. The state-variables, x, are associated with the capacitor voltage
and the inductor current. The solution for x is given by
t
x(t )= _(t)x(O) + _ O(t- "c)Bv('r)d'¢
0
where _(t) is the state transition matrix and x(0) is a vector containing
initial values for the state variables. This solution for x (t) is valid if the
matrices A and B are constant.
The dc-dc converter is a switching converter and is modeled by two
different linear networks (Figures 1 and 2). Therefore, the matrices A and B
are not constant over the interval [0, T ]. The limits of integration must be
changed to an interval over which A and B are constant. Also, v will contain
entries that are unknown functions of time. All of these conditions will
make the convolution integral very difficult to evaluate.
To evaluate the convolution integral it was assumed that the inputs, v,
were constant or clamped over the interval of the integration. Therefore a
recursive relationship for x can be developed if the appropriate interval, T, is
selected for matrices A and B to be constant.
x(k + 1)=¢b(T)x(k)+®(T)Bv(k)
In this equation, x(k+l) and x(k ) are the values of the state vector at
t=(k+l)T and t=kT, respectively; v(k ) is the value of the voltage at t=kT.
The variable T is referred to as the time step, and it must be selected such
that A and B are constant and the inputs, v, does not change appreciably
6-4
over the interval [0, T ]. The variable _(T) is the state transition matrix,
which is calculated using the following power series;
O (T ) = IT + AT----_2+ A 2 T3 + A 3T4 _-...
2! 3! 4!
The variables _(T) and O(T) are related by the following equation;
O(T)=AO(T)+I
Simulation 1
The _(T) and
O(T) are calculated for
each time step. A set of
these matrices is required
for each linear network
model of the boost circuit.
The solution of x(k+l) is a
function of x(k) and u(k),
and the O(T) and ®(T)
for that particular linear
network. The calculation
of x(k+l) requires one
vector addition and two
multiplications of a matrix
and a vector. This
simulation procedure
requires two different sets
of matrices to be
maintained, one set for
each linear network. The
simulation procedure is
summarized in the
flowchart of Figure 3.
An example de
power system is simulated
using the state variable
approach. This system
Calculate O(T) and O(7) ]for each Linear Network. J
Calcu,x  +l ,orone Linear Network.
Calculate u(k+ 1)from x(k+ 1).
Calculate x(k+l) for ]other Linear Network.
Calculate u(k+l)from x(k+ 1).
ofSimu,a on 
Hg. 3. Flowchart for Simulation 1.
consists of a dc power source connected to a boost network with a resistive-
inductive load. The results of this simulation are given in Figure 4.
6-5
O325
!iiiiiiiiiiiiiilll iiiiiiiiiiiiiiiiii
_0o.............._ ! _ .....
125
100
75
50
25
0
0.00 0.01 0.02 0.03
Time in Seconds
Fig. 4. Output voltage for Simulation 1.
.......
¢
0.04
Simulation 2
The objective is to replace the state-space description of the two linear
circuits by a single state-space description that represents the approximate
behavior of the circuit across the period;
r, =Ta +Ta,
To represent the approximate behavior of the circuit across the period, Ts,
the following averaging method was used. The average was taken by
summing the equations for the intervals of Ta and Ta, and multiplying each
set of equations by d and d' respectively. The following linear continuous
systems results;
x = (dA1 + d;42) x + (dbl + d'b2) vs
y = (dcl + d'c2) x
This model is the basic average model that is the starting model for all
other derivations.
6-6
The model represented is an averaged model over a single period Ts.
I f the duty ratio d is assumed to be constant from cycle to cycle, the d = D
(the steady-state de duty ratio). Then the averaged model becomes;
x =Ax +Bvs
y=Cx
where
A = (DA1 + D_2) B= (DB1 + D'B2) c = (DCl + D'C2)
The averaged model is a linear system.
be applied.
The O(T) and
O(T) are calculated for
each time step, Is. One set
of matrices is required for
the averaged linear model _[
of the boost circuit. The
solution of x(k+l) is still a
function of x(k) and u(k),
and O(T) and O(T). The
calculation of x(k+l)
requires one vector
addition and two
multiplications of a matrix No I
and a vector. The I
simulation procedure is
summrized in the flowchart
of Figure 5.
Therefore, superposition holds can
Calculate O(T) and O(7).
Calculate x(k+l).
I Calculate u(k+l)from x(k+l).
End of Simulation?
Fig. 5. Flowchart for Simulation 2.
An example dc power system is simulated using the state variable
approach. This system consists of a dc power source connected to a boost
network with a resistive-inductive load. The results of this simulation are
given in Figure 6.
6-7
I
Vo-2
,, , , ,
50 [
25 /
0
0.00 0.01 0.02 0.03
Time in Seconds
Output voltage for Simulation 2.
0.04
CONCLUSION
Two different simulation methods are presented. The method in
Simulation 1 is a more traditional approach that relies on the modeling of
each mode of operation of the dc-dc power converter. This is an effective
method of modeling and has been used by several of authors [2,3,4].
However, for a large power system the number of models needed to describe
the different modes of a variety of switching converters could become very
large [1 ]. The modeling method presented in Simulation 2 tries to produce a
single model of a switching converter. This method of modeling is done by
producing a weighted average of the different linear models. Anytime an
average is taken some distortion in time will occur because averaging is a
form of high pass filtering. The comparison of the two different modeling
methods in Figure 7 does show time distortion but the voltage magnitudes
appear to be basically the same. More work in developing the theory and
many more simulations must be done but the average method does appear to
have some merit.
6-8
[1]
[2]
[3]
[4]
325
,¢'x
300 ..........t __t__]'l / ......._,_\..................... ' Vo-1
275 l Y ._ vo-2--
250 __
.i
175 ....] ......._____/.i........X .../
150 .....
125 ]
10075 ] ......... ......................
,o 11_..............LLI ...................................
25 r/
0
0.00 0.01 0.02 0.03
Time in Seconds
Fig. 7. Output voltage for Simulation 1 and 2.
0.04
REFERENCES
R.M. Nelms and L.L. Grigsby, "Simulation of DC Spacecraft Power
Systems," IEEE Transactions on Aerospace and Electronic Systems,
Vol. 25, No. 1, January 1989.
B.H. Cho, J.R. Lee, and F.C. Lee, "Large-Signal Stability Analysis of
Spacecraft Power Processing Systems," IEEE Transactions on Power
Electronics, Vol. 5, No. 1, January 1990.
B.H. Cho and F.C. Lee, "Modeling and Analysis of Spacecraft Power
Systems," IEEE Transactions on Power Electronics, Vol. 3, No. 1,
January 1988.
J.R. Lee, B.H. Cho, S.J. Kim, and F.C. Lee, "Modeling and
Simulation of Spacecraft Power Systems," IEEE Transactions on
Aerospace and Electronic Systems, Vol. 24, No. 3, May 1988.
6-9


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