A Benders Decomposition Approach to Corrective
Security Constrained OPF with Power Flow Control
Devices
Javad Mohammadi, Gabriela Hug, Soummya Kar
Department of Electrical and Computer Engineering
Carnegie Mellon University
Pittsburgh, PA, USA
Email: jmohamma@ece.cmu.edu, ghug@ece.cmu.edu, soummyak@andrew.cmu.edu
Abstract—This paper studies a DC Corrective SecurityConstrained OPF (SCOPF) in electricity networks in which the
transmission lines are potentially instrumented with Distributed
Flexible AC Transmission Systems (DFACTS). Hence, in the
considered SCOPF, the transmission lines’ reactance along with
the generators’ output are considered as control variables. The
objective of this SCOPF model is to obtain minimum generation
cost while maintaining the system N-1 secure. Benders decomposition is applied to solve the discussed DC SCOPF problem.
Case studies for the IEEE 30-bus test system, the IEEE 39-bus
test system and the IEEE 118-bus test system are presented. The
performance of Benders decomposition is compared to a brute
force approach, in which the DC SCOPF is solved as a single
optimization problem, with respect to run-time and accuracy of
the results.
Index Terms—DFACTS, SCOPF, Benders Decomposition.
N OMENCLATURE
A. Parameters and Constants
Di
Pi
Pi
ai , b i , c i
Xij
F ij
ΔX ij
ΔX ij
NB
NG
load at bus i
Maximum generation of generator at bus i
Minimum generation of generator at bus i
Generator cost function parameters
Reactance of line ij
Line flow limit of line ij
Minimum reactance of DFACTS in line ij
Maximum reactance of DFACTS in line ij
Number of buses in the system
Number of generators in the system
B. Variables
Pi
DF
Fij
ΔXij
Xij,T ot
Power generated by generator at bus i
Distribution Factor matrix
line flow of line ij
Reactance of DFACTS in line ij
Reactance of line ij with DFACTS
This work has been supported by Smart Wire Grid, Inc. and is part of the
ARPA-E GENI project, “Distributed Power Flow Control using Smart Wires
for Energy Routing”.
978-1-4799-1303-9/13/$31.00 ©2013 IEEE
I. I NTRODUCTION
A. Motivation
Increased penetration of variable renewable resources and
market driven generation dispatch in combination with a lack
of investment in transmission assets has caused an increase in
transmission bottlenecks [1]. Bottlenecks create congestions
that decrease reliability, restrict competition, and increase
prices to consumers due to the resulting suboptimal generation
dispatch. However, it is increasingly difficult to site and build
new transmission lines to enhance the transmission capacity
of the existing system.
Ensuring secure operation of the power system is probably
the most important task of a system operator. Generally, the
system is operated according to the N-1 security paradigm
in which generation is dispatched such that no operational
constraints are violated in normal operation or in any credible
contingency case. Increased occurrences of congestions and
closer operation to the limits of the system puts the system at
higher risks and makes the task of keeping the system secure
more difficult.
In this paper, we focus on the opportunities provided by
power flow control to improve the usage of the existing
transmission assets. Redirecting power flow by controlling
parameters such as the reactance of a line can increase the
loading margin of the overall transmission grid by pushing
power to lines with spare capacity. Flexible AC Transmission
Systems (FACTS) [2] or the newly being developed distributed
version called DFACTS [1], [3], [4] provide such power flow
control capabilities and improve the ability of the system
operator to ensure secure system operation. This paper investigates the incorporation of DFACTS into the SCOPF and
uses Benders decomposition to solve the resulting optimization
problem.
B. Related Work
Benders decomposition is a particularly attractive technique
for solving the SCOPF problem due to the structure of the
problem. The contingency or N-1 cases are coupled to normal
operation only via the common control variables. Hence,
Benders decomposition lends itself to decompose the SCOPF
problem into a master problem corresponding to normal operation and N subproblems each corresponding to a contingency
cases. This facilitates parallel computing and managing the
large scale of the original optimization problem.
Benders decomposition has been mostly used to solve
SCOPF problems to determine the most cost effective generation dispatch fulfilling N-1 security constraints, e.g. [5], [6]. In
[7], Benders decomposition is used to determine the security
constrained unit commitment, in which generation redispatch
is considered as a corrective action. Several implementation
techniques of Benders decomposition for solving the DC
security constrained unit commitment are compared in [8] with
respect to run-time and result accuracy. Also, [9] proposes
two decomposition algorithms based on the Benders cut and
alternating direction method of multipliers for solving the
SCOPF problem. However, in these references no power flow
control devices are considered.
Benders decomposition’s applications with regards to power
flow control devices include the determination of the optimal
placement of FACTS devices, e.g. [10]. By adopting a multi
starting point approach, it is suggested to use Benders decomposition to maximize the loading margin of the network.
Examples for approaches to determine the optimal settings of
power flow control devices in which no Benders Decomposition is employed include e.g. [11], [12]. In [11], a linearized
model to incorporate a Unified Power Flow controller as power
flow controller into the Security Constraint Optimal Power
Flow (SCOPF) is suggested whereas in [12] a control region
for compact and reduced SCOPF formulation that utilizes
FACTS is defined in a way in which a global optimum can be
achieved.
In this paper, we incorporate power flow control devices into
a SCOPF and use Benders decomposition to solve the resulting
optimization problem. We study the performance of Benders
decomposition compared to a brute force approach and also
compare the effect of ubiquitous power flow controllers on
run-time and achieved solution compared to the case without
such devices. Furthermore, we study the effect of diminishing
return of increasing the possible operating range of such power
flow control devices.
C. Paper Organization
The rest of the paper is structured as follows: DFACTS
and SCOPF modeling approaches are discussed in Section II.
Section III presents the problem formulation and how Benders
decomposition is employed to solve this problem. Section IV
describes the test case system and presents simulation results.
Finally, Section V concludes the paper.
ΔXij
Fig. 1.
In this paper, we consider Distributed Series Reactance
(DSR) and Distributed Series Impedance (DSI) as the devices
providing the capability for power flow control [3]. Both of
these devices effectively influence the reactance of the line
in which they are placed. Hence, we model them as variable
DFACTS modules on power line.
reactances ΔXij in series with the line reactance Xij (see
Fig. 1) resulting in a total line reactance of
Xij,T ot = Xij + ΔXij
(1)
The difference between the two types of devices is that the
DSR can only provide inductive compensation, i.e.,
0 ≤ ΔXij,DSR ≤ ΔX ij
(2)
and therefore can only push power to parallel lines. The DSI
on the other hand is capable of providing inductive as well as
capacitive compensation, i.e.,
ΔX ij ≤ ΔXij,DSI ≤ ΔX ij
(3)
where ΔX ij is a negative value. Hence, these devices are able
to increase or decrease the power through the line in question
by pushing power to or drawing power from the rest of the
system.
B. Corrective Control
Traditionally, the system is operated according to N-1
security constraints, i.e. a single element failure may not
lead to any operational constraint violations. Hence, the set
of N-1 security constraints needs to be taken into account
when determining the generation dispatch to ensure that if a
contingency occurs no element is overloaded. We refer to the
case in which the N-1 security constraints need to be fulfilled
without any corrective actions once the contingency occurs
as the preventive approach. The approach in which corrective
actions are allowed is referred to as corrective approach.
Corrective SCOPF is based on the ability of the system
to tolerate short-term violations and perform fast corrective
actions after a fault occurred. In other words, in the case of
a contingency the system should be able to reach a secure
operation state before the violations of operational constraints
have any consequences. Due to the fast response of power
flow control devices, these devices may provide such corrective actions and hence reduce the cost of ensuring system’s
security.
The compact formulation of corrective SCOPF assuming
corrective actions provided by the DFACTS is given by
min
II. M ODELING
A. Power Flow Control
Xij
Pi ,XT0 ,...,XTN
NG
Ci (Pi )
(4)
i=1
s.t.
g0 (XT0 , P ) =
h0 (XT0 , P ) ≤
0
h̄
(5)
(6)
gq (XTq , P ) =
hq (XTq , P ) ≤
0 q = 1, . . . , N
h̄ q = 1, . . . , N
(7)
(8)
where q = 1, . . . , N refers to the contingencies considered in
the SCOPF. In this paper, such contingencies correspond to
line failures which do not lead to any islands in the system.
The decision variable P corresponds to the vector of generator
output settings and XT0 and XTq are the DFACTS setting in
normal operation and each of the contingencies, respectively.
Equality constraints (5) and (7) represent the load balance
equation for normal and contingency cases. Constraints on decision variables and operational constraints are included in (6)
and (8) for normal and contingency cases, respectively. Such
constraints correspond to limits on power flow control device
settings as well as limits on generation output. Operational
constraints include limits on line flows.
III. P ROBLEM F ORMULATION
Due to the large size of the electric power system, the
security constrained optimization problem usually results in a
large scale optimization problem. The Benders decomposition
is one of the commonly used decomposition techniques in
power systems to solve the SCOPF problem, that can be
applied to take advantage of the underlaying problem structure.
In applying Benders decomposition the complexity of the
problem is reduced by decomposing the original problem into
a master problem and several subproblems [13].
The algorithmic approach for solving SCOPF using Benders
decomposition is depicted in Fig. 2. In case of the SCOPF
problem, the master problem corresponds to the problem of
minimizing generation dispatch cost by taking into account
only the constraints for normal operation. Each subproblem
then corresponds to examining the feasibility of the master
problem’s solution with respect to the constraints for each
contingency case [13].
The formulation of the master problem (the upper dashed
box in Fig. 2) is given as follows:
min
NG
(ai · Pi2 + bi · Pi + ci )
Pk − Dk
=
(9)
Fkn ,
k = 1, . . . , NB
= DF · (Pk − Dk ), kn ∈ ΩL
ΔX kn ≤
ΔXkn
Pi ≤
−F kn <
Pi
Fkn
Fig. 2.
Structure of SCOPF with Benders decomposition.
By solving the master problem, the optimal values for con∗
trol variables P ∗ and ΔXkn
in the normal case are obtained
Then, the feasibility of the master problem’s solution for each
of the contingency cases is examined in the subproblems.
The goal of this feasibility check is to ensure that the master
problem’s solution satisfies all operational constraints, such as
line limits, in the contingency cases. To perform the feasibility
check, slack variables rkn for the operational constraints are
introduced and the following optimization problem is solved
for each contingency q:
w = min
s.t.
Pk − Dk
(10)
n∈Ωk
Fkn
(11)
≤ ΔX kn , kn ∈ ΩL
(12)
≤ P i,
< F kn ,
(13)
(14)
infeasibility cut
i = 1, . . . , NG
kn ∈ ΩL
ΩL
rkn
(16)
k=1
i=1
s.t.
(15)
where ΔXkn is the variable reactance introduced by the
DFACTS, Ωk are the buses to which bus k is connected and
ΩL are the lines in the system. If no generation and/or no
load is connected to bus k, Pk and/or Dk for that bus are
equal to zero. Line flows Fkn are defined in (11) where DF
is the distribution factor matrix. It should be noted that the
distribution factor matrix is a function of the DFACTS settings
ΔXkn rendering the problem non-convex.
q
Fkn
=
n∈Ωk
q
q
Fkn
,
k = 1, . . . , NB
= DF · (Pk − Dk ), kn ∈ ΩL
ΔX kn ≤
−F kn − rkn <
rkn
P
q
ΔXkn
q
Fkn
≥
=
≤ ΔX kn ,
(17)
(18)
kn ∈ ΩL (19)
< F kn + rkn , kn ∈ ΩL (20)
0
(21)
P∗
: λ1
(22)
where λ1 is the Lagrangian multiplier of the equality constraint
in (22). If the optimal value of the objective function optimal
is non-zero, then the master problem’s optimal values do not
satisfy the constraints in the contingency case. In case that the
determined generation dispatch does not satisfy the operational
constraints for a specific contingency, the following constraint
is added to the master problem as a so-called infeasibility or
Benders cut
w∗ + λ1 (P − P ∗ ) ≤ 0,
(23)
IV. S IMULATION R ESULTS
A. Test Systems
The IEEE 30-bus [14], the IEEE 39-bus [15] and the IEEE
118-bus test systems [14] are used to investigate the effects
of utilizing DFACTS to ensure N-1 secure operation of the
power system. We consider various operational ranges for the
DFACTS devices from a margin of 10% compensation up to
50% compensation of the transmission lines. Also, we assume
that all of the lines are equipped with DFACTS to study the
effect of ubiquitous power flow control on the computational
performance of Benders decomposition. Line limits for both
systems have been chosen such that they reflect likely limits
given the initial operational state of these systems. The cost
parameters for the generators were derived from heat rate
curves given in the IEEE Reliability Test System and recent
fuel costs. The simulations were carried out in MATLAB
using the KNITRO solver of the commercial optimization
package Tomlab on a core i7 3.4 GHz windows platform.
This is especially important for Sect. IV.C where we compare
computation time between the brute force approach, in which
all N-1 constraints are included in a single optimization
problem solved using Tomlab, and the Benders Decomposition
approach in which the smaller master and subproblems are
each solved using the same commercial software.
B. Cost Comparison
Tables I-III show the generation cost of SCOPF calculations
for the various considered levels of compensation ranges of
DFACTS. Based on these results, obviously, as the compensation ratio increases, the SCOPF procurement cost reduces.
In other words, as DFACTS compensation ratio increases, the
system flexibility to handle contingencies increases, thereby
enabling a cheaper generation dispatch.
TABLE I
C OSTS ( IN $) FOR GENERATION DISPATCH FULFILLING N-1 SECURITY
CONSTRAINTS FOR THE IEEE 30-B US TEST SYSTEM
TABLE II
C OSTS ( IN $) FOR GENERATION DISPATCH FULFILLING N-1 SECURITY
CONSTRAINTS FOR THE IEEE 39-B US TEST SYSTEM
ΔX kn ,ΔX kn
0.1
0.2
0.3
0.4
0.5
w/o DFACTS
22134
22134
22134
22134
22134
DFACTS as DSR
22053
21977
21984
21946
21910
DFACTS as DSI
21961
21910
21778
21612
21358
TABLE III
C OSTS ( IN $) FOR GENERATION DISPATCH FULFILLING N-1 SECURITY
CONSTRAINTS FOR THE IEEE 118-B US TEST SYSTEM
ΔX kn ,ΔX kn
0.1
0.2
0.3
0.4
0.5
w/o DFACTS
118300
118300
118300
118300
118300
DFACTS as DSR
117491
116975
116409
115887
115450
DFACTS as DSI
116850
115456
113925
112370
110560
Moreover, the impact of DFACTS on system operation
differs based on their capability to alter the line reactance. DSI
can improve the transfer capacity of the network more than
DSR, because it can decrease the reactance of uncongested
lines while increasing the reactance of congested lines.
Although, the SCOPF cost decreases with increasing compensation ratio, a saturation is observed. This confirms that
there is a limit on the performance improvement that may be
achieved by using DFACTS. This limit is reached when all
parallel lines reach their limits. The saturation trend is also
observed in Fig. 3, which plots the SCOPF calculation cost
as a function of the various considered levels of DFACTS
compensation ranges. However, the saturation point or the
best achievable performance depends heavily on the system
topology and the generator cost functions.
C. Computational Aspect
This section discusses the computational aspects related to
applying Benders decomposition to solve the SCOPF problem.
It should be noted that the incorporation of DFACTS into
the SCOPF problem renders the problem non-convex. On the
other hand, Benders decomposition requires that the objective
function of the considered problem projected on the subspace
of the complicating variables has a convex hull. Unfortunately,
DFACTS as DSR
DFACTS as DSI
5000
4500
SCOPF cost
where the variable w∗ represents the optimal value of subproblem’s objective function (16). The goal of adding this
constraint to the master problem is to force w to be less than
or equal to zero in the next iteration, and therefore, make the
subproblem (16)-(22) feasible.
Iterations between master and subproblems are continued
until the master problem’s optimal solution satisfies all of the
subproblems’ constraints. It should be noted that changing the
DFACTS setting is the only corrective action that is considered
in this paper. Hence, the master problem’s optimal generation
dispatch P ∗ is assumed to be fixed in the subproblems indicated by (22). However, settings of DFACTS may be modified
in the case of each contingency.
4000
3500
3000
ΔX kn ,ΔX kn
0.1
0.2
0.3
0.4
0.5
w/o DFACTS
5029
5029
5029
5029
5029
DFACTS as DSR
4756
4529
4333
4167
4033
DFACTS as DSI
4480
4032
3862
3809
3809
2500
0
0.1
0.2
0.3
0.4
0.5
0.6
Range of DFACTS setting
0.7
0.8
0.9
Fig. 3. SCOPF cost with respect to DFACTS setting range (IEEE 30-Bus
test system).
96
0.6
95.8
0.4
95.6
0.2
95.4
0
0.1
0.15
0.2
0.25
0.3
0.35
Range of DFACTS setting
0.4
0.45
95.2
0.5
Percentage of Difference
Between the Run−Time of BF and BD
Percentage of Difference
Between the Results of BF and BD
0.8
Fig. 4. Comparing run time improvement versus result difference (IEEE
39-Bus test system).
the problem we applied Benders decomposition. We allowed
for corrective actions by the power flow control devices. It is
assumed that once a contingency occurs the settings of these
devices may be adjusted to resolve violations of operational
constraints. The numerical studies show that with DFACTS
devices the system can operate N-1 secure at lower cost.
Moreover, we discuss the Benders decomposition applicability for solving this paper’s problem. Due to the non-convex
nature of problem, the results obtained from the Benders
decomposition may differ form the ones obtained from the
brute force approach results. However, from the run-time
perspective Benders decomposition contribution for solving
this problem is considerable.
R EFERENCES
this is not the case for this paper’s problem due to the nonconvexity of the problem. Hence, it may happen that the
result of Benders decomposition method differs from the brute
force approach [10]. In [5], similar concerns are mentioned
regarding the application of Benders decomposition. This
reference presents a comparison between decomposed and
unified (brute force) implementation of SCOPF with respect
to run-time and result accuracy. The authors concluded that
applying Benders decomposition results in significantly better
computing speed without significantly sacrificing accuracy.
Fig. 4 compares results from the Benders decomposition
(BD) and the brute force (BF) approach and the respective
run-times to obtain these results for the 39 bus system. The
percentage of difference between the discussed approaches is
calculated as:
BF Result − BD Result
× 100
BF Result
Hence, although Benders decomposition seems to compromise
the result by a minimal amount (less than 0.5%) its run-time
improvement is considerable (more than 95%).
Table IV shows the run-times for the IEEE 30-Bus and
the IEEE 118-Bus test systems, our Benders’ Decomposition
approach yields the same results as the Brute Force, though
with a considerable run-time improvement. On the other hand,
it is seen that the Brute Force method can not handle the
SCOPF problem for the IEEE-118 bus test system. It should
be noted that, computation time is dependent on how many
congestions arise and also how it is implemented and what
computer is used.
V. C ONCLUSION
In this paper, we studied the benefits of utilizing power flow
control devices for corrective power flow control with respect
to the generation cost. In order to reduces the complexity of
TABLE IV
C OMPUTING T IME FOR IEEE 30-B US AND IEEE 118-B US TEST SYSTEMS
w/o DFACTS
DFACTS as DSR
DFACTS as DSI
30-Bus test system
BF (s)
BD (s)
0.64
3
367.23
30.32
431.72
21.56
118-Bus
BF (s)
600
NA
NA
test system
BD (s)
632.38
1726.8
1373.8
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unrestricted/Chapt3c.pdf