Article
Optimal Transmission Switching and Grid Reconfiguration for
Transmission Systems via Convex Relaxations
Vineet Jagadeesan Nair
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;
jvineet9@mit.edu
Abstract
In this paper, we formulate optimization problems and successive convex relaxations to
perform optimal transmission switching (OTS) in order to operate power transmission grids
more efficiently. OTS may be crucial in future power grids with much higher penetrations
of renewable energy sources, which will introduce more variability and intermittency
in generation. Similarly, OTS can potentially help mitigate the effects of unpredictable
demand fluctuations (e.g., due to extreme weather). We explore and compare several
different formulations for the OTS problem in terms of the computational performance and
optimality. In particular, we build upon the literature by considering more complex and
accurate power flow formulations for OTS and introducing novel convex relaxations. This
allows us to model the grid physics more accurately than prior works and generalize to
several different types of networks. We also apply our methods to small transmission test
cases as a proof of concept to determine the effects of applying OTS.
Keywords: power grids; smart grids; optimization; power flow
1. Introduction
Rapid decarbonization of the power grid is crucial to reduce greenhouse gas emissions
and mitigate climate change. This will entail a transition away from centralized power
generation sources like coal and natural gas to renewable sources like solar and wind.
Academic Editor: Murilo E. C. Bento
There will also be an increased footprint of batteries and other energy storage resources
Received: 18 February 2025 to help balance supply and demand. Furthermore, we are experiencing exponential load
Revised: 22 June 2025
growth due to the increasing penetration of electric vehicles, heat pumps, and data centers.
Accepted: 24 June 2025
All of these changes place significant stress on the grid, resulting in long interconnec-
Published: 3 July 2025
tion queues [1], congestion, and high curtailment of renewable resources. The aging grid
Citation: Jagadeesan Nair, V.
infrastructure lacks adequate capacity to accommodate such increases in load and gen-
Optimal Transmission Switching and
Grid Reconfiguration for Transmission eration. Capacity additions, upgrades, and retrofits are also capital-intensive and often
Systems via Convex Relaxations. take 10+ years, with numerous regulatory barriers and permission issues, especially in
Electricity 2025, 6, 37. https://doi.org/ the United States. Thus, grid modernization technologies are an attractive alternative to
10.3390/electricity6030037 maximize the utilization of the existing grid capacity. These use real-time monitoring,
Copyright: © 2025 by the author. optimization, and control to improve network operation without new physical investments.
Licensee MDPI, Basel, Switzerland. In a prior work, we studied the impacts of one such technology, i.e., dynamic line ratings,
This article is an open access article on the Texas grid [2]. In this work, we consider another such smart grid technology, optimal
distributed under the terms and
transmission switching (OTS).
conditions of the Creative Commons
In any given electrical network, several of the transmission lines are generally
Attribution (CC BY) license
(https://creativecommons.org/ equipped with switches, circuit breakers, and/or reclosers. The conventional practice
licenses/by/4.0/). is to operate the grid using a static or fixed configuration. However, it may be beneficial to
Electricity 2025, 6, 37 https://doi.org/10.3390/electricity6030037
Electricity 2025, 6, 37 2 of 15
dynamically reconfigure the grid through switching actions in order to respond to real-time
demand and supply conditions. This has the potential to help reduce costs and improve
efficiency. OTS is a grid reconfiguration method that dynamically opens or closes switches,
circuit breakers, or reclosers to change the network topology in an optimal fashion [3]. This
can be performed to achieve several different objectives, like minimizing electrical losses,
generation costs, or congestion. In a highly decarbonized grid, this can help grid operators
and utilities adapt to changing system conditions to avoid renewables curtailment and
minimize the dispatch of natural gas peaker plants.
Prior Work and Contributions
There have been several previous works that have studied the impacts of OTS on
transmission systems [4,5]. The majority of these works have relied on simplified direct
current (DC)-based power flow models [3,6–9] or have used power transfer distribution
factors (PTDFs) [10], thus resulting in mixed integer linear programs (MILPs). However,
DC power flow solutions based on linearized approximation may not always be feasible in
reality since they do not capture all constraints. Thus, it is preferable to use models that
result in guaranteed feasible solutions.
A few works have applied alternating current (AC) power flow models to OTS [11,12],
which resulted in mixed integer nonlinear programs (MINLPS). A subset of these papers
modeled OTS problems as integer second-order cone programs [13,14]. Another class
of papers included non-deterministic aspects through security constraints and contin-
gencies [14], employing techniques like stochastic programming [15,16], robust optimiza-
tion [17], and probabilistic analyses [18]. In order to improve computational traceability for
larger networks, some have proposed simpler and faster heuristics for solving OTS [19,20],
while sacrificing some optimality and risking potential infeasibilities. However, most of
these papers still make some simplifying assumptions, restrict themselves to only certain
types of networks, or ignore certain constraints completely, especially those relating to
voltage angles. In this work, we aim to address this gap in the literature by incorporating
all AC power flow constraints with convex relaxation strategies.
2. Optimal Power Flow Formulations for OTS
Optimal power flow is the key optimization problem that governs the operations of
power systems. It describes the physical laws the system needs to obey, such as Kirchhoff’s
laws, Ohm’s law, line thermal limits, etc. These are specified by the AC optimal power flow
equations, shown below.
2.1. AC Optimal Power Flow
n
Pi + jQi = Vi ∑ YikV̄k (1)
k=1
Splitting these into real and imaginary components, we get
n ∣ ∣( ( ) ( ))
P = |V | ∑ ∣ ∣i i Vj Gij cos θi − θj + Bij sin θi − θj , (2)
j=1
n ∣ ∣( ( ) ( ))
Qi = |Vi| ∑ ∣V ∣j Gij sin θi − θj − Bij cos θi − θj . (3)
j=1
These are clearly nonconvex constraints. Instead, a commonly used approach for
transmission systems is to make several simplifying assumptions that result in the linear
DC optimal power flow equations instead.
Electricity 2025, 6, 37 3 of 15
2.2. DC Optimal Power Flow
min ∑ fi(PGi), (4)
PG ,QG ,|V|,θ i∈G
subject to (5)
n
PGi − PDi = ∑ Bik(θi − θk), ∀i ∈ N, (6)
k=1
where k is the set of all nodes incident to node i (7)
Pmin ≤ P ≤ PmaxGi Gi Gi , ∀i ∈ G (8)
− f ik ≤ Bik(θi − θk) ≤ f ik, ∀(i, k) ∈ L (9)
θ1 = 0 (Slack bus) (10)
|θi − θk| ≤ ∆θik, ∀(i, k) ∈ L (11)
These only account for the power balance (Equation (6)), generator capacities
(Equation (8)), line power flow limits or thermal ratings (Equation (9)), and voltage an-
gle constraints. While this is a reasonable approximation for high-voltage transmission
grids with low line resistances, it has some key disadvantages since it completely ignores
voltage magnitudes and reactive power. It also relies on a small-angle approximation
to linearize all the trigonometric terms. Thus, it cannot be guaranteed that the DC-OPF
solutions are feasible for the real ACOPF problem. Furthermore, the DC approximation is
even worse for lower voltage distribution grids that have more losses (even though we are
restricting ourselves to transmission systems in this paper).
2.3. DC OPF-Based OTS
Most of the studies in the literature have relied on an OTS problem formulation using
DC OPF [3,7], resulting in modifying the power balance constraints in Equation (6) and
line flow limits in Equation (9) accordingly, to indicate whether a line is open or closed:
n
PGi − PDi − ∑ (Bik(θi − θk) + (1 − zik)M) ≥ 0, ∀i ∈ N (12)
k=1
n
PGi − PDi − ∑ (Bik(θi − θk)− (1 − zik)M) ≤ 0, ∀i ∈ N (13)
k=1
− zik f ik ≤ Bik(θi − θk) ≤ zik f ik, ∀(i, k) ∈ L (14)
∑(1 − zk) ≤ Z, zik ∈ {0, 1} (15)
k
Here, the binary variable z indicates whether a line is open (zik = 0) and no power
flows through it, or closed (zik = 1) and all the constraints apply to it.
2.4. Towards ACOPF-Based OTS Formulations
We can reformulate the AC OPF problem by introducing additional auxiliary variables
to represent the nonlinear terms (see Table 1 for details) and transform them into the
following [21–23]:
min ∑ ∑ fi(PGi) (16)
p,q,c,s,θ
i∈I i∈G
pG ≤ pG ≤ p̄G
i i i
, i = 1, 2, · · · , n (17)
qG ≤ qG G
i i
≤ q̄i , i = 1, 2, · · · , n (18)
V2i ≤ ei ≤ V̄
2
i , i = 1, 2, · · · , n (19)
Electricity 2025, 6, 37 4 of 15
n ( )
pGi + p
U
i − p
D
i = Giiei + ∑ Gijcij + Bijsij , i = 1, 2, · · · , n (20)
j=1,j ̸=i
n ( )
qG + qU − qDi i i = −Biiei − ∑ Bijcij − Gijsij , i = 1, 2, · · · , n (21)
j=1,j ̸=i
( )2 ( )
( )
2 2
−Gijei + Gijcij + Bijsij + Bijei − Bijcij + Gijsij ≤ f ij ∀(i, j) ∈ L (22)
c2ij + s
2
ij = eiej, (i, j) ∈ L (23)
( )
θi − θj = arctan sij/cij ∀(i, j) ∈ L (24)
∆θ ≤ θi − θj ≤ ∆θ (25)
Table 1. Definitions of important variables and parameters.
Variable or Parameter Definition
n Number of buses or nodes in network
(i, j) Bus indices
L Set of all transmission line indices
G Set of all generators
aij Binary variable indicating whether line ij is open or closed
θi Voltage angle at bus i
θij = θi − θj Voltage angle difference between buses i and j
Vi Complex voltage at bus i
|Vi| Voltage magnitude at bus i
cij = |Vi||Vj|cos(θij)
sij = |Vi||Vj|sin(θij)
ei ei = V
2
i
Yij Admittance matrix element of line ij
Gij, Bij Real/reactive parts of admittance matrix element of line ij
Gii, Bii Real/reactive parts of admittance matrix diagonal element of bus i
bij Shunt susceptance of line ij
f ij Apparent power flow capacity of line ij
We notice that now, most of the constraints are actually convex, except for
Equations (23) and (24). Note that Equation (22) is already a convex quadratic second-
order conic constraint. For the first constraint Equation (23), we first relax the equality
to the following inequality:
c2 + s2ij ij ≤ eiej ∀(i, j) ∈ L;
then, this can be transformed into a convex quadratic second-order cone programming
(SOCP) constraint by defining additional auxiliary variables D1−4ij in order to represent
the bilinear right-hand side term. Thus, we can replace Equations (22) and (23) with the
following set of constraints:
pij = −Gijei + Gijcij + Bijsij, ∀(i, j) ∈ L (26)
( )
qij = Bij − bij/2 ei − Bijcij + Gijsij, ∀(i, j) ∈ L (27)
( )2 ( )
( )
2 2
pij + qij ≤ f ij , ∀(i, j) ∈ L (28)
D1ij = 2cij, ∀(i, j) ∈ L (29)
D2ij = 2sij, ∀(i, j) ∈ L (30)
D3ij = ei − ej, ∀(i, j) ∈ L (31)
D4ij = ei + ej, ∀(i, j) ∈ L (32)
Electricity 2025, 6, 37 5 of 15
( )2 ( )2 ( )2 ( )2
D1 2ij + Dij + D
3 4
ij ≤ Dij , ∀(i, j) ∈ L (33)
We chose to employ a second-order conic relaxation (SOCP) here as opposed to a
semidefinite programming relaxation (SDP). Although SDP relaxations have been proven
to be exact for certain types of radial networks, they tend to perform poorly in practice
and worse than SOCP, especially for larger problems [24,25]. We expect to see similar
trends with SDP in our case as well—while it may produce more accurate results with
a smaller optimality gap, it would be less computationally efficient, resulting in longer
runtimes [26,27]. Thus, we decided to focus on using the SOCP approach, since it provides
a better balance of accuracy and speed compared to the DCOPF (less accurate, faster
solutions) and SDP-based AC OPF (more accurate, slower solutions). We use the SOCP
formulation for the OTS problem by introducing binary decision variables for which
transmission lines or branches to keep open. While a few other papers have considered
applying ACOPF models for OTS, they also rely on quite restrictive assumptions and
approximations as part of their final approach [4,12,13]. This motivates the goal behind this
paper: to apply as accurate an OPF formulation as possible for the OTS problem.
2.5. Optimal Transmission Switching Problem
min ∑ ∑ fi(PGi) (34)
p,q,c,s,θ
i∈I i∈G
pG ≤ pGi ≤ p̄
G
i , i = 1, 2, · · · , n (35)i
qG ≤ qGi ≤ q̄
G
i , i = 1, 2, · · · , n (36)i
V2i ≤ ei ≤ V̄
2
i , i = 1, 2, · · · , n (37)
n
pG − pDi i = ∑ pij, i = 1, 2, · · · , n (38)
j=1,j ̸=i
n
qG Di − qi = ∑ qij, i = 1, 2, · · · , n (39)
j=1,j ̸=i
( )
pij = aij −Gijei + Gijcij + Bijsij , ∀(i, j) ∈ L (40)
(( ) )
qij = aij Bij − bij/2 ei − Bijcij + Gijsij , ∀(i, j) ∈ L (41)
( )
θi − θj = arctan sij/cij , ∀(i, j) ∈ L (42)
( )2 ( )
( )
2 2
pij + qij ≤ aij f ij , ∀(i, j) ∈ L (43)
D1ij = 2cij, ∀(i, j) ∈ L (44)
D2ij = 2sij, ∀(i, j) ∈ L (45)
D3ij = ei − ej, ∀(i, j) ∈ L (46)
D4ij = ei + ej, ∀(i, j) ∈ L (47)
( )2 ( )2 ( )2 ( )2
D1 + D2 3 4ij ij + Dij ≤ Dij , ∀(i, j) ∈ L (48)
aij ∈ {0, 1} ∀(i, j) ∈ L (49)
∆θ ≤ θi − θj ≤ ∆θ (50)
p, q, ∈ Rn; c, s ∈ Rn×n; a ∈ R|L|θ (51)
This is a mixed integer nonlinear programming problem (MINLP), which is generally
challenging to solve, especially as we consider large network sizes with many buses/nodes
and branches. We thus focus on reformulating the problematic constraints. The nonconvex
constraint Equation (42) is needed to uniquely define the voltage angles. Most prior papers
Electricity 2025, 6, 37 6 of 15
simply discard this constraint entirely—while this is a valid approach for radial networks, it
cannot be ignored for more general meshed networks, which are the dominant topology for
transmission systems. In order to convexify this, we first rewrite this constraint as follows:
( ) ( )
θij = θi − θj = tan
−1 sij/cij =⇒ tan(θij) = sij/cij .
For the stable operation of power systems, the voltage angle difference between
neighboring buses needs to be kept small, e.g., −1.047 = ∆θ ≤ θij ≤ ∆θ = 1.047 rad.
The bus voltage angle differences are small. For instance, voltage differences between buses
are generally maintained at less than 10° [28] in steady state, with the maximum angle
differences generally lying between 5 and 20° under normal operating conditions [29,30].
However, note that for certain large inter-area systems, these can sometimes reach 60° [31].
We focus only on radial and meshed networks in this paper, since they are the most common
types of transmission networks. While ring networks may have larger angle differences,
these are not very common in practice and are mainly relevant only for urban distribution
grids and medium-voltage systems [32].
Thus, for most general transmission networks, we can apply a small-angle approxima-
tion for tan() to relax this constraint:
tan(θ)
lim = 1.
θ→0 θ
Thus, the nonconvex constraint Equation (24) can be reduced to the following bilinear
equality constraint:
sij
θi − θj = =⇒ sij = cijθij.cij
This can further be relaxed by employing McCormick envelope (MCE) convex
relaxations, replacing this bilinear term with a series of linear inequality constraints.
We know
− |Vi||V j| ≤ sij = |Vi||Vj|sin(θij) ≤ |Vi||V j|
− |Vi||V j| ≤ cij = |Vi||Vj|cos(θij) ≤ |Vi||V j|
∵ −1 ≤ sin(θij), cos(θij). ≤ 1
Generally, we aim to maintain all bus voltages within a tight interval as close to
1 p.u. as possible, e.g., 0.9 = |V| p.u. ≤ |Vi| ≤ |V| = 1.1 p.u. This allows us to bound both
[ ] [ ]
cij, sij ∈ −|V|
2, |V|2] and furthermore θij ∈ ∆θ, ∆θ . We can narrow the bounds for cij
[ ]
further to |cij, cij| = |V|
2cos(∆θ) ≤ cij ≤ |V|2 (∵ 0 ∈ ∆θ, ∆θ ). Thus, we can construct the
McCormick envelopes by replacing the bilinear term with an auxiliary variable wij = cijθij
with the following associated constraints:
wij ≥ cijθij + cijθij − cijθij (52)
wij ≥ cijθij + cijθij − cijθij (53)
wij ≤ cijθij + cijθij − cijθij (54)
wij ≤ cijθij + cijθij − cijθij. (55)
Note that we only focus on OTS under nominal or normal operating conditions,
for which the [0.9, 1.1] p.u. range of voltage magnitudes is well within the ±5% limits set
by the American National Standards Institute (ANSI) [33]. Voltages may exceed ±10% for
short periods during disturbances like line faults or load swings, but such contingency
conditions are outside the scope of this work. Of course, the tightness of this MCE relaxation
depends heavily on the variable bounds used to construct the convex underestimators and
concave overestimators. The relaxation could be improved further by iteratively decreasing
Electricity 2025, 6, 37 7 of 15
the upper bounds and increasing the lower bounds, through piecewise MCE relaxations or
other similar approaches [34].
In our case, we note that we can obtain even tighter bounds on cij by noting
that −1.047 = ∆θ ≤ θij ≤ ∆θ = 1.047 rad, and cos(1.047) ≈ 12 ≤ cos(θ) ≤ 1
|V|2
for θ ∈ [−1.047, 1.047]. This implies that 2 ≤ cij ≤ |V|
2. Plugging these into
Equations (52)–(55) gives us
|V|2θ |V|2ij ∆θ
wij ≥ + cij∆θ −2 2
w ≥ |V|2 + c ∆ − |V|2ij θij ij θ ∆θ
w 2ij ≤ |V| θij + c ∆ − |V|2ij θ ∆θ
|V|2θ 2ij |V| ∆θ
wij ≤ cij∆θ + − .2 2
Disjunctive or Conditional Constraints
We introduce additional binary variables aij ∈ {0, 1} for each line ij ∈ L that indicates
whether a line is open or closed. In order to represent switching, we need to enforce
that both the real and reactive power flows on an open line are constrained to zero, since
an open circuit cannot transfer any current. We thus need to modify the constraints
Equations (40) and (41) accordingly:
( )
pij = aij −Gijei + Gijcij + Bijsij , ∀(i, j) ∈ L (56)
(( ) )
qij = aij Bij − bij/2 ei − Bijcij + Gijsij , ∀(i, j) ∈ L. (57)
Since we do not know the exact bounds on the continuous variables (terms inside the
parentheses), an exact linear reformulation of this bilinear constraint is not possible. How-
ever, we can represent this using the following set of disjunctive or conditional equalities:
   
aij ¬aij
   
 pij = −Gijei + Gijcij + Bijsij  ∨  pij = 0 . (58)
( )
qij = Bij − bij/2 ei − Bijcij + Gijsij qij = 0
These can be relaxed to the following linear constraints using the big-M method, allow-
ing us to get rid of the bilinear terms involving products of continuous and binary variables:
−pij − Gijei + Gijcij + Bijsij + (1 − aij)Mij ≥ 0
−pij − Gijei + Gijcij + Bijsij − (1 − aij)Mij ≤ 0
( )
−qij + Bij − bij/2 ei − Bijcij + Gijsij + (1 − aij)Nij ≥ 0
( )
−qij + Bij − bij/2 ei − Bijcij + Gijsij − (1 − aij)Nij ≥ 0
−Mijaij ≤ pij ≤ Mijaij, −Mijaij ≤ qij ≤ Mijaij,
where Mij, Nij are chosen to be sufficiently large positive numbers. They are chosen such
∣ ∣ ∣( ) ∣
that M > ∣ij Gijcij + Bijsij − G ∣ijei and N ∣ij > Bij − bij/2 ei − B c + G s ∣ij ij ij ij . Note that an
alternative approach to the big-M method is to use convex hull reformulations of these
constraints to represent the feasible set [35].
Electricity 2025, 6, 37 8 of 15
We need to ensure that only lines with either switches or circuit breakers can be
opened. Suppose SW is an n × n matrix that indicates whether a line has a switching
device, i.e., SWij = 1 and is 0 otherwise. Then, the following constraint ensures that only
valid switching actions are allowed:
1 − aij ≤ SWij ∀(i, j) ∈ L. (59)
Finally, we can optionally add an upper limit on the total number of lines that can be
opened in the optimized network:
∑ (1 − aij) ≤ Nsw.
(i,j)∈L
This allows the grid operator to control exactly how many switching operations can
be made at each timestep. This may be necessary for real systems where it may not be
feasible to operate all available switches and circuit breakers. Thus, we have relaxed the
original nonconvex mixed integer nonlinear program for OTS-ACOPF to a mixed integer
second-order conic convex program (MISOCP)—these are much more computationally
tractable to solve than MINLPs.
3. Numerical Simulations and Results
We now briefly present some simulations of the proposed formulations on a few
different IEEE standard transmission test cases. We conduct studies for the IEEE 9-bus,
39-bus, and 118-bus systems, all of which are meshed networks. These are relatively small
networks, but they still demonstrate some interesting results that we summarize below.
Here, we provide some notes on the simulations themselves:
• For the larger systems, there were several cases where the solver had difficulties
converging to the optimal solution or reported local infeasibility. This was especially
a challenge for the MINLP formulations and/or while using free solvers like IPOPT
or SCIP.
• In order to deal with this, we warm-started the optimization process by initializing the
variables using solutions from DCOPF, more relaxed versions of the same problem,
or by temporarily removing certain constraints like the line flow limits. This enabled
achieving convergence for all the cases.
• For simplicity, we used a fixed large value of M = 100 for the big-M relaxations. This
worked well, since we already know most of our variables and terms are of the order
of ≈ 1.0 after per-unitization. However, tuning the value of M for different cases could
improve the performance and also avoid potential issues around ill-conditioning and
scaling (although, this did not occur during our simulations).
All the network parameters and nodal load data are the default values specified
in the respective IEEE test case datasheet from the Texas A&M Electric Grid Test Case
Repository [36] and were taken from the MATPOWER library [29]. The problem was
modeled using the JuMP optimization framework in Julia [37], and all the formulations
except the nonconvex ACOPF were solved using Gurobi. The nonconvex ACOPF was
solved using the IPOPT solver. The simulations were run on a 2021 MacBook Pro M1 Max
with 64 GB RAM. The results are summarized in Tables 2 and 3.
Electricity 2025, 6, 37 9 of 15
Table 2. Computational comparison of different OPF and OTS approaches. For a fair comparison,
these were all solved using the free open-source SCIP solver in Julia.
Runtime (s) 9-Bus 39-Bus
Different OPF formulations
(1) DC OPF 0.0033 0.0061
(2) Non-convex AC OPF 0.0249 0.744
(3) SOCP relaxed OPF 0.045 0.639
(4) SOCP relaxed OPF + MCEs 0.0556 0.677
OTS simulations with no constraints on switching actions Nsw
(5) DC OPF OTS 0.0575 0.555
(6) Non-convex ACOPF OTS 0.0806 0.376
(7) OTS with SOCP relaxation + bilinear terms 0.0728 0.402
(8) OTS with SOCP relaxation + Big-M reformulations 0.0437 0.387
(9) MISOCP relaxed OTS with Big-M + MCEs 0.0815 0.527
Table 3. Optimality comparison of different OPF and OTS approaches.
Optimality Gap (%) 9-Bus 39-Bus
Different OPF formulations
(1) DC OPF 0.0891 1.091
(2) Non-convex AC OPF (exact) 0 0
(3) SOCP relaxed OPF 3.712 × 10−10 1.955 × 10−14
(4) SOCP relaxed OPF + MCEs 1.115 × 10−9 4.671 × 10−12
OTS simulations with no constraints on switching actions Nsw
(5) DC OPF OTS 0.0891 0.024
(6) Non-convex ACOPF OTS 0 0
(7) OTS with SOCP relaxation + bilinear terms 7.71 × 10−7 1.87 × 10−14
(8) OTS with SOCP relaxation + Big-M reformulations 1.639 × 10−14 2.307 × 10−12
(9) MISOCP relaxed OTS with Big-M + MCEs 6.988 × 10−8 8.906 × 10−10
3.1. Comparisons of Different Formulations
We first compared the different formulations in terms of their computational perfor-
mance, as shown in the table below. The formulations are defined as follows:
1. DC OPF: the simplest version of the problem, a convex quadratic program (with
linear constraints).
2. Non-convex nonlinear AC OPF: exactly describes the power physics.
3. SOCP relaxed OPF: using second-order conic programming convex relaxation but still
containing the nonconvex trigonometric voltage angle constraint.
4. SOCP relaxed OPF + MCEs: also relax the nonconvex trigonometric voltage angle
constraints using a small angle approximation and McCormick envelopes (MCE).
5. (1) with OTS.
6. (2) with OTS.
7. (3) with OTS, containing bilinear terms involving products of binary and continu-
ous variables.
8. (7) with big-M reformulations to remove the bilinear terms.
9. MISOCP by combining (8) with MCE relaxations of the angle constraint.
In this study, we focus only on the optimality gap and solution times as the two key
measures of computational performance for each algorithm. However, it is important to
note that other metrics could also be considered for a more rigorous comparison. We plan
to analyze other aspects of each algorithm in future work. However, here we consider a
Electricity 2025, 6, 37 10 of 15
simplified analysis as a starting point since runtime and optimality are often the factors
most important to decision makers like grid operators and utilities. Another important
factor to consider is the feasibility gap, since the relaxed OPF or OTS solutions may not
necessarily always be feasible for the exact OPF problem. However, we know that all of
our proposed relaxed formulations (3), (4), and (6)–(9) are based on AC-power flow and
thus will offer better feasibility guarantees than the DC approaches (1) and (5).
As expected intuitively, we see that the optimality gap in Table 3 increases as we
further relax and approximate the problem. Here, we used the relative difference between
optimal solution values to compare the gap between different formulations. The DC OPF
(1) is significantly worse than all the other formulations, indicating that solutions from the
DC OPF may not necessarily be optimal for the real system or may not even be AC feasible,
since it does not take into account all the constraints. The other relaxations work reasonably
well, although their exact performance depends on the specific test case being considered.
For the nominal operation, the SOCP relaxed OPF (3) works best, and further convexifying
the angle constraint via MCEs in (4) only marginally increases the optimality gap. This
suggests that the MCE approach is valid at least for these test cases. In terms of the OTS
formulations, we see that the final convex MISOCP formulation with MCEs (9) still has a
relatively small optimality gap. One interesting point is that the big-M reformulation in (8)
performs much better than (7) on the 9-bus case but worse on the 39-bus case. This will be
investigated further as part of our future work.
The DC OPF is significantly faster than the other methods in most cases, owing to its
simple model, but this comes at the cost of lower accuracy and we may need to check for the
AC feasibility of the obtained solutions Interestingly, we find that some of the relaxations
actually take slightly longer than their nonconvex stricter counterparts. This may be due
to the additional variables we needed to introduce to deal with the nonlinearities, which
increase the dimensionality. However, we expect that the relaxed formulations may scale
better for larger networks. This hypothesis will be tested through more simulations in
future work. Another encouraging trend from Table 2 is that the relaxations seem to offer
more of an incremental benefit (in terms of faster solution times) when applied to the OTS
problem to solve the MINLPs, as opposed to the nominal case without OTS where we are
solving a simpler problem.
While we observed low optimality gaps with the relaxations for these experiments, this
is not guaranteed for all situations. Further refinements can be made to restrict the feasible
space, obtain tighter relaxations, and ensure that the obtained solutions are feasible for
the original nonconvex program. Nevertheless, applying any of the above ACOPF-based
relaxations would still offer better approximations than the crude DCOPF approximation.
3.2. Sensitivity Analysis for Selecting Big-M Value
We conducted a brief sensitivity analysis to understand the impacts of the big-M
values on the run-time and conditioning of the power flow problem. This was achieved
using the IEEE 9-bus, 39-bus, and 118-bus test cases, by running formulation (9) for the
OTS problem, i.e., MISOCP relaxed OTS with Big-M constraints + MCEs (see Table 2). We
estimated the problem conditioning by extracting the condition number of the constraint ‘A’
(left-hand side) matrix. We observed that while the problem conditioning is pretty stable
and relatively insensitive to the big-M value, the run-time needed to solve the optimization
problem is more significantly affected by the chosen big-M value. Looking across the
sensitivity curves for the three test networks in Figure 1, we find that a value of around
M = 100 generally works well across all three networks, in terms of the lowest average
run times. These relatively low M values seem to work well for our application while also
ensuring satisfactory feasibility and optimality levels. One reason for this may be because
Electricity 2025, 6, 37 11 of 15
all of our inputs, parameters, and decision variables have been normalized to per unit
(p.u.) values, with small magnitudes generally of the order of 1. Although higher M values
may potentially further reduce the runtime or improve strict feasibility, these come with
the tradeoff of risking instability and ill-conditioning, especially for larger networks. As
intuitively expected, we also see that the run-times consistently increase as we increase the
network size, across all big-M values.
In general, the exact optimal value of M does depend on a variety of factors such as
the system size, network density, the number of integer decision variables, etc. A more
exhaustive big-M analysis for larger-scale problems will be considered in future work.
For instance, one potential way to study the influence of the number of decision variables
could be by varying the maximum number of lines with switches that can be opened at any
given time, i.e., Nsw.
Figure 1. Run-time versus big-M value. The 3 different colored lines (and circle data points) denote
the run-time results corresponding to each of the 3 networks tested in our simulations.
3.3. Effects of OTS on the System
Although we tested all nine formulations on the test cases, for conciseness, here,
we only report the results for (i) the nominal case while applying approach (4) vs. (ii)
the OTS case by applying (9) on the larger IEEE-118 bus system. We simulated the OPF
dispatch using load and generation data for 1 h. This is because we did not observe
significant impacts of OTS on the smaller 9-bus and 39-bus systems, indicating that they
are not really transmission capacity-constrained. The IEEE-118 bus system consists of
19 generators, 35 synchronous condensers, 177 lines, nine transformers, and 91 loads,
shown in Figure 2. These simulations were run using the Gurobi solver in Julia. We
assessed the impacts of OTS by examining some important metrics. We can obtain the
locational marginal prices (LMP) at each bus as the dual multipliers associated with the
real power (P) balance constraints. These are how electricity prices are set in the wholesale
market as well. The differences in LMPs across different buses in the network indicate that
there is congestion, i.e., the network operation is constrained by the transmission capacity
Electricity 2025, 6, 37 12 of 15
of the lines (power flow limits). The effect of congestion on system performance can be
quantified via the congestion rent:
( ) ( )
nD nG
Congestion rent = CR = ∑ piD · LMPi − ∑ pGj · LMPj .
i=1 j=1
Figure 2. IEEE-118 bus system.
This is the difference between the total amount paid by electricity consumers minus the
costs of procuring it from the generators; thus, CR > 0 indicates that there are inefficiencies
in the market allocation due to the effects of congestion and losses. Another key metric is
the total system-wide cost of meeting the demand, given by the total generation cost. This
can be directly queried as the objective function value at the optimal solution:
( )2
∑ fi(P∗Gi) = ∑ CG1 pG + CG2 pG + CG13i i i i i ,
i∈G i∈G
where we used a quadratic variable cost function, which is characteristic of conventional
fossil fuel-based generators. Finally, we can quantify the extent to which the transmission
capacity is constrained through the flowrate marginal prices (FMPs), which correspond to
the dual variables of the line flow limit inequality constraints Equation (43). These are zero
for lines that are not congested, since the constraint with strict inequality but screening for
non-zero values allows us to identify which lines are congested and to what extent.
We find that applying OTS creates significant benefits, especially in terms of reducing
system-wide congestion. This results in lower LMPs and FMPs and thus lower congestion
rents. The main results in Table 4 highlight the effects of implementing OTS on the IEEE-118
bus test system. Improved dispatch allows us to shift some generation to cheaper sources,
lowering the total costs slightly as well. The lower spatial volatility in LMPs also indicates
lower congestion and more efficient network operation. For simplicity, these simulations
were run with no constraint Nsw. Thus, it is important to note that these benefits may be
less pronounced if there is an upper limit imposed by the operator on the number of lines
allowed to be open or turned ‘OFF’ (Nsw).
Electricity 2025, 6, 37 13 of 15
Table 4. Effects of OTS on the IEEE-118 bus test system.
Nominal OTS
With no Limit on Nsw
Total generation cost ($/h) 62,165 60,018 (−3.45%)
Congestion rent ($/h) 39,433 2579 (−93.45%)
Average LMP ($/MWh) 44.23 33.84 (−23.49%)
Average |FMP| ($/MWh) 0.7404 0.0176 (−97.62%)
LMP standard deviation ($/MWh) 11.89 0.775 (−93.49 %)
No. of congested lines 5 4
Optimal no. of open lines N/A 32 (out of 186 total)
4. Conclusions and Future Work
In this study, we formulated and simulated various different optimal power flow
relaxations and applied them to the optimal transmission switching problem. We built
upon previous work to include more accurate models and constraints while performing
OTS. In addition to comparing the runtime and performance of these approaches on
three different realistic test networks, we also demonstrated the benefits of applying
OTS on a larger system. It helps reduce overall costs and congestion and results in a
more efficient power dispatch system-wide. For future work, we plan to scale up this
approach to much larger transmission grids in order to more rigorously compare the
different formulations. This will also allow us to assess what the true potential of OTS
is. We would also like to consider OTS in the context of coordinated transmission and
distribution grid operation [38,39]. Extensions to other operating conditions, like line
faults and contingencies, will also be considered, where voltage magnitudes may deviate
significantly from the nominal values.
Funding: This research received no external funding.
Data Availability Statement: All the data used in this study for the IEEE 9-bus, 39-bus, and 118-bus
test systems are publicly available in the Electric Grid Test Case Repository from Texas A&M Uni-
versity, at https://electricgrids.engr.tamu.edu/electric-grid-test-cases/ (accessed on 18 February
2025) [36].
Conflicts of Interest: The authors declare no conflicts of interest.
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