The 14th International Conference on Intelligent System Applications to Power Systems, ISAP 2007
November 4 - 8, 2007, Kaohsiung, Taiwan
New Constriction Particle Swarm Optimization for
Security-Constrained Optimal Power Flow Solution
ZWE-LEE GAING, MEMBER, IEEE AND XUN-HAN LIU
Many global optimization techniques known as genetic
algorithms (GA), simulated annealing (SA), evolutionary
programming (EP), and particle swarm optimization (PSO),
have been successfully used to solve the variant OPF
problems [12]-[14]. However, the objective function not
considering contingency constraints or stability constraints
may result in the economy-security control of system not
being properly carried out. In addition, the premature
convergence may result in the local optima by obtaining
[13]-[14].
In this paper, to improve the drawback of traditional
PSO method, a new constriction PSO (NCPSO) technique is
presented for solving the SCOPF problems. The proposed
NCPSO method involves a new cognitive behavior of
particle. The particle can remember its previously visited
best and worst positions, thus it can explore effectively the
search space. In addition, the individual that is the real-value
representation containing a mixture of continuous and
discrete control variables is defined, and two mutation
schemes are proposed to deal with the continuous/discrete
control variables, respectively. The effectiveness of the
proposed method is demonstrated for the IEEE 30-bus and
57-bus systems, and compared with the traditional PSO [13]
and EP [12] in terms of solution quality and convergence
rate.
II. PROBLEM DESCRIPTION
Abstract -- This paper presents a new version of constriction
particle swarm optimization (NCPSO) with mutation mechanism
for solving the security-constrained optimal power flow (OPF) with
both the steady-state security constraints and the transient stability
constraints. The objective of SCOPF in considering the valve-point
loading effect of the unit and the operating limits of FACTS
devices is defined not only to minimize total generation cost but
also to enhance transmission security, reduce transmission loss, and
improve the bus voltage profile under normal or post-contingent
states. The proposed NCPSO method involves a new cognitive
behavior of particle. The particle can remember its previously
visited best and worst positions, thus it can explore effectively the
search space. The effectiveness of the proposed method is
demonstrated for the IEEE 30-bus and 57-bus systems.
Index terms -- optimal power flow, contingency analysis, transient
stability, particle swarm optimization, valve-point loading effect,
FACTS
I.
INTRODUCTION
Currently, the concept of performing the optimal plan of
power system operation with system security assessment
considered is positively presented to ensure that the system
can secure operation without interruption to customer
service even if the system suffers the contingency impact.
For this reason, the pre-protection strategies of the system
and the security constraints should be taken into
consideration including the transmission capacity limit, bus
voltage limit and transient stability constraints that must be
embedded in the objective function or constraints [1]-[11].
In [3], the security-constrained OPF scheduling can be
undertaken to bring the system to the more acceptable level
of security represented by level 1 or 2. Whether the system
is in a normal operation or contingent state, the security
constraints can ensure that the system can secure operation.
Therefore, the aspect of system economy-security control
can be carried out. In [7], the stability-constrained OPF
using step by step integration (SBSI) method is implemented
to ensure that the system is stable during fault occurrence. In
[8], a multi-contingency stability-constrained OPF using
primal-dual Newton interior point method (IPM) is
presented to guarantee system stability when the system
suffers multi-contingency. In [9], the OPF with transient
stability constraints is equivalently converted into an
optimization problem in the Euclidean space and can be
solved by the standard nonlinear programming technique
adopted by the OPF.
The steady-state security constraints and the transient
stability constraints are the fundamental elements of system
security assessment.
A. Steady-State Security Constraints
The steady-state security-constrained OPF formulation
can be stated as:
Min f (u (0 ) , x ( 0 ) )
(1)
u ,x
s.t.
g ( k ) (u ( k ) , x ( 0 ) ) = 0 ,
for k = 0,1,..., N c
(2)
h (u , x ) ≥ 0 , for k = 0,1,..., N c
(3)
where x is the vector of control variables of
pre-contingency, , while u is the vector of state variables.
Objective function (1) is scalar. Equalities (2) are the
conventional power equations, while inequalities (3) are the
limits on the control variables x and state variables u, and
the operating limits on the power system. The superscript
“o” represents the pre-contingency (base-case) state being
optimized, and superscript “k” (k>0) represents the fault
duration and post-contingency states for the Nc contingency
cases. Moreover, the equality constraints g(o) change to g(k) to
reflect the outage equipment.
Figure 1 shows the concept of the feasible operation
region, which is within the oblique line region. x1 is located
(k )
-------------------------------------------------------------------------------Z. L. Gaing and X. H. Liu are with the Department of Electrical
Engineering of Kao Yuan University, Kaohsiung, Taiwan 821 (e-mail:
zlgaing@ms39.hinet.net, Fax: 886-7-6077009).
106
(k )
( 0)
The 14th International Conference on Intelligent System Applications to Power Systems, ISAP 2007
within the region of steady-state operation of
pre-contingency; however, it might be unable to satisfy the
transient stability or security constraints (i.e. bus voltage
limit and line thermal capacity limit) when suffering the
specific contingency impact. x2 is located within the region
of transient stability; however, it might be unable to satisfy
the security constraints of post-contingency. Moreover, if the
x within the practical security region can drive the
minimization of objective function f, it is the optimal
operation point that is the goal to be achieved in this
research.
damaged.
C. Function of Series FACTS Devices
According to their steady-state characteristics of
variant FACTS devices, the function of series controllers,
such as TCPS and TCSC, is mainly used to control the
power flow of the lines close to their thermal limits. The
TCPS device can to adjust the phase-angle φ to control the
active power flow of transmission line. The TCSC device
can permit to modify the reactance X L of transmission line.
However, the operation limits of both series controllers must
be valid. The models of TCPS and TCSC are shown in Fig.
2.
Region of feasible
operation point
u
I
x1
x2
II
November 4 - 8, 2007, Kaohsiung, Taiwan
1 : 1∠φi
III
x3
RL+ jXL
Vi
Vj
(a) TCPS
RL+ jXL
jXc
I Pre-contingency region
II Transient stability region
III Post-contingency region
x
Vi
(b) TCSC
Fig. 2 Models of TCPS and TCSC
Fig. 1 Region of feasible operation point for system security
B. Transient Stability Constraints
Mathematically, the transient stability problem is
described by solving a set of differential-algebraic equations.
The swing equation set with damping neglected is given by
δi = ∆ωi
(4)
D.
Valve-Point Loading Effect
The constraints of valve-point loading effect of thermal
generating units should be considered, the fuel cost function
of unit i is defined as:
Fi ( Pi ) = ai + bi Pi + ci Pi 2 + d i ⋅ sin( ei ( Pi min − Pi ))
(8)
Ng
πf 0
[ Pmi − ∑ E i′ E ′j Yij′ cos(θ ij′ − δ i − δ j )]
(5)
Hi
j =1
where Ei′ = Ei′ ∠δ i is the output voltage of the i-th
∆ω i =
where ai, bi, ci, di, and ei are the cost coefficients of unit i.
E. Objective Function
The objective is to minimize the objective function FT
that is generalized as follows.
generator behind a transient reactance x′d . H i is the
inertia constant of the i-th generator. Yij′ = Yij′ ∠θ ij′ are the
FT =
elements of the reduced bus admittance matrix.
To recognize conveniently the stability performance of
generators and integrate it into the OPF problem, the concept
of inertia center can serve as a reference frame [7]-[10]. The
rotor angles of generators with respect to the center of inertia
(COI) are employed to indicate whether the system is stable
or not. For a NG-generator system with rotor angle of
generator δ i and its inertia constant H i , the position of
COI is defined as (6).
δ COI =
i
i =1
NG
∑
NG
i
+
Vj
j =1
Nl
∑λ
lh
NG
(V j − V jlim ) 2 + ∑ λQi (QGi − QGilim ) 2
i =1
(S lh − S lhlim ) 2 +
h=1
NG
∑ λ (δ
δi
i
− δ ) 2 + λPL Ploss
(9)
i =1
where λVj , λQi , λlh , λδi , λPL are the penalty factors,
which are large positive constants. If a variable is out of its
limit, a penalty factor is multiplied by the difference
between its value and the limit violated, and then added to
the objective function.
F.
i
(6)
Equality Constraints
Pi ( k ) −
Hi
NB
∑Y
(k )
ij
Vi ( k ) V j( k ) cos(δ i( k ) − δ (j k ) − θ ij( k ) ) = 0 ,
(10)
Vi ( k ) V j( k ) sin(δ i( k ) − δ (j k ) − θ ij( k ) ) = 0 .
(11)
j =1
i =1
Qi( k ) −
δ i = δ i − δ COI ≤ δ
(7)
The discriminator of system stability is expressed as (7).
According to previous studies [7]-[9], a real-world power
system is always operated such that any rotor angle of
generator δ i at any time will not be greater than a
threshold δ
NB
∑ F + ∑λ
i =1
NG
∑H δ
Vj
NB
∑Y
(k )
ij
j =1
G. Inequality Constraints
1) Conventional inequality constraints
PGimin ≤ PGi( k ) ≤ PGimax , i ∈ N g
min
Gi
Q
( as δ =100o). If a rotor angle of generator
≤Q
(k )
Gi
≤Q
max
Gi
T pnmin ≤ T pn( k ) ≤ T pnmax
δ i is larger than such a threshold δ , the generator will be
min
hj
Y
tripped off-line by out-of-step relay to protect it from being
107
≤Y
(k )
hj
≤Y
max
hj
(12)
, i ∈ Ng
(13)
, n ∈ N Tp
(14)
, j ∈ N Sh
(15)
The 14th International Conference on Intelligent System Applications to Power Systems, ISAP 2007
2) Operating limits of FACTS devices
φimin ≤ φi( 0 ) ≤ φimax
, i ∈ NTCPS
X cjmin ≤ X cj( 0 ) ≤ X cjmax , j ∈ NTCSC
3) Steady-state security constraints
V jmin ≤ V j( k ) ≤ V jmax
, j ∈ NB
enhance the effectiveness of NCPSO, a mutation operator of
real-valued GA should be integrated with the NCPSO. The
mutation process is employed as follows:
Let the i-th individual xi = [ x1 ,..., xk ,..., x N ]i , and the
gene xk be selected for mutation according to the probability
pm. The new gene xk can be obtained by (22) and
(16)
(17)
(18)
, m ∈ Nl
(19)
4) Transient stability constraints
(k )
δ i( k ) − δ COI
≤ 100 0 , i ∈ N g
(20)
Slm( k ) ≤ Slmmax
November 4 - 8, 2007, Kaohsiung, Taiwan
xk ∈ [ xkmin , xkmax ] . Then the next offspring of the application
of the mutation operator is xi = [ x1 ,..., x k ,..., x N ]i .
If rand () < pm then
xk = xk × (1 + Gaussian(σ )) .
(22)
where Gaussian(σ ) is a Gaussian distribution function,
σ is set to be 0.1.
where the power flow equations (10)-(11) are used as
equality constraints; the active and reactive power
generation limits (12)-(13), transformer-tap setting limits
(14), shunt admittance limits of the switchable
capacitor/reactor devices (15), operating limits of FACTS
devices (16)-(17), bus voltage limits (18), thermal capacity
limits of transmission lines (19) and transient stability
constraints (20) are used as inequality constraints.
Therefore, the security-constrained OPF problem must
be solved subject to both the pre-contingency constraints,
and to the post-contingency constraints of the selected
contingency cases.
IV.
DEVELOPMENT OF THE PROPOSED METHOD
A. Representation of individual
In this paper, the individual is composed of both
continuous control variables x and discrete control variables
u. An individual s is a mixed-integer structure, that is
s = [ x, u ] = [ PG , V , Tp , Yh , φ , X c ] . The encoding of the
physical variables is performed as follows.
1) Continuous variable x i taking the real value in the
III. A CONSTRICTION PARTICLE SWARM OPTIMIZATION
WITH MUTATION MECHANISM
2)
One of the main drawbacks of the traditional PSO is its
premature convergence, especially while handling problems
with more local optima and heavier constraints. To
overcome this disadvantage, the concept of constriction
factor was suggested by [15] to both speed up convergence
and escape local minima. In this paper, a new constriction
PSO (NCPSO) algorithm with a new scheme of velocity
updating is as (21). The proposed NCPSO method involves a
new cognitive behavior of particle; that is, the particle can
remember its previously visited best and worst positions,
thus it can explore effectively the search space.
r
r
vi( t +1) = ψ [vi(t ) + c1a 1i ( pbest ( t ) − xi(t ) ) + c1b 2i ( xi( t ) − pworst ( t ) )
ri
ri
r3i
+ c2 ( gbesti( t ) − xi(t ) )]
(21)
ri
where
pworst i( t ) the previous worst position of particle xi at
iteration t,
r1i , r2 i , r3i
three uniform random numbers in the range
[0,1]. In addition, ri = r1i + r2 i + r3i .
c1a , c2 b
two acceleration constants. c1a
can
accelerate the particle toward its best position;
c1a can accelerate the particle away from its
worst position.
2
ψ
constriction factor. ψ =
,
2 − ϕ − ϕ 2 − 4ϕ
interval [ x imin , x imax ] .
Discrete variable ui taking the decimal integer value
ni in the interval [ 0 ,..., M i ] ,
M i = INT ((uimax − uimin ) / STi ) ,
and
u i = u imin + ni ⋅ STi .
(23)
(24)
where STi is the adjustable step size of the discrete control
variable ui . INT (⋅) is the operator of rounding the
variable to the nearest integer.
B. Fitness function
In this paper, the objective of SCOPF is not only to
minimize total generation cost but also to enhance
transmission security, reduce transmission loss, and improve
the bus voltage profile under pre-contingency or
post-contingency states.
If an individual sj is a feasible solution, its fitness will
be measured by using the fitness function f as (25).
Otherwise, its fitness will be penalized with a very large
positive constant λ that is set to be 106. The infeasible
individual will not be selected by the proposed scheme for
evolution in the next generation, so the proposed method can
rapidly converge.
 FT ( s j ) , s j ∈ feasible
f ( sj ) = 
, s j ∈ unfeasible
 λ
where
Ng
NB
i
n
(25)
FT(sj) = ∑ Fi ( s j ) + ρ L Ploss ( s j ) + ∑ ( ρ nV ⋅ Vn ( s j ) − Vref ) ,
where ϕ = c1a + c1b + c2 , ϕ > 4 . In general,
c1a + c1b = c 2 = 2.05 and ψ =0.73.
This mutation mechanism potentially provides a means
both of escaping local optima and speeding up the search. To
(26)
Ng
ρ L = ∑ Fi ( s j ) / PDt ,
i
108
(27)
The 14th International Conference on Intelligent System Applications to Power Systems, ISAP 2007
executed on a Pentium IV 1.8 GHz personal computer with
512MB RAM.
Ng
ρ nV = ∑ Fi ( s j ) ⋅ PDn ( s j ) / PDt .
(28)
i
V.
ρ is a weight factor that is mainly there for the purpose of
L
According to the boundary limits of
continuous/discrete control variables, establish
randomly the initial population s(t), using (23) and (24)
t=1
Perform NCPSO operations using (21)-(22)
New offspring population s(t+1)
Perform transient stability analysis using selected
contingency event
No
Satisfy the stability constraints (20)?
Yes
• Parameters of Algorithms
Through many experiments, the results revealed that the
appropriate values for c1a, c1b and c2 are 1.55, 0.50 and 2.05,
respectively. They can yield an optimal evaluation value.
Therefore, the following parameters of NCPSO are used:
• individual length=20,
• population size =50,
• c1a=1.55, c1b=0.50, c2=2.05
• continuous variable: v imax = x imax / 2 , discrete variable:
Perform system steady-state contingency analysis
Satisfy the conventional inequality
constraints (12)-(19)?
Yes
Fitness
←λ
Evaluate the fitness
using (25)
NUMERICAL EXAMPLES AND RESULTS
A. IEEE 30-bus System
The system contains six thermal units, 30 buses and 41
transmission lines. In addition, three series FACTS devices
(one TCPS device and two TCSC devices) are installed on it.
The TCPS device was installed on branches 27-28. The
TCSC devices were installed on branches 10-22 and 12-15.
The load demand is P = 283.4 MW and Q = j126.2 MVAR
[3]. Bus 1 is the reference bus.
The system has a total of 20 control variables as follows:
five active outputs of PV-bus units, six PV-bus voltage
magnitudes, four transformer-tap settings, two var-injection
values of shunt capacitors and three parameter values of
FACTS devices. Because the adjustable range of the
transformer-tap is 0.90 pu to 1.1 pu, and the shunt
admittance is 0.0 to j0.1 pu, the adjustable step size is 0.01
pu in the transformer-tap setting, and the changing step size
is j0.005 pu in the shunt admittance. The M values of the
two discrete variables above are both 20. The upper and
lower limits of the generator-bus and load-bus voltages are
0.95 pu and 1.05 pu, respectively. The limits of the installed
TCPS are taken − 50 ≤ φ ≤ 50 and the limits of the installed
TCSC are taken −0.4 X L ≤ X c ≤ 0.2 X L .
transferring the transmission loss into a penalty cost. ρ nV is
a weight factor of voltage deviation at bus n that is mainly
there for the purpose of transferring the voltage deviation
into a penalty cost. PDn is the load demand at bus n. PDt is
the total load demand of system. NB is the number of system
buses. Vref is a magnitude of reference voltage, in general,
Vref = 1.0 pu.
No
November 4 - 8, 2007, Kaohsiung, Taiwan
t=t+1
v imax = M i / 2 ,
• number of iterations=30.
No
Satisfy the stopping rule ?
• Selected Contingency Event
In the IEEE 30-bus system, a three-phase-to-ground
fault that occurs on line 6-7 near bus 7, the fault is cleared
0.370 second later coupled with the removal of line 6-7.
Yes
Select the best individual sgbest and decode
END
• Results and Discussion
Figure 4 shows the convergence tendency of the average
over 20 trials. The simulation results are summarized in
Table I. The optimal settings of control variables that are
obtained by the three proposed methods are shown in Table
II. In Table I, the average fitness obtained by the proposed
NCPSO-based method is always better than that obtained by
the PSO and EP. The NCPSO method has the best average
fitness of 4140. As seen in Table II, the NCPSO method has
the best fitness of 4108, thus implying a total generation cost
of $3996, a transmission loss of 4.98 MW and a summation
of bus voltage deviation of 0.45 pu. Moreover, Fig. 5 shows
the swing curves of all rotor angles of generator δ i during
the line 6-7 fault that is a stable situation. It indicates that
generator G5 has the largest rotor angle δ i = -70.790 at t =
Fig. 3. Operating procedures of the proposed NCPSO-based SCOPF
method
C. NCPSO-based SCOPF
In general, for power systems with a higher X/R ratio of
transmission line, the fast decoupled load flow (FDLF)
method has superior computation efficiency. To enhance the
effectiveness of the proposed method, the FDLF method is
employed to measure the fitness of the individual for the
acceptable solution quality. The search procedures of the
NCPSO-based SCOPF method are shown in Fig.3. The
proposed NCPSO method was compared with the traditional
PSO and EP in terms of solution quality and convergence
rate using the same fitness function and individual
definition.
The software was written in Matlab language and
109
The 14th International Conference on Intelligent System Applications to Power Systems, ISAP 2007
1.36(s) and satisfies the transient stability constraints
δ i <1000. After line 6-7 tripped, the bus voltage profile of
Case I
1.1
1.05
system shown in Fig. 6 reveals that each bus voltage is
within 0.95 pu to 1.05 pu.
1
Volt. (pu)
0.95
TABLE I
COMPARISON BETWEEN THREE METHODS IN IEEE 30-BUS SYSTEM
Method
Fitness
Ave.
NCPSO
PSO
EP
4140
4151
4159
Min.
(best)
Max.
(worst)
CPU time(sec.)
Average
CPU
time /generation
4108
4115
4124
4235
4238
4265
29.99
29.97
34.31
φ 27−28
NCPSO
PSO
EP
163.95
20.00
44.09
10.00
10.00
40.00
1.0421
1.0477
1.0500
0.9739
0.9899
1.0163
0.96
0.90
0.90
0.900
0.100
0.100
5.00
165.30
20.00
48.59
31.43
10.000
12.00
1.0471
1.0500
1.0307
0.9897
1.0500
0.9752
0.90
0.98
0.90
0.92
0.000
0.060
5.00
164.66
20.00
45.56
10.00
10.00
40.00
1.0312
0.9707
1.0500
1.0083
1.0277
1.0500
0.97
0.90
1.05
0.90
0.100
0.075
-5.00
X c (10−22 )
-4.00
-4.00
-4.00
X c (12 −15 )
0.00
-4.00
-4.00
4108
3996
4.98
0.45
4115
4019
4.27
0.41
4124
4008
6.49
0.40
Fitness
f
FT
PLoss
∑V −V
i
ref
0.8
0.75
0.7
0
NCPSO
PSO
EP
Fitness
4300
4250
4200
4150
5
10
15
20
Number of iterations
25
30
Fig. 4 Convergence tendency
Swing curves of generators in Case I
80
60
G11
G1
Rotor angle, deg.
G13
G2
G5
Method
Fitness
Ave.
NCPSO
PSO
EP
16278
16294
16303
-60
-80
-100
0
0.2
0.4
25
30
TABLE III
COMPARISON BETWEEN THREE METHODS IN IEEE 57-BUS SYSTEM
G8
-40
20
voltage profile of the system shown in Fig. 9 reveals that
each bus voltage is within 0.9 pu to 1.1 pu.
20
-20
15
Bus number
constraints δ i < 1000. After line 1-17 tripped, the bus
40
0
10
B. IEEE 57-Bus System
The system contains seven thermal units, 57 buses and
46 transmission lines. In addition, four series FACTS
devices (one TCPS devices and three TCSC devices) are
installed on it. The peak load demand is P = 1275.8 MW and
Q = j343.1 MVAR (i.e. average load × 1.02). Bus 1 is the
reference bus.
The system has a total of 35 control variables. The
upper and lower limits of the generator-bus and load-bus
voltages are 0.9 pu and 1.1 pu, respectively. The limits of
the installed TCPS are taken − 30 ≤ φ ≤ 30 and the limits of
the installed TCSC are taken −0.3 X L ≤ X c ≤ 0.1X L .
• Parameters of Algorithms
In this study system, through many experiments, the
results revealed that the appropriate values for c1a, c1b and c2
are 1.60, 0.45 and 2.05, respectively.
• Selected Contingency Event
In the IEEE 57-bus system, a three-phase-to-ground
fault occurs on line 1-17 near bus 17, the fault is cleared
0.160 second later coupled with the removal of line 1-17.
• Results and Discussion
Through 20 trials, the simulation results are summarized
in Table III. The optimal settings of control variables that are
obtained by the three proposed methods are shown in Table
IV. In terms of convergence rate and solution quality, as seen
in Fig. 7 and Table III, the NCPSO method is still superior to
the PSO and EP. As can be seen in Table III, the average
fitness of 16278 obtained by the proposed NCPSO-based
method is better than that obtained by the PSO and EP.
Moreover, in Table IV, the NCPSO method has the best
fitness of 16129, thus implying a total generation cost of
$15763, a transmission loss of 17.69 MW and a summation
of bus voltage deviation of 1.31 pu. Moreover, Fig. 8 also
shows the swing curves of all rotor angles of generator δ i
during the line 1-17 fault that is a stable situation. It
indicates that G12 generator has the maximum rotor angle
δ i = -34.540 at t = 0.54(s) It satisfies the transient stability
Case I
4100
5
Fig. 6 Bus voltage profile after removing the fault line
4400
4350
0.9
0.85
TABLE II
OPTIMAL SETTINGS OF CONTROL VARIABLES IN IEEE 30-BUS SYSTEM
Control Variable
Pg1
Pg2
Pg5
Pg8
Pg11
Pg13
V1
V2
V5
V8
V11
V13
Tp6-9
Tp6-10
Tp4-12
Tp27-28
Yh10
Yh24
November 4 - 8, 2007, Kaohsiung, Taiwan
0.6
0.8
1
1.2
Time, sec.
1.4
1.6
1.8
2
Fig. 5 Swing curves of all rotor angles of generator during fault occurrence
110
Min.
(best)
Max.
(worst)
CPU time(sec.)
Average
CPU
/generation
16129
16163
16268
16445
16451
16522
41.32
41.30
46.63
time
The 14th International Conference on Intelligent System Applications to Power Systems, ISAP 2007
new cognitive behavior of particle that can explore
effectively the search space and is superior to the PSO and
EP.
TABLE IV
OPTIMAL SETTINGS OF CONTROL VARIABLES IN IEEE 57-BUS SYSTEM
NCPSO
330.07
50.00
116.09
120.00
257.32
120.00
300.00
1.0500
1.0500
1.0441
1.0021
1.0012
1.0286
0.9985
1.00
0.95
1.00
0.95
0.90
0.94
0.93
0.96
1.00
0.96
0.90
1.08
0.90
0.90
0.90
0.065
0.060
0.095
-3.00
PSO
330.07
50.00
116.90
76.90
300.00
120.00
300.00
1.0500
1.0493
1.0500
0.9894
1.0045
0.9991
0.9999
0.99
0.95
0.99
0.96
0.90
0.90
0.95
0.97
0.95
0.94
0.93
1.04
0.90
1.00
1.10
0.025
0.055
0.045
-3.00
EP
333.09
84.82
120.70
83.78
256.17
119.09
300.00
1.0323
1.0497
1.0500
0.9898
1.0293
1.0302
0.9973
1.00
1.00
0.96
1.02
1.04
0.90
0.92
0.96
0.90
1.10
0.94
0.97
0.90
0.90
0.98
0.010
0.040
0.015
-1.00
X c (6 − 8 )
-3.00
-2.00
-2.00
X c ( 9− 55)
0.00
1.00
-2.00
X c ( 38−48 )
1.00
-2.00
1.00
16129
15763
17.69
1.32
16163
15803
18.09
1.14
16268
15799
21.89
1.29
φ 24− 26
f
Fitness
FT
PLoss
∑V −V
i
ref
IEEE 57-bus system
1.1
1
Voltage (p.u.)
Control Variable
PG1
PG2
PG3
PG6
PG8
PG9
PG12
V1
V2
V3
V6
V8
V9
V12
Tp4-18
Tp7-29
Tp9-55
Tp10-51
Tp11-41
Tp11-43
Tp13-49
Tp14-46
Tp15-45
Tp20-21
Tp24-25
Tp24-26
Tp32-34
Tp40-56
Tp39-57
Yh18
Yh25
Yh53
0.7
0.6
5
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
G1
30
[9]
Rotor angle, deg.
20
G2
G3
0
G6
G8
G9
[10]
-30
-50
[11]
G12
-40
0
0.5
1
1.5
[12]
Time, sec.
Fig. 8 Swing curves of all generators’ rotor angle during fault occurrence
VI.
15
20
25
30
35
Bus number
40
45
50
55
REFERENCES
40
-20
10
Fig. 9 Bus voltage profile after removing line 1-17
Swing curves of generators
-10
0.9
0.8
50
10
November 4 - 8, 2007, Kaohsiung, Taiwan
CONCLUSION
[13]
An efficient new constriction particle swarm
optimization (NCPSO) for solving the security-constrained
optimal power flow (SCOPF) with both the steady-state
security constraints and the transient stability constraints is
presented. The objective of SCOPF is defined not only to
minimize the real power loss of network and improve the
bus voltage profile under pre- or post-contingent states, but
also to satisfy the transient stability constraints during fault
occurrence. Therefore, the operation point of the SCOPF
will enhance obviously the security of system operation
even though the system suffers a specific contingency
impact. In addition, the proposed NCPSO method involves a
[14]
[15]
[16]
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