July 2, 2007
1
Optimal Transmission Switching
Emily Bartholomew Fisher, Student Member, IEEE, Richard P. O’Neill, Michael C. Ferris
Abstract--In this paper, we formulate the problem of finding
an optimal generation dispatch and transmission topology to
meet a specific inflexible load as a mixed integer program. Our
model is a mixed-integer linear program because it employs
binary variables to represent the state of the equipment and
linear relationships to describe the physical system. We find that
on the standard 118-bus IEEE test case a savings of 25 percent in
system dispatch cost can be achieved.
Index Terms—Integer programming, Power generation
dispatch, Power system economics, Power transmission control,
Power transmission economics, Transmission switching
I. NOMENCLATURE
Indices
n, m: nodes
k: lines
g: generators
d: loads
j: number of lines allowed to open in the generalized upper
bound constraint
Variables
θ
n
: voltage angle at node n
P
nk
, P
ng
, P
nd
: real power flow to or from line k, generator g, or
load d to node n.
z
k
: binary variable indicating whether transmission line k is
removed from the system (open, z
k
= 0), or in the system
(closed, z
k
= 1)
TC
j
: Total system cost with j lines
Parameters
P
nk
max
, P
nk
min
: maximum and minimum capacity of line k
P
ng
max
, P
ng
min
: maximum and minimum capacity of generator g
θ
n
max
, θ
n
min
: maximum and minimum voltage angle at node n
c
ng
: cost of generating electricity from generator g located at
node n
B
B
k
: electrical susceptance of line k
M: large number
C: upper limit on number of open transmission lines
Manuscript received July 2, 2007. This work was supported in part by the
National Science Foundation under Grants DMS-0427689 and IIS-0511905
and the Air Force Office of Scientific Research Grant FA9550-07-1-0389.
E. B. Fisher is with Federal Energy Regulatory Commission (FERC),
Washington, DC, USA, and the Johns Hopkins University, Baltimore, MD,
USA (phone: 202-502-8093; e-mail:
emily.bartholomew@ferc.gov).
R. P. O’Neill is the Chief Economic Advisor, FERC, (e-mail:
richard.oneill@ferc.gov).
M. C. Ferris is a Professor at the University of Wisconsin, Madison, WI,
USA (e-mail: ferris@cs.wisc.edu).
Sets
Λ: Set of all transmission lines
L: Set of open transmission lines in solution to transmission
switching in which there is no upper limit on the number of
open lines
L
j
: Set of open transmission lines in solution to transmission
switching in which upper limit on open lines is j
II. I
NTRODUCTION
n large electric networks, transmission is traditionally
characterized as a static system. In regulated areas, the
utility dispatches generators over this fixed system to
minimize cost, and in restructured regions generators use this
network as a means to compete with one another. However, it
is acknowledged, both formally and informally, that system
operators can, and do, change the topology of systems to
improve voltage profiles or increase capacity of a flowgate.
1
In this paper we explore the implications of automating this
practice, and meeting demand efficiently by optimizing the
network through control of transmission line circuit breakers
in addition to generator output. One reason this concept is
particularly relevant now is the extreme difficulty in building
transmission to meet growing demand; we propose one way to
address this problem is to make more efficient use of the
existing network.
Kirchhoff’s laws allow the opening of lines to improve
dispatch cost. Opening lines to improve dispatch is already
being done to a small degree. Anecdotal evidence exists that
some system operators switch lines in and out because of
reactive power consumption or production of lines, or other
reasons.
2
The Northeast Power Coordinating Council includes
“switch out internal transmission lines” in the list of possible
actions to avoid abnormal voltage conditions [1] [2]. In
addition, system operators have procedures in place to close
lines quickly in case of emergency. In PJM, these Special
Protection Schemes (SPSs) allow the operator to disconnect a
line during normal operations but return it to service during a
contingency.
Electric transmission planning has long been an important
field in power system engineering, and many planning
techniques have been developed and studied [3] [4] [5]. By
necessity, these techniques focus on long-term decisions about
investments and uncertainty, not operational decisions based
on short-term system conditions. In this paper, we examine the
possibilities of improved system dispatch by altering existing
networks, which is a different problem from planning new
lines or designing new networks. We assume the load is
known with greater certainty, and the decisions are to change
1
personal communication with Andy Ott, Vice President PJM.
2
personal communication with Steve Nauman, Vice President, Exelon.
I
July 2, 2007
2
temporary network structure. The goal is not meeting
probabilistic long-term reliability criteria, like LOLP or
LOLE, or forecasted load growth, but to make each realization
of the transmission topology optimal and reliable given the
actual—or short-term forecasted—network conditions.
Transmission switching has been explored in the
literature as a control method for problems such as over- or
under-voltage situations, line overloads [6], loss and/or cost
reduction [7], improving system security [8], or a combination
of these [9] [10] [11]. Our investigation of optimal
transmission switching extends this literature by probing the
degree to which transmission switching could increase
economic efficiency of power system dispatch. No other
recent paper has explored the ability of transmission line
switching to reduce dispatch cost, and of papers that have
optimized cost none have used a mixed-integer formulation or
presented the degree of savings possible [7].
The concept of optimal transmission switching for
economic reasons was investigated by O’Neill et al. in a
market context [12]. Here, the authors examined the dynamic
operation and compensation of transmission lines (as well as
phase angle regulators, FACTS devices, and other
transmission technologies) on a small example network. In the
current paper we restrict our analysis to transmission lines,
and in particular apply optimal transmission switching to a
well-known engineering test case to gain a better
understanding of its potential impact in large and more
realistic systems. The idea of market-based payments for
transmission is outside the scope of this paper. In fact, the
concepts presented here do not require the existence of a
particular market design. All we assume is that the system
operator or utility uses an optimal power flow technique to
determine dispatch.
We formulate the problem as a mixed-integer linear
program (MIP), based on the traditional DC optimal power
flow (DCOPF) used to dispatch generators to meet load in an
efficient manner. The integer variables are used to represent
the state, closed or open, of each transmission line. Because of
Kirchhoff’s laws and power conservation, extending the
DCOPF to include integer capability is not as straightforward
as formulating a generator unit commitment problem. We then
use this formulation to examine the potential for improving
generation dispatches by optimizing transmission topology for
a well known IEEE test system.
In proposing new operating techniques, it is important to
keep in mind that the reliability of electricity networks is vital.
Indeed, some may argue that optimizing transmission
topology is unthinkable because of potential threats to
reliability. In this paper we do not ignore the importance of
reliability, nor are we suggesting switching transmission at the
expense of reliable network operations. We are simply
examining the potential for cost savings by streamlining and
automating actions operators can currently take (such as SPSs)
and improving network operation by making use of
controllable components while maintaining system security.
Lines that are open in the optimal dispatch of a network may
be available to be switched back into the system as needed, as
in PJM’s SPSs. In cases where this may not be possible,
optimal transmission switching can be conducted in
conjunction with contingency analysis in order to maintain
reliability levels while taking advantage of improved
topology. However reliability is maintained, optimal
transmission switching is not intrinsically incompatible with
reliable operation of the grid.
The paper is organized as follows. Section
III. presents
the MIP formulation for the optimal transmission switching
problem. Section
IV. discusses results from using the new
formulation on the well-known engineering test case. Section
V. explores computational issues in solving this problem.
Section
VI. contains a brief discussion of practical
implications of this method, and Section
VII. concludes. The
appendix explores heuristic methods that may aid in solving
large MIPs, a topic important for eventual practical
implementation.
III. M
IXED-INTEGER FORMULATION
As mentioned above, our formulation of the optimal
transmission switching problem is based on the standard
DCOPF. We use DC approximations [4] so the underlying
problem is linear, but there is no reason an ACOPF could not
be used instead other than the usual problems with
computation time, convergence and proof of optimality in the
nonlinear setting (see [13], [14] and [15] section 15.6).
A. Basic Optimal Power Flow
We start with a basic formulation of a DCOPF problem, as
presented below. Generator costs (which are negative) are
maximized, subject to physical constraints of the system and
Kirchhoff’s laws governing power flow. It is possible, and
probably desirable from an implementation perspective, to
include other costs in the objective function, such as generator
start-up or cost associated with equipment switching or
degradation. Making the objective function more robust
would ensure all appropriate operating costs were being
considered in the solution.
∑
=
g
ngng
PcMaxTC
s.t.:
(1)
maxmin
nnn
θθθ
≤≤
(2)
for all g and n
maxmin
ng
P
ng
P
ng
P ≤≤
(3)
maxmin
n
k
P
nk
P
n
k
P ≤≤
for all k and n
(4) for all n
0=
∑
−
∑
−
∑
−
dgk
nd
P
ng
P
nk
P
(5)
(
)
0=−−
nk
P
mn
k
B
θθ
for all lines k
with endpoints n and m
The constraints represent physical operating limits of the
July 2, 2007
3
network. Voltage angle limits are imposed by (1).
3
Constraint
(2) limits the output of generator g at node n to its physical
capabilities, and (3) limit the power flow across line k at node
n.
4
Power balance at each node is enforced by (4), and
Kirchhoff’s laws are incorporated in constraint (5). P
nk
represents the power flow from line k to node n, and therefore
can be positive or negative depending on whether power is
flowing into or out of the node. P
ng
and P
nd
represent the
power injected by a generator or withdrawn by a load at node
n, respectively. P
ng
is generally positive, whereas P
nd
is
typically negative.
This formulation is very general, and could be expanded to
include any other constraint or objective function that would
represent the system more accurately.
B. Optimal Power Flow with Optimal Transmission
Switching
Now we make changes to this basic formulation to allow
for optimal transmission switching. In this new formulation,
each line is assigned a binary variable, z
k
, that represents
whether the line is included in the system (the circuit breaker
on that line is closed) or not (circuit breaker is open). Because
of the peculiar characteristics of electricity, it is possible to
improve the dispatch cost by removing a line from the
network. Thus the optimization problem will either return the
result that no lines are taken out with no change in the
objective function value, or that one or more lines are opened
with an improved objective function.
In the formation, the power flow of an open line must be
constrained to zero; an open circuit or conductor can transmit
no power. Because of the constraint representing Kirchhoff’s
laws, the formulation of the problem is more complicated than
simply limiting the power flow to P
nk
max
or P
nk
min
times the
binary variable. Whereas this formulation would allow no
flow on an open line, it also limits power flow to zero on all
lines that share terminal nodes with the open line, because the
voltage angle difference is forced to zero. To account for this,
the Kirchhoff’s laws constraints (above, constraint 5) must be
modified, as they are below (see constraints 5a and 5b).
∑
=
k
nknkj
PcMaxTC
s.t.:
(1)
maxmin
nnn
θθθ
≤≤
(2) for all g and n
maxmin
ng
P
ng
P
ng
P ≤≤
3
Upper and lower voltage angle constraints were set to 0.6 and -0.6
radians, respectively. While these bounds are conservative, they aid in finding
solutions in a timely manner. Loosening these bounds may result in improved
solutions.
4
For a DC model with no losses power flow does not have to be enforced
at both ends of the lines; however if losses were modeled or a full AC power
flow were used it would be important to model both ends of each line because
real and reactive flows would be different at each end.
(3) for all k at nodes n
knknkknk
zPPzP
maxmin
≤≤
(4) for all n
0=
∑
−
∑
−
∑
−
dg
nd
P
ng
P
k
nk
P
(5a)
(
)
(
)
01 ≥−+−− M
k
z
nk
P
mn
k
B
θθ
for all lines k
with endpoints n and m
(5b)
(
)
(
)
01 ≤−−−− M
k
z
nk
P
mn
k
B
θθ
(6)
(
)
jz
k
k
≤−
∑
1
Note that M is a large number greater than or equal to
(
)
max min
B
nm
k
θ−θ
, in the constraints (5a) and (5b). In this
formulation we also added a generalized upper bound
constraint (GUB) (constraint (6)) that can limit the number of
open lines in the optimal network. This constraint is used to
gain understanding about the effects of changing the network
topology.
The following set notation will be used throughout the
analysis section of the paper: The set of all lines is denoted as
Λ. The subset of lines that are open in an optimal solution is
denoted as L
j
, where j is the number of lines allowed to be
open (i.e. constraint (6) in the above formulation would have
C = j). Since any quantity of lines can be allowed to open up
to the total number of lines, j can range from one to the total
number of lines (i.e. j = 1,…,|Λ|).
IV. T
EST NETWORK RESULTS
To test the formulation presented above, we used the
well-known and widely-used IEEE 118-bus engineering test
network. We modeled generator costs as linear, assumed
resistance and shunt capacitance of lines were zero, ignored
losses and reactive power, and did not perform any generator
unit commitment, using instead the commitment profile of the
test cases.
5
The traditional DCOPF and modified optimal
transmission switching DCOPF were tested on the case to
assess potential benefits of optimal transmission switching.
Also, this method was run on the system with various load
profiles to highlight the benefits of system flexibility, and a
contingency analysis was performed to assess effects on
reliability.
A. 118-bus model
The IEEE 118-bus test case was used to test and analyze
the optimal transmission switching formulation. Data for the
test system was downloaded from University of Washington
Power System Test Case Archive, and transmission line
characteristics and generator variable costs were taken from
the network as reported in [16]. This test model is based on a
portion of the AEP network from 1962.
The system consists of 118 buses, 186 transmission lines,
19 committed generators with a total capacity of 5,859 MW,
and 99 load buses with a total consumption of 4,519 MW. In
5
In our model, we assume the generation unit commitment problem has
already been solved. It is possible that an improved solution will result from
simultaneously performing generator unit commitment and optimal
transmission switching.
July 2, 2007
4
what will be called the “base case”, in which no transmission
lines are opened, the system cost of meeting this consumption
is $2,054/h. This cost results from the DCOPF model with
generator cost information from [16], where variable cost
ranges from $0.19/MWh to $10/MWh.
6
Two lines out of the
total of 186 are fully loaded, or congested.
First, we ran the MIP as presented above on the 118-bus
model without constraint (6). This allowed the status of each
line to be determined by the optimization, and for an
unconstrained number of lines to open. While we encountered
some difficulties in getting this problem to solve completely
and to optimality (discussed below in Section
V. ), the best
found solution improved the system cost by 25 percent, or
$511/h, over the base case. In this solution, 38 lines are
opened. There is no general trend in physical characteristics of
the optimally-open lines; rather the decision to open a line or
keep it closed depends on specific operating conditions, loads
and generator costs.
To gain insight on how these lines are affecting system
dispatch cost, we limit the number of open lines by including
the GUB constraint. Allowing only one line to open (j = 1)
results in L
1
= {Line 153}, meaning line 153, connecting bus
89 to 91, opens.
7
Opening the line changes the power output
of four generators. Generators at buses 25, 61, and 87
decrease output, and the generator at bus 111 increases output.
As can be see in the table below, the increased generator is
more expensive than two of the decreased generators, but the
total cost savings, primarily driven by reducing output from
the expensive generator at bus 87, is significant. Power flow
on transmission lines also change. The magnitude of power
flow changes on 168 – or 90 percent – of lines, and the
direction of power flow changes on four lines. The two lines
that were constrained in the base case remain congested, but
no additional lines become congested.
Repeating this analysis for j = {1, …,7}, we identified the
sequence of subsets, L
j
, and the objective function values for
each of these topologies. For |L
j
| ≤ |L
j+1
| it is not necessarily
true that , see in particular the change from j = 6
to j = 7. In other words, these sequentially-found subsets are
not necessarily subsets of one another. Also, for each
additional open line, the system cost decreases, but at a
decreasing rate (see
1+
⊂
jj
LL
TABLE II, below).
This implies two things about this network: one is that a
small number of lines have a large impact on dispatch cost,
and a large number of lines have a small impact. The other is
that improving system cost by opening lines is not a linear
process; serially opening the next line that provides the most
6
These costs we used are on the order of 50 to 100 times smaller than
typical generator costs. Realistic savings would be much higher than the
$2,000/h found here. These costs were used because they are consistent with
others found in the literature, and our goal was to remain as faithful to
previous models as possible.
7
This line forms a classic Wheatstone bridge along with lines connecting
bus pairs 89/92, 92/91, 89/90, and 90/91. It has been shown that Wheatstone
bridges can be associated with Braess Paradox, in which adding a line to a
network can increase the cost of using that network [16] [24]. However, in our
analysis not all optimally-open lines are the Wheatstone bridge link in such
traditional network structures; for example the line from bus 4 to bus 5 is
opened but is not the middle link of a Wheatstone bridge structure.
improvement will not necessarily produce the optimal
network. Thus, to gain the majority of economic benefit from
optimal transmission switching a large number of lines need
not be switched, necessarily; however, the number of lines to
be opened would need to be defined a priori in the
optimization model to achieve the most benefit. Running the
full optimization problem and then choosing only some of the
lines to open would not be the best way to optimally operate
the network. One possibility would be to define, in the
optimization problem, a subset of lines eligible for switching
based on reliability studies.
These conclusions have only been shown in the 118-bus
network thus far, and analysis of other networks is needed
before determining if they are general findings. However, if
these findings are found to be true in other networks they may
be useful in making the practice of optimal transmission
switching feasible.
TABLE I
C
HANGES IN GENERATOR OUTPUT AFTER OPENING LINE 153
Generator Bus
Variable Cost
[$/MWh]
Output Change
[MW]
25 0.434 –5.27
61 0.588 –15.69
87 7.142 –32.97
111 2.173 53.93
TABLE II
S
EQUENCE OF LINE OPENINGS
Number of
Open Lines
Allowed
(j)
Open Lines
(L
j
)
System
Dispatch
Cost
(TC
j
)
Percent
Savings in
TC
j
0 - –$2,054 -
1 Line 153 –$1,925 6.3%
2 Lines 132,153 –$1,800 12.4%
3 Lines 132,136, 153 –$1,646 19.9%
4 Lines 132,136,
153,162
–$1,633 20.5%
5 Lines 64,132,136,153,
162
–$1,607 21.8%
6 Lines 64,69,132,136,
153,162
–$1,602 22.0%
7
Lines 64,86,132,136,
146,153,161
–$1,596 22.3%
No restriction
(optimality not
proven yet)
–$1,543 24.9%
To ensure the large percent savings we saw was not due
to the wide range of generator costs, we also ran the optimal
transmission switching model with the three most expensive
generator costs halved. This sensitivity analysis resulted in a
16.2 percent savings with ten lines removed. While this
savings is smaller than that achieved with the original
generator costs taken directly from the standard network
model, it is still substantial and suggests that the high
generator costs do not dominate the savings.
B. Changing Load Profiles
If opening certain lines make the solution more efficient,
does this mean they should be opened permanently? Does the
existence of alternative transmission topologies with improved
dispatch cost mean the system was not optimally planned, that
July 2, 2007
5
the transmission engineers designed a suboptimal system? Not
necessarily. There are two reasons why a given transmission
system may appear suboptimal. One is that in order to have a
reliable network some lines that decrease the economic
optimality may be necessary. Another reason is that different
patterns of loading may benefit from different topologies; the
optimal network for one generator/load scenario may be
different from the optimal network of another.
To simulate different loading scenarios, we created two
additional cases, called Peak and Off Peak, in which each load
in the case above is scaled either up or down by ten percent,
respectively. The load in the case presented above is referred
to as the Shoulder case. Total load in the Peak case is 4,971
MW, and in the Off Peak case is 4,067 MW, compared to total
load of 4,519 MW in the Shoulder case.
For each of these two new cases, the MIP was performed
with the GUB constraint, for j = 0,…,5 and j = 0,…,4. System
cost decreased by 12.2 percent for the Peak load case for j = 5,
and by 17.8 percent for the Off-Peak case for j = 4. The Peak
load case is not feasible without opening at least one line. To
analyze the Peak load case infeasibility, we increased
transmission capacity and found a feasible solution with six
overloaded lines, all of which have a capacity of 220 MW and
are overloaded by 112 to 149 percent. In the j=1 (feasible)
case, three of these lines are loaded to capacity, and the others
are within 1, 7 and 20 percent of being fully loaded.
To gauge how much improvement comes from
determining transmission topology based on actual system
conditions, the optimal transmission topology of the Shoulder
case was applied to the Peak and Off Peak load scenarios. The
resulting system cost was compared to the optimal system cost
determined by running the MIP on the specific load scenarios.
In both the Peak and Off Peak cases, the optimal open lines
are different from those optimally open in the Shoulder case,
and the transmission topology that is best for the Shoulder
load case results in a higher system dispatch cost, by two to
eleven percent, compared to the optimal transmission
topology determined for the Peak or Off Peak scenarios.
These results are shown below in Fig. 1 and Fig. 2. In the
Peak load case, opening the lines optimal to the Shoulder case
results in a degradation of system cost from j = 2 to j = 3 and
from j = 4 to j = 5. This occurs because the set of open lines
was optimized for a different load profile.
TABLE III
P
EAK LOAD RESULTS
j
Open
Lines
System
Cost
Open
Lines:
Shoulder
case
System
Cost
Percent Diff.
in System
Cost
0 Infeas. Infeas.
1 153 –$2,738 153 –$2,738 0%
2 132,157 –$2,574 132,153 –$2,613 2%
3
132,153,
163 –$2,528
132,136,
153
–$2,677 6%
4
78,132,
153,165
–$2,454
132,136,
153,162
–$2,568 5%
5
78,132,
133,153,
165 –$2,404
64,132,
136,153,
162
–$2,657 11%
2300
2400
2500
2600
2700
2800
012345
j (Upper Limit on Open Lines)
System Dispatch Cost [$]
Peak-Optimal lines open
Shoulder-Optimal lines open
infeasible
Fig. 1. Peak Load System Dispatch Cost
TABLE
IV
O
FF-PEAK LOAD RESULTS
j
Open
Lines
System
Cost
Open Lines:
Shoulder
case
System
Cost
Percent
Diff. in
System
Cost
0 –$1,306 –$1,306 0%
1 132 –$1,212 153 –$1,301 7%
2 132,136 –$1,099 132,153 –$1,208 10%
3
64,132,
136
–$1,075 132,136, 153 –$1,097 2%
4
64,131,
132,133
–$1,073
132,136,
153,162
–$1,091 2%
1000
1100
1200
1300
1400
01234
j (Upper Limit on Open Lines)
System Dispatch Cost [$]
Off Peak-Optimal lines open
Shoulder-Optimal lines open
Fig. 2. Off-Peak Load System Dispatch Cost
These results indicate that a transmission network
optimized for one particular pattern of load on a network is
not necessarily optimal for another. Thus, allowing decisions
about network topology to be made based on real-time system
conditions (or daily or weekly forecasts) can result in a lower-
cost dispatch than using a static network optimized for a
multi-period forecast.
C. Contingency Analysis
Reliability of the system paramount, and must be ensured
by system operators. In this section we explore how the
modified transmission networks, found via optimal
transmission switching, perform under an n–1 contingency
