Power system analysis

1. Ibrahim Omar Habiballah Sep. 2013
CHAPTER 2
PROGRAMMING CONSIDERATIONS
2.1 Introduction
The use of digital computers imposes two main limitations: time, and
memory. In some applications, it is possible to trade time (speed) for
memory or vice versa. A programming technique is considered to be an
efficient one when both time and memory requirements cab be improved.
There are three factors that affect the performance of any programming
technique:
- computer type (SUN-Workstations, Dec-Stations, PC, ...etc.),
- programming language (FORTRAN, BASIC, C, ...etc.), and
- application (power systems, VLSI, NASA, ... etc.).
The main issues for any programming technique are
- sparsity programming (to compress the data stored),
- triangular factorization (to solve linear algebraic equations), and
- data management (to improve processing and accessing of data).
Overlaying is a programming technique widely used in FORTRAN and
BASIC. In this method, certain portions of a computer program are held in
high speed memory during all phases of the execution, and the other
portions (stored on lower media) are only brought into high speed memory
when needed.
2.2 Sparsity Programming
Sparsity is a digital programming technique that store sparse matrices in a
compact form. The basis of this programming technique is the storage of
nonzero entries only, and at the same time keep tracking the location of
these entries in the matrix (also called storage/access programming
technique).

2. Ibrahim Omar Habiballah Sep. 2013
Sparsity programming can also be used for processing nonzero data only
(like in the inversion of nonsingular square matrices). This way it does not
only reduce the memory requirements but also increase the speed.
The performance of any sparsity technique depends on
- matrix symmetry,
- storage order,
- percent sparsity, and
- occurrence of blocks of zeros.
There are tow broad classes of sparsity programming techniques:
1) Entry-Row-Column Storage Method.
2) Chained Data Structure Method.
2.2.1) Entry-Row-Column Storage Method
In this method, the nonzeros and the corresponding row/column locations
are stored in three parallel arrays:
- STO (for storing nonzero entries),
- IR (for storing corresponding row locations), and
- IC (for storing corresponding column locations).
Therefore, the total required storage for n x n sparse matrix with ne
nonzeros is 3ne entries (as oppose to n2
entries when storing the entire
matrix).
Example (1):
a) Consider a 3 x 3 matrix with 4 nonzeros, then
no of entries using normal storage method is 9
no of entries using first storage method is 12
b) Consider a 4 x 4 matrix with 6 nonzeros, then
no of entries using normal storage method is 16
no of entries using first storage method is 18
c) Consider a 1000 x 1000 matrix with 5000 nonzeros, then
no of entries using normal storage method is 1 000 000

3. Ibrahim Omar Habiballah Sep. 2013
no of entries using first storage method is 15 000
Above example reveals the extreme advantage of sparsity programming
techniques in storing large scale sparse matrices.
The nonzero entries of a sparse matrix can be stored in either row or
column fashion. The nonzeros can also be stored in arbitrary order; in this
case, the IR and IC arrays must be scanned every time an access is
required.
Insertion of a new entry requires pushing down of other entries located
below the insertion. Similarly, deletion of an existing entry requires
pushing up of other entries located below the deletion.
A major drawback of this method is the long access time it takes whenever
a new entry is inserted to the list or an entry is deleted from the list.
Example (2):
Consider the following 3 x 3 sparse matrix
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
200
003
010
A
The row fashion storage method is
Counter STO IR IC
1 1.0 1 2
2 3.0 2 1
3 2.0 3 3
Assume the insertion of entry 4 in row 1 and column 3. The storage of this
new entry will push down all entries of STO, IR, and IC arrays as
Counter STO IR IC
1 1.0 1 2
2 4.0 1 3
3 3.0 2 1
4 2.0 3 3

4. Ibrahim Omar Habiballah Sep. 2013
Assume, now, the deletion of entry 1 from row 1 and column 2. The
deletion of this entry will push up all entries of STO, IR, and IC arrays as
Counter STO IR IC
1 4.0 1 3
2 3.0 2 1
3 2.0 3 3
Such insertion/deletion requires long access time.
2.2.2) Chained Data Structure Method
In this method, the nonzeros and the corresponding row (or column)
locations are stored in three parallel arrays:
- STO (for storing nonzero entries),
- IR (for storing corresponding row locations), or
- IC (for storing corresponding column locations), and
- NX (for telling how far down the list the next entry will be found).
It is also necessary to employ a forth array NFIRST which tells where a
certain row/column starts in the list. The size of NFIRST array is equal to
the number of rows (if row fashion is used) or the number of columns (if
column fashion is used).
The total required storage for n x n sparse matrix with ne nonzeros is 3ne +
n entries (as oppose to n2
entries when storing the entire matrix).
Example (3)
a) Consider a 3 x 3 matrix with 4 nonzeros, then
no of entries using normal storage method is 9
no of entries using second storage method is 15
b) Consider a 4 x 4 matrix with 6 nonzeros, then
no of entries using normal storage method is 16
no of entries using second storage method is 22
c) Consider a 1000 x 1000 matrix with 5000 nonzeros, then
no of entries using normal storage method is 1 000 000
no of entries using second storage method is 16 000

5. Ibrahim Omar Habiballah Sep. 2013
The nonzeros of this method can be stored in either row or column fashion.
A major advantage of this method is the high speed access time it takes
whenever a new entry is inserted to the list or an entry is to be deleted from
the list.
Insertion of a new entry requires no pushing down of other entries. In this
case, the NX array may need modification. Similarly, deletion of an entry
requires no pushing up of other entries. In this case, the NX array and/or
NFIRST array may need modification.
Example (4):
Consider the 3 x 3 sparse matrix of Example (2). The four arrays required
by this storage method, using the row fashion approach, are
Counter STO IC NX NFIRST
1 1.0 2 0 1
2 3.0 1 0 2
3 2.0 3 0 3
Assume the insertion of entry 4 in row 1 and column 3. The storage of this
new entry will be inserted at the end of the list, NX should be modified.
Counter STO IC NX NFIRST
1 1.0 2 3 1
2 3.0 1 0 2
3 2.0 3 0 3
4 4.0 3 0
Assume, now, the deletion of entry 1 from row 1 and column 2.
Counter STO IC NX NFIRST
1 1.0 2 3 4
2 3.0 1 0 2
3 2.0 3 0 3
4 4.0 3 0
Notice, in this example, that NX remains untouched.

6. Ibrahim Omar Habiballah Sep. 2013
2.2.3) Updating of an Entry
In this case, the old value is replaced with the new value directly on the
STO array (using any of the two storage methods) without affecting the
other arrays.
2.2.4) Increasing/Decreasing the Size of the Matrix
Increasing/decreasing the size of a sparse matrix using the entry-row-
column storage method is treated exactly as insertion/deletion of entries in
the matrix.
Increasing/decreasing the size of a sparse matrix using the chained data
structure storage method is treated similar to the insertion/deletion of
entries in the matrix, however, the size of the NFIRST array change
accordingly.
2.2.5) Symmetry Matrix Storage
In symmetry matrices, only the upper right (or lower left) triangle,
including the diagonal entries, is stored. If the upper right triangle entries
are stored, a lower left triangle position can be accessed by interchanging
the row column subscripts.
To determine the location e of an entry zrc, r ≤ c (i.e., upper right triangle
entries are stored), use the following formula,
rcce +−=
2
1
2
1 2
(1)
for example, the location of entry (3,4) is 9. This equation is valid when all
entries of the upper right triangle matrix are stored.

7. Ibrahim Omar Habiballah Sep. 2013
Example (4):
The following 9 x 7 sparse matrix has 15 nonzero entries out of 63.
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
−
−
=
9000009
8800000
7000000
0600006
0050000
0044000
0000300
2000020
0001001
A
Using row fashion approach, the following arrays can be recorded
Counter STO IC NX NFIRST
1 1.0 1 1 1
2 -1.0 4 0 3
3 2.0 2 1 5
4 -2.0 7 0 6
5 3.0 3 0 8
6 4.0 4 1 9
7 -4.0 5 0 11
8 5.0 5 0 12
9 -6.0 1 1 14
10 6.0 6 0
11 7.0 7 0
12 8.0 6 1
13 -8.0 7 0
14 -9.0 1 1
15 9.0 7 0

8. Ibrahim Omar Habiballah Sep. 2013
Assuming the entry -2.0 (in row 2 and column 7) is changed to -2.1, and the
entry -9.0 (in row 9 and column 1) is changed to -9.1; then the sizes of all
four arrays remain the same. Only the values of revised entries are
changed.
Counter STO IC NX NFIRST
1 1.0 1 1 1
2 -1.0 4 0 3
3 2.0 2 1 5
4 -2.1 7 0 6
5 3.0 3 0 8
6 4.0 4 1 9
7 -4.0 5 0 11
8 5.0 5 0 12
9 -6.0 1 1 14
10 6.0 6 0
11 7.0 7 0
12 8.0 6 1
13 -8.0 7 0
14 -9.1 1 1
15 9.0 7 0
Assume now the addition of a new entry -5.0 (in row 5 and column 1). The
updated four arrays will be as follows.
Counter STO IC NX NFIRST
1 1.0 1 1 1
2 -1.0 4 0 3
3 2.0 2 1 5
4 -2.1 7 0 6
5 3.0 3 0 8
6 4.0 4 1 9
7 -4.0 5 0 11
8 -5.0 1 8 12
9 -6.0 1 1 14
10 6.0 6 0
11 7.0 7 0
12 8.0 6 1

9. Ibrahim Omar Habiballah Sep. 2013
13 -8.0 7 0
14 -9.1 1 1
15 9.0 7 0
16 5.0 5 0
Assume now the removal of an existing entry -2.1 in (in row 2 and column
7). The updated four arrays will be as follows. Notice that in this case, the
access time is very fast [only changing NX(3) from 1 to 0].
Counter STO IC NX NFIRST
1 1.0 1 1 1
2 -1.0 4 0 3
3 2.0 2 0 5
4 -2.1 7 0 6
5 3.0 3 0 8
6 4.0 4 1 9
7 -4.0 5 0 11
8 -5.0 1 8 12
9 -6.0 1 1 14
10 6.0 6 0
11 7.0 7 0
12 8.0 6 1
13 -8.0 7 0
14 -9.1 1 1
15 9.0 7 0
16 5.0 5 0
Assume now the addition of new row with two entries: -10 (in row 10 and
column 3), and 10 (in row 10 and column 6). The updated arrays will be
Counter STO IC NX NFIRST
1 1.0 1 1 1
2 -1.0 4 0 3
3 2.0 2 0 5
4 -2.1 7 0 6
5 3.0 3 0 8
6 4.0 4 1 9
7 -4.0 5 0 11
8 -5.0 1 8 12
9 -6.0 1 1 14
10 6.0 6 0 17
11 7.0 7 0
12 8.0 6 1
13 -8.0 7 0
14 -9.1 1 1

10. Ibrahim Omar Habiballah Sep. 2013
15 9.0 7 0
16 5.0 5 0
17 -10.0 3 1
18 10.0 6 0
Example (5):
The following 4 x 4 symmetric sparse matrix has 8 nonzero entries out of
16.
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
=
4020
0301
2020
0101
A
Using row fashion approach, the following arrays can be recorded. It can
be noticed that only 6 nonzero entries of the upper right triangle are
recorded because of the symmetry.
Counter STO IC NX NFIRST
1 1.0 1 1 1
2 -1.0 3 0 3
3 2.0 2 1 5
4 -2.0 4 0 6
5 3.0 3 0
6 4.0 4 0
If the access to the entry -1.0 (in row 3 and column 1) needs to be accessed,
then the corresponding entry from the upper right triangle in row 1 and
column 3 (i.e., changing the subscript z31 to z13) is used.