Solving Electrical Power Load Flow
Problems using Intervals
by
Eustaquio A. Martínez1,
Tiaraju Asmuz Diverio2
&
Benjamín Barán3
[email protected]
Facultad Politécnica - UNE
Paraguay
[email protected]
Dpto. de Informática - UFRGS
Brazil
[email protected]
CNC - UNA
Paraguay
Validated Computing 2002
Summary
• Motivation
• Electrical Power Load Flow Problem
• Interval approach
• Solving Sequentially
• Solving Parallely
• Experimental Results
• Conclusions
Validated Computing 2002
Motivation
V1 , 1
V2 ,  2
r12  jx12

1

4

2
3
Electrical System
6
5
Load Flow Problem
F(X )  0
Unknown:
X  (V1 ,,Vn ,1 ,, n )
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Proposition
Interval
Arithmetic
Electrical System
Model
F(X )  0
All solutions in a domain
(operating points)
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Electrical Power Load Flow Problem
The Electrical Power Load Flow Problem can be formulated as a quasilinear
equation system
Yx  I (x)
Y  G  jB
I (x )
is the admittance matrix (problem’s parameters) and
the electric current vector and
x  (V1 ,,Vn ,1 ,, n )T
the unknown
Generally, the problem may be written as:
Pk  Vk iK Vi (Gki cos ki  Bki sen  ki )
Qk  Vk iK Vi (Gki sen  ki  Bki cos ki )
 ki  k  i k 1,..., n
K
n is the problem’s size
k
is the group of the bus bars adjacent to and
k
itself .
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Interval Approach
Interval Newton Method
T


F ( X )  f1 ( X ),..., f n ( X )  0
where
F : Rn  Rn , X  x1, x2 ,..., xn   Rn
T
The system
F(X )  0
and
x i  xi  x i
can be written as a linear interval system :
~k
F (X )( X  X k )   F ( X k )
'
k
X k  IR n
is the interval vector where the solutions is expected to be found
X k  Rn
is an inner vector of
~
X k  IR n
is the unknown interval vector which is expected to contain the solutions
Xk
F ' ( X k )  IR nxn is the interval extension of Jacobian matrix of
F
in
Xk .
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~k
Computed X , the iterative formula of the interval Newton Method
for a system with n variables is:
X
If
X k 1  
k 1
~k
 X X
k
there are not a solutions in X k
The problem’s matrix form is:
H k
 k
J
where
H k
 k
J
~
N k    k   k    Pk 
.
  k   k
k  ~k
L    V  V   Q 
~
 k  ~ k
~k  X
V 
Nk 
 F' ( X k )
k 
L 
 Pk 
k
 k   F(X )
Q 
Domain for a known solution
    max ,max 
where
 max  10º
<1
V     1,  1
heuristic for feasible solution
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Solving Sequentially
Low Flow Problem
Interval Newton/Generalized Bisection
Algorithm
Self Validated Results
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Interval Newton/Generalized Bisection Algorithm
F ( X )   f1 ( X ),..., f n ( X )  0;
T
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Solving Parallely
Low Flow Problem
Self Validated Results
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Partition Algorithm
Algorithm 1
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Paralleling Scheme
Master
0
Slave 1
x ( 5)
x ( 5 ) x  Sol.
0
x  Sol.
Esclavo 1
x ( 5)
x ( 5)
0
Slave 4
0
x0
Slave 2
Esclavo 2
x  xk
Slave 3
Esclavo 3
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Master’s Process Algorithm
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Slaves Process
Interval Newton/Generalized Bisection Algorithm - Modified
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Computing Environment
• 10 Mbps local area network;
• 5 personal computers (Pentium II, 400MHz, 32 MB RAM,
Linux SO) ;
• One acts as the master (NFS, NIS and MPI) ;
• Four work as slaves.
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Experimentals Results
Sequential - Punctual (N-R)
TEST PROBLEM
POWER MISMATCH
COMPUTING TIME
F ( X k 1 )
[s]
IEEE-5
4.35E-4
0.001
IEEE-14
9.70E-5
0.010
Monticelli - 30
1.62E-4
0.060
Combined - 88
2.12E-4
0.160
Sequential - Interval (IN/GB)
TEST PROBLEM
POWER
MISMATCH
SOLUTION DIAMETER
diam ( X
k 1
F ( X k 1 )
)
COMPUTING
TIME
[s]
IEEE – 5
8.42E-4
3.55E-4
5.99E-3
IEEE – 14
8.93E-6
8.67E-4
0.68
Monticelli - 30
1.39E-5
1.15E-5
0.82
Combined - 88
2.48E-5
5.08E-6
16.18
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Parallel - 3 processors
TEST PROBLEM
POWER
SOLUTION DIAMETER
COMPUTING
MISMATCH
diam ( X k 1 )
TIME
F ( X k 1 )
IEEE – 14
[s]
Monticelli - 30
2.41E-6
1.99E-5
7.63E-5
8.98E-6
0.405
0.514
Combined - 88
2.43E-5
5.19E-6
11.982
POWER
SOLUTION DIAMETER
COMPUTING
MISMATCH
diam ( X k 1 )
TIME
Parallel - 5 processors
TEST PROBLEM
F ( X k 1 )
IEEE – 14
[s]
Monticelli - 30
4.43E-5
1.58E-5
9.36E-4
9.58E-6
0.244
0.507
Combined - 88
2.25E-5
4.93E-6
11.851
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Speed - Up
TEST PROBLEM
PROCESSOR
1 (REFERENCE)
3
5
IEEE – 14
1.0
1.679
2.786
Monticelli - 30
1.0
1.595
1.617
Combined - 88
1.0
1.350
1.365
Speed Up
3,00
Sp
2,50
2,00
1,50
1,00
1
3
5
Processor
IEEE – 14
Monticelli - 30
Combined - 88
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Conclusions
Though computationally more expensive, this interval solution of the electrical load
flow problem has advantages if compared to traditional methods:



It proves the inexistence of solutions (feasible solutions) in a given domain
without a solution.
If there are several feasibles solutions in a given interval, the method can
find all the solutions.
It allows to control the precision of each solution, directly on the unknown
value, rather than through related variables (such as power mismatch).
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¡¡Thank you Very Much!!
Tiaraju Asmuz Diverio
[email protected]
U.N.E. - Paraguay
Validated Computing 2002