The reader is informed of what was done for the mathematical modeling of the dc-ac inverter for the Space Shuttle. The mathematical modeling of the dc-ac inverter is an essential element in the modeling of the electrical power distribution system of the Space Shuttle. The electrical power distribution system which is present on the Space Shuttle is made up to 3 strings each having a fuel cell which provides dc to those systems which require dc, and the inverters which convert the dc to ac for those elements which require ac. The inverters are units which are 2 wire structures for the main dc inputs and 2 wire structures for the ac output. When 3 are connected together a 4 wire wye connection results on the ac side. The method of modeling is performed by using a Least Squares curve fitting method. A computer program is presented for implementation of the model along with graphs and tables to demonstrate the accuracy of the model

Berry, Frederick C.

English

NASA Technical Reports Server

N88 - 1486 0 _--3_
MATHEMATICAL MODEL FOR THE DC-AC INVERTER //_ _'/L_/'_..__
FOR THE SPACE SHUTTLE ,_!
/
Final Report ___ I_ {'3_! _i i/ :
NASA/ASEE Summer Faculty Fellowship Program--1987
Johnson Space Center
Prepared by:
Academic Rank:
University & Department:
NASA/JSC
Directorate:
Division:
Branch:
JSC Colleague:
Date:
Contract Number:
Frederick C. Berry
Instructor
Louisiana Tech
Electrical Engr. Dep.
Ruston, Louisiana 71270
Engineering
Avionics Systems
Avionics Integration
Bob Hendrix
August 14,1987
NGT 44"001-800
5-i
https://ntrs.nasa.gov/search.jsp?R=19880005478 2020-03-20T09:08:30+00:00Z
ABSTRACT
The purpose of this report is to inform the reader
of what has been done for the mathematical modeling of
the DC-AC inverter for the Space Shuttle. The mathemati-
cal modeling of the DC-AC inverter is an essential
element in the modeling of the electrical power distri-
bution system of the Space Shuttle. The electrical power
distribution system which is present on the Space Shut-
tle is made up of three strings each having a fuel cell
which provides DC to those systems which require DC, and
the inverters which convert the DC to AC for those
elements which require AC. The inverters are units which
are two-wire structures for the main DC inputs and two-
wire structures for the AC output. When three are conne-
cted together a four wire wye connection will result on
the AC side. The method of modeling will be performed by
using a Least Squares curve fitting method. A computer
program will also be presented for implementation of the
model along with graphs and tables to demonstrate the
accuracy of the model.
5--2 "
INTRODUCTION
The purpose of this report is to inform the reader
of what has been done for the mathematical modeling of
the DC-AC inverter for the Space Shuttle. The mathemati-
cal modeling of the DC-AC inverter is an essential
element in the completion of the modeling of the elec-
trical power distribution system of the Space Shuttle
which the in-house support contractor had started by
producing a model for the DC portion of the electrical
power distribution system. The electrical power distri-
bution system which is present on the Space Shuttle is
made up of three strings each having a fuel cell which
provides DC to those systems which require DC, and the
inverters which converts the DC to AC for those elements
which, require AC. The inverters are units which are two-
wire structures for the main DC inputs and two-wire
structures for the AC output. When three are connected
together a four wire wye connection will result on the
AC side. Therefore, with three strings on the Shuttle
there are nine inverters. For this report only three
inverters will be modeled. There was enough input data
taken on the performance of the inverters that it was
not necessary to write any equations describing the
5-3
internal operations of the inverter. Therefore the
method of modeling will be performed by using a Least
Squares curve fitting method. A computer program will
also be presented for implementation of the model along
with graphs and tables to demonstrate the accuracy of
the model. The following table is given as an tool to
help the reader identify any symbols that are used in
this report.
V ..................................... voltage
Hz ................frequency cycles per second
VA ................................ volt ampere
Pu ................................... per unit
Edc ................................ voltage DC
Idc ................................ current DC
Eac ................................ voltage AC
Iac ................................ current AC
PF ............................... power factor
Pin ............................... DC power in
Pout ............................. AC power out
EF & Eff ........................... efficiency
TF ................................ temperature
The first step taken in developing the math model
for the inverter was to find out what function the
inverter performs and what data was available on the
inverter. The inverters are units which can convert
28VDC nominal input to a single phase AC output of II7V,
400Hz, rated at 750VA continuous, and II25VA for 30
minutes with power factor ranging from 0.7 lagging to
0.9 leading including unity. The full rated load at
1.0Pu is:
I=750VA/II7V
5-4
This defines the steady state operation of the inverter
for constant inputs as a single phase unit.
Data on the inverter was available in the form of
written reports and experimental data. The reports that
were found and used are referenced at the end of this
report. An example of the experimental data is given in
Table i. The experimental data shows the input and
output conditions of the inverter keeping the power
factor, input and output voltage, and frequency constant
while changing the load.
%-VA Edc Idc Eac Iac PF Pout TEMP Pin %-EF
0 30 3.14 117 0.00 .0 0.0 112.7
20 30 5.69 117 1.36 .6 96.8 111.6
40 30 8.63 117 2.59 .6 180.6 107.7
50 30 10.59 117 3.30 .6 232.9 108.1
60 30 12.16 117 3.88 .6 272.9 108.4
80 30 15.88 117 5.12 .6 361.6 110.8
i00 30 20.39 117 6.47 .6 449.4 115.1
125 30 26.86 117 8.17 .6 579.5 122.6
150 30 33.53 117 10.05 .6 700.8 133.1
200 30 48.43 117 13.29 .6
94.1 0.0
170.5 56.8
258.2 69.9
316.6 73.3
362.3 75.3
473.7 76.3
607.1 74.0
797.4 72.6
997.9 70.2
926.8 123.8 1424.3 65.0
TABLE-I: EXAMPLE OF EXPERIMENTAL DATA FOR INVERTER A
This procedure was repeated for each of the three inver-
ters for the following conditions: 24, 26, 28, and 30
volts DC input and -0.6, -0.7, -0.85, 1.0, and 0.9 power
factor at each of the DC voltage inputs.
5-5
MODEL
The math model of the inverter was designed to
produce as many input and output conditions as possible
based on a limited number of inputs to the math model.
The inputs to the model are as follows:
i. Percent Load
2. Input Voltage DC
3. Power Factor
the outputs from the math model are as follows:
I. Input Current DC
2. output Current AC
3. Input Power DC
4. Output Power AC
5. Percent Efficiency
6. Temperature In Degrees F
In order to do this, two different modeling methods
were investigated. These modeling methods are as fol-
lows:
i. Least Squares (LS) fitting of the data.
2. Data fitting by use of a set of orthognal
polynomials (OP).
Each of these methods are found in software that is
already available. The least squares fitting was found
in the Statistical Analysis System (SAS) routines and
the method of using orthogonal polynomials for curve
fitting was found in the International Mathematical and
5-6
Statistical Libraries (IMSL) routines.
Based on analysis of the percent efficiency data it
was decided that a fourth order model would best appro-
ximate the data. The reason for using the efficiency
data to determine the order of the model was because
this data went through the largest number of cycles over
the range of 0 to 200 percent load. Using a fourth order
model for each of the two cases listed above produced
results which tracked the data very well. After being
assured that each method produced satisfactory results,
it was decided to use just the least squares fitting of
the data to model the remaining sets of data. The reason
for this decision was based on the observation that the
least squares method of data fitting produced a tighter
fit.
PROGRAM
Using the data that was supplied for each inverter
a fourth order regression equation using the percent
load as the independent variable was produced to model
each of the six variables that were used from each sheet
of the data for the inverters. Each inverter was tested
at 24, 26, 28, and 30 volts DC input. At each voltage
setting, it was tested at -0.6, -0.7, -0.85,
5-7
1.0, and 0.9 power factor. Using the results of these
tests a set of regression equations was produced to
represent the input current DC, input power DC, output
current AC, output power AC, percent efficiency, and
temperature in degrees fahrenheit for each setting of
voltage and power factor. This produced a set of 120
regression equations for each inverter. Twenty regres-
sion equations are used to describe each of the above
referenced output parameters.
Each set of 20 regression equations is used to
produce a matrix for all voltage and power factor set-
tings, for each inverter. An example of the matrices
that represent all of the input paramaters for a load
of i00 percent for inverter C is shown below:
-0.6 -0.7 -0.85 1.0 0.9
24 24.02708 26.80123 33.31841 26.70104 40.88011
26 23.49872 27.47219 31.48396 37.36097 37.46921
28 20.90991 25.11707 28.68474 21.02460 35.56254
30 20.47581 23.31632 27.19581 33.18778 44.52682
DC INPUT CURRENT FOR i00 PERCENT LOAD FOR INVERTER C
-0.6 -0.7 -0.85 1.0 0.9
24 64.79873 63.91091 64.56882 63.93892 63.57912
26 65.91304 64.58717 66.15194 63.23208 63.20073
28 64.79345 64.45176 65.94344 63.48076 63.68013
30 64.49525 65.50586 64.74864 58.59911 62.55650
AC OUTPUT CURRENT FOR i00 PERCENT LOAD FOR INVERTER C
5-8
24
26
28
3O
-0.6 -0.7 -0.85 1.0 0.9
562.0085 635.5890 784.0984 970.1697 960.8156
583.8451 673.9015 788.1099 962.5731 954.8917
587.7390 692.4550 790.6331 978.7694 970.1374
611.5838 690.3112 815.6928 1022.1897 1008.9542
DC INPUT POWER FOR i00 PERCENT LOAD FOR INVERTER C
-0.6 -0.7 -0.85 1.0 0.9
24
26
28
3O
457.3632 522.5542 639.8809 746.2821 669.2880
461.2574 533.8483 636.5393 738.5239 666.9812
450.8919 528.9160 643.1387 741.7915 672.3812
454.0515 539.8164 643.8911 746.7502 670.5568
AC OUTPUT POWER FOR i00 PERCENT LOAD FOR INVERTER C
-0.6 -0.7 -0.85 1.0 0.9
24 76.98697 80.62087 79.11906 74.81590 67.30208
26 76.46641 77.63451 83.97259 74.75730 68.18011
28 76.08212 76.09306 79.57481 74.34234 67.70587
30 74.93172 71.25590 77.79884 71.92826 64.74466
PERCENT EFFICIENCY FOR i00 PERCENT LOAD FOR INVERTER C
-0.6 -0.7 -0.85 1.0 0.9
24 10.61268 11.22281 11.32683 11.76665 13.42805
26 11.28422 12.14870 11.54441 12.57221 12.17178
28 10.99731 11.23885 12.34068 10.67528 12.72728
30 11.89918 12.72210 12.58744 13.61134 13.33829
TEMPERATURE FOR i00 PERCENT LOAD FOR INVERTER C
5-9
This method of reproducing the data was chosen over
having the data stored in a data file and reading the
desired portion of the data file into the program.
At this point the program will perform a least
squares on each matrix for each voltage setting using
the power factor as the independent variable. This will
produce four regression equations which will then be
used to calculate four points, one for each voltage
setting. With these four points another least squares
will be performed using the voltage as the independent
variable. This will produce one equation which will then
give the desired value of current for the input of
the percent load, input voltage DC, and power factor.
This procedure was repeated for each of the three
inverters. Once this point is reached the following
output is produced:
Input Current DC..
Output Current AC.
Input Power DC....
Output Power AC...
Inverter % Elf ....
Inverter Temp. F..
INVERTER-A INVERTER-B INVERTER-C
27.19581
6.47486
815.69282
643.89115
77.79884
125.87446
27.05088
6.27590
803.43621
624.95208
76.52319
117.18629
27.21824
6.35214
808.26576
631.83659
75.55965
107.85057
OUTPUT FOR EACH INVERTER AS A SINGLE PHASE UNIT
This represents the output for three single phase inver-
ters. The next step is to combine these three inverters
into a 4 wire wye connection with the
5-10
following phase sequence: inverter A, 0 degrees; inver-
ter B, 120 degrees; and inverter C, 240 degrees. Using
this phase sequence the next set of output that is
supplied is the voltage line to neutral, phase current,
neutral current, and power. This output is now represen-
ted as follows:
Inverter-A Output Voltage
Inverter-B Output Voltage
Inverter-C Output Voltage
Total Three Phase Voltage
REAL IMAG.
117.00000 0.00000
- 58.49374 101.32858
- 58.51251 -101.31774
- 0.00625 0.01084
Inverter-A Output Current
Inverter-B Output Current
Inverter-C Output Current
Total Return Current
4.58541 4.67805
6.06423 1.01328
1.71725 - 6.16466
0.23842 0.07286
Inverter-A Output Power
Inverter-B Output Power
Inverter-C Output Power
Total Three Phase Power
377.87152 385.50606
361.95492 369.26788
367.28991 374.71066
1107.11637 1129.48461
THREE PHASE OUTPUT FOR THE INVERTERS
A complete set of graphs is provided in appendix A for
inverter A which shows the results of the math model
calculated values against the experimental values for
the following set of pramaters:
EXPERIMENTAL
Input Current DC IDC
Output Current AC IAC
Power In PIN
Power Out POUT
Percent Efficiency EF
Temperature TF
CALCULATED
IDCC
IACC
PINC
POUTC
EFC
TFC
5-11
CONCLUSION
Thus far three inverters have been modeled by using
the input/output data that was supplied. This data was
modeled by use of a Least Squares fitting technique to
produce a set of equations which approximates the output
conditions and also some of the input conditions, given
certain inputs to the model like percent load, input
voltage DC, and power factor.
At this point only a steady state model for a
balanced three phase system exists for one set of inver-
ters. It is proposed to complete the steady state mode-
ling by performing the following tasks:
i. To increase the accuracy of the model by
increasing the order of the system or by
use of other modeling methods.
2. To give the model the ability to handle
unbalanced loads.
3. To include into the model an option that
would allow for dropping of an inverter
and then calculating the output of the
remaining inverters.
With these additions to the present model a very good
representation of the DC-AC Inverter for the steady
state can be produced.
5-12
REFERENCES
IMPLEMENTATION AND OPERATION PROCEDURES
Prepared by Lockheed Engineering, Houston, Texas
Contract NAS9-15800
SPECIFICATION-INVERTER, POWER STATIC, 117 VOLT, SINGLE
PHASE, 400 Hz
Prepared by Rockwell International, Downey, California
QUALIFICATION TEST PROCEDURE SPACE SHUTTLE POWER STATIC
INVERTER, 117 VOLTS, SINGLE PHASE, 400Hz
Prepared by Westinghouse, Lima, Ohio
Document N5-I18
FINAL TEST REPORT FOR 0V-101 EPDC BREADBOARD EVALUATION
TEST SERIES
Internal Note JSC 77-EG-12
INTERIM TEST REPORT FOR 0V-102 BREADBOARD EVALUATION
TEST SERIES
Internal Note ASD 78-EH-16
5-13
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Mathematical model for the dc-ac inverter for the Space Shuttle

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