Study of using particle swarm for optimal power flow

STUDY OF USING PARTICLE SWARM FOR OPTIMAL POWER FLOW IN IEEE
BENCHMARK SYSTEMS INCLUDING WIND POWER GENERATORS
by
Mohamed A. Abuella
A Thesis
Submitted in Partial Fulfillment of the Requirements for the
Master of Science Degree
Department of Electrical and Computer Engineering
Southern Illinois University Carbondale
2012
1

The main aim of the thesis is to obtain the optimal economic dispatch of real power
in systems that include wind power.
Model considers intermittency nature of wind power.
 Solve OPF by PSO.
 IEEE 30-bus test system.
6-bus system with wind-powered generators.
3

MOTIVATION:
The Growing Importance of Wind Energy:
4

MOTIVATION:
5
•Fluctuations of oil prices;
• One of the most competitive renewable resources.
• Abundance of wind power everywhere;
The Cost of Wind Energy:

MOTIVATION:
6
The Optimal Economic Dispatch of Wind Power:
It’s a challenging topic to search about.
ECE580
Wind Energy Power
Systems
ECE488
Power Systems Engineering
Spring 2011

STATEMENT OF THE PROBLEM
2
i i i i i iC a P b P c   ,w i i iC d w
Conventional-thermal generators Wind-powered generators
7
The objective function of the optimization problem in the thesis is to minimize the
operating cost of the real power generation.

In similar fashion,
,
, , ,
,
(underestimation)( )
( ) ( )
r i
i
p i p i i av i
w
p i i W
w
C k W w
k w w f w dw
 
 
Since Weibull probability distribution function (pdf) of wind power.( )Wf w
, , ,
,
0
( ) (overestimation)
( ) ( )
i
r i r i i i av
w
r i i W
C k w W
k w w f w dw
 
 
8

The model of OPF for systems including thermal and wind-powered generators:
,min ,max
,
min max
m
,
ax
,
, ,
:
( ) ( ) ( ) ( )
:
0
r
M N N N
i i i i i
i i i i
i i i
i r i
M N
i i
i i
i i i
line i lin i
i ipw i
e
Minimize
J C p w w w
Subjectto
p p p
w w
p w L
V V V
C
S
C
S
C   
 
 
 
 

   
 
Particle Swarm Optimization (PSO) algorithm is used for solving this optimization
problem.
9
2
i i i i i iC a P b P c  
, iw i iC d w
,
,, (underestimation)( ) ( )
r i
i
w
p i i W
w
p i k w w f w dwC  
,
0
, ( ) ( ) (overestimation)
i
r i
w
r i i Wk w w f w dwC  
Where:

PROBABILITY ANALYSIS OF WIND POWER:
The wind speeds in particular place take a form of Weibull distribution over time,
as following:
Weibull probability density functions (pdf) of wind speed for three values of scale factor c
( / )
( 1)
( ) ( )
kv c
k
V
k v
f v e
c c


  
   
  
0 5 10 15 20 25 30 35 40
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Wind speed v
Probability
C=25
C=15
C=5
10

The cumulative distribution function (cdf) of Weibulll distribution is:
Weibull pdf and cdf of wind speed for c=5 m/s
( / )
0
( ) 1
k
v
v c
v vF f d e  
  
11
0
0.1
1
wind speed v
pdf
0 5 10 15
0
0.5
1
cdf
cdf
pdf

The captured wind power can be assumed to be linear in its curved portion:
The captured wind power
w



 


0; ( )i ov v or v v 
( )
;
( )
i
r
r i
v v
w
v v

 i rv v v 
;rw r ov v v 
3
2
m p RP C A v

The captured wind power 
12

The linear transformation (in terms of probability) from wind speed to wind power
is done as following:
Pr{ 0} 1 exp exp
k k
i ov v
W
c c
      
                    
Pr{ } exp exp
kk
or
r
vv
W w
c c
     
                   
13
While for the continuous portion:
1
(1 ) (1 )
( ) exp
kk
i i i
W
r
klv l v l v
f w
w c c c
 

     
     
    
1
( )W v
w b
f w f
a a
 
  
 
Q
( )
Where : , r i
r i
v vw
l
w v


 

Probability vs. Wind power for C=10, 15 and 20
14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
w/wr
Probability
C=10
C=15
C=20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C=10
W/Wr
Comulative.Probability
C=15
C=20
The probability density
function pdf of wind power
The cumulative distribution
function (cdf) of wind power
,
, , (underestimation)( ) ( )
r i
i
w
p i p i i W
w
C k w w f w dw 
, ,
0
( ) ( ) (overestimation)
iw
r i r i i WC k w w f w dw 

,
, , ( ) ( )
r i
i
w
p i p i i W
w
fC k w w dww  , ,
0
( () )
iw
r i r i i WC k w w f dww 
, ,: ( ) ( ) ( ) ( )
M N N N
i i wi i p i i r i i
i i i i
Minimize J C p C w C w C w      
1 expPr{ 0} exp
k k
i ov v
c c
W
      
                  


exp xPr{ } e p
k
r
k
or vv
c c
W w
     
                 


1
(1 ) (1
)
)
exp(
kk
i
W
i i
r
klv l v l
w
v
c
f
w c c
 

     
     
    
Then, plug the transformed wind power equations in the model:
15

OPTIMAL POWER FLOW (OPF)
1
1
cos( ) 0
sin( ) 0
n
i i j ij i j ij
j
n
i i j ij i j ij
j
P V V Y
Q V V Y
  
  


   
   


: ( )
Subject to:
( ) 0
( ) 0
Min J
g
h


x,u
x,u
x,u
The optimal power flow (OPF) is a mathematical optimization problem set up to minimize
an objective function subject to equality and inequality constraints.
While: ( ) 0g  x,u
1 1 , 1 , 1 ,[ , ,..., , ,..., , ,...., ]T
G L L NL G G NG line line nlP V V Q Q S Sx
Where:
16
, ,( ) ( ) ( ) ( ) ( )
M N N N
i i wi i p i i r i i
i i i i
J C p C w C w C w      x,u
1 2 , 1 1 ,[ ,..., , ,..., , ,...., , ,... ]T
G NG G G NG NT C C NCV V P P T T Q Qu

min max
min max
min max
1,.....,
1,.....,
1,.....,
Gi Gi Gi
Gi Gi Gi
Gi Gi Gi
P P P i NG
V V V i NG
Q Q Q i NG
  
  
  
Where: Generator Constraints:
Whereas: Transformer Constraints:
min max
1,.....,i i iT T T i NT  
While: Shunt VAR Constraints:
min max
1,.....,Ci Ci CiQ Q Q i NC  
Finally: Security Constraints:
min max
max
, ,
1,.....,
1,.....,
Li Li Li
line i line i
V V V i NL
S S i nl
  
 
On the other hand, for inequality constraints ( ) 0 :h x,u
17

2 2 2
lim 2 lim lim max
1 1 , ,
1 1 1
( ) ( ) ( ) ( )
NL NG nl
P G G V Li Li Q Gi Gi S line i liau ng e i
i i i
J J P P V V Q Q S S   
  
          
By using penalty function principle, all constraints are included in the objective
function ( ):J x,u
, , ,andP V Q S   Where: are penalty factors.
18

PARTICLE SWARM OPTIMIZATION (PSO)
PSO suggested by Kennedy and Eberhart (1997).

1
1 2* * ( ) * ( )i i i i i
k k k k k k
v w v c U pbest x c U gbest x
    
PSO Searching Mechanism:
1 1
i i i
k k k
x x v 
 
max min
max
max
( )
*
w w
w w iter
iter

 
Where:
20

PSO Algorithm Flowchart
21

( , )Min J x u
Implementation of PSO for Solving OPF:
. . ( , ) 0S T g x u
1[ , , , ]G G L lineP Q V Sx
Where:
( , ) 0h x u
[ , , , ]G G cP V T Qu
1 1
( ) * ( , )
NG NS
i
i
a g Gu i i
i
J F P h
 
 
    
 
  x u
1; ( , ) 0
0; ( , ) 0
i
h
h


 

x u
x u
22
Since:
2 2 2
lim 2 lim lim max
1 1 , ,
1 1 1
( ) ( ) ( ) ( )
NL NG nl
P G G V Li Li Q Gi Gi S line i line i
i i
aug
i
J P P V V QJ Q S S   
  
          

STUDY CASES AND SIMULATION RESULTS
OPF Main Flowchart
23

Using Sensitivity Matrices to Select the Most Control Variables:
24

Using Power Flow algorithm to satisfy the equality constraints g(x,u)=0
25

Using PSO algorithm to find the minimum cost
26

IEEE 30-BUS TEST SYSTEM:
27

Study of Base Case:
VL3 VL4 VL6 VL7 VL9 VL10 VL12 VL14 VL15 VL16 VL17 VL18 VL19 VL20 VL21 VL22 VL23 VL24 VL25 VL26 VL27 VL28 VL29 VL30
0.9
0.95
1
1.05
1.1
1.15
P.U.
Load Bus Voltages
Upper limit
Lower limit
PG1
(MW)
PG2
(MW)
PG3
(MW)
PG4
(MW)
PG5
(MW)
PG6
(MW)
Losses
(MW)
Cost
($/hr)
176.94 48.71 21.27 21.09 11.83 12.00 8.4382 798.43
100 200 300 400 500 600 700
797.5
798
798.5
799
799.5
800
800.5
801
Iterations
Cost
28

Application of Sensitivity Analysis for OPF :
29
The order of most effective control variables is:

Using combinations of the most effective control variables for Base Case OPF:
30
The order of most effective control variables is:

6-BUS SYSTEM INCLUDING WIND-POWERED GENERATORS
31

6-BUS SYSTEM INCLUDING WIND-POWERED GENERATORS
1
(1 ) (1
)
)
exp(
kk
i
W
i i
r
klv l v l
w
v
c
f
w c c
 

     
     
    
32
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
w/wr
Probability
C=10
C=15
C=20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C=10
W/Wr
Comulative.Probability
C=15
C=20
, ,: ( ) ( ) ( ) ( )
M N N N
i i wi i i i
i i
p r ii
i i
Minimize J C p C w w C wC      

The Effect of Wind Power Cost Coefficients
The Effects of Reserve Cost Coefficient kr (and kp=0):
33

The Effects of Penalty Cost Coefficient kp (and kr=0):
34

The Effects of Reserve Cost Coefficient kr and Penalty Cost Coefficient kp:
35
(a) kr=20

The Effects of Reserve Cost Coefficient kr and Penalty Cost Coefficient kp:
36

Conclusion and Further Work
• The implementation of PSO algorithm to find OPF solution is useful and worth of
investigation.
• PSO algorithm is easy to apply and simple since it has less number of parameters to deal with
comparing to other modern optimization algorithms.
• PSO algorithm is proper for optimal dispatch of real power of generators that include wind-
powered generators.
• The used model of real power optimal dispatch for systems that include wind power is
contains possibilities of underestimation and overestimation of available wind power plus to
whether the utility owns wind turbines or not; these are the main features of this model.
Conclusion
37

• The probability manipulation of wind speed and wind power of the model is suitable since
wind speed itself is hard of being expected and hence the wind power as well.
• In IEEE 30-bus test system, OPF has been achieved by using PSO and gives the minimum
cost for several load cases.
• Using OPF sensitivity analysis can give an indication to which of control variables have most
effect to adjust violations of operating constraints.
• The variations of wind speed parameters and their impacts on total cost investigated by 6-bus
system.
Conclusion
38

• PSO algorithm needs some work on selecting its parameters and its convergence.
• PSO can be applied in wind power bid marketing between electric power operators.
• The same model can be adopted for larger power systems with wind power.
• The environment effects and security or risk of wind power penetration can be included in the
proposed model and it becomes multi-objective model of optimal dispatch.
• Fuzzy logic is worth of investigation to be used instead probability concept which is used in
the proposed model, especially when security of wind power penetration is included in the
model.
Further Work
39

• Using the most effective control variables to adjust violations in OPF needs more study.
• The incremental reserve and penalty costs of available wind power can be compared with
incremental cost of conventional-thermal quadratic cost; this comparison could lead to useful
simplifications of an economic dispatch model that includes thermal and wind power.
Further Work
40

Thank You for Listening
Any Question
?
41

Study of using particle swarm for optimal power flow

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