Download Tracing Network flows due to Sources/Sinks using Virtual flow Concept

16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010
503
Tracing Network flows due to Sources/Sinks using
Virtual flow Concept
Surendra S, D Thukaram
Department of Electrical Engineering
Indian Institute of Science, Bangalore-12
Email: {dtram, surendra18}@ee.iisc.ernet.in
Abstract—The main component of modern utilities geared
up for open access scenario is determine marginal pricing for
transmission network usage by the different entities involved in
the power market. One important aspect is power tracing which
in turn used to determine the charge allocation for the extent
of service utilized and make up for the loss component in the
network. The methods available in the literature have certain
shortcomings. This paper presents a concept called ”Virtual
flows”, which is based on superposition principle for tracing real
and reactive power flows. method evaluates the injection of each
source on the network when it is considered to act alone with
the specified power as obtained from a solved load flow or from
the output of on-line state estimator. The combined action of all
these gives the resultant state of the network. Modern power
systems incorporate HVDC links [1] for unsynchronized bulk
power transmission. It is aimed to extend the proposed concept to
systems incorporating DC links and further to evaluate network
usage charges based on the outcome of this tracing.
I. I NTRODUCTION
Power flow tracing is very much essential from the point
of transmission charges allocation and to make up the loss
component. System losses are associated with power transfer
in the network. In the interconnected system, determination
of the lines involved in providing transmission services for a
given transaction period among the various entities is to be
determined for allocating charges based on the actual usage,
so that the network cost can be recovered in a equitable manner. Marginal pricing requires the knowledge of transmission
facility availed by each generation and demand present in the
system. Most of the power flow tracing methods have some
amount of arbitrariness. Proportional sharing principle based
loss allocation and its variations are explained in references
[2], [3], [4], [5], [6]. The tracing is done from the net flows
obtained from load flow / output of on line state estimator.
The network parameters have no role in tracing the flow.
Bialek el. [2], [3] explains the flow tracing from each source
to load or vice-versa using upstream and downstream looking
matrix. Loss allocation to generating buses is still a problem
and to be relatively specified with respect to load buses.
The tracing technique defined by Kirschen et.al [4] uses
domain of generation and commons to specify the real power
contributions. Further Kirschen et.al [5], using the information
about complex network currents, the flow tracing is refined for
both real and reactive powers. Graph theory based approach
for tracing is carried out for network flows devoid of loops is
given by Wu et.al [6]. Using the network Z-bus, Coneja et.al
[7] proposed loss allocation by using load flow information.
Here system losses are attributed to buses which takes care of
relative position of generation and loads in the network along
with the capacity but contribution to loads from a given source
is still a problem. Further if line shunts are not present then
Z-bus formation is difficult & charges are also allocated to
counter flows. The shortcomings of Postage stamp and Pro rata
methods are discussed in reference [8]. Postage stamp method
practised in some markets even though simple to adopt, fails to
give proper indication due to which large loads get the benefit
over small loads. Pro rata method ignores the generation/load
location in the network and remote generations/loads benefit
at the cost of others. Cost allocation to HVDC links present in
the system is not easy. This is due to the fact that the marginal
cost is zero for these lines because of automatic control loop
maintains the desired power flow irrespective of variations in
the system conditions in normal state of operation.
This paper propose to use the virtual flows concept for
tracing proposed by Thukaram et.al[9], in order to obtain
source contributions to loads. Principle of superposition is
applied to evaluate the virtual flows based on actual network
conditions. This is suited for on-line application in a given
transactional period. In principle this method can be equally
applied to network with HVDC links. The proposed method
is detailed but preceded by brief look into the computational
aspects for obtaining power flow to AC system incorporating
DC links. Case studies on sample system and a real life
equivalent 33-bus system incorporating an HVDC link, are
presented. Pricing aspect is dealt for the sample system,
lastly the salient points of this approach and conclusions are
presented.
II. P ROPOSED A PPROACH
The initial data as a base case is to be obtained from the
output of load flow / on line state estimator along with the
topology of network at the given transactional period. The bus
power injections(generations/loads) along with the bus voltage
profiles is considered as a priori. From this data the virtual
contribution to loads from each source is evaluated using
superposition. The following section briefs about performing
AC/DC load flow in order to get the base case data.
Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.
16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010
504
A. AC/DC Load Flow
System specified data
Vac∟ψ
Idc
→
1:a
AC System
Pac
Qac Iac ∟ξ
Fig. 2.
AC System solution
Converter representation
bridge is fed from AC side through the secondary of a feeding
transformer whose output can be adjusted through tap control.
Similar converter is operated as a inverter on the other side of
the DC link. Their roles can be interchanged depending upon
the direction of power transfer desired. With the selection of
proper bases for the combined system, the following performance equations for the converter are specified. DC voltage
and power at the converter are,
Vdc = aVac cos α − Rc Idc
(1)
Pdc = Vdc Idc
(2)
Pac = Pdc
YES
NO
Perform
Optimization if
necessary
Obtain final AC/DC
combined power
flow
NO
Readjust DC system
initial settings
Fig. 1.
(3)
From the above, the expression for power factor angle at the
AC side is obtained by,
Vdc = aVac cos(ψ − ξ)
(4)
The reactive power requirement at the converter terminal is
then calculated by,
YES
Check for satisfactory
solution of combined
AC/DC system
_
where α as the firing angle, a feeding transformer tap setting
and Rc , the commutation resistance of the converter which is
a pure reactance term. With losses neglected the expressions
for real power at the AC side is,
DC System Solution
Check for satisfactory
solution of combined
AC/DC system
+
Vdc
The presence of DC links in AC system requires load
flow to be performed in sequential steps as per the reference
[10]. Load flow program developed by Thukaram et.al[11] is
incorporated in this regard. Firstly the condition for DC system
is specified based on the control mode selected i.e. constant
power or constant current or constant voltage. From this, the
voltage along with active and reactive power requirements at
the input of DC terminal is obtained which in turn specifies the
AC side requirement at the secondary of transformer feeding
the converter bridge. With this AC power flow solution is
obtained as in usual case. The power fed at the converter
side is represented as a equivalent load and at the AC side
of inverter as a equivalent generation. The taps of terminal
transformers at the respective converter bridges are computed
and checked against practical available settings so as to obtain
satisfactory solution for the combined system. If the settings
are not satisfactory, the initial conditions of the DC system are
recast by changing the firing/extinction angles and the process
is repeated. If transformer settings are within the satisfactory
limits, settings are selected to the nearest practical values at the
AC-DC terminals and final power flow solution is obtained on
the combined system. Figure 1 shows the major computational
blocks for the specified approach.
To virtual flows
evaluation block
AC/DC Power flow computation blocks
B. Converter Representation
Figure 2 shows the general representation of a converter
station [1] present in the DC link of the AC sub system.
The converter comprises of single or series connected bridges
as per voltage requirement. Mono polar or bipolar operation
is possible depending upon the configuration. The converter
Qdc = Pac tan(ψ − ξ)
(5)
the power factor angle between AC side voltage and current
is (ψ − ξ = θ). For inverter present at the other end of the
converter, the expression for DC voltage is same as in equation
(1) but α being replaced by extinction angle γ, whose polarity
opposite to the converter voltage. For proper operation of the
link such that no commutation failure occurs, the condition to
be met is,
α + µ ≤ 1800 − γ0
(6)
where, µ being the commutation overlap angle and γ0 is the
minimum required extinction angle. The AC power requirement at the input of the inverter is similarly calculated by the
valid assumptions made as earlier wherein AC active power
only being negative and the expressions Pac and Qac being
expressed in terms π − ϕ, where ϕ being the power factor at
the AC side of inverter transformer.
Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.
16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010
C. Concept of virtual flows
As mentioned in the previous section the virtual flows are
to be determined from the output of on-line load flow / state
estimator. After obtaining satisfactory AC/DC load flow solution for the network, the virtual flow block is activated. The
required parameters for computation are the bus voltage profiles, real and reactive power injections/withdrawals from all
the buses along with the parameters of the network elements
such as transformer tap positions and line parameters of the
connected AC network for which the power flow is obtained.
The DC link is now equivalently represented as two loads on
the either side of the AC system to which it is connected such
that converter power flowing into the DC system is taken as a
load. At the inverter side the received power being represented
as power injection to the respective system bus (negative load).
Figure 3 represents the steps involved in computing the virtual
flows from the sources present in the network. The computing
1: Get the required data from the o/p of load
flow / state estimator
505
case is [V 0 ] and the admittance matrix of the network
being [Ybus ].
2) Convert all the complex loads into equivalent admittances.
∗
YLr = SLr
/|VLr |2
(7)
r = 1 .... d (number of loads) where SLr , VLr :load
power and voltage at rth bus.
3) Include all the loads admittances into network [Ybus ]
mod
matrix to form modified matrix say [Ybus
].
4) Convert all the complex generator power injections into
equivalent currents.
IGi = [SGi /VGi ]∗
i = 1 .... g (number of injections) where SGi , VGi :bus
injected power and voltage at ith bus.
5) Form injection current vector [IGi ] with all elements
zero except the one corresponding to ith generator being
non zero. Find the partial bus voltages developed in the
network due to this injected current alone by solving the
following network equation by LU decomposition.
mod
[IGi ] = [Ybus
][Vgi ]
2: Obtain admittance matrix of the
connected network [Ybus]
3: Convert all the loads to equivalent admittances
Convert all the negative loads to equivalent
negative admittances
4 : Include all the load admittances into the
network [Ybus ] matrix to form modified
[Ybus ]mod matrix
5: Convert all the bus power
injections from each source to
equivalent currents
6: Inject equivalent current from ith generator
into modified network
7: Obtain the voltage profile developed in the network
due to this injection & calculate the resulting currents in
all the branches of the network
8 : Evaluate the virtual flows contributed from
ith generator using the base case bus voltage
profile and corresponding branch currents
obtained from previous step
NO
9: All generators
covered
10: Select next
generator
Fig. 3.
D. Illustrative sample systems
To demonstrate the virtual flow concept, a sample 5 bus system comprising of 2 generators and 3 loads as shown in Figure
4 is considered. Shunt reactor of 2.0 p.u(400kV,100MVA base)
is connected to load bus 5.
~
Major computational blocks for proposed approach
blocks are briefly explained with network having g generator
buses, r load buses and remaining as switching buses with
zero generation/load.
1) The required data from the output of load flow / state
estimator is read. The bus voltage vector for this base
(9)
With [Vgi ] the partial bus voltage vector for ith generator
current injection(virtual action). Using this, partial currents in the all network branches due to corresponding
generator can be found by usual manner.
6) The contributions to line flows (partial contributions)
from the given generator is found out by using the base
case bus voltage profile and by corresponding network
currents determined from previous step.
7) Repeat steps from 6 to 8 so as to cover all generators.
8) Contribution to a particular load from a given generator
is evaluated by summing the partial contributions in the
lines incident on the node to which load is connected.
Here the flows determined are defined to be virtual. The
superposition of individual contribution to network flows and
loads matches with the base case result.
YES
Stop
(8)
~
1
2
3
4
5
Fig. 4.
Sample 5 bus system with 3 loads
Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.
16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010
The network parameters are given in Table I. Table II shows
base case data as obtained from solved power flow along with
the bus voltage profiles(p.u). Table III indicates the power flow
in the lines of the network for base case. The partial voltages
developed at network buses, when each of the source injecting
equivalent current(individual action) into the modified network
is given in Table IV. It can be noted that the respective partial
bus voltages developed from each source sums to original bus
voltage profile.
TABLE I
T RANSFORMER ,
Bus
1-3
2-4
3-5
4-5
LINE DATA FOR THE SAMPLE NETWORK IN ( P. U )
Element
Transformer
Transformer
Line
Line
(R & X)
(0.002, 0.02)
(0.002, 0.02)
(0.004, 0.04)
(0.004, 0.04)
B/2
—–
—–
0.5
0.5
Tap
1.000
1.000
—–
—–
Generation/Load
(305.40, 82.60)
(200.00, 79.70)
(-100.00, -50.00)
(-100.00, -50.00)
(-300.00, -150.00)
BASE
CASE
Line
1-3
2-4
3-5
4-5
Forward flow
(305.44, 82.58)
(199.99, 79.71)
(203.35, 12.67)
(99.08, 20.49)
Reverse flow
(-303.44, -62.56)
(-199.06, -70.44)
(-201.47, -86.91)
(-98.48, -107.64)
Loss
(2.00, 20.02)
(0.93, 9.27)
(1.88, -74.24)
(0.60, -87.14)
TABLE IV
PARTIAL BUS VOLTAGE PROFILES DEVELOPED DUE TO INDIVIDUAL
CURRENT INJECTION
Bus
1
2
3
4
5
Partial voltage developed all in p.u
Injection at Bus-1(p.u)
Injection at Bus-2(p.u)
(3.054-j0.826)
(1.942-j0.929)
(0.633+j0.113)
(0.367-j0.111)
(0.560-j0.063)
(0.437-j0.003)
(0.610+j0.053)
(0.367-j0.111)
(0.560-j0.063)
(0.415-j0.040)
(0.566-j0.040)
(0.374-j0.096)
TABLE V
C ONTRIBUTIONS TO LINE F LOWS (MW & MVAR) IN THE LINES OF THE
NETWORK FROM SOURCE AT BUS 1
Line
1-3
2-4
3-5
4-5
Forward flow
(305.40, 82.60)
( 0.00, 0.00)
(236.94, 40.84)
(-57.31, -29.08)
Reverse flow
(-303.40, -62.58)
( 0.00, 0.00)
(-241.02, -75.10)
( 53.64, -27.66)
TABLE VI
C ONTRIBUTIONS TO LINE F LOWS (MW & MVAR) IN THE LINES OF THE
NETWORK FROM SOURCE AT BUS 2
Line
1-3
2-4
3-5
4-5
Forward flow
(0.00, 0.00)
(200.00, 79.70)
(-33.59, -28.12)
(156.34, 49.66)
Reverse flow
(0.00, 0.00)
(-199.07, -70.43)
( 39.42, -11.83)
(-152.19, -80.02)
Loss
(0.00, 0.00)
(0.93, 9.27)
(5.83, -39.95)
(4.15, -30.36)
FROM INDIVIDUAL
GENERATORS
Voltage(p.u)
(1.000+j 0.000)
(0.998-j0.067)
(0.977-j0.059)
(0.975-j0.104)
(0.939-j0.138)
TABLE III
F LOWS (MW & MVAR) IN ALL THE LINES OF THE NETWORK FOR
loss term in these tables gives information about counter flows
from the corresponding generators. It can be inferred that the
superposition of voltage, line currents and power holds good.
TABLE VII
C ONTRIBUTIONS TO LOADS (MW & MVAR)
TABLE II
G ENERATION , L OAD (MW& MVAR) BUS VOLTAGES FOR BASE CASE
Bus
1
2
3
4
5
506
Loss
(2.00, 20.02)
(0.00, 0.00)
(-4.08, -34.26)
(-3.67, -56.75)
The partial contributions to line flows from the two sources
are shown in Table V and Table VI respectively. The negative
Source ↓
Gen. bus-1
Gen. bus-2
Total
Load bus-3
(-66.45, -21.74)
(-33.59, -28.12)
(-100.05, -49.86)
Load bus-4
(-57.31, -29.08)
(-42.73, -20.77)
(-100.04, -49.86)
Load bus-5
(-187.38, -102.76)
(-112.77, -91.85)
(-300.15, -194.61)
Finally the contribution to loads from each sources are
presented in Table VII (loads indicated as negative power
injections at respective buses). It can be inferred that all the
generators will share some portion of each and every load
of the network in some proportion which may be significant
or otherwise. This is evident from the Table VII. The tracing
techniques detailed so far in literature traces the output of generators to only loads which are in its reach. Some generators
may not contribute to a particular load. Table VIII compares
the proposed method with the technique explained by Kirschen
[4], w.r.t real power contributions from each source to loads.
The compared method, traces the output of source-1(bus-1)
to loads at buses 3 and 5 & loads at buses 4 and 5 gets
contributions from source-2(bus-2).
TABLE VIII
C ONTRIBUTIONS TO LOAD FROM SOURCES A
Source
Gen. bus-1
Gen. bus-2
Total
By proposed method
L3
L4
L5
66.4
57.3
187.3
33.6
42.7
112.7
100.0
100.0
300.0
COMPARISON
By Kirschen Method
L3
L4
L5
100.0
—201.5
—100.0
98.5
100.0
100.0
300.0
E. Transmission Usage Charge Allocation
In this section the pricing aspect has been dealt from the outcome of tracing using virtual flows. Charges for transmission
facilities availed by each generating unit based on real power
flows is evaluated and compared with the tracing obtained
from Kirschen method. Table IX and Table X indicates the
charges for each generator. Here the reference is the net flow
direction in each lines as per the base case load flow. Negative
values indicates counter flow components. Lines/transformers
of the sample network are identical in every aspect and with
the assumption of equal transmission cost the results are
presented. The cost of transmission facility for both lines are
Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.
16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010
TABLE IX
L INE USAGE (MW)
Lines
1-3
2-4
3-5
4-5
Total charge
TABLE XII
D ETAILS OF HVDC LINK
AND CHARGES FOR SOURCE AT BUS -1
Proposed method
305.40
00.00
236.94
-57.31
255980.00
Kirschen method
305.40
00.00
203.35
00.00
279700.00
MVA
Transformer
rating
Tap
Vdc
Resistance
Angles
TABLE X
L INE USAGE (MW)
Lines
1-3
2-4
3-5
4-5
Total charge
507
Sending end
Receiving end
(-1547.9-j887.0)
(1504.2+j915.8)
465MVA
460MVA
219kV
216kV
0.9625
0.975
1030.3kV
1001.4kV
9.60960 Ohm
α = 15.78o , γ = 18.00o
AND CHARGES FOR SOURCE AT BUS -2
Proposed method
00.00
200.00
-33.59
156.34
173050.00
Kirschen method
00.00
200.00
00.00
99.08
149080.00
taken at Rs.1000/MW and transformer cost is 25% of the line
cost on similar terms. The proposed method allocates more
charges to generating unit at bus-2. Some portion of the load
at bus-4 is allocated to this generator and extra allocated charge
is compensated. Further, sources which are producing counter
flows in a given line are not charged. As for as virtual flows
are concerned, if there is any counter flow/flows in a given
line, then these counter flows are made zero and non counter
flows are accordingly modified as explained. Assuming in any
line the partial flow components due to four sources being pl1 ,
pl2 , −pl3 & −pl4 , then the non counter components pl1 & pl2
are respectively modified as per equation (10) and line usage
charge are allocated to these two components only.
power, reactive power is required at the input of bridges for
proper operation. Pricing aspect can be dealt on similar lines
as explained in the previous section. The virtual flows are
evaluated as explained in previous section. The net flows as
well as virtual flows in the AC lines connected to HVDC links
are presented in Tables XIII and XIV. It can be verified from
these two tables that the net flows in the lines matches with the
superposition of corresponding virtual flows. The contribution
to DC link from either side of the terminal from all sources
of the system is presented in Table XV. This gives power
injection to DC line by each sources. Only the values of virtual
contributions from the sending has to be taken into account for
pricing purpose. The loads specified in the tables also includes
compensation and shunt reactor MVA if any.
TABLE XIV
N ET FLOWS (MW & MVAR) IN THE LINES ADJACENT TO DC
Lines
1-17
11-18
32-17
17-25
18-33
18-8
′
pl1 = pl1 {1 − (|pl3 + pl4 |)/(pl1 + pl2 )}
′
pl2 = pl2 {1 − (|pl3 + pl4 |)/(pl1 + pl2 )}
(10)
LINK
Flows
(1800.3, 467.8)
(756.0, 440.5)
(473.5, 10.3)
(715.0, -115.0)
(459.9, 138.2)
(1795.0, 31.0)
F. 33 Bus UPSEB equivalent system with an HVDC link
The determination of contributions to loads is applied to real
life 400kV, 33 bus equivalent of UPSEB (Utter Pradesh State)
with a DC link between buses 17(Rihand-400)and 18(NCR400). It comprises 7 transformers and 37 transmission lines.
Two series connected bridges are located on either side of the
link which is configured for bipolar operation. The single line
diagram of the system is shown in Figure 5. The details of
system generation/loads, HVDC system results from on-line
load flow are given in Tables XI & XII respectively.
TABLE XI
G ENERATION / LOAD DETAILS FOR THE UPSEB
MVA
Shunt reactors
Generating units
7
(7399.5+j2326.0)
15
TABLE XV
S OURCE CONTRIBUTIONS (MW & MVAR)
Source↓
RHND-GEN
OBRE-GEN
ANPR-GEN
SNGRLGEN
LAKHWGEN
SRINGGEN
NCR–GEN
Total
Converter end
(at bus 17)
( -486.9, -4.3)
( -214.2, -57.3)
( -371.8, -51.6)
( -472.2, -18.8)
( -5.9, -76.4)
( -10.3, -53.8)
( 13.5, -148.5)
(-1547.8, -410.7)
TO EITHER SIDE OF
DC
LINK
Inverter End
(at bus 18)
(310.2, 7.8)
(153.1, 16.8)
(254.8, 8.7)
(310.6, -0.4)
(118.6, -42.8)
( 80.2, -35.8)
(276.3, -120.6)
(1503.8, -166.3)
SYSTEM
Loads
21
(-7169.0-j4279.2)
-1666.3 MVA
For determining virtual flows, the DC link is expressed
equivalently by two complex loads on either side of the buses
to which it is connected. The sending end power is equivalent
to a positive load, whereas at the receiving end it is taken as
negative load (injection). Though the DC line carry only real
III. C ONCLUSION
A approach for evaluating source contributions to line flows
and loads is presented. By using the network modified admittance matrix and by sparse LU decomposition it is possible to
evaluate virtual flows from the on-line load flow data. Partial
contributions are to be determined for each sources present
in the network. Repeated evaluation of virtual flows can be
avoided if there is no considerable change in load/generation
pattern. Generating units producing counter flows are identified in a transaction. Each generating unit will be entitled
Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.
16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010
Fig. 5.
508
33 bus 400 kV equivalent UPSEB system with single DC link
TABLE XIII
V IRTUAL FLOWS (MW & MVAR) IN THE LINES ADJACENT TO DC
Source↓
RHND-GEN
OBRE-GEN
ANPR-GEN
SNGRLGEN
LAKHWGEN
SRINGGEN
NCR–GEN
Total
1-17
(1800.0, 468.6))
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(1800.0, 468.6)
11-18
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
(756.0, 440.0)
(756.0, 440.0)
Lines
32-17
(-1052.0, -274.1)
(288.8, 55.9)
(518.1, 45.4)
(685.8, 11.5)
( 12.4, 46.9)
( 13.0, 32.7)
(7.0, 91.4)
(473.2,10.1)
to collect charges from each and every loads. The counter
flow information is useful in managing network congestion to
some possible extent. In the presence of HVDC links in the
AC system, this technique can be employed after obtaining the
load flow / state estimation results for the combined AC/DC
system for obtaining the extent of utilization of links by the
sources. From the results it can be inferred that each and every
source is responsible to meet the load demand to some extent.
This finds potential applications in areas such as bilateral &
multilateral transaction evaluation, loss allocation and reactive
power optimization.
R EFERENCES
[1] J. Arrillaga and N. R. Watson, Computer Modelling of Electrical Power
Systems. John Wiley & Sons, Ltd, 2 ed., 2003.
[2] J. Bialek, “Tracing the flow of electricity,” IEE Proceedings-Generation
Transmission Distribution, vol. 143, no. 4, pp. 313–320, 1996.
[3] J. Bialek, “Topological generation and load distribution factors for
supplementary charge allocation in transmission open access,” IEEE
Transaction on Power Systems, vol. 12, no. 3, pp. 1185–1193, 1997.
17-25
(251.5, 4.8)
(74.6, -0.1)
(146.7, -5.5)
(214.2, -6.7)
(6.0,-29.3)
(2.3, -20.9)
(19.7, -57.0)
(715.0,-114.7)
LINK
18-33
(-53.7,-48.1)
(-24.5,-26.0)
(-43.7,-39.9)
(-55.0,-46.7)
(-27.4,-10.3)
(-19.6, -5.7)
(683.7,315.1)
(459.8, 138.4)
18-8
(363.9, 55.9)
(177.6, 42.8)
(298.5, 48.7)
(365.6, 46.2)
(146.1,-32.5)
( 99.8,-30.1)
(343.4, -100.4)
(1794.9, 30.6)
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Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.