1
Coordination of Transmission Path Transfers
Yuan Li and Vaithianathan Venkatasubramanian
Abstract-- The transfer capability on a transmission path is
limited by constraints on acceptability, voltage security, smallsignal stability and transient stability. For a large interconnected
power grid, these constraints are influenced significantly by the
interactions among path flows in different control areas. When a
critical transmission path capability is limited by one of these
constraints, it may be necessary to coordinate the interarea power
transfers so as to improve the transfer capability on the
constrained path without compromising on the security criteria.
Based on such considerations, this paper presents a novel
multiobjective methodology in which global strategies are
developed for the improvement and coordination of transmission
path transfers. The problem is formulated with respect to various
constraints into suitable optimization problems. An efficient
nonlinear programming algorithm with sufficient line search step
is incorporated for finding optimal solutions while also
incorporating security and stability constraints. The MW benefits
for the transfer capability from the coordination procedure are
explicitly demonstrated after the optimization process. The
effectiveness of the methodology is illustrated by case studies on
improving the capability of the California-Oregon Intertie (COI)
for large-scale WECC western American power system models.
Index Terms-- Transfer capability, transient stability, voltage
security, small signal stability, nonlinear optimization,
multiobjective programming
I. INTRODUCTION
Wtransactions are processed across different control areas
ITHIN the context of deregulated power market, more
as the power system gets more stressed with increasing loads.
However, the power flows between these areas are often
limited by various mechanisms, such as stability constraints,
voltage bounds and thermal limits. The maximum power that
could be transferred in a reliable fashion over any transmission
line is described by NERC (North American Electric
Reliability Council) as transfer capability [1].
Owing to the inherent trade-off between increasing
This work was supported by funding from Power Systems Engineering
Research Center (PSerc), and by Consortium for Reliability Technology
Solutions (CERTS), funded by the Assistant Secretary of Energy Efficiency
and Renewable Energy, Office of Distributed Energy and Electricty
Reliability, Transmision Reliability Program of the U.S. Department of
Energy under Interagency Agreement No. DE-AI-99EE35075 with the
National Science Foundation. Partial funding of the work from Bonneville
Power Administration is also gratefully acknowledged.
Yuan Li is with the School of Electrical Engineering and Computer
Science, Washington State University, Pullman, WA 99163 USA (Email:
yli1@eecs.wsu.edu).
Vaithiananthan “Mani” Venkatasubramanian is with the School of
Electrical Engineering and Computer Science, Washington State University,
Pullman, WA 99163 USA (Email: mani@eecs.wsu.edu).
utilization of the grid and security of operation, many research
efforts have been carried out to enhance the transfer capability
without violating those operation constraints. Generally, some
approaches are categorized as Voltage Security-Constrained
Optimal Power Flow (VSC-OPF) [2,3] or Dynamic Stability
Constrained Optimal Power Flow (DSC-OPF) [4] problems
regarding dynamic security or stability limitations. Other
approaches dealing with thermal constraints are usually
associated with traditional OPF problems. Studies in [2,3]
incorporated voltage stability criteria into OPF formulation by
virtue of maximum loading distance. Linearly combined
objective functions and goal programming are implemented to
solve this multiobjective programming problem. Reference [4]
designed preventive control and correction control in DSCOPF through a three level hierarchical decomposition scheme
where each sub-problem is solved by interior point method. In
[5], the authors presented a framework to assess MW transfer
limit concerning both the small-signal stability constraint and
environmental objectives using a bicriterion programming
approach. Eigenvalue sensitivity based constraints are
evaluated into simulated annealing optimization scheme. In
[6], the authors developed a nonlinear optimization based
methodology to assess transfer capability in presence of static
and dynamic constraints. Regarding the thermal limits, various
approaches [7, 8] arise from congestion management to relieve
the heavy loading situations on congested transmission
flowgates.
Unfortunately, most existing methodologies only address
some specific transmission paths within a single control area.
Coordination across several areas is rarely taken into account
owing to the different ownership of these paths. Since modern
power grid consists of multiple entities interconnected tightly
with each other, the flows on one transmission provider’s
system will interact with other flows and will change the
resulting transfer capability of another transmission system by
impacting on the operation and security of other systems [9].
Therefore, it is attractive to study the path transfer capability
while also considering the interactions with other parts of the
entire power grid.
In this paper, a global framework is proposed for
coordinating the MW transfers of several transmission paths,
while also meeting the regulatory requirements on voltage
security and dynamic security. As an example, we focus on
maximizing the transmission capacity of the California-Oregon
AC Inter-tie (COI), by coordinating other path-flows that have
an impact on the COI capacity. We show that substantial
improvements in the COI MW transfer can be achieved with
reasonable rescheduling of neighboring tie-line flows using the
2
optimization algorithms presented. The optimal solutions
suggest which other transmission paths significantly impact on
the transmission capability of a specific path under
consideration. The procedure also suggests potential
redispatch of other transmission paths under highly stressed
operating conditions to increase the transfer capability of a
critical path such as COI. We want to emphasize the fact the
capability coordination is carried out while also incorporating
the constraints on security, which makes the formulation a
challenging optimization problem for the large system.
We discuss the formulations and optimizations in the
following sections of the paper by illustrating the procedure on
the COI path in WECC. The optimization methods presented
below can be applied to general large power systems.
II. PROBLEM DESCRIPTION AND FORMULATION
Typically, all individual transmission paths and subsystems
should comply with four constraints, i.e., transient stability,
small-signal stability, voltage security and thermal limits.
Energy flows over COI plays such an indispensable role in
supplying power to California that all the stability and security
indices will be investigated respectively. The system topology
is demonstrated in Fig.1, where COI flow is transmitted from
Northwest area (NW) to Pacific Gas and Electric (PG&E).
Other four main transfer paths from Northwest to its adjacent
control areas B.C.HYDRO (BC), WEST KOOTENAY (WK),
IDAHO (ID) and MONTANA (MT) should also be taken into
account because the interactions among these interarea tie line
flows will affect the transfer capability over COI directly. In
the following studies, we have kept the power transfer over the
Pacific HVDC inter-tie (PDCI) at constant values to
emphasize the interactions of the AC transmission paths. PDCI
path-flow can also be readily incorporated into the
optimization formulations if so required.
Fig 1 Illustration of Northwest and neighboring power grids
A. Problem Objectives
The primary concern is to transfer more real power over
COI as long as the operation constraints are not violated. The
COI power could be easily controlled by directly adjusting the
generations from NW and PG&E. However, due to the
complexity of large power network, there is no closed form
expression for the stability or security constraints and hence
direct optimization schemes cannot be applied.
Alternatively, a novel methodology is proposed in our work
by dividing the overall task into two stages. First, certain
stability or security indices are maximized by coordinating the
tie line flows from NW to adjacent control regions while the
COI MW flow is kept constant. Then, with the new
dispatching schedule in BC, WK, ID and MT, as suggested by
the optimal solution in the first stage, the COI MW flow is
increased until the system stability or security level reaches the
lowest allowable operation limit. This way, the improvement
in COI transmission capability from the coordination process
can be found by calculating the difference of COI flows for the
original system and the optimized system.
There are basic differences between our coordination
strategy and congestion management. Our strategy originates
from optimal dispatching and copes with several stability
constraints while improving the transmission capability over
critical paths or interfaces. In contrast, congestion management
attempts to adjust transactions systematically to ensure that
security constraints are met [20]. Especially, most existing
methods are proposed for thermal congestion management. In
regulated industry, generations are dispatched by using
distribution factors to estimate the effect of various units on
the line constraints, while in deregulated industry, the
transmission constraints are reflected in congestion prices to
manage the transactions in a reliable range.
B. Voltage Security Constraints
Voltage stability can be related to saddle-node bifurcations.
In [2,3], bifurcation parameter such as “loading factor” is
optimized to improve the stability scenarios. Additionally,
reactive power margins (QV margin) at certain critical buses
also indicate the voltage security level to some extent [10].
WECC voltage security criteria require post-contingency
reactive power margins to be above certain values at critical
buses. Hence, our problem is formulated to maximize QV
margin at the Malin bus, which is at the border of California
and Oregon on the COI intertie. From operating experience,
QV margin at Malin is a good criterion for voltage security for
contingencies affecting COI capability. Therefore, we try to
maximize Malin QV margin using the interarea transfer path
flows as the control variables, while keeping the COI flow as
constant. The neighboring MW transfer flows into NW are
controlled by redispatching the net area exports in the
respective control regions. NW area export is treated as a
dependant variable to keep the COI power flow at a constant
value. The objective during this stage is to improve the voltage
security at Malin by adjusting the neighboring area exchange
levels to NW. The optimization suggests which transfers
significantly influence the QV margin at Malin. Once the
transfers are adjusted to the new values determined from the
optimization process, the COI transfer capability can be
increased to higher values while maintaining the same QV
margin at Malin as before the optimization process. We denote
the resulting improvement in COI transfer as MW benefit, a
standard terminology in WECC, and examples are shown in
the section IV.
3
C. Small-Signal Stability Constraints
In large interconnected power systems, undamped or poorly
damped interarea oscillations can jeopardize the operation
significantly. The western blackout that occurred in WECC on
August 10, 1996 [11] is a classic example of small-signal
instability when the 0.25 Hz interarea mode became negatively
damped. Therefore, to improve the COI capacity with respect
to small-signal stability limitations, we maximize the damping
ratio of the 0.25 Hz interarea mode by coordinating MW
transmissions from neighboring areas. The method can be
modified to handle other interarea modes as relevant for
specific systems.
D. Transient Stability Constraints
COI capability is limited by transient stability constraints
for certain operating conditions. Based on past experiences,
the simultaneous outage of two nuclear units at Palo Verde
(denoted PV2) is one of the most severe contingencies that
impact on COI capability. WECC criterion requires the bus
voltages to stay above certain critical values in the transient
stability simulations of PV2. Based on observations from
typical cases, the transient stability performance is evaluated
quantitatively by investigating the lowest voltage magnitude at
Malin during the first swing after the PV2 contingency. Like in
the previous constraints, we will maximize the bus voltage dip
at Malin during the first swing following PV2, by formulating
the adjacent area transfer flows into NW as the control
variables. We will keep the COI flow constant during this
optimization by adjusting the NW net export as a dependant
variable like before. The proposed strategy of maximizing the
Malin voltage dip as a transient stability optimization index is
a novel contribution of this paper, and it proves to be an
effective formulation for improving transient stability
constraints on COI. We show that this strategy works well by
computing the COI capability MW benefits after the
redispatch as suggested by the optimization procedure.
E. Thermal Limit
Thermal limit or rating, which is also referred to as
“transmission capacity”, describes the maximum power flow
or current over a particular transmission component [1]. In our
coordination strategy, it can be handled with less difficulty by
taking advantage of traditional OPF algorithms. Furthermore,
observing that the thermal limit is not the crucial factor
restricting the COI flow capability in the western power grid, it
is not discussed in our research.
F. Economic Concerns
In power markets with open transmission access, MW
generations with less price or cost will be more attractive for
the customers. Assuming there are different generation costs in
each area, the overall expense associated with redispatching
will fluctuate with the new generation schedules. Possibly, the
overall generation costs will increase when we carry out the
optimization of QV margin in Part B by the redispatching of
transfer path flows. To address these concerns, a bicriterion
optimization scheme is proposed later so that the additional
expenses over the existing contracts can be minimized,
together with the maximization of voltage security level as the
second objective simultaneously.
G. Software Applications
The computations involve four commercial software
engines: 1) Bonneville Power Administration (BPA) Powerflow program pf, 2) Electric Power Research Institute (EPRI)
midterm transient stability program ETMSP, 3) EPRI output
and PRONY analysis program OAP, and 4) EPRI small-signal
stability program PEALS. The optimization algorithms have
been developed as tailor made solutions to this severe implicit
nonlinear problem. The optimization is implemented in the
form of Unix Shell routines, which interface with the
commercial engines mentioned above. The programs were
tested on few standard WSCC planning cases. These are
realistic large-scale representations of the western grid
consisting of about 840 generators and 6300 buses. The
project objective has been to demonstrate the feasibility of the
computations for large-scale systems. Computational
efficiency was not a priority. Typically, the algorithms
converged to near optimal solutions in a few iterations. The
results are dependent on power-flow scenarios and on
contingency cases being considered.
III. OPTIMIZATION ALGORITHMS
A. Optimization Model
Problems formulated regarding voltage security or dynamic
stability constraint can be solved with typical nonlinear
programming algorithms.
max E ( p ) or min f ( p ) = − E ( p )
p∈ P
s.t.
p∈ P
F ( x, p ) = 0
x min ≤ x ≤ x max
p min ≤ p ≤ p max
(1)
where the objective function E(p), i.e., lowest voltage
magnitude, QV margin or critical damping ratios will be
maximized respectively on condition that the equality
constraints such as power flow equations are satisfied.
Vector p and x represents the control variables and other state
variables.
In BPA-pf, the interarea tie line flow can be changed
conveniently by altering the net area exports of corresponding
areas. Thus the control vector p consists of four components
indicating the MW exports from BC, ID, MT and WK.
Moreover, the export from NW is considered as a dependent
variable in order to keep the COI flow constant and the total
MW output of the system balanced.
B. BFGS (Broyden-Fletcher-Goldfarb-Shanno) Algorithm
Owing to enormous computational burden involved in
solving the optimization problems for the large power system,
algorithms with less complexity and better convergence are
taken into account in higher priority. For such implicit
objective functions, some derivative-free algorithms, such as
downhill simplex method [12] and multiple direction search
algorithm [13] seem to be more competitive by avoiding the
heavy computation in gradient vectors. However, such
4
advantages are cancelled out by the difficulties in proper initial
value determination and the unaffordable searching process.
Therefore, BFGS algorithm, an efficient and robust method of
quasi-Newton family [14], is employed with the updating
scheme given as:
(2)
p k +1 = p k − α k H k ∇ f ( p k )
Here, ∇f(pk) is the gradient vector of the objective function (1)
at k-th iteration computed by center differentiation (3).
f ( p + ∆p i ) − f ( p )
∂f ( p )
(3)
∇f ( p) =
=
∂ p i n ×1
∆p i
n ×1
∆pi is selected to be from 1% to 5% perturbation value on ith entry pi of the control variable vector. αk is the optimal step
length determined by line search at the k-th iteration. The
approximation of Hessian matrix Hk is a positive definite
matrix that can be adjusted according to (4).
(4)
H k +1 = ( I − ρ k s k y kT ) H k ( I − ρ k y k s kT ) + ρ k s k s kT
Where
s k = p k +1 − p k
(5)
y k = ∇f k +1 − ∇f k
1
ρk = T
y k sk
(6)
(7)
H0 is an arbitrary positive definite matrix. In the updating
scheme, the curvature condition should also be satisfied:
(8)
s kT y k > 0
Another attractive property of BFGS is its effective selfcorrecting performance. If Hk incorrectly estimates the
curvature in the objective function in which the bad estimation
slows down the iteration, then the Hessian approximation will
tend to correct itself within a few steps.
C. Line Search Step
The merits of BFGS algorithm, such as convergence and
self-correcting property, hold only when an adequate line
search is performed. In particular, as long as the Wolfe or
strong Wolfe conditions (9a) and (9b) are imposed in the line
search part, all the requirements to implement Quasi-Newton
algorithms are met automatically.
With the same denotation of f and p as objective function
and control vector, let d be a descent direction for f, then the
line search step αk would be acceptable if it meets the two
inequalities of Wolfe conditions:
(9a)
f ( p k + α k d k ) ≤ f ( p k ) + c1α k ∇f kT d k
(9b)
d ∇f ( p k + α k d k ) ≥ c 2 d ∇f k
for some constants c1 ∈ (0,1) and c 2 ∈ (c1 ,1)
Condition (9a) forces a sufficient decrease in the function.
However, this condition is not sufficient to guarantee
convergence, because it allows arbitrarily small choices of any
positive αk. Condition (9b), also called curvature condition,
rules out such small choices and usually guarantees that αk is
near a local minimizer of f(pk+αk•dk). To avoid getting lost in
the details and to save space, the idea of the algorithm is
presented briefly in this paper. The details are referred to the
study in [15] by Mor and Thuente.
T
k
T
k
The algorithm consists of a two-phase process: A search
phase generates an interval containing desirable step lengths,
then the interval is updated with a trial value which is selected
by interpolation or extrapolation to approach a good step
length. The basic steps can be stated as follows:
• Search Algorithm
Set I 0 = [0, ∞]
For k = 0,1...
Choose α k ∈ I k ∩ [α min,α max ] .
Test for convergence.
Update the interval.
• Updating Algorithm
Define Φ (α ) = f ( p + α ⋅ d ) . Given a trial value α t in I ,
the end points α l+ and α t+ of the updated interval I + are
determined with following procedure:
Case 1: If Φ (α t ) > Φ (α l ) , then α l+ = α t and α u+ = α u ;
Case 2: If Φ (α t ) ≤ Φ (α l ) , and Φ '(α t )(α l − α t ) > 0 , then
α l+ = α t and α u+ = α u ;
Case 3: If Φ (α t ) ≤ Φ (α l ) , and Φ '(α t )(α l − α t ) < 0 , then
α l+ = α t and α u+ = α l ;
• Trial Value Selection
Assuming that besides the endpoints α l and α u , and current
trial point α t , function values f l , f u , f t and derivatives
g l , g u , g t are also available. The new trial value is solved in
terms of α c (the minimizer of the cubic that interpolates
f l , f t , g l and g t ), α q (the minimizer of the quadratic that
interpolates f l , f t and g l ), and α s (the minimizer of the
quadratic that interpolates f l , g l and g t ):
Case 1: f t > f l
α t+ =
αc
if α c − α l < α q − α l
(α q + α c ) / 2
o/w
Case 2: f t ≤ f l and g l g t < 0
α t+ =
α c if α c − α t ≥ α s − α t
αs
o/w
Case 3: f t ≤ f l , g l g t ≥ 0 and g t ≤ g l
α t+ =
α c if α c − α t < α s − α t
αs
o/w
Case 4: f t ≤ f l , g l g t ≥ 0 and g t ≥ g l
α t+ is chosen as the minimizer of the cubic that interpolates
f u , f t and g u , g t
The Wolfe line search conditions ensure that the gradients
are sampled at points that allow the model to capture
appropriate curvature information. Hence condition (8) holds
automatically. Moreover, Hk+1 is positive definite whenever Hk
is positive definite. Typically we can simply set H0 to be
identity matrix, or a multiple of the identity matrix, where the
multiple is chosen to reflect the scaling or sensitivities of the
5
variables. As a result the convergence of the algorithm is
guaranteed. Of all known Quasi-Newton updates, the BFGS
algorithm with Wolfe line search step has so far the best
performance in practice.
D. Multiobjective Programming
The combination of voltage security and generation expense
objectives call for the solution from Pareto set [16] in the
decision making. To determine this set, one traditional
approach [17] is to optimize a super-objective function created
by each conflict objective functions with various weights:
min w1 E1 + w2 E 2
(10)
where E1 and E2 represent the two decision objectives Q
margin and economical expense, while weights w1 and w2 of
the objectives are consistent with decision maker’s interest. In
particular, we set w1 + w2 = 1 to normalize the objectives.
With different setting of weights, BFGS algorithm is applied to
find the optimal value respectively. The final solution chosen
from the Pareto set will totally depend on the operator’s
preference and experience, which may lead to arbitrary
decisions resulting from the complex nature of the multiple
criteria and difficulty in evaluating performance of power
system precisely.
E. Penalty Function
Once “soft constraints” are taken into account, such as the
upper or lower limit of the MW generation of certain areas or
some power plant, adjustment has to be applied by additional
penalty term to the objective function:
(11)
E p ( p) = ( p − pl )′Dl ( p − pl ) + ( pu − p)′Du ( pu − p)
TABLE 1
ORIGINAL SCHEDULED AREA EXPORTS
AND OPTIMIZED AREA EXPORTS
Area
BC
ID
MT
WK
NW
Original
Plan(MW)
1700
399
595
70.2
6520
Optimized Generation Plan (MW)
Base case
PV2
DB2
1743
1548
1758
864
597
774
830
584
860
160
-207
337
5687
6763
5556
Fig. 2. QV Curve in the base case
where
p: control variable vector
pl: lower bound vector for control variables
pu:upper bound vector for control variables
Du, Dl: diagonal weighting matrix
As a matter of fact, the penalty function in (11) will only
influence the objective function when the constraints are
violated and thus prevent the control variables from an
unacceptable level.
IV. CASE STUDY
A. Voltage Security (1998 Spring Peak WECC Case)
For improving voltage security, QV margin at Malin bus is
maximized by using the net area exports in the areas BC, ID,
MT and WK as the control variables. COI flow is kept
constant by adjusting the value of dependent variable, i.e., the
area export in the Northwest area. Therefore, we are
effectively modifying the power transfers from BC, ID, MT
and WK to NW as the control variables in the optimization
while COI flow remains unchanged. Table 1 summarizes the
adjustment of area exports after the optimization procedure.
Fig. 2 and Fig. 3 illustrate the corresponding improvements in
the Malin QV margin from the coordination of the
transmission path flows to Northwest.
Fig. 3. QV Curve in the contingency cases
For the base case, QV margin can be improved by 170
MVAR. In contingency studies, we consider two cases,
double Palo Verde outage (PV2) discussed earlier, and double
Diablo outage (DB2) contingency which is the simultaneous
outage of two Diablo nuclear units in California. BPA powerflow program pf is used for evaluating the QV margin at Malin
and governor control actions are included in the postcontingency power-flow solutions so that the other generators
will pickup the dropped generation with their reserves after the
contingency events. For the two contingencies, PV2 and DB2,
the Malin QV margins are improved by 140 MVAR and 220
MVAR, respectively. Although the improvement appears to be
only a small proportion of the total QV margin, it is of great
importance to enhance the system voltage security under such
6
severe contingency situations. Hence, our optimization
algorithm has proved to be successful and the efficiency is
verified.
We also recognize that different tie-line flows (or regional
energy generations) will impose different effects on power
system performance. For example, the importance of Idaho is
observed since it contributes the most among all the control
areas, both in the base case and in contingency cases, while the
activity in BC.Hydro is not that significant to improve the
entire system performance.
Next, we calculate the MW benefits for the COI transfer
from the optimization process when it is constrained by
voltage security. We assume that the minimum acceptable QV
margin at Malin is 400 MVAR for all the cases, which is the
criterion used in WECC for all double contingencies. We
compute the COI transfer limit when the Malin QV margin
goes below 400 MVAR for the original set of area exchange
values and the optimized transfer values from the optimization
process. The improvement in COI transfer capability is then
the net MW benefits from the optimization procedure, while
maintaining the same level of voltage security. The computed
benefits for the PV2 contigency cases with different single
component out of service scenarios are tabulated in Table 2.
The first row in Table 2 corresponds to the PV2
contingency starting from the base case that was shown in
Table 1. The other cases assume one equipment to be out-ofservice first as listed below, and then, the PV2 contingency is
applied. Table 2 only summarizes the net MW benefits and the
optimized changes to the area interchanges for each of these
cases are not listed to save space.
TABLE 2
MW BENEFITS FOR COI CAPABILITY
WITH VOLTAGE SECURITY CONSTRAINTS
Case
0
1
2
3
4
5
6
COI limit(MW)
4689
4738
4705
4653
4578
4537
4527
COI limit (New)(MW)
4765
4828
4742
4696
4655
4746
4639
Benefits (MW)
76
90
37
43
77
109
112
The cases tabulated in Table 2 are:
Case 0: Base case
Case 1: Chief Joseph generator out of service
Case 2: McNary generator out of service
Case 3: Lower Monumental generator out of service
Case 4: Ashe to Lower Monumental 500 kV transmission line
out of service
Case 5: Hanford to Coulee 500 kV transmission line out of
service
Case 6: McNary to Sacajawea 500 kV transmission line out of
service
In summary, the benefits may vary with different system
conditions. However, for most of the contingency cases, it is
still possible to make considerable improvement on COI
capability limit from the coordination procedure.
B. Small-signal Stability (1998 Spring Peak WECC Case)
In this section, we will coordinate the area transfers to
improve the damping of the 0.25 Hz COI mode. The high
dimension, strong nonlinearity and complexity of
interconnected power system make it impossible to carry out a
full eigenvalue solution for WECC large power system
models. Furthermore, as far as critical interarea modes are
concerned, existing commercial software for small-signal
stability analysis, such as PEALS, appear to be inconsistent in
computing the damping of the 0.25 Hz WECC mode [18] for
different COI transfer levels. Alternatively, we estimate the
0.25 Hz damping level using the Prony analysis [19, 21] by
simulating a small disturbance on ETMSP as follows:
1) Calculate a power flow solution with BPA-pf to set up the
initial condition for the ETMSP simulation.
2) Perform an ETMSP transient stability type simulation for
say 30 seconds.
3) Identify the damping of the 0.25 Hz COI mode from
PRONY analysis in the EPRI program OAP over a fixed
time window. Adjustments need to be made when
PRONY provides two modes of nearly same frequency,
and the details can be seen in [19].
In our study, two types of disturbance and two types of
control options are adopted on the system individually as
follows.
(1) Case 0: Single 500 kV COI transmission line Malin to
Round Mountain is opened and is reclosed after a short outage
duration to simulate a small disturbance.
(2) Case 1: Double Palo Verde outage contingency (PV2) is
explored as a large disturbance. Area exports for BC, ID, MT,
WK, and NW are used as the optimization variables in this
case. Again, the net exports in BC, ID, MT and WK are set up
as control variables while NW net export is a dependant
variable to keep COI flow constant.
(3) Case 2: To illustrate the effectiveness of the proposed
optimization for redispatching generations within a single area,
specific generator outputs of NW area are altered for the PV2
contingency case. Besides McNary, Chief Jo and John Day,
three other plants selected are Centralia, Wanapum and Lower
Monumental plants.
Optimization algorithm is applied and subsequently, the
MW benefits are obtained for different disturbances and
control options. The optimization process is terminated when
either the convergence criterion is reached or when stability
constraints are violated.
Optimized generation plans are suggested in Table 3 from
the optimization process. Soft constraints on overall regional
generation capacity are counted.
It should be noticed from Table 3 that the critical damping
mode has been improved significantly through the
optimization and coordination process. The outcome also
shows that the interarea damping mode is very sensitive to
MW transmissions. The data in Table 3 also indicates that a
control action such as generation redispatch within the NW
area itself can influence the system small-signal stability
performance significantly. The main generators in NW, such
as McNary, Chief Jo and John Day have larger capacity and
closer electrical distance to COI than those generators in MT
7
and ID. Thus, they appear to be more critical in determining
the damping of the 0.25 Hz interarea damping mode.
TABLE 3
ORIGINAL SCHEDULED AREA GENERATION AND OPTIMIZED
GENERATION PLANS
Case
0
1
2
Control
Variables
BC
ID
MT
WK
NW
BC
ID
MT
WK
NW
McNary
Chief Jo
Low.Mon.
Wanapum
John Day
Centralia
Before
Optim
(MW)
1700
399
595
70.2
6520
1700
399
595
70.2
5520
900
550
700
474
2325
670
After
Optim
(MW)
2028
707
659
-230
6125
1637
639
554
-10
5463
1002
531
712
478
2507
708
Damping
Ratio
(Before)
Damping
Ratio
(After)
0.06475
0.07697
0.04997
0.06185
0.04997
0.06334
coordination of generation dispatches within NW (Case 2) are
designed to maximize the voltage level. Case 1 and Case 2 are
studied with the same definition as those in the Part B.
The new generation schedules following the optimization
are listed in Table 5. In Fig. 4, the plot illustrates the transient
simulation result from original base case and after three new
coordination plans. It is clear that COI ratings restricted by
transient stability constraint are also improved greatly (Table
6). Full details on the implementation and the methodology
can be seen in [22].
TABLE 4
MW BENEFITS FOR COI CAPABILITY
FOR SMALL-SIGNAL STABILITY CONSTRAINT
Case
0
1
2
COI rating(MW)
4935
3725
3725
COI rating (New)(MW)
5257
3923
4008
Benefit (MW)
322
198
283
COI MW benefits from suggested generation redispatching
are listed in Table 4. Here, we assume that COI capability limit
is reached when the damping of the 0.25 Hz mode decreases to
5% as the COI transfer is increased. However, system
scenarios and other constraints can limit the actual benefits.
During the optimization process, the transient stability
constraints may be reached prior to the small-signal stability
constraints. That is, when the real power through COI is
maximized in small-signal stability sense (First row in Table
4), the system becomes so fragile that transient stability cannot
be maintained for even small disturbances. Cases 1 and 2 in
Table 4 are more realistic in that they denote actual
improvements in COI capability for the critical PV2
contingency while also maintaining a stringent 5% damping
requirement in the transient stability simulation. These
simulations show that the proposed optimization procedure can
indeed be a powerful tool for addressing small-signal stability
constraints.
C. Transient Stability
A typical test case, double Palo Verde outage contingency
case PV2 is chosen to investigate the transient stability
constraints that limit the COI transfer capability. The lowest
voltage magnitude of first swing at Malin bus, which is also
one of the end points of COI lines at the California border, is
used as an indicator of system performance during the
optimization process. In this way, two different control plans,
that is, the coordination of area exchanges to NW (Case 1) and
Fig. 4. Transient simulation
It can be observed that different regions or generators play
different roles in enhancing the transient stability problem
during PV2 outage contingency. In Case 1, the generation in
BC is reduced and generation in ID is increased toward better
swing shape. That means that decreasing BC-NW or
increasing ID-NW is helpful to improve COI transient stability
margin. In Case 2, the sensitivities of generators to the stability
can be identified even though they are all located in the NW
area. Roughly speaking, the generators at Chief Jo, McNary,
John Day and Lower Monumental have more significant
effects, as compared to Wanapum and Centralia.
TABLE 5
ORIGINAL SCHEDULED AREA GENERATIONS
AND OPTIMIZED GENERATIONS
Case
1
2
Control
Variables
BC
ID
MT
WK
NW
McNary
Chief Jo
Low.Mon.
Wanapum
JohnDay
Centralia
Before
Optim
1700
399
595
70.2
6520
900
550
700
474
2325
670
After
Optim
1536
836
784
438
5089
1037
700
559
491
2529
678
Lowest
Voltage
Damping
Ratio
0.8677
0.06272
0.8338
0.06120
TABLE 6
MW BENEFITS FOR COI CAPABILITY
FOR TRANSIENT STABILITY CONSTRAINT
Case
1
COI rating (MW)
4090
COI rating (New)(MW)
4350
Benefit (MW)
260
8
2
4090
4213
133
It is also notable that the results also meet the requirements
for small-signal stability. In each case, additional Prony
analysis is performed on the time windows 20 – 30 seconds
and thus the damping of the 0.25 Hz mode can also be
obtained. Actually, when first swing voltage profile is
improved, the small-signal stability is also better off with
higher damping ratio in Table 5. In the base case, the damping
ratio is only 0.05670, and the damping value does improve for
both optimized solutions in Table 5.
In this section, we have proposed a novel method for
improving the COI capability when limited by transient
stability constraints using the coordination optimization
algorithms. The algorithms automate the process of finding
optimal redispatching of either area exchanges or NW
generation schedules for improving the COI transfer
capability.
D. Economic Concerns
We assume that there exist specified generation costs over
different control regions. For simplicity, only slack bus
generations are taken into account. The coordination
procedures suggested in earlier sections of the paper will result
in changes to the overall cost of generating power at these
plants. We can study interesting optimization scenarios for
minimizing these costs and many different formulations are
possible. We illustrate one such example in this section by
considering the economic costs together with voltage security
objectives. The generation costs are assumed as follows.
TABLE 7
COST OF GENERATIONS
Area
Price
BC
0.8
ID
1.5
MT
1.6
WK
1.35
NW
1
We can then compute the total extra cost from redispatching
by calculating the extra cost associated with each generation
beyond the original scheduled generation and by summing
them up together. Hence, a bicriterion programming process is
carried out in Table 8 with different weights on the
improvement of Malin QV margin (w1) and on the total extra
cost (w2).
TABLE 8
GENERATIONS WITH ECONOMIC CONCERNS
Generation
BC
ID
MT
WK
NW
Extra Cost
QV(Mvar)
Base Case (MW)
1700
399
595
70.2
6520
0
1944
Case I (MW)
1750
695
654
86
6097
179
2039
Case II (MW)
1785
625
549
13
6311
48
2005
Case I: w1 = 0.9 , w2 = 0.1
Case II: w1 = 0.8 , w2 = 0.2
The weights reflect the operators’ point of view in decisionmaking. Compared with Case I, we put more emphasis on
economical cost in Case II. Hence, it is reasonable that the
solution of Case II will lead to less cost than that of Case I.
Likewise, since the weight corresponding to QV margin in
Case I is larger, the optimization result in Case I will have
better voltage security performance than that of Case II. As for
the system operators or the market players, they only need to
alter the weight vector based on their preferences. The weight
w1 and w2 are chosen between 0 and 1. Consequently, a
different set of solution indicating the relative significance of
multiple objectives will be created. Similarly, in order to
investigate the effect from multiple constraints, some other
objectives can also be included into the optimization in this
way, such as small-signal stability constraint. The multiobjective optimization procedures are far more challenging.
They require further research and investigation.
V. CONCLUSIONS
The coordination strategies would be of vital importance in
stressed power-flow scenarios when there is a need to increase
the capability of one or more critical transmission lines by
rescheduling of other paths, while also satisfying strict
requirements on system security and stability. We have shown
that the problem can be formulated into constrained
optimization
problems.
The
resulting
optimization
formulations while being highly severe, can indeed be solved
for large-scale systems such as WECC and the results can be
useful for system operation and planning. The nonlinear
optimization results could suggest operating procedures, which
could be entered into the transmission contracts appropriately.
Also, by introducing the economic costs associated with
specific path-flows, the optimization can be used for
maximizing the profits of a specific path owner by
approaching the problem in a global sense. This is because the
optimization specifically points to those neighboring path
flows, which are limiting the capacity of the critical
transmission path under consideration. Moreover, the
coordination strategies can be combined together with
economic concerns into multi-objective optimization problems
which are very challenging technically. The interactions
among different transmission paths and related pricing
mechanisms can lead to sophisticated games from the market
view-point, which can be pursued in future research.
VI. REFERENCES
[1]
[2]
[3]
[4]
[5]
North American Electric Reliability Council (NERC), “Available
transfer capability definitions an ddetermination,” June 1996.
W. Rosehart, C. Ca izares, and V. Quintana, “Costs of voltage security
in electricity markets” in Proc. 2000 IEEE Power Engineering Society
Summer Meeting, Seattle, WA, July 2000, pp. 2115-2120.
W. Rosehart, C. Ca izares, and V. Quintana, “Multiobjective optimal
power flows to evaluate voltage security costs in power networks”, IEEE
Trans. Power System, vol. 18, pp. 578-587, May 2003.
E. Vaahedi, Y. Mansour, and C. Fuchs, etc, “Dynamic security
constrained optimal power flow/VAR planning”, IEEE Trans. Power
System, vol. 16, pp. 38-43, Feb. 2001.
C. S. Chang, A. C. Liew, and J. X. Xu, etc, "Dynamic-securityconstrained multiobjective generation dispatch of longitudinally
interconnected power systems using bicriterion global optimization,"
IEEE Trans. Power System, vol. 11, pp. 1009-1016, May 1996.
9
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
E. D. Tuglie, M. Dicorato and M. L. Scala, etc., “A static optimization
approach to assess dynamic available transfer capability,” IEEE Trans.
Power System, vol. 15, pp. 1069-1076, August 2000.
M. I. Alomoush, S. M. Shahidehpour, “Contingency-constrained
congestion management with a minimum number of adjustments in
preferred schedules,” Electrical Power and Engergy Systems, vol. 22,
pp. 277-290, 2000.
R. S. Fang, A. K. David, “Transmission congestion management in an
electricity market,” IEEE Trans. Power System, vol. 14, pp.877-883,
Aug. 1999.
North American Electric Reliability Council (NERC), “Report of
available transfer capability coordination task force,” July 2000.
E. Vaahedi, Y. Mansour, and D. Sun, etc, “Large scale voltage stability
constrained optimal VAR planning and voltage stability applications
using existing OPF/optimal VAR planning tools”, IEEE Trans. Power
System, vol. 14, pp. 65-74, Feb.1999.
D. N. Kosterev, C. Taylor and W. A. Mittelstadt, “Model validation for
the August 10, 1996 WSCC system outage”, IEEE Trans. Power
System, vol. 14, pp. 967-979, Aug. 1999.
J.-S. R. Jang, C. T. Sun, and E.Mizutani, “Neuto-fuzzy and soft
computing,” New Jersey: Prentice Hall, 1997, pp 189-196.
V. Torczon, “On the convergence of the multidirectional search
algorithm”, SIAM J. Optimization, vol. 1, pp123-145, Feb. 1991.
J. Nocedal, S. J. Wright, "Numerical Optimization," New York:
Springer-Verlag, 1999, pp. 194-201.
J. J. Mor , D. J. Thuente, "Line search algorithms with guaranteed
sufficient decrease”, ACM Trans. Math. Software, vol. 20, pp 286-307,
Sep. 1994.
J. L. Cohon, “Multiobjective programming and planning”, New York:
Academic Press, 1978, pp. 219-223.
C. L. Hwang, A. S. M. Masud, “Multiple objective decision makingmethods and applications,” Berlin: Springer-Verlag, 1979, pp.12-23.
V. Venkatasubramanian, Y. Li, "Coordination of transmisson line
transfer capabilities," PSerc Report, May. 2002, [Online]. Available:
http://www.pserc.wisc.edu/ecow/get/publicatio/pserconlyp/venkata_coo
rdination_report.pdf
K. W. Schneider, “Application of power swing damping control (PSDC)
on an SVC to improve transient stability of the WSCC interconnected
system,” M.S. thesis, School of EECS, Washington State Univ.,
Pullman, 1999.
M. Ilic, J. Zaborszky, “Dynamics and control of large electric power
system”, John Wiley & Sons, 2000,pp.759-767.
T. J. Trudnowski, J. M. Johnson and J. F. Hauer, “Making Prony
analysis more accurate using multiple signals”, IEEE Trans. Power
System, vol.14, pp. 226-231, Feb. 1999.
Y. Li, “Coordination of transmission path transfers in large electric
power systems”, Ph.D. dissertation, School of EECS, Washington State
Univ., Pullman, 2003.
VII. BIOGRAPHIES
Yuan Li is a Ph.D. student in Electrical Engineering at Washington State
University, Pullman, WA. He received his M.S. degree in Electric Power
Systems at Shanghai Jiao Tong University in 1997 and B.S. degree at Xi’an
Jiaotong University in 1994 respectively. His research interests include power
system control and operation.
Vaithaianathan “Mani” Venkatasubramanian is presently an Associate
Professor in Electrical Engineering at Washington State University, Pullman,
WA. His research interests include the stability analysis and control designs
for large scale power system models.