Security-Constrained Economic Scheduling of
Generation Considering Generator Constraints
Zwe-Lee Gaing, Member, IEEE
Abstract-- This paper proposes an efficient constriction particle
swarm optimization (CPSO) with mutation mechanism for solving
the steady-state economic dispatch (ED) problem with contingency
constraints and operating limits of series FACTS devices in power
systems. The objective of security-constrained economic dispatch
(SCED) is defined to not only minimize total generation cost but
also to enhance transmission security, reduce transmission loss, and
improve the bus voltage profile under pre-contingent and
post-contingent states. Many non-linear characteristics of the
generator, such as ramp rate limits and valve-point loading effects
are considered using the proposed method for practical generator
operation. The effectiveness of the proposed method is
demonstrated for the IEEE 118-bus system with series FACTS
devices, and it is compared with the other stochastic optimization
methods in terms of solution quality and convergence rate. The
experimental results show that the proposed CPSO method was
indeed capable of efficiently obtaining higher quality solutions in
SCED problems.
Rung-Fang Chang
Previous efforts in solving traditional ED problems have
employed various mathematical programming methods and
optimization techniques. In conventional numerical methods
for the solution of ED problems, such as gradient-based,
linear-programming, interior-point methods, an essential
assumption is that the incremental cost curves of the units
are monotonically increasing piecewise-linear functions.
Unfortunately, this assumption may render these methods
infeasible because of its non-linear characteristics in
practical systems. These non-linear characteristics of a
generator include discontinuous prohibited zones, ramp rate
limits, valve-point loading effect, whose cost functions are
not smooth or convex [12]-[14]. Recently, many global
optimization techniques known as genetic algorithms (GA),
simulated annealing (SA), evolutionary programming (EP),
and particle swarm optimization (PSO), has been
successfully used to solve the variant ED problems
[2][12]-[17]. However, the objective function is usually not
integrated with the security constraints or operating limits of
FACTS devices, thus resulting in the solution perhaps being
unsuitable for practical operations. In addition, premature
convergence may result in the local optima by obtaining
[18]-[22].
In this paper, an efficient constriction PSO (CPSO)
method with mutation mechanism for solving the
steady-state ED problem with security constraints and
operating limits of series FACTS devices is proposed. The
objective of SCED is defined to not only minimize total
generation cost but also to enhance transmission security,
reduce transmission loss, and improve the bus voltage
profile under pre-contingency and post-contingency states.
Many non-linear characteristics of the generator, such as
ramp rate limits and valve-point loading effects are
considered using the proposed method for practical
generator operation. The effectiveness of the proposed
method is demonstrated for the IEEE 118-bus system with
series FACTS devices, and it is compared with the other
stochastic optimization methods in terms of solution quality
and convergence rate.
Index terms-- economic dispatch, contingency analysis,
ramp-rate limit, flexible ac transmission systems (FACTS),
particle swarm optimization
I.
and
INTRODUCTION
C
urrently, the concept of performing the optimal plan
of power system operation with considering system
security assessment is positively presented to ensure
the system can secure operation without interruption to
customer service even though the system suffered the
contingency impact [1]-[8]. For the reason, installing the
suitable FACTS (Flexible AC Transmission Systems)
devices at key locations to increase the power-transfer
capability of transmission system and keep power-flow over
designed routes has been developed actively [9]-[11].
Therefore, to perform the optimal economic dispatch (ED)
scheduling, the control variables should include the power
output of generators and the parameters setting of FACTS
devices. In addition, the pre-protection strategy of system
and the security constraints also should be taken into
consideration for enhancing the system security. The
constraints included the transmission thermal limit and the
bus voltage limit, to expect an economy-security operation
model, regardless of whether the system is in a normal
operation state or a contingent state. Because the
contingency constraints are a fundamental element of
economy-security control, therefore, the security and
optimality of system operation should be treated
simultaneously for a power system economy-security control,
thus would add to the complexity of system operation
[3]-[8].
II.
PROBLEM DESCRIPTION
A. Security-Constrained ED
The steady-state SCED formulation can be stated as:
Min f (u ( 0 ) , x ( 0 ) )
(1)
u ,x
g ( k ) (u ( k ) , x ( 0 ) ) 0 ,
s.t.
for k 0,1,..., N c
(2)
h (u , x ) 0 , for k 0,1,..., N c
(3)
where is x a vector of control variables of pre-contingency,
such as generation of generator ( Pg ), while u is the vector
(k )
(k )
(0)
(o)
----------------------------------------------Z. L. Gaing and R. F. Chang are with the Department of Electrical
Engineering of Kao Yuan University, Kaohsiung, Taiwan 821 (e-mail:
zlgaing@ms39.hinet.net, Fax: 886-7-6077009).
of state variables, such as bus voltage ( V ) and reactive
1
TCSC device can permit to modify the reactance X L of
transmission line. The models of TCPS and TCSC are shown
in Fig. 1.
power output of generator ( Q g ) . Objective function (1) is
scalar. Equalities (2) are the conventional power equations.
Inequalities (3) are the limits on the control variables x, 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
post-contingency states for the Nc contingency cases. So, the
equality constraints g(o) change to g(k) to reflect the outage
equipment.
Performing the SCED scheduling, if a vector of control
variables x of pre-contingency while satisfying all equality
and inequality constraints during whole operation period,
regardless of whether the system is in a pre-contingent state
or a post-contingent state, is a feasible operation point that
can ensure the system to locate in the security region. In
addition, if the x within the security region can drive the
minimization of objective function f, it is the optimal
operation point that is the pursuing goal of this paper.
1 : 1ij
Vi
jXc
where Pgi is the current output power, and P
RL+ jXL
Vi
Vj
(b) TCSC
Fig. 1 Models of TCPS and TCSC
D. Objective Function
As mentioned previously, the objective of SCED is to
consider simultaneously the pre-protection strategy of
system, security constraints and operating limits of FACTS
devices. The control variables x ( 0 ) must be solved subject
to both the pre-contingency constraints ( u ( 0 ) , g ( 0 ) , h ( 0 ) ) and
the post-contingency constraints ( u ( k ) , g ( k ) , h ( k ) ) of the
selected contingency events. Hence, the SCED is expressed
as a non-convex programming problem.
Min f ( x ( 0 ) )
(7)
s.t.
i) power balance
NB
Y
Pi ( k ) 
(k )
ij
Vi ( k ) V j( k ) cos(i( k ) j( k ) ij( k ) ) 0 ,
(8)
j
1
NB
Y
Qi( k ) 
(k )
ij
Vi ( k ) V j( k ) sin(i( k ) j( k ) ij( k ) ) 0 , j N B
j
1
(9)
ii) unit operation constraints
max( Pgimin , Pgi( 0 ) DRi ) Pgi( 0 ) min( Pgimax , Pgi( 0 ) URi ) (10)
(5)
( 0)
gi
Vj
(a) TCPS
B. Operation Constraints of Generator
The unit generation output is usually assumed to be able
to be adjusted smoothly and instantaneously. Practically, the
operating range of all on-line units is restricted by their ramp
rate limits for forcing the units operation continually
between two adjacent specific operation periods [9]-[10][13].
In addition, the valve-point loading effect in the input-output
curve of unit is due to steam valve operation or vibration in a
shaft bearing. Hence, the two constraints of the unit
operation must be taken into account to achieve true
economic operation.
˙Ramp Rate Limit
According to [12] and [14], the inequality constraints
due to ramp rate limits for unit generation changes are given:
1) as generation increases
Pgi Pgi( 0 ) URi
(4)
2) as generation decreases
Pgi( 0 ) Pgi DRi
RL+ jXL
is the
previous output power. URi is the up ramp limit of the i-th
generator (MW/time-period); and DRi is the down ramp
limit of the i-th generator (MW/time period).
Qgimin Qgi( k ) Qgimax , i N g
iii) security constraints
V jmin V j( k ) V jmax , j N B
(11)
, m N l
(13)
S
(k )
Lm
S
max
Lm
iv) operating limits of FACTS devices
imin i( 0 ) imax
, i N TCPS
˙Valve-Point Loading Effect
The valve-point loading effect of thermal units should
be taken into consideration where the fuel cost function of
unit i is as follows.
Fi ai bi Pgi ci Pgi2 d i 
sin(ei ( Pgimin Pgi ))
(6)
X
min
cj
X
( 0)
cj
X
max
cj
, j N TCSC
(12)
(14)
(15)
where the power flow equations (8)-(9) are used as equality
constraints; the active and reactive power generation limits
(10)-(11), bus voltage limits (12), thermal capacity limits of
transmission lines (13) and operating limits of FACTS
devices (14)-(15) are used as inequality constraints.
where ai, bi, ci, di, and ei are the cost coefficients of unit i.
C. Function of Series FACTS Devices
According to their steady-state characteristics of
variant FACTS devices, the function of series controllers,
such as TCPS ( Thyristor-controlled phase shifter) and
TCSC( Thyristor-controlled series capacitor), 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  to
control the active power flow of transmission line. The
III.
CONSTRICTION PARTICLE SWARM OPTIMIZATION
WITH MUTATION MECHANISM
Let xi and vi denote the positions and the corresponding
flight speed (velocity) of the particle i in a continuous search
space, respectively. In a traditional PSO algorithm, the
particles are manipulated according to the following
2
If rand () pm then
equations [17].
vi( t 1) w 
vi(t ) c1r1i ( pbest ( t ) xi( t ) ) c2 r2i ( gbesti( t ) xi(t ) ) )
(16)
xi( t 1) xi( t ) vi( t 1)
(17)
where
t
: pointer of iterations (generations),
w
: inertia weight factor,
c1 , c2 : acceleration constant,
r1i, r2i : uniform random value in the range [0,1],
vi(t )
: velocity of particle i at iteration t,
x i( t )
pbest
xk xk (1 Gaussian()) .
IV.
: current position of particle i at iteration t,
: the previous best position of particle xi at
iteration t,
gbest (t ) : the best position among all individuals in the
population at iteration t,
vi( t 1) : new velocity of particle i that is limited to a
maximum
velocity
v max ,
DEVELOPMENT OF THE PROPOSED METHOD
Before employing the CPSO method to solve the SCED
problem, two definitions must be made as follows.
A. Individual String
In this paper, the individual is composed of continuous
control variable (as power output of generator) and discrete
control variable (as the value of FACTS parameters). The
individual x (j0 ) is defined as follows:
(t )
i
predefined
(21)
where Gaussian() is a Gaussian distribution function,
 is set to be 0.1.
x (j0 ) [ Pg(10 ) ,..., PgN( 0 ) , 1( 0 ) ,..., N( 0 ) , X c(10 ) ,..., X cN( 0 ) ] j
(22)
According to the power limits of the on-line
units randomly establish the initial
population x(0).
t=1
i.e.
vi( t 1) vimax ,
xi( t 1 ) : new position of particle i.
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 [21] to gain both speed up
convergence and escape local minima. The constriction PSO
(CPSO) algorithm with a new scheme of velocity updating
is as (18). The new scheme will replace the (16). And, in
(18), the relationship between the parameters is given by (19)
and (20). In (18),  is the constriction factor.
r
r
vi( t 1) 
(vi( t ) c1 1i ( pbest ( t ) xi( t ) ) c2 2i ( gbesti( t ) xi( t ) ) )
ri
ri
(18)
where
ri r1i r2i ,
(19)
2

, c1 c2 , 4 .
(20)
2  2 4
Perform CPSO operations using (18)- (20),
and (17)
New offspring population
Perform system steady-state contingency analysis
using security (N-1) criterion
No
Fitness
Satisfy all constraints
(8)-(15)?

Yes
t=t+1
Evaluate the fitness
using (23)
Stopping rule is satisfied ?
No
Yes
Select the best individual
Because the global best individual attracts all members
of the swarm in PSO, it is possible to lead the swarm away
from a current location by mutating a single individual if the
mutated individual becomes the new global best. This
mutation mechanism potentially provides a means both of
escaping local optima and speeding up the search. Therefore,
to enhance the effectiveness of CPSO, a mutation operator
of real-valued GA should be integrated with the CPSO [22].
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 (21) and
PGj( 0 )
END
Fig. 2 Operating procedures of the proposed CPSO-based SCED method
B. Fitness Function
In this paper, the objective of steady-state SCED is 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 fitness function is as (23).
If an individual x (j0 ) is a feasible solution and satisfies
xk [ xkmin , xkmax ] . Then the next offspring of the application
all constraints, its fitness will be measured by using the
fitness function f as in (23). Otherwise, its fitness will be
penalized with a very large positive constant (i.e. violates
the equality constraints (8)-(9) or the control variable
of the mutation operator is xi [ x1 ,..., xk ,..., x N ]i .
3
violates the inequality constraints (10)-(15)). The infeasible
individual will not be selected by the proposed scheme for
evolution in the next generation, so the proposed method can
converge rapidly.
FT ( x (j0 ) )
, x (j0 ) Security
f ( x (j0 ) ) 
, x (j0 ) Un sec urity
 
where
i
i
(23)
TABLE I
SYSTEM STATUS UNDER NORMAL OPERATION AND POST-CONTINGENCY.
NB
Ng
F ( x
FT 
The proposed method was compared with traditional
PSO and GA, in terms of solution quality and convergence
rate using the same fitness function and individual definition.
The software was written in Matlab language and executed
on a Pentium IV 1.8 GHz personal computer with 512MB
RAM.
( 0)
j
) w L Ploss ( x (j0 ) ) ( wnV 
Vk ( x (j0 ) ) Vref )
k
Study
Case
Load
(MW)
Case 1
3668.0
(100%)
3668.0
(100%)
4034.8
(110%)
(24)
Case 2
(25)
Case 3
Ng
w L Fi ( x (j0 ) ) / PDt
i
Normal operation
(Pre-contingency)
Line
Line flow (Mva)
L17-18
100.64
Post-contingency
Line Outage
L17-18
Overload
L15-17, L15-19
L8-30
124.36
L8-30
L5-8, L23-24
L8-30
121.26
L8-30
L5-8
Ng
wVn Fi ( x (j0 ) ) 
PDn ( x (j0 ) ) / PDt
B. Selected Contingency Events
From the result of contingency selection, three of the
most critical faults are proven such as line 17-18 outage, line
8-30 outage, and a heavy load demand (110%) with line
8-30 outage, respectively. Table I shows that the status of the
system under normal operation state (pre-contingency) and
post-contingency state. In Case 1, when line 17-18 faulted,
two lines were overloaded (line 15-17 and line 15-19). In
Case 2, when line 8-30 faulted, two lines were overloaded
(line 5-8 and line 23-24). In Case 3, when the heavy load
(3668.0*110%=4034.8MW) with line 8-30 outage, the, one
line was overloaded (line 5-8).
In this paper, we employed these selected contingency
events to test the performance of the proposed method.
C. Parameters of Algorithm
Through many experiments, the results revealed that the
appropriate values for c1, c2, and Pm are 1.65, 2.45, and 0.01,
respectively. They can yield an optimal evaluation value.
Therefore, the following parameters of CPSO are used:
• individual length= 41,
• population size= 30,
• c1=1.65, c2=2.45
• vimax Pgimax / 2 ,
(26)
i
wL is a weight factor, the purpose of which is mainly to
transfer the transmission loss into a penalty cost. wnV is a
weight factor of voltage deviation at bus n, the purpose of
which is mainly to transfer the voltage deviation into a
penalty cost. PDn is the load demand at bus n. PDt is the total
load demand of the system. Vref is a magnitude of reference
voltage, in general, Vref = 1.0 p.u.
C. CPSO-based SCED
Operating procedures of the proposed CPSO-based
SCED method is shown in Fig. 2. For a power system 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 maximum number of iterations is set at 30, and the
power mismatch accuracy is 0.001 p.u. in the FDLF method.
V.
NUMERICAL EXAMPLES AND RESULTS
To verify the effectiveness of the proposed method, the
IEEE 118-bus system with series FACTS devices was tested.
The test system mainly contains 36 thermal units, 118 buses
and 179 transmission lines. In addition, six series FACTS
devices (three TCPS devices and three TCSC devices) are
installed on it. The TCPS devices were installed on branches
24-72, 71-72 and 82-83. The TCSC devices were installed
on branches 5-8, 23-24 and 92-100.
The detailed characteristics of the 36 thermal units with
the ramp-rate limit and valve-point loading effect are shown
in Table AI. Bus 69 is the reference bus. The system has 41
control variables as the active outputs of 35 PV-bus units
and six parameter values of FACTS devices. The limits of
the installed TCPS are taken 5 0 50 and the limits of
• mutation rate is Pm=0.01,
• number of iterations=30.
D. Results and Discussions
First, to prove the effectiveness of the proposed
CPSO-based SCED method is superior to traditional PSO
and GA, the IEEE 118-bus system in normal operating state
is used to test and verify. Figure 2 shows the convergence
tendency of the average over 20 trials. The simulation results
are summarized in Table II, satisfying the all constraints of
the system. The optimal settings of the control variables that
are obtained by the three proposed methods are shown in
Table III.
the installed TCSC are taken 0.8 X L X c 0.2 X L .
The upper and lower limits of the PV-bus and load-bus
voltages are 0.9 p.u. and 1.1 p.u., respectively. In a stable
operation state, the load demand progressively increases
from 3204.1 MW (at am 10:00) to 3668.0 MW (at am 11:00).
The previous output power of unit Pgi( 0 ) is also shown in
TABLE II
COMPARISON BETWEEN THREE METHODS IN NORMAL OPERATING STATE
Table AI in Appendix.
4
Method
Ave.
Fitness
Min.
Max.
(best)
(worst)
CPSO
PSO
GA
48670
48680
48757
48373
48421
48555
48919
49028
49035
Ave. CPU time
/generation
(sec.)
26.31
26.02
27.78
4
loss of 73.99MW, and a summation of bus voltage deviation
of 1.73 p.u. This is because the CPSO with mutation
mechanism can avoid the premature convergence so that the
average fitness obtained by the proposed CPSO method is
superior to that obtained by the PSO and GA.
In regard to the convergence rate, as seen in Fig. 2 and
Table II, the CPSO method is also superior to the PSO and
GA. This is also an evidence that the CPSO method can
avoid the phenomenon of premature convergence.
Figure 3 shows that the bus voltage profile obtained by
the proposed CPSO method satisfies the bus voltage
constraints (0.9 p.u.~1.1 p.u.) while the system is suffering
the contingency impact.
118-bus system
x 10
CPSO
PSO
GA
5
4.98
Fitness
4.96
4.94
4.92
4.9
4.88
4.86
0
5
10
15
20
Number of generations
25
30
TABLE IV
COMPARISON OF VARIED CONTINGENCY EVENTS
Fitness
Ave. CPU time
Selected
/generation
Min.
Max.
Contingency
Ave.
(sec.)
(best)
(worst)
Events
Fig. 2. Convergence tendency of the evaluation value
TABLE III
OPTIMAL SETTINGS OF CONTROL VARIABLES IN NORMAL OPERATING STATE
USING THREE PROPOSED METHODS
Normal Operating State
Control variable
Pg1
Pg4
Pg8
Pg10
Pg12
Pg24
Pg25
Pg26
Pg27
Pg31
Pg40
Pg42
Pg46
Pg49
Pg54
Pg59
Pg61
Pg65
Pg66
Pg69
Pg70
Pg72
Pg73
Pg80
Pg87
Pg89
Pg90
Pg91
Pg99
Pg100
Pg103
Pg107
Pg111
Pg112
Pg113
Pg116
2472
CPSO
25.00
68.95
282.85
68.95
67.77
253.78
54.25
68.95
68.95
56.29
250.66
68.95
51.12
69.87
197.00
162.51
68.95
223.96
140.00
265.26
17.87
29.12
76.00
158.85
25.00
140.00
224.09
25.00
77.80
68.95
51.12
68.95
25.00
25.00
68.95
140.00
-2.00
PSO
80.12
68.95
264.86
68.95
40.94
259.00
54.25
68.95
68.95
59.06
258.85
68.95
64.63
74.59
197.00
68.95
72.58
204.38
140.00
262.79
76.00
15.20
76.00
154.90
25.00
140.00
256.20
25.00
78.01
68.95
64.63
68.95
25.00
25.00
68.95
140.00
-5.00
7172
5.00
4.00
5.00
8283
-5.00
5.00
-5.00
Case 1
Case 2
Case 3
GA
28.71
68.95
295.29
72.27
79.36
257.28
54.25
68.95
78.63
54.25
257.15
68.95
27.72
90.09
197.00
125.86
123.65
140.00
140.00
257.62
44.60
26.16
76.00
140.00
25.00
140.00
239.84
25.99
68.49
68.95
27.72
68.95
77.42
25.00
68.95
140.00
-5.00
48684
48825
53705
48476
48557
53396
48960
49211
54082
40.27
40.61
39.68
TABLE V
OPTIMAL SETTINGS OF CONTROL VARIABLE USING CPSO METHOD
Selected Contingency Events
Control variable
Case 1
Case 2
Case 3
0.20
0.20
0.20
X c ( 2324)
-0.60
-0.80
-0.80
Pg1
Pg4
Pg8
Pg10
Pg12
Pg24
Pg25
Pg26
Pg27
Pg31
Pg40
Pg42
Pg46
Pg49
Pg54
Pg59
Pg61
Pg65
Pg66
Pg69
Pg70
Pg72
Pg73
Pg80
Pg87
Pg89
Pg90
Pg91
Pg99
Pg100
Pg103
Pg107
Pg111
Pg112
Pg113
Pg116
2472
X c (92100)
-0.60
0.20
-0.80
7172
5.00
5.00
4.00
8283
0.00
1.00
-5.00
X c ( 58)
-0.40
0.10
-0.50
X c ( 2324)
-0.40
-0.60
-0.60
X c (92100)
0.00
-0.50
-0.10
f
48476
46748
80.66
1.74
48557
46849
78.64
1.74
53396
51516
81.20
1.86
X c ( 58)
f
Fitness
FT
PLoss
V V
i
ref
48373
46732
73.99
1.73
48421
46680
82.30
1.74
48555
46838
79.56
1.72
As can be seen in Table II, under a normal operating
state, the average fitness that obtained by the CPSO, PSO
and GA are 48670, 48680 and 48757, respectively. In Table
III, the CPSO method has the best fitness of 48373, thus
implying a total generation cost of $46732, a transmission
Fitness
FT
PLoss
V V
i
5
ref
36.92
68.95
271.03
68.95
25.00
257.40
61.42
68.95
68.95
54.25
259.16
68.95
54.92
72.68
197.00
154.67
83.54
162.81
170.31
262.42
69.17
16.57
76.00
162.08
25.00
140.00
242.72
25.00
74.48
77.06
54.92
68.95
36.60
25.00
68.95
140.00
-1.00
25.00
75.02
267.96
68.95
34.83
249.24
54.25
68.95
83.19
54.25
249.50
68.95
41.07
194.97
179.75
132.08
68.95
191.21
140.00
253.47
49.82
58.32
76.00
140.00
25.00
140.00
226.65
25.00
25.00
68.95
41.07
68.95
36.56
25.90
68.95
140.00
-5.00
26.75
73.26
273.98
68.95
78.64
259.43
65.92
83.26
76.22
123.17
261.49
68.95
65.60
172.54
197.00
156.47
68.95
226.69
140.00
273.44
30.97
15.20
76.00
182.90
25.00
140.00
216.36
36.60
77.88
68.95
65.60
81.19
28.36
26.45
81.17
194.00
-5.00
As evaluated previously, the proposed CPSO method
has the best performance in solving the SCED problem in
normal operating state that has been proved. Furthermore,
we also employed the method to solve the SCED problems
with varied contingency constraints. The simulation results
are summarized in Tables IV-V.
As with Case 1, the best fitness that obtained by the
CPSO method is 48476, thus implying a total generation
cost of $46748, a transmission loss of 80.66MW, and a
summation of bus voltage deviation of 1.74 p.u. As with
Case 2, the best fitness that obtained by the CPSO method is
48557, thus implying a total generation cost of $46849, a
transmission loss of 78.64MW, and a summation of bus
voltage deviation of 1.74 p.u. As with Case 3, the best
fitness that obtained by the CPSO method is 53396, thus
implying a total generation cost of $51516, a transmission
loss of 81.20MW, and a summation of bus voltage deviation
of 1.86 p.u.. These results are quite reasonable, thus
verifying the advantages of the proposed CPSO method.
VI.
limits of series FACTS devices is proposed. The objective of
SCED is defined to not only to minimize total generation
cost but also to enhance transmission security, reduce
transmission loss, and improve the bus voltage profile.
Many non-linear characteristics of the generator, such as
ramp rate limits and valve-point loading effects are
considered using the proposed method for practical
generator operation. Therefore, the solution obtained by the
proposed CPSO-based SCED method not only can minimize
the objective function in pre-contingent state but can also
enhance the security of the system even if the system suffers
one transmission line outage. The experimental results show
that the advantages of the CPSO-based method are greater
than the PSO and GA.
Due to the FACTS devices in a modern meshed network
can be an alternative to reduce the flows in heavily loading
lines, resulting in an increased loadability, low system loss,
improved stability of the network, reduced cost of
production, and fulfilled the security requirements by
controlling the power flows in the network. Therefore, a
steady-state SCED problem with considering the parameter
settings of FACTS devices will be a more valuable
investigation.
CONCLUSION
An efficient constriction PSO (CPSO) method with
Gaussian mutation mechanism for solving the steady-state
SCED problem with contingency constraints and operating
Voltage profile (Pre-contingency)
1.1
Volt. (p.u.)
1
0.9
0.8
0.7
0.6
10
20
30
40
50
60
70
Bus Number
80
90
100
110
Fig.3. Bus voltage profile of system using CPSO method
1997.
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Zwe-Lee Gaing(M’
02) received his M.S. and Ph.D. degree from National
Sun Yat-Sen University, Kaohsiung, Taiwan in 1992 and 1996, respectively.
Currently, he works as a professor in the Electrical Engineering Department
at Kao Yuan University, Kaohsiung, Taiwan, R.O.C. His research interests
are in the field of artificial intelligence with application to power system
operation and control.
Rung-Fang Chang received his B.S. degree in electrical engineering from
the Chung Yuan Christian University, Chung-Li, Taiwan in 1990 and
received the M.S. degree and the Ph.D. degree from the National Sun
Yat-Sen University, Kaohsiung, Taiwan in 1992 and 2002 respectively. He
is an associate professor of the Department of the Electrical Engineering,
Kao Yuan University, Lu-Chu, Taiwan. His research interests are
optimization of power system and load research.
Appendix
TABLE AI
GENERATING UNIT CAPACITY AND COEFFICIENTS IN IEEE 118-BUS SYSTEM
Bus
No.
1
4
8
10
12
24
25
26
27
31
40
42
46
49
54
59
61
65
66
69
70
72
73
80
87
89
90
91
99
100
103
107
111
112
113
116
Pgi( 0 )
25.00
70.00
220.00
60.00
65.00
225.00
54.25
68.95
68.95
54.25
212.00
68.95
20.00
68.95
140.00
80.00
68.95
140.00
140.00
270.00
30.00
30.00
50.00
140.00
25.00
140.00
165.00
50.00
40.00
70.00
16.00
68.95
25.00
25.00
68.95
140.00
Pgimin
25.00
68.95
100.00
68.95
25.00
100.00
54.25
68.95
68.95
54.25
100.00
68.95
15.20
68.95
68.95
68.95
68.95
140.00
140.00
100.00
15.20
15.20
15.20
140.00
25.00
140.00
100.00
25.00
25.00
68.95
15.20
68.95
25.00
25.00
68.95
140.00
Pgimax
100.00
197.00
400.00
197.00
100.00
400.00
155.00
197.00
197.00
155.00
400.00
197.00
76.00
197.00
197.00
197.00
197.00
350.00
350.00
400.00
76.00
76.00
76.00
350.00
100.00
350.00
400.00
100.00
100.00
197.00
76.00
197.00
100.00
100.00
197.00
350.00
Qgimin
-55.00
-120.00
-300.00
-120.00
-55.00
-300.00
-50.00
-120.00
-120.00
-50.00
-300.00
-120.00
-50.00
-120.00
-120.00
-120.00
-120.00
-200.00
-200.00
-300.00
-50.00
-50.00
-50.00
-200.00
-55.00
-200.00
-300.00
-55.00
-55.00
-120.00
-50.00
-120.00
-55.00
-55.00
-120.00
-200.00
Qgimax
100.00
180.00
300.00
180.00
100.00
300.00
140.00
180.00
180.00
140.00
300.00
180.00
50.00
180.00
180.00
180.00
180.00
200.00
200.00
300.00
50.00
50.00
50.00
200.00
100.00
200.00
300.00
100.00
100.00
180.00
50.00
180.00
100.00
100.00
180.00
200.00
S gimax
110.0
200.0
420.0
200.0
110.0
420.0
160.0
200.0
200.0
160.0
420.0
200.0
80.0
200.0
200.0
200.0
200.0
360.0
360.0
420.0
80.0
80.0
80.0
360.0
110.0
360.0
420.0
110.0
110.0
200.0
80.0
200.0
110.0
110.0
200.0
360.0
7
URi
60
50
80
50
60
80
50
50
50
50
80
50
76
50
50
50
50
70
70
80
76
76
76
70
60
70
80
60
60
60
76
50
60
60
50
70
DRi
100
90
120
90
100
120
80
90
90
80
120
90
76
90
90
90
90
110
110
120
76
76
76
110
100
110
120
100
100
100
76
90
100
100
90
110
ai
bi
ci
190.00
220.00
240.00
220.00
190.00
240.00
200.00
220.00
220.00
200.00
240.00
220.00
200.00
220.00
220.00
220.00
220.00
220.00
220.00
240.00
200.00
200.00
200.00
220.00
190.00
220.00
240.00
190.00
190.00
220.00
200.00
220.00
190.00
190.00
220.00
220.00
12.10
10.50
7.10
10.52
12.10
7.10
11.20
10.50
10.53
11.21
7.10
10.50
11.10
10.50
10.50
10.50
10.50
8.50
8.50
7.10
11.20
11.20
11.20
8.50
12.10
8.50
7.10
12.10
12.10
10.50
11.20
10.50
12.10
12.10
10.50
8.50
0.0075
0.0080
0.0072
0.0081
0.0074
0.0071
0.0096
0.0080
0.0082
0.0095
0.0071
0.0080
0.0091
0.0080
0.0080
0.0080
0.0080
0.0090
0.0090
0.0072
0.0091
0.0092
0.0092
0.0090
0.0074
0.0090
0.0071
0.0075
0.0076
0.0080
0.0091
0.0080
0.0074
0.0075
0.0080
0.0090
di
51.1331
61.2111
80.3210
61.2111
51.1331
80.3210
32.7233
61.2111
61.2111
32.7233
80.3210
61.2111
20.1012
61.2111
61.2111
61.2111
61.2111
42.5122
42.5122
80.3210
20.1012
20.1012
20.1012
42.5122
51.1331
42.5122
80.3210
51.1331
51.1331
61.2111
20.1012
61.2111
51.1331
51.1331
61.2111
42.5122
ei
0.0600
0.0210
0.0200
0.0210
0.0600
0.0200
0.0410
0.0210
0.0210
0.0410
0.0200
0.0210
0.0110
0.0210
0.0210
0.0210
0.0210
0.0120
0.0120
0.0200
0.0110
0.0110
0.0110
0.0120
0.0600
0.0120
0.0200
0.0600
0.0600
0.0210
0.0110
0.0210
0.0600
0.0600
0.0210
0.0120