Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
Probabilistic Transfer Capability Assessment in a Deregulated Environment
A. P. Sakis Meliopoulos, Sun Wook Kang
School of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, GA 30332-0250
[email protected]
George Cokkinides
Department of Electrical and Computer
Engineering
University of South Carolina
Columbia, SC 29208
[email protected]
transfer capability. As interchanges and power markets
increase, interconnections may be used at their capacity.
This reality has brought into focus the practical
limitations of interconnections and the associated problem
of transfer capability. Transfer capability which refers to
the capacity and ability of a transmission network to
allow for the reliable movement of electric power from
the areas of supply to the areas of need is an issue of
concern to both system planners as well as system
operators.
In a deregulated environment, the interconnected
power system may comprise many areas corresponding to
owners/utilities. The operation of the system is entrusted
to an Independent System Operator (ISO). The ISO may
receive bids for generation in real time. Bids may be
accepted as long as the transfer capability of each one of
the areas is not exceeded. Therefore it is important that
the ISO computes in real time the available transfer
capability of all the areas under its jurisdiction. At the
same time it is important to evaluate the risk of violating
the transfer capability, because of random events, such as
random failures of equipment. In other words, it is
important to compute the probability that the transfer
capability will not exceed the required. Another issue is
created by the existence of interruptible loads. In a
competitive market it is conceivable to accept a
generation bid and at the same time to interrupt load to
meet transfer constraints, if such action is economically
justified.
Recent events have made this scenario
plausible. In general, in a deregulated environment, there
is increased uncertainty because of: (a) uncertainty in bid
acceptance procedures, (b) customer response to prices,
(c) control of interruptible loads, and other. This
uncertainty must be assessed in real time and the effects
of uncertainty must be quantified for the next few hours.
There are many proposed approaches to determining
the available transfer capability. Most of the approaches
are deterministic.
Specifically, they determine the
available transfer capability using power distribution
factors to project the amount of power that can be
transmitted before the capacity of a transmission corridor
or line will be exceeded. A semi-probabilistic approach
Abstract
A methodology for available (simultaneous) transfer
capability (ATC) analysis based on a probabilistic
approach is presented. It is postulated that the system is
operated by an ISO. The traditional concept of “area” is
extended to include a utility, an individual IPP, a large
customer, etc. All areas are divided into three groups: (a)
study area or area under the ISO control, (b) transfer
participating areas, and (c) external areas which have no
direct transactions or they have fixed transactions with
the study area. A performance index based contingency
selection procedure is applied within the study and
transfer participating areas to rank those contingencies
that will affect simultaneous transfer capability. The
contingency ranking order is utilized by a variation of the
Wind Chime diagram to selected contingencies that are
then evaluated by an optimal power flow algorithm.
Subsequently, the probability distribution of simultaneous
transfer capability is computed based on the electric load,
circuit and unit outage Markov models. The 24x3 bus
IEEE RTS is utilized to evaluate the proposed method.
The performance of the proposed method is also
demonstrated on an actual large scale system (2182 bus,
8 area system).
Keywords: Simultaneous transfer capability, transfer
interface, contingency ranking, optimal power flow, Wind
Chime scheme, expected transfer capability.
1. Introduction
Modern power systems are interconnected to share
their resources and assist in emergencies. The major
objectives of interconnections are (a) economics and (b)
security/reliability. The beneficial economic effects of
interconnections as well as their favorable impact on
system security/reliability have been long recognized.
Recent trends such as competition, non-utility generation,
and transmission access resulted in increased power
transfers among utilities. In addition, power markets
impose new constraints and uncertainties on the available
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
Network constraints
Generation real and reactive power constraints
Transmission circuit loading constraints
(1)
Bus voltage constraints
Security constraints, and
Committed unilateral power transfers, bid offers, etc.
extends this concept to include a number of
contingencies.
This paper presents a flexible and rigorous formulation
of the probabilistic transfer capability problem that
accounts for operating practices of utilities/areas, the
typical variation of the electric load, the new uncertainties
under a deregulated environment and the operating
practices of an ISO.
Where Pij is the real power flow on the line ij of the
transfer interface TI.
The network constraints are formulated in a way as to
fit the problem of the transfer capability. Specifically,
Figure 2 illustrates some of the innovations for
formulating the network constraints. Specifically, the
generating unit output of area k is defined with a variable
zk that represents the total change of generation in area k.
The total generation change in the area is allocated to the
various units of the area via participation factors pi that
are computed with a usual economic dispatch procedure
within the area. In this way, the need to have a slack bus
and all the complications of a slack bus are avoided. It
should be also apparent that in the case of a single
generating unit (IPP, etc.), the variable z becomes the
output of that single unit.
The variation of the electric load at a bus, m, is
assumed to be a linear function of a small number of
variables that can vary randomly. Figure 2 illustrates the
load at bus m to be a linear function of two random
variables, v1 and v2. In general the vector of bus loads is
given by:
2. Model Description
Figure 1 illustrates a general structure of a multi-area
interconnected power system, operated by an ISO. Here
the concept of an area is used in its general term: an area
can be a specific utility, a single IPP, a load center, etc.
Consider area S. Tie lines connecting area S with other
areas provide the means for power transactions between
the area S and outside areas. Real power meters are
placed on tie lines. A set of power meters determines the
transfer interface over which the transfer is to be
evaluated. The purpose of simultaneous transfer
capability analysis is to determine the maximum
simultaneous power transfer through this interface, i.e. to
or from the area S under a certain set of postulated system
load and contingency conditions.
External Areas (E)
E2
E1
P = P 0 + av1 + bv2 + ..
Transfer Participating Areas (P)
.........
area k
P2
P1
Pgi = Pgi + pi zk
....
Pp2
Pp1
(2)
Pbub
.....
M
M
S
Ppn
Em
Pgj = Pgj + pj zk
Pn
M
M
Study Area (S)
Pwuw
System of Interest
Figure 1. A Simplified Interconnected Electric Power
System with Multiple Areas
Mathematically, the simultaneous transfer capability
(STC) problem is represented as an optimization problem
that determines the maximum simultaneous power
transfer through a transfer interface TI (TI may represent
power transfer to or from the area S).
Maximize
Pdm = Pdm + am v1 + bm v2
Pwuw
Figure 2. Definition of Network Model for Transfer
Capability Analysis
STC = Σ Pij
Note that the variables v1, v2, control the loads at all
buses of area k. In case that the number of variables is
ij∈TI
Subject to:
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
(OPF) which yields the maximum STC for a specific
contingency and load. Finally, the probabilistic index
computation provides expectation, frequency and duration
indices of ATC. Figure 4 illustrates the overall procedure.
First, a loading condition is selected. For this loading
condition, initially the base case transfer capability is
determined by using an optimal power flow model and
assuming all circuits and units are available.
Subsequently, contingencies are selected which will
adversely affect simultaneous transfer capability using a
contingency selection model (see section 4). The selected
contingencies are simulated with an optimal power flow
to determine the simultaneous transfer capability yielding
a value of STC for each contingency. The procedure is
repeated for all loading conditions. Finally, the results are
assembled into a probabilistic distribution function of
simultaneous transfer capability. A concise description of
the three major modules, contingency selection,
contingency simulation, and index computations is
provided next.
only one, then we have what is known as conforming
load, i.e. the load at the buses varies proportionally to a
single variable. Obviously, two or more variable can
simulate the randomness of load variation and the
correlation that exists in these variations.
In addition, some of the load may be interruptible. This
generation and load model is embedded in the network
constraints.
The figure also illustrates a number of market
variables, Pb, ub, Pw, uw, price, etc. The variables Pb, ub,
represent an accepted bid for power generation, Pb is the
power delivered and ub, is a 0 or 1 variable representing
the availability of the transaction. The variable ub is a
random variable with a certain probability of being 1 and
the complimentary probability of being 0. Similarly, the
variables Pw, uw, represent a committed bilateral wheeling
transaction.
The control variables in above problem are (1) net real
power generation in each area, (2) generation bus voltage
magnitudes, (3) transformer phase shift adjustments, (4)
switchable capacitor or reactors, (5) accepted bids, (6)
committed bilateral transactions, etc. The above
formulation represents a complex optimization problem
that can be solved by two approaches, i.e. deterministic
approach and probabilistic approach. The two approaches
are vastly different and the computational requirements of
the probabilistic approach are much higher.
Deterministic approach: In this approach a security
constrained optimal power flow model is utilized. The
results of the OPF can be presented in two different ways:
(a) maximum transfer capability through a user selected
transfer interface, and (b) sensitivity of the power flow on
a user specified transfer interface versus net generation of
each area and electric load of each entity (customer
groups). Figure 3 illustrates typical results of this
computational procedure. The results are useful for
determining the ability of the system to transfer load
under the present operating conditions. This approach can
be also extended to include a number of user selected
contingencies.
System State: Base Case
Xfer
Generation
Interf
Area 1
Area 2
1
0.0012
0.0013
2
0.0020
0.0015
…
…
…
Start
Define Study (S),
Transfer Participating (P)
and External (E) Areas
Select Loading Condition 1
Select Next
Loading Condition
Yes
Determine Base Case
Transfer Capability
Optimal Power Flow
Select Contingencies Which
Will Affect Transfer Capability
Within P & S Areas
Contingency Selection
Simulate the Selected
Contingenies and Compute
the Transfer Capability for
Each Contingency
Optimal Power Flow
Any More Loading
Conditions ?
No
Generate Probability
Distribution of Transfer
Capability and Report Results
Stop
…
…
…
…
Load
Area 1
0.0032
0.0019
…
Area 2
0.0037
0.0029
…
…
…
…
…
Figure 4. Flow Chart of Probabilistic Simultaneous
Transfer Capability Analysis
Figure 3. Typical Transfer Capability Sensitivity
Results
3. Contingency Selection Model
A comprehensive presentation of contingency
selection models can be found in [13]. Here we present a
performance index (PI) based method for ranking
contingencies in terms of their effects on simultaneous
transfer capability. The proposed method is based on four
concepts: (1) selection of proper performance indices, (2)
component outage models, (3) computation of
Probabilistic approach: In this approach the problem is
decomposed into three manageable sub-problems: (a)
contingency selection, (b) effects analysis (contingency
simulation), and (c) probabilistic index computations. The
purpose of contingency selection is to select the
contingencies that will adversely affect STC. Contingency
simulation is performed with an optimal power flow
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
performance index changes, and (4) Wind Chime
diagram. These are discussed next.
Performance
indices:
To
effectively
select
contingencies with adverse effects on STC, two types of
performance indices are proposed.
that in turn may cause reduction of simultaneous transfer
capability.
Component outage model: The outage model used in
the proposed contingency ranking method has been
presented in [13]. Specifically, the outage is represented
with one control variable, the outage control variable uc
which has the following property:
(1) Jtc, is defined as the absolute value of the real power
export of the study area, expressed with:
J tc = Σ Pij
. ,
 10
uc = 
 0.0,
(3)
ij∈TI
Figures 5(a) and 5(b) illustrate the use of outage
control variables to represent circuit and unit outages
respectively. Note that the variable uc enables a rigorous
representation of a contingency. For example, consider
the outage of circuit ij. The power flow is expressed in
terms of the control variable, uc:
A negative change in the performance index Jtc due to a
contingency indicates an adverse effect on transfer
capability.
(2) In this category, two performance indices are
proposed: (a) circuit MVA loading performance index,
JP, and (b) bus voltage performance index, JV. These
indices are expressed in equations (4) and (5). More
details of these indices can be found in [13].
 S 
 ! 
W
∑
JP =
!  max 
! ∈S, P
 S!

if the component is in operation
if the component is outaged
Pij = { g ijVi2 − Vi Vj [g ij cos(δ i − δ j ) + b ij sin(δ i − δ j )] }u c (6)
Q ij = { − [b ij + b sij ]Vi2 + Vi Vj [b ij cos(δ i − δ j ) − g ij sin(δ i − δ j )] }u c
2n
Similar expressions can be applied to transmission
lines, transformers, phase shifters, etc.
(4)
i
(g ij + b ij )u c
j
where:
jb sij u c
jb sji u c
is the circuit MVA power flow of circuit !
S!
max
S!
W!
is a weighting factor of circuit !
n
is a selected integer
(a)
is the circuit MVA capacity rating
 Vi − Vi, av 

JV = ∑ Wi 

 Vi, st 
i ∈S, P
o
o
P g2 + (1-u c )p I2 P gi
o
o
P g1 + (1-u c )p I1 P gi
2n
Area J
o
(5)
u c P gi
o
o
P gk + (1-u c )p Ik P gi
i
where:
Area I
Vi
is the voltage magnitude at bus i
Vi,av
= 12 (Vi
Vi,st
= 12 (Vi
max
max
min
+ Vi )
min
Area K
- Vi )
max
Vi
is the maximum allowable magnitude at bus i
(b)
Vi
Wi
is the minimum allowable magnitude at bus i
is a weighting factor of bus i
Figure 5. Illustration of Component Outage Modeling
with Outage Control variable uc
(a) Circuit Outage, (b) Unit Outage in Area I
min
A large positive change in the two performance indices
JP and JV may indicate violation of operating constraints
The outage of a generating unit, i, in area I, will cause the
loss of generation, Pogi, which is absorbed by other units
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
ATC for the base case and specified loading condition.
Subsequently, all contingencies are ranked using the three
performance indices. The basic idea here is to adaptively
select only those contingencies that have adverse effects
on base case STC. All other contingencies are assumed to
have the same ATC as the starting base case. This
procedure drastically reduces the total number of required
simulation cases.
in the same area according to their participation factors,
yielding (see Figure 5b):
Pgi = uc Pogi
i ∈I
(7a)
o
Pgk = P gk + (1 - uc) pIk Pogi
k ∈ I, k ≠ i
(7b)
where pIk is the participation factor for unit k. This outage
model can also represent a multi-component outage with
only one outage control variable uc.
Performance index changes: Contingency ranking is
now performed by simply computing the derivative of the
performance index with respect to uc: dJ/duc. The costate
vector method is used for this computation [15], which is
summarized below:
∂g
dJ ∂J
=
− x" T
∂u
du ∂u
x" T =
∂J  ∂g 
 
∂x  ∂x 
4. ATC Effects Analysis Model
Each contingency is analyzed with an OPF method to
determine the available transfer capability over a
specified interface. The proposed OPF must have the
following properties: (a) provide a solution under any
conditions (non-divergent) and (b) efficiency. An OPF
with these properties has been developed and described in
this section. The non-divergent property of the OPF is
achieved with the introduction of the mismatch variables.
They are defined as follows. Let vector x represent system
states and let vector u represent the available controls.
Assume a given operating state xo and setting of controls
uo. Further, consider bus k as is illustrated in Figure 6.
Unless xo and uo represent a power flow solution, there
will be a power mismatch at bus k equal to Pomk+jQomk.
Now assume that a fictitious generating unit is placed at
bus k as in Figure 6. Let the output of the fictitious unit at
o
o
bus k be Pomk+jQomk. In this case x and u represent
the present operating condition of the system which
includes the fictitious units. The actual operating
condition of the system can be obtained by gradually
reducing the output of the fictitious generating units to
zero while at the same time computing the changes in the
system variables x and u. The solution trajectory
maintains feasibility and optimality. Mathematically, this
procedure is formulated as an optimization problem with
the objective of maximizing the simultaneous transfer
capability:
(8)
−1
(9)
where: J
is the performance index (Jtc, JP, or JV)
g
is the set of power flow equations
x
is the state vector (voltage magnitudes
and phase angles)
x"
is the costate vector.
Since the outage control variable uc is normalized, the
above derivative expresses the first order change of the
performance index due to the contingency. The ranking
order is based on the values of the derivative dJ/du. More
details of the PI approach can be found in [13].
Wind Chime diagram: The Wind Chime diagram is
used to select contingencies (single independent outages
or multiple (common mode) outages). This method has
been extensively used for reliability analysis [18].
However, the Wind Chime diagram used in the
simultaneous transfer capability analysis is different from
that applied in reliability analysis in two respects. First, in
reliability analysis if a contingency is identified causing
system problems, then all its combinative contingencies
are immediately assumed to cause system problems and
detailed simulation of the contingency is not required.
However, for the simultaneous transfer capability
problem, if a contingency is identified to have adverse
effects on STC, then all its combinative contingencies
must be evaluated in order to compute the actual STC.
This procedure makes the required simulation set very
large. Second, for on-line applications, one is concerned
with the operation of the system for the next few hours. In
this case the probability of contingencies is relatively
small and multiple contingencies need not be considered.
This reduces the required simulation set. The required
simulation set is further reduced using a variation of the
Wind Chime scheme: Specifically, first we compute the
Pgk+ jQ gk
Pmk+ jQ mk
bus k
bus i
y
ki
Other
circuits
y
SLk
ZLk yk
ski
y
sik
Electric Load
Figure 6. Illustration of a General Bus k of an Electric
Power System
Minimize:
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
(
)
µ ∑ ∆Pmk + ∆Q mk +
k
∑ Pij
(10a)
∑
(10b)
i ∈ S, j∉ S
Each component (circuit or unit) is modeled with a two
state Markov model. Load levels are modeled with a
multi-state Markov model. For example, Figure 7
illustrates transitions from one load level to another (λij)
and transitions from one contingency to another (λjk).
More details on the load model can be found in [15].
Utilizing these models, a contingency at certain load level
(which is referred to as a Markov state) is characterized
with a certain probability and transition rates to other
Markov states as it is shown in Figure 7. Utilizing this
information the following probabilistic descriptions of
STC can be derived:
or Maximize:
− µ ∑ ( ∆Pmk + ∆Qmk ) +
k
i∈ S , j∉ S
Pij
Subject to:
(11)
Power balance equation
Voltage constraints
Circuit flow constraints (MVA,MW or A)
Unit real and reactive power constraints
Other pertinent constraints, and:
(a) Cumulative probability distribution of STC,
(b) Expected STC,
(c) Probability of STC being at a selected range (i.e.
1000 MW to 1200 MW),
(d) Frequency and duration of STC being at a selected
range.
(12)
o
o
o
-P mksign(P mk) < ∆Pmksign(P mk) < 0, k=1,...,n
o
o
o
-Q mksign(Q mk) < ∆Qmksign(Q mk)< 0, k=1,...,n
C o n tin g e n c y
Load
Level
o
where ∆Pmk = Pmk
mk, ∆Qmk = Qmk- Q mk, m is
an appropriately selected constant, and sign(x) is a sign
function with:
- Po
i
λ ij
Sr
k
 1;
sign(x) = 
−1;
λ jk
j
If x ≥ 0
If x < 0
Constraints (12) are added to guarantee that two
successive solutions will be characterized with nonincreasing mismatches. Thus, these constraints guarantee
non-divergence. The first term of the objective function
(10a or 10b) is a penalty function that tends to reduce the
fictitious mismatches to zero. The solution is obtained in
two steps. In the first step, the objective is to minimize the
penalty expressed in terms of the ficticious mismatch
variables. If a solution to this problem does not exist, it
means that the system cannot support the present loads
and generation schedules without violating operating
limits. In this case the available transfer capability is zero.
If a solution exists, it means that there is some available
transfer capability. The second step of the solution
determines the available transfer capability for this
condition.
Evaluated Markov State
Not Evaluated Markov State
Figure 7. Symbolic Map of Markov States
The computation of above descriptions will be explained
with the aid of Figure 7. Specifically, each Markov state
in Figure 7 represents a contingency at a certain load level
which may have been evaluated or not evaluated. Each
evaluated state is described with a certain transfer
capability in MW, a certain probability and transition
rates to any other state. To each not evaluated
contingency, a STC value is assigned equal to the STC
value of the parent contingency with the same load level.
The not evaluated contingencies are also characterized
with a certain probability and transition rates. The
cumulative probability distribution is constructed by first
constructing a histogram of probability versus STC for all
states in Figure 7. Subsequent integration of the histogram
provides the cumulative probability distribution, denoted
with FTC(t). By definition:
5. Probabilistic Description of Simultaneous
Transfer Capability
The results of the enumeration approach are in the
form of a set of simulated contingencies. For each
contingency and load level a value for the simultaneous
transfer capability has been computed. These data are
combined with the Markov model of electric load, circuits
and units to yield a probabilistic description of the
simultaneous transfer capability.
FTC ( t ) = Pr[TC ≤ t ]
i.e. Pr[TC ≤ t ] is the probability that the transfer capability,
TC, is less or equal to the value t. The expected transfer
capability, TC , is computed by evaluating the integral:
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
TC =
∞
∫t = 0 tdFTC ( t ) =
∑ p jTC j
ranked first level contingencies using Jtc as PI. For
comparison, the OPF results are also listed in columns 6,
7, and 8. The table shows that the proposed power transfer
performance index, Jtc, can effectively select the
contingencies that may cause reduction of STC from the
base case. Note that the derivative of Jtc provides a rather
accurate prediction of the STC reduction. Table 3 also
lists the number of OPF iterations to convergence. Note
also that, in general, the contingency OPF requires less
iterations than the base case OPF.
(13)
j
where:
pj is the probability of Markov state j
TCj is the transfer capability of Markov state j (for a non
evaluated state j, the transfer capability is assumed to be
equal to the base case with the same load level as state j).
The probability, frequency and duration indices being
at a user-selected range are computed as follows. First, all
states in Figure 7 that have a STC in the specified range
are identified. The process yields a set, Sr, as it is
illustrated in Figure 7. The probability Pr, frequency, fr,
and duration, Tr, of STC being at the selected range are
given by:
Pr =
∑p
BUS 307
BUS 302
BUS 301
138 kV
BUS 305
BUS 308
F
BUS 304
BUS 310
BUS 303
BUS 309
BUS 306
BUS 312
Phase
Shifter
BUS 324
BUS 311
Area 30
E
(14)
j
BUS 313
BUS 315
BUS 314
j∈ S r
fr =
220 kV
∑ p ∑λ
j
j∈ S r
BUS 320
BUS 319
BUS 316
D
BUS 323
jk
(15)
A
B
k∉ S r
BUS 322
G
BUS 321
C
BUS 317
BUS 318
P
Tr = r
fr
Area 20
Area 10
(16)
BUS 118
BUS 117
BUS 218
C
BUS 121
G
6. Method Evaluation
BUS 116
C
BUS 221
BUS 222
B BUS 123
A
BUS 119
D
BUS 217
BUS 122
G
B
A
BUS 216
BUS 120
BUS 223
D
BUS 219
BUS 220
220 kV
BUS 114
This section presents example applications of the
method to two test systems. The first test system is the
24x3 bus, three areas system which is proposed as new
IEEE RTS and the second system is a 2182 bus, 8 area
system. The first system is illustrated in Figure 8. It
consists of three identical IEEE 24 bus RTS [19]
interconnected with five tie lines. The data for each RTS
can be found in [19]. The overall system consists of 72
buses, 96 generation units and 119 circuits. Area 10 is
defined as the study area and areas 20 and 30 are transfer
participating areas. The power transfer to the study area is
of concern.
The base case STC yielded a maximum net import of
1404.88 MW to the study area 10. The OPF convergence
performance to compute the base case STC is listed in
Table 1 and the final binding operating constraints are
listed in Table 2. From Table 1, we can observe that as the
OPF algorithm progressed, the mismatches were
gradually reduced, the binding operating constraints were
gradually introduced, and the power transfer gradually
increased. At the optimal solution, the mismatches were
reduced to below the convergence criteria (0.01 p.u.) and
the simultaneous power transfer to the study area was
limited by the tie line capacity as shown in Table 2.
Typical results of the proposed contingency ranking
procedure are shown in Table 3. The table lists the top 15
BUS 113
BUS 115
BUS 214
BUS 215
BUS 213
E
BUS 124
Phase
Shifter
BUS 103
E
BUS 111
BUS 112
BUS 109
BUS 110
F
BUS 104
BUS 106
BUS 224
Phase
Shifter
BUS 211
BUS 203
BUS 209
BUS 212
BUS 206
BUS 210
BUS 204
F
BUS 105
BUS 205
BUS 208
BUS 108
138 kV
BUS 101
BUS 102
BUS 107
BUS 201
BUS 202
BUS 207
Figure 8. Interconnection of RTS 24x3 Bus System
Table 4 provides the top 15 ranked first level
contingencies using performance index JP. We can
observe that the performance index, JP, is also effective in
identifying contingencies that lead to reduced STC. It
should be noted that the performance index, JP, can
identify some contingencies which can not be selected by
the performance index, Jtc. For example, the outage of a
unit at bus 315, which caused 29.52 MW reduction of
simultaneous transfer capability, was ranked 44th using
performance index Jtc, but it was ranked 13th by using JP.
Similar phenomena were observed using performance
index JV. The sequential use of all proposed performance
indices provides a high confidence level that all
contingencies that affect STC were captured.
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Table 4. Top 15 Ranked Contingency Using JP as
Performance Index
Table 1. Convergence Performance of the Proposed
OPF Model to Compute the Base Case STC
Iter
No
Total
Const.
In LP
0
1
2
3
4
5
6
7
8
9
10
11
0
1
1
1
1
1
2
4
6
7
7
7
Maximum
P
Mismatch
(p.u)
5.8657
5.2793
4.2265
2.9571
1.7758
0.8905
0.3587
0.1428
0.0572
0.0230
0.0100
0.0040
Maximum
Q
Mismatch
(p.u)
5.4313
4.8886
3.9181
2.7392
1.6456
0.8263
0.3336
0.1337
0.0523
0.0202
0.0110
0.0064
Rank Comp
Order n
Type
1
UNIT
2
CIR
3
CIR
4
CIR
5
UNIT
6
CIR
7
UNIT
8
UNIT
9
UNIT
10 UNIT
11 UNIT
12 UNIT
13 UNIT
14 UNIT
15 UNIT
Transfer
Power
(MW)
-13.92
132.70
387.95
684.98
964.35
1195.24
1363.98
1456.76
1406.11
1406.44
1405.46
1404.88
Circui
t
From
Bus
108
121
113
123
207
Circuit
To
Bus
Circuit
Rating
(MVA)
Circuit
Flow
(MVA)
Circuit
Overload
(MVA)
203
323
215
217
208
175.00
500.00
400.00
400.00
175.00
175.30
499.81
400.15
399.94
175.17
0.30
-0.19
0.15
-0.06
0.17
STC ≥ 1200 MW STC ≥ 1300 MW
Probability
0.89804 0.82036
-1
Frequency (yrs ) 5.3515 8.5090
Duration (yrs)
0.1678 0.0964
Table 3. Top 15 Ranked Contingencies Using Jtc as
Performance Index
Rnk Comp
Ord n
er Type
1 CIR
2 CIR
3 CIR
4 UNIT
5 UNIT
6 CIR
7 UNIT
8 UNIT
9 UNIT
10 UNIT
11 UNIT
12 UNIT
13 CIR
15 CIR
From To
OPF Compute
Bus Bus ID Iter d STC
(MW)
121 323 1 7 925.74
123 217 1 7 1017.48
113 215 1 7 1015.22
218
1 8 1101.78
318
1 7 1135.15
108 203 1 7 1248.36
221
1 8 1264.70
321
1 7 1313.23
323
3 7 1294.67
223
3 7 1346.32
215
6 10 1372.25
216
1 8 1373.60
314 316 1 8 1399.83
212 223 1 7 1404.17
Change
of STC
(MW)
-479.13
-387.39
-389.65
-303.09
-269.72
-156.51
-140.17
-91.64
-110.20
-58.55
-32.62
-31.27
-5.04
-0.70
Derivative
of JP
(MW)
3.691
3.470
2.816
2.798
2.730
1.987
1.965
1.473
1.249
1.237
1.237
1.237
0.919
0.763
0.606
The load model consisted of three load levels at 100%,
80% and 60% of peak load respectively. Combining the
Markov models of circuits, units and load levels and the
results of the OPF analysis, the probability distribution of
STC for area 10 was computed and illustrated in Figure 9.
For all evaluated contingencies, the STC ranged from 204
MW to 1405 MW. The expected STC is 1239.39 MW.
The simulation results can be utilized to compute many
other quantities of interest. As an example, The
probability, frequency and duration of simultaneous
transfer capability being greater than or equal to 1200
MW and 1300 MW are computed to be:
Table 2. Binding Operating Constraints (Base Case
OPF)
Cn
str
No
.
1
2
3
4
5
From To
OPF Computed Change
Bus Bus ID Iter STC
of STC
(MW)
(MW)
318
1 7 1135.15 -269.72
121 323 1 7 925.74
-479.13
123 217 1 7 1017.48 -387.39
113 215 1 7 1015.22 -389.65
218
1 8 1101.78 -303.09
108 203 1 7 1248.36 -156.51
321
1 7 1313.23 -91.64
223
3 7 1346.32 -58.55
221
1 8 1264.70 -140.17
207
1 6 1404.90 0.02
207
2 6 1404.87 -0.01
207
3 6 1404.87 -0.01
315
6 7 1375.41 -29.46
316
1 7 1369.08 -35.79
213
1 7 1404.88 0.01
Derivative
of Jtc
(MW)
-498.52
-408.50
-404.27
-286.23
-249.06
-189.59
-104.46
-79.20
-56.96
-49.11
-21.72
-18.25
-6.99
-4.15
The proposed method was also tested on an actual
large-scale system, a 2182 bus, eight-area system that
represents the Southern Company System and its
neighbor utilities. The total system peak load is 143535.9
MW. This system consists of 280 generation plants with
570 units. There are 3791 circuits in the system. Among
them 358 are transformers. The study system (Southern
Company system) consists of 1607 buses and has total
summer peak load of 30537.1 MW. For the given data,
the peak area export of the Southern Company system
was 2751.3 MW and the normal scheduled net area export
was 1725.1 MW.
The base case STC was computed by the proposed OPF to
be 3071.44 MW (exported from the Southern Company
system to the other 7 areas). The maximum STC was
limited by the generation capacity of the Southern
Company system while the tie line capacity was
sufficient.
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Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
proposed method to an actual large-scale power system
has demonstrated that the proposed probabilistic
simultaneous transfer capability analysis method is
practical and suitable for practical systems.
0.8
0.6
1.0
Probability of Exceeding Abscissa
Probability of Exceeding Abscissa
1.0
0.4
0.2
0.0
176
351
527
702
878
1054
1229
1405
Simultaneous Transfer Capability (MW)
Figure 9. Cumulative Probability Distribution of STC
for the RTS 24x3 System
0.8
0.6
0.4
0.2
0
384
767
1151
1535
1919
2302
2686
3070
Simultaneous Transfer Capability (MW)
The proposed method was implemented to compute the
probabilistic description of STC for this system. A
contingency depth of two was used (any combinations of
two units or one unit and one circuit outages). The cut-off
probability was set to 0.00000001 and the summer peak
load level was used (one load level). The cumulative
probability distribution of STC was computed and it is
illustrated in Figure 10. The probability, frequency and
duration of STC being greater than or equal to the normal
net export (1725 MW) and peak net export (2751 MW)
were computed to be:
Figure 10. Cumulative Probability Distribution of
STC for the 2182 bus, eight-area System
The proposed probabilistic transfer capability analysis is
useful for assessing the impact of component availability
uncertainty and market uncertainties on the Available
Transfer Capability and the security of the system.
Component availability uncertainty includes the
traditional causes of independent and common mode
failures. The proposed method is also useful for assessing
the effects of the uncertainty associated with the market
participants, i.e. (a) uncertainty in bid acceptance
procedures, (b) customer response to prices, (c) control of
interruptible loads, etc. The uncertainty must be assessed
in real time and the effects of uncertainty must be
quantified for the next few hours. Quantitative results are
important for two reasons: (a) control of ATC and
minimization of associated risks and (b) cost/benefit
analysis of future transmission projects.
STC ≥ 1725 MW STC ≥ 2751 MW
Probability
0.95422 0.48004
-1
Frequency (yrs ) 3.1228 6.8060
Duration (yrs)
0.3055 0.0705
7. Conclusions
This paper presented a formulation of the available
transfer capability in a deregulated environment. A
rigorous formulation has been presented that determines
the ATC through a user specified transfer interface by
modeling all pertinent aspects of power system operation
under an ISO. A deterministic as well as a probabilistic
approach is proposed. The deterministic approach
provides the ATC for a specific operating condition of the
system. The probabilistic approach is based on a Markov
model of system components and provides the expected
value of ATC as well as frequency and duration of
specific ATC values. The Markov states (contingencies)
are selected via multiple performance indices. Selected
contingencies are simulated via an optimal power flow
that determines the maximum simultaneous transfer
capability for a given contingency and load level. Both
procedures, contingency selection and OPF are
computationally efficient. Results on two nontrivial test
systems are given in the paper. The implementation of the
8. References
[1] G. L. Landgren, H. L. Terhune and R. K. Angel,
"Transmission Interchange Capability - Analysis by Computer,"
IEEE Transactions on Power Apparatus and Systems, Vol.
PAS-91, No.6, Nov/Dec, 1972, pp. 2405-2413.
[2] A. P. Sakis Meliopoulos and Feng Xia, "Simultaneous
Transfer Capability Analysis: A Probabilistic Approach,”
Proceedings of the 11th Power System Computation
Conference, Vol. 1, pp. 569~576, Avignon, France, August 30 ~
September 3, 1993.
[3] G. L. Landgren and S. W. Anderson, "Simultaneous Power
Interchange Capability Analysis," IEEE Transactions on Power
Apparatus and Systems, Vol. PAS-92, No.6, Nov/Dec., 1973,
pp.1973-1986.
9
0-7695-0493-0/00 $10.00 (c) 2000 IEEE
9
Proceedings of the 33rd Hawaii International Conference on System Sciences - 2000
[4] G. T. Heydt and B. M. Katz, "A Stochastic Model in
Simultaneous Interchange Capacity Calculations," IEEE
Transactions on Power Apparatus and Systems, Vol. PAS-94,
No. 2, March/April 1975, pp. 350-359.
[17] Xingyong H. Chao, Nondivergent and Optimal Power Flow
- A Unified Approach, Ph. D. Thesis, Georgia Institute of
Technology, Atlanta, Georgia, September, 1991.
[18] EPRI Report, Reliability Evaluation for Large-Scale Bulk
Transmission Systems, Volume 1: Comparative Evaluation,
Method Development, and Recommendation, Project 1530-2,
EPRI EL-5291, January 1988.
[5] PJM Transmission Reliability Task Force, "Bulk Power Area
Reliability Evaluation Consider Probabilistic Transfer
Capability," IEEE Transactions on Power Apparatus and
Systems, Vol. PAS-101, No. 9, September 1989, pp. 3551-3562.
[19] IEEE Committee Report, "IEEE Reliability Test System,"
IEEE Transactions on Power Apparatus and Systems, Vol.
PAS-98, No. 6, November/December 1979, pp. 2047-2054.
[6] M. G. Lauby, J. H. Douda, R. W. Polesky, P. J. Lehman and
D. D. Klempel, "The Procedure Used in the Probabilistic
Transfer Capability Analysis of the MAPP Region Bulk
transmission System," IEEE Transactions on Power Apparatus
and Systems, Vol. PAS-104, No. 11, November 1985, pp. 30133019.
[20] Feng Xia and A. P. Sakis Meliopoulos, ‘A Methodology for
Probabilistic Simultaneous Transfer Capability Analysis,' IEEE
Transactions on Power Systems, Vol. 11, No. 3, pp. 1269-1278,
August 1996.
[7] IEEE Committee Report by the Current Operatinal Problems
Working Group, "Problems in Coping with the Proliferation of
Interchange Schedules," IEEE Transactions on Power Systems,
Vol. PWRS-2, No.4, November 1987.
9. Biographies
A. P. Sakis Meliopoulos (M '76, SM '83, F '93) was born
in Katerini, Greece, in 1949. He received the M.E. and
E.E. diploma from the National Technical University of
Athens, Greece, in 1972; the M.S.E.E. and Ph.D. degrees
from the Georgia Institute of Technology in 1974 and
1976, respectively. In 1971, he worked for Western
Electric in Atlanta, Georgia. In 1976, he joined the
Faculty of Electrical Engineering, Georgia Institute of
Technology, where he is presently a professor. He is
active in teaching and research in the general areas of
modeling, analysis, and control of power systems. He has
made significant contributions to power system
grounding, harmonics, and reliability assessment of power
systems. He is the author of the books, Power Systems
Grounding and Transients, Marcel Dekker, June 1988,
Ligthning and Overvoltage Protection, Section 27,
Standard Handbook for Electrical Engineers, McGraw
Hill, 1993, and the monograph, Numerical Solution
Methods of Algebraic Equations, EPRI monograph series.
Dr. Meliopoulos is a member of the Hellenic Society of
Professional Engineering and the Sigma Xi.
[8] S. L. Daniel, Jr., M. G. Lauby and G. Maillant, "Multiarea
Transfer Capability: Is a Methodology Needed?," IEEE
Computer Applications in Power, October, 1989, pp. 18-21.
[9] EPRI Report, "Proceedings: Power System Planning and
Engineering - Research Needs and Priorities", EL-6503, Final
Report, October 1989.
[10] EPRI Report, "Simultaneous Transfer Capability Project:
Direction for Software Development," Final Report to RP31401, January 1991.
[11] P. Sandrin, L. Dubost and L. Feltin, “Evaluation of Transfer
Capability Between Interconnected Utilities,” Proceedings of
the 11th Power System Computation Conference, Vol. 1, pp.
981~985, Avignon, France, August 30 ~ September 3, 1993.
[12] G. C. Ejebe and B. F. Wollenberg, "Automatic
Contingency
Selection",
IEEE Transactions on Power
Apparatus and Systems, Vol. PAS-98, No. 1, pp. 92-104,
Jan./Feb. 1979.
[13] A. P. Sakis Meliopoulos and C. Cheng, "A hybrid
contingency selection method", Proceedings of the 10th Power
Systems Computation Conference, Graz, Austria, pp. 605-612,
Aug. 1990.
George Cokkinides (M '85) was born in Athens, Greece,
in 1955. He obtained the B.S., M.S., and Ph.D. degrees at
the Georgia Institute of Technology in 1978, 1980, and
1985, respectively. From 1983 to 1985, he was a research
engineering at the Georgia Tech Research Institute. Since
1985, he has been with the University of South Carolina
where he is presently an Associate Professor of Electrical
Engineering. His research interests include power system
modeling and simulation, power electronics applications,
power
system
harmonics,
and
measurement
instrumentation. Dr. Cokkinides is a member of the
IEEE/PES and the Sigma Xi.
[14] O. Alsac, J. Bright, M. Paris, and B. Stott, "Further
Developments in LP-Based Optimal Power Flow," IEEE
Transaction on Power Systems, Vol. PWRS-5, No. 3, pp. 697711, August 1990.
[15] A. P. Meliopoulos, G. J. Cokkinides, and Xing Yong Chao,
"A New Probabilistic Power Flow Analysis Method," IEEE
Transactions on Power Systems, vol. 5, no. 1, pp. 182-190,
February 1990.
[16] A. P. Meliopoulos, G. Contaxis, R.P. Kovacs, N.D. Reppen,
and N. Balu, "Power System Remedial Action Methodology",
IEEE Transaction on Power Systems, Vol. PWRS-3, No. 2, pp.
500 - 509, May 1988.
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