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Transmission Line Performance Indices Calculation Based on Voltage Stability Criterion Komson Daroj

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Transmission Line Performance Indices Calculation Based on Voltage Stability Criterion Komson Daroj
70
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007
Transmission Line Performance Indices
Calculation Based on Voltage Stability
Criterion
Komson Daroj1 , Bundhit Eua-Arporn2 , and Sotdhipong Phichaisawat3 , Non-members
ABSTRACT
In this paper, three performance indices of a transmission line i.e., line security margin, line security index and line severity index are defined and proposed
to use for transmission planning purpose. These indices can be obtained with slight modification from
the power system loadability limit calculation, which
is normally limited by a static voltage stability constraint. A repetitive power flow is employed without loss of generality. Sensitivity factors of apparent
power flow at the receiving end bus are adopted to
predict power flow direction of each transmission line
corresponding to a specified scenario of generation
and demand in-creased. The proposed performance
indices provide information regarding severe transmission paths that need for expansion or upgrading.
A six-bus system is used to test with satisfactory results.
Keywords: Security Margin, Security Index, Severity Index, Loadability, Voltage Stability, Sensitivity
Factors, Pre-dictor-Corrector scheme, P-Q Plane
1. INTRODUCTION
Optimal expansion of transmission systems is a
crucial issue for power system planning. The main
objective is not only to provide transmission path
to transfer the energy from generation to load areas. The challenge is to compromise between system
reliability and costs, e.g. short term operation and investment costs [1]-[3]. In another word, it should be
planned to cope with the future demand with high
quality of services. However, satisfactory degree between reliability and revenue return rate from investment and operating costs is often questionable. This
can be formulated as a multiobjective optimization
problem and can be solved using some techniques
e.g., linear programming [4] dynamic programming
Manuscript received on July 6, 2006 ; revised on September
6, 2006.
This work is supported by The Center of Excellence in Electrical Power Technology (CEPT) Chulalongkorn University,
Thailand.
1 The author is with Ubonratchathani University, Ubonratchathani, Thailand; E-mail: [email protected]hoo.com
2,3 The authors are with Chulalongkorn University,
Thailand;
E-mail:
[email protected] and [email protected]
[5]-[6], and nonlinear programming [7]. For countries whose electric supply industry has not been fully
deregulated, e.g., Thailand, generation and transmission system expansion is conducted under a Power
Development Plan (PDP), which is studied and revised annually according to a long term load forecast
program [8]. For a restructured system, pricing of
transmission services and congestion charges cannot
provide a signal to convince investors for investing in
these infrastructures. In extremis, inadequate or delay of investment in new infrastructures may cause
a serious event as the case of 2003 blackout in the
North of US [9]-[10]. Therefore, transmission system
security is a very essential issue in power system operation.
To verify security of a system, loadability limit of a
power system is an index generally used to serve such
purpose. Loadability limit of a power system can be
normally defined as the maximum demand that can
be supplied by the system corresponding to specified
scenarios of increasing generation and demand, meanwhile satisfying system con-straints [11]-[12]. Traditionally, continuation power flow [13]-[16], sequential
power flow [17], and bifurcation theory [18]-[21] are
the methodologies used to calculate the system limit.
However, this calculation always limits by voltage
stability criterion occurred on transmission system.
Thus, adequacies of transmission paths to transfer the
energy from generation to load areas play an important role to enhance loading capability of a system.
However, the results obtained from the traditional
methods e.g., the system load-ability limit and the
weakest bus, cannot provides information regarding
severity of a transmission system. Hence, they cannot be adopted for transmission planning purpose.
For planning purposes, severity of a transmission system should be verified and presented through some
performance indices containing useful information regarding loading capability. In addition, it is beneficial
if these indices can be obtained with slightly modify
from the traditional methods.
In this paper, we proposed three transmission line
performance indices; line security margin, line security index and line severity index, to use for transmission planning. These performance indices can be
calculated directly from a current operating point.
Thus, the traditional methods e.g., a continuation or
repetitive power flow and bifurcation analysis can be
Transmission Line Performance Indices Calculation Based on Voltage Stability Criterion
used with slightly modification. The proposed indices
provide information concerning loading condition of
each transmission line, which is useful information
for transmission expansion or upgrading study. Formulation of the proposed method adopts the voltage
instability condition, presented on each transmission
line P-Q plane. The method is tested with a six-bus
system with satisfied results.
2. BACKGROUND THEORY
71
(b1 + jb2 )(Pijr − jQijr )
Vj ∠ − δ j
(4)
Rearrange (4) resulting in (5).
Vi ∠δi = (a1 + ja2 )Vj ∠δj +
Vi Vj = (a1 Vj 2 + b1 Pijr + b2 Qijr ) + j(a2 Vj 2 + b2Pijr
−b1Qijr )
2. 1 Voltage Stability Limit on a Transmission
Line
Consider the equivalent π model of a transmission
line connected between bus − i and bus − j as shown
in Fig.1.
(5) can be rewritten as shown in (6).
c1 Vj 4 +(c2 Pijr +c3 Qijr −Vi2 )Vj 2 +c4 (Pijr2 +Qijr2 ) = 0
(6)
where c1 = a12 + a22 , c2 = 2(a1 b1 + a2 b2 ), c3 =
2(a1 b2 + a2 b1 ) and c4 = b12 + b22
a(Vj 2 )2 + b(Vj 2 ) + c = 0
Fig.1: The π model transmission line connected between bus − i and bus − j
We define Vi ∠δi and Vj ∠δj as voltage magnitudes
of bus − i and bus − j respectively. The series
impedance is denoted by Z, whereas half of the line
charging susceptance is denoted by Y c. The apparent
power flows from bus−i to bus−j and vice versa is denoted by Pij + jQij and Pji + jQji respectively. For
an interested line, we assume that real power flows
from bus − i to bus − j. Thus, fictitious load current and apparent power at the receiving end, r, are
Iijr and Pijr + jQijr respectively. Obviously, the fictitious load is equal to the apparent power flowing into
bus − j.
We can formulate the relationship between injected current and voltage at any buses based on generalized ABCD parameters as follows:
Vi ∠δi = AVi ∠δj + BIj
(1)
where A = 1 + ZY c and B = Z. The complex
form of A and B can be expressed as shown in (2).
A = a1 + ja2 and B = b1 + jb2
(2)
The receiving end current, Iijr , can be expressed in
(3).
Iijr = (Pijr − jQijr )/V j∠ − δj
(3)
Substitute A and B from (2) and Iijr from (3) into
(1) resulting in (4).
(5)
(7)
where a = c1 , b = c2 Pijr + c3 Qijr − Vi2 and c4 (Pijr2 +
Qijr2 ).
The solution of (6) is the square of the receiving
end voltage. Thus the receiving end voltage can be
calculated from (8).
s
√
−b ± b2 − 4ac
Vj =
(8)
2a
There are two solutions for (8) i.e., the lower solution lies on the lower part of the P-V curve and is
unstable [22]. Thus, the available solution is a stable
one on the upper half, which can be expressed as in
(9).
s
√
−b − b2 − 4ac
Vj =
(9)
2a
The point where the two trajectories, i.e. stable
and unstable lines, are joined is the nose or bifurcation point. In addition, this is the point where the
maximum power can be transferred, which is the condition in (10).
b2 − 4ac = 0
(10)
Substitute coefficients of the quadratic equation
from (7) into (10) and rearrange, we obtain (11).
(c22 − 4c1 c4 )Pijr2 + (c32 − 4c1 c4 )Qijr2
−2c2 Vi2 Pijr − 2c3 Vi2 Qijr + 2c2 c3 Pijr Qijr + Vi4 . (11)
The relationship between Pijr and Qijr of (11) is a
locus of the collapsing point on the P-Q plane which
separates the operating points into feasible and infeasible regions. To clarify, we rewrite (11) as shown in
(12)
72
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007
E1 (P
c
ijr
)2 +E2 (Q
c
ijr
)+E3 P
c
ijr
+E4 Q
c
ijr
+E5 P
c
ijr
Q
E6 = 0
c
ijr
+
(12)
where the superscript c is denoted for the collapsing point and E1 = (c22 − 4c1 c4 ), E2 =
(c32 − 4c1 c4 ), E3 = −2c2 Vi2 , E4 = −2c3 Vi2 , E5 =
2c2 c3 , E6 = Vi4 .
In [23], for a given P c we can calculate Q c ,
ijr
ijr
or vice versa, which satisfies (12) from (13) and (14)
respectively.
c5 (Q
c8 (P
c
ijr
c
ijr
)2 + c6 Q
c
+ c7 = 0
(13)
+ c10 = 0
(14)
ijr
)2 + c9 P
c
ijr
where
c5 = (c32 − 4c1 c4 ), c6 = 2c3 (c2 P
c
ijr
c7 = (c32 − 4c1 c4 )(P
c
ijr
)2 − 2c3 Vi2 P
c8 = (c32 − 4c1 c4 ), c9 = 2c2 (c3 Q
c
ijr
c10 = (c32 − 4c1 c4 )(Q
− Vi 2 )
c
ijr
c
ijr
+ Vi 4
− Vi 2 )
)2 − 2c3 Vi2 Q
c
ijr
+ Vi 4
Occasionally, the term c5 or c8 in (13) and (14) is
zero. In such case, the above equations are reduced to
linear form. Then Q c = −c7 /c6 and P c = −c10 /c9 .
ijr
ijr
3. CALCULATION FRAMEWORK
The proposed method consists of four main parts,
i.e. in-creasing of generation and demand scenarios,
gradient vec-tor formulation, collapsing point calculation and definitions of transmission line performance
indices. Details of each part are presented below.
3. 1 Scenarios of Generation and Demand Increased
We consider firstly a set of buses in the system
which is defined as ∆ = {s, G, L}, where s refers to
the slack bus, G is a set of generation buses, and L is a
set of load buses. ∆PG and ∆QG are defined as a set
of an incremental change of real and reactive power
injected at generation buses, ∆PL and ∆QL are a set
of an incremental change of real and reactive power
injected to load buses. An incre-mental change of real
and reactive transmission loss is de-noted by ∆Sloss .
If the load change of ∆PL + j∆QL occurs, all the
genera-tion buses must contribute their power for the
consequences. This relationship can be expressed in
(15).
∆PG +j∆QG +∆PL +j∆QL +∆Ps +j∆Qs = ∆Sloss
(15)
We define λn as a vector indicating the contribution in-dices for the increase of generation and demand as described in (16).
λn = (λs , λG , λL ) = (λs , λ1 , . . . , λG , λG+1 , . . . , λL )
(16)
The contribution indices represent the change of
generation and demand of any bus in a system. If we
assume the sum of contribution indices of generation
and demand sides is zero, it implies that the change
of real power loss is compensated by a selected slack
bus.
3. 2 Gradient Vector of the Increased Line
Flow
Based on Fig.1, the real and reactive power flow
at the receiving end bus can be expressed in (17) and
(18) respectively.
Pijr = Vj 2 Yij cos θij − Vi Vj Yij cos (θij + δi − δj )
(17)
Qijr = Vi Vj Yij sin (θij + δi − δj )−Vj 2 (Yij sin θij − Yc )
(18)
where the terms Yij ∠θij represents elements of the
bus admittance matrix.
The terms need to be calculated first are the real
and reactive power sensitivity factors flowing into the
receiving end bus compared to the real and reactive power injected into the generation bus and load
bus, i.e. ∂Pijr /∂Pm , ∂Qijr /∂Pm , ∂Pijr /∂Qk and
∂Qijr /∂Qk . The subscripts m represents all buses
except the slack bus, and k represents all load buses.
All the sensitivity factors can be expressed by (19)(22).
X · ∂Pijr ∂Pu ¸
∂Pijr
=
·
∂δm
∂Pu ∂δm
S
u,m²G L
·
¸
X
∂Pijr ∂Qw
+
·
(19)
∂Qw ∂δm
S
m²G
L,w²L
∂Pijr
=
∂Vk
X
u²G
S
·
L,k²L
∂Pijr ∂Pu
·
∂Pu ∂Vk
¸
∂Qw
·
∂Vk
¸
X · ∂Pijr
+
∂Qw
w,k²L
X · ∂Qijr ∂Pu ¸
∂Qijr
=
·
∂δk
∂Pu ∂δm
S
u,m²G L
·
¸
X
∂Qijr ∂Qw
+
·
∂Qw ∂δm
S
m²G
L,w²L
∂Qijr
=
∂Vk
X
u²G
S
L,k²L
·
X · ∂Qijr
+
∂Qw
w,k²L
∂Qijr ∂Pu
·
∂Pu ∂Vk
¸
∂Qw
·
∂Vk
(20)
(21)
¸
(22)
Transmission Line Performance Indices Calculation Based on Voltage Stability Criterion
Equations (19) and (20) can be represented by (23).
#
"
" ∂P
#
∂Pijr
∂δm
∂Pijr
∂Vk
J3 is a k×m matrix. The diagonal and off-diagonal
elements are shown in (35) and (36) respectively.
ijr
= [J]T
∂Pm
∂Pijr
∂Qk
(23)
Equations (21) and (22) can be represented by
(24).
"
#
" ∂Q
#
∂Qijr
∂δm
∂Qijr
∂Vk
73
n
X
[
∂Qi
=
Vi Vj Yij cos(θij −δi +δj ), i²G L (35)
∂δi
j=1,6=i
∂Qi
= −Vi Vj Yij cos(θij − δi + δj ),
∂δi
ijr
= [J]T
∂Pm
∂Qijr
∂Qk
(24)
The terms ∂Pijr /∂δm , ∂Pijr /∂Vk , ∂Qijr /∂δm and
∂Qijr /∂Vk are zero for every bus except bus − i and
bus − j, of which the values can be obtained from
(25)-(30).
(36)
J4 is a k × k matrix, of which the diagonal and
off-diagonal elements are expressed by (37) and (38).
n
X
∂Pi
= 2Vi Yij sin(θii )−
Vj Yij sin(θij −δi +δj ), i²L
∂Vi
j=1,6=i
∂Pijr
∂Pijr
=−
= Vi Vj Yij sin(θij + δi + δj )
∂δi
∂δj
∂Pijr
= −Vj Yij cos(θij + δi + δj )
∂δi
(37)
(25)
∂Qi
= −Vi Yij sin(θij − δi + δj )
∂Vj
(26)
∂Pijr
= 2Vj V ij cos(θij )−Vj Yij cos(θij +δi +δj ) (27)
∂Vi
Using (31)-(38), we can rewrite (23) and (24) into
(39) and (40) respectively.
" ∂P
# ·
¸−1 " ∂Pijr #
ijr
J
J
1
3
∂Pm
∂δm
=
(39)
∂Pijr
∂Pijr
J2 J4
∂Q
∂Vk
k
∂Qijr
∂Qijr
=−
= Vi Vj Yij cos(θij + δi + δj ) (28)
∂δi
∂δj
∂Qijr
= Vj Yij sin(θij + δi + δj )
∂Vi
(29)
∂Qijr
= Vi Vij sin(θij +δi −δj )−2Vj (Yij sin(θij )−Yc )
∂Vj
(30)
[J] is a conventional Jacobian matrix, which can be
divided into J1, J2, J3 and J4. J1 is an mxm matrix,
of which the diagonal and off-diagonal elements are
presented in (31) and (32).
(38)
"
#
∂Qijr
∂Pm
∂Qijr
∂Qk
·
=
J1
J2
J3
J4
¸−1 "
∂Qijr
∂δm
∂Qijr
∂Vk
#
(40)
We can solve (39) and (40) to obtain ∂Pijr /∂Pm ,
∂Qijr /∂Pm , ∂Pijr /∂Qk and ∂Qijr /∂Qk , which can
be used to calculate the gradient vector of each receiving end line flow, denoted by ∆Pijr + j∆Qijr , of
which the real and imaginary parts can be expressed
in (41) and (42).
∆Pijr =
·
X
m²G
S
L
¸ X·
¸
∂Pijr
∂Pijr
· ∆Pm +
· ∆Qk
∂Pm
∂Qk
k²L
(41)
∂Pi
=
∂δi
n
X
Vi Vj Yij sin(θij − δi + δj ), i²G
[
L (31)
∆Qijr =
j=1,6=i
∂Pi
= −Vi Vj Yij sin(θij − δi + δj ),
∂δi
m²G
(32)
J2 is a m × k matrix, of which the diagonal and
off-diagonal elements are shown in (33) and (34) respectively.
n
X
∂Pi
= 2Vi Yii cos(θii )+
Vj Yij cos(θij −δi +δj ), i²L
∂Vi
j=1,6=i
(33)
∂Pi
= −Vi Yij cos(θij − δi + δj )
∂Vj
(34)
·
X
S
L
¸ X·
¸
∂Qijr
∂Qijr
· ∆Pm +
· ∆Qk
∂Pm
∂Qk
k²L
(42)
The terms ∆Pm and ∆Qk in (41) and (42) are
referred to an incremental change of the real and reactive power at buses m and k respectively. They can
be calculated by (43) and (44).
[
∆Pm = λm · ∆PL , m²G L
(43)
∆Qk = λk · ∆QL , k²L
(44)
Substitute ∆Pm and ∆Qk from (43) and (44) into
(41) and (42), then rearrange these equations to result
in (45) and (46).
74
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007
∆Pijr =
·
X
m²G
S
L
¸
∂Pijr
· λm ·∆PL +
∂Pm
X · ∂Pijr
∂Qk
k²L
¸
· λm
·∆QL
(45)
3. 4 Transmission Line Performance Indices
The transmission line performance indices proposed in this paper i.e., line security margin, line security index and line severity index, can be defined
as below.
¸
¸ 3.4...1 The Line Security Margin
X · ∂Qijr
∂Qijr
The line security margin is defined as the norm of
· λm ·∆PL +
· λm
∆Qijr =
∂Pm
∂Qk
S
the
vector starting from the current operating point
k²L
m²G L
to the predicted collapsing point as illustrated in
·∆QL
(46)
Fig.3. The line security margin can be stated as
It should be noted again that the gradient vector; (51).
∆Pijr +j∆Qijr , is calculated for each iteration, from
°
°
which ∆Spredict in (16) can be obtained.
°
°
ijr
c − P t , Q c − Q t )°
LSMij = °
(P
(51)
° ijr
ijr
ijr
ijr °
3. 3 The Predicted Collapsing Point
³
´
The predicted collapsing point, Pc , Qc , is the
·
X
ijr
ijr
intersec-tion between the voltage stability curve, defined in (11), and the line extended from the gradient
vector, which is ex-pressed in (47).

Qc
ijr
=


∆Qt
ijr 
∆Pt
Pc
ijr
+ Qt
ijr
ijr
∆Qt
−
ijr
∆Pt
ijr

Pt 
ijr
(47)
The (47) can be simplified to (48)
Qc
ijr
where M =
∆Qt
ijr
∆Pt
= M Pc
ijr
+C
and C = Qt
ijr
ijr
(48)
−
∆Qt
ijr
∆Pt
ijr
Pt .
Fig.3: The security margin on P-Q plane
3. 4...2 Line and switch models
ijr
The calculation method to obtain the predicted
collapsing point can be illustrated in Fig.2.
The line security index is defined as the ratio of
the line security margin to the apparent power of the
predicted col-lapsing point, which can be expressed
by (52).
°
°
°
°
°P c − P t , Q c Q t °
° ijr
ijr ijr °
ijr
°
°
LSIij =
(52)
°
°
°P c , Q c °
ijr
ijr
3.4...3 The Line Severity Index
Fig.2: The predicted collapsing point calculation
For convenience, we rewrite (11) to (49).
E1 P 2 + E2 Q2 + E3 P + E4 Q + E5 P Q + E6 = 0 (49)
The line severity index can be defined as ratio of
the norm of the gradient vector to the line security
margin as shown in (53).
°
°
°
°
°P t , Q t °
° ijr ijr °
°
LSvIij = °
(53)
°
°
°P c − P t , Q c Q t °
°
°
ijr
ijr
ijr
ijr
Substitute (48) into (49) resulting in (50).
3. 5 Flowchart
2
2
(E1 +E1 M +E5 M )P +(2E2 M C+E3 +E4 M +E5 C)P
+(E2 C 2 + E4 C + E6 )
(50)
Substitute the feasible solution of (50) into (48),
finally we obtain the predicted collapse point.
The overall calculation procedures presented in
3.1-3.4 can be summarized as follow. First, we calculate the base-case power flow. For a current operating point, generation and load scenarios must be
defined beforehand before go to the next step. Then,
Transmission Line Performance Indices Calculation Based on Voltage Stability Criterion
for a specified line, a P-Q curve, a gradient vector,
the predicted collapsing point, and the transmission
line performance indices can be calculated sequentially. A P-Q curve and gradient vector of each line
can be conducted using (12) and (45) and (46) respectively. Then, a predicted collapsing point can
be calculated as present in section 3.3. Finally, three
transmission line performance indices can be obtained
from (51)-(53). It should be noted that, these indices
are calculated at the current operating point according to a defined scenario. To perform calculation at
the further operating point, power flow solution must
be solved and used as a base-case of the next current
operating point. The process may be repeated until
diverging of the power flow calculation is found. The
calculation procedures presented above can be shown
in Fig.4.
75
The simulation starts from a base-case with 100%
load. Then total generation and demand are increased proportionally until reaching the collapsing
point. For simplicity, a repetitive power flow is
adopted. Generation and transmission thermal limit
are not considered. This is a traditional assumption
to smooth the loading trajectories by neglecting the
automatic actions of controllers and protective equipments. The obtained results at some loading points
are presented. At the beginning, it found that the
gradient vector and the proposed performance indices
change continuously. However, at the verge to the collapsing point line No.5 connected between bus-2 and
4 hits the limit first and then returns to the lower
portion of the P-V curve. This incident results in the
fluctuation of gradient vector direction, which is calculated from sensitivity factors as described in section 3.2. The line security margin, security index
and severity index defined by (51), (52) and (53) are
shown in Tables 1, 2 and 3 respectively.
It is clear that bus − 4 is the weakest bus in this
study due to the reason that some lines, e.g. lines
No. 2 and 5 hit voltage stability limit before the others. In addition, it can be seen from all indices. From
Table 1, security margin of these two lines are drop
close to zero, 0.03 and 0.13 MVA respec-tively. Similar results are also shown in Table 2 for line security
index, which are 0.0002 and 0.0005. It can be found
that line No.2 is more severe than line No.5 since
the severity index of line No.2 is 10772.7 whereas line
No.5 is 2513.8 as shown in Table 3. For the lines connected be-tween generator buses, i.e. lines No.1 and
4 are safe from hitting voltage stability limit especially in the light load condition. It should be noted
here that, at the proximity to the collapsing point,
direction of the gradient vector cannot be calculated
precisely. Thus, most of the security margin for all
the lines increase rapidly. Accordingly, the security
margin in the direction of gradient vector alone is
not suit-able to evaluate the capability of line loading. However, this effect is less impact to line security
and line severity indices.
5. CONCLUSIONS
Fig.4: Flowchart of the overal calculation procesures
4. SIMULATION AND RESULTS
The proposed method is tested with a modified 6bus system [24]. In this system, buses−1, 2 and 3 are
generator buses, whereas buses−4, 5 and 6 are load
buses. All the required bus and line data are available
in the appendix.
Three performance indices of a transmission line
i.e., line security margin, line security and line severity indices are defined in this paper. These indices
are formulated based on a static voltage stability criteria defined on a transmission line P-Q and can be
computed from a process of calculating the maximum
loading capability of power system with slightly modification from the traditional methods. Additionally,
the proposed method provides information to evaluate severity of a transmission system, which can be
used for transmission planning or upgrading purpose.
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ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007
Table 1: The Line Security Margin
Line
No.
1
2
3
4
5
6
7
8
9
10
11
conecting
bus - bus
1
2
1
4
1
5
2
3
2
4
2
5
2
6
3
5
3
6
4
5
6
5
100
221.83
68.23
48.35
186.95
127.40
48.14
91.75
57.15
227.09
54.48
60.95
110
221.55
61.29
44.07
185.36
114.46
43.87
88.02
52.11
218.61
52.35
57.48
120
221.23
54.36
39.82
183.56
101.53
39.61
84.66
47.04
209.97
50.01
53.90
Line Security Margin (MVA) at% of base - case
130
140
150
160
170
220.86
220.43 219.90
219.27
218.46
47.46
40.57
33.70
26.86
20.04
35.59
31.39
27.22
23.09
19.00
181.50
179.12 176.34
173.03
168.96
88.62
75.72
62.84
49.99
37.17
35.36
31.12
26.87
22.62
18.35
81.80
79.63
78.40
2772.80 1326.90
41.92
36.76
31.55
26.29
20.96
201.16
192.14 182.87
173.28
163.33
47.44
44.60
41.45
37.90
33.84
50.20
46.34
42.30
38.04
33.48
Loading
180
217.39
13.27
14.96
163.72
24.42
14.03
743.34
15.51
152.79
29.01
28.49
190
215.79
6.57
11.03
156.23
11.81
9.62
452.01
9.83
141.39
22.86
22.74
200
211.64
1242.20
315.57
138.08
0.06
861.40
248.66
2.39
129.73
11.60
14.02
200.21
210.86
0.03
219.88
134.84
0.13
476.70
232.88
1.58
130.50
10.10
13.13
Table 2: The Line Security Index
Line
No.
1
2
3
4
5
6
7
8
9
10
11
conecting
bus - bus
1
2
1
4
1
5
2
3
2
4
2
5
2
6
3
5
3
6
4
5
6
5
100
0.9882
0.4982
0.5435
0.9173
0.4966
0.5416
0.7494
0.5154
0.7123
0.9306
0.7386
110
0.9857
0.4480
0.4974
0.9025
0.4460
0.4950
0.7263
0.4672
0.6820
0.9197
0.7056
120
0.9829
0.3978
0.4513
0.8859
0.3956
0.4483
0.7051
0.4191
0.6512
0.9070
0.6706
Line
130
0.9796
0.3476
0.4052
0.8674
0.3452
0.4015
0.6867
0.3712
0.6200
0.8922
0.6334
Security Index at%
140
150
0.9758 0.9712
0.2974 0.2474
0.3591 0.3129
0.8464 0.8224
0.2948 0.2446
0.3545 0.3072
0.6721 0.6628
0.3233 0.2755
0.5882 0.5556
0.8746 0.8533
0.5936 0.5505
of base - case Loading
160
170
180
0.9657 0.9588 0.9496
0.1974 0.1475 0.0978
0.2668 0.2208 0.1750
0.7947 0.7618 0.7211
0.1945 0.1446 0.0949
0.2596 0.2115 0.1625
1.0129 1.0255 1.0412
0.2278 0.1801 0.1320
0.5220 0.4871 0.4500
0.8268 0.7923 0.7440
0.5036 0.4516 0.3924
190
0.9361
0.0485
0.1300
0.6661
0.0459
0.1122
1.0590
0.0826
0.4092
0.6669
0.3210
200
0.9018
1.0363
1.0488
0.5465
0.0002
1.0416
1.0894
0.0196
0.3571
0.4466
0.2051
200.21
0.8956
0.0002
1.0123
0.5269
0.0005
1.0456
1.0948
0.0129
0.3549
0.4044
0.1927
190
0.0250
1.2878
0.6180
0.1599
1.3959
0.6477
0.0287
1.1418
0.1945
0.0891
0.2646
200
0.2176
0.0087
0.1291
1.4395
186.64
0.0330
0.4586
22.061
1.1551
1.3247
1.9784
200.21
6.3104
10772.7
5.3836
42.380
2513.8
1.6854
14.387
903.87
31.222
45.386
55.386
Table 3: The Line Severity Index
Line
No.
1
2
3
4
5
6
7
8
9
10
11
conecting
bus - bus
1
2
1
4
1
5
2
3
2
4
2
5
2
6
3
5
3
6
4
5
6
5
100
0.0058
0.2469
0.2031
0.0394
0.2518
0.2073
0.0817
0.2465
0.1102
0.0238
0.0966
110
0.0060
0.2492
0.2017
0.0410
0.2547
0.2062
0.0777
0.2491
0.1069
0.0240
0.0960
120
0.0064
0.2569
0.2040
0.0433
0.2631
0.2089
0.0751
0.2569
0.1051
0.0247
0.0970
Line
130
0.0069
0.2708
0.2103
0.0465
0.2781
0.2158
0.0737
0.2707
0.1050
0.0259
0.0998
Severity
140
0.0076
0.2932
0.2216
0.0508
0.3020
0.2279
0.0736
0.2925
0.1065
0.0277
0.1048
Index at% of base - case Loading
150
160
170
180
0.0086 0.0100 0.0122 0.0160
0.3281 0.3840 0.4823 0.6828
0.2397 0.2681 0.3158 0.4035
0.0568 0.0654 0.0791 0.1032
0.3393 0.3990 0.5046 0.7224
0.2470 0.2771 0.3274 0.4202
0.0752 0.0022 0.0053 0.0116
0.3261 0.3790 0.4697 0.6475
0.1100 0.1162 0.1273 0.1475
0.0303 0.0343 0.0412 0.0542
0.1127 0.1249 0.1451 0.1809
APPENDIX
Table A1: Bus Data of a Modified 6-Bus System
No.
1
2
3
4
5
6
Bus
Type
PV
PV
SLACK
PQ
PQ
PQ
Voltage
(pu)
1.00
1.00
1.00
-
Pg
(MW)
0
0
0
Pl
(MW)
90
180
180
133
90
Ql
(MVar)
90
67.5
45
Qlim
(MVar)
NL
NL
NL
-
Table A2: Bus Data of a Modified 6-Bus System
Bus
From
To
1
2
1
4
1
5
2
3
2
4
2
5
2
6
3
5
3
6
4
5
5
6
NL:No limit
R
(p.u.)
0.1
0.05
0.08
0.05
0.05
0.1
0.07
0.012
0.02
0.2
0.1
X
(p.u.)
0.2
0.2
0.3
0.25
0.1
0.3
0.2
0.26
0.1
0.4
0.3
Yc
(p.u.)
0.02
0.02
0.03
0.03
0.01
0.02
0.02
0.02
0.01
0.04
0.03
Thermal Limit
(MW)
NL
NL
NL
NL
NL
NL
NL
NL
NL
NL
NL
Fig. A1: Topology of a modified 6-bus system
Transmission Line Performance Indices Calculation Based on Voltage Stability Criterion
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Komson Daroj received the B. E.
and M. E. degree in electrical engineering from Chulalongkorn University,
Bangkok, Thailand, in 1993 and 1999
respec-tively. Currently, he is pursuing
the Ph.D. degree at Chulalongkorn University. His areas of research are power
system operation control, power system deregulation and power quality.He
joined the Department of Electric Engineering Ubonratchathani University in
1998 and 2000.
Bundhit Eua-arporn received both
Bachelor and Master Degrees from Elec.
Eng. Dept., Faculty of Engineering,
Chulalongkorn University.
In 1992,
he received Ph.D. degree from Imperial College of Science Technology and
Medicine, London, UK. He is currently
an associate professor at the Elec. Eng.
Dept., and deputy director for Research
affair at the Energy Research Institute,
Chulalongkorn University.
Sotdhipong Phichaisawat received
his B.Eng. in 1994 and M.Eng in 1996
from Chulalongkorn University, Thailand. He is currently a lecturer at Chulalongkorn University, Thailand. His
research interests include the power
system planning, operations and economics, reactive power/voltage control,
and the electricity industry.
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